Household Size, Economies of Scale and Public Goods in Consumption: A Proposal to resolve the Food Share Paradox

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1 1 Household Sze, Economes of Scale and Publc Goods n Consumpton: A Proposal to resolve the Food Share Paradox Ferdoon Kooh-Kamal 2014* ferdoon.kooh@emory.edu, Department of Economcs, Emory Unversty 1602 Fshburne Dr., Rch Buldng, Atlanta, GA, 30322, USA Abstract Ths paper s addressed to an explanaton of the food share paradox encountered n (Deaton and Paxson 1998), and suggests a proposal to resolve t. The paper examnes the effect of famly sze on the demand for food, a commodty group beleved to be partcularly responsve to changes n economes avalable to households. It argues that the sze effect on the budget share of food s negatve; ths has only become a paradox because the sources of economes are assumed to be confned to manly famly publc goods. The paradox s lkely to dsappear once wthn food group prvate sources of economes are acknowledged. I. Economes n consumpton and the budget share of food It seems obvous that the sze of a household should be a major nfluence on ts standard of lvng, yet how that nfluence manfests tself s far from clear. (Deaton and Paxson 1998)-henceforth D-P, s a promnent study nvestgatng ths queston. They have proposed a model of the sze effect on the food share based on a suggeston, due to Dreze, that the economes of scale effect s the result of a relatve fall n the prce of non-food famly publc goods as sze expands 1. Ths approach requres the exstence of a prvate * I wsh to express my specal thanks to John Muellbauer for hs nsghts and valuable suggestons. I am also very grateful to Angus Deaton for hs comments. None are responsble for the vews expressed here; all remanng errors are entrely mne.

2 2 good, whch dsplays a lmted substtuton effect wth respect to a fall n the relatve prce of publc goods resultng from an ncrease n sze. Aggregate food s commonly taken as such a commodty. It has few substtutes outsde the group, f not for the wealthy, then at least for poor households n developng economes, where average food consumpton s closer to subsstence. Thus, wth ncome per head constant, a fall n the prce of publc goods, followng an ncrease n sze, wll release more ncome, whch, wth a fall n food consumpton due to substtuton beng relatvely small, wll result manly n an ncome effect on food, ncreasng avalable per capta food expendture. Thus, f the sze effect were to nfluence food expendture n ths manner, one would expect a postve correlaton between the food share n household budget and household sze. Ths suggests that regressng the food share on per capta ncome and an ndependent sze varable, should produce a postve coeffcent estmate for sze, that s, the food share curve of the larger household wll be above that of the smaller one 2. They admt the possblty of other sources, ncludng some scope for substtuton effects, but dsmss them as lkely to be emprcally small. The mplementaton of ths approach n D-P has resulted n a paradox of strongly negatve sze effects observed n practce 3. That s, ther evdence 1 A descrpton of ths method can be found n (Deaton 1997, 264-8). 2 An excepton may occur f there are fxed costs to household publc goods consumpton, the most mportant beng that for housng. If the fxed cost s already met as sze expands, the predcted effect s ncrease n per head food demand; there s no problem n ths case. However, f the sze expanson makes t more lkely for the household to ncur the fxed cost, then economes of scale, and hence per capta food expendture are reduced. Evdence n D-P shows that n the move from one-person to two-person households, ths counter effect s of some mportance n developed, but not developng countres, presumably because housng s a relatvely less mportant expendture category for the latter. 3 D-P s a large scale study on consumpton effects of sze, contanng parametrc

3 3 suggests that per capta food expendture falls wth household sze for every one of the developed and developng countres n ther study, n sharp contrast to the predcted postve sze effect by ther model. Moreover, the negatve sze effect s stronger for developng countres, whch deepens the paradox, gven that ther food consumpton s closer to subsstence levels, and they have larger average budget shares of food. In the next secton, I attempt to demonstrate below that a postve sze effect, not a negatve one, s the expected outcome. II. An alternatve explanaton of D-P outcomes. Not all economes of scale effects are generated through consumpton of publc goods. There are two other notons of the economes generated through non-shared prvate goods: that based on household technology and that on dscount on bulk purchases; see also Davd (1962) for the dstncton between the two. The prvate sources of the scale are from wthn the group and reduce aggregate food expendture. Let us examne the case of bulk dscount for food, lkely to be a more promnent source of prvate economes n consumpton 4. The prce pad n ths case s a functon of food quantty and non-parametrc evdence from seven budget surveys of developed (US, UK, and France) and developng (Tawan, Thaland, by urban and rural separately, Pakstan, and South Afrca (blacks only)) countres. The parametrc evdence, also backed by non-parametrc results, s based on a food demand functon wth the same demographc specfcaton as (1), consstently producng a negatve sgn for sze n all ther results, n exactly the opposte drecton to that predcted by the publc goods approach. 4 The smlar effect can result from a reducton n food wastage as sze expands. Another source s due to return to scale from household technology. Ths noton s based on the dea that ncome rses less than proportonately wth household sze as addtonal members supply less labour tme to the market. If ths were the outcome, the household s rato of non-market tme to the quantty of goods purchased would rse. Larger households would thus tend to utlze ther relatvely cheaper household

4 4 purchased, so by defnton, q q2 f pq pq pq q1 pq q2,.e expendture on the food declnes wth bulk dscount 5. If ths prvate source of the scale were the domnant one, gven constant PCE, the food share would declne as sze expands, n conformty wth the Engel law and contrary to the outcome predcted f publc goods were the only source. Ths would then provde an alternatve predcton of a negatve effect as the expected drecton of the household sze nfluence on the food share 6. Stll, even f the prvate source of the scale were subordnate to publc goods, ts presence wll modfy the scope from the latter. The absence of substtuton between food and non-food groups does not necessarly mply that the only change n food expendture comes from publc goods outsde the food group. To reach that concluson requres nformaton not provded by the nput, namely non-market tme, to produce goods below ther market prces, e.g. cookng, so also reducng food expendture. However, D-P pont out, t s hard to examne ths effect f one has to condton food expendture solely on total expendture because tme data s unavalable. 5 (Pras and Houthakkar 1955) general measure of economes of scale s obtaned from estmaton of a qualty effect equaton (see note below); one that suggests, wth constant per capta expendture, the prce per unt ncreases wth household sze. There s also emprcal evdence from developng countres that the amount pad per unt ncreases for households that buy more, usually the better off households, see for example (Kooh-Kamal 2005). However, much of ths s probably qualty rather not prce effects. 6 Such a possblty s mentoned n D-P that ctes a study on the French food market, whch supports such a domnant role for bulk dscount; and n addton, some of ts own estmates for France provde support for ths concluson. However, the more common evdence n D-P for other countres shows the French evdence on the promnent role of bulk dscount effect s not typcal, so ths alternatve s lkely to have a lmted scope as an explanaton of the negatve sze outcome.

5 5 publc goods model. Snce t does not appear possble to source separately components of the measured scale effect, the coeffcent estmate on the logarthm of household sze n the model of economes n consumpton represents a compound parameter value 7. In general, there are therefore lkely to be two changes to food expendture from an expanson n sze: an ndrect, spll over effect from outsde the food group, and a drect effect from wthn t. Evdence on the subordnate role of bulk dscount does not resolve the dentfcaton problem snce t does not follow that that effect s necessarly neglgble, suggestng the publc goods measure of economes underestmates the true (publc) sze effect 8. The dentfcaton ssue can be more clearly explaned n terms of the Pras-Houthakker (P-H) model. (Pras and Houthakker 1955, ch. 10) frst proposed and examned the hypothess that an ncrease n household sze leads to an ncrease n economes of scale n consumpton avalable to the household, wthout restrctng the sources of economes to ether publc or prvate goods. The Engel equvalence scale model can be regarded as a specal case of that model, wth dentcal scales across goods, thus resolvng the problem of the lack of dentfcaton between the commodty-specfc scales 7 (Nelson 1988, esp. table III dscusson) proposes a measure of economes of scale based on the product of m a n ). b ( n ) for each commodty for household h of ( h h sze n n a Barten-type model. If s purely prvate, a ( nh nh and 1 ) 1 purely publc b ( n ) n ; 0 1. When the scale has both prvate and publc h h sources, addtonal dentfyng assumptons are requred to separate the effects of the two sources on the scale estmate. 8 (Pollak and Wales 1981) have ponted out that prvate goods economes of scale do not nvolve demographcs at all. If so, ther scope could be measured n physcal quanttes. There s, however, no model, whch would allow solatng prvate economes by a quantty-based measure n order to dentfy the sze-based measure of publc scale. ; f

6 6 and the general ncome scale 9. However, f one were wllng to make an dentfyng assumpton, for nstance, that the food and ncome scales are dentcal, or that they are dentcal across all commodtes, as n the Engel model, then the D-P paradox of a negatve sze effect dsappears, as I demonstrate below. Therefore, t s mportant to emphasze that the explanaton of a negatve sze effect on the food share s the consequence of makng such an dentfyng assumpton. In order to see ths, let us consder 9 More specfcally, ths method s an extenson of ther equvalence scale model, relatng commodty-specfc scales to a general scale. In ts smple form, p q f n x ( n 0 ) s the th commodty Engel curve. (1- ) measures economes for, (1-0 ) does so for all goods, x s total expendture or ncome, and n number of persons n the household. As wth ther equvalence scale model, 0 s a weghted average of the s. n s the only demographc measure, though ths can be modfed to take the composton effect nto account. Gven unchanged PCE, the general scale effect, wth 0 <1, allows a larger household to enjoy a hgher standard of lvng than a smaller one, suggestng ts effect s smlar to a change n qualty. So an estmate for 0 s obtaned from a unt-value equaton; the th Engel curve s estmated subject to ˆ 0, provdng a commodty-specfc measure of the scale, ˆ, for each. However, (Cramer 1969) frst ponted out the lack of dentfcaton of the P-H scales. Once the addng-up restrcton s mposed, the model breaks down. The reason for ths s that we need n equatons to estmate the m s, but due to the budget constrant equaton, we have only (n-1) Engel curves for ths purpose. Therefore, the model falls short by one equaton: the ratos of specfc scales are dentfed, but not ther absolute values, see (Muellbauer 1980). Note, however, that the crucal dentfcaton problem wth the P-H equvalence scale model poses an nsurmountable hurdle here too. In partcular, t s not possble to estmate the Engel economes of scales, whch requres the pror dentfcaton of 0, unless t s assumed that for food s equal to 0, or unless some alternatve dentfyng assumpton s made. The Engel scale can therefore be seen as a specal case of the P-H scale, wth s equal across all goods.

7 7 some smple cases. Thnk of the P-H model as the Barten wthout substtuton. Good 1 s food; good 2 s non-food. q m f ( y / m0 ) where s the commodty scale, m 0 s the scale for ncome y (or weghted average of m 1 and m 2 ). Here the food share s w 1 p1 m1 f1 m ( y / m0 ) / y p1 ( m1 / m0 ) f1 ( y / m0 ) / ( y / m0 ) ( m1 / m0 ) g1 ( y / 0 ) Now, suppose there are economes of scale n non-food but none for food. Then, as sze expands wth y / sze held constant, y / m 0 rses, snce m 0 rses less than n proporton to sze s. One would expect food expendture per capta q 1 / s to rse snce m 1 wll be proportonal to sze s and, wth y / m 0 hgher and food a normal good, f 1 ( ) wll rse. D-P show that q 1 / s actually falls n emprcal practce, where s stands for sze wth y / s s held constant and sze ncreases. If q 1 / s actually falls, then the food share w1 p1 q1 / y ( p1 q1 / s ) / ( y / s ) clearly also falls, snce y / s s constant by assumpton. Consder some cases: (A) Suppose economes of scale affect food more than non-food. Ths s the opposte of D-P s assumpton. Then as s rses wth y / s constant, f 1 ( y / m 0 ) wll rse. But m 1 wll rse less than n proporton: q1 s m s 1 f1 ( y / m0 ) So f m 1 / s falls proportonally more than f 1 ( ) rses, q 1 / s wll fall, as D-P say s true n practce. More generally, consder constant / s ) [ ( log m 1 ) log f 1 ( y / m 0 ) ] wth y / s

8 8 log m1 log f 1 1 wth y / s constant log f1 g ((log y ) ( log m0 )) so log f1 log m0 log m ( 1) (1 0 ) where = ncome elastcty for food, e.g. 0.4; log m0 log m1 log m2 w (1 w) where w = food share. For example, suppose log m1 w 0.6, 0.7, log m Then / s ) ( ) Here one easly gets the negatve effect of sze on food consumpton per head. (B) Suppose the same economes of scale for both food and non-food, e.g. log m1 log m2 0.7 Then / s ) Effectvely, ths bols down to the Engel model. And agan, one easly gets the negatve effect of sze on food consumpton per head. (C) Suppose economes to be hgher for non-food: even then, t s possble to get D-P result, e.g. log m1 log m 0.9, Then

9 9 / s ) ( 0.9 1) 0.4 ( ) Ths s negatve agan. The equvalence scale, w.r.t., s s gven by However, ths could / s ) not be deduced from the data. An elastcty of s also log m1 log m2 consstent wth the Engel model where Fnally, f substtuton s allowed, as n Barten s model, the negatve drecton of the sze effect s renforced. Suppose case (C): as s rses, m 1 rses relatve to m 2, so demand for food per capta drops a lttle more snce food s now relatvely more expensve. It s then even easer to obtan D-P s negatve sze effect. Note that case (C) can also explan the D-P dfference n the sze effect between developed and developng countres: reduce w 1 to 0.3, then / s ) 0.004,.e. a less negatve sze effect 10. Some of the above ssues are mplct n D-P, for nstance ther rejecton of the second Engel law, see (Deaton and Paxson 2003) response to (Gan and Vernon 2003). Furthermore, D-P offer a lst of potental explanatons for ther negatve sze effect, the frst three of whch, especally that based on drect food economes, do take nto account the effect of prvate economes. However, they reject all such explanatons as unlkely to be emprcally mportant enough to resolve the D-P paradox. If the P-H model s true, dentfyng assumptons are requred to make welfare comparson across households of dfferent sze and composton, 10 Cases A and B are less lkely than C. However, they suggest a negatve, not a postve, sze effect should be the expected outcome under a wde varety of assumptons about the scope for economes. The mportance of case B s, of course, n makng the hdden dentfyng assumpton of Engel s model explct.

10 10 though, gven prce varaton, the problem s, n prncple, solvable for the Barten model. Wth no substtuton permtted, no amount of extra prce nformaton can dentfy the P-H, or the Engel, scales, see (Muellbauer 1977 & 1980). The basc ssue that emerges from the above dscusson s that to make welfare comparson across households, one needs to make dentfyng assumpton n order to be able to estmate the P-H, or Engel scales; such assumptons cannot be emprcally tested. To ths extent, D-P have a case n questonng welfare nterpretatons of food share equatons. III. Concluson The effect of sze expanson on the food share wth constant per capta total expendture remans negatve. D-P regard ths outcome, n the context of a (non-food) publc goods model of economes of scale, as a paradox. Such a concluson s based on the mplct assumpton that scope for drect economes n food consumpton s neglgble. Ths s unlkely. Even a moderate degree of economes n food consumpton can explan the D-P negatve sze effect. Such possble explanatons are mentoned n D-P, who, n addton, provde a few peces of suggestve evdence n ther support. Perhaps D-P dsmss some of ther suggested explanatons too readly. Such explanatons can resolve the D- P paradox f they are further examned. References: Cramer, Jan Emprcal Econometrcs. Amsterdam: North-Holland. Davd, Martn Hedenhan Famly Composton and Consumpton. Amsterdam: North-Holland. Deaton, Angus The Analyss of Household Surveys, A Mcroeconometrc Approach to Development Polcy. London: Johns Hopkns Unversty Press. Deaton, Angus and Chrstna Paxson Economy of Scale, household sze, and the demand for food. Journal of Poltcal Economy. 106(5):

11 11 Deaton, Angus and Chrstna Paxson Engel s What? A Response to Gan and Vernon, Journal of Poltcal Economy. 3(6): Gan, L and Vctora Vernon Testng the Barten Model of Economes of Scale n Household Consumpton: Toward Resolvng a Paradox of Deaton and Paxson. Journal of Poltcal Economy. 3(6): Kooh-Kamal, Ferdoon. 2005, Welfare and Consumpton Ratonng: A Study n Behavour Based on a Wartme Iranan Household Expendture Survey, Unpublshed D. Phl thess, Oxford Unversty Muellbauer, John Testng the Barten Model of Household Composton Effects and the Cost of Chldren. Economc Journal, 87(Sept.): Muellbauer, John The Estmaton of Pras-Houthakker Model of Equvalence Scales. Econometrca, 48(1): Nelson, Jule Household Economes of Scale n Consumpton: Theory and Evdence. Econometrca, 56(6): Pollak Robert and Terence Wales Demographc Varables n Demand Analyss. Econometrca, 49(Nov.): Pras, Sg. and Hendrk Houthakker The Analyss of Famly Budgets, Cambrdge: Cambrdge Unversty Press, 2nd ed