Problem 1. Each of 5000 consumers that are buyers of goods X and Y per period has the utility function:
U = 56X0.3Y0.7
where X is the number of units of good X consumed and Y represents all other goods that the consumer purchases. The income of each consumer is $1000. You may assume the price of Y is $1 per unit of other goods consumed. Keep in mind that there are 5000 consumers in this market and the utility function given above is for a single, representative consumer.
On the supply side of this market, the short run production function for each of 5 firms producing good X is:
X = 16.L1/2K1/3
where L and K are the number of labor and capital units, respectively, used by each firm per period. The wage rate paid to labor is $166.66667 (i.e. 166 & 2/3) per unit and the price of capital is $0.25 per unit. Assume there are 15,625 units of capital (which includes machinery) being used by each firm.
Using this information and assuming there are 5 firms in this market, determine the equilibrium level of output in this market, the equilibrium price paid by consumers in this market, the profits to the industry, and the amount of labor used by the industry.
(Continuing above) If a 10% tax is imposed on the buyers in this market (assume there are still 5 firms), determine the equilibrium level of output in this market, the equilibrium price paid by consumers in this market, the profits to the industry, the amount of labor used by the industry and the deadweight loss, if any, due to the tax.
(Continuing above) If a 5% tax is imposed on the sellers in this market (assume there are still 5 firms), determine the equilibrium level of output in this market, the equilibrium price paid by consumers in this market, the profits to the industry, the amount of labor used by the industry and the deadweight loss, if any, due to the tax.
Problem 2. A person supplies labor to his/her workplace according to the following utility function:
U = 100C0.7R0.3,
where C denotes per day consumption "expenditures" and R denotes his/her leisure time per day, that is, the time spent working. Assume that the price of goods consumed is $1 per unit (one dollar buys one dollar's worth of consumption goods), and make special note that this person has $166 per day. In other words, this person has $166 per day to spend regardless of how many hours s/he works.
In addition, suppose the demand for labor at this company is
L = 1.8 + (32.7/w)
where L is the number of hours per worker per day and w is the wage rate per hour.
Given this information and assuming the above utility function is representative of that of the average worker at the plant, what will be
A. The average number of hours worked per employee per day?
B. The wage paid per hour to each worker?
C. The excess demand (or supply) of labor in the plant if the government sets a minimum wage of $7.50 per hour?
D. The excess demand (or supply) of labor in the plant if the government imposes a 7.5 per cent tax on both workers and the employer. (Assume no minimum wage for this part of the problem.)
Lastly, assuming no minimum wage is in effect, then
E. If employers are required to pay time-and-a-half for overtime, how many overtime hours will the average person work?