Stochastic optimal growth model with risk sensitive preferences

Abstract

This paper studies a one-sector optimal growth model with i.i.d. productivity shocks that are allowed to be unbounded. The utility function is assumed to be non-negative and unbounded from above. The novel feature in our framework is that the agent has risk sensitive preferences in sense of Hansen and Sargent (1995). Under mild assumptions imposed on the productivity and utility functions we prove that the maximal discounted non-expected utility in the infinite time horizon satisfies the optimality equation and the agent possesses a stationary optimal policy. A new point used in our analysis is an inequality for the so-called associated random variables. We also establish the Euler equation that incorporates the solution to the optimality equation.Keywords. stochastic growth model, entropic risk measure, unbounded utility, unbounded shocks.onward. Here, E t stands for the expectation operator with respect to period t information. The parameter γ affects consumer's attitude towards risk in future utility. The form of preferences in (1) is due to Hansen and Sargent (1995), who used them to deal with a linear quadratic Gaussian control model. The preferences defined in (1) has several advantages. First of all, they are not time-additive in future utility. Time-additivity, however, requires an agent to be risk neutral in future utility. Risk sensitive preferences, on the other hand, allow the agent to be risk averse in future utility in addition to being risk averse in future consumption. This fact results in partial separation between risk aversion and elasticity of intertemporal substitution. 1 Moreover, as argued by ? risk sensitive preferences are also attractive, because they can be used to model preferences for robustness. It is worth emphasising that risk sensitive preferences of form (1) have found several applications, for instance, in the problems dealing with Pareto optimal allocations (see Anderson (2005)) or small noise expansions (see Anderson et al. (2012)).Our main results are two-fold. First we establish the optimality equation for the nonexpected utility in the infinite time horizon, when the agent has risk sensitive preferences (1). The proof as in the standard expected utility case is based on the Banach contraction principle, see Stokey et al. (1989). However, in oder to show that the dynamic programming operator maps a space of certain functions into itself, we have to confine our consideration to concave, non-decreasing and non-negative functions that are bounded in the weighted supremum norm. A novel feature in this analysis is an application of some inequality for the so-called associated random variables. This inequality also plays a crucial role in proving that the a fixed point of dynamic programming operator is indeed the value function. Moreover, it naturally fits into our model, in which the production and utility functions satisfy some mild conditions such as monotonicity and concavity. Here, we would like to emphasise that similarly as in Kamihigashi (2007), we do not assume the Inada...

Existence of Nash equilibria in stochastic games of resource extraction with risk-sensitive players

“…On the other hand, our result can also be viewed as an extension of the optimization problem (one player case), studied in Asienkiewicz and Jaśkiewicz (2017) and Bäuerle and Jaśkiewicz (2018), to a strategic version of a one-sector optimal growth model. In contrast to Bäuerle and Jaśkiewicz (2018), we examine, as mentioned above, a model with bounded felicity functions. The crucial role played in a study of the unbounded case is the fact that both investment and consumption functions are non-decreasing.…”

“…For instance, Hansen and Sargent (1995) applied them to a linear quadratic Gaussian control model, and Weil (1993) used them to examine precautionary savings and permanent income hypothesis. Moreover, these preferences found applications in the problems of Pareto optimal allocations (Anderson 2005) as well as in the study of Markov decision processes (Asienkiewicz and Jaśkiewicz 2017) or in one-sector optimal growth model with an unbounded felicity function (Bäuerle and Jaśkiewicz (2018)) . As argued by Hansen and Sargent (1995) the preferences in (1) are also attractive, because they can viewed as the robustness preferences.…”

Risk-Sensitive Preferences and Age-Dependent Risk Aversion

“…which resembles risk-sensitive preferences of Hansen & Sargent (1995) and Weil (1993) where the risk-sensitivity parameter ψ j measures degrees of risk aversion toward future utility of age j agents. Unlike EZW, this particular form of preferences satisfies translation-invariant property, making it monotonic and suitable for the study of additive risk such as labor income risk It is important to distinguish between ψ j which denotes the agent's risk attitude toward future utility and the variable γ j in (8), which represents agent's risk aversion in future consumption (Bäuerle & Jaśkiewicz (2018), Anderson (2005)). Agents are risk-averse toward future utilities when ψ j > 0 and are risk-loving when ψ j < 0.…”

“…More risk-averse agents will save more in the presence of income uncertainty to reduce the dispersion in lifetime utilities. Agents with risksensitive preferences also possess a stationary optimal policy function as shown by Bäuerle & Jaśkiewicz (2018), using a one-sector infinite horizon growth model under mild assumptions on productivity and utility functions.…”

On recursive utilities with non-affine aggregator and conditional certainty equivalent

“…A large body of the literature has already established extensions of the standard discounted utility model toward nonadditive aggregators under both deterministic and stochastic settings. Deterministic utility functions based on Koopmans equations can be found in numerous papers in the literature, including works by Boyd-III (2006), Bich et al (2018), Duran (2000), Le-Van and Vailakis (2005), Martins-da-Rocha and Vailakis (2010), Jaśkiewicz et al (2014), among others, while the utility function based on Epstein and Zin (1989) equations can be found in the work of Weil (1993), Skiadas (2015), Bäuerle and Jaskiewicz (2018), Marinacci and Montrucchio (2010), Ozaki and Streufert (1996), and Bloise and Vailakis (2018).…”