Capital externalities in OLG economies

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Abstract

The recent literature has stressed that externalities, however small, may lead to indeterminacy and endogenous fluctuations while, on the contrary, intertemporal substitution in consumption leads to local uniqueness. This paper introduces increasing returns, through aggregate capital externalities, into the overlapping generations model with endogenous labor and consumption in both periods of life. We show that local determinacy of the steady state prevails, when externalities are arbitrarily small, as long as the fraction of young-age consumption out of wage income is large enough. Conversely, local indeterminacy with small externalities requires both labor supply to be close to indivisible and irrealistic values of the propensity to save out of the wage income. More surprising is the fact that increasing the size of externalities indeed reduces the range of values of the consumption-to-wage ratio associated with multiple equilibria, because of two conflicting effects on savings that operate through wage and interest rate.

Introduction

It is well known that endogenous labor supply coupled with aggregate externalities may lead to local indeterminacy, sunspots and endogenous cycles in a variety of dynamic models. This is, for instance, the case in the Ramsey model, as modified by Benhabib and Farmer (1994); in models with capital market imperfections, see Cazzavillan et al. (1998); in multi-sector frameworks, e.g. Ladron-de-Guevarra et al. (1999); in the overlapping generations model (OLG thereafter), as shown in Cazzavillan (2001). In particular, the latter contribution has proved that local indeterminacy and endogenous fluctations may occur in the OLG model when externalities originated by the average capital stock are arbitrarily small. On the other hand, Cazzavillan and Pintus (2004) have pointed out, in the context of OLG economies without externalities, that intertemporal substitution in consumption is a critical mechanism which enables agents to arbitrage away expectation-driven fluctuations when the ratio between savings and wage is reasonably low.

To summarize, the current literature stresses the fact that externalities, however small, may lead to indeterminacy when consumption only occurs in the second period of life (Cazzavillan, 2001), while, on the contrary, intertemporal substitution in consumption leads to local uniqueness (Cazzavillan and Pintus, 2004). Therefore, one naturally wonders whether intertemporal equilibria are locally (in)determinate when arbitrarily small externalities are introduced in OLG economies with consumption in both periods of life. In particular, it remains unclear if local indeterminacy occurs for reasonable parameter values, including both (close to) Cobb–Douglas technology and a not too small propensity to consume out of wage income. Moreover, it is still an open question to understand whether introducing capital externalities enlarges or reduces the range of parameter values compatible with endogenous fluctuations.

This paper aims at answering these questions. We focus on the average capital stock spillover effects, following the standard Arrow–Frankel–Romer like learning-by-doing argument, which we view as consistent with the length of the period implied by the OLG setting. We show that, in the presence of arbitrarily small capital externalities, assuming that the young generation consumes a realistically large fraction of his wage income leads to an important prediction: exactly as in the standard optimal growth model, there is, when it exists, a unique equilibrium path converging to the steady state. Therefore, intertemporal substitution in consumption across periods is shown to be a critical mechanism which is not weakened by the presence of externalities, as it enables short-lived agents to rule out, by arbitrage, expectation-driven fluctuations when the ratio between savings and wage is reasonably low. Most importantly, we show that the larger capital externalities, the lower the values of the consumption/wage ratio that are associated with local indeterminacy and expectation-driven fluctuations. The following intuitive description may help the reader to clarify the mechanisms at work and their interplay. In this model, cyclical paths (be they deterministic or stochastic) arise because of the interaction of two conflicting effects: when the capital stock increases (say, from its steady-state value), this triggers an increase in wage and, therefore, an increase in savings which leads to a higher capital stock in the next period. However, capital accumulation is followed by a decrease in the interest rate (when externalities are small) that will eventually depress savings and, thereby, capital accumulation: when the interest rate is low enough, the agent is now willing to increase leisure, first-period consumption and to reduce second-period consumption (see Eq. (3)). To summarize, the initial wage increase will be eventually offset by a decrease of the interest rate. This may help to understand why, absent the externalities, endogenous cycles and sunspots occur in the OLG model without first period consumption (see, e.g., Grandmont, 1993, pp. 17–23). Now, two forces tend to dampen the conflicting effects of wage and interest rate movements: first, the larger the share of consumption out of wage income in the first period of life, the lower the rise in savings, as consumption by the young also increases with the wage income (see Eq. (3)); second, increasing capital externalities makes the interest rate less and less negatively sensitive to variations in the capital stock (see Eqs. (7), (15)) and, therefore, this makes less likely the cyclical reversal that occurs when externalities are absent. Therefore, it is expected that the larger the externalities (that is, the less elastic the interest rate), the higher the impact of the wage variation on savings (that is, the lower the consumption-to-wage ratio) that is required for cyclical equilibria to occur. In particular, this intuitive interpretation may help to understand why small externalities are favorable to local indeterminacy in the OLG model without first-period consumption, as shown by Cazzavillan (2001): in that context, savings equal wage income, by definition. Therefore, increasing the wage has a large positive effect on savings and leads to a much higher next-period capital stock, which in turn greatly reduces the interest rate even when it is not very elastic (i.e. when externalities are present).

We shall demonstrate that assuming arbitrarily small externalities yields predictions that are very close to those of the model with constant returns to scale. Conversely, we show that, as in the Ramsey economy (see, for instance, Benhabib and Farmer, 1994, and, more recently, Pintus, 2003), significant externalities are required for indeterminacy to arise when labor supply elasticity is constrained to be not too large as predicted by most empirical studies. Moreover, we show that a key condition for indeterminacy is that the consumption-to-wage proportion has to be smaller than a critical value and, most importantly, that the latter upper bound decreases with the size of externalities. Section 5 further illustrates, by means of numerical examples, that local determinacy prevails when personal consumption represents more than half of output, in agreement with averages for the OECD countries.

Our results may be of some interest in view of the many uses of the OLG model as a benchmark framework to study, for instance, intergenerational transfers: we provide conditions that are easy to check and that lead to local uniqueness, thereby allowing one to perform some comparative dynamics exercises. For instance, our analysis may be used as a starting point to extend some results obtained by Hu, 1976, Hu, 1979 (by adding endogenous labor supply in the first period of life), and Rios-Rull (1996), Fuster (1999) (by providing an analytical characterization).

Finally, our results are in contrast with what has been shown to hold in the infinite-horizon model, see Benhabib and Farmer (1994) (and also, among others, Aiyagari, 1995; Schmitt-Grohe, 1997; Wen, 2001; Harrison and Weder, 2002; Pintus, 2003). Well-known (even though implicit) in this literature is the fact that capital externalities only are not enough to lead to local indeterminacy: a higher (than steady state) capital stock implies monotonic convergence to (resp. divergence from) the steady state when externalities are small enough (resp. large enough).

The paper is organized as follows. Section 2 introduces the model and derives the competitive intertemporal equilibria with perfect foresight, while Section 3 establishes existence of a normalized steady state. Section 4 determines the qualitative behavior of the orbits around the normalized steady state, whereas Section 5 presents some numerical examples and Section 6 gathers the concluding remarks.

Section snippets

The general model

We consider a competitive, non-monetary, overlapping generations model with production. The framework involves a unique perishable good, which can be either consumed or saved as investment, a large number of identical competitive firms all facing the same technology, and a constant population composed of households living two periods. Agents, who are identical within each generation, consume in both periods, supply labor and save when young. When old, their saved income is rented as physical

Steady-state existence

An interior steady-state equilibrium is a stationary sequence (kt,t)=(k*,*) with (k*,*) in ++2, which satisfies, for all t0, Eqs. (8), (9). In view of Eq. (7), a steady-state equilibrium (k*,*) must be a solution of the stationary systemk=Aψ(k)ω(k/)-B(U1)-1(BU3()Aψ(k)ω(k/)),Aψ(k)ρ(k/)+1-δ=u2-1(U3()Aψ(k)ω(k/)k)/k.

Existence can be easily established by selecting appropriately the two scaling parameters A and B and imposing the necessary and sufficient conditions so as to obtain a

Local dynamics analysis

To characterize the local dynamics generated by the dynamical system (8) and (9), we shall study the linear map associated with the Jacobian matrix evaluated at steady state (k*,*)=(1,1) the existence of which will be assumed throughout the section (see Proposition 3.1). Let εΩ,k, εΩ, define the elasticities of the function Ω(k,) with respect to k and evaluated at the steady state (1,1). The elasticities of the function Π(k,) are defined, similarly, as εΠ,k, εΠ,. In addition, let θΩ(k*,

Numerical examples

We now illustrate, by means of plausible numerical examples, our main conclusions—first, that local determinacy prevails, in the presence of small externalitites, whenever the propensity to consume out of wage income is close to National Accounts evidence (i.e. (θ-1)/θ close to one half) and, second, that increasing externalities may even reduce further that range of parameter values. As a benchmark case, we note that the annual ratio of personal consumption expenditures over GDP averages at

Concluding remarks

We have studied a version of Diamond's OLG model modified to account for elastic labor supply and capital externalities. We have shown that local determinacy of the steady state prevails, when externalities are arbitrarily small, as long as the fraction of young-age consumption out of wage income is large enough. Conversely, local indeterminacy with small externalities requires both labor supply to be close to indivisible and large values of the propensity to save out of the wage income. More

Acknowledgments

The authors thank Jean-Michel Grandmont for helpful suggestions, and two anonymous referees for their comments that helped to improve both the exposition and the content of the paper. A first draft of this paper was written while Guido Cazzavillan was visiting the Université de la Méditerranée at GREQAM, whose hospitality is gratefully acknowledged.

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