# Robinson Crusoe Production Economy [closed]

Robinson Crusoe’s preferences over coconut consumption, C, and leisure, R, are represented by the utility function U(C, R) = CR. There are 48 hours available for Robinson to allocate between labor and leisure. If he works L hours, he will produce the square root of L of coconuts. He will choose to work.

The answer is 16 and I known this but confused to its working out. Can some give me a step by step?

## closed as off-topic by Giskard, FooBar, cc7768, Alecos Papadopoulos, JamzyMar 29 '16 at 0:46

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not meet the standards for homework questions as spelled out in the relevant meta posts. For more information, see our policy on homework question and the general FAQ." – Giskard, FooBar, cc7768, Alecos Papadopoulos, Jamzy

The utility function is $U(C,R)=CR$ and the time is restricted: $48=R+L$. Now we know that $C=\sqrt L$. $C$ can be replaced by $\sqrt L$. Therefore the langrarian is

$\mathcal L=\sqrt L\cdot R+\lambda (48-L-R)$

The (partial) derivatives are the following. They have to be set equal to zero.

$\frac{\partial \mathcal L}{\partial L}=\frac12 L^{-0.5} R-\lambda=0$

$\frac{\partial \mathcal L}{\partial R}= L^{0.5} -\lambda=0$

Putting $\lambda$ on the RHS

$\frac12 L^{-0.5} R=\lambda \quad (1)$

$L^{0.5} =\lambda \quad (2)$

Dividing (1) by (2):

$\frac12\cdot \frac{R}{L} =1 \Rightarrow R=2L$

The expression for R can be insert in the time restriction

$48=2L+L$

• That´s fine. You´re welcome. – callculus Mar 29 '16 at 10:14