Price dispersion

Last updated June 10, 2019

In economics, price dispersion is variation in prices across sellers of the same item, holding fixed the item's characteristics. Price dispersion can be viewed as a measure of trading frictions (or, tautologically, as a violation of the law of one price). It is often attributed to consumer search costs or unmeasured attributes (such as the reputation) of the retailing outlets involved. There is a difference between price dispersion and price discrimination. The latter concept involves a single provider charging different prices to different customers for an identical good. Price dispersion, on the other hand, is best thought of as the outcome of many firms potentially charging different prices, where customers of one firm find it difficult to patronize (or are perhaps unaware of) other firms due to the existence of search costs.

Economics is the social science that studies the production, distribution, and consumption of goods and services.

The law of one price (LOOP) states that in the absence of trade frictions, and under conditions of free competition and price flexibility, identical goods sold in different locations must sell for the same price when prices are expressed in a common currency. This law is derived from the assumption of the inevitable elimination of all arbitrage.

In microeconomics, search theory studies buyers or sellers who cannot instantly find a trading partner, and must therefore search for a partner prior to transacting.

In statistics, the standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean. Informally, it measures how far a set of (random) numbers are spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by $,, or .$

In most theoretical literature, price dispersion is argued as result from spatial difference and the existence of significant search cost. With the development of internet and shopping agent programs, conventional wisdom tells that price dispersion should be alleviated and may eventually disappear in the online market due to the reduced search cost for both price and product features. However, recent studies found a surprisingly high level of price dispersion online, even for standardized items such as books, CDs and DVDs. There is some evidence of a shrinking of this online price dispersion, but it remains significant. Recently, work has also been done in the area of e-commerce, specifically the Semantic Web, and its effects on price dispersion.

Hal Varian, an economist at U. C. Berkeley, argued in a 1980 article that price dispersion may be an intentional marketing technique to encourage shoppers to explore their options.^[1]

Hal Ronald Varian is an economist specializing in microeconomics and information economics. He is the chief economist at Google and he holds the title of emeritus professor at the University of California, Berkeley where he was founding dean of the School of Information.

A related concept is that of wage dispersion.

Consumer search and price dispersion

Search alone is insufficient

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 necessary condition

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.

Price dispersion in a non-sequential search model

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:

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

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

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

Price dispersion in a sequential search model

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:

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

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

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References

↑ Varian, H. R. (1980). A model of sales. The American Economic Review, 70(4), 651-659.
↑ Diamond, P. A. (1971). A model of price adjustment. Journal of economic theory, 3(2), 156-168.
↑ Burdett, K., & Judd, K. L. (1983). Equilibrium price dispersion. Econometrica: Journal of the Econometric Society, 955-969.
↑ Stahl, D. O. (1989). Oligopolistic pricing with sequential consumer search. The American Economic Review, 700-712.

Dahlby, Bev and Douglas West, (1986), "Price Dispersion in an Automobile Market," Journal of Political Economy, 94(2): 418-438.
Venkatesh Shankar, Xing Pan, and Brian T. Ratchford, (2002), "Do Drivers of Online Price Dispersion Change as Online Markets Grow?," working paper, December, University of Maryland.
Cooper, Sean, "Why You Can't Get iPods At a Discount", Slate.
Gupta, Tanya, and Abir Qasem,(2002), "Reduction of Price Dispersion through Semantic E-commerce," (in Workshop at WWW2002 International Workshop on the Semantic Web, Hawaii, May 7, 2002)
Thiel, Stuart E., "A New Model of Persistent Retail Price Dispersion" (July 6, 2005). Available at SSRN: http://ssrn.com/abstract=757357

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[Varian_1980-1] Varian, H. R. (1980). A model of sales. The American Economic Review, 70(4), 651-659.

[Diamond_1971-2] Diamond, P. A. (1971). A model of price adjustment. Journal of economic theory, 3(2), 156-168.

[Burdett_1983-3] Burdett, K., & Judd, K. L. (1983). Equilibrium price dispersion. Econometrica: Journal of the Econometric Society, 955-969.

[Stahl-4] Stahl, D. O. (1989). Oligopolistic pricing with sequential consumer search. The American Economic Review, 700-712.