A nonsmooth, nonconvex model of optimal growth

Abstract

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 provide sufficient conditions for boundedness, extinction (convergence to zero), survival (boundedness away from zero), and unbounded growth. These conditions reveal that boundedness and survival are symmetrical phenomena, so are extinction and unbounded growth. Since many of the conditions are only local, it is possible that extinction occurs from small capital stocks, while unbounded growth occurs from large capital stocks. Despite such nonclassical results and nonclassical features such as nonconvexity and discontinuity, the model behaves much like a classical one as the discount factor approaches unity. In particular, we show that in most cases, if the discount factor is close to one, any optimal path from a given initial capital stock converges to a small neighborhood of what we define as the golden rule capital stock. If this stock is not finite, i.e., if sustainable consumption is maximized at infinity, then as the discount factor approaches one, unbounded growth at least almost occurs.1 Various results are known for nondifferentiable, convex growth models (e.g., McKenzie, 1986). The discrete-choice problems studied in Kamihigashi (2000a, b) have discontinuous features. More relevant works are discussed below.2 The analysis was extended to a multisector model by Dolmas (1996). See De Hek and Roy (2001) for results on stochastic models.3 A notable exception is Romer (1986, Sec. V.B), who analyzed a one-sector optimal growth model with a special form of increasing returns. The technology in his model,

On sustained economic growth with wealth effects

“…3 3 The analysis has been extended to multi‐sector growth (Dolmas 1996), growth with non‐convex technology (Kamihigashi and Roy 2007) and stochastic growth (see e.g. De Hek and Roy 2001).…”

Non-convex aggregate technology and optimal economic growth

“…Remark 5: In the proposition 4, (i) and (iii) own much to Kamihigashi and Roy (2007). Uniqueness of the steady state in (ii) and (iv) , and the critical value k c in (v) are peculiar to our model of concave-concave technology.…”

“…They obtained for the case of linear utility that any optimal path, which is strictly monotone, either converges to zero or reaches a positive steady state in …nite time and possibly jumps among different steady states. Furthermore, Kamihigashi and Roy (2007) gave general conditions in a Nonsmooth, Nonconvex model of optimal growth to have a steady state, to obtain that an optimal path converges either to zero or to a well determined steady state, or to in…nity.…”

Elementary results on solutions to the bellman equation of dynamic programming: existence, uniqueness, and convergence

“…In this subsection we consider a one-sector growth model with this feature. The example here is a special case of the model studied by Kamihigashi and Roy (2007).…”

“…In this subsection we consider an optimal growth model with "roughly increasing" returns, which is again a special case of the model studied by Kamihigashi and Roy (2007). In particular we assume that the production function f is continuous and strictly increasing, and that there exist θ, θ ∈ R ++ and α ∈ (1, 1/β) such that Figure 2 illustrates a production function of this form, which roughly exhibits increasing returns.…”