Heterogeneous-cost Cournot Competition - Revision history

imported>Ed at 22:29, 11 November 2013

2013-11-11T22:29:20Z

imported>Ed at 02:44, 11 November 2013

2013-11-11T02:44:42Z

imported>Ed: /* Market clearing price and profits */

2013-11-11T02:36:46Z

Market clearing price and profits

imported>Ed at 02:35, 11 November 2013

2013-11-11T02:35:38Z

imported>Ed: /* Solving for optimum quantities */

2013-11-11T02:34:44Z

Solving for optimum quantities

imported>Ed at 02:33, 11 November 2013

2013-11-11T02:33:35Z

imported>Ed: New page: Solving the heterogeneous-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. ==The Set-up...

2013-11-11T02:13:20Z

New page: Solving the heterogeneous-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. ==The Set-up...

New page

Solving the heterogeneous-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.

==The Set-up==

Suppose that we have

==Solving for optimum quantities==

==Market clearing price and profits==

==Comparison to other solutions==

@@ Line 44: / Line 44: @@
 Prices always come from the demand function. Firm's don't make profits by setting prices, they 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>
+:<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:

@@ Line 38: / Line 38: @@
 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>
+:<math>q_i^* = \frac{ A - c_i(n+1) + \sum_{i=1}^n c_i }{B (n+1)}</math>
 ==Market clearing price and profits==
@@ Line 55: / Line 55: @@
 ==Comparison to other solutions==

@@ Line 42: / Line 42: @@
 ==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:
+Prices always come from the demand function. Firm's don't make profits by setting prices, they 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>

@@ Line 38: / Line 38: @@
 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>
+:<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==
@@ Line 48: / Line 48: @@
 And then substitute into <math>\;p=A-BQ</math> to get:
-:<math>p^* & = \frac{A + \sum_{i=1}^n c_i}{n+1}</math>
+:<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>
+:<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==

@@ Line 15: / Line 15: @@
 :<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:
+We now need two constraints to find the solution. First, we need to solve for <math>\sum_{j\ne i}\;q_j^*</math>.
 :<math>
@@ Line 37: / Line 39: @@
 :<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==

@@ Line 1: / Line 1: @@
-Solving the heterogeneous-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.
+==The Set-up==
-==The Set-up==
+Suppose that we have inverse-demand given by
-Suppose that we have
+:<math>\pi_i = q_i \left ( p(Q) - c_i \right)</math>
 ==Solving for optimum quantities==
 ==Market clearing price and profits==
 ==Comparison to other solutions==