In 2022–23 the real interest rate rose by 3.4 points, and unemployment did not budge

3.4 points The rise in a market real interest rate from December 2021 to October 2023; unemployment was 3.9% in both months

Textbooks say that higher interest rates cool the economy. Why did this rise not cause a slump? And in 2009, when Congress spent $797 billion to fight one, why did unemployment end up higher than the forecasts made without any stimulus?

Open with the puzzle. The numbers are from the notebook’s practice problem 3: the 10-year inflation-indexed Treasury yield (FRED series DFII10), a market measure of the real interest rate, averaged about −1.0 percent in December 2021 and about 2.4 percent in October 2023. Between those months the Federal Reserve raised its target range for the federal funds rate from 0–0.25 percent (March 2022) to 5.25–5.50 percent (July 2023). The U.S. unemployment rate was 3.9 percent in both months.

Do not answer the puzzle now. By the end of Act II the room will be able to compute what the simplest IS curve predicts for this rise (a slump of almost 7 percent of output), and the Devil’s-advocate slide comes back to it.

The second question, the 2009 stimulus, is the subject of Act III and of Fig. 6.

Ask the room: when the Fed raises rates, who in your family or your firm changes plans? A car loan, a mortgage, a new machine? That is the channel this chapter models.

Today: three questions

• How does the interest rate reach output? Investment, the six equations of demand, and the IS curve.
• What moves the economy along the curve, and what shifts it? Rate changes, demand shocks, a new technology and the multiplier.
• Why is consumption so smooth, and what can fiscal policy do? The life cycle, permanent income, the 2009 stimulus and three policy questions.

These are the three acts of the deck and the three branches of the mind map on the title slide.

The headline result: a real interest rate above the return on capital cuts investment and pulls output below potential. Demand shocks shift the curve, the multiplier amplifies them, and how spending is paid for decides how far fiscal policy moves it.

Where we are in the book: Chapter 9 measured the short run (the output gap, the Phillips curve, Okun’s law). This chapter builds the first block of the short-run model. Chapter 12 adds how the central bank sets the real interest rate (the MP curve) and puts the pieces together.

Building the IS curve

Act I

Rates, investment and output

Why should an interest rate chosen in Washington change how much the whole country produces?

The real interest rate rises from 5% to 7% and nothing else changes. Write down what happens to investment and to output, and the chain of decisions that links them.

Ask the room first; collect two or three chains before moving on. Typical answers: “loans get dearer, firms buy fewer machines”; “mortgages cost more, fewer houses get built”; “people save more”. All are on the right track. The first two are investment, which is the channel of this chapter. The third is a channel through consumption that the textbook’s exercise 10 adds as an extension.

Ask for a direction, not a number: the size depends on a parameter we have not met yet.

A higher real interest rate means less investment and less output

Model figure: the IS curve with \(\bar{a} = 0\), \(\bar{b} = 2\), \(\bar{r} = 5\%\) and an investment share of 17% at \(R = \bar{r}\). Adapted from Fig. 11.1 in Jones (2024), Macroeconomics, 6th ed.

How to read it: short-run output Ỹ (percent above or below potential) across, the real interest rate R up. The line is the IS curve: each point pairs an interest rate with the output it produces. The slider picks the interest rate; the dot moves to it, and the title traces the chain from R to investment (as a share of potential output Ȳ) to output.

Do live:

1. Start at R = 5 percent: investment is 17 percent of potential output, and Ỹ = 0.
2. Move to 7: investment 13 percent, output 4 percent below potential. This answers the prediction.
3. Move down to 3: investment 21 percent, output 4 percent above potential.

Point at the convention: the equation makes output a function of the interest rate, yet output sits on the horizontal axis. Economists put the “price” (here, the price of borrowing) on the vertical axis, and in Chapter 12 the interest rate becomes something the model determines.

Each point on the interest rate costs 2 points of investment, and so of output

• From 5% to 7%: investment falls from 17% to 13% of potential output, and output to −4%.
• From 5% to 3%: investment rises to 21%, and output to +4%.
• The chain runs through firms’ machines and factories, and through households’ new houses: both count as investment.

Higher rates make borrowing dearer, so fewer projects pay off, spending falls and firms produce less.

Numbers: the slider titles of Fig. 1 (R = 2 to 8 percent: investment 23 to 11 percent of potential, output +6 to −6 percent).

The prediction: investment falls and output falls. Some students say “output falls because firms produce less with less capital”. That is a long-run effect. In the short run, the effect is through demand: fewer orders for machines and buildings, so the firms that make them produce less.

The slope here is 2: one percentage point on the interest rate moves investment by 2 points of potential output, and output by the same 2 points. The next slides show where that number comes from.

Six equations: one identity and one rule for each use of output

\[Y_t = C_t + I_t + G_t + EX_t - IM_t\]

\[C_t = \bar{a}_c\bar{Y}_t \qquad G_t = \bar{a}_g\bar{Y}_t \qquad EX_t = \bar{a}_{ex}\bar{Y}_t \qquad IM_t = \bar{a}_{im}\bar{Y}_t\]

\[\frac{I_t}{\bar{Y}_t} = \bar{a}_i - \bar{b}\,(R_t - \bar{r})\]

Four uses of output follow potential output; only investment reacts to the interest rate.

Six endogenous variables (Y, C, I, G, EX, IM) and six equations. A bar marks something the short-run model takes as given: the parameters, potential output Ȳ and the marginal product of capital r̄, both handed over by the long-run model (Chapters 5 and 6). R has no bar because Chapter 12 will make it endogenous; here we simply take it as given.

Why tie consumption to potential output rather than to actual output? Potential output is far smoother than actual output. In a recession that people expect to be temporary, Y falls but Ȳ hardly moves, and neither does consumption: households keep spending, for instance by running down savings. Act III gives the reason: the life-cycle and permanent-income theories.

The book’s baseline shares, in percent of potential output: consumption 65, investment 17, government purchases 20, exports 12, imports 14. As rough U.S. benchmarks, consumption is close to two thirds of GDP and government purchases close to a fifth.

A firm invests more when a machine earns more than a bond

\[\frac{I_t}{\bar{Y}_t} = \bar{a}_i - \bar{b}\,(R_t - \bar{r})\]

• \(\bar{r}\), the marginal product of capital: what a new machine yields. \(R_t\): what a bond pays, or a loan costs.
• \(\bar{r} = 5\%\), \(R_t = 3\%\): borrow 100, earn 5, pay 3, keep 2. Firms invest more than \(\bar{a}_i\).
• \(\bar{b}\): points of investment per point of gap.

In the long run \(R = \bar{r}\); in the short run a central bank can open a gap within a day.

The worked example is the notebook’s (r̄ = 5, R = 3); the textbook uses 15 and 10 percent.

Why can the two rates differ? In the long run firms keep adding capital until its marginal product falls to the interest rate (the Solow model of Chapter 5). But building factories takes time. A central bank can move R overnight, while r̄ changes only as new capital is built and put to work.

r̄ carries no time subscript: on a balanced growth path the return on capital does not change. Ȳ keeps its subscript because potential output grows.

Ask: a firm sitting on cash does not need to borrow. Does the interest rate still matter for it? (Yes: the cash could be lent at R instead. R is the opportunity cost of using one’s own funds.)

What matters for investment is the gap, not the level of the rate

A high interest rate always holds back investment.

Only the gap \(R - \bar{r}\) enters. At \(R = 10\%\), firms invest more than usual if new capital earns 15%; at \(R = 2\%\), they invest less than usual if it earns 1%.

This is a common mistake when students compare countries or decades: “rates were high in the 1990s, so investment must have been weak”. The model says to compare the interest rate with the return on new capital.

The 10 and 15 percent example is the textbook’s own illustration of the investment equation; the 2 and 1 percent case is its mirror image.

The policy version: a central bank that wants to know whether its rate is tight or loose has to estimate r̄, which cannot be observed. Central bankers call the rate that keeps output at potential r-star (Act III).

Divide by potential output, and six equations collapse into one line

\[\frac{Y_t}{\bar{Y}_t} = \bar{a}_c + \big[\bar{a}_i - \bar{b}\,(R_t - \bar{r})\big] + \bar{a}_g + \bar{a}_{ex} - \bar{a}_{im}\]

\[\widetilde{Y}\!_t \equiv \frac{Y_t - \bar{Y}_t}{\bar{Y}_t} = \frac{Y_t}{\bar{Y}_t} - 1\]

\[\widetilde{Y}\!_t = \bar{a} - \bar{b}\,(R_t - \bar{r}), \qquad \bar{a} \equiv \bar{a}_c + \bar{a}_i + \bar{a}_g + \bar{a}_{ex} - \bar{a}_{im} - 1\]

The IS curve: output depends on demand shocks and on the gap between the two rates.

The full derivation is in the book chapter’s Model overview, steps 1 to 4. In words:

1. Divide the identity by Ȳ, so every use of output becomes a share.
2. Substitute the five demand equations. Five of the six unknowns drop out.
3. Short-run output Ỹ, from Chapter 9, is the percent gap between actual and potential output, which equals Y/Ȳ − 1.
4. Subtract 1 from both sides and give the constant terms one name, ā.

ā is the aggregate demand shock: it packs the five demand shares into one number that says how far total spending at normal interest rates sits from potential output.

All quantities are in percent, so Ỹ = −2 means output 2 percent below potential.

The demand shock is zero in the long run, and the curve gives output in one step

Share of \(\bar{Y}\), %	Baseline
consumption \(\bar{a}_c\)	65
investment \(\bar{a}_i\)	17
government \(\bar{a}_g\)	20
exports \(\bar{a}_{ex}\)	12
imports \(\bar{a}_{im}\)	−14
total	100

\(R\)	\(I/\bar{Y}\)	\(Y/\bar{Y}\)	\(\widetilde{Y}\)
4%	19%	102%	+2%
5%	17%	100%	0%
7%	13%	96%	−4%

In the long run \(\widetilde{Y} = 0\) and \(R = \bar{r}\), so \(\bar{a} = 0\): the shares add up to exactly 100%.

Numbers: the notebook’s check 1. For each interest rate it computes every use of output from its own equation, adds them with the identity, and compares with the IS curve. Both give +2, 0 and −4 percent.

Step 5 of the Model overview: put Ỹ = 0 and R = r̄ into the IS curve, and ā = 0. Read through its definition, that means the five shares sum to one at potential: 65 + 17 + 20 + 12 − 14 = 100. So ā = 0 is the normal case, and shocks push it away from zero for a while.

The demand shares are illustrative; the textbook’s own diagrams use ā = 0, b̄ = 2 and r̄ = 5 percent, which is our baseline.

The IS curve’s three parameters

What sets the slope of the IS curve, and what sets where it sits?

Investment becomes twice as sensitive to the interest rate: \(\bar{b}\) rises from 2 to 4. Does the curve get steeper or flatter? Through which point does it still pass?

Ask the room to sketch the answer on paper. Many students expect “more sensitive” to mean “steeper”. Ask them to think in terms of the graph’s axes: the interest rate is up, output is across. If output reacts more to each point of R, the line covers more horizontal distance for each vertical step.

The sensitivity of investment turns the curve around one point

Model figure: \(\widetilde{Y} = \bar{a} - \bar{b}(R - \bar{r})\) with \(\bar{a} = 0\) and \(\bar{r} = 5\%\). Adapted from Fig. 11.2 in Jones (2024), Macroeconomics, 6th ed.

How to read it: the same axes as Fig. 1. The black dot is the point where R = r̄: there the curve gives Ỹ = ā, here 0. The slider changes b̄, the sensitivity of investment. When b̄ leaves its baseline of 2, the baseline curve stays behind as a dashed gray line and the new curve is labelled IS′. The title writes out the equation.

Do live:

1. Start at b̄ = 2: slope −1/b̄ = −0.5 in the diagram.
2. Move to 4: the line turns around the dot and becomes flatter. This answers the prediction.
3. Move to 0.5: a steep line. Output hardly reacts to the interest rate.

Point at the dot: whatever b̄ is, the curve passes through (ā, r̄).

A larger \(\bar{b}\) flattens the curve; \(\bar{a}\) and \(\bar{r}\) move it

• In the diagram \(R = \bar{r} + (\bar{a} - \widetilde{Y})/\bar{b}\): a line with slope \(-1/\bar{b}\) through the point \((\bar{a}, \bar{r})\).
• With \(\bar{b} = 4\), a rise of \(R\) from 5% to 6% costs 4 points of output instead of 2: the curve is flatter.
• \(\bar{a}\) moves the whole curve left or right, one for one. \(\bar{r}\) moves it up or down.

The slope measures how strongly spending reacts to the interest rate; the position measures demand.

Numbers: the slider titles of Fig. 2 and the notebook’s check 3 (slope dR/dỸ = −1/b = −0.50 at the baseline).

The prediction: flatter, through the same point (0, 5). The common error is to read “more sensitive” as “steeper”. In this diagram a flat curve means a powerful interest rate: a small move in R takes output a long way.

Why it matters for policy: a central bank facing a flat IS curve can steer output with small rate changes, but a mistake about the rate is also costly. Act III returns to this with the problem of not knowing b̄.

“IS” stands for investment equals saving

\[\underset{\text{private saving}}{(Y_t - T_t - C_t)} + \underset{\text{government saving}}{(T_t - G_t)} + \underset{\text{foreign saving}}{(IM_t - EX_t)} = I_t\]

• Rearrange the identity and add and subtract taxes \(T_t\): the three kinds of saving must fund investment.
• At potential, with taxes of 18% of \(\bar{Y}\): private saving 17, government saving −2 (a deficit), foreign saving +2. Total 17, the investment share.
• John Hicks introduced the curve in 1937 as an algebraic reading of Keynes’s General Theory.

Foreigners save in our economy when they sell us more than they buy, and lend us the difference.

Numbers: the notebook’s check 2. Private saving 100 − 18 − 65 = 17; government saving 18 − 20 = −2; foreign saving 14 − 12 = +2. The sum, 17, equals the investment share at R = r̄.

The equation is an identity, not a theory. It holds in every period. What the IS curve adds is behavior: how investment responds to the interest rate.

For the policy students, a preview of Chapter 18: if the government deficit grows while private saving and investment stay put, foreign saving must rise one for one, and the trade deficit widens. That is the “twin deficits” link of the notebook’s practice problem 1 (U.S. 2024: investment 21.5 percent of GDP, national saving 16.6 percent, so foreign saving 4.9 percent).

Moving along or shifting

Act II

Experiment: the real interest rate rises

The economy starts at potential, with \(R = \bar{r} = 5\%\), \(\bar{a} = 0\) and \(\bar{b} = 2\). The real interest rate rises to \(R' = 6\%\). Where does output end up? Does the IS curve move?

Ask for a number and for a sketch. The number should come quickly from the equation: Ỹ = 0 − 2 × (6 − 5) = −2.

The second question is the one students get wrong: many draw a new curve shifted to the left. Hold that thought; the figure settles it.

A higher rate moves the economy up along an unchanged curve

Model figure: the IS curve with \(\bar{a} = 0\), \(\bar{b} = 2\), \(\bar{r} = 5\%\); the slider sets \(R'\). Adapted from Fig. 11.3 in Jones (2024), Macroeconomics, 6th ed.

How to read it: point A is the starting point, R = r̄ = 5 percent and Ỹ = 0. Point B is the economy at the new rate R′ chosen on the slider, with dashed guides to the axes. The title lists the parameters and the move of output from A to B.

Do live:

1. Start at R′ = 6: B sits at Ỹ = −2. The curve has not moved.
2. Move to 7, then 8: −4 and −6 percent.
3. Move back to 5: A and B coincide.

This is the textbook’s own example: ā = 0, b̄ = 2, r̄ = 5 and R′ = 6.

Each point on the rate costs 2 points of output, and nothing shifts

• At \(R' = 6\%\), output falls to −2%; at 7%, to −4%. Only the point moves.
• The IS curve is the list of outputs that go with each interest rate. A new rate is read off the same curve: a movement along it.
• With \(\bar{b} = 3\) the same rise costs 3 points: a flatter curve, a larger fall.

A change in \(R\) never shifts the IS curve; it picks a different point on it.

Numbers: the notebook’s “Key numbers: the textbook’s example”: R = 5, 6, 7 percent gives Ỹ = 0, −2, −4 percent, and with b̄ = 3: 0, −3, −6.

The prediction: −2 percent, and no shift. The intuition is familiar by now: dearer borrowing, less investment, output below potential.

Ask: in Fig. 1 we called this the basic story of the chapter. What is new here? (Only the language: “movement along” versus “shift”. It matters because every later diagram depends on it.)

Experiment: firms become optimistic

Better information technology makes firms optimistic: they raise investment by 2 points of potential output at every interest rate (\(\bar{a}_i\) up by 2). The rate stays at \(\bar{r} = 5\%\). Shift or movement along? Where does output go?

Ask the room to commit: shift or move along, left or right, and by how much. Then ask how the answer would change if the shock came from consumers or from foreigners buying more of our exports.

An aggregate demand shock shifts the whole curve

Model figure: the IS curve with \(\bar{b} = 2\), \(\bar{r} = 5\%\), \(R = \bar{r}\); the slider sets the new demand shock \(\bar{a}'\). Adapted from Fig. 11.4 in Jones (2024), Macroeconomics, 6th ed.

How to read it: the dashed gray line is the old curve IS; the solid line is the new curve IS′ after the demand shock ā′ chosen on the slider. A is the starting point and B the new one, both at R = r̄.

Do live:

1. Start at ā′ = 2: IS′ lies 2 points to the right of IS at every interest rate, and B sits at Ỹ = 2.
2. Move to 4: Ỹ = 4.
3. Move to −2: the curve shifts in, and output falls 2 percent below potential.

The textbook’s figure gives no number for ā′; the book starts at 2, the shock of the textbook’s multiplier example.

Demand shocks pass into output one for one at a given rate

• \(\bar{a}' = 2\): at \(R = \bar{r}\), output rises from 0 to +2%. At every other rate it is also 2 points higher.
• Any demand share works the same way: an investment boom (\(\bar{a}_i\)), confident consumers (\(\bar{a}_c\)), more government purchases (\(\bar{a}_g\)), a foreign boom (\(\bar{a}_{ex}\)).
• A recession abroad lowers \(\bar{a}_{ex}\) and shifts our curve in: that is how recessions travel between countries.

A demand shock changes the output that goes with every interest rate, so the whole curve moves.

Numbers: the slider titles of Fig. 4 and check 3 of the notebook (a 1-point rise in ā_c raises Ỹ by +1.0; a 1-point rise in ā_im lowers it by −1.0).

The prediction: a shift to the right by 2 points; output +2 percent.

Imports enter ā with a minus sign: if consumers switch from home-made to imported goods, the IS curve shifts in. Exercise 3 of the textbook has a fun version: a sudden taste for goods from New Zealand.

Why “at a given rate” matters: in Chapter 12 the central bank will respond to the shock by moving R, and then output need not move one for one.

Only the interest rate moves the economy along the curve

A rise of 1 point in …	effect on \(\widetilde{Y}\)	baseline	the IS curve
the real interest rate \(R\)	\(-\bar{b}\)	−2	movement along (up)
a demand share \(\bar{a}_c\), \(\bar{a}_i\), \(\bar{a}_g\), \(\bar{a}_{ex}\)	\(+1\)	+1	shifts right by 1
imports \(\bar{a}_{im}\)	\(-1\)	−1	shifts left by 1
the marginal product of capital \(\bar{r}\)	\(+\bar{b}\)	+2	shifts up by 1
the sensitivity \(\bar{b}\)	\(-(R - \bar{r})\)	0 at \(R = \bar{r}\)	rotates, flatter
potential output \(\bar{Y}\)	\(0\)	0	nothing

Ask what the graph plots: a change in \(R\) is read off the curve; a change in anything else redraws it.

Numbers: the notebook’s check 3 (“R up 1 point: −2.0; r_bar up 1 point: +2.0; a_c up 1 point: +1.0; a_im up 1 point: −1.0; b doubled, at R = r_bar: +0.0”).

The IS curve is linear, so each effect is a constant: the derivative of Ỹ with respect to that input. Each number reads “percent of potential output per percentage point”.

Two rows deserve a pause. A rise in r̄ shifts the curve up by 1 point, which is a shift to the right by b̄ points (Fig. 9). And potential output is not in the equation at all; a misconception slide later in this act comes back to it.

A rate hike does not shift the IS curve

When the Fed raises interest rates, the IS curve shifts to the left.

The curve already contains every interest rate. A hike moves the economy up along it, from \(A\) to \(B\). The curve shifts only when \(\bar{a}\) or \(\bar{r}\) changes, or rotates when \(\bar{b}\) does.

This is the single most common error in exams on this chapter. Its source is the habit, from supply and demand, of shifting a curve whenever “something happens”.

The test to teach: ask which variable changed, and whether it is on one of the axes. R is on the vertical axis, so a change in R is a point on the curve. ā, r̄ and b̄ are not on the axes, so changes in them redraw the curve.

A related trap for Chapter 12: there the central bank’s rate becomes the MP curve, and a rate hike shifts the MP curve, not the IS curve.

Experiment: the Fed cuts the rate

The Fed pushes the real interest rate below the return on capital, from 5% to 4%. Where does output go? And if demand had just fallen by 2 points (\(\bar{a}' = -2\)), what rate would bring output back to potential?

This is Exercise 2(a) of the textbook. Ask for both answers. The first is the mirror image of Fig. 3. The second needs the equation: 0 = −2 − 2 × (R′ − 5), so R′ = 4.

A cut moves the economy down the curve, above potential

Model figure: the IS curve with \(\bar{a} = 0\), \(\bar{b} = 2\), \(\bar{r} = 5\%\); the slider sets \(R'\) below \(\bar{r}\). Adapted from Fig. 11.7 in Jones (2024), the answer to its Exercise 2(a), Macroeconomics, 6th ed.

How to read it: the same diagram as Fig. 3, with the slider now running from 5 down to 2 percent. A is the start, B the economy at the new rate R′.

Do live:

1. Start at R′ = 4: B sits at Ỹ = +2.
2. Move to 2: Ỹ = +6. A cut of 3 points takes output 6 percent above potential.
3. Back to 5: no change.

The textbook gives no number for R′; the book starts at 4.

A one-point cut adds 2% to output, enough to offset a 2-point fall in demand

• At \(R' = 4\%\) output rises to +2%: firms can borrow below what new capital earns, so they invest more.
• Again a movement along: nothing shifts.
• If demand had fallen to \(\bar{a}' = -2\), the same cut to 4% would bring output back to 0: \(\;0 = -2 - 2\,(4 - 5)\).

A rate cut can offset a fall in demand. That is how monetary policy fights a recession.

Numbers: the slider titles of Fig. 7 and the notebook’s “What to notice” (combine the cut with ā′ = −2 and output is back at potential).

The general rule, from the IS curve: to keep Ỹ = 0 after a demand shock ā′, set R′ = r̄ + ā′/b̄. With ā′ = −2 and b̄ = 2, R′ = 5 − 1 = 4.

Chapter 12 builds on exactly this: the central bank chooses R to steer output. Chapter 14 asks what happens when the needed R would be below what the central bank can reach (the zero lower bound).

Experiment: consumers turn pessimistic

Consumers grow gloomy about the economy and cut their spending by 2 points of potential output (\(\bar{a}_c\) down by 2). The interest rate stays at 5%. Which way does the curve move, and where does output end up?

Exercise 2(b) of the textbook. Most students now get the direction right. Ask the follow-up: is this recession caused by a fall in what the economy can produce, or by a fall in what people want to buy? (Demand: potential output has not changed.)

Pessimistic consumers shift the curve in

Model figure: the IS curve with \(\bar{b} = 2\), \(\bar{r} = 5\%\), \(R = \bar{r}\); the slider sets the demand shock \(\bar{a}'\) from 0 down to −4. Adapted from Fig. 11.8 in Jones (2024), the answer to its Exercise 2(b), Macroeconomics, 6th ed.

How to read it: the dashed gray line is the old curve, the solid IS′ the new one after the fall in demand. A and B are at the same interest rate, R = r̄.

Do live:

1. Start at ā′ = −2: IS′ lies 2 points to the left; B at Ỹ = −2.
2. Move to −4: Ỹ = −4.
3. Back to 0: the two curves coincide.

A recession can start from demand alone

• A 2-point fall in the consumption share lowers \(\bar{a}\) to −2 and output to −2%, at an unchanged interest rate.
• Nothing has happened to the economy’s capacity: potential output is the same. Spending fell.
• A cut in government purchases or a rise in imports would shift the curve in the same way.

With the rate held still, every point of lost demand is a point of lost output.

Numbers: the notebook’s check after Fig. 8 (“a = −2.0: at R = r_bar, Y~ = −2.0%”) and the slider titles.

Connect to Chapter 10: the Great Recession began with a collapse of housing investment and of consumer confidence, a fall in ā_i and ā_c. The Covid-19 shock was a huge but short fall in spending on services. In the language of this chapter, both were large inward shifts of the IS curve, of very different lengths.

Ask: what could policy do here? (Cut R, as in Fig. 7; or raise ā_g, as in Act III.)

Experiment: a better technology

New information technology raises the return on new capital, \(\bar{r}\), from 5% to 6%. The real interest rate stays at 5%. What happens to the IS curve and to output?

Exercise 2(c) of the textbook. The tricky part: r̄ is inside the bracket of the IS curve. Students who answer “the curve shifts right by 1” have treated r̄ like ā. Ask them to plug in: Ỹ = 0 − 2 × (5 − 6).

A higher return on capital shifts the curve up

Model figure: the IS curve with \(\bar{a} = 0\), \(\bar{b} = 2\), \(R = 5\%\); the slider sets the new marginal product of capital \(\bar{r}'\). Adapted from Fig. 11.9 in Jones (2024), the answer to its Exercise 2(c), Macroeconomics, 6th ed.

How to read it: the new curve IS′ crosses Ỹ = 0 at the new, higher r̄′, marked by a dotted line, so it lies above the old curve. The interest rate has not changed, so B is at R = 5 on the new curve.

Do live:

1. Start at r̄′ = 6: B at Ỹ = +2.
2. Move to 7: Ỹ = +4.
3. Back to 5: no change.

Point out the vertical distance between the curves (1 point of interest rate) and the horizontal one (b̄ = 2 points of output).

One more point on the return to capital raises output by \(\bar{b} = 2\) points

• \(\widetilde{Y}' = \bar{b}\,(\bar{r}' - \bar{r}) = 2 \times (6 - 5) =\) +2%: capital now earns more than it costs, so firms borrow and invest.
• It looks like the demand shock of Fig. 4, but its size depends on \(\bar{b}\): with \(\bar{b} = 3\), the same rise gives +3%.
• An earthquake can do the same: it destroys capital, lowers actual and potential output alike, and the capital that is left earns more.

A new technology moves the IS curve through investment, not through potential output.

Numbers: the slider titles of Fig. 9 and the notebook’s “What to notice”.

The earthquake example is the textbook’s exercise 6: a large earthquake on the West Coast destroys houses and buildings. Actual and potential output fall together, so Ỹ does not change on that account; but rebuilding raises the marginal product of capital, investment jumps, and short-run output can rise above potential.

For the policy students: the rate that keeps output at potential is r̄ itself. If r̄ rises and the central bank keeps R where it was, policy has become looser without anyone deciding it. Central bankers call that rate r-star (Act III).

Potential output does not appear in the IS curve

A new technology raises potential output, so it raises short-run output too.

\(\bar{Y}\) is not in \(\widetilde{Y} = \bar{a} - \bar{b}(R - \bar{r})\). A technology that lifts potential output lifts actual output in the same proportion, so the gap is unchanged. Only if it also raises \(\bar{r}\) does the curve shift.

Students confuse the level of output with the gap. Short-run output is a percentage gap, so anything that moves Y and Ȳ together leaves it alone.

The notebook’s check 3 includes this row: a change in potential output changes Ỹ by 0.

The same reasoning explains why the four non-investment demand equations are written as shares of Ȳ: as potential output grows, consumption, government purchases and trade grow with it, and the gap is unaffected.

When consumption follows income, every shock is multiplied

\[\frac{C_t}{\bar{Y}_t} = \bar{a}_c + \bar{x}\,\widetilde{Y}\!_t \quad\Longrightarrow\quad \widetilde{Y}\!_t = \frac{1}{1 - \bar{x}}\,\big[\bar{a} - \bar{b}\,(R_t - \bar{r})\big]\]

• With \(\bar{x} = 1/3\), the multiplier \(1/(1-\bar{x})\) is 1.5.
• A demand shock of 2% raises output by 3%, not 2%.
• A rate rise that would cost 1% of output costs 1.5%.

Lost spending lowers incomes, which lowers spending again.

Numbers: the notebook’s “Key numbers: the multiplier” (x = 1/3: multiplier 1.50; shock +2: 2.0 percent without, 3.0 percent with; R up 0.5 point: −1.0 without, −1.5 with). These are the textbook’s numbers.

The derivation, in words: redo the IS curve with the new consumption share. Ỹ appears on both sides; collect it on the left and divide by 1 − x̄. Full steps in the book chapter’s Model overview.

x̄ stands for households that spend part of a temporary rise in income instead of smoothing it away. Act III shows why such households exist, and the stimulus-check evidence measures how many there are.

A second route to a multiplier: cash flow. If firms invest more when the economy booms because they have more internal funds, investment also rises with Ỹ, and the multiplier becomes 1/(1 − x̄ − z̄), in the book chapter’s own notation.

Round by round, the spending chain adds up to 1.5 times the first push

After round	\(\widetilde{Y}\)
1	−2.0000%
2	−2.6667%
3	−2.8889%
30	−3.0000%

• \(R\) rises by 1 point: investment falls by 2, a first round of −2%.
• Each later round is \(\bar{x} = 1/3\) of the one before: \(-2\,(1 + \bar{x} + \bar{x}^2 + \cdots)\).
• The series sums to \(1/(1-\bar{x})\) only because \(\bar{x} < 1\).

The two routes agree: solving the model and adding up the rounds both give −3%.

Numbers: the notebook’s check 4 (“After round 1: −2.0000%; 2: −2.6667%; 3: −2.8889%; 30: −3.0000%; Equation (5): −3.0000%, multiplier 1.50”).

The geometric series: call the bracket S. Then x̄S is the same series shifted one place, so S − x̄S = 1 and S = 1/(1 − x̄). If every lost dollar of income cut spending by a dollar or more, the rounds would never die out, which is why the model needs x̄ < 1.

Ask: what would make x̄ smaller in a real economy? (Imports: part of each extra dollar is spent abroad. Taxes and unemployment benefits that cushion income. A central bank that leans against the cycle.) These are the reasons real multipliers are smaller than this simple one.

The multiplier makes the curve flatter but keeps its form

Effect	IS curve	with the multiplier	baseline, \(\bar{x} = 1/3\)
a 1-point demand shock \(\bar{a}\)	\(+1\)	\(+1/(1-\bar{x})\)	+1 → +1.5
a 1-point rise in \(R\)	\(-\bar{b}\)	\(-\bar{b}/(1-\bar{x})\)	−2 → −3
a 1-point rise in \(\bar{r}\)	\(+\bar{b}\)	\(+\bar{b}/(1-\bar{x})\)	+2 → +3
slope in the diagram, \(dR/d\widetilde{Y}\)	\(-1/\bar{b}\)	\(-(1-\bar{x})/\bar{b}\)	−0.5 → −0.33

Write \(\bar{a}^m = \bar{a}/(1-\bar{x})\) and \(\bar{b}^m = \bar{b}/(1-\bar{x})\), and it is the same IS curve with larger coefficients.

This is the multiplier table of the book chapter’s Model overview.

The last row answers the textbook’s exercise 8: the curve with the multiplier is flatter, because output moves more for the same change in R.

Why the rest of the book keeps writing Ỹ = ā − b̄(R − r̄): ā and b̄ can be read as already including any multiplier. That is a modelling convenience, not a claim that the multiplier is 1.

Bridge to Act III: the multiplier depends on how much households spend out of a temporary change in income. The permanent-income theory says: very little. The evidence says: more than the theory predicts.

Consumption and fiscal policy

Act III

The life cycle of income and consumption

Income is low at 18, peaks at 50 and falls at retirement. Why does consumption not follow it?

People now expect to live 10 years longer, to 90 instead of 80, with the same income at each age. Does the steady level of consumption they can afford rise or fall?

Ask the room to commit. A common intuition: “more years, more income, so more consumption”. The model says the opposite for an unchanged retirement age: the extra years come with low retirement income, but they must be paid for like any other year.

Warm-up question: would you rather have one coffee every morning, or a whole week’s worth on Saturday and none on the other days? That preference for smoothness is diminishing marginal utility.

Over a life, consumption stays flat while income rises and falls

Model figure: a stylized income profile and the constant consumption it pays for with no interest (average lifetime income). Adapted from Fig. 11.5 in Jones (2024), Macroeconomics, 6th ed.

How to read it: age across, income and consumption up, in illustrative units (only relative sizes matter). Income is low until 20, rises to a peak at 50, stays there until five years before retirement, and drops back at 65. The flat line is consumption. The shaded areas are the gaps: green where people save, blue where they borrow when young or live off savings when old. The slider sets the length of life T; the title gives consumption and the saving years.

Do live:

1. Start at T = 80: consumption 3.23 a year; saving from age 30 to 63.
2. Move to 90: consumption falls to 2.99. This answers the prediction.
3. Move to 76: consumption 3.35.

Consumption equals average lifetime income: 3.23 a year

• Income ranges from 1 to 7.5 units over the life; consumption is 3.23 in every year.
• Saved in the working years: 91.58. Borrowed when young plus spent when old: 91.58. The budget balances over the life.
• Live to 90 and consumption falls to 2.99: the same income must cover more years.

This is why the IS curve ties consumption to potential, not actual, output: a temporary slump barely moves lifetime income.

Numbers: the notebook’s check before Fig. 5 (“Consumption every year: 3.23; saved over the working years 91.58; borrowed when young + dissaved when old 91.58”) and the slider titles (T = 76: 3.35; 80: 3.23; 85: 3.10; 90: 2.99; 95: 2.88).

The theory: the life-cycle model of Franco Modigliani and the permanent-income hypothesis of Milton Friedman, both from the 1950s.

For the policy students (the notebook’s practice problem 7): with life rising from 80 to 88 years and retirement still at 65, consumption falls from 3.23 to 3.03. Raising the retirement age to 68 brings it back to 3.25. Working three of the eight extra years is enough, because each turns a year of low retirement income into a year of high income.

Permanent income is the steady stream with the same present value

\[\sum_{t=0}^{T-1}\frac{y^p}{(1+R)^t} = \sum_{t=0}^{T-1}\frac{y_t}{(1+R)^t} \quad\Longrightarrow\quad y^p = \frac{\sum_t y_t/(1+R)^t}{\sum_t 1/(1+R)^t}\]

• A household that wants smooth consumption spends \(c_t = y^p\) every period; borrowing and saving make the timing work.
• \(y^p\) is a weighted average of income, with less weight on far-off years. With \(R = 0\) it is plain average income: the life-cycle model of Fig. 5.

Consumption reacts to news about lifetime income, not to the income of this month.

Equation (6) of the book chapter’s Model overview; present values first appear in Chapter 7.

The notebook’s check 6 on a short income stream: permanent income 34.074 against an average income of 34.0; the present value of the stream and of a constant y^p are both 163.817.

What the evidence says: large, predictable changes in income are smoothed well; small surprises are partly spent. Hsieh (2003) is the textbook’s example: Alaskans smoothed the large, well-known Permanent Fund payments, but spent part of their federal tax refunds when they arrived.

A windfall raises consumption by its interest, every year

I just inherited $50,000, so I will spend $50,000 more this year.

With \(R = 2\%\), keeping the gift and spending the interest raises consumption by $1,000 a year, forever. The present value of that stream is exactly $50,000.

The algebra (step 4 of the permanent-income block): spend R × W more in every future period. The present value of that stream is R × W × (1/R) = W, so the plan uses the whole windfall without ever touching it.

Numbers: the notebook’s check 6 (“Windfall 50,000: consume 1,000 more per year; present value of that = 50,000”).

The textbook’s exercise 11 uses an interest rate of 10 percent: a 100,000-dollar bequest raises consumption by 10,000 a year. Ask: a lottery that pays 10 million five years from now: does consumption rise today? (Yes, by the interest on its present value.)

Taxing now or borrowing and taxing later costs the same in present value

\[\text{tax today: } \tau \qquad \text{or} \qquad \text{tax next year: } \frac{(1+R)\,\tau}{1+R} = \tau\]

• At \(R = 2\%\), a tax of $1,000 today and one of $1,020 next year both have a present value of $1,000.
• Same permanent income, same consumption: only the present value of taxes matters. This is Ricardian equivalence.
• So more government purchases paid for by future taxes lower \(\bar{a}_c\) and offset part of the rise in \(\bar{a}_g\).

How spending is financed decides how far it shifts the IS curve.

Numbers: the notebook’s check 6 (“Tax of 1,000 today vs 1,020 next year: present values 1,000 and 1,000”).

The government’s budget constraint, which the textbook calls the no-free-lunch principle: spending more today means spending less later or taxing more at some point.

Two cases from the book chapter. If households expect higher taxes to pay for new purchases, their permanent income falls and consumption falls, which offsets part of the outward shift. If the extra spending today is simply spending moved forward from the future (a road built now instead of in five years), there is no extra tax burden, consumption does not fall, and the IS curve shifts out by the full amount.

The idea is named after David Ricardo; its modern statement is Barro (1974). Whether it holds is an empirical question, and the stimulus-check evidence suggests that it holds only in part.

The 2009 forecasts

In February 2009 Congress passed a $797 billion fiscal stimulus, and its budget office forecast unemployment with and without it. What happened next?

Did actual unemployment come in below the forecasts with the stimulus, between the forecasts, or above even the forecast with no stimulus at all?

Background: the American Recovery and Reinvestment Act (ARRA), signed in February 2009, spent on highways, education, research and grants to state governments, and cut taxes. In the model, more government purchases raise ā_g and shift the IS curve out. How far out was hotly debated, so the Congressional Budget Office (CBO), Congress’s nonpartisan budget office, published three forecasts: no stimulus, a “low” estimate of its effect and a “high” estimate.

Ask for the commitment and for the reason. Then ask: if the answer is “above”, what are the two possible explanations?

Unemployment rose above even the no-stimulus forecast

Data: BLS, unemployment rate (UNRATE), via FRED; forecasts: Congressional Budget Office, letter of 11 February 2009 to Senator Judd Gregg on the effects of H.R. 1, Table 1. Chart inspired by Fig. 11.6 in Jones (2024), Macroeconomics, 6th ed.

How to read it: the U.S. unemployment rate, in percent. The open circles are the monthly data; the dashed lines are the three CBO forecasts, one per year for the fourth quarter. The dotted line at 4.8 percent is the CBO’s long-run unemployment rate, which all three forecasts reach by 2015. The buttons switch between the textbook’s window, 2009–2016, and all the data to 2026.

Do live:

1. “2009–2016”: the actual rate shoots above every forecast in 2009 and stays above the no-stimulus line for years.
2. “2009–2026”: after 2016 unemployment kept falling below 4.8 percent, until the Covid-19 shock of April 2020 took it to 14.8 percent.

Data note: October 2025 has no value; the household survey was not run during the federal government shutdown, and the book leaves the gap rather than invent a number.

For seven years, unemployment beat even the worst forecast

• Without the stimulus the CBO expected 9% in 2009; with it, 8.5% (low) or 7.7% (high) at the end of the year.
• Actual: 10.0% in October 2009, and 9.28% on average over 2009, the textbook’s “about 9.3”.
• In the fourth quarter of every year from 2009 to 2015 the actual rate was above the no-stimulus forecast, by 0.23 to 1.43 points. It reached 4.8% only in January 2016.

The stimulus came with worse outcomes than the no-stimulus forecast. That alone does not say it failed.

Numbers: the notebook’s “Key numbers: forecasts against outcomes” (2009 average 9.28; CBO no-stimulus 9.0; highest 10.0 in October 2009; first month at or below 4.8 percent January 2016) and its fourth-quarter table (actual minus no stimulus: 0.93, 0.80, 1.13, 1.40, 1.43, 0.70, 0.23 for 2009–2015; −0.03 in 2016; negative after).

To compare like with like, the book averages October to December of each year, because the CBO’s forecasts are fourth-quarter values.

The data alone cannot tell a weak stimulus from an optimistic forecast

Reading 1: the stimulus did little

• Spending worked much less well than the CBO assumed.
• Or households offset it, expecting future taxes.

Reading 2: the crisis was deeper

• In February 2009 no one knew how deep the financial crisis was.
• Without the stimulus, unemployment would have been higher still.

A test in the notebook: the shortfall grew while the stimulus faded, peaking in 2013. That timing fits reading 2 better.

The two readings are the textbook’s. The test is the notebook’s practice problem 5. Turn unemployment into output gaps with Okun’s law (Ỹ = −2(u − 4.8)). In the CBO’s high estimate the stimulus did the most at the end of 2010 (3.80 points of output) and had almost faded by 2013 (0.40). The shortfall of actual output against the no-stimulus path was largest in 2013 (−2.87 points), three years later. The correlation of the two columns over 2009–2015 is −0.08, close to zero.

If the stimulus had itself lowered output, the shortfall should have been largest in 2010 and shrunk afterwards. Instead it grew as the stimulus faded: more like private demand that stayed weak for longer than anyone expected, as in reading 2. The shortfall against the high-estimate path does move with the stimulus (correlation −0.87), which leaves room for a middle reading: the stimulus helped, but by less than the high estimate. Seven yearly numbers point one way without proving it.

How much output does a dollar of government spending buy?

The case for the 2009 stimulus rested on a multiplier above 1. Should a government count on one when it fights a recession with spending?

What the model says

• More purchases raise \(\bar{a}_g\); output rises by \(1/(1-\bar{x})\) per point.
• Purchases paid for by future taxes lower \(\bar{a}_c\): a smaller shift.
• The rate is held fixed.

What the evidence adds

• Estimates range from below 1 to about 1.5.
• Larger when interest rates are stuck at zero.
• Timing: laws pass late.

The research, in our words; these are contested estimates.

Nakamura and Steinsson (2014, American Economic Review) compare U.S. regions that received different amounts of military spending and estimate a local multiplier of about 1.5, the “one study” of the book chapter: each dollar of federal spending raises GDP by 1.50 dollars in the short run. A regional multiplier is not the national one, because a region shares one central bank and one tax base with the rest of the country.

Ramey (2019, Journal of Economic Perspectives) surveys a decade of research and concludes that most aggregate estimates of the spending multiplier lie between about 0.6 and 1. Ramey and Zubairy (2018, Journal of Political Economy) find, in a century of U.S. data, little evidence that multipliers are larger in times of slack.

Auerbach and Gorodnichenko (2012, American Economic Journal: Economic Policy) find larger multipliers in recessions than in expansions. Christiano, Eichenbaum and Rebelo (2011, Journal of Political Economy) show in a model that the multiplier can be large when the central bank’s rate is stuck at zero, because the bank does not raise R to offset the extra demand. The book chapter mentions this “zero lower bound” result; Chapter 14 studies it.

The model’s link: in this chapter the rate is held fixed, which is the case most favorable to fiscal policy. If the central bank raises R in response, the economy moves up the new IS curve and part of the effect is undone.

Timing: discretionary programs need drafting, a vote and signatures; the money may arrive after the recession ends. Automatic stabilizers (taxes that fall and benefits that rise in a slump) act without a vote. The notebook’s practice problem 10 uses Dolls, Fuest and Peichl (2012, Journal of Public Economics): stabilizers absorb 47 to 48 percent of the income lost in an unemployment shock in the EU (published and discussion-paper versions), against 34 percent in the U.S. (discussion paper).

Should a stimulus be sent as checks?

In 2008, 2020 and 2021 the U.S. mailed one-time payments to households. The theory says they would be mostly saved. Were they?

What the theory predicts

• A one-time check barely moves permanent income.
• So \(\bar{a}_c\) rises by little.
• With 40 years left, about 0.6% of it is spent in three months.

What households did (2008)

• 12–30% spent on nondurables within three months.
• 50–90% once cars and other durables count.
• Many cannot borrow: they spend cash when it comes.

The numbers: Parker, Souleles, Johnson and McClelland (2013, American Economic Review) use the random timing of the 2008 economic stimulus payments: households spent 12 to 30 percent of a payment on nondurable goods and services in the three months after it arrived, and 50 to 90 percent including durables. The notebook’s practice problem 2: with 40 years left, the life-cycle model predicts 1,000/40/4 = 6.25 dollars in three months, 0.6 percent of a 1,000-dollar payment; the lowest estimate is 19.2 times that.

Earlier evidence: Johnson, Parker and Souleles (2006, American Economic Review) on the 2001 tax rebates find similar spending of a sizable share within a few months. Kaplan and Violante (2014, Econometrica) explain it with “wealthy hand-to-mouth” households, who own houses or pensions but hold little cash.

For the IS curve (practice problem 2c): if households spend a share m of any temporary income, a payment of 1 percent of potential output raises output by m/(1 − m): 0.14 percent for m = 0.12, 0.43 for m = 0.30, 1.00 for m = 0.50.

Durables are a subtle case: a car bought with the check gives services for years, so consumption in the sense of services stays smooth, but the national accounts count the full purchase as spending this quarter, and that is what moves ā_c.

Ricardian equivalence (Barro, 1974, Journal of Political Economy) would predict no response at all to a debt-financed check. The evidence rejects the strict version, but the response is far from one for one. Contested points: how much of the 2020–21 checks was spent, and how much went to paying down debt.

How far should a central bank move when it does not know \(\bar{b}\)?

A boom has pushed demand to \(\bar{a}' = 3\). If \(\bar{b} = 2\), raising the rate from 5% to 6.5% brings output back to potential. But \(\bar{b}\) could be anywhere from 1 to 3.

At \(R' = 6.5\%\), if the true \(\bar{b}\) is…

• 1: the boom goes on, \(\widetilde{Y} = +1.5\%\).
• 2: output at potential.
• 3: a slump, \(\widetilde{Y} = -1.5\%\).

The cautious answer

• With \(\bar{b}\) equally likely to be 1, 2 or 3, the best of 6.0, 6.25 and 6.5 is 6.25%.
• A smaller hike than the certain case.

This is the notebook’s practice problem 8. Certainty: 0 = 3 − 2 × (R′ − 5), so R′ = 6.5. At R′ = 6.5 the gap is 3 − 1.5 b̄: +1.5, +0.75, 0, −0.75, −1.5 for b̄ = 1, 1.5, 2, 2.5, 3. Minimizing the average squared gap over b̄ ∈ {1, 2, 3}: the hike d = R′ − r̄ that minimizes it is 3 Σb̄ / Σb̄² = 18/14 = 1.29, so R′ = 6.29 percent. Of the three candidates, 6.25 is the best.

Why smaller: b̄ multiplies the hike, so a bigger hike also makes the outcome more uncertain. William Brainard (1967, American Economic Review) first showed this; Bernanke (2007, a speech on monetary policy under uncertainty) cites it as one reason central banks move gradually. The result is not universal: uncertainty about the persistence of a shock, or about where inflation is heading, can argue for moving faster.

The companion problem (practice problem 9): uncertainty about r̄ itself, which central bankers call r-star and which the New York Fed estimates (Laubach and Williams, 2003, Review of Economics and Statistics, and later updates). To keep output at potential the bank must set R = r̄, whatever b̄ is. A half-point mistake about r-star costs 0.5, 1.0 or 1.5 percent of output for b̄ = 1, 2, 3: most costly when the curve is flat.

“The rate hikes of 2022–23 prove the IS curve wrong”

A skeptic says: “The real rate rose 3.4 points. With \(\bar{b} = 2\) the IS curve predicts output 6.8% below potential, and by Okun’s law unemployment near 7.3%. It stayed at 3.9%. A curve that misses by that much is useless.”

Reply: the prediction assumed that only \(R\) changed. The curve itself names what else could have moved: demand (\(\bar{a}\) up, with pandemic savings and large deficits), the return on capital (\(\bar{r}\) up), or a smaller or slower \(\bar{b}\) (loans locked in at low fixed rates). Each calls for a different path of rates.

Numbers: the notebook’s practice problem 3. ΔR = 2.4 − (−1.0) = 3.4 points; along the IS curve ΔỸ = −2 × 3.4 = −6.8 percent. Okun’s law from Chapter 9, u − ū = −½ Ỹ, adds 3.4 points to unemployment: from 3.9 to about 7.3 percent. It stayed at 3.9 percent.

Take the objection seriously. Each explanation is a change in a parameter of the IS curve that worked against the higher R:

1. ā rose: households spent savings built up in 2020–21 (ā_c up), and the government ran large deficits (ā_g up; the U.S. general-government deficit was 7.8 percent of GDP in 2024). The hikes then cooled a boom instead of causing a slump.
2. r̄ rose: if new investment opportunities raised the return on capital, the gap R − r̄ grew less than R did.
3. b̄ was smaller for a while: many households and firms had locked in long-term loans at low fixed rates, so higher rates reached their spending slowly. Part of the effect may simply come later.

Why it matters for the Fed (problem 3d): if ā rose only for a while, rates must come down in time; if r̄ rose for good, rates have to stay higher for longer; if the effects are only delayed, the bank should wait before raising further. The Fed cannot observe ā, b̄ or r̄; it has to infer them as data come in.

The deeper point: a model that holds R fixed and changes one thing at a time is a tool for organizing such arguments, not a forecast. Chapter 12 makes R a choice of the central bank.

What to remember from this chapter

• The IS curve, \(\widetilde{Y} = \bar{a} - \bar{b}(R - \bar{r})\), comes from the national income identity and an investment rule. It slopes down: a higher rate means less investment and less output.
• A change in \(R\) is a movement along the curve; a demand shock \(\bar{a}\) or a new return on capital \(\bar{r}\) shifts it. Potential output does not enter.
• “IS”: private, government and foreign saving together fund investment.
• When consumption follows current income, the multiplier \(1/(1-\bar{x})\) amplifies every shock; with \(\bar{x} = 1/3\) it is 1.5.
• Consumption tracks permanent income, so only the present value of taxes matters; in 2009 unemployment stayed above even the no-stimulus forecast, and the size of fiscal effects is still debated.

Ask each student to write one sentence answering the opening question: how does the real interest rate move output in the short run? A good answer: when the real interest rate rises above the return on new capital, firms invest less, demand falls and output drops below potential, by b̄ points for each point of gap, more with a multiplier.

For the policy students, three habits: ask whether a change moves the economy along the curve or shifts it; compare the interest rate with the return on capital, not with its own past; and ask how any new spending will be paid for.

Walk the three branches of the mind map in about a minute each: building the IS curve (cyan), moving along or shifting (lavender), consumption and fiscal policy (green). Click the image to zoom in.

A real rate above the return on capital pulls output below potential; demand shocks shift that link

Next: Chapter 12, Monetary Policy and the Phillips Curve

The thesis of the chapter: a real interest rate above the return on capital cuts investment and pulls output below potential. Demand shocks shift the IS curve, the multiplier amplifies them, and how spending is financed decides how far fiscal policy moves it.

Chapter 12 explains how the central bank sets the real interest rate, the MP curve, and joins it with the IS curve and the Phillips curve to complete the short-run model.

Sources: figures and models

• Figs. 1–4 and 7–9, the IS curve and its experiments. Model figures with the textbook’s baseline \(\bar{a} = 0\), \(\bar{b} = 2\), \(\bar{r} = 5\%\) and the book chapter’s illustrative demand shares; adapted from Figs. 11.1–11.4 and 11.7–11.9 in Jones (2024).
• Fig. 5, the life-cycle model. A stylized income profile and the constant consumption it pays for; adapted from Fig. 11.5 in Jones (2024).
• Fig. 6, the 2009 forecasts. BLS, Current Population Survey, unemployment rate (UNRATE), via FRED; Congressional Budget Office, letter to Senator Judd Gregg, 11 February 2009, Table 1. Inspired by Fig. 11.6 in Jones (2024).
• The 2022–23 puzzle. Federal Reserve statements of 16 March 2022 and 26 July 2023; 10-year inflation-indexed Treasury yield (DFII10), via FRED.
• Textbook. Jones, C. I. (2024). Macroeconomics, 6th ed. W. W. Norton.
• Book chapter, with the data, code and every figure. Data Science and Macroeconomic Models, Chapter 11.

Sources: research cited

• The IS curve and consumption. Hicks, J. R. (1937), Econometrica 5(2); Barro, R. J. (1974), Journal of Political Economy 82(6); Hsieh, C.-T. (2003), American Economic Review 93(1).
• Fiscal multipliers. Nakamura, E. and J. Steinsson (2014), American Economic Review 104(3); Ramey, V. A. (2019), Journal of Economic Perspectives 33(2); Ramey, V. A. and S. Zubairy (2018), Journal of Political Economy 126(2); Auerbach, A. and Y. Gorodnichenko (2012), American Economic Journal: Economic Policy 4(2); Christiano, L., M. Eichenbaum and S. Rebelo (2011), Journal of Political Economy 119(1); Dolls, M., C. Fuest and A. Peichl (2012), Journal of Public Economics 96(3–4).
• Stimulus payments. Johnson, D. S., J. A. Parker and N. S. Souleles (2006), American Economic Review 96(5); Parker, J. A., N. S. Souleles, D. S. Johnson and R. McClelland (2013), American Economic Review 103(6); Kaplan, G. and G. L. Violante (2014), Econometrica 82(4).
• Policy under uncertainty. Brainard, W. (1967), American Economic Review 57(2); Bernanke, B. S. (2007), “Monetary Policy under Uncertainty”, speech, 19 October; Laubach, T. and J. C. Williams (2003), Review of Economics and Statistics 85(4).