Class D3 of Utility Functions and Investors

The class D3 is more restrictive than both classes D1 and D2. Here the U(x) satisfies three inequalities involving the first three derivatives of U(x). The D3 class requires not only that the MU evaluated at higher levels of its argument (wealth) should decline, but also that the decline should become more pronounced. We saw in the previous subsection that the desirable NIARA property is satisfied if U'" > (U")2/U' holds true. If U(x) belongs to class D2, we know that U > 0 and U" < 0, that is, (U")2/U' > 0. After substituting this on the right-hand side of the inequality for NIARA, note that the third inequality involving the third derivative is simply U" > 0. Thus the necessary requirement for a U(x) to belong to class D3 is that all three inequalities: U > 0, U" < 0, and U" > 0 are satisfied.

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