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\noindent
  {\bf  Posted Prices vs. Haggling: The Economics of
Isoperfect Price Discrimination },  2 April  2009\\
 David P.  Myatt  and  Eric B.  Rasmusen\footnote{
 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;    Erasmuse@indiana.edu.
  \url{http://www.rasmusen.org}.}

  
A monopolist's marginal cost  at  output $z$ is  $c(z)$.

Demand arises from
a unit mass of consumers, where a consumer's willingness to pay
$v$ for a single unit   is drawn from
$F(\cdot)$ with positive density $f(\cdot)$ over  a support bounded
above by $\overline{v}> c(0)$.

The  price $p$
yields demand quantity $z(p)=1-F(p)$.


 The monopolist may sell either by posting a single   price or  by
bargaining with individual consumers.    Bargaining
splits the surplus,  with fraction $\lambda$ going to the monopolist
and
$(1-\lambda)$ to the consumer.

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


In the  basic
model,   transaction costs are zero, bargaining power is equal,
marginal cost is constant,   demand is linear, and consumers are fully
informed of their tastes:  $t_p=t_b=0$, $\lambda =.5$,     $c(z)=c$,
and
$p''(z)=0$.



\noindent
{\it Definition:} Under   {\bf  ``monopoly pricing''   }:  the
monopolist
  posts a single price, which buyers may only accept or reject.
  Under   {\bf  ``Pigouvian perfect price
discrimination''   }:  the monopolist
  bargains with each buyer
separately, and captures    the entire  surplus.
  Under   {\bf  ``isoperfect price discrimination'' or ``IPD''
}:  the monopolist
  bargains with each buyer
separately, and captures   half of the surplus from each
buyer.
   Under   {\bf  ``generalized isoperfect price
discrimination''  }:  the monopolist
  bargains with each buyer
separately, and captures    fraction $\lambda$ of the   surplus from each buyer.  


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\noindent
{\bf Proposition 1:} {\it  In the basic model the monopolist is
indifferent between  monopoly pricing and    isoperfect price
discrimination, earnings profits in each case of half of the total
surplus.  }



 \noindent
{\bf
 Proposition 2:}{\it    If
marginal cost is increasing  ($c'(z)>0$),  the monopolist prefers
balanced isoperfect price discrimination to monopoly pricing. If
marginal cost is decreasing ($c'(z)<0$), he prefers monopoly pricing.
}


\noindent
{\bf Corollary. } {\it   Isoperfect
price discrimination is
more profitable  than monopoly pricing   if
demand
is convex,  and less profitable   if  demand  is
concave.
 }



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