Imperfect competition and quality signalling.

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where [E.sub.-i] denotes the expectation with respect to the vector of rivals' product quality [[theta].sub.-i]. Because the second term above is linear in the rivals' types and prices, and the types are identically and independently drawn, this term is simply a - b(1 - [[??].sub.i])[delta] + g(n - 1)(1 - [lambda])[delta] - [bp.sub.i] + g(n 1)E([p.sup.*]), where E([p.sup.*]) [equivalent to] [p.sup.*](1) + (1 - [lambda])[p.sup.*](0). Thus, firm i's expected profits can be written as [[PI].sub.i]([p.sub.i], [[theta].sub.i], [[??].sub.i]|E([p.sup.*])) [equivalent to] ([p.sub.i] - k[[theta].sub.i])(a - b(1 - [[??].sub.i])[delta] + g(n - 1)(1 - [lambda])[delta]-[bp.sub.i] + g(n - 1)E([p.sup.*])).

Note that for any given price, it is always more profitable to be perceived as type H, regardless of true type. Thus, no type will incur a signalling distortion in order to be perceived as type L.

To formalize the notion of the consumer's perception of quality, we define a belief function which specifies the type (high- or low-quality) that the consumer assigns to a firm. The firms (of a given type) are completely symmetric, so it is natural to impose a symmetry assumption. Moreover, firm types are independently drawn and no firm has any information about its rivals' types when it chooses its price, so its price cannot vary with the rivals' types, and hence its price cannot convey any information about the rivals' types. Thus, it is also natural to assume that the consumer's beliefs about firm i's type do not vary with the price charged by firm j, j [not equal to] i. Fudenberg and Tirole (1991) incorporate this restriction (which they refer to as "no signalling what you don't know") into their definition of perfect Bayesian equilibrium for a general class of abstract games of which ours is a special case. Let B(p) be the belief function; thus, if firm i charges [p.sub.i], then it is inferred to be of type B([p.sub.i]). This belief function depends only on the price chosen by firm i, and it is the same function for all firms, so it reflects the preceding discussion of symmetry and invariance with respect to rivals' prices.

In equilibrium, each firm maximizes its expected profits, given the pricing functions of its rivals and given the consumer's belief function, and the consumer's beliefs are consistent with the equilibrium pricing function used by the firms. This is formalized in the following definition.

Definition 1. A symmetric separating perfect Bayesian equilibrium consists of a pair of prices ([p.sup.*](0),[p.sup.*](1)) [equivalent to] ([P.sub.L], [P.sub.H]) and beliefs [B.sup.*](p) such that, for i = 1, 2, ..., n,

(i) [[PI].sub.i]([P.sub.L], 0, 0 [absolute value of E([p.sup.*])) [greater than or equal to] [max.sub.p] [[PI].sub.p](p, 0, [B.sup.*](p)] E([p.sup.*]));

(ii) [[PI].sub.i]([P.sub.H], 1, [absolute value of E([p.sup.*])) [greater than or equal to] [max.sub.p] [[PI].sub.i](p, 1, [B.sup.*](p)] E([p.sup.*]));

(iii) [B.sup.*]([P.sub.L]) = 0, [B.sup.*]([P.sub.H]) = 1;

(iv) E([p.sup.*]) = [lambda][P.sub.H] + (1 - [lambda])[P.sub.L].

Part (i) says that a firm that has a low-quality product, and is perceived to have a low-quality product, would prefer to charge [P.sub.L] rather than its best alternative price, given the consumer's belief function and the expected rival price. Part (ii) says that a firm that has a high-quality product, and is perceived to have a high-quality product, would prefer to charge [P.sub.H] rather than its best alternative price, given the consumer's belief function and the expected rival price. These conditions reflect both a firm's best-response behavior vis-a-vis its rivals and its incentive compatibility conditions vis-a-vis its own alter ego. Part (iii) says that the consumer's beliefs are correct in equilibrium; she believes the product is of low quality when the firm charges [P.sub.L] and she believes the product is of high quality when the firm charges [P.sub.H]. Finally, part (iv) says that the prior-weighted expectation of [P.sub.H] and [P.sub.L] equals the expected rival price; that is, it is a mutual best response for all firms to play the price strategy ([P.sub.L], [P.sub.H]). Let ([Q.sub.L], [Q.sub.H]) denote the associated quantities and let ([[PI].sub.L], [[PI].sub.H]) denote the associated profit levels.

Results. In this section, we provide the primary results for the model above, along with a discussion of the intuition for the results. The comparative statics results (which are explored in Propositions 2, 5, and 6) are summarized in Table 1 near the end of the section. Unless otherwise specified, all references to equilibrium entities (i.e., prices, quantities, and profits) are interim versions (that is, each firm knows its type, but not the type of any rival).

We need to restrict the parameters for the analysis so as to guarantee an interior equilibrium, both ex ante and ex post, because the derivations above implicitly assume this. First we consider ex ante restrictions. Define the firm's true-type-dependent marginal costs: [c.sub.s] [equivalent to] k[[theta].sub.s], where s = L,H. The firm's demand (based on its perceived type) is ([d.sub.t]-p)b, where [d.sub.t] [equivalent to] {a - b(1 - [[theta].sub.t])[delta] + g(n - 1)(1 - [lambda])[delta] + g(n - 1)E([p.sup.*])}/b, t = L, H. Thus, we can use the short-hand notation [[PI].sub.st], = (p - [c.sub.s])b([d.sub.t] - p), for s, t = L, H.

For some parameter combinations, one or both types might not be able to choose prices that yield positive expected profits; under such parameter combinations, agents would potentially conjecture that the prior probability of a firm being an H-type was not [lambda], and should be updated, something that is not of particular interest to the current analysis. To ensure that there is always a profitable price for a firm, regardless of the consumer's perceptions of quality and regardless of the rival firm's expected price, we need [d.sub.t] > [c.sub.s] for all s, t. The tightest such constraint is [d.sub.L] > [c.sub.H]; that is, {a - b(1 - [[theta].sub.t])[delta] + g(n - 1)(1 - [lambda])[delta] + g(n - 1)E([p.sup.*])}/b > k. Recognizing that [lambda] may be arbitrarily close to 1, E([p.sup.*]) may be arbitrarily close to zero, and k (though smaller) may be arbitrarily close to [delta], we employ the following sufficient condition, which we maintain throughout the article.

Assumption 1. a > 2[delta]b; that is, [alpha]/[delta] > 2 + 2(n - 1)[[gamma]/([beta] - [gamma])].

Assumption 1 requires that the product be of sufficiently "high value" (in terms of the maximum willingness to pay, [alpha]) relative to the possible loss, [delta]; in the case of a monopoly, we would require [alpha]/[delta] > 2. The second term on the right-hand side reflects the intensity of competition and combines both the number of firms and the degree of product substitution. Thus, for this assumption to hold, we require that competition must be "sufficiently imperfect" in the sense that the extent of substitutability [gamma] (relative to [beta]) or the number of firms n must be sufficiently small. Notice that Assumption 1 is a strong sufficient condition for [d.sub.L] > [c.sub.H], not a necessary condition.

We also need the equilibrium realized demand to be positive, so a second assumption addresses ex post interiority. Below we provide a proposition concerning the separating equilibrium prices ([P.sub.L], [P.sub.H]), which are specified in the Appendix. Using those defined values, note that [P.sub.H] > [P.sub.L] + [delta]. As a consequence, 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, that is, when q = a - b[P.sub.H] + g(n - 1)[delta] + g(n - 1)[P.sub.L]. As shown in the Appendix, this lowest realized demand is positive if a is sufficiently high. In the sequel, we maintain this assumption. Unfortunately, due to the complexity of the algebra involved, it is unclear whether this required level is greater or less than that implied by Assumption 1 (in our later computations, we make sure that the results respect both assumptions).

Assumption 2. [alpha] is sufficiently large such that a - b[P.sub.H] + g(n - 1)[delta] + g(n - 1)[P.sub.L] > 0.

The following proposition indicates that (under Assumptions 1 and 2) there always exists a symmetric perfect Bayesian equilibrium (PBE) in which the two types separate. Moreover, by employing a refinement (the Intuitive Criterion; see the Appendix), we select one separating equilibrium and show that this is the only symmetric equilibrium (separating or pure pooling) which survives refinement. Due to the complexity of the algebra of expressing the equilibrium prices, quantities, and profits, we leave these details to the Appendix.

Proposition 1 (Existence of a unique refined equilibrium). There is a unique (refined) symmetric separating perfect Bayesian equilibrium consisting of a pair of prices ([P.sub.L], [P.sub.H]), with [P.sub.H] > [P.sub.L], and supporting beliefs [B.sup.*](p), with [B.sup.*](p) = 0 whenp < [P.sub.H], and [B.sup.*](p) = 1 when p [greater than or equal to] [P.sub.H]. Moreover, [Q.sub.L] > [Q.sub.H]; that is, a low-quality firm produces a higher equilibrium output than a high-quality firm.