Question

A sequential game A local bee-keeper derives net revenue z(h) from choosing a level of production (number of hives) h given by z(h) = 4h − 2h 2 .

A nearby apple farmer also benefits from being next to the bee-keeper, because the bees help to polinate fruit trees. The value of the bees to the apple farmer is given by y(h) = 12h − 2h 2 .

(a) Show that the bee-keeper maximizes their net benefit by choosing h = 1, while the sum of benefits, y(h) + z(h), is maximized at h = 2.

(b) Suppose that the government has the power to impose a transfer from the farmer to the bee-keeper in amount t. The level of t is chosen after the bee-keeper decides on h. Additionally, the government wishes to maximize the sum of the logarithms of the payoffs of the the two parties, so that it solves max t ln(12h − 2h 2 − t) + ln(4h − 2h 2 + t).

i. Show that by choosing t taking h as given, the government optimally sets t = 4h.

ii. Show that this transfer scheme leads to the bee-keeper wishing to maximize z˜(h) = 8h − 2h 2 .

iii. What level of h will be chosen by the bee-keeper if they anticipate this payment? Show your reasoning.

(c) Would your answer to part (b)(iii) change if the government put less weight on the bee-keeper’s utility so that it solved, instead, max t ln(12h − 2h 2 − t) + 1 2 ln(4h − 2h 2 + t) ?

Justify your answer.

Answer 1