Introduction

1.1 Preface

The objective of this study is to consider short-term Indian economic policy from the viewpoint of modern systems and control theory. The principal concern of policy planners in India has been to regulate the economy such that it progresses in a satisfactory manner. However, thirty-five years of macroeconomic planning has indicated, beyond the shadow of a doubt, that large scale perturbations have occurred very frequently in the Indian economy and, as our knowledge about the functioning of the economy and the effects of instruments has been far from perfect, it has been found impossible to prescribe precise compensatory action. In the absence of such countercyclical policy, the spontaneous regaining of equilibrium has been ruled out and the economy has thereby suffered swings of considerable amplitudes at great costs. This study is principally an attempt to try and use stochastic control theory for macroeconomic regulation, so that the inherent pitfalls of adopting policies of an intuitive nature are ruled out.

1.2 Short-Term Economic Policy And Optimal Control Theory

The principal objective of control theory has been to try and improve system performance through the regulator concept, especially when uncertainty is involved. The feedback control policy leads to a simple method for determining optimal control actions, given appropriate statistics based on available information. The determination of these statistics, namely, the conditional mean and the error covariance matrix of the system state, takes place separately. The relationship between the system state and the information data is explicitly kept in view. The observations of economic activity are assumed to contain observation errors, including changing or fragmentary information, and the incorporation of such indicators into the system is achieved through the Kalman filter. The filter provides minimum-variance, unbiased estimates of the system state, conditional on the available information.

The strategy of, what has been termed in the literature, feedback control is adopted in order to obtain the optimal control actions. This type of control has been defined as the policy where controls are ‘some deterministic function of the current and past observations on the system state variables and of past employed controls’ (Aoki 1967). In such a policy, the information comprising previous control actions and system observations available uptil the present moment when control action is to be specified is utilized in computing the control actions. Such a determination of the optimal control trajectory is achieved in two separate and sequential steps. In the first, the estimation of the system state based on available information is obtained, and subsequently, the optimal control action is determined from a deterministic system, which is obtained from the corresponding stochastic system by invoking the Certainty Equivalence Principle (Simon 1956, Theil 1964), implying thereby the replacing of all the random quantities by their conditional expectations.

As more information accumulates, the conditional expectations need to be updated in order to apply the Certainty Equivalence Principle. The Kalman filter (Kalman 1960, Kalman and Bucy 1961) is a very convenient technique to revise optimally the estimates based on past information in the light of new information alone. In effect, the conditional mean and the error covariance of the system state summarize all the accumulated information, as far as the determination of the optimal control is concerned. The Kalman filter enables their updating recursively and past information need not be used again nor stored, since its effect is summarized in the earlier estimates. Similarly, the results of Meditch (1967) facilitate the recursive revision of the past conditional statistics with the coming of fresh information. The optimal control actions resulting from the closed-loop policy depend not only on the latest observation but also on the past observations. The policy attaches appropriate weights to the sequence of observations, and the outcome is that the optimal control action is a weighted average of the latest and past observed errors in prediction. This feature is technically called the proportional-plus-integral control, and it results in smoother control actions and adds to overall stability.

1.3 Scope Of The Study

Under the present Indian practice, economic regulatory policies tend to be heavily biased towards current rather than current and past observations of the state of the economy. Moreover, they have a tendency to overlook the fact that more decisions will have to be taken later on in the light of new information. These drawbacks are overcome under the present framework. Moreover, while the usual economic literature assumes, rather naively, that the current state of the system is completely known, so that all uncertainty is concentrated in the future, the use of stochastic control theory allows for the systematic treatment of the more realistic case when information is scarce, contradictory and inexact. It is here that the Kalman filter comes in to make the most of the available information. Vishwakarma (1974) was one of the first to apply these elegant prediction and control algorithms developed by Kalman and Bucy to a macroeconomic regulatory problem, within a sort of quasi-Monte Carlo framework. This study is an attempt to apply it in a more rigourous and extensive manner well suited for the Indian context.

The mathematical analysis of linear closed-loop control leads to important results such as the separability of prediction from policy determination and certainty equivalence, even if the poor quality of the data is taken into account. This formally justifies the similar separation carried out by Tinbergen (1956) and his concentration on a deterministic analysis.

The application of optimal control techniques to solve short-term economic problems presupposes that it is possible to construct and operate a plausible mathematical model of the economy. Based on the analogy between the structures of certain economic and physical problems, there is a prima facie case for applying optimal control to such a model of the economy to help analyze, probe and ultimately control the dynamic behaviour of the economic system (Ball 1978). The very essence of the optimal, as opposed to the automatic, control problem is that one does not know in advance what exactly one would like to happen; and since economic analysis is concerned for the most part with the exercise of choice, given constraints, the possibility of transferring optimal choice technology from the physical to economic systems via optimal control theory appears to be very appealing. This is the basic essence of the study.

1.4 Optimal Control Theory

1.4.1 Problem formulation

A macroeconomic model attempts to describe the dynamics of an economy over T periods. The model consists of a system of n difference equations relating; n endogenous variables x(i,t), denoted by the vector x(t) ∈ Rn and describing the state of the economy; s control variables u1(j,t), denoted by the vector u1(t) ∈ Rs and describing the instruments of economic policy at the disposal of the planner to guide the trajectory of the economy towards a specified optimal state; m exogenous variables u2(k,t), denoted by the vector u2(t) ∈ Rm and describing those variables whose values are uncontrollable but are assumed to be known (the existence of such uncontrollable exogenous variables is invariably the case with most econometric models) and which are expected to affect the values of the endogenous variables; parameters describing the structure of the economy and its relationship with the environment; and residual random variables (which from now on we assume specified and incorporated in the functional form of the equations).

To explain the value x(i,t) taken by the ith endogenous variable at period t from the series of past historical data, econometricians estimate equations taking into account the values taken by other variables at period t as well as earlier periods. The foremost past period taken into account in the equation explaining the ith endogenous variable is called the lag of the ith equation. In general, a macroeconomic model comprises behavioural (or reaction) equations explaining endogenous variables and accounting identities expressing ex-post equilibrium conditions enforced at each period. The latter are not explicitly solved for one endogenous variable in terms of the others. Thus, a model in structural form can be described by the system of equations

(1.1)

(1.2a)

(1.2b)

(1.2c)

where p,q and r are the maximal lags with respect to the endogenous, control and exogenous variables, respectively; and h(i) are given parameters such that (1, if the ith equation is solved with respect to x(i,t) h(i)=(

(0, otherwise.

We also assume that eq.(1.1) satisfies the following regularity conditions:

(i) The function f(i,t) is continually differentiable with...