Stabilization Policy Under Adverse Supply Shocks

Why an Adverse Supply Shock Creates a Policy Trade-Off

An adverse supply shock, such as a hypothetical increase in a key input cost, can raise inflation while reducing output. A policymaker who tries to bring inflation back to target quickly may need to restrain aggregate demand, deepening a short-run output decline. Supporting activity instead can leave more of the shock’s inflationary pressure in place. The decision is not simply whether to respond; it is how much weight to place on the two costs.

The earlier material on inflation and cyclical slack establishes why weaker activity can reduce inflation pressure. Here, that relationship is used to compare policy choices after a supply shock. The focus is the short-run trade-off, not a claim that policymakers can permanently choose a higher level of output by accepting higher inflation.

Definition: Let $x$ be the output gap, measured as a percentage deviation of actual output from potential output. Let $\pi$ be the inflation rate and $\pi^*$ the inflation target; the inflation gap is $\pi-\pi^*$, measured in percentage points. In the short-run model used here, $$ \pi-\pi^* = e+\kappa x+u, $$ where $e=\pi^e-\pi^*$ is the expected-inflation gap, $\pi^e$ is expected inflation, $\kappa>0$ measures how inflation responds to the output gap, and $u>0$ is an adverse supply shock expressed in percentage points. The equation assumes expectations and the shock are given over the policy decision period, and policy influences inflation through its effect on the output gap. It is a short-run approximation, not a long-run supply relationship.

The signs matter. With $u>0$, the shock raises inflation at any fixed output gap. With $\kappa>0$, a negative output gap reduces inflation relative to the case where output is at potential. If expected inflation is above target, $e>0$, that also adds to current inflation pressure. Thus $e+u$ is the inflation gap that would occur at $x=0$.

Three Policy Outcomes

For a given $e$, $u$, and $\kappa$, a policymaker can compare three useful cases. They are analytical benchmarks, not necessarily descriptions of every available policy instrument.

Support output at potential: Set $x=0$. The inflation gap is $e+u$; the immediate output gap is avoided, but the inflation pressure remains.
Return inflation to target within the model: Set $\pi-\pi^*=0$. Solving the inflation equation gives $x=-(e+u)/\kappa$. If $e+u>0$, this requires a negative output gap.
Compromise: Choose a negative output gap that reduces inflation pressure but does not force the inflation gap all the way to zero.

Full price stabilization can be costly in output terms. Conversely, leaving output at potential does not remove the shock’s inflation effect. The size of the output cost required for exact stabilization depends on both the total inflation pressure $e+u$ and the sensitivity $\kappa$. These comparisons are conditional on the model: they do not imply that the policymaker can directly set inflation independently of expectations, supply conditions, or output.

Worked Example: Comparing Price Stabilization With Output Support

A hypothetical economy has expected inflation at its 2% target, an adverse supply shock that adds 1.5 percentage points to inflation, and $\kappa=0.5$. Compare the output and inflation outcomes if the policymaker (a) holds output at potential or (b) brings inflation to target in the short-run model. Treat $x$ as a percentage gap and report gaps to two decimals.

Method. First identify $e$ and $u$, then apply the inflation relationship. Because expected inflation equals target, $e=2\%-2\%=0$ percentage points. The shock is $u=1.5$ percentage points.

(a) Output support. Holding output at potential means $x=0$. Therefore, $$ \pi-\pi^*=e+\kappa x+u =0+(0.5)(0)+1.5 =1.5\text{ percentage points}. $$ The policy leaves inflation 1.5 percentage points above target and the output gap at 0.00%.

(b) Exact inflation stabilization. Set the inflation gap to zero and solve: $$ 0=0+(0.5)x+1.5,\qquad (0.5)x=-1.5,\qquad x=-3.00\%. $$ The model therefore requires output to be 3.00% below potential to offset the shock’s inflation effect.

Check. Substituting $x=-3$ gives $0+(0.5)(-3)+1.5=-1.5+1.5=0$, as required. The common mistake is to use a positive output gap to counteract a positive supply shock. The correction is to track the sign: with $\kappa>0$, only a negative $x$ lowers inflation in this equation.

A Quadratic Policy Objective

A simple way to formalize the compromise is to assign a cost to both inflation deviations and output deviations. Let $\lambda>0$ measure the relative weight placed on the output gap. For this model, the policymaker selects $x$ to minimize:

L=(\pi-\pi^*)^2+\lambda x^2

The loss is measured in squared percentage-point units when both gaps are entered in percentage points. A larger $\lambda$ makes output stabilization more important relative to inflation stabilization. This is a modeling device for comparing priorities, not an empirical estimate of a policymaker’s preferences.

Theorem: Under the stated short-run inflation equation, with $e$, $u$, $\kappa>0$, and $\lambda>0$ fixed, the output gap that minimizes $L$ is $$ x^*=-\frac{\kappa(e+u)}{\kappa^2+\lambda}. $$ The resulting inflation gap is $$ \pi-\pi^*=\frac{\lambda(e+u)}{\kappa^2+\lambda}. $$

The result follows by substituting the inflation equation into the objective. Write $q=e+u$, which is fixed for the decision. Then $L=(q+\kappa x)^2+\lambda x^2$. Differentiating with respect to $x$ and setting the derivative to zero gives the first-order condition below. Since the second derivative is positive, this stationary point is the unique minimum.

\frac{dL}{dx}=2\kappa(q+\kappa x)+2\lambda x=0 $$ $$ \kappa q+(\kappa^2+\lambda)x=0 \quad\Longrightarrow\quad x^*=-\frac{\kappa q}{\kappa^2+\lambda}.

For the inflation gap at this choice, substitute $x^*$ into $q+\kappa x$: $$ q+\kappa\left(-\frac{\kappa q}{\kappa^2+\lambda}\right) =q\left(1-\frac{\kappa^2}{\kappa^2+\lambda}\right) =\frac{\lambda q}{\kappa^2+\lambda}. $$ The second derivative is $2(\kappa^2+\lambda)>0$, so the minimum is confirmed. If $q>0$, the optimal gap is negative and the resulting inflation gap remains positive: the compromise reduces, but does not eliminate, the inflation pressure. As $\lambda$ increases, the chosen output gap moves closer to zero and more inflation pressure remains. If $\lambda$ approaches zero, the model puts almost all weight on inflation stabilization.

Measure the inflation pressure at potential output.
Calculate $q=e+u$, keeping expected inflation and the supply shock in percentage points.

Find the compromise output gap.
Use $x^*=-\kappa q/(\kappa^2+\lambda)$. A positive $q$ should produce a negative $x^*$.

Recover inflation and evaluate the objective.
Calculate $q+\kappa x^*$, then calculate $L=(q+\kappa x^*)^2+\lambda(x^*)^2$. Check that the output gap and inflation gap have the expected signs.

Worked Example: Finding the Loss-Minimizing Compromise

In a hypothetical economy, expected inflation is at target, the adverse supply shock is $u=1.0$ percentage point, $\kappa=0.5$, and $\lambda=0.75$. Find the loss-minimizing output gap, the resulting inflation gap, and the loss. Report gaps to two decimals and loss to two decimals.

Method. Use the quadratic-loss solution, then verify the outcome in the original inflation equation and objective. Here $e=0$, so $q=e+u=1.0$ percentage point. The denominator is $$ \kappa^2+\lambda=(0.5)^2+0.75=0.25+0.75=1.00. $$

The output gap is $$ x^*=-\frac{(0.5)(1.0)}{1.00}=-0.50\%. $$ The resulting inflation gap is $$ e+\kappa x^*+u=0+(0.5)(-0.50)+1.0=-0.25+1.0=0.75 $$ percentage points.

The loss is $$ L=(0.75)^2+(0.75)(-0.50)^2 =0.5625+0.75(0.25) =0.5625+0.1875 =0.75 $$ squared percentage-point units.

Check. The first-order condition is satisfied because $\kappa(q+\kappa x^*)+\lambda x^*=(0.5)(0.75)+(0.75)(-0.50)=0.375-0.375=0$. The common mistake is to calculate the loss using the wrong output gap or to omit the weight $\lambda$ from the output term. Here the output contribution is $0.75(0.25)=0.1875$, not $0.25$.

Expectations and the Weight on Output

A supply shock is not the only source of inflation pressure. If expected inflation is above target, then $e>0$ and the pressure at potential output is $e+u$. The same optimization applies, but the policymaker’s calculated response depends on the combined pressure, not on $u$ alone. This matters because a policy choice based only on the measured supply shock can understate the inflation gap when expectations are elevated.

The value of $\lambda$ changes the compromise rather than changing the underlying shock. A relatively high $\lambda$ assigns greater cost to an output shortfall, leading to a smaller contraction and a larger remaining inflation gap. A relatively low $\lambda$ gives inflation control more weight, leading to a larger output shortfall and a smaller inflation gap. Neither choice makes the supply shock disappear.

Worked Example: Accounting for Above-Target Expectations

In a hypothetical economy, the inflation target is 3.0%. Expected inflation is 3.5%, an adverse supply shock adds 1.0 percentage point, $\kappa=0.6$, and $\lambda=0.54$. Find the compromise output gap, inflation gap, and loss. Report gaps to two decimals and loss to two decimals.

Method. Express expected inflation relative to target before applying the solution. Thus $e=3.5\%-3.0\%=0.5$ percentage point, and $q=e+u=0.5+1.0=1.5$ percentage points. The denominator is $$ \kappa^2+\lambda=(0.6)^2+0.54=0.36+0.54=0.90. $$

The output gap is $$ x^*=-\frac{(0.6)(1.5)}{0.90} =-\frac{0.90}{0.90} =-1.00\%. $$ The inflation gap is $$ q+\kappa x^*=1.5+(0.6)(-1.00)=1.5-0.6=0.90 $$ percentage points. Therefore inflation is $3.0\%+0.90\%=3.90\%$.

The loss is $$ L=(0.90)^2+(0.54)(-1.00)^2=0.81+0.54=1.35 $$ squared percentage-point units.

Check. The inflation equation gives $0.5+(0.6)(-1.00)+1.0=0.5-0.6+1.0=0.90$, matching the calculated gap. Using $u=1.0$ alone would miss the 0.5 percentage-point expected-inflation gap and understate the pressure to be addressed.

Common Traps and Limits of the Model

Reversing the sign of the output gap. Under $\pi-\pi^*=e+\kappa x+u$, a negative output gap reduces inflation. A positive gap adds to it. Check the sign by substituting the chosen $x$ into the equation.
Confusing inflation with the inflation gap. The model calculates $\pi-\pi^*$. Add the target back only if the question asks for the inflation rate itself.
Mixing percentage points and percentages. An inflation rate moving from 3% to 4% has risen by 1 percentage point. Keep $e$, $u$, and the resulting inflation gap in percentage points, and keep $x$ in percentage points of output gap.
Treating exact stabilization as costless. When $e+u>0$, setting the inflation gap to zero requires $x=-(e+u)/\kappa$, which is negative. The output cost should be stated alongside the inflation result.
Using the short-run equation beyond its assumptions. The linear model treats expectations and the supply shock as given and summarizes inflation’s response to the output gap. It does not describe all channels of monetary or fiscal policy, supply capacity, changing expectations, or long-run output. If those factors change materially, the calculated compromise need not describe the new situation.

The practical lesson is to report both sides of the choice: the implied inflation gap and the output gap. The loss-minimizing formula makes the trade-off explicit, but the parameter $\lambda$ reflects a policy priority rather than an objective law of the economy. Different priorities can yield different short-run responses to the same adverse shock.

Check Your Understanding

For each problem, use the short-run model and state the relevant inflation and output gaps. Assume the quadratic objective applies when $\lambda$ is specified.

A hypothetical economy has $e=0$, $u=1.2$, $\kappa=0.4$, and $\lambda=0.6$. Find the loss-minimizing output gap and resulting inflation gap. (Answer: $x^*\approx-0.63\%$; inflation gap $\approx0.95$ percentage points.)
Expected inflation is 2.3% against a 2.0% target, and a supply shock adds 1.2 percentage points. If $\kappa=0.5$, find the output gap that exactly returns inflation to target. (Answer: $x=-3.00\%$.)
A hypothetical economy has $e=0.3$, $u=0.9$, $\kappa=0.6$, and $\lambda=0.8$. Find the loss-minimizing output gap and resulting inflation gap. (Answer: $x^*\approx-0.62\%$; inflation gap $\approx0.83$ percentage points.)
At $x=0$, inflation pressure is $q=0.8$ percentage point. Let $\kappa=0.4$ and $\lambda=0.16$. Calculate the loss-minimizing output gap, the inflation gap, and the loss. (Answer: $x^*=-1.00\%$; inflation gap $=0.40$ percentage point; $L=0.32$ squared percentage-point units.)
Expected inflation is at target, a favorable supply shock is $u=-0.6$ percentage point, and $\kappa=0.5$, $\lambda=0.75$. Find the loss-minimizing output gap and resulting inflation gap. (Answer: $x^*=+0.30\%$; inflation gap $=-0.45$ percentage point.)

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