Quasi-linear preferences in the macroeconomy: Indeterminacy, heterogeneity and the representative consumer

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1 Span. Econ. Rev. 5, ) DOI: /s c Springer-Verlag 2003 Quasi-linear preferences in he macroeconomy: Indeerminacy, heerogeneiy and he represenaive consumer Lilia Maliar, Serguei Maliar Deparameno de Fundamenos del Análisis Económico, Universidad de Alicane, Campus San Vicene del Raspeig, Ap. Correos 99, Alicane, Spain {maliarl,maliars}@merlin.fae.ua.es) Absrac. We use aggregaion heory o invesigae he link beween one-consumer and muli-consumer economies under a quasi-linear class of preferences. Our sudy is carried ou in he conex of he neoclassical growh model. The quasi-linear preferences considered are addiive in consumpion and leisure and linear in leisure. We firs show ha in a homogeneous agens economy, he individual hours worked are no uniquely deermined. We hen demonsrae ha he indeerminacy can be resolved by inroducing heerogeneiy. For example, idiosyncraic shocks o produciviies or imperfec subsiuabiliy of labor resore he uniqueness of equilibrium. As a special case, our analysis includes he indivisible labor model by Hansen 1985). JEL Classificaion C73, D90, E21 Key words: Quasi-linear preferences, heerogeneiy, represenaive consumer, indeerminacy, idiosyncraic shocks, indivisible labor 1 Inroducion Much of he research in macroeconomics is buil around he represenaive agen absracion. The hope is ha he replacemen of many heerogeneous consumers wih jus one consumer will no be essenial for he resuls. In some cases, he effec of heerogeneiy on he aggregae predicions of he models is, indeed, eiher absen, as is under Gorman s 1953) exac aggregaion, or quaniaively unimporan, as in, We are graeful o Moren Ravn for his guidance. We have benefied from he commens of an anonymous referee, Jordi Caballé, Finn Kydland, Franck Porier, Michael Reier, Xavier Sala-i-Marin, William Schworm and Andrew Sco. Any remaining errors are ours. This research was suppored by he Insiuo Valenciano de Invesigaciones Económicas and he Miniserio de Ciencia y Tecnología de España, he Ramón y Cajal program, and BEC

2 252 L. Maliar, S. Maliar e.g., Krusell and Smih 1998). However, here are also examples in he lieraure in which he implicaions of one-consumer models do no carry over o heerogeneous agens economies, see Akeson and Ogaki 1996), Maliar and Maliar 2001, 2003). To idenify he cases in which he represenaive agen assumpion can be safely used, one needs o gain some undersanding of how heerogeneiy can affec aggregae dynamics. Apar from he aggregae behavior of he acual economies, economiss are ineresed in disribuive issues. Several papers were able o gain useful insighs ino he deerminans of he disribuional dynamics by exploiing he link beween one-consumer and muli-consumer economies, see Chaerjee 1994), Caselli and Venura 2000), Maliar and Maliar 2000, 2001). Such lieraure proceeds by consrucing a represenaive consumer and uses he knowledge of he aggregae dynamics o resore he evoluion of disribuions. This paper employes aggregaion heory o invesigae he connecion beween one-consumer and muli-consumer economies under one paricular class of preferences, namely, he quasi-linear one. Quasi-linear preferences are hose ha are addiive in all commodiies and linear in some commodiies. The assumpion of quasi-linear preferences is common in incenive lieraure and in public finance. Furhermore, i is essenially he basis of he indivisible labor model by Hansen 1985) and Rogerson 1988), which is one of he benchmark macroeconomic models. A characerisic feaure of quasi-linear preferences is ha hey are no sricly convex. Under such preferences, he exisence and uniqueness of an inerior opimal allocaion is no, in general, guaraneed. Our analysis is carried ou in a general equilibrium framework. We place ourselves in a producion economy wih infiniely lived agens, who derive uiliy from wo commodiies ha we inerpre as consumpion and leisure. The momenary uiliy funcion of each agen is quasi-linear: i is addiive in consumpion and leisure, and linear in leisure. We begin by considering he economy wih homogeneous agens. We show ha such an economy has an indeerminacy of equilibrium, in he sense ha one can consruc infiniely many allocaions for individual hours worked, saisfying all he equilibrium condiions. I is herefore possible ha agens who have idenical fundamenals i.e., endowmens, preferences and produciviies), and who are placed in he same environmen, make differen labor-leisure decisions. The assumpion of muliple consumers is imporan for he indeerminacy resul. Wha disinguishes a muli-consumer from a one-consumer economy is essenially ha, in he former case, he producion possibiliy fronier is linear from he privae perspecive, whereas in he laer case, i is sricly convex. We shall also emphasize ha he indeerminacy described above arises in an opimal economy. 1 Furhermore, he indeerminacy occurs in our model, even if he number of agens is finie. 2 1 In his respec, our paper differs from he exising lieraure, where indeerminacy occurs as a resul of marke imperfecions, e.g., increasing reurns, incomplee markes, asymmeric informaion see Benhabib and Farmer, 1999, for a survey). 2 Kehoe and Levine 1985) argue ha he indeerminacy of equilibrium may arise because of he assumpion of infinie number of an infiniely lived consumers.

3 Quasi-linear preferences in he macroeconomy 253 Our main finding is ha indeerminacy can be ruled ou by he inroducion of heerogeneiy in labor. The role of heerogeneiy is, perhaps, bes undersood by looking a he following wo examples: Firs, we assume ha individual labor produciviies are subjec o shocks. If markes are complee, hen he opimal allocaions for individual working hours are a he corners: an agen works he maximum number of hours possible when produciviy and, consequenly, he wage) is high, and she spends all of her ime endowmen on leisure when i is low. As a resul, he individual hours worked are uniquely deermined. Secondly, we assume ha labor of differen individuals is no perfecly subsiuable in producion. The individual budge consrains, which are linear under he assumpion of perfec subsiues, hen become sricly convex. Thus, he indeerminacy is again resolved. I is worh emphasizing ha no every ype of heerogeneiy resores he uniqueness of equilibrium. For example, indeerminacy is presen in an economy where agens are heerogeneous in iniial endowmens, preference parameers and produciviies, if he relaive produciviies do no change over ime. The propery ha is common o all heerogeneous agens economies considered is ha heir aggregae dynamics can be characerized by represenaive consumer models. However, his is no aggregaion in he sense of Gorman 1953): he srucure of he represenaive consumer models, which we consruc in he paper, depends on specific assumpions abou heerogeneiy and does no, in general, coincide wih he srucure of he corresponding homogeneous agens counerpars. 3 In paricular, i is possible ha he preferences of he represenaive consumer are sricly convex even hough he preferences of all heerogeneous agens are quasilinear i.e., no sricly convex). The aggregaion resuls presened in his paper can be viewed as furher examples of imperfec aggregaion, described in Maliar and Maliar 2003). 4 The paper is organized as follows: Secion 2 describes a compeiive equilibrium economy in which agens have a quasi-linear ype of preferences. Secion 3 presens he corresponding planner s economy. Secion 4 esablishes indeerminacy of equilibrium under he assumpion of homogeneous agens. Secion 5 discusses wo examples in which indeerminacy is ruled ou by inroducing heerogeneiy, and finally, Secion 6 concludes. 2 The model Time is discree and he horizon is infinie, T, where T = {0, 1, 2,...}. The economy consiss of a represenaive firm and infiniely-lived agens wih names in he se S, which is normalized by ds =1. The agens differ in hree dimensions: S endowmens, preferences and produciviies skills). 3 As is known from Blackorby and Schworm 1993), he assumpion of a quasi-linear uiliy funcion is inconsisen wih Gorman s 1953) ype of aggregaion, unless he uiliy funcion is linear in all commodiies. 4 This ype of aggregaion is based on a represenaion of he social uiliy funcion proposed by Consaninides 1982); and i is employed in Akeson and Ogaki 1996), and in Maliar and Maliar 2001, 2003).

4 254 L. Maliar, S. Maliar We denoe he skills of agen s in period by β s. The disribuion of skills in period is B = {β s } s S I R S +. We assume ha B follows a saionary firs-order Markov process. Specifically, le R be he Borel σ-algebra on I. Define a ransiion funcion for he disribuion of skills Π : I R [0, 1] on he measurable space I, R) such ha: for each z I, Π z, ) is a probabiliy measure on I, R), and for each Z R, Π,Z) is a R-measurable funcion. We shall inerpre he funcion Π z,z) as he probabiliy ha he nex period s disribuion of skills lies in he se Z given ha he curren disribuion of skills is z, i.e., Π z,z) =Pr{B +1 Z B = z}. The iniial disribuion of skills B 0 Iis given. 5 The disribuion of skills B is he only exogenous sae variable of our economy. We assume ha here is a complee se of markes, i.e., ha he agens can rade Arrow securiies, coningen on all possible realizaions of he disribuion of skills. An agen s S derives uiliy from consumpion, c s, and leisure, l s. We focus exclusively on he case in which he momenary uiliy funcions of all agens are quasi-linear: addiive in consumpion and leisure, and linear in leisure. Tha is, for each agen s S, we assume: u s c s,l s )=u s c s, n n s ) v s c s )+A s n s, where he variables n s and n are working hours and he oal ime endowmen, respecively. 6 We assume ha v s is sricly increasing and sricly concave and ha A s < 0. The agen s solves he following uiliy maximizaion problem: max {c s,ns,ks +1} T E 0 [ ] δ v s c s )+A s n s ) =0 s.. c s + k+1 s + p Z) m s +1 Z) dz = w s n s +1 d + r )k s + m s B ), R 2) where E 0 is he operaor of condiional expecaion. The discoun facor is δ 0, 1). The wage per uni of labor supplied by agen s is w s. The agen owns capial sock k s and rens i o he firm a he renal price r. Capial depreciaes a he rae d 0, 1]. The porfolio of Arrow securiies bough by he agen in period is { m s +1 Z) } Z R. The price of securiy p Z) is o be paid in period for he delivery of one uni of he consumpion good in period +1if B +1 Z. Iniial endowmens of capial, k s 0, and Arrow securiies, m s 0 B 0 ), are given. The oal endowmen is 1 d + r 0 )k s 0 + m s 0 B 0 ) κ s 0 R +. The represenaive firm rens capial and hires labor o maximize period-byperiod profis. Capial inpu is given by k = S ks ds. Labor inpu is deermined 5 Noe ha he above formulaion allows for aggregae uncerainy in he disribuion of skills, as idiosyncraic disurbances can be correlaed across agens. 6 Here, and furher on in he paper, means is idenical up o an addiive consan. 1)

5 Quasi-linear preferences in he macroeconomy 255 by a funcion h, which depends on boh, individual hours worked and skills. Consequenly, he problem solved by he firm is max π = f k,h ) r k w k, {n s s n s ds 3) }s S S s.. h = h {n s,β s } s S). 4) The producion funcion, f, has consan reurns o scale, is sricly increasing in boh argumens, coninuously differeniable, sricly concave, and saisfies he appropriae Inada condiions. The labor inpu funcion, h, has consan reurns o scale, is sricly increasing in all argumens, coninuously differeniable and concave. Definiion. A compeiive equilibrium in he economy 1) 4) is a sequence of coningency plans for he consumers allocaion, he firm s allocaion and he prices, such ha, given he prices, he allocaion of each consumer solves he uiliy maximizaion problem 1), 2), he allocaion of he firm solves he profi maximizaion problem 3), 4); also, capial and labor markes clear and he economy s resource consrain is saisfied. The equilibrium quaniies are o be such ha c s, w s,k,r 0 and 0 n s n for all, s. 7 3 Planner s economy According o he Firs Welfare Theorem, in an economy wih complee markes and wihou disorions, like ours, a compeiive equilibrium is Pareo opimal and can be resored as he opimal choice of a social planner, who maximizes social welfare. Specifically, le us define he social uiliy funcion 8 U c,h, {λ s,β s } s S) max λ s v s c s {c s )+A s n s ) ds,ns }s S S h S cs ds = c {n s,β s } s S) = h 0 n s n, 5) where c is he aggregae consumpion, and {λ s } s S R S + is he disribuion of welfare weighs, normalized o one, S λs ds =1. A compeiive equilibrium in he heerogeneous agens economy 1) 4) can herefore be resored by solving he following planner s problem max {c,h,k +1} T E 0 δ U =0 c,h, {λ s,β s } s S) 6) 7 The formulaed seup is a heerogeneous agens varian of Kydland and Presco s 1982) model. Under he assumpion of sricly convex preferences, similar heerogeneous agens seings have been sudied in, e.g., Kydland 1984, 1995), Akeson and Ogaki 1996), Maliar and Maliar 2001, 2003). 8 The consrain 0 n s n plays an imporan role in our analysis because he model has, in general, corner soluions for hours worked.

6 256 L. Maliar, S. Maliar s.. c + k +1 =1 d) k + f k,h ). 7) To be precise, for any given disribuion of endowmens in he decenralized economy 1) 4), here exiss a se of welfare weighs in he planner s economy 5) 7) such ha a compeiive equilibrium allocaion is a soluion o he planner s problem. We should poin ou ha he noion of he social uiliy funcion 5) does no, in general, imply aggregaion in he sense of Gorman 1953), since U is allowed o depend no only on aggregae quaniies bu also on he disribuions of welfare weighs and skills. 9 Wih he assumpions of convex preferences and producion ses, we have he Second Welfare Theorem, which is he converse of he Firs Welfare Theorem. Specifically, any Pareo opimal allocaion can be reached as a compeiive equilibrium wih appropriae ransfers of wealh i.e., iniial endowmens). In our model, he correspondence beween Pareo opimal and compeiive equilibrium allocaions is idenified by he expeced lifeime budge consrains. Such consrains are obained by applying forward recursion o individual budge consrains 2) and by imposing he ransversaliy condiion, 10 [ ] E 0 δ τ vs ) c s τ ) v s ) c s 0 ) cs τ n s τ wτ s ) = κ s 0. 8) τ=0 Given any Pareo opimal allocaion, we can compue he lef side of 8) for each s. As a resul, we obain uniquely deermined ransfers of wealh supporing a given Pareo opimal allocaion. I is convenien o characerize he social uiliy funcion U by using he Kuhn- Tucker condiions. Proposiion 1. For he economy 1) 4), U akes he form U c,h, {λ s,β s } s S) = V c, {λ s } s S) + W h, {λ s,β s } s S), 9) where V is defined by S cs ds = c and c s = v s ) ) 1 1 λ s V c, {λ s } s S)) ; 10) W is defined by h {n s,β s } s S) = h and A s λ s h n s W 1 h, {λ s,β s } s S) < 0 n s =0 > 0 n s = n ; 11) =0 0 n s n and where V 1 and W 1 denoe he firs-order parial derivaives of V and W wih respec o c and h, correspondingly. 9 See Maliar and Maliar 2003) for a discussion. 10 For he derivaion of his consrain, see Maliar and Maliar 2001).

7 Quasi-linear preferences in he macroeconomy 257 Proof. See Appendix. As follows from Proposiion 1, he opimal allocaion for he individual working hours in he planner s economy can be eiher inerior or a corners. Wih he following proposiion, we esablish ha he inerioriy of he planner s allocaion for working hours implies he inerioriy of he corresponding allocaion in he decenralized economy, and vice versa. Proposiion 2. The opimal allocaion for individual hours worked is inerior in he decenralized economy 1) 4) if, and only if, i is inerior in he associaed planner s problem 5) 7). Proof. See Appendix. 4 Homogeneous agens: Indeerminacy We firs consider he economy 1) 4) populaed by agens wih idenical fundamenals i.e., endowmens, preferences and skills) such ha v s = v, A s = A, κ s 0 = k 0 and β s =1for all s. We assume ha he labor inpu is given by he sum of individual hours worked, h = S ns ds, and ha he economy consiss of more han one agen. As follows from Proposiion 1, if an equilibrium is inerior, hen 11) holds wih equaliy. Given ha S λs ds =1,wehaveλ s =1for all s. From 10), we obain ha c s = c for all s. Moreover, he economy admis a represenaive consumer wih a quasi-linear uiliy funcion U c,h, {1, 1} s S) v c )+Ah. 12) As has already been shown in Hansen 1985), he aggregae equilibrium allocaion {c,h,k +1 } T is uniquely deermined by 6), 7), 12). Neverheless, individual hours worked, {n s } s S T, are no uniquely deermined. Indeed, opimaliy condiion 11) does no conain n s and, hus, imposes no resricion on he choice of working hours excep for 0 n s n for all s and. The oher condiions o be saisfied in equilibrium are marke clearing in labor, h = S ns ds, and expeced lifeime budge consrain 8). The former resrics he sum of individual working hours wihin each period, whereas he laer does so across periods. We demonsrae ha hese wo condiions are no sufficien o idenify individual labor choice. Proposiion 3. If an inerior equilibrium in he homogeneous agens economy 1) 4) exiss, hen an infinie number of equilibrium allocaions exiss for individual hours worked. Proof. See Appendix. Wha is he source of he indeerminacy of he disribuion of working hours? A he individual level, working hours ener linearly in boh he preferences and

8 258 L. Maliar, S. Maliar he budge consrain. From he perspecive of he agen, hours worked in differen periods are perfec subsiues. Consequenly, he agen is indifferen beween any sequences of working hours leading o he same lifeime disuiliy from working. A he same ime, he working hours of differen individuals are perfecly subsiuable in producion. From he poin of view of he economy as a whole, any subdivision of working hours among agens is opimal as long as i leads o he opimal aggregae labor inpu in all periods. The assumpions ha are essenial for he exisence of indeerminacy are ha he economy lass for more han one period and ha i consiss of more han one agen. In a one-period economy, individual working hours are uniquely deermined by budge consrain 8): all agens consume he same amoun, c s = c, and spend he res of heir endowmen on leisure. In urn, in a one-agen economy, he equilibrium is unique because he individual working hours suppled by he agen coincide wih he labor inpu demanded by he firm, n s = h for all. 11 The individual budge consrain, which is linear in he muli-consumer model, becomes sricly convex in he one-consumer seup. Our analysis has a direc implicaion for he indivisible labor model by Hansen 1985). In such a model, he ime allocaed o he marke job can only have one of wo values, i.e., some fixed number of hours or zero hours. Insead of choosing hours worked, an agen decides on he probabiliy of being employed. Wheher he agen ges he job or no is deermined by a loery, which is won wih he probabiliy chosen by he agen. Regardless of he realizaion of he employmen loery, he agen is paid he expeced labor income he one corresponding o he probabiliy of employmen chosen). I is assumed ha here is a coninuum of exane idenical agens whose uiliy funcions are addiive in consumpion and leisure and logarihmic in boh consumpion and leisure. As formulaed in Hansen 1985), he problem of an individual is max E 0 δ [log c )+Aα log 1 h 0 )] 13) {c,α,k +1} T =0 s.. c + k +1 1 d + r ) k + w α h 0, 14) where α is he probabiliy of being employed, he oal ime endowmen of he agen is equal o one, and h 0 is he indivisible number of hours supplied o he firm by an employed agen. As shown in Hansen 1985), he aggregae equilibrium allocaion in he indivisible labor model 13), 14) is described by he planner s problem 6), 7), 12) and i is uniquely deermined. Regarding he individual opimal allocaion, however, our analysis suggess ha he opimal choice for individual probabiliy of employmen, α, is no uniquely deermined. A he individual level, he probabiliy eners linearly in boh he preferences 13) and he budge consrain 14). This feaure gives rise o an indeerminae equilibrium. 11 Indeed, consider he Firs Order Condiion FOC) of 1), 2) wih respec o working hours. Wih idenical agens, we have v c s ) w = A, where w = f k,h)/ k. If here is only one agen, hen h = n s, which allows us o idenify ns. However, if here is more han one agen, we only know h = S ns ds, which is no sufficien o idenify he disribuion {ns }s S.

9 Quasi-linear preferences in he macroeconomy Ruling ou indeerminacy by inroducing heerogeneiy in labor In his secion, we analyze he implicaions of heerogeneous agens versions of he model. We firs show ha, similarly o he case of idenical agens, he opimal consumpion allocaions are uniquely deermined in heerogeneous agens economies. We subsequenly demonsrae ha inroducing labor heerogeneiy can rule ou he indeerminacy of he individual labor choice. 5.1 Consumpion choice We firs sudy he consumpion choice. Inegraing 10) across agens yields c = S v s ) ) 1 1 λ s V 1 c, {λ s } s S)) ds. 15) For a given se of weighs {λ s } s S and aggregae consumpion c, here is a unique value of V 1 c, {λ s } s S) solving 15). Given he funcion V 1, he opimal consumpion of each agen is uniquely deermined by 10). Below, we show an example in which V can be derived explicily. Example 1. Assume ha v s c s )=c s ) γ, where γ 0, 1). Equaions 10) and 15), respecively, become c s = 1 λ s γ V 1 c, {λ s } s S)) 1/γ 1) 1, and c = γξ V 1 c, {λ s } s S)) 1/γ 1), 1 γ. where ξ = S λs ) ds) 1/1 γ) Hence, we have V c, {λ s } s S) ξc γ. Furhermore, combining he above expressions for c s and c,wege c s =λ s /ξ) 1/1 γ) c. This formula deermines he opimal disribuion of consumpion. 5.2 Labor choice To characerize he opimal labor choice, we are o make specific assumpions on how working effors of differen individuals are aggregaed ino he labor inpu. We begin by discussing he case in which he agens effors are perfecly subsiuable in producion. Laer in he secion, we will consider he case of imperfec subsiues.

10 260 L. Maliar, S. Maliar Perfec subsiues Assume ha he working hours of differen agens are perfec subsiues 12 h = n s β s ds. 16) S We firs consider he case in which he decenralized economy 1) 4), 16) has an inerior equilibrium, or equivalenly, he planner s problem 5) 7), 16) has an inerior soluion see Proposiion 2). In such a case, we have he following resul. Proposiion 4. a). In order an inerior soluion o he planner s problem 5) 7), 16) exiss, i is necessary ha β s = β b s for all and s. b). The welfare weighs in 5) saisfy λ s = bs /A s for all s. S bs /A s ds c). W h, {λ s,β s } s S) A h, where A = β S bs /A s ds ) 1. d). There exiss an infinie number of opimal allocaions for individual hours worked. Proof. See Appendix. The resul ha only one se of weighs is consisen wih an inerior equilibrium is relaed o he well-known propery of quasi-linear uiliy funcion ha a change in wealh affecs he demand for commodiies ha ener he uiliy funcion linearly bu does no affec he demand for any oher commodiies. In he conex of our model, his propery means ha he opimal choice of individual consumpion is independen of he disribuion of iniial endowmens. In he one-period version of our model, for example, he Engel curves are parallel zero-sloped sraigh lines wih he origin consumpion inercep) being deermined by v s,a s and b s. 13 Given ha he opimal consumpion of agens is he same for all disribuions of iniial endowmens, we always have he same se of welfare weighs. The exisence of he represenaive consumer may seem surprising since, as i is known from Blackorby and Schworm 1993), he assumpion of quasi-linear uiliy funcion is inconsisen wih aggregaion unless he uiliy funcion is linear in all commodiies. We are able o escape his negaive implicaion of Blackorby and Schworm s 1993) analysis because our aggregaion requiremen is weaker han he one hey use. Specifically, we allow for he case in which he social uiliy funcion depends no only on aggregae quaniies bu also on he disribuion of welfare weighs. We herefore consruc a represenaive consumer, no for all, bu only for he unique se of welfare weighs ha corresponds o an inerior equilibrium. The 12 This assumpion is common in he heerogeneous agens lieraure, e.g., Kydland 1984, 1995), Aiyagari 1994), Krusell and Smih 1998), Casañeda, Díaz-Giménez and Ríos-Rull 1998), Maliar and Maliar 2001). 13 A class of uiliy funcions leading o parallel Engel curves is called similarly quasi-homoheic. The propery of quasi-homoheiciy is known o be boh necessary and sufficien for aggregaion in he sense of Gorman 1953). Therefore, if he economy sudied in his secion has an inerior equilibrium, hen i admis a represenaive consumer, independenly of specific assumpions abou individual characerisics {v s,a s,b s } s S.

11 Quasi-linear preferences in he macroeconomy 261 definiion of a represenaive consumer used in Blackorby and Schworm 1993) is more resricive as i requires he preferences of he represenaive consumer o be he same for all ses of welfare weighs. As follows from Proposiion 4, if we resric our aenion o inerior equilibrium, he indeerminacy does no disappear afer he inroducion of heerogeneiy. 14 We shall poin ou ha he exisence of an inerior equilibrium requires imposing he specific resricion ha he relaive skills of agens do no change over ime. If such a resricion is no saisfied for some s and, hen he corresponding opimal allocaions are no inerior. Below, we argue ha having he opimal allocaions for working hours a corners can help o resore he uniqueness of equilibrium. According o 11), he agen s opimal labor choice in he presence of he corner soluions is as follows: o work he maximum possible number of hours n if he curren labor produciviy β s is sricly higher han A s λ s /W 1, o work zero hours if β s is sricly lower han A s λ s /W 1, and o work a cerain number of hours n s [0, n], ifβ s is equal o A s λ s /W 1. The aggregae labor inpu is herefore given by h = n β s > As λ s W 1 β s ds + β s = As λ s W 1 n s β s ds. 17) In order o have a unique equilibrium, i is necessary o insure ha he se of agens, who have inerior equilibrium allocaions, has a measure zero. This can be done by assuming ha in each period, individual skills are randomly drawn from some disribuion wih a coninuous densiy. Example 2. Consider an economy populaed by ex-ane idenical agens wih names in he inerval S =[0, 1]. Suppose ha he individual skills are β s = β b s, where β is he aggregae componen, and b s is he idiosyncraic componen drawn from he uniform disribuion [ 0, 2 ]. 15 Noe firs ha ex-ane idenical agens have idenical welfare weighs, λ s =1 for all s. Denoe by b he value of idiosyncraic componen ha makes he agens indifferen beween working or no, i.e., b = A/ β W 1 ), where A s = A for all s. Aggregae labor inpu 17) hen becomes: 2 ) h = nβ b s db s = nβ 1 b2. 18) b 2 Noe ha he se of agens whose opimal allocaions for working hours are inerior i.e., whose skills are b s = b ) has a measure zero. Combining 18) wih he hreshold condiion b = A/ β W 1 ) yields W 1 h, {λ s,β s } s S) A = ) β 2. 1 h nβ 14 Kydland 1984) sudies a similar wo-agen model under he Cobb-Douglas sricly concave) uiliy funcion. He finds ha he model reproduces empirical observaions ha high-produciviy agens work more and experience less volailiy of labor han he low-produciviy. Because of he indeerminacy, similar quesions canno be asked if he uiliy funcion is quasi-linear. Neiher can hey be addressed in he indivisible labor framework. 15 A similar example is considered in Cho 1995).

12 262 L. Maliar, S. Maliar Therefore, he funcion W is W h, {λ s,β s } s S) An h ). 19) nβ The planner s problem 6), 7) wih he social uiliy funcion, defined by 15), 19), yields he aggregae equilibrium allocaion. If such an allocaion is given, he hreshold condiion uniquely deermines who works and who does no. Noe ha he funcion W is sricly concave, even hough he uiliy funcions of all heerogeneous agens are quasi-linear. We should also menion ha he assumpion ha agens are ex-ane idenical is no essenial for resolving he indeerminacy. The opimal allocaions for individual hours worked will be unique, independenly of a specific disribuion of welfare weighs across agens Imperfec subsiues Suppose now ha he individual working effors are no perfecly subsiuable in producion so ha he marginal labor inpu of an agen s can depend on boh skills and quaniies of labor supplied by all agens in he economy: h n s = n s h {n s,β s } s S). 20) In his case, opimaliy condiion 11) provides S equaions conaining S unknown individual hours worked, {n s } s S. Below, we show an example in which his sysem of equaions 11) has a unique soluion. Thus, he assumpion of imperfec subsiues can help us o resolve he indeerminacy of he individual labor decisions. 16 Example 3. Assume ha h is given by he CES funcion 1/ε h = β s n s ) ds) ε, ε, 1). S The parameer ε deermines he degree of subsiuabiliy and complemenariy among he labor inpus of differen agens. In he limis, ε =1and ε,we have perfec subsiues 16) and perfec complimens, respecively. Suppose ha equilibrium is inerior, i.e., condiion 11) holds wih equaliy. By finding he derivaive h n of he CES funcion, subsiuing i ino 11) and solving s wih respec o n s,wege n s = β s A s λ s W 1 h, {λ s,β s } s S)) 1/1 ε) h. 21) 16 If he economy is populaed by an infinie number of agens, hen he equilibrium is unique up o he allocaion of a zero-measure subse of agens. This is because such a subse of agens has no effec on he aggregae equilibrium allocaion, see Kehoe and Levine 1985).

13 Quasi-linear preferences in he macroeconomy 263 Inegraing β s n s ) ε across agens and rearranging he erms yields [ W 1 h, {λ s,β s } s S) ) ] β = β s s ε/1 ε) 1 ε)/ε A s λ s ds A. S Therefore, we have W h, {λ s,β s } s S) A h. The aggregae equilibrium allocaion can again be compued by solving he planner s problem 6), 7). The individual hours worked are uniquely deermined by 21). The propery ha helps us resore he uniqueness of equilibrium in his case is he sric convexiy of individual budge consrains. 6 Concluding commens This paper invesigaes he consequences of quasi-linear preferences addiive in consumpion and leisure, and linear in leisure) in a dynamic general equilibrium model wih muliple consumers. To characerize he disribuional dynamics, we employ he aggregaion heory. Our main findings are as follows: If consumers are homogeneous, here is indeerminacy of equilibrium: he model does no produce sharp predicions abou he individual hours worked. However, he indeerminacy does no, in general, survive he inroducion of heerogeneiy in labor. Two examples of labor heerogeneiy ha resore he uniqueness of equilibrium are idiosyncraic shocks o skills and imperfec subsiuabiliy of labor in producion. All he above resuls also apply o he indivisible labor model by Hansen 1985). Appendix Proof of Proposiion 1 The definiion of he social uiliy funcion 5) implies λ s v s ) c s )+µ =0, 22) A s λ s h + η + ς s ζ s =0, 23) n s ς s 0 and ς s n s =0, 24) ζ s 0 and ζ s n n s )=0, 25) U + µ =0, c 26) U + η =0, h 27) where µ,η,ς s, ζ s are he Lagrange mulipliers associaed wih he consrains S cs ds = c, h {n s,β s }) =h, n s 0 and n s n, respecively. By subsiuing 22) ino 26) and solving wih respec o c s, we obain 10). Condiions 23) 25) ogeher wih 27) yield 11). The fac ha he individual uiliy funcions are

14 264 L. Maliar, S. Maliar addiive in consumpion and working hours implies ha he social uiliy funcion, U, is addiive in aggregae consumpion and labor, i.e., U can be wrien as he sum of wo subfuncions, V and W, as is in 9). This can be shown by subsiuing 10) and 11) ino he objecive funcion in 5). Proof of Proposiion 2 The FOCs of he consumer s uiliy maximizaion problem 1), 2) wih respec o Arrow securiies, capial, consumpion and hours worked, respecively, are λ s p Z) =δλ s +1 Pr {B +1 Z B = z} Z R,z I, 28) λ s [ = δe λ s +1 1 d + r +1 ) ], 29) v s ) c s )+λ s =0, 30) A s + λ s h w n s + ς s ζ s =0, 31) wih ς s and ζ s saisfying ς s 0 and ς s n s =0, 32) ζ s 0 and ζs n n s )=0. 33) Here, λ s, ς s, ζ s are he Lagrange mulipliers corresponding o budge consrain 2) and resricions n s 0 and n s n, respecively, and w s h = w n wih s w fk,h) h. Noe ha equaion 28) implies ha for any wo agens s,s S, we have λ s λ s = λs +1 for all Z R. 34) λ s +1 Therefore, we can rewrie he Lagrange muliplier λ s as λ s = λ /λ s. This is he sandard implicaion of he complee markes assumpion ha he raio of marginal uiliies of any wo agens remains consan in all periods and saes of naure. Wihou loss of generaliy, we normalize he mulipliers by S λs ds =1. Assume ha he equilibrium allocaion for individual working hours in he decenralized economy is inerior, i.e., ς s =0and ζ s =0for all s. Using he resul λ s = λ /λ s, we have ha for any wo agens s,s S, A s λ s A s λ s = h / n s h / n s. 35) For he planner s soluion for individual working hours o saisfy 35), we mus have ha, in 23), ς s =0and ζ s =0for all s. Hence, he planner s allocaion for individual hours worked is inerior. The same ype of argumens can be used o prove he converse saemen: assuming ha he planner s soluion is inerior, ς s =0and ζ s =0for all s, we obain ha he individual working hours saisfy 35), which implies ha for all s, ς s =0and ζ s =0in 31), i.e., he equilibrium allocaion for working hours in he decenralized economy is inerior.

15 Quasi-linear preferences in he macroeconomy 265 Proof of Proposiion 3 We assume ha he economy consiss of more han one agen. Le {c,h,k +1 } T be he aggregae opimal allocaion. Firs of all, noe ha he symmeric allocaion such ha all agens behave idenically, c s = c and n s = h for all s, saisfies all he opimaliy condiions. Le us show ha here exiss also an infinie number of opimal allocaions wih asymmeric labor choices. Consider an allocaion { c s, ñ s } s S T such ha cs = c for all s and ñ s = h for all s excep of agens s and s, whose labor decisions are as follows: {ñi } = {h [0,...,τ 1,τ+2,..., ) } [0,...,τ 1,τ+2,..., ), i {s,s }, ñ s τ = h τ + ε, ñ s τ = h τ ε, ñ s τ+1 = h τ+1 ε δ, ñs τ+1 = h τ+1 + ε δ, where ε>0is any number such ha ñ s τ, ñ s τ, ñ s τ+1, ñ s τ+1 [0, n]. The above consrucion is always possible under our assumpion of inerior equilibrium, which guaranees ha he equilibrium allocaions for he aggregae labor inpu, h, are inerior for all. Observe ha he asymmeric allocaion leads o he same aggregae labor inpu as he symmeric one, since we have ñ s +ñ s = h +h for {τ,τ +1}.Given ha he level of oupu does no change, he equal consumpion, c s = c for all s, is feasible in he asymmeric case. Furhermore, noe ha in he asymmeric and symmeric cases, all agens, including i {s,s }, ge he same lifeime disuiliy from working E 0 {... + δ τ A ñ s τ + δñ s τ+1) +... } = E0 {... + δ τ A h τ + δh τ+1 )+...}. As he asymmeric allocaion implies he same expeced lifeime uiliy for all agens as he symmeric one, his allocaion is opimal. Proof of Proposiion 4 a). According o 11), for any wo agens s,s S, we have A s λ s A s λ s = βs β s for all. 36) Therefore, an inerior equilibrium is possible only if he relaive skills of agens do no change over ime, i.e., β s = β β s for all s. b). The formula for λ s follows from 36) and normalizaion S λs ds =1. c). By subsiuing h n = β s s ino 11), expressing λ s and using he normalizaion S λs ds =1, we obain W 1 = A and, hus, W A h. d). The proof is parallel o ha of Proposiion 3. We again assume ha he economy consiss of more han one agen. Le {c,h,k +1 } T be an aggregae opimal

16 266 L. Maliar, S. Maliar allocaion. Noe ha in he economy wih heerogeneous agens, he symmeric allocaion is, in general, no a soluion. The opimal disribuion of consumpion across agens, {c s } s S T, is he one which saisfies condiion 10). The opimal disribuion of working hours, {n s } s S T, is, however, no idenified by he corresponding opimaliy condiion 11). We firs consruc one disribuion {n s } s S T saisfying all of he opimaliy condiions, and hen show ha an infinie number of oher opimal disribuions exiss. One opimal disribuion of working hours can be consruced by assuming ha each agen works a fixed share of aggregae labor inpu, n s = n s h for all and s. The opimal shares are idenified by individual expeced lifeime budge consrains 8) κ s n s 0 E 0 δ τ v s ) c s τ ) v τ=0 s ) c s 0) cs τ =. β s E 0 δ τ vs ) c s τ ) v s ) c s 0) h τ w τ β τ τ=0 Consider, now, he allocaion { c s, ñ s } s S T such ha cs = c s for all s and ñ s = n s for all s excep for agens s and s, whose labor decisions are {ñi } } [0,...,τ 1,τ+2,..., ) = { n i ñ s τ = n s τ + ε β s, ñ s τ+1 = n s τ+1 ε δβ s,, i [0,...,τ 1,τ+2,..., ) {s,s }, ñ s τ = n s τ ε, β s ñ s τ+1 = n s τ+1 + ε, δβ s where ε>0is a number such ha ñ s τ, ñ s τ, ñ s τ+1, ñ s τ+1 [0, n]. As in Proposiion 3, he disribuions {n s } s S T and {ñs } s S T lead o he same aggregae labor inpu in all periods, so ha oupu and consumpion are also he same. Furhermore, {n s } s S T and {ñs } s S T imply he same lifeime disuiliy from working for all consumers. Consequenly, he allocaion { c s, ñ s } s S T is opimal.

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