Entropy methods for identifying hedonic models
aa r X i v : . [ ec on . E M ] F e b Entropy methods for identifying hedonicmodels
Arnaud Dupuy ∗ CEPS/INSTEAD Alfred Galichon † Sciences PoMarc Henry ‡ Penn State UniversityJune 12, 2014Dedicated to Ivar Ekeland on his 70th birthday § . Abstract
This paper contributes to the literature on hedonic models intwo ways. First, it makes use of Queyranne’s reformulation of ahedonic model in the discrete case as a network flow problem inorder to provide a proof of existence and integrality of a hedo-nic equilibrium and efficient computation of hedonic prices. Sec-ond, elaborating on entropic methods developed in Galichon andSalani´e (2014), this paper proposes a new identification strategyfor hedonic models in a single market. This methodology allowsone to introduce heterogeneities in both consumers’ and produc-ers’ attributes and to recover producers’ profits and consumers’utilities based on the observation of production and consumptionpatterns and the set of hedonic prices. ∗ E-mail: [email protected]. † Corresponding author. E-mail: [email protected]. Galichon’s re-search has received funding from the European Research Council under the EuropeanUnion’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no313699, and from FiME, Laboratoire de Finance des March´es de l’Energie. ‡ E-mail: [email protected]. Henry’s research is supported by SSHRC Grant435-2013-0292 and NSERC Grant 356491-2013. § This paper has benefited from insightful conversations with Ivar Ekeland andBernard Salani´e. We would like to thank an anonymous referee for comments on anearlier version of the paper. Introduction
Starting with Court (1941), Griliches (1961) and Lancaster (1966), alarge literature has aimed at providing a theoretical framework for pricingthe attributes of highly differentiated goods. While this literature was ini-tially mainly empirical in nature and early contributions lacked a propertheoretical setting, the first theoretical treatments of hedonic models ap-peared in Tinbergen (1956) and Rosen (1974). Tinbergen (1956) presentsa stylized model in which preferences are quadratic and attributes nor-mally distributed. Rosen (1974) showed the theoretical relation of he-donic prices to marginal willingness to produce and marginal willingnessto consume. Hedonic models have also been used to study the pricing ofhighly differentiated products such as houses (Kain and Quigley, 1970),wine and champagne (Golan and Shalit, 1993), automobiles (Triplett,1969) among others, but also set forth a new literature on the Value ofStatistical Life following Thaler and Rosen’s (1976) original idea of see-ing jobs attributes and in particular “risk taken on the job” as a vectorof hedonic attributes valued on the labor market. More recently, signif-icant progress on the understanding of the properties of hedonic models(properties of an equilibrium, identification of deep parameters etc.) hasbeen achieved. These developments are to a large extent due to Ivar Eke-land’s contributions, see e.g. Ekeland et al. (2004) and Ekeland (2010a,2010b), and it is a pleasure to dedicate to him the present piece of workin recognition of our intellectual debt to him.In this paper we contribute to the hedonic literature in two ways.First, we elaborate on an idea of Maurice Queyranne who reformulatedthe hedonic model in the discrete case as a network flow problem. Thisreformulation allows us to derive results on the existence of a hedonicequilibrium in the discrete case, and it allows the use of powerful com-putational techniques to solve for the equilibrium. Second, building onrecent development in the matching model literature and in particularthe seminal contribution due to Choo and Siow (2006) generalized byGalichon and Salani´e (2014), we introduce heterogeneities (unobservedby the econometrician) in producer and consumer types. This formal-ism has two advantages: (i) it allows for the incorporation of unobservedheterogeneity in the producers and consumers characteristics, and (ii) itprovides straightforward identification results. Indeed, we follow Gali-chon and Salani´e in making use of the convex duality in discrete choiceproblems to recover utilities from choice probabilities on both side of themarket. 2he remainder of the paper is organized as follows. Section 2 discussesthe properties of an equilibrium in hedonic models and its reformulationas a network flow problem. Section 3 introduces a model with unobservedheterogeneities on both sides of the market and studies the identificationof preference parameters. The discussion in Section 4 concludes the pa-per.
The model . Throughout this paper, X is the set of observable types ofproducers of a given good, and Y the set of observable types of consumersof that good. This good comes in various qualities; let Z be the set ofthe good’s qualities. The sets X , Y and Z are assumed to be finite.It is assumed that there is a supply n x (resp. m y ) of producers (resp.consumers) of type x (resp. y ). It is assumed that producers (resp.consumers) can produce (consume) at most one unit of good. They havethe option not to participate in the market, in which case they choose z = 0.For example, hedonic models can be used to model the market forfine wines . In that case, X may be the set of observable characteristicsof wine producers (say, grapes used, average amount of sunshine, andharvesting technology), and Y may be the set of observable characteris-tics of wine consumers (say country and purchasing channel). Z will bethe quality of the wine (say acidity, sugar content, expert rating).Let p z be the price of the good of quality z . If a producer of type x produces the good in quality z , the payoff to the producer is α xz + p z ,where α xz ∈ R ∪ {−∞} is the producer’s productivity (the opposite of aproduction cost). Similarly, if the consumer of type y consumes the goodin quality z , the payoff to the consumer is γ zy − p z , where γ zy ∈ R ∪{−∞} is the utility of the consumer . Producers and consumers who do notparticipate in the market get a surplus of zero. Supply and demand . Let µ xz be the supply function, that is the We are confident Ivar will approve of this choice of example. Note that in this setup, the utility of agents on each side of the market does notdepend directly on the type of the agent with whom they match, only through thetype of the contract. A more general framework where α and γ depend simultaneouslyon x , y and z is investigated in Dupuy, Galichon and Zhao (2014). x offering quality z ; similarly, µ zy is thedemand function, the number of consumers of type y demanding quality z . One has X z ∈Z µ xz ≤ n x , X z ∈Z µ zy ≤ m y where the difference between the right-hand side and the left-hand side ofthese inequalities is the number of producers of type x (resp. consumersof type y ) deciding to opt out of the market. The market clearing con-dition for quality z expresses that the total quantity of good of quality z produced is equal to the total quantity consumed, that is X x ∈X µ xz = X y ∈Y µ zy (it is assumed that there is no free disposal; if free disposal is assumedthe equality is replaced by ≥ in the expression). Equilibrium prices . At equilibrium, each producer x will optimizeits production behavior given the price vector ( p z ); hence if producingquality z ′ yields strictly more profit than producing quality z , then qual-ity z will not be produced at all; that is α xz + p z < α xz ′ + p z ′ for some z ′ implies µ xz = 0. A similar condition holds for consumers.One can now state a formal definition. Definition 2.1 (Hedonic equilibrium) . Let ( p z ) z ∈Z be a price vector, µ xz a supply function, and µ zy a demand function. Then:(a) ( p, µ ) is called a hedonic equilibrium whenever the following threeconditions are all verified:(i) People counting: the number of producers of type x actually par-ticipating in the market does not exceed the total number of agents of type x , and similarly for consumers of type y . That is, for any x and y , X z µ xz ≤ n x , X z µ zy ≤ m y . (2.1) (ii) Market clearing: for any z , supply for quality z will equate de-mand, that is X x ∈X µ xz = X y ∈Y µ zy . (2.2) (iii) Rationality: no producer or consumer chooses a quality that issub-optimal. That is, given ( x, y, z, z ′ ) , then α xz + p z < α xz ′ + p z ′ implies µ xz = 0 γ zy − p z < γ z ′ y − p z ′ implies µ zy = 0 . b) If n x and m y are integer, ( p, µ ) is called an integral equilibrium whenever ( p, µ ) is a hedonic equilibrium and all the entries µ are integers. The indirect utility u x of a producer of type x and the indirect utility v y of a consumer of type y are given by u x = max z ( α xz + p z ,
0) and v y = max z (cid:0) γ zy − p z , (cid:1) .As a result, if p z is an equilibrium price, then for all x , y and z , u x ≥ α xz + p z and v y ≥ γ zy − p z , thus γ zy − v y ≤ p z ≤ u x − α xz .Therefore: Proposition 2.1.
For a given optimal solution u and v , the set of equi-librium prices are the prices p z such that p max z ≥ p z ≥ p min z . (2.3) where p min z = max y (cid:0) γ zy − v y (cid:1) and p max z = min x ( u x − α xz ) . (2.4)As a result, u x + v y ≥ α xz + γ zy , hence u x + v y ≥ max z (cid:0) α xz + γ zy (cid:1) , (2.5)thus, as observed by Chiappori, McCann and Nesheim (2010), u and v are the stable payoffs of the assignment game in transferable utilitywith surplus Φ xy = max z (cid:0) α xz + γ zy (cid:1) . In the next paragraph, we shall gobeyond this equivalence through a reformulation of the hedonic model asa network flow problem. Interestingly, as understood by Maurice Queyranne, the hedonic equi-librium problem can be reformulated as a network flow problem. Thisreformulation is of particular interest since, as we show below, it helps usestablish the existence of a hedonic equilibrium and provides the build-ing blocks to compute an equilibrium. While the present exposition is asself-contained as possible, a good reference for network flow problems isAhuja, Magnanti and Orlin (1993).
The network . Define a set of nodes by N = X ∪ Z ∪ Y , and a setof arcs A which is a subset of N × N and is such that if ww ′ ∈ A , then w ′ w / ∈ A . Here, the set of arcs is A = ( X × Z ) ∪ ( Z × Y ).5 vector is defined as an element of R A . Here, we introduce thefollowing direct surplus vector φ ww ′ : = α xz if w = x and w ′ = z (2.6a) φ ww ′ : = γ zy if w = z and w ′ = y. (2.6b)For two nodes w and w ′ , a path from w to w ′ is a chain( w w ) , ( w w ) , ..., ( w T − w T − ) , ( w T − w T )such that w i w i +1 ∈ A for each i , w = w and w T = w ′ . T is the length ofthe path. Here, the only nontrivial paths are of length 2 and are of theform ( xz ) , ( zy ) where x ∈ X , z ∈ Z and y ∈ Y .For two nodes w and w ′ , we define the reduced surplus , or indirectsurplus as the surplus associated with the optimal path from w to w ′ .Here, for x ∈ X , y ∈ Y , the indirect suplus Φ xy of producer x andconsumer y is Φ xy := max z ∈Z (cid:0) α xz + γ zy (cid:1) . (2.7)For w ∈ N , we let N w be the algebraic quantity of mass leaving thenetwork at w . Hence N w is the flow of mass being consumed ( N w > N w <
0) at w . The nodes such that N w < N w = 0and N w >
0) are called the source nodes, whose set is denoted S (resp.intermediate nodes I and target nodes T ). Here, for x ∈ X , y ∈ Y , and z ∈ Z , we set N x := − n x , N y := m y , N z := 0 (2.8)so that the set of source nodes is S := X , the set of intermediate nodesis I := Z , and the set of target nodes is T := Y . Gradient, flow . We define a potential as an element of R N . Wedefine the gradient matrix as the matrix ∇ of general term ∇ aw , a ∈ A , w ∈ N such that ∇ aw = − a = ww ′ for some w ′ ∈ N , ∇ aw = 1 if a = w ′ w for some w ′ ∈ N ,so that, for a potential f ∈ R N , ∇ f is the vector such that for a = ww ′ ∈A , one has ( ∇ f ) ww ′ = f w ′ − f w . Here, set the potential of surpluses U as U x := − u x , U z := − p z , U y := v y , (2.9)6nd ( ∇ U ) xz = u x − p z and ( ∇ U ) zy = v y + p z . (2.10)We define the divergence matrix ∇ ∗ (sometimes also called node-edge ,or incidence matrix ) as the transpose of the gradient matrix: ∇ ∗ xa := ∇ ax . As a result, for a vector v ,( ∇ ∗ v ) ww ′ = X z v zw ′ − X z v wz . A flow is a nonnegative vector µ ∈ R A + that satisfies the balance ofmass equation , that is( N − ∇ ∗ µ ) w ≥ , w ∈ S (2.11)( N − ∇ ∗ µ ) w = 0 , w ∈ I (2.12)( N − ∇ ∗ µ ) w ≤ , w ∈ T (2.13)Here, µ : (cid:0) µ xz , µ zy (cid:1) is a flow if and only if µ xz and µ zy satisfy thepeople counting and market clearing equations, that is X z µ xz ≤ n x , X z µ zy ≤ m y and X x ∈X µ xz = X y ∈Y µ zy . Maximum surplus flow . We now consider the maximum surplus flow problem ,that is max µ ∈ R A + X a ∈ A µ a φ a (2.14) s.t. µ satisfies (2.11), (2.12), (2.13),whose value coincides with the value of its dual version, that ismin U ∈ R N X w ∈N U w N w (2.15) s.t. U w ≥ , ∀ w ∈ S ∪ T∇ U ≥ φ, The node-edge matrix is usually denoted A ; our notations ∇ ∗ and terminologyare chosen to stress the analogy with the corresponding differential operators in thecontinuous case. In most physical systems, mass is conserved and the balance equation has the moreusual form of
Kirchoff ’s law ∇ ∗ µ = N . However, in the present setting, producersand consumers have an option not to participate in the market, hence ∇ ∗ µ = N isreplaced by Eqs. (2.11)-(2.13). w ∈ S ∪ T , U w > N w =( ∇ ∗ µ ) w . A standard result is that if N has only integral entries, then(2.14) has an integral solution µ .Here the solution U of (2.15) is related to the solution to the hedonicmodel by Equations (2.9), that is u x = − U x , p z = − U z , v y = U y . Using(2.10) and (2.6), ∇ U ≥ φ implies u x − p z = U z − U x ≥ φ xz = α xz and v y + p z = U y − U z ≥ φ zy = γ zy , thus, using complementary slackness onerecovers u x = max z ( α xz + p z ) + and v y = max z (cid:0) γ zy − p z (cid:1) + . Further, if n and m have only integral entries, then there is an integralsolution µ to (2.14). Therefore: Theorem 2.1 (Queyranne) . The hedonic equilibrium problem of Defi-nition 2.1 can be reformulated as a maximum surplus flow problem asdescribed above.
As announced above, this reformulation has several advantages. First,it establishes the existence of a hedonic equilibrium, and its integrality.
Theorem 2.2 (Existence) . Consider a market given by n x producers oftype x , m y consumers of type y , and where productivity of producer x isgiven by α xz , and utility of consumer y is γ zy .Then:(i) There exists a hedonic equilibrium (cid:0) p z , µ xz , µ zy (cid:1) ;(ii) (cid:0) µ xz , µ zy (cid:1) are solution to the primal problem of the expression ofthe social welfare max µ xz ,µ zy ≥ X xz µ xz α xz + X yz µ zy γ zy (2.16) X z µ xz ≤ n x and X z µ zy ≤ m y and X x µ xz = X y µ zy , while ( p z ) is obtained from the solution of the dual expression of the socialwelfare min u x ,v y ≥ p z X x n x u x + X y m y v y (2.17) u x ≥ α xz + p z and v y ≥ γ zy − p z . xpressed equivalently as min p z { X x n x max z ( α xz + p z ,
0) + X y m y max z (cid:0) γ zy − p z , (cid:1) } . (iii) If n x and m y are integral for each x and y , then µ xz and µ zy canbe taken integral. Second, on the practical side, Theorem 2.2 also has a useful con-sequence in terms of computation of the equilibrium, as shown in thefollowing corollary.
Corollary 2.1.
The equilibrium prices ( p z ) as well as the quantities µ xz , µ zy supplied at equilibrium can be determined using one of the manyminimum cost flow algorithms, see for instance Ahuja, Magnanti andOrlin (1993). Example 2.1.
Assume that there are four sellers and three buyers, eachof whom is unique among her type, and three qualities. Participation isendogenous but there is no free disposal. Assume that the technology andpreference parameters are given by ( α xz ) = and (cid:0) γ zy (cid:1) = . The indirect utilities of the buyers and the sellers are determined bylinear programming. One finds u min x = (0 0 4 0) and v max y = (8 9 10) ,and u max x = (3 0 4 0) , and v min y = (8 6 10) , and the optimal matchingwill consist in matching x with y , which produce together quality 2,and any other two remaining producers with the two other remainingconsumers, producing two units of quality of quality 3. Hence the optimalnumber of goods produced in each quality, denoted l , is given by l x = 0 , l x = 1 and l x = 2 . Making use of p min z = max y (cid:0) γ zy − v max y (cid:1) and p max z =min x ( u max x − α xz ) , one finds that if u = (0 0 4 0) and v = (8 9 10) , then p ∈ [ − , − × [ − , − × {− } . In the spirit of Galichon and Salani´e (2014), who extended the modelof Choo and Siow (2006), we are now going to introduce heterogeneities9n producers’ and consumers’ characteristics. As before, we consider theset X of observable types of producers, the set Y of observable types ofconsumers, and the set Z of qualities, and the sets X , Y and Z are finite .In the sequel, i will denote an individual producer, and j will denote anindividual consumer. The analyst observes the “observable type” x i ∈ X of producer i , and the “observable type” y j ∈ Y of consumer j . Twoproducers (resp. consumers) sharing the same observable type may differin some additional heterogeneity term that will affect their profitability(resp. utility) function. This heterogeneity is observed by the consumersbut not by the analyst. It is assumed that the quality z ∈ Z is fullyobservable by all parties and the analyst.If the price of quality z is p z , then the profit of an individual producer i selling quality z is defined as ˜ α iz + p z ∈ R ∪ {−∞} , and the utility of anindividual consumer j purchasing z is defined as ˜ γ jz − p z ∈ R ∪ {−∞} .If producer i (resp. consumer j ) does not participate in the market, shegets a surplus of ˜ α i (resp. ˜ γ j ). The tilde notation in ˜ α and ˜ γ indicatesthat these terms characterize the invididual level, which will be randomfrom the point of view of the observer. Note that the utility of agentson each side of the market still does not depend directly on the type ofthe agent with whom they match, but only indirectly via the type of thecontract. We introduce an structural assumption regarding the structure of unob-served heterogeneity.
Assumption 3.1.
Assume that the pre-transfer profitability and utilityterms have structure ˜ α iz = α x i z + ε iz and ˜ γ jz = γ y j z + η jz ˜ α i = ε i and ˜ γ j = η j where:a) The surplus shock, or unobserved heterogeneity component ε i ofall producers of a given observable characteristics x are drawn from thesame distribution P x . However, the ideas presented here extend to the continuous case, see Dupuy andGalichon (2014) for a continuous logit approach and Chernozhukov, Galichon andHenry (2014) for an approach based on multivariate quantile maps. ) The surplus shock, or unobserved heterogeneity component η j ofall consumers of a given observable characteristics y are drawn from thesame distribution Q y .c) The distributions P x and Q y have full support. Part a) and b) of this assumption are not very restrictive. They es-sentially express that the quality z is fully observed. Part c) is morerestrictive. It implies that for each type of producer or consumer, andfor any quality, some individual of this type will produce or consumethis quality. This assumption does not hold if, say, some technologicalconstraint prevents some producers from producing a given quality. Al-though this assumption is not required, and is not needed in Galichonand Salani´e (2014), it greatly simplifies the results on identification andwe will maintain it for the purposes of this paper.We will also assume that: Assumption 3.2.
There is a large number of producers and consumersof each given observable type, and each of them are price takers.
This assumption has two virtues. First, it implies that we can have astatistical description of the producers and the consumer of a given typeand we do not need to worry about sample variation. Second, it rules outany strategic behaviour by agents: the market here is assumed perfectlycompetitive.
We now investigate the social welfare, understood as the sum of the pro-ducers’ and consumers’ surpluses. We first focus on the side of producers.At equilibrium, producer i will get utility U x i z + ε iz from producing quality z , where U xz = α xz + p z . The sum of the ex-ante indirect surpluses of the producers of observ-able type x is n x G x ( U x · ), where G x ( U x · ) is the expected indirect utilityof a consumer of type x , that is G x ( U x · ) = E P x (cid:20) max z ∈Z ( U xz + ε iz , ε i ) | x i = x (cid:21) (3.1)11here the argument of G x is the |Z| -dimensional vector of ( U xz ) z ∈Z ,which is denoted U x · , and where the expectation is taken with respect tothe distribution P x of unobserved heterogeneity component ε i . We referto Galichon and Salani´e (2014) for mathematical properties of G andexamples. By the Envelope theorem, the number of producers of type x choosing quality z , denoted µ z | x , is given by µ z | x = µ xz n x = P x ( x chooses z )= ∂G x ( U x · ) ∂U xz . (3.2)This result sheds light on the equilibrium characterization problem :based on the vector of producer surpluses U , this allows to deduce theproduction patterns µ , and a similar picture holds on the consumers’side. However, the identification problem consists in recovering utilityparameters, here U x · based on the observation of producer’ choices, heresummarized by µ xz , the number of producers of observable type x whochoose to sell quality z . This requires inverting relation (3.2). To dothis, still following Galichon and Salani´e (2014), introduce the Legendre-Fenchel transform G ∗ x of G x as G ∗ x ( µ ·| x ) = max U xz X z ∈Z µ z | x U xz − G x ( U x · ) ! if X z ∈Z µ z | x ≤ ∞ otherwise.where µ ·| x is the vector of choice probabilities (cid:0) µ z | x (cid:1) z ∈Z . By the Envelopetheorem, one has U xz = ∂G ∗ x ( µ ·| x ) ∂µ z | x . (3.4)Hence U xz is identified from µ x. by equation (3.4). Galichon andSalani´e (2014) have shown that G ∗ can be very efficiently computed asthe solution to an optimal matching problem.Similarly to the producers’ side of the market, denote V zy = γ zy − p z the deterministic part of the consumer’s payoff from buying good quality z , and write V · y for the |Z| -dimensional vector with z -th component V zy .The sum of expected utilities of consumers with observable characteristics y is given by m y H y ( V · y ), where H y ( V · y ) is the expected indirect utility of12 consumer of type y , that is H y ( V · y ) = E Q y (cid:20) max z ∈Z ( V zy + η jz , η j ) | y j = y (cid:21) , and Q y is the distribution of the unobserved heterogeneity component η j for a consumer indexed by j , with observable characteristics y = y j .Hence, as in the producer’s case, we obtain identification of V zy throughthe following relation. V zy = ∂H ∗ y ( µ ·| y ) ∂µ z | y , (3.5)where H ∗ y is the convex conjugate of H y , defined by a formula similar to(3.3).Recall that the social welfare W is the sum of the producers andconsumers surpluses. We are now able to state the following result. Theorem 3.1. (i) The optimal social welfare in this economy is givenby W = min p X x ∈X n x G x ( α x · + p · ) + X y ∈Y m y H y (cid:0) γ · y − p · (cid:1) . (3.6) (ii) Alternatively, W can be expressed as W = max µ ≥ X x ∈X ,z ∈Z µ xz α xz + X y ∈Y ,z ∈Z µ zy γ zy − E ( µ ) (3.7) s.t. µ satisfies (2.1) and (2.2),where E ( µ ) is a generalized entropy function, defined by E ( µ ) = X x ∈X n x G ∗ x ( µ x. ) + X y ∈Y m y H ∗ y (cid:0) µ · y (cid:1) . (iii) Further the equilibrium (cid:0) p z , µ xz , µ zy (cid:1) is unique and is such that ( p z ) is a minimizer for (3.6) and (cid:0) µ xz , µ zy (cid:1) is a maximizer for (3.7). The terminology “generalized entropy” comes from the fact, that inthe Logit case where the utility shocks ε and η are i.i.d. and have aGumbel distribution, then E ( µ ) is a regular entropy function, namely E ( µ ) = X x ∈X , y ∈Y µ xy log µ xy n x m y + X x ∈X µ xy log µ x n x + X y ∈Y µ xy log µ y m y where µ x = n x − P z ∈Y µ xz and µ y = m y − P z ∈Y µ zy .13 .3 Identification As a result of the first order conditions in the previous theorem, themodel is exactly identified from the observation of the hedonic prices p z ,along with the production and consumption patterns µ xz and µ zy . Theorem 3.2.
The producers and consumers systematic surpluses atequilibrium U and V are identified from µ xz and µ zy by U xz = ∂G ∗ x ( µ ·| x ) ∂µ z | x and V zy = ∂H ∗ y ( µ ·| y ) ∂µ z | y . Hence α and γ are identified from µ xz , µ zy and p z by α xz = ∂G ∗ x ( µ ·| x ) ∂µ z | x − p z and γ zy = ∂H ∗ y ( µ ·| y ) ∂µ z | y + p z . In the Logit case, these formulas become α xz = log( µ xz /µ x ) − p z and γ zy = log( µ zy /µ y ) + p z , where µ x and µ y have been defined at theprevious paragraph.Note that (as it frequently is found in various situations in the econo-metrics literature), the introduction of heterogeneity has allowed to iden-tify simultaneously α xz and γ zy . When there is no heterogeneity, it is wellknown that simultaneous identification of these parameters is not pos-sible. This is due to the fact that, in the absence of heterogeneity, thesolution µ of the problem is no longer an interior point, thus many entries µ xz and µ zy are forced to be equal to zero. The results presented in this paper are applicable to many different em-pirical settings. Returning to the market for fine wines for example, theanalyst will typically have access to data about the share of consumerswith observable characteristics y purchasing wine of quality z and theshare of producers of type x selling wine of quality z . Our methodologyallows to identify the surpluses of consumers and producers from thesedata. If in addition, the price of wine of various qualities are observed,then the utility α of consumers and technology γ of producers are iden-tified as well. 14ext, consider the marriage market example. In classical models ofsorting on the marriage market, following Becker (1973) and Shapleyand Shubik (1972), the matching surplus between a man of type x anda woman of type y is Φ xy = α xy + γ xy where α and γ are the man and the woman’s surplus for being marriedto each other. However, this analysis misses the fact that the partnersin the marriage market also need to make a number of joint decisions,such as whether/when/how to raise children, where to live, how to spendtheir spare time together, etc. This has the flavour of a hedonic model.For the sake of discussion, consider (on the other extreme) a frameworkwhere the observed characteristics is, say, the date of birth of each agent,and where the only variable agents care about is, say, the date of birthof their first child. In this context, the matching surplus is nowΦ xy = sup z ( α xz + γ zy )and the methodology developed in this paper can identify the surplus ofa man born in x = 1985 to have his first child in say z = 2012 and thesurplus of a woman born in y = 1986 to have her first child in z = 2013.The required data are the shares of men and women born in a givenyear who had their first child in a given year. This example, however,is peculiar as men and women are likely to form preferences not onlyover the hedonic attribute z , i.e. the year of birth of first child, but alsoover their spouse’s attributes x and y . One therefore needs to considera model encompassing the hedonic model a la Rosen (1974) with thesorting model `a la Becker (1973). In this model, developed and studiedin Dupuy and Galichon and Zhao (2014) who apply it to the study ofmigration in China, the matching surplus isΦ xy = sup z ( α xzy + γ xzy )and this model embeds both the classical sorting model ( α xzy = α xy and γ xzy = γ xy ) and the hedonic model ( α xzy = α xz and γ xzy = γ zy ). Theempirically interesting question there is to assess which of the “sortingeffect” or “hedonic effect” is strongest.15 eferences [1] Ahuja, R., T. Magnanti, and J. Orlin (1993): Network Flows: The-ory, Algorithms and Applications , Prentice-Hall.[2] Brown, J., and H. Rosen (1982): “On the estimation of structuralhedonic price models,”
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