Our discussion of Cournot in the lecture focused on the case with homoge- nous products. In many markets, products produced by competing firms are not perfect but imperfect substitutes. We consider such a case here. There are in total N firms, each producing a unique variety which are imperfect substitutes to each other. Since they are different, each variety i with i = 1, 2, ..., N, has a price p₁. Let the symmetric inverse demand of each variety be Pi(gi, Q-i) = abgi – oQ-is where Q = 9; is the total output produced by all firms other than firm i. Parameters a, b, and o are all strictly positive. The parameter a is the price when no firms supply to the market. The parameter b measures how sensitive p; is to a change of firm i's own supply, while o measures such sensitivity to other firms' total supply. Notice that in a Cournot model, we would have b = o. An assumption o [-b/(N-1), b] is imposed to ensure the problem is well-defined (this assumption is only for completeness of the problem and is irrelevant for your derivation). Each firm has a constant marginal cost c> 0. We assume that firms compete by choosing quantities, just like in a standard Cournot model with homogenous products. Write down the profit maximization problem for each firm i. Write down each firm's best response function. Are quantities strategic complements or substitutes? Find the symmetric equilibrium quantity q". a. b. C.
Our discussion of Cournot in the lecture focused on the case with homoge- nous products. In many markets, products produced by competing firms are not perfect but imperfect substitutes. We consider such a case here. There are in total N firms, each producing a unique variety which are imperfect substitutes to each other. Since they are different, each variety i with i = 1, 2, ..., N, has a price p₁. Let the symmetric inverse demand of each variety be Pi(gi, Q-i) = abgi – oQ-is where Q = 9; is the total output produced by all firms other than firm i. Parameters a, b, and o are all strictly positive. The parameter a is the price when no firms supply to the market. The parameter b measures how sensitive p; is to a change of firm i's own supply, while o measures such sensitivity to other firms' total supply. Notice that in a Cournot model, we would have b = o. An assumption o [-b/(N-1), b] is imposed to ensure the problem is well-defined (this assumption is only for completeness of the problem and is irrelevant for your derivation). Each firm has a constant marginal cost c> 0. We assume that firms compete by choosing quantities, just like in a standard Cournot model with homogenous products. Write down the profit maximization problem for each firm i. Write down each firm's best response function. Are quantities strategic complements or substitutes? Find the symmetric equilibrium quantity q". a. b. C.
Chapter15: Imperfect Competition
Section: Chapter Questions
Problem 15.3P
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Question
![Our discussion of Cournot in the lecture focused on the case with homoge-
nous products. In many markets, products produced by competing firms
are not perfect but imperfect substitutes. We consider such a case here.
There are in total N firms, each producing a unique variety which are
imperfect substitutes to each other. Since they are different, each variety
i with i = 1, 2, ..., N, has a price pi. Let the symmetric inverse demand of
each variety be
Pi(gi, Q-i) = a - bqi - oQ-i,
-i
where Q_₁ = [q; is the total output produced by all firms other than firm
i. Parameters a, b, and o are all strictly positive. The parameter a is the
price when no firms supply to the market. The parameter b measures how
sensitive p; is to a change of firm i's own supply, while o measures such
sensitivity to other firms' total supply. Notice that in a Cournot model, we
would have b = o. An assumption o € [-b/(N-1), b] is imposed to ensure
the problem is well-defined (this assumption is only for completeness of the
problem and is irrelevant for your derivation).
Each firm has a constant marginal cost c> 0. We assume that firms
compete by choosing quantities, just like in a standard Cournot model with
homogenous products.
Write down the profit maximization problem for each firm i.
a.
b.
C.
d.
e.
f.
g.
Write down each firm's best response function. Are quantities
strategic complements or substitutes?
Find the symmetric equilibrium quantity q*.
Find the symmetric equilibrium price p* for each variety.
How does q* change when N increases? Explain why.
How does q* change when o increases? Explain why.
Suppose o is instead of a negative value. How does q* change
when N increases? Explain why.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a4decfa-c86a-4508-ad17-cd6b643e587a%2Ffc26f7d3-e179-4046-9567-4f9a4cc5debc%2Fmhyo1js_processed.png&w=3840&q=75)
Transcribed Image Text:Our discussion of Cournot in the lecture focused on the case with homoge-
nous products. In many markets, products produced by competing firms
are not perfect but imperfect substitutes. We consider such a case here.
There are in total N firms, each producing a unique variety which are
imperfect substitutes to each other. Since they are different, each variety
i with i = 1, 2, ..., N, has a price pi. Let the symmetric inverse demand of
each variety be
Pi(gi, Q-i) = a - bqi - oQ-i,
-i
where Q_₁ = [q; is the total output produced by all firms other than firm
i. Parameters a, b, and o are all strictly positive. The parameter a is the
price when no firms supply to the market. The parameter b measures how
sensitive p; is to a change of firm i's own supply, while o measures such
sensitivity to other firms' total supply. Notice that in a Cournot model, we
would have b = o. An assumption o € [-b/(N-1), b] is imposed to ensure
the problem is well-defined (this assumption is only for completeness of the
problem and is irrelevant for your derivation).
Each firm has a constant marginal cost c> 0. We assume that firms
compete by choosing quantities, just like in a standard Cournot model with
homogenous products.
Write down the profit maximization problem for each firm i.
a.
b.
C.
d.
e.
f.
g.
Write down each firm's best response function. Are quantities
strategic complements or substitutes?
Find the symmetric equilibrium quantity q*.
Find the symmetric equilibrium price p* for each variety.
How does q* change when N increases? Explain why.
How does q* change when o increases? Explain why.
Suppose o is instead of a negative value. How does q* change
when N increases? Explain why.
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