Why Trace a Demand Shock Through More Than One Point?
A change in spending can move an economy through the business cycle by changing demand for currently produced goods and services. But the effects depend on the time horizon. In the short run, an increase in aggregate demand can raise both output and the price level. If the demand shift persists, output may later return to its potential level while the price level adjusts. The path and the direction of that later price change require care: they depend on where the economy started.
This tutorial uses a simple linear aggregate-demand and aggregate-supply model to trace those outcomes. It focuses on a demand shift, not on the distinct mechanisms behind a supply shock. The model is useful for disciplined comparisons of an initial equilibrium, a new short-run equilibrium, and a long-run endpoint.
A Linear Model for Tracing the Shock
Let \(Y\) denote real output and \(P\) the price-level index. Suppose aggregate demand and short-run aggregate supply are
where \(A\) and \(C\) are fixed intercepts and \(b>0\) and \(s>0\) are fixed slope parameters. The AD curve slopes downward: a higher price level corresponds to lower quantity of output demanded, all else equal. Short-run aggregate supply (SRAS) slopes upward: a higher price level corresponds to higher output supplied in this simplified short-run model. For the calculations, assume the initial equilibrium is where AD and SRAS intersect, the demand shock changes \(A\) but leaves \(b,C,s\) unchanged, and potential output \(Y_p\) is fixed.
A positive demand shock of size \(\Delta A>0\) shifts AD to \(Y=A+\Delta A-bP\). A negative shock has \(\Delta A<0\). The first intersection after the shift is the new short-run equilibrium. If the shift persists and the model’s long-run adjustment is completed, output is \(Y_p\); the price level at that endpoint is found by putting \(Y_p\) into the shifted AD equation. This is a long-run endpoint, not an additional short-run intersection.
Why the Short-Run Changes Have These Sizes
The short-run equilibrium changes can be derived directly. Before the shock, equilibrium requires \(A-bP=C+sP\), so \(P=(A-C)/(b+s)\). After a shift of \(\Delta A\), the new equilibrium satisfies \(A+\Delta A-bP=C+sP\). Subtracting the original equilibrium equation gives
Since short-run supply is \(Y=C+sP\), its change is \(\Delta Y=s\Delta P\). Therefore,
The positive slopes and \(b+s>0\) imply that a positive demand shock raises both short-run output and the price level; a negative shock lowers both. The size of each change depends on the slopes. For example, with \(b=2\) and \(s=2\), half of a demand shift’s size appears as a change in output: \(\Delta Y=2\Delta A/4\). This result describes the new short-run intersection only; it does not by itself determine the long-run price level.
Worked Example: A Positive Shock Starting at Potential Output
A hypothetical economy has AD \(Y=600-2P\), SRAS \(Y=400+2P\), and potential output \(Y_p=500\). A persistent positive demand shock raises the AD intercept by 40, so the new AD curve is \(Y=640-2P\). Find the initial equilibrium, the new short-run equilibrium, and the long-run endpoint.
Method. Find the initial intersection, solve for the new short-run intersection, and then use potential output in the shifted AD equation. The common mistake to avoid is treating the short-run output after the shock as the long-run outcome.
Initial equilibrium. Set AD equal to SRAS:
Substitute \(P_0=50\) into either curve: \(Y_0=600-2(50)=500\). This matches potential output, so the initial economy is at \(Y_p=500\).
New short-run equilibrium. Equate shifted AD and unchanged SRAS:
Then \(Y_{SR}=640-2(60)=520\). Checking supply gives \(400+2(60)=520\), so the intersection is consistent. Relative to the initial equilibrium, short-run output rises by 20 and the price level rises by 10.
Long-run endpoint. Set output equal to potential in the shifted AD curve: \(500=640-2P_{LR}\), so \(2P_{LR}=140\) and \(P_{LR}=70\). The endpoint is \(Y=500,\ P=70\). Output returns to its initial potential level, while the price level is 20 above its initial value of 50. Because this example starts at potential and the positive shift persists, the short-run price level of 60 is below the long-run price level of 70.
The Long-Run Price Level Depends on the Starting Point
For shifted AD, the long-run endpoint is found from \(Y_p=A+\Delta A-bP_{LR}\). Thus,
This equation gives the price level at potential output; it does not say that this level must be higher than the price level in the short-run equilibrium, or higher than the price level before the shock. If the initial equilibrium was at potential, then \(A-Y_p=bP_0\), and the endpoint becomes \(P_{LR}=P_0+\Delta A/b\). Under that specific assumption, a positive shift puts the long-run price level above its initial level, and a negative shift puts it below.
Without the initial-at-potential assumption, compare the relevant equilibrium values using their equations rather than relying on a general verbal rule. In particular, after a positive shock the short-run price level rises relative to the initial short-run price level, but the later price-level movement can be upward or downward. The next example shows why.
Worked Example: A Positive Shock When Output Starts Below Potential
Suppose initial AD is \(Y=200-P\), SRAS is \(Y=20+P\), and potential output is \(Y_p=120\). A persistent positive demand shock shifts AD to \(Y=205-P\). Calculate both intersections and the long-run endpoint, then interpret the price-level changes.
Method. Solve each short-run intersection, then evaluate shifted AD at potential output. The common mistake is to assume that a positive shock necessarily means the price level will keep rising after the new short-run equilibrium.
Initial equilibrium. Equating the curves gives \(200-P=20+P\), so \(180=2P\), \(P_0=90\), and \(Y_0=200-90=110\). This is 10 below potential output.
New short-run equilibrium. Using shifted AD, \(205-P=20+P\), so \(185=2P\), \(P_{SR}=92.5\), and \(Y_{SR}=205-92.5=112.5\). Checking SRAS gives \(20+92.5=112.5\). The demand shock raises short-run output by 2.5 and the price level by 2.5.
Long-run endpoint. At potential output, \(120=205-P_{LR}\), so \(P_{LR}=85\). Thus, as output reaches 120, the price level at the endpoint is 7.5 below the post-shock short-run level of 92.5. It is also 5 below the initial price level of 90. The result is consistent with the shifted AD curve: \(205-85=120\). The positive shock raises output in the short run, but it does not guarantee that the eventual price level exceeds either comparison level when the initial economy is below potential.
Tracing a Negative Demand Shock
A negative demand shock works in the opposite direction at the first short-run intersection: with upward-sloping SRAS, output and the price level both fall. If the shock persists, the long-run endpoint again places output at potential. When the economy starts at potential, the long-run price level is lower than the initial price level after a negative shift. The size of that eventual price change is determined by the shifted AD curve and its slope, not by carrying the short-run price change forward unchanged.
Worked Example: A Negative Shock Starting at Potential Output
A hypothetical economy has AD \(Y=900-2P\), SRAS \(Y=500+2P\), and potential output \(Y_p=700\). A persistent negative demand shock lowers the AD intercept by 40, producing \(Y=860-2P\). Find the initial, short-run, and long-run outcomes.
Method. Use intersections for the initial and new short-run outcomes, then evaluate shifted AD at potential output. The common mistake is to report the short-run decline in output as the long-run output level.
Initial equilibrium. Solve \(900-2P=500+2P\): \(400=4P\), so \(P_0=100\). Output is \(Y_0=900-2(100)=700\), which equals potential output.
New short-run equilibrium. Solve \(860-2P=500+2P\): \(360=4P\), so \(P_{SR}=90\). Output is \(Y_{SR}=860-2(90)=680\). Supply also gives \(500+2(90)=680\). The short-run output gap relative to potential is \(680-700=-20\).
Long-run endpoint. Set shifted AD equal to potential output: \(700=860-2P_{LR}\). Hence \(2P_{LR}=160\), and \(P_{LR}=80\). The endpoint is \(Y=700,\ P=80\). Output returns to potential, while the price level is below both the initial value of 100 and the new short-run value of 90. Each calculated point satisfies the equation used to obtain it.
A Reliable Sequence for Cycle Analysis
A demand-shock question often gives enough information to calculate several outcomes but not enough to infer every movement between them. Keep the initial equilibrium, the new short-run equilibrium, and the long-run endpoint separate. A disciplined sequence prevents an assumption about initial conditions from being smuggled into the answer.
Find the initial AD–SRAS intersection and compare its output with \(Y_p\). Do not assume that initial output equals potential unless the information establishes it.
Represent the demand shock by changing the AD intercept or other stated demand determinant. Keep the unshocked curves’ parameters fixed for this calculation.
Set shifted AD equal to SRAS. For the stated linear model, a positive shift raises short-run output and the price level; a negative shift lowers both.
If the shift persists and long-run adjustment is assumed complete, set output equal to \(Y_p\) in shifted AD. Compare this price level with the initial and short-run levels only after calculating it.
What the Model Does—and Does Not—Establish
The model explains why an AD shift can affect output as well as the price level in the short run: the new intersection occurs on an upward-sloping SRAS curve. It also gives a precise long-run endpoint under the assumption that output returns to potential while the demand shift persists. It does not establish the timing or exact path of adjustment, nor does it imply that the price level must move monotonically between the short-run equilibrium and the endpoint.
These distinctions matter when connecting demand changes with the business cycle. A short-run increase in output can narrow a negative output gap or create a positive one; a short-run decrease can deepen a downturn. But a shock’s sign alone does not identify the initial output gap, the size of the resulting gap, or every later price-level comparison. The equations and the stated assumptions determine those results.
- Do not confuse a movement along AD with a shift of AD. A change in the price level is represented by movement along a given AD curve; a demand shock changes the curve itself.
- Do not assume the economy starts at potential. If initial output differs from \(Y_p\), the long-run price level after a persistent shift need not be above the initial price level after a positive shock.
- Do not infer the later price movement from the shock’s sign alone. Compare the calculated short-run price with the price implied by shifted AD at potential output.
- Keep the time horizon explicit. The short-run intersection and the long-run endpoint answer different questions and generally have different output levels.
Check Your Understanding
For each problem, distinguish the new short-run equilibrium from the long-run endpoint where one is requested. Price levels and output are in the units specified by the model.
- AD is \(Y=420-2P\), SRAS is \(Y=300+2P\), and the AD intercept rises by 24. Find the initial and new short-run equilibria. (Answer: Initial \(P=30,\ Y=360\); new short run \(P=36,\ Y=372\).)
- AD is \(Y=700-3P\), SRAS is \(Y=400+2P\), and potential output is 520. The AD intercept rises by 30 and the shock persists. Find the new short-run equilibrium and the long-run endpoint. (Answer: Short run \(P=66,\ Y=532\); long run \(P=70,\ Y=520\).)
- Initial AD is \(Y=240-P\), SRAS is \(Y=60+2P\), and potential output is 210. The AD intercept rises by 15. Find the initial equilibrium, new short-run equilibrium, and long-run endpoint. Is the long-run price level above the post-shock short-run level? (Answer: Initial \(P=60,\ Y=180\); short run \(P=65,\ Y=190\); long run \(P=45,\ Y=210\); no, the long-run price level is lower.)
- AD is \(Y=900-2P\), SRAS is \(Y=500+2P\), and potential output is 700. The AD intercept falls by 40. Calculate the new short-run output gap relative to potential and the long-run price level. (Answer: New short run \(P=90,\ Y=680\), so the output gap is \(-20\); long-run \(P=80,\ Y=700\).)
- In the linear model, \(b=4,\ s=2\), and a demand shock has \(\Delta A=-36\). Assuming the initial economy is at potential and the shock persists, calculate the short-run changes in output and price level and the long-run price-level change from the initial value. (Answer: Short-run \(\Delta Y=-12,\ \Delta P=-6\); long-run price-level change \(=-9\).)