ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E21-4

General Equilibrium Effects andLabor Market Fluctuations
byNoritaka Kudoh
Hiroaki Miyamoto

June 2021

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

General Equilibrium Effects and Labor Market
Fluctuations
Noritaka Kudoh†

Hiroaki Miyamoto‡

Nagoya University

Tokyo Metropolitan University
March 20, 2021

Abstract
Business cycle models with search-matching frictions are studied to evaluate theimportance of general equilibrium effects generated by movements in the stochastic discount factor and the income effect on labor supply. Without variable hours ofwork, the general equilibrium effect works only through the stochastic discount factor and is quantitatively very weak. With variable hours of work, the income effectoperates to generate procyclical movements in the value of leisure and the marginalhourly wage rate. This effect is sizable and dampens labor market fluctuations. We
also study discount factor shocks and find that capital formation strongly enhances
labor market fluctuations.
JEL classification: E32, J20, J64.
Keywords: labor market search, stochastic discount factor, unemployment volatility.

Part of this research is financially supported by KAKENHI (17K03619, 20H01476).
† E-mail: kudoh@soec.nagoya-u.ac.jp‡ E-mail: hmiyamoto@tmu.ac.jp1

1

Introduction

Explaining unemployment had been one of the greatest challenges in economics, until thesearch-matching model of the labor market, often referred to as the DMP model, was developed by Diamond (1982), Mortensen and Pissarides (1994), and Pissarides (2000). The
subsequent research such as Shimer (2005) explores labor market fluctuations over thebusiness cycle using the DMP model.1 Turning to the main-stream business cycle literature, Merz (1995) and Andolfatto (1996) integrate search-matching frictions of DMP-type
into an otherwise standard dynamic stochastic general equilibrium (DSGE) frameworkand show that search-matching frictions can help improve the DSGE model.2
While the textbook DMP model such as Shimer (2005) and the DSGE model withsearch-matching frictions such as Merz (1995) and Andolfatto (1996) are seemingly identical, they employ different aggregation strategies. To avoid the potential difficulty inmodeling wealth distribution across individuals and over time, the textbook DMP model
assumes risk neutral individuals to rule out wealth accumulation altogether, while theDSGE model with search-matching frictions assumes a representative “big family” tosuppress the issue of wealth distribution while keeping the intertemporal consumption
choice.3 As a result, the discount factor (i.e., the real interest rate) is endogenous andtime-varying in the DSGE model, whereas it is exogenous and constant over time in the
textbook DMP model.4
By directly comparing these aggregation strategies, this paper evaluates the importance of general equilibrium effects over the business cycle generated by intertemporal consumption choice. In the textbook DMP model, each agent discounts the future using a
1 Cole and Rogerson (1999) also bring the DMP model to the business cycle analysis, with special atten-

tion given to job creation and job destruction over the business cycle.
2 Related contributions include den Haan et al. (2000), Trigari (2006, 2009), Krause and Lubik (2007,2013), Gertler and Trigari (2009), and Christiano et al. (2016), to name a few. Ichiue et al. (2013) use
a Bayesian DSGE framework to show that the labor market search model is superior to the sticky wage
model often employed in the DSGE literature.
3 In both models, consumption insurance is perfect because markets are complete. Krusell et al. (2010)
and Nakajima (2012) introduce incomplete markets into the DSGE model with search-matching frictions.
Krusell et al. (2010) ignore hours per worker to focus on the extensive margin of labor adjustment. Nakajima (2012) considers the intensive margin using the Greenwood-Hercowitz-Huffman (GHH) utility function to rule out the income effect. While the incomplete-markets approach is beyond the scope of this paper,
it provides an important direction for further study.
4 For convenience, in what follows the DMP model is referred to as the one with linear utility in consumption, and the DSGE model is referred to as the one with the big family assumption.

2

constant discount factor and (because of linear utility) the aggregate level of consumption
is determined as a residual in the resource constraint. On the other hand, in the DSGE
model, each individual agent discounts the future using the stochastic discount factor,
which interacts with the aggregate level of consumption. This interaction is one of thegeneral equilibrium effects we consider.
With variable hours of work, there is an additional general equilibrium effect—theincome effect on labor supply. Because of concave utility, an increase in the level of consumption decreases the relative price of consumption within a big family, increasing therelative marginal disutility from work (Trigari, 2006, 2009). This effect is absent in models
without the big family assumption.
How important is the income effect? This question is particularly relevant in countries in which the intensive margin of labor adjustment is sizable. While the conventional
search-matching model focuses on the extensive margin, the importance of the intensive
margin has increased in many major countries (Ohanian and Raffo, 2012). For instance,
the intensive margin accounts for about 79 percent of variations in total labor input inJapan (Kudoh et al., 2019). Dossche et al. (2019) document the importance of the intensive
margin in Europe.5 In this paper, we target the Japanese economy for our quantitative
analysis.
Throughout, we evaluate the importance of the general equilibrium effects in understanding the aggregate labor market dynamics. To address this question, we compare
stripped-down versions of the DMP model, which is without the income effect, and theDSGE model, which captures the income effect among other effects. This analytical strategy is important because the modern DSGE models are often “medium scaled,” including nominal price stickiness and monetary policy. To identify the income effect over thebusiness cycle, we consider minimum model elements that are common across the DMP
model and the DSGE model, rather than introducing an unexplored new component.
We consider four models: the DMP model without variable hours of work, which is
close to the textbook DMP model (Shimer, 2005; Hagedorn and Manovskii, 2008; Amaral
Tasci, 2016); the DMP model with variable hours of work (Cooper et al., 2007; Trapeznikova,
2017; Kudoh et al., 2019); the DSGE model without variable hours of work (den Haan etal., 2000; Krause and Lubik, 2007; Costain and Reiter, 2008; Gertler and Trigari, 2009;
Christiano et al., 2016; Atolia et al., 2018); and the DSGE model with variable hours of
5 A partial list of the recent literature on the intensive margin of labor adjustment is Wesselbaum (2016),

Trapeznikova (2017), Kudoh et al. (2019), Dossche et al. (2019), Cacciatore et al. (2020), and Kolasa et al.
(2021).

3

work (Trigari, 2006, 2009; Fang and Rogerson, 2009; Wesselbaum, 2016; Dossche et al.,2019; Cacciatore et al., 2020; Kolasa et al., 2021).6 We compare these four models under
two types of shocks: total factor productivity (TFP) shocks and discount factor shocks.
The latter is motivated by Mukoyama (2009) and Hall (2017).
When labor market fluctuations are driven by TFP shocks, without variable hours ofwork per employee, the two models are identical in the steady state, and their businesscycle properties are nearly identical. The intuition is as follows. An increase in TFP increases the marginal product of capital in both models. In the DSGE model, part of theincrease in the demand for capital is offset by an increase in the rental price of capital
because the loan market becomes tight. However, we find that this effect is quantitatively weak. Thus, we conclude that the general equilibrium effect through the stochastic
discount factor is not very important in accounting for labor market fluctuations.
However, with variable hours of work, the two models generate significantly different
outcomes, both in and out of steady-state. For instance, the unemployment volatility
in response to TFP shocks in the DSGE model is about 2/3 of that of the DMP model.
Although the general equilibrium effect through the stochastic discount factor is weak,
the income effect on labor supply, which works through a decline in the relative priceof consumption, is quantitatively strong. Interestingly, this result is not explained by
the surplus argument (Hagedorn and Manovskii, 2008; Ljungqvist and Sargent, 2017).
Indeed, under the same calibration strategy and targets, the steady-state value of theopportunity cost of employment is exactly the same for the two models.
Our results are complementary to those of Chodorow-Reich and Karabarbounis (2016),
who measure the opportunity cost of employment (denoted by z) over the business cycle for the U.S. and find that it is procyclical and volatile, and argue that the cyclicalityof z dampens unemployment fluctuations because “[r]elative to the constant z case, aprocyclical z increases the surplus from accepting a job at a given wage during a recession, which puts downward pressure on equilibrium wages and ameliorates the increasein unemployment” (p.1567). However, we show that the procyclicality of z as well as
its volatility are both stronger in the DMP model than those in the DSGE model, while
the DMP model generates a higher (not lower) unemployment volatility than the DSGE
model does. Further, Kudoh et al. (2019) find that the DMP model with a constant z and6 Kudoh et al.

(2019) show that the DMP model with variable hours of work naturally implements
Hagedorn Manovskii’s (2008) small surplus calibration through a low Frisch elasticity. This is in sharp
contrast with the standard real business cycle literature, in which a high Frisch elasticity is necessary forlabor market fluctuations of a realistic magnitude.

4

the one with a procyclical z (due to variable hours) generate the same unemployment
volatility. On the surface, these results are at odd with Chodorow-Reich and Karabarbounis (2016).
Our key finding is that the procyclicality of z alone does not dampen unemployment volatility. The income effect is essential for this channel. The procyclicality of
consumption implies that the relative weight on disutility of work is also procyclical.
As a result, the marginal hourly wage rate increases in response to a positive productivity shock, which dampens the increase in hours of work during expansions. This effectplays a crucial role, and is missing in the DMP model, which is without the intertemporalconsumption choice. While Chodorow-Reich and Karabarbounis (2016) consider a rich
DSGE model including the income effect that we study, they treat the procyclicality of z
as the key driving force. Our contribution is to highlight the role of the income effect ingenerating their result by comparing the models with and without general equilibrium
effects.
Mukoyama (2009) and Hall (2017) ask whether variations in the discount factor canaccount for labor market fluctuations.7 This questions is important on its own because
financial turbulence is often associated with a massive job loss (Reinhart and Rogoff, 2009;
Hall, 2017).8 Motivated by these contributions, we study discount factor shocks as analternative driver of business cycle. With discount factor shocks, the general equilibrium
effects significantly dampen labor market fluctuations, with or without variable hours
of work. Without variable hours of work, the volatility of unemployment in the DSGE
model is about 1/6 of that of the DMP model.
The remainder of the paper is organized as follows. Section 2 presents the four models
we consider, followed by characterization of the equilibrium conditions in Section 3. In
Section 4, we calibrate the model parameters, and the quantitative results are presented
in Section 5. Section 6 concludes.
7 Using a variant of the textbook DMP model, Mukoyama (2009) shows that discount factor shocks can-not account for labor market fluctuations in the U.S., and Hall (2017) shows that with wage stickiness,fluctuations in the discount factor can be a major source of U.S. labor market fluctuations. We find thatthe absence of capital explains the difficultly in accounting for discount-factor-driven labor market fluctuations.
8 Note that the long-run relationship between the discount factor and unemployment has been extensively investigated in the literature. A change in the growth rate of technological progress influences job
creation through its impact on the effective discount factor, known as the “capitalization effect” (Pissarides,
2000). It is well-known that the capitalization effect is quantitatively weak (Pissarides and Vallanti, 2007;
Miyamoto and Takahashi, 2011).

52Models

We consider two closely related models for understanding the aggregate labor market,
the DMP model and the DSGE model with search-matching frictions. To facilitate comparisons, we consider a common model environment as much as possible. As in thestandard DSGE model, we consider total factor productivity (TFP) as the driver of business cycles. This requires the standard neoclassical production technology with capital.
To introduce capital, we assume that the representative final-good firm accumulates capital in the DMP model and the representative family accumulates capital in the DSGE
model.9 While large-firms models with intra-firm bargaining are useful for understanding the composition of labor demand over the business cycle (Kudoh et al., 2019), intrafirm bargaining is not an essential component of our analysis as we focus on the generalequilibrium effects from intertemporal consumption choice. Thus, we assume that the
production unit (i.e., the bargaining party) is each worker-vacancy pair.10

2.1

DMP Model

The standard DMP model and its variants emphasize the disaggregated aspect of themarket economy and assume no representative household. To rule out the issue of wealthdistribution over time, DMP models assume risk neutral individuals. Thus, in our firstmodel, which we refer to as the DMP model, each individual has the following objectivefunction:∞

E0 ∑ Dt [ Ite (ht )] ,

(1)

t =0

where Dt is the cumulative discount factor, It is income, and e(h) represents the level ofdisutility from working for h hours. The (common) cumulative discount factor evolves
according to Dt+1 = β t Dt with D0 = 1, where β t = 1/(1 + ρt ). Evidently, Dt = βt when
β t = β for all t.
9 In the DMP model, risk-neutral individuals do not have incentives to hold capital. In the DSGE model,on the other hand, the representative family needs capital (i.e., means of savings) to smooth consumption
over time.
10 Kudoh and Sasaki (2011) and Kudoh et al. (2019) consider multi-worker firms to study hours of workin the context of intra-firm bargaining. Kudoh and Sasaki (2011) show that because of the well-known
overhiring effect, the equilibrium level of hours of work is below the optimal level and this inefficiency
cannot be corrected by the Hosios condition. For a quantitative investigation on this issue, see Dossche etal. (2019).

6

Throughout, we specify the disutility function ase ( h ) = e0h 1+ µ,1+µ(2)

where e00 and 1/µ is the Frisch elasticity parameter. We shall set e0 = 0 to study aversion of the model without variable hours of work per employee.
We introduce a representative final consumption good firm with the production technology
yt = At Xtα k1t α ,(3)where yt is output (i.e., consumption good), At is total factor productivity (TFP), and Xt
is the amount of intermediate inputs supplied by intermediate goods firms.
We assume that the final output firm accumulates capital, as in Cahuc et al. (2008) andKudoh et al. (2019). This assumption is necessary because the risk-neutral individuals do
not make saving. Another important role of the firm is to aggregate intermediate inputs.
Each intermediate input is produced by an intermediate firm, which employs a single
worker and bargains over the terms of trade. This market structure conveniently rulesout the issue of intra-firm bargaining.11
The final output firm behaves competitively. The price of the final consumption good
is normalized to unity. Taking the (sequence of) input price pt as given, the firm solves
the following problem,h∞max E0 ∑ Dt At Xtα k1t α

f Xt ,it g∞
t =0

t =0

p t Xtiit ,subject to k t+1 = (1 δ) k t + it , where δ is the capital depreciation rate and it is the levelof capital formation in period t. Let F (k ) be the value of the firm. Thus, the problem canbe stated recursively ashi
F (k ) = max AX α k1 α pX k0 + (1 δ) k + EβF k0 ,X,k0from which the first-order conditions and the envelope condition are αAX α 1 k1 α = p,
1 = EβF 0 (k0 ), and F 0 (k ) = (1 α) AX α k α + 1 δ. Thus, we obtain αAt Xtα 1 k1t α = pt
and Et β t [(1 α) At+1 Xtα+1 k t+α1 + 1 δ] = 1. In any equilibrium, Xt = ht lt holds because lt
is the number of firms (which equals the number of employees) and ht is the production
11 See Cahuc et al. (2008), Kudoh and Sasaki (2011), Krause and Lubik (2013), and Kudoh et al. (2019) for

models with intra-firm baragining.

7

intensity in each production unit. In a model without hours of work per employee, we
replace Xt = ht lt with Xt = lt .
ξ 1 ξ
The number of matches in period t is determined by m0 Ut Vt , where m0 > 0 and
0 < ξ < 1 are parameters, Ut is the total number of job seekers, and Vt is the number ofaggregate job vacancies. The vacancy filling rate is given byξ1 ξ

qt = m0 Ut Vt/Vt = m0 θtξq (θt ) ,

(4)
ξ

1 ξ

1 ξ

where Vt /Utθt . Similarly, the job finding rate is given by m0 Ut Vt /Ut = m0 θt=θt q(θt ). The labor force is normalized to unity, so the unemployment rate ut satisfies
ut = Ut .
The values of employment and unemployment, denoted respectively by JtE and JtU ,are standard:
JtE = W (ht )e (ht ) + λEt β t JtU+1 + (1JtU = b + θt q (θt ) Et β t JtE+1 + [1λ) Et β t JtE+1 ,θt q (θt )] Et β t JtU+1 ,

(5)
(6)

where W (ht ) is the amount of earnings, λ is the separation rate, and b is the unemployment benefit. In a model without variable hours per employee, we replace W (ht ) e(ht )with the hourly wage rate wt .
The values of a filled job and a vacancy are given respectively as JtF = pt ht W (ht ) +λEt β t JtV+1 + (1 λ)Et β t JtF+1 and JtV = c + q(θt )Et β t JtF+1 + [1 q(θt )]Et β t JtV+1 , where c isthe cost of posting a vacancy. Note that the firm’s revenue is pt ht . Thus, each intermediate
firm sells its employee’s labor input to the final output firm at the competitive price pt .
In a model without variable hours per employee, we replace pt ht W (ht ) with pt wt .
The equilibrium number of vacancies is determined by the free entry condition JtV =0, from which we obtain c = q (θt ) Et β t JtF+1 . Thus, we obtain the job-creation condition:c= Et β t p t +1 h t +1
q (θt )W ( h t +1 ) +(1 λ ) c,q ( θ t +1 )(7)

where W (ht+1 ) is replaced with wt+1 in a model without variable hours of work.
As in the textbook DMP model, we assume Nash bargaining to determine both earnings and hours of work. We follow Cooper et al. (2007) and Kudoh et al. (2019) to assume
that a firm and a worker bargain over a state-contingent contract that specifies an earnings schedule, which maps hours of work into an amount of compensation.
A caveat is that with variable hours of work, we need to distinguish between earnings (denoted by Wt ) and the wage rate (denoted by wt ). As is made clear by Kudoh and8Sasaki (2011) and Kudoh et al. (2019), the (average) hourly wage rate, wt = Wt /ht , has
no allocative role because under any bilateral bargaining protocol, what matters is the division of total surplus, which is earnings Wt , not the (average) wage rate that can only becalculated ex-post by dividing Wt by ht .12 This is in sharp contrast with the perfectly competitive labor market, in which all agents take the wage rate as a given numerical value
when make choices. To allow for a general earnings structure, we specify Wt = W (ht )rather than Wt = wt ht .
Consider the following bargaining problem:
max JtF
Wt ,ht

JtV

1 η

JtE

JtU

η

,

where η is the exogenous bargaining power for the worker. The first order conditions
imply (1 η )( JtE JtU ) = η ( JtF JtV ) and pt = W 0 (ht ), from which we obtain W (ht ) =η pt ht + (1 η )[e(ht ) + b] + ηcθt . Hours per employee are determined in bargaining soµthat pt = η pt + (1 η )e0 (ht ), from which we obtain αAt (ht lt )α 1 k1t α = e0 ht = e0 (ht ).
It is important to note that in our model, who chooses hours of work is irrelevant
for equilibrium conditions because the earnings schedule W (h) guarantees the efficientbargaining outcome for each level of h.13 With intra-firm bargaining and concave production, however, it matters who makes the choice of hours (Kudoh and Sasaki, 2011; Kudoh
et al., 2019; Dossche et al., 2019). The vertical industry structure employed in this paper
and in many recent DSGE models is a useful device that brides the aggregate production
technology and each production unit.

2.2

DSGE Model

DSGE models with search-matching frictions typically allow for a richer preference structure such as concavity and non-separability. However, to rule out the issue of wealthdistribution, all agents are assumed to be members of a single large family (Merz 1995;
Andolfatto, 1996). All incomes of all individuals are pooled in each period and intertemporal consumption choice is made by the big family so that all individuals consume thesame units of the consumption good (i.e., perfect risk sharing among individuals).
12 Kudoh and Sasaki (2011) and Kudoh et al. (2019) therefore emphasize the marginal hourly wage rate,∂Wt /∂ht .
13 This is in sharp contrast with the right-to-manage literature, in which the firm chooses hours of work
given the predetermined hourly wage rate as a numerical value by implicitly assuming a linear relationshipbetween hours and earnings (Trigari, 2006; Sunakawa, 2012). Inefficiency arises because the true relationship between hours and earnings is nonlinear.

9Thus, in our second model, the objective function for the large family is given by
#
"
∞
Ct1 σ
(8)
e ( ht ) lt ,E0 ∑ Dt1 σ
t =0
where Ct is the level of consumption and 1/σ is the intertemporal elasticity of substitution (and σ is the coefficient of relative risk aversion). This family structure works as
a risk-sharing device among the individuals. With concave utility in consumption, thefamily chooses to make saving. Thus, unlike the previous model, capital is held by thefamily and lent to the final output firm.
Since all incomes are pooled within the family, the budget constraint for the family is
Ct + k t+1

k t = Wt lt + (1lt ) b + (r tδ) k t + Tt + Πt ,(9)where k t is the stock of capital (so that k t+1 k t is capital formation in period t), rt is therental price of capital (so that rt δ is that rate of return on capital), Wt denotes earnings
per employed, lt is the proportion of employed individuals, b is the unemployment benefit from the government, Tt is transfer, Πt is dividends (profits). In equilibrium, transfer
Tt is determined so that the government’s budget is balanced. Thus, Tt + (1 lt ) b = 0.
Given that employment, earnings, the rate of returns on capital, and dividends are determined as part of search equilibrium, the big family chooses the paths of consumption
and capital to maximize (8) subject to (9). Let Λt be the Lagrange multiplier associatedwith (9). Then, the first-order conditions areΛt = Ct σ ,(10)Λ t = Et β t Λ t +1 [ 1δ + r t +1 ] .

(11)The transversality condition is limt!∞ Λt Dt k t = 0.
To evaluate all decentralized variables in terms of the marginal utility of the final
consumption good, we shall introduce the stochastic discount factor, defined by
St = β tΛ t +1,Λt(12)

δ + r t +1 ] .

(13)from which (11) reduces to
1 = Et St [1

In any steady state, (13) implies β = 1/(1 + r̃ ), where r̃ is the market real interest rate netof capital depreciation. In contrast to the DMP model, the real interest rate in the DSGE
model is endogenous.
10

The final consumption good is produced by the (perfectly competitive) final output
firm with production technology (3). The price of the final consumption good is normalized to one. Since the big family owns the entire stock of capital, the final output firmuses capital at the rental price rt . Thus, the final output firm’s problem essentially static,
and is given byhimax At Xtα k1t α pt Xt rt k t ,Xt ,k tfrom which we obtain the first-order conditions αAt Xtα 1 k1t α = pt and (1 α) At Xtα k t α =rt . In equilibrium, Xt = ht lt in the model with variable hours of work and Xt = lt in themodel without variable hours per employee.
The value functions for workers are given as
JtE = W (ht )Λt 1 e (ht ) + λEt St JtU+1 + (1JtU = b + θt q (θt ) Et St JtE+1 + [1λ) Et St JtE+1 ,θt q (θt )] Et St JtU+1 ,

(14)
(15)

which are nearly identical to (5) and (6), respectively. The big family’s concave utility
introduces two important differences. One is that workers discount the future at rate St ,which is the stochastic discount factor. To see how it introduces the general equilibrium
effect, suppose for now that utility from consumption is ln C so that St = β t Ct /Ct+1 . Thisimplies that an increase in the consumption growth rate increases the discount factor
because future consumption becomes less costly. This mechanism is absent in the modelwith linear utility.
The other difference the big family assumption introduces is the evaluation of disutility from longer hours, Λt 1 e (ht ). The adjustment through Λt is necessary because thevalue of employment JtE is measured in terms of the utility units of the final consumption
good (Trigari, 2006, 2009; Sunakawa, 2015; Wesselbaum, 2016; Dossche et al., 2019). What
this implies is that as the marginal utility from consumption goes up (when the level ofconsumption declines), the relative value of disutility decreases, inducing longer hours
of work. This mechanism is absent in the textbook DMP model because Λt 1 = 1 under
the linear utility.14 The term Λt 1 captures the strength of the wealth effect on the supplyof hours.
The value of a filled job and the value of a vacancy are JtF = pt ht W (ht ) + λEt St JtV+1 +(1 λ)Et St JtF+1 and JtV = c + q(θt )Et St JtF+1 + [1 q(θt )]Et St JtV+1 , respectively. Free en14 Cooley and Quadrini (1999) develop a monetary general equilibrium model with variable hours inwhich the household’s utility is linear in consumption. We view that the Cooley-Quadrini model belongs
to the DMP paradigm because the discount factor is constant over time.

11

try of jobs implies JtV = 0, from which we obtain the job-creation condition:c= Et S t p t +1 h t +1
q (θt )W ( h t +1 ) +(1 λ ) c.
q ( θ t +1 )(16)

The bargaining outcome is given by W (ht ) = η pt ht + (1 η )[Λt 1 e(ht ) + b] + ηcθt .
µThe optimal level of hours satisfies αAt (ht lt )α 1 k1t α = Λt 1 e0 ht = Λt 1 e0 (ht ) = pt .

3

Equilibria

3.1

Models without Variable Hours of Work

Consider first the models without variable hours of work. The equilibrium conditions forthe DSGE model are summarized by
1 = Et St 1

(1 λ ) c,q ( θ t +1 )(17)c
= Et St αAt+1 ltα+11 k1t+1αq (θt )

w t +1 +wt = ηαAt ltα 1 k1t α + (1η ) b + ηcθt ,(19)ut ) ,(20)

u t +1 = u tytα) At+1 ltα+1 k t+α1 ,δ + (1

θ t q ( θ t ) u t + λ (1cθt ut = Ct + k t+1(1

and
St = β tδ) k t ,
Ct+σ1
Ct σ

(18)

(21)

.

(22)

For the DMP model, the stochastic discount factor St in (17) and (18) is replaced with thetime-varying discount factor β t .
In the DSGE model, the level of consumption Ct is jointly determined by the resource
constraint (21) and stochastic discount factor (22). In contrast, in the DMP model, theequilibrium is determined by (17)-(20), and Ct is determined by the recourse constraint(21) as a residual.
Because S = β in any steady state, the steady-state conditions for the two models areexactly the same:(1 α) AK α = ρ + δ,h(ρ + λ) c+ ηcθ = (1 η ) αAK1 αq (θ )
12

(23)
i
b ,

(24)

with u = λ/[λ + θq (θ )], l = 1 u, K = k/l, and y cθu = C + δk. It is easy to establish
the uniqueness of the steady-state equilibrium from these expressions. Thus, the twomodels are identical in any steady state.

3.2

Models with Variable Hours of Work

Consider the models with variable hours of work. The equilibrium conditions for the
DSGE model are summarized by
1 = Et St 1α) At+1 (ht+1 lt+1 )α k t+α1 ,δ + (1(25)

αAt (ht lt )α 1 k1t α = Λt 1 e0 ht ,
µ

(26)

c
= Et St αAt+1 (ht+1 lt+1 )α 1 k1t+1α ht+1
q (θt )"
Wt = ηαAt (ht lt )α 1 k1t α ht + (1Wt+1 +1+ µη ) Λt1(1 λ ) c,q ( θ t +1 )#h
+ b + ηcθt ,
e0 t
1+µ

(27)
(28)

with (20), (21), and (22). For the DMP model, the stochastic discount factor St is replacedwith the time-varying discount factor β t , and Λt is replaced with 1.
The steady state of the DMP model satisfies(1α) AK α = ρ + δ,(29)

αAK1 α = e0 hµ = e0 (h) ,(30)(ρ + λ) c= (1q (θ )η ) e0 ( h) h

e (h)

b

(31)

ηcθ,

with u = λ/[λ + θq (θ )], l = 1 u, and K = k/hl. The uniqueness of the steady state iseasily verified. First, (29) determines the capital-labor ratio K. Given K, (30) determines
hours of work per employee, h. Finally, given h, (31) determines tightness θ.
The steady state of the DSGE model is a pair (h, θ) satisfying
αAK1 α = [C (h, θ )]σ e0 hµ ,(ρ + λ) c+ ηcθ = (1q (θ )

η)(32)µ
αAK1 α h
1+µ

b ,

(33)

where, from resource constraint,
C=y

cθu

δk = AK1 αδKθq (θ )hλ + θq (θ )13cθλλ + θq (θ )C (h, θ ) .

Figure 1: Vacancy Cost and Equilibrium Hours of WorkInterestingly, through the term (Λ 1 ), we now have a rich interaction between h and θ.
As shown in Figure 1, an increase in c increases h.15 The same result has been shown ina related environment by Fang and Rogerson (2009). As is evident from (30), in the DMP
model, an increase in c has no effect on h, even though (31) implies a positive relationshipbetween h and θ.

4

Calibration

We calibrate our models to match Japanese labor market facts. We use Japan’s economy
as our target because the intensive margin adjustment over the business cycle is sizable inJapan (Kudoh et al., 2019). Our calibration procedure closely follows Kudoh et al. (2019)
and therefore we keep the description here to be brief.16
We choose the model period to be a quarter and set the steady-state subjective discount rate at ρ = 0.01, or β = 1/(1 + ρ) = 0.99. In the production function, we set
α = 2/3. The deprecation rate is set at δ = 0.0094 3. The matching elasticity withrespect to the number of job seekers is set at ξ = 0.6. Following the convention, we setη = ξ = 0.6.
15 The parameter values are those calibrated in Section 4. We plot the equilibrium levels of hours for cfrom 20% below to 20% above its calibrated value.
16 Kudoh et al. (2019) also provide some empirical facts on the Japanese labor market. Miyamoto (2011)and Lin and Miyamoto (2012) study worker flow data to estimate various transition probabilities necessary
for calibration.

14

We target the labor market tightness of 0.78 and the quarterly job finding rate of
0.142 3. These targets with ξ = 0.6 pin down the scale parameter m0 = 0.471. We
also set the exogenous separation rate to be λ = 0.0048 3. As in Kudoh et al. (2019), we
set µ = 1.8 or the Frisch elasticity 1/µ to be 0.56. We target the unemployment benefit b tosatisfy b = 0.6W, which is consistent with Japan’s replacement ratio reported in Nickell(1997). We target the steady-state working hours per employee to be 1/3. With θ = 0.78,
h = 1/3, and b = 0.6W, we determine the remaining parameters b, c, and e0 by solving
the steady-state equilibrium conditions.
We set the relative risk aversion parameter to be σ = 2. Robustness analysis for thisparameter value will be presented.

Parameter

Table 1: Fixed Parameters
Description

Value

ρ
β
σ
α
ξ
λ
δ
µ
η
A

Steady-state discount rate
Steady-state discount factor 1/(1 + ρ)
Relative risk aversion
Parameter in production function
Matching elasticity
Exogenous separation rate
Depreciation rate
Inverse of Frisch elasticity
Worker’s bargaining power
Steady-state productivity level

0.01
0.99
2.0
2/3
0.6
0.014
0.028
1.8
0.6
1.0

Table 1 summarizes the fixed parameters and Table 2 presents the calibrated parameter values. Without variable hours of work, the two models share the same parametervalues because the steady-state equations for the two models are identical. With variable
hours, for the same set of calibration targets, the two models require distinct sets of calibrated parameters. In the DMP model, e0 = 14.23 while in the DSGE model e0 = 27.64.
The calibrated value for the vacancy cost differs significantly for the models with andwithout variable hours. Without hours, c is calibrated to be 0.628 and this amounts to
c/Y is 22.0% of output. On the other hand, with hours of work, c = 0.022 and c/Y is 2.4%
of output. The latter figure is more consistent with the evidence.

15

Parameter

Table 2: Calibrated Parameters
Description
Models w/o h DMP with h

DSGE with h

m0
b
c
e0

Matching efficiency
0.471
Unemployment benefits 1.168
Vacancy cost
0.628
Disutility
-

0.471
0.393
0.022
27.64

5

Results

5.1

TFP Shocks

0.471
0.393
0.022
14.23

TFP follows a first order autoregressive process: log At log A = ρ A (log At 1 log A) +2 ). We set ρ
ε t ,where ε t
N (0, σAA = 0.612 and σA = 0.009 to match the first-order
autocorrelation and standard deviation of TFP in the data.
Table 3 reports the standard deviations of the unemployment rate, the vacancy rate,
employment, hours of work per employee, the (average) hourly wage rate, and the aggregate consumption, scaled by the standard deviation of TFP, which is 0.011. The standard
deviation of the stochastic discount factor St is 0.0003 for Model 2 and 0.0004 for Model4, and it is 0.002 in the data.
Table 3: TFP Shocks
Relative standard deviations
Û
V̂
lˆ
ĥ
ŵ
Ĉ
Data
Model 1: DMP w/o hours
Model 2: DSGE w/o hours
Model 3: DMP with hours
Model 4: DSGE with hours
DSGE with hours (σ = 10.0)

5.45
0.46
0.50
4.33
2.77
2.30

8.92 0.35 0.70 0.91
1.37 0.02
0.97
1.13 0.02
0.88
12.81 0.15 0.63 0.91
7.29 0.09 0.39 0.84
6.53 0.08 0.49 0.93

0.89
8.88
0.42
1.57
0.41
0.14

Table 3 gives us important insights. First, without the intensive margin, the DMP
model and the DSGE model with search-matching frictions generate nearly identicalquantitative results. Thus, we conclude that the general equilibrium effect through thestochastic discount factor does not matter in accounting for labor market fluctuations.
Further, in both models, fluctuations in unemployment, vacancies and employment are16

significantly smaller than those in the data (Shimer, 2005). This is primarily explainedby the small opportunity cost of employment b in the models without hours of work per
employee (Hagedorn and Manovskii, 2008; Chodorow-Reich and Karabarbounis, 2016).
Second, with the intensive margin, the DMP model generates realistic magnitudes offluctuations in these labor market variables (Kudoh et al., 2019). Interestingly, while theDSGE model with the intensive margin outperforms the one without, the magnitudes offluctuations are significantly below those of the DMP model with hours.
It is important to observe that the surplus argument (Hagedorn and Manovskii, 2008;
Ljungqvist and Sargent, 2017) cannot explain the differences in the magnitudes of labormarket fluctuations between the two models because the two models imply the samelevel of surplus under the same calibration strategy and targets. To see this, we computethe opportunity cost of employment. In the DMP model, it is given byz

DMP(1/3)1+1.8= 0.628.
1 + 1.8

h 1+ µ= 0.39 + 14.23= b + e ( h ) = b + e0
1+µIn the DSGE model, the opportunity cost of employment is
z

DSGE

= b+Λ

1

h 1+ µ= 0.39 + 0.722
e ( h ) = b + C e0
1+µ
σ

27.64

(1/3)1+1.8= 0.628.
1 + 1.8

The key to understanding the result is the propagation of shocks. Consider a positive
TFP shock. An increase in TFP increases the marginal product of capital. In the DMP
model, the demand for capital increases significantly because the rental price of capital is
assumed to be constant. In the DSGE model, part of the increase in the demand for capital
is offset by an increase in the rental price of capital because the loan market becomes tight.
However, comparison of Model 1 and Model 2 suggests that this channel is quantitatively
very weak.
The other general equilibrium effect, which works through a decline in the relative
price of consumption Λt , induces individuals to work less in response to an increase in
TFP by increasing the relative weight on disutility of work. The earnings functions forthe two models W (ht ) = η pt ht + (1 η )[e(ht ) + b] + ηcθt and W (ht ) = η pt ht + (1η )[Λt 1 e(ht ) + b] + ηcθt imply that the marginal hourly wage rate is W 0 (ht ) = η pt + (1µη )e0 (ht ) for the DMP model and W 0 (ht ) = η pt + (1 η )Λt 1 e0 (ht ) = η pt + (1 η )Ctσ e0 ht
for the DSGE model. The procyclicality of Ct implies that the marginal hourly wage rate
to be higher and hence hours and output to be lower during expansions in the DSGE model
than those in the DMP model. Through this effect, the DSGE model with the intensive
17

margin has the income effect of labor supply that dampens labor market fluctuations.
Comparison of Model 3 and Model 4 suggests that this effect is sizable.
Chodorow-Reich and Karabarbounis (2016) measure the opportunity cost of employment, which is z in our model, over the business cycle and find that it is procyclical andvolatile. They argue that the cyclicality of z dampens unemployment fluctuations in the
search and bargaining model of unemployment. However, Kudoh et al. (2019) find thatthe DMP model with a constant level of hours and the one with variable hours generate
the same unemployment volatility. This implies that the procyclicality of z alone does not
dampen unemployment volatility. Our results reveal that the income effect is essential
for this channel.
Table 4: The Opportunity Cost of Employmentz

std(ẑ)std(ẑ)/std( Â)corr (ẑ, Û )corr (ẑ, ŷ)

ztDMP = b + e (ht )ztDSGE = b + Λt 1 e(ht )0.007169
0.005147

0.662
0.475

-0.956-0.8470.99850.9983

Table 4 presents some statistics that characterize the business cycle properties of theopportunity cost of employment z for the DMP model and the DSGE model. Consistent
with Chodorow-Reich and Karabarbounis (2016), z is highly procyclical in both models.
A caveat is that while the procyclicality of z as well as its volatility are both stronger
in the DMP model than those in the DSGE model, the DMP model generates a higher
unemployment volatility than the DSGE model does. This result supports our claim thatthe procyclicality of z alone cannot dampen unemployment volatility.
Finally, consider how our results depend on the risk aversion parameter σ. Whenσ = 0, the DSGE model becomes identical to the DMP model. With a risk averse household, consumption smoothing motive becomes strong and hence consumption fluctuations are smaller under a higher risk aversion. This implies that under a higher risk
aversion, the stochastic discount factor fluctuates less, inducing less fluctuations in job
creation. However, as Table 3 suggests, the change in the level of σ does not generate aquantitatively large impact on the results.

5.2

Discount Factor Shocks

To focus on variations in the discount factor (i.e., the real interest rate) as the driver ofbusiness cycles, we shut down TFP fluctuations altogether and set At = 1 for all t. We
18

construct a quarterly series of the real interest rate to obtain a series of discount factor. We
assume that the discount factor follows log St log S = ρS (log St 1 log S) + ε S,t ,where
ε S,tN (0, σS2 ). We set ρS = 0.913 and σS = 0.002 based on the basic loan rate obtained
from the Bank of Japan.17
Table 5: Discount Factor Shocks
Relative standard deviations
Û
V̂
lˆ
ĥ
ŵ
Data
Model 1: DMP w/o hours
No direct impact on capital
Model 2: DSGE w/o hours
Model 3: DMP with hours
Model 4: DSGE with hours
DSGE with hours (σ = 10.0)

9.99
12.13
1.20
2.06
103.0916.28
11.44

16.38 0.64 1.29 1.67
23.71 0.41
12.87
2.36
0.04
0.50
3.31
0.07
1.97
201.51 3.48 7.34 12.28
28.55 0.55 2.86 2.76
21.60 0.39 3.22 2.40

Ĉ
1.63
193.76
0.03
3.65
11.46
3.39
0.76

Table 5 reports the standard deviations of the unemployment rate, the vacancy rate,
employment, hours of work per employee, the (average) hourly wage rate, and the aggregate consumption, scaled by the standard deviation of the discount factor, which is
0.002.
Consider the models without variable hours of work. Contrary to the findings in
Mukoyama (2009) and Hall (2017), our DMP model generates sufficiently large fluctuations in unemployment and vacancies. This is because our model includes capital formation. To see the importance of capital, consider the steady-state equilibrium conditions(23) and (24). According to (24), a reduction in the discount rate directly stimulates job
creation through the capitalization effect. In our model, there is an additional effect atwork: (23) implies that a reduction in the discount rate increases capital, raising the marginal product of labor and stimulates job creation. Once we drop the latter effect fromour model, we confirm Mukoyama (2009) and Hall (2017) that discount factor shocks
cannot account for labor market fluctuations. Thus, a quick remedy for the puzzle posedby Mukoyama (2009) and Hall (2017) is to introduce capital.
Even though introduction of capital enhances the model’s responsiveness to varia17 Providing the empirical analysis on the time series properties of the real interest rate in Japan is beyond

the scope of this paper as we focus on the implications of the presence/absence of the general equilibrium
effects.

19

tions in the discount factor, the general equilibrium effect through the endogenous realinterest rate significantly reduces the magnitudes of labor market fluctuations. Because
the loanable funds are constrained by the household’s intertemporal consumption choice,any increase in the demand for capital is associated with an increase in the real interest
rate, dampening the job-creation incentive. The same reasoning applies to the models
with variable hours of work. In the DMP model, all variables fluctuate too much to
explain the data. However, with the general equilibrium effects, the magnitudes of fluctuations are significantly reduced and we obtain somewhat reasonable results. As in the
results for productivity-driven business cycles, the risk-aversion parameter plays a minor
role in determining the magnitude of labor market fluctuations.

6

Conclusion

The general equilibrium effect through the stochastic discount factor (or the real interest rate) that potentially dampens TFP-driven labor market fluctuations is quantitatively
very weak. With variable hours of work, however, there is an additional general equilibrium effect, namely the income effect on labor supply, and this effect is quantitatively
strong. The cyclicality of the opportunity cost of employment is procyclical and volatile,
consistent with Chodorow-Reich and Karabarbounis (2016). However, it is not the cyclicality of the opportunity cost of employment itself that dampens unemployment volatility. The key is the response of the marginal hourly wage rate driven by the income effect. Our analysis highlights the importance of the marginal hourly wage rate in themacro-labor literature. An important direction of future research is to build an appropriate notion of wage rigidity in models with the intensive margin to further explore theimportance of the cyclicality of the opportunity cost of employment for unemployment
volatility.18
The general equilibrium effects matter very much in accounting for labor market dynamics driven by variations in the discount factor. We showed that, with or without
variable hours of work, the general equilibrium effects significantly dampen labor market fluctuations. Even without variable hours, the volatility of unemployment in the
DSGE model is about 1/6 of that of the DMP model. We also found that capital for18 With variable hours of work, there is a clear distinction between rigid wage rate and rigid earnings.

Kudoh et al. (2019) introduce the notion of contract rigidity to allow the earnings schedule to imperfectlyreflect the current level of TFP.

20

mation enhances the (otherwise weak) short-run capitalization effect. Our analysis suggests the importance of capital in magnifying labor market dynamics. This observation
is somewhat new because capital itself is known to play very little in accounting for output fluctuations (Cooley and Prescott, 1995) and influential DMP models of labor market
fluctuations such as Shimer (2005) do not consider capital. An important avenue for future research is to explore the role of capital over the business cycle.19 For instance, theinteraction between capital utilization and labor utilization over the business cycle is an
open question.

19 Earlier contributions in this direction are Hornstein et al. (2007) and Miyamoto (2011).

21

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