Question

Terry’s utility function over leisure (L) and other goods (Y ) is U(L, Y ) = Y + LY. The associated marginal utilities are MU_Y = 1 + L and MU_L = Y. He purchases other goods at a price of $1, out of the income he earns from working. Show that, no matter what Terry’s wage rate, the optimal number of hours of leisure that he consumes is always the same. 

Answer 1

If Terry's hourly wage rate is w, his total earnings are $(24-L)w$. $PY=1$, so $Y=(24-L)w$ is the number of units of other products he buys.

Terry's $MRS_L,_Y$ must now equal the price ratio $\frac{w}{PY}=w$ at the optimal bundle. From the tangency condition we know that $\frac{Y}{1+L}=w$ As a result of the tangency condition, we can conclude that both of these conditions imply. This means that $L=11.5\%$ is the ideal quantity of leisure. As you can see, this is independent of the wage rate.