Firms hire labor to maximize profits subject to the technology Y . All firms are identical, and we can focus on one representative firm for the analysis. Suppose that the economy is populated by 50 identical individuals. Their income consists of profits ? and wage income, where w denotes the wage paid per hour worked. In addition, consumers must pay taxes lump-sum taxes T to the government. The government uses taxes to pay for expenses G = 10. Their utility function is

. Write down the maximization problem of consumers, specifying what are the endoge nous and what are the exogenous variables. Find the optimal labor supply of the household and denote it by as. What is n5 a function of? How does this compare with the benchmark case discussed in class? . What happens to the individual labor supply as when the wage rate 21) increases? An- swer explaining how the income effect and the substitution effect affect consumption, leisure, and labor choices. . Let the aggregate labor supply be denoted by N5. Assuming that v = 2 and A = 10, plot the aggregate labor supply, making sure to label your axis and being as precise as possible (for example, show the point where w = 1 computed before). Feel free to draw it by hand or use Excel to do this. . Dene a competitive equilibrium. Make sure to list all the variables that are deter- mined in equilibrium. . Plot the labor market equilibrium (e.g. putting demand and supply of labor in the same plot, and showing the equilibrium values of N "' and w'\") . Solve for the equilibrium allocations and prices. Show all intermediate steps for partial credit. Consider a static (one-period) economy. The technology that transforms labor N into output 1" is given by Y = ZNC'KI'\" with 0 1 and A > 0 are constants. Individuals are endowed with 1 unit of time. 1. Write down the rm's maximization problem and nd the optimal labor demand of the rm as a function of to, K, a, and z. Denote the labor demand by Nd. 2. Assuming that z = 1.5, K = 20, and a = 0.6, compute Nd when w = 1. Show all intermediate steps for partial credit. 3. Now plot the labor demand, making sure to label your axis and being as precise as possible (for example, show the point where w = 1 computed before). Feel free to draw it by hand or use Excel to do this