<p>Time series on rental price of capital can be estimated using <span class="math-container">$$r=\frac{P_k}{P}(i-inf+\delta)$$</span>
here, <span class="math-container">$P_k$</span> is the price of capital goods (price index for capital goods), <span class="math-container">$P$</span> is a deflator, <span class="math-container">$i$</span> is nominal interest rate, <span class="math-container">$inf$</span> is an inflation rate and <span class="math-container">$\delta$</span> is depreciation rate of physical capital stock.</p>

CC BY-SA 4.0
2019-07-19T10:54:04.393Z
2019-07-19T11:04:03.520Z
2019-07-19T11:04:03.520Z
london
<p>Take the classic optimization problem of the neo-classical firm:</p>

<p>\begin{equation}
 \begin{array}{*2{&gt;{\displaystyle}r}}
 \mbox{maximize (over $K, L$)} &amp; f(K, L) - RK - WL 
 \end{array}
\end{equation}</p>

<p>The first order condition equates the marginal product of capital with the rental rate of capital $R$.</p>

<p>Which raises the question...</p>

<h3>How do macroeconomists typically estimate a time series for the rental rate of capital using U.S. data?</h3>

<p>(Disclaimer: I don't do macro, but my curiosity has been sparked.)</p>

<p>One approach is to use financial market data to get $R_t$, use another first order condition that the rental rate of capital equals the nominal interest rate plus depreciation (i.e. Hall and Jorgensen). Backing a rental rate out of financial market data is not obvious though! In financial markets, prices vary based upon risk and time.</p>

<ul>
<li>time dimension: Long-term rates are typically higher than short-term rates. When macro-economists and macro-models talk about the rental rate of capital, what's the time frame?</li>
<li>risk dimension: Eg. callable bonds have high yields than non-callable bonds. Debt and equity may have different expected returns based upon risk. </li>
<li>inflation dimension: A fixed nominal rate is a stochastic real rate depending on realized inflation, and the expected real rate is the nominal rate minus inflation expectations. The real rental rate would add back inflation expectations.</li>
</ul>

<p>A completely different direction is taken by Casey Mulligan where he sticks entirely with Bureau of Economic Analysis (BEA) data.</p>

<p>I don't follow this literature, and I don't have a sense of the range of approaches that are considered sensible in modern, empirical macro.</p>

CC BY-SA 3.0
2018-02-08T21:41:13.197Z
Really late but why not have $MPK=r$?
EconJohn
CC BY-SA 3.0
2018-02-08T22:05:08.787Z
@EconJohn Could you expand what you mean?
Matthew Gunn
CC BY-SA 3.0
2018-02-09T03:51:57.073Z
Well, considering classic marginal productivity theory of wages which states the relationship of $\text{MPL}=\text{w}$, a simple extention can be drawn to capital rental rate where $\text{MPK}=\text{r}$.
EconJohn
CC BY-SA 3.0
2018-02-09T04:26:45.773Z
@EconJohn So then to measure marginal productivity of capital, you have to go through an exercise like [this](https://www.imf.org/external/np/res/seminars/2006/arc/pdf/cas.pdf)?
Matthew Gunn
CC BY-SA 3.0
2018-02-09T17:55:48.050Z
I'd say so. This paper is excellent, however they don't relate MPK to the price of capital- I'm suggesting (in a lack of an actual rental data) that solving where $\text{MPK=r}$ is not a bad idea. the formula of having $MPK=\alpha \frac{Y}{K}$ seems appealing.
EconJohn
CC BY-SA 3.0
2018-02-11T20:27:27.190Z
Any thoughts @Matthew Gunn?
EconJohn
<macroeconomics><capital-returns>
Estimating the rental rate of capital from data
Matthew Gunn
# Estimating the rental rate of capital from data

(Asked by: username_0 on 2016-10-05)

Take the classic optimization problem of the neo-classical firm:

\begin{equation}
\begin{array}{\*2{>{\displaystyle}r}}
\mbox{maximize (over $K, L$)} & f(K, L) - RK - WL
\end{array}
\end{equation}

The first order condition equates the marginal product of capital with the rental rate of capital $R$.

Which raises the question...

### How do macroeconomists typically estimate a time series for the rental rate of capital using U.S. data?

(Disclaimer: I don't do macro, but my curiosity has been sparked.)

One approach is to use financial market data to get $R\_t$, use another first order condition that the rental rate of capital equals the nominal interest rate plus depreciation (i.e. Hall and Jorgensen). Backing a rental rate out of financial market data is not obvious though! In financial markets, prices vary based upon risk and time.

* time dimension: Long-term rates are typically higher than short-term rates. When macro-economists and macro-models talk about the rental rate of capital, what's the time frame?
* risk dimension: Eg. callable bonds have high yields than non-callable bonds. Debt and equity may have different expected returns based upon risk.
* inflation dimension: A fixed nominal rate is a stochastic real rate depending on realized inflation, and the expected real rate is the nominal rate minus inflation expectations. The real rental rate would add back inflation expectations.

A completely different direction is taken by Casey Mulligan where he sticks entirely with Bureau of Economic Analysis (BEA) data.

I don't follow this literature, and I don't have a sense of the range of approaches that are considered sensible in modern, empirical macro.

**Tags:** macroeconomics, capital-returns

> **Comments:**
> **username_1**: Really late but why not have $MPK=r$?  
> **username_0**: @username_1 Could you expand what you mean?  
> **username_1**: Well, considering classic marginal productivity theory of wages which states the relationship of $\text{MPL}=\text{w}$, a simple extention can be drawn to capital rental rate where $\text{MPK}=\text{r}$.  
> **username_0**: @username_1 So then to measure marginal productivity of capital, you have to go through an exercise like [this](https://www.imf.org/external/np/res/seminars/2006/arc/pdf/cas.pdf)?  
> **username_1**: I'd say so. This paper is excellent, however they don't relate MPK to the price of capital- I'm suggesting (in a lack of an actual rental data) that solving where $\text{MPK=r}$ is not a bad idea. the formula of having $MPK=\alpha \frac{Y}{K}$ seems appealing.  
> **username_1**: Any thoughts @username_0?  

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## 1 answers

### Answer by username_2 on 2019-07-19. Score: 1

Time series on rental price of capital can be estimated using $$r=\frac{P\_k}{P}(i-inf+\delta)$$
here, $P\_k$ is the price of capital goods (price index for capital goods), $P$ is a deflator, $i$ is nominal interest rate, $inf$ is an inflation rate and $\delta$ is depreciation rate of physical capital stock.


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