Charles Goodhart et al., Support for small businesses amid COVID‐19 (2023)
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An original article by Charles A.E. Goodhart of the London School of Economics, Dimitrios P. Tsomocos of the University of Oxford, and Xuan Wang of Vrije Universiteit Amsterdam and the Tinbergen Institute, Support for Small Businesses Amid COVID-19, published in Economica, Vol. 90, No. 358, 2023, pages 612–652 (DOI 10.1111/ecca.12460). The paper develops a two-sector equilibrium model with private information and a screening contract to analyze how governments balance short-run stabilization against long-run allocative efficiency when offering pandemic loans and credit guarantees to small and medium-sized enterprises. It notes that small and medium-sized enterprises account for over 99% of European Union businesses and over 65% of the workforce, and examines how the interest rate effective lower bound and firm markups affect optimal default sanctions.
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Goodhart, Charles A.E., Tsomocos, Dimitrios P. & Wang, Xuan (2023) Support for
small businesses amid COVID-19. Economica, 90(358), 612 - 652.
https://doi.org/10.1111/ecca.12460
https://researchonline.lse.ac.uk/id/eprint/118164/
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Accepted: 5 January 2023
DOI: 10.1111/ecca.12460
ORIGINAL ARTICLE
Support for Small Businesses Amid COVID-19
Charles A.E. Goodhart1 Dimitrios P. Tsomocos2 Xuan Wang3
1 LSE and CEPR, UK
Abstract
2 Saïd Business School and University of
How should the government support small and
Oxford, UK
medium-sized enterprises amid a pandemic crisis while
3 Vrije Universiteit Amsterdam and Tinbergen
Institute, The Netherlands
balancing the trade-off between short-run stabiliza-
tion and long-run allocative efficiency? We develop a
Correspondence two-sector equilibrium model featuring small businesses
Xuan Wang, Vrije Universiteit Amsterdam and with private information on their likely future suc-
Tinbergen Institute, The Netherlands.
Email: xuan.wang@vu.nl
cess and a screening contract. Businesses in the sector
adversely affected by a pandemic can apply for govern-
ment loans to stay afloat. A pro-allocation government
sets a harsh default sanction to deter entrepreneurs with
poorer projects, thereby improving long-run produc-
tivity at the cost of persistent unemployment, whereas
a pro-stabilization government sets a lenient default
sanction. Interest rate effective lower bound leads to
involuntary unemployment in the other open sector and
shifts the optimal default sanction to a lenient stance. The
rise in firm markups exerts the opposite effect. A high
creative destruction wedge polarizes the government’s
hawkish and dovish stances, and optimal default sanc-
tion is more lenient, exacerbating resource misallocation.
The model illuminates how credit guarantees might be
structured in future crises.
1 INTRODUCTION
The combination of the Coronavirus pandemic and the policy measures of lockdown and
quarantine in response has had a drastic effect on the cash flows and solvency of businesses in
the many countries affected, in particular on small and medium-sized enterprises (SMEs) (see
Fairlie 2020). If the country was to avert an economic collapse, with a large proportion of its
SME population being forced to shut up shop, the need was to get external financial assistance
to them quickly. Given the massive numbers of SMEs, for example, over 99% of all businesses in
the European Union, accounting for over 65% of the workforce according to European Commis-
sion statistics, this could hardly be done, at least not quickly enough, directly from a government
office; rather, it had to be done via the existing relationship between SMEs and their main bank.
This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits
use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or
adaptations are made.
© 2023 The Authors. Economica published by John Wiley & Sons Ltd on behalf of London School of Economics and Political Science.
Economica. 2023;1–41. wileyonlinelibrary.com/journal/ecca 1
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2 ECONOMICA
If the banks themselves were to be left carrying the can for any significant share of the losses
arising from non-performing loans (NPLs), then they would have wished to be extremely care-
ful in monitoring, which, however, takes time and effort. Moreover, the banks might be more
conservative in their own interests than would be socially or politically desirable. For example,
Chodorow-Reich et al. (2022) show that in 2020Q1 and 2020Q2 in the USA, small firms had no
net drawdown of bank credit lines, whereas large firms did. For all these reasons, many countries
in Europe, as well as the USA, implemented credit guarantees for their loan support to SMEs. For
example, in the UK, the government then decided that such emergency loans to SMEs, known
as ‘bounce-back loans’ (BBLs), would henceforth be 100% guaranteed by the government—that
is, they would not count as NPLs or cause losses to the banks. Similarly, in Germany, the credit
guarantee covers from 90% to 100%, and in the USA, the government credit guarantee ranges
from 95% to 100%. However, the Netherlands and Spain seem more cautious and strict, and
the credit guarantee in the Netherlands can be as limited as 67.5%; see European Central Bank
(ECB) (2020,b).
The government credit guarantees have triggered a series of public comments on the likely
massive defaults on emergency loans, or ‘soft default’ in the form of future inflation.1 One major
concern is the likely misallocation of resources that will come from keeping alive businesses that
are sub-par, even zombie companies. While there is empirical work such as Bachas et al. (2021)
suggesting that lending supply to SMEs is responsive to government guarantees, the welfare and
allocative effects of these programmes remain to be seen. Assuming that the government puts
some weight on allocative efficiency, in order to raise productivity and output over future years,
it will need to try to screen out unprofitable and less profitable potential borrowers. So, should
the government take a lenient or harsh stance?
Our objective is to assess the normative issue of whether and what degree of screening would
have been socially optimal against the backdrop of the state of the world during the pandemic. In
particular, advanced economies have been characterized by rising markups and declining inter-
est rates until very recently. (We do not focus on the post-pandemic period inflation where the
central banks are raising rates.) Figure 1 plots the rising aggregate markup in the USA estimated
by De Loecker et al. (2020) against the Federal funds rate and the 3-month Treasury bill sec-
ondary market rate since 1980. These trends are not US-specific. Del Negro et al. (2019) show
that the world interest rate of safe and liquid assets has dropped significantly over the past three
decades. Parallel to this trend, Díez et al. (2021) provide empirical evidence suggesting a decline in
20.00 1.7
18.00
16.00 1.6
14.00
12.00 1.5
10.00
8.00 1.4
6.00 F I G U R E 1 Aggregate markup
4.00 1.3
and interest rates in the USA. Sources:
The markup data are from the De
2.00 Loecker et al. (2020) estimation. Data
Federal funds rate and the 3-month
0.00 1.2
80 982 984 986 988 990 992 994 996 998 000 002 004 006 008 010 012 014 016 018 020
Treasury bill secondary market rate are
9
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 from the Board of Governors of the
Federal F unds Rate (LHS %) Federal Reserve System (USA),
3-Month Treasury Bill Se condary Ma rke t Rate (LHS %) retrieved from FRED, Federal Reserve
Mar kup (RHS) Bank of St Louis.
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 3
competition at the global level. We ask whether the optimal level of screening for the government
loan support to SMEs might change when the competitiveness of the affected industry differs,
and how the interest rate effective lower bound (ELB) might shift the toughness or leniency of
optimal screening.
To this end, we develop a two-sector equilibrium model of small businesses with imperfect
competition in the presence of a pandemic shock. The sectors of the economy locked down by
the COVID-19 pandemic were mainly service sectors, such as hospitality and entertainment. In
these cases the standard small enterprise will still normally have some degree of market power
as a result of location, customer familiarity and reputation. The COVID-19 pandemic shock
forces the adversely shocked sector to close while the other sector remains open. The rationale of
modelling a two-sector economy rather than a single-sector economy is twofold: first, in terms
of realism, the pandemic lockdown policy forced only certain sectors to close but kept key sec-
tors open; second, a two-sector setup allows us to show that the aggregate demand externality
between sectors affects the optimal level of screening in a non-trivial way, which particularly mat-
ters for the low interest rate environment during the pandemic. This channel would not be there
if the model had only a single sector. The government provides BBLs to the adversely shocked
sector to retain at least some key workers, so that these firms avoid large extra costs associ-
ated with matching and training a completely new workforce when production resumes in the
future. Potential applicants for government loans have private information about their expected
profitability. Those with lower profitability are likely to default on government loans. The govern-
ment can choose to implement a default sanction to ameliorate such adverse selection. Although
the context of the model is the pandemic crisis, the model could be applied in more general set-
tings, namely, for an analysis of financial contracting in conjunction with an aggregate demand
externality between sectors.
In the benchmark model, the default penalty—i.e. the default sanction—is modelled as a
pecuniary deduction from the defaulter’s residual income, which is in effect a personal guarantee
pledged on future output.2 This default penalty is used as a screening device, and if borrow-
ers default, then they will lose the personal guarantee. While we acknowledge that in practice,
banks provide loans to SME borrowers, and the loan support instrument that the government
uses is credit guarantee, which affects banks’ monitoring incentive, here by abstracting away
banks’ monitoring incentive problem and using the default sanction as the screening device in
the benchmark model, we obtain analytic tractability. Nevertheless, we extend the benchmark
model in Section VI to include the banking sector by considering its monitoring incentive à la
Martinez-Miera and Repullo (2017), and we change the policy instrument from the default sanc-
tion to credit guarantee. We find that the extended model is isomorphic to the benchmark model,
and that our results go through. In the extended model, it is up to the lender to pass through the
effect of credit guarantees to the borrower through its monitoring effort, and we show that the
credit guarantee is one implementation of the default sanction of the benchmark model. When
the government sets 100% credit guarantee, the default sanction is shown to be zero, attracting
those with potentially less profitable businesses that stay afloat going forward. As the govern-
ment reduces credit guarantee, the default sanction becomes harsher. Essentially, increasing the
default sanction is equivalent to lowering the degree of credit guarantees, and vice versa. The
higher the default sanction, the lower the credit guarantee, so lenders have a stronger incentive
to pursue defaulters and garnish their residual income.
If the government is pro-allocation, then we show analytically that the government can
choose to be hawkish and implement a harsh default sanction to deter potentially less prof-
itable entrepreneurs from applying for loans, but this pro-allocation policy leads to unem-
ployment in the short run, and demand shortage is persistent. If the government is dovish
and pro-stabilization, then we show that the government can choose to implement a lenient
default sanction or even no sanctions (i.e. 100% credit guarantees) to fully stabilize short-run
employment. In this case, demand shortage is short-lived, but the economy is shifted to a
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4 ECONOMICA
lower equilibrium in the future due to misallocation. Moreover, we develop an analytic measure
‘stabilization proclivity’ to characterize the conditions under which the government is likely to be
more pro-allocation or pro-stabilization. In particular, we show that the market power of small
businesses affects stabilization proclivity negatively. When the market power is high, stabiliza-
tion proclivity is low, which signals a harsher stance, assuming that the government has the same
discount factor as households. Indeed, as we show in our numerical solution, when competition
increases, it causes a noticeable shift of the optimal default sanction to a lenient stance. This is
because when the industry is more competitive, the labour income share accounts for a larger
fraction of total output. So the social utility loss associated with unemployment commands a
much higher weighting, hence the government leans towards employment stabilization.
The two-sector setup enables us to uncover the aggregate demand externality mechanism à la
Guerrieri et al. (2022) to inform our optimal screening contract. Assuming that the sectors are
complementary to each other, we show that the interest rate ELB causes involuntary unemploy-
ment even in the sector that is allowed to remain open during the pandemic crisis. Numerically,
we demonstrate that the optimal default sanction is more lenient than the case when the econ-
omy is away from the ELB. This is because the complementarity of the goods sector means
that the spread between the inter-temporal elasticity of the households’ utility function and the
intra-temporal elasticity is positive. With a negative supply shock, the interest rate decreases but
the ELB prevents the interest rate from flexible adjustments, causing inefficiency in employment
across sectors. To assess how sensitive the interest rate changes are to the changes of these two
elasticities, we vary the inter-temporal elasticity and the intra-temporal elasticity individually
while keeping the spread unchanged. We find that the unbounded interest rate changes are more
sensitive to the intra-temporal elasticity. Typically, the interest rate dynamics should be more
sensitive to the inter-temporal elasticity, but here it is the opposite. This is because the negative
supply shock and goods complementarity cause the interest rate dynamics to reflect Keynesian
demand shortage in Guerrieri et al. (2022).
Therefore, at the interest ELB, involuntary unemployment leads to a more lenient default
sanction as optimal policy, which, in terms of implementation, corresponds to a more gener-
ous credit guarantee scheme. This has the opposite implication of congestion externalities on
resource misallocation. To assess how the interaction with ELB and market power jointly affect
involuntary unemployment, we vary both the price elasticity of demand and industry size. We
find that the changes in the price elasticity on the optimal default sanction are asymmetric. When
we decrease the price elasticity from 1.2 of our benchmark case to 1.05, which means that firm
markups and rents increase, we find involuntary unemployment in the open sector increases
across the whole spectrum of default sanction. In particular, when the default sanction is at the
harsh stance that deters all inefficient projects, involuntary unemployment in the open sector
increases to 7.5%. This is striking because the open sector is allowed to operate, and pandemic
policies and the elasticity change are applied only in the closed sector. This reflects the aggregate
demand externality due to complementarity across sectors in the interest rate ELB environment.
This decrease in price elasticity pushes the optimal default sanction to be more lenient, and in
our numerical simulations, the optimal default sanction becomes 4.37% more lenient than that
of the benchmark parameter space.
In contrast, when we increase the price elasticity from 1.2 to 1.4, involuntary unemployment
in the open sector still exists, but it is noticeably less across all spectrums of the default sanction.
This would mean that the change in the leniency of the optimal default sanction is small (only
0.5% in our numerical simulation). Interestingly, the change of industry size almost has little
effect on the open sector’s involuntary unemployment even though it affects firm markup. This
is because the industry size does not have a first-order effect on interest rate dynamics, whereas
the elasticity parameters do.
Since the key trade-off on which we focus is between short-run employment stabiliza-
tion and long-run allocative efficiency, which makes a link to creative destruction, we define
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 5
a wedge to reflect the extent to which unproductive incumbents can be replaced promptly
by productive entrants. More precisely, the wedge is the difference between the fraction of
unproductive incumbents and the fraction of productive new entrants. The creative destruc-
tion wedge is a manifestation of weak property rights and/or high bureaucratic costs to set
up businesses, which can deter creative destruction. We solve the closed-form solution to show
how this wedge affects the two extreme cases of default sanctions, that is, the harsh default
sanction that cares only about long-run allocative efficiency, and the lenient default sanction
that cares only about short-run employment stabilization. We show that the former increases
with the wedge, while the latter decreases with the wedge. This is because when the creative
destruction wedge is large, unproductive incumbents cannot be replaced promptly by produc-
tive entrants. To fully stabilize short-term employment, the threshold of the lenient sanction has
to decrease. And to remove resource misallocation in the long run, the threshold of the harsh
sanction has to increase, since the lack of business dynamism implies that the borrowers’ out-
side option is weak, and the government has to be more hawkish to keep out the inefficient ones.
Therefore the lack of creative destruction (higher wedge) polarizes the two extreme cases even
more.
As the wedge reflects business dynamism, to assess its quantitative effects, we need a dynamic
setting. Thus in our numerical illustration, we adapt our stylized model to a simple dynamic
model. We show that indeed, when we vary the parameter space to generate a wide range of
the creative destruction wedge, the optimal policy leniency responds positively to the wedge.
The higher the wedge, the more lenient the optimal default sanction. This implies that for coun-
tries where the business environment is less dynamic, the government rationally chooses a higher
degree of forbearance, thereby exacerbating resource allocation. Furthermore, when the econ-
omy is at the ELB, the creative destruction wedge in the sector that is forced to close also spills
over to the open sector, exacerbating its involuntary unemployment. This suggests that the ELB
environment further exacerbates the distortion due to the creative destruction wedge. Neverthe-
less, the quantitative effect is small compared with that due to changes in the intra/inter-temporal
elasticities and price elasticity.
Finally, we extend the model to consider banks and model explicitly how the government
credit guarantee affects banks’ monitoring incentives. We show that the quality of banks’ moni-
toring is a decreasing function of credit guarantee, which is consistent with the empirical evidence
in Lelarge et al. (2010). In equilibrium, as the credit guarantee becomes more generous, the aggre-
gate lending also increases, in line with Bachas et al. (2021). Moreover, we show that lowering
credit guarantees in the extended model is equivalent to increasing the default sanction in the
benchmark model.
The remainder of the paper is structured as follows. In Section I, we review the extremely
rapidly growing associated literature on the effects of the pandemic on the economy, especially in
relation to the objectives of stabilization and allocative efficiency. Then in Section II, we set out a
model, simplified as far as possible, in which we aim to explore the effects of introducing a screen-
ing contract into a government scheme for financing companies adversely affected by enforced
closure, on the twin objectives of stabilization and allocative efficiency. Section III provides equi-
librium characterization and analysis. Section IV solves for the optimal default sanction and
its interaction with interest rate ELB. Section Vconducts comparative statics on the interaction
between ELB, markup and creative destruction. Section VI is a model extension, and Section VII
considers other numerical experiments. Section VIIIconcludes.
2 RELATED LITERATURE
First and foremost, our paper relates to the group of COVID-19 literature on public liquidity
provision (see, for example, Bachas et al. 2021; Chodorow-Reich et al. 2022; Elenev et al. 2020;
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6 ECONOMICA
Kahn and Wagner 2020; Minoiu et al. 2021; Segura and Villacorta 2020; Philippon 2021). On
the empirical front, Minoiu et al. (2021) document that the Main Street Lending Program in the
USA increased banks’ willingness to lend, and that participating banks were less likely to tighten
lending standards. Chodorow-Reich et al. (2022) present a framework on bank liquidity provision
across firms and develop testable predictions, and they find that the US-government-sponsored
Paycheck Protection Program alleviated liquidity shortfalls to small firms at a cost to the gov-
ernment. Bachas et al. (2021) find that lending supply is highly sensitive to government credit
guarantees. These empirical findings focus mostly on the effectiveness of government support on
lending supply, but less is clear about its welfare trade-off, so our normative assessment com-
plements these empirical analyses. In this sense, our paper is related to that of Gale (1991) who
develops a theoretical framework à la Stiglitz and Weiss (1981) to analyse the quantitative effects
of federal lending on credit allocation and economic efficiency. Gale (1991) shows that federal
credit programmes are effective in the allocation of credit, but the estimated efficiency cost of the
actual credit policy is high. The model maintains a simple direct link between credit and real activ-
ity in that an increase in credit leads to increased real activity, whereas our model incorporates
the composition of real activity so that credit contractual terms change the marginal incentive
on whether to become an entrepreneur or stay out as a worker. Therefore our contribution lies in
the trade-off between unemployment and misallocation, and the interplay between sectors, with
implications on the policy effectiveness in a low interest rate environment.
On the theoretical side of COVID-19 support programmes, Philippon (2021) analyses
how government interventions can improve efficiency when the decentralized economy amid
COVID-19 is distorted by wage rigidity. Our paper complements Philippon (2021) by focusing
on the government’s long-run and short-run trade-off, and differs in the following distinct ways.
First, the key trade-off in our paper is between short-run employment stabilization and long-run
allocative efficiency, whereas Philippon (2021) presents a one-period model and hence does not
highlight the short-run and long-run trade-off. Second, the short-run unemployment in our paper
is a direct result of the government’s default sanction, but the unemployment in Philippon (2021)
arises due to downward wage rigidity. Thus our paper can show analytically the effect of the
government’s screening policy on short-run employment stabilization. Third, Philippon (2021)
assumes that private creditors can observe firms’ project productivity, so that the government can
implement optimal policy based on the government’s observation of private creditors’ actions.
This is similar to the conditional loan bridge schemes considered in Elenev et al. (2020). Our
paper takes a different view, that private creditors (e.g. banks) would not be able to monitor bor-
rowers or acquire information. This is not just due to the scale and speed of the pandemic crisis,
but is also because government credit guarantees would leave little incentive for banks to make
efforts to monitor and acquire information.
Kahn and Wagner (2020) develop a theory to underpin the conditions under which direct
provision of liquidity is preferable to the traditional distribution of liquidity, and vice versa. The
main trade-off there is between externalities and informational advantages. In a similar spirit
but with a different friction, Segura and Villacorta (2020) analyse different types of govern-
ment interventions to support firms, and develop a pecking order between direct transfers and
indirect support through guarantees to new loans or reductions in the capital requirement. The
critical friction in their paper is the moral hazard due to the borrower’s unobserved effort cost.
Our paper differs from Kahn and Wagner (2020) and Segura and Villacorta (2020) in that we
focus on setting the contractual terms of government loans to ameliorate adverse selection while
endogenizing its impact on persistent unemployment, which is not present in Kahn and Wagner
(2020) and Segura and Villacorta (2020). More specifically, in Segura and Villacorta (2020), the
optimal policy is derived from the deployment of funding through the workings of financial inter-
mediaries to reduce moral hazard. However, our paper focuses on adverse selection, and the
optimal policy stems from the trade-off between allocation (reducing agency cost) and stabiliza-
tion (reducing unemployment). Another distinction between our paper and these two papers is
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 7
that we model multiple goods and sectors, offering an additional perspective on the aggregate
demand externality and its interaction with optimal policy. Of course, there are other ways to
design screening to address adverse selection issues. For example, if the government can exploit
information in observables (e.g. employment at t = 0), then the loan support policy can better
target more efficient borrowers. A recent paper by Wang and Wang (2021) shares a similar spirit
to this idea. Wang and Wang (2021) show that the government can exploit the information dis-
persed within the private banking sector to improve policy targeting; this is because the demand
for loans mirrors the borrower’s funding deficit and reflects the net present value.
Relatedly, Elenev et al. (2020) build a structural model calibrated with the US data to evaluate
three government policies aimed at short-circuiting the interplay between corporate defaults and
banking fragility. They find that the government loan schemes in the USA are preventing the
bulk of firm bankruptcies. Our paper differs from theirs in that we consider explicitly the agency
problem between the government and the borrowing entrepreneurs or firms that possess private
information on their expected profitability. Moreover, we consider the potential adverse effect of
government loan schemes and guarantees that cause zombification and the drop in long-term
productivity.
Therefore our paper contributes to the fast-growing COVID-19 literature by emphasizing and
modelling explicitly the policy trade-off between short-run stabilization and long-run allocation,
with the optimal policy provided given various features of the underlying economy. On stabiliza-
tion, Barrero et al. (2020) estimate that that COVID-19 shock caused 3 new hires for every 10
layoffs, and that 32–42% of COVID-induced layoffs will be permanent, suggesting a slow absorp-
tion of labour into new jobs. Fairlie (2020) also provides timely and early evidence on the impacts
of social distancing restrictions and demand shifts from COVID-19 on small businesses. In con-
trast, on allocation, Acharya et al. (2020) document the effect of cheap credit and zombification
on firm markups and inflation. Jordà et al. (2020) provide long-run evidence on the economic
costs of corporate debt booms and inefficient debt restructuring. Our paper contributes to these
two groups of literature by presenting a model on how to balance the policy trade-off between
them and search for the social optimum.
Sharing a similar spirit but with a different mechanism, Acharya et al. (2021) build a the-
ory on zombie lending in a broader context. The authors find that aggressive unconventional
policy can induce low-capitalized banks to supply credit to low-productivity firms and solve the
optimal policy response. However, in Acharya et al. (2021), risk-shifting induced by regulatory
forbearance is the primitive economic force pushing towards zombie lending, so the authors do
not explicitly model asymmetric information or screening, which is the mechanism of this present
paper.
In terms of the macro-financial environment, we introduce financial contracting in the
presence of asymmetric information to the multi-sector economy along the lines of Guerrieri
et al. (2022), who build a multi-sector infinite horizon model and show how a negative supply
shock due to shutdown can translate into a demand shortage. Similar to Guerrieri et al. (2022),
the pandemic shock of our model is modelled as a shutdown of the adversely affected sec-
tor while the other sector stays open, and we make use of the demand shortage and aggregate
demand externality in the Guerrieri et al. (2022) framework. However, the main friction in our
paper is the adverse selection stemming from the borrowing entrepreneurs’ private information,
which is not present in Guerrieri et al. (2022). Furthermore, the central question in Guerrieri
et al. (2022) is under what conditions a negative supply shock can cause aggregate demand short-
age, while in our paper, we are investigating specifically government loans and credit guarantees
that support small businesses during a pandemic. This is a policy issue in which the trade-off
between reducing unemployment and reducing the agency cost of adverse selection emerges nat-
urally. The theoretical contribution of our paper is to combine financial contracting and insights
from the industrial organization (see Shapiro 1989) with the aggregate demand externality of the
macroeconomy.
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8 ECONOMICA
Finally, given the information friction in our model and the use of screening, our model
connects with the vast literature on private information and screening in financial contract-
ing that follows a rich tradition; see Jensen and Meckling (1976), Fama (1978), Rothschild
and Stiglitz (1978), Smith and Warner (1979), Stiglitz and Weiss (1981), Webb (1984) and
Bester (1985) as classic examples, and more recently, Dubey and Geanakoplos (2002) and Lester
et al. (2019). Nevertheless, the concern of our paper is not only on designing the financial con-
tract to reduce the agency cost, but also about the unintended and undesirable consequences
of reducing the agency cost in a macroeconomic setting. After all, the focus of our paper is on
how to trade off the pandemic-induced stabilization issues as a result of reducing the agency
cost against long-run productivity. We endogenize the social cost of reducing the agency costs
as the near-term surge in unemployment and reduction in production, and bring to the fore-
front the policy trade-off between reducing near-term unemployment and increasing long-term
productivity.
3 THE MODEL
3.1 Environment
The economy has three types of agents: entrepreneurs, workers and the government. It lasts for
three periods (t = 0, 1, 2). Date t = 0 is when the pandemic occurs, date t = 1 refers to the imme-
diate short run, and date t = 2 is interpreted as the long run. There are two sectors, I and J,
that produce different goods. Each sector has N entrepreneurs who hire workers to carry out a
project to produce sector-specific goods. Entrepreneurs in each sector choose how much to pro-
duce and take into account the impact on setting prices. There are Q workers in the economy,
and Q∕2 workers specialize in each sector. Each worker is endowed with 1 unit of labour sup-
plied to an entrepreneur inelastically. Each entrepreneur’s maximum production capacity is to
employ Q∕2N workers. In the absence of the pandemic shock, full employment would be achieved
in both sectors. Both entrepreneurs and workers consume goods from both sectors, and exhibit
the same constant elasticity of substitution (CES) preferences for consumption goods. Note that
we abstract away physical capital investment for production. The reason for this is that during
crises, governments typically provide loans to small businesses to retain workers rather than to
fund investment. Market loans for capital investment are relatively accessible since capital, unlike
workers, can be pledged as collateral, which may obviate the need for government loans (see
Gonzalez-Uribe and Wang 2020).
In times of pandemics—for example, COVID-19—sector I is adversely shocked and forced
to close. The government provides BBLs to the adversely shocked sector to retain work-
ers, so that firms can avoid large extra costs associated with matching and training a com-
pletely new workforce when production resumes in the future.3 If the borrowing entrepreneurs
default, then the government can choose to implement a default sanction. If the government
sets zero default sanctions, then the government assumes all the cost of default and effec-
tively provides full credit guarantees for the BBLs. Figure 2 outlines the flow of funds in the
economy.
Figure 3 illustrates the timeline, and we divide each of the first two dates into two sub-periods.
At the start of t = 0, entrepreneurs in sector I use their previous sales revenues to pay wages
for the labour hired for production in period t = 0. Then an unanticipated short-lived pandemic
shock occurs at the end of t = 0 and hits sector I. Sector I is forced to close, while sector J remains
open. Thus at the end of t = 0, the entrepreneurs in sector I have not produced anything, and the
economy’s spending falls onto sector J at t = 0.
During the pandemic, the government provides loans for businesses to retain at least some
key workers to avoid large additional training and matching costs in the future. The prior
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 9
FIGURE 2 Flow of funds.
Government
Borrows loans from
the government
Repay subject to possible
default sanctions
Sector I Sector J
(shut due to (remains open)
COVID-19)
Purchase sector I and
sector J products
Pay wages Pay wages
Sector I workers Sector J workers
FIGURE 3 Timeline. Sector I agents choose to borrow from
Entrepreneurs in sectors gov to retain workers during crisis
I and J pay workers for (Borrowers have private info of
planned production expected profitability)
t=0 t=1 t=2
Unanticipated COVID shock Sector I entrepreneurs
Sector I is shut, repay or default
Sector J remains open
entrepreneurs can choose to borrow loans and continue operating their businesses, or choose to
become workers in this sector. If they choose to borrow government loans, then they pay wages
immediately at the start of t = 1 to retain key workers in this sector. The prior entrepreneurs have
projects with different productivity levels, and they have private information on the project pro-
ductivity. Those with lower profitability are more likely to default if their revenue is low at the
end of date t = 1 when production is resumed.4
Given the scale and the speed of the pandemic, we assume that the government is unable to
acquire information on the entrepreneurs’ prospective profitability in time. However, the gov-
ernment may implement, should it wish, a sanction if the borrowing entrepreneurs default. As
one might expect, if the government implements a lenient default sanction or no sanctions at all,
then inefficient entrepreneurs with lower expected profitability may try to borrow. If the govern-
ment implements a harsh default sanction, then it is possible to deter those with low profitability,
but this policy may increase the unemployment rate. We will discuss the policy trade-off in more
detail shortly, after we set up the model formally.
3.2 Entrepreneurs
We first characterize the entrepreneurs’ optimization in the absence of the pandemic shock, when
the entrepreneurs simply solve a static profit maximization problem, while taking into account
their price impact.
We set the following notation: WE is the nominal wage paid by entrepreneurs in sec-
tor E, for E ∈ {I, J}; PE is the price of goods in sector E; qE (n) is the quantity of goods
produced by entrepreneur n in sector E; qE is the total quantity of goods in sector E; hE (n)
is the labour demand by entrepreneur n in sector E; and production technology is given as
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10 ECONOMICA
qE (n) = 𝜎 hE (n). We call 𝜎 the project productivity, and 𝜎 = 1 in the absence of the pandemic
shock.
Formally, entrepreneurs maximize the profits of ΠE (n) subject to the production technology
as below. With a finite number of firms in the industry, competition is imperfect. Entrepreneur n
takes into account the impact of their
( production plan on) price setting. Therefore, in their objec-
∑
tive function, the price PE —i.e. PE qE (n) + m≠n qE (m) —is a function of production quantity.
We have
( )
∑
max ΠE (n) = PE qE (n) + qE (m) (qE (n)) − WE hE (n),
qE (n),hE (n)
m≠n
subject to
qE (n) = 𝜎 hE (n).
We introduce 𝜀E as the price elasticity of demand, and by definition, it is expressed as
PE 𝜕qE
𝜀E = − .
qE 𝜕PE
As we assume CES preferences, 𝜀E is the same across goods, and we will drop the indexing E
hereafter (so 𝜀E = 𝜀). We focus on symmetric equilibria so that each entrepreneur sets qEt (n) =
qEt ∕N. The optimality condition for entrepreneur n leads to
( )
1
WE = 𝜎 1 − PE ,
N𝜀
where we assume the parameter space 𝜀 > 1. Substituting in 𝜎 = 1, it follows that WE = (1 −
1∕N𝜀)PE . In the subsequent equilibrium characterization, we normalize the price of goods J to 1.
The profits of entrepreneur n are positive as long as the total number of businesses is not infi-
nite or the price elasticity of demand is not infinite. In the symmetric equilibrium, entrepreneur
n employs the labour of hE (n) = Q∕2N, so we can now drop the indexing for labour and
denote it as h instead. The real profits of entrepreneur n, 𝜋E (n), can be expressed as 𝜋E (n) =
𝜎h∕N𝜀. Entrepreneurs use profits to purchase goods from the two sectors. Let 𝜋(E) be the total
entrepreneurial profits in sector E, let pI be the relative price of goods I, and let cI (E), cJ (E) be
sector E entrepreneurs’ consumption of the two goods, so pI cI (E) + cJ (E) = 𝜋E . Entrepreneurs’
preferences for the two goods are assumed to be the same as those of the workers, to be specified
in a subsection below.
Pandemic shock
We now turn to an unanticipated short-lived pandemic shock that occurs towards the end of t = 0.
At the start of t = 0, entrepreneur n in sector I uses their previous sales revenues to pay wages for
the labour hired for the planned production in t = 0. Then, after the pandemic shock, sector I is
forced to close while sector J remains open. Thus sector I has no production materialized.
As said above, during the pandemic, the government provides loans for businesses to retain
workers in order to avoid large additional training and matching costs in the future. The prior
entrepreneurs totalling N differ in their expected profitability, and they can choose to borrow
and continue business, or they can choose to be a worker instead. If they choose to borrow, then
they pay wages immediately at the beginning of t = 1 to retain workers in this sector. To capture
potential crowding out, a new set of workers with new ideas totalling a fraction 𝜙 of the industry
cap can also borrow and become entrepreneurs. Since sector I was completely shut at t = 0, the
industry gap at t = 1 is simply N. In the future at t = 2, we take a benchmark assumption that the
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 11
industry gap is closed by new entries. Later, we will relax this assumption to capture the long-run
scarring effect of the pandemic.
If this set of workers and the prior entrepreneurs choose to borrow the government’s BBLs
of F1 , then they use the money to retain workers immediately at the start of t = 1. Among the
N prior entrepreneurs, a fraction of them totalling (1 − 𝛼)N have inefficient projects of different
productivity levels that are lower than the pre-pandemic level (i.e. 𝜎(n) < 1). The rest of these
prior entrepreneurs (𝛼N) and the normal set of workers with new ideas (𝜙N) are profitable; they
have good projects of the same productivity as the pre-pandemic level. We assume 𝜙 + 𝛼 < 1 for
two reasons. First, this assumption reflects that the pandemic worsens the overall profitability of
the adversely shocked sector in the short run. Second, this assumption is equivalent to 𝜙 < 1 − 𝛼,
which also implies limited scope for creative destruction. We define the wedge between 1 − 𝛼
and 𝜙 as Δ, and Δ = (1 − 𝛼)∕𝜙. So the limited scope for the creative destruction assumption
is essentially Δ > 1. The higher the creative destruction wedge, the less promptly unproductive
incumbents can be replaced by productive entrants.
When the fraction of prior entrepreneurs with inefficient projects manages to obtain govern-
ment loans to stay afloat, aggregate productivity will drop due to misallocation, that is, inefficient
production. The issue at hand is to design a mechanism that can deter borrowers with ineffi-
cient projects while taking into account the impact on unemployment. Specifically, if the prior
entrepreneurs decide not to borrow, then they simply become workers and receive wages. If
they borrow at the start of t = 1 to stay afloat, then they can carry out a project that produces
output as 𝜎(n) h1 once the shut-down restrictions are lifted. And 𝜎(n) is the project produc-
tivity of entrepreneur n, which represents their profitability. For the profitable entrepreneurs,
their project productivity is equal to 1, the same as the pre-pandemic level. For the less prof-
itable entrepreneurs, we assume that their project productivity 𝜎(n) follows a uniform distribution
𝜎(n) ∼ (𝜎B , 𝜎), where
1 2N − 2∕𝜀
𝜎 =1− , 𝜎B = . (1)
N𝜀 (1 − 𝛾)Q
The parametrization in equations (1) is not necessary for the main result but is used in the
benchmark case for ease of analytical exposition. As we will show in the equilibrium characteriza-
tion, the parameters 𝜎B and 𝜎 ensure the following. In the absence of sanctions, the entrepreneur
with the lowest profitability among the inefficient entrepreneurs will try to borrow government
loans to retain workers and stay afloat. However, after production resumes in the future, their low
revenues are simply insufficient for loan repayment. The entrepreneur with the highest profitabil-
ity among the inefficient entrepreneurs will try to borrow government loans, but after production
resumes in the future, their revenues are just enough to repay the loan obligations (in our numer-
ical simulations, we also vary 𝜎 and 𝜎B widely for robustness checks). However, as we will explain
shortly, because they can divert a fraction of their funds due to an imperfect monitoring and
verification technology, they will default on the government loans nevertheless.
Importantly, 𝜎(n) is the private information of the borrowing entrepreneurs. Different types
of entrepreneurs borrow the same amount to retain workers, so ex ante they are indistinguishable.
Furthermore, when the project productivity is realized at the end of t = 1 and the entrepreneur
defaults, the government takes away only a fraction 𝛾 of the entrepreneurs’ remaining funds.5 We
interpret 𝛾 as the quality of the government’s verification or monitoring technology.
First, we characterize the default decision of entrepreneur n assuming zero sanctions from
the government. When the project productivity is sufficiently high—so that if entrepreneur n
were to declare default after production, then the amount of funds that the government gar-
nishes would be higher than or equal to the loan obligations, i.e. 𝛾 𝜎(n) h1 ≥ F1 —the entrepreneur
would choose to repay fully. When the project productivity is sufficiently low that the revenues
are insufficient to repay loans—i.e. 𝜎(n) h1 < F1 —they are simply unable to repay the loans and
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12 ECONOMICA
have to default after production. However, there is an intermediate region for the project pro-
ductivity where the entrepreneurs will default even when they have enough revenue. We call this
type of default strategic default. With a relatively high project productivity, even if the revenues
are large enough to repay the loans in full, but due to limited commitment the government takes
away a relatively small amount of funds in the case of default such that 𝛾 𝜎(n) h1 < F1 ≤ 𝜎(n) h1 ,
the entrepreneur will nevertheless default strategically. Let 𝕀 be the indicator for default; that is,
𝕀 = 1 means default, and 𝕀 = 0 means repayment. Lemma 1 summarizes the endogenous choice
of default.
Lemma 1. (Default decision) Assuming zero default sanctions, the decision to default is summarized
by
⎧0, if F1 ≤ 𝛾 𝜎(n) h1 ,
⎪
𝕀 = ⎨1 (strategic), if 𝛾 𝜎(n) h1 < F1 ≤ 𝜎(n) h1 ,
⎪1, if 𝜎(n) h1 < F1 .
⎩
Lemma 1 indicates that the higher the project productivity 𝜎(n), the lower the likelihood
of default. To deter the inefficient entrepreneurs from borrowing and, in turn, reduce default,
the government may implement a default sanction as a screening contract such that when the
entrepreneurs default at the end of t = 1, a pecuniary deduction is taken from their residual
income. Because the borrowing entrepreneurs have the outside option of becoming a worker earn-
ing wage income, the residual income of any defaulting entrepreneur is positive (no less than the
outside option), and a pecuniary deduction is possible.
Let 𝜆1 be the default sanction. At the start of t = 1, entrepreneur n forms a conditional
expectation of their proceeds and evaluates them against their outside option. Suppose that
the borrowing entrepreneur n defaults. They can keep a fraction of the revenues totalling
(1 − 𝛾) 𝜎(n) h1 as their residual income (as in Rampini and Viswanathan 2013). And due to
the sanction, the total amount of money that they will receive at the end of t = 1 amounts to
(1 − 𝛾) 𝜎(n) h1 − 𝜆1 . However, had they chosen not to borrow and instead become a worker at the
start of t = 1, then they would have received wages of 𝔼1 (WI1 |𝕀d = 1, 𝜆1 ), where 𝕀d = 1 indicates
remaining as a worker. Therefore, if the incentive constraint
(1 − 𝛾) 𝜎(n) h1 − 𝜆1 ≥ 𝔼1 (WI1 |𝕀d = 1, 𝜆1 ) (2)
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
default outside option
holds, then they will choose to apply for government loans and stay afloat. In the case of the
equality sign, we assume that the entrepreneur chooses to borrow, and we call this entrepreneur
the marginal entrepreneur.
The left-hand side of the incentive constraint (2) states the gains if the entrepreneur defaults,
and the right-hand side is the value of her outside option. The outside option is the expected
wage if she were to remain as a worker.6
As inequality (2) is conditional on her choosing to default, her benefits of repayment must be
smaller or at most equal to her gains in the default case, so the following must hold:
(1 − 𝛾) 𝜎(n) h1 − 𝜆1 ≥ 𝜎(n) h1 − F1 .
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟
default repay
Note that as the government increases the sanction 𝜆1 , the inefficient entrepreneurs with low
profitability may be deterred from borrowing, reducing the aggregate default. This relationship
is proved to be monotonic in Proposition 1 below, after we define the equilibrium.
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 13
3.3 Government
The government choice variable is the default sanction 𝜆1 , and the government commits to the
sanction. The government provides loans of F1 to each borrowing entrepreneur to retain workers
and stay afloat,7 where
WI1 Q
F1 = .
2N
At the end of t = 1, some borrowing entrepreneurs default. Let df1 be the total amount of default,
and let Λ1 be the total money collected from sanctions. The government uses the money collected
via sanctions plus any borrowing if needed to cover default as in
df1 ≤ B1 + Λ1 .
If the money collected via default sanctions is insufficient to cover default—i.e. df1 > Λ1 —then
the government borrows money by issuing a one-shot undated consol of B1 to the workers and
entrepreneurs, and the government pays only the interest in future periods by raising an equiv-
alent amount of taxation. As we show shortly, after defining the equilibrium, the agents in the
economy have sufficient savings to lend to the government after the pandemic. Note that if the
government chooses to implement zero default sanctions, then the government essentially pro-
vides full guarantees for its loan scheme. As the government increases the default sanctions, the
credit guarantees decrease accordingly.
3.4 Workers
Workers consume goods in both sectors. Let us label workers by i ∈ Q. Let cIt (i) be the consump-
tion of sector I goods, and let cJt (i) be the consumption of sector J goods. Their preferences are
represented by the utility function
∑
2
𝔼t 𝛽 t U (cIt (i), cJt (i)) , (3)
t=0
where
( ( )𝜈 ( )𝜈 )(1−𝛿)∕(1−𝜈)
1 1 1
U (cIt (i), cJt (i)) = cIt (i)1−𝜈 + cJt (i)1−𝜈 . (4)
1−𝛿 2 2
The utility function satisfies U ′ > 0, U ′′ < 0, and it features CES 1∕𝜈 (𝜈 < 1) between the two
sectors’ goods bundles and constant inter-temporal elasticity of substitution 1∕𝛿.
In our benchmark case, we assume that workers have access to real, zero net supply,
one-period bonds, paying interest rate rt to share labour income risks. The approximate
real-world mapping of this assumption often takes the form of the government’s unemployment
benefits and various types of tax transfers. Risk sharing provides analytical convenience without
the loss of generality; moreover, it allows us to focus exclusively on the short-run and long-run
trade-offs of the government’s support loan scheme due to asymmetric information without intro-
ducing additional frictions. The policy question, therefore, is how much more the government
should care about short-run stabilization in addition to providing unemployment benefits.
First, we characterize the workers’ maximization. Due to the equal weighting of goods in
CES preferences, exploiting symmetry, the relative goods price is 1 and wages are the same across
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14 ECONOMICA
sectors. Let at (i) be the one-period bonds held by each worker, let pIt be the relative price of
goods I, and let rt be the interest rate.8 Each worker i maximizes function (3) subject to the budget
constraint
pIt cIt (i) + cJt (i) + at (i) ≤ wt ht (i) + (1 + rt−1 ) at−1 (i).
The optimality condition gives the Euler equation for consumption goods J. Let UcJ be the partial
derivative of U with respect to cJt . Given homothetic preferences, we have Gorman aggregation,
so the individual’s marginal rate of substitution between goods equals the relative price, which
is a macro variable. Hence if the Euler equation holds individually, then it also holds for the
aggregate consumption of the workers denoted as c(W ), as in
UcJ (cIt (W ), cJt (W )) = 𝛽(1 + rt ) UcJ (cIt+1 (W ), cJt+1 (W )).
In the absence of the pandemic, cIt (W ) = cIt+1 (W ) and cJt (W ) = cJt+1 (W ), so the interest rate is
r = 1∕𝛽 − 1.
When the pandemic shock hits at date t = 0, sector I shuts down, so cI0 = 0. The workers’
consumption falls onto sector J, and
Q( 1
)
cJ0 (W ) = 1− .
2 N𝜀
We define the natural interest rate in this context as the interest rate in the Euler equation in the
hypothetical case of no private information, and enough profitable entrepreneurs borrow at the
start of t = 1 to stay afloat. Thus the natural interest rate is the interest rate when the economy
operates as if in full potential from date t = 1 onwards.
Let r∗0 be the natural interest rate for date t = 0, so
1 UcJ (0, cJ0 )
1 + r∗0 =
𝛽 Uc∗J (cI1 , cJ1 )
1 UcJ (0, (1 − (1∕N𝜀))Q∕2)
=
𝛽 UcJ ((1 − (1∕N𝜀))Q∕2, (1 − (1∕N𝜀))Q∕2)
( )
1 1 (𝜈−𝛿)∕(1−𝜈)
= . (5)
𝛽 2
Lemma 2. With complete markets, and given the pandemic shock, r∗0 < 1∕𝛽 − 1, and the supply
shock causes a demand shortage at date t = 0 if and only if
𝜈 > 𝛿. (6)
Lemma 2 follows directly from equation (5); it is an insight from Guerrieri et al. (2022). It
states that when the two sectors’ goods are complements, the supply shock due to the pandemic
causes a demand shortage. In a single-sector economy, a negative supply shock typically increases
the current-period marginal utility of consumption, which increases the natural interest rate.
Because a negative supply shock causes a demand boom in a single sector at t = 0, the interest rate
increases to equilibrate the economy. However, in our environment and also in the multi-sector
case of Guerrieri et al. (2022), inequality (6) implies that the two goods are complements, and
interestingly, the natural rate decreases. The reason is that a negative supply shock decreases the
current-period marginal utility of consumption due to goods being complements. So the negative
supply shock causes a demand shortage for goods in the unshocked sector, and the interest rate
needs to decrease to equilibrate the economy. This characteristic, thanks to a multi-sector setup,
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 15
will have interesting implications for designing the screening contract and setting the optimal
default sanction.
If the interest rate is downward rigid, for example, at the ELB where the interest rate cannot
adjust downwards sufficiently, then it leads to a decline in the demand for goods J, and hence
involuntary unemployment in sector J. Indeed, akin to the Drèze equilibrium, when prices are
downward rigid, the supply is rationed (Drèze 1975).
3.5 Equilibrium
The two-sector equilibrium with imperfect competition is defined as an allocation with prices,
given the screening contract 𝜆1 such that:
(i) entrepreneurs engage in imperfect competition and choose their actions simultaneously
taking into account their price impact;
(ii) agents maximize subject to borrowing frictions, the incentive constraint and budget con-
straints;
(iii) goods markets, labour markets and loan markets clear, and expectations are rational—in
particular, goods market clearing conditions are that the output of each sector is consumed
by workers and the entrepreneurs of both sectors, that is,
∑ ∑
cIt (W ) + cIt (E) = qIt , cJt (W ) + cJt (E) = qJt .
E=I,J E=I,J
4 EQUILIBRIUM CHARACTERIZATION
With our equilibrium definition, first, Lemma 3 shows that the agents in the economy have suf-
ficient savings to lend to the government after the pandemic. Due to the pandemic shock, there
is zero production in sector I, and the economy is unable to spend all of its nominal income on
goods J alone at the end of t = 0. Consequently, agents in the economy end up having extra money
as savings at the end of t = 0, and carry it forward. As proved in the Appendix, this amount
of savings is more than enough to invest in the government’s one-shot issuance of the undated
consol.
Lemma 3. At the end of t = 1, agents in the economy have sufficient savings to finance the
government borrowing of B1 .
Proof. See the Appendix. ▪
Now let us suppose that the government cares only about resource allocation and sets the
default sanction 𝜆1 sufficiently harshly to deter less profitable borrowers. We derive the harsh
default sanction 𝜆A1 so that it is a Nash equilibrium for the entrepreneurs with profitable projects
to apply for loans, while those with inefficient projects become workers. Then we suppose that
the government wants to ensure full employment by setting a lenient default sanction. We solve
for such a lenient sanction 𝜆B1 so that it is a Nash equilibrium for the profitable entrepreneurs
along with a fraction of inefficient entrepreneurs to apply for loans, and the rest become workers,
while ensuring that full employment is achieved from t = 1 onwards. To derive the expressions
for the harsh and lenient default sanctions, we first need to check that monotonicity holds—put
differently, that the harsher the sanction, the higher the project productivity of the marginal
entrepreneur, and the lower the number of defaults.
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16 ECONOMICA
Let M (≤ N) be the number of borrowing entrepreneurs. We define uI1 (M) as the unemploy-
ment rate in sector I at date t = 1 as follows, and Proposition 1 proves monotonicity:
QM∕2N
uI1 (M) = 1 − .
(Q∕2) + N − M
Proposition 1. (Monotonicity) Whenever df1 > 0 and M ≤ N, an increase in the default sanction
𝜆1 leads to higher profitability of the marginal entrepreneur and fewer defaults.
Proof. See the Appendix. ▪
Given monotonicity, Theorem 1 below derives the harsh default sanction 𝜆A1 and the lenient
default sanction 𝜆B1 . A harsh sanction indicates a pro-allocation government, and a lenient
sanction indicates a pro-stabilization government.
Theorem 1. (Screening, default and unemployment)
(a) Suppose that the government sets the sanction 𝜆1 > 𝜆A1 , where
( )( )
1 Q
𝜆A1 = 1 − (1 − 𝛾) − (1 − uI (𝜙N + 𝛼N)) .
N𝜀 2N
Then the (𝜙 + 𝛼)N profitable entrepreneurs will borrow, and all the inefficient entrepreneurs
will stay out, and no one defaults.
The sector I short-run unemployment rate is uI1 (𝜙N + 𝛼N). Full employment obtains only in
the long run.
(b) Suppose that the government sets the sanction as 𝜆B1 , where
( ) ( )
𝜙 1−𝜙−𝛼 Q 1
𝜆B1 = (1 − 𝛾) 𝜎+ 𝜎B − 1− (1 − uI (N − 1)) .
1−𝛼 1−𝛼 2N N𝜀
Then (1 − 𝜙 − 𝛼)N inefficient entrepreneurs will borrow and then default, but sector I achieves
full employment in both the short run and the long run.
(c) With no sanctions, all the prior entrepreneurs will apply for loans, and a fraction of profitable
applicants are crowded out.
Proof. See the Appendix. ▪
Theorem 1 states that when the default sanction is sufficiently harsh, the government can
keep all the (1 − 𝛼)N inefficient projects out. In this scenario, the economy suffers persistent
unemployment. However, if the government chooses a sufficiently lenient default sanction, then
it can restore full employment immediately at date t = 1 and beyond. However, in this case,
the entrepreneurs with low profitability stay afloat, harming aggregate productivity in sector I
at both t = 1 and t = 2. Furthermore, if the government does not impose default sanctions at
all, then all the (1 − 𝛼)N inefficient entrepreneurs will try to borrow. This scenario will cause
some of the (𝜙 + 𝛼)N profitable entrepreneurs to be excluded, and further dampen aggregate
productivity. In Theorem 1, the harsh default sanction 𝜆1 > 𝜆A1 and the lenient default sanction
𝜆1 = 𝜆B1 outline two extreme cases. It is possible that the optimal default sanction could be an
intermediate case.
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 17
Furthermore, two interesting observations emerge when we rewrite the harsh sanction 𝜆A1
and the lenient sanction 𝜆B1 in terms of the creative destruction wedge Δ. Note that
( )( ( ( ) ))
1 Q 1−𝛼
𝜆A1 = 1 − (1 − 𝛾) − 1 − uI +𝛼 N
N𝜀 2N Δ
and
( ) ( )
𝜎 − 𝜎B Q 1
𝜆B1 = (1 − 𝛾) + 𝜎B − 1− (1 − uI (N − 1)) .
Δ 2N N𝜀
We can see that 𝜆A1 increases with Δ, whereas 𝜆B1 decreases with Δ. When the creative destruction
wedge is large, unproductive incumbents cannot be replaced promptly by productive entrants. To
fully stabilize short-term employment, the threshold of the lenient sanction has to decrease. To
remove resource misallocation in the long run, the threshold of the harsh sanction has to increase,
since the lack of business dynamism implies that the outside option is weak, and the government
has to be more hawkish to keep out the inefficient ones. Therefore the lack of creative destruction
(higher Δ) polarizes the two extreme cases even more.
Before moving on to solve for the optimal default sanction, Proposition III characterizes the
interest rate dynamics via the aggregate Euler equations of these two polar cases.
Proposition 2. (Sanctions, interest rates and demand shortage) Suppose that inequality (6) holds.
(a) If 𝜆1 > 𝜆A1 , r0 < 1∕𝛽 − 1 and
( )(𝜈−𝛿)∕(1−𝜈)
1 1
1 + r0 = ,
𝛽 (𝜙 + 𝛼)1−𝜈 + 1
then the interest rate remains below 1∕𝛽 − 1 for t = 1, and demand shortage is persistent.
(b) If 𝜆1 = 𝜆B1 , then r0 < 1∕𝛽 − 1 also holds, but from t = 1 onwards, the interest rate immediately
returns to 1∕𝛽 − 1. Demand shortage is short-lived.
Proof. See the Appendix. ▪
Proposition states that when the government is pro-allocation and sets a harsh default
sanction, the interest rate goes down and remains low persistently. This harsh sanction stifles
production in the short run, and the supply shortage leads to a demand shortage due to the com-
plementarity of the two sectors. In equilibrium, the consumption for goods I increases gradually
while the consumption for goods J remains the same. Given 𝜈 > 𝛿, goods I and goods J are com-
plements, so the marginal utility of consumption for goods J immediately drops before it increases
gradually. Demand shortage is persistent. However, if the government is pro-stabilization and
sets a lenient sanction, then the interest rate goes down only for date t = 0 due to the pandemic
shock, and then it immediately returns to 1∕𝛽 − 1. Demand shortage is short-lived. The reason is
that the cheap loan programme attracts enough entrepreneurs to stay afloat, and full employment
is reached immediately, at the possible cost of lowering future productivity.
5 OPTIMAL DEFAULT SANCTIONS
To solve for the optimal sanction, we first define the social welfare function and then allow the
government to choose the number of borrowing entrepreneurs M, where M ∈ [(𝜙 + 𝛼)N, N],
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18 ECONOMICA
subject to the decentralized optimality conditions of the entrepreneurs and workers. We will show
shortly that M has a one-to-one mapping with the default sanction 𝜆1 . The M ∗ that maximizes
the social welfare corresponds to the optimal default sanction 𝜆∗1 .
5.1 Analytics
We assume that the government assigns equal weights to every worker and every entrepreneur in
the economy, so the social welfare function takes the form of the sum of the consumption utilities
of all agents in the economy. Let cIt represent the aggregate consumption of goods I, and let cJt
represent the aggregate consumption of goods J. Recall that workers and entrepreneurs all exhibit
the same CES preferences for consumption goods, and their utility function is homogeneous of
degree 1. Then the social welfare function at date t = 1 takes the form
( ( )𝜈 ( )𝜈 )(1−𝛿)∕(1−𝜈)
∑
2
1 1 1
V1 = 𝔼1 𝛽 t−1 1−𝜈
cIt + 1−𝜈
cJt . (7)
t=1
1−𝛿 2 2
Let 𝜎(n′ ) denote the marginal entrepreneur’s project productivity when the number of loan
applicants is M. Conditional on M, there is no uncertainty from date t = 1 onwards, so we drop
the expectation sign hereafter. Given that 𝜎(n) follows a uniform distribution, 𝜎(n) ∼ (𝜎B , 𝜎),
it follows that
( )
M − (𝛼 + 𝜙)N M − (𝛼 + 𝜙)N
𝜎(n′ ) = 1− 𝜎+ 𝜎B , (8)
(1 − 𝛼)N (1 − 𝛼)N
so the default sanction that corresponds to M is
(( ) )
M − (𝛼 + 𝜙)N M − (𝛼 + 𝜙)N Q
𝜆M1 = (1 − 𝛾) 1− 𝜎+ 𝜎B
(1 − 𝛼)N (1 − 𝛼)N 2N
( )
1
− 1− (1 − uI1 (M − 1)) . (9)
N𝜀
Thus M and 𝜆M1 have a one-to-one mapping.
First, given M, we solve the aggregate consumption of the two goods by substituting in the
decentralized optimality conditions of the entrepreneurs and the workers. Thus we can express
the aggregate consumptions cIt and cJt as functions of exogenous parameters and the policy vari-
able M, and hence the social welfare function can be re-expressed as functions of M with other
exogenous parameters. What then remains to be done is to search for the M that maximizes the
social welfare function. Given equation (9), which expresses the default sanction 𝜆M as a func-
tion of exogenous parameters that include the number of borrowing entrepreneurs M, as long as
the optimal M is obtained, we can derive the optimal sanction.
At t = 1, there are (𝛼 + 𝜙)N profitable entrepreneurs borrowing, and the number of ineffi-
cient borrowers amounts to M − (𝛼 + 𝜙)N. Thus the aggregate consumption of sector I goods in
equilibrium is given as
( )
Q 𝜎(n′ ) + 𝜎
cI1 = (M − 𝜙N − 𝛼N) + (𝛼 + 𝜙)N , (10)
2N 2
where 𝜎(n′ ) is given in equation (8).
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 19
At t = 2, the industry gap closes and the number of entries at t = 2 amounts to N − M. The
aggregate consumption of goods I at t = 2 is thus expressed as
( )
Q 𝜎(n′ ) + 𝜎
cI2 = (M − 𝜙N − 𝛼N) + (𝛼 + 𝜙)N + (N − M) . (11)
2N 2
The aggregate consumption of goods J remains Q∕2 throughout when the interest rate is
unconstrained from the ELB.
Therefore substituting equations (10) and (11), and cJt = Q∕2, in the social welfare func-
tion (7), the social welfare function can be expressed as a function of exogenous parameters and
the policy variable M. Before we solve for the optimal M ∗ , we first develop an analytic measure
conditional on M, which we call ‘stabilization proclivity’ (SP) to characterize the government’s
policy stance: the higher the stabilization proclivity, the more likely the government will set a
lenient sanction (i.e. being closer to 𝜆B1 ); the lower the stabilization proclivity, the more likely the
government will set a harsh sanction (i.e. being closer to 𝜆A1 ). Formally, SP is expressed as
( )−1
1 2
SP = −1 .
𝛽 𝜎(n ) + 𝜎 + (𝜎B − 𝜎)(M − 𝜙N − 𝛼N)∕(1 − 𝛼)N
′
To understand the economic intuition of our measure, let us use U(cIt , cJt ) to denote the
single-period utility, that is,
(( )𝜈 ( )𝜈 )(1−𝛿)∕(1−𝜈)
1 1 1
U(cIt , cJt ) = 1−𝜈
cIt + 1−𝜈
cJt .
1−𝛿 2 2
Then the first-order derivative of V1 (M) for M is expressed as
𝜕V1 (M) 𝜕U(cI1 , cJ1 ) 𝜕U(cI2 , cJ2 )
= +𝛽 . (12)
𝜕M 𝜕M 𝜕M
As we show in Proposition IV below, the first term on the right-hand side of equation (12),
the short-run social utility, increases with M, that is, 𝜕U(cI1 , cJ1 )∕𝜕M > 0, and the second term
̃ < 0.
̃ I2 , cJ2 )∕𝜕 M
on the right-hand side, the long-run social utility, decreases withM, that is, 𝜕 U(c
When the first term dominates the second term, the government may be more pro-stabilization,
as it would prefer a larger M; when the second term dominates the first term, the government
may be more pro-allocation. We develop the measure SP, borrowing the concept of marginal
rate of substitution. The marginal rate of substitution between short-run and long-run social
utility is
( )/( )
̃ I1 , cJ1 )
𝜕 U(c ̃ I2 , cJ2 )
𝜕 U(c
− 𝛽 .
𝜕M̃ 𝜕M̃
Let qIt be the sector I production at date t, and as we will show in Proposition IV, the measure SP
̃
is derived from −(𝜕qI1 ∕𝜕 M)∕(𝛽 ̃ and it offers an indication of whether the government
𝜕qI2 ∕𝜕 M),
prefers to be more pro-allocation or more pro-stabilization.
Proposition 3. (Stabilization proclivity) Given equations (10) and (11), and M ∈ [(𝜙 + 𝛼)N, N], the
short-run social utility increases with M, and the long-run social utility decreases with M. That is,
𝜕U(cIt , cJt ) 𝜕U(cI2 , cJ2 )
> 0, < 0,
𝜕M 𝜕M
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20 ECONOMICA
and SP is derived from
( )−1
𝜕qI1 ∕𝜕M 1 2
− = −1 .
𝛽 𝜕qI2 ∕𝜕M 𝛽 𝜎(n′ ) + 𝜎 + (𝜎B − 𝜎)(M − 𝜙N − 𝛼N)∕(1 − 𝛼)N
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
SP
Proof. See the Appendix. ▪
SP depends on exogenous parameters. By investigating how SP responds to changes in these
parameters, Proposition 4 characterizes the government’s policy stance on the trade-off between
allocation and stabilization.
Proposition 4. (Policy stance) Keeping all other parameters unchanged, we have the following.
(a) An improvement in the monitoring technology 𝛾 increases SP.
(b) An increase in 𝜀 or N decreases entrepreneurial rents and increases SP.
(c) An increase in 𝜎B increases SP.
(d) An increase in 𝛽 decreases SP.
Proof. See the Appendix. ▪
When the monitoring/verification technology improves, the government tends to be
pro-stabilization and set a more lenient default sanction. If the monitoring technology is poor,
then the government tends to set a harsher sanction to deter inefficient entrepreneurs. If a gov-
ernment is myopic—e.g. the government is more impatient than the public—then the government
discounts the future utilities more; consequently, the government will take a very pro-stabilization
stance. For example, if the government cares about immediate re-elections, then it is likely to set
a lenient default sanction or no sanctions at all, to promote short-term employment and sacrifice
long-term productivity. The overall toll on social welfare could be substantial.
When the price elasticity of demand is high or the size of the entrepreneurial pool is large,
the industry is more competitive, so entrepreneurial rents decrease and workers’ wages increase.
Consequently, the government tends to be more pro-stabilization. On the contrary, with declining
competition and rising markup, the government tends to be pro-allocation. Furthermore, an
increase in 𝜎B increases SP, and a decrease in 𝜎B decreases SP. The reason is that as the lowest
profitability 𝜎B decreases, the overall quality of the prior entrepreneurs worsens. Consequently,
the government tends to be less pro-stabilization and more pro-allocation.
5.2 Analytics of the ELB
The interest rate ELB limits monetary policy and constrains it from falling to prop up demand
in the unshocked sector. Goods J produced by the unshocked sector are complements to goods I
that are produced by the adversely shocked sector, so the marginal utility of consumption for
goods J decreases, leading to a demand shortage. The interest rate should decrease to equilibrate
the economy, leaving the supply of goods J unconstrained. When the interest rate cannot adjust
downwards, however, it causes an inward shift of the supply curve, and in turn, involuntary unem-
ployment in sector J. This is an aggregate demand externality that the government would want to
avoid. Therefore the social cost of a harsh default sanction may outweigh that of our benchmark.
Proposition 5 formalizes our argument.
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 21
Proposition 5. (ELB, default sanctions and involuntary unemployment) Suppose that the economy
is at the ELB where the interest rate cannot go below 1∕𝛽 − 1. Then demand shortage leads to
involuntary unemployment at sector J.
Proof. See the Appendix. ▪
Remark. At the interest rate ELB, the optimal default sanction may be more lenient than its
counterpart when the interest rate is unconstrained.
To see why this is, let us observe in
𝜕U(cIt , cJt ) 𝜕U(cIt , cJt ) 𝜕cIt 𝜕U(cIt , cJt ) 𝜕cJt
= +
𝜕M 𝜕cIt 𝜕M 𝜕cJt 𝜕M
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟
X Y
how the single-period utility of consumption U(cIt , cJt ) changes with respect to the number of
loan applicants M when full employment is not achieved in sector I.
As shown in Proposition 3, the increase of M, or equivalently the leniency of the default
sanction, leads to an increase in the single-period utility via increasing the marginal utility of
consuming goods I, that is, X > 0. At the ELB, the demand shortage causes the consumption of
goods J to decrease, so the marginal utility of consuming goods I, 𝜕U(cIt , cJt )∕𝜕cIt , is lower than
its counterpart when the interest rate is unconstrained, thus X is lower than in the unconstrained
case.
However, in our benchmark case, the increase of M causes no externality to sector J, so it does
not change the single-period utility via the marginal utility of consuming goods J, that is, Y = 0.
In contrast, at the ELB, as we have shown in Proposition 5, an increase in the consumption of
goods I leads to an increase in the consumption of goods J, and it follows that the leniency of
the default sanction increases the consumption of goods J, via the aggregate demand externality
channel, that is, 𝜕cJt ∕𝜕M > 0. Therefore an increase in M, or equivalently the leniency of the
default sanction, leads to an increase in the single-period utility via increasing the marginal utility
of consuming goods J. So at the ELB, it follows that Y > 0, countervailing the reduction in X .
Overall, the increase of M, or equivalently the leniency of the default sanction, may contribute
to a higher utility gain at the ELB than the economy away from the ELB, suggesting that the
government may take a more lenient stance in the ultra-low interest rate environment.
6 NUMERICAL ANALYSIS
In this section, we assign numerical values to deep parameters in Table 1 as the benchmark
case, and we run numerical simulations to illustrate the optimal default sanctions with different
features of the underlying economy.
The number of entrepreneurs in each sector is set to be 10 in normal times, and the total
number of workers is 800. Thus each small business firm employs 40 workers in normal times.
According to the UK government’s 2021 national statistics, 5.5 million businesses are small,
which means that they have fewer than 50 employees. Thus we believe that our benchmark
employee number of 40 is a reasonable size for small businesses. Later, we also vary N widely
to change the number of workers per firm for robustness checks. The price elasticity of demand
is chosen to be 1.2, so that in normal times, the entrepreneur’s profit is around 3.6 times the
worker’s wages. The CES 𝜈 is therefore calibrated internally as 0.83, and the inter-temporal elas-
ticity of substitution (1∕𝛿) is set to be 1.43, so that 𝜈 > 𝛿 holds; that is, goods in sectors I and J are
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22 ECONOMICA
TA B L E 1 Parametrization
Discount factor 𝛽 = 0.97
Monitoring technology 𝛾 = 0.7
Price elasticity of demand 𝜀 = 1.2
Inter-temporal elasticity of substitution 1∕𝛿 = 1.43
Fraction of new workers entering 𝜙 = 0.1
Fraction of profitable prior entrepreneurs 𝛼 = 0.5
Number of entrepreneurs in each sector N = 10
Total number of workers Q = 800
FIGURE 4 Output and default
sanctions.
complements. In the subsequent sections, we vary 𝜈 and 𝛿 widely for robustness checks and also
for assessing the sensitivity of interest rate changes to these parameters.
The fraction of new workers with new ideas as potential entrepreneurs in each period is set
to be 0.1, and the fraction of profitable prior entrepreneurs is set to be 0.5, so that if we were to
incorporate the model to an infinite setting where each period would correspond to one year, then
it would take 5 years for sector I to reach full employment and fully restore production if the gov-
ernment sets a harsh default sanction to keep all inefficient projects out. In the second subsection
of this section, we extend the model to a dynamic setting and explain why this reaches a 5-year
target. Given the severity of the COVID-19 pandemic, we think 5 years is a plausible figure. For
robustness checks, in the later sections we also vary this year number. The discount factor 𝛽 is
set to 0.97, which is close to a quarterly discount factor of 0.99, the same as in Ottonello and
Winberry (2020). And finally, we set the monitoring technology 𝛾 to be 0.7. The parametrization
in Table 1 and our assumption of the distribution of the inefficient entrepreneurs’ project pro-
ductivity as in equations (1) implies that the highest project productivity 𝜎 among the inefficient
entrepreneurs is around 0.92, and that the lowest project productivity 𝜎B among them is around
0.076.
Figure 4 illustrates the responses of sector I output to the default sanction in the bench-
mark case. Its horizontal axis indicates the time that starts at t = −1, which corresponds to the
pre-pandemic equilibrium. The dashed line corresponds to a pro-allocation government that sets
the harsh default sanction 𝜆A1 , the dotted line corresponds to a pro-stabilization government that
sets the lenient default sanction 𝜆B1 , and the solid line corresponds to the optimal default sanction
that maximizes the social welfare. Let us observe the case of the pro-stabilization government. As
the pandemic occurs at date t = 0, illustrated by the dotted line, the sector I output drops to zero
before it goes back up at t = 1 at a lower level than the pre-pandemic equilibrium, and remains
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 23
FIGURE 5 Optimal default.
there ever since. This is because the pro-stabilization government uses a lenient default sanction
to mop up the unemployment immediately after the pandemic passes.
Let us now turn to the dashed line in Figure 4, which indicates a pro-allocation govern-
ment. Compared with the pro-stabilization government, the output does not rebound as much
at t = 1, because of short-run unemployment. Output gradually increases and overtakes the
pro-stabilization case only at t = 2. At t = 2, the economy bounces back to a long-run equilibrium
at exactly the same pre-pandemic level. The reason is simple: the pro-allocation government uses
a harsh default sanction to deter all the inefficient entrepreneurs so that the long-run productiv-
ity can be restored fully. The case of the optimal default sanction is illustrated by the solid line.
It is intermediate and is a result of balancing the trade-off between short-term employment and
long-run productivity. The optimal case does not suggest a lenient default sanction to mop up
unemployment completely, and it also tolerates some productivity loss in the long run so that the
short-run unemployment rate is lower. The implication for defaults, almost by definition, varies
with the harshness of the sanction. As can be seen in Figure 5, the harsh default sanction that
corresponds to M = 6 rules out default completely, and the lenient default sanction that corre-
sponds to M = 10 leads to 40% of borrowers defaulting. The optimal default corresponding to
M ∗ is again intermediate.9
Then we increase the number of entrepreneurs in each sector from the benchmark case of 10
to 40.10 This parameter change increases industry competitiveness and reduces entrepreneurial
rents. In turn, labour income share accounts for a larger fraction of total output. Let us turn to
Figure 6. The vertical axis is the value of total social utility denoted as V , which is defined in
equation (7). The horizontal axis is the number of borrowing agents, each of which corresponds
to a default sanction. In particular, the number of borrowers M = 6 corresponds to the harsh
default sanction 𝜆A1 , and the number of overall borrowers M = 10 corresponds to the lenient
default sanction 𝜆B1 . The number of borrowers decreases monotonically with the harshness of
the default sanction. As we can see in the second plot of Figure 6, the optimal default sanction
becomes a more lenient stance as stabilization has become more pertinent.
We also increase the price elasticity of demand 𝜀 to 4 from the benchmark case 1.2, which
decreases the entrepreneurial rents. As we show in Figure A3 in the Appendix, it causes a shift
to a more lenient stance, since lower rents imply a higher wage, which worsens the utility loss
associated with unemployment.
6.1 Interest rate ELB and firm markup
In this subsection, we assess numerically the implication of the interest rate ELB for the optimal
default sanction and its interaction with firm markups. As our numerical example illustrates in
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24 ECONOMICA
V V
47.8 V*
V*
47.6
47.4
47.2
47
46.8
M* M*
46.6
M M
6 6.4 6.8 7.2 7.6 8 8.4 8.8 9.2 9.6 10 24 25.6 27.2 28.8 30.4 32 33.6 35.2 36.8 38.4 40
(Benchmark)
FIGURE 6 Social welfare, optimal sanctions and N .
V
47.8 V
47.8
47.6 V*
47.6 V*
47.4
47.4
47.2
47.2
47 47
46.8 46.8
M* M*
46.6 46.6
6 6.4 6.8 7.2 7.6 8 8.4 8.8 9.2 9.6 10 M 6 6.4 6.8 7.2 7.6 8 8.4 8.8 9.2 9.6 10 M
Benchmark ELB
FIGURE 7 Social welfare, optimal sanctions and ELB.
Figure 7, at the ELB, the overall welfare is lower than the benchmark case due to involuntary
unemployment in sector J, and the optimal default sanction turns out to be more lenient than
the benchmark case. With our benchmark parameters, the unbounded interest rate would fall to
−39%. This fall in the interest rate is stark because we model only two sectors in the economy,
and 50% is forced to close during t = 0. As we see in Table 2, this means that the optimal default
sanction becomes 24% more lenient than the case without the ELB. Furthermore, as shown
in Guerrieri et al. (2022), market incompleteness relaxes the conditions for a negative supply
shock to cause the demand shortage; therefore, when the government fails to provide unemploy-
ment benefits to insure workers against labour income risks, the ELB is likely to exacerbate the
aggregate demand externality, and the optimal default sanction would turn out to be even more
lenient.
In the benchmark parameter space, the large difference between the intra-temporal elasticity
1∕𝜈 and the inter-temporal elasticity 1∕𝛿 implies a large drop in the unbounded interest rate
during the pandemic, which generates the noticeable shift in the leniency stance of the optimal
default sanction. The unbounded interest rate reduction is key. To understand its sensitivity, in
our comparative statics in Table 2, we reduce the spread between 𝜈 and 𝛿 to generate a modest
fall in the unbounded interest rate. When we increase 𝛿 to 0.82 while keeping 𝜈 the same as the
benchmark case, the difference between them decreases from 0.13 to 0.01. This produces the
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 25
TA B L E 2 Interest Rate Changes, Intra/Inter-temporal Elasticities, Optimal Policies
𝜈 𝛿 𝜈−𝛿 Unbounded r0 Change in leniency
0.83 0.7 0.13 −39% 24%
0.83 0.82 0.01 −2% 0.57%
0.71 0.7 0.01 −0.06% 0.51%
F I G U R E 8 Sector J 0.05
involuntary unemployment
and firm markup. Notes: The 0.04
horizontal axis is the number
of borrowers that participate,
0.03
normalized by industry size.
The further to the right, the
more lenient the default 0.02
sanction. The vertical axis is
the involuntary 0.01
unemployment rate in
sector J.
0
0.6 0.64 0.68 0.72 0.76 0.8 0.84 0.88 0.92 0.96 1 M/N
Benchmark parameters epsilon =1.4 N=40 epsilon =1.05
unbounded interest rate as −2% per annum. In equilibrium, the optimal default sanction becomes
0.57% more lenient.
Then we decrease 𝜈 to reduce the spread of 𝜈 − 𝛿 to 0.01 as well, but compared with the pre-
vious case, the unbounded interest rate falls to only −0.06%, a larger change compared with the
benchmark case. This is because the interest rate dynamics is more sensitive to the intra-temporal
elasticity and less to the inter-temporal elasticity. This leads to the optimal default sanction being
0.51% more lenient than the benchmark case. Interestingly, the change in the optimal sanction in
this case is not so different from the previous case where the unbounded interest rate falls more.
This is because although the unbounded interest rate falls less in this case, which should imply a
more modest change in leniency, the decrease in 𝜈 increases the price elasticity of demand, which
reduces the firm profits, causing the change in leniency to increase.
The above numerical sensitivity analysis suggests that the price elasticity and firms’ markup
power would affect the leniency of the optimal default sanction in a low interest rate environment.
Motivated by the stylized fact that we have documented pre-pandemic on the fall in interest rates
and the rise in markup, we conduct the following experiment by changing the industry size and
the price elasticity of demand such that firm markup changes in equilibrium. Figure 8 documents
how involuntary unemployment in sector J changes accordingly. As can be seen, when the price
elasticity 𝜀 decreases from 1.2 (benchmark) to 1.05, which means that firm markup increases
and rents increase, involuntary unemployment in sector J increases across the whole spectrum
of default sanction. In particular, when the default sanction is at the harsh stance that deters
all inefficient projects, involuntary unemployment in sector J increases to 7.5%. This is striking
because sector J is allowed to open and operate, and pandemic policies are applied only in sector I,
and the parameter change is also applied only in sector I, rather than sector J. This reflects the
aggregate demand externality due to complementarity between sector I and sector J in the interest
rate ELB environment. This decrease in 𝜀 should push the optimal default sanction to be more
lenient. In our numerical simulations, the optimal default sanction becomes 4.37% more lenient
than the benchmark parameter space.
Similarly, when we increase the price elasticity from 1.2 to 1.4, the involuntary unemployment
in sector J still exists, but it is noticeably less across all spectrums of the default sanction. This
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26 ECONOMICA
would mean that the change in leniency of the optimal default sanction is small (only 0.5% in
our numerical simulation). Finally, the change in industry size almost has little effect on sector J
involuntary unemployment because the industry size does not have a first-order effect on interest
rate dynamics, whereas the elasticity parameters do.
One qualification of the above aggregate demand externality is that it has the opposite impli-
cation of congestion externalities commonly assumed in the literature of credit misallocation.
As in the seminal contribution of Caballero et al. (2008), subsidized credit to poorly performing
firms affects aggregate productivity and output not only because it attracts more inefficient firms
to stay afloat, but also because of congestion externalities in input and output markets due to
the presence of zombie firms, which reduce the productivity or profitability of healthy firms. For
example, in Acharya et al. (2021), the authors model the productivity of profitable firms being
negatively affected by the extent of zombie lending in the previous period, such that policies
aimed at avoiding short-term recessions can be trapped into protracted low rates and excessive
forbearance, possibility leading to permanent output losses. Therefore such congestion exter-
nalities would tilt the optimal default sanction in our model towards a harsher stance, exerting
opposite effects as opposed to the aggregate demand externality.
6.2 Creative destruction
The creative destruction wedge Δ that we have defined reflects the extent to which unproductive
incumbents can be replaced promptly by productive entrants. Since the wedge reflects business
dynamism, to capture this meaningfully, we modify the benchmark model slightly by introducing
a simple dynamic dimension as follows.
During normal times in every period, a proportion 𝜙 of the N existing entrepreneurs find their
technology outdated and exit. Meanwhile, an equal number of workers totalling 𝜙N (< Q) enter
and become entrepreneurs. The economy is in the efficient steady state. When the pandemic shock
forces sector I to close at the end of t, the normal set of workers with new ideas totalling 𝜙N, and
the prior entrepreneurs totalling N, can choose to borrow government support loans and retain
workers, as in the static case. The difference is now that it takes multiple periods for sector I to
recover because each normal period has only a fraction 𝜙 of new entries. Figure 9 illustrates this
point intuitively. With our benchmark parameters, the fraction of new workers entering in each
period is set to be 0.1, and the fraction of profitable prior entrepreneurs is set to be 0.5, so that
it would take 5 years for sector I to reach full employment and fully restore production if the
government sets a harsh default sanction to keep all unprofitable projects out. Given the severity
of the COVID-19 pandemic, we think 5 years is a plausible figure, and we also vary this figure
for sensitivity analysis.
Now we vary the composition of 𝛼 and 𝜙, both of which matter for the wedge Δ and the
number of years before sector I recovers. By varying the composition of the creative destruction
wedge, we can obtain the corresponding change in optimal default sanction, and we define the
F I G U R E 9 Time periods before
sector I reaches full employment. Notes:
The horizontal axis is the number of firms
cumulatively in sector I. The vertical axis is
the time periods before sector I reaches full
employment.
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 27
changes as the normalized distance from the hawkish stance 𝜆A , that is,
M ∗ − (M|𝜆A )
,
(M|𝜆B ) − (M|𝜆A )
so the further the distance, the more lenient the optimal policy.
As Table 3 shows, the benchmark wedge with 𝛼 = 0.5 and 𝜙 = 0.1 is equal to 5, and the number
of years before sector I employment recovery is 5. The wedge turns out to be equal to the number
of years before sector I employment recovery. In this case, the distance from 𝜆A is 7.5%. When
we increase 𝜙 to 0.124 or increase 𝛼 to 0.6, the wedges in both cases are equal to 4: the distance
from 𝜆A becomes smaller compared with the benchmark case. When we decrease 𝛼 or decrease 𝜙
to decrease the wedge, the distance from 𝜆A becomes larger compared with the benchmark case.
This means that as the creative destruction wedge increases, the optimal policy becomes more
lenient because higher wedges imply that the economy takes a longer time for productive ones to
enter. Thus higher wedges lead to higher levels of resource misallocation.
Furthermore, we find that the optimal policy tends to be slightly more sensitive to the changes
in 𝛼 than to changes in 𝜙. For example, keeping the wedge the same at 6, we can either reduce 𝛼
TA B L E 3 Creative Destruction Wedge and Composition of 𝛼 and 𝜙
Years 𝛼 𝜙 Wedge Δ % from 𝜆A
4 0.5 0.124 4 6.67%
4 0.6 0.1 4 6.67%
5a 0.5a 0.1a 5 7.5%
6 0.4 0.1 6 10%
6 0.5 0.083 6 8%
6.5 0.35 0.1 6.5 11%
6.5 0.5 0.077 6.5 9%
7 0.3 0.1 7 11.7%
7 0.5 0.071 7 10%
Notes a Indicates benchmark value.
0.05
0.04
0.03
0.02
0.01
0 M/N
0.6 0.64 0.68 0.72 0.76 0.8 0.84 0.88 0.92 0.96 1
Benchmark parameters alph a=0.3 Delta=50
F I G U R E 10 Sector J involuntary unemployment and creative destruction. Notes: The horizontal axis is the
number of borrowers that participate, normalized by industry size. The further to the right, the more lenient the default
sanction. The vertical axis is the involuntary unemployment rate in sector J.
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28 ECONOMICA
or reduce 𝜙, but the distance from 𝜆A by reducing 𝛼 is 10%, whereas that by reducing 𝜙 is 8%. The
pattern is the same for the wedges 6.5 and 7. This implies that if the driving force of the changes
in the wedge is due to the crisis severity 𝛼, then the change in optimal policy is larger, and the
resource misallocation due to a higher wedge is larger.
When the economy is at the ELB, the creative destruction wedge in sector I also affects sec-
tor J involuntary unemployment through the aggregate demand externality, although the effect
is mild. For ease of exposition, we use the stylized setup to illustrate this point numerically. As
Figure 10 shows, as the wedge increases to 50 from 5 due to the decrease in 𝜙, sector J involun-
tary unemployment goes up in all spectrums of the default sanction. Similarly, when we decrease
the proportion of profitable prior entrepreneurs from 0.5 to 0.3, which also leads the wedge to
increase, we see an increase in sector J involuntary unemployment. Therefore the optimal default
sanction becomes even more lenient. This exercise suggests that the ELB environment further
exacerbates the resource misallocation due to the creative destruction wedge.
7 EXTENSION: CREDIT GUARANTEE AS IMPLEMENTATION
So far, we have abstracted away banks’ incentive problem for analytical tractability, and the policy
instrument in the above benchmark model is the default sanction 𝜆1 . One might reasonably ques-
tion the micro-foundation of 𝜆1 and how it would interact with banks’ incentives. As we know, in
practice, government does not lend directly to SME borrowers, whereas banks do, and the gov-
ernment’s policy instrument is the credit guarantee, which affects banks’ monitoring incentives.
Therefore, in this section, we extend the model to consider banks that provide loans to SMEs,
rather than the government. The government’s loan support policy instrument is credit guaran-
tee g, rather than the default sanction 𝜆1 . We show that the model with banks is isomorphic to
the benchmark model, the credit guarantee provides a micro-foundation for the default sanction,
and our results still go through.
Following Martinez-Miera and Repullo (2017), let us consider banks that are lenders but
exert private monitoring efforts. Let 𝜂(e) be a bank’s monitoring technology, where e is the bank’s
monitoring effort, and 𝜂(e) is monotonically increasing in e. In this paper, we take a short-cut by
not estimating the functional form of 𝜂(e), as the aim here is to illustrate the intuition. This is also
the reason why we position banks with credit guarantees as a model extension, rather than as the
benchmark main model. When banks monitor and pursue the defaulters, if there are remaining
funds, then banks can take away a fraction 𝜂(e) of the remaining funds, and defaulters divert
away a fraction 1 − 𝜂(e), as in the benchmark model. Thus a higher 𝜂(e) corresponds to a more
superior monitoring technology.
We further assume 𝜂(0) = 𝛾 and 𝜂(+∞) = 1. This assumption implies that when banks do not
exert monitoring effort, the defaulters can privately divert away a fraction 1 − 𝛾 of their remaining
funds, if any, and when banks exert an extremely high level of monitoring effort, their monitor-
ing technology is near-perfect, and the amount of diverted funds approaches zero. What could
affect banks’ monitoring effort is the government’s credit guarantee g. If g = 100%, then the gov-
ernment fully makes up the loss of the defaulted amount to the banks; if g = 0%, then the banks
fully take on the loss of the defaulted amount.
Suppose that an entrepreneur announces default; then the bank monitors and pursues the
defaulter’s remaining funds. The bank’s total proceeds from monitoring and pursuing the default-
ers include the fraction 𝜂(e) of remaining funds that the bank is able to take away, plus the
money received from the government credit guarantee on the defaulted amount, so the bank
earns 𝜂(e) + g(1 − 𝜂(e)) per unit of funds. Meanwhile, the bank exerts effort cost 𝜄e2 ∕2 per unit
of funds à la Martinez-Miera and Repullo (2017). Thus banks maximize the objective func-
tion 𝜂(e) + g(1 − 𝜂(e)) − 𝜄e2 ∕2 by choosing the monitoring effort e. The following proposition
summarizes the main result.
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 29
Proposition 6. (Credit guarantee and default sanction) In equilibrium, we have the following.
(a) Banks’ monitoring technology is a decreasing function of credit guarantee.
(b) Credit guarantee has a one-to-one mapping with default sanction.
(c) When credit guarantee g = 100%, monitoring effort e = 0, monitoring technology 𝜂 = 𝛾, and
default sanction 𝜆1 = 0.
Proof. See the Appendix. ▪
The intuition of Proposition 6 is as follows. As the government increases the level of credit
guarantee, banks’ marginal benefits of exerting monitoring effort decrease. Because banks exert
effort privately, they choose the effort such that the marginal benefits of exerting effort equate
to its marginal cost; in equilibrium, as the level of credit guarantee increases, banks exert less
effort, which leads to an inferior monitoring technology. The inferior monitoring technology
increases the marginal entrepreneur’s benefits of borrowing, compared with her outside option
of staying as a worker, thus attracting more inefficient entrepreneurs to borrow. Similarly, as
the government reduces credit guarantees, the inefficient entrepreneur with low profitability
may be deterred from borrowing, reducing the aggregate default. As we establish in the proof
of Proposition 6, the credit guarantee in this enriched setup has a one-to-one mapping with
the default sanction in the benchmark model. When the government provides 100% credit
guarantee, banks’ monitoring technology becomes 𝛾 of the benchmark model, and the default
sanction 𝜆1 corresponds to zero; the lower the credit guarantee, the larger the default sanction
becomes.
8 OTHER NUMERICAL EXPERIMENTS
In this section, we vary other parameters to assess how the optimal sanction changes its stance
with the figures displayed in the Appendix. Figure A1 displays the responses of the social util-
ity to the number of loan applicants, or equivalently, the default sanction. The first plot is our
benchmark case where the discount factor 𝛽 is set to 0.97; the second plot sets 𝛽 = 0.9; and the
third subplot sets 𝛽 = 0.5. The decrease in 𝛽 can be interpreted as a more impatient or myopic
government.
The first plot of Figure A1 shows that in our benchmark case, the optimal default sanction is
intermediate; it is more lenient than the harsh default sanction 𝜆A1 , but harsher than the lenient
default sanction 𝜆B1 . The second plot of Figure A1 assumes a lower discount factor, 𝛽 = 0.9. The
optimal default sanction is also intermediate, but compared with the case 𝛽 = 0.97, it moves fur-
ther away from the harsh default sanction 𝜆A1 . The third plot of Figure A1 assumes an extremely
low discount factor 𝛽 = 0.5. This result suggests that if the government is myopic,—i.e. it cares
more about the short-term gains than the long-run productivity—then it will choose a more
lenient default sanction than our benchmark case, and attract more borrowers at t = 1 to reduce
unemployment.
Next, we vary parameters to see how they alter the optimal default sanction. First, we increase
𝛾 from 0.7 to 0.95, which means an improvement in the lender monitoring technology. As the
second plot of Figure A2 in the Appendix illustrates, the optimal default sanction becomes the
most lenient stance of 𝜆B1 . Indeed, a near-perfect monitoring technology could justify the govern-
ment’s most generous credit guarantees for its BBLs to SMEs. However, this scenario is unlikely
to hold during the pandemic. Given the scale and the speed of the crisis, lenders are unlikely to
carry out meticulous monitoring in haste. We then increase 𝜎B and 𝜙 to improve the overall qual-
ity of the inefficient entrepreneurial pool. This also results in a shift to a more lenient optimal
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30 ECONOMICA
default sanction (see Figures A4 and A5 in the Appendix), as the utility loss associated with
future productivity slowdown becomes less severe.
Furthermore, we consider the effects of long-run scarring by considering a scarring function.
Specifically, we do not need the industry gap to close at t = 2. If there is an industry gap at the
end of t = 1 (i.e. N > M), then at the start of t = 2, (N − M)∕𝜅 workers with new ideas could
fill in the industry gap, where 𝜅 (≥ 1) is an increasing and concave function of short-run unem-
ployment that reflects long-run scarring of the pandemic: a large 𝜅 could result in insufficient
new entries in the long run, hence long-run structural unemployment is a possibility. We assume
that the scarring function takes the form 𝜅 = ln(UI (M1 ) + e), where UI (M1 ) is the short-run
unemployment rate. This implies that the short-run unemployment leads to insufficient entries
in the long run and structural unemployment, but if the short run obtains full employment, then
the long-run scarring disappears and date t = 2 sees no structural unemployment. Figure A6 in
the Appendix illustrates the comparative statics between our benchmark and the long-run scar-
ring case. The overall welfare with long-run scarring is lower than the benchmark case; moreover,
the optimal default sanction becomes more lenient, so the government will care more about
short-run stabilization rather than allocative efficiency.
9 CONCLUSION
This paper has assessed government loan support for small businesses during a pandemic from a
normative angle. We apply adverse selection in financial contracting in a macroeconomic frame-
work. A two-sector equilibrium model has been developed, featuring screening in the presence
of a pandemic shock and aggregate demand externality. To characterize the government’s pol-
icy stance, we have developed an analytic measure ‘stabilization proclivity’. Numerically, we have
demonstrated that the optimal default sanction is intermediate. The interest rate lower bound
shifts the optimal default sanction to a more lenient stance due to the aggregate demand external-
ity channel. Lower price elasticity of demand increases firm markups, exacerbating involuntary
unemployment at the ELB, and the optimal default sanction becomes even more lenient. The
creative destruction wedge polarizes the government’s hawkish and lenient stances, and a higher
creative destruction wedge implies a more lenient optimal default sanction, worsening long-run
allocative efficiency. Although we study SMEs, our results on firm markups shed light on the
macro-financial policies for the rise of superstar firms characterized by small labour shares and
large profit shares (see Autor et al. 2020). Rents due to imperfect competition alter firms’ incentive
constraints. Using macro-financial policies to shift firms’ incentive constraints should naturally
consider profit shares, as we do in this paper.
One simplifying assumption that we have made is that both incumbents and new entrants
can benefit from government guarantees. We acknowledge that policy interventions in response
to the COVID-19 shock were skewed disproportionatel towards incumbent firms. If the guar-
antee scheme were available only for the incumbents of our model, then the implication would
be as follows. First, when the government sets a harsh default sanction to deter all the ineffi-
cient ones from borrowing, the sector I short-run unemployment rate will be greater, the fall in
interest rate is greater, and demand shortage is greater. With the issue of effective lower bound,
this will cause further involuntary unemployment in sector J, pushing the optimal default sanc-
tion to a more lenient realm. Second, when the government sets a lenient default sanction to
stabilize short-run employment, with the guarantee scheme available only to the incumbent,
the lenient sanction has to be lower than the benchmark case. This means that more inefficient
entrepreneurs have to be granted access to loan support, worsening allocative efficiency. Thus
withholding the loan support from new entrants polarizes the government’s hawkish and dovish
stances.
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 31
While the aggregate demand externality that we highlight in this paper lends more support
to a lenient policy stance, some caveats remain. On the one hand, when the credit support is
scaled back in the future, it may cause banks to roll over legacy lending rather than recogniz-
ing potential losses (see the evergreening channel in Acharya et al. 2021). On the other hand,
given the pressure from incumbents to keep them afloat (see Buera et al. 2013), it is unclear
whether such credit support would be scaled back soon or become protracted. Therefore the
ultimate costs due to resource misallocation might be larger than expected, which would lend
more support to a harsher policy stance. We have kept the model simple, and in the future it
would be useful to embed the model mechanism in a large quantitative setup to evaluate policy
effectiveness.
ACKNOWLEDGMENTS
We thank the editor and three anonymous referees for exceptionally thorough and helpful com-
ments. We thank Venky Venkateswaran, Lukas Vogel, Benjamin Born, Alexander Rodnyansky,
Juuso Vanhala, Remco Zwinkels, Tim Eisert, Rex Wang, and seminar participants at the 2022
American Economic Association Meetings, 2022 Second Finance and Productivity Conference
(CompNet, EBRD, IWH), 2021 Annual Congress of the European Economic Association,
the European Commission Directorate General for Economic and Financial Affairs, the 2020
European Winter Meetings of the Econometric Society, the 2021 Midwest Finance Associa-
tion Annual Conference, the 2021 Royal Economic Society Annual Conference, the 2020 Dutch
Economists Week, Vrije Universiteit Amsterdam Finance Seminars, Rotterdam School of Man-
agement Seminar Series, and School of Banking and Finance seminar of the University of
International Business and Economics for valuable comments.
NOTES
1
See, for example, Morris et al. (2020), Treanor (2020), Elliott (2020) and Thomas (2021).
2
Some credit schemes encourage lenders to request personal guarantees (see, for example, Ono and Uesugi 2009; May-
ordomo et al. (2021), which aligns incentives of the lender to screen borrowers. We thank an anonymous referee for
raising this point.
3
Of course, in reality, output is produced also using equipment and property, and there are overhead payments
to be made—for example, of local and national taxes, rents, minimal utility costs, etc.—if the business is not
to default when it cannot produce. Some of these overhead costs were deferred during the pandemic, but not
all. Our approach here, using a requirement to maintain at least some key workers, is a reasonable way of
dealing with such continuing overhead costs within the context of the simplest possible model to analyse such
circumstances. Indeed, governments have the objective to help firms to retain workers and stabilize employment
when implementing loan support programmes; see, for example, the US Department of the Treasury’s article
on the US Paycheck Protection Program (https://home.treasury.gov/policy-issues/coronavirus/assistance-for-small-
businesses/paycheck-protection-program, accessed 2 December 2022), as well as the UK HM Treasury’s article
on government-backed financial support schemes (https://www.gov.uk/government/news/government-backed-loans-
help-thousands-of-businesses-to-protect-jobs-during-pandemic, accessed 4 January 2023).
4
We are aware that the pandemic can evolve over time and restrictions might be re-introduced if circumstances require.
Our modelling assumption is that the very end of t = 1 is when the vaccination programme is expected to take effect,
so that the economy could function with production resumed. Our model is stylized to keep the analytics tractable, so
it does not incorporate the evolving dynamics of the pandemic, which would require a fully-fledged dynamic model
with inputs of medical expertise on the evolving nature of the virus. Nevertheless, we believe that our mapping captures
the substance of the pandemic episode in 2020–21.
5
That the lender obtains only a fraction of the remaining funds in the case of default is in line with a common
assumption in the financial contracting literature (e.g. Lorenzoni 2008; Rampini and Viswanathan 2013; Dávila and
Korinek 2018; Dávila 2020) that a financial contract is subject to a form of limited commitment. In Dávila (2020), this
fraction is interpreted as the efficiency of the bankruptcy process.
6
Precisely, E1 (WI1 |𝕀d = 1, 𝜆1 ) can be expressed as follows, as can be seen in the derivations in the Appendix:
( )
1
E1 (W1 |𝕀d = 1, 𝜆1 ) = (1 − u1 (M − 1)) 1 −
N𝜀
Q(M − 1)∕2N ( )
1
= 1− ,
(Q∕2) + N − M + 1 N𝜀
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32 ECONOMICA
where M is the number of borrowing entrepreneurs, and u1 (⋅) is the unemployment at t = 1 as a function of borrowing
entrepreneurs. As we can see from the above equation, if the entrepreneur is to remain as a worker, then the labour
supply has to adjust by u1 (M − 1), which in turn affects the conditional expected wage.
7
In practice, such timely and almost real-time government loan support involves the banking sector issuing inside
money against an offsetting credit, with the government providing guarantees. Indeed, in response to the COVID-19
crisis, many governments worldwide have unveiled large-scale loan stimulus programmes through the banking
system.
8
The interest rate is in terms of goods J, rather than the real interest rate obtained by deflating the nominal interest
rate by the expected inflation rate from the price index of the two types of goods. Since during the pandemic sector I
goods are not traded, we cannot observe their price, nor can we measure such a price index. Indeed, as estimated in
Cavallo (2020), the official CPI does not reflect the rapid changes in prices in various sectors due to COVID-19. So
both for simplicity and in line with the most current developments, the interest rate in our context refers to the interest
payment in terms of goods J.
9
We assume that for the less profitable entrepreneurs, their type 𝜎(n) follows a uniform distribution. The assumption of
the distribution would make a quantitative difference to the optimal policy. Suppose that the left tail of the distribution
is more dense than the right tail of the distribution. Then the lenient default sanction 𝜆B1 that stabilizes short-run
employment will lead to greater loss in allocative efficiency, shifting the optimal policy to a harsher stance, and vice
versa. Nevertheless, there are still quantitative effects on incentives by changing 𝜆1 . Suppose that n∗ is far in the left
tail of the distribution. By increasing 𝜆1 , this firm n∗ incentive constraint will no longer be satisfied, thus it will be
deterred from borrowing, but by decreasing 𝜆1 a tiny bit, the quantitative effect on the incentives is small because n∗
is ready far in the left tail. Thus the quantitative effects of changing 𝜆1 would be asymmetric.
10
This means that Q∕N has decreased from 80 to 20. When we increase Q proportionally such that Q∕N remains
unchanged, the optimal default sanction will change as well. This means that individual levels of Q and N matter,
not just their ratio. This is because individual levels of Q matter for the outside option in the incentive constraint (2)
through the aggregate labour supply, and it also matters for the calculation of unemployment rates, whereas N , rather
than Q, matters for markup.
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12460
APPENDIX
Proof of Lemma 3.
Since PI(t=−1) = PJ(t=−1) = 1, at the start of t = 0, workers in each sector get wages totalling (1 −
1∕(N𝜀))(Q∕2), and entrepreneurs in each sector have profits from last period totalling Q∕(2N𝜀).
They spend half of the sum on goods J at the end of t = 1. Consequently, at the start of t = 1,
workers in each sector have extra money totalling (1 − 1∕(N𝜀))(Q∕4), sector I entrepreneurs have
extra money totalling Q(4N𝜀), and sector J entrepreneurs have extra money totalling Q∕(4N𝜀).
Therefore the extra sum of money due to no spending on goods I amounts to Q∕2.
Since max(df1 ) ≤ F1 N, that is, max(df1 ) ≤ (1 − 1∕(N𝜀))(Q∕2), and Λ1 > 0, given that df1 =
B1 + Λ1 , it follows that
Q Q
B1 < − ,
2 2N𝜀
so the extra money that agents carry over from t = 0 is more than enough to invest in B1 . ▪
Proof of Proposition 1.
Conditional on df1 > 0, for the marginal entrepreneur n, the incentive constraint takes the
equality sign and can be re-expressed as
𝜆1 = (1 − 𝛾) 𝜎(n) h1 − 𝔼1 (WI1 |𝕀d = 1, 𝜆1 ). (A1)
F I G U R E A1 Social utility, optimal sanctions and 𝛽 .
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 35
F I G U R E A2 Social welfare, optimal sanctions and 𝛾 .
F I G U R E A3 Social welfare, optimal sanctions and 𝜀.
Suppose that M (≤ N) entrepreneurs-to-be apply for government loans. Of these, (𝜙 + 𝛼)N
are profitable entrepreneurs, and M − 𝜙N − 𝛼N are inefficient ones. Given the uniform distribu-
tion assumption, it follows that
( )
M − (𝛼 + 𝜙)N M − (𝛼 + 𝜙)N
𝜎(n) = 1 − 𝜎+ 𝜎B . (A2)
(1 − 𝛼)N (1 − 𝛼)N
The expected wage conditional on deviation at t = 1 needs to be adjusted by the
out-of-equilibrium unemployment rate, that is,
( )
1
E1 (W1 |𝕀d = 1, 𝜆1 ) = (1 − u1 (M − 1)) 1 −
N𝜀
Q(M − 1)∕2N ( 1
)
= 1− .
(Q∕2) + N − M + 1 N𝜀
We can see that 𝜕E1 (W1 |𝕀d = 1, 𝜆1 )∕𝜕M > 0, and since 𝜕M∕𝜕𝜎(n) < 0 and given equation (A2),
it follows that
𝜕E1 (W1 |𝕀d = 1, 𝜆1 )∕𝜕𝜎(n) < 0. (A3)
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36 ECONOMICA
F I G U R E A4 Social welfare, optimal sanctions and 𝜎B .
F I G U R E A5 Social welfare, optimal sanctions and 𝜙.
Given equation (A1) and inequality (A3), we can see that 𝜕𝜎(n)∕𝜕𝜆1 > 0.
Thus when 𝜆1 increases, the number of borrowing entrepreneurs M decreases, and fewer
entrepreneurs with bad projects borrow, so the number of defaulters decreases. ▪
Proof of Theorem 1.
Part (a) Given 1 − 1∕(N𝜀) = 𝜎, F1 = 𝜎h1 , the inefficient entrepreneur with the highest profitabil-
ity is on the verge of strategic default in the absence of sanctions. Among the prior entrepreneurs,
(1 − 𝛼)N will default conditional on borrowing.
Now let us set 𝜆1 > 𝜆A1 , where
( )( Q
)
1
𝜆A1 = 1 − (1 − 𝛾) − (1 − uI (𝜙N + 𝛼N)) .
N𝜀 2N
Then for the entrepreneur with 𝜎, their benefits of default are smaller than their out-
side option. They are deterred from borrowing. By monotonicity, all the other inefficient
entrepreneurs are deterred from borrowing as well. At date t = 1, the unemployment rate
is uI1 (𝜙N + 𝛼N). In the benchmark assuming away scarring, full employment obtains in the
long run.
Part (b) Since the government wants to obtain full employment at t = 1, it needs to attract (1 −
𝜙 − 𝛼)N prior entrepreneurs to apply for loans. By our uniform distribution specification, the
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 37
F I G U R E A6 Social welfare, optimal sanctions and long-run scarring.
marginal entrepreneur’s 𝜎(n) must satisfy
𝜎 − 𝜎(n) 𝜎 − 𝜎B
= ,
(1 − 𝜙 − 𝛼)N (1 − 𝛼)N
which is equivalent to
𝜙 1−𝜙−𝛼
𝜎(n) = 𝜎+ 𝜎B .
1−𝛼 1−𝛼
Suppose that 𝜆B1 just satisfies the marginal entrepreneur’s incentive constraint, that is,
( )
1
(1 − 𝛾) 𝜎(n) h1 − 𝜆B1 = 1 − (1 − uI (N − 1)) .
N𝜀
Then
( ) ( )
𝜙 1−𝜙−𝛼 Q 1
𝜆B1 = (1 − 𝛾) 𝜎+ 𝜎B − 1− (1 − uI (N − 1)) .
1−𝛼 1−𝛼 2N N𝜀
By monotonicity, the inefficient entrepreneurs with profitability higher than
𝜙 1−𝜙−𝛼
𝜎+ 𝜎B
1−𝛼 1−𝛼
borrow and default, and those with lower profitability stay out.
Part (c) Given the parameter
2N − 2∕𝜀
𝜎B = ,
(1 − 𝛾)Q
it follows that (1 − 𝛾)𝜎B h1 = WI1 and 𝜎B h1 < F1 . Thus the inefficient entrepreneur with the lowest
profitability will try to borrow at the start of t = 1 in the absence of sanctions, and then default
at the end of t = 1.
Given that 𝜎 = 1 − 1∕(N𝜀), it follows that 𝛾𝜎h1 < F1 = 𝜎h1 . Thus the inefficient entrepreneur
with the highest probability will try to borrow in the absence of sanctions, and then strategically
default at the end of t = 1. Among the prior entrepreneurs, conditional on getting the loans, a
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38 ECONOMICA
fraction 1 − 𝛼 of them will default. As the total number of applicants amounts to (𝜙 + 1)N and
the industry size is N, some profitable applicants are crowded out. ▪
Proof of Proposition 2.
Part (a) The following Euler equations for the workers’ consumption hold for date t:
UcJ (cIt (W ), cJt (W )) = 𝛽(1 + rt ) UcJ (cIt+1 (W ), cJt+1 (W )),
which is equivalent to
1 UcJ (cIt (W ), cJt (W ))
1 + rt = .
𝛽 UcJ (cIt+1 (W ), cJt+1 (W ))
Given the CES preference (3), we have
1 (cIt (W )1−𝜈 + cJt (W )1−𝜈 )(𝜈−𝛿)∕(1−𝜈) cJt (W )−𝜈
1 + rt = . (A4)
𝛽 (cIt+1 (W )1−𝜈 + cJt+1 (W )1−𝜈 )(𝜈−𝛿)∕(1−𝜈) cJt+1 (W )−𝜈
We know that cI0 (W ) = 0 and cJ0 (W ) = (1 − 1∕(N𝜀)) (Q∕2). Given 𝜆1 > 𝜆A1 , at date t = 1,
only (𝛼 + 𝜙)N good entrepreneurs produce in sector I. It follows that
( ) ( )
1 (𝜙 + 𝛼)Q 1 Q
cI1 (W ) = 1 − and cJ1 (W ) = 1 − .
N𝜀 2 N𝜀 2
Substituting these values into equation (A4), we obtain
( )(𝜈−𝛿)∕(1−𝜈)
1 1
1 + r0 = . (A5)
𝛽 (𝜙 + 𝛼)1−𝜈 + 1
Therefore, given 𝜈 > 𝛿, we have 1 + r0 < 𝛽1 .
Moving onto t = 2, as the industry gap closes, it follows that
( )(𝜈−𝛿)∕(1−𝜈)
1 (𝛼 + 𝜙)1−𝜈 + 1
1 + r1 = , (A6)
𝛽 2
so we can see that r1 < (1∕𝛽) − 1. Low interest rate is persistent.
Part (b) Given 𝜆1 = 𝜆B1 , full employment obtains from t = 1 onwards. By logic similar to that in
the proof of Proposition 2(a), we can show that r0 < (1∕𝛽) − 1. As production remains constant
from t = 1 onwards, it follows that r1 = (1∕𝛽) − 1. ▪
Proof of Proposition 3.
First, we work out the partial derivative of the single-period utility as follows. For t = 1,
(( )𝜈 ( )𝜈 )(𝜈−𝛿)∕(1−𝜈) ( )𝜈
𝜕U(cI1 , cJ1 ) 1 1 1 1
= 1−𝜈
cI1 + 1−𝜈
cJ1 (1 − 𝜈)c−𝜈
𝜕M 1−𝜈 2 2 2 I1
( )
Q M − 𝜙N − 𝛼N
× 𝜎(n′ ) + 𝜎 + (𝜎B − 𝜎) > 0.
4N (1 − 𝛼)N
At date t = 2, given equation (8),
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 39
(( )𝜈 ( )𝜈 )(𝜈−𝛿)∕(1−𝜈) ( )𝜈
𝜕U(cI2 , cJ2 ) 1 1 1 1
= c1−𝜈 + c1−𝜈
(1 − 𝜈)c−𝜈
𝜕M 1−𝜈 2 I2 2 J2 2 I2
( ( ) )
Q M̃ − 𝜙N − 𝛼N Q
× 𝜎(n ) + 𝜎 + (𝜎B − 𝜎)
′
− < 0.
4N (1 − 𝛼)N 2N
It is straightforward to derive that
( )−1
𝜕qI1 ∕𝜕M 1 2
− = −1 .
𝛽𝜕qI2 ∕𝜕M 𝛽 𝜎(n′ ) + 𝜎 + (𝜎B − 𝜎)(M − 𝜙N − 𝛼N)∕(1 − 𝛼)N
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
SP ▪
Proof of Proposition 4.
Let
( )
1 M − 𝜙N − 𝛼N
X= 𝜎(n ) + 𝜎 + (𝜎B − 𝜎)
′
.
2 (1 − 𝛼)N
Given equation (8), we can further simplify X as
( )
M 𝛼+𝜙
X =𝜎+ − (𝜎B − 𝜎).
(1 − 𝛼)N 1−𝛼
For M ∈ ((𝜙 + 𝛼)N, N), we have
M 𝛼+𝜙
− > 0.
(1 − 𝛼)N 1−𝛼
Given 𝜎, the lower 𝜎B is, the lower X is, and the smaller SP is.
Note that
1 2N − 2∕𝜀
𝜎 =1− , 𝜎B = ,
N𝜀 (1 − 𝛾)Q
so X can be simplified further as
( )( )( )
1 M 𝛼+𝜙 1 2N
X =1− − − 1− 1− .
N𝜀 (1 − 𝛼)N 1−𝛼 N𝜀 (1 − 𝛾)Q
Thus 𝜕X ∕𝜕𝜀 > 0, 𝜕X ∕𝜕𝛾 > 0 and 𝜕X ∕𝜕N > 0.
Since
1
SP = ,
𝛽((1∕X ) − 1)
SP increases with X and decreases with 𝛽; that is, an increase in 𝜀, 𝛾, 𝜎B or N leads to an increase
in SP, whereas an increase in 𝛽 decreases SP. ▪
Proof of Proposition 5.
As shown in the proof of Proposition III, the Euler equation gives
1 UcJ (cIt , cJt )
1 + rt = .
𝛽 UcJ (cIt+1 , cJt+1 )
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40 ECONOMICA
Suppose that t is when sector J reaches full employment, so
1 UcJ (cIt−1 , cJt−1 )
1 + rt−1 = .
𝛽 UcJ (cIt , cJt )
Since at the ELB the interest rate is bound by 1∕𝛽 − 1, it follows that
UcJ (cIt−1 , cJt−1 )
1= . (A7)
UcJ (cIt , cJt )
And because cIt−1 < cIt , 𝜕UcJ (cIt , cJt )∕𝜕cIt > 0 and 𝜕UcJ (cJt , cJt )∕𝜕cJt < 0, for equation (A7) to
hold, cJt−1 < cJt has to hold. Before t, a decrease in the consumption of goods I corresponds to a
decrease in consumption of goods J. Accordingly, solving backwards, it follows that cJ0 < cJ1 <
· · · < cJt−1 . Moreover, cJt = Q∕2, which corresponds to full employment in sector J, so for periods
before t, sector J suffers involuntary unemployment. ▪
Proof of Proposition 6.
Let 𝜂(e) be a bank’s monitoring technology, where e is the bank’s monitoring effort. We assume
that 𝜂(e) is monotonically increasing in the monitoring effort e, and 𝜂(0) = 𝛾 and 𝜂(+∞) = 1.
The government’s credit guarantee is g. If g = 100%, then the government fully makes up the
loss of the defaulted amount to the banks; if g = 0%, then the banks fully take on the loss of the
defaulted amount.
When entrepreneurs announce default, banks monitor and pursue the defaulters’ remaining
funds. Banks’ total proceeds from monitoring and pursuing the defaulters include the fraction
𝜂(e) of remaining funds that banks are able to take away, plus the money received from the govern-
ment credit guarantee on the defaulted amount, so banks earn 𝜂(e) + g(1 − 𝜂(e)) per unit of funds.
Meanwhile, banks exert effort cost 𝜄e2 ∕2 per unit of funds. Thus banks maximize the objective
function
𝜄 2
𝜂(e) + g (1 − 𝜂(e)) − e.
2
Banks choose the monitoring effort e, and the optimality condition is
(1 − g) 𝜂 ′ (e) = 𝜄e.
Total differentiating this optimality condition, it follows that
𝜕g 𝜂 ′ (e) − e 𝜂 ′′ (e)
= −𝜄 .
𝜕e (𝜂 ′ (e))2
Because 𝜂 ′ (e) > 0 and 𝜂 ′′ (e) < 0, we obtain 𝜕e∕𝜕g < 0, and using the chain rule, 𝜕𝜂∕𝜕g < 0. Thus
the banks’ monitoring technology is a decreasing function of the government’s credit guarantee.
Now we can write 𝜂 as 𝜂(g).
Next, we show that the default sanction 𝜆1 in our benchmark model is a decreasing function
of the government credit guarantee.
Note that the marginal entrepreneur’s participation constraint in this model extension is
simply
(1 − 𝜂(g)) 𝜎(n) h1 ≥ 𝔼1 (WI1 |𝕀d = 1, 𝜆1 ),
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
default outside option
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SUPPORT FOR SMALL BUSINESSES AMID COVID-19 41
and recall the participation constraint in the benchmark case,
(1 − 𝛾) 𝜎(n) h1 − 𝜆1 ≥ 𝔼1 (WI1 |𝕀d = 1, 𝜆1 ).
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
default outside option
Equating the left-hand sides of the above participation constraints, we get
𝜆1 = (𝜂(g) − 𝛾) 𝜎(n) h1 .
Using the chain rule, it follows that 𝜕𝜆1 ∕𝜕g < 0. And moreover, when g = 1, e = 0 and 𝜂 = 𝛾, and
𝜆1 = 0.
Other comparative statics ▪
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