Rescue Policies for Small Businesses in the COVID-19 Recession
Summary
SAFE Working Paper No. 343, Rescue Policies for Small Businesses in the COVID-19 Recession, by Alessandro Di Nola, Leo Kaas and Haomin Wang, dated March 2022. The paper builds a general equilibrium model with heterogeneous, financially constrained firms, calibrated to the U.S. economy, and treats the Paycheck Protection Program (PPP) as a grant policy. The authors report that the PPP prevents 35% of small business exits at the onset of the pandemic but has only a small impact on aggregate employment, and that it shifts resources toward less productive firms. They compare a counterfactual policy targeted to impacted firms and find its cost of raising average employment is 63% of the cost of the PPP, while it largely prevents the creation of zombie firms. The paper closes with a list of recent SAFE working papers.
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Alessandro Di Nola | Leo Kaas | Haomin Wang
Rescue Policies for Small Businesses
in the COVID-19 Recession
SAFE Working Paper No. 343 | March 2022
Electronic copy available at: https://ssrn.com/abstract=4064899
Rescue Policies for Small Businesses
in the COVID-19 Recession∗
Alessandro Di Nola†
Leo Kaas‡
Haomin Wang§
March 2022
Abstract
While the COVID-19 pandemic had a large and asymmetric impact on firms, many countries quickly enacted massive business rescue programs which are specifically targeted to
smaller firms. Little is known about the effects of such policies on business entry and exit,
factor reallocation, and macroeconomic outcomes. This paper builds a general equilibrium model with heterogeneous and financially constrained firms in order to evaluate the
short- and long-term consequences of small firm rescue programs in a pandemic recession.
We calibrate the stationary equilibrium and the pandemic shock to the U.S. economy,
taking into account the factual Paycheck Protection Program (PPP) as a specific grant
policy. We find that the policy has only a small impact on aggregate employment because (i) jobs are saved predominately in less productive firms that account for a small
share of employment and (ii) the grant induces a reallocation of resources away from
larger and less impacted firms. Much of this reallocation happens in the aftermath of
the pandemic episode. While a universal grant reduces the firm exit rate substantially,
a targeted policy is not only more cost-effective, it also largely prevents the creation of
“zombie firms” whose survival is socially inefficient.
JEL classification: E22, E65, G38, H25
Keywords: COVID-19, Heterogeneous Firms, Business Subsidies, Paycheck Protection Program
∗
We are grateful to Ivo Bakota, Basile Grassi, Marek Ignaszak, Matthias Kredler, Alexander Ludwig,
Mathias Trabandt and seminar participants at the University of Barcelona, the Munich Center for the Economics of Ageing and conference audiences at the IMF-TARC Conference and the Econometric Society Winter
Meeting for insightful comments. Alessandro Di Nola thanks the German Research Foundation (grant No.
SCHO 1442/2) for financial support.
†
University of Konstanz, alessandro.di-nola@uni-konstanz.de
‡
Goethe University Frankfurt and SAFE, kaas@wiwi.uni-frankfurt.de
§
University of Konstanz, haomin.wang@uni-konstanz.de
Electronic copy available at: https://ssrn.com/abstract=4064899
1
Introduction
The 2020-21 recession induced by the COVID-19 pandemic differs from regular business-cycle
downturns in important ways. Government-mandated shutdown policies, individual demand
adjustments and disruptions of global production chains had a large and asymmetric impact
on private businesses. In particular, the magnitude of output and employment declines were
larger for smaller firms (see Bloom et al., 2021; Cajner et al., 2020).1 Furthermore, business
closures in the U.S. have increased sharply, again with much variation by firm size (Crane
et al., 2021; Chetty et al., 2020). To stabilize income losses in the short-term and to prevent a
severe and long-lasting impact on production capacities, governments in many countries swiftly
implemented small business rescue programs in the form of grants or conditional loans. For
instance, in March 2020 the U.S. enacted the Coronavirus Aid, Relief, and Economic Security
(CARES) Act that allocated over $600 billion for the Paycheck Protection Program (PPP).
In 2020, over three quarters of U.S. small businesses received the PPP loan and most are
eventually forgiven (Autor et al., 2022).
Little is known about the effects of such a large-scale small business rescue policy from a
macroeconomic perspective. On the one hand, offering liquidity to small businesses can prevent
productive firms from permanently shutting down, impeding inefficient capital liquidation and
facilitating a quicker economic recovery once the pandemic terminates. On the other hand,
such rescue plans can inadvertently prolong the lives of unproductive (“zombie”) firms, thus
hampering efficient capital reallocation. Furthermore, the PPP program was designed to favor
timeliness over targeting (Autor et al., 2022), resulting in an unprecedented fiscal cost as
firms that are not impacted or at risk of liquidation also received the forgivable loan. Given
that targeting financial aid to impacted firms requires a greater administrative burden, it is
important to understand the cost-effectiveness of a targeted rescue policy compared to the
rather universal PPP program.
The goal of this paper is to quantify the short- and long-run macroeconomic effects of the
small business rescue policy enacted in the COVID-19 pandemic, and to evaluate a counterfactual targeted rescue grant. To this end, we build a general equilibrium model with firms
that differ in productivity, the level of fixed capital, and financial assets or debt. Firms face
financial constraints and capital adjustment frictions due to partial irreversibility of fixed in1
Using anonymized administrative data provided by ADP (a large private provider of payroll services),
Cajner et al. (2020) show that businesses with fewer than 50 employees reduced employment by more than
25 percent from March to April 2020, whereas larger firms saw declines of 15-20 percent over the same time
period.
1
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vestments. We interpret these entities as “small firms” in a non-corporate sector that can
only borrow against collateral, have no access to capital markets and cannot easily liquidate
a portion of their capital. Small firms incur overhead expenses including the maintenance
cost of capital and payroll expenses. Our model also includes a corporate sector where firms
are not financially constrained. The pandemic has an asymmetric impact on firms in the two
sectors. Although the corporate sector is not the center of our analysis, it is important for
factor reallocation and hence for macroeconomic adjustment following the pandemic shock.
We first calibrate our model such that the stationary equilibrium matches relevant aggregate and firm-level moments of the pre-pandemic U.S. economy. We draw data from various
sources including semi-aggregate tables of the Statistics of U.S. Businesses (SUSB), the Business Dynamics Survey (BDS), as well as micro data from the Kaufman Firm Survey (KFS) to
inform us about the balance sheets of small firms in the U.S. Our model closely replicates the
heavily skewed firm size distribution observed in the SUSB, the pattern that the firm exit rate
decreases in firm size observed in the BDS, and the debt-asset ratio and the share of indebted
firms observed in the KFS. In our calibrated model, potential entrants with low productivity
would not enter, while continuing firms exit if their productivity is too low or debt is too high.
We model the pandemic shock with four components: A shut-down shock that affects a
fraction of small firms, a TFP shock on the corporate sector, a demand shock affecting the
marginal utility of consumption, and a labor supply shock affecting the marginal utility of
leisure. The shut-down shock captures the impact of government mandated lock-down policies
that forced businesses offering “social” goods and services to temporarily close at the beginning
of the pandemic. The demand shock is important for explaining the observed sharp drop in
consumption, and the labor supply shock captures the drop in employment, possibly due to
health risks of in-person working. We calibrate the pandemic shock to match the changes of
U.S. output, consumption, aggregate employment and employment in small firms, while taking
into account the PPP policy and their take-up rates by small firms.
Based on our calibrated model, we compare the PPP to the laissez-faire economy, a counterfactual scenario with no government intervention. We find that the PPP prevents 35%
of small business exits at the onset of the pandemic. Despite being successful at preventing
many businesses from permanently shutting down, the PPP is mostly ineffective in improving
aggregate output and employment. The reason is twofold: (i) The PPP induces inefficient
reallocation of resources towards the more impacted small-firm sector, away from larger and
less impacted firms in the corporate sector. Further, within the small-firm sector, there is also
a reallocation toward less productive firms. (ii) The PPP prevents business exits in less pro-
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ductive firms that account for only a small fraction of total employment. The lack of aggregate
impact echoes previous findings in the literature (see Crouzet and Mehrotra, 2020) showing
that although smaller firms are more exposed to aggregate volatility, the difference has only
modest implications on aggregate fluctuations. Therefore, policies aiming to stabilize smaller
firms should not be expected to have large macroeconomic consequences.
Next, we simulate a counterfactual policy that is similar to the PPP but only gives financial
aids to impacted firms. Since the share of impacted firms is calibrated to be 11%, the fiscal
cost of the targeted policy is only one seventh of the cost of the PPP. The targeted policy leads
to a smaller employment improvement compared to the PPP over a 10 year period, but it is
more cost-effective. Specifically, we compare the cost of an average employment increase by
1% over a 10-year period under the two policies and find that the cost of the targeted policy
is 63% of the cost of the PPP.
Targeting rescue aids to impacted firms has long-run implications. We decompose the
effects of the PPP and the targeted policy over ten years into the short-run (the first two
quarters), the medium run (quarter 3 to the end of year 3), and the long run (years 4–10). We
find that the PPP has not only a short-term, but also a highly persistent effect in reducing firm
exit because it improves the balance sheet of its recipients. As a result, there is a persistent
reallocation of resources from the corporate sector to small firms. By contrast, the targeted
policy largely eliminates the short-run increase in the exit rate without generating persistent
sectoral reallocation effects. Further, the targeted policy drastically reduces the emergence of
“zombie firms” whose survival is socially inefficient.
Related literature Our work relates to different strands of the macroeconomic literature.
Several studies analyze the impact of health policies in the COVID-19 pandemic by integrating
epidemiological dynamics into macroeconomic general-equilibrium models (e.g. Eichenbaum
et al., 2020; Glover et al., 2020) or demand and supply spillovers in multi-sector models (e.g.
Guerrieri et al., 2020; Baqaee and Farhi, 2020). By focusing on the effects of business rescue
policies, our model features a representative household and treats the pandemic shock as
an exogenous event which impacts both the productivity of firms and household preferences
for consumption and leisure in order to generate the factual employment, consumption and
investment responses during the 2020 recession. While simplifying our model analysis, this
modeling choice obviously rules out potential feedback effects of business rescue policies on the
health sector.
Other recent work evaluates the macroeconomic and distributional impacts of fiscal policy
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in the COVID-19 recession. Bayer et al. (2020) and Bigio et al. (2020) analyze conditional
and unconditional transfers to heterogeneous households. Faria-e-Castro (2021) studies the
effectiveness of different types of fiscal policy, including transfers to firms, without considering
firm heterogeneity. Complementary to these studies, our work focuses on the macroeconomic
and welfare consequences of business rescue policies, while abstracting from distributional
implications.
Further contributions examine the role of firms in the pandemic recession. Bilbiie and
Melitz (2020) show how price rigidity amplifies the entry and exit dynamics, and Elenev et al.
(2020) study the impact of different firm bailout policies, focusing on the linkages between the
financial intermediation and production sectors. Gourinchas et al. (2022) calibrate a static
model with heterogeneous firms and find that government grants were quite effective in reducing business exits but also costly due to the lack of targeting. These papers do not allow for
persistent firm heterogeneity by productivity or financial assets and thus they do not examine
the reallocation of production factors across firms and over time.
Most closely related to our work are Buera et al. (2021a) and Jo et al. (2021). Buera
et al. (2021a) examine the impact of a pandemic shock on heterogeneous firms facing financial
frictions, also including occupational choice and labor market frictions. Jo et al. (2021) use
a model setting similar to ours which additionally features households with different health
status (and hence endogenous pandemic dynamics). Different from our work, both papers do
not analyze the role of government grants to small firms for firm selection and macroeconomic
dynamics, and their models do not feature the partial irreversibility of capital investments that
is central for our study.
Finally, we build on a large literature that incorporates heterogeneous, financially constrained firms into macroeconomic models. Our model is based on Khan and Thomas (2013),
where we simplify their setup by featuring fixed, partially irreversible capital investments. As
in Khan et al. (2016), entry and exit are endogenous, yet all debt is secured by collateral so
that default does not occur. The long-term macroeconomic impact of credit-subsidy policies
on heterogeneous firms has been studied by Buera et al. (2013), Buera et al. (2021b) and Jo
and Senga (2019). While they focus on stationary environments, the reallocation effects via
extensive (entry and exit) and intensive (factor intensity) margins are common to our work.
The rest of this paper is organized as follows. Section 2 briefly reviews the response of the
U.S. economy in the COVID-19 recession and the PPP policy. Section 3 presents the model
and Section 4 the calibration. Section 5 shows findings of the paper, and Section 6 concludes.
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2
COVID-19 Pandemic in the U.S. and Rescue Policies
85
Index (2020Q1 = 100)
90
95
100
105
The recession induced by the COVID-19 pandemic differs from past recessions in important
ways. The pandemic shock to the macroeconomy is deep but short-lived. Figure 1 shows the
macroeconomic impact of the COVID-19 pandemic on the U.S. economy. Compared to the
first quarter of 2020, the aggregate economy took a dramatic downturn in the second quarter
of 2020: total non-farm output fell by 10.9%, employment by 12.9%, consumption by 9.7%
and private domestic investment by 15.4%.
2019q4
2020q1
2020q2
Output
Private Investment
2020q3
2020q4
2021q1
Consumption
Employment−Population Ratio
Figure 1: Macroeconomic Impacts of the COVID-19 Pandemic
To mitigate the spread of the coronavirus, many governments imposed strict shutdown
and social-distancing policies at the beginning of the pandemic. The economic impact of
the pandemic was felt disproportionately by small businesses as they face tighter borrowing
constraints and may experience greater difficulties to liquidate their fixed capital. Bartik
et al. (2020) report that small businesses are more likely to experience closures (temporary
or permanent) than larger businesses. Bloom et al. (2021) show that small firms experience
a larger drop in sales. Based on data from the ADP Research Institute, Cajner et al. (2020)
show that firms with fewer employees experience greater employment losses compared to larger
firms. Based on the their data, employment in firms with fewer than 500 employees drops by
23% relative to the pre-pandemic level, whereas employment in larger firms drops by only 18%.
Businesses offering “social goods” are particularly vulnerable. Bartik et al. (2020) report
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that industries with an above-average business closure rate include personal services (86%),
arts and entertainment (71%), tourism (61.5%), restaurants (54%) and retail (except grocery,
52%).
In response to the potentially devastating impacts of the pandemic on small businesses,
the U.S. allocated over $600 billion for the Paycheck Protection Program (PPP) starting in
April 2020.2 The program offers forgivable loans up to 2.5 times the average monthly payroll
to small businesses with up to 500 employees. PPP loans feature an attractive interest rate of
1% p.a. and can be forgiven when certain requirements are met. Appendix A provides details
on the terms of PPP loans and forgiveness requirements. According to the U.S. Small Business
Administration, as of November 2021, 92% of all PPP loans issued in 2020 have been fully or
partially forgiven.
Despite the high degree of heterogeneity in the pandemic impact on businesses, the distribution of PPP loans is largely untargeted and prioritizes speedy loan disbursement (Autor
et al., 2022). According to Autor et al. (2020), initially about 70% of eligible firms applied for
PPP loans, and Borawski and Schweitzer (2021) report that by the end of 2020 PPP loans had
been taken up by 76% of U.S. small businesses.3 It is not clear what the effect of targeting
rescue policies to impacted firms is, and more generally, how effective small-business rescue
policies are for macroeconomic outcomes.
A few empirical studies estimate the initial employment effect of the PPP program. Utilizing the PPP eligibility size threshold to differentiate treatment and control groups, Autor
et al. (2020, 2022) find that the policy increased employment at eligible firms by about three
percent at a high cost of around $200,000 per job-year. Using different data sources, Chetty
et al. (2020) and Hubbard and Strain (2020) find somewhat smaller employment effects in the
range of 1-2%. Bartlett III and Morse (2020) and Hubbard and Strain (2020) also find that
the PPP had a significant impact on the survival probability of smaller businesses. Finally,
Kurmann et al. (2021) use real-time establishment data and find that pandemic policies, such
as the PPP, are effective in mitigating the negative employment effect on small businesses by
relaxing financial constraints.
To study the impact of a business rescue policies in the short- and long-run, including spillover effects and reallocation of production factors in general equilibrium, we consider a model
in which financially constrained small firms and unconstrained large firms are differentially
impacted by a pandemic shock.
2
3
In this paper, we only focus on the small firm rescue program issued in the year 2020.
The total take-up rate of PPP loans in both 2020 and 2021 is around 94% (Autor et al., 2022).
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3
The Model
We consider a discrete-time general equilibrium model of a closed economy. A single consumption/investment good is produced by firms that belong either to the corporate or to the
non-corporate sector.4 While firms in the corporate sector face no financial frictions, noncorporate firms (termed “small firms” in the following) can only borrow against the collateral
value of their capital and they cannot raise equity from outside investors. Entry and exit into
the non-corporate sector are endogenous outcomes. In particular, small firms face idiosyncratic productivity risk and may decide to liquidate their firm if they are unable or unwilling
to continue operating.
While there is no aggregate risk, we consider the economy’s response to a one-time unexpected shock and the transition path back to the unique steady-state equilibrium. To simplify
notation, we index several variables by the time index t, indicating dependence on the aggregate state vector which is either constant (in steady state) or converges deterministically
after the one-time shock.5 The one-time shock includes a shut-down shock on a fraction of
small firms, a TFP shock on the corporate sector, and shocks to household preferences over
consumption and leisure.
3.1
Environment
3.1.1
Small Firms
Small firms operate a fixed amount of capital κ which is drawn from a finite set K upon entry
and constant over time.6 Capital is partly irreversible: upon exit only the fraction θ < 1 of
capital can be liquidated. Every period, the fraction δ of capital depreciates and is replaced
by the firm’s owner. Small firms further incur fixed operating cost cf (κ) which capture general
and administrative expenses.
A small firm with capital κ hires labor services ` to produce output xf (κ, `) where f is a
strictly increasing, strictly concave, and decreasing returns to scale production function and x
4
To keep our model reasonably simple, we do not differentiate between the goods produced for social and
non-social consumption. Given that many social goods have close non-social substitutes (e.g. restaurants vs
food at home, health clubs vs home gyms, or movie theaters vs streaming services), this seems a reasonable
short-cut.
5
Further simplifying notation, the time index is dropped from the firms’ state and decision variables in
which case we use the prime superscript to denote next period values of these variables.
6
As common with the putty-clay literature (e.g. Gilchrist and Williams, 2000; Gourio, 2011; Sorkin, 2015),
this assumption limits capital-labor substitutability in the short-run, while capital reallocations occur via the
business entry and exit margin.
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is exogenous, idiosyncratic productivity which follows a monotone Markov chain on finite set
X with transition probabilities g(x0 |x). Write
πt (x, κ) ≡ max xf (κ, `) − wt ` − δκ − cf (κ)
`≥0
for the operating profit of a small firm with capital κ and with productivity x in period t,
where wt is the real wage.
Small firms can save and borrow in the capital market at safe gross interest rate 1/qt .
Borrowing is secured by the collateral value of capital;7 hence the debt issued in any period is
constrained by the liquidation value of capital,
b0 ≤ θκ .
(1)
Furthermore, small firms cannot raise equity which implies that the dividend income in period
t must be non-negative:
πt (x, κ) − b + qt b0 ≥ 0 ,
(2)
where b is the debt due in period t (or the negative value of savings when b < 0).
At the beginning of every period, a firm that was active in the previous period may exit due
to two distinct events. First, low-productivity and high-debt firms may not be able to fulfill
the two financial constraints (1) and (2); these illiquid firms are then forced to exit. Second,
the firm may voluntarily decide to exit. Regardless of the cause of exit, the firm’s capital
is liquidated and the owner retains the liquidation value net of debt repayment (or financial
savings when b < 0) which is θκ − b.
Also at the beginning of every period, a mass M of potential entrants draw initial productivity, debt levels, and capital (x, b, κ) from the joint distribution Φ(x, b, κ) and decide to enter
whenever the value of the new firm exceeds the installation cost net of debt, κ − b.
3.1.2
Corporate Firms
Firms in the corporate sector use capital K c and labor services Lc to produce output F (K c , Lc )
with strictly increasing, concave, constant returns production function F . Write Πt (K c ) =
maxL F (K c , Lc ) − wt Lc for operating profits in the corporate sector in period t. Capital in the
7
The price of collateral in our one-sector economy is identical to the price of final output, and hence does not
respond to aggregate shocks. As in Kiyotaki and Moore (1997), a fall in the price of collateral would amplify
a recessionary shock through a tightening of financial constraints. While commercial property prices declined
during the COVID-19 recession, the drop was not nearly as severe as during the Great Recession 2008-09.
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corporate sector also depreciates at rate δ.
3.1.3
Households
There is a unit mass of households with unit time endowment who derive period utility
u(C, 1 − L) which is strictly increasing in consumption C and leisure 1 − L, with L denoting
labor supply. Future utility is discounted with factor β < 1 per period. Households own all
firms.
Households can perfectly insure against all idiosyncratic business risks. As a result, their
consumption, labor supply and investment decisions are described by the utility maximization
problem of a representative household. In period t, the representative household takes as given
the aggregate state vector Xt = (Ktc , At , µ0t ) where Ktc is the capital stock in the corporate
sector, At are financial assets, and µ0t is the measure of small firms over idiosyncratic states
(x, b, κ) prior to entry and exit. The household decides about consumption Ct , labor supply
Lt , gross investment Itc in the corporate sector, and financial assets for the next period At+1
which are traded at price qt . Further, the household decides the entry of small firms det (x, b, κ)
which equals one when potential firm (x, b, κ) enters and zero otherwise, liquidation of small
firms dlt (x, b, κ) which equals one when firm (x, b, κ) is liquidated and zero otherwise, and
borrowing/savings decisions b0t (x, b, κ) of active firms. In recursive notation, the representative
household’s problem is
Vt (Xt ) = max u(Ct , 1 − Lt ) + βVt+1 (Xt+1 ) ,
subject to the budget constraint
Z
c
Ct + It + qt At+1 + M [κ − b]det (x, b, κ) dΦ(x, b, κ) ≤ wt Lt + At + Πt (Ktc )
Z
+ πt (x, κ) − b + qt b0t (x, b, κ) dµt (x, b, κ)
Z
+ (θκ − b)dlt (x, b, κ) dµ0t (x, b, κ) ,
(3)
(4)
where
µt = (1 − dlt )µ0t + M det Φ
(5)
is the measure of active firms in period t (i.e., incumbent firms that are not liquidated plus
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entrants), the accumulation equation for capital in corporate firms,
c
Kt+1
= (1 − δ)Ktc + Itc ,
(6)
financial constraints for continuing small firms,
πt (x, κ) − b + qt b0t (x, b, κ) ≥ 0 ,
b0t (x, b, κ) ≤ θκ,
(7)
and the dynamic evolution of the distribution measure of small firms which requires that the
measure of small firms at the beginning of the next period satisfies
Z
0
µt+1 (A) = I(x0 ,b0t (x,b,κ),κ)∈A g(x0 |x) dµt (x, b, κ) ,
(8)
for all Borel sets A ⊂ X × IR ×K.
The budget constraint (4) says that the household’s expenditures for consumption, investment in corporate firms, financial assets and the investment expenditures of entrant firms
(left-hand side) do not exceed the sum of labor income, the stock of financial assets at the
beginning of the period, profit incomes of corporate and small firms, and the liquidation values
of exiting small firms (right-hand side).
3.2
Competitive Equilibrium
3.2.1
Definition
R
Given an initial state (K0c , A0 , µ00 ) with A0 = b dµ00 (x, b, κ), a competitive equilibrium is
a sequence of market prices (wt , qt ), consumption, labor supply and investment decisions
c
(Ct , Lt , Itc , Kt+1
, At+1 ), entry, exit and borrowing decisions (det , dlt , b0t ) and distribution measures µ0t+1 , µt of small firms, for all t ≥ 0, such that
(i) The representative household solves the utility maximization problem (3)–(8).
(ii) The markets for labor and financial assets clear, i.e. in all periods t,
Z
Lt =
Z
At+1 =
`t (x, κ) dµt (x, b, κ) + Lct ,
b0t (x, b, κ) dµt (x, b, κ) ,
with labor demand `t (x, κ) = argmax` xf (κ, `)−wt ` and Lct = argmaxLc F (Ktc , Lc )−wt Lc .
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A stationary competitive equilibrium is a competitive equilibrium with constant state vector
(K c , A, µ0 ) and constant market prices (w, q).
The goods market is in equilibrium because of Walras’s law: the binding budget constraint
together with the other market-clearing conditions implies that consumption plus investment
R
equals aggregate output, Ct + It = F (Ktc , Lct ) + [xf (κ, `t (x)) − cf (κ)] dµt (x, b, κ), where
It = Itc + M
Z
κdet (x, b, κ) dΦ(x, b, κ) −
Z
θκdlt (x, b, κ) dµ0t (x, b, κ) + δ
Z
κ dµt (x, b, κ)
is aggregate gross investment in corporate firms and in entrant firms. The accumulation
R
equation for aggregate capital, denoted by Kt = Ktc + κ dµ0t (x, b, κ), is
Kt+1 = Kt − Dt + It ,
where
Dt = δKtc + δ
Z
Z
κ dµt (x, b, κ) + (1 − θ)
κdlt (x, b, κ) dµ0t (x, b, κ)
is aggregate depreciation. Note that (1 − θ)κ is the loss of capital when a firm is liquidated.
3.2.2
Equilibrium Characterization
The first-order condition of the household’s financial savings problem implies that the asset
price is
uC (Ct+1 , 1 − Lt+1 )
qt = β
.
(9)
uC (Ct , 1 − Lt )
Let vt0 (x, b, κ) be the value of a small firm before entry and exit decisions and let vt (x, b, κ)
be the value after these decisions. These values represent the marginal contributions of the
small firm to the household’s utility, measured in units of the period-t consumption/investment
good. That is, payments accruing in the next period t+1 are priced with the financial discount
factor qt . Therefore, the value of a continuing small firm satisfies the Bellman equation
0
(x0 , b0 , κ) ,
vt (x, b, κ) = max
πt (x, κ) − b + qt b0 + qt Ex0 |x vt+1
0
b
(10)
s.t. b0 ≤ θκ and πt (x, κ) − b + qt b0 ≥ 0 .
0
At the beginning of the next period, the continuation value is Ex0 |x vt+1
(x0 , b0 , κ) where the
expectation is taken over realizations of next period’s productivity x0 conditional on current
productivity x.
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The value of a firm at the beginning of period t and prior to exit decisions, vt0 (x, b, κ), equals
θκ − b when the firm is illiquid, i.e. if πt (x, κ) − b + qθκ < 0, or when vt (x, b, κ) < θκ − b in
which case the firm is voluntarily liquidated. Otherwise the firm remains active. This implies
that
(
θκ − b
if πt (x, κ) − b + qθκ < 0 or vt (x, b, κ) < θκ − b ,
vt0 (x, b, κ) =
(11)
vt (x, b, κ) else ,
with liquidation policy function
(
dlt (x, b, κ) =
1
0
if πt (x, κ) − b + qθκ < 0 or vt (x, b, κ) < θκ − b ,
else.
(12)
Regarding entry decisions, it is optimal to invest into a new firm (x, b, κ) in period t if the firm
value is greater than the installation cost net of debt, which leads to the entry policy function
(
det (x, b, κ) =
1
0
if vt (x, b, κ) ≥ κ − b ,
else.
(13)
Finally, the first-order conditions for labor supply and investment in corporate firms are
0 = uC (Ct , 1 − Lt )wt − u1−L (Ct , 1 − Lt ) ,
c
1 = qt 1 − δ + FK (Kt+1
, Lct+1 ) .
3.2.3
(14)
(15)
Firm Policies in Stationary Equilibrium
The borrowing and savings policies of small firms can be characterized as in Khan and Thomas
(2013). Firms with sufficiently high savings (low debt) are not threatened by illiquidity in any
future state. These unconstrained firms are able to pay positive dividends while keeping the
buffer stock of financial savings sufficiently high. On the other hand, if financial savings are
low (or debt is high), a firm may expect a future state of illiquidity (and hence forced exit) with
positive probability. These constrained firms value retained earnings higher than dividends;
therefore they pay no dividends until they build up enough savings and become unconstrained,
unless they exit before.
We describe the savings policies and value functions of both firm types in a stationary
equilibrium. The same logic applies for the transitional dynamics and is described in Appendix
E. In stationary equilibrium, the financial discount factor is q = β and the time index is dropped
12
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from all variables. Consider first the Bellman equations of unconstrained firms:
v(x, b, κ) = π(x, κ) − b + qb0 (x, b, κ) + qEx0 |x v 0 (x0 , b0 (x, b, κ), κ) ,
(16)
v 0 (x, b, κ) = max[θκ − b, v(x, b, κ)] ,
(17)
where the policy function b0 (x, b, κ) satisfies the two financial constraints (7) and ensures that
the firm remains unconstrained in the next period (as verified below).8 These two equations
demonstrate that value functions of unconstrained firms take the form v 0 (x, b, κ) = V 0 (x, κ)−b
and v(x, b, κ) = V (x, κ) − b (before and after exit). In words, the marginal value of financial
assets held by the firm equals one. This is because a marginal payout today has the same
value for the household owner as keeping these funds in the firm and receiving the payout
later on. It further follows from the first Bellman equation that unconstrained firms are
indifferent regarding the level of savings −b0 (x, b, κ), as long as they ensure that the firm
remains unconstrained in the future. Dropping financial savings from the Bellman equation
obtains
V (x, κ) = π(x, κ) + qEx0 |x max[θκ, V (x0 , κ)] .
(18)
The unique solution of this equation is strictly increasing in x (because π is strictly increasing in x and since the Markov process for x is monotone). This defines a cutoff productivity
level
x̃ (κ) = min{x ∈ X|V (x, κ) ≥ θκ}
such that all unconstrained firms with capital κ and productivity below x̃ (κ) liquidate the
firm, while all others stay.
To become and remain unconstrained, a firm needs to reduce financial debt below
b̃ (κ) =
π(x̃ (κ) , κ)
1−q
(or build savings exceeding −b̃(κ)). To see this, consider a firm with capital κ entering the
period with b ≤ b̃(κ). If this firm’s productivity is x ≥ x̃(κ), the firm can enter the next
period with debt b0 = b̃(κ) and pay non-negative dividends π(x, κ) − b + q b̃(κ) ≥ 0. If the
firm’s productivity is x < x̃(κ), the firm is voluntarily liquidated.9 In any case, the firm is
8
Equation (17) permits voluntary exit. By definition, unconstrained firms never become illiquid and hence
are never forced to exit.
9
For unconstrained firms, this follows from the above arguments. For constrained firms, v(x, b, κ) ≤
V (x, κ) − b < θκ − b, and hence liquidation is also optimal if x < x̃(κ).
13
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either voluntarily liquidated or the firm remains unconstrained in the next period (because of
b0 = b̃(κ)).
If a firm’s debt is larger than b̃(κ) (or savings are smaller than −b̃(κ)), the firm may become
illiquid in the future with positive probability. To see this, suppose that productivity remains at
x̃(κ) for a sufficiently long time. Then the zero-dividend borrowing policy b0 = 1q (b−π(x̃(κ), κ))
will over time lead to exploding levels of debt and eventually violate the borrowing constraint
(as would any other policy with positive dividends). For such constrained firms it is optimal to
pay zero dividends until savings exceed −b̃(κ) because the value of retained earnings exceeds
the one for a dividend payout. These considerations imply that the value and policy functions
of constrained firms are
v(x, b, κ) = max[0, π(x, κ) − b + q b̃ (κ)] + qEx0 |x v 0 (x0 , b0 (x, b, κ), κ) ,
(19)
v 0 (x, b, κ) = max[θκ − b, v(x, b, κ)] ,
1
0
b (x, b, κ) = max b̃ (κ), (b − π(x, κ)) ,
q
(20)
(21)
if π(x, κ) − b + qθκ ≥ 0, and v(x, b, κ) = v 0 (x, b, κ) = θκ − b otherwise.
3.3
Pandemic Shock and Rescue Policies
Suppose that the economy is in stationary equilibrium and that the COVID-19 pandemic shock
hits in period t = 0. The shock is a one-time unexpected event that fades out over time.10 We
assume that the pandemic not only affects productivity, but also the demand for goods and
services and the willingness of households to participate in the labor market.
Specifically, the shock has four components. First, a TFP shock νtc on the corporate sector
such that the corporate production function becomes (1 + νtc )F (Ktc , Lct ). Second, a shut-down
shock νtn on a fraction ηi of impacted small firms such that the firm productivity of impacted
firms becomes (1 + νtn )x. We interpret the impacted firms as those small firms producing social
goods and services that are particularly vulnerable to government-induced lockdown measures.
Third, a demand shock νtd such that marginal utility of consumption becomes
(1 + νtd )uC (Ct , 1 − Lt ). Such a preference shock captures the observed drop in aggregate
consumption at the onset of the pandemic in response to stay-at-home orders and increased
health risks of consuming social goods or services. Finally, a labor supply shock νt` such that
the marginal utility of leisure becomes (1 + νt` )u1−L (Ct , 1 − Lt ). This reflects employment
10
Since the shock is completely unforeseen, it has no impact on households and firms’ behavior ex-ante.
14
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adjustments possibly due to increased heath risks associated with in-person work.
We also take into account the PPP policy in the calibration. Because of the high forgiveness
rate of PPP loans (see Section 2), we model the rescue policy as a grant rather than a loan.
We assume that an exogenous η = 0.76 share of firms receive the grant in period t = 0 and
that the probability of receiving the grant is independent of whether the firm is impacted, thus
capturing the lack of targeting of the PPP policy.11 Moreover, the grant is unconditional and
does not need to be repaid. As a robustness check, we show in Appendix C that our main
results hold when the grant is conditional on payroll spending.
Let bp (x0 , κ) be the amount of the grant offered to a firm with capital κ and whose absentof-shock productivity is x0 in the impact period (t = 0). In line with the actual PPP policy,
we assume that the grant amount is equal to 10 weeks of payroll as follows:
bp (x0 , κ) = Xp w∗ `∗ (x0 , κ) ,
(22)
where Xp = 2.5/3 (our model period is a quarter), w∗ is the wage rate in the steady state, and
`∗ (x0 , κ) is labor demand in the steady state.
To model the grant, we only need to modify the profit of small firms on the transition path.
In t = 0, the profit function reads
π0 (x, κ, ι, s) = max(1 + ν0n ι)xf (κ, `) + sbp (x, κ) − w0 ` − δκ − cf (κ) ,
`≥0
(23)
where ι is a dummy variable indicating whether the firm is impacted by the pandemic and s
is a dummy variable for receipt of the grant. In t ≥ 1, we have
πt (x, κ, ι) = max(1 + νtn ι)xf (κ, `) − wt ` − δκ − cf (κ) .
`≥0
Note that it is always optimal for firms to choose to take up the maximum grant because
grants prevent liquidations (for constrained firms) or raise dividends (for unconstrained firms).
We provide further details and describe how we solve the model on the transition path in
Appendix E.2.
The government finances the rescue grant by imposing a lump-sum tax on households.
Since households own small firms who benefit from the rescue grant and since households are
perfectly insured, the fiscal cost of the grant has no direct bearing on household consumption
11
Using data from the Small Business Administration, Borawski and Schweitzer (2021) document that 76%
of U.S. small businesses received the PPP loan in 2020. In addition, PPP loans reached small businesses in all
industries without substantial variation.
15
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and labor supply decisions apart from the effects on equilibrium prices. In addition, since the
representative household is financially unconstrained and there is no distortionary taxation,
the timing of lump-sum taxation is irrelevant as Ricardian equivalence applies in our model.12
4
Calibration
4.1
Steady-State Calibration
We calibrate the model in steady state to the U.S. economy prior to the COVID-19 pandemic.
Each period in our model corresponds to a quarter of a calendar year. We define small firms
as businesses with fewer than 500 employees, which is the threshold for eligibility of the PPP
loans.
4.1.1
Functional Form Assumptions
In the corporate sector, the production function is
F (K c , Lc ) = A(K c )α (Lc )1−α .
In the non-corporate sector, each firm has the production technology
f (κ, `) = A(κγ1 `1−γ1 )γ2 ,
where γ1 ∈ (0, 1) is the capital share and γ2 ∈ (0, 1) is a span-of-control parameter. The
production function exhibits decreasing returns to scale, possibly due to diminishing returns
of managerial supervision as in Lucas (1978). The log productivity ln(x) follows an AR(1)
process with mean ln(x̄), standard deviation εx and autocorrelation ρx .
Among potential firm entrants, the distribution of initial productivity is independent from
the initial distribution of debt and capital. We assume that, conditional on κ, the initial debt
distribution is such that θκ − b is exponentially distributed with λ for all κ ∈ K.
We assume that the initial productivity is drawn from a log-normal distribution such that
ln(x0 x̄) and ε2x /(1 − ρ2x ) are, respectively, the mean and variance of ln(x0 x̄). x0 < 1 is a shift
parameter that captures a smaller size of startups compared to incumbent firms.
12
Of course, Ricardian equivalence can fail for many reasons, such as distortionary taxation. While studying
the timing of taxation and the implications of rising public debt during the COVID-19 pandemic is an interesting
issue, its analysis is beyond the scope of this paper.
16
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The utility function of the representative household is
C 1−σ
+ ζ(1 − L) .
U (C, 1 − L) =
1−σ
4.1.2
Calibration Strategy and Data
Table 1 shows the values of parameters that are determined outside of the model. The resale
value of capital θ is taken from Lanteri and Rampini (2021), which is in the range of estimated
values in the literature (Li et al., 2016). Following Jo and Senga (2019), we assume that
γ1 = 0.3182 and γ2 = 0.88.
Parameter
Description
Value
Source
β
α
δ
γ1
γ2
A
θ
x̄
σ
Subjective discount factor
Capital Share corporate sector
Capital depreciation rate
Capital Share small firms
Span of control
TFP shifter
Resale value of capital
Mean of ln(x)
CRRA utility coefficient
0.989
0.300
0.015
0.318
0.880
0.250
0.500
1.000
2.000
Annual interest rate of 4%
Standard
Annual depreciation rate of 6%
Jo and Senga (2019)
Jo and Senga (2019)
Normalization
Lanteri and Rampini (2021)
Normalization
Standard
Table 1: External Parameters
Table 2 lists parameters calibrated internally and the main data targets that help identifying
each parameter. It is well understood that all these parameters jointly take an impact on
various model statistics, but we can nonetheless determine which parameter mostly affects
which target. To calibrate the parameters, we compute the model counterparts of these targets
and choose parameter values to minimize the sum of squared percentage distances between
model and data moments. Appendix F provides additional information on how some of the
moments are calculated.
Most data targets are obtained from publicly available tables from the Statistics of U.S.
Businesses (SUSB) and the Business Dynamics Statistics (BDS) in 2010-2018. We use the
confidential data from the Kauffman Firm Survey (KFS) to calculate statistics involving firms’
balance sheet information and firm dynamics. The KFS is a single-cohort longitudinal dataset.
The first survey of KFS was conducted in 2004 on a representative sample of new firms. A
follow-up survey is conducted every year until 2011. We obtain fixed expenses from an analysis
by Sageworks based on Census 2007 data.
17
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18
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Marginal utility of leisure
Description
Mass of potential entrants
Productivity shifter of entrants
Initial debt distribution
Standard deviation
Autocorrelation
Capital levels
0.242
0.43; 4.92
15.3; 215.2
43.490
0.101
0.314
0.146
0.969
27.719
Value
Standard
Data Source
Average employment in small firms
Employment share by firm size
Firm size distribution
Fixed expense to revenue ratio
Firm exit rate by firm size
Small firm share of employment
Average employment in entrants
Debt to asset ratio, share of firms with debt
SUSB
SUSB
SUSB
Sageworks
BDS
SUSB
BDS
KFS
Small firm exit rate
BDS
Autocorrelation of employment in continuing small firms KFS
Time spent in market work
Main Target
Table 2: Internal Parameters and Calibration Targets
Notes: Our sources include the Business Dynamics Statistics (BDS), Statistics of U.S. Businesses (SUSB), Kauffman Firm Survey (KFS) and
data from Sageworks.
Φκ
Prob. of κ1
cf (κ1 ); cf (κ2 ) Fixed costs
κ1 ; κ2
Small firm technology
M
x0
λ
Entry and exit
εx
ρx
AR(1) of idiosyncratic productivity x
ζ
Preferences
Parameter
4.1.3
Parameters and Model Fit
Table 2 shows the calibrated parameter values, and Table 3 and Figure 2 show the fit of
targeted moments. Our model fits the data well. Although we use only two levels of fixed
capital, we generate variation of firm shares, employment shares and exit rates across four
firm-size classes. In particular, we match the fact that the distribution of small businesses
is heavily skewed towards firms with fewer employees (see Figure 2.a): Around 80% of small
businesses have less than 10 employees and only a small fraction, 1.7%, has more than 100
employees.13 We also match relatively well the employment shares by firm size bins, which are
more evenly distributed across the four bins. In addition, the exit rate in our model decreases
in firm size, as the data indicates. We further replicate the pattern that entering firms have
fewer employees.
Moment
Data
Model
Average employment in small firms 9.2516
Small firm share of employment
0.4895
Small firm exit rate
0.0198
Average employment in entrants
5.2935
Fixed expense to revenue ratio
0.2448
Autocorr. employment
0.9667
Debt to asset ratio
0.0820
Time spent in market work
0.3300
Share of firms with debt
0.3288
Exit rate by employment size bins
0 to 9
0.0246
10 to 19
0.0048
20 to 99
0.0033
100 to 499
0.0016
9.5054
0.5836
0.0244
5.6323
0.1821
0.9294
0.0831
0.2947
0.2732
0.0297
0.0040
0.0031
0.0000
Notes: The table shows some model statistics of the benchmark economy and the empirical counterparts based
on data from BDS, SUSB and KFS. Firm and employment shares by employment size are shown in Figure 2.
Table 3: Model Fit
To illustrate the dependence of exit and entry decisions on productivity and debt, we show
the optimal policy for the small firms with κ = κ2 in Figure 3.14 Incumbent firms with high
debt and low productivity are forced to liquidate because they are unable to pay a positive
dividend. Remaining low productivity firms voluntarily liquidate because the value of the
13
14
Note that we abstract from non-employer firms.
The policies are qualitatively similar for κ1 . We omit them for the sake of conciseness.
19
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0.8
0.4
Model
Data
0.7
Model
Data
0.35
0.6
0.3
0.5
0.25
0.4
0.2
0.3
0.15
0.2
0.1
0.1
0.05
0
0
0-9
10-19
20-99
100+
0-9
10-19
Size Bins
20-99
100+
Size Bins
(a) Firm Size Distribution
(b) Employment Distribution
Figure 2: Model Fit. Firm Size and Employment Distributions in the Small Firms Sector
(b) Entry Decision
7
7
6
6
5
Productivity, x
Productivity, x
(a) Exit Decision
Stay
4
3
Forced exit
2
5
Entry
4
3
2
1
Voluntary exit
0
10
20
30
40
50
60
70
No Entry
1
80
90
100
0
Debt, b
10
20
30
40
50
60
70
80
90
100
Debt, b
Figure 3: Small Firm Exit and Entry (κ = κ2 )
firm is below the liquidation value. The entry decision depends mainly on firm productivity.
Because of the partial irreversibility of capital investment, the productivity threshold for entry
is higher compared to the productivity thresholds for staying.
20
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4.2
Calibrating the COVID-19 Shock
The calibration of pandemic shocks is based on the scenario with the PPP grant (see Section
3.3). We assume that the shock decays exponentially with parameter ρ ∈ (0, 1). That is, for
each component of the shock s ∈ {c, n, d, `},
νts = ρt ν s for all t = 0, 1, ...
Since some small firms had to shut down under lockdown policies at the beginning of the
pandemic, we assume that the shock to impacted small firms is ν n = −1. We calibrate the
initial magnitudes of the other pandemic shocks (ν c , ν d , ν ` ) and the share of impacted small
firms ηi to match the magnitudes of impacts of the COVID-19 pandemic on total output,
consumption, employment and employment in small firms in the second quarter of 2020 (which
corresponds to the first period of the transition). We calibrate the decay parameter ρ to match
the change in total output in the third quarter of 2020.
Table 4 shows the values of calibrated shock parameters and Table 5 compares the pandemic
impact in the data to those in the model. Our calibration reveals that 11% of small firms
are impacted, so that the TFP shock on the corporate sector is smaller than the average
productivity shock in the non-corporate sector. There is a sizable preference shock such that
the marginal utility of consumption drops and the marginal disutility of working increases upon
impact. The baseline grant economy tracks the observed pandemic impact closely, including
the change in private investment and output in small firms, which we do not target in the
calibration procedure.
Parameter
Description
Value
ηi
νc
νd
νl
ρ
Fraction of impacted small firms
0.110
Productivity shock the corporate sector -0.007
Preference shock
-0.184
Labor supply shock
0.250
Autocorrelation
0.112
Table 4: Calibrated Pandemic Shock Parameters
Figure 4 shows the paths of calibrated shocks and aggregate variables in our model. The
most salient feature of Figure 4b is the speed of the recovery. With a calibrated persistence
ρ = 0.11, output bounces back to just 2.2% below the initial steady state level in the second
period of the pandemic after falling by almost 11% on impact. The figure also shows that our
21
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Description
Data
Grant
(Baseline)
Targeted
Output, 2020Q2:
Output, 2020Q3:
Consumption, 2020Q2:
Total employment, 2020Q2:
Employment small, 2020Q2:
-10.857
-2.246
-9.667
-12.850
-16.021
-11.351
-3.114
-9.781
-11.760
-14.600
Untargeted
Private investment, 2020Q2:
Small firm output, 2020Q2 :
-15.398
-15.650
-19.953
-14.938
Notes: The pandemic shocks are calibrated so that the “Grant baseline” economy matches the data. Data
sources: GDP, consumption, investment and aggregate employment are taken from FRED (fred.stlouisfed.org).
Employment by firm size comes from data provided by Cajner et al. (2020), who compute employment changes
by firm size using data from ADP. Small firm output is from Bloom et al. (2021).
Table 5: Pandemic Impact and Rescue Policies (% Change from 2020Q1)
(a) Calibrated Shocks
(b) Path of Aggregate Variables
0.4
2
0.2
0
-2
% Change
0
-0.2
-0.4
-0.6
-0.8
TFP shock, corp.
c
TFP shock, impacted small firms
Demand shock
2020Q3
-6
-8
-14
Output
Consumption
Employment
Employment in small firms
-16
2020Q2
2020Q3
-10
d
Labor supply shock
-1
2020Q2
n
-4
-12
l
2020Q4
2021Q1
Quarter
2020Q4
2021Q1
Quarter
Figure 4: Pandemic Shock Calibration
model captures the notable feature of the pandemic recession that consumption drops almost
as much as output, in line with the factual dynamics shown in Figure 1.
22
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10
2
5
0
-2
% change
% change
0
-5
-10
-15
-4
Baseline grant
Laissez-faire
-6
-8
-20
Baseline grant
Laissez-faire
-25
-10
-12
-30
0
2
4
6
8
10
12
14
16
18
0
20
2
4
(a) Small Firm Exit Rate
8
10
12
14
16
18
20
(b) Aggregate Employment
2
0.3
Baseline grant
Laissez-faire
0.2
0.1
0
-2
0
% change
% change
6
Time in transition, t
Time in transition, t
-0.1
-0.2
-0.3
-0.4
-4
Baseline grant
Laissez-faire
-6
-8
-0.5
-10
-0.6
-12
-0.7
0
2
4
6
8
10
12
14
16
18
0
20
2
4
6
8
10
12
14
Time in transition, t
Time in transition, t
(c) Aggregate Capital
(d) Aggregate Output
16
18
20
Figure 5: Impulse Response to the Pandemic Shock
5
Findings
5.1
Impact of the Rescue Grant
To understand the macroeconomic impacts of the rescue grant, we consider a counterfactual
laissez-faire economy which is absent of any government intervention. Figure 5 shows the
impulse response of the aggregate economy under the baseline policy environment with the
rescue grant and under the laissez-faire environment over a 20-quarter horizon.
Based on our calibrated model, in the absence of the rescue grant, the pandemic shock
induces an increase in the firm exit rate by almost 10% upon impact (Figure 5a). The baseline
grant reduces the exit rate by 25% compared to the pre-pandemic period. Overall, the baseline
grant is successful in preventing 35% of business exits. The reduction in business exits leads to
an increase of aggregate capital on impact (Figure 5c). This is driven by an increase in capital
in small firms, many of which decide to stay rather than exit, while the stock of corporate
23
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2
0
0
-1
-2
-2
-4
% change
% change
1
-3
-4
Baseline grant
Laissez-faire
-5
-6
-8
-6
-12
-7
-14
-8
Baseline grant
Laissez-faire
-10
-16
0
2
4
6
8
10
12
14
16
18
20
0
2
4
Time in transition, t
(a) Employment in the Corporate Sector
8
10
12
14
16
18
20
(b) Employment in Small Firms
0
1
-0.2
0.8
0.6
% change
-0.4
% change
6
Time in transition, t
-0.6
Baseline grant
Laissez-faire
-0.8
-1
Baseline grant
Laissez-faire
0.4
0.2
0
-0.2
-1.2
-0.4
-1.4
-0.6
0
2
4
6
8
10
12
14
16
18
20
0
Time in transition, t
2
4
6
8
10
12
14
16
18
20
Time in transition, t
(c) Capital in the Corporate Sector
(d) Capital in Small Firms
Figure 6: Impulse Response to the Pandemic Shock by Sector
capital is predetermined and does not respond immediately to the pandemic shock. However,
in subsequent quarters, aggregate capital falls below the pre-pandemic level and the recovery is
notably slower compared to the laissez-faire scenario. This suggests that the grant has mostly
a temporary benefit but has persistent adverse consequences on aggregate capital.
The reduction in business exits does not translate into significant improvements in aggregate
output or employment (Figures 5b and 5d) for two reasons. First, the grant reallocates labor
and capital away from the corporate sector. Figure 6 shows that, relative to the laissezfaire economy, the rescue grant leads to persistent declines of capital and employment in the
corporate sector and persistent increases of both factors in the non-corporate sector. Thus,
although the grant saves many small firms, it prevents the reallocation of production factors
towards the less impacted corporate sector, which ultimately mutes improvements in aggregate
output and employment.
The second reason for the modest responses of aggregate macro variables is that the grant
rescues mainly low-productive firms which employ only few workers. Figure 7 shows the change
24
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0
0.5
-0.005
Employment share in steady state
Average effect on exit rate
0.4
-0.01
0.3
-0.015
0.2
-0.02
0.1
-0.025
0
Average effect on exit rate
Employment share in steady state
0.6
-0.03
1
2
3
4
5
6
7
8
9
10
Deciles of x
Notes: Average effect on the exit rate (top bars, inverted scale, right axis) measures the difference in the
average quarterly exit rate between firms that receive the baseline grant and firms that do not receive the
baseline grant in the impact period (t = 0) in the baseline grant environment. Deciles of x are determined
according to the steady state distribution of all incumbent firms. Bottom bars show the employment shares in
steady state (left axis).
Figure 7: Impact of the Grant on the Exit Rate and Employment Shares by Firm Productivity
in the exit rate induced by the grant (relative to laissez-faire) for each decile of firm productivity
x and the corresponding share of total small firm employment in the steady state. While the
rescue grant reduces the exit rate in all firms, the effect is much stronger for low-productivity
firms that account for a disproportionately small share of aggregate employment. For example,
the rescue grant reduces the exit rate of firms in the first productivity decile by 2.6 percentage
points, but these firms account for only 1% of employment. As a result, the grant leads to a
reallocation of employment and capital within the small firm sector in favor of less productive
firms.
Figure 8 shows that small firms in the least productive decile gain the most in terms of
employment and capital under the baseline grant environment (top panels) compared to the
laissez-faire environment (bottom panels). As shown by the left panels, the grant policy moves
the firm distribution towards the least productive firms. Much of this shift does not happen
on impact but over several quarters and years following the pandemic shock. The other two
panels demonstrate that this shift goes along with a long-term reallocation of production
factors towards less productive businesses.
Another way to understand why saving firms has small aggregate consequences is that the
25
Electronic copy available at: https://ssrn.com/abstract=4064899
Decile 1
Decile 5
Decile 10
2.5
% change
% change
2.5
2
1.5
1
-5
-10
Decile 1
Decile 5
Decile 10
0.5
5
10
15
0
5
Time in transition, t
10
0
-0.1
-0.3
0
0
-2
-0.1
-6
-8
-10
Decile 1
Decile 5
Decile 10
-0.5
10
15
Time in transition, t
(d) Mass of Firms, Laissez-Faire
Decile 1
Decile 5
Decile 10
-12
-14
0
5
10
15
(c) Capital, Baseline Grant
% change
% change
% change
-0.2
10
Time in transition, t
-4
5
5
Time in transition, t
0
0
1
0
15
(a) Mass of Firms, Baseline Grant (b) Employment, Baseline Grant
-0.4
2
1.5
0.5
0
0
Decile 1
Decile 5
Decile 10
3
0
% change
3
-0.2
-0.3
-0.4
Decile 1
Decile 5
Decile 10
-0.5
15
0
5
Time in transition, t
10
15
Time in transition, t
(e) Employment, Laissez-Faire
(f) Capital, Laissez-Faire
Figure 8: Impulse Response by Decile of Small Firm Productivity x
pandemic impact on output is mainly channeled through the fall in firm productivity and labor
demand in impacted firms, while firm exits play a minor role on output. To illustrate this,
we decompose the output change in small firms in the first two years following the pandemic
shock (t = 0, ..., 7), denoted ∆Y , into three factors: (1) firm productivity change ∆T F P , (2)
labor demand adjustment ∆L and (3) changes in firm exits and entry ∆Exit . That is,
∆Y = ∆T F P + ∆L + ∆Exit ,
where
∆Y =
7 Z
X
[xt (ι)f (κ, `t (xt (ι), κ)) − cf (κ)] dµt (x, b, κ, ι) −
Z
[xf (κ, `(x, κ)) − cf (κ)] dµ(x, b, κ) ,
t=0
∆T F P =
7
X Z
[ηi xt (1) + (1 − ηi )xt (0) − x]f (κ, `(x, κ)) dµ(x, b, κ) ,
t=0
∆L =
7 Z
X
ηi xt (1)[f (κ, `t (xt (1), κ)) − f (κ, `(x, κ))]
t=0
+(1 − ηi )xt (0)[f (κ, `t (xt (0), κ)) − f (κ, `(x, κ))] dµ(x, b, κ)] ,
26
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∆Exit =
7 Z
X
[xt (ι)f (κ, `t (xt (ι), κ)) − cf (κ)] dµt (x, b, κ, ι)
t=0
Z
−
[ηi xt (1)f (κ, `t (xt (1), κ)) + (1 − ηi )xt (0)f (κ, `t (xt (0), κ)) − c (κ)] dµ(x, b, κ) ,
f
with xt (ι) ≡ (1 + νtn ι)x defined as the productivity of firm x in period t on the transition path
given the impact status ι. `t and µt are, respectively, labor demand and the firm distribution
in period t on the transition path, whereas ` and µ are the steady-state counterparts. ∆T F P
is the effect of the shutdown shock (νtn ) on small-firm output while holding labor demand and
the firm distribution at their steady state levels. ∆L is the effect of labor demand adjustment
while holding firm productivity at the pandemic levels and the firm distribution at the steady
state level. Finally, ∆Exit is the effect of changes in the firm distribution, which is in turn
driven by changes in firm exits and entry, while holding firm productivity and labor demand
at their pandemic levels.
We conduct the decomposition exercise in the laissez-faire environment and find that ∆T F P
and ∆L account for 77% and 13% of total pandemic impact on small firm output, respectively.
By contrast, ∆Exit accounts for just below 10% of the total impact. This leaves little room
for a pandemic rescue grant to improve small firm output because the main channel through
which the grant impacts small firms is by reducing firm exits.15
In our baseline policy scenario, the grant is available without conditions. We consider this
as a reasonably approximation to the factual PPP policy whose forgiveness requirements were
quite loose from the beginning (see Section 2). As a robustness experiment, we also analyze the
case of a conditional grant which imposes a minimum employment requirement in the period
the grant is received. Results are presented in Appendix C. In comparison to the baseline
grant, the conditional grant mitigates the employment loss in the impact quarter while the
initial decline of the firm exit rate is a bit smaller. Intuitively, while jobs in impacted firms are
saved under the conditional grant, other firms now prefer to exit. Importantly, the differences
between the two policies are rather small and the main reallocation effects are also observed
in the alternative scenario.
5.2
Targeted Rescue Policy
The PPP aids were dispersed in a highly timely fashion, but there is a lack of targeting such
that the take-up of the loans is almost universal (Autor et al., 2022). The lack of targeting
15
The grant also has a small effect on the real wage and on the interest rate (see Figure 15 in Appendix
D.1), which in turn affects labor demand of small firms. However, this is of secondary importance relative to
the direct labor demand response to the pandemic shock.
27
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has been criticized for incurring an unprecedented fiscal cost and the potential of tying up
resources in unproductive firms that could find more productive use elsewhere in the economy.
In this subsection, we quantify the economic tradeoff of targeting rescue grants.16 To this end,
we consider a counterfactual rescue grant that targets only impacted firms. Other aspects of
the policy, including grant amount and timing, remain the same as the baseline rescue grant.
Since only a fraction ηi = 11% of small firms is shut down in the first period, the fiscal cost of
the grant is now substantially reduced. The first row of Table 6 shows that the baseline grant
is roughly 7 times more expensive than the targeted grant.
Baseline grant
Fiscal cost (frac. GDP)
Small-firm emp. saved over 10 years
Cost per perc. emp. saved (frac. GDP)
6.28%
0.53%
11.8%
Targeted grant Targeted/Baseline
0.91%
0.12%
7.4%
14.5%
22.6%
62.7%
Notes: The fiscal cost is computed as a fraction of GDP in the steady state. Employment saved is computed as
the per-quarter difference in small-firm employment relative to the laissez-faire economy over a 10-year period
from the onset of the pandemic. The last column shows the ratio between the first two columns.
Table 6: Cost per Job Saved in Small Firms
The targeted grant leads to a more limited improvement in small firm employment over the
laissez-faire economy: over a 10-year horizon, the average employment improvement under the
targeted grant is only 0.12%, less than a quarter of that under the baseline grant. To compare
the cost-effectiveness of the two rescue grants, we compute the cost of saving one percent of
job-quarters under the two rescue policies by taking the ratio of the first two rows of Table 6.
We find that the cost of saving 1% of jobs (over a 10-year period) under the targeted grant is
63% of the cost under the baseline grant.
As another robustness experiment, we vary the size of the targeted grant in order to examine
how the number of jobs saved and the cost effectiveness change with the size of the grant. While
a much larger targeted grant saves more jobs, it becomes much less cost effective. This is
because a larger grant provides windfall gains to those impacted firms which are already saved
with a smaller grant while the number of saved firms diminishes at the margin. Conversely, a
targeted grant half as large as considered in the benchmark is a bit more cost effective, but it
saves about 40% fewer jobs (see Appendix B for details).
16
Our analysis does not account for the potential administrative costs associated with granting targeted aids
in a timely fashion. In addition, we only focus on employment in small firms here.
28
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5.3
Zombie Firms
The rescue policy prevents the liquidation of otherwise viable firms that do not have access to
external funding at the time of the shock. On the other hand, it might inadvertently prolong
the life of unproductive firms whose capital may be more valuable in other (corporate or noncorporate) firms. Such adverse consequences might be expected, especially because the rescue
grant is untargeted. In this section we use our model to quantify the risk of generating such
zombie firms.
Specifically, a firm is saved by a grant in period t if the firm chooses to exit in the laissezfaire environment but stays active in the grant environment. We call a saved firm a zombie
if the firm’s liquidation value is larger than its hypothetical value if financial constraints were
not present. In other words, if the rescue policy not merely alleviates financial constraints but
instead prevents a more efficient reallocation of capital, the firm survives although it is socially
inefficient that it does.
Formally, a firm with productivity x, capital κ, and impact status ι ∈ {0, 1} is a zombie
firm in t if Vt (x, κ, ι) < θκ, where Vt is the value of an unconstrained firm (net of financial
assets) in period t. As further outlined in Section 3.2.3 and in Appendix E.2, this value satisfies
the recursion
0
Vt (x, κ, ι) = max (1 + ινtn )xf (x, `) − wt ` − δκ − cf (κ) + qt Ex0 |x Vt+1
(x0 , κ, ι) .
`≥0
The zombie condition Vt (x, κ, ι) < θκ says that the value of liquidated capital exceeds the
expected discounted profit value of the firm, so that the household would be better off if
capital were installed somewhere else.
Table 7 shows the fraction of saved firms and the fraction of zombies among saved firms
on impact under the baseline grant policy and under the counterfactual targeted grant policy.
Upon impact, 16.6% of firms saved by the baseline grant are zombies. In contrast, the targeted
grant saves as many firms as the baseline grant, while it only creates a negligible number of
zombies.
The reason why the targeted grant policy creates substantially less zombie firms is that
the targeted policy saves in proportion many more high-productivity firms, namely all those
impacted firms which would exit without the grant. While these firms are also saved with
the untargeted grant, the latter policy also saves many low-productivity firms that are not
impacted by the pandemic, a non-negligible fraction of which turn out to be zombie firms.
29
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Baseline grant Targeted grant
Fraction of saved firms
Fraction of zombie firms
42.5%
16.6%
43.0%
1.3%
Notes: Fraction of saved firms is the measure of saved firms in period t = 0 divided by the measure of firms
that exit under the laissez-faire environment. Fraction of zombie firms is the measure of zombie firms divided
by the measure of saved firms. See the text for definitions of saved firms and zombie firms.
Table 7: Zombie Firms
5.4
Short- and Long-Run Effects
The impulse response functions in Section 5.1 highlight the persistent effects of the rescue grant
on aggregate capital and on the sectoral allocation of labor and capital. To better understand
the impact of the rescue grant over time, we decompose the cumulative effect of the pandemic
into short-run, medium-run and long-run effects and compare the baseline policy environment
with the two counterfactual environments: laissez-faire and targeted grant.
Figure 9 shows the result of the decomposition. For each policy, the net-height of the bar
represents the overall average quarterly effect of the pandemic over a ten-year period. The
three components representing short-run, medium-run and long-run display the decomposition
of the overall average.17
Figures 9a and 9b show that, in the absence of government intervention, the pandemic
leads to a short-run spike in firm exits and a persistent negative impact on the mass of small
firms. While the targeted grant largely eliminates these impacts, the baseline grant reduces
firm exits not only on impact, but also in the medium and long run, resulting in a highly
persistent increase in the mass of small firms. The long-run effect on firm survival stems from
the fact that the rescue grant improves the balance sheet of its recipients, including those that
do not face the immediate risk of exiting.
Focusing on output in the two sectors, Figure 9e shows that the short-run output loss in
small firms is similar across the three policy environments. This is because the output loss
is mainly due to the productivity loss and labor demand adjustments, while firm exits play a
less important role (see Section 5.1). Figure 9h shows that the short-run output loss in the
corporate sector is also common across the three policy environments. Here the reason is that
the short-run response is mostly due to falling TFP and employment, while capital adjusts
only little in the first two quarters.
In contrast, the medium- and long-run output effects of the pandemic exhibit quite different
patterns under the three policy environments. In the laissez-faire environment, the forced
17
A detailed explanation of how this decomposition is performed is given in Appendix D.2.
30
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0.5
0.8
Short-Run
Medium-Run
Long-Run
-0.5
% Impact
0
% Impact
Short-Run
Medium-Run
Long-Run
0.6
-1
0.4
0.2
0
-1.5
ire
-Fa
ez
L
s
ais
eg
elin
as
ran
-0.2
t
t
Ta
B
-Fa
ez
iss
La
(a) Small Firm Exit Rate
line
se
n
gra
Ba
(b) Mass of Small Firms
0.6
Short-Run
Medium-Run
Long-Run
0.4
% Impact
0.4
0.2
0
Short-Run
Medium-Run
Long-Run
0.6
0.2
0
-0.2
-0.2
ire
-Fa
ez
iss
La
eg
elin
as
B
ran
t
g
ted
rge
ran
-0.6
t
0.2
0
-0.2
-0.4
-0.4
-0.4
-0.6
nt
nt
gra
gra
ted
line
e
e
s
rg
Ba
Ta
e
ire
air
-F
ez
iss
La
Ta
(c) Small Firm Capital
(d) Small Firm Employment
0.2
0
0
-0.2
-0.2
% Impact
-0.8
-1
% Impact
0.2
Short-Run
Medium-Run
Long-Run
-0.4
-0.6
Short-Run
Medium-Run
Long-Run
-0.8
-1.2
ire
-Fa
ez
iss
La
B
eg
elin
as
ran
t
g
ted
rge
ran
-1
t
nt
nt
gra
gra
ted
line
e
e
s
rg
Ba
Ta
Ba
iss
La
Ta
(f) Corporate Capital
ran
dg
te
rge
Ta
-0.6
Short-Run
Medium-Run
Long-Run
-0.8
-1
e
(g) Corporate Employment
t
nt
gra
-0.4
ire
air
-F
ez
line
se
(e) Small Firm Output
0
-0.4
-Fa
ez
iss
La
-0.2
-0.6
Short-Run
Medium-Run
Long-Run
0.4
% Impact
0.6
% Impact
dg
ete
arg
T
0.8
% Impact
t
ran
t
ire
ran
g
ted
rge
-Fa
ez
iss
La
line
se
Ba
t
nt
gra
ran
dg
te
rge
Ta
(h) Corporate Output
Notes: The net height of the bars represents the average quarterly effect of the pandemic over 10 years.
Short-run: first two quarters; medium-run = quarters 3 to 12; long-run = quarters 13 to 40.
Figure 9: Short- and Long-Run Effects
liquidation of small firms and the slow recovery of the non-corporate sector allow the corporate
sector to absorb more labor to expand its production (see Figure 9g), which explains the longrun positive effect of the pandemic on corporate output. Furthermore, the exit of small firms
in the laissez-faire scenario has long-lasting consequences since it takes time for new firms to
enter, which slows down the build-up of small firm capital.
The baseline grant results in markedly larger long-run impacts compared to the targeted
31
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grant environment. Specifically, the baseline grant leads to a persistent positive effect on small
firm output and an opposite impact on corporate output. This is because the grant, due to
its improvement on balance sheets of financially constrained small firms, preserves capital in
the non-corporate sector at the expense of capital in the corporate sector. This reallocation of
capital plays out in the medium- and long-run, and hence has long-lasting consequences which
are also seen in the employment and output changes.
Turning our attention to aggregate variables, Figure 10 shows that the pandemic shock
has mostly a short-run impact on output, employment and consumption under all scenarios
considered. Most of the drop in consumption happens in the short-run in all three cases,
largely due to the sizable but short-lived demand shock. The baseline grant has a positive,
albeit small, effect on consumption in the medium- and long-run. The reason is the following:
the baseline grant prevents firm liquidation and therefore the household suffers less from losses
of capital liquidation, leaving more resources for consumption and savings. At the same time,
the discount factor falls by less (i.e., the increase of the interest rate is dampened), thus
inducing the household to save less and consume more in comparison to the laissez-faire (and
targeted grant) scenarios.
For these reasons, aggregate investment is considerably lower in the medium- and longrun, compared to the laissez-faire environment. The improvement in firms’ balance sheets has
long-lasting dampening effects on firm exit. As the household continues to hold and replace
depreciated capital in these firms, investment in the corporate sector or for the build up of new
small firms is reduced. In turn, the baseline grant reduces aggregate employment and output
in the medium- and long-term because lower investment in the corporate sector eventually
outweighs the capital preservation in smaller firms.
6
Conclusions
The COVID-19 pandemic caused a deep but short-lived economic recession, which prompted
many governments to enact massive rescue policies targeted at small firms. In order to evaluate
the macroeconomic impact of such policies, we build a general equilibrium model where firms
are subject to financial constraints and can only liquidate their capital at a loss.
Based on our calibrated model, we find that an unconditional grant policy, such as the PPP
enacted in the U.S. in the year 2020, is highly effective in preventing the exit of small businesses,
but has only a modest impact on aggregate outcomes such as consumption, employment and
output. This happens for two reasons. First, the grant mostly prevents less productive small
firms from exiting. Second, it reallocates resources towards firms that have lower productivity
32
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0.002
0.05
0
% Impact
% Impact
0
-0.05
Short-Run
Medium-Run
Long-Run
-0.1
-0.15
-Fa
ez
elin
as
s
is
La
B
T
e
arg
te
line
se
Ba
gra
ted
0.1
% Impact
0
Short-Run
Medium-Run
Long-Run
-0.3
0
-0.1
-0.2
Short-Run
Medium-Run
Long-Run
-0.3
-0.4
-0.4
t
t
ran
ran
dg
eg
ete
elin
g
s
r
Ba
Ta
ire
-Fa
ez
s
is
La
nt
gra
rge
Ta
(b) Price of Financial Asset, qt
0.2
-0.2
nt
e
air
-F
ez
iss
La
0.1
-0.1
Short-Run
Medium-Run
Long-Run
-0.008
ran
dg
(a) Wage, wt
% Impact
-0.004
-0.006
t
t
ran
eg
ire
-0.002
e
air
z-F
e
iss
La
(c) Aggregate Output
line
se
Ba
t
nt
gra
ted
n
gra
e
arg
T
(d) Aggregate Employment
1
0.5
-0.05
0
-0.15
-0.2
% Impact
Short-Run
Medium-Run
Long-Run
% Impact
% Impact
-0.1
0
-0.5
-0.25
Short-Run
Medium-Run
Long-Run
-0.3
-0.35
-Fa
ez
iss
La
ire
Ba
ran
se
g
line
T
t
e
arg
ran
t
g
ted
(e) Aggregate Consumption
-1
t
nt
ran
gra
eg
ed
n
i
t
l
e
se
rg
Ba
Ta
Short-Run
Medium-Run
Long-Run
-1
-2
-3
e
air
-F
ez
iss
La
z-F
L
se
ais
t
t
e
ran
ran
air
eg
lin
se
Ba
dg
te
rge
Ta
(f) Aggregate Investment, It (g) Corporate Investment, Itc
Figure 10: Short- and Long-Run Effects (Other Variables)
at the expense of more productive and less impacted corporate and non-corporate firms. This
result echoes previous findings in the literature (see Crouzet and Mehrotra, 2020) showing that
while small firms are typically hit harder during recessions, their volatility has only a small
effect on aggregate fluctuations and that policies relaxing financing constraints are unlikely to
generate a sizable macroeconomic impact.
We also find that much of the reallocation of capital and labor across firms occur after
the pandemic episode is over, although the grant policy is only applied when the shock hits
33
Electronic copy available at: https://ssrn.com/abstract=4064899
the economy. This is because the grant has a lasting impact on the financial position of small
firms, thus preventing firm exits for many years. As a consequence, the reallocation of capital
and employment across firms is very persistent due to the policy.
This is much different when the universal grant is replaced by a targeted grant policy that
specifically targets impacted firms. With a targeted grant, there is no persistent decline of firm
exit, and the reallocation of capital across firms is much more muted. Moreover, the targeted
grant policy largely prevents the creation of zombie firms which are those businesses whose
liquidation would be socially beneficial but is prevented due to the grant.
By focusing on the impact of business grants on firm entry, exit and factor reallocation,
our paper does not analyze how government-subsidized loans, possibly in combinations with
different conditions for partial loan forgiveness, would affect business dynamics and macroeconomic aggregates. We also abstract from public debt and the timing of taxation by assuming
that the government grants are financed by lump-sum taxes. Both are highly-relevant issues
for future research.
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Appendix
A
PPP Loan Terms
Eligibility The following requirements apply to first draw PPP loans:
• The business was operational before February 15, 2020, and is still open and operational.
• There are no more than 500 employees. If a business has multiple locations, there must
be no more than 500 employees per location.
Loan Amount The maximum amount of a PPP loan is determined based on the average
monthly payroll up to a threshold. The maximum amount of first draw PPP loans is 2.5 times
the average monthly payroll up to $10 million. In the Accommodation and Food services
sector, it is 3.5 times the average monthly payroll up to $10 million. For non-employers, the
maximum amount is based on the business net profit.
Interest Rate PPP loans have an interest rate of 1%.
Maturity PPP loans issued after June 5, 2020, have a maturity of five years. PPP loans
issued prior to June 5, 2020, have a maturity of two years. However, by mutual agreement,
the maturity can be extended to 5 years.
Covered Period The covered period is the 8- or 24-week after loan disbursement. The
24-week period applies to all borrowers that received forgiveness prior to December 27, 2020,
but borrowers that received an SBA loan number before June 5, 2020, have the option to use
an eight-week period.
Conditions for Loan Forgiveness Full loan forgiveness is possible if, during the 8- or
24-week covered period following loan disbursement, all of the following are achieved:
• Employee and compensation levels are maintained (temporary layoff is allowed if the
workers are rehired),
• all loan proceeds are spent on payroll costs and other eligible expenses, and
• at least 60% of the proceeds are spent on payroll costs.
38
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Partial Forgiveness “If a borrower uses less than 60% of the loan amount for payroll costs
during the forgiveness covered period, the borrower will continue to be eligible for partial loan
forgiveness, subject to at least 60 percent of the loan forgiveness amount having been used for
payroll costs.”18
B
Targeted Grant
In addition to the baseline grant and the targeted grant described in Section 5.2, we consider
two additional targeted grant policies with different grant amounts. In “targeted grant small”,
we set Xpsmall = 0.5Xp and in “targeted grant large”, we set Xplarge = 5Xp , where Xp is amount
of the grant as a fraction of the quarterly payroll in the baseline grant and the targeted grant
scenarios in Section 5.2.
Table 8 compares the cost-effectiveness of the baseline grant to three targeted grants. As
the grant amount increases, the targeted grant becomes less cost-effective. In particular, a
targeted grant which is almost as costly as the untargeted baseline grant turns out to be much
less cost effective. Intuitively, while a larger targeted grant saves more impacted firms (and
jobs in these firms), it does so with a diminishing impact at the margin. At the same time,
larger grants are merely windfall gains for those firms that are already saved with much smaller
government transfers.
Cost (Frac. GDP)
Emp. save (Frac. Emp)
Cost per perc. jobs saved
Baseline grant
Targeted grant
small
Targeted grant
Targeted grant
large
6.28%
0.53%
11.8%
0.45%
0.07%
6.1%
0.91%
0.12%
7.4%
4.55%
0.30%
15.3%
Notes: The fiscal cost is computed as a fraction of GDP in the steady state. Employment saved is computed as
the per-quarter difference in small-firm employment relative to the laissez-faire economy over a 10 year period
from the onset of the pandemic.
Table 8: Cost per Job Saved in Small Firms
C
Conditional Grant
As a robustness check, we assume that the grant comes with a minimum employment requirement in period t = 0 that resembles the condition for forgiveness in the PPP program (see
18
Source: https://home.treasury.gov/news/press-releases/sm1026
39
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Appendix A).19 That is, firms that choose to take up the grant must maintain employment at
60% of the pre-pandemic level. The grant is only given in period t = 0, as in our benchmark
model, and the minimum employment condition only applies in period t = 0. This is consistent
with the length of the covered period in the PPP program that ranges from 8 to 24 weeks.
Other than the minimum employment requirement, all policy assumptions are the same as in
Section 3.3.
Specifically, an η fraction of small firms have the opportunity to take up the grant. They
choose to do so if v0 (x, b, κ, ι, s = 1) ≥ v0 (x, b, κ, ι, s = 0), where s ∈ {0, 1} indicates grant
take-up and ι ∈ {0, 1} is the impact status. The profit of a firm that takes up the grant in
t = 0 changes from equation (23) to
π0 (x, b, κ, ι, s = 1) = max(1 + ν0n ι)xf (κ, `) + bp (x, κ) − w0 ` − δκ − cf (κ)
`
such that
` ≥ 0.6 × `(x, κ)
where bp (x, κ) is the grant amount defined in equation (22), `(x, κ) is the labor demand in the
steady state, and 0.6 × `(x, κ) is the minimum employment requirement.
We simulate the economy with the conditional grant using the same calibrated parameter
values reported in Section 4.2. The results indicate that, although the minimum employment
condition stimulates short-run employment in small firms, the conditional grant delivers similar
medium- and long-run impacts on aggregate output, employment, and capital as the baseline,
unconditional grant. Thus, the main conclusions of the paper carry over. The conditional
grant modeled here can be viewed as an extreme case because, in practice, the conditionality
of the PPP loan forgiveness is not strictly enforced and partial forgiveness is possible.
Figures 11 and 12 show the impulse responses to a number of variables of interest, comparing the laissez-faire to the conditional grant. Figures 13 and 14 show the short- and long-run
impact of the conditional grant policy. In comparison to the impulse responses under the
unconditional grant, the conditional grant dampens the initial employment decline while mitigating the initial decline in firm exits. This is because employment in staying impacted firms
remains higher while other impacted firms prefer to exit rather than accepting the conditional
grant. From period t = 1 onward, however, the observed reallocation effects are similar to
those under the unconditional grant scenario.
19
For simplicity, we do not model the option of not fulfilling the condition, thus abandoning forgiveness.
40
Electronic copy available at: https://ssrn.com/abstract=4064899
10
2
Conditional grant
Laissez-faire
5
0
Conditional grant
Laissez-faire
-2
% change
% change
0
-5
-10
-4
-6
-15
-8
-20
-10
-25
-12
0
2
4
6
8
10
12
14
16
18
20
0
2
4
Time in transition, t
6
8
10
(a) Small Firm Exit Rate
14
16
18
20
(b) Aggregate Employment
2
0.2
Conditional grant
Laissez-faire
0.1
0
0
Conditional grant
Laissez-faire
-2
% change
-0.1
% change
12
Time in transition, t
-0.2
-0.3
-4
-6
-0.4
-8
-0.5
-10
-0.6
-12
-0.7
0
2
4
6
8
10
12
14
16
18
0
20
2
4
6
8
10
12
14
16
18
20
Time in transition, t
Time in transition, t
(c) Aggregate Capital
(d) Aggregate Output
Figure 11: Impulse Response to the Pandemic Shock (Conditional Grant)
41
Electronic copy available at: https://ssrn.com/abstract=4064899
1
2
0
0
-1
-2
Conditional grant
Laissez-faire
Conditional grant
Laissez-faire
-4
% change
% change
-2
-3
-4
-6
-8
-5
-10
-6
-12
-7
-14
-8
-16
0
2
4
6
8
10
12
14
16
18
20
0
2
4
Time in transition, t
6
8
10
12
14
16
18
20
Time in transition, t
(a) Employment in the Corporate Sector
(b) Employment in Small Firms
1
0
0.8
-0.2
0.6
Conditional grant
Laissez-faire
-0.4
-0.6
% change
% change
0.4
Conditional grant
Laissez-faire
-0.8
0.2
0
-0.2
-1
-0.4
-1.2
-0.6
-1.4
-0.8
0
2
4
6
8
10
12
14
16
18
20
0
Time in transition, t
2
4
6
8
10
12
14
16
18
20
Time in transition, t
(c) Capital in the Corporate Sector
(d) Capital in Small Firms
Figure 12: Impulse Response to the Pandemic Shock by Sector (Conditional Grant)
42
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0.4
0.6
0.2
0.5
Short-Run
Medium-Run
Long-Run
0.4
-0.2
% Impact
% Impact
0
-0.4
Short-Run
Medium-Run
Long-Run
-0.6
0.3
0.2
0.1
-0.8
0
-1
-0.1
-1.2
-0.2
e
air
z-F
se
ais
L
nt
al
n
itio
nd
Co
gra
iss
La
(a) Small Firm Exit Rate
ez
-Fa
t
ire
C
(b) Mass of Small Firms
0.6
0.6
0.4
0.5
Short-Run
Medium-Run
Long-Run
Short-Run
Medium-Run
Long-Run
0.2
% Impact
0.2
% Impact
% Impact
0.4
0
0
-0.2
0
Short-Run
Medium-Run
Long-Run
-0.2
-0.4
-0.6
ire
-Fa
ez
iss
La
l
na
n
gra
-0.5
t
itio
nd
Co
z
sse
-Fa
ire
i
La
(c) Small Firm Capital
l
na
gra
nt
iss
io
dit
n
Co
La
(d) Small Firm Employment
ez
-Fa
ire
al
ion
dit
n
Co
gra
nt
(e) Small Firm Output
0.4
0
Short-Run
Medium-Run
Long-Run
0.2
-0.2
0
Short-Run
Medium-Run
Long-Run
-0.6
-0.2
% Impact
-0.4
% Impact
0
% Impact
ran
lg
na
itio
d
on
-0.2
-0.4
-0.4
Short-Run
Medium-Run
Long-Run
-0.6
-0.6
-0.8
-0.8
-1
-0.8
-1
-F
ez
iss
La
e
air
l
na
itio
nd
Co
nt
gra
(f) Corporate Capital
iss
La
ez
e
air
-F
ion
Co
it
nd
al
gra
nt
(g) Corporate Employment
iss
La
ez
-Fa
ire
al
ion
Co
it
nd
gra
nt
(h) Corporate Output
Notes: The net height of the bars represents the average quarterly effect of the pandemic over 10 years.
Short-run: first two quarters; medium-run = quarter 3 to 12; long-run = quarter 13 to 40.
Figure 13: Short- and Long-Run Effects (Conditional Grant)
43
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0.05
Short-Run
Medium-Run
Long-Run
0
-0.002
% Impact
% Impact
0
-0.05
-0.004
-0.006
-0.1
La
z-F
-0.008
t
e
air
e
iss
nd
Co
na
itio
lg
Short-Run
Medium-Run
Long-Run
t
ire
ran
(a) Wage, wt
ran
-Fa
ez
iss
La
ion
it
nd
g
al
Co
(b) Price of financial asset, qt
0.2
0
0.1
-0.05
Short-Run
Medium-Run
Long-Run
0
% Impact
% Impact
-0.1
-0.15
-0.2
-0.1
Short-Run
Medium-Run
Long-Run
-0.2
-0.25
-0.3
-0.3
-0.35
-0.4
-0.4
ire
-Fa
ez
s
ais
L
C
d
on
n
itio
al
gra
nt
iss
La
(c) Aggregate Output
ez
-Fa
d
on
-0.2
0.5
0
-0.2
-0.4
0
-0.5
-1
-0.25
-0.6
-0.3
-0.8
-2
-0.35
-1
-2.5
z-F
-1.5
t
e
air
se
is
La
a
ion
dit
lg
air
ran
n
Co
(e) Aggregate Consumption
Short-Run
Medium-Run
Long-Run
1
% Impact
-0.15
% Impact
% Impact
Short-Run
Medium-Run
Long-Run
ran
1.5
Short-Run
Medium-Run
Long-Run
0.2
-0.1
lg
(d) Aggregate Employment
0.4
-0.05
na
itio
C
0.6
0
t
ire
z-F
t
e
Co
n
io
dit
t
e
air
ran
lg
na
se
is
La
z-F
na
se
is
La
Co
io
dit
lg
ran
n
(f) Aggregate Investment, It (g) Corporate Investment, Itc
Figure 14: Short- and Long-Run Effects (Conditional Grant)
44
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D
Further Details
D.1
Additional Figures
10
5
0
0
% change
% change
-10
-20
Baseline grant
Laissez-faire
-30
-40
-5
Baseline grant
Laissez-faire
-10
-15
-50
-60
-20
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
Time in transition, t
(a) Corporate Sector Investment, Itc
10
12
14
16
18
20
18
20
(b) Aggregate Investment, It
2
0.04
Baseline grant
Laissez-faire
Baseline grant
Laissez-faire
0.03
% change
1.5
% change
8
Time in transition, t
1
0.5
0
0.02
0.01
0
-0.01
-0.5
-0.02
0
2
4
6
8
10
12
14
16
18
20
0
2
Time in transition, t
4
6
8
10
12
14
16
Time in transition, t
(c) Wage, wt
(d) Price of Financial Asset, qt
Figure 15: IRFs: Investment and Prices
D.2
Short- and Long-Run Decomposition
We decompose the total cumulative effect of the pandemic shock in a 10-year period into shortss
run, medium-run, and long-run effects. Specifically, let irft = yty−y
be the percentage effect
ss
of the pandemic on variable y in period t, and let irf be the average effect of the pandemic
on a variable over a period of TLR periods, i.e.
irf =
TLR −1
1 X
irft .
TLR t=0
45
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We compute irf SR , irf M R and irf LR such that
irf = irf SR + irf M R + irf LR ,
where
TX
−1
1 SR
irft ,
irf SR =
TLR t=0
T R −1
1 MX
irft ,
irf M R =
TLR t=T
SR
irf LR =
TLR −1
1 X
irft .
TLR t=T
MR
We consider the short-run to be the first two quarters post-pandemic, the medium-run to
be quarter 3 to year 3, and the long-run to be year 4 to year 10.
E
Computational Appendix
E.1
Computation of the Stationary Equilibrium
The financial discount factor is q = β. The capital-labor ratio in the corporate sector is pinned
down from 1 = β[1 − δ + FK ]. Using the functional form F (K c , Lc ) = A(K c )α (Lc )1−α , we
obtain
"
c α−1 #
K
1 = q 1 − δ + Aα
,
Lc
from which we obtain the capital-labor ratio in corporate firms
k∗ =
c
K
=
Lc
Aα
1
! 1−α
1
−1+δ
q
.
The real wage follows from w = FL , i.e.
w = A (1 − α) (k ∗ )α .
Consumption follows from the first order condition uC (C, 1 − L) w = u1−L (C, 1 − L), i.e.
Cσ =
w
.
ζ
46
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This permits the calculation of π (x, κ) = max`≥0 xf (κ, `) − w` − δκ − cf (κ). The optimal labor
demand can be derived as
` (x, κ) =
w
xA (1 − γ1 ) γ2
(1−γ 1)γ −1
1
2
−
γ1 γ2
κ (1−γ1 )γ2 −1 .
Given π (x, κ), V (x, κ) can be obtained from (18), which allows us to compute x̃(κ), and
b̃(κ) = π(x̃(κ), κ)/(1 − q). Discretize the interval of relevant debt levels for each κ as [b̃(κ), θκ]
such that b = b̃(κ) (unconstrained firms) is one point, while (b̃(κ), θκ] is cut into intervals of
equal lengths. Value and policy functions are then defined on the finite set X × B × K. Value
functions for unconstrained firms are v(x, b, κ) = V (x, κ) − b; for constrained firms they follow
from (19)–(21). Liquidation and entry policy functions follow from (12) and (13).
This permits calculation of the stationary measure of small firms. Combining (5) and (8)
yields one equation for the stationary measure µ0 :
µ0 (A) =
Z
I(x0 ,b0 (x,b),κ0 )∈A g (x0 |x) 1 − dl (x, b, κ) dµ0 (x, b, κ)+M
Z
I(x0 ,b0 (x,b),κ0 )∈A g (x0 |x) de (x, b, κ) dΦ(x, b, κ) ,
for all Borel subsets A ⊂ X × [b̃(κ), θκ] × K. Again, note that small firm capital is fixed, i.e.
κ0 = κ. After discretization, µ0 is a vector with dimension equal to the cardinality of X×B ×K,
and the above equation can be solved by matrix inversion or by fixed point iteration. This also
defines the measure of active firms µ(x, b, κ) = (1−dl (x, b, κ))µ0 (x, b, κ)+M de (x, b, κ)Φ(x, b, κ)
for (x, b, κ) ∈ X×B ×K. Note that both µ0 and µ are linear in M ; hence parameter M linearly
scales the size of the non-corporate sector.
The capital stock in the corporate sector can be backed out from the goods-market equilibrium condition
Z
C+δ
Z
κ dµ(x, b, κ)+M
e
Z
κd (x, b, κ)dΦ(x, b, κ)−
f
Z
[xf (`(x, κ))−c (κ)] dµ(x, b, κ)−θ
κdl (x, b, κ) dµ0 (x, b, κ)
= K c [F (1, 1/k ∗ ) − δ] ,
with capital-labor ratio k ∗ obtained before. Finally, employment in the corporate sector is
R
Lc = K c /k ∗ and labor supply is L = Lc + `(x, κ) dµ(x, b, κ).
E.2
Computation of the Transition Path
We describe how the transition path after the unexpected pandemic shock at t = 0 back to the
original steady state can be calculated numerically. Choose large enough T and suppose that
the economy has approximately reached the steady state from period T + 1 onward. That is,
all endogenous variables attain their steady state values for t ≥ T + 1.
47
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1. Start with a guess for initial consumption C0 . Then use the following first-order conditions of the household and corporate sector decision problems:
Dt+1
qt = β
Dt
Ct
Ct+1
σ
,
(24)
wt = ζt Ctσ ,
(25)
wt = (1 − α)At ktα ,
(26)
α−1
],
1 = qt [1 − δ + αAt+1 kt+1
(27)
which hold for all t = 0, . . . , T , where At = A (1 + νtc ), Dt = 1 + νtd , and ζt = 1 + νt`
are time-varying multipliers of corporate productivity, utility of consumption, and utility
of leisure, respectively. kt is the capital-labor ratio in the corporate sector. The guess for
C0 directly yields w0 and k0 . Then use all four equations to obtain a dynamic equation
for kt :
βktα =
α
kt+1
At+1 Dt ζt
·
α−1 .
At Dt+1 ζt+1 1 − δ + αAt+1 kt+1
This equation can be inverted numerically to obtain iteratively k1 , . . . , kT .
Given this solution, the above equations can be used to back out Ct , wt and qt for all
t = 0, 1, . . . , T . This obtains operating profits of small firms πt (x, κ, ι) for all periods
t = 1, . . . , T where dummy variable ι indicates whether the firm is impacted by the
shock. In period 0, write π0 (x, κ, ι, s), where index s = 1 indicates that these firms
receive the grant in t = 0.
2. Iterate (10) and (11) (adjusted for the pandemic shock) backwards from steady state
value functions yields value functions vt (x, b, κ, ι) and vt0 (x, b, κ, ι) for t = 0, . . . , T and
ι ∈ {0, 1}. See below for a description how this can be done without optimization over
assets. Regarding period t = 0, calculate v0 (x, b, κ, ι, s) and v00 (x, b, κ, ι, s) separately for
firms with and without grant as these have different profits in t = 0.
Calculate entry and exit decision rules de0 (x, b, κ, ι) and dl0 (x, b, κ, ι, s) and det (x, b, κ, ι),
dlt (x, b, κ, ι) for t = 1, . . . , T . Note that the grant is only paid in period t = 0 to incumbent
firms which is why s enters only the exit policy function dl0 .
Calculate the distribution of active firms in t = 0, µ0 (x, b, κ, ι, s) from (5) and the given
steady-state distribution of incumbent firms at the beginning of the period µ0 (x, b, κ).
Here we impose that fraction ηi of incumbent firms and potential entrants are hit by the
48
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pandemic and that fraction η of continuing firms in period t = 0 receive the grant, while
entrants in period t = 0 are not eligible for the grant. Hence,
0
l
ηi ηµ
, ι = 1, s = 1,
h (x, b, κ)(1 − d0 (x, b, κ, 1, 1))
0
l
ηi (1 − η)µ (x, b, κ)(1 − d0 (x, b, κ, 1, 0))
i
+M de0 (x, b, κ, 1)Φ(x, b, κ)
, ι = 1, s = 0,
µt=0 (x, b, κ, ι, s) =
0
l
(1 − ηi )ηµ
, ι = 0, s = 1,
h (x, b, κ)(1 − d0 (x, b, κ, 0, 1))
0
l
(1 − ηi ) (1 − η)µ (x, b, κ)(1 − d0 (x, b, κ, 0, 0))
i
+M de0 (x, b, κ, 0)Φ(x, b, κ)
, ι = 0, s = 0.
Continue to iterate over (5) and (8), adjusted for the additional state variable ι, to obtain
firm distributions before and after entry/exit in all subsequent periods t = 1, . . . , T ,
µ0t (x, b, κ, ι) and µt (x, b, κ, ι) (note that state variable s only matters in t = 0). Calculate
the net output of small firms:
Z
n
Y0 =
[(1 − ν0n ι)xf (κ, `((1 − ν n )x, κ)) − δκ − cf (κ)] dµ0 (x, b, κ, ι, s)
Z h
+ θκ
ηi ηdl0 (x, b, κ, 1, 1) + (1 − η)dl0 (x, b, κ, 1, 0)
i
+ (1 − ηi ) ηdl0 (x, b, κ, 0, 1) + (1 − η)dl0 (x, b, κ, 0, 0) dµ0 (x, b, κ)
Z
e
−M
ηi d0 (x, b, κ, 1) + (1 − ηi )de0 (x, b, κ, 0) κdΦ(x, b, κ) ,
Z
Z
n
n
Yt =
(1 − νt ι)xf (κ, `(x, κ)) − δκ dµt (x, b, κ, ι) + θκ dlt (x, b, κ, ι) dµ0t (x, b, κ, ι)
Z
− M [ηi det (x, b, κ, 1) + (1 − ηi )det (x, b, κ, 0)]κdΦ(x, b, κ) , t = 1, . . . , T.
Starting at t = 0, given the initial steady-state capital stock in the corporate sector
K0c = K c∗ and the previously obtained path for kt , calculate Lct , output Ytc , investment
c
Itc , and Kt+1
iteratively for all t = 0, . . . , T :
Lct = Ktc /kt ⇒ Ytc = At F (Ktc , Lct )
c
⇒ Itc = Ytn + Ytc − Ct ⇒ Kt+1
= (1 − δ)Ktc + Itc .
If KTc +1 > K c∗ , increase C0 ; if KTc +1 < K c∗ , decrease C0 . Repeat until KTc +1 is reasonably
close to K c∗ (saddle path).
49
Electronic copy available at: https://ssrn.com/abstract=4064899
Regarding value function iteration (step 2), borrowing/savings policies can be directly obtained
using a similar logic as in the stationary equilibrium. Firms become unconstrained when their
financial savings exceed a certain threshold level after which they are able to pay positive
dividends. The value function of these unconstrained firms takes the form vt (x, b, κ, ι) =
Vt (x, κ, ι) − b where
Vt (x, κ, ι) = πt (x, κ, ι) + qt Ex0 |x max[θκ, Vt+1 (x0 , κ, ι)] .
(28)
This equation can be solved backwards starting from VT +1 = V (steady state). Then define
cutoff productivity levels x̃t (κ, ι) = min{x ∈ X|Vt (x, κ, ι) ≥ θκ} such that unconstrained firms
with productivity below x̃t (κ, ι) liquidate the firm in period t, while all others stay.
To determine the level of debt below which a firm becomes unconstrained, calculate backward the debt levels b̃t (κ, ι) = πt (x̃t (κ, ι), κ, ι) + qt b̃t+1 (κ, ι) for t = 0, . . . , T with b̃T +1 (κ, ι) =
b̃(κ) for ι ∈ {0, 1} (steady state). Verify that b̃t (κ, ι) < θκ (otherwise there cannot be unconstrained firms in t − 1). Then, in any period t, a firm is unconstrained if b ≤ b̃t (κ, ι) in
which case it sets b0t (x, b, κ, ι) = b̃t+1 (κ, ι) if x ≥ x̃t (κ, ι). Otherwise, if x < x̃t (κ, ι), the firm is
liquidated, dlt (x, b, κ, ι) = 1. Note that in the pandemic period t = 0, b̃0 (κ, ι) differs between
firms with grant and firms without grant so that the cutoff that differentiates constrained from
unconstrained firms depends on the grand receipt indicator s.
Any firm with b > b̃t (κ, ι) is constrained. This firm is also liquidated if x < x̃t (κ, ι).
Otherwise it pays zero dividends until savings exceed −b̃t+1 (κ, ι). This implies that the value
and policy functions of constrained firms are
vt (x, b, κ, ι) = max[0, πt (x, κ, ι) − b − qt b̃t+1 (κ, ι)] + qt Ex vt0 (x0 , b0t (x, b, κ, ι), κ, ι) ,
(29)
vt0 (x, b, κ, ι) = max[θκ − b, vt (x, b, κ, ι)] ,
1
0
bt (x, b, κ, ι) = max b̃t+1 (κ, ι), (b − πt (x, κ, ι)) ,
qt
(30)
(31)
if πt (x, κ, ι) − b + qt θκ ≥ 0, and vt (x, b, κ, ι) = vt0 (x, b, κ, ι) = θκ − b otherwise. Again, in
period t = 0, the value function of constrained firms differs between firms with grant and firms
without grant via the profit function π0 (x, κ, ι, s) in equation (23).
F
Moment Computation
Firm exit rate. The firm exit rate is the quarterly rate at which small firms permanently
exit. In the model, we calculate the firm exit rate as the measure of firms that exit in a period
50
Electronic copy available at: https://ssrn.com/abstract=4064899
divided by the measure of firms at the beginning of the period. In the steady state, the exit
rate is
R
dl (x, b, κ) dµ0 (x, b, κ)
x,b,κ
R
rexit =
.
dµ0 (x, b, κ)
x,b,κ
The data counterpart is computed based on data from the BDS. Since the BDS data is annual,
we first compute the annual exit rate by dividing the number of establishment death in year
t, and then convert them into the quarterly rate. The relationship between annual exit rate
y
q
(rexit
) and quarterly exit rate (rexit
) is as follows.
y
rexit
= 1−
4
Y
q
(1 − rexit
).
q=1
We use the establishment exit rate to proxy for the firm exit rate since we do not have highquality micro data on firm exits.
Fixed expense to revenue ratio. Fixed expenses are firms’ non-payroll overhead expense. We compute the data target based on a table prepared by Sageworks published in the
Washington Post, which is constructed based on the 2007 U.S. Economic Census.20 We weight
the expense-to-revenue ratios of the nine types of small businesses by their yearly revenues.
The model counterpart of fixed expenses of small firms include the maintenance cost of
capital and the additional fixed cost, and the model counterpart of small firm revenue is their
output. We compute fixed expense to revenue ratio in the steady state as
x,b,κ
δκ + cf (κ) dµ(x, b, κ)
x,b,κ
xf (κ, `(x, κ)) dµ(x, b, κ)
R
R
.
Debt to Asset Ratio. The data target for the debt-to-asset ratio is computed based
on the KFS. It is computed as the ratio between the sum of positive net debt of small firms
divided by the sum of their real assets. The net debt of a firm is its total debt minus financial
assets; we treat firms with a negative net debt (positive financial asset) as having zero debt.
The model counterpart of the debt to asset ratio is
R
max{b, 0} dµ(x, b, κ)
x,b,κ
R
.
κ dµ(x, b, κ)
x,b,κ
20
Link: https://www.washingtonpost.com/wp-srv/special/business/costofrunningabusiness.html, accessed
on May 28, 2021.
51
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