Inflation and Cyclical Slack

Why Inflation and Labor-Market Slack Can Move Together

Inflation often slows when economic activity weakens and unemployment rises. The connection is not mechanical, however: inflation expectations may change, and a supply disruption can raise firms’ costs even as labor markets soften. The expectations-augmented Phillips curve organizes these forces so that cyclical pressure is not confused with other sources of inflation.

The central distinction is between the level of inflation and the change in inflation. If inflation falls from 4% to 3%, prices are still rising, but more slowly. That is disinflation, not deflation. A positive inflation rate can therefore coexist with substantial labor-market slack.

Definition: The unemployment gap is the difference between the actual unemployment rate and the natural rate of unemployment: $$ g^u_t = u_t-u_{n,t}. $$ A positive gap means actual unemployment is above the natural rate, indicating labor-market slack in this model. A negative gap means unemployment is below the natural rate, indicating a relatively tight labor market. The natural rate is a benchmark, not a claim that unemployment is zero or that it never changes.

The Expectations-Augmented Phillips Curve

For a short-run analysis, use the following expectations-augmented Phillips curve:

\pi_t = \pi^e_t-\alpha(u_t-u_{n,t})+s_t.

Here, $\pi_t$ is the inflation rate over the period, $\pi^e_t$ is expected inflation for that period, $u_t$ is the unemployment rate, $u_{n,t}$ is the natural rate, $\alpha$ is a positive coefficient, and $s_t$ is a supply-shock term. In this convention, a positive $s_t$ adds to inflation pressure, as can occur with an adverse cost shock. A negative $s_t$ reduces inflation pressure.

The rates in the equation must use consistent units. If inflation and unemployment are entered as percentages, their gaps are measured in percentage points. The coefficient $\alpha$ then measures the change in inflation, in percentage points, associated with a one-percentage-point change in the unemployment gap. For example, if $\alpha=0.4$, a one-point increase in the unemployment gap lowers predicted inflation by 0.4 percentage points, all else held fixed.

The equation is a short-run model, not an unconditional law. For its direct interpretation, treat $\alpha$ as positive and stable over the comparison, and treat expected inflation and the supply-shock term as separately specified. It does not say that unemployment alone determines inflation, or that the same relationship must hold over every time horizon. The earlier material on Unemployment, Output Gaps, and Okun’s Law helps interpret cyclical slack; here the focus is how that slack enters an inflation relationship.

Interpretation: Holding expected inflation, the natural rate, and the supply shock fixed, a larger unemployment gap lowers predicted inflation. An adverse supply shock can push inflation upward at the same time that a positive unemployment gap is pushing it downward.

Worked Applications

Worked Example: Calculating Inflation With Labor-Market Slack

In a hypothetical economy, expected annual inflation is 3.2%, actual unemployment is 7.0%, and the natural rate is 5.0%. The estimated coefficient is $\alpha=0.4$, and a temporary input-cost shock contributes $s_t=0.3$ percentage points. Calculate the model’s inflation rate.

Method. Use the unemployment-gap form of the expectations-augmented Phillips curve. All rates and the shock are expressed in percentage points.

First calculate the unemployment gap:

g^u_t=u_t-u_{n,t}=7.0-5.0=2.0\text{ percentage points}.

The cyclical contribution is $-\alpha g^u_t=-0.4(2.0)=-0.8$ percentage points. Substitute the given values:

\pi_t=3.2-0.8+0.3=2.7\%.

Check. The positive unemployment gap reduces expected inflation by 0.8 points; the positive supply shock restores 0.3 points. The net adjustment is $-0.5$ points, and $3.2-0.5=2.7\%$. The result is positive inflation, not a falling price level.

Common mistake avoided: The shock term is added with its stated sign. Treating the positive 0.3-point cost shock as a negative term would incorrectly produce 2.1%.

Worked Example: Measuring Cyclical Disinflation

In two successive years, expected inflation remains at 2.8%, the natural unemployment rate is 5.0%, and there is no supply shock. The coefficient is $\alpha=0.5$. Unemployment is 4.0% in Year 1 and 6.0% in Year 2. Calculate inflation in each year and the change between them.

Method. Calculate each year’s unemployment gap first, then apply the same model and parameters to both years.

For Year 1, the gap is $4.0-5.0=-1.0$ percentage point. Therefore:

\pi_1=2.8-0.5(-1.0)+0=2.8+0.5=3.3\%.

For Year 2, the gap is $6.0-5.0=1.0$ percentage point. Therefore:

\pi_2=2.8-0.5(1.0)+0=2.8-0.5=2.3\%.

The change in inflation is $2.3-3.3=-1.0$ percentage point. Inflation has declined by 1.0 point, but it remains positive at 2.3%. This is disinflation, not deflation.

Check. The unemployment gap increased by 2.0 points, from $-1.0$ to $1.0$. At $\alpha=0.5$, that change lowers predicted inflation by $0.5(2.0)=1.0$ point, matching the calculated difference.

Common mistake avoided: Calling Year 2 deflation because inflation fell confuses a lower inflation rate with a falling price level. Deflation would require $\pi_2<0$, which is not the case.

Worked Example: Identifying an Inflationary Supply Shock

A hypothetical economy has expected inflation of 2.4%, actual unemployment of 6.5%, and a natural rate of 5.5%. The coefficient is $\alpha=0.3$, while measured inflation is 3.0%. Use the model to infer the supply-shock term.

Method. Rearrange the Phillips curve to isolate the shock term after calculating the cyclical contribution.

The unemployment gap is:

g^u_t=6.5-5.5=1.0\text{ percentage point}.

Without a supply shock, the model would predict:

\pi^e_t-\alpha g^u_t=2.4-0.3(1.0)=2.1\%.

The observed inflation rate exceeds that value by $3.0-2.1=0.9$ percentage point. Equivalently:

s_t=\pi_t-\pi^e_t+\alpha g^u_t =3.0-2.4+0.3(1.0)=0.9\text{ percentage point}.

The inferred residual is a positive 0.9-point shock contribution. Slack is lowering inflation relative to expected inflation, but the positive shock more than offsets that cyclical effect: $2.4-0.3+0.9=3.0\%$.

Check and limitation. Substitution reproduces the observed 3.0%. The calculation identifies the residual required by this model; by itself, it does not prove which real-world event caused the residual.

Common mistake avoided: Concluding that rising unemployment must mean falling observed inflation ignores $s_t$. The model’s cyclical contribution is downward, but the total inflation rate also includes expectations and shocks.

Why the Relationship Has This Form

The negative sign on the unemployment gap captures the model’s short-run cyclical mechanism: when unemployment is relatively low, labor markets are tighter and wage and cost pressures may be stronger; when unemployment is relatively high, those pressures may ease. Expected inflation provides a baseline, while the supply-shock term accounts for inflation pressure not represented by the gap. The equation summarizes this relationship rather than deriving every wage- and price-setting decision.

A useful comparison follows by writing the equation for two periods and subtracting. If the natural rate and coefficient are stable across the comparison, then:

Write the first-period equation.
$\pi_1=\pi^e_1-\alpha g^u_1+s_1$.

Write the second-period equation.
$\pi_2=\pi^e_2-\alpha g^u_2+s_2$.

Subtract the first equation from the second.
$\pi_2-\pi_1=(\pi^e_2-\pi^e_1)-\alpha(g^u_2-g^u_1)+(s_2-s_1)$.

Interpret the change.
The change in inflation is the change in expected inflation, minus the coefficient times the change in the unemployment gap, plus the change in the supply shock.

This subtraction explains why slack is not enough to predict a change in observed inflation. If expectations and shocks are unchanged, a rising unemployment gap lowers inflation. If the supply shock rises sufficiently, however, it can offset or exceed that downward pressure. Similarly, a change in expected inflation can alter the comparison even when labor-market slack is unchanged.

The relationship can also be expressed using the output gap, defined as actual output relative to trend or potential output. A common short-run form is $\pi_t=\pi^e_t+\kappa x_t+s_t$, where $x_t$ is the output gap and $\kappa>0$. A negative output gap then contributes downward pressure on inflation. This is an alternative representation of cyclical slack, not a reason to add the unemployment-gap and output-gap contributions together without a model specifying how they relate.

Reading the Pattern Without Overclaiming

The Phillips curve is most useful as a disciplined way to separate components. It clarifies why a slowdown in inflation may reflect weaker demand and more labor-market slack, while an increase in inflation during a weak labor market may point to rising cost pressure or changed expectations. It does not establish that every observed movement has one unique cause: measured inflation, expectations, the natural rate, and the shock term may all be uncertain.

Slack rises and inflation slows: consistent with the cyclical channel, especially if expectations and supply shocks are broadly stable.
Slack rises but inflation accelerates: possible if positive supply pressure or rising expected inflation outweighs the downward cyclical contribution.
Inflation remains positive while declining: disinflation; the price level is still rising, just at a slower rate.
Inflation is negative: deflation; the general price level is falling over the measured period.

A common exam trap is to treat a correlation between unemployment and inflation as a complete explanation. The equation requires a comparison of the unemployment gap, expectations, and the shock term. Another trap is to use the wrong sign: a positive unemployment gap is multiplied by $-\alpha$, so it reduces predicted inflation. A negative unemployment gap makes the cyclical term positive.

Key takeaway: In the expectations-augmented Phillips curve, labor-market slack lowers inflation relative to expected inflation, other things equal. A supply shock can offset that effect, so observed inflation must be interpreted by separating the cyclical contribution from expectations and non-cyclical pressure.

Check Your Understanding

For each problem, use percentage-point units and the expectations-augmented Phillips curve unless stated otherwise.

An economy has expected inflation of 2.0%, unemployment of 6.4%, a natural rate of 5.4%, $\alpha=0.25$, and a supply shock of $+0.2$ percentage point. Calculate predicted inflation. (Answer: 1.95%)
Expected inflation is 3.1%, unemployment is 4.8%, the natural rate is 5.8%, $\alpha=0.4$, and the supply shock is zero. Calculate predicted inflation and state whether the unemployment gap contributes upward or downward pressure. (Answer: 3.5%; upward pressure)
In Year 1, expected inflation is 2.5%, the unemployment gap is 0, and the shock is $+0.4$ point. In Year 2, expected inflation is 2.7%, the unemployment gap is $+1.0$ point, and the shock is $+0.1$ point. If $\alpha=0.4$, calculate inflation in each year and the change. (Answer: Year 1, 2.9%; Year 2, 2.4%; change, −0.5 percentage point)
Measured inflation is 4.2%, expected inflation is 2.6%, unemployment is 7.0%, the natural rate is 6.0%, and $\alpha=0.5$. Infer the supply-shock term. (Answer: +2.1 percentage points)
In the output-gap form, expected inflation is 3.0%, $\kappa=0.4$ percentage point per percentage point of output gap, the output gap is −2.0%, and the shock is $+0.6$ point. Calculate inflation. Then state the predicted inflation if the output gap improves to −1.5%, with all other inputs unchanged. (Answer: 2.8%; then 3.0%)

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