## 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.

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For instance, Kocherlakota (2010) argues that bailouts are inevitable if the default of firms causes systemic failure. For this reason, he proposes a Pigouvian tax, aimed at offsetting negative externalities arising from financial instability.

For simplicity, we assume symmetric information and full interest deductibility. For a detailed analysis on business taxation under asymmetric information, see Cohen et al. (2016) and the articles cited therein. Partial interest deductibility does not change the quality of our results.

Assuming that the capital structure changes over time, due to the variability of market conditions, would make our model complex (see, e.g., Strebulaev 2007; De Marzo and Sannikov 2007; Bolton et al. 2019) and would lead to a quite difficult interpretation of tax effects. For this reason, we leave this point for further research.

McGowan and Andrews (2018) provide a comprehensive analysis of insolvency procedures across OECD countries and find that they are quite heterogeneous. In particular, they state that “[a] comparison of the 2010 and 2016 values suggests that recent reform efforts have been largest for prevention and streamlining, with reforms observable in 11 countries, especially European countries (e.g., Portugal). This may reflect the fact that such measures have been recently endorsed by the European Commission and the IMF, in response to the crisis [...]. Barriers to restructuring have also declined in 10 countries, while reform activity affecting the personal costs to failed entrepreneurs has been less ambitious, with only Chile, Greece and Spain undertaking reforms since 2010.” Of course, heterogeneous rules can lead to heterogeneous default costs.

Here, we deal with one policy-maker who can implement both monetary and fiscal policies, although we are aware that separate entities deal with them. As pointed out by Sinn (2018) however, the separation of roles sometimes vanishes and a central bank can affect fiscal policy. The reverse may also be true when bank taxes are levied (De Mooij and Keen 2016).

The general form of the geometric Brownian motion is \({\rm{d}}\Pi =\mu \Pi {\rm{d}}t+\sigma \Pi {\rm{d}}z\) where \(\mu\) is the expected rate of growth. With no loss of generality, here we set \(\mu =0.\)

As pointed out by Estrin et al. (2017), economic agents are sensitive to different elements of the default codes. Moreover, the authors show that some countries are more debt-friendly than others. All of these features are here summarized by our parameter cost \(\alpha .\) If therefore countries are debtor-friendly (-unfriendly), \(\alpha\) is expected to be lower (higher). For a detailed (and economic) analysis of default procedures, see also Claesens et al. (2001).

This model, characterized by a one-off default, has a widespread use since Goldstein et al. (2001). Of course, in a dynamic model where the stochastic process of EBIT may change over time, and the optimal coupon is both state- and time-dependent, a good company might borrow and have some probability of default. We leave this topic for future research.

This maximization implies that there not conflicts of interest between shareholders and lenders.

Taking the log of (10) and differentiating it with respect to \(\tau\) give:

$$\begin{aligned} \frac{\partial \log C}{\partial \tau }=-\frac{1}{\beta _{2}}\left[ \frac{1}{\tau }-\frac{1-\alpha \frac{\beta _{2}}{\beta _{2}-1}}{\left( 1-\tau \right) \alpha \frac{\beta _{2}}{\beta _{2}-1}+\tau }\right] =-\frac{1}{\beta _{2}}\frac{\alpha \frac{\beta _{2}}{\beta _{2}-1}}{\tau \left( \left( 1-\tau \right) \alpha \frac{\beta _{2}}{\beta _{2}-1}+\tau \right) }>0. \end{aligned}$$Of course, this result holds under symmetric information. If, otherwise, information was asymmetric and there were agency costs of monitoring corporate managers, debt finance could have beneficial effects. Indeed, before lending, financial institutions would implement a due diligence on the firm which could considerably reduce equityholders’ monitoring costs. In this case, welfare effects might be different. See, e.g., Cohen et al. (2016).

The numerator can be rewritten as \(\zeta h\left( \tau \right) ^{\xi }\), where \(\zeta \equiv \alpha \frac{\Pi }{r}>0\), \(\xi \equiv \frac{\beta _{2}-1}{\beta _{2}}>0\) and:

$$\begin{aligned} h\left( \tau \right) =\tau \left[ \left( 1-\beta _{2}\right) \left[ \left( 1-\tau \right) \alpha \frac{\beta _{2}}{\beta _{2}-1}+\tau \right] \right] ^{-1}. \end{aligned}$$Since \(\frac{\partial }{\partial \tau }h\left( \tau \right) =-\alpha \beta _{2}\left[ \tau \left( \alpha \beta _{2}-\beta _{2}+1\right) -\alpha \beta _{2}\right] ^{-2}>0\), \(\zeta >0\), and \(\xi >0\), the welfare cost is increasing in \(\tau\).

We have also run a robustness check with \(r=5\%.\) Of course, the quality of results does not change. Results are available upon request.

In a tax-free context, the welfare value is equal to 100.

For instance, time-consuming default procedures are expected to increase \(\alpha\) and vice versa.

Notice that leverage ratios implicit in the following simulations are in line with those collected by Aswath Damodaran. These data are available at: http://pages.stern.nyu.edu/~adamodar/.

Figures about equity and debt value (as a function of \(\tau\)) are available upon request.

The welfare loss is defined as the difference between the tax-free welfare function, i.e., 100, and its effective value with \(\tau >0\).

This relation is made explicit in Eq. (12).

We have also run some robustness checks with different values of \(\Pi\) and

*r*(which are available upon request). The quality of results does not change.As pointed out, our results have been obtained with a fairly simple model. Of course, the use of a more general framework, where, e.g., investment decisions, credit constraints, agency costs and a dynamic capital structure are considered, is left for further research.

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## Acknowledgements

We wish to thank the participants at the 6th MaTax conference, held in Mannheim (DE) on September 12–13, 2019, for their useful comments. We are responsible for any error.

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## Appendices

### Derivations of (4) and (6)

### 1.1 The value of debt

Applying Itô’s Lemma to (3) gives:

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:

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:

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:

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

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:

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:

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:

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:

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

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

Rearranging therefore gives:

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:

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

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:

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:

and hence

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.

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Comincioli, N., Panteghini, P.M. & Vergalli, S. Welfare effects of business taxation under default risk.
*Int Tax Public Finance* **28**, 1412–1429 (2021). https://doi.org/10.1007/s10797-020-09650-1

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DOI: https://doi.org/10.1007/s10797-020-09650-1