Public Spending in a Model of Endogenous Growth with Habit Formation

Abstract

This paper introduces habit-forming preferences in a Barro-type endogenous growth model with productive public services. Government expenditure, which may be subject to congestion, is financed by distortionary income taxation. Different from the standard time-separable model, the presence of habits makes the economy feature transitional dynamics, which are solved in closed form. Setting the income tax so as to equate the elasticity of public services in production is shown to maximize both long-run growth and welfare as in the standard model. This second-best solution coincides with the first-best outcome only in the presence of proportional congestion.

1. Introduction

Following the seminal work of Barro [1], the effect of productive government expenditure has been the subject of active research in the theoretical literature on endogenous growth (see, e.g., Irmen and Kuehnel [2], for a recent survey). This interest has been stimulated by empirical work, starting from Aschauer [3], providing compelling evidence that productive public spending has a positive impact on growth (see, e.g., the review by Romp and de Haan [4]).

Barro [1] introduces the flow of government expenditure, which is fully financed by a distortionary income tax, as a productive input to private production. (Another strand of the literature, starting from Futagami et al. [5], considers that it is the accumulated stock of public capital, rather than the flow of current expenditure, that enters in the production function (see, e.g., Irmen and Kuehnel [2]).) Thus, public policy has two opposite effects on growth, which offset each other if the tax rate on income is equal to the elasticity of public spending in production. Under Cobb-Douglas technology, this growth-maximizing tax rate also maximizes intertemporal utility in a second-best sense. The ensuing literature has enriched the original Barro's setup to incorporate other relevant features such as, among others, adjustment costs [6], congestion [7, 8], public consumption services [9, 10], elastic labor supply [11], alternative government financing [9, 12], progressive taxation [13], rent-seeking [14, 15], finite lifetimes [16, 17], two countries [18], or stochastic growth [19, 20].

This paper adds to this literature by extending the Barro [1] model to incorporate habit-persistent preferences. In models with habit formation, individual's utility depends not only on her level of current consumption but also on how it compares to a reference level–-the habits stock. Recent empirical works by Chen and Ludvigson [21], Korniotis [22], and Grishchenko [23] have reported supporting evidence for the presence of habits that enter utility in a subtractive manner. Furthermore, subtractive habits have been widely incorporated into dynamic equilibrium models in order to address several empirical facts that are difficult to explain under standard time-separable preferences. (A partial list includes the equity premium puzzle, [24, 25] , time-varying expected returns [26] , the term structure of interest rates [27] , the foreign exchange risk premium [28] , some stylized facts of business cycles [29] , current account dynamics [30] , and the effects of monetary policy [31]. Subtractive habits have also become an important feature of many dynamic stochastic general equilibrium models [32, 33].) Based on these works, we assume that habits enter subtractively in utility. The literature distinguishes between internal habits (IH), when individual's habits depend on her own past consumption, and external habits (EH), when habits are formed from average past consumption in the economy. However, whether habits are formed in an internal, external, or hybrid internal-external form appears to be an open empirical question yet. Therefore, we consider a fairly general specification of the habit formation process that nests the cases of internal, external, and hybrid internal-external habits. Besides its own interest, the introduction of habit-persistent preferences is also relevant because it allows to overcome one important limitation of the flow models based on the Barro [1] setup, namely, that they lack transitional dynamics, so that the economy is always on its balanced growth path and all variables grow at a constant rate. (Actually, this also occurs in stock models in which public capital fully depreciates in one period (e.g., [34–36]).)

This paper develops an endogenous growth model with habit persistence in utility and productive government expenditure in production, which is financed by distortionary income taxation. As Barro and Sala-i-Martin [7] argue, almost all public services are subject to some degree of congestion. Therefore, public spending is allowed to be subject to congestion, so that the service of public expenditure to the agent depends on the usage of her capital stock relative to the aggregate capital stock (e.g., [6, 7]). First, we analyze the equilibrium dynamics of the decentralized economy, and characterize the tax rate on income that maximizes the representative agent's utility; that is, the second-best solution. Next, we study the first-best outcome that a benevolent central planner would implement, and compare it with the second-best solution. Finally, some numerical simulations are presented.

The main results of this paper are twofold. First, we show that, unlike the standard time-separable Barro [1] model, the introduction of habit formation makes that, more realistically, the model features transitional dynamics. Furthermore, we give closed-form solutions for the dynamics of both the market and the socially-planned economies. Thus, this paper also adds to a recent literature that investigates the existence of closed-form solutions to growth models as, e.g., Mehlum [37], Smith [38, 39], and Guerrini [40] for the Ramsey model, and Bethmann [41], Ruiz-Tamarit [42], Boucekkine and Ruiz-Tamarit [43], Chilarescu [44, 45], and Hiraguchi [46], among others, for the two-sector Lucas [47] model. This is interesting because such explicit solutions are analytically tractable, and numerical simulations can be performed without error. Second, we find that the results obtained by Barro [1] are robust to the inclusion of habit formation and, consequently, to the existence of transition dynamics. On one hand, within a decentralized economy set-up, the growth-maximizing and welfare-maximizing (in a second-best sense) income tax rates coincide, and are equal to the elasticity of public spending in production. On the other hand, in a first-best setting, the optimal ratio of government expenditure to output is equal to the elasticity of public services in production. Furthermore, the first-best solution can be decentralized as a market equilibrium only in the presence of proportional congestion, when the second-best outcome coincides with the first-best solution. In any other case, the first-best solution cannot be decentralized. The numerical results illustrate the implications for the dynamics of the economy of the introduction of habit formation.

The rest of the paper is organized as follows. Section 2 presents the model. Section 3 analyzes the equilibrium dynamics of a decentralized economy. Section 4 studies the first-best solution attainable by a social planner. Some numerical results are presented in Section 5. Section 6 concludes.

2. The Model

We study a closed economy populated by a large but fixed number 𝑁 of identical infinitely-lived individuals, each of whom has an infinite planning horizon and possesses perfect foresight. Each individual is the owner of a unique firm that produces the single output in the economy, which can be devoted to consumption or investment. The government provides productive services to firms, which are financed by an income tax. With all agents being identical, aggregate private quantities are simply multiples of individual quantities. We will denote individual quantities by lower case letters and aggregate quantities by corresponding upper case letters, so that 𝑋=𝑁𝑥.

2.1. Preferences

The intertemporal utility derived by the agent depends both on its current consumption, 𝑐_𝑡, and a reference consumption level or habits stock, ℎ_𝑡, according to𝑈=^∞_𝑡=0𝑢(𝑐_𝑡−𝛾ℎ_𝑡)𝛽^𝑡,0≤𝛾<1,0<𝛽<1, (2.1) where 𝑢 is the instantaneous utility function, and 𝛽 is the time discount rate. (Another commonly-used specification is the “multiplicative” one (e.g., [48–51]), in which habits-adjusted consumption is defined as the geometric mean of absolute consumption and relative consumption as a ratio, 𝑧_𝑡=𝑐_𝑡^1−𝛾(𝑐_𝑡/ℎ_𝑡)^𝛾.) The habits-adjusted consumption, 𝑧_𝑡≡𝑐_𝑡−𝛾ℎ_𝑡, can be rewritten as 𝑧_𝑡=(1−𝛾)𝑐_𝑡+𝛾(𝑐_𝑡−ℎ_𝑡). Hence, the parameter 𝛾 reflects the importance of absolute versus relative-to-habits consumption in utility: the higher 𝛾, the greater the importance of relative consumption. The case 𝛾=0 corresponds to the standard time-separable model in which only the absolute level of consumption matters. We will assume that the instantaneous utility function 𝑢 is twice differentiable, with 𝑢^′(𝑧)>0 and 𝑢^′′(𝑧)<0 for all 𝑧>0, and that 𝑢^′ is homogeneous of degree −𝜀<0. Therefore, the utility function is isoelastic. This last assumption is imposed for guaranteeing the existence of a balanced growth path (see, e.g., Acemoglu [52] and Alonso-Carrera et al. [53]). We also assume that 𝑢 satisfies standard Inada conditions, lim_𝑧→0𝑢^′(𝑧)=∞ and lim_𝑧→∞𝑢^′(𝑧)=0.

Following Gόmez [54], the habits stock evolves according toℎ_𝑡+1=ℎ_𝑡+𝜌[𝑚(𝑐_𝑡,𝑐_𝑡)−ℎ_𝑡],0<𝜌≤1, (2.2) where 𝑐_𝑡 denotes the economy-wide average level of consumption. The parameter 𝜌, which governs the speed with which the habits stock adjusts to previous-period consumption, determines the relative weight of consumption at different dates: the larger 𝜌, the more important is agent's consumption in the recent past and the smaller is the importance of the old values of the habits stock. If 𝜌=1, then ℎ_𝑡+1=𝑚(𝑐_𝑡,𝑐_𝑡), so that the habits stock depends only on previous-period consumption. Note that if 𝜌=0, then ℎ_𝑡=ℎ₀ for all 𝑡, so that the stock of habits would remain constant at its initial value and would not depend on past consumption levels.

In the internal-habits model, the reference consumption stock depends only on individual's own past consumption, so that 𝑚(𝑐_𝑡,𝑐_𝑡)=𝑐_𝑡. In the external-habits model, habits are formed only from average past consumption in the economy, so that 𝑚(𝑐_𝑡,𝑐_𝑡)=𝑐_𝑡. Thus, in this case average past consumption, which is taken as given by the individual agent, exerts an external effect in utility. In the hybrid internal-external case, the habits stock is determined by both its own consumption and the economy-wide average level of consumption, which are combined by means of a continuously differentiable homogeneous mean. Thus, in this case 𝑚 is a continuously differentiable function such that 𝑚(𝑐,𝑐)=𝑐 for all 𝑐>0, strictly increasing in its components, and (positively) homogeneous of degree one. Differentiating 𝑚(𝑐,𝑐)=𝑐, and using its linear homogeneity, we get that 𝑚₁(𝑐,𝑐)=𝜙 and 𝑚₂(𝑐,𝑐)=1−𝜙 for all 𝑐>0, with 0≤𝜙≤1, where 𝑚_𝑖 denotes the partial derivative of 𝑚 with respect to its 𝑖th argument. Hence, the case 𝜙=1 corresponds to the internal-habits model; the case 𝜙=0 to the external-habits model, and the case 0<𝜙<1 to the model with hybrid internal-external habits. This specification comprises as particular cases the weighted geometric mean, 𝑚(𝑐_𝑡,𝑐_𝑡)=𝑐^𝜙_𝑡𝑐_𝑡^1−𝜙, proposed by Abel [48], and the weighted arithmetic mean, 𝑚(𝑐_𝑡,𝑐_𝑡)=𝜙𝑐_𝑡+(1−𝜙)𝑐_𝑡, considered, for example, by Marrero and Novales [36]. Most of the literature has considered that habits are formed in an external or internal way, and the papers that introduce hybrid internal-external habits use a weighted geometric mean or (to a much lesser extent) a weighted arithmetic mean. All of them are encompassed by the specification assumed which, therefore, is fairly general.

2.2. Production

At each moment of time, the representative agent is endowed with a fixed and constant stock of labor, 𝑙_𝑡=𝑙. Each individual firm produces output, 𝑦_𝑡, using the inelastically supplied labor input 𝑙_𝑡=𝑙, the capital stock 𝑘_𝑡, and public services provided by the government, 𝑃_𝑡, in accordance with the Cobb-Douglas production function:𝑦_𝑡=𝐴𝑘^𝛼_𝑡𝑙^1−𝛼𝑃_𝑡^1−𝛼=𝐴𝑘^𝛼_𝑡𝑃_𝑡^1−𝛼, (2.3) where 𝐴=𝐴𝑙^1−𝛼>0.

Following Turnovsky [6, 55], the services derived by the agent from public investment are represented by𝑃_𝑡=𝐺_𝑡(𝑘_𝑡𝐾_𝑡)^1−𝜎,0≤𝜎≤1, (2.4) where 𝐾_𝑡 denotes the aggregate capital stock and 𝐺_𝑡 denotes the aggregate public expenditure. The former formulation incorporates the potential for public investment to be associated with relative congestion. The case 𝜎=1, so that 𝑃_𝑡=𝐺_𝑡, corresponds to a nonrival nonexcludable public good that is available equally to each individual firm, independent of the usage of others. Thus, there is no congestion. However, as Barro and Sala-i-Martin [7] argue, almost all public services are subject to some degree of congestion, so that the pure public good should be viewed as only a benchmark. As 𝜎 deviates from 1, the nonexcludable public service losses its nonrival nature, and as this occurs the level of public services enjoyed by the individual firm from a given level of public expenditure is enhanced as his individual capital stock increases relative to the aggregate. In the case 𝜎=0, the public good is like a private good in that, since 𝑘_𝑡/𝐾_𝑡=1/𝑁, the individual receives his proportional share 𝑃_𝑡=𝐺_𝑡/𝑁. This case is referred to as proportional relative congestion.

Combining (2.3) and (2.4), firm's output can be expressed as𝑦_𝑡=𝐴𝑘_𝑡^{1−𝜎(1−𝛼)}𝐺_𝑡^1−𝛼𝐾_𝑡^{(𝜎−1)(1−𝛼)}. (2.5) Thus, the productivity of individual capital depends upon the usual elasticity of private capital, 𝛼, and a component, (1−𝜎)(1−𝛼), which reflects the fact that, from the perspective of the individual agent, increasing her stock of capital will increase the level of government services she derives in the presence of relative congestion. Note that the presence of congestion introduces a distortion because the individual firm takes aggregate capital 𝐾_𝑡 as given in her production function (2.5). However, a central planner would take into account that 𝐾_𝑡=𝑁𝑘_𝑡.

2.3. Government

The government levies a tax on income at a constant rate 𝜏. Tax revenues are used to finance productive public expenditure,𝐺_𝑡=𝜏𝑌_𝑡, (2.6) so that the government runs a balanced budget. In particular, this means that lump-sum taxes (or transfers) are not available.

2.4. Individual's Optimization

Output can be used for consumption or investment. Thus, the agent's budget constraint isk_𝑡+1=𝑘_𝑡+(1−𝜏)𝑦_𝑡−𝑐_𝑡. (2.7)

At time 𝑡=0, the agent chooses {𝑐_𝑡,𝑘_𝑡+1,ℎ_𝑡+1}^∞_𝑡=0 to maximize the lifetime utility (2.1) subject to the budget constraint (2.7)–-where 𝑦_𝑡 is given by (2.5)–-and the constraint on the habits stock accumulation (2.2), taking as given the paths of economy-wide average consumption, {𝑐_𝑡}^∞_𝑡=0, public expenditure, {𝐺_𝑡}^∞_𝑡=0, and aggregate capital, {𝐾_𝑡}^∞_𝑡=0, the income tax rate, 𝜏, and the initial conditions on capital 𝑘₀>0 and habits stock ℎ₀>0.

To derive the optimization conditions, let us set up the Lagrangian of the agent's maximization problem, 𝐿=^∞_𝑡=0𝛽^𝑡{𝑢(𝑐_𝑡−𝛾ℎ_𝑡)+𝜆_𝑡[𝑘_𝑡+(1−𝜏)𝑦_𝑡−𝑐_𝑡−𝑘_𝑡+1]+𝜇_𝑡[(1−𝜌)ℎ_𝑡+𝜌𝑚(𝑐_𝑡,𝑐_𝑡)−ℎ_𝑡+1]}, (2.8) where 𝜆_𝑡 and 𝜇_𝑡 are the shadow values of capital and habits stock, respectively. The first-order conditions with respect to 𝑐_𝑡, 𝑘_𝑡+1, and ℎ_𝑡+1 are given by𝑐_𝑡∶𝑢^′(𝑐_𝑡−𝛾ℎ_𝑡)+𝜌𝜇_𝑡𝑚₁(𝑐_𝑡,𝑐_𝑡)=𝜆_𝑡𝑘_𝑡+1,∶𝛽[1+(1−𝜏)(1−𝜎(1−𝛼))𝑦_𝑡/𝑘_𝑡]𝜆_𝑡+1=𝜆_𝑡ℎ_𝑡+1,∶𝛽(1−𝜌)𝜇_𝑡+1−𝛽𝛾𝑢^′(𝑐_𝑡+1−𝛾ℎ_𝑡+1)=𝜇_𝑡, (2.9)(2.10)(2.11) together with the transversality conditionlim_𝑡→∞𝛽^𝑡𝜆_𝑡𝑘_𝑡+1=lim_𝑡→∞𝛽^𝑡𝜇_𝑡ℎ_𝑡+1=0. (2.12)

Equation (2.9) equates the marginal utility of consumption, adjusted by its effect on the future stock of habits, to the shadow price of capital. Equation (2.10) equates the rate of return on capital to the rate of return on consumption. From (2.11) and (2.12), we get𝜇_𝑡=−𝛽𝛾𝑢^′(𝑐_𝑡+1−𝛾ℎ_𝑡+1)−𝛽^∞_𝑗=𝑡+1[𝛽(1−𝜌)]^𝑗−𝑡𝛾𝑢^′(𝑐_𝑗+1−𝛾ℎ_𝑗+1). (2.13) This condition states that the shadow cost of the habits stock is determined as the present discounted value of the stream of extra utils that would be lost by a marginal unit of habits, which depreciates at the rate 𝜌. Note that (2.13) entails that the shadow value of the habits stock is negative, 𝜇_𝑡<0.

3. Equilibrium

An equilibrium for this economy is as a set of paths {𝑐_𝑡,𝑘_𝑡+1,ℎ_𝑡+1}^∞_𝑡=0 that solves the agent's utility maximization problem when 𝑐_𝑡=𝑐_𝑡 and 𝐾_𝑡=𝑁𝑘_𝑡 for all 𝑡, and such that the government obeys its budget constraint.

3.1. Dynamics in the Aggregate Economy

We will focus on the equilibrium dynamics of aggregate variables, 𝐶_𝑡=𝑁𝑐_𝑡, 𝐾_𝑡=𝑁𝑘_𝑡, and 𝐻_𝑡=𝑁𝐻_𝑡. Furthermore, let 𝑍_𝑡≡𝐶_𝑡−𝛾𝐻_𝑡=𝑁𝑧_𝑡 denote the aggregate habit-adjusted consumption, and 𝑞_𝑡≡𝜇_𝑡/𝜆_𝑡 the relative shadow cost of habits. Note first that, using the government budget constraint (2.6) to substitute for 𝐺_𝑡 in the production function (2.5), individual output, 𝑦_𝑡, and aggregate output, 𝑌_𝑡=𝑁𝑦_𝑡, can be expressed as𝑦_𝑡=𝜏^{(1−𝛼)/𝛼}𝐴^1/𝛼𝑁^{𝜎(1−𝛼)/𝛼}𝑘_𝑡=𝐵𝑘_𝑡𝑌_𝑡,=𝜏^{(1−𝛼)/𝛼}𝐴^1/𝛼𝑁^{𝜎(1−𝛼)/𝛼}𝐾_𝑡=𝐵𝐾_𝑡, (3.1)(3.2) where 𝐵=𝜏^{(1−𝛼)/𝛼}𝐴^1/𝛼𝑁^{𝜎(1−𝛼)/𝛼}.

Along a balanced growth path (BGP), 𝐶_𝑡, 𝐻_𝑡, 𝐾_𝑡, and therefore 𝑍_𝑡, all grow at the same constant rate 𝑔. Appendix A derives the following system, which drives the dynamics of the economy in terms of the variables 𝑍_𝑡≡𝑍_𝑡(1+𝑔)^−𝑡, 𝐾_𝑡≡𝐾_𝑡(1+𝑔)^−𝑡, 𝐻_𝑡≡𝐻_𝑡(1+𝑔)^−𝑡, and 𝑞_𝑡≡𝜇_𝑡/𝜆_𝑡, which are constant along a BGP:(1+𝑔)𝑍_𝑡+1𝑞={1−𝜌𝜙_𝑡1−𝜌𝜙𝑞_𝑡+1𝛽[1+(1−𝜏)(1−𝜎(1−𝛼))𝐵]}^1/𝜀𝑍_𝑡,(1+𝑔)𝐾_𝑡+1=(1+(1−𝜏)𝐵)𝐾_𝑡−𝑍_𝑡−𝛾𝐻_𝑡𝐻,(1+𝑔)_𝑡+1=𝐻_𝑡+𝜌(𝑍_𝑡−(1−𝛾)𝐻_𝑡𝑞),_𝑡+1=[1+(1−𝜏)(1−𝜎(1−𝛼))𝐵]𝑞_𝑡+𝛾. (3.3)(3.4)(3.5)(3.6)

The following proposition, which is proved in Appendix B, yields a closed-form expression for the equilibrium dynamics of the decentralized economy.

Proposition 3.1. The decentralized economy has a unique feasible equilibrium with positive long-run growth which is described by 𝐻_𝑡−𝐻_∞=(1−𝜌(1−𝛾)𝑔)1+^𝑡(𝐻₀−𝐻_∞𝐶),_𝑡−𝐶_∞=𝛾(𝐻_𝑡−𝐻_∞𝐾),_𝑡−𝐾_∞=𝛾(𝐻(1−𝜏)𝐵+𝜌(1−𝛾)^𝑡−𝐻_∞), (3.7)(3.8)(3.9) which converges to a BGP given by 𝐶_∞=(𝑔+𝜌)𝑍_∞,𝐾𝑔+𝜌(1−𝛾)_∞=(𝑔+𝜌)𝑍_∞𝐻((1−𝜏)𝐵−𝑔)(𝑔+𝜌(1−𝛾))_∞,=𝜌𝑍_∞,𝑔+𝜌(1−𝛾) (3.10)(3.11)(3.12) where 𝑍_∞=((1−𝜏)𝐵−𝑔)[𝐾₀−𝛾(𝜌𝐾₀+𝐻₀)(1−𝜏)𝐵+𝜌], (3.13) and the long-run growth rate is 𝑔={𝛽[1+(1−𝜏)(1−𝜎(1−𝛼))𝐵]}^1/𝜀−1, (3.14) if and only if parameter values and initial conditions are that 𝑔>0>𝛽(1+𝑔)^1−𝜀−1,𝛾𝐻₀𝐾₀<(1−𝜏)𝐵+𝜌(1−𝛾). (3.15)(3.16)

The equilibrium paths of aggregate consumption, capital, and habits stock are then given by 𝐶_𝑡=𝐶_𝑡(1+𝑔)^𝑡, 𝐾_𝑡=𝐾_𝑡(1+𝑔)^𝑡, and 𝐻_𝑡=𝐻_𝑡(1+𝑔)^𝑡. Note, in particular, that the initial value of consumption is given by 𝐶₀=𝐶₀=𝑍_∞+𝛾𝐻₀.

One important consequence of Proposition 3.1 is that the equilibrium dynamics of the economy is invariant to the specific homogeneous mean 𝑚 chosen; that is, to the implied value of 𝜙. Hence, the dynamics are invariant to whether habits are internally or externally formed and, therefore, the presence of externalities associated to past consumption in the external-habits model does not provoke equilibrium inefficiency. This theoretical finding seems to accord with empirical evidence supporting the habit-formation hypothesis but inconclusive about whether habits are formed in an internal or external way.

If 𝐻₀=𝐻₀>𝐻_∞, from (3.7)–(3.12) in Proposition 3.1, we can find that 𝐻_𝑡𝐻_∞>𝐶_𝑡𝐶_∞>𝐾_𝑡𝐾_∞>1. (3.17) Furthermore, as the economy evolves, 𝐻_𝑡, 𝐶_𝑡, and 𝐾_𝑡 decrease monotonically toward their respective steady-state values. Intuitively, to maintain a flat pattern of detrended adjusted-consumption, given the relatively high level of the habits stock (with respect to its stationary value), the agent starts with a relatively high level of consumption, 𝐶_𝑡, and a low level of investment. Therefore, the capital stock, 𝐾_𝑡, decreases. As the economy evolves, in order to drive the de-trended habits stock to its long-run level, consumption decreases steadily towards its stationary value. These results should be reversed in the case that 𝐻₀=𝐻₀<𝐻_∞.

The explicit expressions derived in Proposition 3.1 allow us, for example, to compute analytically the speed of convergence towards the balanced growth path. Adapting the measure proposed by Eicher and Turnovsky [56] (see also Papageorgiou and Perez-sebastian [57]), the speed of convergence of a variable 𝑥_𝑡 is defined asΦ_𝑡=−(𝑥_𝑡+1−𝑥_𝑡)−(𝑥_∞,𝑡+1−𝑥_∞,𝑡)𝑥_𝑡−𝑥_∞,𝑡, (3.18) where 𝑥_∞,𝑡 is the equilibrium balanced growth path, which may or may not be stationary. In particular, if 𝑥_𝑡 converges to a (constant) stationary value, then 𝑥_∞,𝑡+1=𝑥_∞,𝑡=𝑥_∞. If 𝛾≠0, the convergence speed of the aggregate variables, 𝐻, 𝐾 and 𝐶, is given by Φ=𝜌(1−𝛾). Hence, the convergence speed is increasing in the speed of adjustment of habits to consumption, 𝜌, and decreasing in the strength of habits in utility, 𝛾. If 𝛾=0, we recover the standard time-separable Barro [1] model. As it is wellknown, in this case the adjustment is instantaneous: at the initial time, consumption jumps to its BGP value, 𝐶₀=𝐶_∞=((1−𝜏)𝐵−𝑔)𝐾₀, and thereafter consumption and capital both increase at the constant rate 𝑔.

3.2. Second-Best Solution

Among the different competitive equilibria indexed by the value of the income tax rate, 𝜏, this section determines the one that maximizes individual welfare; that is, the second-best outcome.

Using that 𝑧_𝑡=𝑐_𝑡−𝛾ℎ_𝑡=𝑍_∞(1+𝑔)^𝑡/𝑁, the lifetime utility obtained is𝑈=^∞_𝑡=0𝑍𝑢[^∞(1+𝑔)^𝑡𝑁]𝛽^𝑡. (3.19) Therefore, we are looking for the value of the tax rate on income, 𝜏, that maximizes (3.19). Using (3.14), we can obtain that(1−𝜏)𝐵=(1+𝑔)^𝜀−𝛽,𝛽(1−𝜎(1−𝛼)) (3.20) which can be used to substitute (1−𝜏)𝐵 as a function of 𝑔 in (3.19) by means of (3.13), and allows us to express the lifetime utility 𝑈 as a function of 𝑔. Now, we have that 𝑑𝑈/𝑑𝜏=(𝑑𝑈/𝑑𝑔)(𝑑𝑔/𝑑𝜏). Appendix D shows that 𝑑𝑈/𝑑𝑔>0. Hence, the first-order condition to maximize the lifetime utility 𝑈 with respect to 𝜏 becomes 𝑑𝑔=𝑑𝜏(1+𝑔)^𝜀−1𝐵𝛽(1−𝜎(1−𝛼))(1−𝛼−𝜏)𝛼𝜀𝜏=0, (3.21) which yields the familiar result that the optimal income tax is equal to the elasticity of public services in production,𝜏=1−𝛼. (3.22) Given that 𝑑²𝑔𝑑𝜏²(𝜏)=−(1+𝑔)^𝜀−1𝐵𝛽(1−𝜎(1−𝛼))(1−𝛼)𝛼𝜀𝜏²<0, (3.23) lifetime utility 𝑈 is maximized at the income tax rate 𝜏=1−𝛼, which also maximizes long-run growth.

This result is similar to that obtained by Barro [1] in the time-separable model. There is a hump-shaped relationship between growth (and welfare) and the income tax rate or, equivalently, the ratio of public spending to output: long-run growth first increases, reaches its maximum value, and then decreases as the tax rate increases. This shape results from the combination of two opposing forces on the marginal product of capital, which determines long-run growth: the positive effect caused by the increased provision of productive public services, and the negative effect of a higher tax rate needed to finance them. The growth- and welfare-maximizing tax rate on income is equal to the elasticity of public services in goods production.

4. The Socially-Planned Economy

We now turn to the first-best solution that a benevolent social planner would implement. The planner takes into account that private and aggregate outputs are related by 𝐾_𝑡=𝑁𝑘_𝑡 in the production function, and so,𝑌_𝑡=𝐴𝑁^{𝜎(1−𝛼)}𝐾^𝛼_𝑡𝐺_𝑡^1−𝛼. (4.1) The planner also takes into account that 𝑐_𝑡=𝑐_𝑡 or, in aggregate terms, 𝑁𝑐_𝑡=𝐶_t, and, therefore, the constraint on the habits stock accumulation becomes𝐻_𝑡+1=𝐻_𝑡+𝜌(𝐶_𝑡−𝐻_𝑡). (4.2)

Thus, given the initial conditions on capital, 𝐾₀>0, and habits stock, 𝐻₀>0, the planner chooses {𝐶_𝑡,𝐺_𝑡,𝐾_𝑡+1,𝐻_𝑡+1}^∞_𝑡=0 to maximize the lifetime utility ^∞_𝑡=0𝑢[(𝐶_𝑡−𝛾𝐻_𝑡)𝑁]𝛽^𝑡𝑑𝑡, (4.3)

subject to the resource's constraint 𝐾_𝑡+1=𝐾_𝑡+𝑌_𝑡−𝐶_𝑡−𝐺_𝑡, (4.4) where 𝑌_𝑡 is given by (4.1), and the constraint on the habits stock accumulation (4.2).

4.1. First-Best Solution

The Lagrangian of the planner's problem is 𝐿=^∞_𝑡=0𝛽^𝑡{𝑢[(𝐶_𝑡−𝛾𝐻_𝑡)𝑁]+𝜆_𝑡[𝐾_𝑡+𝑌_𝑡−𝐶_𝑡−𝐺_𝑡−𝐾_𝑡+1]+𝜇_𝑡[(1−𝜌)𝐻_𝑡+𝜌𝐶_𝑡−𝐻_𝑡+1]}, (4.5) where 𝜆_𝑡 and 𝜇_𝑡 are the shadow values of capital and habits stocks, respectively. The first-order conditions with respect to 𝐶_𝑡, 𝐺_𝑡, 𝐾_𝑡+1, and 𝐻_𝑡+1 are given by𝐶_𝑡∶𝑢^′(𝐶_𝑡−𝛾𝐻_𝑡)𝑁^𝜀−1+𝜌𝜇_𝑡=𝜆_𝑡𝐺_𝑡,∶𝜆_𝑡((1−𝛼)𝑌_𝑡𝐺_𝑡𝐾−1)=0,_𝑡+1∶𝛽(1+𝛼𝑌_𝑡𝐾_𝑡)𝜆_𝑡+1=𝜆_𝑡𝐻_𝑡+1,∶𝛽(1−𝜌)𝜇_𝑡+1−𝛽𝛾𝑢^′(𝐶_𝑡+1−𝛾𝐻_𝑡+1)𝑁^𝜀−1=𝜇_𝑡, (4.6)(4.7)(4.8)(4.9) where it has been used that 𝑢^′ is homogeneous of degree−𝜀 in (4.6) and (4.9), together with the transversality conditionlim_𝑡→∞𝛽^𝑡𝜆_𝑡𝐾_𝑡+1=lim_𝑡→∞𝛽^𝑡𝜇_𝑡𝐻_𝑡+1=0. (4.10)

Let 𝜋_𝑡≡𝐺_𝑡/𝑌_𝑡 denote the ratio of public expenditure to output. Equation (4.7) yields the standard rule that the optimal ratio of public expenditure to output is the (constant) elasticity of public services in production,𝜋_𝑡=𝐺𝜋=^𝑡𝑌_𝑡=1−𝛼. (4.11) Substituting (4.11) into (4.1), we obtain the following expression for aggregate output:𝑌_𝑡=(1−𝛼)^{(1−𝛼)/𝛼}𝐴^1/𝛼𝑁^{𝜎(1−𝛼)/𝛼}𝐾_𝑡=𝐵𝐾_𝑡, (4.12) where 𝐵≡(1−𝛼)^{(1−𝛼)/𝛼}𝐴^1/𝛼𝑁^{𝜎(1−𝛼)/𝛼}.

Let 𝑔 denote the long-run growth rate, and let 𝑞_𝑡≡𝜇_𝑡/𝜆_𝑡 denote the relative shadow cost of habits. Proceeding in a similar manner as in the market economy, taking into account (4.11), we can derive the system that drives the dynamics of the centralized economy in terms of the variables 𝑍_𝑡≡𝑍_𝑡(1+𝑔)^−𝑡, 𝐾_𝑡≡𝐾_𝑡(1+𝑔)^−𝑡, 𝐻_𝑡≡𝐻_𝑡(1+𝑔)^−𝑡, and 𝑞_𝑡, which are constant along a BGP(1+𝑔)𝑍_𝑡+1=[1−𝜌𝑞_𝑡1−𝜌𝑞_𝑡+1𝛽(1+𝛼𝐵)]^1/𝜀𝑍_𝑡,(1+𝑔)𝐾_𝑡+1=(1+𝛼𝐵)𝐾_𝑡−𝑍_𝑡−𝛾𝐻_𝑡,(1+𝑔)𝐻_𝑡+1=𝐻_𝑡+𝜌(𝑍_𝑡−(1−𝛾)𝐻_𝑡),𝑞_𝑡+1=(1+𝛼𝐵)𝑞_𝑡+𝛾. (4.13)(4.14)(4.15)(4.16)

The following proposition, which is proved in Appendix C, yields a closed-form solution for the dynamics of the efficient (first-best) solution.

Proposition 4.1. The socially-planned economy has a unique feasible optimal solution with positive long-run growth which is described by 𝐻_𝑡−𝐻_∞=(1−𝜌(1−𝛾)1+𝑔)^𝑡(𝐻₀−𝐻_∞),𝐶_𝑡−𝐶_∞=𝛾(𝐻_𝑡−𝐻_∞),𝐾_𝑡−𝐾_∞=𝛾𝛼(𝐵+𝜌(1−𝛾)𝐻_𝑡−𝐻_∞), (4.17)(4.18)(4.19) where the (constant) ratio of public expenditure to output is 𝜋_𝑡=𝐺_𝑡𝑌_𝑡=1−𝛼, (4.20) which converges to a BGP given by 𝐶_∞=(𝑔+𝜌)𝑍_∞,𝑔+𝜌(1−𝛾)𝐾_∞=(𝑔+𝜌)𝑍_∞(𝛼𝐵−𝑔)(𝑔+𝜌(1−𝛾)),𝐻_∞=𝜌𝑍_∞,𝑔+𝜌(1−𝛾) (4.21)(4.22)(4.23) where 𝑍_∞=(𝛼𝐵−𝑔)[𝐾₀−𝛾(𝜌𝐾₀+𝐻₀)𝛼𝐵+𝜌], (4.24) and the long-run growth rate is 𝑔=[𝛽(1+𝛼𝐵)]^1/𝜀−1 (4.25) if and only if parameter values and initial conditions are that 𝑔>0>𝛽(1+𝑔)^1−𝜀−1,𝛾𝐻₀𝐾₀<𝛼𝐵+𝜌(1−𝛾). (4.26)(4.27)

The efficient paths of aggregate consumption, capital, and habits stock can be easily computed as 𝐶_𝑡=𝐶_𝑡(1+𝑔)^𝑡, 𝐾_𝑡=𝐾_𝑡(1+𝑔)^𝑡, and 𝐻_𝑡=𝐻_𝑡(1+𝑔)^𝑡. In particular, the initial value of consumption is given by 𝐶₀=𝐶₀=𝑍_∞+𝛾𝐻₀.

As in the Barro [1] model, the share of output claimed to finance public services that maximizes welfare is equal to the elasticity of public services in production. The intuition is simple. On one hand, an increase in public expenditure raises the marginal product of capital and this has positive welfare effects. On the other hand, an increase in public expenditure also decreases the amount of output available for consumption. The optimal size of government balances off these two effects.

If 𝛾≠0, the convergence speed of 𝐾, 𝐶, and 𝐻 towards their respective BGPs in the socially-planned economy can be easily shown to be constant and equal to the one in the decentralized economy, Φ=Φ=𝜌(1−𝛾). The case 𝛾=0 corresponds to the standard time-separable Barro [1] model, in which the economy displays no transitional dynamics: at the initial time, consumption jumps to its balanced growth path, 𝐶₀=𝐶_∞=𝑍_∞=(𝛼𝐵−𝑔)𝐾₀, and thereafter consumption and capital both increase at a constant rate 𝑔.

4.2. First-Best versus Second-Best Solution

In the second-best solution, the income tax rate is set according to (3.22) at 𝜏=1−𝛼, and so, 𝐵=𝐵. Comparing the transition paths of 𝐻_𝑡, 𝐶_𝑡, and 𝐾_𝑡 in the market economy given by (3.7)–(3.9) when 𝜏=𝜏=1−𝛼 with their counterparts 𝐻_𝑡, 𝐶_𝑡, and 𝐾_𝑡 in the centrally-planned economy given by (4.17)–(4.19), it is immediate to see that they coincide if and only if the long-run growth rates in the market and the centralized economies coincide, 𝑔=𝑔. Comparing (3.14) with (4.11), we observe that the optimal and equilibrium long-run growth rates are related through 𝑔≤𝑔, with equality if and only if 𝜎=0. Thus, the first-best and the second-best solutions coincide if and only if there is proportional congestion associated to public expenditure. With proportional congestion, 𝜎=0, setting the income tax rate according to (3.22) ensures that the social marginal return to capital, as viewed by the central planner, 𝛼𝐵, coincides with the private after-tax return, as viewed by the individual agent, (1−𝜏)𝐵. Intuitively, the agent ignores the negative externality caused by congestion and overaccumulates capital relative to the optimum. Thus, an income tax can restore efficiency by increasing the effective marginal product of capital, which discourages capital accumulation.

The dynamics of consumption in the first-best and second-best solutions can be easily compared by using the closed-form expressions derived in Propositions 3.1 and 4.1. Comparison of (4.24) and (3.13) when 𝜏=𝜏=1−𝛼, and so, 𝐵=𝐵, reveals that 𝑍_∞𝑍_∞=(𝛼𝐵−𝑔)(𝛼𝐵−𝑔)≤1, (4.28) with equality if and only if 𝑔=𝑔. Hence, the initial value of consumption in the second-best solution, 𝐶₀, is greater than that in the centrally-planned economy, 𝐶₀. However, as the optimal long-run growth rate of consumption is higher than the equilibrium one, consumption in the first-best solution eventually catches up the corresponding one in the second-best solution.

5. Some Numerical Results

This section illustrates the implications for the dynamics of the economy of introducing habits in the Barro [1] model. To this end, we assume that the instantaneous utility function is given by𝑢(𝑐_𝑡,ℎ_𝑡)=(𝑐_𝑡−𝛾ℎ_𝑡)^1−𝜀−11−𝜀if𝜀≠1,ln(𝑐_𝑡−𝛾ℎ_𝑡)if𝜀=1, (5.1) where 𝜀 is the relative risk aversion.

The baseline parameterization is shown in Table 1. The values of the relative risk aversion, 𝜀, and the time discount rate, 𝛽, are standard. As usual in AK-type models, the stock of capital, 𝐾_𝑡, comprises both physical and human capital, which explains the value of 𝛼=0.9. The tax rate on income is set at its (second-best) optimal value. As a benchmark, we assume that there is no congestion, 𝜎=1. Given these parameter values, the value of 𝐵 is chosen so that the long-run growth rate in the market economy, 𝑔, is 2 percent. For example, normalizing 𝑁=1, this entails a value of 𝐴=0.1832. This calibration yields a plausible share of consumption in output of 72.8 percent. In the baseline, we set the strength of habits in utility to 𝛾=0.9, following empirical work suggesting a value of 𝛾 near unity. The value 𝜌=0.25 is then chosen so as to obtain a realistic rate of convergence, Φ, of 2.5 percent. Notwithstanding, we will illustrate the effect of assuming different values of 𝜌 and 𝛾 on the equilibrium dynamics.

Table 1 

Parameter and steady-state values.

We perform an exercise similar to that in Carroll et al. [49], and consider the effect of a 10 percent unanticipated reduction in the capital stock of an economy that was initially on its balanced growth path. It should be noted that this shock does not affect the long-run growth rate. Figure 1(a) illustrates the time paths of consumption, capital and habits relative to their respective balanced growth path values. At 𝑡=0, the capital stock experiments a reduction of 10 percent. Hence, income also falls a 10 percent at the outset, which causes a reduction in consumption. The percentage reduction in consumption is lower than the one in capital in order to keep a flat shape of adjusted-consumption. Hereafter, 𝐾_𝑡, 𝐶_𝑡, and, consequently, 𝐻_𝑡 decrease steadily towards their respective steady-state values, which are a 28.3 percent lower than the initial ones. Figure 1(b) depicts the evolution of the growth rates of aggregate consumption, capital, and habits. At the initial time, the growth rates fall below their common preshock stationary value. Subsequently, they increase monotonically towards their common steady sate. Note that in the standard Barro [1] model, the economy would jump immediately to its new steady state, while the long-run growth rate would keep constant at 2 percent.

(a) Time paths of 𝐾𝑡, 𝐶𝑡, and 𝐻𝑡.

(b) Growth rates of 𝐶𝑡, 𝐾𝑡, and 𝐻𝑡.

Figure 1 

Transition dynamics after a 10% reduction in capital.

Figure 2 depicts the growth rate of consumption for different values of 𝜌 and 𝛾. As (3.14) shows, variations of these parameters do not affect the long-run growth rate of the economy. The behaviour of the growth rate of consumption reflects the fact that the convergence speed, Φ=𝜌(1−𝛾), decreases as 𝜌 decreases and/or 𝛾 increases.

(a) Effect of 𝜌.

(b) Effect of 𝛾.

Figure 2 

Growth rate of consumption after a 10% reduction in capital.

Figure 3 compares the transition dynamics of the growth rates of consumption and capital in the first-best and second-best outcomes. We assume that the economy was initially at the BGP of the decentralized economy, and at time 𝑡=0, it experiments a shock that reduces its capital stock a 10 percent. Figure 3 depicts the time paths of the growth rates in the socially-planned and the decentralized economies. Note that in the standard time-separable Barro [1] model the adjustment would be instantaneous; that is, the equilibrium growth rates of the market economy would keep constant at their common value 𝑔, and the optimal growth rates of the socially-planned economy would jump at the initial time to their new common value, 𝑔.

Figure 3 

Growth rates of capital and consumption after a 10% reduction in capital: First-best versus second-best solution.

6. Conclusion

This paper develops an endogenous growth model with habit-persistent preferences and productive public services. Government expenditure, which may be subject to congestion, is financed by an income tax. Unlike the standard time-separable Barro [1] model, the introduction of habit formation makes the model to feature transitional dynamics, which are solved in closed form. As in the Barro [1] model, within a decentralized economy setup, the growth- and welfare-maximizing (in a second-best sense), income tax rates coincide and are equal to the elasticity of public services in production. This is also the optimal share of government expenditure on income in a first-best setting. The first-best solution can be decentralized as a market equilibrium only in the presence of proportional congestion.

In this paper, we have considered that it is the current flow of public investment which is productive. Another strand of the literature assumes instead that the accumulated stock, rather than the current flow, is the source of contribution to productive capacity. One interesting extension would be to analyze a model with public capital or with a mix of productive capital and public expenditure. Another valuable extension would be to consider a more general parameterization of technology, such as CES production. These topics will be the subject of future research.

Appendices

A. Derivation of the Dynamic System (3.3)–(3.6)

Since 𝑢^′ is homogeneous of degree −𝜀, and using the fact that 𝑐_𝑡=𝑐_𝑡, (2.9) can be rewritten as𝑢^′(𝑐_𝑡−𝛾ℎ_𝑡)=𝑢^′(𝑍_𝑡𝑁)=𝑁^𝜀𝑍_𝑡^−𝜀𝑢^′(1)=𝜆_𝑡(1−𝜌𝜙𝑞_𝑡). (A.1)

Given that (3.1) entails that 𝑦_𝑡/𝑘_𝑡=𝐵, (2.10) becomes𝜆_𝑡+1=𝜆_𝑡𝛽[1+(1−𝜏)(1−𝜎(1−𝛼))𝐵]. (A.2)

Now, evaluating (A.1) at 𝑡=𝑡+1, dividing the result by (A.1), and using (A.2) to eliminate 𝜆_𝑡+1/𝜆_𝑡 from the ensuing expression, we get(𝑍_𝑡+1𝑍_𝑡)^−𝜀=𝜆_𝑡+1(1−𝜌𝜙𝑞_𝑡+1)𝜆_𝑡(1−𝜌𝜙𝑞_𝑡)=(1−𝜌𝜙𝑞_𝑡+1)/(1−𝜌𝜙𝑞_𝑡).𝛽[1+(1−𝜏)(1−𝜎(1−𝛼))𝐵] (A.3)

Using (A.1) evaluated at 𝑡=𝑡+1 to eliminate 𝑢^′(𝑐_𝑡+1−𝛾ℎ_𝑡+1) from (2.11), we get𝛽(1−𝜌)𝜇_𝑡+1−𝛽𝛾𝜆_𝑡+1(1−𝜌𝜙𝑞_𝑡+1)=𝜇_𝑡. (A.4)

Now, (3.3) follows immediately from (A.3). Equations (3.4) and (3.5) result from aggregating the agent's budget constraint (2.7) and the habits stock accumulation law (2.2), respectively, over the 𝑁 identical individuals and multiplying both sides of the resulting expressions by (1+𝑔)^−𝑡. Equation (3.6) can be obtained after dividing (A.4) by (A.2) and solving out for 𝑞_𝑡+1 in the ensuing expression.

B. Proof of Proposition 3.1

Firstly, the long-run growth rate (3.14) can be immediately derived from (3.3), using that 𝑞_𝑡, and 𝑍_𝑡 are constant at the BGP. Taking into account the homogeneity of degree −𝜀 of 𝑢^′, the transversality condition (2.12) is equivalent to 𝛽(1+𝑔)^1−𝜀<1, which combined with the requirement of positive growth yields (3.15). Note that, if 𝑔>0, the transversality condition is automatically satisfied if 𝜀≥1.

Equation (3.6) is a linear difference equation that depends solely on 𝑞_𝑡, which is a jump variable. Its steady state, 𝑞_∞𝛾=−,𝜌(1−𝛾𝜙)+(1−𝜏)(1−𝜎(1−𝛼))𝐵 (B.1) is unstable because 𝑑𝑞_𝑡+1𝑑𝑞_𝑡=1+(1−𝜏)(1−𝜎(1−𝛼))𝐵1−𝜌(1−𝛾𝜙)>1. (B.2) Hence, 𝑞_𝑡 displays no transition dynamics: it jumps at the initial time to its stationary value 𝑞_∞. Substituting 𝑞_𝑡+1=𝑞_𝑡 into (3.3), and using (3.14), we get that 𝑍_𝑡+1=𝑍_𝑡=𝑍_∞ at any time, where 𝑍_∞=𝑍₀ denotes the stationary (and the initial) value of 𝑍_𝑡. Now, the solution to the linear difference equation (3.5) with initial value 𝐻₀=𝐻₀ is𝐻_𝑡=𝜌𝑍_∞𝑔+𝜌(1−𝛾)+[𝐻₀−𝜌𝑍_∞𝑔+𝜌(1−𝛾)](1−𝜌(1−𝛾)𝑔)1+^𝑡, (B.3) and so,𝐶_𝑡=𝑍_𝑡+𝛾𝐻_𝑡=(𝑔+𝜌)𝑍_∞𝑔+𝜌(1−𝛾)+𝛾[𝐻₀−𝜌𝑍_∞𝑔+𝜌(1−𝛾)](1−𝜌(1−𝛾)𝑔)1+^𝑡. (B.4)

Substituting 𝐻_𝑡 from (B.3) and 𝑍_𝑡=𝑍_∞ into (3.4), the solution to the resulting difference equation is 𝐾_𝑡=(𝑔+𝜌)𝑍_∞((1−𝜏)𝐵−𝑔)(𝑔+𝜌(1−𝛾))+Ω(1+(1−𝜏)𝐵𝑔)1+^𝑡+𝛾(1−𝜏)𝐵+𝜌(1−𝛾)[𝐻₀−𝜌𝑍_∞𝑔+𝜌(1−𝛾)](1−𝜌(1−𝛾)𝑔)1+^𝑡, (B.5) where Ω is a constant to be determined. As condition (3.15) implies that (1−𝜏)𝐵−𝑔>0, convergence of 𝐾_𝑡 to its stationary value 𝐾_∞ as 𝑡 tends to infinity requires that Ω=0 and, therefore,𝐾_𝑡=(𝑔+𝜌)𝑍_∞+𝛾((1−𝜏)𝐵−𝑔)(𝑔+𝜌(1−𝛾))(1−𝜏)𝐵+𝜌(1−𝛾)[𝐻₀−𝜌𝑍_∞𝑔+𝜌(1−𝛾)](1−𝜌(1−𝛾)𝑔)1+^𝑡. (B.6)

The stationary value 𝑍_∞ in (3.13) can now be derived by evaluating (B.6) at 𝑡=0, taking into account that 𝐾₀=𝐾₀. From (3.13), the feasibility condition 𝑍₀=𝑍_∞>0 is satisfied if and only if condition (3.16) holds, which also ensures that the equilibrium paths are feasible. The BGP of the market economy given by (3.10), (3.11), and (3.12) can be easily computed by computing the limit as 𝑡 goes to infinity of expressions (B.4), (B.6), and (B.3), respectively. Substituting the BGP values into (B.4), (B.6), and (B.3), we can readily obtain (3.8), (3.9), and (3.7).

C. Proof of Proposition 4.1

The proof is similar to that in Proposition 3.1. Firstly, the long-run growth rate (4.25) can be immediately obtained from (4.13), using that 𝑞_𝑡 and 𝑍_𝑡 are constant at the BGP. Taking into account the homogeneity of degree −𝜀 of 𝑢^′, the transversality condition (4.10) is equivalent to 𝛽(1+𝑔)^1−𝜀<1, which combined with the requirement of positive growth yields (4.26).

Equation (4.16) is a linear difference equation that depends solely on 𝑞_𝑡, a jump variable. Its steady state, 𝑞_∞𝛾=−𝜌(1−𝛾)+𝛼𝐵, (C.1) is unstable because 𝑑𝑞_𝑡+1𝑑𝑞_𝑡=1+𝛼𝐵1−𝜌(1−𝛾)>1. (C.2) Hence, 𝑞_𝑡 displays no transition dynamics: it jumps at the initial time to its stationary value 𝑞_∞. Substituting 𝑞_𝑡+1=𝑞_𝑡 into (4.13), we get 𝑍_𝑡+1=𝑍_𝑡=𝑍_∞ at any time, where 𝑍_∞=𝑍₀ denotes the stationary (and the initial) value of 𝑍_𝑡. Now, the solution to the linear difference equation (4.15) with initial value 𝐻₀=𝐻₀ is𝐻_𝑡=𝜌𝑍_∞𝑔+𝜌(1−𝛾)+[𝐻₀−𝜌𝑍_∞𝑔+𝜌(1−𝛾)](1−𝜌(1−𝛾)1+𝑔)^𝑡, (C.3) and so,𝐶_𝑡=𝑍_𝑡+𝛾𝐻_𝑡=(𝑔+𝜌)𝑍_∞𝑔+𝜌(1−𝛾)+𝛾[𝐻₀−𝜌𝑍_∞𝑔+𝜌(1−𝛾)](1−𝜌(1−𝛾)1+𝑔)^𝑡. (C.4)

As in Proposition 3.1, since condition (4.26) entails that 𝛼𝐵−𝑔>0, the solution to (4.14) with terminal condition lim_𝑡→∞𝐾_𝑡=𝐾_∞ is𝐾_𝑡=(𝑔+𝜌)𝑍_∞(𝛼𝐵−𝑔)(+𝛾𝑔+𝜌(1−𝛾))𝛼𝐵+𝜌(1−𝛾)[𝐻₀−𝜌𝑍_∞𝑔+𝜌(1−𝛾)](1−𝜌(1−𝛾)1+𝑔)^𝑡. (C.5)

The stationary value 𝑍_∞ in (4.24) can now be derived from making 𝐾₀=𝐾₀ in (C.5). The feasibility condition 𝑍₀=𝑍_∞>0 is satisfied if and only if the initial values of capital and habits stock are such that condition (4.27) holds, which also ensures that the optimal growth paths are feasible. The BGP of the socially-planned economy given by (4.21), (4.22), and (4.23) can be easily computed by computing the limit as 𝑡 goes to infinity of expressions (C.4), (C.5), and (C.3), respectively. Substituting these BGP values into (C.4), (C.5), and (C.3), we can readily obtain (4.18), (4.19), and (4.17).

D. Second-Best Solution

This appendix shows that 𝑑𝑈/𝑑𝑔>0. Using that 𝑢^′ is homogeneous of degree −𝜀, and taking into account the condition (3.15), we get that𝑑𝑈𝑑𝑔=1𝑁^∞_𝑡=0(𝑑𝑍_∞𝑑𝑔(1+𝑔)^𝑡+𝑍_∞(1+𝑔)^𝑡−1𝑡)𝑢^′[𝑍_∞(1+𝑔)^𝑡𝑁]𝛽^𝑡=𝑁^𝜀−1𝑢^′(1)𝑍^𝜀_∞^∞_𝑡=0(𝑑𝑍_∞𝑑𝑔(1+𝑔)^{−(𝜀−1)𝑡}+𝑍_∞(1+𝑔)^{−(𝜀−1)𝑡−1}𝑡)𝛽^𝑡=𝑁^𝜀−1𝑢^′(1)𝑍^𝜀_∞(1−𝛽(1+𝑔)^1−𝜀)[𝑑𝑍_∞𝑑𝑔+𝛽(1+𝑔)^−𝜀𝑍_∞1−𝛽(1+𝑔)^1−𝜀]. (D.1) Using (3.20) to substitute (1−𝜏)𝐵 as a function of 𝑔 into (3.13), we can get that 𝑑𝑍_∞𝑑𝑔=𝜀(1+𝑔)^𝜀−1𝛾(𝜌𝐾₀+𝐻₀)((1−𝜏)𝐵−𝑔)𝛽(1−𝜎(1−𝛼))((1−𝜏)𝐵+𝜌)²+𝜀(1+𝑔)^𝜀−1−𝛽(1−𝜎(1−𝛼))𝑍𝛽(1−𝜎(1−𝛼))((1−𝜏)𝐵−𝑔)^∞, (D.2) which substituted into (D.1) yields, after simplification, 𝑑𝑈𝑑𝑔=𝑁^𝜀−1𝑢^′(1)𝑍^𝜀_∞(1−𝛽(1+𝑔)^1−𝜀){((1−𝜏)𝐵−𝑔)𝜀(1+𝑔)^𝜀−1𝛾(𝜌𝐾₀+𝐻₀)𝛽(1−𝜎(1−𝛼))((1−𝜏)𝐵+𝜌)²+[((1+𝑔)^𝜀−1−𝛽)𝜀+(1−𝛼)𝛽(1−𝛽(1+𝑔)^−𝜀)𝜎]((1−𝜏)𝐵−𝑔)(1−𝛽(1+𝑔)^1−𝜀𝑍)𝛽(1−𝜎(1−𝛼))^∞}>0, (D.3) where we have used condition (3.15) to determine its sign.

Acknowledgment

Financial support from the Spanish Ministry of Science and Innovation through Grant ECO2008-04180 is gratefully acknowledged.