رشد خروجی پایدار تحت عدم قطعیت : مدل ساده با سرمایه های انسانی

In a model where agents use their labour/education choice to adjust their consumption profile over time, I show that the impact of uncertainty on growth depends, critically, on agents’ attitudes towards risk, reflected by the coefficient of relative risk aversion. In this respect, the well known result from the literature on ‘saving under uncertainty’ can be extended into a broader context, whereby the intertemporal profile of consumption is determined via human capital accumulation rather than saving and physical capital investment.

During the late 1960s and early 1970s, a variety of theoretical analyses began exploring the impact of aggregate uncertainty on saving decisions (e.g., Levhari and Srinivasan, 1969, Sandmo, 1970, Mirman, 1971 and Rothschild and Stiglitz, 1971). Under different settings, all these analyses seemed to reach a consensus on the importance of attitudes towards risk on determining the reaction of saving rates to higher degrees of future uncertainty. Specifically, the main conclusion derived from the aforementioned analyses is that, in response to higher degrees of uncertainty, saving rates increase (decrease) if the coefficient of relative risk aversion is above (below) unity. When the coefficient of relative risk aversion is equal to one (i.e., the case of logarithmic utility) saving is unresponsive to aggregate uncertainty. More recently, the work of Romer (1986) revived an important idea (originally proposed by Frankel, 1962) within a context of a production economy with intertemporal consumer maximisation. He showed that if the investment activity that adds to the aggregate stock of capital can generate and spread additional knowledge, and if the relative importance of knowledge on productivity is sufficiently high, then the economy can reach an equilibrium with ever increasing levels of output (or, equivalently, a sustainable and endogenously determined growth rate of output). The upshot from Romer’s analysis was that the factors normally impinging on saving rates (and, therefore, aggregate investment) can improve our understanding of the differences in growth rates and, to some extent, potential standards of living across economies. Of course, it was not long before theorists made the apparent connection and understood that, as long as uncertainty is an important consideration behind saving motives and behaviour, higher degrees of uncertainty may have significant long-term implications in terms of output growth trends. In particular, the theoretical analyses by Smith, 1996, de Hek, 1999 and Jones et al., 2005 addressed the issue of the interaction between uncertainty and long-run output growth within the context of dynamic, general equilibrium models with endogenous mechanisms for productivity improvements and stochastic elements arising from the presence of technology (or productivity) shocks. Their results verify the importance of the coefficient of relative risk aversion as this was described within the various analytical frameworks of the literature on optimal savings under uncertainty – a literature to which I alluded earlier. 1 The models constructed by Smith, 1996, de Hek, 1999 and Jones et al., 2005, despite being different in terms of their overall structure, share one common future: all types of capital accumulate through savings – that is, agents decide to sacrifice their current consumption and devote a certain fraction of their produced output with the purpose building up some capital stock that will facilitate future production and consumption. Nevertheless, pecuniary elements need not be the only ones to serve in the accumulation of capital. As the work of Uzawa, 1965, Razin, 1972 and Lucas, 1988 suggested, another important aspect in the formation of capital (especially human capital), the accumulation of knowledge and, therefore, the driving force behind long-run growth involves the various human resources, like time or effort, that individuals devote with the purpose of improving their future productive capacity. Naturally, in such scenarios the nature of the trade-offs between current and/or future benefits are slightly different from the standard consumption–saving choice. For example, we devote more time towards human capital accumulation in order to improve our future consumption possibilities, rather than working in order to achieve more current consumption. We may even choose to devote more time/effort towards both labour and (human capital) investment at the expense of our leisure. The question emerging is the following: to what extent do the aforementioned results on saving and growth under uncertainty survive within a framework in which the endogenous process behind sustainable growth resembles the one put forward by Uzawa, 1965, Razin, 1972 and Lucas, 1988? In this paper I construct a simple model in which I show that, indeed, the main implication of the papers comprising the literature on ‘growth under uncertainty’ is not just a mere extension of the conclusions reached from the literature on ‘optimal saving under uncertainty’. My model shows that the basic premise of uncertainty promoting (impeding) trend growth whenever the coefficient of relative risk aversion is above (below) unity may still emerge in an environment where there is no actual saving involved and growth is driven through purposeful time/effort devoted towards human capital accumulation. The rest of the paper is organised as follows: Section 2 presents the basic model. In section 3 I define and derive the dynamic equilibrium and in Section 4 I show the impact of uncertainty on economic growth. Section 5 discusses and concludes.

In this paper I have recovered a well known result – that is, the outcome whereby the preference parameter indicating relative risk aversion is crucial in determining the impact of uncertainty on output growth – within a context of an economy where the trade-off between labour and education is crucial for the evolution of human capital (and therefore sustainable growth). To complete the analysis, there is a need to provide sufficient intuition on why uncertainty impinges on the optimal human capital investment decisions, in the first place, and on why such investments may either be enhanced or inhibited by uncertainty, hence determining the impact that the latter bears on economic growth. In general, the expectation of higher future productivity has two conflicting effects on the equilibrium allocation of time/effort between different activities: on the one hand, it induces agents to provide more effort towards learning activities, permanently, at the expense of labour, as they try to reap the relatively higher expected future benefits by accumulating human capital – i.e., the substitution effect; on the other hand, the expectation of enhanced future productivity raises lifetime income, thus generating an incentive for increasing the pattern of consumption in all periods, including the current one – i.e., the income effect that works by inducing a permanent increase in labour effort brought about at the expense of learning activities. Nevertheless, in order to understand the effect of uncertainty about future productivity on the optimal allocation of time, we need some knowledge in addition to the impact of productivity per se. In particular, we need to identify and compare the extent of an individual’s average reaction to the expectation of both an increase and a decrease – with both being of equal magnitude (i.e., a mean-preserving spread) – in productivity. Knowledge of which effect (i.e., substitution or income) dominates is useful because it can tell us how individuals adjust their time allocation when they expect future productivity to be either lower or higher. Given these adjustments, we need the characteristic that identifies their magnitude, in order to balance them and find the average impact resulting from mean-preserving spreads. This additional characteristic derives from the fact that the future productivity has a non-linear impact on the optimal decisions of individuals – particularly, from the non-linearity of the stochastic component of productivity inside the expectation term. The latter becomes apparent in the presence of the parameter Θ. First consider the case where ρ ∈ (0, 1). This is the scenario whereby the substitution effect dominates – meaning that View the MathML sourceχ¯ will rise as a result of an expected increase in αt+1 and fall as a result an expected decrease in αt+1. The effect of uncertainty is then determined by the fact that, for ρ ∈ (0, 1), Θ is a concave function of αt+1. This indicates that the rise in View the MathML sourceχ¯ as a result of an expected increase in αt+1 is less pronounced than the fall in View the MathML sourceχ¯ resulting from an expected decrease in αt+1 of equal magnitude (i.e., a mean-preserving spread). Consequently, greater uncertainty leads to a reduction (on average) of the time/effort an individual will be willing to devote on the accumulation of human capital – an effect that results in lower trend growth. Next consider the case where ρ > 1 – i.e., the situation whereby the income effect dominates. In this case, View the MathML sourceχ¯ will fall as a result of an expected increase in αt+1 and rise as a result an expected decrease in αt+1. Now, the effect of uncertainty will be determined by the fact that, for ρ > 1, Θ is a convex function of αt+1. This indicates that the fall in View the MathML sourceχ¯ as a result of an expected increase in αt+1 is less pronounced than the rise in View the MathML sourceχ¯ resulting from an expected decrease in αt+1 of equal magnitude (i.e., a mean-preserving spread). As a result higher uncertainty will boost (on average) an individual’s incentive to spend more effort on the accumulation of human capital. It also means that uncertainty is associated with higher trend growth. Obviously, as long as ρ = 1, utility is logarithmic. In this case, income and substitution effects cancel each other out; therefore uncertainty has no effect on the growth rate of output.