EMPIRICAL FINDINGS

In this section we review some empirical work based on the collective model and empha­sizing the identification of the sharing rule.

The first-generation models used information on private and assignable goods such as consumption of clothing or individual leisure to identify the sharing rule up to a constant.

To identify the way overall resources are allocated and thus measure inequality, one needs more information, either in terms of identifying assumptions on the behavior of the sharing rule (such as nonnegativity conditions discussed earlier) or assumptions on pref­erences. One possibility is to compare the behavior of married and single individuals by making assumptions about the way preferences change with marriage. Other approaches involve specific restrictions on preferences. We show how some of these approaches have been used in the literature. Finally, we also consider the information content of revealed preference restrictions. These extend the revealed preference arguments for individual choice to the case of collective households. Clearly this is a much more complicated setup than standard revealed preference restrictions for individuals or for unitary households because the aggregate household does not necessarily behave like a rational single agent.

However, the issue of identification of the sharing rule is deeper than what is sug­gested by the use of the restrictions above and has to do with the way people make agree­ments at the point of marriage and the level of commitment associated with these agreements. In other words, fundamentally the sharing rule is identified from behavior without having to impose possibly ad hoc restrictions. Identification requires extending the model to include marital decisions in an equilibrium context. Indeed a marriage mar­ket equilibrium will define the sharing rule, and conditions in the marriage market can allow us to identify it. This effectively introduces dynamics, which then allows one to delve deeper into the extent of commitment and what this means about within- household inequality. Characterizing the theoretical and empirical power of using marriage-market data to better understand intrahousehold allocations is a relatively new and active area of research, particularly when limited commitment is allowed for.

Before we discuss the empirical literature we need to introduce a distinction between the concept of identifiably of preferences and the sharing rule on the one hand and econometric identification on the other. The identifiability results discussed earlier in the chapter relate to our ability to recover individual preferences and the sharing rule given we know the household level demand functions exactly. Empirical analysis is concerned with estimating these household demands from empirical data to be able to then recover the sharing rule. This issue brings forth all the standard econometric concerns, such as the role of unobserved heterogeneity, the endogeneity of wages, prices and income, corner solutions (particularly in labor supply), and the like. One of the hardest issues concerns the way that unobserved heterogeneity enters household demands, particularly if such unob­servables are correlated with observables.

16.6.1 "Pure" Identification of the Sharing Rule

In this first generation of collective models we can point to three main empirical studies. The first is by Browning et al. (1994); the second is by Chiappori et al. (2002); and the third is by Blundell et al. (2007). All three share a similar approach to identification: they assume efficiency and an assignable good. However, the details of the empirical approach differ.

In Browning et al. (1994), the authors use a sample of couples drawn from the Canadian FAMEX and estimate a model for the demand of men’s and women’s clothing, and identify the sharing rule, up to a constant. Identification relies on two assumptions: first, clothing is an assignable good, which effectively means that we can observe male and female clothing and that only the person using the clothing derives any utility from it. In other words, clothing does not include a public good element. Second, they assume that the distribution of a partner’s income does not affect preferences, but they may enter the sharing rule, reflecting bargaining positions. Given these assumptions, they identify a sharing rule as a function of the age difference of the partners, total household expendi­ture (thus allowing wealth effects in the way resources are shared), and most importantly, the share of income attributable to the female partner.

¹⁶ For recent attempts in this direction (including a discussion of the specific difficulties it raises), see for instance Lewbel and Pendakur (2013) and Chiappori and Kim (2013). the share of household expenditure by a significant but small 2.3%. The age difference and the level of expenditure also matter with relatively older individuals gaining more and wealthier households allocating more to the wife.

The Browning et al. (1994) paper shows the potential of the approach and the richness of the empirical results that can be obtained by judicious use of information reflecting bargaining power of households. However, the main determinant of female bargaining power in their model is the relative magnitude of female income. A higher share of income may reflect her relative skills or, alternatively, it may reflect her decision to forgo leisure and work more; in other words, this distribution factor is indeed endogenous. In principle, this fact does not harm identification provided that labor supply is separable from consumption: Controlling for total expenditures, individual consumption should then be independent of labor supply (therefore of labor income). However, separability is a strong assumption, that has been empirically criticized. The next two papers address exactly this issue by endogenizing labor supply.

The empirical relevance of the discussion above for within-household inequality and allocation of resources is illustrated by Chiappori et al. (2002). They use data from the PSID to estimate a collective labor supply model, where the sharing rule is identified (up to a constant) based on distribution factors. These include the sex ratio (males/ females) in the state as measured by the 1990 census as well as by dummy variables indi­cating the nature of divorce laws.

¹⁷ The intuition underlying the CFL paper—that a relative scarcity of women and/or more favorable divorce laws should improve the wife’s Pareto weight—can be supported by an explicit matching model, with some nuances (e.g., changes in divorce laws affect differently women already married and women getting married after the change in law). On these issues, see Chiappori et al. (2013). fact that the restrictions from the collective model are not rejected strengthens this interpretation.

The results suggest that marriage and labor market conditions can lead to large dif­ferences in the allocation of household resources within a couple.

These results are important because they show the extent to which within-household allocation of resources can be sensitive to external conditions affecting the bargaining power of the members of the couple. Noting, for example, that average household income in this data is $48,000, the change that can be induced just because of (admittedly extreme) changes in the sex ratio can amount to almost half of household income.

However, there are a number of empirical issues that were not addressed by the papers already discussed. First, we need to be concerned that the allocation of women across states with different sex ratios is not random with respect to their unobserved preferences for labor supply. This can bias the results if women who live in areas abundant with men tend to have lower labor market attachment. Second, we need to address the issue of precision in the estimation of wage effects, an issue that persists in the Blundell et al. (2007) paper we will discuss below. CLF instrument wages, but the instruments are nec­essarily quite weak: they rely on a polynomial in age and education as an instrument while (correctly) controlling linearly for education and age in the labor supply function. This leaves higher-order nonlinearity in the profile of wages with respect to age and education to act as excluded instruments that is both difficult to justify theoretically and, at the same time, is not very informative. To solve these empirical issues we will require exogenous events that change wages and the marriage market, something that a newer generation of collective models is now addressing, such as the paper by Attanasio and Lechene (2014) who use experimental variation in female income induced by the Conditional Cash Transfer (PROGRESA) experiment in Mexico to obtain exogenous variation in the rel­ative bargaining position of males and females.

Beyond these difficulties there is one further important issue that the papers we dis­cussed fail to address, namely nonparticipation of women. Given that many women do not work allowing for this possibility and understanding how resources are allocated despite the fact she is not producing in the formal market is a key concern. The Blundell et al. (2007) paper addresses the question of identification and estimation of a collective labor supply model that allows for male and female non participation in the labour market. In addition, it considers the case where the male labor supply decision is discrete (work or not). This restriction is imposed to accommodate the fact that in the United Kingdom (where the data is drawn from) the male hours of work distribution seem discontinuous between 0 and about 35 h per week, with the entire mass of workers concentrated in the full time range. This restriction is not entirely satisfactory, but it may do better justice to the data than assuming hours of work are freely chosen. Thus, the resulting model is one where females make choices both on the intensive and the exten­sive margin, whereas males choose only on the extensive margin. The authors prove identification of the sharing rule; however, this is only identified (nonparametrically) if at least one of the two household members work. Parametric restrictions provide the rest. In the empirical implementation Blundell et al. (2007) deal with the endogeneity of the wage rate by exploiting the changes in wage inequality across cohorts and education groups. Econometric identification relies on the assumption that whereas the structure of wages changed across education groups and cohorts—a testable assumption—preferences remained unchanged. This implies that changes in work behavior across cohorts and education groups can be attributed to changes in the incen­tive structure, which is the identification strategy employed by Blundell et al. (1998).

The empirical analysis is conducted on a sample of married couples, observed between 1978 and 2001 in the UK Family Expenditure Survey. The assumptions imposed for identification (over and above efficiency) required only private goods and one assignable good. The assignable good is leisure. Because expenditures on children are not separately observable in the data, and because these are effectively public, the authors exclude all couples with children and then assume that the observed aggregate household consumption reflects the sum of private consumption of each of the two members of the household.

The model implies two different sharing rules depending on whether the husband works or not. They differ by a monotonic transformation, which in their empirical spec­ification acts as an attenuation factor, implying that the husband only gets a fraction of transfers when he is not working. This fraction is 0.71, implying that the derivatives of the sharing rule (as well as the level) are attenuated by that amount when he is not working. Their empirical approach does not use any distribution factors that can be excluded from preferences: the sharing rule depends on male wages, female wages, and unearned income as well as education and age. It turns out that empirically the effect of the female wage on the sharing rule is not well identified. However, the effect of the male wage is precisely estimated. It implies that 88% of an increase in male-market earn­ings translates into a transfer to the husband if he is working. Inasmuch as there is no intensive margin for the male decision, this translates to a direct impact on his consump­tion, if he continues to work. Ifhe does not work, the same change in potential earnings translates to a transfer equal to 62% of the potential increase (0.71 ? 0.88). These results imply that when the earnings of a working husband increase, the resulting increase in the consumption of the wife is only small; if potential earnings increase (and he is not work­ing) her consumption declines substantially and he enjoys more of the household resources. Finally, the wife keeps 73% of increases in unearned income. Nevertheless, unearned income is a relatively low fraction of household income.

These results again illustrate that external factors (here the relative wages) can influ­ence the allocation of resources substantially. Unfortunately Blundell et al. (2007) do not provide precise estimates of the effects of female wages, and this hinders an understanding of how the change in the wage structure affected within household allocations. The source of lack of precision is the relatively small sample size where the man does not work. Moreover, allowing both wages and nonlabor income to be endogenous, while important for obtaining consistent estimates that make sense, does affect precision sub­stantially. The paper does demonstrate that one does not need (in principle) distribution factors for identification. However, looking at the empirical problem from the perspec­tive of Chiappori et al. (2002), other environmental factors may be very important in determining allocations and, if they are omitted, they could bias the results. On the other hand, if included they can be allowed to affect preferences as well. Identification does not require they affect the sharing rule alone.

This first generation of models showed the potential of the collective model for iden­tifying allocations of resources within the black box of the household. However there are key issues that had not been dealt with. First, taxes and welfare were ignored. At one level this is an empirical specification issue because ignoring taxes can bias the estimates of the preference parameters. But at a more fundamental level by not taking into account the tax and welfare system we omit one of the most important factors affecting (and sometimes designed to affect) within household allocations. Estimating models that allow for taxes and welfare can then explain how changes in the policy and the market environment can affect the allocation of resources.

The next fundamental issue is that the models described above can only identify the derivatives of the sharing rule, that is, how sharing changes when distribution factors, prices, and unearned income change. This precludes any discussion of the levels of inequality of resources and hence does not allow us to put into perspective the implica­tion of changes that occur over time.

Adding taxes and welfare does not pose any important conceptual problems. In prac­tice it involves allowing for more complex budget sets and solving the model to take into account nonlinear budget sets. An interesting issue is that the welfare and tax system may create a further interdependence in the decisions of husband and wife, over and above that induced by the sharing rule. These issues are considered, for instance, in Donni (2003), who uses a “pure” identification strategy of the type just described, and by Beninger et al. (2006), Myck et al. (2006), and Vermeulen et al. (2006) who use infor­mation from singles and couples.

Extending the model to allow identification of the level of the sharing rule does, how­ever, pose conceptual problems. Fundamentally, the sharing rule is identified by the equi­librium in the marriage market. However, barring the use of a complete marriage market equilibrium model one can obtain information on the level of inequality with alternative auxiliary assumptions. One possibility is to use information on singles. This involves restricting the way preferences change with marriage. This is an approach used by Lise and Seitz in an early version of their paper. Another possibility is to assume something about the sharing rule at one point of the wage space, for example, that all resources are shared equally when wages are equal, which is the assumption made in the published version of Lise and Seitz (2011). Finally, one can make assumptions about the functional forms of demand, as in Dunbar et al. (2013). We now look into these empirical studies.

16.6.2 Intrahousehold Inequality Over Time and the Sharing Rule:

Lise and Seitz (2011)

Lise and Seitz (2011) use the collective model to first estimate overall consumption inequality (at the individual level) and to then decompose this to between household and within household. The important economic fact is that the distribution of wages in the United Kingdom changed dramatically over the period they consider (1968-2001) both within and between education groups (see Gosling et al., 2000). Moreover, the structure of the marriage market has also changed with increased degrees of marital sorting over time. They thus set up a model of male and female labor supply with many (but discrete) choices of hours worked for both members of the household. Hours can take values from 0 to 65 in 5-h intervals. In many ways their empirical frame­work is similar to that of Blundell et al. (2007): They use couples with no children drawn from the UK Family Expenditure Survey over many years. However, they depart in a number of important ways. First, they allow for taxes and account for the impact of joint taxation over the period that this was in effect in the UK (up to 1989). Also they allow for a richer choice set for the male and they impose further structure so as to identify the level of the sharing rule as well as its derivatives. Finally they account for public goods when they define consumption, although they are taken as separable from private consumption and leisure.

Whereas the logic underlying the identification of the derivatives of the sharing rule is similar to that of Blundell et al. (2007), identification of the location (level) of the sharing rule empirically is based on the identifying assumption that when individuals have the same potential earnings they share resources equally. In earlier versions of the paper it was instead assumed that preferences of married and single individuals are identical; both these assumptions can identify the model. The point at which one pins down the sharing rule is welfare irrelevant, because the preference specification adapts to leave welfare unchanged when the location of sharing is fixed. In principle, just normalizing the loca­tion parameter will not cause any bias but will, of course, lead to a specific level of inequality. On the other hand, using information from singles has the advantage that it uses a restriction grounded in some explicit assumption on preferences (marriage does not affect marginal utilities) but, if wrong, will bias all results.

Over the period considered in the paper (1968-2001), earnings inequality increased rapidly. There has been a steady increase in both the potential earnings’ and actual earn­ings’ share of women relative to men and a decline in male employment while female employment increased at the start of the period later remaining constant. Consumption inequality increased rapidly in the period between 1980 and 1990, but was basically stable the rest of the time. When Lise and Seitz interpret these results under the prism of their collective model, they uncover some interesting facts: while between-household inequality of consumption increases, within-household inequality of consumption declines to such an extent that the overall inequality of consumption remains more or less the same over time. When they consider a different measure of resources, namely full consumption, which includes the value of leisure enjoyed by each member, they find similar but less stark results. First, between-household inequality still increases, but much less dramatically because the decline in consumption for those households who have workless members is compensated by the value of leisure. Second again they find that within-household inequality declines as before, but much less. Obviously none of these consumption measures is ideal and a money-metric measure of welfare may be better. However, these results illustrate exactly the potential importance of finding cred­ible ways to understand inequality (and poverty) within households. This is more so given that who marries whom is endogenous and in part drives the way that within-household inequality is determined and has implications for between-household inequality is determined.

16.6.3 Intrahousehold Inequalityand Children

While intrahousehold inequality may be of general interest because it tells us about allo­cation of resources within a household and can reveal hidden poverty and inequality, the whole issue acquires special importance when it comes to allocations of consumption to children. Thus is because child consumption and more generally investments in children have long-term implications for the intergenerational transmission of poverty. Yet little or no empirical work had been done to understand how resources are allocated to chil­dren and the extent to which reallocations of income from the male spouse to the female can affect the shares directed to children. A theoretical framework for the analysis of this question has been developed by Blundell et al. (2005). In a recent important paper, Dunbar et al. (2013) address this issue empirically using data from Malawi. In their model, each child is represented as having his or her own utility function. This creates a very special difficulty regarding the assumption, used for identification in studies such as Browning et al. (2013), namely that preferences of singles and married individuals are the same. Here, such a strategy is no longer available because children are never seen liv­ing as singles. Moreover, in data from Malawi that the authors use, there is not enough price variation—another requirement of the Browning et al. approach. Thus, identifica­tion is obtained by making assumptions on the structure and shape of the Engel curves.

The identification strategy first requires either one assignable private good or one exclusive good per person. Remember that an exclusive good is exclusively consumed by one household member type (for example, child clothing is consumed only by chil­dren), whereas an assignable good is such that each member’s consumption of this good is observable.^[56] Of course, there can be many other purely private goods (such as food) for which we do not observe the amounts of individual consumption: This fact does not hamper identification.

The assignability assumption is not sufficient to identify the share of resources of each household member; additional assumptions are therefore needed. Dunbar et al. (2013) assume, first, that resource shares are invariant to total expenditure. In addition, they make two alternative assumptions on preferences: either the demand for goods is similar across household types (i.e., households with one, two, or more children) or they are similar across types of goods within a household type. An extreme form of the assumption is that preferences do not vary across types of household; since shadow prices vary across households because of the partially public nature of goods, this extreme assumption is essentially equivalent to assuming that the assignable good used for identification is irre­sponsive to prices. Another extreme form of this assumption is that preferences over the assignable good are identical across different household member types (male, female, and children). However, Dunbar et al. (2013) show that identification only requires that some aspect of the demand functions be the same either across household member types or across household types. Thus, in one case, they assume that all household members share the same shape of Engel curves for the assignable good. In another case they assume that preferences are the same across types of household (number of children), conditional on a deflator of income. This deflator reflects the different shadow prices that different­sized households face and is the way that preferences for the assignable good are allowed to vary across types. The key point is that the authors need to define similarity so that identification is delivered without sacrificing theoretical consistency (integrability) of the demand functions.

Dunbar et al. (2013) estimate their model on data from Malawi, probably in one of the first such studies with development data. In a sense their framework is very well adapted to this context because wages and/or prices, which are at the heart of some other iden­tification strategies and are not observed in that case. Their approach relies on measuring expenditures and having an assignable good for which they use clothing and footwear. The results they obtain are both astounding and an excellent illustration of the impor­tance of looking within the household. They find that the male obtains about 45—50% of household resources. His share seems to be insensitive to the number of chil­dren present. The mother’s share declines with the second child, but then remains more or less constant, with the consumption share of children declining.

Even more pertinent are the implied poverty rates. Male poverty rates are at their highest in one-child households and seem to decline in households with more children. However, the important result concerns poverty rates for women and children: Com­pared to the male poverty rate of around 69%, there are 79% poor women and 95% poor children in one-child households. In larger households, the male poverty rate is about 55% whereas the female poverty rate is 89% and nearly all children are poor. Hence their approach not only offers a more complete picture of poverty but reveals the extent of child poverty, which is crucial to development. Without such an approach, child poverty would not be apparent to the extent that it is in reality. Although the authors did not focus on gender differences between children, which may be another important dimension, this line of research can easily be extended in that direction; it offers an obvious mech­anism for trying to understand how resources are allocated by gender.

A potential limitation of this approach is the fixed nature of the sharing rule. While the authors spend a lot of time explaining the upsides of not relying on distribution factors (essentially, they avoid having to take a position on whether they affect preferences or not), the absence of an underlying model of what the resource share should depend on and how it can be affected by exogenous driving forces may in some cases be prob­lematic. In models where the sharing rule is allowed to depend on wages or institutional features we have some understanding of how policy can be used to target individuals. In the Dunbar et al. model this aspect is missing. However, this is not an integral part of the approach, and richer models can be identified.

16.6.4 Revealed Preference Restrictions and the Identification of the Sharing Rule

The approach to the identification of the sharing rule has exploited the structure of the demand functions and the way that income affects observed outcomes when the collec­tive model is true. This leads to a set of differential equations that when solved provide the derivatives of the sharing rule. As already discussed, this is not sufficient for identifying the level of the sharing rule.

A different approach is that of revealed preference. In the context of the single-agent utility maximization model the axioms of revealed preference allow one to test nonpar- ametrically whether a particular set of choices can be rationalized by utility maximization and if they can, to bound the underlying demand functions. Such an approach has been developed and implemented for the unitary model by Blundell et al. (2003, 2008) and is based on the original work of Afriat (1973) and Varian (1982). In the collective frame­work the aggregate household demands will in general violate the revealed preference restrictions corresponding to the unitary model simply because as the budget constraint changes (wages, prices, incomes, etc.) individuals make different choices and the Pareto weights change. This insight was developed by Browning and Chiappori (1998), who showed that the aggregate household demands have to possess a Slutsky matrix that can be decomposed into a symmetric matrix plus a matrix with rank equal to the number of decision makers (whose demands are aggregated) minus 1. The fact that the pattern of choices is restricted implies that there should also be revealed preference type restrictions, as noticed by Chiappori (1988), who provides an early example in a labor supply context. Indeed these restrictions have been fully developed by Cherchye et al. (2007). In a further development Cherchye et al. (2012) show how the revealed preference restrictions can be used to bound the sharing rule without imposing any restrictions other than Pareto efficiency of intrahousehold allocations. The main result is based on the following prin­ciple: Suppose that a set of observed demands are collectively rationalizable in the sense that the observed choices are consistent with the existence of admissible individual demand functions. Then it has to be that any alternative choices that could lead to a Pareto improvement within the household should be infeasible at current market prices and for any allocation of income within the household such that each person receives a nonnegative share. More specifically, consider the set of demands that individual 1 reveals, based on all possible admissible demand functions for that person. They must cost more than person 1’s share of total household income; similarly for person 2. The least- costly bundle that would lead to a Pareto improvement provides the upper bound for a person’s share. Adding up the shares to total income and the assumption that the shares cannot be negative determines the lower bound. The difficulty in implementing this principle is the fact that we need to search over all possible admissible individual demand functions.

This principle turns out to generate nontrivial upper and lower bounds for the sharing rule. Importantly, no restriction is needed for such bounds other than Pareto optimality: All or some goods may be either private, in part public, and in part private or completely private. Moreover, we do not need to specify which goods (if any) are purely private, but if such information were to be available it can be used to tighten the bounds.

Cherchye et al. apply their approach to the PSID from 1999, when expenditures on individual consumption goods became available, until 2009. The sample consists of child­less couples where both are working. Utility depends on leisure, food and other goods, which include health and transportation. Leisure is assumed to be assignable, but no assumption is made on the other goods. This is important because in this case, at least in general, neither the level nor the derivatives of the sharing rule are point identified.

To implement their approach they start by estimating three different versions of an aggregate household demand system: a nonparametric system, the QUAIDS demand sys­tem (Banks et al., 1997) and a QUAIDS demand system where the substitution matrix is restricted to be symmetric plus rank one, which imposes that the demands are consistent with the collective model. Given this demand system they apply their algorithm to bound the sharing rule for different values of the full household income, wages, and prices. Their empirical results are remarkable. First, the bounds are very narrow with the nonparamet­ric demand system implying 12% median difference between upper and lower bounds and the fully restrictive demand system implying only 3%. Going from the nonparametric demand system to the unrestricted QUAIDS system, the tightening is due to imposing the parametric restrictions that may or may not be valid—the authors provide no evi­dence on that matter. However, assuming the parametric restrictions are valid, the further step of going from QUAIDS to restricted QUAIDS is just imposing restrictions that are implied by the problem and hence serve only to make the bounds sharp(er). Thus, when Pareto efficiency is imposed, the median difference between the upper and lower bound tightens from about 9% to 3%, a substantial improvement. It would have been useful to use a shape-constrained nonparametric demand system (see Blundell et al., 2012) avoid­ing the parametric restrictions, but using the Pareto constraints as implied by the model.

Using their bounds they establish that the female share is a normal good, that is, as full­household income grows so does the female share; interestingly, this finding confirms results previously derived in different contexts. Moreover they show that in percentage terms the average female share is very closely bounded around 50%, although there is substantial heterogeneity around that point. However, it is impressive how tightly bounded the sharing rule is throughout the distribution. In interpreting this result, one needs to be careful because it is full income that is being shared equally. This measure of income includes both leisure and consumption. Thus the share of a woman with a high wage who does not work will include her leisure and her consumption; hence a 50% share may in certain cases hide very unequal levels of consumption of all other goods.

In the final part of the analysis, the authors use their estimates to carry out a poverty analysis. The idea here is similar to that in Dunbar et al. (2013) described earlier: They compare poverty rates implied by household-level income and those implied by individual allocations. The household poverty line is 60% of median household income whereas the individual poverty line is set at half this amount. This, of course, is an income-based and not a welfare-based measure and ignores any household economies of scale. This point not­withstanding, the individual rates are higher: while household poverty is 11%, individual poverty is bounded between 16% and 21%, the lower bound being above the household number. Interestingly the bounds do not differ by gender by any substantive amount.

The Cherchye et al. study breaks new ground and shows the power of the collective approach. Specifically it reinforces the identifiability results substantially by showing not only that the levels of the sharing rule can be identified, but more importantly in our view, that the entire sharing rule can also be bounded without much more than within- household Pareto efficiency. Nevertheless, there is still a long and important agenda in this research. First, empirically we need to understand better how to deal with heteroge­neity in preferences within such a nonparametric framework as well as with endogeneity of prices and wages. The entire analysis of Cherchye et al. is based on the assumption that wages and prices are exogenous. This is internally consistent with the absence of hetero­geneity and shocks, but is broadly unsatisfactory. For example, there is a vast labor supply literature dealing with endogenous wage rates. Moreover, prices of goods may not be exogenous if there are aggregate shocks to the demand functions. While these seem to be side issues as far as the central identifiability of the collective model is concerned, they are important for the ultimate empirical credibility of the approach.

16.7.