<p>There's something quite subtle going on here that means your final line is wrong (but it's an easy mistake to make and a tough one to spot).</p>

<p>To compute the MPK, we must differentiate the production function with respect to the current level of capital: <span class="math-container">$\partial F/\partial K$</span>.</p>

<p>But in your final line, you are not differentiating with respect to the current level of capital (which is <span class="math-container">$\tilde{K}=\lambda K$</span>). You are instead differentiating with respect to <span class="math-container">$K$</span>, which is a fraction <span class="math-container">$1/\lambda$</span> of the current amount of capital. </p>

<p>If we compute the derivative with respect to <span class="math-container">$\tilde{K}\equiv \lambda K$</span> instead of just <span class="math-container">$K$</span> then everything works as it should:</p>

<p><span class="math-container">$$\frac{\partial F(\lambda K,\lambda L)}{\partial \lambda K}=\frac{\partial (8(\lambda K)^{1/2}(\lambda L)^{1/2})}{\partial \lambda K}=4(\lambda K)^{-1/2}(\lambda L)^{1/2}=4\frac{\sqrt{L}}{\sqrt{K}}.$$</span></p>

<p>This does not depend on <span class="math-container">$\lambda$</span> so MPK is indeed independent of the scale of the economy. QED</p>

CC BY-SA 4.0
2018-10-21T18:58:01.410Z
2018-10-21T19:03:42.253Z
2018-10-21T19:03:42.253Z
Ubiquitous
<p><a href="https://i.sstatic.net/1yibE.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/1yibE.png" alt="enter image description here"></a></p>

<p>The image is pretty much self-explanatory. To add some context, I'm learning Solow-Swan Growth Theory and my professor said that the marginal product will not change if both capital and labor increase at the same scale. </p>

<p>I can intuitively understand that it makes sense but trying to apply a simple equation (the blue one, the definition of constant marginal product) is just not working. </p>

<p>It's either I'm not partial differentiating correctly or the whole theory is wrong. </p>

<p>I don't see anything wrong with what I've done but why are they not the same? </p>

CC BY-SA 4.0
2018-10-21T20:01:07.827Z
The trick is to show the aggregate production function is homogenous of degree 1
Pedro Cavalcante
<macroeconomics><mathematical-economics>
Showing marginal product of capital is independent of the scale of production
zebralamy
# Showing marginal product of capital is independent of the scale of production

(Asked by: username_0 on 2018-10-21)

[![enter image description here](https://i.sstatic.net/1yibE.png)](https://i.sstatic.net/1yibE.png)

The image is pretty much self-explanatory. To add some context, I'm learning Solow-Swan Growth Theory and my professor said that the marginal product will not change if both capital and labor increase at the same scale.

I can intuitively understand that it makes sense but trying to apply a simple equation (the blue one, the definition of constant marginal product) is just not working.

It's either I'm not partial differentiating correctly or the whole theory is wrong.

I don't see anything wrong with what I've done but why are they not the same?

**Tags:** macroeconomics, mathematical-economics
> **Comments:**
> **username_1**: The trick is to show the aggregate production function is homogenous of degree 1  

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## 1 answers

### Answer by username_2 on 2018-10-21. Score: 4 | Accepted answer

There's something quite subtle going on here that means your final line is wrong (but it's an easy mistake to make and a tough one to spot).

To compute the MPK, we must differentiate the production function with respect to the current level of capital: $\partial F/\partial K$.

But in your final line, you are not differentiating with respect to the current level of capital (which is $\tilde{K}=\lambda K$). You are instead differentiating with respect to $K$, which is a fraction $1/\lambda$ of the current amount of capital.

If we compute the derivative with respect to $\tilde{K}\equiv \lambda K$ instead of just $K$ then everything works as it should:

$$\frac{\partial F(\lambda K,\lambda L)}{\partial \lambda K}=\frac{\partial (8(\lambda K)^{1/2}(\lambda L)^{1/2})}{\partial \lambda K}=4(\lambda K)^{-1/2}(\lambda L)^{1/2}=4\frac{\sqrt{L}}{\sqrt{K}}.$$

This does not depend on $\lambda$ so MPK is indeed independent of the scale of the economy. QED


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