Stochastic Optimal Growth Research Articles

Machine learning for high-dimensional dynamic stochastic economies

We present a novel computational framework that can compute global solutions to high-dimensional dynamic stochastic economic models on irregular state space geometries. This framework can also resolve value and policy functions’ local features and perform uncertainty quantification, in a single model evaluation. We achieve this by combining Gaussian process machine learning with active subspaces; we then embed this into a parallelized discrete-time dynamic programming algorithm. To demonstrate the broad applicability of our method, we compute solutions to stochastic optimal growth models of up to 500 continuous dimensions. We also show that our framework can address parameter uncertainty and can provide predictive confidence intervals for policies that correspond to the epistemic uncertainty induced by limited data. Finally, we propose an algorithm that, based on combining this framework with Bayesian Gaussian mixture models, is capable of learning irregularly shaped ergodic sets as well as performing dynamic programming on them.

Stochastic optimal growth model with risk sensitive preferences

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 the 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 so-called associated random variables. We also establish the Euler equation that incorporates the solution to the optimality equation.

Optimality of Ramsey–Euler policy in the stochastic growth model

For the canonical one sector stochastic optimal growth model, we outline a new set of conditions for a policy function that satisfies the Ramsey–Euler equation to be optimal. An interior Ramsey–Euler policy function is optimal if, and only if, it is continuous or alternatively, if, and only if, both consumption and investment are non-decreasing in output. In particular, we show that under these conditions, the stochastic paths generated by the policy must satisfy the transversality condition; the implication is that in applying our result, one does not need to verify the transversality condition when checking for optimality of a policy function.

Machine Learning for High-Dimensional Dynamic Stochastic Economies

We present the first computational framework that can compute global solutions to very-high-dimensional dynamic stochastic economic models on arbitrary state space geometries. This framework can also resolve value and policy functions' local features and perform uncertainty quantification, in a self-consistent manner. We achieve this by combining Gaussian process machine learning with the active subspace method; we then embed this into a massively parallelized discrete-time dynamic programming algorithm. To demonstrate the broad applicability of our method, we compute solutions to stochastic optimal growth models of up to 500 continuous dimensions. We also show that our framework can address parameter uncertainty and can provide predictive confidence intervals for policies that correspond to the epistemic uncertainty induced by limited data. Finally, we propose an algorithm, based on this framework, that is capable of learning irregularly shaped ergodic sets as well as performing dynamic programming on them.

Stochastic optimal growth with risky labor supply

Production takes time, and labor supply and profit maximization decisions that relate to current production are typically made before all shocks affecting that production have been realized. In this paper we re-examine the problem of stochastic optimal growth with aggregate risk where the timing of the model conforms to this information structure. We provide a set of conditions under which the economy has a unique, nontrivial and stable stationary distribution. In addition, we verify key optimality properties in the presence of unbounded shocks and rewards, and provide the sample path laws necessary for consistent estimation and simulation.

Income inequality and economic growth

Despite the extensive existing literature on income inequality and economic growth, there remains considerable disagreement on the effect of inequality on economic growth. Existing literatures find either a positive or a negative relationship. In this paper, we attempt to theoretically examine that relationship with a stochastic optimal growth model. We make the disagreement clear within a single model. We conclude (i) that both are possible – that is, higher inequality can retard growth in the early stage of economic development, and can encourage growth in a near steady state, (ii) that income redistribution by high income tax does not always reduce income inequality. Income inequality can be reduced by higher income tax in a near steady state, but it cannot be reduced in the early stage of economic development, and (iii) that two government polices – rapid economic growth and low income inequality – can be achieved by low income tax in the early stage of economic development, but both cannot be achieved simultaneously in a near steady state.

Stochastic growth with short-run prediction of shocks

We study a variation of the one-sector stochastic optimal growth model with independent and identically distributed shocks where agents acquire information that enables them to accurately predict the next period’s productivity shock (but not shocks in later periods). Optimal policy depends on the forthcoming shock. A “better” predicted realization of the shock that increases both marginal and total product always increases next period’s optimal output. We derive conditions on the degree of relative risk aversion and the elasticity of marginal product under which optimal investment increases or decreases with a better shock. Under fairly regular restrictions, optimal outputs converge in distribution to a unique invariant distribution whose support is bounded away from zero. We derive explicit solutions to the optimal policy for three well-known families of production and utility functions and use these to show that volatility of output, sensitivity of output to shocks, and expected total investment may be higher or lower than in the standard model where no new information is acquired over time; the limiting steady state may also differ significantly from that in the standard model.

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

In a stochastic economy, long run consumption and output may not be bounded away from zero even when productivity is arbitrarily high near zero and uncertainty is arbitrarily small. In the one-sector stochastic optimal growth model with i.i.d. production shocks, we characterize the nature of preferences that lead to this phenomenon for a stochastic Cobb–Douglas technology. For the general version of the model, we outline sufficient conditions under which the economy expands its capital stock near zero and long run consumption is bounded away from zero with certainty. Our conditions highlight the important role played by risk aversion for small consumption levels.

On discounted dynamic programming with unbounded returns

In this paper, we apply the idea of k-local contraction of Rincon-Zapatero and Rodriguez-Palmero (Econometrica 71:1519–1555, 2003; Econ Theory 33:381–391, 2007) to study discounted stochastic dynamic programming models with unbounded returns. Our main results concern the existence of a unique solution to the Bellman equation and are applied to the theory of stochastic optimal growth. Also a discussion of some subtle issues concerning k-local and global contractions is included.

A stochastic optimal growth model with a depreciation factor

This paper is devoted to the study of a one-sector stochasticgrowth model with the depreciation factor of the output and withbounded and unbounded utility, in which the shocks are allowed tobe bounded or unbounded. Under certain assumptions, the existenceof a unique optimal policy function for the model is shown to betrue and the existence of an invariant distribution for the outputprocess is confirmed.

On Lipschitz continuity of the iterated function system in a stochastic optimal growth model

This paper provides qualitative properties of the iterated function system (IFS) generated by the optimal policy function for a class of stochastic one-sector optimal growth models. We obtain, explicitly in terms of the primitives of the model (i) a compact interval (not including the zero stock) in which the support of the invariant distribution of output must lie, and (ii) a Lipschitz property of the iterated function system on this interval. As applications, we are able to present parameter configurations under which (a) the support of the invariant distribution of the IFS is a generalized Cantor set, and (b) the invariant distribution is singular.

Stochastic technical change, non-renewable resource and optimal sustainable growth

We develop a stochastic endogenous growth model involving a non-renewable resource, in which innovation arrivals are governed by a non-stationary Poisson process. Using a CRRA analytical example, we characterize the optimal trajectories of the model and analyze the effects of uncertainty in the sense of Rothschild and Stiglitz by computing a mean-preserving spread. We show that increased variability in the innovation process always implies a smaller optimal R&D effort, since this leads to a reduced marginal rate of return. Effects on the other variables of the model may also be unambiguously identified depending upon the relative risk aversion of agents, the social discount rate and the marginal arrival rate of innovations. Finally, we investigate the conditions under which, on average, the economy reaches a sustainable growth path.

Loss aversion in aggregate macroeconomic time series

Prospect theory has been the focus of increasing attention in many fields of economics. However, it has scarcely been addressed in macroeconomic growth models—neither on theoretical nor on empirical grounds. In this paper we use prospect theory in a stochastic optimal growth model. Thereafter, the focus lies on linking the Euler equation obtained from a prospect theory growth model of this kind to real macroeconomic data. We will use generalized method of moments (GMM) estimation to test the implications of such a non-linear prospect utility Euler equation. Our results indicate that loss aversion can be traced in aggregate macroeconomic time series.

Stochastic optimal growth with a non-compact state space

This paper studies the stability of a stochastic optimal growth economy introduced by Brock and Mirman [Brock, W.A., Mirman, L., 1972. Optimal economic growth and uncertainty: the discounted case. Journal of Economic Theory 4, 479–513] by utilizing stochastic monotonicity in a dynamic system. The construction of two boundary distributions leads to a new method of studying systems with non-compact state space. The paper shows the existence of a unique invariant distribution. It also shows the equivalence between the stability and the uniqueness of the invariant distribution in this dynamic system.

Stochastic optimal growth with bounded or unbounded utility and with bounded or unbounded shocks

This paper studies a one-sector stochastic optimal growth model with i.i.d. productivity shocks in which utility is allowed to be bounded or unbounded, the shocks are allowed to be bounded or unbounded, and the production function is not required to satisfy the Inada conditions at zero and infinity. Our main results are three-fold. First, we confirm the Euler equation as well as the existence of a continuous optimal policy function under a minimal set of assumptions. Second, we establish the existence of an invariant distribution under quite general assumptions. Third, we show that the density of optimal output converges to a unique invariant density independently of initial output under the assumption that the shock distribution has a density whose support is an interval, bounded or unbounded. In addition, we provide existence and stability results for general one-dimensional Markov processes.

Stochastic optimal economic growth model with natural resources

The paper examines an economic growth problem how social planners reasonably open up and retain natural resources. The objective is to maximize the total expected discounted utility of comsumption. Social planners' optimal devision and optimal expected rates at the steady state are derived. At last, how productivity and productivity shock affect on the expected growth rate, consumption-resources ratio and the fraction of exploited resources, are analyzed.

Stochastic optimal growth with nonconvexities

This paper studies optimal investment and dynamic behavior in stochastically growing economies. We assume neither convex technology nor bounded support of the productivity shocks. A number of basic results concerning the investment policy and the Ramsey–Euler equation are established. We also prove a fundamental dichotomy pertaining to optimal growth models perturbed by standard econometric shocks: either an economy is globally stable or it is globally collapsing to the origin.

Cantor type attractors in stochastic growth models

We study a one-sector stochastic optimal growth model where production is affected by a shock taking one of two values. Such exogenous shock may enter multiplicatively or additively. A result is presented which provides sufficient conditions to ensure that the attractor of the iterated function system (IFS) representing the optimal policy, is a generalized topological Cantor set. To indicate the role of the strict monotonicity condition on the IFS in this result, examples of attractors, which are not of the Cantor type, are constructed with iterated function systems, whose maps are contractions and satisfy a no overlap property.

Theory of Stochastic Optimal Economic Growth

Stochastic optimal growth involves the study of the optimal intertemporal allocation of capital and consumption in an economy where production is subject to random disturbances. The theory traces its roots to the seminal work on deterministic optimal growth by Ramsey [106], Cass [21] and Koopmans [55]. Its influence has been enhanced by research that shows how the convex stochastic growth model can be decentralized to represent the behavior of consumers and firms in a dynamic competitive equilibrium of a productive economy ([102], [115], [15]). This makes the stochastic optimal growth model useful both as a normative exercise and in the development of positive theories of how the economy works. As a consequence, the theory has emerged as one of the central paradigms of dynamic economics. It is based on a simple, yet powerful model that encompasses fundamental questions that are basic to any theory of dynamic economic behavior: What are the characteristics and determinants of optimal policies? What are the economic incentives that govern the optimal intertemporal allocation of resources? What is the transient and long run behavior of variables in the model? Under different assumptions the model admits a rich set of answers to these questions. Historically, the main focal point of the theory has been issues of aggregate economic growth. At the same time its primary variable, capital, has a flexible interpretation that allows the model and its extensions to represent a wide variety of economic problems ranging from the study of business cycles ([59], [63]) and asset pricing ([14], [15]) to the allocation of renewable natural resources ([77], [82], [83]). Equally important, the model provides a strong theoretical foundation for applied analysis of these problems. The model can be solved numerically and has proved a testing ground for many numerical techniques used today in the analysis of dynamic economic problems. This chapter provides an overview of key results the theory of discounted stochastic optimal growth in discrete time.1 The paper begins with an analysis of

Stability of stochastic optimal growth models: a new approach

The paper proposes an Euler equation technique for analyzing the stability of differentiable stochastic programs. The main innovation is to use marginal reward directly as a Foster–Lyapunov function. This allows us to extend known stability results for stochastic optimal growth models, both weakening hypotheses and strengthening conclusions.