Temporary and Persistent Shocks in the Recovery Path

Why Shock Duration Matters

Two economies can experience the same initial fall in output and then follow very different recovery paths. In one, the shock fades quickly and the output gap closes soon. In the other, the shock's effects persist, so output remains below trend for longer. A shock that continues to arrive can deepen the gap even when each period's shock is no larger than the first.

To compare these paths, we need to keep three ideas separate: the shock, the output gap, and the trend path. The output gap describes actual output relative to trend at a point in time. Shock duration describes how long the disturbance itself continues. Persistence describes how much of the gap carries into later periods. These distinctions help explain why the end of a shock does not necessarily mean an immediate return to trend.

Definition: Let $Y_t$ be actual output and $Y_t^*$ be trend (or potential) output in period $t$. The percentage output gap is $$ x_t=100\frac{Y_t-Y_t^*}{Y_t^*}. $$ A negative $x_t$ means output is below trend; a positive $x_t$ means output is above trend. The gap is measured in percentage points. For example, $x_t=-3$ means actual output is 3% below trend.

A Simple Model of Persistence

A compact way to describe the path of a gap is a linear, reduced-form recurrence:

x_t=\rho x_{t-1}+u_t.

Here $u_t$ is the new shock in period $t$, measured in percentage points of the output gap, and $\rho$ is the persistence parameter. This model holds the definition of trend fixed for the comparison and treats the gap as responding linearly. For a stable return toward trend, assume $|\rho|<1$. In the examples, $0\leq\rho<1$, so a gap keeps its sign as it fades. A larger $\rho$ means more of the previous period's gap carries forward.

A temporary one-period shock has a nonzero $u_t$ for just one period, followed by $u_t=0$. A persistent shock process may instead mean that the gap fades slowly because $\rho$ is high, or that new shocks continue to arrive over several periods. Those are different mechanisms, even though either can make the gap last longer. In particular, when new shocks stop, persistence in the gap can still delay recovery.

This recurrence is a teaching model for comparing paths, not a complete explanation of the business cycle. It does not specify why a shock occurs or include every possible response by households, firms, or policymakers. It is most useful when the question supplies a persistence rate and asks how a gap evolves. If trend output changes, if the model's linear relationship is inappropriate, or if shocks have lasting effects on trend itself, the recurrence alone does not describe the whole economy.

Key result: If a one-time shock leaves an initial gap $x_0$ and no further shocks occur, then for $t\geq 0$, $$ x_t=\rho^t x_0. $$ For $0\leq\rho<1$, the gap approaches zero over time. A higher $\rho$ means slower convergence, not necessarily a larger initial shock.

Why a One-Time Gap Fades Geometrically

The geometric path follows directly from the recurrence. Once the one-time shock has occurred, all later $u_t$ values are zero, so each period's gap is just $\rho$ times the previous gap. Applying that rule repeatedly gives the result:

Start with the gap.
At period 0 the gap is $x_0$, the value left by the initial shock.

Apply the no-new-shock rule.
Since later $u_t=0$, $x_1=\rho x_0$.

Repeat the rule.
Then $x_2=\rho x_1=\rho^2x_0$, and $x_3=\rho x_2=\rho^3x_0$.

Generalize.
After $t$ periods, the repeated multiplication gives $x_t=\rho^t x_0$. Since $|\rho|<1$, $\rho^t$ approaches zero, so the gap approaches zero too.

This derivation explains why recovery is gradual in this model. When the shock stops, it sets no new $u_t$; it does not erase the existing $x_t$. With $\rho=0$, the gap vanishes after one transition. With $\rho$ close to 1, each period removes only a small fraction of the remaining gap.

Worked Example: Comparing Fast and Slow Recovery

Two hypothetical economies each begin with an output gap of $-4$ percentage points after a one-time shock. No further shocks occur. Economy A has $\rho=0.25$, and Economy B has $\rho=0.75$. Calculate each gap from period 0 through period 4 and compare the paths.

Method. Use $x_t=\rho^t x_0$, with $x_0=-4$. The negative sign indicates output below trend.

For Economy A, the successive calculations are:

\begin{aligned} x_0&=-4,\\ x_1&=0.25(-4)=-1,\\ x_2&=0.25(-1)=-0.25,\\ x_3&=0.25(-0.25)=-0.0625,\\ x_4&=0.25(-0.0625)=-0.015625. \end{aligned}

For Economy B:

\begin{aligned} x_0&=-4,\\ x_1&=0.75(-4)=-3,\\ x_2&=0.75(-3)=-2.25,\\ x_3&=0.75(-2.25)=-1.6875,\\ x_4&=0.75(-1.6875)=-1.265625. \end{aligned}

Both gaps start at $-4$, but Economy B remains further below trend at each later period because its persistence parameter is higher. In period 4, the gaps are about $-0.0156$ and $-1.2656$ percentage points, respectively. The calculation is checked by applying each economy's $\rho$ to its period 3 gap; the results equal the period 4 gaps shown.

As an additional measure of duration, the sum of the future gaps after a one-time initial gap is a geometric series. For $|\rho|<1$, including period 0, its value is $x_0/(1-\rho)$. Here it is $-4/(1-0.25)=-5.3333$ percentage-point-periods for A, compared with $-4/(1-0.75)=-16$ for B. This sum measures the accumulated gap in the model; it is not a dollar loss.

When the Shock Continues

A shock that lasts for several periods is not the same as a one-time shock with a high $\rho$. In the recurrence, a continuing shock adds a new $u_t$ each period. The gap at any period reflects both the carried-forward gap and the new disturbance. For a sequence of shocks, repeated substitution gives:

x_t=\rho^t x_0+\sum_{j=1}^{t}\rho^{t-j}u_j.

The first term is the part of the initial gap that remains. Each term in the sum is a shock from a particular period, carried forward according to how many periods have passed since it occurred. This expression makes clear why a sequence of shocks can create a deeper gap than a single shock of the same size.

Worked Example: A Shock Lasting Two Periods

A hypothetical economy starts at trend, so $x_0=0$. Let $\rho=0.5$. A negative shock of $-2$ percentage points arrives in each of periods 1 and 2, and no further shocks arrive. Find the gap through period 5 and identify the trough.

Method. Apply $x_t=0.5x_{t-1}+u_t$ period by period. Keep the new shock separate from the carried-forward gap.

\begin{aligned} x_1&=0.5(0)+(-2)=-2,\\ x_2&=0.5(-2)+(-2)=-1-2=-3,\\ x_3&=0.5(-3)+0=-1.5,\\ x_4&=0.5(-1.5)+0=-0.75,\\ x_5&=0.5(-0.75)+0=-0.375. \end{aligned}

The trough is $-3$ percentage points in period 2. When the shock stops after period 2, the gap does not immediately become zero: it halves each period, reaching $-1.5$, $-0.75$, and then $-0.375$. The final check is that each post-shock value is exactly half the preceding gap, as required by $\rho=0.5$.

A common mistake is to treat the period 2 shock as if it replaced the previous gap, giving $x_2=-2$. That omits the carry-forward term. The correct calculation is $0.5(-2)-2=-3$: period 2 inherits half of period 1's gap and then receives the new shock.

Gap Recovery and Trend Growth Are Different

A closing negative output gap does not require actual output to fall or remain constant. Actual output can rise while staying below trend, provided it rises more slowly than trend over the comparison. Conversely, a gap can narrow while trend output itself changes. To calculate actual output from a gap, rearrange its definition:

Y_t=Y_t^*\left(1+\frac{x_t}{100}\right).

The percentage gap is relative to the trend level in the same period. Therefore, when trend grows, it is not correct to apply a later gap percentage to the initial trend level. The trend level must first be advanced to that period.

Worked Example: Output Rises While the Negative Gap Closes

A hypothetical economy has trend output of $200$ billion dollars in period 0. Trend grows by 2% per period. The initial gap is $-3$ percentage points, no further shocks occur, and $\rho=0.5$. Calculate trend output, the gap, and actual output for periods 0 through 3. Report output to the nearest $0.01$ billion dollars.

Method. First grow trend by 2% each period. Then use $x_t=0.5^t(-3)$. Finally calculate actual output as trend multiplied by $1+x_t/100$.

\begin{array}{c|c|c|c} t & Y_t^*\text{ (billions)} & x_t\text{ (percentage points)} & Y_t\text{ (billions)}\\ \hline 0 & 200 & -3 & 200(0.97)=194\\ 1 & 200(1.02)=204 & -1.5 & 204(0.985)=200.94\\ 2 & 204(1.02)=208.08 & -0.75 & 208.08(0.9925)=206.5194\\ 3 & 208.08(1.02)=212.2416 & -0.375 & 212.2416(0.99625)=211.445694 \end{array}

Rounded to the nearest $0.01$ billion dollars, actual output is $194.00$, $200.94$, $206.52$, and $211.45$ billion dollars. It rises in each period, while the gap remains negative and approaches zero. For example, period 3 actual output is still below its trend value of $212.2416$ billion dollars. The check is that $211.445694/212.2416-1=-0.00375$, or a gap of $-0.375\%$.

The common error is to calculate period 3 output using the period 0 trend value: $200(0.99625)$. That uses the correct gap with the wrong trend level. The gap is defined relative to $Y_3^*$, not $Y_0^*$.

Interpreting the Paths Carefully

A useful comparison asks not only whether output is below trend, but also how deep the gap becomes, how long it remains substantial, and whether new shocks continue. In the simple model, higher persistence makes an existing gap decay more slowly. A sequence of same-sign shocks can deepen the gap before recovery begins. These are distinct reasons that the observed recovery path may be prolonged.

The stable-return result depends on $|\rho|<1$ and on the absence of new shocks after the initial disturbance. If $\rho=1$, then a gap with no new shocks remains unchanged: $x_t=x_0$. There is no convergence in that case. If $|\rho|>1$, the gap grows in magnitude rather than fading. Those cases are outside the stable recovery model. A lasting change in productive capacity or trend output also requires a revised trend path; it should not be treated automatically as a temporary cyclical gap.

In applications, measured output and estimated trend may be revised, and the gap need not follow one constant $\rho$ in every period. The recurrence is therefore a disciplined way to interpret a stated model or compare hypothetical paths, not a guarantee that all recoveries have the same shape. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Common exam trap: Do not confuse “the shock has ended” with “the gap has ended.” When $u_t$ returns to zero, the recurrence becomes $x_t=\rho x_{t-1}$; the inherited gap remains unless $\rho=0$. Also distinguish a smaller negative gap from a positive output gap: moving from $-3$ to $-1$ is recovery toward trend, but output is still below trend.

Check Your Understanding

Use the recurrence and the stated trend values where needed. Each answer is given at the end of its question for checking.

A one-time shock leaves $x_0=-5$ percentage points. If $\rho=0.2$ and no further shocks occur, calculate $x_1$ and $x_2$. (Answer: $x_1=-1$, $x_2=-0.2$ percentage points.)
A one-time shock leaves a gap of $-3$ percentage points, and $\rho=0.6$. What is the gap after two periods with no new shocks? (Answer: $x_2=-1.08$ percentage points.)
An economy starts at trend, has $\rho=0.4$, and receives shocks $u_1=-1.5$ and $u_2=-1.5$ percentage points, followed by no further shocks. Calculate $x_1,x_2,x_3,$ and $x_4$, and state the trough. (Answer: $-1.5,-2.1,-0.84,-0.336$ percentage points; trough $=-2.1$ in period 2.)
Trend output is $500$ billion dollars in period 0 and grows by 4% in period 1. The period 0 gap is $-2$ percentage points, and the period 1 gap is $-1$ percentage point. Calculate actual output in both periods. (Answer: period 0: $490.00$ billion dollars; period 1: $514.80$ billion dollars.)
A one-time shock creates a gap of $-6$ percentage points. With no new shocks, the observed next gap is $-4.5$ percentage points. Find $\rho$ and the gap three periods after the initial gap. (Answer: $\rho=0.75$; three periods after the initial gap, $x_3=0.75^3(-6)=-2.53125$, or approximately $-2.53$ percentage points.)

What You'll Learn