<p>This is a misuse of terms.</p>

<p>The marginal product of capital net of depreciation is indeed $MP_k - \delta$, there is no doubt about it.  </p>

<p>What the authors mean  is that </p>

<p>$$R = 1 + (MP_k - \delta)  $$</p>

<p>is the "gross return on capital" factor. In fact a few lines below the equation they call it exactly that, which is the correct term. Here "gross" reflects the inclusion of $1$.  </p>

CC BY-SA 3.0
2017-10-12T11:56:15.873Z
Is this a common thing? Calling $1+X$ the gross return of an asset paying $X$? For instance, it the interest rate of a loan is 3%, is 1.03 the "gross" return on loans? Looks very odd to me.
luchonacho
CC BY-SA 3.0
2017-10-12T12:25:42.343Z
@luchonacho The proper term is the "gross return _factor_" and not the "gross return rate".
Alecos Papadopoulos
CC BY-SA 3.0
2017-10-12T13:00:00.353Z
That is an odd term. [Google](https://www.google.co.uk/search?q=%22gross+return+factor%22&client=ubuntu&hs=u3K&channel=fs&dcr=0&filter=0&biw=1920&bih=917) returns only ten results!
luchonacho
CC BY-SA 3.0
2017-10-12T21:43:02.190Z
@luchonacho Indeed. It returns twenty thousand for "gross return on capital". The problem with this last one, is that it is sometimes used to mean $R$ as above, and sometimes is used to mean "gross interest _rate_" (i.e. prior to subtract depreciation).  Using the word "factor" helps avoid the confusion.
Alecos Papadopoulos
CC BY-SA 3.0
2017-10-12T11:29:15.193Z
2017-10-12T11:34:47.507Z
2017-10-12T11:34:47.507Z
Alecos Papadopoulos
<p>I am trying to understand how marginal product of capital net of depreciation is the following:</p>

<p><a href="https://i.sstatic.net/cfzxI.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/cfzxI.png" alt="enter image description here"></a></p>

<p>Given that the production function is quite standard</p>

<p><a href="https://i.sstatic.net/l6FzV.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/l6FzV.png" alt="enter image description here"></a></p>

<p>I understand the first term of the marginal product, but I am asking about the depreciation part.</p>

CC BY-SA 3.0
2017-10-12T09:15:16.410Z
I think there is an error in the formula. As far as I know the net MPK is simply $MP_K - \delta$. If you google "net marginal product of capital" or so you get that definition everytime. For example, in Mankiw's book [here](http://aida.econ.yale.edu/~nordhaus/homepage/documents/189-220_Mankiw7e_CH07.pdf). Your definition does not nest with the simpler case of $\delta=0$. There is an extra 1 floting around.
luchonacho
CC BY-SA 3.0
2017-10-12T10:31:00.357Z
Thank you. That is exactly why I am confused (about the extra "1"), but I don't think it is a mistake as it is from a published peer-reviewed paper.
Bird_1991
CC BY-SA 3.0
2017-10-12T10:54:23.680Z
Can you provide the link to the paper please?
luchonacho
CC BY-SA 3.0
2017-10-12T11:03:42.213Z
Eq. (2) and (3): http://www.nber.org/papers/w13602.pdf
Bird_1991
CC BY-SA 3.0
2017-10-12T13:10:19.403Z
Notice that NBER papers are not peer-reviewed (beyond informal commentary from seminars and conferences).
luchonacho
CC BY-SA 3.0
2017-10-12T13:42:26.710Z
Yes, included the nber link because it is mostly accessible without a login/payment. But the published version in RES (2013) uses the same equations (2) and (3)
Bird_1991
CC BY-SA 3.0
2017-10-13T08:06:59.343Z
As a further explainer, and disregarding the issue of terminology that Alecos' clarified, notice that the authors use $R$ in the Euler equation as in $C_t=RC_{t+1}$. The "true" Euler equation is $C_t=(1+r)C_{t+1}$, which is why they define $R$ with an extra 1, and where $r$ refers to the net MPK.
luchonacho
<cobb-douglas>
Marginal product of capital net of depreciation
Bird_1991
# Marginal product of capital net of depreciation

(Asked by: username_0 on 2017-10-12)

I am trying to understand how marginal product of capital net of depreciation is the following:

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

Given that the production function is quite standard

[![enter image description here](https://i.sstatic.net/l6FzV.png)](https://i.sstatic.net/l6FzV.png)

I understand the first term of the marginal product, but I am asking about the depreciation part.

**Tags:** cobb-douglas
> **Comments:**
> **username_1**: I think there is an error in the formula. As far as I know the net MPK is simply $MP_K - \delta$. If you google "net marginal product of capital" or so you get that definition everytime. For example, in Mankiw's book [here](http://aida.econ.yale.edu/~nordhaus/homepage/documents/189-220_Mankiw7e_CH07.pdf). Your definition does not nest with the simpler case of $\delta=0$. There is an extra 1 floting around.  
> **username_0**: Thank you. That is exactly why I am confused (about the extra "1"), but I don't think it is a mistake as it is from a published peer-reviewed paper.  
> **username_1**: Can you provide the link to the paper please?  
> **username_0**: Eq. (2) and (3): http://www.nber.org/papers/w13602.pdf  
> **username_1**: Notice that NBER papers are not peer-reviewed (beyond informal commentary from seminars and conferences).  
> **username_0**: Yes, included the nber link because it is mostly accessible without a login/payment. But the published version in RES (2013) uses the same equations (2) and (3)  
> **username_1**: As a further explainer, and disregarding the issue of terminology that Alecos' clarified, notice that the authors use $R$ in the Euler equation as in $C_t=RC_{t+1}$. The "true" Euler equation is $C_t=(1+r)C_{t+1}$, which is why they define $R$ with an extra 1, and where $r$ refers to the net MPK.  

---

## 1 answers

### Answer by username_2 on 2017-10-12. Score: 0 | Accepted answer

This is a misuse of terms.

The marginal product of capital net of depreciation is indeed $MP\_k - \delta$, there is no doubt about it.

What the authors mean is that

$$R = 1 + (MP\_k - \delta) $$

is the "gross return on capital" factor. In fact a few lines below the equation they call it exactly that, which is the correct term. Here "gross" reflects the inclusion of $1$.

> **Comments:**
> **username_1**: Is this a common thing? Calling $1+X$ the gross return of an asset paying $X$? For instance, it the interest rate of a loan is 3%, is 1.03 the "gross" return on loans? Looks very odd to me.  
> **username_2**: @luchonacho The proper term is the "gross return _factor_" and not the "gross return rate".  
> **username_1**: That is an odd term. [Google](https://www.google.co.uk/search?q=%22gross+return+factor%22&client=ubuntu&hs=u3K&channel=fs&dcr=0&filter=0&biw=1920&bih=917) returns only ten results!  
> **username_2**: @luchonacho Indeed. It returns twenty thousand for "gross return on capital". The problem with this last one, is that it is sometimes used to mean $R$ as above, and sometimes is used to mean "gross interest _rate_" (i.e. prior to subtract depreciation).  Using the word "factor" helps avoid the confusion.  

---

