Labor Economics
This problem focuses on the labor supply effects of taxes. Assume there is a worker (Cosmo) who has a utility function over money income m and leisure l is u(m, l) = √ ml. Cosmo chooses how many hours to supply to the labor market where h = 16 − l subject to the market hourly wage w. The feasibility constraint is such that neither leisure nor labor supply can exceed 16 hours and cannot be negative. Cosmo has unearned income a = 8.
1. It can be shown (with calculus) that if utility is u(m, l) = √ ml, the marginal utility of leisure is MU(l) = 1/2 ( m/l )^1/2 and MU(m) = 1/2 ( l/m )^1/2 . Assuming that the interior solution for labor supply will dominate the corner solutions, what is Cosmo’s optimal labor supply as a function of the market wage w?
2. Now suppose the government levies a positive tax of t on each dollar that Cosmo earns. That is, Cosmo’s after tax wage becomes w(1−t). What is the effect of this tax on Cosmo’s budget constraint? Draw a graph showing both the new and old budget constraints (for any positive t).
3. Assuming that the interior solution for labor supply will dominate the corner solutions, NOW what is Cosmo’s optimal labor supply as a function of the market wage w and the tax rate t?
4. Does the number of hours Cosmo supplies to the labor market increase or decrease as t gets bigger (closer to 1)? Based on this, when the tax rate increases, does the income or substitution effect dominate?