Question 1. We reconsider the production economy of Lecture 20. Robinson Crusoe's preferences are as in the lecture, Cobb-Douglas preferences U RC (f,c) = fc . The production technology is different from Lecture 20: when RC divides his time, spending tf time fishing and t_c= 1 t_fcollecting coconuts, he catches 2sqrt(t_f) fish and collects tc coconuts.

(a) How much fish does RC catch? How many coconuts does he collect?

(b) Suppose there are two firms on the island, one fishery (with production function f = 2sqrt(t_f) , where t_fis the labor hired by the fishery) and one coconut farm (with production function c = t_c, where t_cis the labor hired by the farm). RC is the sole shareholder, but runs each firm as a price-taking profit-maximizing firm. Carefully describe the competitive island economy, including the profit functions for the firms, and the budget constraint for RC. Define fully a Walrasian (competitive) equilibrium. [Hint: Example 48 on page 203 of the Course text (Vohra draft) is relevant here.]

(c) Calculate the Walrasian equilibrium of the competitive island economy. Compare the resulting allocation with that found in part (1a).

(d) Consider now the n-fold replica economy of the economy described in part (1b): There are n households, n fisheries, and n coconut farms (all cloned from the RC economy). Suppose all households own an equal share of all firms. What are the market clearing conditions that must be satisfied in a Walrasian equilibrium? Calculate the Walrasian equilibrium of the competitive island economy. Compare the resulting allocation with that found in part (1c).

(e) Suppose now that while there are n households (clones of RC) in the economy, there are m fisheries and m coconut farms (in particular, m may not equal n). How does this change your answer to part (1d)? Which world do the households prefer, one with a large m, or one with a small m?