Heterogenous Workers and Optimal Inflation

Tack Yun
Seoul National University
This Draft: November, 2023

Abstract
It is shown in this paper that interactions between the staggered price-setting of firms and skill ladder ofworkers can lead to the optimality of a positive inflation in the presence of nominal price rigidity. Specifically,a social benefit of inflation arises with the assignment of workers with low skill levels to price-adjusting firms
and workers with high-skill levels to non-adjusting firms. It is also shown that the magnitude and sign ofthe optimal inflation rate depend on the nature of skill - either general or firm-specific or both as well as thepattern of price adjustment. Finally, theoretic models of this paper imply that the long-term U.S. optimal
inflation rate is slightly above 2 percent under baseline parameter values, while numerical results can respondsensitively to changes in skill and productivity distributions.

JEL classification: E31, E52, E58
Keywords: Optimal Inflation; Heterogenous Workers; Staggered Price-Setting; Assignment ofWorkers to Firms; Relative Price Distortion

E-mail address: tackyun@snu.ac.kr.

1Introduction

A lot of academic and practical research works have been devoted to address the issue of what
inflation rate is socially desirable while actual central banks usually set positive inflation targets. A
recent trend in macroeconomics can be characterized by the emphasis on the importance of microeconomic heterogeneity for practical and normative macroeconomic policy prescriptions including
central bank’s inflation target. But little has been discussed on the potential role of interactions
between the staggered price-setting of firms and skill ladder of workers in the determination of the
optimal inflation rate. To fill such a gap, this paper analyzes the impact of assignment of heterogenous workers to firms on the optimal inflation rate in otherwise standard staggered price-setting
models with nominal price rigidity.
It is shown in this paper that a positive inflation rate can be optimal when a social benefit
of inflation is created by the assignment of workers with low skill levels to price-adjusting firms
and workers with high-skill levels to non-adjusting firms. The mechanism behind this result is
two-fold. First, a positive inflation affects relative output levels of individual firms whose prices are
set in a non-synchronized fashion. Second, the attainment of efficient resource allocation depends
on the assignment of workers to firms in the presence of output dispersion of individual firms.
In particular, the aggregate output level is increased for given amounts of production inputs asworkers with higher skill levels are assigned to firms with larger output levels. Hence inflation canbe socially beneficial if workers are efficiently assigned to firms when a positive inflation rate leads
to output dispersion of individual firms. In addition, it should be noted that such a mechanism
does not exist without output dispersion where each firm produces the same amount of output. In
this sense, a novel point of this paper is that interactions between the staggered price-setting offirms and skill ladder of workers play an important role in the optimality of a positive inflation in
equilibrium models of nominal price rigidity.
It might be possible to explain this result heuristically by relying on the well-known optimality
condition of welfare economics that the marginal rate of transformation should be equal to themarginal rate of substitution at the optimal allocation as follows. The steady-state ratio of relative
prices between price-adjusting and non-price adjusting firms is equal to the aggregate gross inflationin staggered price-setting models of nominal price rigidity. The steady-state ratio of marginal
products of labor between price-adjusting and non-price adjusting firms is the ratio of skill levels
assigned to these firms. The steady-state optimal allocation thus can be attained when the aggregate
gross inflation is equal to the ratio of skill levels between non-price adjusting and price-adjusting
firms. It follows from this condition that when workers with low skill levels are assigned to priceadjusting firms and workers with high-skill levels to non-adjusting firms, the optimal gross inflation
1

should be equal to the ratio of skill levels between these workers. To the extent which the ratio ofskill levels is greater than one, it implies the optimality of a positive inflation rate in the presenceof skill heterogeneity between workers.
The magnitude of the optimal inflation rate depends on the nature of skill - either general or
firm-specific or both as well as the pattern of price adjustment. In the case of general skill that canbe widely used across different individual firms, one should allow for the possibility of mismatch
between supply and demand of workers with a skill level. For example, when the measure of priceadjusting firms is larger than that of workers with lowest skill level, the skill level of price-adjusting
firms should be defined as a weighted average of different skill levels. It means that the ratio of skill
levels between non-price adjusting and price-adjusting firms is affected by both skill distribution ofworkers and price distribution of firms. But this result does hold in the case of firm-specific skill
where there is one-to-one correspondence between price and skill distributions. It is shown in thispaper that the optimal inflation rate in the case of general skill tends to be higher than that in thecase of firm-specific skill.
The equality between gross inflation and ratio of skill levels alone does not exclude the possibility
that a negative inflation rate is optimal at the steady state. The reason for this argument is thatwhen workers with high skill levels are assigned to price-adjusting firms and workers with low-skill
levels to non-adjusting firms, the optimal gross inflation still should be equal to the ratio of skill
levels between these workers. In this case, the equality condition for the optimal steady-state grossinflation implies the optimality of a negative steady-state inflation rate. For this reason, one might
wonder if a negative inflation rate is optimal when the social planner is supposed to choose boththe aggregate inflation and the assignment of workers to firms at the same time. It is shown inthis paper that there are both a negative inflation rate and a positive inflation rate that attains theminimum relative price distortion in a symmetric two-period Taylor-type pricing model. But thisresult does not hold in a non-symmetric staggered price-setting model such as Calvo-type pricing
model.
Finally, theoretic models of this paper imply the long-term U.S. optimal inflation rate is slightly
above 2 percent under baseline parameter values. In this sense, the present paper would present atheoretical framework to help rationalize actual inflation targets on the basis of a standard staggered price-setting model in which the theoretical rationalization of a substantially positive optimal
inflation rate has been deemed to be a challenging task. But a caveat is that the magnitude of
the optimal inflation rate varies within a relatively wide range starting from less than 2 percent
through about 8 percent depending on distributional assumptions of skill and price distributions.
By and large, this result can be regarded as a product of two independent factors. The first factor isthe fact that the magnitude of productivity dispersion statistics varies substantially across different2related studies. Specifically, the interquartile range of firm-level productivity (defined as the log
difference between measured values of productivity at three quarter and one quarter) ranges from
0.3 through 0.9. In particular, this feature results into a relatively wide range of possible valuesof each skill ladder. The second one is that the survival probability of prices is much smaller thanthe probability of skill expansion because the average life-span of individual prices is much shorter
than the average employment tenure of individual workers.
The rest of this paper is organized as follows. The following section presents a simple modelwith non-degenerate price and skill distributions to discuss how workers with different skill levels
are assigned to firms with different output levels and its consequences on aggregate production. The
relative price distortion is also defined as the part of the aggregate output that is foregone becauseof price dispersion. Section 3 analyzes the impact of assignment of heterogenous workers to firms
on the optimal inflation on the basis of the social planner’s optimization to minimize relative pricedistortion given a pair of price and skill distributions. In section 4, theoretic results of previous
sections are used to compute the optimal U.S. steady-state inflation rate under different sets ofplausible parameter values. Section 5 shows how the optimal inflation rate can be implemented ina ‘perpetual youth’ overlapping-generations model of Blanchard (1985) and Yarri (1965) (hereafter
Blanchard-Yarri model). Section 6 concludes.

2

Model

In the economy, there are heterogeneous workers with different levels of production skill and firms
with different nominal prices of goods. The whole population of workers is partitioned into (m̄ + 1)
different skill cohorts, m = 0, · · · , m̄. The whole set of firms is partitioned into (k̄ + 1) different
price cohorts, k = 0, · · · , k̄. In addition, values of m̄ and k̄ are either finite or infinite. The skill
distribution at period t of workers is represented by a list of skill levels and corresponding measures,Pm̄{Zm,t , Γm }m̄
m=0 with restrictions of
m=0 Γm = 1 and Zm,t < Zm+1,t for m = 0, · · · , m̄ − 1. The
price distribution at period t of firms is represented by a list of nominal prices and correspondingPk̄∗ , Φ }k̄∗measures, {Pt−kk k=0 with a restriction of
k=0 Φk = 1, where Pt−k is the nominal price reset
optimally at period t − k reflecting the staggered price-setting of firms.

2.1

Goods Market

The distribution of individual demands is straightforwardly inherited from that of individual nominal prices when each firm’s demand is determined by only two variables such as its own relative
price and aggregate demand, following the model of Dixit and Stiglitz (1977). Let Dk,t denotethe demand at period t of a type k firm and Dt the aggregate demand at period t. The model of3

Dixit and Stiglitz then implies that the demand curve facing a type k firm is given by the following
equation:
Dk,t = (∗Pt−k)−ϵ DtPt(2.1)

where ϵ is a positive constant greater than one and Pt is the aggregate price index at period t. The
aggregate demand at period t is defined as
Dt =

k̄
X

Φk

k=0

∗ D
Pt−k
k,t
Pt

(2.2)

The aggregate price index at period t is defined ask̄X∗
Pt = (
Φk (Pt−k)1−ϵ )1/(1−ϵ) .

(2.3)

k=0

In addition, it follows from equation (2.1) that the probability mass function of demands should be
identical to that of nominal prices for k = 0, · · · , k̄.
Each period, production of consumption goods requires the assignment of heterogenous workers
to firms reflecting the presence of skill cohorts of workers and price cohorts of firms. A skill cohort
m consists of type m workers with their marginal product of labor Zm,t (> 0) at period t. A price∗cohort k consists of type k firms whose nominal price at period t is Pt−k
optimally determined atperiod t − k. The output of consumption goods produced by a pair of a type m worker and a typek firm, Yt (m, k), is determined by the following linear function:
Yt (m, k) = Zm,t Ht (m, k)(2.4)

where Ht (m, k) is the amount of hours a type m worker works for a type k firm at period t.

2.2

Assignment of Workers to Firms

Let us begin with the definition of assignment function that records the measure of type m workers
who are assigned to type k firms as follows.
Definition 2.1 (Assignment Function) An assignment function is a function Vt (m, k) : N0 ×
N0 → [0, 1] satisfying the following conditionsPm̄Φk =Vt (m, k)Pk̄m=0Γm =Vt (m, k)
Pk=0
Pk̄
m̄
1
=
m=0
k=0 Vt (m, k)
where N0 is the set of nonnegative integers given sets of {Φk }k̄k=0 and {Γm }m̄
m=0 .

4

(2.5)

The first condition of equation (2.5) requires that the measure of type k firms should equal thesum of all types of workers who work for type k firms in the absence of inactive firms. The
second condition requires that the measure of type m workers should equal the sum of all typesof firms that hire type m workers in the absence of unemployment. The third condition is thatthe double summation of assignment function across all types of workers and firms is equal to one.
The assignment function Vt (m, k) thus can be interpreted as a joint probability mass function ofindividual households in terms of skill and workplace.1
It should be noted that if the magnitude of Dk,t is regarded as the difficulty level of a type kfirm’s production task, the ratio of firm’s demand level to worker’s skill level, Ft (m, k) = (Dk,t /Zm,t ),can be interpreted as a measure of assignment function’s performance because it records the lengthof time taken by a type m worker in the completion of a type k firm’s production task. Given thisinterpretation, the assignment model of this section exhibits the absence of comparative advantage
in light of Sattinger (1975, 1993) when log Ft (m, k) is additively separable between m and k.
It is emphasized in this paper that assignment function depends on government’s inflation policy
as well as the nature of skill. In order to show how government’s inflation policy affects assignment
function, let us assume that government can achieve its inflation target together with the followingclassification of its inflation policy.
Definition 2.2 (Inflation Policy) A zero inflation policy corresponds to Π̄t−k,t = 1 for all k ineach period t = 0, · · · , ∞. A nonzero inflation policy corresponds to Π̄t−k,t ̸= 1 for all k in eachperiod t = 0, · · · , ∞ where Π̄t−k,t represents a geometric mean of gross inflations between periods
Q(t − k) and t: Π̄t−k,t = ( ki=1 Πt−k+i )1/k . A positive inflation policy corresponds to Π̄t−k,t > 1 anda negative inflation policy corresponds to Π̄t−k,t < 1 for all k in each period t = 0, · · · , ∞.
Equation (2.1) is then used to show that relative output levels of individual firms are affected bygovernment’s inflation policy as follows.
Yk,t
∗
−ϵ
= Π̄ϵkt−k,t (pt−k )
Yt

(2.6)

for k = 0, · · · , and where Yk,t = Dk,t are obtained from market clearing conditions for individual∗ /P
goods and p∗t−k (= Pt−kt−k ) is the ratio of the nominal price determined at period t − k to theaggregate price index at period t − k. It follows from equation (2.6) that the equality of Yk,t =∗Yt holds under a zero inflation policy when individual firms set prices Pt−k
= Pt−k for all k. Itmeans that output levels of all individual firms are the same even with the staggered price-settingof firms. To the extent which variations of p∗t−k are not large enough, it follows from equation (2.6)
1

In this case, the first and second conditions require that marginal probability mass functions of skill and workplace
implied by a given assignment function correspond to probability mass functions of skill and firm cohorts respectively.

5

Figure 1: Inflation Policy and Assignment of Workers to Firms

Zero Inflation Policy

Non-Zero Inflation Policythat a positive inflation policy leads each firm’s output to increase with the time length of fixing
price from the last price change, whereas a negative inflation policy leads each firm’s output todecreases with the time length of fixing price from the last price change. Specifically, a positive
inflation policy leads to Yk,t < Yi,t if k < i and a negative inflation policy leads to Yk,t > Yi,t if k
< i for all non-negative integers i and k not greater than k̄.
The reason why assignment function is affected by the nature of skill is that a firm-specific skill
is productive for only one firm whereas a general skill is useful for all firms, while such a distinction
between firm-specific and general skills can be found in the related literature such as Acemoglu andPischke (1999) and Lazear (2009). In the case of general skill, there are two distinct possibilities of
either a single skill cohort being assigned to multiple skill cohorts or multiple skill cohorts working
at a single firm cohort. But a single skill cohort should be assigned to a single firm cohort in the caseof firm-specific skill with the assumption that skill levels required for production can be achieved
only with continuation of employment.
<Figure 1> illustrates how the assignment of workers to firms is affected by inflation policy
and the nature of worker’s skill. The right panel corresponds to the case of a zero inflation policy
and the left panel corresponds to a non-zero inflation policy. In each column of the right panel, a
black box is included to indicate the impact of a zero inflation policy on the assignment of workers
to firms where output levels of individual firms are the same across different firms.
In order to distinguish between firm-specific and general skills, each panel contains three different
columns. The first two columns correspond to general skill and the third one is firm-specific skill.
6

In addition, a single type firms can hire different types of workers in the first column while a single
type workers can spread across different types of firms in the second column. The third column
shows that distributions of skill levels and price levels are the same when skills of households are
purely firm-specific.

2.3

Aggregation and Relative Price Distortion

The aggregate production function can be obtained from the linear aggregation of individual firms’
outputs given an assignment function. Specifically, the aggregate output is expressed as a functionof the aggregate labor input, the aggregate labor productivity, and relative price distortion. In
addition, relative price distortion can be regarded as a part of the aggregate output that is foregone
because of a non-degenerate distribution of nominal prices set by individual firms. Specifically, let∆t represent relative price distortion at period t that is defined as a weighted average of relative
outputs of individual firms:
∆t =

k̄ X
m̄
X

Vt (m, k)

k=0 m=0∗
Zt Pt−k()−ϵ
Zm,t Pt(2.7)

The aggregate labor input can be derived from the linear aggregation of working hours of all
workers:
Ht =

k̄ X
m̄
X

Vt (m, k)Ht (m, k)(2.8)

k=0 m=0

where Ht is the aggregate hours worked at period t. The aggregate productivity of workers isdefined as a weighted harmonic mean of skill levels of individual workers:
k̄ X
m̄
X
−1 −1
Zt = (Vt (m, k)Zm,t
)

(2.9)

k=0 m=0

where Zt is the aggregate labor-augmenting productivity at period t. In particular, such a definitionof the aggregate labor productivity is derived from the requirement that relative price distortion
should a weighted average of relative output levels of individual firms as can be seen in equation(2.7).The substitution of equation (2.4) into equation (2.8) together with the definition of relative
price distortion leads to the aggregate production function as follows.
Yt = ∆−1t Zt Ht(2.10)

where Yt is the aggregate real output at period t.
It should be also emphasized that the impact of aggregate inflation on relative price distortion
arises through its impact on the distribution of relative output levels as discussed above. In line
7

Table 3.1: Social Planner’s Optimization Problem
Description

Equation

Minimization ofmin{Πt ,p∗ ,{{Vt (m,k)}m̄

Objective Function

subject tok̄m=0 }k=0 }tConstraint 1
Constraint 2
Constraint 3
Constraint 4
Constraint 5

Pm̄

m=0

k
Zt Π̄t−k,t ϵ
k=0 Vt (m, k) Zm,t ( p∗t−k )

Pk̄

k−1
1
Π̄t−k,t−1
ϵ−1 Pk̄ϵ−1 ))− ϵ−1p∗t = (Φ−1
0 (1 − Πt
k=1 Φk ( p∗t−k )P
Vt (m, k) for k = 0, 1, · · · , k̄Φk = m̄Pm=0k̄Γm = k=0 Vt (m, k) for m = 0, 1, · · · , m̄Pk̄P
1 = m̄
m=0Q k=0 Vt (m, k)Π̄t−k,t = ( ki=1 Πt−k+i )1/k for k = 1, · · · , k̄

Note: Each period, the social planner takes as given sets of predetermined variables, exogenousk̄−1k̄variables, and probability mass functions such as {p∗t−k }k̄k=1 , {Πt−k̄+i }i=1
, {Zm,t }m̄
m=0 , {Φk }k=0 , and{Γm }m̄
m=0 . In addition, Π̄t,t = Π̄t−1,t−1 = 1.

with this feature, equation (2.7) implies that there is an explicit relationship between aggregate
inflation and relative price distortion as can be seen below.
∆t =

k̄ X
m̄
X

Vt (m, k)

k=0 m=0Zt ϵk(p∗ )−ϵΠ̄
Zm,t t−k,t t−k(2.11)

It then follows from equation (2.11) that ∆t = 1 under a zero inflation policy in light of definition∗
2.2 when individual firms set prices Pt−k
= Pt−k for all k. As a result, relative price distortion

disappears under a zero inflation policy in light of definition 2.2, whereas it deviates from one undera nonzero inflation policy.

3

Social Planner’s Problem<Table 3.1> summarizes the social planner’s optimization problem formulated on the basis of themodel presented in the previous section. The social planner is supposed to choose an assignment
function and aggregate gross inflation by minimizing relative price distortion given skill and pricedistributions of workers and firms. The social planner’s objective function in this table is relative
price distortion specified in equation (2.11). The first constraint is derived from the definition
equation of the aggregate price index (2.3). The second constraint through the fourth constraint
reflect three restrictions on assignment function specified in (2.5). The fifth constraint includes thegeometric mean of gross inflations starting from the reset time period of type k firms through thecurrent period. The social planner is also supposed to take as given probability mass functions ofindividual goods prices and predetermined goods prices as well as skill levels and their probability
mass functions.
8But this result is no longer true when the government deviates from a zero inflation policy. In
particular, simple examples of <Figure 1> can be used to show that a positive inflation policy
makes it optimal to assign more efficient workers to firms with larger outputs. In order to showthat this argument is correct, one needs to show that the assignment of more efficient workers tofirms with larger outputs under a positive inflation policy is consistent with the solution of the
optimization problem specified in <Table 3.1>.

3.1

Optimal Assignment of Workers to Firms

The aim of this subsection is to address the issue of how workers are efficiently assigned to firms
given a set of relative output levels {Yk,t /Yt }k̄k=0 . In order to address this issue, let us illustrate
simple examples with simplifying assumptions specified below. In this light, let us use simple
examples of <Figure 1>. In addition, the following two sets of conditions are imposed on the twocases of general skill.
Condition 3.1 A: If k̄ = 1 and m̄ > 1, then there exist positive integers m0 and m̂0 satisfying
Pm0
Pm̂0
m=0 Γm = Φ0 and
m=0 Γm = Φ1 respectively (General Skill, Case I).
P 0
Φk = Γ0 and
B: If k̄ > 1 and m̄ = 1, then there exist positive integers k0 and k̂0 satisfying kk=0
Pk̂0
k=0 Φk = Γ1 respectively (General Skill, Case II).
Lemma 3.1 Optimal Assignment Function If condition 3.1.A holds, then the solution to thesocial planner’s optimization problem specified in <Table 3.1> leads to the following assignment
function:
Y0,t < Y1,t
Vt (m, 0) = Γm if m ∈ [0, m0 ]Vt (m, 1) = Γm if m ∈ [m0 + 1, m̄]

Y0,t > Y1,t
Vt (m, 0) = Γm if m ∈ [0, m̂0 ]Vt (m, 1) = Γm if m ∈ [m̂0 + 1, m̄](3.1)

If condition 3.1.B holds, then the solution to the social planner’s optimization problem specified in<Table 3.1> leads to the following assignment function:
Y0,t < Y1,t · · · < Yk̄,tVt (0, k) = Φk if k ∈ [0, k0 ]Vt (1, k) = Φk if k ∈ [k0 + 1, k̄]

Y0,t > Y1,t · · · > Yk̄,tVt (0, k) = Φk if k ∈ [0, k̂0 ]Vt (1, k) = Φk if k ∈ [k̂0 + 1, k̄](3.2)

Lemma 3.1 summarizes how the optimal assignment function is determined in four different
cases: two different orderings of output levels for condition 3.1.A and condition 3.1.B respectively.
The key point of lemma 3.1 is that workers with higher skill levels should be assigned to firms
with larger output levels. For this reason, the ordering of relative output levels of firms plays animportant role in the determination of the optimal assignment function.
9

Table 3.2: Determination of Optimal Assignment Function: Simple Cases
General Skill: Case I

General Skill: Case II

Y

Ẑ

Y

Ẑ

Y

Z̄

Y

Z̄

Ŷ

Z

Ŷ

Z

Ȳ

Z

Ȳ

Z

∆◦t = Φ0 Y0,t
( Z1,t)−1 + Φ1 Y1,t
( Z0,t)−1tttt∆◦t = Γ1 Y0,t
( Z1,t)−1 + Γ0 Y1,t
( Z0,t)−1tttt∆⋆t = Φ0 Y0,t
( Z0,t)−1 + Φ1 Y1,t
( Z1,t
)−1
t
t
t
t
P 0
−1 −1
Φ−1
Ẑ0,t = ( m̂
1 Γm Zm,t )
Pm=0
m̄
−1
−1 −1
Ẑ1,t = ( m=m̂0 +1 Φ0 Γm Zm,t
)
Pm0
−1
−1 −1
Z̄0,t = ( m=0 Φ0 Γm Zm,t )
P
−1 −1
Φ−1
Z̄1,t = ( m̄
1 Γm Zm,t )0 +1Pm̂m=m0Γk
Φ1 = Pm=0m0
Φ0 =m=0 Γk∆⋆t = Γ0 Y0,t
( Z0,t)−1 + Γ1 Y1,t
( Z1,t
)−1
t
t
t
t
Pk̂0
Γ−1
Ŷ0,t =
1 Φk Yk,t
Pk=0k̄
Ŷ1,t =
Γ−1 Φk Yk,tk̂0 +1 0Pk=k0
Γ−1Ȳ0,t =
0 Φk Yk,t
Pk=0
k̄
−1
Ȳ1,t =
k=k +1 Γ1 Φm Ym,tPk̂0 0
Φk
Γ1 =
Pk=0
k0
Γ0 =
k=0 Φk

Ẑ

Ẑ

Z̄

Z̄

Ŷ

Ŷ

Ȳ

Ȳ

1 = Φ0 ( Z0,t)−1 + Φ1 ( Z1,t)−1tt

1 = Γ1 Y0,t+ Γ0 Y1,ttt

1 = Φ0 ( Z0,t)−1 + Φ1 ( Z1,t)−1tt

1 = Γ0 Y0,t
+ Γ1 Y1,ttt

Note: Ẑ0,t and Ẑ1,t corresponds to averages skill levels of workers who work for type 1 and type 0
firms respectively, while Z̄0,t and Z̄1,t corresponds to averages skill levels of workers who work fortype 0 and type 1 firms respectively. In addition, Ŷ0,t and Ŷ1,t denote average output levels of firmsthat hire type 1 and type 0 workers respectively, while Ȳ0,t and Ȳ1,t denote average output levels offirms that hire type 0 and type 1 workers.

<Table 3.2> summarizes how to compute values of relative price distortion under condition
3.1.A and condition 3.1.B respectively. In two columns of this table, ∆◦t is the value of relative
price distortion under sub-optimal assignment and ∆⋆t is that of relative price distortion under
optimal assignment. In the left column of <Table 3.2>, the difference between two values ofrelative price distortion is∆◦t − ∆⋆t ==Y0,tẐ1,t −1Z̄YẐZ̄− Φ0 ( Z0,t)−1 ) + Y1,t(Φ1 ( Z0,t)−1 − Φ1 ( Z1,t)−1 )
Yt (Φ0 ( Zt )
t
t
t
t
Y
Y
Ẑ
Z̄
Φ0 ( Y0,t− Y1,t)(( Z1,t)−1 − ( Z0,t)−1 ) > 0
t
t
t
t

(3.3)

where the second equality reflects final two lines in the left column of <Table 3.2> and strict
inequality holds with the assignment function specified in equation (3.1).
In the right column of <Table 3.2>, the difference between two values of relative price distortion
is
Z

Ŷ

Ȳ

Z

Ŷ

Ȳ

∆◦t − ∆⋆t = ( Z0,t)−1 (Γ0 Y1,t
− Γ0 Y0,t) + ( Z1,t)−1 (Γ1 Y0,t
− Γ1 Y1,t
)
t
t
t
t
t
t
Ŷ

Ȳ

Z

Z

= Γ0 ( Y0,t
− Y1,t)(( Z1,t)−1 − ( Z0,t)−1 ) > 0
t
t
t
t

(3.4)

where the second equality reflects final two lines in the right column of Table 3.2 and strict inequality
holds with the assignment function specified in equation (3.2). In sum, equations (3.3) and (3.4)
imply that statements of lemma 3.1 are correct.2
2
The proof of lemma 3.1 relies on the assumption that relative output levels of individual firms are independent of
how to assign workers to firms. In relation to this assumption, one can find from equation (2.1) that relative output
levels are determined by relative prices.

10

Table 3.3: Social Planner’s Optimization Problem: Simple Cases
General Skill: Case I
Price Distribution
Φ0 = Φ1 = 1/2
Optimization Problem
Condition 3.1.A with Y0 ≤ Y1ϵϵminΠ ( 1+Π2ϵ−1 ) ϵ−1 ( γ+Πγ+1 )subject toΠ≥1
Optimization Problem
Condition 3.1.A with Y0 > Y1ϵ−1ϵ)minΠ ( 1+Π2ϵ−1 ) ϵ−1 ( γγ −1+Π+1subject toΠ<1General Skill: Case II
Price Distribution
Φk = (1 − α)αk , k = 0, · · ·Optimization Problem
Condition 3.1.B with Y0 ≤ · · · ≤ Yk̄
ϵminΠ 1−α(1−γ)Π
1−α(1−γ) ∆psubject to
ϵ
ϵ−1
1−α
ϵ−1
∆p = ( 1−αΠϵ )( 1−αΠ
1−α )
Π≥1
Optimization Problem
Condition 3.1.B with Y0 > · · · > Yk̄
−1 )Πϵ∆pminΠ 1−α(1−γ
1−α(1−γ −1 )subject toϵ1−α1−αΠϵ−1 ϵ−1∆p = ( 1−αΠ
ϵ )(
1−α )
Π<1

Note: ∆p is the steady-state relative price distortion in models of homogenous workers
and γ is the skill ratio between skilled and unskilled workers: γ = Z1 /Z0 .

3.2

Optimal Inflation

The analysis of this subsection is focused on the steady-state relation between inflation and assignment function because it permits closed-form solutions of social planner’s optimization problems.
But a potential caveat is that the resulting solution to the social planner’s optimization problem
can be different from that of the corresponding dynamic model.
Let us move onto the discussion of the steady-state optimal inflation. The steady-state optimal
inflation is defined as the aggregate inflation that minimizes the steady-state value of relative pricedistortion. The determination of the steady-state optimal inflation rate is summarized in <Table
3.3>. Tow columns of this table include two different optimization problems facing the social
planner, each of which corresponds to case I and case II of general skill respectively. In particular,
these two optimization problems are obtained by substituting the optimal assignment function
presented in lemma 3.1 into the steady-state version of the social planner’s optimization problem
presented in <Table 3.1>.
It should be noted that cases I and II of general skill in <Table 3.3> correspond to different
staggered price-setting models. In the left column of this table, price distribution is set to be Φ0 =
Φ1 = 1/2, which means that individual firms reset prices every two periods as is done in a symmetric
Taylor-type staggered price-setting model. In the right column of this table, price distribution is
set to be Φk = (1 − α)αk (with 0 < α < 1) for k = 0, · · · as is done in a Calvo-type staggered11

Figure 2: Objective Functions of Optimization Problems in Table 3.3

Note: In each graph, the x-axis is the steady-state aggregate gross inflation and the y-axis
is the steady-state relative price distortion.

price-setting model. In this case, only a fraction (1 − α) of firms set a new price while the otherfraction of firms do not change their prices in each period t = 0, · · · .
Comparing the second line with the third line in each column, one can find two different optimization problems facing the social planner respectively. The second-line optimization problem
satisfies condition 3.1.A that leads to an ascending order of individual firms’ output levels suchas Y0 ≤ Y1 ≤ · · · . The third-line optimization problem satisfies condition 3.1.B that leads to andescending order of individual firms’ output levels such as Y0 ≥ Y1 ≥ · · · . It should be notedthat this difference is reflected in their constraints. Specifically, the steady-state optimal inflation
rate should take a positive value in the second-line optimization problem, whereas it should take anegative value in the third-line optimization problem.
The reason why this distinction should be made can be explained as follows. The first pointis that the sign of the aggregate inflation rate determines the ordering of individual firms’ output
levels. The second one is that the ordering of individual firms’ output levels affects the assignmentof workers to firms. The third one is that the assignment of workers to firms affects steady-state
relative price distortion.
In order to check if the social planner’s optimization problems are well-defined optimization
problems, one might want to present graphs of objective functions drawn under a set of plausible parameter values rather than going through mathematical equations.3 <Figure 2> illustrates
3
The price elasticity of demand for individual firms is set to be ϵ = 7 following a set of papers such as Coibion,
Gorodinichenko, and Wieland (2012) and Nakamura, Steinsson, Sun, and Villar (2018). The parameter of nominal
price rigidity is chosen to be α = (0.593)4 , which is consistent with estimation results of Ascari and Sbordone (2014)
but slightly higher than the numerical value set by Coibion, Gorodinichenko, and Wieland (2012) and Adam and12

graphs of objective functions that correspond to optimization problems in <Table 3.3>. The twographs in the left panel are objective functions of the left column in <Table 3.3> and the two graphs
in the right panel are objective functions of the right column in <Table 3.3>. The blue horizontal
lines in the two panels mark minimum values in a range of Π ≥ 1 that correspond to second-line
optimization problems. The red dotted lines in the two panels mark minimum values in a range ofΠ < 1 that correspond to third-line optimization problems. As a result, one can conclude that thesocial planner’s optimization problems in <Table 3.3> are all well-defined optimization problems.
The implication of <Figure 2> for the steady-state optimal inflation can be summarized asfollows. The black dotted (horizontal) lines are minimum values of red dotted lines and the black
straight (horizontal) lines are minimum values of blue straight lines. In this figure, one can findthat both black dotted and straight lines coincide in the left panel, whereas straight line stays below
dotted line in the right panel. The left panel implies the existence of two different values of thesteady-state optimal inflation rate in a symmetric two-period Taylor-type staggered price-setting
model: One is positive and the other is negative. The right panel implies the existence of a uniquepositive steady-state optimal inflation rate in a Calvo-type staggered price-setting model. In sum,the following lemma summarize results of the social planner’s optimization problem specified in<Table 3.3>.
Lemma 3.2 Suppose that γ (= Z1 /Z0 ) ≥ 1. If condition 3.1.A holds with Φ0 = Φ1 = 1/2, thentwo different values of the steady-state optimal inflation Π = γ and Π = γ −1 attains the minimized
value of relative price distortion:
∆=

1
2
2
(
) 1−ϵ
γ+1 γ+1(3.5)

If condition 3.1.B holds with k0 = 0 and Φk = (1 − α)αk , then there is a unique steady-state
optimal inflation (greater than one) satisfying the following condition(1 − ϕ(1 − α))(Π − 1) + (1 − ϕ)(1 − αΠϵ )2 = 0(3.6)

The proof of lemma 3.2 is provided in appendix. The implication of this lemma for the steadystate optimal inflation can be summarized as follows. The first point is that the steady-state
optimal gross inflation is equal to one when all workers are homogenous, which is consistent withresults of the related literature such as Woodford (2003) and Yun (2005). The second point is thatthe steady-state optimal gross inflation is indeterminate in a symmetric two-period Taylor-type
Weber (2019). In addition, the skill ratio between skilled and unskilled workers is set to be γ = 1.1, which meansthat the amount of hours worked by a skilled worker is less than that of unskilled workers by 10 percent in order tocomplete the same production task.

13

staggered price-setting model. In this case, the steady-state optimal inflation rate can be either
positive (Π = γ − 1) or negative (Π = γ −1 − 1) that gives the same value of relative price distortion.
The third point is that the steady-state optimal gross inflation is greater than one in a Calvo-style
staggered price-setting model with k0 = 0.

4

The Optimal Inflation Rate in the United States

In this section, a theoretic model of the previous section is used to analyze the optimal inflation
rate in the United States. The first part of this section is to show that a sufficient condition for the
optimality of a positive steady-state inflation in the case of general production skill (of workers)
is the presence of a constraint to ensure that higher types of firms are supposed to employ higher
skill levels of workers. The second part is to analyze the implication of this theoretic result for thesteady-state U.S. optimal inflation rate on the basis of plausible parameter values.

4.1

Assortative Assignment Constraint and Social Planner’s Problem

In this section, assortative assignment constraint acts as a constraint facing the social planner
who is supposed to minimize relative price distortion. The reason for the inclusion of assortative
assignment constraint is to guarantee the optimality of a positive inflation rate, reflecting resultsof lemma 3.2. In particular, the introduction of assortative assignment constraint into models withstaggered price-setting of firms helps eliminate the possibility of a negative steady-state optimal
inflation rate. The analysis of this section is therefore focused on finding a positive optimal inflation
rate under the requirement of assortative assignment.
Definition 4.1 (Assortative Assignment Constraint) Let mk denote an upper bound on skill
levels of workers assigned to a type k firm. An assortative assignment of workers to firms is defined as
an assignment scheme under which mk is an increasing function of k. The corresponding assignment
function is defined as follows.
Vt (m, k) = Γm if m ∈ [mk−1 + 1, mk ](4.1)

for k = 0, 1, · · · , ∞ and where m−1 = −1. In addition, a random selection device is used to
implement a random assignment of type (mk−1 + 1) workers through type mk workers to type kfirms.
In light of definition 4.1, firm-specific skill can be regarded as a subset of the assortative assignment constraint. The case of firm-specific skill means that a type k firm must hire a type k worker,
which in turn implies that mk = k. Hence the case of firm-specific skill satisfies the requirement
14Figure 3: Assortative Assignment

𝑘
𝑛−1

𝑛

Firm-Specific Skill
𝑘=𝑚Assortative Assignment

3
2
1

0

𝑚0

𝑚1

𝑚3

𝑚2

𝑚

that mk should be an increasing function of k. But it is possible to show that there are casesof mk ̸= k among which mk is an increasing function of k. The following set of conditions helps
ensure the existence of such cases. It also can be interpreted as a subset of sufficient conditions
to guarantee the existence of a closed-form solution to the social planner’s optimization problem
summarized in <Table 3.1> as will be seen.
Condition 4.1 A:

Pmk
m=0 Γm , k = 0, · · · , ∞} exists.
i=0 Φi == (1 − α)αk for k = 0, · · · , ∞ where 0 < α < ω < 1.

An infinite set of integers {mk :B: Γm = (1 − ω)ω m for m = 0, · · · , ∞ and Φk

Pk

Condition 4.1.A can be viewed as an integer constraint for mk that helps obtain a closed-formcharacterization of how relative price distortion evolves over time under the assortative assignment
constraint in light of definition 4.1. Condition 4.1.B is that both skill and price distributions follow
geometric distributions. Specifically, Φk can be interpreted as a probability of k failures of having
a new price after the last price change and Γm as a probability of m updates of skill after being
employed for the first time. In addition, a strict inequality of α < ω in condition 4.1.B meansthat the survival probability of existing prices is less than the skill update probability of currently
employed workers, which will be shown later to be consistent with actual data.
It will be shown that mk is a linear function of k under conditions 4.1.A and 4.1.B. Specifically,
the substitution of condition 4.1.B into condition 4.1.A leads to a linear representation of mk :mk = nk + n − 1(4.2)

for k = 0, 1, · · · , ∞ and where n (= log α/ log ω) is a positive integer greater than one. It follows
15

from equation (4.2) that n = 1 in the case of α = ω, which corresponds to the case of firm-specificskill.
In order to help understand how workers are assigned to firms under the assortative assignment
constraint consistent with condition 4.1, <Figure 3> illustrates graphs of the inverse function of
mk . The horizontal axis represents skill levels of workers m = 0, 1, · · · , while the vertical axis
records type numbers of firms k = 0, 1, · · · . The dotted line with its slope 45 degree startingfrom the origin corresponds to the case of firm-specific skill. The blue straight line displays theassortative assignment constraint that is implied by equation (4.2). In this figure, workers withn different skill levels are randomly assigned to each (positive) k type firms for k = 1, · · · , while
workers with n − 1 different skill levels are assigned to type 0 firms.
It is possible to obtain a closed-form representation of relative price distortion when two conditions 4.1.A and 4.1.B hold. An immediate consequence of this result is a simplified closed-formcharacterization of the social planner’s problem as well as a closed-form solution. It will be thenshown that the optimal gross inflation is proportional to the ratio of current relative price distortion
to lagged relative price distortion, while its constant of proportionality depends on parameters ofboth skill and price distributions as summarized in the following proposition.
Proposition 4.1 Suppose that conditions 4.1.A and 4.1.B hold. The assortative assignment constraint then leads to a recursive representation of relative price distortion:
∆t =

ϵ
α ϵ
α 1 − αΠϵ−1t
Πt ∆t−1 + (1 − )(
) ϵ−1
γ̄
γ̄
1−α

(4.3)

where γ̄ = γ n . The gross inflation that minimizes relative price distortion in period t can be written
as follows.
Πt = γ̄

∆t
∆t−1

(4.4)

In addition, current-period’s relative price distortion is determined by the following equation11−ϵ∆t = γ̄ −1 (α∆t−1+ (1 − α)ϵ (γ̄ − α)1−ϵ )− ϵ−1(4.5)

for a given lagged value of relative price distortion, ∆t−1 .
Proposition 4.1 summarizes important results for the optimal inflation rate that can be derived
from the theoretic model of this section, while its proof can be found in appendix. It would be
now worthwhile to summarize implications of proposition 4.1 for the steady-state optimal inflation
rate. It follows from equation (4.4) that the steady-state optimal gross inflation is Π = γ̄. To theextent which γ > 1 and n > 0, it means that the steady-state optimal inflation rate is positive. In
addition, the magnitude of the steady-state optimal inflation rate depends on parameters of skill
and price distributions.
16

For the purpose of comparison, the case of γ = 1 corresponds to the case of technologicallyhomogenous firms and workers, which in turn leads to a zero steady-state optimal inflation rate as
shown in Woodford (2003) and Yun (2005). As a result, one can confirm that the introduction of
the assignment of heterogenous workers with different skill levels to firms into an otherwise standard
New Keynesian model with technologically homogenous firms and workers leads to the optimalityof a positive steady-state inflation rate.

4.2

The Optimal Steady-State Inflation Rate in the United States

Having presented a theoretic model of the optimal steady-state inflation rate, the next topic is to
discuss how one can use it to compute the optimal steady-state inflation rate in the United States.
In particular, such a computation relies on model calibration that requires numerical values forparameters of skill and price distributions, which determines the magnitude of the steady-state
optimal inflation rate.
Let us begin with the discussion of how to assign numerical values to parameters of skill distribution such as γ and ω. In doing so, one can exploit the feature of theoretical models that
productivity levels of individual firms are solely determined by skill levels of their workers. But
this feature does not necessarily require the equality between productivity and skill distributions,while productivity distribution can have useful information about skill distribution. In order toclarify conditions under which productivity distribution can have useful information about skill
distribution, the following set of conditions is added to the model.
Condition 4.2 A: Individual firms survive with a constant probability of θ in each period.4
B: Conditional on survival, firms whose minimum skill level of workers at period t is Zm,t are subject
to a minimum skill level of workers at period t + 1 of Zm+1,t+1 .
C: Let N (x) denote the measure of firms whose minimum skill level is x. N (x) is a log-linear
function of x as can be seen below:
log N (x) = κ − ϕ log x(4.6)

where κ and ϕ are positive constants.
Condition 4.2.A implies that the average life-span of individual firms is finite, while θ can beinterpreted as a transition probability from current period’s survival state to next period’s survival
4

In order to be make firm’s optimization consistent with this condition, two mutually independent exogenous
signals are supposed to affect the transition of firm’s state between two consecutive time periods. Each period, anexit signal with a probability of (1 − θ) and then an adjustment signal conditional upon survivals with a probabilityof (1 − α̃) are sent to individual firms. Hence the probability of each firm’s fixing price is α = θα̃ as will be seen in
figure.

17

state. The empirical rationale behind this condition is ‘Establishment Age and Survival BED(Business Employment Dynamics) Data’ published at the home page of Bureau of Labor Statistics.
Specifically, the sample average (for the period of 2011-2022) of ‘survival rates of previous year’ssurvivors’ for firms whose opening years belong to the period of 1994-2003 is 0.963. The average
life-span of individual firms is about 27 years when θ = 0.963.
Condition 4.2.B reflects the assumption that individual firms are subject to minimum skill levels
for workers in order to attain efficient operations of production facilities, while these minimum skill
levels increase over time conditional upon their survivals. Condition 4.2.C is associated with therecent literature on international trade such as Melitz and Redding (2014 and 2015). In these works,
productivity levels of individual firms are determined by a Pareto distribution whose cumulative
distribution function F (x) with a positive shape parameter ϕ isx
F (x) = 1 − ( )ϕ
x

(4.7)

for a positive value x ∈ [x, ∞] and x > 0.
Condition 4.2.C can be interpreted as a sufficient condition to guarantee that the productivity
distribution of individual firms is a Pareto distribution specified in equation (4.7). In order to show
how this result works, let S(x) (= 1 − F (x)) denote on the survival function of productivity distribution F (x). The survival function can be interpreted as the fraction of firms whose productivity
is greater than or equal to x. Looking at definitions of two functions N (x) and S(x), one can findthat there is a relation between these two functions such as S(x) = N (x)/N (x). If this relation
holds, then the log-linearity of N (x) as specified in equation (4.6) leads to equation (4.7) becauseof S(x) = (x/x)ϕ .
Lemma 4.1 Suppose that conditions 4.2.A, 4.2.B, and 4.2.C hold. In addition, suppose that skill
levels of workers at the steady state are determined by a constant log-difference model such as Zm+1
= γZm for m = 0, · · · . If all of these conditions are satisfied, then survival functions of productivity
distribution (4.7) imply that the following relation for parameters holds:
θ = γ −ϕ(4.8)

Proof: Both of condition 4.2.A and condition 4.2.B imply that S(Zm+1,t+1 ) = θS(Zm,t ) for m= 0, · · · . Condition 4.2.C also leads to S(Zm+1,t+1 ) = (Zm+1,t+1 /Zm,t )−ϕ S(Zm,t ) for m = 0, · · · .
Hence steady-state versions of these two equations can be summarized as follows.
S(Zm+1 ) = θS(Zm )S(Zm+1 ) = γ −ϕ S(Zm )(4.9)

Dividing both sides of the first-line equation by those of the second-line equation leads to θ = γ −ϕ ,which is identical to equation (4.8). QED
18

Figure 4: Evolution of Household and Firms’s States

Household’s State

Firm’s State

Lemma 4.1 implies that it is possible to obtain a numerical value of γ once values of θ and ϕ are
known. The value of ϕ is set to be ϕ = 4.25 in Melitz and Redding (2014 and 2015). Recall that θ= 0.963 as discussed above. Hence the value of γ is set to be γ = 1.009.
Let us turn to the discussion of how to choose a value of n. Recall that n = log α/ log ω. Hencesetting a value of n amounts to setting values of two parameters such as α and ω. For this reason,
let us begin with the value of α that determines the degree of nominal price rigidity. The value of
α is set to be α = (0.593)4 , which means that firms tend to fix prices on average for 7.4 months.
This value is consistent with estimation results of Ascari and Sbordone (2014) but slightly higher
than the value used in Coibion, Gorodinichenko, and Wieland (2012) and Adam and Weber (2019).
The determination of ω depends on the nature of worker’s skill, either firm-specific or generalor both. When skill is equally useful to all firms, a worker’s skill level does not need to depend
on where he or she works. In this case, an experienced worker remains at the same skill level nomatter whether he and she moves into a new workplace. But workers have both firm-specific andgeneral skills on average in actual real world. So one can argue that a realistic skill distribution
should reflect this feature. The left panel of <Figure 4> shows how a household’s individual stateevolves over time. In this figure, three independent exogenous random events are supposed to affectthe transition of a household’s state between time periods t and t + 1. The first exogenous random
event determines survivals of individual households by sending a death signal to each householdwith a constant (exogenously determined) probability of (1 − ωs ). The second one determines a setof workers who separate from current workplaces by sending a separation signal to each worker witha constant (exogenously determined) probability of (1 − ωe ). The third exogenous random event
determines skill losses among separated workers by sending a skill-loss signal to each separated
19

worker with a constant (exogenously determined) probability of (1 − ωr ). Specifically, a fractionof workers ωr switch jobs whose production technologies are not fundamentally different without
having any change in skill level, whereas the other fraction of workers (1 − ωr ) relocate to jobs atwhich previous job experiences are no longer useful. In the latter, worker’s skill level returns tothat of the youngest workers in the labor force. The measure of type m workers is then determined
as a solution of a linear difference equation as can be seen below.
Γm = ωs ωe Γm−1 + ωs (1 − ωe )ωn Γm(4.10)In this case, equation (4.10) implies that ω = ωs ωe /(1 − ωs (1 − ωe )ωr ) and Γ0 = (1 − ωs (1 − (1 −ωe )(1 − ωr )))/(1 − ωs (1 − ωe )ωr ).5
Let us move onto the discussion of how to choose parameter values. It should be noted thata typical worker is supposed to spend 1/(1 − ωs ) years in the labor force with 1/(1 − ωe ) years of
average employee tenure at a workplace. Hence, if a typical US worker tends to spend 45 years in
the labor market starting from 20 years old through 65 years old, then it means that ωs = 0.978,which is consistent with the value used in Gertler (1987). In addition, if a typical US worker’s
employee tenure is 4 years, then one can set ωe = 0.75, which is consistent with observed values of
employee tenure published by the Bureau of Labor Statistics.6 Hence values of ωs and ωe are set
to be ωs = 0.978 and ωe = 0.75 respectively.
In order to choose a numerical value of ωr , it might be possible to exploit the idea that alonger-spell of non-employment experience can reduce a worker’s production efficiency, leading to alarger-decrease of wage. On the basis of this idea, ωr can be interpreted as the fraction of separated
workers with both zero and a short duration of non-employment. In this context, one can rely
on recent empirical works on job-to-job flows including Hahn, Hyatt, Janicki, and Tibbets (2017),
Hyatt and McEntarfer (2012), and Fallick, Haltiwanger and McEntarfer (2012) that emphasize theimportance of non-employment durations among job changers for their earnings outcomes.
In theses studies, job-to-job separations are defined as main job separations where workers find
new jobs within the same quarter and in the following quarter after their separations take place.
In particular, Fallick, Haltiwanger and McEntarfer (2012) report that the fraction of job-to-job
separations among all job separations is slightly higher than 60% for the years of 1995, 1999, and2001. Reflecting empirical results discussed above, the value of ωr is set to be ωr = 0.6 in thispaper. The numerical values of parameters discussed above are summarized in <Table 4.1>.
5

The case of purely firm-specific skill leads to a simpler expression for the measure of type m workers: Γm =ωs ωe Γm−1 . In this case, Γ0 = 1 − ωs ωe so that ω = ωs ωe .
6Specifically, employee tenure is defined as the median number of years that wage and salary workers had beenwith their current employer. Refer to ‘Employee Tenure in 2020, New Release, Bureau of Labor Statistics’ for thedetailed discussion of how to estimate employee tenure.

20

Table 4.1: Benchmark Parameter Values
Parametersϕθα(1 − ωs )−1(1 − ωe )−1ωrϵbValues4.250
0.963(0.593)4
45
4
0.6
7
1

Description and Definition
Shape parameter of productivity distribution
Survival probability of firms
Fraction of non price-adjusting firms
Average years of being in the labor force
Average years of employee tenure
Fraction of job-to-job separations
Price elasticity of demand
Coefficient of leisure utility function

The next topic is how to compute values of parameters associated with skill distribution andassignment of workers to firms such as ω and n. First, the value of ω is set by substituting numericalvalues of ωs , ωe , and ωr into equation (4.10), which in turn leads to ω = 0.314. Second, the valueof n is set on the basis of equation (4.2) i.e. n = log α/ log ω, which also leads to n = 2.54.
Having completed the discussion of how to fix parameter values, let us move onto numericalpredictions of the theoretic model of this section for the U.S. optimal inflation rate. The key pointin this light is that the solution to the social planner’s optimization problem can be summarized
as a simple one-line equation: (Π = γ̄ with γ̄ = γ n ). By substituting two parameter values of γ
and n into this equation, one can find that the U.S. steady-state optimal inflation rate is 2.29%,
which is slightly higher than the Federal Reserve’s 2% inflation target. A caveat of this numericalresult is to ignore an integer constraint for values of n associated with equation (4.2). In order torespect such an integer constraint, one can choose a natural number closest to n = log α/ log ω. In
this case, the U.S. steady-state optimal inflation rate turns out to be 2.70%.

4.3

Robustness of Numerical Result

Having computed the U.S optimal steady-state inflation rate under the baseline set of parametervalues, the next topic is to check the robustness of the baseline numerical result shown above by
deviating from baseline parameter values summarized in <Table 4.1>. The first case is to keep
track of how the optimal inflation rate responds to a change in the specification of productivity
distribution. In this case, productivity distribution is assumed to be a geometric distribution.
Lemma 4.1 is no longer valid when productivity distribution is a geometric distribution. Hence oneneeds an alternative calibration strategy to fix a value of γ. For this reason, the following lemma
replaces lemma 4.1.

21

Table 4.2: U.S. Optimal Inflation Rate
Degree of Nominal
Price Rigidity
Specification of
Productivity Distribution
Benchmark Specification: Pareto Distribution
Alternative Specification: Geometric Distribution
Lower Bound: LIQRT F P
Upper Bound: LIQRAP L

α = (0.593)4

α = (0.7292)4

2.26%

1.36%

4.06%
8.08%

2.75%
4.80%

Note: LIQRT F P means the log interquartile range of total factor productivity and LIQRAP L means
the log interquartile range of average product of labor.

Lemma 4.2 Suppose that conditions 4.2.A and 4.2.B hold. In addition, the measure of firms
whose minimum skill of workers is Zm is (1 − θ)θm with Zm+1 = γZm for m = 0, · · · . If all of these
conditions are satisfied, then the following relation holds:
γ = θ−LIQR/ log 3(4.11)

where LIQR denotes the log interquartile range of productivity distribution:
LIQR = log Zm0.75 − log Zm0.25 .

(4.12)

Proof: The measure of firms whose minimum skill level at period t of workers is higher than Zm,t
is θm+1 , while the measure of firms whose minimum skill level at period t of workers is lower thanor equal to Zm,t is 1 − θm+1 . Hence the value of m that corresponds the third quartile of firms (=m0.75 ) is m0.75 = −2 log 2/ log θ − 1. The value of m that corresponds to the first quartile of firms(= m0.25 ) is m0.25 = m0.75 + log 3/ log θ. The substitution of these two equations into the definitionof log interquartile range specified in equation (4.12) then leads to equation (4.11). QED.
Lemma 4.2 implies that estimates of LIQR can be used to obtain a numerical value of γ givena value of θ. In order to measure firm-level productivity dispersion within and across industries,
estimates of LIQR are included in Dispersion Statistics on Productivity (DiSP) jointly publishedby U.S. Bureau of Labor Statistics and U.S. Census Bureau. For example, Cunningham et al.
(2021) report (<Table 3> in their discussion paper) that estimates of LIQR within industries are
0.520 for total factor productivity (hereafter LIQRT F P ) and 0.828 for average product of labor(hereafter LIQRAP L ) respectively during the sample period of 1997-2016. The substitution ofthese estimates into equation (4.11) leads to γ = 1.018 in the former case and γ = 1.031 in thelatter case respectively. Hence the optimal steady-state inflation rate in the United States is 4.60%
22

in the case of LIQRT F P and 8.08% in the case of LIQRAP L respectively.7
The second case is to see how the degree of nominal price rigidity affects the optimal inflation
rate. The optimal steady-state inflation rate lowers as the degree of nominal price rigidity increases.
It is because the value of n decreases as the degree of nominal price rigidity increases. In addition,
it is possible to find a wide range of estimates for the degree of nominal price rigidity reflecting thefact that there are a lot of microeconomic empirical studies on frequencies of price changes sinceseminal works of Bils and Klenow (2004) and Nakamura and Steinsson (2008).
In this literature, estimates of price duration are affected by empirical definitions of price changes
as well as price data sets. For example, it is reported in Eichenbaum, Jaimovich, and Rebelo (2011)that reference prices, most quoted prices within a given time period, have an average durationof 3.70 quarters, which in turn leads to α = 0.7297. Moreover, if the value of α is set to be
α = 0.7297, then the U.S. optimal steady-state inflation rate is 1.36% in the case of the Pareto
distribution specified in equation (4.7), 2.75% in the case of LIQRT F P , and 4.80% in the case of
LIQRAP L respectively as can be seen in <Table 4.2>.
The third case is to allow for the possibility that the probability of price adjustment for individual
firms fluctuates over time. The motivation behind this one is that a model with time-varying
probabilities of price adjustment is consistent with empirical studies on price-adjustment frequencies
of individual firms including Nakamura, Steisson, Sun, and Villar (2018). In this case, let αt denotethe probability of price adjustment in period t, while the social planner is supposed to take it as
given over time.
The result of such a modification of the benchmark model is reflected in the law of motion for
relative price distortion as can be seen below.
∆t =

ϵ
αt ϵ
αt 1 − αt Πϵ−1t
Πt ∆t−1 + (1 − )(
) ϵ−1
γ̄t
γ̄t
1 − αt(4.13)In this equation, γ̄t is defined as follows.
γ̄t = γ nt , nt =log αtlog ω(4.14)

where one can find a detailed discussion of how to obtain equations (4.13) and (4.14) in appendix.
The social planner is now supposed to find the aggregate inflation rate by minimizing relative
price distortion in each period t = 0, · · · . The result of the social planner’s minimization problem
is summarized in the following proposition whose proof is included in appendix.
7

It follows from lemma 4.1 and lemma 4.2 that values of γ are determined by productivity dispersion and survival
probability of individual firms. As a result, determinants of the optimal U.S. inflation in the present analysis differ
from those in Adam and Weber (2019) where the optimal steady-state inflation equals the ratio of the average
productivity growth of old incumbent firms to that of new entrant firms.

23

Figure 5: Optimal Inflation Dynamics: Counterfactual Simulation
4.5
Optimal Inflation
Core PCE

Inflation Rate (Annual %)

4

3.5

3

2.5

2

1.5

1

1990

1995

2000

2005

2010

Year

Proposition 4.2 Given that the social planner takes as given the probability of price adjustment(= αt ), the social planner’s minimization at period t of relative price distortion specified in equation(4.13) leads to the following determination of inflation and relative price distortion.
Πt = γ̄t∆t∆t−1(4.15)−1ϵ
1−ϵ ϵ−1)∆t = γ̄t−1 (αt ∆1−ϵt−1 + (1 − αt ) (γ̄t − αt )(4.16)

given a series of {αt }∞t=0 with initial values of ∆−1 and s−1 .
A system of two equations (4.15) and (4.16) (presented in proposition 4.2) can be interpreted
as a model of the optimal inflation rate that allows for the possibility that the probability of priceadjustment varies over time. In order to simulate such a model, a time-series of αt is taken fromthe empirical work of Nakamura, Steisson, Sun, and Villar (2018) on price-adjustment frequencies
of firms for the period of 1987 - 2014, while the other parameter values are the same as those in<Table 4.1>. But it should be admitted that the simulation used in the section does not require
numerical values of nt to be positive integers. For this reason, its numerical values are approximate
values obtained on the basis of a log-linear interpolation between two positive integer points.
<Figure 5> summarizes the optimal inflation path generated by simulating the model of two
equations presented in proposition 4.2. The black solid and dotted lines correspond to the optimal
inflation rate and the blue dotted line is the U.S. core PCE inflation rate. The black solid line
fluctuates around 2 percent, while the black dotted line shows a persistent decline over the sample
period from slightly above 3 percent to around 2 percent. The core PCE inflation rate also displaysa persistent decline over the sample period starting from significantly higher than 3 percent and24then falling down to around 1.5 percent in early 2010s. It should be noted that the key difference
between the two black lines depends on the existence of a trend in the skill level of workers whoenter the labor market for the first time. The black solid line corresponds to the absence of such atrend, whereas the black dotted line allows for the impact of such a trend on the optimal inflation
path.

5

Implementation of the Optimal Inflation Rate

Reflecting that theoretical results of the previous section are obtained without having a full description of how private agents make their decisions, this section presents a small-scale dynamic
stochastic general equilibrium model that leads to the same optimal inflation rate presented in theprevious section.

5.1

Economic Environment

The model of this section builds on the following three distinct features. The first one is that all
workers have finite lives in the similar way as is done in Blanchard (1985) and Yarri (1965). The
second one is that the whole set of workers is partitioned into an infinite number of distinct skill
cohorts whose marginal products of labor are not the same. Hence there are an infinite numberof age cohorts indexed by a ∈ {0, 1, 2, · · · , ∞} and an infinite number of skill cohorts indexed bym ∈ {0, 1, 2, · · · , m̄}, where the value of m̄ is either finite or infinite. The third feature is thatthe whole set of monopolistically competitive firms can be partitioned into an infinite number ofdifferent groups indexed by k ∈ {0, 1, 2, · · · , ∞} because firms set nominal prices at different times.

5.2

Equilibrium Conditions

In this model, all individual households are identified with their names under the restriction thatthe same name should not be given to different persons. Specifically all names of those who arealive at period t is stored in Ht whose elements are not identical over time with its measure fixed to
be one.8 The instantaneous utility function u(Ch,t , Hh,t ) of a household h ∈ Ht at period t is thenassumed to be a twice continuously differentiable function of its arguments where Ch,t and Hh,t are
its consumption and labor supply at period t respectively. The first-order partial derivative of thefirst variable is positive, and that of the second is negative.
8

In this model, individual households are not identical even within the same age cohort because they can havedifferent skill level, Hence it is not possible to use a cohort-by-cohort identification of households widely used in theliterature such as Nistico (2012) and Del Negro, Giannoi, Patterson (2023). In addition, Gertler (1999) allows forthe possibility that an active worker becomes a retiree with a constant probability in each period. In this case, acohort-by-cohort identification can be used for workers because workers within the same age cohorts have the marginal
product of labor as can be seen in Fujiwara and Yuki (2008), Carvalho, Ferrero and Nechio (2016), Fujiwara, Hori,and Wari (2019), and Gali (2021).

25

The optimization problem at period t of a household h can be written as follows.
max

Ch,t ,Hh,t ,Ah,t{u(Ch,t , Hh,t ) + βωs Et [v(Ah,t )]}(5.1)

subject to the following budget constraint and where v(Ah,t ) is the value function evaluated at the
beginning of period t + 1:
Ch,t + (1 − θA )QM,t Ah,t = (1 + θH )wmh,t ,t Hh,t + ωs−1 (QM,t + DM,t )Ah,t−1 − Tt

(5.2)

where Ah,t is a worker h’s mutual fund investment at period t, QM,t is the real price at period tof the mutual fund’share, DM,t is the real dividend at period t of the mutual fund’share, wmh,t ,t is
the real wage at period t for worker h who belongs to a mh,t skill cohort. In this representation ofhousehold’s budget constraint, there are investment tax credit (proportional to investment cost),
employment subsidy (proportional to labor income), and lump-sum tax, where θA is the ratio ofinvestment tax credit to household’s investment cost, θH is the ratio of employment subsidy to labor
income, and Tt is the lump-sum tax at period t.9 In this budget constraint, assets of households
are managed by a mutual fund under annuity contracts which are supposed to distribute the total
wealth of non-survivals equally to survivors in each period, following Blanchard (1985),
In order to provide a concrete example of household’s optimization problem, household h’s
instantaneous utility function is assumed to be additively separable between consumption andleisure and linear in leisure:u(Ch,t , Hh,t ) = log Ch,t + b(H̄ − Hh,t )(5.3)

where H̄ denotes a limit on the amount of hours worked by individual households and b is a positive
constant. In this case, the corresponding utility-maximization condition for labor supply can be
rewritten as follows.
bCm,t = (1 + θH )wm,t(5.4)

The optimization condition for mutual-fund investment is
1 = Et [Λh,t→t+1 RM,t+1]
1 − θA(5.5)

where Λh,t→t+1 is the household h’s inter-temporal marginal rate of substitution between periods t
and t + 1 and RM,t (= (QM,t + DM,t )/QM,t−1 ) represent the realized gross rate of return at periodt on household’s investment on mutual funds
9

It is shown in Sattinger (1975) that relative abilities of two workers must be equal to their relative wages in theabsence of comparative advantage. In line with this result, the ratio of wages paid to workers with different skill
levels equals the ratio of their skill levels as can be seen from budget constraint (5.2). But it does not mean thatlabor incomes of individual workers depends solely on their skill levels as will be seen later.

26

Reflecting that transitions of three individual states such as survival, employment, and skill
affect the household’s inter-temporal marginal rate of substitution, it is represented as a weighted
average of three ratios of next-period’s and current-period’s marginal utilities as can be seen below.
Λh,t→t+1 = ωe Λ̃h,t→t+1 + (1 − ωe )ωr Λ̂h,t→t+1 + (1 − ωe )(1 − ωr )Λ̄h,t→t+1(5.6)

The first term in the right-hand side of this equation corresponds to the ratio of marginal utilities
conditional on the continuation of employment between periods t and t + 1 with an increase in theskill level:Λ̃h,t→t+1 = ωs β

u1 (c(ah,t + 1, mh,t + 1, kh,t + 1, sa,t+1 ), h(ah,t + 1, mh,t + 1, kh,t + 1, sa,t+1 )
u1 (c(ah,t , mh,t , kh,t , sa,t ), c(ah,t , mh,t , kh,t , sa,t ))The second one corresponds to the ratio of marginal utilities conditional on the absence of skill
change between periods t and t + 1:Λ̂h,t→t+1 = ωs βk̄X

Vt+1 (ah,t + 1, mh,t , k̂)k̂=0

u1 (c(ah,t + 1, mh,t , k̂, sa,t+1 ), h(ah,t + 1, mh,t , k̂, sa,t+1 ))
u1 (c(ah,t , mh,t , kh,t , sa,t ), h(ah,t , mh,t , kh,t , sa,t ))where Vt+1 (mh,t , k̂) is the probability at period t + 1 that household h with skill level mh,t at periodt + 1 works at a k̂ type firm at period t + 1. The third one corresponds to the ratio of marginalutilities conditional on skill loss between periods t and t + 1:Λ̄h,t→t+1 = ωs βk̄X

Vt+1 (0, k̂)k̂=0

u1 (c(ah,t + 1, 0, k̂, sa,t+1 ), h(ah,t + 1, 0, k̂, sa,t+1 ))
u1 (c(ah,t , mh,t , kh,t , sa,t ), h(ah,t , mh,t , kh,t , sa,t ))where Vt+1 (0, k̂) is the probability at period t + 1 that household h with skill level 0 at period t + 1
works at a k̂ type firm at period t + 1.
Let us define the aggregate inter-temporal marginal rate of substitution (= Λt,t+1 ) as a weighted
average of individual households’ intertemporal marginal rates of substitution divided by (1 − θA )as follows:
Λt,t+1 = (1 − θA )−1

k̄ X
m̄
X

Vt (m, k)Λt,t+1 (m, k)(5.7)

k=0 m=0

In particular, Λt,t+1 acts as the stochastic discount factor to measure the value at period t of one unitof consumption good at period t + 1, which is proportional to the price at period t of a contingent
claim to give one unit of consumption good conditional on the occurrence of an aggregate stateat period t + 1. Hence the gross rate of return on household’s investment on mutual funds should
satisfy the following equation:
1 = Et [Λt,t+1 RM,t+1 ](5.8)

Let us turn to the discussion of mutual funds. First, there are a lot of perfectly competitive
mutual funds. Second, the absence of arbitrage profits is also used to determine the market value
27

of mutual funds in the centralized asset market with a complete set of (one-period ahead) statecontingent claims conditional on the whole set of aggregate states in each period t = 0, · · · . The
zero-profit condition at period t for perfectly competitive mutual funds requires the equality between
cash inflows and outflows:
QM,t At + DF,t = (QM,t + DM,t )At−1(5.9)

where At is the total number of shares held at period t by households, and DF,t is the aggregate
dividend income from firms.
Before going further, it would be worthwhile to address the issue of how to determine themagnitude of sales subsidies used for the elimination of steady-state economic distortion caused bymonopolistic competition in goods markets following Woodford (2003). For this reason, one should
discuss whether the optimality of a positive inflation rate has any impact on the specification ofsales subsidies to fix the economic distortion associated with monopolistic competition in goods
markets. Specifically, subsidies at period t of individual firms whose prices are set at period t − k
are determined as follows.
St−k,t = ((1 + θF )Πk − 1)Dt−k,t(5.10)

where θF is a positive constant, Π is steady-state gross inflation, and St−k,t represents sales subsidies
at period t of firms whose prices are set at period t−k. The decentralization of the optimal allocation
requires (steady-state) inflation-indexed sales subsidies in the case of a positive optimal steady-state
inflation rate, whereas it disappears in the case of a zero optimal steady-state inflation rate.
Taking into account the impact of sales subsidies, one can present the following profit maximization problem at period t of firms that rest prices at period t.
max
∗
Pt

∞
X

αk Et [Λt,t+k {(1 + θF )Πk (k=0

Pt∗ 1−ϵP∗)− mct+k ( t )−ϵ }Yt+k ]
Pt+k
Pt+k

(5.11)

The first-order condition of the profit maximization problem (5.11) with respect to the nominal
reset price can be written as follows.

Pt∗
St
=
Pt
Mt

(5.12)

where Mt and St are respectively defined as
Mt = (1 + θF )Yt + αEt [Λt,t+1 (Πϵ−1t+1 Π)Mt+1 ](5.13)St=ϵϵ
ϵ−1 mct Yt + αEt [Λt,t+1 Πt+1 St+1 ]Having described optimizing behaviors of firms and households, let us turn to fiscal and monetary
policies. In this model, the role of fiscal policy is to help implement the optimal allocation in adecentralized equilibrium by using employment subsidies, sales subsidies, and investment credits,
while lump sum taxes are imposed on households. In addition, government’s budget is supposed to28

be balanced in each period. Hence the government’s budget constraint at period t can be writtenas
Tt = TF,t + TH,t + TA,t(5.14)

where aggregate sums of sales subsidies TF,t , employment subsidies TH,t and mutual-fund investment
credits TA,t are respectively defined as follows.
Pk
TF,t = (1 − α) ∞k=0 α St−k,tTH,t = θH wt HtTA,t = θA QM,t At(5.15)

It would be worthwhile to emphasize two different roles of fiscal policy measures {TA,t , TH,t , TF,t }∞t=0
in the implementation process of the optimal allocation. The first one is to deal with the steady-state
distortion associated with monopolistic competition at goods markets. The second one is to deal
with the impact of heterogenous workers with different skill levels on the aggregate inter-temporal
marginal rate of substitution and marginal cost of production. The motivation behind this one isthat aggregation bias arises in the decentralization of the optimal allocation because of different
time preferences and constraints facing individual households and the social planner. It also should
be noted that the first role is well-known in the literature on the optimal monetary policy, whereas
the second one is relatively new to the literature. The central bank’s role is to achieve a sequenceof gross aggregate inflations {Πt }∞t=0 , while households and firms take as inflation targets.
Having described how optimizing behaviors of firms and households are affected by aggregate
variables in previous subsections, the focus is now given on the characterization of aggregate equilibrium conditions. <Table 5.1> contains 8 aggregate equilibrium conditions for 8 aggregate endogenous variables such as {Mt , St , Ht , Yt , p∗t , mct , ∆t , Λt,t+1 }, while the derivation of these equilibrium
conditions is explained in appendix. The number of endogenous variables thus matches the numberof equilibrium conditions given a series of exogenous aggregate productivity levels {Zt }∞t=0 , and aset of fiscal and monetary policies such as (θH , θF , θA ), and Π, {Πt }∞t=0 . It also should be noted that
this table includes a minimal self-sufficient set of equilibrium conditions for aggregate quantities
and prices, rather than including all equilibrium conditions discussed in previous subsections.

5.3

Optimal Inflation Target

The analysis of this section is focused on closed-form solutions to a benevolent government’s optimization problem without having any approximation, while the existence of closed-form solutions
requires a set of simplifying assumptions as can be seen below. The first assumption is that an
infinitely-lived benevolent government exists. The second assumption is that government’s instantaneous utility function is defined as an equally weighted sum of instantaneous utilities of households.
The government’s life-time utility function at period 0 is then defined as the expected discounted
29

Table 5.1: Collection of Equilibrium Conditions
Description

Equation

Production Functiont Ht
Yt = Z∆t

Relative Price Distortion∆t = αγ̄ Πϵt ∆t−1 + (1 − αγ̄ )(Price Level
Real Reset Price
Profit Maximization A
Profit Maximization B
Labor Market
IMRSp∗t = ( 1−αt ) 1−ϵStp∗t = Mt
Mt = (1 + θF )Yt + αEt [Λt,t+1 Πϵ−1
t+1 ΠMt+1 ]ϵϵ
St = ϵ−1 mct Yt + αEt [Λt,t+1 Πt+1 St+1 ]Yt = ac mct Zt
aλ β YtΛt,t+1 = 1−θ
A Yt+11−αΠϵ−1ϵ
1−αΠϵ−1
t
ϵ−1
1−α )

1

Note: IMRS means the aggregate inter-temporal marginal rate of substitution. ac =(1+θH )(1−ω)2 γb(γ−ω)(1−ωγ)

2e )ωr ))+γ(1−ωe )(1−ωr ))and aλ = (1−ω) ωs ((1−ωγ)(ωe +γ(1−ω.
γ(1−ωγ)sum of its instantaneous utility functions with its time discount factor β. The third assumption isthat an equally weighted average of instantaneous utility functions of all workers (who are alive at
period t) is required to satisfy the following relation:Zu(Ct (h), Ht (h))dh =
Ht

∞ X
∞
X

Vt (m, k)u(Ct (m, k), Ht (m, k))(5.16)

k=0 m=0

The third assumption helps obtain the existence of a closed-form solution to the government’s
optimization problem because it facilitates to aggregate preferences of individual households without
taking into account survivals, skill levels and workplaces of individual households in future periods
as will be seen below.
Given the restriction specified in equation (5.16), the infinitely-lived benevolent government’s
objective function at period 0 is defined as follows.
V0 =

∞
X

∞ X
∞
X
β t E0 [Vt (m, k)u(Ct (m, k), Ht (m, k))]t=0(5.17)

k=0 m=0

Here the government’s objective function specified in equation (5.17) does not include the survival
probability of an individual household as a discount factor. A concrete example of equation (5.17)
is that household’ instantaneous utility function (5.3) leads to the following representation of thesocial planner’s objective function at period 0.
V0 =

∞
X

β t E0 [log Ct + b(H̄ − Ht )] + V̂

t=0

where a constant term V̂ is defined as follows.
V̂ = (1 − β)−1 (1 − ωγω log γ+ log(
))
1−ω
1−ω
30

(5.18)

In this representation, government’s instantaneous utility function is defined as an equally weighted
average of instantaneous utility functions of individual households, while its derivation is included
in appendix.
Having discussed how to define government’s objective function, let us move onto the characterization of constraints. The set of equilibrium conditions included in <Table 5.1> can be regarded
as the set of constraints facing the benevolent government. But it should be noted that only a subset of these conditions are directly included in the government’s optimization problem. The otherset of equilibrium conditions should be used when the government should find a set of equilibrium
prices to implement its solution in a decentralized equilibrium. In sum, the benevolent government’s
optimization problem can be written as follows.
V (sa,t ) = max{Ct ,Ht ,Πt ,∆t } {log Ct + b(H̄ − Ht ) + βEt [V (sa,t+1 )]}subject to∆t = αγ̄ Πϵt ∆t−1 + (1 − αγ̄ )(Ct = ∆−1t Zt Htϵ1−αΠtϵ−1 ϵ−11−α )(5.19)′

given a value of the aggregate state vector sa,t = [∆t−1 Zt ] and a conditional probability density
function of the aggregate productivity level g(Zt+1 |Zt ).
The following proposition summarizes the solution of the optimization problem specified inequation (5.19) while its proof is provided in appendix. The optimal inflation that is implied by thesolution of the optimization problem specified in equation (5.19) is the same as that of proposition
4.1. On the basis of this result, the model of this section can be regarded as a dynamic generalequilibrium model that produces the same optimal inflation rate as obtained in the previous section.
In addition, a key point of this proposition is that the solution to the optimization problem specified
in equation (5.19) can be attained in a decentralized equilibrium that reflects optimizing behaviors
of households and firms under optimal fiscal and monetary policies.
Proposition 5.1 Inflation, real output, and relative price distortion are determined by the following dynamic system of three equations under the solution to the government’s optimization problem
specified in equation (5.19).
ΠtYtt= γ̄ ∆∆t−1
Zt= ac ∆
t∆t =(5.20)−11−ϵγ̄ −1 (α∆t−1+ (1 − α)ϵ (γ̄ − α)1−ϵ ) ϵ−1

given a series of {Zt }∞t=0 and an initial value ∆−1 . In addition, there are series of {TA,t , TH,t ,TF,t }∞
t=0 and values of (θH , θF , θA ) that implement inflation and real output specified in equation(5.20) at a decentralized equilibrium.
It should be noted that the determination of inflation and relative price distortion specified
in equation (5.20) is identical to that of two equations (4.4) and (4.5), which in turn means the31same result for the optimal inflation rate in proposition 4.1 and proposition 5.1. It also should
be noted that proposition 5.1 provides a policy prescription with the central bank in terms of its
target variables such as inflation and output. Given that the optimal output in the case of full-priceflexibility is defined to be Yt∗ = ac Zt and the output gap is defined to be xt = log Yt − log Yt∗ , therelation between inflation and output that is implied by equation (5.20) can be written as
xt = xt−1 − (πt − π)(5.21)

where πt (= log Πt ) is the aggregate inflation rate and π (= log Π) is the steady-state inflation rate.
If the central bank’s long-term target is set equal to the steady-state optimal inflation, equation(5.21) implies that the central bank should contract current-period’s aggregate demand relative
to previous period’s aggregate demand when inflation rate is higher than its long-term target andexpand when inflation rate is lower than its long-term target. In this sense, the policy prescriptionof proposition 5.1 supports an inflation targeting regime.
Finally, turning to the model with time-varying probabilities of price adjustment, one mightask if results of lemma 4.2 can be attained in a decentralized equilibrium in the similar way as isshown in the model with a constant probability of price adjustment. The following position showsthat the corresponding government’s optimization problem leads to the same result for the optimal
inflation, while its proof is included in appendix.
Proposition 5.2 Inflation, real output, and relative price distortion are determined by the following dynamic system of three equations under the solution to a benevolent government’s optimization
problem.
Πt
Ytt= γ̄t ∆∆t−1
Zt= ac,t ∆t∆t =(5.22)−11−ϵγ̄t−1 (αt ∆t−1
+ (1 − αt )ϵ (γ̄t − αt )1−ϵ ) ϵ−1

given a series of {Zt , αt }∞t=0 and an initial value ∆−1 .

In addition, there are series of {TA,t , TH,t ,θH,t , TF,t }∞t=0 and values of (θF , θA ) that implement inflation and real output specified in equation(5.22) at a decentralized equilibrium.

6

Discussion of Related Literature

The first topic of this section is a review of recent studies on the optimality of a positive inflation
rate in recent DSGE models. It should be admitted that a lot of important works exist in theliterature on the optimal inflation rate as discussed in Schmitt-Grohe and Uribe (2010) and Guido
and Sbordone (2014).10 But the focus is given on a set of recent works whose numerical results for
10

The optimality of a positive inflation rate also can be obtained in models with nominal-wage downward rigidities
such as Kim and Ruge-Murcia (2009), Benigno and Ricci (2011), Carlsson and Westmark (2016), Schmitt-Grohe and32the optimal inflation rate are relatively close to actual inflation targets of central banks in advanced
countries.
In this direction, two sets of recent works emerge as follows. The first set shares in common theemphasis on the importance of an effective lower bound on the short-term nominal interest rate to
generate the optimality of a positive inflation rate. The main mechanism behind this result is thata positive inflation target helps reduce the macroeconomic risk of being stuck at such an effective
lower bound in the presence of significant declines in the natural real interest rate. In this case, anegative relationship between the natural real interest rate and the optimal inflation target arises
because the welfare cost of the effective lower bound creates the welfare cost of a low inflation rate,as can be seen in Coibion, Gorodnichenko, and Wieland (2012) and Andrade, Gali, Le Bihan, andMatheron (2019).
The second set shares in common the emphasis on the importance of firm heterogeneity togenerate the optimality of a positive inflation rate such as different productivity growth patterns
between price-adjusting and non-adjusting firms in Adam and Weber (2018) and quality bias in anonlinear aggregation of individual prices in Schmitt-Grohe and Uribe (2012). The set-up of thepresent paper is closer to that of the second set than the first one in the sense that it permits
different labor productivity levels between price-adjusting and non-adjusting firms. But a novel
feature of the present paper is that the optimality of a positive inflation rate is associated withinteractions between firm heterogeneity and household heterogeneity through assignment of workers
with different skill levels to firms with different output levels.
It also should be noted that numerical predictions on the optimal inflation rate remain within arange between 2 percent and 3 percent in the work of Andrade, Gali, Le Bihan, and Matheron andbetween 1 percent and 3 percent in the work of Adam and Weber. The present paper’s numericalpredictions on the optimal inflation rate also fluctuate within a range between 2 percent and 3
percent under the benchmark calibration as can be seen in Figure 5. It means that actual inflation
targets of central banks in advanced countries are embedded in the optimal range of inflation target
prescribed by the theoretical model of the present paper.
It would be worthwhile to mention that the productivity dispersion of individual firms has been
widely incorporated into recent DSGE models for various reasons. The related literature in thisdirection can be classified into three distinct sets. The first one is firm-specific production input
factors to improve the empirical performance of theoretical models such as Woodford (2005), de
Walque, Smets, and Wouters (2006), Matheron (2006), and Altig, Christiano, Eichenbaum, andUribe (2013), models with endogenous entries of firms such as Bilbiie, Fujwari, and Ghironi (2014) and in models
with financial frictions such as Brunnermeier and Sannikov (2016). But the present paper is focused on nominal pricerigidity and exogenous entries and exits of firms.

33

Linde (2011). The second one is idiosyncratic productivity shocks in both state-dependent pricing
models and time-dependent pricing models to help match observed frequencies and sizes of pricechanges in microeconomic data such as Dotsey, King, Wolman (1999), Golosov and Lucas (2007),
Gertler and Leahy (2008), and Nakamura and Steisson (2008). The third one is to study impacts
of asymmetries in sectoral disturbances and firm-level productivity trends (between new and oldfirms) on optimal policy prescriptions such as Aoki (2001), Wolman (2011), and Adam and Weber(2019).
The main difference between the present paper and these works mentioned above lies in themechanism to create the productivity dispersion of firms. The reason behind this argument is thatthe skill dispersion of workers acts as the main source of the productivity dispersion of firms in themodel of this paper. In particular, this mechanism works through the assignment of workers withdifferent skill levels to firms with different output levels when production skills of workers are notfirm-specific. As a result, technologically ex-ante homogeneous firms become heterogenous firms
with different productivity levels, which in turn leads to the optimality of a positive inflation rate.
The theoretical prescription obtained from the minimization of relative price distortion is thatworkers with higher skill levels should work at firms with higher output levels. It means thatworker’s skill level should increase with firm size measured in the unit of value-added if government
wants to attain an efficient allocation. Given this theoretic result, one might wonder how thisnormative prescription is associated with results of microeconomic empirical studies. The reason
for this one reflects a concern that the theoretical result of this paper must be hardly implementablein actual economies if negative correlations between firm’s size and worker’s skill are widely observedin microeconomic data. But it should be pointed out that a set of empirical works report positive
correlations between firm’s size and firm’s productivity and between firm’s size and worker’s wage
such as Idson and Oi (1999) and Berlingieri, Calligaris, and Criscuolo (2018), Bloom, Guvenen,
Smith, Song, and Wachter (2018).

7

Conclusion

It has been shown in this paper that the optimality of a positive inflation rate emerges in models
with nominal price rigidity when workers with higher skill levels are assigned to firms larger output
levels. The theoretic contribution of this paper is two-fold. The first one is to show that the central
bank’s inflation policy can play an active role in the determination of resource allocation throughits impact on the assignment of workers to firms. In particular, it has been shown in this paperthat inflation can be used as a government’s tool to generate the issue of how to assign workers tofirms even when ex-ante technologically homogenous firms set prices in a non-synchronized fashion.
34

The second one is closely associated with the first one in the sense that the optimal steady-state
inflation rate is a function of real economic variables including productivity and skill distributionsof firms and workers as well as frequencies of price changes. In this sense, the model of this paper
can be regarded as a real economic model of the optimal long-term inflation rate that emphasizes
the importance of productivity dispersion or skill dispersion, average life-span of firms, and average
employment tenure as well as average price duration.
As a practical policy prescription of this theoretical result, it also has been shown that the longterm U.S. optimal inflation rate is slightly above 2 percent under the benchmark model calibration.
In this sense, the present paper would present a theoretical framework to help rationalize a 2
percent inflation target of actual central banks on the basis of a standard staggered price-setting
model in which the theoretical rationalization of a substantially positive optimal inflation rate has
been deemed to be a challenging task. But a caveat is that the magnitude of the optimal inflation
rate varies within a relatively wide range starting from less than 2 percent through about 8 percent
depending on distributional assumptions of skill and price distributions.
It should be admitted that inefficient matches between workers and firms can weaken the social
benefit of inflation discussed above. The reason for this argument is that more firms with large
output levels tend to hire more workers with low skill levels as inefficient matches between workers
and firms increase, while the model of this paper abstracts from such mismatches between workers
and firms as well as unemployed workers and vacant jobs. The introduction of such frictions into
the present theoretical framework thus can be a realistic extension to explore. Given the workof Shimer (2005) where an assortative matching between workers and firms arises as the resultof the social planner’s problem who is constrained by anonymity restrictions, one might want toconjecture that mismatch between workers and firms will not eliminates all the social benefit of
inflation discussed above. However, the size of reduction in the benefit of inflation is still an open
question. As a result, the extension of the present analysis in this direction can be regarded as aninteresting future research topic normatively and positively as well.

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39

Appendix
A

Proof of Lemma 3.2 in Chapter 3

A.1

Optimal Steady-State Inflation in a Symmetric Two-Period Taylor Model

It has been shown in the text that there are two different representations of relative price distortion
depending on how workers are assigned to firms in a symmetric two-period Taylor model. Let us
start with the first representation that corresponds to the case where skilled workers are assigned∗
to firms with Pt−1
and unskilled workers are assigned to firms with Pt∗ . In this case, relative pricedistortion together with the determination of the aggregate price level can be summarized as follows.
∆t =
1

=

∗
P∗
1 Zt Pt −ϵt+ 12 ZZ1,t( Pt−1)−ϵ
2 Z0,t ( Pt )t∗
P∗
1 Pt 1−ϵ+ 12 ( Pt−1
)1−ϵ2 ( Pt )t(A.1)

The substitution of the second-line equation into the first-line equation and then the elimination of∗ /P ) in the resulting equation leads to the following representation of relative price distortion.
(Pt−1
t

∆t =

ϵ
1 Zt ∗ −ϵ 1 Zt
(pt ) +
(2 − (p∗t )1−ϵ ) ϵ−1
2 Z0,t2 Z1,t(A.2)

where p∗t = Pt∗ /Pt . The partial differentiation of relative price distortion with respect to real reset
price then leads to the following optimization conditionp∗t = (

2Z
1 + Z1,t
0,t

1

) 1−ϵ

(A.3)

By substituting equation (A.3) into equation (A.2), the resulting optimized value of relative pricedistortion in the first case is
ϵ

ϵ

∆t =

1 Zt
2
2
ϵ−1 + 1 Zt (2 −) ϵ−1Z
2 Z0,t ( 1+ Z1,t )2 Z1,t1+ 1,t

=

ϵ
ϵ
2Z0,t
2Z1,t
1 Zt
ϵ−1 + 1 Zt (ϵ−1
2 Z0,t ( Z0,t +Z1,t )2 Z1,t Z0,t +Z1,t )

Z0,t

Z0,t

(A.4)

Turning to the second case, the second representation corresponds to the case where unskilled∗workers are assigned to firms with Pt−1
and skilled workers are assigned to firms with Pt∗ . In thiscase, relative price distortion together with the determination of the aggregate price level can besummarized as follows.
∆t =
1

=

∗
P∗
1 Zt Pt −ϵt+ 12 ZZ0,t( Pt−1)−ϵ
2 Z1,t ( Pt )t∗∗
P
1 Pt 1−ϵ+ 12 ( Pt−1
)1−ϵ2 ( Pt )t(A.5)

The substitution of the second-line equation into the first-line equation and then the elimination of∗ /P ) in the resulting equation leads to the following representation of relative price distortion.
(Pt−1
t

∆t =

ϵ
1 Zt ∗ −ϵ 1 Zt
(pt ) +
(2 − (p∗t )1−ϵ ) ϵ−1
2 Z1,t2 Z0,t40(A.6)

The partial differentiation of relative price distortion with respect to real reset price then leads tothe following optimization conditionp∗t = (

2Z
1 + Z0,t
1,t

1

(A.7)

) 1−ϵ

By substituting equation (A.7) into equation (A.6), the resulting optimized value of relative pricedistortion in the second case is
ϵ

ϵ

∆t =

1 Zt
2
2
ϵ−1 + 1 Zt (2 −) ϵ−1Z
2 Z1,t ( 1+ Z0,t )2 Z0,t1+ 0,t

=

ϵ
ϵ
2Z1,t
2Z0,t
1 Zt
ϵ−1 + 1 Zt (ϵ−1
2 Z1,t ( Z0,t +Z1,t )2 Z0,t Z0,t +Z1,t )

Z1,t

(A.8)

Z1,t

Comparing the second-line of equation (A.4) with that of equation (A.8), one can confirm the same
optimized value of relative price distortion under the two cases, which is also identical to equation(3.5). This result therefore proves the first part of Lemma 3.2.

A.2

Optimal Steady-State Inflation in a Calvo Model

Let us turn to the second equation of Lemma 3.2. Let us start with the first case where unskilledworkers are assigned to price-adjusting firms and skilled workers are assigned to non-adjusting firms
in the case of a positive inflation. In this case, current and lagged relative price distortions can be
written as follows.
∗

∆t

=

αΠϵt ∆t−1 =∗P∞ k Pt−k −ϵ(1−α)Zt Pt∗ −ϵ−1 ( Pt−1 )−ϵ + γ −1
)
+
αγ
((
k=2 α ( Pt ) )
Z0,t
Pt
Pt
∗
∗
P∞ k Pt−kPt−1(1−α)Zt−1
−ϵ
−1
−ϵ
+γ
k=2 α ( Pt ) )Z0,t−1 (α( Pt )(A.9)

In the presence of a stationary skill distribution, the definition of the aggregate labor productivity
leads to the following relations.11−1(1 − α + γ −1 α)Zt−1 = Z0,t−1
−1
(1 − α + γ −1 α)
Zt−1= Z0,t−1(A.10)

By substituting equation (A.10) into equation (A.9) and then subtracting the second-line from thefirst-line in the resulting equation leads to the following representation of relative price distortion.
∆t = αΠϵt ∆t−1 +

∗
Pt−1
1−α
Pt∗ −ϵ
−1
((
)
+
α(γ
−
1)(
)−ϵ )
1 − α + γ −1 α Pt
Pt

In this case, steady-state relative price distortion is∆=11ϵ(1 − α)(1 + α(γ −1 − 1)Πϵ ) 1 − αΠϵ−1 ϵ−1()ϵ−1(1 − αΠ )(αγ + 1 − α)1−αFor the simplicity of the analysis, measures of firms and workers are assumed to be the same.

41(A.11)

Let us turn to the second case where skilled workers are assigned to price-adjusting firms andunskilled workers are assigned to non-adjusting firms in the case of a negative inflation rate. In thiscase, current and lagged relative price distortions are given by∆t=

αΠϵt ∆t−1 =∗∗P∞ k Pt−k
Pt−1
Pt∗ −ϵZt−ϵ + (1 − α)γ
−ϵ
α
(
((1
−
α)(
)
+
α(1
−
α)γ(
)
k=2
Z1,t
Pt
Pt
Pt ) )∗∗
P
Pt−1 −ϵZt−1
k Pt−k −ϵ+ (1 − α)γ ∞k=2 α ( Pt ) )Z1,t−1 (α(1 − α)( Pt )(A.12)In the presence of a stationary skill distribution, the definition of the aggregate labor productivity
leads to the following relations.
−1
Zt−1 = Z1,t
(1 − α + γα)−1−1
Zt−1 = Z1,t−1 (1 − α + γα)(A.13)

By substituting equation (A.13) into equation (A.12) and then subtracting the second-line from thefirst-line in the resulting equation leads to the following representation of relative price distortion.
∆t = αΠϵt ∆t−1 +

P∗
1−α
P∗
(( t )−ϵ + α(γ − 1)( t−1 )−ϵ )αγ + 1 − α Pt
Pt

In this case, steady-state relative price distortion isϵ(1 − α)(1 + α(γ − 1)Πϵ ) 1 − αΠϵ−1 ϵ−1()(1 − αΠϵ )(αγ + 1 − α)1−α∆=(A.14)

It follows from equations (A.11) and (A.14) that one can obtain a unified representation of steadystate relative price distortion as follows∆=

ϵ(1 − α)(ϕ + (1 − ϕ)Πϵ ) 1 − αΠϵ−1 ϵ−1()ϵ
1 − αΠ
1−α(A.15)

where ϕ is defined as
ϕ = 1 − α(γ −1 − 1)
= 1 − α(γ − 1)

if Π > 1
if Π < 1(A.16)The requirement of (1 < γ < α−1 + 1) leads to ϕ > 1 when Π > 1 and 1 > ϕ > 0 when Π < 1.
In addition, the value of ϕ does not affect the value of relative price distortion when Π = 1. The
partial differentiation of steady-state relative price distortion with respect to gross inflation leads
to the following equation.
∂∆= 0 → (1 − ϕ(1 − α))(Π − 1) + (1 − ϕ)(1 − αΠϵ )2 = 0∂Π(A.17)

Comparing equation (A.17) with equation (3.6), one can confirm that the two equations are thesame. Hence this result proves the second part of lemma 3.2.

42

B

Proofs of Lemmas and Propositions in Chapter 4: The Optimal
Inflation Rate in the United States

This appendix summarizes proofs of propositions and lemmas used to calibrate the model in chapter
4 where there are two propositions that summarize the law of motion for relative price distortion
under a assortative assignment and the corresponding solution to the social planner’s optimization
problem in models with constant and time-varying probabilities of price adjustment. In addition,
there are two lemmas to be used for the model calibration of the U.S. optimal steady-state inflation
rate.

B.1

Proof of Proposition 4.1

The first part of this proof is to derive the law of motion for relative price distortion under assortative assignment specified in equation (4.3). In particular, condition 4.1.A. implies the following
representation of relative price distortion.
∆t = (1 − α)

∞
X

αk

k=0

∗
Zt Pt−k()−ϵ
Z̄k,t Pt

(B.1)

where Z̄k,t is defined as a harmonic mean of skill levels assigned to all type k firms:
mkXZ̄k,t = ((k) −1 −1
Zm,t ) ,ω̄m

m=mk−1 +1

Γm
(k)
ω̄m
= Pi=mki=mk−1 +1 Γi

(B.2)

(k)

where ω̄m is the weight given to the skill level of type m workers who work at type k firms.
Let us begin with the discussion of how Zt /Z̄k,t is determined. The substitution of equation(k)(4.2) into the definition of ω̄m leads to the following equation.
(k)
ω̄m
=

(1 − ω)ω m
ω nk (1 − ω n )(B.3)

The substitution of equation (B.3) into equation (B.2) implies that the average skill level of workers
who work at type k firms is determined as follows.
Z̄k,t =

(1 − ω n )(1 − ωγ −1 ) kγ̄ Z0,t(1 − ω)(1 − ω n γ̄ −1 )(B.4)

where γ̄ = γ n . In addition, the substitution of condition 4.1.B into the definition of the aggregate
labor productivity leads to the following relation between the aggregate productivity and initialskill level:

1 − ωγ −1
Z0,t
(B.5)
1−ω
The substitution of equations (B.4) and (B.5) into equation (B.1) leads to the following decompoZt =sition of relative price distortion:∞∗(1 − α)(1 − ω n γ̄ −1 ) Pt∗ −ϵ X (1 − α)(1 − ω n γ̄ −1 ) α k Pt−k
∆t =
(
)
+
(
)
(
)−ϵ .
1 − ωnPt
1 − ωn
γ̄
Pt
k=1

43

The coefficient of the first-term in the right-hand side of this equation can be rewritten as
ωn = α →(1 − α)(1 − ω n γ̄ −1 )
α
=1− .
n
1−ω
γ̄

The substitution of this equation into the decomposition of relative price distortion specified above
leads to the following representation.
∞∗
α Pt∗ −ϵ X (1 − α)(1 − ω n γ̄ −1 ) α k Pt−k∆t = (1 − )( ) +
)−ϵ .
(
)
(
γ̄ Pt
1 − ωn
γ̄
Pt

(B.6)

k=1

The first-term in the right-hand side of equation (B.6) reflects the impact of current-period’s real
reset price on relative price distortion and the second-term summarizes the impact of lagged real
reset prices on relative price distortion. Meanwhile it should be noted that the second-term in
the right-hand side of equation (B.6) is simplified by the use of one-period lagged relative pricedistortion as can be seen below.
∞
∗
X
α ϵ
(1 − α)(1 − ω n γ̄ −1 ) α k Pt−kΠt ∆t−1 =
(
)
(
)−ϵ .
γ̄
1 − ωn
γ̄
Pt

(B.7)

k=1

By subtracting equation (B.7) from equation (B.6) and then substituting the definition of theaggregate price level into the resulting equation, one can obtain a recursive representation of relative
price distortion as follows.
∆t =

ϵ
α ϵ
α 1 − αΠϵ−1t
Πt ∆t−1 + (1 − )(
) ϵ−1
γ̄
γ̄
1−α

(B.8)

Now one can confirm that equation (B.8) is equal to the first equation of proposition 4.1 specified
in equation (4.3).
Having completed the discussion of how to specify the law of motion for relative price distortion,
let us move onto the minimization of relative price distortion. This one corresponds to the second
part of proposition 4.1. The first-order partial differentiation of relative price distortion with respect
to the aggregate gross inflation isΠt ∆t−1 =1γ̄ − α 1 − αΠϵ−1
t
(
) ϵ−1
1−α
1−α

(B.9)

In order to compute the impact of lagged relative price distortion on current-period’s relative pricedistortion, let us substitute the first-order condition (B.9) into equation (B.8), which in turn leads
to the following representation.
∆t =1γ̄ − α 1 − αΠϵ−1
t
(
) ϵ−1
γ̄(1 − α)1−α(B.10)The substitution of equation (B.10) into the first-order condition (B.9) leads to the following relation
between the aggregate gross inflation and growth of relative price distortion
∆t
∆t−1
44

Πt = γ̄(B.11)

In particular, this equation is equal to the second equation of proposition 4.1 specified in equation(4.4).
Let us turn to the third equation of proposition 4.1. By substituting equation (B.11) into thelaw of motion for relative price distortion (B.8) (to eliminate gross inflation) and then rearranging
the resulting equation, one can obtain the optimal law of motion for relative price distortion as
follows.
1−ϵ∆1−ϵ= γ̄ ϵ−1 (α∆t−1+ (1 − α)ϵ (γ̄ − α)1−ϵ )t(B.12)

Here this equation is equal to the third equation of proposition 4.1. This warps up the derivationof three equations of proposition 4.1. In addition, the substitution of equation (B.10) into thedefinition of the aggregate price index (2.3) together with Φk = (1 − α)αk also leads to the following
representation of the optimal real reset price.
p∗t =

B.2

γ̄ − α∆−1
γ̄(1 − α) t

(B.13)

Proof of Proposition 4.2

The first part of this appendix is to show how to obtain the law of motion for relative price distortionin the model with time-varying probabilities of price adjustment. The second part is to present theproof of proposition 4.2 on the basis of the first part’s result. Recall that a fraction of firms (1 − αt )reset prices and the other fraction of firms αt do not in each period t = 0, · · · . In this case, relative
price distortion is defined as follows.
∆t =

∞ k−1
∗
(1 − αt )Zt Pt∗ −ϵ X Y(1 − αt−k )Zt Pt−k
( ) +
(
(
)−ϵ
αt−i )
Pt
P
Z̄0,t
Z̄
t
k,t
i=0

(B.14)

k=1

In the same way as is done in the model with a constant probability of price adjustment, the averageskill level at period t of workers who work at type k firms, Z̄k,t , is defined as a harmonic mean of
their skill levels:mk,tZ̄k,t = (

X

(k)

−1 −1
ω̄m,t Zm,t
) ,m=mk−1,t +1

Γm(k)ω̄m,t = Pi=mk,t(B.15)

i=mk−1,t +1 Γi(k)where ω̄m,t is the weight at period t given to the skill level of type m workers who work at type kfirms and mk,t is the upper bound at period t of skill levels of workers who work at type k firms:
ω mk,t +1 =

k
Y

αt−i

(B.16)

i=0

for k = 0, · · · .
Having specified the definition of relative price distortion in the model with time-varying probabilities of price adjustment, let us move onto the discussion of how to obtain the law of motion for45

relative price distortion. In doing so, the first task is to obtain the law of motion for the averageskill level of firms that do not reset prices, which amounts to finding an a relation between Z̄k,t andZ̄k−1,t−1 . It follows from equation (B.15) that weights of type m workers who work for type k firms
at period t and type k − 1 firms at period t − 1 can be written as
(k)

ωm,t

=

(k−1)

ωm,t−1 =ω(1−ω)ωm
mk−1,t +1
m−mk−1,t(1−ω k,t)(1−ω)ωmm+1
m−mk−2,t−1
ω k−2,t−1 (1−ω k−1,t−1)(B.17)In the same way as is done in the model with a constant probability of price adjustment, the valueof nt−i (= log αt−i / log ω) is a positive constant for i = 0, · · · . In addition, equation (B.16) is used
to show that the following relations hold.
mk−i,t − mk−(i+1),t = nt−kmk−i,t − mk−(i+1),t−1 = nt

(B.18)

for k = 2, · · · and a non-negative integer i less than k − 1. The substitution of equations (B.17) and(B.18) into equation (B.15) leads to the following representations of average skill levels at periodst − 1 and t of workers who work at firms that reset prices at period t − k.
−1
Z̄k,t

=

−1
Z̄k−1,t−1=

Pnt−k ω m(1−ω)( m=0( γ ) ) −1
Zmk−1,t +11−αt−kPnt−k ω m(1−ω)( m=0 ( γ ) ) −1
Zmk−2,t−1 +1
1−αt−k

(B.19)

Dividing the first-line equation of (B.17) by the second-line equation and then rearranging theresulting equation, one obtain a law of motion for the average skill of workers for firms whose prices
are reset at period t − k.
Z̄k,t = vt Z̄k−1,t−1(B.20)

for k = 1, · · · and where vt is defined as follows.
vt = γ n tZ0,t.
Z0,t−1(B.21)

Having described how the average skill of workers who work at each type firms evolves overtime, let us turn to the discussion of how to obtain the recursive representation of relative pricedistortion. Equation (B.14) implies that current-period’s relative price distortion is∗P∗
αt Z̄0,t(1 − αt−2 )αt−1 Z̄1,t Pt−2
Z̄0,t ∆tP∗
= (1 − αt )( t )−ϵ +((1 − αt−1 )( t−1 )−ϵ +()−ϵ + · · · ) (B.22)
Zt
Pt
Pt
Pt
Z̄1,t
Z̄2,t

The substitution of equation (B.20) into equation (B.22) leads to the following equation.
∗P∗Z̄0,t ∆tαt Z̄0,t(1 − αt−2 )αt−1 Z̄0,t−1 Pt−2P∗
= (1−αt )( t )−ϵ +((1−αt−1 )( t−1 )−ϵ +()−ϵ +· · · ) (B.23)
Zt
Pt
Pt
Pt
Z̄1,t
Z̄1,t−1

Equation (B.14) also implies that one-period lagged relative price distortion is∗P∗(1 − αt−2 )αt−1 Z̄0,t−1 Pt−2
Πϵt Z̄0,t−1 ∆t−1
= (1 − αt−1 )( t−1 )−ϵ +()−ϵ + · · ·
Zt−1
Pt
P
Z̄1,t−1t
46(B.24)

The substitution of equation (B.24) into equation (B.23) leads to the following equation.
αt Z̄0,t Πϵt Z̄0,t−1 ∆t−1
Z̄0,t ∆tP∗
= (1 − αt )( t )−ϵ +
Zt
Pt
Zt−1
Z̄1,t

(B.25)

Dividing both sides of equation (B.25) by Z̄0,t /Zt and then rearranging the resulting equation, one
can obtain the following representation of relative price distortion.
∆t =

αt ϵ
1 − αt Pt∗ −ϵΠt ∆t−1 +
( )
γ̄t
st
Pt

(B.26)

Z̄0,t
Zt
st Z̄1,tst−1 Z̄0,t(B.27)

where st and γ̄t are defined as follows
st

=

γ̄t =

In order to simplify the expression of relative price distortion in equation (B.26), one can use an
equilibrium relation between st and γt as will shown below. The price distribution is used to showthat the aggregate labor productivity can be written as follows.
1
Zt

t−1 )
t−2 )αt−1= (1 − αt ) Z̄1 + αt ( (1−α+ (1−αZ̄
+ ···)
Z̄
0,t

=
=

1,t

2,t

t−1 )
(1 − αt ) Z̄1 + αvtt ( (1−α+ (1−αZ̄t−2 )αt−1 + · · · )Z̄0,t−10,t1,t−1t(1 − αt ) Z̄1 + vt Zαt−10,t(B.28)

By substituting equation (B.20) into the final line of equation (B.28) and then substituting thedefinitions of st into the resulting equation, one can obtain a law of motion for st as can be seen
below.
Z̄0,t
Z̄0,t Z̄0,t−1Z̄0,t= 1 − αt + αt
→ st = 1 − αt + αt
st−1Zt
Z̄1,t Zt−1Z̄1,t(B.29)

By dividing both sides of equation (B.28) by st and then substituting the definition of γ̄t into theresulting equation, one can find that the coefficient of the second term in the right-hand side ofequation (B.26) can be written as follows.
st = 1 − αt + αtZ̄0,tst−1 →
Z̄1,t1 − αt
αt
=1−
st
γ̄t

The substitution of this equation into the right-hand side of equation (B.26) leads to the following
representation of relative price distortion.
∆t =

αt ϵ
αt P ∗
Πt ∆t−1 + (1 − )( t )−ϵγ̄tγ̄t Pt(B.30)In order to replace real reset price by gross inflation in the right-hand side of equation (B.30), let
us turn to the definition of the aggregate price index in the model with time-varying probabilities
of price adjustment as follows.
∗ )1−ϵ + (1 − α∗
1−ϵ + · · · )Pt1−ϵ = (1 − αt )(Pt∗ )1−ϵ + αt ((1 − αt−1 )(Pt−1t−2 )αt−1 (Pt−2 )
1−ϵ
∗
1−ϵ
= (1 − αt )(Pt )+ αt Pt−1

47

Diving both sides of this equation by Pt1−ϵ leads to the following equation.
1Pt∗
1 − αt Πϵ−1
t
=(
) 1−ϵ
Pt
1 − αt(B.31)

By substituting equation (B.31) into equation (B.30), one can obtain the following representationof relative price distortion.
∆t =

ϵ
αt ϵ
αt 1 − αt Πϵ−1t
Πt ∆t−1 + (1 − )(
) ϵ−1
γ̄t
γ̄t
1 − αt(B.32)

given an initial value of ∆−1 and a series of {αt }∞t=0 . By substituting equations (B.20) and (B.21)
into the second line of equation (B.27), it is also possible to obtain a simplified representation of γ̄t .
γ̄t = vt

Zt−1
Zt

→ γ̄t = γ nts0,ts0,t−1

where s0,t (= Z0,t /Zt ) is the ratio of the lowest skill level to the aggregate average skill level atperiod t. The presence of a stationary skill distribution of workers implies that s0,t = s0,t−1 as canbe seen from equation (B.4). In this case, γ̄t is determined as follows.
γ̄t = γ nt

(B.33)Let us move onto the minimization of relative price distortion. The minimization condition canbe simplified as follows.
Πt ∆t−1 =1γ̄t − αt 1 − αt Πϵ−1
t
(
) ϵ−1
1 − αt1 − αt(B.34)The substitution of (B.34) into equation (B.32) leads to the following equation.
∆t =1γ̄t − αt 1 − αt Πϵ−1
t
(
) ϵ−1
γ̄t (1 − αt )1 − αt(B.35)

Dividing both sides of equation (B.34) by those of equation (B.35), one can find that the optimal
gross inflation is proportional to the growth of relative price distortion as can be seen below.
Πt = γ̄t∆t∆t−1(B.36)In addition, substituting equation (B.36) into equation (B.32) in order to eliminate gross inflationand then rearranging the resulting equation, one can obtain the optimal law of motion for relative
price distortion.
ϵ
1−ϵ
∆1−ϵ
= γ̄tϵ−1 (αt ∆1−ϵ)tt−1 + (1 − αt ) (γ̄t − αt )(B.37)The substitution of equation (B.36) into the definition of the aggregate price index (B.31) leads tothe following representation of the optimal real reset price.
p∗t =γ̄t − αt∆−1
γ̄t (1 − αt ) t
48(B.38)

C

Aggregate Equilibrium Conditions and Implementation of Optimal Allocation

C.1

Aggregate Equilibrium Conditions: Constant Probability of Price Adjustment

Let us begin with the derivation of 8 equilibrium conditions included in <Table 5.1>. The first line
corresponds to the aggregate production function (2.10). The second line is the law of motion for
relative price distortion specified in equation (4.3). The third line comes from the determination of
the aggregate price index specified in equation (2.3) under the assumption that Φk = (1 − α)αk for
k = 0, · · · . Since the discussion of how to obtain these three equations can be found in the text, thisappendix begins with the fourth line of <Table 5.1>. The fourth line through sixth line of Table
5.1 are derived from the profit-maximization of firms that reset prices in period t. In this table, theprofit-maximization condition consists of three equations with a recursive representation, while itis also possible to use the alternative one-line representation of the profit-maximization conditionin terms of an expected discounted sum of an infinite number of future terms. It will be shown in
the next subsection that firm’s marginal production cost is independent of its output level.

C.2

Derivation of Marginal Production Cost

The key reason why marginal production costs of individual firms are independent of their output
levels is that real wages of individual workers are proportional to their skill levels. In order toprove that this argument is correct, one can start with the definition of the aggregate labor income.
The aggregate labor income is defined as the aggregate sum of labor incomes of individual workers
in each period. It is possible to rely on either household (labor) income surveys or establishment
payroll surveys in the aggregation of labor incomes of individual workers. For this reason, one cancome up with the following relation for the definition of the aggregate real income.
R
wt Ht = Ht wh,t Hh,t dh
P∞ P∞
=
m=0
k=0 Vt (m, k)wt (m, k)Ht (m, k)(C.1)

It should be noted that equation (C.1) can be used to define the aggregate real wage given thedefinition of the aggregate hours worked by workers.
Now one can rely on a “guess-and-verification” method to show how real wages of individual
workers are determined under the assumption that their real wages are proportional to skill levels
as can be seen below.
wt (m, k) = Zm,t Xt(C.2)

where Xt does not depend on m and k in each period t = 0, · · · . The substitution of equation(C.2) into the second-line equation of equation (C.1) together with equations (2.4) and (2.10) leads49

to the following representation of Xt
Xt =wt ∆ tZt ∆p,t(C.3)

By substituting equation (C.3) into equation (C.2), one can find that real wages of individual workers
are affected by aggregate variables such as the aggregate real wage, aggregate labor productivity,
and relative price distortion as well as their skill levels:
wt (m, k) =Zm,t ∆t
Zm,t ∆t
wt → wm,t =wt
Zt ∆p,tZt ∆p,t(C.4)

where wm,t is the real wage at period t of type m workers.
An important consequence of equation (C.4) is that marginal costs of production are independentof output levels of individual firms. Since labor is the only one production input in the model ofthis paper, the substitution of equation (C.4) into the total production cost for type k firms leads
to the following representation.
wm,t Ht (m, k) =wt ∆ tYk,tZt ∆p,t(C.5)

It follows from equation (C.5) that real marginal costs (= mct ) is independent of output levels of
individual firms.
mct =

C.3

wt ∆ tZt ∆p,t(C.6)

Derivation of Labor-Market Equilibrium Condition

Let us move onto the equilibrium condition of the labor market. In general, the aggregate consumption is defined as follows.
Ct =

∞ X
∞
X

Vt (m, k)Ct (m, k)(C.7)

k=0 m=0

Before going further, recall that the definition of the aggregate labor productivity leads to thefollowing equation:
Zt =

γ−ω
Z0,t
γ(1 − ω)(C.8)

Given that Zm,t = γ m Z0,t , equation (C.8) implies that the skill level of type m workers can be
written as follows.
Zm,t =

γ m+1 (1 − ω)
Zt
γ−ω

(C.9)

Let us turn to the utility optimization condition of households. By substituting equation (C.4)
into equation (5.4), one can show that a type m worker’ consumption is affected by his or her skill
level.
Cm,t =(1 + θH )Zm,t ∆twt
bZt ∆p,t50(C.10)

The substitution of equation (C.9) into equation (C.10) also leads to the following equation:
Cm,t =

γ m+1 (1 + θH )(1 − ω)∆twtb(γ − ω)∆p,t(C.11)

By substituting equation (C.11) into equation (C.7), one can find that the aggregate consumption
is linear to the aggregate real wage.
∆t
wt
∆p,t

(C.12)γ(1 + θH )(1 − ω)2.
b(γ − ω)(1 − ωγ)(C.13)

Ct = a c
where coefficient ac is defined as follows.
ac =In this representation of the aggregate consumption function, the positivity of the aggregate consumption is guaranteed by the condition of 0 < γω < 1 together with ω < 1 < γ. The substitutionof equation (C.6) into equation (C.12) leads to the seventh equation of <Table 5.1>.
Ct = ac mct Zt

C.4(C.14)

Derivation of Aggregate Inter-temporal Marginal Rate of Substitution

Let us move onto the final equilibrium condition of <Table 5.1>, which corresponds to the characterization of the aggregate inter-temporal marginal rate of substitution in terms of the growth of
the aggregate consumption. In order to do this one, equations (C.11) and (C.12) are used to showthat a type m household’s consumption level is proportional to the aggregate consumption level as
can be seen below.
Cm,t =γ m (1 − ωγ)Ct
1−ω

(C.15)

Now let us pick a household h ∈ Ht whose skill level and workplace are mh,t = m and kh,t = k. By
substituting equation (C.15) into definitions of IMRS (inter-temporal marginal rate of substitution)
included in the right-hand side of equation (5.6), one can obtain three different representations ofhousehold h’s IMRS reflecting three different cases:Λ̃h,t→t+1 =Λ̂h,t→t+1 =Λ̄h,t→t+1 =(1−ω)ωs βCt
γCt+1(1−ω)ωs βCt
Ct+1(1−ω)ωs βγ m Ct
Ct+1

(C.16)

By substituting equation (C.16) into equation (5.6), a type m worker’s IMRS can be written as
follows.
Λm,t→t+1 = β(1 − ω)ωs (γ −1 ωe + (1 − ωe )ωr + (1 − ωe )(1 − ωr )γ m )

51

Ct
Ct+1

(C.17)

The substitution of equation (C.17) into the definition of the aggregate inter-temporal marginal rateof substitution (5.7) leads to a closed-form representation of the aggregate intertemporal marginal
rate of substitution as can be seen below.
Λt,t+1 =a λ β Ct
1 − θA Ct+1

(C.18)

where aλ is defined asaλ =(1 − ω)2 ωs((1 − ωγ)(ωe + γ(1 − ωe )ωr )) + γ(1 − ωe )(1 − ωr ))γ(1 − ωγ)

By comparing equation (C.18) with the eighth equation of <Table 5.1>, one can confirm that these
two equations are the same.

C.5

Implementation of Government’s Optimal Allocation: Constant Probability of Price Adjustment

In order to formulate a benevolent government’s optimization problem, let us begin with the characterization of its objective function. The government’s objective function is obtained by theaggregation of preferences of individual households. In the model of this paper, the aggregationof preferences can be facilitated by the result that consumption levels of individual households
are determined solely by their skill levels. The substitution of equation (C.15) into the definition of the instantaneous social welfare function (5.16) implies that an equally weighted average of
instantaneous utility functions of individual households can be written as follows.
∞XΓm (log Cm,t + b(H̄ − Hm,t )) = log Ct + b(H̄ − Ht ) +m=0

ω log γ1 − ωγ+ log()
1−ω
1−ω

(C.19)Let us turn to the optimality conditions obtained by using the social welfare function specified
in equation (5.19). The optimization condition for the aggregate consumption is
bCt =

Zt
∆t

(C.20)

By substituting this condition into the aggregate production function, one can find that the optimal
amount of the aggregate hours worked at period t is
Ht = b−1(C.21)The optimization condition for gross inflation isΠt = γ̄

∆t
∆t−1

52

(C.22)

Table C.1: Optimal Fiscal Policy and Monetary Policies
Description
Sales Subsidies

Equation
TF,t = (τF,t − 1)Yt

Employment Subsidies

τF,t = α(Πϵ−1t Π)τF,t−1 + θF (1 − α)(TH,t = θH ∆p,t mct Yt∆p,t = αΠϵt ∆p,t−1 + (1 − α)(TA,t = θA Wt
W+DF,t+1]
Wt = Et [ t+1
RM,t+1

Investment Creditsϵ
1−αΠϵ−1
t
ϵ−1
1−α )

ϵ
1−αΠtϵ−1 ϵ−1
1−α )

Y R

Government’s Budget
Inflation

M,t+1
βEt [ t Yt+1]=1
DF,t = Yt (1 − ∆p,t mct ) + TF,t
Tt = TA,t + TF,t + TH,ttΠt = γ̄ ∆∆t−1

Note: In this table, initial values of τF,−1 , ∆−1 and ∆p,−1 are taken as given and thesteady-state value of gross inflation is Π = γ̄. The determination of Yt and ∆t is specified
in equation (5.20). The real marginal cost is determined by mct = ac ∆−1
t . Wt = QM,t At
is the market value of mutual funds.

C.6

Optimal Prices Consistent with Government’s Solution: Constant Probability of Price Adjustment

The substitution of equation (B.38) into profit-maximization condition (5.12) implies that the
profit-maximization condition consistent with the optimal allocation can be written as
p∗t =

St
Mt

St
γ̄ − α=∆−1Mtγ̄(1 − α) t→(C.23)

By substituting equations (C.22) and (C.23) into a pair of forward-looking conditions specified in(5.13) and then subtracting both sides of resulting two equations, the real marginal cost consistent
with the optimal allocation can be written as follows.
mct = (1 + θF )(ϵ − 1)(γ̄ − α) −1∆tϵγ̄(1 − α)(C.24)

By substituting equation (C.24) into equation (C.6), the real wage consistent with the optimal
allocation can be written as follows.
wt = (1 + θF )(ϵ − 1)(γ̄ − α) ∆p,t Zt
ϵγ̄(1 − α)∆t ∆t(C.25)The substitution of equation (C.24) into equation (C.14) implies that the corresponding aggregate
equilibrium consumption is
Ct = (1 + θF )ac (ϵ − 1)(γ̄ − α) Ztϵγ̄(1 − α)∆t53(C.26)

Comparing equation (C.20) with equation (C.26), one can find that the following condition is needed
to attain the optimal allocation as a result of a decentralized equilibrium.
b−1 = (1 + θF )ac (ϵ − 1)(γ̄ − α)ϵγ̄(1 − α)(C.27)Let us move onto the discussion of how to determine θA , θF , θH . In the case of the intertemporal marginal rate of substitution (IMRS), the solution to government’s optimization problem
implies that the optimal IMRS is
Λt,t+1 = β

Ct
Ct+1

(C.28)

Comparing equation (C.18) with equation (C.28), one can find that the following condition is needed
to ensure the equality between equilibrium and optimal IMRS:aλ=1
1 − θA(C.29)The optimal fiscal policies are defined as a set of taxes and subsidies (θA , θH , θF ) to help
implement the optimal allocation in a decentralized equilibrium. First, equation (C.29) can be used
to choose a value of θA as can be seen below.
θA = 1 − a λ(C.30)

Second, the substitution of equation (C.13) into (C.27) (for the elimination of ac ) leads to thefollowing equation for pairs of (θH , θF ):ϵ(1 + θF )γ̄ − α (1 + θH )(1 − ω)2 γ=
ϵ−1γ̄(1 − α) (γ − ω)(1 − ωγ)(C.31)

Third, the role of sales subsidies is to fix the steady-state distortion associated with the existenceof monopolistically competitive firms in goods markets in the same as is done in the literature suchas Woodford (2003). In particular, these sales subsidies are used to restore the equality of ‘price= marginal cost’ in the model of full-price flexibility. Recall that α = 0 corresponds to the case offull-price flexibility. In the case of α = 0, the profit-maximization condition (5.12) is reduced to asimple representation: Pt∗ /M Ct = ϵ/((ϵ − 1)(1 + θF )). In this case, Pt∗ = M Ct is satisfied when thefollowing condition holds.
ϵ=1(1 + θF )(ϵ − 1)

(C.32)

In sum, equations (C.30), (C.31), and (C.32) can be used to show how to set values of fiscal policy
measures necessary to implement the optimal allocation in a decentralized equilibrium as can beseen below.
θA
θH
θF

= 1 − aλ= (1−α)(γ−ω)(1−ωγ)γ̄−1γ(γ̄−α)(1−ω)2
1
= ϵ−1
54

(C.33)

Let us move onto the discussion of how subsidies and taxes evolve over time in the model with aconstant probability of price adjustment. Let us begin with sales subsidies of <Table C.1>. Recall
that sales subsidies are defined as follows.
TF,t = Yt ((1 + θF )T
τF,t = YF,t+1
t∗k Pt−k 1−ϵ − 1)
k=0 (1 − α)(αΠ) ( Pt )P∞

Dividing both side of the first-line equation by Yt and then substituting the second-line equation
into the resulting equation leads to the following equation.
P∞P∗

k t−k 1−ϵk=0 (αΠ) ( Pt )∗∗PPk Pt−k 1−ϵ )= (1 + θF )(1 − α)(( Ptt )1−ϵ + ∞
k=1 (αΠ) ( Pt )

τF,t = (1 + θF )(1 − α)(C.34)In addition, one-period lagged version of the first-line of equation (C.34) is(αΠ)Πϵ−1t τF,t−1 = (1 + θF )(1 − α)∞XP∗(αΠ)k ( t−k )1−ϵPt

(C.35)

k=1

Subtracting equation (C.35) from the second-line of equation (C.34) leads to the following representation.
τF,t = (αΠ)Πϵ−1t τF,t−1 + (1 − α)(1 + θF )(Pt∗ 1−ϵ) Yt
Pt(C.36)The substitution of the definition of the aggregate price index into equation (C.36) the leads thesame equation as the second equation of <Table C.1>.
Turning to employment subsidies of <Table C.1>, recall that TH,t = θH wt Ht as specified inequation (5.15). Given this equation, equation (C.6) implies the third equation of <Table C.1> ascan be seen below.
TH,t = θH mct

Zt Ht∆p,t → TH,t = θH ∆p,t mct Yt∆t(C.37)

Finally, the determination of mutual-fund investment credits consists of two forward-looking
difference equations. One is the law of motion for the market value of mutual fund and the otheris the asset pricing equation that holds for mutual fund’s market price in the absence of arbitrage
profits. In addition, by using equation (C.6) together with the definition of the aggregate production
function, the aggregate dividend can be written as follows.
DF,t = Yt − wt Ht + TF,t → DF,t = Yt (1 − ∆p,t mct ) + TF,t

C.7

(C.38)

Aggregate Equilibrium Conditions: Time-Varying Probabilities of Price
Adjustment

Let us move onto the discussion of how to obtain aggregate equilibrium conditions in the modelwith time-varying probabilities of price adjustment, which are comparable to those of <Table 5.1>
in the model with a constant probability of price adjustment. The introduction of time-varying
55

probabilities of price adjustment into the benchmark model alters specifications of subsidies (given
to firms and households) that are needed to implement the optimal allocation obtained from thebenevolent government’s optimization. For example, sale subsidies given to firms are determined
as follows.

k−1
YSt−k,t = ((1 + θF )(Ψt−k+i ) − 1)(

i=0

∗
Pt−k
)Dt−k,tPt(C.39)

for k = 1, · · · and St,t = θF Pt∗ Dt−k,t /Pt . The aggregate sum at period t of sale subsidies TF,t isalso defined as follows.
TF,t = (1 − αt )St,t +∞kXY(1 − αt−k )( αt−k+i )St−k,t(C.40)

i=1

k=1

In addition, employment subsidies for households vary over time as well. For this reason, the ratioof the aggregate sum of employment subsidies to the aggregate labor income is specified as θH,t =TH,t /(wt Ht ) in each period t = 0, · · · .
Given these changes of subsidies discussed above, it is also necessary to modify optimization
conditions of individual households. Despite of this result, one can check a set of equations (C.2),(C.3), (C.4), (C.5), and (C.6) to confirm that the labor market condition leads to the same relation
between aggregate consumption and marginal cost of production except for the time-varying coefficient of marginal cost. Specifically, the substitution of Ct = Yt into the labor-market equilibrium
condition leads to the following condition in the model with time-varying probabilities of priceadjustment.
Yt = ac,t mct Zt(C.41)

where ac,t is defined as follows.
ac,t =(1 + θH,t )(1 − ω)2 γb(γ − ω)(1 − ωγ)Moreover, the functional form of the aggregate production function is the same as that of the modelwith a constant probability of price adjustment:
Yt =

Zt Ht
∆t

(C.42)

Recall that the determination of relative price distortion ∆t becomes more complicated in the modelwith time-varying probabilities of price adjustment as shown in equation (B.31).
Let us turn to the profit-maximization problem of firms in the model with time-varying probabilities of price adjustment. Given sale subsidies for firms, the profit-maximization problem atperiod t of firms can be written as follows.
∞
k
∗
X
Y
Pt∗
∗ Pt
∗
max
{Φ
(
)
+
E
[Λ
(
α
)Φ
(
)]}
t
t+i
t,t+k
t
t+k
Pt∗
Pt+k
Pt+k
i=1
k=1

56

(C.43)

Table C.2: Collection of Equilibrium Conditions:Time-varying Probabilities of Price Adjustment
Description

Equation

Production Functiont Ht
Yt = Z∆t

Relative Price Distortion∆t = αγ̄tt Πϵt ∆t−1 + (1 − αγ̄tt )(Price Level
Real Reset Price
Profit Maximization A
Profit Maximization B
Labor Market
IMRS

t t
) 1−ϵ
p∗t = ( 1−αtSt∗
p t = Mt
Mt = (1 + θF )Yt + Et [αt+1 Λt,t+1 Πϵ−1
t+1 Ψt+1 Mt+1 ]ϵ
mct Yt + Et [αt+1 Λt,t+1 Πϵt+1 St+1 ]St = ϵ−1
Yt = ac,t mct Zt
aλ β YtΛt,t+1 = 1−θ
A Yt+11−α Πϵ−1ϵ1−αt Πϵ−1t
ϵ−11−αt )

1

Note: IMRS means the aggregate inter-temporal marginal rate of substitution. ac,t =(1+θH,t )(1−ω)2 γb(γ−ω)(1−ωγ)

2e )ωr ))+γ(1−ωe )(1−ωr ))and aλ = (1−ω) ωs ((1−ωγ)(ωe +γ(1−ω.
γ(1−ωγ)P∗tHere the instantaneous profit flows at period t + k, represented by Φ∗t+k ( Pt+k), is defined as follows.

Φ∗t+k (

k
Y
Pt∗
P∗
P∗
) = ((1 + θF )( Ψt+i )( t )1−ϵ − mct+k ( t )−ϵ )Yt+k
Pt+k
Pt+k
Pt+k

(C.44)

i=1

for k = 1, · · · , and Φ∗t (Pt∗ /Pt ) = ((1 + θF )(Pt∗ /Pt )1−ϵ − mct (Pt∗ /Pt )−ϵ )Yt .
The first-order condition of the profit maximization problem (C.43) can be summarized asfollows.

Pt∗
St
=
Pt
Mt

(C.45)

where Mt and St are defined as
Mt = (1 + θF )Yt + Et [αt+1 Λt,t+1 (Πϵ−1t+1 Ψt+1 )Mt+1 ](C.46)St=ϵϵ
ϵ−1 mct Yt + Et [αt+1 Λt,t+1 Πt+1 St+1 ]It also should be noted that an equilibrium relation between profit-maximizing price and aggregate
inflation can be obtained by using the definition of the aggregate price index as can be seen inequation (B.30).
<Table C.2> contains 8 aggregate equilibrium conditions in the model with time-varying probabilities of price adjustment that are comparable to those of <Table 5.1> for 8 aggregate endogenous
variables such as {Mt , St , Ht , Yt , p∗t , mct , ∆t , Λt,t+1 }. In this table, the number of endogenous
variables matches the number of equilibrium conditions given series of {Zt , αt , θH,t , Πt }∞t=0 with(θF , θA ). The benevolent government’s optimization problem is
V (sa,t ) = max{Ct ,Ht ,Πt ,∆t } {log Ct + b(H̄ − Ht ) + βEt [V (sa,t+1 )]}subject to∆t = αγ̄tt Πϵt ∆t−1 + (1 − αγ̄tt )(Ct = ∆−1t Zt Ht57

ϵ1−αt Πtϵ−1 ϵ−1)1−αt(C.47)

Table C.3: Optimal Fiscal Policy and Monetary Policies:
Time-varying Probabilities of Price Adjustment
Description
Sales Subsidies

Equation
TF,t = τF,t Yt

Employment Subsidies

τF,t = αt Ψt Πϵ−1t τF,t−1 + (1 + θF )(1 − αt )(TH,t = θH,t ∆p,t mct Yt∆p,t = αt Πϵt ∆p,t−1 + (1 − αt )(TA,t = θA Wt
W+DF,t+1
Wt = Et [ t+1]
RM,t+1

Investment Creditsϵ1−αt Πϵ−1t
ϵ−11−αt )

ϵ1−αt Πtϵ−1 ϵ−11−αt )Y R

Government’s Budget

M,t+1
]=1
βEt [ t Yt+1
DF,t = Yt (1 − ∆p,t mct ) + TF,t
Tt = TA,t + TF,t + TH,t′given the aggregate state vector sa,t = [∆t−1 Zt αt ] and the conditional joint probability density
function of aggregate productivity level and price-adjustment probability g(Zt+1 , αt+1 |Zt , αt ). The
partial differentiation of the social welfare function included in the social planner’s optimization
problem (C.47) with respect to consumption gives the following optimization condition:
bCt =

Zt
∆t

→ Yt =

Ztb∆t(C.48)The substitution of equation (C.48) into the aggregate production function specified in the first lineof <Table C.2> implies that the optimal amount of the aggregate hours worked at period t is
Ht = b−1(C.49)

Recall that two optimality conditions for the aggregate gross inflation and relative price distortion
are included in equations (B.36) and (B.37) respectively, while the optimality condition for the real
reset price at period t is also included in equation (B.38).
Let us turn to the decentralization of the optimal allocation in the model with time-varying
probabilities of price adjustment. The substitution of equation (B.38) into the profit maximization
condition specified in the fourth line of <Table C.2> leads to the following equation.
p∗t =γ̄t − αt∆−1 →
γ̄t (1 − αt ) tStγ̄t − αt=∆−1Mt
γ̄t (1 − αt ) t(C.50)The substitution of equation (C.50) into the sixth line of <Table C.2> and then subtracting theresulting equation from the fifth line leads to the following equation.
mct=Ψt+1 =(1+θF )(ϵ−1)(γ̄t −αt ) −1∆t
ϵγ̄t (1−αt )−1 (1−αt )(γ̄t+1 −αt+1 )γ̄t (1−αt+1 )(γ̄t −αt )58(C.51)

By substituting the first line of equation (C.50) into the seventh line of <Table C.2> and thecomparing the resulting equation with the aggregate production function with Ht = b−1 leads tothe following equation.
b−1 = ac,t

(1 + θF )(ϵ − 1)(γ̄t − αt )ϵγ̄t (1 − αt )(C.52)

In sum, equations (C.30), (C.32), and (C.52) can be used to show how to set values of fiscal policy
measures necessary to implement the optimal allocation in a decentralized equilibrium as can beseen below.
θA= 1 − aλt )γ̄t (γ−ω)(1−ωγ)θH,t = (1−α−1γ̄t −αt(1−ω)2 γ
1
θF
= ϵ−1

59

(C.53)