Introduction
A fiscal demand shock can move real output and the price level in the short run, but those effects need not persist. The distinction matters: an increase in government purchases may raise production when prices and wages adjust slowly, while over time wage and price adjustments can move output back toward its potential level.
This tutorial traces that adjustment in a simple aggregate-demand and aggregate-supply model. It separates the initial short-run response from the later long-run equilibrium and shows why a demand shift, by itself, does not permanently change potential output. The treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
A Simple Model for Tracing the Shock
Let \(Y\) denote real output, \(P\) the price level, and \(Y^p\) potential output: the level of production the economy can sustain when resources are used at normal rates. In the short run, assume the expected price level \(P^e\) is fixed. Short-run aggregate supply (SRAS) is upward sloping: when the actual price level is higher than expected, firms have an incentive to produce more. A linear representation is:
For this tutorial, represent aggregate demand (AD) as:
Here \(A\) is the demand intercept, expressed in output units, and \(\beta\) describes how real output demanded responds to the price level in this simplified model. A fiscal expansion raises \(A\); a fiscal contraction lowers it. The equations are stylized relationships, not a complete model of every channel through which fiscal policy works. They assume, in particular, that there is no monetary-policy offset or other change in demand that cancels the fiscal shock.
The short-run equilibrium is where the quantities of output demanded and supplied are equal. In the long run, wages and price expectations have time to adjust. In this model, long-run aggregate supply is vertical at \(Y^p\): a change in the price level alone does not permanently raise or lower potential output. This conclusion assumes the fiscal shock does not itself change productive capacity, such as through a lasting change in technology or the labor force.
Deriving the Short-Run and Long-Run Effects
The short-run price and output changes can be derived by substituting the AD equation into the SRAS equation. At equilibrium:
Collecting the terms in \(P\) gives the equilibrium price. Substituting that price back into either curve gives output:
Now hold \(Y^p\), \(P^e\), \(\beta\), and \(\gamma\) fixed, and let a fiscal demand shift change \(A\) by \(\Delta A\). Subtracting the old equilibrium from the new equilibrium gives:
The price level and output move in the same direction as the demand shift in the short run. In this particular model, some of the shift appears as a higher or lower price level, so the short-run output change is not generally equal to the entire change in the AD intercept.
For the long run, expected prices adjust until actual and expected prices agree, so \(P=P^e\). The SRAS equation then gives \(Y=Y^p\). Substituting potential output into AD gives:
Thus, when the fiscal shock changes demand but not productive capacity, it changes the long-run price level in this model but not long-run output.
Use the initial AD and SRAS equations to determine output and the price level before the fiscal change.
Keep \(P^e\) fixed for this calculation. The immediate output gap is measured relative to \(Y^p\).
If output is above potential, upward wage and price pressure shifts SRAS up and left. If output is below potential, weaker wage and price pressure shifts SRAS down and right.
Set output equal to \(Y^p\), find the price level consistent with AD, and set expected prices equal to that price level.
Worked Examples
Worked Example: Expansion from an Initial Potential-Output Equilibrium
Suppose potential output is \(1{,}000\), the initial expected price level is \(100\), and the curves are \(Y=1{,}200-2P\) for AD and \(Y=600+4P\) for SRAS. A fiscal expansion raises the AD intercept by \(120\). Find the initial equilibrium, the new short-run equilibrium, and the long-run equilibrium. Output is measured in billions of dollars; the price level is an index.
Method and assumptions. Solve the intersection of AD and SRAS. In the short-run calculation, hold \(P^e=100\) fixed. For the long-run calculation, hold potential output at \(1{,}000\) and allow expected prices to adjust.
Initial equilibrium. Set the initial curves equal: \(1{,}200-2P=600+4P\). Therefore, \(600=6P\), so \(P=100\). Substitution gives \(Y=1{,}200-2(100)=1{,}000\), which equals potential output.
Short-run equilibrium after the expansion. The new AD curve is \(Y=1{,}320-2P\). Equating it to SRAS gives \(1{,}320-2P=600+4P\), so \(720=6P\) and \(P=120\). Output is \(Y=1{,}320-2(120)=1{,}080\). The short-run output gap is \(1{,}080-1{,}000=80\) billion dollars above potential.
Long-run equilibrium. Set output equal to potential in the new AD curve: \(1{,}000=1{,}320-2P\). Thus \(2P=320\), so \(P=160\). Expected prices adjust to \(P^e=160\). At that endpoint, SRAS gives \(Y=1{,}000+4(160-160)=1{,}000\).
Check. In the long run, AD gives \(1{,}320-2(160)=1{,}000\), and SRAS gives \(1{,}000\). The initial expansion raises output to \(1{,}080\) in the short run, but output returns to potential as expected prices and wages adjust. The common mistake this method avoids is treating the short-run output increase as a permanent increase in potential output.
Worked Example: From a Fiscal Multiplier to the AD–AS Response
In an expenditure model, let the marginal propensity to consume be \(0.80\), the proportional tax rate be \(0.25\), and the marginal propensity to import be \(0.10\). A government-purchase increase of \(20\) billion dollars shifts the AD intercept by the spending multiplier times the purchase change. The initial AD curve is \(Y=1{,}000-2P\), and SRAS is \(Y=600+2P\). Potential output is \(800\). Find the multiplier and the short-run and long-run equilibria after the fiscal change.
Method and assumptions. Use the open-economy spending multiplier with proportional taxes and income-related imports, \(k_G=1/[1-c(1-t)+m]\), where \(c\) is the marginal propensity to consume, \(t\) the tax rate, and \(m\) the marginal propensity to import. Assume those expenditure-model parameters are fixed. The resulting demand shift is then traced through the stated AD–AS curves.
Calculate the spending multiplier. First, \(c(1-t)=0.80(1-0.25)=0.80(0.75)=0.60\). The denominator is \(1-0.60+0.10=0.50\), so \(k_G=1/0.50=2\). The AD-intercept change is \(2(20)=40\) billion dollars.
Check the initial equilibrium. Set \(1{,}000-2P=600+2P\). Then \(400=4P\), so \(P=100\). Output is \(1{,}000-2(100)=800\), equal to potential output.
Find the new short-run equilibrium. The new AD curve is \(Y=1{,}040-2P\). Equating it to SRAS gives \(1{,}040-2P=600+2P\), so \(440=4P\) and \(P=110\). Output is \(1{,}040-2(110)=820\), or \(20\) billion dollars above potential.
Find the long-run equilibrium. Set \(Y=800\) in the new AD curve: \(800=1{,}040-2P\). Thus \(2P=240\), so \(P=120\). Expected prices adjust to \(120\), and SRAS gives \(Y=800+2(120-120)=800\).
Check. Both curves give output \(800\) at \(P=120\) in the long run. The multiplier calculates the change in the AD intercept under the stated expenditure-model assumptions; it is not, by itself, the final short-run change in output. Here the intercept rises by \(40\), while short-run output rises by \(20\).
Worked Example: A Contractionary Shift and a Recessionary Gap
Suppose potential output is \(1{,}000\), expected prices are initially \(100\), AD is \(Y=1{,}200-2P\), and SRAS is \(Y=600+4P\). A fiscal contraction lowers the AD intercept by \(120\). Calculate the immediate equilibrium and the long-run endpoint.
Method and assumptions. Solve the intersection after the intercept change while holding expected prices fixed. Then use potential output and the new AD curve to find the long-run price level. Assume the contraction does not change potential output.
Check the initial equilibrium. Equating the initial curves gives \(1{,}200-2P=600+4P\), so \(600=6P\) and \(P=100\). Output is \(1{,}200-2(100)=1{,}000\), exactly potential output.
Find the short-run equilibrium. After the contraction, AD is \(Y=1{,}080-2P\). Set this equal to SRAS: \(1{,}080-2P=600+4P\). Hence \(480=6P\), giving \(P=80\). Output is \(1{,}080-2(80)=920\). The output gap is \(920-1{,}000=-80\) billion dollars.
Find the long-run equilibrium. Set output equal to potential: \(1{,}000=1{,}080-2P\), so \(2P=80\) and \(P=40\). Expected prices adjust to \(40\); SRAS then gives \(Y=1{,}000+4(40-40)=1{,}000\).
Check. The new AD curve gives \(1{,}080-2(40)=1{,}000\), matching potential output. In this example the contraction moves output below potential at first; subsequent downward wage and price pressure shifts SRAS down and right, allowing output to recover. The common mistake avoided is to assume that the short-run fall in output is the long-run result.
How Wage and Price Adjustment Closes the Output Gap
After an expansionary demand shock, the short-run equilibrium may lie above potential output. Firms are trying to produce more than the economy can sustain at normal resource utilization. Competition for workers and other inputs can place upward pressure on nominal wages and costs. As wages and expected prices adjust, firms require a higher price level to supply a given amount of output. In the model, this is an upward or leftward shift of SRAS. Output moves back toward \(Y^p\), while the price level continues to adjust.
After a contractionary demand shock, the short-run equilibrium may lie below potential output. Weaker demand for labor and other inputs tends to reduce wage and cost pressure. As nominal wages and expected prices adjust downward relative to the price level, firms can supply more at a given price level. SRAS shifts downward or rightward, and output moves back toward potential. The speed of this adjustment depends on how quickly wages, prices, and expectations respond; the model does not imply that the transition is instantaneous.
The long-run endpoint follows directly from the model equations. With \(P=P^e\), SRAS implies \(Y=Y^p\). AD then determines the price level that makes planned demand equal to potential output. This is why a demand shock can have a lasting effect on the price level without permanently changing output, given the assumption that potential output is unchanged.
Common Mistake: Confusing the Multiplier with the Long-Run Output Change
A common incorrect calculation is to take a positive change in the fiscal intercept, \(\Delta A=120\), and conclude that long-run output must rise by \(120\). That treats a demand change as though it directly changed productive capacity. In the model above, the short-run output change is instead \(\gamma\Delta A/(\beta+\gamma)\), with expected prices held fixed. The long-run output change is zero if potential output is unchanged; the long-run price level adjusts to reconcile AD with \(Y^p\).
Another error is to shift SRAS in the same direction as the output gap. If output is above potential, rising wage and cost pressure shifts SRAS up and left, not down and right. If output is below potential, weaker cost pressure shifts SRAS down and right. Keep the stages separate: first solve the new short-run intersection, then describe the adjustment toward the long-run endpoint.
Check Your Understanding
For each problem, state the relevant short-run or long-run result. Treat output and AD intercepts as being in the units specified in the question.
- Potential output is \(1{,}050\), expected prices are \(150\), AD is \(Y=1{,}500-3P\), and SRAS is \(Y=600+3P\). The AD intercept rises by \(90\). Find the new short-run equilibrium and long-run price level. (Answer: Short run \(P=165,\ Y=1{,}095\); long run \(P=180,\ Y=1{,}050\).)
- Let \(c=0.80\), \(t=0.25\), and \(m=0.10\) in the spending-multiplier model. Government purchases rise by \(15\) billion dollars. Find the multiplier and the change in the AD intercept. (Answer: Multiplier \(k_G=2\); AD intercept rises by \(30\) billion dollars.)
- Potential output is \(1{,}000\), expected prices are \(100\), AD is \(Y=1{,}200-2P\), and SRAS is \(Y=600+4P\). The AD intercept falls by \(80\). Find the new short-run equilibrium and long-run price level. (Answer: Short run \(P=86.6667,\ Y=946.6667\); long run \(P=60,\ Y=1{,}000\).)
- Potential output is \(1{,}200\), expected prices are \(100\), SRAS is \(Y=800+4P\), and initial AD is \(Y=1{,}600-4P\). The AD intercept falls by \(80\). Find short-run output and the long-run price level. (Answer: Short-run output \(Y=1{,}160\), which is \(40\) below potential; long-run price level \(P=80\).)