Problem 3. Longer problem. Consider 2 firms F1 and F2 that produce identical products and have identical cost functions c₁ (1) y and c₂ (y2) = y. The demand function is p(T) = 24 - YT where Ty1+y2 is the output produced by the two firms. i) Find a competitive equilibrium including the price, quantity and profit in which price marginal cost. ii) Find the monopoly solution including the price, quantity and profit in which marginal revenue marginal cost. iii) Find the oligopoly (duopoly) Nash equilibrium including the price, quantity and profit in which both firms engage in Cournot competition. iv) Suppose that the oligopolists form a cartel and play a repeated game. Assume that they maximize infinite discounted profits for periods t = 0,1,0,..., specifically, To+++ 2 (1+r)² The duopolists can remain in cartel forever and share the monopoly profit equally so each of them gets + + 2 2(1+r) 2(1+r)² +... Alternatively, one of them can decide to break the agreement (cheat) to get a higher profit at t=0 (find such profit from the best response function) and to get Nash equilibrium profit for the rest of the time forever. NE NE cheating+ + +... (1+r) (1+r)² Find the interest rate at which the duopolists decide to break the agreement. (Hint: use the formula for infinite geometric sequence (1+r) + 2(1+r)² +...= ½). v) Find the oligopoly (duopoly) solution to the sequential Stackelberg competition including the price, quantity and profit.
Problem 3. Longer problem. Consider 2 firms F1 and F2 that produce identical products and have identical cost functions c₁ (1) y and c₂ (y2) = y. The demand function is p(T) = 24 - YT where Ty1+y2 is the output produced by the two firms. i) Find a competitive equilibrium including the price, quantity and profit in which price marginal cost. ii) Find the monopoly solution including the price, quantity and profit in which marginal revenue marginal cost. iii) Find the oligopoly (duopoly) Nash equilibrium including the price, quantity and profit in which both firms engage in Cournot competition. iv) Suppose that the oligopolists form a cartel and play a repeated game. Assume that they maximize infinite discounted profits for periods t = 0,1,0,..., specifically, To+++ 2 (1+r)² The duopolists can remain in cartel forever and share the monopoly profit equally so each of them gets + + 2 2(1+r) 2(1+r)² +... Alternatively, one of them can decide to break the agreement (cheat) to get a higher profit at t=0 (find such profit from the best response function) and to get Nash equilibrium profit for the rest of the time forever. NE NE cheating+ + +... (1+r) (1+r)² Find the interest rate at which the duopolists decide to break the agreement. (Hint: use the formula for infinite geometric sequence (1+r) + 2(1+r)² +...= ½). v) Find the oligopoly (duopoly) solution to the sequential Stackelberg competition including the price, quantity and profit.
Chapter1: Making Economics Decisions
Section: Chapter Questions
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Transcribed Image Text:Problem 3. Longer problem.
Consider 2 firms F1 and F2 that produce identical products and have identical cost
functions c₁ (31) = y2 and c₂ (y2) = y2. The demand function is p (yr) = 24 - yr where
yry1 + y2 is the output produced by the two firms.
i) Find a competitive equilibrium including the price, quantity and profit in which price
= marginal cost.
ii) Find the monopoly solution including the price, quantity and profit in which marginal
revenue marginal cost.
iii) Find the oligopoly (duopoly) Nash equilibrium including the price, quantity and
profit in which both firms engage in Cournot competition.
iv) Suppose that the oligopolists form a cartel and play a repeated game. Assume that
they maximize infinite discounted profits for periods t = 0, 1, 0, ..., specifically, To ++
(1+r)² + .... The duopolists can remain in cartel forever and share the monopoly profit
equally so each of them gets
Tm
2
+
2
77m
2(1+r) 2(1+r)²
cheating +
+
Alternatively, one of them can decide to break the agreement (cheat) to get a higher profit
at t= 0 (find such profit from the best response function) and to get Nash equilibrium
profit for the rest of the time forever.
NE
NE
(1+r) (1+r)²
+...
+
+...
Find the interest rate at which the duopolists decide to break the agreement. (Hint: use
the formula for infinite geometric sequence (1+r) +
+... = =).
2(1+r)²
v) Find the oligopoly (duopoly) solution to the sequential Stackelberg competition
including the price, quantity and profit.
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