INCENTIVES AND NUTRITION FOR ROTTEN KIDS: INTRAHOUSEHOLD FOOD ALLOCATION IN...

INCENTIVES AND NUTRITION FOR ROTTEN KIDS:

INTRAHOUSEHOLD FOOD ALLOCATION IN THE

PHILIPPINES

PIERRE DUBOIS AND ETHAN LIGON

Abstract. Using data on individual consumption expendituresfrom a sample of farm households in the Philippines, we constructa direct test of the risk-sharing implications of the collective house-hold model. We are able to contrast the efficient outcomes pre-dicted by the collective household model with the outcomes wemight expect in environments in which food consumption deliversnot only utils, but also nutrients which affect future productivity.Finally, we are able to contrast each of these two models with athird, involving a hidden action problem within the household; inthis case, the efficient provision of incentives implies that the con-sumption of each household member depends on their (stochastic)productivity.

The efficiency conditions which characterize the within-householdallocation of food under the collective household model are vio-lated, as consumption shares respond to earnings shocks. If futureproductivity depends on current nutrition, then this can explainsome but not all of the response, as it appears that the quality ofcurrent consumption depends on past earnings. This suggests thatsome actions taken by household members are private, giving riseto a moral hazard problem within the household.

1. Introduction

In recent years, a variety of authors have sought to test the hypothe-sis that intra-household allocations are efficient. Often these have beenconstrued as tests of the “collective household” model. Special casesof this model are associated with Samuelson (1956) and Becker (1974);

Date: July 24, 2008.We thank IFPRI for providing us with the main dataset used in this paper,

and Lourdes Wong from Xavier University for her help and guidance in collectingweather data. This paper has benefitted from comments and corrections providedby Costas Meghir, Mark Rosenzweig, Howarth Bouis, Emmanuel Skoufias, RafaelFlores, and seminar participants at IFPRI, Cornell, DELTA Paris, Berkeley, theUniversity College of London, the London School of Economics, Stanford, MIT,Harvard, Michigan State University, Toulouse and the 2005 World Congress of theEconometric Society. Muzhe Yang provided able research assistance.

1

INCENTIVES AND NUTRITION FOR ROTTEN KIDS 2

more recent formulations are associated with work by McElroy, Chiap-pori, and others (e.g. McElroy, 1990; Browning and Chiappori, 1998;Chiappori, 1992). Full intra-household efficiency implies both produc-tive efficiency, as well as allocational efficiency. Other authors whohave conducted tests of intra-household efficiency have tested only oneor another of these. Udry (1996), for example, focuses on productiveefficiency, while a much larger number of authors have focused on allo-cational efficiency (e.g., Thomas, 1990; Lundberg et al., 1997; Brown-ing and Chiappori, 1998). One important difficulty (which the previousauthors each address in distinct ingenious but indirect ways) involvedin testing intra-household allocational efficiency is that intra-householdallocations are seldom observed—ordinarily the best an econometriciancan hope for is carefully recorded data on household-level consumption.In this paper we exploit a carefully collected dataset which records ex-penditures for each individual within a household, and thus are able toconduct the first direct test of intra-household allocational efficiency ofwhich we are aware.

By allocational efficiency we mean, in effect, that the marginal rateof substitution between any two commodities will be equated acrosshousehold members. Importantly, we follow the Arrow-Debreu con-vention of indexing commodities not only by their physical charac-teristics, but also by the date and state in which the commodity isdelivered. Thus, allocational efficiency implies not only that peoplewithin a household consume apples and oranges in the correct propor-tion, but also that within the household there is full insurance. Thetests we conduct here are really a joint test of these two sorts of allo-cational efficiency (allocation of ordinary commodities, and allocationof state-date contingent commodities).

Consider referring to discussion of collective allocationsin dynamic models in conclusion of ?.

Add ¶here on how importance of individual characteris-tics and nutritional investment makes this exercise dif-ferent from, e.g. Townsend (1994).

Without pretending any sort of exhaustive comparison of our pa-per with existing literature, we will briefly describe two papers, eachof which shares (different) points of similarity with the present paper.Dercon and Krishnan (2000) test the hypothesis of full intra-householdrisk-sharing in Ethiopia by looking at the response of individual nu-tritional status to illness shocks. In order to deal with limitations oftheir data, they assume that utility depends on food consumption only


via anthropometric status. So, for example, children are implicitly as-sumed to be indifferent between consuming a varied diet with fruit,meat, and vegetables and a subsistence diet of beans, provided thateither diet results in similar weight-for-height outcomes. With thisassumption, Dercon and Krishnan reject intra-household efficiency, atleast for poorer households, but their results are also consistent withefficient intra-household allocation if people derive utility directly fromfood consumption. Our data allow us to distinguish between thesepossibilities, and so we allow individual utility functions to dependon consumption both directly and via the influence of consumptionon nutritional outcomes. Using the same dataset as we do, Foster andRosenzweig (1994) doesn’t address the question of intra-household allo-cation at all, but rather asks whether or not individual anthropometricmeasures depend on the nature of the contract governing compensa-tion for off-farm work, interpreting this as a test for the importance ofincentives. As in Dercon and Krishnan (2000), Foster and Rosenzweigassume that food only influences utility to the extent that it influ-ences measures of weight for height, but find that indeed incentivesprovided in the workplace outside the household influence consump-tion and physical status. In contrast to Foster and Rosenzweig, ourfocus is on the allocation of goods within the household, and on therole that food consumption may play in providing incentives above andbeyond the determination of weight and height.

We proceed as follows. First, we provide an extended descriptionof the data in Section 2. We describe some patterns observed in thesharing rules of Philippino households, including expected levels of con-sumption, and both individual and household-level measures of risk inboth consumption and income.

Second, in Section 3 we formulate a sequence of simple models, eachcorresponding to a dynamic program which characterizes the problemfacing the household head in different environments. The first modelis a simple ‘naive collective’ program, in which utility depends on con-sumption, but productivity does not. The head allocates consump-tion goods, makes investment decisions, and assigns activities to otherhousehold members. From this model we derive simple restrictionson household members’ intertemporal marginal rates of substitution.Working with a parametric representation of individuals’ utility func-tions, we exploit these restrictions to estimate a vector of preferenceparameters, which allows us to characterize changes in intra-householdsharing rules as a function of observable individual characteristics suchas age and sex.


Our second model generalizes the first, in that we allow for the pos-sibility that food consumption may influence future productivity. Inparticular, while food consumption produces both direct utility (whichdepends on the quantity and quality of different kinds of foodstuffs),and also represents a sort of human capital investment which influenceslabor productivity (but this investment depends only on the quantityand nutritional content of foodstuffs, and not food quality). This leadsus to consider a model of nutritional investments, which reproducessome of the features of models formulated by, e.g., Pitt et al. (1990);Pitt and Rosenzweig (1985). In this model there is no private informa-tion and hence no need to provide incentives, but the head takes intoaccount the effects that consumption has on both utility and productiv-ity. This model also implies a set of restrictions on household members’intertemporal marginal rates of substitution which distinguish it fromthe ‘naive collective’ model. A key testable prediction of this modelwhich distinguishes it from the naive collective model is that if there’san anticipated increase in the marginal product of labor for householdmember i, then nutritional investment in this member will increase atthe same time that the quality of food consumed by i decreases.

Finally, our third model extends the model with nutritional invest-ment so that the off-farm labor effort of other household members isn’tnecessarily observed by the household head.1 Accordingly, the intra-household sharing rule must be incentive compatible. The key differ-ence between this model and the naive collective model or the nutri-tional investment model is that household members must be providedwith appropriate incentives to induce them to take the actions recom-mended by the household head. We show that in this model of efficientintra-household incentives food quality should respond to unpredictedindividual earnings shocks. Section 5 concludes.

2. The Data

The main data used in this paper are drawn from a survey conductedby the International Food Policy Research Institute and the ResearchInstitute for Mindanao Culture in the Southern region of the Bukid-non Province of Mindanao Island in the Philippines during 1984–1985.These data are described in greater detail by Bouis and Haddad (1990)and in the references contained therein. Additional data on weather

1“Off-farm labor” in this context means agricultural work on somebody else’sfarm. Using the same dataset, Foster and Rosenzweig (1994) find evidence whichsuggests that the managers of such workers can’t observe labor effort, so that pre-sumably the geographical remote household head can’t observe this effort either.

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Expend. Rice Corn Staples Meat Veg. Snacks Calories ProteinSample 5.914 0.724 1.203 0.3 1.876 0.477 0.809 1823.626 54.664Male 6.428 0.802 1.271 0.295 1.984 0.484 1.068 1926.152 57.508Female 5.361 0.64 1.13 0.306 1.76 0.47 0.53 1713.438 51.606≤ 5 years 3.802 0.398 0.727 0.226 1.407 0.241 0.386 1137.078 34.8866–10 years 4.792 0.607 1.044 0.276 1.629 0.362 0.422 1603.433 48.10211–15 years 6.878 0.872 1.51 0.37 2.239 0.62 0.632 2232.415 66.05616–25 years 7.929 1.061 1.606 0.362 2.271 0.758 1.238 2412.472 71.97526–50 years 8.879 1.009 1.612 0.359 2.491 0.693 2.074 2416.091 72.495> 50 years 7.119 0.877 1.44 0.247 1.875 0.567 1.363 2136.464 61.653≤ 5 years (Male) 3.719 0.419 0.733 0.23 1.405 0.216 0.293 1166.963 35.4766–10 years (Male) 4.943 0.671 1.04 0.269 1.712 0.366 0.444 1638.98 48.92611–15 years (Male) 7.107 0.924 1.669 0.379 2.279 0.582 0.719 2360.029 69.03616–25 years (Male) 9.465 1.21 1.828 0.355 2.623 0.837 1.956 2688.411 80.57426–50 years (Male) 10.411 1.18 1.769 0.348 2.69 0.74 3.007 2653.379 79.364> 50 years (Male) 7.96 1.039 1.497 0.229 1.944 0.626 1.732 2300.727 66.907≤ 5 years (Female) 3.901 0.372 0.72 0.221 1.409 0.272 0.497 1101.531 34.1846–10 years (Female) 4.638 0.54 1.048 0.283 1.544 0.357 0.399 1566.915 47.25611–15 years (Female) 6.657 0.822 1.357 0.362 2.201 0.657 0.549 2109.172 63.17916–25 years (Female) 6.614 0.934 1.415 0.367 1.97 0.69 0.623 2176.366 64.61726–50 years (Female) 6.858 0.783 1.406 0.374 2.228 0.63 0.844 2102.985 63.43> 50 years (Female) 4.573 0.39 1.269 0.302 1.667 0.391 0.248 1639.464 45.756

Table 1. Mean Daily Food Consumption. The first column reports mean total food expendituresper person (in constant Philippine pesos). The next six columns report means for particular sortsof food expenditures (differences between total food expenditures and the sum of its constituents isaccounted for by “other non-staple” foods). The final two columns report individual calories andprotein derived from individual-level food consumption.


used in this paper were collected by the first author from the weatherstation of Malay-Balay in Bukidnon.

Bukidnon is a poor rural and mainly agricultural area of the Philip-pines. Early in 1984, a random sample of 2039 households was drawnfrom 18 villages in the area of interest. A preliminary survey wasadministered to each household to elicit information used to developcriteria for a stratified random sample later selected for more detailedstudy. The preliminary survey indicated that farms larger than 15hectares amounted to less than 3 per cent of all households, a figurecorresponding closely to the 1980 agricultural census. Only householdsfarming less than 15 hectares and having at least one child under fiveyears old were eligible for selection. Based on this preliminary survey, astratified random sample of 510 households from ten villages was cho-sen. Some attrition (mostly because of outmigration) occurred duringthe study and a total of 448 households from ten villages finally partic-ipated in the four surveys conducted at four month intervals beginningin July 1984 and ending in August 1985. The total number of personsin the survey is 3294.

The nutritional component of the survey interviewed respondents toelicit a 24-hour recall of individual food intakes, as well as one monthand four month interviews to measure household level food and non-food expenditures. Food intakes include quantity information for ahighly disaggregated set of food items. Individual food expenditurescan be computed using direct information on the prices and quantitiesof foods purchased, and on quantities consumed out of own-productionand in-kind transactions.

Later in the paper we will concern ourselves with changes in in-dividuals’ shares of consumption, intentionally neglecting to explaindifferences in levels of consumption, where theory has less to say. How-ever, some of these differences are interesting, and so some informationon levels of individual expenditures along with caloric and protein in-takes are given in Table 1. Turning to the final columns of the table,we first note that the average individual in our sample is not terriblywell-fed. Comparing the figures in Table 1 to standard guidelines forenergy-protein requirements (WHO, 1985) reveals that even the aver-age person in our sample faces something of an energy deficit.

When we consider the average consumption of different age-sex groups,it becomes clear that particular groups are particularly malnourished.Also, these figures show clearly that the relationship between consump-tions and age follows consistently an inverse U shaped pattern whichis quite reassuring about the reliability of these measures.


The picture of inequality drawn by our attention to energy andprotein intakes is, if anything, exacerbated by closer attention to thesources of nutrition. While all of the foods considered here are sourcesof calories and protein, it also seems likely that food consumption isvalued not just for its nutritive content, but that individuals also de-rive some direct utility from certain kinds of consumption. This pointreceives some striking support from Table 1. Consider, for example, av-erage daily expenditures by males aged 26–50, compared with the samecategory of expenditures by women of the same age. The value of ex-penditures on male consumption of all staples is 28 per cent greaterthan that of females of the same age. This difference seems smallenough that it could easily be attributed to differences in activity ormetabolic rate. However, compare expenditures on what are presum-ably superior goods: expenditures on male consumption of meat (andfish), vegetables, snacks (including fruit) is 424 per cent greater thanthe corresponding expenditures by women in the same age group. Sincenothing like a difference of this size shows up in calories or protein, thisseems like very strong evidence that intra-household allocation mech-anisms are designed to put a particularly high weight on the utility ofprime-age males relative to other household members, quite indepen-dent of those prime-age males’ greater energy-protein requirements.Note that although these differences in consumption seem to point toan inegalitarian allocation, these differences provide no evidence tosuggest that household allocations are inefficient.

3. Some Simple Models

Consider a household having n members, indexed by i = 1, 2, . . . n,where an index of 1 is understood to refer to the household head. Timeis indexed by t = 0, 1, 2, . . . . During each period, member i consumes aquantity of nutrients cit. These nutrients have a corresponding qualityϕit, assumed to be bounded away from zero by ϕ > 0. At the sametime, i supplies some labor ait.

Household member i derives direct utility from both the quantityand quality of his consumption and disutility from his labor. Fur-ther, at time t person i possesses a set of characteristics (e.g., sex,health, age) which we denote by the L-vector bit. These characteristicsmay have an influence on the utility he derives from both consump-tion and activities. Thus, we write his momentary utility at t as someU(ϕit, cit, bit) + Zi(ait, bit), where the function U is assumed to be in-creasing, concave, and continuously differentiable in both the quantityand quality of nutrients, and where Zi is the disutility of labor ait for


an individual with characteristics bit. Future utility is discounted viaa common discount factor β ∈ (0, 1).

3.1. Stochastic Structure. Households face two basic sources of un-certainty. First, in each period t an index of some public shock θt ∈Θ = 1, 2, . . . , S is realized; among other things, the realization of θt

determines the prices faced by the household. Note that while pub-licly observed, this shock need not be common, including e.g., infor-mation on expected wages for different individuals. The probabilityof some particular θ′ occurring at time t may depend on the previ-ous period’s realization θt−1 via a collection of Markovian transitionprobabilities Pr(θt|θt−1), while the cumulative probability that the re-alization of θt will be less than or equal to some value θ′ is written

G(θ′|θ) =∑θ′

r=1 Pr(r|θ).The second source of uncertainty faced by households is produc-

tion or earnings uncertainty. Each period t, each household memberi = 1, . . . , n supplies labor ait, and produces some quantity of thenumeraire good yit ∈ R at t. Each of these outputs is drawn from acontinuous distribution F i(yi|a, z, θ′) which depends on the labor a andinvestment z supplied and on the public shock θ′. We assume that forevery (a, z, θ′) the corresponding density f i(yi|a, z, θ′) exists, and thatthe support of yi doesn’t vary with a. We further assume that f i is acontinuously differentiable function of a, and denote such derivativesby f i

a(yi|a, z, θ′).As the notation is meant to suggest, output yit is assumed to be

conditionally independent of yjs for all (i, t) 6= (j, s). Accordingly, usingbold characters to denote lists of variables for each family member,let y = (y1, y2, . . . , yn) denote the list of outputs for each member ofthe family, with joint distribution F (y|a,z, θ′) =

∏n

i=1 F i(yi|ai, zi, θ′)

and joint density f(y|a,z, θ′). Denote by ω the collection of randomelements which affect household i at the end of a period, so that ω =(y, θ′); for the sake of brevity, let H(ω|a,z, θ) = F (y|a,z, θ′)G(θ′|θ).

The individual characteristics of an individual are allowed to evolveover time. Let Mi be a law of motion for the vector of characteristicsof individual i, with bit+1 = Mi(bit, cit, θt), and let M ℓ

i (bit, cit, θt) denotethe law of motion for the ℓth characteristic. Similarly, let bt denotethe list of individual characteristics at t, with M the law of motion forcharacteristics for all n family members. The idea here is that e.g.,person i’s weight at t + 1 may depend on his weight in the previousperiod as well as on his consumption. Note that the evolution of bit

is assumed not to depend on the quality of consumption, but only onits quantity. We’ll think of consumption c as nutrients. A variety of


different diets could plausibly provide the same quantity of nutrientsc; however, not all of these diets will provide the same level of utility.

The implicit price of nutrients may depend on quality and the publicshock. When the public shock is θ′, we write the the price of a unit ofnutrients having quality ϕ as p(ϕ, θ′).

Assumption 1. Let Φ = [ϕ, ϕ] ⊂ R. We make the following assump-tions regarding the manner in which prices vary with quality:

(1) p : Φ×Θ is a continuously differentiable function of quality; wedenote this derivative by p′(ϕ, θ′).

(2) p is a strictly increasing function of quality.(3) p(ϕ, θ′) > 0 for all θ′ ∈ Θ.(4) The quality elasticity η(ϕ, θ′) ≡ p′(ϕ, θ′)/p(ϕ, θ′) is either strictly

increasing in ϕ or is strictly decreasing in ϕ.(5) For any θ′ ∈ Θ there exists some ϕ ∈ Φ such that η(ϕ, θ′) = 1.

The first three parts of this assumption are commonsensical: Theprices of even low quality nutrients must always be positive, and higherquality nutrients must cost more. The last two parts are more restric-tive, but turn out to be important if wishes to avoid corner solutionsto the problem of choosing nutrient quality. A simple example pricefunction which satisfies Assumption 1 is p(ϕ, θ′) = g(θ′)(ϕ + ϕ)ρ, withboth g(θ′) and ρ positive.

3.2. The Household Head’s Problem. The household head decidesthe labor each household member should supply, how much the house-hold should collectively save or invest, how to allocate the remaininghousehold resources across household members, and what resourcesshould be promised to individual family members in the future.2 Weformulate the problem as a dynamic program, adopting an approachsimilar to e.g., Spear and Srivastava (1987) in which future ‘utilitypromises’ are made by the head to individual family members and ap-pear as state variables in the head’s dynamic programming problem.

The head chooses the allocation of consumption only after observingthe realization of ω, so that consumptions are assigned after individualoutputs are determined and the public shock θ′ is observed. Allocating

2There is a small literature devoted to the subject of division of responsibilitiesbetween adult males and females in Philippine households (e.g., Illo, 1995; Eder,2006). Though unfortunately we can’t offer independent evidence from our dataset,this list of tasks seems to be consistent with the kinds of responsibilities typicallyundertaken by adult females, rather than males. Accordingly, we assume below thatthe senior female in the household is the household head unless noted otherwise.


consumption involves choosing both a vector of nutrients ci(ω) to awardto person i in state ω as well as corresponding qualities ϕi(ω).

If in period t the public shock was θt, then for any time t + 1 real-ization of ω the head must satisfy the budget constraint

(1)n∑

i=1

p(ϕi,t+1(ωt+1), θt+1)ci,t+1(ωt+1) ≤n∑

i=1

yi,t+1 −n∑

i=1

zi,t+1(ωt).

At the same time that the head allocates contemporaneous consump-tion, she also makes promises to other household members about theirfuture levels of (discounted, expected) utility. Past promises must alsobe honored. Thus, if the head promised future utils wit to person i atdate t − 1, then honoring that promise requires that(2)∫

[U(ϕit(ωt), cit(ωt), bit) + Zi(ait, bit) + βwit+1(ωt)]dH(ωt|at, θt) = wit.

We formulate the problem facing the head recursively. At the be-ginning of a period, the head takes as given an n-vector reflecting hercurrent utility promises to other household members (w), investmentsfrom the previous period (z), a list of the characteristics of householdmembers (b), and the previous period’s public shock θ. Given her pref-erences, the head then assigns labor, decides how much to save/investfor subsequent periods, and makes another set of utility promises, allsubject to the constraints implied by these prices and resources. Inparticular, let V (w,z, b, θ) denote the discounted, expected utility ofthe head given the current state, and let this function satisfy

Program 1.

(3)

V (w,z, b, θ) = maxai,z

′

i,ϕi(ω),ci(ω),w′

i(ω)ω∈Ωn

i=1

∫

[U(ϕ1(ω), c1(ω), b1) + Z1(a1, b1)

+βV (w′(ω),z′,M (b, c(ω), θ′))] dH(ω|a,z, θ)

subject to the household budget constraint

(4)n∑

i=1

p(ϕi(ω), θ′)Tci(ω) ≤n∑

i=1

yi −

n∑

i=1

z′i

for all ω = (y, θ′) ∈ Ω, and subject also to a set of promise-keepingconstraints

(5)

∫

[U(ϕi(ω), ci(ω), bi) + Zi(ai, bi) + βw′i(ω)]dH(ω|a,z, θ) = wi,


for all i = 1, . . . , n; and subject finally to the requirement ϕi(ω) ∈ Φfor all i = 1, . . . , n and all states ω ∈ Ω.

In this recursive formulation of the problem we’re able to drop thetime subscripts from variables: all variables may be assumed to bedated t unless adorned by the notation ′, in which case these are datedt + 1.

Lemma 1. Let αit = ∂V∂wi

(wt,zt, bt, θt) in Program 1. If U(ϕ, c, b) is a

concave function of (ϕ, c) for all b, then αit = αit+1 for all dates t.

Proof. The envelope condition with respect to wi in Program 1 impliesthat ∂V

∂wi

(wt,zt, bt, θt) ≡ αit is equal to the Lagrange multiplier on the

promise-keeping constraint (5). This fact along with the first ordercondition for the head’s problem with respect to wi(ωt+1) then implythat αit = αit+1. The concavity of U is a sufficient condition for thisfirst order condition to characterize the solution to the head’s problem.

The lemma simply reflects the point that in environments such asthat described here, optimal allocations of consumption will keep theratio of marginal utilities of different household members constantacross both dates and states; this is basically a consequence of riskaversion and our assumption of time separable expected utility.

We want to consider the outcomes predicted by this simple modelunder a variety of different assumptions regarding preferences and otheraspects of the environment facing the household. We begin by assuminga particular parametric form for the utility function U , adopting

Assumption 2. We place the following restrictions on the form of theutility function for person i:

(1) U(ϕ, c, b) = h(b)ui(ϕc);(2) ui : R → R is strictly increasing, strictly concave, and continu-

ously differentiable; and(3) limxց0 u′

i(x) = ∞.

The first part of this assumption implies that the utility functionsatisfies a form of separability between characteristics b and the qualityand quantity of consumption; also, that the effect that characteristicshave on the marginal utility of consumption take as common formacross different people. The first part of Assumption 2 also implies thatthe utility of quantity and quality of nutrients depends on the product


of these two dimensions of the consumption bundle.3 The second partof the assumption is entirely standard, while the third is one of the‘Inada’ conditions which allows us to avoid having to worry about thepossibility that the household head might assign zero consumption toany household member.

We now turn our attention to characterizing the relationship be-tween consumption, expenditures, and quality for various householdmembers relative to the household head, under a variety of alternativeassumptions. These are summarized by

Proposition 1. If preferences satisfy Assumption 2 then the consump-tion allocations which solve Program 1 will satisfy

(6) ∆ logu′

1(ϕ1tc1t)

u′i(ϕitcit)

= ∆ logh(bit)

h(b1t)− ∆ log

p′(ϕit, θt+1)

p′(ϕ1t, θt+1)

for all i = 1, . . . , n, and all dates t.

Proof. For any i = 2, . . . , n, let αi denote the Lagrange multiplier as-sociated with the head of household’s corresponding promise-keepingconstraint (5). We adopt the convention that α1 ≡ 1.

Then the first order conditions from the head’s problem for ϕi can berearranged to yield the following expression for person i’s consumptionquantity in state ω:

(7) ci(ω) =αi

µ(ω)

∂U(ϕi(ω), ci(ω), bi)

∂ϕ

1

p′(ϕi(ω), θ′)

where µ(ω) is the Lagrange multiplier of the collective household bud-get constraint when the public shock is ω, and where p′(·, θ′) is thepartial derivative of the price function with respect to quality ϕ.

Dividing the expression given by (7) for the consumption quantityof person i > 1 by the corresponding expression for i = 1 yields

ci(ω)

c1(ω)= αi

∂U(ϕi(ω), ci(ω), bi)/∂ϕ

∂U(ϕ1(ω), c1(ω), b1)/∂ϕ

p′(ϕ1(ω), θ′)

p′(ϕi(ω), θ′).

Fix a particular period t; Lemma 1 implies that αi is invariant overtime. Then taking logarithms and differences across adjacent time

3Assuming that utility depends on the product of quantity and quality is naturalbut also restrictive. It’s possible to relax this assumption considerably, by assum-ing, for example, that utility depends on an idiosyncratic Cobb-Douglas function ofquantity and quality. The addition of this additional idiosyncratic element of indi-vidual preferences requires only modest changes to our results, which are availablefrom the authors on request.


periods gives us

(8) ∆ logcit

c1t

= ∆ log∂U(ϕit, cit, bit)/∂ϕ

∂U(ϕ1t, c1t, b1t)/∂ϕ+ ∆ log

p′(ϕ1t, θt+1)

p′(ϕit, θt+1).

Using Assumption 2, the first term on the right-hand side of this ex-pression is equal to

∆ logh(bit)

h(b1t)+ ∆ log

cit

c1t

+ ∆ logu′

i(ϕitcit)

u′1(ϕ1tc1t)

.

Substituting this into (8) yields the result.

When additional nutrients influence productivity, one can think ofconsumption as a sort of nutritional investment. The returns to thisinvestment depend both on the marginal product of characteristics band on the marginal impact that nutrition has on these characteris-tics. The marginal return to a nutritional investment in person i willgenerally depend on the utilities promised to all the members of thehousehold w, other investments z, current characteristics b, the shockθ, i’s consumption ci, and the household head’s current marginal utilityof income, which we write as µ. Then the marginal return to additionalnutritional investment in person i can be written

Ri(ci,w,z, b, θ) =1

µ

L∑

ℓ=1

∂V

∂bℓi

(w,z,M (b, c, θ), θ)∂M ℓ

i

∂ci

(b, c, θ).

Now consider the case in which there’s no return nutritional invest-ment (food consumption doesn’t influence future characteristics). Weexpress this case using the following

Assumption 3. The law of motion for characteristics M(b, c, θ′) =M(b, c, θ′) for all (b, θ′) and all c, c ∈ R.

Note that when Assumption 3 is satisfied, the returns to nutritionalinvestment will be zero.

Proposition 2. If the price function satisfies Assumption 1, prefer-ences satisfy Assumption 2 and Assumption 3 is satisfied, then the al-located quality of nutrients which solves Program 1 will not vary acrossindividuals in the household, and the allocated quantities of nutrientswill satisfy

(9) ∆ log cit =γ1

γi

∆ log c1t +1

γi

∆ logh(bit)

h(b1t)

for all i = 1, . . . , n, and all dates t.


Proof. We begin by showing that ϕi = ϕ1. The first order condition ofProgram 1 with respect to ci(ω) is

αi

∂U

∂c(ϕi(ω), ci(ω), bi)−µ(ω)p(ϕi(ω), θ′)+βµ(ω)Ri(ci(ω),w(ω),z′, b, θ′) = 0.

However, Assumption 3 implies that the term involving Ri is equal tozero. Using this fact along with Assumption 2 and re-arranging yields

(10) αih(bi)ϕi(ω)

u′i(ϕi(ω)ci(ω))

= µ(ω)p(ϕi(ω), θ′).

The first order condition with respect to ϕi(ω) is

αi

∂U

∂ϕ(ϕi(ω), ci(ω), bi) − µ(ω)p′(ϕi(ω), θ′)ci(ω) = 0,

which, so long as ci(ω) > 0, can be similarly re-arranged to yield

(11) αih(bi)1

u′i(ϕi(ω)ci(ω))

= µ(ω)p′(ϕi(ω), θ′).

Assumption 1 guarantees that the price p(ϕ, θ′) and its partial deriva-tive with respect to ϕ will always be positive. Then dividing (10) by(11) and rearranging yields the result that

p′(ϕi(ω), θ′)

p(ϕi(ω), θ′)ϕi(ω) = 1.

The last two parts of Assumption 1 guarantee that this equation hasa unique solution. Then since the price function is common to allhousehold members, the solution must be the same for all householdmembers.

Next, we prove that (9) characterizes the allocation of quantitiesof nutrients. But since qualities are identical within the household,the ratio p′(ϕi, θ

′)/p′(ϕ1, θ′) = 1. Using this fact along with (6) from

Proposition 1 yields (9).

We next turn our attention to characterizing within-household con-sumption allocations when the labor of members can’t be directly ob-served by the household head. In this situation, the problem facing thehead can be expressed as

Program 2.

(12) V (w,z, b, θ) = maxai,z

′


i(ω)ω∈Ωn

i=1

∫

[U(ϕ1(ω), c1(ω), b1)

+Z1(a1, b1) + βV (w′(ω),z′,M (b, c(ω), θ′))] dH(ω|a,z, θ)


subject to the household budget constraint (4) for all ω = (y, θ′) ∈ Ω;and subject also to the set of promise-keeping constraints (5). In addi-tion, the household head must choose allocations and labor assignmentswhich satisfy a set of incentive compatibility constraints

(13) ai ∈ arg maxa

∫

[U(ϕi(ω), ci(ω), bi) + Zi(a, bi) + βwi(ω)]

· d

(

F i(yi|a, zi, θ′)∏

j 6=i

F j(yj|aj, zj, θ′)G(θ′|θ)

)

for all i = 1, . . . , n; and subject finally to a requirement that ϕki (ω) ∈ Φ

for all i = 1, . . . , n; all nutrients k = 1, . . . , K; and all states ω ∈ Ω.

Program 2 is identical to Program 1 save for the addition of the in-centive compatibility constraint (13), which requires that person i willhave no incentive to deviate from the action ai recommended by thehead. Unfortunately, it’s difficult to characterize the effects of this kindof hidden action when the requirement of incentive compatibility is ex-pressed in the form of (13). Accordingly, we also provide an alternative‘relaxed’ program, which will have the same solutions so long as theso-called ‘first order approach’ is valid (Arpad Abraham and Pavoni,2008).

Program 3.

(14) V (w,z, b, θ) = maxai,z

′


i(ω)ω∈Ωn

i=1

∫

[U(ϕ1(ω), c1(ω), b1)

+Z1(a1, b1) + βV (w′(ω),z′,M (b, c(ω), θ′))] dH(ω|a,z, θ)

subject to the household budget constraint (4) for all ω = (y, θ′) ∈ Ω;and subject also to the set of promise-keeping constraints (5). In addi-tion, the household head must choose allocations and labor assignmentswhich satisfy the first-order conditions from the agent’s choice of labor,

(15)

∫

[U(ϕi(ω), ci(ω), bi) + Zi(a, bi) + βwi(ω)]f i

a(yi|ai, zi, θ′)

f i(yi|ai, zi, θ′)

· d (F (y|a,z, θ′)G(θ′|θ)) = −∂Zi(ai, bi)

∂ai

for all i = 1, . . . , n; and subject finally to a requirement that ϕki (ω) ∈ Φ

for all i = 1, . . . , n; all nutrients k = 1, . . . , K; and all states ω ∈ Ω.

Assumption 4. Any solution to the “relaxed” Program 3 is also asolution to Program 2.


Lemma 2. Let αit = ∂V∂wi

(wt,zt, bt, θt) in Program 3. If Assumption 4

is satisfied, then αit = αit−1 + νitf i

a(yit|ait,zit,θt+1)f i(yit|ait,zit,θt+1)

for all dates t.

Proof. The envelope condition with respect to wi in Program 1 im-plies that ∂V

∂wi

(wt,zt, bt, θt) ≡ αit is equal to the Lagrange multiplier

on the promise-keeping constraint (5). This fact along with the firstorder condition for the head’s problem with respect to wi(ωt+1) then

imply that αit+1 = αit +νitf i

a(yit|ait,zit,θt+1)f i(yit|ait,zit,θt+1)

. By Assumption 4, these first

order conditions to characterize the solution to the head’s problem inProgram 2.

Proposition 3. If preferences satisfy Assumption 2 and Assumption 4then the consumption allocation for person i at time t solving Program2 will satisfy

(16) ∆ logu′

1(ϕ1tc1t)

u′i(ϕitcit)

= ∆ logh(bit)

h(b1t)− ∆ log

p′(ϕit, θt+1)

p′(ϕ1t, θt+1)

+ log

[

1 +νit

αit

f ia(yit|ait, zit, θt+1)

f i(yit|ait, zit, θt+1)

]

for some non-negative scalar νit.

Proof. Let αit denote the Lagrange multiplier associated with the promise-keeping constraint (5) for person i in period t. Then the first orderconditions from the head’s problem for ϕi and ϕ1 imply that(17)

∂U(ϕit, cit, bit)/∂ϕ

∂U(ϕ1t, c1t, b1t)/∂ϕ

[

αit + νit

f ia(yit|ait, zit, θt+1)

f i(yit|ait, zit, θt+1)

]

=p′(ϕit, θt+1)

p′(ϕ1t, θt+1)

cit

c1t

,

where νit is the Lagrange multiplier associated with the constraint (??).Lemma 2 implies that the expression in brackets on the left-hand handside of (18) is equal to αit+1; using this fact, we can write

(18) αit+1 =∂U(ϕ1t, c1t, b1t)/∂ϕ

∂U(ϕit, cit, bit)/∂ϕ

p′(ϕit, θt+1)

p′(ϕ1t, θt+1)

cit

c1t

.

Using Assumption 2, this becomes

αit+1 =u′

1(ϕ1tc1t)

u′i(ϕitcit)

p′(ϕit, θt+1)

p′(ϕ1t, θt+1)

h(b1t)

h(bit).


Combining this with (??) it follows that

u′1(ϕ1tc1t)

u′i(ϕitcit)

p′(ϕit, θt+1)

p′(ϕ1t, θt+1)

h(b1t)

h(bit)=

u′1(ϕ1t−1c1t−1)

u′i(ϕit−1cit−1)

p′(ϕit−1, θt)

p′(ϕ1t−1, θt)

h(b1t−1)

h(bit−1)

+ νit−1f i

a(yit|ait, zit, θt+1)

f i(yit|ait, zit, θt+1).

Dividing this expression on both sides by αit, taking logarithms, andrearranging yields the result, with νit = νit/αit−1.

We also have a corresponding characterization governing the assign-ment of quality, given by the following proposition.

Proposition 4. If prices satisfy Assumption 1, preferences satisfy As-sumption 2, and Assumption 4 holds then any interior quality assign-ment which solves Program 2 will satisfy

(19) ϕi(ω) =p(ϕi(ω), θ′)

p′(ϕi(ω), θ′)−

β

p′(ϕi(ω), θ′)Ri(ci(ω),w(ω),z′, b, θ′).

Proof. The first order condition of Program 2 with respect to ci(ω) is

∂U

∂c(ϕi(ω), ci(ω), bi)

[

αi + νi

f ia(yi|ai, zi, θ

′)


]

− µ(ω)p(ϕi(ω), θ′)

+ βµ(ω)Ri(ci(ω),w(ω),z′, b, θ′) = 0.

Using Assumption 2 and re-arranging, this yields

(20) h(bi)ϕi(ω)

[ϕi(ω)ci(ω)]γi

[

αi + νi

f ia(yi|ai, zi, θ

′)


]

= µ(ω)p(ϕi(ω), θ′) − βµ(ω)Ri(ci(ω),w(ω),z′, b, θ′).

The first order condition with respect to ϕi(ω) is

∂U

∂ϕ(ϕi(ω), ci(ω), bi)

[

αi + νi

f ia(yi|ai, zi, θ

′)


]

−µ(ω)p′(ϕi(ω), θ′) = 0,

which, so long as ci(ω) > 0, can be similarly re-arranged to yield

(21) h(bi)1

[ϕi(ω)ci(ω)]γi

[

αi + νi

f ia(yi|ai, zi, θ

′)


]

= µ(ω)p′(ϕi(ω), θ′).

Dividing (20) by (21) then yields the result.

Proposition 4 allows us to characterize the assignment of qualitywithin the household when there is both nutritional investment andalso hidden actions. What if there are hidden actions but no nutritionalinvestment?


Information

Full information Hidden actionsConsumption influ- No Naive Collective Incentivesences productivity Yes Nutritional Investment Incentives & Investment

Table 2. Connection between regimes and model characteristics.

Predicted Earnings Unpredicted EarningsModel Quantity Quality Quantity Quality

Naive Collective 0 0 0 0Incentives − 0 + 0

Nutritional Investment + − + −Incentives & Investment + − + +/−

Table 3. Correlations between predicted or unpre-dicted changes in earnings and the quantity and qualityof nutrients in different regimes.

Corollary 1. If the assumptions of Proposition 4 are satisfied andin addition Assumption 3 holds, then assigned quality will not varyacross members of the household, and the provision of incentives willbe achieved only via changes in the quantity of nutrients, not theirquality.

4. Empirical Tests

4.1. Distinguishing among regimes. The model we’ve describedabove gives rise to four different possible regimes, depending on whetheror not consumption influences subsequent productivity (Assumption 3)and on whether or not household members take actions which are hid-den from the household head. Table 2 summarizes the connectionsbetween model characteristics and regimes, while Table 3 describes thepattern of correlations between both predicted and unpredicted changesin earnings with both the quality and quantity of nutrients. It’s ulti-mately differences in the pattern of these correlations predicted whichallows us to advance a claim regarding the regime which actually pre-vails in the real-world environment which generates the data describedin Section 2.

In the naive collective regime current consumption contributes onlyto utility, and doesn’t influence productivity or other characteristics.This regime is analogous to that envisioned in most research on in-ter -household risk-sharing, such as Townsend (1994). Idiosyncratic


earnings shocks will be efficently shared in the naive collective regime,with the consequence that these shocks will have no influence on per-son i’s consumption relative to the consumption of the household head(per Proposition 1). Further, since in this regime consumption has noinfluence on productivity or other characteristics of household mem-bers, then the assigned quality of nutrients will be the same across allmembers of the household.

In the incentives regime current consumption contributes only toutility, and doesn’t influence productivity or individual characteristics.However, because actions are hidden, the household head may awardadditional consumption to members with unexpectedly high earningsas a way of preventing shirking or reducing consumption for householdmembers with earnings which are less than expected. Thus, idiosyn-cratic earnings shocks may be correlated with changes to quantities ofindividual consumption (relative to the household head), via Proposi-tion 3. Perhaps suprisingly, the quality of nutrients will be constantacross household members, as in the naive collective model—it’s opti-mal to provide incentives only by varying the quantity of nutrients, notquality (cf. Proposition 4).

The third regime, nutritional investment, is one in which the allo-cation of nutrients affects not only the utility of different householdmembers, but also the production possibility set of the household. Inthis model, the allocation of energy and protein in the household mayrespond to changes in the expected marginal productivity of actionsfor a particular household member. The most obvious example mighthave to do with the additional energy required by some household mem-bers during different seasons: household members who engage in heavyagricultural labor may be assigned a disproportionate share of caloriesduring the harvest season, for example, or these same people may re-ceive a greater share of protein in advance of a period of hard labor.Thus, in the nutritional investment regime our model predicts that thequantity of nutrients consumed will be positively correlated with earn-ings shocks. Perhaps more surprisingly, the model predicts that in thisregime the quality of nutrients will be negatively correlated with thoseshocks, per Proposition ??. The basic idea is that since the householdmember is being given additional quantities of food (which increasesnot only productivity but also utility), the household head can reducethe quality of food while still making good on ex ante utility promises.

The fourth regime combines the attributes of the nutritional invest-ment regime and the incentives regime, and so we title it incentivesand investment. In this regime consumption is allocated both as aninvestment and to provide incentives for household members to exert


themselves. In this regime our model predicts that unpredictable earn-ings shocks will be positively correlated with quantities of nutrients.Further, when one considers the sign of correlation between predictableearnings shocks and quality, the prediction of the model in the in-vestment and incentives regime is unambiguous—predictable earningsshocks should be negatively correlated with quality.

Though in the investment and incentive regime predictable earn-ings shocks will be negatively correlated with consumption quality, thedirection of the effect of unpredictable earnings shocks on quality isambiguous. Increased quantities of nutrition assigned to a worker forinvestment purposes will lead the household to reduce the quality as-signed, but the efficient provision of incentives in this regime calls foran increase in quality, so the ultimate sign of the correlation betweenunexpected earnings shocks and quality depends on which of the in-vestment or incentives effects dominate.

4.2. Testing for Nutritional Investment. Proposition 3 and Propo-sition 4 provide us with the basis for the estimating equations whichpermit us to distinguish among regimes and to estimate the parametersof the intra-household demand system when there is either nutritionalinvestment or the provision of incentives in the face of hidden actions.

We first consider how to test the hypothesis that there is no nutri-tional investment, by using the result of of Proposition 4. Assumption3 holds by assumption, so that (19) implies that

(22) ϕit = ϕ1t,

where the subscripts t indicate that the relationship holds given thestate ω realized at t. (22) takes advantage of our result Corollary ??

that in the absence of any motive for making nutritional investmentsincentives are most efficiently provided by varying the quantity of con-sumption rather than its quality.

In the absence of nutritional investments, quality-adjusted pricesp(ϕit, θt) will not vary within the household. However, the quality-adjusted prices pit which we actually observe in the data (e.g., expen-ditures per Calorie) are measured with a multiplicative error having anunknown mean. We assume that the distribution of the measurementerror is stationary and independent over time; also, that its logarithmis mean independent of other observables. Then taking logarithms anddifferences of (22) gives the system of estimating equations

(23) ∆ log pit = β0 + β1∆ log p1t + eit,


Calorie cost Protein cost F (p-value)Head’s quality × i’s sex: Male 3.8164∗ 1.8719∗ 1615.2035

(0.1087) (0.0365) (0.0000)Head’s quality × i’s sex: Female 1.5882∗ 1.0857∗ 470.3805

(0.0985) (0.0381) (0.0000)Age male 0.0036∗ −0.0183 10.1705

(0.0017) (0.0257) (0.0000)Age female 0.0000 0.0140 0.2853

(0.0020) (0.0297) (0.7518)Days sick, male (minus head) 0.0001∗ 0.0009 5.4003

(0.0000) (0.0006) (0.0045)Days sick, female (minus head) 0.0000 0.0010 1.1505

(0.0001) (0.0008) (0.3165)Pregnant (minus head pregnant) 0.0003 −0.0016 0.1688

(0.0011) (0.0159) (0.8447)Minutes Nursing (minus head’s minutes nursing) −0.0000 −0.0007 0.1451

(0.0001) (0.0019) (0.8649)Second quarter 0.0004 0.0048 1.1122

(0.0003) (0.0039) (0.3289)Third quarter −0.0007∗ −0.0112∗ 4.0745

(0.0003) (0.0040) (0.0170)Fourth quarter 0.0011∗ 0.0147∗ 8.0336

(0.0003) (0.0041) (0.0003)

Table 4. Changes in the allocation of food quality.

where the null hypothesis of no nutritional investment implies thetestable restrictions that β1 = 1 and that E(eit|∆ log ζjt, ∆ log yjt, θt) =0 for all j = 1, . . . , n.

Here, for time-varying individual characteristics log ζit, we use a setof time effects, interactions between sex and the logarithm of age inyears and the number of days sick in the most recent period, an indica-tor with the value of one if person i is in the second or third trimesterof pregnancy, and a measure of lactation (the number of minutes spentnursing per day). For each of these measures we compute the differ-ence between the value of the measure for person i and the value of thesame measure for the household head. For fixed individual characteris-tics, we’ve simply used the person’s sex. We report results using boththe changes in the unit cost per Calorie and the unit cost per gram ofprotein. Results are reported in Table 4.


Either of the regimes without nutritional investment implies that thecoefficient associated with the quality of the head’s consumption shouldbe one. In Table 4 we interact the (change in the logarithm of) thehead’s consumption quality with the sex of person i, allowing us to testwhether changes in the allocation of quality within the household de-pend on sex. Variables used to estimate (23) are all transformed in sucha way that the estimated coefficients can be estimated as elasticities.

The first two rows of Table 4 provide a dramatic rejection of thehypothesis that nutritional investment doesn’t matter. For every oneper cent increase in the cost per Calorie for the head, we estimatethat males in the household will receive an increase of 3.8 per cent,while other females in the household will receive an increase of 1.5per cent. Each of these estimated coefficients is highly significant, andsignificantly different from one. We infer that the assignment of foodwithin households in our dataset depends to some extent on nutritionalinvestments, and assert with a very high degree of confidence thatneither the naive unitary regime nor the pure incentive regime describesthe mechanism used to assign food quality in the Philippine settingwhich generated our data.

In the pure nutritional investment regime there there are no unob-served actions, and hence no need to provide incentives. In this case(19) implies that quality will vary negatively with the expected mar-ginal benefit of nutritional investments. However, a rejection of thehypothesis that quality is perfectly correlated in the household is alsoconsistent with the investment and incentives regime. To distinguishbetween these two regimes we need to consider the possibly distincteffects of

In addition, we’ll find it convenient to think of there being three dif-ferent sorts of individual characteristics. Accordingly, we partition thevector of personal characteristics bit into three distinct parts. Let υi

denote time invariant characteristics of person i (such as sex), whichmay or may not be observed by the econometrician. Let ζit denoteobserved time-varying characteristics of the same person (such as ageand health). In contrast, let ξit denote time-varying characteristics ofperson i at time t which aren’t observed in the data. Then withoutadditional loss of generality, let (ι1, δ, ι3) be a triple of vectors whichselect and weight characteristics which influence the utility of consump-tion of nutrient k such that log h(bit) = ι1Tυi + δTζit + ι3T ξit, or morecompactly log h(bit) = υi + δT ζit + ξit.


5. Conclusion

In this paper we’ve constructed a direct test of the hypothesis thatfood is efficiently allocated within households in part of the rural Philip-pines. Conditional on our specification of preferences (a generalizationof CES utility), we’re able to reject this hypothesis, as the allocation offood expenditures, calories, and protein is significantly related to therealization of each individual’s off-farm earnings.

We then turn to two alternative explanations of this feature of thedata. We first consider a model in which the off-farm efforts of individ-ual family members can’t be observed, so that the allocation of food isdesigned to provide incentives to these workers. Second, we considera model in which food consumption produces not only utils but alsofunctions as a form of nutritional investment, which may be used todirectly influence the marginal productivity of workers. A third modelsupposes that food is used to provide incentives for workers, whoselabor effort may be hidden from the household head. Of these two mo-tives (investment and incentives), we are able to reject the hypothesisthat changes in the allocation of food are used solely to provide incen-tives, and are similarly able to reject the hypothesis that changes in theallocation of food are used solely as a form of nutritional investment.We’re left with evidence that households in this setting allocate foodboth to provide incentives and as a form of nutritional investment.


Appendix A. Data

Details on the consumption data:

(1) Data on individual food intake comes from 24 hour recall in-terviews. Some eighty different sorts of foodstuffs are foundin the data, but only 49 appear with sufficient frequency tobe usefully categorized as anything but “other.” These includecorn (boiled/grits/meal), soft drinks, alcoholic drinks, rice andrice products, corn products, bread products, kamote, pota-toes, cassava and cassava products, other root crops, sugar,cooking oil, mantika, fresh fish, dried fish, shrimps and othershellfish, cooked meat, organ meat, processed meat, chicken, ba-goong, patis, buro, sardines, pork (lean), beef (lean), carabeefand goat meat, eggs (all types), milk (all types), mongo, soy-beans, other dried beans, kamote tops, kangkong, malonggay,other leafy greens, squashes, tomatoes, mangoes and papayas,bananas, other fruit, and other vegetables.

(2) Calorie and protein individual intakes are computed using equiv-alence tables of these quantitative food intakes on each of the49 food categories.

(3) Individual expenditures of food are computed using these foodintakes valued at a household-specific price.

Details on the weather data:

Appendix B. Estimator

In this appendix we devise an estimator with which to estimate thesystem of equations

∆ log(

xkit

)

= ∆ log(

xk1t

) θ′kυ1

θ′kυi

+ (∆ζit − ∆ζ1t)δ

θ′kυi

+ νkit

for k = 1, 2, 3 and with νkit =

ǫk

it

θ′kυi

.

cov(

νkit, v

k′

i′t

)

= cov

(

ǫkit

θ′kυi

,ǫk′

i′t

θ′k′υi′

)

=1

θ′kυiθ′k′υi′cov(ǫk

it, ǫk′

i′t)

(1) One can assume that

cov(

vkit, v

k′

i′t

)

= σkk′ if i = i′

= 0 if i 6= i′

and

E(

vkit | ∆ log

(

xk1t

)

, (∆ζit − ∆ζ1t))

= 0


In this case, FGLS allows to estimated consistently all param-

eters, for all kθ′kυ1

θ′kυi

, δθ′kυi

(2) One can still have

cov(

vkit, v

k′

i′t

)


= 0 if i 6= i′

but that

E(

vkit | ∆ log

(

xk1t

)

, (∆ζit − ∆ζ1t))

6= 0

because of an endogeneity problem, that is that shocks vkit are

correlated with the head’s shock and thus with the head’s changesof log-consumption (or log-protein, or log-calories).Then, taking advantage of the availability of household leveldata on consumption measures, coming from a different mod-ule an measurement method of the survey, we use these dataas instrumental variables for household head’s changes in con-sumption or food intakes.Denoting by ck

it the household level consumption of good k byhousehold of individual i at period t, we assume that

E(

vkit | ck

it, (∆ζit − ∆ζ1t))

= 0

Then we can estimate parameters, for all kθ′kυ1

θ′kυi

, δθ′kυi

by 3SLS.

(3) We could also argue that the correlation of residuals of foodequations across members of a same household should not bezero. Then, we can choose to specify the correlation structuresuch that for some vector z of observable variables:

cov(

vkit, v

k′

i′t

)

= σkk′λkk′ (zi − zi′)

where λkk′ (0) = 1, z can be a vector.A special case constitutes the case where λkk′ (.) = λ (.), ∀k, k′.

cov(

vkit, v

k′

i′t

)

= σkσk′λ (zi − zi′)

λ (0) = 1

(4) Assuming that

cov(

ǫkit, ǫ

k′

i′t

)


= 0 if i 6= i′


then

cov(

vkit, v

k′

i′t

)

= cov

(

ǫkit

θ′kυi

,ǫk′

i′t

θ′k′υi′

)

=σkk′

(θ′kυi) (θ′k′υi′)if i = i′ and = 0 if i 6= i′

Notations:

ǫk =

ǫk11

.

.ǫkit

.

.ǫkNT

, Xk′ =

Xk11

.

.Xk

it

.

.Xk

NT

=

(

Xk11(1), .., Xk

11(p))

.

.(

Xkit(1), .., Xk

it(p))

.

.(

XkNT (1), .., Xk

NT (p))

(dim : NT ∗ p) ,

Zk =

Zk11

.

.Zk

it

.

.Zk

NT

=

(

Zk11(1), .., Zk

11(k))

.

.(

Zkit(1), .., Zk

it(k))

.

.(

ZkNT (1), .., Zk

NT (k))

(dim : NT ∗ k) , Y k =

Y k11

.

.Y k

it

.

.Y k

NT

(dim : 3NT ∗ 1)

ǫ =

ǫ1

ǫ2

ǫ3

(dim : 3NT ∗ 1) , X =

X1 0 00 X2 00 0 X3

(dim : 3p ∗ 3NT ) , β =

β1

β2

β3

(dim : 3p ∗ 1)

Z =

Z1

Z2

Z3

(dim : 3NT ∗ k) , Y =

Y 1

Y 2

Y 3

(dim : 3NT ∗ 1)


Then

Y = X ′β + ǫ

dim : (3NT ∗ 1) = (3NT ∗ 3p) ∗ (3p ∗ 1) + (3NT ∗ 1)

Denoting

PZ = Z(Z ′Z)−1Z ′

dim : (3NT ∗ 3NT )

we have

βOLS = (X ′X)−1X ′Y

βIV = (X ′PZX)−1X ′PZY

βGLS = (X ′Ω−1X)−1X ′Ω−1Y

βIV GLS = (X ′PZΩ−1PZX)−1X ′PZΩ−1Y

var (βOLS) = (X ′X)−1

var (βIV ) = (X ′PZX)−1

var (βGLS) = (X ′Ω−1X)−1

var (βIV GLS) = (PZXΩ−1X ′PZ)−1

where Ω = E (ǫǫ′)

Ω = E (ǫǫ′) = E

ǫ1

ǫ2

ǫ3

ǫ1

ǫ2

ǫ3

′

=

Eǫ1ǫ1′ Eǫ2′ǫ1 Eǫ1ǫ3′

Eǫ2ǫ1′ Eǫ2ǫ2′ Eǫ2ǫ3′

Eǫ3ǫ1′ Eǫ3ǫ2′ Eǫ3ǫ3′

(dim : 3NT ∗ 3NT )

Using the matrix notations:

• When

cov(

ǫkit, ǫ

k′

i′t

)


= 0 if i 6= i′

thenΩ = Σ3 ⊗ INT

with

Σ3 =

σ11 σ12 σ13

σ12 σ22 σ23

σ13 σ23 σ33

Ω−1 = Σ−13 ⊗ INT

Thus

βGLS = (X ′(

Σ−13 ⊗ INT

)

X)−1X ′(

Σ−13 ⊗ INT

)

Y

βIV GLS = (X ′PZ

(

Σ−13 ⊗ INT

)

PZX)−1X ′PZ

(

Σ−13 ⊗ INT

)

Y


• When

cov(

ǫkit, ǫ

k′

i′t

)

= σkk′λ (zit − zi′t′)

then

Ω = Σ3 ⊗ Σ(z)

with

Σ(z) = INT + λ[

(z ∗ J1,NT ) − (z ∗ J1,NT )′]

Ω−1 = Σ−13 ⊗ Σ(z)−1

• When

cov(

ǫkit, ǫ

k′

i′t

)

= σkk′λ (zit − zi′t′)

cov

(

ǫkit

θ′kυi

,ǫk′

i′t

θ′k′υi′

)

= σkk′

λ (zit − zi′t′)

(θ′kυi) (θ′k′υi′)

then

Ω = Σ3 ⊗ Σ(z)

with

Σ(z) =(

INT + λ[

(z ∗ J1,NT ) − (z ∗ J1,NT )′])

· / (A)

where ·/ is for the element by element division and

A [it, i′t′] = (θ′kυi) (θ′k′υi′)

Ω−1 = Σ−13 ⊗ Σ(z)−1

References

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University of Toulouse (IDEI, INRA)

University of California, Berkeley

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