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\begin{LARGE}
 \begin{center}
 {\bf  14 Pricing }
 \end{center}
\end{LARGE}

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

\noindent
{\bf   14.1 Quantities as Strategies: Cournot Equilibrium Revisited}

 \noindent
 Chapter 14 is about how firms with market power set prices.  Section  14.1
extends the Cournot Game of Section 3.5 in which   two firms choose the
quantities they sell, while Section 14.2 extends  the Bertrand model in which
they choose   prices to the case where capacity is limited.   Section 14.3 goes
back to the origins of product differentiation, and develops two Hotelling
location models. Section 14.4 shows how to do comparative statics in games,
using the  differentiated Bertrand  model  as an example  and  supermodularity
and the implicit function theorem  as tools. Section 14.5 looks at another sort
of differentiation: choice of ``vertical'' quality, from good to bad, by
monopoly or duopoly.    Section 14.6   concludes  this book with the problem
facing a  firm selling a durable good  because buyers foresee that it will be
tempted to reduce the price over time to   price-discriminate among them. At
that point, perhaps you'll wonder how much this  book will cost next year!



\bigskip \noindent
 {\bf Cournot Behavior with General Cost and Demand Functions}

\noindent
   In the next few sections, sellers compete against each other  while moving
simultaneously.  We will start by generalizing the  Cournot Game of Section 3.5
from linear demand and zero costs to a wider class of functions. The two players
are firms Apex and Brydox, and their strategies are their choices of the
quantities $q_a$ and $q_b$.  The payoffs are based on the total cost functions,
$c(q_a)$ and $c(q_b)$, and the demand function, $p(q)$, where $q=q_a + q_b$.
This specification  says that only the sum of the outputs  affects  the price.
The implication is that the firms produce an identical product, because whether
it is Apex or Brydox that produces an extra unit, the effect on  the price is
the same.

 Let us take the point of view of Apex. In the Cournot-Nash analysis, Apex
chooses its output of $q_a$ for a given level of $q_b$ as if its choice did not
affect $q_b$. From its point of view, $q_a$ is a function of $q_b$, but $q_b$ is
exogenous. Apex sees the effect of its output on price as
 \begin{equation} \label{e13.1}
 \frac{ \partial p}{\partial q_a}=  \left( \frac{dp}{ d q} \right)  \left(
\frac{\partial q}{ \partial q_a} \right) = \frac{dp}{ d q}.
\end{equation}
 Apex's payoff function is
\begin{equation} \label{e13.2}
 \pi_a = p(q) q_a - c(q_a).
  \end{equation}
To find Apex's reaction function, we  differentiate with respect to its strategy
to obtain
 \begin{equation} \label{e13.3}
  \frac{d\pi_a}{dq_a} = p +  \left(\frac{dp}{dq}  \right)q_a - \frac{dc}{d q_a}
= 0, \end{equation} which implies
\begin{equation} \label{e13.4}
  q_a = \frac{\frac{dc}{dq_a}  - p}{\frac{dp}{dq}}, \end{equation} or,
simplifying the notation,
 \begin{equation} \label{e13.5}
 q_a = \frac{c'  - p}{p'}.
  \end{equation}

If particular functional forms for $p(q)$ and $c(q_a)$ are available, equation
(\ref{e13.5}) can be solved to find $q_a$ as a function of $q_b$.  More
generally, to find the change in Apex's best response for an exogenous change in
Brydox's output, differentiate (\ref{e13.5}) with respect to $q_b$, remembering
that $q_b$ exerts not only a direct effect on $p(q_a+q_b)$, but possibly an
indirect effect  via  $q_a$.
 \begin{equation} \label{e13.6}
\frac{d q_a}{d q_b} = \frac{ \left(p-c' \right) \left(p'' + p'' \left( \frac{d
q_a}{d q_b}  \right) \right)}{p'^2} + \frac{ c'' \left( \frac{d q_a}{d q_b}
\right) - p' - p'  \left(\frac{d q_a}{d q_b} \right)}{p'}.
 \end{equation}
 Equation (\ref{e13.6}) can be solved for $\frac{d q_a}{d q_b}$ to obtain the
slope of the reaction function,
 \begin{equation} \label{e13.7}
 \frac{d q_a}{d q_b} = \frac{(p-c')p''- p'^2} {2p'^2 - c''p' - (p- c')p''}
  \end{equation}
 If both costs and demand are linear, as in section 3.5, then $c''=0$ and $p''=
0$, so equation (\ref{e13.7}) becomes
\begin{equation} \label{e13.8}
 \frac{d q_a}{d q_b} = - \frac{p'^2 }{2p'^2 } = -\frac{1}{2}.
\end{equation}

\includegraphics[width=150mm]{fig14-01.jpg}


\begin{center}
 {\bf Figure  1:  Different Demand Curves }
 \end{center}

The general model faces two problems that did not arise in the linear model:
nonuniqueness and nonexistence. If demand is concave and costs are convex, which
implies that $p'' < 0$ and $c''> 0$, then all is well as far as existence goes.
Since price is greater than marginal cost ($p >c'$), equation (\ref{e13.7})
tells us that the reaction functions are downward sloping, because $2p'^2 -
c''p' - (p-c')p''$ is positive and both $(p-c')p''$ and $-p'^2$ are negative.
If the reaction curves are downward sloping, they cross and an equilibrium
exists, as was shown in   Figure  1a for the linear case represented by equation
(\ref{e13.8}).    We usually do assume that costs are at least weakly convex,
since that is the result of diminishing or constant returns, but there is no
reason to believe that demand is either concave as in Figure 1b or convex, as in
Figure 1c.  If the demand curves are not linear, the contorted reaction
functions of equation (\ref{e13.7}) might give rise to multiple Cournot
equilibria as in   Figure  2.

\includegraphics[width=150mm]{fig14-02.jpg}

\begin{center}
 {\bf Figure  2: Multiple Cournot-Nash Equilibria }
 \end{center}

  If demand is convex or costs are concave, so   $p'' > 0$ or $c''<0$, the
reaction functions can be upward sloping, in which case they might never cross
and no equilibrium would exist.  The problem can also be seen from Apex's payoff
function, equation (\ref{e13.2}). If $p(q)$ is convex, the payoff function might
not be concave, in which case standard maximization techniques break down.  The
problems of the general Cournot model teach a lesson to modellers: sometimes
simple assumptions  such as linearity  generate atypical results.

\bigskip \noindent
{\bf Many Oligopolists}

\noindent
   Let us return to the simpler game in which production costs are zero and
demand is linear. For concreteness, we will use the same specific inverse demand
function  as in Chapter 3,
\begin{equation} \label{e13.9}
p(q) = 120 - q.
 \end{equation}
Using (\ref{e13.9}), the  payoff function, (\ref{e13.2}), becomes
 \begin{equation} \label{e13.10}
 \pi_a  = 120q_a - q_a^2 - q_b q_a.
  \end{equation}
In section 3.5,  firms picked outputs of 40 apiece given demand function
(\ref{e13.9}). This  generated a price of 40. With $n$ firms instead of two, the
demand function is
 \begin{equation} \label{e13.11}
 p \left(\sum_{i=1}^n q_i \right) = 120 - \sum_{i=1}^n q_i,
\end{equation} and  firm $j$'s payoff function is
 \begin{equation} \label{e13.12}
 \pi_j  = 120q_j - q_j^2 - q_j\sum_{i \neq j} q_i. \end{equation}
Differentiating $j$'s payoff function with respect to $q_j$  yields
\begin{equation} \label{e13.13}
 \frac{d\pi_j}{dq_j}  = 120 - 2q_j - \sum_{i \neq j} q_i= 0.
 \end{equation}
The first step in finding  the equilibrium is to  guess that it is symmetric, so
that $q_j = q_i,( i = 1,\ldots, n)$. This is  an educated guess, since every
player faces a first-order condition like (\ref{e13.13}).  By symmetry, equation
(\ref{e13.13}) becomes $120 - (n+1)q_j = 0$, so that
 \begin{equation}\label {e13.14}
 q_j = \frac{120}{n+1}.
\end{equation}
 Consider several different values for $n$. If $n=1$, then $q_j = 60$, the
monopoly optimum; and if $n=2$ then $q_j = 40$, the Cournot output found in
section 3.5.  If $n = 5$, $q_j = 20$; and as $n$ rises, individual output
shrinks to zero. Moreover, the total output of $ nq_j = \left(\frac{ n}{n+1}
\right) 120$ gradually approaches 120, the competitive output, and the market
price falls to zero, the marginal cost of production.  As the number of firms
increases, profits fall.

  \vspace{.5in}

 \noindent
 {\bf  14.2  Capacity Constraints: The Edgeworth Paradox }

\noindent
   In the last section we assumed constant marginal costs (of zero),  and we
assumed constant marginal costs of 12 in Chapter 3 when we first discussed
Cournot and Bertrand equilibrium.
    What if it were increasing, either gradually,  or abruptly rising to
infinity at a fixed capacity?

 In the Cournot model, where firms compete in quantities,  increasing marginal
costs or a capacity constraint complicate the equations   but do not  change the
model's features dramatically. Increasing marginal cost would reduce output as
one might expect.  If one firm had a capacity   that was less than   the
ordinary Cournot   output, that firm would produce only  up to  its capacity and
the other firm would   produce more than the ordinary Cournot output, since
their outputs are strategic substitutes.

   What happens in the Bertrand model, where firms  compete in prices, is less
straightforward.  In Chapter 3's game, the demand curve was  $p(q) = 120 - q$,
which we also used in the previous section of this chapter, and the constant
marginal cost of firms Apex and Brydox  was $c=12$. In equilibrium, $p_a=p_b=12$
and $q_a=q_b=54$. If Apex deviated to  a  higher price  such as  $p_a=20$, its
quantity would fall to zero, since all customers would prefer Brydox's low
price.

 What happens if we constrain each firm to sell no more than its capacity of
$K_a=K_b=70$? The industry capacity of 140 continues to exceed the demand of 108
at $p_a=p_b=12$. If, however, Apex deviates to the higher price of  $p_a=20$,
it can still  get customers. All 108  customers would prefer  to buy from
Brydox, but Brydox could only serve 70 of them, and the rest would have to go
unhappily to Apex.

   To discover what deviation is most profitable for Apex  when $p_a=p_b=12$,
however, we need to
know what Apex's exact payoff would be from deviation. That means we need to
know not only that 38 (108 minus 70) of the customers are turned away by Brydox,
but which 38 customers.  If they are the customers at the top of the demand
curve,
who are willing to pay prices near 100, Apex's optimal deviation will be much
different than if they are ones towards the bottom, who are only willing to pay
prices a little above  12.


 Thus,  in order to set up the payoff functions for the game, we need to specify
a {\bf rationing rule}  to tell us  which consumers are served at the low price
and which must buy from the high-price firm. The rationing rule is unimportant
to the payoff of the low-price firm, but crucial to the high-price firm.

\noindent
   One possible rule is

\noindent
{\bf Intensity rationing} (or {\bf efficient rationing}, or {\bf high-to-low
rationing} ).  { \it The consumers able to buy from the firm with the lower
price are those who value the product most.}

 The inverse demand function from equation (\ref{e13.9}) is $p = 120- q$, and
under intensity rationing  the $K$ consumers with the strongest demand buy from
the low-price firm. Suppose that  Brydox is the low-price firm, charging (for
illustration) a price of $p_b=30$,    so   90 consumers wish to buy from it
though only $K$ can do so, and Apex is charging some higher price $p_a$. The
residual demand facing Apex is   either 0 (if $p_a >120-K$) or
  \begin{equation}\label{e13.18}
 q_a = 120 - p_a - K.
 \end{equation}
That is the  demand curve in Figure  3(a).

\includegraphics[width=150mm]{fig14-03.jpg}

\begin{center}
 {\bf Figure  3: Rationing Rules when $p_b =30$, $p_a >30$, and $K = 70$ }
 \end{center}

\noindent
 Under {\bf  intensity rationing}, if $K=70$  the demand function for Apex
(Brydox's is analogous) is
 \begin{equation}\label{e13.19}
 \begin{array}{ll}
q_a = & \left\{ \begin{array}{llr}
    Min \{ 120 - p_a, 70 \} &{\rm if} \;p_a < p_b  & (a) \\
 & & \\
 \frac{  120 - p_a }{2}  & {\rm if }\; p_a = p_b & (b)  \\
 & & \\
   Max \{(120 - p_a- 70),0 \}  & {\rm if}\; p_a > p_b, p_b < 50  & (c) \\
  & & \\
0        & {\rm if }\; p_a > p_b, p_b \geq 50 & (d)\\
 \end{array} \right.
 \end{array}
 \end{equation}

Equation (\ref{e13.19}a) is true because if  Apex has the lower price, all
consumers will want to buy from Apex if they buy at all. All  of the $(120-p_a)$
customers  who want to buy at that price  will be satisfied   if there are 70 or
less; otherwise only 70.    Equation (\ref{e13.19}b) simply says the two firms
split the market equally if prices are equal.
Equation (\ref{e13.19}c) is true because if    Brydox's price is the lowest and
is  less than 50,   Brydox will sell 70 units, and the residual demand curve
facing Apex will be as in equation  (\ref{e13.18}).  If Brydox's price is  the
lowest but exceeds 50, then less than 70 customers  will want to buy at all, so
Brydox   will satisfy all of them and zero will be left for Apex -- which is
equation (\ref{e13.19}d).


 The appropriate rationing rule depends on what is being modelled. Intensity
rationing is appropriate if buyers with more intense demand make greater efforts
to obtain low prices. If the intense buyers are wealthy people who are unwilling
to wait in   line, the least intense buyers might end up at the low-price firm
which is the case of the {\bf inverse-intensity rationing}  (or {\bf low-to-high
rationing}) in Figure 3b.  An intermediate rule is proportional rationing, under
which every type of consumer is equally likely to be able to buy at the low
price.

\noindent
 {\bf Proportional rationing.} {\it Each consumer has the same probability of
being able to buy from the low-price firm.}

  Under proportional rationing, if $K= 70$ and 90 consumers wanted to buy from
Brydox, 2/9 ($=\frac{q(p_b)-K}{q(p_b)}$) of each type of consumer will be forced
to buy from Apex (for example, 2/9 of the type willing to pay 120).  The
residual demand curve facing Apex, shown in Figure 3c and equation
(\ref{e13.20}), intercepts the price axis at 120, but slopes down at a rate
three times as fast as market demand because there are only 2/9  as many
remaining consumers of each type.
  \begin{equation}\label{e13.20}
  q_a = (120 - p_a) \left(\frac{120- p_b - K}{120 - p_b} \right)
 \end{equation}

    We thus have three choices for rationing rules, with no clear way to know
which to use. Let's use intensity rationing. That is the rule which makes
deviation to high prices least attractive, since the low-price firm keeps the
best customers for itself, so if  we find that the normal Bertrand equilibrium
breaks down there, we will know it would break down under the other rationing
rules too.

\begin{center} {\bf The { Bertrand Game} with Capacity Constraints }
  \end{center}
  {\bf Players}\\
 Firms Apex and Brydox

 \noindent
  {\bf The Order of Play}\\
 Apex  and Brydox simultaneously choose   prices  $p_a$    and $p_b$ from the
set  $[0, \infty)$.

\noindent {\bf Payoffs}\\
 Marginal cost is constant at $c=12$. Demand is a function of the total quantity
sold, $ Q(p) = 120-p.$ The payoff function for Apex (Brydox's would be
analogous) is, using equation (\ref{e13.19}) for $q_a$,
 \begin{equation}\label{e13.20a}
 \begin{array}{ll}
\pi_a = & \left\{ \begin{array}{llr}
   (p_a-c) \cdot  Min \{ 120 - p_a, 70 \} &{\rm if} \;p_a < p_b  & (a) \\
 & & \\
 \left(p_a-c\right)  \left(   \frac{120 - p_a  }{2} \right)  & {\rm if }\; p_a =
p_b & (b)  \\
 & & \\
 (p_a-c)  \cdot  Max \{(120 - p_a- 70), 0 \} & {\rm if}\; p_a > p_b, p_b < 50  &
(c) \\
  & & \\
0        & {\rm if }\; p_a > p_b, p_b \geq 50 & (d)\\
 \end{array} \right.
 \end{array}
 \end{equation}

     The capacity constraint has a very important effect: $(p_a=12, p_b=12)$ is
no longer a Nash equilibrium in prices, even though the industry capacity of 140
is well over the market demand of 108  when  price equals marginal cost. Apex's
profit would be zero in that strategy profile.   If Apex increased its price to
$p_a=20$,   Brydox would immediately sell $q_b=70$, and to the most intense  70
of buyers.  Apex would be left with all the buyers between  $p_a=20$ and $p_a=
12$ on the demand curve for  sales of $q_a=30$ and a  payoff  of 240 from
equation (\ref{e13.20a}c).     So deviation by Apex is profitable. (Of course,
$p_a=20$ is not necessarily  the most profitable deviation --- but we do not
need to check that;  any profitable deviation is enough to refute the proposed
equilibrium.)

Equilibrium prices must be lower than 120, because that price yields a zero
payoff under any circumstance. There are  three remaining possibilities (now
that we have ruled out $p_a=p_b=12$)  for prices chosen in the  open interval
[12,120).


(i)  Equal prices      with $p_a = p_b>12$ are not an equilibrium.      Even if
the price is close to 12, Apex would sell at most 54 units as its half of the
market, which is less than its capacity of 70.    Apex could profitably deviate
to  just below $p_b$     and have a discontinuous jump in   sales  for an
increase in profit, just as in the basic Bertrand game.

(ii) Unequal prices  with one equal to 12 are  not an equilibrium.  Without loss
of generality, suppose  $p_a > p_b=12$.  Apex could not profitably deviate, but
Brydox could deviate to $p_b = p_a-\epsilon$ and make positive instead of zero
profit.

 (iii)   Finally, unequal prices of  $(p_a, p_b)$ with both  greater than 12
are not an equilibrium. Without loss of generality, suppose  $p_a > p_b>12$.
Apex's profits are shown in equation (\ref{e13.20a}c). If $\pi_a=0$, it can
  gain by deviating to   $p_a=p_b-\epsilon$.   If $\pi_a= (p_a-c)(50 - p_a)$, it
can
  gain by deviating to   $p_a=p_b-\epsilon$, because equation  (\ref{e13.20a}c)
tells us that Apex's payoff will rise to either   $\pi_a= (p_a-c)(70)$ (if $p_b
\geq 50$) or $\pi_a= (p_a-c)(120-p_a )$
(if $p_b < 50$).

  Thus,    no equilibrium exists in pure strategies under intensity rationing,
and similar arguments rule out pure-strategy equilibria under other forms of
rationing.     This is known as the {\bf Edgeworth paradox}  after Edgeworth
(1897, 1922).

   Nowadays we know that the resolution to many a paradox is mixed strategies,
and that is the case here too.
A mixed-strategy   equilibrium does exist, calculated using intensity rationing
and linear demand  by Levitan \& Shubik (1972).  Expected profits are positive,
because the firms charge   prices above marginal cost.  In the symmetric
equilibrium, the firms mix using  distribution $F(p)$ with   a support
$[\underline{p}, \bar{p}]$,  where  $\underline{p}>c$ and $\bar{p}$ is the
monopoly price for the residual demand curve (\ref{e13.18}), which happens to be
$\bar{p}=36$ in our example. The upper bound $\bar{p}$ is that monopoly price
because $F(\bar{p}$ and the firm choosing that price certainly is the one with
the highest price and so should maximize its profit using the residual demand
curve. The payoff from playing the lower bound, $\underline{p}$, is
$(\underline{p}-c)(70)$ from equation (\ref{e13.20a}c),  so since that payoff
must
equal the payoff of 336 (=(p-c)q= (36-12) (50-36)) from $\bar{p}$, we can
conclude that $(\underline{p} = 16.8$. The mixing distribution $F(p)$ could then
be found by  setting $\pi(p) = 336 = F(p) (p-c) (50-p) + (1-F(p)) (p-c)(70)$ and
solving for $F(p)$.






If capacities are large enough--- above  the capacity of  $   K=Q(c) =108$ in
this
example--- the Edgeworth paradox disappears.    The argument made above  for why
equal prices of $c$ is not an equilibrium fails, because if Apex  were to
deviate to a positive price, Brydox would be fully capable of serving the entire
market, leaving Apex with no consumers.

If capacities are small enough--- less than $K=36$ in our example--- the
Edgeworth
paradox also  disappears, but so does the Bertrand paradox. The equilibrium is
in pure strategies, with each firm using its entire capacity, so $q_a=q_b =K$
and charging the same price. There is no point in a firm reducing its price,
since it cannot sell any greater quantity. How about a deviation to increasing
its price and reducing its quantity?  Its best deviation is to the price which
maximizes profit using the residual demand curve $(120-K-p)$. This turns out to
be
  $p^* = (120-K+c)/2$, in which case
 \begin{equation}  \label{e13.20c}
 q^* =   \frac{120-K-c}{2} .
\end{equation}
 But if
 \begin{equation}  \label{e13.20b}
K <  \frac{120-c}{3},
\end{equation}
then
 $q^*>K$ in equation (\ref{e13.20c})  and  is infeasible---   the profit-
maximizing price   is from using all the capacity. The critical level  from
inequaity (\ref{e13.20b}) is $K=36$ in our example.  For any lower capacities,
firms simply dump their entire capacity onto the market and the price, $p_a=p_b=
120 - 2K$,  exceeds marginal cost.




 \vspace{.5in}

\noindent
 {\bf  14.3 Location Models }

\noindent
   In Chapter 3 we analyzed the Bertrand model with differentiated products
using demand functions whose arguments were the prices of both firms. Such a
model is suspect because it is not based on primitive assumptions.  In
particular, the demand functions might not be generated by maximizing any
possible utility function. A demand curve with a constant elasticity less than
one, for example, is impossible because as the price goes to zero, the amount
spent on the commodity goes to infinity. Also, the  demand curves   were
restricted to prices below a certain level, and it would be good to be able to
justify that restriction.

   Location models construct demand functions like those in Chapter 3   from
primitive assumptions.  In location models, a differentiated product's
characteristics are points in a space. If cars differ only in their mileage, the
space is a one-dimensional line.  If acceleration is also important, the space
is a two-dimensional plane. An easy way to think about this approach is to
consider the location where a product is sold.  The product ``gasoline sold at
the corner of Wilshire and Westwood,'' is different from ``gasoline sold at the
corner of Wilshire and Fourth.''  Depending on where consumers live, they have
different preferences over the two, but, if prices diverge enough, they will be
willing to switch from one gas station to the other.

        Location models form a literature in themselves. We will look at the
first two models analyzed in the classic article of Hotelling (1929), a model of
price choice and a model of location choice. Figure 4   shows what is common to
both. Two firms are located at points $x_a$ and $x_b$ along a line running from
zero to one, with a constant density of consumers throughout. In the Hotelling
Pricing Game, firms choose prices for given locations.  In the Hotelling
Location Game, prices are fixed and the firms choose the locations.

\includegraphics[width=150mm]{fig14-04.jpg}

 \begin{center}
 {\bf Figure  4: Location Models  }
\end{center}

 \begin{center}
{\bf  The Hotelling Pricing Game }\\
 (Hotelling [1929])
 \end{center}
{\bf Players}\\
Sellers Apex and Brydox, located at $x_a$ and $x_b,$ where $x_a < x_b$, and a
continuum of buyers indexed by location $x \in [0,1]$.

\noindent
{\bf The Order of Play }\\
 1 The sellers simultaneously choose  prices $p_a$ and $p_b$.\\
 2 Each buyer chooses a seller.

\noindent
{\bf Payoffs}\\
  Demand is uniformly distributed on the interval [0,1] with a density equal to
one (think of each consumer as buying one unit). Production costs are zero. Each
consumer always buys, so his problem is to minimize the sum of the price plus
the linear transport cost, which is $\theta$ per unit distance travelled.
 \begin{equation} \label{e13.33}
 \pi_{buyer \;at \;x} =V  -Min\{ \theta |x_a -x| + p_a, \; \theta |x_b - x| +
p_b \}.
  \end{equation}

\begin{equation} \label{e13.33aa}
\pi_a = \left\{ \begin{tabular}{llr}
 $p_a (0) =0$ & if   $p_a - p_b  >  \theta (x_b - x_a)$   & (a)\\ &    (Brydox
captures entire market) & \\
 & & \\ $ p_a (1) =p_a$  & if   $p_b - p_a  > \theta (x_b - x_a)$   & (b)\\
& (Apex captures entire market)  & \\ & & \\ $p_a ( \frac{1}{2\theta} \left[
(p_b - p_a) + \theta(x_a + x_b) \right] ) $ & otherwise (the market is divided)&
(c) \\
\end{tabular} \right.
 \end{equation}

\noindent
 Brydox has analogous payoffs.

 The payoffs result from buyer behavior.   A buyer's utility depends on the
price he pays and the distance he travels.  Price aside, Apex is most attractive
of the two sellers to the consumer at $x=0$ (``consumer 0'')  and least
attractive to the consumer at $x = 1 $ (``consumer 1''). Consumer   0  will buy
from Apex so long as
\begin{equation}\label{e13.34}
 V-(\theta x_a + p_a) >V- (\theta x_b + p_b),
 \end{equation} which implies that
 \begin{equation} \label{e13.35}
 p_a - p_b  < \theta (x_b - x_a), \end{equation}
 which yields payoff
(\ref{e13.33aa}a) for Apex.  Consumer    1  will buy from Brydox if
\begin{equation}\label{e13.36}
 V-[\theta(1- x_a) +p_a] < V- [\theta (1-x_b) + p_b],
 \end{equation} which implies that
 \begin{equation}\label{e13.37}
 p_b - p_a  < \theta (x_b - x_a),
 \end{equation}
 which  yields payoff (\ref{e13.33aa}b) for Apex.

  Very likely, inequalities  (\ref{e13.35}) and (\ref{e13.37}) are both
satisfied, in which case Consumer 0 goes to Apex and Consumer 1 goes to Brydox.
This is the case represented by payoff (\ref{e13.33aa}c), and the next task is
to find the location of consumer $x^*$,   defined as the consumer who is at the
boundary between the two markets, indifferent between  Apex and Brydox.  First,
notice that if Apex attracts Consumer $x_b$, he also attracts all $x > x_b$,
because beyond $x_b$ the consumers' distances from both sellers increase at the
same rate. So we know that if there is an indifferent consumer he is between
$x_a$ and $x_b$. Knowing this, the consumer's payoff equation,  (\ref{e13.33}),
tells us that
 \begin{equation}\label{e13.38}
 V- [ \theta(x^*- x_a) + p_a] = V-  [\theta (x_b -x^*) + p_b],
 \end{equation}
 so that
 \begin{equation}\label{e13.39}
 p_b - p_a = \theta (2x^*- x_a - x_b ),
 \end{equation} and
 \begin{equation}\label{e13.40}
 {\displaystyle x^* = \frac{1}{2\theta} \left[ (p_b - p_a) + \theta(x_a + x_b)
\right]},
\end{equation}
which generates demand curve (\ref{e13.33aa}c)-- a differentiated Bertrand
demand curve.

Remember, however,  that equation (\ref{e13.40}) is valid only if there really
does exist a consumer who is indifferent -- if such a consumer does not exist,
equation (\ref{e13.40}) will generates a number for $x^*$, but that number is
meaningless.

 Since Apex keeps all the consumers between 0 and $x^*$, equation (\ref{e13.40})
is the demand function facing Apex so long as he does not set his price so far
above Brydox's that he loses even consumer 0. The demand facing Brydox equals
$(1 - x^*)$.  Note that if $p_b = p_a$, then from (\ref{e13.40}), $x^* =
\frac{x_a + x_b}{2}$, independent of $\theta$, which is just what we would
expect. Demand is linear in the prices of both firms, and looks similar to the
demand curves   used in Section 3.6 for the Bertrand game with differentiated
products.

 Now that we have found the demand functions, the Nash equilibrium can be
calculated in the same way as in Section 14.2, by setting up the profit
functions for each firm, differentiating with respect to the price of each, and
solving the two first-order conditions for the two prices. If there exists an
equilibrium in which the firms are willing to pick prices to satisfy
inequalities (\ref{e13.35}) and (\ref{e13.37}), then it  is
 \begin{equation}\label{e13.41}
p_a = \frac{(2 + x_a + x_b)\theta}{3}, \;\;p_b = \frac{(4 - x_a - x_b)\theta}
{3}.
 \end{equation}
 From (\ref{e13.41}) one can see that Apex charges a higher price if a large
$x_a$ gives it more safe consumers or a large $x_b$ makes the number of
contestable consumers greater. The simplest case is when $x_a = 0$ and $x_b =1$,
when  (\ref{e13.41}) tells us that both firms charge a price equal to $\theta$.
Profits are positive and increasing in the transportation cost.

  We cannot rest satisfied with the neat equilibrium of equation (\ref{e13.41}),
because the assumption that  there exists an equilibrium in which the firms
choose prices so as to split the market on each side of some boundary  consumer
$x^*$ is often violated. Hotelling did not notice this, and fell into a common
mathematical trap. Economists are used to models in which the calculus approach
gives an answer that is both the local optimum and the global optimum. In games
like this one, however, the local optimum is not global, because of the
discontinuity in the objective function. Vickrey (1964) and  D'Aspremont,
Gabszewicz  \& Thisse (1979) have shown that if $x_a$ and $x_b$ are close
together, no pure-strategy equilibrium exists, for reasons similar to why none
exists in the Bertrand model with capacity constraints.  If both firms charge
nonrandom prices, neither would deviate to a slightly different price, but one
might deviate to a much lower price that would capture every single consumer.
But if both firms charged that low price, each would deviate by raising his
price slightly.  It turns out that if, for example,   Apex and Brydox are
located symmetrically around the center of the interval, $x_a \geq 0.25$, and
$x_b \leq 0.75$, no pure-strategy equilibrium exists (although a mixed-strategy
equilibrium does,  as Dasgupta \& Maskin [1986b] show).

 Hotelling should  have done some numerical examples.  And he should have
thought about the comparative statics carefully.   Equation (\ref{e13.41})
implies  that Apex should choose a higher price if both $x_a$ and $x_b$
increase, but it is odd that if the firms are locating closer together, say at
0.90 and  0.91, that Apex should be able to charge a higher price, rather  than
suffering from more intense competition. This kind of  odd result is a typical
clue that the result has a logical flaw somewhere. Until  the modeller can
figure out an intuitive reason for his odd result, he should suspect an error.
For practice, let us try a few numerical examples, illustrated in Figure  5.

\includegraphics[width=150mm]{fig14-05.jpg}

 \begin{center}
{\bf Figure  5: Numerical Examples for Hotelling Pricing  }
 \end{center}

\noindent
 {\bf  Example 1. Everything works out simply}\\
 Try    $x_a = 0, x_b = 0.7$ and $\theta =0.5$.  Then equation (\ref{e13.41})
says $ p_a= (2+0+0.7)0.5/3 =0.45 $ and $p_b= (4-0-0.7)0.5/3 =  0.55$. Equation
(\ref{e13.40}) says that $  x^* = \frac{1}{2*0.5} \left[ (0.55-0.45) +
0.5(0.0+0.7)\right]= 0.45$.

 In Example 1, there is a pure strategy equilibrium and the equations generated
sensible numbers given the parameters we chose.  But it is not enough to
calculate just one numerical example.

\noindent
 {\bf Example 2. Same location -- but different prices?}\\
 Try  $x_a = 0.9, x_b = 0.9$ and $\theta =0.5$.  Then equation (\ref{e13.41})
says $ p_a= (2.0+0.9+0.9)0.5/3 \approx 0.63$ and     $p_b= (4.0-0.9-0.9) 0.5/3
\approx 0.37$.

 Example 2  shows  something odd   happening.    The  equations generate
numbers that seem innocuous until one realizes that   if both firms are located
at 0.9,  but  $p_a =0 .63$ and  $p_b = 0.37$, then  Brydox will    capture the
entire market! The result is nonsense, because  equation (\ref{e13.41})'s
derivation relied on the assumption that $x_a < x_b$, which is false in this
example.

\noindent
{\bf  Example 3. Locations too near  each other.  }\\
$x^* < x_a < x_b$.  Try  $x_a = 0.7, x_b = 0.9$ and $\theta =0.5$. Then equation
(\ref{e13.41}) says that   $ p_a= (2.0+0.7+0.9)0.5/3 =0.6$ and     $p_b=
(4-0.7-0.9)0.5/3 =0.4$.  As for the split of the market, equation (\ref{e13.40})
says that $    x^* = \frac{1}{2*0.5} \left[ (0.4-0.6) + 0.5(0.7+0.9) \right]=
0.6$.

 Example 3 shows a serious problem.  If the market splits at  $x^* = 0.6$ but
$x_a=0.7$ and $x_b=0.9$,  the result violates   our implicit assumption that the
players split the market.    Equation (\ref{e13.40}) is based on the premise
that there does exist some indifferent consumer, and when that is a false
premise, as under the parameters of Example 3,   equation  (\ref{e13.40}) will
still  spit out a value of $x^*$, but the value  will not mean anything.    In
fact  the consumer at  $x=0.6$ is not really indifferent between Apex and
Brydox.    He   could buy from Apex at  a total cost of  0.6 + 0.1(0.5) = 0.65
or from Brydox, at a total cost of  0.4 + 0.3 (0.5) = 0.55.     There exists no
consumer   who  strictly prefers Apex.   Even Apex's `home' consumer at $x= 0.7$
would have a total cost of buying from Brydox of  $0.4 + 0.5 (0.9-0.7) =0.5$
and would prefer Brydox.   Similarly, the consumer at $x=0$ would have a total
cost of buying from Brydox of  $0.4 + 0.5 (0.9-0.0) =  0.85$, compared to a
cost   from Apex of  $0.6 + 0.5(0.7-0.0)= 0.95$, and he, too, would prefer
Brydox.

 The problem in Examples 2 and 3 is that the firm with the higher price would do
better to deviate with a discontinuous price cut,  to just below the  other
firm's price.  Equation  (\ref{e13.41}) was derived by calculus,  with the
implicit assumption that  a local profit  maximum was also a global profit
maximum, or, put differently, that  if no small change could raise a firm's
payoff, then it had found the optimal strategy.  Sometimes a big change will
increase a player's payoff even though a small change would not. Perhaps this is
what they mean in business by the importance of  ``nonlinear thinking'' or
``thinking out of the envelope.'' The  everyday manager or  scientist as
described by Schumpeter (1934) and Kuhn (1970) concentrates on  analyzing
incremental  changes  and  only the entrepreneur or  genius  breaks through with
a  discontinuously  new idea, the  profit source or  paradigm shift.

\bigskip

    Let us now turn to the choice of location.  We will simplify the model by
pushing consumers into the background and imposing a single exogenous price on
all firms.

\bigskip

 \begin{center} {\bf   The  Hotelling Location Game }\\
 (Hotelling [1929])
 \end{center}
 {\bf Players}\\
$n$ Sellers.

 \noindent
 {\bf The Order of Play }\\
The sellers simultaneously choose locations $x_i \in [0,1].$

 \noindent {\bf Payoffs}\\
 Consumers are distributed along the interval [0,1] with a uniform density equal
to one. The price equals one, and production costs are zero.  The sellers are
ordered by their location so $x_1 \leq x_2 \leq \ldots \leq x_n$, $x_0 \equiv 0$
and $x_{n+1} \equiv 1.$ Seller $i$ attracts half the consumers from the gaps on
each side of him,  as shown in Figure 14.6, so that his payoff is
\begin{equation}\label{e13.42}
 \pi_1 = x_1 + \frac{x_2 - x_1}{2},
\end{equation}
\begin{equation}\label{e13.43}
 \pi_n =    \frac{x_n - x_{n-1}}{2} + 1 - x_n,
 \end{equation}
or, for $i = 2, \ldots n-1$,
 \begin{equation}\label{e13.44}
\pi_i = \frac{x_i - x_{i-1}}{2} + \frac{x_{i+1} - x_i}{2}.
 \end{equation}

\includegraphics[width=150mm]{fig14-06.jpg}

 \begin{center}
  {\bf Figure 6: Payoffs in the Hotelling Location Game }
 \end{center}

\bigskip

  With {\bf one seller}, the location does not matter in this model, since the
consumers are captive. If price were a choice variable and demand were elastic,
we would expect the monopolist to locate at $x=0.5$.

  With {\bf two sellers}, both firms locate at $x= 0.5$, regardless of whether
or not demand is elastic.  This is a stable Nash equilibrium, as can be seen by
inspecting Figure  4 and imagining best responses to each other's   location.
The best response is always to locate $\varepsilon$ closer to the center of the
interval than one's rival. When both firms do this, they end up splitting the
market since both of them end up exactly at the center.

\includegraphics[width=150mm]{fig14-07.jpg}

\begin{center}
 {\bf Figure  7: Nonexistence of pure strategies with three players }
\end{center}

     With {\bf three sellers} the model does not have a Nash equilibrium in pure
strategies.  Consider any strategy profile in which each player locates at a
separate point. Such a strategy profile is not an equilibrium, because the two
players nearest the ends would edge in to squeeze the middle player's market
share. But if a strategy profile has any two players at the same point $a$, as
in Figure  7, the third player would be able to acquire a share of at least
$(0.5 - \epsilon)$ by moving next to them at $b$; and if the third player's
share is that large, one of the doubled-up players would deviate by jumping to
his other side and capturing his entire market share.  The only equilibrium is
in mixed strategies.


\includegraphics[width=150mm]{fig14-08.jpg}

 \begin{center}
{\bf Figure  8:  The Equilibrium Mixed-Strategy Density in the Three-Player
Location Game  }
 \end{center}

Suppose all three players use the same mixing density,  with $m(x)$ the
probability density for location $x$, and positive density on  the support
$[g,h]$, as depicted in Figure  8. We will need the density for the distribution
of the minimum of the locations of Players 2 and 3.  Player 2 has location $x$
with density  $m(x)$, and Player 3's location is greater than that with
probability  $1-M(x)$, letting $M$ denote the cumulative distribution, so the
density for Player 2 having location  $x$ and it being   smaller is  $m(x)[1-
M(x)]$. The  density for either Player 2 or Player 3 choosing $x$ and it being
smaller than the other firm's location is then $2m(x)[1-M(x)]$.

If Player 1 chooses $x=g$  then his  expected payoff is
 \begin{equation}\label{e13.new1}
 \pi_1(x_1=g) =  g + \int_g^h   2m(x)[1-M(x)]   \left( \frac{  x-g } {2} \right)
dx,
 \end{equation}
 where   $g$ is  the  safe set of consumers to his left, $2m(x)[1- M(x)]$ is the
density for $x$ being the next biggest   location of a firm, and $\frac{  x-g }
{2}$ is Player 1's share of the consumers between his own location of $g$ and
the next biggest location.

If Player 1 chooses $x=h$  then his  expected payoff is, similarly,
 \begin{equation}\label{e13.new1a}
  \pi_1(x_1=h) =  (1-h)  + \int_g^h   2m(x) M(x)   \left( \frac{  h-x } {2}
\right) dx,
 \end{equation} where $(1-h)$ is the set of safe consumers to his right

   In a mixed strategy equilibrium, Player 1's payoffs from  these two pure
strategies must be equal, and they are also equal to his payoff from  a location
of 0.5, which we can plausibly guess is in the support of his mixing
distribution. Going on from this point, the algebra and calculus start  to
become fierce.    Shaked (1982) has computed the symmetric mixing probability
density $m(x)$ to be as shown in Figure  9,
\begin{equation}\label{e13.45}
m(x)=  \left\{
 \begin{array}{ll}
 2 &{\rm if}\;\; \frac{1}{4} \leq x \leq \frac{3}{4}  \\
 & \\
 0 &{\rm  otherwise} \\
\end{array}
\right. \end{equation}

 You can check  this equilibrium  by seeing that with the mixing density
(\ref{e13.45}) (depicted in Figure 9)  the payoffs in equation (\ref{e13.new1})
and (\ref{e13.new1a}) do equal each other.  This method has only shown what the
symmetric equilibrium is like, however; it turns out that asymmetric equilibria
also exist (Osborne \& Pitchik [1986]).

\includegraphics[width=150mm]{fig14-09.jpg}

\begin{center}
 {\bf Figure  9: The Equilibrium Mixing Density for Location }
 \end{center}

  Strangely enough, three is a special number. With {\bf more than three
sellers}, an equilibrium in pure strategies does exist if the consumers are
uniformly distributed, but this is a delicate result  (Eaton \& Lipsey [1975]).
Dasgupta \& Maskin (1986b), as amended by Simon (1987), have also shown that an
equilibrium, possibly in mixed strategies, exists for any number of players $n$
in a space of any dimension $m$.

  Since prices are inflexible, the competitive market does not achieve
efficiency.  A benevolent social planner or a monopolist who could charge higher
prices if he located his outlets closer to more consumers would choose different
locations than competing firms.  In particular, when two competing firms both
locate in the center of the line, consumers are no better off than if there were
just one firm. As shown in Figure 10, the  average distance of a consumer from a
seller would be minimized by setting $x_1 = 0.25$ and $x_2 = 0.75$, the
locations that would be chosen either by the social planner or the monopolist.

\includegraphics[width=150mm]{fig14-10.jpg}

\begin{center}
 {\bf Figure  10:  Equilibrium versus Efficiency }
 \end{center}


    The Hotelling Location Model, however,  is very well suited to politics.
Often there is just one dimension of importance in political races, and  voters
will vote for the candidate closest to their own position, so there is no analog
to price.  The Hotelling Location Model predicts that the two candidates will
both choose the same position, right on top of the median voter.   This seems
descriptively realistic; it accords with the common complaint that all
politicians are pretty much the same.

  \vspace{1in}

\noindent
{\bf  14.4 Comparative Statics and   Supermodular Games}

   Comparative statics  is the  analysis of  what happens to endogenous
variables in a model when the exogenous variables change. This is a central part
of economics. When   wages rise, for example, we wish to know how the price of
steel will change in response. Game theory presents special problems for
comparative statics, because when a parameter changes, not only does  Smith's
equilibrium strategy change in response, but Jones's strategy changes as a
result of Smith's change as well.  A small change in the parameter might produce
a large change in the equilibrium   because of feedback between the different
players' strategies.

Let us use a differentiated Bertrand game as an example. Suppose there are $N$
firms, and for firm $j$ the demand curve is
\begin{equation} \label{e13.46}
 Q_j = Max \{ \alpha -  \beta_j p_j + \sum_{i \neq j} \gamma_i p_i, 0\},
 \end{equation}
   with $\alpha \in (0, \infty)$, $\beta_i \in (0, \infty)$,  and $\gamma_i \in
(0, \infty)$ for $i = 1, \ldots, N$. Assume that   the effect of $p_j$ on firm
$j$'s sales is larger than the effect of   the other firms'  prices,  so that
\begin{equation} \label{e.47}
 \beta_j > \sum_{i \neq j} \gamma_i. \end{equation} Let firm $i$ have constant
marginal cost $\kappa   c_i$, where $\kappa \in \{1,2\}$ and  $c_i \in (0,
\infty)$, and let us assume that each firm's costs are low enough that it does
operate in equilibrium.   (The shift variable $\kappa$ could represent  the
effect of the political regime on costs.)

 \noindent
The payoff function for firm $j$ is
 \begin{equation} \label{e13.48}
 \pi_j = (p_j  - \kappa c_j)(\alpha  - \beta_j p_j + \sum_{i \neq j} \gamma_i
p_i).
  \end{equation}
   Firms choose prices simultaneously.

   Does this game have an equilibrium? Does it have several equilibria? What
happens to the equilibrium price if a parameter such as $c_j$ or $\kappa$
changes? These are difficult questions because if $c_j$ increases, the immediate
effect is to change firm $j$'s price, but the other firms will react to the
price change, which in turn will affect $j$'s price. Moreover, this is not a
symmetric game -- the costs and demand curves differ from firm to firm, which
could make algebraic solutions of the Nash equilibrium quite messy. It is not
even clear whether the equilibrium is unique.

 Two approaches to comparative statics can be used here: the implicit function
theorem, and supermodularity. We will look at each in turn.

 \bigskip

\noindent
  {\bf  The Implicit Function Theorem}

\noindent
The implicit-function theorem says that if $f(y,z) = 0$, where $y$ is endogenous
and $z$ is exogenous,  then
\begin{equation} \label{e13.49}
 \frac{ d y} {  d z } = - \left( \frac{ \frac{\partial f}{ \partial z} } {
\frac{\partial f}{  \partial y} } \right).
 \end{equation}

It is worth knowing how to derive this. We start with $f(y,z) = 0$, which can be
rewritten as $f(y(z), z) ) = 0$, since  $y$ is endogenous. Using the calculus
chain rule,
 \begin{equation} \label{e14.49a}
\frac{ d f} {  d z } = \frac{\partial f}{ \partial z}  + \left( \frac{\partial
f}{   \partial y} \right)  \left(\frac{ d y} {  d z }\right) = 0.
 \end{equation}
 where the expression equals zero because    after a small change in $z$, $f$
will still equal zero after   $y$ adjusts. Solving for $\frac{ d y} {  d z }$
yields equation (\ref{e13.49}).

  The implicit function theorem is especially useful   if $y$ is a choice
variable and $z$ a parameter, because then we can use the first-order condition
to set $f(y, z ) \equiv  \frac{\partial \pi}{\partial y}  =0$ and
 the second-order condition tells us that   $\frac{\partial f}{\partial y}=
\frac{\partial^2
\pi}{\partial y^2} \leq 0$. One only   has to make certain that the solution is
an interior solution, so the first- and second-order conditions are valid, and
keep in mind that if the solution is only a local maximum, not  a global one,
the maximizing choice  might ``jump'' up or down when an exogenous variable
changes.

  We do have a complication if the model is strategic: there will be more than
one endogenous variable, because more than one player is choosing variable
values.  Suppose that instead of simply $f(y,z) = 0$, our implicit equation has
two endogenous and two exogenous variables, so $f(y_1, y_2,z_1, z_2) = 0$.   The
extra $z_2$ is no problem; in comparative statics we are holding all but one
exogenous variable constant. But the $y_2$ does add some complexity to the mix.
Now, using the calculus chain rule yields not equation (\ref{e14.49a})  but
\begin{equation} \label{e14.49b}
 \frac{ d f} {  d z_1 } = \frac{\partial f}{ \partial z_1}  + \left(
\frac{\partial f}{   \partial y_1}\right)  \left(  \frac{ d y_1} {  d z_1 }
\right) +  \left(\frac{\partial f}{   \partial y_2}\right)  \left( \frac{ d y_2}
{  d z_1 } \right) = 0.
 \end{equation}
 Solving for $\frac{ d y_1} {  d z_1 }$ yields
\begin{equation} \label{e14.49c}
 \frac{ d y_1} {  d z_1 } = - \left( \frac{  \frac{\partial f}{ \partial z_1}  +
\left(\frac{\partial f}{   \partial y_2}\right)  \left(  \frac{ d y_2} {  d z_1}
\right)}{ \frac{\partial f}{   \partial y_1}}   \right).
 \end{equation}
It is  often unsatisfactory to solve out for  $\frac{ d y_1} {  d z_1 }$ as a
function of both the exogenous variables $z_1$ and $z_2$ and the endogenous
variable $y_2$  (though it is  okay if all you want is to discover whether the
change is positive or negative), but  ordinarily the modeller will also have
available an optimality condition for Player 2 also: $g(y_1, y_2,z_1, z_2) = 0$.
This second condition yields  an equation similar to   (\ref{e14.49c}), so that
two equations can be solved for the two unknowns.

We can use the differentiated Bertrand game to see how this works out.
Equilibrium prices  will  lie inside the interval  ($c_j, \overline {p} $) for
some large number $\overline {p}$, because a price of  $c_j$ would yield zero
profits, rather than the positive profits of a slightly higher price,   and
$\overline{p}$ can be chosen to    yield zero quantity demanded  and hence zero
profits. The equilibrium  or equilibria are, therefore, interior solutions, in
which case    they satisfy the first-order condition
\begin{equation} \label{e.50}
\frac{\partial  \pi_j }{ \partial p_j  } =  \alpha -  2\beta_j p_j+ \sum_{i\neq
j} \gamma_i p_i + \kappa c_j \beta_j  = 0, \end{equation} and the second-order
condition, \begin{equation} \label{e.51} \frac{\partial^2  \pi_j }{ \partial
p_j^2  } = -2 \beta_j   < 0.
\end{equation}

Next, apply the implicit function theorem by   using $p_i$ and $c_i$, $i=1,
\ldots, N,  $ instead of   $y_i$ and $z_i$,  $i=1,2$, and by letting
$\frac{\partial \pi_j}{ \partial p_j  } = 0$ from   equation (\ref{e.50}) be our
$f(y_1, y_2, z_1, z_2)  = 0$. The chain rule yields \begin{equation}
\label{e.51a}
 \frac{ d  f} {  d c_j } =  -  2\beta_j \left(  \frac{  d p_j} { d c_j } \right)
+ \sum_{i\neq j} \gamma_i  \left( \frac{  d p_i} { d c_j } \right) + \kappa
\beta_j  = 0, \end{equation}
  so
\begin{equation}\label{e.51b}
 \frac{  d p_j} { d c_j }   = \frac{ \sum_{i\neq j} \gamma_i  \left( \frac{  d
p_i} { d c_j } \right) + \kappa  \beta_j }{ 2\beta_j}.
 \end{equation}

 Just what is $\frac{  d p_i} { d c_j}$?   For each $i$, we need to  find  the
first-order condition for firm $i$ and then use the chain rule again. The first-
order condition for Player $i$ is that  the derivative of $\pi_i$ with respect
to $p_i$ ({\it not} $p_j$) equals zero, so
 \begin{equation} \label{e.51c}
 g^i  \equiv \frac{\partial  \pi_i }{ \partial p_i } =  \alpha - 2\beta_i p_i +
\sum_{k\neq i} \gamma_k p_k + \kappa c_i \beta_i  = 0.
\end{equation}
 The chain rule yields (keeping in mind that it is a change in $c_j$ that
interests us, {\it not} a change in $c_i$),
 \begin{equation}\label{e.51d}
 \frac{ d  g^i} {  d c_j } =  - 2\beta_i  \left(\frac{d p_i }{d c_j}\right)
+\sum_{k\neq i} \gamma_k  \left( \frac{d p_k }{d c_j}\right)   = 0.
\end{equation}
  With equation (\ref{e.51b}), the $(N-1)$ equations (\ref{e.51d}) give us $N$
equations for the  $N$ unknowns  $\frac{d p_i }{d c_j}$, $i = 1, \ldots, N$.

 It is easier to see what is going on if there are just two firms, $j$ and $i$.
Equations (\ref{e.51b}) and  (\ref{e.51d}) are then \begin{equation}
\label{e.51e}
 \frac{  d p_j} { d c_j }   =  \frac{ \gamma_i  \left(\frac{  d p_i} { d c_j }
\right)+ \kappa  \beta_j }{ 2\beta_j}. \end{equation}
  and
 \begin{equation}\label{e.51f}
 - 2\beta_i \left( \frac{d p_i }{dc_j} \right)  +  \gamma_j  \left( \frac{d p_j
}{dc_j}\right) = 0.
 \end{equation}

 Solving  these two equations for   $\frac{  d p_j} { d c_j }$  and $\frac{d p_i
}{dc_j}$ yields
\begin{equation}\label{e.51g}
 \frac{  d p_j} { d c_j }   = \frac{2 \beta_i \beta_j \kappa   }{ 4 \beta_i
\beta_j - \gamma_i \gamma_j  }
 \end{equation}
and
 \begin{equation}\label{e.51h}
 \frac{d p_i }{dc_j}      = \frac{ \gamma_j \beta_j \kappa   }{  4 \beta_i
\beta_j - \gamma_i \gamma_j  }.
\end{equation}

 Keep in mind that  the implicit function theorem only tells about infinitesimal
changes, not finite changes. If $c_n$ increases enough, then the nature of the
equilibrium changes drastically, because firm $n$ goes out of business. Even if
$c_n$ increases a finite amount, the implicit function theorem is not
applicable, because then the change in $p_n$ will cause changes in the prices of
other firms, which will in turn change $p_n$ again.

 We cannot go on to discover the effect of changing $\kappa$ on $p_n$, because
$\kappa$ is a discrete variable, and the implicit function theorem only applies
to continuous variables.  The implicit function theorem is none the less very
useful when it does apply. This is a simple example, but the approach can be
used even when the functions involved are very complicated. In complicated
cases, knowing that the second-order condition  holds allows the modeller to
avoid having to determine the sign of the denominator if all that interests him
is the sign of the relationship between the two variables.

\bigskip

\noindent
 {\bf Supermodularity}

The second approach uses the idea of the supermodular game, an idea related to
that of strategic complements (Chapter 3.6).  Suppose that there are $N$ players
in a game, subscripted by $i$ and $j$, and  that player $i$ has a strategy
consisting of $\overline{s}^i$  elements, subscripted by $s$ and $t$, so his
strategy is the vector  $ y^i = (y^i_{1}, \ldots, y^i_{ \overline{s}^i})$. Let
his strategy set be $S^i$ and his payoff function be $\pi^i(y^i, y^{-i};  z)$,
where $z$   represents a fixed parameter.  We say that the game is a {\bf smooth
supermodular game} if the following four conditions are satisfied for every
player $i = 1, \ldots N$:

\noindent
 {\bf  A1$'$} The strategy set is an interval in $\boldmath{R^{\overline{s}^i}}
$:
\begin{equation} \label{e13.53}
 S^i = [\underline{y^i}, \overline{y^i}].
\end{equation}

\noindent
{\bf A2$'$} $\pi^i$  is twice continuously differentiable on $S^i$.

\noindent
 {\bf A3$'$ (Supermodularity) } Increasing one component of  player $i$'s
strategy does not decrease the net marginal benefit of any other component: for
all $i$, and  all $s$  and $t$  such that $ 1 \leq s < t  \leq \overline{s}^i$,
 \begin{equation} \label{e13.54}
 \frac{\partial^2 \pi^i}{ \partial y^i_{s} \partial y^i_{t}} \geq 0.
 \end{equation}

\noindent
 {\bf  A4$'$ (Increasing differences in   strategies)} Increasing one component
of $j$'s strategy does not decrease the net marginal  benefit of increasing any
component of player $i$'s strategy: for all $j \neq i$, and all   $s$ and $t$
such that $1  \leq s \leq \overline{s}^i$ and  $1  \leq   t \leq \overline{s}^j
$,
\begin{equation} \label{e13.55}
\frac{\partial^2 \pi^i}{\partial y^i_{s} \partial y^j_{t}} \geq 0 .
 \end{equation}

In addition, we will be able to talk about the comparative statics of smooth
supermodular games if a fifth condition is satisfied,   increasing differences
in parameters.

\noindent
 {\bf  A5$'$: (Increasing differences in parameters)}  Increasing parameter $z$
does not decrease the net marginal benefit to player $i$ of any   component of
his own strategy: for all $i$, and all $s$    such that $ 1 \leq s   \leq
\overline{s}^i$,
\begin{equation} \label{e13.56a}
\frac{\partial^2 \pi^i}{ \partial y^i_{s} \partial z } \geq 0 .
 \end{equation}

The heart of supermodularity is in assumptions $A3'$ and $A4'$. Assumption $A3'$
says that the components of player $i$'s strategies are all {\bf complementary
inputs};  when one component increases, it is worth increasing the other
components too. This means that  even if a strategy is a complicated one, one
can still arrive at qualitative results about the strategy,  because all the
components of the optimal strategy will move in the same direction together.
Assumption A4$'$ says that the strategies of players $i$ and $j$ are {\bf
strategic complements}; when  player $i$ increases a component of his strategy,
player $j$ will want to do so also. When the strategies of the players reinforce
each other in this way, the feedback between them is less tangled than if they
undermined each other.

I have put primes on the assumptions because they are the special cases, for
smooth games, of the general definition of supermodular games in the
Mathematical Appendix.  Smooth games use differentiable functions, but the
supermodularity theorems apply more generally.    One condition that is relevant
here is condition A5:

\noindent
{\bf  A5:  (Increasing differences in parameters)}  $\pi^i$ has increasing
differences in $y^i$ and $z$  for fixed $y^{- i}$; for all $y^i \geq y^{i}'$,
the difference $\pi^i(y^i, y^{-i},z) - \pi^i(y^{i}', y^{-i}, z)$ is
nondecreasing with respect to $z$.

\bigskip

 \noindent
Is the differentiated Bertrand game  supermodular? The strategy set can be
restricted to  [$c_i$, $\overline{p}$] for player $i$, so A1$'$ is satisfied.
$\pi_i$ is twice continuously differentiable on the interval[$c_i, \overline{p}
$], so A2$'$ is satisfied. A player's strategy has just one component, $p_i$, so
A3$'$ is immediately satisfied. The following inequality is true,
 \begin{equation} \label{e13.57}
 \frac {\partial^2 \pi_i} {\partial p_i \partial p_j} = \gamma_j >0,
\end{equation} so A4$'$ is satisfied. And it is also true that
\begin{equation} \label{e13.58}
\frac {\partial^2 \pi_i} {\partial p_i \partial c_i} = \kappa \beta_i > 0,
\end{equation}
so A5$'$ is satisfied  for $c_i$.

From equation (\ref{e.50}), $\frac{\partial \pi_i}{\partial p_i}$ is increasing
in $\kappa$,  so    $\pi_i(p_i, p_{-i}, \kappa  ) - \pi_i( p_i', p_{-i},  \kappa
) $  is nondecreasing in $\kappa$ for $p_i> p_i'$, and A5 is satisfied for
$\kappa$.

Thus, all the assumptions are satisfied.  This being the case,   a number of
theorems   can be applied, including the following two.

\noindent
{\bf Theorem  1.} {\it  If the game is supermodular, there exists a largest and
a smallest Nash equilibrium in pure strategies.}

 \noindent
 {\bf Theorem  2.} {\it  If the game is supermodular and assumption (A5) or
(A5$'$)  is satisfied, then the largest and smallest equilibrium are
nondecreasing functions of the parameter $z$.}

Applying Theorems  1 and  2 yields the following results for the differentiated
Bertrand game:

\noindent
1. There exists a largest and a smallest Nash equilibrium in pure strategies
(Theorem  1).

\noindent
2. The largest and smallest equilibrium prices for   firm $i$ are nondecreasing
functions of the cost parameters $c_i$  and $\kappa$ (Theorem  2).

Supermodularity, unlike the implicit function theorem,  has yielded comparative
statics on $\kappa$, the discrete exogenous variable. It yields weaker
comparative statics on $c_i$, however, because it just finds the effect of $c_i$
on $p_i^*$ to be nondecreasing, rather than telling us its value or whether it
is actually increasing.

For more on supermodularity, see Milgrom \& Roberts (1990),     Fudenberg \&
Tirole (1991, pp. 489-497), or  Vives's 2005  survey.

   \vspace{1in}

 \noindent
   {\bf   *14.5  Vertical Differentiation}

In previous sections of this chapter we have been looking at product
differentiaton, but differentiation in dimensions that cannot be called ``good''
versus ``bad''. Rather, location along a line is a matter of taste  and ``de
gustibus non est disputandum''.   Another form of product differentiation is
from better to worse, as analyzed in Shaked \& Sutton (1983) and around pages
150 and 296 of Tirole (1988).  Here, we will look at that in a simpler game in
which there are just two types of buyers and two levels  of quality, but we will
compare a monopoly to  a duopoly under various circumstances.



\begin{center}
{\bf   Vertical Differentiation I: Monopoly Quality Choice }
\end{center}
{\bf Players}\\
  A seller and a continuum of buyers.

\noindent
  {\bf The Order of Play }\\
 0  Nature assigns quality values to a continuum of buyers of length 1. Half  of
them  are ``weak'' buyers ($\theta=0$) who value high quality at 20 and low
quality at 10.  Half  of them  are ``strong'' buyers ($\theta=1$) who value high
quality at 50 and low quality at 15.       \\
1 The seller  picks    quality  $s$   to be either  $s_L=0$ or $s_H=1$. \\
 2 The seller  picks price   $p$   from the interval $[0,\infty)$.\\
 3 Each buyer    chooses one unit of  a good, or refrains from buying.  The
seller produces   at  constant marginal cost $c=1$, which does not vary with
quality. \\

\noindent
  {\bf Payoffs}\\
 \begin{equation} \label{e100}
 \pi_{seller} =  (p -1) q.
\end{equation}
and
\begin{equation} \label{e101}
 \pi_{buyer}  = (10 + 5\theta)  + (10+ 25\theta)  s - p.
 \end{equation}

  The seller  should clearly set the quality to be high,since then he can charge
more to the  buyer (though note that this runs contrary to a common
misimpression that a monopoly will result in lower quality than a competitive
market.)  The price should be either 50, which is the most the strong  buyers
would pay, or 20, the most the weak buyers would pay. Since $\pi (50) = 0.5
(50-1) = 24.5$ and $\pi (20) = 0.5 (20-1)+ 0.5 (20-1)=19$, the seller should
choose $p=50$.  Separation (by inducing only the strong buyer to buy) is better
for the seller than pooling.





Next we will allow the seller  to use two quality levels. A social planner would
just use one-- the maximal one of $s= s^*$--- since it is no cheaper to produce
lower quality. The monopoly seller might use two, however, because it   helps
him to price-discriminate.

     \begin{center}
 {\bf   Vertical Differentiation II: Crimping the Product  }
 \end{center}
{\bf Players}\\
 A seller and a continuum of buyers.

\noindent
  {\bf The Order of Play }\\
 0  Nature assigns quality values to a continuum of buyers of length 1. Half  of
them  are ``weak'' buyers ($\theta=0$) who value high quality at 20 and low
quality at 10.  Half  of them  are ``strong'' buyers ($\theta=1$) who value high
quality at 50 and low quality at 15.       \\
1 The seller  decides to sell both qualities      $s_L=0$  and $s_H=1$ or just
one of them. \\
 2 The seller  picks prices   $p_L$   and $p_H$ from the interval $[0,\infty)
$.\\
 3 Each buyer    chooses one unit of  a good, or refrains from buying.  The
seller produces   at  constant marginal cost $c=1$, which does not vary with
quality.

\noindent
  {\bf Payoffs}\\
 \begin{equation} \label{e102}
 \pi_{seller} =  (p_L -1) q_L + (p_H -1) q_H.
\end{equation}
 and
\begin{equation} \label{e103}
 \pi_{buyer}  = (10 + 5\theta)  + (10+ 25\theta)  s - p.
 \end{equation}



 This is a problem of mechanism design. The seller needs to pick  $p_1$, and
$p_2$ to satisfy incentive compatibility and participation constraints if he
wants to offer two qualities with positive sales of both, and he also needs to
decide if that is more profitable than offering just one quality.

We already solved the    one-quality problem  in Vertical Differentiation I,
yielding profit of 24.5.   The monopolist cannot simply add a second, low-
quality, low-price good for the weak buyers, because the strong buyers, who
derive zero payoff from the high-quality good, would switch to the low-quality
good, which would give them a positive payoff. In equilibrium, the monopolist
will have to give the strong buyers  a positive payoff. Their participation
constraint will be non-binding, as we have found so many times before for the
``good'' type.

   Following the usual pattern, the participation constraint for the weak buyers
will be binding, so $p_L=10$. The self-selection constraint for the strong
buyers will also be binding, so
 \begin{equation} \label{e104}
 \pi_{strong} (L)  = 15 - p_L   =  50 - p_H.
 \end{equation}
  Since $p_L=10$, this results in $p_H=45$.  The price for high quality must be
at least 35 higher than the price for low quality to induce separation of the
buyer types.

\noindent
 Profits will now be:
    \begin{equation} \label{e105}
 \pi_{seller} =  (10 -1)(0.5)  + (44-1)  (0.5)  = 26.
\end{equation}
 This exceeds the one-quality profit of 24.5, so it is optimal for the seller to
sell two qualities.

  This result, is, of course, dependent on the parameters chosen, but it is
nonetheless a fascinating special case, and one which is perhaps no more special
than the other special case, in which the seller finds that profits  are
maximized with just one quality. The outcome of allowing price discrimination is
a pareto improvement. The seller is better off,  because profit has risen from
24.5 to 26. The strong buyers are better off, because the price they pay has
fallen from 50 to 45. And the weak buyers are no worse off. In Vertical
Differentiation I their payoff was zero because they chose not to buy; in
Vertical Differentiation I their payoffs are zero because they buy at a price
exactly equal to their value for the good.

 Indeed, we can go further. Suppose the cost for the low-quality good was
actually {\it higher} than for the high-quality good,  e.g.,  $p_L=3$ and $p_H=
1$, because the good is normally
produced as high quality and needs to be purposely damaged before it becomes low
quality.  The price-discrimination profit in
(\ref{e105}) would then be $\pi_{seller} =  (10 -3)(0.5)  + (44-1)  (0.5)  =
25$. Since that is still higher than 24.5, the seller would still price-
discriminate. The buyers' payoffs would be unaffected. Thus, allowing the seller
to damage some of the good at a cost in real resources of 2 per unit, converting
it from high to low quality,  can result in a pareto improvement!

This is the point made in
 Deneckere \& McAfee (1996), which illustrates the theory   with  real-world
examples  of computer chips and   printers purposely damaged  to
allow price discrimination. See too
 McAfee (2002, p. 265), which  tells us,  for example, that Sony made two sizes
of minidisc
in 2002,  a 60-minute and a 74-minute version. Production of both starts with a
capacity of 74 minutes, but   Sony added code to the 60-minute disc to make 14
minutes of it unusable. That code is an  extra fixed cost, but IBM's  1990
Laserprinter E is an example of a damaged product with extra marginal cost. The
Laserprinter E was a   version of the original  Laserprinter that was only half
as fast. The reason? IBM added five extra chips to the Laserprinter E to slow it
down.



   \bigskip

We will analyze one more version of the product differentiation game: with two
sellers instead of one. This will show how the  product differentiation which
increases profits in the way we have seen in the Hotelling games   can occur
vertically as well as horizontally.

\begin{center}
{\bf   Vertical Differentiation III: Duopoly Quality Choice }
  \end{center}
   {\bf Players}\\
 Two  sellers and a continuum of buyers.

 \noindent
  {\bf The Order of Play }\\
 0  Nature assigns quality values to a continuum of buyers of length 1. Half  of
them  are ``weak'' buyers ($\theta=0$) who value high quality at 20 and low
quality at 10.  Half  of them  are ``strong'' buyers ($\theta=1$) who value high
quality at 50 and low quality at 15.       \\
  1 Sellers 1 and 2 simultaneously choose values for   $s_1$ and $s_2$ from  the
set $\{    s_L=0, s_H=1 \}$. They may both choose the same value.  \\
 2 Sellers 1 and 2 simultaneously choose  prices   $p_1$   and $p_2$ from the
interval $[0,\infty)$.\\
 3 Each buyer    chooses one unit of  a good, or refrains from buying.  The
sellers produce    at  constant marginal cost $c=1$, which does not vary with
quality. \\

\noindent
  {\bf Payoffs}\\
 \begin{equation} \label{e106}
 \pi_{seller} =  (p  -1) q
\end{equation}
 and
\begin{equation} \label{e107}
 \pi_{buyer}  = (10 + 5 \theta)  + (10+ 25 \theta)  s - p.
 \end{equation}

  If both sellers both choose the same quality level, their profits will be
zero, but if they choose different quality levels, profits will be positive.
Thus, there are three possible equilibria in the quality stage of the game:
(Low, High), (High, Low), and a symmetric mixed-strategy equilibrium.   Let us
consider the pure-strategy equilibria first, and  without loss of generality
suppose that Seller 1 is the low-quality seller  and Seller 2 is the high-
quality seller.

\noindent
(1) The equilibrium prices of Vertical Differentiation II, $(p_L=10, p_H=45$),
will no longer be equilibrium prices.  The problem is that the low-quality
seller would deviate to $p_L=9$, doubling his sales for a small reduction in
price.

\noindent
 (2) Indeed, there is no pure-strategy equilibrium in prices. We have seen that
$(p_L=10, p_H=45$) is not an equilibrium, even though $p_H=45$ is the high-
quality seller's best response to $p_L=10$. $P_L >10$ will attract no buyers, so
that cannot be part of an equilibrium.   Suppose $P_L \in (1, 10)$.  The
response of the high-quality seller will be to set $p_H=p_L+35$, in which case
the low-quality seller can increase his profits by slightly reducing $p_L$ and
doubling his sales. The only   price  left  for the low-quality seller that does
not generate negative profits is $p_L=1$, but that yields zero profits, and so
is worse than $p_L=10$.  So no choice of $p_L$ is part of a pure-strategy
equilibrium.

\noindent
(3)  As always, an equilibrium does exist, so it must be in mixed strategies, as
shown below.

\bigskip
\noindent
  {\bf The Asymmetric  Equilibrium: Pure Strategies for Quality, Mixed for
Price}


\noindent
  The low-quality  seller picks $p_L$   on the support $[5.5, 10]$ using the
cumulative
distribution
\begin{equation} \label{e108}
  F(p_L) =     1-  \left( \frac{39.5}{  p_L +34 }\right)
\end{equation}
   with an atom of probability $  \frac{39.5}{ 44} $ at $p_L=10$.

\noindent
The  high-quality seller picks $p_H$ on the support $[40.5, 45]$ using the
cumulative
distribution
\begin{equation} \label{e109}
 G(p_H) = 2-  \left(\frac{9}{ p_H-36 }  \right)
\end{equation}

\noindent
Weak buyers    from  the low-quality seller if  $10-p_L \geq 20 - p_H$, which is
always true in equilibrium.  Strong buyers buy from the low-quality seller if
$15-p_L > 50 - p_H$, which has positive probability, and otherwise
from the
high-quality seller.

\bigskip

  This equilibrium is noteworthy because it includes a probability atom in the
mixed-strategy distribution, something not uncommon in pricing games. The low-
quality seller usually chooses $p_L=10$, but with some probability he mixes
between 5.5 and 10.  The intuition for why this  happens is   that  for the low-
quality seller the weak buyers are ``safe'' customers, for whom the monopoly
price is 10, but unless the low-quality seller chooses to shade the price with
some probability to try to attract the strong customers, the high-quality seller
will maintain such a high price ($p_H=45$) as to make such shading irresistable.


 To start deriving this equilibrium,
let us  conjecture  that   the low-quality seller will  not include any prices
above 10 in his mixing support but will include $p_L=10$ itself. That is
plausible  because  he
would lose all the low-quality buyers at prices above 10, but   $p_L=10$ yields
maximal
profits whenever $p_H$ is low enough that only weak consumers buy low quality.

 The low-quality seller's profit from $p_L=10$ is
$\pi_L (p=10) = 0.5 (10-1) = 4.5. $ Thus, the lower bound   of the support
of his mixing distribution (denote it by $a_L$) must also yield a profit of 4.5.
There is no point in charging a price less than the price which would capture
even  the
  strong consumers with probability one,   in which case
 \begin{equation} \label{e110}
 \pi_L (a_L) = 0.5 (a_L-1) + 0.5 (a_L-1)  =4.5,
\end{equation}
 and  $a_L=5.5$.
Thus, the   low-quality seller mixes on $[5.5, 10]$.

On that mixing support, the low-quality seller's profit must equal 4.5 for any
price. Thus,
 \begin{equation} \label{e111}
\begin{array}{ll}
 \pi_L (p_L)   =  4.5& = 0.5 (p_L-1) + 0.5 (p_L-1) Prob (15-p_L> 50 - p_H) \\
 & \\
 &= 0.5 (p_L-1) + 0.5 (p_L-1) Prob  (p_H > 35+ p_L) \\
 & \\
 & =  0.5 (p_L-1) + 0.5 (p_L-1)[ 1-G(  35+ p_L)]
\end{array}
\end{equation}
 Thus, the $G(p_H)$ function   is such that
 \begin{equation} \label{e112}
   1-G(  35+ p_L) = \frac{4.5}{0.5(p_L-1)} -1
\end{equation}
and
 \begin{equation} \label{e113}
 G(35+ p_L) = 2- \left(\frac{4.5}{0.5(p_L-1)} \right).
\end{equation}
   We want a $G$ function with the argument $p_H$, not $(35+ p_L)$, so let's
 \begin{equation} \label{e114}
G(p_H)  = 2-\left( \frac{4.5}{0.5([p_H-35] -1)}\right) = 2-\left( \frac{9}{
p_H-36}\right).
\end{equation}
  As explained in Chapter 3, what we have just done is to find the strategy for
the high-quality seller  that makes the low-quality seller indifferent among all
the values of $p_L$ in his mixing support.

We can find the support of the high-quality seller's mixing distribution by
finding values $a_H$ and $b_H$ such that $G(a_H) =0$ and $G(b_H) =1$, so
\begin{equation} \label{e115}
 G(a_H) = 2-\left( \frac{9}{ a_H-36 }\right) =0,
\end{equation}
which yields $a_H=40.5$,  and
\begin{equation} \label{e116}
 G(b_H) = 2- \left(\frac{9}{(0) \cdot b_H-36 }\right) =1,
\end{equation}
which yields $b_H = 45$.  Thus the support of the high-quality seller's mixing
distribution is $[40.5, 45]$.

 Now let us find the low-quality seller's mixing distribution, $F(p_L)$. At
$p_H=40.5$, the high-quality seller  has zero probability of losing the strong
buyers to the low-quality seller, so  his profit is $0.5(40.5-1) = 19.75. $ Now
comes  the tricky step. At $p_h=45$, if  the high-quality seller had
probability one  of losing the strong buyers to the low-quality seller, his his
profit would be zero, and he would strictly prefer $p_H=40.5$. Thus, it must be
that at  $p_h=45$  there is strictly positive probability that $p_L=10$--- not
just a positive density.      So let us continue, using our finding that the
profit of the high-quality seller must be 19.75 from any price in the mixing
support. Then,
 \begin{equation} \label{e117}
\begin{array}{ll}
 \pi_H (p_H)   = 19.75& =   0.5 (p_H-1) Prob (15-p_L< 50 - p_H) \\
 & \\
 &=  0.5 (p_H-1) Prob (p_H-35< p_L  ) \\
 & \\
 & =   0.5 (p_H-1) [1-F( p_H-35   )] \\
\end{array}
\end{equation}
so
 \begin{equation} \label{e118}
 F( p_H-35) = 1- \left( \frac{19.75}{0.5(p_H-1)}\right).
\end{equation}
  Using the same substitution trick as in equation (\ref{e114}),  putting
$p_L$  instead of   $(p_H-35)$ as  the  argument for $F$, we get
\begin{equation} \label{e119}
\begin{array}{ll}
   F(p_L) &=   1- \left(\frac{19.75}{0.5(p_L+35-1)}\right)
  = 1- \left(\frac{39.5}{ p_L+34 } \right)\\
\end{array}
\end{equation}
   In particular, note that
 \begin{equation} \label{e120}
 F(5.5) =  1- \left(\frac{39.5}{ 5.5+34 }\right) =0,
\end{equation}
 confirming our earlier finding that  the minimum $p_L$ used is 5.5, and
\begin{equation} \label{e121}
 F(10) =  1- \left(\frac{39.5}{ 10+34 }\right) = 1-\frac{39.5}{44} <1.
\end{equation}
   Equation (\ref{e121}) shows that at the upper bound of the low-quality
seller's mixing support  the
cumulative mixing  distribution   does not equal 1, an oddity we
usually do not see in mixing distributions. What it implies is that there is an
atom of probability at  $p_L=10$,  soaking up
all the remaining probability beyond what equation (\ref{e121}) yields for the
prices below 10. The atom must equal     $\frac{39.5}{44} \approx 0.9$.

  Happily, this solves our paradox of zero high-quality seller profit at $p_H=
45$. If $p_L=10$ has probability $\frac{39.5}{44} $, the profit from $p_H=45$ is
$0.5 (\frac{39.5}{44} )(45-1) = 19.75.  $   Thus,
the profit from $p_H=45$ is the same as from $p_H=40.5$, and the seller is
willing to mix between them.



  One of the technical lessons of Chapter 3 was that if your attempt to
calculate mixing probability results in probabilities of less than zero or more
than one, then probably  the  equilibrium is not in mixed strategies (algebra
mistakes being another possibility). The lesson here is that if your attempt to
calculate the support of a mixing distribution results in impossible bounds,
then  you should consider the possibility that  the distribution has atoms of
probability.

\bigskip

 The  duopoly sellers' profits are  4.5 (for low-quality) and 19.75 (for high
quality) in the asymmetric equilibrium of
Vertical Differentiation III,  a total of 24.25 for the industry.   This is less
than either the 24.5 earned by the nondiscriminating  monopolist of Vertical
Differentiation I or  the 26 earned by the discriminating monopolist of Vertical
Differentiation II.  But what about the  mixed-strategy equilibrium for Vertical
Differentiation III?


\bigskip
\noindent
  {\bf The Symmetric  Equilibrium: Mixed Strategies for Both  Quality and Price}

Each player chooses low quality with probability $\alpha = 4.5/24.25$ and high
quality otherwise. If they choose the same quality, they next  both choose a
price equal to 1, marginal cost. If they choose different qualities, they choose
prices according to the mixing distributions in the asymmetric equilibrium.

    This equilibrium is easier to explain. Working back from the end, if they
choose the same qualities, the two firms are in undifferentiated  price
competition and will choose prices equal to marginal cost, with  payoffs of
zero. If they choose different qualities, they are in the same situation as they
would be in the asymmetric equilibrium, with expected  payoffs of 4.5 for the
low-quality firm and 19.75 for the high-quality firm. As for choice of product
quality, the expected payoffs from each quality must be equal in equilibrium, so
there must be a higher probability of both choosing high-quality:
\begin{equation} \label{e123}
 \pi(Low)  =   \alpha (0)    + (1-\alpha) 4.5   = \pi(High) =  \alpha (19.75)
+ (1-\alpha)  (0).
\end{equation}
Solving equation (\ref{e123}) yields $\alpha = 4.5/24.25 \approx 0.17$, in which
case each player's payoff is  about 3.75.
Thus, even if a player is stuck in the role of low-quality seller in the pure-
strategy equilibrium,  with an expected payoff of 4.5, that is better than the
expected  payoff he
would get in the ``fairer'' symmetric equilibrium.


\bigskip

We can conclude that if the players could somehow arrange what equilibrium would
be played out, they would arrange for a pure-strategy equilibrium, perhaps by
use of cheap talk and some random focal point variable.


 Or, perhaps they could change the rules of the game so that they would choose
qualities sequentially.   Suppose one  seller  gets to choose
quality first. He would of course choose high quality, for a payoff of 19.75.
The second-mover, hwoever,
choosing low-quality, would  have  a payoff of 4.5,   better than the expected
payoff
in the symmetric mixed-strategy equilibrium of the simultaneous quality-choice
game. This is the same phenomenon as the pareto superiority of a sequential
version of the Battle of the Sexes over the symmetric mixed-strategy equilibrium
of the simultaneous-move game.

What if Seller 1 chooses both quality and price first, and   Seller 2 responds
with   quality and price? If Seller 1 chooses low quality,  then his optimal
price is $p_L=10$, since the second player will choose high quality and  a price
low enough to attract the strong buyers--- $p_H=45$, in equilibrium--- so Seller
1's payoff would be 0.5(10-1) = 4.5. If Seller 1 chooses high quality,  then his
optimal price is $p_H=40.5$, since the second player will choose low  quality
and would choose a price high enough to lure away the strong buyers if
$p_H<40.5$. If, however, $p_H=40.5$, Seller 2 would give up on attracting the
strong buyers and pick $p_L=10$. Thus, if Seller 1 chooses both quality and
price first, he will choose high quality and $p_H=40.5$ while Seller 2 will
choose low quality and $p_L=10$, resulting in the same payoffs  as in the
asymmetric equilibrium of the simultaneous-move game, though no longer in mixed
strategies.


  What Product Differentiation III shows us is that product differentiation can
take place  in oligopoly vertically as well as horizontally. Head-to-head
competition reduces profits, so firms will try to differentiate in any way that
they can. This increases  their profits, but it can also benefit consumers---
though  more obviously in the case of  horizontal differentiation than in
vertical. Keep in mind, though, that in our games here we have assumed that high
quality costs no more than low quality. Usually high quality is more expensive,
which means that having more than one quality level can be efficient. Often poor
people prefer  lower quality, given the cost of higher quality, and even a
social planner would provide a variety of quality levels. Here, we see that even
when only high quality would be provided in the first-best, it is better that a
monopolist provide two qualities than one, and a duopoly is still better for
consumers.




   \vspace{1in}

\noindent
 {\bf   *14.6 Durable Monopoly}

 \noindent
    Introductory economics  courses are   vague on the issue of the time period
over which transactions take place. When a   diagram    shows the supply and
demand for widgets, the $x$-axis is labelled ``widgets,'' not ``widgets per
week'' or ``widgets per year.''  Also, the diagram splits off one time period
from future time periods, using the implicit assumption that supply and demand
in one period is unaffected by events of future periods. One problem with this
on the demand side is that the purchase of a good which lasts for more than one
use  is an investment;  although the price is paid now, the utility from the
good continues into the future. If Smith buys a house, he is buying not just the
right to live in the house tomorrow, but the right to live in it for many years
to come, or even to  live in it for a few years and then sell the remaining
years to someone else.  The continuing  utility he receives from this durable
good is called its {\bf service flow}.    Even though he may not intend to rent
out the house, it is an investment decision for him because it trades off
present  expenditure  for future utility.    Since  even a shirt produces a
service flow over more than an instant of time, the durability of goods presents
difficult definitional problems for national income accounts. Houses are counted
as part of national investment (and an estimate of their service flow as part of
services consumption),  automobiles as durable goods consumption, and shirts as
nondurable goods consumption, but all are  to some extent durable investments.

  In microeconomic theory,     ``durable monopoly'' refers not to monopolies
that last a long time, but to monopolies that sell durable goods.  These present
a curious problem.        When a monopolist sells something like a refrigerator
to a consumer, that consumer drops out of the market   until the refrigerator
wears out. The demand curve is, therefore, changing over time as a result of the
monopolist's choice of price, which means that the modeller  should not make his
decisions in one period  and ignore future periods.  Demand is not {\bf time
separable}, because a rise in price at time $t_1$ affects the quantity demanded
at time $t_2$.

  The  durable monopolist has a special problem because in a sense he does have
a competitor -- himself in the later periods. If he were to set  a high price in
the first period, thereby removing   high-demand buyers from the market, he
would be tempted to set a lower price in the next period to take advantage of
the remaining consumers. But if it were  expected that he would lower the price,
the high-demand buyers would not buy at a high price in the first period.  The
threat of the future low price forces the monopolist to keep his current price
low.

    This presents another aspect of product differentiation:  the durability of
a good. Will a monopolist produce a shoddier, less durable product? Durability
is different from the vertical differentiation we have already analyzed because
durability has temporal implications. The buyer of a less durable product will
return to the market sooner than the buyer of a more durable one, regardless of
other aspects of product quality.

 To formalize this situation,  let  the   seller have a monopoly on a durable
good  which lasts two periods. He must set a price for each period, and the
buyer must decide what quantity  to buy in each period.  Because this one buyer
is meant to represent the entire market demand, the moves are ordered so that he
has no market power, as in the principal-agent models in Chapter 7 and onwards.
Alternatively, the buyer can be viewed as representing a continuum of consumers
(see Coase [1972] and Bulow [1982]).  In this interpretation,  instead of  ``the
buyer'' buying  $q_1$ in  the  first period, $q_1$ of the buyers each buy one
unit in the first period.

\begin{center}
{\bf   Durable Monopoly }
\end{center}
 {\bf Players}\\
 A buyer and a seller.

\noindent {\bf The Order of Play }\\
 1 The seller picks  the first-period    price, $p_1$.\\
 2 The buyer buys quantity  $q_1$ and consumes service flow $q_1$.\\ 3 The
seller picks  the second-period price, $p_2$.\\
 4 The buyer buys   additional quantity  $q_2$ and consumes service flow
$(q_1+q_2)$.

\noindent
  {\bf Payoffs}\\
 Production cost is zero and there is no discounting. The seller's payoff is his
revenue, and the buyer's payoff is the sum across periods of his benefits  from
consumption minus his expenditure.  The buyer's  benefits arise from his being
willing to pay as much as
 \begin{equation} \label{e13.64}
  B(q_t) = 60 - \frac{q_t}{2}
 \end{equation}
 for the marginal unit  service flow  consumed in period $t$, as shown in Figure
10. The payoffs are therefore
 \begin{equation} \label{e13.65}
 \begin{array}{lllr}
 \pi_{seller} & = & q_1 p_1 + q_2p_2  &  \\
 \end{array}
\end{equation}
  and, since a consumer's total benefit is the sum of a triangle plus a
rectangle of benefit, as shown in Figure 10,
 \begin{equation}\label{e13.66}
 \begin{array}{lllr}
  \pi_{buyer} & =& [consumer \;surplus_1] + [consumer \;surplus_2]\\ & & & \\ &
=& [total \;benefit_1 - expenditure_1]+  [total \;benefit_2 - expenditure_2] &\\
 & & & \\
 & =& \left[ \left( \frac{(60-B(q_1))q_1}{2}   + B(q_1)q_1 \right) - p_1q_1
\right] &\\
 & & & \\
 & & + \left[ \left(\frac{60-B(q_1 + q_2)}{2} \left( q_1 + q_2 \right) +
B(q_1+q_2)(q_1+q_2) \right)  - p_2q_2 \right] &
\end{array}
\end{equation}

 Thinking about durable monopoly is hard   because we are used to one-period
models in which the demand curve, which relates the price to the quantity
demanded, is identical to  the marginal-benefit curve, which relates the
marginal benefit to the quantity consumed.  Here, the two curves are different.
The marginal benefit curve is the same each period,  since it is part of the
rules of the game, relating consumption to utility.   The  demand curve will
change over time and depends on the equilibrium strategies, depending as it does
on   the number of  periods left in which to consume the good's services,
expected future prices, and the quantity already owned. Marginal benefit is a
given for the buyer; quantity demanded is his strategy.

   The  buyer's  total benefit  in period 1 is the dollar value of his utility
from his purchase of $q_1$, which equals the  amount he would have been willing
to pay to rent $q_1$. This is composed of the two areas shown in Figure 11a, the
upper triangle of area $ \left(\frac{1}{2} \right)\left( q_1 + q_2 \right)
\left( 60-B(q_1 + q_2) \right)  $ and the lower rectangle of area $(q_1+q_2)
B(q_1+q_2)$. From this must be subtracted his expenditure in period 1,
$p_1q_1$,  to obtain what we might call his consumer surplus in the first
period. Note that $p_1 q_1$ will not be the lower rectangle, unless by some
strange accident, and the ``consumer   surplus'' might easily  be negative,
since the expenditure in period  1 will also yield utility in period 2 because
the good is durable.

\includegraphics[width=150mm]{fig14-11.jpg}

\begin{center}
 {\bf Figure  11:  The Buyer's Marginal Benefit per Period in  Durable Monopoly}
\end{center}

   To find the equilibrium price path one cannot simply differentiate the
seller's utility with respect to $p_1$ and $p_2$, because that would violate
the sequential rationality of the seller and the rational response  of the
buyer.  Instead, one must look for a subgame perfect equilibrium, which means
starting in the second period and discovering how much the buyer would purchase
given his first-period purchase of $q_1$, and what  second-period price the
seller would charge given the buyer's second-period demand function.

   In the first period, the marginal unit  consumed was the $q_1 -th.$ In the
second period, it will be the $(q_1+q_2) -th$.   The residual demand curve
after the first period's purchases is shown in Figure  11b. It is a demand curve
very much like the demand curve resulting from intensity rationing in the
capacity-constrained Bertrand game of Section 14.2, as shown in Figure 11a. The
most intense portion of the buyer's demand, up to $q_1$ units, has already been
satisfied, and what is left begins with a marginal benefit of $B(q_1)$,  and
falls at the same slope as the original marginal benefit curve. The equation for
the residual demand is therefore, using  equation (\ref{e13.64}),
  \begin{equation}\label{e13.67}
p_2    = B(q_1) - \frac{q_2}{2} = 60 - \left(\frac{1}{2}\right) q_1  -\left(
\frac{1}{2} \right) q_2 .
 \end{equation}
Solving for the monopoly quantity, $q_2^*$, the seller  maximizes $q_2p_2$,
solving the problem
\begin{equation}\label{e13.68}
\stackrel{Maximize}{q_2}  q_2\left(60 - \left(\frac{1 }{2}\right)  \left( q_1 +
q_2  \right)\right),
 \end{equation}
which generates the first-order condition
\begin{equation}\label{e13.56}
60 - q_2 - \left(\frac{1}{2} \right) q_1 = 0,
\end{equation}
 so that
\begin{equation}\label{e13.70}
 q_2^* = 60 - \left( \frac{1}{2}\right) q_1.
 \end{equation}
 From equations (\ref{e13.67}) and (\ref{e13.70}), it can be seen that $ p_2^* =
30 -q_1/4$.

   We must now find $q_1^*$.  In period one,  the buyer looks ahead to the
possibility of buying in period two at a lower price. Buying in the first period
has two benefits: consumption of the service flow in the first period and
consumption of the service flow in the second period.  The price he would pay
for a unit in period one cannot exceed the marginal benefit from the first-
period service flow in period one plus the foreseen value of $p_2$, which from
(\ref{e13.70})  is $30-q_1/4$. If the seller chooses to sell $q_1$ in the first
period, therefore,   he can do so at the price
 \begin{equation}\label{e13.71}
  \begin{array}{ll}
 p_1 (q_1)&= B(q_1) + p_2  \\ & \\ & = ( 60 - \left(\frac{1}{2}\right)q_1) +( 30
- \left( \frac{1}{4}\right)q_1),\\ & \\ &= 90 -  \left( \frac{3}{4}\right)q_1.\\
\end{array}
\end{equation}
 Knowing that in the second period he will choose $q_2$ according to
(\ref{e13.70}), the seller combines (\ref{e13.70})  with  (\ref{e13.71})  to
give the maximand in the problem of choosing $q_1$ to maximize profit over the
two periods, which is
\begin{equation}\label{e13.72}
 \begin{array}{ll}
  \pi_{seller}= \left( p_1 q_1 + p_2q_2 \right)  & = \left(90 - \frac{3q_1}{4}
\right)q_1 + \left( 30 - \frac{q_1}{4} \right)(60 - \frac{q_1}{2} )\\
  & \\
 & = 1800 + 60q_1 - \frac{5q_1^2}{8},
  \end{array}
 \end{equation}
  which has the first-order condition
 \begin{equation}\label{e13.73}
 60 - \frac{5q_1}{4}  = 0,
 \end{equation}
 so that
\begin{equation}\label{e13.74}
 q_1^* = 48
  \end{equation} and, making use of (\ref{e13.71}), $p_1^* = 54$.

  It follows from  (\ref{e13.70})  that $q_2^*=36$ and $p_2 = 18.$ The seller's
profits over the two periods are  $\pi_s = 3,240$ ($ = 54(48) + 18(36))$.

    The purpose of these calculations is to compare the situation with three
other market structures: a competitive market, a monopolist who rents instead of
selling, and a monopolist who commits to selling only in the first period.

 A {\it competitive market} bids down the price to the marginal cost of zero.
Then,  $p_1 = 0$ and $q_1 = 120$ from (\ref{e13.64}) because buyers buy till
their marginal benefit is zero, and profits equal zero also.

  If the monopolist {\it rents } instead of selling, then equation
(\ref{e13.64}) is like an ordinary demand equation, because the monopolist is
effectively selling the good's services separately each period. He could rent a
quantity of 60 each period at a rental fee of $30$ and his profits would sum to
$\pi_s = 3,600$.  That is higher than 3,240, so profits are higher from renting
than from selling outright.    The problem  with selling outright  is that the
first-period price cannot be very high or  the buyer knows that the seller will
be tempted to lower the price once the buyer has bought in the first period.
Renting avoids this problem.

  If the monopolist can {\it commit to not producing in the second period}, he
will do just as well as the monopolist who rents, since he can sell a quantity
of 60 at a price of 60, the sum of the rents for the two periods.   An example
is the artist who breaks the plates for his engravings after a production run of
announced size.   We must also assume that the artist can convince the market
that he has broken the plates.  People joke that the best way an artist can
increase the value of his work is by dying, and that, too, fits the  model.

 If the modeller ignored sequential rationality and simply looked for the Nash
equilibrium that maximized the payoff of the seller by his choice of $p_1$ and
$p_2$, he would come to the commitment result. An example of such an equilibrium
is ($p_1=60$, $p_2=200$, {\it Buyer purchases according to $q_1 = 120-p_1$, and
$q_2=0$}). This is Nash because neither player has incentive to deviate given
the other's strategy, but it fails to be subgame perfect, because the  seller
should realize that if he deviates and  chooses a  lower price once the second
period is reached, the buyer will respond by  deviating from $q_2=0$ and will
buy more units.

 With more than two periods, the difficulties of the durable-goods monopolist
become even more striking. In an infinite-period model without discounting, if
the marginal cost of production is zero, the equilibrium price for outright sale
instead of renting is constant -- at zero!  Think about this in the context of a
model with many buyers.  Early consumers foresee that the monopolist has an
incentive to cut the price after they buy, in order to sell to the remaining
consumers who value the product less. In fact, the monopolist would continue to
cut the price and sell more and more units to consumers with weaker and weaker
demand until the price fell to marginal cost.  Without discounting, even the
high-valuation consumers refuse to buy at a high price, because they know they
could wait until the price falls to zero. And this is not a trick of infinity: a
large number of periods generates a price close to zero.

  We can also use the durable monopoly model to think about the durability of
the product.  If the seller can develop a product so flimsy that it only lasts
one period, that is equivalent to renting. A consumer is willing to pay the same
price to own a one-hoss shay that he knows will break down in one year as he
would pay to rent it for a year.  Low durability leads to the same output and
profits as renting, which explains why a firm with market power might produce
goods that wear out quickly. The explanation is not that the monopolist can use
his market power to inflict lower quality on consumers-- after all, the price he
receives is lower too-- but that the lower durability makes it credible to high-
valuation buyers that the seller expects their business in the future and will
not reduce his price.

\vspace * {1in}

    With durable-goods monopoly,  this book is concluded. Is this book itself a
durable good?  As I am now writing its fourth edition, I cannot say that it is
perfectly durable, because it has improved with each edition, and I can honestly
say that a  rational consumer who liked the first edition should have bought
each successive edition. If you can benefit from this book, your time is
valuable enough that you should substitute book reading for solitary thinking
even at the expensive prices my publisher and I charge.

Yet although I have added new material, and improved my presentation  of the old
material, the
basic ideas remain the same. The central idea is that in modern economic
modelling
the modeller starts by thinking about players, actions, information, and
payoffs, stripping a situation down to its essentials. Having done that, he sees
what payoff-maximizing  equilibrium  behavior arises from the assumptions.  This
book teaches a  variety of  common ways that   assumptions  link  to conclusions
just as a book on
chess strategy teaches how  variety of common configurations of a chessboard
lead to winning or losing. Just
as with a book on chess, however, the important thing is not just to know common
tricks and simplifications, but  to be able to recognize
the general features of a situation and know what tricks to apply.  Chess is
just a game, but game
theory, I hope, will provide you with tools for improving your life and the
policies you  recommend to  others.




\newpage

\begin{small}

\bigskip \noindent
  {\bf Notes}

\noindent
{\bf N14.1} {\bf Quantities as Strategies: The Cournot Equilibrium Revisited}
\begin{itemize}
\item Articles on the existence and uniqueness of a pure-strategy equilibrium in
the Cournot model include   Roberts \& Sonnenschein (1976), Novshek (1985), and
  Gaudet \& Salant (1991).

 \item
{\bf Merger in a Cournot model.} A problem with the Cournot model is that a
firm's best policy is often to split up into separate firms. Apex gets half the
industry profits in a duopoly game.  If Apex split into firms $Apex_1$ and
$Apex_2$, it would get two thirds of the profit in the Cournot triopoly game,
even though industry profit falls.

$\;\;\;$ This point was made by Salant, Switzer  \& Reynolds (1983)  and is the
subject of problem 14.2. It is interesting that nobody noted this earlier, given
the intense interest in Cournot models. The insight comes from approaching the
problem from asking whether a player could improve his lot if his strategy space
were expanded in reasonable ways.

\item
An ingenious look at how the number of firms in a market affects the price is
Bresnahan \& Reiss (1991), which looks  empirically at a number of  very small
markets with one, two, three or more competing firms. They find a big decline in
the price from one to two firms, a smaller decline from two to three, and not
much change thereafter.

Exemplifying theory, as discussed in the Introduction to this book, lends itself
to explaining particular cases, but it is much less useful for making
generalizations across industries.  Empirical work associated with exemplifying
theory tends to  consist of historical anecdote rather than the linear
regressions to which economics has become accustomed.  Generalization and
econometrics are still often useful in industrial organization, however, as
Bresnahan \& Reiss (1991)   shows. The most ambitious attempt to  connect
general data with the modern theory of industrial organization is Sutton's 1991
book, {\it Sunk Costs and Market Structure}, which is an extraordinarily well-
balanced mix of theory, history, and numerical data.

  \end{itemize}

\bigskip

\noindent
 {\bf N14.2} {\bf Prices as Strategies:  The  Bertrand Equilibrium}
\begin{itemize}
 \item
As Morrison (1998) points out,   Cournot actually does (in Chapter 7)  analyze
the case of  price competition with imperfect substitutes, as well as the
quantity competition that bears his name.  It is convenient to continue to
contrast ``Bertrand'' and ``Cournot'' competition, however, though  a case can
be made for  simplifying terminology to  ``price'' and ``quantity'' competition
instead.  For the history of how the Bertrand name came to be attached to price
competition, see Dimand \& Dore (1999).

 \item Intensity rationing has also been called {\bf efficient rationing}.
Sometimes, however,   this rationing rule is inefficient. Some low-intensity
consumers left facing the high price decide not to buy the product even though
their benefit is greater than its marginal cost. The reason intensity rationing
has been thought to be efficient is that it is efficient if the rationed-out
consumers are unable to buy at any price.

 \item
 OPEC has tried both price and quantity controls (``OPEC, Seeking Flexibility,
May Choose Not to Set Oil Prices, but to Fix Output,'' {\it Wall Street
Journal}, October 8, 1987, p. 2; ``Saudi King Fahd is Urged by Aides To Link Oil
Prices to Spot Markets,'' {\it Wall Street Journal}, October 7, 1987, p. 2).
Weitzman (1974) is the classic reference on price versus quantity control by
regulators, although he does not use the context of oligopoly.  The decision
rests partly on enforceability, and OPEC has also hired accounting firms to
monitor prices (``Dutch Accountants Take On a Formidable Task: Ferreting Out
`Cheaters' in the Ranks of OPEC,'' {\it Wall Street Journal}, February 26, 1985,
p.  39).

\item Kreps \& Scheinkman (1983) show how capacity choice and Bertrand pricing
can lead to a Cournot outcome.  Two firms face downward-sloping market demand.
In the first stage of the game, they simultaneously choose capacities, and in
the second stage they simultaneously choose prices (possibly by mixed
strategies).  If a firm cannot satisfy the demand facing it in the second stage
(because of the capacity limit), it uses intensity rationing  (the results
depend on this).   The unique subgame perfect equilibrium is for each firm to
choose the Cournot capacity and price.

   \item Haltiwanger \& Waldman (1991) have suggested a dichotomy applicable to
many different games between players who are {\bf responders}, choosing their
actions flexibly, and those who are {\bf nonresponders}, who are inflexible. A
player might be a nonresponder because he is irrational, because he moves first,
or simply because his strategy set is small. The categories are used in a second
dichotomy, between games exhibiting {\bf synergism}, in which responders choose
to do whatever the majority do (upward sloping reaction curves), and games
exhibiting {\bf congestion}, in which responders want to join the minority
(downward sloping reaction curves).  Under synergism, the equilibrium is more
like what it would be if all the players were nonresponders; under congestion,
the responders have more influence.  Haltiwanger and  Waldman apply the
dichotomies to network externalities, efficiency wages, and reputation.



 \item
There are many ways to specify product differentation.  This chapter looks at
horizontal differentiation where   all consumers agree that products  A and B
are more alike than A and C, but they disagree as to which is best.   Another
way horizontal differentiation might work is for  each consumer to like a
particular product best, but to consider all others as equivalent. See Dixit \&
Stiglitz (1977) for a model along those lines. Or, differentiation might be
vertical: all consumers agree that  A is better than B and B is better than C
but they disagree as to how {\it much } better A is than B.    Firms therefore
offer different qualities at different prices.  Shaked \& Sutton (1983) have
explored this kind of vertical differentation.

   \end{itemize}

\bigskip
 \noindent
{\bf N14.3} {\bf Location models}
 \begin{itemize}
  \item For a booklength treatment of location models, see Greenhut \& Ohta
(1975).

 \item
Vickrey notes the  possible absence of a pure-strategy equilibrium in
Hotelling's model in pp.323-324 of his 1964 book {\it Microstatics}.
D'Aspremont, Gabszewicz \& Thirse (1979) work out  the  mixed-strategy
equilibrium for the case of quadratic transportation costs, and       Osborne \&
Pitchik (1987)  do the same for Hotelling's original model.

  \item Location models and switching cost models are attempts to go beyond the
notion of a market price.  Antitrust cases are good sources for descriptions of
the complexities of pricing in particular markets. See, for example, Sultan's
1974 book on electrical equipment in the 1950s, or antitrust opinions such as
{\it US v. Addyston Pipe \& Steel Co.}, 85 F. 271 (1898).

\item
It is important in location models whether the positions of the players on the
line are moveable. See, for example, Lane (1980).

\item
The location games in this chapter model use a one-dimensional space with end
points, i.e., a line segment. Another kind of one-dimensional space is a circle
(not to be confused with a disk). The difference is that no point on a circle is
distinctive, so no consumer preference can be called extreme.  It is, if you
like, Peoria versus Berkeley.  The circle might be used for modelling
convenience or because it fits a situation: e.g., airline flights spread over
the 24 hours of the day.  With two players, the Hotelling location game on a
circle has a continuum of pure-strategy equilibria that are one of two types:
both players locating at the same spot, versus players separated from each other
by 180$^\circ$. The three-player model also has a continuum of pure-strategy
equilibria, each player separated from another by 120$^\circ$, in contrast to
the nonexistence of a pure-strategy equilibrium when the game is played on a
line segment.

 \item
Characteristics such as the color of cars could be modelled as location, but
only on a player-by-player basis, because they have no natural ordering. While
Smith's ranking of (red=1, yellow=2, blue=10) could be depicted on a line, if
Brown's ranking is (red=1, blue=5, yellow=6) we cannot use the same line for
him. In the text, the characteristic was something like physical location, about
which people may have different preferences but agree on what positions are
close to what other positions. \end{itemize}

\bigskip
\noindent
 {\bf N14.6} {\bf Durable Monopoly}
 \begin{itemize}
 \item The proposition that price falls to marginal cost in a durable monopoly
with no discounting and infinite time is called the ``Coase Conjecture,'' after
Coase (1972).  It is really a proposition and not a conjecture, but alliteration
was too strong to resist.

\item
 Gaskins (1974) has written a well-known article on the problem of the durable
monopolist who foresees that he will be creating his own future competition in
the future because his product can be recycled, using the context of the
aluminum market.

\item
Leasing by a durable monopoly was the main issue in the antitrust case {\it US
v. United Shoe Machinery Corporation}, 110 F. Supp. 295 (1953), but not because
it increased monopoly profits. The complaint was rather that long-term leasing
impeded entry by new sellers of shoe machinery, a curious idea when the proposed
alternative was outright sale.  More likely, leasing was used as a form of
financing for the machinery consumers; by leasing, they did not need to borrow
as they would have to do if it was a matter of financing a purchase. See Wiley,
Ramseyer, and Rasmusen (1990).

\item Another way out of the durable monopolist's problem is to give best-price
guarantees to consumers, promising to refund part of the purchase price if any
future consumer gets a lower price. Perversely, this hurts consumers, because it
stops the seller from being tempted to lower his price.  The ``most-favored-
consumer'' contract, which is the analogous contract in markets with several
sellers, is analyzed by Holt \& Scheffman (1987), for example, who demonstrate
how it can maintain high prices, and Png \& Hirshleifer (1987), who show how it
can be used to price discriminate between different types of buyers.

\item The durable monopoly model should remind you of bargaining under
incomplete information.  Both situations can be modelled using two periods, and
in both situations the problem for the seller is that he is tempted to offer a
low price in the second period after having offered  a high price in the first
period.  In the durable monopoly model this would happen if the high-valuation
buyers bought in the first period and thus were absent from consideration by the
second period. In  the bargaining model  this would happen if the  buyer
rejected the first-period offer and the seller   could conclude that he must
have a low valuation and act accordingly in the second period. With a rational
buyer, neither of these things can happen,  and the models' complications arise
from the attempt of the seller to get around the problem.

In the durable-monopoly model this would happen if the high-valuation buyers
bought in the first period and thus were absent from consideration by the second
period.  In the bargaining model this would happen if the buyer rejected the
first-period offer and the seller could conclude that he must have a low
valuation and act accordingly in the second period.  For further discussion, see
the survey by Kennan \& Wilson (1993).

 \end{itemize}

 \newpage \noindent
  {\bf Problems}

 \bigskip

\noindent
 {\bf 14.1. Differentiated Bertrand with Advertising} (medium)\\
Two firms that produce substitutes are competing with demand curves
\begin{equation} \label{e13.78}
 q_1= 10 - \alpha p_1 + \beta p_2 \end{equation} and \begin{equation}
\label{e13.79}
    q_2= 10 - \alpha p_2 + \beta p_1.
 \end{equation} Marginal cost is constant at $c=3$.  A player's strategy is his
price. Assume that $ \alpha > \beta/2.$

 \begin{enumerate}

 \item[(a)]
 What is the reaction function for firm 1? Draw the reaction curves for both
firms.

   \item[(b)] What is the equilibrium? What is the equilibrium quantity for firm
1?

   \item[(c)] Show how firm 2's reaction function changes when $\beta$
increases. What happens to the reaction curves in the diagram?

\item[(d)] Suppose that an advertising campaign could increase the value of
$\beta$ by one, and that this would increase the profits of each firm by more
than the cost of the campaign. What does this mean? If either firm could pay for
this campaign, what game would result between them?

 \end{enumerate}

\bigskip \noindent {\bf 14.2. Cournot Mergers}  (easy) (See  Salant, Switzer, \&
Reynolds [1983])\\ There are three identical firms in an industry with demand
given by $P = 1-Q$, where $Q = q_1+q_2+q_3$.  The marginal cost is zero.

 \begin{enumerate}

 \item[(a)] Compute the Cournot equilibrium price and quantities.

   \item[(b)]
 How do you know that there are no asymmetric Cournot equilibria, in which one
firm produces a different amount than the others?

   \item[(c)]
Show that if two of the firms merge, their shareholders are worse off.

 \end{enumerate}

\bigskip
 \noindent {\bf 14.3. Differentiated Bertrand} (medium) \\ 
 Two firms that produce
substitutes  have the   demand curves
 \begin{equation} \label{e13.80}
q_1=  1 - \alpha p_1 + \beta (p_2-p_1)
 \end{equation}
and \begin{equation} \label{e13.81}
 q_2= 1 -   \alpha p_2 + \beta (p_1-p_2), \end{equation} where $\alpha > \beta$.
Marginal cost is constant at $c$, where $c < 1/\alpha$.  A player's strategy is
his price.

 \begin{enumerate}

 \item[(a)]
 What are the equations for the reaction curves $p_1(p_2)$ and $p_2(p_1)$? Draw
them.

   \item[(b)]
 What is the pure-strategy equilibrium for this game?

   \item[(c)] What happens to prices if $\alpha$, $\beta$, or $c$ increase?

\item[(d)] What happens to each firm's price if $\alpha$ increases, but only
firm 2 realizes it (and firm 2 knows that firm 1 is uninformed)? Would firm 2
reveal the change to firm 1?

 \end{enumerate}

\bigskip \noindent
  {\bf  Problem 14.4. Asymmetric Cournot Duopoly} (easy)\\
  Apex    has  variable costs
of $q_a^2$  and a fixed cost of 1000, while  Brydox has variables costs of
$2q_b^2$ and  no fixed cost. Demand is $p = 115 - q_a - q_b$.

 \begin{enumerate}

 \item[(a)]
What is the equation for Apex's Cournot reaction function?

   \item[(b)] What is the equation for Brydox' Cournot reaction function?

   \item[(c)] What are the outputs and profits in the Cournot equilibrium?

 \end{enumerate}



\bigskip \noindent
 {\bf  Problem 14.5.  Price Discrimination} (medium) \\ 
 A seller faces a  large number of
buyers whose market demand is given by  $P= \alpha - \beta Q$. Production
marginal  cost is constant at $c$.

 \begin{enumerate}

 \item[(a)] What is the monopoly price and profit?

   \item[(b)] What are the prices under perfect price discrimination if the
seller can make take-it-or-leave-it offers?  What is the profit?

   \item[(c)] What are the prices under perfect price discrimination if the
buyer and sellers bargain over the price and split the surplus evenly?  What is
the profit?

 \end{enumerate}
%---------------------------------------------------------------


 \newpage

\begin{center}
{\bf The   Kleit Oligopoly Game:  A Classroom Game for  Chapter 14}
 \end{center}



  The  widget industry in  Smallsville  has $N$ firms.  Each firm produces 150
widgets  per month.  All costs are fixed, because labor is contracted for on a
yearly basis, so we can ignore production cost for the purposes of this case.
Widgets are perishable;   if they are not sold within the month,  they explode
in flames.

  There are two   markets for   widgets,   the  national  market, and the local
market.  The price in the national market is \$20 per widget, with the customers
paying for delivery, but the price in the local market depends on how many are
for sale there in a given month. The price is given by the following market
demand curve:
$$
P = 100 - \frac{Q}{N},
 $$
where $Q$ is the total output of widgets sold in the local market. If, however,
this equation would yield a negative price, the price is just zero, since the
excess widgets can be easily destroyed.

    \$20 is the    opportunity cost   of  selling a widget locally-- it is what
the firm loses by making that decision.  The benefit from the decision depends
on what other firms do.  All firms make their decisions at the same time  on
whether  to ship widgets out of town to the national market. The train only
comes to Smallsville once a month, so firms cannot retract their decisions.   If
a firm delays making its decision till too late, then it misses the train, and
all its output will have to be sold in Smallsville.

\bigskip
  \noindent
{\bf    General  Procedures }

  For the first seven months,  each of you will be a separate firm. You will
write down two things on an index card: (1) the number of the month, and (2)
your  LOCAL-market   sales  for that month.  Also record your local and national
market sales  on your Scoresheet.  The instructor will collect the index cards
and then announce the price for that month.  You should then calculate your
profit for the month and add it to  your cumulative total, recording both
numbers on your Scoresheet.

 For the last five months, you will be  organized into five different firms.
Each firm has a capacity of 150, and submits a single index card. The card
should have the number of the firm on it, as well as the month and the local
output.  The instructor will then calculate the market price, rounding it to the
nearest dollar to make computations easier.  Your own computations will be
easier if you pick round numbers for your output.

  If you do not turn in an index card by the deadline, you  have missed the
train and all  150 of your units must be sold locally. You can change your
decision up until the deadline  by handing in  a new card noting both your old
and your new output, e.g., ``I want to change from 40 to 90.''


\bigskip
\noindent
 {\bf   Procedures Each Month}

1.   Each student is one firm.  No  talking.

2. Each student is one firm.  No talking.

3. Each student is one firm.  No talking.

4. Each student is one firm.  No talking.

5. Each student is one firm.  No talking.

6.  Each student is one firm. You can talk with each other, but then you write
down your own output and hand  all outputs in separately.

7.  Each student is one firm. You can talk with each other, but then you write
down your own output and hand  all outputs in separately.

\vspace*{12pt}

8.  You are organized into  Firms 1 through 5, so N=5.  People can talk within
the firms, but firms cannot talk to each other.  The outputs of the firms are
secret.

9.   You are organized into  Firms 1 through 5, so N=5.  People can talk within
the firms, but firms cannot talk to each other.  The outputs of the firms are
secret.

10. You are organized into  Firms 1 through 5, so N=5.   You can talk to anyone
you like, but when the talking is done,    each firm  writes down its output
secretly  and hands it in.

 11.  You are organized into  Firms 1 through 5, so N=5.   You can talk to
anyone you like, but when the talking is done,    each firm  writes down its
output secretly  and hands it in. Write the number of your firm with your
output. This number will be made public once all the outputs have been received.


  You may be wondering about the  ``Kleit''.  Andrew Kleit is an economics
professor  Pennsylvania  State  University who  originated the ancestor of this 
oligopoly game   game for classroom use.

\end{small}

  \end{document}










