Question

Robinson has a utility of u(c, r) = cr where c = coconuts and r = leisure. In order to produce coconuts, technology dictates the production, given by c = a(L^1/2) where a = 8, and L = labour. The time constraint is r+L = 12. How many units of labour will be used?

Answer 1

u=cr......(1) where c=a(L^1/2)

so substituting the value of c in equation 1 we get,

u= a(L^1/2) r

or, u=8(L^1/2) r subject to the time constraint r+L=12.

To find out the units of labour we need to maximize u with respect to the constraint.

z= 8(L^1/2) r + (r+L-12)......(2)

the first order condition of z with respect to L is,

dz/dL= 4L^1/2 r + =0.......(3)

the first order condition of z with respect to r is,

dz/dr=8(L^1/2)+=0..........(4)

by diving equation 3 by 4 we get, 1/2 L^1/2 r/ L^1/2=1

or, r=2

putting the value of r in the time constraint r+L=12

or, L=12-2 = 10