# Estimating the rental rate of capital from data

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

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

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.

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

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.