Takashi Kamihigashi

A Nonsmooth, Nonconvex Model of Optimal Growth

This paper analyzes the nature of economic dynamics in a one-sector optimal growth model in which the technology is generally nonconvex, nondifferentiable, and discontinuous. The model also allows for irreversible investment and unbounded growth. We develop various tools to overcome the technical difficulties posed by the generality of the model. We provide sufficient conditions for optimal paths to be bounded, to converge to zero, to be bounded away from zero, and to grow unboundedly. We also show that under certain conditions, if the discount factor is close to one, any optimal path from a given initial capital stock converges to a small neighborhood of the golden rule capital stock, at which sustainable consumption is maximized. If it is maximized at infinity, then as the discount factor approaches one, any optimal path either grows unboundedly or converges to an arbitrarily large capital stock.

Necessity of Transversality Conditions for Stochastic Problems

This paper establishes (i) an extension of Michels (1990, Theorem 1) necessity result to an abstract reduced-form model, (ii) a generalization of the results of Weitzman (1973) and Ekeland and Scheinkman (1986), and (iii) a new result that is useful particularly in the case of homogeneous returns. These results are shown for an extremely general discrete-time reduced-form model that does not assume differentiability, continuity, or concavity, and that imposes virtually no restriction on the state spaces. The three results are further extended to a stochastic reduced-form model. The stochastic extensions are easily accomplished since our deterministic model is so general that the stochastic model is in fact a special case of the deterministic model. We apply our stochastic results to a stochastic reduced-form model with homogeneous returns and a general type of stochastic growth model with CRRA utility.

A Simple Proof of the Necessity of the Transversality Condition

This note provides a simple proof of the necessity of the transversality condition for the differentiable reduced-form model. The proof uses only an elementary perturbation argument without relying on dynamic programming. The proof makes it clear that, contrary to common belief, the necessity of the transversality condition can be shown in a straightforward way.

Real business cycles and sunspot fluctuations are observationally equivalent

Abstract This paper points out that real business cycle models are observationally equivalent to externality models, which we define as models with externalities and sunspots but without productivity shocks. As far as standard first-order systems are concerned, however, externality models turn out to be more flexible than RBC models. This explains, at least partly, why previous studies have found that externality models perform better than standard RBC models at generating realistic business cycle behavior. Our results cast doubt on the arguments of Farmer and Guo (1994) and Caballero and Lyons (1992).

Stochastic Stability in Monotone Economies

This paper extends a family of well-known stability theorems for monotone economies to a significantly larger class of models. We provide a set of general conditions for existence, uniqueness and stability of stationary distributions when monotonicity holds. The conditions in our main result are both necessary and sufficient for global stability of monotone economies that satisfy a weak mixing condition introduced in the paper. Through our analysis we develop new insights into the nature and causes of stability and instability.

The Policy Function of a Discrete-Choice Problem is a Random Number Generator

This paper studies a life-cycle model in which the consumption good is assumed to be indivisible. This assumption requires the number of units of the good purchased in each period to be an integer. It is shown that, if the discount factor is sufficiently small, the policy function takes the form of a pseudo-random number generator or, more precisely, a linear congruential generator. It is also shown that optimal plans are almost always asymptotically non-periodic regardless of the discount factor. Various numerical examples of the value function and the policy function are provided.

Chaotic dynamics in quasi-static systems: theory and applications1

Abstract By a dynamical system, we mean a system of N ∈ N equations that depend on xt−μ, …, xt+ν ∈ R N, where μ, ν ∈ N . We define a static system as a dynamical system that depends only on xt. We define a quasi-static system as a dynamical system that is in a certain sense relatively close to a static system. We show that under additional conditions, a quasi-static system is chaotic in a generalized sense of Li and Yorke. This result provides easy-to-verify sufficient conditions for chaos for general multidimensional dynamical systems, including maps. We show that these conditions are stable under small C1 perturbations. We apply these results to two types of growth models with externalities. We show that the models display chaotic dynamics for certain parameter values. We also construct a numerical example in which utility is logarithmic and the dynamics are chaotic (and the discount rate is small). Our conditions for chaos are particularly useful in analyzing dynamic versions of static models with multiple equilibria, as well as dynamic models with multiple steady states.

Externalities and nonlinear discounting: Indeterminacy

Abstract This paper studies a one-sector model with externalities and nonlinear discounting. Loosely, if the Bellman equation for a standard model is written as V ( k )=max{ u + βV ( k ′)}, then the Bellman equation for our model is written as V ( k )=max{ u + B ( V ( k ′))} for some concave function B . Our model assumes that period utility ( u ) is linear in consumption and leisure and the production function exhibits increasing social returns and nonincreasing private returns. We establish the existence of a steady state under fairly general conditions and give a necessary and sufficient condition for local indeterminacy. We show that regardless of the degrees of increasing returns and externalities, there exists a discount function B such that indeterminacy arises. We also show that in a certain sense, as the degree of increasing returns converges to zero, the variability of discounting (the second derivative of B ) required to generate indeterminacy also converges to zero. Numerical examples are provided. The main results are also extended to a one-sector model with fixed labor supply.

Increasing marginal impatience and intertemporal substitution

For infinite-horizon models with recursive preferences the condition known as increasing marginal impatience is often adopted, but the condition is not fully understood in the literature. This paper shows that increasing marginal impatience is equivalent to the intuitive property that the substitutability between the consumption levels in two different periods is a decreasing function of the distance between the periods.

Seeking ergodicity in dynamic economies

In estimation and calibration studies, the convergence of time series sample averages plays a central role. At the same time, a significant number of economic models do not satisfy the classical ergodicity conditions. Motivated by existing work on asymptotics of stochastic economic models, we develop a new set of results on limits of sample moments and other sample averages using an order-theoretic approach. Our results include a condition that is necessary and sufficient for convergence over a broad class of moment functions. We discuss implications, sufficient conditions and a range of economic applications.