Okun's Law and the Output–Inflation Chain

Okun’s law

Output and unemployment move together over the business cycle. The empirical regularity, due to Arthur Okun:

$$u - u_n \approx -k\,(Y - Y_n)/Y_n,$$

with $k \approx 0.5$ for the US (each percentage-point of output gap corresponds to about half a percentage-point change in unemployment in the opposite direction). When output runs above its natural level, firms hire more; when below, they lay off.

The Phillips curve, again

The expectations-augmented Phillips curve:

$$\pi \;=\; \pi^e - \beta\,(u - u_n) + \varepsilon.$$

Low unemployment relative to the natural rate pushes inflation above expectations. Conversely high unemployment pulls inflation down.

Stringing them together

Chaining Okun and Phillips, an output gap directly produces an inflation response:

$$\pi \;=\; \pi^e + \beta k\,(Y - Y_n)/Y_n + \varepsilon.$$

The product $\beta k$ is the sacrifice ratio’s inverse: roughly how much extra inflation a percentage point of over-heating output produces in the short run.

Output gap (Y−Yₙ)/Yₙ: 0.00%Natural rate uₙ: 4.5%Expected inflation πᵉ: 2.5%Phillips slope β: 0.50Okun coefficient k: 0.50

u = 4.50%, π = 2.50%

Okun: gap → unemployment

Phillips: unemployment → inflation

Positive output gap (boom) ⇒ lower unemployment via Okun ⇒ higher inflation via the Phillips curve. Try sliding the output gap from −3% to +3% — that's roughly the swing the US economy made between the 2020 trough and the 2022 over-heating.

The 2020-2022 US path

Try sliding the output gap from −3% (the 2020 trough) up through +2% (the 2022 over-heating) and watch the chain:

1. Output collapses (gap = −3%) ⇒ unemployment spikes (Okun) ⇒ inflation falls toward expectations.
2. Stimulus + recovery push the gap to +2% by mid-2022 ⇒ unemployment falls to ~3.5% (well below the natural rate) ⇒ inflation overshoots expectations dramatically.
3. Aggressive Fed tightening through 2023 closes the gap back toward zero ⇒ unemployment normalizes ⇒ inflation eases toward target.

The framework doesn’t tell you why the gap moved — supply shocks, fiscal stimulus, monetary loosening, and post-pandemic re-opening all contributed. But once you know the gap, Okun + Phillips give you a rough prediction of unemployment and inflation.