Economics 201bSpring 2010Problem Set 6Due Thursday April 291. There are many foreign embassies in the Washington, DC. In fact, there is anarea of the city where quite a few of them are very close to each other. As youwalk along one of the streets you observe three embassies standing next to eachother. Each embassy has a flagpole with its national flag flying in the wind. Youknow that the flagpole height chosen by each embassy depends continuously onthe heights chosen by other two embassies. (For instance, having too tall a polecompared to the neighbors would be ostentatious, whereas having one too shortwould look stingy.) Moreover, having observed heights chosen by the neighbors,each embassy has a single, most favorite height to set. DC ordinance imposesan upper limit of 100 feet on flagpole heights of the embassies. The choices offlagpole heights are in equilibrium when no one wishes to change the height oftheir flagpole. Prove that there exists an equilibrium.2. Consider a two-person, two-good exchange economy where all agents have thesame utility function, i = 1, 2:u(x1i, x2i) = max{2 min{2x1i, x2i}, min{x1i, 4x2i}}.(a) Draw the indifference curves for one of the consumer. Are this consumer’spreferences convex?(b) Draw the Edgeworth box for this economy, denoting Pareto set, individu-ally rational and core allocations.(c) Now suppose we have an economy of I ∈ N identical consumers with I ≥ 2,each of which has the same preferences as the consumers described aboveand endowments are (ω1, ω2) = (1 + 3α, 2 − α) where α ∈ (0, 1). Find anecessary and sufficient condition on α that must be satisfied for there toexist a Walrasian equilibrium of this economy. Show that as I increases,the set of a ∈ (0, 1) that satisfy the condition you found increases in size.(d) Explain in a few sentences how these results relate to Theorem 2 in theLecture Notes 11. That is, relate your above results to the fact that, inthis economy, we can show that ∀α ∈ (0, 1) ∃p∗ 0, p∗∈ ∆0with 0 ∈conE(p∗) and x∗i∈ Di(p∗) such that1ILX`=1p∗` IXi=1x∗i−IXi=1ωi!l≤2LImax{kωik∞: i = 1, ..., I}Fix α =13and compute an explicit bound for the market value of thesurpluses and shortages in the economy. Verify that the bound providedby the theorem is tight enough.13. Consider four-person, two-good pure exchange economy where agents have en-dowments ω1= ω2= (10, 10) and ω3= ω4= (10, 30) and the same utilityfunction Ui(x1i, x2i) = log x1i+ log x2i, i = 1, 2, . . . , 4. For each allocation vec-tor given below show whether the it is Pareto optimal; is in the core (if not,provide a blocking coalition); can be supported as a competitive equilibria forsome price vector. Explain your reasoning.(a) x1= x2= (7.5, 15) and x3= x4= (12.5, 25).(b) x1= x2= (√50, 2√50) and x3= x4= (20 −√50, 40 − 2√50).(c) x1= (8, 12), x2= (9, 11), x3= (12, 23) and x4= (11, 29).4. Give an example of a three-person, two-good pure exchange economy where allagents have the same utility function Ui(x1i, x2i) = log x1i+log x2i,. Find a set ofinteger endowments for these agents along with a Pareto optimal, individuallyrational integer allocation that is not in the core.5. Consider a pure exchange economy with H = 2 consumers and L goods, withsocial endowment ¯ω ∈ RL++. In this question, we will consider the n-fold replicaof this economy. In the n-fold replica, there are 2n agents, of whom n (referredto as type 1 agents) have preferences and endowments identical to those of agent1 in the original economy, and n (referred to as type 2 agents) have preferencesand endowments identical to those of agent 2 in the original economy.(a) Let p∗be an equilibrium price vector for the original economy. Show thatp∗is also an equilibrium price vector for the (larger) n-fold replica economy.(b) Now, lets consider a special case where there are two commodities and twotype of agents. Type 1 is characterized asU1(x11, x21) = x11x21, ω1= (10, 0)and type 2 is characterized asU2(x12, x22) = (x12)12(x22)12, ω2= (0, 10).Show that the allocation x1= x2= (5, 5), (x1to agents of type 1 and x2to agents of type 2) is in the core for all levels of replication n.(c) Continue to assume two-good, two-agent type economy given above. Showthat the allocation x1= (9, 9), x2= (1, 1), is in the core for the originaleconomy with one agent of the each type and is not in the core for then-fold replica with n ≥ 2. Discuss.6. Give an example of acyclic preference relation that is not