The effects of income taxation on entrepreneurial investment: A puzzle?

Abstract

We investigate how personal income taxes affect the portfolio share of personal wealth that entrepreneurs invest in their own business. In a portfolio choice model that allows for tax sheltering, we show that lower tax rates may increase investment in entrepreneurial equity at the intensive margin, but decrease it at the extensive margin. Using German panel data, we identify tax effects on the portfolio shares of six asset classes by exploiting tax and entry regulation reforms. Our results indicate that lower taxes drive out businesses that are viable only due to tax sheltering, but increase investment in productive entrepreneurial businesses.

Appendices

Appendix 1: Necessary and sufficient conditions

Necessary and sufficient conditions under which we predict opposite signs of the effects of a tax change at the extensive and intensive margins are, for the Slemrod (2001) model:

There exists a critical value $k_{C}>0$ such that at the given tax rate:

$$\begin{aligned}&{\bar{U}}(k_{C})\equiv E[u(W_{0}+(1-t)(rW_{0}+({\tilde{e}}-r)k_{C})+t\gamma (k_{C}{\tilde{e}},t)-a({\tilde{c}},k_{C}{\tilde{e}}))]= u((1+(1-t)r)W_{0}) \end{aligned}$$

(15)

$$\begin{aligned}&\frac{\partial {\bar{U}}}{\partial k}\bigg |_{k=k_{C}}>0. \end{aligned}$$

(16)

where ${\tilde{c}}^{*}=\gamma (k_{C}{\tilde{e}},t),$ and the expectation is taken with respect to the distribution of ${\tilde{e}}$.

In words, there exists a positive k-value $(k_{C})$ at which expected utility is equal to that at $k=0$ and is strictly increasing at that point. Intuitively, the tax gain from tax sheltering for all positive realizations of ${\tilde{e}}$ must be sufficient to compensate for the negative net returns in some states.

If condition (15) is satisfied, using the certainty equivalent of the left hand side, the entrepreneur will have a risk premium $\rho _{C}>0$ such that

$$\begin{aligned} E[(1-t)(rW_{0}+({\tilde{e}}-r)k_{C})+t\gamma (k_{C}{\tilde{e}},t)-a({\tilde{c}},k_{C}{\tilde{e}})]=(1-t)rW_{0}+\rho _{C} \end{aligned}$$

(17)

implying

$$\begin{aligned} (1-t)E({\tilde{e}}-r)+\frac{tE[{\tilde{c}}-a({\tilde{c}},k_{C}{\tilde{e}})]}{k_{C}}= \frac{\rho _{C}}{k_{C}} \end{aligned}$$

(18)

with $E[{\tilde{c}}-a({\tilde{c}},k_{C}{\tilde{e}})]>0.$ This tells us that this case is more likely to arise the higher the tax rate, the greater the expected value of sheltered income net of transactions costs, the less risk averse the entrepreneur, and the smaller the absolute value of the (negative) expected net return.

Note that if $t=1,$ in this model, as long as the net return from sheltering any business income is positive, we must have $k>0,$ since then

$$\begin{aligned} E[u(W_{0}+{\tilde{c}}-a({\tilde{c}},k{\tilde{e}}))]>u(W_{0}). \end{aligned}$$

(19)

Therefore, by continuity of ${\tilde{y}}_{T}$ in t,  there must exist an interval of t-values sufficiently close to 1 for which condition (15) is satisfied. On the other hand, at $t=0$ these conditions cannot be satisfied, and again by continuity there will be an interval of t-values at which the corner solution is optimal. How large these respective intervals are is determined by the parameters of the model.

Appendix 2: Derivation of the estimation equations

1.1 Selection correction

Equation (11) describes portfolio shares at the intensive margin, where Eqs. (12) and (13) are the equations of selection into ownership of a particular asset. To avoid clutter, we suppress the asset class indices in this subsection and assume that the individual fixed effects have already been eliminated by partialling out from the linear selection and share equations. $X_{{{it}}}$ and $Z_{{{it}}}$ are row vectors which conform to the column vectors of unknown coefficients $\beta$ and $\gamma$, respectively. The X’s and Z’s are assumed to be exogenous in this appendix to focus on selection.

Assume that the expected value of the error of the intensive regression is zero, $E(u_{{{it}}})=0$, and its variance is $E(u_{{{it}}}u_{jt})=\sigma ^2_u {\text { for }} i=j$, and zero otherwise. The expected value of the selection threshold is equal to $E(\nu _{{{it}}})=\mu _{\nu }$, its variance is $E[(\nu _{{{it}}}-\mu _{\nu })(\nu _{jt}-\mu _{\nu })]=\sigma ^2_{\nu } {\text { for }} i=j$, and zero otherwise. The covariance between the error of the intensive regression and the selection threshold is $Cov(u_{{{it}}},\nu _{jt})=E(u_{{{it}}}\nu _{{{it}}})-E(u_{{{it}}})E(\nu _{{{it}}})=\rho \sigma _{\nu } \sigma _u {\text { for }} i=j$, and zero otherwise. Assume the expected value of the error of the intensive regression conditional on the value of the selection threshold is $E(u_{{{it}}}|\nu _{{{it}}})=\rho (\nu _{{{it}}}-\mu _{\nu })\sigma _u/\sigma _{\nu }.$

By assuming the conditional expectation of $u_{{{it}}}$ given $\nu _{{{it}}}$ is linear in $\nu _{{{it}}}$ we can use the decomposition

$$\begin{aligned} u_{{{it}}}=\rho (\nu _{{{it}}}-\mu _{\nu })\sigma _u/\sigma _{\nu }+\varepsilon _{{{it}}}, \end{aligned}$$

where $\varepsilon _{{{it}}}$ and $\nu _{{{it}}}$ are uncorrelated. Substituting this in $y_{{{it}}}=X_{{{it}}}\beta +u_{{{it}}}$ gives

$$\begin{aligned} y_{{{it}}}=X_{{{it}}}\beta +\rho (\nu _{{{it}}}-\mu _{\nu })\sigma _u/\sigma _{\nu }+\mu _{i}+\varepsilon _{{{it}}}. \end{aligned}$$

Then, the conditional mean is

$$\begin{aligned} E(y_{{{it}}}|X_{{{it}}}, \nu _{{{it}}}<Z_{{{it}}}\gamma )=X_{{{it}}}\beta +\rho \sigma _u E(\nu _{{{it}}}|\nu _{{{it}}}<Z_{{{it}}}\gamma )/\sigma _{\nu }-\rho \sigma _u\mu _{\nu }/\sigma _{\nu }. \end{aligned}$$

If $\nu _{{{it}}}$ is a standard normally distributed random variable with mean $\mu _{\nu }=0$ and variance $\sigma ^2_{\nu }=1$, then it follows (Heckman 1979) that

$$\begin{aligned} E(\nu _{{{it}}}|\nu _{{{it}}}<Z_{{{it}}}\gamma ) = -\frac{\phi (Z_{{{it}}}\gamma )}{\varPhi (Z_{{{it}}}\gamma )} \text { (Inverse Mill's Ratio)} \end{aligned}$$

and the estimation equation is:

$$\begin{aligned} E(y_{{{it}}}|X_{{{it}}},\nu _{{{it}}}<Z_{{{it}}}\gamma )=X_{{{it}}}\beta -\underbrace{\rho \sigma _u}_{\delta } \frac{ \phi (Z_{{{it}}}\gamma )}{\varPhi (Z_{{{it}}}\gamma )}, \end{aligned}$$

where $\delta$ and $\beta$ are the parameters to be estimated.

Following Olsen (1980) instead, if $\nu _{{{it}}}$ is uniformly distributed over the interval [0, 1], then $E(\nu _{{{it}}})=\mu _{\nu }=\frac{1}{2}$ and $V(\nu _{{{it}}})=\frac{1}{12}$, so $\sigma _{\nu }=\frac{1}{2\sqrt{3} }.$ Using the equation for the conditional mean as above with these values gives

$$\begin{aligned} E(y_{{{it}}}|X_{{{it}}},\nu _{{{it}}}<Z_{{{it}}}\gamma )&= X_{{{it}}}\beta +\rho \sigma _u E(\nu _{{{it}}}|\nu _{{{it}}}<Z_{{{it}}}\gamma )/\sigma _{\nu }-\rho \sigma _u\mu _{\nu }/\sigma _{\nu } \\&= X_{{{it}}}\beta +2\sqrt{3}\rho \sigma _u E(\nu _{{{it}}}|\nu _{{{it}}}<Z_{{{it}}}\gamma )- \sqrt{3}\rho \sigma _u. \end{aligned}$$

Using $E(\nu _{{{it}}}|\nu _{{{it}}}<Z_{{{it}}}\gamma )= Z_{{{it}}}\gamma E(\nu _{{{it}}}) = Z_{{{it}}}\gamma /2$ we can write

$$\begin{aligned} E(y_{{{it}}}|X_{{{it}}},\nu _{{{it}}}<Z_{{{it}}}\gamma )=X_{{{it}}}\beta +\sqrt{3}\rho \sigma _u (Z_{{{it}}}\gamma )-\sqrt{3}\rho \sigma _u \end{aligned}$$

$$\begin{aligned} =X_{{{it}}}\beta +\sqrt{3}\rho \sigma _u(Z_{{{it}}}\gamma -1). \end{aligned}$$

From this follows

$$\begin{aligned} E(y_{{{it}}}|X_{{{it}}},\nu _{{{it}}}<Z_{{{it}}}\gamma )=X_{{{it}}}\beta +\underbrace{\sqrt{3}\rho \sigma _u} _{\delta } (Z_{{{it}}}\gamma -1). \end{aligned}$$

1.2 System estimation

Based on the assumption of a normally distributed error term in the selection equation, Shonkwiler and Yen (1999) show that the conditional mean of $y_{{{mit}}}$ for individual i in equation $m=1,\ldots ,M$ is

$$\begin{aligned} E(y_{{{mit}}}|X_{{{it}}},\nu _{{{mit}}}<Z_{{{it}}}\gamma _{m})=X_{{{it}}}\beta _{m}+\delta _m\frac{ \phi (Z_{{{it}}}\gamma _{m})}{\varPhi (Z_{{{it}}}\gamma _{m})}. \end{aligned}$$

Because $E(y_{{{mit}}}|X_{{{it}}},\nu _{{{mit}}}\ge Z_{{{it}}}\gamma _{m})=0$, the unconditional mean of $y_{{{mit}}}$ for the mth equation, which can be estimated based on the full sample, is

$$\begin{aligned} E(y_{{{mit}}}|X_{{{it}}})=\varPhi (Z_{{{it}}}\gamma _{m})X_{{{it}}} \beta _{m}+\delta _m\phi (Z_{{{it}}}\gamma _{m}). \end{aligned}$$

In our case, we have analogously for the uniform distribution

$$\begin{aligned} E(y_{{{mit}}}|X_{{{it}}},\nu _{{{mit}}}<Z_{{{it}}}\gamma _{m})=X_{{{it}}} \beta _{m}+\delta _m(Z_{{{it}}}\gamma _{m}-1) \end{aligned}$$

(20)

and

$$\begin{aligned} E(y_{{{mit}}}|X_{{{it}}})=(Z_{{{it}}}\gamma _{m}) X_{{{it}}} \beta _{m}+\delta _m((Z_{{{it}}}\gamma _{m})^2-Z_{{{it}}}\gamma _{m}). \end{aligned}$$

(21)

1.3 Marginal effects

Under the assumptions listed above, the marginal effects for a variable $x_{itk}$ that is an element of both $Z_{{{it}}}$ and $X_{{{it}}}$ conditional on selection are

$$\begin{aligned} \frac{\partial E(y_{{{mit}}}|X_{{{it}}},\nu _{{{mit}}}<Z_{{{it}}}\gamma _{m})}{\partial x_{itk}} =\beta _{mk}+\delta _m\gamma _{mk}, \end{aligned}$$

and the unconditional marginal effects are

$$\begin{aligned} \frac{\partial E(y_{{{mit}}}|X_{{{it}}})}{\partial x_{itk}}=\gamma _{mk} (X_{{{it}}}\beta _{m})+(Z_{{{it}}}\gamma _{m}) \beta _{mk}+2\delta _m\gamma _{mk} (Z_{{{it}}}\gamma _{m}) - \delta _m\gamma _{mk}. \end{aligned}$$

Appendix 3: Supplementary tables

See Tables 6, 7, 8 and 9.

Table 6 Portfolio shares of asset classes (with cluster robust standard errors).

Table 7 Portfolio shares of asset classes (equation-by-equation 2SLS estimation).

Table 8 Ownership probabilities of asset classes with base year income splines.

Table 9 Portfolio shares of asset classes with base year income splines.