<p>You can't just say that from the diminishing marginal product assumption. <span class="math-container">$F_{KK}&lt;0$</span> and <span class="math-container">$F_{LL}&lt;0$</span> does not imply that <span class="math-container">$F_{LK} &lt;0$</span>.</p>
<p>In fact, a standard textbook Cobb-Douglas production function <span class="math-container">$F= AK^{\alpha}L^{1-\alpha}; 0&lt;\alpha&lt;1$</span>, which is often used together with Solow Swan model will have positive cross derivative, given the parameter restrictions, if any capital and labor is employed:</p>
<p><span class="math-container">$$F_{LK}=  (1-\alpha)\alpha\frac{A}{(KL)^\alpha} &gt;0 $$</span></p>
<p>This is despite that the same function has diminishing marginal product for both capital and labor (you can verify that yourself).</p>
<p>Of course, you can play with the model by using different production functions, however, assuming <span class="math-container">$F_{LK}&lt;0$</span> is from economic perspective very strange.</p>
<p><span class="math-container">$F_{LK}&lt;0$</span> says that when you increase both the amount of labor and capital in your factory at the same time marginal output declines. For example, if you have sewing factory adding both more sewing machines and more seamsters to operate the sewing machine would result in less shirts produced on the margin. This is very implausible, perhaps there are some special circumstances where you want to assume that but it will not be generally assumed in macroeconomic models that approximate production of whole economy.</p>

CC BY-SA 4.0
2024-03-23T12:30:08.870Z
"$F_{LK}<0$
*says that when you increase both the amount of labor and capital in your factory at the same time output declines.*" Can you please explain why this says what you claim it says, and why it is not just a second order effect? By the same logic $F_{LK}=0$ would imply that increasing both inputs has no effect on the output, but $F(K,L) = K + L$ is a trivial counterexample to this. In fact, the less conventional $F(K,L) = K + L - 1/(KL)$ is a counterexample to your claim under most parameter combinations.
Giskard
CC BY-SA 4.0
2024-03-23T12:37:20.580Z
@Giskard cross-derivative $f_{xy}$ tells you how the slope of x changes with respect to changes in y. $f_{xy}<0$ implies that with more y the slope of x becomes less steep. In a context of production that means that when we add more labor the marginal productivity of capital declines
1muflon1
CC BY-SA 4.0
2024-03-23T12:39:58.997Z
I also think this is what it means, but you wrote something completely different: "*when you increase both the amount of labor and capital in your factory at the same time output declines.*"
Giskard
CC BY-SA 4.0
2024-03-23T12:41:43.387Z
@Giskard oh I think I know what you mean, I was thinking on the margin, but I will edit it to the text to make it clear
1muflon1
CC BY-SA 4.0
2024-03-23T12:42:50.280Z
I think you meant to write "oh the way I wrote it was incorrect, I will make changes so that it reflects my thinking and is not a false statement", because it was not "unclear" it was incorrect. Happy to have helped!
Giskard
CC BY-SA 4.0
2024-03-23T12:44:18.897Z
@Giskard I don't see any reason why you need to assume bad faith
1muflon1
CC BY-SA 4.0
2024-03-23T12:45:04.690Z
? What do you mean? I am just pointing out it was not a clarity issue, the statement was clear and false.
Giskard
CC BY-SA 4.0
2024-03-23T12:45:58.683Z
New version of the statement however is unclear, as there are several possible margins you can think of. Losing the final paragraph is the simplest solution to the entire problem.
Giskard
CC BY-SA 4.0
2024-03-23T13:05:21.370Z
@Giskard sigh, you can't have it both ways. Output could refer to both marginal or total output. Yes there are several possible margins like there are several possible outputs that can be referred to. So how can one be clear and another is not clear? Yes you assume bad faith, because when I clarified the former you assumed that I am lying to you and not what I was thinking of in my head.
1muflon1
CC BY-SA 4.0
2024-03-23T14:36:07.133Z
Thank you 1muflon1 and @Giskard. The answer and the discussion that followed was helpful to me.
guest
CC BY-SA 4.0
2024-03-23T15:49:26.923Z
@1muflon1 Your final comment is pure satire, I don't know what to write. Have a nice day.
Giskard
CC BY-SA 4.0
2024-03-24T12:21:55.867Z
@1muflon1  I think you are probably right that $F_{LK}<0$ would be very strange, but don't find the explanation in your final paragraph convincing.  A couple of suggestions: a) see Maths SE [here](https://math.stackexchange.com/questions/163860/class-of-functions-with-negative-mixed-partial-derivatives), b) it might be helpful to consider what sort of isoquants would be implied.
Adam Bailey
CC BY-SA 4.0
2024-03-23T12:27:03.810Z
2024-03-23T12:43:14.440Z
2024-03-23T12:43:14.440Z
1muflon1
<p>I was studying about the Solow Model from Acemoglu. One of the properties of the production function <span class="math-container">$Y = F(A,K,L)$</span> is that it exhibits diminishing marginal products. That is, <span class="math-container">$F_{KK} &lt; 0$</span> and <span class="math-container">$F_{LL} &lt; 0$</span>.</p>
<p>Can we also say that <span class="math-container">$F_{LK} &lt; 0$</span>?</p>

<macroeconomics><solow>
Production function in Solow Model
guest
# Production function in Solow Model

(Asked by: username_0 on 2024-03-23)

I was studying about the Solow Model from Acemoglu. One of the properties of the production function $Y = F(A,K,L)$ is that it exhibits diminishing marginal products. That is, $F\_{KK} < 0$ and $F\_{LL} < 0$.

Can we also say that $F\_{LK} < 0$?

**Tags:** macroeconomics, solow

---

## 1 answers

### Answer by username_1 on 2024-03-23. Score: 1 | Accepted answer

You can't just say that from the diminishing marginal product assumption. $F\_{KK}<0$ and $F\_{LL}<0$ does not imply that $F\_{LK} <0$.

In fact, a standard textbook Cobb-Douglas production function $F= AK^{\alpha}L^{1-\alpha}; 0<\alpha<1$, which is often used together with Solow Swan model will have positive cross derivative, given the parameter restrictions, if any capital and labor is employed:

$$F\_{LK}= (1-\alpha)\alpha\frac{A}{(KL)^\alpha} >0 $$

This is despite that the same function has diminishing marginal product for both capital and labor (you can verify that yourself).

Of course, you can play with the model by using different production functions, however, assuming $F\_{LK}<0$ is from economic perspective very strange.

$F\_{LK}<0$ says that when you increase both the amount of labor and capital in your factory at the same time marginal output declines. For example, if you have sewing factory adding both more sewing machines and more seamsters to operate the sewing machine would result in less shirts produced on the margin. This is very implausible, perhaps there are some special circumstances where you want to assume that but it will not be generally assumed in macroeconomic models that approximate production of whole economy.

> **Comments:**
> **username_2**: "$F_{LK}<0$ *says that when you increase both the amount of labor and capital in your factory at the same time output declines.*" Can you please explain why this says what you claim it says, and why it is not just a second order effect? By the same logic $F_{LK}=0$ would imply that increasing both inputs has no effect on the output, but $F(K,L) = K + L$ is a trivial counterexample to this. In fact, the less conventional $F(K,L) = K + L - 1/(KL)$ is a counterexample to your claim under most parameter combinations.  
> **username_1**: @Giskard cross-derivative $f_{xy}$ tells you how the slope of x changes with respect to changes in y. $f_{xy}<0$ implies that with more y the slope of x becomes less steep. In a context of production that means that when we add more labor the marginal productivity of capital declines  
> **username_2**: I also think this is what it means, but you wrote something completely different: "*when you increase both the amount of labor and capital in your factory at the same time output declines.*"  
> **username_1**: @Giskard oh I think I know what you mean, I was thinking on the margin, but I will edit it to the text to make it clear  
> **username_2**: I think you meant to write "oh the way I wrote it was incorrect, I will make changes so that it reflects my thinking and is not a false statement", because it was not "unclear" it was incorrect. Happy to have helped!  
> **username_1**: @Giskard I don't see any reason why you need to assume bad faith  
> **username_2**: ? What do you mean? I am just pointing out it was not a clarity issue, the statement was clear and false.  
> **username_2**: New version of the statement however is unclear, as there are several possible margins you can think of. Losing the final paragraph is the simplest solution to the entire problem.  
> **username_1**: @Giskard sigh, you can't have it both ways. Output could refer to both marginal or total output. Yes there are several possible margins like there are several possible outputs that can be referred to. So how can one be clear and another is not clear? Yes you assume bad faith, because when I clarified the former you assumed that I am lying to you and not what I was thinking of in my head.  
> **username_0**: Thank you 1muflon1 and @Giskard. The answer and the discussion that followed was helpful to me.  
> **username_2**: @1muflon1 Your final comment is pure satire, I don't know what to write. Have a nice day.  
> **username_3**: @1muflon1  I think you are probably right that $F_{LK}<0$ would be very strange, but don't find the explanation in your final paragraph convincing.  A couple of suggestions: a) see Maths SE [here](https://math.stackexchange.com/questions/163860/class-of-functions-with-negative-mixed-partial-derivatives), b) it might be helpful to consider what sort of isoquants would be implied.  

---

