Michael R. Caputo

How to do comparative dynamics on the back of an envelope in optimal control theory

Abstract The dynamic primal-dual methodology of Caputo (1988) is extended and applied to a general nonautonomous optimal control problem with a fixed and finite time horizon, a fixed vector of initial states, a free vector of terminal states, a time-independent vector of parameters, and a vector of nondifferential constraints on the state and control variables. A symmetric negative semidefinite matrix subject to constraint is shown to characterize the qualitative properties of the model as implied by the dynamic maximization assertion.

Comparative Dynamics via Envelope Methods in Variational Calculus

The primal-dual methodology of Samuelson (1965) is extended and applied to a nonautonomous variational calculus problem with a fixed vector of initial stocks, fixed initial and terminal time values, a free vector of terminal stocks, and a time-independent vector of parameters. It is shown that if the solution of the variational problem is smooth enough, the qualitative effects of parameter perturbations on the entire optimal arcs can be represented by a generalized Slutsky-type matrix, which holds in integral form and is symmetric negative semidefinite. Sufficient conditions for the optimal value function to be convex in the parameters are also given.

Price Changes, Maintenance, and the Rate of Depreciation

This study estimates rates of deterioration and depreciation for a sample of used privately owned single- and twin-engine aircraft over the period 19711991. The adoption of a strict liability standard in the 1970s lead to a 775 increase in liability expenses for the manufacturers of private planes between 1977 and 1985, resulting in sharp increases in the prices of new and used planes throughout the late 1970s and 1980s. This period of rapid price inflation coincides with a decrease in the depreciation rates for used single- and twin-engine aircraft after 1975. In addition, our results indicate that the rate of deterioration is positively related to the required cost of engine maintenance. These findings call into question the commonly invoked assumption that depreciation rates may be treated as exogenously determined constants, and lend support to the hypothesis that deterioration and depreciation rates respond systematically to key economic variables.

The comparative dynamics of closed-loop controls for discounted infinite horizon optimal control problems

Abstract A dual view of the Hamilton–Jacobi–Bellman equation corresponding to the class of discounted infinite horizon optimal control problems yields a concise proof of the envelope theorem and an exhaustive comparative dynamics characterization of the closed-loop optimal control vector in the form of a symmetric and positive semidefinite matrix. The theorems are applied to the Hotelling model of the nonrenewable resource extracting firm. Not only do they uncover numerous heretofore unknown qualitative properties of the model, they also yield a new and tractable method for empirically implementing the model in a manner fully consistent with the underlying theory.

The envelope theorem for locally differentiable Nash equilibria of finite horizon differential games

Abstract Envelope theorems are established for a ubiquitous class of finite horizon differential games. The theorems cover open-loop and feedback information patterns in which the corresponding Nash equilibria are locally differentiable with respect to the parameters of the game. Their relationship with extant envelope results is discussed and an application of them to a generalized capital accumulation game is provided. An important implication of the theorems is that, in general, the archetypal economic interpretation of the costate vector, namely, as the shadow value of the state vector along the Nash equilibrium, is valid for feedback Nash equilibria, but not for open-loop Nash equilibria.

A Complete Theory of Comparative Statics for Differentiable Optimization Problems

A new comparative statics formalism using generalized compensated derivatives is presented that, in contrast to existing methodologies, directly yields constraint-free semidefiniteness results for any differentiable, constrained optimization problem. The formalism provides a natural and powerful method of constructing comparative statics results, free of constraints and unrestricted in scope. New results on envelope relations, invariance conditions, rank inequalities and non-uniqueness are derived that greatly extend their utility and reach. The methodology is illustrated by deriving the comparative statics of multiple linear constraint utility maximization models and the principal-agent problem with hidden actions, both highly nontrivial and hitherto unsolved problems.

A qualitative characterization of the competitive nonrenewable resource extracting firm

Abstract Using variational differential equations, a systematic qualitative analysis is carried out for the competitive nonrenewable resource extracting firm. The parameters of interest are the output price, variable input price, discount rate, lease length, initial resource stock, and various tax rates. The effects of changes in these parameters on the entire optimal time paths of the current shadow value of the stock, the resource stock, and the extraction rate are characterized in the perturbed phase plane.

Comparative statics of the generalized maximum entropy estimator of the general linear model

The generalized maximum entropy method of information recovery requires that an analyst provides prior information in the form of finite bounds on the permissible values of the regression coefficients and error values for its implementation. Using a new development in the method of comparative statics, the sensitivity of the resulting coefficient and error estimates to the prior information is investigated. A negative semidefinite matrix reminiscent of the Slutsky-matrix of neoclassical microeconomic theory is shown to characterize the said sensitivity, and an upper bound for the rank of the matrix is derived.

SENSITIVITY OF THE GME ESTIMATES TO SUPPORT BOUNDS

The claim has been made that the Generalized Maximum Entropy (GME) estimator of Golan, Judge and Miller is not sensitive to variations in the support bounds of either the parameters or the error terms. In this paper, we scrutinized this claim by means of Monte Carlo experiments and found that the parameter estimates are impacted in a substantial way by these changes. We also analyzed the famous data sample on the US manufacturing industry used by Cobb and Douglas in 1934 and found that the GME estimator is very sensitive to changes in support bounds. We conclude with a general result by Caputo and Paris according to which any support bound variation produces unexpected responses in the parameter estimates.

The qualitative structure of a class of infinite horizon optimal control problems

Many papers in economics have been written using optimal control theory with what appear to be diverse models. The qualitative results developed in the papers have been related only to the specific model at hand. This paper shows that most of the useful qualitative results occur because the same small number of crucial assumptions are being made about the mathematical structure of the integrand and/or state equation. The taxonomy of models is examined and the seemingly diverse results are unified in this research for the class of one-dimensional, discounted infinite horizon optimal control problems. This is achieved by explicitly linking the local stability of the steady state, the steady state comparative statics and the local comparative dynamics of the optimal control problem.