On Joan Robinson’s Output Symmetry Theorems with Various Taxations under Third-Degree Price Discrimination: A Generalization

Abstract

This paper incorporates more general cases with a new class of constantly adjusted concavity demand curves and includes three types of taxes. To verify the output symmetry under various forms of taxation, we simulate both linear and constant elasticity demand functions under the unit, demand ad valorem, and cost ad valorem taxes. If all the demand functions in the submarkets are linear, the total outputs are identical under both uniform pricing and third-degree price discrimination. Furthermore, if all the weak market demand curves are strictly “Robinson-concave” and all the strong market demand curves are strictly “Robinson-convex” or linear, then the total output under price discrimination exceeds that under uniform pricing, and vice versa. While different taxes lead to higher costs, the cost pass-through changes the prices of the products, and the change of total output still depends on the curvature of the demand curve. Therefore, the curvature of the demand curve remains the main determinant of changes in output. Our study provides a theoretical basis for market intervention in price discrimination.

1. Introduction

The use of discriminatory pricing by firms to maximize their profits in imperfect markets is a well-studied topic. In general, a first-degree price discrimination model involves different prices being charged for each individual, requiring detailed personal information while gaining more popularity as big data has become quite plausible. Second-degree price discrimination charges different prices for different volumes of usage. Higher unit prices are charged for the first block of usage, and thereafter, the price is gradually lowered as the consumer/client uses more of the good. In third-degree price discrimination, a monopoly firm can charge different markets different prices according to each market’s income and price elasticity (dependent on consumers’ loyalty). For example, the price of Glucophage (500 mg, 30 tablets) sells for $40.20 in the US but only $8.70 in Ecuador.

It is well- known that third-degree price discrimination is widely applied in the business world. The impact of third-degree price discrimination on market performance is widely discussed in the literature; for example, see Chao and Nahata [1], Miklós-Thal and Shaffer [2], Bergemann et al. [3], and Zhang et al. [4]. This is not a trivial theoretical interest; rather, it erodes consumer household disposable income significantly. For example, according to the 2020 International Living report, the prices the brand-name drug Lipitor (40 mg, 30 tablets) in the US were $452.8 and in Mexico $15.87; the prices of Ventolin HFA (90 mcg, 1 inhaler) were $65.21 (US), $10.04 (Panama), and $3.23 (Spain); and the prices of Synthroid (100 mcg, 100 tablets) were $131.92(US), $3.40 (Thailand), and $2.08 (Vietnam). Similar examples can be found in international arms sales: much higher prices are charged to rich countries such as Saudi Arabia or needy governments such as Japan or India; this is not petty cash but rather in tens or hundreds of billions.

In addition, the impact of taxation on output is also an important research focus. Generally speaking, the most commonly used tax instruments are: per unit tax, demand ad valorem tax, and cost ad valorem tax. In the gasoline market, an example of a per unit tax is a 25 cents per gallon gasoline tax; taking a demand ad valorem tax for example, there is a 6% sales tax on the price that consumers pay; finally, an example of a cost ad valorem tax is a 10% tax on the cost that suppliers pay. The impact of different tax instruments on market output and its welfare effects has always been an important policy comparison in economic analysis, with third-degree price discrimination and its effects on social welfare featured as a debating topic for economists, firms, consumers, antitrust bodies, and regulators.

This paper studies the economic consequences of third-degree price discrimination (Pigou, Robinson) and examines whether the output theorems first analyzed by Robinson are preserved and verify the output symmetry under various types of taxes.

It is organized as follows: Section 2 provides a critical review of the literature. Section 3 examines the output theorem, based on the mean-value theorem approach by Shih et al. [5], under the unit tax. Section 4 and Section 5 explore the output theorems under the demand and cost ad valorem taxes, respectively; and the final section contains concluding remarks.

2. Literature Review

There have been extensive analyses of the output effect under the Pigouvian third-degree price discrimination (Pigou [6]) in the literature. As is well known, the adjusted concavity criterion established by Joan Robinson [7] can lead to erroneous conclusions. Subsequent revisions include the slope ratio criterion by Edwards [8], the Lagrange techniques by Silberberg [9], Finn [10], Löfgren [11], and as suggested by Leontief [12], and the criterion of the SMR-DMR by Smith and Formby [13]. In addition, Schmalensee [14] observed that if all demand curves possess the same type of curvature, there is no simple way to tell if the discrimination will raise or lower total outputs. Greenhut and Ohta [15] provided a counterexample consisting of two constant elasticity demand functions. Willig [16] showed that all consumers and the producer are at least as well off with a particular pricing scheme than with flat rate pricing. Moreover, Formby et al. [17] demonstrated that monopolistic discrimination can increase output over a wide range of elasticity values for the n-market case. To a somewhat lesser extent, several papers extended the analysis in the spatial context (Greenhut and Ohta [18]; Yang and Cheng [19]). Specifically, Ikeda and Toshimitsu. [20] showed that total output and welfare can increase with discrimination in the content of endogenous quality choice. Zhang et al. [4] provide a critical literature review on third-degree price discrimination.

A strong (weak) market is defined as one in which, when the discrimination pricing is employed, the price will rise (fall) in the market. That is, the strong market is the market in which the price under third-degree price discrimination is greater than that under uniform pricing. Layson [21] pointed out that a price discrimination will be socially beneficial if the demand curve is highly convex in the weak market or if the demand curve is highly concave in the strong market. Aguirre et al. [22] showed that if the discriminatory prices are sufficiently close and the inverse demand curvature is more convex in the weak market than in the strong market, price discrimination will lead the aggregate output and social welfare to increase. Adachi and Matsushima [23] pointed out that social welfare improves only from the increase in the firms’ profit, but consumer surplus never improves under third-degree price discrimination in oligopolistic competition with horizontal product differentiation. Chen and Schwartz [24] found that while third-degree price discrimination without cost differences would increase average prices, discrimination pricing with cost differences would not. Aguirre [25] argued that competitive pressure and the number of firms are more important than the relative demand curvature in the influence of price discrimination on social welfare. Recently, Adachi [26] has shown that a third-degree price discrimination may raise or lower the aggregate output in a differentiated oligopoly.

However, Shih et al. [5], applying the mean-value theorem, were able to show that the adjusted concavity or slope ratio alone cannot determine the output effect. They delineate the EV approach in which the shape of the marginal revenue curves also plays an important role. Since total output must increase under third-degree price discrimination in order for welfare to increase (Schmalensee [14]; Varian [27]; and Schwartz [28]), a study on output effect is warranted, especially with the presence of different taxes. Yang [29] studies the impact of different taxations on the output and welfare under third-degree price discrimination. The purposes of this paper are twofold: (i) to examine if the output theorems first analyzed by Robinson are preserved and (ii) to verify the output symmetry under various types of taxes.

3. The Output Theorem under the Unit Tax

The mean-value theorem approach is considered to be more general than either Robinson’s adjusted concavity or Edward’s slope ratio criterion. It covers a wide class of demand functions, such as constant adjusted concavity demand relations including linear and constant elasticity demand functions (Shih et al. [5], p. 156). Therefore, we employ their approach to validate the theorems in this paper.

Following Shih et al. [5], let the profit function in the ith submarket with a constant marginal cost (the constancy of the marginal cost is adopted here for the purpose of simplicity and comparison; as argued by Robinson [7] and proven by Silberberg [9], the slope of the marginal cost curve affects only the magnitudes but not the direction of the output change) be given by

$${\pi }_i\left(q_i\right)=\left[p_i\left(q_i\right)-c-u\right]q_i$$

(1)

where $p_i\left(q_i\right)$ is the inverse demand function of the ith market; $q_i$ is the corresponding unit sales; $c$ is a constant unit cost; and $u$ is a per unit tax.

Assuming that all sales are positive in $n$ markets under monopolistic discrimination, we have

$${\pi }_i^{\prime }\left(q_i^{\ast }\right)=p_i^{\prime }\left(q_i^{\ast }\right)q_i^{\ast }+p_i\left(q_i^{\ast }\right)-c-u=0$$

(2)

and hence

$$\begin{array}{cc}{q}_i^{\ast }=\left[c+u-p_i\left(q_i^{\ast }\right)\right]/p_i^{\prime }\left(q_i^{\ast }\right)& \forall i\in N\end{array}$$

(3)

where N = {1,2,…,n} is an integer set. Alternatively, under uniform pricing, the monopolist maximizes the after-tax profit by choosing a single price $p^o$ with the first-order condition:

$$d\left[{\displaystyle \sum _{i\in N}{\pi }_i\left(q_i\right)}\right]/d{p}^o={\displaystyle \sum _{i\in N}\left\{q_i^o+\left[\left(p_i\left(q_i^o\right)-c-u\right)/p_i^{\prime }\left(q_i^o\right)\right]\right\}}=0$$

(4)

where $q_i^o$ is the optimum output in market i under uniform pricing, and p_i = p^o $\forall i\in N$. Solving for $q_i^o$ yields:

$$\begin{array}{cc}{q}_i^o=\left[c+u-p_i\left(q_i^o\right)\right]/p_i^{\prime }\left(q_i^o\right)& \forall i\in N\end{array}$$

(5)

Summing Equations (3) and (5) over $n$ markets and subtracting between them, we have

$$Q^o-Q^{\ast }={\displaystyle \sum _{i\in N}\left\{\left[\left(c+u-p_i\left(q_i^o\right)\right)/p_i^{\prime }\left(q_i^o\right)\right]-\left[\left(c+u-p_i\left(q_i^{\ast }\right)\right)/p_i^{\prime }\left(q_i^{\ast }\right)\right]\right\}}$$

(6)

It follows immediately from the mean-value theorem that for a given value of ${\overline{q}}_i$ between $q_i^o$ and $q_i^{\ast }$, there exists:

$$\begin{gathered} \left[\left(c+u-p_i\left(q_i^o\right)\right)/p_i^{\prime }\left(q_i^o\right)\right]-\left[\left(c+u-p_i\left(q_i^{\ast }\right)\right)/p_i^{\prime }\left(q_i^{\ast }\right)\right] \\ =\left(q_i^o-q_i^{\ast }\right)\left\{\left[p_i\left({\overline{q}}_i\right)-c-u\right]\left[p_i^{″}\left({\overline{q}}_i\right)/{\left[p_i^{\prime }\left({\overline{q}}_i\right)\right]}^2\right]-1\right\} \end{gathered}$$

(7)

Substituting Equation (7) into Equation (6) and making use of ${\displaystyle \sum _{i\in N}{q}_i^o=Q^o}$ and ${\displaystyle \sum _{i\in N}{q}_i^{\ast }=Q^{\ast }}$ gives:

$$Q^o-Q^{\ast }=\left(1/2\right){\displaystyle \sum _{i\in N}\left(q_i^o-q_i^{\ast }\right)\left\{\left[p_i\left({\overline{q}}_i\right)-c-u\right]\left[p_i^{″}\left({\overline{q}}_i\right)/{\left[p_i^{\prime }\left({\overline{q}}_i\right)\right]}^2\right]\right\}}$$

(8)

Now, we define the strong market as the market in which the price under third-degree price discrimination is greater than that under uniform pricing. The converse holds for the weak market. Note that such a definition may not be viable for a class of demand functions prescribed by Nahata et al. [30], DeGraba [31], and Malueg [32]. This class of demand functions is more general since it allows for a class of polynomial demand functions with some positive slopes of marginal revenue and multiple profit equilibria (Formby et al. [33]). Such a distinct possibility cannot be easily dismissed. For instance, Nahata et al. [30] presented the cubic demand functions to show that the presence of third-degree price discrimination can either raise or lower all prices in the submarkets. In such a case, the outputs, hence the prices, are not analytically solvable for a polynomial degree greater than 3. We discuss this possibility briefly in the Conclusion.

If the unit tax $u$ is not prohibitive, it is reasonable to assume $p_i\left({\overline{q}}_i\right)-c-u>0$ to ensure that the profit of the firms is always non-negative. Under this assumption, we state the following Proposition 1.

Proposition 1.

If all the demand functions in submarkets are linear in the presence of a unit tax, the total outputs are identical under both uniform pricing and third-degree price discrimination. Furthermore, if all weak market demand curves are strictly “Robinson-concave” (or $p_i^{″}\left(\overline{q}\right)>0$) and all strong market demand curves are strictly “Robinson-convex” (or) or linear, then the total output under the discrimination exceeds that under uniform pricing, and vice versa.

The proof is straightforward since $p_i^{″}\left(\overline{q}\right)=0$ in the case of linear demand functions. Moreover, in the strong markets, $q_i^o>{\overline{q}}_i>q_i^{\ast }$ must hold, whereas $q_i^o<{\overline{q}}_i<q_i^{\ast }$ holds in all weak markets. The equation must be negative in sign.

For the scenario in which all the demand functions are either strictly Robinson concave or strictly Robinson convex, we can expand the marginal profit function ${\pi }_i^{\prime }\left(q_i\right)$ by the mean-value theorem to yield:

$${\pi }_i^{\prime }\left(q_i^o\right)={\pi }_i^{\prime }\left(q_i^{\ast }\right)+\left(q_i^o-q_i^{\ast }\right){\pi }_i^{″}\left({\tilde{q}}_i\right)$$

(9)

where ${\tilde{q}}_i\in \left(q_i^o,q_i^{\ast }\right)$. Dividing Equation (9) by $p_i{}^{\prime }\left(q_i^o\right)\ne 0$, using ${\pi }_i^{\prime }\left(q_i\right)=0$ and summing it over $n$ markets, we have

$${\displaystyle \sum _{i\in N}\left(q_i^o-q_i^{\ast }\right)\left[{\pi }_i^{″}\left({\tilde{q}}_i\right)/p_i^{\prime }\left(q_i^o\right)\right]}={\displaystyle \sum _{i\in N}\left(q_i^o-q_i^{\ast }\right)\left(1/E_i^o{V}_i\right)}=0$$

(10)

where $E_i^o=p_i^{\prime }\left(q_i^o\right)/{\pi }_i^{″}\left(q_i^o\right)$ and $V_i={\pi }_i^{″}\left(q_i^o\right)/{\pi }_i^{″}\left({\tilde{q}}_i\right)$ for all ${\pi }_i^{″}\ne 0$.

Equation (10) can be partitioned according to the strong market S and the weak market W such that $\mathrm{S}\cup \mathrm{W}=N$ and $\mathrm{S}\cap \mathrm{W}=\varphi$, or

$${\displaystyle \sum _{j\in \mathrm{S}}\left(q_j^o-q_j^{\ast }\right)}\left(1/E_j^o{V}_j\right)={\displaystyle \sum _{k\in \mathrm{W}}\left(q_k^{\ast }-q_k^o\right)}\left(1/E_k^o{V}_k\right)$$

(11)

The result is exactly the same as that established by Shih et al. [5] (p. 153). $E_i^o$ is the slope ratio and $V_i$ is completely determined by ${\pi }_i^{″}\left(q_i\right)<0$ (in their analysis, the sign of ${\pi }_i^{″}\left(q_i\right)<0$ is assumed; obviously, it does not hold in the case of demand functions presented by Nahata et al. [30]) or the slope of the marginal revenue curves. It is easy to see, as shown by Shih et al. [5] (Equations (16) and (17)), that

$$E_i^o\frac{>}{<}1/2 \mathrm{according} \mathrm{to} p_i^{″}\left(q_i\right)\frac{>}{<}0,$$

(12)

$$V_j\frac{<}{>}1 \mathrm{if} \begin{array}{cc}{\pi }_j^{‴}\left(q_j\right)\frac{>}{<}0& \forall j\in \mathrm{S}\end{array}$$

(13)

$$V_k\frac{>}{<}1 \mathrm{if} \begin{array}{cc}{\pi }_k^{‴}\left(q_k\right)\frac{>}{<}0& \forall k\in \mathrm{W}\end{array}.$$

(14)

where ${\pi }_i^{‴}\left(q_i\right)=p_i^{‴}\left(q_i\right)q_i+3p_i^{″}\left(q_i\right)$. Within this framework, it can be seen that $E_i^o=1/2$ and $V_j=V_k=1$ in the case of linear demand functions. Hence, from Equation (11), it follows that $Q^o=Q^{\ast }$. In light of this, we state the following Proposition 2:

Proposition 2.

In the case of the unit tax, if all the values of $E_j^o{V}_j$and$E_k^o{V}_k$are the same, the total output is identical under both uniform pricing and price discrimination. Furthermore, if the value of $E_k^o{V}_k$in each of the weak markets is greater than that of $E_j^o{V}_j$in all of the strong markets, the total output is greater under price discrimination than that under uniform pricing, and vice versa.

The results obtained by Shih et al. [5] are preserved in the case of the unit tax as long as the tax rate is not prohibitive. In the next section, the output theorem under the demand ad valorem tax will be presented.

4. The Output Theorem under the Demand Ad Valorem Tax

The objective function of a discriminatory monopolist is to maximize the after-tax profit in each market or

$$\mathrm{Maximize} {\pi }_i\left(q_i\right)=\left[\left(1-v\right)p_i\left(q_i\right)-c\right]q_i$$

(15)

where $v$ is an ad valorem tax rate with $0<v<1$.

The first-order condition and the resulting optimum output are:

$$\begin{gathered} {\pi }_i^{\prime }\left(q_i\right)=\left(1-v\right)\left[p_i^{\prime }\left(q_i^{\ast }\right)q_i^{\ast }+p_i\left(q_i^{\ast }\right)\right]-c=0 \\ \mathrm{and} \begin{array}{cc}{q}_i^{\ast }=\left[c-\left(1-v\right)p_i\left(q_i^{\ast }\right)\right]/\left(1-v\right)p_i^{\prime }\left(q_i^{\ast }\right)& \forall i\in N\end{array} \end{gathered}$$

(16)

Similarly, the optimum output under uniform pricing can be shown as:

$$\begin{array}{cc}{q}_i^o=\left[c-\left(1-v\right)p_i\left(q_i^o\right)\right]/\left(1-v\right)p_i^{\prime }\left(q_i^o\right)& \forall i\in N\end{array}$$

(17)

where $p_i=p^o \forall i\in N$. Summing Equations (16) and (17) over $n$ markets and taking the difference between them produces

$$Q^o-Q^{\ast }={\displaystyle \sum _{i\in N}\left\{\left[\left(c-\left(1-v\right)p_i\left(q_i^o\right)\right)/\left(1-v\right)p_i^{\prime }\left(q_i^o\right)\right]-\left[\left(c-\left(1-v\right)p_i\left(q_i^{\ast }\right)\right)/\left(1-v\right)p_i^{\prime }\left(q_i^{\ast }\right)\right]\right\}}$$

(18)

Invoking the mean-value theorem for a ${\overline{q}}_i$ between $q_i^o$ and $q_i^{\ast }$, we have

$$\begin{gathered} \left[\left(c-\left(1-v\right)p_i\left(q_i^o\right)\right)/\left(1-v\right)p_i^{\prime }\left(q_i^o\right)\right]-\left[\left(c-\left(1-v\right)p_i\left(q_i^{\ast }\right)\right)/\left(1-v\right){p^{\prime }}_i\left(q_i^{\ast }\right)\right] \\ =\left(q_i^o-q_i^{\ast }\right)\left\{\left[\left(\left(1-v\right)p_i\left({\overline{q}}_i\right)-c\right){p^{″}}_i\left({\overline{q}}_i\right)/\left(1-v\right){\left[p_i^{\prime }\left({\overline{q}}_i\right)\right]}^2\right]-1\right\} \end{gathered}$$

(19)

Substituting Equation (19) into Equation (18) yields:

$$Q^o-Q^{\ast }=\left(1/2\right){\displaystyle \sum _{i\in N}\left(q_i^o-q_i^{\ast }\right)\left\{\left[\left(1-v\right)p_i\left({\overline{q}}_i\right)-c\right]\left[p_i^{″}\left({\overline{q}}_i\right)/\left(1-v\right){\left[p_i^{\prime }\left({\overline{q}}_i\right)\right]}^2\right]\right\}}$$

(20)

Once again, as long as the ad valorem tax rate $v$ is not too high, we can reasonably assume that  $(1-v)p_i\left({\overline{q}}_i\right)-c>0$ to ensure that the profit of the firms is always non-negative. Thus, we state the following Proposition 3.

Proposition 3.

In the case of the demand ad valorem tax, if all the demand schedules in all the submarkets are linear, the total outputs are identical under both uniform pricing and third-degree price discrimination. Moreover, if all the weak market demand curves are strictly “Robinson-concave” and all the strong market demand curves are strictly “Robinson-convex” or linear, the total output under discrimination exceeds that under uniform pricing, and vice versa.

The proof is exactly the same as that in the previous section. Note that if one submarket suffers a profit/loss due to the ad valorem tax, Proposition 3 may not be valid.

If all the demand schedules are either Robinson concave or Robinson convex, expanding the marginal profit function yields:

$${\pi }_i^{\prime }\left(q_i^o\right)={\pi }_i^{\prime }\left(q_i^{\ast }\right)+\left(q_i^o-q_i^{\ast }\right){\pi }_i^{″}\left({\tilde{q}}_i\right)$$

(21)

By the exact same procedure as that of the demand ad valorem tax, we have

$${\displaystyle \sum _{j\in \mathrm{S}}\left(q_j^o-q_j^{\ast }\right)}\left(1/E_j^o{V}_j\right)={\displaystyle \sum _{k\in \mathrm{W}}\left(q_k^{\ast }-q_k^o\right)}\left(1/E_k^o{V}_k\right)$$

(22)

Again, $E_i^o=1/2$ and $V_i=1$ in the case of a linear demand curve, and $V_i$ is always positive due to ${\pi }_i^{″}\left(q_i\right)<0$. We state the following Proposition 4.

Proposition 4.

In the case of the demand ad valorem tax, if all $E_i^o{V}_i$s are equal, the total output remains unchanged under both pricing modes. If the product of $E_k^o{V}_k$in each of the weak markets is greater than that of$E_j^o{V}_j$in all of the strong markets, the total output under price discrimination exceeds that under uniform pricing, and vice versa.

The case of the cost ad valorem tax is presented in the next section.

5. The Output Theorem under the Cost Ad Valorem Tax

The net profit function of a monopolist can be shown as:

$${\pi }_i\left(q_i\right)=\left[p_i\left(q_i\right)-\left(1+t\right)c\right]q_i$$

(23)

where $t$ is a cost ad valorem tax rate.

The optimum outputs under both pricing models are:

$$q_i^{\ast }=\left[\left(1+t\right)c-p_i\left(q_i^{\ast }\right)\right]/p_i^{\prime }\left(q_i^{\ast }\right)$$

(24)

$$q_i^o=\left[\left(1+t\right)c-p_i\left(q_i^o\right)\right]/p_i^{\prime }\left(q_i^o\right)$$

(25)

By exactly the same procedure, we can compare the difference in the total output:

$$Q^o-Q^{\ast }=\left(1/2\right){\displaystyle \sum _{i\in N}\left(q_i^o-q_i^{\ast }\right)\left\{\left[\left(p_i\left({\overline{q}}_i\right)-\left(1+t\right)c\right)\right]\left[p_i^{″}\left({\overline{q}}_i\right)/p_i^{\prime }{\left({\overline{q}}_i\right)}^2\right]\right\}}$$

(26)

As long as the tax rate $t$ is not too prohibitive, we can assume that $p_i\left({\overline{q}}_i\right)-\left(1+t\right)c>0$ to ensure that the profit of the firms is always non-negative. As a result, the same conclusion can be reached as in Proposition 3.

In the case of all demand curves sharing the same general curvature, we can similarly expand the marginal profit function and rearrange to obtain:

$${\displaystyle \sum _{j\in \mathrm{S}}\left(q_j^o-q_j^{\ast }\right)}\left(1/E_j^o{V}_j\right)={\displaystyle \sum _{k\in \mathrm{W}}\left(q_k^{\ast }-q_k^o\right)}\left(1/E_k^o{V}_k\right)$$

(27)

The same conclusion can be reached as in Proposition 4. Hence, the output theorems are preserved under the cost ad valorem tax. To verify the output theorems, we simulate both linear and constant elasticity demand functions under the unit, demand ad valorem, and cost ad valorem taxes (Liebman et al. [34]). The results are shown in

Table 1. Optimum Solution to the Demand, Cost Ad Valorem, and Unit Taxes: Linear Demand Functions (in thousands) $q_i=a_i-b_i{p}_i$ ($i=1,2$) *.

	Tax Rate $\mathit{u}$, v, $\mathbf{or} \mathit{t}$	0	10	20	30	40	50
TPD ** Demand Ad-Valorem Tax	$q_1$	20	19.722	19.375	18.929	18.333	17.5
	$q_2$	40	38.889	37.5	35.714	33.333	30
	$Q$	60	58.611	56.875	54.643	51.666	47.5
SPM *** Demand Ad-Valorem Tax	$q_1$	28	27.722	27.375	26.929	26.333	25.5
	$q_2$	32	30.889	29.5	27.714	25.333	22
	$Q$	60	58.611	56.875	54.643	51.666	47.5
TPD Cost Ad-Valorem Tax	$q_1$	20	19.75	19.5	19.25	19	18.75
	$q_2$	40	39	38	37	36	35
	$Q$	60	58.75	57.5	56.25	55	53.75
SPM Cost Ad-Valorem Tax	$q_1$	28	27.75	27.5	27.25	27	26.75
	$q_2$	32	31	30	29	28	27
	$Q$	60	58.75	57.5	56.25	55	53.75
TPD Demand Unit Tax	$q_1$	20	19	18	17	16	15
	$q_2$	40	36	32	28	24	20
	$Q$	60	55	50	45	40	35
SPM Demand Unit Tax	$q_1$	28	27	26	25	24	23
	$q_2$	32	28	24	20	16	12
	$Q$	60	55	50	45	40	35

* $a_1=45$, $b_1=20$, $a_2=100$, $b_2=80$, and $c=0.25$; ** TPD = Third-degree price discrimination; *** SPM = Single price monopoly.

and

Table 2. Optimum Solution to the Demand, Cost Ad Valorem, and Unit Taxes: Constant Elasticity Demand (in thousands) $p_i=a_i{q}_i^{-1/{\alpha }_i}$ ($i=1,2$ ) *.

	Tax Rate u$, \mathit{v}$$, \mathbf{or} \mathit{t}$	0	0.1	0.2	0.3	0.4	0.5
TPD ** Demand Ad-Valorem Tax	$q_1$	0.2963	0.216	0.1517	0.1016	0.064	0.037
	$q_2$	0.2789	0.2143	0.1596	0.1143	0.0778	0.0493
	$Q$	0.5752	0.4303	0.3113	0.2159	0.1418	0.0863
SPM *** Demand Ad-Valorem Tax	$q_1$	0.2531	0.1837	0.1284	0.0856	0.0536	0.0308
	$q_2$	0.3182	0.2437	0.1808	0.1289	0.0872	0.055
	$Q$	0.5713	0.4274	0.3092	0.2145	0.1408	0.0858
TPD Cost Ad-Valorem Tax	$q_1$	0.2963	0.2226	0.1715	0.1349	0.108	0.0878
	$q_2$	0.2789	0.2197	0.1767	0.1447	0.1202	0.1012
	$Q$	0.5752	0.4423	0.3483	0.2796	0.2282	0.189
SPM Cost Ad-Valorem Tax	$q_1$	0.2531	0.1894	0.1454	0.114	0.091	0.0738
	$q_2$	0.3182	0.25	0.2005	0.1637	0.1357	0.1139
	$Q$	0.5713	0.4394	0.3459	0.2777	0.2267	0.1877
TPD Demand Unit Tax	$q_1$	0.2963	0.2226	0.1715	0.1349	0.108	0.0878
	$q_2$	0.2789	0.2197	0.1768	0.1447	0.1202	0.1012
	$Q$	0.5752	0.4423	0.3482	0.2796	0.2282	0.189
SPM Demand Unit Tax	$q_1$	0.2531	0.1894	0.1454	0.114	0.091	0.0738
	$q_2$	0.3182	0.25	0.2005	0.1637	0.1357	0.1139
	$Q$	0.5713	0.4394	0.3459	0.2777	0.2267	0.1877

* Following Greenhut and Ohta [15], let $a_1=a_2=1$, ${\alpha }_1=3$, ${\alpha }_2=2.5$, and c = 1; ** TPD = Third-degree price discrimination; *** SPM = Single price monopoly.

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In Table 1, the simulations of three types of tax are provided. If all the demand schedules in all the submarkets are linear, the total outputs are identical under both uniform pricing and third-degree price discrimination. In Table 2, following Greenhut and Ohta [15], let $a_1=a_2=1$, ${\alpha }_1=3$, ${\alpha }_2=2.5$, and $c=1$; we obtain that the outputs under third-degree price discrimination are larger than in the case of uniform price monopoly. We conclude that if all the weak market demand curves are strictly “Robinson-concave” and all the strong market demand curves are strictly “Robinson-convex”, the total output under discrimination exceeds that under uniform pricing.

6. Concluding Remarks

The economic consequences of third-degree price discrimination, along the lines of Pigou and Robinson, have been examined extensively, although most of these studies are analyzed with the assumption of zero taxes—a highly unlikely case. We follow the mean-value theorem approach by Shih et al. [5] to incorporate more general cases with a new class of constantly adjusted concavity demand curves. The scientific novelty can shed valuable light on the output and welfare effects of the third-degree price discrimination model. We find that no matter whether under the per unit tax, demand ad valorem tax, or cost ad valorem tax, if all the weak market demand curves are strictly “Robinson-concave” and all the strong market demand curves are strictly “Robinson-convex” or linear, the total output under price discrimination exceeds that under uniform pricing, and vice versa. While different taxes lead to higher costs, the cost pass-through changes the prices of the products, and the change in total output still depends on the curvature of the demand curve. Therefore, the curvature of the demand curve remains the main determinant of changes in output. Our study provides a theoretical basis for market intervention in price discrimination.

However, for the class of polynomial demand functions presented by Nahata et al. [30], this analysis does not apply, since no easy and meaningful way can be derived to examine the output effect. For instance, in the two cubic demand functions used by Nahata et al. [30], the total output under price discrimination exceeds that under uniform pricing in one case but falls short of it in another case. In the case of polynomial demand functions with non-declining marginal revenues, we ought to employ the sufficient condition that leads to the increased output under discrimination: both discriminatory prices are lower than the uniform price under various forms of taxation. In conclusion, the output theorems first investigated by Joan Robinson are generally preserved with very reasonable assumptions under various forms of taxes.

As indicated, the first-degree price discrimination model involves charging different prices for each individual, thereby requiring detailed personal information but gaining in popularity as big data becomes quite plausible. The connection between the taxation, output theorems, and digitalization trade market could become a future research direction. The assumptions on third-degree price discrimination models include no possibility for arbitrage—that is, simultaneous purchase and sale of the same asset in different markets—although a more flexible model could include limited arbitrage, as in the example where consumers in Detroit frequently travel across the US border to Canada to purchase (cheaper) goods. We do not discuss a class of demand curves that exhibit extreme price inelasticity, which remains an interesting topic for future research.