Matching with Transfers The Economics of Love and Marriage课件

,Matching with TransfersThe Economics of Love and Marriageby Pierre-Andre ChiapporiColumbia UniversityExcerpts by:James J. HeckmanUniversity of ChicagoEcon 350, Winter 2017,Chiappori,Matching with Transfers,2. Matching with transfers: basic notions,Chiappori,Matching with Transfers,2.1 Bilateral, one-to-one matching: common framework,Chiappori,Matching with Transfers, Consider two compact, separable metric spaces X , Y : space offemale and male characteristics Other interpretations (workers and rms, lenders andborrowers, etc.) are possible. The corresponding vectors of characteristics fully describe theagents; i.e., for any x X , two women with the same vector ofcharacteristics x are perfect substitutes for the matching game(and similarly for men).,Chiappori,Matching with Transfers, Spaces endowed with measures F and G respectively Both F ( X ) and G ( Y ) are nite. Spaces are nite dimensional; specically, X R n andY R m , endowed with a Borel measure. Spaces may be multidimensional- that is, n 1 and m 1 ingeneral.,Chiappori,Matching with Transfers, Some agents may remain unmatched. In order to capture this situation within the same globalnotation, a standard trick is to augment the spaces byincluding an isolated point in each: a dummy partner X forany unmatched man and a dummy partner Y for anyunmatched woman. Therefore, from now on we consider the spaces X := X X and Y := Y Y . The point X (resp. Y ) is endowed with a mass measure equalto the total measure of Y ( X ). In particular, it is possible (although not ecient in general) toconsider a matching in which all women (all men) remainsingle, by posing that they are all matched with Y ( X ).,Chiappori,Matching with Transfers,Denition of a matching Matching: a measure h on X Y ; Intuitively, one can think of h ( x , y ) as the probability that x ismatched to y in this matching.,Chiappori,Matching with Transfers,dh ( x , y ) = F ( x ) and,x X,dh ( x , y ) = G ( y ),(1),y Y Constraint is linear .,Chiappori,Matching with Transfers, Matching is pure if the support of the measure is included inthe graph of some function - that is, if for almost all ( x , y ),h ( x , y ) = 0 unless y = ( x ). Purity prohibits randomization: if the matching is pure thenthere exists a mapping such that y = ( x ) almosteverywhere.,Chiappori,Matching with Transfers,2.2 The three types of models,Chiappori,Matching with Transfers,2.2.1 Dening the Problem: NTU; TU; ITU,Chiappori,Matching with Transfers, Two spaces X and Y , together with the correspondingmeasures. Starting with the NTU case, the matching of x X andy Y generates two utilities, one for each spousesay,u ( x , y ) and v ( x , y ). The utility, for Mrs. x , of being matched with Mr. y (as well asthat of Mr. y for being matched with Mrs. x ) is exogenouslygiven; both are part of the statement of the problem.,Chiappori,Matching with Transfers, Note also that u and v are dened over the entire set X Y ; Individual utilities are dened for any possible match, and alsofor singlehood ( u ( x , Y ) being the utility of x when single, andsimilarly for v ( X , y ). Finally, one can see that only the ordinal representation ofutilitiesi.e., the structure of individual preferences overpotential matesmatters. Replacing u ( x , y ) with u ( x , y ) , x for some mapping thatis strictly increasing in its rst argument does not change thegame.,Chiappori,Matching with Transfers, The TU case is dierent. Here, what is given is one function, say S ( x , y ), representingthe total gain generated by the matching. How this gain will be divided between the spouses isendogenous, therefore part of the solution. The TU assumption simply means that for any matched couple,what she gets and what he gets add up to the surplus theygenerate.,Chiappori,Matching with Transfers, Technically, the match denes a Pareto frontier, the equationof which is,u ( x ) + v ( y ) = S ( x , y ) .,(2), Here, u ( x ) (resp. v ( y ) is the utility she (he) obtains, and aregenerally called the (male and female) payo functions ;,Chiappori,Matching with Transfers, Normalize payo to singlehood to zero As we shall see below, there is a standard distinction betweenthe gain from marriage, dened as the maximum sum ofutilities any given pair can achieve, and the surplus , which isthe dierence between the gain generated by the matching of xand y and the sum of utilities that x and y could respectivelyachieve as singles. Posit:S ( x , Y ) = S ( X , y ) = 0 x , y (3),Chiappori,Matching with Transfers, S is the surplus generated by a marriageover and abovewhichever utility levels the spouses could achieve when single. Note, however, that this leads to a corresponding interpretationfor u and v ; namely, u ( x ) represents the additional utility xderives from marriage, over and above what she could get as asingle (and the same interpretation is obviously valid for v ( y ).,Chiappori,Matching with Transfers, Unlike the NTU case, what matters here is the cardinalrepresentation of S . Replacing S ( x , y ) with ( S ( x , y ), where is strictlyincreasing (but not necessarily ane), will in general changethe outcome of the game.,Chiappori,Matching with Transfers, The ITU case generalizes the TU framework by relaxing thelinearity assumption in (2). Again, there exists a feasibility constraint that limits the pair( u ( x ) , v ( y ) of utilities that x and y could reach if matchedtogether. This constraint is not assumed linear; rather, its equation is ofthe type:u ( x ) = H ( x , y , v ( y ) (4)where the function H is decreasing and concave in v . Equation (4) requires a particular, cardinal representation ofindividual utilities.,Chiappori,Matching with Transfers, A natural representation of these three cases uses the Paretofrontier they each generate. The Pareto set is dened as the set of pairs of utilities that agiven couple may reach (as a function of their respectivecharacteristics). The Pareto frontier is the subset of the Pareto set consistingonly of points that are not strictly pareto-dominated; it isusually represented in a two-dimensional graph, with u on thehorizontal axis and v on the vertical one.,Chiappori,Matching with Transfers,Figure 1: Shape of the Pareto frontier (solid) and set (shaded),Chiappori,Matching with Transfers, To summarize, a matching problem is dened: in the NTU case, by two sets X and Y , with their measures,and two functions u and v mapping X Y to R ; in the TU case, by two sets X and Y , with their measures, andone function S mapping X Y to R ; in the ITU case, by two sets X and Y , with their measures,and a function F mapping X Y R to R .,Chiappori,Matching with Transfers,2.2.2 Dening the solution,Chiappori,Matching with Transfers, Similar dierences appear in the denition of an equilibrium. In all cases, the basic equilibrium concept is stability . A matching is stable if: no matched individual would rather be single no pair of individuals would both like being matched togetherbetter than their current situation.,Chiappori,Matching with Transfers, It follows that any matching belonging to the core of the gamemust be stable; for otherwise a deviating couple would be ablocking coalition. Conversely, any stable match is in the core; the intuition forthis result is that, given the structure of the game, the onlymeaningful deviations are by individuals or pairs, so a matchingthat is robust to such deviations is robust to any deviation.,Chiappori,Matching with Transfers,Non Transferable Utility,Chiappori,Matching with Transfers,(5), A matching is stable if one cannot nd x , y , x , y such thath ( x , y ) 0 , h ( x , y ) 0 andu ( x , y ) u ( x , y ) , v ( x , y ) v ( x , y )with one inequality at least being strict.,Chiappori,Matching with Transfers, Case of randomization (i.e., x being matched to both y and ywith positive probability), then it must be the case thatu ( x , y ) = u ( x , y ) (if, for instance, u ( x , y ) u ( x , y ) then theprevious property is violated for x = x ). In particular, for any stable matching, let u ( x ) (resp. v ( y )denote the utility reached by Mrs. x (Mr. y ). Then it must be the case that:u ( x ) = max u ( x , z ) | v ( x , z ) v ( z ) z Yandv ( y ) = max v ( z , y ) | u ( z , y ) u ( z ) z X,Chiappori,Matching with Transfers,Transferable Utility Determining, for couples who are matched with positiveprobability, how the surplus will be shared. Dene two functions u ( x ) and v ( y ) such thatu ( x ) + v ( y ) = S ( x , y ),(6),for any ( x , y ) belonging to the support of h (i.e., such thath ( x , y ) 0, implying that x and y are matched with positiveprobability; Mathematically, the equality (6) should hold h almosteverywhere , meaning that the measure, by h , of set of couplesfor which it is not satised must be zero).,Chiappori,Matching with Transfers,Transferable Utility Here, u ( x ) (resp. v ( y ) is the utility reached by Mrs. x (Mr.y ) at this particular matching; it is often called the payo ofMrs. x (Mr. y ). (6) simply states that whenever a couple is matched withpositive probability, the sum of what they each get equals thetotal surplus they generate.,Chiappori,Matching with Transfers,Transferable Utility The matching is stable if and only if these functions satisfy:u ( x ) + v ( y ) S ( x , y ) ( x , y ) X Y,(7), To see why, assume for a moment that one can nd( x , y ) X Y such that this inequality is not satised - i.e.,such thatu ( x ) + v ( y ) S ( p , y ) , S ( p , y ) = S ( p , y ) orS ( p , y ) S ( p , y ). In the rst case, the Pareto frontier P corresponding to ( p , y )is interior to the frontier P , corresponding to ( p , y ); for anypoint on P , there is a point on P that strictly dominates it. The same is true, mutatis mutandis , in the third case; and inthe second case, P and P are equal the two Pareto frontiers(and sets) exactly coincide. What cannot happen, however, is that the two frontier intersectin some point(s) without being identical everywhere.,Chiappori,Matching with Transfers, We conclude that under TU, whatever the cardinalization,either the Pareto sets corresponding to two dierent,price-income bundles coincide, or one is totally included in theother but the frontiers cannot intersect. The conclusion is illustrated in Figure 2; under TU, both a) andb) are possible, but c) is not.,Chiappori,Matching with Transfers,Figure 2: Possible and impossible patterns for Pareto sets under TU,Chiappori,Matching with Transfers, This property, in turn, has an important implication: The choice between the two bundles ( p , y ) and ( p , y ) doesnot depend on the distribution of power within the pair,because agents are always unanimous regarding the best choice(obviously, the one that generates the largest Pareto set),provided they can freely renegotiate: they will always end up ona point located on the exterior Pareto frontier, no matter whatthe property rights may have been.,Chiappori,Matching with Transfers, In other words, there is a complete separation between thecouples choice of a price-income bundle (and generally itschoice between any two possible