Robinson Crusoe lives alone on a desert island. He works 9 hours per day gathering

coconuts and hunting fish. His production functions for the two goods are:

C = 2 Lc and F = Lf 1/2

where C and F are the quantities of coconuts and fish produced (and consumed) per day and

Lc and Lf are the number of hours Crusoe spends producing coconuts and fish respectively.

Crusoe's utility function from consumption is

U(C, F) = CF.

(a) Find Crusoe's production possibilities frontier. Find a formula for the marginal rate of

transformation.

(b) How many hours will Crusoe spend each day on each of the two production activities

in order to maximize his utility? (Allow fractional quantities of goods to be produced.)

(c) Show that, at the optimum you found in (b), the marginal rate of transformation is equal

to the marginal rate of substitution. What price ratio would support this optimum as a

competitive equilibrium?