Sustained positive consumption in a model of stochastic growth: The role of risk aversion

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Abstract An intriguing problem in stochastic growth theory is as follows: even when the return on investment is arbitrarily high near zero and discounting is arbitrarily mild, long run capital and consumption may be arbitrarily close to zero with probability one. In a convex one-sector model of optimal stochastic growth with i.i.d. shocks, we relate this phenomenon to risk aversion near zero. For a Cobb-Douglas production function with multiplicative uniformly distributed shock, the phenomenon occurs with high discounting if, and only if, risk aversion diverges to infinity sufficiently fast as consumption goes to zero. We specify utility functions for which the phenomenon occurs even when discounting is arbitrarily mild. For the general version of the model, we outline sufficient conditions under which capital and consumption are bounded away from zero almost surely, as well as conditions under which growth occurs almost surely near zero; the latter ensures a uniform positive lower bound on long run consumption (independent of initial capital). These conditions require the expected marginal productivity at zero to be above the discount rate by a factor that depends on the degree of risk aversion near zero. Terms of use: Documents inJEL Classification: D9, E2, O41.

On a safe capital stock for consumption maintenance in a convex model of stochastic growth

“…Note that configuration (i) includes the situation where capital and consumption grow near zero even under the worst realization of the random shock. The existing literature on stochastic growth contains various sufficient conditions on preferences and technology under which the optimal policy satisfies configuration (i); see, among others, Brock and Mirman (1972), Chatterjee and Shukayev (2008), Mitra and Roy (2006, 2010) and Olson and Roy (2000).…”

“…It has been shown that in this case consumption and capital fall below any strictly positive level infinitely often with probability one, no matter how large the initial stock (Mitra and Roy 2007). While it is easy to see that configuration (ii) can arise if the marginal productivity at zero is sufficiently small relative to the discount rate, Mirman and Zilcha (1975) presented an example to show that the optimal policy may lead to configuration (ii) even if the marginal productivity is infinite at zero for all realizations of the random shock; Mitra and Roy (2010) develop necessary and sufficient conditions for this phenomenon and relate it to the nature of risk aversion displayed by the utility function.…”

“…The literature on stochastic growth with convex technology characterizes conditions under which capital and consumption paths from all initial stocks are bounded away from zero (see, among others, Brock and Mirman 1972; Chatterjee and Shukayev 2008; Mitra and Roy 2010). 2 A smaller literature has focused on the possibility that capital and consumption paths are arbitrarily close to zero no matter how large the initial stock, and even if the marginal productivity at zero is infinitely large (see, Mirman and Zilcha 1976; Mitra and Roy 2010).…”

“…The literature on stochastic growth with convex technology characterizes conditions under which capital and consumption paths from all initial stocks are bounded away from zero (see, among others, Brock and Mirman 1972; Chatterjee and Shukayev 2008; Mitra and Roy 2010). 2 A smaller literature has focused on the possibility that capital and consumption paths are arbitrarily close to zero no matter how large the initial stock, and even if the marginal productivity at zero is infinitely large (see, Mirman and Zilcha 1976; Mitra and Roy 2010). 3 The literature on non‐convex stochastic models of optimal resource management has discussed the existence of a critical safe stock for maintaining capital and consumption (referred to as a “safe standard of conservation”); see, among others, Olson and Roy (2000) and Mitra and Roy (2006).…”

The stochastic Mitra–Wan forestry model: risk neutral and risk averse cases

“…In most cases results show differences in the optimal policies when the objective function is no longer the expected value. Interest in understanding the effects of risk aversion in finance and economics has increased significantly in the last years, specially in dynamic models (see, for example, Blomvall and Shapiro (2006);Denuit et al (2006); Mitra and Roy (2012); Riedel (2004)). In this paper, risk aversion is represented by the Conditional Value-at-Risk (CVaR), a risk measure widely used in finance that was popularized by Rockafellar and Uryasev (2000).…”