Turnpike property and convergence rate for an investment model with general utility functions

In this paper we aim to address two questions faced by a long-term investor with a power-type utility at high levels of wealth: one is whether the turnpike property still holds for a general utility that is not necessarily differentiable or strictly concave, the other is whether the error and the convergence rate of the turnpike property can be estimated. We give positive answers to both questions. To achieve these results, we first show that there is a classical solution to the HJB equation and give a representation of the solution in terms of the dual function of the solution to the dual HJB equation. We demonstrate the usefulness of that representation with some nontrivial examples that would be difficult to solve with the trial and error method. We then combine the dual method and the partial differential equation method to give a direct proof to the turnpike property and to estimate the error and the convergence rate of the optimal policy when the utility function is continuously differentiable and strictly concave. We finally relax the conditions of the utility function and provide some sufficient conditions that guarantee the turnpike property and the convergence rate in terms of both primal and dual utility functions.     

A ‘funnel’ turnpike theorem for optimal growth problems with discounting

A turnpike theorem for the optimal control problem with discounting is given. The optimal trajectory is shown to lie in an exponentially bounded region of the optimal steady-state. This region, referred to as a funnel, is determined by the discount rate of the problem. The funnel theorem reduces to the classical turnpike theorem when the discount rate is zero.     

Optimal control on an infinite time horizon: The turnpike approach

The optimal control problem with infinite horizon is defined. Under assumptions motivated by the theory of optimal economic growth the controllability of the system, the turnpike property, the asymptotic behavior and the existence of optimal trajectories are fully explored.     

A continuous-time portfolio turnpike theorem

We prove a continuous-time portfolio turnpike theorem. The proof uses the theory of martin-gales and is more intuitively appealing than the usual discrete-time mode of proof using dynamic programming. When the interest rate is strictly positive, the present value of any contingent claim having payoffs bounded from above can be made arbitrarily small when the investment horizon increases. Thus an investor concentrates his wealth in buying contingent claims that have payoffs unbounded from above at the very beginning of his horizon. As a consequence, it is the asymptotic property of his utility function as wealth goes to infinity that determines his optimal investment strategy at the very beginning of his horizon.     

A turnpike theorem for rolling plans

This paper analyzes the asymptotic properties of rolling plans in a multi-sector growth model with time-independent preferences and technology. Assuming that the model under consideration has a unique turnpike towards which all the finite optimal programs bend, the paper shows that, if plans are constantly revised with a fixed but sufficiently long planninghorizon, the resulting growth path converges to a neighborhood of the turnpike.     

The efficient paths to infinity in closed leontief models

For a simple Leontief technology, the results of finite-horizon turnpike theorems are extended to the case of infinite horizon. Further qualitative information is obtained about the set of efficient programs. Uniqueness is discussed, and shown by an example to be not true in general.     

The turnpike of dynamic general equilibrium paths and its insensitivity to initial conditions

Asymptotic turnpike theorems in the case where the future is discounted are traditionally derived under the assumptions that a utility function is twice differentiable and that is Hessian is negative definite. Replacing these assumptions by a weaker one. I give a proof of the theorem simpler than existing ones.     

The primal route to the turnpike and asymptotic stability

Asymptotic turnpike theorems in the case where the future is discounted are traditionally derived under the assumptions that a utility function is twice differentiable and that is Hessian is negative definite. Replacing these assumptions by a weaker one. I give a proof of the theorem simpler than existing ones.     

A global robustness measure for input–output projections from ESA and SNA tables

Input–output (interindustry) data are in wide use in empirical research and constitute an integral part of the European System of Accounts (ESA) and of the System of National Accounts (SNA). In a strict sense, however, these data are merely estimates of the true economic relationships. Therefore, we suggest a measure of robustness of input–output projections with respect to errors or changes in the underlying Leontief matrix. Our measure is based on the mathematical theory of norms and characterizes a complete Leontief matrix. Thereby, no assumptions are required on the distribution of the matrix elements. We discuss alternative numerical-computing algorithms and provide useful bounds and approximation formulas. The paper concludes with a large set of empirical sample applications.     

THE ROLE OF SUPPLY CONSTRAINTS IN MULTIPLIER ANALYSIS

Multiplier analysis based upon the information contained in Leontief's inverse is undoubtedly part of the core of the input–output methodology and numerous applications and extensions have been developed that exploit its informational content, both at the national and regional levels. Nonetheless there are some implicit theoretical assumptions whose policy implications need to be assessed. This is the case for the ‘excess capacity’ assumption, which implies that resources are available as needed to adjust production to new equilibrium states. In an actual economy, however, new resources are often scarce and always costly. When supply constraints intervene, the assessment of the effects of government demand policies may be substantially different from that of the standard Leontief multiplier matrix. Using a closed general equilibrium model that incorporates supply constraints, we perform some simple numerical exercises and proceed to derive two ‘constrained’ multiplier matrices, based upon the implicit Jacobian matrix, that can be compared with the standard ‘unconstrained’ Leontief matrix.     

Econometric analysis of linearized singular dynamic stochastic general equilibrium models

In this paper I propose an alternative to calibration of linearized singular dynamic stochastic general equilibrium models. Given an a-theoretical econometric model as a representative of the data generating process, I will construct an information measure which compares the conditional distribution of the econometric model variables with the corresponding singular conditional distribution of the theoretical model variables. The singularity problem will be solved by using convolutions of both distributions with a non-singular distribution. This information measure will then be maximized to the deep parameters of the theoretical model, which links these parameters to the parameters of the econometric model and provides an alternative to calibration. This approach will be illustrated by an application to a linearized version of the stochastic growth model of King, Plosser and Rebelo.     

On the smoothness of optimal paths II: some local turnpike results

We study the local turnpike property for two classes of infinite-horizon discrete-time deterministic maximization problems which have common applications, e.g., optimal growth theory. We follow a functional-analytic approach and rely on an implicit function theorem for the space of the sequences which converge to zero. We shall assume the existence of an optimal path which is not necessarily a steady-state. Relying on material developped in Blot and Crettez (Decis Econo Finance 27:1–34, 2004), “On the smoothness of optimal paths” Decis Econ Finance, 21:1–34, 2004), we provide conditions under which a variation in the initial conditions (i.e., capital stock and discount rate) yields an optimal solution which converges toward a reference solution when time becomes infinite. We also provide new results on bounded solutions of difference equations. We gratefully thank the editor, Silvano Holzer, and two anonymous referees for remarks and advices on a previous version of this paper.     

Optimization in a Dynamic Leontief Model with Restricted Value Growth

The purpose of this paper is to develop a dynamic Leontief model of an msector economic system in which the production of all goods requires one time period and one primary factor, but no capital stocks of any good, and in which the total value of outputs from all sectors is required to grow at a specified rate in each period. The requirement of a fixed rate of total value growth is less restrictive than the familiar condition of balanced growth across all sectors, and it permits the definition and analysis of interesting finite-period optimization problems. Specific results of the paper include the following: (1) the proof that a value-added maximization problem with an unrestricted initial state will experience consumption in exactly one sector in each time period, and willyield an optimal value function which is linear in the variables that describe the terminal state of the system; (2) the development of an efficient Dantzig-Wolfe procedure for analysis of the total value-added maximization problem where both the initial and terminal states are specified; (3) the derivation of testable properties that will guarantee the attainability of a specified target state from a specified initial state of the system; (4) a formal comparison of some basic characteristics of total value growth and balanced value growth.     

Epistemological reasons for preferring component analysis to factor analysis

Spearman's and Thomson's mathematical controversy over factor theory was forgotten when it became evident that empirical tetrad-differences bound away from zero (and when empirical evidence argued the need for extracting more factors from a matrix). In fact, both their models lead to zero tetraddifferences. Being more interested in the psychological than in the mathematical aspects of Spearman's model. Thompson remained indifferent to mathematical aspects of multiple factor analysis when Thurstone theorized it. Thus, he did not perceive that his counter-example negated the assumption Thrustone shared the rank of the matrix. The idea that components to be extracted must be equal to the rank of the matrix is not assumed in Hotelling's component model: as a result, this is the first epistemological reason for preferring component analysis to factor analysis. A second epistemological reason is the central theorem of Thurstone's multiple-factor model, which can be criticized because it is an assumption that, the rank of a complete matrix being n, it becomes k when commonalities are in the principal diagonal. This assumption goes against common sense, a fact demonstrated through comparison between residuals after k components have been extracted and after k principal factors have been extracted.     

The First Two Eigenvalues of Large Random Matrices and Brody's Hypothesis on the Stability of Large Input–Output Systems

Bródy (1997) notices that for large random Leontief matrices, namely non-negative square matrices with all entries i.i.d., the ratio between the subdominant eigenvalue (in modulus) and the dominant eigenvalue declines generically to zero at a speed of the square root of the size of the matrix as the matrix size goes to infinity. Since then, several studies have been published in this journal in attempting to rigorously verify Bródy's conjecture. This short article, drawing upon some theorems obtained in recent years in the literature on empirical spectral distribution of random matrices, offers a short proof of Bródy's conjecture, and discusses briefly some related issues.     

A New Method for the Identification of Patterns in Input–Output Matrices

This paper suggests a new algorithm for selecting the input–output (IO) coefficients of a Leontief matrix in order of importance, so providing an analytical method for decomposing an IO matrix. It avoids the choice of arbitrary thresholds for eliminating flows or coefficients, and allows for circular relationships. For this purpose, a simple inverse-important criterion has been chosen, which is consistent with the logic of the Leontief model. A procedure that greatly reduces the computational burden is then devised. This method permits new comparisons of IO structures of different countries or regions, for identifying their different degree of internal integration and their reciprocal influence through the exchange of intermediate goods. An application to an IO model for seven European Community countries for 1980 is then presented.     

Economic Impact of the Structural Changes of Production Technology: A Robust Approach

In this article, edge theorem and boundary theorem, one of the most important results obtained recently in the field of robust control theory, is applied to analyze the economic impact of structural change of production technology in one economic sector, by means of the famous dynamic Leontief input–output model. Two theorems, parallel to edge theorem and boundary theorem, are given in terms of economics in this paper. It is argued that in a sense the complex analysis about the economic impact of production technological changes is equivalent to a one-dimensional computation problem, which is easy to solve. A case study is given in the last section of this paper.     

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.     

Solvability of perturbation solutions in DSGE models

We prove that the undetermined Taylor series coefficients of local approximations to the policy function of arbitrary order in a wide class of discrete time dynamic stochastic general equilibrium (DSGE) models are solvable by standard DSGE perturbation methods under regularity and saddle point stability assumptions on first order approximations. Extending the approach to nonstationary models, we provide necessary and sufficient conditions for solvability, as well as an example in the neoclassical growth model where solvability fails. Finally, we eliminate the assumption of solvability needed for the local existence theorem of perturbation solutions, complete the proof that the policy function is invariant to first order changes in risk, and attribute the loss of numerical accuracy in progressively higher order terms to the compounding of errors from the first order transition matrix.     

ON THE ECONOMIC INTERPRETATION OF THE BRÓDY CONJECTURE

The aim of the paper is to study the economic aspects of the Bródy conjecture: an increase in the size of a (random) input matrix causes a decline in the ratio of its subdominant and dominant eigenvalues and implies faster convergence to equilibrium [Bródy, A. (1997) The Second Eigenvalue of the Leontief Matrix. Economic Systems Research, 9, 253–258]. Simulation results provide evidence that this ratio depends inversely on the level of data aggregation and can therefore not be a good indicator of the speed of convergence of an economy to its equilibrium path. We show that this is consistent with findings based on actual input–output tables of EU member states. These results imply that theorems about the speed of convergence of random matrices are not useful in describing the cyclical dynamics of real economies.