77-1Cover.qxd 12/4/09 11:38 AM Page 77(1) No 270 pp 1–416 THE REVIEW OF ECONOMIC STUDIES Vol 77(1) No 270 THE REVIEW OF ECONOMIC STUDIES Jane Martin: Special Tribute Dynamic Matching and Evolving Reputations Axel Anderson and Lones Smith Competitive Non-linear Pricing and Bundling Mark Armstrong and John Vickers 30 The Swing Voter’s Curse in the Laboratory Marco Battaglini, Rebecca B Morton and Thomas R Palfrey 61 Managerial Skills Acquisition and the Theory of Economic Development Paul Beaudry and Patrick Francois 90 Non-Parametric Identification and Estimation of Truncated Regression Models Songnian Chen 127 Millian Efficiency with Endogenous Fertility J Ignacio Conde-Ruiz, Eduardo L Giménez and Mikel Pérez-Nievas 154 Multi-Product Firms and Flexible Manufacturing in the Global Economy Carsten Eckel and J Peter Neary 188 Network Games Andrea Galeotti, Sanjeev Goyal, Matthew O Jackson, Fernando Vega-Redondo and Leeat Yariv 218 On-the-Job Search, Mismatch and Efficiency Pieter A Gautier, Coen N Teulings and Aico van Vuuren 245 Pairwise-Difference Estimation of a Dynamic Optimization Model Han Hong and Matthew Shum 273 Optimal Monetary Policy with Uncertain Fundamentals and Dispersed Information Guido Lorenzoni 305 Quantile Maximization in Decision Theory Marzena Rostek 339 M Utku Ünver 372 Dynamic Kidney Exchange Erratum January 2010 January 2010 415 Review of Economic Studies (2010) 77, 1–2 © 2010 The Review of Economic Studies Limited 0034-6527/09/00411011$02.00 doi: 10.1111/j.1467-937X.2009.00595.x Special Tribute JANE MARTIN Since 1997, Jane was the administrator and production editor for the Review of Economic Studies In that post she blossomed, and with her literary and technical skills, her goodwill, quick wit, helpfulness and sense of humour became the hub for the ever-changing cast of editors, referees and authors I knew Jane more or less from when she joined the journal, first as one of her editors and more recently as Chairman of the journal Although physically frail, Jane had a strong and unflappable personality She must have corresponded with an astonishing number of people over the years, many of whom had large egos and—if they had received a rejection letter from the editors, say—were not necessarily on their best behaviour Jane invariably calmed the stormy waters The fact that the journal Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 We are very sorry to report that Jane Martin, the Review’s administrator for many years, passed away on 26 September 2009 As a tribute to her, we reproduce here a short extract from a reading at her funeral service: REVIEW OF ECONOMIC STUDIES MARK ARMSTRONG St Giles-in-the-Fields, London 15 October 2009 © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 has such a loyal community of board members, authors and referees is due in very large part to her sure touch at the helm I never did hear a critical word about Jane from anyone By chance, many of us at the journal had the chance to say goodbye to Jane recently, although we did not know that that is what we were doing We had our 2009 annual meeting on 25 and 26 September in London, which Jane organized with her customary efficiency and warmth One thing the two of us discussed beforehand was the location of the dinner She had the imaginative idea of us going to the Royal Air Force Club for a change I timidly opted for something more anodyne, mainly because I was not sure that having paintings of spitfires bearing down on us would make for a fully relaxing evening (especially for some of our colleagues from the Continent) Our world will surely be a duller and colder one without her Let me mention just a couple of extracts from the many messages I received from people when they heard about Jane The editor who originally recruited her in Oxford wrote: “Jane had a good sense of what academic work was about and valued being associated with the Review She settled into her role smoothly from the very beginning Over the years, Jane became the face of the Review, and we were very lucky to have her.” Another editor: “I just remember her charm and warmth She had a beautifully cultured voice and way of expressing herself.” Our publisher: “I’ve known Jane for about ten years, having first worked with her on the production side, and always found her to be a wonderful person to work with Everyone here who came into contact with Jane was I think touched by her combination of graciousness and professionalism.” A foreign editor: “I never met Jane, but I just wanted to express that I had so many pleasant interactions with her over the years that I somehow thought of her as a dear friend She was very highly appreciated, I’m sure, not just by me but by all the people she communicated with over the years.” Finally, a friend and colleague wrote: “I would like to say that Jane was a loyal and generous friend, someone who enjoyed listening and helping others if she could She could also be very funny, and her love for her family always showed Jane loved writing and liked to share her pieces of work with me Her commitment to the journal was total, even when she was in hospital after an accident in 2007, and though she was in a lot of pain she was still replying to journal emails.” Review of Economic Studies (2010) 77, 3–29 © 2009 The Review of Economic Studies Limited 0034-6527/09/00411011$02.00 doi: 10.1111/j.1467-937X.2009.00567.x Dynamic Matching and Evolving Reputations Georgetown University, Washington, DC and LONES SMITH University of Michigan First version received June 2005; final version accepted March 2009 (Eds.) This paper introduces a general model of matching that includes evolving public Bayesian reputations and stochastic production Despite productive complementarity, assortative matching robustly fails for high discount factors, unlike in Becker (1973) This failure holds around the highest (lowest) reputation agents for “high skill” (“low skill”) technologies We find that matches of likes eventually dissolve In another life-cycle finding, young workers are paid less than their marginal product, and old workers more Also, wages rise with tenure but need not reflect marginal products: information rents produce non-monotone and discontinuous wage profiles INTRODUCTION Consider a static Walrasian pairwise matching economy where output depends solely on exogenous abilities Becker (1973) showed that positive assortative matching (PAM) arises when abilities are productive complements This is the foundational paper in the noncooperative theory of decentralized matching markets, and has established PAM as the benchmark allocation in the matching literature Shimer and Smith (2000) and Atakan (2006) have since found complementarity conditions under which PAM still obtains in this fixed type framework with random matching and search frictions In a static world, productively complementary individuals assortatively match by their expected abilities We introduce and explore a recursively solvable continuum agent matching model where agents have slowly evolving characteristics In this dynamic model we prove existence of a steady state equilibrium and the welfare theorems quite generally We then specialize to a world where all abilities are simply “high” or “low” We assume unobserved abilities, and stochastic but publicly observable output, where the separate contributions to joint production are unseen Everyone is then summarized by the public posterior chance that he is “high”–namely, his reputation is his characteristic Within this general learning framework we consider two specific models We focus on the partnership model, in which workers with unobserved abilities are matched in pairs to produce output In the employment model, these workers are matched one-to-one with jobs whose characteristics are known The partnership model can be interpreted literally as a model of production partnerships, or as a parable for production in teams within-firms, or finally as a model of within firm Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 AXEL ANDERSON REVIEW OF ECONOMIC STUDIES task assignment Output in many organizations is largely produced by teams: academic coauthoring, movie production, advertising, the legal profession, consulting, or team sports The O-Ring example of Kremer (1993) illustrates the role of stochastic joint production in high-tech industrial production Employment model We next specialize our model to one where workers are matched to jobs whose types are known Workers still have unknown abilities revealed over time via © 2009 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 The partnership model Our analysis of the partnership model begins with a two period setting Becker’s result yields PAM in the final period This yields a fixed convex continuation value function We then deduce that the fixed expected continuation values are strictly convex in the reputation of one’s partner We show that this induces strict gains from rematching any assortatively matched interior agents with or (i.e surely low or surely high individuals), or both, opposing production complementarity Despite this informational gain to non-assortative matching, PAM will again obtain in the first period with sufficient weight on the current period However, since the static production losses from non-assortative matching in the first period are bounded, PAM cannot be optimal with sufficient weight on the future (Proposition 2) Finite horizon models can have drastically different predictions than their infinite horizon counterparts Is our two period analysis representative of the general setting? While our findings hang in the balance, we rescue a failure of PAM that turns on a trade-off between value convexity due to learning and static input complementarity To see where our earlier logic goes wrong, we observe that the two period analysis critically relies on fixed continuation values With an infinite horizon, the continuation value is endogenous to the discount factor, and in a troubling fashion: as is well known, it “flattens out” with rising patience So as the discount factor rises to 1, current production and information acquired in a match both become vanishingly important A flattening value function is well understood, but we find a more subtle change While it is true that the value function becomes less convex for any fixed reputation, it becomes more convex in a neighbourhood of the extremes and 1; thus, we are led once again to check whether PAM fails near these extremes Our analysis requires a very precise characterization of the extremal behaviour of the value function to resolve the knife-edged tradeoff between information and productive efficiency as patience rises The paper then turns to a labour economics story Call the technology high skill if matches of one or two “low” agents are statistically similar For example, the production function in Kremer (1993) (in which project success requires success in all subtasks) is a high skill technology Proposition shows that efficient matching depends on the nature of the technology: PAM fails for high (low) reputations when production is sufficiently high (low) skill Not all technologies are high or low skill The information effect may reinforce the static output effect near and 1, yielding PAM for any level of patience In general, the PAM failure is quite robust Proposition shows that for randomly chosen production technologies, the chance of both a high and low skill technology tends to one, as the number of production outcomes grows We also offer simulation evidence that these conditions are extremely likely to hold in practice with few production outcomes Unlike other matching models with fixed types, ours affords an economically compelling micro-story as well While the market is in steady-state, individuals proceed through their lifecycle, and their reputations randomly change, converging towards the underlying true abilities So, with enough patience, if two genuinely high abilities are paired, then we should expect their reputations to rise as time passes Eventually, they enter the region where PAM fails, and the partnership will dissolve ANDERSON & SMITH DYNAMIC MATCHING REPUTATIONS stochastic production outcomes We assume that workers’ and firms’ types are productive complements, and so ideally should sort by type But with incomplete information, a worker’s job assignment determines both his expected output and the quality of information revealed in production We then arrive at a much different PAM result: workers near the reputational extremes will always match assortatively (Proposition 6), since the productive effects there are strongest This difference is the key empirical distinction between the partnership and employment models Related work PAM fails in Kremer and Maskin’s (1996) complete information matching model–but so does productive complementarity In Serfes (2005) and Wright (2004), negative assortative matching arises in a principal–agent framework There is a small literature of equilibrium matching with incomplete information Jovanovic (1979) considers a model where slow revelation of information about worker abilities causes turnover Niederle and Roth (2004) match three key features of our model: complementarity, uncertain types, and publicly revealed signals Chade (2006) extends Becker’s work to © 2009 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 A parsimonious model for labour economics Our partnership and employment models together provide a single coherent framework for understanding a variety of stylized facts in labour economics Wages Drift Up Wages generally rise with work experience Our model delivers this prediction, since expected values rise over time by Corollary 2, and so on average wages rise But also consistent with the reality, wages sometimes fall from period-to-period Both facts are true of our partnership and employment models Job Tenure, Mobility and Wages Wages rise with job tenure, separation rates fall with job tenure, and high current wages are correlated with low subsequent mobility (see Jovanovic, 1979; Moscarini, 2005) Just as in MacDonald (1982), our employment model with discrete known jobs matches these stylized facts To see why, note that workers at the reputational extremes are assortatively matched Since a worker’s wage equals his expected output, these workers receive the highest wages Finally, over time workers’ reputations are pushed to the extremes as their true types are revealed Thus, the longer a worker is with the same firm, the closer its reputation will be to the extremes and the higher its wage Finally, the closer a worker’s reputation to the extremes, the longer until its type crosses an interior threshold for job changing Life Cycle Marginal Products versus Wages Several empirical studies (e.g Medoff and Abraham, 1980; Hutchens, 1987; Kotlikoff and Gokhale, 1992) have found evidence for an increasing relationship between wages and productivity over the life cycle: young workers earn less than their marginal product and old workers more In our partnership model, workers at the reputational extremes are paid an informational premium, and others sacrifice for type revelation But if we follow a cohort of workers over time, their reputations move toward the extremes as their types are revealed So on average, younger workers will see their wages lag their productivity, while the reverse holds for older workers Observe how this result in our partnership model is entwined with our PAM failure With assortative matching, the two partners each receive half the output in wages, and there is no wage productivity gap Wage Dispersion by Cohort Huggett et al (2006) find that earnings dispersion across individuals within a cohort increases with age This is consistent with both our partnership and employment models Agents who have been around longer should have more accurate reputations than those at the beginning of their careers, and thus their reputations are more dispersed REVIEW OF ECONOMIC STUDIES Paper outline In Section 2, we set up our general model, define a Pareto optimum and competitive equilibrium, and establish the welfare theorems and existence Our theory thereby applies both to the efficient and equilibrium analyses; however, our interest in the planner’s problem is for the information it provides us about individual agents, since the planner’s multipliers are precisely the agents’ private present values of wages In Section 3, we develop Becker’s model for workers with uncertain abilities, explore the tradeoff between static complementarity and dynamic information gathering, and prove our PAM failure result In Section 4, we analyse the employment model A technical appendix follows THE MATCHING ECONOMY 2.1 The static matching model We consider a matching model with a continuum of agents, each described by a scalar human capital x belonging to [0, 1] Let Q(x, y) denote the static output of the match of types x and y We assume that Q(x, y) is symmetric, twice smooth, increasing in x and y, with a nonzero cross partial, lest matching trivialize As we assume everyone is risk neutral, Q can be either a deterministic output function or the expected output from stochastic production A twice differentiable function Q is strictly supermodular iff Q12 > 0, and strictly submodular when Q12 < Although we not require any special assumptions on Q for our existence and welfare theorems, the following assumption is used in some characterization results Assumption (Supermodularity) Q12 (x, y) > Assume a distribution G over human capital x ∈ [0, 1] The social planner maximizes the expected value of output For now, let F (x, y) be the measure of matches inside [0, x] × [0, y] As the planner cannot match more of any type than available, he solves: V(G) = max F s.t Q(x, y)F (dx, y)dy (1) [0,1]2 Feasibility: F (x, y) ≤ G(x) ∀y (2) The matching set is the support of F (x, y) Positive assortative matching (PAM) obtains if the matching set coincides with the 45◦ line, so that F (x, y) = G(min(x, y)) Negative assortative © 2009 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 uncertain abilities, but assumes private information, reinforcing PAM, by way of a new “acceptance curse” MacDonald (1982) also considers matching with incomplete information But in his model, the information revelation is invariant to the match Unlike these papers, we show that Becker’s finding robustly unravels given an informational friction that depends on match assignment Our model is also related to the learning paper by Easley and Kiefer (1988), who ask when the decision maker eventually learns the true state Incomplete learning requires that a myopically optimal action be uninformative at some belief Easley and Kiefer show that no such action is dynamically optimal for a patient enough decision maker Here, the statically optimal action (PAM) is not chosen given sufficient patience Bergemann and Văalimăaki (1996) and Felli and Harris (1996) are related in that an element of the static price is information value, as with our wages ANDERSON & SMITH DYNAMIC MATCHING REPUTATIONS matching (NAM) obtains when every reputation x matches only with the opposite reputation y(x) solving G(y(x)) = − G(x) Then:1 Proposition (Becker, 1973) Given supermodularity, PAM solves the planner’s static maximization problem.2 NAM is efficient given submodularity • Worker maximization: v(x) = w(x|y) and v(y) = w(y|x) for all (x, y) ∈ supp F • Value maximization: v(x) = maxy w(x|y) • Output shares: w(x|y) + w(y|x) = Q(x, y) (3) Becker proved the welfare theorems which Theorem revisits in a dynamic setting Theorem (Becker, 1973) The First and Second Welfare Theorems obtain, and the competitive equilibrium wage is w(x | y) = Q(x, y) − v(y) for any matched pair 2.2 Dynamically evolving human capital We now develop our model in a stationary infinite horizon context over periods 0, 1, 2, Crucially, human capital evolves with each match For instance, when junior and senior colleagues match, each is changed from the experience We capture these dynamic effects by positing a transition function τ (s|x, y), which is the sum of the transition chances that x updates to at most s, and that y updates to at most s, when x matches with y Let X ∗ be the space of matching measures on [0, 1]2 , with generic cdf F For any z ∈ [0, 1], let B : X ∗ → Z be the posterior cdf B(F )(z) = z [0,1]2 τ (s|x, y)F (dx, dy)ds Towards a nondegenerate steady state, we assume that agents live to the next period with survival chance σ , and to maintain a constant mass of agents, posit a − σ weight on the inflow cdf G To properly align incentives, we assume that the agents’ implicit rate of time preference equals the planner’s discount factor γ < scaled by the survival chance, namely δ ≡ σ γ To avoid trivialities, G does not place all weight on and Given an initial type cdf G, the planner chooses the matching cdf F in each period to maximize the average present value of output, respecting feasibility Let (G) be the feasibility set in (2) For any F ∈ (G), define the policy operator TF V(G) = (1 − δ) Q(x, y)F (dx, dy) + δV((1 − σ )G + σ B(F )) (4) Our propositions are descriptive matching results, and theorems are technical equilibrium results Becker proved this for the discrete case For our purposes, Lorentz (1953) is more appropriate as he proved the formal result in the continuum case (albeit without providing any economic context) © 2009 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 In a competitive equilibrium, each worker x chooses the partner y that maximizes his (expected) wage w(x|y), achieving his value v(x) Also, wages of matched workers exhaust output, and the market clears Altogether, a competitive equilibrium (CE) is a triple (F, v, w) where F obeys the feasibility constraint (2), while F, v, w satisfy: REVIEW OF ECONOMIC STUDIES Here, (1 − σ )G + σ B(F ) is next period’s type distribution Thus, the planner solves for the Bellman value V, namely a fixed point of the operator T V(G) = maxF ∈ (G) TF V(G) The planner trades off more output today for a more profitable measure over types tomorrow This trade-off lies at the heart of our paper A steady state Pareto optimum (PO) is a triple (G, F, v) such that (F, v) solves the planner’s problem given G, and G = (1 − σ )G + σ B(F ) Just as in the analysis of the modified golden rule in growth models, the social planner does not maximize across steady states Instead, she chooses an optimal matching in each period, after which the steady state requirement is imposed While our results obtain both in and out of steady state, we focus on the steady state for simplicity Theorem (Pareto optimum) A steady state Pareto optimum exists The appendix proves this The first order conditions (FOC) for this problem are: (x, y) ∈ supp(F ) ⇒ v(x) + v(y) − (1 − δ)Q(x, y) − δ v (x, y) = (5) v(x) + v(y) − (1 − δ)Q(x, y) − δ v (x, y) ≥ 0, (6) where v(x) is the multiplier on the constraint (2), i.e the shadow value of an agent x, and v (x, y) = ψ v (x|y) + ψ v (y|x) is the sum of the expected continuation values ψ v (x|y) = E[v(x )|x, y] So the sum of the shadow values in any matched pair (a) equals the planner’s total value of matching them, and (b) weakly exceeds their alternative value in other matches In a competitive equilibrium (CE), let w(x|y) be the wage that agent x earns if matched with y Anticipating a welfare theorem to come, we overuse notation, letting v(x) denote the maximum discounted sum of wages that x can earn–the private value A steady state CE is a 4-tuple (G, F, v, w), where G = (1 − σ )G + σ B(F ), F obeys constraint (2), wages w(x|y) are output shares (3), dynamic maximization obtains: • Worker maximization: v(x) = max[(1 − δ)w(x|y) + δψ v (x|y)], y (7) and finally y is a maximizer of (7) whenever (x, y) ∈ supp (F ) Theorem (Welfare theorems) If (G, F, v, w) is a steady state CE, then (G, F, v) is a steady state PO Conversely, if (G, F, v) is a steady state PO, then (G, F, v, w) is a steady state CE, where for all matched pairs (x, y), the wage w(x|y) of x satisfies: static wage w(x|y) = Q(x, y) − v(y) + dynamic rent (y to x) δ [ψ v (y|x) − v(y)] 1−δ (8) See how we assert that the planner’s shadow values and the private values coincide These welfare theorems are greatly complicated by the evolution of types Fortunately, continuation values are linear, and therefore convex, in measures of matched agents The competitive wage has two components First is the static wage, or the difference between match output and one’s partner’s outside option Second is the dynamic rent, or the discounted excess of one’s partner’s continuation value over his outside option That the dynamic benefits are publicly observed sustains the welfare theorems, since they can be © 2009 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 2.3 Existence and welfare analysis ANDERSON & SMITH DYNAMIC MATCHING REPUTATIONS compensated For instance, in our Bayesian model the public reputations serve as the types Here, dynamic rents will be positive by convexity even when identical agents match, and reputations near and will earn greater dynamic rents For some insight into why this wage decentralizes the Pareto optimum, consider a pair (x, y) matched in equilibrium Worker maximization (7) requires that v(x) equal (1 − δ)w(x|y) + δψ v (x|y) = (1 − δ) Q(x, y) − v(y) + δ[ψ v (y|x) − v(y) + ψ v (x|y)] using our computed wage (8) With some simplification, we get: which holds if (x, y) are matched in the Pareto optimum, by the planner’s FOC (6) Finally, we consider existence Theorem proved that a steady state PO exists; also, any such PO can be decentralized as a CE, by Theorem Altogether: Corollary There exists a steady state competitive equilibrium 2.4 Values, shadow values, and dynamic rents We next exploit the equivalence between the competitive equilibrium and Pareto optimum, and prove that agents’ private values v(x) are convex The convexity of the multipliers is a separate new contribution Theorem (MPS and convexity) Assume bilinear, strictly supermodular output Q(x, y), z with τ (s|x, y)ds convex in x and y, and convex along the diagonal y = x (a) The planner’s value V strictly rises in mean-preserving spreads (MPS) of types (b) The shadow value v(x) is everywhere convex (i.e convex and nowhere locally flat) (c) The expected continuation value function ψ v (x|y) is separately convex in x and y Proof of (a) V strictly rises in mean preserving spreads Let’s consider monotonicity of the planner’s value V in mean preserving spreads: ˆ ≥ V(G) whenever G ˆ is a mean preserving spread of G (P) V(G) We prove below that if V obeys P, then T V obeys P, and because P is closed under the sup norm, the fixed point V = T V obeys P In fact, we prove that T V obeys the stronger property P+ , where strict inequality obtains, so that V strictly rises in MPS ˆ be an MPS of G (the premise of P) Write ζ (x) = G(x) ˆ Let G − G(x), where xdζ (x) = 0, and ζ does not almost surely vanish Let F ∈ (G) be optimal for G, and define a new matching Fˆ (x, y) = F (x, y) + min(ζ (x), ζ (y)) So Fˆ differs from F insofar as it places all ˆ along the diagonal Since G ˆ is an MPS of G, and Q(x, x) is weight not common to G and G everywhere convex, being bilinear and strictly supermodular: Qd Fˆ − QdF = Q(x, x)dζ (x) = ˆ Q(x, x)d G(x) − Q(x, x)dG(x) > For the same reason, and since B(F ) is a linear operator, we have B(Fˆ )(s) − B(F )(s) = ˆ τ (s|x, x)dε(x) = τ (s|x, x)d G(x) − τ (s|x, x)dG Changing the order of integration: z B(Fˆ )(s)ds − z B(F )(s)ds = which is non-negative because z z ˆ τ (s|x, x)ds d G(x) − z τ (s|x, x)ds dG(x) ˆ is an MPS of G τ (s|x, x)ds is convex and G © 2009 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 v(x) + v(y) = (1 − δ)Q(x, y) + δ ψ v (x|y) + v (y|x) , ă UNVER DYNAMIC KIDNEY EXCHANGE 401 CONCLUSIONS APPENDIX A PROOFS OF RESULTS Proof of Proposition Suppose that pair i of type X–Y∈ P O is the only overdemanded pair in the pool, j = i is a type Z1 –Z2 ∈ P pair in the pool, and E = (j, j1 , , jk ) is an exchange that matches pair j Let A –O be the type of each pair j in E We have Z2 A1 , O A +1 for all ∈ {1, , k − 1}, and Ak Z1 Two cases are possible: A and • Type of j ∈ P U : Since Z1 Z2 and Z2 = Z1 , by acyclicity of , there exists one pair j with O O = A , i.e there exists an overdemanded pair j ∈ E Since i is the single overdemanded pair in the pool, j = i By transitivity of and by the fact that there are at most two object types at a compatibility level of , we have (1a) Z2 X or (1b) Z2 X and X Z2 , and yet there exists some jm ∈ E with m < such that jm is of type Z2 -X (so that E is individually rational) Similarly, we have (2a) Y Z1 or (2b) Z1 Y 25 Observe that announcing Y Z or W X may result with individually irrational exchanges Hence, we assume that such manipulations are not possible 26 In the context of kidney exchange, since blood types exclusively determine the compatibility between a recipient and a donor of another pair, and since manipulating blood types is extremely difficult, it is almost impossible for a pair to use “compatibility” as a strategic tool to manipulate the dynamic system © 2009 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Having a partial order compatibility structure (which is not a linear order) is the necessary requirement for multi-way dynamically efficient mechanisms having state-dependent features and being different from statically efficient mechanisms We use a minimal partial order structure to derive dynamically efficient exchange mechanisms in a general exchange model We observe three important properties of dynamically efficient mechanisms They (for both two-way and multi-way matching) are not affected by the magnitude of the unit waiting cost c They conduct at most one exchange at a time Moreover, whenever an exchange becomes feasible, they conduct it immediately ă In a static setting, Roth, Săonmez and Unver (2007) showed that n-way exchanges usually suffice to obtain all benefits from an exchange domain with n object types under a partial order compatibility relation and mild assumptions In our study, for kidney exchanges, when self-demanded type pairs participate in exchange, the largest possible exchange size is instead of as predicted by the above result, since in a dynamic setting some of the assumptions of the above study not hold In the simulations conducted, we showed that exchanges larger than four-way are extremely rare in a dynamic setting The policy simulations show that the threshold values of the efficient kidney exchange mechanism are quite sensitive to the changes in arrival probabilities of A–B and B–A type pairs Therefore, for our mechanism to have a realistic application, the health authority should measure these arrival rates, precisely, and these rates should be close to each other A final note about incentive properties of dynamically efficient mechanisms will be useful We can refine the definition of efficient mechanisms as follows If an X–Y type pair is going to be matched in an exchange and there are multiple X–Y type pairs available in the pool, then the mechanism selects the earliest arriving pair Suppose that a pair of type X–Y∈ P can manipulate its type and announce it as W–Z∈ P with W X and Y Z.25 It is easy to show that announcing X–Y is the weakly dominant strategy for the pair, i.e the mechanisms are strategy-proof.26 Entry timing can be another strategic tool Suppose that each pair, after becoming available, can delay its entry to the pool as a strategic variable In this case, the above dynamically efficient two-way and multi-way matching mechanisms are delay-proof, i.e no pair will benefit by delaying its entry to the pool 402 REVIEW OF ECONOMIC STUDIES and Y Z1 , and yet there exists some jm ∈ E with m > necessity – – – – If If If If such that jm is of type Y–Z1 27 This proves (1a) and (2a) are satisfied, we can always choose E (in terms of types of pairs) as (Z1 –Z2 , X–Y) (1b) and (2a) are satisfied, we can choose E as (Z1 –Z2 , Z2 –X, X–Y) (1a) and (2b) are satisfied, we can choose E as (Z1 –Z2 , X–Y, Y–Z1 ) (1b) and (2b) are satisfied, we can choose E as (Z1 –Z2 , Z2 –X, X–Y, Y–Z1 ) – – – – If If If If (1a) and (2a) are satisfied, we can always choose E (in terms of types of pairs) as (Z1 –Z1 , X–Y) (1b) and (2a) are satisfied, we can choose E as (Z1 –Z1 , Z1 –X, X–Y) (1a) and (2b) are satisfied, we can choose E as (Z1 –Z1 , X–Y, Y–Z1 ) (1b) and (2b) are satisfied, we can choose E as (Z1 –Z1 , Z1 –X, X–Y, Y–Z1 ) These prove sufficiency • Type of j ∈ PR : We have Z2 Z1 and Z1 Z2 Suppose there is a pair h of type Z2 –Z1 Then a two-way exchange consisting of types (Z1 –Z2 , Z2 –Z1 ) is an individually rational exchange, and E could be chosen using these types Suppose that there is no pair of type Z2 –Z1 in the pool Then by acyclicity of , there exists some pair j with O A with O = A , i.e there exists an overdemanded pair j ∈ E Since i is the single overdemanded pair in the pool, j = i By the fact that there are only two types in the compatibility levels of Z1 and Z2 , there are no possible reciprocal types other than Z1 –Z2 and Z2 –Z1 Since Z2 –Z1 type pair does not exist, there is no other pair of the same compatibility level with in Z1 –Z2 in the exchange E,and by acyclicity we have Z2 X and Y Z1 i.e Z–Y and Z1 –Z2 are mutually compatible types, proving necessity The types of pairs in E can be chosen as (Z1 –Z2 , X–Y) whenever Z2 X and Y Z1 , proving sufficiency Proposition (Maximal exchange composition using overdemanded types) Under Assumption 1, suppose that X–Y∈ P O is the type of an overdemanded pair that arrives at the exchange pool Then, we can conduct an (n + k + + 1)-way exchange serving • the overdemanded pair of type X–Y; • a maximum of n = LX − LY underdemanded pairs; • one pair from each of the distinct reciprocally demanded types W1 –W2 ,W3 –W4 , , W2k−1 –W2k ∈ PR such that W1 , W2 W3 , W4 · · · W2k−1 , W2k (i.e these pair types are not reciprocal of another and are ordered according to their compatibility levels), Y W1 and W2k X • one pair from each of the distinct self-demanded types V1 –V1 , , V –V ∈ PS such that (1) V1 V2 X or W2k−1 –W2k = V -X, and (4) if there exists some · · · V , (2) Y V1 or W1 –W2 = Y–V1 , (3) V d ∈ {1, , − 1} such that Vd and Vd+1 are at the same level then there exists some index cd ∈ {1, , k} such that W2cd −1 –W2cd =Vd –Vd+1 ; whenever such reciprocally demanded and self-demanded pairs exist in the pool Proof of Proposition Suppose the hypothesis of the proposition holds For notational purposes, let Z−1 = X, Z0 = Y, Z2n+1 = X, and Z2n+2 = Y Under Assumption 1, there exists n underdemanded pairs belonging to pair types Z1 –Z2 , Z3 –Z4 , , Z2n−1 –Z2n ∈ P U 27 Equivalently, we could have written Condition (1) as “X Z2 and if Z2 X then there exists some jm ∈ E with m < such that jm is of type Z2 –X” as in the hypothesis of the proposition A similar equivalence is also valid for Condition (2) Thus, these are equivalent to the conditions given in the hypothesis of the proposition © 2009 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 These prove sufficiency • Type of j ∈ PS : Since Z2 = Z1 , when there is another pair h of type Z1 –Z1 , a two-way exchange consisting of types (Z1 –Z1 , Z1 –Z1 ) is an individually rational exchange, and E could be chosen using these types Suppose that there is no other pair of type Z1 –Z1 in the pool Then by acyclicity of , there exists some pair j with O A with O = A , i.e there exists an overdemanded pair j ∈ E Since i is the single overdemanded pair in the pool, j = i By transitivity of and by the fact that there are at most two object types at a compatibility level of , we have (1a) Z1 X or (1b) Z1 X and X Z1 , and yet there exists some jm ∈ E with m < such that jm is of type Z1 –X (so that E is individually rational) Similarly, we have (2a) Y Z1 or (2b) Z1 Y and Y Z1 , and yet there exists some jm ∈ E with m > such that jm is of type Y–Z1 These prove necessity ă UNVER DYNAMIC KIDNEY EXCHANGE 403 such that • for each m ∈ {1, , n}, there are m levels between Z2m and Y, that is LZ2m = LY + m; • for each c ∈ {1, , k} , there exists an index mc ∈ {1, , n} such that Z2mc = W2c−1 and Z2mc +1 = W2c ; • for each d ∈ {1, , } with Vd−1 Vd (whenever Vd−1 exists) and Vd Vd+1 (whenever Vd+1 exists), there exists an index md ∈ {1, , k} such that Z2m = Vd ; and d • for each m ∈ {1, , n − 1} \ {m1 , , mk }, Z2m = Z2m+1 Thus, the exchange E consisting of pairs belonging to pair types ⎞ ⎛ =Y =Z2 =W2c−1 =W2c =Z2n−2 =X for all c = 1, 2, , k is a feasible (k + n + 1)-way exchange We can enlarge exchange E by inserting the given self-demanded type pairs in order to obtain exchange E as follows: • Recall that for each d ∈ {1, , − 1} with same level Vd+1 and Vd , there exists some index cd ∈ {1, , k} such that W2cd −1 –W2cd = Vd –Vd+1 In the exchange E above, we can insert – the Vd − Vd type pair between the Z2mcd −1 − Z2mcd type pair, and W2cd −1 − W2cd type reciprocally =W2c −1 d demanded pair; and – the Vd+1 − Vd+1 type pair between the W2cd −1 − W2cd type reciprocally demanded pair and the Z2mcd +1 − Z2mcd +2 type pair =W2c d • For each d ∈ {1, , } with Vd−1 Vd (whenever Vd−1 exists) and Vd Vd+1 (whenever Vd+1 exists), since the object type Z2m is chosen as Z2m = Vd , we can insert the Vd –Vd type self-demanded pair, in exchange d d E , between the pairs Z2m −1 –Z2m and Z2m +1 –Z2m +2 d d d =Z d 2md Thus, the newly formed exchange E serves all of the n + k + + pairs including the ones given in the hypothesis of the proposition Proof of Proposition Suppose Assumption holds Let an overdemanded pair i of type X–Y ∈ PO arrive at the exchange pool We will show that the opportunity cost of holding onto the X–Y type pair and underdemanded pairs, which could be matched immediately, with the expectation of creating a larger exchange in the future is larger than any alternative decision First note that pair i will not be used in matching an underdemanded pair that will arrive in the future Since all underdemanded pairs exist abundantly by Assumption 1, by Corollary we can use it to match LX − LY underdemanded pairs in an exchange E immediately Moreover suppose that we can match in total n pairs in this exchange (possibly including some reciprocally demanded and self-demanded pairs) In the future, LX − LY is the most underdemanded pairs we can match through pair i, thus we not hold onto i to match future underdemanded pairs We will show that pair i will not be used in matching a self-demanded type pair that will arrive in the future, either Suppose that V–V is a self-demanded type and a pair of this type can be inserted in exchange E (see Proposition and its proof) Hence, if pair i is used to wait for a V–V type pair to arrive, n pairs in exchange E will wait until the V–V type pair arrives, instead of being matched immediately A V–V type pair can be matched in several ways It can be matched with another V–V type pair in a two-way exchange Or it can be inserted in other exchanges between two pairs such that the object of the first pair is compatible with V and the requirement of the agent of the second pair is also compatible with V Consider the case in which we match it exclusively with a future V–V type pair j For the same expected duration that pairs in exchange E wait for pair j to arrive, pair j will wait until the next V–V type pair arrives Thus, the cost of this exchange is making a future V–V type pair j wait for the same expected duration for a new V–V type pair This second alternative is less costly than making n (which is larger than 1) pairs © 2009 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 ⎟ ⎜ ⎜X–Y, Z1 –Z2 , Z3 –Z4 , , Z2m – Z2mc , W2c−1 –W2c , Z2mc +1 –Z2mc +2 , , Z2n−1 – Z2n ⎟ c−1 ⎠ ⎝ 404 REVIEW OF ECONOMIC STUDIES is E e−ρτ = −λpW −W τ ∞ −ρτ λpW −W e dτ e W2 –W1 type arriving is λpW −W λpW −W +ρ , = λpW −W λpW −W +ρ Similarly, the expected discounting until a and a W1 –W2 or W2 –W1 type arriving is λ pW −W +pW −W 2 λ pW −W +pW −W +ρ 2 (since the arrival of a W1 –W2 or W2 –W1 type pair is a Poisson with rate λpW1 −W2 + λpW2 −W1 ) For simplicity of notation, until the end of the proof, we use λ1 ≡ λpW1 −W2 and λ2 ≡ λpW2 −W1 Also observe that the upper-bound of total expected surplus assuming that all W1 –W2 type pairs are matched as soon as they arrive is given as: λ1 λ1 + ρ c + ρ λ1 λ1 + ρ c + ··· + ρ λ1 λ1 + ρ m λ1 c c + ··· = ρ ρ ρ Similarly, the upper-bound of total expected surplus assuming that all W2 –W1 type pairs are matched as soon as they λ arrive is given as ρ2 ρc • First, we find an upper-bound of surplus (regarding pairs in E and all future W1 –W2 and W2 –W1 type pairs) from waiting and not conducting exchange E immediately: λ1 ρ λ1 + λ2 λ1 λ2 ES = + (n + 1) + c λ1 + λ2 + ρ λ1 + λ2 ρ ρ λ1 + λ2 λ2 λ1 = + (n + 1) + λ1 + λ2 + ρ ρ ρ where in the first line and λ1 +λ2 λ1 +λ2 +ρ + λ2 λ1 + λ2 n+1+ λ2 λ1 + ρ ρ is the discounting that occurs until either a W1 –W2 or W2 –W1 pair arrives λ – probability λ +λ refers to the pair arriving being of type W1 –W2 and normalized (by ρc ) surplus n + refers to the fact that we conduct the (n + 1)-way exchange using the W1 –W2 type pair and the λ λ waiting n pairs, normalized surplus ρ1 + ρ2 is an upper-bound of all future matches regarding W1 –W2 and W2 –W1 type pairs; λ2 refers to the pair arriving being of type W2 –W1 and normalized surplus n refers to – probability λ +λ λ λ the fact that we conduct exchange E at that instance, and + ρ1 + ρ2 is the surplus of matching the W2 –W1 pair immediately (an upper-bound assumption) and all other future W1 –W2 and W2 –W1 type pairs as soon as they arrive (another upper-bound assumption).28 • Second, we find a lower-bound for the efficient surplus (regarding pairs in E and all future W1 –W2 and W2 –W1 type pairs) when we conduct exchange E immediately: ρ {λ1 , λ2 } ES = n + c ρ 28 This is not a tight upper-bound and smaller upper-bounds can be found © 2009 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 of the exchange E wait for the same expected duration; therefore an X–Y type pair i will not be used to match an expected V–V type pair in the future Next, we show that pair i will not be used in matching a reciprocally demanded type pair that will arrive in the future Suppose that instead of exchange E, we use the type X–Y pair to match one reciprocally demanded pair k of type W1 –W2 that will arrive in the future Moreover, by Proposition 2, pairs in E (other than pair i) cannot be matched without i Thus, suppose that we would like to use pair i to serve also this first pair k which will arrive in the future This causes the exchange E not to be conducted immediately and forces n pairs to wait By Corollary 1, we can match n + pairs (including k) immediately when k arrives, if we not conduct exchange E now We will find an upper-bound for exchange surplus regarding these n pairs and all W1 –W2 and W2 –W1 type pairs that will arrive in the future Then, we will find a lower-bound for exchange surplus when n pairs are instantly matched within exchange E and the overdemanded type pair i is not held on to Then, we will show that when pW1 −W2 and pW2 −W1 are sufficiently close to each other, the second surplus is greater than the the first one Observe that the expected time difference between arrival of W1 –W2 type pairs, τ , follows an exponential −λp τ distribution with density function λpW1 −W2 e W1 −W2 , and the expected discounting between those two arrivals ă UNVER DYNAMIC KIDNEY EXCHANGE 405 where n refers to the normalized immediate exchange surplus regarding n pairs in conducted exchange E min{λ1 ,λ2 } refers to the lower-bound of surplus found by matching W1 –W2 type pairs exclusively with and ρ W2 –W1 type pairs in the future.29 We observe that when λ1 is sufficiently close to λ2 , ρc ES > immediately matched in n-way exhange E for efficient matching.30 ρ c ES Thus, overdemanded pair i will be Theorem (The Existence and Uniqueness Theorem) Let Z be a countable state set Let F be a finite action set Let V be the set of bounded functions defined from Z to R Let ≤ δ < For any z ∈ Z, let f ∈F (A1) σ ∈Z where (i) for all σ ∈ Z and all f ∈ F , p (σ |z, f ) ≥ 0, and for all f ∈ F , r (z, f ) ∈ R is bounded Then: σ ∈C p (σ |z, f ) = 1, and (ii) for all f ∈ F , Function v ∈ V exists and is uniquely defined as the limit of the sequence {v m } ⊆ V (under the sup norm),31 where v is arbitrary, and for any m > 0, p (σ |z, f ) v m−1 (σ ) v m (z) = δ max r (z, f ) + f ∈F (A2) σ ∈Z There exists a (deterministic) Markovian mechanism φ : Z → F such that for all z ∈ Z, p (σ |z, φ (z) ) v (σ ) v (z) = δ r (z, φ (z)) + (A3) σ ∈Z We will use the above theorem in our proof of Proposition ∗ Proof of Proposition Let W1 –W2 ∈ PR Let F = {do-not-match, match} and f1 =do-not-match, f2 = match Consider the Bellman equations given in equations (17), (18), and (19) Let the normalized surplus for choosing the smaller exchanges (action f1 ) regarding W1 –W2 be given by ⎧ ⎨2pW2 –W1 if sW1 –W2 > 0 if sW1 –W2 = , r sW1 –W2 , f1 = (A4) ⎩ 2pW1 –W2 if sW1 –W2 < and the normalized surplus for choosing larger exchanges (action f2 ) be given by r sW1 –W2 , f2 ⎧ 2pW2 –W1 + pX–Y ⎪ ⎪ ⎪ ⎪ ⎪ X–Y∈P O (W1 –W2 ) ⎨ = ⎪ ⎪ ⎪ pX–Y 2pW1 –W2 + ⎪ ⎪ ⎩ X–Y∈P O (W2 –W1 ) if sW1 –W2 > if sW1 –W2 = if sW1 –W2 < (A5) When smaller exchanges (action f1 ) are chosen, the transition probabilities are given by p sW1 –W2 − | sW1 –W2 , f1 = pW2 –W1 , p sW1 –W2 | sW1 –W2 , f1 = − pW1 –W2 − pW2 –W1 , (A6) p sW1 –W2 + | sW1 –W2 , f1 = pW1 –W2 29 This is not a tight lower-bound, and bigger lower-bounds can be found 30 The situations in which pair i can be used in matching multiple reciprocal type pairs in different levels is very similar to this case and skipped for brevity 31 For all v ∈ ν, v = sups∈S |v (s)| is the sup norm of v © 2009 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 p (σ |z, f ) v (σ ) , v (z) = δ max r (z, f ) + 406 REVIEW OF ECONOMIC STUDIES , if sW1 –W2 > if sW1 –W2 = , if sW1 –W2 < (A7) Let V = {v : Z → R+ such that v is bounded} be the set of Markov surplus functions for W1 –W2 types Let v ∈ V For all m ∈ {1, 2, 3, } (≡ Z++ ), let v m ∈ V be defined through the following recursive system, v m sW1 –W2 = max f ∈{f1 ,f2 } w m sW1 –W2 , f (A8) with w m : Z × {f1 , f2 } → R+ defined for all f ∈ {f1 , f2 } as follows: w m sW1 –W2 , f = λ [r sW1 –W2 , f + λ+ρ sW –W +1 σ =sW –W −1 p σ sW1 –W2 , f v m−1 (σ ) ] (A9) The state component space for W1 –W2 and W2 –W1 types, Z, is countable Action space F = {f1 , f2 } is finite λ < Observe that by equations (A6) and (A7), for any sW1 –W2 ∈ Z and Since λ > and ρ > 0, we have < λ+ρ sW –W σ =sW –W −1 f ∈ F , p σ |sW1 –W2 , f ≥ for all σ ∈ Z, and, p σ |sW1 –W2 , f = By equations (A4) and (A5), for any sW1 –W2 ∈ Z and f ∈ F , r sW1 –W2 , f is bounded Since equations (A4)–(A9) are directly obtained from the ∗ ∈V Bellman equations (17), (18), and (19), by the Existence and Uniqueness Theorem, there is a unique ESW –W2 such that under the sup norm, for all s ∈ S, ∗ ESW –W2 sW1 –W2 = lim v m sW1 –W2 (A10) m→∞ The following Lemmata prove Theorem 2: Lemma There exist s ∗ ≥ and s ∗ ≤ for some s ∗ , s ∗ ∈ S such that φ s matching mechanism Proof of Lemma match Let ∗ ,s ∗ is a dynamically efficient multi-way ∗ Fix W1 –W2 ∈ PR Let F = {do-not-match, match} and f1 = do-not-match, f2 = ∗ h∗ ≡ ESW –W2 and z ≡ sW1 –W2 for notational convenience The state component space regarding W1 –W2 and W2 –W1 types is given by Z We rewrite the Bellman equations (17), (18), and (19) as follows: For any z ∈ Z, h∗ (z) = © 2009 The Review of Economic Studies Limited max f ∈{f1 ,f2 } w (z, f ) , (A11) Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 When larger exchanges (action f2 ) are chosen, the transition probabilities are given by ⎧ ⎪ pX–Y if sW1 –W2 > ⎨pW2 –W1 + p sW1 –W2 − | sW1 –W2 , f2 = X–Y∈P O (W1 –W2 ) ⎪ ⎩ pW2 –W1 if sW1 –W2 ≤ ⎧ − pW1 –W2 − pW2 –W1 − pX–Y ⎪ ⎪ ⎪ ⎪ ⎪ X–Y∈P O (W1 –W2 ) ⎨ − pW1 –W2 − pW2 –W1 p sW1 –W2 | sW1 –W2 , f2 = ⎪ ⎪ ⎪1 − pW –W − pW –W − pX–Y ⎪ 2 ⎪ ⎩ X–Y∈P O (W2 –W1 ) ⎧ pW1 –W2 if sW1 –W2 ≥ ⎪ ⎨ pX–Y if sW1 –W2 < p sW1 –W2 + | sW1 –W2 , f2 = pW1 –W2 + ⎪ ⎩ X–Y∈P O (W2 W1 ) ă UNVER DYNAMIC KIDNEY EXCHANGE 407 where w (z, f ) = λ [r (z, f ) + λ+ρ z+1 p (σ |z, f ) h∗ (σ )], (A12) σ =z−1 and r (z, f ) is defined by equations (A4) and (A5), and p (σ |z, f ) is defined by equations (A6) and (A7) For all z ∈ Z, f z = arg max f ∈{f1 ,f2 } (A13) w (z, f ) such that if w (z, f1 ) = w (z, f2 ) , then f2 = arg max (A14) w (z, f ) For all z ∈ Z, let h∗ (z) = h∗ (z) − h∗ (z − 1) (A15) We prove Lemma using the following four claims: Claim Suppose that z > is such that f z = f2 , and f z+1 = f1 Then there is no k ≥ such that f z+k+1 = f2 Proof of Claim Let z > be such that f z = f2 , and f z+1 = f1 We prove the claim by contradiction Suppose there exists some k ≥ such that f z+2 = f1 , , f z+k = f1 , f z+k+1 = f2 Therefore, by Observation and definitions in equations (A13), (A14), and (A15), h∗ (z) ≤ 1, h∗ (z + 1) > 1, , h∗ (z + k) > 1, and h∗ (z + k + 1) ≤ (A16) By definitions in equations (A11), (A12), (A13), and (A15); for r (z, f ) in equations (A4) and (A5); and for p (σ |z, f ) in equations (A6) and (A7), we obtain h∗ (z + 1) = h∗ (z + 1) − h∗ (z) = w(z + 1, f1 ) − w(z, f2 ) ⎡ pX–Y ( h∗ (z) − 1) + pW2 –W1 λ ⎢ = ⎣ X–Y∈P O (W1 –W2 ) λ+ρ h∗ (z + 1) + pW1 –W2 + − pW1 –W2 − pW2 –W1 h∗ (z) h∗ ⎤ ⎥ ⎦ (z + 2) λ pW2 –W1 + − pW1 –W2 − pW2 –W1 h∗ (z + 1) + pW1 –W2 h∗ (z + 2) ≤ λ+ρ h∗ (z) ≤ (in Inequality System (A16)) since, by f z = f2 , we have < pW2 –W1 + − pW1 –W2 − pW2 –W1 λ 1, by definitions in equations (A11), (A12), (A13), and (A15); for r (z, f ) in equations (A4) and (A5); and for p (σ |z, f ) in equations (A6) and (A7), we obtain h∗ (z + ) = h∗ (z + ) − h∗ (z + − 1) = w(z + , f1 ) − w(z + + 1, f1 ) (A19) λ pW2 –W1 h∗ (z + − 1) + − pW1 –W2 − pW2 –W1 = h∗ (z + ) + pW1 –W2 h∗ (z + + 1) λ+ρ h∗ (z + ) + pW1 –W2 h∗ (z + + 1) (A20) < pW2 –W1 h∗ (z + − 1) + − pW1 –W2 − pW2 –W1 λ (stated in Inequality System (A16)) We showed that for any z > whenever f z = f2 and f z+1 = f1 , there is no k ≥ such that f z+k+1 = f2 , completing the proof of Claim Claim There exists z ≥ such that for all z > z we have f z = f2 Proof of Claim Consider a scenario in which there are infinitely many W1 –W2 type pairs available at the exchange pool That is, the state component is z = ∞ Every incoming overdemanded pair of one of the types in P O (W1 –W2 ) can be used in an exchange that matches a W1 –W2 type pair After such an exchange, there will still be infinitely many W1 –W2 type pairs, implying that incoming W2 –W1 type pairs are not affected by the previous decision of choosing largest possible exchanges Therefore, at state component z = ∞, the efficient action is f2 (option match, conducting largest possible exchanges) Therefore, every incoming W2 –W1 type pair will be matched in a two-way exchange with a W1 –W2 type pair, and every incoming pair of one of the types in P O (W1 –W2 ) will be matched efficiently serving a W1 –W2 type pair Since we are discussing the next incoming pairs, this surplus should λ be discounted with E e−ρτ = λ+ρ The exchange surplus for the first matched W1 –W2 or W2 –W1 type pair in this scenario is ⎡ ⎤ h = λ ⎢ ⎣ λ+ρ c ρ pX–Y X–Y∈P O (W1 –W2 ) + pW2 –W1 c ⎥ ⎦ ρ (A36) Similarly, the current value of the exchange surplus for the second matched W1 –W2 or W2 –W1 type pair is λ h = λ+ρ h , , and the current value of the exchange surplus for the k th matched W1 –W2 or W2 –W1 type pair is k λ k−1 h = ( λ+ρ ) h Therefore, the total exchange surplus of state component ∞ is ∞ h (∞) = k=1 = k ∞ h = ⎛ λc ⎜ ⎝ ρ2 k=1 λ λ+ρ X–Y∈P O (W1 –W2 ) By normalizing h (∞) by c ρ, k−1 h = 1 h λ − ( λ+ρ ) ⎞ ⎟ pX–Y + 2pW2 –W1 ⎠ (A37) we obtain ⎛ λ⎜ h∗ (∞) = ⎝ ρ ⎞ X–Y∈P O (W1 –W2 ) ⎟ pX–Y + 2pW2 –W1 ⎠ (A38) © 2009 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Using the definition of g (in equation (A25)), we observe that 410 REVIEW OF ECONOMIC STUDIES Clearly, the normalized exchange surplus at state component ∞ is an upper-bound for the normalized efficient exchange surplus for z → ∞ Suppose that there is no z > such that for all z > z , f z = f2 By Claim 1, there exists some z > such that for all z > z , f z = f1 and h∗ (z) ≥ h∗ (z − 1) + (by Observation 2) Therefore, for any z > z , h∗ (z) ≥ z − z + h∗ z (A39) Then as z → ∞, h∗ (z) → ∞, contradicting the fact that h∗ (∞) is bounded This and Claim imply that there exists some z > such that for all z > z , f z = f2 We state the following two claims, whose proofs are symmetric versions of the proofs of Claims and 2: Claim Suppose that z < is such that f z = f2 , and f z−1 = f1 Then there is no k ≥ such that f z−k−1 = f2 By Claims and 2, there exists s ∗W –W ≥ such that f z = f2 for all z > s ∗W –W and f z = f1 for all 2 ≤ z < s ∗W –W By Claims and there exists s ∗W –W ≤ such that f z = f2 for all z < s ∗W –W and f z = f1 ∗ ∗ ,s ∗ for all ≥ z ≥ s ∗W –W Since W1 –W2 ∈ PR is arbitrary, the threshold mechanism φ s mechanism is an efficient matching ∗ For each W1 –W2 ∈ PR , Lemma ∗ ESW –W2 (0) < λ pW1 –W2 + pW2 –W1 ρ (A40) ∗ Proof of Lemma Fix W1 –W2 ∈ PR Consider the state component z = If W1 –W2 and W2 –W1 type pairs could be matched as soon as they arrived at the exchange pool, the decision problem of the health authority would be trivial and it would match the overdemanded type pairs in the largest possible exchanges That is, since no W1 –W2 or W2 –W1 type pairs remain in the pool unmatched, whenever an X–Y ∈ P O (W1 –W2 ) ∪ P O (W2 –W1 ) type overdemanded pair arrives at the exchange pool, it will be matched in an exchange without a W1 –W2 or W2 –W1 type pair that matches the maximum number of underdemanded pairs possible and possibly some other reciprocally demanded pairs Let the associated exchange surplus with this process be ESW1 –W2 (0) Since in reality W1 –W2 and W2 –W1 type pairs are not matched as soon as they arrive, ESW1 –W2 (0) > ESW1 –W2 (0) The exchange surplus related to a pair is ρc Since we are discussing the next incoming pair, this surplus should be discounted with λ , implying that the associated exchange surplus is E e−ρτ = λ+ρ ESW1 –W2 = λ λ+ρ pW1 –W2 + pW2 –W1 c ρ (A41) λ λ+ρ ESW1 –W2 , , λ k−1 ( λ+ρ ) ESW1 –W2 Therefore, Similarly, the exchange surplus associated with the second incoming pair is ESW1 –W2 = k the exchange surplus associated with the k th incoming pair is ESW1 –W2 = ∞ k ∞ ESW1 –W2 = ESW1 –W2 (0) = k=1 k=1 λ λ+ρ k−1 ESW1 –W2 = 1 ESW1 –W2 λ − ( λ+ρ ) λc = pW1 –W2 + pW2 –W1 ρ ∗ Recall that ESW (0) = –W2 ∗ ESW –W2 Lemma ρ c ESW1 –W2 (0) = and (A42) (0) Hence, ρ ρ λ pW1 –W2 + pW2 –W1 ESW1 –W2 (0) < ESW1 –W2 (0) = c c ρ ∗ For each W1 –W2 ∈ PR , we have s ∗W © 2009 The Review of Economic Studies Limited –W2 ≥ and s ∗W –W2 = 0, or s ∗W –W2 (A43) = and s ∗W –W2 ≤ Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Claim There exists z ≤ such that for all z < z we have f z = f2 ă UNVER DYNAMIC KIDNEY EXCHANGE 411 Proof of Lemma Fix W1 –W2 ∈ PR We prove the lemma by contradiction Suppose that there exist some ∗ ∗ s ∗W –W > and s ∗W –W < such that φ s ,s is efficient Since s ∗W –W > 0, action f1 (do-not-match W1 –W2 2 type pair and choose the smaller exchange option) is chosen at state component 1, whenever an action needs to ∗ be taken By the Bellman equation (17), the normalized exchange surplus related to action f1 is ESW (1), –W2 the normalized exchange surplus related to action f2 (match W1 –W2 type pair and choose the larger exchange ∗ ∗ ∗ option) is ESW (0) + 1, and we have ESW (1) ≥ ESW (0) + (by Observation and since in case –W2 –W2 –W2 of equality f2 is chosen) Similarly, since s ∗W –W < 0, action f1 , that is, the smaller exchange, is chosen at state component –1, whenever an action needs to be taken By the Bellman equation (18), the normalized exchange surplus ∗ related to action f1 is ESW (−1), the normalized exchange surplus related to action f2 (the larger exchange) –W2 ∗ ∗ ∗ is ESW –W (0) + 1, and we have ESW (−1) ≥ ESW (0) + (by Observation 3) We recall the Bellman –W2 –W2 equation for state component as follows (equation (19)): –W2 (0) = λ λ+ρ ⎥ ⎟ ⎢⎜ ∗ pX–Y ⎠ ESW (0)⎥ ⎢⎝ –W2 ⎥ ⎢ ⎥ ⎢ X–Y∈P O (W1 –W2 )∪P U ∪PR \{W1 –W2 ,W2 –W1 } ⎦ ⎣ ∗ ∗ +pW1 –W2 ESW –W (1) + pW2 –W1 ESW –W (−1) ES ∗ ES ∗ We replace (1) by the smaller number (0) + and above expression to obtain the following inequality: ES ∗ (A44) (−1) by the smaller number ES ∗ (0) + in the ⎡⎛ ⎞ ⎤ ⎜ ⎟ ⎥ ∗ λ ⎢ pX–Y ⎠ ESW (0)⎥ ⎢⎝ ∗ –W2 ESW (0) ≥ ⎢ ⎥ –W ⎦ λ + ρ ⎣ X–Y∈P O (W1 –W2 )∪P U ∪PR \{W1 –W2 ,W2 –W1 } ∗ ∗ +pW1 –W2 (ES (0) + 1) + pW2 –W1 (ES (0) + 1) (A45) Arranging the terms in the above inequality, we obtain ∗ ESW –W2 contradicting Lemma Therefore, we have s ∗W Proof of Theorem (0) ≥ –W2 λ pW1 –W2 + pW2 –W1 , ρ ≥ and s ∗W –W2 = 0, or s ∗W (A46) –W2 = and s ∗W –W2 ≤ The proof follows directly from Lemmata 2, 3, and APPENDIX B ON THE EFFICIENT KIDNEY EXCHANGE MECHANISM WHEN SELF-DEMANDED TYPES PARTICIPATE IN EXCHANGE In this appendix, we retain Assumptions 1, 2, and 4, and relax Assumption 3, that is, we assume that selfdemanded type pairs also participate in exchange When there are self-demanded types in the exchange pool, under Assumptions 1, 2, and 4, the full state of the matching mechanism should be denoted not only by the difference between the number of A–B and B–A type pairs but also by four other variables that denote the number of O–O, A–A, B–B, and AB–AB type pairs Next, we outline the intuition behind the derivation of the structure of the efficient mechanism under Assumptions 1, 2, and A formal derivation using Bellman equations is complicated because of the high dimensionality of the state space However, we can make use of the underlying structure of the problem and our results in the previous subsection in explaining the intuition: ∗ ∗ Let φ s ,s be the efficient matching mechanism under Assumptions 1, 2, and Without loss of generality, ∗ let s ≥ and s ∗ = Suppose Assumptions 1, 2, and still apply, while self-demanded types can participate in exchange Two cases can arise in the pool: • When a self-demanded type pair arrives: Suppose this pair is of type X–X If there is another type X–X pair available in the exchange pool, then we obtain a two-way exchange by matching these two pairs together © 2009 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 ∗ ESW ⎤ ⎞ ⎡⎛ 412 REVIEW OF ECONOMIC STUDIES State s s0 (AB–O, O–A, A–AB), (AB–O, O–B, B–AB) (B–O, O–B) (AB–O, O–B, B–A, A–AB) (B–O, O–A, A–B) When s A–B type (AB–A, A–AB) (AB–A, A–B, B–AB) pairs exist (AB–O, O–A, A–AB), (AB–O, O–B, B–AB) (AB–O, O–A, A–B, B–AB) Three cases are possible: – X–X ∈ {O–O, AB–AB}: Observe that type X–X pair can be inserted in ES if and only if it can be inserted in EL in each case Therefore, the existence of X–X type pair or the absence of X–X type pairs has no effect on the thresholds, since in either case the marginal gain of the larger exchange is only one pair Therefore, whenever such an X–X type pair exists, inserting the X–X type pair in ES or EL , whichever is chosen under the thresholds s ∗ and s ∗ , is the efficient action – X–X = A–A: Consider the case when there is A–A type pair If s ∗ < let the pair types of ES be (AB–B, B–AB) and the pair types of EL be (AB–B, B–A, A–AB), and if s ∗ > 0, let the pair types of ES be (B–O, O–B) and the pair types of EL be (B–O, O–A, A–B) In each case, the A–A pair cannot be inserted in the smaller exchange, but it can be inserted in the larger exchange Therefore, the marginal gain of the larger exchange is two pairs (with A–A type pair) Recall that when no A–A pair exists, the marginal gain of the larger exchange is only one pair Therefore, the threshold for smaller exchanges with an A–A type pair cannot exceed the threshold without an A–A type pair in absolute value, s ∗ or s ∗ , whichever applies For all other possibilities for ES and EL pair types, the A–A type pair can be inserted in ES if and only if it can be inserted in EL ; hence the threshold s ∗ or s ∗ is still valid for smaller exchanges – X–X = B–B: The symmetric argument for the case X–X = A–A applies by interchanging the roles of A and B blood types Based on this intuition, we state the following remark: Remark Suppose Assumptions 1, 2, and hold, i.e self-demanded type pairs can also participate in exchange ∗ ∗ Let φ s ,s be the dynamically efficient kidney exchange mechanism under Assumptions 1, 2, 3, and Consider the ∗ case s > and s ∗ = Then, there exist thresholds ≤ s ∗A–A ≤ s ∗ and ≤ s ∗B–B ≤ s ∗ such that under an efficient mechanism whenever a decision is required between two exchanges–the largest exchange with an A–B type pair (option match) or the largest exchange without an A–B type pair (option do-not-match)–the smaller exchange is chosen if and only if the number of A–B type pairs, s, satisfies • s ∗A–A ≥ s ≥ 0, if an A–A type pair exists and a B–O type pair arrives, © 2009 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 immediately Observe that this exchange is efficient, since self-demanded pairs cannot save any underdemanded pairs by Proposition Therefore, under the efficient mechanism, there will be or self-demanded type pairs in the pool • When a non-self-demanded type pair arrives: Let E = (i1 , , ik ) be a feasible exchange without any selfdemanded types (let ik+1 ≡ i1 ) If there exists a self-demanded X–X type pair i available in the exchange pool such that there are pairs i and i +1 with X blood-type donor and X blood-type recipient, respectively, then we can insert pair i between pairs i and i +1 and obtain a feasible exchange E This exchange is better than E, since (1) self-demanded types cannot save any underdemanded types, (2) overdemanded types are most efficiently used in saving underdemanded types, and finally, (3) self-demanded types can otherwise be matched with only same-type pairs if they are not inserted in larger exchanges So, we need to enlarge the exchanges as much as possible by inserting all possible existing self-demanded type pairs By the above argument, and given the fact that efficient mechanism under Assumptions 1, 2, 3, and is a threshold mechanism, the efficient mechanism under Assumptions 1, 2, and is a generalized threshold mechanism, with a threshold number of A–B (or B–A type pairs) to conduct smaller exchanges (the largest exchanges without the A–B or B–A type pairs), where the threshold number depends on the existence or absence of self-demanded type pairs at each state Let ES be the possible smaller exchange and EL be the possible larger exchange without any self-demanded type pairs We have the following possibilities for the pair types in ES and EL : ă UNVER DYNAMIC KIDNEY EXCHANGE 413 • s ∗B–B ≥ s ≥ 0, if a B–B type pair exists and an AB–A type pair arrives, • s ∗ ≥ s ≥ 0, otherwise If these conditions are not satisfied, the largest exchanges are conducted as soon as they become feasible The efficient mechanism is symmetrically defined for the case s ∗ = and s ∗ < with thresholds s ∗ , ≥ s ∗A–A ≥ s ∗ , and ≥ s ∗B–B ≥ s ∗ 32 We can state a single state variable approximation of the efficient mechanism as follows under Assumptions 1, 2, and with s ∗A–A = s ∗B–B = s ∗ and s ∗A–A = s ∗B–B = s ∗ : s Let this mechanism be called φˆ ∗ ,s ∗ We conduct policy simulations using this mechanism Acknowledgements I would like to thank especially Tayfun Săonmez, and also David Abraham, Murat Fadılo˘glu, Fikri Karaesmen, David Kaufman, Onur Kesten, Fuhito Kojima, Alvin E Roth, Andrew Schaefer, ˙Insan Tunalı, and Nes¸e Yıldız, participants at SAET Conference at Kos, Matching Workshops at Barcelona and Caltech, seminars at British Columbia, Boston College, Carnegie Mellon, Koc¸, and Bilkent for comments and suggestions I would also like to thank the editor, an associate editor, and anonymous referees of the journal for their excellent comments I am grateful to NSF for financial support REFERENCES ˘ ABDULKADIROGLU, A and LOERTSCHER, S (2006), “Dynamic House Allocation” (Working Paper, Duke University and Columbia University) ă ABDULKADIROGLU, A and SONMEZ, T (1999), “House 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Studies, 76 (4), 1295–1318 415 Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Another related branch of the literature considers sovereign risk arising when a government takes an action to expropriate foreigners who are in bilateral private contracts with domestic agents Tirole (2003) investigates a government’s incentives to take a costly action that affects the ability of domestic corporations to repay foreign debt Since domestic firms not internalize their effect on the government’s action, the level of foreign borrowing may be suboptimal Broner and Ventura (2008a,b) analyse a welfare-maximizing government that takes a decision whether to enforce contracts where the party receiving payment can be either a foreign or a domestic resident, assuming that it is impossible to discriminate between the two The government’s decision is then determined by the following trade-off Not enforcing payments can benefit the average domestic agent by expropriating the foreigners, but it can also hurt him by eliminating domestic transfers That might undermine ex ante risk sharing or investment efficiency; see Broner and Ventura (2008a,b), respectively Like Kremer and Mehta (2000), who also use the nondiscrimination assumption, they conclude that globalization may reduce domestic welfare by increasing foreign holdings, which in turn increase the incentive for a government not to enforce payments