2.1 Workers

The model is populated by a unit mass of identical hand-to-mouth workers whose preferences are represented by the utility function

where c_t denotes consumption and U(c) = c^(1-σ))/(1 — σ) is a standard CES utility function with a risk aversion parameter σ. Each worker is endowed with $\overline{h}\in \{\overline{h},\overline{h}\}>0$ units of labor that are supplied inelastically, with h_n denoting the labor supply in normal times, while h_p denoting the labor supply during the pandemic. Since agents are hand-to-mouth consumers and supply all their endowed labor, their consumption equals their labor income $w{\overline{h}}_t$ with w_t denoting wages.

We model the COVID-19 shock following Guerrieri, Lorenzoni, Straub and Werning (2020) as a sudden drop in the labor supply. This suggests that due to the pandemic, it is unsafe for some workers to report to work because their job requires close interaction with the public; hence they stay at home either by choice or by governments’ imposing quarantines or social distancing policies. Capturing the economic effects of COVID-19 through a labor supply shock makes sense since (i) lockdowns to deal with the pandemic are a direct shock on labor supply, (ii) its persistence, which can be set from low to moderate, determines the speed with which the economy can bounce back (given that the pandemic is of transitory nature), (iii) the reduction in employment is the source rather than the outcome of the recession (unlike a regular business cycle or financial recession), and (iv) the drop in unemployment is larger than in a standard recession.

2.2 Entrepreneurs

The model is populated by a unit mass of identical entrepreneurs whose preferences are denoted by the utility function

where x_t is consumption. Their utility function takes the same form as the one of workers.

Entrepreneurs produce y_t = F(z_t, k_t, l_t,v_t) each period. F(•) is a Cobb-Douglas production function, which combines labor, l_t, with the stock of capital purchased in the previous period, k_t, and an intermediate good, v_t; z_t is an aggregate productivity shock. Aggregate capital is in unit fixed supply: K_t = 1. The intermediate good is traded in competitive world markets at a fixed exogenous price, p^v. The flow budget constraint of entrepreneurs is given by

where b_t denotes the beginning-of-period borrowing from one-period non-state contingent bonds issued last period (a negative b_t implies positive net holdings of bonds), q_t is the price of capital, and R is the world-determined gross real interest rate taken as given in the small open economy. Entrepreneurs’ consumption, x_t, equals output net of the outlays for the factors of production, v_t and l_t, the net capital expenditure, q_t(k_t+1 – k_t), and the net debt issuance, b_t+1/R – b_t.

We assume that entrepreneurs cannot raise equity and that their borrowing decision is limited by a collateral constraint, which can endogenously be derived from a limited commitment problem similar to Jermann and Quadrini (2012) and Bianchi and Mendoza (2018). Entrepreneurs obtain two types of loans: An inter-temporal (b_t+1/R) and an intra-temporal loan. They need the latter to finance ahead of production a portion θ ≤ 1 of the intermediate good purchases and labor wages. Hence, the total liabilities at the beginning of the period comprise of θ(p^vv_t + W_tl_t) + b_t+1/R. While b_t+1 is an inter-temporal loan and bears an interest payment, θp^vv_t + θw_tl_t does not as it is repaid within the same period. All borrowed funds can be diverted, a situation which is precluded by imposing the following collateral constraint

Constraint (4) limits the size of total borrowing to a fraction κ_t of the beginning of the period asset holdings.

Entrepreneurs maximize (2) subject to (3) and (4). This maximization problem leads to the following optimality conditions for each date t = 0,..., ∞

where U_x,tµ_t denotes the Lagrange multiplier on the collateral constraint scaled by entrepreneurs’ marginal utility and X_i,t denote the first derivatives of a function X(i) with respect to a variable i at time t.

The presence of the collateral constraint distorts both the optimal inter- and intra-temporal marginal decisions when binding. Conditions (5) and (6), defining entrepreneurs’ optimal choice of the intermediate good and labor, embed an additional cost, i.e. the cost of collateral financing equal to θµ_tp^v and θµ_tw_t, respectively . In addition, both Euler equations are distorted. The Euler equation for borrowing (7) implies that the marginal benefit from increasing borrowing today outweighs the expected future marginal cost by an amount equal to the shadow price, µ_t, of relaxing the collateral constraint. Similarly, the Euler equation with respect to capital (8), equating the marginal cost of an extra unit of capital with its marginal benefit, embeds an additional benefit obtained by relaxing the collateral constraint, valued at κ_t+1 U_x,t+1 µ_t+1 q_t+1. As we will show and discuss later, this equation is at the core of the mechanism through which the pecuniary externality operates: The choice of borrowing and consumption today influence the price of the asset used as collateral, which in turn affects the tightness of the collateral constraint.

2.3 Competitive Equilibrium

The competitive equilibrium in this economy is defined as follows.

Definition 1. For given initial values of the endogenous state variable, b₀, and exogenous processes ${\{z_t,{\kappa }_t,h_t\}}_{t=0}^{\infty }$, a competitive equilibrium for the economy with a collateral constraint is a sequence of allocations ${\{c_t,x_t,v_t\}}_{t=0}^{\infty }$, an asset profile ${\{k_{t+1},b_{t+1}\}}_{t=0}^{\infty }$, and a price system ${\{q_t,w_t,p^v\}}_{t=0}^{\infty }$, such that

1. Given the price system ${\{q_t,w_t,p^v\}}_{t=0}^{\infty }$, the allocations and the asset profile solve workers’ and entrepreneurs’ optimization problems, and

2. Labor, asset and goods markets clear, satisfying conditions

3.1 Planner’s Economy

The policy design follows the Ramsey approach, which consists of the planner choosing policies, prices, and allocations in order to maximize the economy’s wide social welfare function. In doing so, the planner has to respect all equilibrium conditions of the competitive economy to ensure that the allocations chosen can be implemented as allocations in the competitive economy.

Unlike in the standard Ramsey literature, where the planner optimally chooses distortionary policies intended to finance government expenditure, the planner here chooses policies to alleviate the inefficiencies arising from pecuniary externalities.

We assume that the planner has access to two types of policy instruments: A tax/subsidy on borrowing (as in Bianchi and Mendoza, 2018) and tax/subsidy on labor. Both instruments are Pigouvian in nature with the tax revenues being rebated lump-sum back to the private agents, T_t. This way the policy has a role to correct a distorted decision that arises because of the presence of the financial frictions.⁹

With the policy instruments in place, the budget constraint of entrepreneurs in the decentralized economy takes the following form

where ${\tau }_t^b$ is the tax on new borrowing b_t+1 determined at t but levied at t + 1 when debt is repaid, ${\tau }_t^h$ is the labor tax, and T_t is the lump-sum transfer, which in equilibrium is equal to ${\tau }_{t-1}^b{b}_t+{\tau }_t^h{w}_t{l}_t$.

Moreover, the Euler conditions with respect to borrowing and labor become, respectively,

All other equilibrium conditions remain the same as outlined in section 2.

In the formulation of the Ramsey problem, we do not impose as constraints the policy distorted optimality conditions, (13) and (14), in the planner’s optimization problem since it can easily be shown that ${\tau }_t^{b}and{\tau }_t^h$ are chosen such that the Lagrange multipliers on these two conditions are zero.¹⁰ Also, note that the optimality condition, (5), with respect to the intermediate good is not distorted by a tax instrument and will hold with equality in the planner’s problem. Thus, we use it to solve for the Lagrange multiplier µ_t,

and substitute its value directly into the planner’s problem to solve for µ_t+1 in equation (8).

Finally, we assume that the planner does not have the technology to commit to future policies.¹¹ Therefore, we solve for the optimal time-consistent macroprudential policy, taking into account the effects of the planner’s current period choices on future planners’ choices. As a result, the planner does not have an incentive to deviate from policy rules of previous social planners.

We follow the utilitarian approach under which the planner wants to maximize the infinite weighted-sum of agents future discounted utilities, ${\sum }_{t=0}^{\infty }{\beta }^t\left[\omega U\right(c_t)+U(x_t\left)\right]$. The relative welfare weight on the worker, ω, is assigned exogenously and we consider alternative values for it in our quantitative exercises.

The planner’s maximization problem is given by

where the Lagrange multipliers associated with each constraint are given in parentheses.

Equation (16) denotes the budget constraint of entrepreneurs after accounting for the Pigouvian taxes/subsidies and lump-sum transfers. Similarly equation (17) denotes the budget constraint of workers. Equation (18) is the economy’s collateral constraint, and (19) is the implementability condition of the planner, which reflects the fact that the planner has to respect competitive asset pricing in the economy. It is through this equation that the planner internalizes how private agents’ choices affect equilibrium asset pricing. Note that we have substituted for µ_t+1 appearing in the right-hand side of (19) using (15) for t + 1.¹² Finally, capital and labor are set to their equilibrium aggregate values, K = 1 and $$l_t={\overline{h}}_t$$, respectively.

The first order optimality conditions of the planner take the following form

where Ω_t+1 collects all partial derivatives with respect to b_t+1 on the right-hand side of the capital-Euler equation, capturing the impact of the planner’s choice of b_t+1 on the actions of future planners (reflecting the “time-consistency” nature of the policy rule).¹³

The allocations of the planner and the competitive economy differ in two main respects. First, unlike the private agents, the planner internalizes how consumption and borrowing choices affect asset prices and hence the borrowing ability in states in which the collateral constraint binds. Second, the planner internalizes the difference in the shadow costs of wealth between workers and entrepreneurs and can improve on the allocations by affecting the relative price, i.e. wages, at which agents trade. To outline these differences, we compare the optimality conditions in the two economies.

First we compare the first order condition with respect to entrepreneurs’ consumption in the competitive and the planner’s economy. The competitive economy condition is U_x,t = λ_t, where λ_t is the Lagrange multiplier on the budget constraint of entrepreneurs (3), while the corresponding condition of the planner is given by equation (20). The key difference between these two equations is that the shadow value of wealth in the planner’s solution do not only incorporate the marginal utility from current consumption, but also the amount by which an additional unit of consumption relaxes the collateral constraint through its effect on prices (-ξ_tU_xx,tq_t). The latter is not accounted for in the competitive equilibrium.¹⁴ Hence the private agents do not internalize how their consumption choice affect the asset price, q_t, as well as the tightness of the collateral constraint. This equation is at the core of the pecuniary externality present in the model.

Next we compare the planner’s optimality condition with respect to the intermediate good (23) and the corresponding condition in the competitive economy (5). The only difference between the two is that the former is not scaled by the shadow cost of wealth, while the latter is, since the shadow cost of wealth in the planner’s condition incorporates the pecuniary externality. Using (15) and (23) we get the following condition that connects the Lagrange multipliers on the collateral constrained between the competitive economy and the planner’s solution

Hence, µ_t and ${\mu }_t^{SP}$ are either both positive or zero.

Next we compare the Euler equation for bonds in the competitive economy to the corresponding equation of the planner, which can be written as follows by combining entrepreneurs’ optimal consumption and borrowing decisions, (20) and (22),

This comparison highlights the pecuniary externalities that operate through the future and current price of capital.

Consider that the collateral constraint does not bind at t, such that the Lagrange multipliers on the collateral constraint in the competitive, µ_t = 0, and the planner’s economy, ${\mu }_t^{SP}={\xi }_t=0$, equal zero. In this case, based on the Euler equations for borrowing, (7) and (27), the planner’s economy features a higher marginal cost of borrowing at t than the competitive economy by an amount βRE_t [ξ_t + U_cc^e,t+1qt+1]. This term implies that the planner, through the implementability constraint (19), internalizes the impact that a larger debt at t has on reducing the borrowing capacity at t + 1 by lowering the price of capital, q_t+1, when the t + 1 constraint binds.¹⁵ In other words, the planner understands that more borrowing at t will need to be repaid at t + 1, which would reduce consumption if the future borrowing capacity is curtailed. This would result in higher future marginal utility and, thus, a lower asset price q_t+1, which further tightens the collateral constraint, reduces the borrowing capacity, and requires an even bigger drop in consumption and asset prices.

Now consider that the collateral constraint binds at t (and may also bind at t + 1). Then there are two opposing effects resulting from the borrowing decision that the planner needs to consider. One the one hand, higher borrowing accompanied by higher consumption at t, increases the price of capital q_t and relaxes the collateral constraint. On the other hand, more borrowing and higher consumption at t may result in lower consumption and lower price of capital at t + 1 if the collateral constraint continues to bind in the future. Hence, the planner faces a trade off between choosing allocations such that she increases current prices, q_t, at the cost of potentially decreasing future prices, q_t+1.

Finally, we consider the optimality condition (25) with respect to wages, linking the shadow cost of labor income and labor cost between workers and entrepreneurs. Note that this condition is absent from the competitive economy; yet the planner would like to equalize the shadow values of the two agents, while accounting for the possibility of a binding collateral constraint. The reason why this condition does not hold in the competitive economy is twofold. First, private agents do not internalize that the shadow value of labor income/cost depends on the pecuniary externality (i.e. ${\lambda }_t^{SP,e}$ is a function of ξ_t). Needless to say, this will only be the case as long as the collateral constraint binds. Second, even absent a binding collateral constraint, the shadow values of labor income/cost will still not be equalized between the two agents as there is no equation in the competitive economy equalizing the two. The unequalized shadow values of labor income/cost generate the distributive externality, augmented by a pecuniary externality when constraints bind. Hence, the planner chooses allocations in a way to redistribute income such that the shadow costs/benefits of labor income/cost of the two agents get closer.

3.2 Optimal Tax Rates

This section derives the optimal credit and labor tax rates. The optimal tax on credit can be derived by combining the Euler equation for borrowing of the planner (27) with the corresponding equation of the agents incorporating the credit tax (13), and takes the following form

The optimal credit tax consists of two components that match the pecuniary externalities operating via the prices of capital, q_t and q_t+1, identified in the planner’s Euler equation. For simplicity, first assume that the collateral constraint does not bind at t in both economies, i.e.

${\mu }_t={\mu }_t^{SP}={\xi }_t=0$. Then, the tax rate reduces to

which can easily be shown to be always positive as long as the collateral constraint binds in expectation. The tax rate tackles the pecuniary externality operating via q_t+1, and has a macro-prudential interpretation as it is levied during good times (i.e. when the collateral constraint does not bind), to allow for more borrowing during bad times (i.e. when the collateral constraint binds in the future).

Now assume that the collateral constraint also binds at t. In this case, $1/\left(\beta R{E}_t{U}_{x,t}\right)[{\mu }_t^{SP}-U_{x,t}{\mu }_t+{\xi }_t{U}_{xx,t}{q}_t-\beta R{\xi }_t{\Omega }_{t+1}]$ is non-zero, tackling the externality operating via q_t. This part of the tax rate pushes for a subsidy on credit as higher borrowing at t supports higher asset prices q_t and relaxes the collateral constraint. However, more borrowing at t would require a repayment at t+1, resulting in lower asset prices q_t+1 and a tighter collateral constraint. When choosing the optimal tax rate, the planner balances this two effects.

Although we cannot show analytically how the labor shock affects the pecuniary externality, we discuss its implications intuitively first and corroborate the discussion with our quantitative results presented in section 4. The labor supply shock will operate distinctly through the period t and t+1 channels described above. If the collateral constraint does not bind at t, the first order effect of the labor supply shock will be lower current production and higher borrowing to smooth consumption. The planner understands that higher indebtedness can exacerbate the pecuniary externality when constraints bind in the future and, thus, levies a higher macroprudential tax at t than otherwise via equation (29). If the collateral constraint binds at t, the first order effect of the labor supply shock will not only lower production, but also considerably lower consumption given the inability to increase borrowing. In turn, this suppresses asset prices resulting in even tighter collateral constraint and borrowing. As a result, the role of the pecuniary externality from binding constraints today—the first term in (28)—is enhanced, while the role of the pecuniary externality from binding constraints in the future—the second term in (28)—is weakened given that current borrowing and indebtedness go down. These considerations would urge the planner to levy a higher credit subsidy at t than otherwise. Overall, the labor supply shock exacerbates the policy response in opposite directions depending on whether the collateral constraint binds in the present. We verify this line of thinking when we present the quantitative results in section 4 and discuss in more detail the policy implications.

Now we turn to the optimal tax on labor. This tax rate can be derived by combining the planner s optimal decision or wages (25), which can be rewritten as $1+\theta {\mu }_t={\lambda }_t^{SP,w}/{\lambda }_t^{SP,e}$ using (26), and the privately optimal labor decision (14) that incorporates the tax rate. Moreover, from (20), (24), and (2), ${\lambda }_t^{SP,e}$ can be re-written as ${\lambda }_t^{SP,e}=U_{x,t}/(1+{\mu }_t{\kappa }_t{U}_{xx,t}/U_{x,t}{q}_t)$, while ${\lambda }_t^{SP,w}=\omega U_{c,t}$ from (21). Using the above conditions, substituting for wages from the budget constraint of workers (17), and employing market clearing in the labor market $(l_t=\overline{h}{}_t)$, the tax on labor takes the following form

The labor tax consists of two terms, the first is the labor share over workers’ consumption, and the second term is the ratio of the marginal utilities of workers and entrepreneurs augmented by a term that captures the pecuniary externality. The planner uses the tax (subsidy) on labor to tackle the distributive externality described in section 3.1. To reiterate the intuition, the objective of the planner is to manipulate the wage such that to equate the shadow values of labor income and cost between agents while taking into account binding collateral constraints.

To fix ideas, assume for now that the collateral constraint does not bind, i.e. ${\mu }_t^{SP}={\mu }_t=0$. Then, equation (25) reduces to ${\lambda }_t^{SP,e}={\lambda }_t^{SP,w}$, and the first order conditions with respect to entrepreneurs’ and workers’ consumption in the planner’s economy, (20) and (21), respectively, imply U_x,t = ωU_c,t. This means that the planner would like to redistribute income between the two agents in order to equate their marginal utilities. She will do so by manipulating wages. In turn, this means that the wage is not equal to the marginal product of labor in the planner’s economy, hence the ratio $F_{l,t}{\overline{h}}_t\ne c_t$.¹⁶ If the labor share is lower than workers’ consumption, $F_{l,t}{\overline{h}}_t<c_t$, then the planner levies a higher tax (lower subsidy) on labor, which suppresses wages and helps entrepreneurs. The opposite is true, i.e. the planner levies a lower tax (higher subsidy) on labor, if $F_{l,t}{\overline{h}}_t>c_t$. Hence, the labor tax is levied to implement the wage the planner chooses to address this distributive externality¹⁷

Next, suppose that ${\mu }_t^{SP}>0$, and µ_t > 0 from (26).¹⁸ Then, in addition to manipulating the ratio between labor share and workers’ consumption to balance the marginal utilities of consumption between the two agents, the planner also needs take into consideration the effect of the pecuniary externality. A binding collateral constraint pushes the tax rate in opposite directions. On the one hand, it pushes for a lower tax (higher subsidy) on labor, as ωU_c,t > U_x,t when constraints bind and the planner would like to redistribute more resources towards workers.¹⁹ On the other, it pushes for a higher tax (lower subsidy) on labor due to the presence of the pecuniary externality, captured by the term µ_tκU_xx,t/U_x,tq_t < 0, resulting in a higher redistribution of resources towards entrepreneurs.

Finally, it is important to understand how the labor supply shock affects the tax rate. It is straightforward that workers would suffer more from the labor shocks as they do not have any other means to consume. As long as the collateral constraint does not bind, the planner would reduce labor taxes or implement a subsidy to redistribute more resources to workers. However, if the collateral constraint binds, then the labor shock may more severely affect entrepreneurs, who could experience a rapid decrease in their consumption. This would not only exacerbate the pecuniary externality, but would also push the distributive externality more in favor of entrepreneurs. Thus, the planner would want to increase the labor tax not only to tackle the pecuniary externality, but also to redistribute more resources to entrepreneurs. We verify this line of thinking when we present the quantitative results in section 4 and discuss in more detail the policy implications.

4.1 Calibration

Given the similarity of the model, the majority of the model parameters are calibrated following Bianchi and Mendoza (2018), who calibrate their model to OECD data.²⁰ Each time period should be interpreted as a year. We deviate from their calibration for a subset of the parameters. First, the global interest rate in their model follows an AR(1) process. In order to limit the exogenous number of states in the model, we opt for a fixed interest rate R = 1.01, set at the long-term average level of the rate in Bianchi and Mendoza (2018). Second, we calibrate the fixed labor supply in normal times to the long-run average value in Bianchi and Mendoza (2018), i.e. ${\overline{h}}_n=0.6$. The rest of the parameters are reported in Table 1.

^{Table (1)}

Calibration

^{Table (1)}

Calibration

Parameter	Value
Risk aversion	σ = 1
Share of intermediate good in output	αv = 0.45
Share of labor in output	αl = 0.352
Share of assets in output	αk = 0.008
Interest rate	R = 1.01
TFP process	ρz = 0.78
	σε = 0.01
Discount factor	β = 0.95
Working capital coefficient	θ = 0.16
Tight credit regime	κl = 0.75
Normal credit regime	κh = 0.90
Transition probability, κh to κl	Ph,l = 0.10
Transition probability, κl to κl	Pl,l = 0.00
Labor supply in normal times	${\overline{h}}_n=0.60$
Labor supply in pandemic times	${\overline{h}}_p=0.51$
Transition probability, ${\overline{h}}_{n}to{\overline{h}}_p$	Pn,p = 0.01
Transition probability, ${\overline{h}}_{p}to{\overline{h}}_p$	Pp,p = 0.20

View Table

^{Table (1)}

Calibration

Parameter	Value
Risk aversion	σ = 1
Share of intermediate good in output	αv = 0.45
Share of labor in output	αl = 0.352
Share of assets in output	αk = 0.008
Interest rate	R = 1.01
TFP process	ρz = 0.78
	σε = 0.01
Discount factor	β = 0.95
Working capital coefficient	θ = 0.16
Tight credit regime	κl = 0.75
Normal credit regime	κh = 0.90
Transition probability, κh to κl	Ph,l = 0.10
Transition probability, κl to κl	Pl,l = 0.00
Labor supply in normal times	${\overline{h}}_n=0.60$
Labor supply in pandemic times	${\overline{h}}_p=0.51$
Transition probability, ${\overline{h}}_{n}to{\overline{h}}_p$	Pn,p = 0.01
Transition probability, ${\overline{h}}_{p}to{\overline{h}}_p$	Pp,p = 0.20

The COVID-19 shock represents a 15 percent drop in labor supply, i.e. ${\overline{h}}_p=85{\overline{h}}_n=0.51$. Brinca, Duarte and Faria-e Castro (2020) show that the labor supply shock in April 2020 resulted in a 12 percent drop in the growth rate of total private hours worked in the U.S. Using high-frequency Automatic Data Processing (ADP), Autor, Cho, Crane, Goldar, Lutz, Montes, Peterman, Ratner, Villar and Yildirmaz (2020) show that employment declined by about 14 percent for both big and small firms through the beginning of the crisis into April 2020. During the time while preparing this draft, data for hours worked were not yet available for the OECD average, but unemployment is forecasted to climb to about 13 percent under the adverse scenario from 5 percent before the COVID-19 outbreak.²¹ The qualitative results and the optimal policy we derive are not affected by the level of the labor supply shock. We have opted to calibrate the labor supply shock to about 15 percent based on the studies using high-frequency data of hours worked. The probability of such shock materializing is set to once in one hundred years. Moreover, the labor shock persists in the following period with probability 20 percent which implies a 14-month length for the pandemic since its outbreak.

4.2 Financial Amplification Dynamics

In order to assess the model’s ability to generate crisis and the effectiveness of the policy instruments at reducing the severity of the crisis, we perform an event analysis of the competitive and planner’s economies. We do so by examining long-term averages across three events: (I) when the collateral constraint binds, (II) when a COVID-19 shock hits, and (III) when the collateral constraint binds and a COVID-19 shock hits. Event (II) could be interpreted as the first phase of the pandemic, during which financing constraints do not bind as the drop in output due to the labor supply shock has still not weakened firms’ ability to borrow either due to deteriorated firm balance sheet or tighter lending standards; and event (III) could be interpreted as the second phase of the pandemic, during which the pandemic continues and financing constraints become binding.

The events are constructed as follows. First, the competitive and the planner’s economy are simulated for 100,000 periods and the three events are identified following the approach of event study analysis in the empirical literature.²² Next, we construct 5-year event windows centered around the year when the event materializes, i.e. T. We compute averages for each variable across the cross section of crisis events at each date. These steps generate the dynamics plotted in figures 1 and 2. The competitive and the planner’s economies start from the same level of borrowing in the initial period and go through the same simulated path of shocks for the exogenous variables. The results are presented in terms of deviations from the long-term mean.

Figure 1 shows the dynamics for new borrowing, output and asset prices in the competitive economy under the three scenarios. First, the solid line shows the responses of the variables when an adverse shock, other than a COVID-19, results in binding collateral constraints and curtails the ability to borrow. Second, the dashed line shows the response of the variables to a COVID-19 shock when collateral constraints do not bind. Finally, the dotted line shows the responses of the variables to a COVID-19 shock and binding collateral constraints.

When collateral constraints start to bind (solid line in Figure 1), the borrowing ability of agents is curtailed, which leads to a drop in asset prices and output. The reduction in output is not very severe as labor and capital supply are not impeded. On the contrary, when the COVID-19 shock hits (akin to phase one of the pandemic), output falls by about 15 percent within the same year (dashed line in Figure 1). Agents smooth consumption by borrowing, which needs to be repaid in the following period, resulting in a decrease in consumption and asset prices in the following period. Finally, when collateral constraints bind and the COVID-19 shock hits (akin to phase two of the pandemic), agents are no longer able to smooth consumption as borrowing is constrained (dotted line Figure 1), leading to a more severe drop in asset prices, output and borrowing.

Comparison of competitive economy across events

Citation: IMF Working Papers 2020, 189; 10.5089/9781513557311.001.A001

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Comparison of competitive economy across events

Citation: IMF Working Papers 2020, 189; 10.5089/9781513557311.001.A001

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Comparison of competitive economy across events

Citation: IMF Working Papers 2020, 189; 10.5089/9781513557311.001.A001

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Figure 2 compares the dynamics of the same set of variables in the competitive (solid line) and the social planner’s (dashed line) economy under the three scenarios. The first row corresponds to event (I), the second corresponds to event (II), and the third corresponds to event (III). Three observations are worth noting.

Comparison competitive economy and social planner

Citation: IMF Working Papers 2020, 189; 10.5089/9781513557311.001.A001

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Comparison competitive economy and social planner

Citation: IMF Working Papers 2020, 189; 10.5089/9781513557311.001.A001

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Comparison competitive economy and social planner

Citation: IMF Working Papers 2020, 189; 10.5089/9781513557311.001.A001

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First, Figure 2 shows that the planner’s economy is characterized by a lower volatility than the competitive economy across all crisis events. This can be seen by the lower deviation from the long-term averages for borrowing and asset prices for the planner than for the private agents.

Second, when the collateral constraint binds (event I and III), the planner’s economy experiences a much lower drop in borrowing as she manages to alleviate the negative effects of the pecuniary externality operating via the price of capital. As a result, also asset prices during these crisis events experience a smaller drop. The planner achieves this by providing credit subsidies (see Table 3 and discussion in section 4.3 for details).

^{Table (2)}

Welfare gains

Note: Welfare gains are reported in terms of average compensating consumption variations.

^{Table (2)}

Welfare gains

Unconditional	4.19%
Conditional on CC	4.17%
Conditional on COVID	4.20%
Conditional on CC & COVID	4.23%

Note: Welfare gains are reported in terms of average compensating consumption variations.

View Table

^{Table (2)}

Welfare gains

Unconditional	4.19%
Conditional on CC	4.17%
Conditional on COVID	4.20%
Conditional on CC & COVID	4.23%

Note: Welfare gains are reported in terms of average compensating consumption variations.

^{Table (3)}

Tax rates across events

^{Table (3)}

Tax rates across events

	CC		COVID		CC & COVID
	κh	κl	κh	κl	κh	κl
τb	-0.3%	-6.2%	-0.4%		0.8%	-6.8%
τh	30.4%	57.1%	34.9%		30.6%	67.0%

View Table

^{Table (3)}

Tax rates across events

	CC		COVID		CC & COVID
	κh	κl	κh	κl	κh	κl
τb	-0.3%	-6.2%	-0.4%		0.8%	-6.8%
τh	30.4%	57.1%	34.9%		30.6%	67.0%

Third, when the COVID-19 shock hits but the collateral constraint does not bind, private agents would like to borrow to smooth consumption. The planner would like to do the same, but unlike the private agents she takes into account that higher borrowing today limits the borrowing ability tomorrow. Conditional on a COVID-19 shock, the probability that the collateral constraint binds is higher once the COVID-19 shock hits. The planner internalizes this, hence she restricts the increase in current period’s borrowing (compared to the competitive economy) to avoid becoming too indebted such that pecuniary externalities are exacerbated in the future. The planner achieves this by imposing a minimal credit subsidy. However, when the COVID-19 shock is accompanied by binding collateral constraints, the planner would levy a bigger credit subsidy to support borrowing and tackle not only the adverse consequences of the COVID-19 shock, but also the pecuniary externalities from binding constraints in the present (see Table 3 and discussion in section 4.3 for details).

Before turning to the implementation of the planner’s solution with tax instruments, we present the welfare gains achieved by the planner. Table 2 shows the percentage compensating consumption variation, γ^Event, equally obtained by both agents such that the social welfare of the competitive economy equals the welfare in the planner’s economy. We calculate both unconditional and conditional on the three events welfare gains using the following formula:

where Event corresponds to (i) all observations in the simulations (unconditional welfare gain), or (ii) one of the three events (conditional welfare gain). Then mean welfare gain is the average γ^Event(B, s) computed with the ergodic distribution.

Table 2 shows that the planner’s economy always achieves a higher welfare than the competitive economy (with both agents realizing positive gains).

4.3 Tax Instruments

In this section, we examine the quantitative features of the policy instruments and their welfare implications.

The top-left panel in Figure 3 shows the optimal borrowing tax as a function of the current level of debt for looser lending standards, i.e. κ_t = κ^h , and a low productivity realization.²³ The chart verifies the findings in section 3.2, confirming an established result in the literature: For low levels of debt, the collateral constraint does not bind either today or tomorrow and the tax on borrowing is zero. As debt levels increase, the probability of a binding collateral constraint tomorrow increases, requiring a positive tax on borrowing to tackle the pecuniary externalities (operating via q_t+1) from binding collateral constraints in the future (Bianchi and Mendoza, 2018). However, for higher levels of debt, binding collateral constraints today reverse the direction of policy, requiring a subsidy on borrowing to tackle the pecuniary externalities (operating via q_t) from binding collateral constraints in the present.

Tax Policy Functions

Citation: IMF Working Papers 2020, 189; 10.5089/9781513557311.001.A001

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Tax Policy Functions

Citation: IMF Working Papers 2020, 189; 10.5089/9781513557311.001.A001

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Tax Policy Functions

Citation: IMF Working Papers 2020, 189; 10.5089/9781513557311.001.A001

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These qualitative results continue to hold under a COVID-19 shock, but the quantitative response differs in interesting ways. First, the macroprudential tax used to lean against pecuniary externalities in the future is higher, which can be interpreted as a means to preempt the financial amplification induced by the COVID-19 shock should the collateral constraints bind in the future. Second, the borrowing subsidy when collateral constraints bind in the present is higher when accompanied by a COVID-19 shock. This result can be interpreted as a way to deal with the current financial amplification of the COVID-19 shock. Third, both the borrowing tax and subsidies kick in for much lower levels of existing debt suggesting a higher need for policy intervention during the pandemic.

The top right chart in Figure 3 presents the optimal borrowing tax for tighter lending standards, i.e. κ_t = κ^l, and a low productivity realization. As above, the borrowing subsidy is higher and kicks in faster under a COVID-19 shock. An important difference is that there is no scope for preemptive macroprudential policy, which is reasonable given the curtailed ability to borrow due to the unfavorable financial conditions. This result could potentially suggests that preemptive policy is only possible during the first phase of the pandemic when firms can still obtain credit at more favorable conditions. But during the second phase, accommodative policies are needed even for lower levels of indebtedness.

These results are verified in the average borrowing tax across the three events in the simulated economy presented in Table 3. When financial conditions are favorable (κ^h), the tax on borrowing is very close to zero prescribing a minimal role for intervention.²⁴ This result implies that, across all three events, it is optimal to balance the negative impact of current versus future externalities from borrowing decisions. The former dominates when current financial conditions are unfavorable (κ^l), pushing for a higher borrowing subsidy, and vice versa.

The bottom left chart in Figure 3 presents the optimal labor tax for favorable financial conditions and low realization of productivity. As discussed in the analytical part in section 3.2, the labor tax effectively represents a redistribution across agents by affecting the wage rate.²⁵ In general, if workers’ (entrepreneurs’) consumption is lower, the planner will levy a lower (higher) labor tax. Moreover, when a COVID-19 shock hits the economy, the planner further reduced the tax on labor.²⁶. This result is reasonable given that workers experience a large drop in income by not being able to supply labor. However, for high levels of debt, the ability of entrepreneurs to borrow and produce is lower, resulting in lower entrepreneurs’ profits compared to workers’ labor income, which manifests in a higher labor tax. Due to the presence of pecuniary externalities, these tax rates are amplified if collateral constraints are binding in the present, as we show analytically in section 3.2. As discussed, this amplification comes from the desire to tackle the pecuniary externality on top of engaging in redistribution between workers and entrepreneurs. The amplification is much more severe under a COVID-19 shock, resulting in a labor policy that is even less favorable for workers as entrepreneurs are constrained to borrow.

Tax policy functions for different values of ω

Citation: IMF Working Papers 2020, 189; 10.5089/9781513557311.001.A001

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Tax policy functions for different values of ω

Citation: IMF Working Papers 2020, 189; 10.5089/9781513557311.001.A001

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Tax policy functions for different values of ω

Citation: IMF Working Papers 2020, 189; 10.5089/9781513557311.001.A001

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The bottom right chart in Figure 3 presents the optimal labor tax for unfavorable financial conditions and low realizations of productivity. The same qualitative results, as under favorable financial conditions, continue to hold. Quantitatively the difference is that financial amplification is more pronounced under a COVID-19 shock calling for higher labor taxes to support entrepreneurs, especially for higher levels of indebtedness. This finding is important for comparing the first and second phase of the pandemic as policies to support labor income may be tougher to implement in the latter. The average tax rates presented in Table 3 confirm these findings: When financial conditions are unfavorable, the tax on labor is always higher than when they are positive.