5. Consider a production economy with two consumers/workers, one type of labor and one type of consumption good. Both agents A and B have utility functions of the form U(cl) = a Inc+ (1 - a) In where c and represent the consumption levels of the consumer good and of leisure respectively and a (0,1). Assume that both agents are endowed with one unit of labor, which may be supplied to a firm or consumed as leisure. The agent's leisure cannot exceed his endowment of labor. The consumption good is produced by a firm with F(0) = 2V as its production function. This firm is wholly owned by Agent B. 3 Normalize the price of labor (i.e., the wage) at 1 and denote the price of the con- sumption good by p. Assume that both agents, as well as the firm, are price-takers. (a) Find the firm's demand for labor and its profit as functions of p. (b) Show that agent A's optimal choice of (c, 2) as a function of p is (e) Find agent B's optimal choice of (c,) as a function of p. Show that this agent's consumption of leisure is increasing in p and equals 1 if and only if pp= - Draw a diagram depicting B's choice in the case where p p. Your diagrams should have leisure on the horizontal axis and the produced good on the vertical axis. Budget lines, endowments points, and utility-maximizing indifference curves must be clearly indicated. (d) Assuming that both agents are supplying labor to the firm in strictly positive quantities, find the equilibrium value of p. (e) Is your answer to (d) the only equilibrium in this economy? 5. Consider a production economy with two consumers/workers, one type of labor and one type of consumption good. Both agents A and B have utility functions of the form U(cl) = a Inc+ (1 - a) In where c and represent the consumption levels of the consumer good and of leisure respectively and a (0,1). Assume that both agents are endowed with one unit of labor, which may be supplied to a firm or consumed as leisure. The agent's leisure cannot exceed his endowment of labor. The consumption good is produced by a firm with F(0) = 2V as its production function. This firm is wholly owned by Agent B. 3 Normalize the price of labor (i.e., the wage) at 1 and denote the price of the con- sumption good by p. Assume that both agents, as well as the firm, are price-takers. (a) Find the firm's demand for labor and its profit as functions of p. (b) Show that agent A's optimal choice of (c, 2) as a function of p is (e) Find agent B's optimal choice of (c,) as a function of p. Show that this agent's consumption of leisure is increasing in p and equals 1 if and only if pp= - Draw a diagram depicting B's choice in the case where p p. Your diagrams should have leisure on the horizontal axis and the produced good on the vertical axis. Budget lines, endowments points, and utility-maximizing indifference curves must be clearly indicated. (d) Assuming that both agents are supplying labor to the firm in strictly positive quantities, find the equilibrium value of p. (e) Is your answer to (d) the only equilibrium in this economy