Does Home Production Drive Structural Transformation?

Transcription

1 Does Home Producion Drive Srucural Transformaion? Alessio Moro Universiy of Cagliari Solmaz Moslehi Monash Universiy March 17, 2015 Saoshi Tanaka Universiy of Queensland Absrac Using new home producion daa for he U.S., we esimae a model of srucural ransformaion wih a home producion secor, allowing for boh non-homoheiciy of preferences and differenial produciviy growh in each secor. We repor hree main findings. Firs, he daa suppor a specificaion wih differen income elasiciies of marke and home services. Second, non-homoheiciy can alone accoun for he decline in he home services share, while nonhomoheiciy and price effecs ogeher are responsible for he rise of marke services. Third, he slowdown in home labor produciviy, sared in he lae 70s, is a key deerminan of he lae acceleraion of he marke services secor. We use he esimaed model o run a couner-facual experimen and find ha, by keeping he growh rae of home labor produciviy as i was before 1978, he model displays he share of marke services in consumpion expendiure lowered by 6.9% in We hank Benjamin Bridgman for providing he home producion daa for he U.S. We also hank Chris Edmond, Bruce Prenson, Begoña Dominguez, Yuichiro Waki, and oher seminar paricipans a Universiy of Melbourne, Monash Universiy, and Queensland Macro Workshop for heir helpful commens and suggesions. The usual disclaimers apply. Conac: amoro@unica.i. Conac: solmaz.moslehi@monash.edu.au. Conac: s.anaka@uq.edu.au. 1

2 Share in Exended Toal Consumpion Home and Marke Service Shares Marke Service Home Producion Year Labor Produciviy (Uni: 2005 USD per Hour) Home Labor Produciviy Year Figure 1: Home and Marke Service Value Added Shares in Exended Toal Consumpion (Lef) and Home Labor Produciviy (Righ) Noe: Daa are from Bridgman (2013) and Herrendorf, Rogerson, and Valeninyi (2013). 1 Inroducion How imporan is he role of he home producion secor for he process of srucural ransformaion? In his paper, we address his issue by esimaing a model of srucural change using new home producion daa for he U.S. for he period 1947 o Our paper is moivaed by he coincidence of he acceleraion of he service secor growh, and he slowdown in home labor produciviy. Figure 1 repors he shares of marke and home services in exended oal consumpion and home labor produciviy from Bridgman (2013). 1 The lef panel shows ha he growh rae of he share of marke services acceleraes around 1978, while he share of home producion is fla unil ha year and i declines aferwards. The righ panel shows ha labor produciviy a home mainains susained growh (around 2.5% per-year on average) beween 1947 and 1978, bu undergoes a marked slowdown in he remaining par of he sample period. 2 These observaions appear o sugges a role for home producion in shaping srucural change, somehing ha has been noed in he previous lieraure. 3 In his paper, we propose and esimae a model of srucural ransformaion wih a home producion secor ha can accoun for he movemen of marke and home services shares. We hen assess wheher he lae acceleraion of services can be accouned for by he slowdown in home labor produciviy hrough couner-facual experimens. We sar from a sandard model of srucural ransformaion wih non-homoheiciy of preferences and differenial produciviy growh in hree marke secors as in Buera and Kaboski (2009). We hen exend he model o include a home 1 We define exended oal consumpion as he value of home producion plus oal consumpion. 2 The slowdown is saisically significan a 1% level according o he es proposed in Bai and Perron (1998, 2003) and break dae is beween 1978 and See for insance Rogerson (2008), and Ngai and Pissarides (2008) among ohers. 2

3 producion secor, allowing for a differen income elasiciy beween marke and home services in household s demand. Since he iner-emporal and he inra-emporal problems can be solved independenly in his model, we can re-wrie he laer as a saic, consumpion choice problem of he household, which depends on he prices of he hree marke secors, home labor produciviy, exended oal consumpion, and exended oal value added. This version of he model allows us o esimae he implied share equaions, using he home producion daa from Bridgman (2013). In he esimaion, we allow for hree differen specificaions of household preference. In he firs specificaion, we consider he non-homoheic erm for aggregae services o be zero, and also assume marke and home services have he same income elasiciy. In he lieraure, he nonhomoheic erm for aggregae services is inerpreed as home producion; so by having an explici home secor in he model, i is naural o se i o zero. 4 In he second specificaion, we esimae he model by allowing he non-homoheic erm for aggregae services o be non-zero, keeping he assumpion ha marke and home services have he same income elasiciy. The inerpreaion we give here is ha he parameer simply reflecs a non-homoheic naure of services, which canno be explained by home producion. Finally, in he hird specificaion, we assume he non-homoheic erm for aggregae services o be zero, bu allow he income elasiciy of services o be differen beween home and marke. We highligh hree main resuls. Firs, in he esimaion, he bes fi of he daa is given by he hird specificaion, implying ha he daa suppor a differen income elasiciy beween home and marke services in household demand. Previous lieraure explained he movemen of marke and home service shares hrough differences in echnologies. 5 Our esimaion resuls indicae he model is no able o generae he movemen of marke and home service shares only hrough differences in he raes of echnological progress. Second, we assess he role of price and income effecs in generaing srucural change hrough couner-facual experimens. We find ha he nonhomoheiciy alone can accoun for he whole decline in he share of he home good and for a par of he rise of marke services. The remaining par of he increase in marke services is due o relaive prices effecs. Finally, he hird resul is obained by running a couner-facual experimen in which we le home labor produciviy grow a he consan rae of 2.5% afer 1978, he average pace before he slowdown. We find ha in his case he share of services is 0.78 in 2010, compared o 0.84 in he daa, which represens a change of 6.9%. Tha is, wihou he slowdown in home produciviy, he exen of srucural change would be considerably lower han he acual daa. This resul indicaes ha he behavior of he home producion secor can have quaniaively imporan implicaions for he srucural change observed in marke secors. Our paper relaes o he lieraure ha pus forh home producion as a key deerminan of he process of srucural ransformaion, such as Rogerson (2008) and Ngai and Pissarides (2008). In paricular, we focus on an aspec of he daa which has no received exensive aenion in he lieraure, namely he lae acceleraion of services. The empirical evidence for his fac is repored 4 See Kongsamu, Rebelo, and Xie (2001). 5 For example, see Ngai and Pissarides (2008) and Buera and Kaboski (2012b). 3

4 in Buera and Kaboski (2012a) for he U.S., and in Buera and Kaboski (2012b) and Eichengreen and Gupa (2013) for a large se of counries. Theories proposed o address his fac include hose of Buera and Kaboski (2012a), who focus on human capial differences, and Buera and Kaboski (2012b), who sudy differences in producion scale beween he home and he marke service secor. Our sraegy here is insead o esimae a model ha depars minimally from he sandard model of srucural change, and quanify he role of home producion for he acceleraion of marke services hrough couner-facual experimens. In paricular, we show ha he marked slowdown of home labor produciviy is largely responsible for he lae rise of services in our model. This resul calls for a cross-counry invesigaion of he role of home producion in generaing srucural change. We also relae o he lieraure ha esimaes subsiuabiliy beween marke and home services. One se of sudies uses flucuaions of aggregae home hours over he business cycles for esimaion. For insance, McGraan, Rogerson, and Wrigh (1997) find a value beween 1.49 and 1.75 for he elasiciy of subsiuion beween marke and home services, while Chang and Schorfheide (2003) esimae i as 2.3. Anoher se of works employs household micro daa on home hours, insead. Ruper, Rogerson, and Wrigh (1995) find a value in he range beween 1.60 and 2.00 while Aguiar and Hurs (2006) esimae i as Our paper differs from hese sudies in ha we esimae subsiuabiliy by exploiing variaions in secoral shares when prices change. In our mos preferred specificaion, we esimae a value of 2.75, which is somewha larger han he values obained in he differen approaches of he previous sudies. The remaining of he paper is as follows. Secion 2 presens he model; Secion 3 presens he esimaion mehodology and he daa employed, while Secion 4 repors he resul and he couner-facual exercises. In Secion 5 we presen some robusness exercises and in Secion 6 we conclude. 2 Model This secion presens a model of srucural change wih a home producion secor. 2.1 Seup Time is discree. There is a represenaive household, whose objecive is o maximize her uiliy. There are five ypes of good produced in his economy: four consumpion goods (agriculure, manufacuring, marke services, and home services) and one invesmen good. The household s preference is given by u = β ln C, =0 where β is he subjecive discoun facor. The composie consumpion index C is defined as C = i=a,m,s (ω i) 1 σ ( c i + c 4 σ 1 i) σ σ σ 1. (1)

5 where c i denoes consumpion of each good i {a, m, s}. In (1), he parameer ω i deermines he weigh on each good in he household s preference; he parameer c i conrols non-homoheiciy in preference; and he parameer σ governs he elasiciy of subsiuion beween hree goods. Service consumpion is a composie of marke services, c sm [ c s = ψ(c sm ) γ 1 γ + (1 ψ)(c sh, and home produced services, c sh, as ] γ + c sh ) γ 1 γ γ 1. (2) In (2), he parameer γ governs he elasiciy of subsiuion beween marke and home services and ψ is he share parameer in he service aggregaor. Noe ha we allow for a differen income elasiciy beween marke and home services hrough he parameer c sh. We provide a discussion on his parameer in Secion 2.5 and in he esimaion secion. In our seup, for each period, he household is endowed wih l = 1 uni of labor ha she splis ino working ime in he marke, l mk, paid a wage w and working ime a home, l sh. Also, he household holds he capial sock k in he economy, and decides how much o ren in he marke, k mk, a rae r, and how much o use in home producion, k sh. Then, he household s consrains are given by p a c a + p m c m + p sm c sm + k+1 mk (1 δ) k mk + k+1 sh (1 δ) k sh = r k mk + w l mk, (3) l mk + l sh where p j is he price of good j {a, m, sm} and δ is he depreciaion rae. We normalize he price of he invesmen goods o be equal o one. The oal amoun of capial is defined as k k mk = l, + k sh. The household produces home services hrough he following echnology, c sh ( = A sh k sh ) α ( l sh ) 1 α. In his economy, here is a perfecly compeiive firm in each marke secor j {a, m, sm} wih echnology, Y j = A j ( ) K j α ( ) L j 1 α. Finally, here is also a perfecly compeiive firm operaing in he invesmen good secor wih echnology, 2.2 Household s Problem Y x = A x (K x ) α (L x ) 1 α. Nex, we re-wrie he previous seup by reaing he home producion secor as being operaed by a perfecly compeiive firm. This allows us o consider he home producion secor as similar o 5

6 he oher secors, which helps us o simplify he problem. We firs define an implici price index for he home good by p sh Using he above price, we can show ha r α w 1 α A sh α α 1 α. (4) (1 α) p sh c sh = w l sh + r k sh. (5) We now add up (5) o he budge consrain of he household, (3), and obain p a c a + p m c m + p sm c sm + p sh c sh + k +1 (1 δ) k = r k + w l. Thus, we rewrie he household problem as max β ln C =0 (P1) subjec o C = [ c s = ψ(c sm i=a,m,s ) γ 1 γ (ω i) 1 σ ( c i + c + (1 ψ)(c sh σ 1 i) σ σ σ 1 ] γ + c sh ) γ 1 γ 1 γ,, p a c a + p m c m + p sm c sm + p sh c sh + k +1 (1 δ) k = r k + w l. Given he definiion of he implici home price, (4), i is sraigh-forward o show ha he problem (P1) is equivalen o he original seup in Secion Separaing Iner-Temporal and Inra-Temporal Problems As he final sep oward he esimaion, we separae he iner-emporal problem from he main problem. Specifically, we show (in he appendix) ha he household s problem (P1) can be decomposed ino he following wo problems. 1. Iner-Temporal Problem: The household solves: max β ln C {C,k +1 } =0 (P2) subjec o where P P C + k +1 (1 δ) k = r k + w l + p sh c sh + [ i=a,m,s ωi ( p i ) 1 σ ] 1 1 σ i=a,m,s p i c i, [ ( and p s ψ γ (p sm ) 1 γ + (1 ψ) γ p sh ) 1 γ ] 1 1 γ. 6

7 2. Inra-Temporal Problem: The household solves: max {c a,cm,csm,c sh } i=a,m,s (ω i) 1 σ ( c i + c σ 1 i) σ σ σ 1 (P3) subjec o [ c s = ψ(c sm ) γ 1 γ + (1 ψ)(c sh ] γ + c sh ) γ 1 γ 1 γ, and p a c a + p m c m + p sm c sm + p sh c sh = P C p i c i p sh c sh E, i=a,m,s where E sands for he exended oal expendiure on consumpion - ha is, oal consumpion plus home producion. The above decomposiion indicaes ha he iner-emporal problem (P2) and he inra-emporal problem (P3) can be solved separaely. Also, noe ha he inra-emporal problem (P3) is he one ha causes secoral ransformaion among four consumpion good secors in our seup. In Secion 3, we esimae he inra-emporal problem (P3) using ime series daa for prices { } p a, p m, p sm, p sh and exended oal consumpion, E. 6 We choose o esimae (P3) insead of he full model (P1) for wo reasons. Firs, we are ineresed in esimaing preference parameers in a model of srucural ransformaion wih home producion. Given he separaion of he wo problems shown in his secion, i is sufficien o esimae (P3) o obain consisen esimaors of he relevan parameers. Second, o esimae he full model (P1), we would need o ake a sand on how o bring he invesmen secor o he daa. We know ha, in he daa, aggregae invesmen comes from he hree marke secors (agriculure, manufacuring, and services) and ha he composiion has been changing over ime. Thus, depending on he modeling choice for he invesmen secor, esimaes could be differen. Wih his in mind, we avoid o make his modeling choice in he iner-emporal problem, and focus insead on he esimaion of he inra-emporal problem (P3). 2.4 Alernaive Preference Specificaions The model presened in he previous subsecions encompasses he sandard models of srucural ransformaion, namely hose of Kongsamu, Rebelo, and Xie (2001) and Ngai and Pissarides (2007), wih he addiion of home producion. Since our purpose is o sudy he effec of home producion on srucural ransformaion, we esimae he following hree differen specificaions, which imply differen ineracion mechanisms of he home and he marke secors. 7 Model 1: We firs impose c s = c sh = 0. As discussed in Kongsamu, Rebelo, and Xie (2001), he parameer c s > 0 can be inerpreed as home producion of services. Thus, when adding an explici 6 Regarding he implici price of home good, p sh, we discuss how o link i o labor produciviy daa in Secion Noe ha in all specificaions we resric c m o be zero as sandard in he lieraure. 7

8 home producion secor o he model, a naural resricion is o impose c s = 0. In his way we can asses wheher he home secor can replicae he role played by he non-homoheic parameer in he sandard model. Model 2: In he second specificaion, we only impose c sh = 0. This implies ha we are allowing for boh an explici home producion secor and he sandard non-homoheiciy effec for services. One can consider ha he parameer c s simply reflecs a non-homoheic naure of services, which is no fully measured by home producion. This is he inerpreaion ha we ake in esimaing his version of he model. Model 3: Finally, we esimae he specificaion in which c s = 0. In his case we are allowing he non-homoheiciy o be specific o he home services hrough c sh. Puing i differenly, we are allowing he non-homoheiciy o be differen beween marke and home services. 8 This is done because he empirical evidence suggess ha services caegories can have differen income elasiciies. For insance, Eichengreen and Gupa (2013) show ha he share of modern marke services rises faser wih income compared o ha of he more radiional marke services, which can also be produced a home. Alhough his evidence does no provide insighs on he income elasiciy of home services, i is reasonable o suppose ha home and marke services have differen income elasiciies. If he laer have a larger elasiciy, as suggesed in Eichengreen and Gupa (2013), we should expec a parameer c sh < 0. In his case, he parameer can be inerpreed as a minimum requiremen of home producion ha he household has o provide (for insance mainenance and cleaning) before enjoying he res of home services produced. 2.5 Implici Price for Home Services In order o link he implici home price, p sh, o he home labor produciviy, we consider he firs order condiion wih respec o labor in he home secor: p sh = w ( (1 α) A sh K sh L sh ) α = (1 α) w ( ) = Y sh L sh w (1 α) A sh (6) where A sh is he labor produciviy of he home secor. Thus, he implici price for home producion is a funcion of he wage rae and he home labor produciviy. To find he wage rae, w, noe ha we have L a + L m + L sm + L sh + L x = l = 1. Then, we obain w = (1 α)egdp, (7) 8 Noe ha in a CES aggregaor wih wo goods, he presence of a non-homoheic erm associaed wih one of he goods implies ha he oher good will also display a non-homoheic behavior. This is he case here for home services and marke services. On his poin, see Moro (forhcoming). 8

9 where EGDP denoes he exended oal value added. The above equaion means ha he wage rae is equal o he labor share in he exended oal value added, given ha he oal amoun of labor is normalized o one. From (6) and (7), we can derive 9 p sh = EGDP A sh. (8) We use he above implici price for home services o solve he problem (P3). 3 Esimaion This secion explains how we esimae our model. 3.1 Procedure To esimae he model, we firs derive equaions for he share of each secor in he exended oal consumpion. Given he implici home price equaion, (8), and he se of (pre-deermined) variables, ( ) x p a, p m, p sm, A sh, E, EGDP, problem (P3) can be solved for four shares, pi ci E, where i {a, m, sm, sh}. The se of parameers o be esimaed in he model is θ ( ) σ, c a, c s, c sh, ω a, ω m, ω s, ψ, γ. Since secoral shares sum up o one, he error covariance marix becomes singular wih four share equaions. Thus, we drop one share equaion, and finally have he hree non-linear equaions o be esimaed: p a c a = f 1 (x ; θ) + ɛ 1, E p m c m = f 2 (x ; θ) + ɛ 2, E p sm c sm E = f 3 (x ; θ) + ɛ 3. In he appendix, we show he derivaion of (f 1, f 2, f 3 ). For he esimaion procedure, we closely follow previous works in he lieraure: Deaon (1986) and Herrendorf, Rogerson, and Valeninyi (2013). Specifically, we employ ieraed feasible generalized nonlinear leas square o esimae he share equaions. For he parameers wih consrains (σ 0, ω a + ω m + ω s = 1, ω j 0, 0 ψ 1, γ 0), we ransform hem ino unconsrained 9 Here, we are using he assumpion ha he labor share parameer (α) is he same in he marke secors and he home secor. In Secion 5.1, we relax his assumpion. 9

10 parameers as follows; σ = e b 1, ω a = e b 2 + e b 3, ω m = e b e b 2 + e b 3, ω s = e b e b 2 + e b 3, ψ = eb4 1 + e b 4, γ = eb 5. Afer esimaing he unconsrained parameers, we ransform hose back o compue poin esimaes and sandard errors for he original parameers. 3.2 Daa One of he conribuions of his paper is o esimae he srucural change model using newlycreaed home producion daa for he U.S. by Bridgman (2013). 10 Since he consrucion of home producion values in his paper is based on he value-added mehod, we focus on consumpion value added shares for our esimaion. Here, we lis he se of he daa we use for our esimaion: Value Added Consumpion and Price Index: The daa for value added consumpion and consisen price index for agriculure, manufacuring, and services is from Herrendorf, Rogerson, and Valeninyi (2013). The advanage of using heir daa is ha hey compue value-added consumpion from final consumpion expendiure by using an inpu-oupu marix, in order o avoid invesmen componens included in consumpion value-added daa. 11 Toal Value Added: We need he oal value added daa for he calculaion of he implici home price, (8). We obain he daa from BEA. Value Added and Labor Produciviy in he Home Secor: We use he value added of he home secor and home labor produciviy daa from Bridgman (2013). In our esimaion, we focus on he ime period beween 1947 and 2010, due o he availabiliy of he daa lised above. 12 In order o calculae four secor shares (agriculure, manufacuring, services, and home) in exended consumpion value added, we combine consumpion value added daa wih value added of he home secor. One imporan assumpion made here is ha goods produced a home are no used for invesmens. To derive he implici home price, (8), we use he exended oal value added and home labor produciviy. The exended oal value added is obained by combining he oal value added wih value added of he home secor. 4 Resuls In his secion, we describe our esimaion resul. 10 Bridgman (2013) consruced home producion daa by using he equaion, Y sh = w L sh + ( j r j + δ j) Q j, where Q j is he value of capial good j a home. 11 Oher previous works in his lieraure assume invesmens are all made from manufacuring goods, creaing a problem because invesmen exceeds manufacuring from 1999 onward in BEA s daa. 12 Our daa for he esimaion sars from 1947 because we canno consruc value-added consumpion from final consumpion expendiure before he period due o he unavailabiliy of an inpu-oupu marix. Our daa ends in 2010, because he daa for value added and labor produciviy in he home secor is no released afer he ime. 10

11 Table 1: Secoral Share Esimaion Resul (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1a 1b 1c 2a 2b 2c 3a 3b 3c 3d σ (0.0265) (0.0280) (0.0212) (0.0276) (0.0249) (0.0243) (0.0009) (0.0012) (0.0009) c a (4.0798) (4.0280) (5.7545) (3.3737) (3.6396) (3.4175) (4.8018) ( ) (9.5673) (6.2414) c s ( ) ( ) ( ) c sh ( ) ( ) ( ) ( ) ω a (0.0001) (0.0001) (0.0005) (0.0001) (0.0000) (0.0001) (0.0005) (0.0010) (0.0009) (0.0006) ω m (0.0014) (0.0019) (0.0013) (0.0017) (0.0022) (0.0014) (0.0021) (0.0024) (0.0022) (0.0021) ω s (0.0014) (0.0019) (0.0013) (0.0017) (0.0022) (0.0014) (0.0024) (0.0030) (0.0026) (0.0024) ψ (0.0020) (0.0023) (0.0018) (0.0016) (0.0024) (0.0015) (0.0011) (0.0072) (0.0019) (0.0010) γ (0.0763) (0.0828) (0.0331) (0.0318) N AIC BIC RMSE a RMSE m RMSE s RMSE h Robus sandard errors in parenheses p < 0.10, p < 0.05, p < 0.01 Noe: N sands for he number of he sample in he esimaion. AIC is Akaike Informaion Crierion. BIC is Bayesian informaion Crierion. RMSE j is he Roo Mean Squared Error for j-secor s share equaion. 11

12 Model 1a Model 2a Share in Exended Toal Consumpion Service Home Manufacuring Agriculure Share in Exended Toal Consumpion Service Home Manufacuring Agriculure Year Year Figure 2: Fied Secoral Shares in Exended Toal Consumpion for Model 1a ( c s = c sh = 0) and Model 2a ( c sh = 0) Noe: Daa (Blue and Solid Line) and Model (Red and Dashed Line) 4.1 Esimaes Table 1 summarizes all he esimaion resuls for our hree specificaions of he model. In columns (1) o (3), Model 1 ( c s = c sh = 0) is esimaed for differen ypes of resricions on γ. In column (1) we se no resricions on γ. In column (2) we se γ o 1.5, he smalles value used in he lieraure, while in column (3) γ is equal o 2.3, he larges value in he lieraure. Similarly, we esimae Model 2 ((4) no resricion, (5) γ = 1.5, and (6) γ = 2.3), and also Model 3 (c s = 0) ((7) no resricion, (8) γ = 1.5, and (9) γ = 2.3). 13 From he perspecives of he Akaike and Bayesian Informaion Crieria (AIC and BIC henceforh) repored in Table 1, i is eviden ha neiher of he specificaions of Model 1 and 2 (he column (1) hrough (6)) has beer performance han Model 3a (column (7)), which has lower values of he AIC and he BIC. The failures of he fi of Model 1a and 2a are also visually presened in Figure 2, indicaing ha neiher models can correcly capure he increasing rend of marke services and he decreasing rend of home services. These facs imply ha he common specificaion in he lieraure, which assumes he same income elasiciy of marke and home services (i.e. c sh = 0), canno explain why he demand for marke services has increased relaive o home services over he period. Also, from he esimaion of Model 2, i is clear ha he non-homoheiciy erm on aggregae services ( c s ) doesn help o solve he issue. We now urn o discuss Model 3. There are wo poins which are worh emphasizing here. Firs, by comparing he performance of 3a, 3b, and 3c, we noe ha he subsiuabiliy parameer (γ) plays an imporan role in deermining he model s fi. This is also visually shown in Figure 3. When he subsiuabiliy parameer (γ) is se o 1.5 (Model 3b), he model canno accoun for he slowdown in he growh of he service share in 1970s, and is rapid growh hereafer. The 13 We ake 1.5 from McGraan, Rogerson, and Wrigh (1997) and 2.3 from Chang and Schorfheide (2003). 12

13 Model 3b Model 3c Share in Exended Toal Consumpion Service Home Manufacuring Agriculure Share in Exended Toal Consumpion Service Home Manufacuring Agriculure Year Year Figure 3: Fied Secoral Shares in Exended Toal Consumpion for Model 3b (γ = 1.5) and Model 3c (γ = 2.3) Noe: Daa (Blue and Solid Line) and Model (Red and Dashed Line) model s performance improves when he value of he subsiuabiliy parameer (γ) becomes larger (see Model 3c in Figure 3). When he value is unresriced, we esimae is value o be 2.75 (Model 3a in he column (7) of Table 1), which is somewha larger han oher esimaes in he lieraure as, for insance, hose of McGraan, Rogerson, and Wrigh (1997), Chang and Schorfheide (2003), Ruper, Rogerson, and Wrigh (1995) and Aguiar and Hurs (2006). 14 Second, once he non-homoheic erm for home service is inroduced, he value of σ is no longer saisically significanly differen from zero. The poin esimaor of σ is , and he value of he heeroscedasiciy-robus sandard error is This implies ha he uiliy funcion akes a Leonief specificaion in erms of agriculural, manufacuring, and aggregae services. Noably, his resul for σ is similar o ha of Buera and Kaboski (2009) and of Herrendorf, Rogerson, and Valeninyi (2013). Given ha he poin esimaor of σ is no saisically significanly differen from zero, we resric he value of σ o zero, and run he esimaion of Model 3d (column (10) in Table 1). The resul shows ha, while he roo mean squared errors are unchanged, he AIC and he BIC decrease, implying ha his specificaion is he mos preferable in erms of hose measures. Therefore, we use Model 3d for our couner-facual experimens in he subsequen subsecions Noe also ha he esimaed value for he share parameer in he services aggregaor (ψ) is wihin he range obained in previous work. For insance, McGraan, Rogerson, and Wrigh (1997), Rogerson (2008) and Rendall (2011) repor values beween 0.4 and To inerpre he esimaed non-homoheic erms c a and c sh for Model 3d, we compue heir values relaive o he consumpion level of each good in The value of ( c a /c a ) in 1947 is 0.67, while ha of ( c sh /c sh ) in 1947 is

14 Model 3a Model 3d Share in Exended Toal Consumpion Service Home Manufacuring Agriculure Share in Exended Toal Consumpion Service Home Manufacuring Agriculure Year Year Figure 4: Fied Secoral Shares in Exended Toal Consumpion for Model 3a (σ unresriced) and Model 3d (σ = 0) Noe: Daa (Blue and Solid Line) and Model (Red and Dashed Line) 4.2 Price and Income Effecs In his secion we analyze price and income effecs in our model. We sar by presening a parial equilibrium exercise in which here is a change in he price of eiher manufacuring or marke services. We work in parial equilibrium because we wan o analyze he variaion in shares due o he reacion of he household o he change in price, given oher condiions equal. We compare our model wih he sandard model of srucural ransformaion in Herrendorf, Rogerson, and Valeninyi (2013), in which here are only hree consumpion good secors, and no home producion secor. These resuls are shown in Figure 5 and Figure 6. Comparing he responses of he wo models o price shocks, firs noe ha, in boh models, σ is zero, meaning ha he uiliy funcion akes a Leonief specificaion in erms of agriculural, manufacuring, and aggregae services. This, in he model wihou home producion, implies ha quaniies in equilibrium are lile affeced by changes in prices, so ha he share of manufacuring (or services) increases and he oher shares decline afer he rise in he price of manufacuring (or services) (see HRV in Figure 5 and Figure 6). This is also he same for he model wih home producion, when he shock is on he manufacuring price (Model 3d in Figure 5). However, when he shock is on he service price (Model 3d in Figure 6), he rise in he share of marke services is smaller, because he household subsiues marke services wih home services. This is refleced in he decline of oal consumpion expendiure in he marke (las panel in Figure 6). As a resul, he variaion of all marke shares is smaller, compared o he case wih no home producion. In summary, when here is a shock o service price, our model exhibis a share movemen which is subsanially differen from he one in he previous lieraure. Nex, we perform a couner-facual exercise in which we shu down price and income effecs one a a ime, and compare he evoluion of he shares as implied by he model wih hose in 14

15 Manufacuring Price Agriculure Share Manufacuring Share Parcen Change Model 3d HRV Time Parcen Change Time Parcen Change Time Service Share Home Share Toal Marke Consumpion Parcen Change Time Parcen Change Time Parcen Change Time Figure 5: Effec of an Increase in Manufacuring Price on Secoral Shares, Comparison wih HRV (2013) Noe: For he purposes of comparison, we replicae he resuls of he specificaion (2) in Table 3 in HRV (2013). All he shares are calculaed relaive o oal marke consumpion, in order o make he resuls in our model and HRV (2013) comparable. 15

16 Service Price Agriculure Share Manufacuring Share Parcen Change Model 3d HRV Parcen Change Parcen Change Time Time Time Service Share Home Share Toal Marke Consumpion Parcen Change Parcen Change Parcen Change Time Time Time Figure 6: Effec of an Increase in Service Price on Secoral Shares, Comparison wih HRV (2013) Noe: For he purposes of comparison, we replicae he resuls of he specificaion (2) in Table 3 in HRV (2013). All he shares are calculaed relaive o oal marke consumpion, in order o make he resuls in our model and HRV (2013) comparable. 16

17 Prices Fixed a 1947 Value Income Fixed a 1947 Value Share in Exended Toal Consumpion Service Home Manufacuring Agriculure Share in Exended Toal Consumpion Service Home Manufacuring Agriculure Year Year Figure 7: Secoral Share Movemens when Prices are Fixed a 1947 Value (Lef) and Income is Fixed a 1947 Value (Righ) Noe: Daa (Blue and Solid Line) and Model (Red and Dashed Line) he daa. The resuls of his exercise are repored in Figure 7. The lesson we can draw is ha boh income and price effecs are imporan o accoun for acual shares also in he model wih home producion. However, here seems o be a key difference beween marke and home services. Wih income effecs only, he model is able o accoun for he enire decline of he home share over he period, while i canno explain he enire rise of marke services (lef panel). The res of he changes in marke services are generaed by he price changes (righ panel). To sum up, he decline of he home secor is explained by he difference in degrees of non-homoheiciy beween home and marke services, while he rise of marke services is he resul of boh non-homoheiciy and price effecs. 4.3 Slow Down in Home Labor Produciviy afer 1978 As documened in Bridgman (2013), he growh rae of home labor produciviy ou-paces ha of he marke economy during he period (2.5% versus 2.1%). Afer ha period, he growh rae of labor produciviy flucuaes around zero (see Figure 8, lef panel). To precisely dae he slowdown, we es for muliple srucural breaks using he approach in Bai and Perron (1998, 2003). A 1% significance level, here is a unique break beween 1978 and 1979, afer which he mean growh rae of home labor produciviy decreases by 2.5%. 16 This is a remarkable slowdown, boh for magniude and for is long lasing behavior. As home services could be subsiues for services in he marke, i is reasonable o ask how large he quaniaive effec of his slowdown is for he process of srucural ransformaion. We address his quesion by running a couner-facual experimen in which, oher condiions equal, home labor produciviy keeps growing a a consan rae from 1978 o More precisely, we assume ha in he household problem, all marke prices 16 See he appendix for deails. 17

18 Couner Facual Home Labor Produciviy Couner Facual Share Movemen Labor Produciviy (Uni: 2005 USD per Hour) Daa Couner Facual Share in Exended Toal Consumpion Service Home Manufacuring Agriculure Year Year Figure 8: Couner-Facual Experimen: No Slow-Down in Home Labor Produciviy afer 1978: Labor Produciviy (Lef), Movemen of Shares (Righ) Table 2: Couner-Facual Experimen: No Slow-Down in Home Labor Produciviy Exended Consumpion Share Consumpion Share Consumpion per Capia Bench Couner-Facual Bench Couner-Facual Bench Couner-Facual Agriculure (9.1%) (28.6%) (9.4%) Manufacuring (17.1%) (37.5%) (17.1%) Service (-20.7%) (-6.9%) (-26.1%) Home (33.7%) (33.6%) Noe: Consumpion per capia is in 2005 U.S. dollars. The numbers in brackes are percen changes from he benchmark fied value of Model 3d. and oal expendiure evolve as in he daa, while he price of he home good, given by (8), evolves now in a differen fashion due o he couner-facual evoluion of home produciviy A sh. To run he couner-facual we use model 3d, which provides he bes fi of he daa. The oucome of he exercise is displayed in he righ panel of Figure 8. Wihou he slowdown in home produciviy here is almos no divergence beween he marke services share and he home share in he exended oal consumpion over he period. The exercise hus suggess ha, given oher condiions equal, he slowdown in home produciviy is crucial o accoun for he lae acceleraion of he marke services share. Models ha do no display an explici home producion secor migh overlook an imporan facor which deermines he rise of he marke services secor. Table 2 repors he level of shares in he fied value in he benchmark esimaion and in he couner-facual for he year Regarding he exended consumpion shares, he marke services share is 0.59 in he benchmark and 0.46 in he couner-facual experimen. For he mos par, his difference is compensaed for by he home share, which is 0.31 in he benchmark and 0.41 in he couner-facual. A similar resul holds for he services share when considering marke consumpion 18

19 DLS: Model 3a DLS: Model 3d Share Service Home Manufacuring Agriculure Share in Exended Toal Consumpion Service Home Manufacuring Agriculure Year Year Figure 9: Fied Secoral Shares in Exended Toal Consumpion for Model 3a and Model 3d wih New Implici Home Price Definiion Noe: Daa (Blue and Solid Line) and Model (Red and Dashed Line) shares (i.e. consumpion shares which appear in GDP): services is 0.84 in he benchmark and 0.78 in he couner-facual. The las wo columns of Table 2 show he difference beween per capia consumpion in he benchmark and in he couner-facual experimen. Wihou he slowdown in home produciviy, agriculure and manufacuring consumpion is larger han in he benchmark, while marke services are a a subsanially lower level han in he benchmark. This is due o he large amoun of home producion in he couner-facual experimen wih respec o he benchmark, which depends on he susained growh of home labor produciviy. This resul suggess ha he slowdown in home produciviy is also crucial for he evoluion of real shares of GDP. 5 Robusness Here, we discuss he robusness of he resuls in Secion Differen Labor Shares beween Marke and Home When we derive he implici price for home services in Secion 2.5, we assume ha he share parameer (α) is he same beween he marke secors and he home secor. During he period 1947 o 2010, he mean labor share in GDP, (1 α mk ), is 0.702, while he mean labor share in he home secor, (1 α sh ), is 0.632, showing a difference of 7% poins. Therefore, in his subsecion, we relax he assumpion ha he marke and he home have he same share parameer (α), and check wheher he resuls in Secion 4 change or no. When he share parameers are differen beween he marke secors and he home secor, we 19

20 NG: Model 3a NG: Model 3d Share in Exended Toal Consumpion Service Home Manufacuring Agriculure Share in Exended Toal Consumpion Service Home Manufacuring Agriculure Year Year Figure 10: Fied Secoral Shares in Exended Toal Consumpion for Model 3a and 3d wih No Governmen Consumpion Noe: Daa (Blue and Solid Line) and Model (Red and Dashed Line) can calculae he wage rae as w = ( 1 α mk) ( GDP + 1 α sh) Y sh. where GDP and Y sh denoe value added in he marke and a home, respecively. Then, he implici price for home services is defined as: p sh = ( 1 α mk) ( GDP + (1 α sh ) A sh 1 α sh) Y sh Using he above implici price for home services, we run he esimaion and couner-facual experimen again. Esimaion resuls are repored in he firs four columns of Table 3, while Figure 9 displays he fied values. The resuls indicae ha he esimaed parameer values for all four specificaions, 1a, 2a, 3a, and 3d, are no significanly differen from hose in Table 1. Again, noe ha Model 3a and 3d perform beer han Model 1a and 2a in erms of he AIC and he BIC saisics, implying a differen income elasiciy in marke and home services. In he middle secion of Table 4, we repor he resuls for he couner-facual experimen for he model wih differen labor shares, in which home labor produciviy keeps growing a a consan rae from 1978 o Noe ha he majoriy of quaniaive resuls only show small differences wih respec o he benchmark case. Therefore, we conclude ha he difference in he labor share observed in he daa doesn really affec he resuls in he couner-facual experimen.. 20

21 Table 3: Secoral Share Esimaion Resul for Robusness (1) (2) (3) (4) (5) (6) (7) (8) DLS: 1a DLS: 2a DLS: 3a DLS: 3d NG: 1a NG: 2a NG: 3a NG: 3d σ (0.0306) (0.0320) (0.0007) (0.0277) (0.0229) (0.0190) c a (3.4615) (6.3239) (7.8216) (6.0989) (2.7966) (7.1123) (6.0650) (6.8808) c s ( ) ( ) c sh ( ) ( ) ( ) ( ) ω a (0.0002) (0.0004) (0.0007) (0.0006) (0.0000) (0.0006) (0.0007) (0.0007) ω m (0.0015) (0.0020) (0.0021) (0.0022) (0.0019) (0.0022) (0.0021) (0.0023) ω s (0.0015) (0.0020) (0.0025) (0.0026) (0.0019) (0.0020) (0.0023) (0.0027) ψ (0.0015) (0.0013) (0.0010) (0.0010) (0.0014) (0.0012) (0.0012) (0.0012) γ (0.0827) (0.0965) (0.0331) (0.0297) (0.0590) (0.0600) (0.0174) (0.0198) N AIC BIC RMSE a RMSE m RMSE s RMSE h Robus sandard errors in parenheses p < 0.10, p < 0.05, p < 0.01 Noe: N sands for he number of he sample in he esimaion. AIC is Akaike Informaion Crierion. BIC is Bayesian informaion Crierion. RMSE j is he Roo Mean Squared Error for j-secor s share equaion. 21

22 Table 4: Couner-Facual Experimen for Robusness: No Slow-Down in Home Labor Produciviy Exended Consumpion Share Consumpion Share Consumpion per Capia Bench Couner-Facual Bench Couner-Facual Bench Couner-Facual Baseline Resul Agriculure (9.1%) (28.6%) (9.4%) Manufacuring (17.1%) (37.5%) (17.1%) Service (-20.7%) (-6.9%) (-26.1%) Home (33.7%) (33.6%) Differen Labor Share Agriculure (9.3%) (27.4%) (8.4%) Manufacuring (17.1%) (37.2%) (17.0%) Service (-20.5%) (-6.8%) (-20.5%) Home (33.3%) (33.3%) No Governmen Agriculure (8.4%) (27.3%) (7.9%) Manufacuring (20.4%) (39.0%) (19.5%) Service (-20.1%) (-7.2%) (-20.2%) Home (25.9%) (25.8%) Noe: Consumpion per capia is in 2005 U.S. dollars. The numbers in brackes are percen changes from he benchmark fied value of Model 3d. 5.2 Excluding Governmen Consumpion In his subsecion we perform he esimaion of he model by subracing governmen value added from he hree marke secors. Governmen consumpion is exernally imposed on he household, and here is no a price a which households decide how much quaniy o purchase. For his reason, we re-esimae he model by removing he governmen secor from boh consumpion and expendiure daa. By doing his we are assuming ha he household is axed by he governmen o run a balanced budge, and ha governmen spending does no provide uiliy o he household. Esimaion resuls are repored in he las four columns of Table 3, while Figure 10 displays he fied values. Noe ha he share of marke services is now lower wih respec o Figure 4. As for he esimaion including governmen, he bes fi is provided by Models 3a and 3d, which show he lowes values of he AIC and he BIC saisics. Ineresingly, he esimaed value for σ in Model 3a is 0.1, which is larger ha in he benchmark esimaion, bu sill close o zero, which implies Leonieff preferences. As before, in Model 3d, we impose σ = 0. In conras wih he benchmark esimaion, he AIC and he BIC now provide conrasing resuls on he bes model. The firs crierion favors Model 3a, while he second favors Model 3d. The lowes secion of Table 4 repors he resuls for he couner-facual experimen in which home labor produciviy keeps growing a a consan rae from 1978 o 2010, when we exclude he governmen secor. Quaniaive resuls are similar o baseline ones. One noable difference is observed in he consumpion per capia of marke services, which decreases by 20.2%, compared o he fall of 26.1% in he baseline experimen, when including governmen. Also, consumpion 22

23 per-capia of he home good displays he larges change among he four ypes of goods wih respec o he baseline experimen, 25.8% versus 33.3%. 6 Conclusion In his paper we sudy a model of srucural ransformaion wih home producion and esimae i, using U.S. daa. We find ha he specificaion of he model wih a differen degree of nonhomoheiciy beween home and marke services provides he bes fi of he daa. In paricular, he esimaion provides an income elasiciy of home services lower han ha of marke services. This is in line wih recen empirical evidence suggesing ha he share of marke services ha can be produced also a home, grows slower wih income compared o ha of marke services which don have home counerpars. The esimaed model is hen used o run couner-facual experimens. In paricular, we measure he conribuion of he slowdown in home produciviy growh o he lae acceleraion of he marke services share in he U.S. We find ha wihou he slowdown, he model does no produce a quaniaively significan srucural change. This resul suggess ha home produciviy represens an imporan source of srucural change. As he observed lae acceleraion of services appears o be a feaure common o mos high income counries, our resul calls for a cross-counry analysis of he role of home produciviy for srucural change. 23

24 References Aguiar, M., and E. Hurs (2006): Measuring Trends in Leisure: The Allocaion of Time over Five Decades, NBER Working Paper, w Bai, J., and P. Perron (1998): Esimaing and Tesing Linear Models wih Muliple Srucural Changes, Economerica, 66(1), (2003): Compuaion and Analysis of Muliple Srucural Change Models, Journal of Applied Economerics, 18(1), Bridgman, B. (2013): Home Produciviy, Working Paper, Bureau of Economic Analysis. Buera, F. J., and J. P. Kaboski (2009): Can Tradiional Theories of Srucural Change Fi The Daa?, Journal of he European Economic Associaion, 7(2-3), (2012a): The Rise of he Service Economy, American Economic Review, 102(6), (2012b): Scale and he Origins of Srucural Change, Journal of Economic Theory, 147(2), Chang, Y., and F. Schorfheide (2003): Labor-Supply Shifs and Economic Flucuaions, Journal of Moneary Economics, 50, Deaon, A. (1986): Demand Analysis, Handbook of Economerics, 3, Eichengreen, B., and P. Gupa (2013): The Two Waves of Service-Secor Growh, Oxford Economic Papers, 65(1), Hansen, B. E. (2001): The New Economerics of Srucural Change: Daing Breaks in US Labour Produciviy, Journal of Economic Perspecives, 15(4), Herrendorf, B., R. Rogerson, and Á. Valeninyi (2013): Two Perspecives on Preferences and Srucural Transformaion, American Economic Review, 103(7), Kongsamu, P., S. Rebelo, and D. Xie (2001): Beyond Balanced Growh, The Review of Economic Sudies, 68(4), McGraan, E. R., R. Rogerson, and R. Wrigh (1997): An Equilibrium Model of he Business Cycle wih Household Producion and Fiscal Policy, Inernaional Economic Review, pp Moro, A. (forhcoming): Srucural Change, Growh and Volailiy, American Economic Journal: Macroeconomics. Ngai, L. R., and C. A. Pissarides (2007): Srucural Change in a Mulisecor Model of Growh, The American Economic Review, 97(1),

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26 Appendix A. Separaing Iner-Temporal and Inra-Temporal Problem In his appendix, we show how o separae he iner-emporal problem, in which he household decides aggregae consumpion and invesmen across ime, from he inra-emporal one, in which he household decides consumpion levels of he four goods, given resources allocaed o consumpion in ha period. We re-wrie here he household equivalen problem (P1): max β ln C =0 subjec o C = [ c s = ψ(c sm i=a,m,s ) γ 1 γ (ω i) 1 σ ( c i + c + (1 ψ)(c sh σ 1 i) σ σ σ 1 ] γ + c sh ) γ 1 γ 1 γ,, p a c a + p m c m + p sm c sm + p sh c sh + k +1 (1 δ) k = r k + w l, The firs order condiions for he four consumpion goods are L c a L c m L c sm =0 = β (ω a ) 1/σ (c a + c a ) 1 σ (C ) 1 σ = λ p a (9) C =0 = β (ω m ) 1/σ (c m + c m ) 1 σ (C ) 1 σ = λ p m (10) C =0 = β (ω s ) 1/σ ψ (c sm ) 1 γ L c sh =0 = β (ω s ) 1/σ (1 ψ) (c s ) 1 γ (c s + c s ) 1 σ Raise (11) and (12) o 1 γ, sum hem and raise o 1 1 γ C (C ) 1 σ ( 1 c sh + c sh) γ (c s ) 1 γ (c s + c s ) 1 σ (C ) 1 σ C o obain = λ p sm (11) = λ p sh (12) β (ω s ) 1/σ (c s + c s ) 1 σ C (C ) 1 σ = λ [ ( (p sm ) 1 γ ψ γ + p sh ) 1 γ (1 ψ) γ] 1 1 γ. (13) As λ is he marginal uiliy of one addiional uni of good j divided by he price of ha good, we can define [ ( ) ] 1 p s ψ γ (p sm ) 1 γ 1 γ + (1 ψ) γ p sh 1 γ, (14) 26

27 ha is, one uni of he services consumpion bundle coss p s. Noe ha by using (14) we can wrie β (ω s ) 1/σ (c s + c s ) 1 σ (C ) 1 σ = λ p s C. (15) Now sum FOCs (9) and (10) and use he definiion of p s o obain β (ω s ) 1/σ (c s + c s ) 1 σ (C ) 1 σ [ c s = λ C p sm c sm + p sh c sh + p sh c sh] (16) Recall now from (15) ha β (ω s ) 1/σ (c s + c s ) 1 σ p s C so we can use he las expression in (16) o obain (C ) 1 σ = λ, p s c s p sh c sh = p sm c sm + p sh c sh. (17) Now raise each condiion (9), (10) and (15) o 1 σ and sum across condiions raise o 1 σ 1 β (1 σ) C 1 σ σ C 1 σ i=a,m,s and simplify o obain β (ω i) 1 σ ( c i + c = λ C σ 1 i) σ = λ 1 σ ( 1 1 σ ) 1 σ ω i p i i=a,m,s ( ) 1 σ ω i p i, i=a,m,s As λ is he marginal uiliy of one addiional uni of he consumpion aggregaor C in unis of ha good, and β C as is he marginal uiliy of consumpion, we can define he implici price index P P ( 1 1 σ ) 1 σ ω i p i i=a,m,s Now sum across condiions (9), (10) and (15) o obain Use (17) o subsiue for p sm c sm P C = i=a,m,s p i c i + i=a,m,s.. p i c i. (18) + p sh c sh in he budge consrain of he household o obain p a c a + p m c m + p s c s + k +1 (1 δ) k = r k + w l + p sh c sh 27