Spring 2018 Recitation 1 Notes

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1 Spring 2018 Recitation 1 Notes Arda Gitmez Feburary 16, 2018 Today s paper: Assigning Resources to Budget-Constrained Agents by Che, Gale and Kim (REStud, 2013). Why is this paper important? 1. Structure-wise: it stands at a nice point between the market design and mechanism design literatures. Market Design: Let s see if we can improve this market. Mechanism Design: What s the optimal mechanism given constraints? These two sometimes call for different approaches to the same problem. (Remember the economist as an engineer/plumber debate.) Here: a little of both approaches. First, analysis and comparison of some popular mechanisms; second, derivation of the optimal mechanism from a mechanism design perspective. 2. The main question: it s really about the Coase Theorem. If property rights are clearly specified and there are no transaction costs, the initial allocation does not matter: we will always get an efficient outcome. Sort of a blow to the whole idea of market design. (A brief detour.) It s not the only result of this sort: revenue equivalence theorem, efficient market hypothesis, Modigliani-Miller Theorem... Why are these results still useful? Because they point to directions on where to go. In this particular case, we have a sense that the initial allocation should matter, so it would either require unclear property rights or existence of transaction cost. Milgrom s book, p.20: The zero transaction cost on which the Coasian argument is based, however, is not the one that Coase ever advocated as a description of reality. Rather, it was advanced as part of a thought experiment to emphasize the importance of understanding actual transaction costs. The other extreme of this is impossibility results : Arrow, Gibbard-Satterthwaite, Myerson- Satterthwaite, Holmström... They also give an idea on which assumption to relax. For instance, we don t give up looking for Social Welfare Functions because of Arrow s impossibility result. Instead, it motivates restricting attention to single-peaked preferences and deriving the Median Voter Theorem. This paper focuses on a particular transaction cost: budget constraints (or imperfect capital markets). 1

2 3. It deals with some ideas that will keep appearing later in the course. The trade-off between fairness and efficiency: as we will see later, almost all market design literature is built around understanding this trade-off. Cash vs. in-kind subsidies: again, related to the question of why not do everything through a monetary market? The authors have sort of an idiosyncratic view on this matter: relaxing budget constraints (i.e. giving subsidies) is useful, but not for the reason you think. Subsidize the poor for the sake of efficiency, not fairness. Indeed, give cash subsidies to people with low valuations to keep them away from the mechanism. The Model Preliminaries Assume that there is a mass S (0, 1) of indivisible good to allocate. There is a unit mass of buyers who each demand one unit and has an unobservable type (w, v). w W = [0, 1], w G(w). wealth v V = [0, 1], v F (v). valuation w and v are distributed independently, but the authors argue that the results generalize. Basic assumptions: a buyer w, v can pay at most w. Budget Constraint. enjoys utility vx + w t when obtaining the good with prob x and paying t in expectation. Quasi-linear and risk-neutral preferences, in line with mechanism design literature. Examples: pollution permits, public housing, organ donation, school vouchers... The Question & Overview of Results Main question: how should one allocate the good? Welfare Criterion: ex post utilitarian efficiency, i.e. max (total realized value). Fairness is sort of implicit in this criterion, right? First-Best Allocation (i.e. if types were observable): just give the good to people with highest v. Three popular assignment methods: 1. Competitive Market (CM): same thing as an auction. Only those with high valuation and high wealth will be able to buy. 2. Random Assignment without Resale (RwoR): price is capped, a lottery assigns the good. Government announces p, agents enter the lottery voluntarily. If won, an agent pays p and receives the good. 3. Random Assignment with Resale (RwR): same as RwoR, except that the agents are allowed to resell the good. 2

3 The first result (Proposition 1) can be summarized as follows: RwR > CM > RwoR. The authors then say: But can we do better than RwR? They take a mechanism design approach to answer this question. The second result (Proposition 2) suggests that the optimal mechanism looks pretty much like RwR. The authors then take a simpler, 2-by-2 model to fully characterize the optimal mechanism (Proposition 3). Results Comparing Salient Methods The key thing to notice about this setup is that, in any of the methods considered, the agents who are at the north-eastern part of (w, v) plane will end up with the good. This means that there will be cutoff values w and v such that only agents with w w and v v will end up with the good. Up to some normalization, then, the utilitarian welfare under this allocation is: E[v v v, w w ] But independence of w and v implies that this equals: E[v v v ] Which is increasing in v. The problem then reduces to a simpler one: find what the allocation in each method looks like. Characterize the cutoffs and compare them. A simple visual argument 1 provides the basic intuition about the following result, then: Proposition 1. RwR yields higher welfare than CM, and the welfare rises as p falls. RwoR yields lower welfare than CM, and the welfare falls as p falls. Why does RwR do better than CM? Essentially it is because it allows some high-valuation-low-wealth guys to obtain the good by giving it at a capped price (i.e. it gives an in-kind subsidy). This is a feature that CM does not have. The cost of such a feature, however, is that many low-valuation agents who would not receive the good in the first-best solution also receive the good. (This is exactly the reason why RwoR does worse than CM.) To solve this issue, RwR allows for resale, so that there will be a reshuffling of goods between low-valuation agents and high-valuation-high-wealth agents in the aftermarket. At the end of the day, many low-valuation agents leave the market with some positive profits from resale (i.e. RwR gives them a cash subsidy). These features of RwR (in-kind and cash subsidies) are not without problems, however. In particular, there is a general tendency in economics to believe that in-kind subsidies are relatively inefficient compared to cash subsidies. Cash subsidies, on the other hand, attract a bunch of speculators to the market and possibly decrease the welfare. Can we do better than RwR, then? The next section of this paper attempts to answer this question using a mechanism design framework. Mechanism Design Definition 1. A direct mechanism in this problem is Γ = (x, t) : W V [0, 1] R. Here, if an agent (w, v) reports type (w, v ), she receives the good with probability x(w, v ) and makes the expected payment t(w, v ). 1 Look at the board! :) 3

4 Q: Why direct mechanisms? A: Revelation Principle. We re looking for direct mechanisms which solve the following constrained optimization problem: subject to: Aggregate Supply: E[x(w, v)] S Budget Balance: E[t(w, v)] 0 Budget Constraint: t(w, v) wx(w, v), w, v Individual Rationality: vx(w, v) t(w, v) 0, w, v max E[vx(w, v)]s.t. x,t Incentive Compatibility: vx(w, v) t(w, v) vx(w, v ) t(w, v ), w, v such that t(w, v ) x(w, v )w Q: Do the mechanisms considered in the previous section satisfy these? Proposition 2. The optimal mechanism involves: Cash transfers to a positive measure of agents (i.e. t(w, v) < 0 for a positive measure of (w, v)), and, Random assignment to a positive measure of agents (i.e. 0 < x(w, v) < 1 for a positive measure of (w, v)). As one can see, the optimal mechanism involves a cash subsidy and an in-kind subsidy, which are the two salient features of RwR. Nevertheless, it is difficult to fully characterize the optimal mechanism, so the authors focus on a 2-by-2 case. 2-by-2 model Let s assume W = {w H, w L } and V = {v H, v L }, with w H > v H > v L > w L 0. Proposition 3. Whenever first best is not achievable, the optimal mechanism involves a menu of three contracts. 4

5 The Rest Rest of the paper includes, in my opinion, relatively secondary stuff, so I m going to leave to you. The authors talk about how to implement the optimal mechanism (Proposition 4), They discuss an extension where there is an observable/verifiable signal about type (Proposition 5), They consider the limit when there are a lot of speculators (Proposition 6). 5