# 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?

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