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Academic Commons Search Resultsen-usPartial Identification of Finite Mixtures in Econometric Models
https://academiccommons.columbia.edu/catalog/ac:184350
Henry, Marc; Kitamura, Yuichi; Salanie, Bernard10.7916/D8959GD4Thu, 29 Jun 2017 03:42:22 +0000We consider partial identification of finite mixture models in the presence of an observable source of variation in the mixture weights that leaves component distributions unchanged, as is the case in large classes of econometric models. We first show that when the number J of component distributions is known a priori, the family of mixture models compatible with the data is a subset of a J(J1)-dimensional space. When the outcome variable is continuous, this subset is defined by linear constraints, which we characterize exactly. Our identifying assumption has testable implications, which we spell out for J=2. We also extend our results to the case when the analyst does not know the true number of component distributions and to models with discrete outcomes. Keywords. Partial identification, finite mixture models. JEL classification. C24.Economicsmh530, bs2237EconomicsArticlesIdentifying Preferences under Risk from Discrete Choices
https://academiccommons.columbia.edu/catalog/ac:183938
Chiappori, Pierre-Andre; Gandhi, Amit; Salanie, Bernard; Salanie, Francois10.7916/D8JH3K1SThu, 29 Jun 2017 03:39:06 +0000When studying consumption choices, economists have often relied on the abstraction of a representative agent. Such an agent can indeed be shown to exist and to replicate the aggregate consumers’ demand, but only under very strong (and actually quite unrealistic) assumptions (Alan P. Kirman 1992). There was also a justifiable reluctance to introduce heterogeneous preferences, as such a step might seem ad hoc when trying to explain different consumption behaviors. The rise of empirical studies based on microdata has opened new perspectives. The microeconomic importance of uninsurable risks is now recognized, and threatens the foundations of the representative agent hypothesis often used in macroeconomics. The continuing controversies surrounding the question of individual attitudes toward risk have motivated many empirical studies and observations; most of them find a bewildering diversity of individual preferences (Robert B. Barsky et al. 1997; Alma Cohen and Liran Einav 2002; Luigi Guiso and Monica Paiella 2008; Syngjoo Choi et al. 2007; Chiappori and Paiella 2007). Clearly, the identifiability of the heterogeneous distribution of preferences becomes a crucial issue in this perspective. This paper proposes conditions under which heterogeneous individual attitudes toward risk can be nonparametrically identified from individual- or market-level data on the choices made by agents over risk prospects. Our main result establishes that given data that is usually available (essentially market shares of the different risky prospects present within a market, plus the realizations of the final outcomes of agents), the analyst can recover the whole distribution of individual preferences so long as preferences can be indexed by a one-dimensional parameter that satisfies a fairly weak single-crossing condition. We then discuss several applications of our general methodology.Economics, Commercepc2167, bs2237EconomicsArticlesOn Competitive Equilibria with Asymmetric Information
https://academiccommons.columbia.edu/catalog/ac:183950
Pouyet, Jerome; Salanie, Bernard; Salanie, Francois10.7916/D8S181CWThu, 29 Jun 2017 03:38:55 +0000Asymmetric information is widely supposed to impair the functioning of markets. We show that the presence of competition may invalidate this intuition. Consider a market in which principals compete for attracting heterogeneous agents by offering contracts. Suppose contracts are exclusive, and there are constant returns to trade. When the agents' types are publicly observed under mild conditions, competitive equilibria are efficient. Efficiency is also obtained when types are privately observed, provided that principals do not directly care about the agents' private information (the private value case). Thus hidden information only matters in competitive markets if it affects common values.Economics, Communication, Commercebs2237EconomicsArticlesModeling Competition and Market Equilibrium in Insurance: Empirical Issues
https://academiccommons.columbia.edu/catalog/ac:183947
Chiappori, Pierre-Andre; Salanie, Bernard10.7916/D89022MGThu, 29 Jun 2017 03:38:53 +0000In the last decade or so, numerous papers have been devoted to empirical investigations based on contract theory. Many contributions use insurance data, and specifically files provided by firms. A typical paper would analyze the relationship between individual characteristics, the contracts chosen and the corresponding “outcome,” as measured by claims. The natural next step in this research agenda is to model empirically market equilibrium on insurance markets. Empirical models of competitive insurance markets are important in many respects. First, such models are an indispensable first step for the empirical analysis of existing markets. The discussion of optimal pricing strategies or the definition of new insurance contract would greatly benefit from such models. From a policy perspective, the design of any regulation requires estimating its likely impact on the market allocation. For instance, while a ban on specific pricing options (based, say, on gender or age) is often advocated on ethical grounds, a precise assessment of its impact on insurance markets is needed before any decision is made; and an empirical model is required to provide such an assessment.
From a purely theoretical perspective, any description of insurance markets that aims at a modicum of realism needs to come to terms with a host of complex features (horizontal differentiation of products, unobserved heterogeneity of preferences, frictions of various types), the theoretical analysis of which may be forbiddingly complex. A simple model that can be solved or at least numerically simulated may, in that case, be particularly helpful. Finally, a tractable model of insurance equilibrium can be used to run experiments, which should help us understand individual behavior in such strategic settings as competition under asymmetric information. On the other hand, modeling insurance markets raises several theoretical and empirical issues, starting, of course, with the well-known pitfalls in modeling equilibrium in contracts. The goal of the present paper is to discuss these problems and summarize the knowledge acquired so far. We successively discuss modeling of the demand side, the supply side, and the equilibrium itself.Economics, Commercepc2167, bs2237EconomicsArticlesAsymmetric information in insurance: general testable implications
https://academiccommons.columbia.edu/catalog/ac:184371
Chiappori, Pierre-Andre; Jullien, Bruno; Salanie, Bernard; Salanie, Francois10.7916/D8R78D25Thu, 29 Jun 2017 03:38:16 +0000Several recent articles on empirical contract theory and insurance have tested for a positive correlation between coverage and ex post risk, as predicted by standard models of pure adverse selection or pure moral hazard. We show here that the positive correlation property can be extended to general setups: competitive insurance markets and cases where risk aversion is public. We test our results on a French dataset. Our tests confirm that the estimated correlation is positive; they also suggest the presence of market power.Economics, Commercepc2167, bs2237EconomicsArticlesTesting for Asymmetric Information in Insurance Markets
https://academiccommons.columbia.edu/catalog/ac:184175
Chiappori, Pierre-Andre; Salanie, Bernard10.7916/D81R6PDVThu, 29 Jun 2017 03:36:36 +0000The first goal of this paper is to provide a simple and general test of the presence of asymmetric information in contractual relationships within a competitive context. We also argue that insurance data are particularly well suited to such empirical investigations. To illustrate this claim, we use data on contracts and accidents to investigate the extent of asymmetric information in the French market for automobile insurance. Using various parametric and nonparametric methods, we find no evidence for the presence of asymmetric information in this market.Economics, Commerce, Political sciencepc2167, bs2237EconomicsArticlesEstimating Preferences under Risk: The Case of Racetrack Bettors
https://academiccommons.columbia.edu/catalog/ac:184178
Jullien, Bruno; Salanie, Bernard10.7916/D8S75F6JThu, 29 Jun 2017 03:36:35 +0000In this paper we investigate the attitudes toward risk of bettors in British horse races. The model we use allows us to go beyond the expected utility framework and to explore various alternative proposals by estimating a multinomial model on a 34,443‐race data set. We find that rank‐dependent utility models do not fit the data noticeably better than expected utility models. On the other hand, cumulative prospect theory has higher explanatory power. Our preferred estimates suggest a pattern of local risk aversion similar to that proposed by Friedman and Savage.Economics, Statisticsbs2237EconomicsArticlesEarly Starters versus Late Beginners
https://academiccommons.columbia.edu/catalog/ac:184181
Chiappori, Pierre‐Andre; Salanie, Bernard; Valentin, Julie10.7916/D8HQ3XR3Thu, 29 Jun 2017 03:36:22 +0000We consider a model of wage formation characterized by two features, learning and downward rigidity. We show that wages should exhibit a late‐beginning property: when one controls for the wage at date t, the wage at date t + 1 should be negatively correlated with the wage at date 6–1. We test this property on a sample of about 1,000 executives of a French state‐owned firm whose careers we observe for 15 years. This organization exhibits the features that charecterize internal labor markets; in particular, careers consist of sequences of discrete promotions, a fact that generates specific econometric problems. The results confirm the prediction.Economics, Business, Labor economics, Political sciencepc2167, bs2237EconomicsArticlesPrélèvements et transferts sociaux: une analyse descriptive des incitations financières au travail
https://academiccommons.columbia.edu/catalog/ac:184353
Laroque, Guy; Salanie, Bernard10.7916/D88914Q2Thu, 29 Jun 2017 03:36:21 +0000Un ensemble complexe de prélèvements et de transferts sociaux s’interpose entre la rémunération versée aux ménages et le revenu dont ils disposeront effectivement. D’un côté, cotisations sociales, impôts et taxes viennent grever ce revenu ; de l’autre, prestations sociales et allocations l’augmentent. Mais le fonctionnement de ce système a des conséquences variables sur le niveau du revenu disponible d’un ménage en fonction des caractéristiques de ce ménage (situation du conjoint, nombre d’enfants) et du niveau de ses revenus (RMI, bas salaires). Jusqu’à présent, ce fonctionnement n’était décrit qu’à travers l’analyse de cas-types. L’application de ce système à un échantillon représentatif d’une partie de la population française (près de 20 millions d’individus) permet, en plus, d’étudier la répartition des taux nets de prélèvement dans cette sous-population.
Des exercices de simulation réalisés, il ressort que ce sont les ménages ayant les revenus les plus bas qui ont les taux marginaux de prélèvement les plus hauts, ce qui peut avoir pour effet de limiter les effets des incitations financières à la reprise d’un emploi. En particulier, l’incitation financière à reprendre un emploi payé au Smic paraît faible pour nombre des chômeurs et des inactifs.Economics, Statistics, Labor economics, Social sciences--Researchbs2237EconomicsArticlesHigher-order Properties of Approximate Estimators
https://academiccommons.columbia.edu/catalog/ac:199766
Kristensen, Dennis; Salanie, Bernard10.7916/D8KK9BVXWed, 21 Jun 2017 14:27:44 +0000Many modern estimation methods in econometrics approximate an objective function, for instance, through simulation or discretization. These approximations typically affect both bias and variance of the resulting estimator. We first provide a higher-order expansion of such “approximate” estimators that takes into account the errors due to the use of approximations. We show how a Newton-Raphson adjustment can reduce the impact of approximations. Then we use our expansions to develop inferential tools that take into account approximation errors: we propose adjustments of the approximate estimator that remove its first-order bias and adjust its standard errors. These corrections apply to a class of approximate estimators that includes all known simulation-based procedures. A Monte Carlo simulation on the mixed logic model shows that our proposed adjustments can yield significant improvements at a low computational cost.Econometrics, Estimation theory, Statistics, Economics, Mathematics, Computer sciencedk2313, bs2237EconomicsReportsInference On Two-Component Mixtures Under Tail Restrictions
https://academiccommons.columbia.edu/catalog/ac:199769
Jochmans, Koen; Henry, Marc; Salanie, Bernard10.7916/D8B27VCRWed, 21 Jun 2017 14:27:41 +0000Many econometric models can be analyzed as finite mixtures. We focus on two-component mixtures and we show that they are nonparametrically point identified by a combination of an exclusion restriction and tail restrictions. Our identification analysis suggests simple closed-form estimators of the component distributions and mixing proportions, as well as a specification test. We derive their asymptotic properties using results on tail empirical processes and we present a simulation study that documents their finite-sample performance.Estimation theory--Asymptotic theory, Econometrics--Mathematical models, Economics, Mathematicsmh530, bs2237EconomicsReportsDivorce and the Duality of Marital Payoff
https://academiccommons.columbia.edu/catalog/ac:199616
Chiappori, Pierre A.; Radchenko, Natalia; Salanie, Bernard10.7916/D8X0674CWed, 21 Jun 2017 14:27:36 +0000Empirical studies on the determinants of divorce are scarce in economics. One reason is the duality of the value of marriage, which combines economic gains for which proxies can be found and non-pecuniary gains that are much harder to measure. The literature on marital stability has therefore focused on the impact of income differentials between partners, omitting shocks to the non-economic components of the value of the marriage. We fill in the gap by extending the model of marriage dissolution to account for a time-varying non-pecuniary quality of the match. To explore its importance, we exploit a unique data set from the Russia Longitudinal Monitoring Survey (RLMS) which provides both labor market outcomes of the couples and subjective well-being data. Our results suggest that the monetary and non-monetary components enter the joint surplus additively, with gender-specific marginal rates of substitution. The valuation of the monetary components also reveals gender asymmetry, which we link to differences in remarriage prospects.Divorce--Economic aspects, Equality, Marriage--Economic aspects, Economicspc2167, bs2237EconomicsReportsIdentification in Separable Matching with Observed Transfers
https://academiccommons.columbia.edu/catalog/ac:188774
Salanie, Bernard10.7916/D847498HWed, 21 Jun 2017 13:58:08 +0000Imposing a separability assumption on the joint surplus in transferable utility matching models has proved very useful in empirical work. Yet when only “who matches whom” is observed, the distributions of unobserved heterogeneity cannot be identified separately. This note derives the distribution of equiilibrium transfers and shows that if the distribution of transfers within cells is observed, the distribution of heterogeneity can often be recovered, separability can be tested, and complementarities in surplus inferred.Economics, Mathematicsbs2237EconomicsReportsIdentifying Effects of Multivalued Treatments
https://academiccommons.columbia.edu/catalog/ac:191654
Lee, Sobkae; Salanie, Bernard10.7916/D8N58M1KWed, 21 Jun 2017 13:58:05 +0000Multivalued treatment models have only been studied so far under restrictive assumptions: ordered choice, or more recently unordered monotonicity. We show how marginal treatment effects can be identified in a more general class of models. Our results rely on two main assumptions: treatment assignment must be a measurable function of threshold-crossing rules; and enough continuous instruments must be available. On the other hand, we do not require any kind of monotonicity condition. We illustrate our approach on several commonly used models; and we also discuss the identification power of discrete instruments.Economics, Mathematical, Distribution (Probability theory), Economics--Mathematical models, Economicsbs2237EconomicsReportsHigher-order Properties of Approximate Estimators
https://academiccommons.columbia.edu/catalog/ac:188409
Kristensen, Dennis; Salanie, Bernard10.7916/D89886BKWed, 21 Jun 2017 13:58:02 +0000Many modern estimation methods in econometrics approximate an objective function, for instance, through simulation or discretization. These approximations typically affect both bias and variance of the resulting estimator. We first provide a higher-order expansion of such "approximate" estimators that takes into account the errors due to the use of approximations. We show how a Newton-Raphson adjustment can reduce the impact of approximations. Then we use our expansions to develop inferential tools that take into account approximation errors: we propose adjustments of the approximate estimator that remove its first-order bias and adjust its standard errors. These corrections apply to a class of approximate estimators that includes all known simulation-based procedures. A Monte Carlo simulation on the mixed logit model shows that our proposed adjustments can yield spectacular improvements at a low computational cost.Statistics, Economics, Mathematics, Computer sciencedk2313, bs2237EconomicsReportsCupid’s Invisible Hand: Social Surplus and Identification in Matching Models
https://academiccommons.columbia.edu/catalog/ac:186702
Galichon, Alfred; Salanie, Bernard10.7916/D8S181NGWed, 21 Jun 2017 13:57:57 +0000We investigate a model of one-to-one matching with transferable utility when some of the characteristics of the players are unobservable to the analyst. We allow for a wide class of distributions of unobserved heterogeneity, subject only to a separability assumption that generalizes Choo and Siow (2006). We first show that the stable matching maximizes a social gain function that trades off exploiting complementarities in observable characteristic sand matching on unobserved characteristics. We use this result to derive simple closed-form formulæ that identify the joint surplus in every possible match and the equilibrium utilities of all participants, given any known distribution of unobserved heterogeneity. If transfers are observed, then the pre-transfer utilities of both partners are also identified. We discuss computational issues and provide an algorithm that is extremely efficient in important instances. Finally, we present two estimators of the joint surplus and we revisit Choo and Siow’s empirical application to illustrate the potential of our more general approach.Economicsbs2237EconomicsReportsThe Econometrics of Matching Models
https://academiccommons.columbia.edu/catalog/ac:186690
Chiappori, Pierre A.; Salanie, Bernard10.7916/D8T152SVWed, 21 Jun 2017 13:57:50 +0000In October 2012 the Nobel prize was attributed to Al Roth and Lloyd Shapley for their work on matching. Both the seminal Gale-Shapley (1962) paper and most of Roth’s work were concerned with allocation mechanisms when prices or other transfers cannot be used—what we will call non-transferable utility (NTU) in this survey. Gale and Shapley used college admissions, marriage, and roommate assignments as examples; and Roth’s fundamental work in market design has led to major improvements in the National Resident Matching Program (Roth and Peranson 1999) and to the creation of a mechanism for kidney exchange (Roth, Sönmez and Ünver 2004.) The resulting insights have been applied to a host of issues, including the allocation of students to schools, the marriage market with unbalanced gender distributions, the role of marital prospects in human capital investment decisions, the social impact of improved birth control technologies and many others. The econometrics of matching models have recently been reconsidered, from different and equally innovative perspectives. The goal of the present project will be to survey these methodological advances. We shall describe the main difficulties at stake, the various answers provided so far, and the issues that remain open.Economicspc2167, bs2237EconomicsReportsOn Human Capital and Team Stability
https://academiccommons.columbia.edu/catalog/ac:154183
Chiappori, Pierre A.; Galichon, Alfred; Salanie, Bernard10.7916/D80Z7BH6Mon, 19 Jun 2017 16:04:10 +0000In many economic contexts, agents from a same population team up to better exploit their human capital. In such contexts (often called “roommate matching problems”), stable matchings may fail to exist even when utility is transferable. We show that when each individual has a close substitute, a stable matching can be implemented with minimal policy intervention. Our results shed light on the stability of partnerships on the labor market. Moreover, they imply that the tools crafted in empirical studies of the marriage problem can easily be adapted to many roommate problems.Human capital, Matching theory, Economicspc2167, bs2237EconomicsReportsFrom Aggregate Betting Data to Individual Risk Preferences
https://academiccommons.columbia.edu/catalog/ac:154180
Chiappori, Pierre A.; Salanie, Bernard; Salanie, Francois; Gandhi, Amit10.7916/D88G8TXDMon, 19 Jun 2017 16:04:08 +0000As a textbook model of contingent markets, horse races are an attractive environment to study the attitudes towards risk of bettors. We innovate on the literature by explicitly considering heterogeneous bettors and allowing for very general risk preferences, including non-expected utility. We build on a standard single-crossing condition on preferences to derive testable implications; and we show how parimutuel data allow us to uniquely identify the distribution of preferences among the population of bettors. We then estimate the model on data from US races. Within the expected utility class, the most usual specfications (CARA and CRRA) fit the data very badly. Our results show evidence for both heterogeneity and nonlinear probability weighting.Economicspc2167, bs2237EconomicsReportsCupid's Invisible Hand: Social Surplus and Identification in Matching Models
https://academiccommons.columbia.edu/catalog/ac:133162
Salanie, Bernard; Galichon, Alfred10.7916/D8B2835BFri, 16 Jun 2017 17:07:06 +0000We investigate a matching game with transferable utility when some of the characteristics of the players are unobservable to the analyst. We allow for a wide class of distributions of unobserved heterogeneity, subject only to a separability assumption that generalizes Choo and Siow (2006). We first show that the stable matching maximizes a social gain function that trades of two terms. The first term is simply the average surplus due to the observable characteristics; and the second one can be interpreted as a generalized entropy function that reflects the impact of the unobserved characteristics. We use this result to derive simple closed-form formulÃ¦ that identify the joint surplus in every possible match and the equilibrium utilities of all participants, given any known distribution of unobserved heterogeneity. Moreover, we show that if transfers are observed, then the pre-transfer utilities of both partners are also identified. We conclude by discussing some empirical approaches suggested by these results for the study of marriage markets, hedonic prices, and the market for CEOs.Economicsbs2237EconomicsReportsPartner Choice and the Marital College Premium
https://academiccommons.columbia.edu/catalog/ac:133169
Chiappori, Pierre A.; Salanie, Bernard; Weiss, Yoram10.7916/D8QF91RQFri, 16 Jun 2017 17:06:46 +0000Several theoretical contributions have argued that the returns to schooling within marriage play a crucial role for human capital investments. Our paper quantifies the evolution of these returns over the last decades. We consider a frictionless matching framework á la Becker-Shapley-Shubik, in which the gain generated by a match between two individuals is the sum of a systematic effect that only depends on the spouses' education classes and a match-specific term that we treat as random; following Choo and Siow (2006), we assume the latter component has an additively separable structure. We derive a complete, theoretical characterization of the model. We show that if the supermodularity of the surplus function is invariant over time and errors have extreme value distributions with time-invariant but education-dependent variances, the model is overidentified. We apply our method to US data on individuals born between 1943 and 1972. Our model fits the data very closely; moreover, we find that the deterministic part of the surplus is indeed supermodular and that, in line with theoretical predictions, the "marital college premium" has increased for women but not for men over the period.Economics, Educationpc2167, bs2237EconomicsReportsIdentifying Finite Mixtures in Econometric Models
https://academiccommons.columbia.edu/catalog/ac:128353
Salanie, Bernard; Henry, Marc; Kitamura, Yuichi10.7916/D8TH8ZXCMon, 12 Jun 2017 20:49:40 +0000Mixtures of distributions are present in many econometric models, such as models with unobserved heterogeneity. It is thus crucial to have a general approach to identify them nonparametrically. Yet the literature so far only contains isolated examples, applied to specific models. We derive the identifying implications of a conditional independence assumption in finite mixture models. It applies for instance to models with unobserved heterogeneity, regime switching models, and models with mismeasured discrete regressors. Under this assumption, we derive sharp bounds on the mixture weights and components. For models with two mixture components, we show that if in addition the components behave differently in the tails of their distributions, then components and weights are fully nonparametrically identified. We apply our findings to the nonparametric identification and estimation of outcome distributions with a misclassified binary regressor. This provides a simple estimator that does not require instrumental variables, auxiliary data, symmetric error distributions or other shape restrictions.Economicsbs2237, mh530EconomicsReportsHigher Order Improvements for Approximate Estimators
https://academiccommons.columbia.edu/catalog/ac:127318
Kristensen, Dennis; Salanie, Bernard10.7916/D83X8JV4Mon, 12 Jun 2017 20:49:30 +0000Many modern estimation methods in econometrics approximate an objective function, through simulation or discretization for instance. The resulting "approximate" estimator is often biased; and it always incurs an efficiency loss. We here propose three methods to improve the properties of such approximate estimators at a low computational cost. The first two methods correct the objective function so as to remove the leading term of the bias due to the approximation. One variant provides an analytical bias adjustment, but it only works for estimators based on stochastic approximators, such as simulation-based estimators. Our second bias correction is based on ideas from the resampling literature; it eliminates the leading bias term for non-stochastic as well as stochastic approximators. Finally, we propose an iterative procedure where we use Newton-Raphson (NR) iterations based on a much finer degree of approximation. The NR step removes some or all of the additional bias and variance of the initial approximate estimator. A Monte Carlo simulation on the mixed logit model shows that noticeable improvements can be obtained rather cheaply.Economicsdk2313, bs2237EconomicsReportsMatching with Trade-offs: Preferences over Competing Characteristics
https://academiccommons.columbia.edu/catalog/ac:127315
Galichon, Alfred; Salanie, Bernard10.7916/D8H70T2XMon, 12 Jun 2017 20:49:27 +0000We investigate in this paper the theory and econometrics of optimal matchings with competing criteria. The surplus from a marriage match, for instance, may depend both on the incomes and on the educations of the partners, as well as on characteristics that the analyst does not observe. The social optimum must therefore trade off matching on incomes and matching on educations. Given a flexible specification of the surplus function, we characterize under mild assumptions the properties of the set of feasible matchings and of the socially optimal matching. Then we show how data on the covariation of the types of the partners in observed matches can be used to estimate the parameters that define social preferences over matches. We provide both nonparametric and parametric procedures that are very easy to use in applications.Economicsbs2237EconomicsReportsDoes Fertility Respond to Financial Incentives?
https://academiccommons.columbia.edu/catalog/ac:125201
Laroque, Guy; Salanie, Bernard10.7916/D8G166PFFri, 02 Jun 2017 13:03:26 +0000There has been little empirical work evaluating the sensitivity of fertility to financial incentives at the household level. We put forward an identification strategy that relies on the fact that variation of wages induces variation in benefits and tax credits among "comparable" households. We implement this approach by estimating a discrete choice model of female participation and fertility, using individual data from the French Labor Force Survey and a fairly detailed representation of the French tax-benefit system. Our results suggest that financial incentives play a notable role in determining fertility decisions in France, both for the first and for the third child. As an example, an unconditional child benefit with a direct cost of 0.3% of GDP might raise total fertility by about 0.3 point.Managerial economics, Economicsbs2237EconomicsReports