frictional labor markets, bargaining wedges · view of labor markets, tax-smoothing ceases to be...

Frictional Labor Markets, Bargaining Wedges, and

Optimal Tax-Rate Volatility ∗

David M. Arseneau †

Federal Reserve Board

Sanjay K. Chugh‡

University of Maryland

First Draft: January 2008

This Draft: April 18, 2008

Abstract

We re-examine the optimality of tax smoothing from the point of view of frictional labor

markets. In frictional labor markets, unlike in Walrasian labor markets, wages can play various

roles and can be determined in various ways. Our first main result is that if wages are deter-

mined by ex-post Nash bargaining, tax smoothing in the face of business cycle shocks is not

optimal. Quantitatively, in what has emerged as the standard DSGE labor search and bargain-

ing model, the optimal labor tax rate is one to two orders of magnitude more volatile than in

standard Ramsey models. Tax volatility partially offsets cyclical wedges due to bargaining. Our

second main result is that if wages are instead posted before workers and firms meet and search

is directed by wages — that is, labor markets are governed by competitive search equilibrium —

the optimality of tax smoothing is restored because bargaining wedges are absent. Our results

are robust to a number of alternative features of the environment that govern the the severity

of search frictions. Thus, our main conclusion is that one has to accept a largely Walrasian,

or competitive, view of wage determination in order to accept the prescription of tax smoothing.

Keywords: labor market frictions, optimal taxation

JEL Classification: E24, E50, E62, E63

∗We thank seminar participants at the University of Delaware and the Federal Reserve Board’s International

Finance Workshop for comments, and especially David Stockman and Andrea Raffo for helpful discussions. The

views expressed here are solely those of the authors and should not be interpreted as reflecting the views of the Board

of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. An

early draft of this paper circulated under the title “Tax Smoothing May Not Be As Important As You Think.”†E-mail address: [email protected].‡E-mail address: [email protected].

1

Contents

1 Introduction 4

2 Baseline Model 7

2.1 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Wage Bargaining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Government . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Matching Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Private-Sector Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Ramsey Problem in Baseline Model 13

4 Optimal Taxation in Baseline Model 14

4.1 Parameterization and Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Main Result: The Optimality of Tax Volatility . . . . . . . . . . . . . . . . . . . . . 15

4.3 Recovering Tax Smoothing in the Baseline Model . . . . . . . . . . . . . . . . . . . . 19

5 Alternative Views of the Labor Market 21

5.1 Instantaneous Hiring (Model 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.1.1 Modifications of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.1.2 Optimal Taxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 Endogenous Labor Force Participation . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2.1 Baseline Timing Assumption (Model 3) . . . . . . . . . . . . . . . . . . . . . 25

5.2.2 Instantaneous Hiring (Model 4) . . . . . . . . . . . . . . . . . . . . . . . . . . 27


5.3 Competitive Search Equilibrium (Model 5) . . . . . . . . . . . . . . . . . . . . . . . 31

5.3.1 Modifications of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3.2 Static Tax Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


6 Summary and Discussion 35

7 Conclusion 39

A Nash Bargaining in Model with Standard Timing 41

B Nash Bargaining in Model with Instantaneous Hiring 43

2

C Elasticity of Market Tightness to Labor Tax Rate 45

D Derivation of Implementability Constraint 47

E Dynamic Bargaining Power Effect 48

3

1 Introduction

We re-examine the optimality of tax smoothing from the point of view of frictional labor markets.

Since Barro’s (1979) partial-equilibrium intuition, Lucas and Stokey’s (1983) general-equilibrium

analysis, and continuing through to today’s quantitative DSGE models used to study optimal

fiscal policy, the prescription that governments ought to hold labor tax rates virtually constant

in the face of aggregate shocks is well-known to macroeconomists. We show that this cornerstone

optimal-policy prescription and the intuition underlying it depend crucially on a Walrasian view

of labor markets. If one instead takes what has emerged as the standard search and bargaining

view of labor markets, tax-smoothing ceases to be important for empirically-relevant labor market

parameters. If wages are determined in bilateral bargaining after workers and firms meet, not only

is tax-smoothing unimportant, but purposeful tax-rate volatility is actually welfare-enhancing by

partially offsetting cyclical bargaining-induced wedges.

Our baseline search and bargaining environment is identical to the one that has come into

widespread use in recent DSGE modeling efforts. We quantitatively demonstrate the optimality

of labor tax-rate volatility in this environment. In an effort to recover tax smoothing, we then

incrementally alter the environment in a number of ways, each of which in principle reduces the

severity of search and bargaining frictions. In particular, we change the timing of labor market

flows, introduce a labor-force participation margin, and allow for wages to be determined in a

competitive fashion, rather than through ex-post bilateral bargaining. Changing the timing of

labor market flows and allowing a labor-force participation choice each, as well as together, only

modestly reduces the degree of optimal tax-rate volatility.

However, allowing for competitive determination of wages — using Moen’s (1997) concept of

competitive search equilibrium, in which ex-ante posted wages direct search activity — is a critical

change in market structure that reinstates the optimality of tax smoothing. For the competitive

search economy, we show analytically that it is only labor taxes that create relatively-standard static

wedges between marginal rates of substitution between consumption and leisure and corresponding

marginal rates of transformation. The usual reasons for tax smoothing — that wedges between

this MRS and MRT should be kept (nearly) constant over time — then apply. As a methodological

by-product of our analysis, we are, to the best of our knowledge, the first to provide a simple

MRT interpretation for a labor-search model, one that differs from the notion of MRT in standard

neoclassical model of the labor market.

If wages are determined by ex-post bilateral bargaining, proportional labor taxes affect the

labor market in a dramatically different way. We identify two distinct roles played by the labor tax

in our bargaining environments: the usual static role, in which a positive labor tax wedge is needed

period-by-period in order to raise revenue for the government, and a novel dynamic role, in which

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changes in tax rates affect private-sector search activity. Thus, for any given level of the labor tax

rate in period t, the change in the tax rate is a distinct lever that the Ramsey government can use

as a way of directing labor-market outcomes over the business cycle. This dynamic role of labor

taxes — which we refer to as a dynamic bargaining power effect — only arises in the environment

with bilateral bargaining and can only be revealed in an explicitly dynamic analysis.

The key difference between the two market structures, which drives the stark difference in

optimal-policy prescriptions, is the fundamental forces underlying wage determination. In the

competitive search economy, posted wages allow unemployed individuals to optimally direct their

search activity in the labor market, which in turn generates competition amongst wage-setting

firms. This competition ensures private-sector labor-market activity is efficient — up to the static

tax wedge — over the business cycle, much like in standard Walrasian-based Ramsey models; hence

the prescription to smooth tax rates over time.

In contrast, if wages are determined by ex-post bilateral bargaining, such competitive forces are

absent. We show that ex-post bargaining in and of itself introduces a time-varying wedge between

the notions of MRS and MRT that we develop. One interpretation of these bargaining-induced

wedges is the holdup problems inherent in the bilateral monopoly of ex-post bargaining, a feature

of bargaining that Caballero (2007) emphasizes. A related interpretation of these wedges is that

they reflect incomplete-contracting problems. Regardless of the interpretation of the wedge, in

order to partially offset these cyclical bargaining-induced wedges, the Ramsey government exploits

the dynamic bargaining power effect we identify, and optimal labor tax rates are volatile. Our

results show that bargaining frictions can dramatically alter conventional thinking regarding one

of the most basic optimal-policy prescriptions. One has to accept a competitive view of wage

determination — one devoid of bargaining frictions — in order to accept the prescription of tax

smoothing.

The conclusion that the optimality of tax smoothing depends critically on the market structure

that determines real wages in and of itself may not be surprising. What is surprising are the

quantitative magnitudes we find. In our baseline search and bargaining model, optimal tax rate

volatility is between 1.5 percent and 7 percent for empirically-relevant calibrations. For comparison,

the Ramsey literature’s conventional tax-smoothing result entails optimal tax rate volatility near

0.1 percent or less — for example, see the overview in Chari and Kehoe (1999). These numbers all

refer to the standard deviation of the level of the Ramsey-optimal labor tax rate, and all of them

refer to fluctuations around average tax rates in the range 20-60 percent. Thus, labor tax rates are

one to two orders of magnitude more volatile in our baseline search and bargaining model than in

simple Walrasian-based Ramsey environments. Our results suggest that optimal-policy theorists

perhaps ought not to be so enchanted with the cornerstone Ramsey dynamic tax-smoothing result.

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Werning (2007) recently shed new analytical insight on the cornerstone Ramsey dynamic tax-

smoothing result. By bringing distributional issues between different types of consumers to the

foreground, Werning (2007) connects results in the Ramsey literature to insights from the new

dynamic public finance literature, which emphasizes tradeoffs between distributions and efficiency.

Just like the standard Ramsey intuition for tax smoothing, however, Werning’s (2007) analysis and

insights depend — as, indeed, do many results in the new dynamic public finance literature — on

fundamentally spot views of all markets. Labor markets — indeed, perhaps goods markets and

capital markets as well — seem to be ill-characterized as spot markets. Given the broad macro

literature’s recent embrace of labor search models, we think it worthwhile to begin exploring how

such models might change conventional thinking regarding optimal policy.

Our conclusions that the details of market structure and the precise notion of equilibrium in

search models can matter for issues in macroeconomics connect with those of Rocheteau and Wright

(2005). Rocheteau and Wright (2005) show that in search-based monetary models, the precise no-

tion of equilibrium can matter a great deal for efficiency. Like us, they show that whether frictional

markets are best characterized by a bargaining equilibrium or a competitive search equilibrium can

have quite important implications for policy, in their case monetary policy. They do not study

Ramsey-optimal policy or consider stochastic environments, but some of the essence of our results

echoes their insights.

Finally, we also note that tax volatility in our model is not driven by any incompleteness of

government debt markets, which is a well-understood point in Ramsey models since Aiyagari,

Marcet, Marimon, and Sargent (2002). Rather, our point of departure from a standard Ramsey

setup is that we model labor trades as governed by primitive search and matching frictions. While

search-based DSGE models have become commonplace in recent years, their ability to shed new

insights on optimal fiscal policy has not yet been much explored.1 Our model features no capital

accumulation, in order to highlight the dynamics of labor taxes. We see no reason why our basic

intuition and results would not extend to the classic case of Ramsey taxation of both labor and

capital income.

The rest of our work is organized as follows. Section 2 lays out the baseline model, which hews

very closely to the recent vintage of DSGE labor-search models, in which we study optimal fiscal

policy. Section 3 lays out the Ramsey problem in our baseline model. Section 4 presents our main

result that tax smoothing is unimportant in the presence of search frictions in the labor market.

The rest of our analysis, in Section 5, explores modifications to the baseline search model that may

be important in recovering the optimality of tax smoothing; we in turn allow for match formation1To our knowledge, the only work investigating aspects of optimal fiscal policy in labor-search dynamic general

equilibrium models is Domeij (2005) and our own previous work, Arseneau and Chugh (2006, 2007); in none of these

is the focus explicitly on tax smoothing.

6

to occur instantly, an endogenous labor force participation decision, and for wages to be determined

in a competitive manner, rather than through bargaining. Section 6 provides several alternative

ways to understand our results, and Section 7 concludes.

2 Baseline Model

We establish our main result in a simple model featuring labor search and matching frictions. As

many other recent studies have done, we embed the Pissarides (2000) textbook search model into a

dynamic stochastic general equilibrium framework. We present in turn the choice problems of the

representative firm, the representative household, the determination of wage payments, the actions

of the government, and the definition of equilibrium. As an aid to understanding the timing of

events in our model, Figure 1 accompanies the ensuing description of the environment.

2.1 Production

The production side of the economy features a representative firm that must open vacancies, which

entail costs, in order to hire workers and produce. The representative firm is “large” in the sense

that it operates many jobs and consequently has many individual workers attached to it through

those jobs.

The firm requires only labor to produce its output. The firm must engage in costly search for

a worker to fill each of its job openings. In each job k that will produce output, the worker and

firm bargain over the pre-tax real wage wkt paid in that position. Output of any job k is given by

ykt = zt, which is subject to a common technology realization zt.

Any two jobs ka and kb at the firm are identical, so from here on we suppress the second

subscript and denote by wt the real wage in any job, and so on. Total output of the firm thus

depends on the technology realization and the measure of matches nft that produce,

yt = ztnft . (1)

The total real wage bill of the firm is the sum of wages paid at all of its positions, nft wt.

The firm begins period t with employment stock nft . Its future employment stock depends on its

current choices as well as the random matching process described below. With probability kf (θ),

taken as given by the firm, a vacancy will be filled by a worker. Labor-market tightness is θ ≡ v/u,

where u denotes the number of individuals searching for jobs. Matching probabilities depend only

on market tightness given the Cobb-Douglas matching function we will assume.

Wages are determined through bargaining, as we describe below. In the firm’s profit maximiza-

tion problem, the wage-setting protocol is taken as given.2 The firm thus chooses vacancies to post2This assumption is without loss of generality in the standard Pissarides-type model because even if the firm

7

vt and future employment stock nft+1 to maximize discounted nominal profits starting at date t,

Et

∞∑t=0

βt

Ξt|0[ztn

ft − wtn

ft − γvt

]. (2)

where Ξt|0 is the period-0 value to the representative household of period-t goods, which we assume

the firm uses to discount profit flows because households are the ultimate owners of firms.3 In

period t, the firm’s problem is thus to choose vt and nft+1 to maximize (2) subject to a sequence of

perceived laws of motion for its employment level,

nft+1 = (1− ρx)(nft + vtkf (θt)). (3)

Firms incur the real cost γ for each vacancy created, and job separation occurs with exogenous

fixed probability ρx. Note the timing of events embodied in the law of motion (3) and shown in

Figure 1. Period t begins with a stock nt that is used for period-t production. Following production,

a measure ρx of jobs end and the labor matching process occurs. A measure ρx of newly-formed

matches are also destroyed before ever becoming productive, thus determining the new employment

stock nt+1.

The firm’s first-order conditions with respect to vt and nft+1 yield the standard job-creation

conditionγ

kf (θt)= (1− ρx)Et

[Ξt+1|t

(zt+1 − wt+1 +

γ

kf (θt+1)

)], (4)

where Ξt+1|t ≡ Ξt+1|0/Ξt|0 is the household discount factor (again, technically, the real interest

rate) between period t and t+ 1. The job-creation condition states that at the optimal choice, the

vacancy-creation cost incurred by the firm is equated to the discounted expected value of profits

from a match. Profits from a match take into account future marginal revenue product from the

match, the wage cost of the match, and the asset value of having a pre-existing relationship with

an employee in period t + 1. This condition is a free-entry condition in the creation of vacancies

and is a standard equilibrium condition in a labor search and matching model.

believed it could opportunistically manipulate the wages it paid by under- or over-hiring, the fact that labor’s

marginal product is independent of total employment prevents such opportunistic manipulation of wage-bargaining

sets. Thus, in the standard exogenous-productivity Pissarides model, holdup problems are in principle present, but

there is no lever by which firms can strategically react to them. If firm output exhibited diminishing marginal product

in its total employment level, then the firm would have an incentive to over-hire. See, for example, Cahuc, Marque,

and Wasmer (2008) and Krause and Lubik (2006).3Technically, of course, it is the real interest rate with which firms discount profits, and in equilibrium the real

interest rate between time zero and time t is measured by Ξt|0. Because there will be no confusion using this

equilibrium result “too early,” we skip this intermediate level of notation and structure.

8

Period t-1 Period t+1Period tAggregate

state realized

nt nt+1 = (1-ρx)(nt + m(ut, vt))

Bargaining occurs (i.e., asset values

defined here)

Production (using ntemployees), goods markets and asset

markets meet and clear

Employment separation occurs (ρxnt employees

separate)

Search and matching in labor market

Fraction ρx of new matches

separateyields

Figure 1: Timing of events in baseline model.

2.2 Households

There is a representative household in the economy. Each household consists of a continuum of

measure one of family members. In our baseline model, there is no labor-force participation decision,

hence each member of the household either works during a given time period or is unemployed and

searching for a job. There is a measure nht of employed individuals in the household and a measure

uht = 1− nht of unemployed individuals. We assume that total household income is divided evenly

amongst all individuals, so each individual has the same consumption.

The representative household thus maximizes expected lifetime discounted utility

E0

∞∑t=0

βtu(ct) (5)

subject to the sequence of flow budget constraints

ct + bt = nht (1− τnt )wt + (1− nht )χ+Rtbt−1 + dt, (6)

where χ is the flow of unemployment benefits each unemployed individual receives, (1−τnt )wt is the

after-tax wage rate each employed individual earns, dt is firm dividends received lump-sum by the

household, and bt−1 is the household’s holdings of a state-contingent one-period real government

bond at the end of period t− 1, which has gross payoff Rt at the beginning of period t. Important

to note is that the government is able to issue fully state-contingent debt; thus, none of our optimal

policy results will be driven by an inability on the part of the government to use debt as a shock

absorber. Incompleteness of government debt markets can be an important driver of results in

9

Ramsey models — see, for example, Aiyagari, Marcet, Marimon, and Sargent (2002). Finally,

because this is a Ramsey-taxation model, no lump-sum taxes or transfers exist.

With no labor-force participation decision, the only equilibrium condition stemming from house-

hold optimization is the usual consumption-savings condition,

u′(ct) = Et[βu′(ct+1)Rt+1

]. (7)

In equilibrium, firms discount profit flows according to Ξt+1|t = βu′(ct+1)/u′(ct).

2.3 Wage Bargaining

We assume period-by-period Nash negotiations over the wage payment. Important to note in

Figure 1 is the point in time at which bargaining occurs. As is standard in this class of models,

period-t asset values are defined at the point in time at which period-t bargaining occurs, which

is after labor-market matching (of period t − 1) has taken place. We think this timing is natural

because negotiations over the wage can take place only after the two parties meet. With ex-post

wage negotiation, each vacancy opening in which a firm invests is thus undertaken without perfect

knowledge of the wage that the firm will negotiate with the worker that fills that position. The

same is true for search activity on the part of individuals: while engaged in the search process,

individuals do not know the wage they might receive if search is successful. Caballero (2007)

interprets the inefficiencies that arise in such an environment as holdup problems. In their survey

of labor-search theory, Mortensen and Pissarides (1999) interpret these inefficiencies as due to

incomplete-contracting problems. Regardless of the precise interpretation, inefficiencies due to

ex-post bargaining arise naturally in search environments.4

Here, we simply present the Nash wage-bargaining outcome; a detailed derivation is presented

in Appendix A. Assuming that η is a worker’s Nash bargaining power and 1 − η a firm’s Nash

bargaining power, the Nash wage outcome is given by

wt = η

[zt +

γ

kf (θt)

]+

(1− η)χ1− τn

t

− η(

(1− ρx)(1− kh(θt))1− τn

t

)Et

[Ξt+1|t(1− τn

t+1)(zt+1 − wt+1 +

γ

kf (θt+1)

)].

(8)

The wage outcome (8) is a variation of the typical Nash wage solution in labor-search models.

The first term on the right-hand-side of (8) is standard: it represents the firm’s flow return from

production and the asset value from the formation of an additional job-match. This term essentially

is the upper bound of the wage-bargaining set. The second term on the right-hand-side also seems

relatively standard: it represents the outside option (χ) of a worker appropriately deflated by the4When we change the timing of markets in Section 5, the precise point in time at which bargaining occurs will

change, resulting in a slight change in the wage bargaining outcome, but bargaining-induced frictions will still be

present.

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take-home rate 1− τnt . This term essentially is the lower bound of the wage-bargaining set. From

the point of view of the Ramsey planner in our environment, the possibility that τn, and hence

the lower end of the wage-bargaining set, can be purposefully manipulated over time is important.

Finally, the third term on the right-hand-side of (8) also shows how variations in labor tax rates

affect outcomes. It reveals that both current (period-t) and expected future (period-t+1) labor tax

rates affect the current wage bargain. Intuitively, the third term reflects part of the continuation

value of a match. A similar term arises in a standard labor search model; what is different, here,

however, is that this component of the continuation value is influenced by changes in tax rates.

By cyclically altering ex-post wage-bargaining sets, tax rate variations influence the ex-ante

incentives for parties to invest in the costly matching process in the first place. Thus, tax-rate

variability has the potential to mitigate inefficiencies stemming from bargaining over the business

cycle. We could also describe the effects of policy by saying that tax-rate variations alter effective

(i.e., inclusive of tax) bargaining shares. Arseneau and Chugh (2007) dubbed variations in effective

bargaining shares stemming from cyclical policy changes a dynamic bargaining power effect, and

this effect is also present in our model here. The dynamic bargaining power effect is the source of

our optimal-policy results.5

If labor taxes were constant at τnt = τn ∀t — in which case the dynamic bargaining power

effect of course disappears — the wage outcome would be wt = η(zt + γθt) + (1−η)χ1−τ .6 In this case,

the presence of the labor tax only changes firms’ effective bargaining power in a static manner:

τn > 0 causes (1− η)/(1− τn) > 1− η. This kind of static bargaining wedge underpins the results

in Arseneau and Chugh (2006); in our model here, this static effect would be unable to correct

variations in the severity of bargaining-induced wedges along the business cycle. If furthermore

τnt = 0 ∀t, the bargained wage collapses to the even more familiar wt = η(zt + γθt) + (1 − η)χ,

which can be found in, for example, Pissarides (2000, p. 17).

2.4 Government

The government finances an exogenous stream of spending gt by collecting labor income taxes

and issuing real state-contingent debt. The period-t government budget constraint is

τnt wtnt + bt = gt +Rtbt−1 + (1− nt)χ. (9)5In standard models, the third term on the right-hand-side of (8) can be simplified using the job-creation condition

and then condensed with the first-term on the right-hand-side of (8); these standard manipulations yield a wage

equation that depends on only period-t variables. However, the presence of the stochastic tax rate τnt+1 prevents such

simplification here and means that wage-setting is explicitly forward-looking.6To obtain this, note from the job-creation condition (4) thatγ


[Ξt+1|t

(zt+1 − wt+1 + γ

kf (θt+1)

)].

11

As we noted when we presented the problem of the representative household, the fact that the

government is able to issue fully state-contingent real debt means that none of our results are

driven by incompleteness of debt markets.

We include payment of unemployment benefits as a government activity for two reasons. First,

we think it empirically descriptive to view the government as providing such insurance.7 Second,

from a technical standpoint, including (1−nt)χ in the government budget constraint means that χ

does not appear in the economy-wide resource constraint (presented below). In DSGE labor-search

models, it is common to include unemployment benefits in the household budget constraint but yet

exclude them from the economy-wide resource constraint — see, for example, Krause and Lubik

(2007) or Faia (2007). In such models, the government budget constraint is a residual object due to

the presence of a lump-sum tax. In contrast, we rule out lump-sum taxes in order to conduct our

Ramsey analysis and thus cannot treat the government’s budget as residual. A Ramsey problem

requires specifying both the resource constraint and either the government or household budget

constraint as equilibrium objects, and this requires us to take a more precise stand on the source of

unemployment benefits than usually taken in the literature. To make our model setup as close as

possible to existing ones, we must assert that payment of unemployment benefits is a government

activity.

2.5 Matching Technology

In equilibrium, nt = nft = nht , so we now refer to employment simply as nt. Matches between

unemployed individuals searching for jobs and firms searching to fill vacancies are formed according

to a constant-returns matching technology, m(ut, vt), where ut is the number of searching individuals

and vt is the number of posted vacancies. A fraction ρx of matches that produce in period t are

exogenously destroyed before period-t + 1, and a fraction ρx of newly-formed matches in period

t are destroyed before ever becoming productive. This timing in DSGE labor-search models is

conventional. The evolution of aggregate employment is thus given by

nt+1 = (1− ρx)(nt +m(ut, vt)). (10)

2.6 Private-Sector Equilibrium

A symmetric private-sector equilibrium is made up of endogenous processes ct, wt, nt, θt, ut, Rt∞t=0

that satisfy the job-creation condition (4), the consumption-savings optimality condition (7), the

equilibrium wage condition (8), the law of motion for the aggregate stock of employment (10), the7Notwithstanding the home-production explanation often offered for χ in labor-search models.

12

resource constraint on the total size of the labor force

nt + ut = 1, (11)

and the aggregate resource constraint of the economy

ct + gt + γutθt = ztnt. (12)

In (12), total costs of posting vacancies γutθt are a resource cost for the economy; note that we

have made the substitution vt = utθt, eliminating vt from the set of endogenous processes. As we

discussed above, unemployment benefits χ do not absorb any part of market output. The private

sector takes as given stochastic processes zt, gt, τnt ∞t=0.

3 Ramsey Problem in Baseline Model

In standard Ramsey models with flexible prices, a well-known result is that all equilibrium condi-

tions, apart from the resource frontier, can be encoded by a single, present-value implementability

constraint (PVIC) that must be respected by Ramsey allocations. In more complicated environ-

ments, such as Schmitt-Grohe and Uribe (2005), Chugh (2006), and Arseneau and Chugh (2007),

it is not always possible to construct such a single constraint, meaning that, in principle, all of the

equilibrium conditions must be imposed explicitly as constraints on the Ramsey problem.

Our model presents an environment in which it is instructive to construct a PVIC but nonethe-

less leave some equilibrium conditions as separate constraints on the Ramsey planner. As we show

in Appendix D, we can construct a PVIC starting from the household flow budget constraint (6)

and using the standard household optimality conditions (7); the PVIC for our model is given by

E0

∞∑t=0

βtu′(ct) [ct − (zt − τnwt)nt − (1− nt)χ+ γvt]

= u′(c0)R0b−1. (13)

But we refrain from substituting other equilibrium conditions into the PVIC and instead impose

them separately as explicit constraints on the Ramsey optimization.

The Ramsey problem is thus to choose state-contingent processes

ct, wt, nt, τnt , θt, ut∞t=0 to maximize (5) subject to the PVIC (13), the job-creation condition (4),

the Nash wage outcome (8), the aggregate law of motion for the employment stock (10), the restric-

tion on the size of the labor force (11), and the aggregate resource constraint (12). Note that in our

formulation we leave the policy process τnt as an explicit object of Ramsey optimization. The

labor tax rate cannot be eliminated from the set of Ramsey choice variables because it appears in

the time t−1 forward-looking wage expression (8). Because τnt cannot be eliminated through simple

static conditions as in basic Ramsey models, we view it as a fundamental part of the allocation.

13

4 Optimal Taxation in Baseline Model

We characterize both the Ramsey steady state and dynamics numerically. Before presenting quan-

titative results, we describe our baseline parameterization.

4.1 Parameterization and Solution Strategy

Well-understood since Hagedorn and Manovskii (2007) is that the (net-of-tax) social flow gain

from employment, z− χ1−τn , is important for dynamics in search models. In Appendix C, we derive

the steady-state elasticity of labor-market tightness to the labor tax rate, which shows that the

social flow gain from unemployment is likely to be important for the optimality of tax smoothing

as well.8 Because we normalize the steady-state level of technology to z = 1 and τn is of course

endogenous under the Ramsey policy, it is thus χ that is crucial. We report and discuss our main

dynamic results for calibrations of χ such that unemployment benefits constitute 40 percent and

95 percent of after-tax real wages. The 95-percent replacement rate corresponds to the Hagedorn

and Manovskii (2007) — hereafter, HM — calibration, while the 40-percent replacement rate is

the Shimer (2005) value. To gain deeper understanding of how the model works, we also document

results for alternative values of unemployment benefits, corresponding to 70 percent, 90 percent,

and 99 percent replacement rates.

The rest of our calibration is relatively standard in this class of models. We assume a quarterly

subjective discount factor β = 0.99 and instantaneous household utility u(c) = ln c. The matching

function is Cobb-Douglas, m(u, v) = ψuξut v1−ξut , with ξu = 0.4, in line with the evidence in Blan-

chard and Diamond (1989), and ψ set so that the quarterly job-finding rate of a searching individual

is 60 percent in the model with a 50-percent replacement rate. The resulting value is ψ = 0.66,

which we hold constant as we vary the replacement rate as well when we consider extensions of

our model along several dimensions. Given our Cobb-Douglas specification, this corresponds to

calibrating kh(θ) = 0.60 in the Ramsey equilibrium with a 40-percent replacement rate. The fixed

cost of opening a vacancy is set so that posting costs absorb 5 percent of total output in the Ramsey

equilibrium with a 40-percent replacement rate.

We fix the Nash bargaining weight at η = 0.40 so that it ostensibly satisfies the well-known

Hosios (1990) condition η = ξu for static search efficiency. We use this as our main guidepost,

although we point out that because τn 6= 0 in the Ramsey equilibrium, the naive calibration η = ξu

actually does not deliver search efficiency in the steady state. Rather, the proper setting must also

take into account the steady-state distortionary tax rate.9 Our main focus, though, is not on steady-8Specifically, see expression (75) in Appendix C.9Recall we pointed out when we discussed wage determination that if τnt = 0 ∀t that the Nash bargain collapses

to the usual one in Pissarides (2000), which is the basis for the standard Hosios (1990) condition.

14

state holdup and other inefficiencies, but rather, as we mentioned above, possible inefficiencies in

the cyclical fluctuations of search behavior; setting η = ξu does not obscure this. Arseneau and

Chugh (2006) provide an extensive analysis of how labor taxation drives a steady-state wedge in

the usual Hosios efficiency condition.

Finally, the exogenous productivity and government spending shocks follow AR(1) processes in

logs,

ln zt = ρz ln zt−1 + εzt , (14)

ln gt = (1− ρg) ln g + ρg ln gt−1 + εgt , (15)

where g denotes the steady-state level of government spending, which we calibrate in our baseline

model with a 40-percent replacement rate to constitute 17 percent of steady-state output in the

Ramsey allocation. The resulting value is g = 0.07, which we hold constant across all experi-

ments and all specifications of our model. The innovations εzt and εgt are distributed N(0, σ2εz)

and N(0, σ2εg), respectively, and are independent of each other. We choose parameters ρz = 0.95,

ρg = 0.97, σεz = 0.006, and σεg = 0.027, consistent with the RBC literature and Chari and Kehoe

(1999). Also regarding policy, we assume that the steady-state government debt-to-GDP ratio (at

an annual frequency) is 0.5, in line with evidence for the U.S. economy and with the calibrations

of Schmitt-Grohe and Uribe (2005) and Chugh (2006, 2007).

To study dynamics, we approximate our model by linearizing in levels the Ramsey first-order

conditions for time t > 0 around the non-stochastic steady-state of these conditions. We use

our approximated decision rules to simulate time-paths of the Ramsey equilibrium in the face

of a complete set of TFP and government spending realizations, the shocks to which we draw

according to the parameters of the laws of motion described above. Our numerical method is our

own implementation of the perturbation algorithm described by Schmitt-Grohe and Uribe (2004).

As is common when focusing on asymptotic policy dynamics, we assume that the initial state of

the economy is the asymptotic Ramsey steady state, thus adopting the timeless perspective. As

we mentioned above, we assume throughout, as is also typical in the literature, that the first-

order conditions of the Ramsey problem are necessary and sufficient and that Ramsey allocations

are interior. We conduct 5000 simulations, each 200 periods long. For each simulation, we then

compute first and second moments and report the medians of these moments across the 5000

simulations.

4.2 Main Result: The Optimality of Tax Volatility

Table 1 presents our main result. Our focus is on the dynamics of the optimal labor tax rate,

displayed in the first row of each panel. The top panel shows that at the Shimer calibration (in

which the usual Hosios parameterization η = ξu is satisfied and unemployment benefits constitute

15

40 percent of after-tax wages on average), the optimal labor tax rate is extremely volatile, with a

standard deviation of 7 percent around a mean of about 16 percent.10 To appreciate how startlingly

volatile this tax rate is, compare our result with that of Chari and Kehoe (1999, p. 1710), which

has become the quantitative benchmark for modern stochastic Ramsey models. Chari and Kehoe

(1999) report a standard deviation of τn of 0.10 percentage point around a mean of 23.87 percentage

points. That is, in their benchmark Ramsey model, the optimal labor tax rate remains between

23.77 percent and 23.97 percent two-thirds of the time in the face of aggregate business cycle

shocks. A long line of other studies — for example, Schmitt-Grohe and Uribe (2005), Siu (2004),

and Chugh (2006, 2007) — have confirmed this result. Werning (2007, p. 944) confirms such

magnitudes analytically.

A large part of the DSGE labor search literature has embraced something close to the Shimer

calibration, so we think it worthwhile to first discuss the intuition behind our results for this case.

The basic reason that tax smoothing is not part of the optimal policy prescription is that labor

tax rates affect allocations in a very different way than they do in a Walrasian labor market. In

the standard frictionless view, the labor tax rate is the only potentially time-varying component

of wedges between labor demand and labor supply. Time-variation in the labor wedges caused by

a positive average tax rate is avoided by the Ramsey planner because total discounted deadweight

losses over time are convex in labor wedges — this is Barro’s (1979) basic intuition, one that has

carried over to general-equilibrium Ramsey environments.

In contrast, in our search and bargaining model, the labor tax rate can play a quite different role.

As we described in Section 2.3, a dynamic bargaining power effect arises through which variations

in tax rates, because they affect ex-post wage-bargaining sets, can affect parties’ ex-ante search

incentives. By cyclically altering ex-ante search incentives, tax-rate volatility can ease variations in

the magnitude of bargaining-induced inefficiencies along the business cycle. Or, in language similar

to Mortensen and Pissarides (1999) or Boone and Bovenberg (2002), because tax rates are known

before any private-sector agents make their search decisions (even though pre-tax wages are not),

labor-tax policy can be thought of as easing the incomplete contracting problems inherent in an

environment of ex-ante search but ex-post bargaining. In Section 6, we develop this idea further,

along with several other, related, ways that one can understand optimal tax-rate volatility in our

model. The simplest way to intuitively understand the result is that ex-post bargaining in and of10In the interest of documenting the properties of our model, we also present our model’s dynamics for several other

salient variables, but we do not dwell on them. The one aspect of our results to which we draw the reader’s attention

is that tax-rate volatility does not stem from unreasonably-high aggregate volatility of our model: the volatility of

total output is fairly stable, between about 1.3 and 1.7 percentage points, as unemployment benefits rise from 40

percent to 95 percent of after-tax wages, well within the range of calibrated models and in line with estimates of U.S.

GDP volatility.

16

Variable Mean Std. Dev. Auto corr. Corr(x, Y ) Corr(x, Z) Corr(x,G)

Shimer calibration (η = 0.4, ue benefits 40 percent of after-tax wage)

τn 0.1625 0.0701 0.5327 0.6832 0.6443 -0.2310

gdp 0.8446 0.0148 0.9293 1.0000 0.9955 -0.0757

c 0.6727 0.0071 0.9465 0.9302 0.9104 -0.3861

N 0.8439 0.0028 0.9529 0.9109 0.8685 -0.1779

w 0.8685 0.0161 0.9101 0.9341 0.9065 -0.2095

θ 0.8579 0.0159 0.9061 0.9846 0.9815 -0.1935

v 0.1339 0.0011 0.3292 0.2128 0.2946 -0.0476

u 0.1561 0.0028 0.9529 -0.9109 -0.8685 0.1779

Exact HM calibration (η = 0.052, ue benefits 95 percent of after-tax wage)

τn 0.6114 0.0166 0.3609 -0.4796 0.5212 -0.1520

gdp 0.4620 0.0065 0.5225 1.0000 0.1104 0.6938

c 0.3759 0.0036 0.0657 0.5506 0.2557 -0.1656

N 0.4618 0.0087 0.7886 0.6314 -0.6702 0.5208

w 0.9620 0.0375 0.3730 -0.4716 0.5335 -0.1516

θ 0.0400 0.0037 0.2384 -0.3502 -0.3276 0.2456

v 0.5382 0.0087 0.7886 -0.6314 0.6702 -0.5208

Modified HM calibration (η = 0.4, ue benefits 95 percent of after-tax wage)

τn 0.5929 0.0141 0.9385 0.3569 0.8450 -0.2484

gdp 0.4714 0.0060 0.8495 1.0000 0.6825 0.4735

c 0.3843 0.0031 0.8074 0.5700 0.7829 -0.4018

N 0.4711 0.0050 0.9715 0.2274 -0.5064 0.6149

w 0.9610 0.0191 0.9530 0.4357 0.8987 -0.2198

θ 0.0425 0.0010 0.8579 -0.1250 -0.5414 0.6274

v 0.0225 0.0005 0.7874 -0.2521 -0.4209 0.4741

u 0.5289 0.0050 0.9715 -0.2274 0.5064 -0.6149

Table 1: Baseline model results at the Shimer calibration, the Hagedorn and Manovskii (HM) calibration,

and a modified HM calibration.

17

itself gives rise to a time-varying “wedge” in the labor market that tax-rate volatility is designed

to offset. We defer a more complete discussion of this idea to Section 6 because it will be easier to

understand this channel once we have presented all of our models and results.

By now, it is well-known that under the Shimer parameterization, the standard labor search

model does not deliver the large degree of volatility in unemployment observed in the U.S. data.

One resolution to this puzzle that has received a fair amount of attention is the HM calibration,

which contends that for studying aggregate cyclical dynamics, one needs to model the flow gain from

employment as being much smaller than a typical calibration assumes. With the HM calibration,

the standard search and matching model generates dynamics that are consistent with the cyclical

behavior of unemployment and vacancies. In light of this, we think it reasonable to test the

robustness of our main result to the HM calibration. Another motivation to quantitatively assess

our model with the HM calibration is the analytics presented in Appendix C, which suggests that

the sensitivity of the labor market to tax rates becomes large as the flow gain from employment

becomes small.

The middle panel of Table 1 shows that even at the HM calibration of very low worker bargaining

power (η = 0.052) and a 95-percent replacement rate, we still find substantial tax rate volatility

— a standard deviation of 1.6 percent, while lower than under the Shimer calibration, is still an

order of magnitude larger than Chari and Kehoe (1999). The HM calibration assigns a larger

allocative role to the (after-tax) real wage because the flow gain of employment is relatively small

at the margin, in the spirit of an RBC model. This larger allocative role of the after-tax real wage

is reflected in the drop in tax rate volatility to 1.6 percent from the 7 percent under the Shimer

calibration, but we would still not call this “tax smoothing” in the sense understood in the Ramsey

literature.

The HM calibration differs from the Shimer calibration in two respects: it features both a

higher replacement rate and a lower bargaining power for workers. In the lower panel of Table 1,

we keep η fixed at the Shimer value but use HM’s 95-percent replacement rate. At 1.4 percent,

tax-rate volatility under this modified HM calibration is quite similar to that under the exact HM

calibration. Quantitatively, then, optimal tax-rate variability is much more sensitive to the flow

gain from employment than to η. To keep the number of models and cases we analyze further below

manageable, we fix attention from here on to the parameterization η = ξu = 0.40. Thus, when we

speak of the “HM calibration” from here on, really what we refer to is a modified calibration that

retains HM’s 95-percent replacement rate but fixes η = ξu.

In summary, then, we find it makes no difference for our main qualitative results whether one

employs the Shimer calibration or the HM (or, at least, HM-style) calibration, although it does

matter quantitatively. Hence, we have our main result — tax smoothing is not an important

18

optimal policy prescription in the presence of search and bargaining frictions in the labor market

as typically modeled.

4.3 Recovering Tax Smoothing in the Baseline Model

A conclusion of the preceding analysis is that the most important parameter governing the degree

of tax smoothing is unemployment benefits — or, equivalently, the flow gain from employment. We

thus present in Table 2 Ramsey dynamics for several other replacement rates, holding, as justified

at the close of the previous section, η fixed at 0.4. Reading down Table 2, the average labor tax

rate rises sharply as unemployment benefits rise, while tax-rate volatility declines.

The rise in the average tax rate reflects two forces: a sharp fall in the labor tax base and the

larger unemployment benefits that the government must finance. First, note that the employment

stock n is only about 60 percent as large at a 95-percent replacement rate as it is at a 50-percent

replacement rate. That ∂n/∂χ < 0 is quite intuitive: the larger is the flow benefit from unem-

ployment, the less incentive to work. More precisely, because our model (so far) lacks a labor

force participation margin, the mechanism that leads to ∂n/∂χ < 0 is that the Nash-bargained

wage rises close to the marginal product of labor as χ rises, which diminishes firms’ incentives to

create vacancies. The resulting lower job-finding rate for individuals, kh(θ), leads to fewer jobs

in equilibrium. For a given level of government spending, the Ramsey government must impose a

higher τn as the tax base shrinks. Recall also from (9) that we have assumed that the government

bankrolls unemployment benefits. Larger (exogenous) χ, ceteris paribus, also requires a higher

average tax rate; coupled with lower (endogenous) n, the average tax rate unambiguously rises as

unemployment benefits become more generous.

Costain and Reiter (2007) have recently argued that “solutions,” such as the HM calibration,

to Shimer’s (2005) volatility puzzle lead to quite unrealistic predictions regarding policy. One

could view the implausibly high average tax rates our Ramsey model predicts as replacement rates

become very large as being subject to their critique. We do not view this as a damning criticism

of our results for two reasons. First, our work is more a theoretical exploration — what sorts

of labor-market models and mechanisms are important for the cornerstone Ramsey result of tax-

smoothing — rather than a data-matching exercise. Second, as Rogerson, Visschers, and Wright

(2007) counter, one could introduce other realistic features, such as home production, to circumvent

the Costain and Reiter (2007) critique. The exact quantitative effects of introducing such a feature

or features into our model is an open question that we leave for future research.

As unemployment benefits rise, the decline in optimal labor tax rate volatility our model predicts

reflects the fact that the range of wage-bargaining outcomes acceptable to both firms and workers

shrinks because the upper end of the bargaining interval (the marginal product of labor) is fixed

19


ue benefits 50 percent of after-tax wage

τn 0.1853 0.0482 0.9645 0.8048 0.7493 -0.2427

gdp 0.8288 0.0147 0.9306 1.0000 0.9952 -0.0753

c 0.6662 0.0071 0.9497 0.9298 0.9096 -0.3870

N 0.8281 0.0029 0.9546 0.9149 0.8720 -0.1701

w 0.8782 0.0161 0.9719 0.9593 0.9311 -0.2156

θ 0.7083 0.0128 0.9002 0.9855 0.9871 -0.1845

v 0.1217 0.0010 0.4030 0.2553 0.3403 -0.0529

u 0.1719 0.0029 0.9546 -0.9149 -0.8720 0.1701


τn 0.2551 0.0344 0.9747 0.8564 0.8097 -0.2406

gdp 0.7761 0.0139 0.9316 1.0000 0.9952 -0.0648

c 0.6366 0.0070 0.9484 0.9239 0.9082 -0.3915

N 0.7755 0.0028 0.9534 0.9211 0.8791 -0.1148

w 0.9026 0.0171 0.9747 0.9502 0.9217 -0.2109

θ 0.4068 0.0062 0.8474 0.9658 0.9830 -0.1194

v 0.0913 0.0008 0.4277 0.3441 0.4294 -0.0428

u 0.2245 0.0028 0.9534 -0.9211 -0.8791 0.1148


τn 0.4436 0.0213 0.9684 0.7655 0.8486 -0.2479

gdp 0.6119 0.0092 0.9021 1.0000 0.9727 0.0899

c 0.5094 0.0051 0.9166 0.8366 0.8988 -0.4139

N 0.6115 0.0020 0.9366 0.2467 0.0382 0.6593

w 0.9427 0.0189 0.9726 0.8378 0.9102 -0.2171

θ 0.1098 0.0012 0.6541 0.1373 0.0276 0.6016

v 0.0427 0.0004 0.4514 0.0233 0.0125 0.3306

u 0.3885 0.0020 0.9366 -0.2467 -0.0382 -0.6593

ue benefits 95 percent of after-tax wage (modified HM calibration)

τn 0.5929 0.0141 0.9385 0.3569 0.8450 -0.2484

gdp 0.4714 0.0060 0.8495 1.0000 0.6825 0.4735

c 0.3843 0.0031 0.8074 0.5700 0.7829 -0.4018

N 0.4711 0.0050 0.9715 0.2274 -0.5064 0.6149

w 0.9610 0.0191 0.9530 0.4357 0.8987 -0.2198

θ 0.0425 0.0010 0.8579 -0.1250 -0.5414 0.6274

v 0.0225 0.0005 0.7874 -0.2521 -0.4209 0.4741

u 0.5289 0.0050 0.9715 -0.2274 0.5064 -0.6149


τn 0.9308 0.0029 0.1000 -0.3384 0.5232 -0.2075

gdp 0.1409 0.0052 0.8465 1.0000 0.0027 0.8979

c 0.0694 0.0013 -0.1621 0.1939 0.1373 -0.2415

N 0.1409 0.0055 0.8783 0.9173 -0.3548 0.8432

w 0.9878 0.0246 0.2465 -0.3086 0.6188 -0.2040

θ 0.0025 0.0004 -0.0417 -0.0341 -0.1710 0.3929

v 0.0022 0.0003 -0.0656 -0.0750 -0.1550 0.3550

u 0.8591 0.0055 0.8783 -0.9173 0.3548 -0.8432

Table 2: Baseline model results at various replacement rates.

20

by construction while the lower end of the bargaining interval (unemployment benefits) rises by

assumption. In a rational equilibrium (which is guaranteed by Nash bargaining because it always

picks an outcome inside the bargaining set), the wage outcome must become less volatile as the

bargaining set shrinks. Table 2 shows that the pre-tax wage w does not become less volatile

as χ rises, seemingly in contradiction with this intuition. What is relevant in our environment,

though, is the after-tax wage rate, (1−τnt )wt. We have confirmed that the after-tax wage rate does

indeed become smoother as χ rises, and, comparing with our results in Table 2, virtually all of this

smoothing stems from the decline in τn volatility.

Quantitatively, the result that really stands out in Table 2 is that as we drive unemployment

benefits incredibly high, to a 99-percent replacement rate, we essentially recover tax smoothing.

Granted, the mean tax rate rises to a stunning 93 percent — but, as just described, the high average

tax rate directly reflects the rise in χ and the associated fall in n.

Conceptually, what is quite interesting about the very low tax volatility at a 99-percent replace-

ment rate is that it arises in an environment that is nearly Walrasian in the sense that individuals

are very nearly indifferent between work and non-work, even moreso than in HM. In a Walrasian

labor market, individuals are of course exactly indifferent between work and nonwork in equilib-

rium. Thus, an extremely high replacement rate — even more extreme than HM — can be viewed

as a partial proxy for a competitive labor market. In Section 5, we pursue further this idea of

making our model “more Walrasian” by reducing the severity of search and bargaining frictions

through a variety of mechanisms. The most important of these model variations in terms of its

implications for tax smoothing is the one in which competitive search replaces Nash bargaining as

the economy’s wage-determination mechanism.

5 Alternative Views of the Labor Market

We just showed that as we almost completely close the gap between the after-tax flow value of

work and non-work (which, we emphasize, requires an even more extreme view than HM), mak-

ing households virtually indifferent to employment versus unemployment, an essentially standard

tax-smoothing result emerges. Absent such a calibration, however, the model predicts that tax

smoothing is not a critical feature of optimal policy. In this section, we explore a number of al-

ternative mechanisms, each designed to reduce the severity of search and/or bargaining frictions,

thereby moving our environment closer to a frictionless labor market. Our goal is to evaluate each

of these mechanisms, both individually and together, to obtain a sense of how far one has to depart

from the standard DSGE labor search model for tax smoothing to (re-)emerge as an optimal policy.

21


state realized

nt-1 nt

Bargaining occurs (i.e., asset values

defined here)



Employment separation occurs (ρxnt-1 employees

separate)


nt = (1-ρx)nt-1 + m(ut, vt)yields

Figure 2: Timing of events in instantaneous-hiring model.

5.1 Instantaneous Hiring (Model 2)

In our baseline model, matches made in period t do not begin producing until period t + 1. This

timing has become standard in DSGE labor-search models. In a frictionless labor market, though,

a “match” produces immediately.11 This latter feature of frictionless markets can be captured in

our model simply by reorganizing the timing of the law of motion of employment, as Blanchard and

Gali (2006) and Krause, Lopez-Salido, and Lubik (2007) have also done. We label this environment

Model 2.

5.1.1 Modifications of the Model

Suppose that aggregate employment now evolves according to

nt = (1− ρx)nt−1 +m(ut, vt), (16)

which should be compared with (10). New matches formed in period t produce in period t; the

new timing of events is depicted in Figure 2.

The new timing of events has no impact on household optimization. It does, however, modify

the firm problem. The representative firm’s problem now includes the perceived law of motion

nft = (1− ρx)nft−1 + vtkf (θt), which replaces (3). This alters the job-creation condition to

γ

kf (θt)= zt − wt + (1− ρx)Et

Ξt+1|t

γ

kf (θt+1)

, (17)

which obviously replaces (4) as an equilibrium condition. The flow benefit to a firm of a new match

zt − wt is now contemporaneously realized.11Again, we use this terminology loosely because there is no formal notion of bilateral exchange in a Walrasian

market.

22

Because the new timing changes the flow benefit of a match to a firm, the wage-bargaining

outcome is affected. As in the baseline model, we define asset values at the time that bargaining

occurs, which is just before period-t production — compare Figures 1 and 2. Holdup problems are

still present in this environment because it is still the case that bargaining determines wages after

matching has taken place. The fact that both events (matching and bargaining) now occur in the

same time period, rather than in different periods as in the baseline model, is inconsequential for

the presence of the holdup problem. Left to assess, of course, is whether the holdup problem is less

severe.

We provide details in Appendix B, but the baseline model’s wage equation (8) is replaced by

wt =χ

1− τnt+

η

1− η

[γ

kf (θt)− 1

1− τntEt

[Ξt+1|t(1− ρx)(1− kh(θt+1))(1− τnt+1)

γ

kf (θt+1)

]]. (18)

The wage outcome (18) differs from (8) in that, because a match produces instantaneously, a

worker is paid the value of his outside option (accounting for the tax rate) plus some fraction of the

continuation value of the match. A portion of the continuation value accrues contemporaneously

(the first term in the square brackets) while there is some uncertainty — due to possible match

destruction — over the continuation value in period t+ 1 (the second term in the square brackets).

We point out that because both τnt and τnt+1 appear in (18), the dynamic bargaining power effect

channel of policy that we discussed above remains in force.

Summarizing, the change to the timing of match formation leads to (16) replacing (10), (17)

replacing (4), and (18) replacing (8), while all other equilibrium conditions are as in the baseline

model. The Ramsey problem thus modifies in the obvious way.

5.1.2 Optimal Taxation

To facilitate comparison with the baseline model, we hold all parameter values fixed at their baseline

values. Table 3 displays results for the modified model comparable to those in Tables 1 and 2 for

the baseline model. Comparing the first rows of each panel of Tables 1, 2, and 3, it is obvious that

tax rates are optimally smoother over time when new matches become productive in the period

in which they are formed. Except at the 99-percent replacement rate, tax rates are one-half to

three-fourths as volatile with the altered timing of labor flows than in the baseline model. Under

the Shimer calibration and the HM calibration, the tax rate is, respectively, 86 percent and 49

percent as volatile with the altered timing of labor flows compared to the baseline model.

The instantaneity of match output thus by itself goes a way, but not all the way, towards

restoring the classic tax-smoothing result. In standard Walrasian labor markets, employment “re-

lationships” obviously become productive in the period they are “formed.” Viewed in this light,

the modified timing assumption reduces, but does not eliminate, search frictions, and it takes us

23


ue benefits 40 percent of after-tax wage (Shimer calibration)

τn 0.1527 0.0603 -0.8999 0.1090 0.1280 -0.0156

gdp 0.8656 0.0138 0.9069 1.0000 0.9993 -0.0508

c 0.6976 0.0070 0.9171 0.9240 0.9234 -0.3828

N 0.8649 0.0013 0.9526 0.8198 0.8008 -0.0463

w 0.8774 0.0077 0.7285 0.9042 0.9123 -0.0571

θ 0.9551 0.0092 0.7933 0.9492 0.9429 -0.0507

v 0.1290 0.0004 0.2523 0.1906 0.2097 -0.0094

u 0.1351 0.0012 0.9511 -0.9292 -0.9153 0.0520


τn 0.2136 0.0100 0.8646 0.7060 0.7150 -0.1172

gdp 0.8191 0.0133 0.9089 1.0000 0.9992 -0.0464

c 0.6756 0.0069 0.9191 0.9193 0.9208 -0.3863

N 0.8185 0.0015 0.9104 0.8302 0.8148 -0.0089

w 0.9032 0.0075 0.9003 0.9457 0.9501 -0.0622

θ 0.5321 0.0055 0.7997 0.8560 0.8496 -0.0072

v 0.0966 0.0005 0.3161 0.2561 0.2625 -0.0042

u 0.1815 0.0015 0.9678 -0.9468 -0.9337 0.0126


τn 0.4681 0.0077 0.9312 0.4794 0.9059 -0.2398

gdp 0.5920 0.0061 0.7947 1.0000 0.7442 0.4688

c 0.4972 0.0033 0.7843 0.5767 0.8480 -0.3926

N 0.5917 0.0055 0.9749 -0.0217 -0.6309 0.5158

w 0.9547 0.0096 0.9420 0.5904 0.9579 -0.1504

θ 0.0803 0.0021 0.8836 -0.32719 -0.7148 0.5592

v 0.0328 0.0005 0.7918 -0.3407 -0.5356 0.4060

u 0.4083 0.0055 0.9749 0.0977 0.7014 -0.5641

ue benefits 95 percent of after-tax wage (modified HM calibration)

τn 0.5996 0.0069 0.7546 -0.3759 0.7820 -0.2369

gdp 0.4651 0.0062 0.8145 1.0000 0.0084 0.8267

c 0.3814 0.0020 0.4536 0.4053 0.2558 -0.1253

N 0.4646 0.0095 0.9396 0.6685 -0.6245 0.4905

w 0.9677 0.0114 0.8342 -0.2668 0.8755 -0.1793

θ 0.0344 0.0020 0.7802 0.1883 -0.5448 0.4130

v 0.0184 0.0010 0.7460 0.0048 -0.3828 0.2836

u 0.5351 0.0091 0.9427 -0.6390 0.7269 -0.5624


τn 0.9307 0.0041 -0.7100 -0.6093 0.1550 -0.0614

gdp 0.1409 0.0065 0.6462 1.0000 -0.0551 0.8083

c 0.0694 0.0010 -0.2438 0.5801 -0.0167 0.0268

N 0.1401 0.0099 0.0102 -0.0493 -0.2167 0.4863

w 0.9875 0.0330 -0.5459 -0.5968 0.2466 -0.0592

θ 0.0022 0.0014 -0.8088 0.6304 -0.0429 0.0946

v 0.0019 0.0011 -0.8124 0.6228 -0.0401 0.0883

u 0.8592 0.0069 0.6751 -0.9508 0.3370 -0.7650

Table 3: Model 2 results at various replacement rates.

24

at least part-way towards recovering tax smoothing. Of course, labor markets are still frictional —

even though new matches produce immediately, search is still fundamentally a costly activity.

“How quickly” newly-formed employment relationships become productive is an empirical ques-

tion, on which our results obviously do not shed any light. If one’s calibration is a yearly frequency,

instantaneous production seems a palatable assumption — in very few cases would such a long de-

lay between match formation and match output occur. If one’s calibration is a monthly frequency,

instantaneous production may be a lot less realistic — inherent delays in moving, setting up office

space, etc., may often preclude match formation and the start of match output from occurring

within a few weeks of each other. Macro models are typically calibrated as quarterly, as we have

done — at this frequency, intuition seems less helpful in guiding choice of timing. Regardless,

though, we think it important to know that such timing issues may leave very different impres-

sions about optimal policy prescriptions. We next reduce the severity of our model’s search and

bargaining frictions along a different dimension.

5.2 Endogenous Labor Force Participation

In Walrasian labor markets, the optimal labor supply condition obviously plays a critical role in

achieving efficiency in the labor market. In the models we have considered so far, all (a measure one)

household family members were assumed to be in the labor force. We now allow the household

to choose how many of its members are in the labor force. Including a participation margin is

yet another dimension along which we can move our model towards a standard Walrasian model.

Of course, search frictions still imply that participation is distinct from employment — no such

distinction exists in a Walrasian labor market. Introducing a labor-force-participation margin is also

a step towards the competitive search model we consider in Section 5.3, in which we drop bargaining

as the wage-determination mechanism. We allow here for choice over labor force participation using

both our baseline timing assumption and the instantaneous hiring assumption.

5.2.1 Baseline Timing Assumption (Model 3)

The modifications to the baseline model are straightforward. The representative household maxi-

mizes

E0

∞∑t=0

βt[u(ct) + g(1− uht − nht )

], (19)

where g(.) is instantaneous utility from the fraction of the household that enjoys leisure — that

is, instantaneous utility from the fraction of the household that is outside the labor force. The

sequence of flow budget constraints is just as before,

ct + bt = (1− τnt )wtnht + uht χ+Rtbt−1 + dt, (20)

25

and now the household also faces a sequence of perceived laws of motion for its employment level,

nht+1 = (1− ρx)(nht + uht kh(θt)). (21)

Note we have reverted to the timing assumption of the baseline model. We label this environment

Model 3.

The household’s consumption-savings optimality condition (7) is unchanged from the baseline

model. Household optimization with respect to uht and nht+1 yields the household’s labor-force

participation condition

g′(1− uht − nht )− u′(ct)χkh(θt)

= β(1− ρx)Et

[u′(ct+1)(1− τnt+1)wt+1 − g′(1− uht+1 − n

ht+1) +

g′(1− uht+1 − nht+1)− u′(ct+1)χ

kh(θt+1)

],

(22)

in which we have used the fact that, under optimal choices, u′(ct) equals the marginal value to

the household of a unit of wealth. The optimal participation decision has a straightforward in-

terpretation: at the optimum, the household sends a number uht of family members to search for

jobs until the expected cost of searching — the left-hand-side of (22) — is equated to the expected

benefit of search — the right-hand-side of (22). The expected cost of search is measured by the

marginal utility of leisure (each unit of search involves forgoing one unit of leisure) net of the direct

unemployment benefits obtained by search (converted into appropriate units using the marginal

utility of wealth). The expected benefit of search involves the marginal utility value of expected

future after-tax wage income (recall the one period lag between match formation and production)

and the future marginal disutility of work, along with the asset value to the household of having

an additional family member engaged in an ongoing employment relationship. The latter reflects

the value to the household of sending one fewer family member out to look for a job in the future.

Aside from household optimization, the rest of the model is unchanged compared to the base-

line model without a labor-force participation margin. Compared to the definition of private-sector

equilibrium in the baseline model, condition (22) replaces condition (11). Other than this replace-

ment of an exogenous size of the labor force with an endogenous size of the labor force, the definition

of private-sector equilibrium is exactly as in the baseline model. For the Ramsey problem, we now

write the PVIC slightly more precisely as

E0

∞∑t=0

βtu′(ct) [ct − (zt − τnt wt)nt − utχ+ γvt]

= A0. (23)

Compared to (13), utχ replaces (1 − nt)χ because nt + ut 6= 1 due to endogenous labor-force

participation. Besides these straightforward modifications, the Ramsey problem is identical to that

in our baseline model.

26

5.2.2 Instantaneous Hiring (Model 4)

We are also interested in how the combination of endogenous labor force participation and in-

stantaneous hiring influences the optimal policy prescription. The reason for the interest in this

combination of models is that both endogenous labor force participation and instantaneous pro-

duction are features of Walrasian markets. We label this environment Model 4. For the sake of

brevity, we only point out that the labor-force participation decision is given by

g′(1− uht − nht )− u′(ct)χkh(θt)

= u′(ct)(1− τnt )wt − g′(1− uht − nht ) + β(1− ρx)Et

[g′(1− uht+1 − n

ht+1)− u′(ct+1)χ

kh(θt+1)

], (24)

in place of the participation condition (22). The most important difference between (24) and (22)

is that the contemporaneous after-tax wage affects the former, but the future after-tax wage affects

the latter. All other equilibrium conditions are as in Model 2.


We choose a fairly standard RBC-type specification for the subutility function g(x) = κ1+φx

1+φ,

and set φ = 0.7 and κ = 4; the latter delivers a labor-force participation rate of 76 percent at a

Shimer replacement rate of 40 percent. Because the model with labor force participation is quite

different than the model without it, we recalibrate in Model 3 the value of χ to deliver a given

replacement rate, and then hold these constant as we move to Model 4.12 All other parameter

values are held fixed at their baseline values to facilitate comparison.

Table 4 displays results using the standard timing (Model 3). Comparing the results with

those obtained in the baseline model (Tables 1 and 2), two features stand out. First, holding

constant the replacement rate, the average tax rate is higher in the model with endogenous labor

force participation. This is due to the smaller tax base — compare the mean value of n across

Tables 1, 2, and 4 — which requires a higher average tax rate to finance the same level of government

spending. Second, introducing endogenous labor force participation has a broadly similar effect on

tax-rate volatility (compared to the baseline model) as does the instantaneous hiring model with

an exogenous labor force (Model 2), although the tax rate is not uniformly more or less volatile in

either Model 2 or Model 3.

Table 5 displays results for the model with instantaneous hiring (Model 4). Comparing tax

rate volatility in Table 5 with that in Table 4, we see that, except at the extreme calibration of

a 99-percent replacement rate, instantaneous production does add something over and above the

endogenous labor force margin in its implications for labor tax smoothing. Except at the 99-percent

replacement rate, tax rates are uniformly smoother in Model 4 than in Model 3.12Recall that the replacement rate in our model is not simply χ, but rather χ

(1−τn)w.

27



τn 0.2787 0.0176 0.3352 0.2696 0.3591 -0.1530

gdp 0.3667 0.0081 0.9541 1.0000 0.9091 0.3370

c 0.2525 0.0035 0.9255 0.6318 0.8333 -0.4673

N 0.3663 0.0035 0.9865 0.6825 0.3435 0.8070

w 0.8685 0.0138 0.9547 0.8045 0.9582 -0.2288

θ 0.8587 0.0291 0.9370 0.7742 0.9324 -0.2872

v 0.0580 0.0011 0.8263 0.1631 -0.0093 0.7911

u 0.0677 0.0030 0.9292 -0.5231 -0.7160 0.5488


τn 0.3639 0.0123 0.9119 0.1948 0.8465 -0.2268

gdp 0.3382 0.0059 0.9130 1.0000 0.5597 0.7195

c 0.2381 0.0022 0.7290 0.2152 0.6794 -0.4916

N 0.3379 0.0053 0.9811 0.4729 -0.4133 0.7948

w 0.9024 0.0116 0.9322 0.4152 0.9771 -0.1683

θ 0.4071 0.0123 0.7859 0.4966 0.8760 -0.2037

v 0.0397 0.0014 0.7035 -0.0710 -0.4729 0.5868

u 0.0977 0.0062 0.7305 -0.2776 -0.6877 0.4311


τn 0.5447 0.0148 0.9375 -0.2544 0.8694 -0.3116

gdp 0.2676 0.0060 0.9052 1.0000 0.1465 0.8952

c 0.1836 0.0017 0.3325 0.0739 0.3326 -0.3657

N 0.2674 0.0069 0.9622 0.7536 -0.5009 0.7630

w 0.9427 0.0169 0.9571 -0.1327 0.9370 -0.2412

θ 0.1098 0.0021 0.4078 0.7885 0.3431 0.4506

v 0.0186 0.0013 0.3978 -0.0069 -0.3526 0.4253

u 0.1699 0.0140 0.3211 -0.1870 -0.3823 0.2561

ue benefits 95 percent of after-tax wage (HM calibration)

τn 0.6910 0.0112 0.8289 -0.3186 0.7949 -0.3273

gdp 0.2061 0.0061 0.9266 1.0000 0.0878 0.9144

c 0.1289 0.0015 0.2905 0.0709 0.2442 -0.3273

N 0.2060 0.0067 0.9612 0.8472 -0.4188 0.8179

w 0.9612 0.0200 0.8904 -0.2332 0.8724 -0.2761

θ 0.0425 0.0011 0.8417 0.8606 -0.1675 0.9096

v 0.0098 0.0008 0.3954 0.0648 -0.2823 0.4488

u 0.2313 0.0179 0.2492 -0.2111 -0.2527 0.1906


τn 0.8993 0.0042 0.4559 -0.3336 0.6308 -0.3095

gdp 0.1159 0.0061 0.9372 1.0000 0.0399 0.9353

c 0.0437 0.0012 0.0856 0.0604 0.1244 -0.2870

N 0.1158 0.0063 0.9495 0.9459 -0.2602 0.8968

w 0.9789 0.0238 0.5964 -0.2858 0.7296 -0.2842

θ 0.0097 0.0005 0.6960 0.6660 -0.1879 0.8775

v 0.0031 0.0005 0.1713 0.1092 -0.1534 0.4445

u 0.3163 0.0369 0.0617 -0.1518 -0.1178 0.1977

Table 4: Model 3 results for various replacement rates.

28

Intuitively, the inclusion of a participation margin does not explicitly reduce search frictions.

Instead, it gives households an additional margin along which it can optimally manage the friction.

In this sense, allowing for a labor supply decision allows the (after-tax) wage to play a greater

allocative role — because after-tax wages of course appear in the labor force participation condi-

tion — hence tax smoothing becomes more desirable than in the comparable models without the

participation margin. Nevertheless, at roughly 1 percent across model specifications and replace-

ment rates, we still would not characterize either endogenous labor supply or instantaneous match

production, or the combination of these two features, as features that restore the optimality of

tax smoothing — recall that the benchmark tax smoothing result of Chari and Kehoe (1999) and

confirmed by Werning (2007) is tax-rate variability of 0.1 percentage point. Although we have in

some sense reduced the search frictions with the modifications we have introduced so far, we have

not reduced the severity of bargaining-induced frictions. Caballero’s (2007, p. 119) position is that

bargaining-induced frictions are likely to be more important for business-cycle issues than search

frictions per se; by eliminating bargaining, the competitive search model we turn to next allows

one assessment of this contention.

29



τn 0.3593 0.0142 0.4425 -0.1605 0.4354 -0.1928

gdp 0.2546 0.0037 0.8400 1.0000 0.5051 0.7445

c 0.1554 0.0021 0.6860 -0.1410 0.5554 -0.6538

N 0.2543 0.0038 0.8354 0.3936 -0.3954 0.6926

w 0.8754 0.0093 0.7766 0.2825 0.8150 -0.1586

θ 0.9842 0.0096 0.8258 0.4828 0.9597 -0.0995

v 0.0386 0.0028 0.5589 0.1498 -0.1884 0.2730

u 0.0392 0.0031 0.5703 0.0772 -0.2966 0.2661


τn 0.4273 0.0088 0.7678 0.0042 0.6739 -0.3614

gdp 0.2374 0.0035 0.8999 1.0000 0.5044 0.7827

c 0.1473 0.0019 0.7537 -0.2088 0.5971 -0.7248

N 0.2372 0.0034 0.9312 0.4560 -0.4106 0.7784

w 0.9075 0.0086 0.8690 0.3186 0.9098 -0.1925

θ 0.4688 0.0048 0.7728 0.6611 0.7404 0.2398

v 0.0267 0.0012 0.5490 0.1391 -0.2849 0.4418

u 0.0569 0.0026 0.5342 -0.0108 -0.4660 0.3955


τn 0.5845 0.0099 0.7125 -0.4652 0.5920 -0.4637

gdp 0.1937 0.0046 0.9289 1.0000 0.1171 0.9351

c 0.1140 0.0013 0.5650 -0.2990 0.4463 -0.5830

N 0.1935 0.0051 0.9405 0.7775 -0.3854 0.7641

w 0.9447 0.0123 0.7906 -0.2692 0.7623 -0.3165

θ 0.1287 0.0070 0.8292 0.6389 -0.2847 0.7167

v 0.0130 0.0009 0.6841 0.2386 -0.2387 0.4244

u 0.1010 0.0040 0.5665 -0.4510 -0.0387 -0.2118


τn 0.7217 0.0109 0.4097 -0.5846 0.3762 -0.3396

gdp 0.1540 0.0060 0.8853 1.0000 -0.0723 0.9321

c 0.0787 0.0013 0.2398 0.2161 0.0533 -0.0644

N 0.1537 0.0071 0.8727 0.8207 -0.3337 0.7173

w 0.9613 0.0207 0.5106 -0.4954 0.4983 -0.2795

θ 0.0511 0.0078 0.7032 0.5158 -0.2366 0.5079

v 0.0072 0.0013 0.6664 0.1828 -0.1312 0.2657

u 0.1407 0.0087 0.6002 -0.7074 0.2117 -0.4892


τn 0.9354 0.0143 -0.0032 -0.7413 0.0516 -0.0483

gdp 0.0913 0.0080 0.5632 1.0000 -0.0470 0.6751

c 0.0193 0.0015 -0.1926 0.4586 -0.0030 0.0006

N 0.0905 0.0117 -0.0907 -0.1060 -0.1303 0.4251

w 0.9706 0.1205 -0.0844 -0.7202 0.0770 -0.0451

θ 0.0129 0.0152 -0.2760 0.7015 -0.0282 0.0868

v 0.0024 0.0023 -0.2532 0.6786 -0.0272 0.0822

u 0.1957 0.0276 -0.0961 -0.6169 0.0604 -0.1880

Table 5: Model 4 results for various replacement rates.

30

5.3 Competitive Search Equilibrium (Model 5)

The final step in our exploration is to push our frictional labor market as close as possible to

a Walrasian one by employing a competitive scheme, rather than Nash bargaining, to determine

wages. We use Moen’s (1997) concept of competitive search equilibrium, in which wages are deter-

mined by decentralized forces and taken as given by all labor market participants. A crucial aspect

of the idea of competitive search equilibrium is that wages direct search behavior, even though

matching is still random.13 Thus, the key conceptual step we are taking here is to eliminate the

problems inherent in ex-post Nash bargaining per se. As we will show, competitive search restores

a relatively-standard purely static wedge due to labor income taxation, in contrast to the dynamic

wedge due to labor income taxation inherent in Nash bargaining. Despite this, competitive search

equilibrium does not circumvent the primitive coordination frictions inherent in the environment;

indeed, market tightness is still a fundamentally-important allocative signal. A competitive view

of wage determination is not incompatible with a frictional, search-based, view of labor markets

5.3.1 Modifications of the Model

We follow the implementation of competitive search equilibrium presented in Arseneau and Chugh

(2008), who adapt Moen’s (1997) framework for a fully-specified DSGE environment. This imple-

mentation requires both endogenous labor-force participation and instantaneous match production.

Hence, Model 4 is a building block for the competitive search model we label as Model 5.

Proceeding straight to optimality conditions, the household labor-force participation condition

remains (24). However, it is useful to rearrange this expression a bit; dividing (24) through by u′(ct),

combining terms, and using our definition of the stochastic discount factor Ξt+1|t = βu′(ct+1)/u′(ct),

we have(1 + kh(θt)kh(θt)

)g′(1− uht − nht )

u′(ct)= (1−τnt )wt+

χ

kh(θt)+(1−ρx)Et

Ξt+1|t

g′(1−uht+1−nht+1)

u′(ct+1) − χkh(θt+1)

.

(25)

The job-creation condition remains that in Model 2 (condition (17)), which we rewrite in slightly

different form:

γ = kf (θt)(zt − wt + (1− ρx)Et

Ξt+1|t

γ

kf (θt+1)

). (26)

As we show in Arseneau and Chugh (2008), the competitive price and market tightness are

determined by maximizing the right-hand-side of (26) taking as constraint (25). The objects of13That is, search and matching can each be random or non-random. Each has been random in our models so far.

Here, search is not random — it is directed by market wages — but matching is. Also see Rogerson, Shimer, and

Wright (2005, p. 972-976) for more on competitive search equilibrium.

31

maximization are wt and θt.14 In symmetric equilibrium, the wage and market tightness are now

determined jointly by (25) and

kf′(θt) [zt − wt + (1− ρx)Γt] = −ϕtkh

′(θt)

[u′(ct)(1− τnt )wt − g′(1− ut − nt) + (1− ρx)Λt

], (27)

where ϕt ≡ kf (θt)kh(θt)u′(ct)(1−τnt )

, Γt = Et

Ξt+1|tγ

kf (θt+1)

, and Λt ≡ βEt

g′(1−ut+1−nt+1)−u′(ct+1)χ

kh(θt+1)

.

With regard to policy implications, the most important thing to note about (27) is that only τnt

appears. This is in contrast to the wage equations (8) and (18), in which both τnt and τnt+1 appeared.

Thus, we see right away that the dynamic bargaining effect channel of policy, which relies on both

current and future tax rates, is not present in competitive search equilibrium. More important,

however, is the fact that under competitive search, the dynamic bargaining effect is not needed

by the Ramsey government because private-sector search activity is endogenously efficient, up to a

static tax wedge; we describe this next.

5.3.2 Static Tax Wedge

We can see this last point much more clearly by simplifying the wage outcome (27) to cast it in a

form that shows that labor taxes create only static wedges under competitive search equilibrium,

in contrast to the dynamic wedges they create under Nash bargaining. Given the Cobb-Douglas

specification m(u, v) = ψuξuv1−ξu , we have kf (θ)/kh(θ) = θ−1, kh′(θ)/kf

′(θ) = −(1− ξu)θ/ξu, and

thus (kf (θ)/kh(θ))(kh′(θt)/kf

′(θt)) = −(1− ξu)/ξu. Using these, we can simplify (27) to

u′(ct)(1− τnt )[zt − wt + (1− ρx)Et

Ξt+1|t

γ

kf (θt+1)

](28)

=1− ξuξu

[u′(ct)(1− τnt )wt − g′(1− ut − nt) + (1− ρx)βEt

g′(1− ut+1 − nt+1)− u′(ct+1)χ

kh(θt+1)

].

Next, use the job-creation condition (17) to substitute for the term in square brackets on the

first line, and use the labor-force participation condition (24) to substitute for the term in square

brackets on the second line; doing so yields

u′(ct)(1− τnt )γ

kf (θt)=(

1− ξuξu

)(g′(1− ut − nt)− u′(ct)χ

kh(θt)

). (29)

Expression (29) defines wt — the wage is implicit in the time-t asset values of a match for firms

and households that appear on the left- and right-hand-sides, respectively.14We are abusing notation a bit by writing the problem this way. The precise formulation of the problem would

feature, on the right-hand-side of (26), kf (θit) and wit outside the expectations operator and kf (θt+1) inside the

expectations operator, where the index i denotes a particular (type of) firm’s wage-market tightness pair and non-

indexed wages and tightness denote aggregates. Also, the constraint (25) would feature kh(θit) and wit as far as

contemporaneous variables go. With this more precise formulation, the maximization is with respect to wit and θit.

See Arseneau and Chugh (2008) for more details and discussion of formulating a competitive search equilibrium in a

DSGE model; Moen (1997) is the original reference on the theory of competitive search equilibrium.

32

Again using the result kh(θ)/kf (θ) = θ, one more rearrangement gives

g′(1− ut − nt)− u′(ct)χu′(ct)

= (1− τnt )γθt(

ξu1− ξu

), (30)

which highlights the static nature of the tax wedge. We interpret (30) as follows: the left-hand-

side measures the marginal rate of substitution (MRS) between leisure and consumption, and the

right-hand-side measures the after-tax marginal rate of transformation (MRT) between search (a

unit of unemployment) and consumption. That the left-hand-side of (30) is an MRS is obvious. To

understand why the right-hand-side measures the corresponding MRT, consider the algebraic units

of each term. The units of γ is goods per vacancy posting, the units of θt (which, recall, ≡ vt/ut) is

vacancy postings per unit of unemployment, and ξu/(1 − ξu), because it is the ratio of elasticities

in the Cobb-Douglas matching function, is a unitless number. The latter term measures the social

(technological) contribution of a unit of unemployment to the aggregate production of jobs. Thus,

γθtξu/(1− ξu) measures how many goods are produced for each unit of search unemployment and

has the interpretation of the economy’s MRT between search and goods.15

To the best of our knowledge, this MRT interpretation is unique in the modern labor search

literature. Expression (30) shows that under competitive search equilibrium the labor tax throws

only a static wedge between the economy’s MRS and MRT between leisure and goods, just as in

a standard Walrasian model of labor markets. Given this purely static wedge, it seems likely that

tax smoothing may reemerge under this market arrangement; we now quantitatively assess whether

this hypothesis is true.


Table 6 presents Ramsey dynamics under competitive search equilibrium. The top panel shows

that, under the Shimer calibration, we virtually recover the classic Ramsey tax-smoothing result

— a standard deviation of τn of about 0.25 percentage points is obviously higher than Chari and

Kehoe (1999), but of course Model 5’s labor market is not completely frictionless. Interestingly,

as the replacement rate rises, tax-rate volatility rises, seemingly questioning whether competitive

search equilibrium delivers tax smoothing.

Our understanding of these two results — with competitive search, tax smoothing essentially

re-emerges at low replacement rates but not at high replacement rates — is as follows. To aid15Another way to see this is to first recognize that the marginal rate of technical substitution (MRTS) between

unemployment and vacancies (i.e., holding fixed the number of job matches) implied by the Cobb-Douglas matching

technology is θtξu/(1− ξu). Then, as is apparent from the resource frontier (12), the marginal rate of transformation

between vacancies and final goods is γ. Thus, the overall MRT between unemployment and final goods is γθtξu/(1−ξu).

33



τn 0.2646 0.0026 0.8911 -0.2085 0.1906 -0.8666

gdp 0.3748 0.0077 0.9411 1.0000 0.8973 0.4008

c 0.2636 0.0034 0.9107 0.6020 0.8682 -0.4397

N 0.3744 0.0033 0.9806 0.5522 0.1709 0.9014

w 0.8808 0.0079 0.9348 0.8647 0.9953 -0.0706

θ 0.8903 0.0238 0.9145 0.6792 0.9180 -0.3505

v 0.0542 0.0011 0.6351 0.1595 -0.1020 0.7456

u 0.0609 0.0024 0.8097 -0.3731 -0.6607 0.6044


τn 0.3424 0.0077 0.8936 0.0433 0.7765 -0.5342

gdp 0.3486 0.0054 0.9307 1.0000 0.6099 0.7538

c 0.2502 0.0021 0.7906 0.0955 0.7606 -0.5402

N 0.3483 0.0049 0.9695 0.3517 -0.4552 0.7868

w 0.9114 0.0071 0.9260 0.5557 0.9951 -0.0686

θ 0.4247 0.0092 0.8408 0.3520 0.9225 -0.3047

v 0.0375 0.0014 0.5344 0.0479 -0.4242 0.5161

u 0.0884 0.0048 0.6323 -0.1097 -0.6547 0.4708


τn 0.5097 0.0098 0.8942 -0.2382 0.7809 -0.5134

gdp 0.2829 0.0048 0.9417 1.0000 0.3248 0.9253

c 0.1992 0.0015 0.7011 -0.3649 0.5914 -0.6708

N 0.2827 0.0055 0.9643 0.5607 -0.5166 0.7479

w 0.9472 0.0095 0.8786 0.1780 0.9603 -0.1607

θ 0.1174 0.0011 0.7174 0.3171 0.4490 0.2344

v 0.0182 0.0009 0.5140 0.2106 -0.3873 0.4658

u 0.1553 0.0070 0.5032 0.1610 -0.5149 0.4560


τn 0.6492 0.0094 0.8589 -0.5016 0.6766 -0.6207

gdp 0.2243 0.0050 0.9527 1.0000 0.1818 0.9692

c 0.1468 0.0011 0.6269 -0.5245 0.4799 -0.7073

N 0.2241 0.0056 0.9651 0.7273 -0.4506 0.7938

w 0.9636 0.0104 0.8573 -0.0124 0.9285 -0.2073

θ 0.0468 0.0012 0.6158 0.4062 -0.2380 0.5291

v 0.0100 0.0006 0.5185 0.3264 -0.3265 0.4935

u 0.2139 0.0073 0.3750 0.2463 -0.3768 0.4377


τn 0.8573 0.0093 0.7856 -0.7803 0.4282 -0.8023

gdp 0.1342 0.0057 0.9546 1.0000 0.0347 0.9944

c 0.0618 0.0006 0.2751 -0.5739 0.2562 -0.6466

N 0.1341 0.0060 0.9552 0.8840 -0.3030 0.8768

w 0.9796 0.0114 0.7520 -0.2208 0.8375 -0.2605

θ 0.0113 0.0010 0.4747 0.4981 -0.2002 0.5331

v 0.0034 0.0004 0.3831 0.4330 -0.1920 0.4838

u 0.3010 0.0093 -0.0420 0.1758 -0.1335 0.2603

Table 6: Model 5 results for various replacement rates

34

intuition, suppose that ρx = 1, meaning that employment spells last only one period at a time, and

χ = 0. Then, the participation condition (25) simplifies to(1 + kh(θt)kh(θt)

)g′(1− uht − nht )

u′(ct)= (1− τnt )wt, (31)

which is the typical RBC (static) consumption-leisure optimality condition, modified to take ac-

count of the matching friction. At the optimium, the household sets its MRS between consumption

and leisure equal to a function of the after-tax wage rate and its perceived matching probabilities.

Because matching probabilities depend on only the matching technology, one could then view this

condition through the lens of a standard optimal-taxation model built on a standard RBC foun-

dation: the MRS is equated to a relevant (after-tax) function of (both production and matching)

technology. With competitive wage-setting also circumventing bargaining frictions, the Ramsey

planner ends up respecting essentially just a usual static consumption-leisure efficiency condition.

To avoid inducing fluctuations in the wedge between MRS and MRT over time — for the usual

intertemporal deadweight-loss smoothing arguments — the labor tax rate must optimally be quite

stable over time. Relaxing ρx = 1 makes this channel a bit less quantitatively important than in a

simple Ramsey model, but our results show by not very much.

Instead, allowing χ > 0 (given ρx = 1) weakens the link between the MRS and (the combined

effects of both the production and matching) technology because it introduces a constant factor as

part of the relevant price. As χ rises, this “fixed-factor in prices” means the importance of tax-rate

smoothing falls, which explains why with competitive search equilibrium, tax rate volatility rises

as the replacement rate rises.16

6 Summary and Discussion

We have presented an array of models to re-examine the issue of tax smoothing. Our models

conceptually span the spectrum between DSGE labor search models as typically formulated and

standard Walrasian labor markets. Table 7 summarizes the important features of the five models

we have studied. We summarize our main findings regarding optimal tax rates in Table 8. While all

of the mechanisms we consider push optimal policy in the direction of tax smoothing, only under

the competitive search equilibrium (with a sufficiently low setting for χ) does the standard tax

smoothing result, as understood in the basic Ramsey literature, arise.

16We have verified that this type of effect also arises in a simple Ramsey model with Walrasian labor markets.

Allowing for an “unemployment benefit” χ (financed by the government, just as we have assumed in our model)

in the household budget constraint of the simple RBC model makes the household’s consumption-leisure optimality

condition g′(.)/u′(.) = (1 − τn)w + χ. Typical Ramsey models of course have χ = 0. Solving and simulating the

Ramsey problem, we find that tax-rate volatility is indeed increasing in χ.

35

Model Timing Labor-force participation Wage determination

1 Standard No Nash bargaining

2 Instantaneous No Nash bargaining

3 Standard Yes Nash bargaining

4 Instantaneous Yes Nash bargaining

5 Instantaneous Yes Competitive search

Table 7: Features of the five models.

Model Mean Std. Dev.

40-percent replacement rate

1 0.1625 0.0701

2 0.1527 0.0603

3 0.2787 0.0176

4 0.3593 0.0142

5 0.2646 0.0026

95-percent replacement rate

1 0.5929 0.0141

2 0.5996 0.0069

3 0.6910 0.0112

4 0.7217 0.0109

5 0.6492 0.0094

Table 8: Dynamics of Ramsey tax rate across models. Unemployment benefit reported as percentage of

steady-state after-tax wage.

36


state realized

nt-1 nt

Wages posted



Employment separation occurs (ρxnt-1 employees

separate)


nt = (1-ρx)nt-1 + m(ut, vt)yields

Figure 3: Timing of events in competitive search model.

Figure 3 shows that what is critically different about the competitive search economy compared

with all our bargaining environments (compare with Figures 1 and 2) is that wage-determination

happens before, rather than after, search and matching. Coupled with the directed nature of search,

posted wages get around ex-post bargaining frictions by generating ex-ante competition amongst

wage-setting firms. Thus, private-sector search incentives are (absent taxation) already efficient;

the Ramsey government’s role is then simply to minimize the intertemporal distortions stemming

from the purely static labor tax wedge, and this requires, just as in a standard Walrasian market,

smoothing labor tax rates over time.

There are a number of (ultimately related) ways of thinking intuitively about our results. As

discussed by Mortensen and Pissarides (1999, p. 2591), in competitive search equilibrium the private

sector prices market-tightness (our variable θt) efficiently. Efficiency dictates that parties should

arrange wage terms before undertaking their search activities, and competitive search achieves

exactly this by pricing tightness efficiently. In contrast, market tightness is generically mispriced

in bargaining equilibria because it is impossible to arrange wage terms before engaging in search

— each party simply has no idea who it will meet and thus with whom it needs to arrange wage

terms. As we have been alluding to, this is a problem of incomplete markets — in the language of

Mortensen and Pissarides (1999, p. 2589), the missing market is the market for “matching delay.”

Boone and Bovenberg (2002) study optimal taxation in a one-shot variant of the Pissarides

search model and offer the interpretation (p. 64) that tax policy effectively partially creates this

missing market. The extension of their interpretation to our fully dynamic stochastic environment

is that because period-t tax rates are known to the private sector before period-t search activity

occurs, tax rates help direct search activity efficiently over the business cycle. That is, tax-rate

volatility acts as a partial substitute for the market structure of competitive search equilibrium. In

37

terms of the timelines of events described in Figures 1, 2, and 3, tax-rates are announced (“posted”)

by the government before search and matching in all of the environments we consider. This is what

we mean that taxes (and thus tax volatility) acts as a partial substitute for ex-ante posted wages.

Caballero (2007) stresses the holdup problems that incomplete-contracting problems engen-

der. In search environments, parties must unilaterally undertake search decisions — which can be

thought of as investment decisions — but then share the fruits of their search activity with others

in bilateral trades. With no pre-committed terms of trade in meetings, ex-ante search activity

is distorted — hence, bargaining environments naturally lead to holdup problems. Thus, we can

also interpret tax-rate volatility in our bargaining models as a way of combatting cyclical holdup

problems.

Regardless of one’s preferred interpretation, in language familiar to DSGE modelers, we can

think of ex-post bargaining as creating, in and of itself, a wedge between MRS and MRT. In the

competitive search environment, recall, we are able to express the equilibrium wage outcome as

MRSt = (1− τnt )MRTt, (32)

where MRS and MRT are defined in the discussion following expression (30). We found that tax

smoothing is optimal in this environment. In our bargaining models, however, we cannot reduce

the equilibrium wage outcome to such a simple expression.17 Instead, equilibrium in the ex-post

bargaining environments can be represented as

MRSt = (1− τnt )ΩtMRTt. (33)

We think of Ωt as a time-varying “wedge” that emanates from the primitive bargaining friction.

The endogenous wedge Ωt is a complicated object depending on period-t allocations and period-

t expectations of future events; as such, it is not completely independent of τnt in the Ramsey

equilibrium.

Nevertheless, fluctuations in Ωt are not due solely to fluctuations in policy. We have confirmed

this by solving and simulating an exogenous-policy (i.e., non-Ramsey) version of Model 4 in which

we hold the labor tax rate constant. Simulations of this exogenous-policy variant of Model 4 show

that Ωt does fluctuate in the face of either TFP shocks or government purchase shocks — Ωt is

countercyclical with respect to TFP shocks and procyclical with respect to government purchase

shocks. Thus, bargaining frictions in and of themselves cause fluctuations in this wedge. This seems

to be the crux of Caballero’s (2007, p. 119) contention — or, at least, our interpretation of it —

that bargaining-induced inefficiencies in and of themselves likely induce cyclical dynamics.17For Model 4, which is identical to Model 5 except for the wage-determination mechanism, detailed derivations

showing that it is not possible to reduce things to an expression as simple as something akin to (32) are available

from the authors on request.

38

In the Ramsey equilibria of our bargaining models, cyclical movements of τnt seem designed

to partially offset movements in Ωt. For example, in our simulated Ramsey equilibria of Model 4

(we use Model 4 to make our point because it is the closest to Model 5, our competitive search

model), the median correlation between τnt and Ωt is -0.74, which suggests that the goal of the

Ramsey policy is to partially stabilize the total wedge (1 − τnt )Ωt. In this sense, tax volatility

in our bargaining models can be thought of as promoting wedge-smoothing. Such a distinction

between wedges and taxes of course does not arise in simple Ramsey models because labor-market

wedges are almost always due only to labor taxes — wedge-smoothing thus immediately calls for

tax-smoothing in simple Ramsey models. The competitive search economy endogenously achieves

Ωt = 1 ∀t, which restores the equivalence between wedge-smoothing and tax-smoothing.

In closing, we note that the notion of competitive search equilibrium is a quite substantial

departure from the recent literature’s typical DSGE formulations of labor search models. But it

is a well-known and often-exploited equilibrium concept in labor-market theory. Which market

structure — bargaining or competitive search — is the best description of labor markets and which

construct turns out to be the most important for business-cycle matters is an issue that of course our

analysis cannot answer. In the end, our simple point is that provided one is willing to accept that

search and bargaining frictions may be important in labor markets, tax smoothing as a prescriptive

recommendation may not be as important as standard Ramsey theory suggests.

7 Conclusion

Our aim was to re-examine the conventional wisdom regarding the optimality of labor-tax smooth-

ing through the lens of the modern frictional view of labor markets. Using a standard DSGE

labor search model, we showed that tax smoothing is not an important goal for fiscal policy in

the presence of labor search frictions as typically modeled. Indeed, purposeful volatility in labor

income tax rates serves to ease bargaining-induced problems associated with job-creation over the

business cycle. Our result is robust to the two most commonly-used calibrations of the standard

Pissarides (2000) model, the Shimer (2005) calibration and the Hagedorn and Manovksii (2007)

calibration. We then showed that introducing alternative mechanisms into the model that, in one

way or another, push the search model closer to its frictionless counterpart makes tax smoothing

more desirable. However, only in the competitive search economy, in which bargaining problems

are by construction absent, are we able to recapture tax smoothing in the familiar Ramsey sense.

From this, we conclude that whether or not tax smoothing is as important as standard Ramsey

theory suggests critically depends on the wage-formation process. If one believes that ex-post bar-

gaining is the most natural description of wage determination, then tax smoothing as a prescriptive

recommendation may not be as important as policy theorists typically think.

39

There are several extensions that may be interesting to pursue. From the point of view of pure

Ramsey theory, a natural question is to ask what sorts of policy tools other than a labor tax might

ease the holdup problems that lead to our results. One such instrument is a time-varying hiring

subsidy; we take a step towards analyzing this type of policy intervention in Appendix E, but

there are some technical issues about implementing this in a full Ramsey framework. We did not

purse this avenue extensively because our main focus was on how various market structures affect

insights regarding the use of standard policy tools, but further work along these latter lines would

be interesting.

On the theme of market structure, it would be instructive to examine how our findings change

in the face of various bargaining models. Ex-post bargaining implies inefficiencies, and Nash bar-

gaining, which is the only bargaining model we considered, allows the government a channel —

the dynamic bargaining power effect — through which to manage the problem. Other bargaining

protocols, which would presumably still imply inefficiencies, may not feature a dynamic effect of

taxes on equilibrium. Tax-smoothing may arise in such bargaining models, but for a very different

reason than why tax-smoothing arises in the competitive search economy. A number of alternative

bargaining models have begun to appear in the labor literature — for example, Hall and Milgrom’s

(2007) credible-bargaining model and Acemoglu and Hawkins’s (2006) Shapley-value-based multi-

lateral bargaining model. None of these immediately suggest themselves as clearly lacking Nash

bargaining’s dynamic bargaining power effect, but it could be interesting to pursue this idea further.

More broadly, our results here continue to add to the growing sense — also developed in Ar-

seneau and Chugh (2006, 2007), Aruoba and Chugh (2006), and Faia (2007) — that search-based

models may yield novel predictions in answer to some standard optimal-policy questions. Some-

thing that has seemingly escaped notice in the rapidly-growing literature applying labor-search

models to standard macroeconomic policy issues is that a competitive view of wage determination

is not incompatible with a frictional, search-based, view of labor markets. Admitting this more

nuanced view of labor markets is likely to yield very different insights than the strict bargaining

view that has been nearly-universally adopted.

40

A Nash Bargaining in Model with Standard Timing

Here we derive the Nash-bargaining solution for the wage payment in the baseline model.

The marginal value to the household of a family member who is employed is

Wt = (1− τnt )wt + Et[Ξt+1|t ((1− ρx)Wt+1 + ρxUt+1)

], (34)

and the marginal value to the household of a family member who is unemployed and searching for

work is

Ut = χ+ Et[Ξt+1|t

(kh(θt)(1− ρx)Wt+1 + (1− kh(θt)(1− ρx))Ut+1

)], (35)

where kh(θt) = m(ut, vt)/ut is the probability that an unemployed individual finds a match.

The value to a firm of a filled job is

Jt = zt − wt + (1− ρx)Et[Ξt+1|tJt+1

], (36)

and we have, from the job-creation condition,

γ


[Ξt+1|tJt+1

]. (37)

Note for use below that Jt = zt − wt + γkf (θt)

.

The firm and worker choose wt to maximize the Nash product

(Wt −Ut)η Jt

1−η, (38)

with η the bargaining power of the worker. The first-order condition with respect to wt is

η (Wt −Ut)η−1

(∂Wt

∂wt− ∂Ut

∂wt

)J1−ηt + (1− η) (Wt −Ut)

η J−ηt∂Jt

∂wt= 0. (39)

With ∂Wt∂wt

= 1− τnt , ∂Ut∂wt

= 0, and ∂Jt∂wt

= −1, the first-order condition gives the Nash sharing rule

Wt −Ut

1− τnt=

η

1− ηJt. (40)

The labor tax drives a wedge in the Nash sharing rule, so net-of-taxes the worker receives a smaller

share of the surplus than he would absent the tax.

Using the definitions of Wt and Ut,

Wt −Ut = (1− τnt )wt − χ+ Et[Ξt+1|t

((1− ρx)(1− kh(θt))Wt+1 − (1− ρx)(1− kh(θt))Ut+1

)].

(41)

Combine terms on the right-hand-side to get

Wt −Ut = (1− τnt )wt − χ+ Et[Ξt+1|t(1− ρx)(1− kh(θt))(Wt+1 −Ut+1)

]. (42)

41

Using the sharing rule Wt −Ut = (1−τnt )η1−η Jt,

Wt −Ut = (1− τnt )wt − χ+ (1− ρx)(1− kh(θt))(

η

1− η

)Et[Ξt+1|t(1− τnt+1)Jt+1

], (43)

or

Wt −Ut

1− τnt= wt −

χ

1− τnt+

((1− ρx)(1− kh(θt))

1− τnt

)(η

1− η


]. (44)

Next, insert this in the sharing rule to get

wt −χ

1− τnt+

((1− ρx)(1− kh(θt))

1− τnt

)(η

1− η


]=

η

1− ηJt. (45)

Using Jt = zt − wt + γkf (θt)

on the right-hand-side,

wt−χ

1− τnt+

((1− ρx)(1− kh(θt)

1− τnt

)(η

1− η


]=

η

1− η

(zt − wt +

γ

kf (θt)

).

(46)

Solving for wt,

wt = η

[zt +

γ

kf (θt)

]+

(1− η)χ1− τnt

− η(

(1− ρx)(1− kh(θt))1− τnt


]. (47)

Finally, make the substitution Jt+1 = zt+1 − wt+1 + γkf (θt+1)

to write

wt = η

[zt +

γ

kf (θt)

]+

(1− η)χ1− τn

t

− η(

(1− ρx)(1− kh(θt))1− τn

t

)Et

[Ξt+1|t(1− τn

t+1)(zt+1 − wt+1 +

γ

kf (θt+1)

)],

(48)

which is expression (8) in the text.

42

B Nash Bargaining in Model with Instantaneous Hiring

The Nash solution is slightly different in the case when period-t matches become productive in

period t. For completeness, we derive this case in full.

The marginal value to the household of a family member who is employed is

Wt = (1− τnt )wt + Et[Ξt+1|t

((1− ρx + ρxkh(θt+1))Wt+1 + ρx(1− kh(θt+1))Ut+1

)]; (49)

The ρxkh(θt+1) and ρx(1− kh(θt+1)) terms capture the fact that an individual who was employed

in period t and loses his job at the beginning of period t + 1 has probability kh(θt+1) of finding a

new job immediately.

The marginal value to the household of a family member who is unemployed and searching for

work is

Ut = χ+ Et[Ξt+1|t

(kh(θt+1)Wt+1 + (1− kh(θt+1)Ut+1

)], (50)

where kh(θt) = m(ut, vt)/ut is the probability that an unemployed individual finds a match.

The value to a firm of a filled job is

Jt = zt − wt + Et[Ξt+1|t(1− ρx)Jt+1

], (51)

Note for use below that Jt = zt − wt + Et

Ξt+1|t(1−ρx)γkf (θt+1)

.

The firm and worker choose wt to maximize the Nash product

(Wt −Ut)η Jt

1−η, (52)

with η the bargaining power of the worker. The first-order condition with respect to wt is

η (Wt −Ut)η−1

(∂Wt

∂wt− ∂Ut

∂wt

)J1−ηt + (1− η) (Wt −Ut)

η J−ηt∂Jt

∂wt= 0. (53)

With ∂Wt∂wt

= (1− τnt ), ∂Ut∂wt

= 0, and ∂Jt∂wt

= −1, the first-order condition gives the Nash sharing rule

Wt −Ut

1− τnt=

η

1− ηJt. (54)

The labor tax drives a wedge in the Nash sharing rule, so net-of-taxes the worker receives a smaller

share of the surplus than he would absent the tax.

Using the definitions of Wt and Ut,

Wt−Ut = (1− τnt )wt−χ+Et[Ξt+1|t

((1− ρx)(1− kh(θt+1))Wt+1 − (1− ρx − kh(θt+1))Ut+1

)].

(55)

Combine terms on the right-hand-side to get

Wt −Ut = (1− τnt )wt − χ+ Et[Ξt+1|t(1− ρx)(1− kh(θt+1))(Wt+1 −Ut+1)

]. (56)

43

Using the sharing rule Wt −Ut = (1−τnt )η1−η Jt,

Wt −Ut = (1− τnt )wt − χ+(

η

1− η

)Et[Ξt+1|t(1− ρx( 1− kh(θt+1))(1− τnt+1)Jt+1

], (57)

or

Wt −Ut

1− τnt= wt −

χ

1− τnt+(

11− τnt

)(η

1− η

)Et[Ξt+1|t(1− ρx)(1− kh(θt+1))(1− τnt+1)Jt+1

].

(58)

Next, insert this in the sharing rule to get

wt−χ

1− τnt+(

11− τnt

)(η

1− η


]=

η

1− ηJt. (59)

Using Jt = zt − wt + Et


on the right-hand-side,

wt −χ

1− τnt+

(1

1− τnt

)(η

1− η


]=

η

1− η

(zt − wt + Et

Ξt+1|t

(1− ρx)γ

kf (θt+1)

).

(60)

Solving for wt,

wt = η

[zt + Et

Ξt+1|t

(1− ρx)γkf (θt+1)

]+

(1− η)χ1− τnt

−η(

11− τnt


].

(61)

Finally, make the substitution Jt+1 = zt+1 − wt+1 + Et+1

Ξt+2|t+1

(1−ρx)γkf (θt+2)

to write

wt = η

[zt + Et

Ξt+1|t

(1− ρx)γ

kf (θt+1)

]+

(1− η)χ

1− τnt− η(

1

1− τnt

)Et

[Ξt+1|t(1− ρx)(1− kh(θt+1))(1− τnt+1)

(zt+1 − wt+1 +

Ξt+2|t+1

(1− ρx)γ

kf (θt+2)

)].

(62)

This expression looks quite cumbersome, but, using the job-creation condition (17), it simplifies

quite a lot. In the first term on the right-hand-side, use (17) to substitute out zt+Et


,

and do the same with the time-t+ 1 version on the far right-hand-side of as well; this yields

wt = η

[wt +

γ

kf (θt)

]+

(1− η)χ1− τnt

−η(

11− τnt

)Et

[Ξt+1|t(1− ρx)(1− kh(θt+1))(1− τnt+1)

(γ

kf (θt+1)

)].

(63)

Solving for wt gives

wt =χ

1− τnt+

η

1− η

[γ

kf (θt)− 1

1− τntEt

[Ξt+1|t(1− ρx)(1− kh(θt+1))(1− τnt+1)

γ

kf (θt+1)

]], (64)


44

C Elasticity of Market Tightness to Labor Tax Rate

Here, we derive the expression for the steady-state elascticty of labor market tightness θ to the

labor tax rate τn in our baseline model. Start with the steady-state versions of the private-sector

job-creation condition (4) and Nash wage outcome (8):

γθξ =β(1− ρx)(z − w)

1− β(1− ρx)(65)

w = η(z + γθ) +(1− η)χ1− τn

; (66)

In (65), we have used the result that, because we assume a Cobb-Douglas matching function,

kf (θ) = θ−ξ. Inserting (66) into (65), we can define the implicit function

F (θ, τn) ≡ γθξu − β(1− ρx)[(1− η)z − ηγθ − (1− η)χ(1− τn)−1

]1− β(1− ρx)

= 0. (67)

The objective is to construct εθ,τn ≡ dθdτn

τn

θ . By the implicit function theorem, dθdτn = −Fτn

Fθ. We

have

Fτn =β(1− ρx)(1− η)χ

(1− β(1− ρx))(1− τn)2(68)

and

Fθ =ξuγθ

ξu−1(1− β(1− ρx)) + β(1− ρx)ηγ1− β(1− ρx)

. (69)

Constructing the elasticity,

εθ,τn = − τn

(1− τn)2

1θ

β(1− ρx)(1− η)χξuγθξu−1(1− β(1− ρx)) + β(1− ρx)ηγ

. (70)

Use (65) in the denominator of the last term on the right-hand-side to re-express this as

εθ,τn = − τn

(1− τn)2

β(1− ρx)(1− η)χθ

1

β(1− ρx)[ξuθ (z − w) + ηγ

] , (71)

which can be rearranged to

εθ,τn = − τn

(1− τn)2

(1− η)χξu(z − w) + ηγθ

. (72)

Using (66) to eliminate w on the right-hand-side of this last expression and grouping terms,

εθ,τn = − τn

1− τn(1− η)χ

ξu(1− η) [z(1− τn)− χ] + (1− ξu)ηγθ(1− τn). (73)

Finally, divide both the numerator and denominator of the second term on the right-hand-side by

1− τn to obtain

εθ,τn = − τn

(1− τn)2

(1− η)χ

ξu(1− η)[z − χ

1−τn]

+ (1− ξu)ηγθ. (74)

45

Several aspects of this expression are worth noting. First, if χ = 0, taxes do not at all affect θ

and hence do not affect any aspect of the labor market or the rest of the economy. This is because

unemployment benefits are the lever by which labor taxes affect the equilibrium: in steady-state, τn

disappears from the wage equation (8) if χ = 0. Second, the gap between the flow output of a match

and tax-adjusted unemployment flow benefits, z − χ1−τn , is important. The smaller this gap, the

more sensitive is the labor market to the tax rate, and hence the more important is tax-smoothing

likely to be. This argument is a fiscal-policy analog of one of Hagedorn and Manovskii’s (2007)

main conclusions, that the social gain from employment is critical for dynamics in search models.

In our baseline model, the social gain from employment is a key determinant of the importance of

tax smoothing.

Finally, as would be expected, the parameters surrounding the search and matching process

η, ξu, and γ also play a role in this elasticity. Especially interesting to note is the case when η = ξu,

which corresponds to the usual Hosios (1990) condition for search efficiency.18 In this case, the

elasticity simplifies to

εθ,τn = − τn

(1− τn)2

χ

ξu[z + γθ − χ

1−τn] . (75)

The term in square brackets is the after-tax total social gain from a match — it takes into account

both the flow gain z − χ1−τn as well as the asset value as measured by γθ. We conduct our main

quantitative experiments for the parameter setting η = ξu.

18We say “usual” because as we have shown in Arseneau and Chugh (2006, 2007), labor income taxation in and of

itself disrupts the usual Hosios condition for search efficiency. But we proceed here with this language because it is

convenient.

46

D Derivation of Implementability Constraint

The derivation of the implementability constraint follows that laid out in Lucas and Stokey (1983)

and Chari and Kehoe (1999). Start with the household flow budget constraint in equilibrium

ct + bt = (1− τnt )wtnt + (1− nt)χ+Rtbt−1 + dt. (76)

Multiply by βtu′(ct) and sum over dates and states starting from t = 0,

E0

∞∑t=0

βtu′(ct)ct + E0

∞∑t=0

βtu′(ct)bt = E0

∞∑t=0

βtu′(ct)(1− τnt )wtnt

+E0

∞∑t=0

βtu′(ct)(1− nt)χ+ E0

∞∑t=0

βtu′(ct)Rtbt−1 + E0

∞∑t=0

βtu′(ct)dt.

Use the household’s Euler equation, u′(ct) = Et [βu′(ct+1)Rt+1], to substitute for u′(ct) in the term

on the left-hand-side involving bt,

E0

∞∑t=0

βtu′(ct)ct + E0

∞∑t=0

βt+1u′(ct+1)Rt+1bt = E0

∞∑t=0

βtu′(ct)(1− τnt )wtnt

+E0

∞∑t=0

βtu′(ct)(1− nt)χ+ E0

∞∑t=0

βtu′(ct)Rtbt−1 + E0

∞∑t=0

βtu′(ct)dt.

Canceling terms in the second summation on the left-hand-side with the third summation on the

right-hand-side leaves only the time-zero bond position,

E0

∞∑t=0

βtu′(ct)ct = E0

∞∑t=0

βtu′(ct)(1−τnt )wtnt+E0

∞∑t=0

βtu′(ct)(1−nt)χ+E0

∞∑t=0

βtu′(ct)dt+u′(c0)R0b−1.

(77)

Next, insert equilibrium dividends, given by dt = ntzt − wtnt − γvt on the right hand side.

Combining terms yields

E0

∞∑t=0

βtu′(ct) [ct − (zt − τnwt)nt − (1− nt)χ+ γvt]

= u′(c0)R0b−1, (78)


47

E Dynamic Bargaining Power Effect

Further quantitative evidence for this policy channel comes from the following two experiments. We

can solve for the Ramsey steady-state policy as we already have done, but then, when solving for the

dynamics of the Ramsey equilibrium, allow the planner to optimally choose a time-varying vacancy

subsidy. We model this proportional subsidy τ st as in Arseneau and Chugh (2006), assuming that

it makes firms’ vacancy-creation costs (1 − τ st )γ. When solving for the local Ramsey dynamics

around τ s = 0, we find substantial volatility in τ st , and, using the Shimer calibration, labor-tax rate

volatility of 4.58 percentage points, which is noticeably, if not dramatically, lower than the 7.01

percentage point standard deviation reported in Table 1. A related experiment is that, after solving

for the steady state, we adjust the steady-state vacancy subsidy rate from (its hard-coded value

of) τ s = 0 to τ s = 0.35. Solving the Ramsey dynamics around this adjusted steady state, volatility

in the labor tax rate is 2.24 percentage points (compared to 7.01 and 4.58 reported in Table 1 and

in the immediately preceding experiment, respectively), and we again find substantial volatility in

τ st . Thus, allowing the planner access to a time-varying instrument to directly hit search activity

permits lower labor-tax rate volatility.

The value of τ s = 0.35 comes from solving for the Ramsey steady-state allowing the planner

to optimally choose both the long-run labor income tax rate and the long-run vacancy subsidy

rate. We cannot solve for both the Ramsey steady state and the Ramsey dynamics allowing for

optimal choice over the vacancy subsidy, however, because doing so allows the planner to achieve the

socially-efficient outcome in the steady-state (a point we discovered in Arseneau and Chugh (2006)).

Given the local approximation method we employ, if the Ramsey steady-state achieves the first-

best outcome, we cannot approximate the dynamics of the vacancy subsidy because the condition

governing it drops out as an equilibrium condition (because all Ramsey multipliers associated with

non-technological constraints end up being zero if the Pareto optimum is achieved, and we end up

trying to approximate around the expression 0 = 0, which obviously does not permit numerical

analysis). In short, if we want to permit the planner to use a vacancy subsidy, we must assume

that his use of it is not fully optimal in either the long run or in the short run. Nonetheless, these

two experiments go a long way towards showing that optimal labor-tax policy is geared towards

exploiting the dynamic bargaining power effect.

48

References

Acemoglu, Daron and William Hawkins. 2006. “Equilibrium Unemployment in a General-

ized Search Model.” MIT.

Aiyagari, S. Rao, Albert Marcet, Thomas J, Sargent, and Juha Seppala. 2002. “Op-

timal Taxation without State-Contingent Debt.” Journal of Political Economy, Vol. 110, pp.

1220-1254.

Arseneau, David M. and Sanjay K. Chugh. 2006. “Ramsey Meets Hosios: The Optimal

Capital Tax and Labor Market Efficiency.” International Finance Discussion Paper no. 870 ,

Board of Governors of the Federal Reserve System.

Arseneau, David M. and Sanjay K. Chugh. 2007. “Optimal Fiscal and Monetary Policy

with Costly Wage Bargaining.” International Finance Discussion Paper no. 893 , Board of

Governors of the Federal Reserve System.

Arseneau, David M. and Sanjay K. Chugh. 2008. “Competitive Search Equilibrium in a

DSGE Model.” University of Maryland.

Aruoba, S. Boragan and Sanjay K. Chugh. 2006. “Optimal Fiscal and Monetary Pol-

icy When Money is Essential.” International Finance Discussion Paper no. 880 , Board of

Governors of the Federal Reserve System.

Barro, Robert J. 1979. “On the Determination of the Public Debt.” Journal of Political

Economy, Vol. 87, pp. 940-971.

Blanchard, Olivier and Peter Diamond. 1989. “The Beveridge Curve.” Brookings Papers

on Economic Activity, Vol. 1, pp. 1-76.

Boone, Jan and Lans Bovenberg. 2002. “Optimal Labor Taxation and Search.” Journal of

Public Economics, Vol. 85, pp. 53-97.

Caballero, Ricardo J. 2007. Specificity and the Macroeconomics of Restructuring. MIT Press.

Cahuc, Pierre, Francois Marque, and Etienne Wasmer. 2008. “Intrafirm Wage Bargain-

ing in Matching Models: Macroeconomic Implications and Resolutions Methods with Multiple

Labor Inputs.” International Economic Review. Forthcoming.

Chari V. V., and Patrick J. Kehoe. 1999. “Optimal Fiscal and Monetary Policy. In

Handbook of Macroeconomics, edited by John B. Taylor and Michael Woodford, Vol. 1C.

Elsevier.

Chugh, Sanjay K. 2006. “Optimal Fiscal and Monetary Policy with Sticky Wages and Sticky

Prices.” Review of Economic Dynamics, Vol. 9, pp. 683-714.

Costain, James S. and Michael Reiter. 2007. “Business Cycles, Unemployment Insur-

ance, and the Calibration of Matching Models.” Journal of Economic Dynamics and Control.

Forthcoming.

49

Domeij, David. 2005. “Optimal Capital Taxation and Labor Market Search.” Review of Eco-

nomic Dynamics, Vol. 8, pp. 623-650.

Faia, Ester. 2007. “Optimal Monetary Policy Rules with Labor Market Frictions.” Journal

Economic Dynamics and Control. Forthcoming.

Hagedorn, Marcus and Iourii Manovskii. 2007. “The Cyclical Behavior of Equilibrium

Unemployment and Vacancies Revisited.” University of Pennsylvania.

Hall, Robert E. and Paul R. Milgrom. 2007. “The Limited Influence of Unemployment on

the Wage Bargain.” Stanford University.

Hosios, Arthur J. 1990. “On the Efficiency of Matching and Related Models of Search and

Unemployment.” Review of Economic Studies, Vol. 57, pp. 279-298.

Krause, Michael U., David Lopez-Salido, and Thomas A. Lubik. 2007. “Inflation

Dynamics with Search Frictions: A Structural Econometric Analysis.” Federal Reserve Bank of

Richmond.

Krause, Michael U. and Thomas A. Lubik. 2006. “Does Intra-Firm Bargaining Matter for

Business Cycle Dynamics.” Federal Reserve Bank of Richmond.

Mortensen, Dale T. and Christopher A. Pissarides.. 1999. “New Developments in Models

of Search in the Labor Market. In Handbook of Labor Economics, edited by Orley Ashenfelter

and David Card, Vol. 3B. Elsevier.

Pissarides, Christopher A. 2000. Equilibrium Unemployment Theory. MIT Press.

Rocheteau, Guillaume and Randall Wright. 2005. “Money in Search equilibrium, in

Competitive Equilibrium, and in Competitive Search Equilibrium.” Econometrica, Vol. 73, pp.

175-202.

Rogerson, Richard, Lodewijk P. Visschers, and Randall Wright. 2007. “Labor Market

Fluctuations in the Small and in the Large.” University of Pennsylvania.

Schmitt-Grohe, Stephanie and Martin Uribe. 2004. “Solving Dynamic General Equilib-

rium Models Using a Second-Order Approximation to the Policy Function.” Journal of Eco-

nomic Dynamics and Control, Vol. 28, pp. 755-775.

Schmitt-Grohe, Stephanie and Martin Uribe. 2005. “Optimal Fiscal and Monetary Policy

in a Medium-Scale Macroeconomic Model.” NBER Macroeconomics Annual 2005.

Shimer, Robert. 2005. “The Cyclical Behavior of Equilibrium Unemployment and Vacancies.”

American Economic Review, Vol. 95, pp. 25-49.

Siu, Henry E. 2004. “Optimal Fiscal and Monetary Policy with Sticky Prices.” Journal of

Monetary Economics, Vol. 51, pp. 576-607.

Werning Ivan. 2007. “Optimal Fiscal Policy with Redistribution.” Quarterly Journal of Eco-

nomics, Vol. 122, pp. 925-967.

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