Consider a homogeneous-good duopoly in which inverse demand is given by P(Q) = 2 − Q, where Q, homework help

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3. Consider a homogeneous-good duopoly in which inverse demand is given by P(Q) = 2 − Q, where Q

denotes the total output in the market. Firm 1’s marginal cost is c1 = 1 while firm 2’s is c2 = c ∈ [0, 1.5].

(a) (3 points) Determine the Cournot equilibrium outputs and the associated social welfare (as a function

of the parameter c).

(b) (3 points) Determine the Stackelberg equilibrium outputs when firm 1 chooses its output first. Determine

the associated social welfare.

(c) (3 points) For what values of c does the Cournot equilibrium give greater social welfare and for which

does the Stackelberg equilibrium?

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Problem Set #3 Prof. D. Malueg Econ 160 Solutions are due at the beginning of class: November 17, 2016 Instructions: Answer all questions and show your work. Be sure to put all group members’ names on the solution set for your study group. 1. Consider a market in which firms produce a homogeneous good, and market inverse demand is given by P (Q) = 1 − Q, where Q denotes total output of the firms. Suppose there are two firms. Firm 1’s cost function is C1 (q) = q 2 and firm 2’s cost function is C2 (q) = 21 q 2 . (a) (3 points) Calculate the Cournot (Nash) equilibrium outputs for each firm. (b) (3 points) Calculate the outputs for each firm that together maximize the combined profit of the two firms. (c) (3 points) Suppose this is a competitive market. Determine the equilibrium output per firm. (d) (3 points) Determine the deadweight welfare loss at the Cournot duopoly equilibrium. 2. Now suppose there are n firms, each with the cost function C(q) = 1 2 2q . When there are n firms, the market demand is given by Q(p|n) = n (1 − p), which yields constant demand per firm (at a given price) as the number of firms (n) in the market changes. (a) (3 points) For given n, calculate the Cournot (Nash) equilibrium output per firm and the resulting price. (b) (3 points) As n → ∞ (i.e., n gets very large, approaching infinity), what happens to the Cournot equilibrium price? (Calculate the limiting price.) (c) (3 points) Now suppose this market is competitive. For given n, determine each firm’s supply function and the competitive equilibrium price. Determine what happens to this equilibrium price as n → ∞. 3. Consider a homogeneous-good duopoly in which inverse demand is given by P (Q) = 2 − Q, where Q denotes the total output in the market. Firm 1’s marginal cost is c1 = 1 while firm 2’s is c2 = c ∈ [0, 1.5]. (a) (3 points) Determine the Cournot equilibrium outputs and the associated social welfare (as a function of the parameter c). (b) (3 points) Determine the Stackelberg equilibrium outputs when firm 1 chooses its output first. Determine the associated social welfare. (c) (3 points) For what values of c does the Cournot equilibrium give greater social welfare and for which does the Stackelberg equilibrium?
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