ECONOMIC PAPERS, VOL. 32, NO. 1, MARCH 2013, 67–80

Preferences and Policies: An Intra-Household Demand System Michael Malcolm1

I estimate a household demand system with specific focus on allocation to children, adults and joint household goods. The main finding is that marginal dollars are spent disproportionately on adult-assignable goods relative to the way in which the average dollar is spent. The estimation provides both income effects and a complete set of cross-price effects, which informs analysis both of income transfers and of policies that induce relative price changes. Contrasted with earlier work, the demand system in this paper covers nearly all household expenditures. I show bias that results from reduced-form systems. Keywords: intra-household allocation, demand systems, income transfers, price subsidies.

1. Introduction Although economics has traditionally taken the household to be the atomic unit of observation, recent work has increasingly focused on decision making within the household. Demand systems analysis is a powerful empirical tool for revealing policy-relevant features of individual choice problems, and this paper applies demand systems analysis to intra-household decision making. The main finding of the paper is that, for the mean one-child household, a marginal dollar is spent disproportionately on adult-assignable and leisure goods like alcohol, restaurant meals and adult clothing relative to how an average dollar is spent. Conversely, the proportion of the marginal dollar spent on child-assignable and joint household goods is lower than the proportion of the average dollar spent on these goods. The estimation uses microdata with more than 10,000 observations over twelve years of the UK’s Family Expenditure Survey (FES). The contribution is twofold. First, expenditures on children are an important public policy issue. Although most work on intra-household allocation of income focuses on gender differences,2 this paper focuses specifically on allocation to adults versus allocation to children. This question is critical for policy-makers since ensuring adequate investment in children creates a large public good effect for society at large; Heckman and Masterov (2007) argue that there are a whole host of public benefits to parental investment in children, among them permanently higher productivity and lower propensity to commit crimes. In this context, deficiencies in expenditures on children are generally viewed as a social problem.

1

Department of Economics, West Chester University of Pennsylvania. Lundberg et al. (1997) is the most well-known example. JEL classifications: D13, H53, I38 Correspondence: Michael Malcolm, Department of Economics, West Chester University of Pennsylvania, 700 South High St, West Chester, PA 19383. Email: [email protected] 2

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Second, the small literature that exists on allocation within households to children has largely not benefitted from more generalised functional forms for estimating parameters of household choice problems, estimation of which is now possible through large increases in computing power. For example, Gronau (1991) estimated the marginal propensity to consume on children, which is very close to the spirit of this paper, but the estimation in that paper is significantly reduced form. Another approach is that employed in Lundberg et al. (1997), who look at behaviour before and after a discrete policy change to impute particular features of household decision making. In contrast, this paper estimates the parameters of a household demand system with more generality, which exposes some features that are not obvious from partial systems analysis. The implications of adopting this approach are discussed in the next section. In essence, this paper uses well-established empirical tools from the demand systems literature and applies these tools in a new way to the intra-household decision problem over adults and children. The policy implications are manifold. First, understanding how households spend marginal dollars clearly informs the long-standing debate on untied income transfers versus tied income transfers or in-kind benefits. Furthermore, complete demand systems estimation also provides a set of cross-price effects, suggesting that children’s goods and adult goods are net complements. Wealth effects induced by lowering the price of child-assignable goods are likely to increase consumption not only of these goods but also of adult-assignable goods as well. As policy-makers design programmes that lower the relative price of children’s goods (such as the “sales tax holidays” common in the United States) or increase the relative price of adult goods (such as alcohol and tobacco taxes), it is important to take into consideration that wealth effects could at least partially counteract the intended substitution effects. The paper is organised as follows: Part II discusses related literature. Part III presents the model of the household demand system and Part IV presents the data used for estimation. Part V outlines the estimation technique. Results are discussed in Part VI and Part VII concludes.

2. Related Literature This paper addresses a hybrid of two literatures in the sense that I study household allocation questions using well-established empirical techniques in the demand systems literature. Work on household allocation is often significantly reduced form. Conversely, the demand systems literature, with a few exceptions, does not generally address intra-household allocation. Assigning components of a household demand system to various members of the household involve determining which purchases benefit which members of the household. These purchases are the observed outcomes. Some goods are more convincingly assignable than others: clothing, for example, is naturally distinguishable among that which benefits male, female and juvenile members of the household. Alcohol and tobacco purchases are generally understood to benefit adult household members exclusively (e.g., Blow et al., 2012). Some goods, like housing or food, presumably benefit all members of the household. Lundberg et al. (1997) and Browning et al. (1994) study the household allocation problem using partial demand systems analysis, restricted to clothing, where the assignment is most transparent. Since these models are designed so that the budget constraint holds by construction, there are two choices. First, one can define a local income only over the goods of interest. For example, Lundberg et al. (1997) formulate a “clothing expenditure” for part of their analysis. Second, everything other than the assignable good of interest can be lumped into a composite good. Blow et al. (2012) take this approach. Either restriction is somewhat limiting since it does not address how income is allocated into the restricted system in the first place.3 Estimating any demand system requires some kind of variation in exogenous variables across observations. More authors are interested in income responsiveness than in price responsiveness, but this presents a problem: Income is endogenous from the perspective of a household choice problem. Families choose their labour hours, and hence their incomes, in a way that could be correlated with 3

Technically, Gronau (1991) shows that preferences must be separable over adult and child expenditures in order for such a restriction to allow for valid estimation of parameters.

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their preference ordering related to their children.4 Broadly, leisure and home-produced goods can be viewed as unobserved components of a household’s consumption vector, but these objects are themselves mechanically related to income. A common solution is to find some income change that is exogenous to preferences. A well-known and oft-used example is the UK Child Benefit program; Lundberg et al. (1997) and a number of follow-up papers, for example, Blow et al. (2012) and Ward-Batts (2003) use the imposition of this policy change, which resulted in a change in the distribution of income across household members towards mothers and away from fathers, as an identifying instrument. Such exogenous changes reveal interesting features of household behaviour, but are too discretised for estimating a household demand system in more generality. In addition, the latter approach also provides price effects as well as income effects, which may be interesting to policy-makers. There are a plethora of papers that estimate household demand systems, with diverse objectives. Many are in the agricultural economics literature and estimate demand systems over food (e.g., Huang, 1996). Some other examples focus on carbon taxes (Brannlund and Nordstrum, 2002), electricity demand (Reiss and White, 2005) and demand for foodstuffs in developing economies (Kebede, 2003).5 There is surprisingly little application of this technique to intra-household allocation problems. Pollak and Wales (1992) made a number of early methodological contributions and Huffman (2006) considers the relationship between consumption purchases and unpaid household inputs in the context of a household demand system. Among previous work on this topic, the objective of Gronau (1991) is closest to the spirit of this paper – his research objective is to estimate marginal propensity to consume on children. However, like other earlier authors, he uses substantially reduced-form systems. The basis for estimation in this paper is a unitary model of household decision making. Becker’s (1981) well-known rotten kid theorem rationalises the unitary framework, where a household maximises a well-defined household utility function. My primary focus of inquiry is allocation to children versus adults; I do not make an attempt to distinguish between male and female adult consumption, where issues of bargaining and non-neutrality of income by source are known to be serious (Browning et al., 1994).6 A well-defined household demand system provides empirical results that are straightforward in their application to policy questions, but still sufficiently flexible to reveal interesting features of intra-household allocation decisions.

3. Model

The primitive is a unitary household utility-maximisation problem. Throughout, we will let i = 1,…, n index goods and we will let q = 1, … , k index households. Our goal is to recover the parameters of the utility function and thus develop policy-relevant comparative statics. The Almost Ideal Demand System (AIDS) is a very general parametric specification and was proposed by Deaton and Muellbauer, 1980 to approximate any locally non-satiated utility function arbitrarily well up to second order.7 The starting point is a translog expenditure function, with functional forms chosen so that it will locally approximate any well-defined expenditure function up to second order. See Martinez-Garcia (2005) for a proof of this result and a detailed explication of the properties of this class of expenditure functions. One can then apply Shephard’s Lemma to the expenditure function to derive the demand functions. I refer the reader to Deaton and Muellbauer for more details on the derivation of the demand system. For purposes of estimation, it is easiest to write the demand system in budget share form. The budget share sqi for household q and good i is given by the following equation: 4 Preferences placing value on children’s consumption would tend to drive labour hours up, while valuing staying at home to rear children might drive labour hours down (e.g., Chiswick, 1986), either of which would lead to erroneous parameter estimates if income were viewed as exogenous. 5 Kebede (2003) is a particularly interesting example since the price variation is cross section rather than temporal; residents of different villages face different relative prices. 6 There is some inquiry into bargaining over children’s goods. Thomas (1990) finds differential gender preference over male and female children depending on the source of family income. 7 This assumes, of course, that the function is sufficiently smooth that the second derivative exists.

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Sqi ¼ ai þ

Xn

c j¼1 ij

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In pqj þ bi In

mq þ eqi pq

In this equation, mq is household q’s total expenditure. pqj is the price that household q faces for good j. Pq is a household-specific price index, discussed in this paper. Parameters are common across observations and are good specific. Income mq is household specific and is identically equal to total expenditure level. The prices pqj are household specific only in the sense that the price series are quarterly – every household observed in the same quarter faces the same price vector. The price index P in the full AIDS model is a highly non-linear function of parameters, and estimation was computationally infeasible. Other authors have made this observation, and the usual solution – first suggested by Deaton and Muellbauer themselves – is to use Stone’s geometric price index, which is a first-order approximation to the properly specified price index. Here, the price index P for each observation is given by the following equation: Xn In P ¼ s In pi i¼1 i Using Stone’s price index, this model is often called the Linear-Approximate AIDS (LA/AIDS) model. Demand theory implies the following restrictions on parameters (Deaton and Muellbauer): Xn Xn Xn a ¼1 c ¼0 b ¼1 ð1Þ i¼1 i i¼1 ij i¼1 i Xn

c j¼1 ij

¼0

cij ¼ cji

ð2Þ ð3Þ

The restrictions in (I) correspond to budget-balancedness and hold here by construction. Restriction (II) is implied by homogeneity. Restriction (III) is necessary and sufficient for Slutsky symmetry. Notice that (II) is redundant with the second condition of (I) if (III) holds.8 In Part V, I discuss how I deal with these restrictions in estimation.

4. Data Data on household expenditures are drawn from the UK’s FES, collected by the Office of National Statistics (ONS), a public agency. These data are in wide use by economists. It is a continuous-sample survey of several thousand households across the United Kingdom each year. Households are selected randomly from post office databases, and response is voluntary. Responses are distributed across the year and are identified by the quarter and year in which they were taken. This is important because it allows us to hone the relevant price indices more specifically. Each household member maintains a two-week log of all expenditures. All household members aged sixteen or older maintain a log. Beginning in 1998, children aged 7–15 also maintained logs, but I use only adult expenditures since data on young children’s expenditures are not available prior to 1998; these expenditures are very small. The interviewers also collect information on recurring monthly expenditures like rent and car payments, even if the payment was not made during the survey window. The records are very specific in the sense that household expenditures are classified quite narrowly. My sample is composed of all one-child families surveyed in the period 1988-FY 2000.9 Some households’ records are incomplete. After 1994, only complete records are provided, so I use only complete records for the whole time frame. The total sample size is 10,937 households. The restriction to one-child households is common in the literature (e.g., Blow et al., 2012), which means that the conclusions are specific for this household structure. The restriction is in essence an implicit control for the demographic structure of the household. While this enhances the internal 8 These restrictions are necessary, but not sufficient to guarantee that the demand system is consistent with that generated by a well-defined utility maximisation problem. It is also necessary that the slutsky matrix be negative semi-definite; this restriction cannot be imposed ex ante (Deaton and Muelbauer). 9 FY 2000 was the last year of the survey in this form.

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validity of the study because it circumvents having to specify a parametric restriction for the way in which the number of children affects household spending, it limits external validity if there is a selection bias among families who choose to have only one child.10 We do not differentiate by the age of the child, which again is typical in the literature, although adult children are not included – by child, we mean a household member under the age of eighteen. My data classify almost all household expenditures, with a few exceptions: 1 Dynamics – I do not consider any expenditures dealing with choice across time (e.g., savings or contributions to pension funds) or choice over uncertainty (e.g., insurance payments). A demand system dealing with this would need to be more carefully specified to account for notions of intertemporal choice and risk preferences, which is difficult with repeated cross-sectional data since each household is observed only one time. 2 Education expenditures – I do not include education expenditures. Fewer than 5 per cent of the FY 2000 sample record any positive expenditure in this category, and almost all are quite small – under 10. In addition, the data do not reveal which of these expenditures benefit children versus adults (e.g., hobby classes) and there is no price index available. 3 Health care expenditures – Changes in personal health care expenditures over time are more likely the result of changes to government policy than price and income shocks, especially given the public health care framework of the United Kingdom. Again, there is also no price index. 4 A few other very minor and sporadic expenditures, for example, cable TV repairs, mainly due to cross-time differences in definition. Naturally, computational feasibility requires some bundling. Where n is the number of goods over which the demand system is estimated, the number of parameters to be estimated is O(n2) Another practical constraint is that the goods can only be separated into more precisely defined bundles to the degree that price indices are separately available, since good-specific prices are a necessary component of the estimation. Beyond these practical constraints, bundling was guided by ease of intuitive assignability. Alcohol, tobacco and gambling are lumped together as an obviously adult-assignable good, while food eaten in and food eaten out are kept separate since the latter is sometimes thought of as more of a luxury good benefitting adults.11 Bundling was also guided by collinearity of the price series. For example, I bundled non-assignable clothing with other household goods since the correlation between the price indices was quite high (q = 0.993). On the other hand, I kept entertainment goods and entertainment services separate since, although this seems like a natural aggregation, the price series were not highly correlated (q = 0.310). Conversely, although the price indices of motoring and the adult-assignable alcohol, tobacco and gambling good are highly correlated, bundling these together is problematic as it obfuscates the assignability issue. Ultimately, the household demand function is estimated over eleven categories of goods. Appendix Table A1 provides descriptions of each category, in addition to summary statistics on the budget share allocated to this category across households.12 Censoring is an issue for adult clothing and children’s clothing, where the percentage of the sample recording zero expenditures for these categories are 35.9 per cent and 51.7 per cent respectively. In 10 As one referee pointed out, an interesting extension of this paper would be an augmented AIDS model that studies how these results vary across demographic structures. Another approach, taken by Lundberg et al. (1997), is to partition the data and develop multiple, independent sets of estimates. 11 Via time savings in preparing meals, for example. 12 Some of these bundles, like housing, are pre-aggregated by ONS. However, a few adjustments had to be made for consistency. Gardening equipment and supplies were aggregated with entertainment goods prior to 1994, but with household goods after. I adjusted the pre-1994 aggregates to include these in household goods over the whole series. The opposite was true for stationery and writing utensils. Also, underclothing expenditures are separated by adults and children after 1994, but not prior to 1994. Thus, child-assignable and adult-assignable clothing includes only overclothing and footwear, which are identified separately for children and adults over the whole series. All underclothing is bundled with other non-assignable clothing, which in turn is bundled with other household goods and services.

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Section VI, I discuss sensitivity of the results to censoring issues. In all other categories, <5 per cent of the sample record zero expenditures. The ONS maintains Retail Price Indices (RPI) on approximately eighty goods categories. Some are easily available, and others were provided by Crawford and Image (2004). Although all price indices are available by month, expenditure records are identified only by quarter, so the three corresponding monthly price indices were averaged together to derive a quarterly price index. Where price indices had to be combined into a single aggregated price index (e.g., separate price indices are maintained for children’s outer clothing and children’s footwear), I developed a weighted average using the RPI weights, also provided by Crawford and Image, averaged together by quarter. All price indices are normalised to 100 for the first quarter of 1988, the first year of our sample. Tables 1 and 2 give descriptive statistics for and correlations among all the price indices over the time frame under consideration in the paper respectively.

5. Estimation Recall that our demand system is, written in budget share form: Xn mq sqi ¼ ai þ c In pqj þ bi In þ eqi j¼1 ij pq If there were no cross-equation restrictions, the demand system could be estimated equation-byequation using Ordinary Least Squares (OLS) possibly with an instrumental variable for income. However, because the Slutsky symmetry condition cij = cji is imposed as an estimation restriction, the full demand system must be estimated simultaneously. The traditional approach is full information maximum likelihood (FIML), which is a multivariate analogue to Maximum Likelihood Estimator (MLE) specifying a joint distribution for the error terms in the simultaneously estimated system. I modify this approach slightly because the FIML maximisation over the bi terms is problematic since it amounts to an implicit assumption that real income mp is exogenous. This seems reasonable for research that studies demand over various types of foodstuffs, for example, but is not credible in this context since income is generally viewed as a choice variable with respect to household behaviour, as discussed in Part II. To accommodate this difficulty, I use generalised method of moments (GMM). Rather than using all of the moment conditions implied by the standard FIML estimation, I discard the moment conditions related to income and preserve only the moments related to prices. Formally, the moment conditions are E(eqi pqj) = 0 over all goods i and j.13 This gives a vector of n2 moment conditions, which is still sufficient for identification in this case as long as there are n  5 goods, since the number of parameters to be estimated is 12 ðn2 þ 5nÞ. In other words, for our estimation with n = 11, the model is overidentified even after discarding the moment conditions related to income which are imposed in the standard FIML estimation.14 The GMM estimator replaces the n2-vector of moment conditions with sample moments. We use the identity matrix I as the weight matrix. The estimated parameters that result are consistent estimators of the population parameters (Hansen, 1982).

13 At the household level, this seems like a benign assumption – households do not consider the general equilibrium effects of their own purchases. However, the issue of exogeneity of prices in aggregate is ubiquitous in the demand systems literature. Bronsard and Salvas-Bronsard (1984) find as an empirical matter that the price exogeneity assumption is of little consequence when using disaggregated data. Further, observe that any demand systems analysis using the more common FIML technique faces this same issue, but also faces the issue of income exogeneity My estimation eliminates the latter problem, which is by far more serious in the context of a household decision problem. 14 Many prices move in a relatively uncorrelated way, which provides the independence of moment conditions needed for identification. However, note that this is only part of what one needs for a valid instrumental variable. The instrument also needs to have a useful correlation with the variable being instrumented. In the present case, aggregate economic conditions tie prices and incomes. Furthermore, relative prices are related to allocations, which are in turn tied to real income earned by households.

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Table 1. Descriptive Statistics on the Level of the Price Index, Normalised to 100 in Q1, 1988 (n.b. Q4, FY 2000 is the Last Quarter Under Consideration) Good Housing Fuel and power Motoring and fares Food (in home) Food (out of home) Alcohol, tobacco and gambling Adult clothing Children’s clothing Entertainment goods Entertainment services Household goods and services

Mean price index level

SD price index level

Price index level: Q4, FY 2000

160.32 126.23 140.58 126.76 149.93 155.40 109.77 114.55 113.63 154.98 130.99

26.07 10.04 22.15 12.25 28.09 32.86 4.88 4.94 6.17 30.25 15.41

211.51 125.64 171.93 140.76 194.30 208.22 97.89 107.48 106.75 206.72 151.08

Green and Alston (1990, 1991) give the formulas for deriving elasticities from the parameters of the LA/AIDS model. As they are continuous functions of the parameters, the elasticity estimates are themselves consistent estimates of the true elasticities by the continuous mapping theorem. Since the model is non-linear, the magnitude of the marginal effects depends, of course, on where they are evaluated. I here follow convention and report elasticities at the mean budget share. Finally, I use the bootstrap to develop standard errors. I applied the Hall Hall and Horowitz (1996) re-centring technique for the overidentified specifications as the standard error estimates are not consistent otherwise. h The Slutsky equation is useful for interpretation. Letting em i be the Marshallian elasticity, ei be the Hicksian elasticity, gi be the income elasticity and si be the budget share, then the Slutsky equation states that h em i ¼ ei  gi si

Recall the well-known intuitive interpretation that em i represents the “total effect” on demand of a price increase, which can be decomposed: ehi is the “substitution effect” – the part of the realised change in demand that results from relative price changes alone, whereas gisi is the “income effect” – the part of the realised change in demand that results from a change in purchasing power irrespective of the relative price change. This separation is important here since some goods represent sizeable budget shares, leading to potentially large wealth effects. P We will also make use of Engel aggregation (Jehle and Reny, 2001): ni¼1 si gi ¼ 1. This decomposition is intuitively useful in that it shows how a marginal dollar of income is spent across goods, which is an important public policy question.

6. Results and Discussion

Table 3 gives the elasticity estimates,15 with standard errors computed via bootstrap simulation. Table 4 decomposes the income effect gisi into the income elasticity and budget share separately, and also gives 95 per cent bootstrap confidence intervals rather than standard errors. The income effect terms, typically more interesting to policy-makers, are estimated with considerably more precision than either the Marshallian or the Hicksian price elasticities. One reason is mechanical: the budget share si from the income-effect term gisi is not a function of estimated parameters. However, even the income elasticities gi themselves are estimated with more precision than price elasticities. A potential explanation is that income is different for each observation, whereas 15

Note that the whole system is estimated simultaneously. It is incorrect to say that the wealth effects are inferred from the price effects or vice versa since all are estimated jointly.

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House Fuel Motor Food (in) Food (out) Alcohol Clothes (adult) Clothes (kids) Ent. Goods Ent. services Goods & serv.

1 0.513 0.891 0.866 0.896 0.898 0.145 0.282 0.312 0.901 0.884

House

1 0.664 0.798 0.662 0.634 0.521 0.823 0.904 0.648 0.729

Fuel

1 0.963 0.997 0.997 0.098 0.361 0.513 0.997 0.992

Motor

Table 2. Correlations Between the Price Series

1 0.966 0.955 0.089 0.537 0.674 0.958 0.984

Food (in)

1 0.998 0.114 0.347 0.499 0.999 0.992

Food (out)

1 0.149 0.309 0.467 0.999 0.988

Alcohol

1 0.858 0.686 0.127 0.003

Clothes (adult)

1 0.894 0.330 0.444

Clothes (kids)

1 0.481 0.590

Ent. goods

1 0.991

Ent. serv.

1

Goods & serv.

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prices are the same for all observations in a given quarter, so there is more information about income responsiveness. As Engel aggregation implies, the income effects reported in the last column of Table 3 sum to one and give the estimated increase in expenditures on each of the goods resulting from a marginal dollar of income. From the framework of analysing how households allocate resources among members, a useful exercise is to compare the average expenditure share, reported in Appendix Table A1, with the marginal expenditure share. For example, although in-home food accounts for 16.1 per cent of total expenditures, only $0.122 of the marginal dollar is spent on food. On the other hand, although only 8.05 per cent of total income is allocated to the adult-assignable alcohol/tobacco/gambling good, $0.135 of the marginal dollar finances this good; only 4.46 per cent of total expenditures are dedicated to adult clothing, but $0.161 of the marginal dollar finances additional adult clothing purchases.16 This inequality also holds for restaurant meals, entertainment goods, entertainment services – marginal expenditure share on these goods exceeds average expenditure, and this result is robust. For joint household goods and services, marginal and average expenditure shares are approximately equal. This suggests that, even though we observe substantial expenditures on joint household and children’s goods, marginal income finances leisure and adult-assignable goods disproportionately, while income allocated to joint household goods and children’s goods is held relatively constant. One way to interpret this result is that parents maintain some baseline level of consumption for their children (there are constraints on the lower end that are active here – at the very least, there are laws against neglect), but that the share of marginal increases in income that adults allocate to their children is low.17 However, it is important to recall that these are the elasticities evaluated at the mean budget shares. Thus, this same finding is entirely consistent with, for example, Gruber (2000) who shows that low-income single mothers spend a very large share (about 95 per cent) of Aid to Families with Dependent Children (AFDC) income on food and housing, benefiting children. To receive AFDC payments, these are not marginal dollars at the mean income, but marginal dollars at low income. The fact that a greater proportion of the marginal dollar than the average dollar is spent on adult-assignable goods suggests that, at low incomes, parents spend initial resources to finance joint household and child consumption and then finance adult goods with additional discretionary income. Note also that only one-child families are included in the empirical analysis. If there is selection bias among families that choose to have only one child, then these results may not be generalisable across all family structures. The results agree with partial results obtained by other authors. For example, Lundberg et al. (1997) find that rising income is associated with a reduction in the ratio of child to adult male clothing expenditures and that income shifts towards mothers are associated with increases in this ratio. Gronau (1991) also finds that, similar to our result, the marginal propensity to consume on adult goods far exceeds the marginal propensity to consume on children’s goods. As for price responsiveness, the price elasticity estimates are reasonably intuitive: housing, fuel and food consumed in-home are relatively inelastic; adult clothing and joint household goods and services are close to unit elastic; motoring expenses, entertainment services, and particularly entertainment goods are highly elastic. Three of the estimated Hicksian elasticities: food eaten out, the adult-assignable alcohol/tobacco/gambling good and children’s clothing have theoretically implausible positive signs. However, they are not estimated precisely, and plausible estimates are well within the confidence bands. Recalling Appendix Table A1, these three goods have relatively low budget 16 The estimated income elasticity of children’s clothing is actually negative; one possible explanation is that single-mother households have lower family incomes than two-parent households and, from section II, there is evidence that consumption of children increases when women control household expenditures. An interpretation of the inferiority of motoring expenses is that wealthier families move to pricier properties closer to their jobs. 17 An interpretation in terms of bargaining power is problematic. On one hand, it is not clear what it means to say that a young child has bargaining power. On the other hand, spending on children may be a reasonable proxy for the mother’s bargaining share in the household. See Thomas (1990), for example.

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Table 3. Estimated Own-Price Elasticities from the Complete Demand System em i ¼

ehi 

gi si

0.6996 (0.118) 0.0095 (0.670) 1.9366 (0.721) 0.3730 (0.366) 0.6701 (0.547) 0.3006 (0.560) 0.9820 (1.041) 1.0218 (3.023) 4.6327 (1.001) 1.7872 (0.652) 1.0069 (0.429)

0.6109 (0.109) 0.0597 (0.672) 1.8078 (0.729) 0.2511 (0.374) 0.7347 (0.553) 0.4353 (0.564) 0.8210 (1.040) 0.9750 (3.024) 4.4887 (0.995) 1.7220 (0.641) 0.8187 (0.434)

0.0887 (0.024) 0.0502 (0.022) 0.1288 (0.028) 0.1219 (0.019) 0.0645 (0.015) 0.1348 (0.023) 0.1612 (0.019) 0.0468 (0.015) 0.1441 (0.019) 0.0652 (0.024) 0.1822 (0.022)

Good Housing Fuel Motoring Food (in) Food (out) Alcohol/tobacco/gambling Clothes (adult) Clothes (kids) Entertainment goods Entertainment services Goods and services

Table 4. Estimated Own-Price Elasticities from the Complete Demand System, with Income Effect Decomposed and Bootstrap Confidence Intervals Good Housing Fuel Motoring Food (in) Food (out) Alcohol/Tobacco/ Gambling Clothes (adult) Clothes (kids) Entertainment goods Entertainment services Goods and services

em i ¼

ehi 

gi

si

0.6996 [0.898, 0.549] 0.0095 [0.851, 0.914] 1.9366 [3.31, 1.00] 0.3730 [1.35, 0.245] 0.6701 [1.07, 0.750] 0.3006 [0.918, 0.863]

0.6109 [0.798, 0.438] 0.0597 [0.909, 0.870] 1.8078 [3.18, 0.867] 0.2511 [1.23, 0.120] 0.7347 [1.02, 0.825] 0.4353 [0.799, 1.01]

0.5578 [0.329, 0.815] 0.9346 [1.52, 0.403] 0.8792 [0.586, 1.20] 0.7587 [0.597, 0.960] 1.2621 [0.934, 1.87] 1.6740 [1.05, 2.01]

0.1590 0.0537 0.1465 0.1607 0.0511 0.0805

0.9820 [2.30, 0.988] 1.0218 [4.37, 5.54] 4.6327 [6.16, 3.25]

0.8210 [2.14, 1.11] 0.9750 [4.41, 5.51] 4.4887 [6.00, 3.11]

3.6138 [2.55, 3.80] 2.5599 [2.89, 0.461] 2.9291 [2.30, 3.49]

0.0446 0.0183 0.0492

1.7872 [2.92, 0.735]

1.7220 [2.82, 0.678]

1.1215 [0.578, 1.81]

0.0581

1.0069 [1.54, 0.359]

0.8187 [1.38, 0.187]

1.0557 [0.885, 1.26]

0.1783

shares. With small, irregular expenditures, it is more difficult to deduce marginal effects of price changes on these goods than for goods where expenditures are larger and more regular, where our estimates are intuitive. Table 5 reports Marshallian cross-price elasticities between all goods. The diagonal elements, of course, correspond to the own-price elasticities reported in Table 3. One can, from this, deduce how particular price changes are likely to induce changes in demand for other products. For example, estimates suggest that a program lowering the relative price of children’s clothing in an attempt to increase the well-being of children is likely to also induce an increase in consumption of food eaten out-of-home, alcohol and tobacco, and adult’s clothing via the income effect of the ensuing general increase in purchasing power. There are a number of important policy implications that follow from these results. First, policymakers should be aware that average expenditure patterns are a poor indicator of the way in which marginal income is likely to be expended. Thus, if one of the intentions of income transfer programs to poor families is to enhance the welfare of children specifically, then these results lend support to

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Ó 2013 The Economic Society of Australia

Goods & serv.

Ent. services

Ent. goods

Clothes (kids)

Clothes (adult)

Alcohol

Food (out)

Food (in)

Motor

Fuel

House

Fuel 0.09 (0.08) 0.01 (0.67) 0.16 (0.37) 0.06 (0.24) 0.71 (0.44) 0.76 (0.39) 0.28 (0.65) 0.27 (1.50) 0.15 (0.58) 1.64 (0.54) 0.29 (0.24)

House 0.70 (0.12) 0.55 (0.30) 0.53 (0.16) 0.04 (0.12) 0.20 (0.17) 1.26 (0.20) 0.93 (0.28) 3.81 (0.75) 2.43 (0.33) 1.06 (0.28) 0.03 (0.13) 0.54 (0.14) 0.13 (1.04) 1.94 (0.72) 0.77 (0.32) 2.41 (0.70) 1.24 (0.59) 0.24 (1.07) 4.45 (2.87) 2.02 (1.12) 6.19 (0.74) 1.30 (0.33)

Motor 0.02 (0.11) 0.41 (0.73) 0.82 (0.37) 0.37 (0.37) 0.68 (0.52) 1.94 (0.38) 1.49 (0.98) 7.78 (2.37) 1.63 (0.93) 4.14 (0.67) 0.69 (0.38)

Food (in) 0.03 (0.04) 0.58 (0.42) 0.82 (0.25) 0.19 (0.17) 0.67 (0.55) 4.43 (0.30) 0.69 (0.61) 4.50 (1.61) 0.23 (0.56) 3.08 (0.52) 1.29 (0.22)

Food (out) 0.55 (0.09) 0.97 (0.59) 0.61 (0.33) 1.03 (0.19) 1.80 (0.47) 0.30 (0.56) 2.14 (0.84) 0.79 (1.88) 0.98 (0.97) 1.61 (0.55) 0.40 (0.26)

Alcohol 0.12 (0.07) 0.40 (0.55) 0.21 (0.33) 0.52 (0.27) 0.49 (0.54) 1.25 (0.48) 0.98 (1.04) 3.50 (2.24) 0.19 (0.71) 3.60 (0.77) 0.05 (0.40)

Clothes (adult) 0.38 (0.08) 0.10 (0.52) 0.63 (0.36) 0.93 (0.27) 1.67 (0.57) 0.24 (0.43) 1.55 (0.92) 1.02 (3.02) 2.08 (0.85) 2.09 (0.71) 0.57 (0.42)

Clothes (kids) 0.63 (0.09) 0.02 (0.54) 0.79 (0.37) 0.59 (0.29) 0.30 (0.54) 0.56 (0.59) 0.19 (0.78) 5.85 (2.26) 4.63 (1.00) 0.25 (0.76) 0.02 (0.40)

Ent. goods

0.43 (0.09) 1.92 (0.60) 2.47 (0.31) 1.47 (0.25) 3.51 (0.61) 1.21 (0.41) 4.88 (1.00) 6.92 (2.28) 0.44 (0.90) 1.79 (0.65) 0.77 (0.19)

Ent. serv.

0.06 (0.14) 0.56 (0.82) 1.55 (0.40) 0.71 (0.43) 4.47 (0.79) 1.02 (0.58) 0.28 (1.61) 6.25 (4.21) 0.44 (1.50) 2.36 (0.63) 1.01 (0.43)

Goods & serv.

Table 5. Estimated Marshallian Cross-Price Elasticities from the Complete Demand System (n.b. Elasticities are ɛij e.g., a 1% Increase in the Price of Housing Corresponds to a 0.55% Increase in the Demand for Fuel)

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tied transfers or even in-kind transfers rather than untied income transfers. Second, even upper and middle-class families that are not eligible for transfer payments receive indirect subsidies for their children via child tax credits, and indeed investment in children is frequently cited as a rationale for these subsidies. Again, these results suggest that increases in these subsidies are likely to enhance purchases of adult-assignable goods, while child consumption remains relatively constant. Third, regarding relative price changes, several states in the United States offer “sales tax holidays” that lower the effective price of child-assignable goods (mostly clothing and school supplies) for a few days or weeks. While there may be a substitution effect that results from these price changes, the results in this paper suggest that there is also a strong income effect. Ultimately, these price cuts may end up financing additional purchases of adult-assignable goods. Finally, the converse applies to sin taxes on adult-assignable goods like alcohol and tobacco. There may be a substitution away from alcohol and tobacco that results from these taxes, but the income effect estimated here is substantial and suggests that expenditures on child-assignable goods will also be lowered to compensate. The approach adopted by other authors, either to look only at expenditures on assignable goods under study or to lump all other expenditures into an otherwise unspecified composite good leads to substantial bias in estimates of the parameters of interest.18 In the latter case, the composite good takes up a large expenditure share, with estimated income and price elasticities near unity. This emphasises the desirability of defining demand systems with as much specificity as is computationally feasible. As discussed in Section IV, censoring is an issue for the two clothing baskets. To ensure that results are not sensitive to these zeroes, I re-estimate the model twice, dropping all observations with zero expenditures for each of the two clothing baskets in turn. This is a fairly common sensitivity check (e.g., Blow et al., 2012). Results are reported in Tables 6 and 7, estimating the model using only nonzero observations for children’s clothing and adult clothing respectively. The price elasticities are relatively similar, especially for the goods with high expenditure shares. More importantly, although there are small differences in magnitudes, the main qualitative result that the marginal dollar disproportionately finances adult-assignable and leisure goods relative to the average dollar still holds in both cases.

7. Conclusion I have estimated a full household demand system, deriving empirical results that describe policyrelevant features of household behaviour for one-child households. I showed that the allocation of the average dollar is a poor indicator of how the marginal dollar is likely to be expended. Marginal Table 6. Estimated Own-Price Elasticities for the Complete Demand System, Omitting Observations with Zero Expenditure on Children’s Clothing Good Housing Fuel Motoring Food (in) Food (out) Alcohol/tobacco/gambling Clothes (adult) Clothes (kids) Entertainment goods Entertainment services Goods and services

em i ¼

ehi 

gi si

si

0.6403 (0.216) 0.1625 (0.819) 2.1334 (0.950) 0.5087 (0.546) 1.0034 (0.924) 0.3586 (0.930) 1.0635 (1.397) 0.0499 (1.857) 2.8931 (1.338) 1.5911 (1.134) 1.1660 (0.737)

0.6019 (0.196) 0.2328 (0.818) 2.0260 (0.939) 0.4026 (0.551) 1.0963 (0.918) 0.1888 (0.920) 0.9112 (1.378) 0.0032 (1.853) 2.7229 (1.345) 1.5107 (1.116) 0.9603 (0.747)

0.0385 (0.042) 0.0704 (0.031) 0.1074 (0.040) 0.1061 (0.032) 0.0929 (0.023) 0.1698 (0.039) 0.1523 (0.041) 0.0531 (0.030) 0.1703 (0.029) 0.0803 (0.041) 0.2058 (0.035)

0.1562 0.0499 0.1419 0.1536 0.0517 0.0753 0.0462 0.0379 0.0504 0.0567 0.1802

18

Full results are available from the author upon request.

Ó 2013 The Economic Society of Australia

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Table 7. Estimated Own-Price Elasticities for the Complete Demand System, Omitting Observations with Zero Expenditure on Adult Clothing Good Housing Fuel Motoring Food (in) Food (out) Alcohol/tobacco/gambling Clothes (adult) Clothes (kids) Entertainment goods Entertainment services Goods and services

em i ¼

ehi 

gi si

si

0.5950 (0.158) 0.9585 (0.644) 2.0109 (0.703) 0.3234 (0.478) 0.7381 (0.958) 0.1118 (0.755) 1.3250 (0.690) 1.7211 (3.823) 4.8957 (1.237) 1.7031 (1.081) 1.015 (0.936)

0.5839 (0.137) 1.0008 (0.656) 1.8787 (0.700) 0.2068 (0.478) 0.8132 (0.962) 0.2677 (0.750) 1.1601 (0.686) 1.6536 (3.826) 4.6897 (1.244) 1.6717 (1.076) 0.8848 (0.930)

0.0111 (0.044) 0.0422 (0.026) 0.1322 (0.031) 0.1165 (0.027) 0.0751 (0.021) 0.1559 (0.028) 0.1649 (0.031) 0.0675 (0.024) 0.2060 (0.039) 0.0313 (0.039) 0.2167 (0.026)

0.1510 0.0464 0.1487 0.1494 0.0532 0.0771 0.0695 0.0178 0.0504 0.0614 0.1752

household income is disproportionately allocated to leisure and adult-assignable goods. In addition, wealth effects induced by price changes of child-assignable or adult-assignable goods are strong and generate counteractive behaviour. There remain many interesting areas that are unexplored. First, a more complete model would address inter-temporal dynamics. Given concerns over high debt and low savings, taking account in our demand system of choice over present and future consumption generally is an interesting extension. Unfortunately, the present data set is inadequate for studying this problem since it is repeated cross section and each household is observed only one time. Second, endogenising the labour supply decision would give a more complete model, especially with respect to parents making a choice whether to stay at home and rear children. Third, as discussed earlier, it would be an interesting extension of this result to determine how these decisions vary across family structures. Finally, although there is a large literature on estimation issues in demand systems literature, it does not focus on the application of this technique to household problems specifically, and as we discuss in this paper, household choice problems create a unique set of problems in estimating demand systems.

Appendix Table A1. Definition of Goods Good

Housing Fuel and power Motoring and fares Food (in home) Food (out of home) Alcohol, tobacco and gambling Adult clothing

Description

Mean budget share

SD budget shares

Recurring rent and mortgage payments; home maintenance Electricity, heating oil and gas Recurring car payments, gasoline, rail/bus fares Food prepared in-home; no alcohol; includes school lunches by convention Food prepared outside of the home: take-out and eat-in; no alcohol Alcohol in-home and out; tobacco; gross gambling expenses Adult outerwear and footwear

0.159

0.105

0.054 0.147 0.161

0.044 0.120 0.081

0.051

0.039

0.080

0.074

0.045

0.060

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Table A1. (Continued) Good

Children’s clothing Entertainment goods Entertainment services Household goods and services

Description

Children’s outerwear and footwear Electronics, cable and satellite, sports and hobby equipment, books and music, etc. Live entertainment, theatre, sporting events, travel expenses, etc. Durable and nondurable goods, except as defined above; services; non-assignable clothing

Mean budget share

SD budget shares

0.018 0.049

0.033 0.058

0.058

0.083

0.178

0.105

REFERENCES Becker, G.S. (1981), A Treatise on the Family. Harvard Univ. Press, Cambridge. Blow, L., Walker, I. and Zhu, Y. (2012), ‘Who Benefits from Child Benefit?’, Economic Inquiry, 50, 153–70. Brannlund, R. and Nordstrum, J. (2002), ‘Carbon Tax Simulations Using a Household Demand Model’, European Economic Review, 48, 211–33. Bronsard, C. and Salvas-Bronsard, L. (1984), ‘On Price Exogeneity in Complete Demand Systems’, Journal of Econometrics, 24, 235–47. Browning, M., Bourguignon, F., Chiappori, P.-A. and Lechene, V. (1994), ‘Incomes and Outcomes: A Structural Model of Intrahousehold Allocation’, The Journal of Political Economy, 102, 1067–96. Chiswick, B.R. (1986), ‘Labor Supply and Investment in Child Quality: A Study of Jewish and Non-Jewish Women’, The Review of Economics and Statistics, 68, 700–3. Crawford, I. and Image, I. (2004), ‘The RPI and the Cost-of-Living Index: Testing for Consistency Between Theory and Practice’, Fiscal Studies, 25, 79–91. Deaton, A. and Muellbauer, J. (1980), ‘An Almost Ideal Demand System’, The American Economic Review, 70, 312–26. Green, R. and Alston, J.M. (1990), ‘Elasticities in AIDS Models’, American Journal of Agricultural Economics, 72, 442–5. Green, R. and Alston, J.M. (1991), ‘Elasticities in AIDS Models: A Clarification and Extnesion’, American Journal of Agricultural Economics, 73, 874–5. Gronau, R. (1991), ‘The Intrafamily Allocation of Goods – How to Separate the Adult from the Child’, Journal of Labor Economics, 9, 207–35. Gruber, J. (2000), ‘Cash Welfare as a Consumption Smoothing Mechanism for Single Mothers’, Journal of Public Economics, 75, 157–82. Hall, P. and Horowitz, J.L. (1996), ‘Bootstrap Critical Values for Tests based on Generalized Method of Moments Estimation’, Econometrica, 64, 891–916. Hansen, L. (1982), ‘Large Sample Properties of Generalized Method of Moments Estimators’, Econometrica, 50, 1029–54. Heckman, J.J. and Masterov, D.V. (2007), ‘The Productivity Argument for Investing in Young Children’, Applied Economic Perspectives and Policy, 29, 446–93. Huang, K.S. (1996), ‘Nutrient Elasticities in a Complete Food Demand System’, American Journal of Agricultural Economics, 78, 21–9. Huffman, W. (2006), ‘Understanding Post-War Changes in US Household Production: A Full-Income Demand System Perspective’, Iowa State University Working Paper Series. Jehle, G.A. and Reny, P.J. (2001), Advanced Microeconomic Theory. Addison Wesley, Boston. Kebede, B. (2003). ‘Intra-household Distribution of Expenditures in Rural Ethiopia: A Demand Systems Approach’, The Centre for the Study of African Economies Working Paper Series. Lundberg, S.J., Pollak, R.A. and Wales, T.J. (1997), ‘Do Husbands and Wives Pool Their Resources? Evidence from the United Kingdom Child Benefit’, The Journal of Human Resources, 32, 463–80. Martinez-Garcia, E. (2005), ‘The Story Behind the Translog Unit Expenditure Function’, Working paper. ?????. Pollak, R.A. and Wales, T.J. (1992), Demand System Specification & Estimation. Oxford University Press, New York. Reiss, P.C. and White, M.W. (2005), ‘Household Electricity Demand, Revisited’, The Review of Economic Studies, 72, 853–83. Thomas, D. (1990), ‘Intra-Household Resource Allocation: An Inferential Approach’, The Journal of Human Resrouces, 25, 635–64. Ward-Batts, J. (2003), Out of the Wallet and into the Purse: Modeling Family Expenditures to Test Income Pooling. Claremont Colleges Working Paper.

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Preferences and Policies: An IntraHousehold Demand System

Preferences and Policies: An Intra-Household Demand System. Michael Malcolm1. I estimate a household demand system with specific focus on allocation to children, adults and joint household goods. The main finding is that marginal dollars are spent disproportionately on adult-assignable goods relative to the.

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