Taxation and the quality of entrepreneurship

Abstract

We study the effect of taxation on entrepreneurship, investigating how taxes affect both the number of start-ups and their average quality. We show theoretically that even with risk neutral agents and no tax evasion progressive taxes can increase entrepreneurial entry, while reducing average firm quality. So called “success taxes” encourage start-ups with lower value business ideas by reducing the option value of pursuing better projects. This suggests that the most common measure used in the literature, the likelihood of entry into self-employment, may underestimate the adverse effect of taxation.

Appendix

1.1 Existence and uniqueness of equilibrium

Given that both the RHS and the LHS of Eq. (6) are continuous functions of $\theta ^{*}$, sufficient conditions for existence of an equilibrium are:

$$\begin{aligned} w&> \frac{\theta _{l}(1+\beta p)-\beta E(\theta )}{1-\beta (1-p)} \\ w&< \theta ^{h} \end{aligned}$$

These conditions describe a relationship between the market wage and the distribution of the quality of entrepreneurial projects. The second condition simply says that the best possible entrepreneurial idea must yield a return higher than the market wage. The first condition says that the wage rate should not be inferior to a quantity that depends on the distribution of entrepreneurial projects. It is positively related to the lowest entrepreneurial project and negatively related to the average entrepreneurial project. Intuitively if the lowest entrepreneurial project increases, then market wage should generally be higher. However if the lowest bound on the distribution increases the average value of entrepreneurial ideas also increases, counteracting the first effect. Generally the effect depends on the particular form of the distribution. To get a sense of how strict this requirement is consider the following. Since the thetas are defined over a positive interval we can normalize without loss of generality $\theta _{l}=0.$ In this case the first condition simply says that the wage has to be greater than some negative number. Since the wage is non-negative by definition this is not a strict requirement.

Nothing in the structure of the problem ensures a unique solution to Eq. (6). Since the LHS of the equation is decreasing in $\theta ^{*},$ a sufficient condition for uniqueness is:

$$\begin{aligned} F(\theta ^{*})\left[ 1+\frac{\theta ^{*}f(\theta ^{*})}{F(\theta ^{*})}\right] \le \frac{1+\beta p}{\beta } \end{aligned}$$

This condition clearly imposes some restrictions on the shape of the distribution function of thetas that depend on the value of the parameters $\beta$ and $p$.

In our problem multiplicity of equilibria arises from the fact that agents can follow several internally consistent decision rules. Having a decision rule over decision rules will eliminate the multiplicity problem. One such decision rule is the following:

$$\begin{aligned} \theta ^{*}=\max \{\theta \in [\theta _{l},\theta _{h}]:V^{s}(\theta )=V^{e}(\theta )\} \ \end{aligned}$$

This decision rule generates the highest utility (discounted profit). Suppose there is more than one solution to Eq. (6), so that different decision rules are available to the agent. Call these solutions $\theta _{1}^{*},\theta _{2}^{*}\ \ldots \theta _{n}^{*}$ and let’s assume, without loss of generality, that $\theta _{1}^{*}<\theta _{2}^{*}<\cdots <\theta _{n}^{*}$.

Notice that $V^{e}(\theta ^{*})$ is an increasing function of $\theta ^{*}$; it follows that $V^{e}(\theta _{1}^{*})<V^{e}(\theta _{2}^{*})<\cdots <V^{e}(\theta _{n}^{*}).$ Moreover since $\theta _{i}^{*}$ for $i=1,\ldots ,n$ are defined as those points that equate the value of searching and the value of entrepreneurship, it has to be the case that $V^{S}(\theta _{1}^{*})<V^{S}(\theta _{2}^{*})<\cdots <V^{S}(\theta _{n}^{*}).$

Let’s define $V^{e}(\theta |\theta _{i}^{*})$ and $V^{s}(\theta |\theta _{i}^{*})$ as the value functions for the entrepreneur and the worker when the agent chooses the $\theta _{i}^{*}$ as his decision rule. Then $V_{i}(\theta _{i}^{*})=\max \{V^{e}(\theta |\theta _{i}^{*}),V^{s}(\theta |\theta _{i}^{*})\}$ is the value function for the agent when using decision rule $\theta _{i}^{*}$.

Equation (4) implies that if $\theta _{i}^{*}>\theta _{j}^{*}$ then $V^{s}(\theta |\theta _{i}^{*})>V^{s}(\theta |\theta _{j}^{*});$ hence from Eq. (3) we see that $V^{e}(\theta |\theta _{i}^{*})>V^{e}(\theta |\theta _{j}^{*})$ for every $\theta .$ It follows that for every $\theta ,$ $V_{i}(\theta _{i}^{*})\ge V_{j}(\theta _{j}^{*})$ when $\theta _{i}^{*}>\theta _{j}^{*}$. We conclude that the agent will choose the decision rule that will give him at least as much utility as the others. Notice however that different $\theta ^{*}$ imply that the agent will choose to become an entrepreneur with different probabilities each period. In particular, the higher is $\theta ^{*}$ the longer the wait to become an entrepreneur (and enjoy the higher income guaranteed by it). It is possible that an impatient agent will choose a lower $\theta ^{*}$ in order to enjoy the income guaranteed by entrepreneurship sooner. We do not address this issue.

1.2 Entry costs

With entry costs we can re-write (2) as:

$$\begin{aligned} V^{e}(\theta )=\theta -E+\beta \left[ pV^{s}(\theta )+\left( 1-p\right) V^{e}(\theta )\right] \end{aligned}$$

(15)

where $E$ represents the entry costs. Following the same logic used before we obtain that $\theta ^{*}$ is the solution to the following equation:

$$\begin{aligned} w+\gamma \int _{\theta ^{*}}^{\theta _{h}}(\theta -E)dF(\theta )=\frac{\gamma }{\beta }(\theta ^{*}-E)\left[ 1+\beta p-\beta F(\theta ^{*}) \right] \end{aligned}$$

with $\gamma =\frac{\beta }{1-\beta (1-p)}.$ Note that the derivative of $\theta ^{*}$ with respect to $E$ is:

$$\begin{aligned} \frac{d\theta ^{*}}{dE}=1-\frac{\beta (1-F(\theta ^{*}))}{1+\beta p-\beta F(\theta ^{*})} \end{aligned}$$

and with simple algebra one can prove that $\frac{d\theta ^{*}}{dE}>0,$ for all $\theta ^{*}$.

The inclusion of entry costs in the model increases the “reservation entrepreneurial idea.” The inclusion of entry costs in the model affects the rest of the formulas in this article in a similar fashion, changing the level at which in any of the previous cases it is optimal to start a firm. However the qualitative conclusion of the paper does not change: the effect of an increase in the top tax rate on the quality or quantity of entrepreneurship depends on the shape of the tax schedule, i.e. the relative position of $w$, $\theta ^{*}$ and $\hat{\theta }$.

1.3 Proofs of propositions

1.3.1 Proof of Proposition 2

Proof

We can write (9) using the known result:

$$\begin{aligned} E(x)=\int _{a}^{b}xdF(x)=\int _{a}^{b}(1-F(x))dx-\left[ (1-F(b))b-(1-F(a))a \right] \end{aligned}$$

which in our case implies:

$$\begin{aligned} \int _{\theta _{\tau }^{*}}^{\theta _{h}}\theta dF(\theta )=\theta _{h}-\int _{\theta _{\tau }^{*}}^{\theta _{h}}F(\theta )d\theta -F(\theta _{\tau }^{*})\theta _{\tau }^{*} \end{aligned}$$

Using this result, writing $\theta _{\tau }^{*}=\theta _{\tau }^{*}(\tau _{w})$ and differentiating (9) with respect to $\tau _{w}$ we get

$$\begin{aligned} \frac{d\theta _{\tau }^{*}}{d\tau _{w}}=\frac{-w(1-\beta (1-p))}{\left( 1-\tau _{\pi }\right) (1+\beta p-\beta F(\theta _{\tau }^{*}))}<0 \end{aligned}$$

Analogously we can show that:

$$\begin{aligned} \frac{d\theta _{\tau }^{*}}{d\tau _{\pi }}&= \frac{-\beta \int _{\theta _{\tau }^{*}}^{\theta _{h}}\theta dF(\theta )+\theta _{\tau }^{*} \left[ 1+\beta p-\beta F(\theta _{\tau }^{*})\right] }{\left( 1-\tau _{\pi }\right) \left[ 1+\beta p-\beta F(\theta _{\tau }^{*})\right] }= \,\,\, \hbox {using } (9) \\&= \frac{\left[ 1-\beta (1-p)\right] \left( 1-\tau _{w}\right) w}{\left( 1-\tau _{\pi }\right) ^{2}\left[ 1+\beta p-\beta F(\theta _{\tau }^{*}) \right] }>0 \end{aligned}$$

$\square$

1.3.2 Proof of Proposition 3

Proof

As before, using the following expressions for the expected values

$$\begin{aligned} \int _{\theta _{\tau }^{*}}^{\theta _{h}}\theta dF(\theta )&= \theta _{h}-\int _{\theta _{\tau }^{*}}^{\theta _{h}}F(\theta )d\theta -F(\theta _{\tau }^{*})\theta _{\tau }^{*} \\ \int _{\hat{\theta }}^{\theta _{h}}\theta dF(\theta )&= \theta _{h}-\int _{ \hat{\theta }}^{\theta _{h}}F(\theta )d\theta -F(\hat{\theta })\hat{\theta } \end{aligned}$$

writing $\theta _{\tau }^{*}=\theta _{\tau }^{*}(\tau )$ and differentiating (14) with respect to $\tau$ we get:

$$\begin{aligned} \frac{d\theta _{\tau }^{*}}{d\tau }=\frac{\beta \left[ 1-F(\hat{\theta }) \right] }{\left[ 1+\beta p-\beta F(\theta _{\tau }^{*})\right] }\left[ \hat{\theta }-E(\theta |\theta >\hat{\theta })\right] \le 0 \end{aligned}$$

$\square$

1.3.3 Analysis when the marginal entrepreneur’s income is in the top bracket

Case 1: $\theta ^{*}>\hat{\theta }>w$

In this case the equilibrium is defined by:

$$\begin{aligned} w+\gamma \int _{\theta _{\tau }^{*}}^{\theta _{h}}\theta dF(\theta )-\tau \gamma \int _{\theta _{\tau }^{*}}^{\theta _{h}}(\theta -\hat{\theta } )dF(\theta )=\frac{\theta _{\tau }^{*}-\tau (\theta _{\tau }^{*}- \hat{\theta })}{1-\beta (1-p)}\left[ 1+\beta p-\beta F(\theta _{\tau }^{*})\right] \nonumber \\ \end{aligned}$$

(16)

The effect of a top marginal tax rate increase in this case is summarized by the following proposition.

Proposition 4

With a progressive tax schedule when the marginal entrepreneur earns more than the top marginal tax rate bracket and more than the average worker, the effect of an increase in the top marginal tax rate on entrepreneurial activity is uncertain.

Proof

As before we can calculate the total derivative of (16) with respect to the tax rate

$$\begin{aligned} \frac{d\theta _{\tau }^{*}}{d\tau }&= \frac{\beta \left[ 1-F(\theta _{\tau }^{*})\right] }{\left[ 1+\beta p-\beta F(\theta _{\tau }^{*}) \right] \left( 1-\tau \right) }\\&\times \left[ \hat{\theta }-E(\theta |\theta >\theta _{\tau }^{*})+\frac{\left[ 1+\beta p-\beta F(\theta _{\tau }^{*})\right] }{\beta \left[ 1-F(\theta _{\tau }^{*})\right] }(\hat{\theta }-\theta _{\tau }^{*}) \right] \end{aligned}$$

the sign of this derivative is not certain and depends on the relative position of $\hat{\theta }$ and $\theta _{\tau }^{*}$. We cannot determine the relative positions of $\theta _{\tau }^{*}$ and $\theta ^{*}$ either. $\square$

Case 2: $\theta ^{*}>w>\hat{\theta }$

We consider the case in which labor income is taxed. As in the previous section we consider the case in which the kink in the entrepreneurial value function is below the value of searching. The situation when it is above the equilibrium is exactly as in Case 1.

The value function for the entrepreneurs is still the one described in (13) while the value function for the worker becomes

$$\begin{aligned}&V_{\tau }^{s}(\theta )=w-\tau (w-\hat{w})+\beta V_{\tau }^{s}(\theta )F(\theta _{\tau }^{*})\nonumber \\&\quad +\gamma pV^{s}(\theta )+\gamma \int _{\theta _{\tau }^{*}}^{\theta _{h}}\theta dF(\theta )-\tau \gamma \int _{\theta _{\tau }^{*}}^{\theta _{h}}(\theta -\hat{\theta })dF(\theta ) \end{aligned}$$

(17)

$\theta _{\tau }^{*}$ is the solution to

$$\begin{aligned}&w-\tau (w-\hat{w})+\gamma \int _{\theta _{\tau }^{*}}^{\theta _{h}}\theta dF(\theta )-\tau \gamma \int _{\theta _{\tau }^{*}}^{\theta _{h}}(\theta - \hat{\theta })dF(\theta )\nonumber \\&\quad =\frac{\theta _{\tau }^{*}-\tau (\theta _{\tau }^{*}-\hat{\theta })}{1-\beta (1-p)}\left[ 1+\beta p-\beta F(\theta _{\tau }^{*})\right] \end{aligned}$$

(18)

Again the effect of an increase in the top marginal tax rate is summarized by the following proposition.

Proposition 5

With a progressive tax schedule when the marginal entrepreneur earns more than the top marginal tax rate bracket but less than the average worker, the effect of an increase in the top marginal tax rate on entrepreneurial activity is uncertain.

Proof

Once again we can calculate the derivative of $\theta _{\tau }^{*}$ with respect to $\tau$ from (18):

$$\begin{aligned} \frac{d\theta _{\tau }^{*}}{d\tau }&= \frac{\beta \left[ 1-F(\theta _{\tau }^{*})\right] }{\left[ 1+\beta p-\beta F(\theta _{\tau }^{*}) \right] \left( 1-\tau \right) } \\&\times \left[ \hat{\theta }-E(\theta |\theta >\theta _{\tau }^{*})+\frac{\left[ 1-\beta F(\theta _{\tau }^{*})\right] }{\beta \left[ 1-F(\theta _{\tau }^{*})\right] }(\hat{\theta }-\theta _{\tau }^{*})-\frac{(w-\hat{w})}{\gamma \left[ 1-F(\theta _{\tau }^{*})\right] }\right] \end{aligned}$$

the sign of this derivative is not certain. $\square$