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.

(a)   What is the number of hours he would like to have for leisure? (1 marks)

(b)  Determine the MRS of leisure for labour (2 marks)

(c)   Draw a leisure-influenced labor curve (2 marks)

Answer 1

(a) U(L, Y ) = Y + LY

$MU_y=1+L\\ MU_L=Y \\P_y=1$

budget line WL+ Y = 24W.......(1)

At optimality , we we know

$MRS = \frac{MU_y}{MU_L}=\frac{1+L}{Y}=\frac{P_y}{P_L}=\frac{1}{W}$

$\frac{Y}{1+L}=W \ \ \ \ \implies \boxed{Y = W(1+L)}$ put in (1)

WL +W(1+l)=24W

WL+W+WL=24

2WL=23W

$L=\frac{23W}{2W}=11.5$ $\boxed{leisure = 11.5}$ and Y = W(12.5)

(b) $MRS = \frac{1+L}{Y}= \frac{12.5}{Y}=\frac{12.5}{12.5(W)}=\frac{1}{W}$

$\boxed{MRS = \frac{1}{W}}$

(c)

leisure remains same irrespective of wage rate ,

thus labor remains same irrespective of wage rate.