February Abstract

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1 Labor Supply, consumpton and domestc producton: a new method to estmate labor supply elastctes Franços Gardes, Pars School of Economcs, Unversté Pars I Panthéon-Sorbonne 1 Davd N. Margols, Pars School of Economcs-CNRS and IZA 2 February 2015 Abstract Ths artcle derves a new approach for estmatng the elastcty of labor supply usng Becker s allocaton of tme framework and household choces concernng tme and monetary expendtures. Labor supply elastctes wth respect to market wages and the opportunty cost of non-market tme are calculated usng data from France and Canada. They are found to be coherent wth those obtaned by classc methods on ndvdual data whle avodng usual econometrc problems. The estmates of the elastcty of labor supply fare lower for couples wth chldren as expected. Introducton JEL Codes: J22, D11, D13 Estmaton of the elastcty of the labor supply wth respect to market wages has produced abundant lterature n many countres (see Bargan and Pechl, 2013), wth results varyng accordng to the methods and the datasets (country, macro or mcro data) used. Among the econometrc problems encountered n these estmatons, endogenety and observablty of wages and unearned ncome are among the most promnent, whle explct functonal form assumptons or structural models are also sometmes employed. An alternatve approach nvolves recognzng that market labor supply s the complement of domestc producton, ther sum beng equal to total tme avalable (after deductng tme devoted to sleep and other actvtes necessary to survve). Ths suggests that t should be possble to estmate labor supply elastctes ndrectly, through the estmaton of tme use functons for home producton and consumpton. Ths paper explots the framework developed by Gardes (2014), whch combnes monetary expendtures (observable n data such as Famly Expendture Surveys) wth tme use data n a model that generates dstnct values for tme devoted to market work (supposed to depend on, but not be equal to, the household s wage rate net of taxes w) and tme used n domestc producton. Ths model s used n the frst secton to derve the opportunty cost of tme and formulas for elastctes of monetary expendtures and tme use wth respect to the full prces of goods and to the opportunty cost of tme. The second secton presents the dervaton 1 Centre d Econome de la Sorbonne, Boulevard de l Hôptal, 75647, Pars Cedex 13, France. E-mal : gardes@unv-pars1.fr. (Correspondng author) 2 Centre d Econome de la Sorbonne, Boulevard de l Hôptal, 75647, Pars Cedex 13, France. E-mal : Davd. N.Margols@gmal.com. Ths artcle uses the French dataset prepared n collaboraton wth C. Starzec (Gardes, Starzec, Sayad, 2015). 1

2 of the labor supply functon whle the thrd secton presents the dataset. These are used n secton 4 to estmate the elastcty of the labor supply wth respect to the opportunty cost of tme and to the wage rate. Secton 5 concludes. 1. Domestc producton and consumpton In Gardes (2014), the household s supposed to dvde ts monetary ncome and total dsposable domestc tme (excludng ts necessary tme for sleepng and the workng tme) between a set of actvtes (fnal goods) whch are supposed to be produced by the consumpton of x unts of an aggregate market good (wth unt prce p ) and an amount of tme equal to t =τ x, where τ s the tme t takes to covert x unts of the market good nto a unt of fnal good 3. The full prce π for one unt of the fnal good (domestc actvty) s wrtten: p x + ωt = (p + ωτ )x, n whch ω s the opportunty cost of tme spent outsde the labor market. The full prce and ncome budget constrants, wth T beng total tme and t w beng workng tme, can thus be wrtten: (p x + ωt ) = y f + (ω w)(t t w ) = y f + (ω w) τ x (1) In order to smplfy the dervaton of the opportunty cost of tme, a Cobb-Douglas structure s chosen both for the utlty and the domestc producton functons of the fnal goods Q whch depend on the monetary and tme nputs 4. The optmzaton program s, accordng to the assumptons of secton 1 (all varables correspond to a household whose ndex s omtted n the equatons): under the full ncome constrant (1). max m,t u(q) = a Q γ wth Q = m α t β (2) Note that T t w = t and that both the market wage and the shadow wage (.e. the opportunty cost of tme ω) appear n the budget equaton: the shadow wage corresponds to the valuaton of tme n domestc producton, and dffers from the market wage w whenever there exsts mperfectons on the labor market or f the dsutlty of labor s smaller for domestc producton. In order to estmate the opportunty cost for tme, the utlty functon can be re-wrtten: γ u(q ) = a Q = a α γ α γ [ m = am α γ t β γ (3) ] α γ [ t β γ β γ ] β γ 3 The producton of the fnal good s modeled by a Leontef technology wth respect to tme, based on the amount of market good used n producton (or equvalently based on the amount of the fnal good, as each unt of fnal good necesstates, assumng the absence of economy of scale, the same consumpton of the market good). 4 Ther parameters wll be estmated on each pont (for each household n the dataset), so that ths specfcaton just supposes the constancy of each household s elastctes of the domestc productons n the utlty, and the two factors n the producton functons. 2

3 wth m and t the geometrc weghted means of the monetary and tme nputs wth weghts α γ α γ and β γ β γ. Dervng the utlty over ncome Y and total avalable lesure and domestc producton tme T gves the opportunty cost of tme: ω = u T u Y = u t t T u m m Y = m β γ t α γ t T m Y (4) All parameters of the utlty functon are estmated locally (for each household) so that the household s welfare wll depend both on the set of parameters (α, β, γ) and on ts monetary and tme expendtures m and t. In order to calculate the parameters of the utlty and domestc producton functons, Gardes (2014) consders the substtutons between tme and money resources for the producton of each actvty and between money expendtures (or equvalently tme expendtures) for two dfferent actvtes 5. The substtuton between tme and money n the domestc producton functon of actvty generates the frst order condton: α = m ωt +m and β = ωt ωt +m (5) under the constrant of a constant returns to scale for each producton functon: α + β = 1, whle the substtuton between tmes t and t j n the domestc producton of two dfferent fnal goods and j mples another condton between the parameters of the domestc producton functons and the utlty functon: γ = γ 1 t t 1 α 1 α (6) The estmaton of parameters α, β, γ allows one to defne both the opportunty cost of tme ω and the welfare ndex u(q ) at the household level. Appendx A dscusses two methods for the estmaton of these parameters. The estmaton s made on sx actvtes (food consumpton, dwellng expendtures (ncludng mputed rent), clothng expendtures, lesure expendtures, transport expendtures and mscellaneous goods and servces), excludng expendtures on health and educaton, whch contan many zeros. The estmaton of the opportunty cost of tme s made for data grouped nto cells (defned by household sze, the educaton level and the age of the head) n order to obtan robust estmates of the parameters α, β and γ of the utlty and domestc producton functons. Usng equaton (4), an estmate of the opportunty cost of tme s obtaned for each household. It averages 6.72, wth a range between 5.3 and It s thus sgnfcantly smaller than the average net wage rate (9.64), wth 73% of households havng an opportunty cost more than 25 percent lower than ts net wage rate. The average estmate of the opportunty cost of non-market tme s close to the mnmum wage rate (6.92). It corresponds qualtatvely to ndvduals responses n drect surveys of 5 Note that t s suffcent to examne the mplcaton of two over these three types of substtuton. 3

4 ther substtuton between tme and money, whch usually generate an opportunty cost of tme lower than the agent s wage rate net of taxes. Moreover, ω s postvely related to the household s net wage (wth an elastcty of 0.85) and to ncome condtonal to net wage (elastcty of 0.19). It s also postvely related to the household relatve ncome, defned by the rato of the household s average ncome to the average ncome of a reference populaton, defned by the aggregaton used n the estmaton of the parameters of the utlty and domestc producton functons. It ncreases untl the household s head s 45 years old, then decreases 15 years later (just as observed by Aguar and Hurst, 2007, n ther estmaton of the opportunty cost of tme for shoppng actvtes). It also ncreases wth famly sze, especally wth respect to the number of adults, whch may show that home producton s more valued n large famles because of economes of scale (.e. producton of publc goods). Ths estmate of the opportunty cost of tme allows one to defne the full prce of each actvty and to estmate full prce elastctes for these expendtures. As the monetary prce s not observed, the full cost for actvty s proxed by the rato of the full expendture over ts monetary component (see Gardes, 2014, for a dscusson): π = (p +ω h τ h )x h p x h = p +ω h τ h p = 1 + ω hτ h p (7) Usng the full prce elastctes E π estmated wth these proxes, the elastctes wth respect to the monetary prce E p, the tme used for the consumpton actvty E t and the opportunty cost of tme E ω are easly recovered (see for nstance De Vany, ) and can be calculated by means of the expresson for full expendtures and ts monetary and tme components: 2. Labor supply E x /p = E π E x /t = E π E x /ω = E πj j p p = E x π π = E π x π ωτ π ωτ j π j = E π m m +ωt (8) ωt m +ωt (9) ωt j = j E πj (10) m j +ωt j The relaton between the allocaton of tme for market work, home producton, nvestment actvtes (educaton, health care ) and lesure can be wrtten as: t = τ x = T t w (11) Takng the dervatve of (10) wth respect to the opportunty cost of tme gves: t w = t ω ω whch can be wrtten n terms of elastctes as: = (τ x + x τ ω ) (12) ω 6 Note an error n De Vany s formula (11) for the opportunty cost elastcty. 4

5 E tw /ω = τ x (Ex t w ω 5 + Eτ ) (13) ω The elastcty of τ (whch s the tme used per unt of market good ) wth respect to the opportunty cost of tme can be calculated n terms of the elastcty of the market good and the elastcty of substtuton between the monetary (market good) and tme factors used to home produce the fnal good (actvty) : σ = ln (x t ) = Ex ln(ω) Et ω ω so that Eτ = Et ω Ex ω = σ ω. Ths elastcty of substtuton has been estmated by Canelas et al. (2014) as ( σ = 0.031) for Food and (σ = 0.023) for other expendtures 7. We obtan fnally the formula for the elastcty of labor supply wth respect to the opportunty cost of tme: 3. Dataset E tw /ω = τ x (E t x ω σ ) (14) w We use a French dataset from INSEE whch combnes monetary and tme expendtures at the ndvdual level nto a common, unque goods and servces consumpton structure by a statstcal match of the nformaton contaned n two surveys: the Famly Expendture Survey (FES, INSEE BDF 2001) and the Famly Tme Budget (FTB, INSEE BDT 1999). We defne 8 types of actvtes or tme use types compatble wth the avalable data both from the FES and FTB: Eatng and cookng tme (FTB) and food consumpton (FES), cleanng and home mantenance and dwellng expendtures (ncludng mputed rent), clothng mantenance and clothng expendtures, educaton tme and educaton expendtures, health care tme and health expendtures, lesure tme and lesure expendtures, transport tme and transport expendtures, mscellaneous tme use and mscellaneous goods and servces. Two matchng methods were used: frst, by clusterng (nto 40 cells) the whole populaton n terms of age, educaton, and locaton as key varables, one obtans a good treatment of measurement errors and zero expendture problems. For each actvty and for each household we compute the correspondng tme use and then take the cell weghted average 8. Ths allows us to create comparable cells of base nformaton for the full tme and money expendture nomenclature from both surveys. The addton of both wll be possble once the ndvdual tme value s estmated. Ths method was used n Gardes et al. (2013) and was shown to allow for robust estmaton of the full cost of a chld (whch s estmated to be greater than the monetary cost). However, t seems more approprate to use an alternatve method, calculatng the tme component of each actvty at the household level rather than for a cell that groups 7 The estmates for varous expendtures are all greater than for food (from for transport to for Lesure) but seem to be less robust than the estmate for all expendtures less food, so we use the estmate for the aggregate n our calculaton. 8 Weghtng was necessary to take nto account the survey ntervew day (week-end or a work day). The weghts are the proportons of the persons ntervewed n the week (0.74) or durng the week-end (0.26).

6 many dfferent households. The second matchng method, used n ths artcle, s based on an ndvdual matchng by regresson: tme used n each actvty s regressed on household characterstcs (see Gardes, 2014). Tme use equatons for all selected actvtes are estmated on households characterstcs for all observaton unts n the FTB survey and these estmatons serve to predct the tme spent on these actvtes n the correspondng unts n the FES survey. 4. Results Table 1 contans the estmates of the elastctes of monetary expendtures wth respect to full prces and the opportunty cost of tme 9, whch are used to compute (table 2) the elastcty of the labor supply for the whole populaton and for varous sub-populatons (bachelors, couples wthout chldren, couples wth chldren). Monetary ncome Food (0.0207) Housng (0.0167) Clothng (0.0346) Transport (0.0240) Lesure (0.0280) Other (0.0117) Full ncome* (.0082) (.0135) (.0158) (.0151) (.0092) (.0206) Table 1 Income and full prce elastctes Opportunty Cost (0.0170) (0.0204) (0.0327) (0.0222) (0.0144) (0.0182) Full Own-prce elastctes (0.0065) (.0156) (.0058) (.0057) (.0092) (.0128) Elastcty of substtuton (0.031) (0.023) (0.023) (0.023) (0.023) (0.023) Source: Gardes, 2014, tables 1 and 2. See Appendx B and C. N.B. *Formula (12) n Gardes (2014). For the calculaton of full own-prce elastctes: qualty effect corrected by Deaton s method, the monetary budget share and ncome are used, and the full prce s calculated wth the estmated opportunty cost of tme. 9 Appendx A presents detaled results of the estmaton for the whole populaton. The market work tme has been estmated n France as 1750 hours multpled by the average number of workers n the sub-populaton. In Canada, t s drectly observed n the mcro dataset. 6

7 Table 2 Elastcty of the labor supply Whole Populaton Sngles wthout chldren Sngles wth chldren Couple wthout chldren Couple wth chldren τ x t w (Ex ) τ x ω t w (0.0038) (0.0107) (0.0159) (0.0096) (0.0064) σ E tw ω E tw w (0.0032) (0.0091) (0.0135) (0.0082) (0.0054) Table 2 shows that, on average, the estmated elastcty of hours of labor suppled wth respect to the opportunty cost of tme s 0.861, whle the elastcty wth respect to wages s Ths result derves mechancally from the relaton between wages and the opportunty cost of non-market tme (the latter elastcty beng multpled by a factor of 0.85). Ths relaton makes ntutve sense as well, as the dfference between the opportunty cost of tme and wages derves from the other ncome sources condtonal on wages, and ncreasng other avalable ncome allows the household to substtute purchased fnal goods for own-produced fnal goods, freeng up the correspondng tme for more market work. Another result worth notng s that labor supply elastctes are estmated to be systematcally lower when chldren are present, regardless of whether the household s a sngle or a couple. If there s an ncompressble mnmum amount of tme that must be spent on chld care n households wth chldren, ths wll necessarly leave less tme avalable for market work, and ths less possble varaton n the amount of tme spent workng n response to any gven varaton n wages or the opportunty cost of tme. As a pont of reference, Bargan and Pechl (2013) dentfed many studes (ncludng fve studes for France) that estmate the elastcty of labor supply wth respect to wages. Usng a meta-analyss to consoldate nformaton across studes and controllng for a seres of study specfc varables 10, they fnd that average marred women s hour elastctes wth respect to wages are and have been trendng down by per year. Wth an estmated standard error of 0.089, one cannot reject the hypothess that ther estmate s the same as our estmate for couples wth chldren (0.502, wth a standard error of ). The same model has been estmated on a smlar dataset n Canada (matchng a Tme Use survey wth a Famly Expendture survey n ). The elastcty of the labor supply 10 Ther meta-analyss ncludes controls for year, model type, whether the outcome was desred nstead of actual hours, whether jont decson makng was assumed, fxed costs and an ndcator varable for a US-based study. 11 Dataset prepared by P. Merrgan. 7

8 wth respect to the opportunty cost of tme s estmated as 0.17 (s.e ) for the whole populaton (whch corresponds to an elastcty wth respect to the market wage rate around 0.14 usng the proporton between the two elastctes calbrated for France, a value coherent wth the estmaton proposed n the ltterature). Contrary to the estmaton on French data, the elastcty does not dffer sgnfcantly between couples wth one chld and couples wthout chldren. Concluson Labor supply s one use of tme among many, and tme allocaton models provde a means of estmatng the senstvty of consumpton and tme allocaton to wages and the opportunty cost of non-market tme. Usng the Gardes (2014) model, one can use tme and monetary expendtures on consumpton goods to derve ndrectly the elastcty of labor supply wth respect to the opportunty cost of non-market tme and wages, whch are shown to dffer. Ths approach leads to estmates of wage and opportunty cost elastctes that, although slghtly hgher than those found elsewhere n the lterature, are not sgnfcantly dfferent from those found elsewhere. These elastctes, beng derved at the household level, can be decomposed along many dmensons and hghlght mportant behavoral substtutons n terms of tme allocaton that take place when the prce of tme changes. References Aguar, M. A., Hurst, E., 2007, Lfe-Cycle Prces and Producton, Amercan Economc Revew, 97(5): Bargan, O., Pechl, A., 2013, Steady-State Labor Supply Elastctes: A Survey, IZA Dscusson Paper no. 7698, October. Becker G.S., 1965, A Theory of the Allocaton of Tme, The Economc Journal, vol 75, Canelas, C., Gardes, F., Merrgan, P., Salazar, S., 2014, The Elastcty of Substtuton between Tme and Monetary Expendtures: an Estmaton for Canada, Ecuador, France and Guatemala, w.p. CES, Pars School of Economcs, Unversty Pars I Panthéon Sorbonne, January. Deaton, A., 1988, Qualty, Quantty, and Spatal Varaton of Prces, Amercan Economc Revew, vol. 78, 3, June, De Vany, A., 1973, The Revealed Value of Tme n Ar Travel, Revew of Economcs and Statstcs, February, vol. 56, 1, Gardes, 2014, Full prce elastctes and the value of tme: A Trbute to the Beckeran model of the allocaton of tme,, w.p. CES , Pars School of Economcs, Unversty Pars I Panthéon Sorbonne. 8

9 Gardes, F., Sayad, I., Starzec, C., 2015, Les échelles d équvalence complètes: une estmaton ntégrant les dmensons monétare et temporelle des dépenses des ménages, w.p. CES, Pars School of Economcs, Unversty Pars I Panthéon Sorbonne, Revue d Econome Poltque. 9

10 Appendx A Estmaton of the opportunty cost of tme n the domestc producton model In order to calculate the parameters of the utlty and domestc producton functons, we consder the substtutons whch are possble, frst between tme and money resources for the producton of some actvty, second between money expendtures (or equvalently tme expendtures) concernng two dfferent actvtes 12. Frst, the substtuton between tme and money n the domestc producton functon of actvty generates the frst order condton: u t u m = ω α β = m ωt whch mples: α = m ωt +m, β = ωt ωt +m (5) under the constrant of a constant economy of scale for each producton functon: α + β = 1. We suppose also that all margnal productvtes are postve: α, β, γ 0 and we normalze the utlty: γ = 1. Consder now the substtuton between tmes t and t j n the domestc producton of two dfferent fnal goods and j: ths substtuton mples another condton between the parameters of the domestc producton functons and the utlty functon: γ γ j = β jt β t j = α jm α m j so that: γ = γ 1 m m 1 α 1 α for all 1 (6) All other substtutons between monetary and tme resources devoted to dfferent fnal goods can be derved from (9) and (10). In order to estmate these parameters, a possble method (C) conssts n the calbraton of the opportunty cost of tme n a frst stage, for nstance at the mnmum wage rate whch s constant over the populaton. Equatons (5) thus gves an estmate of α and β for each household, whch gves γ by equaton (6). In the second step, an estmate of the opportunty cost of tme ω s gven by equaton (4) whch allows the computaton of the ndvdual values of the parameters α, β, γ for each household usng equatons (5) and (6). These values enter equaton (4) to gve for each household the second step estmate of ω. The estmatons on the French data as well as smulatons 13 tends to show that ths procedure may not converge rapdly to the true value of the opportunty cost of tme. Another method can be drectly based on equatons (5) and (6) whch mply for all actvtes : m γ j = m j γ + ωγ t j ωγ j t (15) 12 See estmates of the elastcty of substtuton between tme and monetary expendtures n Gardes et al., performed by J. Boelaert. 10

11 Ths can be estmated as a system of (n-1) equatons for j=1, =2 to n, calbratng γ j at the full budget share for food 14 or under the homogenety constrant of the utlty functon: γ = 1. In ths system, the opportunty cost of tme s over-dentfed, as well as all γ j, j > Note that takng together equatons 8, 9 and 11 to calculate the value of ω gves rse to a hghly non- lnear equaton n γ and ω whch cannot be solved algebracally. 11

12 Appendx B Econometrc methodology for the estmaton of the demand system (Gardes, 2014) The Almost Ideal Demand System s the most commonly used model to estmate demand elastctes. One of the man advantages of the model s that even f the model s nonlnear, one can use a Stone prce ndex to approxmate the AI model to ts lnear verson LAIDS, so as to facltate estmaton. As ponted out by Pashardes (1993), the errors comng from that approxmaton can result n based parameter estmates, as t can be seen as an omtted varable. The bas s bgger when the AI model s appled to mcro-data, because n ths case the expendture effects are hghly correlated wth the demographc characterstcs of the household and thus very heterogeneous between households. In order to correct ths bas, Pashardes proposes a smple re-parameterzaton of the prce parameter that crcumvents the problem created by the Stone prce ndex. In the estmatons presented n table 3 (3 rd column), the full prce elastctes are computed n a lnear Almost Ideal demand system specfcaton on full budget shares (corrected from the ncome effect by formula A1 below wth ncome proxed by total expendture π x ) and full prces (defned by equaton 13). The own-full prce elastcty for consumpton wrtes: E = γ w + β 1 wth β the ncome coeffcent of the AI demand system equaton for consumpton and γ the coeffcent of the own logarthmc prce logπ. Four specfc problems appear n the estmaton of ths demand system on matched data: frst, supposng that full expendtures follow an ndependent optmzaton scheme, based ether on a utlty functon or a cost functon, mples a total substtuton between tme and monetary household s expendtures. It s also plausble to suppose that two ndependent optmzatons exst for monetary and for tme allocatons, but n ths case the demand system for full expendture cannot n general be smlar to the equatons for monetary and tme expendtures. If for nstance the cost functons for the monetary and the tme expendtures are supposed to be Pglog, both demands are specfed as an Almost Ideal demand system (wth dfferent parameters). But n that case, the budget share for full expendtures w f depends on the monetary and tme budget shares: w f = y mw I +y t w t and the resultng demand equaton for y m +y t full expendture cannot be wrtten under as Almost Ideal specfcaton because of the nonlnearty n the ncome varable. Followng ths hypothess of two separate optmzatons for the components of the full expendture, we can calculate the full ncome elastcty E f n terms of the monetary and tme elastctes E m and E t : E f = E m. w I w f. 1 1+k + E t. w t k. w f 1+k (B1) wth k the dervatve of the temporal ncome over the monetary ncome. The ncome coeffcent s fxed by equaton (A1) n the estmaton of the full expendtures demand system n the second column of Table 3. Second, qualty effects are lkely to exst n full prce and expendture data. Indeed, an ncrease (n the cross-secton dmenson.e. between two households) of the full prce for 12

13 commodty (actvty) may result ether from the dfference (between the two agents) of the opportunty cost ω or from the dfference of ther tme allocated to actvty. Both causes may ncrease the qualty of ths actvty, by means of an ncreased productvty (whch can be supposed to be postvely related to ω) or of the tme devoted to. Ths endogenous qualty appears n the same form as n Deaton s technque to estmate prce-elastctes on local prces after removng the qualty ncorporated n unt values (whch s the rato of expendtures over quanttes consumed). In our matched dataset, local prces are replaced by the ndvdual full prces for each household. Deaton (1988) shows that the elastcty of expendtures Q over ts unt value V wrtes ε p = lnq lnv = E p E p 1+η E y wth E p the true prce-elastcty, E y the ncome-elastcty and η the ncome-elastcty of the unt value. Ths formula allows to calculate the true prce-elastcty n terms of the other parameters. In order to estmate η = lnv lny and 15) s wrtten for household h n cluster C:, the two equatons model (Deaton s equatons 14 w hc = α 1 + β 1 lny hc + Z hc γ 1 + u 1hC lnv hc = α 2 + β 2 lny hc + Z hc γ 2 + u 2hC (B2) (B3) I defne clusters as households whch are supposed to have the same opportunty cost of tme and the same domestc producton (thus the same τ ) for nstance by means of a common age class, locaton and educaton of the head. I thus estmate η= β 2 by (B3) wthn clusters and E p =γ 1 and β 1 =E y by (A2) wth the full ndvdual prces ncluded n Z hc. The thrd problem concerns the correcton of varances necesstated by the fact that budget tmes are generated regressors beng predcted for each household of the Famly Expendtures survey from the tme budgets recorded n the Tme Use survey. These estmated tmes are added to the household s monetary expendtures to form the household s full expendtures. These expendtures serve to calculate ndces of scarcty whch are used as full prces π n the estmaton of the demand system h(x, β, W, π) where W s the set of varables used n the frst step to predct π, α and β and the parameters of the demand functons. Thus, the full prces are generated n a frst step before the estmaton of the demand system, whch necesstate to correct the estmated varances. Murphy and Topel (1985) proposed a method adapted to ths case. Ther theorem states that the second step estmator β s consstent and asymptotcally normal wth an asymptotc covarance matrx (as stated by Greene, 2008): V β = σ 2 V b + V b [CV c C CV c R RV c C ]V b where V b s the covarance matrx gven by the second step of the estmaton, C = n plm 1 1 x n 0 ε 2( h(x,β,w,π) =0 ) and R = n plm 1 1 x π n 0 ε ( g(w,π) =0 ) wth g dependng π on the log-lkehood functon. In the case where π s predcted by a lnear regresson, C can be wrtten: n C = d 2 =1 e wth e the error term of the demand functon n the second step and d the coeffcents n the estmaton of π n the frst step. R s null f the regresson dsturbances of the regresson n the 13 x w

14 frst and second steps are uncorrelated, whch s the case of our model snce full prces are predcted from the Tme Use surveys and used as a regressor on the famly Expendtures survey. Fnally, we obtan C =,j d d j d j dependng on the coeffcents and covarance terms of the frst step regresson. A smple bootstrap procedure s an alternatve to the Murphy-Topel method. Consder that the full prces are estmated usng the actvty tmes predcted by the actual tmes observed n the Tme use survey and can be consdered as nstrumented values of the actual full prces. The usual method to correct the varances n case of nstrumentaton compares the resduals ε^1 = y Xβ^ lnπ IVγ^IV estmated n the second step regresson usng the logarthmc prces defned by the predcted tmes wth the resduals ε^2 computed wth the parameters β^, γ^iv ssued from nstrumentaton and the actual value for full prces lnπ:ε^2 = y Xβ^ lnπγ^iv. In our case, actual full prces π are not observed n the monetary statstcs. We can however smulate them knowng the dstrbuton for the set of prces for household h ssued from the predcton of prces: lnπ h ~N(log [π IV (h), V ] supposed to be normal (as ndcated by the dstrbuton of the logarthmc full prces) wth the estmated prce as the mean of the dstrbuton. The varance V = V(logπ ) of the logarthmc prce for actvty s estmated by the emprcal varance of the logarthmc full prces lnπ IV computed on the monetary expendtures survey. Therefore, ths resdual wrtes for household h: ε^2(h) = ε^1(h) + γ^[lnπ IV (h) lnπ (h)] so that: V(ε^2) = V(ε^1) + γ^2 V. Ths procedure wll be preferred because of ts smplcty and the fact that t takes fully nto account the non-lnear nature of predcted logarthmc full prces. Both procedures gve smlar nflaton of standard errors (for nstance 156% and 165% for housng expendtures, 119% and 152% for transportaton). Fnally, endogenety may appear n the full demand equatons because the opportunty cost of tme (and the unt tme for actvty τ ) appear both n the full expendture for, n the full total expendture and n the vector of full prces for all commodtes. Ths problem exsts because full prces are endogenous, dependng on the household type and characterstcs (n classc demand systems, prces are on the contrary pre-determned and generally supposed to be constant across the populaton). Ths possble endogenety bas can be taken nto account by nstrumentaton of full prces and full total expendture or GMM. Also, t s possble to calbrate the full ncome elastcty by formula (B1), usng monetary and tme elastctes estmated by two demand systems wrtten respectvely on monetary and tme expendtures. In our estmatons, we check that defnng prces by an alternatve valuaton of tme than the opportunty cost of tme used to defne the household s expendtures and full ncome (for nstance, full prces are defned usng the mnmum wage rate as the opportunty cost of tme whle full expendtures are computed wth an econometrc estmate of the household s opportunty cost) gves close estmates (up to a 20% dfference) to those usng the opportunty cost to value all varables. Another way to estmate full prce elastctes conssts n estmatng the demand system on monetary expendtures and full prces. It gves prce elastctes smlar to those obtaned by the estmaton on full expendtures whch we use n the emprcal applcaton (Table 3, 3 th column). 14

15 Appendx C Table C1 ncludes the prce elastctes estmated by LAIDS on monetary budget shares wth qualty correcton (see Gardes, 2014). They are compared to the parameters estmated under separablty restrctons by the Frsch method (Thel et al.,1987, Selvanathan, 1993). The prce elastcty estmates under strong separablty are estmated for two calbraton of the nverse of the ncome flexblty at -0.5 and whch s the estmated value for ths dataset obtaned by the estmaton of a Rotterdam model (GArdes, 2014, secton III). Full prce elastctes estmated on full budget share by a LAIDS system (calbratng the full ncome elastcty by formula B1 n Appendx B) are very smlar to those obtaned on the equaton usng monetary expendtures. The estmaton of prce elastctes over the ndex of scarcty defned by equaton (7) s robust to the specfcaton of the demand system, ether on full expendtures or on ther monetary components and does not dffer much accordng to the method used to compute the opportunty cost. 15

16 Table C1 (table 2, Gardes, 2014) Income, prce and opportunty cost Elastctes Food Monetary ncome (0.0154) Full ncome* (0.0170) Full prce** (0.033) Monetary prce (0.012) Tme (0.021) Opportunty Cost (.0062) Mon. prce : Groupng method *** (0.169) Mon. prce : Strong separablty**** ω (.0111) (.0218) Housng (0.0129) (0.0204) (0.440) (0.301) (0.139) (.0104) (0.150) (.0082) (.0118) Clothng (0.0242) (0.0327) (0.183) (0.117) (0.066) (.0110) (0.066) (.0170) (.0369) Transport (0.0174) (0.0222) (0.055) (0.030) (.025) (.0109) (0.010) (.0234) (.0234) Lesure (0.0190) (0.0144) (0.031) (0.008) (.023) (.0062) (0.032) (.0144) (.0278) Other (0.0082) (0.0182) (0.049) (0.028) (0.021) (.0206) (0.142) (.0178) (.0366) * combnng the monetary ncome and tme elastctes, equaton (A1) n Appendx. ** LAIDS specfcaton wth full budget share; ncome parameter calbrated by formula A1; full prce calculated wth the estmated opportunty cost of tme ω^ (method D, estmaton on 40 cells, table 1). Qualty effect corrected by Deaton s method (see Appendx). ***Hcks-Lewbel groupng method (Ruz, 2006). **** Frsch formulas for the own-prce and cross-prce elastctes E and E j (ωˇ 1 =-0.5 or -1.18; E =ncome elastcty; δ j = 0, j; δ = 1): E j = ωˇ 1 δ j w j E (1 + ωˇ 1 E j ); see a dscusson of the calbraton of the ncome flexblty ωˇ n secton III. Some mportant results come out from these estmatons: frst of all, we observe that all the (compensated) monetary own-prce elastctes are sgnfcantly negatve. The estmates range from -0.6 and -0.2 and average -0.35, perhaps a lttle larger than those of the macroeconomcs estmates that oscllate often between -0.1 and -0.3 for sem-aggregate commodtes. As we already ponted out, elastctes derved from macroeconomc data face measurement errors and aggregaton bas. Second, we observe that the correcton of qualty decreases by 24% n average the magntudes of the elastctes estmates, for both full prce elastctes and monetary 16

17 elastctes. Ths s consstent wth the theory as the qualty s ncluded n the prce, so once the qualty effect s corrected the elastcty s smaller. Thrd, regardng the estmaton under a strong separablty assumpton on utlty, we observe a sgnfcant dstance wth the LAIDS estmatons. The prce elastcty parameters under strong separablty (for ωˇ 1 = 0.5) have a larger magntude than those estmated wthout the latter restrcton for ( among the 6 commodtes. By the way, the estmaton under separablty depends heavly over the calbraton of the ncome flexblty. We can therefore strongly suspect ths hypothess of strong separablty 15. The monetary prce elastctes estmated by our method dffer also from those ndcated by the groupng method 16. Fourth, all tme elastctes are negatve, and ther magntudes are not related to the ncome elastcty nor to prce elastctes. Ther defnton (equatons 7 to 9) shows that ths s caused by the absence of systematc relatonshp between the monetary and tme component of the full expendtures (as shown by Table C1). The elastcty of expendtures as regards the opportunty cost of tme are postve for all tems except lesure expendtures: ths s probably explaned by the fact that tme plays a promnent role for these expendtures (the proporton of tme n the full expendtures for lesure s equal to 81% compared to 57% for all other expendtures). Ffth, prce and tme elastctes change sgnfcantly between dfferent types of households, for nstance between bachelors, couples wthout chldren and famles wth chldren: all prce effect (as well as the elastctes over tme or aganst the opportunty cost for tme) ncrease wth the famly sze. The larger sensblty for prces for large famles can be related to the noton that of household s needs ncrease (condtonal to ncome) wth ther sze. 15 Note however that the Pgou s law (whch relates the drect prce elastcty wth the ncome elastcty E = 0.5E ) apples rather well to the average estmate of ths rato (column 4 over column 1): 0.57, very close to the average Pgou s rato (0.60) for the calbraton of ωˇ 1 at Indvdual prces for detaled tems (such as bread, suggar ) are aggregated nto sem-aggregates (food) usng the houshold s budget-shares. The correspondng prce ndex (for food) thus changes between households (see a crtcal vew on ths method, whch appear as not robust, for nstance n the recent applcaton by Ruz, 2006, n Gardes, 2014). 17

18 Appendx D Estmatons for the whole populaton n France Fnal good Elastcty of Expendture Over the O.C. Food *** (0.0084) Housng *** (0.0191) Clothng *** (0.0305) Transport *** (0.0190) Lesure *** (0.0091) Other *** (0.0279) Number of observatons Standard Errors and Standard Devatons n Parentheses Average Tme Use by Expendture (424.57) (282.44) (100.03) (124.77) (567.12) (86.32) 18