Consider a small ”Robinson Crusoe” economy where there is a firm that produces consumption goods (C, measured in consumption baskets) from labor as the only resource (L, measured in hours) according to the production function C = f(L) = 4√L. The representative consumer (Robinson Crusoe) has preferences described by the utility function u(F, C) = FC, where F denotes the consumers’ leisure in hours and C, the number of consumption baskets. The consumer’s time endowment consists of 24 hours a day. Assume that both the firm and the consumer are price takers. The prices of both goods are denoted by w (nominal hourly wage rate) and p (the price of one consumption basket). Throughout this problem, assume that the price of the consumption basket (price level, cf. Macroeconomics) is 1. Then the relative price w/p with an interpretation of the real wage rate (cf. Macroeconomics) is identical with wage rate w. In other words, w is the only relevant market price in this economy. 

a) What is the (daily) Pareto optimal allocation (labor time, leisure time, and consumption)? In order to answer this question, assume that a social planner attempts to maximize the representative consumer’s utility level subject to the constraint of the available technology and resources. What is the condition for Pareto optimality in general equilibrium with production (see lecture and textbooks)? Represent your result graphically in an (F, C) diagram where you draw the transformation curve (production possibility frontier) and a set of indifference curves. 

b) Consider the firm’s production technology. Does it exhibit increasing, decreasing, or constant returns to scale, and why? Given these results, do you expect this firm’s profits to be positive or zero? Assume that the firm is a competitive price taker that is owned by the representative consumer and attempts to maximize its profit. It hires labor at the wage rate w and sells the produced consumption baskets at the price of 1. Write down this firm’s profit maximization problem and solve it. Find this firm’s labor demand function L^D(w), supply function C^S(w) and the profit function (the optimal profit of the firm at given prices) π(w) as functions of the wage rate. If the representative consumer is the only shareholder of this firm, what will be his profit income depending on the wage rate?

c) Now consider the representative consumer’s consumption decisions given the market conditions. What is his time constraint? Note that he has two sources of income: he can use his time endowment to work and generate labor income but he is also the firm’s shareholder who obtains profits. Given these two sources of income (labor income and profit income), write down Robinson’s complete budget constraint and represent it in standard form with variables F and C.
d) Write down Robinson’s utility maximization problem. What does his utility function tell us about his preferences for leisure and consumption? Deter- mine his demand functions for consumption C^D(w) and for leisure F(w). What is Robinson’s labor supply function L^S(w)? 

e) How many markets are there in this economy? Explain why it is sufficient to consider the market clearing condition on only one of these markets.

 Determine the equilibrium wage rate. (Note: its value may be an irrational number.) In this equilibrium, how many hours per day will Robinson work and how many consumption baskets per day will he consume? Represent the equilibrium price line in your graph. Compare your outcome with the social optimum from (a) and explain whether this market equilibrium allocation is Pareto-efficient.

(f) So far, we have assumed that Robinson Crusoe’s economy is closed and has no access to global labor and goods markets. Now assume that this economy opens and has access to the world market where both goods are available at the wage rate of w = 1. Are labor and consumption goods becoming relatively cheaper or relatively more expensive in the global market than on Crusoe’s island? Given the new opportunity cost of labor, assume that the domestic firm on Crusoe’s island now faces the international wage rate. What will be this firm’s labor demand, production level, and profits?

 Explain why it changes its production decisions compared to the initial equilibrium under autarky. What will be now the domestic consumer’s decision about the demand for consumption, leisure, and supply of labor? How many hours of labor will Robinson Crusoe work for the domestic firm and on the international labor market? By comparing Crusoe’s utility levels under autarky and with international trade, evaluate the domestic economy’s gains from specialization and trade. Represent your results graphically in your diagram that you extend by drawing the international price line (terms of trade) tangent to the new production plant of the domestic firm.