Optimal Growth Model Research Articles

Uncertain horizon and stability: Analysis in an optimal growth model

In a one-sector optimal growth model with uncertainty about production optimal capital stocks converge in distribution to a stochastic modified golden rule [see, for example Brock and Mirman (1972, 1973)]. We show that such a result cannot be obtained, in general, if in addition to the random one-period shocks to production there is also a lasting shock to the production function at some random date in the future; however, the conditional optimal capital stocks ‘bunch together’ over time, i.e., a turnpike result for optimal programs is proved.

Oscillations in optimal growth models

Oscillations in optimal growth models

OPTIMAL REGIONAL GROWTH MODELS: MULTIPLE OBJECTIVES, SINGULAR CONTROLS, AND SUFFICIENCY CONDITIONS

ABSTRACT We develop a generalized Harrod‐Domar type of model for optimal regional growth which allows determination of the optimal allocation of regional public investments and which considers multiple growth objectives, as well as both normal and singular fiscal instruments. This general model is shown to include most of the previous optimal regional growth models as special cases. Necessary conditions for some special cases of the general model are analyzed in detail, and decision rules associated with derived optimal regional growth policies are articulated. These special cases verify that singular controls do exist in certain instances, and that they must be considered for the complete specification of optimal regional growth policies; this is significant because singular controls have not been previously analyzed in the literature. We conclude the paper with a discussion of sufficiency conditions for optimal regional growth models which is more general than that given by previous authors.

INFLATION DEBT AND FISCAL POLICY ATTITUDES

porally optimal growth model, where the financing shares are precommitted away from zero and he shows that the existence of endogenous inflation taxes induces non-neutrality. Dornbusch (1977) also considered the mixedfinancing assumption using a non-optimal growth model and established non-neutrality and the increased possibility of instability when the share of debt finance is raised. Thus another policy concern about fiscal deficits involves the problem of servicing outstanding debt. In a survey of literature on the stability of models of debt finance, Christ (1979) showed that it may be appropriate, even necessary to introduce an endogenous component into the fiscal deficit. Indeed, the contribution of this paper relies on Christ's suggestion of endogenising fiscal policy to avoid instability. The degree to which fiscal deficits have to be and are endogenised, albeit in a model with flexible prices, is parameterised in this paper. The extent to which governments decide to cover themselves against possible instability can be shown to have important short and long run implications for inflation (and by implication, for output). In this context the concept of an excessively cautious or conservative government as opposed to an optimistic or liberal government is introduced, giving the paper a political economy flavour. The paper shows that, at low rates of inflation, a shift in the money-debt finance share will fulfil monetarist predictions in the short and long run if the government is conservative but will not if it is liberal. The monetarist label is used here in the sense that a shift away from debt finance will The author is grateful for helpful discussions with William Scarth and Mono Chatterji. The views expressed herein are solely those of the author.

Monetary Uncertainty and Investment in an Optimizing, Rational Expectations Model with Income Taxes and Government Debt

THE EFFECT OF MONETARY GROWTH on an economy's investment has long been of interest to economists, and monetary growth models provide a convenient tool for the analysis of this question. The standard optimizing growth model postulates that money is injected into the private sector through transfer payments which typically either are of lump sum form or are proportional to wealth, income, or money holdings. (See, for example, Brock (1974), Calvo (1979), Fischer (1979), and Gertler and Grinols (1982).) In these models monetary changes are tied directly to changes in subsidy or tax rates, and greater expected money growth generally either has no effect on investment or increases it through a Tobin (1965) effect. In this paper we analyze the effects that a stochastic monetary policy has on investment in an economy with individual optimization, rational expectations, and a government which makes expenditures and finances them through borrowing, money creation, and proportional income taxes. With such a formulation, money can be injected into the private sector through government purchases and open market operations as well as through transfers, and the link between money growth and subsidy or tax rates is weakened or nonexistent. We assume that the nominal income tax rates are constant over time. Given tax rates and (stochastic) government consumption, the stochastic monetary growth rule followed by the government induces a stochastic government debt policy. The effects that changes in the stochastic monetary growth rate have on investment depend on the nature of the depreciation deduction and on the relationship between the coefficient of relative risk aversion and the tax rate on production income. When depreciation deductions are based on historical costs, an equilibrium is unique so long as the coefficient of relative risk aversion is greater than the tax rate applicable to production income. In this case an increase in the mean rate of money growth decreases equilibrium investment while a mean preserving spread in the money growth rate increases investment. Given Friend and Blume's (1975) finding that the coefficient of relative risk aversion is substantially greater than one, this seems to be the empirically relevant case. If the coefficient of relative risk aversion is less than the tax rate on production income, multiple equilibria are possible, and the effects on investment of changes in the stochastic money growth rate are generally indeterminate. When depreciation deductions are based on replacement costs, investment is independent of the stochastic money growth rate. The paper is organized as follows. Section 2 develops the optimization problem of the representative individual when depreciation deductions are based on historical costs. Section 3 describes the behavior of the government and develops the rational expectations equilibrium for this economy. Section 4 examines the effects on investment of changes in the stochastic money growth rate. Section 5 briefly reconsiders the model when depreciation deductions are based on replacement costs.

Green manure and nitrogenous fertilizer—a two-sector optimal-growth model

AbstractA continuous-time model of a two-sector trading economy with a finitely-saturating production function and constant population is constructed. To apply it to the farm-economy problem of optimal trading of cereal for chemical nitrogenous fertilizer and of optimal allocation of land for green manure the special assumptions are made of similar technologies in both sectors and of a nitrogen-capital loss proportional to the cereal production. Optimal-control theory is applied to get the pair of controls that maximizes an infinite-horizon integral of utility of consumption. The analysis of the model is shown to reduce to that of the Ramsey-Koopmans one-sector neoclassical model of optimal growth. Calculations of the feedback control and optimal time-paths for the standardized dimensionless model are tabulated for a range of 7 utility functions of constant elasticity of marginal utility of magnitude 1 to 4 and of production functions of degree 2 to 4. Two particular analytic solutions are given. The application of the results both to the farm-economy model and to the Ramsey-Koopmans model is illustrated.

On the indeterminacy of capital accumulation paths

In neoclassical optimal growth models the stability of the accumulation paths depends on the discount parameter. We prove that, for discount factors small enough, the policy function which describes an optimal path can be of any type. The result is achieved using the notion of α-concavity. We adopt a constructive approach. Given any twice differentiable map we show how to construct an optimal growth problem which produces that map as the optimal policy function. A consequence is that “chaos” can appear in these models. We also provide bounds on the values of the discount parameter for which “indeterminacy” is possible.

The existence of optimal consumption policies in optimal economic growth models with nonconvex technologies

An existence theorem for a class of continuous time infinite horizon optimal growth models is developed. The underlying technology set is not assumed to be convex, instead the “slices” of the technology set corresponding to a fixed capital stock vector are assumed convex and compact in the consumption and net investment variables. This allows consideration of the case of increasing returns to scale. Existence of an optimal capital stock and consumption policy is proved directly without consideration of the underlying Hamiltonian dynamical system that arises from applying Pontryagin's maximum principle.

The invariance principle and income-wealth conservation laws

Abstract The purpose of this paper is to uncover ‘hidden’ conservation laws in the models with heterogeneous capital goods. The first conservation law uncovered is a pseudo-net-productivity relation which implies that the rate of change in national income is equal to the discount rate multiplied by the utility-value-of-investment. Secondly, the ‘modified income’ conservation law is confirmed when the economy possesses taste change and ⧸ or technical change. From an empirical perspective, knowledge of conservation laws can assist in the econometric testing of optimal growth models. In addition to the capital-labor ratio and saving ratio the empirical validity of the models can be tested by the appropriate income-wealth ratios.

A note on comparative statics and dynamics of stationary states

Certain results are communicated regarding effects of parameter changes, when the stationary state is stable, on the stationary state and an optimal paths in its neighbourhood in discrete time multi-sectoral optimal growth models where period utilities are discounted.

A discrete model of optimal economic growth

Contrary to most of the literature on optimal economic growth a discrete rather than a continuous model is investigated. It is shown that in such a discrete model it is easy to account for relatively freely changing functions and parameters. Thus, the production function, labor, the investment ratio, and the parameters for time preference, marginal utility, and depreciation are all allowed to depend on time. Using discrete dynamic programming methods, optimal investment policies are determined explicitly. These generalize important results from previous literature on optimal economic growth.

Risk-aversely efficient random variables: Characterization and an application to growth under uncertainty

Consider a problem of choice from a set R of multivariate random variables. Let us examine only efficient elements of R which are optimal choices of risk averse decision-makers (whose aim is to maximize expected utility over R). We obtain a price characteristic of all risk-aversely efficient random variables in R. This result has been applied to multi-sector optimal growth model to obtain a characterization by competitive prices of all risk-aversely efficient stationary consumption programs.

A new concept of stability and dynamical economic systems

A new concept of stability, strict set stability, is defined and applied to optimal growth models. Autonomous differential equations have a strictly stable set if there exists a bounded set which the solutions visit within a finite time and in which they remain thereafter. Global asymptotic stability, quasi-stability, and limit cycles are special cases of the strict set stability. It is shown that under certain conditions optimal paths in optimal growth models have a sufficiently small, strictly stable set.

Comparative dynamics in the one-sector optimal growth model

The comparative dynamics properties of the one-sector optimal growth model with discounting are derived. A dynamic programming argument is used to show that an increase in the utility discount factor increases the capital stock at each time, initially lowers consumption and eventually increases consumption as compared to the original program. The present value of the stream of increments in capital is also shown to be positive at the initial support prices. The relationship with the Correspondence Principle is discussed as well.

The maximin objective function and less developed countries

Modifying a traditional optimal growth model by using a maximin objective function and allowing the country to borrow from abroad is a useful process for examining the problems faced by LDC's. For example, an LDC's planners may realize the importance of long-term planning, but, because of the gravity of the country's current situation, be forced to emphasize short-term goals. In the model developed here, the planners' desire to maximize current sustainable consumption precludes the possibility of long-term gorwth. The ability to borrow from abroad eases this dilemma but does not remove it.

Total Utility, Overlapping Generations and Optimal Population

The necessary and sufficient conditions for the solution to an optimal population growth model with overlapping generations, using the sum of total utility per generation as the objective function are derived. The solution for a specific, steady state model and an example are presented, and compared to that for a similar model without overlapping generations. In both cases, a positive, but less than infinite, optimal growth rate is found. Next, since an additively separable individual utility function is used, the differences between a total utility per generation model and a total utility per period model, a la Lerner, are discussed. Finally, the results of the total utility model are compared to those from a model in which the discounted sum of per capita utility is the objective function. An extension to the Samuelson-Lerner Utility Paradox, concerning the optimal rate of population growth, is discussed.

Technological renewal of natural resource stocks

This paper presents a macroeconomic model of optimal growth with stocks of an exhaustible resource. These stocks can be renewed by technological processes, which imply some specific investment costs. The dates, levels and the order of the processes are endogeneous decisions. The main results are: (1) the present value of the exhaustible resource follows a step function in time falling whenever a new stock becomes available and related to the physical return of the process; and (2) shadow prices allow to decentralize the allocation of resources to stock renewal, but they do not permit to decentralize the choice of optimal timing. This would justify the existence of a coordinating agency.

Stability and optimal exploitation over an infinite time horizon of interacting populations

AbstractModels of optimal control of two‐species ecological systems are considered on an infinite time horizon. Conditions are obtained under which the optimal exploitation of the system is stabilizing. Thus, a stationary state toward which all optimal trajectories must converge is defined. This extends to this class of models, the co‐called turnpike property of optimal economic growth models.

Complete comparative static differential equations

Equations having form (1) arise as first-order conditions in microeconomic constrained optimization models, as defining characterizations for general equilibrium and macroeconomic models, and as steady-state solution characterizations for descriptive and optimal growth models. (See Silberberg [12], Hansen [7], Cass and Shell [3], and Brock [l].) If Y :R' -+ is continuously differentiable over a neighborhood of a point (x0, a”) in R n+l for which Y(xO, LX’) = 0 and IY,( x0, or’)] # 0, then the implicit function theorem guarantees the existence of a continuously differentiable function x: N(cr’) + over some open neighborhood N(crO) of CX’ in R such that dx z (ix) = YX(X(X), a)-lYJx(a), X), G! E N(cP). (2)

Certainty Planning in an Uncertain World: A Reconsideration

In a recent paper, Merton (1975) seeks to extend the theory of growth under beyond the customary concern of demonstrating the existence, uniqueness, and stability of the asymptotic stationary distributions of consumption, output, and capital stock. His goal is to derive additional properties of these asymptotic distributions and, in particular, to investigate the (in an expected value sense) induced by assuming a certainty model when, in fact, outcomes are uncertain (Merton (1975, p. 375)). For his (only) source of uncertainty, Merton (1975) models the size of the aggregate labour force as a diffusion process. By further assuming the aggregate savings policy to be linear homogeneous in output, and specifying per worker output as being Cobb-Douglas in capital per worker, Merton (1975) is able to derive closed form expressions for the associated stationary distributions of per worker consumption, output, and capital stock. By comparing the expected values of these distributions with the corresponding steadystate values obtained by applying the identical savings rule to the same model specification, though in the absence of uncertainty, he demonstrates biases in the certainty estimates of the corresponding uncertainty expected values. Specifically, in his model, expected per worker consumption, output, and capital stock under strictly exceed their respective certainty levels. Merton (1975, p. 383) is therefore led to conclude that care must be taken in using the certainty analysis even as a first moment approximation theory. This note is an attempt to replicate Merton's (1975) results in an alternative discrete time version of the one-sector model of optimal stochastic growth. It suggests that diffusion process models have no monopoly, or even apparent advantages, in pursuing characterizations of the asymptotic distributions. Our work confirms the assertion that certainty estimates may be biased approximations of analogous values, although our biases are exactly opposite in sign to those of Merton's (1975) model. In our model, the linear homogeneous savings function, which results from optimizing behaviour under (rather than being imposed as in Merton (1975)), is the same as that arising in the certainty specification. Thus while our results illustrate Mirrlees' (1974, p. 48) contention that the of may not, at the aggregative level, have significantly large influence on optimal policies, the indicated biases nevertheless suggest that the introduction of does significantly alter the description of economies growing along optimal paths.