Imperfect competition and quality signalling.

Elimination of (pure) pooling equilibria. Consider possible pooling equilibria. Firms will pool at a price [P.sup.P] if the following incentive compatibility constraints hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A7)

with beliefs B(P) = 0 for P [not equal to] [P.sup.P]. The above incentive constraints express the profits of firm i at the pooling price, when the consumer believes it is an H-type firm with probability [lambda] supported by beliefs that treat any out-of-equilibrium price as indicating that the firm is type L for sure (less harsh out-of-equilibrium beliefs can also support pure pooling). The pair of inequalities in (A7) generate the following associated inequalities in terms of the parameters of the problem:

(i) [P.sup.P]b([d.sup.P] - [P.sup.P]) [greater than or equal to] b[([d.sub.L]/2).sup.2] and (ii) ([P.sup.P] - k)b([d.sup.P] - [P.sup.P]) [greater than or equal to] b[(([d.sub.L] - k)/2).sup.2],

where [d.sup.P] is the "pooled" version of [d.sub.H] and [d.sub.L], and can be shown to be [d.sup.P] = [d.sub.L] + [lambda][delta].

We need not actually construct a pooling equilibrium, as we need only show that if one exists, then there is a price to which the H-type firm could profitably defect (that would be unprofitable for an L-type firm) if the consumer were to update her beliefs and infer that the signal came from an H-type firm. Thus, [P.sup.P] fails the Intuitive Criterion if there exists [P.sup.*] such that

(i) [P.sup.*]b([d.sub.H - [P.sup.*]) [less than or equal to] [P.sup.P]b([d.sup.P]-[P.sup.P]) and (ii) ([P.sup.*] - k)b([d.sub.H] - [P.sup.*]) [greater than or equal to] ([P.sup.P] - k)b([d.sup.P] - [P.sup.P]). (A8)

In (A8), the left-hand side of each inequality is the profits that would be obtained by (respectively) the L-type and H-type firms by defecting (and being taken to be an H-type firm after the consumer has updated her beliefs), whereas the profits from the pooling equilibrium appear on the right-hand side.

Let us denote the roots to (A8i) as [P.sup.-.sub.L] and [P.sup.+.sub.L] > [P.sup.-.sub.L], and the roots to (A8ii) as [P.sup.-.sub.H] and [P.sup.+.sub.H] > [P.sup.-.sub.H]. Then (A8) is equivalent to asking whether there exists [P.sup.*] such that [P.sup.*] [member of] [[P.sup.-.sub.H], [P.sup.+.sub.H] and [P.sup.*] [not member of] [[P.sup.-.sub.L], [P.sup.+.sub.L]; if so, then [P.sup.P] fails the Intuitive Criterion. With a little algebra one can show that [P.sup.+.sub.H] > [P.sup.+.sub.L], so such a [P.sup.*] exists for any [P.sup.P]. Thus no pooling equilibrium survives refinement.

Full-information price and profit formulas. Let firm i's type be [[theta].sub.i] and firm i's rivals' types be summarized by the vector of types [[theta].sub.-i]. Then the full-information equilibrium price for firm i is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The full-information price for an industry composed only of L-type firms is [P.sub.L.sup.F] [equivalent to] [P.sup.F](0, [0.sub.-i]) and the full-information price for an industry composed only of H-type firms is [P.sub.H.sup.F] = [P.sup.F](1, [1.sub.-i]), where [0.sub.-i] denotes an n - 1 vector of 0s and [1.sub.-i] denotes an n - 1 vector of 1s. Finally, substitution and simplification yields [[PI].sup.F](0, [[theta].sub.-i]) = b[([P.sup.F](0, [[theta].sub.-i])).sup.2] and [[PI].sup.F](1, [[theta].sub.-i]) = b[([P.sup.F](1, [[theta].sub.-i]) - k).sup.2].

Proof of Proposition 3. In each argument below, the first inequality follows from Proposition 2, and the remaining results follow from evaluating and comparing the pricing equations. First consider the L-type firm: for all [lambda] [member of] (0, 1), [P.sub.L] > [lim.sub.[lambda] [right arrow] 0] [P.sub.L] = ([alpha] - [delta]) ([beta] - [gamma])/[2[beta] + (n - 3)[gamma]] = [p.sup.F] (0, [0.sub.-i]) [greater than or equal to] [P.sup.F] (0, [[theta].sub.-i]) for all [[theta].sub.-i]. Now consider the H-type firm: for all [lambda] [member of] (0, 1), [P.sub.H] > [lim.sub.[lambda] [right arrow] 0] [P.sub.H] > ([alpha] - [delta])([beta] - [gamma])/[2[beta] + (n - 3)[gamma]] + [delta] > [P.sup.F](1, [0.sub.-i]) [greater than or equal; to] for all [[theta].sub.-i]). Q.E.D.

Proof of Proposition 4.

(i) [[PI].sub.L] = b[([P.sub.L]).sup.2] and [[PI].sup.F](0, [[theta].sub.-i]) = b[([P.sup.F](0, [[theta].sub.-i])).sup.2], so the claim follows immediately from Proposition 3. (ii) Note that [lim.sub.[lambda] [right arrow] 1] [P.sub.H] = [[alpha] ([beta] - [gamma]) + 2([beta] + (n - 2)[gamma])[W.sup.*]]/[2/[beta] + (n - 3)[gamma]], while [P.sup.F] (1, [1.sub.-i]) = [[alpha]([beta] - [gamma]) + k([beta] + (n - 2)[gamma])]/[2[beta] + (n - 3)[gamma]]. Thus, [lim.sub.[lambda] [right arrow] 1] [P.sub.H] > [P.sup.F] (1, [1.sub.-i] if and only if 2[W.sup.*] > k, which is easily verified. The full-information price for a collusive industry composed only of H-type firms is [P.sub.C.sup.F] = ([alpha] + k)/2. Then [lim.sub.[lambda] [right arrow] 1] [P.sub.H] < ([alpha] + k)/2 if and only if [alpha](n - 1)[gamma] + k(2[beta] + (n - 3)[gamma]) > 4[W.sup.*]([beta] + n - 2)[gamma]). This is certainly true if [alpha] is large enough, because the left-hand side is increasing linearly in [alpha], while the right-hand-side increases at a rate less than the square root of [alpha]. When [lim.sub.[lambda] [right arrow] 1] [P.sub.H] is less than the collusive price, then we can conclude that [lim.sub.[lambda] [right arrow] 1] [[PI].sub.H] > [PI].sup.F] (1, [1.sub.-i]). This is because when all firms charge the same price (whether it is the noncooperative full-information price, the collusive full-information price, or the interim price), each firm's profits are given by (p - k)(a - (b - (n - 1)g)p), which is a quadratic function that reaches its maximum at the collusive price. Thus, for high-value markets, [lim.sub.[lambda] [right arrow] 1] [[PI].sub.H] > [[PI].sup.F] (1, [1.sub.-i] = [lim.sub.[lambda] [right arrow] 1] [E.sub.-i] {[[PI]>sup.F] (1, [[theta].sub.-i)}, where the equality follows because in the limit only the term [[PI].sup.F] (1, [1.sub.-i]) has positive weight. Consequently, [[PI].sub.H] > [E.sup.-i]{[PI].sup.F](1, [[theta].sub.-i])} when [lambda] is sufficiently close to 1. Q.E.D.

Partial proof of Proposition 6. Here we provide a partial proof of Proposition 6 as an illustration of how the other comparative statics results can be proved. Recall that [P.sub.L] = [Y.sup.*] from equation (A6) and [P.sub.H] = [Y.sup.*] + [delta]/2 + [W.sup.*] from equation (A5).

(i) First consider the effect of [delta] on [W.sup.*].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A9)

Note that [partial derivative][W.sup.*]/[partial derivative][delta] > 0 for all [lambda] [member of] (0, 1) if [partial derivative]([delta][[eta].sub.4]/[partial derivative][delta] > 0 for all [lambda] [member of] (0, 1). But

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first line is positive by Assumption 1, and the remaining terms are positive for all [lambda]. Thus, we conclude that [parital derivative][W.sup.*]/[partial derivative][delta] > 0 for all [lambda] [member of] (0, 1). This implies that [P.sub.H] - [P.sub.L] = [delta]/2 + [W.sup.*] is strictly increasing in [delta] for all [lambda] [member of] (0, 1), as claimed in part (i).

(ii) Recall that [P.sub.L] = [Y.sup.sup.*]], [Q.sub.L] = b[Y.sup.*], and [[PI].sub.L] = b[(Y.sup.*]).sup.2]. Thus, to determine the effect of a change in [delta] on each of these expressions, it is sufficient to determine the effect of a change in [delta] on [Y.sup.*].

sgn{[partial derivative][Y.sup.*]/[partial derivative][delta]} = sgn{-([beta] - [gamma]) - [gamma][lambda](n - 1)/2 + [gamma][lambda](n - 1)([partial derivative][W.sup.*]/[partial derivative][delta])}.

The sum of the first two terms is negative for all [lambda] [member of] (0, 1). Because [partial derivative][W.sup.*]/[partial derivative][delta] > 0 for all [lambda] [member of] (0, 1), the last term goes to zero as [lambda][right arrow]0, but it is strictly positive for [lambda] [member of] (0, 1). Thus, [lim.sub.[lambda][right arrow]0] [partial derivative] [Y.sup.*]/[partial derivative] [delta] < 0, which implies the claims made in part (ii).

(iii) The expression [partial derivative][W.sup.*]/[partial derivative][delta] involves a ratio whose numerator is positive and increases linearly in [alpha] and a denominator that is positive and increases at a rate less than the square root of [alpha], while the negative terms are independent of [alpha]. Thus, for any fixed [lambda] [member of] (0, 1), there exists a value [[alpha].sup.*]([alpha]) < [infinity] that is sufficiently large to ensure that the positive term [gamma][lambda](n - 1)([partial derivative][W.sup.*]/[partial derivative][delta]) balances the negative constant terms and, thus, that [partial derivative][Y.sup.*]/[partial derivative][delta] = 0. Let [[alpha].sup.*]](1) [equivalent to] [lim.sub.[lambda][right arrow]1] [[alpha].sup.*] ([lambda]); by construction, at [[alpha]sup.*] (1), [lim.sub.[lambda][right arrow]1] [partial derivative] [Y.sup.*]/[partial derivative] [delta] = 0. Now choose any [[alpha].sub.6iii], > [[alpha].sup.*](1); at [[alpha].sub.6iii], [lim.sub.[lambda][right arrow]1] [partial derivative] [Y.sup.*]/[partial derivative] [delta] > 0. Consequently, there is a neighborhood of [lambda] = 1, denoted ([[lambda].sub.6iii], 1), such that [partial derivative][Y.sup.*]/[partial derivative][delta] > 0 for all [lambda] [member of] ([[lambda].sub.6iii], 1), as claimed in part (iii). Q.E.D.

References