Optimal Growth
Optimal Growth
Optimal growth theory occupies a central part of modern capital theory and dynamic models of planning, macroeconomics, exhaustible resources, natural resources, development economics, finance, and dynamic games. It originated with a classic 1928 paper by the mathematician Frank Ramsey, who investigated what kind of savings policy (equal to investment) would realize the largest social welfare, defined as the sum total of utilities from current and future consumption flows. Since the choice of capital stock at any point in time can only be based upon the consumption streams the stock can yield in the future, the problem necessarily takes an open-ended form with no finite terminal date for the planning horizon.
A brief and abstract statement of the typical mathematical problem in optimal growth theory follows. Let t denote time in discrete units, t = 0, 1, 2 … k t the stock of goods at t, and T t the technology set known at t, composed of pairs of vectors (x,y) such that from stocks x at t it is feasible to reach the stocks y at t + 1. A function ut: Tt → R provides the maximum utility ut (x,y) that is achievable, consistent with the transition of stocks from x to y. Let k denote the historically given initial stock. A path from k is an infinite sequence satisfying k0 = k and (kt, kt+1) ∈ Tt for each t ≥ 0. Loosely speaking, a path is optimal if it maximizes over the set of all possible paths from k.
The reduced-form model stated above, developed by David Gale (1967) and Lionel McKenzie (1968), is flexible enough to allow for changing technology and tastes with very liberal interpretations of stocks, including environmental factors, fixed resources, and unfinished goods in process, to name a few. A substantial part of the literature concentrates attention on the quasi-stationary model where technology and utility is time independent; that is, there is a set T, a function ut: Tt → R, and a constant discount factor δt where 0 < δ < 1, such that Tt= T and ut (x,y ) = δtu(x,y). A special case is the extensively studied one-sector neoclassical growth model, with one consumption cum capital good, the output of which is given by a production function f, and T= {(x,y) : x > 0,0 <y<f(x)}. A utility function w is defined directly over consumption c = f(x) – y; so, here, ut(x,y) = w(c). It is noteworthy that the general reduced-form formulation may allow for direct stock effects over and above the effect of consumption on utilities and is a much richer framework.
Ramsey studied the one-sector model with time as a continuous variable and the infinite sum accordingly replaced by an integral. He obtained a remarkable rule governing the optimal savings rate at each point of time, essentially the Euler equation in the calculus of variations. Ramsey focused primarily on the undiscounted case, that is, δ = 1; much later Tjalling Koopmans (1965) and David Cass (1965) treated the discounted case, that is, δ < 1 thoroughly. The undiscounted case presents the problem that the welfare sum may not converge. Ramsey circumvented the problem by assuming that, due to constraints on tastes or technology, there is an upper bound on the achievable level of utility, and he sought to minimize the divergence of the utility level along paths from this bliss level. This problem received much more careful and rigorous attention later, employing variants of Carl C. von Weizsäcker’s (1965) overtaking criterion, which provides only a partial ranking over paths but can still yield a workable notion of optimality. Simultaneously, following Gale and William Brock (1970), existence of an optimal program was explored carefully in multisector models. The so-called golden rule program or optimal stationary state, where the per period utility along constant feasible programs is maximized, plays a role analogous to that of the bliss level in Ramsey.
TURNPIKE THEORY
A 1945 paper by another mathematician, John von Neumann, on a general equilibrium model primarily stressing production, which established the existence of a balanced growth path with a maximal rate of expansion and associated present value prices at which the activities employed are profit maximizing, led to the turnpike property first explicated in 1958 by Robert Dorfman, Paul Samuelson, and Robert Solow. Several authors followed with rigorous proofs of turnpike theorems, which showed that planning paths, with long horizons and terminal stock accumulation as the goal, would tend to take advantage of the maximal rate of growth of the balanced path, the turnpike, by orienting stocks to those of the turnpike for most of the horizon.
In the context of optimal growth models, Roy Radner’s (1961) value-loss method was adapted by authors such as Gale and McKenzie (1968) to show that the optimal stationary state plays the role of the turnpike and optimal programs converge to the golden rule, albeit with some interesting qualifications when linearities are present in the model in an essential way. This convenient property is, in general, limited to the undiscounted case. José Scheinkman (1976) showed that for large discount factors the turnpike property holds; however, for small discount factors Jess Benhabib and Kazuo Nishimura (1985) showed that optimal cycles are possible. Later Michele Boldrin and Luigi Montruchhio (1986) showed that chaotic optimal paths are possible. These papers gave rise to the literature on the possibility of nontrivial business cycles in deterministic macromodels with a representative agent carrying out the infinite horizon optimization. Also, the effect of variations in parameters, especially of the discount factor, on the optimal policy function became a topic of considerable interest following a remarkable 1992 paper by Gerhard Sorger (in this connection, see also Mitra et al. 2006).
Following Paul Romer (1986), models allowing aspects of externalities or public goods in production, which may have many equilibrium paths, gave rise to the literature of the indeterminacy problem surveyed in Nishimura and Alain Venditti (2006). Integration of turnpike theory with general equilibrium theory began with Robert Becker (1980) who, following up on Ramsey’s conjecture, investigated the long-run wealth distribution that arises in a Ramsey-type general equilibrium model with several infinitely lived agents and related it to the discount factors of the agents.
DUALITY THEORY
Since Edmund Malinvaud (1953), the well-known connection between the static theory of price and efficiency went through a major evolution in the context of efficient allocation of resources in infinite horizon models. In the context of optimal growth models, McKenzie (1986) provides a transparent version of Martin Weitzman’s 1973 proof of the existence of present value prices at which in each period myopic profit maximization takes place, with utility treated as an output. These are likened to familiar individual decision rules in competitive equilibrium, and the optimality conditions are usually called the competitive conditions. The distinctive feature of the infinite horizon aspect of the problem is that, in general, these need to be supplemented by a transversality condition, which requires that an appropriate limit condition be satisfied by the value of stocks, a condition that cannot be interpreted as a myopic decision rule. There has been some success at obtaining alternate price characterizations of optimal paths that require only period-by-period conditions to be verified by myopic agents; that is, the optimal path is “decentralizable” (see Majumdar 1992). Indeed, in a fairly wide class of production models satisfying a reachability condition, the competitive conditions alone characterize optimality (see Dasgupta and Mitra 1999a).
Duality theory played a significant role in grounding and extending the familiar macroeconomic concept of national product, based upon explicit welfare considerations in dynamic models. Weitzman (1976) observed that in a continuous-time framework, national income at any time along an optimal path is a proxy measure of the hypothetical level of constant consumption that, if possible to maintain forever, provides the same level of welfare found along the optimal path. While the same is not necessarily true for competitive equilibrium paths in general, in some models involving exhaustible resources, it is possible to extend such results to paths that are not necessarily optimal but satisfy the competitive equilibrium conditions only (see Dasgupta and Mitra 1999b). The interpretation of stocks in Weitzman is very general and may include exhaustible resources, environmental factors, and the like, leading to this literature developing a strong connection with the concept of “green” national income accounting. Connections with sustainable consumption paths have also been explored following Solow (1986).
Convexity is essential for the necessity side of duality theorems. Nonconvexity arises in many models, such as those with natural resources. The qualitative properties are significantly different (see Majumdar 2006). Nonconvexity is also of importance in endogenous growth models following Romer (1986).
Optimal growth allowing uncertain technology, changing technology, and so forth have been studied. With uncertainty, the possibility of ensuring long-run survival may temper optimality considerations in interesting ways (see Olson and Roy 2006). The additive-separable form of the welfare function is a severe restriction, and the literature on recursive utility functions seeks to address that (see Boyd 2006). Introducing population growth as a control variable raises interesting questions about appropriate forms of the welfare function.
An assessment is needed of the implications, the importance, and the weaknesses of the manifold developments in optimal growth theory, especially of the literature on cycles, chaos, and indeterminacy in the context of macrodynamics and business cycles, bearing in mind that the phenomenon of excess capacity does not appear in these models. An assessment is also needed of the models’ practical significance for planning, bearing in mind that practical planning is expected to be finite horizon in nature and to rely on decentralized mechanisms for its implementation. This may yield more productive directions for future research.
SEE ALSO Economic Growth; Golden Rule in Growth Models; Maximization; Neoclassical Growth Model; Optimizing Behavior; Social Welfare Functions
BIBLIOGRAPHY
Becker, Robert A. 1980. On the Long-Run Steady State in a Simple Dynamic Model of Equilibrium with Heterogeneous Households. Quarterly Journal of Economics 95 (2): 375–382.
Benhabib, Jess, and Kazuo Nishimura. 1985. Competitive Equilibrium Cycles. Journal of Economic Theory 35: 284–306.
Boldrin, Michele, and Luigi Montruchhio. 1986. On the Indeterminacy of Capital Accumulation Paths. Journal of Economic Theory 40: 26–39.
Boyd, John H. 2006. Discrete-Time Recursive Utility. In Handbook on Optimal Growth, eds. Rose-Anne Dana, Cuong Le Van, Tapan Mitra, and Kazuo Nishimura, 251–272.
Berlin and New York: Springer. Brock, William A. 1970. On Existence of Weakly Maximal Programmes in a Multi-sector Economy. Review of Economic Studies 37 (2): 275–280.
Cass, David. 1965. Optimal Growth in an Aggregative Model of Capital Accumulation. Review of Economic Studies 32: 233–240.
Dasgupta, Swapan, and Tapan Mitra. 1999a. Optimal and Competitive Programs in Reachable Multi Sector Models. Economic Theory 14 (3): 565–582.
Dasgupta, Swapan, and Tapan Mitra. 1999b. On the Welfare Significance of National Product for Economic Growth and Sustainable Development. Japanese Economic Review 50 (4): 422–442.
Dorfman, Robert, Paul A. Samuelson, and Robert M. Solow. 1958. Linear Programming and Economic Analysis. New York: McGraw-Hill.
Gale, David. 1967. On Optimal Development in a Multi-sector Economy. Review of Economic Studies 34: 1–18.
Koopmans, Tjalling. 1965. On the Concept of Optimal Economic Growth. Pontificae Acadmiae Scientarum Varia 28: 225–300.
Lucas, Robert E. 1988. On the Mechanics of Economic Development. Journal of Monetary Economics 22: 3–42.
Majumdar, Mukul, ed. 1992. Decentralization in Infinite Horizon Economies. Boulder, CO: Westview.
Majumdar, Mukul. 2006. Intertemporal Allocation with a Non-convex Technology. In Handbook on Optimal Growth, eds. Rose-Anne Dana, Cuong Le Van, Tapan Mitra, and Kazuo Nishimura, 171–202. Berlin and New York: Springer.
Malinvaud, Edmund. 1953. Capital Accumulation and Efficient Allocation of Resources. Econometrica 21: 233–268.
McKenzie, Lionel W. 1968. Accumulation Programs of Maximum Utility and the von Neumann Facet. In Value, Capital, and Growth: Papers in Honour of Sir John Hicks, ed. J. N. Wolfe, 348–357. Edinburgh: Edinburgh University Press.
McKenzie, Lionel W. 1986. Optimal Economic Growth, Turnpike Theorems, and Comparative Dynamics. In Handbook of Mathematical Economics, Vol. 3, eds. Kenneth J. Arrow and Michael D. Intriligator, 1281–1358. New York: North Holland.
Mitra, Tapan. 2006. Duality Theory in Infinite Horizon Optimization Models. In Handbook on Optimal Growth, eds. Rose-Anne Dana, Cuong Le Van, Tapan Mitra, and Kazuo Nishimura, 55–84. Berlin and New York: Springer.
Mitra, Tapan, Kazuo Nishimura, and Gerhard Sorger. 2006. Optimal Cycles and Chaos. In Handbook on Optimal Growth, eds. Rose-Anne Dana, Cuong Le Van, Tapan Mitra, and Kazuo Nishimura, 141–169. Berlin and New York: Springer.
Nishimura, Kazuo, and Alain Venditti. 2006. Indeterminacy in Discrete-Time Infinite-Horizon Models. In Handbook on Optimal Growth, eds. Rose-Anne Dana, Cuong Le Van, Tapan Mitra, and Kazuo Nishimura, 273–296. Berlin and New York: Springer.
Olson, Lars J., and Santanu Roy. 2006. Theory of Stochastic Optimal Growth. In Handbook on Optimal Growth, eds. Rose-Anne Dana, Cuong Le Van, Tapan Mitra, and Kazuo Nishimura, 297–336. Berlin and New York: Springer.
Radner, Roy. 1961. Paths of Economic Growth that Are Optimal with Regard Only to Final States: A Turnpike Theorem. Review of Economic Studies 28 (2): 98–104.
Ramsey, Frank. 1928. A Mathematical Theory of Saving. Economic Journal 38: 543–559.
Romer, Paul. 1986. Increasing Returns and Long-Run Growth. Journal of Political Economy 94: 1002–1037.
Scheinkman, José A. 1976. On Optimal Steady States of n-Sector Growth Models When Utility is Discounted. Journal of Economic Theory 12 (1): 11–30.
Solow, Robert M. 1986. On the Intergenerational Allocation of Natural Resources. Scandinavian Journal of Economics 88: 141–149.
Sorger, Gerhard. 1992. On the Minimum Rate of Impatience for Complicated Optimal Growth Paths. Journal of Economic Theory 56: 160–179.
Sorger, Gerhard. 2006. Rationalizability in Optimal Growth Theory. In Handbook on Optimal Growth, eds. Rose-Anne Dana, Cuong Le Van, Tapan Mitra, and Kazuo Nishimura, 85–113. Berlin and New York: Springer.
Von Neumann, John. 1945. A Model of General Economic Equilibrium. Review of Economic Studies 13: 1–9.
Von Weizsäcker, Carl C. 1965. Existence of Optimal Programs of Accumulation for an Infinite Time Horizon. Review of Economic Studies 32 (2): 85–104.
Weitzman, Martin L. 1973. Duality Theory for Infinite Horizon Convex Models. Management Science 19: 783–789.
Weitzman, Martin L. 1976. On the Welfare Significance of National Product in a Dynamic Economy. Quarterly Journal of Economics 90: 156–162.
Swapan Dasgupta