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3. Consider an economy with two types of agents, both of whom have preferences defined over food (r) and prescription medicine (y). Type A agents each have the differentiable utility function u.(1., y.) and are endowed with w, units of z. Similarly, type B agents each have differentiable utility us(rs, ys) and are endowed with w units of r. Assume this is public information. There are equal numbers of both types. Agents can give up some or all of their r and buy y at price p (in units of r). a. First, write down the decision problem each agent faces in deciding how much r and y to consume and depict a solution. In order to cover administrative costs, the government imposes (lump sum) taxes go and go (in units of r), respectively, on types A and B. (Assume the government can distinguish between the types - say, A agents are rich and B are poor-and can tell which type a given consumer is.) b. Assuming the agents take the taxes as given, how does this affect their choice prob- lems? C. Assume the government must collect a total of G from each A-B pair (i.e., gotg = G). Argue that, from an efficiency point of view, it does not matter how the tax burden is distributed. d It is implicitly assumed in part c that there is equal-treatment- of equals, that is, that all agents of the same type pay the same tax. Is this necessary for efficiency? Explain. Now suppose that instead of covering administrative costs, tax collections were used solely to provide y at the price p , is locally nonsatiated for every i e I. Answer the following questions. (a) Define Walrasian equilibrium and Pareto efficient allocation in this econ- (b) Prove that every Walrasian equilibrium allocation is Pareto efficient. (c) Suppose that / = (1.2,3). Suppose that consumer 1 and consumer 2 decide to trade exclusively with each other, effectively excluding consumer 3 from any exchange. Consumer 1 and 2 negotiate to come up with a pair of consumption vectors r;,2; ( RY such that s; +% 2 en tee. Of course consumer 3 just consumes her endowment es (or a part of it). Let (x*. p*) c RY x RL be any Walrasian equilibrium that would have realized if every consumer Can participate in the market. Clearly consumer 3 is always (weakly) worse off by consuming ea rather than x. But is it possible that consumer 1 and 2 are better off negotiating with each other, i.e. r, _, x; for i = 1, 2 and r; ), at for i = 1 or 2? If so. find such an example. If not, explain why.2. Pareto Efficiency and Externality We consider a pure exchange economy Zeff with consumption externality, where consumer I's utility is directly affected by other consumers' consumptions. Consumer I's preference can be represented by a continuous utility function "(x)= /(x) - (x) , where f is increasing ( > = () > {(,)) and concave and g, is increasing and convex. The preference of consumer : (# 1) is represented by a usual utility function a, (I,) , which is increasing and concave. (a) Let U = tuc R. : Ir feasible, a Su (x) } be the utility possibility set. Show that / is closed and convex. (b) Show that a feasible allocation a in this economy Part is Pareto efficient if and only if it maximizes the weighted sum of utilities with respect to some weight vector (21 ,.... as) ER+ +. (c) Consider the following two-good pure exchange economy with consump- tion externality: / = (1.2) . w (x) = lari,1 + Inme -me,1. we (x1) = Inc,1 + Inmay, and e = ey = (1/2, 1/2). The definition of Walrasian equilibrium (x*, p*)(R xR, is the same as usual, except that of solves maxes: 1 (21.21). s.t. p* rip*vei given r;. Characterize the set of Pareto efficient allocations in R , and show that every Walrasian equilibrium allocation in Rf , is Pareto inefficient.2. Pareto Efficiency and Externality We consider a pure exchange economy Zeff with consumption externality, where consumer I's utility is directly affected by other consumers' consumptions. Consumer I's preference can be represented by a continuous utility function "(x)= /(x) - (x) , where f is increasing ( > = () > {(,)) and concave and g, is increasing and convex. The preference of consumer : (# 1) is represented by a usual utility function a, (I,) , which is increasing and concave. (a) Let U = tuc R. : Ir feasible, a Su (x) } be the utility possibility set. Show that / is closed and convex. (b) Show that a feasible allocation a in this economy Part is Pareto efficient if and only if it maximizes the weighted sum of utilities with respect to some weight vector (21 ,.... as) ER+ +. (c) Consider the following two-good pure exchange economy with consump- tion externality: / = (1.2) . w (x) = lari,1 + Inme -me,1. we (x1) = Inc,1 + Inmay, and e = ey = (1/2, 1/2). The definition of Walrasian equilibrium (x*, p*)(R xR, is the same as usual, except that of solves maxes: 1 (21.21). s.t. p* rip*vei given r;. Characterize the set of Pareto efficient allocations in R , and show that every Walrasian equilibrium allocation in Rf , is Pareto inefficient.5. Signaling with Outside Opportunities Consider the following simple Spencian signaling model. The set of types is O = (01. 02.02) = (1,3.4), which are equally likely. A type t worker has a marginal product of #,. The cost of signaling at level q for type Of worker is ( (Or, q) = A (Of) q, where A(01) = 1. A (02) = 3. A (0a) = 10. A type or worker has an outside payoff (self-employment) of wa (Or), where to (01) = 1. No (02) = 14, us (03) = 25. (a) Explain why it is a Perfect Bayesian Equilibrium (PBE) outcome for type , worker to choose q (0,) = 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 , and 0, to signal with q (@, ) = q (0 ) = 0 and type Of to choose q (03) = 6. (d) Does the PBE satisfy the Intuitive Criterion? (e) Show that there is a PBE that satisfies the Intuitive Criterion (f) BONUS: Suppose henceforth that the outside payoff are as follows: "(01) = 1. "(0) = 1;. wa(04) = 25. Is there a PBE that satisfies the Intuitive Criterion? If not why not. If so, solve for the equilibrium signals and payoffs.5. Signaling with Outside Opportunities Consider the following simple Spencian signaling model. The set of types is O = (01. 02.02) = (1,3.4), which are equally likely. A type t worker has a marginal product of #,. The cost of signaling at level q for type Of worker is ( (Or, q) = A (Of) q, where A(01) = 1. A (02) = 3. A (0a) = 10. A type or worker has an outside payoff (self-employment) of wa (Or), where to (01) = 1. No (02) = 14, us (03) = 25. (a) Explain why it is a Perfect Bayesian Equilibrium (PBE) outcome for type , worker to choose q (0,) = 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 , and 0, to signal with q (@, ) = q (0 ) = 0 and type Of to choose q (03) = 6. (d) Does the PBE satisfy the Intuitive Criterion? (e) Show that there is a PBE that satisfies the Intuitive Criterion (f) BONUS: Suppose henceforth that the outside payoff are as follows: "(01) = 1. "(0) = 1;. wa(04) = 25. Is there a PBE that satisfies the Intuitive Criterion? If not why not. If so, solve for the equilibrium signals and payoffs
