3. Consider a two-stage game of R&D competition. The players are firm i and j, each produces with different marginal cost c; and cj, where i, j = 1, 2 and i ‡ j. The rules are the firms first invest, denoted by I, and I;, in cost-reducing innovation, they then engage in Cournot competition in the output market by deciding qk, k = i, j and i # j simultaneously. The payoffs to the firms are their profits. (a) write out the objective function for firm i and j at the R&D stage and that at the output stage. (b) Assuming that p'(Q) < 0 and p'(Q)+p"(Q)qi <0 for all 0≤ qk ≤ >0 and Q, show that 1 p"Q+3p' (c) dvi dli < 0. əqi p"qj+2pi aci pi(p"Q+3p') = <0 2 əqi acj == p"qi+p' p'(p"Q+3p') aQ aci = show, in the R&D stage, that the first-order conditions are = [p' (Q(C₁, c)) ³9/(c) — 1] qi(c. c) = Jaj(cuci) aci — 1] qi(C₁, Cj) = qp¹ (1i,1, čį, ēj) = 0. And Ij=constant (d)--) In a closed economy, show that the total differentiation of the national welfare function is dW = −p'(q¡dqi+qjdqj) — q₁dc₁ — qjdc¡ – dl¡ – dlj. - -

3. Consider a two-stage game of R&D competition. The players are firm i and j, each produces with different marginal cost c; and cj, where i, j = 1, 2 and i ‡ j. The rules are the firms first invest, denoted by I, and I;, in cost-reducing innovation, they then engage in Cournot competition in the output market by deciding qk, k = i, j and i # j simultaneously. The payoffs to the firms are their profits. (a) write out the objective function for firm i and j at the R&D stage and that at the output stage. (b) Assuming that p'(Q) < 0 and p'(Q)+p"(Q)qi <0 for all 0≤ qk ≤ >0 and Q, show that 1 p"Q+3p' (c) dvi dli < 0. əqi p"qj+2pi aci pi(p"Q+3p') = <0 2 əqi acj == p"qi+p' p'(p"Q+3p') aQ aci = show, in the R&D stage, that the first-order conditions are = [p' (Q(C₁, c)) ³9/(c) — 1] qi(c. c) = Jaj(cuci) aci — 1] qi(C₁, Cj) = qp¹ (1i,1, čį, ēj) = 0. And Ij=constant (d)--) In a closed economy, show that the total differentiation of the national welfare function is dW = −p'(q¡dqi+qjdqj) — q₁dc₁ — qjdc¡ – dl¡ – dlj. - -

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter15: Imperfect Competition

Section: Chapter Questions

Problem 15.8P

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