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This note analyzes the equilibrium dynamics in the neoclassical growth model with habit-forming preferences and elastic labor supply. Habits enter into utility in a multiplicative way. The specification of the habit formation process comprises the particular cases of internal and external habits. Existence, uniqueness and saddle-path stability of the steady state are proved analytically.

This note analyzes the equilibrium dynamics in the neoclassical growth model with habit formation and elastic labor supply. In our model utility is additively separable and CRRA in adjusted consumption and leisure, and habits enter utility in a multiplicative way. These are specifications commonly used in the literature. Specifically, we demonstrate analytically that the steady state is unique and (locally) saddle-path stable, so that the equilibrium is (locally) uniquely determined.

Habit-forming preferences have been widely incorporated to dynamic macroeconomic models. The reason is that they help to explain some empirical facts difficult to accommodate with standard time-separable preferences as, e.g., the equity premium puzzle (e.g., [1,2]), the savings-growth nexus (e.g., [

Previous work has analyzed the equilibrium dynamics of growth models with habit formation, mainly in AKtype growth models (e.g. [6-10]). However, in all these works labor supply is assumed to be inelastically provided. A notable exception is [

The remaining of the paper is organized as follows. Section 2 presents the model. Section 3 analyzes the equilibrium dynamics. Section 4 concludes.

Consider an economy populated by N identical infinitely-lived representative agents that grows at the exogenous rate. The intertemporal utility derived by the agent is

where C_{i} and H_{i} are agent’s i consumption and reference consumption level (habits stock), respectively, L_{i} is agent’s i work time, g reflects the importance of habits in utility, b is the rate of time preference, h denotes the inverse of the labor supply elasticity, and 1/e is the intertemporal elasticity of substitution of consumption in the time-separable case. The assumption that is taken from [8,14], which show that otherwise the optimization problem might not be well-defined in a similar model with inelastic labor supply.

Following [

where denotes the economy-wide average consumption. Setting f = 1 corresponds to the internal habit formation case, in which the reference stock is formed as an exponentially declining average of own past consumption. Setting f = 0 corresponds to the external habit formation case, in which the reference stock is formed as an exponentially declining average of economy-wide average past consumption. The case 0 < f < 1 corresponds to an intermediate case, in which the reference stock is formed as an exponentially declining average of own and average past consumption. The rate of adjustment of the reference stock is then

Individual output, Y_{i}, is determined by the CobbDouglas technology

where K_{i} is the individual’s capital stock. The agent’s budget constraint is

where d is the rate of depreciation of capital.

The agent chooses C_{i}, L_{i}, K_{i}, and H_{i} to maximize individual’s intertemporal utility (1) subject to her budget constraint (5) and the constraint on the accumulation of the habits stock (3). Let J be the current value Hamiltonian of the agent’s optimization problem,

The first-order conditions for an interior optimum are

plus the transversality condition

We focus on a symmetric equilibrium in which, with all agents being identical, . Hence, (6a) yields

Defining, from (7) we get

From (6b) and (8a), we find the following expression of the work time L as a function of K, C, H and q:

Differentiating (7) with respect to time, we get

From (7) and (6b), we obtain

The system that drives the dynamics of the economy is

where L is given by (9). Equation (12d) is obtained from (3), using that. Equation (12a) is obtained by substituting for from (12d), from (6c) and from (11) into (10), and using (8). Equation (12b) is the budget constraint (5). Equation (12c) is obtained by substituting for from (6c) and from (11) into.

Now, we focus on an interior steady state. An overline will denote the steady-state value of a variable. The following proposition states the existence and uniqueness of a steady state.

Proposition 1. The economy has a unique steady state

and the steady-state value of the work time is

Proof. Let. Imposing, the steady state of (12) is the solution of the system

Equation (17) entails that, which substituted into (14) yields

From (16) and (18), we obtain (13c). Now, from (18) we get

From (19) and, we have that, which substituted into (15) yields (13b). Substituting and for (13b) into (9) we get (13d). Substituting for (13d) into, using (19), we get (13a) after simplifycation. The transversality condition (6e) can be easily shown to be equivalent to. ■

The following Lemma will be used to study the stability of the steady state.

Lemma 1. Let the characteristic equation for a matrix B of order 4 ´ 4 be

.

If, the matrix B features two (stable) roots with negative real parts.

Proof. The number of roots of the characteristic equation with negative real parts (stable roots) is equal to the number of roots of the polynomial with positive real parts. Using the Routh-Hurwitz theorem (e.g., [

where and . If then, and so, we have the scheme

Hence, there are two variations in sign. If we have the configuration

where a question mark represents an unknown sign, which could be even zero. Irrespective of the unknown sign (even if it is zero), there are two variations in sign. If, we substitute for a positive constant e than tends to zero, and we obtain the following configuration

Since the sign of the entry to the left of the zero is different to that to the right of it, this indicates a change of sign, and so, there are two variations in sign. Hence, in any case there are two variations in sign, and so, B has two (stable) roots with negative real parts. ■

The following proposition establishes the saddle-path stability of the steady state.

Proposition 2. The steady state of the economy described by (13a)-(13c) is locally saddle-path stable.

Proof. Linearizing (12) around its steady state (13) we obtain

where

with, , and .

The characteristic equation for the matrix B is

where p_{3} is the opposite of the trace of B,; p_{2} is the sum of all the leading principal minors of order 2 of B; p_{1} is the opposite of the sum of all the leading principal minors of order 3 of B, and p_{0} is the determinant of B,. It can be proved by direct computation that

Using Lemma 1, the matrix B has two stable roots. Since the system (12) features two predetermined variables, K and H, the number of stable roots is equal to the number of predetermined variables. Hence, the steady state is locally saddle-path stable. ■

In accordance with the results reported in [

This paper has analyzed the equilibrium dynamics of the neoclassical growth model with multiplicative habits and elastic labor supply. The specification of habit formation comprises the particular cases of internal and external habits. Uniqueness and saddle-path stability of the steady state is proved analytically. The stability analysis shows that the transitional dynamics of the model is represented by a two-dimensional stable saddle-path. This provides a much richer dynamics for the transition paths relative to the standard neoclassical growth model without habits (e.g., [

In this paper we have assumed that leisure and adjusted consumption are additively separable in utility, and that habits enter utility in a multiplicative way. Interesting extensions would be to analyze whether the saddle-point stability result is robust with respect to a non-separable specification of adjusted-consumption and leisure, and with respect to habits entering utility in an subtractive way (e.g., [

The author wishes to thank an anonymous referee for useful comments. Financial support from the Spanish Ministry of Science and Innovation through Grant ECO2011- 25490 is gratefully acknowledged.