Do Habits Generate Endogenous Fluctuations?[(]
Hyun Park[†]
Kyung Hee University
April 22, 2010
Abstract
This article examines endogenous fluctuations under habit persistence in preferences using otherwise a standard one-sector endogenous growth model. I show that indeterminacy of the balanced growth path, in conjunction with a continuum of competitive transitional equilibrium paths, exists when a household’s preferences are characterized by habit persistence in a growing competitive economy with exogenous fiscal policies, namely, income taxes and productive public spending. Indeterminacy also emerges in the socially optimal second-best allocation even when the government intends to internalize habit externalities, including jealousy and admiration, by introducing optimal time-variant income taxes and public capital services. Therefore, in the presence of multiple competitive equilibria and socially optimal allocations and in the absence of continuous exogenous stochastic shocks, self-fulfilling beliefs drive endogenous business cycles in these growing economies.
Keywords: habit formation; indeterminacy; endogenous business cycles; fiscal policies
JEL classification: D91, E62, H21, O41
I. INTRODUCTION
Several recent studies in the business cycles literature explore the implications of habit persistence in preferences not only for equity premiums but also for volatilities related to consumption, investment, output, and employment (Abel, 1990; Campbell and Cochrane, 1999; Constantinides, 1990). Wide-ranging general equilibrium analysis under habit formation becomes an important instrument to acquire their consistency with joint behaviors of asset prices and fluctuations in quantities (Boldrin, Christiano, and Fisher, 2001; Jermann, 1988; Lettau and Uhlig, 2000). Due to the presence of habits, some government interventions take into account the nature of consumption externalities, for example, jealousy or admiration of others. This article examines the role of government policies to determine whether such policies can mitigate business cycles fluctuations (Ljungqvist and Uhlig, 2000; Fuhrer, 2000).
Specifically, I examine the role of habit persistence on aggregate endogenous fluctuations in a simple general equilibrium model with fiscal government policies.[1] In contrast to the traditional arguments that habit stocks and fiscal policies serve as stabilizers,[2] this study posits that external habits can cause indeterminacy. Namely, self-fulfilling aggregate fluctuations can emerge, and an economy can become destabilized in the presence of habits. More specifically, aggregate instability arises in a growing economy where habit persistence in preferences is characterized as either jealousy or admiration and a government provides public capital services for persistent long-run economic growth. In other words, a competitive equilibrium path can be indeterminate, thereby allowing the emergence of sunspot equilibrium paths. The indeterminacy prevails even when fiscal policies are chosen to be socially optimum in a second-best situation. Hence, in the absence of any exogenous stochastic shocks to fundamentals, this sunspot model can generate business cycle fluctuations, which are, in fact, the result of rational expectations. This model suggests an additional reason why habit persistence in preferences can destabilize a growing economy even in the absence of exogenous fluctuations in the determinants of the economic environment, including consumers’ tastes and the state of technology. Endogenous fluctuations, not real business cycles, emerge in a decentralized competitive allocation with exogenous fiscal policies and in a socially optimal allocation with second-best fiscal policies in its planning economy of agents with perfect foresight.
My analysis is based on a dynamic general equilibrium model à la Ramsey, incorporated with habit persistence in preferences and augmented by productive government spending. This model maintains the basic framework of previous business cycles studies (see, e.g., Boldrin, Christiano, and Fisher, 2001; King, Plosser, and Rebelo, 1988). To investigate endogenous aggregate instability in general equilibrium, the model incorporates the following specifications. First, the model contains no exogenous stochastic shock to the economy. Second, all agents are endowed with perfect foresight on the infinite horizon. Third, habit stocks are assumed to be an average of aggregate consumption in the economy. That is, habit stocks are external. Fourth, habit persistence in preferences encompasses both jealousy and admiration and thus affects the consumer’s felicity negatively and positively, respectively. Fifth, habit stocks are integrated with both catching-up-with-the-Joneses and keeping-up-with-the-Joneses preferences. Sixth, the model incorporates productive government expenditures.[3] In other words, public capital services are neither exogenously wasteful nor nondistortive lump-sum transfers.[4] Finally, I consider two alternative fiscal policies in a growing economy: one with exogenous fiscal policies and the other with second-best optimal fiscal policies including income taxes and public capital services over periods.
The main finding of this study is that habit persistence in preferences is a primary source of indeterminacy and thus endogenous business cycles. Local indeterminacy emerges under preferences with external habit stocks characterized as either jealousy or admiration in a growing economy with productive government spending. In the context of a decentralized competitive economy, indeterminacy emerges when the presence of habits plays an equivalent role of consumption externalities; government spending to individual firms can maintain their total factor productivities in the long run. That is, external habit stocks are consumption externalities, thereby causing a market failure in a growing economy where public capital services operate as an engine of long-run economic growth. In the corresponding planning economy, indeterminacy also arises because second-best fiscal policy instruments are not sufficient to internalize the distortive effects of consumption externalities. These policy instruments, including optimal income taxes, fail to alleviate the market failures in the presence of externalities and distortionary tax and spending policies. As a result, aggregate instability and endogenous fluctuations are possible even in the social planning economy. Therefore, aggregate instability in this study is based on the emergence of indeterminacy, and endogenous business cycles arise in the absence of any exogenous stochastic shocks to fundamentals.[5]
Indeterminacy is of a continuum of equilibrium paths for any initial stocks of habits and private capital, each of which is consistent with the convergence to the same balanced growth path. It is well known in the literature that indeterminate balanced growth paths are associated with rational expectations models in which a continuum of self-fulfilling equilibrium paths exists.[6] In contrary to real business cycles models, these equilibrium paths with rational expectations are able to generate business cycle fluctuations in the absence of any shocks to fundamentals. More precisely, an absolutely stable long-run growth path is associated with rational expectations models in which many sets of self-fulfilling beliefs exist, each of which is consistent with a dynamic equilibrium that converges to the same growth rate along its asymptotic balanced growth path but not to the same level of consumption, capital, and welfare. Hence, within rational expectations, these equilibrium paths appear as aggregate instability.
The emergence of indeterminacy is rather intuitive in a growing economy with habit preferences: Consider starting from an equilibrium path, and the agents believe that another equilibrium also exists in which the shadow price of investment is higher than its current price and the future returns support a higher level of investment. This higher shadow price reduces the current consumption and diverts the agent’s income to investment. In a growing economy of public capital services, an increase in the private assets, ceteris paribus, increases or maintains its future returns in both the competitive equilibrium and socially optimal allocation. Under habit persistence in preferences, the shadow prices of investment and consumption are constant in a competitive economy and decline in a social planning economy—either with admiration as positive consumption externalities or jealousy as negative consumption externalities—and, then, eventually returns to its steady state value of the private assets. Hence, the agents’ belief on the existence of another equilibrium path is self-fulfilled.[7]
Despite the breadth of existing literature incorporating the topic of habits, the role of habits as a significant factor in the emergence of indeterminacy, and the associated implications of aggregate instability have not been explored. Previous studies in the literature on indeterminacy and endogenous fluctuations, however, have shown that the inclusion of consumption–production externalities introduces a potential source for the existence of indeterminate balanced growth paths. Among selective examples for indeterminacy, Boldrin and Rustichini (1994) and Benhabib and Farmer (1994, 1999) included increasing returns and market imperfections either through production externalities or monopolistic competition in a one-sector growth model. In multisector growth models with an imperfectly competitive market, human and physical capital externalities continue to be important sources of indeterminacy (see, e.g., Benhabib and Perli, 1994). Cazzavella (1996) and Park and Philippopoulos (2004) showed indeterminacy when the fiscal policy provides consumptive spending externalities. Their conditions for indeterminacy, in the presence of capital–labor–consumption externalities, are closely related with conditions in both decentralized competitive and social planning economies as in this study, which uses stocks of habit externalities. Schmitt-Grohe and Uribe (1997), Farmer and Guo (1996), and Park (2009) showed that governments with distortive fiscal policies, including balanced budget constraints, can generate indeterminacy when they exhibit fiscal increasing returns. Similarly, I posit that a coordination failure of government policies under habit persistence in preferences is sufficient for indeterminacy of socially optimal second-best allocations.[8] However, contrary to the main features of my model, most previous studies rely on the presence of consumption–production externalities and increasing returns technologies for indeterminacy,[9] and none explicitly consider endogenous business fluctuations in reference to preferences for habits and AK–technology in a growing economy.
The remainder of the article is as follows. Section 2 presents the competitive economy with habit stocks and fiscal policies. Section 3 characterizes a dynamic competitive equilibrium and shows the conditions for indeterminacy. Section 4 examines socially optimal allocation fiscal policies and finds existence of the unique balanced growth path. Section 5 shows the possibility of local indeterminacy of a social allocation with optimal fiscal policies, and Section 6 presents concluding remarks.
II. THE COMPETITIVE ECONOMY WITH EXOGENOUS FISCAL POLICIES
I introduce an endogenously growing economy with a private sector and a government sector. The private sector consists of a representative household and a representative firm. All representative agents are endowed with perfect foresight and act competitively in the economy with no uncertainty. The household consumes, supplies labor inelastically, and rents out its assets. The firm produces output competitively by using labor and private capital. Both representative agents have an infinite life. The economy incorporates habit persistence in preferences and fiscal policies including public capital services. That is, the representative household’s preferences are characterized as habit persistence (e.g., Abel, 1990; Ljungqvist and Uhlig, 2000), and the representative firm’s total factor productivities increase in the public capital services (e.g. Barro, 1990; Rebelo, 1991). The government imposes linear taxes on the final output to provide public capital services.[10] Capital does not depreciate, and the population does not grow. Time is continuous on an infinite horizon.
I consider a decentralized competitive economy, in which habit stocks are external to a representative household, and tax rates and government spending are exogenous. The household maximizes intertemporal utility over time, :
, (1)
where is private consumption, is external habit stocks, and the parameter is the rate of time preference; the felicity function is increasing and concave in , twice continuously differentiable in , and satisfies a constant elasticity of intertemporal substitution; the Inada condition in ; and the felicity function is differentiable, increasing, and concave in when an external habit stock is a positive consumption externality, whereas differentiable, decreasing and convex in when an external habit stock is a negative consumption externality. For simplicity, I further assume that is logarithmic in , linear in , and additively separable between and . That is, is specified as
. (2)
This model is a special case of a ratio form of constant of elasticity substitution felicity functions.[11] Clearly, the elasticity of intertemporal substitution for is . The linear form of habit stocks ensures the concavity of the felicity function regardless of the sign of . This specification also satisfies the convexity property of the felicity function . In addition, the elasticity of intertemporal substation for is zero, where and .[12]
When is positive, the stock of habits represents negative, and when is negative, the stock of habits represents positive externalities. Following Turnovsky and Monteiro (2007), implies that an increase in reference habit stocks reduces felicity of current consumption, and implies that an increase in a reference habit stock increases felicity of current consumption. As in Dupor and Liu (2003), Alonso-Carrera, Caballe, and Raurich (2004), reduced felicity denotes jealousy and increased felicity denotes admiration of preferences with consumption spillovers.
The stock of habits is specified as , where is the average consumption in the aggregated economy;[13] is the rate of depreciation of the habit stocks; and, for some positive , is the contribution of one unit of average consumption to the formation of the habit stocks. Hence, the dynamic equation of habit stocks is:
, (3)
where a dot over a variable denotes the time derivative.[14] Along with the felicity function defined in (2), the dynamic habit equation encompasses “keeping-up-with-the-Joneses” and “catching-up-with-the-Joneses” preferences: When goes to with some positive value of , the stock of habits is equivalent to the current consumption; thus, the habit dynamics with the felicity function capture a continuous-time version of “keeping-up-with-the-Joneses” preferences. When is closed to zero with some finite positive value of , the stock of habits consists of accumulation of the past aggregate consumption; thus, the habit dynamics with the felicity function capture a continuous version of “catching-up-with-the-Joneses” preferences. Hence, the felicity function with respect with individual consumption and external habit stocks is a combination of the current value of individual consumption, the current consumption of average aggregate consumption, and the lagged value of average aggregate consumption. As in a discrete-time version, this model is a continuous-time version of interdependent preferences of external habits.[15]
The household saves in the form of assets, denoted by . Then, the household receives interest income , where is the competitive rate of returns to the assets. The household inelastically supplies one unit of labor services so that its wage income is , where is the competitive wage rate. It receives the net dividend from its ownership of the firm’s profit. Thus, the budget constraint of the household, given the initial asset , is