Price dispersion

Even when consumers search, price dispersion is not guaranteed. Consumers may search, yet firms set the same price, negating the mere fact of searching. This is referred to as Diamond's paradox. [2]

Assume that many firms provide a homogeneous good. Consumers will randomly sample only one firm if they expect that all firms charge the same price. Consequently, each firm has an equal share of consumers. Since consumers disregard the competitions, each firm acts as a monopoly on its share of consumers. Firms choose a price that maximizes profit: the monopoly price.

A recurrent observation is that some consumers must sample one firm and only one, while the remaining consumers must sample at least two firms.

If all of them sample only one firm, then the market faces Diamond's Paradox. Firms would ask the same price, and so there would be no price dispersion.

On the contrary, if all consumers sample at least two firms. The most expensive firm will not get any consumer, because consumers know at least another firm that is cheaper. As a result, prices must be as low as possible: equal to marginal costs of production, as in a Bertrand economy.

A non-sequential search strategy consists in choosing a number of prices to compare. If consumers follow a non-sequential search strategy, as long as some consumers sample only one firm, then an equilibrium in price dispersion exists.[3]

There is an equilibrium in price dispersion if some consumers search once, and the remaining consumers search more than one firm. Moreover, the distribution of prices has a closed form if consumers search at most two firms:

${\displaystyle F\left(x\right)={\begin{cases}0,&{\text{if }}p<{\underline {p}}\left(q\right)\\1-\left({\frac {p^{*}-p}{p-r}}\right)\left({\frac {q}{2\left(1-q\right)}}\right),&{\text{if }}{\underline {p}}\left(q\right)<p\leq p^{*}\\1,&{\text{if }}p>p^{*}\end{cases}}}$

where ${\displaystyle {\underline {p}}\left(q\right)=\left(p^{*}-p\right){\frac {q}{2\left(1-q\right)}}+r}$; with ${\displaystyle q}$ the share of consumer who sample only one firm, ${\displaystyle p^{*}}$ consumers' reservation price, and ${\displaystyle r}$ firms' marginal costs of production.

Such an equilibrium in price dispersion occurs when consumers minimize ${\displaystyle \mathbb {E} \left[p_{n}\right]-cn}$, with ${\displaystyle n}$ the sample size, ${\displaystyle c}$ a search cost, and ${\displaystyle p_{n}}$ the smallest price sampled.

A sequential search strategy consists in sampling prices one by one, and stop after identifying a sufficiently low price. In sequential search models, the existence of perfectly informed consumers guarantees the equilibrium in price dispersion if the remaining consumers search once and only one. There is a continuous relationship between the share of informed consumers and the type of competition: from Bertrand competition to Diamond competition as fewer and fewer consumers are initially perfectly informed.[4]

The distribution of price has a closed form:

${\displaystyle F\left(p\right)=1-\left[\left({\frac {1-\mu }{N\mu }}\right)\left({\frac {R\left(P_{r}\right)}{R\left(p\right)}}-1\right)\right]^{\frac {1}{N-1}}}$

on support ${\displaystyle \left[0,P_{r}\right]}$; where ${\displaystyle \mu }$ the share of perfectly informed consumers, ${\displaystyle N}$ the number of firms, ${\displaystyle R\left(.\right)}$ the revenue function that attains its maximum in ${\displaystyle {\hat {p}}}$, ${\displaystyle r}$ consumers' reservation price, and ${\displaystyle P_{r}=\min \left\lbrace r,{\hat {p}}\right\rbrace }$