Two Questions How can such price dispersion persist in markets where consumers behave in a rational manner. that is when consumers search for the lowest price? How do rational consumers optimally search for a low price in a market with dispersed prices
Two Questions • How can such price dispersion persist in markets where consumers behave in a rational manner, that is, when consumers search for the lowest price? • How do rational consumers optimally search for a low price in a market with dispersed prices?
A Model of search price Dispersion Let us consider an economy with a continuum of consumers, indexed by s on the interval [L, h according to their cost for going shopping, where we assume that H>3L>0. Thus, consumers indexed by a high s(s close to H) are high time-valued consumers, whose cost of searching for the lowest price is high. The consumers indexed by a low s(s close to L) are low time-valued consumers for whom the cost of going shopping and searching for the lowest price is small. Figure 16. 1 illustrates how consumers are distributed according to their cost of shopping There are three stores selling a single product that is produced at zero cost. One store, denoted by d is called discount store, selling the product for a unit price of pn. The other two stores, denoted by ND, are expensive(not discount)stores, and are managed by a single ownership that sets a uniform price, PNp, for the two nondiscount stores Searching Buying at random O L H Figure 16 Consumers with variable search cost searching for the lowest price fine p to be the average product price. Formally, PD+ 2pND (161
A Model of Search & Price Dispersion
A Model of search price Dispersion We assume that the consumers do not know which store is a discount and which is expensive unless they conduct a search at a cost of s. However, consumers do know the average store price. Thus, if a consumer does not conduct a search, he knows to expect that random shopping would result in paying an average price of p. Each consumer buys one unit and wishes to minimize the price he or she pays for the product plus the search cost. Formally, denoting by L'(a, P)the loss function of consumer type S. sE [L, H. we assume that L'= PD+as if he or she searches for the lowest price p if he or she purchases from a randomly chosen store (16.2) The parameter a measures the relative importance of the search cost in consumer preferences Clearly, since each consumer s minimizes (16.2), a type s consumer will search for. the lowest price if pD tassp, that is, if the sum of the discount price plus the search cost does not exceed the average price(which equals the expected price of purchasing from a randomly chosen store). In contrast, if PD+as >F, then clearly, buying at random is cheaper for consumer s than searching and buying from the discount store (Note: The expected number of searches to get the lowest price is 1/3 (0+1+2=1.)
A Model of Search & Price Dispersion (Note：The expected number of searches to get the lowest price is 1/3*(0+1+2)=1.)
A Model of search price Dispersion Definition 16.1 A price dispersion equilibrium is the prices PD and PND Such that I The discount store cannot increase its profit by unilaterally deviating from the price Pi 2. The owner of the two expensive stores cannot increase his profit by unilaterally deviating from the price PND 3. For every consumerS, sE[L, H], the consumer searches and buys from the discount store if and only if pi+as sp. Otherwise, the consumer buys from the first available store
A Model of Search & Price Dispersion
A Model of search price Dispersion It follows from Definition 16. I that if some consumers search for the lowest price and some buy at random then there exists a consumer denoted by who is indifferent to the choice between searching and shopping at random. Thus, for the consumer indexed by we have pD+a=p≈p+2p9 (163) Hence 2(pND 30 (164) Consequently, in view of Figure 16.1, for given prices PD<PND, all consumers indexed by sE L, a] pay the cost of s for searching for the lowest price, and all consumers indexed by s E(,Hj buy at random and pay an average price ofp (Note: An alternative way to calculate s is pp+=as=pnd
A Model of Search & Price Dispersion (Note：An alternative way to calculate 𝑠 is 𝑝𝐷 + 3 2 𝛼𝑠 = 𝑝𝑁𝐷.)