| Sumario: | In a recent article in The Review of Economic Studies, Dhrymes [2] discusses some conditions for the differentiability of demand functions in the case where one does not require knowledge of derivatives beyond those of the second order to determine whether a given stationary point is an extremum. Starting from production and utility functions with non-singular Hessian matrices, he develops the well-known salient propositions of the theories of the firm and consumer behaviour (see, e.g. Samuelson [3]), using convenient matrix notation to avoid cumbersome properties of determinants. On the one hand, it is well-known (see, e.g., Theil [4]) that the case of constant returns to scale in the theory of the firm involves a singular Hessian. On the other hand, the assumption of a Hessian with rank less than n-1 must be ruled out, since it would contradict the traditional second-order concavity conditions. In this note we shall show that for some propositions of Dhrymes [2] we are still able to derive the same properties if the Hessian is singular with rank n-1. Extension to the other propositions and their conclusions is obvious and hence deleted. We have adopted the notation used by Dhrymes [2] throughout.
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