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Solving the heterogeneous==The Set-cost Cournot competition model is pretty straight forward and is a good exercise for both undergraduate and graduate students. Here's how it's done.up==
Suppose that we have :<math>\pi_i = q_i \left ( p(Q) - c_i \right)</math>
Heterogeneous-cost Cournot Competition (view source)
Revision as of 22:33, 10 November 2013
, 22:33, 10 November 2013no edit summary
Suppose that we have inverse-demand given by :<math>p==The SetA-up=BQ, \mbox{where }Q =\sum_{i}^{n} q_i </math> and there are <math>\;n</math> firms in the market. Firm profit is then given by:
==Solving for optimum quantities==
Begin by taking a first-order condition:
:<math>2q_i^* = \frac{A - c_i}{B} - \sum_{j \ne i} q_j</math>
We now need two constraints to find the solution. First, we need to solve for <math>\;q_j^*</math>. There are <math>\;n</math> of these first order conditions:
:<math>
\begin{array}{lclllllllllllll}
2q_1^* &=& \frac{A - c_1}{B}& - &(&0 &+ &q_2 &+ &\ldots &+ &q_{n-1} &+ &q_n &)\\
2q_2^* &=& \frac{A - c_2}{B}& - &(&q_1 &+ &0 &+ &\ldots &+ &q_{n-1} &+ &q_n &)\\
\quad\vdots & & \quad\vdots &&&\vdots&&\vdots&&&&\vdots&&\vdots& \\
2q_n^* &=& \frac{A - c_n}{B}& - &(&q_1 &+ &q_2 &+ &\ldots &+ &q_{n-1} &+ &0 &)
\end{array}
</math>
As each of these conditions holds in equilibrium, their sum must all also hold:
:<math>\sum_{i=1}^n 2 q_i^* = \frac{n A - \sum_{i=1}^n c_i - B (n-1) \sum_{i=1}^n q_i}{B} </math>
Noting that <math>\;Q=Q^*</math> in equilibrium, and rearranging gives
:<math>\sum_{j \ne i}^n q_j^* = - q_i^* + \frac{n A - \sum_{i=1}^n c_i}{B (n+1)}</math>
Now we can solve for <math>\;q_i^*</math>. Substituting this into the original first-order condition gives:
:<math>$q_i^* = \frac{\left ( A - c_i(n+1) + \sum_{i=1}^n c_i \right )}{B (n+1)}</math>
==Market clearing price and profits==
Prices always come from the demand function. Firm's don't make profits by setting prices, the make profits by providing less than competitive quantities. Accordingly first find the total quantity provided:
:<math>Q^* = \sum_{i=1}^n q_i^* = \frac{\left ( nA - sum_{i=1}^n c_i \right )}{B (n+1)}</math>
And then substitute into <math>\;p=A-BQ</math> to get:
:<math>p^* & = \frac{A + \sum_{i=1}^n c_i}{n+1}</math>
Now find firm profits by subsituting both <math>\;q_i^*</math> and <math>\;p^*</math> into <math>\;\pi_i = q_i \left ( p(Q) - c_i \right)</math>:
:<math>\pi_i^* &= \frac{1}{B }\left (\frac{A - c_i(n+1) + \sum_{i=1}^n c_i }{(n+1)} \right )^2</math>
==Comparison to other solutions==
