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\begin{center}
              {\huge\bf Indeterminate Output Allocations} \\
 \vspace{2ex} {\huge\bf in Collusive Equilibria} \\
 \vspace{2ex} {\huge\bf and Multi-plant Firms} \\
 \vspace{5ex} {\Large\bf Thomas Mitchell} \\ \mbox{} \\
              {\large    Department of Economics        \\
                         Southern Illinois University at Carbondale  \\
                         Carbondale, Illinois 62901--4515 \\
                         (618) 453--5073 \\
                         {\tt tmitch @ siu.edu}} \end{center}
\vfill
\begin{center} {\Large\bf Abstract}  \end{center}
\begin{quote}
If $n$ colluding oligopolists all have the cost function
\mbox{$C(q_{i}) = c\,q_{i}$}, then it will not be possible to uniquely
allocate among the firms the monopoly output that maximizes their
{\it joint\/} profit.  Similarly, if all plants of an $n$-plant firm
have the cost function \mbox{$C(q_{i}) = c\,q_{i}$}, then it will not be
possible to uniquely allocate the firm's optimal output among the $n$
plants.  This paper identifies the necessary and sufficient condition
for such ``allocative indeterminacies'' to occur.
\end{quote}
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\mbox{ }
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            {\large\bf DRAFT:} {\large Not for quotation} \\
                     Comments welcome \end{center}
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\section{Introduction}
Introductory analysis of oligopoly usually involves a simple example of
Cournot duopoly.  The simple example frequently utilizes a linear market
demand and cost functions that are characterized by identical and
constant average and marginal costs.  These assumptions simplify the
algebra of solving for the Cournot equilibrium and demonstrating that by
colluding---acting as a single production entity so as to maximize their
joint profit---each of the two firms could increase its profit relative
to the Cournot equilibrium.  However, the nature of the equilibrium
under collusion is such that it is not possible to uniquely allocate
outputs (and, therefore,
profits) to each of the two firms.  In other words, as long as the
{\it total\/} output is equal to the unique profit-maximizing monopoly
output, {\it any\/} combination of outputs for the two firms summing to
the monopoly output would be a feasible equilibrium when the firms
collude.  Obviously, this same ``allocative indeterminacy'' will also be
faced by any single firm possessing two or more plants with identical
and constant average and marginal costs for each plant, regardless of
the structure of the market in which the multi-plant firm operates
(competitive, monopolistically competitive, oligopolistic, or
monopolistic).  Recognizing that the linear market demand plays no role
in the occurrence of the allocative indeterminacies, the purpose of this
paper is to identify all functional forms of the cost functions that
lead to this allocative indeterminacy for colluding firms in an
oligopoly or the individual plants in a multi-plant firm.  We will see
that an allocative indeterminacy occurs {\it if and only if\/} each
production unit---a firm that is part of a collusive oligopoly, or a
plant within a multi-plant firm---has the same constant value for
marginal cost.

\section{The Cournot Model of Duopoly}
The standard Cournot model of duopoly has two firms---call them
``Firm~1'' and ``Firm~2''---producing a homogeneous product.  There is
neither entry of new firms, nor exit of existing firms (although an
existing firm could choose to produce nothing).  The firms are assumed
to choose their output levels (rather than prices as in a differentiated
goods model), and they choose their outputs simultaneously, so there
does not exist the possibility of one firm gaining an advantage by
delaying its output choice until its competitor has committed to a
particular output.  (For a discussion of the dynamics of a Cournot
model, see Friedman, 1983.)  The buyers of the homogeneous product have
no transactions costs, and there are no differentials in transportation
costs based on the locations of the sellers or the buyers.  The single
equilibrium price that must prevail in such a market is the price that
equates the market quantity demanded and the [total] quantity supplied
by the two firms.  Further, although it does not affect the results of
this paper in any way, the usual Cournot model of duopoly supposes that
the market demand curve is linear.  Let $p$ denote the single price
prevailing in the market at a particular time; let $q_{1}$ and $q_{2}$
denote the quantities produced by Firm~1 and Firm~2, respectively, and
let $Q$ denote the industry output produced, \mbox{$Q \equiv q_{1} +
q_{2}$}.  Then the usual market demand can be described by the following
inverse demand function,
\begin{equation} p = p(Q) = a - bQ, \;\;\;\; a, b > 0.
\label{eq:demand} \end{equation}
Finally, it is often assumed that each firm has the cost function
\mbox{$C(q) = cq$}, so that we can define the cost function for Firm~1,
$C_{1}$, and the cost function for Firm~2, $C_{2}$, with
\begin{equation} C_{i}(q_{i}) = c\,q_{i}, \;\;\;\; i = 1, 2.
\label{eq:C(q)} \end{equation}
To assure ourselves that a market will exist for the given product, the
parameter $c$ in (\ref{eq:C(q)}) must be less than the parameter $a$ in
(\ref{eq:demand}):  \mbox{$c < a$}.

Before reviewing the Cournot equilibrium for this duopoly, recall the
monopoly outcome.  If there were a single firm with constant average and
marginal cost equal to $c$, we can identify the market equilibrium
quantity and price by equating marginal revenue and marginal cost.
Letting $Q_{m}$ denote the equilibrium market quantity under monopoly,
it is a simple matter to show that \mbox{$Q_{m} = (a - c)/2b$}.
Letting $p_{m}$ denote the equilibrium price under monopoly, from the
inverse demand function we find that \mbox{$p_{m} = (a + c)/2$}.  Then
the monopoly case can be summarized by
\[ p_{m} = \frac{a + c}{2}, \;\; Q_{m} = \frac{a - c}{2b} . \]

We now consider the Cournot equilibrium for the market described by
eqs.~(\ref{eq:demand}) and (\ref{eq:C(q)}).  Each firm is assumed to
pursue maximum profit, and the profit of Firm~$i$, denoted by $\pi_{i}$
$(i = 1, 2)$, is given by
\begin{eqnarray*}
\pi_{i} & = & p(Q) \cdot q_{i} - C_{i}(q_{i}), \;\;\;\; i = 1, 2 \\
        & = & [a - b(q_{1} + q_{2})] q_{i} - c q_{i}, \;\;\;\; i = 1, 2.
\end{eqnarray*}
Note that Firm~1's profit depends on Firm~2's output and Firm~2's profit
depends on Firm~1's output, since the outputs of both firms contribute
to the determination of the market price through the market quantity $Q$
in the inverse demand function $p(Q)$.  Each firm attempts to maximize
its own profit through the choice of its own output.  To identify each
firm's optimal output, we differentiate $\pi_{i}$ with respect to
$q_{i}$ $(i = 1, 2)$:
\begin{eqnarray}
\frac{\partial \pi_{i}}{\partial q_{i}}
      & = & p(Q) + q_{i} \frac{\partial p}{\partial q_{i}} -
                  \frac{dC_{i}}{dq_{i}}, \;\;\;\; i = 1, 2 \nonumber \\
      & = & a - b(q_{1} + q_{2}) + q_{i} (-b) - c, \;\;\;\; i = 1, 2
                                                           \nonumber \\
      & = & a - 2bq_{i} - bq_{j} - c, \;\;\;\; j \neq i, \; i = 1, 2.
\label{eq:conj-var}
\end{eqnarray}
Embodied in (\ref{eq:conj-var}) is the ``Cournot assumption'':  each
firm naively assumes that if it changes its own output, the firm's
competitor will not change the output it is producing; i.e.\ for
Firm~$i$, $\partial q_{j} / \partial q_{i} = 0$ $(j \neq i, i = 1, 2)$.
In other words, the ``conjectural variation'' for each firm is zero.
Solving \mbox{$\partial \pi_{i} / \partial q_{i} = 0$} for $q_{i}$ in
terms of $q_{j}$ $(j \neq i)$, the output of Firm~$i$'s competitor, we
find
\begin{equation} q_{i}^{\;*} =
                  r_{i}(q_{j}) = \frac{a - c}{2b} - \frac{1}{2} q_{j},
                                 \;\;\;\; j \neq i, \; i = 1, 2.
\label{eq:reactions} \end{equation}
The function $r_{i}(q_{j})$, which expresses Firm~$i$'s profit
maximizing output in terms of its competitor's output $q_{j}$, is
Firm~$i$'s ``reaction function,'' and each firm has a reaction function
giving its optimal output in terms of its competitor's output.

A ``Cournot equilibrium'' occurs when neither firm has an incentive to
change its output.  Suppose that Firm~1 is producing $q_{1}^{\;C}$; then
Firm~2 will maximize its profit by producing $r_{2}(q_{1}^{\;C})$.  If
Firm~2 is producing $r_{2}(q_{1}^{\;C})$, then Firm~1 will maximize its
profit by producing $r_{1}[r_{2}(q_{1}^{\;C})]$.  Firm~1 will have no
incentive to change its output if $r_{1}[r_{2}(q_{1}^{\;C})]$ is equal
to what Firm~1 is already producing, namely $q_{1}^{\;C}$.  Then
Firm~1's output in a Cournot equilibrium is found by solving the
following equation for $q_{1}^{\;C}$:
\begin{equation}
               r_{1}[r_{2}(q_{1}^{\;C})] = q_{1}^{\;C}.
\label{eq:C-eq-condition} \end{equation}
Solving (\ref{eq:C-eq-condition}) using (\ref{eq:reactions}) for the
forms of the reaction functions for both Firm~1 and Firm~2 yields
Firm~1's output in a Cournot equilibrium:
\mbox{$q_{1}^{\;C} = (a - c)/3b$}.  Substituting Firm~1's Cournot
equilibrium output into Firm~2's reaction function reveals that Firm~2
produces the same quantity; i.e.\ the Cournot equilibrium is symmetric
with respect to the two firms' outputs in this model:
\mbox{$q_{2}^{\;C} = (a - c)/3b$}.  If we let $Q_{C}$ denote the market
output in a Cournot equilibrium, it is obvious that
\mbox{$Q_{C} = 2(a - c)/3b$}.  Substituting this into (\ref{eq:demand})
we find that the Cournot equilibrium price in this model, denoted by
$p_{C}$, is \mbox{$p_{C} = (a + 2c)/3$}.  Summarizing the Cournot
equilibrium,
\[ p_{C} = \frac{a + 2c}{3}, \;\;\;\; Q_{C} = \frac{2(a - c)}{3b}. \]

\section{Collusion and the Indeterminacy of Individual Outputs}
Once the Cournot equilibrium has been described and found, it is
inevitably pointed out that each of the two firms could increase its
profit by colluding.  That is to say that if the firms maximized their
{\it joint\/} profit, the outcome could mean a higher profit for
{\it each\/} firm.  To demonstrate this, consider the profits earned by
the monopolist and the Cournot duopolists in the previous section.
In the monopoly case, the single firm earns a profit of
\begin{equation}
\pi_{m} = (p_{m} - c) Q_{m} =
  \left( \frac{a+c}{2} - c \right) \left( \frac{a - c}{2b} \right) =
  \frac{(a - c)^{2}}{4b},
\label{eq:mono-profit} \end{equation}
while in the Cournot equilibrium each firm earns a profit of
\[ \pi_{i}^{\;C} = (p_{C} - c)q_{i}^{\;C} =
 \left( \frac{a + 2c}{3} - c \right) \left( \frac{a - c}{3b} \right) =
             \frac{(a - c)^{2}}{9b} ,  \;\; i = 1, 2,  \]
so that industry profits, $\Pi^{C}$, are twice that, but still less than
the monopoly profit:
\begin{equation}
\Pi^{C} = \frac{2(a - c)^{2}}{9b} < \frac{(a - c)^{2}}{4b} = \pi_{m}.
\label{eq:Cournot-profit} \end{equation}
Since each duopolist in the Cournot equilibrium has the same cost
function, and it is the same as that of the monopolist whose profits are
given in eq.\ (\ref{eq:mono-profit}), the duopolists {\it could\/} earn
the monopoly profit by jointly producing the monopoly output of
\mbox{$Q_{m} = (a - c)/2b$}.  Therefore, a feasible collusive
equilibrium is one in which each firm produces \mbox{$q_{i} = (a-c)/4b$}
(\mbox{$i = 1,2$}), which is half the monopoly output, and each firm
earns a profit of \mbox{$\pi_{i} = (a-c)^{2}/8b$} (\mbox{$i = 1,2$}),
which is half the monopoly profit.  Aside from the fact that each
firm will be tempted to ignore the agreement and increase its profit
by moving to its reaction curve, a more fundamental problem lies in
initially allocating the monopoly output between the two duopolists;
since the firms are technologically identical in every way and possess
identical and constant average and marginal costs, there does not exist
a unique way to allocate the monopoly output between them.  In
maximizing their joint profit, the firms want to solve the following
problem,
\[ \max_{q_{1}, q_{2}} \; (\pi_{1} + \pi_{2}) =
 p(Q) \cdot q_{1} - c \cdot q_{1} + p(Q) \cdot q_{2} - c \cdot q_{2}. \]
Unfortunately, however, the above problem can be written in such a way
as to make the individual $q_{i}$'s indistinguishable:
\begin{eqnarray}
\max_{q_{1}, q_{2}} \; (\pi_{1} + \pi_{2}) & = &
      p(Q) \cdot (q_{1} + q_{2}) - c \cdot (q_{1} + q_{2})  \nonumber \\
        & = & p(Q) \cdot Q - c \cdot Q . \label{eq:Pi(Q)} \end{eqnarray}
Clearly, while the above problem may have a unique solution for a single
choice variable $Q$, and we would call that solution value the monopoly
output $Q_{m}$, the problem will not have a unique solution for the two
choice variables $q_{1}$ and $q_{2}$; there would be an infinite number
of solutions, all of which can be described by
\mbox{$q_{1} + q_{2} = Q_{m}$}.  This indeterminacy raises an obvious
question:  since the form of $p(Q)$ obviously plays no significant
role here, for what {\it cost functions} will this indeterminacy
occur?  Is it {\it only\/} when the two firms have identical cost
functions of the form given in (\ref{eq:C(q)}), or are there other
instances as well?

\section{The Indeterminate Cases}
The Cournot model of eqs.~(\ref{eq:demand}) and (\ref{eq:C(q)}) does not
admit a unique allocation of outputs between the two firms when they
collude because of the way in which their cost functions combine in the
expression of their joint profit.  Consider the joint profit of two
duopolists when they have the same cost function, $C(q)$, but it is not
necessarily given by the linear form in eq.~(\ref{eq:C(q)}),
\begin{eqnarray}
\Pi^{C} & = & [p(Q) \cdot q_{1} - C(q_{1})] +
          [p(Q) \cdot q_{2} - C(q_{2})]  \label{eq:general-case} \\
        & = & p(Q) \cdot Q - [C(q_{1}) + C(q_{2})].
\label{eq:general-case,reduced}  \end{eqnarray}
Note that eqs.~(\ref{eq:general-case}) and
(\ref{eq:general-case,reduced}) are perfectly general expressions of
profit for two different situations:  colluding duopolists producing a
homogeneous product, and a two-plant firm producing one good for sale in
one market.  In each situation the firms, or firm, face demand
represented by the inverse demand function $p(Q)$.  It should be
pointed out that we longer assume the inverse demand function is
linear; the results of this paper in no way depend on linear demand.
Two duopolists with identical cost functions as in (\ref{eq:C(q)}) will
be unable to uniquely allocate the monopoly output between them, and one
firm with two plants possessing identical cost functions as in
(\ref{eq:C(q)}) will be unable to uniquely allocate the firm's optimal
output.   These allocative indeterminacies arise because the sum
of the two relevant cost functions, \mbox{$C(q_{1}) + C(q_{2})$}, can
also be written in the form \mbox{$C(q_{1} + q_{2}) = C(Q)$}, which
utilizes the {\it same\/} cost function $C$ but depends only on the
total output of the two firms or plants:
\[        C(q_{1}) + C(q_{2}) = c\,q_{1} + c\,q_{2} =
          c\,(q_{1} + q_{2}) = C(q_{1} + q_{2}) = C(Q).      \]
This makes the variables $q_{1}$ and $q_{2}$ indistinguishable, as can
be seen in eq.~(\ref{eq:Pi(Q)}) as well.  Then the condition that
characterizes an allocative indeterminacy in the collusion and
multi-plant problems is the following:
\begin{equation}
C(q_{1}) + C(q_{2}) = C(q_{1} + q_{2}).
\label{eq:Cauchy} \end{equation}
We would like to know what functions $C$ satisfy eq.~(\ref{eq:Cauchy}),
which is known as the ``Cauchy equation'' in the study of functional
equations.  The study of functional equations attempts to solve
equations for the form of unknown {\it functions}, much as the study of
algebra attempts to solve equations for the values of unknown
{\it variables}.  The functional equation in (\ref{eq:Cauchy}) was
solved in 1821 by A.L.\ Cauchy and the equation is known to be satisfied
[for all real values of $q_{1}$ and $q_{2}$] by {\it only one}
functional form (see Acz\'{e}l, 1966, p.~34), namely
\[ C(q) = c \, q, \]
which is exactly the form we considered in eq.~(\ref{eq:C(q)}).
Although in the previous section we used a linear market demand, it
would not be difficult to show that a cost function with the form given
in eq.~(\ref{eq:C(q)}) is sufficient to lead to an allocative
indeterminacy; we now see that the cost function in
eq.~(\ref{eq:C(q)}) is also {\it necessary\/} to lead to an allocative
indeterminacy.  Furthermore, this result generalizes in a very
straightforward manner to the case of $n$ firms or plants, each
possessing the same cost function.  In this case, the condition that
leads to an allocative indeterminacy in the collusion and multi-plant
problems is just an obvious generalization of the Cauchy equation
(\ref{eq:Cauchy}),
\[ C(q_{1}) + C(q_{2}) + \cdots + C(q_{n}) =
                                 C(q_{1} + q_{2} + \cdots + q_{n}), \]
whose solution form for $C$ is still that of eq.~(\ref{eq:C(q)}),
\mbox{$C(q) = c \, q$}; see Acz\'{e}l, (1966, p.~31).  What we have
seen to this point can now be formally stated.
%------------------------------------------------------------------------
\begin{myprop}
If $n$ colluding firms or plants of a multi-plant firm
\mbox{$(2 \leq n < \infty )$} have identical cost functions, then an
allocative indeterminacy will arise if and only if the cost function for
each firm or plant is \mbox{$C(q) = c \, q$}.
\end{myprop}
%------------------------------------------------------------------------

Unfortunately, the result of Proposition~1 is not very interesting
because the assumption of identical cost functions is potentially quite
restrictive.  Therefore, we now consider the problem without restricting
the cost functions to be identical.

Suppose that $C_{1}(q_{1})$ is Firm or Plant~1's cost function and
$C_{2}(q_{2})$ is Firm or Plant~2's cost function.  The joint profit for
the two firms, or the profit of the two-plant firm, is
\[ \Pi = p(Q) \cdot Q - [C_{1}(q_{1}) + C_{2}(q_{2})]. \]
We will have an allocative indeterminacy if
\mbox{$C_{1}(q_{1}) + C_{2}(q_{2})$} can be written equivalently as some
third function $C^{*}$ depending only on the total output
\mbox{$Q = q_{1} + q_{2}$},
\begin{equation}
C_{1}(q_{1}) + C_{2}(q_{2}) = C^{*}(q_{1} + q_{2}).
\label{eq:Pexider} \end{equation}
Eq.~(\ref{eq:Pexider}) is a generalization of the Cauchy equation
(\ref{eq:Cauchy}) known as the ``Pexider equation.''  From
Acz\'{e}l~(1966, p.~142), the most general system of solutions of
Pexider's equation---i.e.\ the solution functional forms for $C_{1}$,
$C_{2}$, and $C^{*}$---with $C^{*}$ continuous, is given by the
following
\[ C_{1}(q_{1}) = c \, q_{1} + k_{1} , \;\;\;
   C_{2}(q_{2}) = c \, q_{2} + k_{2}  , \;\;\;
   C^{*}(Q) = c \, Q + k_{1} + k_{2} , \]
where $k_{1}$ and $k_{2}$ are arbitrary constants.  This result also
generalizes very easily to $n$ firms or plants, each possessing a
{\it different\/} cost function.  In such a case, the Pexider equation
in (\ref{eq:Pexider}) becomes,
\[ C_{1}(q_{1}) + C_{2}(q_{2}) + \cdots + C_{n}(q_{n}) =
                               C^{*}(q_{1} + q_{2} + \cdots + q_{n}), \]
and the solution functional forms for all of the cost functions is given
by
\begin{eqnarray}
\label{eq:Pexider-soln-1-to-n}
C_{i}(q_{i}) & = & c \, q_{i} + k_{i} , \;\;\; i = 1, 2, \ldots , n, \\
C^{*}(Q)     & = & c \, Q + \sum_{i = 1}^{n} k_{i} . \nonumber
\end{eqnarray}
This result can be formally stated as
%-----------------------------------------------------------------------
\begin{myprop}
If $n$ colluding firms or plants of a multi-plant firm
\mbox{$(2 \leq n < \infty )$} all have different cost functions, then an
allocative indeterminacy will arise if and only if the cost function for
Firm or Plant~$i$ is $C_{i}(q_{i}) = c \, q_{i} + k_{i}$,
for all $i = 1, 2, \ldots , n$.
\end{myprop}
%-----------------------------------------------------------------------
The interpretation of Proposition~2 in terms of economics is obvious:
The $n$ different cost functions all possess the same, constant marginal
cost, given by the parameter $c$, and the only way in which these
``different'' cost functions differ is in the value of the parameter
$k_{i}$, which has the obvious interpretation of Firm or Plant~$i$'s
fixed cost $(i = 1, 2, \ldots , n)$.

\section*{Conclusion}
What is plain to see from Propositions~1 and 2 is a very strong result:
an allocative indeterminacy can occur among $n$ colluding
oligopolists or in an $n$-plant firm \mbox{($2 \leq n < \infty$)}
{\it if and only if\/} all $n$ production units have the same
constant value for marginal cost.  There may be fixed costs present if
the cost functions may differ, but {\it only\/} with respect to the
units' fixed costs may their cost functions differ in order for an
allocative indeterminacy to occur.  This result suggests
that we look separately at the short- and long-run cases.  If we do
that, then Proposition~2 has a very strong result in each case.

In short run analysis, we will find an allocative indeterminacy
{\it if and only if\/} each of the $n$ production units
\mbox{$(2 \leq n < \infty )$} has a cost function of the form given in
eq.~(\ref{eq:Pexider-soln-1-to-n}):
\mbox{$C_{i}(q_{i}) = c\,q_{i} + k_{i}$}, where $k_{i}$ is unit $i$'s
fixed cost \mbox{$(i = 1, 2, \ldots , n)$}.

In long run analysis, we will face an allocative indeterminacy
{\it if and only if\/} each of the $n$ production units
\mbox{$(2 \leq n < \infty )$} has a cost function of the form given in
eq.~(\ref{eq:Pexider-soln-1-to-n}).  But long run cost functions
generally do not include fixed costs, so we are forced to set
\mbox{$k_{i} = 0$} for
all \mbox{$i = 1, 2, \ldots , n$}.  This leaves us with identical cost
functions for each firm or plant, and these identical cost functions are
characterized by identical and constant average and marginal costs, as
given in eq.~(\ref{eq:C(q)}):  \mbox{$C_{i}(q_{i}) = c \, q_{i}$} for
all \mbox{$i = 1, 2, \ldots , n$}.  Therefore, in long run analysis
there is no case in which the firms or plants have different cost
functions and an allocative indeterminacy occurs.

\section*{References}
\begin{description}
      \setlength{\itemsep}{0ex}   \setlength{\parsep}{0ex}
      \setlength{\leftmargin}{0em}
\item Acz\'{e}l, J.D.: {\em Lectures on Functional Equations and}
      {\em Their Applications.}  Academic Press, New York, 1966.
\item Friedman, J.W.: {\it Oligopoly Theory}.  Cambridge University
      Press, Cambridge, 1983.
\end{description}
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