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Article Excerpt
Abstract This paper studies a one-sector optimal growth model with linear utility in which the production function is only required to be increasing and upper semi-continuous. The model also allows for a general form of irreversible investment. We show that every optimal capital path is strictly monotone until it reaches a steady state; further, it either converges to zero, or reaches a positive steady state in finite time and possibly jumps among different steady states afterwards. We establish conditions for extinction (convergence to zero), survival (boundedness away from zero), and the existence of a critical capital stock below which extinction is possible and above which survival is ensured. These conditions generalize those known for the case of S-shaped production functions. We also show that as the discount factor approaches one, optimal paths converge to a small neighborhood of the capital stock that maximizes sustainable consumption.
Keywords Nonconvex, nonsmooth, discontinuous technology * Extinction * Survival * Turnpike * Linear utility
JEL Classification Numbers C61 * D90 * O41 * Q20
1 Introduction
The aggregative model of optimal economic growth is an important theoretical paradigm that is widely used for analyzing economic issues related to intertemporal allocation of resources and capital accumulation. Much of the existing literature focuses on "classical" versions of the model where the underlying technology is smooth and convex. Yet, there exists a large variety of economic settings where the technology exhibits nonconvexities, nonsmoothness, and even discontinuities due to various factors such as fixed costs, increasing returns to scale, economies of scope, and stock effects in biological reproduction of species. In particular, upward discontinuities can be regarded as technological breakthroughs, and are often associated with threshold effects (e.g., Azariadis and Drazen 1990).
This paper presents a comprehensive analysis of a one-sector optimal growth model with linear utility in which the production function is only required to be increasing and upper semicontinuous. In an earlier paper (Kamihigashi and Roy 2005) we analyzed the case of strictly concave utility. In this paper we focus on arguments specific to the linear utility case.
The literature on optimal growth with nonconvex technology dates back to Clark (1971), who analyzed the problem of optimal dynamic consumption of a biological resource where the production function is S-shaped and the objective function is linear in consumption. A full characterization of optimal paths in the context of optimal growth was provided by Majumdar and Mitra (1983). Mitra and Ray (1984) studied a more general model with concave utility in which the production function is only required to be strictly increasing and continuous.
While various results are known on nonconvex one-sector optimal growth models with strictly concave utility (e.g., Skiba 1978; Majumdar and Mitra 1982; Majumdar and Nermuth 1982; Dechert and Nishimura 1983; Kamihigashi and Roy 2005), many of the arguments there cannot readily be applied to the linear utility case. For example, arguments based on the Euler equation cannot directly be applied to the linear utility case, where optimal paths are often not in the interior of the feasible set. In addition, though optimal paths are known to be monotone in the case of strictly concave utility, this is not true in the linear utility case, as shown in Proposition 3.2 of this paper.
We show however that every optimal capital path is strictly monotone until it reaches a steady state; further, it either converges to zero, or reaches a positive steady state in finite time and possibly jumps among different steady states afterwards. This sharpens, in the linear utility case, the result by Mitra and Ray (1984) that every optimal path approaches the set of steady states asymptotically, and extends it to cases with discontinuous production functions and irreversible investment. We also establish conditions for extinction (convergence to zero), survival (boundedness away from zero), and the existence of a critical capital stock below which extinction is possible and above which survival is ensured, known as the "minimum safe standard of conservation" in the bioeconomic literature (e.g., Clark 1971). These conditions generalize those established by Clark (1971) and Majumdar and Mitra (1983) for the case of S-shaped production functions.
Moreover we show that despite the nonclassical features of the model, as the discount factor approaches one, optimal paths converge to a small neighborhood of the capital stock that maximizes sustainable consumption. This result allows us to extend the turnpike theorem of Majumdar and Mitra (1982) to the case of linear utility.
Much of our analysis is based on what we call the partial and total gain functions. Roughly speaking, the partial gain function measures one-period returns on investment; the total gain function measures infinite-horizon returns on investment. In the case of concave utility, the partial gain function was used by Majumdar and Mitra (1982), Dechert and Nishimura (1983), and Mitra and Ray (1984) to study the properties of steady states. In the linear utility case, it was used by Spence and Starrett (1973) and Clark (1990) to study optimality of "most rapid approach" paths. We follow Majumdar and Mitra (1983) in using both partial and total functions, but we utilize them to their full strength. In particular we show that along an optimal path, the partial gain function either becomes larger at some point or stays constant, and the total gain function never decreases as long as it is feasible to repeat the same capital stock. Both functions are also useful in other models whenever the objective function is additively separable in current and future state variables.
The rest of the paper is organized as follows. Section 2 describes the model. Section 3 develops various properties that constitute the essential tools of our analysis. Section 4 shows results on monotonicity and convergence of optimal paths. Section 5 offers conditions for survival, extinction, and the existence of a minimum safe standard of conservation. Section 6 establishes turnpike properties of optimal paths.
2 The model
Consider the following maximization problem:
[max.[{[c.sub.t],[x.sub.t]}[.sub.t=0.sup.[infinity]]]] [[infinity].summation over (t=0)] [[delta].sup.t] [c.sub.t] (2.1)
s.t. [for all]t [member of] [Z.sub.+], [c.sub.t] + [x.sub.t+1] = f([x.sub.t]), (2.2)
[c.sub.t] [greater than or equal to] 0, (2.3)
[x.sub.t+1] [greater than or equal to] r([x.sub.t]), (2.4)
[x.sub.0] given, (2.5)
where [c.sub.t] is consumption in period t, [x.sub.t] is the capital stock at the beginning of period t, [delta] is the discount factor, f is the production function, and (2.4) means that capital cannot be decreased below its depreciated level r([x.sub.t]); we call r the depreciation function for convenience. Our formulation allows for nonlinear depreciation. The standard case of reversible investment is a special case in which r(x) = for all x [greater than or equal to] 0. (1)
We use the following standard definitions. A path {[c.sub.t], [x.sub.t]}[.sub.t=0.sup.[infinity]] is feasible if it satisfies (2.2)-(2.4). A capital path {[x.sub.t]} is feasible if there is a consumption path {[c.sub.t]} such that {[c.sub.t], [x.sub.t]} is feasible. A path...
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