Econ 99 the question is okay.

1. First Welfare Theorem Let E pure = (fXi ; i ; eigi2I ) be the standard pure exchange economy with free disposal, where Xi = R L + and i is locally nonsatiated for every i 2 I: Answer the following questions. (a) Dene Walrasian equilibrium and Pareto e cient allocation in this economy. (b) Prove that every Walrasian equilibrium allocation is Pareto e cient. (c) Suppose that I = f1; 2; 3g. Suppose that consumer 1 and consumer 2 decide to trade exclusively with each other, eectively excluding consumer 3 from any exchange. Consumer 1 and 2 negotiate to come up with a pair of consumption vectors x 0 1 ; x0 2 2 R L + such that x 0 1+x 0 2 e1+e2: Of course consumer 3 just consumes her endowment e3 (or a part of it). Let (x ; p ) 2 R 3L + R L + be any Walrasian equilibrium that would have realized if every consumer can participate in the market. Clearly consumer 3 is always (weakly) worse o by consuming e3 rather than x 3 . But is it possible that consumer 1 and 2 are better o negotiating with each other, i.e. x 0 i i x i for i = 1; 2 and x 0 i i x i for i = 1 or 2? If so, nd such an example. If not, explain why.

2. Pareto E ciency and Externality We consider a pure exchange economy E ext with consumption externality, where consumer 1s utility is directly aected by other consumersconsumptions. Consumer 1s preference can be represented by a continuous utility function u1 (x) = f (x1) X i6=1 gi (xi); where f is increasing (x 00 1 x 0 1 ) f (x 00 1 ) > f (x 0 1 )) and concave and gi is increasing and convex. The preference of consumer i(6= 1) is represented by a usual utility function ui (xi); which is increasing and concave. (a) Let U = fu 2 R I + : 9x feasible; u u (x)g be the utility possibility set. Show that U is closed and convex. (b) Show that a feasible allocation x in this economy E ext is Pareto e cient if and only if it maximizes the weighted sum of utilities with respect to some weight vector (a1; :::; aI ) 2 R I ++: (c)Consider the following two-good pure exchange economy with consumption externality: I = f1; 2g ; u1 (x) = ln x1;1 + ln x1;2 x2;1; u2 (x2) = ln x2;1 + ln x2;2; and e1 = e2 = (1=2; 1=2): The denition of Walrasian equilibrium (x ; p ) 2 R 4 +R 2 + is the same as usual, except that x 1 solves maxx12R2 + u1 (x1; x 2 ); s.t. p x1 5 p e1 given x 2 : Characterize the set of Pareto e cient allocations in R 4 ++ and show that every Walrasian equilibrium allocation in R 4 ++ is Pareto ine cient

4 Public Good Provision The government must decide whether to build a project that is of potential value to two rms. The cost of the project is c; the value to rm 1 is either 1 or 0, the value to rm 2 is either 2 or 0; in each case the probability of a positive value is p (where 0 < p < 1) and the probabilities are independent. Whenever the government decides to build the project it will divide the cost c between the rms but will never make a prot or provide a subsidy. The government wants to use a socially e cient mechanism: that is, a mechanism that causes the project to be built if and only if the cost is less than the total value to the rms. (To avoid complications we will ignore cases where the cost might be exactly equal to the total value to the rms.) Notice that this is not a symmetric problem, so the mechanism(s) need not be symmetric either. (a) If 2 < c < 3, nd a socially e cient mechanism that is incentive compatible and (interim) individually rational for the rms. (That is, the rms are willing to participate in the mechanism after they know their true values.) (b) If 1 < c < 2; nd a socially e cient mechanism that is incentive compatible and (interim) individually rational for the rms. (c) If 0 < c < 1; nd the region in cost c and probability p space for which there is a a socially e cient mechanism that is incentive compatible and (interim) individually rational for the rms. In that region nd such a mechanism. 5 5. Signaling with Outside Opportunities Consider the following simple Spencian signaling model. The set of types is = f1; 2; 3g = f1; 3; 4g; which are equally likely. A type t worker has a marginal product of t: The cost of signaling at level q for type t worker is C (t; q) = A (t) q; where A (1) = 1 2 ; A (2) = 1 3 ; A (3) = 1 10 : A type t worker has an outside payo (self-employment) of uo (t); where uo (1) = 3 4 ; uo (2) = 1 1 4 ; uo (3) = 2 1 4 : (a) Explain why it is a Perfect Bayesian Equilibrium (PBE) outcome for type 1 worker to choose q (1) = 0 and the other types to choose their outside alternatives? (b) What is the Intuitive Criterion? Does this PBE satisfy the Intuitive Criterion? (c) Explain why it is a PBE for types 1 and 2 to signal with q (1) = q (2) = 0 and type 3 to choose q (3) = 6: (d) Does the PBE satisfy the Intuitive Criterion? (e) Show that there is a PBE that satises the Intuitive Criterion (f) BONUS: Suppose henceforth that the outside payos are as follows: uo (1) = 3 4 ; uo (2) = 1 3 4 ; uo (3) = 2 1 2 : Is there a PBE that satises the Intuitive Criterion? If not why not. If so, solve for the equilibrium signals and payos.

6. Monopoly and Product Quality The set of buyer types is = ftg T t=1 : If a type t customer purchases a unit of quality level q and pays R; his benet is B (t; q) = tq and so his payo is u (t; q; R) = tq R + uo: No customer places any value on additional units. Let fq (t); R (t)gt2 be a set of quality levels and prices that satises the participation constraints. (a) Show that a necessary condition for incentive compatibility is that fq (t)gt2 is increasing. Henceforth consider the special three type case. Types are equally likely so that the probability of a type t is f (t) = 1 3 : The cost of each unit of quality q is C (q) = 2q 2 : The monopoly objective is expected prot maximization. Consider the relaxed problem in which only the local downward incentive constraints are satised. (b) Show that for prot maximization there are three binding constraints. (c) Prove that the solution to the relaxed problem is incentive compatible for type 1:(The proof for other types is almost identical). (d) Show that the expected prot of the rm can be written in the following form: (q) = 1 3 X 3 t=1 h Atq (t) 2q (t) 2 : i [Conrm that A2 = 22 3 and solve for A1 and A3]. (e) Hence obtain conditions under which only one quality level will be sold.

solar system. One feature of these aliens is that they can read immediately the utility function of others. (Although this sounds quite useful, it is rather a curse.)1 Anyway, as a proof of this claim we print here Emma's utility function: u(x1, x2, e) = x1 + (a2, ae) x2 e 1 2 (x2, e)B x2 e , (3) where a2, ae > 0 and B = b22 b2e be2 bee is symmetric positive definite. I know, you surely must think "Wow" but let's focus again on the prelim exam. Assume that solutions are interior and that constraints are satisfied with equality. Write out the system of equations from problem c.) for Emma's utility function. e.) Provide an interpretation of the partial derivatives x2(p1,p2,w,ei) ei and e(p1,p2,w,ei) ei and their signs. f.) Compute x2(p1,p2,w,ei) ei and e(p1,p2,w,ei) ei . g.) Assume b2e 0. Derive the signs of x2(p1,p2,w,ei) ei and e(p1,p2,w,ei) ei . h.) Assume now b2e < 0. Show that without additional assumptions, the signs of x2(p1,p2,w,ei) ei and e(p1,p2,w,ei) ei remain ambiguous in this case. Find additional assumptions on matrix B and that allow you to determine the signs of x2(p1,p2,w,ei) ei and e(p1,p2,w,ei) ei . i.) We want to get a better understanding of whether good 2 and the externality are complements or substitutes. In our particular context, do we need to distinguish between gross complements/substitutes and (net) complements/substitutes? j.) To figure out whether good 2 and the externality are complements or substitutes, we can use the Slutsky substitution matrix. To this end, compute first Walrasian demand functions as if the externality has a market price pe. That is, compute Walrasian demand functions x2(p1, p2, pe, w) and e(p1, p2, pe, w). k.) How does the fact that good 2 and the externality are substitutes or complements depend on the sign of b2e? `.) How is the fact that good 2 and the externality are substitutes or complements related to the sign of the derivatives x2(p1,p2,w,ei) ei and e(p1,p2,w,ei) ei discussed earlier?