Unemployment, inflation and the Fisher relation

Rates that count people

The labour force is everyone of working age who is either employed or actively looking for work. Two ratios come from it:

$$\text{Unemployment rate}=\frac{\text{Unemployed}}{\text{Labour force}}\text{Participation rate}=\frac{\text{Labour force}}{\text{Working-age population}}$$

Suppose 1.3 million people are unemployed in a labour force of 21.0 million drawn from a working-age population of 32.5 million. Unemployment is $1.3/21.0=6.19\%$ (=B2/B3) and participation is $21.0/32.5=64.6\%$ (=B3/B4).

The denominator is the trap. A discouraged worker who stops searching leaves the labour force: the numerator and the denominator both fall, and because the numerator was the smaller share, the unemployment rate falls. Participation falls too. Nothing improved.

Four kinds of unemployment, by cause: frictional (between jobs — a graduate searching), structural (skills no longer match — a mill closes for good), cyclical (a contraction — layoffs that return with the recovery), seasonal (a ski resort in June). Only cyclical unemployment is what monetary policy leans against.

Rates that count prices

The Consumer Price Index (CPI) prices a fixed basket; inflation is its rate of change. From 158.4 to 163.2: $163.2/158.4-1=3.03\%$ — =B6/B5-1. Core inflation strips the volatile components (food, energy) so the trend shows. Demand-pull inflation comes from spending outrunning capacity; cost-push from rising input costs — wages, oil, tariffs. A fall in the rate (4 % to 2 %) is disinflation; prices still rise, more slowly. Deflation is a falling price level — a negative rate. Stagflation is inflation with a stagnant economy.

The real rate between them: Fisher

A nominal rate of 5.5 % when inflation is 3 % does not leave 2.5 % of purchasing power. The exact Fisher relation is

$$1+r_{\text{nominal}}=(1+r_{\text{real}})(1+\text{inflation})⟹r_{\text{real}}=\frac{1.055}{1.03}-1=2.427\%$$

=(1+0.055)/(1+0.03)-1. The approximation subtracts: $5.5-3=2.5\%$ — =0.055-0.03. The two differ by the cross-term $r_{\text{real}}\times \text{inflation}$, small at low inflation and material above about 5 %. Real return bonds (Lesson 9.4) are the instrument that pays the real rate directly.

Recompute

1.5 million unemployed in a 22.0 million labour force from 33.0 million: =1.5/22 = 6.82 %; =22/33 = 66.7 %. CPI 163.2 → 168.1: =168.1/163.2-1 = 3.00 %. A 6 % nominal rate with 8 % inflation: exact =(1.06)/(1.08)-1 = −1.85 %; approximate −2.0 %. A negative real rate is what a saver earns when inflation outruns the deposit rate.