Energy Tax and Equilibrium Indeterminacy

Transcription

1 Energy Tax and Equilibrium Indeerminacy Yan Zhang Anai College of Economics and Managemen Shanghai Jiao Tong Universiy Augus 11, 2010 Absrac We sudy he e ec of energy axes (or ari s) in a sandard neoclassical growh model wih impored energy in producion. We nd ha (1) he model may exhibi local indeerminacy and sunspos when energy ax raes are endogenously deermined by a balanced-budge rule wih a consan level of governmen expendiures (or lump-sum ansfer), and (2) indeerminacy disappears if he governmen nances endogenous public spending and ransfers wih xed ax raes. Under he rs ype of balanced budge formulaion, we provide numerical (calibraion) examples o illusrae ha he governmen should no disor he energy price paid by rms wih energy axes in order o avoid aggregae insabiliy. Under he second ype of balanced budge formulaion, we prove ha he economy exhibis equilibrium uniqueness regardless of he exisence of lump-sum ransfers. This is a shorened version of my disseraion a New York Universiy, "Tari and Equilibrium Indeerminacy". I would like o hank my advisers, Jess Benhabib and Jushan Bai, for heir guidance. I likewise hank Boyan Jovanovic, Luis Aguiar-Conraria, Wen Yi, Zhou Lin, Pierpaolo Benigno, Paul Dower, Marin Uribe, Vivian Yue, Alain Vendii, Zheng Xu, Chen Yan, Lu Lei, Vikor Tsyrennikov and seminar paricipans a he 8h Journees Louis-Andre Gerard- Vare conference in public economics held in Marseille in June 2009 for heir valuable commens. Finally, he auhor acknowledges he nancial suppor of New York Universiy (Henry M. MacCracken Fellowship) and Shanghai Jiao Tong Universiy. Address correspondence o Zhang Yan, Deparmen of Economics, Anai College of Economics and Managemen, Shanghai Jiao Tong Universiy, Shanghai, , China. Tel and Fax: ; address: laurencezhang@yahoo.com (Y. Zhang). 1

2 Key Words : Indeerminacy, Endogenous Energy Tax Rae, Balanced-budge Rule JEL Classi caion Number: Q43, F41 1. Inroducion A large body of lieraure on opimal axaion has recenly suggesed ha energy axes (or ari s) have a high e ciency cos when he governmen needs o raise revenue by using ax insrumens. Alhough much of he early research was concerned wih deerminacy cases in dynamic sochasic general equilibrium (DSGE) models wih energy in producion [Roemberg and Woodford (1994); de Miguel and Manzano (2006)], energy axes on inermediae goods (say, impored energy), which ac like a ax on he reurns o facors of producion, may generae indeerminacy in a way similar o facor income axes. The pioneer work of Schmi-Grohe and Uribe (1997, henceforh SGU) shows ha, in a Ramsey model wih a pre-se level of governmen expendiures (or lump-sum ransfers), if he labor income ax raes are endogenously deermined by a balanced budge rule, he model can exhibi indeerminacy for empirically plausible values of income ax raes. Guo and Harrison (2004, henceforh GH) furher show ha consan income ax raes can no be per se a source of local indeerminacy in a Ramsey model. In his paper, we exend heir analysis, sudying he role of public spending nanced by energy axes (or ari s) on he emergence of local indeerminacy, wihin a sandard Ramsey model wih energy in producion, for he same class of scal policy rules. 1 We nd ha energy axes and labor income axes are equivalen in generaing local indeerminacy when scal increasing reurns arise. More precisely, we consider a counercyclical ( a) energy ax policy in which consan (endogenous) governmen expendiures (or lump-sum ransfers) are nanced by endogenous (exogenous) ax raes. 1 For simpliciy, we assume ha he governmen does no impose consumpion axes on he radable goods or facor income axes on he producion facors. The energy ax revenue in his model can also be inerpreed as ari revenue. 2

3 I urns ou ha a counercyclical energy ax policy is required in generaing indeerminacy in his model, while a a energy ax policy can make he economy immune o indeerminacy regardless of he exisence of lump-sum ransfers. In addiion, under he balanced-budge rule wih a counercyclical ax policy, for empirically plausible values of energy ax raes, we provide calibraion examples o illusrae ha he governmen should no disor he energy (oil) price paid by rms wih energy axes in order o avoid aggregae insabiliy. A novel feaure of his model is ha he mechanism behind our indeerminacy resul is essenially he same as ha in SGU. If he represenaive agen expecs fuure energy ax raes o increase, fuure impors of foreign inpus and he marginal produc of capial (for any given capial sock) will be lower. I implies ha he curren demand for foreign inpu will be lower, hus leading o a fall in oal oupu. If he energy ax rae is regressive wih respec o he oupu, he ax rae oday will increase, hus validaing he agen s iniial expecaions. In conras, he mechanism described above does no work in he case of a consan energy ax rae because consan raes wih he diminishing marginal producs of inpu can reduce he higher anicipaed reurns, hus making indeerminacy hard o arise. Calibraed examples show ha when we use Aguiar-Conraria and Wen s (2005, 2007, 2008; henceforh ACW) esimaion of he impored energy share in Denmark and he Neherlands, he high ax raes on energy (oil) in hese counries can lead hem ino desabilizaion. 2 Anoher novel feaure of his model is ha he larger he energy share in gross domesic produc (GDP), he easier i is for he economy o be subjec o muliple equilibria. Here, we consider he case where expecaions of a fuure ax increase shif he labor supply curve up. We nd ha he larger he energy share is, he larger he decline in employmen is because he slope of he labor demand curve decreases in absolue value as he energy share increases. Therefore, he larger he increase in energy ax rae required o balance he budge is, he larger he energy share is. Being 2 Alhough hroughou he paper, we analyze he model for he developed counries, he resul also holds for lessdeveloped counries whose producions are dependen on he impored facors. 3

4 aware of his mechanism exising in his model is crucial o undersand why reliance on foreign energy can increase he likelihood of aggregae insabiliy. Since hese properies described above are obained under he assumpion ha he energy is a capial subsiue, a relevan issue is o undersand wheher indeerminacy can easily arise when he energy is a labor subsiue. In his paper, we brie y discuss his issue and nd ha muliple equilibria become more likely o occur under he capial subsiue assumpion han under he labor subsiue assumpion because he absolue value of he slope of he labor demand curve under he capial subsiue assumpion is smaller han ha obained under he labor subsiue assumpion, he increase in energy ax rae required o balance he budge under he former assumpion should be larger han ha obained under he laer assumpion when expecaions of a fuure ax increase shif he labor supply curve up. In he res of his paper, we presen he model in Secion 2, ha exends he SGU (1997) framework o allow for public spending, which is nanced by energy axes insead of labor axes. We obain he model dynamics and discuss in deail our indeerminacy resuls, providing numerical examples. In Secion 3, we compare our model wih he Benhabib and Farmer (1994), SGU, and ACW models and prove several properies of he indeerminacy resuls. Finally, we conclude he paper in Secion An Economy wih Energy Taxes This paper incorporaes wo di eren formulaions of governmen budge consrains ino a sandard neoclassical growh model ha incorporaes foreign energy as a hird producion facor. We assume ha labor is indivisible (as in Hansen (1985)), and he only source of governmen revenue is an energy ax. In paricular, he balanced-budge rule consiss of exogenous (and/or endogenous) governmen purchases (and/or ransfers) and endogenous (and/or exogenous) ax raes levied on he impored 4

5 inpu Firms We inroduce governmen ax policy ino he coninuous ime neoclassical growh model wih producive impored inpu. A coninuum of idenical compeiive rms exiss, wih he oal number normalized o one. The single good is produced by he represenaive rm wih consan reurns o scale Cobb-Douglas producion funcion: y = k n an o, (1) where y is he oal oupu, k is he aggregae sock of capial, n is he aggregae labor supply, + a n + = 1, and he hird facor in he producion, say oil (o ), is impored. Assuming ha rms are price akers in he facor markes, pro maximizaion by each rm leads o he following rs-order condiions: w = a n y n, (2) r = y k, (3) and p o (1 + ) = y o, (4) where r denoes he marginal produc of capial, w denoes he real wage, p o denoes he real price of oil, and is he ax rae levied on he impored oil and is uniform o all rms. We should emphasize ha (1) p o is he relaive price of he foreign inpu in erms of he single good, which is he numeraire and radable, and (2) he variable represens he endogenous (or exogenous) ax 5

6 rae levied on he foreign inpu, and we require ha 0 o rule ou he exisence of impor subsidies. Since we assume ha he economy is open o imporing energy, he agen can use he radable good o buy he foreign inpu. The energy price is assumed o be exogenous, and he foreign inpu is assumed o be perfecly elasically supplied. 3 These imply ha he energy price, p o, is independen of he facor demand for o and deermined on a world marke ha is no in uenced by he domesic economy. Hence, by subsiuing o in he producion funcion wih o = he following reduced-form producion funcion: y p o (1+ ), we can obain y = A k 1 an 1 a n 0. (5) The erm A = [ p 0 (1+ ] a0 ) 1 acs as he "Solow residual" in a neoclassical growh model, which is inversely relaed o he foreign facor price and. In his reduced-form producion funcion, he "e ecive reurns o scale" are measured by + a n 1 = 1. (6) 2.2. Households The economy is populaed by a uni measure of idenical, in niely lived households. Each household is endowed wih one uni of ime and maximizes he ineremporal uiliy funcion: Z 1 0 e (log c bn )d, b > 0, (7) 3 The model is based on he sandard DSGE models ha incorporae foreign energy as a hird producion facor. This class of models (such as hose of Kim and Loungani (1992), Finn (2000), Roemberg and Woodford (1996), Wei (2003), and ACW (2005, 2007, and 2008)) has been used widely o sudy he business-cycle e ecs of oil price shocks. The empirical jusi caion for he exogeneiy of p o is provided by Hamilon (1983, 1985). 6

7 where c and n are individual household s consumpion and hours worked, and 2 (0; 1) is he subjecive discoun rae. We assume ha here are no inrinsic uncerainies presen in he model. The budge consrain of he represenaive agen is given by : k = (r )k + w n c + T, k 0 > 0 given, (8) where : k denoes he ne invesmen, 2 (0; 1) denoes he depreciaion rae of capial, and T 0 is he lump-sum ransfers. The rs-order condiions for he household s problem are 1 c =, (9) b = w, (10) : = ( + r ), (11) where denoes he marginal uiliy of income. Equaions (9) and (10) require ha he household s marginal rae of subsiuion beween consumpion and leisure be equal, ha is, b = w c. Equaions (9) and (11) imply he consumpion Euler equaion Governmen The governmen chooses he ax/ransfer policy f ; T g, and balances is budge in each period. A each poin in ime, he budge consrain of he governmen can be saed as follows: 7

8 p o o = y (1 + ) = G + T, (12) where G 0 represens governmen expendiures. Finally, marke clearing requires ha aggregae demand equal aggregae supply: c + G + : k + k + o p o = y. (13) Noe ha inernaional rade balances each period. Equaion (13) shows ha domesic producion is divided among consumpion, invesmen, impors, and governmen expendiures (c +i +p o o + G = y, i = : k + k ). In oher words, in order o receive energy, he domesic rm pays he amoun p o o in erms of oupu o foreign counry. And he ax revenue p o o is divided beween governmen expendiures and lump-sum ransfers Analysis of he model dynamics Similar o GH (2004), we assume ha ax revenues can eiher be consumed by he governmen (i.e., G 0 for all ) or reurned o households as ransfers (i.e., T 0, for all ). Verifying ha he economy in which he governmen nances endogenous public spending and/or ransfers wih xed ax raes is immune o indeerminacy is rivial. This is due o he following proposiion: Proposiion 1. If he ax rae is exogenous, producion does no exhibi increasing reurns o scale since he A erm is a consan for all. (In his case, governmen expendiures are endogenous under he balanced budge rule.) Therefore, he economy exhibis saddle pah sabiliy regardless of he exisence of lump-sum ransfers. GH prove ha under perfec compeiion and consan reurns-o-scale, if he governmen nances endogenous public spending and ransfers wih xed income ax raes, a one-secor real business cycle 8

9 model exhibis deerminacy regardless of he exisence of lump-sum ransfers. In his model, we have he same resul. Wih consan ax raes, he model does no display increasing reurns o scale. Therefore, indeerminacy canno arise. To remain comparable wih SGU s analysis, we focus on he cases where he governmen eiher consumes all ax revenues (i.e., T = 0) or ransfers he revenue o he household in a lump-sum way (i.e., G = 0). In he following secions, we mainly discuss he case where T = 0 holds for all. Under his speci c assumpion T = 0, we replace consumpion wih 1 and ransform he wage rae and he renal rae ino funcions of capial and labor. Then he equilibrium condiions can be reduced o he following ve equaions: b = a n A k 1 an 1 1 n, (14) : = + A k 1 1 an 1 a n 0, (15) : k = (1 1 )y k G, (16) 1 + G = y (1 + ), (17) and y = A k 1 an 1 a n 0. (18) 9

10 Firs, we claim ha for a given ax rae, a unique inerior seady sae exiss in he he dynamical sysem. Lemma 1. The dynamical sysem possesses a unique inerior seady sae when he governmen consumes all ax revenues, and he ax rae is exogenous, ha is, =, for all. (In his case, A ( ) = A() holds for all.) Proof. To nd he unique seady sae, we rs se : in (15) equal o zero and solve he capial/labor raio in he seady sae. We nd ha ( k n ) ss = [ + A() ] 1 an is dependen on he energy ax rae and unique for he given ax rae. Second, a unique and posiive value of in he seady sae, which is ss = b a [ + na() A() ]=a n, can be derived from equaion (14). Using he seady sae value of, he governmen budge consrain (17), and he seady sae condiion : k = 0, he marke-clearing condiion (16) can be wrien as (1 )k ss [A()( k n ) an 1 ss ] = a na() [ + b A() ] =a n. (kss) Since ( k n ) ss is known given he ax rae, we can nd ha k ss (he seady sae value of he capial sock) is unique and posiive. Since he seady sae values of he capial sock and he capial/labor raio are posiive and unique, n ss (he seady sae value of he labor supply) is also posiive and unique. Finally, he seady sae level of governmen purchases given by (17) is also unique. Sraighforward compuaions show ha i can be wrien as G ss = 1 + A()k ss ( k an n ) 1 ss, (g) where k ss is he soluion o he (kss) equaion. Therefore, G ss is coninuous in. 10

11 From (kss) and (g), one can show ha when is equal o zero, G ss is also equal o zero because k ss is posiive and nie in his case. If he ax rae is exogenous, we can prove ha here exiss a unique ax rae ha maximizes G ss. I is m = an. Second, for a given level of governmen expendiures, a (seady-sae) La er curve-ype relaionship beween he ax rae and ax revenue exiss, which means ha he number of seady sae ax raes ha generaes enough revenue o nance he pre-se level of governmen purchases will generally be eiher 0 or 2. We prove his in he following lemma: Lemma 2. When ax raes are endogenously deermined by a balanced-budge rule wih a consan level of governmen expendiures, he seady sae in he dynamical sysem ha consiss of (14)-(18) may exis, and he number of seady sae ax raes ( ss ) ha generaes enough revenue o nance he pre-se level of governmen purchases will generally be eiher 0 or 2. If wo seady saes exis in he model, we only focus on he seady sae associaed wih he low seady sae ax rae since he seady sae associaed wih he high seady sae ax rae is always locally deerminae. Proof. We derive he seady sae values of he variables k n = [ and k = ana(ss) + [ b A(ss) ] [ 1 (+) ] an + A( ] 1 ss) an, = b a [ + na( ss) A( ] an ss),, where A( ss ) denoes he seady sae value of A as is equal o is seady sae value ss. We nd ha in he seady sae, G = holds, and he consan is ( p o ) an (+)a n( + ) b[ 1 (+) ] an ss an+ (1+ ss) an consan F ( ss ). F ( ss ) is herefore clearly non-monoone, and he number of posiive seady sae ax raes ha generaes enough revenue o nance a pre-se level of governmen purchases will generally be eiher 0 or 2. The second ineresing nding is ha if is exogenous as in he = 0 implies ha here exiss a unique exogenous ax rae ha maximizes G ss. Is value is an. This is due o G ss being equal o an+ (1+) an consan. Inser Figure 1 here. Figure capion: Seady-sae La er curve. 11

12 Third, we show ha when governmen expendiures are exogenous, he ax rae is counercyclical wih respec o he ax base or he oupu under he balanced budge rule. The following proposiion is he key o indeerminacy in his model: Proposiion 2. If governmen expendiures are exogenous, he ax rae is regressive wih respec o he ax base (p o o ) or he oupu under he balanced budge rule, (counercyclical) ax labor. < 0. The regressive < 0) can induce increasing reurns o scale wih respec o capial and Proof. p o o = (1+ ) = G implies equaions around he seady sae G = y (1+, A ) = [ o verify ha ^y = ^ 1 (1+ ss) k + a n ^ 1 (1+ ss) < 0. Considering he log-linearizaion of he following n, where k ^, p 0 (1+ ] a0 ) 1, and y = A k 1 n an 1, i is easy ^ n, and ^y denoe he log deviaions of k, n, and y from heir respecive seady saes (i.e., k ss, n ss and y ss ). This means ha producion exhibis increasing reurns o scale wih respec o capial and labor, ha is, an endogenous energy ax rae could be a source of scal increasing reurns. +a n 1 (1+ ss) > 1. Thus, GH illusrae ha in a sandard neoclassical growh model, SGU s indeerminacy resul depends on a scal policy rule in which he ax rae decreases wih household s axable income. In his model, we obain a similar resul ha requires he counercyclical rae o generae indeerminacy. To faciliae he analysis of model dynamics, we consider he log linear approximaion of he equilibrium condiions around he seady sae. Le, ^ k, ^, and ^n denoe he log deviaions of, k,, and n from heir respecive seady saes. The dynamics of he economy can be summarized by he sysem of di erenial equaions : : ^ k = a n ( + ) ss ( + ) ( + ) (1 ) 1 ( ss+1) ss ( + ) ss ss 1 ss ^ k (19) 12

13 Noe ha he race and he deerminan of he Jacobian marix ha deermines he local dynamics around he seady sae (denoed by J) are saed as follows: r(j) = de(j) = ( + ), (20) ss ( + ) ( + ) 1 f(a n ss ) [a n (1 ) ss (1 (1 + ss ))]g. (21) ss ss Proposiion 3. The necessary and su cien condiion for he model o exhibi indeerminacy is r(j) < 0 < de(j), or, < ss < an. Since he capial sock (k ) is a predeermined variable, he model is indeerminae if and only if boh eigenvalues of he Jacobian marix have negaive real pars. This is equivalen o requiring ha he deerminan be posiive and he race, negaive. I is easy o verify ha r(j) = ) < 0 if and only if ss >. If he race condiion is sais ed, he erm ss ( + (+) ss on he righ side of he deerminan is negaive. de(j) > 0 if and only if G( ss ) = [ (+)a2 0 (1 ) a 2 0 ] 2 ss ss [ (+)(1 ) 2 (1 )]+[(+)a n (1 ) a n ] < 0. Furher, showing ha G( ) = 0 and G(0) > 0 is rivial. Then he necessary and su cien condiion for he model o exhibi indeerminacy is equivalen o G < 0, or, < ss <, where = [(+)an(1 ) a n ] [(+) (1 ) ] = an >. Similar o SGU, if he se of ax raes saisfying he necessary and su cien condiion is no empy, a su cien condiion is ha he labor share should be larger han he capial share (i.e., a n > ). If he seady-sae ax raes are smaller han or greaer han an, he deerminan is negaive, and herefore, he model exhibis local deerminacy. Tha SGU shows ha he revenue maximizing ax rae is he leas upper bound of he se of ax raes for which he model is indeerminae should be emphasized; his propery also holds in our case. The inuiion behind he indeerminacy resul is quie sraighforward. If agens expec fuure ax raes o increase, hen for any given capial sock, fuure impors will be lower. Since he marginal 13

14 produc of capial increases in he oil inpu, he rae of reurn on capial will be lower as well. The decrease in he expeced rae of reurn on capial may lower he curren oil demand, leading o a decrease in curren oupu. Since he ax rae is he curren ax rae o increase, hus validaing agens iniial expecaions. < 0), budge balance can cause To help undersand he inuiion beer, we consider he consumpion Euler equaion (in discree ime for ease of inerpreaion). c +1 c = (1 + y +1 k +1 ) = [1 + ( ) where denoes he discoun facor, r b +1 ( p 0 ) a0 1 1 a 1 k 0 +1 n an 1 1 r b +1], (22) +1 he before-ax reurn on capial and +1 he ax rae in period ( + 1). When he agen s opimisic expecaions lead o a higher invesmen, he lef-hand side of his equaion will increase bu he before-ax reurn on capial r b +1 will be lower due o he diminishing marginal producs. The counercyclical ax rae can increase he righ-hand side of he equaion, hus making he iniial expecaions become self-ful lling. If he ax rae is a consan under he scal policy rule wih endogenous public spending and/or ransfers, he righ-hand side of (22) falls. As a consequence, indeerminacy canno arise in he laer case regardless of he exisence of lump-sum ransfers. Capial accumulaion is crucial in generaing indeerminacy in his economy. I is easy o show ha wihou capial accumulaion, he equilibrium is locally deerminae. 4 In addiion, we can exend he basic model o consider he case where G = 0 and T =consan hold for all. I can be shown ha considering a consan level of lump-sum ransfers does no aler our main resul, and a similar condiion o ha obained in proposiion 3 on endogenous ax raes is all ha is needed for indeerminacy. In ha case, we canno explicily derive he necessary and su cien condiion 4 Wihou capial, (14) becomes b= = a n[ G=n = ( 1 p o ) a0 1 ( 1+ ) 1 ] a0 p o (1+ ) 1, (16) becomes 1= = (1 an. From hese equaions, locally unique soluions for, n, and exis. 1+ )y G, and (17) becomes 14

15 for he balanced budge rule o generae indeerminacy. This is because relaxing he assumpions of public spending will make he deerminan of he Jacobian marix become more complicaed up o a hird-order polynomial. However our indeerminacy resul is robus o his exension Calibraed Examples In his secion, following SGU (1997), we calibrae he model using srucural parameers ha are sandard in he real business cycle lieraure. We se he ime period in he model o be one year, he annual real ineres rae = 0:04, and he annual depreciaion rae = 0:1. Esimaes of he labor income share (a n ) for he Neherlands economy range from 0:684 o 0:71 according o OECD. Sa. Based on inpu-oupu ables from OECD (1995) repors, ACW (2005) esimae impored energy s share in GDP for he Neherlands economy o be abou 0:21 (a o ). This implies ha he capial share ( ) ranges from 0:08 o 0:106. Therefore, he lower bound of he indeerminacy region ranges from 38:1% o 50:5%. The energy ax raes repored by he Inernaional Energy Agency (1998) for he Neherlands economy range from 61:3% o 74:9%, which clearly fall wihin he indeerminacy region. 5 Nex, we consider he energy ax policy ha is implemened in Denmark. As he opimal ari argumen, he energy axes are relaively high in his counry. 6 From 1990 o 1995, he energy ax raes on leaded and unleaded gasoline range from 60:3% o 72:2%. 7 We calibrae he model s srucural parameers following SGU (1997) and ACW (2005). We se he ime period in he model o be one year, he annual real ineres rae = 0:04, he energy share a o = 0:2, and he annual depreciaion rae = 0:1. Esimaes of he labor income share in Denmark (a n ) range from 0:662 o 5 From 1990 o 1996, he energy ax raes for he leaded and unleaded gasoline range from 61.3% o 74.9% (see Tables 8, 10 and 11, pp , 4h Quarer 1998, Energy prices and axes). The parameer value of a n is aken from 1993 o As in SGU, if he only source of governmen revenues is an endogenous consumpion ax, he model will exhibi local deerminacy. Similar o Giannissarou (2007), we conjecure ha he possibiliy of aggregae insabiliy caused by energy axes is reduced if we add an endogenous consumpion ax ino his model. 6 The energy ax revenue is overwhelmingly oil ax revenue in some EU counries, see Newbery (2005). 7 See Tables 8, 9, and 10, pp , 4h Quarer 1998, Energy prices and axes. 15

16 0:684, which implies ha he capial share in Denmark ( ) ranges from 0:116 o 0:138. The oil ax raes in Denmark fall wihin he indeerminacy region. 8 Inser Figure 2 here. Figure capion: Energy ax raes and labor share. 3. Comparison wih he Benhabib and Farmer, SGU, and ACW Models In his secion, we rs show ha a close correspondence exiss beween he indeerminacy condiion of our model and ha of he Benhabib and Farmer (1994) model wih producive increasing reurns. Tha is, he necessary condiion for local indeerminacy is ha he "equilibrium labor demand schedule" can be upward sloping and seeper han he labor supply schedule. Unlike he model of Benhabib and Farmer (1994), his model does no rely on increasing reurns in producion o make he "equilibrium labor demand schedule" upward sloping. In fac, he equilibrium labor demand schedule in our model is upward sloping because increases in employmen can decrease he equilibrium ax raes and increase he afer-ax reurn on labor. To see his, we wrie he afer-ax labor demand funcion as follows (in log deviaions from he seady sae): ^ w = 1 ^ k 1 ^n 1 ss 1 + ss ^, (23) where ^w = ^b w 1 ss 1+ ss ^ is he log deviaion of he afer-ax wage rae from he seady sae. 9 We see ha he rm s labor demand schedule is a decreasing funcion of ^n. However, when we replace ^ wih ^ k and ^n using he balanced-budge equaion, obaining he equilibrium labor demand schedule is rivial: 8 The parameer value of a n is also aken from OECD. Sa from 1993 o The values of and are aken from SGU (1997). The lower bound of he indeerminacy region ranges from 0:58 o 0:69. 9 ^b w denoes he log deviaion of he before-ax wage rae from he seady sae. 16

17 As our case, ^ w = a ^ k k + ( ss ) ^ n. (24) 1 (1 + ss ) 1 (1 + ss ) < ss < an, he equilibrium labor demand funcion is upward sloping since ( ss) 1 (1+ ss) > 0. In ^ w = ^c, so he aggregae labor supply is horizonal and he labor demand schedule will be seeper han he labor supply schedule if < ss < an. Tha our economy can easily be shown o be equivalen o he SGU model mus be emphasized because in boh cases, he price-o-cos markup is counercyclical wih respec o he oupu, which is he key o generaing indeerminacy (see he Appendix). Second, we compare our model wih he SGU model. SGU prove ha in a Ramsey model, when he governmen relies on changes in labor income axes o balance he budge, his scal policy rule can realize expecaions of higher ax raes. If he impor facor is assumed o be a labor subsiue, he endogenous ax rae levied on he impored oil may no make indeerminacy arise more easily. Alhough in he above secions, we follow ACW o assume ha he impored facor is mainly a subsiue for capial, we canno eliminae he possibiliy ha he impored facor is a subsiue for labor. We hus come up wih he following proposiion: Proposiion 4. If we assume ha he impored facor is mainly a labor subsiue insead of a capial subsiue, which means ha we x a a given level (say, = 0:3), and le vary in he inerval (0; 1 a n ), indeerminacy may no easily arise under he labor subsiue assumpion. 10 Proof. A formal proof can be saed as follows. We consider an economy wih capial share (a) and labor share (1 a). When we inroduce he foreign inpu wih share b as a labor subsiue ino he model, he indeerminacy region becomes a b < ss < 1 a b b. When we inroduce he 10 Under he capial subsiue assumpion, we x a n a a given level, and le vary in he inerval (0; 1 a n). 17

18 foreign inpu wih share b as a capial subsiue ino he model, he indeerminacy region becomes a b b < ss < 1 a b. The lower bound of he region under he labor subsiue assumpion is clearly larger han ha obained under he capial subsiue assumpion. From his proposiion, we nd ha alhough energy axes share a similar mechanism for indeerminacy wih facor income axes, hey have di eren implicaions in generaing indeerminacy. Tha is, he "equivalence" relaionship beween hem only holds hrough scal increasing reurns by endogenizing raes and making governmen revenue exogenous. ACW (2008, p. 721) nd ha if he impored facor is a subsiue for labor, a larger oil share ( ) implies a smaller hreshold value of he producion exernaliy alhough he reducion in he laer is less dramaic. In his model, we nd ha for he same oil share ( ), under he labor subsiue assumpion, he hreshold value of he (seady sae) ax rae needed o generae indeerminacy (i.e., he lower bound of he indeerminacy region) can be larger han ha obained under he capial subsiue assumpion. We provide he economic inuiion behind his resul by considering he equilibrium condiion in he labor marke. Suppose ha he expecaions of a fuure ax increase shif he labor supply schedule up (since he rm will impor more oil oday o produce more oupu). The slope of he labor demand schedule is equal o 1, so he absolue value of he slope under he capial subsiue a b assumpion (j 1 b j = a b 1 b ) is smaller han ha obained under he labor subsiue assumpion a (j 1 b j = a 1 b ). This implies ha he decline in employmen under he capial subsiue assumpion should be larger han ha obained under he labor subsiue assumpion. As a resul, he increase in energy ax rae required o balance he budge under he capial subsiue assumpion should be larger han ha obained under he labor subsiue assumpion. Hence, muliple equilibria become more likely o occur under he capial subsiue assumpion han under he labor subsiue assumpion (see Fig. 3a, where LS represens he aggregae labor supply, LD represens he aggregae labor demand under he capial subsiue assumpion, LD* represens he aggregae labor demand 18

19 under he labor subsiue assumpion, and WH represens he equilibrium wage-hour locus). Inser Figure 3. here. Figure capion: Indeerminacy and he labor marke. Finally, we compare our model wih he ACW model. ACW (2008) show ha heavy reliance on impored energy can induce aggregae insabiliy in he presence of increasing reurns o scale: he larger he impored energy share in GDP, he easier i is for he economy o be indeerminae and unsable. We have a similar proposiion as follows: Proposiion 5. We x a n a a given level (say, a n = 0:7), which implies ha he impored inpu is a capial subsiue (i.e., + = (1 a n ) is xed). The larger he impored energy share in GDP, he easier i is for he economy o be indeerminae. The lower bound of he indeerminacy region < ss < an empirical ax raes he larger is. decreases as increases, so indeerminacy occurs more easily in he range of When he energy share ( ) increases, he lower bound of he seady sae ax rae ha generaes indeerminacy decreases (given ha a n is xed). We provide he economic inuiion by considering he equilibrium condiion in he labor marke. If he agen expecs ha here is a fuure ax increase, his will shif he labor supply schedule up. The slope of he labor demand schedule is 1 (= 1 1+ an ). Therefore, he smaller is, he larger he decline in employmen is (since he slope of he labor demand schedule decreases in absolue value as decreases). As a consequence, in order o balance he budge, he required increase in ax rae is larger he smaller is; hence, i is easier for muliple equilibria o occur he larger is (see Fig. 3b, where LS represens he aggregae labor supply, LD represens he aggregae labor demand wih a large energy share, LD represens he aggregae labor demand wih a small energy share, and WH represens he equilibrium wage-hour locus). 19

20 4. Conclusion We explore he "channel equivalence" beween facor income axes and energy axes o generae indeerminacy. 11 The channel is hrough scal increasing reurns by endogenizing raes and making he governmen spending (or lump-sum ransfers) exogenous. We show ha, in he presence of scal increasing reurns caused by endogenous energy axes, indeerminacy easily occurs in open economies ha impor foreign energy. The required seady sae energy ax raes can be empirically realisic. An implicaion of his paper is ha economies largely dependen on non-reproducible naural resources may be vulnerable o sunspo ucuaions if he governmen nances public spending wih endogenous energy axes. One fuure research direcion is o deermine under wha circumsances, energy axes and capial income axes are equivalen in generaing indeerminacy since he essenial elemen for indeerminacy in he SGU model is an endogenous labor income ax rae. 5. Appendix: We summarize he equilibrium condiions of he model wih a balanced-budge rule, endogenous energy ax raes, and consan governmen purchases shown in his paper. We consider he discree ime case of an energy ax. The balanced-budge rule is as follows G = y (1 + ). The following equilibrium condiions hold for all, 11 In he working paper, Zhang (2009) brie y discusses he robusness of our indeerminacy resul in he economy wih endogenously deermined energy ax raes. As in SGU, income-elasic governmen spending and more general preferences are allowed. In addiion, we hink ha allowing for public deb and predeermined ax raes will no change our main resuls. 20

21 U c (c ; n ) =, U n (c ; n ) = w, Y = c + k +1 (1 )k, and 1 = +1 (1 + r +1 ), where is he Lagrangian muliplier of he budge consain of he agen. In his model, disposable income, Y ; is Y = (1 )y = y p 0 o G, where G denoes a xed cos ha guaranees ha he domesic rms do no earn pure pro s in he long run (given ha he foreign rms ake away heir paymens). The afer-ax wage rae w, and he afer-ax renal rae r are r b = ( p o ) a0 1 1 a 1 k 0 an 1 a n 0 = r, and w b = a n ( p o ) a0 1 a 1 k 0 an 1 1 n = w, 21

22 where r b and w b denoe he before-ax renal rae and he before-ax wage rae respecively. In his model, denoes he wedge beween marginal produc and afer-ax facor prices. I is easy o verify ha he markup is counercyclical wih respec o y since = (1 + ) 1 = (1 1 G Y ) 1 = ( G Y ). References [1] Aguiar-Conraria, L., Wen, Y., Foreign rade and equilibrium indeerminacy. Federal Reserve bank of S. Louis, a. [2] Aguiar-Conraria, L., Wen, Y., Undersanding he large negaive impac of oil shocks. Journal of Money, Credi, and Banking 39, [3] Aguiar-Conraria, L., Wen, Y., A noe on oil dependence and economic insabiliy. Macroeconomic Dynamics 12, [4] Benhabib, J., Farmer, R.E., Indeerminacy and increasing reurns. Journal of Economic Theory 63, [5] De Miguel, C., Manzano, B., Opimal oil axaion in a small open economy. Review of Economic Dynamics 9, [6] Finn, M., Perfec compeiion and he e ecs of energy price increases on economic aciviy. Journal of Money, Credi, and Banking 32, [7] Giannisarou, C., Balanced-budge rules and aggregae insabiliy: he role of consumpion axes. The Economic Journal 117,

23 [8] Guo, J.T., Harrison, S.G., Balanced-budge rules and macroeconomic (in)sabiliy. Journal of Economic Theory 119, [9] Hamilon, J., Oil and he macroeconomy since World War II. Journal of Poliical Economy, 91:2, [10] Hamilon, J., Hisorical causes of poswar oil shocks and recessions. Energy Journal, 6, [11] Hansen, G., Indivisible labor and he business cycles. Journal of Moneary Economics 16, [12] Inernaional Energy Agency, Energy prices and axes, 4h Quarer. [13] Kim I.M., Loungani, P., The role of energy in real business cycle models. Journal of Moneary Economics 29, [14] Newbery, D., Why ax energy? owards a more raional policy. The Energy Journal 26 (3), [15] OECD, The OECD inpu-oupu daabase. OECD Publicaion and Informaion Cener. [16] Roemberg, J., Woodford, M., Energy axes and aggregae economic aciviy. In J. Poerba (ed.), Tax Policy and he Economy, vol. 8, Cambridge, MA: MIT Press. pp [17] Roemberg, J., Woodford, M., Imperfec compeiion and he e ecs of energy price increases on economic aciviy. Journal of Money, Credi, and Banking 28, [18] Schmi-Grohe, S., Uribe, M., Balanced-budge rules, disorionary axes, and aggregae insabiliy. Journal of Poliical Economy 105,

24 [19] Wei, C., Energy, he sock marke, and he puy-clay invesmen model. American Economic Review 93, [20] Zhang, Y., Tari and equilibrium indeerminacy. Mimeo, New York Universiy. hp://mpra.ub.uni-muenchen.de/13099/. 24