Journal Article Professor Hicks' Index Number Theorem Get access R. L. Marris R. L. Marris Cambridge Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 25, Issue 1, October 1957, Pages 25–40, https://doi.org/10.2307/2296119 Published: 01 October 1957

COMPARATIVE dynamics is a relatively unworked but highly significant area of economic analysis. It is important to appreciate the implications of changes in the structure of a dynamic system and, thus, the extent to which the analytical results depend on the particular, restricted forms employed. These are the problems with which comparative dynamics deals in general 1 and with which this paper is centrally concerned, with particular respect to the stability of difference-equation models of income determination. Dynamic difference-equation models of economic fluctuations have been useful tools for the economist. Though not intended to provide, in themselves, a complete description of aggregate economic activity, such models have led to a better appreciation of cycle-producing forces and the character of cyclical movements. These models have been developed in a variety of forms based on different assumptions about time sequences and using different types of expenditure functions as components. Inevitably, however, each of the models represents a special case whose analysis provides conclusions of only limited applicability. There has not, I believe, been adequate recognition of the important qualifications which must as a result be applied to many of the existing dynamic models of economic fluctuations. In this paper greater generality will be sought in two ways: first, by the construction of a number of alternative systems based on different assumptions about the lag sequences involved; second, by detailing somewhat more than has been done the expenditure functions which are the components of the models. I believe that it will be possible in this way to provide further insights which are important in themselves and in qualifying previous dynamic analyses. The variety of results which can be obtained will be illustrated rather than treated exhaustively. The procedure of this paper will be to construct and compare a series of alternative models. In section I the general method of analysis will be illustrated in presenting models based on the Robertsonian lag of expenditures behind the receipt of income.2 Section II will begin to break new ground in the analysis of models which use the Lundberg lag of output behind sales.3 Finally, in section III models will be developed using both types of lag. The investigation is confined to models of effective demand in a closed economy. The expenditure functions used are, in general, adaptations of a simple consumption function and the acceleration principle -admittedly gross, but useful, simplifications of reality. Even though thus confined to simple lag relations and spending functions, the analysis becomes quite complicated. After constructing each model, the types of movements which it may generate and the conditions which it must satisfy for stability will be investigated.4 The systems constructed will be subject to the well-known limitations of linear models. It would not be difficult, however, to apply simple types of nonlinear restraints to the systems, and the effects of such restraints will be noted at several places. The mathematical techniques of stability analysis which are the basis of this paper have been introduced into economics by Professor P. A. Samuelson. Their usefulness has, perhaps, not been fully appreciated, and this paper provides an example of their application. A general comment on the use of stability conditions may be in order here. We want to investigate the economic implications of a number of alternative dynamic systems; we know that such systems are able to produce a variety of motions consisting of monotonic and

ONE of the more conceptually mysterious aspects of linear programming is the problem of degeneracy the breaking down of the simplex calculation method under certain circumstances. Although a set of rules for dealing with degeneracy is well known, in the absence of an understanding of the nature of the problem the rules must be followed by rote. This note presents a simple geometrical explanation of the problem, the solution, and explains why certain types of programming problems frequently lead to degeneracy. It is hoped that the presentation will be of use both to students and teachers of linear programming. Before presenting the geometry of degeneracy, it is necessary to review briefly the geometry of linear programming and the simplex method of computing optimum linear programs.

The description of politics as art of the applies likewise to public policy in general and to monetary policy in particular. To say that central banking is an art rather than a science, as both students and practitioners of central banking have been accustomed to do, is not to deny that, in this as in every other branch of applied economics, scientific methods and a scientific attitude can be extraordinarily fruitful. But the fact does remain that central banking is an art. The principal weakness of central bankers lies not in any failure to recognize that fact but in deciding what constitutes the possible when it comes to the application of their art. The main difficulty confronting central bankers is that what is possible for central bank policy is in no sense an absolute. It is not that what is possible is merely a matter of expediency, although expediency undoubtedly has to be considered; the policy-maker who disregards what is realistic is only a little less ridiculous than the one who forgets what is ideal.' The more baffling consideration is that what it is possible for central banking to accomplish, and by what means, is relative to many things. These include such internal considerations as the objectives to which it is committed, the guides available to it, the instruments at its disposal. They also include such outside factors as the state of the economy, the international climate both political and economic, and the attitude of the public. While some of these endogenous and exogenous elements are subject to accurate determination or control, others clearly are not. To suggest that the problem of the practitioners of the art of central banking is difficult is not to imply that they do as good a job as can legitimately be asked. A better job can be expected, however, only through heightened sensitivity to the changing environment in which central bankers operate and increased willingness to modify central bank actions in the light of these changes. And it is highly probable that the pressure to effect these adaptations will have to come mainly from students of the art who are outside the system itself.2

IN order to overcome the harmful effects on regression and correlation estimates of using highly collinear time series observations, and in order to obtain more accurate estimates of income elasticities of demand, economic statisticians have turned increasingly to the device of extraneous estimators. This technique has been most commonly employed in demand studies but also could be used in other applications. In the case of demand functions the procedure has been to obtain the income coefficient from cross-section budget data for which the price variables are presumably constant. Thus an estimate of the partial regression of quantity on income, with price given, is obtained. This cross-section estimate of the income regression-coefficient is then multiplied by the time-series aggregate of income, and the product is in turn subtracted from the annual time series of quantity demanded, to form a new dependent variable. This new dependent variable, as a possibly unintended consequence, usually has a larger variance than the original dependent series, which often displays little variation beyond the simplest trend component. Having been thus corrected, the dependent series is then regressed against the time series of the price variables to obtain an estimate of the price elasticity of demand.' It should be fairly clear that when the purpose is to make short-run forecasts, the described techniques often may be unnecessary and in many instances could actually prove harmful.2 Specifically, someone making forecasts need not be especially worried about multicollinearity. If some of the explanatory variables are multicollinear, the prediction interval obtained from such a set of observations will be quite large. By eliminating a number of the collinear variables it will usually be possible to substantially reduce the prediction interval for given values of included independent variables. Of course, while the elimination of collinear explanatory variables will tend to reduce the prediction interval, the actual prediction, by hypothesis, will change very little. Hence the pragmatic forecaster might be indifferent to the extent of collinearity, while the more sophisticated forecaster will not be indifferent; both will make similar forecasts and the actual errors of the forecast will be approximately the same. The combined use of cross-section and timeseries data is therefore intended to overcome multicollinearity (which entails the arbitrary splitting up of the influence of the explanatory variables) in order to obtain structurally mnore accurate estimates of the various coefficients. The question, however, can legitimately be asked: Exactly what structure does the statistician seek to estimate? Insofar as demand studies are concerned, it is quite possible, as will shortly be argued at length, that the kind of behavior measured from cross-section * For helpful comments and discussions on an earlier draft of this paper, we are indebted to John S. Chipman, Gregory Chow, James S. Duesenberry, John Lintner, Guy H. Orcutt, Robert Solow, and Charles Zwick. The authors were aided in preparing this paper by research grants from the School of Industrial Management, Massachusetts Institute of Technology, and Division of Research, Harvard Business School (under a Rockefeller Foundation grant for a Study of Profits and the Functioning of the Economy). 'While the techniques used differ in some important respects, the rationale is fully explained in each of the following sources: Richard Stone, The Measurement of Consumers' Expenditure and Behavior in the United Kingdom 1920-1938 (Cambridge, England, I954); and particularly J. Durbin, Note on Regression When There is Extraneous Information About One of the Coefficients, Journal of the American Statistical Association, xLvm (December I953), 799-808; Herman Wold and Lars Jureen, Demand Analysis (New York, I953). This method has also been used by J. Tobin, Statistical Demand Function for Food in the U.S.A., Journal of the Royal Statistical Society, Series A, cxiII (Part II I950), II3-4I. A comprehensive review article, William C. Hood, Empirical Studies of Demand, Canadian Journal of Economics and Political Science, xxi (August I955), 309-27, provides a worthwhile reference on the subject. The distinction between longand short-run estimates is to be found in the useful paper by Richard J. Foote, Price Elasticity of Demand for Nondurable Goods with Emphasis on Food, Agricultural Marketing Service Bulletin 96, USDA (Washington, D.C., I956). 'Although none of the people who have used the combined techniques has had short-run forecasting as his immediate goal, it seems useful to indicate how such prediction fits into the scheme of possible objectives.