Credibility and the Costs of Disinflation

Why Credibility Matters for Disinflation

Disinflation is a reduction in the rate of inflation; it does not necessarily mean that the general price level is falling. A central bank can seek lower inflation by restraining aggregate demand, but the output cost depends in part on how households and firms form their expectations. If they expect high inflation to persist, wage and price decisions may continue to reflect it even after policy changes. A believable commitment to lower inflation can bring expectations down sooner and reduce the contraction needed to achieve the new rate.

The key distinction is between announcing a lower inflation objective and being credible enough that people incorporate it into their decisions. Credibility is not a mechanical guarantee that inflation falls without cost. It changes one part of the transmission process: expected inflation. The output cost still depends on the relationship between inflation and economic slack, how expectations adjust, and whether other shocks are present. This tutorial develops simple models to make those relationships explicit, aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: Disinflation is a decline in the inflation rate. Credibility is the degree to which the public believes that the policymaker will follow through on its stated inflation objective. Anchored expectations are expectations that remain close to the announced objective rather than moving substantially with recent inflation.

A useful measure of the output cost is the sacrifice ratio: the cumulative loss of output, relative to potential output, associated with reducing inflation by one percentage point. In a simple calculation, output losses are measured as percentage-point-years of negative output gap. The ratio is a model-based summary, not a universal constant: different expectation formation, supply conditions, and policy paths can produce different costs.

A Simple Expectations-Augmented Phillips Curve

Let \(\pi_t\) be inflation in period \(t\), \(\pi_t^e\) expected inflation for that period, and \(x_t\) the output gap, measured as actual output minus potential output as a percentage of potential output. A positive output gap means output is above potential; a negative gap means it is below potential. Consider the following reduced-form Phillips curve:

\pi_t=\pi_t^e+\kappa x_t

Here \(\kappa>0\) measures how much inflation responds to a one-percentage-point change in the output gap. For example, if \(\kappa=0.5\), a one-percentage-point negative output gap reduces inflation by 0.5 percentage points relative to expected inflation. This is a simplified model: it assumes a stable \(\kappa\), abstracts from supply shocks, and treats the output gap as the relevant source of inflation pressure. It should not be applied as an exact forecasting equation when these assumptions are not reasonable.

Model assumption: Under backward-looking expectations, expected inflation equals the previous period's inflation, so \(\pi_t^e=\pi_{t-1}\). In this special case, the Phillips curve implies \(\pi_t-\pi_{t-1}=\kappa x_t\). This assumption is useful for deriving a basic sacrifice ratio, but it does not describe all ways expectations can form.

The backward-looking model explains why a rapid reduction in inflation can require a negative output gap. If expectations adjust only after observing past inflation, current inflation will not fall merely because a lower rate has been announced. Policy must create enough slack to bring actual inflation below expected inflation. If the announcement is trusted, however, expected inflation may fall in advance, reducing the required gap.

Why the Basic Sacrifice-Ratio Formula Holds

Under backward-looking expectations, substitute \(\pi_t^e=\pi_{t-1}\) into the Phillips curve and rearrange:

\pi_t-\pi_{t-1}=\kappa x_t

Add this equation across periods \(1\) through \(T\). The intermediate inflation terms cancel: \((\pi_1-\pi_0)+(\pi_2-\pi_1)+\cdots+(\pi_T-\pi_{T-1})=\pi_T-\pi_0\). Therefore:

\sum_{t=1}^{T}x_t=\frac{\pi_T-\pi_0}{\kappa}

If inflation falls, then \(\pi_T-\pi_0<0\), so cumulative output gaps are negative. Define the sacrifice ratio as the magnitude of the cumulative output gap divided by the percentage-point decline in inflation. In this particular model it equals \(1/\kappa\). The result depends on the stated assumptions: it need not hold when expectations respond to a credible announcement, \(\kappa\) changes, or supply shocks affect inflation.

\text{Sacrifice ratio} =\frac{-\sum_{t=1}^{T}x_t}{\pi_0-\pi_T} =\frac{1}{\kappa}

Worked Example: Calculating a Backward-Looking Sacrifice Ratio

A hypothetical economy has inflation of 9% before a disinflation. Over the next three periods, inflation is 7%, 5%, and 3%. Assume the Phillips curve has \(\kappa=0.5\), expected inflation in each period equals the previous period's actual inflation, and there are no supply shocks. Calculate each output gap and the sacrifice ratio.

Method. Use \(\pi_t-\pi_{t-1}=\kappa x_t\), so \(x_t=(\pi_t-\pi_{t-1})/\kappa\). The inflation changes are \(7-9=-2\), \(5-7=-2\), and \(3-5=-2\) percentage points.

For period 1, \(x_1=(-2)/0.5=-4\%\). For period 2, \(x_2=(-2)/0.5=-4\%\). For period 3, \(x_3=(-2)/0.5=-4\%\). Cumulative output gap is \(-4-4-4=-12\) percentage-point-years. Inflation declines by \(9-3=6\) percentage points, so the sacrifice ratio is \(12/6=2\).

Check. The formula \(1/\kappa=1/0.5=2\) gives the same ratio. The calculation is internally consistent. The common mistake is to divide the inflation reduction by cumulative output loss, which gives the reciprocal; the sacrifice ratio is output loss per percentage point of disinflation.

How Credibility Changes Expected Inflation

To represent credibility in a simple way, suppose the policymaker announces a permanent target \(\pi^*\). Let \(\gamma\), between 0 and 1, measure how quickly expected inflation adjusts toward that target:

\pi_t^e=\pi_{t-1}^e+\gamma(\pi^*-\pi_{t-1}^e)

A larger \(\gamma\) means expectations move more quickly toward the target. In this illustrative model, stronger credibility corresponds to a larger adjustment parameter. It does not mean the public believes every announcement in every circumstance; it is a compact way to compare expectation adjustment under different degrees of trust. If the policymaker aims to deliver \(\pi_t=\pi^*\) each period, the output gap required by the Phillips curve is:

x_t=\frac{\pi^*-\pi_t^e}{\kappa}

So long as expected inflation remains above target, the required output gap is negative. With faster adjustment, the gap between expected inflation and the target shrinks more rapidly, reducing the negative output gap. The model assumes that policy can bring actual inflation to the target in each period; it isolates the effect of expectations rather than describing every operational or timing constraint.

Worked Example: Comparing Strong and Weak Credibility

A hypothetical central bank has brought inflation from 6% to a target of 3%. Before the target is announced, expected inflation is 6%. Assume \(\kappa=1\), no supply shocks, and that actual inflation is 3% in each of the next three periods. Compare the cumulative output-gap cost for strong credibility, \(\gamma=2/3\), and weak credibility, \(\gamma=1/3\). Expectations adjust toward the 3% target before the output gap for each period is determined.

Method. In each period, first update expected inflation using the credibility equation, then calculate \(x_t=(3-\pi_t^e)/1\). With strong credibility, expected inflation is \(6+(2/3)(3-6)=4\) in period 1, \(4+(2/3)(3-4)=10/3\) in period 2, and \(10/3+(2/3)(3-10/3)=28/9\) in period 3. The output gaps are therefore \(-1\), \(-1/3\), and \(-1/9\) percentage points. Their sum is \(-13/9\), approximately \(-1.444\) percentage-point-years.

With weak credibility, expected inflation is \(6+(1/3)(3-6)=5\) in period 1, \(5+(1/3)(3-5)=13/3\) in period 2, and \(13/3+(1/3)(3-13/3)=35/9\) in period 3. The output gaps are \(-2\), \(-4/3\), and \(-8/9\). Their sum is \(-38/9\), approximately \(-4.222\) percentage-point-years.

Check. Over the three-period horizon, the weak-credibility path has \(38/13\), or about 2.923, times the cumulative output-gap loss of the strong-credibility path. The horizon matters: expectations are still above target at the end of period 3, so this is not a claim about the total cost over every future period. A common mistake is to set expected inflation equal to 3% immediately in both cases. That would erase the very difference in credibility the example is designed to measure.

Credibility Along a Gradual Disinflation Path

Disinflation need not aim to reach the final target in one period. A policymaker may announce a sequence of lower inflation rates. For this case, let \(\pi_t^{ann}\) be the announced rate for period \(t\), and use a gradual-path expectation rule:

\pi_t^e=\pi_{t-1}^e+c(\pi_t^{ann}-\pi_{t-1}^e)

Here \(c\) measures how much current expectations respond to the announced rate for that period. Assume actual inflation equals the announced rate and there are no supply shocks. Then \(x_t=(\pi_t^{ann}-\pi_t^e)/\kappa\). When expectations adjust only partly, they remain above the announced rate during disinflation, requiring a negative output gap. This rule and the permanent-target rule above are alternative illustrative specifications; they should not be combined as though they were one universal model.

Worked Example: A Credible Two-Step Disinflation

A hypothetical economy begins with expected inflation of 9%. The policymaker announces inflation of 7% in period 1 and 5% in period 2. Assume actual inflation equals those announced rates, \(\kappa=2\), and no supply shocks. Compare a stronger-credibility case with \(c=0.6\) and a weaker-credibility case with \(c=0.25\). Calculate the output gaps and cumulative gap in each case.

Method. Update expectations toward each period's announced rate, then use \(x_t=(\pi_t^{ann}-\pi_t^e)/2\). With stronger credibility, period 1 expectations are \(9+0.6(7-9)=7.8\), giving \(x_1=(7-7.8)/2=-0.4\). Period 2 expectations are \(7.8+0.6(5-7.8)=6.12\), giving \(x_2=(5-6.12)/2=-0.56\). Cumulative output gap is \(-0.4-0.56=-0.96\) percentage-point-years.

With weaker credibility, period 1 expectations are \(9+0.25(7-9)=8.5\), giving \(x_1=(7-8.5)/2=-0.75\). Period 2 expectations are \(8.5+0.25(5-8.5)=7.625\), giving \(x_2=(5-7.625)/2=-1.3125\). Cumulative output gap is \(-0.75-1.3125=-2.0625\) percentage-point-years.

Check. The weaker-credibility case has the larger contraction in both periods. The figures follow from the stated expectation rule and target path. The common mistake is to update expectations toward the final 5% target in period 1; the rule instead uses the announced rate for each period, so period 1 expectations respond toward 7%.

Interpretation, Limits, and Exam Traps

The examples show the central mechanism: if a credible commitment lowers expected inflation, actual inflation can be brought down with less negative output relative to potential. A gradual path can still involve output losses when expectations lag the announced rates, but stronger adjustment can reduce those losses. This is why a sacrifice ratio estimated under backward-looking expectations should not automatically be applied to a different policy regime.

The calculations are conditional, not guarantees. Actual outcomes can differ if the output-inflation relationship changes, if expectations respond in a different way, or if supply shocks raise or lower inflation independently of the output gap. The simple models also assume the announced path is delivered. They do not establish that an announcement alone can control inflation or that credibility is costless to maintain.

Do not confuse disinflation with deflation. Inflation falling from 6% to 3% means prices are still rising, just more slowly.
Keep percentage points and percent changes distinct. A decline from 6% inflation to 3% is a fall of 3 percentage points.
Track the expectation rule period by period. In a gradual path, the relevant announcement may be the current period's rate, not the final target.
Keep the sign of the output gap. If expected inflation exceeds actual inflation in the Phillips curve used here, the implied output gap is negative.
State the horizon and assumptions. A finite-horizon cumulative gap is not necessarily the lifetime cost, and \(1/\kappa\) is not a universal sacrifice ratio.

Key takeaway: In an expectations-augmented Phillips curve, the output cost of disinflation depends on how expected inflation adjusts. A credible commitment can lower expectations sooner and reduce the contraction required, but the size of the effect depends on the model, the policy path, and the assumptions used.

Check Your Understanding

For each problem, use the specified model and keep track of the output-gap sign.

Under backward-looking expectations, inflation falls from 7% to 6% and then 5%. If \(\kappa=0.8\), find each period's output gap and the sacrifice ratio. (Answer: output gaps of \(-1.25\%\) in each period; sacrifice ratio \(1.25/1=1.25\).)
Expected inflation begins at 10%. A permanent target of 4% is announced. In the first two periods, expectations adjust by \(\gamma=0.5\) toward the target. If actual inflation is 4% in both periods and \(\kappa=1.5\), find the two output gaps and their cumulative total. (Answer: \(-2\%\) and \(-1\%\); cumulative output gap \(-3\) percentage-point-years.)
Expected inflation starts at 9%. Announced and actual inflation are 7% in period 1 and 5% in period 2. Expectations move a fraction \(c=0.6\) toward each period's announced rate, and \(\kappa=2\). Find the cumulative output gap. (Answer: \(-0.96\) percentage-point-years.)
Under backward-looking expectations and a constant \(\kappa=0.4\), inflation is reduced by 3 percentage points. What cumulative output loss does the model imply, in percentage-point-years? (Answer: \(7.5\) percentage-point-years of output loss; the cumulative output gap is \(-7.5\).)
Expected inflation begins at 7%, and the permanent target is 3%. Actual inflation reaches 3% in the first period. If \(\kappa=2\), compare the first-period output gaps for \(\gamma=0.8\) and \(\gamma=0.2\). (Answer: \(-0.4\%\) with \(\gamma=0.8\), and \(-1.6\%\) with \(\gamma=0.2\); the weaker-credibility case is 1.2 percentage points more negative.)

What You'll Learn