Competition of Two Firms as Cournot Duopolists

Use Wolfram Mathematica to solve the following problems:

Consider an industry consisting of two firms, competing as Cournot duopolists. The two firms produce an homogenous good with demand given by p = A − Q, where Q = q1 + q2 is the sum of individual outputs. Each firm’s marginal costofproductionisgivenbyci =C−xi−βxj,i,j=1,2andj̸=i,wherexi andxj
are the firms’ R&D investment. The term β ∈ [0, 1] is the R&D spillover parameter, which determines the size of the spillover from the R&D investment made by the rival firm. The cost of R&D is given by γx2i /2. The decision marking sequence is as follows: (1) firms choose R&D effort, then (2) firms choose their output.

1. Suppose that the firms behave non-cooperatively at both stages, find the equilibrium R&D and output.

2. Suppose that the firms only compete in the final goods market, but choose to cooperate when choosing their R&D investment, find find equilibrium R&D and output.

3. Compare the R&D investment and output levels in the two models for the entire ranges of R&D spillover, when does the non-cooperative R&D model dominate the cooperative model? Are there any parameter values for which they are equal? In which model is welfare greater and is this true for all values of γ.

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