Limited Memory Consumers and Price Dispersion

Abstract

We examine the effects of limited consumer memory on the pricing strategies of competing firms. We show that when the valuations of consumers are heterogeneous, it is possible to observe price dispersion even when each firm charges a single price.

Appendix

Before proving our proposition, we provide the profit functions of the two firms. Let $p_{i}$ be the price for Firm $i=1,2$. The profit of Firm $1$ is characterized as follows (the profit of Firm $2$ is specified in an analogous way):

$$\pi _{1} \left( {p_{1} ,p_{2} } \right) = \left\{ {\begin{array}{*{20}l} {\frac{1}{2}\left( {1 - p_{1} } \right)p_{1} } \hfill & {{\text{if}}} \hfill & {p_{2} \le p_{1} \le 1} \hfill \\ {\left( {\frac{1}{2}\left( {1 - p_{2} } \right) + \left( {p_{2} - p_{1} } \right)} \right)p_{1} } \hfill & {{\text{if}}} \hfill & {p_{1} < p_{2} \le 1} \hfill \\ {\left( {1 - p_{1} } \right)p_{1} } \hfill & {{\text{if}}} \hfill & {p_{1} \le 1 < p_{2} } \hfill \\ 0 \hfill & {{\text{if}}} \hfill & {1 \le p_{1} } \hfill \\ \end{array} .} \right.$$

If Firm 2 charges a price that is too high, then the demand for Firm 1 would be equivalent to that of a monopoly: $\max \left\{ \left( 1-p_{1}\right) p_{1},0\right\}$. When firms charge prices that are lower than 1, the profit of each firm depends on the relative magnitudes of the prices.

If Firm 1 charges a price that is higher than Firm $2$’s price, then the demand that corresponds to the consumers with valuations that are higher than $p_{1}$ is split so that the demand for Firm $1$ would be equal to $\frac{1}{2}\left( 1-p_{1}\right)$. If Firm 1 charges lower than Firm 2, then it gets half of the consumers that are willing to buy from either of the firms – $\frac{1}{2}\left( 1-p_{2}\right)$ – as well as the rest of the consumers who are not willing to buy from high price firm but are willing to buy from the low price firm, $p_{2}-p_{1}$.

For the $p_{1}<p_{2}$ case, $p_{2}-p_{1}$ portion of the demand for Firm 1 comes from the consumers that are certain that they would not buy from Firm 2. On the other hand, $\frac{1}{2}\left( 1-p_{2}\right)$ portion of the demand comes from the consumers that do not know which price is higher.

Proof

(Proposition 1) Assume that the consumers are indifferent between all prices below their valuations. Hence, as long as the price of the product is below their valuations, the consumers buy the product. The equilibrium outcome would be equivalent to the case where the consumers are ignorant about the market so that each consumer randomly picks a firm and buys the product if the price is below her valuation. If the price is above the consumer’s valuation, she checks the price that is charged by the other firm. Given $p_{2}$, Firm 1 can either set a price higher than (or equal to) $p_{2}$ and get $\pi _{1}= \frac{1}{2}\left( 1-p_{1}\right) p_{1}$ or set a price smaller than $p_{2}$ and get $\pi _{1}=\left( \frac{1}{2}\left( 1-p_{2}\right) +\left( p_{2}-p_{1}\right) \right) p_{1}$. We refer to these strategies as up ($U$) and down ($D$), respectively. The optimal prices for up and down strategies are $p_{1}=\max \left\{ \frac{1}{2},p_{2}\right\}$ and $p_{1}=\min \left\{ \frac{1+p_{2}}{4},p_{2}\right\}$, respectively. When presenting the profits, we mention whether the relevant prices correspond to up or down strategies. There are three cases to consider.

Case 1:

$$\begin{aligned} p_{2} \ge \frac{1}{2}\Rightarrow \pi _{1}\left( \frac{1+p_{2}}{4},p_{2};D\right) -\pi _{1}\left( p_{2},p_{2};U\right) =\frac{1}{16}\left( 3p_{2}-1\right) ^{2}>0\text {.} \end{aligned}$$

Hence, firm $1$ plays $p_{1}=\frac{1+p_{2}}{4}$.

Case 2:

$$\begin{aligned} p_{2} \le \frac{1}{3}\Rightarrow \pi _{1}\left( \frac{1}{2},p_{2};U\right) -\pi _{1}\left( p_{2},p_{2};D\right) =\frac{1}{8}\left( 1-2p_{2}\right) ^{2}>0 \text {.} \end{aligned}$$

Hence, firm $1$ plays $p_{1}=\frac{1}{2}$.

Case 3:

$$\begin{aligned} \frac{1}{3} < p_{2}<\frac{1}{2}\Rightarrow \pi _{1}\left( \frac{1}{2},p_{2};U\right) -\pi _{1}\left( \frac{1+p_{2}}{4} ,p_{2};D\right) =\frac{1}{16}\left( -p_{2}^{2}-2p_{2}+1\right) >0 \Leftrightarrow p_{2}<\sqrt{2}-1\text {.} \end{aligned}$$

Therefore, if $p_{2}<\sqrt{2}-1$, firm $1$ plays $p_{1}=\frac{1}{2}$; and if $p_{2}>\sqrt{2}-1$, firm plays $p_{1}=\frac{1+p_{2}}{4}$. The best response function of Firm 2 is similar. These findings indicate that the pure strategy equilibria are asymmetric and such that $\left( p_{1},p_{2}\right) =\left( \frac{1}{2},\frac{3}{8}\right)$ and $\left( p_{1},p_{2}\right) =\left( \frac{3}{8},\frac{1}{2}\right)$. Hence, one of the firms plays the up strategy and the other plays the down strategy. The profit for the firm that plays the up strategy is $\frac{1}{8}$ and the profit for the firm that plays the down strategy is $\frac{9}{64}$.