Question

Two firms compete in Stackelberg. The leader, firm L, has no production costs. The marginal cost of the follower, firm F, is equal to 3 with probability 0.25, and to 7 with probability 0.75. The follower knows its cost, but the leader only knows the probability distribution and associated costs. The demand function is given by p = 100 – 2 yL – 2 yF . Calculate quantities and prices in the Stackelberg equilibrium.

Answer 1

If the marginal cost of the follower, firm F, is equal to 3 with probability 0.25, and to 7 with probability 0.75, then the expected marginal cost is $MC = 3*0.25 + 7*0.75 = 6,$ so the total cost TCF = 6yF.

First, we should find the follower's reaction function.

The total profit equation of the follower is:

$TPF = TRF - TCF = (100 - 2yL - 2yF)*yF - 6yF = 94yF - 2yL*yF - 2yF^2,$

$TPF'(yF) = 0, so:$

$(94yF - 2yL*yF - 2yF^2)' = 0,$

$94 - 2yL - 4yF = 0,$

$yF = 23.5 - 0.5yL.$

The total profit equation of the leader is:

$TPF = TRF - TCF = (100 - 2yL - 2yF)*yL - 0 = 100yL - 2yL^2 - 2yL*(23.5 - 0.5yL) = 53yL - yL^2,$

$TPF'(yL) = 0, so:$

$53 - 2yL = 0,$

$yL = 26.5$  units,

$yF = 23.5 - 0.5*26.5 = 10.25$ units.

$p = 100 - 2*26.5 - 2*10.25 = 26.5.$