Question

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.PLEASE write clear

Answer 1

Answer:

$Q=nq=\frac{n}{n+1}\frac{a-c}{b}$

if n=1 (monopoly) we have $Q^M=\frac{1}{2}\frac{a-c}{b}$

if n goes to infinity (approaching competitive level), we get the competition quantity that would be $Q^c=\frac{a-c}{b}$

Explanation:

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

$Max_{q_i} \,\, \Pi_i=(a-b(\sum_{i=1}^n q_i)-m)q_i$

from the FPC we have that

$a-b\sum_{i=1}^n q_i -m -b q_i=0$

$q_i=\frac{a-b \sum_{i=2}^n q_i-m}{2b}$

since all firms are homogeneous this means that $q_i=q \forall i$

then $q=\frac{a-b (n-1) q-m}{2b}=\frac{a-m}{(n+1)b}$

the industry output is then

$Q=nq=\frac{n}{n+1}\frac{a-c}{b}$

if n=1 (monopoly) we have $Q^M=\frac{1}{2}\frac{a-c}{b}$

if n goes to infinity (approaching competitive level), we get the competition quantity that would be $Q^c=\frac{a-c}{b}$