From a Demand Shock to a New Long-Run Equilibrium
An aggregate demand shock changes the position of aggregate demand, so the economy generally moves away from potential output in the short run. That short-run equilibrium is not necessarily permanent. As households and firms revise their expectations and wage-setting decisions respond, short-run aggregate supply shifts. In the standard AD-AS account, this adjustment brings output back to potential, while the price level settles at a new value.
The direction of the adjustment depends on the demand shock. After an expansionary demand shock, output initially rises above potential and the price level rises. As expected prices and nominal wages adjust upward, short-run aggregate supply shifts left or upward; output falls back to potential, and the price level rises further. After a contractionary demand shock, the short-run output gap is negative. Expectations and wages adjust downward, shifting short-run aggregate supply right or downward; output returns to potential, and the price level falls further.
A Simple Model of the Adjustment
To make the sequence precise, use a simplified linear AD-AS model. Let \(x=Y-Y^*\) denote the output gap, where \(Y\) is actual output and \(Y^*\) is fixed potential output. Let \(P\) be the price-level index and \(P^e\) the price level expected by wage and price setters. Represent aggregate demand and short-run aggregate supply as
where \(A\) is the aggregate-demand intercept, \(b>0\) and \(\lambda>0\) are fixed slope parameters. A demand shift changes \(A\); a higher \(A\) means aggregate demand is higher at each output gap. A change in \(P^e\) shifts the short-run aggregate supply relation. These equations are a simplified model, not a universal numerical law: they assume linear curves, a fixed \(Y^*\), and no additional shocks or policy changes during the adjustment.
For the long-run conclusion, the model assumes that expectations eventually match the actual price level, so \(P=P^e\). The SRAS equation then implies \(x=0\), because \(P=P^e+\lambda x\) and \(\lambda>0\). Thus output is at potential. Aggregate demand determines the long-run price level at that output: setting \(x=0\) in AD gives \(P=A\). This conclusion applies when the demand shift persists and no other shift changes the model during adjustment.
Why Expectations Adjustment Restores Potential Output
The mechanism is a sequence of short-run equilibria. Immediately after a demand shock, expected prices and preset nominal wages have not fully adjusted. The new aggregate-demand curve therefore intersects the existing SRAS curve at an output gap. If the shock is expansionary, stronger demand raises output and the price level above their levels before the shock. As actual prices rise, wage and price setters revise expectations upward. Higher expected prices and related nominal costs shift SRAS upward: at each output gap, firms require a higher price level to supply output.
The upward shift reduces the positive output gap. The process continues while expected prices are below the price level consistent with the current demand curve at potential output. In the long run, expectations align with the actual price level and output returns to potential. Following a contractionary shock, the same logic runs in the opposite direction: the initial shortfall in output puts downward pressure on prices and expectations, and the downward shift in SRAS closes the negative output gap.
The long-run conclusion follows directly from the equations, not from the idea that the original aggregate-demand shift disappears. The shift remains. Rather, SRAS adjusts until output is at \(Y^*\); at that output, the shifted AD curve determines the new price level. This distinction is important: a return to potential output does not mean a return to the initial price level.
Determine whether aggregate demand rose or fell, and hold the new AD curve fixed for the adjustment analysis.
Use the original expected price level and the new AD curve to determine the output gap and price level.
Rising actual prices after an expansionary shock tend to raise expected prices and nominal wages; falling prices after a contractionary shock tend to lower them. SRAS shifts accordingly.
Set output equal to potential, so \(x=0\), and use the shifted AD curve to find the final price level.
Worked Applications
Worked Example: An Expansionary Demand Shock
An economy has potential output \(Y^*=500\) index units. Its initial expected price level is \(P^e=100\). After an expansionary demand shock, aggregate demand is \(P=112-2x\), while short-run aggregate supply is initially \(P=100+4x\). Find the short-run equilibrium and the long-run equilibrium. Compare each price level with the initial equilibrium.
Method. Solve the new AD and initial SRAS equations together for the short-run equilibrium. For the long run, set \(x=0\) and use the same shifted AD curve.
At the initial equilibrium, \(x=0\) and \(P=100\). In the short run, equate aggregate demand and short-run aggregate supply:
Subtract 100 from both sides and add \(2x\) to both sides: \(12=6x\), so \(x=2\). Therefore actual output is \(Y=Y^*+x=500+2=502\). Substituting into either curve gives \(P=112-2(2)=108\); the SRAS calculation gives \(100+4(2)=108\), confirming the intersection.
In the long run, \(x=0\), so output returns to \(Y=500\). The shifted AD curve then gives \(P=112-2(0)=112\). The price level rises from 100 initially to 108 in the short run and to 112 in the long run. These are price-level index points, not percentage inflation rates.
Common mistake avoided. Returning output to potential does not restore the original price level. The correct long-run price is found on the shifted AD curve, not the initial one.
Worked Example: A Contractionary Demand Shock
A different economy has potential output \(Y^*=800\), an initial expected price level of 100, and initial equilibrium output at potential. After a contractionary demand shock, AD is \(P=88-2x\); initial SRAS is \(P=100+4x\). Find the short-run output gap and price level, then the long-run price level.
Method. Equate the new AD curve with the existing SRAS curve for the short run, then evaluate AD at \(x=0\) for the long run.
Subtract 100 and add \(2x\): \(-12=6x\), hence \(x=-2\). Actual output is \(Y=800+(-2)=798\), which is below potential. The short-run price level is \(P=88-2(-2)=88+4=92\). Checking with SRAS gives \(100+4(-2)=92\), so the two curves agree.
In the long run, \(x=0\), so output returns to 800 and the price level on the shifted AD curve is \(P=88\). The price level falls from 100 initially to 92 in the short run and to 88 in the long run.
Common mistake avoided. A contractionary demand shock does not initially move the economy along the old AD curve; it shifts AD. The short-run equilibrium must use the new intercept, 88.
Worked Example: Tracking Gradual Expectation Adjustment
Suppose AD is \(P=112-2x\), SRAS is \(P=P^e+4x\), and the initial expected price level is 100. For this example, assume expectations are updated after each period according to \(P^e_{\text{next}}=P^e+\frac12(P-P^e)\), where \(P\) is the current equilibrium price level. Find the first two short-run equilibria and show whether the output gap moves toward zero.
Method. Solve the two equilibrium equations for a given \(P^e\), then update expectations using the stated rule. This particular updating rule is an illustrative assumption; it is not required for the general long-run result.
In the first period, \(P^e=100\), so SRAS is \(P=100+4x\). Equating it to AD gives \(112-2x=100+4x\), hence \(12=6x\) and \(x=2\). The price level is \(P=108\).
Update expectations: \(P^e_{\text{next}}=100+\frac12(108-100)=100+4=104\). In the second period, set \(P^e=104\). Equating \(112-2x=104+4x\) gives \(8=6x\), so \(x=\frac43\). The price level is \(P=112-2(\frac43)=112-\frac83=\frac{328}{3}\), or 109.333 when rounded to three decimals.
The output gap has fallen from \(2\) to \(\frac43\), while the price level has risen from 108 to approximately 109.333. To verify the direction more generally for this example, the equilibrium equations imply \(x=(112-P^e)/6\) and \(P-P^e=4x\). Therefore the update rule gives \(P^e_{\text{next}}=P^e+2x\), and the next output gap is \((112-P^e_{\text{next}})/6=(2/3)x\). Each update reduces the positive gap by a factor of \(2/3\); it approaches zero, and the price level approaches 112.
Common mistake avoided. The expectation update uses the difference \(P-P^e\), not the difference between the new price level and the initial price level. In period one that difference is \(108-100=8\), so the update is 4.
Derivation: The Long-Run Price Level
The key long-run result can be justified directly. With the demand shock in place, write AD as \(P=A-bx\), where \(A\) is the new intercept. In long-run equilibrium, output is at potential, which means \(x=0\). Substitution into AD gives \(P=A\). The long-run SRAS condition is consistent with that value when expected prices have adjusted to actual prices: \(P=P^e\), and then \(P=P^e+\lambda x=P^e\). Thus the output gap is zero and the long-run price level is the shifted AD intercept.
This derivation also shows why a temporary output gap and a lasting price-level change can occur together. Short-run output depends on where the new AD curve intersects SRAS before expectations have fully adjusted. Long-run output is pinned at potential in this model, but the new AD curve still determines a different price level there.
Common Interpretation Traps
- Stopping at the first intersection. The first short-run equilibrium is not the long-run result if output is away from potential and expectations are still adjusting.
- Moving AD during the adjustment without a stated cause. In this analysis, the demand shock shifts AD once; the later movement is a shift in SRAS as expectations and wages adjust.
- Assuming output returns to its old level and prices do too. Output returns to potential, but the demand curve has shifted, so the long-run price level may differ from its initial value.
- Confusing a price-level index with an inflation rate. A comparison such as 100 to 112 is a change in the level of the index. It does not by itself state an annual inflation rate or the time over which that change occurred.
For applied economic analysis, the distinction between the immediate gap and the eventual equilibrium matters when interpreting a demand-driven expansion or contraction. A short-run rise in output does not establish a permanent increase in potential output, and a return to potential does not mean that the price level is unchanged. This treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
Check Your Understanding
For each problem, keep the shifted aggregate-demand curve in place during adjustment. Unless a problem states otherwise, use the long-run condition \(x=0\).
- An economy has \(Y^*=300\), initial price level 50, and after an expansionary shock AD is \(P=59-x\). Initial SRAS is \(P=50+2x\). Find the short-run output gap and price level, then the long-run price level. (Answer: Short run \(x=3,\ P=56\); long run \(P=59\).)
- After a contractionary shock, AD is \(P=70-3x\), and initial SRAS is \(P=82+3x\). Find the short-run output gap and price level, then the long-run price level. (Answer: Short run \(x=-2,\ P=76\); long run \(P=70\).)
- Potential output is 1,000. A demand shock changes AD to \(P=105-2x\); initial SRAS is \(P=99+4x\). Calculate short-run output and price level, then state how the long-run price level compares with the initial price level of 99. (Answer: Short run \(x=1,\ Y=1{,}001,\ P=103\); long-run \(P=105\), six index points above the initial level.)
- AD is \(P=90-x\), SRAS is \(P=P^e+3x\), and \(P^e=84\). Find the current output gap and price level. If expectations next rise by half of \(P-P^e\), find the next expected price level. (Answer: Current \(x=1.5,\ P=88.5\); next \(P^e=86.25\).)
- After a demand shift, AD is \(P=130-4x\), while initial SRAS is \(P=118+2x\). Find the short-run equilibrium and the long-run price level. Explain whether the shock is expansionary or contractionary relative to the initial equilibrium at \(x=0,\ P=118\). (Answer: Short run \(x=2,\ P=122\); long-run \(P=130\); expansionary, with output above potential initially.)