Published in the American Economic Review Volume 102, Issue 1, February 2012, pages 594-601. doi:10.1257/aer.102.1.594 CONTRACTS VS. SALARIES IN MATCHING FEDERICO ECHENIQUE Abstract. Firms and workers may sign complex contracts that govern many aspects o their interactions. I show that when irms regard contracts as substitutes, bargaining over contracts can be understood as bargaining only over wages. Substitutes is the assumption commonly used to guarantee the existence o stable matchings o workers and irms. Workers and irms may bargain over general, multi-dimensional contracts; they may negotiate over health beneits, housing, retirement plans, etc. Substitutes, on the other hand, is the assumption commonly placed on irms preerences to guarantee the existence o stable matchings o workers and irms. In this note I show that, when irms regard contracts as substitutes, bargaining over contracts can be embedded into a model o bargaining over wages. The economics o the embedding is straightorward, except or a small twist. When a irm and a worker negotiate over a contract, they may bargain over many dimensions. However, the Pareto rontier o contracts is, in a sense, one-dimensional: what is better or the worker is worse or the irm. So Pareto optimal contracts may be viewed as salaries, with the better contracts or the irm meaning lower salaries, and the better contracts or the worker meaning higher salaries. The twist is that a irm s ranking over contracts might be aected by the irm s other hires. For example, health plan A may be better than B or a irm i it has many employees, but B beats A i it has ew. When contracts are substitutes, it turns out that the ranking is not aected in this way. Hatield and Milgrom (2005) present a model o two-sided workerirm matching with contracts. A irm will hire a collection o workers, and will negotiate a contract with each one o them. The model is a generalization o Kelso and Craword (1982), where each irm and each Division o the Humanities and Social Sciences, Caliornia Institute o Technology. Email: ede@caltech.edu I thank Vince Craword, Flip Klijn, Scott Kominers, and Michael Ostrovsky or very useul comments. I am also very grateul to three anonymous reerees or their thoughtul questions and suggestions. 1
2 FEDERICO ECHENIQUE worker negotiate a wage. 1 Kelso and Craword show that, when irms demands satisy gross substitutes, the core o the matching market is nonempty. Hatield and Milgrom show that, when the irms preerences over contracts satisy their notion o substitutes, the core o the market is nonempty. I show that Hatield and Milgrom s model can be embedded into the model o Kelso and Craword. Hatield and Milgrom s assumption o substitutability enables an embedding where irms demands or workers satisy Kelso and Craword s notion o gross substitutes. As a result, the nonemptiness o the core ollows rom the argument in Kelso and Craword, and their salary adjustment algorithm inds a stable matching o workers to irms, and a vector o supporting salaries. Hatield and Milgrom s paper is an elegant analysis o two-sided matching. It simpliies matching models, and makes their relationship to auction models more transparent. Their paper contributes much more than showing the nonemptiness o the core when irms and workers can sign general contracts; and my result does not diminish their contribution in the least. I believe, however, that there is value in clariying the relationship between contracts and salaries. One step in that direction is taken by Hatield and Kojima (2010), who investigate conditions on preerences over contracts that are weaker than substitutes and still generate stable matchings. My embedding does not work under Hatield and Kojima s weaker conditions (see 2.3 below). Future research should explain the consequences o the added generality o contracts over salaries in dierent economic environments. A paper that seeks to extend some classical result on matching to matching with contracts would need to sort out to what extent allowing or contracts provides a more general result. It may be that contracts are not more general; which would not by itsel invalidate the exercise, but would be an important eature to understand. 1. Embedding 1.1. Deinitions. I shall describe two models. The model o a matching market with contracts with substitutable choices is due to Hatield and Milgrom (2005). The model o a matching market with salaries and gross substitutes in demand is due to Kelso and Craword (1982). 1.1.1. Contracts. A matching market with contracts is described by: 1 Kelso and Craword build on the analysis o Craword and Knoer (1981); see Roth and Sotomayor (1990) or a description o the models.
CONTRACTS VS. SALARIES IN MATCHING 3 (inite, disjoint) sets W o workers, F o irms and X o contracts; each contract x X is assigned one worker x W W and one irm x F F ; or each worker w W, a utility unction u w : X { } R; and or each irm F, a utility unction u : 2 X R; all utility unctions are one-to-one (preerences are strict). A irm s utility unction determines a choice rule C : or A X, C (A) is the maximal subset o A according to u. Note that since u is one-to-one, C (A) is uniquely deined. The empty set represents or the option o hiring no workers. For notational convenience, I have not restricted the domain o u to contracts with = x F, but o course we want to sign contracts only in its own name; assume then that x C (A) implies = x F. Assume also that x, x C (A) implies x W x W. For a worker w, represents an outside option: a contract that is always available to her i she chooses to reject the contract some irm oers her. Suppose that i x W w then u w (x) < u w ( ). A set o contracts A is easible i, or all workers w, there is at most one x A with w = x W. A irm s utility satisies substitutability i, or any set o contracts A, and any two contracts x and x, x / C (A {x}) implies x / C (A {x, x }). The tuple (F, W, X, (u ), (u w )) describes a a matching market with contracts. A set o contracts A X is individually rational i, or all x A, u xw (x) u xw ( ); and or all irms, C (A) = {x A : = x F }. A set o contracts A X is stable i it is individually rational and i or any irm and set o contracts A A with A = C (A A ), there is one contract x A such that either u x W (x ) < u x W ( ) or there is x A with x W = x W and u x (x ) < u W x W (x). 1.1.2. Salaries. A matching market with salaries is described by: (inite, disjoint) sets W o workers, F o irms, and S R + o salaries; or each worker w W, a utility unction v w : F { } S R; and or each irm F, a utility unction v : A W A S R; all utility unctions are one-to-one (preerences are strict). We can suppose that S is the set {0, 1,... L} o the irst L + 1 nonnegative integers, or some L. For a irm, v deines a demand unction D : S W 2 W by D (s) = argmax A W v ({(w, s w ) : w A}).
4 FEDERICO ECHENIQUE I restrict attention to demand unctions, not correspondences, by virtue o Hatield and Milgrom s assumption o strict preerences. Say that D satisies gross substitutes i, or any two vectors o salaries, s and s, i s s and s w = s w then w D(s) implies that w D(s ). A matching is a unction µ : W F { } S. A matching assigns to each worker a irm and a salary; I use the µ(w) = (, 0) notation or when w is unmatched (unemployed). A matching speciies, or each irm a collection o workers with their corresponding salaries: µ 0 () = {(w, s) : (, s) = µ(w)}. The set µ 0 () is thus the set o workers employed by, and their salaries, in the matching µ. The tuple (F, W, S, (v ), (v w )) describes a matching market with salaries. A matching µ is individually rational i, or every and w, v (µ 0 ()) v (B) or all B µ 0 () and v w (µ(w)) v w (, 0). A matching µ is stable i it is individually rational and i, or any irm and A W, i there is a vector o salaries (s w ) w A with v ({(w, s w ) : w A}) > v (µ 0 ()) then there is w A with v w (µ(w)) > v w (, s w ). 1.2. Embedding. Let (F, W, X, (u ), (u w )) be a matching market with contracts, and (F, W, S, (v ), (v w )) a matching market with salaries. An embedding o (F, W, X, (u ), (u w )) into (F, W, S, (v ), (v w )) is a oneto-one unction g which maps each x X into a triple (x F, x W, s) F W S. Let g be such an embedding and A X. Say that g(a) deines a matching i or any w there is at most one s and with (, w, s) g(a). The matching deined by A under g is the unction µ : W F S deined by µ(w) = (, s) i g 1 (, w, s) A and µ(w) = (, 0) otherwise. Theorem 1. Let (F, W, X, (u ), (u w )) be a matching market with contracts. I irms choices satisy substitutability, then there is a matching market with salaries (F, W, S, (v ), (v w )), and an embedding g o (F, W, X, (u ), (u w )) into (F, W, S, (v ), (v w )) such that (1) irms demand unctions in (F, W, S, (v ), (v w )) satisy gross substitutes; (2) A X is a set o stable contracts i and only i g(a) deines a stable matching. The proo o Theorem 1 works by constructing an embedding. As suggested in the introduction, the Pareto rontier o contracts is onedimensional: what is better or the worker is worse or the irm. So
CONTRACTS VS. SALARIES IN MATCHING 5 Pareto optimal contracts may be translated into salaries, with the better contracts or the irm deining lower salaries, and the better contracts or the worker deining higher salaries. The problem is that a irm s ranking over contracts might be aected by the irm s other hires. For example, consider contracts x and x, both involving worker w and irm. Suppose that x is better than x or, and x is better than x or w; so we would map x into a lower salary than x. In the absence o substitutes, this mapping might be aected by the presence o other contracts. However, since x is rejected by when x is available, substitutes make sure that it will continue to be rejected when other workers and contracts are available. So the assumption o substitutes allows or the pairwise mapping o contracts into salaries to work globally, when all workers and irms are considered. The construction uses the salaries 1, 2, 3,...; but any other grid works in exactly the same way. Proo. Say that a contract x is dominated or x F and x W i there is a contract x with x F = x F, x W = x W, u x F (x ) > u xf (x) and u xw (x ) > u xw (x). Let X w be the set o all contracts x with x F = and x W = w that are not dominated or and w. Note that X w can be ordered by u w ; that is, I can enumerate the elements o X w as x 1,..., x Xw with u w(x i ) < u w (x i+1 ). Then I can write X w = {(w, s) : s = 1,..., X w } with the understanding that (w, s) corresponds to oering worker w the contract x s in X w. Note that s < s i and only i u ({w, s}) > u ({w, s }): by deinition, i s < s then u w (, s) < u w (, s ) so u ({w, s}) < u ({w, s }) would imply that (w, s) is dominated. Let K = max{ X w : F, w W } and S = {1, 2,..., K + 1}. For convenience, lets augment the contracts in X w to include (w, s) with X w < s K + 1. Assume that i s > X w then, or all A, (w, s) / C (A). The embedding g is the mapping that takes x X into (, w, s) with (w, s) being the representation o x in X w i x is not dominated, and into (x F, x F, K + 1) i it is. 2 Deine irms and workers utilities as ollows. Let v w (, s) = u w (x), where x X w corresponds to s. Let v ((w 1, s 1 ),..., (w n, s n )) = u ((w 1, s 1 ),..., (w n, s n )). For a vector o wages s = (s w ) w W S W, let X s X be the set o contracts (w, s w ). Deine a demand unction D or irm by D (s) = C (X s ) (this is consistent with our deinition o utility or ). 2 Strictly speaking, or dominated x we would need to set g(x) = (x F, x F, K + l), choosing l 1 so that g remains one-to-one.
6 FEDERICO ECHENIQUE I shall now prove that D satisies the GS property. Let X s+ = {(w, s) W S : s s w }, where s w denotes w s salary in s. I irst prove that C (X s) = C (X s+ ). I Xs+ = X s then there is nothing to prove. Let (w, s) X s+ \X s. There must exist some s, with s < s and (w, s ) X s+. Note that s < s implies (w, s) / C ({(w, s), (w, s )}) because u (w, s) < u (w, s ) and C ({(w, s), (w, s )}) cannot contain two contracts with the same worker. So {(w, s), (w, s )} X s+ implies, by the property o substitutability, that (w, s) / C (X s+ ). Thus I have shown that C (X s+ ) C (X s). Since Xs Xs+, the deinition o C implies that C (X s+ ) = C (X s). Now, let s = (s w ) and s = (s w) be vectors with s s while or w 0 W, s w0 = s w 0 and w 0 D (s). Suppose by way o contradiction that w 0 / D (s ). Then (w 0, s w 0 ) / C (X s ) = C (X s + ). But then X s + X s+ so substitutability implies that (w 0, s w 0 ) / C (X s+ ). Now, C (X s+ ) = C (X s) and (w 0, s w0 ) = (w 0, s w 0 ) implies that (w 0, s w0 ) / C (X s ), a contradiction. The proo that A is stable in (F, W, X, (u ), (u w )) i and only i g(a) is stable in (F, W, S, (v ), (v w )) is straightorward. 2. Discussion 2.1. Antecedents. The identiication o contracts and salaries is not new. Both Craword and Knoer (1981), and Kelso and Craword (1982), mention how salaries can be interpreted as contracts. Roth (1984), who presents an early model o matching with contracts, also identiies contracts with salaries. The new observation here is that substitutability is important or the identiication o contracts and salaries. It is not mentioned in the literature that ollows Hatield and Milgrom (2005). 2.2. Quasilinearity. In Kelso and Craword, irms proits are quasilinear, but their existence proo is more general and does not depend on quasilinearity. One detail is that they require a salary that is high enough so no worker would be hired at that salary (see their Lemma 2). In the embedding in the theorem, we do have such a salary. 2.3. Bilateral substitutes. In a model o matching with contracts, Hatield and Kojima (2010) present a generalization o substitutes, called bilateral substitutes. They show stable matching with contracts exists under bilateral substitutes. Here I show that the model o Hatield-Kojima cannot be embedded into the Kelso-Craword model.
CONTRACTS VS. SALARIES IN MATCHING 7 The ollowing example is rom Hatield and Kojima (2010). Let the set o irms be {, }, the set o workers {w, w } and the set o contracts be {x, x, z, z, z }. Let x and x involve irm and worker w, while z and z involve worker w and irm. Contract z is between w and. Suppose that agents utilities are such that their preerences are: w w {x, z} {z } x z { z} x z { x} z {x} {z}. I have omitted the alternatives that are worse than. Suppose that there is a embedding, where x maps to the salary s x and x to the salary s x. Then x / C ({x, x}) means that s x > s x. This would imply that u (x W, z W, s x, s z ) < u ( x W, z W, s x, s z ), as x W = x W, which is incompatible with the preerences above. I should clariy that a deviation rom substitutes does not by itsel prevent the model with contracts rom being embeddable into a model with salaries. The structure o bilateral substitutes is interesting because it is a deviation rom substitutes or which existence is guaranteed. 2.4. Algorithm. Under the embedding, the algorithm proposed by Hatield and Milgrom (2005) is equivalent to the algorithm proposed by Kelso and Craword (1982). It is easy to see that they are equivalent because the two algorithms ind the same matching: the irm-optimal or the worker-optimal matching, depending on how the algorithms are ormulated. Here, I want to show that the algorithms not only calculate the same matchings, but that they also work in essentially the same way: the algorithms take equivalent routes to a stable matching. The Hatield- Milgrom algorithm starts by each irm oering the best contracts or, and the workers sequentially rejecting oers. In the embedding, the best contracts correspond to the lowest salaries; and, in a similar vein, the Kelso-Craword algorithm starts at the lowest salaries. In Hatield-Milgrom, when a contracts gets rejected, it is as i the salary or that worker is raised. Similarly, the Kelso-Craword algorithm works by raising the salaries o workers who reject an oer. Broadly speaking, the steps taken by the Hatield-Milgrom algorithm represent movements along the Pareto rontier o contracts; movements in
8 FEDERICO ECHENIQUE a direction which corresponds to higher salaries in the Kelso-Craword model. The ollowing example illustrates the point, and hopeully helps understand the nature o the embedding better. Suppose there are two irms and two workers: F = { 1, 2 } and W = {w 1, w 2 }. Let X = {x 1, x 3, x 2, y 1, y 2 } be the set o contracts. The agents preerences are: 1 2 w1 w2 {x 1 } {y 2 } x 3 y 1 {x 2 } {x 3 } x 2 {y 1 } x 1. The Hatield-Milgrom algorithm takes the steps in the ollowing table. I am using the same notation as Hatield-Milgrom: R W and R F are the rejected contracts o, respectively, workers and irms. X W R W (X W ) X F R F (X F ) 0) X {x 2, y 1, x 3 } 1) {x 1, y 2 } {y 2 } X {x 2, y 1, x 3 } 2) {x 1, y 2 } {y 2 } {x 1, x 2, x 3, y 1 } {x 2, y 1 } 3) {x 1, y 2, x 3 } {x 1, y 2 } {x 1, x 2, x 3, y 1 } {x 2, y 1 } 4) {x 1, y 2, x 3 } {x 1, y 2 } {x 2, x 3, y 1 } {y 1 } 5) {x 1, x 2, x 3, y 2 } {x 1, x 2, y 2 } {x 2, x 3, y 1 } {y 1 } 6) {x 1, x 2, x 3, y 2 } {x 1, x 2, y 2 } {x 3, y 1 } 7) X {x 1, x 2, y 2 } {x 3, y 1 } I shall interpret the iterations so as to simpliy the comparison with Kelso-Craword. In (0) irms oer contracts x 1 and y 2 (they reject contracts {x 2, y 1, x 3 }. In (1) the workers respond by w 1 temporarily accepting x 1, while w 2 rejects y 2. In (2) the irms oer x 1 and x 3. In (3) w 1 accepts x 3 and rejects x 2. In (4) 1 oers x 2 and 2 oers x 3. (5) w 1 rejects x 2. (6) 1 oers y 1 and 2 oers x 3. (7) the workers accept y 1 and x 3. We can embed this market with contracts into a market with salaries. Let the set o salaries be {1, 2}. The relevant part o the agents preerences are: 1 : (w 1, 1) (w 1, 2) (w 2, 1) 2 : (w 2, 1) (w 1, 1) w 1 : ( 2, 1) ( 1, 2) ( 1, 1) w 2 : ( 1, 1)
CONTRACTS VS. SALARIES IN MATCHING 9 So, irm 1 preers to hire w 1 at a lower rather than a higher salary; and preers to hire w 1 at either salary over hiring w 2 at a salary o 1. I omit, or example, that or w 1 ( 2, 2) ( 2, 1) because irm 2 is unwilling to hire w 1 at a salary o 2 (she preers to leave the position vacant). Kelso and Craword s algorithm does: w 1 w 2 0) ( 1, 1) ( 2, 1) 1) ( 1, 1)( 2, 1) 2) ( 1, 2)( 2, 1) 3) ( 2, 1) ( 1, 1) The meaning o the iterations is the ollowing. The algorithm starts with all salaries being 1. At his vector o salaries, 1 demands worker w 1 while 2 demands worker w 2. Worker w 2 rejects the oer o irm 2 while w 1 accept the oer rom 1 ; this is indicated by underlining the oer in the table. Since 2 s oer to w 2 was rejected, the salary or that pair increases to 2. At that salary, 2 decides not to oer to w 2 and instead demand w 1 at a salary o 1. The rest o the iterations should be intuitive. In any case, the steps are essentially those o the Hatield-Milgrom algorithm, where workers responses are included within each step in the Kelso-Craword algorithm, and as a separate step in Hatield-Milgrom. A general equivalence result is possible, but requires writing down an algorithm that interprets the steps o one algorithm in terms o the other. 2.5. Law o demand. In a matching model with contracts, a irm satisies the law o aggregate demand i A A implies C (A) C (A ). In a matching model with salaries, a irm satisies the law o aggregate demand i s s implies that D (s ) D (s). Using the embedding o Theorem 1, it is easy to show that i a irm satisies the law o aggregate demand in the matching model with contracts, then it satisies the law o aggregate demand in the embedded matching model with salaries. Reerences Craword, Vincent P. and Elsie M. Knoer, Job Matching with Heterogeneous Firms and Workers, Econometrica, mar 1981, 49 (2), 437 450. Hatield, John and Paul Milgrom, Matching with Contracts, American Economic Review, Sep 2005, 95 (4), 913 935.
10 FEDERICO ECHENIQUE Hatield, J.W. and F. Kojima, Substitutes and stability or matching with contracts, 2010. Forthcoming, Journal o Economic Theory. Kelso, Alexander S. and Vincent P. Craword, Job Matching, Coalition Formation, and Gross Substitutes, Econometrica, 1982, 50, 1483 1504. Roth, Alvin and Marilda Sotomayor, Two-sided Matching: A Study in Game-Theoretic Modelling and Analysis, Vol. 18 o Econometric Society Monographs, Cambridge University Press, Cambridge England, 1990. Roth, Alvin E., Stability and Polarization o Interests in Job Matching, Econometrica, jan 1984, 52 (1), 47 57.