Inequality, fiscal policy and COVID19 restrictions in a demand-determined economy

Abstract

We evaluate the effects of inequality, fiscal policy, and COVID19 restrictions in a model of economic slack with potentially rigid capital operating costs. Rich households satiate their demand for goods/services (and consume an endowment on the margin), whereas poor households’ spending on goods/services is limited by their income (which in turn depends on spending by the rich and on fiscal transfers). The model implies that inequality has large negative effects on output, while also diminishing the effects of demand-side fiscal stimulus. COVID restrictions can reduce current-period GDP by more than is directly associated with the restrictions themselves when rigid capital costs induce firm exit. Higher inequality is associated with larger restriction multipliers. The effectiveness of fiscal policies depends on inequality and the joint distribution of capital operating costs and firm revenues. Furthermore, COVID19 restrictions can cause future inflation, as households tilt their expenditure toward the future.

“Low-income households have experienced, by far, the sharpest drop in employment, while job losses of African-Americans, Hispanics and women have been greater than that of other groups. If not contained and reversed, the downturn could further widen gaps in economic well-being that the long expansion had made some progress in closing” –Jerome Powell, testimony to Senate Banking Committee, June 16th, 2020.

1. Introduction

Rising inequality and COVID-related reductions in consumer spending are among the most prominent features of the current economic environment. Yet the macroeconomic implications of inequality and COVID-19 restrictions are far from clear. With respect to inequality, Summers (2015) and Krugman (2016) have posited that inequality can substantially dampen aggregate demand, while more formal models have not found support for this hypothesis (Auclert and Rognlie, 2020). More generally, the key macroeconomic transmission mechanisms during a pandemic remains an open question. For example, firms are faced with falling revenues and potentially rigid (pre-committed) payments (e.g., debt payments and rent). Which fiscal policies are the most effective in such an environment? Understanding the transmission channels of fiscal policy is as pressing now as ever, with trillions of dollars on the line.

In this paper we offer new insights on the roles of inequality and COVID19-related restrictions, separately and in conjunction in transmitting macroeconomic shocks, and we evaluate the effectiveness of various fiscal stimuli. In particular, we extend a model of economic slack (Murphy, 2017) to an environment in which inequality, capital costs, and multiproduct firms play a central role.1

The mechanism through which inequality affects output is a formalization of the relationship conjectured by Krugman and Summers that declines in the share of income accruing to the majority of the population pull down aggregate demand. Households in our model have non-homothetic preferences over goods/services that are produced with negligible marginal costs (NMC) and an endowed numeraire good. The rich satiate their demand for goods/services in the NMC sector, whereas poor households’ spending on NMC goods/services is limited by their income (which in turn depends on spending by the rich and on fiscal transfers). Therefore, a lower income share for the poor (and hence higher inequality) is associated with lower spending by poor households and hence lower aggregate income and output. Inequality has potentially strong effects even though all households can access credit.

Inequality not only pulls down aggregate output, but it also dampens the multiplier effects of shocks to aggregate demand. Government transfers, for example, have weaker effects the lower is the income share of the poor, as each round of spending leads to a lower increase in income for the poor. This result stands in stark contrast to the predictions of standard models with heterogeneous agents, in which higher inequality is often associated with more credit-constrained households and hence stronger fiscal multipliers. Recent empirical evidence (e.g., Miranda-Pinto et al., 2020a; Yang, 2017) has documented an inverse relationship between inequality and fiscal effects, thus pointing to the potential relevance of the mechanism in our theory.

In addition to offering general insights on the effects of inequality, the model also provides a lens through which to analyze the pandemic. We model the economic restrictions associated with COVID19 as a temporary decrease in the share of varieties of goods/services that can be sold by multi-product firms.2 The effect of COVID19 restrictions depends on the steady-state level of transfers to poor households, as well as the extent to which firms face fixed capital operating costs. If there are no steady-state transfers and capital operating costs are flexible (such that the price of capital adjusts to prevent firm exit), then the output loss is proportional to the fraction of varieties that are directly subject to COVID19 restrictions – that is, the restriction multiplier is unity, and consumption of unrestricted products remains the same. This is because the direct reduction in household spending on restricted products equals the reduction of household income from selling those products (and hence no additional adjustments are necessary).

Positive steady-state transfers to the poor lead to a smaller restriction multiplier. Households allocate their expenditure across varieties and across time. When fewer varieties are available in the current period, households reallocate their transfer income (if any) toward the remaining existing products during the pandemic and toward products available in the future. This higher per-product spending leads to higher output per product in the current period, which mitigates the aggregate output effects of the restrictions.

One of the most pressing policy questions to emerge during the pandemic is how consumer spending will react once the pandemic subsides. Our model predicts that COVID-19 restrictions cause future nominal spending to increase beyond what it would have been in the absence of restrictions (due to the expenditure reallocation across time). This effect is magnified by government transfers. If prices are flexible in the future, the restrictions cause future inflation.

Restriction multipliers are potentially much larger when firms’ fixed operating costs are rigid. Restrictions on a subset of firms’ products pulls down firm revenue, which causes firms for which fixed operating costs are high relative to steady-state revenues to exit and therefore cease production of other unrestricted goods and services. For example, restaurants are restricted from serving customers in the establishment but are able to provide carry-out and delivery services, and airlines shut down some routes (and/or passenger seats) but maintain others. If restrictions cause some restaurants’ revenues to decline below their fixed costs, then these restaurants will cease producing carry-out services. Likewise, airlines with revenues below fixed costs will cease flying entirely. This firm exit channel leads to large indirect (multiplier) effects of economic restrictions and provides a strong rationale for policies aimed at mitigating fixed operating costs. In the absence of these multiplier effects or significant re-entry costs, it might be optimal to allow firms to temporarily exit and then re-enter once restrictions are lifted. But the large multipliers imply that such exit can be very costly.

We examine the effect of various fiscal transfers in this environment: direct transfers to households, direct transfers to firms, and various targeted transfers. The preferred policy depends on inequality, the joint distribution of firm revenues and fixed operating costs, and the extent to which the government can target particular groups of households and firms.

Targeted transfers to low-income households can increase spending on unrestricted items, thus supporting income during the restrictions. However, the transfers have smaller effects during the pandemic period since there are fewer products on which to spend, and they can lead to an inflationary spending boom in the post-pandemic period. Furthermore, the output effect of transfers is falling in inequality, as spending multipliers are increasing in the income share of the poor.

The strongest effect of fiscal stimulus is through targeted transfers to multiproduct firms for which the restrictions push their revenues below their fixed operating costs. Such targeted transfers prevent firm exits that lead to large secondary (multiplier) output declines. In practice it may be difficult to identify and target such firms, although the model offers some guidance. The firms most at risk of exit are those with relatively low profitability and for which fixed operating costs are the largest or most rigid. As documented by Gilje et al. (2020), rigid capital contracts can arise from asymmetric information regarding firms’ ability to cover capital costs. In our context, the asymmetric information friction is perhaps the most severe for smaller businesses that are not subject to the same reporting requirements as public firms. Direct loans and transfers to small private businesses may therefore target the firms on the margin of exit and have large benefits per dollar spent.

While targeted transfers to firms have the largest potential benefit, untargeted transfers to firms have among the least benefit. Not only are some resources spent on firms that are not in danger of exit, but a large share of the transfers received by firms ultimately accrues to high-income households for whom spending is less sensitive to transfers.

This paper is broadly related to emerging work evaluating the indirect economic effects of COVID19, with different papers focusing on different transmission channels. For example, Baqaee and Farhi (2020) focus on the production network, Fornaro and Wolf (2020) focus on productivity growth, and Caballero and Simsek (2020) examine the role of asset prices. Using a large calibrated HANK model, Auclert and Rognlie (2020) find that, for standard business cycles, inequality has relatively mild effects on output, while we find that the effect of inequality can be quite large in the NMC setting.3 Most closely related is Guerrieri et al. (2020), who model COVID19 as a restriction on labor supplied to a subset of firms. They argue that COVID19 restrictions can cause a further fall in output in the presence of strong complementarities between restricted goods and other goods (low elasticity of substitution across products, EOS), a large intertemporal elasticity of substitution (IES), and large shares of credit-constrained households–parameters for which there is strong disagreement in the literature. If these conditions are sufficiently strong, the economy can exhibit a multiplier whereby output falls by more than the size of the direct supply restrictions. Our approach to modeling the COVID19 restrictions is similar in that a subset of firms cannot sell output to consumers but the transmission mechanisms in our model are quite different (e.g., our model does not rely on credit constraints, high IES, or low EOS).4 In particular, the transmission mechanisms in our model stem from a subset of households for which current and future spending are limited by their income (which is determined by autonomous government and rich-household spending). The link between spending and income for these households is based not on credit constraints but rather on interactions between preferences and permanent income heterogeneity. Furthermore, our tractable household demand system along with negligible marginal costs permits an analytical assessment of the inflationary effects of macroeconomic shocks, as firms’ prices depend directly on household income rather than only indirectly through marginal costs.5

We also contribute to the rapidly growing literature emphasizing the effects of micro-level heterogeneity for macroeconomic outcomes. This line of work typically focuses on differences in liquidity constraints and idiosyncratic income risk. For example, Bilbiie (2020) provides a particularly illuminating analysis of how a New Keynesian cross can arise in a two-agent economy (liquidity-constrained and unconstrainted households) and more generally in a New Keynesian economy with heterogenous agents. This literature also examines how inequality can amplify or attenuate business cycles. Similar to this earlier work, we also stress micro-level heterogeneity but at the same time we abstract from liquidity constraints, treat inequality as exogenous, and focus on differences in preferences and permanent income.

Finally, the role of firm exit in amplifying shocks shares features of the mechanism in Bilbiie and Melitz (2020), whereby a negative supply shock induces exit among firms that cannot fully adjust prices in response to the increase in marginal costs. In our model, the driving force for exit is a reduction in revenues due to restrictions on a subset of products (rather than broad-based changes in marginal costs). Both models imply amplification of shocks through the firm exit channel in the presence of nominal rigidities.

At attractive feature of our model environment is that it yields an analytic analysis of the interactions among COVID-19 restrictions, inequality, multiproduct firms, fixed operating costs, and fiscal policy. As a result, the model delivers new insights, including (as discussed above) the potentially large effects of inequality on GDP, post-pandemic nominal spending booms, the importance of mitigating fixed operating costs during the pandemic, and the cost effectiveness of fiscal transfers targeted to firms on the margin of exit.

2. Baseline model

2.1. Motivation and overview

Our model is motivated both by a long-term trend of rising inequality as well as the recent pandemic crisis of 2020 (and the corresponding fiscal policy response). Early in the pandemic many local governments imposed restrictions on travel and other contact-intensive services. Meanwhile, many rich households voluntarily cut back on purchases of contact-intensive services even in the absence of government restrictions (e.g., Alexander and Karger, 2020, Chetty et al., 2020, Goolsbee and Syverson, 2020). These restrictions and cutbacks led to a massive recession, including a rapid decline in the number of small businesses (Fairlie, 2020) and sharp declines in household wages and income, especially for households with low income (Cajner et al., 2020).

In response to the recession, the U.S. government passed the CARES Act, which authorized over $2 trillion in fiscal transfers including rebate payments to households of up to $1200 and transfers to businesses that maintained their payroll (the Paycheck Protection Program). Given the unprecedented scale of the recession and fiscal stimulus, understanding which policy levers are the most effective is imperative. One view is that fiscal stimulus was unnecessary, since the recession was an efficient response to a global health crisis. Another view is that spending plummeted by more than would be implied by private spending cuts directly associated with health concerns, justifying government intervention to stimulate aggregate demand (and raising the question of which interventions are most effective).

2.2. Model

To formally study these issues, we develop the heterogeneous-household version of the negligible-marginal-cost (NMC) model in Murphy (2017). This version of the model features rich and poor households, denoted by $h\in \left\{\mathbb{R},\mathbb{P}\right\}$, each of which receives different shares of income from the NMC sector and consumes services from the NMC sector. We further extend the model to incorporate government transfers, multiproduct firms, and features that map into the pandemic crisis.

The model also features an endowment good that is owned and consumed by the rich. The endowment represents land or other factors of production that are used to produce goods consumed primarily by the rich (e.g., beach homes and other luxury items). One should interpret rich households as those who are sufficiently wealthy that, on the margin, they spend additional income on items (e.g., high-end properties) that are not consumed by the non-wealthy. For example, a millionaire likely directs additional income toward purchasing a beachfront home rather than spending more on restaurant meals. Note that, by construction, the endowment good is in fixed supply, and hence additional spending on it does not contribute directly to real GDP. As discussed below, changes in the demand for the endowment good may affect nominal GDP through the impact on the endowment's market-clearing price. A key aspect of this modeling assumption is that the wealthy will have a negligible marginal propensity to consume out of increases in income on NMC-sector commodities, which is consistent with the evidence that very wealthy households have marginal propensities to spend additional income on typical goods and services of effectively zero (Ganong et al., 2020).

Poor households in the model should be interpreted as including middle-income households and more generally any household that has not satiated its demand for standard goods and services (e.g., haircuts, restaurant meals, etc.). For simplicity, and without loss of generality, we aggregate across all non-rich households and refer to them as poor.

The endowment pins down the interest rate and the consumption path of the rich household (and hence delivers a unique equilibrium). Agents trade bonds to satisfy their desired time paths of consumption, subject to a no-Ponzi constraint that the present value of their asset position must be weakly greater than zero.

We assume that time can be split into two periods: $t=0$, which captures the crisis, and $t=1$, which, without loss of generality, corresponds to all post-crisis time. We evaluate policy responses to a one-time, exogenous restriction in spending at date $t=0$.

2.3. Households

There is a unit mass of homogenous varieties in the NMC sector. Households inelastically supply labor to the NMC sector, and there are zero marginal costs of labor associated with increasing output.6 In this sense there is firm-level slack. In the initial period, a share $1-\xi$ of the varieties is restricted from being sold. We interpret a reduction in $\xi$ as either direct restrictions imposed by the government or choices by households to avoid certain services due to health concerns.7

Household type $h$ maximizes

$$U^h=\sum _{t=0}^1{\beta }^t\left(y_t^h+\underset{0}{\overset{{\psi }_t}{\int }}\underset{0}{\overset{{\xi }_t}{\int }}\left({\theta }_t^h{q}_{jkt}^h-\frac{\gamma }{2}{\left(q_{jkt}^h\right)}^2\right)dkdj\right),$$ (1)

subject to the budget constraints

$$\underset{0}{\overset{{\psi }_0}{\int }}\underset{0}{\overset{{\xi }_0}{\int }}{p}_{jk0}{q}_{jk0}^{h}dkdj+y_0^h+QB={\Pi }_0^h+e_0^h+T_0^h,$$ (2)

$$\underset{0}{\overset{{\psi }_1}{\int }}\underset{0}{\overset{{\xi }_1}{\int }}{p}_{jk1}{q}_{jk1}^{h}dkdj+y_1^h={\Pi }_1^h+e_1^h+T_1^h+B,$$ (3)

where $q_{jkt}^h$ is type $h$’s consumption of variety $k\in \left[0,1\right]$ from firm $j\in \left[0,1\right]$ from the NMC sector in period $t$. The household's preferences are over each producer-commodity ($jk$) element. ${\xi }_t$ is the fraction of goods/services that can be sold without restriction and ${\psi }_t\le 1$ is the endogenously determined number of firms in the economy. We will assume that ${\xi }_t=1$ and ${\psi }_t=1$ in the absence of COVID-related restrictions, and that the restrictions imply ${\xi }_0\equiv \xi <1,{\xi }_1=1$ (and potentially ${\psi }_t<1$).8 For simplicity we assume that reductions in $\xi$ are associated with equal restrictions for poor and rich households, although there is some evidence that rich households may have been more likely to avoid spending due to health concerns (Chetty et al., 2020).9 ${\Pi }_t^h$ is agent $h$’s income from the NMC sector of the economy, $e_t^h$and $y_t^h$ are $h$’s endowment and consumption of the numeraire, where $e_t^{\mathbb{P}}=0$. $T_t^h$ is net transfers from the government. $Q$ is the price of a bond $B$ that pays a unit of the numeraire in period 1. Since agents can smooth consumption (and hence the effect per unit of the present value of future net transfers is the same as that of present-period net transfers), we will write the present value of total net transfers as $T^h\equiv T_0^h+Q{T}_1^h$.

A convenient feature of the quasilinear utility function is that agents consume only the good from the NMC sector when their income is sufficiently low (depending on $\theta$ and $\gamma$).10 This feature, along with the assumption that poor agents are not endowed with the numeraire, $e_t^{\mathbb{P}}=0\forall t$, greatly simplifies the analysis and maintains the focus on demand-determined output in the NMC sector. We assume parameter values such that the rich household consumes the NMC-sector goods and the numeraire endowment good, while the poor household consumes only the NMC-sector goods. One implication of this assumption is that, similar to the Lucas-tree model, variation in endowments $e$ pins down $Q$ to the discount factor $\beta$ of the rich, that is $Q=\beta$. This assumption is a reduced-form attempt to model the economy when interest rates are fixed at some level (for example, the effective lower bound).

2.4. Firms

Output in the NMC sector is produced by firms who hire workers as fixed costs and pay a fixed capital operating cost $f_{jt}$. Firm $j$ faces demand for product $jk$ from household type $h$

$$q_{jkt}^h=\frac{1}{\gamma }\left({\theta }_t^h-{\lambda }_{ht}{p}_{jkt}^h\right),$$ (4)

where ${\lambda }_{ht}$ is household $h$’s budget multiplier at time $t$.

We assume that prices for NMC goods/services in period $t=0$ are fixed at the respective levels for the poor and the rich, ${\overline{p}}_{jk0}^{\mathbb{P}}$ and ${\overline{p}}_{jk0}^{\mathbb{R}}$, that we would observe if firms set their prices on their expectation that $\xi =1$. These prices remain fixed in the presence of shocks in period $t=0$. We write ${\overline{p}}_{jk0}^{\mathbb{P}}$ and ${\overline{p}}_{jk0}^{\mathbb{R}}$ with an overbar to emphasize that these prices are rigid.11 Prices in the post-crisis period $t=1$ are fully flexible.12

For analytic convenience, we assume that firms can discriminate between the rich and the poor when setting prices. The profit-maximizing price charged to household type $h$ is

$$p_{jkt}^h=\frac{{\theta }_t^h}{2{\lambda }_{ht}^S},$$ (5)

where due to rigid initial-period prices ${\lambda }_{h0}^S$ is the household $h$’s period-0 budget multiplier in the state of the world in which there are no shocks ($\xi =1$) and ${\lambda }_{h1}^S$is the household's period-1 budget multiplier adjusted for the realization of shocks. The rich household's budget multiplier is pinned down by the marginal utility of the numeraire, ${\lambda }_{\mathbb{R}t}=1$, so prices charged to the rich are invariant to all shocks other than the rich household's preference for NMC-sector goods/services ${\theta }^{\mathbb{R}}.$

Given prices in Eq. (5) and imposing ${\lambda }_{\mathbb{R}t}=1$ we can write quantities demanded as

$$q_{jk0}^h=\frac{1}{\gamma }\left({\theta }^h-{\lambda }_{h0}{\overline{p}}_{jk0}^h\right),q_{jk1}^h=\frac{{\theta }^h}{2\gamma }$$ (6)

and expenditure by each household type $h$ on each good $jk$ as

$$c_{jkt}^h\equiv p_{jkt}^h{q}_{jkt}^h.$$ (7)

Let ${\overline{\theta }}^{\mathbb{R}}$ be the firms’ expected rich-household demand parameter prior to the realization of any shocks. Then we can write $c_{jk0}^{\mathbb{R}}=\frac{{\overline{\theta }}^{\mathbb{R}}}{2\gamma }\left({\theta }_0^{\mathbb{R}}-\frac{{\overline{\theta }}^{\mathbb{R}}}{2}\right)$ and $c_{jk1}^{\mathbb{R}}={\left({\theta }_1^{\mathbb{R}}\right)}^2/4\gamma$. Rich-household expenditure on any given firm-commodity is a function only of exogenous parameters and we therefore treat $c_{jkt}^{\mathbb{R}}$as exogenous for the remainder of the analysis. This invariance of rich-household expenditure to other macroeconomic conditions greatly simplifies the analytic derivation of results. One can interpret $c_{jkt}^{\mathbb{R}}$ as “autonomous” spending in the economy.

A firm $j$’s revenues are equal to expenditures across households and products: $R_{jt}=\underset{0}{\overset{{\xi }_t}{\int }}\left(c_{jkt}^{\mathbb{R}}+c_{jkt}^{\mathbb{P}}\right)dk$. By symmetry of varieties (all firm-commodity combinations that continue to be produced in equilibrium have the same revenue), we can write $R_{jt}={\xi }_t\left(c_{jkt}^{\mathbb{R}}+c_{jkt}^{\mathbb{P}}\right)$. Firm $j$ pays a fixed capital operating cost $f_{jt}$ in period $t$.13 We assume that households own capital in the same proportion to their share of firm profits and so we roll capital income into profits (i.e., $\Pi$ includes profits and $f_{jt}$). A firm exits for period $t$ if $R_{jt}<f_{jt}$. We assume that the distribution of fixed costs is such that the unit mass of firms all produce if $\xi =1$ and that only a share ${\psi }_0\left(\xi \right)<1$ continue to produce in the initial period if $\xi <1$. If there are additional costs to re-entry once restrictions are lifted, then ${\psi }_1<1$. In the absence of such costs to re-entry, ${\psi }_1=1$.

The poor household receives a share $\kappa$ of the revenues from the NMC sector in each period, while the rich household receives the remaining $1-\kappa$ share.14 The poor household also owns a share $\kappa$ of the capital stock (and therefore earns a share $\kappa$ of the payments from firms for fixed capital operating costs). It can be shown that there exists a threshold value $\overline{\kappa }$ such that $\forall 0<\kappa <\overline{\kappa }$, the poor consume output only from the NMC sector. $\overline{\kappa }$ depends on model parameters and fiscal policy. We assume parameter values such that $\kappa <\overline{\kappa }$.

Output (real GDP) $Y_t$ is defined as the product of quantities consumed per product and total mass of available products:

$$Y_0=\xi {\psi }_0\left(q_{jk0}^{\mathbb{P}}+q_{jk0}^{\mathbb{R}}\right),Y_1={\psi }_1\left(q_{jk1}^{\mathbb{P}}+q_{jk1}^{\mathbb{R}}\right).$$ (8)

2.5. Equilibrium

Equilibrium consists of prices and quantities such that households maximize utility (1) subject to budget constraints (2) and (3), firms’ prices are given by Eq. (5), and ${\psi }_t$ is determined by the number of firms for which revenues exceed fixed capital operating costs (specified below).15

The interesting aspects of the equilibrium are based on the expenditure of poor households (since the rich household's real expenditure is effectively exogenous). Total expenditure by household $h$ in period $t$ is the sum of expenditure on the varieties. Given the assumptions about ${\xi }_t$, we can write

$$c_0^h=\underset{0}{\overset{{\psi }_0}{\int }}\underset{0}{\overset{{\xi }_0}{\int }}{c}_{jk0}^{h}dkdj={\psi }_0\xi c_{jk0}^h,c_1^h=\underset{0}{\overset{{\psi }_1}{\int }}\underset{0}{\overset{{\xi }_1}{\int }}{c}_{jk1}^{h}dkdj={\psi }_1{c}_{jk1}^h.$$ (9)

Let $C^{\mathbb{P}}$ be the present value of the poor household's total lifetime expenditure. Then substituting (9) into (2) and (3) and simplifying implies that the present value of the poor household's total lifetime expenditure is

$$C^{\mathbb{P}}=c_0^{\mathbb{P}}+Q{c}_1^{\mathbb{P}}={\psi }_0\xi c_{jk0}^{\mathbb{P}}+Q{\psi }_1{c}_{jk1}^{\mathbb{P}}.$$ (10)

To be clear, $c_{jkt}^h$ represents the equilibrium level of spending, which is the same for any $jk$ produced in equilibrium. To save notation, from now on, $jk$ denotes spending on any variety. The poor household's lifetime income $I^{\mathbb{P}}$ is

$$\begin{array}{c}{I}^{\mathbb{P}}=\kappa {\psi }_0\xi R_{j0}+Q\kappa {\psi }_1{R}_{j0}+T^{\mathbb{P}}\hfill \\ =\kappa {\psi }_0\xi \left(c_{jk0}^{\mathbb{P}}+c_{jk0}^{\mathbb{R}}\right)+Q\left(\kappa {\psi }_1\left(c_{jk1}^{\mathbb{P}}+c_{jk1}^{\mathbb{R}}\right)\right)+T^{\mathbb{P}},\hfill \end{array}$$ (11)

which reflects the fact that the poor household earns a share $\kappa$ of total expenditure. Since households own capital in the same proportion to their share of firm profits and households (as firm owners) are both liable for firms’ capital operating costs and receive income from payments to capital, capital costs and income are netted out of household income.

Setting lifetime expenditure $C^{\mathbb{P}}$ equal to lifetime income $I^{\mathbb{P}}$ and collecting terms yields

$$c_{jk0}^{\mathbb{P}}{\psi }_0\xi \left(1-\kappa \right)+c_{jk1}^{\mathbb{P}}{\psi }_1\left(Q-Q\kappa \right)=c_{jk0}^{\mathbb{R}}{\psi }_0\xi \kappa +c_{jk1}^{\mathbb{R}}Q{\psi }_1\kappa +T^{\mathbb{P}}.$$ (12)

Inspection of Eq. (12) implies that increases in transfers and rich-household spending will increase poor-household spending (and hence increase GDP in periods in which prices are rigid). Indeed, many of the results derived below follow directly from Eq. (12).

We can represent the equilibrium condition graphically by writing $C^{\mathbb{P}}$ as a function of $I^{\mathbb{P}}$ using Eqs. (10) and (11):

$$C^{\mathbb{P}}=\left[\frac{I^{\mathbb{P}}}{\kappa }-\left({\psi }_0\xi c_{jk0}^{\mathbb{R}}+Q{\psi }_1{c}_{jk1}^{\mathbb{R}}+\frac{T^{\mathbb{P}}}{\kappa }\right)\right].$$ (13)

This expression, along with the equilibrium condition that lifetime expenditure equals lifetime income (represented by Eq. (12)), pins down the equilibrium in a quasi-Keynesian cross, as depicted in Fig. 1 . It is immediately apparent that increases in $\kappa$ (declines in inequality) flatten the consumption line, thus increasing equilibrium lifetime income. Increases in transfers, rich-household consumption, and the number of available products lead to increases in equilibrium income, as they reduce the y-intercept and shift out the consumption line. Since the slope of the consumption line is greater than unity, these shifts lead to even larger income effects in equilibrium. An increase in transfers, for example, leads to an increase in desired consumption, which increases income (and hence consumption), leading to large multiplier effects. Propositions 1 and 6 below follow from this graphical depiction of the equilibrium.

Fig. 1.

Equilibrium determination.

The quasi-Keynesian cross depicted in Fig. 1 implies larger multipliers than a traditional Keynesian cross, in which the consumption function is flatter than the equilibrium (income=expenditure) line. In this sense our cross is similar to the Keynesian cross implied by the two-agent model in Bilbiie (2008) and Bilbiie (2020).16 While both our model and Bilbiie's model imply large multipliers, the mechanisms are quite different: in Bilbiie's model, large multipliers are driven by credit-constrained households whose income shares rise with aggregate output. Here, large multipliers are driven by increases in permanent income that cause higher desired spending (and hence higher income and so on) among households who can smooth consumption. In this sense, the model is most similar to the Keynesian cross implied by Murphy (2015), in which autonomous (government) spending raises expectations of permanent income due to imperfect information, which induces additional private spending and income.17 One of the distinguishing features of our model is that large consumption multipliers occur even in the presence of perfect information and the absence of credit constraints.

2.6. NK vs. NMC frameworks

To draw contrast between the NMC framework and a New Keynesian (NK) approach, note that a simple way of capturing the mechanics of a New Keynesian model is to assume

$$Y_0^{NK}=C_0,Y_1^{NK}=\overline{Y},$$ (14)

where the superscript indicates the New Keynesian representation of the model. Here, future output $Y_1$ is determined by the endowment $\overline{Y}$, reflecting the supply-side dominance of New Keynesian models at horizons after which price rigidities have dissipated. To solve the model, one must simply determine $C_0$, which in general will be based on consumption smoothing and an intertemporal budget constraint. A simple version of consumption smoothing can be written as

$$C_0=C_1,$$ (15)

and the budget constraint can be written (assuming $\beta =1$) as

$$C_0+C_1=Y_0^{NK}+Y_1^{NK}.$$ (16)

Substituting the equilibrium conditions from (14) and solving for $C_0$ yields

$$C_0=\overline{Y}\Rightarrow Y_0=\overline{Y}.$$ (17)

Therefore, in the presence of consumption smoothing (the absence of credit constraints), output in the demand-determined period depends on the future supply side of the economy. In short, in the absence of credit constraints, the supply side dominates. As a result, credit constraints (and associated high MPCs) and the strength of intertemporal substitution are key considerations for policymakers in thinking about the macroeconomic effect of the restrictions (e.g., Guerrieri et al., 2020). If one is persuaded by recent evidence that many middle-to-low-income households are not credit-constrained but rather have low MPCs (see, e.g., Miranda-Pinto et al. 2020b for a survey), or by evidence that the elasticity of intertemporal substitution is well below unity (e.g., Cashin and Unayama, 2016; Schmidt and Toda, 2019), then one may conclude that output effects of the restrictions are not a reason for policy intervention.

Now consider a situation in which future output is demand-determined.18 In this case, the equilibrium conditions can be written as

$$C_0=\frac{1}{2}\left(Y_0+Y_1\right),$$ (18)

where $Y_0=C_0$ and $Y_1=C_0$(by consumption smoothing). Here, any level of desired consumption is a potential equilibrium. This is similar to the indeterminacy of equilibria in some NK models featuring liquidity traps (e.g., Benhabib et al., 2002). Our NMC model avoids this indeterminacy problem because a share of spending (in particular, that of rich households on the NMC sector) is determined by exogenous parameters and is independent of income.

As discussed above in relation to Fig. 1, the poor-household income share parameter in our model, $\kappa$, affects GDP. As we discuss below, it also affects fiscal multipliers. In this sense $\kappa$ plays a similar role to credit constraints in New Keynesian models with heterogeneous agents. However, the economic interpretation is quite different. Here, higher inequality is associated with lower GDP and lower multipliers. In standard New Keynesian models, higher inequality is associated with higher shares of credit-constrained households and hence higher fiscal multipliers.

Our model also differs from New Keynesian models in that our model abstracts from monetary policy, a clear limitation of our analysis. Specifically, the real interest rate in our model is pinned down by the rich household's linear utility over the numeraire (which is traded at a flexible price). Given that the real rate is fixed, one can interpret our setting as an economy in a liquidity trap. A useful extension of our model (especially for quantitative analyses) would be to incorporate a role for monetary policy, which would require introducing curvature in utility of the numeraire and either a) introducing sticky prices in the numeraire sector or b) interpreting the numeraire as money, which can be created by a central bank. These extensions would come at the cost of analytic convenience, which we perceive to be a useful feature of our current setup. In particular, these extensions would imply that rich-household spending would adjust in response to macroeconomic shocks and interest rate changes set by monetary policy. We conjecture that such an extension, while adding significant complexity and nonlinearity to the model, would not substantially alter the qualitative predictions that we derive from our more analytically tractable setting.19

3. Demand shocks and fiscal policy in the NMC model

To study properties of the model described in the previous section, we linearize the model around the steady state with no shocks (i.e., $\xi =1$, ${\psi }_0={\psi }_1=1$) and no transfers to the poor (i.e., $T^{\mathbb{P}}=0$). For some exercises, it will be instructive to consider cases where $T^{\mathbb{P}}>0$ in the steady state. As a first step, we explore how structural parameters such as the share of income going to the poor (which also controls the level of inequality in the economy), spending by the rich (“autonomous” spending), and transfers to the poor affect key endogenous variables in the model. Then we introduce the COVID19 shock to the model and investigate how this shock propagates in the economy. Finally, we study how various fiscal policies can counter the COVID19 shock.

The effect of different transfers depends on how they are financed. It is clear that taxing low-income households (which decreases $T^{\mathbb{P}}$) will reduce GDP (all else equal). An alternative source of funding is to tax the rich exclusively. As long as the rich maintain enough post-tax resources to satiate their demand for NMC commodities and spend on the numeraire at the margin, there will be no effect of this taxation on GDP in the NMC sector for either period. There is also the possibility that the transfers could be financed through money creation by the central bank (e.g., Galí, 2020). While this process is not specified in our model, one could, for example, adopt an alternative interpretation of the numeraire as money, consistent with models of monetary non-neutrality driven by money in the utility function. In our model money-financed transfers would have the same effect as taxing the rich (via an inflation tax). For the remainder of the analysis we assume that transfers are financed through taxing the rich, and we will examine the relative effectiveness of different types of spending.20

3.1. Inequality, rich-household spending, and government transfers

Inspection of Eq. (12) implies that poor-household expenditure (and hence aggregate expenditure on NMC goods and hence GDP) is falling in inequality and increasing in spending by rich households and in transfer income. In general, any factor that increases the income of the poor generates an increase in spending and aggregate output, as output is limited only by poor households’ spending (which is limited by their income):

Proposition 1

GDP and poor-household expenditure are increasing in the income share of the poor (falling in inequality), spending by the rich, and transfers. The effects of rich-household spending and transfers are increasing in the income share of the poor. In particular, in the absence of steady-state transfers $T^{\mathbb{P}}$ :

$$\begin{array}{c}{\frac{d{c}_{jk0}^{\mathbb{P}}}{d\kappa }|}_{T^{\mathbb{P}}=0}=\frac{c_{jk0}^{\mathbb{P}}+c_{jk0}^{\mathbb{R}}}{1-\kappa }\Rightarrow {\frac{d{q}_{jk0}^{\mathbb{P}}}{d\kappa }|}_{T^{\mathbb{P}}=0}=\frac{1}{{\overline{p}}_{jk0}^{\mathbb{P}}}\frac{c_{jk0}^{\mathbb{P}}+c_{jk0}^{\mathbb{R}}}{\left(1-\kappa \right)}\hfill \\ {\frac{d{c}_{jk0}^{\mathbb{P}}}{d{c}_{jk0}^{\mathbb{R}}}|}_{T^{\mathbb{P}}=0}=\frac{\kappa }{1-\kappa }\Rightarrow {\frac{d{q}_{jk0}^{\mathbb{P}}}{d{c}_{jk0}^{\mathbb{R}}}|}_{T^{\mathbb{P}}=0}=\frac{1}{{\overline{p}}_{jk0}^{\mathbb{P}}}\frac{\kappa }{1-\kappa }\hfill \\ {\frac{d{c}_{jk0}^{\mathbb{P}}}{d{T}^{\mathbb{P}}}|}_{T^{\mathbb{P}}=0}=\frac{1}{\left(1+Q\right)\left(1-\kappa \right)}\Rightarrow {\frac{d{q}_{jk0}^{\mathbb{P}}}{d{T}^{\mathbb{P}}}|}_{T^{\mathbb{P}}=0}=\frac{1}{{\overline{p}}_{jk0}^{\mathbb{P}}}\frac{1}{\left(1+Q\right)\left(1-\kappa \right)}.\hfill \end{array}$$ (19)

Proof: Appendix. See Fig. 2 for a graphical proof.

Fig. 2.

Graphical proof of Proposition 1.

As the income share of the poor $\kappa$ increases (inequality falls), poor households spend more in both periods. Due to rigid initial-period prices, this additional spending translates into higher initial-period consumption by the poor (and hence higher real GDP). The effect of inequality is quantitatively large. One can show that $\frac{d\log c_{jk0}^{\mathbb{P}}}{d\kappa }{|}_{T^{\mathbb{P}}=0}=\frac{1}{\kappa \left(1-\kappa \right)}$, which is bounded below by $\frac{1}{0.5\left(1-0.5\right)}=4$.

Rich-household spending and transfers from the government also increase income for the poor, which in turn induces higher spending by the poor and higher GDP. These relationships are consistent with recent evidence from Chetty et al. (2020) and Coibion et al. (2020), documenting that fiscal transfers associated with the CARES Act increased spending by low-income households. Furthermore, spending cuts during the pandemic were largest among low-income households working in areas that were most exposed to the decline in rich-household spending.

The effect of transfers on GDP is higher the larger is the income share of the poor (the lower is inequality). The relationship between the fiscal transfer multiplier and inequality reflects the fact that the general-equilibrium effects implied by the model are much larger than would be implied by examining the partial equilibrium response of poor-household spending alone, since in general equilibrium the initial spending causes additional poor-household income, which generates additional spending, and so on. For example, the partial-equilibrium effect of transfers on poor-household income is $\frac{\partial I^{\mathbb{P}}}{\partial T^{\mathbb{P}}}=1$. But the general-equilibrium effect – accounting for the effect of poor-household spending on their own income – is $\frac{d{I}^{\mathbb{P}}}{d{T}^{\mathbb{P}}}=1+\kappa \frac{d{C}^{\mathbb{P}}}{d{T}^{\mathbb{P}}}$, which is rising in the income share of the poor. We discuss this relationship in more detail below when we compare alternative fiscal policies.

3.2. COVID19 shock

The social distancing restrictions associated with COVID19 can be modeled as a decrease in $\xi$ from an initial value of 1, reflecting the restrictions on the exchange of services such as restaurant meals, movie theaters, and sporting events. The size of the restriction multiplier – the net effect of restrictions ($d{Y}_0/d\xi$) relative to the direct effect ($\partial Y_0/\partial \xi$) – depends on the extent to which restrictions cause firms to exit. In the absence of firm exit (e.g., due to flexible capital costs), the restriction multiplier equals unity.

Proposition 2

In the absence of a firm exit margin, the decline in output is bounded by the share of products that are restricted (the restriction multiplier is unity). Firm exit causes a larger fall in output – a restriction multiplier greater than unity:

$$\frac{d{Y}_0}{d\xi }{|}_{T^{\mathbb{P}}=0}=Y_0\left(1+\frac{d{\psi }_0}{d\xi }\right),\frac{\partial Y_0}{\partial \xi }=Y_0.$$ (20)

Proof: Appendix.

In the absence of transfers, restrictions decrease poor-household aggregate spending to a degree that perfectly balances the decrease in poor-household aggregate income. Given product symmetry, there is no adjustment within product categories.21 This also implies that in the absence of a firm-entry margin, there are no multiplier effects from product-level restrictions. In other words, GDP falls by an amount proportional to the share of products that are restricted, $\frac{\partial Y_0}{\partial \xi }=Y_0$. If restrictions force firms to exit ($\frac{d{\psi }_0}{d\xi }>0$), the restriction multiplier is $\frac{d{Y}_0}{d\xi }/\frac{\partial Y_0}{\partial \xi }{|}_{T^{\mathbb{P}}=0}=1+\frac{d{\psi }_0}{d\xi }>1$. Chetty et al. (2020) document higher rates of small business closure in places in which spending was cut the most, which suggests that $\frac{d{\psi }_0}{d\xi }>0$. Workers in these locations experienced larger income declines and cut their spending by more, consistent with model's prediction of the adverse effect of firm exit on consumer income and spending.22

The multiplier effects of COVID restrictions are smaller in the presence of positive steady-state transfers. This is because, in the presence of positive transfers, the poor household responds to the restrictions by spreading its transfer wealth over the fewer available initial-period products and the products available in the future. This per-product spending increase is associated with higher output per product in the initial period (when prices are fixed) and higher prices in the future period (when prices are flexible). We formalize this point in the following proposition.

Proposition 3

(Economic Restrictions and Future Inflation). In the presence of positive steady-state transfers, spending restrictions cause future inflation, as households reallocate spending across the remaining set of goods/services available in the current and future periods:

$$\frac{d{c}_{jk0}^{\mathbb{P}}}{d\xi }=\frac{d{c}_{jk1}^{\mathbb{P}}}{d\xi }=-\frac{T^{\mathbb{P}}\left(1+\frac{d{\psi }_0}{d\xi }\right)}{{\left(1+Q\right)}^2\left(1-\kappa \right)},\frac{d{p}_{jk1}^{\mathbb{P}}}{d\xi }=\frac{2}{{\theta }^{\mathbb{P}}}\frac{d{c}_{jk0}^{\mathbb{P}}}{d\xi }.$$ (21)

Proof: Appendix.

Intuitively, with positive transfers, the poor households’ reduction in income is proportionally smaller than the reduction in the number of commodities they buy, so it increases their demand for all remaining commodities. When households must forgo spending on a subset of products, they reallocate their wealth (transfers) across the remaining available products in the initial period and in the future. The increase in per-product expenditure causes higher per-product output in the current period (when prices are rigid) and higher prices in the future (when prices are fully flexible). If we were to expand the model to include an intermediate period $t=\left\{0\to 1\right\}$ in which prices are only partially flexible, then the initial-period restrictions would be followed by a boom in output, as households would consume more per product and (if $d{\psi }_{\left\{0\to 1\right\}}=0$) would have the full set of products available to purchase. This prediction of the model is consistent with the behavior of the U.S. economy following World War II. Inflation surged to nearly 20 percent within a year-and-a-half of the end of the war, consistent with households transferring spending power from during the war (when spending was restricted) to after the war.

The inflationary effects of the pandemic restrictions and fiscal stimulus are driven by higher future consumer spending (which induces less-elastic product-level demand curves). Stepping outside of the model, one could imagine forces that exacerbate or mitigate inflation. Inflation could be amplified by firms that run into capacity constraints (positive marginal costs), which we have abstracted from for simplicity. Inflation could be attenuated in a setting with an active role for monetary policy. The net effect of these forces depends on specific assumptions and parameter values.

Note that Eq. (21) implies that restrictions increase per-product spending (initial-period output and future-period prices) by more the greater the income share of the poor $\kappa$ (lower inequality), holding fixed the number of firms in the economy. This is because the poor household recycles its purchasing power more the higher is its income share. COVID restrictions increase per-product spending on unrestricted products (e.g., streaming video services, computer games), and this increased spending is multiplied more the larger is the income share of the poor. Therefore, economic restrictions reduce aggregate output by less (more) in the presence of higher (lower) poor-household income shares, which correspond to lower (higher) inequality. Even accounting for the endogenous response of firm entry, the output response to COVID restrictions can be shown to be increasing in inequality (see Corollary 1 in Appendix).

3.3. The response of firm exit

The discussion above makes clear that COVID restrictions have large multipliers (>1) in our model only if firms exit. To study this margin, we must specify $f_j$, the fixed cost of operating firm $j$. Let the PDF of the distribution of $f$ be $v$ and the CDF be $V$. Then

$${\psi }_0=\underset{0}{\overset{R_{j0}}{\int }}v\left(f\right)df=V\left(R_{j0}\right)$$

and

$$d{\psi }_0=v\left(R_{j0}\right)d{R}_{j0},$$

where $d{R}_{j0}=\left(c_{jk0}^{\mathbb{P}}+c_{jk0}^{\mathbb{R}}\right)d\xi +\xi d{c}_{jk0}^{\mathbb{P}}$. The COVID shock thus affects the number of firms directly: COVID restrictions ($\xi <1$) reduce the number of products that firms can sell and thus push some firms into the red, forcing them to exit. There is also an indirect channel: the COVID shock increases the spending of the poor household on remaining products, which helps to mitigate firm exit. In the absence of steady-state transfers to poor households ($T^{\mathbb{P}}=0$), the indirect channel is not operational and firm exit is entirely determined by the distribution of firms’ fixed costs:

$$\frac{d{\psi }_0}{d\xi }{|}_{T^{\mathbb{P}}=0}=v\left(R_{j0}\right)\left(c_{jk0}^{\mathbb{P}}+c_{jk0}^{\mathbb{R}}\right).$$ (22)

As discussed in Section 3.2 (after Proposition 2), in the absence of transfers, restrictions decrease poor-household aggregate spending to a degree that perfectly balances the decrease in poor-household aggregate income. Given product symmetry, there is no adjustment within product categories.

If transfers $T^{\mathbb{P}}$ are positive in the steady state, the increase in per-product household spending induced by the restrictions can help mitigate firm exit in response to COVID restrictions. The poor households spread their transfer wealth over the fewer remaining products, which helps to support firm revenues and mitigate firm exit.

If we interpret fixed costs as the rental costs of the existing capital stock, then $f_{j0}$ is the rental price of capital and firm exit is associated with a reduction in demand for the existing capital stock. If prices $f_{j0}$ are rigid, there will be excess supply of capital. However, if prices $f_{j0}$ are flexible, then they will adjust downward to mitigate the effect of falling revenues on firm profits. Given the inelastic supply of capital and the symmetry of firms, the capital market clears once the mass of operating firms is at its steady-state level ($\psi =1$). We collect these results in the following proposition.

Proposition 4

If fixed costs are rigid, COVID restrictions induce firm exit. If fixed costs are flexible, there is no exit (and hence the restriction multiplier equals unity).

Proof: Appendix.

Clearly, if costs of operation can be adjusted in response to the COVID shock (e.g., set $v\left(R_{j0}\right)$as low as zero), firm exit can be avoided entirely and thus the adverse effects of the COVID shock minimized. However, there are plenty of reasons to expect that capital costs may not be flexible, at least in the short run. Asymmetric information between capital owners and the firms that rent the capital is among the reasons for rigid capital prices. If capital is imperfectly substitutable such that owners have pricing power, then capital owners may be reluctant to adjust if they cannot identify which firms can pay and which cannot. Indeed, recent empirical evidence documents a strong role for asymmetric information in preventing renegotiations between capital owners and firms even when such renegotiations would otherwise benefit both (Gilje et al., 2020).

The experience of the pandemic to date is consistent with rigid costs. The number of small business owners plummeted at the fastest rate on record between February and April 2020 (Fairlie, 2020). The adverse experiences of small businesses led to a sharp fall in household wages and income, especially for households with low income (Cajner et al., 2020). Household evictions also accelerated, according to data from the Eviction Lab, indicative of rigid housing rental prices.23

Note that the disproportionate fall in income for poor households documented by Canjer et al. (2020) is consistent with the model. Rich households receive income from their ownership of the numeraire and from the NMC sector, whereas poor households receive income only from the NMC sector. Therefore, adverse shocks to the NMC sector disproportionately reduce the income of poor households.24

3.4. Inequality and firm exit

The effect of restrictions on firm exit is stronger the higher is the income share of the poor, unless the distribution of fixed costs is strongly decreasing at $R_{j0}$ – that is, unless the elasticity of the density function $\frac{R_{j0}{v}^{\prime }\left(R_{j0}\right)}{v\left(R_{j0}\right)}$ is less than −1:

$$\frac{d^2{\psi }_0}{d\xi d\kappa }{|}_{T^{\mathbb{P}}=0}=\left(1+\frac{R_{j0}{v}^{\prime }\left(R_{j0}\right)}{v\left(R_{j0}\right)}\right)v\left(R_{j0}\right)\frac{d{c}_{jk0}^{\mathbb{P}}}{d\kappa }.$$

Higher income of the poor is associated with higher spending on each product and hence a greater revenue loss for each product that is restricted. If the higher spending does not sufficiently reduce the number of firms on the margin of exit, then in the face of high poor-household-income-shares, restrictions pull down revenues more and induce more firm exit. Therefore, higher inequality can mitigate the adverse effect of restrictions on firm exit.

3.5. Fiscal policy

Government transfers to households and/or firms can mitigate the adverse effects of the restrictions.

3.5.1. Transfers to households

Consider first transfers to low-income households. One can show that the effect on real GDP is

$$\frac{d{Y}_0}{d{T}^{\mathbb{P}}}=Y_{0}v\left(R_{j0}\right)\xi \frac{d{c}_{jk0}^P}{d{T}^{\mathbb{P}}}+\frac{d{q}_{jk0}^{\mathbb{P}}}{d{T}^{\mathbb{P}}}>0.$$ (23)

Transfers to low-income households of sufficient size can in principle fully offset secondary economic effects of the COVID-related restrictions. Transfers stimulate output through two channels. First, they increase spending on products sold by existing firms. Second, they induce firm entry, and this entry causes additional private-sector spending on the products of the entering firms. This firm entry margin is consistent with recent empirical evidence of the effects of fiscal stimulus (Auerbach et al., 2020b) and more generally with the effect of aggregate demand shocks (Campbell and Lapham, 2004).

Inspection of Eq. (19) implies that the effects of demand shocks on poor-household spending and consumption are falling in inequality. This, along with Eq. (23), implies that the effect of transfers on GDP is falling in inequality: the smaller is the income share of low-income households, the less spending circulates back as income to low-income households (and hence the less they can spend).

Proposition 5

(Fiscal multipliers and inequality): Transfers induce firm entry, amplifying the fiscal multiplier. Furthermore, the fiscal transfer multiplier is falling in inequality (rising in the income share of the poor$\kappa$):$\frac{d^2{Y}_0}{d{T}^{\mathbb{P}}d\kappa }>0$.

Proof: See Appendix.

Intuitively, lower inequality increases the multiplier because the larger is the income share of low-income households, the more spending circulates back as income to low-income households (and hence the more they can spend). This can be seen graphically through the quasi-Keynesian cross (Fig. 3 ). A higher income share of the poor $\kappa$ implies a flatter consumption line, which yields a stronger equilibrium response of consumption (and hence output). This is a rather surprising result, given that inequality has often been associated with large shares of credit-constrained households (and hence potentially large fiscal multipliers, as in Brinca et al., 2016 and Lee, 2020). However, recent empirical evidence documents an inverse relationship between fiscal effects and inequality (Miranda-Pinto et al., 2020a; Yang, 2017). Our theory and this evidence imply that household-level transfers are less effective during the recent episode of rising inequality.

Fig. 3.

Higher Income Share of the Poor Implies higher Fiscal Multipliers.

3.5.2. Transfers to firms

An alternative to household-level transfers is to provide transfers $T^{\mathbb{F}}$ to firms. If the government cannot target the firm transfers but instead must allocate across all firms, then firm-level transfers are unambiguously less effective than household-level transfers: for a firm-level transfer, low-income households (which drive spending multipliers) only end up with a share of the transfer $\kappa$. More formally,

$$\frac{d{Y}_0}{d{T}^{\mathbb{F}:All}}=\kappa \frac{d{Y}_0}{d{T}^{\mathbb{P}}}<\frac{d{Y}_0}{d{T}^{\mathbb{P}}},$$

where $T^{\mathbb{F}:All}$ is untargeted transfers across firms. Such transfers stimulate spending and firm entry, but their effect is diminished because a share $1-\kappa$ of the transfer ends up with the rich households, who do not contribute to spending multipliers.

While untargeted firm-level transfers are not an effective form of stimulus, targeted firm-level transfers are potentially very effective. In particular, transfers that are targeted to firms on the margin of exit (those for which fixed costs are a large share of their revenues, $R_{j0}\approx f_{j0}$) prevent firm exit, which as discussed above is the mechanism though with COVID restrictions cause large multiplier effects. In particular, for each dollar targeted to marginal firms, the government would create $d{\psi }_0=\frac{1}{v\left(R_{j0}\right)}$ firms. Equivalently, if the mass of marginal firms is $v\left(R_{j0}\right)$, the government must spend that amount to keep them alive. So $\frac{d{\psi }_0}{d{T}^{\mathbb{F}:Target}}=\frac{1}{v\left(R_{j0}\right)}$. If the government can target such firms, the extra multiplier from targeted transfers $T^{\mathbb{F}:Target}$ (relative to untargeted firm-level transfers $T^{\mathbb{F}:All}$) is (see the Appendix for derivations)

$$\frac{d{Y}_0}{d{T}^{\mathbb{F}:Target}}-\frac{d{Y}_0}{d{T}^{\mathbb{F}:All}}=\left(\kappa \frac{d{Y}_0}{d{T}^{\mathbb{P}}}+\frac{\partial Y_0}{\partial {\psi }_0}\frac{1}{v\left(R_{j0}\right)}\right)-\left(\kappa \frac{d{Y}_0}{d{T}^{\mathbb{P}}}\right)=\frac{\partial Y_0}{\partial {\psi }_0}\frac{1}{v\left(R_{j0}\right)},$$ (24)

where

$$\frac{\partial Y_0}{\partial {\psi }_0}{|}_{T^{\mathbb{F}}=0}=Y_0.$$ (25)

For example, if fixed costs $f$ are uniformly distributed on $\left[0,U\right]$, the marginal targeted tax dollar creates $Y_0/U$ additional units of GDP compared to the marginal untargeted tax dollar.

In the absence of large changes in the distribution of fixed costs as revenues change, lower inequality (higher $\kappa$) is associated with larger relative benefits of targeted transfers. Each firm is associated with higher GDP the larger is the income share (and hence spending) of poor households. Therefore, saving these marginal firms is associated with larger net output gains.

The relative benefit (in terms of GDP per dollar spent) of targeted transfers to firms versus transfers to low-income households depends on how many firms are kept afloat with each dollar spent:

$$\frac{d{Y}_0}{d{T}^{\mathbb{F}:Target}}-\frac{d{Y}_0}{d{T}^{\mathbb{P}}}=\left(\kappa \frac{d{Y}_0}{d{T}^{\mathbb{P}}}+\frac{\partial Y_0}{\partial {\psi }_0}\frac{1}{v\left(R_{j0}\right)}\right)-\left(\frac{d{Y}_0}{d{T}^{\mathbb{P}}}\right)=\frac{\partial Y_0}{\partial {\psi }_0}\frac{1}{v\left(R_{j0}\right)}-\left(1-\kappa \right)\frac{d{Y}_0}{d{T}^{\mathbb{P}}}.$$ (26)

In this sense, the benefits of targeted transfers to firms (relative to transfers to the poor) are proportional to the indirect costs of the COVID19 restrictions (i.e., endogenous firm exit). If there are large restriction multipliers $\left(1+\frac{d{\psi }_0}{d\xi }\right)$ (based on the joint distribution of fixed capital costs and firm revenues), then the relative benefits of targeted transfers are large and these benefits could be even larger if there are costs of reentry. Furthermore, lower inequality (higher $\kappa$) is associated with larger relative benefits of targeted firm-level transfers because as discussed above $Y_0$ is increasing in $\kappa$. Poor households also receive a higher initial (direct) share of the firm-level transfer, although this effect is offset by the fact that the effect of household transfers on output is increasing in $\kappa$.

Proposition 6

(The optimal composition of transfers): Targeted transfers to firms can be the most cost-effective method of using transfers to mitigate a restriction multiplier above unity. The relative benefit of targeted transfers depends on the joint distribution of firm revenues and capital operating costs. The relative benefit is higher the greater is the income share of the poor (lower inequality) as long as$\left(\frac{v^{\prime }\left(R_{j0}\right)}{v\left(R_{j0}\right)}\right)$is not too large.

Proof: Appendix.

In practice it may be difficult to identify and target marginal firms, although the model offers some guidance. The firms most at risk of exit are those with relatively low profitability and for which capital operating costs are the largest or most rigid. Because of low profit margins, small businesses are likely to be particularly prone to exit (consistent with the evidence in Fairlie, 2020), therefore implying an important role of targeted transfers to firms. If the government attempts to target firms but can do so only imperfectly, the multiplier will be in the range $\left[\frac{d{Y}_0}{d{T}^{\mathbb{F}:All}},\frac{d{Y}_0}{d{T}^{\mathbb{F}:Target}}\right]$, with a larger effect the more of the stimulus goes to marginal firms.

An alternative policy to firm-level transfers is government loans to firms. But firms still need to cover their future-period fixed costs. Firms for which the present value of revenues in both periods falls below the present value of fixed costs will not be helped by loans (specifically, ${\psi }_0$ and ${\psi }_1$ can fall below 1 even if the government offers loans). Loans are only effective for the firms that cannot cover their fixed costs in the initial period but nonetheless earn profits in present value. Chetty et al. (2020) document a limited impact of loans to small businesses on firm employment and suggest that liquidity injections are insufficient for restoring employment at small businesses. Their evidence, interpreted through the lens of our model, is consistent with a decline in the present value of revenues sufficient to push firms to exit.

4. Conclusion

The COVID crisis has both raised immediate policy questions and highlighted key structural relationships in the economy. Inequality has risen, rich households have cut back spending on services, and firms have been pushed to the brink of exit in the face of rigid capital operating costs. We develop a model capable of addressing the roles of inequality and other key features of the pandemic economy. Our results have general implications for the macroeconomic effects of the interactions of inequality and fiscal policy, while also providing guidance on the relative merits of alternative fiscal policies in the face of COVID restrictions on economic activity.

Our framework implies that rising inequality will drag down GDP, as will any additional reallocation of spending by rich households away from service sectors in which low-income households work. In the absence of these developments, the strongest macroeconomic threat associated with COVID19 is firm exit resulting from restrictions on the exchange of services and rigid capital costs, a pattern clearly observed in the data. Our model suggests that the adverse effects may be offset by transfers to households and firms. Furthermore, we show that transfers to firms on the margin of exit are particularly effective in mitigating economic contraction.

Our framework indicates a number of measures that will be useful to monitor as the COVID19 crisis evolves. In the absence of rising inequality or reductions in spending by high-income households, nominal GDP will rebound to a level at or potentially beyond what it would have been in the absence of COVID. Rising inequality or reductions in spending by high-income households can mitigate this boom or cause a prolonged slump. Fiscal stimulus will be especially useful in the event of a slump, although its effect per dollar spent is decreasing in inequality. Another important measure is the rental prices of firms’ operating capital, especially for firms that have large fixed operating costs relative to revenues and for multiproduct firms. Downward adjustment of capital prices can mitigate large restriction multipliers.

Appendix

The following relationships are referenced throughout the proofs. First, in equilibrium the bond price $Q$ equals the discount factor $\beta$. This follows from the rich household's first-order-condition with respect to the bond and the fact that ${\lambda }_{\mathbb{R}t}$ equals the marginal utility of the numeraire. In particular, the first-order condition for either household is $Q{\lambda }_0^h=\beta {\lambda }_1^h.$ Since ${\lambda }_{\mathbb{R}t}=1$, it follows that $Q=\beta$. Furthermore, it follows that ${\lambda }_{\mathbb{P}0}={\lambda }_{\mathbb{P}1}$.

Second, in the steady-state (in the absence of shocks) households smooth their expenditure: $c_{jk1}^h=c_{jk0}^h$. Plugging firms’ prices (Eq. (5)) into the household's demand (Eq. (4)) implies that equilibrium steady-state quantities are $q_{jkt}^h=\frac{{\theta }^h}{2\gamma }$ and equilibrium expenditure is

$$c_{jkt}^h=\frac{{\left({\theta }^h\right)}^2}{4\gamma {\lambda }_{ht}}.$$ (27)

Expenditure smoothing (in steady-state) follows from ${\lambda }_{h0}={\lambda }_{h1}$.

Third, the responses to shocks of product-level poor-household consumption at time t = 0, prices at t = 1, and poor-household expenditure in either period all move in the same direction. This is because the response of each can be captured by the response of the budget multiplier:

$$d{c}_{jk1}^{\mathbb{P}}=d{c}_{jk0}^{\mathbb{P}}=\frac{{\left({\theta }^{\mathbb{P}}\right)}^2}{4\gamma {\lambda }_{\mathbb{P}0}^2}d{\lambda }_{\mathbb{P}0}$$ (28)

$$d{q}_{jk0}^{\mathbb{P}}=-\frac{1}{\gamma }{p}_{jk0}d{\lambda }_{\mathbb{P}0}$$ (29)

$$d{p}_{jk1}^{\mathbb{P}}=-\frac{{\theta }^{\mathbb{P}}}{2{\lambda }_{\mathbb{P}0}^2}d{\lambda }_{\mathbb{P}0}$$ (30)

The results follow from $d{c}_{jk0}^{\mathbb{P}}=p_{jk0}^{\mathbb{P}}d{q}_{jk0}^{\mathbb{P}}$ due to fixed initial-period prices, $q_{jk0}^{\mathbb{P}}=-\frac{1}{\gamma }{p}_{jk0}d{\lambda }_{\mathbb{P}0}$, and $d{c}_{jkt}^{\mathbb{P}}=d\left(\frac{{\theta }^2}{4\gamma {\lambda }_{\mathbb{P}1}}\right)=d\left(\frac{{\theta }^2}{4\gamma {\lambda }_{\mathbb{P}0}}\right)=-\frac{{\theta }^2}{4\gamma {\lambda }_{\mathbb{P}0}^2}d{\lambda }_{\mathbb{P}0}$. These relationships imply that we can infer the direction of output per product in the initial period and the direction of prices in the future from the direction of spending in either period (or equivalently, from the response of the budget multiplier).

Proof ofProposition 1: Total differentiation of (12), after imposing the steady-state values $\xi ,{\psi }_0,{\psi }_1=1$ and $d{c}_{jk1}^{\mathbb{P}}=d{c}_{jk0}^{\mathbb{P}}$, yields

$$\frac{T^{\mathbb{P}}}{1+Q}d\xi +\frac{T^{\mathbb{P}}}{1+Q}d{\psi }_0+\left(c_{jk1}^{\mathbb{P}}\left(1-Q\kappa \right)-Q{c}_{jk0}^{\mathbb{R}}\kappa \right)d{\psi }_1+\left(1+Q\right)\left(1-\kappa \right)d{c}_{jk0}^{\mathbb{P}}=\left(1+Q\right)\kappa d{c}_{jk0}^{\mathbb{R}}+d{T}^{\mathbb{P}}+\left(1+Q\right)\left[c_{jk0}^{\mathbb{P}}+c_{jk0}^{\mathbb{R}}\right]d\kappa .$$ (31)

It follows that around a steady-state in which $T^{\mathbb{P}}=0$ (and for simplicity setting $d{\psi }_1=0$):

$$\begin{array}{c}\frac{d{c}_{jk0}^{\mathbb{P}}}{d\kappa }=\frac{c_{jk0}^{\mathbb{P}}+c_{jk0}^{\mathbb{R}}}{1-\kappa }\Rightarrow \frac{d{q}_{jk0}^{\mathbb{P}}}{d\kappa }=\frac{1}{{\overline{p}}_{jk0}^{\mathbb{P}}}\frac{c_{jk0}^{\mathbb{P}}+c_{jk0}^{\mathbb{R}}}{\left(1-\kappa \right)}\hfill \\ \frac{d{c}_{jk0}^{\mathbb{P}}}{d{c}_{jk0}^{\mathbb{R}}}=\frac{\kappa }{1-\kappa }\Rightarrow \frac{d{q}_{jk0}^{\mathbb{P}}}{d{c}_{jk0}^{\mathbb{R}}}=\frac{1}{{\overline{p}}_{jk0}^{\mathbb{P}}}\frac{\kappa }{1-\kappa }\hfill \\ \frac{d{c}_{jk0}^{\mathbb{P}}}{d{T}^{\mathbb{P}}}=\frac{1}{\left(1+Q\right)\left(1-\kappa \right)}\Rightarrow \frac{d{q}_{jk0}^{\mathbb{P}}}{d{T}^{\mathbb{P}}}=\frac{1}{{\overline{p}}_{jk0}^{\mathbb{P}}}\frac{1}{\left(1+Q\right)\left(1-\kappa \right)}.\hfill \end{array}$$ (32)

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Proof ofProposition 2: The total derivative of initial-period GDP (from (8)) is

$$d{Y}_0=Y_0\left(d\xi +d{\psi }_0\right)+d{q}_{jk0}^{\mathbb{P}}+d{q}_{jk0}^{\mathbb{R}}.$$ (33)

It follows that (imposing $\frac{d{q}_{jk0}^{\mathbb{P}}}{d\xi }{|}_{T^{\mathbb{P}}=0}=0$ from (31) and the steady-state relationship $c_0^{\mathbb{P}}\left(1-\kappa \right)-c_{jk0}^{\mathbb{R}}\kappa =\frac{T^{\mathbb{P}}}{\left(1+Q\right)}$, as well as imposing $\frac{d{q}_{jk0}^{\mathbb{R}}}{d\xi }=0$ from (6))

$$\frac{d{Y}_0}{d\xi }{|}_{T^{\mathbb{P}}=0}=Y_0\left(1+\frac{d{\psi }_0}{d\xi }\right).$$

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Proof ofProposition 3:$d{c}_{jk0}^{\mathbb{P}}$ and $d{p}_{jk1}^{\mathbb{P}}$ are related by substituting $d{\lambda }_{\mathbb{P}0}$ out of (28) and (30):

$$d{p}_{jk1}^{\mathbb{P}}=\frac{2}{{\theta }^{\mathbb{P}}}d{c}_{jk0}^{\mathbb{P}}.$$

From (31), we can write

$$\frac{d{c}_{jk0}^{\mathbb{P}}}{d\xi }=-\frac{T^{\mathbb{P}}\left(1+\frac{d{\psi }_0}{d\xi }\right)}{{\left(1+Q\right)}^2\left(1-\kappa \right)}.$$ (34)

Therefore, $\frac{d{p}_{jk1}^{\mathbb{P}}}{d\xi }=-\frac{2}{{\theta }^{\mathbb{P}}}\frac{T^{\mathbb{P}}\left(1+\frac{d{\psi }_0}{d\xi }\right)}{{\left(1+Q\right)}^2\left(1-\kappa \right)}$. When $\xi$ falls, future prices increase. ■

Corollary 1

In the presence of steady-state transfers, inequality is associated with stronger output effects of COVID restrictions.

$$\frac{d^2{q}_{jk0}^{\mathbb{P}}}{d\xi d\kappa }=-\frac{T^{\mathbb{P}}}{p_{jk0}^{\mathbb{P}}}\left[\frac{\left(1-\kappa \right)\frac{d^2{\psi }_0}{d\xi d\kappa }+\left(1+\frac{d{\psi }_0}{d\xi }\right)}{{\left(1+Q\right)}^2{\left(1-\kappa \right)}^2}\right],$$

which is strictly negative as $\left(1-\kappa \right)\frac{d^2{\psi }_0}{d\xi d\kappa }+\left(1+\frac{d{\psi }_0}{d\xi }\right)>0$.

Proof ofCorollary 1: Note that by (34) and $d{c}_{jk0}^{\mathbb{P}}=p_{jk0}^{\mathbb{P}}d{q}_{jk0}^{\mathbb{P}}$ we can write

$$\frac{d{q}_{jk0}^{\mathbb{P}}}{d\xi }=-\frac{1}{p_{jk0}^{\mathbb{P}}}\frac{T^{\mathbb{P}}\left(1+\frac{d{\psi }_0}{d\xi }\right)}{{\left(1+Q\right)}^2\left(1-\kappa \right)}.$$

Taking the derivative with respect to $\kappa$ yields

$$\frac{d^2{q}_{jk0}^{\mathbb{P}}}{d\xi d\kappa }=-\frac{T^{\mathbb{P}}}{p_{jk0}^{\mathbb{P}}}\left[\frac{\left(1-\kappa \right)\frac{d^2{\psi }_0}{d\xi d\kappa }+\left(1+\frac{d{\psi }_0}{d\xi }\right)}{{\left(1+Q\right)}^2{\left(1-\kappa \right)}^2}\right],$$

which is positive if $\left(1-\kappa \right)\frac{d^2{\psi }_0}{d\xi d\kappa }+\left(1+\frac{d{\psi }_0}{d\xi }\right)>0$. First, we must derive a closed-form expression for

$$\frac{d{\psi }_0}{d\xi }=v\left(R_{j0}\right)\left[\left(c_{jk0}^{\mathbb{P}}+c_{jk0}^{\mathbb{R}}\right)+\frac{d{c}_{jk0}^{\mathbb{P}}}{d\xi }\right],$$

Substituting in $\frac{d{c}_{jk0}^{\mathbb{P}}}{d\xi }=-\frac{T^{\mathbb{P}}\left(1+\frac{d{\psi }_0}{d\xi }\right)}{{\left(1+Q\right)}^2\left(1-\kappa \right)}$ from Eq. (31) and rearranging yields

$$\frac{d{\psi }_0}{d\xi }=v\left(R_{j0}\right)\left[\left(c_{jk0}^{\mathbb{P}}+c_{jk0}^{\mathbb{R}}\right)-\frac{T^{\mathbb{P}}\left(1+\frac{d{\psi }_0}{d\xi }\right)}{{\left(1+Q\right)}^2\left(1-\kappa \right)}\right]$$

$$\frac{d{\psi }_0}{d\xi }=\frac{v\left(R_{j0}\right)}{\left[1+\xi \frac{T^{\mathbb{P}}}{{\left(1+Q\right)}^2\left(1-\kappa \right)}\right]}\left[\left(c_{jk0}^{\mathbb{P}}+c_{jk0}^{\mathbb{R}}\right)-\frac{T^{\mathbb{P}}}{{\left(1+Q\right)}^2\left(1-\kappa \right)}\right]$$

Using the steady-state relationship $c_{jk0}^{\mathbb{P}}\left(1-\kappa \right)-c_{jk0}^{\mathbb{R}}\kappa =\frac{T^{\mathbb{P}}}{\left(1+Q\right)}$, this expression can be rewritten as

$$\frac{d{\psi }_0}{d\xi }=\frac{v\left(R_{j0}\right)}{\left[1+\xi \frac{T^{\mathbb{P}}}{{\left(1+Q\right)}^2\left(1-\kappa \right)}\right]}\left[\frac{T^{\mathbb{P}}Q}{{\left(1+Q\right)}^2\left(1-\kappa \right)}+\frac{c_{jk0}^{\mathbb{R}}}{\left(1-\kappa \right)}\right].$$ (35)

which is strictly positive. The next step is to derive $\frac{d^2{\psi }_0}{d\xi d\kappa }$. Taking the derivative of (35) with respect to $\kappa$ yields

$$\begin{array}{c}\left(\frac{d^2{\psi }_0}{d\xi d\kappa }\right)=\frac{v\left(R_{j0}\right)}{\left[1+\frac{T^{\mathbb{P}}}{{\left(1+Q\right)}^2\left(1-\kappa \right)}\right]}\left[\frac{T^{\mathbb{P}}Q}{{\left(1+Q\right)}^2{\left(1-\kappa \right)}^2}+\frac{c_{jk0}^{\mathbb{R}}}{{\left(1-\kappa \right)}^2}\right]\hfill \\ +\frac{\left[\frac{T^{\mathbb{P}}Q}{{\left(1+Q\right)}^2\left(1-\kappa \right)}+\frac{c_{jk0}^{\mathbb{R}}}{\left(1-\kappa \right)}\right]}{{\left[1+\frac{T^{\mathbb{P}}}{{\left(1+Q\right)}^2\left(1-\kappa \right)}\right]}^2}\left(\left(1+\frac{T^{\mathbb{P}}}{{\left(1+Q\right)}^2\left(1-\kappa \right)}\right)v^{\prime }\left(R_{j0}\right)\frac{d{R}_{j0}}{d\kappa }-v\left(R_{j0}\right)\frac{T^{\mathbb{P}}}{{\left(1+Q\right)}^2{\left(1-\kappa \right)}^2}\right).\hfill \end{array}$$

It can be shown that this reduces to

$$\left(\frac{d^2{\psi }_0}{d\xi d\kappa }\right)=\frac{1}{1-\kappa }\frac{d{\psi }_0}{d\xi }\left\{1+\frac{1}{\left[1+\frac{T^{\mathbb{P}}}{{\left(1+Q\right)}^2\left(1-\kappa \right)}\right]}\left(\left(1+\frac{T^{\mathbb{P}}}{{\left(1+Q\right)}^2}\right)\frac{v^{\prime }\left(R_{j0}\right)}{v\left(R_{j0}\right)}\frac{d{R}_{j0}}{d\kappa }-\frac{T^{\mathbb{P}}}{{\left(1+Q\right)}^2\left(1-\kappa \right)}\right)\right\}.$$

Since $\frac{d{R}_{j0}}{d\kappa }>0$, this expression is strictly greater than $\Psi \equiv \frac{1}{1-\kappa }\frac{d{\psi }_0}{d\xi }\left\{1-\frac{A}{\left[1+A\right]}\right\},$ where $A\equiv \frac{T^{\mathbb{P}}}{{\left(1+Q\right)}^2\left(1-\kappa \right)}$.

Therefore, it is sufficient to prove that $\left(1-\kappa \right)\Psi +\left(1+\frac{d{\psi }_0}{d\xi }\right)>0$. Indeed, we have

$$\frac{d{\psi }_0}{d\xi }\left\{1-\frac{A}{\left[1+A\right]}\right\}+\left(1+\frac{d{\psi }_0}{d\xi }\right)=\frac{d{\psi }_0}{d\xi }\left(2-\frac{A}{\left[1+A\right]}\right)+1>0.$$

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Proof ofProposition 4: The effect of restrictions in the presence of rigid capital costs follows from (22). When capital costs are flexible, the price of capital will adjust to clear the capital market: Let $r$ be the rate at which capital is rented out to firms, and let ${\gamma }_j$ be the fixed capital requirement of firm $j$ (so that $f_j=r{\gamma }_j$). If the capital market is flexible, then the rate will adjust so that the rental rate equals the revenues of the marginal firm. Without loss of generality, we can assume that the rental rate in the absence of COVID restrictions is such that there is a unit mass of firms: $\underset{0}{\overset{1}{\int }}{\gamma }_{j}dj=K$. Covid restrictions shifts in the demand for capital (as firms’ revenues fall). Given the inelastic supply of capital, a flexible capital market implies that $r$ adjusts so that in equilibrium there remains a unit mass of firms, with the price determined by the firm on the margin of exit. ■

Proof ofProposition 5: Substitute for $\frac{d{c}_{jk0}^P}{d{T}^{\mathbb{P}}}$ and $\frac{d{q}_{jk0}^{\mathbb{P}}}{d{T}^{\mathbb{P}}}$ from (19) into the expression for $\frac{d{Y}_0}{d{T}^{\mathbb{P}}}$ in (23):

$$\frac{d{Y}_0}{d{T}^{\mathbb{P}}}=Y_{0}v\left(R_{j0}\right)\xi \frac{1}{\left(1+Q\right)\left(1-\kappa \right)}+\frac{1}{p_{jk0}^{\mathbb{P}}}\frac{1}{\left(1+Q\right)\left(1-\kappa \right)}$$

Taking the total derivative yields

$$\frac{d^2{Y}_0}{d{T}^{\mathbb{P}}}=Y_{0}v\left(R_{j0}\right)\xi \frac{1}{\left(1+Q\right)\left(1-\kappa \right)}\left[d{Y}_0+v^{\prime }\left(R_{j0}\right)d{R}_{j0}\right]+\frac{1}{\left(1+Q\right)}\left(Y_{0}v\left(R_{j0}\right)\xi +\frac{1}{p_{jk0}^{\mathbb{P}}}\right)\frac{1}{{\left(1-\kappa \right)}^2}d\kappa$$

$\frac{d^2{Y}_0}{d{T}^{P}d\kappa }>0$follows from $\frac{d{R}_{j0}}{d\kappa },\frac{d{Y}_0}{d\kappa }>0$. ■

Proof ofProposition 6(Effect of Targeted Firm-Level Transfers):

For each dollar targeted to marginal firms, the government would create $d{\psi }_0=\frac{1}{v\left(R_{jt}\right)}$ firms. Equivalently, if the mass of marginal firms is $v\left(R_{jt}\right)$, the government must spend that amount to keep them alive. So $\frac{d{\psi }_0}{d{T}^{Target}}=\frac{1}{v\left(R_{jt}\right)}$.

$\kappa d{T}^{\mathbb{F}:Target}$ would also be transferred to households (as they own a share $\kappa$ of capital).

Therefore

$$\frac{d{Y}_0}{d{T}^{\mathbb{F}:Target}}=\frac{\partial Y_0}{\partial T^{\mathbb{P}}}\frac{d{T}^{\mathbb{P}}}{d{T}^{\mathbb{F}:Target}}+\frac{\partial Y_0}{\partial {\psi }_0}\frac{d{\psi }_0}{d{T}^{\mathbb{F}:Target}}$$

$$\frac{d{Y}_0}{d{T}^{\mathbb{F}:Target}}=\kappa \frac{d{Y}_0}{d{T}^{\mathbb{P}}}+\frac{\partial Y_0}{\partial {\psi }_0}\frac{1}{v\left(R_{j0}\right)}$$

$$\frac{d{Y}_0}{d{T}^{\mathbb{F}:Target}}=\kappa \frac{d{Y}_0}{d{T}^{\mathbb{P}}}+\frac{\partial Y_0}{\partial {\psi }_0}\frac{1}{v\left(R_{j0}\right)}$$

If the government could not target firms – but rather spent across all firms, it would create only

$$\frac{d{Y}_0}{d{T}^{\mathbb{F}:Firms}}=\frac{\partial Y_0}{\partial T^{\mathbb{P}}}\frac{d{T}^{\mathbb{P}}}{d{T}^{\mathbb{F}:Firms}}=\kappa \frac{d{Y}_0}{d{T}^{\mathbb{P}}}.$$

Targeted firm transfers have an additional multiplier effect given by

$$\frac{d{Y}_0}{d{T}^{\mathbb{F}:Target}}-\frac{d{Y}_0}{d{T}^{\mathbb{F}:Firms}}=\frac{\partial Y_0}{\partial {\psi }_0}\frac{1}{v\left(R_{j0}\right)},$$

where

$$\frac{\partial Y_0}{\partial {\psi }_0}{|}_{T^{\mathbb{P}}=0}=Y_0.$$

The relationship between the net benefit of targeted transfers and inequality is

$$\frac{d}{d\kappa }\left(Y_0\frac{1}{v\left(R_{j0}\right)}\right)=\frac{1}{v\left(R_{j0}\right)}\frac{d{Y}_0}{d\kappa }-Y_0\frac{v\left(R_{j0}\right)}{v^2\left(R_{j0}\right)}\frac{d{R}_{j0}}{d\kappa }=\frac{1}{v\left(R_{j0}\right)}\frac{1}{{\overline{p}}_{jk0}^{\mathbb{P}}}\frac{c_{jk0}^{\mathbb{P}}+c_{jk0}^{\mathbb{R}}}{\left(1-\kappa \right)}-Y_0\frac{v^{\prime }\left(R_{j0}\right)}{v^2\left(R_{j0}\right)}\frac{d{c}_{jk0}^{\mathbb{P}}}{d\kappa }.$$

Substituting in the steady-state relationship $c_{jk0}^{\mathbb{P}}{|}_{T^{\mathbb{P}}=0}\frac{\kappa }{\left(1-\kappa \right)}{c}_{jk0}^{\mathbb{R}}$, as well as expressions for ${\overline{p}}_{jk0}^{\mathbb{P}}$ and $\frac{d{c}_{jk0}^{\mathbb{P}}}{d\kappa }$, this relationship can be written as (see Lemma 1 below for details)

$$\frac{d}{d\kappa }\left(Y_0\frac{1}{v\left(R_{j0}\right)}\right)=\frac{c_{jk0}^{\mathbb{R}}}{v\left(R_{j0}\right){\left(1-\kappa \right)}^2}\left[\frac{\kappa }{\left(1-\kappa \right)}\frac{{\left({\theta }^{\mathbb{R}}\right)}^2}{{\left({\theta }^{\mathbb{P}}\right)}^2}-\frac{\left({\theta }^{\mathbb{P}}+{\theta }^{\mathbb{R}}\right)}{2\gamma }\frac{v^{\prime }\left(R_{j0}\right)}{v\left(R_{j0}\right)}\right].$$

As long as the percent change of the distribution $v^{\prime }/v$ is not too high (specifically, as long as $\frac{v^{\prime }}{v}<\frac{\kappa }{\left(1-\kappa \right)}\frac{{\left({\theta }^{\mathbb{R}}\right)}^2}{{\left({\theta }^{\mathbb{P}}\right)}^2}\frac{2\gamma }{{\theta }^{\mathbb{P}}+{\theta }^{\mathbb{R}}}$), a higher income share of the poor (lower inequality) is associated with a higher net benefit of targeted transfers. Note that the same result applies to the net benefit of target firm-level transfers relative to transfers to poor households,

$$\frac{d{Y}_0}{d{T}^{\mathbb{F}:Target}}-\frac{d{Y}_0}{d{T}^{\mathbb{P}}}=Y_0\frac{1}{v\left(R_{j0}\right)}-\left(1-\kappa \right)\frac{d{Y}_0}{d{T}^{\mathbb{P}}},$$

since it is straightforward to show that $\frac{d}{d\kappa }\left(-\left(1-\kappa \right)\frac{d{Y}_0}{d{T}^{\mathbb{P}}}\right)=0$.■

Lemma 1

$$\frac{d}{d\kappa }\left(Y_0\frac{1}{v\left(R_{j0}\right)}\right)=\frac{1}{v\left(R_{j0}\right)}\frac{d{Y}_0}{d\kappa }-Y_0\frac{v\left(R_{j0}\right)}{v^2\left(R_{j0}\right)}\frac{d{R}_{j0}}{d\kappa }=\frac{1}{v\left(R_{j0}\right)}\frac{1}{{\overline{p}}_{jk0}^{\mathbb{P}}}\frac{c_{jk0}^{\mathbb{P}}+c_{jk0}^{\mathbb{R}}}{\left(1-\kappa \right)}-Y_0\frac{v^{\prime }\left(R_{j0}\right)}{v^2\left(R_{j0}\right)}\frac{d{c}_{jk0}^{\mathbb{P}}}{d\kappa }$$

$$\text{Sub}\text{in}{c}_{jk0}^{\mathbb{P}}{|}_{T^{\mathbb{P}}=0}\frac{\kappa }{\left(1-\kappa \right)}{c}_{jk0}^{\mathbb{R}}\text{and}\frac{d{c}_{jk0}^{\mathbb{P}}}{d\kappa }{|}_{T^{\mathbb{P}}=0}=\frac{c_{jk0}^{\mathbb{P}}+c_{jk0}^{\mathbb{R}}}{1-\kappa }$$

$$\frac{d}{d\kappa }\left(Y_0\frac{1}{v\left(R_{j0}\right)}\right)=\frac{c_{jk0}^{\mathbb{R}}}{v\left(R_{j0}\right){\left(1-\kappa \right)}^2}\left[\frac{1}{{\overline{p}}_{jk0}^{\mathbb{P}}}-Y_0\frac{v^{\prime }\left(R_{j0}\right)}{v\left(R_{j0}\right)}\right]$$

Substitute in ${\overline{p}}_{jk0}^{\mathbb{P}}=\frac{{\theta }^{\mathbb{P}}}{2{\lambda }_{\mathbb{P}}}$and$Y_0=\frac{{\theta }^{\mathbb{P}}}{2\gamma }+\frac{{\theta }^R}{2\gamma }$. Note we can solve for $\lambda$ from $c_{jk0}^{\mathbb{P}}{|}_{T^{\mathbb{P}}=0}\frac{\kappa }{\left(1-\kappa \right)}{c}_{jk0}^{\mathbb{R}}$, where $c_{jk0}^{\mathbb{P}}=\frac{{\theta }^2}{4\gamma \lambda }$. Specifically, ${\lambda }_{\mathbb{P}}=\frac{{\theta }^2}{4\gamma c_{jk0}^{\mathbb{R}}}\frac{1-\kappa }{\kappa }$, where $c_{jk0}^{\mathbb{R}}=\frac{{\left({\theta }^{\mathbb{R}}\right)}^2}{4\gamma }$.

Then

$$\frac{d}{d\kappa }\left(Y_0\frac{1}{v\left(R_{j0}\right)}\right)=\frac{c_{jk0}^{\mathbb{R}}}{v\left(R_{j0}\right){\left(1-\kappa \right)}^2}\left[\frac{\kappa }{\left(1-\kappa \right)}\frac{{\left({\theta }^{\mathbb{R}}\right)}^2}{{\left({\theta }^{\mathbb{P}}\right)}^2}-\frac{\left({\theta }^{\mathbb{P}}+{\theta }^{\mathbb{R}}\right)}{2\gamma }\frac{v^{\prime }\left(R_{j0}\right)}{v\left(R_{j0}\right)}\right]$$

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