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Unit 13: Oligopoly




          The second order condition for equilibrium requires that                              Notes

                                      2
                                     ∂Π  i  ∂ 2 R i  −  ∂ 2 C i
                                     ∂ X 2 i  =  ∂ X 2 i  ∂ X 2 i   < 0 (i=1,2)
          or

                                            ∂  2 R  ∂ 2 C
                                               i  <  i
                                            ∂ X 2  ∂ X 2
                                               i    i
          Each duopolist’s MR must be increasing less rapidly than his MC, that is, the MC must cut the
          MR from below, for both duopolists.

                 Example: Assume that the market demand and the cost of the duopolists are

                           P  =  100–0.5(X +X )
                                       1  2
                           C   =  5X 1
                            1
                           C   =  0.5X 2
                            2      2

          The profits of the duopolists are
          1.   π  = PX –C  = [100–0.5(X +X )]X –5X 1
                                  1
                     1
                       1
                1
                                     2
                                        1
               or
               π  = 100X –0.5X –0.5X X –5X
                           2
                1     1     1   1  2  1
          2.   π  = PX –C  = [100–0.5(X +X )]X –0.5X   2
                                        2
                                             2
                     2
                       2
                                  1
                                     2
                2
               or
               π  = 100X –0.5X –0.5X X –0.5X 2
                           2
                2     2     2   1  2    2
               Collecting terms we have
               π  = 95X –0.5X –0.5X X 2
                          2
                                1
                           1
                     1
                1
               and
               π  = 100X –X –0.5X X
                         2
                2     2  2    1  2
          For profit maximisation under the Cournot assumption we have

                                      ⎧ ∂Π 1               ⎫
                                                −
                                      ⎪   = 0=95 X − 0.5X 2  ⎪
                                                  1
                                      ⎪ ∂ X 2              ⎪
                                      ⎨ ∂Π                 ⎬                     ...... (3)
                                      ⎪  2  = 0=100 2X − 0.5X  ⎪
                                                 −
                                      ⎪ ∂ X 2       2     1 ⎪ ⎭
                                      ⎩
          The reaction functions are:
                                           X  = 95–0.5X 2
                                            1
                                           X  = 50–0.25X
                                            2        1
          The graphical solution of Cournot’s model is found by the intersection of the two reaction curves
          which are plotted in Figure.
          Mathematically the solution of system (3) yields
                −
          X =95 0.5X 2 ⎤
            1
                −
          X =50 0.25X 1⎦ ⎥
            2
          X  = 95–0.5(50–0.25X )
                          1
           1
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