Cournot%20Example - @ & 1 ( q 1 ; q 2 ) =@q 1 =...

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Economics 3130 Cournot Duopoly: a numerical example The Model p = 100 (0 : 5) ± [ q 1 + q 2 ] : the inverse demand function. C 1 = 5 ± q 1 : the cost function of duopolist #1; C 2 = (0 : 5) ± q 2 2 : the cost function of duopolist #2; Verify that the reaction functions of the duopolists are given by: q 1 = 95 (0 : 5) ± q 2 :::::::: (1) and q 2 = 50 (0 : 25) ± q 1 : :::::::: (2) The Cournot equilibrium solution is given by ( q 1 ; q 2 ) = (80 ; 30) : The equilibrium price p = 45 1 = 3200 2 = 900 Given q 2 ; 1 ( q 1 ; q 2 ) = p ± q 1 5 ± q 1 = f 100 (0 : 5) ± [ q 1 + q 2 ] g ± q 1 5 ± q 1 :::: (3) 1 ( q 1 ; q 2 ) =@q 1 = 0 ; @ 2 1 ( q 1 ; q 2 ) =@ 2 q 2 1 < 0 : Now, from (3), we have
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Unformatted text preview: @ &amp; 1 ( q 1 ; q 2 ) =@q 1 = [100 &amp; (0 : 5) [ q 1 + q 2 ]] 1 &amp; (0 : 5) q 1 &amp; 5 = 0 :::::: (4) Simplify and get (1). Verify the second order condition @ 2 &amp; 1 ( q 1 ; q 2 ) =@ 2 q 2 1 = &amp; 1 &lt; : Given (1) and (2), q 2 = 50 &amp; (0 : 25) [95 &amp; (0 : 5) q 2 ] Solve and get q &amp; 2 = 30 : Now, from (1) get q &amp; 1 = 80 : p &amp; = 100 &amp; (0 : 5) [ q &amp; 1 + q &amp; 2 ] = 45 &amp; &amp; 1 = [100 &amp; (0 : 5) 110] 80 &amp; 5 80 = 3200 : Complete the other calculations. 1...
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