Exercise 1: Stackelberg in quantities Suppose we have two firms, a leader and a follower. The leader's cost function is given by CL(qL) = 10 + qL and the follower's cost function is C (q,) = 5q, + 5. Market demand is given by P(Q) = 250 -5Q, where Q = qf + qL. a) (5 points) Find equilibrium price and quantities b) (5 points) Find profits, Consumer surplus and Total surplus Exercise 2: Static Cournot Suppose a market has two firms (Cournot duopoly) each of them with its respective cost function: Ci(q1) = qi + 2 and C2(q2) = 2q2 + 4. Each firm sets its output simultaneously. Market demand function is given by P(Q) = 200 - Q where Q = q1 + 92. a) (5 points) Obtain each firm's individual level of output b) (5 points) What is the market price? c) (5 points) Compute the HHI for this marketExercise 3: Stackelberg Vs Cournot Consider an industry producing a homogeneous good. Let q be the output for firm 1 and q2 the output for firm 2. Market demand is given by P(Q) = 200 - }0 where as usual Q = q1 + 42. Firm 1 has a cost function given by Ci(q1) = qi and firm 2 has cost function given by C2(q2) = 4q; + 10. For parts (a)-(d) assume that firm 1 sets its output first (leader) and firm 2 responds after that, by setting its own output (follower). For parts (e)-(f), assume firms choose their output simultaneously. a) (5 points) Set up firms 2's problem and find its best response function q2 = BR(q1) b) (10 points) Set up firm I's problem and calculate firm 1's profit as a function of its own output. Find firm 1 and firm 2's optimal outputs c) (5 points) Find the market price, the profits for both firms and consumer surplus. d) (5 points) In the Stackelberg setting, the leader firm produces more than the follower. Are there any features of the market that accentuate the output difference between the two firms? e) (10 points) Find the Cournot equilibrium price. Do consumers prefer firms competing Cournot style or Stackelberg style (Hint: Do not calculate consumer surplus to answer this question) f) (10 points) Suppose firms are competing in a market structure like the one in e), and a tax of T per dollar is placed on firm 2's total revenue. How would this affect firm 2's output and the output gap between both firms? Derive the best response function for firm 2. Exercise 4: Stackelberg in prices Suppose there are two firms in the market, a leader and a follower. Both have the same cost function: C(q) = cq and they face the following individual demands. qL(PL. P/) = & - PL - Y(PL - Pf) q,(PL. P;) = 0 - pf - Y(Pf - PL) where a = 3, y = 0.5 and c = 1 a) (10 points) Find the equilibrium prices, quantities and profits
