Imperfect competition and quality signalling.

Because [delta] > k, this entire interval involves prices in excess of [[rho].sub.HH] = ([d.sub.H] + [c.sub.H])/2; thus, the type H firm distorts its price upward from the best-response function it would follow if it were known to be of type H. Note that the type H firm could also deter mimicry by the type L firm by using a downward-distorted price but type H would prefer to give up and be taken as a type L firm rather than use such a low price to distinguish itself.

Refinement of best-response functions. We have identified an interval of candidates for the type H firm's best response. We now apply the Intuitive Criterion (see Mas-Colell, Whinston, and Green, 1995, for a discussion of equilibrium domination and the Intuitive Criterion of Cho and Kreps, 1987). It is appropriate to apply this refinement at this stage in the game because, conditional on any common conjecture (common to firm i and the consumer) about the strategy being employed by all other firms (including the equilibrium strategy), what remains is simply a signalling game between firm i's two types and the consumer. The Intuitive Criterion says that the consumer should infer type H from firm i's price p so long as type H would be willing to charge p, yet mimicry by type L would be deterred, even under this most-favorable inference. Thus, the firm of type H distorts its best response to the minimum extent necessary to deter mimicry by its alter ego (type L). Formally, this means that firm i can convince the consumer that it is of type H by playing the separating best response [[rho].sub.H](E([p.sup.*])) = .5{[d.sub.H] + [c.sub.L] + ([([d.sub.H] - [c.sub.L]).sup.2] - [([d.sub.L] - [c.sub.L]).sup.2])1/2}. As argued above, type L's best response is [[rho].sub.L](E([p.sup.*])) =([d.sub.L] + [c.sub.L])/2. Note that E([p.sup.*]) enters these functions through the terms [d.sub.H] and [d.sub.L]. Recalling that [c.sub.L] = 0, we can simplify to obtain [[rho].sub.H](E([p.sup.*])) = .5{[d.sub.H] + [(([d.sub.H] - [d.sub.L])([d.sub.H] + [d.sub.L])).sup.1/2]} and [[rho].sub.L](E([p.sup.*])) = [d.sub.L]/2. Both of these best-response functions are increasing in E([p.sup.*]); thus, own-price and expected rival-price are strategic complements.

Derivation of refined equilibrium prices. Each type of firm i plays a best response to the common rival separating strategy (which is summarized, for firm i's purposes, by its expected value). Then in a symmetric equilibrium, the equilibrium expected price, E([p.sup.*]), is a solution to the equation

X = [lambda][[rho].sub.H](X) + (1 - [lambda])[[rho].sub.L](X) = [lambda][d.sub.H]/2 + (1 - [lambda])[d.sub.L]/2 + ([lambda]/2)(([d.sub.H] - [d.sub.L])[([d.sub.H] + [d.sub.L])).sup.1/2]. (A1)

Upon substitution, we obtain

X = [U + g(n - 1)X]/2b + [lambda][delta]/2 + [lambda]{[delta][(U + g[(n - 1)X)/2b] + [[delta].sup.2]/4}.sup.1/2], (A2)

where U [equivalent to] a - b[delta] + g(n - 1)(1 - [lambda])[delta] > 0 by Assumption 1. Let Y [equivalent to] [U + g(n - 1)X]/2b; then X = (2bY - U)/g(n - 1), and (A2) becomes

Y[2b - g(n - 1)]/g(n - 1) = U/g(n - 1) + [lambda][delta]/2 + [lambda][{[delta](Y + [delta]/4)}.sup.1/2]. (A3)

For later purposes, note that [[rho].sub.L](E([p.sup.*])) = [d.sub.L]/2 = Y.

Finally, let W [equivalent to] [{[delta](Y + [delta]/4)}.sup.1/2]; then Y = [W.sup.2]/[delta] - [delta]/4. For later purposes, note that [p.sub.H](E([p.sup.*])) = [d.sub.H]/2 + W and [[rho].sub.H](E([p.sup.*])) - [[rho].sub.L](E([p.sup.*])) = [d.sub.H]/2 + W - [d.sub.L]/2 = [delta]/2 + W. Substitution of Y in terms of W into (A3) and simplification yields the following quadratic in IV:

[[eta].sub.1] [W.sup.2] + [[eta].sub.2] W + [[eta].sub.3] = 0, (A4)

where [[eta].sub.1] [equivalent to] [2b - (n - 1)g]/[delta], [[eta].sub.2] [equivalent to] - [lambda]g(n - 1), and [[eta].sub.3] [equivalent to] - U - [delta]g(n - 1)[delta]/2 - ([delta]/4)[2b - (n - 1)g]. The coefficient [[eta].sub.1] is positive, while [[eta].sub.2] and [[eta].sub.3] are negative. Thus, equation (A4) has one positive and one negative root; given that W is defined as a square root, the solution we seek is the positive root of equation (A4). Solving and substituting back the original parameters, [W.sup.*] can be written as

[W.sup.*] = [[lambda][delta][gamma](n - 1) + {[([lambda][delta][gamma](n - 1)).sup.2] + 4[delta](2[beta] + [(n - 3)[gamma])[[eta].sub.4]}.sup.1/2]]/2(2[beta] + (n - 3)[gamma]), (A5)

where [[eta].sub.4] [equivalent to] [alpha]([beta] - [gamma]) - [delta]([beta] + (n - 2)[gamma]) + [gamma][delta] (n - 1)(1 - [lambda]) + [lambda][delta][gamma](n - 1)/2 + ([delta]/4)(2[beta] + (n - 3)[gamma]). Notice that [W.sup.*] depends upon [alpha] (and therefore upon a, as a = [alpha]/([beta] + (n - 1)[gamma])) only through the expression [[eta].sub.4], which appears within the square root term in [W.sup.*]. Thus, [W.sup.*] increases in a (similarly, [W.sup.*] increases in a), but at a rate less than [[alpha].sup.1/2] (similarly, at a rate less than [[alpha].sup.1/2]). Using the fact that W = [{[delta](Y + [delta]/4)}.sup.1/2] and (A3), we obtain

[Y.sup.*] = [U + [lambda][delta]g(n - 1)/2 + [lambda]g(n - 1)[W.sup.*]]/[2b - (n - 1)g]. (A6)

Using the fact that X = (2b Y - U)/g(n - 1), we obtain [X.sup.*] = [U + b[lambda][delta] + 2b[lambda][W.sup.*]]/ [2b - (n - 1 )g]. (A6)

We noted above that [[rho].sub.L](E([p.sup.*])) = Y and [[rho].sub.H](E([p.sup.*])) - [[rho].sub.L](E([p.sup.*])) = [delta]/2 + W. Thus, the equilibrium interim prices [P.sub.L] and [P.sub.H] are simply [P.sub.L] = [[rho].sub.L]([X.sup.*]) = [Y.sup.*] and [P.sub.H] = [[rho].sub.H] ([X.sup.*]) = [Y.sup.*] + [delta]/2 + [W.sup.*]. The equilibrium interim quantities are [Q.sub.L] = b[Y.sup.*] and [Q.sub.H] = b([Y.sup.*] + [delta]/2 - [W.sup.*]). Finally, the equilibrium interim profits are [[PI].sub.L] = b[([Y.sup.*]).sup.2] and [[PI].sub.H] = ([Y.sup.*] + [delta]/2 + [W.sup.*] - k)b([Y.sup.*] + [delta]/2 - [W.sup.*]).

Claim in text regarding positive realized equilibrium demand. In the text before the statement of Proposition 1, it was claimed that (i) [P.sub.H] > [P.sub.L] + [delta], and thus, when firms use the prices ([P.sub.L], [P.sub.H]), a firm's realized demand is lowest when it is of type H and all of its rivals are of type L. Moreover, it was claimed that (ii) there exists [[alpha].sub.A2] < [infinity] such that for all [alpha] > [[alpha].sub.A2], this lowest realized demand is positive.

Proof.

(i) [P.sub.H] - [P.sub.L] = [delta]/2 + [W.sup.*], and it is straightforward to show that [W.sup.*] > [delta]/2. Realized demand for firm i is given by [q.sub.i] = a - b(1 - [[theta].sub.i])[delta] + g [[summation].sub.j [not equal] i] (1 - [[theta].sub.j])[delta] - [bp.sub.i] + g [[summation].sub.j [not equal to] i] [p.sub.j]. This is smallest when the coefficient of b (that is, (1 - [[theta].sub.i])[delta] + [p.sub.i]) is largest and the coefficient of g (that is, [[summation].sub.j [not equal to] i] (1 - [[theta].sub.])[delta] + [[summation].sub.j [not equal to] i] [p.sub.j]) is smallest. Because [P.sub.H] > [P.sub.L] + [delta], this occurs when firm i is of type H and all other firms are of type L.

(ii) Thus, the lowest realized demand is given by q = a + g(n - 1)[delta] - b[P.sub.H] + g(n - 1)[P.sub.L]. Substitution of [P.sub.L] = [Y.sup.*] and [P.sub.H] = [Y.sup.*] + [delta]/2 + [W.sup.*] into this formula, followed by substitution of [Y.sup.*] in terms of [W.sup.*] from equation (A6), and collecting terms yields q = ab/(2b - (n - 1)g) + [K.sub.1] - [K.sub.2] W*(a), where [K.sub.1] is a positive constant independent of a and [K.sup.2] is a positive constant independent of a. Because equation (A5) provides [W.sup.*] as a function of [alpha] and [alpha] = a([beta] + (n - 1)[gamma], we indicate the dependence of [W.sup.*] on a by [W.sup.*](a). The first term in the expression for q increases linearly in a, while a appears only under a square root in [W.sup.*](a). Thus, there is a sufficiently high value of a, call it [[alpha].sub.A2], such that the sum of the first two terms will exceed the last term for all a > [a.sub.A2]. Because a = [alpha]/([beta] + (n - 1)[gamma]), the corresponding required value of [alpha] is [[alpha].sub.A2] = [a.sub.A2]([beta] + (n - 1)[gamma]). Q.E.D.

Results and proofs of selected propositions.

Proof of Proposition 1. We have identified a unique (refined) candidate for a symmetric separating equilibrium. To verify that the strategies and beliefs do provide a separating equilibrium, suppose that all firms but firm i play the strategy ([P.sub.L], [P.sub.H]) given above, with expected value [X.sup.*], and that the consumer maintains the beliefs [B.sup.*] (p) = 0 whenp < [P.sub.H], and [B.sup.*](p) = 1 when p [greater than or equal to] [P.sub.H]. Then, by construction, the type L firm i would be unwilling to charge a price at or above [P.sub.H] (which is equal to [[rho].sub.H]([X.sup.*])) in order to be taken for type H. Rather, it will prefer to be taken for type L and to charge the price [P.sub.L] (which is equal to [[rho].sub.H]([X.sup.*])). On the other hand, the type H firm i would be willing to charge a price at or somewhat above [P.sub.H] (which is equal to [[rho].sub.H]([X.sup.*])) in order to be taken for type H, but among these it prefers the lowest price; that is, [P.sub.H]. The consumer's beliefs are correct in equilibrium, and [X.sup.*] = [delta][P.sub.H] + (1 - [lambda])[P.sub.L]. Finally, note that [P.sub.H] - [P.sub.L] = [delta]/2 + [W.sup.*] > 0, and [Q.sub.L] - [Q.sub.H] = b([W.sup.*] - [delta]/2), which is easily shown to be positive. Q.E.D.