Optimal Fiscal Policy in a Model of Firm Entry and Financial Frictions∗ Dudley Cooke† University of Exeter Tatiana Damjanovic‡ Durham University Abstract: This paper studies optimal dividend and labor-income taxation in a model of endogenous firm entry and financial frictions. We identify a novel trade-off for fiscal policy. On the one hand, fiscal policy can counteract a reduction in firm entry associated with financial frictions. On the other hand, fiscal policy can reduce the societal costs associated with firm default. We find that optimal fiscal policy should limit firm entry relative to the case without financial frictions and this implies a switch away from subsidizing dividendincome and towards subsidizing labor-income. Changes in the volatility of firm-level shocks imply the optimal labor-income tax is countercyclical. JEL Classification: E32, E44, E62 Keywords: Financial Frictions, Firm Entry, Optimal Fiscal Policy.

∗

We thank seminar participants at Carlos III for suggestions when the paper was at an early stage of

development and Tiziano Ropele for his discussion of the paper.

We also thank Julian Neira, Christian

Siegel and Rish Singhania for comments. † Department of Economics, University of Exeter, Streatham Court, Rennes Drive, Exeter EX4 4PU, United Kingdom. Email: [email protected] ‡ Department of Economics, Durham University, Mill Hill Lane, Durham DH1 3LB, United Kingdom. Email: [email protected].

1. Introduction This paper studies optimal dividend and labor-income taxation in a model of endogenous firm entry and financial frictions. The presence of financial frictions and the costs of firm default generate a novel trade-off for fiscal policy.

Financial frictions lead to a reduction

in firm entry which the policymaker would like to mitigate.

In doing so, however, the

policymaker must also account for the societal costs of default, and these costs increase with entry. A main result of the paper is that optimal policy should limit firm entry relative to the case without financial frictions and this involves a switch away from subsidizing dividendincome and towards subsidizing labor-income. Changes in the volatility of firm-level shocks - which are consistent with procyclical firm entry and a countercyclical default rate - imply the optimal labor-income tax is countercyclical. The model of endogenous firm entry we develop builds on the analysis of Bilbiie et al. (2012) insofar as each firm produces a differentiated good under conditions of monopolistic competition.1 We amend this setup in two directions. We suppose each firm receives an idiosyncratic demand shock which occurs after labor has been hired and production has taken place. This generates uncertainty over the revenue a firm can generate from the sale of its product. We also assume that firms finance their labor requirements by borrowing working capital from financial intermediaries who operate an imperfect monitoring technology similar to Calstrom and Fuerst (1997).2

In this environment, firms that produce goods with a

relatively low level of ex-post demand default and agency costs mean that default is costly. To better understand the mechanism in our model consider the production decision of an individual firm as the volatility of idiosyncratic demand shocks rises. In an attempt to take 1

Chatterjee and Cooper (1993) and Devereux et al. (1996) also develop general equilibrium models with

procyclical firm entry and monopolistic competition. 2 Financial frictions in this paper are similar to those in Bernanke et al. (1999) but the monitoring technology interacts with the working capital constraint (Jermann and Quadrini, 2012).

2

advantage of a potentially good realization of demand a firm will hire more labor and expand production.

It is optimal for the firm to do this because the market is monopolistically

competitive and firm-level revenue is unknown when the hiring decision is made.3 Expanding production, however, amounts to committing to a greater level of borrowing, in advance, and increased borrowing requires each firm to generate more revenue to avoid default. As the volatility of firm-level demand shocks rises, so does the minimum level of demand required for an individual firm to be able to repay its loan, and this leads to an increase in the rate of firm default.4 New firms enter each period by paying a one-time cost. Firms enter until their expected profit, conditional on not defaulting, is sufficient to cover this cost. The volatility of firmlevel shocks affects the entry decision through two off-setting channels. In a more volatile economy, because there is an increased probability of default, there is a reduction in conditional expected profits, and this discourages firm entry. However, there is also the possibility of increased profits via a good realization of demand, and this encourages firm entry.5 Overall, the default channel dominates and a more volatile economy is one with lower aggregate profits and fewer operating firms. We begin by analyzing optimal taxation when a lump-sum transfer is available to balance the budget.6 We show analytically that the government faces two trade-offs when deciding 3

With uncertainty over the idiosyncratic level of demand the price at which a good is sold is also unknown.

The effect of price uncertainty on factor inputs, for a competitive firm, is analysed in, for example, Sandmo (1971) and Hartman (1972). What matters for our analysis is that revenue (profit) is concave in prices. 4 Giesecke et al. (2011) report that the value-weighted default rate for US non-financial firms rises in recessions.

Historical default rates, such as those published by Moody’s, and which use issuer-weighted

default rates, show a similar pattern. See Exhibit 5 in Ou et al. (2011). 5 Lee and Mukoyama (2015) find that the firm entry rate is more cyclical than the firm exit rate (based on US manufacturing data). 6 In doing so we abstract from the public finance aspects of fiscal policy.

3

on dividend-income taxation. The first trade-off stems from changes in profit per-firm (a profit destruction effect) and in product variety (a consumer surplus effect) when firm entry is endogenous.

Without financial frictions, dividend-income should be subsidized - and

firm entry encouraged - because the profit destruction effect is relatively weaker than the consumer surplus effect. This result has been discussed previously in Bilbiie et al. (2008, 2016) and Chugh and Ghironi (2015). The second trade-off we identify is a consequence of financial frictions and costly default. This is the novel part of our analysis. We characterize this new trade-off via the endogenous default-threshold level of demand and the rate of firm default. Whilst financial frictions reduce firm entry such that dividend-income should subsidized the agency costs associated with firm default imply dividend-income should be taxed. Thus, as the default rate rises, the subsidy to dividend-income should be lowered, and similarly, when we consider an exogenous increase in the volatility of firm-level demand shocks, the subsidy (tax) to dividend-income should fall (rise). Overall, the strength of taxation applied to dividend-income is determined by the relative costs of firm default such that as the costs of default rise so does the rate of taxation. Our model also has a second margin: labor supply. As Bilbiie et al. (2008) demonstrate, when firm entry is endogenous, it is necessary to preserve firm profits, and this implies laborincome should receive a subsidy equal to price-markup. The same basic idea holds in our analysis but there are two important differences. First, each firm sells it’s good at a different price because each product has a different level of demand. This means that the aggregate price-markup depends on the endogenous default threshold. Second, each firm borrows to finance its labor bill which means that marginal costs depend on the interest rate applied to working capital. Taken together, the trade-off that characterizes dividend-income taxation also applies to the tax on labor-income.

We show labor-income should be increasingly

4

subsidized as the costs of default rise such that optimal fiscal policy involves a switch away from subsidizing dividend-income and towards subsidizing labor-income. We then extend the baseline model to a dynamic setting.

We calibrate the model and

consider a persistent increase in the volatility of shocks to firm-level demand. Changes in the volatility of firm-level demand shocks have meaningful quantitative implications in our framework. A positive one-time one-standard-deviation shock to volatility generates a 0.23 percentage point rise in the default rate on impact and a 0.2 percent drop in output after three periods. Using our calibrated model we revisit optimal fiscal policy and study the mix of taxes on dividend and labor-income when the government issues state-contingent real debt. We show that a positive one-standard-deviation shock to volatility implies a rise in the tax on dividend-income of over 1 percent and a drop in tax on labor-income of 0.6 percent. These changes in policy are persistent and imply strong departures from tax-smoothing. Our paper contributes to the literature on the design of fiscal policy when firm entry is endogenous and our results on optimal fiscal policy are closely related to Chugh and Ghironi (2015). An important finding in the context of their analysis is that dividend and laborincome should not respond to aggregate shocks when preferences are of the Dixit-Stiglitz type we consider.7 This result allows us to provide a clear analytical link between optimal fiscal policy and the rate of firm default. In general, we find that financial frictions imply taxes should be time-varying. The design of fiscal policy with firm entry has also been studied in environments with physical capital (Coto-Martinez et al., 2007), long-run risk (Croce et al., 2013), and oligopolistic competition (Colciago, 2016).8 7

In Chugh and Ghironi (2015), the extent to which profits should be taxed (in the long-run and short-run)

is also discussed in the context of preference aggregation.

We choose to work with a form of preferences

that lead to constant long-run taxes to focus on the role of financial frictions. 8 Lewis and Winkler (2015) analyse tax policy in a static model with firm entry. structure of demand and costs of firms entry.

5

They focus on the

We also consider changes in the volatility of firm-level shocks as a potential source of aggregate fluctuations consistent with the analysis presented in a number of recent papers.9 For example, in Christiano et al. (2014) a widening in the distribution of productivity shocks increases the fraction of loan defaults, and in Gilchrist et al. (2014), financial frictions magnify shocks to firm-level uncertainty through movements in credit spreads. Arellano et al. (2012) argue that the majority of the decline in employment during the 2007-09 recession can be explained by an increase in firm-level volatility.

Our analytical and quantitative

results imply that changes in firm-level volatility not only help explain firm entry but that financial frictions and microeconomic volatility also play a role in shaping policy decisions.10 Finally, this paper also contributes to research on endogenous firm entry and exit more broadly. Our approach is most similar to Bilbiie et al. (2012). To their model of firm entry we allow for endogenous exit by incorporating ex-post firm-level heterogeneity, a working capital constraint, and financial frictions. A complementary approach to studying firm entry and exit, which amends Hopenhayn’s (1992) model with ex-ante heterogeneous firms to allow for investment in physical capital and aggregate shocks, is developed by Clementi and Palazzo (2015). Our modelling choices - which imply a symmetric employment decision by firms in equilibrium - are driven by the desire to generate relatively simple policy implications. A general point, however, is that, in either setting, firm entry is a form of investment in which up-front costs incurred to start a business generate expected future profits. The remainder of the paper is organized as follows. In section 2 we develop a general equilibrium model of firm entry and financial frictions. 9

In section 3 we derive analytical

Empirical evidence on the role of firm-level shocks is presented in Bloom et al. (2014), Caldara et al.

(2016) and Jurado et al. (2015). Our approach is also consistent with a literature aims at generating business cycle fluctuations without large fluctuations in aggregate productivity such as Guerrieri and Lorenzoni (2011), Perri and Quadrini (2011), and Midrigan and Philippon (2016). 10 It is well-established that firm entry plays an important role in influencing aggregate fluctuations. For example, Gourio et al. (2016) show that reduced firm entry leads to persistent negative effects on GDP.

6

expressions for the optimal mix of taxes on dividend and labor-income when a lump-sum tax is also available and we show how shocks to firm-level volatility affect the cyclicality of taxes. In sections 4 and 5 we develop a quantitative version of the model and revisit the results on the optimal mix of taxes by allowing the government to issue state-contingent real debt. A final section concludes. 2. Analytical Model In this section we develop a static general equilibrium model of firm entry and financial frictions. We discuss the setup of the model, specify the optimization problem for households and firms, and the definition of equilibrium. 2.1. Setup The economy is populated by a measure nt > 0 of firms and a measure one of households and financial intermediaries. Each firm has a linear production technology and supplies a differentiated good with an idiosyncratic level of demand, ε ≥ 0. New firms are created each period by paying a one-time entry cost. Households consume a basket of goods and supply labor.

Financial intermediaries hold household deposits and issue intra-period working

capital loans to firms. If a firm has sufficient revenue it repays it’s working capital loan to the financial intermediary. If not, the firm defaults, and the intermediary repossesses the assets of the firm, subject to a cost of receivership.11 The timing of the model is as follows.

At the beginning of the period new firms are

created and households place deposits with financial intermediaries. Firms then make an employment decision and sign a contract with a financial intermediary to cover their working capital requirements. 11

Production takes place and idiosyncratic demand (and revenue) is

Our formulation is equivalent to all firms selling their production and the financial intermediary bearing

the burden of unpaid loans.

7

realized.12 Firms with a sufficiently high level of demand, ε ∈ [ε?t , ∞), sell their goods to households. Firms with a low level of demand, ε ∈ [0, ε?t ), default. Households receive netof-tax dividend and labor-income, interest payments on deposits, and a lump-sum transfer from the government. At the end of the period all non-defaulting firms exit. Households Each household draws utility from a composite of goods Ct and dis-utility from aggregate labor Lt , according to the following additively separable function, u (Ct , Lt )

(1)

which is strictly increasing and strictly concave in Ct and strictly decreasing and strictly convex in Lt . Total consumption is, 1/θ Z θ Ct = [ε (i) × ct (i)] di

(2)

i∈Ω

where ct (i) is the consumption of good i ∈ Ω and 1/ (1 − θ) > 1 is the elasticity of substituR tion. The integration over the probability space Ω is nt dG (ε) and G(ε) is the cumulative distribution function of idiosyncratic demand shocks.13

Demand shocks are lognormally

distributed with probability density function, " # (ln ε − m)2 1 g (ε) = √ exp − 2σ 2 ε 2π

(3)

where m and σ are the location and scale parameters. Firms Each firm produces a differentiated good with technology, yt (i) = lt (i)

(4)

where yt (i) is the output and lt (i) is the employment level of firm i.

Firms use working

capital to finance their labor requirements. Working capital requires a loan, at gross rate 12 13

The timing restriction we place on the firm is similar to Neumeyer and Perri (2005). Similar to Bernard et al. (2011) the firm-level shock reflects product attributes or product appeal.

Midrigan (2011) refers to this shock as a quality shock.

8

rt ≥ 1, equal to wt lt (i).14 The profit of firm i, with demand level ε, is written as, πt (i, ε) = pt (i, ε) yt (i) − wt rt lt (i)

(5)

where pt (i, ε) is the price of good i in units of consumption, pt (i, ε) yt (i) is firm revenue, and wt rt lt (i) is the debt of firm i. Throughout the analysis we assume firms operate under limited liability and act as though profit is bounded from below at zero. This implies a threshold level of demand, ε?t , determines the mass of firms unable to meet their debt obligations expost. This default-threshold level of demand is defined as ε?t ≡ inf {ε (i) : πt (i, ε) > 0} and R ε? the probability of default is G (ε?t ) = 0 t dG (ε). There is an unbounded mass of potential entrants.

The creation of a new firm is sub-

ject to a one-time entry cost. Firms enter until conditional expected profits, π (ε?t ) ≡ R∞ π (i, ε) dG (ε); that is, expected profit conditional on not defaulting, net of taxation, ε? t

τt < 1, is equal to the cost of entry. The free entry condition reads, (1 − τt ) π (ε?t ) = fe

(6)

where the cost of entry, fe > 0, is specified in units of output.15 Financial Intermediaries Each financial intermediary receives deposits from households and issues working capital loans to firms. The expected assets of a financial intermediary are the revenue from the repayment of loans and the assets from liquidated firms, less the cost of receivership, fm > 0. Financial intermediaries are competitive and earn zero profit, 14

The interest rate on loans, rt , is strictly greater than the interest rate on deposits. The deposit rate is

exogenous in this version of the model and assumed equal to unity. 15 Firms face entry costs before starting production (for example, see Restuccia and Rogerson (2008)). As emphasized by Djankov et al. (2002), entry costs not only reflect the time and effort of the entrepreneur, but also bureaucratic and transactions costs required for setting up a business. Higher taxes reduce entry rates. Da Rin et al. (2008) present evidence on taxes an entry rates in European countries.

9

which leads to, "Z

#

∞

ε?t

dG (ε) + η (ε?t ) rt = 1 + fm [G (ε?t ) /wt lt ]

where

R∞ ε?t

dG (ε) is the survival probability of a firm and η (ε?t ) ≡

ratio of assets-to-loans of defaulting firms.

(7) R ε?t 0

(ε/ε?t )θ dG (εt ) is the

The liabilities of financial intermediaries are

given by rtd wt lt , where rtd = 1 is the normalized interest rate on deposits.

Equation (7)

defines the interest rate on working capital loans. 2.2. Optimization
Each household maximizes utility subject to the budget constraint Ct = 1 − τtL wt Lt + Tt where τtL < 1 is a labor-income tax and Tt is a lump-sum transfer. This leads to a standard labor supply equation,
−uL (t) 1 − τtL wt = uC (t)

(8)

where uC (t) and uL (t) denote the marginal utility functions, evaluated at time-t arguments. Each household also chooses the consumption level, ct (i), to minimize the cost of acquiring Ct , taking prices and income as given. This leads to a downward-sloped demand curve for each good, " ct (i) =

pt (i, ε) ε (i)θ

#−1/(1−θ) Yt

(9)

where Yt is aggregate output. Each firm chooses an employment level, lt (i), subject to demand and technological constraints, given by equations (4) and (9), and market clearing, ct (i) = yt (i). Proposition 1 characterizes the optimal employment decision of firm i. To economize on notation, in what follows we drop the i index.

10

Lemma 1 Profit maximization implies firm-level employment, lt , and the default-threshold level of demand, ε?t , are determined by the following conditions. " #1−θ Z ∞ ∆(0) ? θ εθt dG(εt ) = [1 − G (ε?t )] (ε?t )θ (εt ) = wt rt and θ 1/θ lt ε?t where ∆ (0) ≡

R ∞ 0

(10)

1/θ εθ dG (ε) .

Proof See Appendix A.  The first equation in Lemma 1 determines firm-level employment as a function of the marginal cost of production - the wage rate multiplied by the interest rate on working capital loans wt rt . For given marginal costs, there is a positive relationship between the default-threshold level of demand and firm level employment.

This is because an increase in employment,

at the firm-level, requires more debt, with working capital loans equal to wt rt lt . In turn, to avoid default, a more indebted firm needs to generate more revenue, and this implies a higher default-threshold level of demand. The second expression in Lemma 1 determines the default-threshold level of demand. Products with demand εt ∈ [0, ε?t ) are taken into receivership and products with demand εt ∈ [ε?t , ∞) are sold directly to consumers. An important simplifying property of this expression is that it alone determines the default-threshold and we use it to generate the following result. Lemma 2 When idiosyncratic shocks have a log normal distribution there exists a unique default-threshold level of demand, ε?t > 0. Proof See Appendix A.  An immediate implication of Lemma 2 is that the default threshold is unaffected by the firms optimal employment decision and we can treat this variable as independent of the rest of the h i1/θ R∞ θ 1 economy. At this point it is useful to define ∆ (ε?t ) ≡ 1−G(ε ε dG (ε) such that the ? ) ε? t

11

t

default-threshold level of demand is implicitly determined by [ε?t /∆ (ε?t )]θ = θ.

This new

variable captures the relative dispersion of demand across non-defaulting firms, is a measure of conditional expected revenue, and acts as an endogenous revenue shifter.

We use this

new expression to relate the mass of available products for consumption to marginal costs in h iθ h iθ (1−θ)/θ ∆(0) ∆(0) 1 wt rt the following way nt = [∆(0)] where < 1 is the aggregate markup ? ? ∆(ε ) θ ∆(ε ) t

t

directly induced by financial frictions. Since ∆0 (ε?t ) > 0, then that markup is falling in the default rate.16 2.3. Equilibrium Labor is used for the production of goods and labor market equilibrium requires, Lt = nt lt

(11)

Equation (11) implies that increased firm-level employment, lt , translates directly into fewer operating firms, nt , for given levels of aggregate employment. We also know that firm-level employment rises with the default rate and so financial frictions lead to reduced firm entry. The resource constraint of the economy is, Yt = Ct + fe nt + fm [nt G (ε?t )] (1−θ)/θ

where Yt = nt

(12)

∆(0) is aggregate output, fe nt represents investment at the extensive

margin, and fm [nt G (ε?t )] is the resource cost of firm defaults. Finally, the government budget constraint is, Tt = τt nt π (ε?t ) + τtL wt Lt 16

Formally, we can show,

(13) θ ∂[∆(ε? t )] ∂ε? t

=

dG(ε? t) 2 [1−G(ε? t )]

i R∞h θ ? θ ε − (ε ) dG (ε) > 0 where ε > ε?t and ∆ (ε?t ) > ? t ε t

∆ (0).

12

where Tt is a lump-sum transfer to the household and the right-hand side of equation (13) is total government revenue. 3. Optimal Fiscal Policy and the Role of Firm-Level Volatility In this section we study optimal fiscal policy.

We also examine the role played by the

volatility of firm-level shocks. 3.1. Optimal Fiscal Policy In this section we demonstrate that financial frictions and costly default imply a novel tradeoff for fiscal policy. We characterize optimal fiscal policy in the following proposition. Proposition 1 1. The optimal dividend-income tax is, τt = 1 −

1 1 ? θD (εt ) 1 + f G (ε?t )

(14)

where f ≡ fm /fe is a positive constant. 2. The optimal labor-income tax is, τtL = 1 −

1 − G (ε?t ) rt θD (ε?t )

(15)

where rt ≥ 1 is determined by equation (7). 3. The terms, G (ε?t )

Z =

R∞

ε?t

dG (ε)

and

D (ε?t )

0

=

εθ dG (ε) ε?t R∞ εθ dG (ε) 0

both lie between zero and one with G0 (ε?t ) > 0 and D0 (ε?t ) < 0. Proof See Appendix A. 

13

(16)

We begin by assuming there are no financial frictions. In this case, G (ε?t ) → 0 and D (ε?t ) → 1. It is then immediate from Proposition 1 that both dividend and labor-income should be subsidized at the monopolistic markup and 1 − τt = 1 − τtL = 1θ . Subsidies are required, in this case, for the following reasons. Dividend income should be subsidized because the returns-to-variety outweigh the reduction in profit per-firm implied by additional firm entry. This is a result of a profit destruction effect and a consumer surplus effect when firm entry is endogenous.17

Labor-income should be subsidized because leisure is not subject to a

markup and there is a wedge between the marginal rates of substitution and transformation of consumption and leisure. From the perspective of our analysis, the relevant property of equations (14) and (15), is that, without financial frictions, there is no role for short-run stabilization policy. With financial frictions the government faces a novel trade-off when deciding on fiscal policy. Consider the tax on dividend-income. The optimal dividend-income tax equalizes the social and private margins on firm entry whilst accounting for the cost of default - this is the financial-frictions trade-off when firm entry is endogenous. To understand the implications of this new trade-off we define two wedges. At the societal level, using the resource constraint given by equation (12), the ratio of the marginal cost to marginal benefit of firm entry is defined as, Λst ≡

1 π (ε?t ) ∂Yt /∂nt = fe + fm G (ε?t ) 1 + f G (ε?t ) θfe

(17)

where πt = (Yt /nt ) (1 − θ) is expected profit.

At the decentralized level, because each

potential entrant does not internalize the cost of default the marginal benefit of firm entry is the expected profit and the cost of entry is fe > 0. 17

The ratio of the marginal cost to

This terminology is taken from Grossman and Helpman (1991) and the trade-off is discussed in Bilbiie et

al. (2008, 2016) and Chugh and Ghironi (2015) both of which analyse the steady-state of a dynamic model without default. The exact specification of the profit tax depends entry costs and the form of preferences

14

marginal benefit at the decentralized level is defined as, Λdt ≡ (1 − τt )

D (ε?t ) π (ε?t ) fe

(18)

Notice that the trade-off generated by financial frictions and costly default is captured by a single statistic: the default-threshold level of demand. This statistic is independent of the rest of the economy (Lemma 1). As such, we can analyze how a change in ε?t affects the optimal tax on dividend-income.

An increase in ε?t implies a higher default rate, G (ε?t ).

As the default rate rises, the subsidy to dividend-income falls and it is optimal to restrict entry, relative to the case without financial frictions.18 However, an increase in ε?t also leads to a rise in

1 . D(ε?t )

This has the opposite effect to the change in the default rate as far as

optimal taxation is concerned. The term D (ε?t ) is equal to π (ε?t ) /πt , which is the ratio of conditional to expected profits (and equals unity without financial frictions). As

1 D(ε?t )

rises,

profits from firm creation fall relative to the case without frictions, firm entry is depressed, and dividend-income should be subsidized. The optimal labor-income tax also targets the distortions created by financial frictions. h iθ 1−G(ε? ) ∆(0) The key term is D(ε? )t = ∆(ε < 1 which is the price-markup attributable directly to ?) t

t

the presence of financial frictions ( 1θ > 1 is the markup due only to the assumed form of preferences in equation (2)). This is decreasing in ε?t . Although the tax on labor-income depends directly on the price-markup it is not independent of the resource costs of default. The presence of the interest rate on working capital loans in equation (15) introduces a role for resource costs and, all else equal, rt is increasing with fm - see equation (7). When fm is relatively small τtL increases with ε?t because the effect of limited liability is to reduce the negative welfare effects of the aggregate mark up. However, as the distortions associated 18

We also note that since entry costs also matter for tax policy, any regulatory policy aimed at encouraging

entry should be partly offset with taxation.

It is well-documented that costs of firm entry vary across

countries and that these have important macroeconomic implications (Poschke, 2010 and Barseghyan and DiCecio, 2011).

15

with default grow (i.e., when fm is relatively large), the impact of a rise in ε?t , which also causes the interest rate to rise, leads to a drop in τtL .

This feature of the optimal labor-

income tax is therefore very similar to the optimal tax applied to dividend-income. 3.2. Movements in the Volatility of Firm-Level Demand Shocks In this section we study an exogenous movement in the volatility of firm-level demand shocks. We are motivated to do so based on evidence presented in, for example, Christiano et al. (2014) and Arellano et al. (2016), that suggests such shocks play an important role aggregate fluctuations. We are also interested in linking changes in the default rate - studied in the context of optimal fiscal policy - to the macroeconomy. Lemma 3 When idiosyncratic shocks have a log normal distribution the default threshold and the probability of default increase with the volatility of firm-level shocks for G(ε?t ) < 1/2. Proof See Appendix A.  Lemma 3 shows that a rise in volatility leads to a rise in ε?t and G(ε?t ).

There are two

opposing effects of increased volatility on aggregate output. Expected revenue, ∆(0) = 1/θ R ∞ θ ε dG (ε) , which increases with volatility, raises aggregate output, whereas the mass 0 of operating firms, nt , falls, and this reduces output. The size of the fall in nt is determined by the firm employment decision (see Lemma 1) and its strength onto output is greater the larger are the returns-to-variety (and the larger the monopolistic markup).19 Overall, more volatility leads to lower output - which we refer to as a volatility-induced recession - with procyclical firm entry and a countercyclical default rate. It should be clear that financial frictions play a crucial role in allowing increased volatility to generate recessions. Without financial frictions volatility shocks would have the opposite 19

We focus on the case in which the returns-to-variety are less than unity.

The returns-to-variety are

given by (1 − θ) /θ > 0; the markup minus one, and there is increasing returns to an expansion in variety with the degree 1/θ. We make this restriction because firm entry costs are specified in units of output.

16

effect on the economy. This is because, with monopolistic competition (such that profits are concave in prices) and flexibility in the factor input (the distribution of demand is known before employment decisions are made) firms expand to take advantage of a potentially good realization of demand and there is a positive shift in conditional expected revenue. Such a change in factor demand under increased uncertainty - in our case via demand shocks is consistent with an Oi-Hartman-Abel effect as emphasized in Bloom (2014).20 However, because an increase in conditional expected revenue is also associated with a rising defaultthreshold firms are increasingly unlikely to generate sufficient revenue to repay their working capital loans. This is why the increase in volatility also implies an increase in the proportion of firms that default. We can also determine analytically how optimal dividend-income taxation reacts to changes in firm-level volatility. Proposition 2 The dividend-income tax is pro-cyclical when there are shocks to firm-level volatility. Proof See Appendix A.  To understand the response of the tax on firm profits, let fm → 0, such that only the financial frictions wedge matters for the optimal policy decision. Define x?t ≡ [ln (ε?t ) − µ] /σt , where σt is the measure of firm-level volatility and µ is the location parameter of the lognormal distribution. Following an exogenous change in the volatility of firm-level shocks, the optimal change in the tax on dividend-income is given by, τbt = −

Φ0 (− (x? − θσ)) × [x? x b?t − θ (σb σt )] θ

(19)

where a caret denotes the deviation of a variable from its long-run value (with τbt ≡ τt − τ ) 20

As Bloom (2014) discusses, the Oi-Hartman-Abel effect implies that firms can expand to exploit good

outcomes and contract to insure against bad outcomes, making them potentially risk loving.

17

and Φ(x? ) is the CDF of the normal distribution.21

Since τt is negative when fm → 0,

the sign of the change in the profit tax that results from a change in firm-level volatility depends on the sign of the term in square brackets on the right-hand side of equation (19). In this term, x?t is negative and x b?t is positive (since εb?t > 0). Thus, for a given increase in volatility - such that σ bt > 0 - the subsidy to profits increases. However, as fm > 0 rises, so do the resource implications of default. Thus, when default is sufficiently costly, shocks to firm-level volatility are associated with pro-cyclical tax policy. 4. Dynamic Model In this section we develop a dynamic version of the model in which firms are long-lived. We also drop the assumption that the government has access to a lump-sum transfer and introduce state-contingent government debt.

Taxation is used to finance an exogenous

stream of public spending. 4.1. Firm and Household Optimization We define the net worth of a firm as zt (ε) ≡ max [(1 − τt ) Et πt (ε) , 0] + vt (εt ), where vt (ε) is the price of the firm at the end-of-period t, after the realization of uncertainty. Under this formulation, once a firm defaults, its value is retained by the firm and sold in the following period. The instantaneous profit function is now written as, o n 1−θ θ  lt − wt rt (lt + fo ) , 0 πt (ε) = max εθ × n1/θ ∆(0)lt

(20)

which is a generalization of equation (5).22 In equation (20), the term fo > 0 is a quasi-fixed overhead cost, and lt is average firm-level employment, which is taken as given by the firm   R x? Specifically, Φ(x? ) = √12π −∞ exp −(x)2 /2 dx and Φ0 (− (x? − θσ)) > 0. 22 In our specification the entire wage bill is borrowed in advance. Evidence for this assumption is presented 21

in Lewis and Poilly (2012).

18

when maximizing net worth. employment.  Z ∞ θ ? θ θε − (εt ) ε?t

lt lt + fo

Profit maximization implies the following optimal level of

 dG(ε) = 0

(21)

h i1−θ 1/θ lt − wt rt (lt + fo ) = 0. for ε > ε?t where ε?t is determined by (ε?t )θ nt ∆(0) Household intertemporal utility is E0 discount factor.

P∞

t=0

β t u (Ct , Lt ) where β ∈ (0, 1) is the subjective

Households place deposits, dt , with financial intermediaries, and hold

shares, xt , in firms. They also have access to a complete set of state-contingent government s . Households maximize their lifetime utility, subject to the following flow budget bonds, Bt+1

constraint, dt +

X 1
nt+1 s d L B + C + x v = r d + 1 − τ wt Lt + Bts + nt xt−1 z (ε?t ) t t t t−1 t+1 t−1 t s r 1 − δ t+1 s

(22)

s are the rates of return on deposits and bonds, respectively, and z (ε?t ) = where rtd and rt+1 R∞ z (ε) dG (ε) is conditional expected net worth. Household decisions over bonds and ε? t

deposits are characterized by the following consumption Euler equations, s uC (t) = βEt rt+1 uC (t + 1)

and uC (t) = βEt rtd uC (t + 1)

(23)

The optimality condition for equity is,  
 vt = (1 − δ) Et Mt+1,t (1 − τt+1 ) π ε?t+1 + vt+1

(24)

where Mt+1,t = [β (1 − δ)]t uC (t + 1) /uC (t) is a stochastic discount factor. 4.2. Firm Entry and Government Budget Constraint The free entry condition reads, E0

∞ X

M0,t (1 − τt ) πt = fe

(25)

t=1

19

where v0 ≡ E0

P∞

t=1

M0,t (1 − τt ) π (ε?t ) is the expected present discounted value of after-tax

dividend payments and δ ∈ (0, 1) is the exogenous probability of firm exit.

The law of

motion for the mass of firms is, nt+1 = (1 − δ) (nt + ne,t )

(26)

which reflects the assumption that there is a one-period lag between entry and production. The government collects taxes and issues state-contingent real debt to finance an exogenous constant stream of government spending, G > 0. The flow government budget constraint is, τtL wt Lt + τt nt xt−1 π (ε?t ) +

X 1 s Bt+1 = Bt + G s r t+1 s

(27)

where τtL wt Lt + τt nt xt−1 π (ε?t ) is government income from taxation. 4.3. Model Summary and Optimal Fiscal Policy In this section we present a model summary and provide a definition for optimal fiscal policy. Model Summary Table 1 presents the equations for the model economy,

===== Table 1 Here ===== where the ε?t index has been suppressed where possible. The conditions in Table 1 form a sys d tem of equations which can be used to solve for {Ct , Yt , Lt , lt , nt , ne,t , zt , πt } and wt , rt−1 , rt and {ε?t }, for given government expenditure, G > 0. Optimal Fiscal Policy We solve for optimal dividend and labor-income taxes as specified in the following reduced policy problem.

20

∞ Definition 1 Given the exogenous process {σt }∞ t=0 plans Ωt ≡ {ne,t , nt+1 , Ct , Lt , lt }t=0 and

{ε?t }∞ t=0 represent the optimal allocation if they solve the following problem. max E0

{Ωt ,ε?t }∞ t=0

∞ X

β t U(Ct , Lt , ξ) + β t λ1,t [Yt − G − Ct − fe ne,t − fm nt G (ε?t )]

t=0

t

+β E0 λ2,t [Lt − nt (lt + fo )] + β t λ3,t [(1 − δ) (nt + ne,t ) − nt+1 ] ) (Z    ∞ l t dG(εt ) − ξA +β t E0 λ4,t θεθt − (ε?t )θ ? lt + fo εt

(28)

where, U(Ct , Lt ) ≡ u(Ct , Lt ) + ξ [uC (t)Ct + uL (t)Lt ]

(29)

and given,  d  A ≡ uC (0) r−1 d−1 + b0 + n0 z (ε?0 )

(30)

where {λj,t }4j=1 are lagrange multipliers associated with constraints, and ξ is a (constant) lagrange multiplier associated with the implementability constraint. A detailed derivation of this reduced policy problem - as stated in equations (28)-(30) - is presented in Appendix B.23 The substantive point is that relative to the case without default the only additional constraint placed on the policy maker is the labor demand equation. Given the structure of the problem in equations (28)-(30) there is also a tight link between the default-threshold level of demand and the mass of operating firms. Once the government picks ε?t > 0, for a given level of aggregate employment, Lt , the mass of operating firms is determined as a function of the underlying shocks.

The remaining constraints - over

resources, both in goods and labor markets, the law of motion for firms, and the present value implementability constraint - are the same as when financial frictions are absent. 23

As in the standard Ramsey taxation problem, the government is assumed to commit, as of period zero,

to time invariant policy functions for t ≥ 1. Following Chugh and Ghironi (2016), we also assume that the schedule of state-contingent profit taxes is posted one period in advance.

21

Exogenous Shock The only source of aggregate uncertainty in our model is firm-level volatility. We assume firm-level volatility follows an AR(1) processes, ln (σt ) = (1 − ρ) σ + ρ ln (σt−1 ) + ωt

(31)

where ωt is a normally distributed mean-zero shock. 5. Numerical Exercises In this section we undertake a numerical analysis of the model developed in section 4. We first outline the calibration of the steady-state and then compute impulse responses for endogenous variables for a one-time shock to firm-level volatility. Finally, we study optimal response of dividend and labor-income taxes to changes in firm-level volatility. 5.1. Parameterization and Calibration This section discusses the parameterization and calibration of the steady-state of the model.24 We start with standard technology and preference parameters, followed by fiscal variables, and finally the distribution of firm-level shocks and fixed cost parameters. A period in the model is a year. The discount rate is set at β = 0.98 and we adopt the following functional form for period utility, u (Ct , Lt ) = ln Ct + χ

(1 − Lt )1−υ − 1 1−υ

(32)

The scale parameter χ > 0 in equation (32) is set such that that households allocate 20 percent of their time to work in the steady-state. The Frisch elasticity of labor supply with respect to wages - here equal to uL /uLL L > 0 - is assumed to be 0.72, based on the empirical evidence in Heathcote et al. (2010). The elasticity of substitution between differentiated goods is set at 3.8. This value is taken from Bernard et al. (2003) and, without financial 24

In doing so we also revert to the assumption that the government has a lump-sum transfer available to

balance its budget for the reasons outlined in Chugh and Ghironi (2015).

22

frictions, implies a markup of 35.7 percent. Finally, we set δ = 0.1 to match an annual exit rate for firms of 10 percent. Fiscal variables - government expenditure-to-output and labor-income and firm profit taxes  are set at g, τ L , τ = {0.2, 0.2, 0.25}, respectively. When we consider optimal fiscal policy in section 4 we set the steady-state debt-to-output ratio at b ≡ B/Y = 0.42. The values for g and b are taken from Schmitt-Groh`e and Uribe (2005) and the values for τ L and τ are taken from Arseneau and Chugh (2010) and Gourio and Miao (2010), respectively. We also assign a value to long-run firm-level volatility. Firm-level demand is assumed to have a log-normal distribution as defined above in equation (3).

Empirical estimates of

firm-level volatility range from 0.09, used in Bachmann and Bayer (2012), to 0.23, used in Christiano et al. (2013). We use an intermediate value of σ = 0.135 based on Comin and Mulani (2006) and then set the mean to m = −σ 2 /2 such that E (ε) = 1.25 Given long-run micro volatility, we then calibrate overhead and default costs to match two features of the data. Normalizing the parameter governing entry costs to unity (fe = 1), we set overhead costs parameter (f0 > 0) such that the default rate is 1.0%.

We base this calibration on

historical value-weighted default rates of non-financials reported in Giesecke et al. (2011).26 We then set the default cost parameter (fm > 0) to generate an average credit spread of 204 basis points as reported in Gilchrist and Zakrajˇsek (2012). Table 2 presents the parameters we use in our calibration. ===== Table 2 Here ===== 25

Comin and Mulani (2006) use annual data on net sales from COMPUSTAT and find that since 1980

average (weighted) volatility is around 0.135. Arellano et al. (2012) - who also focus on demand shocks use sales growth data and find 0.18. 26 Giesecke et al. (2011) report that the mean (median) default rate for US nonfinancial is around 1.52% (0.54%). Over the long term, the also argue that credit spreads are twice as large as default losses, such that the average credit risk premium is 0.8%.

23

Our calibration has implications for variables other than those presented in Table 2. The average price-markup in our model is given by [θ∆(0)]−1 [∆(0)/∆ (ε?t )]θ .

Our calibration

implies a price-markup of 21.7. Also, although the use of overhead labor in our economy is used to match default rates, it implies overhead costs account for 6.44 percent of total employment. Bartelsman et al. (2013) suggest that firms’ use of overhead labor accounts for approximately 14 percent of total employment in U.S. manufacturing establishments. We can match this figure, but keeping default rates below 1 percent requires lower firmlevel volatility. Finally, business investment as a fraction of aggregate output across OECD countries is between 10 and 15 percent, and our calibration implies a value of 13.13 percent and the cost of default is 1.55 percent of GDP. 5.2. Shocks to Firm-Level Volatility with Given Fiscal Policy In this section we consider the aggregate implications of shocks to the volatility of firm-level demand.

The process for volatility is parameterized following Chugh (2016).

Based on

empirical estimates using annual US manufacturing data Chugh (2016) estimates innovations to firm-level volatility to be 3.15%. With persistence parameter ρ = 0.48 - recall equation (31) which specifies the path of volatility - this implies V ar (ωt ) = 0.0282 .27 Figure 2 shows the response of selected endogenous variables for a one-time one standard deviation shock to volatility.

The horizontal axis measures years and the vertical axis

measures the percentage deviation from the steady-state, unless otherwise stated.

===== Figure 1 Here ===== 27

As Chugh (2016) discusses, since it is not possible to distinguish cost shocks from revenue shocks, his

results can be intrepreted as being driven by both supply and demand shifts. We also note that Alessandria et al. (2015) assume ρ = 0 and V ar (ωt ) = 0.12 in their analysis.

24

A one-standard-deviation increase in the volatility of firm-level demand (a volatility-induced recession) leads to a 23 basis points rise in the firm default rate and a 22 basis points rise in the credit interest rate premium on impact. Increased volatility leads to a rise in the firm default rate in our economy because it is optimal for firms to expand and take advantage of potentially good realizations of demand. They can do so by hiring more labor but this requires that they take on more debt due to working capital constraints. In a more volatile economy, firms therefore need to generate a greater level of revenue to avoid default, and ε?t rises (not drawn).

The impulse responses also show that, from year 3 onwards, both

the default rate and credit interest rate premium are lower than their pre-shock levels and afterwards slowly increase. This is a result of an interaction between the default-threshold level of demand and firm-level employment when there are overhead costs.28 The response of the mass of operating firms to a change in firm-level volatility is humpshaped.

Greater volatility reduces firm entry, and over time, less entry translates into a

gradual reduction in the number of available goods. This effect peaks after 4 years by which time the initial rise in volatility has dropped by around 90 percent. Whilst the change in the mass of operating firms is relatively small (dropping by 0.3 percent) the transition back to the steady state is persistent. The model is relatively less successful at capturing labor market dynamics. Aggregate employment drops by around 0.1 percent in response to an increase in volatility. Given the mechanism in the model this is perhaps not too surprising - recall that aggregate employment is Lt = nt (lt + fo ) - because the only way for firms to take advantage of greater volatility is to expand production. Thus, a sufficiently large fall in the mass of operating firms is required for aggregate labor supply to fall. In the calibrated economy, the long-run default rate is 1 percent, with an investment premium of 104 b.p., and the size of the innovations to volatility we consider are relatively 28

In section 2 we characterize the response of the default rate analytically assuming no overhead costs.

25

conservative. Despite this, the interaction of firm-level volatility shocks and financial frictions have important aggregate implications.

As a simple robustness check, in the first

instance, we maintained the premium and reduced firm default to 0.4 percent (low default). We then maintained the default rate of 1 percent but reduced the premium to 30 b.p. (low premium). In both cases, we kept long-run volatility at σ = 0.135. In a second case, we set σ = 0.1 and used the baseline specification of 1 percent default with a 60 b.p. premium. Our main finding is that lowering the long-run volatility generates a larger response of firm entry, on impact, but firm-level employment is less persistent. 5.3. Optimal Dividend and Labor-Income Taxation We now consider short-run stabilization policy - specifically, the optimal response of taxes in a recession.29 Figure 3 plots the response of taxes on dividend and labor-income (alongside other endogenous variables) to a one-time change in volatility.

===== Figure 2 Here =====

First consider the response of taxes.

Following a one-standard deviation shock to firm-

level volatility there is a near 1.5 percentage points rise in the tax on dividend income (implemented in periods t ≥ 1) and 0.6 percentage points drop in the tax on labor-income on impact. Qualitatively, the change in taxation is consistent with the analytical results of section 3, where an exogenous increase in the threshold level of demand, i.e., εb?t > 0, implied a switch away from subsidizing dividend-income and towards subsidizing labor-income. Here, the shock to volatility, which drives the rise in the threshold level of demand and rate of default implies persistent changes in taxation. Comparing the response of firm entry and firm default it is clear that tax policy acts to smooth changes in both of these variables, 29

The impluse responses we report are based on an optimal stabilization policy and represent deviations

from the steady-state of the model as calibrated in section 4.

26

although only by a quantitatively small amount. On the contrary, however, consumption and and total employment change considerably under optimal policy, with consumption initially above it’s pre-shock level and employment persistently above it’s pre-shock level. It is useful to compare our results on short-run stabilization policy to those reported in Chugh and Ghironi (2015) since they adopt a similar mechanism for firm entry and also consider Dixit-Stiglitz preferences.

They find that taxes on dividend and labor-income should be

invariant over the business cycle.30

The reason is that the price-markup is also constant

along the business cycle and thus so are inefficiency wedges.

This is not the case once

we allow for financial frictions. As we discuss above, the aggregate price-markup depends explicitly on the threshold level of demand and is therefore connected to the rate of firm default.

This means it is no longer possible to map markups into optimal tax responses

without accounting for the resource implications of default and this is what creates the dual role of fiscal policy in our analysis. 6. Conclusion This paper studies optimal fiscal policy in a general equilibrium model of firm entry and financial frictions. We provide analytical expressions for optimal fiscal policy and show that the government faces two trade-offs. The first arises from a profit destruction and a consumer surplus effect when firm entry is endogenous. The second is novel and arises because financial frictions reduce firm entry and default is costly. We also study quantitatively the optimal mix of taxes on dividend and labor-income in a dynamic version of our model with government debt and shocks to the volatility of firm-level demand.

30

This tax-smoothing result is first presented in Chari et al. (1994) in the context of an RBC model.

27

Appendix A In this Appendix we present proofs omitted in the text. Appendix A.1 (Proof of Lemma 1) Firms maximize conditional expected profit,

R∞ ε?t

π (i, εt ) dG(εt ), choosing employment level,

lt (i), subject to technology, demand, and market clearing. In units of consumption, profit  −1/(1−θ) is given by, π (i, εt ) = p (i, εt ) yt (i) − wt rt lt (i), where yt (i) = lt (i) = Yt p (i, εt ) /εθt and ε?t = inf {εt : π (i, εt ) > 0}. The unconstrained problem is, Z max?

lt (i),εt

∞

n o (εt )θ [lt (i)/Yt ]θ−1 lt (i) − wt rt lt (i) dG(ε)

(33)

ε?t

with prices and Yt > 0 given. The first order conditions implies, Z

∞

ε?t

θ (εt )θ (lt (i)/Yt )θ−1 dG(ε) − [1 − G (ε?t )] wt rt = 0

(34)

for all i. The threshold level of demand is determined by, π (i, ε?t ) = 0. We determine ε?t using the expression, π (i, ε?t ) = (ε?t )θ [lt (i)/Yt ]θ−1 lt (i)−wt rt lt (i) = 0. Incorporating the above R∞ first-order condition, we then find, ε? θεθt dG(εt ) − [1 − G (ε?t )] ε?θ t = 0. The labor demand t

expression also pins-down the price of a good in our model for given costs of production. 1−G(ε? ) R ∞ Using the demand curve, the average price is, p (ε?t ) = wt rt θ t / ε? εθt dG(ε). Finally, we hR t i1−θ  ∞ θ (θ−1)/θ R ∞ θ ? θ ? 1/θ ε dG(ε) , so (ε ) ε dG(ε)/l (ε ) = solve for employment using, Yt = lt t t t t 0 0 wt rt . Appendix A.2 (Proof of Lemma 2) We drop time subscripts and define the following function, ?

Z

∞

f (ε , σ) ≡

θεθ dG(ε, σ) − [1 − G (ε? , σ)] (ε? )θ = 0

ε?

28

(35)

This implicitly determines ε? . Assume ε has a lognormal distribution with PDF, g(ε, σ) = h i −(ln σ−µ)2 1 √ exp , and define, x ≡ (ln ε − µ) /σ ⇔ ε = exp (xσ + µ). The default2σ 2 εσ 2π threshold level of demand is determined by,    Z ∞ 1 −(x)2 ? ? f1 (x , σ) = √ dx = 0 [θ (exp θσx) − (exp θσx )] exp 2 2π x? Rx We express this condition in terms of a normal CDF, Φ(ε) ≡ √12π −∞ exp [−(x)2 /2] dx, which gives, 
 f1 (x? , σ) = θ exp (σθ)2 /2 Φ(−(x? − θσ))) − [exp (θσx? )] Φ(−(x? ))

(36)

We prove Proposition 2 in following steps: (i), that x? exists and is unique, (ii), that ∂ f ∂x 1

(x? , σ) < 0.

Lemma 2a There exist and x? which satisfies (36) and

∂ f ∂x 1

(x? , σ) < 0.

Proof lim Φ(ε) = 1 and lim exp (ε) Φ(ε) = 0 implies ?lim f1 (x? , σ) = θ exp ((σθ)2 /2) > ε→+∞

0 and

x →−∞ ?

ε→−∞

?

?

lim f1 (x , σ) = − [exp (θσx )] < 0. Since f1 (x , σ) is a continuous function and

x∗ →+∞

it changes sign from positive to negative, a solution exists. We know f1 (x, σ) > 0 at some small x and f1 (x, σ) crosses zero at least once. Let x? be the first point where f1 (x? , σ) = 0. As f1 (x? , σ) approaches the line from above,

∂ f ∂x 1

(x? , σ) ≤ 0.

To prove uniqueness, recall that the lognormal distribution has strictly decreasing hazard ratio (Thomas, 1971), and therefore,     Z ∞ −(x)2 −(y)2 Φ(−x? ) f2 (x) = exp exp dy = 0 ? 2 2 Φ (x ) x

(37)

is a strictly increasing function. In order to have multiple solutions there should be a x?? > x? such that

∂ f ∂x 1

(x?? , σ) ≥ 0 and f1 (x?? , σ) = 0. We will show that this is impossible. To

prove our result we make use of the following lemma. Lemma 2b If ∂ f ∂x 1

∂ f ∂x 1

(x? , σ) ≤ 0, then for any x?? such that x?? > x? , it is true that

(x?? , σ) < 0. 29

Proof By contradiction. Assume there is an x?? > x? such that

∂ f ∂x 1

(x?? , σ) ≥ 0. This

implies,    1 ∂ −(x?? )2 ?? ?? f1 (x , σ) = (1 − θ) √ [exp θσx ] exp − θσ [exp θσx?? ] Φ(−(x?? )) ≥ 0 ∂x 2 2π and,   ?? 2  √ ∂ (x ) ?? × f1 (x?? , σ) = (1 − θ) − θσf2 (x?? ) ≥ 0 2π [exp (−θσx )] exp 2 ∂x However,

∂ f ∂x 1

(38)

(x? , σ) ≤ 0 implies (1 − θ)−θσf2 (x? ) ≤ 0. Combining this with the preceding

expression we get f2 (x? ) ≥ f2 (x?? ).

This contradicts the lemma of Thomas (1971) as

presented in equation (37). Lemma 2a and Lemma 2b prove the existence and uniqueness of x? , the solution to (36) and that

∂ f ∂x 1

(x? , σ) < 0.

Appendix A.3 (Proof of Proposition 1)  The policy problem is to choose τt , τtL to maximize consumption subject to the equilibrium conditions of the model. In what follows we suppress the ε?t index where possible. First note that ε?t > 0 is given and the labor demand equation is not a constraint faced by the government. The policymaker chooses the following allocations and prices: {Ct , nt , lt , Lt , wt , rt }. We then use the default threshold wt =

1 rt

(1−θ)/θ

(ε?t )θ [∆ (0)]1−θ lt

to eliminate wages and note

that Lt = nt lt . The policy problem reduces to, o n 1/θ−1 ? max u (Ct , Lt ) + λ1,t Lt nt ∆ (0) − Ct − nt [fe + fm G (εt )] Ct ,Lt ,nt ,rt   "Z #" #1−θ Z ε?t  θ   ∞ ε 1 ∆ (0) ? dG (ε) + +λ2,t (ε?t )θ dG (ε) − − f n G (ε ) m t t 1/θ   ε?t rt nt ε?t 0 It is immediate that λ2,t = 0 by the choice of rt .

After eliminating λ1,t the first-order

conditions imply, 1/θ−1

−uL (t) /uC (t) = nt

∆(0) and

[fe n + fm G (ε?t )] nt = 30

1−θ 1/θ−1 L t nt ∆(0) θ

(39)

We start with the labor-income tax. Consider the labor leisure equation which is given
by 1 − τtL wt = −uL (t) /uC (t). Wages are determined by the default threshold, which h iθ 1/θ−1 ∆(0) wt 1 r and where ρt = nt . This we re-express in terms of prices as, ρt = [∆(0)] ∆(ε? ) θ t t

condition determines equilibrium in the labor market as;  θ
−uL (t) rt ∆(0) 1/θ−1 L nt 1 − τt = θuC (t) ∆(0) ∆ (ε?t )

(40)

Using the first equation in (39) we arrive at τtL = 1 − (rt /θ) [∆(0)/∆ (ε?t )]θ which is the second equation in Proposition 3. To determine the optimal dividend-income tax we start with the free entry condition, which is, τt = 1 − fe /πt , where profit is equal to, (1−θ)/θ

πt = [1 − G (ε?t )] [∆ (ε?t ) /∆(0)]θ (1 − θ) nt

[∆(0)] lt

1/θ

and Yt = nt

[∆(0)] lt

(41)

Using the second equation in (39) we arrive at the expression for τt reported in the text. Appendix A.4 (Proof of Lemma 3) We wish to show that if Φ(x? ) < 1/2, then x? increases with volatility and D (ε? ) = R∞ θ R∞ ε dG (ε) / 0 εθ dG (ε) falls with volatility. Using the definitions introduced above, ε? D (x? , σ) = 1 − Φ(x? − θσ), and, dD (x? , σ) ∂D (x? , σ) ∂D (x? , σ) dx? = + dσ ∂σ ∂x? dσ    ? 1 −(x − θσ)2 1 −(x? − θσ)2 dx? = θ √ exp − √ exp 2 2 dσ 2π 2π    ? 2 ? 1 −(x − θσ) dx = √ exp θ− 2 dσ 2π In order to determine the change in D (x? , σ) when volatility increases, we require, dx? /dσ < θ. We use definition of x? , f1 (x? , σ) = 0. From the implicit function theorem,   ∂f1 (·) dx? ∂f1 (·) ∂f1 (·) − −θ = +θ ∂x dσ ∂σ ∂x where, 
 f1 (x? , σ) = θ exp (σθ)2 /2 Φ(−(x? − θσ))) − [exp (θσx? )] Φ(−(x? )) 31

which leads to,  
∂f1 (·) 1 −(x? − θσ)2 exp −(σθ) /2 = (1 − θ) √ exp − θ2 σΦ(−(x? − θσ)) ∂x 2 2π 2

and,  
∂f1 (·) θ2 −(x? − θσ)2 2 ? ? exp −(σθ) /2 × = θ (σθ − x ) Φ(−(x − θσ)) + √ exp ∂σ 2 2π 2

(42)

This leads to,    
∂f1 (·) ∂1 f (·) 1 −(x? − θσ)2 exp −(σθ) /2 +θ = √ exp − x? Φ(−(x? − θσ)) ∂σ ∂x 2 2π 2

This expression is positive if x? < 0, which is equivalent to the probability of default being less than one half. As such, D (ε? ) falls with volatility under the same conditions as Proposition   ? ∂f1 (·) ∂f1 (·) 1 (·) > 0 when x? < 0. Therefore dx = / − > 0, 2. Note that (42) implies ∂f∂σ dσ ∂σ ∂x and x? , and the probability of default Φ(x? ) = G (ε? ) increases with volatility.

32

Appendix B In this Appendix we detail the Ramsey policy problem and derive the first-order conditions for optimal policy in the dynamic model. Appendix B.1 (Derivation of the Present Value Constraint) Recall that the household budget constraint - given by equation (22) in the text.

We

first multiply this constraint by the marginal utility of consumption, uC (t), impose the equilibrium condition xt−1 = 1, and integrate forward. We then use the labour supply     and the dynamic Euler equations, uC (t) = βEt uC (t + 1)rtj ; uC (t) = βEt uC (t + 1)rtd and   vt uC (t) = β(1 − δ)Et z(ε?t+1 )uC (t + 1) . Finally, we use dynamic equation for product creation, nt = (1 − δ) (nt−1 + ne,t−1 ), to write the present value constraint as, E0

∞ X

β t [uC (t)Ct + uL (t)Lt ] = A

(43)

t=0

 d  where A ≡ uC (0) r−1 d−1 + b0 + n0 z(ε?0 ) is assumed exogenous. Appendix B.2 (Definition of the Ramsey Problem) Following Chugh and Ghironi (2015), the Ramsey policy maker picks τtL and commits to pick P d t in period t. The problem can be written as one of maximizing E0 ∞ τt+1 t=0 β u(Ct , Lt ), subject to all the conditions presented in Table 1, the present value constraint, given by equation (43), and a constraint that ε?t > 0. Plans are made over {ne,t , nt+1 , lt , Ct , Lt , π (ε?t )}∞ t=0 , prices  ∞  ∞ d wt , rt , rt−1 , tax rates, τt+1 , τtL t=0 and the default threshold, {ε?t }∞ By choosing t=0 . t=0 tax rates, however, the constraints on the labor-leisure and the Euler equation for shares (i.e., product creation) do not bind. Similarly, by picking wages and interest rates directly, the constraints on firm pricing, the zero profit condition for financial intermediaries, and the Euler equation for deposits do not bind.

This allows us to re-write the reduced Ramsey

policy problem as in the text where ε?t > 0. 33

Appendix B.3 (Optimality Conditions for the Ramsey Problem) Consider the Ramsey problem defined in the text. max E0

∞ X

β t u(Ct , Lt ) + β t λ1,t {pt [∆(0)] lt nt − Ct − fe ne,t − G − fm nt G (ε?t )}

t=0 t

+β λ2,t [Lt − nt (lt + fo )] + β t λ3,t [(1 − δ) (nt + ne,t ) − nt+1 ] ) ( Z ∞ εθ dG (ε) (lt + fo ) − (ε?t )θ × lt [1 − G (ε?t )] +β t λ4,t θ ε?t

( +ξ

A0 − E0

∞ X

) β t [uC (t)Ct + uL (t)Lt ]

t=0

Differentiating with respect to ε?t , we find, n o 0 = λ4 at [(1 − θ)lt − θfo ] (ε?t )θ g (ε?t ) − θlt (ε?t )θ−1 [1 − G (ε?t )] − λ1,t [fm nt g (ε?t )]

(44)

where λ1,t the lagrange multiplier associated with the resources (output) equation.31 It is clear from this expression that λ4,t > 0.

The remaining first-order conditions (obtained

from simply differentiating with respect to {ne,t , nt+1 , lt , Ct , Lt }) are, λ1,t fe = (1 − δ) λ3,t

(45)

      Yt+1 Lt+1 β Et λ1,t+1 − βEt λ2,t+1 = λ3,t − β (1 − δ) Et λ3,t+1 (46) θ nt+1 nt+1   Z ∞ fo Yt (ε?t )θ dG(ε) 0 = λ1,t Et − λ2,t Et nt + λ4,t (47) 2 Et ? lt (lt + fo ) εt       uCC (t)Ct uLL (t)Lt λ1,t = uC (t) 1 + ξ 1 + ; − λ2,t = uL (t) 1 + ξ 1 + (48) uC (t) uL (t)  h ? iθ  (1−θ)/θ ε where pt = nt . Without aggregate uncertainty, we have ... + λ4,t θ − ∆(εt ? ) in t

equation (47).

31

Note that, without uncertainty, we can show the final term in this expression is equal to h ? iθ ε λ4,t lt Et ∆(εt ? ) θ (∆ε? − 1), where ∆ε? ≡ ε? [∆0 (ε? ) /∆ (ε? )] < 1. t

34

References ARELLANO, C., BAI, Y. and KEHOE, P. (2012), “Financial Markets and Fluctuations in Uncertainty”, working paper, University of Minnesota. ARSENEAU, D. and CHUGH, S. (2010), “Tax Smoothing in Frictional Labor Markets”, Journal of Political Economy 120, 926-985. BACHMANN, R. and BAYER, C. (2014), “Investment Dispersion and the Business Cycle, American Economic Review 104, 1392-1416. BARSEGHYAN, L. and DICECIO, R. (2011), “Entry Costs, Industry Structure, and CrossCountry Income and TFP Differences”, Journal of Economic Theory 146, 1828-1851. BARTELSMAN, E., HALTIWANGER, L. and SCARPETTA, S. (2013), “Cross-Country Differences in Productivity: The Role of Allocation and Selection”, American Economic Review 103, 305-34. BERNAKE, B., GERTLER, M. and GILCHRIST, S. (1999), “The Financial Accelerator in a Quantitative Business Cycle Framework”, in John B. Taylor and Michael Woodford, eds., Handbook of Macroeconomics, Vol. 1, Part C, Elsevier. BERNARD, A. EATON, J., JENSEN, B. and KORTUM, S. (2003), “Plants and Productivity in International Trade”, American Economic Review 93, 1268-1290 BERNARD, A., REDDING, S. and SCHOTT, P. (2011), “Multiproduct Firms and Trade Liberalization”, Quarterly Journal of Economics 126, 1271-1318. BILBIIE, F., GHIRONI, F. and FUJIWARA, I. (2014), “Optimal Monetary Policy with Endogenous Entry and Product Variety”, Journal of Monetary Economics 64, 1-20. BILBIIE, F., GHIRONI, F. and MELITZ, M. (2008), “Distortions and Remedies”, NBER working paper, 14383.

35

BILBIIE, F., GHIRONI, F. and MELITZ, M. (2012), “Endogenous Entry, Product Variety, and Business Cycles”, Journal of Political Economy 120, 304-345. BLOOM, N. (2014), “Fluctuations in Uncertainty”, Journal of Economic Perspectives 28, 153-176. CALDARA, D., FUENTES-ALBERO, C., GILCHRIST, S. and ZAKRAJSEK, E. (2016), The Macroeconomic Impact of Financial and Uncertainty Shocks, European Economic Review, forthcoming. CALSTROM, C. and FUERST, T. (1997), “Agency Costs, Net Worth, and Business Fluctuations: A Computable General Equilibrium Analysis”, American Economic Review 87, p. 893-910. CHARI, V., CHRISTIANO, L. and KEHOE, P. (1994), “Optimal Fiscal Policy in a Business Cycle Model”, Journal of Political Economy 102, 617-652. CHATTERJEE, S. and COOPER, R. (1993), “Entry and Exit, Product Variety and the Business Cycle”, NBER Working Papers 4562. CHRISTIANO, L., MOTTO, R. and ROSTANGNO, M. (2009), “Risk Shocks”, American Economic Review 104, 27-65. CHUGH, S. (2016), “Firm Risk and Leverage-Based Business Cycles”, Review of Economic Dynamics 20, 111-131. CHUGH, S. and GHIRONI, F. (2015), “Optimal Fiscal Policy with Endogenous Product Variety”, working paper, Boston College. CLEMENTI, G. and PALAZZO, D. (2016), “Entry, Exit, Firm Dynamics, and Aggregate Fluctuations”, American Economic Journal: Macroeconomics, forthcoming. COLCIAGO, A. (2016), “Endogenous Market Structures and Optimal Taxation”, forthcoming, Economic Journal. 36

COMIN, D. and MULANI, S. (2006), “Diverging Trends in Macro and Micro Volatility”, Review of Economics and Statistics 88, 374-383. COTO-MARTINEZ, J., GARRIGA, C. and SANCHEZ-LOSADA, F. (2007), “Optimal taxation with imperfect competition and aggregate returns to specialization”, Journal of the European Economic Association 5, 1269-1299. CROCE, M., NGUYEN, T. and SCHMID, L. (2013), “Fiscal Policy and the Distribution of Consumption Risk”, working paper, UNC at Chapel Hill. DECKER, R., D’ERASMO, P. and MOSCOSO BOEDO, H. (2015), “Market Exposure and Endogenous Firm Volatility over the Business Cycle”, American Economic Journal: Macroeconomics, forthcoming. DEVEREUX, M., HEAD, A. and LAPHAM, B. (1996), “Aggregate Fluctuations with Increasing Returns to Specialization and Scale”, Journal of Economic Dynamics and Control 20, 627-656. DJANKOV, S., LA PORTA, R., LOPES DE SILANES, F. and SHLEIFER, A. (2002), “The Regulation of Entry”, Quarterly Journal of Economics 117, 1-37. GIESECKE, K., LONGSTAFF, F., SCHAEFER, S. and STREBULAEV, I. (2011), “Corporate Bond Default Risk: A 150-Year Perspective”, Journal of Financial Economics 102, 233-250. GILCHRIST, S. and ZAKRAJSEK, E. (2012), “Credit spreads and business cycle fluctuations”, American Economic Review 102, 1692-1720. GILCHRIST, S., SIM, J. and ZAKRAJSEK, E. (2014), “Uncertainty, Financial Frictions and Investment Dynamics”, working paper, Boston University. GOMES, J., YARON, A. and ZHANG, L. (2003), “Asset Prices and Business Cycles with Costly External Finance”, Review of Economic Dynamics 6, 767-788. 37

GOURIO, F. and MIAO, J. (2010), “Transitional Dynamics of Dividend Tax and Capital Gains Tax Cuts”, Review of Economic Dynamics 14, 368-383. GOURIO, F., MESSER, T. and SIEMER, M. (2016), “Firm Entry and Macroeconomic Dynamics: A State-Level Analysis”, American Economic Review: Papers & Proceedings 106,214-218. DE GRAEVE, F. (2008), “The External Finance Premium and the Macroeconomy: US post-WWII Evidence”, Journal of Economic Dynamics and Control 32, 3415–3440. GROSSMAN, G. and HELPMAN, E. (1991), “Innovation and Growth in the Global Economy”, MIT Press, Cambridge, MA. JERMANN, U. and QUADRINI, V. (2012), “Macroeconomic Effects of Financial Shocks”, American Economic Review, 102, 238-71. HARTMAN, R. (1972), “The Effects of Price and Cost Uncertainty on Investment”, Journal of Economic Theory 5, 258-266. HEATHCOTE, J., STORESLETTEN, K. and VIOLANTE, G. (2010), “The Macroeconomic Implications of Rising Wage Inequality in the United States”, Journal of Political Economy 118, 681-722. MUKOYAMA, T. and LEE, Y. (2005), “Entry and Exit of Manufacturing Plants over the Business Cycle”, European Economic Review 77, 20-27. LEWIS, V. and POILLY, C., (2012), “Firm Entry, Markups and the Monetary Transmission Mechanism”, Journal of Monetary Economics 59, 670-685. LEWIS, V. and WINKLER, R. (2015), “Product Diversity, Demand Structures and Optimal Taxation”, Economic Inquiry 53, 979-1003. LJUNGQVIST, L. and UHLIG, H. (2000), “Tax Policy and Aggregate Demand Management under Catching Up with the Joneses”, American Economic Review 90, 356-366. 38

MIDRIGAN, V. (2011), “Menu Costs, Multi-Product Firms and Aggregate Fluctuations”, Econometrica 79, 1139-1180. NEUMEYER, P. and PERRI, F. (2006), “Business Cycles in Emerging Economies: The Role of Interest Rates”, Journal of Monetary Economics 52, 345-380. OU, S., CHUI, D. and METZ, A. (2011), “Corporate Defaults and Recovery Rates 1920– 2009”, Technical Report, Moody’s Investors Service (February). POSCHKE, M. (2010), “The Regulation of Entry and Aggregate Productivity”, Economic Journal 120, 1175-1200. DA RIN, M., DI GIACOMO, M. and SEMBENELLI, A. (2011), “Entrepreneurship, Firm Entry, and the Taxation of Corporate Income: Evidence from Europe, Journal of Public Economics 95, 1048-1066. RESTUCCIA, D. and ROGERSON, R. (2008), “Policy Distortions and Aggregate Productivity with Heterogeneous Establishments”, Review of Economic Dynamics 11, 707-720. SANDMO, A. (1971), “On the Theory of the Competitive Firm under Price Uncertainty”, American Economic Review 16, 65-73. SCHAAL, E. (2015), “Uncertainty and Unemployment”, working paper, New York University. SCHMITT-GROHE, S. and URIBE, M. (2005), “Optimal Fiscal and Monetary Policy in a Medium Scale Macroecomic Model”, NBER Macroeconomics Annual.

39

Table 1: Model Summary

Description

Equation

Labor Market Clearing and Production

Lt = nt (lt + fo ) and Yt = ∆ (0) nt lt

Resource Constraint

Labor Supply

Yt − G = Ct + fe ne,t + fm nt G (ε?t )  i R∞h lt dG(ε) = 0 Et ε? θεθt − (ε?t )θ lt +f o t
wt 1 − τtL = − uuCL (t) (t)

Net Worth

zt = (1 − τt ) D (ε?t ) πt + vt

Expected Profit

πt = ∆(0)nt

Mass of Firms

nt+1 = (1 − δ) (nt + ne,t ) i1−θ h 1/θ lt − wt rt (lt + fo ) = 0 (ε?t )θ × ∆(0)nt h i d wt (lt + fo ) rt−1 + fm G (ε?t ) = 1 − G (ε?t ) + (εη?t)θ (lt + fo ) wt rt t i h uC (t+1) fe = β(1 − δ)Et uC (t) zt+1 h h i i uC (t+1) d s and 1 = βE 1 = βEt uCuC(t+1) r rt+1 t t (t) uC (t)

Labor Demand

Default Threshold Financial Intermediaries Euler Equation (equity) and Entry Euler Equation (deposits and bonds)

1/θ

(1−θ)/θ

40

[(1 − θ) lt − θfo ]

Table 2: Exogenous Parameters and Calibration

Parameters Set Exogenously Statistic

Parameter

Value

Target/Source

Firm exit rate

δ

0.1

10%

Markup

θ

0.74

Bernard et al. (2003)

Discount factor

β

0.98

2% risk-free rate

0.72

Heathcote et al. (2010)

1

Normalization

Frisch elasticity

υ

Sunk cost

fe


1−L L

Calibrated Parameters Statistic

Parameter

Value

Target

Source

Volatility (long-run)

σ

0.135

-

Comin and Mulani (2006)

Interest rate spread

fm

1.314

204 b.p.

Gilchrist and Zakrajˇsek (2012)

Default rate

fo

0.135

1%

Giesecke et al. (2011)

Hours worked

χ

1.341

20%

-

41

Figure 1: Impulse Responses to a Volatility Shock with Given Fiscal Policy

Output

0

Consumption

0

-0.05

Firm Entry

1 0

-0.1

-0.1

-1 -0.2

-0.15 -0.2

-2

-0.3 0

5

10

15

Employment

0

-3 0

-0.1

-0.1

-0.2

10

15

Operating Firms

0

-0.05

5

0

5

10

15

Aggregate Profit

0 -0.1 -0.2

-0.15

-0.3

-0.3 0

5

10

15

Profit Wedge

0.1

-0.4 0

-0.1

-0.2

10

15

Default Rate (p.p.)

0.3

0

5

0.2

40

0.1

20

0

0

5

10

15

5

10

15

Spread (b.p.)

60

-0.1 0

0

-20 0

5

42

10

15

0

5

10

15

Figure 2: Optimal Tax Responses to a Volatility Shock

Dividend tax, τ 1.5

Dividend tax, τ L

t

t

0

1

Consumption

0.2 0.1

-0.2

0.5

0 -0.4

0 -0.5

-0.1

-0.6 0

5

10

15

Firm Entry

1

-0.2 0

-1

-2

10

15

Employment

0.2

0

5

0.15

0

0.1

-0.1

0.05

-0.2

5

10

15

Aggregate Profit

0.06

0.02

-0.1

5

10

15

Profit Wedge

0.1

0

10

15

-0.3 0

0.04

5

Operating Firms

0.1

0 0

0

0

5

10

15

Default Rate (p.p.)

0.3 0.2 0.1

0

0

-0.2 0

5

10

15

-0.1 0

5

43

10

15

0

5

10

15