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\begin{center}

{\large {\bf    Isoperfect Price Discrimination in a Hotelling
Duopoly}
}

15  August  2008 . Pdf'd \today.

\bigskip



 David P.  Myatt  and  Eric B.  Rasmusen

{\it Abstract}

\end{center}
\begin{small}

 When duopolists compete by haggling with consumers, the form of the
bargaining model is very important, whether in a Bertrand model or a
Hotelling model.




 \noindent
 Myatt:  Nuffield College, Room L.  University of Oxford, New Road,
Oxford, England, OX1 1NF. 011-44-1865 278-578 or (01865) 278-578.
David.myatt@economics.ox.ac.uk.

\noindent
Rasmusen:  Dan R. and Catherine M. Dalton Professor, Department of
Business Economics and Public Policy, Kelley School of Business,
Indiana
University,BU 438,  1309 East 10th St., Bloomington, Indiana, USA
47405;    Erasmuse@indiana.edu.
  \url{http://www.rasmusen.org}.

{\small We would like to thank xxx, and  xxx  and seminar participants
at xxx  for helpful comments.  }





\end{small}

  {\small \textbf{\emph{Private research notes.  Not for circulation.
 }}}


\newpage

\noindent
{\bf 1. Introduction}





There is a literature on posted pricing versus bargaining: Wang (1995,
EER). Gill  \&   Thanassoulis, Anderson   \&   Renault (2003) JET,
Camera  \&  Delacroix  (2004)    {\it Review of Economic Dynamic.},
Cason,    Friedman, . and Milam,  (2003)  IJIO. Desai  \&  Purohit
(2004)    {\it Marketing Science.}



\bigskip

\noindent
{\bf 2. The Model}


A    Hotelling (1929) city has linear transportation costs. The city
consists of a single street,  the unit interval $[0,1]$. Consumers are
uniformly distributed along this street. Two firms located at 0 and 1
sell a product with a constant marginal cost that we normalize to
zero. Each consumer has a maximum willingness-to-pay of the same
number $v>0$ for a single unit of a product, but  there are linear
transportation costs, so a     consumer located at $x\in [0,1]$
purchasing from a firm located at $y\in [0,1]$ is willing to pay at
most $v - t|x-y|$.





 All functions and parameters are common knowledge.  The firms
know  each consumer's reservation price, can identify each  consumer,
and can prevent resale.



\noindent
 Let us define terms as follows:

\noindent
{\it Definition:} Under   {\bf  ``posted pricing''   }:  the
firm
  posts a single price, which buyers may only accept or reject.


\noindent
{\it  Definition:} Under   {\bf  ``Pigouvian perfect price
discrimination''   }:  a firm
  bargains with each buyer
separately, and captures    the entire  surplus.

\noindent
{\it  Definition:} Under   {\bf  ``isoperfect price discrimination''
}:  a firm
  bargains with each buyer
separately, and captures   fraction $\lambda$ of the surplus from each
buyer.




\noindent
{\it Definition:}  Under   {\bf  ``balanced isoperfect price
discrimination'' or ``IPD'' }:  a firm
  bargains with each buyer
separately, and captures   half the   surplus from each buyer. i.e.,
$\lambda=.5$.



 We have just described a reduced-form model of the  pricing process.
This flows easily from  natural structural models.    The monopolist
engages in
simultaneous discrete-time   bargaining games with all
the consumers.





In each period, the consumer picks a firm. He bargains with that firm
according to our bargaining game. Then he can switch.

 With probability  $ \lambda $ the seller makes the offer in a period,
and with probability   $(1-\lambda)$ the buyer  does. The buyer offers
price $p_b$ and the seller offers $p_s$. The non-offeror accepts or
rejects. If agreement is not reached, a period elapses. At the end of
the period the buyer can go to the other seller if  he wishes, and
start bargaining there.

 This is a model in the style of Sutton (1986),
except with a probabilistic offeror instead of  alternating  offerors.
(so it  is more like Baron et al.)

  There are two  ways that bargaining  between two players might break
down: in such a way as to end the game entirely, or in such a way that
the players are free to seek alternatives to the current potential
surplus.

\noindent
{\it Definition:}   {\bf (1)  Sudden-Death  Breakdown}: {\it This
occurs when bargaining breaks down exogenously and the players cannot
replace the gains from trade by dealing with someone else instead.}


\noindent
{\it Definition:}    {\bf (2)  Bargain-Specific Breakdown}: {\it This
occurs when bargaining breaks down exogenously and the players can
replace  part of the gains from trade by dealing with someone else
instead.}


With    probability $1-\beta$, bargaining  between the particular
buyer and seller breaks down. With probability  $\gamma$ this is
bargain-specific breakdown and the buyer can  visit another seller
instead.   With probability  $1-\gamma$ this is sudden-death breakdown
and the  game ends with zero payoffs for all players.





Sudden-death  breakdown  can    represent time
preference, with the time preference rate $r$ such that  $\beta =
\frac{1}{1+r}$.


 In general, the difference between the two kinds of breakdown is in
the way they affect disagreement points and outside options. After
sudden-death breakdown, the players receive their disagreement
payoffs, usually normalized to zero.  Like the bargaining surplus,
their outside options suddenly vanish.  After bargain-specific
breakdown, the players receive their outside options as payoffs.

 We describe
a   bargaining game here in which the monopolist bargains with all
consumers simultaneously. We could instead have used a
model in which alternating-offer games were played with    consumers
in
an exogenous sequence or in a sequence chosen by the monopolist.


\bigskip

\noindent
{\bf    Equilibrium under Monopoly }


Consider a monopolist located at $y=0$.  If he charges  price $p$ then
a consumer located at $x$ is willing to buy one unit if and only if
$v-tx\geq p$. Hence a monopolist supplying $z$ units faces the  demand
curve
\begin{equation} \label{e20}
p(z) = v - tz.
\end{equation}

Let us write  $z^{*}$ for
the efficient quantity, where if  $t  \geq v  $   then
$z^{*} = v/t$,  and if $t  \leq v  $ then $z^{*} = 1.$

  The monopolist faces a competitive fringe which charges the price
$\bar{p}$ for a product  with the location   $y =1$.

The market demand curve for the monopolist's product  is  linear  up
to the reservation price $\bar{p}$  if marginal cost is high enough
that not
all consumers are served when price equals marginal cost.



The  profit-maximizing  quantity,  monopoly price, and profits  under
posted
pricing  are, if the competitive fringe's price is not a binding
constraint,
\begin{equation} \label{e21}
z^{m} =  \frac{v}{ 2t}
\quad\Rightarrow\quad
p^m = .5v,
\,\,\text{and}\,\,
\pi^m =    \frac{v^2}{ 4t}
.
\end{equation}
 If $.5v  > \bar{p}$ then the presence of the competitive fringe
constrains the  monopolist and he  sets his price equal to  $\bar{p}$
instead of $.5v$.



This yields a solution so long as $z^{m}\leq 1$, or, equivalently,
$t\leq .5v$.  If $t < .5v$ then the constraint $z\leq 1$ binds.  the
monopolist chooses to supply the entire market, setting a price
$p^{m}=v-t$ to ensure that the consumer located at $x=1$ is willing
to buy.  Hence:
\begin{equation} \label{e22}
t < .5v
\quad\Rightarrow\quad
z^{m} = 1
\quad\Rightarrow\quad
p^{m} = v-t
\,\,\text{and}\,\,
\pi^{m} = v-t
.
\end{equation}

Under posted pricing the    profits are thus
\begin{equation} \label{e23}
\pi^{m} =
\begin{cases}
   vt-t^2    &    t   \leq .5v
\\
 .25v^2  &  t  \geq .5v
\end{cases}
\end{equation}

\bigskip


\noindent
{\bf Monopoly Price Discrimination}

If  the  monopolist uses  price discrimination,   he sells the
efficient
quantity $z^*$.


Under our assumptions, with probability $ 1-\lambda $ the buyer makes
the offer in a period, and with probability $ \lambda $ the seller
does. The buyer offers price $p_b$ and the seller offers $p_s$. The
non-offeror accepts or rejects. If agreement is not reached, a period
elapses. At the end of the period,  with probability   $1-\beta$
breakdown occurs. With probability $\gamma$ the buyer can  purchase
from the competitive fringe at price $\bar{p}$, for a payoff of
$v-\bar{p}$. With probability $1-\gamma$ the buyer cannot purchase
from the competitive fringe and his payoff is zero.

 In equilibrium, there is immediate agreement. Hence:
\begin{equation} \label{e23a}
\begin{array}{lll}
 \pi_s^{offer}& =& p_s \\
 & & \\
 \pi_s^{accept} & =& p_b \\
 & & \\
   \pi_b^{offer}& =& v- p_b \\
 & & \\
  \pi_b^{accept} & =& v- p_s .\\
\end{array}
\end{equation}



The  seller's payoff from rejecting is
\begin{equation} \label{e25a}
 \pi_s^{reject} =     (1-\beta)(0) +  \beta [
   (1-\lambda)  \pi_s^{accept}  +   \lambda  \pi_s^{offer}
],
\end{equation}
   because in the next period, if the bargaining does not fall
through,  the seller will accept the buyer's offer in equilibrium if
the buyer offers.



 Substituting  $\pi_s^{accept}  =  p_b$ and  $\pi_s^{offer}  =  p_s$
from \eqref{e23a}   yields
 \begin{equation} \label{e25}
 \pi_s^{reject } =           \beta  ( (1-\lambda) p_b  +   \lambda
p_s     ).
\end{equation}


If the buyer makes the offer, he does it to make the seller
indifferent between accepting and rejecting, so $\pi_s^{reject} =
\pi_s^{accept}= p_b$. Thus,
\begin{equation} \label{e28}
p_b =    \beta      (1-\lambda)  p_b + \beta  \lambda  p_s
\end{equation}

Then,
\begin{equation} \label{e30}
p_s =   p_b   \left(    \frac{ 1   -  \beta  (1-\lambda)}
{\beta  \lambda   }   \right)
\end{equation}

\bigskip



\noindent
  The buyer's payoff from rejecting is
\begin{equation} \label{e27a}
 \pi_b^{reject}  =(1-\beta) (1-\gamma)(0) +    (1- \beta)  \gamma
(v-\bar{p}) +  \beta
 (
(1-\lambda)  \pi_b^{offer} +  \lambda   \pi_b^{accept }   )  .
\end{equation}

The form of equation \eqref{e27a} is  different from the seller's
analog, equation \eqref{e25a},  because after bargain-specific
breakdown (probability $  (1-\beta) \gamma $) the	buyer can
still buy from the competitive fringe at price $\bar{p}$.

We can substitute $ \pi_b^{accept } = v - p_s $ and $ \pi_b^{offer} =
v - p_b $  from \eqref{e23a} to get
  \begin{equation} \label{e27}
 \pi_b^{reject}  =   (1- \beta)   \gamma  (v-\bar{p}) +  \beta
 (
(1-\lambda)  (v - p_b) +  \lambda   (v - p_s)    )  .
\end{equation}

The seller chooses $p_s$ to make the buyer indifferent between
accepting and rejecting, so $\pi_b^{reject} = \pi_b^{accept }  = v -
p_s $ and we can write
\begin{equation} \label{e29}
  v-p_s  =  (1-\beta)  \gamma    v   -   (1-\beta)    \gamma  \bar{p}
+
\beta  v -  \beta  (1-\lambda)  p_b -  \beta   \lambda  p_s
\end{equation}

Then
\begin{equation} \label{e29a}
 v - (1-\beta)   \gamma    v -  \beta  v +   (1-\beta)
\gamma\bar{p}
+  \beta  (1-\lambda)  p_b  = p_s -  \beta   \lambda  p_s
\end{equation}

 and
\begin{equation} \label{e31a}
 p_s =    \frac{   v - (1-\beta)   \gamma    v -  \beta  v +
(1-\beta)    \gamma\bar{p}
+  \beta  (1-\lambda)  p_b }{ 1-  \beta   \lambda}.
\end{equation}




Equating our two expressions for $p_s$ from  \eqref{e30} and
\eqref{e31a}  yields
\begin{equation} \label{e32}
  p_b   \left(    \frac{ 1   -  \beta  (1-\lambda)}
{\beta  \lambda   }   \right) =   \frac{   v - (1-\beta)   \gamma    v
-  \beta  v +   (1-\beta)    \gamma \bar{p}
+  \beta  (1-\lambda)  p_b }{ 1-  \beta   \lambda}
\end{equation}

 and
\begin{equation} \label{e32a}
  [ 1-\beta \lambda] [1   -  \beta  (1-\lambda)] p_b  =
\beta  \lambda   [ v - (1-\beta)   \gamma    v
-  \beta  v +  (1-\beta)    \gamma\bar{p}
+  \beta  (1-\lambda)  p_b   ]
\end{equation}


 and
\begin{equation} \label{e32b}
   [    1-\beta \lambda   -  \beta  (1-\lambda)  +  \beta^2 \lambda
(1-\lambda) ] p_b =
\beta  \lambda v - \beta  \lambda (1-\beta)   \gamma    v
-  \beta^2  \lambda v +   \beta   \lambda
(1-\beta)    \gamma \bar{p}
+  \beta^2  \lambda  (1-\lambda)  p_b
\end{equation}


 and
\begin{equation} \label{e32c}
 [    1-\beta \lambda   -  \beta  (1-\lambda)   ] p_b =
\beta  \lambda v - \beta  \lambda (1-\beta)   \gamma    v
-  \beta^2  \lambda v +   \beta   \lambda
(1-\beta)    \gamma \bar{p}
\end{equation}




 and
\begin{equation} \label{e32e}
   p_b        =   \frac{\beta  \lambda v - \beta  \lambda (1-\beta)
\gamma    v
-  \beta^2  \lambda v +      \beta   \lambda
(1-\beta)    \gamma \bar{p}} { 1-\beta     }
\end{equation}


 It follows from \eqref{e30}  that
\begin{equation} \label{e33}
 \begin{array}{lll}
p_s &=  & \frac{\beta  \lambda v - \beta  \lambda (1-\beta) \gamma
v -  \beta^2  \lambda v +      \beta   \lambda (1-\beta)    \gamma
\bar{p}} { 1-\beta     }   \left(    \frac{ 1   - \beta  (1-\lambda)}
{\beta  \lambda   }   \right)\\
 &&\\
 &= & \frac{[1   - \beta  (1-\lambda)] [ \beta  \lambda v - \beta
\lambda (1-\beta) \gamma    v -  \beta^2  \lambda v +      \beta
\lambda (1-\beta)    \gamma \bar{p}] } {\beta  \lambda ( 1-\beta)
} \\
 &&\\
 &= & \frac{[1   - \beta  (1-\lambda)] [    v -
 (1-\beta) \gamma    v -  \beta    v +
 (1-\beta)    \gamma \bar{p}] } {    1-\beta
} \\
 &&\\
 &= & \frac{      v  -
  (1-\beta) \gamma    v -  \beta    v +
  (1-\beta)    \gamma \bar{p}   - \beta   (1-\lambda)   v
+ \beta   (1-\lambda)  (1-\beta) \gamma    v +  \beta^2
 (1-\lambda)v  -  \beta   (1-\lambda)   (1-\beta)
\gamma \bar{p}     } {    1-\beta  } \\
 &&\\
 &= & \frac{      v  - (1-\beta) \gamma    v -  \beta     v  - \beta
(1-\lambda)   v + \beta   (1-\lambda)   (1-\beta) \gamma    v +
\beta^2 (1-\lambda)v    + (1-\beta)    \gamma \bar{p}   -  \beta
(1-\lambda) (1-\beta) \gamma \bar{p}     } {     1-\beta  } \\
\end{array}
\end{equation}






We are interested in what happens in the limit, as the probability
$\beta$ of breakdown goes to one.  This represents what happens as
time periods get very short. Both the numerator and denominator of
\eqref{e32e}  and \eqref{e33} approach zero, so we need to use
L'Hopital's rule, which
gives us:
\begin{equation} \label{e34}
\begin{array}{lll}
 \stackrel{ p_b} {\beta \rightarrow 1}   &= & \lim_{\beta\to 1}
\frac{
\lambda
v -
\lambda \gamma    v + 2 \beta \lambda \gamma v
-  2\beta   \lambda v +  \lambda  \gamma \bar{p}  -   2 \beta
\lambda
    \gamma \bar{p}} { -1    }\\
 && \\
 &=& - \lambda v +
\lambda \gamma    v - 2 \lambda \gamma v
+ 2    \lambda v - \lambda  \gamma \bar{p}  +   2
\lambda
    \gamma \bar{p}\\
 & & \\
 &=& \fbox{ $\lambda  (1-\gamma)    v + \lambda  \gamma \bar{p}$}
\end{array}
\end{equation}
and
\begin{equation} \label{e33}
 \begin{array}{lll}
 \stackrel{ p_s} {\beta \rightarrow 1}   &=  &
 \frac{       \gamma    v -       v  -
(1-\lambda)   v + \gamma  (1-\lambda)    v - 2\beta     \gamma
(1-\lambda)    v + 2
\beta  (1-\lambda)v    - \gamma \bar{p}    -  \gamma (1-\lambda)
\bar{p}    +2  \beta
  \gamma (1-\lambda) \bar{p}     } {     -1 } \\
 &&\\
    &=  &
   -  \gamma    v +      v  + (1-\lambda)   v - \gamma  (1-\lambda)
v + 2     \gamma (1-\lambda)    v -2 (1-\lambda)v    + \gamma \bar{p}
+ \gamma (1-\lambda) \bar{p}    -2 \gamma (1-\lambda) \bar{p}       \\
 &&\\
  -  \gamma    v +      v      +      \gamma (1-\lambda)    v -
(1-\lambda)v    + \gamma \bar{p}       -  \gamma (1-\lambda) \bar{p}
\\
 &&\\
 (1-   \gamma)    v      - (1-     \gamma) (1-\lambda)    v    +
\gamma  \lambda  \bar{p}       \\
 &&\\
  \lambda (1-   \gamma)    v         + \gamma  \lambda  \bar{p}
\\
\end{array}
\end{equation}
  These are the same values, because if the time periods are very
short there is  infinitesimal  advantage to offering first.


Suppose the buyer is offering a price and the time period is short. If
breakdown is sudden-death ($\gamma=0$), then $p_b= \lambda v$; the
buyer pays a share of $v$  proportional to the seller's bargaining
power, and the competitive fringe doesn't matter. $p_s=\lambda    v $
also.



 If breakdown is bargain-specific  ($\gamma=1$), then $p_b= \lambda
\bar{p}$; the
buyer pays a share of $\bar{p} $  proportional to the seller's
bargaining
power, so the competitive fringe certainly does matter to the price.

  These two polar situations illustrate the ambiguity of the term
``bargaining surplus''. One way to view it is as $v$, the gains from
trade between buyer and seller. The other way to view it is as $v -
\bar{p}$, the  increase  in social surplus from having the transaction
be between this buyer and seller rather than this buyer and some other
seller.

   In the context of bilateral monopoly with a competitive fringe,
surely  bargain-specific breakdown is the correct model. [argue that
here xxx] Buyer and monopolist are haggling over gains from the buyer
buying from the monopolist rather than the competitive fringe.



 \bigskip




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4.2	0.864864865
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  \end{center}
  \caption{Sales Strategies for a Hotelling Monopolist}
  \label{fig:hotelling-monopolist}
\end{figure}

%\end{comment}


\bigskip


\noindent
 {\bf Duopoly}

If  we have two firms located at  0 and 1  and the marginal cost is
zero,   the efficient outputs are those that would result when both
firms' prices equal zero. A consumer located at $x$ would be willing
to buy from Firm 1 if $v-tx \geq 0$ and from Firm 2 if $v-t(1-x) \geq
0$. The consumer at $x=.5$ will buy if $v-.5t \geq 0$; that is, if $t
\leq 2v$. In that case, the two firms split the market and $z_1=.5,
z_2 = .5$. Otherwise, $v- tz_1 =0$ and $v-t(1-z_2) =0$, so $z_1 =
\frac{v}{t} $,   $z_1 = 1-\frac{v}{t}$,    and $z=z_1+z_2  <1$.

 We must specify the order of play carefully:

\noindent
1. The firms simultaneously choose policies. A firm can either choose
posted pricing with a particular price $p_1$ or $p_2$, or price
discrimination.

\noindent
2. Consumers choose firms.  Switching firms incurs a small cost
$\varepsilon$.

\noindent
3. Consumer purchase or haggle.  A haggling consumer may switch to the
other firm.

\noindent
4. The game ends when all consumers have agreed to a price or decided
not to buy.


\bigskip

\noindent
 {\bf Two  Firms Using Posted Pricing}

 Suppose both firms use posted pricing.
   A consumer located at $x$ has payoff $(v-tx-p_1)$ buying from
Firm 1, $(v-t(1-x) -
p_2)$    from Firm 2, and 0 not buying at all (ignoring the switching
cost $\varepsilon$).   If  he is willing to buy, Consumer $x$
  is indifferent between the suppliers if and only if
   \begin{equation} \label{a24}
v - tx - p_1 = v - t(1-x) - p_2,
\end{equation}
 which is equivalent to
 \begin{equation} \label{a24a}
x = .5 + \frac{p_2-p_1}{2t}.
\end{equation}

For $t$ small enough, Firms 1 and 2  sell amounts $x$ and $1-x$. For
larger $t$,  there will be consumers in a  ``no man's land''  unserved
by  either firm. There are thus three cases to consider.


\noindent
(3)    $t \in (  0, \frac{2 }{3}v )$. From \eqref{a24a}, if there is a
consumer  $x$ indifferent between the two firms and willing to buy,
Firm 1's profit of $xp_1$ equals  $p_1/2 + \frac{p_1(p_2-p_1)}{2t}$.
Maximizing this  with respect to $p_1$  and    solving  with the
analogous condition for Firm 2 yields
 $p_1=p_2 = t$.    Consumer $x=.5 $ is willing to buy if $v-t/2 - t
\geq 0$, which requires that $t \leq\frac{2 }{3}v$.
   Hence:
\begin{equation} \label{a26}
z^{pp} =  \frac{ 1}{2 } , \,\, p^{pp} = t \,\,\text{and}\,\, \pi_1
(pp,pp)  = \frac{ t}{2 } .
\end{equation}



\bigskip




\noindent
(2)     $t \in (  \frac{2 }{3}v, v)$.  The duopolists split the
market evenly and the central customer is indifferent about buying.
With transportation costs below $v$, a duopolist does not have as much
incentive as a monopolist to reduce price, because competition
constrains his ability to increase quantity by selling to distant
consumers. Instead, his best strategy may be to use a high price to
extract surplus from nearby consumers, where he faces less
competition.
 If $p_1 = p_2 =  v -t/2$, then each firm will sell to exactly half
the market and Consumer $x=.5$ will be indifferent about buying from
Firm 1, buying from Firm 2, and not buying at all.
    Hence:
\begin{equation} \label{a27}
z^{pp}= .5 , \,\, p^{pp} = v - .5t \,\,\text{and}\,\, \pi_1 (pp,pp) =
.5\left[v-.5t  \right].
\end{equation}

\bigskip



\noindent
(3)   $t \in [v, \infty]$. In this case,    $z^{pp}\leq .5$ and there
is
a no man's land;  the two
  firms act as local monopolists, using the monopoly solution from
\eqref{e21},
\begin{equation} \label{a25}
z^{pp} =    \frac{ v}{2t},
p^{pp} =\frac{ v}{2 }
\,\,\text{and}\,\,
  \pi_1 (pp,pp)  =  \frac{    v^2}{4 t}
.
\end{equation}

In summary,
a posted-pricing duopolist enjoys profits of
\begin{equation} \label{a28}
  \pi_1 (pp,pp)  =
\begin{cases}
    \frac{  v^2}{4t}     &   t \in [v, \infty]
\\
 .5v -.25t   & t \in (  \frac{2 }{3}v,v  )
\\
  .5t & t \in [ 0,\frac{2  }{3}v )
\end{cases}
\end{equation}









\bigskip

\noindent
 {\bf   Two Firms  Using Price Discrimination}

Now let both firms use  price discrimination (we are not yet
permitting
deviation by one to  posted pricing).



\noindent
 (1) $t     \leq 2v$.
Now  $z^{*}>.5 $  and  efficiency dictates that all consumers be
served. The  duopolists split the market evenly.

 xxx

\noindent
 (2) $t     \geq 2v$.  In this case,  $z^{*}<.5$, and  efficiency
dictates that centrally located consumers will not be served.  Hence
each duopolist earns the profits of a  monopolist:
\begin{equation} \label{a32}
 \pi_1 (pd,pd)  = \pi_1 (pp,pp)  =   \frac{\lambda v^2}{ 2t}
.
\end{equation}







\begin{figure}[t]
%  \includegraphics[width=6in]{pdisc10a.jpg}
  \caption{Both Firms Use Isoperfect Price Discrimination and $t < 2v$
}
  \label{fig:10}
\end{figure}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\bigskip

\noindent
 {\bf Mixed Duopoly: A Price Poster and a Price Discriminator   }

Now let us  consider  mixed duopoly.  Firm 1 uses a posted price,
$p_1$.  Firm 2 uses price discrimination,   so a consumer buying from
Firm 2 enjoys a fraction $ 1-\lambda $ of the  surplus from their
transaction. In our model  this is a Rubinstein bargaining game
between Firm 2 and the consumer, where the consumer may  purchase
from Firm 1 at price $p_1$ as an outside option.


\bigskip

\noindent
 {\bf (1)  $t \in [1.5v, \infty]$. The Small Outputs/Dual Monopoly
Case}

Ideally Firm 1 would supply $z^{pp}$ consumers and Firm 2 would haggle
with $z^{pd}$.  If $z^{pp}+z^{pd}<1$, then these ideals are mutually
compatible.  That is,
\begin{equation} \label{e44}
z^{pp}+z^{pd}\leq 1
\quad\Leftrightarrow\quad
\frac{v}{2t} + \frac{v}{t}\leq 1
\quad\Leftrightarrow\quad
t  \geq 1.5v
.
\end{equation}

Hence in this circumstance it is straightforward to calculate profits:
\begin{equation} \label{e45}
\pi_1 (pp, pd) =    \frac{v^2}{4t}
\quad\text{and}\quad
\pi_2 (pp, pd) =    \frac{\lambda v^2}{2t}
.
\end{equation}

\bigskip

\noindent
 {\bf Large Outputs: Potential Conflict between the Price Poster and
the
Price Discriminator   }

When $  t <1.5v   $,    the positions of the two
duopolists may conflict, since the ``target'' outputs of the price
poster and  the price discriminator might add to more than 1.
Suppose
the price poster   serves $z$ consumers in equilibrium. If $z$ does
not
equal 0 or 1, the    consumer located at $z$ must be   indifferent, so
\begin{equation} \label{e46}
CS(z, pd) = (1-\lambda)[v-t(1-z)]
= CS(z, pp) =
v - tz - p
\end{equation}
which implies that
\begin{equation} \label{e46a}
p = \lambda v +(1-\lambda)t - (2-\lambda)tz .
\end{equation}



\noindent
The revenue  $zp(z)$ is strictly concave in $z$, and maximized by
\begin{equation} \label{e47}
\begin{array}{lll}
\tilde{z}& =& \frac{\lambda v +(1-\lambda)t}{2t(2-\lambda)}\\
 & & \\
&=&
\frac{ \lambda \left( \frac{v}{t}  \right)
+(1-\lambda)}{4-2\lambda}
.\\
\end{array}
\end{equation}

There are are various cases to consider, depending on whether the
price poster captures the entire market, none of the market,   part of
the market where some consumers go unserved, or part of the market
where
all consumers are served.  Let us figure out the best responses of the
price discriminator in these various cases.



\bigskip

\noindent
 {\bf (2) $t \in [\frac{(4-\lambda)v}{3-\lambda}, 1.5v]$.  Constrained
Monopoly}

This case is when the price setter chooses a low enough $p$ that the
price discriminator can have sales of $v/t$. This requires $t>v$. The
price poster's sales will be $z_{pp}=1-v/t$. Thus,  it must be that
the solution to the price poster's maximization problem in \eqref{xx}
is
\begin{equation} \label{e48}
\frac{\lambda \left( \frac{v}{t}  \right)
+(1-\lambda)}{4-2\lambda}
\leq
1-  \frac{v}{t}
,  which \; implies\;
 t  \geq \frac{(4-\lambda)v}{3-\lambda}.
\end{equation}

A necessary condition for this last inequality to hold is
$   t  > (4/3) v$, and hence the ``constrained monopolist'' case
can apply
only when
\begin{equation} \label{e49}
\frac{4v}{3} <  t   \leq 1.5v
.
\end{equation}
 tho it might not apply for some of that range, depending on
$\lambda$'s value.

For this case, the  price-discriminating duopolist avoids any
competition and hence
\begin{equation} \label{e50}
  \pi_1 (pd,pp) =   \frac{\lambda v^2 }{2t} .
\end{equation}

The  price-posting duopolist sells $z^{pp} = 1-  \frac{v}{t}
 $ by setting the price
  $p=v-tz^{pp}$.  Hence
\begin{equation} \label{e51}
\pi_1 (  pp,pd) =  2 (v-t)\left( 1-
\frac{v}{t}  \right)) = 4v -2t -\frac{2v^2}{t}
.
\end{equation}



 \bigskip

\noindent
 {\bf (3)  $t \in (\frac{\lambda v}{3-\lambda}, \frac{(4-\lambda)v}
{3-\lambda})$. The True Mixed-Duopoly Case}



  \bigskip

\noindent
 {\bf (4)  $t \in [0,  \frac{\lambda v}{3-\lambda}]$. The Price Poster
Captures the Entire Market}


  The solution in  \eqref{e47} for the price-posting firm is $z^{pp}=
1$, and hence the  price-discriminating firm makes no sales and earns
no profits.

 This is the second place where the assumption of $\varepsilon>0$
comes into play. Suppose a consumer deviated by going to the price-
discriminating Firm 2  instead of the price-posting Firm 1.  Consumer
x's payoff from Firm 1 is $(v- tx - p_1)$.  If he visits Firm 2, then
his payoff is the bargaining solution giving him either  $(1-\lambda)$
of the surplus $(v- t(1-x))$ or the outside option of $[(v- tx - p_1)-
\varepsilon]$, whichever is greater.

 The consumer with the greatest surplus from trade with Firm 2 is   at
$x=1$. If he visits Firm 1 his payoff is $v - t - p_1$.  If he visits
Firm 2,  his payoff is   $Max(1-\lambda)v , v-t - p_1 -  \varepsilon)
$.  In this parameter range, the outside option is binding, so his
payoff from choosing Firm 2 is $v-t - p_1 -  \varepsilon$, which is
lower than from visiting Firm 1.

 Figure \ref{fig:11} illustrates this  for  the case of $\lambda =.5$.
Consumer $x=1$ could get payoff of $s_1$ from visiting Firm 1.
Visiting Firm 2, his Rubinstein payoff would be $v/2$, a smaller
amount. His outside option would be a payoff of $s_1-\varepsilon$,
more than $v/2$ so the bargaining game would yield him a  payoff of
$s_1-\varepsilon$, but that is still worse than buying from Firm 1.


\begin{figure}[t]
%  \includegraphics[width=6in]{pdisc11a.jpg} needs replacing
  \caption{Firm 1 Posts a Price; Firm 2 Uses Isoperfect Price
Discrimination, Low Transportation Cost}
  \label{fig:11}
\end{figure}





 \begin{equation} \label{e58}
       \pi_1(pp, pd) =  \lambda  v    - t \,\,\text{and}\,\,
\pi_2(pp,pd  ) = 0 .
\end{equation}

Since the maximum value that  $\lambda$ can take is 1, this case
requires that  $t < .5v$.


\bigskip

\section*{ The Equilibrium.   }

 We will now  assume
 balanced isoperfect price discrimination: $\lambda =.5$


    If $t>2v$, then the industry is in effect two separate monopolies,
not a duopoly, since it is unprofitable for either firm to serve the
consumer at $x=.5$. Similarly, if $t \in (1.5v, 2v]$ and one firm
decides to use posted pricing, some consumers will go unserved whom
that firm decides not to serve and whom it would be unprofitable for
the price-discriminating firm to serve. Both cases lack true
competition, so the question of how the presence  of another firm
affects the choice of pricing policy is vacuous.

What if $ t  < 1.5v$?  Then there is at least one consumer whose
custom is solicited by both firms, and competition does constrain a
firm's pricing.

First, note that there will exist no mixed equilibrium in which one
firm
uses  price discrimination and the other  uses a posted price.  For
that to happen would require that there be a consumer  indifferent
between the two firm, given the prices he would be offered at each.
If, however, that consumer is indifferent, then if he chooses to go to
the price-discriminating firm, the price-discriminating firm would
deviate from its equilibrium price. It would raise it by
$\varepsilon/2$, which, given the switching cost of $\varepsilon$,
would not repulse the consumer but would yield higher profit.
Foreseeing this, however, the consumer would not go to the price-
discriminating firm in the first place. Thus, there cannot exist a
consumer indifferent between the two firms, which contradicts our
original supposition.

 This leaves as possible equilibria both firms using posted prices or
both firms using price discrimination.


\noindent
 {\bf Profits with Both Using Posted Pricing versus Both Using Price
Discrimination}

 xxx


\bigskip

 \noindent
  {\bf 8. Concluding Remarks}

xxx not written yet

\newpage




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