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

`Q=nq=(n)/(n+1)(a-c)/(b)`

if n=1 (monopoly) we have
`Q^M=(1)/(2)(a-c)/(b)`

if n goes to infinity (approaching competitive level), we get the competition quantity that would be
`Q^c=(a-c)/(b)`

Step-by-step 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=(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=(a-b (n-1) q-m)/(2b)=(a-m)/((n+1)b)`

the industry output is then

`Q=nq=(n)/(n+1)(a-c)/(b)`

if n=1 (monopoly) we have
`Q^M=(1)/(2)(a-c)/(b)`

if n goes to infinity (approaching competitive level), we get the competition quantity that would be
`Q^c=(a-c)/(b)`