Gifts to government

Abstract

Gifts to government might provide warm glow to some citizens, especially if they can be earmarked toward specific government activities. We develop a model in which such gifts may be privately worthwhile, even for those people who evade taxes, and describe the conditions under which this will be the case. The latter can occur when the warm glow of transferring money to the government via gifts is higher than the warm glow from transferring the money via paying taxes, and additionally, the marginal rate of substitution of warm glow from “donating” to the public good for private good is sufficiently high. We then conduct empirical analyses of explicit gifts to the US federal government over the last century. Although small compared to either federal taxes and expenditures or donations to charitable organizations, we show that they are systematically, although fairly weakly, related to measures of government fiscal activity. The war years and their immediate aftermath dominate the systematic relationships we uncover. This suggests that these gifts are not simply the random, and randomly timed, behavior of an unrepresentative sample of Americans and that this behavior might warrant further empirical analysis.

Appendices

Appendix

A The calculation of the budget set

To determine the maximal budget set for x and n, we first reorganize the constraints. From the last constraint in (1), it follows that $g=\frac{n-bt(y-E)}{a-bDt}$. Then, the constraint $g\ge 0$ implies $E\ge y-\frac{n}{bt}$. Substituting the expression for g into the first constraint in (1) gives

$$\begin{aligned} x+\frac{1-Dt}{a-bDt}n=y(1-t\frac{a-b}{a-bDt})+t\frac{a-b}{a-bDt}E-c(E). \end{aligned}$$

(8)

Substituting the expression for g into the constraint $Dg+E\le y$ gives $E\le y-D\frac{n}{a}$. Thus, the maximum budget line can be defined as

$$\begin{aligned} x+\frac{1-Dt}{a-bDt}n=\underset{E\in {\mathcal{B}}(y,n)}{\max}\left\{ y(1-t\frac{a-b}{a-bDt})+t\frac{a-b}{a-bDt}E-c(E)\right\} , \end{aligned}$$

(9)

where ${\mathcal{B}}(y,n)=\{E\,|\,max\{0,y-\frac{n}{bt}\}\le E\le y-D\frac{n}{a}\}$.

1.1 A.1 If $\varvec{a\le b}$

If $a\le b$, then the constrained optimal evasion, $E^{c}=\underset{E\in {\mathcal{B}}(y,n)}{{\rm argmax}}\left\{ -t\frac{b-a}{a-bDt}E-c(E)\right\} =max\{0,y-\frac{n}{bt}\}$, is determined as the left boundary solution in this case. Substituting into (9) gives the expression for the budget line

$$\begin{aligned} x={\left\{ \begin{array}{ll} y-\frac{n}{b}-c(y-\frac{n}{bt}) &{} {\rm if}\,\,0\le n\le bty\\ y(1-t\frac{a-b}{a-bDt})-\frac{1-Dt}{a-bDt}n &{} {\rm if}\,\,bty<n\le \frac{ay}{D}. \end{array}\right. } \end{aligned}$$

(10)

Note that, here and later, we allow for brevity sake expressions that include division on D for $D\in [0,1]$. To proper understand this notation, please treat it as equal to infinity when $D=0$. For example, expression $\frac{ay}{D}$ for $D\in [0,1]$ is equal to $\frac{ay}{D}$ when $D\in (0,1]$, and is equal to infinity when $D=0$.

Also note that the value of $\frac{ay}{D}$ could be very large when the deductibility, D, is close to zero, so that the corresponding value of x will be negative. To exclude this situation, we denote the maximum n that can be achieved if $x=0$ and $E=0$ by ${\hat{n}}$, which is equal to $y\left( bt+\frac{1-t}{1-Dt}(a-bDt)\right)$.

Hence, the expression for the budget line is

$$\begin{aligned} x={\left\{ \begin{array}{ll} y-\frac{n}{b}-c(y-\frac{n}{bt}) &{} {\rm if}\,\,0\le n\le bty\\ y(1-t\frac{a-b}{a-bDt})-\frac{1-Dt}{a-bDt}n &{} {\rm if}\,\,bty<n\le {\hat{n}}. \end{array}\right. } \end{aligned}$$

(11)

1.2 A.2 If $\varvec{a>b}$

If $a>b$, then the optimal evasion level depends on the marginal cost of evasion. Define the unconstrained optimal level of evasion as

$$\begin{aligned} {\hat{E}}=\underset{E}{{\rm argmax}}\left\{ t\frac{a-b}{a-bDt}E-c(E)\right\} =(c')^{-1}(t\frac{a-b}{a-bDt}). \end{aligned}$$

(12)

To find the constrained optimal evasion level $E^{c}=\underset{E\in B(n,y)}{{\rm argmax}}\left\{ t\frac{a-b}{a-bDt}E-c(E)\right\} ,$ define the following threshold levels of n as ${\underline{n}}=bt(y-{\hat{E}})$ and ${\overline{n}}=\frac{a}{D}(y-{\hat{E}})$. It will be assumed that the cost of evasion is high enough so that ${\hat{E}}<y$. Then, the constrained optimal evasion level is given by

$$\begin{aligned} E^{c}={\left\{ \begin{array}{ll} y-\frac{n}{bt}&\begin{array}{cc} {\rm if}\, &{} \,0\le n<{\underline{n}}\end{array}\\ {\hat{E}} &{} \begin{array}{cc} {\rm if}\, &{} \,{\underline{n}}\le n\le {\overline{n}}\end{array}\\ y-D\frac{n}{a} &{} \begin{array}{cc} {\rm if}\, &{} \,{\overline{n}}<n\le \frac{ay}{D}.\end{array} \end{array}\right. } \end{aligned}$$

(13)

The intuition for equation (13) is simple. If $y-\frac{n}{bt}\le {\hat{E}}\le y-D\frac{n}{a},$ then the constrained optimal evasion equals the unconstrained optimal evasion, or else boundary solutions are obtained.

Note that the threshold ${\overline{n}}=\frac{a}{D}(y-{\hat{E}})$ could be very large when the deductibility, D, is close to zero, so that the corresponding value of x will be negative. To exclude this situation, we define the value $n_{H}$ such that in (8) $x=0$ at $E={\hat{E}}$, which is $n_{H}=\frac{(a-bDt)}{1-Dt}(y-c({\hat{E}}))-\frac{t(a-b)}{1-Dt}(y-{\hat{E}})$.

The budget set can be defined by substituting the solution for the constrained optimal evasion (13) into equation (9), which gives

$$\begin{aligned} \begin{array}{c} x={\left\{ \begin{array}{ll} y-\frac{n}{b}-c(y-\frac{n}{bt}) &{} {\rm if}\,\,0\le n<{\underline{n}}\\ y(1-t\frac{a-b}{a-bDt})-\frac{1-Dt}{a-bDt}n+t\frac{a-b}{a-bDt}{\hat{E}}-c({\hat{E}}) &{} {\rm if}\,\,{\underline{n}}\le n<min\{{\overline{n}},n_{H}\}\\ y-\frac{n}{a}-c(y-D\frac{n}{a}) &{} {\rm if}\,\,min\{{\overline{n}},n_{H}\}\le n\le n_{H}. \end{array}\right. }\end{array} \end{aligned}$$

(14)

In the case, when ${\overline{n}}>n_{H}$, the third region of the budget line does not exist. This happens if $D<{\overline{D}}$, where ${\overline{D}}$ is a solution of $y(1-D)-{\hat{E}}-Dc({\hat{E}})=0$ and ${\hat{E}}=(c')^{-1}(t\frac{a-b}{a-bDt})$.

B Proofs

Proof of Proposition 2

If $a<b$ or $a=b$ and $c^{'}(\cdot )>0$, then it can be seen that (4) and (5) cannot be equal to zero simultaneously. If (4) equals zero, then automatically (5) is negative, and vice versa. Therefore, the optimal gift and optimal evasion cannot be positive at the same time. $\square$

Proof of Proposition 3

As Proposition 2 states, the optimal gift and optimal evasion cannot be positive at the same time if $a\le b$.

If $a>b$, as can be seen from the budget set (7), two cases are possible: $g^{*}=0$ and $E^{*}>0,$ or $g^{*}>0$ and $E^{*}>0$. Note that $g^{*}>0$ and $E^{*}=0$ is impossible, because this would imply $x\le 0$. For the case $g^{*}>0$ and $E^{*}>0$ to be chosen in the optimum, it should be that the slope of the indifference curve at the delimiting point $g=0$ and $E={\hat{E}}=(c')^{-1}(t\frac{a-b}{a-bDt})$ should be steeper than the slope of the budget line, which is equal to $\frac{1-Dt}{a-bDt}$. Thus, in this case the condition for the optimal gift to be positive is $\left. \frac{U_{n}^{'}}{U_{x}^{'}}\right| _{g=0,E={\hat{E}}}>\frac{1-Dt}{a-bDt}$. $\square$

Proof of Proposition 4

We focus on the case of co-existence of a positive gift and evasion, the conditions for which are described in Proposition 3. When $a>b$ and the constraint $Dg+E\le y$ is non-binding, the optimal evasion is $E^{*}={\hat{E}}=(c')^{-1}(t\frac{a-b}{a-bDt})$ and the optimal gift is $g^{*}=\frac{n^{*}-bt(y-{\hat{E}})}{a-bDt}$.

The derivative of the optimal evasion with respect to the tax rate is

$$\begin{aligned} \frac{\mathrm{d}{\hat{E}}}{Dt}=\frac{a(a-b)}{c''(E)(a-bDt)^{2}}>0. \end{aligned}$$

(15)

Thus, optimal evasion increases with t.

The derivative of $n^{*}$ with respect to t can be expressed as

$$\begin{aligned} \frac{\mathrm{d}n^{*}}{\mathrm{d}t}=\frac{1}{S}\frac{(a-b)}{(a-bDt)^{2}}\left( DU_{x}^{'}-(a-bDt)(y-Dg-{\hat{E}})\left[ U_{xn}^{''}-pU_{xx}^{''}\right] \right) , \end{aligned}$$

(16)

where $p=\frac{1-Dt}{a-bDt}$ is the “price” of n in terms of x, where $S=-(a-bDt)^{2}\frac{\partial ^{2}L}{\partial g^{2}}=-p^{2}U_{xx}^{''}+2pU_{xn}^{''}-U_{nn}^{''}>0$ is positive, because the second derivative of the Lagrangian with respect to g is negative.

The first term in (16) represents a substitution effect, which is positive, because the relative price of n in terms of x decreases with t. The second term, $-(a-bDt)(y-Dg-{\hat{E}})\left[ U_{xn}^{''}-pU_{xx}^{''}\right]$, is the income effect and is negative, because an increase in t decreases total income. Thus, the sign of $\frac{\mathrm{d}n^{*}}{\mathrm{d}t}$ depends on the relative magnitude of the income and substitution effect. If substitution effect dominates, then $\frac{\mathrm{d}n^{*}}{\mathrm{d}t}$ is positive.

The derivative of the optimal gift with respect to the tax rate is

$$\begin{aligned} \frac{\mathrm{d}g^{*}}{\mathrm{d}t}=\frac{1}{a-bDt}\left[ \frac{\mathrm{d}n^{*}}{\mathrm{d}t}+bt\frac{\mathrm{d}{\hat{E}}}{\mathrm{d}t}+b(Dg^{*}+{\hat{E}}-y)\right] . \end{aligned}$$

(17)

The third term in brackets is negative, and the second term defined above in (15) is positive. The first term is defined by (16). Hence, there are three effects of an increase in the tax rate on the gift, and the total effect of the tax rate increase depends on the relative magnitude of these three effects. First, an increase in the tax rate changes the optimal choice of n, where the magnitude of this change depends on the relative size of the income and substitution effects. Second, an increase in the tax rate results in an increase in optimal evasion, which increases the optimal gift all else equal. Third, an increase in the tax rate mechanically increases the tax payment contribution toward warm glow, which crowds out gifts.

When $a>b$ and the constraint $Dg+E\le y$ is binding, the optimal gift is $g^{*}=\frac{n^{*}}{a}$ and the optimal evasion is $E^{*}=y-D\frac{n^{*}}{a}$. Neither the optimal gift nor the optimal evasion depend on the tax rate, because the budget constraint in this case becomes $x=y-\frac{n}{a}-c(y-D\frac{n}{a})$ and, hence, the optimal $n^{*}$ does not depend on the tax rate. $\square$

Proof of Proposition 6

Define ${\widetilde{a}}\equiv a(1-Dt)$. When ${\widetilde{a}}=b$, the optimal gift is positive only if the optimal evasion is zero, and the budget constraints in (1) can be rewritten as

$$\begin{aligned} \begin{array}{c} x^{*}=y(1-t)-{\widetilde{g}}^{*}\\ n^{*}=\frac{{\widetilde{a}}-bDt}{1-Dt}{\widetilde{g}}^{*}+bty \end{array},\;{\rm where}\;{\tilde{g}}=(1-Dt)g. \end{aligned}$$

(18)

From the first constraint, $\frac{d{\widetilde{g}}^{*}}{dD}=-\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$. Substituting the first constraint into the second one, it follows that $x^{*}+pn^{*}=\alpha y,$ where $p=\frac{1-Dt}{{\widetilde{a}}-bDt}$ and $\alpha =1-t+bt\frac{1-Dt}{{\widetilde{a}}-bDt}$. Now, using this constraint and the FOC $\frac{U_{n}^{'}}{U_{x}^{'}}=p$, the derivative of the optimal $x^{*}$ with respect to the deductibility, D, can be calculated and expressed as

$$\begin{aligned} \frac{\mathrm{d}x^{*}}{\mathrm{d}D}=\frac{1}{S}\frac{bt}{({\widetilde{a}}-bDt)^{2}}\left( pU_{x}^{'}-{\widetilde{g}}^{*}\left[ U_{xn}^{''}-\frac{1}{p}U_{nn}^{''}\right] \right) , \end{aligned}$$

where $S=-({\widetilde{a}}-bDt)^{2}\frac{\partial ^{2}L}{\partial g^{2}}=-p^{2}U_{xx}^{''}+2pU_{xn}^{''}-U_{nn}^{''}>0$ is positive.

The first term in this formula, $pU_{x}^{'},$ represents the substitution effect, and the second term, $-{\widetilde{g}}^{*}\left[ U_{xn}^{''}-\frac{1}{p}U_{nn}^{''}\right] ,$ represents the income effect. The substitution effect is positive in this case, while the income effect is negative (assuming $U_{xn}^{''}\ge 0$). Therefore, the sign of $\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$ depends on the relative magnitude of the income and substitution effects. If the substitution effect dominates, then $\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$ is positive.

Because $\frac{d{\widetilde{g}}^{*}}{\mathrm{d}D}=-\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$, the sign of $\frac{d{\widetilde{g}}^{*}}{\mathrm{d}D}$ might be positive or negative. The sign of $\frac{\mathrm{d}g^{*}}{\mathrm{d}D}=\frac{t{\widetilde{g}}}{(1-Dt)^{2}}+\frac{1}{1-Dt}\frac{d{\widetilde{g}}^{*}}{\mathrm{d}D}$ is also undetermined and could be positive or negative. $\square$

C The effect of deductibility

The effect of deductibility on gifts to government when $N=0$ (warm glow depends on the gross sacrifice) is described by the following proposition.

Proposition C.1

(a) If $a=b$, an increase in the deductibility of gifts to government increases the gross gift, g, but does not affect the net gift, ${\tilde{g}}=(1-Dt)g$.

(b) If $a>b$, an increase in the deductibility of gifts to government always increases the gross gift, g. It also increases the net gift, ${\tilde{g}}=(1-Dt)g$, if the substitution effect on x due to the decrease in the relative price of n is big enough, i.e., if

$$\begin{aligned} \frac{pU_{x}^{'}}{S}>\frac{g^{*}\left[ pU_{xn}^{''}-U_{nn}^{''}\right] }{S}-\frac{b^{2}t^{2}}{c^{''}({\hat{E}})}, \end{aligned}$$

(19)

where $p=\frac{1-Dt}{a-bDt}$ is the “price” of n it terms of x, $S=-(a-bDt)^{2}\frac{\partial ^{2}L}{\partial g^{2}}=-p^{2}U_{xx}^{''}+2pU_{xn}^{''}-U_{nn}^{''}>0$

(c) If $a<b$, an increase in the deductibility of gifts to government may increase or decrease the gross gift and may increase or decrease the net gift.

Proof of Proposition C.1

(a) When $a=b$, the budget constraints in (1) can be rewritten as

$$\begin{aligned} \begin{array}{c} x=y(1-t)-tE-c(E)-{\tilde{g}}\\ n=bt(y-E)+b{\tilde{g}} \end{array},\;where\;{\tilde{g}}=(1-Dt)g. \end{aligned}$$

(20)

As can be seen, both x and n depend only on the net gift ${\tilde{g}}$ and do not depend on the gross gift g. Thus, the deductibility does not enter directly into the FOCs, it enters only indirectly through the dependence of the net gift ${\tilde{g}}$ on D.

The only constraint that depends on deductibility is $Dg+E\le y$, which is not binding here and, therefore, does not affect the solution. Indeed, the FOCs in this case are

$$\begin{aligned}&\begin{array}{c} \frac{\mathrm{d}L}{\mathrm{d}g}=-(1-Dt)[U_{x}^{'}-bU_{n}^{'}]-\lambda D={\left\{ \begin{array}{ll} 0 &{} \begin{array}{cc} if &{} g>0\end{array}\\ <0 &{} \begin{array}{cc} if &{} g=0,\end{array} \end{array}\right. }\end{array} \end{aligned}$$

(21)

$$\begin{aligned}&\begin{array}{c} \frac{\mathrm{d}L}{\mathrm{d}E}=t[U_{x}^{'}-bU_{n}^{'}]-U_{x}^{'}c'(E)-\lambda ={\left\{ \begin{array}{ll} 0 &{} \begin{array}{cc} if &{} E>0\end{array}\\ <0 &{} \begin{array}{cc} if &{} E=0.\end{array} \end{array}\right. }\end{array} \end{aligned}$$

(22)

Equations (21) and (22) cannot be equal to zero simultaneously. Therefore, the optimal gift and evasion cannot be positive at the same time, as shown in Fig. 1.

We are interested in the case when the gift is positive, so that the optimal evasion is equal to zero, $E^{*}=0$. Holding $E=0$ and expressing ${\tilde{g}}$ from the second constraint in (20) gives ${\tilde{g}}=\frac{n}{b}-ty,$ and substituting it into the first constraint in (20) gives $x+\frac{n}{b}=y$. Then, the constraint $x\ge 0$ is equivalent to $n\le by$, which implies that ${\tilde{g}}\le (1-t)y$. The last inequality guarantees that $Dg\le y$, meaning that $\lambda =0$ in the above FOCs. Therefore, the optimal solution is determined by the following condition:

$$\begin{aligned} U_{x}^{'}-bU_{n}^{'}=0\quad if\,x^{*}\ge 0,\,n^{*}\ge bty. \end{aligned}$$

Thus, the optimal $n^{*}$ does not depend on D, and so the optimal net gift ${\tilde{g}}^{*}=\frac{n^{*}}{b}-ty$ is not affected by the deductibility D. Consequently, the gross gift which can be expressed as $g=\frac{{\tilde{g}}}{1-Dt}$, is increasing in the deductibility D.

(b) If $a>b$, then both the gross gift, g, and the net gift, ${\tilde{g}}=(1-Dt)g$, are affected by an increase of the deductibility of gifts.

Consider the case when $a>b$ and the constraint $Dg+E\le y$ is non-binding, i.e., $D>{\bar{D}}$ and ${\underline{n}}<n^{*}<{\overline{n}}$ or $D\le {\overline{D}}$. Then, optimal evasion is $E^{*}={\hat{E}}=(c')^{-1}(t\frac{a-b}{a-bDt})$, and the optimal gift is $g^{*}=\frac{n^{*}-bt(y-{\hat{E}})}{a-bDt}$ . The derivative of optimal evasion with respect to the deductibility, D, is

$$\begin{aligned} \frac{d{\hat{E}}}{\mathrm{d}D}=\frac{bt^{2}(a-b)}{c''({\hat{E}})(a-bDt)^{2}}>0, \end{aligned}$$

(23)

so that optimal evasion is increasing with the deductibility of gifts.

The intuition for this is the following. Both gifts and remitting taxes (i.e., restraining evasion) are ways of acquiring warm glow at the cost of private goods. When the deductibility of gifts increases, it becomes cheaper to transfer money by making a gift than by remitting taxes. Therefore, it is efficient to remit less tax, i.e., evade more.

The derivative of the optimal $n^{*}$ with respect to the deductibility, D, can be calculated and expressed as

$$\begin{aligned} \frac{\mathrm{d}n^{*}}{\mathrm{d}D}=\frac{1}{S}\frac{t(a-b)}{(a-bDt)^{2}}\left( g^{*}\left[ U_{xn}^{''}-pU_{xx}^{''}\right] +U_{x}^{'}\right) >0, \end{aligned}$$

where $p=\frac{1-Dt}{a-bDt}$ is the “price” of n it terms of x, where $S=-(a-bDt)^{2}\frac{\partial ^{2}L}{\partial g^{2}}=-p^{2}U_{xx}^{''}+2pU_{xn}^{''}-U_{nn}^{''}>0$ is positive, because the second derivative of the Lagrangian with respect to g is negative.

In the formula above, the first term, $g^{*}\left[ U_{xn}^{''}-pU_{xx}^{''}\right] ,$ represents the income effect and the second term, $U_{x}^{'},$ represents the substitution effect. Because both the income and substitution effects are positive, as long as $U_{xn}^{''}\ge 0$, $n^{*}$ increases with the deductibility.

The derivative of the optimal gross gift with respect to the deductibility, D, is

$$\begin{aligned} \frac{\mathrm{d}g^{*}}{\mathrm{d}D}=\frac{1}{a-bDt}\left[ \frac{\mathrm{d}n^{*}}{\mathrm{d}D}+bt\frac{\mathrm{d}{\hat{E}}}{\mathrm{d}D}\right] +\frac{btg^{*}}{(a-bDt)^{2}}>0. \end{aligned}$$

This derivative is positive, because the last term is obviously positive, and the first and the second terms are positive as we just have shown. Thus, the optimal gross gift increases with deductibility due to the following three effects. First, the gift increases because the optimal consumption of good n increases with deductibility. Second, as described above, when the deductibility of gifts increases, it becomes cheaper to transfer money by making a gift than by paying taxes, which increases the gift and increases evasion, all other things being equal. Third, the deductibility of gifts mechanically decreases paid taxes and thus requires a bigger gift to reach a chosen amount of n.

The derivative of the optimal net gift with respect to the deductibility, D, can be calculated by differentiating the first constraint in the maximization problem (1), and can be expressed as

$$\begin{aligned} \frac{\mathrm{d}{\tilde{g}}}{\mathrm{d}D}=\frac{bt(1-Dt)}{a-bDt}\frac{\mathrm{d}{\hat{E}}}{\mathrm{d}D}-\frac{\mathrm{d}x^{*}}{\mathrm{d}D}. \end{aligned}$$

The sign of $\frac{\mathrm{d}{\tilde{g}}}{\mathrm{d}D}$ depends on the sign of $\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$. The derivative of $x^{*}$ with respect to the deductibility, D, is

$$\begin{aligned} \frac{\mathrm{d}x^{*}}{\mathrm{d}D}=\frac{1}{S}\frac{t(a-b)}{(a-bDt)^{2}}\left( (a-bDt)g^{*}\left[ pU_{xn}^{''}-U_{nn}^{''}\right] -pU_{x}^{'}\right) . \end{aligned}$$

The first term in this formula $\sim (a-bDt)g^{*}\left[ pU_{xn}^{''}-U_{nn}^{''}\right]$ represents the income effect, and the second term $-pU_{x}^{'}$ represents the substitution effect. The income effect is positive in this case (assuming $U_{xn}^{''}\ge 0$), while the substitution effect is negative. Therefore, the sign of $\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$ depends on the magnitude of the income and substitution effects. If the substitution effect dominates, then $\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$ is negative.

When $a>b$ and the constraint $Dg+E\le y$ is binding, i.e., $D>{\bar{D}}$ and $n^{*}>{\overline{n}}$, the optimal gift is $g^{*}=\frac{n^{*}}{a}$ and the optimal evasion is $E^{*}=y-D\frac{n^{*}}{a}$. The budget constraint in this case becomes $x=y-\frac{n}{a}-c(y-D\frac{n}{a})$.

The derivative of the optimal $n^{*}$ with respect to the deductibility, D, can be calculated and expressed as

$$\begin{aligned} \frac{\mathrm{d}n^{*}}{\mathrm{d}D}=\frac{1}{S}\left( g^{*}c^{'}(y-Dg^{*})\left[ U_{xn}^{''}-pU_{xx}^{''}\right] +\frac{1}{a}\left( c^{'}(y-Dg^{*})-Dg^{*}c^{''}(y-Dg^{*})\right) U_{x}^{'}\right) , \end{aligned}$$

where $p=\frac{1}{a}-\frac{D}{a}c^{'}(y-Dg^{*})$ is the “price” of n it terms of x, where $S=-a^{2}\frac{\partial ^{2}L}{\partial g^{2}}=-p^{2}U_{xx}^{''}+2pU_{xn}^{''}-U_{nn}^{''}+\left( \frac{D}{a}\right) ^{2}c^{''}(y-\frac{Dn^{*}}{a})U_{x}^{'}>0$.

The derivative $\frac{\mathrm{d}n^{*}}{DD}$ is positive if the substitution effect is positive, i.e., if $c^{'}(y-Dg^{*})-Dg^{*}c^{''}(y-Dg^{*})>0$. Assuming so, the optimal gross gift increases with the deductibility.

c) When $a<b$, the optimal gift is positive only if the optimal evasion is zero, and then, the budget constraints in (1) can be rewritten as

$$\begin{aligned} \begin{array}{c} x^{*}=y(1-t)-{\tilde{g}}^{*}\\ n^{*}=\frac{a-bDt}{1-Dt}{\widetilde{g}}^{*}+bty \end{array},\;where\;{\tilde{g}}=(1-Dt)g. \end{aligned}$$

(24)

From the first constraint, $\frac{\mathrm{d}{\widetilde{g}}^{*}}{\mathrm{d}D}=-\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$. Substituting the first constraint into the second one, it follows that $x^{*}+pn^{*}=\alpha y$ where $p=\frac{1-Dt}{a-bDt}$ and $\alpha =1-t+bt\frac{1-Dt}{a-bDt}$. Now, using this constraint and the FOC $\frac{U_{n}^{'}}{U_{x}^{'}}=p$, the derivative of the optimal $x^{*}$ with respect to the deductibility, D, can be calculated and expressed as

$$\begin{aligned} \frac{\mathrm{d}x^{*}}{\mathrm{d}D}=\frac{1}{S}\frac{(b-a)tp}{(1-Dt)^{2}}\left( p^{2}U_{x}^{'}-{\widetilde{g}}^{*}\left[ pU_{xn}^{''}-U_{nn}^{''}\right] \right) , \end{aligned}$$

where $S=-(a-bDt)^{2}\frac{\partial ^{2}L}{\partial g^{2}}=-p^{2}U_{xx}^{''}+2pU_{xn}^{''}-U_{nn}^{''}>0$ is positive.

The first term in this formula $\left( p^{2}U_{x}^{'}\right)$ represents the substitution effect, and the second term $\left( {\widetilde{g}}^{*}\left[ pU_{xn}^{''}-U_{nn}^{''}\right] \right)$ represents the income effect. The substitution effect is positive in this case, while the income effect is negative (assuming $U_{xn}^{''}\ge 0$). Therefore, the sign of $\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$ depends on the relative magnitude of the income and substitution effects. If the substitution effect dominates, then $\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$ is positive.

Because $\frac{\mathrm{d}{\widetilde{g}}^{*}}{\mathrm{d}D}=-\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$, the sign of $\frac{\mathrm{d}{\widetilde{g}}^{*}}{\mathrm{d}D}$ might be positive or negative. The sign of $\frac{\mathrm{d}g^{*}}{\mathrm{d}D}=\frac{t{\widetilde{g}}}{(1-Dt)^{2}}+\frac{1}{1-Dt}\frac{\mathrm{d}{\widetilde{g}}^{*}}{\mathrm{d}D}$ is also undetermined and could be positive or negative. $\square$

So, when $a=b$, i.e., the amounts of warm glow provided by a dollar of tax remittance and gross gift are the same, the extent of deductibility does not affect the net gift and mechanically increases the gross gift.

In the case where $a>b$, the derivative of the optimal gross gift with respect to deductibility can be expressed as $\frac{\mathrm{d}g^{*}}{\mathrm{d}D}=\frac{1}{a-bDt}\left[ \frac{\mathrm{d}n^{*}}{DD}+bt\frac{\mathrm{d}{\hat{E}}}{\mathrm{d}D}\right] +\frac{btg^{*}}{(a-bDt)^{2}}>0$ and is composed of the following three effects. First, the optimal gift increases because the optimal consumption of good n increases with deductibility. This is because both income and substitution effects are positive: An increase in D increases the after-tax income and decreases the relative price of $n^{*}$. Second, when the deductibility of gifts increases, it becomes cheaper to transfer money by making a gift than by paying taxes, which increases evasion and hence increases the gift, all other things being equal. Third, the deductibility of gifts mechanically decreases paid taxes and thus requires a bigger gift to reach a chosen amount of n. Thus, all three effects lead to an increase in the optimal gift as deductibility increases.

The derivative of the optimal net gift with respect to deductibility is easy to analyze when it expressed as $\frac{\mathrm{d}{\tilde{g}}}{\mathrm{d}D}=\frac{bt(1-Dt)}{a-bDt}\frac{\mathrm{d}{\hat{E}}}{\mathrm{d}D}-\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$. As was explained earlier, the optimal amount of evasion increases with deductibility, so that the first term in the expression is positive. Therefore, the sign of $\frac{\mathrm{d}{\tilde{g}}}{\mathrm{d}D}$ depends on the sign of $\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$, which is not determinate. Indeed, the derivative of $x^{*}$ with respect to D is $\frac{\mathrm{d}x^{*}}{\mathrm{d}D}=\frac{1}{S}\frac{t(a-b)}{(a-bDt)^{2}}\left( (a-bDt)g^{*}\left[ pU_{xn}^{''}-U_{nn}^{''}\right] -pU_{x}^{'}\right)$, where the first term $\sim (a-bDt)g^{*}\left[ pU_{xn}^{''}-U_{nn}^{''}\right]$ represents the income effect, and the second term $\sim -pU_{x}^{'}$ represents the substitution effect. The income effect is positive in this case, while the substitution effect is negative. Therefore, the sign of $\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$ depends on the relative magnitude of the income and substitution effects. If the substitution effect dominates, i.e., if $\frac{pU_{x}^{'}}{S}>\frac{g^{*}\left[ pU_{xn}^{''}-U_{nn}^{''}\right] }{S}-\frac{b^{2}t^{2}}{c^{''}({\hat{E}})}$, then $\frac{\mathrm{d}x^{*}}{\mathrm{d}D}$ is negative and, hence, the optimal net gift increases with the deductibility.

When $a<b$, the warm glow from tax remittance is bigger than the warm glow from a gift to government. Therefore, a positive gift will be made only when the maximal tax is already paid, i.e., when evasion is zero. Then, when deductibility increases, getting warm glow becomes more expensive in terms of consumption and the available income²² becomes low. The first effect might result in decreasing the net gift and the second effect might cause an increase in the net gift.

D Model with two public goods, with conditional and unconditional giving

Now we allow for two kinds of public good, both of which benefit individuals only through warm glow. The difference is that taxes go to a mix of the two public goods in a fixed proportion, as do unconditional gifts. In contrast, conditional gifts can be designated to the higher warm-glow purpose. Whether the conditional gifts actually increase the ultimate allocation to this purpose (i.e., are hard-earmarked) is irrelevant because we have assumed that it is warm glow rather than altruism that motivates behavior.

Individuals choose unconditional gifts, g, and conditional gifts, h, and as before can also evade an amount E, which comes at a cost of c(E). The individual problem can be written as

$$\begin{aligned} \begin{array}{c} \underset{x,g,h,E}{\max}U(x,n,s)\\ s.t.\;x=y-t(y-Dg-Dh-E)-c(E)-g-h,\\ n=pbt(y-Dg-Dh-E)+pag,\\ s=(1-p)bt(y-Dg-Dh-E)+(1-p)ag+a_{s}h. \end{array} \end{aligned}$$

(25)

We denote as n (and as s) the quantity of warm glow from “contributing” to the first public good (to the second public good). Here, $a_{s}$ is the relative value of the conditional gift, where $a_{s}>a>b$, and $(1-p)$ is the fraction of taxes that go toward funding the specific public good that the conditional gift targets completely.

To develop the solution of the model, it is instructive to consider constraints on g, h, E, x, n, and s guaranteeing that the budget set is bounded. They are

$$\begin{aligned} \begin{array}{c} g\ge 0,\\ h\ge 0,\\ E\ge 0,\\ x\ge 0,\\ n\ge 0,\\ s\ge 0,\\ Dg+Dh+E\le y. \end{array} \end{aligned}$$

(26)

Let us now start from reorganizing the constraints in order to determine the maximal budget set. From the second and third constraints in the maximization problem (25), it follows that $s=\frac{1-p}{p}n+a_{s}h$. Because $h\ge 0$, the value of s is partly determined by n, meaning that $s=\frac{1-p}{p}n$ when $h=0$. As a result, in this case with two public goods it is simpler to consider the budget set as a function of x, n, and h instead of as a function of x, n, and s, with utility being maximized equal to $U(x,n,\frac{1-p}{p}n+a_{s}h)$. From the second constraint in the maximization problem (25), it follows that

$$\begin{aligned} g=\frac{n-pbt(y-E-Dh)}{p(a-bDt)}. \end{aligned}$$

(27)

Then, the constraint $g\ge 0$ implies $E\ge y-(\frac{n}{pbt}+Dh)$. Additionally, the constraint $h\ge 0$ corresponds to $s\ge \frac{1-p}{p}n$. Plugging the expression for g into the first constraint in the maximization problem (25) gives

$$\begin{aligned} x+\frac{1-Dt}{p(a-bDt)}n+\frac{a(1-Dt)}{a-bDt}h=y(1-t\frac{a-b}{a-bDt})+t\frac{a-b}{a-bDt}E-c(E). \end{aligned}$$

(28)

Plugging the expression for g into $Dg+Dh+E\le y$ gives $E\le y-(D\frac{n}{pa}+Dh)$.

Thus, the maximum budget line can be defined as

$$\begin{aligned}&x+\frac{1-Dt}{p(a-bDt)}n+\frac{a(1-Dt)}{a-bDt}\nonumber \\ h\qquad =\underset{E\in {\mathcal{B}}(y,n,h)}{\max}\left\{ y(1-t\frac{a-b}{a-bDt})+t\frac{a-b}{a-bDt}E-c(E)\right\} , \end{aligned}$$

(29)

where ${\mathcal{B}}(y,n,h)=\{E\,|\,max\{0,y-(\frac{n}{pbt}+Dh)\}\le E\le y-(D\frac{n}{pa}+Dh)\}$.

Let us denote the unconstrained optimal level of evasion as ${\hat{E}}=\underset{E}{{\rm argmax}}\left\{ t\frac{a-b}{a-bDt}E-c(E)\right\} =(c')^{-1}(t\frac{a-b}{a-bDt})$. To find the constrained optimal evasion level $E^{c}=\underset{E\in B(y,n,h)}{{\rm argmax}}\left\{ t\frac{a-b}{a-bDt}E-c(E)\right\}$, let us introduce the following notations. Define the threshold levels of n as ${\underline{n}}=bt(y-{\hat{E}})$ and ${\overline{n}}=\frac{a}{D}(y-{\hat{E}})$, and the threshold functions of h from n as ${\underline{h}}(n)=\frac{1}{D}(y-{\hat{E}}-D\frac{n}{pbt})$ and ${\overline{h}}(n)=\frac{1}{D}(y-{\hat{E}}-D\frac{n}{pa})$. These threshold functions determine three regions in (n, h) surface and are helpful to find the solution for the constrained optimal evasion $E^{c}$, which are shown in Fig. 3. Then, assuming ${\hat{E}}<y$, the constrained optimal evasion level is

$$\begin{aligned} E^{c}={\left\{ \begin{array}{ll} y-(\frac{n}{pbt}+Dh)&\begin{array}{cccc} {\rm if} &{} 0\le n<{\underline{n}} &{} and &{} 0\le h<{\underline{h}}(n),\end{array}\\ {\hat{E}} &{} \begin{array}{cccc} {\rm if} &{} 0\le n\le {\overline{n}} &{} and &{} {\underline{h}}(n)\le h\le {\overline{h}}(n),\end{array}\\ y-(D\frac{n}{pa}+Dh) &{} \begin{array}{cccc} {\rm if} &{} 0\le n\le \frac{pay}{D} &{} and &{} {\overline{h}}(n)<h\le \frac{y}{D}-\frac{n}{pa}\end{array}. \end{array}\right. } \end{aligned}$$

(30)

The budget set can be defined by substituting the solution for the constrained optimal evasion (30) into Eq. (29), which gives

$$\begin{aligned} x={\left\{ \begin{array}{ll} y-\frac{n}{pb}-h-c(y-\frac{n}{pbt}-Dh)&\begin{array}{cccc} {\rm if} &{} 0\le n<{\underline{n}} &{} and &{} 0\le h<{\underline{h}}(n),\end{array}\\ y-t\frac{a-b}{a-bDt}(y-{\hat{E}})-\frac{1-Dt}{(a-bDt)}(\frac{n}{p}+ah)-c({\hat{E}}) &{} \begin{array}{cccc} {\rm if} &{} 0\le n\le {\overline{n}} &{} and &{} {\underline{h}}(n)\le h\le {\overline{h}}(n),\end{array}\\ y-\frac{n}{pa}-h-c(y-D\frac{n}{pa}-Dh) &{} \begin{array}{cc} {\rm if} &{} 0\le n\le \frac{pay}{D}\,and\,{\overline{h}}(n)<h\le \frac{y}{D}-\frac{n}{pa}.\end{array} \end{array}\right. } \end{aligned}$$

(31)

Fig. 6

The budget set regions in (n, h) surface for two public goods model

Figure 6 shows regions corresponding to different cases of the optimal solutions for choices of gifts $g^{*}$, $h^{*}$ and evasion $E^{*}$ given the optimal choice of $n^{*}$ and $h^{*}$. For both $h^{*}=0$ and $h^{*}>0$, there are three regions which define conditions determining optimal solution for gift $g^{*}$ and evasion $E^{*}$. In the first region (I), i.e., $0\le n\le {\underline{n}}$ and $0\le h<{\underline{h}}(n)$, the condition $g\ge 0$ is binding, so the optimal gift is zero and optimal evasion is $E^{*}=y-(\frac{n^{*}}{pbt}+Dh^{*})$. In the second region (II), i.e., $0\le n<{\overline{n}}$ and ${\underline{h}}(n)\le h\le {\overline{h}}(n)$, the optimal evasion $E^{*}$ is constant and equal to ${\hat{E}}$, the optimal gift $g^{*}$ is equal to $\frac{n^{*}-pbt(y-{\hat{E}}-Dh^{*})}{p(a-bDt)}$. In the third region (III), i.e., $0\le n\le \frac{pay}{D}$ and ${\overline{h}}(n)<h\le \frac{y}{D}-\frac{n}{pa}$, the constraint $Dg+Dh+E\le y$ is binding and the optimal gift and evasion are equal to $g^{*}=\frac{n^{*}}{pa}$ and $E^{*}=y-(D\frac{n^{*}}{a}+Dh^{*})$ correspondingly. Note that given optimal $n^{*}$ and $h^{*}$ the optimal $s^{*}=\frac{1-p}{p}n^{*}+a_{s}h^{*}$.

To build on further results, we supplement this graphical solution with the standard Lagrangian approach. The Lagrangian for the maximization problem (25) which directly includes the last constraint from (26) is

$$\begin{aligned} \begin{array}{c} L=U\left\{ y-t(y-Dg-Dh-E)-c(E)-g-h,\,pbt(y-Dg-Dh-E)+pag,\,\right. \\ \left. (1-p)bt(y-Dg-Dh-E)+(1-p)ag+a_{s}h\right\} +\lambda (y-Dg-Dh-E). \end{array} \end{aligned}$$

The constraints $x\ge 0$, $n\ge 0$, and $s\ge 0$ are ignored, assuming that they are satisfied.

The FOCs for $\underset{g\ge 0,h\ge 0,E\ge 0}{\max}L$ are

$$\begin{aligned}&\begin{array}{c} \frac{\partial L}{\partial g}=-U_{x}^{'}(1-Dt)+(a-bDt)(pU_{n}^{'}+(1-p)U_{s}^{'})-\lambda D=\qquad \qquad \qquad \qquad \\ =-(1-Dt)[U_{x}^{'}-b(pU_{n}^{'}+(1-p)U_{s}^{'})]+(a-b)(pU_{n}^{'}+(1-p)U_{s}^{'})-\lambda D={\left\{ \begin{array}{ll} 0 &{} \begin{array}{cc} {\rm if} &{} g>0\end{array}\\ <0 &{} \begin{array}{cc} {\rm if} &{} g=0\end{array}, \end{array}\right. } \end{array} \end{aligned}$$

(32)

$$\begin{aligned}&\begin{array}{c} \frac{\partial L}{\partial h}=-U_{x}^{'}(1-Dt)-bDt(pU_{n}^{'}+(1-p)U_{s}^{'})+a_{s}U_{s}^{'}-\lambda D=\qquad \qquad \qquad \qquad \\ -(1-Dt)[U_{x}^{'}-b(pU_{n}^{'}+(1-p)U_{s}^{'})]-b(pU_{n}^{'}+(1-p)U_{s}^{'})+a_{s}U_{s}^{'}-\lambda D={\left\{ \begin{array}{ll} 0 &{} \begin{array}{cc} {\rm if} &{} h>0\end{array}\\ <0 &{} \begin{array}{cc} {\rm if} &{} h=0\end{array}, \end{array}\right. } \end{array} \end{aligned}$$

(33)

$$\begin{aligned}&\begin{array}{c} \frac{\partial L}{\partial E}=U_{x}^{'}(t-c^{'}(E))-bt(pU_{n}^{'}+(1-p)U_{s}^{'})-\lambda =\qquad \qquad \qquad \qquad \\ =t[U_{x}^{'}-b(pU_{n}^{'}+(1-p)U_{s}^{'})]-U_{x}^{'}c'(E)-\lambda ={\left\{ \begin{array}{ll} 0 &{} \begin{array}{cc} {\rm if} &{} E>0\end{array}\\ <0 &{} \begin{array}{cc} {\rm if} &{} E=0\end{array}. \end{array}\right. } \end{array} \end{aligned}$$

(34)

The complementary slackness condition is $\lambda (y-Dg-Dh-E)=0$.

Below we will proceed by considering various cases when the constraint is binding or non-binding. These cases are similar to those for one public good model.

If the constraint $Dg+Dh+E\le y$ is non-binding, i.e., $\lambda =0$, then $g^{*}>0$ and $E^{*}>0$ simultaneously only if $E^{*}={\hat{E}}$, i.e., $c'(E^{*})=t\frac{a-b}{a-bDt}$. Indeed, in this case FOCs (32) and (34) should be equal to zero. Expressing $pU_{n}^{'}+(1-p)U_{s}^{'}$ from Eq. (34) and substituting into Eq. (32) give

$$\begin{aligned} U_{x}^{'}(t-\frac{a-bDt}{a-b}c'(E))=0, \end{aligned}$$

(35)

which can be satisfied only if $c'(E^{*})=t\frac{a-b}{a-bDt}$.

If the constraint $Dg+Dh+E\le y$ is non-binding, i.e., $\lambda =0$, and the marginal cost of evasion is zero at $E=0$, then the case when $g^{*}=0$ and $E^{*}>0$ is possible, while the case when $g^{*}>0$ and $E^{*}=0$ is impossible. This is because FOC (34) cannot be negative at $E=0$ if $c'(0)=0$. In case when $g^{*}=0$, optimal evasion is $E^{*}=y-Dh^{*}-\frac{n^{*}}{bt}$.

If the constraint $Dg+Dh+E\le y$ is binding, i.e., $\lambda >0$, then from the last constraint in maximization problem (1) optimal $g^{*}=\frac{n^{*}}{pa}$ and, hence, $E^{*}=y-D\frac{n^{*}}{pa}-Dh^{*}$.

FOC (33) determines when optimal $h^{*}$ is zero or positive.

If the optimal $g^{*}$ is positive, then combining FOC (32) satisfied as an equality and FOC (33) satisfied as an inequality provides the following condition for $h^{*}=0$:

$$\begin{aligned} \frac{U_{n}^{'}}{U_{s}^{'}}\ge \frac{a_{s}-a(1-p)}{ap}. \end{aligned}$$

(36)

If the optimal $g^{*}$ is zero, then FOC (34) for optimal evasion can be used to derive the condition for $h^{*}=0$, which is

$$\begin{aligned} \frac{U_{n}^{'}}{U_{s}^{'}}\ge \frac{a_{s}}{pbt}\frac{t-c^{'}(y-\frac{n}{pbt})}{1-Dc^{'}(y-\frac{n}{pbt})}-\frac{1-p}{p}. \end{aligned}$$

(37)

The following propositions describe the formal conditions for conditional/unconditional gift to be positive and to co-exist with evasion.

Proposition 7

• (i) In the absence of tax evasion, the unconditional gifts to government may occur as long as $\left. \frac{pU_{n}^{'}+(1-p)U_{s}^{'}}{U_{x}^{'}}\right| _{g=0,h=0,E=0}>\frac{1-Dt}{a-bDt}$.

• (ii) The conditional gifts to government may occur as long as $\left. \frac{U_{s}^{'}}{(1-Dt)U_{x}^{'}+pbDtU_{n}^{'}}\right| _{g=0,h=0,E=0}>\frac{1}{a_{s}-(1-p)bDt}$.

• (iii) These conditions are sufficient if $\frac{\partial ^{2}L}{\partial g\partial h}>0$ .

Proof

Again, the tax evasion would not exist if $c^{'}(0)>t$ and $U_{x}^{'}>0,\;U_{n}^{'}>0,\;U_{s}^{'}>0$. This is immediately follows from FOC (34). Unless FOC (32) for g depended on h, the condition $\left. \frac{\partial L}{\partial g}\right| _{g=0}>0$ would guarantee that the optimal unconditional gift is positive. However, FOCs (32) and (33) depend on both g and h. Therefor, we need to assume $\frac{\partial ^{2}L}{\partial g\partial h}>0$ for the condition $\left. \frac{\partial L}{\partial g}\right| _{g=0,h=0}>0$ to be sufficient to guarantee that the optimal unconditional gift is positive. The condition $\left. \frac{\partial L}{\partial g}\right| _{g=0,h=0}>0$ can be expressed as

$$\begin{aligned} \left. \frac{pU_{n}^{'}+(1-p)U_{s}^{'}}{U_{x}^{'}}\right| _{g=0,h=0}>\frac{1-Dt}{a-bDt}. \end{aligned}$$

(38)

Similar, assuming $\frac{\partial ^{2}L}{\partial g\partial h}>0$ the condition $\left. \frac{\partial L}{\partial h}\right| _{g=0,h=0}>0$ is sufficient to guarantee that the optimal conditional gift is positive, and it can be expressed as

$$\begin{aligned} \left. \frac{U_{s}^{'}}{(1-Dt)U_{x}^{'}+pbDtU_{n}^{'}}\right| _{g=0,h=0}>\frac{1}{a_{s}-(1-p)bDt}, \end{aligned}$$

(39)

which concludes the proof. $\square$

Proposition 8

• (i) Unconditional gifts to government may co-exist with tax evasion as long as $\left. \frac{pU_{n}^{'}+(1-p)U_{s}^{'}}{U_{x}^{'}}\right| _{g=0,h\in {\underline{H}},E={\hat{E}}}>\frac{1-Dt}{a-bDt}$, where ${\underline{H}}=\{h={\underline{h}}(n),\;0\le n\le {\underline{n}}\}$ and ${\hat{E}}=(c')^{-1}(t\frac{a-b}{a-bDt})$.

• (ii) Conditional gifts to government may co-exist with tax evasion as long as $\left. \frac{U_{s}^{'}}{U_{n}^{'}}\right| _{g\in G^{+},h=0,E={\hat{E}}}>\frac{ap}{a_{s}-a(1-p)}$ and $\left. \frac{U_{s}^{'}}{U_{n}^{'}}\right| _{g=0,h=0,E\in \Sigma }>\frac{pbt(1-Dc^{'}(E)}{a_{s}(t-c^{'}(E))-(1-p)(1-Dc^{'}(E))}$, where $G^{+}=\{g=\frac{n-pbt(y-{\hat{E}})}{p(a-bDt)},\;{\underline{n}}\le n\le \frac{pay}{D}\}$ and $\Sigma =\{E=y-\frac{n}{pbt},\;0\le n\le {\underline{n}}\}$.

• (iii) These conditions are sufficient if $\frac{\partial ^{2}L}{\partial g\partial h}>0$ and $\frac{\partial ^{2}L}{\partial g\partial E}<0$.

Proof

Note that if the marginal cost of evasion is zero at $E=0$, then $E^{*}=0$ is impossible. This is because FOC (34) cannot be negative at $E=0$ if $c'(0)=0$.

When evasion is positive, $E^{*}>0$, the optimal conditional gifts is zero, $g^{*}=0$, if FOC (32) is less or equal to zero and FOC (34) is equal zero. Expressing $pU_{n}^{'}+(1-p)U_{s}^{'}$ from Eq. (34) and substituting into Eq. (32) give

$$\begin{aligned} U_{x}^{'}(t-\frac{a-bDt}{a-b}c'(E))\ge 0. \end{aligned}$$

(40)

This means that $E^{*}\ge {\hat{E}}=(c')^{-1}(t\frac{a-b}{a-bDt})$. Solutions of such type arise in area I depicted in Fig. 3, i.e., $0\le n\le {\underline{n}}$ and $0\le h<{\underline{h}}(n)$.

Thus, assuming that $\frac{\partial ^{2}L}{\partial g\partial h}>0$ and $\frac{\partial ^{2}L}{\partial g\partial E}<0$ the condition $\left. \frac{\partial L}{\partial g}\right| _{g=0,h\in {\underline{H}},E={\hat{E}}}>0$, where ${\underline{H}}=\{h={\underline{h}}(n),\;0\le n\le {\overline{n}}\}$ guarantees that $g^{*}$ is positive and co-exist with positive tax evasion. This conditions can be expressed

$$\begin{aligned} \left. \frac{pU_{n}^{'}+(1-p)U_{s}^{'}}{U_{x}^{'}}\right| _{g=0,h\in {\underline{H}},E={\hat{E}}}>\frac{1-Dt}{a-bDt}. \end{aligned}$$

(41)

FOC (33) determines when optimal $h^{*}$ is zero or positive. In the case, when the optimal $g^{*}$ is positive, combining FOC (32) satisfied as equality and FOC (33) satisfied as an inequality gives $\left. \frac{U_{s}^{'}}{U_{n}^{'}}\right| _{g^{*},h=0,E^{*}}\le \frac{ap}{a_{s}-a(1-p)}$ as the condition for $h^{*}=0$.

Therefore, if we assume $\frac{\partial ^{2}L}{\partial g\partial h}>0$ and $\frac{\partial ^{2}L}{\partial h\partial E}<0$, the condition $\left. \frac{\partial L}{\partial h}\right| _{g\in G^{+},h=0,E={\hat{E}}}>0$, where $G^{+}=\{g=\frac{n-pbt(y-{\hat{E}})}{p(a-bDt)},\;{\underline{n}}\le n\le \frac{pay}{D}\}$ guarantees that $h^{*}$ is positive and co-exists with positive tax evasion. This condition can be expressed

$$\begin{aligned} \left. \frac{U_{s}^{'}}{U_{n}^{'}}\right| _{g\in G^{+},h=0,E={\hat{E}}}>\frac{ap}{a_{s}-a(1-p)}. \end{aligned}$$

(42)

If the optimal $g^{*}$ is zero, then instead of FOC (32) FOC (34) for optimal evasion should be used to derive the condition for $h^{*}=0$, which is $\left. \frac{U_{s}^{'}}{U_{n}^{'}}\right| _{g*=0,h=0,E^{*}=y-\frac{n^{*}}{pbt}}\le \frac{pbt(1-Dc^{'}(y-\frac{n}{pbt})}{a_{s}(t-c^{'}(y-\frac{n}{pbt}))-(1-p)(1-Dc^{'}(y-\frac{n}{pbt}))}$. Again, if we assume $\frac{\partial ^{2}L}{\partial g\partial h}>0$ and $\frac{\partial ^{2}L}{\partial h\partial E}<0$, the condition $\left. \frac{\partial L}{\partial h}\right| _{g=0,h=0,E\in \Sigma }>0$, where $\Sigma =\{E=y-\frac{n}{pbt},\;0\le n\le {\underline{n}}\}$, guarantees that $h^{*}$ is positive and co-exists with positive tax evasion. This condition can be expressed

$$\begin{aligned} \left. \frac{U_{s}^{'}}{U_{n}^{'}}\right| _{g=0,h=0,E\in \Sigma }>\frac{pbt(1-Dc^{'}(E)}{a_{s}(t-c^{'}(E))-(1-p)(1-Dc^{'}(E))}, \end{aligned}$$

(43)

which concludes the proof. $\square$

E Tables of summary statistics and sources of the variables

See Tables 3, 4 and 5.

Table 3 Summary statistics for regression variables, 1914–2012

Table 4 Summary statistics for regression variables, 1961–2012

Table 5 Data sources for analyses of gifts to the federal government