Suppose that the coefficients of an input-output matrix, A, are random variables but that we have ascertained their expected values, EA. What will be the relation of the Leontief inverse of EA, (I EA) ', to the expected value of the inverse, E(I A) ? Will one or the other be uniformly greater? We will show that if all coefficients of A are independent, then the expected value of the inverse is uniformly greater than or equal to the inverse of the expected value. If, on the other hand, the column and row sums of the coefficient matrix are fixed, and smaller than one, so that the variables are not independent, then, in the two-by-two case, the opposite is true of the off-diagonal elements.

IN AN EARLIER PAPER1 I derived conditions for the stability of equilibrium in monopolistic competition for two competitors. The extension of that analysis to cover more than two competitors is by no means obvious, and it is a matter of importance to know how the number of competitors affects the question of stability. It is therefore to the solution of the problem for n competitors that the present paper will be devoted. Although the economic problem will be limited to the question of monopolistic price competition, the methods employed can be used to test the stability of any equilibrium determined by the-solution of a system of linear equations. In Section I we shall formulate a demand function for n competitors in an imperfect market, and also their cost functions. In Section II we shall derive the conditions for the existence of equilibrium and in Section III we shall determine the conditions for its stability in the cases both of noncontinuous and continuous adjustment on the basis of a given set of expectations. In Section IV we shall consider the implications of our results for the general cases of two, three, and n competitors, while in Section V we solve the problem completely for n identical competitors. Finally in Section VI, we shall adumbrate the problems involved when the stability of the expectations themselves is brought into question.