The Encyclopedia of Economic Principals

Chapter 55

Business Cycles, Propagation, Hysteresis, and Stagnation

Trace how shocks turn into cycles, linger, and leave scars.

Four of the chapter's worked examples, made interactive: the accelerator's investment spike, a multiplier-accelerator cycle, real-business-cycle propagation through capital, and hysteresis in unemployment. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Investment jumps on a demand rise

Why does a modest rise in expected output cause a large, temporary burst of investment?

Investment responds to the change in expected output, not its level. A rise forces the whole capacity gap to be bought at once; when output stops rising, investment falls back to replacement even though demand stays high.

Equation, written in LaTeX: K_0^*=6(8)=48

Equation, written in LaTeX: K_1^*=6(9.5)=57.

Equation, written in LaTeX: I_1^g=(57-48)+0.10(48)=13.8

Equation, written in LaTeX: I_t^g=K_t^*-K_{t-1}+\delta K_{t-1}.

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Output is in units of 10,000 cases and capital in equipment units costing $50,000 each. Desired capital is K* = v Y, where v is the capital-output ratio; delta = 0.10 is depreciation. The plant starts with 48 units matching expected output 8.

Predict first. If expected output stays at 9.5 in the following period, does investment spending stay high?

Your prediction

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Figure: Investment jumps on a demand rise. Gross investment spending: baseline $240,000, adjustment period $690,000, following period $285,000.
New expected output (10,000 cases): 9.5, Capital-output ratio: 6
Constructed example: the chapter's hypothetical food processor (v = 6, delta = 0.10, output 8 then 9.5, $50,000 a unit); expected outputs of 8, 9 and 10 and ratios of 4 and 8 are added for comparison.

Calculated values

Desired capital before
48.0
Desired capital after
57.0
Baseline spending
$240,000
Adjustment spending
$690,000
Following-period spending
$285,000
Drop after the gap closes
$405,000
Rise in expected output
18.75%

K* = 6 x 8 = 48.0 units, so baseline investment is 0.10 x 48.0 = 4.80 units, or $240,000. With expected output 9.5, K* = 6 x 9.5 = 57.0, and I = (57.0 - 48.0) + 4.80 = 13.80 units, or $690,000. Once capital is 57.0, spending falls back to replacement, 0.10 x 57.0 = 5.70 units or $285,000, a drop of $405,000 while expected output stays at 9.5.

Worked steps

  1. K0* = 6 x 8 = 48.0; baseline = 0.10 x 48.0 = 4.80 units
  2. K1* = 6 x 9.5 = 57.0
  3. I1 = (57.0 - 48.0) + 4.80 = 13.80 units
  4. Spending = 13.80 x 50,000 = $690,000
  5. Next period = 0.10 x 57.0 = 5.70 units = $285,000

Use the idea

When demand growth slows, expect capital spending to fall even if demand itself holds steady.

Where the conclusion applies

Immediate adjustment to desired capital, a fixed capital-output ratio and no capacity constraints on suppliers; real firms spread the adjustment over time.

Check your understanding: What is gross investment in the period after the gap closes?
0.10 x 57 = 5.7 units, and 5.7 x 50,000 = $285,000.

Chapter 55 source: section "Accelerator principle".

Demonstration 2 of 4

A single shock that cycles

Can one burst of spending set off a cycle with no further shocks?

Consumption follows income with a lag and investment follows the change in consumption. When consumption growth slows, investment collapses even though consumption is still high, which turns output down and starts an overshoot below the steady state.

Equation, written in LaTeX: \bar Y=\frac{24}{1-0.5}=48.

Equation, written in LaTeX: Y_2=30+7.2+24=61.2.

Equation, written in LaTeX: Y_3=30.6+0.72+24=55.32.

Equation, written in LaTeX: Y_t=c(1+v)Y_{t-1}-cvY_{t-2}+A_t.

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C_t = c Y_{t-1} is consumption from last period's income, I_t = v (C_t - C_{t-1}) is induced net investment and A_t is autonomous spending, 24 except for 36 in period 1. Y_t = C_t + I_t + A_t.

Predict first. After the impulse ends in period 1, does output fall immediately in period 2?

Your prediction

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Figure: A single shock that cycles. Income path over 12 periods after a one-period rise in autonomous spending, with c = 0.50 and v = 1.2. Income starts at 48.00, reaches 60.00 in period 1 and peaks at 61.20 in period 2.
Marginal propensity to consume: 0.5, Accelerator coefficient: 1.2
Constructed example: the chapter's hypothetical economy (c = 0.5, v = 1.2, A = 24 with 36 in period 1); propensities 0.6 and 0.75 and accelerator coefficients 0.5 and 2.0 are added for comparison.

Calculated values

Steady state
48.0000
Y1
60.0000
Y2
61.2000
Y3
55.3200
Y4
48.1320
Y5
43.7532
Peak period
2

Steady state Y = 24 / (1 - 0.50) = 48.0000. Period 1: Y1 = 24.0000 + 0 + 36 = 60.0000. Period 2: C2 = 0.50 x 60.0000 = 30.0000, I2 = 1.2 x (30.0000 - 24.0000) = 7.2000, so Y2 = 30.0000 + 7.2000 + 24 = 61.2000. Output keeps rising after the impulse ends, because investment responds to the rise in consumption.

Worked steps

  1. Y = 24 / (1 - 0.50) = 48.0000
  2. Y1 = 24.0000 + 0 + 36 = 60.0000
  3. C2 = 0.50 x 60.0000 = 30.0000; I2 = 1.2 x (30.0000 - 24.0000) = 7.2000
  4. Y2 = 30.0000 + 7.2000 + 24 = 61.2000
  5. C3 = 0.50 x 61.2000 = 30.6000; I3 = 1.2 x (30.6000 - 30.0000) = 0.7200
  6. Y3 = 30.6000 + 0.7200 + 24 = 55.3200

Use the idea

Read a turning point as the result of slowing growth, not falling levels: induced investment reverses when demand merely stops accelerating.

Where the conclusion applies

Fixed propensities, one-period lags, no capacity limits and negative net investment allowed. Larger c v makes the cycle more persistent or explosive.

Check your understanding: What is induced investment in period 4?
C4 = 0.5 x 55.32 = 27.66, so I4 = 1.2 x (27.66 - 30.6) = 1.2 x (-2.94) = -3.528.

Chapter 55 source: section "Multiplier-accelerator interaction".

Demonstration 3 of 4

A productivity shock that lingers

How does a one-time productivity shock keep output high after it fades?

The shock raises output now; part of the gain is saved as investment, so capital is higher next period. Persistent productivity plus the larger capital stock carry the boom forward.

Equation, written in LaTeX: K_{t+1}^{0}=0.92(160)+24=171.2.

Equation, written in LaTeX: \frac{Y_t^1}{Y_t^0}=1.04(1.03)^{0.65}\approx1.0602.

Equation, written in LaTeX: \frac{Y_{t+1}^{1}}{Y_{t+1}^{0}}=1.02(\frac{175.2}{171.2})^{0.35}\approx1.0283.

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Output is Cobb-Douglas with capital share alpha = 0.35. Baseline output is 100, capital 160, investment 24 and depreciation 8 percent. Productivity rises by the shock this period and by half of it next period; labor responds this period only; investment rises to 28.

Predict first. Next period, with productivity only 2 percent up and labor back to normal, is output more than 2 percent above trend?

Your prediction

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Figure: A productivity shock that lingers. Left: output is 6.02 percent above the no-shock path this period and 2.83 percent next period. Right: next period's gain splits into 2.00 points of productivity and 0.83 points from extra capital.
Productivity shock: 4%, Labor response: 3%
Constructed example: the chapter's hypothetical economy (alpha 0.35, output 100, capital 160, investment 24 then 28, shock 4 percent then 2, labor plus 3 percent); shocks of 2 and 6 percent and labor responses of 0 and 5 percent are added for comparison.

Calculated values

Current output ratio
1.0602
Output
106.02
Consumption change
2.65%
Investment change
16.67%
Next-period capital
175.2 vs 171.2
Next-period output ratio
1.0283

Capital is fixed this period, so Y1 / Y0 = 1.04 x 1.03^0.65 = 1.0602 and output is 106.02. With investment at 28, consumption is 106.0175 - 28 = 78.0175 against 76, a change of 2.65 percent, while investment rises 16.67 percent. Next period capital is 0.92 x 160 + 28 = 175.2 against 171.2, and with productivity 2 percent up the ratio is 1.02 x (175.2 / 171.2)^0.35 = 1.0283, more than the 2 percent productivity gain alone.

Worked steps

  1. K next (no shock) = 0.92 x 160 + 24 = 171.2
  2. Y1 / Y0 = 1.04 x 1.03^0.65 = 1.0602
  3. C = 106.0175 - 28 = 78.0175; change = 78.0175 / 76 - 1 = 2.65%
  4. K next (shock) = 0.92 x 160 + 28 = 175.2
  5. Next ratio = 1.02 x (175.2 / 171.2)^0.35 = 1.0283

Use the idea

When judging how long a supply shock lasts, add the capital it leaves behind to the persistence of the shock itself.

Where the conclusion applies

Investment fixed at 28 in every state (the book's choice), productivity next period half the shock (the book's 4 then 2 percent), and labor back at its reference level next period. The book divides the rounded output 106.02 and prints a 2.66 percent rise in consumption; the unrounded 78.0175 / 76 gives 2.65 percent, shown here.

Check your understanding: With no labor response, what is the current output ratio?
1.04 x 1.00^0.65 = 1.04.

Chapter 55 source: section "Real-business-cycle propagation".

Demonstration 4 of 4

Recessions that leave scars

How much of a recession's unemployment stays after demand recovers?

With hysteresis, the recession path changes the equilibrium rate itself. Part of the excess unemployment turns into lasting barriers, so restoring demand does not restore the old rate.

Equation, written in LaTeX: u_{t+1}^*=\bar u+\rho(u_t^*-\bar u)+\gamma(u_t-u_t^*),

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u_t is observed unemployment, u_t* the equilibrium rate and u bar = 5 percent the pre-recession benchmark, in a labor force of 24,000. rho = 0.9 is persistence and gamma is the share of excess unemployment that becomes embedded.

Predict first. Does a program cutting embedding by two thirds (gamma = 1/12) bring unemployment back to 5 percent?

Your prediction

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Figure: Recessions that leave scars. Unemployment rate before the recession 5 percent, in the recession 12.5 percent and after recovery 6.875 percent, with 1,650 unemployed.
Embedding coefficient gamma: 0.25, Recession unemployment (%): 12.5
Constructed example: the chapter's hypothetical region (24,000 workers, 5 percent before, 12.5 percent in recession, rho 0.9, gamma 0.25 or 1/12 with the program); gamma of 0 and 0.5 and recession rates of 10 and 15 percent are added for comparison.

Calculated values

Equilibrium rate after recovery
6.875%
Unemployed after recovery
1,650
Embedded workers
450
Displaced in the recession
1,800

u1* = 5 + 0.9 x (5 - 5) + 0.25 x (12.5 - 5) = 5 + 1.875 = 6.875 percent. That is 0.06875 x 24,000 = 1,650 unemployed after demand recovers, 450 more than the original 1,200: the recession has left a scar.

Worked steps

  1. u1* = 5 + 0.9 x 0 + 0.25 x (12.5 - 5) = 6.875%
  2. Unemployed = 0.06875 x 24,000 = 1,650
  3. Embedded = 1,650 - 1,200 = 450

Use the idea

Weigh the cost of a deep recession by its lasting effect on the equilibrium rate, and judge retention programs by how much of that embedding they prevent.

Where the conclusion applies

A reduced-form law with an assumed gamma; the chapter stresses that it does not supply an empirical value or identify which channel operates.

Check your understanding: With gamma = 1/12, how many are unemployed after recovery?
5 + 7.5 / 12 = 5.625 percent, and 0.05625 x 24,000 = 1,350.

Chapter 55 source: section "Hysteresis in unemployment".