Welfare effects of business taxation under default risk

Abstract

In this article, we use a stochastic model with a representative firm to study business tax policy under default risk. We will show that, for a given tax rate, tax revenue and welfare are crucially affected by default risk and its costs, as long as interest expenses are deductible. Thus, an evaluation made without accounting for default may be dramatically biased. Moreover, we show that the “debt bias” due to the tax treatment of debt finance causes a quite relevant deadweight loss.

Appendices

Derivations of (4) and (6)

1.1 The value of debt

Applying Itô’s Lemma to (3) gives:

$$\begin{aligned} rD(\Pi )=L+\frac{\sigma ^{2}}{2}\Pi ^{2}D_{\Pi \Pi }(\Pi ), \end{aligned}$$

(14)

where $L=\left( 1-\alpha \right) \left( 1-\tau \right) \Pi ,C,$ and $D_{\Pi \Pi }(\Pi )\equiv \frac{\partial ^{2}D(\Pi )}{\partial \Pi ^{2}}.$ The general closed-form solution of function (14) is therefore equal to:

$$\begin{aligned} D\left( \Pi \right) =\left\{ \begin{array}{ll} \frac{\left( 1-\alpha \right) \left( 1-\tau \right) \overline{\Pi }}{r}+\mathop {\sum }\nolimits _{i=1}^{2}B_{i}\Pi ^{\beta _{i}} &{} \text {after default,}\\ \frac{C}{r}+\mathop {\sum }\nolimits _{i=1}^{2}D_{i}\Pi ^{\beta _{i}} &{} \text {before default,} \end{array}\right. \end{aligned}$$

(15)

where $\beta _{1}=\frac{1}{2}+\sqrt{\left( \frac{1}{2}\right) ^{2}+\frac{2r}{\sigma ^{2}}}>1,$ and $\beta _{2}=\frac{1}{2}-\sqrt{\left( \frac{1}{2}\right) ^{2}+\frac{2r}{\sigma ^{2}}}<0$ are the two roots of the characteristic equation $\Psi (\beta )\equiv \frac{1}{2}\sigma ^{2}\beta (\beta -1)-r=0.$ To calculate $B_{i}$ and $D_{i}$ for $i=1,2$, we need three boundary conditions. Firstly, we assume that, whenever $\Pi$ goes to zero, the lender’s claim is nil, namely condition $D\left( 0\right) =0$ holds: this implies that $B_{2}=0.$ Secondly, we assume that financial bubbles do not exist: this means that $B_{1}=D_{1}=0.$ Thirdly, we must consider that at point $\Pi =\overline{\Pi }$, the predefault value of debt must be equal to the post-default one, net of the default cost. Using the two branches of (15), we thus obtain:

$$\begin{aligned} \left( 1-\alpha \right) \frac{\left( 1-\tau \right) \overline{\Pi }}{r}=\frac{C}{r}+D_{2}\overline{\Pi }^{^{\beta _{2}}}. \end{aligned}$$

Rearranging gives $D_{2}=\left[ \frac{\left( 1-\alpha \right) \left( 1-\tau \right) \overline{\Pi }-C}{r}\right] \overline{\Pi }^{^{-\beta _{2}}}.$ Hence, the value of debt is (4).

1.2 The value of equity

Using (5) and Itô’s Lemma, we obtain the following non-arbitrage condition:

$$\begin{aligned} rE\left( \Pi \right) =\left( 1-\tau \right) \left( \Pi -C\right) +\frac{\sigma ^{2}}{2}\Pi ^{2}E_{\Pi \Pi }\left( \Pi \right) , \end{aligned}$$

(16)

before default since, after default, the general-form solution of (16) is:

$$\begin{aligned} E\left( \Pi \right) =\left\{ \begin{array}{ll} 0 &{} \text {after default,}\\ \left( 1-\tau \right) \left( \frac{\Pi -C}{r}\right) +\mathop {\sum }\nolimits _{i=1}^{2}A_{i}\Pi ^{\beta _{i}} &{} \text {before default.} \end{array}\right. \end{aligned}$$

(17)

In the absence of financial bubbles, $A_{1}$ is nil. To calculate $A_{2}$, we recall that default occurs when $\Pi =\overline{\Pi }.$ In this case, the value of equity falls to zero, namely:

$$\begin{aligned} E\left( \overline{\Pi }\right) =0. \end{aligned}$$

(18)

Substituting (17) into (18), and solving for $A_{2}$ gives $A_{2}=-\frac{\left( 1-\tau \right) \left( \overline{\Pi }-C\right) }{r}\overline{\Pi }^{^{-\beta _{2}}}.$ Hence, the predefault value of equity is equal to:

$$\begin{aligned} E\left( \Pi \right) =\frac{\left( 1-\tau \right) \left( \Pi -C\right) }{r}-\frac{\left( 1-\tau \right) \left( \overline{\Pi }-C\right) }{r}\left( \frac{\Pi }{\overline{\Pi }}\right) ^{\beta _{2}} \end{aligned}$$

(19)

and zero otherwise. These results give (6).

Proof of Lemma 1

Assume ab absurdo that a second default can take place because, after the first default, the firm survives and borrows. In this case, the value of equity will be:

$$\begin{aligned}&E^{AD}\left( \overline{\Pi }\right) \nonumber \\&\quad =\left\{ \begin{array}{ll} 0 &{} \text {after the second default}\\ \left( 1-\alpha \right) \left[ \frac{\left( 1-\tau \right) \left( \Pi -C\right) }{r}-\frac{\left( 1-\tau \right) \left( \Pi ^{AD}-C\right) }{r}\left( \frac{\overline{\Pi }}{\overline{\Pi }^{AD}}\right) ^{\beta _{2}}\right] &{} \text {before the second default} \end{array}\right. \end{aligned}$$

(20)

where $\Pi ^{AD}$ is the current value of EBIT after the first default. As shown in (20), before default the equity value is given by $(1-\alpha )$ times the summation between the perpetual rent $\frac{\left( 1-\tau \right) \left( \Pi ^{AD}-C\right) }{r}$ and the loss contingent on the event of default, -$\frac{\left( 1-\tau \right) \left( \Pi ^{AD}-C\right) }{r}\left( \frac{\overline{\Pi }}{\overline{\Pi }^{AD}}\right) ^{\beta _{2}}$. Given (20), we can now calculate the default threshold point under debt financing. Following Leland (1994) and Goldstein et al. (2001), shareholders are assumed to solve the following problem:

$$\begin{aligned} \max _{\overline{\Pi }^{AD}}E^{AD}\left( \overline{\Pi }^{AD}\right) . \end{aligned}$$

(21)

Using (20) and rearranging the F.O.C. of (21) give the optimal threshold value of EBIT, after the first default:

$$\begin{aligned} \overline{\Pi }^{AD}=\frac{\beta _{2}}{\beta _{2}-1}C<C. \end{aligned}$$

(22)

As can be seen, the equality $\overline{\Pi }=\overline{\Pi }^{AD}$ holds. Hence, we have:

$$\begin{aligned}&E^{AD}\left( \overline{\Pi }^{AD}\right) \\&\quad =\left\{ \begin{array}{ll} 0 &{} \text {after the second default}\\ \left( 1-\alpha \right) \left[ \frac{\left( 1-\tau \right) \left( \overline{\Pi }-C\right) }{r}-\frac{\left( 1-\tau \right) \left( \Pi ^{AD}-C\right) }{r}\left( \frac{\overline{\Pi }}{\overline{\Pi }^{AD}}\right) ^{\beta _{2}}\right] &{} \text {before the second default} \end{array}\right. \end{aligned}$$

Rearranging therefore gives:

$$\begin{aligned} E^{AD}\left( \overline{\Pi }^{AD}\right) =\left\{ \begin{array}{ll} 0 &{} \text {after default}\\ 0 &{} \text {before default} \end{array}\right. \end{aligned}$$

(23)

Like the first one, the second default causes the value of equity to be null. Moreover, the two default events coincide, because $\overline{\Pi }=\overline{\Pi }^{AD}$.

Let us next turn to debt. Since the (sunk) default cost is equal to a percentage $\alpha$ of the defaulted firm, the lender will own $(1-\alpha )$ of the defaulted firm. Using dynamic programming, we can therefore write debt as follows:

$$\begin{aligned}&D^{AD}\left( \Pi ^{AD}\right) \nonumber \\&\quad =\left\{ \begin{array}{ll} \left( 1-\alpha \right) \left( 1-\tau \right) \overline{\Pi }^{AD}{\rm{d}}t+e^{-r{\rm{d}}t}{\mathbb{E}}\left[ D(\Pi +{\rm{d}}\Pi )\right] &{} \text {after the second default,}\\ C{\rm{d}}t+e^{-r{\rm{d}}t}{\mathbb{E}}\left[ D(\Pi +{\rm{d}}\Pi )\right] &{} \text {before the second default,} \end{array}\right. \end{aligned}$$

(24)

where ${\mathbb{E}}$ is the expected value operator. Given $\left( \frac{\overline{\Pi }}{\overline{\Pi }^{AD}}\right) ^{\beta _{2}}=1,$ rearranging (24) gives:

$$\begin{aligned}&D^{AD}\left( \Pi ^{AD}\right) \nonumber \\&\quad = \left\{ \begin{array}{ll} \frac{\left( 1-\alpha \right) \left( 1-\tau \right) \overline{\Pi }}{r} &{} \text {after the second default,}\\ \frac{C}{r}+\left[ \frac{\left( 1-\alpha \right) \left( 1-\tau \right) \overline{\Pi }-C}{r}\right] \left( \frac{\overline{\Pi }}{\overline{\Pi }^{AD}}\right) ^{\beta _{2}}= &{} \frac{\left( 1-\alpha \right) \left( 1-\tau \right) \overline{\Pi }}{r}\text { before the second default,} \end{array}\right. \end{aligned}$$

(25)

where $\beta _{2}=\frac{1}{2}-\sqrt{\left( \frac{1}{2}\right) ^{2}+\frac{2r}{\sigma ^{2}}}<0.$ As shown in (25), before default the debt value consists of two terms. The first one, $\frac{C}{r}$, is a perpetual rent which measures the debt value without default, while the second term accounts for the default effects. In particular, the term $\left( \frac{\overline{\Pi }}{\overline{\Pi }^{AD}}\right) ^{\beta _{2}}$, which measures the present value of 1 Euro contingent on the default event, is equal to 1. After default, the lender becomes shareholder and the value of his/her claim is equal to $\frac{\left( 1-\alpha \right) \left( 1-\tau \right) \overline{\Pi }}{r}$. We can now calculate the firm’s value and find the optimal coupon $C^{AD}.$ Using (25) and (23) gives the value of the firm after the first default and before the second one:

$$\begin{aligned} V^{AD}\left( \Pi ^{AD}\right)= & {} \frac{\left( 1-\alpha \right) \left( 1-\tau \right) \Pi ^{AD}}{r}. \end{aligned}$$

(26)

To find the optimal coupon, we maximize $V^{AD}\left( \Pi ^{AD}\right)$ with respect to C. Differentiating $V^{AD}\left( \Pi ^{AD}\right)$ with respect to C thus gives:

$$\begin{aligned} \frac{\partial V^{AD}\left( \Pi ^{AD}\right) }{\partial C}=\frac{\partial ^{2}V^{AD}\left( \Pi ^{AD}\right) }{\partial C^{2}}=0, \end{aligned}$$

and hence

$$\begin{aligned} C^{AD}=0. \end{aligned}$$

Since the optimal coupon (after the first default) is zero, the second default never occurs and the defaulted firm is fully equity financed. This contradicts the starting assumption. Q.E.D.