Question

Labor supply. 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) = c 2/3 l 1/3 . 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

Santi's utility function is , income I is M, price of c is 1 while price of leisure is w, and .

1. Santi's budget constraint can be given as for c is the amount of goods he consumes and is Santi's total income, and since price of good is 1, Santi's income represents the amount of good Santi consumes. The budget constraint can be further arranged as, since or , then or or . The graph is as below.

In the above figure, MAO is the budget constraint. MA is the amount of consumption if one takes full leisure of 24, and have M income only, in which case c is M. But, after A, Santi has to substitute leisure to consume more, with the rate od w (ie -w).

2. Santi's utility maximization problem can be stated as below.

subjected to , and .

The Lagrangian function will be .

3. The FOC's are as below.

or or . Equating , we have or , the constraint itself.

or or . Equating , we have or .

or or . Equating , we have or or .

As the RHS of last two FOC is lambda, hence equating both, we have or or or . This is the optimal consdition of consumption and leisure. Putting it in the budget constraint, we have or or . Also, as , we have or .

Hence, the consumption demand is , and the leisure demand is .

4. Supposing M = 100 and w = 10, the consumption demand is or or (approx), the leisure demand is or or or .

5. Santi's elasticity of leisure demand with respect to w is or or .

6. Santi's elasticity of leisure demand with respect to M is or or .