Question

A firm’s output is K(100 − K). The price of the product is 1 per
unit. The capital provider charge a rent of r per unit of capital, K, where
r is between 0 and 100. The firm either accepts or rejects the offer. If the firm
accepts the offer, it chooses the amount K of capital (which you should take to
be a continuous variable, not an integer); if it rejects the offer, no production
takes place (K = 0). The firm’s preferences are represented by its profit; the
capital provider’s preferences are represented by the value of rK.
(i) Formulate situation as a sequential game. (what are the players? which
one is first mover? what are the strategies and payoff functions?)
(ii) Find subgame perfect equilibrium of the game.
(iii) Is there a (K, r) which will generate a higher total pay-off (sum of firm and
capital provider’s pay-offs) than the total pay-off at the SPE.
(iv) Find Nash equilibrium for which the outcome differs from any subgame perfect equilibrium outcome.

Answer 1

(i) The players are provider and the firm.

The firm has a dominant strategy, and is a first mover.

(ii) $profit = k(100-K) - Kr =100k-k^2-kr,$

You can maximise profit by differentiating profit with respect to K

$100-2k-r=0,$

$k=50-0.5r$ , so this gives how much capital they will rent.

Now the capital owner has a profit function of

$rk =r(50-0.5r)=50r-0.5r^2$

Differentiate with respect to r = 0

$50-r=0,$

r = 50.

(iii) There are no (K, r) which will generate a higher total pay-off, than the total pay-off at the SPE.

(iv) There is no Nash equilibrium for which the outcome differs from any subgame perfect equilibrium outcome.