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Okun's Law
Model

An empirical regularity linking the unemployment gap to the output gap: u - u* = -beta * (Y - Y*)/Y*. When output exceeds potential, unemployment falls below its natural rate, and vice versa.

Derivation

Step-by-step mathematical derivation with typeset equations and expandable detail.

Sections

Output gap and unemployment gapOkun's coefficient and the empirical relationship

Output gap and unemployment gap

Okun's law is an empirical regularity connecting the output gap to the unemployment gap. The output gap is the percentage deviation of actual real GDP YYY from potential output Y∗Y^*Y∗: (Y−Y∗)/Y∗(Y - Y^*)/Y^*(Y−Y∗)/Y∗. Potential output is the level consistent with full employment of resources and stable inflation. The unemployment gap is the difference between the actual unemployment rate uuu and the natural rate u∗u^*u∗ (the rate consistent with stable inflation, sometimes called the NAIRU). When output is above potential, unemployment tends to be below the natural rate, and vice versa.

Arthur Okun documented this relationship in 1962 using U.S. data, finding that a one-percentage-point increase in unemployment above the natural rate was associated with roughly a two-to-three percent shortfall in output below potential. The relationship is not one-for-one because labor is not the only input: when demand falls, firms also reduce hours, cut overtime, and hoard workers (keeping employees on payroll despite lower production), so output drops by more than the unemployment rate rises.

Y~=Y−Y∗Y∗×100\tilde{Y} = \frac{Y - Y^*}{Y^*} \times 100Y~=Y∗Y−Y∗​×100

The output gap in percentage terms: positive when the economy operates above potential.

u~=u−u∗\tilde{u} = u - u^*u~=u−u∗

The unemployment gap: positive when the actual unemployment rate exceeds the natural rate.

Okun's coefficient and the empirical relationship

Okun's law takes the form Y~=−β(u−u∗)\tilde{Y} = -\beta(u - u^*)Y~=−β(u−u∗), where β>0\beta > 0β>0 is the Okun coefficient. The negative sign reflects the inverse relationship: unemployment above the natural rate is associated with output below potential. Empirical estimates for the United States typically place β\betaβ in the range of 2 to 3, meaning each percentage point of excess unemployment corresponds to a 2-3% output gap. The relationship can equivalently be written in changes: ΔY/Y≈3%−2Δu\Delta Y/Y \approx 3\% - 2\Delta uΔY/Y≈3%−2Δu, where 3% is a rough estimate of normal output growth. If the unemployment rate rises by one point, output growth falls about two points below trend.

The coefficient β\betaβ exceeds one because of several amplifying channels. When firms face lower demand, they reduce not just headcount but also hours per worker, effort intensity, and overtime. Productivity tends to fall in recessions (labor hoarding means more workers per unit of output), which amplifies the output loss per unit of unemployment increase. Conversely, in booms, firms increase hours and effort before hiring, so output rises faster than unemployment falls. The Okun coefficient captures all of these margins and is best interpreted as a summary statistic of the labor market's response to demand fluctuations, not a structural parameter.

Y~=−β (u−u∗)\tilde{Y} = -\beta \,(u - u^*)Y~=−β(u−u∗)

Okun's law: the output gap is proportional to the unemployment gap, with coefficient β≈2\beta \approx 2β≈2-333 for the U.S.

ΔYY≈g∗−β Δu\frac{\Delta Y}{Y} \approx g^* - \beta \,\Delta uYΔY​≈g∗−βΔu

Growth-rate form: actual GDP growth equals trend growth minus the Okun coefficient times the change in unemployment.

β≈2(U.S. postwar average)\beta \approx 2 \quad (\text{U.S. postwar average})β≈2(U.S. postwar average)

A one-percentage-point rise in unemployment is associated with roughly a 2% decline in output relative to potential.

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