Question

The Swedish manufacturer Bofors and the British manufacturer Rock Island Arsenal (producer of the Howitzer cannon) are competing for orders of field cannons in Saudi Arabia. Saudi Arabia’s inverse demand curve for cannons is given by p = 10,000 – Q, Q=qB+qH, qB where, is the number of cannons it buys from Bofors, and qH the number it buys from Rock Island Arsenal. The marginal cost for Bofors is $250 and for Rock Island is $200 respectively to produce each cannon. Write down the profit functions of the firms. How many cannons should each company produce in a Stackelberg equilibrium with Rock Island the leader and Bofors the follower firm? What is the price of a cannon? I need actual numerical values for qB, qH, and p.

Answer 1

Market demand P = 10000 - qH - qB. Given that Rock Island is the leader and Bofors is the follower firm, first find the profit function of follower

?B = Revenue - cost = 10000qB - qHqB - qB^2 - 250qB

= 9750qB - qHqB - qB^2

Maximize profit for Bofors to get its best response function

9750 - qH - 2qB = 0

qB = 4875 - 0.5qH

Incorporate the same in profit function of Rock Island

?H = Revenue - cost = 10000qH - qHqB - qH^2 - 200qB

= 9800qH - qHqB - qH^2

= 9800qH - qH(4875 - 0.5qH) - qH^2

= 9800qH - 4875qH + 0.5qH^2 - qH^2

= 4925qH - 0.5qH^2

Maximize the leader's profit

4925 = qH

and qB =  4875 - 0.5*4925 = 2412.50

Price P = 10000 - (2412.50 + 4925) = 2662.50. These are the required values.