Secondorder Partial Derivatives 475

The exercise of determining the economic interpretation of first- and seeond-ordcr partial derivatives is illustrated in the following example:

Example 11.12 Find and interpret the second-order partial derivatives of the Cobb-Douglas production function with two inputs.

Solution

The general form of the Cobb-Douglas production function with two inputs is

>• = /(je,, *->) = Ax';4, vi. *2 > o wliere .rj and x2 are input levels, y is the output level, and a, P, A > 0, are technological parameters. We usually add the restrictions that u < I and p < \ , for reasons that are described below.

The (first-order) partial derivatives of this function arc

which is the marginal product of input I, and h = pAxfxt1

which is the marginal product of input 2, The conditions a > 0 and P > 0 ensure that the marginal products are positive, which implies that adding more of either input leads to a greater level of output, as one would expect. The second-order partial derivatives are

Given the assumptions made about the parameters. 0 < u, p < 1, A > 0. it follows that /n and f22 are negative (since a < 1 =>a - I < (land/) < I <()).. which implies diminishing marginal productivity of each input. (See example 5.15 for a discussion of this phenomenon in the context of a single input.) /ij and f2i arc positive. The cross-partial derivative J): is the rale at which the marginal product of input 1 changes as more of input 2 is added. fi2 > 0 implies that as more of input 2 is added (e.g.. capital), an additional unit of input I (e.g.. labor) becomes more productive; that is. the marginal productivity of one input is enhanced by having more of the other input available. ■

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