Economics 201bSpring 2010Problem Set 2Due Thursday April 11. Competitive Equilibrium(-a) When Preferences Are Kinked. Recentlythere have been a surge in decision theory models that are non-differentiablein nature. For example, popular models incorporating loss aversion in prospecttheory, or ambiguity aversion as illustrated by Ellsberg Paradox, have kinkedindifference curves.In this exercise we are going to take a reduced form of these preferences andexamine the implications of “kinkiness” on equilibrium prices and allocationsin our simplest 2 × 2 exchange economy, where agents’ utility functions are∀i ∈ {1, 2},Ui(x1i, x2i) =√x1i+12√x2iif x1i≤ x2i12√x1i+√x2iif x1i> x2i.(1)(a) Suppose the initial endowments are ωi= (4, 4), ∀i ∈ {1, 2}. Draw theEdgeworth box for this economy. Find the Pareto optimal allocations.Verify that the initial endowment is an equilibrium allocation. Find thesupporting equilibrium price(s).(b) Now suppose the endowments are instead ω01= (5, 3), ω02= (3, 5). Findthe individual and market excess demand functions (notice that the utilityfunction is not differentiable). Find the competitive equilibrium prices andallocations. Is there a unique equilibrium?(c) Now suppose the endowments are instead ω001= (8, 2), ω002= (3, 5). Repeatthe calculations that you have done in (b). (Note that the Edgeworth boxis different in this case). Comment on how the kinkiness of preferencesaffect the size of competitive equilibria.(d) Does the First Welfare Theorem holds in this economy? What about theSecond Welfare Theorem?2. More Fun with Offer Curves! Consider simple two-person, two-good econ-omy in which agents’ utility functions are given byU1(x11, x21) = min{x11, x21}, and U2(x12, x22) = min{4x12, x22}. (2)and endowments are ω01= (30, 0), ω02= (0, 20).(a) If neither agents can have negative consumption of either good, what isWalrasian equilibrium?1(b) Now suppose the first agent starts only with 10 units of good 1 instead of30 and none of the second. What is Walrasian equilibrium in this case?Explain briefly your results. Hint: be sure to find all Walrasian equilibria.(c) Suppose that an agent decides to throw away part of her endowment tochange the equilibrium prices in the economy. Can agent be better offin the new equilibrium than in the equilibrium with the original endow-ment? Provide an example and explain. (The example does not have tobe analytic, however, it must be described clearly and coherently).3. Equilibrium with “Bads.” Consider an exchange economy that contains twoconsumers with utility function of the form ∀i ∈ {1, 2}:Ui(x1i, x2i) = x1i(4 − x2i) (3)defined over consumption set [0, 5] × [0, 3] ⊂ R2+. Notice that the second com-modity is “bad.” Endowments are given by ω1= (1, 3) and ω2= (3, 1).(a) Show that feasible allocation x is Pareto optimal if and only if x11+x21= 4.(b) Compute excess demand functions and find the Walrasian equilibrium.Illustrate it with Edgeworth box diagram.(c) What happens to the Walrasian equilibrium if the first consumer has theright to dump all of her endowment of the second commodity onto thesecond consumer?(d) What happens to the Walrasian equilibrium if there is an ad valorem taxt on any sale of good 2 paid by the seller, which is then shared equallybetween two agents?4. Importance of Assumptions. Give examples of the following, and illustratethem using an Edgeworth box. Please be clear and precise.(a) A Pareto optimal allocation that can’t be sustained as a Walrasian equi-librium with transfers.(b) A Walrasian equilibrium that is not Pareto optimal (Please do not useexternalities).5. Computing the Transfers. Consider again the exchange economy from thequestion 1 of the problem set 1: a two-person, two-good exchange economywhere the agents’ utility functions are U1(x11, x21) = x11x21and U2(x12, x22) =x12x22, and the initial endowments are ω1= (1, 3) and ω2= (3, 1). Showdirectly that every interior Pareto optimal allocation in this economy is a priceequilibrium with transfers by finding the associated prices and