Problem 3: General Equilibrium in an Economy with Production (35 points) Consider a small "Robinson Crueoc" economy where there is a firm that produces consumption goods (C, measured in consumption bushets) from labor as the only resouroe ( L. mcasured in houns) according to the production function C=f(L)=4L The representative conesaner (ftobinson Crusoe) has preferences described by the utility fanetion u(F,C)=FC, where F denotes the consumers' leisure in hours and C, the number of consumption baskets. The cotsutter's time endowtuent consists of 24 hours an day. Assume that both the firm and the consumer are price talors. The prices of both goods are chenoted by w (nominal hourly wage rate) and P (price of ane consumption basket). Throtughoat this probhein, aseume that the price of the cossumption basket fprice level. cf. Macroeconotnics) is. I. Then the relative price w/p with an interpretation of the real wage rate (ef. Macrocoonomics) is identical with wage rate ar, In other words, w is the only relesant market price in this cconotuy. (a) (7 points) What is the (daily) Pareto optimal allocation (habor time, beisure time and consumption)? In order to answer this question, aveume that a social planser attempts to musximize the representative cousumer's utility level subject to the constraint of the atualable tochnology and resources. What is the condition for Pareto optimality is a geacral equilihriam with prodoction (ave loctare and textbooko)? Represent your result graphically in a (F,C) diagram where you draw the transformation curve (production possibility fronticr) and a sct of indifference curver. (b) (7 points) Consider the firm's production technology. Does it exhibit increasing. decreasing or constant returns to scale and why? Given this results, do you cxpect this firm's profits to be positive of zero? Asume 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 LD(se), supply function C ( e ) and the profit function (the optimal profit of the firm at given prices) (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) (7 points) 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: lse 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) (7 points) Write down Robinson's utility maximization problen. What does his utility function tell ns about his preferences for leisure and consumption? Determine his demand functions for consumption CD(w) and for leisure F(w). What is Robinsou's labor supply function LS(w) ? (e) (7 points) Ilow many markets are there in this economy? Explain why it is suffcient 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 constmption 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