Question

The television market in South Africa has two major suppliers producing
differentiated television models. The monthly demand functions of the two suppliers is as follows:
Samsung: Q1 = 450-2P1+P2
LG: Q2 = 450-2P2+P1
The marginal costs of production are R60 per unit and fixed costs are equal to zero.
Part 1
Suppose the two producers compete by setting their quantities in accordance with the Cournot model.
a) Derive the two reaction functions of the two firms
b) Determine the output levels of the two producers at the Nash equilibrium
c) Determine the price and the profits made by each producer

Answer 1

a) Pr-profit

$P1=-\frac{Q1-450-P2}{2}$

$P2=-\frac{Q2-450-P1}{2}$

$Pr1=-\frac{Q1-450-P2}{2}*Q1-60*Q1$

$Pr2=-\frac{Q2-450-P1}{2}*Q2-60*Q2$

$Pr1'=\frac{-2Q1+450+P2}{2}-60=0$

$Pr2'=\frac{-2Q2+450+P1}{2}-60=0$

$Q1=225+0.5P2-60$

$Q2=225+0.5P1-60$

b)$Q1=225+0.5*(-0.5Q2+225+0.5P1)-60$

$Q1=225-0.25*(225+0.5P1-60)+112.5+0.25P1-60=277.5-56.25-0.125P1+15+0.25P1=236.25+0.125P1$

$Q2=236.25+0.125P2$

The Nash equilibrium: P1=P2; Q1=Q2

c) Price and profit depend of the television quantity demended. At the Nash equilibrium: P1=P2; Q1=Q2. Suppose we have the equllibrium. In break even point:

$Pr1=-\frac{Q1-450-P2}{2}*Q1-60*Q1=0$

Q1=0 or Q1=660+P2

Q2=0 or Q2=660+P1