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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.

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  • $\begingroup$ Really late but why not have $MPK=r$? $\endgroup$ – EconJohn Feb 8 '18 at 21:41
  • $\begingroup$ @EconJohn Could you expand what you mean? $\endgroup$ – Matthew Gunn Feb 8 '18 at 22:05
  • $\begingroup$ 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 $\text{MPK}=\text{r}$. $\endgroup$ – EconJohn Feb 9 '18 at 3:51
  • $\begingroup$ @EconJohn So then to measure marginal productivity of capital, you have to go through an exercise like this? $\endgroup$ – Matthew Gunn Feb 9 '18 at 4:26
  • $\begingroup$ 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. $\endgroup$ – EconJohn Feb 9 '18 at 17:55
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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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