Question

When you solve a household labor supply problem if the optimal leisure time, l, is more than 1, which is impossible due to time constraint, we have a corner solution. The best a household can do is to set l = 1. In this case a household does not work at all. Use this little expansion to answer the following questions.

(a) Rose used to have a utility function u(c, l) = √ c + 6√ l. She faces a wage rate of 8 and a non-wage income of 8. Does Rose work at all?

Answer 1

MRS for Rose is -0.5c^-0.5 / 6*0.5l^-0.5

= -(1/6)(l/c)^0.5

Slope of the budget line is -1/w = -1/8

At the optimal choice MRS = slope of budget

(1/6)(l/c)^0.5 = 1/8

l = 0.5625c

Budget equation is c + 8l = 16

c + 8*0.5625c = 16

c = 2.91

l = 1.64

It is given that  if the optimal leisure time, l, is more than 1, which is impossible due to time constraint, we have a corner solution. The best a household can do is to set l = 1.

Here also we have l = 1 which means Rose does not work

Hence Rose works for 0.14 unit of time and has a leisure of 0.86 units