A Cournot Model with Product Differentiation

3. A Cournot Model with Product Differentiation Consider two firms competing in quantities a la Cournot selling differentiated goods. The inverse demand function for every firm i is p.(qi, q,) = a - bq; - dq; where i, j E 1, 2 and i / j. This means that the inverse demand functions for Firm 1 and Firm 2 are p (91, 92) = a-bq1 -dq2 and pz(91, 92) = a-bq2 -dq1, respectively. In this model b and d represent the degree of product differentiation for every firm i and j. b is assumed to be greater than zero (i.e. b > 0), and for simplicity lets assume that d can take on any value between zero and b (i.e. b 2 d 2 0). The products are considered to be differentiated (also known as heterogeneous goods) if b / d, and if b = d the firms' products are identical (also known as homogeneous goods). Each firm has the same marginal cost of c, and all firm's fixed costs are assumed to be equal to zero (i.e. F = 0). We formulate each firm's Profit Maximization Problem (i.e. PMP) as: CALCULUS PART: max m, = [a - (bq: + dq,)] q: - cq. 4:20 ami (qi, q)) aqi = a - 2bq - dq; - c = 0 (5) And through symmetry we know that firm j's PMP is max a; = [a - (bq; + dq.)] q; - cq; 9/20 On (qi, 9)) = a - 2bq; - dq; - c =0 (6) where we now have two equations ((5) and (6)), and two choice variables (q; and q,) to solve for. CALCULUS PART FINISHED. YOUR CALCULATIONS START HERE. (a) Please find the Best Response Functions for each firm (i.e. BRF: = q:(q;) and BRF, = q,(q:)). How does firm i respond with it's own quantity with an increase in a, b, c, d and q,? (b) Find the optimal equilibrium allocation for each firm when they are competing a la Cournot with differentiated products. That is, find q, and q;. What happens to q, as d increases towards b (i.e. d -+ b)? What happens to q, as d decreases towards zero (i.e. d -+ 0)? What is the intuition behind this? (c) Find the optimal price for each firm (i.e. p; = a - bq; - dq; and p; = a - bq; - dq;). (d) Find the equilibrium profits of each firm (i.e. ; and #;)