# labor leisure trade off

Walras has available 24 hours per day. He has to alloacate this 24 hours between leisure $(L)$ and work. His utility function depends on leisure and the composite good $(C)$ and is given by $U=LC$. Work pays him a wage of 1 dollar per hour and this is his only income.

What is his optimal choice of $L$ and $C$ ?

I understand that $MRS=C/L$ and it should be equal to the price ratio. Is it 1 or 2? I am confused whether budget constraint is $C+L=24-L$ (in this case price ratio is 2). Can someone clarify this?

## 1 Answer

My suggestion would be to take the habit of fully writing down and then solving formally even the simplest model. Yours is a static one,

$$\max_{C,L} U(C,L) = CL,\;\;\;\; s.t. C = (24-L)w$$

since we silently assume that the individual cannot consume more than his income. Here $L$ is leisure, not work.

Inserting the constraint into the objective function,

$$\max_L U[C(L), L] = (24-L)w\cdot L = 24wL - wL^2 = w(24L - L^2)$$

This permits you to see that the wage doesn't matter for the optimal allocation (it will matter, for the level of utility attained). This may sound counter-intuitive... It is, and it has to do with the category of functions to which the specific utility function belongs to. In other words, it is an unrealistic aspect of the model introduced by a mathematical choice we made.

The above has a straightforward solution for a maximum.

(PS: Has Walras been notified that his name is used in such a manner?)