Demonstration 1 of 4
Chasing low unemployment
What inflation path is needed to hold unemployment below its natural rate?
Unemployment falls below the natural rate only while inflation beats expectations. Adaptive expectations catch up each year, so keeping the gain requires ever higher inflation.
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pi is inflation and u unemployment, in percent. The natural rate is 5 percent, alpha = 1 and expectations are fully adaptive: next year's expected inflation equals this year's actual inflation. Inflation and expectations start at 2 percent.
Predict first. If the authority stops accelerating in year 3 and holds inflation at 4 percent, does unemployment stay at 4?
Choose an example
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Constructed example: the chapter's hypothetical economy (natural rate 5, target 4, expectations starting at 2, and its hold case); targets of 3 and 5 percent are added for comparison.
Calculated values
- Inflation years 1 to 3
- 3, 4, 5
- Unemployment year 3
- 4%
- Year-3 policy
- keep accelerating
Year 1: pi = 2 - (4 - 5) = 2 + 1 = 3; Year 2: pi = 3 - (4 - 5) = 3 + 1 = 4; Year 3: pi = 4 - (4 - 5) = 4 + 1 = 5. Each year inflation must exceed the rate already expected, so it keeps rising.
Worked steps
- Year 0: 2 = 2 - (u - 5), so u = 5
- Year 1: pi = 2 - (4 - 5) = 2 + 1 = 3
- Year 2: pi = 3 - (4 - 5) = 3 + 1 = 4
- Year 3: pi = 4 - (4 - 5) = 4 + 1 = 5
Use the idea
Treat a lasting unemployment target below the natural rate as a promise of rising inflation, not a one-time cost.
Where the conclusion applies
A known natural rate, alpha = 1, no supply shocks and full one-year adaptive expectations.
Check your understanding: With a 3 percent target, what inflation is required in year 2?
Chapter 54 source: section "Natural-rate hypothesis".
Demonstration 2 of 4
One rule, two signals
How does a Taylor rule combine inflation and the output gap into one policy rate?
The rule moves the policy rate more than one for one with inflation (pi plus half the gap), so the real rate rises when inflation rises; slack pulls the rate down.
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i is the policy rate, pi inflation and x the output gap (percent of potential), with r* = 2, target inflation 2 and weights a = b = 0.5: i = r* + pi + a(pi - 2) + b x.
Predict first. With 3 percent inflation but output 2 percent below potential, is the rate above or below 6.5?
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Constructed example: the chapter's hypothetical rule (r* 2, target 2, weights 0.5) at its cases (2, 0), (3, 2) and (3, -2); inflation of 1 and 4 percent is added for comparison.
Calculated values
- Policy rate i
- 6.50%
- Change from 4% baseline
- +2.50 points
- Real rate i - pi
- 3.50%
i = 2 + 3 + 0.5(3 - 2) + 0.5(2) = 2 + 3 + 0.50 + 1 = 6.50 percent, 2.50 points above the 4 percent baseline. The real rate is 6.50 - 3 = 3.50.
Worked steps
- i = 2 + 3 + 0.5(3 - 2) + 0.5(2) = 2 + 3 + 0.50 + 1 = 6.50%
- Change = 6.50 - 4 = 2.50 points
- Real rate = 6.50 - 3 = 3.50
Use the idea
Use the rule as a benchmark: compute it from current inflation and the gap, and ask why actual policy differs.
Where the conclusion applies
Known r*, potential output and targets; the weights 0.5 are the book's illustration, not a recommendation for any central bank.
Check your understanding: What does the rule give at pi = 3, x = -2?
Chapter 54 source: section "Taylor rule".
Demonstration 3 of 4
Prices that reset a few at a time
How fast does the average price reach a new desired price when only some firms can reset?
Each period only the share 1 - theta of remaining firms resets, so the gap to the new price shrinks by the factor theta each period.
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theta is the probability a firm cannot reset its price in a period. All firms start at 100 dollars and the new desired price is 108. The share still at 100 after k periods is theta^k; the book uses a simple weighted average for the aggregate price.
Predict first. After two periods at theta = 0.75, is the price more or less than halfway to 108?
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Constructed example: the chapter's hypothetical firms (theta 0.75 and the flexible case 0, prices 100 and 108, periods 1 and 2); theta of 0.5 and 0.9 and four periods are added for comparison.
Calculated values
- P after 2 periods
- 103.50
- Share still at 100
- 0.5625
- Progress to 108
- less than halfway
P1 = 0.25(108) + 0.75(100) = 27.00 + 75.00 = 102.00; Share still at 100 after 2 = 0.75^2 = 0.5625; P2 = 0.5625(100) + 0.4375(108) = 56.25 + 47.25 = 103.50. After 2 periods the average price is less than halfway from 100 to 108.
Worked steps
- P1 = 0.25(108) + 0.75(100) = 27.00 + 75.00 = 102.00
- Share still at 100 after 2 = 0.75^2 = 0.5625
- P2 = 0.5625(100) + 0.4375(108) = 56.25 + 47.25 = 103.50
Use the idea
Read theta as price stickiness: the expected time to reset is 1 / (1 - theta) periods, which sets how slowly a cost change reaches measured prices.
Where the conclusion applies
A one-time permanent change in the desired price, equal firm weights and the book's linear average rather than the exact model index.
Check your understanding: At theta = 0.5, what is P2?
Chapter 54 source: section "Calvo staggered pricing".
Demonstration 4 of 4
Tight money now, more printing later
If seigniorage is cut today while the deficit continues, how much is needed tomorrow?
Debt carries interest. Seigniorage withheld today becomes debt that must be serviced, so with a fixed deficit and a debt ceiling the monetary authority ends up printing more later.
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B is real public debt (billion dollars), r the real rate, D = 5 the primary deficit and S real seigniorage. The baseline plan uses 10 of seigniorage each year; investors cap debt at 130.
Predict first. Does postponing 10 billion of seigniorage cost exactly 10 billion later?
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Constructed example: the chapter's hypothetical budget (debt 100, rate 10 percent, deficit 5, seigniorage 10, ceiling 130, tight case of 0); rates of 5 and 15 percent and year-1 seigniorage of 5 are added for comparison.
Calculated values
- B1
- 115.00
- Year-2 need
- 131.50
- Minimum seigniorage for the ceiling
- 1.50
- Seigniorage to return to baseline
- 21.00
- Above baseline year-2 seigniorage
- 11.00
B1 = 1.10(100) + 5 - 0 = 115.00; Year-2 need = 1.10(115.00) + 5 = 126.50 + 5 = 131.50; Seigniorage to return to baseline = 131.50 - 110.50 = 21.00. Postponing 10 of seigniorage costs 11.00 in year 2: the 10 plus 10 percent interest, 10 x 1.10 = 11.00.
Worked steps
- B1 = 1.10(100) + 5 - 0 = 115.00
- Year-2 need = 1.10(115.00) + 5 = 126.50 + 5 = 131.50
- Baseline B2 = 1.10(105.00) + 5 - 10 = 110.50
- Seigniorage to return to baseline = 131.50 - 110.50 = 21.00
- Minimum for the ceiling = 131.50 - 130 = 1.50
Use the idea
Judge a monetary tightening together with the fiscal path; without future surpluses it may only delay and enlarge money creation.
Where the conclusion applies
A fixed primary deficit, no growth, a hard debt ceiling and a real rate above growth. A future primary surplus would change the conclusion.
Check your understanding: Why is the year-2 requirement 11 billion above baseline?
Chapter 54 source: section "Unpleasant monetarist arithmetic".