Tracing a Shock Through a Full Business Cycle

From an Initial Shock to a Recovery

A business-cycle shock rarely ends with its first effect. A fall in spending can reduce output, weaken employment and alter inflation; those changes can then affect expectations, investment and policy decisions. To assess the whole cycle, it helps to trace the links in order and distinguish the initiating shock from the mechanisms that prolong or offset it.

The earlier material on spending feedback, inventories, investment, unemployment, inflation, expectations, rigidities, financial conditions and stabilization policy supplies the mechanisms used here. The purpose is to assemble them into a coherent timeline. The equations below are deliberately simple local models: they organize a hypothetical scenario, not universal forecasts.

Definition: The output gap is the percentage difference between actual output and potential output: $x_t=100(Y_t-Y_t^*)/Y_t^*$, where $Y_t$ is actual output and $Y_t^*$ is potential output in period $t$. A negative gap means output is below potential. A business-cycle recovery is an improvement in cyclical conditions after a downturn; it need not mean that output has returned to its former trend or that all effects of the downturn have disappeared.

A Compact Model for Tracing the Cycle

For a numerical trace, suppose output-gap dynamics follow $x_t=z_t+c x_{t-1}$. Here, $z_t$ is the period’s new net demand impulse, measured in percentage points of potential output, and $c$ is the fraction of the previous period’s gap that persists. Assume $0\leq c<1$, with no other output influences omitted from the scenario. This is a reduced-form description: $z_t$ may summarize a shock and any policy support, while persistence may reflect spending feedback, slow adjustment or other propagation mechanisms.

To connect output to the labor market, use a stated Okun-type calibration, $u_t-u^*=-\beta x_t$, where $u_t$ is the unemployment rate, $u^*$ its baseline rate and $\beta>0$ is a scenario-specific coefficient. For inflation, use a short-run relation $\pi_t=\pi_t^e+\kappa x_t+s_t$, where $\pi_t$ is inflation, $\pi_t^e$ is expected inflation, $\kappa>0$, and $s_t$ is a supply-driven inflation contribution. A negative output gap therefore reduces inflation relative to expectations when $s_t=0$; a positive supply shock can instead raise inflation even as output falls.

One possible expectations rule is adaptive updating: $\pi_t^e=\lambda\pi_{t-1}^e+(1-\lambda)\pi_{t-1}$, where $0\leq\lambda\leq1$ measures the weight on the previous expectation, and $\pi_{t-1}$ is the previous period’s actual inflation. Each equation requires its own assumptions. These linear relationships are illustrative approximations over a limited range, not identities; they need not apply unchanged across a large shock, a structural change or a different policy regime.

Worked Example: A Demand Shock and Its Gradual Offset

Method: Apply the output recurrence period by period, then use the stated unemployment and inflation relationships. A hypothetical economy has potential output of $1,000 billion, an initial output gap of zero, a baseline unemployment rate of 5%, and an initial inflation rate and expectation of 2%. Let $c=0.5$, $\beta=0.4$, $\kappa=0.25$, and $\lambda=0.6$. A one-time demand shock gives $z_1=-2$ percentage points. There are no new impulses in periods 2 and 3. In period 4, policy support contributes $z_4=+0.25$ percentage points. Assume $s_t=0$ throughout.

Output. Period 1 has $x_1=-2+0.5(0)=-2\%$. Then $x_2=0+0.5(-2)=-1\%$, and $x_3=0+0.5(-1)=-0.5\%$. With policy support, $x_4=0.25+0.5(-0.5)=0\%$. The corresponding output levels are $1{,}000(1-0.02)=\$980$ billion, $1{,}000(1-0.01)=\$990$ billion, $1{,}000(1-0.005)=\$995$ billion and $1,000 billion.

Unemployment. The unemployment gap is $-0.4x_t$, with $x_t$ entered as a percentage-point number. Thus the unemployment rates are $5-0.4(-2)=5.8\%$, $5-0.4(-1)=5.4\%$, $5-0.4(-0.5)=5.2\%$, and $5-0.4(0)=5.0\%$.

Inflation and expectations. In period 1, $\pi_1^e=2\%$ and $\pi_1=2+0.25(-2)=1.5\%$. In period 2, $\pi_2^e=0.6(2)+0.4(1.5)=1.8\%$, so $\pi_2=1.8+0.25(-1)=1.55\%$. In period 3, $\pi_3^e=0.6(1.8)+0.4(1.55)=1.70\%$, so $\pi_3=1.70+0.25(-0.5)=1.575\%$. In period 4, $\pi_4^e=0.6(1.70)+0.4(1.575)=1.65\%$; because $x_4=0$, $\pi_4=1.65\%$.

Check. Output rises from $980 billion to $1,000 billion as the gap closes, and unemployment returns to its baseline. Inflation is still below its initial 2% rate at the end because expectations adjusted downward during the downturn. The common mistake is to assume that closing the output gap instantly restores inflation expectations; under the stated updating rule, expectations respond with a lag.

Why Persistence Shapes the Cycle

The recurrence makes the role of persistence explicit. Suppose there is only one demand impulse $z_1$, with no later impulses. Then $x_1=z_1$, $x_2=cz_1$, $x_3=c^2z_1$, and, by repeated substitution, $x_t=c^{t-1}z_1$. If $0\leq c<1$, each successive gap is smaller in absolute value, so the gap eventually approaches zero.

The total of the successive gaps also has a simple form. For a one-time impulse, summing over periods gives $z_1+cz_1+c^2z_1+\cdots$. Multiplying this infinite series by $c$ gives $cz_1+c^2z_1+c^3z_1+\cdots$. Subtracting the second expression from the first leaves $z_1$, so the sum is $z_1/(1-c)$. This result requires $|c|<1$ and no further impulses. It describes cumulative percentage-point gaps across periods—not a dollar loss or a forecast of total GDP. If persistence is one or greater, or shocks continue, this convergence argument does not establish recovery.

Identify the initiating shock.
Classify it as a demand-side or supply-side impulse and state its size, timing and assumed duration.

Trace output and labor-market effects.
Use the scenario’s output-gap assumptions to find output, then apply the stated labor-market relationship to find the unemployment response.

Separate inflation pressures.
Distinguish the effect of cyclical slack from the supply contribution and update expectations only according to the specified rule.

Assess policy and recovery.
Consider how policy changes demand, what trade-offs it creates, and whether output is recovering relative to current potential or an earlier trend.

Supply Shocks and the Policy Trade-Off

A demand downturn and an adverse supply shock need not produce the same inflation pattern. In the model above, weaker demand lowers $x_t$, putting downward pressure on inflation. An adverse supply shock raises $s_t$, putting upward pressure on inflation directly. If the two effects occur together, the net inflation result depends on their sizes. Policy that supports demand can narrow a negative output gap, but in the stated inflation relationship that also removes some of the downward pressure from slack.

Worked Example: Stabilizing Output After a Supply Shock

Method: Compare the same hypothetical supply shock with and without a demand-support policy. Potential output is $800 billion, the baseline unemployment rate is 5%, expected inflation is 2%, and the previous actual inflation rate is 2%. Let $\beta=0.4$, $\kappa=0.25$, and $\lambda=0.6$. The shock produces an output gap of $-1.5\%$ and a supply contribution $s_t=+1$ percentage point. Assume the policy response raises the output gap by exactly 1 percentage point, to $-0.5\%$, in this period.

Without the policy response, output is $800(1-0.015)=\$788$ billion. Unemployment is $5-0.4(-1.5)=5.6\%$. Inflation expectations are $0.6(2)+0.4(2)=2\%$, so actual inflation is $2+0.25(-1.5)+1=2.625\%$.

With the policy response, output is $800(1-0.005)=\$796$ billion. Unemployment is $5-0.4(-0.5)=5.2\%$. Given the same initial inflation history, expectations remain 2% in this period, and actual inflation is $2+0.25(-0.5)+1=2.875\%$.

Check. The response raises output by $8 billion and lowers unemployment by 0.4 percentage points relative to no response. Inflation is 0.25 percentage points higher because the smaller negative gap provides less downward pressure. The supply contribution remains one percentage point in both cases. The common mistake is to infer that output support must reduce inflation; here, the direct supply pressure and the cyclical effect are distinct, and the latter moves against disinflation when the gap narrows.

A policy comparison like this is conditional, not a universal prescription. It holds the supply contribution and the initial expectations history fixed, and assumes the stated output effect of policy. A full evaluation would also consider policy lags, the persistence of the shock, expectations, financial conditions and the costs of unemployment. Monetary policy and fiscal responses may work through different channels and time horizons, as covered in the earlier material on stabilization policy.

Cyclical Recovery Versus Lasting Scarring

A shrinking negative output gap does not by itself show that output has returned to its former path. Potential output can change if a prolonged downturn affects investment, skills, labor-force attachment or productive capacity. This possibility is associated with hysteresis: cyclical weakness may have persistent effects on the economy’s capacity. It should not be assumed from a single period of weak output; the scenario must specify a lasting change in potential.

Worked Example: Recovery After a Loss of Potential Output

Method: Compare output against both the original and revised potential level, then calculate unemployment using the stated gap relationship. A hypothetical economy initially has potential output of $500 billion, a 5% baseline unemployment rate, and an output gap of $-3\%$. Let $\beta=0.4$. Assume a stabilization measure improves the output gap by 1.2 percentage points in the initial period. Also assume that a prolonged downturn reduces next period’s potential output by 1% from its original level; this lasting reduction is given, not inferred from the initial gap.

Initially, actual output is $500(1-0.03)=\$485$ billion. Unemployment is $5-0.4(-3)=6.2\%$. After the assumed policy improvement, the gap is $-3+1.2=-1.8\%$, so output is $500(1-0.018)=\$491$ billion and unemployment is $5-0.4(-1.8)=5.72\%$.

Next period, potential output is $500(1-0.01)=\$495$ billion. Suppose actual output reaches $495 billion. The gap relative to revised potential is $100(495-495)/495=0\%$, so the unemployment rate under this model returns to 5%. Yet output remains $5 billion, or 1%, below the original $500 billion potential level.

Check. The economy has closed its cyclical gap relative to its new potential but has not recovered its original productive capacity. A mistaken conclusion would be that a zero output gap proves the entire downturn has been undone. The comparison must state which potential-output path is being used.

Common Traps When Tracing a Shock

Mixing up a shock and its propagation. A one-time impulse can generate several periods of effects when the gap is persistent. Do not count each later effect as a new initiating shock unless the scenario specifies one.
Reversing the unemployment relationship. With $u_t-u^*=-\beta x_t$, a negative output gap raises unemployment. Using a positive sign would incorrectly predict lower unemployment in a downturn.
Treating inflation as a single-channel outcome. Cyclical slack, supply pressure and expectations can move inflation in different directions. State each contribution before adding them.
Confusing output recovery with restoration of potential. A closed gap can coexist with output below an earlier trend if potential has fallen.
Applying a local equation outside its assumptions. These linear relationships use the given calibrations and period definitions. A different shock, policy regime or structural change may require different relationships.

Key takeaway: A complete cycle analysis follows the shock through output, employment, inflation, expectations and policy, while keeping cyclical movements separate from changes in potential output. Recovery depends on the persistence of the shock, the response of private decisions and policy, and whether productive capacity has been impaired.

This integrated way of reasoning is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. In a scenario, make the assumptions explicit, keep percentage points distinct from percentage changes, and check whether the proposed recovery is relative to current potential or to the economy’s earlier path.

Check Your Understanding

For each question, use only the assumptions stated in its scenario and trace the requested effects in order.

A hypothetical economy has $x_0=0$, $c=0.5$, and $x_t=z_t+cx_{t-1}$. A demand impulse of $-1.6$ percentage points occurs in period 1. There is no new impulse in period 2, and policy contributes $+0.2$ percentage points in period 3. Find the output gaps in periods 1, 2 and 3. (Answer: $-1.6\%$, $-0.8\%$, and $-0.2\%$)
Potential output is $1,200 billion, the output gap is $-1.25\%$, baseline unemployment is 4.5%, and $\beta=0.3$. Find actual output and unemployment. (Answer: $1,185 billion and 4.875%)
Expected inflation is 2.4%, the output gap is $-2\%$, the supply contribution is $+0.6$ percentage points, and $\kappa=0.25$. Find actual inflation. (Answer: 2.5%)
Potential output is $700 billion and the initial output gap is $-2\%$. A policy response improves the gap by 0.5 percentage points. Find output before and after the response. (Answer: $686 billion before and $689.5 billion after)
Under adaptive expectations, $\pi_t^e=0.6\pi_{t-1}^e+0.4\pi_{t-1}$. Previous expected inflation is 2.0% and previous actual inflation is 1.5%. In the current period, $x_t=-0.4\%$, $s_t=0$, and $\kappa=0.25$. Find current expected and actual inflation. (Answer: expected inflation is 1.8%; actual inflation is 1.7%)

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