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)

Draw a leisure-influenced labor curve 

Answer 1

(a) Utility function$=U(L, Y)=Y+LY$ 

Marginal Utility of good

Y$=MU_Y=1+L$

Marginal Utility of Leisure $=MU_L=Y$

Price of good Y $=P_Y=\$1$

The wage of Terry=w, \ income will therefore be $Y=(24-L)w$

Optimal condition$=MRS(L,Y)=\frac{P_L}{P_Y}$

When we substitute the values from the optimal condition, we get $=\frac{Y}{1+L}=\frac{w}{1}$

$Y=w(1+L)$

We already know that $Y=(24-L)w$ so we substitute $Y$ from the above equation and get $w(24-L)=w(1+L)$

$(24-L)=(1+L)$

Therefore, $L=\frac{23}{2}=11.5$

Hours of leisure will be 11.5 hours.

(b)  $MRS=\frac{P_L}{P_Y}$

$\frac{11.5}{1}$ $=11.5$

The wage rate does not affect the amount of leisure and therefore there will be no marginal rate of substitution.

(c.)