Why Supply Shocks Create a Policy Trade-Off
An adverse supply shock raises firms’ costs or reduces the economy’s productive capacity. Inflation can rise even as output weakens, leaving a central bank with an uncomfortable choice: tightening demand may reduce inflation pressure, but it can deepen the output gap. The appropriate response depends in part on how long the shock is expected to last and whether inflation expectations remain near the target.
The key distinction is between the direct effect of the shock on inflation and the effect of monetary policy on demand. A central bank cannot directly reverse a disrupted supply chain or a jump in input costs. It can influence demand, and therefore the output gap, but using that channel to offset inflation immediately may carry a substantial output cost. This tutorial uses a simple short-run model to make that trade-off explicit.
Use the following short-run Phillips curve. Inflation and the inflation target are expressed in percentage points, as is the output gap; \(\kappa>0\) converts a one-percentage-point output gap into inflation pressure. Let \(e\) be expected inflation’s deviation from the target and \(u\) the supply shock’s inflation effect.
Here, \(\pi\) is inflation, \(\pi^*\) is the target, and \(e=\pi^e-\pi^*\) is the deviation of expected inflation from target. An adverse shock has \(u>0\). The model is a simplified one-period relationship: it treats \(e\) and \(u\) as given when policy chooses \(x\). It does not describe how expectations form or how supply capacity changes over time. Those features must be assessed separately when applying the model to a persistent shock.
A One-Period Flexible-Targeting Model
Suppose the central bank chooses the output gap to minimize a quadratic loss that penalizes both deviations of inflation from target and output from potential:
The parameter \(\lambda\) measures the relative weight placed on output-gap stability. A larger \(\lambda\) makes the central bank less willing, in this model, to create a negative output gap to reduce inflation. This loss function is a modeling assumption, not a claim that all central banks use one fixed numerical weight.
These expressions describe a compromise. If expected inflation and the supply shock together put inflation above target, the best output gap in this model is negative. But, unless the output-gap weight is zero, the minimizing choice does not eliminate the inflation deviation completely.
Why the Flexible-Targeting Formula Holds
Substitute the Phillips curve into the loss function. This expresses the loss entirely in terms of the policy choice \(x\). Differentiating and setting the first derivative equal to zero gives the minimizing choice:
Since \(\pi-\pi^*=e+u+\kappa x\), substitute this expression into \(L(x)\).
The derivative is \(\kappa(e+u+\kappa x)+\lambda x\).
Rearranging gives \((\kappa^2+\lambda)x=-\kappa(e+u)\).
The second derivative is \(\frac{d^2L}{dx^2}=\kappa^2+\lambda>0\), so this stationary point is a minimum. Substituting \(x^*\) into the Phillips curve gives the inflation deviation:
The result applies only to the stated one-period model with the shock and expected inflation treated as given. It does not by itself determine the right policy path over several periods. In particular, if today’s policy affects future expectations or if the shock changes over time, those paths must be considered explicitly.
Worked Applications
Worked Example: A Supply Shock with Above-Target Expectations
In a one-period model, let \(\kappa=0.4\), \(\lambda=0.6\), \(e=0.3\) percentage points, and \(u=1.2\) percentage points. Find the loss-minimizing output gap and the resulting inflation deviation. State the method and identify the calculation trap.
Method. Use the flexible-targeting formulas. First combine the expected-inflation deviation and the supply shock: \(e+u=0.3+1.2=1.5\) percentage points. The denominator is \(\kappa^2+\lambda=(0.4)^2+0.6=0.16+0.6=0.76\).
The output gap is \(x^*=-0.4(1.5)/0.76=-0.6/0.76=-15/19\approx-0.7895\%\). The inflation deviation is \(0.6(1.5)/0.76=0.9/0.76=45/38\approx1.1842\) percentage points above target.
Check. Substituting the output gap into the Phillips curve gives \(1.5+0.4(-15/19)=1.5-6/19=45/38\), which matches the calculated inflation deviation. Common mistake avoided: Do not use \(\kappa\) in the numerator of the inflation-deviation formula. The output-gap formula has \(\kappa\) in its numerator; the inflation-deviation formula has \(\lambda\).
Worked Example: A Temporary Shock with Expectations at Target
A central bank faces an adverse supply shock of \(u_1=2\) percentage points in the current period and expects it to disappear in the next period, so \(u_2=0\). Expectations remain at target in both periods, so \(e_1=e_2=0\). Use \(\kappa=0.5\) and \(\lambda=1\). Find the flexible-targeting choices and inflation deviations in each period. Then compare the current-period choice with full inflation stabilization.
Method. Apply the one-period formulas separately to each period, treating that period’s shock and expectations as given. The common denominator is \((0.5)^2+1=0.25+1=1.25\).
In period 1, \(e_1+u_1=2\), so \(x_1^*=-0.5(2)/1.25=-1/1.25=-0.8\%\). Inflation is \(2+0.5(-0.8)=2-0.4=1.6\) percentage points above target.
In period 2, \(e_2+u_2=0\), so \(x_2^*=0\), and inflation is at target. The shock has faded, so the model no longer calls for an output gap to counter it.
To eliminate the period-1 inflation deviation completely, set \(0=2+0.5x_1\), which gives \(x_1=-4\%\). That is much more negative than the flexible-targeting choice of \(-0.8\%\). At \(-4\%\), the loss is \(\frac12[0^2+1(4^2)]=8\). At the flexible-targeting choice, the loss is \(\frac12[1.6^2+1(0.8^2)]=\frac12[2.56+0.64]=1.6\).
Check. The flexible choice accepts a 1.6-percentage-point inflation deviation while avoiding the much larger output loss associated with a \(-4\%\) gap. Common mistake avoided: Full inflation stabilization is not automatically the lowest-loss response. It ignores the output cost in the assumed objective.
Worked Example: A Persistent Shock and Rising Expectations
A supply disturbance of \(1.5\) percentage points persists for two periods. In period 1, expectations are at target. In period 2, expected inflation is \(0.5\) percentage points above target. Let \(\kappa=0.5\) and \(\lambda=1\). Find the period-by-period flexible-targeting output gaps, inflation deviations, and total two-period loss, assuming the same loss function applies in each period and the central bank minimizes each period’s loss separately.
Method. Calculate each period’s \(e+u\), then apply the one-period formulas. The denominator is \(0.25+1=1.25\).
In period 1, \(e_1+u_1=0+1.5=1.5\). Thus \(x_1^*=-0.5(1.5)/1.25=-0.75/1.25=-0.6\%\). Inflation is \(1.5+0.5(-0.6)=1.2\) percentage points above target. The loss is \(\frac12[1.2^2+(-0.6)^2]=\frac12[1.44+0.36]=0.9\).
In period 2, \(e_2+u_2=0.5+1.5=2\). Thus \(x_2^*=-0.5(2)/1.25=-1/1.25=-0.8\%\). Inflation is \(2+0.5(-0.8)=1.6\) percentage points above target. The loss is \(\frac12[1.6^2+(-0.8)^2]=\frac12[2.56+0.64]=1.6\). The two-period loss is \(0.9+1.6=2.5\).
Check. The second-period policy response is stronger because the shock persists and expected inflation is also above target. The resulting inflation deviation, \(1.6\), is greater than in period 1, \(1.2\), despite the more negative output gap. Common mistake avoided: Do not treat the period-1 shock as though it had disappeared, or omit the period-2 expectations deviation from \(e+u\).
When Gradual Adjustment Is Preferable
A temporary supply shock with expectations still near target can make gradual adjustment preferable. The shock raises inflation directly, but a forceful contraction in demand may impose a persistent output cost even if the supply disturbance soon fades. In the model, the flexible-targeting choice shares the burden: it reduces the output gap enough to moderate inflation, without requiring the gap that would eliminate the shock’s inflation effect immediately.
Gradual adjustment does not mean ignoring inflation. It means considering how the shock is likely to evolve, how expectations are behaving, and how costly further output weakness would be. If evidence indicates that the shock is fading and expectations remain anchored, a central bank may have reason to avoid responding to every temporary inflation increase with a large contraction in demand. A smaller initial response can be reassessed as new information arrives.
A persistent shock changes the assessment. If it continues to raise prices, or if expected inflation moves above target, the inflation pressure is no longer just a one-off disturbance. The worked example shows the arithmetic: the relevant pressure in the one-period formula is \(e+u\), so a higher expectations deviation increases the negative output gap implied by the model. A timely, firmer response may then be needed to limit continuing inflation pressure. The model does not specify the exact path of policy rates; that requires additional information about how policy instruments affect demand over time.
| Condition | What the model indicates | Policy interpretation |
|---|---|---|
| Shock expected to fade; expectations near target | The temporary \(u\) raises inflation, but its direct effect may soon recede. | A measured response can avoid creating an unnecessarily deep output gap. |
| Shock persists; expectations remain anchored | The inflation pressure continues, even if \(e\) stays near zero. | Reassess the response as evidence on persistence accumulates. |
| Shock persists; expectations rise above target | Both \(u\) and \(e\) contribute to above-target inflation. | A more forceful response may be warranted to limit ongoing pressure. |
Common Mistakes and Limits of the Model
- Confusing the shock with the output gap. The shock \(u\) raises inflation at a given output gap. The output gap \(x\) is a policy-relevant measure of demand relative to potential output.
- Assuming every inflation increase is evidence of excess demand. In this model, \(u>0\) can raise inflation even when \(x=0\) or \(x<0\).
- Setting inflation deviation to zero without pricing the output cost. Full stabilization requires \(x=-(e+u)/\kappa\); that choice need not minimize the stated loss.
- Using the one-period result as a complete dynamic policy rule. The formulas treat \(e\) and \(u\) as given. They do not model how expectations respond to policy, the time path of the shock, or policy lags.
- Treating “gradual” as “always optimal.” If a shock proves persistent or expectations rise, the basis for a restrained response weakens. The policy choice must be reconsidered as the evidence changes.
The practical value of the framework is not that it supplies a mechanical answer for every shock. It helps separate the channels: the supply disturbance raises inflation directly, the output gap affects inflation through \(\kappa x\), and expectations add another source of pressure. Keeping those channels distinct makes it easier to explain why a measured response can suit a temporary shock, while persistence or rising expectations can justify firmer action. This treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
Check Your Understanding
For each problem, use the stated one-period model unless a multi-period assumption is specified. Report output gaps in percent and inflation deviations in percentage points.
- A central bank has \(\kappa=0.3\), \(\lambda=0.7\), expected inflation deviation \(e=0.2\), and adverse supply shock \(u=0.8\). Calculate the flexible-targeting output gap and inflation deviation. (Answer: \(x^*\approx-0.3797\%\); inflation is approximately \(0.8861\) percentage points above target.)
- Let \(\kappa=0.4\), \(\lambda=0.6\), \(e=0\), and \(u=1\). Calculate the flexible-targeting output gap and inflation deviation. Then calculate the output gap that would exactly stabilize inflation. (Answer: Flexible targeting gives \(x^*\approx-0.5263\%\) and inflation deviation \(0.7895\) percentage points; exact stabilization requires \(x=-2.5\%\).)
- A shock of \(u=1\) persists for two periods. Let \(\kappa=0.5\), \(\lambda=1\), with \(e_1=0\) and \(e_2=0.2\). Find each period’s flexible-targeting output gap and inflation deviation, then add the two period losses using \(L_t=\frac12[(\pi_t-\pi^*)^2+x_t^2]\). (Answer: Period 1: \(x_1^*=-0.4\%\), inflation deviation \(0.8\), loss \(0.4000\). Period 2: \(x_2^*=-0.48\%\), inflation deviation \(0.96\), loss \(0.5760\). Total loss \(0.9760\).)
- Suppose \(\kappa=0.5\), \(\lambda=1\), \(e=0.1\), and \(u=1.4\). Find the flexible-targeting output gap and inflation deviation, then find the gap that would set inflation exactly at target. (Answer: Flexible targeting gives \(x^*=-0.6\%\) and inflation deviation \(1.2\) percentage points; exact stabilization requires \(x=-3\%\).)