Long-Run Adjustment After An Aggregate Supply Shock

Why the Duration of a Supply Shock Matters

An adverse supply shock can raise the price level while reducing output in the short run. What happens next depends on both the shock’s persistence and whether it changes the economy’s productive capacity. A temporary disruption that eventually ends can leave output and the price level returning to their original levels. A permanent reduction in potential output instead changes the long-run equilibrium: output settles at a lower level, while the price level adjusts to be consistent with aggregate demand.

This tutorial uses a deliberately simple linear AD-AS model to make those paths precise. Its adjustment rule is an assumption for tracking the transition; it is not a claim that all economies adjust at the same speed. The analysis is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: A temporary supply shock shifts short-run aggregate supply for a limited time but does not permanently change potential output. A permanent change in potential output changes the amount the economy can produce in the long run and therefore shifts long-run aggregate supply. In this tutorial, an adverse temporary cost disturbance is represented by \(c_t>0\); a change in potential output is represented by \(z\), which may be positive or negative.

A Reduced-Form Model for Adjustment

Let \(y\) be output’s deviation from its initial level, measured in output units, and let \(p\) be the price level’s deviation from its initial level, measured in price-index points. The initial equilibrium is normalized to \(y=0\) and \(p=0\). Aggregate demand is unchanged throughout the examples and is represented by

$$ y=-2p. $$

Thus, a one-point increase in the price level is associated with a two-unit decrease in output along aggregate demand. The short-run aggregate supply relation is

$$ y=z+(p-e_t)-c_t. $$

Here, \(z\) is the permanent change in potential output, \(e_t\) is the price-level reference used in the short-run supply relation during period \(t\), and \(c_t\) is a temporary adverse cost disturbance. A positive \(c_t\) shifts short-run aggregate supply left. A negative \(c_t\) represents a favorable supply disturbance. The model assumes the disturbance eventually ends, so \(c_t=0\) in the long run.

To describe a gradual adjustment path, assume the reference for the next period equals the current price-level deviation: \(e_{t+1}=p_t\). Initially, \(e_0=0\). This rule is a simplifying device: it lets us calculate how the short-run supply relation changes after each period. It does not require us to model the wage-setting or expectation-formation processes behind that change.

Short-run equilibrium: Combining the two relations gives

p_t=\frac{e_t+c_t-z}{3}, \qquad y_t=-2p_t.

The denominator is \(3\) because substituting aggregate demand into short-run aggregate supply gives \(-2p_t=z+p_t-e_t-c_t\), or \(-3p_t=z-e_t-c_t\). In the long run, the temporary disturbance is gone and the price-level reference has adjusted to the actual price level: \(c=0\) and \(e=p\). The supply relation then gives \(y=z\). Aggregate demand determines the price level consistent with that output:

p=-\frac{z}{2}, \qquad y=z.

These long-run results apply when \(z\) represents the permanent change in productive capacity. They do not imply that a temporary disruption immediately disappears. The short-run path depends on how long that disruption lasts.

Temporary Shocks: Recovery After the Disruption Ends

For a temporary adverse cost shock, the initial short-run effect is a higher price level and lower output. Once the disturbance ends, the price-level reference still reflects the previous period’s price level. Consequently, the short-run supply curve does not instantly return to its initial position. With the adjustment rule above and \(z=0\), after the shock ends the equations imply \(p_{t+1}=p_t/3\). The price-level deviation and output gap therefore shrink over time, returning toward their initial values.

Worked Example: A One-Period Cost Disruption

Suppose the economy begins at \(y=0\), \(p=0\), and \(e_0=0\). An adverse supply disruption of \(c_0=6\) output units lasts for one period. Potential output is unchanged, so \(z=0\). Calculate the impact equilibrium and the equilibrium in the following period, when the disruption has ended.

Method. Use the short-run equilibrium equations for each period. After the first period, update the reference using \(e_1=p_0\). This method keeps the temporary shock separate from a change in potential output.

Impact period. Substitute \(e_0=0\), \(c_0=6\), and \(z=0\):

p_0=\frac{0+6-0}{3}=2, \qquad y_0=-2(2)=-4.

The impact price level is 2 index points above its initial level, and output is 4 units below its initial level. The price reference for the next period becomes \(e_1=p_0=2\).

Following period. The disruption has ended, so \(c_1=0\):

p_1=\frac{2+0-0}{3}=\frac{2}{3}, \qquad y_1=-2\left(\frac{2}{3}\right)=-\frac{4}{3}.

The price-level deviation has fallen from \(2\) to \(2/3\), and the output gap has narrowed from \(-4\) to \(-4/3\). With no further disturbance, the next price deviation is \(p_2=e_2/3=p_1/3=2/9\), so both deviations continue toward zero.

Check and common mistake. At impact, short-run supply gives \(y_0=0+(2-0)-6=-4\), matching aggregate demand \(y_0=-2(2)=-4\). A common error is to treat the adverse supply shock as a leftward shift of aggregate demand and assign a negative price change. Here the shock shifts short-run aggregate supply; the correct impact price change is positive.

Persistence Changes the Transition Path

A temporary disturbance need not last only one period. The longer it remains, the longer it continues to shift short-run aggregate supply left. Even after it ends, the price-level reference may still be above its initial value, so recovery is gradual rather than instantaneous. The long-run destination is still the initial equilibrium if potential output is unchanged and the temporary disturbance eventually disappears.

Worked Example: A Disruption That Persists for Two Periods

Use the same model, with \(z=0\) and \(e_0=0\). This time, an adverse cost disturbance of \(c_0=c_1=9\) lasts through periods 0 and 1, then ends so that \(c_2=0\). Calculate output and the price-level deviation in each of the first three periods.

Method. Solve the equilibrium separately for each period, carrying forward the reference \(e_{t+1}=p_t\). The key is not to set \(c_t\) to zero until the period when the scenario says the disruption has ended.

Period 0. With \(e_0=0\), \(c_0=9\), and \(z=0\):

p_0=\frac{0+9-0}{3}=3, \qquad y_0=-2(3)=-6.

Therefore \(e_1=p_0=3\).

Period 1. The cost disturbance is still present, so \(c_1=9\):

p_1=\frac{3+9-0}{3}=4, \qquad y_1=-2(4)=-8.

The price reference for period 2 becomes \(e_2=p_1=4\).

Period 2. The disruption has ended, so \(c_2=0\):

p_2=\frac{4+0-0}{3}=\frac{4}{3}, \qquad y_2=-2\left(\frac{4}{3}\right)=-\frac{8}{3}.

The results show that the second period of disruption deepens the short-run output loss from 6 to 8 units and raises the price-level deviation from 3 to 4 points. After the shock ends, the output gap narrows to \(-8/3\); with no further cost disturbance, subsequent deviations move toward zero.

Check and common mistake. In period 1, short-run supply gives \(y_1=(4-3)-9=-8\), which equals aggregate demand, \(-2(4)=-8\). A common mistake is to stop applying the shock in period 1 because it is called “temporary.” It remains active for both periods specified in the problem; only period 2 has \(c_2=0\).

Permanent Changes in Potential Output

A lasting change in productive capacity has a different long-run result from a temporary cost disruption. In the model, a permanent reduction in potential output is represented by \(z<0\). Long-run output then settles at \(y=z\), not at its original level. With aggregate demand unchanged, the price level must also settle at a different value. This distinction matters when deciding whether a shock describes a temporary interruption to production or a lasting change in what the economy can produce.

Worked Example: A Permanent Reduction in Capacity

Suppose productive capacity falls permanently by 6 output units, so \(z=-6\). There is no temporary cost disturbance, \(c_t=0\), aggregate demand remains \(y=-2p\), and the economy begins with \(e_0=0\). Calculate the first three short-run equilibria and the new long-run equilibrium.

Method. Treat the permanent capacity loss as \(z=-6\), not as a temporary \(c_t\). Solve period by period using the adjustment rule, then impose the long-run condition \(e=p\).

Period 0. The initial reference is \(e_0=0\):

p_0=\frac{0+0-(-6)}{3}=2, \qquad y_0=-2(2)=-4.

Thus \(e_1=p_0=2\).

Period 1. With \(e_1=2\):

p_1=\frac{2+0-(-6)}{3}=\frac{8}{3}, \qquad y_1=-2\left(\frac{8}{3}\right)=-\frac{16}{3}.

Thus \(e_2=p_1=8/3\).

Period 2. With \(e_2=8/3\):

p_2=\frac{\frac{8}{3}+0-(-6)}{3} =\frac{\frac{26}{3}}{3} =\frac{26}{9}, \qquad y_2=-2\left(\frac{26}{9}\right)=-\frac{52}{9}.

Long-run equilibrium. In the long run, \(e=p\) and \(c=0\). Output is the permanent potential-output change, \(y=z=-6\). Aggregate demand then requires \(-6=-2p\), so \(p=3\). The long-run equilibrium is therefore \(y=-6\) and \(p=3\): output is 6 units below its original level, and the price level is 3 index points above its original level.

Check and common mistake. The short-run results satisfy aggregate demand: for example, period 2 has \(y_2=-52/9=-2(26/9)\). A common mistake is to claim that output eventually returns to zero because “the economy adjusts.” Here adjustment does not remove the capacity loss. The long-run output level is \(z=-6\), so output remains below its original level.

Deriving the Long-Run Result

The long-run result follows directly from the model’s assumptions. Once a temporary cost disturbance has ended, \(c=0\). Once the reference has adjusted to the actual price level, \(e=p\). Substituting both conditions into short-run aggregate supply eliminates the price-level term:

$$ y=z+(p-p)-0=z. $$

This is the model’s long-run output result: output equals the permanent change in potential output. Substituting \(y=z\) into unchanged aggregate demand gives \(z=-2p\), hence \(p=-z/2\). A temporary shock with \(z=0\) therefore has a long-run destination of \(y=0,\ p=0\). A permanent capacity loss with \(z<0\) has a lower long-run output level and a higher price level in this model.

Key takeaway: Track two separate questions. Does the disturbance eventually disappear, and does it permanently change potential output? If the disturbance disappears and potential output is unchanged, the long-run equilibrium returns to its starting point. If potential output changes, the long-run output level changes with it.

Common Exam Traps

Confusing a supply shift with movement along aggregate demand. The supply shock shifts short-run aggregate supply. The price and output changes at equilibrium are found where the shifted supply curve intersects unchanged aggregate demand.
Assuming every supply shock has the same long-run outcome. A temporary cost disturbance and a permanent change in potential output have different long-run destinations.
Ignoring persistence. If a shock remains active for multiple periods, include it in each of those periods before applying the recovery calculation.
Reversing the signs. In this model, an adverse short-run supply shock raises \(p\) and lowers \(y\); a favorable one lowers \(p\) and raises \(y\).
Confusing the price-level deviation with an inflation rate. The variable \(p\) here measures the price level relative to its initial level; it is not an inflation rate.

Check Your Understanding

Use the model and adjustment rule stated above. Unless a question says otherwise, the economy begins with \(e_0=0\), aggregate demand is \(y=-2p\), and each period’s reference updates by \(e_{t+1}=p_t\).

A one-period adverse cost disturbance has \(c_0=12\), with \(z=0\). Find the impact values of \(p_0\) and \(y_0\). (Answer: \(p_0=4,\ y_0=-8\).)
A one-period adverse disturbance has \(c_0=3\), with \(z=0\). Find the impact equilibrium and the next period’s equilibrium after \(c_1=0\). (Answer: Period 0: \(p_0=1,\ y_0=-2\); period 1: \(p_1=1/3,\ y_1=-2/3\).)
An adverse disturbance has \(c_0=c_1=6\), then ends with \(c_2=0\); potential output is unchanged. Find the equilibrium in periods 0, 1, and 2. (Answer: Period 0: \(p_0=2,\ y_0=-4\); period 1: \(p_1=8/3,\ y_1=-16/3\); period 2: \(p_2=8/9,\ y_2=-16/9\).)
Potential output falls permanently by 4 units, so \(z=-4\), and there is no temporary disturbance. Find the impact equilibrium and the new long-run equilibrium. (Answer: Impact: \(p_0=4/3,\ y_0=-8/3\); long run: \(p=2,\ y=-4\).)
A one-period favorable supply disturbance has \(c_0=-6\), with \(z=0\). Find the impact equilibrium and the next period’s equilibrium after the disturbance ends. (Answer: Period 0: \(p_0=-2,\ y_0=4\); period 1: \(p_1=-2/3,\ y_1=4/3\).)

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