a quantitative assessment* El coste en términos de bienestar de los hábitos externos: un análisis cuantitativo

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1 The Estudios welfare de Eonomía. ost of external Vol. 37 habits - Nº 1, / Junio Manuel 21. A. Gómez Págs The welfare ost of external habits: a quantitative assessment* El oste en términos de bienestar de los hábitos externos: un análisis uantitativo Manuel A. Gómez** Abstrat This paper quantifies the welfare ost of onsumption externalities in an endogenous growth model with habit formation. Agent s utility depends on both urrent onsumption and a referene onsumption level determined by eonomy-wide average past onsumption. Although utility may be lower in the presene of onsumption externalities, the welfare loss relative to the effiient growth path is found to be very small. This result is extremely robust to variations in the parameter values. However, there are relatively important differenes in the timing of the welfare loss and in generational welfare. Key words: Welfare; Habit formation; Endogenous growth. Resumen Este trabajo uantifia la pérdida de bienestar en presenia de externalidades asoiadas al onsumo en un modelo de reimiento on formaión de hábitos. La utilidad depende del onsumo atual y de un nivel de onsumo de referenia determinado por el nivel medio de onsumo pasado en la eonomía. Aunque la utilidad puede ser menor en presenia de externalidades asoiadas al onsumo, la pérdida de bienestar on respeto a la soluión efiiente es muy reduida. Este resultado es robusto a variaiones paramétrias. Sin embargo, hay diferenias relativamente importantes en el timing de la pérdida de bienestar y en el bienestar generaional. Palabras lave: Bienestar, Formaión de hábitos, reimiento endógeno. JEL lassifiation: E21, D62, O41. * The author wishes to thank the editor, Rómulo humaero, and an anonymous referee for their helpful omments. Finanial support from the Spanish Ministry of Siene and Innovation and FEDER through Grant EO is gratefully aknowledged. ** Faultad de ienias Eonómias y Empresariales, Universidad de A oruña. ampus de Elviña, 1571 A oruña, Spain. Phone: , Fax: mago@ud.es

2 6 Estudios de Eonomía, Vol Nº 1 1. Introdution Dynami equilibrium models with habit formation have reently attrated a great attention in the literature beause of their ability to aount for some empirial fats that are diffiult to explain under standard time-separable preferenes. 1 In models with habit formation, individuals utility depends not only on their level of urrent onsumption but also on how their urrent onsumption ompares to a referene level the habits stok. Two extreme speifiations have been onsidered aording to how the habits stok is formed. In the model with internal habits (IH), the referene stok is formed from own past onsumption levels (e.g., onstantinides, 199, Fuhrer, 2, and Boldrin et al., 21). In the model with external habits (EH), habits arise from eonomy-wide average past onsumption (e.g., ampbell and ohrane, 1999, Lettau and Uhlig, 2, and Turnovsky and Monteiro, 27). The introdution of habits formed from some external benhmark taken as given by individuals may render the ompetitive equilibrium ineffiient. An interesting issue this raises is to quantitatively assess what is the welfare ost of onsumption spillovers assoiated to the presene of external habits. This paper assesses the welfare ost of onsumption externalities in an endogenous growth model with habit formation. To this end, we ompute the welfare loss that happens in the external-habits eonomy relative to the effiient externality-free internal-habits eonomy after a shok that redues the apital stok. The fous of the paper is on the welfare impliations of the introdution of habits into utility. Hene, we keep the prodution side of the eonomy as simple as possible, and onsider that output is produed with an AK tehnology. 2 This simplifiation allows isolating the effet of habits on the eonomy dynamis beause, as it is well-known (e.g., Rebelo, 1991), the AK model with standard time-separable utility does not exhibit transitional dynamis. Therefore, the dynamis of the eonomy would be driven exlusively by preferenes; i.e., by the presene of habits. In ontrast with the simpliity of the prodution side of the eonomy, we propose a general speifiation of preferenes whih enompasses, as partiular ases, the most ommonly used speifiations in the literature: the subtrative model (e.g., onstantinides, 199, ampbell and ohrane, 1999), in whih individuals derive utility from the differene between urrent onsumption and the habits stok, and the multipliative speifiation (e.g., Abel, 199, and arroll et al., 1997, 2), in whih individuals derive utility from the ratio between urrent onsumption and the habits stok. 1 For example, Abel (199), onstantinides (199) and ampbell and ohrane (1999) try to explain the equity premium puzzle; arroll et al. (2) study the observed relationship between savings and growth, Lettau and Uhlig (2) and Boldrin et al. (21) try to fit some stylized fats of business yles; Fuhrer (2) studies monetary poliy, Diaz et al. (23) analyze the determination of preautionary savings and the shape of the wealth distribution, and Burashi and Jiltsov (27) try to explain the term struture of interest rates. 2 Previous works that have taken this approah inlude, e.g., arroll et al. (1997, 2), Shieh et al. (2), Ferraguto and Pagano (23), Mansoorian and Mihelis (25), Gómez (26, 28) and Tsoukis (27).

3 The welfare ost of external habits / Manuel A. Gómez 7 We find that the welfare loss aused by the presene of onsumption externalities is very small. This result is extremely robust to variations in the parameters governing the habit-formation proess: the speed of adjustment, the strength of habits in utility, the externality parameter that reflets the weight of past average onsumption in the habit formation proess, and the elastiity of intratemporal substitution between onsumption and the habits stok. We also find that there is no welfare loss in the subtrative model, so that the ompetitive equilibrium of the external or mixed habits eonomy is effiient, for all the parameter onfigurations onsidered. Two additional related issues are addressed. First, we analyze the welfare timing. In the multipliative model, in whih the elastiity of substitution between onsumption and habits is unitary, welfare in the external-habits eonomy is always lower than that in the internal-habits eonomy, and the differene may be signifiant. The highest welfare loss is generated on impat and, thereafter, it dereases steadily toward its stationary value. However, different values of the elastiity of intratemporal substitution between onsumption and habits may lead to a ompletely different behavior, in whih the highest welfare gain is generated on impat and, thereafter, the welfare gain dereases, eventually beomes zero, and then the welfare loss onverges to its stationary value. Seond, we analyze the effet of the shok on the welfare of different generations. In the multipliative model, only individuals born in the very first generations are made worse off in the external-habits eonomy than the orresponding ones in the internal-habits eonomy. Latter generations may obtain a relatively important welfare gain. Again, the qualitative behavior of generational welfare is extremely dependent on the elastiity of substitution between onsumption and habits. hanging this parameter, all generations in the external-habits eonomy may be made worse off than the orresponding ones in the internal-habits eonomy, and the latter the generation the larger the welfare loss. The rest of the paper is organized as follows. Setion 2 presents the model. Setion 3 analyzes the ompetitive equilibrium of the market eonomy, and Setion 4, the optimal growth path of the entrally planned eonomy. Setion 5 omputes the welfare ost of external and mixed habits, and performs an extensive sensitivity analysis. Setion 6 presents some onluding remarks. 2. The Model We study a losed eonomy populated by N idential infinitely-lived representative agents that grows at the exogenous rate N N= n Preferenes The instantaneous utility derived by the agent depends both on urrent onsumption,, and a referene onsumption level or habits stok, H. The intertemporal utility derived by the agent is represented by βt (1) U ( ) = ut ( ( ), H()) t e dt β >,

4 8 Estudios de Eonomía, Vol Nº 1 where β is the rate of time preferene. We onsider a general speifiation of the instantaneous utility funtion, ( 1 ε) ϕ ϕ ϕ 1 γ H (2) uh (, ) =,, ε > 1 < γ < 1 ϕ. 1 ε 1 γ The parameter γ reflets the strength of habits, ε is the oeffiient of relative risk aversion when γ =, and determines the elastiity of (intratemporal) substitution between onsumption and habits, 1 ( 1 ϕ ). 3 If ϕ =1, the utility funtion (2) yields the subtrative speifiation 1 ε 1 γ H 1 ( 1 γ) + γ( H) (3) uh (, ) = = 1 ε 1 γ 1 ε 1 γ in whih individuals derive utility from a weighted arithmeti mean of urrent onsumption and the differene between urrent onsumption and the habits stok. As goes to zero, the utility funtion (2) onverges to the multipliative speifiation 1 ( 1 ) ( 1 ) 1 1 (4) uh (, ) = ( H ) = [ ( H) ] 1 ε 1 ε 1 ε, γ ε γ γ γ ( 1 ε) ( 1 γ ), in whih individuals derive utility from a weighted geometri mean of urrent onsumption and the ratio between urrent onsumption and the habits stok. Substituting ε = γ + ( 1 γσ ), (4) may be rewritten as (5) uh (, ) = 1 H ( )( ) ( 1 γ 1 σ ) γ 1 σ. The habits stok is formed as an exponentially delining average of past onsumption, t ρ( s t) 1 (6) Ht () = ρ e s () s () ds ρ>, 1, 3 A similar ES utility funtion has been employed by Dupor and Liu (23), with H representing externalities assoiated to urrent onsumption. They assume that < 1. However, this assumption may lead to the lak of onavity of the instantaneous utility ϕ ϕ funtion (2) with respet to onsumption, whih requires that ε γ( 1 ϕ) H for ϕ ϕ all feasible (, H); i.e., suh that γ H >. Given that ε > 1, the assumption ensures strit onavity of u with respet to. Furthermore, u is onave in (, H) if and only if 1.

5 The welfare ost of external habits / Manuel A. Gómez 9 where denotes the eonomy-wide average onsumption. Differentiating (6) with respet to time, the rate of adjustment of the habits stok is ρ 1 (7) H = ( H ). The parameter ρ >, whih governs the speed with whih the habits stok adjusts to urrent onsumption, determines the relative weight of onsumption at different dates: the larger is ρ, the more important is onsumption in 1 the reent past. As ρ then H so that agent s utility depends only on urrent onsumption. The parameter ϕ reflets the weight of own past onsumption in habits. Setting ϕ = orresponds to the EH model, in whih habits are formed from the eonomy-wide average past onsumption. Setting ϕ = 1 orresponds to the externality-free IH model, in whih habits arise from own past onsumption Prodution Eah individual inelastially supplies one unit of labor in eah period. Individual s gross output, Y, is determined by the AK tehnology (8) Y = BK, B>, where K is the individual s apital stok The agent s problem The single good of the eonomy an be either onsumed or invested. The agent s budget onstraint is, then, (9) K = BK ( n+ δ ) K = AK, where δ is the rate of depreiation of apital, and we define A= B n δ. The agent hooses, K and H to maximize the lifetime utility (1) subjet to her budget onstraint (9) and the onstraint on the habits stok aumulation (7), taking as given the path of average onsumption,, and the initial onditions on apital, K() >, and habits stok, H() >. 3. The Market Eonomy A ompetitive equilibrium is as a set of paths { t ( ), Kt (), Ht ( )} t= that solves the agent s utility maximization problem, and t () = t () at any time. A balaned growth path (BGP) or steady-state equilibrium is a ompetitive equilibrium along whih, K and H grow at onstant rates. Let J be the urrent value Hamiltonian of the agent s maximization problem:

6 1 Estudios de Eonomía, Vol Nº 1 J = u( H, ) + λ ( AK + µρ( 1 ) H), where λ and µ are the shadow values of apital and habits stok, respetively. Appendix shows that the dynami behavior of the eonomy along an interior solution an be desribed by the following third-order dynamial system in the variables / H, h H / K and q µ / λ: u (1) (,) 1 ρq = A + + u β ερ( 1) (,) ρq, (11) h = h[ ρ( 1 ) A+ h ], (12) q A q q u H (,) 1 = ( + ρ) + ( 1+ ρ ) u (,) 1, where u (,) 1 u (,) 1 = ( γ) [ ε γ( 1 ϕ )], and u (,) 1 u (,) 1 = γ ϕ ϕ H 1 ϕ. Now, we fous on the BGP at whih onsumption, apital and habits stok all grow at the same rate. A hat ^ over a variable will denote its stationary value. We an state the following proposition. Proposition 1. The deentralized eonomy has a unique steady-state equilibrium with positive long-run growth, A β ρ gˆ (13) ˆ = 1+ = + ερ ρ, (14) ˆ A ρ(ˆ 1) h = = ˆ A gˆ, ˆ u (ˆ, ) ( ˆ (15) ˆ H 1 γ g + ρ) q = =, ( A+ ρ) u (ˆ, 1) + ρu (ˆ, 1) ( ) ˆ ϕ A+ ρρ ( gˆ + ρργ ) H where the long-run growth rate of onsumption, apital and output per apita is A β (16) ĝ =, ε if and only if (17) A > β.

7 The welfare ost of external habits / Manuel A. Gómez 11 Proof. Eqs. (13), (14) and (15) are obtained from equating (1), (11) and (12) to zero and solving the resulting system. Eq. (16) is obtained from gˆ = ρ(ˆ 1 ). Eq. (16) shows that ĝ > if and only if A > B, whih yields ĉ >1. Furthermore, ĥ > if and only if A > g ˆ, whih is satisfied beause it is equivalent to β > ( 1 ε) A. The transversality onditions (A.4) and (A.5) are also satisfied beause they an be equivalently expressed as β+ λ ˆ λ ˆ + ˆ ˆ = β + ˆ µ ˆ µ + ˆ K K H Hˆ = A+ gˆ <. Proposition 1 proves that the steady state is unique and, therefore, the possibility of multiple steady states is ruled out. This result is standard in the AK model with habit formation (e.g., arroll et al., 1997, 2, Alonso-arrera et al., 25, 26, Gómez, 28), and also in the neolassial growth model with habits (e.g., Alonso-arrera et al., 25, Álvarez-uadrado et al., 24, and Gómez, 27) when labor supply is inelasti. 4 In ontrast, multiple steady states may emerge in the presene of external habits when labor supply is elasti (see, e.g., hen et al., 29). The system (1)-(12) is aessible to phase-diagram analysis. Note first ϕ ϕ that u is well-defined for all (, H) suh that > γ H ; i.e., for all suh that > =γ 1 ϕ. If ϕ = the = lous is vertial at = ˆ and stable. If < 1, the = lous is 1 A β ερ( 1) + ργ (18) q= l () = ρ β+ ρ+ ερ( 1) ργ 1 ϕ 1 ϕ Sine the denominator is stritly inreasing for all > γ 1 ϕ, the = lous has at most one vertial asymptote in = L ( γ 1 ϕ, ). Furthermore, L < ˆ beause the denominator is positive for all ˆ. The = lous is stritly dereasing sine l () <, and stable beause d dq <. Thus, in (, q) spae, the arrows point west (east) above (below) the = lous. Furthermore, lim l () = 1 ( ρ ) = q. From (12), the q = lous is γ (19) q= lq() =, ϕ 1 ( A+ ρ) ργ and its slope is given by. 4 Although hen (27) seems to show that multiple steady states may arise in an AK model with habit persistene, Tohe (29) shows that really hen s model features a unique steady state.

8 12 (2) dl q ϕ d γ( A+ ρ)( 1 ϕ) () = ϕ [( A+ ρ) ργ] Estudios de Eonomía, Vol Nº Furthermore, we have that (21) dq dq = A+ ρ ργ 1 ϕ If ϕ = 1, the q = lous is onstant at q= qˆ and unstable. If ϕ > 1, the q = lous is stritly dereasing in ( γ 1 ϕ, ), and unstable beause dq dq > A + ρ( 1 ) γ >. 1 It has an asymptote at = M = [ ( A + )] ( ϕ 1 ργ ρ ) ϕ < γ 1, beause the denominator of (19) is stritly inreasing and ( A+ ρ) ργ > [ A + ρ( 1 )] γ > for ϕ 1 all > γ 1 ϕ, with lim l q() =. If ϕ < 1, the q = lous is stritly inreasing 1 ( ϕ 1) and if it has an asymptote at = M = [ ργ ( A+ ρ)] > ˆ, beause the denominator of (19) is stritly dereasing and tends to ργ < as goes to infinity. The q = lous is positive and unstable if < M, and negative and stable if > M, with lim l q() = 1 ( ρ ). If < 1, it an be easily obtained that A β ερ( 1) sgn[ l() lq()] = sgn ϕ 1 ϕ, 1 [( A+ ρ) ργ][ β+ ρ+ ερ( 1) ργ ] whih entails that the unique intersetion between both loi ours at = ˆ. The top left, top right and bottom left panels of Figure 1 depit phase diagrams in (, q) spae if ϕ >1, ϕ < 1 and ϕ = 1, respetively. 5 Given the onfiguration of the two loi, the steady state (ˆ, q ˆ) is saddle-path stable. The bottom right panel of Figure 1 is a phase diagram in (, h) spae. Given that the eonomy is on its saddle path in (, q) spae, onverges monotonially, and so, the = lous is vertial and stable in (, h) spae. The h = lous is A (22) lh ()= + ρ ρ, whih is dereasing, onvex and unstable, with lim lh() = > ˆ > lim lh () = ρ. Given the onfiguration of the two loi, the (unique) steady state (ˆ, h ˆ) is saddlepath stable. Thus we an state the following proposition.. 5 To depit Figure 1, we have onsidered that < 1. If =, the phase diagrams are similar, but now the = lous is vertial.

9 The welfare ost of external habits / Manuel A. Gómez 13 FIGURE 1 PHASE DIAGRAM onvergene in (, q) spae q = > 1 q = < 1 q = qˆ q = qˆ q q L = ĉ q ĉ M q = q = 1 h onvergene in (, h) spae = qˆ q = h ĥ = ĉ ĉ h = Proposition 2. Let A > β. The steady state of the deentralized eonomy desribed by (13), (14) and (15) is a saddle-point. The loal stability analysis onfirms the former result. Linearizing the system (1)-(12) around the steady state (13)-(15), we get (23) d h = hh ˆ( ˆ + ) h ˆ ˆ ρ q d d d ˆ ˆ h hˆ = D h ˆ h, q qˆ q qˆ where d u (ˆ, 1 ) ρ = ερ + u q d (ˆ, 1) 1+ ρ ˆ 11 31,

10 14 Estudios de Eonomía, Vol Nº 1 d d u (ˆ, 1) ρ = u (ˆ, 1) 1+ ρqˆ d uh (ˆ, 1) u(ˆ, 1) u(ˆ, 1) uh(ˆ,) 1 = ( + ρqˆ) [ u (ˆ, 1)] 2 = γ( ε 1)( ϕ 1) = ( 1+ ρqˆ) 2ˆ(ˆ ϕ γ ) 31 1 d 33 1 uh (ˆ, 1) ˆ ϕ γ = = >, qu ˆ (ˆ, 1) qˆ The last inequality follows from ˆq > beause, using A> gˆ and ĉ > 1, we have that ϕ ( A+ ρρ ) ˆ ( gˆ + ρργ ) > (ˆ g+ ρρ ) (ˆ γ) > (ˆ g+ ρρ ) ( 1 γ ) >. Given the struture of the matrix D, the seond diagonal element, ˆ ˆ h, is an unstable root. The other two roots are those of the submatrix D 13 ϕ d11 d13 = d d The determinant of D 13 is det = ερd33u(ˆ, 1) u (ˆ, 1) <. Hene, D has one stable (real) root and two instable (real) roots, whih proves that the steady state is loally saddle-point stable., 4. The entrally Planned Eonomy The entral planner possesses omplete information and hooses all quantities diretly, taking all the relevant information into aount. The agent hooses, K and H to maximize the agent s lifetime utility (1) subjet to the resoures onstraint (9) and the onstraint on the habits stok aumulation (24) H = ρ( H), taking as given K() > and H() >. An optimal growth path is as a set of paths { t ( ), Kt (), Ht ( )} that solves the planner s utility maximization problem. t= It an be readily observed that the planner s problem in the entrally planned eonomy oinides with the agent s problem in the market eonomy when =1. Hene, the optimal growth path of the entralized eonomy oinides with the ompetitive equilibrium of the (effiient) externality-free IH eonomy, and the system that drives the dynamis of the entrally planned eonomy in terms of the variables, h and q is given by (1), (11) and (12) with =1. Propositions 1 and 2 an be readily speialized to the ase of the

11 The welfare ost of external habits / Manuel A. Gómez 15 entrally planned eonomy. A bar over a variable will denote its steady state in the entralized eonomy. Proposition 3. The entrally planned eonomy has a unique and loally saddlepath stable steady-state equilibrium with positive long-run growth, A β ρ g = 1+ = + ερ ρ, h = A ρ( 1) = A g, uh (, 1) γ( g + ρ) q = =, ( A+ ρ) u (, 1) + ρu (,) 1 ϕ ( A + ρ) ρ ( g+ ρ) ργ H where the long-run growth rate of onsumption, apital and output per apita is if and only if A > β. A β g =, ε 5. Welfare Analysis This setion provides a quantitative assessment of the welfare ost of external (or mixed) habits. To this end, we perform an exerise similar to that in arroll et al. (1997), and onsider the effet of a 1 perent unantiipated redution in the apital stok of an eonomy that was initially on its balaned growth path. For omparative purposes, we also study the effet of a larger redution of 25 perent in the apital stok. We ompute the welfare loss in the EH/MH market eonomy relative to the entrally planned eonomy i.e., the effiient externalityfree IH market eonomy after the shok. The welfare loss is measured as the perentage redution in the flow of onsumption in the IH eonomy required to equate the level of welfare to that in the EH/MH eonomy. Let E () t and HE () t denote the time paths of onsumption and habits in the EH/MH eonomy after the shok, and let I () t and HI () t denote the orresponding ones in the IH eonomy. If the shok that redues the apital stok ours at time t =, the welfare ost of external habits is the value κ suh that I I βt u (( 1 κ ) ( t ), H ()) t e dt = u ( (), t H ( t )) e βt dt. It should be noted that after the shok, both the market and the entrally planned eonomies onverge to their ommon pre-shok steady state. Rather than relying on linear approximations, we expliitly solve for the non-linear E E

12 16 Estudios de Eonomía, Vol Nº 1 transitional dynamis by using the bakward integration method (Brunner and Strulik, 22, Atolia and Buffie, 29). The baseline parameter values displayed in Table 1 are those employed by arroll et al. (1997). 6 They onsider the utility funtion (5) with σ = 2 and, therefore, the orresponding value of ε in Table 1 for the equivalent speifiation (4) is ε= γ + ( 1 γσ ) = The baseline parameterization entails that habits are purely external. The implied long-run growth rate is gˆ = g=2.. In the following Figures, the solid (dotted) line orresponds to the ase of a 1% (25%) drop in the apital stok. Table 1 Benhmark alibration Parameter Value Interpretation A.8 Net sale parameter in the goods setor β.5 Rate of time preferene ε 1.5 Relative risk aversion γ.5 Strength of habits in utility ρ.2 Speed of adjustment of habits ϕ Weight of own past onsumption in habits Elastiity of intratemporal substitution parameter The top panels of Figure 2 display the welfare ost of external habits as the speed of adjustment ρ varies from to 4, while the remaining parameter values are maintained at their baseline values reported in Table 1. We display the results for the multipliative ase, =, and the ase that = 2. 7 As ρ inreases, the welfare loss first inreases from zero to its maximum value, and then dereases steadily toward zero as ρ goes to infinity. If ρ =, the habit stok is onstant and, therefore, both the EH and IH models beome equivalent to the onventional AK model. Hene, onsumption externalities do not provoke ineffiieny and the welfare loss is zero. As ρ tends to infinity, arroll et al. (1997) show that the habits stok tends to a weighted geometri average of own and eonomy-wide 1 average urrent onsumption, H=, so that the model ollapses to a model with ontemporaneous onsumption externalities. This kind of externalities does not provoke ineffiieny (e.g., Liu and Turnovsky, 25), and so the welfare loss tends to zero as ρ goes to infinity (not displayed in the Figure). 6 Eqs. (1)-(12) and (13)-(15) show that the parameters B, n and δ affet the dynamis and the steady state of the eonomy only through the value of A = B n δ whih, therefore, is the relevant parameter. Different ombinations of B, n and δ an lead to the value of A =.8 shown in Table 1. For example, arroll et al. (1997) onsider B =.13, n = and δ =.5. 7 In a similar disrete-time multipliative model, =, Alonso-arrera et al. (25) show that the interior solution haraterized by the first-order onditions will indeed be a maximum if ε > 1. If = 2, the Hamiltonian is onave in (, H) and, therefore, the neessary onditions are also suffiient.

13 The welfare ost of external habits / Manuel A. Gómez 17 Figure 2 Welfare loss in the EH eonomy relative to the IH eonomy (In perent).1 = Speed of adjustment, ρ.2 = Strength of habits, γ Elastiity of substitution parameter, 1% redution in apital stok = Speed of adjustment, ρ = Strength of habits, γ Elastiity of substitution parameter, 25% redution in apital stok The medium panels of Figure 2 display the welfare loss after the apital shok as the parameter of strength of habits in utility γ varies from to 1, while the remaining parameters are maintained at their baseline values displayed in Table 1. 8 If γ =, habits do not enter utility. Hene, the equilibrium dynamis 8 arroll et al. (2) analyze the dynamis of the IH eonomy for different values of γ, where utility is speified as (5). In their simulations, for eah value of γ, the value of σ is hosen to keep the steady state whih the eonomy onverges to unhanged; i.e., suh that ε = γ + (1 γ)σ is onstant. This is equivalent to use the multipliative speifiation (4).

14 18 Estudios de Eonomía, Vol Nº 1 of the IH and EH eonomies oinide and, therefore, the welfare ost is zero. As γ inreases, the welfare loss first inreases from zero to its maximum value, and then dereases steadily as γ goes to 1. The bottom panels of Figure 2 display the welfare ost of external habits as the elastiity of substitution parameter inreases, keeping onstant the remaining parameters at their baseline values shown in Table 1. As inreases from to 1, the welfare loss dereases and it even beomes zero in the subtrative model, = 1. This entails that in the subtrative eonomy, the ompetitive equilibrium is effiient. 9 As inreases from 1 the welfare loss inreases until it reahes its maximum level at around = 15 (resp., = 23) and then dereases steadily toward as goes to infinity. The reason is that as goes to infinity, the utility funtion (2) tends to 1 ε uh (, ) = ( 1 ε ). Hene, utility does not depend on the habits stok and, therefore, the welfare loss is zero. Although the overall effet on welfare of the apital shok is quite similar in the EH and IH eonomies, there ould be signifiant differenes in its timing. Let V (T, ) denote the utility attained at time T following a apital shok at time t =, for a given value of the parameter, βt (25) VT (, ϕ) = ut ( (), H( t)) e dt. T Now, we shall fous on the two extreme ases of internal and external habits. Figure 3 illustrates the time path of the welfare loss in the EH eonomy relative to the IH eonomy when = and = 2, while the remaining parameter values are maintained at their baseline values shown in Table 1. When =, after a 1% (resp., 25%) destrution of the stok of apital, welfare in the EH eonomy is always lower than that in the IH eonomy. The highest welfare loss of.91% (resp., 2.38%) is generated on impat. Thereafter, it dereases steadily toward its stationary value of.1% (resp.,.1%) as time goes to infinity. In ontrast, when = 2, after a 1% (resp., 25%) destrution of the stok of apital, the EH eonomy has a welfare gain relative to the IH eonomy of.35% (resp.,.76%) on impat. Thereafter, the welfare gain dereases steadily, beomes negative (i.e., a loss) and then the welfare loss onverges toward its stationary value of.6% (resp.,.4%) as time goes to infinity. Another interesting question is whether generational welfare is affeted by the speifiation of habits. Following Grüner and Heer (2), the integral βt (26) WT (, ϕ) = ut ( (), H( t)) e dt T an be interpreted as the utility of an individual born at time T for a given value of. 9 In the subtrative model, = 1, the welfare loss is found to be zero for every parameter onfiguration. The ompetitive equilibrium of the EH/MH eonomy oinides with the optimal growth path of the entrally planned eonomy, and so, it is effiient. This is the reason why the subtrative ase is not depited in the Figures.

15 The welfare ost of external habits / Manuel A. Gómez Figure 3 Time path of welfare loss in the EH eonomy relative to the IH eonomy (In perent) = time, T time, T 1% redution in apital stok 25% redution in apital stok = 2 Figure 4 Generational welfare loss in the EH eonomy relative to the IH eonomy (In perent) = = time, T time, T 1% redution in apital stok 25% redution in apital stok Figure 4 depits the generational welfare loss (or gain) in the EH eonomy relative to the IH eonomy when = and = 2, while the remaining parameter values are maintained at their baseline values shown in Table 1. When =, after a 1% (resp., 25%) redution in the apital stok, only individuals born in the initial.24 (resp.,.67) periods in the IH eonomy are made better off than the orresponding ones in the EH eonomy. From then on, generational welfare is higher in the EH eonomy than that in the IH eonomy, and the welfare gain is relatively important. For example, for individuals born in period T = 25, the welfare gain in the EH eonomy relative to the IH eonomy amounts to.27% (resp.,.76%). This results ontrast markedly with those obtained when = 2. In this ase, after a 1% (resp., 25%) redution in the apital stok, generational welfare in the EH eonomy is lower than that in the IH eonomy whatever the

16 2 Estudios de Eonomía, Vol Nº 1 Figure 5 onsumption in the EH eonomy relative to the IH eonomy (Ratio) = time, T time, T 1% redution in apital stok 25% redution in apital stok = 2 generation onsidered. Furthermore, the generational welfare loss inreases steadily as the time T inreases, and it ould be relatively important. For example, for individuals born in period T = 25, the welfare loss in the EH eonomy relative to the IH eonomy amounts to.25% (resp.,.73%). The former results reveal that, although the overall welfare ost of external habits is quantitatively unimportant, there are differenes in welfare timing and generational welfare that an be rather signifiant from a quantitative viewpoint. This qualifies somewhat the onlusion that onsumption externalities in habit formation models are irrelevant for welfare. Furthermore, the elastiity of substitution between onsumption and habits plays an important role in the qualitative behavior of welfare timing and generational welfare. The dynami path of welfare and the behavior of generational welfare an be more easily understood by looking at Figure 5, whih depits the ratio of onsumption in the EH eonomy relative to that in the IH eonomy. The time path of onsumption is similar in both eonomies. In the multipliative model, =, after a 1 perent (resp., 25 perent) redution in the apital stok, onsumption in the EH eonomy is lower than that in the IH eonomy during the first 9.7 periods (resp., 9.8 periods) and, from that time on, is higher than that. Sine the greatest differene in onsumption levels happens at the initial time, the highest welfare loss in the EH eonomy relative to the IH eonomy is generated on impat. As the eonomy evolves, onsumption in the EH eonomy athes up and, eventually, exeeds onsumption in the IH eonomy. As a onsequene, the welfare loss in the EH eonomy relative to the IH eonomy dereases steadily toward its stationary value. This also explains that generational welfare in the EH eonomy is higher than that in the IH eonomy for nearly all generations, exept for the very first ones. These results are in sharp ontrast with the ones obtained in the ase that = 2. Now, onsumption in the EH eonomy is higher than that in the IH eonomy during the first 12.8 periods (resp, 13.4 periods) and, from that time on, is lower than that. Sine the greatest differene in onsumption levels ours at the initial time, the highest welfare gain in the EH

17 The welfare ost of external habits / Manuel A. Gómez 21 eonomy relative to the IH eonomy is generated on impat. As the eonomy evolves, onsumption in the IH eonomy athes up and, eventually exeeds onsumption in the EH eonomy. As a onsequene, the welfare gain in the EH eonomy relative to the IH eonomy dereases steadily and, eventually, beomes a welfare loss that onverges toward its stationary value. Furthermore, generational welfare in the EH eonomy is lower than that in the IH eonomy for all generations. 1 The different dynami behavior of onsumption in the EH eonomy relative to the IH eonomy between the ases ϕ = 2 and ϕ = shown in Figure 5 an be explained by the relationship between the intertemporal marginal rates of substitution (MRS) of onsumption in the IH and EH eonomies. The MRS between onsumption at times t 1 and t 2, with t > t, is U t2 MRS t 1, = ( ) ( ) t = 2 U ( ) t ( ) βt ρt ( β+ ρ) t t2 H βt1 + ρ ρ t 1 ( β + ρ ) t t1 H 2 2 u ( ( t2), H( t2)) e + ρe e u ( t ( ), Ht ()) dt =, u ( ( t ), H( t )) e e e u ( t ( ), Ht ()) dt 1 1 where U ( ) t () denotes the Volterra derivative of U() defined in (1) with respet to onsumption at time t. Using (A.6) and (A.7) we get u ( ( t2), H( t2)) [ 1+ ρqt ( 1)] (27) MRS t 1, t = e 2 u ( ( t ), H( t )) [ 1+ ρqt ( )] β( t t ) onsider first that ϕ >1. After a redution in apital stok, the eonomy starts with a level of h above its steady-state value. Figure 1 shows that as the eonomy evolves h dereases and inreases monotonially toward their respetive steady-state values. As arroll et al. (1997) argue, the agent maintains a low level of / H relative to its steady-state value beause this allows dereasing h H / K by fostering apital aumulation and by slowing the growth of the habits stok. As h approahes its stationary value, onsumption needs to be depressed less, so grows toward its steady state. Figure 1 also shows that the relative ost of habits, q, dereases steadily toward its stationary value. Hene, (27) entails that the MRS in the EH eonomy would be lower than that in the IH EH IH eonomy, MRS < MRS, when evaluated along the equilibrium path of t1, t2 t1, t Following arroll et al. (2), we have also omputed the effet of an unantiipated but permanent inrease in the produtivity parameter B (or, equivalently, in A) suh that the steady-state growth rate of the eonomy inreases from 1 perent to 2 perent. The results obtained are qualitatively similar, but quantitatively even of less magnitude, to those obtained in the present paper for a apital drop. Results are available upon request.

18 22 Estudios de Eonomía, Vol Nº 1 the IH eonomy. A lower MRS means that the agent is less willing to substitute present by future onsumption. Thus the transition in the EH eonomy is haraterized by initial over-onsumption relative to the IH eonomy, followed by subsequent under-onsumption, as shown in Figure 5 when ϕ = 2. The opposite ours when ϕ <1 (see the ase ϕ = in Figure 5). Now, q is inreasing and, EH IH therefore, MRSt t MRS 1, > 2 t1, t. Hene, the desire to substitute intertemporally 2 in the EH eonomy would be higher than that in the IH eonomy. 6. onluding Remarks This paper quantifies the welfare ost of onsumption externalities in an endogenous growth model with habit formation. Although utility is lower in the model with external or mixed habits than in the model with internal habits beause of the presene of a negative externality, the welfare loss is insignifiant. This result is extremely robust to parameter variations. Hene, the speifiation of habits as external or internal ould seem to be irrelevant from a welfare perspetive. However, this onlusion should be somewhat qualified beause of the relatively important differenes arising in the welfare timing and in generational welfare. Referenes Abel, A. (199). Asset pries under habit formation and athing up with the Joneses, Amerian Eonomi Review, Vol. 8; Alonso-arrera, J.; aballé, J. and Raurih, X. (25). Growth, habit formation, and athing-up with the Joneses, European Eonomi Review, Vol. 49; Alonso-arrera, J.; aballé, J. and Raurih, X. (26). Welfare impliations of the interation between habits and onsumption externalities, International Eonomi Review, Vol. 47; Álvarez-uadrado, F.; Monteiro, G. and Turnovsky, S.J. (24). Habit formation, athing-up with the Joneses, and eonomi growth, Journal of Eonomi Growth, Vol. 9; Atolia, M. and Buffie, E. (29). Reverse shooting made easy: Automating the searh for the global nonlinear saddle path, omputational Eonomis, Vol. 34; Boldrin, M.; hristiano, L.J. and Fisher, J.D.M. (21). Habit persistene, asset returns and the business yle, Amerian Eonomi Review, Vol. 91; Brunner, M. and Strulik, H. (22). Solution of perfet foresight models: a simple method and appliations, Journal of Eonomi Dynamis and ontrol, Vol. 26;

19 The welfare ost of external habits / Manuel A. Gómez 23 Burashi, A. and Jiltsov, A. (27). Habit formation and maroeonomi models of the term struture of interest rates, Journal of Finane, Vol. 62; ampbell, J. and ohrane, J. (1999). By fore of habit: A onsumption-based explanation of aggregate stok market behaviour, Journal of Politial Eonomy, Vol. 17; arroll,.d.; Overland, J. and Weil, D.N. (1997). omparison utility in a growth model, Journal of Eonomi Growth, Vol. 2; arroll,.d.; Overland, J. and Weil, D.N. (2). Saving, growth and habit formation, Amerian Eonomi Review, Vol. 9; hen, B.-L. (27). Multiple BGPs in a growth model with habit persistene, Journal of Money, redit and Banking, Vol. 39; hen, B.-L.; Hsu, Y.-H. and Mino, K. (29). Are spillovers of onsumption standards really a soure of equilibrium indeterminay?, Working Paper. onstantinides, G.M. (199). Habit formation: A resolution of the equity premium puzzle, Journal of Politial Eonomy, Vol. 98; Diaz, A.; Pijoan-Mas, J. and Rios-Rull, J.V. (23). Preautionary savings and wealth distribution under habit formation preferenes, Journal of Monetary Eonomis, Vol. 5; Dupor, B. and Liu, W.-F. (23). Jealousy and equilibrium overonsumption, Amerian Eonomi Review, Vol. 93; Ferraguto, G. and Pagano, P. (23). Endogenous growth with intertemporally dependent preferenes, ontributions to Maroeonomis, Vol. 3 (1), artile 1. Fuhrer, J.. (2). Habit formation in onsumption and its impliations for monetary poliy models, Amerian Eonomi Review, Vol. 9; Gómez, M.A. (26). Optimal onsumption taxation in a model of endogenous growth with external habit formation, Eonomis Letters, Vol. 93; Gómez, M.A. (27). Equilibrium effiieny in the Ramsey model with habit formation, Studies in Nonlinear Dynamis & Eonometris, Vol. 11 (2), artile 2. Gómez, M.A. (28). onvergene speed in the AK endogenous growth model with habit formation, Eonomis Letters, Vol. 1; Grüner, H. S. and Heer, B. (2). Optimal flat-rate taxes on apital a reexamination of Luas supply side model, Oxford Eonomi Papers, Vol. 52; Lettau, M. and Uhlig, H. (2). an habit formation be reoniled with business yles fats?, Review of Eonomi Dynamis, Vol. 3; Liu, W.F. and Turnovsky, S.J. (25). onsumption externalities, prodution externalities, and long-run maroeonomi effiieny, Journal of Publi Eonomis, Vol. 89; Mansoorian, A. and Mihelis, L. (25). Money, habits and growth, Journal of Eonomi Dynamis and ontrol, Vol. 29;

20 24 Estudios de Eonomía, Vol Nº 1 Rebelo, S. (1991). Long-run poliy and long-run growth, Journal of Politial Eonomy, Vol. 99; Shieh, J.; Lai,. and hang, W. (2). Additive behavior and endogenous growth, Journal of Eonomis, Vol. 72; Tohe, P. (29). Multiple BGPs in a growth model with habit persistene: A omment, Journal of Money, redit and Banking, Vol. 41; Tsoukis,. (27). Keeping up with the Joneses, growth, and distribution, Sottish Journal of Politial Eonomy, Vol. 54; Turnovsky, S.J. and Monteiro, G. (27). onsumption externalities, prodution externalities, and effiient apital aumulation under time-dependent preferenes, European Eonomi Review, Vol. 51;

21 The welfare ost of external habits / Manuel A. Gómez 25 Appendix Derivation of the ompetitive equilibrium Let J be the urrent value Hamiltonian of the agent s maximization problem: J = u( H, ) + λ ( AK ) + µρ( H) 1. The first-order onditions for an interior optimum are 1 1 (A.1) u (, H ) + ρµ = λ, (A.2) A = β λ λ, (A.3) uh (, H ) µ ρ= β µ µ, and the transversality onditions (A.4) (A.5) lim e t lim e t t β λk =, t β µ H =. Eq. (A.1) equates the marginal utility of onsumption, adjusted by its effet on the future habits stok, to the shadow prie of apital. Eq. (A.2) equates the rate of return on apital to the rate of return on onsumption. From (A.3) and (A.5), we an get ( β+ ρ)( s t) (A.6) µ () t = e uh ( ( s), H()) s ds. t This ondition states that the shadow value of the habits stok is determined as the present disounted value of the stream of extra utils that would be gained (or lost) by a marginal unit of habits, whih depreiates at the rate ρ. Heneforth, we take into aount that = in a symmetri equilibrium. Eq. (A.1) an be rewritten as (A.7) u (, H ) + ρµ = λ. Defining q µ / λ, using (A.7) we an get (A.8) λ = u (, H ) ( 1 + ρq ), (A.9) µ = qu ( H, ) ( 1 + ρq ).

22 26 Estudios de Eonomía, Vol Nº 1 Differentiating (A.7) with respet to time, we get (A.1) u (, H) + u (, H) H = λ ( 1+ ρq) + λρq = H ρ q = β A + ρq u 1+ (, H ), where the last equality follows from using (A.2) to eliminate λ, and then, substituting λ for (A.8). Rearranging terms, Eq. (A.1) an be rewritten as (A.11) u (, H) ρq uh (, H) = β A + u (, H) 1 + ρq u ( H, ) H. The system (1), (11) and (12) that drives the dynamis of the market eonomy in terms of / H, h H / K and q an be obtained as follows. Sine = H H, from (A.11) and using the homogeneity of degree ε of u, we get u (,) 1 ρq uh (,) 1 H H = β A + u (,) ρq u (,) 1 H, H whih using that u (,) 1 + u (,) 1 = ε u (,) 1, an be rewritten as H u (,) 1 ρq H = A + + u β ε (,) ρq H. Using (7), we finally get Eq. (1). Sine hh = H H K K, Eq. (11) results from (7) and (9). Sine q q= µ µ λ λ, Eq. (12) results from (A.3) and (A.2), taking into aount (A.8) and (A.9).