Search and Price dispersion

Economics of Information George J. Stigler: The Economics of Information, JPE, June 1961. George Stigler 1911-1991 Nobel Prize 1982 One should hardly have to tell academicians that information is a valuable resource: knowledge is power. And yet it occupies a slum dwelling in the town of economics."

The Diamond paradox (1971) Firms produce a homogeneous product at constant marginal cost. Consumers are informed about the price of one firm - equally distributed among firms. Consumers may choose to get information about the price of another firm - chosen at random – at a cost z ,which may differ between consumers, with the lowest k inf(k) > 0. Firms choose their prices so as to maximize their profit given consumers strategies. Peter Diamond Nobel prize 2010

The only equilibrium is when all firms charge the monopoly price. The Diamond paradox ________* When a consumer approaches a firm, he learns its price, he may then buy at this price or continue his search. Consumers know the distribution of prices and will decide to continue their search if their expected gain from search is higher than its cost z . The only equilibrium is when all firms charge the monopoly price. * The firm makes a take-it-or-leave-it offer to the consumer

The Diamond paradox In equilibrium all firms charge the monopoly price. If not, let p be the lowest price. The firm could increase its price to p + k and the customer would still buy the product.

A Simple Model of Price Dispersion A continuum of firms produce a homogeneous good, each produces with cost C(q). There is free-entry to the market. A consumer buys a single unit of the good and is willing to pay up to w for it. A measure L of the consumers are Locals and buy from one of the cheapest firms. (Their cost of search is ZERO) A measure T of the consumers are Tourists and will buy at one of the firms chosen at random. (Their cost of search is INFINITE) Let a measure n of firms enter the market with measure k ≤ n of them charging the lowest price.

The Tourists are equally divided among all the firms, each firm will have T/n tourist customers. If k of the firms charge the lowest price each will sell to T/n +L/k customers. Clearly, the firms selling at the lowest price make 0 profits, otherwise, a firm will enter with a slightly lower price and capture all Locals. Also, the firms selling only to tourist will charge w, and will make 0 profit. There will therefore be only two types of firms. Those selling to tourists only, and those selling cheapest.

The Average Cost curve C(q)/q AC w Assume w > p0 . Let AC( q1 ) = w . at the minimum p0 q q1 q0

The Average Cost curve C(q)/q n - all firms k - firms charging min price AC w Find n, k such that: q1 = T/n q0 = T/n +L/k n = T /q1 k = L/ (q0 - q1 ) p0 q q1 q0 But k should be < n

Otherwise n = k and there will be only one price p0. n = T /q1 But k should be < n k = L/ (q0 - q1 ) AC k = L/ (q0 - q1 ) < T /q1 = n w To support two prices there should be enough Tourists in the population. p0 q q1 q0 Otherwise n = k and there will be only one price p0.

w No firm can gain by changing its behavior. This is therefore an equilibrium with two prices: w , p0. p0 q1 q0

Sequential Search Taking one random sample from a distribution F( ) costs a searcher k. A sample is an opportunity for the searcher and has some value for him. A sample y has the value y. a consumer After drawing the sample the searcher can opt for it, or he may continue the search by paying k again. If he takes y his payoff is: Let the searcher use an optimal strategy which gives him expected value of V* He will clearly not take any opportunity with value y < V* and accept any with y ≥ V*.

Thus: This assumes stationarity. But are k, F( ) unchanged over time? For a worker, k may increase after a long unemployment period. Aging may change the opportunities and F( ) with it If the searcher discards options that he already seen (no replacement) then F( ) changes over time. Or if the searcher learns about the distribution during his search.

The optimum x* satisfies V(x*)=V* If the searcher uses a reservation value rule x, accepting only opportunities whose value is ≥ x then his expected value is: Now choose the reservation value x to maximize V(x): The optimum x* satisfies V(x*)=V*

If the searcher found x* He can either keep it and leave, or he may run one more search Which cost k and may bring him extra profit y-x* or else he just keeps consumes x*. This can also be written as:

In some models the searcher may determine the intensity of search. Let time be continuous, to get intensity s of search opportunities (Poisson Process), i.e. To have one search opportunity at a short time interval Δ with probability sΔ, the searcher has to pay k(s)Δ cost If the searcher uses a reservation rule x, his expected value is: Probability of search and finding something better than x Probability of no search opportunity or finding < x Discount rate

The marginal cost of increasing search intensity equals the expected gain from an additional search Then: Let x* maximize V(s,x), and let s* maximize V(s,x*), then: