Question

Santi derives utility from the hours of leisure (l) and from the amount of goods (c) he consumes. In order to maximize utility, he needs to allocate the 24 hours in the day between leisure hours (l) and work hours (h). Santi has a Cobb-Douglas utility function, u(c,l) = c2/3l1/3. Assume that all hours not spent working are leisure hours, i.e, h + l = 24. The price of a good is equal to 1 and the price of leisure is equal the hourly wage, w. Santi also has passive income of M per month from his asset.

1. Write Santi’s budget equation and draw his budget constriant with consumption on the x-axis and leisure on the y-axis.

2. Set up Santi’s utility maximization problem.

3. Find the first order condition for optimal consumption and leisure. Derive his consumption and leisure demand functions from the first order condition and budget constraint.

4.Suppose Santi has non-labor income from this return in asset of $100 per month and he makes an hourly wage of $10. What is his consumption, leisure, and work hours per day?

5. Find Santi’s elasticity of leisure demand with respect to hourly wage. (Ed = ∂l w ) ∂w l

6. Find Santi’s elasticity of leisure demand with respect to non-labor income. (EI = ∂l M ) ∂M l

7. How do his leisure, and work hours change when his wage increases? Explain the effects in 2-3 sentences.

8. Suppose now the return on asset is taxed, which results in a decrease in his non labor income, M. Explain the impact that the tax on asset returns has on his leisure, and work hours.

9. Given that his leisure hour l cannot be more than 24 hours, l ≤ 24, and the hourly wage is equal to $10. Find the minimum non-labor income that will guarantee Santi to no longer work, i.e work hour =0, and l = 24.

Answer 1