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\begin{LARGE}


   
\parindent 24pt

 \parskip 10pt

 
  

\noindent  24 October 2006 .  Eric Rasmusen, Erasmuse@indiana.edu.  \\
 Http://www.rasmusen/org/GI/chap10\_mechanisms.pdf.

\begin{LARGE}
  \begin{center}
  { \bf  10   Mechanism Design  and Post-Contractual Hidden Knowledge  }
 \end{center}
 \end{LARGE}

 This is just for price discrimination.

\newpage

 \noindent
{\bf  10.5:  Price Discrimination }

   Suppose there are two consumers, willing to pay 45  and 36  for a car with a marginal cost of 10  thousand dollars.   

A single-price monopolist would charge 36, for profit of 2(36-10) = 52 instead of 45 with its profit of (45-10) = 35. 

A two-price monopolist would charge 36 and 44, increasing his profits to 26+35= 61.  

If the monopolist doesn't know the types, he'd like to design a mechanism to get the buyers to reveal them. He can't do much here, though. He could use an indirect mechanism like this:

p(m=36) = 36, p(m=45) = 45

By the Revelation Principle, we can find a  direct mechanism  which  achieves the same outcome with no lying: 

p(m=36) = 36, p(m=45) = 36.

In either case, the consumer willing to pay 45   can hide
under the guise of being a less intense consumer  and despite facing a
monopolist he can end  up retaining consumer surplus -- an {\bf informational
rent}, a return to the consumer's private information about his own
type. 

\newpage

\noindent
 TYPES OF PRICE DISCRIMINATION

\noindent
{\bf  1 Interbuyer Price Discrimination.  } Different prices  for  different  buyers.

 Smith's price  for a hamburger  is \$4
per burger, but Jones's is  \$6.  (Close to Pigou's 3rd degree--coarse buyer segmentation) 

\noindent
{\bf  2 Interquantity Price Discrimination}  or  {\bf  Nonlinear Pricing}.
Different  unit prices for different
quantities.

   A consumer can buy a  first   sausage for \$9, a  second sausage
for  \$4, and  a third sausage for \$3.  Rather than  paying  the ``linear''
total price of  \$9 for one sausage, \$18 for two, and \$27 for three,  he  thus
pays  the  nonlinear price of  \$9 for one sausage, \$13 for two, and \$16 for
three, the concave price path shown in Figure   3. (Pigou's 2nd degree)

\includegraphics[width= 50mm]{fig10-03.jpg}

 

\noindent
 {\bf  3 Perfect Price Discrimination.}   This combines interbuyer and
interquantity price discrimination. 

   Smith might end up paying  \$50 for his first hot dog and  \$20
for his second,   while next to him Jones pays \$4 for his first and \$3 for his
second. (Pigou's first degree)

 



 \newpage

 \begin{center}
  {\bf  Varian's Nonlinear Pricing Game}
 \end{center}
 {\bf Players}\\
One seller and one buyer.

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

0 Nature assigns   the   buyer a type, $s$. The buyer is  ``unenthusiastic''
with utility function  $u$    or   ``valuing''   with utility function  $v$,
with equal probability.  The seller does not observe Nature's move, but the
buyer does. \\

1 The  seller offers  mechanism  $\{ w_m, q_m \}$ under which the buyer can
announce his type as $m$ and buy amount  $q_m$ for lump sum $w_m$.\\

2 The buyer   chooses a message $m$ or rejects the mechanism entirely and does
not buy at all.  \\

\newpage

\noindent
{\bf Payoffs}\\
 The seller  has a constant marginal cost of $c$, so his payoff  is
 \begin{equation}  \label{e6a}
  w_u + w_v - c \cdot (q_u+q_v).
 \end{equation}


 The buyers' payoffs are  $\pi_u=u (q_u) - w_u$ and $\pi_v=v (q_v) - w_v$ if $q$
is positive, and 0 if $q=0$, with $u', v'>0$ and  $u'', v'' <0$. The    marginal
willingness  to pay is greater  for  the valuing buyer: for  any   $q$,

 \begin{equation}  \label{e7}
  u' (q ) < v '(q )
 \end{equation}


This second is an example of    {\bf the single-crossing property}.
  Combined with the assumption
that $v(0)=u(0) =0$, it also implies that
 \begin{equation}  \label{e7a}
 u  (q ) < v (q )
 \end{equation}
 for any value of $q$.     

\newpage

  

 
 \noindent
{\bf Perfect Price Discrimination}

 The game would allow  perfect price discrimination if the seller did know
which buyer had which utility function.  He  can then  just maximize  profit
subject to the participation constraints for the two buyers:
 \begin{equation}  \label{e8}
 \stackrel{ Maximize}{w_u  ,w_v, q_u, q_v}  \;\;\;   w_u + w_v - c \cdot
(q_u+q_v).
 \end{equation}
 subject to
\begin{equation}  \label{e9}
 \begin{array}{ll}
 (a) & u (q_u) - w_u \geq 0 \;\; {\rm and} \\
 & \\
(b)&v (q_v) - w_v \geq 0.
 \end{array}
 \end{equation}
The constraints will be satisfied as equalities, since the seller will charge
all that the buyers will pay. Substituting  for $w_u$ and $w_v$ into the
maximand, the first order conditions become
\begin{equation}  \label{e10}
 \begin{array}{ll}
 (a) &u' (q_u^*) -c= 0 \;\;\; {\rm and} \\
 & \\
 (b)&v ' (q_v^*)-c=0.
\end{array}
\end{equation}

 

\includegraphics[width=100mm]{fig10-04.jpg}
 


\newpage

\noindent
{\bf Interbuyer Price Discrimination}

Now suppose the seller can charge the two players different prices ({\bf and prevent resale}, but they have to be  linear,  per-unit, prices. $p$, not $w$. 
\begin{equation}  \label{e11}
 \stackrel{ Maximize}{q_u,q_v, p_u  ,p_v}  \;\;\;   p_u q_u  + p_v q_v - c \cdot
(q_u+q_v),
 \end{equation}
 subject to the participation constraints
 \begin{equation}   \label{e12}
 \begin{array}{ll}
  & u (q_u) - p_u  q_u  \geq 0  \;\;\; {\rm and} \\
 & \\
 &v (q_v) - p_v q_v \geq 0
 \end{array}
\end{equation}
 and the incentive compatibility constraints
 \begin{equation}   \label{e13}
\begin{array}{ll}
  & q_u = argmax [u (q_u) - p_u  q_u]   \;\;\;  {\rm and} \\
 & \\
 & q_v = argmax [v (q_v) - p_vq_v] .
 \end{array}
 \end{equation}

This is like  moral hazard with two contracts for  two agents.

\newpage

\noindent
 The  agents will solve their quantity choice problems in (\ref{e13}), yielding
\begin{equation}   \label{e14}
 \begin{array}{ll}
  &  u' (q_u) - p_u  =0 \;\;\;  {\rm and} \\
 & \\
  & v ' (q_v) - p_v=0.
 \end{array}
\end{equation}
Thus, we can simplify the original problem in (\ref{e11})  to

 \begin{equation}  \label{e15}
 \stackrel{ Maximize}{q_u,q_v}  \;\;\;  u'(q_u)  q_u  +  v '(q_v) q_v - c \cdot
(q_u+q_v),
 \end{equation}
 subject to the participation constraints
\begin{equation}  \label{e16}%   \label{e10.10}
 \begin{array}{ll}
   & u (q_u) - u' (q_u)  q_u  \geq 0\;\;\; {\rm and} \\
 & \\
 &v (q_v) - v ' (q_v) q_v \geq 0.
 \end{array}
\end{equation}



   The participation constraints cannot be binding. If they were, then $ u(q)
/q  =  u'(q)$, but since $u''<0$ there is diminishing utility of consumption and
the average utility, $U(q)/q$, will be greater than the  marginal utility,
$u'(q)$. Thus we can solve problem (\ref{e15}) as if there were no constraints.

 The first-order conditions are
\begin{equation}   \label{e17}
 \begin{array}{ll}
   &  u''(q_u)  q_u   + u' =c \;\;\; {\rm and} \\
 & \\
  &v ''(q_v) q_v+v '  = c.
\end{array}
 \end{equation}

 This is  just  the `marginal revenue equals marginal cost' condition that any
monopolist uses,  but  one for each buyer instead of one for the entire market.
 


\newpage

  \noindent
 {\bf Back to  Nonlinear Pricing and Mechanism Design}

 Now, the seller does not know the buyer's type, {\bf but he can prevent resale}.  The seller's problem is


 \begin{equation}   \label{e18}
 \stackrel{ Maximize}{q_u,q_v, w_u,  w_v}  \;\;\; w_u  + w_v  - c \cdot
(q_u+q_v),
 \end{equation}


 subject to  the participation constraints,
\begin{equation}   \label{e19}
 \begin{array}{l}
 (a)\;\;\;u (q_u) - w_u \geq 0\;\;\;  {\rm and} \\ (b)\;\;\;v (q_v) - w_v
\geq 0,
 \end{array}
 \end{equation}


and  the self-selection constraints,
 \begin{equation}  \label{e20}
 \begin{array}{l}
 (a) \;\;\;  u (q_u) - w_u \geq   u (q_v) - w_v   \\
 \\
(b) \;\;\;   v (q_v) - w_v  \geq   v (q_u) - w_u.  \\
 \end{array}
 \end{equation}

Which constraints are binding? 

 \newpage

\noindent {\bf
Which constraints are binding? }

 

(1)  Suppose  the optimal contract is in fact separating, and also that both types
accept a  contract.    At least one type  will  have a binding participation
constraint.

(2)  Since the valuing consumer gets more consumer surplus from a given
$w$ and $q$ than an unenthusiastic consumer, it must be the unenthusiastic
consumer who is driven down to zero surplus for $(w_u, q_u)$.


 The valuing
consumer would get positive surplus from accepting that same contract.

(3)  To persuade the valuing consumer to
accept $(w_v, q_v)$ instead, the seller must give him that same positive surplus
from it.

 The seller will not be any more generous than he has to, though, so the
valuing consumer's self-selection constraint will be binding.


\newpage

\noindent
{\bf Efficiency in the Second-Best}

\noindent
 Rearranging our  two binding constraints and setting them out as equalities
we can 
  reformulate the seller's problem from (\ref{e18})   as
\begin{equation} \label{e22}
  \stackrel{Maximize}{q_u,q_v}  \;\;\; u(q_u)  + u(q_u) -v (q_u)   +v (q_v)   -
c \cdot (q_u+q_v),
  \end{equation}
 which has the first-order conditions
\begin{equation}  \label{e23}
\begin{array}{ll}
(a) &  u'(q_u)  - c  + [u'(q_u)  - v '(q_u)]  =0\\
 & \\
(b)  &   v' (q_v) -c =0\\
 \end{array}
\end{equation}

   Equation (b)  tells us that the
valuing  type of buyer buys a quantity such that his last unit's marginal
utility exactly equals the marginal cost of production; his consumption is at
the efficient level.  

 The unenthusiastic type, however,   buys less than his
first-best amount.

From the single-crossing  property,    $u'(q) < v'(q)$, which implies from (\ref{e23}a)
that $u'(q_u)  - c >0$ and the unenthusiastic type has not bought enough to
drive his marginal utility down to marginal cost. 

 The
seller must sell less than   first-best optimal to   the unenthusiastic type so
as not to make that contract  too attractive to the  valuing type. On the other
hand, making the valuing type's contract more valuable to him actually  helps
separation, so $q_v$ is chosen to maximize social surplus.

\newpage

\noindent
 {\bf Quantities Bought}

   Substituting
from first-order condition (b)  into first-order condition (a)
yields
\begin{equation}   \label{e24}%  \label{e10.18}
[u'(q_u)  - v' (q_v)]  + [u'(q_u)  - v '(q_u)]  =0
 \end{equation}

   The second term in square brackets  is  negative  by the single-crossing
property.

 Thus, the first term must be positive. 

 But since the single-crossing
property tells us that  $[u'(q_u)  - v' (q_u)] <0$,  it must be true, since
$v''<0$, that  if  $q_u  \geq q_v$  then   $[u'(q_u)  - v' (q_v)] <0$ -- that
is, that the first  term is negative. 

We cannot have that without contradiction,
so it must be that   $q_u  < q_v$. No pooling.   Unenthusiastic buyer buys less. 


\newpage

 \noindent
 {\bf Pooling vs. Separating Contracts}

 The  separating contract in which both types buy positive amounts is not necessarily optimal. 

Drop the assumption that the two types are of equal probability. 

(1) Would the seller ever prefer a contract in which only the Enthustiastic buyer bought anything? 

First-- what would such a contract look like? 

\vspace*{1in}

(2) Would the seller ever prefer a contract in which only the Unenthusiastic buyer bought anything? 

First, what would that contract look like? 




\newpage

\noindent
 {\bf A  Graphical Approach to the Same Problem}

  Here, just draw the diagram. 

\includegraphics[width=100mm]{fig10-05.jpg}

\hspace*{-6in}
\includegraphics[width=100mm]{fig10-05.jpg}


Diagram A: 

PERFECT PRICE DISC, or INTERBUYER with lump sum prices:  

 A+B

A+B+J+K+L

SECOND BEST SEPARATING, BOTH BUY:
wu  A

wv = A+B+K+L

rent: J

deadweight loss: B

SECOND BEST SEP: JUST HIGH BUYS: 
w = A+B+J+K+L

  
\newpage

\noindent
 {\bf  The Single-Crossing Property}

Condition (\ref{e7}) is an example of    {\bf the single-crossing property},
since it implies that the indifference curves of the two agents cross  at most
one time. 

 Combined with the assumption that $v(0)=u(0) =0$, it   implies that
$u  (q ) < v (q )$ for any value of $q$.

When we say that   Buyer V's demand is stronger than Buyer U's, however,  there
are  two things we might mean:

1. Buyer V's {\it average demand} is stronger: $  \frac{v (q )}{q}>\frac{u  (q )
}{q} $. Buyer V would pay more for quantity $q$ than Buyer U would.

2. Buyer V's {\it marginal demand} is stronger: $  v' (q )> u'  (q )$. Buyer V
would pay more for an additional unit than Buyer U would.

\newpage

  Definitions (1) and (2)  are not equivalent. In Figure 6a,   Buyer U is
willing to pay 5 per unit up to $q=4$, but only 1 per unit thereafter. Buyer V
is willing to pay only 2 per unit up to $q=10$, and 1 per unit thereafter. As a
result $u  (q ) > v (q )$, and U has the stronger demand by definition (1).  But
for $q \in [4, 10]$, Buyer V is willing to pay 2 per new unit while Buyer U is
only willing to pay 1, so in that interval Buyer V has the  stronger demand by
definition (2).

\hspace*{-1.5in}
\includegraphics[width=250mm]{fig10-06.jpg}



\begin{center} {\bf Figure 6: Marginal versus Average Demand} \end{center}

\newpage


 It is the marginal demand that is more important to economic behavior and which
forms the basis for the single-crossing property.  The utility functions in
Figure 6b   satisfy the single-crossing property, even though it is not true
that $u  (q ) < v (q )$ ( because they don't satisfy $u(0)=v(0)=0$.)  

\hspace*{-1.5in}
\includegraphics[width=250mm]{fig10-06.jpg}



\begin{center} {\bf Figure 6: Marginal versus Average Demand} \end{center}

\newpage

 We wrote utility as $\pi = u(q) -w(q) $ before. 
This is the same as the quasilinear payoff function:

 $\pi_u(q, money) = money + u(q)$,
 where $money = wealth-w(q)
$.

 Figure  (a) satsifies single crossing.  $u = \sqrt{q}$ and  $v = 2\sqrt{q}$. 

 

\hspace*{-1.8in}
\includegraphics[width=250mm]{fig10-07.jpg}

 
 


\newpage

\hspace*{-4in}
\includegraphics[width=300mm]{fig10-07.jpg}

  
 Utility is abstract, so it's nice to look at a graph without it on the axis. 
   

Figure (b)  shows how the buyers trade off money and the commodity.

 
 

\end{LARGE}

\end{document}




