Question

firm has the production function: TC=120-0.1Q^2 , and sells its output in two separate markets with demand functions: Q1=800-2p1

Q2=750-2.5p2

;

(i) Find the profit-maximizing output and sales in each market [8 marks]

(ii) Use the Hessian matrix to check if the second-order conditions for a maximum a

Answer 1

$TC=120-0.1Q^2$

MC= 0.2Q

$Q1=800-2P_1$

$P_1= 400-0.5Q_1$

TR= P.Q= Q(400-0.5Q$_1$ )

=400Q-0.5Q₁$^2$

MR$_1=400-0.1Q_1$

Equilibrium at market 1

MR=MC

0.2Q₁=400-0.1Q₁

0.3Q₁= 400

Q₁= 1333.33

$P_1= 400-0.5(1333.33)= 266.67$

Equilibrium at market 1

$Q_2=750-2.5P_2$

$P_2= 300-0.4Q_2$

MR= 300- 0.8Q₂

300- 0.8Q₂=0.2Q

Q= 300

$P_2= 300-0.4(300)= 180$