Question #143853

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.

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.

Expert's answer

(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.

Learn more about our help with Assignments: Engineering

## Comments

## Leave a comment