We propose a novel nonparametric method to empirically identify economies of scale in multiperson household consumption. We assume consumption technologies that define the public and private nature of expenditures through Barten scales. Our method (solely) exploits preference information revealed by a cross-section of household observations while accounting for fully unobserved preference heterogeneity. An application to data drawn from the US Panel Study of Income Dynamics shows that the method yields informative results on scale economies and intrahousehold allocation patterns. In addition, it allows us to define individual compensation schemes required to preserve the same consumption level in case of marriage dissolution or spousal death.
A DEFINING characteristic of multiperson households is that some goods are partly or completely publicly consumed, which gives rise to economies of scale. Motivating examples include housing, transportation, and commodities produced by household work. The level of these scale economies generally depends on both the household technology, which defines the (public versus private) nature of goods, and the individual preferences of household members, which define the allocation of household expenditures to the different goods. Understanding the nature of scale economies allows for addressing a variety of questions on interpersonal and interhousehold comparisons of well-being (Chiappori & Meghir, 2014; Chiappori, 2016). For example, what are the consumption shares of husbands and wives in alternative household types? What is the income compensation a women should receive to guarantee the same material well-being after her husband passes away? How should this compensation vary with the number of dependent children? How relevant are scale economies for the assessment of inequality and poverty at the level of individual household members?
In this paper, we propose a structural method to empirically identify economies of scale in household consumption. Our method recovers the consumption technology by solely exploiting preference information revealed by households' consumption behavior. We assume a household consumption model that has three main components. First, we follow Chiappori (1988, 1992) by assuming collective households that consist of individuals with heterogeneous preferences, who reach Pareto-efficient intrahousehold allocations. Second, we adopt the framework of Browning, Chiappori, and Lewbel (2013) and use Barten scales to define the public versus private nature of the goods consumed by the household (see also Barten, 1964, and Muellbauer, 1977). Finally, we exploit marriage market implications to identify households' scale economies. Particularly, we use stability of observed marriages as our key identifying assumption. Our empirical application will show that assuming marital stability significantly benefits the identification results. In this respect, our analysis fits within the economics perspective on marriage that was initiated by Becker (1973, 1974) and Becker, Landes, and Michael (1977). These authors argue that individuals behave as rational utility maximizers when choosing their partners on the marriage market. We exploit this argument empirically and use the observed marital decisions to learn about underlying individual preferences, household technologies, and intrahousehold allocations, while explicitly accounting for economies of scale.
We extend the revealed preference methodology recently developed by Cherchye, Demuynck, De Rock, and Vermeulen (2017), who derived the testable implications of stable marriage for observed household consumption patterns. They showed that these testable implications allow for identifying the within-household decision structure that underlies the observed household consumption behavior. An important difference between our study and the original one is that these authors assumed the public or private nature of goods to be known a priori to the empirical analyst. As an implication, their methodology cannot fully assess the impact of scale economies on the welfare-related questions mentioned above. By contrast, our method will define the nature of goods a posteriori by empirically identifying good-specific Barten scales under the maintained assumption of stable marriage. It will account for the possibility that some goods are partly privately and partly publicly consumed. The basic intuition behind our identification strategy is that higher economies of scale imply more gains from marriage, which leads to more competition in the marriage market. Conversely, lower economies of scale lead to fewer gains from marriage, which reduces the incentive to be married. By assuming marital stability for the observed households we can define informative upper and lower bounds on good-specific Barten scales for different households. This “set” identifies the level of household-specific economies of scale.
Our identification method has a number of additional features that are worth emphasizing. First, it does not impose any functional structure on the within-household decision process, which makes it intrinsically nonparametric. Next, the method allows for fully unobserved preference heterogeneity across individuals in different households, and requires only a single consumption observation per household. Interestingly, we will show that we do obtain informative results on households' scale economies even under these minimalistic priors. In their empirical analysis, Browning et al. (2013) assumed that men and women in households have the same preferences as single men and women. We show that it is possible to obtain informative identification results without that assumption by exploiting the testable implications of marriage stability. We believe that this is an attractive finding, as Browning et al.'s (2013) assumption of preference similarity is often regarded to be overly restrictive.1
Our methodological extension of Cherchye, Demuynck, De Rock, and Vermeulen (2017) is particularly relevant from a practical perspective. Admittedly, some data sets do contain fairly detailed information on the public and private nature of household consumption—for example, the Danish, Dutch and Japanese data that have been studied by, respectively, Bonke and Browning (2009), Cherchye, De Rock, and Vermeulen (2012b), and Lise and Yamada (2019). However, the more frequently used data sets (like the one of our own application) typically do not contain this information. Moreover, the public and private nature of expenditures (e.g., on transportation and household work) is often difficult to define. This paper opens the possibility of exploiting marital stability for the identification of within-household allocation patterns in such empirical settings.
To show its practical usefulness, we apply our method to a cross-sectional household data set drawn from the 2013 wave of the US Panel Study of Income Dynamics (PSID). In this application, households allocate their full income (i.e., the sum of both spouses' maximum labor income and nonlabor income) to both spouses' leisure, two commodities produced through the spouses' household work and the consumption of a Hicksian aggregate good.2 We build on the observation that household technologies are closely related to observable household characteristics. For example, it is often argued that the presence of children has a significant impact on households' demand patterns (Browning, 1992). For our own sample of households, we find that households' consumption patterns vary substantially depending on the number of children, age, education level, and region of residence (see appendix C.3).
Our novel methodology allows investigating how these diverging consumption patterns relate to households' economies of scale and intrahousehold allocations. For example, what is the effect of children on public consumption in households? Does it matter whether the husband has a college degree? Is the pattern of intrahousehold consumption sharing different according to the region of residence or the age category? Should we model household work as fully publicly consumed or also as partly private? To meaningfully analyze these questions, we will assume that similar households (in terms of, e.g., age, education, or region of residence) operate on the same marriage market and are characterized by a homogeneous consumption technology. Our method then yields informative results on the nature of scale economies and intrahousehold allocation patterns for alternative household types. In turn, we can address the well-being questions that we mentioned above. As a specific illustration, we will discuss the possibility of computing individual compensation schemes required to preserve the same material well-being in case of marriage dissolution or spousal death. In addition, we can show the importance of explicitly accounting for scale economies and associated intrahousehold allocation patterns when assessing poverty at the individual level.
The rest of this paper unfolds as follows. Section II introduces our notation and the structural components of our household consumption model. Section III formally defines our concept of stable marriage. Section IV presents the testable implications of our model assumptions for observed household consumption patterns. Here, we will also indicate that these implications allow us to (set) identify households' economies of scale (i.e., Barten scales). Section V introduces the setup of our empirical application. Section VI presents our empirical findings regarding economies of scale for our sample of households and section VII the associated results on the intrahousehold allocation of resources. Section VIII provides some concluding discussion.
II. Household Consumption
We study households that consist of two decision makers, a men and a women . Our application consider, households that allocate their full income to spouses' leisure, household work, and consumption of a Hicksian aggregate good. In what follows, we provide more formal details on the household decision setting we have in mind. Subsequently, we introduce our concept of consumption technology (with Barten scales). Finally, we show how our setup allows us to analyze households' economies of scale and intrahousehold allocation patterns.
We assume that each individual spends his or her total time on leisure , market work , and household work . The price of time for each individual is his or her wage () from market work. The time constraint for each individual is .
Let be a -dimensional (column) vector denoting the observed aggregate consumption bundle for the pair . In our following empirical application, this vector will contain goods bought on the market (captured by a Hicksian aggregate good), as well as time spent on leisure and household production by both individuals, which implies . Each individual's time spent on household production actually represents an input and not an output that is consumed inside the household (see Becker, 1965). Under the assumption that each individual produces a different household good by means of an efficient one-input technology characterized by constant returns-to-scale; however, the individual's input value can serve as the output value. Note that this implies specialization with respect to the production of household goods rather than specialization with respect to market work versus household work (see also Pollak & Wachter, 1975, and Pollak, 2013). We return to the possibility of considering more sophisticated intrahousehold production technologies in the concluding section VIII.
Next, we let represent the (row) vector of prices faced by the pair and the (row) vectors of prices faced by and when they are single. In our application, the price of the Hicksian market good will be normalized at unity. The prices for leisure and household production will equal the observed individual wages. We assume that individuals' wages are unaffected by marital status or spousal characteristics (i.e., there is no marriage premium or penalty), which implies that they remain the same as in the current marriage when individuals become single or remarry some other potential partner.3
B. Consumption Technology
For any matched couple , the consumption bundle consists of a public part that is shared by the husband and the wife. We define these public quantities from the aggregate consumption quantities by using Barten scales. Specifically, we let denote a diagonal matrix that represents the degree of publicness for each individual good, with the th diagonal entry representing the fraction of good that is used for public consumption. If the th good is consumed entirely privately, then Similarly, if the th good is consumed entirely publicly, then . In general, all entries of the technology matrix are bounded between 0 and 1. The Barten scale is given by for each good ; by construction, its value is between 1 (full private consumption) and 2 (full public consumption).4
If the pair buys the bundle , then the public quantities can be represented as and gives the corresponding private quantities. The private consumption bundle is shared between the partners. In particular, let and denote the spouses' private shares that satisfy the adding-up constraint .
For a given consumption bundle , the household allocation is given by . So far, we did not put any restriction on the technology matrix . In our empirical application, we assume that married couples that are observationally similar are characterized by the same degree of publicness of the consumed goods. We do so by conditioning the value of on observable household characteristics. In particular, we assume that a household's consumption technology for matched couples can vary with the number of children in the household, the region of residence, and the husband's age and education level.5 As we discuss in sections VI and VII, this assumption is sufficient to obtain informative empirical results when using cross-sectional household data (containing only a single observation per individual household). In principle, if we used a panel household data set (with a time series of observations for each household), then we could account for unobserved heterogeneity of the household technologies as well. We briefly return to this point in our concluding discussion in section VIII.
C. Economies of Scale
By construction, we will have that . If everything is consumed privately (i.e., for all ), then will equal 1, which means that there are no economies of scale. At the other extreme, if all goods are consumed entirely publicly (i.e., for all ), then equals 2. If the household is characterized by both public and private consumption, then will be strictly between 1 and 2. Generally, our measure of scale economies quantifies a household's gains from sharing consumption. To take a specific example, let us assume that the measure equals 1.30 for some household. This means that the two individuals together would need more income as singles to buy exactly the same aggregate bundle as in the household.
At this point, it is useful to remark that our scale economies measure is close in spirit to the popular equivalence scale concept, which aims at quantifying the cost for a household with given size and composition to achieve the same living standard as some reference household. For instance, if a single-adult household needs dollars to reach a given standard of living and the corresponding equivalence scale for a household with two adults and one child is 2.2, then this last household needs dollars to achieve the same living standard. Similar to our scale economies concept, equivalence scales capture the basic idea that households with multiple members benefit from consumption sharing. In our example, the three-member household needs less than to reach the same standard of living as the one-member household. Equivalence scales are often used to address policy-relevant questions related to poverty, inequality, the cost of children, the income compensation for spousal death, or alimony rights, to name only a few.
However, the conceptual underpinnings of the equivalence scale concept are arguably very weak (see, e.g., Chiappori, 2016). Most notably, it implicitly assumes that households (instead of individuals) have utilities that are comparable across household types, and it ignores the importance of intrahousehold inequality. Further, by construction, it only accounts for (observable) interhousehold heterogeneity in size and composition, and so ignores other possible sources of (unobservable) heterogeneity.6 Browning et al. (2013) proposed the so-called indifference scale notion as a better-grounded alternative to assess policy questions associated with household welfare. Indifference scales define the incomes that individuals would need to be equally well off (in utility terms) when living alone as in their current (multimember) household. In contrast to equivalence scales, indifference scales acknowledge the fact that individuals (and not households) have utilities. In addition, they naturally account for the presence of intrahousehold inequality. We briefly return to the computation of indifference scales by using our nonparametric methodology in the concluding section VIII.
D. Intrahousehold Allocation
The interpretation is similar to the scale economies measure . Specifically, these RICEBs capture the fractions of household expenditures that a men (women) would need as a single to achieve the same consumption level as under marriage at the new prices (resp. . The RICEBs also describe the allocation of expenditures to the men and women in a given household. Given our particular setting, this allocation is defined by the household's economies of scale, as well as the intrahousehold sharing pattern, which essentially reflects the individuals' bargaining positions. We will illustrate the importance of these two channels when interpreting the results for and in our empirical application.
Our RICEB measures are closely related to the sharing rule concept that is frequently used in the collective household literature. The sharing rule defines the individuals' shares of total household expenditures and is often used as an indicator of individuals' within-household bargaining positions. In a setting with public goods, the within-household sharing rule will evaluate the publicly consumed quantities at individual-specific Lindahl prices to define the individuals' expenditure shares (see, e.g., Cherchye, De Rock, & Vermeulen, 2011). These Lindahl prices reflect the individuals' willingness to pay for public consumption and must add up to the observed market prices. This implies a main difference with the RICEB measures and in equations (3) and (4), which use the market prices and for the public quantities . This last feature effectively makes that our RICEB measures give the expenditures that individuals would face as singles (for the prices and ) when consuming the same bundles as in their current marriage. The use of market prices (instead of Lindahl prices) also implies that these measures naturally capture scale economies that are related to marriage: the sum of the individual RICEBs and may well exceed 1, indicating that the total value of consumption (summed over the two household members and evaluated at the market prices and ) exceeds the household expenditures .
A final issue pertains to the prices and to be used for the absent spouse's household work in case one becomes a single. In our application, we will assume that exactly the same public good produced by the absent spouse will be bought on the market. Given the earlier discussed production technology, this implies that we can use this spouse's wage as the price for the household work that serves as an input in the production process. Sometimes other options may be available, though. More detailed information on the time use of spouses, for example, would make it possible to use market prices for marketable commodities like formal child care, cleaning the house, or gardening. Our current data set contains only an aggregate of the spouses' time spent on household work, which rules out such an approach. Still, as a robustness check, we have redone our following empirical analysis by using the sample averages of female and male wages (instead of the current spousal wages) to define and . Reassuringly, this extra analysis yielded the same empirical conclusions (see appendix E.1).
III. Marital Stability
This study considers only married couples: for any and for any (which implies ). In principle, it is relatively easy to include singles in our framework. However, our following application will show that our method gives informative results even if we do not use information on singles. Therefore, and also to simplify our exposition, we have chosen to only use couples' information in our main analysis. As a robustness check, in appendix E.5, we show the results of an additional analysis based on a data set that also includes singles. This resulted in a qualitatively similar empirical analysis.
For a given matching function , the set represents the collection of household allocations defined over all matched pairs. In what follows, we will say that a matching allocationis stable if it is Pareto efficient, individually rational, and has no blocking pairs. Essentially, this means that the allocation belongs to the core of all possible marriage allocations. To formally define our stability criteria, we will assume that every individual is endowed with a utility function . These utility functions are individual-specific (i.e., fully unobserved heterogeneity) and egoistic in the sense that each individual is assumed to get utility only from his or her own private and public consumption. We further assume that the utility functions for all individuals are nonnegative, increasing, continuous, and concave. Finally, we make the technical assumption that (with the public consumption), that is, each individual needs at least some private consumption (e.g., food) to achieve a positive utility level.
A. Pareto Efficiency
B. Individual Rationality
C. No Blocking Pairs
IV. Revealed Preference Conditions
In what follows, we first specify the type of data set that we will use in our following application, and we define what we mean by “rationalizability” by a stable matching. Subsequently, we present our testable revealed preference conditions for a data set to be rationalizable. We also show that these conditions can be relaxed by accounting for divorce costs (e.g., representing unobserved aspects of match quality or irrational behavior). Our conditions are linear in unknowns, which makes them easy to use in practice. Finally, we indicate how our conditions enable (set) identification of households' economies of scale and intrahousehold allocation patterns.
A. Rationalizability by a Stable Matching
We observe a data set on men and women that contains the following information:
The matching function
The consumption bundles for all matched couples
The prices for all and
Total nonlabor incomes for all matched couples
Obviously, to verify if a given marriage allocation is stable, the analyst needs to know who is married to whom (). Next, we observe the aggregate consumption demand () of the matched pairs but not the associated intrahousehold allocation of this consumption. Similarly, we do not observe the aggregate consumption demand of the unmatched pairs (with ). In our following conditions, we treat the vector for as an unknown variable representing the potential consumption of (). By contrast, we observe the prices for all decision situations, that is, for observed marriages but also for unobserved singles and unobserved potential couples. We recall from section II that the quantity vectors contain a Hicksian aggregate good and time spent on leisure as well as on household production, and, correspondingly, the price vectors contain the price of the aggregate good (which we normalize at unity) and individual wages. Finally, for the observed/married couples , we use a consumption-based measure of total nonlabor income: nonlabor income equals reported consumption expenditures minus full income. Then we treat individual nonlabor incomes as unknowns that are subject to the restriction that they must add up to the observed (consumption-based) total nonlabor income, , and, for a given specification of the individual incomes and , we obtain the full incomes , and as in equation (1).
We say that the data set is rationalizable by a stable matching if there exist nonlabor incomes and (defining , , and ), utility functions and , a diagonal matrix (with diagonal entries ) and individual quantities , with such that the matching allocation is stable. As discussed before, stability means that we can represent the observed consumption and marriage behavior as Pareto efficient, individually rational, and without blocking pairs for some specification of the individual utilities and household technologies (i.e., Barten scales).
B. Testable Implications
We can now define testable conditions for rationalizability by a stable matching. The main innovative feature of our current setup is that we consider that the public consumption of the matched couples could be represented by an unknown technology matrix defining the public versus private nature of household expenditures. As motivated in section I, this specific extension is particularly attractive from a practical perspective, as it opens the possibility of studying data sets in which the public and private nature of household expenditures are unknown or hard to define.
Our testable conditions use only information that is contained in the data set and do not require any (nonverifiable) functional structure on the within-household decision process, which minimizes the risk of specification error. In addition, the conditions avoid any preference homogeneity assumption for individuals in different households. Moreover, they use only a single consumption observation per household, which makes them applicable to cross-sectional household data sets. The conditions are stated in the next result. See appendix A for the proof.
The data set is rationalizable by a stable matching only if there exists a diagonal matrix with diagonal entries (for all ) and, for each matched pair :
Nonlabor incomes with
And individual quantities with
that meet, for all men and women ,
- The individual rationality restrictions
- and the no blocking pair restrictions
Interestingly, the testable implications in proposition 1 are linear in the unknown technology matrix , the nonlabor incomes and , and the individual quantities and . This makes it easy to verify them in practice. The explanation of the different conditions is as follows. First, the proposition requires the construction of a technology matrix of which the diagonal entries capture the degree of publicness in each consumption good, ranging from entirely private () to entirely public (). Next, conditions a and b specify the adding up restrictions for matched couples that we discussed above, which pertain to the unknown individual nonlabor incomes and privately consumed quantities.
Further, conditions i and ii impose the individual rationality and no blocking pair restrictions that apply to a stable marriage allocation. They have intuitive revealed preference interpretations. More specifically, condition i requires, for each individual men and women, that the total income and prices faced under single status ( and for men and and for women ) cannot afford a bundle that is strictly more expensive than the one consumed under the current marriage ( for and for ). Indeed, if this condition were not satisfied for some individual, then he or she would be strictly better off as a single. Similarly, condition ii imposes, for each potentially blocking (i.e., currently unmatched) pair , that the total income () and prices () cannot afford a bundle that is strictly more expensive than the sum of the individuals' private bundles () and the public bundle that is composed of the highest quantities consumed in the current marriages (which is defined as ).9 Intuitively, if this condition is not met, then men and women can allocate their joint income so that they are both better off (with at least one strictly better off) than with their current partners.
C. Divorce Costs
So far, we have assumed that marriage decisions are driven only by material payoffs captured by the individual consumption bundles for men and for women Implicitly, we assumed that individuals are perfectly rational in their consumption and marriage behavior and that there are no gains from marriage originating from unobserved match quality (such as love or companionship). We have also abstracted from frictions on the marriage market and costs associated with marriage formation and dissolution.
In our empirical application, we follow Cherchye, Demuynck, De Rock, and Vermeulen (2017) and include the possibility that these different aspects may give rise to costs of divorce, which means that the observed consumption behavior (captured by the observed data set ) may violate the strict rationality requirements in proposition 1. In particular, we make use of “stability indices” to weaken these strict constraints. Intuitively, these indices represent income losses associated with the different exit options from marriage (becoming single or remarrying a different partner). We represent these postdivorce losses as percentages of potential labor incomes.10 Alternatively, these indices can also be interpreted as quantifying how close the observed household behavior is to “exactly stable” behavior as characterized by the conditions in proposition 1; they allow us to account for deviations from such exact stability in the empirical analysis. In that sense, the stability indices are similar in spirit to the nonparametric “goodness-of-fit” indices (interchangeably referred to as critical cost efficiency indices and Afriat indices in the literature) that Afriat (1972, 1973) and Varian (1990) proposed in the context of revealed preference analysis.
We also add the restriction Generally a lower stability index corresponds to a greater income loss associated with a particular option to exit marriage.
D. Set Identification
To address identification, we first need to check whether the data set satisfies the testable restrictions in proposition 1. We do so by solving the already discussed linear program with objective (7). If this program yields a solution value of 1 for all stability indices , , and , we conclude that the observed consumption behavior satisfies our strict requirements for marital stability. In the other case, the program calculates minimal divorce costs (captured by the indices , , and ) that are required to rationalize the observed behavior by a stable matching allocation. In our application, we will use the computed values of , , and to rescale the original potential labor incomes (, , and ), which will define an adjusted data set that is rationalizable by a stable matching. For this new data set, we can address alternative identification questions by starting from our rationalizability conditions.
In the following sections, we specifically focus on the scale economies measure in equation (2) and the associated RICEB measures in equation (3) and in (4). Particularly, we obtain “set” identification by defining upper and lower bounds for these measures subject to our maintained assumption of marital stability; these bounds define intervals of feasible values for the measures that are compatible with our rationalizability restrictions. From an operational perspective, an attractive feature of the measures , , and is that they are also linear in the unknown matrix and individual quantities and . As a result, we obtain our upper/lower bounds for these measures by maximizing/minimizing these linear functions subject to our linear rationalizability restrictions in proposition 1. This set effectively identifies the households' economies of scale and intrahousehold allocation patterns through linear programming. This set identification essentially only exploits marital stability as our key identifying assumption, without any further parametric structure for intrahousehold decision processes or homogeneity assumptions regarding individual preferences.
As a final remark, we note that the stability indices may also be seen as indicating an incentive to divorce. In that reasoning, divorce costs signal unstable marriages, which makes the assumption of marriage stability useless for the identification of intrahousehold decision processes. In our following empirical analysis, we account for this concern by performing a robustness check in which the empirical identification analysis includes only couples who do not require a divorce cost for any exit option to rationalize the observed household consumption. (See our discussion in appendixes D.1 and E.2 for more details.)
V. Empirical Application: Setup
We consider households that spend their full income (potential labor income and nonlabor income) on a Hicksian aggregate market good, time for household production and time for leisure. Our data set includes information on individuals' time use for household work and for leisure. Apps and Rees (1997) and Donni (2008) have emphasized the importance of considering home production for identifying intrahousehold allocations and conducting individual welfare analysis. In particular, ignoring time spent on household production means that all time not spent on market labor will be considered as pure leisure. In such a case, an individual with low market labor supply (e.g., a part-time working mother) will be regarded as consuming a lot of leisure, even if in fact (s)he spends a large amount of time on home production (e.g., child care). In our model, we only associate (potential) economies of scale with consumption goods that have market substitutes; these scale economies can effectively be compensated in case of spousal death or marriage dissolution. As an implication, we allow the Hicksian market good and time spent on household production to be characterized by a public component, while time spent on leisure is modeled as purely private.11
We use household data drawn from the 2013 wave of the Panel Study of Income Dynamics (PSID). The PSID data collection began in 1968 with a nationally representative sample of over 18,000 individuals living in 5,000 families in the United States. The data set contains a rich set of information on households' labor supply, income, wealth, health, and other sociodemographic variables. From 1999 onward, the panel data are supplemented by detailed information on households' consumption expenditures. The 2013 wave includes data on 9,063 households.
In our empirical analysis, we focus on couples with or without children and no other family member living in the household. Because we need wage information, we consider only households in which both spouses work at least ten hours per week on the labor market.12 After removing observations with missing information (e.g., on time use) and outliers, we end up with a sample of 1,322 households.13 In appendixes C.1 to C.3, we provide further information on our data set.
To implement the rationalizability restrictions in proposition 1, we need to define the prices and incomes that apply to the different exit options from marriage (becoming single or remarrying). In what follows, the price of individuals' time use (leisure and household work) equals their wage rate, and we will assume that wages are unaffected by marital status. This implies that we can use the observed wages as the price of own time use in any counterfactual situation. Next, for spousal household work, we use the wage rate of the potential spouse when evaluating the exit option of remarriage (in the no blocking pair restrictions) and the wage rate of the current spouse when evaluating the exit option of becoming single (in the individual rationality restrictions).14 Further, we set the price of the Hicksian market good equal to 1 in all counterfactual scenarios. Finally, we need to define the individuals' potential labor and nonlabor incomes to construct full potential incomes that correspond to the alternative postdivorce scenarios. Using our assumption that labor productivity is independent of marital status, we obtain individuals' maximal labor incomes for any exit option as total available time ((24 8) 7 112 hours per week) multiplied by their wage rates. Next, as discussed in section IV, we treat the individuals' nonlabor postdivorce incomes as unknowns in our rationalizability restrictions. Following Cherchye, Demuynck, De Rock, and Vermeulen (2017), we exclude unrealistic scenarios by imposing that individual nonlabor incomes after divorce must lie between 40% and 60% of the total nonlabor income under marriage.
B. Marriage Markets
We let household technologies vary with observable household characteristics (age, education, number of children, and region of residence) and use those same characteristics to define households' marriage markets. As an implication, while our analysis accounts for fully (unobservably) heterogeneous individual preferences (as explained before), we do consider that all potential couples on the same marriage market are characterized by a homogeneous consumption technology (defining the public versus private nature of goods). Thus, we specifically focus on marriage matchings on the basis of individuals' preferences for the public and private goods that are consumed within the households, and we build on this premise to learn about the underlying household technology from the observed matchings.
Evidently, in real life, individuals may well account for remarriage possibilities that are characterized by different technologies (for different household characteristics). In addition, they may consider repartnering with other individuals who are currently single. Including information on these additional repartnering options would increase the number of potentially blocking pairs, and this can only improve our identification analysis.15 From this perspective, our following empirical analysis adopts a conservative approach and only uses largely uncontroversial assumptions on individuals' remarriage options. We show that even this minimalistic setup leads to insightful empirical conclusions. (In appendixes C.4 to C.5, we discuss in more detail the construction of our marriage markets.)
VI. Economies of Scale
When checking the strict rationalizability conditions in proposition 1, we found that our data satisfy these conditions for 69 of the 128 marriage markets. For the remaining 59 markets, we computed the divorce costs that we need to rationalize the observed consumption and marriage behavior. As explained in section IV, for each different exit option (becoming single or remarrying), this computes a minimal divorce cost that makes the observed data set consistent with the sharp restrictions in proposition 1. These divorce costs can be interpreted in terms of unobserved aspects that drive (re)marriage decisions, such as imperfect rationality, match quality, and frictions on the marriage markets.
As we discuss in detail in appendix D.1, we only need to mildly adjust the postdivorce incomes to rationalize the observed consumption behavior in terms of a stable matching allocation. Therefore, we use the divorce costs summarized in table 12 (in appendix D.1) to construct a new data set that is rationalizable by a stable matching.16 In turn, this allows us to set identify the decision structure underlying the observed stable marriage behavior. We begin by considering the upper- and lower-bound estimates for the scale economies measure in equation (2). In doing so, we also consider the associated good-specific Barten scales (the diagonal entries of the household technology matrix ). In our application, these Barten scales capture the degree of publicness of spouses' household work and couples' consumption of market goods.
As a first step, we compare our estimated upper and lower bounds with so-called naive bounds. These naive bounds do not make use of the (theoretical) restrictions associated with the assumption that marriage markets are stable. In this respect, we remark that the sole assumption of Pareto-efficient intrahousehold allocations (without marital stability) imposes no empirical restriction on observed household consumption when allowing for fully heterogeneous individual preferences (see Cherchye, Demuynck, De Rock, and Vermeulen, 2017). More specifically, the naive bounds are defined as follows. The lower bound corresponds to a situation in which equals the 0 matrix, which means that there is no public consumption at all. By contrast, the naive upper bound complies with the other extreme scenario in which spouses' household work and market goods are entirely publicly consumed, which corresponds to a value of unity for the diagonal elements of the matrix . Note that the private consumption of leisure implies that this upper bound will in general be different from 2, which would be the upper bound in case all commodities are purely publicly consumed. In what follows, we call the bounds that we obtain by our methodology “stable” bounds, as they correspond to a stable matching allocation on the marriage market. Comparing these stable bounds with the naive bounds provides insight into the identifying power of our key identifying assumption, that is, stability of observed marriages.
The results of this comparison are summarized in table 1A.17 Columns 2 to 4 describe the bounds for that we estimate by our method, and columns 5 to 7 report on the associated naive bounds. We also give summary statistics on the percentage point differences between the (stable and naive) upper and lower bounds (see the “Difference” columns); these differences indicate the tightness of the bounds for the different households in our sample. To interpret these results, we recall that leisure is assumed to be fully privately consumed. However, as extensively discussed above, we do not impose any assumption regarding the public or private nature of the remaining expenditure categories (household work and market goods). Even under our minimalistic setup, our identification method yields informative results. Specifically, the average lower bound on equals 1.06, while the upper bound amounts to 1.18, corresponding to an average difference of only 12 percentage points. Importantly, these stable bounds are substantially tighter than the naive bounds. The naive lower bound is 1.00 by construction, and the upper bound equals 1.36 on average, which implies a difference of no less than 36 percentage points. Moreover, for 50% of the observed households, we obtain a difference of less than 3 percentage points, which is substantially tighter than for the naive bounds.
|A. Economies of Scale|
|B. Degree of Publicness|
|Housework by Female||Housework by Male||Market Good|
|A. Economies of Scale|
|B. Degree of Publicness|
|Housework by Female||Housework by Male||Market Good|
Table 1B reports on our estimates of the diagonal entries (for each good ) of the technology matrix that underlies the scale economies results in table 1. For the spouses' household work and the Hicksian market good, the “Min” columns 2, 5, and 8 correspond to the lower stable bounds in table 1; the “Max” columns 3, 6, and 9 to the upper stable bounds; and the “Avg” columns 4, 7, and 10 to the average of the Min and Max estimates. We note that the associated naive estimates of the -entries (underlying the naive bounds in table 1) trivially equal 0 for the minimum scenario and 1 for the maximum scenario, by construction.
Table 1 again shows the informative nature of the bounds that we obtain. On average, there seems to be some difference in publicness of household work by women or by men: the respective lower bounds equal 0.25 and 0.14, and the associated upper bounds amount to 0.51 and 0.38. Interestingly, our results do reveal quite some variation across households: in some households, all household work is privately consumed (the minimum value for the upper bound on equals 0), while in other households, the consumption is fully public (the maximum value for the lower bound on equals 1).
Next, we find that the average -estimate for the Hicksian market good is situated between 0.15 (lower bound) and 0.47 (upper bound), which implies that the Barten scale for market goods (defined as ) is situated between 1.15 and 1.47. These estimates are reasonably close to other estimates that have been reported in the literature (for different household samples, without leisure and using a parametric methodology), thus providing external validation for the results obtained through our novel method. For example, Browning et al. (2013) measure scale economies for Canadian households that correspond to an average Barten scale of 1.52 for market goods, and Cherchye, De Rock, and Vermeulen (2012a) compute an average Barten scale that equals 1.38 for the market consumption of Dutch elderly couples. Once more, we observe quite some heterogeneity in the -estimates across households (ranging from a minimum value for the upper bound of 0 to a maximum value for the lower bound of 0.66).
The results in table 1 show the potential of our identification method to obtain informative results, even if we make minimal assumptions regarding the data at hand. Moreover, our findings reveal some interhousehold heterogeneity in the patterns of scale economies. We investigate this further in appendix D.3 by relating the estimates summarized in table 1 to observable household characteristics. Our results reveal that public consumption increases with the number of children in the household, and that particular households situated in the Northeast region of the United States are characterized by more scale economies, while richer households generally experience less economies of scale.
VII. Intrahousehold Allocation
As explained in section II, we can also use our methodology to calculate bounds on the male and female “relative individual costs of equivalent bundles” (RICEBs) and (see equations  and ). Basically, these individual RICEBs quantify who consumes what relative to the household's full income. In what follows, we investigate these RICEBs in more detail, which will provide specific insights into intrahousehold allocation patterns. We also indicate how we can use these results to compute individual compensation schemes needed to preserve the same consumption level in case of marriage dissolution or spousal death. More generally, this illustrates the usefulness of our methodology to address the well-being questions that we listed in section I.
Similar to before, we start by comparing the “stable” RICEB bounds, which we obtain through our identification method, with naive bounds. For a given individual, the naive lower bound equals the fraction of the budget share of the individual's leisure consumption (which is assignable and private), while the naive upper bound equals this lower bound plus the budget share of the household's nonleisure consumption (which is nonassignable). The results of this exercise are summarized in table 2. We also report on the percentage point differences between the (stable and naive) upper and lower bounds (see the “Diff” columns).18
|Min||Women Max||Diff||Min||Men Max||Diff||Min||Women Max||Diff||Min||Men Max||Diff|
|Min||Women Max||Diff||Min||Men Max||Diff||Min||Women Max||Diff||Min||Men Max||Diff|
Once more, we conclude that our method has substantial identifying power. The stable bounds are considerably tighter than the naive bounds, with the average difference between upper and lower bounds narrowing from 36 percentage points (for the naive bounds) to no more than 9 to 11 percentage points (for the stable bounds). The stable bounds are also informatively tight. For example, we learn that, on average, men seem to have more control over household expenditures than women: the average male RICEB is situated between and , while the average female RICEB is only between and . As before, however, there is quite a bit of heterogeneity between households: lower bounds for women (resp. men) range from to (respectively, to ) and upper bounds from to (respectively, to ).19
In appendix D.5, we relate the revealed interhousehold heterogeneity in individual RICEBs to observed household characteristics. It appears that the RICEBs bear significant relations with the intrahousehold wage ratio, the household's full income, the number of children, the interspousal age difference, and the region of residence. Subsequently, we use these regression results to compute individual compensation schemes that guarantee the same consumption level in case of marriage dissolution or spousal death. This provides answers to some of the questions raised in section I. We refer to tables 18 to 20 and the corresponding discussion in appendix D.5 for more details.
B. Individual Poverty
Our RICEB estimates allow us to conduct a poverty analysis directly at the level of individuals in households rather than at the level of aggregate households. Given our particular setup, such a poverty analysis can simultaneously account for both economies of scale in consumption (through public goods) and within-household sharing patterns (reflecting individuals' bargaining positions). To clearly expose the impact of these two mechanisms, we perform three exercises. In our first exercise, we compute the poverty rate defined in the usual way: as the percentage of households having full income that falls below the poverty line, which we fix at 60% of the median full income in our sample of households.20 This also equals the individual poverty rates if there would be equal sharing and no economies of scale. The results of this exercise are given in table 3 under the heading, “No economies of scale and equal sharing.” We would label of the individuals (and couples) as poor if we ignored scale economies and assumed that household resources are shared equally between men and women.
|No economies of scale, equal sharing||12.48||12.48||12.48|
|With economies of scale and equal sharing||Lower bound||5.14||5.14||5.14|
|With economies of scale and unequal sharing||Lower bound||–||8.32||11.72|
|No economies of scale, equal sharing||12.48||12.48||12.48|
|With economies of scale and equal sharing||Lower bound||5.14||5.14||5.14|
|With economies of scale and unequal sharing||Lower bound||–||8.32||11.72|
In a following exercise, we use the same household poverty line but now account for the possibility that household consumption exceeds the expenditures because of economies of scale. In particular, we increase the households' aggregate consumption levels by using the (lower and upper) scale economies estimates that we summarized in table 1. Again, we assume equal sharing within households. Then we can compute lower and upper bounds on individual poverty rates while accounting for the specific impact of households' scale economies. We report these results under the heading, “With economies of scale and equal sharing” in table 3. Not surprisingly, we see that poverty rates decrease when compared to the calculations that ignore intrahousehold scale economies; the estimated poverty rate is now between (lower bound) and (upper bound).
So far, we have computed poverty rates under the counterfactual of equal sharing within households. However, households typically do not split consumption perfectly equally. Therefore, in our third exercise, we compute poverty rates on the basis of our RICEB results summarized in table 2. Here, we label an individual as poor if his or her RICEB-based estimate falls below the individual poverty line, which we define as half of the poverty line for couples that we used above. As before, we can compute upper- and lower-bound estimates for the individual poverty rates. The outcomes are summarized under the heading, “With economies of scale and unequal sharing” in table 3. It is interesting to compare these results with the ones that account for scale economies but assume equal intrahousehold sharing. We conclude that unequal sharing considerably deteriorates the poverty rates for both the women and men in our sample. In particular women seem to suffer the most: the lower and upper rates of female poverty are 11.72% and 24.06%, well above the upper bound of 10.89%. In appendix D.6, we provide some further insights in these poverty rates by differentiating households with different characteristics. It illustrates that our method can be used to analyze poverty differences between women and men, depending on, for example, the number of children or region of residence.
These results fall in line with the findings of Cherchye, De Rock, Lewbel, and Vermeulen (2015), who also showed that due to unequal sharing of resources within households, the fraction of individuals living below the poverty line may be considerably greater than the fraction obtained by standard measures that ignore intrahousehold allocations. A main novelty of our analysis is that we also highlight the importance of households' scale economies in assessing individual poverty. For some households and individuals, publicness of consumption may partly offset the negative effect of unequal sharing. As we have shown, our method effectively allows us to disentangle the impact of the two channels.21
We have presented a novel structural method to empirically identify households' economies of scale that originate from public consumption (defined by Barten scales). We take it that these economies of scale imply gains from marriage and use the observed marriage behavior to identify households' scale economies under the maintained assumption of marital stability. Our method is intrinsically nonparametric and requires only a single consumption observation per household. In addition, the method can be implemented through simple linear programming, which is attractive from a practical point of view.22 Our method produces informative empirical results that provide insight into the structure of scale economies for alternative household types. In turn, these findings can be used to address a variety of follow-up questions (e.g., on intrahousehold allocation patterns and individual income compensations in case of marriage dissolution or spousal death).
We have demonstrated alternative uses of our method through an empirical application to consumption data drawn from the PSID, for which we assume that similar households (in terms of age, education, number of children, and region of residence) operate on the same marriage market and are characterized by a homogeneous consumption technology. We found that public consumption increases with the number of children living in the household and that particularly households in the Northeast United States experience more economies of scale, while richer households are generally characterized by lower-scale economies. Next, we have analyzed intrahousehold allocation patterns of expenditures by computing the relative costs of equivalent bundles (RICEBs) for the men and women in our sample, and we showed the relevance of these RICEBs for individual poverty analysis (revealing substantial inequalities between men and women in households with dependent children). We found that the individual RICEBs are significantly related to the intrahousehold wage ratio, the household's full income, the number of children, the interspousal age difference, and the region of residence. As an implication, the same variables also have an impact on the individual compensation schemes required to guarantee the same consumption level in case of marriage dissolution or wrongful death.
In our application, we have made a number of simplifying modeling choices. Weakening these assumptions can enrich the empirical investigation and the insights that are drawn from it. For example, as we discussed in section II, we have assumed a fairly simple household production setting, in which each individual produces a single domestic good. An interesting avenue for follow-up research consists of including more sophisticated production processes, in which the domestic goods are produced by the two spouses simultaneously. See, for example, Goussé, Jacquemet, and Robin (2017), who also consider a marriage matching context. By extending our methodology to also identify a more complicated within-household production structure, we will obtain a tool kit that can empirically address research questions related to, for example, marriage matching on productivity and specialization in marriage. Such an extension can also provide a fruitful ground to explicitly include the (welfare of) children in the structural identification analysis. Next, because our method uses information on individual wages, we have restricted our analysis to couples in which both partners are active on the labor market. Obviously, an interesting further development consists of including couples with inactive partner(s) (and unobserved wage(s)). In such instances, we can proceed by using shadow wages, which can also be identified on the basis of a structural household production model. For example, following a similar nonparametric approach, Cherchye, De Rock, Vermeulen, and Walther (2017) showed how to infer shadow prices under the assumption of efficient household production and constant returns to scale. Clearly, integrating these insights in our methodological framework would significantly widen the range of empirical questions that can be addressed.
At the empirical level, a specific feature of our analysis is that we used only a single consumption observation per household. This shows the empirical usefulness of our method even if only cross-sectional household data can be used. In practice, however, panel data sets containing time series of observations for multiple households are increasingly available. The use of household-specific time series would allow us to additionally exploit the specific testable implications of our assumption that collective households realize Pareto-efficient intrahousehold allocations (under the assumption of time-invariant individual preferences; see Cherchye, De Rock, & Vermeulen, 2007, 2011, for detailed discussions). Obviously, this can only enrich the analysis. For example, it would allow us to recover individual indifference curves, which enables the computation of indifference scales as defined by Browning et al. (2013). These indifference scales can be used to compute Hicksian-type income compensations (i.e., for fixed utility levels) in case of divorce or spousal death, which constitute useful complements to the (Slutsky-type) RICEB-based compensations (with fixed consumption levels) that we consider in our study. In addition, the use of household-specific time series could also allow us to relax our assumption that observationally similar households are characterized by a homogeneous consumption technology, and thus to account for unobserved heterogeneity of the household technologies.
Finally, to operationalize the no-blocking-pairs condition in our empirical method we need to define individuals' marriage markets. As discussed in section III, our empirical application adopted a minimalistic approach by focusing on small marriage markets containing observationally similar households. In addition, our results appeared to be robust for the assumption that individuals consider only a subset of the potential partners in the marriage markets that we constructed. Still, we do see a more refined modeling of individuals' marriage markets as a useful extension of the method that we propose in this paper. Intuitively, this boils down to constructing individual-specific consideration sets for the particular context of marital matching.23 Such a construction may use insights from the literature on structurally explaining observed marriage patterns (see, e.g., Choo & Siow, 2006, and, more recently, Dupuy & Galichon, 2014).
Given the overidentification of the basic model of Browning et al. (2013), there is room to parameterize preference changes due to marriage. Dunbar, Lewbel, and Pendakur (2013) suggested an identification approach that no longer assumes that individuals in couples have the same preferences as singles. Their approach needs to assume either that preferences are similar across people for a given household type or, alternatively, that preferences are similar across household types for a given person. In our method, we account for fully unobserved preference heterogeneity across individuals in different households.
We implicitly consider two types of household technologies. The focus of this paper is on household technologies along the lines of Browning et al. (2013), which are associated with economies of scale. The other type of household technologies is related to the transformation of time spent on domestic work to commodities consumed inside the household in a Becker (1965) fashion. Under appropriate assumptions, a spouse's time spent on domestic production can serve as the output of the home produced good by this spouse. We will come back to this in section II.
In principle, it is possible to relax this assumption of exogenous wages for the revealed preference method that we introduce below, along the lines suggested by Cherchye, Demuynck, et al. (2017). For example, an alternative is to impute these postdivorce wages and incomes based on regressions that take account of the so-called marriage premium. To facilitate our exposition, we abstract from this extension in our current analysis. Moreover, for our identification method with the marital stability assumption, it can be argued that the wage rate inside marriage is probably a good benchmark when individuals compare their opportunity sets inside their current marriage and outside marriage as a single or with a different partner.
As discussed in section I, our use of Barten scales to represent public versus private consumption follows Browning et al. (2013). In their theoretical discussion, these authors also considered a more general setting in which households buy the bundle and consume the bundle such that , where is a nonsingular matrix and is a vector. We discuss this more general setting in appendix B. Our main empirical analysis will focus on a special case of this general type of linear household technologies.
For our data set, we could also have conditioned the household technology on the age and education of the wife. We have chosen not to do so because the observed marriage matchings are largely positively assortative for these individual characteristics. For example, the sample correlation between the ages of husband and wife amounts to and the correlation between education levels is . See appendix C.5. for more details.
The scale economies measure that we use in this study avoids these weaknesses and allows for heterogeneity across households. The findings of our empirical application show the importance of accounting for interhousehold heterogeneity. See, for example, table 1.
Browning, Chiappori, and Weiss (2014) define the relative cost of an equivalent bundle at the couple's level, which coincides with the economies-of-scale measure in equation (2). We define the relative cost at the individual level, which allows us to analyze the intrahousehold allocation of resources, as we will show in our empirical application.
In our application, “marriage” stands for legal marriage as well as cohabitation.
The expression represents the element-by-element maximum, that is, indicates for all goods .
We consider adjustment in labor incomes because nonlabor incomes are unknown variables in our conditions in proposition 1. By considering only postdivorce adjustments of labor incomes, we preserve linearity in unknowns when treating the stability indices as unknown variables. This allows us to use linear programming to compute these indices.
As a further robustness check, we also consider the scenario in which a fraction of leisure is allowed to be publicly consumed (reflecting externalities). Specifically, instead of assuming that all leisure is privately consumed, we now put upper bounds of 5%, 10%, and 15% on the degree of publicness of male and female leisure (i.e., we set , and ). Evidently, because this allows for more public consumption, our scale economies estimates and individual RICEBs generally increase, by construction. However, and importantly, the main qualitative conclusions of our empirical analysis remain intact (see appendix E.3).
We see two possible approaches to account for spouses who are not active on the labor market. First, we could exogenously define the wage of the inactive spouses on the basis of their observable characteristics (like education and age). Second, we could use the method of Cherchye, De Rock, Vermeulen, and Walther (2017) and define shadow prices endogenously under the assumption of efficient household production and constant returns to scale. To simplify our discussion, we chose not to follow these approaches in this paper. But the extensions are fairly easy.
We dropped 4,429 households with an unmarried head of family. We did not consider 2,640 households because of missing information (mainly on individuals' education, time use, and wages), and another 617 households had household members different from husband, wife, and children. Finally, we lost 55 households because of data trimming (leaving out the households in the 1st and 99th percentiles of the male and female wage distributions).
Cherchye, Demuynck, De Rock, and Vermeulen (2017) used similar assumptions to define the price of leisure in their empirical application. As these authors point out, an alternative possibility is to impute the counterfactual wages and incomes for the different exit options (e.g., based on reduced-form analysis). For time spent on household work, another alternative option is to use the prices of marketable commodities like formal child care, cleaning the house, and gardening. By lack of detailed information on the spouses' time use, we do not follow this route in the paper. However, in appendix E.1 we report on a robustness check in which we use the average wage of men and women in the sample to evaluate spousal domestic work in the counterfactual situation of singlehood.
Technically, including additional blocking pair constraints will lead to smaller feasible sets characterized by the rationalizability constraints in proposition 1. In turn, this will lead to sharper upper and lower bounds (i.e., tighter set identification). We illustrate this point in appendix E.5, which presents a robustness analysis in which we also include singles as potentially blocking pairs.
As indicated before, in appendix E.2, we present a robustness check in which we focus only on the couples that satisfy the sharp restrictions. The results are qualitatively the same.
Appendix D.2 shows the empirical cumulative distribution of the stable upper and lower bounds on our scale economies measure .
Appendix D.2 shows the empirical cumulative distribution of the stable upper and lower bounds on the individual RICEBs. Appendix D.4 gives complementary results on public and private components of the individual RICEBs reported in table 2.
Recall that the lower bound of the RICEB and upper bound of the RICEB need not necessarily add up to unity. The reason is that these RICEBs divide the value of individual consumption by total household expenditures (see equations  and ). Because of scale economies, the total value of consumption (summed over the two spouses) can exceed the household expenditures. See also our discussion of the measures and in equations (3) and (4) in section II.
While 60% of the median income is a standard measure of relative poverty (e.g., used in the definition of OECD poverty rates), in our case, the poverty rate is calculated on the basis of full income instead of (the more commonly used) earnings or total expenditures. Also, our data set pertains to couples where both spouses participate in the labor market, and so our poverty line will be different from a line based on data that include households with singles, unemployed, or retired members.
For the sake of brevity, we focused on the importance of economies of scale in assessing individual poverty. However, our method would also allow us to investigate the role played by economies of scale and unequal sharing in assessing between and within-household consumption inequality (Lise & Seitz, 2011 and Greenwood et al., 2014, 2016, for alternative methods and applications).
This linear programming structure can also be useful from an inferential point of view. For example, Kaido, Molinari, and Stoye (2019) introduce a bootstrap-based procedure to do inference on the value of a linear program with estimated constraints. We see the adaptation of this work to our identification method as an interesting avenue for follow-up research.
The use of consideration sets received substantial attention in the recent literature on revealed preferences (see, e.g., Manzini & Mariotti, 2014). This existing work can provide a useful starting point to develop this question further.
This paper is a tribute to Professor Anton Barten (1930–2016), who introduced us to structural demand econometrics at the KU Leuven. We thank editor Brigitte Madrian, two anonymous referees, Richard Blundell, Bart Capéau, Ian Crawford, Jeremy Fox, Marion Leturcq, François Maniquet, Sujoy Mukerji, Krishna Pendakur, Aloysius Siow, Jörg Stoye, and seminar participants in CORE, Institute for Fiscal Studies, Queen Mary University of London, Rice University, Royal Holloway University of London, Tilburg University, University of Toronto, Universidad del Pacífico, University of Luxembourg, University of Copenhagen, BI Norwegian Business School, ETH Zürich, University of Ljubljana, the SEHO workshop (Paris), the HCEO Family Inequality Workshop Spring 2018 (Leuven), the CEMMAP Workshop on Heterogeneity in Supply and Demand (Boston), and the WOLFE Workshop (York) for helpful comments. L.C. gratefully acknowledges the European Research Council for his consolidator grant 614221. Part of this research is also funded by the FWO (Research Fund-Flanders). B.D.R. gratefully acknowledges FWO and FNRS for their financial support. K.S. gratefully acknowledges financial support from the FWO through grant V402718N. F.V. gratefully acknowledges financial support from the Research Fund KU Leuven through grant STRT/12/001 and from the FWO through grant G057314N.
A supplemental appendix is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/rest_a_00829.