Question

Consider a consumer who has preferences over consumption (x) and leisure (L) represented

by u(L, x) = 10 ln L + 5 ln x. The consumer has 24 hours in the day (T = 24) to divide

between work and leisure. The consumer can choose however many hours they want to

work. For each hour of work they are paid a wage given by w = 10. Consumption (x) costs

1 per unit.

(a) Initially suppose that the consumer has no income aside from the wages they get paid

to work (m = 0). How many hours would the consumer work and how many units of

x would they consume?

(b) Now suppose that the consumer has some outside income (m > 0). What is the

minimum level of m at which the consumer would choose to work zero hours? (Hint:

you only need to use the wage and an expression for the consumer’s MRS at the corner

solution, which itself is a function of m.)

Answer 1

the consumer would work 8 hours, consume 80 units and spend 16 hours in leisure.

b) The minimum level of non wage income to make the consumer stop working is $80 because that is where he maximized his utility when he had no non wage income. If he gets non wage income of the same amount then he has no incentive to work.