There are 800 consumers in an economy that each have the same utility function given by U(c,

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Question:

There are 800 consumers in an economy that each have the same utility function given by U(c, l) =
32c^1/2−(24−l)^2 where c is their consumption and l is the number of hours they spend for leisure. A
single firm serves the market with production function Y = 32L^1/2K^1/2. The firm cannot choose
its capital stock, which is fixed at K = 1600. You can assume the price level is equal to 1 so real
and nominal wages are equivalent.
a. Solve for an individual consumer’s labor supply as a function of the real wage
b. What is the total supply of labor hours for the economy in one day as a function of the real
wage?
c. Solve for the firm’s labor demand as a function of the real wage
d. What is the equilibrium real wage and equilibrium total number of hours worked per day?
How many hours does each consumer work per day?
e. What is total output for the economy? What is consumption? (challenge: why isn’t the
consumption you calculated equal to output?)
f. The government wants to increase the equilibrium real wage, so it mandates a minimum
wage of $15 per hour. What happens to labor supply and demand? How many total hours
will be worked? What is total output?
g. Now the government gets rid of the minimum wage and tries a wage subsidy instead. For
every hour worked, they pay the consumer some benefit b. How large does b have to be
in order to reach $15/hr of total compensation (w + b)? (don’t forget that the equilibrium
wage will change with the subsidy) Now how many hours will be worked and what is total
output?
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h. Bernie Sanders is worried that the wage subsidy is actually subsidizing the firm. The government
passes a 100% tax paid by the firm for every dollar of benefit the firm’s employees
receive from the government (in other words, for every hour of labor the firm hires the worker,
it needs to pay the wage plus the benefit its workers receive). Solve for total compensation
received by the worker (you will not be able to solve for the wage and benefit separately,
solve for w + b). Does Sanders’s proposal help workers? Explain the intuition.