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         \begin{document}

\begin{LARGE}

 
\begin{center}
 {\bf 11 Signalling}
 \end{center}
 

 \noindent
10 November  2005. 3 November 2006.   Eric Rasmusen, Erasmuse@indiana.edu.   Http://www.rasmusen.org.

 This is for two 75-minute sessions. I skip section 11.5 (two-signal models) 


In the first session, I covered 11.1-11.4, but didn't have time for the separating equilibrium of the continuous-signal model. 

 I did that in the second session, but then didnt' have time to do more than lay out the idea and model of Countersignalling (I didn't solve out the model).

Interesting digressions that came up: 

1. What if there is a monopoly screener? 

2. What if signalling productive? How would you model that in Education I? 

In the first printing of the 4th edition, I messed up in section 11.6 on the value of v*. It is 4F/q, not 2F/q. I fix that here. 


\newpage

 
 
\begin{center} \noindent
 {\bf  Education I} \end{center}
  

\noindent
{\bf The Order of Play}\\
 0 Nature chooses the worker's ability $a \in \{2,5.5\}$, the $Low$ and $High$
ability each having probability 0.5. The variable $a$ is observed by the worker,
but not by the employers.\\ 1 The worker chooses education level $s \in \{0,1\}
$.\\
 2 The employers each offer a wage contract $w(s)$.\\ 3 The worker accepts a
contract, or rejects both of them.\\
4 Output equals $a$.

\bigskip

\noindent
{\bf Payoffs}\\
 The worker's payoff is his wage minus his cost of education, and the employer's
is his profit. 

\bigskip


$ \pi_{worker} = $ $\left\{ \begin{tabular}{ll}$w - 8s/a$ &  if the worker
accepts contract $w$.\\
 0  &  if the worker rejects both contracts.\\ \end{tabular} \right.$

\bigskip

    $ \pi_{employer} = $ $\left\{ \begin{tabular}{ll} $a - w$ & for the accepted-contract employer,
 \\
0  & for the other employer. \end{tabular} \right.$

  \newpage

\begin{center}
  {\bf Equilibria}
 \end{center}

\begin{quotation}
\noindent
\begin{tabular}{cl}
 {\bf Pooling Equilibrium 1.1} & $\left\{ \begin{tabular}{l} $ s(Low)= s(High) =
0$\\ $ w(0)=w(1) = 3.75$\\
$Prob(a=Low|s=1) = 0.5$ \end{tabular} \right.$\\ & \\ \end{tabular}
\end{quotation}

  

If $Prob(a=Low|s=1) = 0$ the pooling equilibrium breaks down. 

\begin{quotation}
\noindent \begin{tabular}{cl}
 {\bf Separating Equilibrium 1.2} & $\left\{ \begin{tabular}{l} $s(Low) = 0,
s(High) =1$ \\ $ w(0) = 2, w(1) = 5.5 $\\ \end{tabular} 
\right.$ \\
 & \\
 \end{tabular} \end{quotation}

 The Intuitive Criterion rules out the beliefs of the pooling equilibrium. 

\newpage

\begin{center}
  {\bf The Constraints}
 \end{center}

    The participation constraints for the
employers require that
 \begin{equation} \label{e10.1} w(0)
    \leq a_L = 2 \;\; {\rm and} \;\; w(1) \leq a_H=5.5.
 \end{equation} 
 Competition between the employers makes the expressions in
(\ref{e10.1}) hold as equalities.

  The self-selection constraint of the $Lows$
is \begin{equation} \label{e10.2} U_L (s = 0) \geq U_L(s = 1), \end{equation}
which in Education I is \begin{equation} \label{e10.3} w(0) - 0 \geq w(1) -
\frac{8(1) }{2}. \end{equation} Since in Separating Equilibrium  1.2 the
separating wage of the $Low$s is 2 and the separating wage of the $High$s is 5.5
from (\ref{e10.1}), the self-selection constraint (\ref{e10.3}) is satisfied.

   The self-selection  constraint of the $High$s is
\begin{equation} \label{e10.4} U_H(s = 1) \geq U_H(s=0),
 \end{equation} which in Education I is \begin{equation} \label{e10.5}
 w(1) - \frac{8 (1)}{5.5} \geq w(0) - 0. \end{equation}  

\newpage

\begin{center}
 {\bf  Another Conceivable Equilibrium }
 \end{center}

 There is another conceivable pooling equilibrium for Education I, in which $
s(Low)= s(High) = 1$, but this turns out not to be an equilibrium, because the
$Low$s would deviate to zero education.

 Even if such a deviation caused the
employer to believe they were low - ability with probability 1 and reduce their
wage to 2, the low - ability workers would still prefer to deviate, because
\begin{equation} \label{e10.0a}
 U_L (s = 0)= 2  \geq U_L(s = 1) = 3.75 - \frac{8(1)}{2}.
 \end{equation}


 

\newpage

\begin{center}
 {\bf  The Single Crossing Property}
 \end{center}

 \includegraphics[width=150mm]{fig11-01.jpg}

  \begin{center}
{\bf Figure  1:  Education VI:  No Pooling   Equilibrium in a Screening Game }
 \end{center}

 The indifference curves in Education I are like this too.

 They cross a single time--- the single crossing property. 

 
   \newpage

 

 
 \begin{center} {\bf   Education II: Modelling Trembles so Nothing Is Out of Equilibrium}
 \end{center}

 \noindent {\bf The Order of Play}\\
 0 Nature chooses worker ability $a \in \{2,5.5\}$, each ability having
probability 0.5. ($a$ is observed by the worker, but not by the employer.)  With
probability 0.001, Nature endows a worker with free education of $s=1$. \\
$\ldots$

 \noindent
 {\bf Payoffs} \\ 
 $\pi_{worker} =$ $\left\{ \begin{tabular}{ll} $ w  - 8s/a$ &
Accepts contract $w$ (ordinarily)\\ $ w  $ & Accepts
contract $w$ (with free education) \\ $0$ & Refuses 
contract\\ \end{tabular} \right.$\\

   
Out-of-equilibrium, if the employer observed $s=1$, he could use Bayes's Rule:
\begin{equation}\label{e200}
\begin{array}{ll}
Prob (a=Low|s=1) &= \frac{Prob (s=1|a=L) Prob (L)}{Prob (s=1|a=L) Prob (L) +
Prob (s=1|a=H) Prob (H)}\\
 & \\
 &  = \frac{(0.001)(0.5)}{(0.001)(0.5) + (0.001)(0.5)}\\
 & \\
 &  = 0.5\\
 \end{array}
\end{equation}
  Normal workers would not deviate from $s=0$ because it would not increase the
employer's estimate of their ability.



\newpage


\begin{center}
  {\bf Education III: No Separating Equilibrium, Two Pooling Equilibria}
 \end{center}
  


\noindent 
  the  worker
abilities from $\{2, 5.5\}$ to $\{2, 12\}$.   



{\bf Pooling Equilibrium 3.1 }

\begin{quotation} \noindent \begin{tabular}{cl}  &
$\left\{ \begin{tabular}{l} $ s(Low)=s(High) = 0$\\ $ w(0)=w(1)=7$\\ $Prob(a=
Low|s=1) = 0.5$   \\ 
 \end{tabular} \right.$\\ \end{tabular}

\vspace{1in}



{\bf Pooling Equilibrium 3.2 }
\noindent \begin{tabular}{cl} 
  & $\left\{
\begin{tabular}{l} $s(Low)=s(High) = 1$\\ $ w(0) = 2, w(1)=7$\\ $Prob(a=Low|s=0)
= 1$ \\ \end{tabular} \right.$\\ \end{tabular} \end{quotation}

   

\newpage

 

\begin{center} \noindent
 {\bf  Education IV:  Continuous Signals and a Continua of Equilibria} \end{center}
 

\noindent
{\bf The Order of Play}\\
 0 Nature chooses the worker's ability $a \in \{2,5.5\}$, the $Low$ and $High$
ability each having probability 0.5. The variable $a$ is observed by the worker,
but not by the employers.\\
  1 The worker chooses education level $s \in [0,\infty$.\\
 2 The employers each offer a wage contract $w(s)$.\\
 3 The worker accepts a contract, or rejects both of them.\\
4 Output equals $a$.

\noindent
{\bf Payoffs}\\
 The worker's payoff is his wage minus his cost of education, and the employer's
is his profit.\\
$ \pi_{worker} = $ $\left\{ \begin{tabular}{ll}$w - 8s/a$ &   
Accepts contract $w$.\\
 0  &  Rejects both contracts.\\
  \end{tabular} \right.$

    $ \pi_{employer} = $ $\left\{ \begin{tabular}{ll} $a - w$ &  
Contract is accepted.\\
0  & Contract rejected. \end{tabular} \right.$

\newpage


\begin{center}
 {\bf  The Continua of Equilibria}
 \end{center}

Each $s^*$ in the interval $
[0,\overline{s}]$ supports a different equilibrium. 

\begin{quotation}
\noindent \begin{tabular}{cl} {\bf Pooling Equilibrium 4.1} & $\left\{
\begin{tabular}{l} $ s(Low)= s(High) = s^*$\\ $ w(s^*) = 3.75$\\ $ w(s \neq s^*)
= 2$\\ $Prob(a=Low|s \neq s^*) = 1$ \end{tabular} \right.$\\ \end{tabular}
\end{quotation}

The critical value $\overline{s}$ can be discovered from the incentive
compatibility constraint of the $Low$ type, which is binding if $s^* =
\overline{s}$.

The most tempting deviation is to zero education, so that is the
deviation that appears in the constraint.
  \begin{equation} \label{e10.1a} U_L(s
= 0)=2 \leq  U_L(s = \overline{s})= 3.75  - \frac{8  \overline{s}} {2}.
\end{equation} Equation (\ref{e10.1a}) yields $\overline{s} = \frac{7}{16}$. Any
value of $s^*$ less than $\frac{7}{16}$ will also support a pooling equilibrium.


Note that the incentive-compatibility constraint of the $High$ type is not
binding. If a $High$ deviates to $s=0$, he, too, will be thought to be a $Low$,
so \begin{equation} \label{e10.1b} U_H(s = 0)=2 \leq U_H(s = \frac{7}{16} )=
3.75 - \frac{8 \overline{s}}{5.5} \approx 3.1. \end{equation}

\newpage


\begin{center}
  {\bf Separating Equilibrium 4.2}
 \end{center}
   

\begin{quotation}
\noindent
\begin{tabular}{cl}
 & $\left\{ \begin{tabular}{l} $ s(Low)=
0,\;\;\;  s(High) = s^*$\\ $ w(s^*) =  5.5$\\ $ w(s \neq s^*) = 2$\\ $Prob(a=
Low|s \not\in \{0,s^*\}) = 1$ \end{tabular} \right.$\\
 \end{tabular} \end{quotation}
 The critical value $\overline{s}$ can be discovered from the incentive-
compatibility constraint of the $Low$, which is binding if $s^* = \overline{s}$.
\begin{equation} \label{e10.9}
 U_L(s = 0)=2 \geq  U_L(s = \overline{s})= 5.5  - \frac{8  \overline{s}} {2}.
\end{equation}
 Equation  (\ref{e10.9}) yields $\overline{s} = \frac{7}{8}$. 

 If the education needed for the wage of 5.5 is too great, the $High$
workers will give up on education too. Their incentive compatibility constraint
requires that
   \begin{equation} \label{e10.1d}
 U_H(s = 0)=2 \leq U_H(s = \overline{\overline{s}} )= 5.5 - \frac{8
\overline{\overline{s}}}{5.5} .
 \end{equation}
 Equation  (\ref{e10.1d}) yields $\overline{\overline{s}} = \frac{77}{32}$. 

  \newpage

 
\begin{center}
  {\bf Problems in Applying Signalling to Education}
 \end{center}
 
 

\noindent On the empirical level, the first question to ask of a signalling
model of education is, ``What is education?''. 

For operational purposes this
means, ``In what units is education measured?''.

 Two possible answers are
``years of education'' and ``grade point average.'' 


  
         Layard \& Psacharopoulos (1974) give three rationales for rejecting
signalling as an important motive for education.

\noindent
1. Dropouts get as high a
rate of return on education as those who complete degrees, so the signal is not
the diploma, although it might be the years of education. 

\noindent
2. Wage
differentials between different education levels rise with age, although one
would expect the signal to be less important after the employer has acquired
more observations on the worker's output. 

\noindent
3. Testing is not widely used for
hiring, despite its low cost relative to education. 

 \newpage

\begin{center}
  {\bf Efficient Uses of  Signalling}
 \end{center}

 


\noindent
1.  it allows the employer to match
workers with jobs suited to their talents.  

\noindent
2. Signalling keeps talented workers from moving to jobs where their
productivity is lower  but their talent is known. 

  
\noindent
3. If ability is endogenous --- moral hazard rather than adverse selection
--- signalling encourages workers to acquire ability.  

\newpage

\begin{center}
  {\bf   Education V: Screening with a Discrete Signal }
 \end{center}
 
 
\noindent {\bf The Order of Play}\\
 0 Nature chooses worker ability $a \in \{2,5.5\}$, each ability having
probability 0.5. Employers do not observe ability, but the worker does.\\ 1 Each
employer offers a wage contract $w(s)$.\\
2 The worker chooses education level $s \in \{0,1\}.$\\
 3 The worker accepts a contract, or rejects both of them.\\ 4 Output equals
$a$.

 \noindent
{\bf Payoffs} \\
 $ \pi_{worker} =$  $\left\{ \begin{tabular}{ll} $w - \frac{8s}{ a}$ & if the
worker accepts contract $w$.\\ 0 &if the worker rejects both contracts.\\
\end{tabular} \right.$

\noindent
$ \pi_{employer} =$   $\left\{ \begin{tabular}{ll} $ a - w$ &  
Contract is accepted.\\ 0 & Contract is rejected.\\ \end{tabular}
\right.$

\newpage

\begin{center}
 {\bf Separating Equilibrium 5.1}
 \end{center}

     Education V    has no   pooling equilibrium, because if one employer tried
to offer the zero profit pooling contract, $w(0)=3.75$, the other employer would
offer $w(1) = 5.5$ and draw away all the $High$s. The unique equilibrium is


\bigskip

\noindent
 \begin{tabular}{cl}
 & $\left\{ \begin{tabular}{l} $s(Low) = 0,
s(High) = 1$\\
$ w(0) =2, w(1) = 5.5$\\
 \end{tabular} \right. $\\
  \end{tabular}

 Beliefs do not need to be specified.



\newpage

 

  
\begin{center}
 {\bf   Education VI: Screening with a Continuous Signal }
\end{center}
 {\bf Players}\\
A worker and two employers.

 \noindent {\bf The Order of Play} \\
 0 Nature chooses worker ability $a \in \{2,5.5\}$, each ability having
probability 0.5. Employers do not observe ability, but the worker does.\\
 1 Each employer offers a wage contract $w(s)$.\\
 2 The worker choose education level $s \in [0, 1].$\\
3 The worker chooses a contract, or rejects both of them.\\
  4 Output equals $a$.

\noindent {\bf Payoffs }\\
 $ \pi_{worker} =  \left\{ \begin{tabular}{ll} $w - 8s/a$ & if the worker
accepts contract $w$.\\
 0 & if the worker rejects both contracts.\\ \end{tabular} \right.$

\noindent $ \pi_{employer} =  \left\{ \begin{tabular}{ll} $ a - w$ & Contract is accepted.\\
 0 & Contract is rejected.\\ \end{tabular} \right.$ 

\newpage

 \begin{center}
 {\bf Separating Equilibrium 6.1} 
 \end{center}


\noindent
 \begin{tabular}{cl}
& $\left\{ \begin{tabular}{l} $s(Low) = 0,
s(High) =0.875$\\
 \\
 $w =\left\{ \begin{tabular}{ll} 2 & if $s < 0.875$\\ 5.5 & if $s \geq 0.875$
\end{tabular} \right.$\\ \end{tabular} \right.$\\ \end{tabular}

    The participation constraints for the employers require that
\begin{equation} \label{e10.11}
 w(0) \leq a_L=2 \;\; {\rm and} \;\; w(s^*) \leq a_H=5.5,
 \end{equation}
 where $s^*$ is the separating value of education that we are trying to find.
 
 The
self selection constraint for the low-ability workers is
  \begin{equation} \label{e10.12}
  U_L (s = 0) \geq U_L(s = s^*),
 \end{equation}
 which in Education VI    is
\begin{equation} \label{e10.13}
 w(0) - 0 \geq w(s^*) - \frac{8s^*}{2}.
\end{equation}
 Since the separating wage is 2 for the $Low$s and 5.5 for the $High$s,
constraint  (\ref{e10.13}) is satisfied as an equality if $s^* = 0.875$, which
is the crucial education level in Separating Equilibrium 6.1.
\begin{equation} \label{e10.14}
 U_H (s = 0)= w(0) \leq U_H(s = s^*) = w(s^*) - \frac{8s^*}{5.5}.
\end{equation}
 If $s^* = 0.875$, inequality (\ref{e10.14}) is true, and it would also be true
for higher values of $s^*$.   But competition makes $s^* = 0.875$.


\newpage


\begin{center}
  {\bf Competition between Firms}
 \end{center}

 Competition in offering attractive contracts rules out pooling
contracts. The nonpooling constraint, required by competition between employers,
is
 \begin{equation} \label{e10.15}
 U_H(s = s^*) \geq U_H(pooling),
\end{equation} 
 which, for Education VI, is, using the most attractive possible
pooling contract,
\begin{equation} \label{e10.16}
 w(s^*) - \frac{8 s^*}{5.5}  \geq 3.75.
 \end{equation}
 Since the payoff of $High$s in the separating contract is 4.23 ($= 5.5 - 8
\cdot 0.875/5.5$, rounded), the nonpooling constraint is satisfied.

\newpage

\begin{center}
  {\bf No Pooling Equilibrium   in Screening: Education VI}
 \end{center}

 


 \noindent
 The screening game
Education VI    lacks a pooling equilibrium, which would require the outcome
$\{s= 0, w(0) = 3.75\}$, shown as $C_1$ here.

\includegraphics[width=150mm]{fig11-01.jpg}

  

No
employer can offer a pooling contract, because the other employer could always
profit by offering a separating contract paying more to the educated, e.g., 
  $C_2$.

$C_2$:  Pay 5 to workers with an
education of $s = 0.5$,  a payoff of 4.89  to the $High$s ($= 5 -
[8\cdot 0.5]/5.5$, rounded) and 3  to the $Low$s ($= 5 - 8\cdot 0.5/2$). Positive payoff for employer. 

  


 
\newpage

\begin{center}
 {\bf Nonexistence of a Pure-Strategy Equilibrium }
 \end{center}

 Let the proportion of $High$s be 0.9
instead of 0.5 

  The pair
$(C_3, C_4)$ is the most attractive pair of contracts that separates $High$s
from $Low$s. $Low$ workers accept contract $C_3$, obtain $s=0$, and receive a
wage of 2, their ability.  $High$s accept contract $C_4$, obtain $s= 0.875$, and
receive a wage of 5.5, their ability.  Education is not attractive to $Low$s.

\includegraphics[width=150mm]{fig11-02.jpg}

 
  The wage of the pooling contract $C_5$ is 5.15, so   even  $High$s
  prefer $C_5$ to $(C_3,C_4)$.  But our reasoning that no pooling
equilibrium exists is still valid.  No pure-strategy Nash equilibrium exists. 

 \newpage

\begin{center}
  {\bf Ways to Communicate}
 \end{center}
  
 
\noindent
{\bf 1. Cheap Talk Games. } The Sender's message is costless and there is no penalty for lying. 

   
  
\noindent
{\bf 2. Truthful Announcement Games. }  The Sender's message is costless. He  may be silent instead of sending a message, but if he sends a message it must be truthful. 


 
 
\noindent
{\bf 3. Auditing Games. } The Sender's message  might or might not be costly. The Receiver may audit the message at some cost and discover if the Sender was lying. 

   
 
\noindent
{\bf 4. Mechanism Games. }  The Sender's message  might or might not be costly. Before he sends it, he commits to a contract with the Receiver, with their decisions based on what they can observe and enforcement based on    what can be verified by the courts.  

     
 


\bigskip
\noindent
{\bf 5. Signalling Games. } The Sender's message is  costly when he lies, and more costly when he lies than when he tells the truth.  He sends it before the Receiver takes any action. 
 

 
 
\noindent
{\bf 6.  Expensive-Talk Games. } The Sender's message is costly,  but the cost is the same   regardless of his type. There is no penalty for lying. 

 
 
\noindent
{\bf 7. Screening Games. } The Sender's message is  costly when he lies, and more costly when he lies than when he tells the truth.  He sends it in response to an offer by the Receiver. 

  
 \newpage

  

\begin{center}
 {\bf   Limit Pricing as Signal Jamming }
 \end{center}

 
  
 \noindent
   {\bf The Order of Play}\\
 0 Nature chooses the reservation price  $v$ using the continuous density $h(v)$
on the support $[c, d]$,  observed only by the
incumbent.  (The parameter $c$ is marginal cost.)  \\
  1 The incumbent chooses  the first-period price $p_1$, generating sales of $q
$ if $p_1 \leq v$ and $ 0$ otherwise.The variables $p_1$ and $q$  are observed by  both
players.  \\
 2  The rival  decides whether to enter  at cost $F$ or to  stay out.\\
3  If the rival did not enter, the  incumbent chooses the second period price,
$p_{2,monopoly}$,  generating sales of $q $ if $p_{2m} \leq v$ and $ 0$
otherwise.\\
     If the rival did enter, the duopoly price is $p_{2}(v)$, with $p_{2} \geq c$ and
$\frac{d p_{2}}{dv} >0$.      \\

 
\noindent
 {\bf Payoffs}\\
   If the rival does not enter and $p_1$  and $p_{2m}$ are    no greater than
$v$, the payoffs are
\begin{equation} \label{e100}
\begin{array}{ll}
\pi_{incumbent}& =   (p_1-c) q    +   (p_2-c) q    \\
 & \\
\pi_{rival} & = 0.\\
 \end{array}
\end{equation}
  If    the rival does   enter and  $p_1$ are no greater than $v$, the payoffs
are
\begin{equation} \label{e101}
\begin{array}{ll}
  \pi_{incumbent}& =   (p_1-c) q   +   (p_{2}(v)-c) \left( \frac{q}{2}\right)
\\
 & \\
\pi_{rival}& = -F +   (p_{2}(v)-c) \left( \frac{q}{2} \right).\\
\end{array}
 \end{equation}

\newpage

\begin{center}
 {\bf   Assumptions on Functional Form}
 \end{center}
  
Assume that the second-period duopoly price is halfway between marginal cost and
the monopoly price, so
\begin{equation} \label{e108}
   p_2(v)  = \frac{c+  v }{2},
 \end{equation}
  the distribution of reservation prices is uniform, so
 \begin{equation} \label{e109}
  h(v) = \frac{1}{d-c},
 \end{equation}
  that $$
c=0
$$
  and that $ d > 6.4F/q$.  


\newpage

\begin{center}
 {\bf  Finding the Equilibrium}
 \end{center}

Work back from the end.

 
The second period price is
 $p_{2,monopoly}=v$,
since there is no threat of future entry.

  The rival's expected  payoff from entry depends on his beliefs about $v$,
which in turn depend upon which of the multiple equilibria of this game is being
played out.

 If the incumbent were nonstrategic, he would charge   $p_1= v$, maximizing his
first-period profit. 

The rival would deduce the value $v$ and would enter in
the second period if $p_1 $ exceeded the  critical value $v^* =
c + \frac{4F}{q}$, a value which  yields zero profits because
 under our special assumptions, 
$$
p_2(v) = \frac{v}{2}  
$$
 and 
 \begin{equation} \label{e102}
\begin{array}{ll}
  \pi_{rival, enter} & =  -F + \left[ \left(p_2(v^*)-c\right)\left( \frac{q}{2} 
\right) \right] \\
 & =  -F + \left[ \left(\frac{v^*}{2} \right)\left( \frac{q}{2} 
\right) \right]\\
 &  =   -F + \left[ \left(\frac{4F/q }{2} \right)\left( \frac{q}{2} 
\right) \right]\\
  & = 0  \\
\end{array}
\end{equation}

 If the incumbent is willing to charge a lower $p_1$, though,
and accept lower first-period profits,   he may be able to deter entry.

 
\newpage

\begin{center}
 {\bf A Limit Pricing  Equilibrium  } 
 \end{center}


\noindent
 {\bf Incumbent:} $p_2 =v$.   
 \begin{equation}  \nonumber
 p_1 = \left\{
 \begin{array}  {lll}
  v & if &   v <a   \\
 a & if &  v \in [a,b] \\
 v & if &  v >b \\
 \end{array}
\right.
\end{equation}

\noindent
{\bf Rival:}  Enter if $p_1 >a$. Otherwise, stay out.

 If the rival observes $p_1 \in (a,b]$, his out-of-equilibrium belief is that $ v=  \frac{h(v)}{
\int_{p_1}^b h(v) dv}$, the expected value of $v$ if it lies between $p_1$ and $b$.   \\

$$
  a =  \left( \frac{ 2}{5}  \right)    \left( \frac{ 4F}{q}  \right) 
$$

$$
b =  \left( \frac{ 8}{5}  \right)    \left( \frac{ 4F}{q}  \right) 
$$


\newpage

\begin{center}
 {\bf  Will the Incumbent Deviate? }
 \end{center}

\noindent
If the incumbent deters entry, his payoff is
\begin{equation} \label{e104}
  \pi_{incumbent} (no\; entry)    =  (p_1-c) q    +   (v-c)q
 \end{equation}
and if he does not it is
\begin{equation} \label{e105}
 {\displaystyle  \pi_{incumbent} (entry)  =  (v-c) q    +   (p_{2}(v)-
c)  \left( \frac{q}{2}\right).  }
 \end{equation}

 The incumbent's advantage from limit pricing is  the difference between these when $p_1=a$, which is 
\begin{equation} \label{e106}
 {\displaystyle  [(a-c) q    +   (v-c)q ]  -
    \left[(v-c) q    +  (p_{2 }(v)-c) \left( \frac{q}{2} \right) \right]. }
 \end{equation}
   This advantage is declining in $v$, and $b$ is defined as the  value at which it equals zero. Choosing $v=b$ in expression (\ref{e106}),  equating  to zero, and solving for $a$ yields
\begin{equation} \label{e107}
   a = \frac{c+  p_2(b) }{2} = \frac{b}{4}
 \end{equation}
using our specific functional forms. 

 

\newpage

\begin{center}
  {\bf Will  the Rival Deviate? }
 \end{center}

\noindent
The rival's   payoff if he enters is
\begin{equation} \label{e103}
 \begin{array}{ll}
  \pi_{rival, enter}& = {\displaystyle  -F +  (p_{2}(v)-c) \left(  \frac{q}{2}
\right)}\\
 & \\
 & ={\displaystyle   -F +   \left( \int_a^b p_{2}(v)  \left( \frac{h(v)}{
\int_a^b h(v) dv} \right) dv     -c \right) \left( \frac{q}{2}\right),}
\end{array}
 \end{equation}
 where the density for the expectation $Ep_{2}(v)$ is $   \frac{h(v)}{
\int_a^b h(v) dv}  $ instead of just $h(v)$ because it is conditional on
$v$ being between $a$ and $b$, rather than $v$ taking any of its possible
values.

\newpage

\begin{center}
  {\bf Deriving  the value of a and b }
 \end{center}

The entry payoff of the rival equals zero in equilibrium for the entry-deterring
values of $a$ and $b$. That is how we derive those values. 

Given our specific assumptions on $h(v)$  and $p_2(v)$,   $p_2(b) = b/2$ and $h(v) =
1/d$. Recall that $a = b/4$
 Since $\int_a^b h(v)dv = \int_a^b  (1/d)dv= \frac{b}{d} - \frac{a}{d}$,
\begin{equation} \label{e111}
 \begin{array}{ll}
   \pi_{rival}& =  {\displaystyle -F +   \left( \int_a^b \left(  \frac{v}{2}
\right) \left(  \frac{ \frac{1}{d}}{\frac{b}{d} - \frac{a}{d}}    \right)    dv
\right) \left( \frac{q}{2}\right)} \\
 & \\
 & =  {\displaystyle -F +     \bigg|_{v=a}^b \left(  \frac{v^2}{4 (b-a)}\right)
\left( \frac{q}{2}\right)} \\
& \\
 & =  {\displaystyle -F +      \left(  \frac{b^2}{4 (b-a)} - \frac{a^2}{4(b-a)}
\right)   \left( \frac{q}{2}\right)} \\
& \\
 & =  {\displaystyle -F +      \left(   b^2  - \frac{b^2}{16}  \right)   \left(
\frac{q}{8(b- \frac{b}{4})}\right)} \\
\end{array}
 \end{equation}
so
 \begin{equation} \label{e112}
   b   =  \left( \frac{ 8}{5}  \right)    \left( \frac{ 4F}{q}  \right), 
 \end{equation}
where $b<d$ because of our special assumption that $ d > 6.4F/q$. 


\newpage

\begin{center}
  {\bf The Values in the Signal-Jamming Range }
 \end{center}
 
 $$
   b   =  \left( \frac{ 8}{5}  \right)    \left( \frac{ 4F}{q}  \right), 
$$
Since $a =b/4$, 
\begin{equation} \label{e113}
   a =  \left( \frac{ 2}{5}  \right)    \left( \frac{ 4F}{q}  \right).
 \end{equation}
  It is useful to compare the limits $a$ and $b$  with $v^*$, the   value of $v$ for which entry  
yields a   payoff of zero to the rival. That value, which we found above using
equation (\ref{e102}), is
   \begin{equation} \label{e113a}
   v^* =  \frac{ 4F}{q}.
 \end{equation}

\newpage

\begin{center}
  {\bf General Comments on Limit Pricing}
 \end{center}

 The values of $a$ and $b$ were chosen
so that, in effect,  $v^*$ was the expected value of $v$ on $[a,b]$, and the
rival would be deterred even though he did not know the exact value of $v$.



If $a \geq v^*$, then the expected value of $v$ on $[a,b]$ would be greater than
$v^*$, and the rival would feel safe in entering if the incumbent charged $p_1=
a$.

 Thus, to use limit pricing, the incumbent must charge  strictly less  than the
monopoly price appropriate if the  duopoly market would yield zero
profits to an rival. 

 This kind of signal jamming   reduces the information that reaches the rival compared to nonstrategic behavior, since the rival learns the precise value of $v$ only if $v$ is less than $a$ or greater than $b$. 

\newpage

 
\begin{center}
 {\bf   Countersignalling}
 \end{center}

 This idea is modelled in: 


    Feltovich, Nick, Richmond
Harbaugh \&  Ted To    (2002) ``Too Cool for School? Signalling and
Countersignalling,''   {\it RAND Journal of Economics},   33(4): 630-649
(Winter 2002)

  This is the idea that only mid-quality people signal. The highest quality do not. Why? 

Think about restaurants. Would a    restaurant of very high quality benefit from posting a notice of its high hygiene level? See:



Jin, Ginger \& Philip Leslie (2003)  ``The Effect of Information on Product
Quality:
Evidence from
Restaurant Hygiene Grade Cards,'' {\it  Quarterly Journal of Economics}, 118(2):
409-451 (2003).

\newpage

\begin{center}
 {\bf   Countersignalling: The Bank Game}
 \end{center}

\noindent
 {\bf Players}\\
   Banks and depositors.

 \noindent
   {\bf The Order of Play}\\

 0 Nature chooses the  solvency  $\theta_i$  of each bank on a continuum of
length 1,  using cumulative  distribution $F(\theta)$ on the support $[-10,10]$ and $F(0) =0.2$.
\\

 1 Bank $i$ chooses to spend  $s_i$ on its building, where it must spend at
least $\bar{s} =9$ to operate at all, and otherwise must exit.  \\

  2 Depositors on a continuum of length $D=1$  observe $\hat{\theta}=
\theta_i+u_i$, where the $u_i$ values are chosen independently and take values
of $-5, 0$, and $+5$ with equal probability.      \\

 3  Each depositor chooses a bank.\\

 

\newpage

\begin{center}
 {\bf  Payoffs in  the Bank Game}
 \end{center}

\noindent
 {\bf Payoffs}\\
  Bank $i$'s payoff if it attracts depositors   is
\begin{equation} \label{e114}
 \pi_{i} =  \frac{D}{B} -  \left(  \frac{s_i}{10+\theta_i} \right),
\end{equation}
 where $B$ is the interval of banks that attract depositors.  If the bank
attracts no depositors,  its payoff is $-\left( \frac{s_i}{10+\theta_i} \right)
$.

\noindent
  A depositor's  payoff is 1 if he picks a bank with quality  at or above 0, and
$0$ if he picks quality below 0.

\newpage


\begin{center}
 {\bf An Equilibrium} 
 \end{center}


\noindent
 Banks with solvency   $\theta \in [-10,0]$  choose $s=0$.\\
 They do not signal,
and attract no depositors.\\

  Banks with solvency   $\theta \in [0, 5)$  choose 
$$
 s= s^*=12.5.
$$    
 They attract depositors.

 Banks with solvency   $\theta \in [5,10]$  choose $s=9$.\\
 They    attract
depositors.  

Depositors choose banks with either $\hat{\theta} \geq 5$\\
  or $s \geq s^*$ or both.\\



\newpage


 In this   equilibrium, $B = 1 - F(0)=0.8$  because depositors choose the banks
which are solvent. 
 
 If $\hat{\theta} \geq 5$, a depositor  can be sure the bank is
solvent, because the minimum possible true level of $\theta$ is 0, with $u=+5$.
Otherwise, they   rely on the signal.


Note: setting the minimum level of s equal to 9, and the fraction of insolvent potential banks to 0.2 are just for concreteness. 

  
\newpage

\begin{center}
 {\bf  The Self Selection Constraints}
 \end{center}

The self-selection constraint requires that banks with $\theta<0$ not signal.
Thus, the crucial   signal level, $s^*$, requires that $\pi_{0} (s=0)  = \pi_{0}
(s=s^*)$, so
\begin{equation} \label{e115}
  0 =   \frac{D}{B} -  \left( \frac{s^*}{10+0}  \right)
\end{equation}


 Solving equation (\ref{e115}) yields
\begin{equation} \label{e115a}
 s^* = \frac{10D}{1-F(0)} =12.5.
 \end{equation}

 

\newpage

\begin{center}
 {\bf  Bank's Payoffs} 
 \end{center}

\noindent
The payoff of a bank  in $ \theta=[0,5)$ if it enters and signals $s^*$ is
\begin{equation} \label{e115b}
  \pi(s=s^*) =   \frac{D}{B} -  \left( \frac{s^*}{10+ \theta} \right).
\end{equation}
 This  is at least as high as the  zero payoff from not entering,  because
substituting in for $B$ and $s^*$ yields
\begin{equation} \label{e115c}
  \pi(s=s^*) =   \frac{D}{ 1-F(0)} -   \frac{ \frac{10D}{1-F(0)}}{10+ \theta}
\geq   \frac{D}{ 1-F(0)} -     \frac{10D }{10  [1-F(0)] } = 0.
\end{equation}
 
\newpage

\begin{center}
 {\bf  Mid-Quality Banks}
 \end{center}


 The mid-quality banks could also pretend to be high-quality banks by  entering,
not signalling and hoping to have a positive measurement error $u$. The payoff
from that would be composed of a 2/3 probability of attracting no customers and
a 1/3 probability of attracting $D/B$:
\begin{equation} \label{e115d}
\begin{array}{ll}
  \pi(s=9) = \left( \frac{2}{3}  \right) \left( 0 \right)  + \left( \frac{1}{3}
\right)  \left( \frac{D}{B}  \right)  -    \left( \frac{4}{10+ \theta}\right)
\\
\end{array}
\end{equation}
 

The payoff of the high-quality  banks in $ \theta=[5,10]$ is highest from entry
without signalling, when it is $\pi(\theta) = D/B - 9/(10+\theta)$, which is
positive. Signalling $s^*$  merely
adds costs, while not entering yields a payoff of zero.


 
Other equilibria   exist with higher levels of signal, and fewer banks entering.  Consumers might believe that the mid-range of bank quality     signals with  $s=\dot{s}>s^*$. In that case, the mid-range shrinks, since for $\theta=0$ signalling no longer  yields positive profits. 

\newpage

\begin{center}
  {\bf  Countersignalling where  Lack of a Signal Can Be Good News }
 \end{center}


In some  countersignalling models   the absence of a signal can have a positive,
additional, effect on the uninformed player's  estimate  of the  informed
player's quality.

 In our banking game, that can't happen because  if the noisy
observation is  $\hat{\theta} \geq 5$,  depositors have the highest possible
opinion of the bank's safety.

 Thus, the
high-quality banks abstain from signalling simply to save money, not because the
countersignal actually helps them.


Suppose, though, that
      we added   new depositors to the model, an amount small   enough to be
 measure zero so they would not affect the equilibrium, and these new depositors
had the special features that: 

\noindent
(a) they     could not observe even $\hat{\theta}$, and

\noindent
(b) their payoffs were 0  not just for insolvent banks  but for any bank with
$\theta <7$. 

They would  observe  only  that some banks have $s=9$
and some have $s=s^*$. Since they would be  looking for especially high-quality banks,
they would actually prefer to choose a bank with  the lower signal value of $s=
9$, because they could be sure that if $s=s^* $ then  $\theta <7$. 


 


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

 


\end{LARGE}

\end{document}

