Frictionless matching, directed search and frictional transfers in the marriage market

Transcription

1 Lecture at the Becker Friedman Institute Frictionless matching, directed search and frictional transfers in the marriage market Aloysius Siow October 2012

2 Two questions that an empirical model of the marriage market should answer: 1. How do the marriage distribution changes as population vectors change? 2. Can we estimate e ects of policies which a ect the gains to marriage? E.g. legalization of abortion. Two empirical problems: 1. No universal ranking for spousal characteristics (E.g. Individuals di er by age, education, religion, ethnicity). But no structure means loss of identi cation. 2. Marital output and marital transfers to clear the marriage market are usually not observed.

3 1 Identi cation problem I types of men and J types of women For each type of woman, I preference parameters to characterize her utility from each type of spouse and remaining single. For men and women, there are 2 I J preference parameters. Researcher observes the quantity of each type of women (men) in the marriage market, f j (m i ) for type j (i) women (men) (I + J exogenous observations), and the quantity of type i men married to type j women, ij (I J endogenous observations). Total number of endogenous observables are I J.

4 For I; J 2, number of endogenous observables are less than the number of unknown preference parameters. Beware of marriage matching models which claim to identify male and female preferences for spouses.

5 2 Marriage matching function Every empirical model of the marriage market generates a marriage matching function. M vector with element m i. F vector with element f j. vector of parameters. A marriage matching function is an I J matrix (M; F ; ) whose i; j element is ij :

6 X l X k X lj + ik + IX i=1 JX j=1 ij = f j 8 j (1) ij = m i 8 i (2) lj ; X ik ; ij 0 8 i; j (3) l k There are few behavioral marriage matching functions.

7 3 Marriage rate regressions For an increasing G(:), a marriage rate regression for type j women: G "P # i ij f j = X j 0 j + u j Interpretation: The marriage rate of a particular type of individual is positively related to factors which increase the welfare gain to marriage for that type. Hard to impose marriage matching restrictions on marriage rate regressions for men and women. See Angrist QJE for inconsistent behavioral estimates.

8 4 CS marriage matching function CS is an empirical transferable utility model of the marriage market with frictionless matching. Marital output of an i; j pair depends on i and j. I J marital outputs plus I + J outputs of types being single. Transferable utility models maximize the sum of marital output in the society. McFadden s (1974) additive random utility functions with idiosyncratic type I extreme value tastes for choices over spouses.

9 Our marriage matching function: ij q (mi Pk ik )(f j Pl lj ) = ij ij are the parameters of the model. They are known as the total gains to an (i; j) marriage. Non-parametric marriage matching function with spillover e ects which will t any observed marriage distribution. The CS model is just identifed with a single marriage market. Given population vectors M and F, and parameters matrix, the equilibrium marriage matching matrix is unique (Decker, Lieb, McCann, Stephens).

10 k > 0 k < 0 8j Galichon and Salanie show how one can derive marriage matching functions for a large class of additive random utility models (beyond the logit class) using stability arguments and also by a utilitarian social planner maximizing total marital output of the society. Then, identi cation rests on (i) reducing the rank of and/or (ii) having multi-market data. I discuss the Chiappori, Salanie and Weiss case below.

11 5 The CS model Let the utility of male g of type i who marries a female of type j be: V ijg = e ij ij + " ijg ; where (4) e ij : Systematic gross return to male of type i married to female of type j. ij : Equilibrium transfer made by male of type i to spouse of type j. " ijg : idiosyncratic i.i.d. random variable speci c to g.

12 1. V ijg is known as the additive random utility model. It consists of a systematic component which is common to husbands in fi; jg marriages and an idiosycratic component which is speci c to g. 2. Neither systematic return not transfer, i.e. systematic component, depends on the speci c woman chosen. 3. The idiosyncratic component, " ijg, is also independent of any particular woman of type j. I.e. no particular woman can hold the man up for additional transfers. Is this a strong assumption? The payo to g from remaining unmarried, denoted by j = 0, is: where " i0g is also an i.i.d. random variable. V i0g = e i0 + " i0g (5)

13 Individual g will choose according to: V ig = max[v i0g ; ::; V ijg ; ::; V ijg ] (6) j Before he sees his idiosyncratic realizations, the probability that g will choose type j female is: Pr(V ijg V ikg 0 8k = 0; ::; J) = Pr(e ij ij (e ik ik ) " ikg " ijg 8k = 0; ::; (7) For large m i, we can estimate Pr(V ijg V ikg 0 8k = 0; ::; J) as: Pr(V ijg V ikg 0 8k = 0; ::; J) = d ij m i (8)

14 Let " ijg be a Type 1 extreme value random variable. It has density: h(" ijg ) = e( " ijg) exp(e( " ijg)); 1 < " ijg < 1 Using (8) and (7), Pr(V ijg V ikg 0 8k = 0; ::; J) = d ij m i = exp(e ij ij ) P Jk=0 exp(e ik ik ) Obtain quasi-demand function for j type spouse relative to remaining unmarried is: ln d ij ln d i0 = e ij e i0 ij (9) = ij ij

15 The above is McFadden s celebrated logit choice model which is standard in IO discrete choice models. Derivation in CS and many econometric textbooks. Expected gain to entering marriage market and marriage rate: V i = EV ig = c + e i0 + ln(1 + X j exp( ij ij )) n i = ln(1 + X j exp( ij ij )) = ln( m i d ) i0 P j d ij m i = r i V i = c + e i0 + ln( m i d ) i0

16 Assume women have the same additive random utility functions for husbands. Then we can dervide the quasi supply function of j type spouse to fi; jg market: ln s ij ln s 0j = ij + ij (10) where ij = e ij 0j is the systematic payo to a type j woman married to a type i man relative to remaining unmarried. Marriage market clearing for all fi:jg submarkets: ij = d ij = s ij (11) CS marriage matching function obtained by summing (9) and (10), and imposing (11): ln ij ln i0 + ln 0j 2 = ij + ij 2 = ij

17 ij p i0 0j = ij CS calls ij the total gains to an fi; jg marriage. It can be estimated by p ij i0. With a single marriage market, the model is just identi ed. Note that 0j there is only I J endogenous data points and there are I J parameters,. With multiple marriage marriage markets, the model will be overidenti ed. The marriage matching function has constant returns to scale. By construction, all accounting identities of a marriage matching function are satistisfed.

18 It can t any observed marriage matching distribution. Note that ij and ij are not separately identi ed even if you have multiple market data. 6 Positive assortative matching There is positive assortative matching by many spousal attributes. I.e. Husbands and wives tend to have similar years of education, age of birth, religion, ethnicity, etc.. If there is gains to specialization in household production, should men and women match positively by labor market earnings potential?

19 There are two non-exclusive reasons for positive assortative matching in the marriage market. (1) Preference for own type in spousal choice. (2) Market equilibrium outcome when the spousal attribute in question have the same ranking for all individuals. Consider the following case of the CS model where i denotes the one dimensional schooling level of men and j denote the one dimensional schooling level of women. So di erent types of individuals are individuals with di erent levels of by schooling attainment. Sytematic marital output ij is increasing in i and j: ij = ij + ij 2 i+1;j > ij i;j+1 > ij

20 There is complementarity in spousal educational attainment. For all fi; j}: i+1;j+1 + ij > i+1;j + i;j+1 (12) Consider two men with di erent levels of education and two women with di erent levels of education. Complementarity means that the sum of systematic marital output is higher with positive assortative matching than with negative assortative matching by education. Recalling the CS marriage matching function: implies ln ij ln i0 + ln 0j 2 = ij 0 ij l ij = ln i+1;j+1 ij i+1;j i;j+1 = i+1;j+1 + ij i+1;j i;j+1 (13)

21 The left hand side of (13) is known as the local odds ratio of the marriage distribution. Equation (13) says that the local odds ratio measures the degree of local complementarity of the systematic marital output function. The above result is remarkable in the sense that the local odds ratio do not depend on the distributions of men and women by educational attainment, i.e. M and F. This validates some demographers who study marriage matching which ignores population supplies. If there is complementarity in spousal educational attainment, i.e. (12) holds, then l ij = ln i+1;j+1 ij i+1;j i;j+1 > 0 8i; j

22 Generalization to general additive random utility model (Graham; Galichon Salanie): Let H(" ijg ) and G( ijk ) be the cumulative distribution functions of " ijg and ijk respectively, where " ijg are the idiosyncratic payo to man g (woman k) of type i (j) in an fi; jg marriage. Assume H( " ijg ) and G( ijk ) are both strictly increasing on the entire real line with continuous, bounded derivatives. Then sign of i+1;j+1 + ij i+1;j i;j+1 is equal to the sign of l ij. The generalization raises the question as to what other properties of also generalizes beyond the Type 1 extreme value distribution for the additive idiosyncratic payo s to marriage.

23 CS application: Brandt, Siow and Vogel What were the marital outcomes of famine a ected cohorts? Complicated quantity and quality e ects. 1. Famine-born cohorts should enjoy an increase in their scarcity values in the marriage market. 2. Relative scarcity in the labor market should increase their wages, and therefore marital attractiveness. 3. Bad health e ects to famine born cohort reduced their labor and marital attractiveness.

24 4. Survival of pre famine cohort implies tter cohort. Paper presents two kinds of estimates: 1. Non-parametric reduced form estimates of total e ects of the famine on a ected cohorts. NO functional form assumption. Includes all general equilibrium e ects. Estimator is a rst di erence estimator. Consistency depends on suitable choice of treatment and control. 2. Use reduced form estimates to estimate CS, a non-parametric structural model.

25 Use CS to decompose total e ects into quantity versus quality effects. Quality estimates based on residual accounting. Consistency is model dependent.

26 Data: Rural Sichuan Compare marital behavior of famine-a ected cohorts in 1990 to their same age peers in In 1990, the post-famine cohort was Most women of 26 and older would have acquired their permanent marital status. Except for 26 and 27 year olds, also true for men. 2. The pre and post famine-born cohorts were between and in 1990, respectively. 3. Compare the marital behavior of 26 to 34 year olds in 1990 to their peers in 1982.

27 7 Identi cation of frictionless vs directed search (non-transferable utility) marriage market models using multimarket data CS benchmark One type of boys and one type of girls. m k is supply of boys and f k is supply of girls in school k. n and s are mean gross gains to boys for non-sexual vs sexual relationship. n and s are mean gross gains to girls for non-sexual vs sexual relationship.

28 Mean gross gain to remaining unattached is zero. n k and s k are the transfers from boys to girls for a non-sexual vs sexual relationship in school k. n k and s k are the number of non-sexual vs sexual relationships in school k. Assume utility in a relationship is additive in net gross gains and an i.i.d. extreme value Type 1 random variable. Then utility maximizing boys will generate a quasi demand for relationship r: ln r k m k n k s k = r r k (14)

29 Utility maximizing girls will generate a quasi demand for relationship r: ln r k f k n k s k = r + r k (15) Summing the two equations imply marriage market clearing: ln r q k (mk n k s k )(f k n k s k ) = r + r 2 = r Also: ln n k s k = n s (16) r is identi ed from data from one school. r and r are not separately identi ed. Data from other schools are overidentifying.

30 If ln n k s is inversely correlated with the sex ratiom, ln m k k f, then CS is wrong. k More than that, you can likely reject the CS class ala Graham above. Identifying assumption: With many types of students, this model explains the varying fractions of unattached students by type as due to voluntary choice. No one is unattached involuntarily. What if some students are unattached involuntarily. Then it becomes an identi cation problem because we cannot compute the number of voluntarily unattached.

31 ABM directed search non-transferable utility model (Arcidiacono, Beauchamp, McElroy) Individuals rst choose a market, n or s to join.! r k boys join market r where m k =! n k +!s k. r k girls join market r and f k = n k + s k. Number of type r relationships in school k: r k = (!r k ; r k ; ) (17) Individuals who do not nd a relationship will remain unattached and get a utility of zero.

32 let the expected utility of a boy g in school k entering relationship r be: EU rg k = pr k er +" rg k " rg k is a standard Type 1 extreme value random variable. Take logs and then utility maximizing boys will generate a quasi demand for relationship n relative to s : ln m k! n k! n k = ln p n k + n (ln p s k + s ) (18) p r k = r k! r k (19)

33 Then utility maximizing girls will generate a quasi supply for relationship n relative to s : ln n k f k n k = ln P n k + n (ln P s k + s ) (20) P r k = r k r k (21) Do comparative statics of ln n s w.r.t. sex ratio. In each school, ABM can observe f n k ; s k ; m k; f k g. So two endogenous observables per school. They say they can identify, r and r which are at least 5 unknowns. So you will need at least three schools.

34 No choice of remaining unattached. If students choose not to participate, then no way to estimate number of willing participants. With many type of students, this model explains the varying fractions of unattached students by type as due to search frictions. Can we nest and test the two di erent models? 8 CS with frictional transfers Let r k be the transfer which the boy makes to the girl in relationship r in market k.

35 Let the value of the transfer to the girl be r r k where r is the girl s marginal utility of income in relationship r. Note that we can extend to the case where r also depends on the type of boy and girl. Why is this plausible? The boy may say that he went to the romantic comedy movie to make her happy. She says that he enjoys it himself and therefore its not a real transfer. Put another way, its easy to see why giver and receiver values transfers di erently. This di erent valuation does raise two interesting questions. Does the competitive solution still coincide with a planner s solution ala Galichon and Salanie? Do frictions in marital transfers reduce the marriage rate? So the quasi demand by boys for girls become: ln r k m k n k s k = r r k (22)

36 utility maximizing girls will generate a quasi demand for relationship r: ln r k f k n k s k = r + r r k (23) Assuming market equilibrium and solving out r k : r ln r k (m k n k s k ) + ln r k (f k n k s k ) = r r + r = r (24) (24) is an equilibrium relationship.

37 Let n = s +, then (1 + s ) ln n k s k + ln n k (m k n k s k ) = n s ln n k s k = n s n (1 + s ) + n k (1 + s ) If > 0, when the sex ratio increases, girls have more bargaining power and we expect n k to increase which will lead to an increase in ln n k s which is the k right comparative static. Since we can estimate r, we can see if it gives sensible comparative static. If = 0, then ln n k s is invariant to changes in the sex ratio. So assuming k r = is not innocuous. This assumption can be tested.

38 For a single school, there are four unobservables, r, r and two equations in (24), one for each type of relationship. With multiple schools data, we can identify r and r : r = " ln r k 0 (f k n k s k ) (f k 0 n k 0 s k 0 ) r k # " ln r k (m k 0 n k 0 s k 0 ) (m k n k s k )r k 0 # 1 There are still lots of overidentifying restrictions with multiple schools because we only increased the number of parameters over CS by 2. There are 4 quasi demand and supply relationships. But two more additional unknowns, the transfers. So you dont buy more identi cation by stick with just the equilibrium relationships.

39 8.1 Chiappori, Salanie and Weiss (CSW) They extend CS to heteroscedastic idiosyncratic errors and derive the marriage matching function using stability arguments. Applying to our context, let the additive idioyscratic component of a particular boy g in relationship r in school k be r " r kg where "r kg is a Type 1 extreme value random variable where r > 0. Before he sees his idiosyncratic realizations, the probability that g will choose relationship r is: Pr(V r kg V r0 kg 0 8r0 = 0; n; s) = Pr( r r k r r0 r0 k r0 " r0 kg " r kg 8r0 )

40 Then Pr(V r kg kg 0 8r0 = 0; n; s) = rd k = exp(r r k )(r ) 1 m k Pr 0 exp(r0 r0 k )(r0 ) 1 V r0 Obtain quasi-demand function of boys in school k for r type relationship relative to remaining unattached is: r (ln rd k ln 0d k ) = r Similarly, let the additive idioyscratic component of a particular girl j in relationship r in school k be r " rj kg where "r kj is a Type 1 extreme value random variable where r > 0. Obtain quasi-supply function of girls in school k for r type relationship relative to remaining unattached is: r (ln rs k r k ln 0s k ) = r + r k

41 Market equilibrium implies rd k = rs k = r k and r ln r k (m k n k s k ) + r ln r k (f k n k s k ) = r + r (25) We may as well normalize r = 1. Then CSW is observationally equivalent to CS with frictional transfers. What is the di erence in interpretation between the two models? Can we distinguish between di erent kinds of frictional transfers versus di erent idiosyncratic error structures ala Galichon and Salanie? Finally, can we nest transferable and non-transferable utility models in estimation? Can we tell the CS with frictional transfers apart from ABM?

42 References Arcidiacono, Beauchamp, McElroy (2010). Terms of Endearment: An Equilibrium Model of Sex and Matching. NBER working paper. Brandt, Siow and Vogel (2009), Large shocks and small changes in the marriage market. IZA discussion paper. Chiappori, Salanie and Weiss (2011). Partner choice and the college marriage premium. Columbia working paper. Eugene Choo and Aloysius Siow (2006), Who Marries Whom and Why. JPE. Colin Decker, Elliott H. Lieb, Robert M. McCann, and Benjamin K. Stephens (2012), Unique Equilibria and Substitution E ects in a Stochastic Model of the Marriage Market JET forthcoming.

43 Alfred Galichon and Bernard Salanie (2012), Cupid s Invisible Hand: Social Surplus and Identi cation in Matching Models. Columbia working paper. Aloysius Siow (2009), Testing Becker s Theory of Positive Assortative Matching. University of Toronto working paper. Bryan Graham (2011), Econometric Methods for the Analysis of Assignment Problems in the Presence of Complementarity and Social Spillovers. Handbook of Social Economics.