Question

Robinson Crusoe obtains utility from the quantity of fish he consumes in one day (F), the quantity of coconuts he consumes that day (C), and the hours of leisure time he has during the day (L) according to the utility function: U + F^1/4C^1/4L^1/2

Robinson's production of fish is given by: F = (N_F)^1/2 where N_F is the hours he spends fishing.

Robinson's production of coconuts is given by C + (N_C)^1/2, where N_C is the hours he spends picking coconuts.

Robinson is also bound by the constraint: N_F + N_C + L = 24.

what are Robinson's optimal hours of fishing (N_F*), picking coconuts (N_C*), and leisure, (L*)?

Answer 1

According to the given informations,

Robinson Crusoe obtains utility from the quantity of fish he consumes in one day (F), the quantity of coconuts he consumes that day (C), and the hours of leisure time he has during the day (L) according to the utility function:

Robinson's production of fish is given by:

F = (NF)1/2 where NF is the hours he spends fishing.

Robinson's production of coconuts is given by

C = (NC)1/2, where NC is the hours he spends picking coconuts.

Robinson is also bound by the constraint: NF + NC + L = 24.

Now, from the production functions, we will calculate the PPF of his economy. The calculations are shown below.

Now, we will use the lagrange's method to maximize the utility subject to the PPF. Hence, the problem looks like

Hence, the Lagrange's equation is written below and solved for F*, C* and L*.

Now please look at the four equations above.

Deviding (4) by (5) we get

C/F=F/C

or, ........(8)

Also, deviding (4) by (6) we get

(1/2)×(L/F)=2F

or, .......(9)

Now putting the (8) and (9) values in equation (7), we get

or,

or,

or, F* = 2

Hence, from (8) we get

C* = F* = 2

Also from (9) we get

L = 4×(2)^2 = 16

Putting the values of F* and C* in the production functions, we get NF* and NC*.

Hence, from equation (1)

as F*=2, hence

NF* = 4

And, from equation (2)

as C*=2, hence

NC* = 4

Hence, Robinson's optimal hours of fishing is 4 hours i.e. NF*=4, optimal hours for picking coconuts is 4 hours i.e. NC*= 4, and optimal leisure is 16 hours i.e. L*=16.

Hope the solution is clear to you my friend.