. Let's study the Cournot model more generally. Assume that there are n firms, where n is exogenously given. The output of the i th firm is qi and the total output Q is the sum of the output of each firm: Q q1 + q2 +. . . + qn. The (inverse) market demand function is given by p(Q) = a - bQ and each Firm i's total cost is given by C( qi) = mqi, where a, b, and m are all positive constants. (a) We know that Firm 1 tries =
. Let's study the Cournot model more generally. Assume that there are n firms, where n is exogenously given. The output of the i th firm is qi and the total output Q is the sum of the output of each firm: Q q1 + q2 +. . . + qn. The (inverse) market demand function is given by p(Q) = a - bQ and each Firm i's total cost is given by C( qi) = mqi, where a, b, and m are all positive constants. (a) We know that Firm 1 tries =
Chapter1: Making Economics Decisions
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![Let's study the Cournot model more generally. Assume that there are n firms, where
n is exogenously given. The output of the i th firm is qi and the total output Q is the
sum of the output of each firm: Q = q1 + q2 + ... + qn. The (inverse) market
demand function is given by p(Q) = a - bQ and each Firm i's total cost is given by C(
qi) = mqi, where a, b, and m are all positive constants. (a) We know that Firm 1 tries
to maximize its profit through its choice of q1, i.e. Firm 1 chooses q1 to solve max q
1 \pi 1(q1, q2, . .., qn) = q1·p(Q) - C(q1) Write down the profit maximizing
condition for Firm 1, and obtain the best - response function q* 1 (q2, q3, . .., qn)
for Firm 1. (b) From part (a), we know that each Firm i 0 s best response function is
given by q * i (q-i) = am 2b - Pj6 = iqj 2, where q-i denotes the quantities
chosen by all firms besides Firm i and Pj6 = iqj = q1 + q2 + ... qi-1 + qi
+1 + . . . qn. Find the symmetric Nash equilibrium, i.e. find the Nash equilibrium in
which all firms choose the same quantity: q* 1 = q*2 = ... = q*n = q*.
What is the corresponding market price in this equilibrium? (c) For the symmetric Nash
equilibrium found in part (b), if we set n = 1, we know that the equilibrium price and
quantity correspond to the monopoly price and quantity. What then would happen to
the equilibrium price and quantity as n approaches infinity?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F696d38cd-963d-415c-83b1-46a2700ebc8d%2Ff6ef375f-394a-48f7-9a49-60bf89a1ac27%2Fqr3w08r_processed.png&w=3840&q=75)
Transcribed Image Text:Let's study the Cournot model more generally. Assume that there are n firms, where
n is exogenously given. The output of the i th firm is qi and the total output Q is the
sum of the output of each firm: Q = q1 + q2 + ... + qn. The (inverse) market
demand function is given by p(Q) = a - bQ and each Firm i's total cost is given by C(
qi) = mqi, where a, b, and m are all positive constants. (a) We know that Firm 1 tries
to maximize its profit through its choice of q1, i.e. Firm 1 chooses q1 to solve max q
1 \pi 1(q1, q2, . .., qn) = q1·p(Q) - C(q1) Write down the profit maximizing
condition for Firm 1, and obtain the best - response function q* 1 (q2, q3, . .., qn)
for Firm 1. (b) From part (a), we know that each Firm i 0 s best response function is
given by q * i (q-i) = am 2b - Pj6 = iqj 2, where q-i denotes the quantities
chosen by all firms besides Firm i and Pj6 = iqj = q1 + q2 + ... qi-1 + qi
+1 + . . . qn. Find the symmetric Nash equilibrium, i.e. find the Nash equilibrium in
which all firms choose the same quantity: q* 1 = q*2 = ... = q*n = q*.
What is the corresponding market price in this equilibrium? (c) For the symmetric Nash
equilibrium found in part (b), if we set n = 1, we know that the equilibrium price and
quantity correspond to the monopoly price and quantity. What then would happen to
the equilibrium price and quantity as n approaches infinity?
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