% vs. Share Factors…and…Exponential vs. Linear

One can discuss a p.c improve in a revenue margin… however that basically solely serves to confuse.

Suppose I needed to take a look at an after tax revenue margin (on this case “Revenue per unit of actual gross worth added of nonfinancial company enterprise: Company income after tax with IVA and CCAdj (unit income from present manufacturing)” (A466RD3Q052SBEA)).

Determine 1: Revenue per unit actual gross worth added in nonfinancial company , after tax (blue). NBER outlined peak-to-trough recession dates shaded grey. Crimson dashed traces at 1992Q4-2022Q4; arrows denote p.c and share level development between these dates. Supply: BEA through FRED, NBER, and writer’s calculations.

One may discuss a share development price of a revenue margin (4.5%/yr in Determine 1), nevertheless it’s bizarre to calculate a share change on a p.c. That’s why sometimes, when contemplating modifications in ratios, one speaks of share level modifications (0.4 ppts/yr in Determine 1), thereby avoiding useless confusion. (Those that intentionally need to confuse would possibly need to use “p.c change” of ratios, then).

When to make use of p.c change? Nicely, when discussing one thing in ranges. E.g., Determine 2 beneath.

Determine 2: Revenue for nonfinancial company enterprise sector, after tax, in bn.Ch2012$ SAAR, on log scale (blue). Earnings deflated by GDP deflator. NBER outlined peak-to-trough recession dates shaded grey. Crimson dashed traces at 1992Q4-2022Q4; arrows denote share level development between these dates. Supply: BEA through FRED, NBER, and writer’s calculations.

The opposite level I need to make is that, technically, one thing rising exponentially doesn’t imply it’s essentially rising quick. Could be, may not be. As an illustration one thing rising at 0.01% per yr is perhaps growing rather a lot slower than one thing rising at 0.01 models per yr… Exponential simply implies that if the sequence is logged, the logged sequence grows linearly.