How do companies change their capital use when an economy is going through a business cycle?

Economies often go through business cycles, where output expands and then contracts and then expands again. The variations in output during business cycles are quite small, perhaps a few percent, and they only last a few years. Businesses could respond to the downturn by getting rid of their capital or labour, but that would incur sales or redundancy costs, and when the economy turned up again they would probably have to buy or hire again, which would incur new costs. Sales and redundancies do happen, but much of the time it is cheaper for companies to keep their labour and capital the same, and just work them easier or harder when the economy is contracting or expanding, respectively.

Economists using data on capital and labour in estimation face a problem as a result. For example, they often calculate capital manually from investment data or from company purchase costs or in other ways that do not measure how hard the capital is being used. If an economist wants to estimate capital's effect on output, and if we are not allowing for usage intensity, then capital will seem to have a different relation with output that it really does.

Economists would like to know how hard capital is being worked. It is not something that is readily available in statistics. An ad hoc adjustment can be made instead to capture some of the changes in usage intensity, albeit imperfectly. One way is to calculate some simple trend in output, then work out how much higher or lower output is at every time. If output is higher than is trend, then capital is assumed to be working harder than usual, and if output is below trend, then capital is assumed to be working easier than usual and adjustments are made. For example, the paper here (on page 13) uses a simple time trend. It is not a big chore to make the adjustments. For example, the following pseudo-code generates the adjusted capital:

regress gdp on year
predict time_trend=regression_estimates
generate adjusted_capital=capital*(1+(gdp-time_trend)/time_trend)

The adjustment matters. For example, an estimation of output on capital and labour (in log form) gives

ln output = 0.59*ln capital + -0.09*ln education + other terms
(details: world countries, panel data for the 1990s, fixed effect estimation with time dummies)

while an estimation of output with adjusted capital gives more likely coefficients

ln output = 0.46*ln capital + 0.14*ln education + other terms.