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