Question

1. Consider an individual with the following utility function:

U = min{c, h}

Where c is consumption and h is leisure. The wage rate is $15 per hour, and the total number of hours available to the individual is normalized to 1.

1. Find out the optimal level of consumption, leisure and labor.
2. The government imposes a tax of $2 per hour of labor. Find out the new optimal values of consumption, leisure and labor.
3. Comment on the form of the utility function above. In general, what does it tell you regarding the effect of taxes on labor supply? What will be your policy recommendation?

Note: question c is very important to be answered well

Answer 1

a) Given a leisure level h, the consumption c would be (1-h)*15. The optimal h would be such that (1-h)*15=h which gives h=15/16. Thus, the optimal level of consumption is 15/16, leisure is 15/16 and labor is 1/16.

b) Given a leisure level h, the consumption c would be (1-h)*13. The optimal h would be such that (1-h)*13=h which gives h=13/14. Thus, the optimal level of consumption is 13/14, leisure is 13/14 and labor is 1/14.

c) This is the Leontief utility function which has a zero substitution effect. Therefore, the final price effect is decribed by the income effect. As wage falls due to increase in taxes, both consumption and leisure will fall due to lower income levels.

Note that since the leisure falls due to increase in taxes, the labour hours increase. In particular, if t is the tax rate, then the labour supplied will be 1/(16-t) and therefore, the tax revenue for the government will be t/(16-t). Also, the consumer's surplus at tax rate t is (15-t)/(16-t). Thus, the sum of consumer surplus and government revenue is 15/(16-t) which is increasing in t. Thus, the policy recommndation would be to increase taxes.