Question

1. Consider two Cournot duopolists. Each firm sells a homogenous product and has a MC = c per unit, and no fixed costs. Market demand is P = a−bQ, where market quantity sold Q = q1 +q2, where q1 is firm 1’s output and q2 is firm 2’s output. Each firm simultaneously chooses its quantity to sell, then lets price clear the market. a. What is firm 1’s best response function (or reaction function)? b. Solve for the profit maximising level of q1? c. What is firm 1’s profit and what is the market price?

Answer 1

Market demand, P = a-bQ, where Q= q₁ + q₂

and MC = MC₁ = MC₂ = c

a) Response Function of firm 1- the response of firm 1 in producing the profit maximizing level of output for firm 1 given the level of output produced by firm 2.

max profit, = pq₁ - cq₁

= aq₁ - b(q₁ + q₂)q₁ - cq₁

First order condition for profit maximization,

= a - 2bq₁ -  bq₂ - c = 0

or, a-c- q₂ = 2bq₁

or, q₁ =(a-c- bq₂)/ 2b ............ (i)

= -2b 0 maxima

Similarly, we can calculate q₂ =(a-c- bq₁)/ 2b

Hence, the best response function is

=    if q₂ (a-c)/b and 0 otherwise

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b) The profit maximizing level of output for Firm 1 is

q₁ =(a-c- bq₂)/ 2b from (i)

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c) Firm 1 profit = = aq₁ - b(q₁ + q₂)q₁ - cq₁

= q₁ (a- bq₁ - bq₂- c)............. (ii)

Substituting q₁ =(a-c- bq₂)/ 2b and q₂ =(a-c- bq₁)/ 2b in (ii)

solving this equation will give

  

and market price is

P = a- b(q₁ + q₂)

is the market price.

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