Problem 1. Andy (consumer A) and Red (consumer B) growing old together on a remote island on which only fish (good 1) and coconut (good 2) are available for consumption. Both Andy and Red have Cobb-Douglas preferences. That is, consumer i has u(xi, z) = (a)¹/2 (1)¹/2, where x is consumer i' s good j consumption with j = 1, 2. Also, Andy's ini- tial endowment is w (w, w) (1,3) while Red's initial endowment is w = (wi, wz) = = = = (3, 1). (a) Draw the Edgeworth box for this economy. Mark the point indicating the initial endowment of each consumer. (b) Draw the contract curve for this economy in an Edgeworth box (a graphical represen- tation is sufficient). Explain if it is Pareto efficient for Andy and Red to consume their endowments. (c) What is the set of allocations that could be the outcome under barter in this economy? (d) Let the price of fish be p₁ while the price of coconut be normalized to 1 without loss of generality. For each consumer, solve the utility maximization problem, i.e., derive z for j= 1, 2 and i = A, B as a function of the price p₁. (e) Suppose p₁2. Draw the budget set for both Andy and Red, and show their optimal consumption bundles on the Edgeworth box. Can the economy be in equilibrium when P₁ = 2? (f) Calculate the equilibrium prices and demands for the economy. Show the equilibrium outcome on the Edgeworth box. (g) Plot the offer curve for consumer A. What happens to his consumption of good 1 and good 2 as the price p₁ increases? Plot also the offer curve of consumer B. Graphically, find the intersection of the two offer curves. What does this intersection signify? (h) Can you calculate the equilibrium price as a function of the initial endowments w's?

Tom, Dick, and Harry live in the same apartment building in downtown Los Angeles. Tom and Dick work at local auto parts stores, and each of them has an income of y dollars per week. Harry is less fortunate. He used to have a good job at the LAPD, but his penchant for firing large caliber weapons in crowded public places led to his dismissal. He currently has no income. Tom and Dick (who are originally from Texas) firmly believe in a man’s right to draw his gun in the defence of just about anything, and are happy to financially support Harry. Tom gives Harry zT dollars each week, and Dick gives Harry zD dollars each week Tom’s utility UT depends upon the dollar value of his own weekly consumption, cT , and of Harry’s weekly consumption, cH : Likewise, Dick’s utility UD depends upon the dollar value of his own weekly consumption, cD, and of Harry’s weekly consumption: Harry spends all of the money that he receives from Tom and Dick. Tom and Dick spend…

The utility possibility frontier is defined as the set of maximum utility levels for all consumers that is feasible in the economy, given resource constraints and preferences. For a two-consumer economy, it is defined formally as the solution to: u (ūB) = max u A (xA) subject to ug(xB) > ūg and (xA, XB)feasible. For each of the economies above, find an equation for the utility possibilities frontier, and graph it.

Two individuals (A and B) live on an island where they consume coconuts (good x) and bananas (good y) for survival. A owns the coconut producing part of the island and B owns the banana part. Their utility functions and endowments are given by UA (TA,YA) = A+YA +min {A, YA}, UB (TB, YB) = TB +YB + min {ïß, YB}, (ww) = (0, 12) B B Assume that the price of coconuts is p and that of bananas is 1, so relative prices are denoted by p. (ww) = (10,0) A 1. What are their respective budget constraints? 2. What are their individual demand functions? (Hint: you have to consider multiple cases for the value of p) 3. The individuals trade their endowments with each other. What is the competitive equilibrium price in the market? What are their respective utilities at equilibrium?