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Consider a Cobb-Douglas Prod. Function $$Y=AL^{a}K^{1-\alpha}$$ This has the cross-partial: $$\frac{\partial^2 Y}{\partial K\partial L}=(1-\alpha)\alpha AL^{\alpha-1}K^{-\alpha}$$

Is the interpretation of this the change in the return to labour, with a change in capital (and by Schwarz's theorem, also the change in the return the capital, with a change in labour)?

Then, is it correct to say that the return to labour increases as $\frac{1}{K^a}$ with capital?

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  • $\begingroup$ shouldn't your cross partial be $\frac{\partial^2 Y}{\partial K\partial L}=(1-\alpha)\alpha AL^{\alpha-1}K^{-\alpha}$ $\endgroup$
    – EconJohn ♦
    Commented May 30, 2018 at 4:21
  • $\begingroup$ @EconJohn Yep, corrected. $\endgroup$
    – user526463
    Commented May 30, 2018 at 16:29

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The interpretation of the cross partial here is the change in the return (or marginal productivity) of labor when capital increases marginally. Similarly, it can be interpreted as the change in the return of capital, when labor increases marginally.

We often only want to interpret the sign of the cross-partial and not the exact value. Here we see that the productivity of one factor increases with the other factor. Interpreting the exact value is more difficult, but also not really interesting for most economic applications.

Furthermore, it is not correct to say that the return to labour increases as $\frac{1}{K^a}$ with capital. How did you arrive at this conclusion?

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  • $\begingroup$ By simply observing thtat the cross partial scales as $K^{-a}$. $\endgroup$
    – user526463
    Commented May 31, 2018 at 16:20
  • $\begingroup$ The whole expression gives you the amount that the labor retunr increases with K, not just the k variable in the cross-partial. $\endgroup$
    – BB King
    Commented May 31, 2018 at 19:00

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