# Changing Capital Values and Output

The BEA publishes data on the fixed assets (structures, equipment and software).  I took the ratio of the current value private fixed asset series, divided by the chained quantity index to get a time series proxy for changing capital values that is more stable than measures of stock market prices.

Assuming the no-arbitrage condition:

$q_{n+1}/q_n + r_n/q_n = b_n + \sigma_n$

Where $q_n$ is the price of capital goods, $b_n$ is the period bond return, $\sigma_n$ is a risk term, and $r_n$ is the capital rental payment. All quantities are nominal, but as this is a spread of rates, the implied yield is a real quantity.

The difference between the (short term) bond yield and the change in price of capital goods should be the capital rental yield minus a risk term. Holding the risk term constant, I define the difference to be the implied capital rental yield.

BEA fixed asset data goes back to 1925, and my source for short term bond rates and real GDP growth was from measuring worth (rates were measured by the short term ordinary funds consistent series).

I ran a VAR(2) model on the implied yield, y/y Real GDP Growth, and a dummy variable for WW2.  I’m not a VAR expert, but the model looks reasonably clean (e.g. passing serial correlation, kurtosis tests, etc.)

Here are the impulse response functions for a change in implied yields:

As expected, an increase in the implied yield gives rise to a negative shock to GDP growth. Also as expected, a change in GDP gives rise to an an initial increase in the implied yield, followed by a decrease.

This highlights a zero bound problem faced by the central bank when adjusting rates. In those cases when (nominal) bond yields are already low or the expected decline in capital goods prices is significant, the central bank may not be able to prevent the rental rate of capital from increasing even if it reduces the bond rate.  The central bank does not control the capital rental rate, it only controls one input into this rate. It cannot convince households to bear losses just because it lowers the bond rate.