Firms and workers may sign complex contracts that govern many aspects of their interactions. I show that when firms 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 of stable matchings of workers and firms

Echenique, Federico

English

Caltech Authors - Main

DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA 91125
CONTRACTS VS. SALARIES IN MATCHING
Federico Echenique
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SOCIAL SCIENCE WORKING PAPER 1322
May 2010
Contracts vs. Salaries in Matching
Federico Echenique
Abstract
Firms and workers may sign complex contracts that govern many aspects of their
interactions. I show that when firms 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 of stable matchings of workers and firms.
JEL classification numbers: C78
Key words: Contracts, Substitutes, Gross Substitutes, Stable Matching
Contracts vs. Salaries in Matching∗
Federico Echenique
Workers and firms may bargain over general, multi-dimensional contracts; they may
negotiate over health benefits, housing, retirement plans, etc. In this note I show that,
when firms regard contracts as substitutes, bargaining over contracts can be embedded
into a model of bargaining over wages. Substitutes, on the other hand, is the assumption
commonly used to guarantee the existence of stable matchings of workers and firms.
The economics of the embedding are straightforward, except for a small twist. When
a firm and a worker negotiate over a contract, they may bargain over many dimensions.
However, the Pareto frontier of contracts is, in a sense, “one-dimensional:” what is
better for the worker is worse for the firm. So Pareto optimal contracts may be viewed
as salaries, with the better contracts for the firm meaning lower salaries, and the better
contracts for the worker meaning higher salaries. The twist is that the firm’s ranking
over contracts might be affected by the firm’s other hires. For example, health plan A
may be better than B for a firm if it has many employees, but B beats A if it has few.
When contracts are substitutes, it turns out that the ranking is not affected in this way.
Hatfield and Milgrom (2005) present a model of two-sided worker-firm matching with
contracts. A firm will hire a collection of workers, and will negotiate a contract with each
one of them. The model is a generalization of Kelso and Crawford (1982), where each
firm and each worker negotiate a wage.1 Kelso and Crawford show that, when firms’
demands satisfy gross substitutes, the core of the matching market is nonempty. Hatfield
and Milgrom show that, when the firms’ preferences over contracts satisfy their notion
of substitutes, the core of the market is nonempty.
I show that Hatfield and Milgrom’s model can be embedded into the model of Kelso
and Crawford. Hatfield and Milgrom’s assumption of substitutability enables an em-
bedding where firms’ demands for workers satisfy Kelso and Crawford’s notion of gross
substitutes. As a result, the nonemptiness of the core follows from the argument in Kelso
and Crawford, and their salary adjustment algorithm finds a stable matching of workers
to firms, and a vector of supporting salaries.
∗I thank Vince Crawford, Flip Klijn and Michael Ostrovsky for very useful comments.
1Kelso and Crawford build on the analysis of Crawford and Knoer (1981); see Roth and Sotomayor
(1990) for a description of the models.
Hatfield and Milgrom’s paper is an elegant analysis of two-sided matching. It con-
tributes much more than showing the nonemptiness of the core when firms and workers
can sign general contracts, and my observation does not diminish their contribution in
the least. I believe, however, that there is value in clarifying the relationship between
contracts and salaries.
One step in that direction is taken by Hatfield and Kojima (2010), who investigate
conditions on preferences over contracts that are weaker than substitutes and still gener-
ate stable matchings. My embedding does not work under Hatfield and Kojima’s weaker
conditions (see 1.3.3 below). Future research should explain the consequences of the
added generality of contracts over salaries in different economic environments.
1 Embedding
1.1 Definitions
I shall describe two models. The model of a matching market with contracts with substi-
tutable choices is due to Hatfield and Milgrom (2005). The model of a matching market
with salaries and gross-substitutes in demand is due to Kelso and Crawford (1982).
1.1.1 Contracts
A matching market with contracts is described by:
• (finite, disjoint) sets W of workers, F of firms and X of contracts; each contract
x ∈ X is assigned one worker xW ∈ W and one firm xF ∈ F ;
• for each worker w ∈ W , a utility function uw : X ∪ {∅} → R; and for each
firm f ∈ F , a utility function uf : 2X → R; all utility functions are one-to-one
(preferences are strict).
A firm f ’s utility function determines a choice rule Cf : for A ⊆ X, Cf (A) is the
maximal subset of A according to uf . Note that since uf is one-to-one, Cf (A) is uniquely
defined. The empty set ∅ represents for f the option of hiring no workers. For notational
convenience, I have not restricted the domain of uf to contracts with f = xF , but of
course we want f to sign contracts only in its own name; assume then that x ∈ Cf (A)
implies f = xF . Assume also that x, x
′ ∈ Cf (A) implies xW 6= x′W .
For a worker w, ∅ represents an outside option: a contract that is always available to
her if she chooses to reject the contract some firm offers her. Suppose that if xW 6= w
then uw(x) < uw(∅).
A set of contracts A is feasible if, for all workers w, there is at most one x ∈ A with
w = xW .
2
A firm f ’s utility satisfies substitutability if, for any set of contracts A, and any
two contracts x and x′, x /∈ Cf (A ∪ {x}) implies x /∈ Cf (A ∪ {x, x′}).
The tuple (F,W,X, (uf ), (uw)) describes a a matching market with contracts.
A set of contracts A ⊆ X is individually rational if, for all x ∈ A, uxW (x) ≥
uxW (∅); and for all firms f , Cf (A) = {x ∈ A : f = xF}.
A set of contracts A ⊆ X is stable if it is individually rational and if for any firm
f and set of contracts A′ 6= A with A′ = Cf (A ∪ A′), there is one contract x′ ∈ A′ such
that either ux′W (x
′) < ux′W (∅) or there is x ∈ A with xW = x′W and ux′W (x′) < ux′W (x).
1.1.2 Salaries
A matching market with salaries is described by:
• (finite, disjoint) sets W of workers, F of firms and S ⊆ R+ of salaries;
• for each worker w ∈ W , a utility function vw : F ∪ {∅}× S → R; and for each firm
f ∈ F , a utility function vf : ∪A⊆WA× S → R; all utility functions are one-to-one
(preferences are strict).
We can suppose that S is the set {0, 1, . . . L} of the first L+ 1 non-negative integers, for
some L.
For a firm f , uf defines a demand function Df : S
W → 2W by
Df (s) = argmaxA⊆Wvf ({(w, sw) : w ∈ A}).
Say that D satisfies gross substitutes if, for any two vectors of salaries, s and s′, if
s ≤ s′ and sw = s′w then w ∈ D(s) implies that w ∈ D(s′).
A matching is a function µ : W → F ×S. A matching assigns to each worker a firm
and a salary; I use the µ(w) = (∅, 0) notation for when w is unmatched (unemployed).
A matching specifies, for each firm f a collection of workers with their corresponding
salaries: µ0(f) = {(w, s) : (f, s) = µ(w)}. The set µ0(f) is thus the set of workers em-
ployed by f , and their salaries, in the matching µ.
The tuple (F,W, S, (vf ), (vw)) describes an matching market with salaries.
A matching µ is individually rational if, for every f and w, vf (µ
0(f)) ≥ vf (B) for
all B ⊆ µ0(f) and uw(µ(w)) ≥ uw(∅, 0).
A matching µ is stable if it is individually rational and if, for any firm f and A ⊆ W ,
if there is a vector of salaries (sw)w∈A with vf ({(w, sw) : w ∈ A}) > vf (µ0(f)) then there
is w ∈ A with uw(µ(w)) > uw(f, sw).
3
1.2 Embedding
Let (F,W,X, (uf ), (uw)) be a matching market with contracts, and (F,W, S, (vf ), (vw)) a
matching market with salaries. An embedding of (F,W,X, (uf ), (uw)) into (F,W, S, (vf ), (vw))
is a one-to-one function g which maps each x ∈ X into a triple (xF , xW , s) ∈ F ×W ×S.
Let g be such an embedding and A ⊆ X. Say that g(A) defines a matching if for
any w there is at most one s and f with (f, w, s) ∈ g(A). The matching defined by A
under g is the function µ : W → F × S defined by µ(w) = (f, s) if g−1(f, w, s) ∈ A and
µ(w) = (∅, 0) otherwise.
Let (F,W,X, (uf ), (uw)) be a matching market with contracts.
Theorem 1. If firms’ choices satisfy substitutability, then there is a matching mar-
ket with salaries (F,W, S, (vf ), (vw)), and an embedding g of (F,W,X, (uf ), (uw)) into
(F,W, S, (vf ), (vw)) such that
1. firms’ demand functions in (F,W, S, (vf ), (vw)) satisfy gross substitutes;
2. A ⊆ X is a set of stable contracts if and only if g(A) defines a stable matching.
Proof. Say that a contract x is dominated for xF and xW if there is a contract x
′ with
xF = x
′
F , xW = x
′
W , uxF (x
′) > uxF (x) and uxW (x
′) > uxW (x). Let Xfw be the set of all
contracts x with xF = f and xW = w that are not dominated for f and w. Note that
Xfw can be ordered by uw; that is I can enumerate the elements of Xfw as x1, . . . , x|Xfw|
with uw(xi) < uw(xi+1). Then I can write Xfw = {(w, s) : s = 1, . . . , |Xfw|} with the
understanding that (w, s) correspond to offering worker w the contract xs in Xfw. Note
that s < s′ if and only if uf ({w, s}) > uf ({w, s′}): by definition, if s < s′ then uw(f, s) <
uw(f, s
′) so uf ({w, s}) < uf ({w, s′}) would imply that (w, s) is dominated.
Let K = max{|Xfw| : f ∈ F,w ∈ W} and S = {1, 2, . . . , K + 1}. For convenience,
we augment the contracts in Xfw to include (w, s) with |Xfw| < s ≤ K + 1 and assume
that uw(w, s) < uw(∅) and (w, s) /∈ Cf (A) for any A if s > |Xfw|. The embedding g is
the mapping that takes x ∈ X into (f, w, s) with (w, s) being the representation of x in
Xfw if x is not dominated, and into (xF , xF , K + 1) if it is.
2
Define firms’ and workers’ utilities as follows. Let vw(f, s) = uw(x), where x ∈ Xfw
corresponds to s. Let vf ((w1, s1), . . . , (wn, sn)) = uf ((w1, s1), . . . , (wn, sn)). For a vector
of wages s = (sw)w∈W ∈ SW , let Xsf ⊆ Xf be the set of contracts (w, sw). Define a
demand function Df for firm f by Df (s) = Cf (X
s
f ) (this is consistent with our definition
of utility for f).
I shall now prove thatD satisfies the GS property. LetXs+f = {(w, s) ∈ W × S : s ≥ sw},
where sw denotes w’s salary in s. I first prove that Cf (X
s
f ) = Cf (X
s+
f ). Let (w, s) ∈
2Strictly speaking, for dominated x we would need to set g(x) = (xF , xF ,K + l), choosing l ≥ 1 so
that g remains one-to-one.
4
Xs+f \ Xsf . There must exist some s′, with s′ < s and (w, s′) ∈ Xs+f . Note that s′ < s
implies (w, s) /∈ Cf ({(w, s), (w, s′)}); so {(w, s), (w, s′)} ⊆ Xs+f implies, by the property
of substitutability, that (w, s) /∈ Cf (Xs+f ). Thus I have shown that Cf (Xs+f ) ⊆ Cf (Xsf ).
Since Xsf ⊆ Xs+f , the definition of Cf implies that Cf (Xs+f ) = Cf (Xsf ).
Now, let s = (sw) and s
′ = (s′w) be vectors with s ≤ s′ while for w0 ∈ W , sw0 = s′w0
and w0 ∈ D(s). Suppose by way of contradiction that w0 /∈ D(s′). Then (w0, s′w0) /∈
Cf (X
s′
f ) = Cf (X
s′+
f ). But then X
s′+
f ⊆ Xs+f so substitutability implies that (w0, s′w0) /∈
Cf (X
s+
f ). Now, Cf (X
s+
f ) = Cf (X
s
f ) and (w0, sw0) = (w0, s
′
w0
) implies that (w0, sw0) /∈
Cf (X
s
f ), a contradiction.
The proof that A is stable in (F,W,X, (uf ), (uw)) if and only if g(A) is stable in
(F,W, S, (vf ), (vw)) is straightforward.
1.3 Discussion
1.3.1 Antecedents
The identification of contracts and salaries is not new. Kelso and Crawford (1982) men-
tion how salaries can be interpreted as contracts. Roth (1984), who presents an early
model of matching with contracts, also identifies contracts with salaries. The observation
that substitutability is important for this identification to hold seems to be new. It is
not mentioned in the literature that follows Hatfield and Milgrom (2005).
1.3.2 Quasilinearity
In Kelso and Crawford, firms’ profits are quasilinear, but their existence proof 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.
1.3.3 Bilateral substitutes
In a model of matching with contracts, Hatfield and Kojima (2010) present a generaliza-
tion of substitutes, called bilateral substitutes. They show stable matching with contracts
exists under bilateral substitutes. Here I show that the model of Hatfield-Kojima cannot
be embedded into the Kelso-Crawford model.
The following example is from Hatfield and Kojima (2010). Let the set of firms be
{f, f ′}, the set of workers {w,w′} and the set of contracts be {x, x˜, z, z˜, z′}. Let x and x˜
involve firm f and worker w, while z and z˜ involve worker w′ and firm f . Contract z′ is
5
between w′ and f ′. Suppose that agents’ utilities are such that their preferences are:
f f ′ 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 sx and x˜ to the salary sx˜. Then x /∈ Cf ({x, x˜}) means that
sx > sx˜. This would imply that
uf (xW , zW , sx, sz) < uf (x˜W , zW , sx˜, sz),
as xW = x˜W , which is incompatible with the preferences above.
References
Crawford, V. P., and E. M. Knoer (1981): “Job Matching with Heterogeneous
Firms and Workers,” Econometrica, 49(2), 437–450.
Hatfield, J., and F. Kojima (2010): “Substitutes and stability for matching with
contracts,” Forthcoming, Journal of Economic Theory.
Hatfield, J., and P. Milgrom (2005): “Matching with Contracts,” American Eco-
nomic Review, 95(4), 913–935.
Kelso, A. S., and V. P. Crawford (1982): “Job Matching, Coalition Formation,
and Gross Substitutes,” Econometrica, 50, 1483–1504.
Roth, A., and M. Sotomayor (1990): Two-sided Matching: A Study in Game-
Theoretic Modelling and Analysis, vol. 18 of Econometric Society Monographs. Cam-
bridge University Press, Cambridge England.
Roth, A. E. (1984): “Stability and Polarization of Interests in Job Matching,” Econo-
metrica, 52(1), 47–57.
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