Internal habits in an endogenous growth model with elastic labor supply

Economic Modelling 51 (2015) 583–595

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Internal habits in an endogenous growth model with elastic labor supply☆ Manuel A. Gómez a,⁎, Goncalo Monteiro b,1 a b

Departamento de Economía Aplicada II, Universidade da Coruña, Campus de Elviña, 15071 A Coruña, Spain Department of Economics, University at Buffalo, Buffalo, NY 14260, USA

a r t i c l e

i n f o

Article history: Accepted 3 September 2015 Available online 29 September 2015 Keywords: Endogenous growth Habits Leisure

a b s t r a c t This paper studies the implications of introducing internal versus external habits in an endogenous growth model with elastic labor supply. We first show that the comparative-static effects of a shock in habits parameters are qualitatively different depending on how habits are specified. An increase in the weight of habits in utility raises long-run growth and labor supply in the external-habits model, whereas it reduces labor supply and has an ambiguous effect on long-run growth in the internal-habits model. Increasing the speed of adjustment of habits to current consumption has no effect on long-run values with external habits, but has a negative effect on long-run growth and labor supply with internal habits. Numerical simulations reveal that the qualitative differences are also quantitatively important. Finally, we illustrate the dynamic effects of an increase in productivity. On impact and in the long-run this shock has a positive effect on growth, labor supply and the savings rate in both models. However, along the transition labor supply exhibits a procyclical response to the productivity increase in the internal-habits model, but a countercyclical response in the external-habits model. These results could be helpful on the still open debate on whether habits are internally or externally formed. © 2015 Elsevier B.V. All rights reserved.

1. Introduction One way to generate smoothness in consumption, and in its rate of change, is by assuming the habit formation hypothesis in consumption, so that individual's utility depends on current consumption as well as on how it compares to a reference level — the habits stock. Therefore, this hypothesis has been widely used in various fields to explain, among other issues, the equity premium puzzle (Abel, 1990), the behavior of the savings rate (Carroll et al., 2000), the effects of monetary policy (Fuhrer, 2000) or, more recently, the life-cycle household allocations (Aydilek, 2013), the dynamics effects of oil price shocks (Schubert, 2014), and some stylized facts of business cycles (Khorunzhina, 2015). The literature distinguishes between internal habits (IH) formed from individual's own past consumption, and external habits (EH) formed from average economy-wide past consumption. Although recent empirical evidence provides support to the habit formation hypothesis (e.g., Chen and Ludvigson, 2009; Korniotis, 2010), whether habits are internally or externally formed appears to be still an open question. This, together with the important role of habits in the recent macroeconomics literature, makes it interesting to analyze the different

☆ Detailed comments of two anonymous referees and the Editor of the journal are gratefully acknowledged. Manuel Gómez acknowledges the financial support from the Spanish Ministry of Science and Innovation through Grant ECO2011-25490. ⁎ Corresponding author at: Facultad de Economía y Empresa, Universidade da Coruña, 15071 A Coruña, Spain. Tel.: +34 981167000; fax: +34 981167070. 1 Tel.: +1 716 645 8678; fax: +1 716 645 2127.

http://dx.doi.org/10.1016/j.econmod.2015.09.007 0264-9993/© 2015 Elsevier B.V. All rights reserved.

implications that introducing habits internal or externally formed could have on the dynamics and long-run performance of the economy. This paper studies the implications of introducing internal habits in an endogenous growth model with elastic labor supply, and compares them with the implications of introducing external habits, which have been studied in Gómez (2015). As Gómez (2015), who in turn follows Carroll et al. (1997), we keep the production side of the economy as simple as possible and consider an AK-type technology à la Romer (1986). As it is well-known, standard AK-type models with timeseparable utility do not exhibit transition dynamics. Hence, this simplification allows to isolate the effect of habits on the dynamics of the economy, so that the dynamics of the economy is driven exclusively by preferences; i.e., by the presence of habits. First, we prove that the economy has a unique feasible steady-state equilibrium, which is locally saddle-path stable. Although not surprising, this result is interesting because the complexity of the involved dynamic systems prevented an analytical study of the stability properties of the steady state in related settings (e.g., Alonso-Carrera et al., 2004; Alvarez-Cuadrado et al., 2004; Turnovsky and Monteiro, 2007).2 2 Gómez (2012) proves the existence, uniqueness and saddle-path stability of steadystate equilibrium in the neoclassical growth model with habit formation and elastic labor supply. However, he assumes that utility is additively separable in adjusted consumption and leisure; an assumption that reduces significantly the complexity of the dynamic system. In particular, this allows computing the steady-state value of leisure time explicitly as a function of the parameters of the model, and not implicitly as it happens in the present model (see Section 3). Furthermore, he does not compare the implications of assuming internal or external habits on the transition dynamics or the long-run equilibrium.

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Furthermore, habits can generate multiplicity and indeterminacy of equilibrium or endogenous fluctuations when labor supply is elastic (e.g., Chen et al., 2013; Gori and Sodini, 2014), so local determinacy should not be taken for granted. Second, we characterize the comparative-static effects of a shock in preference parameters in the IH model, and compare them with the ensuing ones in the EH model analyzed in Gómez (2015). An increase in the weight of habits in utility has a negative effect on long-run work time in the IH model. Intuitively, as the agent takes into account the negative indirect effect that current consumption has on future utility through its effect on a higher habits stock, an increase in the weight of habits in utility makes consumption less desirable relative to leisure which, therefore, increases. This is in stark contrast with the EH model, where an increase in the weight of habits in utility causes an increase in the long-run work time (Gómez, 2015). Furthermore an increase in the weight of habits in utility has a direct positive effect on growth and savings through its positive impact on the effective EIS, and an indirect effect through its impact on leisure. In the IH model, both effects have opposite signs, and so, the overall effect on long-run savings rate and growth is ambiguous. In contrast, Gómez (2015) shows that both effects are positive in the EH model, so long-run savings and growth rates increase. Third, we find that an increase in the relative importance of current consumption in the formation of the habit stock (or adjustment speed of habits) has a negative effect on long-run growth, savings rate and work time in the IH model, whereas Gómez (2015) showed that it has no effect in the EH model. Intuitively, in the IH case, the agents takes into account the effect that his decisions about consumption have on the habit stock and indirectly on future utility. Therefore, an increase in how fast habits adjust to current consumption, increases the weight that current consumption has on the formation of the habit stock, which in turn increases the negative indirect effect it will have on utility. This leads agents to substitute consumption for leisure reducing the time spent working and consequently the savings and the long-run growth rate. In contrast, in the EH case, the agent does not take into account how his current decisions about consumption affect the habit stock, and thus changes in how fast the habit sock adjust to current consumption has no effect on the long-run values. We also show that long-run growth, work time and the savings rate are higher with external habits than with internal habits. Finally, we present some numerical simulations that confirm and supplement the theoretical findings in several dimensions. The numerical simulations show that the different effects are also quantitatively important both in the balanced growth path and along the transition. An increase in productivity leads to an immediate increase in consumption and hours worked relative to their pre-shock levels. The agent in the EH economy does not take into account the negative impact of larger consumption on the reference stock. In contrast, the agent in the IH economy is aware of that, and uses leisure to compensate for the negative impact that current consumption increase will have on future welfare. Thus, on impact, the increase in consumption and hours worked is lower in the IH case than it is in the EH case. Over time the agent in the IH economy goes on increasing hours worked to maintain the higher growth rate of consumption, which started the initial jump up. In the EH economy, because of their complementarity, the higher consumption is accompanied with an increase in leisure time along the transition. Several authors have studied the equilibrium dynamics in growth models with internal and external habit-forming preferences. However, most of them have considered that labor supply is inelastic (e.g., Carroll et al., 1997; Alonso-Carrera et al., 2006; Gómez, 2006, 2010, in the AK model, Alonso-Carrera et al., 2004, 2005; Alvarez-Cuadrado et al., 2004; Gómez, 2007, in the neoclassical growth model). Turnovsky and Monteiro (2007) analyze the equilibrium dynamics in a non-scale growth model with elastic labor supply, and compare the EH and IH versions of the model. However, its semi-endogenous growth nature entails that preferences – in particular, the presence of habits – have no role on

the determination of long-run growth. Recently, Gómez (2015) provides a comprehensive analysis of the AK model with elastic labor supply. However, with the aim of obtaining analytical results, he focuses on the model with external habits. Thus, he does not consider the case when habits are internally formed and, therefore, he does not compare the different implications that specifying habits as internal or external could have. This paper tries to fill this gap. The rest of the paper is organized as follows. Section 2 presents the model. Section 3 analyzes the equilibrium dynamics. Section 4 studies the comparative-static effects of introducing habits into utility. Sections 5 and 6 present some numerical results on the effects of changes on the habits parameters on the steady state and the transitional dynamics, respectively. Section 7 concludes. 2. The model Consider an economy populated by a continuum of mass one of identical individuals, who own shares of a mass one of identical firms. Each individual is endowed with a unit of time, part of which, (1 − L), can be supplied as labor input and the remainder, L, consumed as leisure. 2.1. Preferences At any instant of time each individual derives utility from her current consumption, C, leisure, L, and also from the current level of the reference consumption level (habits stock), H. Thus the agent's utility is represented by an iso-elastic function of the type employed by Abel (1990), Carroll et al. (1997), and others: U¼

1 1−ε

Z

∞ 0

CLη =Hγ

1−ε

e−βt dt

β N 0; ε N 1; 0 b γ b 1; η N 0;

ð1Þ

where γ reflects the importance of habits in utility, β is the rate of time preference, η reflects the importance of leisure in utility, and 1/ε is the elasticity of intertemporal substitution (EIS) in the timeseparable case (γ = 0).3 Following Fisher and Hof (2000), we shall term σ = 1/[γ + ε(1 − γ)] as the “effective” elasticity of intertemporal substitution (effective EIS). Following Alvarez-Cuadrado et al. (2004), we consider that the rate of adjustment of the habits stock, is given by   ^ 1−ξ −H : H¼ ρ C ξ C 

ð2Þ

where Ĉ = ∫10C(i)di denotes the economy-wide average consumption, and ρ reflects the relative importance of recent consumption in determining the stock of habits. As Carroll et al. (1997) point out, the case of externalities associated with current consumption is obtained as a limiting case when ρ → + ∞, so that H → C ξĈ1− ξ. This expression (2) encompasses the two extreme specifications typically considered in the literature, which are identified by the different values of ξ. Setting ξ = 0 corresponds to the external-habits (EH) case in which habits are formed from economy-wide average past consumption alone, whereas ξ = 1 corresponds to the internal-habits (IH) case in which habits arise from own past consumption. It is the latter case we are mostly interested in, but we also want to compare the results of the internal-habits model with the ones derived by Gómez (2015) for the external-habits model. Hence, we consider a formulation of habits that encompasses as particular cases the IH and EH models. 3 King et al. (1988) have shown that in a Ramsey model without habits this specific form is consistent with balanced growth. The assumption that ε N 1 is borrowed from Alonso-Carrera et al. (2005) and Hiraguchi (2008), who show that otherwise the agent's optimization problem might not be well-defined in a similar IH model with inelastic labor supply.

M.A. Gómez, G. Monteiro / Economic Modelling 51 (2015) 583–595

The optimality conditions for an interior optimum are

2.2. Production To ease comparability, we follow Gómez (2015) and assume that each firm produces output, Y, according to Y ¼ AK

1−ψ

585

ϕ ^ψ

ð1−LÞ K

∂J=∂C ¼ 0

^ ⇒ C −ε Lηð1−εÞ H−γð1−εÞ þ μξρC ξ−1 C

1−ξ

¼ λ;

∂J=∂L ¼ 0 ⇒ ηC 1−ε Lηð1−εÞ−1 H −γð1−εÞ ¼ λw; A N 0; 0 b ϕ ≤ ψ b 1;

ð3Þ

^ ¼ ∫ 1 KðiÞdi is the avwhere K is the individual's firm capital stock and K 0 erage stock of capital in the economy, which is taken as given by the firm.4 There are constant returns to scale in the reproducible input – capital – at the social level, which ensures the existence of endogenous growth. If ϕ ≤ ψ the production function exhibits non-increasing returns to scale at the private level, whereas if ϕ = ψ the production function will exhibit constant returns to scale to private inputs. The firm rents capital at the rate of interest r and hires labor at the wage rate w. Taking as given the interest rate, the wage rate, and the av^ the firm chooses capital and labor used erage capital in the economy, K, to maximize its profits: max π ¼ Y−rK−wð1−LÞ:

K;1−L

Profits, if any, are distributed back to individuals as dividends.

⇒

λ¼ ðβ−r Þλ;

ð7Þ

μ ¼ βμ−∂J=∂H

⇒

μ ¼ ðβ þ ρÞμ þ γC 1−ε Lηð1−εÞ H −γð1−εÞ−1 ;

ð8Þ





together with the transversality conditions lim e−βt λðt ÞK ðt Þ ¼ lim e−βt μ ðt ÞHðt Þ ¼ 0:

t→∞

t→∞

ð10Þ ψ

^ ; w ¼ ∂Y=∂ð1−LÞ ¼ ϕAK 1−ψ ð1−LÞϕ−1 K

ð4Þ

and the initial values of the capital stock, K(0) = K0, and the habits stock, H(0) = H0. In doing so, the agent takes as given the path of aver^ and average consumption, Ĉ. age capital, K, 3. Equilibrium A symmetric competitive equilibrium for this economy is a set of paths for the quantities {C(t), L(t), K(t), H(t)}∞ t=0 and prices {r(t), w(t)}∞ t = 0 such that (i) the agent maximizes her intertemporal ^ welfare, (ii) firms maximize profits, (iii) Ĉ(t) = C(t) and KðtÞ ¼ KðtÞ at each point in time, and (iv) all markets clear. A balanced growth path (BGP) (or steady-state equilibrium) is a symmetric competitive equilibrium along which C, K and H grow at constant rates, and L is constant.

ð11Þ

and the implied profits are ψ

^ ¼ ðψ−ϕÞY: π ¼ Y−rK−wL ¼ ðψ−ϕÞAK 1−ψ ð1−LÞϕ K

Let J be the current value Hamiltonian of the agent's optimization problem, and let λ denote the agent's shadow value of capital, and μ denote the shadow value of the agent's reference stock:  J¼

1−ε   CLη =H γ ^ 1−ξ −H : þ λ½rK þ wð1−LÞ þ π−C  þ μρ C ξ C 1−ε

ð12Þ

^ ¼ K in a symmetric Hereafter, we take into account that Ĉ = C and K equilibrium. Furthermore, we will use (10), (11) and (12) to substitute for r, w and π. From Eq. (4) we get the resources' constraint K ¼ Að1−LÞϕ K−C: 

ð13Þ

Defining c ≡ C/H and h ≡ H/K, together with q ≡ −μ/λ, we can express the dynamics of the economy in terms of these variables, which are constant along a balanced growth path, as (see Appendix A): 

c 1−ϕL fð1−ψÞch−β−½ðε−1Þð1−γ Þ þ ψρðc−1Þg ¼ ΩðLÞ c ðε−1Þηð1−LÞ þ ð1−ϕLÞð1−ψÞ h ðε−1Þηð1−LÞ þ ð1−ϕLÞ − − ΩðLÞ ΩðLÞ h   ρξ q ; ð14Þ × 1 þ ρξq 





h ¼ −Að1−LÞϕ þ ρðc−1Þ þ ch; h h i q¼ ð1−ψÞAð1−LÞϕ þ ρ q−ð1 þ ρξqÞγc; 

3.1. Transitional dynamics

ð9Þ

The interpretation of the optimality conditions is standard. Eq. (6) equates the utility of an additional unit of consumption, adjusted by its impact on the future consumption reference stock (or habits), to the shadow value of capital. Eq. (6) equates the marginal utility of leisure to its opportunity cost, which is the real wage rate at the shadow value of capital. Eq. (7) equates the marginal return to capital to the rate of return on consumption. Profit maximization entails that ψ

The single good of the economy can be either consumed or invested. We abstract from the depreciation of capital because it simplifies the derivations and has no qualitative implications on the results. Therefore, the agent in this economy chooses consumption, labor supply, reference consumption stock and the rate of capital accumulation to maximize the lifetime utility function (1) subject to the evolution of the reference stock (2), the capital accumulation equation 



ð6Þ

λ¼ βλ−∂J=∂K

^ ; r ¼ ∂Y=∂K ¼ ð1−ψÞAK −ψ ð1−LÞϕ K

2.3. The agent's problem

K ¼ rK þ wð1−LÞ þ π−C;



ð5Þ

ð15Þ ð16Þ

where L is a function of c, h and q defined implicitly by Lð1−LÞϕ−1 ¼

ð1 þ ρξqÞη ch: ϕA

ð17Þ

3.2. Balanced growth path Now, we focus on an interior steady state that can be obtained by 

4 Turnovsky and Monteiro (2007) consider instead the technology Y ¼ AK 1−ψ ð1−LÞψ ϕ K , and assume that ϕ + ψ b 1, which rules out the case of endogenous growth.



solving the system c =c ¼h =h ¼q¼ 0, while taking into account (17). A bar ‘¯’ over a variable will denote its steady-state value. We can state the following Proposition. 

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M.A. Gómez, G. Monteiro / Economic Modelling 51 (2015) 583–595

Proposition 1. The economy has a unique feasible steady-state equilibrium,

equilibrium. The long-run effects of an increase in γ on leisure times in the IH model is given by

 ϕ ð1−ψÞA 1−L −β ; c¼1þ ½γ þ εð1−γÞρ

ð18Þ

 ∂G=∂γξ¼1 dL ¼− N0; dγ ∂G=∂L

n o  ϕ ρ ½ðε−1Þð1−γ Þ þ ψA 1−L þ β h¼ ;  ϕ ð1−ψÞA 1−L −β þ ½γ þ εð1−γÞρ

ð19Þ

q¼

 ϕ ð1−ψÞA 1−L −β þ ½γ þ εð1−γÞρ γ  ; ρ γξβ þ ½γð1−ξÞ þ εð1−γ Þð1−ψÞA1−Lϕ þ ð1−ξγÞ½γ þ εð1−γÞρ

ð20Þ and the savings rate is 2

3

ch 1 β s ¼ 1−C=Y ¼ 1−  ϕ ¼ γ þ εð1−γ Þ 41−ψ−  ϕ 5; A 1−L A 1−L

ð21Þ

ð25Þ

implying that an increase in γ leads to a decrease in the time spent working. Intuitively, when habits are formed in an internal way, the agent takes into account the negative indirect effect that current consumption has on future utility through its effect on a higher habits stock. Hence, as the weight of habits in utility γ increases, consumption becomes less desirable relative to leisure which, therefore, increases. The effects of an increase in γ on long-run savings rate and growth are given by  ϕ−1 A 1−L ð1−ψÞϕ dL dg ∂g ∂g dL ðε−1Þg ¼ ¼ − þ dγ ∂γ ∂L dγ γ þ εð1−γÞ dγ γ þ εð1−γÞ

ð26Þ

ds ∂s ∂s dL ðε−1Þs βϕ dL ¼ ¼ − : þ dγ ∂γ ∂L dγ γ þ εð1−γ Þ ½γ þ εð1−γ ÞA1−Lϕþ1 dγ

ð27Þ

where L is defined implicitly by the following equation    ϕ−1 G L; γ; ρ; ξ ≡ ϕAL 1−L −ηð1 þ ρξqÞch ¼ 0;

ð22Þ

with positive long-run growth rate of output, capital stock, habits stock and consumption per capita g¼

 ϕ ð1−ψÞA 1−L −β ; γ þ εð1−γ Þ

ð23Þ

if and only if  ð1−ψÞA N β 1 þ

ηðβ þ ρÞ ϕ½β þ ð1−ξγÞρ

ϕ

:

ð24Þ

Proof. See Appendix A. One immediate consequence of Proposition 1 is that the steady-state values of c = C/H, h = H/K, the savings rate s, and the growth rate g, do not depend directly on the parameter ξ; i.e., on habits being external or internal. Thus, the choice of habits as internal or external (or intermediate) only affects these stationary values indirectly through its effect on the steady-state leisure time (see Eq. (22)). This result is not surprising as the steady-state values of these variables coincide in the AK model with internal and external habits when labor supply is inelastic (e.g., Carroll et al., 1997). The next proposition establishes the saddle-path stability of the steady state. Proposition 2. The steady state of the economy ðc; h; qÞ described by Eqs. (18), (19) and (20) – where L is defined implicitly by Eq. (22) – is locally saddle-path stable.

The former expressions, which are valid both in the EH and IH models, show that the different effects of an increase in γ in the IH and EH models would be due to the different impact on leisure time in both models. In the IH model, the effect of an increase in γ on long-run growth and the savings rate is ambiguous. Looking at expression (26) we see that increasing the weight of habits has a direct effect on growth through its positive impact on the effective EIS, σ = 1/[γ + ε(1 − γ)], and an indirect effect through its impact on leisure. On the one hand, given that ε N 1 an increase in γ represents an increase in the long-horizon EIS in consumption, and thus must lead to higher growth rates. In other words, a higher value of γ means consumers are more willing to substitute intertemporally because the gain or loss in utility associated with a given increase or decrease in consumption will be diminished by the adjustment of the reference stock.5 On the other hand, an increase in γ provokes an increase in long-run leisure. Hence, both effects have opposite signs, so the overall effect can be positive or negative. The effect on savings is also ambiguous.6 In sharp contrast to the case when habits are internal, Gómez (2015) shows that an increase in γ provokes an unambiguous increase in longrun growth, work time, and savings rate in the EH model.7 As in the IH case, increasing the weight of habits has a direct positive effect on growth through the impact on the effective EIS, σ = 1/[γ + ε(1 − γ)], and an indirect effect through its impact on leisure. As the agent does not take into account the negative indirect effect that current consumption has on future utility through its effect on a higher habits stock, both effects are positive in the EH model (Gómez, 2015). In order to shed more light on the channels through which γ affects the steady-state equilibrium, we disentangle its direct effect from its indirect effect through a higher effective EIS.8 Thus, we examine the effect of a change in γ when the relative risk aversion ε is adjusted to keep constant the effective EIS, γ + ε(1 − γ). Appendix A shows that a compensated

Proof. See Appendix A. 5

Gali (1994) makes a similar point in the context of asset pricing. For example, if A = 0.5, β = 0.04, ε = 1.25, γ = 0.5, η = 2, ϕ = 0.2, ρ = 0.01 and ψ = 0.6 we have dg=dγ ¼ −0:01483 b 0 and ds=dγ ¼ −0:01826 b 0 , so that an increase in the weight of habits decreases long-run growth and savings. If we set ε = 2, we have dg=dγ ¼ 0:00431 N 0 and ds=dγ ¼ 0:05502 N 0, so that an increase in the weight of habits increases long-run growth and savings. 7 This can also be observed from Eqs. (25), (26) and (27). In particular, using that ∂G/∂γ| ξ = 0 N 0, from Eq. (22) we can get that dL=dγ ¼ −ð∂G=∂γjξ¼0 Þ=ð∂G=∂LÞ N0 in the EH model. 8 Note that if habits enter utility in an additive manner, U = [1/(1 − ε)]∫∞ 0 [(C − γH)L η ]1 − εe − βtdt, a change in γ would not affect the “effective” EIS – which would be equal to the standard EIS, 1/ε (e.g., Gómez, 2010) –, so this channel would be absent. Hence, the effect of a compensated change could also be interesting if some kind of comparison with the additive model is intended. 6

4. Long-run effects of different habit specifications The goal of this section is twofold. First, we analyze the effects of considering internal habits, on the long-run equilibrium values. Second, we contrast these results with the external-habits case obtained by Gómez (2015). We will use that differentiation of Eq. (22) implies that (see Eqs. (A.17)–(A.20) in Appendix A) that ∂G=∂L N0, ∂G/∂γ|ξ = 1 b 0, ∂G/∂ρ|ξ = 1 b 0 and ∂G/∂ρ|ξ = 0 = 0. First, we study the effect of an increase in the importance of habits utility, which is determined by the parameter γ, on the long-run

M.A. Gómez, G. Monteiro / Economic Modelling 51 (2015) 583–595

increase in γ provokes a reduction in labor supply and the long-run growth rate in the IH model. The reason is that the positive impact on growth (and savings rate) through the impact on the effective EIS is now absent, and so, the negative effect of a lower labor supply implies a lower growth rate. Appendix A also shows that a compensated increase in γ in the EH model, which has not been analyzed in Gómez (2015), has no effect on the long-run equilibrium. Therefore, the unique way in which the introduction of habits affects the steady state in the EH model is through its effect on the effective elasticity of intertemporal substitution. Once this effect is compensated, habits do not affect the long-run values. Finally, given that ρ parameterizes the relative importance of recent consumption in determining the reference stock (or how fast habits adjust to current consumption), it is interesting to look at how changes in ρ affect the long-run equilibrium. Looking at the effect of an increase in the speed of adjustment of habits to current consumption in the IH model, we find that it affects negatively the long-run growth rate, work time and the savings rate (see Appendix A). Intuitively, the agent in the IH economy takes fully into account the negative indirect effect that current consumption has on future utility through its effect on a higher habits stock. As ρ increases, the influence that current consumption has on the determination of the future reference stock (or habits) increases, and so does its negative indirect effect. Thus, leisure becomes more desirable relative to consumption, and so, leisure time increases and, consequently, the long-run growth rate and savings rate decrease. In contrast, Gómez (2015) shows that the speed of adjustment of habits, ρ, does not affect the long-run equilibrium in the EH model. The reason is that in the EH model the agent does not take into account the externality associated with habits. Hence, whether consumption externalities are associated with contemporaneous consumption (ρ → ∞) or past consumption (ρ b ∞) is irrelevant for long-run equilibrium. For comparison with the inelastic-labor-supply (ILS) model ILS

(e.g., Carroll et al., 1997; Gómez, 2006), so thatdL =dγ ¼ 0, let us assume that leisure is initially at the level corresponding to the steady-state value ILS

IH

in the model with internal habits, L ¼ L . It is immediate that an increase in the weight of habits causes long-run growth and the savings rate to unambiguously increase if labor supply is inelastic. In contrast, growth and savings can decrease if labor supply is elastic and, even if the effect is positive, we have that dg ILS =dγ Ndg IH =dγ and dsILS =dγ NdsIH = dγ , so the effect is weaker with elastic labor supply. Given that ILS

dL =dγjσ¼constant ¼ 0, an increase in the weight of habits compensated so that the effective EIS is kept constant has no long-run effect with inelasILS

tic labor supply. Given that dL =dρ ¼ 0, the speed of adjustment has no long-run effect with inelastic labor supply either. The following Proposition summarizes the comparative-static results derived so far for the IH model. Proposition 3. In the IH model: i) An increase in the weight of habits in utility, γ, has a negative effect on long-run work time, whereas the effect on long-run growth and the savings rate is ambiguous. ii) An increase in the weight of habits in utility, γ, while keeping constant the effective EIS has a negative effect on long-run growth, the savings rate and work time. iii) An increase in the speed of adjustment of habits, ρ, has a negative effect on long-run growth, the savings rate and work time. Finally, we compare the stationary equilibria in the IH and EH EH

587

Table 1 Benchmark parameters. Production parameters Preference parameters

A = 0.389, ϕ = 0.6, ψ = 0.6 β = 0.05, γ = 0.5, ρ = 0.2, η = 0.82 (IH), η = 1.35 (EH)

Plugging this value into Eq. (22) with ξ = 1 for the IH economy, we get EH





EH EH EH EH ϕ−1 G L ; γ; ρ; 1 ¼ ϕAL 1−L −η 1 þ ρqEH cEH h ¼ −ηρqEH cEH h b 0:

IH

EH

Given that ∂G=∂Ljξ¼1 N0 , it must be that L NL , as shown in Turnovsky and Monteiro (2007) for the non-scale growth model. Therefore, we can state the following proposition. Proposition 4. Long-run growth, work time and the savings rate in the external-habits model are higher than their counterparts in the internalIH EH habits model; i.e., g EH Ng IH , L NL and sEH NsIH . Hence, for given parameter values, long-run growth, work time and the savings rate are all higher in the model with external habits than their counterparts in the model with internal habits. 5. Numerical analysis: long-run equilibrium This section presents some numerical results to get an insight on what are the effects of changes in the parameters of interest, γ and ρ, on the long-run equilibrium in the models with external- and internalhabit formation. Table 1 summarizes the benchmark parameters values. We follow Carroll et al. (1997) for the common parameters: the rate of time preference, β = 0.05, the instantaneous EIS, 1/ε = 0.5, the weight of habits in utility, γ = 0.5, and the speed of adjustment of the habits stock, ρ = 0.2. Following Carroll et al. (1997) we also set a target long-run growth rate of 2% in the calibration. Unlike Carroll et al. (1997), we consider that labor supply is elastic, so there are two additional parameters to calibrate: the elasticity of labor in output and the elasticity of leisure in utility. The elasticity of labor in output, ϕ = 0.6, is standard (e.g., Cooley and Prescott, 1995), and we consider that there are constant returns to scale in private inputs, so that 1 − ψ = 0.4. The productivity parameter, A, and the elasticity of leisure in utility, η, are set so that the long-run growth rate is 2% (e.g., Carroll et al., 1997) and the steady-state fraction of time devoted to leisure is 0.67, so that work time is about one third of the total, which is a standard choice in the literature (see, e.g., Cooley and Prescott, 1995 — which rely on data from Ghez and Becker, 1975; Juster and Stafford, 1991; Turnovsky and Monteiro, 2007). For comparison purposes, it is convenient that both models (EH and IH) have the same long-run equilibrium values in the baseline (see, e.g., Engen et al., 1997). We get that by setting η = 0.82 in the IH model and η = Table 2 Effects of parameter variations on the steady-state equilibrium. (A) Internal habits with slow adjustment (ρ = 0.2, η = 0.82)

(B) Habits with inelastic labor supply (independent of ρ and η)

γ

g%

L

s

g%

L

s

0 0.50 0.75

2.3 2 1.29

0.5533 0.67 0.7594

0.0958 0.10 0.0783

1.5 2 2.4

0.67 0.67 0.67

0.075 0.10 0.12

(C) External habits (independent of ρ, η = 1.35)

(D) Internal contemporaneous habits (ρ → ∝, η = 0.68)

models. The steady-state leisure in the EH economy, denoted by L , can be obtained by setting ξ = 0 in Eq. (22):

γ

g%

L

s

g%

L

s



EH

EH EH EH ϕ−1 G L ; γ; ρ; 0 ¼ ϕAL 1−L −ηcEH h ¼ 0:

0 0.50 0.75

1.45 2 2.46

0.6764 0.67 0.6645

0.0736 0.10 0.1220

2.61 2 0.56

0.5031 0.67 0.8126

0.1022 0.10 0.0392

588

M.A. Gómez, G. Monteiro / Economic Modelling 51 (2015) 583–595

Table 3 20% increase in productivity A. A. Levels Impact

External habits Internal habits Contemporaneous habits

After 50 years

%ΔK

%ΔC

%ΔY

%ΔS

%ΔL

%ΔK

%ΔC

%ΔY

%ΔS

%ΔL

0 0 0

16.42 14.98 17.61

22.43 21.31 21.62

76.57 78.30 57.64

−1.68 −0.90 −1.11

81.68 84.15 77.97

113.7 117.0 109.3

121.0 124.5 116.4

186.4 191.8 180.6

−1.11 −1.32 −1.11

B. Growth rates and ratios Impact

External habits Internal habits Contemporaneous habits

After 50 years

%ptΔgK

%ptΔgY

%ptΔgC

%ptΔs

%ptΔgK

%ptΔgY

%ptΔgC

%ptΔs

1.53 1.57 1.15

1.40 1.64 1.15

1.73 1.96 1.15

4.42 4.70 2.96

1.15 1.17 1.15

1.15 1.17 1.15

1.15 1.17 1.15

2.96 3.00 2.96

C. Welfare evaluation

External habits Internal habits Contemporaneous habits

Impact (%Δ)

After 50 Years (%Δ)

Intertemporal (%Δ)

13.78 14.13 7.63

47.71 49.55 43.59

22.24 23.35 17.97

1.35 in the EH model, while the productivity parameter is A = 0.389 in both models.9 Furthermore, we consider the IH model with contemporaneous habits, ρ → ∞, when we get that the calibrated parameter values are η = 0.68 and A = 0.389. The four panels of Table 2 summarize the long-run values of the growth rate, the fraction of time devoted to leisure and the savings rate as the parameter γ changes. Our simulations allow γ to take on three values namely: i) γ = 0 corresponding to the standard timeseparable utility case, ii) γ = 0.5 which is our benchmark case, and iii) γ = 0.75. In particular, we note the following: 1. In the model with inelastic labor supply (Panel B), growth and the savings rate increase as the weight of habits increase, reflecting the effect of a higher effective EIS. These results are independent of the speed of adjustment ρ. 2. In panels A, C and D: The baseline case is indicated in bold face. (i) In the IH model with slow adjustment (Panel A), ρ = 0.2, the long-run share of time devoted to leisure increases noticeably by about 20 percentage points, and the long-run growth falls from 2.3% to 1.29%, as γ increases from zero to 0.75. Intuitively, the increase in γ reduces instantaneous welfare, which the agent internalizes and tries to counteract by increasing leisure and, thus, the growth rate of the economy decreases. Panel D shows that similar qualitative conclusions can be drawn in the case of contemporaneous habits. However, the negative effects on growth and work time are substantially weaker when the speed of adjustment is lower. Interestingly, in the model with internal habits the savings rate can evolve in a non-monotonic fashion as γ increases, as shown in Panel A.10 (ii) In the EH model (panel C), the long-run growth rate is increasing in γ in a noticeable way from 1.45% to 2.46%. Leisure time decreases as the weight of habits increases, but because agents do not internalize the effect of the consumption reference on welfare the variation is small, which indicates that the stronger growth effect is the one associated with the increase in the

effective EIS. Comparing panels B and C, we can observe that the long-run effects in the model with external habits and with inelastic labor supply are quite similar, reflecting the small effect on leisure time. In summary, the numerical results confirm the theoretical predictions and illustrate that the choice between internal or external habits may have significant quantitative (and not only qualitative) implications. 6. Numerical analysis: transitional dynamics This section presents some simulation results to get an insight on the transitional dynamics of the IH and EH economies. To this end we study the effect of an increase in productivity as, e.g., in Turnovsky and Monteiro (2007). The welfare effect of the shock is measured as the constant permanent percentage variation in the flow of consumption that leaves the household indifferent between remaining in the pre-shock balanced growth equilibrium or experiencing the shock. Let the subscript “B” denote the time path of a variable before the shock and let the subscript “A” denote the time path of a variable after the shock happens at time t = 0. Hence the welfare gain of the shock is measured as the value of κ such that 1 1−ε

Z

∞

1−ε −βt η γ ð1 þ κ ÞC B ðt ÞLB ðt Þ=HB ðt Þ e dt Z ∞  1−ε −βt 1 η γ C A ðt ÞLA ðt Þ=H A ðt Þ e dt: ¼ 1−ε 0 0

The welfare accumulated up to time T is WA(T) = [1/(1 − ε)]∫T0 η γ [CA(t)LA(t)/HA (t)]1 − εe− βtdt, and WB(T) = [1/(1 − ε)]∫T0[CB(t)LηB(t)/ γ 1 − ε − βt HB (t)] e dt, respectively. Therefore, the (overall) intertemporal welfare gain of the shock is given by κ ¼ ½W A ð∞Þ=W B ð∞Þ1=ð1−εÞ −1:

9

This approach is particularly convenient for when we analyze a 10% increase in productivity. 10 Numerical simulations (not reported in the paper) show that this may also happen with the long-run growth rate for other parameterizations.

Analogously, the intertemporal welfare gain of the shock up to time T can be measured by κ ðT Þ ¼ ½W A ðT Þ=W B ðT Þ1=ð1−εÞ −1:

M.A. Gómez, G. Monteiro / Economic Modelling 51 (2015) 583–595

Following Alvarez-Cuadrado et al. (2004), the relative welfare gain at any time t along the transitional path (over the base level at the corresponding time) can be computed as ξðt Þ ¼ ½Z A ðt Þ=Z B ðt Þ1=ð1−εÞ −1; where ZA(t) = [CA(t)LηA(t)/HAγ(t)]1− ε and ZB(t) = [CB(t)LηB(t)/HBγ(t)]1− ε. 6.1. Productivity shock Table 3 summarizes the effect of a 20 percent permanent and unanticipated increase in productivity. Fig. 1 displays the transitional dynamics of several variables of interest after the shock. Looking at the contemporaneous habits case, we see that the longrun value of capital, consumption, output, and savings increase by 77.97, 109.3, 180.6% respectively, whereas leisure decreases by 1.11. Also because of the way we are measuring the increase in productivity (increase in A), output increases immediately by 21.62%. This increase, associated with the fact that agents are forward looking, allows them to increase consumption level and savings on impact by 17.61 and 57.64% respectively.

589

From Table 3 and Fig. 1, we can see that a 20 percent increase in productivity increases the steady state growth rates of per capita variables to 3.15% in the EH economy and the economy with contemporaneous habits, and 3.17% in the IH economy, while their pre-shock long-run growth rate was 2%. After 50 years the higher growth rates accumulate to substantial effects on capital, consumption, output, savings and leisure. The immediate response to a 20% increase in productivity is an increase of about 16 and 15% in consumption relative to the pre-shock level for both the EH and the IH case. In addition the savings rate goes up by 76 and 78% respectively, allowing the growth rate of consumption to go up. After that consumption adjusts monotonically towards a new balanced growth path as shown in Fig. 1b, and the growth rate of consumption declines to a higher growth rate than what the economy would have experienced if the shock had not taken place, as shown in Fig. 1a. This behavior adjustment is similar to what happens to the savings rate, as shown in Fig. 1c. An interesting difference between IH and EH case is the adjustment of leisure (or equivalently, of labor supply) over time. Leisure falls on impact by 1.68% to 0.658 in the EH economy and by 0.9% to 0.664 in the IH economy relative to their pre-shock level of 0.67. After that, leisure time adjusts to its new long-run value from below in the EH

(b) Consumption relative to pre-shock

(a) Consumption growth

2.4

0.040

2.2

0.038

2.0 0.036

1.8

0.034

1.6

0.032

1.4 1.2

0.030 0

10

20

30 Time

40

50

0

60

10

(c) Savings rate

20

30 Time

40

50

60

50

60

(d) Leisure time

0.150

0.666

0.145

0.664

0.140 0.662 0.135 0.660

0.130 0.125

0

10

20

30

40

50

60

0.658

Time

30 Time

(e) Intertemporal welfare gain,

(f) Instantaneous relative welfare gain,

0.25

0

10

20

40

0.6

0.20

0.4

0.15 0.2 0.10 0.05

0 0

10

20

30

40

50

60

0

Time

20

30

40

50

Time

IH slow adjustment ( = 0.2) IH-EH contemporaneous habits (

10

EH slow adjustment ( = 0.2) )

Fig. 1. Dynamics of several variables after a 20 percent increase in productivity A.

60

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M.A. Gómez, G. Monteiro / Economic Modelling 51 (2015) 583–595

(a)

0.664

(b)

= 0.5, = 0.5

= 0.5, = 0.25

0.666

0.662

0.664

0.660 0.662

0.658

0.660

0.656 0.654

0

10

20

(c)

30 Time

40

50

60

0.658

0

10

20

30

40

50

60

50

60

Time

(d)

= 0.8, = 0.5

0.664

0.664

0.662

0.660

0.660

= 0.8, = 0.75

0.656

0.658 0.652

0.656 0.654

0.648 0

10

20

IH

30 Time

40

50

60

EH

0

10

20

30 Time

40

IH-EH contemporaneous habits (

)

Fig. 2. Dynamics of leisure time after a productivity shock: sensitivity analysis.

case and from above in the IH case. This can be explained by the fact that the agent in the EH economy does not take into account the negative effect that current consumption increase has on future utility, through its effect on a greater stock of habits. In contrast, the agent in the IH economy is aware of that, and uses leisure to compensate for the negative impact that current consumption increase will have on future welfare. Thus, on impact, the increase in consumption and hours worked is lower in the IH case than it is in the EH model. Over time the agent in the IH economy goes on increasing hours worked to maintain the higher growth rate of consumption, which started the initial jump up. This process ends when leisure has dropped by 1.32% to 0.66. In contrast the EH case does not take into account the negative impact of larger consumption on the reference stock. Therefore, the increase in consumption is associated with a decrease in leisure. This initial behavior leads to a transition that is characterized by initial overconsumption relative to the IH economy followed by subsequent under-consumption, during later phases of the transition. Leisure increases during the transition towards a new steady state of 0.662 as shown in Fig. 1d. This happens because the initial over-consumption is associated with initial under-saving relative to the IH case, thus reducing the growth rate of consumption over-time. These different responses are interesting in the light of the discussion on real business-cycle models with habits (see, for instance, Khorunzhina, 2015). In the RBC literature it has been pointed that the introduction of habits in consumption generally produces the countercyclical response of hours worked to a positive productivity shock, and this is at odds with the observed procyclicality of hours worked. We show that a positive response from labor supply is possible – both on impact, in the long-run and along the transition – in the presence of internal habits, without having to resort to the alternatives presented in Khorunzhina (2015), such as adding habits in leisure or capital adjustment costs. In contrast, labor supply increases on impact and in the long-run after a positive productivity shock, but exhibits a countercyclical behavior along the transition in the model with external habits. Thus, this could be seen as an advantage of the internal versus the external specification of habits.

Looking at welfare we can see it increases by 13.78 and 14.13% in EH and IH case and by 7.63% in the contemporaneous habit case. After 50 years welfare has increased by almost 50% in the IH case and by 47.71% in the EH case, relative to what would have been if there was no increase in productivity. The different intertemporal gain reflects the differences along the transitional time paths. Once more the gain of 23.35% observed in the IH case is slightly bigger than the gain of 22.24% in the EH case, and 17.97% in the economy with contemporaneous habits. One interesting implication from the analysis, is that although agents in the IH case enjoy smaller short-run absolute consumption gain than agents in the EH case, they nevertheless enjoy larger shortrun utility gains. This is because they compensate for this with a lower decrease in leisure on impact. In the long-run IH agents revert this combination by letting leisure drop by more than in the EH case, but have a larger gain in consumption. 6.2. Sensitivity analysis In the previous subsection we have shown that, after a positive productivity shock, labor supply evolves in a qualitatively different fashion in the IH and EH models. It increases on impact and in the long-run in both cases, but it increases along the transition in the IH model and (counter-factually) declines in the EH model. In this subsection, we study whether this different behavior is due to the choice of the parameter values; in particular, the speed of adjustment ρ and the weight of habits in utility γ, for which evidence is sparse. Following AlvarezCuadrado et al. (2004), who in turn rely on Fuhrer (2000), we consider faster speeds of adjustment, ρ = 0.5 and ρ = 0.8, and a higher weight of habits in utility, γ = 0.75. We also consider a lower weight of habits, γ = 0.25. Fig. 2 illustrates the time path of leisure time after a 20% increase in productivity for representative parameter values.11 11 The model has been simulated considering values of ρ between 0.005 and 5, and γ between 0.1 and 0.9. For the sake of space, only the most representative simulations are depicted in Fig. 2, but detailed results are available from the authors upon request.

M.A. Gómez, G. Monteiro / Economic Modelling 51 (2015) 583–595

Our results show that the response of labor supply to the productivity shock in the EH model is qualitatively similar irrespective of the parameterization chosen: it declines on impact and then (counterfactually) increases towards its stationary value, which is below its before-shock value. However, the qualitative behavior of labor supply after the shock in the IH model depends on the parameter values: in any case, it increases both on impact and in the longrun after the shock, but it can exhibit a pro-cyclical (increasing) or counter-cyclical (decreasing) behavior. Our simulation results show that the faster the speed of adjustment and/or the lower the weight of habits in utility the more likely labor supply exhibits a counter-cyclical behavior along the transition. Intuitively, the agent in the IH economy is aware of the negative effect that an increase in current consumption has on future utility through a higher habits stock, and uses leisure to compensate it. A faster adjustment speed entails that current consumption has a greater influence on the determination of the reference stock, and so does its negative indirect effect. Thus, leisure becomes more desirable relative to consumption and, eventually, leisure time may be increasing – and labor supply decreasing – along the transition to the after-shock steady state. A lower weight of habits entails a lower contribution of relative consumption to utility, so greater increases in the level of consumption are needed to achieve an increase in instantaneous welfare (e.g., Alvarez-Cuadrado et al., 2004, p. 71). Therefore, leisure becomes more desirable relative to consumption. 7. Conclusion Whether habits are externally or internally formed appears to be still an open empirical question, so the literature uses both

591

specifications interchangeably. However, we have shown that, in the presence of elastic labor supply, assuming internal or external habits has qualitatively different implications. A similar conclusion has been obtained by Ikefuji and Mino (2009) in a quite different framework — an overlapping generation model with inelastic labor supply. First we show that the unique feasible steady state is locally saddle path stable. Second we do a comparative static analysis of the effect that the habit parameters have on the long run equilibrium. An increase in the weight of habits has different implications for both model specifications. Under the internal-habits case, it increases leisure and has an ambiguous effect on the growth and the savings rate, whereas under the external-habits case has a positive effect on longrun growth, work time and savings rate. On the other hand an increase in the speed of adjustment of the reference stock does not affect long-run equilibrium in the external-habits case, but has a negative effect on the growth, the savings rate and the time spent working in the model with internal habits. Finally, we illustrate the dynamic effects of an increase in productivity. On impact and in the long-run, this shock has a positive effect on consumption, labor supply, welfare and the savings rate. Along the transition, labor supply exhibits a procyclical response to the productivity increase in the IH model, but a countercyclical response in the EH model. This sharply contrasts conventional wisdom and data (e.g., Khorunzhina, 2015), and might be an advantage of the internal- versus the external-habits specification. The countercyclical evolution of labor supply in the EH model remains robust in the sensitivity analysis. In the IH model, however, the faster the speed of adjustment and/ or the lower the weight of habits in utility the less likely labor supply exhibits a pro-cyclical behavior along the transition.

Appendix A Derivation of system (14)–(16). Eq. (5) can be rewritten as C −ε Lηð1−εÞ H −γð1−εÞ ¼ λ−μξρ ¼ λð1 þ ρξqÞ; so that λ ¼ C −ε Lηð1−εÞ H −γð1−εÞ =ð1 þ ρξqÞ;

ðA:1Þ

μ ¼ −C −ε Lηð1−εÞ H −γð1−εÞ q=ð1 þ ρξqÞ:

ðA:2Þ

Log-differentiating (A.1) with respect to time yields 







C L H λ ρξ q ¼ 0: −ε þ ηð1−εÞ −γð1−εÞ − − C L H λ 1 þ ρξq 



ðA:3Þ

From Eq. (8), using Eq. (A.2), we get μ C ð1 þ ρξqÞ : ¼ β þ ρ−γ H q μ 

ðA:4Þ

From Eqs. (5) and (6) we have that Lð1−LÞϕ−1 ¼ ð1 þ ρξqÞηC=ðϕAK Þ:

ðA:5Þ

Log-differentiating (A.5) with respect to time, we find that ! L 1−L C K ρξ q ¼ − þ : L 1−ϕL C K 1 þ ρξq 







ðA:6Þ

þ

592

M.A. Gómez, G. Monteiro / Economic Modelling 51 (2015) 583–595 



Let Ω(L) = (ε − 1)η(1 − L) + ε(1 − ϕL). Substituting for λ from Eq. (7), and L from (A.6) into (A.3), we obtain that the growth rate of consumption is ( ) !   C H 1−ϕL C H ðε−1Þηð1−LÞ þ ð1−ϕLÞð1−ψÞ H K ðε−1Þηð1−LÞ þ ð1−ϕLÞ ρξ q ð1−ψÞ −β−½ðε−1Þð1−γ Þ þ ψ ¼ þ − − : − ΩðLÞ K ΩðLÞ ΩðLÞ C H H H K 1 þ ρξq 











ðA:7Þ Eq. (14) is obtained from Eqs. (A.7) and (2); Eq. (15) is obtained from Eqs. (2) and (13); Eq. (16) is obtained from Eqs. (7) and (A.4), and Eq. (17) is obtained from Eq. (A.5). Proof of Proposition 1. From Eqs. (14)–(15), solving the system ċ = ḣ = 0, taking into account that q¼ 0, we get Eqs. (18) and (19). Now, Eq. (20) is obtained from q¼ 0, using Eq. (18). Eq. (22) is obtained from Eq. (17). The long-run growth rate (23) results from Eq. (18) into 





g ¼H =H ¼ ρðc−1Þ. Using Eqs. (18), (19), (20) and (22) we can obtain that c¼

gþρ ; ρ

ðA:8Þ

h¼

fg ½ðε−1Þð1−γ Þ þ ψ þ βgρ ; ð1−ψÞðg þ ρÞ

ðA:9Þ

q¼

γðg þ ρÞ : β þ g ½εð1−γ Þ þ γ ð1−ξÞ þ ρð1−ξγÞ

ðA:10Þ

Hence, if g N0 then we have that c N0, hN0 and qN0. Using Eqs. (18), (19) and (20), from (22) we can get that L ¼ ΨðgÞ, where L is defined implicitly as a function of g, L = Ψ(g), by the functional equation P ðL; g Þ ≡ ϕALð1−LÞϕ−1 −

 η γξðg þ ρÞ 1þ fβ þ g ½ðε−1Þð1−γÞ þ ψg ¼ 0: 1−ψ β þ g ½εð1−γÞ þ γð1−ξÞ þ ρð1−ξγ Þ

ðA:11Þ

ðL; gÞN0 for all L ∈ (0, 1). Hence, Eq. (A.11) has a unique solution Note that for all g ≥ 0, we have that P(0, g) b 0, limL → 1P(L, g) = + ∞, and ∂P ∂L L ∈ (0, 1) for every value of g ∈ [0, + ∞). In particular, this entails that Ψ(0) N 0. Using the implicit function theorem, after simplification we can obtain that for all g ≥ 0, " #  2−ϕ 2   ϕ  ϕ ð1−ψÞA 1−L þ ρ η½ðε−1Þð1−γ Þ þ ψ 1−L q dL βξγ ðg Þ ¼ þ ð1−ξγÞρ þ  fð1−ψÞA 1−L Ψ ðg Þ ¼   dg γ þ εð1−γÞ 1 þ ρξq Aϕ 1−ϕL γ2 c2  βξγ½ð1−ψÞβ þ ðγ þ εð1−γÞÞρψ N0: þð1−ξγÞρ2 þ ½ðε−1Þð1−γÞ þ ψ½γ þ εð1−γÞ 0

Hence, L = Ψ(g) is monotonic increasing for all g ≥ 0. Recalling Eq. (23), we also have that L ¼ ΓðgÞ, where   β þ ½γ þ εð1−γÞg 1=ϕ : Γ ðg Þ ¼ 1− ð1−ψÞA

ðA:12Þ

It can be easily shown that Γ(0) = 1 − {β/[(1 − ψ)A]}1/ϕ, limg → + ∞Γ(g) = − ∞, and for all g ≥ 0 Γ 0 ðg Þ ¼ −

  ½γ þ εð1−γ Þ β þ ½γ þ εð1−γ Þg ð1−ϕÞ=ϕ b 0: ð1−ψÞϕA ð1−ψÞA

Hence, L = Γ(g) is monotonic decreasing for all g ≥ 0. Therefore, there exists a unique g ∈ ð0; þ∞Þ such that L ¼ ΨðgÞ ¼ ΓðgÞ if and only if Ψ(0) b Γ(0), where L0 ¼ Ψð0Þ is the solution to P ðL; 0Þ ¼ Lð1−LÞϕ−1 −

ηðβ þ ρÞβ ¼ 0: ϕA½β þ ð1−ξγ Þρ

ðA:13Þ

Let L0 = Γ(0) = 1 − {β/[(1 − ψ)A]}1/ϕ. Since P(L, 0) is monotonic increasing with respect to L, we have that Ψ(0) b Γ(0) = L0 if and only if P(L0, 0) N 0. This conditions means that the solution to the Eq. (A.13) must be to the left of L0 and, after simplification, it yields the condition (24). Finally, the transversality condition is equivalent to 







−βþ K =Kþ λ =λ ¼ −βþ H =Hþ μ =μ ¼ −β−g ðε−1Þð1−γ Þ b 0; which therefore is fulfilled.

M.A. Gómez, G. Monteiro / Economic Modelling 51 (2015) 583–595

593

Proof of Proposition 2. In order to derive the coefficients of the linearization of the system (14)–(16) around its steady state ðc; q; hÞ we have to take into account that L is defined implicitly as a function of c, h and q by means of Eq. (22). Thus, for example, we have that  ∂ c ∂ c ¼ ∂h ∂h L 



∂ c ∂L ; ∂L ∂h 

þ constant

and the other derivatives have to be calculated in the same way. We shall use that Eq. (14) can be rewritten as: 

c¼ Λ 1 Γ þ Λ 2 h þΛ 3 q 

ðA:14Þ



where ð1−ϕLÞc ; ΩðLÞ ½ðε−1Þηð1−LÞ þ 1−ϕLρξc Λ3 ¼ − : ΩðLÞð1 þ ρξqÞ

Γ ¼ ð1−ψÞch−β−½ðε−1Þð1−γÞ þ ψρðc−1Þ; Λ2 ¼ −

Λ1 ¼

½ðε−1Þηð1−LÞ þ ð1−ψÞð1−ϕLÞc ; ΩðLÞh

Hence, for x = c, h, q we have that at the steady state ðc; q; hÞ 

∂c ∂Γ ∂h ∂q ¼ Λ1 þ Λ3 þ Λ2 ∂x ∂x ∂x ∂x 



where Λ 1 , Λ 2 and Λ 3 are Λ1, Λ2 and Λ3 evaluated at the steady state. Linearizing the system (14)–(16) around its steady state ðc; q; hÞ we get 0 1 0 b11 c @ h A ¼ @ b21 b31 q 





b12 b22 b32

1 0 1 10 c−c c−c b13 @ A @ A h−h ¼ B h−h A; b23 q−q q−q b33

ðA:15Þ

where n o b11 ¼ Λ 1 ð1−ψÞh−½ðε−1Þð1−γÞ þ ψρ þ Λ 2 b21 þ Λ 3 b31 ; b12 ¼ Λ 1 ð1−ψÞc þ Λ 2 b22 þ Λ 3 b32 ; b13 ¼ Λ"2 b23 þ Λ 3b33 ;  # η 1−L ð1 þ ρξqÞh b21 ¼ h þ ρ þ h; 1−ϕL   η 1−L ξρ 2 ch ; b23 ¼ − 1−ϕL   ηð1−ψÞ 1−L cq b32 ¼ ð1 þ ρξqÞ; 1−ϕL

" b22 ¼ 1 þ " b31 b33

#   η 1−L ð1 þ ρξqÞ

ch; 1−ϕL   # ð1−ψÞη 1−L hq ¼− γþ ð1 þ ρξqÞ; 1−ϕL   γc ηð1−ψÞ 1−L ρ − ¼ qch: q 1−ϕL

After simplification, it can be proved by direct computation that 8  9
ð1−ψÞc b22 b32

 0  b23 ; b 

ðA:16Þ

33

which after simplification yields that   ργη 1−L hc2 ½ðε−1Þð1−γÞ þ ψ detðBÞ ργhc2 ½γ þ εð1−γ Þ−  ¼−  2 q Λ1 q 1−ϕL fβ þ ρð1−γξÞ þ g ½γð1−ξÞ þ εð1−γ Þg  f½γ þ εð1−γÞg ½ðγ ð1−ξÞ þ ε ð1−γ ÞÞg þ 2β þ 2ð1−ξγÞρ þ ð1−ξγ Þρðβ þ ρÞg o   n ργη 1−L hc2 ½ðε−1Þð1−γ Þ þ γξ þ ψβ2 þ ½ðε−1Þð1−γ Þð1−ξγÞ þ ψβρ − b 0:   2 q 1−ϕL fβ þ ρð1−γξÞ þ g ½γ ð1−ξÞ þ εð1−γ Þg Given that Λ 1 N0, we have that det(B) b 0. Let λ1, λ2 and λ3 be the (real or complex) eigenvalues of the coefficient matrix B of Eq. (A.15). Note that if there are complex eigenvalues, they must appear in conjugate pairs. We have that det(B) = λ1λ2λ3 b 0, so that there are one or three eigenvalues with negative real parts (i.e., stable

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roots). Given that tr(B) = λ1 + λ2 + λ3 N 0, there cannot be three eigenvalues with negative real parts. Hence, matrix B must have exactly one negative real (stable) root and, as the system (14)–(16) features one state variable h, the steady state ðc; q; hÞ is locally saddle-path stable. Q.E.D. Computation of the partial derivatives of G in Eq. (22). After simplification, we get  ϕ−1   ϕ−2 A 1−L ηϕ½ðε−1Þð1−γÞ þ ψq2 ∂G ϕþ ¼ A 1−ϕL 1−L ∂L γ2 ½γ þ εð1−γÞc2 ( " #  ϕ   ϕ ð1−ψÞA 1−L þ ρ βγξ βγξ½ð1−ψÞβ þ ðγ þ εð1−γÞÞρψ þ ð1−γξÞρ þð1−γξÞρ2 þ N 0; þ  ð1−ψÞA 1−L γ þ εð1−γÞ ½ðε−1Þð1−γ Þ þ ψ½γ þ εð1−γÞ 1 þ ρξq

ðA:17Þ

 ϕ−1  n h i A 1−L ϕL ∂G g 2 2 ð ð ε−1 Þ ð 1−γ Þ þ ψ Þ  βρ þ ψεg þ ¼ − ρ þ β þ β ð ε−1 Þψ þ ð 1−ψ Þψρ b 0; ½β þ ð1−γÞðεg þ ρÞfβ þ ½ðε−1Þð1−γ Þ þ ψgg γ þ ε ð1−γ Þ ∂γ ξ¼1

ðA:18Þ

  ϕ−1 ðε−1Þg ∂G N 0; ¼ A 1−L ϕL ∂γ ξ¼0 ½γ þ εð1−γÞch

ðA:19Þ

h i  ϕ−1  ϕ ξγϕA 1−L Lð1 þ ρξqÞ ð1−γ Þðε−1Þð1−ψÞA 1−L þ β ∂G ; ¼− h i2  ϕ ∂ρ ½γ þ εð1−γÞ ð1−ψÞA 1−L þ ρ

ðA:20Þ

h i  ϕ  ξηρ ð1−ψÞA 1−L þ ρ ∂G ∂G ∂G dε ∂G ðε−1Þ ∂G ¼ ¼ hq2 : þ þ ¼− γ2 ∂γ σ ¼constant ∂γ ∂ε dγ ∂γ ð1−γ Þ ∂ε

ðA:21Þ

Effect of a compensated increase in the weight of habits in utility, γ. Note first that when the relative risk aversion ε is adjusted to keep constant the effective EIS, γ + ε(1 − γ), we have that (1 − γ)dε + (1 − ε)dγ = 0; i.e., dε/dγ = (ε − 1)/(1 − γ). Substituting ξ = 1 into Eq. (A.21) we get that ∂G/∂γ|σ = constant b 0 in the IH model, and so, the ‘compensated’ effects are given by    

IH 

dL dγ

¼−

 ∂G=∂γσ¼constant

σ ¼constant

N0; ∂G=∂L

 IH ϕ−1 A 1−L ð1−ψÞϕ dLIH  ¼−  dγ  γ þ εð1−γ Þ

   IH dg IH  ∂g IH  ∂g IH dL  ¼ þ    dγ  ∂γ  ∂L dγ σ¼constant  σ ¼constant   σ ¼constant IH dsIH  ∂sIH  ∂sIH dL  ¼ þ ¼−    dγ  ∂γ  ∂L dγ  σ¼constant

σ ¼constant

σ ¼constant

b 0;

σ ¼constant IH 

βϕ dL  b 0: 

IH ϕþ1 dγ  σ ¼constant ½γ þ εð1−γ ÞA 1−L

Substituting ξ = 0 into (A.21) we get that ∂G/∂γ|σ = constant = 0 in the EH model, and so, the ‘compensated’ effects are given by

 dgEH   dγ  σ ¼constant dsEH   dγ 

σ¼constant

 EH dL  ¼ 0;  dγ    σ¼constant EH ∂g EH  ∂g EH dL  ¼ þ ¼ 0;   ∂γ  ∂L dγ σ ¼constant   σ ¼constant EH ∂sEH  ∂sEH dL  ¼ þ ¼ 0;   ∂γ  ∂L dγ  σ¼constant

σ ¼constant

where it has been used that  ∂g EH   ∂γ 

σ ¼constant

¼

∂gEH ∂g EH dε ∂g EH ðε−1Þ ∂g EH ¼ þ þ ¼ 0: ð1−γ Þ ∂ε ∂γ ∂ε dγ ∂γ

M.A. Gómez, G. Monteiro / Economic Modelling 51 (2015) 583–595

595

Effect of an increase in the speed of adjustment of habits to current consumption, ρ. We have that  ∂G=∂ρξ¼1 dL ¼− N0; dρ ∂G=∂L  ϕ−1 A 1−L ð1−ψÞϕ dL dg ∂g ∂g dL ¼ ¼− b 0; þ dρ ∂ρ ∂L dρ dρ γ þ ε ð1−γ Þ ds ∂s ∂s dL βϕ dL ¼ ¼− þ  ϕþ1 dρ b 0: dρ ∂ρ ∂L dρ ½γ þ εð1−γÞA 1−L References Abel, A., 1990. Asset prices under habit formation and catching up with the Joneses. Am. Econ. Rev. 80, 38–42. Alonso-Carrera, J., Caballé, J., Raurich, X., 2004. Consumption externalities, habit formation and equilibrium efficiency. Scand. J. Econ. 106, 231–251. Alonso-Carrera, J., Caballé, J., Raurich, X., 2005. Growth, habit formation, and catching-up with the Joneses. Eur. Econ. Rev. 49, 1665–1691. Alonso-Carrera, J., Caballé, J., Raurich, X., 2006. Welfare implications of the interaction between habits and consumption externalities. Int. Econ. Rev. 47, 557–571. Alvarez-Cuadrado, F., Monteiro, G., Turnovsky, S.J., 2004. Habit formation, catching-up with the Joneses, and economic growth. J. Econ. Growth 9, 47–80. Aydilek, A., 2013. Habit formation and housing over the life cycle. Econ. Model. 33, 858–866. Carroll, C.D., Overland, J., Weil, D.N., 1997. Comparison utility in a growth model. J. Econ. Growth 2, 339–367. Carroll, C.D., Overland, J., Weil, D.N., 2000. Saving, growth and habit formation. Am. Econ. Rev. 90, 341–355. Chen, X., Ludvigson, S.C., 2009. Land of addicts? An empirical investigation of habit-based asset pricing models. J. Appl. Econ. 24, 1057–1093. Chen, B.-L., Hsu, Y.-S., Mino, K., 2013. Can consumption habit spillovers be a source of equilibrium indeterminacy? J. Econ. 109, 245–269. Cooley, T., Prescott, E.C., 1995. Economic growth and business cycles. In: Cooley, T. (Ed.), Frontiers of Business Cycle Research. Princeton University Press, Princeton, NJ, pp. 1–38. Engen, R., Gravelle, J., Smetters, K., 1997. Dynamic tax models: why they do the things they do. Natl. Tax J. 50, 657–682. Fisher, W., Hof, F., 2000. Relative consumption, economic growth, and taxation. J. Econ. 72, 241–262. Fuhrer, J.C., 2000. Habit formation in consumption and its implications for monetary policy models. Am. Econ. Rev. 90, 367–390. Gali, J., 1994. Keeping up with the Joneses: consumption externalities, portfolio choice, and asset prices. J. Money Credit Bank. 26, 1–8.

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