Economic Characterizations of the Second-Order Sufficient Conditions

The second-order sufficient conditions for optimization in terms of Hessian matrix is equivalent to the law of diminishing marginal productivity for one factor case, and the second-order conditions in terms of the bordered Hessian matrix is equivalent to the law of diminishing marginal rate of substitution for the two-factor case. But in general, the whole implications of the second-order conditions cannot be replaced by these laws. In this paper, we introduce the conditions of generalized diminishing marginal productivity and of generalized diminishing marginal rate of substitution, and show that these conditions are equivalent to the second order sufficient conditions.