In a homogeneous-good cornet model where each of the n firms has a constant marginal cost m and the market demand curve is p = a - bq, show that the nash-cournot equilibrium output of a typical firm is q=a-m/(n+1)b . show that industry output, q (= nq), equals the monopoly level if n = 1 and approaches the competitive level as n gets very large. write clear

if n=1 (monopoly) we have

if n goes to infinity (approaching competitive level), we get the competition quantity that would be

Explanation:

In the case of a homogeneous-good Cournot model we have that firm i will solve the following profit maximizing problem

from the FPC we have that

since all firms are homogeneous this means that

then

the industry output is then

if n=1 (monopoly) we have

if n goes to infinity (approaching competitive level), we get the competition quantity that would be

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