This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Calculus and Game Theory 2009 Finding Nash equilibria in continuous strategies 1 Cournot duopoly with specific numbers Two firms compete in quantities. The market inverse demand curve is P = 56 2 Q , where Q = q 1 + q 2 . The cost functions are c 1 ( q 1 ) = 20 q 1 and c 2 ( q 2 ) = 20 q 2 . Step 1: formalise payoff functions Firm 1s payoff is her profits; revenue minus costs: 1 ( p 1 , p 2 ) = P q 1 20 q 1 where P = 56 q 1 q 2 = (56 2 ( q 1 q 2 )) q 1 20 q 1 = (56 20 2 q 1 2 q 2 ) q 1 = (36 2 q 1 2 q 2 ) q 1 Step 2: take a firstorder condition Using the product rule: 1 q 1 = ( 2) q 1 + (36 2 q 1 2 q 2 ) 1 = 36 4 q 1 2 q 2 Setting this slope to zero gives us a firstorder condition. Step 3: solve the firstorder condition to get a bestresponse rela tion When the slope of firm 1s payoff function is zero as q 1 is increased, it must be true that: 0 = 36 4 q 1 2 q 2 4 q 1 = 36 2 q 2 q 1 = 36 2 q 2 4 = B 1 ( q 2 ) 1 Calculus and Game Theory 2009 Note that this is a linear function with vertical intercept 36 / 4 = 9 and slope 1 2 . Step 4: simultaneously solve the bestresponses (or the firstorder conditions) to get the Nash equilibrium If the two best response functions are: q 1 = 9 q 2 2 , q 2 = 9 q 1 2 then the Nash equilibrium will be the intersection of these best responses. We can substitute the second one into the first, and then solve for q 1 . q 1 = 9 1 2 bracketleftbigg 9 1 2 q 1 bracketrightbigg q 1 q 1 1 4 = 9 9 2 q 1 parenleftbigg 1 1 4 parenrightbigg = 9 parenleftbigg 1 1 2 parenrightbigg q 1 3 4 = 9 2 q 1 = 4 3 9 2 = 2 3 = 6 The short cut The two firms have symmetric payoff functions, so it is a reasonable conclu...
View
Full
Document
This note was uploaded on 05/24/2011 for the course ECON 201 taught by Professor Paulclacott during the Fall '10 term at Victoria Wellington.
 Fall '10
 PaulClacott
 Microeconomics, Game Theory

Click to edit the document details