University of California, BerkeleyEconomics 201ASpring 2005 Final Exam–May 20, 2005Instructions: You have three hours to do this exam. The exam is out ofa total of 300 points; allocate your time accordingly. Please write yoursolutions to Parts I and II in separate bluebooks.Part I1. (100 points) Define or state and briefly discuss the importance of eachof the following within or for economic theory:(a) Shapley-Folkman Theorem on Sums of Sets(b) local nonsatiation(c) Walrasian Quasiequilibrium(d) Second Welfare Theorem in an Arrow-Debreu Economy(e) Debreu-Gale-Kuhn-Nikaido Lemma2. (80 points) Consider an Edgeworth Box economy, with endowmentsω1= (3, 1), ω2= (1, 3), and utility functions u1(x11, x21) = x1/311x2/321,u2(x12, x22) = x2/312x1/322.(a) Find all the Walrasian Equilibria of this economy.(b) Compute the set of Pareto Optima of this economy.(c) Find prices and transfers that make the allocation x1=³1,167´,x2=³3,127´a Walrasian equilibrium with transfers.(d) Compute the core of this economy. [You should obtain equationsfor the end points of the core, but you needn’t solve these equa-tions explicitly.]1Part II3. (120 points) Consider an exchange economy with I consumers and L =2 goods. The vector of endowments, ω ∈ R2I, is fixed, and ω1> 0. LetU denote the set of utility functions u on R2+satisfying• u is C2(the second partial derivatives all exist and are continuous)• 5u|x>> 0 and the Hessian matrix Hu|xis negative definite forall x ∈ R2++• u(x) = 0 for x ∈ R2+\ R2++• u(x) > 0 for x ∈ R2++The preferences of consumers i = 2, ..., I are fixed and generated byutility functions u2, . . . , uI∈ U. i = 1’s preference is described by aparametrized utility function u1: R2+× ((0, ∞) × (0, 1)) → R given byu1(x1, α) = v(x11, x21) + α1xα211x1−α221, for some v ∈ U.(a) Write down the first-order conditions defining the demand of agent1, and show that they are necessary and sufficient to characterizethe demand.(b) Using the Implicit Function Theorem, show that the demand ofagent 1 is a C1function of (p, α).(c) Using the Transversality Theorem, show that for almost all α, theeconomy is regular.(d) Using the Implicit Function Theorem, show that for almost allα, the economy has finitely many equilibria which move in a C1fashion in resp onse to changes in