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\noindent
  \textbf{Answers to Odd-Numbered Problems, 4th Edition of Games and
Information, Rasmusen }

\begin{center}
  {\bf PROBLEMS FOR CHAPTER 11: Signalling} 
\end{center}

\noindent    10 June 2007.     16 November 2006.   Erasmuse@indiana.edu.
Http://www.rasmusen.org.

\noindent This  contains answers to the odd-numbered problems in the
fourth edition of \textit{Games and Information} by Eric Rasmusen.  
 
\newpage

\begin{center}
 {\bf PROBLEMS FOR CHAPTER 11: Signalling}
 \end{center}

\noindent
  \textbf{11.1. Is Lower Ability Better?}\\
 Change Education I so that
the two possible worker abilities are $a \in \{1,4\}$.

\begin{enumerate}

\item[(a)] What are the equilibria of this game? What are the payoffs of the
workers (and the payoffs averaged across workers) in each equilibrium?

\noindent \textit{Answer}. The pooling equilibrium is \begin{equation}
\label{e98} {\ s_L = s_H = 0, w_0=w_1 =2.5, Pr(L|s=1) =0.5}, \end{equation}
which uses  passive conjectures:    . The payoffs are ${U_L = U_H = 2.5}$, for an
average payoff of \textrm{2.5}.

The separating equilibrium is 
\begin{equation}  \label{e99} {\ s_L = 0, s_H = 1,
w_0=1, w_1 =4. } \end{equation} The payoffs are ${U_L =1}$ and ${U_H = 2}$, for
an \textrm{average payoff of 1.5 }. This equilibrium can be justified by the
self selection constraints
 \begin{equation}  \label{e100} U_L(s=0) = 1 > U_L(s=
1)= 4 - 8/1 =-4 \end{equation} and 
\begin{equation}  \label{e101} U_H(s=0) = 1 <
U_H(s=1)= 4 - 8/4= 2.
 \end{equation}
 Thus, the payoff averaged across workers is 1.5 (=.5[1] + .5[2]).  

\item[(b)] Apply the Intuitive Criterion (see N6.2). Are the equilibria the
same?

\noindent 
\textit{Answer}. \textrm{Yes.} The intuitive criterion does not rule
out the pooling equilibrium in the game with $a_h=4$. There is no incentive for
\textit{either} type to deviate from $s=0$ even if the deviation makes the
employers think that the deviator is high-ability. The payoff to a persuasive
high-ability deviator is only 2, compared the 2.5 that he can get in the pooling
equilibrium.

\item[(c)]
 What happens to the equilibrium worker payoffs if the high ability is
5 instead of 4?

\noindent \textit{Answer}. The pooling equilibrium is
 \begin{equation}
\label{e102} {\ s_L = s_H = 0, w_0=w_1 =, Pr(L|s=1) =0.5},
 \end{equation} which
uses passive conjectures. The payoffs are ${U_L = U_H = 3}$, with an
\textrm{average payoff of 3}.

The separating equilibrium is
 \begin{equation}  \label{e103}
 {\ s_L =0, s_H = 1,
w_0= 1, w_1 =5.} \end{equation}
 The payoffs are ${U_L=1}$ and ${U_H=3.4}$ with
an \textrm{average payoff of 2.2}. The self-selection constraints are
\begin{equation}  \label{e104}
 U_H(s=0) = 1 < U_H(s=1)= 5 - \frac{8}{5} =3.4
\end{equation}
 and
 \begin{equation}  \label{e105}
 U_L(s=0) = 1 > U_L(s=1)= 5 -
\frac{8}{1} =-3. 
\end{equation}

\item[(d)] Apply the Intuitive Criterion to the new game. Are the equilibria the
same?

\noindent
 \textit{Answer}.   {No.} The strategy of choosing $s=1$ is dominated
for the Lows, since its maximum payoff is $-3$, even if the employer is
persuaded that he is High. So only the separating equilibrium survives.

\item[(e)] Could it be that a rise in the maximum ability reduces the average
worker's payoff? Can it hurt all the workers?

\noindent 
\underline{\textit{Answer}.}  {Yes.} Rising ability would reduce the
average worker payoff if the shift was from a pooling equilibrium when $ a_h=4 $
to a separating equilibrium when $a_h=5$. Since the Intuitive Criterion rules
out the pooling equilibrium when $a_h=5$, it is plausible that the equilibrium
is separating when $a_h=5$. Since the pooling equilibrium is pareto-dominant
when $a_h=4$, it is plausible that it is the equilibrium played out. So the
average payoff may well fall from 2.5 to 2.2 when the high ability rises from 4
to 5.  {This cannot make every player worse off}, however; the high-ability
workers see their payoffs rise from 2.5 to 3.4.


[Below are some additional notes that are in first draft, since they came up in a recent class-- be on the lookout for errors] 

 What if the signal were continuous instead of having to equal either 1 or 4? Then, a lower signal could still induce separation. Would that remove the perverse result of higher ability reducing average payoffs? The basic problem is that any  signalling that goes on can't increase output. All the signalling does is to reduce  the average payoff. On the other hand, increasing ability does increase the average payoff, as a direct effect. So the question is whether the increase in signalling cost outweighs the increase in output. Let's see what happens here: 

 The high-ability self-selection constraint would be 
\begin{equation}  \label{e104a}
 U_H(s=0) = 1  \leq  U_H(s=s^*) = 5- \frac{8s^*}{5}, 
\end{equation}
 which means we would need $s^* \leq 2.5$. 

 The 
low-ability self-selection constraint would be 
\begin{equation}  \label{e104b}
U_L(s=0) = 1  \geq  U_L(s=1)= 5 -
\frac{8 s^*}{1}, 
\end{equation}
so $s^*$ must be at least 0.5. If it is, then the separating payoffs are 1 for the low-ability and $5- \frac{8 (0.5)}{5} = 4 \; 1/5$ for the  high-ability, an average payoff of   2.6.

 The Intuitive Criterion results in the minimum necessary signal being used. Thus, use of it would argue for a shift from an average payoff of 2.5 to one of 2.6 when ability increased from 4 to 5 if the equilibrium shifted from pooling to separating. But the Intuitive Criterion would say that the pooling equilibrium would break down even when ability was 4, if the signal can be as low as 0.5. That's because the high-ability worker could deviate to signalling $s=0.5$ and get payoff $4-\frac{8*0.5}{4} = 4-1 =3$, which is better than the 2.5 from pooling. In fact, the value of $s^*$   when abilities are 1 and 4 can  be even lower: solving  $U_L(s=0) = 1  \geq  U_L(s=1)= 4 -
\frac{8 s^*}{1}$  yields $s^*=3/8$. That gives an average payoff of $(3.625+1)/2 =  2.3125$. So applying the Intuitive Criterion, higher ability now helps. If we don't apply it, though, a shift from pooling to separating might well reduce average welfare, and could even reduce the welfare of both players.  

 




\end{enumerate}

\bigskip \noindent \textbf{11.3. Price and Quality }\\
 Consumers have prior beliefs that Apex produces low-quality goods with
probability 0.4 and high quality with probability 0.6. A unit of output costs 1
to produce in either case, and it is worth 10 to the consumer if it is high-
quality and 0 if low-quality. The consumer, who is risk neutral, decides whether
to buy in each of two periods, but he does not know the quality until he buys.
There is no discounting.

\begin{enumerate}

\item[(a)]
 What is Apex' price and profit if it must choose one price, $ p^*$,
for both periods?

\noindent
 \underline{\textit{Answer}.} A consumer's expected consumer surplus is
\begin{equation}  \label{e109}
 CS = 0.4 (0-p^*) + 0.6 (10-p^*) + 0.6(10-p^*) = -1.6p^* + 12.
\end{equation}
 Apex maximizes its profits by setting $CS=0$, in which case ${  p^* = 7.5 }$
and profit is ${  \pi_H = 13 }$ ( = 2(7.5 - 1)) or ${  \pi_L= 6.5}$ ( (=7.5-1).

\item[(b)] 
What is Apex' price and profit if it can choose two prices, $ p_1$
and $p_2$, for the two periods, but it cannot commit ahead to $p_2$?

\noindent
 \underline{\it Answer}. If Apex is high quality, it will choose ${
p_2 = 10 }$, since the consumer, having learned the quality first period, is
willing to pay that much. Thus consumer surplus is
 \begin{equation}  \label{e110} CS = 0.4 (0-p_1) + 0.6 (10-p_1) + 0.6(10-10) = -
p_1 + 6, \end{equation} and, setting this equal to zero, ${\ p_1=6}$, for a
profit of ${\ \pi_H =14}$ (= (6-1) + (10-1) ) or ${\ \pi_L =5}$ (= 6-1 ).

\item[(c)] What is the answer to part (b) if the discount rate is $r = 0.1$?

\noindent \underline{\it Answer}. Apex cannot do better than the prices
suggested in part (b).

\item[(d)] Returning to $r=0$, what if Apex can commit to $p_2$?

\noindent \underline{\textit{Answer}.} Commitment makes no difference in this
problem, since Apex wants to charge a higher price in the second period anyway
if it has high quality--- a high price in the first period would benefit the
low-quality Apex too, at the expense of the high-quality Apex.

\item[(e)] How do the answers to (a) and (b) change if the probability of low
quality is 0.95 instead of 0.4? (There is a twist to this question.)

\noindent \underline{\textit{Answer}.} With a constant price, a consumer's
expected consumer surplus is
\begin{equation}  \label{e111}
  CS = 0.95 (0-p^*) + 0.05 (10-p^*) + 0.05(10-p^*) = -1.05p^* + 0.5
\end{equation}
 Apex would set $CS=0$, in which case $p^* =\frac{10}{21}$, but since this is
less than cost, Apex in fact would not sell anything at all, and would earn zero
profit.

With changing prices, high-quality Apex will choose ${\ p_2 = 10}$, since the
consumer, having learned the quality first period, is willing to pay that much.
Thus consumer surplus is \begin{equation}  \label{e112} CS = 0.95 (0-p_1) + 0.05
(10-p_1) + 0.05(10-10) = -p_1 + 0.5. \end{equation} and, setting this equal to
zero, you might think that $p_1= 0.5$, for a profit of $\pi_H = 8.5 (= (0.5-1) +
(10-1))$. But notice that if the low-quality Apex tries to follow this strategy,
his payoff is $\pi_L = 0.5-1 <0$. Hence, only the high-quality Apex will try it.
But then the consumers know the product is high-quality, and they are willing to
pay 10 even in the first period. What the high-quality Apex can do is charge up
to ${\ p_1 = 1}$ in the first period, for profits of 9 (=$(1-1) + (10-1))$.

\end{enumerate}

\bigskip

 \noindent
 \textbf{11.5. Advertising  }\\
Brydox introduces a new shampoo which is actually very good, but is believed by
consumers to be good with only a probability of 0.5. A consumer would pay 10 for
high quality and 0 for low quality, and the shampoo costs 6 per unit to produce.
The firm may spend as much as it likes on stupid TV commercials showing happy
people washing their hair, but the potential market consists of 100 cold-
blooded economists who are not taken in by psychological tricks. The market can
be divided into two periods.

\begin{enumerate}

\item[(a)] If advertising is banned, will Brydox go out of business?

\noindent \underline{\it Answer}. \textrm{No.} It can sell at a price of 5 in
the first period and 10 in the second period. This would yield profits of 300 (=
(100)(5-6) +(100) (10-6)).

\item[(b)]  If there are two periods of consumer purchase, and consumers
discover the quality of the shampoo if they purchase in the first period, show
that Brydox might spend substantial amounts on stupid commercials.

\noindent \underline{\textit{Answer}.} If the seller produces high quality, it
can expect repeat purchases. This makes expenditure on advertising useful if it
increases the number of initial purchases, even if the firm earns losses in the
first period. If the seller produces low quality, there will be no repeat
purchases. Hence, \textrm{advertising expenditure can act as a signal} of
quality: consumers can view it as a signal that the seller intends to stay in
business two periods.

\item[(c)]  What is the minimum and maximum that Brydox might spend on
advertising, if it spends a positive amount?

\noindent \underline{\textit{Answer}.} If there is a separating signalling
equilibrium, it will be as follows. Brydox would spend nothing on advertising if
its shampoo is low quality, and consumers will not buy from any company that
advertises less than some amount X, because such a company is believed to
produce low quality. Brydox would spend X on advertising if its quality is high,
and charge a price of 10 in both periods.

\textrm{Amount X is between 400 and 500. } If a low-quality firm spends X on
advertising, consumers do buy from it for one period, and it earns profits of
(100)(10-6)-X = 400-X. Thus, the high-quality firm must spend at least 400 to
distinguish itself. If a high-quality firm spends X on advertising, consumers
buy from it for both periods, and it earns profits of (2) (100)(10-6)-X = 800-X.
Since it can make profits of 300 even without advertising, a high-quality firm
will spend up to 500 on advertising.

\end{enumerate}

% -----------------------------------------------------------

  \bigskip 

\noindent {\bf    11.7.  Salesman Clothing    } \\
  Suppose  a salesman's ability might be either $x= 1$  (with probability
$\theta$) or $x=4$,  and that if he dresses well, his output is greater, so that
his total output is $x + 2s$ where $s$ equals 1 if he dresses well and 0 if he
dresses badly. The utility of the salesman is $U= w - \frac{8s}{x}$, where $w$
is his wage. Employers compete for salesmen.

\begin{enumerate}

 \item[(a)] Under full information, what will the wage be for a salesman with
low ability?

\noindent 
 \underline{\it Answer.}      Salesmen with low ability would not dress
well. Dressing well would raise their output to 3, but their utility at a wage
of 3 would be -5, whereas if they dress poorly their utility is 1. Thus, the
wage is 1.

 \item[(b)] Show the self selection contraints that must be satisfied in a
separating equilibrium under incomplete information.

\noindent \underline{\it Answer.} In a separating equilibrium, the low-ability
salemen must be satisfied with a contract in which they dress poorly, so  it
must be true that $$ \pi_L(poorly)  =  w(poorly)   \geq   \pi_L(well) = w(well)
-  8. $$ The high-ability salemen must be satisfied with a contract in which
they dress well, so  it must be true that $$ \pi_H(poorly)  =  w(poorly)   \leq
\pi_H(well) = w(well) - 2. $$

 \item[(c)] Find all the equilibria for   this game if information is
incomplete.

\noindent \underline{\it Answer.}   In the separating equilibrium, $w(poorly)=
1$ and $w(well)=6$. This satisfies the self selection constraints of part (b)
and yield zero profits to the employers.

In  one pooling equilibrium,  $ w(poorly)=  \theta  +  4(1-\theta) $  and
$w(well)= 3$ and all salesmen dress poorly, where $\theta$ is the percentage of
low-ability salesmen.  This is supported by the out-of-equilibrium belief   that
anyone who dresses well has low ability.

There is no pooling equilibrium in which everyone dresses well. That would
require that  $w(poorly)= 1$ and $w(well)=\theta + 4(1-\theta) + 2$, and that
$$ \pi_L(poorly)  =  w(poorly)   \leq   \pi_L(well) = w(well) -  8, $$ so $$
\pi_L(poorly)  =  1   \leq   \pi_L(well) = \theta + 4(1-\theta) + 2 -  8, $$ but
regardless of how close $\theta$ is to 0, this is impossible.

\end{enumerate}





\bigskip



\noindent 
 \textbf{11.9. Crazy Predators } (adapted from Gintis [2000], Problem
12.10 )\\
  Apex has a monopoly in the market for widgets, earning profits of $m$
per period, but Brydox has just entered the market. There are two periods and no
discounting. Apex can either $Prey$ on Brydox with a low price or accept
$Duopoly$ with a high price, resulting in profits to Apex of $-p_a$ or $d_a$ and
to Brydox of $-p_b$ or $d_b$. Brydox must then decide whether to stay in the
market for the second period, when Brydox will make the same choices. If,
however, Professor Apex, who owns 60 percent of the company's stock, is crazy,
he thinks he will earn an amount $p^*> d_a$ from preying on Brydox (and he does
not learn from experience). Brydox initially assesses the probability that Apex
is crazy at $\theta$.

\begin{enumerate}

\item[(a)]
  Show that under the following condition, the equilibrium will be
separating, i.e., Apex will behave differently in the first period depending on
whether the Professor is crazy or not: 
\begin{equation}   \label{e15.a1}
  -p_a +m
< 2d_a 
\end{equation}

\noindent \underline{\textit{Answer.}} In any equilibrium, Apex will choose
$Prey$ both periods if the Professor is crazy. In any equilibrium, Apex will
choose $Duopoly$ in the second period if the Professor is not crazy, by subgame
perfectness.

If the equilibrium is separating, Apex will choose $Duopoly$ in the first period
if the Professor is not crazy, and Brydox will respond by staying in for the
second period. This will yield Apex an equilibrium payoff of $2d_a$. The
alternative is to deviate to $Prey$. The best this can do is to induce Brydox to
exit, leaving Apex an overall payoff of $- p_a +m$ for the two periods, but if
$- p_a +m < 2d_a,$ deviation is not profitable. (And if Brydox would
\textit{not} exit in response to $Prey$, $Prey$ is even less profitable.)

\item[(b)] Show that under the following condition, the equilibrium can be
pooling, i.e., Apex will behave the same in the first period whether the
Professor is crazy or not:
 \begin{equation}  \label{e15.a2}
 \theta \geq \frac{d_b}{p_b+d_b}
 \end{equation}

\noindent 
 \underline{\textit{Answer.}} The only reason for Apex to choose $Prey$
in the first period if the Professor is not crazy is to induce Brydox to choose
$Exit$. Thus, we should focus on Brydox's decision. Brydox's payoff from $Exit$
is 0. Its payoff from staying in is $  \theta (-p_b) + (1-\theta) d_b.  $
Exiting is as profitable as staying in if $  0 \geq \theta (-p_b) + (1-\theta)
d_b, $ which implies that $  (p_b + d_b) \theta \geq d_b,\;\mathrm{and\; thus}
\; \theta \geq \frac{ d_b}{ p_b + d_b}.  $

\item[(c)] If neither  condition (\ref{e15.a1}) nor (\ref{e15.a2}) apply, the
equilibrium is hybrid, i.e., Apex will use a mixed strategy and Brydox may or
may not be able to tell whether the Professor is crazy at the end of the first
period. Let $\alpha$ be the probability that a sane Apex preys on Brydox in the
first period, and let $\beta$ be the probability that Brydox stays in the market
in the second period after observing that Apex chose $Prey$ in the first period.
Show that the equilibrium values of $ \alpha $ and $\beta$ are:
  \begin{equation}  \label{e15.a3}
 \alpha = \frac{ \theta p_b }{(1-\theta)d_b} \end{equation}
 \begin{equation}  \label{e15.a4} \beta =\frac{-p_a +m -2d_a}{m-d_a}
 \end{equation}

\noindent \underline{\textit{Answer.}}   An equilibrium mixing probability
equates the payoffs from its two pure strategy components. First, consider Apex.
Apex's two pure-strategy payoffs are: \begin{equation}  \label{e15.a3a}
 \pi_a(Prey) = -p_a + \beta d_a + (1-\beta)m = d_a + d_a = \pi_a(Duopoly),
\end{equation} so $\beta (d_a -m) = -m + p_a + 2d_a$ and we reach equation
(\ref{e15.a4}).

Note that we know the numerator of equation (\ref{e15.a4}) is positive, because
we have ruled out a separating equilibrium by not having the inequality (
\ref{e15.a1}) hold. Also, the mixing probability is less than one because the
numerator is less than the denominator.

Now consider Brydox. Brydox's prior that Apex is crazy is $\theta$, but on
observing $Prey$, it must modify its beliefs. There was some chance that Apex,
if sane (which has probability $(1-\theta)$, would have chosen $ Duopoly $, but
that didn't happen. That had probability $(1- \alpha) (1-\theta)$ ex ante. Using
Bayes' Rule, the posterior probability that Apex is crazy is \begin{equation}
\label{e15.a3c} \frac{ \theta} {1- (1-\alpha) (1-\theta)}, \end{equation} and
the probability that Apex is sane is \begin{equation}  \label{e15.a3d} \frac{ (
\alpha)(1-\theta)} {1-(1-\alpha) (1-\theta)}, \end{equation}

Brydox's two pure-strategy payoffs after observing $Prey$ are therefore
\begin{equation}  \label{e15.a3b} \pi_b(Exit) = -p_b = -p_b + \frac{ \theta}
{1-(1-\alpha) (1- \theta)} (-p_b) + \frac{ ( \alpha)(1-\theta)} {1-(1-\alpha)
(1-\theta)}d_b = \pi_b(Stay \; in), \end{equation} so $0 = \theta (- p_b) + (
\alpha)(1-\theta) d_b$ and \begin{equation}  \label{e15.a3e} \alpha = \frac{
\theta p_b }{(1-\theta)d_b} \end{equation} If condition (\ref{e15.a2}) is false,
then expression (\ref{e15.a3e}) is less than 1, a nice check that we have
calculated the mixing probability correctly (and it is clearly greater than
zero).

\item[(d)]  Is this behavior related to any of the following phenomenon?--
Signalling, Signal Jamming, Reputation, Efficiency Wages.

\noindent \underline{\textit{Answer.}} This is an example of signal jamming.
Apex alters its behavior in the first period so as to avoid conveying
information to Brydox. It is not signalling,because Apex is not trying to signal
its type. It is not reputation, because this is just a two-period model, not an
infinite-period one. In loose language, one might call it reputation, because
Apex is trying to avoid acquiring a reputation for sanity, but it has nothing in
common with Klein- Leffler reputation models. It is not efficiency wages because
no agent is being paid more than his reservation utility so as to maintain
incentives, nor is even any firm being rewarded highly under the threat of
losing the reward if it behaves badly.

\end{enumerate}
 





\bigskip

 \noindent
 {\bf 11.11.  Monopoly Quality  }\\

 A consumer faces a
monopoly. He initially believes that the probability that the monopoly has a
high-quality product is H, and that a high-quality monopoly would be able to
send him an advertisement at zero cost.    With probability (1-H), though, the
monopoly has low  quality, and it would cost the firm A to send an ad.  The firm
does send an ad, offering the product at price P. The consumer's  utility from
a high-quality product is  $X>P$, but from a low quality product it  is 0.  The
production cost is $C$ for the monopolist  regardless of quality, where $C < P-
A$. If the consumer does not buy the product, the seller does not incur the
production cost.

 You may assume that the high-quality firm always sends an ad, that the consumer
will not buy unless he receives an ad, and that $P$ is exogenous.

\begin{enumerate}

 \item[(a)] Draw the extensive form for this game.

\noindent \underline{\it Answer.} xxxx  Answer unavailable now (the old diagram
file is unusable)

 \item[(b)] What is the equilibrium  if H is  sufficiently  high?

\noindent \underline{\it Answer.}      If H is high, then   both types of
monopoly will advertise, and the consumer will buy the product if he gets an
advertisement.

 \item[(c)] If H is low enough,  the equilibrium is in mixed strategies.  The
high-quality firm always advertises,   the    low quality firm  advertises with
probability  M, and the consumer buys with probability  N.   Show  using Bayes
Rule  how the consumer's posterior belief $R$  that  the  firm is high-quality
changes  once he receives an ad.

\noindent \underline{\it Answer.}    The prior is H. The posterior is $$ R =
Prob  (High|Advertise)= \frac{Prob  (Advertise|High)  Prob (High)  }{Prob
(Advertise)} = \frac{  (1) (H) }{    (1) (H)  +  (M) (1-H)}. $$

  \item[(d)] Explain why the equilibrium is not in pure  strategies if H is  too
low (but $H$ is  still positive).

 \noindent \underline{\it Answer.}  If $H$ is low, then it cannot be an
equilibrium for  the Low firm always to  advertise.  Suppose H is close to zero.
Then if the Low firm always enters, almost all advertising firms will have low
quality, and the consumer will not buy.  This  would result negative payoffs for
the Low firms, so they would not want to advertise.

But neither can it   be an equilibrium for no Low firm to advertise. In that
case, the consumer would buy, which would make it profitable for the Low firm to
advertise.

 \item[(e)] Find the equilibrium probability of M.  (You don't have to figure
out N.)

\noindent \underline{\it Answer.} The Low firm's mixing probability M  must be
such that the consumer is indifferent between buying and not buying.  His
expected payoff from not buying is 0. From buying, the payoff  must be computed
using his belief about the probability that the seller has high quality  which
is the  posterior probability $R$. Thus, $$ R(X-P) +  (1-R) (-P) =  RX -P =
\frac{  (X) (H) }{    (1) (H)  + (M) (1-H)}  - P $$ Equating  this to the payoff
of zero from not buying yields $ HX = (H + M - MH)P  $, so  $HX - HP = MP - MHP
$ and  $M = \frac{H(X-P)}{P(1-H)}$.

\end{enumerate}

%-------------------------------------------------------


\noindent
 {\bf 11.x    A Continuum of Pooling Equilibria       }  (medium)\\
Suppose that with equal probability a worker's ability is $a_L=1$ or $a_H=5$,
and that the worker chooses any amount of education $y \in [0,\infty)$. Let
$U_{worker}=w - \frac{8y}{a}$ and $\pi_{employer} = a-w$.

  There is a continuum of pooling equilibria, with different levels of $y^*$,
the amount of education necessary to obtain the high wage.  What education
levels, $y^*$, and wages, $w(y)$, are paid in the pooling equilibria, and what
is a set of out-of-equilibrium beliefs that supports them? What are the self-
selection  constraints?


\noindent
\underline{\it Answer}. A pooling equilibrium for any $y^* \in [0, 0.25]$ is
\begin{equation}  \label{e113}
   w = \left| \begin{array}{l}
  1 \;if \;y \neq y^* \\
     3 \;if \;y = y^* \\
 \end{array} \right.
 \end{equation}
    with the out-of-equilibrium belief that $     {Pr(L|(y \neq y^*))=1}$, and
with $     {y=y^*}$ for both types.

   The self-selection  constraints say that neither High nor Low workers want to
deviate by acquiring other than $y^*$ education. The most tempting deviation is
to zero education, so the constraints are:
 \begin{equation} \label{e114} U_L(y^*) =w(y^*) - 8y^* \geq U_L(0)= w(y \neq
y^*)
\end{equation}
 and \begin{equation}  \label{e115}
U_H(y^*) =w(y^*) - \frac{8y^*}{5}  \geq U_H(0)=  w(y \neq y^*). \end{equation}
The constraint on the Lows requires that $y^* \leq 0.25$ for a pooling
equilibrium.

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\end{document}


