<p>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.</p>

<p>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.</p>

<p>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?</p>

CC BY-SA 4.0
2018-05-31T16:20:35.063Z
By simply observing thtat the cross partial scales as $K^{-a}$.
user526463
CC BY-SA 4.0
2018-05-31T19:00:43.157Z
The whole expression gives you the amount that the labor retunr increases with K, not just the k variable in the cross-partial.
BB King
CC BY-SA 4.0
2018-05-31T10:18:05.093Z
2018-05-31T10:18:05.093Z
BB King
<p>Consider a Cobb-Douglas Prod. Function $$Y=AL^{a}K^{1-\alpha}$$
This has the cross-partial:
$$\frac{\partial^2 Y}{\partial K\partial L}=(1-\alpha)\alpha AL^{\alpha-1}K^{-\alpha}$$</p>

<p>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)?</p>

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

CC BY-SA 4.0
2018-05-30T04:21:01.150Z
shouldn't your cross partial be $\frac{\partial^2 Y}{\partial K\partial L}=(1-\alpha)\alpha AL^{\alpha-1}K^{-\alpha}$
EconJohn
CC BY-SA 4.0
2018-05-30T16:29:04.150Z
@EconJohn Yep, corrected.
user526463
<microeconomics><mathematical-economics><economic-growth><production-function>
Interpretation of the Cross Partials of the Cobb-Douglas
user526463
# Interpretation of the Cross Partials of the Cobb-Douglas

(Asked by: username_0 on 2018-05-29)

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?

**Tags:** microeconomics, mathematical-economics, economic-growth, production-function

> **Comments:**
> **username_1**: shouldn't your cross partial be $\frac{\partial^2 Y}{\partial K\partial L}=(1-\alpha)\alpha AL^{\alpha-1}K^{-\alpha}$  
> **username_0**: @username_1 Yep, corrected.  

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## 1 answers

### Answer by username_2 on 2018-05-31. Score: 3

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?

> **Comments:**
> **username_0**: By simply observing thtat the cross partial scales as $K^{-a}$.  
> **username_2**: The whole expression gives you the amount that the labor retunr increases with K, not just the k variable in the cross-partial.  

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