Endogenous third-degree price discrimination in Hotelling model with elastic demand

Abstract

We relax two common assumptions in the Hotelling model with third-degree price discrimination: inelastic demand and exogenously assumed price discrimination. Based on the constant elasticity of substitution representative consumer model, we allow firms to endogenously choose whether to acquire consumer information and price discriminate. We find that when the information cost is sufficiently low, there exist two symmetric sub-game perfect Nash equilibria irrespective of the demand elasticity: both firms acquiring information and price discriminating, and neither firm acquiring information and charging a uniform price. This implies that the widely discussed prisoners’ dilemma, in which both firms are exogenously assumed to price discriminate, is not in fact a dilemma. A comparison of social welfare shows that when the demand elasticity is large enough, price discrimination improves social welfare. This is in contrast to the finding—price discrimination harms social welfare—in the existing literature assuming perfect inelastic demand.

Appendix

Proof of Lemma 1

In this sub-game, each firm charges a uniform price in the second stage. The marginal consumer thus is defined by Eq. (1). Each firm’s profit can be calculated by Eq. (3). The first order conditions are:

$$\begin{aligned}&\frac{\partial \pi _A}{\partial p_A}=\frac{-1}{2t}\left( t\varepsilon +2p_A^{1-\varepsilon }-p_B^{1-\varepsilon }-t\right) p_A^{-\varepsilon },\\&\quad \frac{\partial \pi _B}{\partial p_B}=\frac{1}{2t}\left( -t\varepsilon +p_A^{1-\varepsilon }-2p_B^{1-\varepsilon }+t\right) p_B^{-\varepsilon }. \end{aligned}$$

Solving the two FOCs, we get the equilibrium prices as follows:

$$\begin{aligned} p_A=p_B=\left( t(1-\varepsilon )\right) ^{\frac{1}{1-\varepsilon }}. \end{aligned}$$

Substitute the prices back into Eq. (2), we get the conditional demands are:

$$\begin{aligned} q_A=q_B=\frac{1}{2}\left( t(1-\varepsilon )\right) ^\frac{\varepsilon }{\varepsilon -1}. \end{aligned}$$

The corresponding profits are,

$$\begin{aligned} \pi _A=\pi _B=\frac{1}{2}t(1-\varepsilon ). \end{aligned}$$

Consumer surplus and social welfare are respectively

$$\begin{aligned} { CS}= & {} \int _0 ^{\hat{x}} V_A dx+\int _{\hat{x}}^1 V_B dx \\= & {} \int _0 ^{\hat{x}} \left( Y+u\left( p_{A}\right) -td_A\left( x\right) \right) dx + \int _{\hat{x}}^{1} \left( Y+u\left( p_{B}\right) -td_B\left( x\right) \right) dx \\= & {} Y+u-\frac{5}{4}t\\ W= & {} { CS}+\pi _A+\pi _B=Y+u-t\left( \frac{1}{4}+\varepsilon \right) . \end{aligned}$$

$\square$

Proof of Lemma 2

When firm A acquires information and firm B does not, firm A charges $p_{1A}$ in market 1 and $p_{2A}$ in market 2, while firm B charges a uniform price $p_B$ in both markets. By substituting the prices into $\hat{x}$, we get two marginal consumers $\hat{x}_1$ and $\hat{x}_2$ as following.

$$\begin{aligned} \hat{x}_1=\frac{1}{2}+\frac{p_B^{1-\varepsilon }-p_{1A}^{1-\varepsilon }}{2t\left( 1-\varepsilon \right) }, \quad \hat{x}_2=\frac{1}{2}+\frac{p_B^{1-\varepsilon }-p_{2A}^{1-\varepsilon }}{2t\left( 1-\varepsilon \right) }. \end{aligned}$$

The profits of firm 1 and firm 2 are

$$\begin{aligned} \pi _A=p_{1A}\hat{x}_1q_{1A}+p_{2A}\left( \hat{x}_2-\frac{1}{2}\right) q_{2A}, \quad \pi _B=p_B\left( \frac{3}{2}-\hat{x}_1-\hat{x}_2\right) q_B. \end{aligned}$$

The FOCs are:

$$\begin{aligned} \frac{\partial \pi _A}{\partial p_{1A}}= & {} \frac{-1}{2t}\left( t\varepsilon +2p_{1A}^{1-\varepsilon }-p_B^{1-\varepsilon }-t\right) p_{1A}^{-\varepsilon }, \quad \frac{\partial \pi _A}{\partial p_{2A}}=\frac{-1}{2t}\left( 2p_{2A}^{1-\varepsilon }-p_B^{1-\varepsilon }\right) p_{2A}^{-\varepsilon },\\ \frac{\partial \pi _B}{\partial p_B}= & {} \frac{1}{2t}\left( -t\varepsilon +p_{2A}^{1-\varepsilon }+p_{1A}^{1-\varepsilon }-4p_B^{1-\varepsilon }+t\right) p_B^{-\varepsilon }. \end{aligned}$$

Solving the three FOCs, we get the equilibrium prices.

$$\begin{aligned} p_{1A}=\left( \frac{3}{4}t\left( 1-\varepsilon \right) \right) ^{\frac{1}{1-\varepsilon }}, \quad p_{2A}=\left( \frac{1}{4}t\left( 1-\varepsilon \right) \right) ^{\frac{1}{1-\varepsilon }}, \quad p_B=\left( \frac{1}{2}t\left( 1-\varepsilon \right) \right) ^{\frac{1}{1-\varepsilon }}. \end{aligned}$$

And the conditional demands are:

$$\begin{aligned} q_A=\frac{3}{8}\left( \frac{3}{4}t\left( 1-\varepsilon \right) \right) ^\frac{\varepsilon }{\varepsilon -1}+\frac{1}{8}\left( \frac{1}{4}t\left( 1-\varepsilon \right) \right) ^\frac{\varepsilon }{\varepsilon -1}, \quad q_B=\frac{1}{2}\left( \frac{1}{2}t\left( 1-\varepsilon \right) \right) ^\frac{\varepsilon }{\varepsilon -1}. \end{aligned}$$

The corresponding profits are,

$$\begin{aligned} \pi _A=\frac{5}{16}t\left( 1-\varepsilon \right) -c, \quad \pi _B=\frac{1}{4}t\left( 1-\varepsilon \right) . \end{aligned}$$

Consumer surplus and social welfare are respectively:

$$\begin{aligned} { CS}= & {} \int _0 ^{\hat{x}_1} \left( Y+u\left( p_{1A}\right) -td_A\left( x\right) \right) dx + \int _{\hat{x}_1}^{1/2} \left( Y+u\left( p_{B}\right) -td_B\left( x\right) \right) dx \\&+ \int _{1/2} ^{\hat{x}_2} \left( Y+u\left( p_{2A}\right) -td_A\left( x\right) \right) dx+\int _{\hat{x}_2}^1 \left( Y+u\left( p_{B}\right) -td_B\left( x\right) \right) dx \\= & {} Y+u-\frac{27}{32}t.\\ W= & {} { CS}+\pi _A+\pi _B=Y+u-\frac{9}{16}t\left( 1+\varepsilon \right) . \end{aligned}$$

$\square$

Proof of Lemma 3

When both firms acquire information in the first stage, they each charge different prices in market 1 and 2 in the second period. Due to symmetry, we only need to consider market 1. In that market, the marginal consumer can be calculated as

$$\begin{aligned} \hat{x}_1=\frac{1}{2}+\frac{p_{1B}^{1-\varepsilon }-p_{1A}^{1-\varepsilon }}{2t(1-\varepsilon )}, \end{aligned}$$

and firms’ profits are:

$$\begin{aligned} \pi _{1A}=p_{1A}\hat{x}_1q_{1A}, \quad \pi _{1B}=p_{1B}\left( \frac{1}{2}-\hat{x}_1\right) q_{1B}. \end{aligned}$$

The FOCs are:

$$\begin{aligned} \frac{\partial \pi _A}{\partial p_A}=\frac{-1}{2t}\left( t\varepsilon +2p_{1A}^{1-\varepsilon }-p_{1B}^{1-\varepsilon }-t\right) p_{1A}^{-\varepsilon },\quad \frac{\partial \pi _B}{\partial p_B}==\frac{1}{2t}\left( p_{1A}^{1-\varepsilon }-2p_{1B}^{1-\varepsilon }\right) p_{1B} ^{-\varepsilon }. \end{aligned}$$

Solving the two FOCs, we get the equilibrium prices:

$$\begin{aligned} p_{1A}=p_{2B}=\left( \frac{2}{3}t\left( 1-\varepsilon \right) \right) ^{\frac{1}{1-\varepsilon }}, \quad p_{2A}=p_{1B}=\left( \frac{1}{3}t\left( 1-\varepsilon \right) \right) ^{\frac{1}{1-\varepsilon }}. \end{aligned}$$

The conditional demands are:

$$\begin{aligned} q_A=q_B=\frac{1}{3}\left( \frac{2}{3}t\left( 1-\varepsilon \right) \right) ^\frac{\varepsilon }{\varepsilon -1} +\frac{1}{6}\left( \frac{1}{3}t\left( 1-\varepsilon \right) \right) ^\frac{\varepsilon }{\varepsilon -1}. \end{aligned}$$

The corresponding profits are,

$$\begin{aligned} \pi _A=\pi _B=\frac{5}{18}t\left( 1-\varepsilon \right) -c. \end{aligned}$$

Consumer surplus and social welfare are respectively:

$$\begin{aligned} { CS}= & {} 2 \left( \int _0 ^{\hat{x}_1} \left( Y+u\left( p_{1A}\right) -td_A\left( x\right) \right) dx + \int _{\hat{x}_1}^{1/2} \left( Y+u\left( p_{1B}\right) -td_B\left( x\right) \right) dx \right) \\= & {} Y+u-\frac{31}{36}t.\\ W= & {} { CS}+\pi _A+\pi _B=Y+u-\frac{1}{36}t\left( 11+20\varepsilon \right) . \end{aligned}$$

$\square$

Proof of Proposition 3

Given the uniform price is $p_u=(t(1-\varepsilon ))^{\frac{1}{1-\varepsilon }}$, the discriminatory prices are $p_{d1}=(\frac{2}{3}t(1-\varepsilon ))^{\frac{1}{1-\varepsilon }}$, $p_{d2}=(\frac{1}{3}t(1-\varepsilon ))^{\frac{1}{1-\varepsilon }}$. The differences between the uniform price and the discriminatory prices are $\Delta p_1=p_u-p_{d1}=(t(1-\varepsilon ))^{\frac{1}{1-\varepsilon }}-(\frac{2}{3}t(1-\varepsilon ))^{\frac{1}{1-\varepsilon }}$ and $\Delta p_2=p_u-p_{d2}=(t(1-\varepsilon ))^{\frac{1}{1-\varepsilon }}-(\frac{1}{3}t(1-\varepsilon ))^{\frac{1}{1-\varepsilon }}$. Both $\Delta p_1$ and $\Delta p_2$ are positive and decrease with the elasticity of demand. $\square$

Proof of Proposition 4

When both firms price discriminate, the consumer surplus for those who buy from a distant firm and those who buy from a close firm are respectively:

$$\begin{aligned} { CS}^{{ ds}}=\frac{1}{3}Y+\frac{1}{3}u-\frac{5}{18}t, \quad { CS}^{{ dn}}=\frac{1}{6}Y+\frac{1}{6}u-\frac{11}{72}t. \end{aligned}$$

When both firms charge uniform price, their consumer surplus are respectively:

$$\begin{aligned} { CS}^{{ us}}=\frac{1}{3}Y+\frac{1}{3}u-\frac{7}{18}t, \quad { CS}^{{ un}}=\frac{1}{6}Y+\frac{1}{6}u-\frac{17}{72}t. \end{aligned}$$

The differences between them are

$$\begin{aligned} \Delta { CS}^{s}={ CS}^{{ ds}}-{ CS}^{{ us}}=\frac{1}{9}t, \quad { CS}^{n}={ CS}^{{ dn}}-{ CS}^{{ un}}=\frac{1}{12}t. \end{aligned}$$

Since both groups of consumers are better with price discrimination, overall consumer surplus is improved. $\square$

Proof of Proposition 6

Given information cost c, when both firms price discriminate: $\pi _A^{I-I}=\pi _B^{I-I}=\frac{5}{18}t(1-\varepsilon )-c$. When firm A price discriminates while firm B does not, the profits are: $\pi _A^{I-N}=\frac{5}{16}t(1-\varepsilon )-c$, $\pi _B^{I-N}=\frac{1}{4}t(1-\varepsilon )$. It can be easily shown that when $c>\frac{1}{36}t(1-\varepsilon )$, $\pi _B^{I-N}>\pi _B^{I-I}$, such that both firms acquire information and price discriminate is not a SPNE.

Besides, when both firms choose price discrimination, the social welfare is:

$$\begin{aligned} W^D=Y+u-\frac{1}{36}t\left( 11+20\varepsilon \right) -2c. \end{aligned}$$

when both firms choose uniform price, the social welfare is:

$$\begin{aligned} W^U=Y+u-t\left( \frac{1}{4}+\varepsilon \right) . \end{aligned}$$

We find that when $c<\frac{t}{36}(8\varepsilon -1)$, $W^D>W^U$. Thus, we can draw a conclusion that price discrimination improves social welfare when the demand elasticity is large enough $\left( \varepsilon >\frac{1}{8}\right)$ and information cost is small enough $\left( c<\frac{t}{36}(8\varepsilon -1)\right)$. $\square$