Quality as vertical Product Differentiation There is a continuum of consumers uniformly distributed on the interval [o, 1]. There are two firms denoted by a and B and located at points a and b(o sbs1) from the origin, respectively. Figure 12.2 illustrates the location of the firms on the 0, 1 interval he utility of a consumer located at point x, t E [0, 1] and buying brand i, i=A, B is defined by Pa i=A U()≡{bx-P=B (122) where pA and pB are the price charged by firm A and B, respectively firm a firm B (b-a) Figure 12.2 Vertical differentiation in a modified Hotelling model We seek to define a two-period game where firms choose location in the first period, and choose price in the second period, after locations have been fixed. Before defining the game, let us solve for a Nash-Bertrand equilibrium in prices, assuming fixed locations as illustrated in Figure 12.2
Quality as Vertical Product Differentiation
Quality as vertical Product Differentiation Let denote a consumer who is indifferent to whether he or she buys from firm A or firm B asas, the location of the indifferent consumer is determinedbres between the two firms, that is assuming that such a consumer exists. and that consumer lo Ua(A)=ai-PA=02-PB=U2(B 123) Thus, the utility of consumer indexed by al from buying brand A equals his utility from buying brand B. Therefore, assuming that a≤金≤b, the number of consumers buying from firm A is≥, whereas the number of consumers buying from firm B is(1-a). Solving for i from(12.3)yields and 1-f=1_PB (124) Figure 12.3 provides a graphic illustration of how a is determined. The left-hand side of Figure 12.3 illustrates the utility for a consumer located at any point 0 s asI when he or she buys brand A and when he or she buys brand b, assuming that pb> pA. by definition, for the consumer located at f the utility from buying a equals the utility from buying b. moreover, Figure 12.3 shows that all consumers located on[0, al gain a higher utility from purchasing brand A (lower quality)than from purchasing brand B. Similarly, all consumers located on E, 1] brand B(higher-quality brand)than from purchasing brandy gain a higher utility from purchasing
Quality as Vertical Product Differentiation
Quality as vertical Product Differentiation U=(B) U U=(A) PA 1 -PB -PB pa Figure 12.3 Determination of the indifferent consumer among brands vertically differentiated on the basis of quality. Left: pA< pB, Right: PA>PB
Quality as Vertical Product Differentiation
Quality as vertical Product Differentiation For given locations of firms(a and b), in the second period each firm takes the price set by its rival firm as given and chooses its price to maximize its profit level. Formally, firm A and B solve PB-PA maxTa(a, b,pA,PB)= PAi=PAb-a (12) maxB(a,b,PA, PB) (1-2) Definition 12.2 The quadruple s ae, be, pA(a, b) pB(a, b)> is said to be a vertically differentiated industry equilibrium if Second period: For(any) given locations of firms(a and b/ pi(a, 6) and ps(a, b)constitute a nash equilibrium First period: Given the second period-price functions of locations PA(a, b)p(a, b)and t(pa(a, b), P B(a,b),(ae, be) is a Nash equilibrium in location
Quality as Vertical Product Differentiation
Quality as vertical Product Differentiation We now proceed to solve the model, starting from the second period The first-order conditions to (2.5)are given b 0- and 0= B =1-2B-P 12 Hence n(c40=23°mh=2。m2 Note that both equilibrium prices exceed marginal cost despite the fact that one firm produces inferior quality
Quality as Vertical Product Differentiation