Tax evasion in a Cournot oligopoly with endogenous entry

Abstract

If an additional competitor reduces output per firm in a homogenous Cournot oligopoly, market entry will be excessive. Taxes can correct the so-called business stealing externality. We investigate how evading a tax on operating profits affects the excessive entry prediction. Tax evasion raises the number of firms in market equilibrium and can alter their welfare-maximising number. In consequence, evasion can aggravate or mitigate excessive entry. Which of these outcomes prevails is determined by the direct welfare consequences of tax evasion and the relationship between evasion and the tax base. We also determine conditions which imply that overall welfare declines with tax evasion.

Appendix

1.1 Impact of higher tax rate

The derivatives of Eqs. (6), (7) and (8) with respect to the tax rate, $\tau$, are $A_{\tau } = 0$ and:

$$\begin{aligned} {B}_{\tau } =\left( {{\alpha }-1} \right) \left( {{P}\left( {X} \right) -{c}} \right) {x}={E}_{\tau } \left( {{\alpha }-1} \right) <0 \end{aligned}$$

(19)

Since, furthermore, $A_{\alpha } = B_{\alpha } = 0$ holds, tax evasion does not alter the effects of a change in the tax rate on output and the number of firms, such that $\hbox {d}x^{0}/\hbox {d}\tau = \hbox {d}x^{+}/\hbox {d}\tau = \hbox {d}x/\hbox {d}\tau , \hbox {d}n^{0}/\hbox {d}\tau = \hbox {d}n^{+}/\hbox {d}\tau = \hbox {d}n/\hbox {d}\tau$ and $\hbox {d}X^{0}/\hbox {d}\tau = \hbox {d}X^{+}/\hbox {d}\tau = \hbox {d}X/\hbox {d}\tau$. The respective derivatives are:

$$\begin{aligned} \frac{{\hbox {d}x}}{{\hbox {d}\tau }}= & {} \frac{{A}_{n} {B}_{\tau } }{{A}_{x} {B}_{n} -{B}_{x} {A}_{n} }>0 \,\hbox {if} \,n+\eta >0\end{aligned}$$

(20)

$$\begin{aligned} \frac{{\hbox {d}n}}{{\hbox {d}\tau }}= & {} \frac{-{A}_{x} {B}_{\tau } }{{A}_{x} {B}_{n} -{B}_{x} {A}_{n} }<0\end{aligned}$$

(21)

$$\begin{aligned} \frac{{\hbox {d}X}}{{\hbox {d}\tau }}= & {} \frac{{\hbox {d}x}}{{\hbox {d}\tau }}{n}+\frac{{\hbox {d}n}}{{\hbox {d}\tau }}{x}=\frac{-{B}_{\tau } {P^{\prime }}\left( {X} \right) {x}}{{A}_{x} {B}_{n} -{B}_{x} {A}_{n} }<0 \end{aligned}$$

(22)

The impact of a higher rate on tax evasion is determined by:

$$\begin{aligned} \frac{{\hbox {d}\alpha }}{{\hbox {d}\tau }}= & {} \frac{{B}_{\tau } \left[ {{A}_{x} {E}_{n} -{E}_{x} {A}_{n} } \right] -{E}_{\tau } \left[ {{A}_{x} {B}_{n} -{B}_{x} {A}_{n} } \right] }{{D}^{+}}\nonumber \\&=\frac{P'(X^+)x^+(2n^+ + \eta )\left[ {B}_{\tau } {E}_{x} -{E}_{\tau } {B}_{x} \right] }{n^{+}(n^{+}-1)D^+}\nonumber \\&= \frac{{{{P'}}\left( {{{{X}}^ + }} \right) \left( {{{P}}\left( {{{{X}}^ + }} \right) - {{c}}} \right) {{\left( {{{{x}}^ + }} \right) }^2}\left( {2{{{n}}^ + } + {{\eta }}} \right) }}{{{{{n}}^ + }\left( {{{{n}}^ + } - 1} \right) {{{D}}^ + }}}\left[ {{{{E}}_{{x}}}\left( {{{\alpha }} - 1} \right) - {{{B}}_{{x}}}} \right] \nonumber \\&={\mathop {\mathop {\mathop {\underbrace{\frac{(P'(X^{+}))^{2} (P(X^{+})-c)(x^{+})^{3}(2n^{+}+\eta )}{n^{+}D^{+}}}}}}\limits _{(-)}}\times \nonumber \\&\left( {{{T}}_2}\left( {{{\alpha }},{{\pi }}} \right) + \left( {1 - {{\alpha }}} \right) {{{T}}_{12}}\left( {{{\alpha }},{{\pi }}} \right) - 1 \right) \end{aligned}$$

(23)

Given $D^{+} < 0$ from (12) and $2n^{+}+\eta > 0$ from the second-order condition (4), $\hbox {d}\alpha /\hbox {d}\tau > 0$ will result if the costs of tax evasion are independent of the tax base, such that $T(\alpha , \pi ) = T(\alpha , 0)$ and, thus, $T_{2}(\alpha , \pi ) = T_{12}(\alpha , \pi ) = 0$. $\hbox {d}\alpha /\hbox {d}\tau > 0$ will also hold if higher operating profits raise the fraction of taxes evaded. This condition implies that $E_{\pi } > 0$ and, hence, $\tau > T_{12}(\alpha , \pi )$ (cf. Eq. (11c)). In consequence, we have $T_{2}(\alpha , \pi ) + (1 - \alpha )T_{12}(\alpha , \pi ) - 1< T_{2}(\alpha , \pi ) + (1 - \alpha )\tau - 1 = -[1 - (1 - \alpha )\tau - T_{2}(\alpha , \pi )] < 0$. If, finally, $E_{\pi } < 0, \hbox {d}\alpha /d\tau > 0$ will hold if, additionally, $0 < T_{2}(\alpha , \pi ) + (1- \alpha )T_{12}(\alpha , \pi )$ is not too large in absolute value.

If the cost-of-evasion function increases with the tax rate as well ($\partial T/\partial \tau > 0$), $B_{\tau }$ continues to be negative. Therefore, $E_{\tau } \ge 0$ is a sufficient condition for $\hbox {d}\alpha /\hbox {d}\tau > 0$, given $E_{x} < 0$. In the case of $E_{x} > 0$, $E_{\tau } > 0$ becomes a necessary condition for $\hbox {d}\alpha /\hbox {d}\tau > 0$.

1.2 Proof of Proposition 1

Using $A_{q} = 0, B_{q} = -S(\alpha , \pi ) < 0$, $E_{q} = -S_{1}(\alpha , \pi ) < 0$, and given $T_{1}(\alpha , \pi ) = qS_{1}(\alpha , \pi )$, the impact of the detection probability q on the optimal fraction, $\alpha$, of the tax burden evaded, is:

$$\begin{aligned} \frac{{\hbox {d}\alpha }}{{\hbox {d}q}}= & {} \frac{{B}_{q} \left[ {{A}_{x} {E}_{n} -{E}_{x} {A}_{n} } \right] -{E}_{q} \left[ {{A}_{x} {B}_{n} -{B}_{x} {A}_{n} } \right] }{{D}^{+}}\nonumber \\&=\frac{P'(X^+)x^+(2n^{+}+\eta )}{n^+(n^{+}-1)D^+}\left( B_{q} E_{x}-B_{q} B_{x}\right) \nonumber \\&=\frac{\left( {P}^{{{\prime }}}\left( {{X}^{+}} \right) \right) ^{2}\left( {{x}^{+}} \right) ^{2}\left( {2{n}^{+}+{\eta }} \right) }{{n}^{+}{D}^{+}}\nonumber \\&\quad \times \left\{ {S}_1 \left( {{\alpha },{\pi }} \right) {\mathop {\mathop {\underbrace{\left( {1-{\tau }\left( {1-{\alpha }} \right) -{T}_2 \left( {{\alpha },{\pi }} \right) } \right) }}}\limits _{(+)}}-{S}\left( {{\alpha },{\pi }} \right) {\mathop {\mathop {\underbrace{\left( {{\tau }-{T}_{12} \left( {{\alpha },{\pi }} \right) } \right) }}}\limits _{={E}_{\pi }}} \right\} \nonumber \\ \end{aligned}$$

(24)

Tax evasion will increase with a decline in q if the term in curly brackets is positive, since $D^{+ }< 0$. This will obviously be the case if higher operating profits decrease the fraction of taxes evaded, $\alpha$, ceteris paribus, i.e. if $E_{\pi }=\tau - T_{12}(\alpha , \pi ) < 0$. Moreover, the last line of (24) will be positive if the costs of tax evasion are independent of the tax base, i. e. if $T = T(\alpha , 0)$ and $T_{2}(\alpha , \pi ) = T_{12}(\alpha , \pi ) = 0$ holds. In this case, the term in curly brackets in (24) can be rewritten as $(1 - \tau )S_{1}(\alpha , 0) + \tau (S_{1}(\alpha , 0)\alpha - S(\alpha , 0))$, which is greater than zero because $S_{1}, S_{11} > 0$ and $S(0, 0) = 0$.

The number of firms characterising the market equilibrium rises with a fall in q.

$$\begin{aligned} \frac{{\hbox {d}n}^{+}}{{\hbox {d}q}}=-\frac{{E}_{\alpha } {A}_{x} {B}_{q} }{{D}^{+}}<0 \end{aligned}$$

(25)

Consequently, if a fall in q increases the fraction of the tax burden evaded $(\hbox {d}\alpha /\hbox {d}q < 0)$, the number of firms will rise. This proves Proposition 1.

1.3 Proof of Proposition 2

Part (a) of the Proposition is obvious from inspection of Eq. (18).

Note that $\eta + 2n \ge 1 + n + \eta$ will hold for $n \ge 1$. If $\beta < 1$, the derivative in (18) will, hence, be positive if the term in curly brackets subsequent to the last equality sign is greater in absolute value than the positive first summand, $T(\alpha , \pi )$. The conditions enlisted in part (b) of Proposition 2 imply that the second and third summands in curly brackets are zero because $T(\alpha , \pi ) = T(\alpha , 0)$ implies that $T_{2}(\alpha , 0) = T_{12}(\alpha , 0) = 0$. If the costs of tax evasion are given by a second- or higher-order polynomial, they may be expressed as $T(\alpha , 0) = q\alpha ^{m} + c_\mathrm{fix}$, where $m \ge 2$. Hence, $T(\alpha , 0) \le (T_{1}(\alpha , 0))^{2}/T_{11}(\alpha , 0) = q^{2}m^{2}\alpha ^{2m-2}/(qm(m - 1)\alpha ^{m-2}) = qm\alpha ^{m}/(m - 1)$ is a sufficient condition for the derivative in (18) to be positive. The inequality $q\alpha ^{m} + c_\mathrm{fix} \le qm\alpha ^{m}/(m - 1)$ will be fulfilled if $(m - 1)c_\mathrm{fix} \le q\alpha ^{m}$, i.e. if the fixed costs of tax evasion are zero or not too high. This proves Proposition 2b).

If $T(\alpha , \pi ) = T(\alpha \pi )$ holds, $T_{1}(\alpha \pi ) = T'\pi , T_{11}(\alpha \pi ) = T''\pi ^{2}$, and $T_{2}(\alpha \pi ) = T'\alpha$. Equation (8) is given by $E = \tau \pi - T_{1}(\alpha \pi )=\tau \pi - T'\pi = 0$, such that $E_{\pi } = -T''\alpha \pi$. Replacing $T_{1}, T_{11}, T_{2}$ and $E_{\pi }$ in the second line of (18) shows that the term in curly bracket is zero. This proves part c.

1.4 Proof of Proposition 4

Employing (24), (25), (13), and $\partial n/\partial q = -(\partial x/\partial q)(A_{x}/A_{n})$, the impact of a rise in the parameter q can be calculated as:

$$\begin{aligned} \frac{{\hbox {d}W}^{+}}{{\hbox {d}q}}= & {} \frac{\partial {W}^{+}}{\partial {n}}\frac{\partial {n}}{\partial {q}}+\frac{\partial {W}^{+}}{\partial {x}}\frac{\partial {x}}{\partial {q}}+\frac{\partial {W}^{+}}{\partial {\alpha }}\frac{\partial {\alpha }}{\partial {q}}+\frac{\partial {W}^{+}}{\partial {q}}{} \nonumber \\&=\frac{\partial {W}^{+}}{\partial {n}}\frac{-{A}_{x} {B}_{q} }{{A}_{x} {B}_{n} -{A}_{n} {B}_{x} }+\frac{\partial {W}^{+}}{\partial {x}}\frac{{A}_{n} {B}_{q} }{{A}_{x} {B}_{n} -{A}_{n} {B}_{x} }\nonumber \\&\quad -\frac{\partial {W}^{+}}{\partial {\alpha }}\frac{\left( {{A}_{n} {E}_{x} -{A}_{x} {E}_{n} } \right) {B}_{q} }{{E}_{\alpha } \left( {{A}_{x} {B}_{n} -{A}_{n} {B}_{x} } \right) }-\frac{\partial {W}^{+}}{\partial {\alpha }}\frac{{E}_{q} }{{E}_{\alpha } }+\frac{\partial {W}^{+}}{\partial {q}}\nonumber \\= & {} \frac{-{A}_{x} {B}_{q} }{{A}_{x} {B}_{n} -{A}_{n} {B}_{x} }\left[ {\frac{\partial {W}^{+}}{\partial {n}}-\frac{\partial {W}^{+}}{\partial {x}}\frac{{A}_{n} }{{A}_{x} }+\frac{\partial {W}^{+}}{\partial {\alpha }}\frac{{A}_{n} {E}_{x} -{A}_{x} {E}_{n} }{{E}_{\alpha } {A}_{x} }} \right] \nonumber \\&-\frac{\partial {W}^{+}}{\partial {\alpha }}\frac{{E}_{q} }{{E}_{\alpha } }+\frac{\partial {W}^{+}}{\partial {q}}\nonumber \\= & {} {\mathop {\underbrace{\frac{{n}^{+}\left( {1+{n}^{+}+{\eta }} \right) {S}\left( {{\alpha },{\pi }} \right) }{\left( {1-{\tau }\left( {1-{\alpha }} \right) } \right) {P^{\prime }}\left( {{X}^{+}} \right) \left( {{x}^{+}} \right) ^{2}\left( {2{n}^{+}+{\eta }} \right) }}}\limits _{<0}} \frac{{dW}^{+}}{{dn}}\nonumber \\&+\left( {1-{\beta }} \right) {n}^{+}\left[ {\frac{\left( {{S}_1 \left( {{\alpha },{\pi }} \right) } \right) ^{2}}{{S}_{11} \left( {{\alpha },{\pi }} \right) }-{S}\left( {{\alpha },{\pi }} \right) } \right] \end{aligned}$$

(26)

Hence, welfare rises with a decline in the parameter q, i.e. the derivative in (26) is negative if, for example, $\hbox {d}W^{+}/\hbox {d}n > 0$ and $\beta = 1$ hold. This establishes part a. Assuming $\beta < 1$, the derivative in (26) is positive if $\hbox {d}W^{+}/\hbox {d}n < 0$ and $(S_{1}(\alpha , \pi ))^{2}/S_{11}(\alpha , \pi ) - S(\alpha , \pi ) > 0$ apply. The inequality $(S_{1}(\alpha , \pi ))^{2}/S_{11}(\alpha , \pi )- S(\alpha , \pi ) > 0$ holds if $T(\alpha , \pi ) = T(\alpha , 0)$ or $T(\alpha , \pi ) = T(\alpha \pi )$, and $T(\alpha , \pi )$ is given by a second- or higher-order polynomial. For $T(\alpha , \pi ) = (\alpha \pi )^{m }+ c_\mathrm{fix}$, where $m \ge 2, (S_{1}(\alpha , \pi ))^{2}/S_{11}(\alpha , \pi )- S(\alpha , \pi ) = (\alpha \pi )^{m}/(m - 1) > 0$ results such that the restriction is fulfilled. If the costs of evasion are a function of the fraction of taxes evaded, such that $T(\alpha , 0) = \alpha ^{m} + c_\mathrm{fix}$, we have $(S_{1}(\alpha , 0))^{2}/S_{11}(\alpha , 0) - S(\alpha , 0) = \alpha ^{m}/(m - 1) > 0$.

1.5 Numerical example

We denote market outcomes in the absence of taxation with the superscript ‘−’, in the presence of taxes but absence of evasion by ‘0’ and in the presence of tax evasion by ‘+’. Constrained-optimal solutions are indexed with the additional superscript ’opt’. Further, we set $c = K = 1, q = 0.4, c_\mathrm{fix} = 0$, $P(X) = 10 - X$ and $\tau$ = 0.2, unless there is no taxation ($\tau$ = 0).

1.5.1 Market outcome in the absence of taxation (Table 1, row 1)

The market outcome in the absence of taxes is, setting $\tau = 0$, determined by Eqs. (6′) and (7) and, thus, by $P - X - x - c = 9 - X - x = 0$ and $(9 - X)x = 1$. Combining both equalities yields $x^{- }= 1, n^{- }= X^{- }= 8$ and $W^{- }= (10 - P(X))X/2 + (P(X) - c)X - n = X^{2}/2 + X - n = 32$.

1.5.2 Market outcome in the presence of taxation but without evasion (Table 1, row 2)

We next turn to a world with taxes but without evasion. Following the same approach as above, that is, using $9 - X = x$ and $(1 - \tau )(9 - X)x = 1$, taxation can be shown to reduce the number of firms in market equilibrium to $n^{0} \approx 7.05$, while output per firm increases to $x^{0} \approx 1.118$, resulting in an aggregate output of $X^{0} \approx 7.882$ and a welfare level of $W^{0} \approx 32.82$.

1.5.3 Market outcome in the presence of taxation and evasion (Table 1, rows 3, 4)

Initially, we assume that the costs of tax evasion are a function of the product of profits and the share of the tax burden evaded, namely $\alpha \pi$. To ensure that restrictions on the functional form for T imposed in Sect. 2.1 hold, we specify T as $T(\alpha , \pi ) = q(\alpha \pi )^{2}$. This assumption implies that $E_{\pi }=\tau - T_{12}(\alpha , \pi )=\tau - 4q\alpha \pi$. As an alternative, we consider $T(\alpha ) = q\alpha ^{2}$, resulting in $E_{\pi }=\tau > 0$. Outcomes in a setting in which $T(\alpha , \pi ) = q(\alpha \pi )^{2 }[T(\alpha , 0) = q\alpha ^{2}]$ holds are subsequently denoted by the additional superscript 1 [2]. This way of differentiating results is used for market and constrained-optimal outcomes.

Given $T^{+,1 }= q(\alpha \pi )^{2}$, the first-order condition for the optimal choice of $\alpha$ (cf. Eq. (8)) yields $\alpha \pi =\tau /(2q) = 1/(10q) = 0.25$. Hence, the costs of tax evasion are constant and equal $T^{+,1 }= q(\alpha \pi )^{2 }= 0.025$. Using this information, it is possible to solve the zero-profit condition (6) to obtain $\alpha ^{+,1 }= 8/(100q - 1) \approx 0.205$. Since $\pi = x^{2}$ from (13a) and using the definition of $\pi$, we, furthermore, find $x^{2} = 1/(10\alpha q) = (100q - 1)/(80q)$, such that x rises in q, while $n = 9/x - 1$ declines in the detection probability, q (see Proposition 1). Solving for x, we obtain $x^{+,1 }\approx 1.104$. Inserting this value into $n = 9/x - 1 ~\mathrm{yields}~~n^{+,1 }\approx 7.15$, such that $X^{+,1 }\approx 7.896$. Furthermore, welfare can be computed as $W^{+,1 }\approx 32.65$.

Under the alternative assumption of $T^{+,2 }= q\alpha ^{2}$, the firm’s optimal evasion choice (cf. Eq. (8)) results in $\pi = 10q\alpha$. Using this information in the zero-profit constraint allows for the derivation of the optimal evasion decision, which is given by ${\alpha }^{+,2}=\sqrt{16+1/{q}}-4{}\approx {}0.301$. This implies $T^{+,2 }\approx 0.036$. Combining $\pi = x^{2}$ from (13a) and $\pi = 10q\alpha$ yields $x^{+,2 }\approx 1.098$ and $n^{+,2 }\approx 7.2$, such that $X^{+,2 }\approx 7.903$. Note that an increase in q reduces $\alpha$ and raises profits, $\pi = 10q\alpha (q)$, and hence, output per firm, x. This, in turn, implies that the number of firms—which declines in output per firm—becomes smaller if tax evasion is reduced. Conversely, a decline in q implies a rise in the number of entrants (see Proposition 1). Finally, welfare can be computed as $W^{+,2 }\approx 32.57$.

1.5.4 Second-best outcome in the absence of tax evasion (Table 1, row 5)

The second-best outcomes are the same in settings in which there are either no taxes or, alternatively, taxes cannot be evaded because taxation has no impact on the firm’s output choice. Hence, $9 - X - x = 0$ holds. The second-best outcome results from the maximisation of welfare, W, subject to the adjustment in output to the number of firms. Combining (15) and (13a) yields:

$$\begin{aligned} \frac{{\hbox {d}W}}{{\hbox {d}n}}=\left( {{P}\left( {X} \right) -{c}} \right) \frac{{x}}{1+{n}}-1=\left( {9-{X}} \right) \frac{{x}}{1+{n}}-1=0 \end{aligned}$$

(27)

Substituting, for example, $X = xn$ into this equation and into $9 - X = x$ and solving the system, we obtain $x^{0,\hbox {opt} }\!=\! 9^{1/3 }\!\approx \! 2.08$, $n^{0,\hbox {opt} }\approx 3.327$, $X^{0,\hbox {opt} }\!=\! 6.92, \pi ^{0,\hbox {opt} }= 4.32$ and $W^{0,\hbox {opt} }\approx 35$.

1.5.5 Second-best outcome in the presence of tax evasion (Table 1, rows 6, 7)

Using Eq. (13a), the constrained optimum can be defined as:

$$\begin{aligned} \frac{{\hbox {d}W}}{{\hbox {d}n}}=\left( {{P}\left( {X} \right) -{c}} \right) \frac{{x}}{1+{n}}-{K}-\left( {1-{\beta }} \right) \left[ {{T}\left( {{\alpha },{\pi }} \right) +{nT}_1 \frac{{d\alpha }}{{\hbox {d}n}}+{nT}_2 \frac{{\hbox {d}\pi }}{{\hbox {d}n}}} \right] =0\nonumber \\ \end{aligned}$$

(28)

Assuming $T^{+,1 }= q(\alpha \pi )^{2}$, using the specific values employed for the numerical example, and also taking into account (13b) and (13c), (28) can be rewritten as:

$$\begin{aligned} \frac{{\hbox {d}W}}{{\hbox {d}n}}{=}\left( {9-{X}} \right) \frac{{x}}{1+{n}}{-}1{-}\left( {1-{\beta }} \right) {q}\left[ {{\alpha }^{2}{\pi }^{2}-{n\alpha }\frac{{E}_{\pi } {x}^{2}}{{q}\left( {1+{n}} \right) }-{n\alpha }^{2}{\pi }\frac{{x}^{2}2}{\left( {1+{n}} \right) }} \right] =0\nonumber \\ \end{aligned}$$

(29)

Using the definition of $E_{\pi }$ (cf. Eq. (11b)) and $\alpha \pi =\tau /(2q)$ from (8) clarifies that last two terms in square brackets sum to zero. Replacing $\alpha \pi$ by $\tau /(2q) = 0.25$ and $(9 - X)x = \pi = x^{2}$, we can solve (29) for n.

$$\begin{aligned} {n}=\frac{{x}^{2}-\left( {1+0.025\left( {1-{\beta }} \right) } \right) }{1+0.025\left( {1-{\beta }} \right) } \end{aligned}$$

(30)

Combining this expression with the first-order condition for the firm’s output choice and replacing $\beta$ by 0.5, we obtain:

$$\begin{aligned} \frac{{x}^{2}-1.0125}{1.0125}=\frac{9-{x}}{{x}} \end{aligned}$$

(31)

Solving this equation yields ${x}^{+,1,\hbox {opt}}=\root 3 \of {9.1125}\approx 2.089$. From this output level per firm, we can calculate $n^{+,1,\hbox {opt} }\approx 3.309$, $X^{+,1,\hbox {opt} }\approx 6.912$ and $\pi ^{+,1,\hbox {opt} }\approx 4.36$. Given $\alpha \pi = 0.25$, the fraction of the tax base evaded equals $\alpha ^{+,1,\hbox {opt} }= 1/(4\pi ) \approx 0.057$. Welfare can, using the standard approach outlined above, be computed as $W^{+,1,\hbox {opt} }\approx 34.97$.

Finally, we calculate $\hbox {d}W/\hbox {d}n = 0$ for $T^{+,2 }= q\alpha ^{2}$. This derivative is given by:

$$\begin{aligned} \frac{{\hbox {d}W}}{{\hbox {d}n}}=\left( {9-{X}} \right) \frac{{x}}{1+{n}}-1-0.5\left[ {{q\alpha }^{2}-{n}2{q\alpha }\frac{{\tau }2{nx}^{2}}{2{qn}\left( {1+{n}} \right) }} \right] =0 \end{aligned}$$

(32)

Since $\pi = (9 - X)x = x^{2}$ and $\alpha =\tau x^{2}/(2q)$ from (8) and $\pi = x^{2}$, we obtain:

$$\begin{aligned} \frac{{\hbox {d}W}}{{\hbox {d}n}}=\frac{{x}^{2}}{1+{n}}-1-0.125\frac{{\tau }^{2}{x}^{4}}{{q}}\frac{1-3{n}}{1+{n}}=0 \end{aligned}$$

(33)

Solving this expression for n, yields:

$$\begin{aligned} {n}=\frac{8{qx}^{2}-8{q}-{\tau }^{2}{x}^{4}}{8{q}-3{\tau }^{2}{x}^{4}} \end{aligned}$$

(34)

Combining (34) and $n = (9 - X)x$, and substituting for q and $\tau$, we obtain a fifth-order polynomial.

$$\begin{aligned} 0=28.8-3.2{x}^{3}-1.08{x}^{4}+0.16{x}^{5} \end{aligned}$$

(35)

Solving this expression for x, we obtain $x^{+,2,\hbox {opt} }\approx 1.837$. Further solutions to (35) lie outside the range $0< x < 4.5$. Since $x > 4.5$ implies that $n < 1$, we can ignore such solutions. Accordingly, the constrained-optimal number of firms equals $n^{+,2,\hbox {opt} }\approx 3.899$, and aggregate output and profits are given by $X^{+,2,\hbox {opt} }\approx 7.164$ and $\pi ^{+,2,\hbox {opt} }\approx 3.37$. The fraction of taxes evaded is $\alpha ^{+,2,\hbox {opt} }\approx 0.844$. Finally, welfare equals $W^{+,2,\hbox {opt} }\approx 34.36$.