allocation represent a Pareto improvement? Explain your answer using indifference curves to illustrate it. 3. (20 points) Suppose Alex and Brianna are consumers in a pure exchange economy, with two goods, thneeds' and zaboomafoos'. Alex has 2 units of thneeds and 4 units of zaboomafoos. Brianna has 8 units of thneeds and 6 units of zaboomafoos. Alex and Brianna have identical preferences given by the utility function U(x) = (x)1 2(x*)? for i = 4;B (Alex and Brianna) a) Suppose the price of zaboomafoos, pr, is normalized to 1. In a perfectly competitive market, solve for the equilibrium price of thneeds, pi and the optimal allocation of consumption for both consumers Alex and Brianna. b) Calculate the utility for Alex and Brianna at the initial endowment allocation and the optimal allocation of consumption. Are Alex and Brianna made better off at the optimal allocation of consumption? c) Show in the Edgeworth box the initial allocation of endowment W, the optimal allocation of consumption M, the price line, and Alex and Brianna's indifference curves which go through the optimal allocation of consumption. d) Is there another allocation of endowment which can achieve the general equilibrium solved in question (b)? If so, what is it? 4. (15 points) Chuck Nolan is on a deserted island off the coast of Brazil (yes, that is the character Tom Hanks plays in Castaway), and is trying to survive by either gathering bacuri (b) or cupuacu (c). The technologies available for these two production processes are given by: and where L: is the amount of labour (time) used for gathering bacuri and _, is the amount of labour (time) used for gathering cupuacu. The total time (1) is $ hours per day, ? = Wez, + 2-Z :9: Chuck Nolan's preferences are given by the utility function, Ub,c) = bc. Note: both bacuri and cupuacu are fruits indigenous to the Amazon rainforest a) Construct the production possibility frontier (PPF) for bacuri and cupuacu and illustrate graphically. b) Find the marginal rate of transformation (MR), and the marginal rate of substitution (MRS) between cupuacu and bacuri. c) Find the optimal combination of bacuri and cupuacu and the optimal allocation of time. Show the competitive equilibrium on the graph. A Thneed is a highly versatile object knitted from the foliage of a Truffula Tree. Zaboomafoo is the name of a lemur in a children's television program