Unformatted text preview: f (and likewise, as we add more z, it changes the productivity of y. The term, , then, captures the net change in productivity due to adding, marginally, additional units of y and z. To see this a bit more closely, suppose we had the separable production function we discussed in class,
,,
,
, where and are concave. In this setting, changes in y and changes in z have no effect on the productivity of the other input, so ≡ 0, and, in this case, 0 as we’d expect. From here, we can find the elasticity. I haven’t done this below (the elasticity was one reason we might want the derivative, ). To find the cross derivatives , , we do similarly as we did above: Set 0; we have 0
1 0 The determinant, Δ hasn’t changed, We have: 0
1
0
Δ Δ Again, the same sort of argument as we made above applies. Let’s look at the separable case, ,, ≡
,
. In this case, 0. Since the function is concave, we know that mean? 0, so 0 provided 0. What does this is the marginal product of x. If the price of the factor y goes up, w...
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 Winter '14
 Economics, δ, Ms. B, Ms. A, first‐order conditions

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