Tax-and-Transfer Progressivity and Business Cycles
Youngsoo Jang

Takeki Sunakawa

Minchul Yum

June 2023

Abstract
This paper studies how tax-and-transfer progressivity in‡uences aggregate ‡uctuations
when interacting with household heterogeneity. Using a simple static model of the extensive
margin labor supply, we analytically characterize how a degree of progressivity in‡uencesdi¤erential labor supply responses to aggregate conditions across heterogeneous households.
We then build a quantitative dynamic general equilibrium model with both idiosyncratic
and aggregate productivity shocks and show that it delivers moderately procyclical averagelabor productivity and a large cyclical volatility of aggregate hours relative to output. Ourquantitative exercises suggest that progressivity at the bottom of the income distribution
shaped by the phasing out of transfers is key for these …ndings. Finally, we provide suggestive empirical evidence on the heterogeneity of employment responses across the wage
distribution.
Keywords: Progressivity, targeted transfers, extensive margin labor supply, businesscycles, redistributive policies
JEL codes: E32, E24, H31, H53, E21
Jang: University of Queensland, Australia, e-mail: youngsoo.jang@uq.edu.au. Sunakawa: Hitotsubashi University, Japan, e-mail: takeki.sunakawa@gmail.com. Yum: University of Southampton and CEPR, United Kingdom, e-mail: m.yum@soton.ac.uk. We thank the editor, Kjetil Storesletten, and three anonymous referees fortheir helpful comments and suggestions that have improved the paper substantially. We are also grateful to
Fabian Kindermann, Alex Ludwig, Eunseong Ma and Yikai Wang for their constructive discussions at various
stages of this project, and to Árpád Ábrahám, Björn Brügemann, Yongsung Chang, Jay Hong, Pantelis Kazakis, Aubhik Khan, Sun-Bin Kim, Tom Krebs, Dirk Krueger, Per Krusell, Claudio Michelacci, Matthias Meier,Benjamin Moll, Fabrizio Perri, Shuhei Takahashi, Michèle Tertilt, Julia Thomas, and seminar participants at the
Bank of Canada, the Bank of Japan, EUI, Federal Reserve Board, Hitotsubashi, KAIST, Korea Univ., Ohio State
Univ., Queen Mary Univ. of London, Sogang, UCL, Univ. of Edinburgh, Univ. of Kent, Univ. of Konstanz,
Univ. of Mannheim, Univ. of Queensland, Yonsei, and participants at many conferences for comments. Financial
support from the German Research Foundation (DFG) through CRC TR 224 (Project A3) and Japan Society
for the Promotion of Science through Grant–in–Aid for Scienti…c Research (KAKENHI) for Young Scientists B(Project 15K20918) is gratefully acknowledged. A previous version of this paper was circulated with the title "IsHousehold Heterogeneity Important for Business Cycles?".

1

Introduction

A large body of literature has investigated the macroeconomic implications of the progressive nature of taxes and transfers.1 A natural yet relatively unexplored question is how the progressivityof taxes and transfers a¤ects aggregate ‡uctuations. This paper asks how the degree of tax-andtransfer progressivity alters the way aggregate shocks are transmitted to the macroeconomy withendogenous labor supply at the extensive margin.2
We begin by considering a simple static model of the extensive margin labor supply, withagents who di¤er in their potential earnings (high or low) and assets according to a distribution
featuring a high concentration near zero. We consider higher progressivity a variation in thetransfer schedule, such that agents with low potential earnings (the low type) receive more thanthose with high potential earnings (the high type). We show that higher progressivity induces
the labor supply of the low type to respond more strongly to the aggregate shifter. This isbecause the threshold asset relevant to the employment decision moves closer to zero, aroundwhich there is a higher density of (marginal) agents. Consequently, higher progressivity leads toa lower cyclicality of average labor productivity through changes in the composition of workers(Bils 1985), and potentially to a greater volatility of aggregate hours driven by the low type.
To explore the role of tax-and-transfer progressivity quantitatively, we then consider a standard incomplete-markets framework with heterogeneous households who make consumptionsavings and extensive margin labor supply decisions in the presence of both idiosyncratic productivity risk and aggregate risk (Chang and Kim 2006). Since various welfare programs for
low-income households are phased out as income rises, overall tax-and-transfer progressivity
may vary over the income distribution with it being particularly high at the bottom of theincome distribution.3 To better replicate this empirical pattern, we use two parsimonious yet‡exible nonlinear functions for taxes and transfers, separately. We calibrate our model economy
to match some salient features in the micro-level data, including the degree of progressivity inwelfare programs in the Survey of Income and Program Participation (SIPP) data.
We …nd that our baseline model delivers aggregate labor market dynamics that di¤er con1

The research questions that have been considered include normative ones (such as optimal progressivity) andpositive ones (such as the role of progressivity in explaining macroeconomic outcomes). For example, see Conesa
et al. (2009), Heathcote et al. (2014), Bick and Fuchs-Schündeln (2018), and Guner et al. (2020) among others.
2
The sizes of welfare programs in the United States— e.g., cash transfers, and food, medical, and childcare support to low-income households— have been steadily growing since the 1970s (Ben-Shalom et al. 2011). Nonetheless,some recent papers found that progressivity has had no clear trend during the post war period, despite a few
drastic ups and downs (e.g., Ferriere and Navarro (2022) who focused on taxes, and Heathcote et al. (2020) who
considered both taxes and transfers among working-age individuals).
3
For example, Fleck et al. (2021) and Ferriere et al. (2022) …nd that a single log-linear progressive taxation
function is not su¢ cient to approximate the observed tax and transfer scheme, especially at the bottom of theincome distribution where progressivity is disproportionately larger.

1

siderably from its nested versions that abstract either from transfers entirely, from di¤erences intransfers across households, or from household heterogeneity. Speci…cally, it generates considerably lower correlations between average labor productivity and output (0:69) compared to all
the nested models (between 0:84 and 0:95, as compared to 0:30 in the data). At the same time,
the cyclical volatility of aggregate hours relative to output is 0:73 in the baseline model, which ishigher than 0:51 and 0:60 in the nested heterogeneous-agent models and is much closer to 0:80
obtained from its representative-agent counterpart: a version of a Hansen–Rogerson economy
that is known to be successful in generating a high volatility of hours.4
It is important to note that the model speci…cations with household heterogeneity do notachieve the quantitative success in the absence of transfers, even though we maintain the degreeof tax progressivity. This suggests that transfers play important roles in delivering the above
results. First, with the transfer schedule entered separately that enables us to better replicate
progressivity at the bottom of the income distribution, our baseline quantitative model can deliver the theoretic mechanism highlighted in the analytical model more e¤ectively. We visualize
this by using impulse response functions at the disaggregated level that show that low productivity households are more responsive to changes in aggregate shocks in our baseline model with
progressive transfers, when compared to one without them. The second role arises due to risk
and market incompleteness. In the absence of any transfers, the labor supply of low productivity
households is highly inelastic irrespective of aggregate conditions for precautionary reasons. The
presence of transfers mitigates this precautionary motive, thereby raising the responsiveness of
their labor supply to aggregate shocks.
Next, we conduct another counterfactual exercise where we raise progressivity by using either
the tax function or the transfer function. We …nd that higher progressivity through a faster phaseout of the transfer system further reduces the correlation between average labor productivity andoutput and raises the volatility of hours. On the other hand, changes in tax progressivity holding
the transfer scheme unchanged has limited e¤ects on aggregate labor market dynamics. These
results reinforce the above …nding that progressivity shaped by the rate at which welfare transfers
are phased out matter for labor market ‡uctuations with labor supply at the extensive margin.
Finally, we use micro data from the Panel Study of Income Dynamics (PSID) to empirically
explore the heterogeneity of employment changes. Speci…cally, we …nd that the individual-levelprobability of adjusting labor at the extensive margin is signi…cantly higher among low-wage
workers, and that the full-time employment rate has fallen more in lower wage quintiles during
4

Hagedorn and Manovskii (2008) …nd that in models with labor search frictions, unemployment bene…ts closer
to the potential wage make the value of working closer to that of being unemployed. This in turn increases labormarket volatilities, and we note similarities with our volatility results in this regard. However, our key model
mechanism relies on heterogeneity across households in terms of wealth and the size of transfers. This enables us
to go beyond labor market volatilities and study issues related to the cyclicality of average labor productivity.

2the most recent recessions.
Although extensive studies have shown the importance of heterogeneity in accounting for
macroeconomic aggregates and equilibrium prices in the absence of aggregate risk (Huggett
1993; Heathcote 2005), the earlier literature that considers aggregate uncertainty often foundthat incorporating micro-level heterogeneity has only limited impacts on the business cycle ‡uctuations of macroeconomic aggregates (e.g., Krusell and Smith 1998; Khan and Thomas 2008;
Chang and Kim 2014). Our main result— that household heterogeneity at the micro level can beimportant for understanding the dynamics of macroeconomic variables— is broadly in line withrecent papers, such as Krueger et al. (2016), and Ahn et al. (2017).
Weak correlations between average labor productivity and output or hours— often referred
to as the Dunlop–Tarshis observation— are known to be di¢ cult to explain using standard realbusiness cycle models. The literature has suggested various mechanisms to dampen strongly
positive correlations, with earlier studies relying on the introduction of additional shocks torepresentative-agent models such as home-production technology shocks (Benhabib et al. 1991),
government spending shocks (Christiano and Eichenbaum 1992), and income tax shocks (Braun
1994). Recently, Takahashi (2020) reduces the correlation between average labor productivity andhours by incorporating uncertainty shocks into a standard heterogeneous-agent model (Chang
and Kim 2007). Our result is distinct from the existing literature because our mechanism relieson the existence of institutional features leading to heterogeneous responses.
Our quantitative model highlights the e¤ect government transfers have on the precautionary
behavior of poor households. Hubbard et al. (1995) show that social insurance discourages precautionary savings among low-income households. Using an incomplete-markets model without
aggregate uncertainty, Yum (2018) …nds that transfers are important in bringing the employment rate of wealth-poor households closer to the data, which has important implications forthe long-run employment e¤ects of labor taxes. Our results suggest that they have important
implications for the dynamics of macroeconomic aggregates over the business cycle as well.
The rest of this paper is organized as follows. Section 2 presents the simple static modeland present the analytic results on its key mechanism. Section 3 introduces the quantitative
dynamic models. Section 4 explains calibration and shows the properties of the quantitativemodels in stationary equilibrium. Section 5 presents the main quantitative results. Section 6presents empirical supporting evidence. Section 7 then presents our conclusions.

3

2

A static model of the extensive margin labor supply

In this section, we present a simple static model of the extensive margin labor supply.5 The
goal of this section is to illustrate the direct e¤ects of …scal policy on aggregate labor market‡uctuations in a tractable way. This tractability is achieved by the simplifying assumptionthat the distribution of wealth is …xed with respect to changes in …scal policy, and that it isindependent of potential earnings. As …scal policy may change the distribution of wealth, andbecause this indirect e¤ect could potentially a¤ect the theoretical predictions of this section, we
will explore this mechanism using a more realistic dynamic model in the subsequent sections.
Our model here considers a continuum of agents in the unit interval. We assume that there
are two types of agents with di¤erent potential earnings (or wage o¤ers). That is, the individualcomponent of the potential wage can be either low or high: xi 2 fxl ; xh g: The mass of each typeis denoted by l and h satisfying the condition l + h = 1: Agents also di¤er in their level ofasset holdings ai , and they can choose to either work full-time or not at all: ni 2 f0; 1g.
The decision problem of each type i is given by:max

ci 0;ni 2f0;1g

flog cibni gsubject to
ci

zxi ni + ai + Ti ;where c denotes consumption and b > 0 captures the disutility of work. We set b = log(2) > 0
without loss of generality. We use z to denote an aggregate shifter of potential earnings andconsider its small perturbations to be the source of aggregate ‡uctuations. To study the role ofprogressivity, we allow Ti to depend on the type i (or potential wages).6
By comparing the utility conditional on working to that on not working, the agent chooses
to work if:log (zxi + Ti + ai )b

5log (Ti + ai ) ;Our analytical framework in this section builds on the theoretical framework of Doepke and Tertilt (2016),
although the focus of our analysis is di¤erent. Whereas their model is based on two gender types and continuous
preference heterogeneity, our model is instead based on two wage-o¤er types and continuous asset heterogeneity.
Moreover, our results cover not only labor supply elasticity but also average labor productivity.
6
We can interpret Ti as transfers net of taxes, or equivalently, negative taxes net of transfers. We opt for
the former for expositional convenience. In this section, we require Ti to be non-negative. If Ti is negative, thiscomplicates the analytical derivations since consumption could become negative especially for those with low aand low x.

4

or if
ai

zxi

Ti :

This decision rule shows that the agent is more likely to choose to work if the aggregate shifter z
or the individual earnings potential x is higher. Also note that the agent is less likely to choose
to work if the size of the transfers is higher.
In this model, aggregate employment is determined by both the decision rule and by the
distribution. Let F (a) be the conditional (di¤erentiable) distribution function of assets with
its probability density being f (a) = F 0 (a). Speci…cally, we use the exponential function in ourfollowing results. For a 0;
F (a) = 1

exp( a),f (a) = F 0 (a) = exp( a):This density function has a long right tail in its asset distribution, with a large fraction holding
low wealth, being in line with the data.
Given the density function and the decision rule, the fraction of agents working (i.e., theemployment rate) for each type is given by:
N = F (ai ) = 1

exp( ai );where
ai = zxi

Ti :

(1)

In other words, the employment rate Ni of the type i is the integral of all those type i agents
whose asset level is lower than the threshold level ai : We now present some theoretical results
based on this model, with all proofs provided in Appendix A.
Proposition 1 Let "i be the labor supply elasticity of the type i agents:
"i

@Ni z
:
@z Ni

Assume Ti = 0. The labor supply elasticity of agents with low potential earnings is greater thanthat of agents with high potential earnings. That is, "l > "h :This shows that our model naturally delivers the heterogeneity of labor supply elasticity. The
shape of the wealth distribution and the relative location of threshold assets are important forthis result. To see this, note that the threshold asset level for the low-potential-wage agents is5

Figure 1: Visual illustration of Proposition 1

Note: The shaded areas denote the relevant density of agents who are a¤ected by perturbations in the aggregate
shifter z.

lower than that for the high-potential-wage agents: al < ah : As shown in Figure 1, the density ofthe distribution around al is greater. Since there are more marginal agents around al , the samechange in the aggregate shifter z— which perturbs both al and ah — will more strongly a¤ect theemployment rate of the low-potential-wage agents.
We now consider the role of government transfers and how they interact with heterogeneity.
To simplify the algebra, we impose symmetry. Speci…cally, we assume that l = h = 0:5: In
addition, xh = 1 + and xl = 1; where 2 (0; 1) measures the cross-sectional dispersion.
To study the e¤ects of progressivity, Ti is assumed to be:
Tl = T (1 + ! ) ;Th = T (1! );where T 2 [0; z(1)=2] captures the scale of transfers and ! 2 [0; 1= ] shapes the degree of7
progressivity. Note that a change in ! does not a¤ect the aggregate size of the transfers.8 Given
the above assumptions, the employment rates for each type are given by Ni = 1 exp( ai );where al = z(1) T T ! and ah = z(1 + ) T + T ! :Proposition 2 Greater progressivity increases the labor supply elasticity of the low-potentialwage agents, while it decreases the labor supply elasticity of the high-potential-wage agents. Thatlhis, @"> 0 and @"
< 0.
@!
@!
7

The maximum values of T and ! ensure that the threshold assets stay non-negative.
Progressivity can increase through two channels: (i) the scale of the transfers, and (ii) the relative size ofthe transfers received by low-income households. A change in ! is meant to capture the second channel (i.e.,
controlling the speed of phasing-out of welfare transfers).
86Intuitively, greater progressivity (or a higher !) shifts al to the left where the distribution isdenser. There, the same change in the aggregate shifter z would induce more agents to change
their employment decision, thereby leading to an even larger elasticity for the low-potential-wageagents. By contrast, greater progressivity shifts ah to the right, around which the distributionof assets is thinner. This implies that the elasticity of the high-potential-wage agents should
become smaller.
Proposition 3 Let N denote the aggregate employment rate: N =aggregate labor supply elasticity:
@N z
"
:
@z N

l Nl +h Nh : Let " be the@"> 0:
The aggregate labor supply elasticity is higher with greater progressivity. That is, @!

The key to this result is that an increase in the elasticity of low-potential-wage agents should
be large enough to outweigh the opposing e¤ects from a decrease in the elasticity of high-potentialwage types. Given that the density function declines at an increasing rate, this condition is satis…ed. This result suggests that progressivity potentially has a role in generating large volatility
in aggregate hours, as observed in the data.
Finally, we consider the implications for the cyclicality of average labor productivity. We
de…ne average labor productivity as output divided by aggregate hours:Pj2fl;hgPi (zxi Ni )j2fl;hg

i Ni

=z

P

j2fl;hgPi (xi Ni )j2fl;hg

i Ni

z 0;

where we separately de…ne the second term as 0 . Here, we can clearly see that a change
in the aggregate shifter z would directly cause average labor productivity to become procyclicalthrough the …rst term z, as is the case in real business cycle models. The second term 0 captures
the e¤ects through worker composition, which indirectly depends on z through heterogeneous
employment responses. The following two propositions focus on this second term.
Proposition 4 A change in the aggregate shifter z has a direct and an indirect e¤ect on average
0 (z)labor productivity z 0 (z). The indirect e¤ect is negative: @ @z
< 0.
Proposition 5 Average labor productivity becomes less positively (or more negatively) correlated@ 0@with z as progressivity increases: @!
< 0:@zProposition 5 shows that progressivity can shape the cyclicality of average labor productivity
through worker composition e¤ects. To see what this means, let us suppose that the aggregate7shifter z increases (i.e., in a boom). While both types of agents are more likely to work, relatively more low-type workers would do so when progressivity is greater. This follows from thedisproportionate rise in low-type labor supply elasticity shown in Proposition 3. This force would
cause lower increases in average labor productivity during booms, thereby dampening the tightpositive link between z and average labor productivity.

3

Quantitative business cycle models

As noted earlier, the key results in Section 2 capture the direct e¤ects of …scal policy changes
since they are derived in a static environment. Therefore, it is a quantitative question whether
this mechanism would be relevant in a more realistic and dynamic model environment. In theremaining sections, we explore the key mechanisms in models that allow endogenous wealthdistributions that can di¤er by productivity types.

3.1

Baseline model

The baseline quantitative model we use here builds on a standard incomplete-markets frameworkwith both idiosyncratic productivity risk (Huggett 1993; Aiyagari 1994) and aggregate risk,
as pioneered by Krusell and Smith (1998). In this model, heterogeneous households make aconsumption-savings choice— which endogenizes the distribution of wealth— and a labor supply
decision at the extensive margin. There are also di¤erences in transfers across households.
Households The model economy is populated by a continuum of in…nitely lived households.
It is convenient to describe the decision problem faced by such households in a recursive manner.
At the beginning of each period, households are distinguished by their asset holdings a andproductivity xi : We assume that xi takes a …nite number of values Nx and follows a Markov
chain with transition probabilities ijx from state i to state j: In addition to the individualstate variables, a and xi , there are also aggregate state variables, including the distribution ofhouseholds (a; xi ) over a and xi ; and aggregate total factor productivity shocks zk : We also
assume that zk takes a …nite number of values Nz following a Markov chain with transitionz
probabilities kl
from state k to state l: We assume that these Markov processes of individual
productivity x and aggregate total factor productivity (TFP) shock z capture the following
continuous AR(1) processes in logarithms:
log x0 =0x log x + "x ;(2)

log z 0 =0z log z + "z ;(3)8N (0; z2 ). We denote a variable with a prime symbol its value
where "xN (0; x2 ) and "zin the next period. Finally, we assume competitive markets: households take as given the wage
rate per e¢ ciency unit of labor w( ; zk ) and the real interest rate r( ; zk ); both of which dependon the aggregate state variables. Households also take government policies as given.
The dynamic decision problem facing households can then be written as the following functional equation:
V (a; xi ; ; zk ) = max V E (a; xi ; ; zk ); V N (a; xi ; ; zk ) ;
where

(

V E (a; xi ; ; zk ) = max
log c
0
a a;
c 0

Bn +

Nx
X

x
ij

j=1

Nz
X

z
0
0
kl V (a ; xj ;0)

; zl0 )

l=1

(4)

subject to
c + a0(5)(e; e)e + (1 + r( ; zk ))a + T

e = w( ; zk )xi n
T = T1 + T2 (m)m = e + r( ; zk ) maxfa; 0g0= ( ; zk ):and(log c +V N (a; xi ; ; zk ) = max
0
a a;
c 0

Nx
X
j=1

x
ij

Nz
X
l=1

z
0
0
kl V (a ; xj ;0)

; zl0 )

(6)subject to
c + a0(1 + r( ; zk ))a + T(7)

T = T1 + T2 (m)m = r( ; zk ) maxfa; 0g0= ( ; zk ):Households maximize utility by choosing their optimal consumption c, their asset holdings in
the next period a0 ; and their labor supply n. Households also face a borrowing limit a0.
Their labor supply decision is discrete (i.e., n 2 f0; ng), and the disutility of work is captured
by B > 0. Households understand that the expected future value (discounted by a discountfactor ) is a¤ected by stochastic processes for individual productivity x0 and aggregate TFP
9productivity z 0 , as well as the whole distribution 0 . The budget constraints state that the sumof spending should be less than or equal to the sum of income. The evolution of is governedby the law of motion, as denoted by 0 = ( ; zk ).
As shown in the budget constraints, our model incorporates a progressive tax and transfer
system, and these two components are captured separately by two nonlinear functions. First,
earnings e are subject to progressive taxation— as is standard in the recent quantitative macroeconomics literature. Speci…cally, for those who have earnings e, progressive taxation leads to a
tax rate of:
o
n
p
;0 :
(8)
(e; e) = max 1
s (e=e)Note that, although this function follows the parametric form of Benabou (2002) and Heathcote
et al. (2014), we restrict (e) to being non-negative. As is well known, p
0 captures thedegree of progressivity and s 0 inversely controls the scale of taxation. As the input into theprogressive tax schedule is earnings normalized by its average e (Guner et al. 2014), a change
in p tilts this schedule around average earnings. This strongly a¤ects tax progressivity yet has
little e¤ect on the size of taxation.
On top of this typical progressive tax schedule, we also separately introduce progressive
transfers. This helps us to better capture progressivity at the bottom of the income distribution,
which is di¢ cult to replicate with a single nonlinear tax function (Fleck et al. 2021; Ferriere et al.
2022). Following Krusell and Rios-Rull (1999), we make the speci…c assumption that transfers
T consist of two components. The …rst component T1 is given to all households equally, whereas
the second component T2 captures the income security aspect of transfers. In the U.S., there arevarious means-tested programs, such as the Supplemental Nutrition Assistance Program (SNAP)(formerly known as food stamps), and the Temporary Assistance for Needy Families (formerly
the Aid to Families with Dependent Children). As shown in Section 4, the existence of these
programs leads us to the observation that the amount of transfers is negatively associated withincome. We assume that T2 depends on total household income m, and use this to replicate themeasured transfer progressivity observed in the U.S. data using the following functional form
(Yum 2018):
T2 (m) = !s (1 + m) !p :(9)
This parametric assumption adds two parameters. First, !s
0 is a scale parameter that
determines the overall size of the non-‡at part of government transfers (i.e., T2 ). The next
parameter, !p
0, governs the degree of progressivity: a higher !p makes T2 decrease fasterwith income.

10

Representative …rm and government Aggregate output Y is produced by a representative
pro…t-maximizing …rm that solves
max fzk F (K; L)
K;L(r( ; zk ) + )Kw( ; zk )Lg(10)

where F (K; L) captures a standard neoclassical production technology in which K denotes aggregate capital, L denotes aggregate e¢ ciency units of labor inputs, and is the capital depreciation
rate. As is standard in the literature, we assume that the aggregate production function follows
a Cobb-Douglas function with constant returns to scale:F (K; L) = K L1

(11)

:

The …rst-order conditions for K and L giver( ; zk ) = zk F1 (K; L);w( ; zk ) = zk F2 (K; L):

(12)
(13)

The government in this economy collects labor taxes from households and uses the tax revenue to …nance total transfers to households. The remaining tax revenue is spent as government
spending G, which is not valued by households. Note that government spending plays no important role in the exercises of this paper.
Equilibrium A recursive competitive equilibrium is a collection of factor prices r( ; zk ) andw( ; zk ); household decision rules ga (a; xi ; ; zk ) and gn (a; xi ; ; zk ); government spending G;
a value function V (a; xi ; ; zk ); a distribution of households (a; xi ) over the state space; theaggregate capital and labor K( ; zk ) and L( ; zk ); and the aggregate law of motion ( ; zk );such that
1. Given factor prices r( ; zk ) and w( ; zk ); the value function V (a; xi ; ; zk ) solves the household decision problems de…ned above, with the associated household decision rules being:a0 = ga (a; xi ; ; zk );(14)

n = gn (a; xi ; ; zk ):(15)

2. Given factor prices r( ; zk ) and w( ; zk ), the …rm optimally chooses K( ; zk ) and L( ; zk )following (12) and (13).

113. Markets clear:K( ; zk ) =L( ; zk ) =

Nx Z
X
i=1

a

i=1

a

Nx Z
X

ad

(16)

xi gn (a; xi ; ; zk )d :(17)

4. Government balances its budget. That is, the sum of government spending G and totaltransfers to households is equal to the total tax revenue.
5. The law of motion for the distribution of households over the state space 0 = ( ; zk ) is
consistent with individual decision rules and the stochastic processes governing xi and zk .

3.2

Alternative model speci…cations

In addition to the baseline model just introduced, we also consider alternative speci…cations
to illustrate the importance of the interplay between household heterogeneity and governmenttransfers.9 For convenience, the baseline model featuring "Heterogeneous Agents" and "Targeted" transfers is called Model (HA-T).
The …rst alternative model speci…cation, denoted as Model (HA-N), is simply a nested speci…cation of the baseline "Heterogeneous-Agent" model with "No" government transfers (i.e.,
T1 = !s = 0). This model roughly corresponds to the standard incomplete-markets real business cycle model of Chang and Kim (2007), with household heterogeneity and endogenous labor
supply at the extensive margin.10
The second alternative model speci…cation also keeps household heterogeneity but removes
di¤erences in transfers across households. We call this model speci…cation Model (HA-F), which
is obtained as a nested "Heterogeneous-Agent" model by making transfers "Flat"— that is, independent of income (!p = 0). Chang et al. (2013) also consider a business cycle model that isclose to this model speci…cation. Note that this form of transfers (‡at lump-sum) is very broadly
used in the quantitative macroeconomics literature.
Our …nal alternative speci…cation shuts down household heterogeneity. This "RepresentativeAgent" version of the model is called Model (RA). Given the indivisible labor supply assumption,9

We have also considered a speci…cation which shuts down tax progressivity only. Because its quantitative roleis minimal, we have placed those results in Appendix H as a sensitivity check. In Section 5:3, we also consider acounterfactual exercise where we alter tax progressivity using the baseline model speci…cation.
10
A noticeable di¤erence between Model (HA-N) in our paper and the model in Chang and Kim (2007) is that
ours includes progressive taxation whereas theirs does not. However, as shown in Section 5 and Appendix H,
the business cycle properties of the model are barely a¤ected by the existence of progressive taxation— except for
output volatility.

12Model (RA) is essentially the business cycle model studied in Hansen (1985) augmented withtaxes and transfers. The key feature of this model speci…cation is that the aggregation of Rogerson (1988) under certain assumptions (such as employment lotteries and consumption insurance)
leads to the introduction of the stand-in household whose disutility from work is linear— a powerful mechanism used to generate the large volatility of aggregate hours, observed in U.S. data
(Hansen 1988). Appendix G includes the detailed model environment and its equilibrium de…nition.

3.3

Solution method

We solve each of the models numerically. Several key features make the numerical solution
method nontrivial for the heterogeneous-agent models. First, the key decision variables in ourmodel are a discrete employment choice and a consumption-savings choice in the presence ofa borrowing constraint. Therefore, our solution method is based on a nonlinear method (i.e.,the value function iteration) applied to the recursive representation presented above. Second,
the aggregate law of motion and the state variables involve an in…nite-dimensional object: thedistribution . This requires us to solve the model by approximating the distribution of wealthas its mean (Krusell and Smith 1998). Since market-clearing is nontrivial in our model withendogenous labor, our solution method also incorporates an additional step to when simulating
the model to …nd market-clearing prices in each period.
We now describe the solution method brie‡y, with more details found in Appendix G. Following Krusell and Smith (1998), we assume that households use a smaller object that approximatesthe in…nite-dimensional distribution when they forecast the future state variables in order tomake current decisions. More precisely, we approximate (a; xi ) by its mean with respect tothe asset distribution K. Furthermore, when determining the aggregate capital in the next period K 0 , real wage per e¢ ciency units w and real interest rate r are assumed to be functions of(K; zk ) instead of ( ; zk ). We impose parametric assumptions to approximate the aggregate lawof motion K 0 = (K; zk ) and w = w(K; zk ) following^ 0 = ^ (K; zk ) = exp (a0 + a1 log K + a2 log zk )K

(18)w^ = w(K;^zk ) = exp (b0 + b1 log K + b2 log zk ) ;(19)

as in Chang and Kim (2006, 2007). Households obtain a forecasted r^ based on these forecasting
rules, as implied by the …rst-order conditions of the pro…t maximization problem facing therepresentative …rm.
The model is solved in two steps. First, given the forecasting rules, we solve for the indi-

13

vidual policy functions using the value function iterations (the inner loop). Then, we update
the forecasting rules by simulating the economy using the individual policy functions (the outerloop). It is important to once again note that, since our model environment with endogenous
labor supply involves non-trivial factor market clearing, we have to incorporate a step to …nd
the market-clearing factor prices in the outer loop (Chang and Kim 2014; Takahashi 2014). We
repeat this procedure until the coe¢ cients in the forecasting rules converge.
It is straightforward to solve the representative-agent version of the model. For the purposesof comparison, we keep the same assumptions on the discretization of the TFP shock process as
used in the heterogeneous-agent model. The steady-state equilibrium can then be obtained analytically. For solutions with aggregate uncertainty, we use the policy function iteration method.

4

Calibration and model properties in steady state

All model speci…cations are calibrated to U.S. data. A period in the model is a quarter, as
is standard in the business cycle literature. We consider all four of our speci…cations: Model(HA-T), Model (HA-N), Model (HA-F), and Model (RA).
Calibrating the baseline model We …rst describe how we calibrate the baseline speci…cation,
which involve two sets of parameters. The …rst set is calibrated externally, in line with thebusiness cycle literature. These parameter values are set in common across all four of our model
speci…cations. The second set of parameters is calibrated to match the same number of relevanttarget statistics.
We begin by describing the …rst set of externally calibrated parameters. Most of these arecommonly used parameters in the literature. The capital share is chosen to be consistent withthe empirical capital share value of 0.36, and the quarterly depreciate rate is set to 2.5%. In
our model speci…cations with a binary labor supply choice, the number of hours worked n canbe arbitrarily set since it simply determines the scale of the calibrated disutility parameter B.
By setting n to 1=3— implying that working individuals spend a third of their time endowment~
on working— we can calibrate B
Bn directly. Furthermore, the borrowing limit a is set toT1 =(1 + r), where r is the equilibrium interest rate in steady state.11
In the literature, tax progressivity p has been estimated using the same functional form weuse. As noted by Holter et al. (2019), the estimate of p varies quite a lot (from 0:05 to 0:18),
depending on the degree of completeness of the data on government transfers used by researchers.
11This is a form of the natural borrowing limit that ensures that non-working agents are able to pay back theirdebts in the next period. In Appendix H, we report a version of the model with a zero borrowing limit, as is
standard in the literature. The main results found in this paper are robust to this variation.

14

Because we model progressive transfers separately in addition to progressive taxes, our taxation
parameters in (8) should ideally only capture tax progressivity. As the Internal Revenue Service(IRS) income tax data used by Guner et al. (2014) do not include welfare transfers, we use
their estimate for p = 0:053 and s = 0:911.12 As discussed below, we then use micro data on
the distribution of welfare transfers across households to calibrate the parameters of the transfer
function in (9).
The broad goal of this paper is to study how progressive transfers alter the transmission of
aggregate shocks in the macroeconomy. As a …rst step, we consider the most standard one— totalfactor productivity shocks (Kydland and Prescott 1982)— as an aggregate risk, and employ thestandard values of z = 0:95 and z = 0:007 (Cooley and Prescott 1995). These values are usefulas we can easily compare our results to those from recent related papers, such as Chang and Kim(2007) and Takahashi (2020), who also use the same TFP shock estimates.13
Next, x captures the persistence of idiosyncratic risk in the productivity of households. We
estimate the persistence of idiosyncratic risk using the PSID following a standard method fromthe literature (Heathcote et al. 2010), as discussed in Appendix F. The quarterly value based on
this estimate is x = 0:9847. The variability of the idiosyncratic risk is calibrated internally andis explained below. We keep the same values for these two parameters, x and x ; for all of thenested model speci…cations using heterogeneous agents to control for the underlying idiosyncraticrisk.
The second set of parameters is jointly calibrated. As shown in Table 1, six parameters arecalibrated by matching the same number of target statistics. We now explain how each parameteris linked to a target statistic.
~ which captures the disutility of work, as de…ned above. The most
The …rst parameter is B;
relevant target moment is the employment rate of 78:2% from the SIPP sample.14 The next parameter captures the discount factor of households and is targeted to match a quarterly interest
rate of 1%. The next parameter x governs the variability of idiosyncratic labor productivity.
We calibrate it to match the overall wage dispersion captured by the Gini index of worker wages.
The target statistic is chosen to be 0:359; which is the average Gini wage in 2000 (Heathcote,
12This is the estimate for when the Earned Income Tax Credit (EITC) is included because we do not consider
it in our calibration of welfare transfers. We also considered alternative values for p , but these did not a¤ect ourquantitative results substantially. This quantitative insigni…cance of tax progressivity can also be seen explicitly
in our counterfactual exercise in Section 5.3.
13
An interesting exercise for the future would be to investigate how our results might carry over in the presenceof other types of aggregate shocks— on top of the standard TFP shocks. The estimation of multiple aggregate
shocks within a model including heterogeneous agents and nonconvexities is an important task yet is di¢ cult at
the present moment due to computational costs.
14
This value is higher than the employment-population ratio of around 60% that was used in the previous
literature (e.g., Chang and Kim 2007) because we focus here on working-age samples.

15

Table 1: Parameter values chosen internally

ValuesParameters
Description~ = .692 Disutility of workB=.985 Subject discount factor.126 S.D. of innovations to ln xx =T1 = .0337 Overall transfer size!s = .117 Scale of non-‡at transfers!p = 3.62 Progressivity of transfers

Model

Data

.777
.010
.360
.044
.0203
3.07

.782
.010
.359
.044
.0201
3.06

Target statistics
Description
Employment rate
Real interest rate
Wage Gini index
Ratio of Avg (T1 + T2 ) to output
Ratio of Avg T2 to output
E(T2 j1st income quintile)/E(T2 )Perri, and Violante 2010).15
The last three parameters— T1 ; !s and !p — govern statistics related to transfers. Recall thatT1 determines the size of universal transfers and !s determines the scale of non-‡at transfers (T2 ).
Therefore, the …rst target statistic regarding transfers is set as the total transfers-output ratioof 4:4%. This is obtained from the time-series average of the ratio of transfers (excluding Social
Security and Medicare) to output over the years 1961-2016 according to the Bureau of Economic
Analysis (BEA) data. The next target is the average government expenditures on social bene…ts
related to income-security (Table 3.12 from the BEA) over the years 1961–2016— that is, 2:0% of
output.16 Next, !p shapes the degree of progressivity in government transfers. Our calibration
strategy is to let the model replicate an empirically reasonable degree of transfer progressivity
through !p , given the value of !s . For this purpose, we measure the degree of progressivity in
the U.S. transfer programs using the SIPP data. We construct a broad measure of governmenttransfers, including means-tested programs and social insurance (as detailed in Appendix E).
Since these welfare programs are highly relevant for poor households, we choose as a target
statistic the ratio of the average amount of means-tested transfers received by the …rst income
quintile to its unconditional mean (3:06) (Yum 2018).
Calibrating alternative model speci…cations Having explained the calibration strategy
of our baseline model, Model (HA-T), we now describe how we calibrate our nested model
speci…cations: Model (HA-N), Model (HA-F), and Model (RA). In general, it would be ideal
to minimize the number of parameters to be recalibrated to ensure that our comparison across15

Inequality has been steadily rising in the U.S. In Appendix H, we also consider di¤erent values for this target.
We select the components in order to be consistent with our measurement of transfers from the SIPP data,
as described below. Our classi…cation of transfers is similar to Krusell and Rios-Rull (1999). See Appendix E for
details.
16

16

di¤erent model speci…cations is not driven by di¤erent values of parameters. We therefore hold asconstant across the nested speci…cations the parameters governing the idiosyncratic productivity
risk: x and x . However, it is necessary to recalibrate a subset of the internally calibrated
parameters to ensure that the di¤erent model speci…cations are similar in terms of their target
statistics (e.g., the employment and interest rates in steady state equilibrium).
First, consider Model (HA-N). Because it abstracts from transfers (T1 = !s = 0), the para~ and to match the employment rate of 78:2% andmeter !p is irrelevant. Thus, we recalibrate B~ = 0:974 and = 0:9833. Next, consider
the real interest rate of 1%. This leads to values of B
Model (HA-F), which shuts down di¤erences in transfers across households. With !p = 0; distinguishing between T1 and !s becomes unnecessary. Therefore, we can calibrate the sum of T1~ andand !s to match the total transfers-output ratio of 4:4%. Aside from this, we recalibrate B
in the same manner, to yield values of 0:714 and 0:9848, respectively. Finally, and unlike theheterogeneous-agent models, Model (RA) can be calibrated analytically, as is shown in Appendix
B. As for the parameters related to tax and transfers, we simply use the average tax rate andthe average transfers because progressivity is irrelevant in Model (RA).
Steady state properties Table 1 reports that the baseline model does a good job of matching
the target statistics, and the other nested model speci…cations do a great job of matching a smaller
number of targets as well. This does not necessarily mean that the model can account for otherrelevant statistics. We therefore present the (non-targeted) distributional aspects in steady state.
First, Table 2 summarizes the share of wealth and the employment rates by wealth quintile fromboth the model and the data.17 Overall, all heterogeneous-agent model speci…cations do a goodjob of accounting for the shares of wealth held by each wealth quintile.
When we look at the employment rate by wealth quintile reported (also reported in Table2), we can clearly see that Model (HA-T) does a signi…cantly better job of accounting for the
cross-sectional employment-wealth relationship. In the U.S., the employment rate of the …rst
wealth quintile is relatively low (70:0%) and is then relatively ‡at across the other wealth quintiles. This weak inverted-U-shape of the employment rates across wealth quintiles in the data
is relatively well captured in Model (HA-T). On the other hand, Model (HA-N) predicts that
employment falls sharply with wealth, consistent with the …ndings of Chang and Kim (2007).
In this class of the incomplete-markets framework, the existence of transfers mitigates the excessively strong precautionary motive for employment among poor households who expect to benear the borrowing limit in the near future (Yum 2018). Although Model (HA-F) mitigates the17

Table 2 presents statistics on wealth distribution obtained from the 1992–2007 Survey of Consumer Finances(SCF), as reported by Yum (2018). These statistics from the SCF show a greater concentration of wealth in the
top wealth quintile because it better captures the top of the distribution by over-sampling the rich.

17

Table 2: Characteristics of wealth distribution

1st

Wealth quintile
2nd 3rd 4th

5th

Share of wealth (%)
U.S. Data (SIPP)
U.S. Data (SCF)
Model (HA-T)
Model (HA-N)
Model (HA-F)

-2.2
-0.4
-0.0
-0.1
-0.0

1.2
1.2
0.9
0.1
0.3

6.8
5.1
5.2
4.8
4.9

18.4
13.6
19.7
20.4
20.2

76.3
80.5
74.3
74.8
74.7

Employment rate (%)
U.S. Data (SIPP)
Model (HA-T)
Model (HA-N)
Model (HA-F)

70.0
85.3
100.0
100.0

77.9
79.3
99.2
92.0

80.9
84.4
74.0
75.2

82.5
75.2
66.0
67.9

79.7
64.2
51.9
54.0

Note: U.S. data are based on the 2001 Survey of Income and Program Participation (SIPP) and the 1992–
2007 Survey of Consumer Finances (SCF) (Yum 2018). Model (HA-T) is the heterogeneous-agent model withtargeted transfers. Model (HA-N) is the heterogeneous-agent model with no transfers. Model (HA-F) is theheterogeneous-agent model with ‡at transfers.

negative wealth gradients in employment seen in Model (HA-N), it does not generate the nearly‡at employment rates across wealth quintiles.
Next, Table 3 shows the micro relationship between income or employment status and transfers in the steady state equilibrium. Speci…cally, the reported numbers are the ratios of average
progressive-component transfers in each income quintile (left panel) or in each employment status (right panel) to the unconditional mean progressive-component transfers. In the U.S., there
is a clear negative relationship between income and the amount of income-security transfers.
Note that, in our model, this is a complicated equilibrium object: it is shaped not only by theparametric assumption on the nonlinear transfer schedule (9) but also by endogenous householdchoices (such as those regarding consumption-saving and labor supply). Despite the relatively
simple functional form (9), we can see that our baseline model does an excellent job of replicating
the degree of transfer progressivity in the U.S. Note that, since di¤erences in transfers acrosshouseholds are removed by design in Model (HA-F), this ratio is one for all income quintiles.
Another way to examine the speed at which transfers phase out is to compute average welfaretransfers by employment status. Although we do not target this directly, our baseline model
generates a substantially higher ratio for non-workers (2.69) relative to workers (0.53), in linewith the data.
18

Table 3: Progressivity of income-security transfers

1stIncome quintile
2nd 3rd 4th

Full-time emp.
Yes
No

5th

Conditional mean/unconditional mean
U.S. Data
3.06 0.99 0.52 0.26 0.17
Model (HA-T) 3.07 1.07 0.56 0.24 0.06
Model (HA-F) 1.00 1.00 1.00 1.00 1.00

0.48
0.53
1.00

2.86
2.69
1.00

Note: The source of U.S. data is the Survey of Income and Program Participation 2001.

We now present results related to the aggregate employment responses in steady state. In
this regard, it is useful to consider the recent empirical evidence of Mui and Schoefer (2020),
who propose a novel concept called the reservation raise. Speci…cally, a reservation raise, , isde…ned as the ratio of the reservation wage, which would make an agent indi¤erent between thetwo choices of working and not working, to the actual (or potential) wage.18 Once computing
these micro reservation raises, we then obtain their cumulative distribution function, denoted by
R( ). As detailed in Mui and Schoefer (2020), if we consider a change in the aggregate raise ,the arc elasticities of extensive-margin labor supply can be de…ned as"( ) =

R(1 + ) R(1)
R(1)

= ;

with R( ) being interpreted as an aggregate labor supply curve along the extensive margin.
Figure 2 shows "( ) from Mui and Schoefer (2020) estimated using the representative U.S.
sample as well as its model counterpart.
We …nd that our baseline model replicates two salient patterns observed in the data. As
noted by Mui and Schoefer (2020), the empirical arc elasticities show that (i) local elasticities
are large with respect to small changes in potential earnings (or reservation raises), and that (ii)elasticities are smaller with respect to large changes in potential earnings. It is worth notingthat our baseline model not only qualitatively reproduces these two salient patterns, but it also
generates empirically reasonable quantitative responses. Speci…cally, the elasticities are as highas three (or above) for very small changes in reservation raises (such as 5%). Moreover, they
become smaller (at around two) when the change in potential earnings approaches -20%, andthey get even smaller again (at around one) when the change in raises approaches +20%. To sum
up, it is reassuring to see that our model generates an empirically reasonable aggregate labor
18

Hence, reservation raises for those who choose to work would be less than one (ornon-workers would be larger than one (or > 1).

19< 1), whereas those forFigure 2: Arc elasticities: data versus model
8
Data
Model

7

Implied Arc Elasticity

6
5
4
3
2
1
0
-0.2

-0.15

-0.1

-0.05

0

0.05

Change in Aggregate Raise (

0.1

0.15

0.2

)

Note: The data plot is from Mui and Schoefer (2020) who report the U.S. estimates of arc elasticities. The arc
elasticities are computed, based on the reservation raise distribution.

supply curve, which is a complicated, non-targeted object.19

5

Quantitative analysis

In this section, we report the main business cycle results and illustrate the mechanism underlying
our main quantitative results.

5.1

Business cycle properties

We …rst compare the business cycle statistics of key macroeconomic variables from model simulations to those from the data. We …lter all the series using the Hodrick–Prescott …lter witha smoothing parameter of 1,600. The U.S. data statistics are computed using the aggregate
data from 1961Q1 to 2016Q4 (see Appendix D for more details). Table 4 summarizes the cyclical volatility of the following key aggregate variables: output Y , consumption C, investment I,
aggregate e¢ ciency unit of labor L, aggregate hours H, and average labor productivity Y =H.
Volatility is measured using the percentage standard deviation. As is standard in the business
19

In Appendix H, we report aggregate labor supply curves and arc elasticities from the other heterogeneousagent model speci…cations.

20

Table 4: Volatility of aggregate variables
Model
U.S. data (HA-T) (HA-N) (HA-F) (RA)
Y
C= Y
I= Y
L= Y
H= Y
Y =H = Y

1.50
0.58
2.96
0.98
0.52

1.27
0.27
2.87
0.50
0.73
0.64

1.48
0.28
2.99
0.64
0.51
0.54

1.46
0.27
2.99
0.62
0.60
0.57

1.83
0.25
3.08
0.80
0.25

Note: See Table 2 or Section 3.2 for the description of the model speci…cations. Each quarterly variable is logged
and detrended using the Hodrick-Prescott …lter with a smoothing parameter of 1600. Volatility is measured by
the percentage standard deviation of each variable. The U.S. statistics are based on aggregate time-series from
1961Q1 to 2016Q4.

cycle literature, our discussion focuses on relative volatility, which is computed as the absolute
volatility of each variable divided by that of output.
The most notable …nding in Table 4 is that the high volatility of aggregate hours relative to
output observed in U.S. data ( H = Y = 0:98) is well accounted for by Model (HA-T). Standard
real business cycle models are known to have di¢ culties in generating a large relative volatilityof hours without relying on a low curvature of the utility function (or a high Frisch elasticity).
When the stand-in household’s utility function in Model (RA) features zero curvature in labor
supply, it indeed generates a substantial relative volatility of hours (0:80). It is striking that ourbaseline model, Model (HA-T), delivers a comparably high volatility of hours (0:73).
The results of Chang and Kim (2006, 2007) suggest that a large relative volatility of hours
obtained through indivisible labor (Rogerson 1988) in Hansen (1985) may not be robust inincomplete markets economies with heterogeneous households. We can also see this point when
we look at the performance of our Model (HA-N), which delivers a substantially smaller volatilityof hours (0:51). This is the case even though our Model (HA-N) has progressive taxes— anadditional feature relative to Chang and Kim (2006, 2007). However, as noted above, once
heterogeneity in transfers is incorporated in line with the observed patterns in the micro data,
our baseline model shows that the heterogeneous-agent incomplete markets model can perform
similarly to the Hansen–Rogerson economy— in terms of a large volatility of hours over thebusiness cycle.
The performance of Model (HA-F) reveals that introducing ‡at transfers into the model canhelp obtain a larger relative volatility of hours (0:60). However, this value is still quite far fromits counterpart in Model (HA-T) of 0:73, suggesting that the phasing-out of transfers plays an
21

Table 5: Cyclicality of aggregate variables
Model
U.S. data (HA-T) (HA-N) (HA-F) (RA)Cor(Y; C)
Cor(Y; I)
Cor(Y; L)
Cor(Y; H)
Cor(Y; Y =H)
Cor(H; Y =H)

0.81
0.90
0.86
0.30
-0.23

0.85
0.99
0.92
0.77
0.69
0.07

0.85
0.99
0.96
0.95
0.95
0.81

0.84
0.99
0.96
0.87
0.85
0.48

0.84
0.99
0.99
0.84
0.74

Note: See Table 2 or Section 3.2 for the description of the model speci…cations. Each quarterly variable is logged
and detrended using the Hodrick-Prescott …lter with a smoothing parameter of 1600. Cyclicality is measured by
the correlation of each variable with output. The statistics are based on aggregate time-series from 1961Q1 to
2016Q4.

important role. This …nding is in line with our analytical results in Proposition 3 of Section 2,
which showed that greater progressivity increases the degree to which aggregate hours ‡uctuateswith respect to an aggregate shifter (TFP in this case).20
We now move on to the cyclicality of macroeconomic variables— a key focus of this paper. The…rst …ve rows of Table 5 show correlations between output and other aggregate variables. The lastrow shows the correlation between aggregate hours and labor productivity. As is well known in theliterature (King and Rebelo 1999), most macroeconomic variables like consumption, investment,
and aggregate hours are highly procyclical in the U.S. Table 5 shows that the strongly positive
correlations with output are fairly well replicated in all of our model speci…cations, regardlessof the presence of heterogeneity or institutional details. Therefore, one might conclude thatheterogeneity or government transfers are irrelevant, at least with respect to the cyclicality ofmacroeconomic variables over the business cycle.
However, we can see that this conclusion is premature when we look at the comovementof average labor productivity and output. In the U.S., strong procyclicality is not a featureof average labor productivity (i.e., Cor(Y; Y =H) = 0:30). A related observation is that thecorrelation between hours and average labor productivity is even weakly negative ( 0:23); often
referred to as the Dunlop–Tarshis observation (Christiano and Eichenbaum 1992). By contrast,
20

There are other interesting di¤erences in the volatility of macroeconomic aggregates. For instance, thevolatility of average labor productivity over the business cycle tends to be more consistent with the data in theheterogeneous-agent models, as compared to Model (RA). Another observation is that the presence of governmenttransfers tends to reduce the volatility of consumption over the business cycle. This suggests that governmenttransfers play a role as stabilizers, e¤ectively providing insurance against aggregate risk (e.g., see McKay andReis 2016).

22

canonical real business cycle models generate highly procyclical average labor productivity, andthus fail to replicate the limited cyclicality of average labor productivity seen in the data. The
high correlation between output and average labor productivity in Model (RA) (0:84) is also amanifestation of this weakness.21
The most notable …nding in Table 5 is that the strong procyclicality of average labor productivity is considerably muted (0:69) in Model (HA-T), and as such it is closer to the data (0:30).
In contrast to the existing literature— which tends to rely on the introduction of additional
exogenous shocks (e.g., Benhabib et al. 1991; Christiano and Eichenbaum 1992; Braun 1994;
and Takahashi 2020)— the key to our result is the interaction between household heterogeneity
and transfer progressivity. This in turn generates heterogeneous labor supply behavior acrosshouseholds, as highlighted in Section 2. The importance of the interplay between household heterogeneity and transfers can be seen by the performance of our nested model speci…cations. Once
we abstract from either household heterogeneity or di¤erences in transfers across households, ourmodel generates highly procyclical average labor productivity (above 0.8). In particular, when
we abstract from transfers in their entirety (as in Chang and Kim, 2006, 2007) with Model (HAN), we generate a very high correlation of 0:95. This implies that heterogeneity per se does not
dampen highly procyclical average labor productivity in real business cycle models.

5.2

Impulse responses

We now investigate the mechanism underlying our quantitative success by using impulse response
functions. Figure 3 shows the impulse responses of the key aggregate variables such as output,
consumption, aggregate hours, average labor productivity, and investment following a persistent
negative 2% shock to z (or TFP) for each of our heterogeneous-agent model speci…cations. We
follow the simulation-based methodology developed by Koop et al. (1996), as described in detailin Appendix G (see also Bloom et al. 2018).
The impulse response of aggregate hours clearly con…rms that Model (HA-T) (solid line) delivers a larger fall in hours than the nested heterogeneous-agent models— Model (HA-N) (dashedline) and Model (HA-F) (dotted line)— despite the fact that its output declines the least strongly
on impact. Another important di¤erence is the impulse responses of average labor productivity.
In Model (HA-N), the dynamics of average labor productivity closely follow the pattern of output
since it falls quite sharply on impact. This explains the very high correlation of Y =H with Y in
Table 5. When ‡at transfers are present in Model (HA-F), we see that the overall decrease inaverage labor productivity is mitigated. In Model (HA-T), the magnitude of the fall in average
21

These correlations would become even higher in models without indivisibility of labor (Hansen 1985) or in
the absence of labor taxes.

23

Figure 3: Impulse responses of macroeconomic aggregates
TFP

Output% deviation from s.s.

0
-0.5

0

-1
-0.5

-1
-2
-1.5

-1
-3

-2
-2.5

-4
0

20

40

60

-1.5
0

Quarter
Total hours

0.5% deviation from s.s.

Consumption

0

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

-2

-2.5

20

40

60

20

40

Quarter

60

20

40

Quarter

Avg labor productivity

Investment

60

0

-5
Model (HA-T)
Model (HA-N)
Model (HA-F)

-10

-2.5
0

0

Quarter

-15
0

20

40

Quarter

60

0

20

40

60

Quarter

Note: TFP denotes total factor productivity. The …gures display the IRFs of macroeconomic aggregates to anegative 2 percent TFP shock with persistence z .

labor productivity is even smaller, despite it having the largest fall in hours.
To understand the underlying cause of these di¤erences in aggregate dynamics, it is useful toinvestigate the impulse responses at a more disaggregated level.22 Speci…cally, in each period, wecategorize households into three almost evenly distributed groups: (i) the low productivity group
fxi g4i=1 ; (ii) the mid productivity group fxi g6i=5 ; and (iii) the high productivity group fxi g10
i=7 :Figure 4 plots the impulse responses of hours by productivity following the same negative shocks,
whereas Figure 5 plots its counterparts with respect to positive TFP shocks.
There are several important patterns worth noting. First, there is a relatively small di¤erencein labor supply responses among the mid productivity group across the di¤erent model speci…cations. On the other hand, the response of the high productivity group is clearly weaker in Model(HA-T) compared to the other heterogeneous-agent models. Second, recall that Proposition 1
from our simple model implies that agents with lower potential earnings tend to be more elastic in
22

Another obvious candidate is the dynamics of equilibrium prices. Figure A5 displays changes in the marketclearing wage per e¢ ciency units of labor and in real interest rates following the same negative TFP shock foreach of our model speci…cations. It appears as though the di¤erence between these speci…cations is not substantial
among the heterogeneous agent models, suggesting that our main results are not driven mainly by the di¤erencein equilibrium price dynamics.

24

Figure 4: Impulse responses of total hours by productivity
Low productivity% deviation from s.s.

2

Mid productivity

2

0

0

0

-2

-2

-2

-4

-4

-4

-6

-6
0

20

40

Quarter

60

High productivity

2

Model (HA-T)
Model (HA-N)
Model (HA-F)

-6
0

20

40

Quarter

60

0

20

40

60

Quarter

Note: Households are grouped into low productivity (below median), mid productivity (median), and high productivity (above median). The …gures display impulse responses for employment in each group to a negative 2
percent TFP shock with persistence z .

their labor supply. In fact, this pattern clearly applies to Model (HA-T), which generates greater
magnitudes of changes in labor supply among low productivity groups. This heterogeneity oflabor supply responses explains why Model (HA-T) is able to reduce the cyclicality of averagelabor productivity.
However, this monotonous relationship between elasticity and individual productivity breaks
down for the low productivity group, especially in Model (HA-N). This exceptionally inelastic
employment response is related to the results of Domeij and Floden (2006) and of Yum (2018).
Speci…cally, both of these papers consider incomplete markets models without public insurance
and show that wealth-poor households who lack self-insurance have precautionary labor supply
motives at the intensive margin (Domeij and Floden 2006) and at the extensive margin (Yum2018). Such precautionary motives can dominate the standard intertemporal substitution motive,
which in turn could weaken the responses of hours with respect to a persistent fall in wages.
This inelastic labor supply among the low productivity group provides a key reason for boththe lower volatility of total hours and the highly procyclical average labor productivity seenin Model (HA-N). This illustrates why heterogeneity per se is not su¢ cient to explain our key
results in incomplete markets environments. The impulse responses from Model (HA-F)— which
provides uniform transfers— show that the low productivity group has now become responsive toaggregate TFP changes though its magnitude is weak.
The precautionary motive would be less relevant when positive aggregate shocks hit theeconomy since they would move agents away from the borrowing limit. Figure 5 shows that,with respect to positive aggregate shocks, low productivity agents in Model (HA-T) do not show

25

Figure 5: Impulse responses of total hours by productivity with respect to positive TFP shocks
Low productivity% deviation from s.s.

6

Mid productivity

6

4

4

4

2

2

2

0

0

0

-2

-2
0

20

40

Quarter

60

High productivity

6

Model (HA-T)
Model (HA-N)
Model (HA-F)

-2
0

20

40

60

Quarter

0

20

40

60

Quarter

Note: Households are grouped into low productivity (below median), mid productivity (median), and high productivity (above median). The …gures display impulse responses for employment in each group to a positive 2
percent TFP shock with persistence z .

muted responses on impact. This is in contrast to their sluggish immediate responses following
negative aggregate shocks in Figure 4. Since negative shocks tend to move agents toward theborrowing limit, they raise precautionary motives and thus weaken intertemporal substitution
motives.23

5.3

Progressivity and aggregate ‡uctuations

We now use our baseline model to conduct counterfactual exercises about how higher progressivitya¤ects aggregate ‡uctuations. The …rst counterfactual adjusts the parameters in the transfer
function so that higher overall tax-and-transfer progressivity is achieved by a¤ecting low-income
households disproportionately. We also consider a case where we raise the progressivity in the tax
function while keeping the transfer function unchanged. To control for the strength of each policy
reform, we ensure that each policy increases the di¤erence between the income Gini coe¢ cients
before and after taxes and transfers by 2 percentage points, as compared to the baseline model.24
Table 6 reveals that the …rst counterfactual exercise indeed increases the amount of transfers
going to low-income households with a higher rate at which transfers are phased out. Conse23

Figure 5 also reveals that the overall magnitude of employment responses features asymmetry. Speci…cally,
the overall responses are stronger with respect to negative shocks, as compared to those with respect to positive
shocks. This is in fact in line with the distribution of arc-elasticities, based on reservation raises: Figure 2 showsthat downward adjustments induce greater elasticities (above two) relative to upward adjustments (around one).
24
Speci…cally, tax progressivity is increased by raising p by 80% (or p = 0:0954). A higher p tends to raiseoverall tax revenues, and this works to increase redistribution. As for transfer progressivity, we adjust both !pand !s simultaneously. This is because a higher !p tends to reduce the overall size of transfers, which works
against an increase in redistribution. The required percentage increase is 25%.

26

Table 6: E¤ects of progressivity on the steady-state economy and aggregate ‡uctuations
Baseline
Higher progressivity
Model Transfer
Tax(HA-T) function
function
Steady state
Employment rate (%)
Overall
77.7
71.2
By wealth quintile
1st
85.3
50.0
2nd
79.3
83.8
3rd
84.4
80.6
4th
75.2
76.8
5th
64.2
64.6

78.4
92.7
75.3
85.1
74.6
64.2

- Cond. mean/uncond. mean of T2 by income quintile
1st
3.07
3.61
3.03
2nd
1.07
0.92
1.09
3rd
0.56
0.34
0.57
4th
0.24
0.12
0.25
5th
0.06
0.02
0.06

Y
H= Y
Cor(Y; Y =H)
Cor(H; Y =H)

Business cycles
1.29
1.37
0.73
1.09
0.69
0.19
0.08
-0.44

1.29
0.75
0.66
0.05

Note: Each counterfactual exercise leads to the same Gini of after-tax-and-transfer income using each policy
instrument.

quently, the steady-state employment rates of wealth-poor households are a¤ected heavily. On
the other hand, the second counterfactual exercise has more balanced e¤ects on the employment
rates across the distribution, and the pattern of transfers across the income distribution remain
nearly unchanged.
Having checked the steady state e¤ects, we now move on to business cycle implications. Table
6 reports that the …rst counterfactual exercise reduces the cyclicality of average labor productivity and raises the volatility of total hours quite substantially. On the other hand, the second
counterfactual exercise has limited e¤ects on aggregate labor market dynamics. Average laborproductivity becomes slightly less procyclical and the volatility of hours increases marginally.25
25

In Figure A3, we plot aggregate labor supply curves (or reservation raise cumulative distributions) and their

27

Overall, the above results reinforce our conclusion in the previous subsection that changes in therate at which transfers are phased out are quantitatively more important than changes in averagetax progressivity when it comes to aggregate labor market ‡uctuations with labor supply at theextensive margin.

6

Microeconomic evidence of heterogeneity in the extensive margin labor supply responses

As shown in the previous sections, the key mechanism of our model relies on heterogeneous labor
supply responses. More precisely, households with low potential earnings are considerably moreelastic in adjusting their labor supply at the extensive margin, which weakens a highly procyclicalaverage labor productivity and enlarges the volatility of aggregate hours worked over the businesscycle. In this section, we empirically document heterogeneity in labor supply responses to verify
whether our key model mechanism exists in the micro data.26
Speci…cally, we exploit the panel structure of the PSID to explore whether extensive marginlabor supply responses di¤er as a function of hourly wage. This panel structure is useful because
we can keep track of the same people and observe their labor supply decisions over time. Because
labor supply changes can be measured in di¤erent ways and can be shaped by forces at di¤erent
levels (i.e., idiosyncratic vs. aggregate), we consider two approaches. The …rst approach computes
the probability of the extensive margin labor supply adjustment for each individual and illustrates
how it di¤ers by wage. The second approach focuses on di¤erences in the magnitude of full-time
employment rate changes across wage groups during the last six recessions.
The …rst approach requires us to have relatively long time-series observations for each individual to obtain a consistent estimate of the adjustment probability, based on individual-level‡ow data.27 First, let us …x the year at j, and denote i as an individual index, and t the year
when the individual is observed. We de…ne the extensive margin adjustment based on a full-time
employment indicator Ei;t : an individual i in year t is in full-time employment (i.e., Ei;t = 1) if
the annual number of hours worked is greater than 1,000.28 Then, we de…ne a binary switchingcorresponding arc elasticities for the two counterfactual experiments, which corroborate our cyclical volatility
results.
26
There is limited empirical evidence of heterogeneity in labor supply responses at the extensive margin acrosswage groups. See Kydland (1984) and Juhn et al. (1991) for earlier evidence. Hoynes et al. (2012) provide
evidence of heterogeneity in employment rates across other (potentially related) dimensions, such as race, gender,
age, and education.
27
Since the frequency of the PSID survey had been annual until 1997 and became biannual from 1999 onward,
we use only samples observed annually from the 1969–1997 waves.
28
The results in this section are quite robust to alternative threshold values for the full-time employment
variable. In Appendix I, we report the results when we use 1,500 hours as a full-time threshold value.

28

Table 7: Probability of extensive margin adjustment, by wage quintile

5 years
Wage quintile
in base year
1st
2nd
3rd
4th
5th
Base years
Avg. no. obs
in base years
Total no. obs.
Avg. age

All
.097
.051
.038
.034
.037

Switches
Pos only Neg only
.061
.036
.030
.020
.020
.018
.016
.018
.018
.019

The length of tracking time T
10 years

All
.075
.042
.032
.028
.032

Switches
Pos only Neg only
.048
.027
.025
.017
.018
.014
.014
.015
.015
.017

15 years

All
.066
.038
.031
.026
.030

Switches
Pos only Neg only
.042
.024
.022
.015
.017
.013
.012
.014
.014
.016

1969–1993 (J = 25)
1,677

1969–1988 (J = 20)
1,189

1969–1983 (J = 15)834

41,92040.2

23,78341.0

12,51441.5

Note: See text for the de…nition of the switching probability reported in this table. Numbers in parentheses show
the number of base years. We use samples whose age is between 22 and 64 (inclusive) and who are heads and are
not self-employed. "All" refers to the baseline estimates when using both positive and negative switches, whereas"pos only" and "neg only" use only positive ones (i.e., Ei;t = 1 and Ei;t 1 = 0) and only negative ones (i.e.,
Ei;t = 0 and Ei;t 1 = 1), respectively.

variable Si;t such that it equals one if Ei;t 6= Ei;t 1 and it equals zero otherwise. We exclude
transitions from Ei;t 1 = 1 to Ei;t = 0 if the individual has a non-zero unemployment spell inperiod t in order to rule out transitions caused by layo¤s.
Note that, given the length of time spent tracking each individual T; there are T 1 countsof Si;t for each individual i. Once we take the average over time, we obtain the individualspeci…c probability of an extensive-margin adjustment with an annual frequency (i.e., pi;j
Pj+T 11
t=j+1 Si;t ). As we are interested in di¤erences across the wage distribution, we computeT 1q
pj , which is de…ned as the conditional mean of pi;j for the wage quintile bin of each individualq 2 f1; 2; :::; 5g determined in the base year j.
Because di¤erent values for the length of time spent tracking each individual entails a tradeo¤, we consider three variants: T 2 f5; 10; 15g. On the one hand, a larger number is bene…cial
because we are more likely to have a consistent estimate of the adjustment probability at theindividual level. On the other hand, a longer tracking time implies a tighter restriction on
the samples— because we keep only samples that were observed for T consecutive years. Given29the value of T , we compute the estimates of fpqj g5q=1 by changing the base year j. That way,
we attempt to mitigate variations due to di¤erences in the initial wage distribution, which ispotentially a¤ected by business cycle ‡uctuations. The reported values in Table 7 are the mean
P qswitching probabilities for each wage quintile averaged across the base years, pq J1
pj ; where
the number of base years J is reported in parentheses.
Table 7 reveals a clear pattern: the individual-level probability of adjusting the extensive
margin is signi…cantly higher among low-wage workers. For instance, when T = 5, the probabilityof switching to or from full-time employment among the …rst wage quintile is 9:7% (the annual
frequency). In particular, we can see that this probability tends to decrease with wage. For
the third to …fth quintiles, this probability is relatively ‡at at approximately 3:5%. When T
increases, we also …nd that the key pattern of extensive margin adjustment probabilities acrosswage quintiles is still present. However, because the samples become slightly older and T becomes
longer, we also see that their switching probabilities generally become lower.
We also compute these statistics using only either positive switches (Ei;t Ei;t 1 > 0) or
negative switches (Ei;t Ei;t 1 < 0). Table 7 shows that negative wage gradients in the probabilityof full-time employment adjustments are present in both cases. Interestingly, positive switchesfeature not only higher adjustment probabilities but also a quantitatively larger negative wage
gradient, showing that the overall gradient is more strongly driven by such positive adjustments.
The above exercise is based on long-run information regarding labor market ‡ows at theindividual level. The next empirical exercise instead uses the di¤erences in magnitude of fulltime employment level changes across wage groups during recessions. Speci…cally, we choose six
recessions and for each recession we choose a peak year and a trough year, as guided by thecyclical component of quarterly real GDP per capita (Figure A1). Our de…nition of peak andtrough years is limited by the frequency of the PSID because the data set was available annually
until 1997 and only biannually since 1999. Therefore, our choice is also based on declines inaggregate employment during each recession event— according to our micro samples from the
PSID. The resulting year combinations for each recession are shown in Table 8.
Next, we compute the conditional mean of full-time employment by wage quintile in the peakP qqyear for each recession by N q1
i Ei;peak ; where Npeak is the number of observations in wage
peak
quintile bin q during the peak year. We then measure the percentage changes in the full-time
employment rate by wage quintile in the corresponding trough year. It is important to note that
we keep the set of households in each wage group …xed by assigning a wage quintile to eachhousehold in the peak year. That way, our measured changes by wage quintile are not a¤ectedby compositional changes, but are rather based on changes from within the same households.
Table 8 clearly shows that the employment rate fell most sharply in the …rst and second
wage quintiles during all of the recessions, and that the magnitude of these declines tends to be30

Table 8: Full-time employment changes in recessions, by wage quintile
Recession
1969–71 1973–76 1980–83 1990–92 2000–02 2006–10
Wage quintile
in peak year
1st
2nd
3rd
4th
5th

-7.1
-3.2
-3.7
-4.7
-0.9

-10.1
-7.4
-7.0
-4.5
-5.7

-9.8
-4.5
-5.2
-5.9
-4.9

-8.7
-6.3
-4.6
-5.4
-4.6

-10.0
-5.6
-3.0
-4.8
-2.2

-17.2
-12.9
-11.2
-10.4
-5.8

No. obs.

1,655

1,756

2,007

2,166

2,924

2,802

Note: The full-time employment threshold is set to 1,000 annual hours. The year ranges denote the peak andtrough years of each recession. Reported values are percentage changes in the full-time employment rate by wage
quintiles (in the peak year of each recession) following the same set of individuals.

smaller among the higher wage quintiles. For example, the full-time employment rate among the…rst wage quintile during the last recession (i.e., the Great Recession) fell by 17:2%, whereas thecounterpart among the …fth wage quintile fell by only 5:8%. This pattern of full-time employmentchanges by wage quintiles is quite robust across di¤erent recessions despite variations in overallmagnitude.29
One may be concerned about the possibility that the wage gradient of full-time employmentchanges found in Table 8 is driven largely by the demand channel of the …rms, and that this maya¤ect household employment status di¤erentially across the wage distribution. To alleviate thisconcern, we also use the information from the PSID data about unemployment spells (available
since the 1976 wave or the year of 1975) and exclude samples that experienced any unemployment
spells in either the peak year or the trough year. We thereby attempt to rule out the e¤ects
caused by di¤erential layo¤ probabilities across the wage distribution, although the number ofobservations in each recession decreases because of this additional sample restriction. Table A5
shows that although the magnitudes of full-time employment changes are somewhat weaker, themagnitude is negatively related to wage quintiles during most recessions.
Although the above two approaches are designed to capture di¤erent aspects of labor supply
adjustments, they yield consistent results: the employment adjustments of lower-wage workers
are more volatile. Both of these empirical …ndings are therefore consistent with the pattern of
29

Note that the overall magnitude of the fall in employment is relatively greater in the recessions of 1973–76,
1980–83 and 2006–10. This …nding is, in fact, consistent with the relatively larger amplitudes of these recessions,as shown in Figure A1. This provides some external validation for our micro samples.

31

heterogeneity in labor supply responses seen in the baseline model. Nevertheless, we would also
like to stress that the results in this section are only suggestive since we cannot rule out otherpossible demand-related e¤ects from being behind the observed heterogeneous patterns. For
example, they could be consistent with intrasectoral changes in the quality of workers made by…rms (Ohanian 2001). Alternatively, there might be disproportionately more low-wage workers
in those industries that are more vulnerable to the e¤ects of the business cycle (Hoynes et al.
2012), or labor hoarding might be di¤erential across wage groups.

7

Conclusion

In this paper, we have explored the interplay of household heterogeneity and tax-and-transfer
progressivity in shaping the dynamics of macroeconomic aggregates over the business cycle. Using
analytical results obtained from a stylized static model of the extensive margin labor supply, we…rst presented the key insight that higher progressivity would strengthen negative wage gradientsin employment responses to aggregate shifters. Using a general equilibrium business cycle modelwith household heterogeneity, we have shown that micro-level heterogeneity substantially shapes
the dynamics of aggregate labor market variables when heterogeneity interacts with progressive
transfers. In particular, our baseline model delivers less procyclical average labor productivity
when compared to the nested models that are similar to standard real business cycle models. At
the same time, it retains the success of the canonical representative-agent indivisible labor supply
model in generating a large volatility of aggregate hours without the assumptions of lotteries andperfect consumption insurance (Rogerson 1988). Our counterfactual analysis shows that in the
U.S. tax and transfer system, the rate at which transfers are phased out is quantitatively very
important for understanding labor supply along the extensive margin.
There are several future research questions that follow naturally from our study. One interesting and novel result that we highlighted in this paper is that the e¤ects of higher progressivity
can be quite di¤erent depending on which part of the income distribution is more a¤ected by
the …scal change. A straightforward application would be to design a tax-and-transfer systemthat takes into account the welfare costs of business cycles in the presence of heterogeneous
agents (e.g., Krusell et al. 2009). Second, although the current paper focuses on the total factor productivity shock, it would be interesting to explore how other types of aggregate shocks(such as monetary policy shocks) would be transmitted di¤erently in our framework. Finally,
our paper introduces some theoretical and quantitative mechanisms suggesting that changes inprogressivity at the bottom of the income distribution (e.g., welfare transfers) might be behindthe vanishing procyclicality of average labor productivity, as documented by Galí and van Rens32

(2021).30 Formal investigations of these changing relationship are out of the scope of the current
paper, but they would nonetheless be highly valuable to address in future work.

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Rogerson, Richard. 1988. "Indivisible Labor, Lotteries and Equilibrium." Journal of Monetary
Economics 21 (1): 3-16.
Takahashi, Shuhei. 2020. "Time-Varying Wage Risk, Incomplete Markets, and Business Cycles."Review of Economic Dynamics 37: 195-213.
— — — . 2014. "Heterogeneity and Aggregation: Implications for Labor-Market Fluctuations:
Comment." American Economic Review 104 (4): 1446-1460.
Yum, Minchul. 2018. "On the Distribution of Wealth and Employment." Review of Economic
Dynamics 30: 86-105.

36

Online Appendix
A

Proofs in Section 2

Proof of Proposition 1 Assume Ti = 0. Then, we can rewrite
ai = zxi :Therefore,
Ni = 1exp( zxi )Given this, note that
"i

@Ni z
= xi exp( zxi )@z Ni
1
zxi exp( zxi )=1 exp( zxi )zexp( zxi )For expositional convenience, assume that x is continuous for now.
"(x) =

zx exp( zx)
1 exp( zx)[z exp( zx) z 2 x exp( zx)] [1 exp( zx)] zx exp( zx) [z exp( zx)]
@"(x)
=
@x
[1 exp( zx)]2exp( zx)z [1 zx] [1 exp( zx)] z 2 x exp( zx) [exp( zx)]=
[1 exp( zx)]2
z exp( zx) f1 zx exp( zx)g=
[1 exp( zx)]2
Since exp( zx) < 1 for all z; x > 0,z exp( zx) (1 zx 1)@"(x)
z exp( zx) (1 zx exp( zx))=<
2@x
[1 exp( zx)]
[1 exp( zx)]2
z exp( zx) ( zx)=< 0:[1 exp( zx)]2

A-1

Proof of Proposition 2 Since@Nl= exp( al )(1
);
@z
@Nh
= exp( ah )(1 + ):@z
we have
@ @Nl
@! @z
@ @Nh
@!
@z

= exp( al )(1=)T > 0;exp( ah )(1 + )T < 0:Also, note that@Nl
= exp( al )T < 0@!
@Nh= exp( al )T > 0:@!
Proof of Proposition 3 Since
"

@N z
@z N
@Nl
+
=
l
@z

h

@Nh
@z

z
l Nl +h Nh

the aggregate labor supply elasticity is given by"=zexp( al )(1) + exp( ah )(1 + )2 exp( al ) exp( ah )where
al = z(1)

ah = z(1 + )

A-2

T

T!

T + T! :Then, we have

@"
=z
@!

= zT

[exp( al )(1)( 1)( T ) + exp( ah )(1 + )( 1)T ] [2 exp( al ) exp( ah )]) + exp( ah )(1 + )] [ exp( al )( 1)( T ) exp( ah )( 1)T ][exp( al )(1[2exp( al )exp( ah )]2[exp( al )(1) exp( ah )(1 + )] [2 exp( al ) exp( ah )]) + exp( ah )(1 + )] [exp( al ) exp( ah )]+ [exp( al )(1[2exp( al )exp( ah )]2@"
The sign of @!
is equal to that of the numerator, which can be rewritten as

Numerator = 2(1) exp( al )(1

) exp( 2al )2(1 + ) exp( ah ) + (1 + ) exp( ah+ (1) exp( 2al )+ (1 + ) exp( ah= 2 [(1Letting(1
al )) exp( al )) exp( ah(1) exp( ahal )al ) + (1 + ) exp( 2ah )al )(1 + ) exp( 2ah )(1 + ) exp( ah ) + 2 exp( ahal )] :)= (1; we can rewrite
(1+ )

(1
)
exp( al )(1 + )

exp( ah ) += 2(1 + ) [ exp( al ) + (1) exp( ah2(1 + )

2exp( ah(1 + )al )al )exp( ah )] :Since exp( x) is convex, we know
exp( al ) + (1) exp( (ah + al )) > exp ( f al + (1= exp ( f(1) (ah + al )g)) ah + al g) :Applying this inequality, we have
Numerator = 2(1 + ) [ exp( al ) + (1
> 2(1 + ) [exp ( f(1A-3) exp( ah) ah + al g)al )exp( ah )]exp( ah )]0

if and only if(1(1 + ) [z(1z(1 + )(1))

(1 + )T(1 + )T) ah + al

ah

al

ah

T! ]

(1

(1 + )T !

) [z(1 + )z(1 + )(1

(1 + )!

(1
1) + (1)

T + T! ](1)T + (1

)T !

)!

!

which is always satis…ed.
Proof of Proposition 4 Note that
0 =

=
=

(1
1
2

exp( al )) + (1 + ) (1 exp( ah ))2 exp( al ) exp( ah )exp( ah )exp( al ) + exp( al ) + 1 +2 exp( al ) exp( ah )(1
) exp( al ) (1 + ) exp( ah ):2 exp( al ) exp( ah )) (1exp( ah )Therefore, we have(1
)2 exp( al ) + (1 + )2 exp( ah ) [2 exp( al ) exp( ah )]@ 0=@z(2 exp( al ) exp( ah ))2[2 (1) exp( al ) (1 + ) exp( ah )] [exp( al ) (1) + exp( ah ) (1 + )]2
(2 exp( al ) exp( ah ))
8
9
2
2
>
>
)
+
2
(1
+
)
exp(
a
)
2
(1
)
exp(
a
>
>
l
h
>
>
>
>
2
2
>
>
>
>
(1
)
exp(
2a
)
(1
+
)
exp(
a
a
)
>
>
l
h
l
>
>
>
>
2
2
<
=
(1
) exp( ah al ) (1 + ) exp( 2ah )
1
=
>
2 (1
) exp( al ) 2 (1 + ) exp( ah )(2 exp( al ) exp( ah ))2 >
>
>
>
>
>
>
2
>
>
>
>
+
(1
)
exp(
2a
)
+
(1
+
)
(1
)
exp(
a
a
)
>
l
h
l >
>
>
>
>
2
: + (1 + ) (1) exp( ah al ) + (1 + ) exp( 2ah ) ;
=

=

2 (

2 f(

1) exp( al ) + 2 ( + 1) exp( ah ) 4 2 exp( ah al )(2 exp( al ) exp( ah ))2
1) exp( al ) + ( + 1) exp( ah ) 2 exp( ah al )g< 0:(2 exp( al ) exp( ah ))2A-4

Proof of Proposition 5 De…ne
(!)

@ 0
@z

log

:

Since the log transformation preserves monotonicity, it su¢ ces to show that(!) = log 2 + log f(
2 log (21) exp( al ) + ( + 1) exp( ah )exp( al )2 exp( ah0(!) < 0. Asal )gexp( ah ))we have
0

(!) =

=

T (
1) exp( al ) + T ( + 1) exp( ah )(1) exp( al ) + ( + 1) exp( ah ) 2 exp( ahT exp( al ) T exp( ah )2
2 exp( al ) exp( ah )T (1) exp( al ) + T ( + 1) exp( ah )|{z}positive(|21) exp( al ) + ( + 1) exp( ah ){z

2 exp( ahnegative

T [exp( al ) exp( ah )]
|
{z
}

al )
}

positive

2
|

< 0:

B

al )

exp( al ) exp( ah ){z}positive

Representative-agent (RA) model

We …rst describe the environment of Model (RA). At the beginning of each period, the stand-in
household holds the assets of that period k. The aggregate state variables are the aggregate
capital K and the aggregate TFP shock zk , with the latter following the same stochastic process
as in the baseline model. Taking the real wage rate w(K; zk ); the real interest rate r(K; zk ), andthe aggregate law of motion (K; zk ) as given, the dynamic decision problem of the representative
household can be written as the following functional equation:
V (k; K; zk ) = 0max

k 0;c 0n2[0;1]

(

log c

Bn +

Nz
X
l=1

A-5

)

z
0
0 0
kl V (k ; K ; zl )

subject to c + k 0(1l )w(K; zk )n + (1 + r(K; zk ))k + T

K 0 = (K; zk )The household maximizes utility by choosing its optimal consumption c, the next period’s capital k 0 ; and its labor supply n. The utility of the stand-in household is linear with respect toemployment n due to the aggregation theory of Rogerson (1988). The budget constraint statesthat the sum of consumption c and the next period’s capital k 0 should be less than or equal to
the sum of net-of-tax labor income (1l )w(K; zk )n, current capital k, capital income r(K; zk )kand government transfers T .
Government then collects taxes on labor earnings l wn to …nance transfers T and government
spending G. We keep the same assumptions on the …rm side as in the heterogeneous-agent models.
The resulting …rst-order conditions for K and L are the same as those presented in (12) and (13).
A recursive competitive equilibrium is a collection of factor prices r(K; zk ), w(K; zk ), household decision rules gk (k; K; zk ), gn (k; K; zk ), government policy variables l , G, T , the household
value function V (k; K; zk ), the aggregate labor L(K; zk ) and the aggregate law of motion for
aggregate capital (K; zk ) such that
1. Given factor prices r(K; zk ), w(K; zk ) and government policy l , G, T , the value function
V (k; K; z) solves the household’s decision problem, and the associated decision rules are
k 0 = gk (k; K; zk )n = gn (k; K; zk ):2. Prices r(K; zk ), w(K; zk ) are competitively determined following (12) and (13).
3. Government balances its budget:
G + T = l w(K; zk )L(K; zk ):4. Consistency is satis…ed: for all K;
K 0 = (K; zk ) = gk (K; K; zk )L(K; zk ) = gn (K; K; zk ):It is straightforward to calibrate the parameters of Model (RA) using the steady state equi-A-6librium equations. First,

is directly obtained by:= (1 + r) 1 :Then, given the targets of T =Y = 0:044, L = 0:782 and
B=

(1
1

l = 0:1111; B is obtained by

l )(1
K
G
Y
Y

)
L

)

T
:
Y

where
K
=
Y
r+
G
= (1
Y

Finally, since Y =K = (K=L) 1 , we can obtain K=L. This in turn gives us K; and thus Y . We
can then obtain T using the calibration target ratio T =Y = :044. The resulting calibrated values
are = 0:9901; B = 1:0164; and T = 0:1277.

C

Heterogeneous-agent models without labor supply indivisibility

Indivisible labor supply is a key feature of our analysis. We illustrate this point by consideringa heterogeneous-agent model with divisible labor. The economic environment in this model ismostly identical to the heterogeneous-agent models in the main text, and includes features suchas idiosyncratic shocks, progressive taxation, and …rm technology. However, one exception isthat households can adjust their hours in a fully ‡exible way under the following period utility
function with constant Frisch elasticity :1

U (c; h) = log c

h1+
:
1+ 1

(A1)

We consider two di¤erent values of 2 f1; 2g: To illustrate the role of transfers for businesscycle ‡uctuations in this alternative environment, we consider two cases: zero transfers and ‡attransfers. In the latter case, we target the same moment (4.4% of output) that is used in themain text. For each speci…cation, we also calibrate and to target the full-time employment
rate of 78.2% and the real interest rate of 1% where full-time is de…ned as hours greater than

A-7

0.2.
The results are summarized in Table A1, with two …ndings in particular being worth highlighting. First, the models without indivisible labor supply have di¢ culty in generating a su¢ ciently
high volatility of hours worked, echoing the performance of representative-agent real businesscycle models (Kydland and Prescott 1982). Even with a relatively large value of = 2, thevolatility of aggregate hours is considerably smaller than in the data. Moreover, these divisible
labor models generate average labor productivity that is almost perfectly correlated with output,
given that labor supply responses are nearly homogeneous across households and thus do notvary negatively with individual productivity. This is in sharp contrast to our baseline models
with labor supply indivisibility (recall Proposition 1 in Section 2). The second notable observation is that the presence of transfers appears almost irrelevant to the cyclicality of average laborproductivity, although it does moderately raise the volatility of hours.
Table A1: Results from models without indivisibility

Y
C= Y
I= Y
L= Y
H= Y
Y =H = Y

Cor(Y; C)
Cor(Y; I)
Cor(Y; L)
Cor(Y; H)
Cor(Y; Y =H)
Cor(H; Y =H)

Indivisible(HA-N) (HA-F)

:00

=1
:044

:00

=2
:044

1.15
0.32
2.77
0.34
0.28
0.73

1.16
0.30
2.77
0.35
0.30
0.70

1.27
0.31
2.79
0.45
0.36
0.65

1.28
0.29
2.79
0.47
0.39
0.62

1.48
0.28
2.99
0.64
0.51
0.54

1.46
0.27
2.99
0.62
0.60
0.57

0.91
0.99
0.98
0.98
1.00
0.96

0.91
0.99
0.98
0.97
1.00
0.97

0.91
0.99
0.98
0.98
0.99
0.96

0.90
0.99
0.98
0.99
0.99
0.96

0.85
0.99
0.96
0.95
0.95
0.81

0.84
0.99
0.96
0.87
0.85
0.81

T =Y =

Note: Each model speci…cation is calibrated to generate the same interest rate and the full-time employment rate.

D

Aggregate data

The business cycle statistics are based on the aggregate time-series data from U.S. Bureau of
Economic Analysis (BEA), National Income and Product Accounts (NIPA) Tables covering theperiod from 1961Q1 to 2016Q4. For output, we use the “Real Gross Domestic Product (millions
A-8

Figure A1: Cyclical component of real GDP per capita

Note: A quarterly series of real GDP per capita is detrended using HP …lter with a smoothing parameter of 1,600.

of chained 2012 dollars)”entry in Table 1.1.6. As for consumption, we use expenditures on nondurable goods and services, as reported in Table 2.3.5 (Personal Consumption Expenditure).
Investment is constructed as the sum of expenditures on durable goods (Table 2.3.5) and private…xed investments (Table 5.3.5). The real values of consumption and investment are calculatedusing the price index for Gross Domestic Product from Table 1.1.4. Data on total hours worked
are obtained from Cociuba et al. (2018). We modi…ed all of the raw time series into per capita
series by dividing the raw data by the quarterly population reported by Cociuba et al. (2018).
A target statistic regarding the size of income-security transfers is based on the aggregate dataobtained also from the BEA NIPA Tables. Speci…cally, we use data from Table 3.12 (Government Social Bene…ts) on the Supplemental Nutrition Assistance Program (SNAP), Supplemental
Security Income, Temporary Disability Insurance, and medical care (Medicaid, General Medical
Assistance, and state child healthcare programs). Note that we do not include large programs
such as Medicare, unemployment insurance, and veterans’bene…ts.

A-9

E

Micro data

For the transfer-related statistics obtained at the micro level, we use data from the Survey of Income and Program Participation (SIPP). This data set is representative of the non-institutionalized
U.S. population and has a monthly survey period. The SIPP covers a wide range of information
on income, wealth, and participation in various transfer programs. We choose samples from the…rst wave to the ninth wave of the SIPP, covering the years 2001 to 2003. The original data set
is composed of a main module and several topical modules. While the main module contains
monthly information on income and transfers, variables such as wealth are reported quarterly inthe topical modules. We combine both modules on a quarterly basis.
We construct all variables at the household level. Data sets in the SIPP contain not only
household variables but also individual variables so, in order to generate a household variable fromits corresponding individual variable, we take the following steps. First, we identify households by
their sample unit identi…er (SSUID) and their sample household address identi…er (SHHADID).
Second, we add up the values of the variable in question for all members of the same household.
The government transfers used to infer the degree of progressivity are based on a broad range oftransfer programs including Supplemental Security Income (SSI), Temporary Assistant for Needy
Family (TANF), the Supplemental Nutrition Assistance Program (SNAP), the Supplemental
Nutrition Program for Women, Infants, and Children (WIC), childcare subsidies and Medicaid.
We do not include age-dependent programs such as Social Security and Medicare. We also
construct a broad household income variable: it consists of labor income, income from …nancial
investments, and property income. We consider households whose head is aged between 23 and65, and the results we presented are almost the same as for alternative age ranges around theselimits. Finally, we convert the nominal values of all these variables to 2001 U.S. dollars using
the CPI-U.
The empirical analysis in Section 6 is based on the PSID data. We choose samples for theperiod of 1969–2010. To avoid the oversampling of low-income household heads, we exclude
households listed in the Survey of Economic Opportunity. We also drop the samples whose wage
is below one half of the minimum wage. The nominal values are again converted into 2001 U.S.
dollars using the CPI-U.

F

Estimation of idiosyncratic productivity risk

We estimate the persistence of idiosyncratic productivity risk in the U.S. using the PSID data,
following Heathcote et al. (2010). Our measure of productivity is de…ned as a worker’s hourly
wage relative to other individuals. We consider household heads between the ages of 18 and 70,
A-10and whose wages were observed for at least four consecutive periods.31 To focus on full-timeworkers, we drop the samples whose annual hours worked was less than 1; 000.
We run the ordinary least squares regression on the logarithm of the productivity (hourly
wages) on a dummy for male, a cubic polynomial in potential experience (age minus years ofeducation minus …ve), a time dummy, and a time dummy interacted with a college education
dummy. We take its residual xi;j as an idiosyncratic productivity variable that contains a wide
range of individual abilities valued by the labor market. This stochastic process is composed of
the summation of a persistent process i;j and a transitory process i;j as described by:
xi;j =
0

i;j =

i;j +

i;j ; i;j

N (0;

0

0

i;j 1 + i;j ; i;j

2

(A2)

);

N (0;

2

):

We use a minimum distance estimator to estimate the parameters of the process. This method
is used to …nd parameters that minimize the distance between the empirical and theoretical
moments. We take the covariance matrix of the residual xi;j as our moments, and denote by
the vector ( ; v ; ). We then let mj;j+n ( ) be the covariance of the labor productivity between
age j and j + n individuals, and de…ne m^ j;j+n as the empirical counterpart of mj;j+n ( ). We use
the following moment conditions:
E [m^ j;j+nmj;j+n ( )] = 0

(A3)

where
m
^ j;j+n =Nj;j+n1
Nj;j+nXxi;j xi;j+n

i=1

The moments can be represented by as an upper triangle matrix:231

3m0;0 ( ) m0;1 ( )m0;J 1 ( )m0;J ( )67
6 0m1;1 ( )m1;J 1 ( )m1;J ( ) 7
6
7
6 0
7
0
m
(
)
m
(
)
m
(
)
2;2
2;J
1
2;J
6
7
m( ) = 6 .
7
..
..
..
..
..
6 ..
7
.
.
.
.
.
6
7
6
7
0
0
mJ 1;J 1 ( ) mJ 1;J ( )5
4 0
0
0
0
0
mJ;J ( )We use a somewhat less restricted age range in order to obtain a large number of samples. Note that we
impose stricter restrictions on wages and hours, which would naturally remove irrelevant samples such as retirees.
Thus, a change in the age band leads to only relatively small changes in the estimated persistence of idiosyncratic
shocks.

A-11

We denote a vector of M ( ) by vectorizing m( ) with length (J + 1)(J + 2)=2. To estimateparameters , we solvehi0 hi^ M( ) W M^ M( )
min M
where the weighting matrix W is set to be an identity matrix.32

G

Numerical methods used for the heterogeneous-agent
modelsG.1

Solving for the equilibrium with aggregate risk

The models which aggregate risk are solved with the following two steps. First, we solve for theindividual policy functions given the forecasting rules (the inner loop). Then, we update theforecasting rules by simulating the economy using those individual policy functions (the outerloop). We iterate the two steps until the forecasting rules converge— that is, when the di¤erence
between the old forecasting rule used in the inner loop and the new forecasting rule generated in
the outer loop becomes small enough.
G.1.1

Inner loop

In the inner loop, we solve for the following value functions: V (a; xi ; K; zk ); V E (a; xi ; K; zk ) andV N (a; xi ; K; zk ). These value functions are stored on a non-evenly spaced grid for a and an evenlyspaced grid for K, with the number of grid points being na = 400 and nK = 40, respectively.
Unlike Chang and Kim (2006, 2007) and Takahashi (2014), we discretize the stochastic processes
for xi and zk by using the Rouwenhorst (1995) method. We …nd that the approximation ofcontinuous AR(1) processes with our estimate featuring very high persistence is considerably
better with the Rouwenhorst method given the same number of grid points.33 Our baseline
results are based on nx = 10 and nz = 5; both of which replicate the true parameters of thecontinuous AR(1) processes very precisely.
To obtain V (a; xi ; K; zk ) = max V E (a; xi ; K; zk ); V N (a; xi ; K; zk ) , we solve the following
32

Using the identity matrix has been common in the literature since Altonji and Segal (1996) show that the
optimal weighting matrix generate severe small sample biases.
33
Speci…cally, we use the simulated data from the methods of Rouwenhorst and Tauchen, and estimate thepersistence and the standard deviation of the error terms in the AR(1) processes for both aggregate productivity
shocks and idiosyncratic shocks (results available upon request).

A-12

problemsE(

V (a; xi ; K; zk ) = max
log c
0
a a;
c 0

Bn +

Nx
X
j=1

x
ij

Nz
X

)

z
0
0 ^0 0kl V (a ; xj ; K ; zl )

l=1

(A4)

subject to
c + a0(e; e)e(w(K;^zk )) + (1 + r^(K; zk ))a + T (w(K;^zk ); r^(K; zk ))and(

V N (a; xi ; K; zk ) = max
log c +
0
a >a;
c>0

c + a0

Nx
X
j=1

x
ij

Nz
X

)

z
0
0 ^0 0kl V (a ; xj ; K ; zl )

l=1

(A5)

(1 + r^(K; zk ))a + T (^r(K; zk ))^ 0 ) which are not on
To evaluate the functional value of the expected value function on (a0 ; K
the grid points, we use the piecewise-linear interpolation. By solving these problems, we obtain the individual policy function for work gn (a; xi ; K; zk ) by comparing V E (a; xi ; K; zk ) with
V N (a; xi ; K; zk ). We also obtain conditional policy functions for the optimal a0 : gaE (a; xi ; K; zk )as the maximizer of the problem (A4) and gaN (a; xi ; K; zk ) as the maximizer of the problem (A5).
G.1.2

Outer loop

In the outer loop, we simulate the model economy based on the information obtained in the
inner loop. We note that a key step is to …nd the market-clearing prices in each period during
the simulation. Although this is computationally burdensome, we …nd that the results without
the market-clearing step are substantially misleading, as is consistent with Takahashi (2014) andChang and Kim (2014).
The measure of households (a; xi ) is approximated by a non-evenly spaced grid on a that is…ner than that used in the inner loop (Rios-Rull 1999) and has 4,000 grid points. The variable
K is then constructed by aggregating individual asset holdings over the measure of households:
R PNx
i=1 a (da; xi ): Following Takahashi (2014), we use a bisection method to obtain the equiliba
rium factor prices in each simulation period as follows:
1. Set an initial range of (wL ; wH ) and calculate the aggregate labor demand Ld = (111
) (zk =w) K implied by the …rm’s FOC for each w. Note that r is obtained by using the11w
relationship r = zk
; implied jointly by (12) and (13).
1
2. Calculate the aggregate e¢ ciency unit of labor supply Ls at each w and make sure thatthe excess labor demand (Ld Ls ) is positive at wL and it is negative at wH .
A-13

H
3. Compute w~ = wL +w
and obtain Ld
2
wH = w.
~

Ls at w.
~ If Ld

4. Continue updating (wL ; wH ) until jwL

Ls > 0, set wL = w;
~ otherwise, set

wH j is small enough.

Taking the measure of households (a; xi ), the aggregate state (K; zk ), and factor prices w
and r as given, we compute the aggregate e¢ ciency unit of labor supply Ls (K; zk ). Speci…cally,
we solve (A4) and (A5) given the expected value function in the next period using interpolation.
Note that we use the valued function obtained in the inner loop and the forecasting rule (18)
^ 0 = (K; zk ) which is not on the grid points of K. Then, the individual household decisionfor K
rules are given by

n = gn (a; xi ; K; zk ) =

8
<n
:0

if V E (a; xi ; K; zk ) > V N (a; xi ; K; zk );otherwise.

By having n = gn (a; xi ; K; zk ) for each grid point (a; xi ) on at hand, the aggregate e¢ ciency
R P x
unit of labor supply is obtained by Ls (K; zk ) = a N
i=1 xi gn (a; xi ; K; zk ) (da; xi ). After …nding
the market-clearing prices, we update the measure of households in the next period by using
a0 = ga (a; xi ; K; zk ) =8<g E (a; x ; K; z )ik:g N (a; x ; K; z )ikif V E (a; xi ; K; zk ) > V N (a; xi ; K; zk );otherwise;

and the stochastic process for xi . We simulate the economy for 10; 000 periods, as in Khan andThomas (2008).
Finally, the coe¢ cients (a0 ; a1 ; a2 ; b0 ; b1 ; b2 ) in the forecasting rules
log K 0 = a0 + a1 log K + a2 log z;(A6)

log w = b0 + b1 log K + a2 log z;(A7)

are updated by ordinary least squares with the simulated sequence of fK 0 ; w; K; zg. Our parametric assumptions regarding the forecasting rules are the same as those made in Chang andKim (2007, 2014) and Takahashi (2014, 2020). We repeat the whole procedure for the inner andouter loops until the coe¢ cients in the forecasting rules converge.
As is clear in the forecasting rules (A6) and (A7), households predict prices and the future
distributions of capital based only on the mean capital stock instead of the entire distribution.
Therefore, it is important to check whether the equilibrium forecast rules are precise or not.
We summarize the results regarding the accuracy of the forecasting rules for the future mean
A-14

capital stock K 0 and for the wage w in Table A2. It is clear that all R2 values are very highin all speci…cations. We also check the accuracy statistic proposed by Den Haan (2010). Since
our dependent variables are logarithmic, we multiply the statistics by 100 to interpret them aspercentage errors. We …nd that the mean errors are su¢ ciently small (considerably less than
0:1% for all cases) and the maximum errors are also reasonably small (not exceeding 0:8% for
all cases).
Table A2: Estimates and accuracy of forecasting rules
Model

Dependent
variable

Coe¢ cient
Const. log K
log z

R

2

Den Haan (2010) error
Mean (%) Max (%)(HA-T)

log K 0log w

0.1193 0.9554 0.0940 0.99997-0.2689 0.4242 0.8037 0.99818

0.088
0.086

0.445
0.749

(HA-N)

log K 0log w

0.1528 0.9413 0.1170 0.99998-0.5117 0.5291 0.6683 0.99918

0.087
0.060

0.499
0.452

(HA-F)

log K 0log w

0.1489 0.9431 0.1154 0.99998-0.4557 0.5045 0.6826 0.99920

0.086
0.058

0.427
0.425

G.2

Impulse response functions

There is no generally accepted way to calculate conditional impulse responses in nonlinear models.
To compute impulse response functions in this paper, we follow the simulation-based proceduredeveloped by Koop et al. (1996) (see also Bloom et al. 2018):
Draw i = 1; :::; Nsim sets of exogenous random variables for aggregate TFP shocks, each ofwhich have t = 1; :::; Tsim periods.34
For each set of i, simulate two sequences, one is from the shock economy and the other isfrom the no-shock economy.
1. In the shock economy, simulate all interested variables Xitshock for t = 1; :::; Tshock 1
as normal (as we do in the outer loop). Then, in period Tshock , impose a disturbance34

We use a random sampling with Markov chains. That is, by taking as given the index for today’s aggregate
z Nz
productivity i and the conditional distribution for tomorrow’s productivity f ij
gj=1 (i.e., the i-th row of theMarkov chain), we draw a random variable u
U [0; 1] to pick up tomorrow’s shock index j. We do so by
Pjzchoosing the highest j satisfying the condition u < k=1 ik.

A-15

on aggregate TFP so that it takes an extreme value (e.g., the lowest one z1 ). Simulate
the economy as normal for the rest of the periods t = Tshock + 1; :::; Tsim .35
2. In the no-shock economy, simulate all interested variables Xitnoshock for all the periods without any restrictions. The two economies are di¤erent only in terms of theimposition of the extreme shock in period Tshock .
The e¤ect of the disturbance on X is given by the average percentage (or percentage point)di¤erence between the two sequences:^ t = 100X
^ t = 100
X

Nsim
1 X
log Xitshock =Xitnoshock
Nsim i=1
Nsim
1 X
Xitshock
Nsim i=1

Xitnoshock(percentage di¤erence)(percentage point di¤erence)

The results are based on Nsim = 2; 000 simulations with each simulation having two sequencesof the variables of interest for Tsim = 150 periods. The responses are equal to zero before Tshockby construction. The disturbance then hits the economy at period Tshock = 50, which we label
as the …rst period in our …gures.

H

Additional model results

Table A3 reports business cycle results for several alternative models recalibrated to match thesame target statistics as in Table 1. First, we replace the progressive taxation system in (8) witha linear taxation system while keeping the average tax constant. This is helpful for understanding
how important the presence of progressive taxation is for business cycles while controlling fortransfer progressivity. We …nd that its impact is very minimal for business cycle ‡uctuations.
The second sensitivity check concerns the borrowing limit. The third column in Table 1 reports
the results from when we set a to zero, and these show that aggregate ‡uctuations are barelya¤ected by this change. Next, we consider a change in target statistics regarding the variabilityof idiosyncratic shocks. Recall that the baseline model targets the Gini wage of 0:36. We …ndthat, although its impact is not sizable, a higher wage variation tends to lower the cyclicality of
average labor productivity and raise the relative volatility of hours.
35

Note that the e¤ect of the disturbance is persistent because we sample aggregate productivity using theconditional distribution of the Markov chain.

A-16

Figure A2: Aggregate labor supply elasticities and arc elasticities for di¤erent model speci…cations

1
0.9
0.8

CDF Raise

0.7
0.6
0.5
0.4
0.3
Model (HA-T)
Model (HA-N)
Model (HA-F)
Model (HA-N) w/ low

0.2
0.1

x

0
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Reservation Raise

7
Model (HA-T)
Model (HA-N)
Model (HA-F)
Model (HA-N) w/ low

6x

Implied Arc Elasticity

5

4

3

2

1

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Change in Aggregate Raise (

0.4

0.6

0.8

1

)

Note: The arc elasticities (bottom panel) are computed, based on the reservation raise distribution that can beinterpreted as a extensive-margin labor supply curve. The latter is smoothed as in Mui and Schoefer (2020).
Speci…cally, we use moving averages with a window length of 5. The reservation raise value of represents agross percentage change in the agent’s potential wage that would make the agent indi¤erent between working andnon-working, divided by 100. The bottom right panel is from a version of Model (HA-N) with x = 0:929 andx = 0:227 (Chang and Kim 2007).

A-17

Figure A3: Aggregate labor supply elasticities and arc elasticities with higher progressivity

1
0.9
0.8

CDF Raise

0.7
0.6
0.5
0.4
0.3
0.2
Baseline
Higher prog. w/ transf er f n.
Higher prog. w/ tax f n.

0.1
0
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Reservation Raise

7
Baseline
Higher prog. w/ transf er f n.
Higher prog. w/ tax f n.

6

Implied Arc Elasticity

5

4

3

2

1

0
-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Change in Aggregate Raise (

0.4

0.6

0.8

1

)

Note: We plot the reservation raise distribution or extensive-margin labor supply (top panel) and the corresponding arc elasticities (bottom panel) for two counterfactual exercises that increase progressivity in Section
5.3.

A-18

Figure A4: Impulse responses of macroeconomic aggregates with respect to positive TFP shocks

TFP% deviation from s.s.

2

Output
3

1.5

2

1

1

0.5

1.5
1
0.5
0

0
0

20

40

60

0
0

20

Quarter

4060

0Quarter

Total hours

2% deviation from s.s.

Consumption

1.5

40

60

Quarter

Avg labor productivity

220

Investment

10

Model (HA-T )Model (HA-N)
Model (HA-F)

8

1.5

6

1
1

4

0.5
0.5

0
-0.5

2
0

0
0

20

40

60

0

20

Quarter

4060

0Quarter

20

40

60

Quarter

Note: TFP denotes total factor productivity. The …gures display the IRFs of macroeconomic aggregates to a
positive 2 percent TFP shock with persistence z .

Figure A5: Impulse responses of equilibrium prices
Wage

Interest rate

0.05

% point deviation from s.s.

% deviation from s.s.

0.5
0
-0.5
-1
-1.5
-2

0
-0.05
-0.1

Model (HA-T)
Model (HA-N)
Model (HA-F)
Model (RA)

-0.15
-0.2

0

20

40

60

Quarter

0

20

40

60

Quarter

Note: The …gures display equilibrium market-clearing price responses, wt and rt , to a negative 2 percent TFP
shock with persistence z . In heterogeneous-agent models, wt captures the aggregate component of wages conditional on the worker selection in each period.

A-19

Table A3: Sensitivity checks
Baseline

Y
C= Y
I= Y
L= Y
H= Y
Y =H = Y

Cor(Y; C)
Cor(Y; I)
Cor(Y; L)
Cor(Y; H)
Cor(Y; Y =H)
Cor(H; Y =H)

LinearGini wage Gini wage
= 0.35= 0.37
Taxation a = 0

1.27
0.27
2.87
0.50
0.73
0.64

1.23
0.28
2.85
0.47
0.66
0.64

1.27
0.27
2.86
0.50
0.73
0.65

1.32
0.26
2.87
0.53
0.80
0.62

1.24
0.27
2.85
0.48
0.66
0.63

0.85
0.99
0.92
0.77
0.69
0.07

0.87
0.99
0.91
0.78
0.76
0.18

0.84
0.99
0.92
0.76
0.68
0.04

0.84
0.99
0.94
0.79
0.60
-0.02

0.85
0.99
0.92
0.79
0.76
0.21

Note: Each alternative model is recalibrated to match the same target statistics as in the baseline model.

I

Additional empirical results

We provide additional results presented in Section 6 for sensitivity checks.

A-20

Table A4: Probability of extensive margin adjustment, by wage quintile

5 years
Wage quintile
in base year
1st
2nd
3rd
4th
5th
Base years
Avg. no. obs
in base years
Total no. obs.
Avg. age

All
.146
.093
.075
.069
.072

Switches
Pos only Neg only
.097
.049
.060
.033
.045
.030
.037
.032
.040
.032

The length of tracking time T
10 years

All
.119
.080
.066
.061
.062

Switches
Pos only Neg only
.079
.039
.052
.029
.040
.027
.033
.028
.033
.029

15 years

All
.103
.072
.063
.055
.060

Switches
Pos only Neg only
.070
.033
.046
.026
.039
.024
.030
.025
.031
.029

1969–1993 (J = 25)
1,677

1969–1988 (J = 20)
1,189

1969–1983 (J = 15)834

41,92040.2

23,78341.0

12,51441.5

Note: The full-time employment threshold is set to 1,500 annual hours. Numbers in parentheses show the numberof base years. We use samples whose age is between 22 and 64 (inclusive) and who are heads and are not selfemployed. "All" refers to the baseline estimates when using both positive and negative switches, whereas "posonly" and "neg only" use only positive ones (i.e., Ei;t = 1 and Ei;t 1 = 0) and only negative ones (i.e., Ei;t = 0
and Ei;t 1 = 1), respectively.

A-21

Table A5: Full-time employment changes in recessions excluding samples with unemployment
spells, by wage quintile
Recession
1973–76 1980–83 1990–92 2000–02 2006–10
Wage quintile
in peak year
1st
2nd
3rd
4th
5th

-10.7
-5.5
-6.6
-4.2
-5.2

-4.7
-0.8
-3.4
-6.1
-5.2

-7.8
-4.9
-4.7
-3.5
-4.1

-5.2
-3.6
-1.6
-4.0
-1.8

-8.6
-8.8
-6.4
-7.2
-4.7

No. obs.

1,547

1,481

1,765

2,454

2,365

Note: The full-time employment threshold is set to 1,000 annual hours. The year ranges denote the peak andtrough years of each recession. Reported values are percentage changes in the full-time employment rate bywage quintiles (in the peak year of each recession) following the same set of individuals. Those who experienced
unemployment spells in either the peak year or the trough year are excluded. The results for the …rst recession
is omitted because the unemployment information is available only since the 1976 wave (or the year of 1975).

Table A6: Full-time employment changes in recessions, by wage quintile
Recession
1969–71 1973–76 1980–83 1990–92 2000–02 2006–10
Wage quintile
in peak year
1st
2nd
3rd
4th
5th

-7.3
-7.0
-5.8
-4.2
-1.0

-10.4
-10.5
-8.2
-4.7
-3.9

-11.1
-10.6
-6.3
-8.0
-5.2

-7.1
-8.3
-7.7
-7.2
-3.3

-8.3
-8.9
-6.7
-5.8
-2.1

-17.9
-16.3
-14.9
-11.1
-7.4

No. obs.

1,655

1,756

2,007

2,166

2,924

2,802

Note: The full-time employment threshold is set to 1,500 annual hours. The year ranges denote the peak andtrough years of each recession. Reported values are percentage changes in the full-time employment rate by wage
quintiles (in the peak year of each recession) following the same set of individuals.

A-22

Table A7: Full-time employment changes in recessions excluding samples with unemployment
spells, by wage quintile
Recession
1973–76 1980–83 1990–92 2000–02 2006–10
Wage quintile
in peak year
1st
2nd
3rd
4th
5th

-8.5
-4.7
-6.1
-4.4
-2.5

-3.0
-4.3
-5.2
-6.2
-6.1

-7.2
-5.7
-6.9
-3.9
-2.4

-7.0
-6.4
-4.8
-5.5
-2.7

-9.0
-11.1
-8.7
-8.7
-5.2

No. obs.

1,547

1,481

1,765

2,454

2,365

Note: The full-time employment threshold is set to 1,500 annual hours. The year ranges denote the peak andtrough years of each recession. Reported values are percentage changes in the full-time employment rate bywage quintiles (in the peak year of each recession) following the same set of individuals. Those who experienced
unemployment spells in either the peak year or the trough year are excluded. The results for the …rst recession
is omitted because the unemployment information is available only since the 1976 wave (or the year of 1975).

A-23

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