January 24, 2001 Eric Rasmusen notes, erasmuse@indiana.edu Http://Php.Indiana.edu/~erasmuse p. 263. This is a model of a monopolist who can either produce high quality or who cannot; he does not get to choose quality. Conceivably, if the high- quality production cost c is high enough, a firm might prefer to be low- quality-- but it does not have that choice. All it can do is be *thought* to be low quality, but it cannot get the zero cost of a low-quality firm unless it starts with it. p. 264. X denotes the ratio of informed to uninformed consumers, and M is the total amount of consumers. This is unfortunate notation. M is not used, and might as well have been normalized to 1. But it wasn't, so let's use it. We have two equations: X = Informed/Uninformed M = Informed + Uninformed Therefore, Uninformed = M - informed = M - uninformed * X. We can rewrite this as Uninformed = M/(1 +X). And then, Informed = XM/(1+X). These are useful things to keep in mind in reading the paper. The important expressions in footnote 4 are based on them. A fraction (1+ P^L - P)/b of uninformed consumers buy when the price is P in analogy to the fraction (1 + P^L-P)/1 of informed consumers who buy when the price is P and the quality is high. The intuitive criterion rules out equilibria in which too high a price fails to signal high quality. Condition (a) rules out an equilibrium in which a firm is afraid to raise its price because it will lose the uninformed consumers, I think. I'm not sure about that-- I should think more. Footnote 4 is actually very important, and ought to be in the text, since it shows the profit for a low-quality firm when it charges P and consumers put probability b(P) on high quality at that price. The notation pi(L,b,P) for a firm charging P for low quality when uninformed consumers believe it is high quality with probability b only applies for P> P^L. It does not cover the case where P=P^L, when even the informed consumers would buy. (Otherwise, the first equation in footnote 4 is wrong.) But this does mean it doesn't properly apply in footnote 6 on the next page. p. 265. Lemma 2's proof seems incomplete. It does not say why P(H)> P^L. But it is not incomplete; if this is a separating equilibrium, then by definition it must be that P(H) does not equal P(L). And profits just fall if the price is reduced *below* P^L. Note that we don't know that a separating equilibrium exists until Corollaries 2 and 3 later in the paper. Lemma 2 just says what one will look like if it does exist. The low-quality firm earns zero profits if P(L) > P^L because nobody would buy a product known to be low quality if the price were above P^L. It would earn positive profits if P(L) = P^L because the product cost is assumed to be 0 for low quality, and P^L >0. In the middle of this page, the set being considered is for the price P such that the low-quality firm could attract *all* the uninformed consumers by charging P. That yields the profit pi(L, 1, P). Footnote 6. Footnote 4 is important to deriving these equations. Consider the optimal choice of P for a low-quality firm under the assumption that all the uninformed consumers believe it has high quality (so b=1). Maximize (1+P^L-P) (P-c) P This is to maximize P -c +P^L P - P^L c -P^2 +Pc which has first order condition 1+P^L -2P +c=0, so P= (1+P^L+c)/2. If P^L > 1+c, this would yield P