Difference between revisions of "Heterogeneous-cost Cournot Competition"
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Now we can solve for <math>\;q_i^*</math>. Substituting this into the original first-order condition gives: | Now we can solve for <math>\;q_i^*</math>. Substituting this into the original first-order condition gives: | ||
| − | :<math> | + | :<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== | ==Market clearing price and profits== | ||
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And then substitute into <math>\;p=A-BQ</math> to get: | And then substitute into <math>\;p=A-BQ</math> to get: | ||
| − | :<math>p^* | + | :<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>: | 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^* | + | :<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== | ==Comparison to other solutions== | ||
Revision as of 21:35, 10 November 2013
Contents
The Set-up
Suppose that we have inverse-demand given by
- [math]p=A-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:
- [math]\pi_i = q_i \left ( p(Q) - c_i \right)[/math]
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]\sum_{j\ne i}\;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]
