Increasing Generalized Correlation: A Definition and Some Economic Consequences

The question 'when is a random variable Y riskier or more variable than another random variable X ? ' has recently been answered in the literature in a manner that is consistent with expected utility theory. This paper provides a similarly natural and theoretically sound definition for the statement that 'the random variables Y1 and Y2 are more correlated or positively interdependent than the random variables X1 and X2. ' The usefulness of the definition is demonstrated by applying it to the determination of the effects of increased correlation on behaviour in some standard economic models. Correlation generalisee croissante: une definition et quelques implications economiques. Recemment on a pu repondre a la question 'quand une variable aleatoire Y est-elle plus aleatoire ou plus variable qu'une variable al6atoire X? ' d'une fagon qui est consistente avec la theorie de l'utilite basee sur l'esperance math6matique. Ce memoire veut construire une definition tout aussi naturelle et theoriquement robuste pour la proposition 'les variables al6atoires Y, et Y2 sont davantage co-reliees ou davantage positivement interdependantes que les variables aleatoires X1 et X2. ' Les auteurs montrent l'utilite de cette d6finition en l'appliquant a la calibration des effets d'une correlation plus grande sur le comportement dans des modeles economiques conventionnels. The question 'when is a random variable Y riskier or more variable than another random variable X? ' has been answered in the literature in a manner consistent with expected utility theory (see for example Hanoch and Levy, 1969; Hadar and Russell, 1969; and especially Rothschild and Stiglitz (RS), 1970). These papers consider only scalar random variables (see, however, Brumelle and Vickson, 1975, and the references therein). Once the analysis is extended to a multivariate, and in particular a bivariate, framework it seems reasonable to ask whether a similarly natural and theoretically sound definition may be provided for the statement that 'the random variables Y1 and Y2 are more correlated (positively interdependent or interrelated) than the random variables X1 and X2. ' The research described in this paper was supported in part by grants from the Canada Council and the National Research Council. This paper was presented at the Canadian Economic Theory Conference, Montreal, May 1978. Canadian Journal of Economics / Revue canadienne d'Economique, XIII, no. I February / fevrier 1980. Printed in Canada / Imprime au Canada. 0008-4085 / 80 / 0000-0016 $01.50 / ? ) 1980 Canadian Economics Association This content downloaded from 157.55. 39.138 on Sun, 26 Jun 2016 05:55:14 UTC All use subject to http://about.jstor. org/terms Increasing generalized correlation / 17 The formulation and theoretical justification of such a definition is the objective of the first part of this paper. In the second part we demonstrate the usefulness of the definition by applying it to the determination of the effects of increased correlation on behaviour in some standard economic models. The structure and approach of the first part are similar to those of RS. We proceed as follows: the key notion of an elementary correlation increasing transformation (CIT) of a given bivariate probability distribution is defined and is used to motivate the first definition of greater correlation. The CIT iS then used to define correlation-averse and correlation-affine utility functions. They in turn are used to formulate the following plausible alternative definition of greater correlation: Y1 and Y2 are more correlated than X1 and X2 if all expected utility-maximizers who are correlation averters (lovers) prefer (disprefer) (X1, X2) to (Y1, Y2). The third section proves that the above two definitions of greater correlation are equivalent, and the fourth section compares them with others used in the literature. In particular, we point out the limited theoretical validity of the linear (Pearsonian) correlation coefficient or the covariance as measures of the positive interdependence of two random variables. Many of the notions and results described may be found in scattered references in the literature. One contribution of this paper is to bring them into focus as the essential components of a natural and theoretically sound definition of greater correlation. In addition, we feel that our approach to Theorem 6, via elementary correlation-increasing transformations, provides further insight and a new perspective regarding the definition of greater correlation. The second part contains a more extensive analysis of the effects of increased correlation in an expected-utility framework than may be found in existing literature. Portfolio diversification is discussed first, and then the analysis of portfolio diversification is extended to the case where future consumption good prices, as well as asset returns, are uncertain. The use of an asset as a hedge against uncertain inflation is considered. Finally, we analyse the effects of correlated price expectations in a two-period model of the behaviour of a competitive firm. Proofs of the principal theorems in the text are collected in an appendix. Proofs of most of the remaining theorems may be found in Epstein and Tanny (1978).