Hitotsubashi University
Institute of Innovation ResearchTime-Varying Employment Risks and Consumption:
A Quantitative General Equilibrium Study

Makoto Nirei
Sanjib Sarker

IIR Working Paper WP#08-04

September 2008

Institute of Innovation Research
Hitotsubashi University
Tokyo, Japan
http://www.iir.hit-u.ac.jp

Time-Varying Employment Risks and Consumption:
A Quantitative General Equilibrium Study
Makoto Nirei∗
Hitotsubashi University
Institute of Innovation Research, 2-1 Naka, Kunitachi, Tokyo 186-8603, Japan

Sanjib Sarker
Utah State University
Department of Economics, 3530 Old Main Hill, Logan, UT 84322, U.S.A.

September 12, 2008

Abstract
This paper quantifies the effect of time-varying employment risks on thefluctuations of aggregate consumption in a dynamic general equilibrium withincomplete markets. A government’s redistribution policy through provision ofunemployment insurance can cause a positive correlation between aggregate consumption and government’s payments due to precautionary savings effects. The∗

Corresponding author. E-mail: nirei@iir.hit-u.ac.jp. Phone: +81 (42) 580-8417. Fax: +81 (42)580-8410.

1

underlying mechanism is that a reduction of unemployment risk increases expected lifetime income substantially across a wide range of asset-holding groups
when the risk reduction is sufficiently persistent. By contrast, the correlation
between consumption and government becomes negative when the government
intervention hampers supply of goods.
Keywords: Time-varying idiosyncratic risk, employment risk, precautionary savings, regime-switching fiscal policy
JEL codes: E21, E62

1

Introduction

In this paper, we ask in what circumstances fiscal policies lead to the comovement of
aggregate consumption and government expenditure. We focus on the comovement
mechanism via time-varying unemployment risks. Consider a government project thatwill hire a fraction of unemployed workers. The reduced unemployment rate decreases
an idiosyncratic unemployment hazard that workers will face. This change in risk environment induces the workers to consume more, since they reduce the precautionarysavings in incomplete markets where consumers cannot insure against their idiosyncratic unemployment risk. The purpose of this paper is to quantify the movement in
consumption when various types of fiscal policies affect the magnitude of idiosyncratic
unemployment risks.
This paper is motivated by recent developments in empirical estimation on the effects of fiscal policy on consumption. Researchers such as Fatás and Mihov [10], Blanchard and Perotti [3], and Galı́, Valleés, and López-Salido [12] have found a positive
correlation between consumption and government spending in identified VAR estimation. Their findings contrast with the negative estimates obtained for war-time events2

by Ramey and Shapiro [17], Edelberg, Eichenbaum, and Fisher [8], and Burnside,
Eichenbaum, and Fisher [4].
Galı́ et al. proposes a model of rule-of-thumb consumers to account for the positive
comovement between consumption and government expenditure, whereas the standard
dynamic general equilibrium model with fiscal shocks such as Baxter and King [2] isconsistent with the negative correlation. We propose a different avenue to account forthe comovement. We highlight the macroeconomic effects of precautionary savings,
which have been analyzed by authors such as Aiyagari [1], Carroll [5], and Huggett[13].
The precautionary motive of savings provides a possible answer to the excess sensitivity of consumption which states that consumption seems to respond to a change inincome more than the permanent income hypothesis predicts. In incomplete markets
where there is no insurance provided for individual employment shocks, households
can partially insure against such hazard by accumulating wealth. The precautionary savings behavior implies a concave consumption function with respect to wealth.
Thus, a windfall of income to the households in a low wealth group would generate
an increase in consumption that is larger than the small wealth effect implied by thepermanent income hypothesis. If this channel of income effect is quantitatively large,
it can have important consequences on macroeconomic predictions such as the impactof the expansionary fiscal policy on the aggregate demand.
Krusell and Smith [14] formally analyze a dynamic stochastic general equilibrium
model with incomplete markets and with aggregate and idiosyncratic shocks. Theyfind that the consumption function in such an economy is almost linear in wealth,and therefore the aggregate consequence of incomplete markets in the business cycles
frequency is limited. Carroll [5] argues that Krusell and Smith’s model can underesti-

3

mate the effect of precautionary savings, because it generates a fairly centered wealthdistribution whereas the curvature of the consumption function concentrates on the
low wealth levels.
This paper focuses on a different channel of precautionary effects on consumption:
time-varying idiosyncratic risk. The demand for precautionary savings are affected by
the magnitude of idiosyncratic employment risk that individual households bear. We
consider that the magnitude of the employment risk fluctuates as the unemployment
rates vary. When the unemployment is high, there is a higher chance of losing jobs
for the households who are currently employed, and a lower chance of finding jobs for
those who are unemployed. Since the unemployment rate fluctuates in the businesscycles frequency, it is possible that the households’ savings decision also fluctuates inthis frequency. Namely, the households consume less when the unemployment rate
is high, because the high unemployment implies high risk for the current and future
employment and thus induces households to save so that they can partially hedge
against the risk.
There has been little formal quantitative analysis on the impact of the changing
unemployment risk on the aggregate consumption. We find that aggregate consumption
can respond strongly to the employment risk under conventional calibration. Consider
the case where a fiscal intervention reduces the unemployment rate. This policy has an
obvious effect on consumption through an increased level of employment: a fraction ofworkers switch from the unemployed status to the employed status, and they increase
consumption. There is a less obvious effect of this policy: all the households perceive
the reduction in the employment risk, and thus they start to dissave their wealth. Thiseffect is observed for a wide range of wealth levels of households. Thus, the wealthdistribution seems to matter less than the case of a windfall income.

4

It is hard to quantify the effect of precautionary savings analytically, and thus we use
the numerical method developed by Krusell and Smith. The original Krusell and Smith
model does contain a time-varying idiosyncratic risk, as the unemployment hazard in
the model depends on the time-varying aggregate state. However, it is hard to identify
the contribution of the time-varying idiosyncratic risk to consumption in their model,
because the magnitude of risk moves along with aggregate productivity. Instead, here
we associate the time-varying risk to switching regimes of fiscal policy. We consider two
kinds of fiscal policy. First, we consider a simple model of unemployment insurance.
In one regime, government is non-interventionist. In the other regime, government
provides a fraction of unemployed workers with unemployment compensation that isfinanced by lump sum taxation. In the second model, we consider a switching rate ofcorporate tax. The employment rate fluctuates along with the corporate tax, and sodoes the unemployment hazard rate.
Our main findings are following. First, a pure transfer of wealth from the employed to the unemployed through the unemployment insurance policy can increase
aggregate consumption, and thus cause a comovement of consumption and government
payments. While the correlation coefficient is substantial, the impact on consumption
is quantitatively small. Secondly, the impact on consumption becomes large, while thecorrelation between the consumption and government payments remains positive, if
the provision of the unemployment compensation comes along with an increase in the
resource available to the economy. We consider two such cases: the case of productive
government activity and the case of foreign trade. Thirdly, we consider the case ofswitching corporate tax rather than the unemployment insurance. A combination of areduced tax and a reduced government spending enhances the employment of private
firms. In this case, the magnitude of consumption response is as large as the case of5

productive government, while the correlation between consumption and government
spending becomes negative in a balanced budget scheme. These findings seem to fit
well with the mixed results on the comovement between consumption and government
spending found in the empirical literature.
Government’s employment is similar to the unemployment insurance in the function that they both transfer wealth from the employed workers to the unemployed.
Finn [11] studies the macroeconomic effects of the government’s employment in a dynamic general equilibrium model with complete markets. Cavallo [6] finds that themodel with government labor purchases helps explain the consumption response in
the Ramey-Shapiro episodes. Our contribution in this context is to provide quantitative assessments of the effect of government labor expenditures on consumption inan economy with incomplete markets, where the precautionary savings effect plays animportant role. The link between the unemployment insurance and the precautionarysavings is pursued by Engen and Gruber [9], who find evidence for the effect that theinsurance reduces the savings in households data.
Next section presents the model economy. Section 3 shows main results, and Section
4 concludes the paper. Details of computation is deferred to Appendix.

2

Model

We consider a dynamic stochastic general equilibrium model with incomplete markets,
uninsurable employment shocks, and aggregate shocks as Krusell and Smith [14] (KShenceforth). The economy is populated by a continuum of households with population

6

normalized to 1. The household maximizes the utility subject to the budget constraints:
max E0ct ,kt+1

s.t.

∞
X

β t c1−σ/(1 − σ)
t

(1)

t=0

ct + kt+1 = (rt + 1 − δ)kt + wt ht − τt ,

∀t

(2)

where ct is the consumption, kt is the capital asset, ht is the labor, τt is the lump sum
tax, rt is the gross return to capital and wt is the real wage where the consumption
good is the numeraire. The capital depreciates at the rate δ, and the future utility is
discounted by β. The household is either unemployed (ht = 0) or employed (ht = 1),
and ht follows an exogenous process as discussed shortly.
The representative firm produces goods with the technology specified by a CobbDouglas production function with constant returns to scale Yt = Ktα Ht1−α where Yt is
the aggregate goods produced and Kt and Ht are the aggregate capital and labor. The
firm maximizes its profit in a competitive market, where the following conditions hold:rt = α(Kt /Ht )α−1(3)wt = (1 − α)(Kt /Ht )α .

(4)

Our model features a fiscal policy as the aggregate shock. We first consider a partial
provision of an unemployment insurance by the government. The fiscal policy zt follows
a Markov process with two states {0, 1} and with a transition matrix [πzz0 ]. The government is inactive in state zt = 0. The lump sum tax is set at zero and the aggregate
unemployment rate stays at a high rate u0 . In state zt = 1, the government provides afull unemployment compensation wt for a fraction of the unemployed households. We
can interpret the unemployment compensation as a “buy-out” of the fraction of theunemployed labor force and thus it constitutes a wage expenditure of the government.
The fraction of the households who are neither employed by the firms nor bought-out
7

by the government is u1 , which is strictly less than u0 . The lump-sum tax is equal
to the aggregate amount of contemporaneous unemployment compensation. Thus, thelump sum tax per worker is:
τt =


 0

if zt = 0(5)

 w (u − u ) if z = 1t 0
1t

In this set up, the aggregate labor supplied for firms is exogenously constant at Ht =1 − u1 for any t regardless of zt , whereas the total workers employed by firms or
government is either 1 − u0 or 1 − u1 depending on zt .
We consider that the aggregate shock zt affects the transition probability of theindividual employment status ht . Let Π denote the transition matrix for the pair of
the individual labor and the fiscal policy status, (ht , zt ). The transition probability from(h, z) to (h0 , z 0 ) is denoted by πhh0 zz0 . In our model, the aggregate shock z determines
both the employment level and the fiscal policy regime. We set the regime switchingprobability so that the average duration of each regime is 8 quarters in the benchmark
calibration, following KS. In an alternative calibration, we set the transition probability
by using Davig’s [7] estimates on the regime switching of the US fiscal policy.
A recursive competitive equilibrium is defined as follows. The household’s maximization problem is written as a dynamic programming with state variables (k, h, z, Γ)
where Γ is the cross-section distribution of (ki , hi ) across households i ∈ [0, 1]. The
law of motion for (h, z) is determined by the exogenous transition matrix Π. Define
a transition function T that maps Γ to the next period distribution Γ0 . The recursivecompetitive equilibrium is defined by the value function V (k, h, z, Γ), the policy function F of the household, and the transition function T , such that V and F solve thehousehold’s problem under T and the competitive factor prices that satisfy (3,4), thatR
R
they are consistent with the market clearing conditions K = ki dΓ and H = hi dΓ,8and that T is consistent with F and Π. The goods market clears by Walras’s law,R
C + K 0 − (1 − δ)K = Y , where C = ci di is the aggregate consumption.
We calibrate parameters largely following KS for the sake of comparison. The
transition matrix Π must satisfy:uz (π00zz0 /πzz0 ) + (1 − uz )(π10zz0 /πzz0 ) = uz0 ,z ∈ {0, 1}(6)

to be compatible with the aggregate labor employed by firms or government, 1 − u. We
also restrict Π so that the mean duration of unemployment is 1.5 quarters for the state
0 and 2.5 for 1, and that π0001 = 0.75π0011 and π0010 = 1.25π0011 following KS. These
restrictions fully determine Π. The calibration of the other parameters draws on KS
as α = 0.36, β = 0.99, δ = 0.025, σ = 1, u0 = 0.1, and u1 = 0.04. We approximate thetransition function T by a linear mapping of log K. Following Mukoyama and Şahin[15], we specify that the slope of the function is common but the constants can varyacross z:
log K 0 = az + b log Kz + ,z ∈ {0, 1}(7)

Simulations show that the linear transition function on the first moment provides a
sufficiently accurate forecast on the future aggregate capital as in KS.

3

Results

3.1

Unemployment insurance

We first consider the model with unemployment insurance which is financed by contemporaneous lump sum tax (5), leaving the government budget balanced all the time.
This is a pure transfer policy that levies lump sum tax and distributes the proceeds toa fraction u0 − u1 of randomly selected unemployed workers. Aggregate production is9not affected by this policy, unless capital level is changed. Table 1 shows the consumption for different states. Cz denotes the average aggregate consumption for policy statez. Cze and Czu denote the average consumption for the employed and the unemployed,
respectively, for z. In parentheses are the standard errors for the estimated moments
obtained by 30 iterated runs. The current model specification corresponds to the policy
regime “UI I” in the table.
In Table 1, we note that the consumption of the unemployed increases by 1.54%
by the provision of unemployment insurance (the policy transition from 0 to 1). Thisshows the precautionary savings effect: since the government policy reduces the unemployment hazard, the households with low current income and wealth is inclined toincrease consumption. Next, we note that the average consumption of the employed isreduced by 0.21%. This is because the employed suffers from the policy that transfersa part of their wealth to the unemployed.
The overall consumption is increased by 0.03% by the policy. Note that the precautionary motives affect all groups of workers. The lump-sum transfer per se also
affects the consumption of all workers negatively. The point of the exercise here is toquantify the difference in the effects of the two forces on the employed and unemployedgroups. We observe a positive overall impact of the transfer policy on the aggregate
consumption, which indicates that the positive precautionary effect outweighs the negative wealth effect.
To distinguish the precautionary effect, we decompose the overall effect as in Table
2. We consider three groups of workers: u1 = 4% of workers who remain unemployed
before and after the policy transition, u0 − u1 = 6% of workers who transit from theunemployed to the employed by the policy, and 1 − u1 = 90% of workers who remain
employed before and after. Note that the composition of each group reshuffles in each

10

UI I

UI II

UI III

z

Cze

Czu

Cz

Cze

Czu

Cz

Cze

Czu

Cz

0

2.47662.40832.46982.52532.46422.51922.38302.30382.3751(0.0001) (0.0006)(0.0000) (0.0018) (0.0025)(0.0019) (0.0018) (0.0025)(0.0019)

2.47152.47042.55392.38252.40831

2.44582.55462.53732.4094(0.0001) (0.0004)(0.0001) (0.0017) (0.0022)(0.0017) (0.0018) (0.0022)(0.0018)-0.00210.00030.01370.0139log diff.

0.01540.01150.02930.01100.0336

Table 1: Consumption changes in policy transition for average workers in different
groups. UI I is the case of unemployment insurance as a means of pure wealth transfer,
UI II is the case of government’s productive employment, and UI III is the case ofstorage or trade by government.

(1 − u0 ) log C1e /C0e

u1 log C1u /C0u

(u0 − u1 ) log C1e /C0u

log C1 /C0

UI I-0.00190.00060.00160.0003

UI II

0.01040.00120.00220.0137UI III

0.00990.00130.00270.0139

Table 2: Contributions to aggregate consumption growth by different groups of workers

11period during the new policy regime, but it does not affect the aggregate property of
the group, because each group consists of uncountably many workers who are ex-ante
identical. Each of the workers in the first group (unemployed to unemployed) increases
consumption by 1.54%, and thus the group as a whole contributes to the rise of total
consumption by 0.06 percentage. Each worker in the second group (unemployed to
employed) increases consumption by 2.59% (log C1e − log C0u ), which amounts to 0.16
percentage rise in total consumption. Finally, each worker in the third group (employed
to employed) reduces the consumption by 0.21%, which results in 0.19 percentage
reduction in total consumption. The net increase in overall consumption is 0.03%.
Note that the fall in consumption by the employed workers in the third group is almost
cancelled out by the increase in consumption by the workers in the second group who
newly receive compensations from the government. This corresponds to the direct
effect of the wealth transfer from the employed to the unemployed. The net increase intotal consumption thus comes from the contribution of the unemployed workers, who
dissave precautionary savings due to the reduced unemployment risk. This shows thatthe precautionary savings effect is present in aggregation, and that its quantitative
importance is limited at least in the case of pure wealth transfer.
Table 3 summarizes the key aggregate statistics obtained by the simulation. Government expenditure Gt is the payment for the unemployed, and satisfies Gt = τt .
We note that Gt correlates positively with aggregate consumption Ct . Thus, the government’s transfer of wealth from the employed to the unemployed causes a positive
correlation with total consumption. The correlation between C and G is fairly high
at 0.43. We should note, however, that the magnitude of the movement is small, as
the standard deviation of the total consumption is only 0.03%. We also observe thatthe output and government spending are almost uncorrelated. This is because the

12

UI I

UI II

UI III

Tax I

Tax II

s.d. Ys.d. Ccorr(Y, C)

corr(Y, I)

corr(Y, G)

corr(C, G)

0.00020.00030.27240.5116-0.04910.4352(0.0000) (0.0000)(0.0167)(0.0311)(0.0225)(0.0228)0.02340.01250.73630.93260.97070.5525(0.0002) (0.0004)(0.0068)(0.0036)(0.0013)(0.0040)0.00610.01290.9642-0.7720

1.00000.9642(0.0002) (0.0005)(0.0030)(0.0149)(0.0000)(0.0030)

0.02350.01230.61570.9240-0.9574-0.3623(0.0002) (0.0004)(0.0124)(0.0034)(0.0016)(0.0094)

0.02540.01890.74580.8529-0.9191-0.4235(0.0004) (0.0007)(0.0110)(0.0073)(0.0033)(0.0074)

Table 3: Second moments for different policy schemes

13

UI I

UI II

UI III

Tax I

Tax IIC e /C u

C

I/Y

k

R̂2

â0

â1

b̂

1.01972.47020.252333.3489

0.99860.00530.0054

0.9985(0.0005) (0.0000)(0.0000) (0.0030)(0.0002)(0.0000)(0.0000)(0.0000)

1.01570.245132.9508

0.99990.13580.1321

0.9616(0.0005) (0.0022)(0.0001) (0.0375)(0.0000)(0.0000)(0.0000)(0.0000)

1.02290.270332.6184

1.00000.13770.13390.9610(0.0006) (0.0025)(0.0004) (0.0412)(0.0000)(0.0000)(0.0000)(0.0000)

1.01530.242932.47111.00000.13220.1281

0.9626(0.0003) (0.0015)(0.0001) (0.0290)(0.0000)(0.0000)(0.0000)(0.0000)

1.0156

0.268532.4567

0.99990.13260.1266

0.9627(0.0001) (0.0705)(0.0000)(0.0000)(0.0000)(0.0000)

2.53672.39152.53012.4438(0.0006) (0.0037)

Table 4: Other estimates
workers hired by the government are not productive in this experiment. Thus, thelevel of productive employment stays fixed during policy transitions. The fluctuationof output is solely caused by the small movement of capital. Table 4 lists the otherestimates. The column C e /C u shows the ratio of average consumptions between theemployed and unemployed. While households partially hedge the unemployment riskby accumulating wealth, a substantial gap (1.97%) remains uninsured. The last threecolumns show the approximated law of motion for the capital distribution. The R2
shows that the approximation is quite accurate.
The benchmark model above indicates the positive effect on consumption causedby the provision of unemployment insurance through precautionary motives of savings. However, the effect is quantitatively small, because the policy does not affect14

production. From now on, we explore the models in which a rise in employment by thegovernment policy increases contemporaneous resources available to the economy.
First we consider a variation of the benchmark model, in which government has alinear technology to produce goods. Namely, the government employs u0 − u1 workers
and produces (u0 − u1 )wt value added. The employment is financed by the sales of thegoods at the goods market, and thus no tax is levied: τt = 0 for all t. This is a modelof productive government activities. A linear production function is natural for thegovernment, because GDP statistics include government’s salary expenditure as value
added. The government’s output adds to the supply side of the goods market, thus the
goods market clearing condition becomes C + K 0 − (1 − δ)K = Y + (u0 − u1 )w.
The simulation results for this model of productive government are reported under“UI II” in the tables. We first note that the output and consumption fluctuates much
more than the benchmark case UI I. In Table 3, the standard deviations of output andconsumption amounts to 2.34% and 1.25%, respectively. This is a direct consequenceof the switching activities of the productive government.
Second, we note that the consumption is increased for the employed as well as for theunemployed in the periods of insurance provision. Table 1 shows that the consumptionof the employed increases by 1.15%. In the benchmark setup, the consumption of theemployed decreases because they have to pay the lump sum tax during the regimeof insurance provision. In the present setup, there is no tax levied for the insurance
policy. Thus, the expected lifetime income of the employed workers increases by theprospect of less unemployment hazard in the near future. This reduces the need forprecautionary savings and thus increases the consumption of the employed. Whilethe increase in average consumption of the employed is modest at 1.15% comparedto 2.93% increase for the unemployed, its contribution to the total consumption is15

large because of its large share among workers. As seen in the decomposition in Table
2, the employed groups contributes more than three quarters of the increase in total
consumption (1.04/1.37). The correlation between C and G is also high at 0.55.
Note that the wage rate is determined only by the capital level and not affectedby the policy regime, because the employment of firms does not change by the policy.
Thus, when the policy switches, the expected lifetime income changes largely throughthe change in perceived employment risks in future. The big impact on the consumptionof the employed workers indicates that the risk environment is important in determining
the consumption demand of the large mass of workers.
We consider another variation of the benchmark model, in which government has
a storage technology. The government collects lump-sum tax in every period andstores the proceeds. The government transfers the storage to the unemployed workers
during the periods when it adopts the unemployment insurance policy. One example ofsuch storage technology is a trade with foreign countries. Government can accumulate
foreign assets during the inactive periods, and use the assets to import goods in the
periods of active policy. To finance the storage/trading activities, the government
collects lump sum tax that is constant across periods. In this setup, workers face a
constant tax burden across time, and the government budget is generally imbalanced
as the policy switches.
The simulation results for this model are reported under “UI III” in the tables.
We observe that the statistics are similar to the case of productive government (UIII): the output and consumption fluctuate much more than the benchmark case UI I,
and the correlation between consumption and government payment is positive. The
consumption of the employed is increased during the active policy periods by 1.1%, andit accounts for 71% of the total consumption increase. The mechanism is similar as the16

productive government model. Since the policy regime does not affect the wage nor
the tax the workers pay in every period, the increase in the expected lifetime income
is largely caused by the prospect of less unemployment hazard in future.

3.2

Corporate tax

In this section, we consider an alternative government intervention. We replace theunemployment compensation program with a regime-switching tax rate as studied in
Davig [7]. We will see that the government-consumption correlation hinges on how
the government intervention affects the contemporaneous resources available to theeconomy.
We consider that the government levies a flat-rate tax on firms’ revenue. The tax
rate ξt fluctuates between two states according to the Markov process specified by Π.
We continue to assume the exogenous aggregate employment process that fluctuates
between two states u0 and u1 along with the policy status z ∈ {0, 1}. An implicit
mechanism underlying the exogenous employment process is that, when the tax rate
is low, labor demand shifts out and employment is increased. The production factors
are paid for their after-tax marginal products: rt = (1 − ξt )α(Kt /(1 − ut ))α−1 and
wt = (1 − ξt )(1 − α)(Kt /(1 − ut ))α . We set the tax rate so that real wage is independentof the policy status. Then the tax rate is set as:
ξ(z) = 1 − (1 − uz )α ,z = 0, 1.

(8)

When zt = 0, the tax is high at ξ(0) and the unemployment is high at u0 . When zt = 1,
the tax is low at ξ(1) and the unemployment is low at u1 .
We consider two cases for the government expenditure. In the first case, which wecall “Tax I”, the tax proceeds are rebated to the households in a lump sum manner. By
17

Tax I

Tax II

z

Cze

Czu

Cz

Cze

Czu

Cz

0

2.52382.46662.51812.42912.37102.4233(0.0015) (0.0020)(0.0015) (0.0032) (0.0040)(0.0033)

2.54252.54172.44792.4642(0.0014) (0.0018)(0.0014) (0.0029) (0.0033)(0.0029)0.00740.00930.01681

log diff.

2.52380.02292.46490.0146

0.0319

Table 5: Consumption changes in policy transition for average workers in different
groups. Tax I is the case of corporate tax with lump sum rebates, and Tax II is thecase of corporate tax and wasteful government spending.
(1 − u1 ) log C1e /C0e

u1 log C1u /C0u

(u0 − u1 ) log C1e /C0u

log C1 /C0

Tax I

0.00660.00090.00180.0093Tax II

0.01320.00130.00230.0168

Table 6: Contributions to aggregate consumption growth by different groups of workers:
Case of corporate tax
abuse of notation, we redefine −τt as the lump sum transfer. Then −τt = ξt Yt . Using
this notation, the household’s budget constraint continues to be written as (2). In thesecond case (“Tax II”), the tax proceeds are used by the government for non-productive
activities (i.e., thrown into the ocean). In this case, the transfer τt is zero for everyt. The government expenditure Gt is equal to the tax proceeds ξt Yt , and it appears in
the demand side of the goods market clearing condition: C + K 0 − (1 − δ)K + G = Y .
Table 5 shows the consumption for various states. We note that the consumption

18

increases in the periods of low tax for both the employed and the unemployed workers
in both models Tax I and II. Table 6 shows the decomposition of the total consumption
growth into the contributions of the groups of workers according to the employment
status. The contribution of the employed workers is substantial in both models: 71%
in Tax I and 79% in Tax II.
In Tax I, the tax proceeds are rebated back to the households, so the tax is adistortionary transfer from the firms to the households. The lowered tax rate induceshigher labor demand and larger output. While the real wage is held fixed, the lumpsum transfer to the households is reduced during the low tax periods. The reduced
transfer income should hurt the consumption demand of the unemployed. Nonetheless,the consumption of the unemployed is increased by 2.29% by the tax reduction. Thisshows that the wealth effect of the prospect of low unemployment hazard overwhelms
the effect of less transfer income.
This wealth effect can be directly observed in Tax II. In Tax II, both the real wage
and the government transfer (zero) are fixed during the policy transitions, and thus
the contemporaneous income of the employed workers is not affected by the policy atall. Table 5 shows that the consumption of the employed is increased considerably by
1.46%.
The magnitude of the fluctuations in consumption and output is as large in thecorporate tax model as the productive government model of the unemployment insurance (UI II) as seen in Table 3. This is because the reduction of the corporate tax
increases productive employments in the firms sector. The corporate tax models andthe productive government model share the similar supply side mechanism and theconsumer behavior, which leads to the similarity in the moment properties in output,
consumption, and investment.

19

The correlations between government expenditure (or transfer) with output andconsumption become negative in the tax models. The reason is that the government’s
intervention is distortionary and suppresses labor input, and thus less government
activities induce more production and consumption. This result contrasts with themodel of unemployment insurance. This shows that the correlation of the government
expenditure and consumption depends on whether the government activity suppresses
production or not.
We conduct two sets of sensitivity analysis. First, we increase the risk aversion
parameter to σ = 3. The results are shown in Appendix C. We note that the consumption effect of pure wealth transfer in UI I is doubled. This is consistent with ourintuition: the households who are sensitive to risks respond more to the reduced risks.
Other statistics are similar to the case of σ = 1. In the second sensitivity analysis, we
set the parameters following Davig’s [7] estimation of the regime-switching fiscal policy.
The results are shown in Appendix D. Overall, we observe more fluctuations and larger
responses to the policy. This is because the average duration of policy regimes is much
longer under this calibration. The results are consistent with our expectation that theeffect of policy is amplified when the policy regime is more persistent.

4

Conclusion

This paper quantitatively studies the dynamic stochastic general equilibrium modelwith idiosyncratic employment risk, when the magnitude of employment risk is changed
over time as fiscal policies switch between two regimes stochastically. In the experiments, we consider two kinds of fiscal policies: unemployment insurance and corporate
tax. The unemployment insurance model provides a simple case which facilitates our20interpretations of the result, whereas the corporate tax model examines the case where
government activities hinder private sector’s production.
We find that the government policy that reduces the unemployment hazard canincrease the aggregate consumption demand by a non-negligible magnitude. In a pure
wealth transfer from the employed to the unemployed workers, we observe a positive
correlation between the aggregate consumption and the government’s transfer payments, but it’s magnitude is small. The correlation is positive and the impact of policy
on consumption becomes large, if the unemployment insurance program enhances theresources available to the economy. We show two examples: the case where the government’s employment generates value added, and the case where the government cantrade with foreign countries. Finally, we find that the correlation between the consumption and government spending is negative, when we consider a distortionary corporate
tax. This shows that the consumption-government correlation depends on how thepolicy affects the supply of goods.
We decompose the impact of the policy on the aggregate consumption into threeeffects: the increased fraction of employed households, the reduced unemployment hazard for the employed workers, and the increased employment chance for the unemployedworkers. We find the effect of reduced employment hazard for the employed workers
considerably large, when the tax burden for the employed workers is held fixed. The
effect of the reduced risk can be large, because it affects not only the unemployed but
also a wide range of the employed households. It makes a contrast with the effect ofa windfall income which affects a relatively small fraction of workers whose borrowing
constraints are binding.

21

Appendix
A

Details of computation

The state space for the household’s capital ki is discretized by 1000 grids equallydivided in the range [−3, 100]. The lower bound is chosen so that the gap in the
consumption growth rates between the low asset and the high asset holders roughlymatches with Zeldes’ estimate [18, 16]. The upper bound is chosen to be high enoughso that households do not reach the upper bound in simulated paths. The number ofthe grids is chosen to be high enough so that the further increase of the grids does notchange simulated mean capital. The state space for the mean capital is discretized by
five grids.
Given the approximated law of motion of the joint distribution of the capital holding
and the employment states, the value function is obtained by the iteration of the
Bellman equation. To evaluate the value function at the forecasted mean capital in thenext period, we interpolate the value function in the dimension of the mean capital by
the spline method.
Once the value function is obtained, we simulate the equilibrium path with 1000
households for 10000 periods. In each period of the simulation, the policy function isinterpolated at the current mean capital level by the spline method, and the interpolated policy function evaluated at the current mean capital and the current aggregate
state is further fitted by a quadratic function for each employment state. Fitting by
the higher-degree polynomial functions does not alter the results. The fitted functionis then used to compute the next-period capital holding for each household. The simulated mean capital path for the last 9000 periods is used to estimate the law of motion

22

of the form (7). The tolerance for the value function iteration is 0.01 in the sup norm.
The tolerance for the law of motion is 0.001 for the coefficients in (7).

23

B

Simulation plots

B.1

Sample paths of output and consumption for different

3.5

3.5

3

3

3

2.5

2.5

2.5

2

2

2

1.5

1.5

1

1
OutputConsumption of an employed
Consumption of an unemployed
z

0.5
0
800

820

840

860

880

Y,C

3.5

Y,C

Y,C

models1.5
1OutputConsumption of an employed
Consumption of an unemployed
z

0.5
0
800

900

820

840

3.5

3.5

3

3

2.5

2.5

2

2

1.5

880

900

0
800

820

840

860
Quarter

1.5

11OutputConsumption of an employed
Consumption of an unemployed
z

0.5
0
800

860
Quarter

Y,C

Y,C

Quarter

OutputConsumption of an employed
Consumption of an unemployed
z

0.5

820

840

860

880

900

OutputConsumption of an employed
Consumption of an unemployed
z

0.5
0
800

820

Quarter

840

860

880

900

Quarter

Figure 1: Sample time series of Y and C. The policy regime z is shown by the dottedline. Top: UI I, II, and III. Bottom: Tax I and II. Note that there is little output
volatility for UI I and UI III.

24

880

900

B.2

Computed functions for UI II when σ = 1
100
95
90

80

next period k

85

v

80
75
70
Least mean kSecond least mean kMiddle mean kSecond largest mean kLargest mean k

65
60
55
0

20

40

60

80

60

40

20
Employed
Unemployed

0

100

0

20

40

k

60

80

100

k

300
3.53
250

3.52

200
Frequency

mean k(t+1)

3.51
3.5
3.49
3.48

150
100

3.47
50
3.46

Actual dynamics of mean kForecasted mean k
3.46

3.47

3.48

3.49 3.5
mean k(t)

3.51

3.52

0
0

3.53

10

20

30

40

50

60

70

k

Figure 2: From top left clockwise, value functions, policy functions at medium capitallevel, cross-section distribution of capital, and the law of motion of capital and its
approximation, for model UI II and σ = 1.

25

B.3

Computed functions for UI II when σ = 3
100
−4

80

next period k

v

−6

−8

−10

Least mean kSecond least mean kMiddle mean kSecond largest mean kLargest mean k

−12
0

20

40

60

80

60

40

20
Employed
Unemployed

0

100

0

20

k

40

60

80

100

k

250
3.48
200

3.44

Frequency

mean k(t+1)

3.46

3.42

150

100

3.4
50

3.38
Actual dynamics of mean kForecasted mean k

3.36
3.36

3.38

3.4

3.42
3.44
mean k(t)

3.46

0
10

3.48

20

30
k

Figure 3: Same plot as Figure 2 when σ = 3

26

40

50

C

Sensitivity analysis I: Higher risk aversion
UI I

UI II

UI III

z

Cze

Czu

Cz

Cze

Czu

Cz

Cze

Czu

Cz

0

2.44422.32502.43232.50462.38582.49272.36882.23352.3553(0.0001) (0.0006)(0.0001) (0.0023) (0.0032)(0.0024) (0.0018) (0.0027)(0.0019)2.43632.43392.52002.31642.38241

2.37492.52232.46632.3852(0.0001) (0.0004)(0.0001) (0.0023) (0.0027)(0.0023) (0.0021) (0.0024)(0.0021)-0.0032

0.00070.01090.0114log diff.

0.02120.00700.03320.0069

Table 7: Same as Table 1 when σ = 3(1 − u0 ) log C1e /C0e

u1 log C1u /C0u

(u0 − u1 ) log C1e /C0u

log C1 /C0

UI I-0.00290.00080.00280.0007

UI II

0.00630.00130.00330.0109UI III

0.00620.00150.00390.0114

Table 8: Same as Table 2 when σ = 3270.0364

UI I

UI II

UI III

Tax I

Tax II

s.d. Ys.d. Ccorr(Y, C)

corr(Y, I)

corr(Y, G)

corr(C, G)

0.00060.0006

0.41250.6351-0.04200.5238(0.0000) (0.0000)(0.0226)(0.0281)(0.0248)(0.0189)0.02420.01180.72870.9442

0.94720.4710(0.0003) (0.0004)(0.0076)(0.0025)(0.0024)(0.0036)

0.00790.01190.9667-0.2466

1.00000.9667(0.0003) (0.0005)(0.0029)(0.0167)(0.0000)(0.0029)0.02420.01160.6922

0.9421-0.9302-0.3796(0.0003) (0.0005)(0.0127)(0.0032)(0.0041)(0.0062)0.02700.01790.79640.8985-0.8678-0.3913(0.0006) (0.0008)(0.0112)(0.0047)(0.0070)(0.0042)

Table 9: Same as Table 3 when σ = 3

28

UI I

UI II

UI III

Tax I

Tax IIC e /C u

C

I/Y

k

R̂2

â0

â1

b̂

1.03852.4331

0.236930.2065

0.99940.00230.00230.9993(0.0005) (0.0001)(0.0000) (0.0067)(0.0001)(0.0000)(0.0000)(0.0000)

1.03610.233930.6059

1.00000.08680.08270.9752(0.0007) (0.0028)(0.0002) (0.0668)(0.0000)(0.0000)(0.0000)(0)

1.04490.259830.5267

1.00000.08950.08550.9744(0.0006) (0.0023)(0.0002) (0.0550)(0.0000)(0.0000)(0.0000)(0)

1.02950.232130.2269

1.00000.08190.0777

0.9766(0.0007) (0.0032)(0.0003) (0.0813)(0.0000)(0.0000)(0)(0.0000)

1.03510.257630.2020

1.00000.08380.0776

0.9763(0.0001) (0.1032)(0.0000)(0)(0)(0.0000)

2.50632.36882.50032.4164(0.0007) (0.0040)

Table 10: Same as Table 4 when σ = 3

Tax I

Tax II

z

Cze

Czu

Cz

Cze

Czu

Cz

0

2.49852.40002.48872.40982.29812.3986(0.0028) (0.0037)(0.0029) (0.0035) (0.0044)(0.0036)

2.51362.51182.38462.4349(0.0026) (0.0030)(0.0026) (0.0037) (0.0044)(0.0037)

0.00600.00930.01501

log diff.

2.46860.02822.43690.01120.0369

Table 11: Same as Table 5 when σ = 329

(1 − u1 ) log C1e /C0e

u1 log C1u /C0u

(u0 − u1 ) log C1e /C0u

log C1 /C0

Tax I

0.00540.00110.00280.0093Tax II

0.01010.00150.00350.0150

Table 12: Same as Table 6 when σ = 330

DSensitivity analysis II: Persistent policy

Parameter values are changed as α = 0.33 and β = 0.9916. Most noticeably, thetransition probability of the policy πzz0 is changed. Following Davig’s estimates, thepolicy switches from the high tax regime to the low tax regime in average 25 years,
and the low to high tax regime in average 11 years. This is much more persistent thanthe benchmark calibration at 8 quarters.
UI I

UI II

UI III

z

Cze

Czu

Cz

Cze

Czu

Cz

Cze

Czu

Cz

0

2.0933

1.98332.0823

2.10601.99282.09471.9924

1.8524

1.9784(0.0002) (0.0009)(0.0002) (0.0018) (0.0033)(0.0019) (0.0016) (0.0025)(0.0017)2.08562.08452.20302.04792.08511

2.05672.20392.17992.0866(0.0001) (0.0004)(0.0001) (0.0008) (0.0012)(0.0008) (0.0008) (0.0011)(0.0008)-0.00370.00110.05040.0525log diff.

0.03630.04540.08970.0462

Table 13: Same as Table 1 for persistent policy(1 − u0 ) log C1e /C0e

u1 log C1u /C0u

(u0 − u1 ) log C1e /C0u

log C1 /C0

UI I-0.00330.00150.00300.0011

UI II

0.04090.00360.00600.0504UI III

0.04160.00400.0071

0.0525

Table 14: Same as Table 2 for persistent policy
31

0.1003

UI I

UI II

UI III

Tax I

Tax II

s.d. Ys.d. Ccorr(Y, C)

corr(Y, I)

corr(Y, G)

corr(C, G)

0.00080.00090.32960.6601-0.59650.5204(0.0000) (0.0000)(0.0177)(0.0236)(0.0230)(0.0382)

0.02310.02350.99180.9162

0.99230.9686(0.0010) (0.0011)(0.0005)(0.0016)(0.0004)(0.0018)0.00470.0246

0.8945-0.81811.00000.8945(0.0002) (0.0010)(0.0127)(0.0201)(0.0000)(0.0127)

0.02840.02300.92450.8762-0.9595-0.7798(0.0011) (0.0010)(0.0061)(0.0013)(0.0015)(0.0133)

0.03080.03350.96830.7425-0.9443-0.8324(0.0013) (0.0017)(0.0033)(0.0034)(0.0027)(0.0118)

Table 15: Same as Table 3 for persistent policy

32

UI I

UI II

UI III

Tax I

Tax IIC e /C u

C

I/Y

k

R̂2

â0

â1

b̂

1.02592.08390.240026.3253

0.99950.0662

0.0664

0.9797(0.0012) (0.0000)(0.0000) (0.0058)(0.0000)(0)(0.0000)(0.0000)

1.02390.233126.3967

0.99960.14210.14010.9568(0.0020) (0.0051)(0.0002) (0.0361)(0.0001)(0.0000)(0.0000)(0.0000)

1.03450.256226.88440.99720.14510.14310.9559(0.0020) (0.0042)(0.0012) (0.0328)(0.0002)(0)(0.0000)(0.0000)

1.0194

0.236626.97540.99990.13350.12970.9597(0.0015) (0.0048)(0.0002) (0.0838)(0.0000)(0.0000)(0)(0.0000)

1.02930.259727.54220.99910.13380.12910.9597(0.0004) (0.1133)(0.0001)(0.0000)(0.0000)(0)2.17072.0536

2.17572.1231(0.0019) (0.0071)

Table 16: Same as Table 4 for persistent policy

Tax I

Tax II

z

Cze

Czu

Cz

Cze

Czu

Cz

0

2.12492.02772.11512.0451

1.92662.0333(0.0034) (0.0043)(0.0035) (0.0050) (0.0063)(0.0051)2.20232.20162.13122.1639(0.0018) (0.0023)(0.0018) (0.0023) (0.0028)(0.0023)0.03580.04000.0623

1

log diff.

2.18460.07452.16530.05710.1009

Table 17: Same as Table 5 for persistent policy33

(1 − u1 ) log C1e /C0e

u1 log C1u /C0u

(u0 − u1 ) log C1e /C0u

log C1 /C0

Tax I

0.03220.00300.00500.0400Tax II

0.05140.00400.00700.0623

Table 18: Same as Table 6 for persistent policy

34

E

Other plots

E.1

Computed functions for the models other than UI II
100

100

80

next period k

90

v

80

70
Least mean kSecond least mean kMiddle mean kSecond largest mean kLargest mean k

60

50
0

20

40

60

80

60

40

20
Employed
Unemployed

0

100

0

20

40

60

80

100

k

3.4845

300

3.484

250

3.4835200
Frequency

mean k(t+1)

k

3.483
3.4825

150
100

3.482
50
3.4815

Actual dynamics of mean kForecasted mean k003.4815 3.482 3.4825 3.483 3.4835 3.484 3.4845
mean k(t)

10

20

30

40

50

k

Figure 4: Same plot as Figure 2 for UI I and σ = 1.

35

60

70

100

95
90

80

85

next period k

80

v

75
70
65
60

Least mean kSecond least mean kMiddle mean kSecond largest mean kLargest mean k

55
50
0

20

40

60

80

60

40

20
Employed
Unemployed

0

100

0

20

40

k

60

80

100

k

300
3.52
250
3.51
200
Frequency

mean k(t+1)

3.5
3.49
3.48

150
100

3.47
3.46

50
Actual dynamics of mean kForecasted mean k

3.45
3.45

3.46

3.47

3.48 3.49
mean k(t)

3.5

3.51

0
0

3.52

10

20

30

40

50

k

Figure 5: Same plot as Figure 2 for UI III and σ = 1.

36

60

70

100
95
90

80

85
next period k

v

80
75
70
65

Least mean kSecond least mean kMiddle mean kSecond largest mean kLargest mean k

60
55
0

20

40

60

80

60

40

20
Employed
Unemployed

0

100

0

20

40

k

60

80

100

40

50

60

k

250
3.52
3.51

200

3.49

Frequency

mean k(t+1)

3.5

3.48
3.47

150

100

3.46
3.45

50

3.44

Actual dynamics of mean kForecasted mean k

3.43
3.44

3.46

3.48
mean k(t)

3.5

0
10

3.52

20

30
k

Figure 6: Same plot as Figure 2 for Tax I and σ = 1.

37

100

95
90

80

85
next period k

80

v

75
70
65
Least mean kSecond least mean kMiddle mean kSecond largest mean kLargest mean k

60
55
50
0

20

40

60

80

60

40

20
Employed
Unemployed

0

100

0

20

40

k

60

80

100

k

350
3.54
300
3.52

Frequency

mean k(t+1)

250
3.5
3.48
3.46

200
150
100

3.44
3.42

50

Actual dynamics of mean kForecasted mean k
3.42

3.44

3.46

3.48
mean k(t)

3.5

3.52

0
0

3.54

10

20

30

40

50

k

Figure 7: Same plot as Figure 2 for Tax II and σ = 1.

38

60

70

100

−2
−4

80

next period k

−6

v

−8
−10
−12

Least mean kSecond least mean kMiddle mean kSecond largest mean kLargest mean k

−14
−16
0

20

40

60

80

60

40

20
Employed
Unemployed

0

100

0

20

40

k

60

80

100

30

40

50

k

250
3.413
3.412

200

3.41

Frequency

mean k(t+1)

3.411

3.409
3.408

150

100

3.407
3.406

50

3.405
3.404
3.404

Actual dynamics of mean kForecasted mean k
3.406

3.408
3.41
mean k(t)

0
0

3.412

10

20
k

Figure 8: Same plot as Figure 2 for UI I and σ = 3.

39

100
−4

80

next period k

v

−6

−8

−10

Least mean kSecond least mean kMiddle mean kSecond largest mean kLargest mean k

−12
0

20

40

60

80

60

40

20
Employed
Unemployed

0

100

0

20

k

40

60

80

100

k

250
3.48
200

3.44

Frequency

mean k(t+1)

3.46

3.42

150

100

3.4
50

3.38
Actual dynamics of mean kForecasted mean k

3.36
3.36

3.38

3.4

3.42
3.44
mean k(t)

3.46

0
10

3.48

20

30
k

Figure 9: Same plot as Figure 2 for UI II and σ = 3.

40

40

50

100
−4
80

−6
next period k

v

−8
−10
−12
Least mean kSecond least mean kMiddle mean kSecond largest mean kLargest mean k

−14
−16
−18
0

20

40

60

80

60

40

20
Employed
Unemployed

0

100

0

20

k

40

60

80

100

k

250
3.46

200

Frequency

mean k(t+1)

3.44

3.42

150

100

3.4
50

3.38
Actual dynamics of mean kForecasted mean k
3.38

3.4

3.42
mean k(t)

3.44

0
10

3.46

20

30
k

40

Figure 10: Same plot as Figure 2 for UI III and σ = 3.

41

50

100
−3
−4

80

next period k

−5

v

−6
−7
−8
−9

Least mean kSecond least mean kMiddle mean kSecond largest mean kLargest mean k

−10
−11
0

20

40

60

80

60

40

20
Employed
Unemployed

0

100

0

20

40

k

60

80

100

k

250
3.46
200

Frequency

mean k(t+1)

3.44
3.42
3.4

150

100

3.38
50
3.36

Actual dynamics of mean kForecasted mean k
3.36

3.38

3.4
3.42
mean k(t)

3.44

0
0

3.46

10

20

30
k

40

Figure 11: Same plot as Figure 2 for Tax I and σ = 3.

42

50

60

100
−4
80

next period k

v

−6
−8
−10
Least mean kSecond least mean kMiddle mean kSecond largest mean kLargest mean k

−12
−14
0

20

40

60

80

60

40

20
Employed
Unemployed

0

100

0

20

40

k

60

80

100

40

50

60

k

250
3.5
200

Frequency

mean k(t+1)

3.45

3.4

3.35

150

100

50
Actual dynamics of mean kForecasted mean k
3.35

3.4
mean k(t)

3.45

0
10

3.5

20

30
k

Figure 12: Same plot as Figure 2 for Tax II and σ = 3.

43

3.5

3

3

2.5

2.5

2.5

2

2

2

1.5

1.5

1

1
OutputConsumption of an employed
Consumption of an unemployed
z

0.5
0
800

820

840

860

880

Y,C

3.5

3

Y,C

Y,C

3.5

1.5
1
OutputConsumption of an employed
Consumption of an unemployed
z

0.5
0
800

900

820

840

3.5

3.5

3

3

2.5

2.5

2

2

1.5

880

900

0
800

820

840

860
Quarter

1.5

11OutputConsumption of an employed
Consumption of an unemployed
z

0.5
0
800

860
Quarter

Y,C

Y,C

Quarter

OutputConsumption of an employed
Consumption of an unemployed
z

0.5

820

840

860

880

900

OutputConsumption of an employed
Consumption of an unemployed
z

0.5
0
800

820

Quarter

840

860

880

900

Quarter

Figure 13: Sample time series of Y and C for the case σ = 3. Top: UI I, II, and III.
Bottom: Tax I and II.

44

880

900

References[1] S. Rao Aiyagari. Uninsured idiosyncratic risk and aggregate saving. Quarterly
Journal of Economics, 109:659–684, 1994.
[2] Marianne Baxter and Robert G. King. Fiscal policy in general equilibrium. American Economic Review, 83:315–334, 1993.
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