Putting Fiscal and Monetary Responses into an AD–AS Case
A stabilization case can involve several forces at once: an economic shock, a fiscal measure that arrives after a lag, private spending displaced by higher interest rates, and a monetary response that either reinforces or restrains demand. The practical challenge is to keep those effects separate until they are combined. Applying a policy measure in the wrong period, or counting a monetary offset twice, can reverse the conclusion.
The model below is a deliberately compact, short-run AD–AS framework for tracking those effects. It produces an output gap and a price-level deviation at a given horizon. It is not a forecast of a particular economy: its coefficients and shocks are hypothetical, and its results depend on the assumptions stated here.
The reference equilibrium has \(x=0\) and \(p=0\) when all shocks and policy shifts are zero. A negative \(x\) is a recessionary output gap; a positive \(x\) is an above-potential gap. A positive \(p\) means the price level is above the reference level. It is a price-level deviation, not an inflation rate. In a multi-period case, compare price-level deviations across periods to describe the change in the price level.
This linear model treats the parameters as fixed over the horizon and assumes expectations and potential output do not change within the calculation. It is a short-run device, not a long-run supply model. The policy terms \(f\), \(c\), and \(m\) are already expressed as shifts in aggregate demand; they are not government spending dollars or interest-rate changes. Convert any given policy description into those units before solving.
Solving the Two Equations
Substitute the supply equation into aggregate demand. This gives one equation in the output gap; once \(x\) is found, substitute it into supply to find \(p\). The derivation also clarifies why a supply shock affects both outcomes.
Calculate \(A=z+f-c+m\), applying only the measures that have taken effect by the period being analyzed.
Since \(p=\kappa x+s\), the demand equation becomes \(x=A-b(\kappa x+s)\).
Rearrange to find \(x=(A-bs)/(1+b\kappa)\), then calculate \(p=\kappa x+s\).
For the worked cases, use \(b=0.5\) and \(\kappa=0.4\). Thus \(1+b\kappa=1+(0.5)(0.4)=1.2\), and the equations simplify to
Keep the signed values in these formulas: a contraction, a negative demand shock, and a negative supply shock are entered as negative numbers. Do not treat a policy announced today as an active demand shift today unless its implementation and impact lags have elapsed.
Worked Cases
Worked Example: A Fiscal Package That Arrives After the Recession Begins
A hypothetical economy experiences a demand shock of \(z=-12\) output units. A fiscal package would add \(10\) demand units once it takes effect, but its implementation and impact lags mean it has no effect in period 1. When the package does take effect in period 2, higher interest rates have displaced \(2\) units of private demand. Assume \(s=0\), \(m=0\), and the model coefficients above in both periods. Find \(x\) and \(p\) in each period, and assess the package’s effect.
Method and timing. First form the demand shift separately for each period. In period 1, the fiscal measure has not taken effect, so \(f=0\) and \(c=0\). In period 2, \(f=10\) and \(c=2\). The common mistake avoided here is counting the announced package before it affects demand.
Period 1. The aggregate-demand shift is $$ A_1=z+f-c+m=-12+0-0+0=-12. $$ With \(s=0\), $$ x_1=\frac{-12-(0.5)(0)}{1.2}=\frac{-12}{1.2}=-10, \qquad p_1=\frac{(0.4)(-12)+0}{1.2}=\frac{-4.8}{1.2}=-4. $$
Period 2. Once the package takes effect, the direct fiscal shift is partly offset by crowding out: $$ A_2=-12+10-2+0=-4. $$ Therefore, $$ x_2=\frac{-4-(0.5)(0)}{1.2}=\frac{-4}{1.2}=-3.333\ldots, \qquad p_2=\frac{(0.4)(-4)+0}{1.2}=\frac{-1.6}{1.2}=-1.333\ldots. $$
Interpretation and check. The recessionary gap is smaller in period 2: \(x\) rises from \(-10\) to about \(-3.333\) output units. The price-level deviation rises from \(-4\) to about \(-1.333\) percentage points. Substitution checks the period 2 result: \(A_2-0.5p_2=-4-0.5(-1.333\ldots)=-3.333\ldots=x_2\), and \(0.4x_2+s=-1.333\ldots=p_2\). The package reduces the shortfall but does not eliminate it; after crowding out, its net demand contribution is \(10-2=8\), not \(10\).
Worked Example: A Delayed Expansion Meets Recovery and Monetary Offset
A fiscal expansion of \(6\) demand units was authorized during a downturn but takes effect only in the next period. By then, the underlying demand shock has changed to \(z=4\). The expansion is accompanied by a monetary offset of \(m=-5\). A supply shock is \(s=2\) price-level percentage points. There is no additional crowding out. Find the resulting output gap and price-level deviation, and compare them with the outcome if monetary policy did not offset the expansion.
Method and timing. In the period being assessed, the delayed fiscal expansion is active, so \(f=6\), while the current shock is \(z=4\), not the earlier downturn shock. The common mistake avoided is carrying the old recession shock into the new period or treating the monetary offset as an additional fiscal contraction.
With monetary offset. Combine the current demand terms: $$ A=4+6-0-5=5. $$ The supply shock reduces the numerator for output by \(bs=(0.5)(2)=1\). Thus $$ x=\frac{5-1}{1.2}=\frac{4}{1.2}=3.333\ldots, \qquad p=\frac{(0.4)(5)+2}{1.2}=\frac{4}{1.2}=3.333\ldots. $$
Without monetary offset. Set \(m=0\), so \(A=4+6=10\). Then $$ x=\frac{10-1}{1.2}=\frac{9}{1.2}=7.5, \qquad p=\frac{(0.4)(10)+2}{1.2}=\frac{6}{1.2}=5. $$
Interpretation and check. Monetary restraint lowers the output gap from \(7.5\) to about \(3.333\) units and the price-level deviation from \(5\) to about \(3.333\) points. It does not make either result zero. For the offset case, \(A-0.5p=5-0.5(3.333\ldots)=3.333\ldots=x\), and \(0.4x+s=0.4(3.333\ldots)+2=3.333\ldots=p\). The result illustrates why a delayed expansion can become procyclical: the underlying economy may have moved into recovery before the measure affects demand.
Worked Example: Supply Pressure, Crowding Out, and Monetary Accommodation
A hypothetical economy has a demand shock \(z=-3\) and a positive supply shock \(s=3\). A fiscal measure adds \(8\) direct demand units, but higher interest rates crowd out \(2\) units of private demand. Consider three monetary choices: no additional response (\(m=0\)), accommodation (\(m=2\)), and offset (\(m=-3\)). Calculate \(x\) and \(p\) under each choice.
Method. For each choice, calculate \(A=z+f-c+m=-3+8-2+m=3+m\), then solve the same AD–AS system. The common mistake avoided is subtracting crowding out from the monetary response: these are separate terms, each included once.
No additional monetary response. Here \(m=0\), so \(A=3\). Therefore, $$ x=\frac{3-(0.5)(3)}{1.2}=\frac{1.5}{1.2}=1.25, \qquad p=\frac{(0.4)(3)+3}{1.2}=\frac{4.2}{1.2}=3.5. $$
Accommodation. With \(m=2\), \(A=3+2=5\). Hence $$ x=\frac{5-1.5}{1.2}=\frac{3.5}{1.2}=2.9166\ldots, \qquad p=\frac{2+3}{1.2}=\frac{5}{1.2}=4.1666\ldots. $$
Offset. With \(m=-3\), \(A=3-3=0\). Hence $$ x=\frac{0-1.5}{1.2}=-1.25, \qquad p=\frac{0+3}{1.2}=2.5. $$
Interpretation and check. Accommodation supports output but adds to the price-level pressure: compared with no response, \(x\) increases from \(1.25\) to about \(2.917\), while \(p\) increases from \(3.5\) to about \(4.167\). Offset restrains demand enough to produce a negative output gap, while the positive supply shock leaves \(p\) above its reference level. For the offset case, \(A-0.5p=0-1.25=-1.25=x\), and \(0.4x+s=-0.5+3=2.5=p\). The direct fiscal shift is \(8\), but its contribution after crowding out is \(8-2=6\), before the monetary choice is added.
Reading the Result and Avoiding Timing Errors
A calculated output gap and price-level deviation describe the model’s equilibrium for the chosen period; neither alone establishes whether policy has successfully stabilized the economy. Compare the outcome with the relevant starting point or counterfactual, and state whether the output gap is smaller or larger and whether the price-level deviation has risen or fallen. A policy can reduce one problem while worsening another, particularly when supply conditions are adverse.
For a multi-period case, build a separate demand shift for each period. A recognition lag delays the point at which policymakers identify the problem; a decision or implementation lag delays the measure; an impact lag delays its effect on demand. The model only uses \(f\) or \(m\) in a period when that instrument affects demand. If the underlying shock changes before implementation, the current \(z\) must also be updated. The earlier material on fiscal policy lags and counterfactual evaluation provides the relevant timing distinctions.
Crowding out and monetary response should be kept conceptually distinct. Crowding out \(c\) is the private demand displaced through the interest-rate effects associated with fiscal expansion; monetary policy \(m\) is a separate response by the monetary authority. Depending on the case, the authority may accommodate the fiscal shift, offset part of it, or make no additional response. Do not assume that every fiscal expansion causes the same amount of crowding out or that a stated monetary response fully cancels a fiscal measure. Use the magnitudes supplied in the case.
| Case feature | Entry in the model | Effect to check |
|---|---|---|
| Demand shock | \(z\), with its stated sign | Changes aggregate demand in the current period |
| Fiscal measure | \(f\), only after it takes effect | Positive for expansion; negative for contraction |
| Crowding out | Subtract \(c\) | Reduces the net demand contribution of fiscal expansion |
| Monetary response | \(m\), positive for accommodation and negative for offset | Shifts aggregate demand separately from fiscal policy |
| Supply shock | \(s\), in the supply equation | Enters both solved outcomes; do not add it to \(A\) |
The most important calculation check is to substitute the reported values back into both original equations: verify \(x=A-bp\) and \(p=\kappa x+s\). This catches sign errors, omitted crowding out, and accidental use of a policy measure before its lag has elapsed. In an answer, report both outcomes with units and explain the direction of the policy effect relative to the stated comparison.
Check Your Understanding
For each case, use \(b=0.5\) and \(\kappa=0.4\). Treat all demand quantities as output units and \(s\) as a price-level deviation in percentage points.
- A current demand shock is \(z=-9\). A fiscal measure now adds \(7\) demand units and crowds out \(1\) unit of private demand. There is no monetary response or supply shock. Find \(x\) and \(p\). (Answer: \(x=-2.5\) output units; \(p=-1.0\) percentage point.)
- A fiscal contraction contributes \(f=-3\), the demand shock is \(z=5\), and monetary policy restrains demand by \(1\) unit. There is no crowding out and \(s=1.2\). Find \(x\) and \(p\). (Answer: \(x=0.3333\) output units; \(p=1.3333\) percentage points.)
- In period 1, \(z=-6\), and a planned fiscal expansion has not yet taken effect. In period 2, the shock is \(z=2\), the expansion contributes \(f=5\), crowding out is \(c=1\), and monetary policy contributes \(m=-2\). Assume \(s=0\) in both periods. Find both periods’ outcomes. (Answer: Period 1, \(x=-5\), \(p=-2\); period 2, \(x=3.3333\), \(p=1.3333\).)
- A case has \(z=-4\), \(f=6\), \(c=1\), and \(s=2\). There is no additional monetary response. Find \(x\) and \(p\), and state whether output is at potential. (Answer: \(x=0\) output units; \(p=2\) percentage points; output is at potential.)
- With \(z=-2\), \(f=5\), \(c=1\), and \(s=-1\), compare no monetary response with accommodation \(m=3\). Find \(x\) and \(p\) in each case. (Answer: No response, \(x=2.0833\), \(p=-0.1667\); accommodation, \(x=4.5833\), \(p=0.8333\).)