ABSTRACT
Title of dissertation: ESSAYS ON FRICTIONS
AND ECONOMIC FLUCTUATIONS
Hsuan Yu
Doctor of Philosophy, 2019
Dissertation directed by: Professor John Shea
Department of Economics
I study how information frictions, in the forms of limited information capacity
or asymmetric information, affects the firm’s production and physical capital accu-
mulation decisions, and how it can help a dynamic general equilibrium model to
account for selected empirical characteristics over business cycle frequencies.
In the first chapter, I explore how limited information capacity affects fished-
good inventory accumulation by firms. I use rational inattention to understand the
responses of inflation and output inventories after nominal shocks. In the data,
output inventories move less than sales in the U.S. manufacturing and trade sectors.
To reconcile the model with the data, some studies have suggested that variation
in price markups, rather than cost rigidities, must account for the bulk of the real
effects of nominal shocks. I propose that this conclusion does not necessarily hold
when the firms decisions are affected by an information capacity constraint. In my
model, firms observe aggregate conditions with idiosyncratic noise. Paying more
attention helps a decision maker to reduce the noise, but also incurs information
costs. Firms allocate their attention between pricing and production decisions, and
their decision rules deviate from the first-best rules. This friction serves as a force
to hinder drastic movements in production and inventory accumulation. I show that
inventories can move less than sales even when the marginal cost of production is
rigid. Numerical results suggest that inattention on the firm side can qualitatively
match empirical impulse responses and business cycle moments. The fit with data
is better than a staggered pricing model with elastic cost pressure, in terms of both
matching impulse responses and simulated moments.
In the second chapter, I study how information asymmetry about the quality of
used capital affects capital reallocation. Empirical studies of business cycle dynamics
indicate that the reallocation of capital, or the movement of capital input from less
productive firms to more efficient firms, is procyclical, whereas the dispersion of
marginal product of capital is countercyclical. I build a model of endogenous partial
capital irreversibility, which stems from both capital specificity and information
asymmetry in the market for used capital. The resale price and average quality of
used capital in the market vary with aggregate productivity shocks. In the model,
imperfect substitution between new and used capital and information asymmetry
interact to generate procyclical reallocation. Preliminary numerical results show
procyclical reallocation and countercyclical dispersion of capital returns.
ESSAYS ON FRICTIONSAND ECONOMIC FLUCTUATIONS
by
Hsuan Yu
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2019
Advisory Committee:
Professor John Shea, Chair/Advisor
Professor S. Boraǧan Aruoba
Professor L. Luminita Stevens
Professor Felipe Saffie
Professor Phillip L. Swagel
©c Copyright by
Hsuan Yu
2019
Dedication
To my parents and my loved one, whose encouragement made this work pos-
sible.
ii
Acknowledgments
I am deeply indebted to my advisors. Professor John Shea gave me invaluable
guidance at all stages of my dissertation. He meticulously read every version of my
humble drafts, even when the ideas were still very immature. He pointed to me
important directions for further development. Professor Shea is always available for
discussion and supports me in every aspect during my graduate life. I am truly
grateful to have him as my committee chair. Professor Luminita Stevens patiently
discussed the punchlines and the logic of my dissertation with me. She taught me im-
portant know-how on how to organize the dissertation and presentation persuasively
and effectively, as well as the correct ways to summarize and generate research ideas
from reading the literature. She gave me analytical tools to conduct macroeconomic
research. The benefit of her guidance will surely extend beyond the completion of
this dissertation. Professor Boraǧan Aruoba points out drawbacks in the empirical
and quantitative parts of my drafts, and gave me insightful suggestions on compu-
tation. He imbued me with a rigorous attitude toward research and academic work.
His knowledge in this profession is what I would strive to achieve in the future.
Professor Felipe Saffie gave me serious comments and intuition during my
presentations and generously agreed to read my drafts. Professor Phillip Swagel
kindly accepted my invitation to be the Dean’s Representative in my dissertation
defense. They both gave me insightful comments on improving my dissertation, and
I greatly appreciate their support and help. I also want to thank Professor John
Haltiwanger for giving me suggestions on my third year paper, which developed into
iii
chapter 2 of my dissertation.
I benefited a lot from discussions with colleagues, especially Xing Hong, Yang
Xu, Diyue Guo, Lixin Tang, Dun Calvin Jia, Jacob Blackwood, Hyung Choi,
Joonkyu Choi, and Seth Murray. I would also like to thank participants at seminars,
particularly the ECON 709 group meetings at the university of Maryland.
I would like to acknowledge the scholarship provided by the Ministry of Edu-
cation in Taiwan. Its financial support emboldened me to pursue a doctoral degree
in the United States.
iv
Table of Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1. The Behavior of Inventories and Marginal Cost under Rational Inattention 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.2 Final Goods Firms . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.3 Intermediate Goods Firms . . . . . . . . . . . . . . . . . . . . 12
1.3.4 Labor Unions and Wage Rigidity . . . . . . . . . . . . . . . . 16
1.3.5 Government . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.6 Resource Constraint . . . . . . . . . . . . . . . . . . . . . . . 20
1.3.7 Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.8 Intermediate Goods Firms : Rational Inattention . . . . . . . 22
1.3.9 Equilibrium under Rational Inattention . . . . . . . . . . . . . 27
1.4 Comparative Statics & Discussion . . . . . . . . . . . . . . . . . . . . 29
1.4.1 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.2 The Role of Information Frictions . . . . . . . . . . . . . . . . 36
1.5 Empirical & Numerical Results . . . . . . . . . . . . . . . . . . . . . 39
1.5.1 Empirical Results: Vector Autoregression . . . . . . . . . . . . 39
1.5.2 Calibration & Estimation . . . . . . . . . . . . . . . . . . . . 43
1.5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.6 Robustness & Extension . . . . . . . . . . . . . . . . . . . . . . . . . 62
1.6.1 Sensitivity to Information Cost . . . . . . . . . . . . . . . . . 62
1.6.2 Larger Interest Rate Shock . . . . . . . . . . . . . . . . . . . . 65
1.6.3 Money Growth Shock . . . . . . . . . . . . . . . . . . . . . . . 67
1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2. Endogenous Irreversibility and Capital Reallocation under Information Asym-
metry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.3 A Simple Model of Capital Reallocation . . . . . . . . . . . . . . . . 79
2.3.1 The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.3.2 Entrepreneur’s Optimal Investment Decision . . . . . . . . . . 86
2.3.3 Equilibrium in the Market for Used Capital . . . . . . . . . . 88
2.4 A Full DSGE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.4.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.4.2 Entrepreneurs . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.4.3 Market Clearing & Recursive Equilibrium . . . . . . . . . . . 96
2.4.4 Analysis & Computation . . . . . . . . . . . . . . . . . . . . . 98
2.4.5 Amplification Mechanism of Reallocation Friction . . . . . . . 104
2.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.5.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.5.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 108
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A. Appendix for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.1 Firm’s Optimality Conditions without Frictions . . . . . . . . . . . . 118
A.2 Aggregate Price Index . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A.3 Non-Stochastic Steady State . . . . . . . . . . . . . . . . . . . . . . . 122
A.4 Proof of Proposition 1 and Quadratic Approximation of Profit Loss . 125
A.5 Algorithm for the Rational Inattention Model . . . . . . . . . . . . . 132
A.6 Solving the Sticky Price Model . . . . . . . . . . . . . . . . . . . . . 134
A.6.1 Conjectured Policy Rules . . . . . . . . . . . . . . . . . . . . . 134
A.6.2 Expected Inventory Stock . . . . . . . . . . . . . . . . . . . . 135
A.6.3 Expected Aggregate Price and Real Marginal Cost . . . . . . 137
A.6.4 Inventory-Sales Ratio . . . . . . . . . . . . . . . . . . . . . . . 139
A.6.5 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
B. Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
B.1 Proof of Corollary 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
B.2 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
vi
List of Tables
1.1 Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.2 Decomposition of Estimation Criterion . . . . . . . . . . . . . . . . . 55
1.3 Comparison of Simulated Moments. . . . . . . . . . . . . . . . . . . . 62
1.4 Sensitivity analysis to marginal information costs. . . . . . . . . . . 63
1.5 Sensitivity analysis to larger standard deviation of interest rate shock. 67
1.6 Comparing simulated moments. . . . . . . . . . . . . . . . . . . . . . 69
2.1 Benchmark parameters. . . . . . . . . . . . . . . . . . . . . . . . . . 107
2.2 Steady state values under benchmark calibration . . . . . . . . . . . . 108
A.1 Policy Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
List of Figures
1.1 Production Decision: Comparative Statics of Higher Real Interest
Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.2 Production Decision: Comparative Statics of Lower Real Marginal
Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.3 Pricing Decision: Comparative Statics of Lower Real Marginal Cost . 33
1.4 Comparative Statics of Higher Interest Rate . . . . . . . . . . . . . . 35
1.5 Comparative Statics of Lower Real Marginal Cost . . . . . . . . . . . 35
1.6 IRF: Shock to Fed Funds Rate. Q1 1970 - Q4 2015 . . . . . . . . . . 44
1.7 IRF: Shock to Fed Funds Rate. Q1 1982 - Q4 2007 . . . . . . . . . . 45
1.8 Impulse Response Comparison: RI, Calvo Pricing, & VAR after Shock
to Fed Funds Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.9 Impulse Response Comparison: Calvo Pricing (sticky marginal cost)
& VAR after Shock to Fed Funds Rate. . . . . . . . . . . . . . . . . 59
1.10 First-Best and Actual Response of Stock for Sale Zjt under rational
inattention. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.11 First-Best and Actual Response of Price pjt under rational inattention. 60
1.12 The Sensitivity of Zjt to Marginal Information Costs µ . . . . . . . . 64
1.13 The Sensitivity of Inventory-Sales Ratio to Marginal Information
Costs µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
1.14 The Sensitivity of Inventory-Sales Ratio to σr . . . . . . . . . . . . . 66
1.15 Impulse Responses to a −0.6% Money Growth Shock, RI Model . . . 69
1.16 Impulse Responses to a −0.6% Money Growth Shock, Calvo Model . 70
2.1 Impulse Responses to Negative TFP Shocks. ( = 2.5, b=0.35). . . . 112
2.2 Impulse Responses to Negative TFP Shocks. ( = 3.5, b=0.35). . . . 113
2.3 Impulse Responses to Negative TFP Shocks. ( = 2.5, b=0.25. . . . . 114
2.4 Impulse Responses of Allocative Efficiencies. ( = 2.5, b=0.35). . . . 115
2.5 Impulse Responses of Allocative Efficiencies. ( = 3.5, b=0.35). . . . 116
2.6 Impulse Responses of Allocative Efficiencies. ( = 2.5, b=0.25). . . . 117
List of Abbreviations
LHS Left hand side
RHS Right hand side
DSGE Dynamic stochastic general equilibrium
IRF Impulse response function
VAR Vector autoregression
ARMA Auto-regressive moving average
MA Moving average
RI Rational inattention
PPP Purchasing Power Parity
FFR Fed Funds Rate
PP&E Property, Plants, and Equipment
TFP Total factor productivity
MPK Marginal product of capital
ix
Chapter 1: The Behavior of Inventories and Marginal Cost under
Rational Inattention
1.1 Introduction
This paper studies inventory and inflation behavior in response to monetary
shocks. I focus on the relation between cost rigidities, inflation and inventory re-
sponses in monetary transmission. Estimates from vector autoregressive (VAR)
models suggest that monetary tightening leads to a persistent decline in output, a
muted and sluggish decline in inflation, and a rise in the inventory-sales ratio. Even
though inventories decline after a monetary tightening, the movement is muted
compared to the decline in sales, so that the inventory-sales ratio increases after a
monetary tightening. In a standard model, the movement of inventory investment
is tied to the elasticity of the marginal cost of production with respect to output.
If marginal cost falls sharply with the fall in output, then the decline in inventories
will indeed be limited, because firms’ lower marginal cost spurs firms to accumulate
more inventories. Therefore, inventories can indeed move less then sales. However,
if the marginal cost of production does not respond much to sales, there is no incen-
tive for intertemporal substitution in firms’ production to counteract the effect of a
spike in the real interest rate, and the latter will kick start a large disinvestment of
inventories, since the real interest rate is a major determinant of the carrying cost
of inventories (along with the depreciation rate).
For this reason, existing monetary business cycle models have a hard time
explaining the joint behavior of finished good inventories relative to sales/shipments,
inertial inflation, and the persistent response of output simultaneously. Models that
assume that marginal cost is sluggish and not very responsive to changes in output
can explain why the decline in the price level is muted and why output falls following
a monetary tightening, but predict that inventories fall much more than sales, as
the spike in the real interest rate translates into an increase in the cost of carrying
inventories. Models in which the marginal cost of production falls sharply after a
monetary tightening can explain why inventories move less than sales, but imply
a more abrupt and temporary downward movement of the inflation rate even with
a high degree of nominal rigidity, unless the model also assumes a high degree of
rigidity (strategic complementarity) among firms’ pricing decisions. Furthermore,
elastic cost pressure and rigid prices translates to countercyclical price markups,
which means firms expand their profit margin after a monetary tightening. However,
in the data we do not find evidence of countercyclical markup or profits.1 This
calls into question viewing countercyclical markup variation as the key transmission
mechanism.
1 See the empirical results in section 2. Also, Loecker and Eeckhout (2018) studies the evolution
of markups based on firm-level data. They find that markups are stable before 1980. After the
1980s the average markup is on a rising trend. Anderson et al. (2018) studies the behavior of
markups in the U.S. retail sector. They find that markups are relatively stable over time and
mildly procyclical.
2
I propose that this conundrum involving the responses of inventories, inflation,
and marginal production cost can be reconciled when I embed information frictions
in the inventory model in place of nominal rigidity in firms’ pricing decisions. I
show that when it takes time (and possibly other resources) for the firm to figure
out the true state of the economy, then inventories can move less than sales even
without a pronounced response of marginal cost, and inflation can be sluggish (due
to both the information friction and the sluggish movement of production cost).
The intuition is simple: information frictions affect both the firm’s pricing and
production decisions, unlike nominal rigidities, which only work through the pricing
decisions. In other words, information frictions on the firm side generate sluggishness
of both inflation and inventories. Compared to a model in which produced goods are
not storable and thus a firm’s relative price directly determines demand for its goods
and its production level, in an inventory model unsold goods are storable and the
firm’s production is not directly pinned down by its relative price (though pricing
and production decisions are still linked through optimality conditions). Under
nominal rigidities and full information, a high responsiveness of production cost
is required to discipline the movement of production (and inventories). Instead,
if I assume limited information capacity, the information friction itself serves to
hinder the impact response of prices and production to shocks. The model can thus
accommodate a rigid cost of production, which adds to the persistence of the real
effects of monetary shocks.
The information friction adopted in this paper follows Mackowiak and Wieder-
3
holt (2015) and Mackowiak and Wiederholt (2009).2 I consider endogenous inat-
tentive behavior, which affects the pricing and production decisions of a firm. In
the model, firms face both aggregate shocks and idiosyncratic noise shocks. Firms
cannot attend perfectly to all information by tracking each source of disturbance
without error, because paying more attention to a particular shock incurs a flow in-
formation cost, which can be thought of as time devoted to information processing.
Firms thus face a constraint on information flow and optimally decide the amount of
attention allocated to different sources of shocks, as well as to their pricing and pro-
duction decisions. One contribution of the paper is to examine the relative attention
allocated to the two decisions and how it affects the inventory behavior. I argue that
the firm pays more attention to the pricing decision because it is more relevant to
profit maximization. Therefore the firm allows the inventory accumulation decision
to deviate more from the first-best action.3 To the best of my knowledge, this is
the first study that explores the joint dynamics of prices and inventories from the
perspective of information flows.
1.2 Literature Review
This paper is related to studies comparing information frictions and stag-
gered pricing frictions à la Calvo, including Trabandt (2007), Korenok and Swanson
2 As in Mackowiak and Wiederholt’s papers, I only consider how inattentive firms’ allocation of
attention affects the firm-level price setting. Recent works such as Pasten and Schoenle (2014) and
Stevens (2019) have explored how endogenous information acquisition affects product-level prices
and how it determines aggregate inflation dynamics.
3 As will become clear later, the amount of goods a firm produces determines its inventory accu-
mulation ex ante. Throughout this paper I use the three terms “production decision”, “inventory
decision”, and “Zj decision” interchangeably.
4
(2007), Collard and Dellas (2010) and Mackowiak et al. (2009). In particular, Tra-
bandt (2007) runs a horse race between sticky information and sticky prices with
dynamic indexation. He concludes that both the sticky information model and the
Calvo time-dependent pricing model with dynamic inflation indexation can account
for inflation inertia, and suggests considering sticky information as a micro founda-
tion for the hard-wired assumption of time-dependent pricing and indexation to past
inflation. In this paper, inattentiveness on the firm side affects both the firm’s price
setting decision and its production decision (which affects inventory accumulation),
and thus information frictions and time-dependent pricing frictions do have very
different implications regarding the underlying force driving inventory movements,
which affects the model’s ability to fit the data.
This paper is also related to the debate about the relative importance of vari-
ations in price markups and cost rigidities. Staggered pricing and countercyclical
price markups have been the key mechanism generating real effects of monetary
shocks in New Keynesian models. Chari et al. (2000) raises the persistence problem
facing staggered pricing models, in that empirically plausible degrees of price rigidity
generate only a modest degree of output persistence in response to monetary shocks,
and cannot account for the estimated output persistence in the U.S. economy. This
persistence problem has led researchers to adopt real rigidities to explain the real
effects of monetary shocks. Cost rigidity is an external source of real rigidity that
reduces the responsiveness of firms’ profit-maximizing prices to variation in aggre-
gate output. Christiano et al. (2005) find that cost rigidity is more important than
nominal price rigidities in accounting for the observed inertia of inflation and per-
5
sistence of output. Dotsey and King (2006) reach a similar conclusion, showing that
supply side frictions, including produced inputs and variable capacity utilization,
that reduce the elasticity of marginal cost with respect to output serve to reduce
the extent of price adjustment and generate persistent output responses to monetary
shocks.
The other class of real rigidity are internal real rigidities, assumptions on pref-
erences and technology that make it costly for a firm to charge prices that are too
different from those of its competitors, and thus reinforce strategic complementar-
ity and countercyclical markups. Examples include a quasi-kinked demand curve,
firm-specific capital, and firm-specific labor. Exactly which class of real rigidity is
more important has been controversial. Karabarbounis (2014) finds that variations
in the gap between the real wage and household’s marginal rate of substitution
between labor and leisure, which may arise from sticky wages, explain 80% of the
countercyclical fluctuations in the labor wedge in the U.S., while variations in price
markups account for a small proportion. Closely related to my paper is Kryvtsov
and Midrigan (2013). They utilize the tight link between inventory movements,
markups and marginal cost in a theoretical model to infer that countercyclical price
markups, rather than cost rigidities, must account for most (as high as 90%) of
the real effects of aggregate demand shocks. I show that information frictions can
weaken this tight link, and thus conclusions about the elasticity of marginal cost
with respect to output drawn from inventory behavior need to be assessed with more
care. In section 5 I apply a negative money growth shock and show that even when
markup variation is relatively low (accounting for only 24% of the real effect), inven-
6
tories can move less than sales because of the information capacity constraint. The
implication of this paper is in line with Basu and Fernald (1997), which finds that
the average 2-digit industry appears to produce with constant or even increasing
returns, so that markups should be stable or even procyclical. This paper is also in
line with the findings of Anderson et al. (2018), which finds that although markups
have regional dispersion, they are relatively stable over time.
The relative importance of cost rigidity and markup variation has policy im-
plications besides the persistence problem. Nominal price rigidity points to the
importance of frictions in the product market that affect the firm’s pricing decision,
such as menu costs of price adjustment, a quasi-kinked demand curve, or firm-
specific labor or capital. Different forms of strategic complementarity have different
welfare implications for steady state inflation or deflation, as pointed out by Levin
et al. (2006). For example, consider factor specificity which gives rise to an upward
sloping marginal cost schedule. With factor specificity, non-adjusting firms suffer
from steady state inflation as the distortion of their lower relative prices increases
their cost of production significantly. Generally, when the distortions exists only in
the product market, complete stabilization of the price level is sufficient to eliminate
inefficient price dispersion and close the output gap, thus meeting the central bank’s
dual mandate.
In contrast, a rigid marginal cost points to the existence of frictions in factor
markets. For example, wage rigidity can result from collective wage bargaining by
trade unions or implicit wage contracts. The frictions in the labor market create a
new source of distortions, and stabilizing the price level alone becomes suboptimal
7
for welfare maximization. Just as variation in the price level creates price dispersion
and inefficient distortions, volatility in wage inflation creates an inefficient distribu-
tion of employment across households. The central bank thus need to stabilize price
inflation, wage inflation, and the output gap. Erceg et al. (2000) points out that it
is impossible for monetary policy to attain a Pareto optimum unless either prices or
nominal wages are completely flexible to keep the real wage at its Pareto optimum
level. The optimal rule will induce greater flexibility in the more flexible nominal
variable. Thomas (2008) studies optimal monetary policy in a setting with labor
search and matching frictions and both staggered price and staggered wage setting.
It reaches a similar conclusion as Erceg et al. (2000). Nominal wage rigidity gives
rise to a rigid real wage, which generates inefficient job creation and inefficient un-
employment fluctuations. A zero inflation policy is no longer optimal, and policy
makers should use price inflation to bring the real wage closer to the flex-wage tar-
get. Generally, frictions in wage setting imply a case against strict price stability.
Thus, the source and extent of cost rigidity matters for normative analysis.
In my model of limited information capacity, the deviation of the actual price
response from the first-best response has significant persistence and dies out after
25 quarters. Relating to the international economics literature discussing the ad-
justment of a country’s exchange rate toward Purchasing Power Parity, Taylor and
Taylor (2004) points out that relative PPP only appears to hold in the long run
(after taking a long-run average). Rogoff (1996) notes that the estimated half-life
of adjustments is between 3 to 5 years, and the slow speed of adjustment is due to
the persistence in nominal variables such as nominal wages and prices. Bergin and
8
Feenstra (2001) shows that trans-log preferences and pricing to market can signif-
icantly increase endogenous persistence. In my model, persistence is generated by
both the limited information capacity and nominal wage rigidity.
1.3 The Model
I construct a DSGE model that incorporates inventory investment. This sec-
tion first lays out the model without pricing frictions or information constraints on
the firm side (in subsection 2.2), and then introduces an information friction in the
form of a limited information capacity for firms (in subsection 2.3). To facilitate the
comparison with Kryvtsov and Midrigan (2013), I assume that the firms’ inventory
holding motive comes from stock-out avoidance. Firms cannot observe the demand
for their goods before production. Since production takes time, inventories are used
as a buffer for higher-than-expected demand. The economy consists of final goods
firms, intermediate goods firms, households, labor unions, and a government sector.
1.3.1 Households
Household i ∈ [0, 1] chooses consumption Ct(i), hours worked Lt(i), and bonds
Bt(i) to maximize its lifetime utility:
∑∞ [ 1 ( ) ]1−σc ψl
max E βt0 Ct(i)− hC 1+σt− l1 − Lt(i)
{Ct(i),Lt(i),Bt(i)} 1− σc 1 + σt=0 l
9
subject to the budget constraint:
Bt(i) T
h
t Bt−1(i) W
C (i) + + = + t
Lt(i) Πt Divt
t + + (1.1)
PtRt Pt Pt Pt Pt Pt
where Tt is a nominal lump-sum tax. The parameter h captures the degree of
external habit formation. Household income comes from wages (with nominal wage
rate W ht ), risk-free interest income from bonds, the profit from the intermediate
goods sector Πt, and dividends rebated from labor unions, Divt.
Since households are homogeneous, they will make the same choices for con-
sumption, hours worked, and bond holdings. The first order conditions are listed as
follows (dropping the i index):
( )−σc
(∂Ct) Ξt = Ct − hCt−1 (1.2)
W h
(∂Lt) ψlL
σl
t = Ξ
t
t [ ] (1.3)PtΞt+1
(∂Bt) Ξt = βRtEt (1.4)
πt+1
where Ξt is the Lagrange multiplier associated with the budget constraint.
1.3.2 Final Goods Firms
There is a unit measure of final goods firms. Final goods firms are competitive.
They combine differentiated intermediate goods into the final good St, which can
be used for consumption by the household sector and government. The final goods
10
firms bundle intermediate goods with the following Dixit-Stiglitz aggregator:
[ ∫ 1 − ] θ1 θ 1
θ−1
S θ θt = vjtSjt , θ > 1 (1.5)
0
where θ is the elasticity of demand for firm j’s good and vjt is an i.i.d. taste shock
2
for type j good that follows a lognormal distribution log(v σv 2jt) ∼ N (− , σv) with2
distribution function G(·). Therefore, the problem of a final goods firm is as follows:
∫ 1
max PtSt − PjtSjtdj
{Sjt}j∈[0,1] 0
subject to equation (1.5) and the no stock out constraint:
Sjt ≤ Zjt ∀j (1.6)
where Zjt is intermediate goods firm j’s quantity of goods available for sale at time
t after production.
Solving the above problem gives the demand for intermediate good j as
[P ]−θjt + γjt
Sjt = vjt St (1.7)
Pt
where γjt is the Lagrange multiplier associated with stock out constraint (1.6).
The aggregate price Pt is:
[ ∫ 1 [ ] ] 11−θ 1−θ
Pt = vjt Pjt + γjt (1.8)
0
11
1.3.3 Intermediate Goods Firms
Intermediate goods firms are monopolistically competitive. They produce with
labor only and are subject to aggregate productivity shocks (as well as the taste
shock vjt introduced in equation (1.5)). The production function is
Yjt = exp
at Lαjt (1.9)
where at is the aggregate productivity. The usual economic logic for specifying
decreasing returns in labor inputs is that physical capital, which this paper abstracts
from, cannot be adjusted smoothly in the short run. The parameter α ∈ (0, 1)
controls the returns to scale of the production technology, which also determines
the sensitivity of real marginal cost to changes in output, and will thus play a
crucial role in our estimation.4
Timing: I assume that an intermediate goods firm cannot see the realization of its
taste shock vjt before its pricing and production decisions are made. After the firm
decides on its price and production quantity, vjt is realized and the firm will try to
meet all its demand, subject to the no stock out constraint (1.6). After production
in period t, the firm will have a quantity of goods available for sale, denoted Zjt,
which is the sum of undepreciated inventory stocks carried from the last period,
4 The purpose of specifying a decreasing returns to scale production technology is to control the
responsiveness of the marginal cost to changes in output. Jung and Yun (2006) showed that the
movement of inventories will be too sharp and severe after a monetary shock compared to the data
without adding an adjustment friction. Alternatively, I assume that the production technology is
decreasing returns to scale, as in Kryvtsov and Midrigan (2013), to increase the elasticity of the
marginal cost.
12
(1 − δz)invj,t−1, and goods produced this period Yjt. Unsold goods become this
period’s inventories. Depending on the realization of vjt, there is a chance that the
firm may run out of goods to supply, i.e., stock out. Given the demand from the
final goods sector (equation (1.7)), the sales of firm j are:
{ (P )−θ }jt
Sjt = min Zjt, vjt St (1.10)
Pt
Thus, firm j will stock out if the quantity demanded exceeds its quantity available
for sale. The stock of goods available for sale, Zjt, and its inventories at the end of
the period, invjt, evolve over time according to the following:
Zjt = (1− δz)invj,t−1 + Yjt (1.11)
invjt = Zjt − Sjt (1.12)
Marginal Cost of Production
Given the production function, the (nominal) marginal cost of production of
intermediate goods firm j, denoted as Λjt, can be derived as:
W 1−αt
Λ αjt = at Yjt (1.13)
α exp α
where Wt is the aggregate nominal wage. Note that the higher is α, the less re-
sponsive is marginal cost to fluctuations in production. In the extreme case when
α is equal to one (constant returns to scale), marginal cost of production is not
13
influenced by the output level.
Intermediate Firm’s Problem without Information Friction
In the absence of pricing and informational frictions, the intermediate firm’s
problem can be characterized as:
∑∞ P Ξ { ∫0 t (Pjt ) }−θ )
max E βt0 Pjt min(vjt St, Zjt dG(v)−WtLjt (1.14)
{Zjt,Pjt} P Ξ P
t=0 t 0 t
subject to
[ (P ]− j,t−1 )−θ )Yjt = Zjt (1− δz) Zj,t−1 −min(v St−1, Zj,t−1 (1.15)
Pt−1
The firm’s expected profit is expected sales revenue net of the cost of production.
Let v∗jt denote the threshold taste shock level beyond which firm j would stock out
given goods available for sale Zjt. Then
v∗
Z
jt = (P )jt− (1.16)jt θS
P tt
Intuitively, when the firm has a higher stock of finished goods Zjt or less demand for
its goods, its threshold taste shock is higher and it is less likely to hit the stock-out
constraint.
14
Expected sales revenue can be derived as:
∫ ( ∫Pjt )−θ ) (Pjt )−θ
Pjt min(v [S∫t, Zjt dG(v) = P
∗
jt St min(v,]vjt)dG(v)( Pt Pt∗P ) v− jt σ2jt θ
= Pjt St vdG(v) + v
∗
jt(1− F[(log v
∗ + v ))
( P
jt
t 0 2
P 2jt )−θ ( σ ) σ2 ]
= P ∗jt StF log vjt − v + PjtZjt 1− F (log v∗ vjt + ) (1.17)Pt 2 2
2
where F (·) is the distribution function of N (0, σ2v), and 1 − F (log v∗ +
σv
jt ) is the2
probability of a stock out.
Decision Rules
Substituting the above expression for expected sales (1.17) into the firm’s prob-
lem and with some algebraic manipulation, I can obtain the optimality conditions
with respect to finished goods stock Z 5jt and price Pjt as:
( ) [ ]σ2 λjt − β(1− δz)E Ξt+1t λΞ j,t+1t
(∂Zjt) 1− F log v∗jt + v = [ ] (1.18)2 Pjt − β(1− δ )E Ξt+1λ
P( z) t Ξ j,t+1t tPjt jt β(1− δz) Ξt+1
(∂Pjt) = Et λj,t+1 (1.19)
Pt jt − 1 α Ξt
where λjt = Λjt/Pt is the real marginal cost of production, and
∫ ∫ v∗θ jt0 vdG(v)jt = ∗ (1.20)min(v, vjt)dG(v)
5 The derivation is relegated to the appendix.
15
is θ times the fraction of sales in all states in which the firm does not stock out.
Note that jt can be viewed as the firm’s effective elasticity of demand.
The economic intuition of the two optimal decision rules (equations (1.18) and
(1.19)) is as follows. The LHS of equation (1.18) is the probability of a stock out,
which decreases in Zjt. Note that the numerator on the RHS of equaion (1.18)
is the net cost of producing one more unit of goods this period (net of the cost
saving from the need to produce next period). The product of the denominator
and the LHS is the net benefit of producing one more unit of goods this period: a
higher Zjt means the firm is less likely to stock out and more likely to increase its
profit made from selling an additional unit of goods, which is the price minus the
expected discounted marginal cost next period. The intuition of (1.19)) is that the
firm’s optimal price is affected by the discounted marginal cost of production in the
next period. Furthermore, the higher is the effective elasticity of substitution, the
lower is the price-cost markup.
1.3.4 Labor Unions and Wage Rigidity
Staggered wage setting is modeled as in Smets and Wouters (2007).6 I as-
sume that households supply their homogeneous labor to intermediate labor unions,
indexed by l ∈ [0, 1]. The labor union differentiates labor services supplied by house-
holds, and sets the nominal wage Wt(l) for differentiated labor service Lt(l), subject
6 The purpose of incorporating sticky wages is to give rise to a delayed response of the real wage
(and thus of real marginal cost) to monetary shocks. Without frictions in wage setting, the real
wage is determined by equation (1.3), which tracks changes in output perfectly.
16
to a Calvo friction.7 Each labor union supplies differentiated labor to a competitive
labor packer, which packages differentiated labor through a composite function:
[ ∫ 1 ] θw
θw−1 θw−1
L θt = Lt(l) w dl , θw > 1
0
and supplies Lt to the intermediate goods firms with composite wage rate Wt. The
labor packer takes Wt and Wt(l) as given and chooses {Lt(l)}l∈[0,1] to maximize its
profit: [ ∫ 1 ] θ ∫w 1
θw−1 θw−1
max Wt Lt(l) θw dl − Wt(l)Lt(l)dl
{Lt(l)}l∈[0,1] 0 0
The FOC of the labor packer gives
(Wt(l))−θw
Lt(l) = Lt (1.21)
Wt
From the zero profit condition for the labor packer one obtains the wage cost for
the intermediate good producers:
[ ∫ 1 ] 1
1−θ 1−θW = W (l) w
w
t t (1.22)
0
Labor unions are subject to a Calvo wage setting friction, which is the source
of nominal wage rigidity in the model. In any given period, each labor union can
readjust Wt(l) with probability 1 − ξw. If the union is not allowed to adjust, the
wage rate it charges is indexed to past inflation rates, with degree of indexation `w.
7 Note that households supply homogenous labor services and receive homogenous wage rate Wht
in return. It is labor unions that differentiate labor services and set wages Wt(l) for differentiated
labor.
17
If the union is allowed to readjust, the problem is to choose an optimal wage W̃t(l)
that maximizes the discounted wage income in all future states in which the union
is not allowed to adjust (taking the wage rate paid to households {W ht+s} as given):
{∑∞ Ξ P [ ] }
maxE ξs βs
t+s t h
t w Wt+s(l)−Wt+s Lt+s(l)
W̃t(l) ΞtPs=0 t+s
subject to
(Wt+s(l))−θw
Lt+s(l) = Lt+s (1.23)
Wt+s
where
Wt+s(l) = W̃t(l)Xt,t+s (1.24)
 1 for s = 0,Xt,t+s =  Πsl=1π`wt+l−1 for s = 1, . . . ,∞
The first order condition is:
∑∞ Ξ P [t+s t ( )
(∂W̃t(l)) E ξ
s s
t wβ Xt,sLt+s(l)− θw W̃t(l)X −W ht,sΞ P t+s
s=0 t t+s ( )−θ −1 ]
· Xt,sW̃t(l)
w Xt,s
( )Lt+s = 0
Wt+s Wt+s
18
Substituting in equation (1.23), the above FOC can be simplified as:
∑∞
s sΞ
[ ]
t+sPt θw
Et ξwβ (θw − 1)L ht+s(l) W −Xt,sW̃t(l) = 0 (1.25)ΞtPt+s θw − 1 t+ss=0
Since all the unions allowed to adjust face the same problem and will set the same
optimal wage rate, I can write W̃t(l) as W̃t. The aggregate wage can be written as:
[
− 1−θ
1
w ]
Wt = (1 ξw)W̃t + ξ
`w 1−θw 1−θw
w(πt−1Wt−1) (1.26)
The above equations (1.25) and (1.26) give rise to the wage Phillips Curve, and
parameters ξw as well as `w control the degree of wage stickiness.
1.3.5 Government
At time t, the government pays back Bt−1, issues new bonds, Bt, collects
lump-sum tax Tt and consumes Gt. The government budget constraint is
Bt
PtGt +Bt−1 = Tt + (1.27)
Rt
where Gt is real government expenditure, assumed to be a constant fraction of
aggregate output:
Gt = g∗Yt
The monetary authority follows a feedback nominal interest rate rule that
responds to the lagged nominal interest rate, current inflation, and the current
19
deviation of output from its steady state level:
R ( )ρ [t Rt−1 R (πt )φ (Y ) ]φ 1−ρt Rπ
= Y exp(εr,t) (1.28)
R∗ R∗ π∗ Y∗
where εr,t is the interest-rate shock that follows
ε 2r,t ∼ N (0, σr)
.
1.3.6 Resource Constraint
Substituting the government budget constraint (1.27) into the household bud-
get constraint (1.1), and integrating across all households, one obtains:
W ht Lt Πt DivtCt +Gt = + + (1.29)
Pt Pt Pt
Aggregate profit earned by the intermediate goods firms, Πt, can be expressed as:
∫ 1 ∫ 1 ∫ 1
Πt = Πtjdj = PjtSjt − WtLjt
0 0 0
Using the inventory evolution equations (1.11) and (1.12), we have:
[ ]
Sjt = Yjt − invjt − (1− δz)invj,t−1
20
Thus, aggregate profit can be written as:
∫ 1 ∫ 1 [ ]
Πt = PjtYjtdj − Pjt invjt − (1− δz)invj,t−1 dj −WtLt (1.30)
0 0
where Lt is total labor supplied by the labor packer, which should be equal to
aggregate labor demand by firms:
∫ 1
Lt = Ljtdj (1.31)
0
Using the labor packer’s zero profit condition
∫ 1
W L = W (l)L (l)dl = W ht t t t t Lt +Divt (1.32)
0
the resource constraint (equation (1.29)) becomes
∫ 1 ∫P 1jt [ ]
Ct +Gt + invjt − −
Pjt
(1 δz)invj,t−1 dj = ( )Yjtdj (1.33)
0 Pt 0 Pt
Thus, market clearing requires aggregate output be equal to the sum of aggregate
consumption, government expenditure, and inventory investment.
1.3.7 Shocks
Other than the i.i.d. (across firms and time) taste shocks that follow lognormal
2
distribution log(v ) ∼ N (−σvjt , σ2v), there are two exogenous aggregate shocks in the2
21
model:
1. The interest rate shock εr,t in the nominal interest rate rule follows a Gaussian
white noise process: εr,t ∼ N (0, σ2r)
2. The aggregate technology shock at follows a first order autoregressive process
with mean zero:
at = ρ
2
aat−1 + εa,t, εa,t ∼ N (0, σa)
The innovations r,t and εa,t are mutually independent.
1.3.8 Intermediate Goods Firms : Rational Inattention
In the model presented in the previous subsection, decision makers in interme-
diate goods firms make optimal pricing and production decisions without frictions.
In the numerical experiments in section 4, I impose either a Calvo pricing friction or
an information friction on the intermediate goods firms, and compare the impulse
responses generated by these two assumptions. The information friction is modeled
as rational inattention, in that intermediate goods firms are inattentive to shocks,
as opposed to a setting in which the firms have perfect information but may not be
allowed to change prices in each period. The two settings are identical in all aspects
except for the exact friction affecting the firm’s decisions.
Inattentive firms are modeled as follows. I assume that intermediate firms
devote limited attention to aggregate shocks hitting the economy, which include
the aggregate TFP shocks and the nominal interest rate shocks.8 They optimally
8 The detailed description of the processes of aggregate TFP and the interest rate shocks is
22
allocate their attention to various sources of shocks, as well as to their pricing and
production decisions, subject to a flow of information constraint.9
Attention Problem
The decision maker in firm j chooses the amount of attention allocated to the
pricing and production decisions. He chooses the laws of motion for the price pjt
and stock of goods for sale Zjt to maximize the expected discounted sum of profit
minus the cost of paying attention:
{ [∑∞ (1
max E βt (x † ′j,−1 jt − xjt) Hx,0(xjt − x
†
jt)
κ,{Bk(L),Ck(L)}k=1,2 2
t=0 )] µ }
+ (xjt − x† ′ †jt) Hx,1(xj,t+1 − xj,t+1) − κ (1.34)1− β
subject to the law of motion for the optimal action under perfect information
x†jt = x
†R †a
jt + xjt (1.35)
= A1(L)εr,t + A2(L)εa,t (1.36)
where x† † † ′jt ≡ [pjt;Zjt] denotes the firm’s optimal actions with full information,10
which depend on current and lagged values of the underlying shocks to the inter-
deferred to section 2.8.
9 Since the taste shocks vjt, which intermediate firms cannot observe before making decisions,
are i.i.d. across firms and time, intermediate firms do not consider taste shocks when making their
decisions.
10 In this particular subsection, as well as in Appendix C, I abuse notation for ease of exposition to
let variables (except for shocks) with time subscripts denote log deviations from the non-stochastic
steady state values of the corresponding variables introduced in subsection 2.2. Therefore, x†jt and
xjt are vectors of log deviations, as is the vector ζt introduced later.
23
est rate and technology, and Hx,0 ≡ Hx ,x and Hx,1 ≡ Hx ,x are matrices ofjt jt jt j,t+1
second derivatives of the lifetime discounted profit function. In the above equa-
tion {Ak(L)}k=1,2 (as well as {Bk(L), Ck(L)}k=1,2 introduced below) are absolutely
summable matrices of lag operators. Note that the firm’s optimal decisions (and
actual decisions) do not need to respond to its idiosyncratic taste shock, since taste
shocks are i.i.d. across time and each period the firm has to make decisions before
the taste shock is realized.
Following Mackowiak and Wiederholt (2015), I assume that the decision maker
inherently cannot observe the realization of structural shocks εr,t and εa,t with full
precision, but faces idiosyncratic noise shocks denoted by νRjt and ν
a
jt respectively.
Those noise shocks are assumed to be mutually independent (and also independent
of the underlying structural shocks) Gaussian white noises with unit variances. The
decision maker chooses the law of motion for the actual actions (signals):
xjt = B︸ 1(L)εr,t︷+︷ C1(L)νRj︸t + ︸B2(L)εa,t︷+︷ C2(L)νaj︸t (1.37)
xR x
a
jt jt
where x ′jt ≡ [pjt;Zjt] denotes the actual decisions, which can be decomposed into
two parts: the actual responses to the interest rate shock xRjt and to the aggre-
gate productivity shock xa 11jt. The term κ is the information capacity limiting the
11 Note that this decomposition holds since I assume the structural shocks and noise shocks are
mutually independent.
24
information flow between the firm’s optimal decisions and actual decisions:
({ } { } )
I x†R †a ∞ R a ∞jt ,xjt ; xt=0 jt,xjt ≤ κ (1.38)t=0
( )
where I {x†jt}, {xjt} is the mutual information between the two stochastic pro-
cesses {x† ∞jt}t=0 and {xjt}∞t=0, and can be defined as the reduction in entropy:
(
I {x†
)
} { } 1
[ ]
jt , xjt = lim I(X
†)− I(X†|X)
T→∞T
1 [1 ( det Ω )]†
= lim log X
T→∞T 2 det ΩX†|X
where I(X†) denotes the entropy of the random vector of optimal actions up to
T − 1, X† = (x†0, . . . ,x
† ); I(X†T−1 |X) denotes the conditional entropy of X
† con-
ditional on the information of X = (x0, . . . ,xT−1); and ΩX† and ΩX†|X denote the
covariance matrix of X† and the conditional covariance matrix of X† conditional on
X respectively. The second equality follows from the assumption that the shocks
affecting the firm’s decision are Gaussian.
The sum of the first and second terms in the firm’s problem (1.34) is the second-
order approximation to the difference of lifetime discounted profit from following
the law of motion of the actual actions rather than that of the optimal actions.
For a decision maker in a firm, the benefit of paying more attention to different
sources of disturbances is that his actual actions (and thus the firm’s profit) become
closer to the optimal actions. However, paying more attention means the decision
maker needs to process more information, which incurs a per-period marginal flow of
25
information cost µ, assumed to be a constant. A decision maker compares the benefit
and cost of paying attention and optimally allocates information flow. Since the
marginal cost of information flow µ is assumed to be constant, and the disturbances
coming from different sources are assumed to be independent, the optimal amount
of attention allocated to each disturbance is independent of the amount of attention
allocated to other shocks. This simplifies the computation process as I can focus on
a particular shock when I simulate the firm’s optimal and actual decisions.
Note that it is implicitly assumed that the signals observed by the decision
maker of a firm will contain some noise, represented by νjt’s. Indeed, if firms can
observe εr,t and εa,t without effort, then the decision maker can easily follow op-
timal decision rules. The essence of inattention is that the decision maker can
endogenously choose the information structure of its signals through choosing the
polynomials of lag operators Bk(L) and Ck(L), k = 1, 2. If there is no information
capacity constraint, the decision maker can potentially make the signals extremely
precise, so that Bk(L) = Ak(L) and Ck(L) = 0, k = 1, 2. However, in the pres-
ence of the capacity constraint, doing so would incur too much information cost.
The firm will thus focus more on the shock that matters the most for its objective.
The following proposition characterizes the optimal decision rules x†jt ≡ [pjt, Z ′jt] as
driven by the vector of state variables exogenous to the firm ζt = [wt, St,Ξt, a ]
′
t (and
its one-period lead and lag), which the firm also cannot observe with full precision.12
12 Appendix C contains the derivation of the firm’s objective function (1.34) and the proof of
this proposition.
26
Proposition 1.
Let ζt = [wt, St,Ξ
′
t, at] denote the vector of variables exogenous to the firm. Optimal
decisions {x† ≡ [p†jt jt, Z
† ′ ∞
jt] }t=−1 of the firm are determined by the firm’s optimal
stock condition (equation (1.18)) and optimal pricing condition (equation (1.19)),
and satisfy:
(i) The economy starts from the non-stochastic steady state: x† ′j,−1 = (0, 0) .
(ii) In each period t ≥ 0, x†jt satisfies
[ ] [ ]
x† = H−1
1 † −1 −1 †
t x,0 Hx,1xt−1 +Hxζ,−1ζβ t−1
+Hx,0Hxζ,0ζt +Hx,0Et Hx,1xt+1 +Hxζ,1ζt+1
(1.39)
where Hx,−1 ≡ Hxjt,xj,t− , H1 xζ,0 ≡ Hxjt,ζ , Ht xζ,−1 ≡ Hx ,ζ − , and Hjt t 1 xζ,1 ≡
Hx ,ζ are Hessian matrices from the firm’s optimality conditions evaluatedjt t+1
at the steady state. Et[·] is the expectation operator conditional on all available
information in period t (including the evolution of {ζt} up to the current period
t).
Equation (1.39) can replace equation (1.36) and help pin down optimal decision
rules in our algorithm to solve for the equilibrium.
1.3.9 Equilibrium under Rational Inattention
Definition. The equilibrium of the model with inattentive firms consists of the
decision rules of households {Ct(i)}, {Lt(i)}, {Bt(i)}; firms {P † †jt},{Zjt}, {Pjt},{Zjt}
27
{Sjt}, {invjt}, {Yjt}, {Ljt}; labor unions {Lt(l)}l∈[0,1]; and the government {Gt},
{Bt}, {T ht}; aggregate allocations {Yt},{Lt},{Πt}, {Divt} and prices {Pt}, {Wt },
{Wt}; the realization of aggregate and idiosyncratic states {Rt}, {at} {vjt}, as well
as the realizations of firm specific signals, such that:
(i) Given the wage rate {W ht }, the nominal interest rate {Rt}, and the inflation
rate {πt}, each household’s decision rule {Ct(i)}, {Lt(i)}, {Bt(i)} satisfies her
optimality conditions (1.2) - (1.4).
(ii) For each intermediate goods firm j, given aggregate sales, St, prices {Pt},
wage {Wt}, and the realization of shocks and signals, the optimal decision
rules {P † †jt} and {Zjt} evolve according to equation (1.39), and the actual
decision rules {Pjt} and {Zjt} evolve according to equation (1.37); and the
absolutely summable coefficients {B(L), C(L)} maximize the firm’s objective
(1.34) subject to the information capacity constraint (1.38). Furthermore,
given the initial inventory holding invj,−1, {Pjt}, {Zjt}, {Sjt}, {invjt}, {Yjt},
and {Ljt} evolve according to equations (1.9)-(1.12).
(iii) Given aggregate labor demand {Lt}, aggregate price {Pt}, aggregate wage
{Wt}, the wage paid to households {W ht }, and the backward indexation scheme,
a labor union’s decisions {W̃t(l)}, {Wt(l)} and {Lt(l)} satisfy equations (1.23)
- (1.25).
(iv) Given initial bond issuance B−1 and initial nominal interest rate R−1, aggre-
gate output {Yt}, and the inflation rate {πt}, variables {Gt}, {Bt} and {Tt}
28
satisfy the government budget constraint (equation (1.27)), and {Rt} satisfies
the interest rate rule (1.28).
(v) The aggregate price {Pt} satisfies equation (1.8), and the aggregate wage {Wt}
satisfies equation (1.26).
(vi) Labor demand equals labor supply:
[ ∫ 1 ] θ ∫w 1
θw−1 θw−1
Lt(l) θw dl = Lt = Ljtdj
0 0
In addition, aggregate profit {Πt} satisfies equation (1.30); {W ht } and {Divt}
satisfy equation (1.32). And the aggregate resource constraint (equation (1.33))
holds (goods market clears).
1.4 Comparative Statics & Discussion
1.4.1 Comparative Statics
The two optimality conditions of the frictionless intermediate firms’ problem,
which are reprinted here for convenience as (1.40) and (1.41), deserve more elabora-
tion, since they will provide useful intuition for the comparison of model dynamics
under the two distinct frictions. Here I present the comparative statics of an interest
29
rate hike and a drop in the real marginal cost.
( [ ]σ2 ) λjt − β(1− δz)E [Ξt+1t λ∗ v Ξ j,t+1− t(∂Zjt) 1 F log vjt + = ] (1.40)2 Pjt − β(1− δ Ξt+1
P( z))Et λΞ j,t+1t tPjt jt β(1− δz) Ξt+1
(∂Pjt) = Et λj,t+1 (1.41)
Pt jt − 1 α Ξt
[ ]
When the real interest rate increases, the discount factor E Ξt+1t decreasesΞt
(through the household’s Euler equation). Therefore, the right-hand side of equation
(1.40) increases, which means that ceteris paribus the net cost of producing goods
becomes higher relative to the net benefit. The firm responds by decreasing Zjt,
increasing the probability of a stock out. Thus, production and the inventory stock
would fall. The intuition is that the real interest rate is the opportunity cost of
accumulating and carrying inventories. As the opportunity cost increases, firms draw
down their inventories. Conversely, if the current marginal cost λjt falls relative to
the expected future marginal cost, then all else equal the right-hand side of equation
(1.40) decreases, the net cost of producing goods decreases relative to the net benefit,
and firms produce more and hold more Zjt. It turns out that these inter-temporal
substitution effects make Zjt very sensitive to movements in the discount factor
and to movements in the relative marginal cost of production. Therefore, when
a monetary shock hits and the real interest rate goes up, for Zjt to decrease by
less than sales, current real marginal cost λjt must fall by more than the expected
discounted marginal cost next period. This is why Kryvtsov and Midrigan (2013)
reach the conclusion that for inventories to move slowly, marginal cost must be very
30
elastic on impact with respect to nominal disturbances.
Figure 1.1 plots the movement of the Zj surface, which I define as the LHS
of equation (1.40) minus the RHS, for different combinations of Zjt and relative
price Pjt/Pt, when there is a rise in the real interest rate, holding all other variables
exogenous to the firm at their steady state values (under a generic set of parameters).
The darker shape is the original Zj surface and the lighter shape is the surface after
the interest rate change. We can see that a higher interest rate shifts the surface
downward. Conversely, figure 1.2 shows the effect of a lower current real marginal
cost on the Zj surface. We can see that lower marginal cost shifts the surface from
its original position upward and to the right. These comparative statics confirm
the intuition described above that marginal cost must be elastic and fall sufficiently
following an interest rate hike to curb the movement of Zjt (and thus the change in
inventories).
It is clear from equation (1.41) that, characteristic of a monopolistically com-
petitive environment, the firm’s optimal price is a markup over the expected value
of discounted real marginal cost in the next period. The difference is that the ef-
fective elasticity of substitution jt is θ times the share of sales in states in which
the firm does not stock out, which reflects the non-linearity coming from the stock
out constraint. The firm’s markup depends on the probability of a stock out, which
decreases with a higher threshold level v∗jt as firm j has higher Zjt for sale. The
optimal price depends on expected next-period marginal cost instead of current
marginal cost. Movements in real marginal cost change the shadow value of goods,
and the firm optimally adjusts its price in the absence of frictions. Figure 1.3 shows
31
Fig. 1.1: Production Decision: Comparative Statics of Higher Real Interest Rate
Fig. 1.2: Production Decision: Comparative Statics of Lower Real Marginal Cost
32
the effect of a lower marginal cost of production on the Pj surface, which I define
as the LHS of equation (1.41) minus its RHS. The darker shape is the original Pj
surface and the lighter shape is the surface after the drop in real marginal cost when
all other variables exogenous to the firm are kept at the steady state levels. We
can see that a drop in marginal cost shifts the surface upward, so ceteris paribus
movements in real marginal cost push the firm’s price in the same direction.
Fig. 1.3: Pricing Decision: Comparative Statics of Lower Real Marginal Cost
Another result is that the quantity of Zjt affects the firm’s optimal markup.
When Zjt falls, the threshold level v
∗
jt decreases, increasing the probability of a
stock out. This reduces the effective elasticity of demand jt, and thus increases
the markup. This fact is reflected by the negative slope of the line representing the
intersection of the Pj surface with the zero plane. When marginal cost is rigid, the
33
optimal price can actually increase after a tightening monetary shock due to the
endogenous change in the desired markup induced by the stock-out constraint.13
Having established the effects of a change in the real interest rate and a change
in the marginal cost on the optimality conditions w.r.t. Zj and Pj, I combine
the two optimality conditions in two-dimensional graphs to show the equilibrium
comparative static effects. Figure 1.4 shows the equilibrium ceteris paribus effect of
a rise in real interest rate, where the darker lines are the (Zj, Pj) combinations that
satisfy optimality condition (1.40) before and after the rise in the real interest rate,
and the lighter lines correspond to the optimality condition (1.41). One can see that
a rise in the real rate pushes both the Zj line and Pj line to the left. This will cause
a drop in Zj (and thus production and ex post inventory stock), but the equilibrium
price does not necessarily drop. Figure 1.5 shows the ceteris paribus effect of a drop
in the real marginal cost. This will push the Zj line to the right and push the Pj
line further to the left, causing a drop in the equilibrium price. This is the reason
that a drop in marginal cost can balance the effect of a rise in the real interest rate
on Zj. However, it also inevitably creates cost pressure for firms to mark down their
price. The information capacity constraint introduced in subsection 2.3 will resolve
this conundrum by making the actual responses of firms’ decisions deviate from the
comparative statics depicted here. In other words, actual movements in Zj and Pj
lines will be smaller under information capacity constraints.
13 This mechanism implies that the rational inattention model, under rigid cost of production,
can potentially account for the “price puzzle”, the perverse temporary rise in the price level after
monetary tightening that has been documented in some empirical studies.
34
1.015  
Zj line
1.01 Pj line
Zj line high ir
P line high ir
1.005 j
1
0.995
0.99
0.985
0.98 
4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.5
Zj
Fig. 1.4: Comparative Statics of Higher Interest Rate
1.015  
Zj line
1.01 Pj line
Zj line low mc
P line low mc
1.005 j
1
0.995
0.99
0.985
0.98 
4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.5
Zj
Fig. 1.5: Comparative Statics of Lower Real Marginal Cost
35
Pj/P Pj/P
1.4.2 The Role of Information Frictions
The above exposition and comparative statics establish that when the real in-
terest rate moves, the firm would adjust Zj abruptly in the opposite direction unless
there is a corresponding offsetting movement in real marginal cost. In other words,
theory predicts a tight connection between inventory dynamics and the movement
of marginal production cost following an interest rate shock. Common frictions in
the literature to regulate the movement of inventory investment, such as decreasing
returns to scale, labor adjustment costs, and the quadratic cost of deviating from a
target stock-sales ratio, all serve to increase the elasticity of the marginal cost to the
quantity of output, and thus all imply strong procyclicality of marginal production
cost. This is why inventory behavior is seen as an evidence for elastic marginal
cost and countercyclical markup variation (Kryvtsov and Midrigan (2013)). In this
paper, I argue that information frictions serve to limit the response of inventories
without requiring marginal cost of production to be elastic. Without perfect infor-
mation about real input prices and the real discount factor, Pjt and Zjt (and thus
the inventory stock) do not fully adjust in the short run to the relative movements
in the real interest rate and the marginal cost of production.
The intuition is quite simple. To make the inventory decision, firms not only
need to observe their current demand and input prices, but they also need to know
the aggregate price level and need to form expectation about future marginal costs.
If firms cannot observe the aggregate variables in real time or cannot disentangle
the exact sources of the shocks pushing aggregate variables, their reactions will be
36
delayed. To fix ideas, consider for the moment a sticky information environment à
la Mankiw and Reis (2002), in which firms update their information sporadically.
Suppose that each period a firm can update its information with probability (1 −
φ), 0 < φ ≤ 1. Therefore, each period there are a mass of (1−φ)φτ firms who make
decisions based on stale information sets with vintage τ, τ = 1, 2, . . . ,∞. Assume
α = 1 so that all intermediate firms face the same marginal cost of production. After
log linearization of the optimality condition w.r.t Zj (equation (1.40)) and summing
over all firms, I get the following expression for the log deviation of the stock-sales
ratio from its steady state value:
∑∞
−η(zt −
1
st) = (1− φ) φτEt−τ (mct)
1− β(1− δz) τ=0
∞
(p∗ −mc∗)β(1− δz) ∑
+ (1− φ) φτE (rr −mc )
[1[− β(1− δz)]∑[p∗ −
t−τ t t+1
β(1− δz)mc∗] τ=0
∞ ]
+ η st − (1− φ) φτEt−τ (st)
τ=0
where lower case letters denote log deviations from the steady state values, and mct
and rrt denote the log deviations of the real marginal cost λt and the real interest
rate respectively. η > 0 is a coefficient whose value depends on the parameters of the
model. The final term on the right hand side is an expectation error term. When
φ = 0 the above equation collapses to its full information counterpart. The larger
is φ, the stickier is the information and Zj responds less to the relative variation in
the real interest rate and the real marginal cost.
Instead of assuming sticky information, this paper follows Sims (2003) in as-
37
suming that firms face an information processing capacity constraint and are ra-
tionally inattentive. Different sources of shocks (such as productivity shocks and
interest rate shocks) may imply different persistence of the variation in marginal
cost, and firms need to disentangle the exact sources and magnitudes of the struc-
tural shocks to correctly forecast future marginal cost. Even if firms can observe
the nominal interest rate Rt, they need to forecast πt+1 to know the real interest
rate they face. In short, intermediate firms observe aggregate variables with noise.
They make decisions Pj and Zj based on imprecise information, and due to the
information capacity constraint their actual decisions deviate from their first-best
actions (defined as the actions under no information capacity constraint, P †j and
Z†j ). Therefore, even if the aggregate conditions dictate a significant drop in Z
†
j (for
example, after an increase in the interest rate), the firm’s actual change in Zj may be
smaller. In the appendix, I derive a linear quadratic representation of the distance
of the firm’s actual profit from its first-best profit, which gives us the firm’s objective
under rational inattention (1.34). The distance depends on P − P †j j , Z
†
j − Zj , and
the interaction between the two decisions. Other than being less ad hoc than the
sticky information assumption, a rational inattention setting enables one to analyze
how the firm allocates attention (information flow) between Pj and Zj.
38
1.5 Empirical & Numerical Results
1.5.1 Empirical Results: Vector Autoregression
I estimate the dynamic response of key aggregates to monetary policy shocks
in the data using a vector autoregression (VAR). The specification is based on a
standard monetary VAR that appears in Christiano et al. (1996) and Evans and
Marshall (1998), augmented by two variables pertaining to inventory accumulation,
namely the inventory-sales ratio and finished good inventories in the manufacturing
and trade sector. It is assumed that the monetary policy instrument is the Fed
Funds rate, denoted as FFt, which is determined by the following relationship
FFt = f(Ωt) + r,t
where f is a linear function that describes the monetary authority’s reaction to the
state of the economy, Ωt is the information set available to the monetary authority
at time t, and r,t is an exogenous shock to the monetary policy reaction function,
which reflects non-systematic factors that affect monetary policy decisions. Define
39
Yt as the 9-dimensional vector of the variables included in the VAR:
  ln(Real GDPt)   ∆ln(PCE Price Index )  t 

ln(Non-Durable Consumption )  t ln(Inventoriest/Sales  t
) 


 ln(Inventories  t
) 
Yt =  ln(Unit Labor Cost ) 
t
 ln(Non-Fin. Corporate Profit/GDP )  t   (PPI Commodity Annual Growth t
) 
  

FF
 t 

(NBR/TRt) 
(M2 Money Annual Growtht)
The last two variables in Yt are the ratio of non-borrowed reserves to total reserves
of depository institutions and the log annual growth rate of monetary aggregate M2.
The information set available to the monetary authority at time t, Ωt, includes time
t values of real GDP, the PCE deflator, the inventory-sales ratio, the finished good
inventory stock, unit labor cost, and the commodity price index. As in Christiano et
al. (1996), r,t is assumed to be orthogonal to all variables in Ωt. Therefore the iden-
tification strategy is recursive because the variables in Ωt are not contemporaneously
affected by r,t.
I use quarterly data and construct a VAR with 3 lags. The lag length is chosen
40
by a corrected Akaike information criterion developed by Hurvich and Tsai (1993)
specifically designed for VARs. The reduced-form VAR is specified as the following:
∑3 ∑3
Yt = Γ0 + ΓkYt−k + ut = ΓkYt−k +Det (1.42)
k=1 k=1
where Γ0 is an 11 × 1 vector of constants; Γ1, Γ2, and Γ3 are 11 × 11 matrices; et
is an 11 × 1 vector of serially uncorrelated structural shocks, and D is an 11 × 11
matrix. Because of the recursive identification assumption, D is assumed to be a
lower triangular matrix, which can be obtained from a Cholesky decomposition of
the variance-covariance matrix :
DD′ = E[u u′t t]
Most data series come from the FRED database, including Real GDP (FRED
code GDPC96), the personal consumption expenditure price index (FRED code
PCEPI), the producer price index for all commodities (FRED code PPIACO),
non-borrowed reserves, total reserves, and monetary aggregate M2. Sales are real
sales of the manufacturing and trade sector, obtained from NIPA underlying de-
tail tables 2AUI and 2BU. Inventories are real finished-goods inventories of the
manufacturing sector plus the inventories of the trade sector, also obtained from
the NIPA database.14 Unit labor cost is constructed following Gali and Gertler
(1999). Non-Financial corporate profits before tax come from the FRED database
14 Inventories of the manufacturing sector and the trade sector account for 85% of U.S. inventory
stocks. The rest are in mining, utilities, and construction.
41
(Code:A464RC1Q027SBEA). I use the augmented Dickey-Fuller test on each series
to check for possible unit roots. For those series for which I cannot reject the null
hypothesis of the existence of a unit root, I distill their transient components as the
2-year ahead projection errors, as suggested by Hamilton (2018). Series that are
free from a unit root include changes in the ln(PCE index), ln (Inventory/Sales),
and the PPI commodity annual growth rate.1516
Figure 1.6 reports the impulse responses to a 60 basis-point (annualized) tight-
ening monetary policy shock estimated using data from the 1st quarter of 1970 to
the 4th quarter of 2007.17 The shaded areas around the point estimates are 95% con-
fidence intervals obtained from 5,000 bootstraps. I also plot the impulse responses
generated by the local projection method of Jordá (2005) (the bubbled line) as a
robustness check. A monetary tightening shock decreases real GDP and pushes the
inventory-sales ratio up. The point estimates for the response of inventory stock
appear to fall after 2 quarters, although this response is not statistically significant
as the confidence interval contains zero. The inventory-to-sales ratio rises and peaks
in the second quarter at around 0.35% above its steady state level. The rise in the
inventory-sales ratio is statistically significant. The inflation rate has a very stag-
nant response. The point estimates indicate a falling inflation rate, with a trough
15 A common practice in the literature to treat possibly non-stationary series is to use the Hodrick-
Prescott Filter. Hamilton (2018) points out possible problems associated with HP-filtering and
suggests constructing projection errors by running linear regression on lagged values to avoid the
problems.
16 An alternative is to take a first difference of the possibly I(1) series. The VAR results thus
obtained are qualitatively the same as the results from HP- filtering the series, but the responses
of output and consumption become less persistent.
17 The 60 basis point magnitude of the Fed Funds Rate shock is in line with the ones adopted in
Christiano et al. (2005) and Christiano et al. (1996).
42
10 quarters after the shock at around −0.03%, but the confidence bands indicate
that the responses are not statistically different from zero. The response of unit
labor cost does not exhibit strong procyclicality (unit labor cost actually appears to
rise following the tightening monetary shock). The point estimates suggest that the
non-financial corporate profit margin decreases after the tightening, and it is statis-
tically significant at the third quarter. In summary, the inventory-sales ratio indeed
goes up, but judging from the responses of unit labor cost and corporate profits, one
cannot find evidence for elastic marginal cost or a rising price-cost markup after the
monetary tightening. As a robustness check for the behavior of the inventory-sales
ratio, I run the same VAR for the post-Volcker period, from Q1 1982 to Q4 2007.
The results are depicted in figure 1.7. The point estimates of the impulse responses
are similar, although the confidence bands are wider compared to figure 1.6 due to
a smaller sample size.
1.5.2 Calibration & Estimation
The parameter values used in the numerical experiments are listed in table 1.1.
The parameters used in either the Calvo Pricing model or the rational inattention
model (abbreviated as RI henceforth) are divided into 2 subsets. The first subset
are either chosen on the basis of existing literature, or are calibrated to match the
steady state values of the model to long-run averages of the U.S. data. They include
{β, µ, θ, θw, ψl, g∗, δz, σv, ρR, φπ, φy, h}. It is worth mentioning that in the numerical
experiments presented below, the Calvo sticky price model and the RI model share
43
 GDP                   Inflation             Consumption           Inventory−Sales Ratio
0.2 0.08
0.5
0.1
0.06 0.40.1
0.04 0 0.30
0.02 0.2
−0.1
0 −0.1 0.1
−0.2 −0.02
−0.2 0
−0.3 −0.04
−0.1
−0.4 −0.06 −0.3 −0.2
−0.08
0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15
 Inventories           Unit Labor Cost       Non−Fin Profits/GDP   PPI Commodity Growth 
0.3 0.2 2 0.4
0.2 0.15
1
0.1 0.1 0.2
0 0.05 0
−0.1 00
−0.2 −0.05 −1
−0.2
−0.3 −0.1
−2
−0.4 −0.15 −0.4
−0.5 −0.2 −3
0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15
Fig. 1.6: This figure displays the impulse responses to a 0.60% (annualized) shock to
the Fed Funds Rate from estimated recursive vector autoregression (VAR). The
shaded areas are 95% confidence bands, obtained by 5,000 bootstraps. The
bubbled lines show the points estimates of impulse responses using the local
projection method of Jordá (2005).
44
 GDP                   Inflation             Consumption           Inventory−Sales Ratio
0.15
0.2 1
0.1 0.1
0.1 0.05
0.05
0 0 0.5
0 −0.05
−0.1
−0.05 −0.1
0
−0.2 −0.15
−0.1
−0.3 −0.2
−0.15 −0.25 −0.5
0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15
 Inventories           Unit Labor Cost       Non−Fin Profits/GDP   PPI Commodity Growth 
 
0.6 0.3 3
0.4 0.2 2 0.5
1
0.2 0.1
0 0
0 0 −1
−0.1 −2 −0.5−0.2
−3
−0.4 −0.2 −4 −1
−0.6 −0.3 −5
 
0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15
Fig. 1.7: This figure displays the impulse responses to a 0.60% (annualized) shock to
the Fed Funds Rate from estimated recursive vector autoregression (VAR). The
shaded areas are 95% confidence bands, obtained by 5,000 bootstraps. The
bubbled lines show the points estimates of impulse responses using the local
projection method of Jordá (2005).
45
the same parameter values for parameters in the first subset (if applicable). The
remaining set of parameters are estimated, and will differ across those two models in
our numerical exercise. Parameters in the second subset are {ξp, `p, ξw, `w, σc, σl, α}.
Those parameters are chosen to be estimated because they either govern the behavior
of firms and households or have implications for model selection and comparison.18
They control the degree of price and wage stickiness, the returns to scale of firms’
production function (which also governs the degree of strategic complementarity),
and the household’s inter-temporal elasticities of substitution and labor supply.
The predetermined parameter values are determined as follows. The discount
factor β is set equal to 0.9898, consistent with long-run averages of the Fed Funds
Rate and the inflation rate. g∗ = 0.16 is consistent with the long run average ratio
of government final consumption expenditure to output in the United States. I set θ
and θw, the elasticities of demand for differentiated goods and labor, to be equal to
5, following Kryvtsov and Midrigan (2013). This implies that the firm’s markup is
20%. The standard deviation σv of the idiosyncratic taste shock is set equal to 0.33
and the quarterly depreciation rate of inventories δz is 1.8%. These two values jointly
determine the quarterly frequency of stock outs as 16.3%, and the steady state end-
of-period inventory-sales ratio as 0.498. Both values, when converted to monthly
terms, roughly match those considered in Kryvtsov and Midrigan (2013). Also, the
inventory depreciation rate δz implies an average change in the ratio of finished good
inventories to output of 0.9%, which roughly matches the corresponding ratio in the
18 The habit persistence parameter h governs the response of consumption and is a source of real
rigidity in both models. To make the comparison between the information friction model and the
staggered pricing model clear, I let both models have the same habit persistence.
46
data for the U.S. manufacturing and trade sectors. For the RI model, the marginal
cost of flow of information µ is set equal to 0.0525% of the intermediate goods firm’s
sales each period. This value is lower than the 0.058% adopted in Mackowiak and
Wiederholt (2015). The monetary policy rule coefficients ρR, φπ, and φy, as well as
exogenous habit formation in consumption h, come from Smets and Wouters (2007).
Estimation
I use U.S. quarterly data to estimate the second set of parameters. The es-
timation strategy is to match model implied impulse response functions of chosen
variables to the impulse responses of the estimated VAR model described in the
previous subsection. The estimation method can be summarized as follows. Let
ϑ = [ξp, `p, ξw, `w, σc, σl, α] denote the vector of estimated parameters, and let Ψ(ϑ)
be the model impulse response functions depending on the parameter vector. Let
Ψ̂ denote the impulse responses generated by the VAR. The estimation criterion is
J = min(Ψ̂−Ψ(ϑ))′ V −1 (Ψ̂−Ψ(ϑ)) (1.43)
ϑ
where V is the sampling uncertainty of Ψ̂. I use a weighted variance-covariance
matrix to construct V . Let n be the number of variables whose impulse responses
are included in Ψ̂, and m denote the number of periods tracked for each variable’s
impulse response. Then Ψ̂ can be written as a 1× nm vector:
Ψ̂ = [τ1,1, . . . , τ1,m, τ2,1, . . . , τ2,m, . . . , τn,1, . . . , τn,m]
47
The matrix V is constructed to be the following:
 Σ1 0 . . . 0
 0 Σ 0 0 

 2 

V =  . 0 0 . . 0 

0 0 . . . Σn
where submatrices Σi along the diagonal are the variance-covariance matrices of
the individual impulse responses for each variable. Therefore, I consider only the
covariance of each individual variable’s impulse responses across periods, but omit
the covariance in the impulse responses between any two different variables. For
each Σi, i = 1, . . . , n, I use a weighted variance-covariance matrix of the following
form:  
σ
 2 σi,12 i,1(m−1) σσ . . . i,1m i,1 2 m−1 m   σi,21 2 σσi,2 . . . i,2m 
Σ = 
2 m
 . 

i ..
. . .
.. . . .. 
 σ σ2 i,(m−1)mi,m−1 m 
σi,1m . . . σ2
m i,m
The weighted variance-covariance matrices account for the idea that it is more im-
portant to match the first periods of the theoretical and empirical impulse responses
rather than the later ones. Each Σi can be estimated by applying the Monte Carlo
method. I use the originally estimated VAR model to simulate a series of artificial
data. Then, I use the artificial data to reestimate the VAR model and generate
the impulse responses. This serves as one Monte Carlo draw. I run 5,000 Monte
48
Carlo draws and Σi, i = 1, . . . , n can be obtained from the variation in the impulse
responses of each variable across all draws.
I estimate the second set of parameters ϑ for the Calvo Sticky Price model.
Due to the computational burden of the RI model, it is not viable to run the im-
pulse response matching exercise. I thus apply the values of the posterior mode of
corresponding parameters reported in Smets and Wouters (2007). In particular, I
estimate parameters to match the Calvo Pricing model and VAR impulse responses
for 6 variables, including output, inflation, non-durable consumption, manufacturing
and trade inventory-sales ratio, finished-good inventories, and the real wage (unit
labor costs). I then apply the estimated parameters to the RI model, provide an
initial guess for the first-best decisions {P † }(1) and {Z† }(1)jt jt , and let the firm choose
the precision of their decisions by choosing B(L) (the response to current and lagged
realizations of structural shocks) and C(L) (the response to current and lagged id-
iosyncratic noises) that maximize the firm’s objective (1.34), and thus obtains the
actual decisions {P (1) † (1)jt} and {Zjt} . Then using equation (1.39) in Proposition 1,
I obtain the second guess for first-best decisions {P † }(2)jt and {Z
† (2)
jt} , and run the
optimization of the firm’s objective again to obtain {P (2) (2)jt} and {Zjt} . I keep
iterating until successive guesses for the first-best decision rules converge and the
equilibrium is obtained. I then generate impulse responses and simulate various
moments from the RI equilibrium. The horizon of each variable’s response being
matched is chosen as m = 16.19
19 Due to the identification assumption for the Fed Funds Rate shock in the VAR, the impulse
responses in the first period cannot be matched, so for each variable I generate 16 periods of
impulse responses and try to match 15 periods, starting from the second period.
49
Note that through this estimation exercise, I give the Calvo Pricing model the
best possible chance to fit the data impulse responses, by searching over the feasible
range of parameters to obtain parameter values that best fit 6 impulse response
functions estimated from the VAR. For computational reasons, I cannot directly
estimate the parameters of the rational inattention model using the same impulse
response matching exercise. Instead, I handicap the rational inattention model by
assigning it parameters that come from Smets and Wouters (2007), which would
give the model higher cost rigidity (through higher values of ξw, `w, and α). This
procedure builds in an advantage for the Calvo Pricing model over the RI model,
since the RI model’s goodness of fit can potentially be improved further by searching
over the parameter space.
The lower half of table 1.1 lists estimated parameter values. From the esti-
mated parameters listed in the lower half of the 2nd column, notice that the impulse
response matching exercise of the Calvo Pricing model obtains a low degree of wage
stickiness, making the real wage and the real marginal cost responsive to the interest
rate shock. This is because the model tries to match the responses of finished-good
inventories and the inventory-sales ratio by incorporating an elastic marginal cost of
production (and thus a countercyclical price markup) to hinder excessive inventory
accumulation, corresponding to the intuition depicted in the comparative statics
(figure 1.4 and figure 1.5). On the other hand, the exercise also obtains an α lower
than 1, meaning that there are decreasing returns in production. Lower α implies
more strategic complementarity in firms’ pricing and production decisions. With
an (individual) elastic marginal cost schedule firms will be more reluctant to adjust
50
prices in response to cost pressure, since a smaller α means that a given magnitude
of price adjustment brings about a larger change in the firm-specific marginal cost
and profit. One may wonder why the model estimates imply a relatively low degree
of price rigidity (ξp = 0.5059 and `p = 0.01). This is because the model also needs
to match the response of unit labor cost. If prices are too sticky, the response of
the real wage will be too dramatic and deviate too much from the empirical impulse
responses, as the nominal wage is very flexible in the estimated model.
RI Calvo Pricing Description
Predetermined:
1 1
β 0.96 4 0.96 4 Discount factor
µ 0.0525% – Flow cost of info. (% of sales)
θ 5.0 5.0 Kryvtsov and Midrigan (2013)
θw 5.0 5.0 Kryvtsov and Midrigan (2013)
δz 1.8% 1.8% Inventory depreciation (quarterly)
σv 33% 33% s.d. of demand shock.
ψl 1.2 1.2 Labor disutility coefficient
g∗ 0.16 0.16 Avg. gov’t consumption/GDP
ρR 0.81 0.81 AR(1) coefficient of interest rate rule
φπ 2.03 2.03 Policy response to πt
φY 0.22 0.22 Policy response to output gap
π∗ 1.0 1.0 Steady state gross inflation
h 0.71 0.71 Consumption habit persistence
SW (2007): Estimated:
ξp – 0.5059 Degree of price stickiness
`p – 0.01 Degree of price indexation
ξw 0.82 0.1398 Degree of wage stickiness
`w 0.69 0.1155 Degree of wage indexation
σc 1.3981 1.2469 Inverse of IES
σl 0.40 0.3823 Inverse Frisch elasticity
α 1.0 0.8633 Returns to scale
Tab. 1.1: Parameter Values
51
1.5.3 Discussion
I apply the parameters listed in table 1.1 for the Calvo Sticky Price model
and RI model, run numerical experiments, and compare the results obtained from
the two different assumptions about the frictions affecting the intermediate goods
firm’s price setting and production decisions. Note that compared to the Calvo
Pricing model, the parameter values applied to the RI model entail a much higher
cost rigidity. Real wages are highly sticky, and there is constant short-run returns to
scale in production. In the first numerical experiment, the economy starts from the
non-stochastic steady state and is perturbed by a positive 0.60% (annualized) shock
to the nominal interest rate. I compare the model generated impulse responses and
the 95% confidence bands of the estimated VAR model (the sample period is from
Q1 1970 to Q4 2015 ) described in section 4.1.
Comparing Theoretical Models to the Estimated VAR: Impulse Re-
sponse Matching
Figure 1.8 compares the impulse responses generated from the RI model and
the Calvo Pricing model to those of the VAR, applying the parameter values listed
in table 1.1. The first thing to note is that the response of output generated by the
RI model is more in line with the empirical impulse response compared to the Calvo
pricing model. The Calvo pricing model generates an immediate decline of output
of around 0.12% on impact of the shock, followed by a rapid rebound back to the
52
steady state value. On the contrary, the response of the RI model is larger and more
persistent, with output falling initially by 0.18% and remaining below steady state
for 15 quarters. The two models also generate significant disparities in the responses
of inflation and the real wage. The response of the real wage is more modest and
stagnant in the RI model, while the real wage falls by more than 0.25% on impact
in the Calvo Pricing model, because of the estimated low degree of wage stickiness
(ξw = 0.1398). This translates to a much more dramatic response of real marginal
cost in the Calvo Pricing model than in the RI model.20 As we can see, the drop in
real marginal cost on impact in the Calvo Pricing model is almost 6 times as large
as in the RI model. Thus, the cost pressure in the Calvo Pricing model makes its
inflation response more pronounced and short-lived. Inflation goes down by about
0.14% in the Calvo Pricing model, while for the RI model the inflation response is
extremely modest and lies within the 95% confidence bands of the VAR impulse
response for most periods, partly due to the stagnant real marginal cost, and partly
due to the rigidity generated by the information processing constraint. In the last
subplot I plot the response of real profits (defined as the price-cost markup times the
amount of sales) generated from the two models. It is clear that the Calvo Pricing
model generates a sharp rise in the firm’s real profit, whereas in the RI model the
response of real profit is mild but much more persistent.
Compared to the Calvo model, the RI model has a larger and more persistent
consumption response, in line with data, even though the parameter of habit per-
20 Since RI model is constant returns to scale (α = 1), the response of the real wage equals the
response of its marginal cost of production.
53
sistence is the same across the two models. Once again, this is due to the disparity
in the elasticities of real marginal cost. As expected, the more elastic marginal cost
in the Calvo model makes the real effects of monetary shocks more short-lived. Re-
garding inventory dynamics, consistent with the data, both models generate a rising
inventory-sales ratio and a declining inventory stock after the monetary tightening
shock, but the RI model’s response is larger and more persistent. Comparing the
overall goodness of fit, the value of the estimation criterion is 221.6 for the Calvo
Pricing model and 146.0 for the RI model. In other words, the deviation from the
VAR impulse responses implied by the Calvo Pricing model is 52% larger than that
of the RI model. Thus, in terms of matching impulse responses, the RI model with
an inelastic response of marginal cost fits the data better than the Calvo Pricing
model. Table 1.2 provides a decomposition of the contribution to the estimation
criterion for the 6 matched series across the two models. The RI model performs
better for 5 out of the 6 series being matched. Not only does it match the inventory-
sales ratio and inventory dynamics better, it also excels significantly in matching
the persistent responses of output, inflation, and consumption. Note that the two
models have the same parameter value for habit persistence, but the slow-moving
marginal production cost increases the overall response persistence in the RI model.
The reason that the Calvo model performs better in matching the unit labor cost
is mainly due to the abrupt and short-lived response of the real wage in the Calvo
model, and due to the limitation that the immediate large deviation when the shock
hits is not accounted for in the estimation criteria because of the identification spec-
ification in the VAR (that the Fed Funds Rate shock does not immediately affect
54
the variables ranked before the FFR).
RI Calvo Pricing
(figure 1.8)
Output 30.90∗ 56.12
Inflation 18.06∗ 38.39
Consumption 8.83∗ 39.45
Inventory-Sales 10.93∗ 22.00
Inventories 36.96∗ 38.78
Unit Labor Cost 40.29 26.87∗
Sum 146.0∗ 221.6
Tab. 1.2: Decomposition of Estimation Criterion
To make the comparison between the two models clearer, I re-estimate the
Calvo Pricing model by matching output, inflation, consumption, and unit labor
cost, dropping inventories and inventory-sales ratio. The results are presented in
figure 1.9 (which also lists the estimated parameter values). We can observe that if
the Calvo Pricing model is not required to match the inventory dynamics, the model
would want to parameterize a high degree of nominal price and wage stickiness.
The degree of wage stickiness even surpasses price stickiness, resulting in inertial
marginal cost. However, the Calvo Pricing model will perform poorly in matching
the inventory dynamics (the 4th and 5th subplots).
The RI model under an acyclical price markup achieves a better fit than the
Calvo model because rational inattention on the firm side makes the firm’s actual
decisions deviate from its first-best decisions. The information friction adopted
in this paper serves as a mechanism to limit the response of inventory invest-
ment/disinvestment and production to nominal shocks, and thus the model does
not need variation in marginal cost and price markups to counteract the effect on
55
inventories of real interest rate movements as in a model of full information. Figure
1.10 plots the actual response of stock for sale Zjt (the solid line) in the RI model
to a positive 0.60% (annualized) interest rate shock and the corresponding first-best
structural response Z†jt (the dash-dotted line). In the absence of information con-
straints, the firm should decrease Zjt by about 1.0% on impact. The firm’s actual
adjustment is much smaller compared to its optimal rule. Figure 1.11 depicts the
same comparison between the actual and the first-best responses for prices. The
deviation of the actual price response from the first-best response has significant
persistence and dies out after 25 quarters. Note that the magnitude of the distance
Zjt −Z†jt is much larger than the magnitude of P
†
jt − Pjt. This is because the firm’s
profit is much more sensitive to its pricing decision than to its Zj decision around
the steady state. With our benchmark parametrization, the value of the Hessian
matrix Hx,0 in the firm’s objective (1.34) is:
 
−3.418 −0.365Hx,0 = 
−0.365 −0.088
The profit deviation is more than 34 times more sensitive to (P − P † )2jt jt than to
(Zjt − Z†jt)2. The reason is that the pricing decision determines the demand for
the firm’s goods and thus both directly determines the firm’s revenue and indirectly
affects the probability of a stock out, while Zjt only affects the firm’s current period
profit indirectly, by affecting the marginal probability of a stock out, multiplied by
the forgone price markup (profit) in the case of a stock out. As the probability of
56
a stock out is not high in the steady state, this term has a small value. Therefore,
firms tend to let Zjt deviate from Z
†
jt by more.
57
Value of Estimation Criterion-
RI: 146.0, Calvo Pricing: 221.6
 Output                Inflation              Consumption          Inventory−Sales Ratio
0.5
0.2
0.1
0.05 0.4
0.1
0 0.30 0
0.2
−0.1 −0.1
−0.05 0.1
−0.2
−0.2 0
−0.3 −0.1
−0.1
−0.4 −0.3
−0.15 −0.2
0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15
 Inventories           Unit Labor Cost       Real Marginal Cost    Real Profit          
0.3 0.2  
0 0.5
0.2
0.1
0.1 −0.05 0.4
0 −0.1
0 0.3
−0.1
−0.15
−0.2 −0.1 0.2
−0.2
−0.3
0.1
−0.4 −0.2 −0.25
−0.5 −0.3 0
 
0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15
 RI model  Calvo Pricing model  VAR
Fig. 1.8: This figure compares the impulse responses generated by the Rational Inat-
tention model (the solid lines), the Calvo Pricing model (dashed line), and the
vector autoregression (VAR) model (the dashed-cross lines). The areas within
the dashed lines are the 95% confidence bands obtained from 5,000 bootstraps.
58
Estimated Parameters (matching 4 variables):
ξp = 0.854, `p = 0.590, ξw = 0.953, `w = 0.882, α = 0.994, σc = 1.102, σl = 0.341
 Output                Inflation              Consumption          Inventory−Sales Ratio
0.2 0.08 0.4
0.1
0 0.06 0.2
0.04 0
−0.2 0
0.02 −0.2
−0.4 0 −0.1 −0.4
−0.6 −0.02 −0.6
−0.2
−0.04 −0.8
−0.8
−0.06 −1−0.3
−1 −0.08 −1.2
0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15
 Inventories           Unit Labor Cost       Real Marginal Cost   
0.2  0.02
0 0.15 0
0.1
−0.02
0.05
−0.5 −0.04
0
−0.06
−0.05
−1
−0.1 −0.08
−0.15 −0.1
−1.5 −0.2  −0.12
0 5 10 15 0 5 10 15 0 5 10 15
 Calvo Pricing model  VAR
Fig. 1.9: This figure compares the impulse responses generated by the Calvo Pricing
model (dashed line) and the vector autoregression (VAR) model (the dashed-
cross lines). The areas within the dashed lines are the 95% confidence bands
obtained from 5,000 bootstraps. The parameter values for the Calvo model are
obtained by matching only 4 variables, including output, inflation, consumption,
and unit labor cost.
59
0.4  
0.2
0
−0.2
−0.4
−0.6
Actual Zjt response to interest rate hike−0.8
†
Frictionless optimal Zjt response to interest rate hike
−1 
0 5 10 15 20 25 30
Periods after shock
Fig. 1.10: First-Best and Actual Response of Stock for Sale Zjt under rational inattention.
0.01  
0
−0.01
−0.02
−0.03
−0.04
−0.05
−0.06
−0.07 Actual pjt response to interest rate hike
†
Frictionless optimal pjt response to interest rate hike
−0.08 
0 5 10 15 20 25 30
Periods after shock
Fig. 1.11: First-Best and Actual Response of Price pjt under rational inattention
The upper and lower figures compare the optimal and actual response of stock
available for sale Zjt and relative price pjt to a positive 0.12% nominal interest
rate shock respectively. The dash-dotted line is the optimal response and the
solid line is the actual response.
60
% Dev. from steady state % Dev. from steady state
Comparing Simulated Moments
Table 1.3 compares various simulated business cycle moments (conditional on
monetary policy shocks) generated from the VAR, the RI model and the Calvo
Pricing model. Overall, the RI model generates business cycle moments that qual-
itatively match the VAR generated moments, except for the correlation between
output and inflation, which is negative in the RI model but positive in the data.
For the Calvo Pricing model, all the generated correlations qualitatively match the
data. In terms of the correlations of the inventory-sales ratio with output and sales
respectively, both the RI model and the Calvo Pricing model match the negative
signs of the correlations seen in the data, although in terms of magnitude of the
correlations, the RI model fits better. Both the RI model and the Calvo Pricing
model can match the high correlation between output and sales, but the RI model
fits slightly better in terms of magnitude. Overall, we observe that the RI model
matches the correlation between inventory dynamics and other variables better then
the Calvo model. In terms of matching the ratios of relative standard deviations
of inflation to output, the real wage to output, and the inventory-sales ratio to
sales, the RI model clearly fits better, as the ratios for the Calvo Pricing model are
far above those in the data, because the Calvo Pricing model generates too much
variation in inflation and the real wage and too little variation in output and sales.
61
VAR RI Calvo Pricing
ρ(ISt, St) −0.898 −0.881 −0.653
ρ(ISt, Yt) −0.942 −0.787 −0.381
ρ(Yt,∆It) 0.812 0.120 0.415
ρ(Yt, St) 0.963 0.970 0.860
ρ(Yt, πt) 0.374 −0.291 0.888
ρ(∆πt,∆πt−1) −0.220 −0.491 −0.232
σ(πt)/σ(yt) 0.095 0.012 1.098
σ(ISt)/σ(St) 0.527 0.981 1.656
σ(wt)/σ(yt) 0.285 0.622 2.095
Tab. 1.3: Reported simulated moments are conditional on interest rate shocks. All series
are quarterly. ISt, St, ∆It, wt, and Yt stand for inventory-sales ratio, real sales,
inventory investment, real wage, and output.
1.6 Robustness & Extension
To further characterize the RI model solution and check its robustness and sen-
sitivity, I perform three experiments. In the first experiment, I check the sensitivity
of Zj and the inventory-sales ratio to different marginal information costs. Next,
I examine the model’s performance in the presence of larger interest rate shocks.
Finally, I consider a shock to the aggregate money supply (an aggregate demand
shock) instead of a shock to the interest rate rule, and then calculate the relative
contribution of cost rigidity and variation in price markups in accounting for the
changes in aggregate consumption, as in Kryvtsov and Midrigan (2013).
1.6.1 Sensitivity to Information Cost
The benchmark parametrization applies a modest marginal information cost
µ, corresponding to 0.0525% of sales revenue each period. Since there is not an
empirical target corresponding to the information processing cost in the model, this
62
parameter is chosen rather arbitrarily. One might be curious how sensitive are in-
ventory dynamics to different values of µ. Figure 1.12 depicts how the response of
the actual stock for sale Zjt is affected by different levels of marginal information
cost µ. Intuitively, a lower information cost loosens the information capacity limit
and brings the response of Zjt closer to the first-best response. As a result, a lower
marginal information cost amplifies the response of Zjt and alters the response of the
inventory-sales ratio as shown in figure 1.13. As we can see, when information pro-
cessing becomes less costly, the stock for sale Zj has a more pronounced downward
movement with inertial marginal production cost following a tightening interest rate
shock, pushing the inventory-sales ratio downward as the theory predicts. Table 1.4
lists the firm’s profit deviation and information flows as a function of the marginal
information cost µ. Consistent with intuition, the lower is the marginal informa-
tion cost, the higher are the information flows to pricing and production decisions,
and the smaller is the absolute value of profit deviation (the difference of the firm’s
profit with the information capacity constraint from its first-best level without such
a constraint). Notice that the firm always allocates more attention to its pricing
decision than to its production decision.
Marginal info. cost µ 0.0525% 0.0263% 0.0132%
Profit deviation 0.0025 0.0014 0.0013
Info. flow to Pj 0.0768 0.2264 0.1779
Info. flow to Zj 0.0596 0.1038 0.1310
Tab. 1.4: Sensitivity analysis to marginal information costs.
63
0  
−0.05
−0.1
−0.15
µ = 0.000525
µ = 0.000263
−0.2 µ = 0.000132
−0.25 
0 5 10 15 20 25 30
Periods after shock
Fig. 1.12: The Sensitivity of Zjt to Marginal Information Costs µ
0.3  
0.2
0.1
0
−0.1
−0.2
µ = 0.000525
−0.3 µ = 0.000263
µ = 0.000132
−0.4 
0 5 10 15 20 25 30
Periods after shock
Fig. 1.13: The Sensitivity of Inventory-Sales Ratio to Marginal Information Costs µ
64
% Dev. from steady state % Dev. from steady state
1.6.2 Larger Interest Rate Shock
One may wonder whether the model can still give rise to a limited movement
of inventories when the standard deviation of the shock becomes larger. From
information theory, one would predict that as the aggregate shocks become more
volatile, firms should pay more attention to aggregate conditions. Paying more
attention may improve the precision of their decisions, and therefore larger shocks
might amplify the response of Zj and dampen the effect of the information capacity
constraint. The standard deviation of the interest rate shock in the benchmark
calibration is set as 60 basis points (in the quarterly interest rate) in the previous
section. Here I vary the standard deviation and plot the inventory-sales ratio in
figure 1.14.
The figure shows that the inventory-sales ratio increases by more when the
magnitude of the shock becomes larger. This is mainly due to a larger drop of sales.
The response of Zj also become larger, but is not as sensitive to the magnitude
of the shock compared to the response of sales. I list the absolute values of the
deviation in profit (from the first-best level) and the information flows allocated to
Pj and Zj decisions respectively in the following table 1.5. The table shows that
even though the firms increase their information flows to both decisions in response
to larger shocks, the deviation in profit widens. Therefore, even though firms pay
more attention to track the aggregate conditions, their actual decisions deviate by
even more from the first-best decisions (which are also amplified by larger shocks).
Notice that in the firm’s problem (1.34), the interaction between Pj and Zj also
65
affects the profit deviation. If overall the firm’s Pj and Zj decisions deviate from
the first-best responses in opposite directions, their interaction makes the firm’s
profit closer to the first best. The intuition is that, from the firm’s optimality
condition equation (1.18), the firm wants to equate the probability of stock out to
the ratio between the net cost and the net benefit of producing one more unit in
period t. A lower Pj (relative to the first best P
†
j ) induces larger demand for the
firm’s product and thus increases the chance of a stock out, and in this case the
firm would want a higher Zj (relative to the first best Z
†
j ) to decrease the chance
of a stock out. The (negative) contribution of the interaction terms to the profit
deviation becomes larger in magnitude with an increased magnitude of the shock,
which means that larger deviations of Pj and Zj from the first-best decisions also
contain a counteracting force in their interaction that benefits the firm’s profit. The
interaction term thus provides a limiting force to the firm’s information allocation.
0.45  
0.4
0.35
0.3
0.25 σr = 15bps
σr = 20bps
0.2
σr = 25bps
0.15
0.1
0.05
0
−0.05 
0 5 10 15 20 25 30
Periods after shock
Fig. 1.14: The Sensitivity of Inventory-Sales Ratio to σr
66
% Dev. from steady state
Standard deviation σr 15 bps 20 bps 22 bps 25 bps
Profit deviation 0.0025 0.0047 0.0052 0.0072
Info. flow to Pj 0.0768 0.1009 0.1055 0.1164
Info. flow to Zj 0.0596 0.0878 0.1009 0.1039
Contribution of interaction
to profit deviation -15.5% -17.2% -18.8% -19.1%
Tab. 1.5: Sensitivity analysis to larger standard deviation of interest rate shock.
1.6.3 Money Growth Shock
The model employed in this paper assumes that monetary policy is represented
by the feedback interest rate rule (equation (1.28)). Kryvtsov and Midrigan (2013)
instead consider a shock to the monetary aggregate. As aggregate nominal spending
equals the nominal money supply, they can decompose the real effects of the nominal
money growth shock into a cost rigidity term and a price markup term:
PtCt = M[t =⇒
∆ ln(Ct) = ︸∆ ln(Mt︷)︷ ︸] ︸ [ ]− ln(Λt) + ∆ ln(Λt) ︷−︷∆ ln(Pt)︸
cost rigidity term markup term
They show that the model with staggered pricing and staggered wage setting can
match the countercyclical inventory-sales ratio only when nominal marginal cost Λt
tracks the changes in money supply Mt closely and is much more responsive than
Pt, or in other words, when the markup is strongly countercyclical. They conclude
that in view of the model, the markup term contributes as much as 90% to changes
in aggregate expenditure ∆ ln(Ct). To provide a direct comparison of the RI model
results to those of Kryvtsov and Midrigan (2013), I set government consumption
67
to zero so that aggregate expenditure is equal to private consumption Ct. Also,
I assume h = 0 (no habit persistence,) as in their paper. I replace the interest
rate rule with a cash-in-advance constraint in the consumer’s problem and let the
monetary authority control the nominal money supply. I assume ξw = 0.90 to give
the model higher cost rigidity. Other parameters are the same as those listed in the
second column of table 1.1.
I shock the economy by a 0.6% decrease in the nominal money supply and
plot the impulse responses in figure 1.15. In the upper-left subplot, one can see
that changes in the price markup (the distance between nominal marginal cost and
the price level) are small compared to the fall in money supply. In the upper right
subplot we see that inventories move less than sales, resulting in a countercyclical
inventory-sales ratio. The RI model stands in sharp contrast to the staggered pric-
ing model of Kryvtsov and Midrigan (2013), which can generate a countercyclical
inventory-sales ratio only when nominal marginal cost closely tracks the shifts in
nominal money supply. In their paper, the markup variation contributes as much
as 90% to the change in aggregate expenditure. Figure 1.16, I apply the param-
eter values listed in table 1.1 (except h = 0) to the Calvo Pricing model with a
money growth shock. One can see stark differences from figure 1.15 in that, nom-
inal marginal cost tracks the change in money supply closely in the first subplot.
In the RI model (figure 1.15), the cost rigidity term accounts for the bulk of the
changes in consumption, whereas the variation in the price markup only accounts
for about 24%. Table 1.6 reports various moments of the RI model and their data
counterpart (which are taken from TABLE 3 in Kryvtsov and Midrigan (2013)).
68
Conditional on a nominal money supply shock, the RI model with rigid marginal
cost does a satisfactory job matching the data moments.
0  0.6  
−0.1 Money 0.4
Nominal Marginal Cost
−0.2 0.2
Price
−0.3 0
−0.4 −0.2
−0.5 −0.4 Inventory
Sales
−0.6 −0.6 IS Ratio
−0.7 −0.8 
0 5 10 15 20 25 0 5 10 15 20 25
0  0.07  
−0.1 0.06
Real Interest Rate
−0.2
Production 0.05 Nominal Interest Rate
−0.3 Consumption
0.04
−0.4
0.03
−0.5
0.02
−0.6
−0.7 0.01
−0.8 0 
0 5 10 15 20 25 0 5 10 15 20 25
Fig. 1.15: Impulse Responses to a −0.6% Money Growth Shock, RI Model
Data RI
ρ(ISt, St) −0.710 −0.8255
ρ(ISt, ISt−1) 0.880 0.876
ρ(Yt,∆It) 0.630 0.246
σ(ISt)/σ(St) 0.93 1.018
σ(yt)/σ(St) 1.110 1.022
Elast. ISt to St -0.660 -0.842
Elast. It to St 0.340 0.158
Tab. 1.6: Simulated moments of the RI model and moments in the data (taken from
Kryvtsov and Midrigan (2013)). ISt, St, ∆It, wt, and Yt stand for the inventory-
sales ratio, real sales, inventory investment, the real wage, and output.
69
% deviation from steady state % change
percentage point change % deviation from steady state
0  0.4  
−0.1
Money 0.2
−0.2 Nominal Marginal Cost
Price
−0.3 0
−0.4 −0.2
−0.5 Inventory
−0.4 Sales
−0.6 IS Ratio
−0.7 −0.6 
0 5 10 15 20 25 0 5 10 15 20 25
0.2  0.15  
0
0.1
Real Interest Rate
−0.2 Nominal Interest Rate
0.05
Production
−0.4 Consumption
0
−0.6
−0.8 −0.05 
0 5 10 15 20 25 0 5 10 15 20 25
Fig. 1.16: Impulse Responses to a −0.6% Money Growth Shock, Calvo Model
1.7 Conclusion
I construct a medium-scale DSGE model with cost rigidities that incorporates
inventory investment, and reexamine what inventory adjustment implies for the elas-
ticity of real marginal cost with respect to output in response to monetary shocks.
I compare a Calvo staggered pricing friction with an information capacity friction
that assumes firms are inattentive to shocks hitting the economy and endogenously
choose the information content in their actions. In a staggered pricing setting with
full information, inventories are highly sensitive to the costs of inventory acquisition
(the real marginal cost of production) and inventory holding (the real interest rate).
Therefore, such a model predicts that real marginal cost needs to be highly elastic
and sensitive to economic fluctuations, to counteract the direct effect of changes
70
% deviation from steady state % change
percentage point change % deviation from steady state
in interest rates and explain why the inventory-sales ratio rises following a mon-
etary tightening. In contrast, this paper finds that with an information capacity
constraint, the impact of the relative movements in the real interest rate and real
marginal cost on the response of inventories is weakened, as the information capacity
constraint implies that firms may deviate from their first-best decision rules. The
capacity constraint serves as a force to limit excessive movements in inventories.
Thus, the model can deliver a countercyclical inventory-sales ratio after a monetary
shock even if marginal cost is rigid. Numerical results demonstrate that a model
with inattentive firms and high cost rigidity generates more persistent real effects,
and fits the empirical VAR impulse responses and simulated moments better than
the Calvo Pricing model with elastic marginal cost.
71
Chapter 2: Endogenous Irreversibility and Capital Reallocation un-
der Information Asymmetry
2.1 Introduction
Some empirical studies of business cycle dynamics indicate that the reallo-
cation of capital, or the movement of capital input from less productive firms to
more efficient firms, is procyclical. Eisfeldt and Rampini (2006) compiled empir-
ical evidence showing strong procyclicality of the reallocation of physical capital
among US firms, as represented by acquisitions and sales of plants, property and
equipment.1 Another related important fact is that the cross-sectional standard
deviation of capital productivity is countercyclical according to several measures,
which means that the benefit of reallocating physical capital across establishments
to more productive uses is potentially counter-cyclical; in other words, the benefit
is greater in periods of recessions. These two facts point to a puzzle: why does the
amount of capital reallocation evolve in almost exactly the opposite way to what
the “creative destruction” or the “cleansing effect” point of view regarding business
cycle downturns would predict? Eisfeldt and Rampini (2006) postulate that there
1 The authors provide an updated online version of their empirical evidence of the comovement
of capital reallocation and the cyclical component of GDP at the following URL https://sites.
google.com/site/andrealeisfeldt/home/capital-reallocation-and-liquidity
are costs or frictions involved in reallocating capital, and that those frictions and
costs are countercyclical. If we define the time-varying reallocation cost as the in-
verse of the liquidity of capital, which might stem from adverse selection or financial
frictions, then liquidity is expected to be procyclical. A calibrated measurement in
Eisfeldt and Rampini (2006) finds that the cost of reallocation is 2.6 times higher in
recessions than on average. On the relative importance of reallocation, defined as
the sum of acquisitions and sales of property, plants and equipment (PP&E), with
respect to aggregate investment activity, Eisfeldt and Rampini (2006) report that
reallocation is about 28.6% on average for publicly traded firms in Compustat data,
with acquisitions accounting for 19.5% and sales of PP&E about 9.1%.2 Using Cen-
sus data, Eisfeldt and Rampini (2007) find that smaller firms and more financially
constrained firms rely more on used capital in total capital expenditure.
This paper attempts to fill the gap in the literature by identifying possible
sources of frictions that impede capital reallocation, and studying the implications
of such frictions for the dynamics of reallocation following a productivity shock. I
model the time-varying cost of reallocating capital as an endogenous partial capital
irreversibility that stems from both capital specificity and information asymmetry
in the used capital market. I extend the idea of Kurlat (2013) and build a general
equilibrium model with heterogeneous entrepreneurs producing an identical final
good. Entrepreneurs hold capital, a fixed proportion of which is subject to an
exogenous separation into capital goods of bad quality, or lemons, which are of
2 Both acquisitions and sales of PP&E appear to be proyclical for publicly traded firms. Acqui-
sitions do exhibits more procyclicality than sales of PP&E. The correlation between acquisitions
and output 0.675, while the correlation between sales of PP&E and output is 0.329. Both statistics
are significant.
73
no use either for production or consumption. Entrepreneurs can directly trade
existing capital in the used-capital market. The market is plagued by information
asymmetry, since only the sellers know the true quality of used capital they bring to
the market, and the buyer cannot verify the quality traded in the market ex ante.
The pooling price of used capital thus depends on the rationally expected proportion
of lemons traded in the market.
As in Kurlat (2013)’s model, heterogeneity is essential here, as it determines
the demand and supply of used capital and thus the market equilibrium. I as-
sume that entrepreneurs can buy and sell their existing capital in the used capital
market, but due to partial specificity of used capital, they must combine used in-
vestment goods with some new investment goods in a bundle to install productive
capital. This assumption, together with information asymmetry, create a gap be-
tween the buying and selling prices of investment goods and thus partial investment
irreversibility, which fluctuates with aggregate shocks.
The Mechanism. The mechanism works as follows. The gap between the
price of new investment goods and the price of used capital creates partial irre-
versibility in plant-level capital accumulation. Because idiosyncratic technological
shocks are persistent, depending on the current period idiosyncratic shock and cap-
ital holdings, an entrepreneur falls into one of three situations. If he is hit by a
low idiosyncratic shock, he will downsize the production scale by selling some good
(non-lemon) capital. Conversely, an entrepreneur with a higher idiosyncratic pro-
ductivity shock expands production scale by acquiring new and used investment
goods and supplies no non-lemon used capital to the market. An entrepreneur with
74
an intermediate idiosyncratic shock is characterized by inaction. It neither acquires
new capital nor sells any non-lemon capital. The threshold that splits the inaction
region from the other two regions depends on the gap between the buying and sell-
ing prices of investment goods, which is endogenously affected by aggregate shocks
because they affect the decisions of the agents. This mechanism could potentially
make capital reallocation procyclical if the inaction region widens during down-
turns. In turn, procyclical reallocation potentially amplifies the effect of a negative
aggregate shock on output, since if less capital flows to the more productive plants,
measured aggregate TFP declines. I also extend the model to include a fixed bor-
rowing limit, which does not exceed the natural borrowing constraint. This credit
market friction compounds capital irreversibility and further exacerbates capital
misallocation, since a borrowing limit hinders the investment undertaken by small
scale entrepreneurs with relatively high productivity, and drags down the demand
for investment goods. This model also creates a countercyclical dispersion of the
marginal product of capital, as small and productive firms expand by less and large
unproductive firms downsize by less in recessions, meaning the inherent benefit of
capital reallocation is countercyclical.
2.2 Literature Review
Kiyotaki and Moore (2012) were the first papers to introduce the concept of
liquidity into a DSGE model. Kiyotaki and Moore (2012) defines the illiquidity of
private assets as the degree of impediment to their transactions. In Kiyotaki and
75
Moore (2012), firms receive an idiosyncratic investment opportunity each period.
To finance favorable investment projects, a firm can either issue financial claims
to the new project (capital stock) or liquidate existing capital stock. A firm can
only issue claims against a fixed fraction of the future returns from the new capital,
so that internal funds are needed. Further, existing private claims issued by other
firms held on the firm’s balance sheet are not fully resalable. Kiyotaki and Moore
(2012) models liquidity shocks as changes in the fraction of existing private claims
that a firm can sell to finance new investments. In Kiyotaki and Moore (2012), this
fraction is exogenously determined and fluctuates randomly. When this fraction
decreases, the amount of resources that can be efficiently transferred is reduced,
causing aggregate investment and output to drop.
One major criticism of Kiyotaki and Moore (2012) is that the exogenous liq-
uidity shock is ad hoc. Some recent studies try to provide a microfoundation to
endogenize fluctuations of asset resaleability. Kurlat (2013) builds a model in which
entrepreneurs face idiosyncratic investment technology shocks and part of their ex-
isting capital becomes lemons each period. Information asymmetry between buyers
and sellers of capital regarding its quality generates an adverse selection problem.
The model can generate an endogenous liquidity crunch following a shift in the dis-
tribution of investment efficiencies. Bigio (2015) builds a related model with two
frictions: a limited enforcement friction either in labor contracts or investment in-
puts (or both), and information asymmetry between sellers and buyers of existing
capital (where the buyers are financial intermediaries) regarding the quality of the
capital. The limited enforcement friction amounts to a working capital constraint.
76
In the spirit of Kiyotaki and Moore (2012), entrepreneurs need to sell existing capital
stocks to finance working capital (to relax the enforcement constraint). Information
asymmetry is also key in this model to generate endogenous liquidity fluctuations.
The major difference from Kurlat (2013) is that the endogenous liquidity crunch in
Bigio (2015) comes from a shock to the dispersion of asset quality.
Another strand of literature deals with capital irreversibility and its implica-
tions. Eisfeldt and Rampini (2006) provide empirical evidence that the amount of
capital reallocation is highly procyclical while the inherent benefit of reallocation is
countercyclical, meaning that some frictions must exist that prevent the resale of
capital in recessions. Cooper and Haltiwanger (2006) point out that in the presence
of capital irreversibility, represented as a gap between the buying and selling price
of capital, the firm will respond to an adverse shock by holding on to capital to
avoid the loss associated with reselling. Further, because of caution the firm will
not build its capital stock as quickly. Two papers closely related to mine are Cui
(2014) and Lanteri (2015). Cui (2014) builds a DSGE model with constant partial
capital irreversibility. Firms hit with negative shocks will hold on to capital at first,
and then start disinvesting when the internal return of capital falls below the re-
sale price. Since disinvesting firms face a constant resale price, Cui (2014)’s model
cannot generate procyclical capital reallocation without introducing an exogenous
tightening of credit constraints. Cui (2014) thus models procyclical capital reallo-
cation as the result of exogenous credit shocks. Lanteri (2015) constructs a model
with heterogeneous firms and endogenously time-varying capital irreversibility that
stems from partial specificity of used capital.
77
My model combines both the specificity of used capital, as in Lanteri (2015),
and information asymmetry in the market for used capital, as in Kurlat (2013).
Similar to Kurlat (2013), this paper has the characteristic that a negative shock af-
fects both the investment/disinvestment decisions of agents and the average quality
of assets traded in the market, which feeds back to affect agents’ decisions. In my
model, however, imperfect substitutability compounds the information asymmetry
problem in the asset market, as specificity makes quantity demanded not sensitive
enough to price changes, exacerbating the adverse selection effect and pushing av-
erage quality of assets to worsen and the equilibrium price to fall more following a
negative shock. Capital specificity interacts with adverse selection to determine the
resale price of capital and the price gap, so that the model can potentially generate
more significant comovement and fluctuation of reallocation. In addition, Lanteri
(2015)’s model features perfect capital markets without frictions, whereas I consider
a fixed borrowing limit. Khan and Thomas (2013) also study the interplay between
credit frictions and capital irreversibility. In their model, capital irreversibility is
exogenously given and they focus on studying the role of exogenous credit shocks.
Caveats. This paper, as other studies cited here, does not distinguish between
acquisitions and sales of PP&E. However, acquisitions tend to be qualitatively dif-
ferent from organic transfers of used capital, since acquisitions can be merely owner-
ship changes without significant physical reshuffling at the establishment level. The
model developed in this paper is more suited to account for physical transfers of
PP&E. Furthermore, establishment entry and exits are not considered in my study.
Endogenous firm exits can be an important factor shaping reallocation fluctuations.
78
In the following section, I use a simple one-period model to study analytically
the qualitative relationship between the aggregate TFP shock, the price of used
capital and capital reallocation. Section 3 develops a dynamic general equilibrium
model with labor input and credit constraints.
2.3 A Simple Model of Capital Reallocation
I first construct a simple one-period model of heterogeneous entrepreneurs
facing idiosyncratic productivity shocks. The model features information asymmetry
in capital quality in the market for used capital, and imperfect substitutability
between new and used investment goods. The model is based jointly on the work of
Kurlat (2013) and Lanteri (2015). For simplicity I abstract from the labor market
and assume that capital is the sole production input. Entrepreneurs own capital
and are heterogeneous in their idiosyncratic productivity. As in Kurlat (2013), a
fixed proportion (λ) of existing capital units turn at the beginning of the period into
bad quality assets or ”lemons”, which are of no use either as a production input or
for consumption. I assume that trading of used capital is conducted in a centralized
market. Only the seller of used capital knows the quality it supplies to the market.
Buyers are not able to verify the quality before the transaction takes place, and
thus the equilibrium price of used capital is a pooling price, which depends both
on the demand and supply of used capital, as well as on the average quality of
capital traded in the market. Firms that draw a low idiosyncratic productivity will
downsize by supplying some good-quality capital on the market. On the other hand,
79
entrepreneurs that wish to expand can do so either by purchasing new investment
goods or by acquiring used capital. Used capital and new investment goods are not
perfect substitutes, but differ along two dimensions. First, they differ in their quality,
as the average quality of used capital depends on the fraction of lemons supplied to
the market. Second, used capital has some degree of specificity (exogenously given)
so that to install used capital bought in the market the entrepreneur has to combine
it with some new investment goods. A negative aggregate shock, such as a TFP
shock, affects both the demand and supply of used capital and changes the average
quality traded in the market and thus its price, which feeds back to affect agents’
decisions. This section serves as a qualitative assessment of how exogenous shocks
affect capital reallocation. I use the simple model to derive some analytical results
on the response of price and aggregate reallocation of capital to exogenous changes
of aggregate productivity.
2.3.1 The Setting
The economy consists of a continuum of entrepreneurs j ∈ [0, 1] normalized
to a unit measure. Each entrepreneur owns the same initial capital level k0. An en-
trepreneur produces a homogeneous final output good with the production function
y αj = zjZkj (2.1)
where Z is aggregate total factor productivity, zj is the entrepreneur’s idiosyncratic
productivity shock with distribution function F (z), and α ∈ (0, 1) is the elasticity of
80
output with respect to capital. The fraction of capital goods of bad quality is fixed
and given by λ ∈ (0, 1). I assume that capital is infinitely divisible and that the
exogenous separation of capital into good and bad quality applies to new investment
goods as well.
At the beginning of the period, the entrepreneur observes his idiosyncratic
productivity shock and aggregate TFP. The entrepreneur can adjust his capital be-
fore production. Apparently, since capital goods of bad quality have no use for
production, all entrepreneurs will sell them on the used-capital market, as long as
the pooling price of used capital is greater than zero. Firms that draw a low id-
iosyncratic productivity will find it optimal to disinvest some non-lemon capital, as
long as the marginal product of keeping an additional unit of capital is lower than
the market price of used capital. On the other hand, an entrepreneur with higher
idiosyncratic productivity would like to expand his production scale. It can do so by
purchasing new investment goods (assumed to be supplied inelastically by the house-
holds) or acquiring used capital from other entrepreneurs. I follow Lanteri (2015)
in assuming that used capital is partially specific to its previous owners. Therefore,
expanding entrepreneurs cannot invest by buying used capital only. Used capital
goods must be bundled with some new investment goods to make them specific to
the entrepreneur. Partial specificity makes new investment goods and used capital
imperfectly substitutable. This assumption serves two purposes. Firstly, it makes
the buying price of capital different from the selling price of capital even if the
entrepreneur expands by purchasing used capital, and thus creates partial capital
irreversibility. Secondly, it also helps the model generate procyclical reallocation
81
and a countercyclical average market fraction of lemon capital. In a recession, more
firms would like to disinvest and downsize, so that the price of used capital tends to
fall. However, when used capital is partially specific, expanding firms cannot fully
take advantage of a falling price, so that the price effect on quantity demanded is
limited by this imperfect substitutability. This makes the price of used capital fall
deeper in recession and the inaction region to widen. This point will become clearer
in later discussion.
Imperfect Substitution. In my model, units of new and used capital do not
enter the capital accumulation function as perfect substitutes even after accounting
for quality, as in equation (3) below. This is a key assumption underpinning the
mechanism in this paper. Imperfect substitution through a CES aggregator is a
crude formulation which reflects some factors not explicitly considered in the model.
First, new and used equipment may have different maintenance schedule. After
acquiring plant and equipment from liquidators, a firm generally still needs to make
maintenance amendments and/or replacements to make the plant ready for use.
Another related consideration is the vintage of used capital. Vintage is not explicitly
modeled here, but used machines may become less substitutable for new machines as
they age, since technological changes may be embedded in new machines. There is
also some empirical evidence in the literature supporting imperfect substitutability
between new and used assets. Edgerton (2011) uses a tax credit affecting only new
capital to test the assumption that used and new capital are perfect substitutes. If
they are indeed perfectly substitutable, they must sell at the same after-tax price,
and tax provisions favoring new capital should imply a lower price for existing used
82
capital. However, his test showed that the estimated effect of the bonus depreciation
on the price of used construction machinery is close to zero, and his estimates
for the elasticity of substitution are between 1 and 10. Furthermore, if perfect
substitutability holds, even a small increase in the price of used capital would cause
investing firms to demand only new capital, which is not consistent with the data.
Perfect substitutability is also hard to reconcile with the fact that the price of used
investment goods is more volatile than the price of new capital. Garvazza (2011)
finds that the resale price of more specific models of aircraft is significantly more
volatile than that of more flexible models.
Investment Technology. I assume that an expanding entrepreneur does not
have the option to buy only new investment goods. He must combine both new and
used investment goods to increase his capital stock. In the model, the price of new
investment goods is normalized to one.
The investment technology is given by a CES aggregator of new investment
and used capital investment:
kj − (1− λ)k0 = g(ij,new, ij,used) (2.2)
{ 1 −1 1 −1} 
g(ij,new, ij,used) = η  [(1− λ)ij,new]  + (1− η)  [(1− λM)i −1j,used]  (2.3)
where ij,new denotes new investment goods and ij,used is used capital purchased by
expanding entrepreneurs; λ is the aggregate fraction of bad quality capital (which
83
applies to new investment goods as well) and λM is the fraction of bad quality
used capital traded in the market. η ∈ (0, 1) is a parameter that determines the
average ratio between new and used investment, and  ∈ (0,∞) is the elasticity
of substitution between new and used investment goods, which also serves as an
inverse measure of capital specificity. When  → ∞, new and used investment
goods are perfectly substitutable; when  = 0, the investment technology is Leontief
and does not allow any substitutability between new and used capital. Note that
new investment does not suffer specificity issues. Investing entrepreneurs bundle new
investment goods (ij,new) with used investment goods (ij,old) via the CES aggregator.
I normalize the price of new investment goods to be 1 and let p denote the
equilibrium price of used capital. The following corollary gives the CES price index
of a bundle of new and used investment goods.
Corollary 1. Given a bundle g of new and used investment goods, the CES
price index associated with the bundle is
[ ( 1 ) ( ) ] 11−
ρ = η + (1− p 1− 1−η) (2.4)
1− λ 1− λM
and the optimal composition of new and used investment goods in the bundle is
i = η(1− λ)−1new ρg (2.5)
i M −1used = (1− η)(1− λ ) p−ρg (2.6)
Proof : See the appendix.
84
The CES aggregator gives a well defined optimal ratio of new to used invest-
ment, given the values of p and λM . From corollary 1, it is clear that the higher
is the price of used capital p, the lower is the weight of used investment goods in
the bundle. On the other hand, as long as the elasticity of substitution of new and
used investment goods  is greater than one, a rise in λM , the fraction of bad quality
capital traded in the market, decreases the relative weight of used investment goods
in the bundle. This means that when the perceived average quality of used invest-
ment goods is higher, an investing entrepreneur is more willing to substitute used
investment goods for new investment goods. Furthermore, note that since λM > λ,
p p
− M > − always holds. As long as p ≤ 1 (which is always true throughout the1 λ 1 λ
paper, since 1 is the price of new investment goods), I can deduce that ρ > p/(1−λ)
and thus ρ > p always holds. This means that there is a gap between the buying
and selling price of investment goods, and this gap endogenously changes with the
price of used capital p and the market fraction of bad quality investment goods λM .
Note that rather than modeling the market for new capital explicitly, I make
the assumption that expanding entrepreneurs are subject to the investment technol-
ogy to bundle new investment goods with old ones, and the price of new investment
goods is fixed at 1. This is a reduced form way to link investment activities to
equilibrium in the market for used capital. It also helps reconcile the model with
the empirical fact that a significant proportion of the investment of production units
consists of investment in used capital (acquisitions and sales of property, plant and
equipment).3 This assumption implicitly posits that the supply of new investment
3 Eisfeldt and Rampini (2007) point out that smaller and more credit constrained entrepreneurs
85
goods is fully elastic, so that the price of new capital goods doesn’t change. In the
real world, the price of new investment goods does fluctuate, although its volatility is
smaller than that of used capital prices.4 Suppose that I allow the entrepreneurs the
option to expand by purchasing new investment goods only, and denote the price
of new capital by PN . Then, equilibrium prices ρ , p , λMt t t , and P
N
t should make
expanding entrepreneurs indifferent between purchasing new capital goods only or
choosing the bundle. This leads to a modified version of equation (2.4):
[ ( PN ) ] 1t 1− ( p ) − PN t 1  1−ρt = η + (1− η) = t
1− λ 1− λM 1− λ
⇒ p P
N
t
= = t
1− λM 1− λ
Hence, the price of new investment goods PNt will likely move in the same
direction as pt and in the same direction as λ
M .
2.3.2 Entrepreneur’s Optimal Investment Decision
In the one-period model, an entrepreneur simply maximizes one period output,
plus the net proceeds from the sales of used capital and the purchase of investment
goods. After observing the aggregate TFP and his own idiosyncratic productivity
tend to invest in a higher proportion of used capital when they make investment. They rationalize
this phenomenon by the timing of cash flows that constitute used capital investment. I abstract
from this consideration here and assume that the same degree of partial specificity applies to all
entrepreneurs.
4 Lanteri (2015) used a dataset on the value of commercial aircraft of different vintages, and
reported that the standard deviation of used aircraft prices is more than two times as large as new
aircraft prices.
86
shock, an entrepreneur chooses his optimal capital level for production. Under
the same assumptions that lead to corollary 1, I can characterize the optimization
problem of a generic entrepreneur j as
max z αjZkj − ρgj + p(sb,j + sg,j)
sb,j ,sg,j ,kj ,gj [ ]
subject to kj = (1− λ)k0 − sg,j + gj
gj ≥ 0, sb,j ∈ [0, λk0], sg,j ∈ [0, (1− λ)k0]
where gj is the entrepreneur’s investment, sb,j are his sales of bad quality capital
and sg,j are his sales of good quality capital. Since capital goods of bad quality
(lemons) are not useful for production, as long as p > 0 the entrepreneur will sell
all of his bad capital to the market. Therefore, sb,j = λk0. Furthermore, since
ρ > p, it is never optimal to sell good capital and buy investment goods at the same
time. An entrepreneur can either set gj > 0 or (sg,j > 0), but not both. If the
entrepreneur chooses neither to invest (gj = 0) nor to disinvest sg,j = 0, the amount
of good capital that enters the production function is (1−λ)k0. As the optimization
problem is static, an entrepreneur’s optimal investment or disinvestment decision
can be easily characterized:
• If the entrepreneur has high enough productivity, he will invest by buying a
bundle of new and used investment goods. The entrepreneur will invest if his
idiosyncratic productivity zj is greater than a threshold z
I ≡ [ ρ ]α−1 .
αZ (1−λ)k0
87
The optimal capital level is given by
(αz Z) 1j 1−α
kj =
ρ
and his investment is given by gj = kj − (1− λ)k0.
• If the entrepreneur is hit by a sufficiently low idiosyncratic productivity, he
will disinvest by selling some good quality capital. The entrepreneur will
disinvest if his idiosyncratic productivity zj is lower than a threshold z
D ≡
[ p ]α−1 . The optimal capital level is given by
αZ (1−λ)k0
(αzjZ) 11−α
kj =
p
and his sales of good capital are given by sg,j = (1− λ)k0 − kj.
• Firms with intermediate idiosyncratic productivity zD ≤ zj < zI will neither
buy investment goods nor sell any good quality capital, as their marginal
product of capital lies between the purchasing price ρ and selling price p.
2.3.3 Equilibrium in the Market for Used Capital
Under Assumption 1, given a chosen amount of total investment, investing
entrepreneurs buy a bundle of new and used capital. The result in corollary 1 gives
a well-defined fraction of used investment goods relative to the whole bundle. The
market demand for used capital can be obtained by integrating over the measure of
88
investing entrepreneurs:
( ∫p) [ ]− (αzjZ ) 1
D = (1− η)(1− λM)−1 1−αused − (1− λ)k0 dF (zj) (2.7)
ρ zI ρ
The total supply of used capital is the sum of the supply of lemon used capital λk0
from all entrepreneurs, and the supply of good-quality used capital:
∫ zD [ (
− − αz
]
jZ ) 1
S 1−αused = λk0 + (1 λ)k0 dF (zj) (2.8)
p
The market fraction of bad-quality capital λM , which must coincide with the en-
trepreneur’s expected value, is given by
M λk0λ = ∫ [ ( ]1 (2.9)zD − − αz Z )λk + (1 λ)k j 1−α0 0 dF (zp j)
The denominator of the above equation is lower than k0, so λ
M > λ must hold. In
other words, the market fraction of bad capital is always larger than the given frac-
tion λ. Market clearing in the used capital market requires that the total demand for
used investment goods is equal to the total supply. This condition defines implicitly
the equilibrium price of used capital p(Z; ) as a function of aggregate productivity
Z and the elasticity of substitution between new and used investment goods . The
following proposition summarizes the relationship between this elasticity and the
effect of aggregate shocks on capital irreversibility and reallocation.
Proposition 1. Provided that a regularity condition holds, there exists an
89
 > 0 such that for  <  the elasticity of p with respect to Z is greater than 1, and
reallocation is increasing in Z.
The proof of this result is in the appendix, and the regularity condition ba-
sically ensures that the market excess demand for used capital is decreasing in the
price of used capital p, which is always satisfied in our quantitative exercise. The
intuition behind proposition 1 is as follows. Consider the case when a negative
aggregate TFP shock hits the economy. At the original price p and λM , quantity
demanded will be lower as fewer firms want to expand, and used capital supplied
will be higher as more firms want to downsize. Therefore, there is excess supply in
the market and the price for used capital tends to fall. However, a higher supply
of good quality used capital also drives down the market fraction of lemons, which
tends to induce entrepreneurs to substitute more used investment goods for new
investment goods because of the higher average quality. This is the indirect effect
of Z on excess demand through λM , and can be represented by a outward shift in
the demand curve for used capital in the price-quantity space. Provided that this
indirect effect is not too large, there is still excess supply in the market, and the
price of used capital has to fall. When the elasticity of substitution between new and
used capital is low (that is, when specificity of used capital is significant), investing
entrepreneurs cannot fully take advantage of the falling price since they are limited
by this imperfect substitutability. Therefore quantity demanded is not very sensitive
to price changes, and p has to fall further to eliminate excess demand. However, a
falling p decreases the supply of non-lemon used capital and increases λM , which
induces market demand to shift inward, reducing the equilibrium quantity of capital
90
reallocated and further exacerbating the decline in price p. The additional thrust
of the average quality λM on market equilibrium is the major difference between
the mechanism in this paper and that of Lanteri (2015). This mechanism of ad-
verse selection can potentially generate more volatile capital reallocation dynamics
in business cycles.
2.4 A Full DSGE Model
In this section, I present an infinite-horizon general equilibrium model with a
representative household and a unit measure of heterogeneous entrepreneurs produc-
ing an identical final good. This model incorporates the static effect of an aggregate
productivity shock on capital reallocation described in the previous section.
2.4.1 Households
The economy is populated by a unit measure of identical households and a unit
measure of heterogeneous entrepreneurs. Households consume and supply labor to
entrepreneurs to earn wage income. Since my focus is on capital reallocation and
investment in the entrepreneurial sector, I keep the Household’s problem simple by
assuming that they do not accumulate wealth and are therefore hand to mouth.
Their period utility function is given by the GHH preference:
1 ( l1+θh )1−τ
U(ch, lh) = ch − ψ h (2.10)
1− τ 1 + θh
subject to ch = w(Z, µ)lh
91
where w(Z, µ) is the equilibrium wage rate as a function of aggegate productivity Z
and the cross-sectional distribution of entrepreneurs over idiosyncratic states, which
households take as given. θh is the reciprocal of the Frisch elasticity of labor supply,
which governs wage stickiness in this model. Let Ch(Z, µ) denote the household
decision rule for current consumption, and Lh(Z, µ) the decision rule for labor supply.
2.4.2 Entrepreneurs
There is a continuum of heterogeneous entrepreneurs with unit measure. En-
trepreneurs differ from one another in three dimensions: their idiosyncratic pro-
ductivity shock z, bond holdings b, and capital at hand k. Let µ(k, b, z) denote
the distribution of entrepreneurs over capital, bond holding, and idiosyncratic pro-
ductivity. As in the one-period model, entrepreneurs hold capital and trade used
capital with one another. Entrepreneurs produce a homogeneous output good with
the following decreasing-returns-to-scale technology:5
yjt = zjtZ k
α
t jtn
γ
jt (2.11)
with α+ γ < 1. The entrepreneur chooses current labor demand, the future level of
capital, and his current consumption to maximize his lifetime utility. I assume that
5 I assume the production technology to be decreasing instead of constant returns to scale for
computational purposes. As will become clear, if the technology is CRS the entrepreneurial profit
will be linear in capital, and so an entrepreneur’s investment/disinvestment decision will depend
only on his idiosyncratic productivity, independent of his capital holdings. In a numerical experi-
ment, since the stochastic process for the entrepreneur’s idiosyncratic shock needs to be discretized,
this property makes the quantity demanded and supplied less sensitive to price adjustments.
92
the idiosyncratic shock follows a first-order autoregressive process:
log zjt = ρz log zj,t−1 + εzt (2.12)
where εzt is an i.i.d. perturbation with mean zero and variance σε.
Timing. At the start of period t, an entrepreneur observes his state variables
kjt, bjt, and zjt, and aggregate productivity Zt. He hires labor and uses existing
capital for production. After production, the markets for used investment goods
and bonds open. Existing capital depreciates after production at a depreciation
rate δ. There is also an exogenous scrappage of bad capital at a given rate λ. For
installed capital, I assume that capital depreciation and scrappage happens at the
same time, and scrapped capital is part of depreciated capital, so that δ ≥ λ holds.
The entrepreneur decides his optimal consumption, investment/disinvestment and
bond holdings, and trades used capital with other entrepreneurs. At the end of the
period all lemon capital and the depreciated capital are retired from the economy
and the entrepreneur carries the remaining capital into the next period t+ 1.
Entrepreneurs take the equilibrium wage rate w(Zt, µ), gross interest rate
R(Zt, µ), the market fraction of bad-quality used investment goods λ
M
t , and the
market price of used capital pt as given. They can borrow with a one-period bond,
but the amount that they can borrow is subject to a fixed borrowing limit b. An
entrepreneur chooses current labor, future capital and bond holding to maximize his
lifetime utility. Since there is no labor market friction, the labor hiring decision is
intratemporal and can be separated from the investment or disinvestment decision
93
in a convenient way. I first solve for the labor decision and derive the implied return
on capital.
Given the amount of capital at hand, an entrepreneur’s labor demand is de-
termined by equating the marginal product of labor to the wage rate:
[γz α ] 1jtZtkjt 1−γ
njt = (2.13)
wt
and entrepreneurial output net of the wage bill is
[(
− γ
) γ ( γ ) ]( ) 11 1−γ
y w n 1−γ 1−γ αjt t jt = − wt
wt ( ) zjtZtkw jtt1
α 1−γ= π(wt) zjtZtkjt (2.14)
[( ) γ ( ) ]1
where π(w ) = γ 1−γ −w γ 1−γt t . Since I assume that the production technol-wt wt
ogy is decreasing returns to scale, i.e. α + γ < 1, the return to capital is concave
in the entrepreneur’s existing capital kjt. Since optimal labor demand is an in-
tratemporal decision, this return on capital function can be incorporated into the
entrepreneur’s intertemporal consumption, saving and investment decisions.
The entrepreneur’s intertemporal problem is recursive in the state vector
(kjt, bjt, zjt), where bjt are his holdings of one-period bonds. After observing the
state, each entrepreneur makes a discrete decision of investing, disinvesting (by sell-
ing some good-quality capital), or inaction. He also chooses current consumption
and next-period bond holding. Let subscript i denote investing, d denote disin-
vesting, and n denote inaction. I describe a generic entrepreneur j’s intertemporal
94
problem in recursive form, suppressing the j subscript for ease of notation:
{ }
V (k, b, z;Z, µ) = max V i(k, b, z;Z, µ), V d(k, b, z;Z, µ), V n(k, b, z;Z, µ) (2.15)
The value of an investing entrepreneur is
[ ]
V i(k, b, z;Z, µ) = max log(c) + βE V (k′, b′, z′;Z ′, µ′)|z, Z (2.16)
c,k′>(1−δ)k,b′,sb
b′ ( ) 11−γ
subject to c+ = π(w ) z Z kαt jt t jt − ρ[k′ − (1− δ)k] + p sb + bR
b′ ≥ −b
sb ∈ [0, λk]
The value of a disinvesting entrepreneur is
[ ]
V d(k, b, z;Z, µ) = max log(c) + βE V (k′, b′, z′;Z ′, µ′)|z, Z (2.17)
c,k′<(1−δ)k,b′,sb
b′ ( ) 11−γ
subject to c+ = π(wt) z
α
jtZtkjt − p[k′ − (1− δ)k] + p sb + bR
b′ ≥ −b
sb ∈ [0, λk]
95
The value of inaction is
[
V n(k, b, z;Z, µ) = max log(c) + βE V ((1− δ)k, b′, z′
]
;Z ′, µ′)|z, Z (2.18)
c,b′
b′ ( ) 11−γ
subject to c+ = π(w αt) zjtZtkjt + p sb + bR
b′ ≥ −b
sb ∈ [0, λk]
where b is an exogenously given borrowing limit that does not exceed the natural
borrowing constraint, and sb is the entrepreneur’s sales of bad-quality capital.
2.4.3 Market Clearing & Recursive Equilibrium
There are four markets in the model: the market for used investment goods,
the market for labor, the market for final goods, and the market for one-period
bonds. Market clearing in the market for used investment goods is imposed in a
similar way to the one-period model in section 2. Investing entrepreneurs build
new investment goods and bundle them with used investment goods via a CES
aggregator. The demand for used investment goods is obtained by solving a CES
expenditure minimization problem. The pooling price of used capital pt equates the
total demand for used investment goods and total supply. The wage rate wt equates
aggregate labor demand from entrepreneurs and labor supply from households. The
gross interest rate Rt clears the market for one-period bonds (such that aggregate
net bond holding of the entrepreneurial sector is zero).
96
The following definition describes a recursive competitive equilibrium.
Definition 1. (Recursive Competitive Equilibrium). A recursive competitive equi-
librium is defined as a set of prices {p, ρ, R,w}; the market fraction of bad capital
λM ; a law of motion Γ for the cross-sectional distribution µ(k, b, z); value functions
of entrepreneurs {V i, V d, V n, V }; decision rules for entrepreneurs {c, k′, b′, n, sb, g, inew,
iused, d} and decision rules for households {Ch, Lh}, such that
(i) Given wage rate w, {Ch, Lh} maximizes period utility of the households.
(ii) The entrepreneur’s labor demand n(k, b, z) satisfies (13).
(iii) The value functions V i, V d, V n and V solve (15)-(18) and c, k′, b′ are the asso-
ciated policy functions.
(iv) For investing entrepreneurs, the investment is g(k, b, z) = k′(k, b, z)− (1− δ)k
where k′ solves (16). g(k, b, z) is allocated to inew and iused according to (5) and (6).
(v) For disinvesting entrepreneurs, the disinvestment (supply of good used capital)
is d(k, b, z) = (1− δ)k − k′(k, b, z), where k′ solves (17).
(vi) The market for final output clears:
∫ ∫ ∫
Ch + c(k, b, z)dµ+ inew(k, b, z)dµ = Z zk
αn(k, b, z)γdµ
∫
(vii) The labor market clears: Lh = n(k∫, b, z)dµ
(viii) Aggregate net bond holding is zero: b′(k, b, z)dµ=0.
(ix) The market for used capital clears:
∫ ∫ ∫
λ kdµ+ d(k, b, z)dµ = iused(k, b, z)dµ
97
(x) The market fraction of bad quality capital satisfies
M ∫ ∫λ ∫kdµλ = .
λ kdµ+ d(k, b, z)dµ
Furthermore, λM , the price of used capital p and the price of investment bundle ρ
satisfy (4).
(xi) The idiosyncratic shock zj follows the AR(1) process log zjt = ρz log zj,t−1 + εzt.
The law of motion Γ defines the evolution of the distribution µ of entrepreneurs over
k, b and idiosyncratic productivity z, given the policy functions and the transition
probability of z.
2.4.4 Analysis & Computation
Households
Since households do not accumulate wealth, their problem is static and the
first order condition of (10) with respect to lh gives the following simple equilibrium
conditions:
(w(Z, µ)) 1θ
l hh = (2.19)
ψ
ch = w(Z, µ)lh (2.20)
Equations (2.19) and (2.20) fully characterize the solution to the household’s
98
problem. Note that the higher is the Frisch elasticity 1/θh, the more sensitive is
labor supply to changes in the wage rate.
Entrepreneurs
The entrepreneur’s problem involves discrete choices arising from investment
irreversibility, three-dimensional heterogeneity in capital holding, saving, and id-
iosyncratic productivity, and an occasionally binding borrowing constraint, and is
therefore too computationally demanding to solve with standard VFI. I apply the
endogenous grid method (EGM) first proposed by Carroll (2006) to solve this dy-
namic programming problem. The essence of EGM is to predetermine the optimal
value of the endogenous state variable and use the first order optimality condition
to directly solve for the decision rule for the control variable, thus circumventing
the most time-consuming procedure of root finding of the first order optimality
conditions in standard VFI. However, one major difficulty in applying EGM to the
entrepreneur’s problem stems from the mixed discrete-continuous choice arising from
partial capital irreversibility. While EGM is more efficient and accurate than stan-
dard VFI, standard EGM cannot be used when the value function is non-concave,
as the first-order condition is necessary but not sufficient to guarantee an optimal
solution, as in the entrepreneur’s problem considered here, which involves a mixed
discrete-continuous choice in capital holdings. I use the EGM algorithm developed
by Fella (2014), which can deal with non-smooth and non-concave problems. In what
follows I briefly describe how the algorithm can be used to solve the entrepreneur’s
99
problem.
All entrepreneurs trivially sell their lemon capital. The investing entrepreneur’s
flow budget constraint can be written as
b ( ) 1t+1 1−γ
ct + = π(w) zjtZtk
α
jt + bt − ρt[kt+1 − (1− δ)kt] + ptλktRt
and the disinvesting entpreneur’s flow budget constraint is
b ( ) 1t+1 α 1−γct + = π(w) zjtZtkjt + bt − pt[kt+1 − (1− δ)kt] + ptλktRt
The borrowing constraint is bt+1 ≥ −b. Apply a change of variable to get:
at+1 = bt+1 + b ≥ 0 (2.21)
where at+1 denotes the entrepreneur’s end-of-period asset holding adjusted for un-
tapped borrowing capacity. Let
( ) 1
1−γ Rt − 1
m(kt+1; at, kt, z
α
jt) = π(w) zjtZtkjt + at − h(kt+1, kt) + ptλkt − bRt
be the total resources available for consumption, conditional on choosing kt+1, where
 ρt[kt+1 − (1− δ)kt] if k t+1
> (1− δ)kt
h(kt+1, kt) =  pt[kt+1 − (1− δ)kt] otherwise
100
I can rewrite the entrepreneur’s budget constraint as:
at+1
ct + = m(kt+1; at, kt, zjt) (2.22)
Rt
An entrepreneur’s recursive problem in a stationary equilibrium without aggregate
uncertainty can be rewritten as:
a′
V(a, k, z) = max log(m(k′; a, k, z)− ) + Ṽ (a′, k′, z) (2.23)
k′≥0,a′∈[0,m(k′;a,k,z)R] R
where V(a, k, z) is the value function of the entrepreneur in state (a, k, z) and
Ṽ (a′, k′, z) = βE V(a′z , k′, z′)
denotes the expectation of the continuation value.
Let Ω(·, ·; a, k, z) denote the feasibility set given state variables a, k, and zt,
with Ω(a′, k′; a, k, z) being one specific point in it and
Ω(·, ·; a, k, z) = {a′, k′ : a′ ∈ [0,m(k′; a, k, z)], k′ > 0}
Denote the per-period utility function as
′
U(a′, k′
a
; a, k, z) = log(m(k′; a, k, z)− )
R
The specification of the per-period utility functions satisfy the following two condi-
101
tions:
Condition 1. The per-period utility function U(a’,k’; a,k,z) is differentiable with
respect to a′ on the interior of Ω(·, k′; a, k, z) and with respect to a on the interior
of Ω(a′, k′; ·, k, z).
Condition 2. The second derivative Ua′a′(a
′, k′; a, k, z) is strictly negative.
The first order condition of (23) with respect to a′ is:
− 1 ′ + Ṽa(a′, k′, z) = 0 (2.24)
R[m(k′; a, k, z)− a ]
R
I state and prove the following corollary that underpins solving the entrepreneur’s
problem in the quantitative exercise:
Corollary 2. The feasibility set Ω(·, ·; a, k, z) is convex, and the first order condition
(24) is a necessary and sufficient condition for a maximum of a′.
Proof : Consider any two points (a′ ′1, k1) and (a
′ ′
2, k2) that belong to the feasi-
bility set Ω(·, ·; a, k, z). To show that the feasibility set is convex, it suffices to show
that any linear combination of (a′1, k
′
1) and (a
′
2, k
′
2) still lies in the feasibility set. In
other words, we want to show that for any θ ∈ (0, 1),
θa′ + (1− θ)a′ ≤ m(θk′ + (1− θ)k′1 2 1 2; a, k, z)R (2.25)
It is straightforward to see that if both k′1 and k
′
2 are both larger than (1− δ)k, then
linear combinations of (a′1, k
′
1) and (a
′ , k′2 2) satisfy the above inequality. The same
holds true if both k′ and k′1 2 are both less than or equal to (1− δ)k. Without loss of
102
generality, let k′1 > (1 − δ)k and k′2 ≤ (1 − δ)k. The proof can be divided into two
cases.
Case 1: θk′1 + (1− θ)k′2 > (1− δ)k; In this case, since in equilibrium 0 < pt < ρt and
since k′2 ≤ (1− δ)k, I have
−θρt[k′1 − (1− δ)k]− (1− θ)p ′t[k2 − (1− δ)k]
≤ −θρt[k′1 − (1− δ)k]− (1− θ)ρt[k′2 − (1− δ)k] = −ρt[θk′1 + (1− θ)k′2 − (1− δ)k]
Thus, inequality (25) is satisfied.
Case 2: θk′1 + (1 − θ)k′2 ≤ (1 − δ)k; In this case, since in equilibrium 0 < pt < ρt
and since k′1 > (1− δ)k, I have
−θρt[k′1 − (1− δ)k]− (1− θ)p [k′t 2 − (1− δ)k]
≤ −θpt[k′1 − (1− δ)k]− (1− θ)p [k′ ′ ′t 2 − (1− δ)k] = −pt[θk1 + (1− θ)k2 − (1− δ)k]
Thus, inequality (25) is still satisfied. This establishes that the feasibility set
Ω(·, ·; a, k, z) is convex. By Theorem 4.11 in Stokey et al. (1989), I know that
the first order condition (24) is a necessary and sufficient condition for a maximum
of a′.
Given corollary 2, I can apply the endogenous grid method (EGM) developed
by Barillas and Fernandez-Villaverde (2007) to run value function iteration more
efficiently.
103
2.4.5 Amplification Mechanism of Reallocation Friction
In this section, I argue that procyclical capital reallocation can amplify an
exogenous TFP shock, in the sense that measured aggregate TFP falls by more
than the magnitude of the exogenous TFP shock itself. Let ZA denote measured
aggregate TFP. An econometrician usually measure aggregate TFP by the residual
method. He would first assume an aggregate production function
Y
Y = ZAK
αNγ =⇒ ZA =
KαNγ
where Y , K, and N denote aggregate capital, labor, and output respectively. The
econometrician would get estimates of α and γ, and use observed Y , K, and N
to back out measured aggregate TFP, ZA. Measured TFP is affected both by the
exogenous TFP shock, Z, and by the allocative efficiency of capital and labor across
heterogeneous entrepreneurs. Suppose, for simplicity, that the entrepreneurs’ id-
iosyncratic productivity can only be either zh or zl, which denote the high and low
idiosyncratic shock respectively, with zh > zl. Further, suppose that the measure of
entrepreneurs facing high and low shocks is equal, and that the production function
is constant returns to scale, α = 1 − γ. Let actual aggregate TFP, Z, be 1. Let
kh and kl denote capital holdings of entrepreneurs facing high and low idiosyncratic
shocks respectively. The following corollary argues heuristically that more severe
capital misallocation lowers measured aggregate TFP.
Corollary 3. Under the above assumptions, the lower is Kh , the lower is Z
K Al
104
Proof : By definition
Y yh + yl
ZA =
KαN1−
=
α (kh + kl)α(nh + n 1−αl)
Labor demand and output are expressed in equations (2.13) and (2.14) as functions
of idiosyncratic productivity, the wage rate, and capital holding. Plugging them
back into the above equation, I get
1−α 1 1 1 1
( γ ) α (z αh kh + z
α
l kl)
(
w z
α
h kh + z
αk )αl
ZA = =
l
1−α 1 1
((kh + kl)α(
γ ) k + kα)(z αk + z αk )1−α h lw h h l l1 11 z α α
= z α − h − zlh kh + 1
kl
Since zh > zl, from the above equation it is clear that when
kh is lower, measured
kl
TFP is lower. This captures the amplification mechanism of procyclical reallocation
frictions. In a recession, the price gap widens and entrepreneurs do not adjust to
their efficient capital level. Entrepreneurs with low idiosyncratic productivity hold
too much capital, which results in an efficiency loss of the economy.
The endogenous allocative component of measured TFP can be measured as
TFPend = log(ZA,t)− log(Zt)
My numerical results in the next section show that TFPend falls following a
negative TFP shock.
105
2.5 Numerical Results
This section presents some numerical results in general equilibrium of the econ-
omy without aggregate uncertainty. The economy is without aggregate uncertainty
in the sense that it stays at the stationary equilibrium until the shock hits. The
shock is unanticipated by the agents and after the perturbation periods agents ex-
pect no more shocks in the future, so that the economy transitions back to the
stationary equilibrium eventually. To arrive at the stationary equilibrium, I keep
aggregate TFP (Z) fixed at 1, and start from an initial cross-sectional distribution
µ0. Market clearing prices p, w, r are solved iteratively with a bisection method until
excess demand/supply is within an error bound, and λM is obtained with adaptive
updates in the inner loop. I numerically solve the entrepreneur’s problem, obtain
his policy functions and compute the equilibrium variables which clear the market
for used capital, the labor market, and the bond market. Then the cross-sectional
distribution is updated and I iterate on this process until the cross-sectional distri-
bution converges to the stationary distribution. After the stationary distribution
is reached, I hit the model with negative TFP shocks and compute the impulse
responses of equilibrium variables.
2.5.1 Parameters
Table 1 lists the key parameters used in the numerical experiment. The pa-
rameters are set generally in correspondence with past numerical exercises. The
investment technology is defined by two parameters: η and . I follow Lanteri
106
(2015) and set η = 0.70. This gives a 31.7% steady-state ratio of used capital to to-
tal capital purchased by investing firms, which is approximately the upper bound of
the estimates found by Eisfeldt and Rampini (2007). The elasticity of substitution
between new and used investment goods  is a key parameter in this model. Edger-
ton (2011) exploits a tax credit (bonus depreciation incentive) which affected only
new investment to provide estimates of this elasticity using data from construction
equipment, aircraft and farming equipment. His estimated values range between 1
and 10. I intentionally set this elasticity to be low in my benchmark case to examine
the effects of a low elasticity of substitution in this model. Since there is limited
empirical evidence on this parameter, I try a higher value of  to see how the results
are affected. I also try different values of the borrowing limit b to numerically assess
the effect of a tighter or looser borrowing constraint.
Parameter Value
β 0.95
γ 0.55
α 0.24
δ 0.065
λ 0.002
η 0.7
 2.5
b 0.35
θh 0.4
ψ 2.0
τ 2.0
ρz 0.9
σε 0.11
Tab. 2.1: Benchmark parameters.
The benchmark calibration gives rise to the following steady-state values:
107
Y C I K N p λM Capital Reallocation
0.557 0.480 0.077 1.190 0.262 0.944 6.84% 0.035
Tab. 2.2: Steady state values under benchmark calibration
The borrowing constraint is about 63% of production in the steady state.
The natural borrowing limit is the production lowest realization of the idiosyncratic
productivity shock divided by the net interest rate (Rt − 1).
2.5.2 Numerical Results
The entrepreneur’s idiosyncratic shock process is discretized using the Tauchen’s
method into 5 different values. I assume that the economy is at the stationary equi-
librium at t = 0, and I perturb the economy with negative aggregate TFP shocks
of −1.00% at t = 1 and −0.90% at t = 2, after which TFP reverts back to Z = 1.
The figures 2.1, 2.2, and 2.3 present the impulse responses of various model
variables in general equilibrium, whereas figures 2.4, 2.5, and 2.6 present impulse
responses of allocative efficiency.6 Besides IRFs under the presence of information
asymmetry, I also plot the impulse responses under no information asymmetry (λ =
0), in which case the model collapses to that of Lanteri. The figures 2.1 and 2.4
are under the benchmark specification with the parameter values listed in table 2.1.
In figures 2.2 and 2.5, I consider a higher elasticity of substitution ( = 3.5), and I
consider the effect of a tighter borrowing limit (b = 0.25) in figures 2.3 and 2.6.
In the benchmark case, as the shock hits, there’s only a small drop in the price
of used investment goods. However, following the fall in p, the market fraction of
6 The IRFs of the market fraction λM and of aggregate sales of good capital Sg are mirror
images, as λ = SbM S +S , and Sb is determined by the aggregate capital stock, which is quite stable.b g
108
bad capital rises substantially, which is consistent with what the model predicts.
The decrease in price and the increase in λM exacerbates the drop in both the
supply of good-quality investment goods and the demand for used capital, causing
capital reallocation to fall. Total reallocation stands for Sb + Sg. It is clear that
with the information asymmetry, the drop in total reallocation is about 2 times
as large as under no information asymmetry, confirming the model’s implication
that adding information asymmetry can substantially increase the procyclicality of
capital reallocation. One can also see that the magnitude of the fall in investment
is significantly larger under information asymmetry, output is marginally lower, and
it takes longer for aggregate capital to climb back to the pre-shock steady state
level. As we can see in figure 2.2, considering a higher elasticity of substitution
does not alter the above observation, but the effects of the negative TFP shocks on
capital reallocation and on aggregate investment are slightly more muted. Figure
2.2 suggests that the procyclicality of capital reallocation may be robust to a range
of the elasticity of substitution. Figure 2.3 considers a tighter borrowing limit. One
can see that the fall in capital reallocation and aggregate investment is now larger,
compared to figure 2.1, both in the case of my model and in that of Lanteri (2015).
The intuition is that some investing entrepreneurs will be credit constrained and
cut back ther capital acquisition, causing their demand for both the new and used
capital to fall. However, the overall qualitative distinction of the responses under
the two cases does not change significantly.
Allocative Efficiency In figures 2.4 - 2.6, I look at the change in two mea-
sures of allocative efficiency following the negative TFP shocks. The first measure is
109
mean MPK ratio, defined as the mean marginal product of capital of entrepreneurs
with the highest idiosyncratic shock divided by the mean MPK of entrepreneurs re-
ceiving the second lowest idiosyncratic shock. The second measure is the endogenous
allocative efficiency component of measured TFP discussed in section 3.5. Figure
2.4 shows that the mean MPK ratio rises on impact of the negative shock, and it
increases more in the presence of information asymmetry. One can make a simi-
lar conclusion by looking at the responses of endogenous TFP. These results show
that the dispersion of the marginal return of capital widens and allocative efficiency
worsens following the shock. Figure 2.5 corresponds to a higher elasticity of substi-
tution. The results are similar to figure 4, but the responses are slightly dampened.
In 2.6, we can observe that, with a tighter borrowing limit, the effect of the shocks
on the mean MPK ratio is much larger, while the responses of endogenous TFP
are almost identical whether or not there is information asymmetry or not. These
results suggests that a tighter borrowing limit might exert a significant effect on
allocative efficiency.
2.6 Conclusion
This paper explores the cyclicality of capital reallocation. The model incor-
porates two key features: imperfect substitutability between new and used capital
and information asymmetry in the resale market of used investment goods. Impulse
response functions of equilibrium variables show that the interaction of these two
ingredients does generate procyclicality of reallocation, countercyclical dispersion of
110
capital returns, and procyclical allocative efficiency. A higher elasticity of substi-
tution  seems to dampen the procyclical property of reallocation, while a tighter
borrowing limit tends to increase the procyclicality of capital reallocation. How-
ever, the effect of information asymmetry per se seems to be robust under different
parameter values considered in my exercise.
111
      p                           λM                     S                        g
1  
0.5 4 0
0
2 −2
−0.5
−1 λ=0.2% 0 −4
λ=0 (Lanteri)
−1.5  
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
t t t
 Total reallocation (Quantity)      Output                        Investment               
1
0
0 0
−2
−2 −1
−2 −4
−4
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
t t t
     Labor                         Capital                       Wage                     
1 1 1
0.5 0.5
0
0 0
−1 −0.5 −0.5
−1 −1
−2
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
t t t
Fig. 2.1: Impulse responses to negative TFP Shocks that last for 2 periods with  = 2.5
and b = 0.35. The magnitudes of the shocks are 1.0% in the first period and
0.9% in the second period.
112
% dev from ss
% dev from ss % dev from ss
% dev from ss
% dev from ss
% dev from ss
% dev from ss
% dev from ss % dev from ss
      p                           λM                     S                        g
1  
0.5 4 0
0
2
−0.5 −2
baseline
−1 λ=0.2%
λ=0 (Lanteri) 0 −4
 
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
t t t
 Total reallocation (Quantity)      Output                        Investment               
1
0
0 0
−2 −1
−2
−2 −4
−4
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
t t t
     Labor                         Capital                       Wage                     
1 1 1
0.5 0.5
0
0 0
−1 −0.5 −0.5
−1 −1
−2
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
t t t
Fig. 2.2: Impulse responses to negative TFP Shocks that last for 2 periods with  = 3.5
and b = 0.35. The magnitudes of the shocks are 1.0% in the first period and
0.9% in the second period.
113
% dev from ss
% dev from ss % dev from ss
% dev from ss
% dev from ss
% dev from ss
% dev from ss
% dev from ss % dev from ss
      p                           λM                     S                        g
1  
4 2
0.5
2 0
0
−2
−0.5 0
baseline
−1 λ=0.2% −4
λ=0 (Lanteri) −2
−1.5  
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
t t t
 Total reallocation (Quantity)      Output                        Investment               
1
0 00
−2
−2 −1
−4
−4 −2
−6
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
t t t
     Labor                         Capital                       Wage                     
1 1 1
0.5 0.5
0
0 0
−0.5
−1 −0.5
−1 −1
−2 −1.5
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
t t t
Fig. 2.3: Impulse responses to negative TFP Shocks that last for 2 periods with  = 2.5
and b = 0.25. The magnitudes of the shocks are 1.0% in the first period and
0.9% in the second period.
114
% dev from ss
% dev from ss % dev from ss
% dev from ss
% dev from ss % dev from ss
% dev from ss
% dev from ss % dev from ss
      Mean MPK ratio          TFP          end
  
0.8
0.7 0.1
0.6
0.05
0.5
0
0.4
0.3
−0.05
0.2
−0.1
0.1
0 −0.15
−0.1 λ=0.2% λ=0.2%
λ=0 (Lanteri) λ=0 (Lanteri)
 −0.2  
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
t t
Fig. 2.4: Impulse responses of allocative efficiencies to negative TFP Shocks that last for
2 periods with  = 2.5 and b = 0.35. The magnitudes of the shocks are 1.0% in
the first period and 0.9% in the second period.
115
% dev from ss
% dev from ss
      Mean MPK ratio          TFP          end
0.8   
0.7
0.1
0.6
0.5 0.05
0.4
0
0.3
0.2 −0.05
0.1
−0.1
0
−0.1 −0.15
λ=0.2% λ=0.2%
−0.2 λ=0 (Lanteri) λ=0 (Lanteri)
 −0.2  
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
t t
Fig. 2.5: Impulse responses of allocative efficiencies to negative TFP Shocks that last for
2 periods with  = 3.5 and b = 0.35. The magnitudes of the shocks are 1.0% in
the first period and 0.9% in the second period.
116
% dev from ss
% dev from ss
      Mean MPK ratio          TFP          end
  
0.15
2
0.1
1.5 0.05
0
1
−0.05
0.5 −0.1
−0.15
0 λ=0.2% λ=0.2%
λ=0 (Lanteri) λ=0 (Lanteri)
  
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14
t t
Fig. 2.6: Impulse responses of allocative efficiencies to negative TFP Shocks that last for
2 periods with  = 2.5 and b = 0.25. The magnitudes of the shocks are 1.0% in
the first period and 0.9% in the second period.
117
% dev from ss
% dev from ss
Chapter A: Appendix for Chapter 1
A.1 Firm’s Optimality Conditions without Frictions
Plug the expression for expected sales:
∫ (Pjt )−θ ) ( ∫Pjt )−θ
min(v St, Zjt dG(v) = S[t min(v, v
∗
jt)dG(v)(Pt PtP 2 2 ]jt )−θ ( σ ) σ
= StF log v
∗ v
jt − + Zjt 1− F (log v∗ vP 2 jt
+ )
t 2
and the firm’s constraint (1.15) back into the firm’s objective (1.14) and take par-
tial derivatives with respect to Zjt yield (note that the terms involving the partial
derivative on the distribution function F (·) would cancel out):
P [ 2 ]jt ( ∗ σ )− v − Ξt ∂(ΛjtYjt) ∂Yjt(∂Zjt) 0 = Ξt 1 F log v
P j[t
+
t 2 Pt ∂Yjt ∂Zj]t
− Ξt+1 ∂(Λj,t+1Yj,t+1) ∂Yj,t+1βEt
Pt+1 ∂Yj,t+1 ∂Zjt
Since ∂(ΛjtYjt)/∂Yjt = Λjt/α, I can rewrite the above equation as
λ P [ ( σ2 )]jt jt
Ξt = Ξt 1− F log v∗jt + vα Pt ( 2 σ2 ) [ λ ]j,t+1
+ β(1− δz)F log v∗jt + v Et Ξt+1 (A.1)2 α
[ ]
λ
Dividing equation (A.1) by Ξ Ξt+1 j,t+1t and then subtracting both sides by β(1−δz)Et Ξt α
yields:
λ [Ξ λ ] [ 2 ]{ [ ]}jt − − t+1 j,t+1 (β(1 δ )E = 1− F log v∗ σ ) Pjt Ξt+1 λj,t+1z t jt + v − β(1− δz)Etα Ξt α 2 Pt Ξt α
which after rearranging gives the expression equation (1.18).
Taking derivative with respect to Pjt yields:
Ξt (Pjt ) ∫− 2θ Ξt (Pjt )−θ ( σ )
(∂Pjt) 0 = St min(v, v
∗ )dG(v)− θ S F log v∗ − vt
P P jt jtt t Pt Pt 2
− ∂(λj,t+1Yj,t+1) ∂Yj,t+1βΞt+1
∂Yj,t+1 ∂Pjt
( ) ∫
⇒ Ξt Pjt −θ Ξt
(Pjt )−θ ( σ2 )
= 0 = St min(v, v
∗
jt)dG(v)− θ StF log v∗ vP P P P jt
−
t t t t 2
θβ(1− δz) λ 2j,t+1 (Pjt )−θ−1 ( σ )
+ Ξt+1 StF log v
∗ v
α P P jt
−
t t 2
119
Ξt ( ∫⇒ Pjt )−θ Ξt= 0 = St min[(v, v
∗
jt)dG(v)− θPt Pt ( ) Pt ] 2
− θ Pjt −θ Pjt − β(1− δz)
(
S Ξ Ξ λ ∗
σ )v
t t t+1 j,t+1 F log v −
Pjt P
jt
t Pt α 2
(A.2)
P
Multiplying equation (A.2) by Pjt and dividing by (
jt )−θSt, and using thePt
fact that ( σ2 ) ∫ v∗jt
F log v∗ − vjt = vdG(v)2 0
I get:
∫ ∫
P [jt ∗ − Pjt − β(1− ( ∗δz) Λ )] vjtt+1Ξt min(v, vjt)dG(v) θ Ξt Et Ξt+1 vdG(v) = 0Pt Pt α Pt+1 0
∫
Dividing the above equation by Ξt min(v, v
∗
jt)dG(v) and rearranging yields:
[∫ ∫ v∗ ∫ v∗P θ jtjt 0 vdG(v) ] θ jt vdG(v) ( )
∗ − 1 = ∫ 0 β(1− δz) Ξt+1 Λj,t+1P min(v, v )dG(v) min(v, v∗ Ett jt jt)dG(v) α Ξt Pt+1
which gives optimal pricing condition equation (1.19).
A.2 Aggregate Price Index
In equilibrium the aggregate price level (equation (1.8)) can be expressed in
terms of v∗jt. One can use the fact that:
[P ]−θ [ ]−θjt + γjt Pjt
Sjt = v
∗ ∗
jt St = v St if vjt > v
P jtt P
jt
t
120
to write [ ] (1−θ v∗ ) θ−1jt θ
Pjt + γ = P
1−θ
jt jt if vjt > v
∗
v jtjt
Substituting the above expression into the aggregate price level (equation (1.8)), we
have:
[ ∫ ∫ v∗ ∫ ∫jt ∞ (v∗ ) θ−1 ]
P 1−θ = P 1−θ 1−θ
jt θ
t v dG(v)dj + P v∫ jt ∫ jt ∫jt jt
dG(v)dj
[ 0 v∗ vjtjtv∗jt ∞ 1 ]
1−θ θ−1= Pjt vjtdG(v) + v
∗ θ θ
jt vjtdG(v) dj∫ { 0( v∗jt
1−θ ∗ − σ
2 ) [ ]}
v ∗ θ− (σ2(1− θ)) ( σ2 21 σ )= ∫ Pjt F log v θjt + vjt exp
v 1− F log v∗ − v + v dj
2 2θ2 jt θ 2
≡ P 1−θjt Φ(v∗jt)dj (A.3)
121
A.3 Non-Stochastic Steady State
I list and show how to solve for the steady state of the economy. For nominal
variables, let the lower case letters denote real variables. For example, wt = Wt/Pt,
w̃t = W̃t/Pt and pjt = Pjt/Pt. Let asterisk subscripts denote steady state values.
The equation for aggregate wage (1.26), together with the first-order condition for
the optimal wage equation (1.25), imply that in the steady state
θw
w̃∗ = w∗ = w
h
∗ (A.4)θw − 1
The inventory evolution equations (1.11) and (1.12) imply
S∗ + δzinv∗ = Y∗ (A.5)
inv∗ = Z∗ − S∗ (A.6)
The production function of intermediate goods firms implies
Y∗ = L
α
∗ (A.7)
122
From the FOCs of the household’s problem one obtains:
[ ]
− −σcΞ∗ = [(1 h)C∗] [ ] (A.8)−σc wh 1
L∗ = (1− h)C σ ∗ σ∗ l l (A.9)
ψl
R∗ 1
r∗ = = (A.10)
π∗ β
Intermediate goods firms’ optimal decisions (A.1) and (A.2) imply the following
equations in the steady state:
( 2 )
− ∗ σv λ∗[1− β(1− δz)]1 F log v∗ + =2 αp∗ − β(1− δz)λ∗
∫ [ (∫ )(A] .11)min(v, v∗)dG(v) θλ ∗∗ min(v, v )dG(v)
(θ − 1)p ∗ ∗∗ ∗ = 1 + β(1− δz) − 1v ∗∗ α v∗
(A.12)
The aggregate price (A.3) implies
{ ( σ2 ) − (σ2(1− θ))[ ( σ2 σ2 )]}θ 1
p1−θ ∗ v ∗ v∗ F log v∗ − + v∗ θ exp 1− F log v∗∗ − v + v = 12 2θ2 θ 2
(A.13)
Finally, the aggregate resource constraint (1.29) in the steady state is:
C∗ + g∗Y∗ + p∗δzinv∗ = p∗Y∗ (A.14)
123
Using (A.11), (A.12), and (A.13) I can solve for v∗, p , and λ . Since v∗ = Z∗∗ ∗ ∗ ∗ p
θ
∗ IS∗
also obtain Z∗ . Using (A.4), (A.7) and (A.9), Y∗ can be expressed as a function ofS∗
C∗:
{[ ] }
− −
σc [(θ − 1)w ] 1 α
Y = (1 h)C σ
w ∗ σ
∗ ∗ l l (A.15)
θwψl
Using (A.5) and (A.6) Y∗ can be expressed as a function of S∗:
[ Z ]∗
Y∗ = 1 + δz( − 1) S∗ (A.16)
S∗
Using the equation for marginal cost (1.13) I get:
α−1
w∗ = αλ∗Y α∗ (A.17)
The aggregate resource constraint can be written as:
(Z∗ )
C∗ + g∗Y∗ + p∗δzS∗ − 1 = p∗Y∗ (A.18)
S∗
I can jointly solve for C∗, S∗, Y∗, and w∗ From (A.15), (A.16), (A.17), and (A.18).
Thus, I also obtain wh∗ and Z∗, as well as inv∗ and L∗.
124
A.4 Proof of Proposition 1 and Quadratic Approximation of Profit
Loss
Proof of Proposition 1:
I derive the linear quadratic approximation to the firm’s lifetime discounted profit
function, as in Mackowiak and Wiederholt (2015). The period profit function of
an intermediate goods firm can be expressed in terms of log-deviations from the
steady state. For convenience and clarity I abuse the notation here to let a variable
with a time subscript denote the log deviation of the original variable (in the main
text) from its non-stochastic steady state value. The period utility function can be
written as:
( 2 )
Ξ eΞtp1−θS e(1−θ)p +S ∗ − σt tF lo v∗ ∗ ∗ [ (g v∗ + Zjt + θpjt − St2 2∗ σ )]+ Ξ∗eΞtp∗Z∗epjt+Zjt 1{− F log v∗ +
v +( Zjt + θpjt − St2
− Ξ wt ∗
1
Ξ e Y ewt− (aα t Z eZjt − (1− δ ) Z eZj,t−1 −θ −θpj,t−1+St−1∗ ∗ ∗
α ( z ∗ − p∗ S∗e2 )
· σF log v∗ − v∗ [ + Zj,t−1 + θpj,t−1 − St−12 (
− Z σ
2 )])} 1
− αZ e j,t 1∗ 1− F log v∗∗ + v + Zj,t−1 + θpj,t−1 − St−12
Let xt = [pjt, Z
′
jt] and let ζt = (wt, St,Ξ
′
t, at) denote the vector of variables
exogenous to the firm. Let h(x−1,x0, ζ0,x1, ζ1, . . .) denote the function that is
obtained by multiplying the period profit function by βt and summing all over t
125
from time zero to infinity. Then
[ ]
Ej,−∑1 h(x−1,x[0, ζ0,x , ζ , . . .)
∼
1 1 = h(0, 0, 0, 0, 0, . . .)
∞
t ′ 1 1 1+ β Ej,−1 hxxt + h
′ ζ + x′H ′ζ t t x,−1xt−1 + xtHx,0xt + x
′
tHx,1xt+12 2 2
t=0
1 ]
+ x′H ζ + x′
1
H ζ + x′
1 ′ 1 ′ 1 ′
( t xζ,−1 t−1 t xζ,0 t tHxζ,1ζ + ζ2 2 t+1 2 tHζx,−1xt−1 + ζtHζx,1xt+1 + ζ2 2 tHζζt1
+ h′
1
x + x′
1 ′ 1 1H x + x H x + x′ H ζ + ζ ′
)
x −1 −1 x,0 −1 −1 x,1 0 −1 xζ,1 0 −1Hζx,1x0β 2 2 2 2
(A.19)
( )
where βthx is the vector of first derivatives of h(x−1,x0,(ζ0,x)1, ζ1, . . .) with re-
spect to xt evaluated at the non-stochastic steady state, β
thζ is the vector of
first de(rivative)s of h(. . .) with respect to ζt evaluated at the non-stochastic steady
state. βtHx,τ is the matrix of second derivative o(f h(. . .) w) ith respect to xt and
xt+τ evaluated at the non-stochastic steady state, β
tHxζ,τ is the matrix of sec-
ond derivative of h((. . .) wi)th respect to xt and ζt+τ evaluated at the non-stochastic
steady state, and βtHζx,τ is the matrix of second derivative of h(. . .) with respect
to ζt and xt+τ evaluated at the non-stochastic steady state.
Note that hxt(x−1,x0, ζ0,x1, ζ1, . . .) = 0 corresponds to the firm’s optimality
conditions in period t. The process of optimal actions {x† ∞t}t=0 should satisfy a
first order approximation to hxt(x−1,x0, ζ0,x1, ζ1, . . .) = 0 around the steady state,
which can be obtained from taking the first derivative of the RHS of (A.19) with
126
respect to xt and set the derivative equal to zero. I get:
[ ]
Et hx +H
† † †
x,−1xt−1 +Hx,0xt +Hx,1xt+1 +Hxζ,−1ζt−1 +Hxζ,0ζt +Hxζ,1ζt+1 = 0
which can be rewritten as:
[ ] [ ]
x† = H−1 −1
1 † −1
t x,0hx +Hx,0 Hx,1xt−1 +Hxζ,−1ζt−1 +Hx,0Hxζ,0ζt +H
−1E H x†
β x,0
t x,1 t+1 +Hxζ,1ζt+1
where Et[·] is the expectation operator conditional on all available information in
period t (including the evolution of {ζt} up to current period t). Thus, the process
of optimal actions {x† ∞t}t=0 can be characterized by the following conditions:
(i) The economy starts from the non-stochastic steady state: x†−1 = [0, 0]
′.
(ii) In each period t ≥ 0, x†t satisfies:
[ ] [ ]
x† = H−1 −1
1 † −1 −1 †
t x,0hx +Hx,0 Hx,1xt−1 +Hxζ,−1ζt−1 +Hx,0Hxζ,0ζt +Hx,0Et Hx,1xt+1 +Hxζ,1ζβ t+1
This establishes Proposition 1.
Quadratic Approximation of Profit Loss:
Using the following equivalence relations:
H ′ ′ ′ ′xζ,0 = Hζx,0, Hxζ,1 = βHζx,−1, Hζx,1 = βHxζ,−1, Hx,1 = βHx,−1
127
I can rewrite the above equation as:
[ ]
Ej,−∑1 h̃(x−1,x[0, ζ0,x1, ζ1, . . .) = h(0, 0, 0, 0, 0, . . .)∞ 1
+ βtE h′ x ′ ′ ′j,−1 x t + hζζt + xtHx,0xt + x2 t
Hx,1xt+1
t=0 ]
+ x′
1
tH
′ ′
xζ,1[ζt+1 + ζtHζx,1xt+1 + xtHxζ,0ζ + ζ
′
t tHζζ2 t
1 1 ]
+ E h′ ′ ′ ′ ′j,−1 −1x−1 + x−1Hx,−1x−1 + x−1Hx,1x0 + x−1Hxζ,1ζ0 + ζ−1Hζx,1x0β 2
(A.20)
Multiplying the above equation by (xt − x† ′ †t) , and using the fact that (xt − xt) is
realized at time t yields
[
Et (xt − x†)′t (hx +Hx,−1x
†
t−1 +H x
†
x,0 t +H
†
x,1]xt+1
+Hxζ,−1ζt−1 +Hxζ,0ζt +Hxζ,1ζt+1) = 0
Using the law of iterated expectations and rearranging yields
[ ]
Ej,−1 (x − x†t [ t)
′(hx +Hxζ,−1ζt−1 +Hxζ,0ζt +Hxζ,1ζt+1)
† ′ † † † ]= −Ej,−1 (xt − xt) (Hx,−1xt−1 +Hx,0xt +Hx,1xt+1) (A.21)
Using (A.20) yields the difference in the firm’s lifetime discounted profit when its
128
actual actions deviates from the optimal actions as:
[ ] [ ]
Ej,−∑1 h(x−1,x[0, ζ0,x1, ζ1, . . .) − Ej,−1 h(x
† † †
−1,x0, ζ0,x1, ζ1, . . .)
∞ ]
∼= βt
1
E x′ ′
1 †′ † †′ †′
j,−1 tHx,0xt + xtHx,1xt+1 xt Hx,0xt − xt Hx,1x2 2 t+1
t=∑0∞ [ ]
+ βtE †j,−1 (xt − xt)(hx +Hxζ,−1ζt−1 +Hxζ,0ζt +Hxζ,1ζt+1)
t=0
1 [ 1 ]
+ Ej,−1[h
′ ′
−1x−1 + x−1Hx,−1x−1 + x
′ ′
−1Hx,1x0 + x−1Hxζ,1ζ +β 2 0
1 ]− E ′ † 1 †′ ′ ′j,−1 h−1x−1 + x−1H x† †x,−1 −1 + x−1Hx,1x†0 + x†β 2 −1Hxζ,1ζ0 + (A.22)
I assume that the economy starts from the non-stochastic steady state. Therefore
x † ′−1 = x−1 = (0, 0) . Substituting (A.21) into (A.22) yields:
[ ] [ ]
E∑j,−1 h(x−[1,x0, ζ0,x1, ζ1, . . .) − Ej,−1 h(x
† †
−1,x0, ζ0,x
†
1, ζ1, . . .)
∞ ]
= βt
1
E x′
1 ′ ′ ′
j,−1 tH
′ † † † †
x,0xt + xtHx,1xt+1 − xt Hx,0x2 2 t
− xt Hx,1xt+1
[ t=0 [ ]
−E (x − x†)′
] 1 ′
j,−1 t t (Hx,−1x
† +H x†t−1 x,0 t +H
† † †
x,1xt+1) + Ej,−1 x−1Fx,1(x0 − x0)β
Using that H ′x,1 = βHx,−1 and rearranging the above equation yields
[ ] [ ]
E † † †∑ j,−1 h̃([x−1,x0, ζ0,x1, ζ1, . . .) − Ej,−1 h̃(x−1,x0, ζ0,x1, ζ1, . . .)∞ ]
= βt
1
E (x − x†)′H † † ′ †j,−1 t t x,0(xt − xt) + (xt − xt) Hx,1(xt+1 − x2 t+1
)
t=0
which forms the firm’s objective function (1.34).
129
The Form of Hessian Matrices: Using the steady-state equations, I obtain:
h ′x =(0, 0) ,  App Apz
Hx,0 =Ξ∗p
−θ  
∗ S∗  
Azp Azz
where ( 2 2
A =(1− θ)2p F log v∗ σ
) [ ( )]
v
pp [ ∗ ∗ − ] (+ p∗v
∗
∗ 1− ∗
σ
F log v v
2 ) ∗ + 2
− β(1− δ
2 2
z) σ ( )
θ2 p − λ f log v∗∗ ∗ ∗ − v( −
2β(1− δz) σθ λ∗F log v∗ − v
α 2 α ∗ 2
− 2β(1− δ )
2
z (1− α) p−θ 2
θ λ ∗
S∗ ∗ σ )v 2
[ ∗α( α σ2 )]
F log v∗ −Y∗[ 2
∗ − ∗ v − − β(1− δz)
] ( σ2 )
Apz =p∗v∗ 1 F log v
∗ v
∗ + θ2 ( p∗ α ) (λ∗ f log v∗ − 2
− β(1− δ
2
z) (1− α) Z 2 2∗ σ σ )
θ λ F log v∗ − v ∗ v∗ F log v +
α α Y ∗ ∗∗ 2 2
Azp =Apz [ β(1− δz) ] ( σ2 )
Azz =− v∗∗ p∗ − λ∗ f log v∗ + vα ∗ 2
− (1− α) ∗ Z∗
[ ( σ2 )2]
v∗λ∗ 1 + β(1− δ )2z F log v∗ + v .α2 Y ∗∗ 2
 0 Bpp
Hx,1 =Ξ∗p
−θ
∗ S∗  
0 Bzz
where
θβ(1− δ 2z) (1− α) λ∗ ( σ )
Bpp = Z∗F log v
∗ − v
α α Y ∗∗ 2
β(1− δz) (1− α) λ∗ ∗ ( σ2 )Bzz = Z∗v∗F log v∗ + vα α Y ∗∗ 2
130
0 0− ( ) 0 0 0

H θ xζ,−1 =Ξ∗p∗ S∗ 
0 − (1−α) (1−δ
2
z) λ∗Z F log v∗ − σv∗ ∗ 0 0 0α α Y∗ 2
 ( 2 )

0 D − θβ(1−δz)λ F log v∗ − σv
− ps ∗
0 0 
H θ
α ∗ 2 
xζ,0 =Ξ∗p∗ S∗  ( ) 
−λ
2
∗v∗ D −β(1−δz)λ v∗∗ zs ∗ ∗F log v∗ +
σv λ∗ ∗ λ∗
∗ 2v∗ 2v
∗
α α 2 α α ∗
where [ ( σ2 )] [ β(1− δ ) ] ( σ2z )
Dps = −p∗v∗∗ 1− F log v∗ + v∗ + θ p∗ − λ∗ f log v∗ − v2 α ∗ 2
θβ(1− δ 2z) (1− α) −θ λ 2∗ ( σ )2+[ p∗ S∗ F log v
∗ − v
α α Y ∗∗ 2
∗ − β(1− δ )
] ( σ2z )
Dzs = v∗ p∗ λ∗ f log v
∗ + v
α (∗ 2β(1− δ )2 (1− α) λ σ2 ) ( σ2z ∗ ∗ ∗ )+ Z F log v + v F log v − v∗ ∗ ∗ .α α Y∗ 2 2
 Ep 0 Ep − 1E − 1p Eα α p
H −θ  xζ,1 =Ξ∗p∗ S∗  
E 0 E − 1z z E 1z − Eα α z
where
θβ(1− δz) ( 2∗ σ )E vp = λ∗F log v −
α ∗ 2
β(1− δz) 2
E = λ v∗
( σ )
z ∗ ∗F log v
∗ + v .
α ∗ 2
131
A.5 Algorithm for the Rational Inattention Model
The rational inattention model is solved using an iteration procedure as in
Mackowiak and Wiederholt (2015), with some slight deviations because in my model
the firm decision rules involve both the pricing decision and the production decision.
I solve the RI model with an outer iteration to pin down the first-best and actual
decision rules and an inner iteration for equilibrium aggregation, divided into the
following 5 steps:
Step 1: I make a guess for the law of motion for the optimal price P †jt and
optimal stock for sale z†jt. This involves guessing the equilibrium vector of first-
best decision rules x†jt in equation (1.36) that enter the firm’s optimization problem
(1.34). A good initial guess will be the equilibrium responses obtained from a full-
information model.
Step 2: I solve for the firm’s optimization problem (1.34) in chapter 1. As
in Mackowiak and Wiederholt (2015), I turn the problem into a finite-dimensional
problem, by parameterizing the infinite-order polynomials Bk(L) and Ck(L) in equa-
tion (1.37) as a lag polynomial of a finite-order ARMA process. Various nonlinear
optimization packages can be used to solve the problem by choosing the coefficients
for the ARMA process. I use the csminwel package developed by Chris Sims, avail-
able on his web page.
Step 3: This step involves the inner iteration for equilibrium aggregate sales.
I first guess a path for the response of aggregate sales, use the firm’s actual pricing
decisions to obtain the aggregate price level using the log-linearized version of the
132
equation for aggregate price (A.3), and use the firm’s actual decision rule Zjt to
obtain the aggregate stock for sale Zt. Given the guess for the sales series, I can
obtain the responses of aggregate inventories and aggregate output. Then, I can use
the interest rate rule (1.28) to obtain the responses of the nominal and real interest
rates. Then, I use the households optimality conditions (1.2) and (1.4) to obtain the
responses of aggregate consumption. Using equation (1.33) I can obtain a new guess
for the aggregate sales series. The inner iteration continues until the aggregate sales
series converges.
Step 4: I can use the wage Phillips curve (derived from equation (1.25) and
equation (1.26)) to obtain the response of the aggregate nominal wage rate.
Step 5: I compute the law of motion for the optimal responses x†jt using
equation (1.39) in Proposition 1 of chapter 1. If the law of motion for the vector
of first-best responses differs from the previous guess, I make the guess in the next
outer iteration a weighted average of the previous guess and the solution obtained by
equation (1.39). The outer iteration continues until a fixed point for x†jt is obtained.
1
1 By assuming a constant marginal cost of attention µ, the RI model can be solved for each
shock separately.
133
A.6 Solving the Sticky Price Model
Since the production function is (likely) decreasing returns to scale and firms
can carry different levels of inventory stock, firms’s pricing and production decisions
will depend on their state variables. Since firms with different inventory stock may
choose different produciton levels and the marginal cost they face would depend on
the quantity of produciton, firms’ marginal cost become firm specific. Therefore,
firms that are allowed to re-optimize do not necessarily choose the same optimal
price and optimal production. The standard technique used to linearize and solve
the representative agent model around the steady state thus could not be completely
applied in the model. In what follows, I describe the model under Calvo pricing and
work out a solution method via the method of undetermined coefficients.
A.6.1 Conjectured Policy Rules
I conjecture (and later verify) two sets of linear policy functions for a firm’s
labor hiring, pricing, and inventory decisions, depending on whether the firm is
allowed to re-optimize price or not. The form of the conjectured policy rules are
listed in the following table:
The conjectured policy rules are log-linear. I use tilde to symbolize the devia-
tion of any variable to symbolize its deviation from the mean value of all firms. For
example, m̃cjt is the log deviation of firm j’s marginal cost to the economy-wide av-
erage marginal cost. Note the optimizing and non-optimizing firms share the same
policy rules except for the price decision, as non-optimizing firms could not reset its
134
Firms: optimize pjt non-optimize pjt
w/ prob. 1− ξp w/ prob. ξp
Policy Rules:
l̃ l ˜ l l ˜ ljt = ψ1invj,t−1 + ψ2p̂jt ψ1invj,t−1 + ψ2p̂jt
p̂jt = ψ
p
1in˜v
∗
j,t−1 + p̂t p̂j,t−1 − πt + `pπt−1
Z̃jt = ψ
z
1in˜v
z z ˜
j,t−1 + ψ2 p̂jt ψ1inv + ψ
z
j,t−1 2 p̂jt
m̃c = ψmcin˜v − + ψ
mcp̂ ψmcin˜ mcjt 1 j,t 1 2 jt 1 vj,t−1 + ψ2 p̂jt
Tab. A.1: Policy Rules
price other than indexation to inflation rate in the previous period. p̂∗t denotes the
average of optimal prices set by optimizing firms.
A.6.2 Expected Inventory Stock
From the log-linearized form of inventory evolution process (1.11) and (1.12)
and after taking expectation, one get:
− Y∗ S∗Etinvjt = (1 δz)invj−1,t + yjt − Etsjt (A.23)
inv∗ inv∗
Note that the expected sales of firm j is
( (P )−θ ) (∗ jt P )−θjtEtSjt = Et min vjt St, Zjt = StΨ(v∗P P jt)t t
where
( σ2 ) [ σ2 ]
Ψ(v∗ ∗ v ∗jt) = F log vjt − + vjt 1− F (log v∗jt + v ) (A.24)2 2
135
Therefore, the log-linearized equation for expected sales could be expressed as
[ ]
Etsjt = Ψ θpjt + zjt − st − θpjt + st
= Ψzjt − θ(1− Ψ)pjt + (1− Ψ)st
where Ψ = Ψ
′(v∗ ∗ ∗∗)v∗/Ψ(v∗) is the elastivity of Ψ(·) w.r.t v∗jt in the steady state.
Putting the above expression for expected sales as well as the log-linearized equations
for yjt and zjt] into (A.23), I get:
Y∗ ( )
Etinvjt =(1− δz){invj[−1,t + at + αljtinv∗
− S∗ inv
] }
∗ Y∗ ( )
Ψ (1− δz) invj,t−1 + at + αljt − θ(1− Ψ)pjt + (1− Ψ)st
inv∗ Z∗ Z∗
After rearranging, the above equation becomes:
( S ) Y ( S )∗ ∗ ∗ ( )
Etinvjt =(1− δz) 1− Ψ invj,t−1 + 1− Ψ at + αljt
Z∗ inv∗ Z∗
S∗
+ θ(1− S∗Ψ)pjt − θ(1− Ψ)st
inv∗ inv∗
Substracting the analogous equation for aggregate economic variables, one obtains:
( S ) Y ( S )( )
E in˜
∗ ∗ ∗
t vjt =(1− δz) 1− Ψ in˜vj,t−1 + 1− Ψ αl̃jt
Z∗ inv∗ Z∗
S∗
+ θ(1− Ψ)p̂jt
inv∗
136
Plugging in the guessed policy rule for l̃jt from table 3, one can write expected
inventory stock as:
Etin˜v = ψ
inv ˜ inv
jt 1 invj,t−1 + ψ2 p̂jt
where  ( ) ( )ψinv 1 = (1− δ ) 1−
S∗
z Ψ +
Y∗ 1− S∗  αψl
 Z inv Z Ψ 1 ∗ ( ∗ ) ∗ψinv = S∗ θ(1−  ) + Y∗2 1− S∗  linv∗ Ψ inv∗ Z∗ Ψ αψ2
The log-linearized equation for stock for sale is
inv∗ Y∗
zjt = (1− δz) invj,t−1 + (at + αljt)
Z∗ Z∗
After subtracting the analogous aggregate equation:
inv∗ Y∗ ( )
z̃jt = (1− δz) invj,t−1 + αl̃jt
Z∗ Z∗
So that  ψz = (1− δ ) inv∗ + Y∗αψl
 1 z Z∗ Z ∗
1
ψz = Y∗αψl2 Z∗ 2
A.6.3 Expected Aggregate Price and Real Marginal Cost
The expression for nominal marginal cost is
W 1−αt Wα tΛ 1−αjt = at Yjt = L
α exp α α expat
jt
137
Therefore, the log-linearized form for real marginal cost is:
mcjt = wt + at + (1− α)ljt
After subtracting aggregate variables:
m̃cjt = (1− α)l̃jt =⇒
 ψmc = (1− α)ψ
l

1 1
ψmc2 = (1− α)ψl2
The expected next-period price is
[ ]
Etp̂j,t+1 = ξpEt(p̂jt − πt+1 + `pπt + p̄t − p̄t+1) + (1− ξ ) ψpp 1Et(in˜vjt) + E ∗t(p̂t+1)
Note that
p̂∗ ∗t+1 = p̃t+1 − p̄t+1
and
∗ ξp − 1
( )
p̃t+1 = (πt+1 `pπt) + p̄t+1 − ξpp̄− − t1 ξp 1 ξp
Therefore the expected next-period price becomes
[ ]
Et(p̂j,t+1) = ξpp̂jt + (1− ξ pp) ψ1Et(in˜vjt)
138
The expected next-period real marginal cost is:
Etm̃ct+1 =ψ
mc
[ 1 Et(in˜vjt) + ψ
mc
2 Et(]p̂j,t+1)
= ψmc + ψmc(1− ξ )ψp E (in˜v ) + ψmc[ 1 2 p 1] t jt 2 ξpp̂jt
= ψmc + ψmc1{[ 2 (1− ξp)ψ
p inv
1 ψ1 ]in˜vj,t−1 }
+ ψmc + ψmc1 2 (1− ξ )ψ
p ψinvp 1 2 + ψ
mc
2 ξp p̂jt (A.25)
A.6.4 Inventory-Sales Ratio
The first order condition w.r.t Zjt gives rise to the following equation:
( )
λ Ξt+1 2jt − β(1− δz)Et( λΞ j,t+1) − ∗ σt = 1 F (log(v vjt + )) = Ψ′(v∗ )Pjt − jtβ(1− δ )E Ξt+1z t λj,t+1 2Pt Ξt
After log-linearization, it becomes:
− mcjt − β(1− δz)Et(dt,t+1 +mcj,t+1)εΨ′(θp̃jt + zjt st) =
1− β(1− δz)
− p∗p̃jt − β(1− δz)mc∗Et(dt,t+1 +mcj,t+1)
p∗ − β(1− δz)mc∗
where
Ψ′′
2
(v∗∗)v
∗
∗ −f(log(v
∗ + σvjt )
ε 2Ψ′ =
Ψ′
=
(v∗ 2∗) 1− F (log(v∗jt +
σv )
2
139
Rearranging the above equation gives:
[ ]
εΨ′(zjt − −
p∗ 1
st) = [θεΨ
′ + p̃ + mc
p∗ −
jt jt
β(1− δz)mc∗ 1−]β(1− δz)β(1− δz)mc∗ − β(1− δz)+ E (d +mc )
p∗ − β(1− δz)mc∗ 1− β(1−
t t,t+1 j,t+1
δz)
Subtracting the analogous aggregate condition I obtain:
[ p ]∗ 1
εΨ′ z̃jt =− [θεΨ
′ + p̂jt + m̃cjt
p∗ − β(1− δz)mc∗ 1−]β(1− δz)β(1− δz)mc∗ β(1− δz)
+ − E (m̃c )
p∗ − β(1−
t j,t+1
δz)mc∗ 1− β(1− δz)
Denote [
− β(1− δz)mc∗ − β(1− δz)
]
B1 =
p∗ − β(1− δz)mc∗ 1− β(1− δz)
Rearranging and using the expression for expected marginal cost (A.25), I obtain
the following equations by matching coefficients:
ψmc [ ]1 = ε ψz +B ψmc + ψmcΨ′ 1 1 1 2 (1− ξ )ψp inv1− β(1− pδ ) 1 ψ1 (A.26)z
ψmc2 p∗=ε zΨ′ψ + θε ′ +
1− Ψβ(1− δz) 2{[ p∗ − β(1− δz)]mc∗ }
+B mc mc1 ψ1 + ψ2 (1− ξ )ψ
p ψinv + ψmcp 1 2 2 ξp (A.27)
140
A.6.5 Pricing
The equation determining re-optimizing firm’s optimal price is:
∑∞ i−1 `pΞ [ ∗t+i Pt+i PjtΠl=0πt+l ( j,t+i Ξt+i+1 )]Ep̄t (ξ β)ip Sj,t+i − β(1− δz)Et+i λj,t+i+1 = 0Ξt P ∗i=0 jt Pt+i j,t+i − 1 Ξt+i
(A.28)
where j,t+i is the effective elasticity of demand as defined in (1.20), which I rewrite
here:
∫ ∫ v∗ ( 2 )θ j,t+i vdG(v) [ v∗ 1− F (log(v∗ − σv ) ] 0 j,t+i j,t+i 2j,t+i = = θ 1−
min(v, v∗j,t+i)dG(v) Ψ(v
∗
j,t+i)
Log-linearizing 
̂j,t+i −1 ∗ − 1[ ̂ 2 ]
= ̂ θ ∗
( σ )∗ v ˆ∗
− − j,t+i
= v̂
 1  1 ∗ j,t+i
+ 1− F (log(vj,t+i − ) −Ψ(v2 j,t+i
)
j,t+i ∗ θ
(A.29)
−1 ∗ − θ( )[ ]
= 1− εΨ + εΨ′ θp̃j,t+i + zj,t+i − st+i (A.30)
∗ −[1 ∗ ]
= −D θp̃j,t+i + zj,t+i − st+i (A.31)
where
≡ 1 ∗ − θ
( )
Dε 1− εΨ + ε− Ψ
′
∗ 1 ∗
141
Log-linearizing Pricing Equation
Log-linearizing (A.28) gives
∑∞ ∑∞ [ ]
Ep̄ i ∗ p̄ i
̂j,t+i Ξ̂t+i+1
t (βξp) p̃j,t+i = Et (βξp) Et+i + −mc− j,t+i+1 1 Ξ
i=0 i=0 j,t+i t+i
Plugging (A.31) into the above equation and rearranging yields:
∑∞ ∑∞ [ ( )]
Ep̄ (βξ )ip̃∗ (1 +D θ) = Ep̄ (βξ )it p j,t+i  t p −D(zj,t+i − st+i) + Et+i dt+i,t+i+1 +mcj,t+i+1
i=0 i=0
(A.32)
Deriving Phillips Curve
Now we work toward solving for the Phillips curve from the above linearized
pricing equation (A.32). Note that from (A.25) I know
 in˜vt+i−1Et+im̃cj,t+i+1 = Φ1 
p̂j,t+i
where  T
 [ ][ ψmc mc p inv1 + ψ2 (1− ξp)ψ1 ψ1 Φ1 ≡ ]  (A.33)
ψmc + ψmc1 2 (1− ξp)ψ
p ψinv + ψmc1 2 2 ξp
142
Define the matrix Ψ1 as:
 
Ψ ≡ ψinv ψinv1 2 1 
0 1
For i ≥ 1,
 in˜vj,t+i−1 {
   in˜vj,t+i−2  0  }
Ep̄t = E
p̄
t Ψ    1  +   (πt+i − `pπt+i−1)
p̂j,t+i p̂j,t+i−1 −1
{      in˜vt+i−3
= Ep̄t Ψ
2 1   0   0+ Ψ1 (πt+i−1 − `pπt+i−2) +  }(πt+i − `pπt+i−1)
p̂j,t+i−2 −1 −1
= . . .
{    in˜vj,t− ii ∑  0  }= Ep̄ i t Ψ1 + Ψi−l1   (πt+l − `pπt+l−1)
p̂ l=1j,t −1
Therefore I can write:
∑∞ ∑ ∞  in˜vj,t+i−1
Ep̄t (βξ
i p̄ i  
p) m̃cj,t+i+1 = Et (βξp) Φ1  
i=0 i=0 p̂j,t+i
∑ {    ∞ in˜vt−i= Ep̄ (βξ )iΦ Ψi  ∑i+ Ψi−l  0  }t p 1 1 1  (πt+l − `pπt+l−1)
i=0  p̂ l=1j,t −1∑∞ in˜vj,t−i
 ∑ ∑    ∞ i−1 0 in˜vj,t−i
 ∑  ∞ ∑∞  0 
= Ep̄ Φ i p̄ it 1(βξpΨ1) + Et Φ1(βξp) Ψ
i−l−1  1  (πt+l+1 − `pπt+l) = Φ1(I− ξpβΨ )−1  + Ep̄ Φ (βξ i i−l−1  1 t 1 p) Ψ1   (πt+l+1 − `pπt+l)
i=0 p̂ i=1 l=0 −1 p̂ l=0 i=l+1j,t j,t −1
=⇒
143
∑  ∞ in˜vj,t−i
Ep̄t (βξp)
im̃cj,t+i+1 = Φ1(I− ξpβΨ −1  1)  
i=0 p̂j,t
∑  ∞  0+ Ep̄ Φ Ψ− − l 1t 1 1 (βξ Ψ l+1p 1) (I− ξpβΨ −1 1)  (πt+l+1 − `pπt+l)
l=0
{ 
−1
  ∑  in˜v ∞j,t−i 0 }
= Φ1(I− ξpβΨ1)−1  + Ep̄t (βξp)l(πt+l − `pπt+l− ) 1 
p̂ l=1j,t −1
(A.34)
Define  Tψz1Φ2 ≡ 
ψz2
By the same token I can derive:
∑ ∞ ∑∞  in˜vj,t+i−iEp̄ (βξ )iz̃ = Ep̄ (βξ )i t p j,t+i t p Φ2 
i=0 i=0 p̂j,t+i{   in˜vj,t−i ∑∞  0− }= Φ (I− ξ βΨ ) 1 + Ep̄2 p 1 t (βξ l p) (πt+l − `pπt+l−1) 
p̂ l=1j,t −1
(A.35)
144
For the price aggregation, I know that:
∑∞ ∑∞ [ ∑i ∑i ]
Ep̄t (βξ )
ip̃ p̄ ip j,t+i = p̃jt + Et (βξp) p̃jt + `p πt+l−1 − πt+i
i=0 i=1 ∑ ∑l=1 l=1∞ i1
= p̃ p̄ ijt − Et (βξp) (πt+l − ` π− p t+l−1
)
1 βξp ∑i=1 l=1∞ ∑∞1
= p̃jt − Ep̄t (πt+l − `pπt+l−1) (βξp)i1− βξp
l=1 i=l
∞
1 ∑ (βξp)l
= p̃ p̄jt − Et (π− t+l
− `pπt+l−1) (A.36)
1 βξp 1− βξp
l=1
Slightly manipulating the RHS of (A.32) gives:
∑∞ ∑∞ [ ]
Ep̄t (βξp)
ip̃∗ p̄ ij,t+i(1 +Dθ) =Et (βξp) Et+im̃cj,t+i+1 −Dz̃j,t+i
i=0 i=0∑∞ [ ( )]
+ Ep̄t (βξ )
i
p −D(zt+i − st+i) + Et+i dt+i,t+i+1 +mct+i+1
i=0
(A.37)
Plugging (A.34) and (A.35) into the RHS, and (A.36) into the LHS of the above
equation gives:
∞
(1 +Dθ) ∑
p̃∗ − (1 +Dθ)Ep̄ l
1− βξ jt 1− βξ t (βξp) (πt+l − `pπt+l−1)p p l=1{ ∑  in˜v ∞ 0 }
= (Φ −DΦ )(I− ξ βΨ )−
 j,t−i  
1  + Ep̄1 2 p 1 t (βξ lp) (πt+l − ` π − ) p t+l 1  
∗ l=1
∑∞ [ p̂jt ( )]
−1
+ Ep̄t (βξ
i
p) −D(zt+i − st+i) + Et+i dt+i,t+i+1 +mct+i+1 (A.38)
i=0
145
Integrating the above equation over all re-optimizing firms gives:
∞
(1 +Dθ) ∗ − (1 +D
∑
θ)
p̃ p̄t Et (βξ
l
p) (πt+l − `pπt+l−1)
1− βξp 1− βξp
l=1
{   0
= (Φ −DΦ )(I− ξ βΨ )−1   ∑∞  0 }p̄1 2 p 1 + Et (βξp)l(π t+l − `pπt+l−1) 
p̂∗ l=1∑ [ t∞ ( )]
−1
+ Ep̄t (βξ
i
p) −D(zt+i − st+i) + Et+i dt+i,t+i+1 +mct+i+1 (A.39)
i=0
Now, subtracting (A.39) from (A.38) yields:
 
˜
(1 +D θ) inv
(p̂∗ − p̂∗) = (Φ −DΦ )(I− ξ βΨ )−  j,t−i1 
1− jt t 1 2 p 1βξ  p
p̂∗jt − p̂∗t
Therefore, by matching coefficients, I have the following two equations:
 
(1 +D θ)  1 
ψp = (Φ1 −DΦ2)(I− ξ βΨ )−1  p 1 (A.40)
1− βξ 1  p
ψp1
Equations (A.26), (A.27), and (A.40) are three equations to solve for three unknown
coefficients ψl , ψl1 2, and ψ
p
1.
146
Chapter B: Appendix for Chapter 2
B.1 Proof of Corollary 1
Consider the CES expenditure minimization problem of choosing new and used
investment goods in a bundle:
min inew + p iused
inew,iused{ 1 −1 1 −1} 
subject to η  [(1− λ)ij,new]  + (1− η) M [(1− λ )i ]  −1j,used = g
Combining the first order conditions, we have:
( i ) 1( 1− λ ) −1(new   η ) 1 1
=
i Mused 1− λ ( 1−) η pη 1− λM −1
=⇒ i −used = inew p (B.1)
1− η 1− λ
Plugging the above equation back into the constraint, we have:
{ M 2
1 −1 1 −1 (1− λ ) (−1) (1− η) − } 1 −1
g = η  [(1− λ){ij,new]
 + (1− η)  [(1− λM)ij,new}]
   p1−
1 1− ( 1− λ η
⇒ 1− λ
M ) −1 −1
= g = (1− λ)i η + (1− η)η p1−new   (B.2)
1− λ
Multiplying both sides of the above equation by η and then (1− λ)−1, we have:
{ (1− λM ) } − −1 −1(1 λ)i 1−new {η + (1− η) p = ηg( 1− λ1− λM ) } ⇒ −1= (1− λ)i η + (1− η) p1− −1{ new = η(1− λ)
−1g
( 1− λ
⇒ 1
) } 1− ( p )1− −1
= inew η + (1− η) = η(1− λ)−1g (B.3)
1− λ 1− λM
{ ( )1− ( ) } 11 − p 1− 1−Let ρ = η − + (1 η) . Combining equations (B.1) and (B.3),1 λ 1−λM
we have:
i = η(1− λ)−1new ρg
i = (1− η)(1− λM)−1p−ρused g
Therefore, the total expenditure can be expressed as
[ ( 1 )1− ( p ) ]1−
inew + piused = η + (1− η) rhog
1− λ 1− λM
= ρ1−ρg = ρg
which verifies that ρ is the CES price index.
148
B.2 Proof of Proposition 1
The market clearing condition can be written as follows:
E(p, Z; ) =∫Du[se(d − Susedαz Z ) ]j 1
= θ(p, λM{ )
1−α
∫ [ − (1− λ)k0 dF](zM j)zI ρ(p, λ )zD (
− αzjZ
) }1
λk0 + (1− λ)k − 1−α0 dF (zj) = 0 (B.4)
p
where
[ (
M 1 ) ] 11− ( p )1− 1−ρ(p, λ ) = η + (1− η) ,
1− λ 1− λM
I [ ρz = ]α−1
αZ (1− λ)k0
D [ pz = ] ,
αZ (1− α−1λ)k0
λM ∫ [ λk0= ( ) ]zD 1− − αzjZλk + (1 λ)k 1−α0 0 dF (zj)p
and ( )
θ(p, λM) = (1− η)(1− p −λM)−1
ρ
is the ratio of used investment to total investment for investing entrepreneurs. Note
that the market fraction λM depends both on p and Z. Equation (B.4) defines
the market clearing price of used investment goods p as an implicit function of
the aggregate productivity Z and the elasticity of substitution between new and
used investment goods . Using total differentiation of equation (B.4) to obtain
149
the derivative of p with respect to Z in equilibrium, we get the relationship of the
derivatives of excess demand with respect to p and Z:
dp EZ
Ep dp+ EZ dZ = 0 =⇒ = −
dZ Ep
with
∫ ( ) ∫ Dθ αzjZ 1 1 z (αz Z ) 1
E = 1−α
j 1−α
Z dF (zj) + dF (zj)
(1− α)Z z∫I [(ρ (1− α])Z p∂λM αzjZ ) 1
+ θ 1−αλM ∫ − (1− λ)k0 dF (zj)∂Z zI ρ
− θρλM ∂λ
M (αzjZ ) 11−αdF (zj)
(1− α)ρ ∂∫Z [z(I ρ∂λM αzjZ ) ]1
+ θ ρ 1−αρ λM − (1− λ)k0 dF (zj)∂Z zI ρ
The last three terms on the RHS of the above equation describe the indirect effect
of Z on market demand of used capital through the market fraction λM .
∫ [( ∫αz ]jZ ) 1 θρ (αz Z ) 1
E =θ 1−αp p ∫ − −
p j
(1 λ)k dF (z )− 1−α0 j dF (zj)
zI ρ ( ) (1− α∫)ρ [(zI ρzD M ) ]1 1
− 1 αzjZ ∂λ αz1−α jZdF (z ) + θ 1−α∫ j λM − (1− λ)k0 dF (zj)(1− α)p p ∂p zI ρ
− θρλM ∂λ
M (αzjZ ) 11−αdF (z
− j
)
(1( α)ρ ∂p zI ∫ ρ∂λM ) [(αzjZ ) ]1
+ θρ ρp + ρ 1−αλM − (1− λ)k0 dF (zj)∂p zI ρ
150
The last three terms on the RHS of the above equation describe the indirect effect
of p on market demand of used capital through the market fraction λM .
Consider the case when  = 0 (new and used investment goods are perfect
complements). In this case, θ = 1−η , θ = 0, θ = 0, and θ = 1−η− M ρ p λM − M 2 > 0. The1 λ (1 λ )
price of the investment bundle is
η (1− η)p (1− η) (1− η)p
ρ = + , ρ = > 0, ρ M = > 0.
1− λ 1− λM p 1− λM λ (1− λM)2
Also, note that
∫
∂λM (λM)2 1 (αzjZ ) 1
= 1−αdF (z
− j
) > 0,
∂Z λk0 (1 α)Z zD p
and ∫
∂λM −(λ
M)2 1 (αzjZ ) 1
= 1−αdF (zj) < 0.
∂p λk0 (1− α)p zD p
So that
∂λM ∂λM
Z = −p .
∂Z ∂p
Hence, when  = 0 the elasticity of p with respect to Z can be written as:
dp Z −EZ Z (1− α)ZEZφp,Z(0) = = = (B.5)
dZ p Ep p −(1− α)pEp
151
where the numerator is
∫ ∫
1− η (αz Z ) 1 zDj (− αzjZ ) 1(1 α)ZE = 1−αZ { dF (z ) +
1−α
∫ [j dF (zj)1− λM zI ρ p∂θ ∂λM (αz Z ) ]j 1
+ (1− α) ∫ Z
1−α
} − (1− λ)k0 dF (zj)∂λM ∂Z zI ρθρ M ∂λM (αzjZ ) 1− λ Z 1−αdF (zj) (B.6)
ρ ∂Z zI ρ
and the denominator is
(1−η)p ∫ ∫ D
− − 1− η 1−λM
(αzjZ ) 1 z (αz Z ) 1
(1 α)pE = 1−α
j 1−α
p { ∫ [ dF (zj) + ] dF (zj)1− λM ρ zI ρM ( p
− − ∂θ ∂λ αzjZ
) 1
(1 α) 1−α∫ p − (1− λ)k0 dF (zj)∂λM ∂p zI ρ
− θρλM ∂λ
M (αzjZ ) }1
p 1−αdF (zj) (B.7)
ρ ∂p zI ρ
Note that the terms inside the big curly brackets in equations (B.6) and (B.7) are
the indirect effect of Z (and p) on the demand of used investment goods through
λM . The indirect terms in the numerator and th(e deno)minator are actually theM M
same since Z ∂λ = −p∂λ . Furthermore, since (1−η)p− M /ρ < 1, the sum of the∂Z ∂p 1 λ
first two terms on the RHS of the denominator equation (B.7) is less than the sum
of the first two terms on the RHS of the numerator equation (B.6). Therefore, as
long as the indirect effect of p on the demand of used investment goods
is smaller (in absolute value) than the direct effect, both the numerator and
the denominator are positive and φp,Z(0) > 1. Now, since φp,Z() is continuous in
, I can establish that there exists an  > 0 such that φp,Z() > 1 for  < . This
152
completes the proof of the first part.
The meaningful reallocati∫on o[f capital is th(e tot)al su]pply of used investmentzD 1αz Z
goods of good quality, Sgused = (1− λ)k −
j 1−α
0 dF (zj). Now, note thatp
by the chain rule
( )
Z
dSg g dused dS p= (use)d
dZ {d Z dZp ∫− Dp z [(αzjZ ) ] } ( )d Z11−α p= dF (z )
(1− jα)Z p dZ
and
( )
d Z
p 1 [ ]
= − Z dp 1= 1− φp,Z() < 0 for  < 
dZ p p2 dZ p
Therefore, reallocation of used capital increasing in Z.
153
Bibliography
Anderson, E., S. Rebelo, and A. Wong, “Markups Across Space and Time,”
2018. NBER Working Paper No. 24434.
Barillas, F. and J. Fernandez-Villaverde, “A Generalization of the Endogenous
Grid Method,” Journal of Economic Dynamics and Control, 2007, 31 (8), 2698–
2712.
Basu, S. and J. G. Fernald, “Returns to Scale in U.S. Production: Estimates
and Implications,” Journal of Political Economy, 1997, 105 (2), 249–283.
Bergin, P. R. and R. C. Feenstra, “Pricing-to-Market, Staggered Contracts,
and Real Exchange Rate Persistence,” Journal of International Economics, 2001,
54 (2), 333–359.
Bigio, S., “Endogenous Liquidity and the Business Cycle,” American Economic
Review, 2015, 105 (6), 1883–1927.
Carroll, C., “The Method of Endogenous Grid Points for Solving Dynamic Stochas-
tic Optimization Problem,” Economic Letters, 2006, 91 (3), 313–320.
Chari, V., P. Kehoe, and E. McGrattan, “Sticky Price Models of The Business
Cycle: Can The Contract Multiplier Solve The Persistence Problem?,” Economet-
rica, 2000, 68 (5), 1151–1179.
Christiano, L. J., M. Eichenbaum, and C. L. Evans, “Identification and the
Effects of Monetary Policy Shocks,” in M. I. Blejer, Z. Eckstein, Z. Hercowitz,
and L. Leiderman, eds., Financial Factors in Economic Stabilization and Growth,
Cambridge University Press, 1996, pp. 36–74.
, , and , “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary
Policy,” Journal of Political Economy, 2005, 113 (1), 1–45.
Collard, F. and H. Dellas, “Monetary Misperceptions, Output and Inflation
Dynamics,,” Journal of Money, Credit and Banking, 2010, 42 (2), 483–502.
Cooper, R. and J. Haltiwanger, “On the Nature of Capital Adjustment Costs,”
Review of Economic Studies, 2006, 73 (3), 611–633.
Cui, W., “Delayed Capital Reallocation,” 2014. Available at SSRN 2482431.
Dotsey, M. and R. G. King, “Prcing, Production, and Persistence,” Journal of
the European Economic Association, 2006, 4 (5), 893–928.
Edgerton, J., “The Effects of Taxation on Business Investment: New Evidence
from Used Equipment,” Working Paper, Federal Reserve Board, 2011.
Eisfeldt, A. L. and A. Rampini, “Capital Reallocation and Liquidity,” Journal
of Monetary Economics, 2006, 53 (3), 369–399.
155
and , “New or Used? Investment with Credit Constraints,” Journal of Mon-
etary Economics, 2007, 54 (8), 2656–2681.
Erceg, C. J., D. W. Hendersen, and A. T. Levin, “Optimal Monetary Policy
with Staggered Wage and Price Contracts,,” Journal of Monetary Economics,
2000, 46 (2), 281–313.
Evans, C. L. and D. A. Marshall, “Monetary Policy and the Term Structure of
Nominal Interest Rates: Evidence and Theory,” Carnegie-Rochester Conference
Series on Public Policy, 1998, 49, 53–111.
Gali, J. and M. Gertler, “Inflation Dynamics: A Structural Econometric Anal-
ysis,” Journal of Monetary Economics, 1999, 44 (2), 195–222.
Garvazza, A., “The Role of Trading Frictions in Real Asset Markets,” American
Economic Review, 2011, 101 (4), 1106–1143.
Hamilton, J. D., “Why You Should Never Use the Hodrick-Prescott Filter,” Re-
view of Economics and Statistics, 2018, 100 (5), 831–843.
Hurvich, C. M. and Chih-Ling Tsai, “A Corrected Akaike Information Criterion
for Vector Autoregressive Model Selection,” Journal of Time Series Analysis,
1993, 14 (3), 271–279.
Jordá, Ó., “Estimation and Inference of Impulse Responses by Local Projections,”
American Economic Review, 2005, 95 (1), 161–182.
156
Jung, Y. and T. Yun, “Monetary Policy Shocks, Inventory Dynamics, and Price-
Setting Behavior,” Working Paper, Federal Reserve Bank of Sanfrancisco, 2006.
Karabarbounis, L., “The Labor Wedge: MRS vs. MPN,” Review of Economic
Dynamics, 2014, 17 (2), 206–223.
Khan, A. and J. Thomas, “Credit Shocks and Aggregate Fluctuations in an
Economy with Production Heterogeneity,” Journal of Political Economy, 2013,
121 (6), 1055–1107.
Kiyotaki, N. and J. Moore, “Liquidity, Business Cycles, and Monetary Policy,”
2012. NBER Working Paper No. 17934.
Korenok, O. and N. R. Swanson, “How Sticky Is Sticky Enough? A Distribu-
tional and Impulse Response Analysis of New Keynesian DSGE Models,” Journal
of Money, Credit and Banking, 2007, 39 (6), 1481–1508.
Kryvtsov, O. and V. Midrigan, “Inventories, Markups, and Real Rigidities in
Menu Cost Models,” Review of Economic Studies, 2013, 80 (1), 249–276.
Kurlat, P., “Lemons Market and the Transmission of Aggregate Shocks,” American
Economic Review, 2013, 103 (4), 1463–1489.
Lanteri, A., “The Market for Used Capital: Endogenous Irreversibility and Re-
allocation over the Business Cycle,” American Economic Review, 2015, 108 (9),
2383–2419.
157
Levin, A. T., D. Lopez-Salido, and T. Yun, “Strategic Complementarities and
Optimal Monetary Policy,” 2006. unpublished.
Loecker, Jan. D. and J. Eeckhout, “The Rise of Market Power and the Macroe-
conomic Implications,” 2018. NBER Working Paper No. 22687.
Mackowiak, B. and M. Wiederholt, “Optimal Sticky Prices under Rational
Inattention,” American Economic Review, 2009, 99 (3), 769–803.
and , “Business Cycle Dynamics under Rational Inattention,” Review of Eco-
nomic Studies, 2015, 82 (4), 1502–1532.
, E. Moench, and M. Wiederholt, “Sectoral Price Data and Models of Price
Setting,” Journal of Monetary Economics, 2009, 56, S78–S99.
Mankiw, N. G. and R. Reis, “Sticky Information versus Sticky Prices: A Pro-
posal to Replace the New Keynesian Phillips Curve,,” The Quarterly Journal of
Economics, 2002, 117 (4), 1295–1328.
Pasten, E. and R. Schoenle, “Rational Inattention, Multi-Product Firms and
the Neutrality of Money,” Journal of Monetary Economics, 2014, 80, 1–16.
Rogoff, K., “The Purchasing Power Parity Puzzle,” Journal of Economic Litera-
ture, 1996, 34 (2), 647–668.
Sims, C. A., “Implications of rational inattention,” Journal of Monetary Eco-
nomics, 2003, 50 (3), 665–690.
158
Smets, F. and R. Wouters, “Shocks and Frictions in US Business Cycles: A
Bayesian DSGE Approach,” American Economic Review, 2007, 97 (3), 586–606.
Stevens, L., “Coarse Pricing Policies,” 2019. Kilts Center for Marketing
at Chicago Booth Nielsen Dataset Paper Series 1-027. Available at SSRN:
https://ssrn.com/abstract=2544681.
Stokey, N. L., R. E. Lucas, and E. C. Prescott, in “Recursive Methods in
Economic Dynamics,” Harvard University Press, 1989.
Taylor, A. M. and M. P. Taylor, “The Purchasing Power Parity Debate,” Jour-
nal of Economic Perspectives, 2004, 18 (4), 135–158.
Thomas, C., “Search and Matching Frictions and Optimal Monetary Policy,” Jour-
nal of Monetary Economics, 2008, 55 (5), 936–956.
Trabandt, M., “Sticky Information vs. Sticky Prices: A Horse Race in a DSGE
Framework,,” 2007. Sveriges Riksbank Working paper Series 209.
159