Unemployment Insurance in Macroeconomic Stabilization (NBER Working Paper 29505)

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NBER Working Paper No. 29505, Unemployment Insurance in Macroeconomic Stabilization, by Rohan Kekre, dated November 2021, Revised August 2022. The abstract states that the author studies unemployment insurance (UI) in general equilibrium with incomplete markets, search frictions, and nominal rigidities, and that an increase in generosity raises the aggregate demand for consumption if the unemployed have a higher marginal propensity to consume than the employed or if agents precautionary save. It states that in an analysis of the U.S. economy over 2008-2014, UI benefit extensions had a contemporaneous output multiplier around 1 and that the unemployment rate would have been as much as 0.4pp higher absent these extensions.

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NBER WORKING PAPER SERIES

UNEMPLOYMENT INSURANCE IN MACROECONOMIC STABILIZATION

Rohan Kekre

Working Paper 29505
http://www.nber.org/papers/w29505

NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
November 2021, Revised August 2022

I thank the editor Dirk Krueger and four anonymous referees. This paper is a substantial revision
of the first chapter of my PhD dissertation at Harvard University. I am indebted to Emmanuel
Farhi, Gita Gopinath, Raj Chetty, and Gabriel Chodorow-Reich for their guidance. I thank Peter
Ganong, Veronica Guerrieri, Erik Hurst, Greg Kaplan, and Joseph Vavra for numerous
discussions, and Robin Li for excellent research assistance. The views expressed herein are those
of the author and do not necessarily reflect the views of the National Bureau of Economic
Research.

NBER working papers are circulated for discussion and comment purposes. They have not been
peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies
official NBER publications.

© 2021 by Rohan Kekre. All rights reserved. Short sections of text, not to exceed two paragraphs,
may be quoted without explicit permission provided that full credit, including © notice, is given
to the source.
Unemployment Insurance in Macroeconomic Stabilization
Rohan Kekre
NBER Working Paper No. 29505
November 2021, Revised August 2022
JEL No. D52,E21,E62,J64,J65

ABSTRACT

I study unemployment insurance (UI) in general equilibrium with incomplete markets, search
frictions, and nominal rigidities. An increase in generosity raises the aggregate demand for
consumption if the unemployed have a higher marginal propensity to consume (MPC) than the
employed or if agents precautionary save in light of future income risk. This raises output and
employment unless monetary policy raises the nominal interest rate. In an analysis of the U.S.
economy over 2008-2014, UI beneﬁt extensions had a contemporaneous output multiplier around
1. The unemployment rate would have been as much as 0.4pp higher absent these extensions.

Rohan Kekre
Booth School of Business
University of Chicago
5807 S. Woodlawn Avenue
Chicago, IL 60637
and NBER
rohan.kekre@chicagobooth.edu

An appendix is available at http://www.nber.org/data-appendix/w29505
1 Introduction
Economists have long viewed unemployment insurance (UI) as an important automatic sta-
bilizer — but should it also serve as a discretionary tool in the stabilization of short-run
fluctuations? Since the 1950s, policymakers in the United States have treated UI generosity
as precisely such an instrument, extending benefits in recessions. This practice was expanded
in unprecedented and controversial fashion during the Great Recession, when benefit dura-
tions were raised almost four-fold at the depth of the downturn. While critics emphasized
the costly supply-side effects of more generous UI, supporters pointed to potential stimulus
benefits of transfers to the unemployed.1 More recently, a similar debate has unfolded with
respect to UI benefit increases during the Covid-19 pandemic.
The existing analysis of UI in the literature cannot speak fully to these debates because
it has largely ignored these potential interactions between UI and aggregate demand. Most
prior work has studied UI in partial equilibrium, while analyses in general equilibrium have
focused on environments in steady-state or in the real business cycle tradition. This paper
studies the output and employment effects of UI in a general equilibrium framework with
macroeconomic shocks and nominal rigidities.
I demonstrate that the effect of UI on aggregate demand makes it expansionary when
monetary policy is constrained. An increase in UI generosity raises aggregate demand if the
unemployed have a higher marginal propensity to consume (MPC) than the employed or if
agents precautionary save in light of future income risk. If monetary policy does not respond
to the demand stimulus by raising the nominal interest rate, this raises equilibrium output
and employment. Calibrating the model to the U.S. economy during the Great Recession
implies an important stabilization role of UI through these channels. With monetary policy
and unemployment matching the data over 2008-2014, the observed extensions in UI duration
had a contemporaneous output multiplier around 1. The unemployment rate would have
been as much as 0.4pp higher absent these extensions.
Several real and nominal frictions interact to set the stage for the paper’s results. First,
search and matching frictions in the tradition of Diamond [1981], Mortensen [1982], and
Pissarides [1984] give rise to unemployment and disincentive effects of UI. Second, mar-
ket incompleteness with respect to unemployment risk, building on Bewley [1983], Huggett
[1993], and Aiyagari [1994], generates a consumption insurance role for publicly provided UI.
Third, nominal rigidities render the level of production partially demand-determined.
In such an environment, I first analytically demonstrate how nominal rigidities and ac-
1
See Summers [2010], Congressional Budget Office [2012], and Blanchard et al. [2013] for examples of
commentary emphasizing the potential stimulus from UI.

1
commodative monetary policy reverse the conventional effects of UI on macroeconomic ag-
gregates. A budget-balanced increase in UI raises the aggregate demand for consumption
in the same period if the unemployed have a higher MPC than the employed. Expectations
of more generous future UI raise aggregate demand by reducing agents’ incentive to precau-
tionary save. Absent nominal rigidity, these impulses are undone by a rise in the equilibrium
real interest rate, and the effects of UI on wages and search intensity drive a reduction in
equilibrium output and employment. The same results obtain with nominal rigidity but
monetary policy which replicates the aforementioned path of real interest rates. In contrast,
with nominal rigidity but monetary policy which maintains a constant real interest rate,
the stimulus to aggregate demand drives an increase in equilibrium output and employment.
These results are only amplified when monetary policy implements a constant nominal rather
than real interest rate — as at the zero lower bound — in which case the supply-side effects of
UI raise inflation expectations, lower real interest rates, and thus further stimulate demand.
I then study these mechanisms in a richer model calibrated to the U.S. economy. Because
the model is consistent with evidence on consumption sensitivities to income by employment
status, the incidence and effects of unemployment, and the cross-sectional distribution of
consumption, we can have confidence in its predictions regarding MPC heterogeneity and
precautionary saving. Because it is consistent with evidence on the disincentive effects of UI
and degree of price rigidity, and because it can flexibly accommodate real wage rigidity as
in much of the literature, we can have confidence in its supply-side predictions.
I first quantitatively characterize the roles of nominal rigidity and accommodative mone-
tary policy in rendering UI extensions expansionary around the model’s steady-state. Absent
nominal rigidity, extending UI benefit duration by three months for one year has a contem-
poraneous output multiplier of -0.5: while the increase in generosity would raise short-run
consumption in partial equilibrium, the real interest rate rises so that output falls in general
equilibrium. With nominal rigidity but assuming that monetary policy follows a conven-
tional Taylor rule, the output multiplier becomes 0.6 as the nominal and thus real rate
rises in response, but not as much as under flexible prices. If monetary policy maintains a
fixed nominal interest rate during the period of benefit extensions and follows a Taylor rule
thereafter, the output multiplier nearly doubles to 1.1.
Quantitatively, MPC heterogeneity and diminished precautionary saving drive the stimu-
lus from UI when monetary policy is constrained. Absent cross-sectional heterogeneity in the
incidence of unemployment which contributes to MPC heterogeneity, the output multiplier
is substantially diminished. Conversely, if UI is expected to be extended over a greater hori-
zon, the multiplier substantially rises due to the feedback loop between lower precautionary
saving and higher job-finding rates. While a greater search or wage response to UI only

2
amplifies the stimulus at a constant nominal interest rate by raising inflation expectations,
the direct effects of UI on aggregate demand via heterogeneity in MPCs and dimininished
precautionary saving account for a majority of the baseline equilibrium effects.
I then study the role of UI benefit extensions during the Great Recession. I first isolate 13
distinct shocks to UI policy during this period. Twelve of these correspond to distinct pieces
of legislation introducing or reauthorizing benefits under the Emergency Unemployment
Compensation Act of 2008 (EUC08). One of these corresponds to the initiation of benefits
under the Extended Benefits (EB) program for the median U.S. state. Given these shocks
to UI policy, I then calibrate a sequence of discount factor shocks to match the dynamics of
U.S. unemployment from May 2008 through December 2014, and I calibrate the degree of
real wage rigidity to match the dynamics in consumer prices over this period. The goal is not
to provide a complete account of the Great Recession, but instead to quantify the effects of
UI extensions in a large downturn of this kind. With that said, untargeted macroeconomic
time series validate the fit of the model despite the parsimony of focusing on a single driving
force. In the simulation, as in the data, the nominal interest rate is at a binding zero lower
bound during almost all of the period when UI benefits are extended.
Comparing the model to a counterfactual economy without these UI extensions, I con-
clude that the unemployment rate would have been as much as 0.4pp higher absent the
extensions. I explore the mechanisms underlying this result in three ways, all of which are
consistent with the steady-state impulse responses described above. First, in an analysis of
each of the 13 shocks to UI policy in isolation, I find output multipliers ranging between
0.6-1.8, with the variation across these explained by agents’ endogenously evolving expecta-
tions regarding the horizon over which the zero lower bound will bind and variation in the
horizon of each UI shock. Second, the accommodative response of monetary policy is indeed
crucial: in a counterfactual in which there is no zero lower bound, UI extensions would have
triggered a monetary policy tightening and the magnitude of transfers themselves would
have been smaller (as the recession would have been less severe). Third, the stimulus from
UI is primarily driven by heterogeneity in MPCs and diminished precautionary saving at my
estimated degree of real wage rigidity. That being said, more flexible real wages would only
amplify the model-implied stimulus from UI at the zero lower bound.

Related literature The results of this paper are distinct from previous models of UI be-
cause I accommodate and focus on the combination of incomplete markets, nominal rigidities,
and constraints on monetary policy. In environments without nominal rigidity, Krusell et al.
[2010], Nakajima [2012a], and Mitman and Rabinovich [2015, 2020] find that increases in
UI are contractionary. I demonstrate that these results are reversed with nominal rigidity

3
and constraints on monetary policy. In New Keynesian models with a zero lower bound but
with complete asset markets, Albertini and Poirier [2015] and Christiano et al. [2016] find
that increases in UI can be expansionary because the induced rise in inflation expectations
lowers the real interest rate. I demonstrate that this channel is complemented by the direct
stimulus to aggregate demand through heterogeneity in MPCs and diminished precautionary
saving when markets are incomplete. Quantitatively, the latter channels are more important
than that through inflation expectations in my simulation of the Great Recession. While
MPC heterogeneity and precautionary saving have featured prominently in the policy debate
regarding UI extensions, to my knowledge this is the first paper to quantify their role in a
dynamic stochastic general equilibrium model.2
In doing so, my paper contributes to the rapidly growing literature on heterogeneous
agent New Keynesian (HANK) models.3 The most closely related strand of this literature
also accounts for endogenous unemployment, including Challe et al. [2017], den Haan et al.
[2018], Gornemann et al. [2021], Heathcote and Perri [2018], McKay and Reis [2021], and
Ravn and Sterk [2017, 2020]. Relative to all of these, my focus is on discretionary changes in
UI rather than its time-invariant level. For this reason, my analysis emphasizes heterogeneity
in MPCs alongside the effects of UI on precautionary savings, whereas the above papers
have largely emphasized the latter alone. Furthermore, constraints on monetary policy play
a crucial role in my analysis. Indeed, to my knowledge, my quantitative analysis of a HANK
model subject to a sequence of shocks gradually pushing it “far” from steady-state with an
endogenous and time-varying duration at the zero lower bound is novel to this literature.
By focusing on benefit extensions in a model calibrated to the Great Recession, my
results also provide a structural interpretation of empirical analyses of the UI extensions
in the U.S during this period. This includes the work of Boone et al. [2019], Chodorow-
Reich et al. [2019], Dieterle et al. [2020], Hagedorn et al. [2016], and Hagedorn et al. [2019].
Researchers in this literature have obtained conflicting results. My findings are consistent
with the upper end of stimulus estimated in this literature; the stimulus remains modest in
2
My results here parallel the (much larger) literature studying conventional government spending in New
Keynesian models. My analytical results in particular follow the conceptual approach of Woodford [2011],
who contrasts the government spending multiplier in a conventional small-scale New Keynesian model in
which monetary policy replicates the natural (flexible price) real interest rate; maintains a constant real
interest rate; or maintains a constant nominal interest rate. In these cases, the multiplier is below one, equal
to one, and greater than one, respectively. My quantitative results parallel those of Christiano et al. [2011],
who quantify the government spending multiplier in a medium-scale model of the Great Recession. In section
5, I also quantify the government spending multiplier in my model, demonstrating that it is consistent with
the empirical evidence and lending credibility to my analysis of the effects of UI.
3
In addition to the papers focused on unemployment risk which I discuss here, this literature has included
analyses of government spending (as in Auclert et al. [2018] and Hagedorn et al. [2019], among others) and
monetary policy (as in Auclert [2019], Kaplan et al. [2018], and Werning [2015], among others).

4
terms of employment because, despite a sizeable output multiplier, transfers to the long-term
unemployed are small versus output. The model can further explain differences in the precise
estimates of researchers as arising from differences in the horizon of UI shocks studied, since
these imply differential stimulus through precautionary saving.

Outline The rest of the paper is structured as follows. In section 2 I analytically char-
acterize the effects of marginal increases in UI generosity in a very simple environment. In
section 3 I introduce the full model and in section 4 I parameterize it to the U.S. economy.
In section 5 I study impulse responses to UI policy, and in section 6 I evaluate the effects of
benefit extensions during the Great Recession. Finally, in section 7 I conclude.

2 UI, nominal rigidity, and monetary policy
I first characterize the marginal effects of UI in a simple setting to frame the quantitative
results which follow. An increase in UI raises aggregate demand if the unemployed have a
higher MPC than the employed or if agents have a precautionary saving motive given future
income risk. With nominal rigidity and monetary policy which does not respond by raising
the nominal interest rate, this raises equilibrium output and employment.

2.1 A simple two-period environment
Consider the following environment with incomplete markets, search frictions, and nomi-
nal rigidity which captures the essence of the full model studied in the rest of the paper.
Appendix A outlines the environment more formally, and here I summarize it.
There are two periods, 0 and 1. In period 0 (the “short run”), firms producing an
intermediate good post vacancies, a measure one of potential workers search, matches occur
randomly, and then production takes place. Monopolistically competitive retailers purchase
intermediate goods and sell them as differentiated final goods subject to adjustment costs in
price-setting. For simplicity, all firm profits are paid to employed agents. Period 1 (the “long
run”) is an endowment economy in which agents receive an identical endowment. Agents
trade a real bond between periods 0 and 1, and have standard concave, separable preferences
over consumption and a convex, separable disutility from searching in period 0.
We can summarize agents’ micro-level optimization as follows. The representative agent’s
optimal search effort at the start of period 0 is s0 (θ0 , y0e , b0 , r0 ), a function of labor market
tightness θ0 , real income when employed y0e , real income when unemployed b0 (UI), and the
real interest rate r0 . An employed agent’s value function and optimal consumption in period

5
0 are vp(y§,70) and c}(yG,7o). For an unemployed agent these are vj (bo, 79) and c§(bo, ro),
respectively. We suppress the dependence of these policies on exogenous period 1 income.
A small set of conditions fully characterize the equilibrium. Goods market clearing in

period 0 requires

P(9o)s0(-)eo(-) + (1 — p(o)so(-))eo(-) = p(Ao)so(-) — kAos0(-), (1)

where the left-hand side is aggregate consumption and the right-hand side is aggregate
production, and I have suppressed the arguments of s9, c§, and c# for brevity.4 Aggregate
consumption depends on the consumption levels of employed and unemployed workers as well
as the employment rate p(0o)50, where p(9) is the job-finding probability per unit search.
Aggregate production depends on the employment rate less the measure of labor used in
recruiting rather than production, k@9s9, where k controls the magnitude of hiring costs in
this frictional labor market.® We normalize productivity to one.

Government budget balance implies that

P(9o)so(-)¥o + (1 — p(Go)S0(-))bo = p(Ao)S0(-) — kAoso(-), (2)

where the left-hand side is aggregate income and the right-hand side is aggregate production.

Optimal vacancy posting by intermediate good firms requires

Wo! (1 55 ) =u (3)

where jo denotes the price of final goods relative to intermediate goods, q(0) is the vacancy-
filling probability,° and wo is the real wage. The left-hand side is the marginal benefit of
hiring a worker, rising in the relative price of intermediate goods ig‘ and falling in the hiring
costs per worker k/q(@9). The right-hand side is the marginal cost of employing a worker,
simply the real wage.

Finally, Nash bargaining implies

+
w'(co(-))

4This condition also uses that the consumption and production of each individual retailer variety is the
same, an implication of the symmetry across retailers which we assume.

°Since tightness 09 is given by the ratio of vacancies to search, k@989 equals k times the measure of
vacancies, so that k formally corresponds to the measure of recruiters per vacancy as in Shimer [2010]. This
interpretation of course makes more sense with incumbent workers as in the quantitative model.

(oC) = 8) = 2 (wot ~ wn), (4)

° As is standard, this is related to the job-finding probability and tightness according to q(09) = ® Po)

a standard surplus-sharing condition in which @ denotes the bargaining power of workers.
Conditional on bp and ro, these constitute 4 equations in 4 unknowns {y§, 00, Ho, wo}-’
The real interest rate rp is in turn determined by the Fisher equation
Po
1+ro = (1+ %)—
which in turn depends on monetary policy {i9, P;} as well as the short-run price level Po,

which reflects the optimizing behavior of retailers subject to adjustment costs in price-setting.

2.2 Effects of a change in UI

I now characterize how a change in UI bo affects equilibrium output yo = p(00)s0 — kS0.
Regardless of the price-setting and monetary policy regime, (1) and (2) together imply

the following result.®

Lemma 1. As k + 0 while 3 remains fixed,

Oct Ocs Oc Ocy rr
dyy (1 P(0)s0) ($2 — $4) + (p(G0)s0e8 + (1 — p(G0)s0) SEE) $e

db — (ce — ce) — 1 2 ’
° 1— (cj — @) P(90)s0 Oy bo

where all partial derivatives refer to the micro-level policy functions defined in the main text.

The first term in the numerator summarizes the direct, contemporaneous effect of UI
on aggregate demand. It says that an increase in UI will raise aggregate demand if the
unemployed have a higher MPC than the employed. This is scaled by the economy’s unem-
ployment rate, which determines the volume of transfers.

The second term in the numerator summarizes the indirect effect of UI on aggregate
demand through the induced change in the real interest rate.

The denominator reflects amplification through the Keynesian cross. The more that
employed agents consume versus unemployed agents, the larger will be the feedback to
aggregate demand when employment rises. The higher is the MPC of employed agents, the
larger will be the feedback to aggregate demand when employment rises and thus employed
agents’ tax burden to finance UI falls.

The assumption that the hiring cost (k) is sufficiently small ensures that the direct
resource cost associated with changes in vacancy posting vanishes from Lemma 1. The

k
T_

assumption that Tg remains fixed in this limit (so workers’ bargaining share ¢ is sufficiently

7By Walras’ Law, these conditions imply that the bond market also clears.
8The proof of this result, like all other results in this section, is provided in appendix A.

7
close to one) ensures that workers’ surplus from employment remains positive even as hiring
costs vanish. Both assumptions are empirically plausible; in the quantitative model studied
later in this paper, k is indeed calibrated to be small and φ is calibrated close to one.
Armed with Lemma 1, we can interpret the main result of this section:

Proposition 1. Suppose k is small, φ is close to one, and agents face a sufficiently tight
borrowing constraint in period 0. Then:

• If prices are fully flexible (and UI b0 is close to optimal), dy0
db0
< 0 and dr
db0
0
> 0.

• If prices are sticky but monetary policy replicates the real interest rate r0 absent nominal
rigidity, dy0
db0
< 0 is identical to that under flexible prices.

• If prices are sticky and monetary policy maintains a constant r0 , then dy0
db0
> 0.

The assumption that agents face a sufficiently tight borrowing constraint in period 0 has
the natural implication that the unemployed will be constrained whereas the employed will
not. Thus, the unemployed have a higher MPC than the employed, so by Lemma 1 the
increase in UI will generate an initial stimulus to aggregate demand.
With fully flexible prices, equilibrium output nonetheless falls with more generous UI
because it raises equilibrium wages, depressing vacancy creation, and further reduces work-
ers’ incentive to search.9 Lemma 1 thus makes clear that the initial stimulus to aggregate
demand must be met by an increase in the real interest rate which is sufficiently strong
∂ce
that it lowers aggregate consumption (since ∂r00 is necessarily negative when the magnitude
of borrowing/lending is small and thus substitution effects dominate income effects). With
sticky prices but monetary policy replicating this real interest rate, the equilibrium effects
of UI are identical to the flexible price case.
Conversely, with nominal rigidity and monetary policy maintaining a constant real inter-
est rate, there is no crowd out of the stimulus to aggregate demand characterized in Lemma
1. As is standard in New Keynesian models, endogenous mark-ups are crucial to this mecha-
nism. Indeed, in firms’ optimal vacancy-posting condition (3), a lower gross mark-up earned
by retailers µ0 is consistent with firms’ increase in real marginal cost.

2.3 Additional insights
The key takeaway from the prior subsection is that if a marginal increase in UI stimulates
aggregate demand, it will raise output given nominal rigidity and a constant real interest rate.
9
The assumption that the initial level of UI is close to optimal is sufficient to sharply sign these general
equilibrium responses of wages and search. The optimal level of UI here refers to the value of b0 which
maximizes utilitarian social welfare given flexible prices in period 0.

8
This contrasts starkly with the equity-efficiency trade-off emphasized in partial equilibrium
analyses in public finance and general equilibrium analyses ignoring nominal rigidity.
Several extensions presented in appendix A provide additional insights on this result
which help to frame the quantitative analysis in the remainder of the paper.

Precautionary saving and dynamic amplification The above results demonstrate
that aggregate demand rises with UI in the same period if the unemployed have a higher
MPC than the employed. Extending the model to an infinite horizon with unemployment
risk in future periods, an expected future increase in UI in any period t ≥ 1 also stimulates
aggregate demand in prior periods by reducing agents’ incentive to precautionary save. This
again raises output given nominal rigidity and a constant real interest rate.
We can also ask how the magnitude of stimulus in period 0 varies with the period t in
which UI is raised. As t rises, there are two offsetting forces. On the one hand, the stimulus
in period 0 is dampened because binding borrowing constraints limit the fraction of agents
which respond to changes in future income. This is consistent with the dampening effects of
forward guidance in models with incomplete markets, as in McKay et al. [2016]. On the other
hand, the stimulus in period 0 is amplified because of the dynamic interplay between lower
income risk, higher aggregate demand, and thus a higher job-finding rate. This mechanism
has been emphasized by Acharya and Dogra [2020], Challe et al. [2017], McKay and Reis
[2016, 2021], and Ravn and Sterk [2017, 2020]. We can prove that this latter effect will
dominate the former effect if the initial generosity of UI is sufficiently low.

Trade in equities and investment These insights are robust to allowing for trade in
equities and a separation rate less than one (as in the quantitative analysis of the rest of
the paper). In previous work, Challe et al. [2017], den Haan et al. [2018], and Krueger
et al. [2016a,b] have emphasized that a decline in savings would reduce the demand for firm
equity and investment, counteracting the stimulus to output. The endogenous rise in the
real interest rate is the key mechanism by which investment — which encompasses hiring in
a frictional labor market — is crowded out. Conditional on the path of real interest rates,
an increase in UI stimulates output. In particular, the decline in desired savings must be
met by an increase in income rather than decline in investment to clear the asset market.

Constant i versus r The above results assume that the monetary authority maintains a
constant real interest rate. In practice, monetary policy more naturally features a constant
nominal interest rate — as when the nominal interest rate is at the zero lower bound, as in
my simulation of the Great Recession later in this paper. In such a context, the rise in firms’

9
marginal cost due to an increase in UI will further amplify its stimulus, as it raises expected
inflation, lowers the ex-ante real interest rate, and thus stimulates output. This mechanism
exists in representative agent economies, having been studied in a standard New Keynesian
model by Eggertsson [2010] and Werning [2012] and in the context of UI in particular by
Albertini and Poirier [2015] and Christiano et al. [2016]. The above analysis complements
this work by demonstrating that with incomplete markets, UI is stimulative even absent this
inflation expectations channel.

2.4 Summing up
Taken together, the effects of a marginal increase in UI depend crucially on the degree of
nominal rigidity and response of monetary policy. Absent nominal rigidity, an increase in UI
is contractionary. This remains with nominal rigidity but monetary policy which replicates
the real interest rate under flexible prices. With nominal rigidity and a constant real interest
rate, a marginal increase in UI is instead expansionary if the unemployed have a higher
MPC than the employed or if agents have a precautionary saving motive. With a constant
nominal interest rate, a marginal increase in UI is further expansionary by raising inflation
expectations. With these mechanisms in mind, the rest of the paper quantifies the effects of
UI extensions in a richer model of the U.S. economy during the Great Recession.

3 Model
The framework integrates workhorse quantitative models of incomplete markets, search and
matching, and nominal rigidities in a unified framework.

3.1 Environment and equilibrium
Timing Each period, firms post vacancies, workers search, and matches occur randomly;
production takes place and agents face a standard consumption-savings problem; and then
a fraction of employed workers exogenously separate.

Workers Each period workers differ in their employment status (i ∈ {e, u}), wealth in
bonds (ztb ) and shares in firm equity (ztf ), and a vector of other idiosyncratic states (ζti )
which evolve exogenously conditional on {e, u} transitions. These latter states will capture
dimensions of heterogeneity such as labor productivity when employed and the duration of
unemployment when unemployed. Conditional on an unemployed worker becoming employed
in period t, they transition according to Γt (ζte |ζtu ). Conditional on an employed worker

10
e
remaining employed or separating between t and t+1, they transition according to Γt (ζt+1 |ζte )
u
and Γt (ζt+1 |ζte ), respectively. Finally, among unemployed workers between t and t + 1 they
transition according to Γt (ζt+1 u
|ζtu ).
We study an environment without aggregate risk (in the transitional dynamics, aggregate
shocks are unanticipated). Hence, except for the initial period when any shock is realized,
in equilibrium we can collapse the state variables (ztb , ztf ) into a worker’s total real wealth

Πt + Qt f
zt ≡ ztb + zt (5)
Pt

where Πt is the dividend paid on firm equity, Qt is its price, and Pt is the price level. To
simplify notation, we thus exposit the model in terms of total wealth zt alone.
At the beginning of period t, incumbent workers’ value functions are

ṽte (zt ; ζte ) = vte (zt ; ζte ), (6)

and initially unemployed workers’ value functions are
Z
ṽtu (zt ; ζtu ) = max (pt (θt ; ζtu )st ) vte (zt ; ζte )Γt (ζte |ζtu )dζte
st ζte (7)
+(1 − pt (θt ; ζtu )st )vtu (zt ; ζtu ) − ψ(st ).

In the latter, unemployed agents’ disutility of search effort is given by ψ(st ) and their job-
finding probability per unit search is

pt (θt ; ζtu ) = m̄1−η
t mt (ζtu )θtη , (8)

where m̄t controls overall match efficiency, mt (ζtu ) controls relative match efficiency, and θt
is labor market tightness characterized further below.
In the middle of period t, the employed face

vte (zt ; ζte ) = max
e e
u(cet )
ct ,zt+1
" Z
e e e
+βt (ζte ) (1 − δt (ζte )) e
ṽt+1 e
(zt+1 ; ζt+1 )Γt (ζt+1 |ζte )dζt+1
e
ζt+1

(9)
Z #
+δt (ζte ) u
ṽt+1 e
(zt+1 u
; ζt+1 u
)Γt (ζt+1 |ζte )dζt+1
u
s.t.
u
ζt+1

Pt cet + (1 + it )−1 Pt+1 zt+1
e
≤ Yte (ζte ) + Pt zt ,
e
zt+1 ≥ zt,

11
and the unemployed face
Z
u
vtu (zt ; ζtu ) = max u(cut ) + βt (ζtu ) u
ṽt+1 u
(zt+1 u
; ζt+1 u
)Γt (ζt+1 |ζtu )dζt+1 s.t.
cet ,zt+1
u
u
ζt+1
(10)
Pt cut + (1 + it )−1 Pt+1 zt+1
u
≤ Ytu (ζtu ) + Pt zt ,
u
zt+1 ≥ zt,

where each agent’s discount factor is βt (ζti ), income inclusive of government taxes and trans-
fers is Yti (ζti ), and when employed, separation rate at the end of the period is δt (ζte ). In asset
markets, agents face the nominal interest rate it and borrowing constraint z t .

Producers A representative producer hires workers to produce a homogenous intermediate
good sold at price PtI . In period t, the producer starts with a stock p̃et of incumbent workers
and can hire more workers by posting νt vacancies which are filled with probability qt (θt ).
Managing a vacancy requires k incumbent workers with the average level of productivity, āt .
The distribution of workers across the idiosyncratic state space is relevant for the firm because
these states affect workers’ productivity, separation rate, and wages; let ϕ̃et (zt ; ζte ) denote
m (ζ u )s (z ;ζ u )
the distribution of state variables among incumbent workers, and t t s̄tt t t ϕ̃ut (zt ; ζtu ) the
distribution of state variables among the pool of unemployed workers matched with the firm.
The latter reflects the raw distribution of unemployed workers ϕ̃ut (·) weighted by workers’
relative match efficiency mt (ζtu ) and search effort st (zt ; ζtu ) relative to weighted average search
Z Z
s̄t ≡ mt (ζtu )st (zt ; ζtu )ϕ̃ut (zt ; ζtu )dzt dζtu . (11)
ζtu zt

Hence, the representative producer faces

JtI (p̃et , ϕ̃et ) = max ΠIt + (1 + it )−1 Jt+1
I
(p̃et+1 , ϕ̃et+1 ) (12)
ΠIt ,νt ,p̃et+1 ,ϕ̃et+1

subject to its flow of funds
Z Z
ΠIt = PtI at (ζte ) − Wt (ζte ) ×

ζe zt
" t #
Z u u
m̄ (ζ
t t )s (z
t t t ; ζ )
p̃et ϕ̃et (zt ; ζte ) + qt (θt )νt Γt (ζte |ζtu ) ϕ̃ut (zt ; ζtu )dζtu dzt dζte − PtI āt kνt ,
u
ζt s̄ t

the evolution of its stock of incumbents

12
Pix = [ [o — 6G) ietteen

mel Gi) sez Gr") 3

taldve f Pecele) (2a Gi)dGe | dexacs

and the evolution of ¢f consistent with Bayes’ Rule. Nominal wages W;(¢’) depends on each
worker’s type and are described further below. Since there is no aggregate risk, the producer

discounts future profits at the nominal interest rate 7;.

Retailers Retailers purchase the intermediate good and sell a differentiated variety to

consumers. When choosing its price P;;, retailer j faces Rotemberg [1982] adjustment costs

v( Pa Vp
AC = 5 ( Po, - i) i Piryindk

given its prior price P,_,; and aggregate nominal output fo Piryipdk. Hence, retailer j faces

Ji (Puaj) = max = Wf + (1+ i) Uh; (Py) 8-4.

v (P “(ft
t-1j 0

Uti = Ty;

_ (Pa\~
Uty = P, Ct,

where y;; is its production using x; units of the intermediate good and a linear technology;
c, is the sum of c!(%;¢!) across the idiosyncratic state space; and rT” is an ad-valorem tax
on inputs, rebated back to retailers via the lump-sum instrument T/*.1° I focus on the case
with symmetric initial prices and thus identical production and consumption of varieties in

equilibrium, so I drop the index 7 going forward.

Policy The government specifies a Taylor rule for 7;, its bond position z/, and its schedule

of real UI benefits b,(¢").1! It balances its budget using a tax on the employed T;.

Wages and income Finally, a union represents workers of each type ¢f and Nash bargains

the wage W/?(¢°) with producers given a bargaining share ¢ and a utilitarian welfare function

10T further assume that the price adjustment costs are paid to the government and rebated back via T/*,
so that the only effect of these costs is on retailers’ price-setting decisions, not on resources used.
1 Following Woodford [2003], I model the “cashless limit” where money serves only as the unit of account.

13
over workers of that type having different levels of wealth.12 In the model’s steady-state, the
W nb (ζ e )
real wage is given by the Nash bargained wage: w(ζte ) = P t . Following Hall [2005] and
Blanchard and Gali [2010], in response to aggregate shocks we then allow the equilibrium
real wage to be a weighted average of the re-bargained wage and the steady-state real wage

Wt (ζte ) W nb (ζ e )
= ιw(ζte ) + (1 − ι) t t , (14)
Pt Pt

allowing us to accommodate real wage rigidity within the bilaterally efficient bargaining set.
By assuming wages are (at least partially) Nash bargained, but are bargained at the level
of worker types rather than individual workers, we retain the desirable axiomatic properties
of Nash bargaining while avoiding the substantial computational difficulties of solving for
the transitional dynamics of a wage schedule over the entire wealth distribution over time.13
I verify that equilibrium wages remain bilaterally efficient for all individual workers in all
calibrations and all transitional dynamics studied in the paper.
Given these wages and the above policy instruments, agents’ incomes are

Yte (ζte ) = Wt (ζte ) − Tt , (15)
Ytu (ζtu ) = Pt bt (ζtu ). (16)

Market clearing As already defined, at the beginning of period t the employment rate is
p̃et and the distributions of incumbent and unemployed workers are ϕ̃et (zt ; ζte ) and ϕ̃ut (zt ; ζtu ),
respectively. Let pet , ϕet (zt ; ζte ), and ϕut (zt ; ζtu ) denote the analogs in the middle of period t.
Then asset market clearing (the sum of bond and equity market clearing) is
Z Z
e
pet zt+1 (zt ; ζte )ϕet (zt ; ζte )dzt dζte
ζte zt
Z Z
e u g Πt+1 + Qt+1
+ (1 − pt ) zt+1 (zt ; ζtu )ϕut (zt ; ζtu )dzt dζtu + zt+1 = , (17)
ζtu zt Pt+1

where aggregate dividends and the price of equity satisfy
Z 1
Πt = ΠIt + ΠR
tj dj, (18)
0
12
We assume the union represents newly matched workers and incumbents then receive the same wage.
13
A similar approach has been used by Costain and Reiter [2005] and Nakajima [2007]. Using a Krusell
and Smith [1998] solution approach, Krusell et al. [2010] and Nakajima [2012b] solve for wages as a function
of individual worker wealth. Using my sequence space solution approach — which affords other advantages,
such as the ability to study a wide variety of aggregate shocks to UI policy and fundamentals — this is
extremely computationally demanding. I leave it to future work to make progress on this important task.

14
Qt = (1 + it )−1 (Πt+1 + Qt+1 ) . (19)

In the labor market, labor market tightness

νt
θt = (20)
(1 − p̃et )s̄t

determines the vacancy-filling probability and job-finding probability per unit search. Hence,
the aggregate number of matches corresponding to the job-finding probabilities in (8) is

m̄1−η
t ((1 − p̃et )s̄t )1−η νtη ,

and the vacancy-filling probability referenced in the producer problem (12) is given by

m̄1−η ((1 − p̃et )s̄t )1−η νtη
qt (θt ) ≡ t
= m̄1−η
t θtη−1 .
νt

Absent heterogeneity in match efficiency (mt (ζtu ) = 1 for all ζtu ), this specification of the
labor market collapses to the Cobb-Douglas case of that in Pissarides [2000].14
Goods market clearing is
Z Z
pet cet (zt ; ζte )ϕet (zt ; ζte )dzt dζte +
ζte zt
Z Z Z Z "
(1 − pet ) cut (zt ; ζtu )ϕut (zt ; ζtu )dzt dζtu = at (ζte ) p̃et ϕ̃et (zt ; ζte )+
ζtu zt ζte zt
#
mt (ζtu )st (zt ; ζtu ) u
Z
q(θt )νt Γt (ζte |ζtu ) ϕ̃t (zt ; ζtu )dζtu dzt dζte − āt kνt . (21)
ζtu s̄t

Finally, budget balance for the government is characterized by
Z Z
g
pet Tt + Pt ztg = Pt (1 − pet ) bt (ζtu )ϕut (zt ; ζtu )dzt dζtu + (1 + it )−1 Pt+1 zt+1 . (22)
ζtu zt
14
Shimer [2004] challenges an equilibrium implication of this matching function that search effort rises
in tightness. Mukoyama et al. [2018] propose a generalized job-finding function where search effort can
fall in tightness, consistent with evidence from the Great Recession. I conjecture that the results in this
paper would be little changed under such a job-finding function, since I find that the aggregate consumption
responses dominate search responses in determining the macroeconomic effects of UI in the presence of
nominal rigidity and accommodative monetary policy. I maintain the Pissarides [2000] approach because it
can easily accommodate an aggregate matching function with heterogeneous match efficiencies, which I need
to quantitatively match negative duration dependence in observed job-finding rates later in this paper.

15
g
Equilibrium Conditional on policy {bt (ζtu ), zt+1 , Tt } and the monetary policy rule, as well
as exogenous aggregates {at (ζte ), δt (ζte ), mt (ζtu ), βt (ζti ), āt , m̄t , z t } and initial prices {P−1 }, the
definition of equilibrium is standard. I characterize the equilibrium in appendix B.

3.2 Functional forms
I now specify the functional forms assumed when parameterizing and quantifying the model.

3.2.1 Heterogeneity beyond employment status and wealth

Beyond employment status and wealth, I assume that agents differ in

ζte ≡ (aPt , aTt , ν β ),
ζtu ≡ (aPt , aTt , ν β , dt , 1U I ),

where {aPt , aTt } are components of productivity, ν β indexes discount factor heterogeneity, dt
is the duration of unemployment, and 1U I is an indicator for UI eligibility and take-up. The
e
transition probabilities Γt (ζte |ζtu ), Γt (ζt+1 u
|ζte ), Γt (ζt+1 u
|ζte ), and Γt (ζt+1 |ζtu ) are induced by the
transitions for each state variable described below.

Shocks to labor productivity To capture income volatility conditional on employment,
I adopt a standard persistent-transitory process for labor productivity at such that

at (ζte ) ≡ aPt aTt . (23)

When a worker is employed in period t, these components evolve as

log aPt = log āt + ηtP ,
ηtP = ρP ηt−1
P
+ εPt , εPt ∼ N (0, (σ P )2 ), (24)

T 1 T 2 T 2
log at ∼ N − (σ ) , (σ ) ,
2

where āt controls the economy-wide average productivity and follows an exogenous process.
When a worker is unemployed in period t, the transitory component aTt continues to evolve
as above, and the persistent component ηtP remains fixed.15
15
Given the specification of UI benefits described later in this section, the latter assumption ensures that
an unemployed worker’s UI benefits remain constant until they expire.

16
Heterogeneous discount factors As argued by Carroll et al. [2017] and Krueger et al.
[2016a], discount factor heterogeneity can further help in matching the empirical distribution
of wealth. I assume that v® indexes this heterogeneity, with a fraction one-third of worker-
consumers having v? = —A*, one-third having v® = 0, and one-third having v? = A? for

some positive dispersion parameter A’. An agent’s period t discount factor is then
Bi(Gt) = Be + v" (25)
where (3; is the economy-wide average discount factor and follows an exogenous process.

Heterogeneous separation rates Separation rate heterogeneity allows the model to ac-
commodate the uneven risk of unemployment across the population, with its attendant

consequences for the wealth distribution. In particular, an agent’s separation rate is

b(GF) = d¢ (1 + (€2/9) (log af’ — log a) + (€5/8)(8:(CF) — B1)) 5 (26)

where 6; controls the economy-wide average separation rate and follows an exogenous process.
In steady-state, 6, = 6 and thus the parameters ¢> and a control the elasticities of an agent’s

separation rate with respect to her productivity and discount factor, respectively.

Incomplete eligibility and take-up In practice, not all newly unemployed workers are
eligible for benefits, and many who are still do not take it up (Blank and Card [1991]).
To capture this pattern in the data, I assume that only with probability ¢ does a newly

unemployed worker begin receiving benefits, a state denoted with indicator 1y,.'®

Structural duration dependence Finally, structural duration dependence allows the
model to better reflect empirical hazard rates out of unemployment, consistent with the
evidence of Ghayad [2013], Kroft et al. [2013], Eriksson and Rooth [2014] and others. I
assume that the relative match efficiency of an unemployed agent with duration d; relevant

for her job-finding probability per unit effort (8) is

mu (CY) = exp(d;A) for d: < 8, (27)

exp(7A) for d; > 8.

16Using the 2001 and 2008 SIPP panels studied in appendix C, I find that household income prior to
unemployment has an association with an indicator for UI receipt which is statistically indistinguishable
from zero once we exclude the bottom 25% of income observations. Hence, for parsimony, I assume that all
workers have an identical probability of receiving UI conditional on job loss.

17
Here, λ controls duration dependence in match efficiencies. I assume furthermore that match
efficiencies are flat after an unemployed agent has been unemployed for 8 months or more.17,18

3.2.2 Functional forms for primitives and policy

I specify the structure of UI and monetary policy to be consistent with U.S. practice, and
choose standard preferences over consumption and search.

Household income during unemployment UI benefits have finite duration and scale
with earnings over a base period prior to job loss (up to a cap).19 Moments from the Survey
of Income and Program Participation (SIPP) in appendix C reveal the importance of non-
UI income through unemployment. I thus assume transfers to an unemployed agent with
productivity {aPt , aTt }, discount factor ν β , duration dt , and UI eligibility/take-up indicator
1U I
(
min{rrt (1 − ω0 )wt (aPt , 1, ν β ), uit } + ω1 wt (aPt , aTt , ν β ) if 1U I = 1, dt < d¯t ,
bt (ζtu ) = (28)
ω2 wt (aPt , aTt , ν β ) if 1U I = 0 or dt ≥ d¯t ,

where the job loser is assumed to have earned a fraction 1 − ω0 of household income prior to
job loss; UI policy parameters in period t are replacement rate rrt , maximum benefit level
uit , and duration d¯t ; and {ω1 , ω2 } control the level of non-UI income through the spell.20
Higher ω1 than ω0 parsimoniously captures the household income implications of an “added
worker effect” after job loss (e.g., Lundberg [1985], Stephens [2002]). Higher ω2 than ω1
captures the crowd-out of non-UI income during UI receipt, another form of moral hazard
associated with UI.

Taylor rule Monetary policy specifies the nominal interest rate according to a standard
Taylor rule with a zero lower bound

it = max r + φΠ ΠPt + φy (yt /y − 1), 0 ,

(29)
17
This is computationally convenient because it limits the state space. But it is also consistent with the
flatter empirical hazards out of unemployment after 8 months reported in Figure 7(A) of Kroft et al. [2016].
18
An alternative literature has argued that dynamic selection among heterogeneous job-seekers better
explains observed duration dependence in job-finding rates (Ahn and Hamilton [2020], Alvarez et al. [2015]).
I refer the reader to the previous drafts of this paper on my website in which I demonstrate that the
macroeconomic effects of changes in UI are robust to this alternative environment.
19
UI benefits are also subject to a floor (a minimum weekly benefit amount), but given the discretized
income process in my calibration this would not bind at observed values, so I ignore it for simplicity.
20
As documented in appendix C, non-UI income is mostly the earnings of other household members.
Modeling this as a transfer is a parsimonious way of accounting for it without extending the framework to
model dual-income households. The results are robust to modeling this income as an endowment instead.

18
where r denotes the steady-state real interest rate, ΠPt ≡ PPt−1
t
−1 denotes aggregate inflation,
yt denotes aggregate output, and y denotes its value in steady-state.

Preferences Finally, among workers, I assume CRRA flow utility from consumption

c1−σ − 1
u(c) =
1−σ

as well as isoelastic disutility of searching when unemployed

ψ(s) = sξ+1 .

4 Stationary recursive competitive equilibrium
In this section I calibrate the model to match patterns in wealth, income, and employment
in U.S. data. The consistency of the stationary recursive competitive equilibrium (RCE)
with available evidence on consumption sensitivities to income by employment status, the
incidence and effects of unemployment, and disincentive effects of UI can give us confidence
in its demand- and supply-side predictions in response to redistribution.

4.1 Calibrating the stationary RCE
I first set a subset of parameters to be consistent with common benchmarks and the broader
literature. Relative risk aversion is σ = 1, allowing us to normalize ā = 1 since the en-
vironment is consistent with balanced growth. The elasticity of job-finding with respect
to tightness is η = 0.7 consistent with the range reported in Petrongolo and Pissarides
[2001], and the average separation rate is δ̄ = 0.034 as calculated by Shimer [2010] at a
monthly frequency. The persistent-transitory process for worker productivity is ρP = 0.997,
σ P = 0.057, and σ T = 0.228, based on Krueger et al. [2016a]’s estimates using post-tax,
per-capita household earnings data conditional on an employed household head, adjusted
to a monthly frequency.21 The fraction of household income earned by the head prior to
job loss is 1 − ω0 = 0.67, consistent with evidence from the SIPP in appendix C. I assume
τ R = −7.6% so that, anticipating the calibration targets described below, government debt
will be roughly 60% of annual output, consistent with total public debt relative to GDP in
21
Using annual data from the Panel Study of Income Dynamics, these authors estimate a persistent-
transitory process of ρP P 2 T 2
a = 0.9695, (σa ) = 0.0384, (σa ) = 0.0522. They then translate these estimates to a
quarterly frequency; using their same approach, I adjust these to a monthly frequency using ρP = (ρPa)
1/12
,
(σ P )2
(σ P )2 = (1 − (ρP )2 ) 1−(ρ
a
P )2 , σ
T
= σaT .
a

19
Moment Target Achieved Parameter Value

Real rate, wealth, and average MPC

Real interest rate (ann.) 2% 2.0% 29/4 -8.3
Mean wealth / monthly HH income 66.0 65.5 8 0.99335
Mean (U-E) wealth / monthly HH income _ -47.5 -47.6 & -4.55
Fraction HH with negative wealth 0.08 0.05 z/a -0.15
Mean quarterly MPC to $500 rebate* 0.21 0.21 Ae 0.0045
Income during unemployment
Share unemployed receiving UI 0.39 0.42 ¢ 0.5
Max UI / mean wage UI recipients 0.6 0.60 ut 0.4
Mean HH income w. UI / pre job loss 0.76 0.75 Wy 0.37
Mean HH income w.o UI / pre job loss 0.55 0.56 We 0.50
Incidence of unemployment
Unemployment rate 5% 5.0% @ 0.957
Fraction w/ duration > 6 mos 0.17 0.17 \ -0.14
EU probability on log wage -0.012 -0.012 é -0.011
Search and the labor market
Duration elasticity to benefit duration 0.1 0.09 E 15
Vacancies per unemployed worker 0.634 0.634 m 0.19
Fraction of monthly wage to hire worker 0.108 0.108 k/a 0.044

Table 1: targeted moments and calibration results

Note: sources for targets are provided in the main text. The table provides the main parameter used to
target each moment. Figure 1 describes the identification of key parameters in particular.
* Among households earning < $75k ann. income (0.92 times average household income in model).

the years immediately preceding the Great Recession.?? In most states, regular UI benefits
are paid for 6 months, and the weekly benefit amount (among those not hitting the cap)
replaces roughly half of wages. I thus set rr = 50% for d = 6 months.

I calibrate the remaining parameters to match salient patterns in wealth, income, and
unemployment in U.S. data prior to the Great Recession.?? The targeted moments and
simulated values are summarized in Table 1. The table indicates the value of the model
parameter which is primarily varied in order to target the given moment.

The first set of parameters target key features of wealth and income. Combining the data
for employed and unemployed households from the 2004 Survey of Consumer Finances (SCF)
summarized in appendix C, and defining wealth as total net worth, I find that mean wealth

equaled 66.0 times average monthly household income, mean wealth of households with an

?2Government debt to output equals aggregate wealth to output, less the discounted present value of firm
profits to output. Absent hiring costs (which will indeed be calibrated to be small, consistent with available
evidence), firm profits are retailer profits. Relative to output, these are 1— ~! = 1— t i = xz. Given ¢ = 6,
the targeted aggregate wealth to output, and targeted real interest rate, I thus set tT accordingly.

?3The computational algorithm used to solve the stationary RCE is discussed in appendix F.

20
unemployed head was 47.5 months of average household income below that of households
with an employed head, and 8% of households had negative wealth.24 I use the average
discount factor β̄ to target mean wealth, the sensitivity of separation rates δβ to target
the mean difference in wealth by employment status, and the borrowing constraint z to
target the fraction with negative wealth. To match a 21% mean quarterly MPC out of an
unanticipated $500 rebate among households earning $75,000 or less annually (the midpoint
of the 12%-30% range reported by Parker et al. [2013]), I use the dispersion in discount
factors ∆β .25 To clear the asset market at a targeted 2% annualized real interest rate, I
use the government’s bond position z g . To match the 39% of unemployed receiving benefits
computed by Chodorow-Reich and Karabarbounis [2016] over 1961Q1-2008Q2, I use the
probability of UI receipt conditional on job loss ζ. To match the 60% ratio between the
maximum statutory level of UI and the average wage (weighted across states),26 I use the
maximum level of benefits ui. Finally, to match the dynamics of household income through
unemployment using the SIPP in appendix C, I use ω1 to target 24% lower income after job
loss and ω2 to target 45% lower income after UI exhaustion.
The second set of parameters target key features of unemployment and job search. Based
on monthly data over 1995-2007 from the Bureau of Labor Statistics, I calculate a 5.0%
average unemployment rate with 17% of the unemployed having duration greater than 26
weeks. I use the bargaining power of workers φ to target the overall unemployment rate and
the duration dependence of match efficiency λ to target the fraction 0.17 of unemployed with
duration greater than 6 months. Since much of my analysis will focus on policy affecting UI
duration, I use the elasticity of workers’ disutility from search ξ to target a micro-elasticity
of unemployment duration to potential duration of benefits of 0.1, within the range surveyed
by Schmieder and von Wachter [2016].27 I use the sensitivity of separation rates δa to match
the negative relationship between EU probabilities and wages which I estimate using the
Current Population Survey (CPS) over 2004-2007 in appendix C. Finally, I use the level of
24
I study an alternative calibration matching features of the liquid wealth distribution, rather than total
wealth distribution, in appendix D. Since I find even larger effects of UI extensions on aggregate demand in
that environment, I present the calibration to total wealth in the main text to be conservative.
25
The tax rebates under study in Parker et al. [2013] phased out rapidly above $75,000 annual income
among single earner households, and $150,000 annual income among dual earner households. I use the former
threshold to compute the comparable moment in my model so that, if anything, my calibration of MPCs is
conservative. I focus on a $500 rebate as in Kaplan and Violante [2014]. I convert both the $500 rebate and
$75,000 annual income cutoff into model scale using average monthly household income from the 2004 SCF,
and then compute MPCs by simulating agents over three months including and after receipt.
26
In the data, I first compute the ratio of the maximum statutory weekly benefit amount to average
weekly wage in each state. I then average across states, weighting by UI claimants. All data comes from the
U.S. Department of Labor for the first quarter of 2008.
27
As in the data, this micro-elasticity in the model is computed using the change in expected duration for
a single unemployed agent able to receive longer UI benefits in an unchanged macroeconomic environment.

21
Mean quarterly MPC to $500 rebate∗ Mean (U-E) wealth / monthly HH income

-42
target=0.21
0.21

-44
0.2

-46
0.19
target=-47.5
0.18 -48

0.17
0.0033 0.0037 0.0041 0.0045 0.0049 -4.75 -4.55 -4.35 -4.15 -3.95
∆β ǫδβ
EU probability on log wage Fraction unemployed w/ duration > 6 mos
-0.008
0.18
-0.009

target=0.17
-0.01 0.17

-0.011 0.16
target=-0.012
-0.012
0.15
-0.013

-0.012 -0.011 -0.01 -0.009 -0.008 -0.16 -0.14 -0.12 -0.1 -0.08
ǫδa λ

Figure 1: identification of selected parameters in stationary RCE
Note: each panel characterizes a given moment in the stationary RCE as I vary a single parameter, keeping
all other parameters unchanged from their values in Table 1. The only exception to this is in the first
panel where I vary ∆β : in that case, I also vary β̄ such that β̄ + ∆β = 0.99785, ensuring a bounded wealth
distribution. In each panel, the horizontal line denotes the targeted value of the moment, and the shaded
marker denotes the calibrated value of the parameter and resulting moment in the stationary RCE.
∗
Among households earning ≤ $75k ann. income (0.92 times average household income in model).

match efficiency m̄ to target 0.634 vacancies per unemployed worker reported by Hagedorn
and Manovskii [2008], and the cost k to target the 10.8% of a recruiter’s monthly wage used
in hiring one worker reported by Silva and Toledo [2009].
Figure 1 illustrates the identification of several parameters deserving further comment.
First, the convexity of MPCs by cash on hand implies that the average MPC is rising in
discount factor heterogeneity. Consistent with Carroll et al. [2017], such heterogeneity is
needed to generate an average MPC high enough to match the data provided that the
average level of wealth is also consistent with the data. Second, separation rates must

22
fall with discount factors (δβ < 0) to match the mean difference in wealth between the
unemployed and employed. Indeed, separation rates which fall only with productivity (δa <
0) to match the observed relationship between EU probabilities and wages cannot rationalize
this large difference in wealth by employment status.28 Notably, this is consistent with
evidence from the SIPP in appendix C that EU probabilities are negatively related to wealth
even conditional on income.29 Finally, match efficiencies must fall with duration (λ < 0) to
match the observed fraction of unemployed agents who are long-term unemployed. Absent
this decline in efficiency, job-finding rates rise with duration as agents search harder through
unemployment, rendering the fraction of long-term unemployed counterfactually low.

4.2 Consumption, unemployment risk, and wealth: data vs. model
The model’s consistency with untargeted moments on consumption, unemployment risk, and
wealth can give us further confidence in its predictions for the effects of redistribution. I
assemble a broad set of related moments from a variety of data sources in Table 2. More
detail on the estimates, including standard errors and sample descriptions, is in appendix C.
The model is first consistent with estimates of consumption sensitivities to income by
employment status. Using the 2010 Survey of Household Income and Wealth (SHIW), unique
in providing household heads’ employment status alongside (self-reported) MPCs, I find that
the annual MPC out of unexpected, transitory income shocks is 25% higher for unemployed
versus employed households.30 The results of Ganong and Noel [2019] using data from
JPMorgan Chase indicate that the consumption sensitivity to income is especially high
among the long-term unemployed: at the month of UI exhaustion, spending falls by 20% of
the reduction in household income. The calibrated model implies corresponding moments
only a bit larger than the empirical counterparts.
The model is also consistent with untargeted moments regarding the incidence and effects
of unemployment. Exploiting the panel structure of the SIPP, I estimate that one additional
month of income in net worth is associated with a 0.02pp decrease in the probability that a
household head becomes unemployed one year in the future. In the model, the same increase
in net worth is associated with a 0.01pp decline in the probability of unemployment one year
in the future. The consequences of unemployment for consumption have been studied by a
28
While the sensitivity of separation rates to discount factors is needed to match these features of the
wealth distribution, note that even without such sensitivity, Table 5 demonstrates that MPCs rise with
unemployment duration.
29
I do not use the EU-wealth relationship from the SIPP to calibrate the model because it is imprecisely
estimated. But I show in the next subsection that the model is consistent with this untargeted moment.
30
While the horizon of spending was not explicitly asked, as Auclert [2019] notes, the consistency with
the annual MPCs elicited in a later 2012 survey suggests that respondents had an annual horizon in mind.

23
Moment Estimate Source Model
Consumption sensitivities to income
Annual MPC unemp. - emp. 0.25 2010 SHIW 0.27
∆ spend. / ∆ inc. at UI exhaustion 0.20 Ganong and Noel [2019] 0.27
Incidence and consumption effects of unemployment
EU prob. on wealth / mo. HH income -0.0002 2004 Panel SIPP -0.0001
During UI receipt / pre job loss 0.91 Ganong and Noel [2019] 0.85
After UI exhaustion / pre job loss 0.80 Ganong and Noel [2019] 0.77
Aggregate consumption shares by wealth
Quintile 5 (highest) 37.2% Krueger et al. [2016a] 33.5%
Quintile 4 22.4% Krueger et al. [2016a] 26.6%
Quintile 3 16.8% Krueger et al. [2016a] 21.1%
Quintile 2 12.4% Krueger et al. [2016a] 11.2%
Quintile 1 (lowest) 11.3% Krueger et al. [2016a] 7.7%
Table 2: untargeted moments on MPCs, unemployment risk, and consumption
Note: further details on the estimates are provided in appendix C. The model moments are computed in the
same way as the estimates except for the annual MPC, which is self-reported in the data but computed in
the model for each agent using the change in average consumption over the year following an unanticipated
rebate equal to average economy-wide monthly income.

literature beginning with Gruber [1997] using progressively richer data. Using the JPMorgan
Chase panel, Ganong and Noel [2019] estimate that the spending of UI exhaustees falls by
9% during UI receipt and a further 11% after exhaustion. As in their partial equilibrium
analysis, agents in the model reduce consumption by more in anticipation of exhaustion
relative to what we observe in the data. However, the model more closely matches observed
consumption after exhaustion, more important for the experiments which follow because of
my primary focus on changes to UI benefit duration affecting the long-term unemployed.
The model is finally consistent with the relative importance of consumption among house-
holds by wealth. Krueger et al. [2016a] sort households in the 2007 PSID into quintiles based
on their total level of net worth, and then compute the share of aggregate consumption by
households in each group. If anything, the model slightly undershoots the share of aggregate
consumption accounted for by households in the bottom two wealth quintiles. Since (as I
show in the next subsection) MPCs are strongly associated with wealth, this implies that the
model does not overstate the importance of high-MPC households in aggregate consumption.

4.3 Long-term unemployed as a “tag” for high MPCs
The structural approach taken in this paper illustrates why, to borrow a phrase from Akerlof
[1978], the long-term unemployed are an especially good “tag” for high MPCs: low wealth,
and low temporary income relative to permanent income.

24
Consumption Invariant marginal distributions
0.3
1.2
0.25

1 0.2

0.15
0.8

e 0.1
0.6 u(d=0)
u(d=3) 0.05
u(d=6)
0.4 0
0 2 4 6 8 10 0 2 4 6 8 10
Assets Assets

Figure 2: consumption policy functions and marginal wealth distributions in stationary RCE
Note: e denotes employed, u denotes unemployed, and d denotes the number of months an agent has been
unemployed in the current spell (not including the current month). The policy functions in the first panel
average over all other state variables using the marginal distributions conditional on employment status.
The dashed vertical line in both panels marks the borrowing constraint.

This is evident from agents’ consumption policy functions and the marginal distributions
over wealth, depicted in Figure 2 for employed agents and unemployed agents in their first
month of unemployment (d = 0), in their fourth month (d = 3), and just after the expiration
of benefits (d = 6), averaging over all other state variables. The marginal distributions are
consistent with the unemployed being drawn disproportionately from the pool of low wealth
agents and then further decumulating assets through the spell, a temporary shock. The pol-
icy functions are consistent with the temporary income losses associated with unemployment
leading agents, even conditional on wealth, to have lower consumption and higher sensitiv-
ities to cash on hand. Taken together, Table 3 illustrates that the long-term unemployed
have a quarterly MPC out of an unexpected rebate of $500 which is over three times that of
the employed. In fact, the long-term unemployed have a quarterly MPC which is more than
one and a half times that of the bottom quintile of households by wealth.31
31
The MPCs by wealth are further useful in validating the model given the consistency between the
model-implied pattern of MPCs declining in wealth and available evidence. For instance, Broda and Parker
[2014] find that the roughly half of households reporting less than two months of income available in liquid
assets have a quarterly MPC at least double that of households reporting sufficient liquidity. The distinction
between liquid and illiquid wealth is of course one that my baseline model does not capture, though I provide
an alternative calibration to the liquid wealth distribution in appendix D.

25
Group Mean quarterly MPC
Overall 0.17
By wealth
Quintile 5 (highest) 0.01
Quintile 4 0.05
Quintile 3 0.17
Quintile 2 0.24
Quintile 1 (lowest) 0.39
By employment status
Employed 0.16
Unemployed 0.42
Short-term (d ∈ {0, 1, 2}) 0.36
Medium-term (d ∈ {3, 4, 5}) 0.47
Long-term (d ≥ 6) 0.59
Table 3: model-generated quarterly MPC to $500 rebate
Note: the $500 rebate is first translated into model scale using average monthly household income of $6,761
from the 2004 SCF. The MPC for each agent is then computed as the change in average consumption over
the three months after an unanticipated receipt of this rebate in a given month.

5 Impulse responses to UI shocks
I now study UI policy starting from the stationary RCE. Consistent with the earlier analytical
results, UI extensions are contractionary absent nominal rigidity but are expansionary given
nominal rigidity and monetary policy which does not raise the interest rate. The latter
stimulus is more pronounced when MPCs rise more sharply by duration of unemployment
or the extensions last for a longer horizon.

5.1 Roles of nominal rigidity and the monetary policy response
I start with an unanticipated extension of UI duration by three months and lasting one year.
This subsection clarifies the key roles of nominal rigidity and the monetary policy response
in shaping the resulting macroeconomic effects.
I assume for now ι = 1, so that real wages do not change in response to the shock,
and all other macroeconomic parameters are fixed, such as z g , so that changes in UI are
fully financed by contemporaneous changes in taxes. I explore the sensitivity to alternative
changes in UI policy and parameters in results which follow.
Since unanticipated shocks lead to a change in the value of firm equity, we must take a
stand on the composition of household portfolios at t = 0. As described in appendix C, I

26
Sticky Sticky
Flex Sticky
prices + prices +
prices prices
fixed r fixed i
Output multiplier -0.5 0.6 1.1 1.1
Avg change in unemp. rate +0.02pp -0.01pp -0.02pp -0.02pp
Table 4: baseline effects of UI
Note: the panels summarize the effects of a one-year extension of UI duration by three months starting from
the stationary RCE under flexible prices, sticky prices and an active Taylor rule, sticky prices and a constant
real interest rate for 18 months, and sticky prices and a constant nominal interest rate for 18 months. The
output multiplier is defined in (30). The average change in unemployment is computed over the year of
extended benefits.

use empirical patterns in household portfolios by wealth to initialize these portfolios.32
I use a partial equilibrium / general equilibrium decomposition to understand the results.
I compute the partial equilibrium impulse using agents’ re-optimized consumption and search
behavior in response to the UI shock, holding fixed the initial value of firm equity, the path of
labor market tightness, and the path of real interest rates, and only updating taxes to balance
the government budget given the change in worker search effort and thus employment. The
resulting impulse is consistent with partial equilibrium analyses in public finance, as in
Baily [1978] and Chetty [2006], which accounts for the fiscal externality associated with
changes in search. I then characterize the full general equilibrium response accounting for
the endogenous adjustment of firm equity, tightness, and real rates.
The partial equilibrium impulse to desired consumption resulting from extended UI re-
flects the effects of MPC heterogeneity, lower precautionary saving, and moral hazard in
search. This impulse is the thin solid line in the first panel of Figure 3. The reallocation
of desired consumption from the future to the present is consistent with UI redistributing
to higher MPC households and reducing the incentive for all households to precautionary
save. Because of moral hazard in search, the change in desired consumption is negative in
net present value terms as more households stay unemployed with more generous UI.
Accounting for the responses in tightness and interest rates in general equilibrium, the
UI extension leads to lower output and employment under flexible prices. The real interest
rate rises to equilibrate the asset (and goods) market, as shown by the thick solid line in the
second panel of Figure 3. This undoes the effects of more generous UI on households’ desired
consumption-savings plan and, consistent with Hall [2017], reduces vacancy creation by rais-
ing the discount rate applied by firms on the surplus from continuing matches. Together with
lower worker search effort due to moral hazard, the net effect on aggregate consumption and
32
The computational algorithm used to characterize each of the transitional dynamics in this section
builds on Guerrieri and Lorenzoni [2017] and Auclert et al. [2021] and is discussed further in appendix F.

27
Aggregate consumption Real interest rate (ann.)
0.05
0.05
0.04

% of SS consumption
0.03
0
0.02
pp
PE 0.01
-0.05 GE, flex prices
GE, sticky prices 0
GE, sticky prices + fixed r
GE, sticky prices + fixed i -0.01
-0.1
0 5 10 15 20 0 5 10 15 20
Month Month

Figure 3: effects of UI starting from steady-state
Note: the panels describe the effects of a one-year extension of UI duration by three months starting from
the stationary RCE with no other macroeconomic shocks. In the left panel, the partial equilibrium impulse
describes the change in aggregate consumption after agents re-optimize consumption and search given the
change in UI benefits and taxes (where the latter balances the government budget), holding fixed the initial
value of firm equity, the path of tightness, and the path of real interest rates. The general equilibrium impulse
describes the change in aggregate consumption (which equals output) accounting for the final response of
firm equity, tightness, and real interest rates.

thus output in the first panel of Figure 3 is negative.33 The first row of Table 4 summarizes
the effect on output by the contemporaneous multiplier
P11
∆ outputt
output multiplier ≡ P11 t=0 , (30)
t=0 ∆ UI paymentst

where the denominator uses the initial distribution of agents across the state space. The
multiplier is -0.5, with the unemployment rate rising by 0.02pp in the year of extended UI.
With nominal rigidity but a standard Taylor rule, real interest rates rise — but not as
much as above — in response to UI extensions. I set the adjustment cost on prices to ψ = 360,
so that (given an elasticity of substitution across retailer varieties of = 6) the log-linear
New Keynesian Phillips curve has slope on marginal cost −1 ψ
= 0.014. This is equivalent
to the log-linear Phillips curve in a model of Calvo pricing with expected price duration 9
months, consistent with the range of estimates surveyed in Nakamura and Steinsson [2013].
I set the coefficients in (29) to φΠ = 1.5 and φy = 1/12, following much of the literature
since Taylor [1999]. Monetary policy tightens in response to the increase in UI such that the
real interest rate, shown by the dashed line in the second panel of Figure 3, rises. Because
33
Appendix D provides the impulse responses of equilibrium tightness, search, and other variables.

28
Baseline Identical 6

Quarterly MPC, employed 0.16 0.17
Quarterly MPC, ST unemployed 0.36 0.27
Quarterly MPC, MT unemployed 0.47 0.35
Quarterly MPC, LT unemployed 0.59 0.43
Output multiplier 1.1 0.8
Avg change in unemp. rate -0.02pp -0.03pp

Table 5: sensitivity of effects of UI under sticky prices and fixed 2

Note: the counterfactual economy sets «> = e = 0 and re-calibrates all other parameters to match the same
moments as in the baseline, as described in appendix D. In all cases the quarterly MPC is computed for an
unexpected $500 rebate.

real interest rates do not rise by as much as in the flexible price case, the resulting output
multiplier becomes positive at 0.6.

Eliminating the rise in the real interest rate, the expansionary effects of UI extensions
nearly double. Suppose first that monetary policy maintains a constant real interest rate for
18 months, after which it follows the Taylor rule. The left panel of Figure 3 demonstrates
that aggregate consumption rises more substantially, and the third column of Table 4 in-
dicates that the UI extension now has an output multiplier of 1.1, generating an average
decline in the unemployment rate of 0.02pp. If monetary policy instead maintains a con-
stant nominal interest rate for 18 months before reverting to the Taylor rule, the left panel of
Figure 3 demonstrates that the increase in consumption is even higher, and the right panel
demonstrates that this is because the real interest rate falls due to an increase in inflation
expectations at a constant nominal rate. Nonetheless, as both this figure and the last two

columns of Table 4 make clear, the latter mechanism is small in these baseline results.

5.2 Roles of MPC heterogeneity and precautionary saving

MPC heterogeneity and reduced precautionary saving drive the aggregate effects of UI ex-
tensions when monetary policy is constrained.

I first study an alternative calibration of the model eliminating heterogeneity in separation
rates, illustrating the importance of heterogeneity in MPCs. With «> = a = (0 and other
parameters re-calibrated to match the same targets as the baseline as described in appendix
D, the first panel of Table 5 indicates that the stationary RCE features a smaller difference
in MPCs between the unemployed and employed. The second panel of Table 5 indicates

that the output multiplier from a three-month extension of UI is diminished.** These results

34The effect on the unemployment rate is slightly amplified, however. This is because the average produc-
tivity of the employed is lower than in the baseline, so a larger change in employment is needed to achieve

29
Output multiplier Output multiplier
2 2

1.5 1.5

1 1

0.5 0.5
extension in 1st month only
extension in 12th month only
i can adjust
0 0
3 6 9 12 15 18 21 24 27 30 33 3 6 9 12
Horizon of UI extension (months) Magnitude of UI extension (months)

Figure 4: sensitivity to horizon and magnitude given sticky prices and fixed i
Note: the left panel depicts the output multipliers corresponding to different UI extension horizons, in each
case extended from 6 to 9 months. In all cases the output multiplier is computed over the horizon of the
extension. The right panel depicts the output multipliers corresponding to different UI extension magnitudes,
in place for one year (circles); in the 1st month of the simulation only (small squares); and in the 12th month
of the simulation only (small diamonds). In all cases the output multiplier is computed over a fixed 12-month
horizon. Shaded markers in both panels depict the baseline three-month extension of UI for one year.

indicate that heterogeneity in the incidence of unemployment, which Figure 1 demonstrated
was needed to rationalize observed variation in EU flows by income and differences in wealth
conditional on these flows, plays a key role in the macroeconomic effects of UI by determining
the MPCs of those receiving the transfers.
Returning to focus on the baseline calibration alone, I next vary the number of calendar
months during which extended benefits are provided, illustrating the importance of pre-
cautionary saving. The first panel of Figure 4 plots the output multiplier associated with
extended UI over horizons of 3 to 33 months, holding fixed the three-month duration by
which benefits are extended. Over the period in which monetary policy is accommodative
(18 months), the stimulus grows as the announced horizon increases. This is consistent
with the dynamic amplification of the stimulus from reduced precautionary saving, higher
aggregate demand, and lower unemployment risk.35 Of course, once UI extensions are ex-
pected to persist beyond the period of a constant nominal interest rate, the tightening which
accompanies the extensions eventually renders them less expansionary.
Varying the number of months of unemployment by which UI is extended illustrates
a particular change in output.
35
Output multipliers rising with the horizon of a spending shock can also rise from the inflation expecta-
tions channel, but that is not dominant here: almost the same multipliers are obtained if monetary policy
maintains a constant real interest rate rather than nominal interest rate over the first 18 months.

30
Lower real Higher dis-
Baseline wage incentive
rigidity elasticity
Flexible prices
Avg change in real rate (ann.) +0.03pp +0.04pp +0.01pp
Output multiplier -0.5 -5.2 -1.8
Avg change in unemp. rate +0.02pp +0.13pp +0.07pp
Sticky prices
Avg change in real rate (ann.) +0.01pp +0.02pp +0.02pp
Output multiplier 0.6 0.6 0.6
Avg change in unemp. rate -0.01pp -0.01pp +0.00pp
Sticky prices + fixed i
Avg change in real rate (ann.) -0.00pp -0.01pp -0.00pp
Output multiplier 1.1 2.2 1.3
Avg change in unemp. rate -0.02pp -0.05pp -0.02pp
Table 6: sensitivity of effects of UI to supply-side responses
Note: in all cases the changes in the annualized interest rate, output, and unemployment rate are computed
over the year of extended benefits.

an interesting tension between the stimulus via heterogeneous MPCs and precautionary
saving.36 The right panel of Figure 4 plots the output multiplier associated with duration
extensions by 3 to 12 months, holding fixed a one-year horizon over which benefits are
extended. It further plots the output multiplier associated with such extensions at t = 0
and t = 11 alone (both announced at t = 0). On the one hand, the marginal recipients
of transfers have higher MPCs as UI is extended by a greater magnitude, explaining why
the output multiplier from UI extensions in the 1st month of the simulation alone rise. On
the other hand, a smaller fraction of agents are affected by the policy change, reducing
the precautionary saving effects of UI and explaining why the output multiplier from UI
extensions in the 12th month of the simulation alone fall. Because these forces offset, the
overall output multiplier from a one-year extension of benefits is remarkably stable with the
magnitude of the extension.

5.3 Sensitivity to supply-side responses
Larger supply-side responses only amplify the stimulus from UI when monetary policy is
constrained, in contrast to the results with flexible prices or an active Taylor rule.
This is first evident in the case of a lower degree of real wage rigidity. In the second column
36
I find a similar tension when considering an alternative calibration of the coefficient of relative risk
aversion σ: raising σ raises agents’ prudence, but compresses the distribution of MPCs by employment
status, such that the effects of UI extensions on output are roughly unchanged from the baseline calibration.

31
of Table 6, I lower ι from its baseline value of 1 to 0.9. In the usual way, this implies that
real wages rise with UI generosity, as this raises workers’ outside option in the Nash bargain.
Under flexible prices, this increase in wages leads to a reduction in firm profits and thus firm
vacancies, generating a large negative multiplier. With sticky prices, the Taylor rule similarly
calls for a larger rise in the nominal and thus real interest rate, though it remains dampened
versus flexible prices. However, with sticky prices and constrained monetary policy, these
results are reversed. The increase in real wages directly increases firms’ marginal costs and
generates inflation, reducing the ex-ante real interest rate when the nominal interest rate is
held fixed. Hence, the stimulus is amplified relative to the baseline.
This mechanism also applies to a larger micro disincentive effect. In the third column of
Table 6, I re-calibrate the steady-state to target a micro elasticity of unemployment duration
to potential benefit duration of 0.4, exceeding the baseline target of 0.1 and at the high end of
U.S. estimates summarized in Schmieder and von Wachter [2016].37 The larger disincentive
elasticity translates into a larger contractionary effect of UI extensions under flexible prices.
In contrast, the stimulus with sticky prices and constrained monetary policy is amplified.

5.4 Other features of UI policy and fiscal policy
In appendix D, I conduct several additional experiments which further elucidate the effects
of UI extensions in the model given sticky prices and constraints on monetary policy. I first
consider an increase in UI eligibility/take-up, which amplifies the stimulus by expanding the
scale of transfers. I then consider deficit-financed increases in UI duration, which amplifies
the stimulus because the model is non-Ricardian. I then compare the effects of raising the
replacement rate to extending UI duration, which underscores that duration extensions have
a larger multiplier because long-term unemployment is a “tag” for particularly high MPCs.
Finally, in appendix D I study the effects of conventional government purchases, rather
than UI (transfers). I augment the model to feature government spending on final goods
which enters separably into households’ utility. The model-implied fiscal multiplier (defined
analogously to (30)) ranges between 0.6-1.4 and is on the higher end of that range when
monetary policy maintains a constant nominal interest rate (as at the zero lower bound) and
government spending is deficit-financed. These magnitudes are in line with estimates of the
fiscal multiplier exploiting time-series and cross-sectional variation as surveyed by Ramey
[2011] and Chodorow-Reich [2019], respectively. This lends further credibility to my model’s
implications for the general equilibrium effects of UI.
37
The re-calibrated parameters are again summarized in appendix D.

32
6 Evaluation of policy during Great Recession
Armed with these insights from steady-state impulse responses, I now evaluate the effects
of UI policy during the Great Recession. The goal is not to provide a complete account of
the Great Recession, but instead to quantify the effects of UI extensions in a large downturn
of this kind. I calibrate a sequence of shocks to agents’ discount factors to replicate the
path of unemployment over 2008-2014 given the observed extensions to UI. I compare the
observed path of unemployment to counterfactual economies without the observed extensions
or without a zero lower bound. I find that the unemployment rate would have been as much
as 0.4pp higher were it not for the benefit extensions, and the binding zero lower bound
is essential for this result. The model-implied stimulus is consistent with the upper end of
estimates of stimulus in the empirical literature.

6.1 UI policy during Great Recession
The Emergency Unemployment Compensation Act of 2008 (EUC08) was legislated by the
U.S. Congress in June 30, 2008 to provide a federally-financed extension of UI across U.S.
states through June 2009. Subsequent extensions and re-authorizations, together with the
triggering of the Extended Benefits (EB) program, meant that UI duration exceeded 26
weeks in most states through 2013, reaching 99 weeks in some states.
Based on the Chronology of Federal Unemployment Compensation Laws and median UI
benefits by state reported by Farber and Valetta [2013], I simulate 13 shocks to UI policy
summarized in Table 7. Each shock is defined by its month of occurrence and the revised
path of UI duration from that month onwards. Twelve shocks correspond to new extensions
or re-authorizations of EUC08. I date each based on when the legislation was passed by the
U.S. Congress, and I define the revised path of UI duration based on the maximum weeks
of UI allowed by each piece of legislation.38 One shock corresponds to the triggering of EB.
I date it based on when EB benefits began for the median U.S. state, and I assume for
simplicity that EB benefits are reauthorized at the same time and over the same horizon as
EUC08 benefits in the subsequent months (until the last EB benefits were paid in August
2012). In all cases, I assume 4.5 weeks per month and round to the nearest month to obtain
UI durations consistent with the model frequency. Figure 5 shows that the realized path of
UI duration in my model corresponds closely to the median UI duration across U.S. states.
38
The exceptions to the latter are shocks 12 and 13, in which case (from September 2012 onwards) I
adjust downward the expected UI duration to match the median realized UI duration across states. As
states started letting extended benefits expire or no longer met the necessary “lookback” provisions, the
maximum available weeks of UI exceeded the realized measure.

33
Date Revised UI extensions Relevant legislation/program
1 7/2008 3 months through 6/2009 P.L. 110-252 (EUC08 Tier 1 announced)
2 12/2008 7 months through 11/2009 P.L. 110-449 (EUC08 Tier 2 announced)
3 3/2009 7 months through 5/2010 P.L. 111-5
4 5/2009 12 months through 5/2010 EB starts for median state
5 11/2009 16 months through 5/2010 P.L. 111-92 (EUC08 Tiers 3, 4 announced)
6 1/2010 16 months through 7/2010 P.L. 111-118
7 3/2010 16 months through 8/2010 P.L. 111-144
8 4/2010 16 months through 10/2010 P.L. 111-157
9 8/2010 16 months through 4/2011 P.L. 111-205
10 1/2011 16 months through 5/2012 P.L. 111-312
11 1/2012 16 months through 8/2012 P.L. 112-78
12 3/2012 16 months through 5/2012, P.L. 112-96
14 months through 8/2012,
8 months through 12/2012
13 1/2013 8 months through 12/2013 P.L. 112-240
Table 7: UI shocks simulated in model based on EUC08 and EB extensions
Note: for all shocks corresponding to EUC08 legislations, if legislation was passed in the first half of the
month, it is dated as that month; if it was passed in the second half of the month, it is dated as the following
month. UI duration is quoted in number of weeks in practice, so I assume 4.5 weeks per month and round
to the nearest month. See text for additional description of how each shock is classified.

My implementation tries to strike a balance between realism and parsimony.39 Impor-
tantly, by simulating 13 distinct shocks to UI policy, I capture the fact that the extended UI
benefits over 2008-13 were not characterized by perfect foresight in 2008, and instead evolved
in real time based on developments in the U.S. labor market and U.S. Congress. At the same
time, I abstract from the state-contingency involved in some of the extensions, namely the
second through fourth tiers of the EUC08 program as well as the EB program, which de-
pended on the evolution of the unemployment rate in each state. I further abstract from the
increase in U.S. government borrowing during this period, instead making the (conservative)
assumption that the benefits are financed via contemporaneous taxes on the employed.

6.2 Model versus data
I then simulate a sequence of shocks to agents’ average discount factor β̄t to match the
observed path of unemployment from May 2008 through December 2014, conditional on the
aforementioned shocks to UI and a value of real wage rigidity ι calibrated to match the
39
I also view it as conservative: since the output multiplier rises with the horizon of UI extensions when
monetary policy maintains a constant nominal interest rate (Figure 4), I expect that I would find even
more stimulative effects of the UI extensions if agents were not surprised by future reauthorizations and had
instead expected them to continue so long as monetary policy was at the zero lower bound.

34
Duration of UI

Months
10

5
Model
Data (median state)
0
12/08 12/09 12/10 12/11 12/12 12/13

Figure 5: realized UI duration in model versus data
Note: vertical line corresponds to initial date of simulation, before which model is in stationary RCE. At
any given month of simulation, agents’ expectations regarding future UI duration are described in Table 7.
Data is from Farber and Valetta [2013], who report weeks of UI by state. I assume 4.5 weeks per month and
round to the nearest month.

dynamics of prices during this period. I assume that the average discount factor follows the
AR(1) process
β̄t = (1 − ρβ̄ )β̄ + ρβ̄ β̄t−1 − β̄ + β̄t .

and I set ρβ̄ = 0.95.40 As in section 5, I assume price adjustment costs ψ = 360 (implying
a Calvo-equivalent frequency of price adjustment of 9 months). Monetary policy follows the
standard Taylor rule subject to zero lower bound in (29) with φΠ = 1.5 and φy = 1/12.
I view discount factor shocks as a way to capture changes in financial conditions which
change households’ desired (or required) saving.41 A shock to the credit constraint z delivers
the same comovements.42 In appendix E I also characterize the effects of productivity, sep-
aration rate, and match efficiency shocks, which imply distinct comovements from discount
factor or credit constraint shocks. In particular, these “supply” shocks generate a rise in
unemployment together with an increase in inflation. Later in this section, I thus consider
40
I find that the assumed value of ρβ̄ has little effect on the results which follow; it simply implies a
different magnitude of shocks to rationalize the observed path of unemployment.
41
A large literature has tried to isolate the driving forces behind the Great Recession. This is of course not
the contribution of my paper. The ability of discount factor shocks to rationalize the empirical comovements
in my model is consistent with the many papers finding a key role for negative demand shocks of this sort in
estimated DSGE models of the Great Recession, such as Christiano et al. [2015] and Del Negro et al. [2015].
42
However, it is quantitatively problematic to match the rapid rise in aggregate unemployment early in
the Great Recession via shocks to z alone: because the credit constraint shock has little effect on high wealth
households, the required size of shocks to z would imply some agents have negative consumption. It is for
this reason that I focus on discount factor shocks.

35
the sensitivity of my findings to an alternative calibration featuring both discount factor and
separation rate shocks; in my baseline calibration, however, I focus on discount factor shocks
alone for parsimony.

The calibration proceeds as follows. For a grid of values of real wage rigidity v, I calibrate
a sequence of shocks {eb } to match unemployment in the data. I choose the unemployment
rate as my calibration target so that the model-implied scale of UI payments is as consistent
with the data as possible. Given a sequence of shocks {eb } calibrated in this way for each
value of 1, I then choose . to minimize the sum of squared differences between the final
goods price index in model and data. I use the final goods price index as the key nominal
calibration target because expected inflation is what enters into agents’ Euler equations and
thus affects aggregate demand while monetary policy is at the zero lower bound.*?

The computational heart of this algorithm is in calibrating the sequence of discount fac-
tor of shocks, conditional on any 1. This proceeds iteratively: conditional on the distribution
of agents over the state space at t — 1, the calibrated 8,_,, and the expected future path of
UI from t onwards, I calibrate & so that in the perfect foresight equilibrium from period t
onwards, the unemployment rate in period t is consistent with the data in the corresponding
period.** I proceed to period t + 1 and repeat the steps. Because I re-solve for equilib-
rium policies, prices, and quantities at each step of this algorithm, I respect the important
nonlinearities induced by the zero lower bound. To my knowledge, the solution of such a
heterogeneous agent model as it gradually travels “far” from the initial steady-state, with
an endogenously time-varying duration at the zero lower bound, is novel to the literature.*®

This algorithm implies . = 0.94, a high degree of real wage rigidity, and the set of discount
factor shocks &P depicted in the first panel of Figure 6. As is evident, the model requires
positive shocks to rationalize the rise and persistence in unemployment in the early part
of the sample period, and negative shocks to rationalize its decline toward the end. The
simulated unemployment rate is depicted in the second panel, almost identical to the data.

Figure 7 compares other model-generated time-series with the data, all of which are

untargeted in the calibration except the final goods price index.*® The first panel demon-

43T use the core PCE as the data counterpart of final goods prices because the PCE index is the one
targeted by the FOMC, and the core index in particular is typically viewed as a more reliable measure of
inflation because it excludes volatile food and energy prices.

447 again use the empirical patterns in household portfolios described in appendix C to revalue households’
relative wealth given the change in the price of equity on impact of a shock at t.

45What makes this possible is a simple algorithm to dynamically update the Jacobian used to solve the
sequence of equilibrium conditions as the simulation proceeds. I describe this further in appendix F.

46The data sources are as follows. The nominal interest rate is the effective funds rate reported by
the Federal Reserve Board, rounded down to the nearest 25bp. Total non-farm job openings, unemployed
persons, unemployment by duration, and average hourly earnings of all private employees are reported by
the Bureau of Labor Statistics (BLS). The PCE index excluding food and energy is from the Bureau of

36
Shocks to average discount factor (×10−3 ) Unemployment rate
1 7
Model
6
Data
0.5 5

4
0 pp
3

2
-0.5
1

0
-1
12/08 12/09 12/10 12/11 12/12 12/13 12/14 12/08 12/09 12/10 12/11 12/12 12/13 12/14

Figure 6: discount factor shocks and unemployment
Note: average discount factor assumed to follow an AR(1) process with persistence 0.95. Shocks are chosen
each period so that, together with the shocks to UI described in Table 7, unemployment in model matches
that in data. Unemployment series is displayed in deviations from steady-state (for model) and April 2008
(in data).

strates that the model generates a binding zero lower bound from late 2008 (as in the data)
through late 2013 (earlier than “lift-off” in the data). The second and third panels plot
vacancies relative to the measure of unemployed and the fraction of the unemployed which
are long-term unemployed; the model generates substantial movements in both variables,
albeit of a smaller magnitude than in the data. The fourth panel plots consumption per
capita: the model and data are quite consistent through mid-2011, though consumption per
capita recovers in the model (consistent with unemployment) whereas in the data it falls
further below trend.47 The fifth and sixth panels focus on nominal wages and prices; the
model generates a decline in both quite consistent with the data.48 Notably, the high degree
of real wage rigidity ensures that nominal prices only fall by less than 3pp below trend by
2014 — amounting to less than 0.5pp below trend per year — despite the substantial rise in
unemployment during the Great Recession.49
Figure 8 plots an important endogenous outcome in the simulation: the horizon over
Economic Analysis (BEA). Consumption per capita is nominal consumption of non-durables plus services
(BEA), divided by the product of the GDP deflator (BEA) and total civilian population over 16 (BLS),
linearly interpolated from a quarterly to monthly frequency.
47
I expect that a change in trend growth could rationalize the model with the data in this dimension.
48
Interestingly, nominal wages rise in the first few months of the simulation, as in the data. In the model,
this occurs because of selection: the decline in job-finding rates disproportionately affects low-productivity
workers who are overrepresented among the unemployed.
49
In this way, the model rationalizes the “missing disinflation” during this period (see, e.g., Harding et al.
[2021] and citations therein).

37
Nominal interest rate (ann.) Vacancies / unemployed
0 20
Model
Data 0
-0.5
-20

-1
pp pp -40

-1.5 -60

-80
-2
-100
12/08 12/09 12/10 12/11 12/12 12/13 12/14 12/08 12/09 12/10 12/11 12/12 12/13 12/14

Fraction long-term unemployed Consumption per capita
30 5

25
0
20

15
pp pp -5
10

5
-10
0

-5 -15
12/08 12/09 12/10 12/11 12/12 12/13 12/14 12/08 12/09 12/10 12/11 12/12 12/13 12/14

Average nominal wage Final good prices
2 2

1 1

0 0

-1 -1
pp pp
-2 -2

-3 -3

-4 -4

-5 -5
12/08 12/09 12/10 12/11 12/12 12/13 12/14 12/08 12/09 12/10 12/11 12/12 12/13 12/14

Figure 7: untargeted macro time-series
Note: series displayed in deviations from steady-state (for model) and April 2008 (in data). In the data,
consumption per capita, the nominal wage index, and the nominal price index are detrended at their average
growth rates over 1990-2019 (1.7%, 2.6%, and 1.9% per year, respectively).

38
Expected horizon at ZLB
15

Months

0
12/08 12/09 12/10 12/11 12/12 12/13 12/14

Figure 8: expected horizon at the zero lower bound
Note: model hits the zero lower bound in the first month of the simulation, as depicted in Figure 7.

which agents expect the zero lower bound to bind, which evolves as agents (rationally)
forecast the economy’s response to shocks. Through August 2011, the model-implied expec-
tations of roughly one year at the zero lower bound are quite consistent with actual expecta-
tions. For instance, as summarized by Swanson and Williams [2014], the Blue Chip survey
of forecasters implied an expected number of quarters at the zero lower bound between 2-5
quarters from January 2009 through August 2011. Yield curve data, interpreted for instance
through the lens of a shadow rate term structure model as in Bauer and Rudebusch [2016],
similarly imply that the zero lower bound was expected to bind for around one year over
this same period. In August 2011, the Federal Reserve began forward guidance,50 dramati-
cally increasing the market’s expectation of the horizon of zero nominal interest rates. The
absence of forward guidance in the model means that it undershoots the expected horizon
at the zero lower bound from this point onwards, and indeed the model economy ultimately
“lifts off” from the zero lower bound at the end of 2013, whereas in practice this only hap-
pened in mid-2015. I do not enrich the model to feature forward guidance for two reasons.
First, as described in Table 7, 10 of the 13 extensions to UI in my model all occur before
August 2011, so that accounting for forward guidance would have a relatively small effect on
my results. Second, accounting for forward guidance would only increase the model-implied
stimulus from UI in the remaining 3 extensions, rendering my results conservative.
50
The Federal Open Market Committee announced that “economic conditions...are likely to warrant
exceptionally low levels for the federal funds rate at least through mid-2013.”

39
6.3 Effects of UI extensions
I now characterize the effects of the UI extensions in this environment.
I first compare the model dynamics to a counterfactual economy subject to the same
discount factor shocks but with no UI extensions. Figure 9 presents the same endogenous
variables as Figures 6 and 7 except for the nominal interest rate; I do not show the latter for
brevity because in the counterfactual economy, as in the baseline, the nominal interest rate
remains at zero through virtually the entire period of UI extensions.
The first panel of Figure 9 demonstrates that the counterfactual economy sees a larger
rise in unemployment. The unemployment rate would have been as much as 0.4pp higher
were it not for the benefit extensions. More generally, the model implies that the benefit
extensions were not a cause of unemployment persistence during the Great Recession.
The remaining panels of Figure 9 demonstrate that the counterfactual economy is more
slack on a number of other dimensions. Labor market tightness and aggregate consumption
are lower, while long-term unemployment is higher. These differences are especially pro-
nounced by 2011. With more slack in the counterfactual economy, there is more disinflation.
Table 8 considers the effects of each of the 13 shocks in Table 7 in isolation. The effect of
each UI shock is computed by comparing the transitional dynamics with the shock to those
without the shock, assuming no future shocks to UI nor fundamentals from that month
onwards. As is evident, all 13 shocks feature a positive output multiplier, around 1. There
is nonetheless obvious heterogeneity across shocks which underscores the key mechanisms
underlying these extensions.
For instance, a comparison of shock 1 and 4 demonstrates the key role of the binding
zero lower bound. Both shocks extend UI over a roughly one year horizon, but at the time
of shock 1 the zero lower bound is not expected to bind, whereas at the time of shock 4 it
is expected to bind over almost the entire period of extended UI. For that reason, shock 4
generates more stimulus.
A comparison of shock 3 and 9 demonstrates the key role of dynamic amplification via
reduced precautionary savings and a higher job-finding rate. Both shocks push out the
horizon of extended benefits by 6 months, but shock 3 does so 8 months in the future
whereas shock 9 does so only 2 months in the future. Consistent with Figure 4, the output
multiplier for shock 3 is larger than for shock 9.
Appendix E finally provides a case study of the expiration of extended benefits in De-
cember 2013. In the analysis above, I assume that upon the final reauthorization of benefits
in January 2013, agents rationally expected that benefits would expire in December 2013.
In the appendix, I consider an alternative scenario in which agents expected benefits to last
through 2014, but then they unexpectedly expire in December 2013. The analysis clarifies

40
Unemployment rate Vacancies / unemployed
10
5
0
4
-10

3 -20
pp pp
2 -30

-40
1
Model -50
0 No UI extensions
-60
12/08 12/09 12/10 12/11 12/12 12/13 12/14 12/08 12/09 12/10 12/11 12/12 12/13 12/14

Fraction long-term unemployed Consumption per capita
20 1

0
15
-1
10
pp pp -2

5 -3

-4
0

-5
12/08 12/09 12/10 12/11 12/12 12/13 12/14 12/08 12/09 12/10 12/11 12/12 12/13 12/14

Average nominal wage Final good prices
0.5 0.5

0 0

-0.5 -0.5

-1 -1

pp -1.5 pp -1.5

-2 -2

-2.5 -2.5

-3 -3

-3.5 -3.5
12/08 12/09 12/10 12/11 12/12 12/13 12/14 12/08 12/09 12/10 12/11 12/12 12/13 12/14

Figure 9: effects of UI shocks
Note: counterfactual environment maintains the same discount factor shocks in Figure 6.

41
Avg change in
Output
Date unemployment
multiplier
rate
1 7/2008 0.6 -0.01pp
2 12/2008 0.8 -0.03pp
3 3/2009 1.8 -0.04pp
4 5/2009 1.2 -0.02pp
5 11/2009 0.8 -0.01pp
6 1/2010 1.0 -0.04pp
7 3/2010 0.9 -0.02pp
8 4/2010 1.0 -0.04pp
9 8/2010 1.1 -0.09pp
10 1/2011 1.5 -0.10pp
11 1/2012 1.0 -0.04pp
12 3/2012 1.2 -0.03pp
13 1/2013 1.0 -0.07pp
Table 8: effects of each UI shock
Note: see Table 7 for description of each UI shock. Effect of shock in period t is computed by comparing
transitional dynamics with the UI shock to those without the UI shock, assuming no future shocks to UI
nor fundamentals from period t + 1 onwards. Output multiplier and effect on unemployment computed over
horizon of UI shock.

first that even if the expiration was unexpected, it need not have been accompanied by a
sharp deterioration in the labor market to the extent that favorable fundamental shocks
supported the economy at the same time. For instance, when I recalibrate discount factor
shocks to match the dynamics of unemployment, the (untargeted) dynamics of vacancies
are virtually indistinguishable between my baseline model and the model with unexpected
expiration. Of course, this does not mean the expiration is neutral: holding fixed the path
of discount factor shocks, but comparing the dynamics with unexpected expiration to those
in a counterfactual without expiration, I find that the real economy would have recovered
faster towards trend in the counterfactual economy.

6.4 Role of zero lower bound
I now further characterize the key role of the zero lower bound for the above results.
I first simulate a counterfactual economy subject to the same discount factor shocks and
UI extensions as the baseline model, but without a zero lower bound on the nominal interest
rate. The left panel of Figure 10 demonstrates that this counterfactual economy sees a
smaller rise in unemployment than the baseline. The right panel demonstrates that what
underlies this result is a substantial decline in the nominal interest rate (below zero), which

42
Unemployment rate Nominal interest rate (ann.)
0
7
Model Model
6 No ZLB No ZLB
No ZLB, no UI extensions -1 No ZLB, no UI extensions
5

4 -2
pp pp
3

2 -3

1
-4
0

12/08 12/09 12/10 12/11 12/12 12/13 12/14 12/08 12/09 12/10 12/11 12/12 12/13 12/14

Figure 10: effects of zero lower bound
Note: counterfactual environments maintain the same discount factor shocks in Figure 6.

helps to undo the recessionary effects of positive discount factor shocks.
I then simulate this same economy except without any UI extensions. The left panel
demonstrates that unemployment is close to indistinguishable from the environment with
the extensions. The right panel demonstrates that monetary policy endogenously tightens
in response to these extensions, explaining why they have smaller effects on economic activity.
Taken together, the zero lower bound on unemployment plays two key roles. First, it
generates a deeper economic recession given the same fundamental shocks, implying a larger
pool of workers affected by UI extensions. Second, it explains why monetary policy does not
tighten in response to the demand stimulus from extended UI.

6.5 Role of real wage rigidity
I now shed further light on the identification of real wage rigidity and its role in these results.
In Figure 11, I first compare the data and baseline path of nominal wages and prices to
alternative calibrations with more rigid real wages (ι = 1) and less rigid real wages (ι = 0.88).
The latter two cases are generated using alternative sequences of discount factor shocks to
match the same unemployment series in Figure 5. As is evident, a lower degree of real wage
rigidity would imply that nominal wages and prices fall by more during this period. This
makes clear that the model requires a high degree of real wage rigidity to rationalize the
absence of substantial deflation despite the rise in unemployment during the Great Recession.
Of course, with more real wage flexibility, the model implies only greater stimulus from
UI extensions at the zero lower bound. In Table 9, I compute the output multipliers from

43
Average nominal wage Final good prices
2 2
Model
1 Data 1
ι=1
0 ι = 0.88 0

-1 -1
pp pp
-2 -2

-3 -3

-4 -4

-5 -5
12/08 12/09 12/10 12/11 12/12 12/13 12/14 12/08 12/09 12/10 12/11 12/12 12/13 12/14

Figure 11: wages and prices with alternative real wage rigidity ι
Note: series displayed in deviations from steady-state (for model) and April 2008 (in data). Model series
under alternative values of ι are generated using alternative sequences of discount factor shocks to match
the same unemployment series in Figure 5.

each of the 13 UI shocks across these calibrations. As is evident, a lower degree of real wage
rigidity only implies more stimulus from UI extensions, consistent with Table 6.
Nonetheless, the case with fully rigid real wages still features substantial stimulus. In
fact, the multipliers are more than half of those obtained with ι = 0.94. Since the inflation
expectations channel is extremely small when real wages are fully rigid — evident from
the comparison of a constant real versus nominal interest rate in Table 4 — I conclude
that around my estimated degree of real wage rigidity, the direct effect of UI extensions on
aggregate demand are more important than the induced effect through inflation expectations
in driving the macroeconomic effects of UI extensions during the Great Recession.
Several other considerations justify the high degree of real wage rigidity in my analysis.
While there continues to be substantial debate in the literature, such high degrees of real wage
rigidity are well within the range consistent with macro evidence on labor market fluctuations
over the business cycle,51 as well as micro evidence on the responsiveness of real wages to
changes in UI.52 I also continue to estimate a high degree of real wage rigidity when I account
51
Shimer [2010], for instance, also focuses on a degree of (monthly) real wage rigidity around or above
0.95. The comovements between the labor wedge, hours, and consumption/output ratios justify this choice.
In the data, the labor wedge and hours are strongly negatively correlated, while the labor wedge and
consumption/output ratio are essentially uncorrelated (see his Table 1.1). Absent real wage rigidity in
a conventional search model with capital and (only) productivity shocks around a deterministic trend, the
labor wedge has a correlation of 0.96 with employment and almost -1 with the consumption/output ratio
(see his Table 3.3). Real wage rigidity of 0.993 is needed to bring both close to zero (see his Figure 4.2).
52
While I have followed Hall [2005], Blanchard and Gali [2010], Shimer [2010], and much of the literature
in assuming that a single parameter (ι) governs the responsiveness of real wages to all shocks, one could

44
Output multiplier
Date ι = 0.88 Model ι=1
1 7/2008 0.6 0.6 0.6
2 12/2008 1.4 0.8 0.6
3 3/2009 2.9 1.8 1.0
4 5/2009 1.7 1.2 0.8
5 11/2009 1.0 0.8 0.8
6 1/2010 1.2 1.0 0.8
7 3/2010 1.0 0.9 0.7
8 4/2010 1.2 1.0 0.8
9 8/2010 1.3 1.1 0.9
10 1/2011 2.3 1.5 0.9
11 1/2012 1.4 1.0 0.8
12 3/2012 1.9 1.2 0.8
13 1/2013 1.4 1.0 0.8
Table 9: effects of each UI shock with alternative real wage rigidity ι
Note: see Table 7 for description of each UI shock. Effect of shock in period t is computed by comparing
transitional dynamics with the UI shock to those without the UI shock, assuming no future shocks to UI nor
fundamentals from period t + 1 onwards. Output multiplier computed over horizon of UI shock.

for other driving forces besides demand shocks during the Great Recession: in appendix E, I
present an alternative calibration featuring both discount factor and separation rate shocks,
where the latter is directly disciplined by the time-series of the aggregate separation rate.
Even in this case, I continue to estimate ι = 0.94.

6.6 Interpretation of empirical evidence
My analysis complements a growing empirical literature studying UI during this period.
This literature has obtained conflicting results. Boone et al. [2019] and Chodorow-Reich
et al. [2019] find small but potentially positive effects of the UI extensions on employment.
Hagedorn et al. [2016], Hagedorn et al. [2019], and Dieterle et al. [2020] find that the UI
extensions lowered employment, with large effects in the first two papers.
The present model implies effects which are consistent with the upper end of estimates
in this literature. Boone et al. [2019] find that a three-month extension of UI raises the
employment-to-population rate between -0.09pp and 0.24pp,53 while Chodorow-Reich et al.
entertain a specification in which real wages are more responsive to some shocks than others. Of most
relevance for my results is the responsiveness of real wages to changes in UI. In this context, my estimated
high degree of real wage rigidity is especially reasonable: Card et al. [2007], Jäger et al. [2020], Lalive [2007],
and van Ours and Vodopivec [2008] have all exploited micro variation in UI and estimated extremely small
effects of UI duration on wages.
53
In Table 2, these authors find that a 73-week extension of benefits increases employment-to-population

45
[2019] find that a three-month extension of UI reduces the unemployment rate between -
0.06pp and 0.09pp.54 The positive ranges of these confidence intervals are consistent with
my baseline results. One reason that my estimates may be at the positive end of these
ranges is that the empirical literature has focused on cross-state identification which may
understate the increase in consumption on tradeables.
The structural approach in this paper provides three other insights on mechanisms and
implications which complement these estimates. First, the modest estimates of stimulus
in terms of unemployment are consistent with meaningfully-sized output multipliers, since
transfers to the long-term unemployed were a small fraction of overall output even during
the Great Recession. Second, differences in identifying variation can rationalize why Boone
et al. [2019] may have estimated greater stimulus than Chodorow-Reich et al. [2019]: since
the former use differences in UI with an average half-life more than a year while the latter
use shocks to UI with an average half-life of three months, the former would be expected
to measure greater stimulus from diminished precautionary saving. Third, the estimates of
stimulus are consistent with the micro evidence on MPC heterogeneity and precautionary
saving used to calibrate and validate my model, and do not rely on an especially large effect
through inflation expectations at the zero lower bound.

7 Conclusion
This paper studies UI in general equilibrium with incomplete markets, search frictions, and
nominal rigidities. An increase in UI raises aggregate demand if the unemployed have a
higher MPC than the employed or if agents have a precautionary saving motive. This raises
equilibrium output and employment if monetary policy does not raise the nominal rate. In a
quantitative analysis of the U.S. economy over 2008-2014, I find that these mechanisms drove
the macroeconomic effects of UI extensions during this period. The observed extensions had
a contemporaneous output multiplier around 1. The unemployment rate would have been
as much as 0.4pp higher were it not for the benefit extensions.
There are two natural directions to build on the analysis of this paper. One direction is
to study the positive effects of other social insurance and cash transfer programs: as with
UI, these programs may be effective “tags” for agents with high MPCs and affect agents’
by 0.43pp with standard error 0.47. This implies that a 13-week extension of benefits increases employment-
to-population by [0.43 − 1.96 × 0.47, 0.43 + 1.96 × 0.47] × (13/73) = [−0.09, 0.24]pp. Scaling by (13/73) can
be justified by my results in Figure 4 that the stimulus from UI is stable across magnitudes of the extension.
54
In their discussion of Table 4, the authors argue that their estimates imply a one-month UI duration
innovation reduces the unemployment rate by [-0.02,0.03]pp. This implies that a three-month UI duration
innovation reduces the unemployment rate by [-0.06,0.09]pp.

46
incentives to precautionary save, giving these programs an aggregate demand stabilization
role in recessions. Another direction is to explore the normative implications of my findings:
the effects of UI and other social insurance programs on aggregate demand will naturally
affect their optimal generosity over the business cycle, particularly if monetary policy is
constrained. I leave the analysis of these exciting questions to future research.

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