Maximizing Tax Revenue for Profit Maximizing Monopolist with the Cobb-Douglas Production Function and Linear Demand as a Bilevel Programming Problem

Abstract

Optimal taxation and profit maximization are two very important problems, naturally related to one another since companies operate under a given tax system. However, in the literature, these two problems are usually considered separately, either by studying optimal taxation or by studying profit maximization. This paper tries to link the two problems together by formulating a bilevel model in which the government acts as a leader and a profit maximizing follower acts as a follower. The exact form of the tax revenue function, as well as optimal tax amount and optimal input levels, are derived in cases when returns to scale take on values 0.5 and 1. Several illustrative numerical examples and accompanying graphical representations are given for decreasing, constant, and increasing returns to scale values.

1. Introduction

Taxation is one of the most important instruments of any government. Its basic function is to fund the government’s expenditure. However, it also has a profound impact on the competitiveness of companies, thus affecting economic growth and investments as well as supply and demand. Therefore, devising optimal tax policy is of key importance to any government. The problem of devising optimal taxation obviously contains hierarchical structure since the government makes tax decisions first and then companies independently operate under a given tax system trying to achieve what is personally best for them. Hence, it is natural to model the problem as a bilevel programming problem. Bilevel programming problems [1,2,3] were first described by von Stackleberg [4,5], with the first mathematical model given by Bracken and McGill [6,7]. They model problems with hierarchical structure, having two players located at different levels of hierarchy. The decision maker on the upper level is called the leader, while the one on the lower level is called the follower. Decisions are made sequentially, assuming perfect information. The leader acts first, with the goal of optimizing her objective. However, in making her decision, she has to anticipate the response of the follower who, once having the leader’s decision, acts independently, trying to optimize the objective of her own. Therefore, one of the constraints of the leader’s optimization problem is the optimization problem of the follower.

By itself, the problem of profit maximizing monopolist is so important that it is covered in all intermediate microeconomics textbooks, as Mas-Colell et al. did in [8], for example. As for production functions, Avvakumov and Rasmussen in [9,10] offer extensive coverage of the properties of the Cobb-Douglas and CES production function. On the other hand, tax revenue is one of the key tools of fiscal policy. Maximization of tax revenue function is also an important and well-studied problem, as in [11,12] by Lott and Gahvari. Tanaka [13] considers optimal commodity taxation under monopolistic competition and shows that optimal commodity taxes are partitioned into tax revenue part, tax shifting part, and product variation part. Auerbach and Hines [14] analyze features of perfect or optimal taxation when one or more private markets is imperfectly competitive and demonstrate the close relationship between the policy rules for correcting externalities and competitive imperfections, and an investigation of how governments how governments should behave in an environment in which the degree of market imperfection is uncertain. Coto-Matrinez et al. [15] explore economies with imperfect competitive markets and find that the optimal fiscal policy depends on the relationship between the index of market power, the returns to specialization, and the government’s ability to control entry. Reinhorn [16] studies optimal taxation in a Dixit–Stiglitz model of monopolistic competition and shows the compensated effect of the optimal tax system on the number of firms in the free entry equilibrium. Takatsuka [17] examines how unit tax and ad valorem tax affect firm location in a monopolistic competition model with asymmetrically sized regions and a quasi-linear preference. Colciago [18] provides optimal labor and dividend income taxation in a general equilibrium model with oligopolistic competition and endogenous firms’ entry. Sachs et al. [19] study the incidence of nonlinear labor income taxes in an economy with a continuum of endogenous wages. Lukač [20] formulates the problem of determining optimal tax policy for a single homogeneous commodity produced by n competing companies located in n different countries with export costs as a Stackelberg game with multiple followers and derives the optimal tax policy.

The main contribution of this paper is the formulation of the two problems, the problem of profit maximizing monopolist and the problem of optimal taxation, together as a single problem in the form of a bilevel programming problem, in which the leader is the government deciding about tax amount with the objective of maximizing her tax revenue. The tax is modelled as an amount per unit product. The follower is the monopolist who, once having the tax decision of the leader, chooses the level of production that maximizes her profit. The model assumes that production is described by the Cobb-Douglas production function and that the market price follows linear demand. An additional contribution of this paper is the closed-form solution for decreasing and constant returns to scale values 0.5 and 1, respectively. Moreover, the numerical solution for the illustrative example with increasing returns to scale value set to 2 showed that the solution can exist under a monopolist assumption instead of a perfect competition assumption.

The paper is organized as follows. After the notation in the second section, the third and the fourth sections formulate and solve the problem for the case of two inputs and provide three illustrative numerical examples. Conclusions are given in the last section.

2. Notation

Table 1. Notation.

Label	Constraint	Description
$x_1,x_2$	$x_1,x_2>0$	inputs (labor and capital)
$w_1,w_2$	$w_1,w_2>0$	input prices
${\alpha }_1,{\alpha }_2$	${\alpha }_1,{\alpha }_2>0$	output elasticities of inputs
$q$	$q>0$	output level
$t$	$t>0$	the tax amount per unit product
$p$	$p>0$	price of the product received by producer
$p+t$	$p+t>0$	market price of the product
$a,b$	$a,b>0$	coefficients of demand function

shows the notation used in this paper.

3. Model

The first assumption in this model is the perfect information assumption. Perfect information means that the government is familiar with the monopolist’s profit function, i.e., the government knows the monopolist reaction: production level q, for every tax decision t chosen by the government.

Other assumptions of the model are as follows. The monopolist’s output quantity is described by the Cobb-Douglas production function of two inputs, labor and capital, denoted as $x_1$ and $x_2$, respectively, with output elasticities ${\alpha }_1$ and ${\alpha }_2$, respectively:

$$q=q\left(x_1,x_2\right)=x_1^{{\alpha }_1}{x}_2^{{\alpha }_2},$$

(1)

According to Zevelev [21], it is known that the firm’s cost function, in the case of the Cobb-Douglas technology, is given by:

$$c\left(q\right)=\omega q^{1/\epsilon },$$

(2)

where

$$\omega =\epsilon {\rho }^{-1/\epsilon }, \epsilon ={\alpha }_1+{\alpha }_2, \rho ={\left({\displaystyle \frac{{\alpha }_1}{w_1}}\right)}^{{\alpha }_1}{\left({\displaystyle \frac{{\alpha }_2}{w_2}}\right)}^{{\alpha }_2}$$

(3)

Additionally, the firm’s conditional input demand functions, in the case of the Cobb-Douglas technology, are given by:

$$x_i={\displaystyle \frac{{\alpha }_i}{w_i}}{\rho }^{-1/\epsilon }{q}^{1/\epsilon }, i=1, 2.$$

(4)

Furthermore, in our model, the monopolist receives price p for the product. However, the government charges a tax amount t per unit product. Therefore, the price buyers pay at the market for the product is equal to p + t. The demand on the market is linear, described by the inverse demand function:

$$p+t=a-bq \Rightarrow p=p\left(q\right)=a-t-bq,$$

(5)

where

$$a-t>0 \mathrm{and} a-t-bq>0.$$

(6)

The leader makes the tax decision per unit product t. However, in doing this, she has to take into account the response of the follower who, given the leader’s tax decision, does what is personally best for her. In other words, the government’s tax decision affects the supply in the market. Therefore, the leader’s tax revenue function is a function of t, as well as of output level q, and it is equal to:

$$T\left(t,q\right)=t\cdot q .$$

(7)

Variable t is controlled by the leader (i.e., government) while variable q is controlled by the follower (i.e., monopolist).

For the given tax decision t, the follower maximizes her profit function. Since the price the monopolist receives is equal to p, from (2) and (5), it follows that her profit function is equal to:

$$\Pi \left(t,q\right)=p\left(q\right)q-c\left(q\right)=\left(a-t\right)q-b{q}^2-\omega q^{1/\epsilon }.$$

(8)

The problem of determining optimal tax policy for profit maximizing monopolist with the Cobb-Douglas production function and linear demand function can now be stated as the following bilevel programming problem:

$$\underset{0<t<a}{\max }T\left(t,q\right)=tq,$$

(9)

$$\mathrm{s}.\mathrm{t}. \underset{q>0}{\max }\Pi \left(t,q\right)=\left(a-t\right)q-b{q}^2-\omega q^{1/\epsilon }.$$

(10)

The tax decision t made by the leader affects the choices available to the follower. Given the leader’s choice t, the follower acts independently and does what is personally best for her and not for the leader, i.e., the monopolist chooses the level of output that maximizes her profit.

4. Main Results

In order to solve the bilevel programming problem (9)–(10), one first needs to solve the follower’s problem for an arbitrary but fixed tax amount t and, thus, obtain the optimal output level as a function of t. Knowing the optimal follower’s decision for any choice of t, the leader’s problem can be solved. This gives the solution to the overall bilevel programming problem.

To solve the follower problem, one needs to find the follower’s problem stationary points, i.e., the optimal output level q for a given tax amount t.

Theorem 1. 

The stationary point of the follower’s problem (10) for an arbitrary but fixed tax amount t is defined implicitly as the solution of the equation:

$$(a-t)-2bq={\displaystyle \frac{\omega }{\epsilon }}{q}^{{\displaystyle \frac{1}{\epsilon }}-1},$$

(11)

where condition

$$(a-t)-2bq>0$$

(12)

must be satisfied.

Proof of Theorem 1. 

Let the tax amount t per unit product be arbitrary but fixed. For a given t, from the first order condition on the follower’s optimization problem given in the form (10), the following equation is obtained:

$${\Pi }_q\left(t,q\right)=(a-t)-2bq-{\displaystyle \frac{\omega }{\epsilon }}{q}^{{\displaystyle \frac{1}{\epsilon }}-1}=0,$$

(13)

from which Equation (11) follows directly. The right hand side of (11), ${\displaystyle \frac{\omega }{\epsilon }}{q}^{{\displaystyle \frac{1}{\epsilon }}-1}$ is positive as it is a product of positive factors $\omega$, $\epsilon$, and $q^{{\displaystyle \frac{1}{\epsilon }}-1}$, and because of the economic reasons. Therefore, the left-hand side of (11), $(a-t)-2bq$, must be positive, too, i.e., condition (12) must hold. This proves the theorem. □

The stationary point of the follower’s problem (10) cannot be obtained explicitly for any choice of the sum of the output elasticities $\epsilon ={\alpha }_1+{\alpha }_2$ because Equation (11) $(a-t)-2bq={\displaystyle \frac{\omega }{\epsilon }}{q}^{{\displaystyle \frac{1}{\epsilon }}-1}$ cannot be solved for q for arbitrary value of $\epsilon$. Instead, the solution is obtained by solving Equation (11) for given elasticities ${\alpha }_1$ and ${\alpha }_2$, i.e., for some special values of $\epsilon$.

Once the solution of the follower’s problem for an arbitrary but fixed tax amount t is known, the leader’s optimization problem can be solved. Let us illustrate the solution of the problem (9)–(10) through the following three numerical cases.

Corollary 1. 

(Constant returns to scale) For the constant returns to scale $\epsilon ={\alpha }_1+{\alpha }_2=1$, the bilevel programming problem (9)–(10) becomes:

$$\underset{0<t<a}{\max }T\left(t,q\right)=tq,$$

(14)

$$\mathrm{s}.\mathrm{t}. \underset{q>0}{\max }\Pi \left(t,q\right)=\left(a-\omega -t\right)q-b{q}^2.$$

(15)

The optimal solution of the problem (14)–(15) is given by:

$$t^{*}={\displaystyle \frac{a-\omega }{2}},$$

(16)

$$q^{*}={\displaystyle \frac{a-\omega }{4b}},$$

(17)

where $t^{*}$ and $q^{*}$  are the optimal leader’s and follower’s decisions, respectively. The optimal leader’s tax amount is equal to:

$$T^{*}={\displaystyle \frac{{\left(a-\omega \right)}^2}{8b}},$$

(18)

while the optimal follower’s profit and input levels are, respectively, equal to:

$${\Pi }^{*}={\displaystyle \frac{{\left(a-\omega \right)}^2}{16b}},$$

(19)

$$x_i^{*}={\displaystyle \frac{{\alpha }_i}{w_i\rho }}\cdot {\displaystyle \frac{a-\omega }{4b}}, i=1, 2.$$

(20)

Proof of Corollary 1. 

For $\epsilon =1$, Equation (13) becomes:

$$a-t-\omega -2bq=0,$$

(21)

which implies

$$q={\displaystyle \frac{a-t-\omega }{2b}}.$$

(22)

Note that (22) together with $q>0$ and $b>0$ imply the following condition:

$$a-t-\omega >0\iff t<a-\omega \iff t\in 〈0, a-\omega 〉.$$

(23)

Furthermore, since $b>0$ and, thus, $-2b<0$:

$${\Pi }_{qq}\left(t,q\right)={\displaystyle \frac{\partial }{\partial q}}\left(a-t-\omega -2bq\right)=-2b<0,$$

(24)

the follower’s profit function is concave. Overall, its domain is in q and the maximum profit is achieved. By substituting (22) into (14), we have:

$$T\left(t,{\displaystyle \frac{a-\omega -t}{2b}}\right)=-{\displaystyle \frac{t^2}{2b}}+{\displaystyle \frac{a-\omega }{2b}}t.$$

(25)

It is easy to see that the maximum of the function (25) equals (18) and that it is obtained for (16). Indeed, $T\prime \left(t,{\displaystyle \frac{a-\omega -t}{2b}}\right)=-{\displaystyle \frac{t}{b}}+{\displaystyle \frac{a-\omega }{2b}}=0$ and the stationary point is $t={\displaystyle \frac{a-\omega }{2}}$. Since $T″\left(t,{\displaystyle \frac{a-\omega -t}{2b}}\right)=-{\displaystyle \frac{1}{b}}<0$, T is concave overall its domain in t and the maximum tax is achieved. That is, the optimal leader’s tax is equal to (18) and it is achieved at level (16). Now, substituting $t={\displaystyle \frac{a-\omega }{2}}$ (16) into $q={\displaystyle \frac{a-t-\omega }{2b}}$ (22), we get $q^{*}={\displaystyle \frac{a-\omega }{4b}}$ (17). Finally, by substituting (16) and (17) into (15), we get (19). Furthermore, for constant returns to scale $\epsilon =1$, from (17) and (4) follows (20). This proves the theorem. □

Let us consider the following Example 1 as an illustration of the theoretical results given in Theorem 1 and Corollary 1.

Example 1. 

Let the input prices be $w_1=4$ and $w_2=9$, respectively, and let the output elasticities of inputs be ${\alpha }_1=0.3$ and ${\alpha }_2=0.7$, respectively. Furthermore, let the coefficients of demand function be $a=50$ and $b=30$, respectively.

Note that the production function (1):

$$q=q\left(x_1,x_2\right)=x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}=x_1^{0.3}{x}_2^{0.7},$$

(26)

has constant returns of scale, $\epsilon ={\alpha }_1+{\alpha }_2=0.3+0.7=1$, i.e., it is linearly homogeneous.

Note, also, that according to (3):

$$\rho ={\left({\displaystyle \frac{{\alpha }_1}{w_1}}\right)}^{{\alpha }_1}{\left({\displaystyle \frac{{\alpha }_2}{w_2}}\right)}^{{\alpha }_2}={\left({\displaystyle \frac{0.3}{4}}\right)}^{0.3}{\left({\displaystyle \frac{0.7}{9}}\right)}^{0.7}\approx 0.077,$$

(27)

and

$$\omega =\epsilon {\rho }^{-1/\epsilon }={\rho }^{-1}\approx 12.998,$$

(28)

since, for $\epsilon =1$:

$$\Pi \left(t,q\right)=\left(a-\omega -t\right)q-b{q}^2=\left(37.002-t\right)q-30q^2,$$

(29)

problem (14)–(15) or (9)–(10) can now be stated as the following bilevel programming problem:

$$\underset{0<t<a}{\max }T\left(t,q\right)=tq,$$

(30)

$$\mathrm{s}.\mathrm{t}. \underset{q>0}{\max }\Pi \left(t,q\right)=\left(37.002-t\right)q-30q^2.$$

(31)

First, we solve the follower’s problem (31) for a given but fixed tax amount $t$. From the first order conditions on (31):

$${\Pi }_q\left(t,q\right)=(37.002-t)-60q=0,$$

(32)

we obtain the optimal output level

$$q^{*}(t)={\displaystyle \frac{37.002-t}{60}}.$$

(33)

Furthermore, since ${\Pi }_{qq}\left(t,q\right)=-60<0$, the follower’s profit function is concave. Overall, its domain is in q and the maximum profit is achieved.

By substituting $q^{*}(t)={\displaystyle \frac{37.002-t}{60}}$ into the leader’s optimization problem (30), that is, $\underset{0<t<a}{\max }T\left(t,q\right)=tq$, we get the following tax revenue function of only tax amount t:

$$T\left(t,{\displaystyle \frac{37.002-t}{60}}\right)={\displaystyle \frac{-t^2+37.002t}{60}}.$$

(34)

The maximum of the leader’s tax revenue function $T(t)$ is obtained for $T\prime (t)=-2t+37.002=0$ because $T″(t)=-2<0$. It is easy to see that $t^{*}\approx 18.501$ and $T^{*}\approx 5.705$ (Figure 1).

Furthermore, the optimal output level (Figure 1) is equal to:

$$q^{*}(t^{*})={\displaystyle \frac{37.002-t^{*}}{60}}={\displaystyle \frac{37.002-18.501}{60}}\approx 0.308.$$

(35)

Finally, by substituting $t^{*}=18.501$ and $q^{*}=0.308$ back into the follower’s profit function, we get the optimal follower’s profit (Figure 1):

$${\Pi }^{*}\left(t^{*},q^{*}\right)=\left(37.002-t^{*}\right)q^{*}-30{(q^{*})}^2\approx 2.852.$$

(36)

Furthermore, for constant returns to scale $\epsilon =1$, from the optimal output level $q^{*}=0.308$ and (4), the optimal inputs (labor and capital) levels follow:

$$x_1^{*}={\displaystyle \frac{{\alpha }_1}{w_1}}{\rho }^{-1/\epsilon }{(q^{*})}^{1/\epsilon }={\displaystyle \frac{0.3}{4}}\cdot {0.077}^{-1}\cdot {0.308}^1\approx 0.301,$$

(37)

and

$$x_2^{*}={\displaystyle \frac{{\alpha }_2}{w_2}}{\rho }^{-1/\epsilon }{(q^{*})}^{1/\epsilon }={\displaystyle \frac{0.7}{9}}\cdot {0.077}^{-1}\cdot {0.308}^1\approx 0.312.$$

(38)

Figure 2 shows production function (26) isoquants and its optimal input levels $x_1^{*}$ and $x_2^{*}$ given in (37) and (38).

Corollary 2. 

(Decreasing returns of scale) For the decreasing returns to scale $\epsilon ={\alpha }_1+{\alpha }_2=0.5$, the bilevel programming problem (9)–(10) becomes:

$$\underset{0<t<50}{\max }T\left(t,q\right)=tq,$$

(39)

$$\mathrm{s}.\mathrm{t}. \underset{q>0}{\max }\Pi \left(t,q\right)=-\left(b+\omega \right)q^2+\left(a-t\right)q.$$

(40)

The optimal solution of the problem (14)–(15) is given by:

$$t^{*}={\displaystyle \frac{a}{2}},$$

(41)

$$q^{*}={\displaystyle \frac{a}{4\left(b+\omega \right)}},$$

(42)

where $t^{*}$ and $q^{*}$ are the optimal leader’s and follower’s decisions, respectively. The optimal leader’s tax is equal to:

$$T^{*}={\displaystyle \frac{a^2}{8\left(b+\omega \right)}},$$

(43)

while the optimal follower’s profit and input levels are, respectively, equal to:

$${\Pi }^{*}={\displaystyle \frac{a^2}{16\left(b+\omega \right)}},$$

(44)

$$x_i^{*}={\displaystyle \frac{{\alpha }_i}{w_i}}\cdot {\displaystyle \frac{a^2}{4^2{\rho }^2{\left(b+\omega \right)}^2}}, i=1, 2.$$

(45)

Proof of Corollary 2. 

For $\epsilon =0.5$ Equation (13) becomes:

$$a-t-2\left(b+\omega \right)q=0,$$

(46)

which implies

$$q={\displaystyle \frac{a-t}{2\left(b+\omega \right)}}.$$

(47)

Note that (47) implies the following condition:

$$a-t>0 \iff t\in 〈0, a〉.$$

(48)

Furthermore, since:

$${\Pi }_{qq}\left(t,q\right)={\displaystyle \frac{\partial }{\partial q}}\left(a-t-2\left(b+\omega \right)q\right)=-2\left(b+\omega \right)<0,$$

(49)

the follower’s profit function is concave. Overall, its domain is in q and the maximum profit is achieved. By substituting (47) into (39), we have:

$$T\left(t,{\displaystyle \frac{a-t}{2\left(b+\omega \right)}}\right)={\displaystyle \frac{1}{2\left(b+\omega \right)}}\left(-t^2+at\right).$$

(50)

Similary to tax function in expression (25), it is easy to see that the maximum of the function (50) equals (43) and that it is obtained for (41). That is, the optimal leader’s tax is equal to (50) and it is achieved at level (41). Now, substituting (41) into (47), we get (42). Finally, by substituting (41) and (42) into (40), we get (44). Furthermore, for decreasing returns to scale $\epsilon =0.5$, from (42) and (4) follows (45). This proves the theorem. □

Let us consider the following Example 2 as an illustration of the theoretical results given in Theorem 1 and Corollary 2.

Example 2. 

Consider the problem used in Example 1. We use the same input prices $w_1=4$ and $w_2=9$, respectively, and the same coefficients of demand function $a=50$ and $b=30$, respectively. In this example, the output elasticity ${\alpha }_1$ is again set to ${\alpha }_1=0.3$, but this time the output elasticity ${\alpha }_2$ is set to ${\alpha }_2=0.2$, which give the decreasing returns to scale $\epsilon ={\alpha }_1+{\alpha }_2=0.3+0.2=0.5$.

Note that the production function (1):

$$q=q\left(x_1,x_2\right)=x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}=x_1^{0.3}{x}_2^{0.2},$$

(51)

is homogeneous with decreasing returns to scale equal to $\epsilon ={\alpha }_1+{\alpha }_2=0.4+0.1=0.5$.

Note also that, according to (3):

$$\rho ={\left({\displaystyle \frac{{\alpha }_1}{w_1}}\right)}^{{\alpha }_1}{\left({\displaystyle \frac{{\alpha }_2}{w_2}}\right)}^{{\alpha }_2}={\left({\displaystyle \frac{0.3}{4}}\right)}^{0.3}{\left({\displaystyle \frac{0.2}{9}}\right)}^{0.2}\approx 0.215,$$

(52)

and

$$\omega =\epsilon {\rho }^{-1/\epsilon }=0.5{\rho }^{-2}\approx 10.845.$$

(53)

Since, for $\epsilon =0.5$:

$$\Pi \left(t,q\right)=-\left(b+\omega \right)q^2+\left(a-t\right)q=-40.845q^2+(50-t)q,$$

(54)

problem (39)–(40) or (9)–(10) can now be stated as the following bilevel programming problem:

$$\underset{0<t<50}{\max }T\left(t,q\right)=tq,$$

(55)

$$\mathrm{s}.\mathrm{t}. \underset{q>0}{\max }\Pi \left(t,q\right)=(50-t)q-40.845q^2.$$

(56)

First, we solve the follower’s problem (56) for a given but fixed tax amount $t$. From the first order conditions on (56):

$${\Pi }_q\left(t,q\right)=-81.690q+50-t=0,$$

(57)

we obtain the optimal output level:

$$q^{*}(t)={\displaystyle \frac{50-t}{81.690}}.$$

(58)

Furthermore, since ${\Pi }_{qq}\left(t,q\right)=-81.690<0$, the follower’s profit function is concave. Overall, its domain is in q and the maximum profit is achieved.

By substituting $q^{*}(t)={\displaystyle \frac{50-t}{81.690}}$ into the leader’s optimization problem (55) $\underset{0<t<a}{\max }T\left(t,q\right)=tq$, we get the following tax revenue function of only tax amount $t$:

$$T\left(t,{\displaystyle \frac{50-t}{81.690}}\right)={\displaystyle \frac{-t^2+50t}{81.690}}.$$

(59)

The maximum of the leader’s tax revenue function $T(t)$ is obtained for $T\prime (t)=-2t+50=0$ because $T″(t)=-2<0$. It is easy to see that $t^{*}=25$ and $T^{*}=7.651$ (Figure 3).

Furthermore, the optimal output level (Figure 3) is equal to:

$$q^{*}(t^{*})={\displaystyle \frac{50-t^{*}}{81.690}}={\displaystyle \frac{50-25}{81.690}}\approx 0.306.$$

(60)

Finally, by substituting $t^{*}=25$ and $q^{*}=0.306$ back into the follower’s profit function, we get the optimal follower’s profit (Figure 3):

$${\Pi }^{*}\left(t^{*},q^{*}\right)=\left(50-t^{*}\right)q^{*}-40.845{(q^{*})}^2=3.825.$$

(61)

Furthermore, for decreasing returns to scale $\epsilon =0.5$, the optimal inputs (labor and capital) levels are derived from the optimal output level $q^{*}=0.306$ and (4):

$$x_1^{*}={\displaystyle \frac{{\alpha }_1}{w_1}}{\rho }^{-1/\epsilon }{(q^{*})}^{1/\epsilon }={\displaystyle \frac{0.3}{4}}\cdot {0.215}^{-2}\cdot {0.306}^2\approx 0.152,$$

(62)

and

$$x_2^{*}={\displaystyle \frac{{\alpha }_2}{w_2}}{\rho }^{-1/\epsilon }{(q^{*})}^{1/\epsilon }={\displaystyle \frac{0.1}{9}}\cdot {0.215}^{-2}\cdot {0.306}^2\approx 0.045.$$

(63)

Figure 4 shows production function (51) isoquants and its optimal input levels $x_1^{*}$ and $x_2^{*}$ given in (62) and (63).

According to Lukač [22], in a case where a follower is a perfect competitor, the follower’s maximum profit cannot be achieved for increasing returns of scale, i.e., for $\epsilon >1$. However, in our model (9)–(10), since the follower is a monopolist with the linear demand (5), the solution of the bilevel programming problem (9)–(10) can be achieved for certain numerical values of parameters in the model. Let us consider the following Example 3 as an illustration of this situation and the theoretical results given in Theorem 1.

Example 3. 

Consider the problem used in Example 1 and Example 2. We use the same input prices $w_1=4$ and $w_2=9$, respectively, and the same coefficients of demand function $a=50$ and $b=30$, respectively. In this example, the output elasticity ${\alpha }_1$ is again set to ${\alpha }_1=0.3$, but this time the output elasticity ${\alpha }_2$ is set to ${\alpha }_2=1.7$, i.e., we consider increasing returns to scale $\epsilon ={\alpha }_1+{\alpha }_2=0.3+1.7=2$.

Note that the production function (1):

$$q=q\left(x_1,x_2\right)=x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}=x_1^{0.3}{x}_2^{1.7},$$

(64)

is homogeneous with increasing returns to scale $\epsilon ={\alpha }_1+{\alpha }_2=0.3+1.7=2$.

According to (3):

$$\rho ={\left({\displaystyle \frac{{\alpha }_1}{w_1}}\right)}^{{\alpha }_1}{\left({\displaystyle \frac{{\alpha }_2}{w_2}}\right)}^{{\alpha }_2}={\left({\displaystyle \frac{0.3}{4}}\right)}^{0.3}{\left({\displaystyle \frac{1.7}{9}}\right)}^{1.7}\approx 0.027,$$

(65)

and

$$\omega =\epsilon {\rho }^{-1/\epsilon }=2{\rho }^{-1/2}\approx 12.162.$$

(66)

Since, for $\epsilon =2$:

$$\Pi \left(t,q\right)=-b{q}^2+\left(a-t\right)q-\omega q^{1/2}=-30q^2+(50-t)q-12.162q^{1/2},$$

(67)

problem (39)–(40) or (9)–(10) can now be stated as the following bilevel programming problem:

$$\underset{0<t<50}{\max }T\left(t,q\right)=tq,$$

(68)

$$\mathrm{s}.\mathrm{t}. \underset{q>0}{\max }\Pi \left(t,q\right)=-30q^2+(50-t)q-12.162q^{1/2}.$$

(69)

First, we solve the follower’s problem (69) for a given but fixed tax amount $t$. From the first order conditions on (69):

$${\Pi }_q\left(t,q\right)=-60q+50-t-6.081q^{-1/2}=0,$$

(70)

we should obtain the optimal output level $q^{*}(t)$. Note that equation (70) cannot be solved for $q$ explicitly. Solving the problem (68)–(69) numerically, we obtain the optimal solution $t^{*}=17.556$ and $q^{*}(t^{*})=0.375$ with $T^{*}=6.589$.

Finally, by substituting $t^{*}=17.556$ and $q^{*}=0.375$ back into the follower’s profit function, we get the optimal follower’s profit:

$${\Pi }^{*}\left(t^{*},q^{*}\right)=-30{(q^{*})}^2+(50-t^{*})q^{*}-12.162{(q^{*})}^{1/2}=0.500.$$

(71)

Furthermore, for increasing returns to scale $\epsilon =2$, the optimal inputs (labor and capital) levels follow from the optimal output level $q^{*}=0.375$ and (4):

$$x_1^{*}={\displaystyle \frac{{\alpha }_1}{w_1}}{\rho }^{-1/\epsilon }{(q^{*})}^{1/\epsilon }={\displaystyle \frac{0.3}{4}}\cdot {0.027}^{-1/2}\cdot {0.375}^{1/2}\approx 0.279,$$

(72)

and

$$x_2^{*}={\displaystyle \frac{{\alpha }_2}{w_2}}{\rho }^{-1/\epsilon }{(q^{*})}^{1/\epsilon }={\displaystyle \frac{0.1}{9}}\cdot {0.027}^{-1/2}\cdot {0.375}^{1/2}\approx 0.704.$$

(73)

5. Conclusions

This paper formulates the bilevel problem of determining the optimal tax amount, that is, of maximizing a government’s tax revenue function in the presence of a profit maximizing monopolist whose output is described by the Cobb-Douglas production function. The necessary conditions are first derived for the case of two inputs. Necessary conditions can be derived only in an implicit form, which needs to be solved for different values of output elasticities with respect to inputs. The exact form of the tax revenue function, as well as optimal tax amount and optimal input levels, are derived for the case when decreasing returns to scale is equal to 0.5 and for the case of the constant returns to scale. Additionally, since our model assumes a monopolist with linear demand as a follower, we illustrated through a numerical example that the optimal solution of our model can be achieved even in the case of increasing returns to scale hold, which is not the case if a follower is a perfect competitor.

In future work, we would like to see more about the solution of increasing returns to scale scenario, check CES (constant elasticity of substitution) and VES (variable elasticity of substitution) production functions instead of the Cobb-Douglas one, use nonlinear demand functions, and find some read data and real-world examples.