A vocabulary of 20,000 words holds an almost endless supply of new texts

10⁴³⁰ The number of different 100-word paragraphs that can be written with 20,000 words

Machines run into diminishing returns, and raw materials are finite. Recipes are neither. Can new recipes keep income growing?

Open with the number. It comes from the notebook’s “Model in code” section: with 20,000 words, there are 20,000^100 possible strings of 100 words, which is 10 to the power 100 × log10(20,000), about 10^430. Most of them are nonsense, of course, but even a tiny fraction of useful ones is a vast store.

The same logic holds for chemicals, genes, machine parts or lines of code: combining a finite set of objects in new ways gives an enormous number of possible recipes.

Link back to Chapter 5: the Solow model showed that piling up machines cannot keep income per person growing, because each machine adds less than the last. Today’s question is whether ideas, the recipes for using objects, can do what machines could not.

Today: three questions

• Why ideas are different. What makes a recipe unlike a machine, and why can’t perfect competition pay for it?
• The Romer model. How do ideas produce sustained growth, and what changes its level rather than its rate?
• Ideas in the data. Have raw materials run out? Is growth slowing down, and why?

These are the three acts of the deck and the three branches of the mind map on the title slide. Act I builds the economics of ideas from one property, nonrivalry. Act II puts that property into a growth model and adds back the capital of Chapter 5. Act III takes the theory to a century of commodity prices and to the growth of the United States and other countries since 1960.

The headline result: growth in income per person comes from growth in the stock of ideas, and that comes from the number of people looking for them.

Why ideas are different

Act I

Objects are rival; ideas are nonrival

Objects

• Land, oil, machines, buildings, workers.
• Rival: if one factory uses a machine, another cannot use it at the same time.
• Scarcity comes from rivalry.

Ideas

• Designs, formulas, recipes, computer code.
• Nonrival: a second factory can follow a design while the first still has all of it.
• Ideas are instructions for using objects.

Nonrival is not the same as free: a patent can make an idea excludable, but it does not make it rival.

Romer (1990) sorted all goods into these two groups. Ask for examples of each: a phone is an object; its design is an idea. A dose of a drug is an object; its chemical formula is an idea. Management methods count as ideas too, such as just-in-time inventories or a checklist that cuts hospital infections.

Excludability is the legal or technical ability to stop others from using a good. Patents, copyrights and trade secrets give it to some ideas for a while. Rivalry is a physical property; excludability is an institution. Keep the two apart: much of the policy debate in this chapter is about excludability.

The production function of Chapters 4 and 5 now has a time subscript on productivity: Y = A_t K^(1/3) L^(2/3), where A_t is the stock of ideas, which keeps growing.

Nonrivalry alone implies increasing returns

\[Y_t = A_tK_t^{1/3}L_t^{2/3} \qquad\Longrightarrow\qquad (\lambda A)(\lambda K)^{1/3}(\lambda L)^{2/3} = \lambda^{2}\,Y\]

Multiply …	by	Output is multiplied by
capital and labor only	2	2
the stock of ideas only	2	2
objects and ideas	2	4
objects and ideas	3	9

To double output, copy the factory and its workers; the blueprint does not need copying. Doubling the blueprints too must more than double output.

This is the replication argument. A second identical factory needs its own machines and its own workers, but not its own design: both factories follow the same one. So there are constant returns to objects (the exponents 1/3 and 2/3 add to one).

Once ideas count as an input, scaling every input by λ multiplies output by λ²: doubling everything quadruples output, tripling it gives nine times as much. The notebook checks the table with numbers (2.00, 4.00, 9.00).

Increasing returns are not an extra assumption here. They follow from nonrivalry. The next slides show the same thing through a fixed cost, which is how firms experience it.

A new antibiotic: research first, then cheap doses

Discovering a new antibiotic costs $2.5 billion. After that, each dose costs $10 to make. If a firm doubles its total spending from $3.6 billion to $7.2 billion, how many times more doses does it make?

Let students compute before the reveal. A common first answer is “twice as many”, which would be right without the research cost.

The calculation: with spending X and research cost F̄, the firm makes (X − F̄)/10 doses. At 3.6 billion: 1.1 billion dollars of production, 110 million doses. At 7.2 billion: 4.7 billion dollars, 470 million doses. That is more than four times as many.

Without the fixed cost, returns are constant; with it, they are increasing

Model figure: an antibiotic with a research cost \(\bar{F}\) and $10 a dose; spending \(X\) in billions of dollars, doses \(Y\) in millions. Adapted from Fig. 6.1 in Jones (2024), Macroeconomics, 6th ed.

How to read it: panel (a) has no fixed cost, Y = X/10. Panel (b) has the research cost, so output is zero until spending covers F̄ and then rises one for one. The black dots mark spending of 3.6 and 7.2 billion. The dashed rays from the origin show the average product, doses per billion dollars. The slider sets F̄ from 0 to 3.5 billion; the title reports the output gain from doubling spending.

Do live:

1. At F̄ = 2.5: doubling spending multiplies output by 2 in (a) and by 4.27 in (b). The annotation gives the ray slopes: 31 million doses per billion dollars at 3.6 billion, 65 at 7.2.
2. Point out that the ray to the larger scale is steeper: output per dollar rises with scale.
3. Move the slider to 0: panel (b) becomes panel (a). Then to 3.5: the gain jumps to 37, because the smaller scale barely covers the research cost.

The larger the market, the cheaper each dose

Research cost \(\bar{F}\)	Doubling spending multiplies doses by
$0	2.00
$1.0 billion	2.38
$2.0 billion	3.25
$2.5 billion	4.27
$3.0 billion	7.00

\[\frac{Y}{X} = \frac{1}{c}\left(1 - \frac{\bar{F}}{X}\right)\]

Output per dollar rises with spending \(X\), because the research cost is spread over more doses: 31 million doses per billion dollars at $3.6 billion, 65 at $7.2 billion.

The increasing returns come entirely from the fixed cost of the idea.

The numbers are the slider titles and the annotation of Fig. 1.

The formula: total spending X buys (X − F̄)/c doses. Divide by X to get the average product. Without the fixed cost it is 1/c, a constant. With it, the term F̄/X shrinks as the firm grows, so each extra billion buys more doses than the last.

Equivalently, average cost per dose, c + F̄/Y, falls with scale. This is why bigger markets make more ideas worth inventing, a point that comes back in the software example and when we ask why the Romer model is about the world’s ideas.

If the price equals marginal cost, nobody pays for the research

\[\pi = (p - c)\,Y - \bar{F} \qquad\Longrightarrow\qquad p = c:\ \ \pi = -\bar{F}\]

\[\text{break even:}\quad p = c + \frac{\bar{F}}{Y}\]

• Spending $5.0 billion makes 250 million doses. Sold at $10, they bring in $2.5 billion: a loss of exactly the $2.5 billion research cost.
• At 470 million doses, the firm needs at least \(10 + 2{,}500/470 \approx\) $15.32 a dose.

Perfect competition drives prices to marginal cost, so it cannot reward invention. Some wedge between price and cost is needed.

The numbers are the notebook’s key numbers and Check 1.

Walk through the timing. After the drug exists, selling at marginal cost looks fine: each dose covers its own cost. But before the research is done, a firm that expects to sell at marginal cost will never spend the 2.5 billion. So the price has to exceed marginal cost at some point, or the research must be paid for some other way.

This is where Adam Smith’s invisible hand runs into trouble. With increasing returns, the competitive price cannot cover all costs.

The wedge has a cost too: a price above marginal cost keeps out buyers who would pay more than the cost of making one more dose, but less than the price. One price cannot both reward invention and spread an existing idea as widely as possible.

Software: a huge fixed cost, almost no cost per copy

A word processor with perfect speech recognition costs $100 million to write, and each copy costs $1 to distribute. How many copies must the firm sell to break even at $20 a copy? At $2?

This is a worked exercise in the textbook. Ask students to set up the break-even condition first: revenue p·Y must cover the fixed cost plus the cost of the copies, 100 + Y (in millions). So Y = 100/(p − 1).

Most students get the $20 case quickly. The $2 case is the point: at a low price, the firm needs a huge market.

The production function curves up from the origin

Model figure: a word processor with a development cost of $100 million and $1 a copy. Adapted from Fig. 6.7 in Jones (2024), Macroeconomics, 6th ed., which illustrates a worked exercise.

How to read it: spending X (millions of dollars) on the horizontal axis, copies Y (millions) on the vertical. The production function is flat at zero up to 100 million and then rises one for one. The dot marks the break-even point at the price set by the slider (2 to 100 dollars).

Do live:

1. At $20: break-even at 5.26 million copies.
2. Slide to $100: only 1.01 million. Slide to $2: 100 million.
3. Point at the shape: spending 200 million gives 100 million copies, but spending half as much gives none. That is increasing returns in its starkest form.

Low prices are possible only with huge markets

Price per copy	$2	$3	$5	$10	$20	$50	$100
Copies to break even (millions)	100	50	25	11.11	5.26	2.04	1.01

\[\text{break even:}\quad pY = 100 + Y \quad\Longrightarrow\quad Y = \frac{100}{p-1}\]

At the marginal cost of $1 the firm loses its whole $100 million. Scale matters: a bigger market lets a firm survive at a lower price.

The numbers are the slider titles of Fig. 7 and the notebook’s key numbers (profit at a price of $1: −100 million dollars).

Each halving of the markup roughly doubles the market the firm needs. This is why software, music, films and drugs are sold far above the cost of one more copy, and why the size of the market matters so much for the incentive to invent.

Ask: open-source software such as Linux is sold at a price of zero. Who pays for it? (Volunteers, companies that use it, people building a reputation. Profit is not the only motive for creating ideas.)

Paying for ideas: patents, public money or prizes?

Research must be paid for somehow, and a price at marginal cost cannot do it. Which institutions should?

Patents and copyrights

• A price above cost rewards the inventor.
• It also prices out buyers who would pay the cost of a copy.
• Their effect on follow-on research is disputed.

Public money and prizes

• Public research funding leads to private patents.
• Prizes and purchase commitments pay for results.
• But someone must pick the targets, and taxes pay.

Left column. Patents give a temporary monopoly in exchange for publishing the idea. Williams (2013, Journal of Political Economy) finds that Celera’s intellectual property on human genes was followed by less later research and fewer products on those genes. Sampat and Williams (2019, American Economic Review) find that gene patents themselves had no clear effect on follow-on innovation. Budish, Roin and Williams (2015, American Economic Review) show that fixed patent lengths tilt cancer research toward drugs that can be tested quickly. Moser (2005, American Economic Review) finds that countries without patent laws in the nineteenth century invented as much, but in industries where secrecy worked.

Right column. Azoulay, Graff Zivin, Li and Sampat (2019, Review of Economic Studies) use quirks in how the U.S. National Institutes of Health allocates grants and find that public biomedical funding produces substantial private-sector patenting. Kremer (1998, Quarterly Journal of Economics) proposes that governments buy out patents and put the ideas in the public domain. Kremer, Levin and Snyder (2020, AEA Papers and Proceedings) review advance market commitments, such as the one for pneumococcal vaccines in poor countries.

The trade-off from the model: any wedge between price and marginal cost rewards invention but limits use; public funding avoids the wedge but needs taxes and someone to choose the research. Ask the policy students: which mix would you choose for vaccines, for software, for basic science?

The Romer model

Act II

The Romer model: goods, ideas and people

What the model assumes	Equation	Parameter
Workers make goods with all the ideas	\(Y_t = A_t^{\gamma}L_{yt}\)	\(\gamma\)
Researchers make new ideas	\(\Delta A_{t+1} = \bar{z}L_{at}\)	\(\bar{z}\)
Everyone makes goods or ideas	\(L_{yt} + L_{at} = N_t\)
Population grows	\(\Delta N_{t+1}/N_t = \bar{n}\)	\(\bar{n}\)
A fixed share does research	\(L_{at} = \bar{\ell}N_t\)	\(\bar{\ell}\)

No capital at all: the model isolates ideas. Capital comes back at the end of the act.

Five equations, five unknowns in every year: output Y, the stock of ideas A, goods workers L_y, researchers L_a and population N.

The parameters: γ is the degree of increasing returns, how much ideas matter in production (scaling ideas and workers by λ multiplies output by λ^(1+γ)). z̄ is research productivity, new ideas per researcher per year. n̄ is population growth, and ℓ̄ is the research share.

The stock of ideas is like a library that is never emptied: each year researchers add z̄ L_a new blueprints, and nothing is lost.

The research share is a rule of thumb, like the investment rate s̄ in the Solow model. In Romer’s original paper, patents and monopoly profits decide how many people do research, and markets tend to choose too few.

Income depends on the total stock of ideas, not ideas per person

\[y_t = \frac{Y_t}{N_t} = A_t^{\gamma}\,(1-\bar{\ell}) \qquad\qquad A_t^* = \frac{\bar{z}\bar{\ell}N_t}{\bar{n}}\]

• One new idea raises the productivity of every worker at once: that is nonrivalry at work.
• Compare Solow: \(y = \bar{A}k^{1/3}\) depends on capital per person; a new machine helps only the worker who uses it.
• On the balanced growth path, the stock of ideas is proportional to the number of researchers, \(\bar{\ell}N_t\).

How to get there. Step 1: L_a = ℓ̄N and L_y = (1 − ℓ̄)N. Step 2: divide output by N. Step 3: the growth rate of ideas is z̄ℓ̄N/A. On a balanced growth path every growth rate is constant, so A must be a constant times N, and ideas grow at n̄. Rearranging gives A* = z̄ℓ̄N/n̄.

The key contrast with Chapter 5: in the Solow model, adding people dilutes capital per person. In the Romer model, adding people adds researchers, and their ideas are shared by everyone.

Long-run growth equals the degree of increasing returns times population growth

\[y_t^* = (1-\bar{\ell})\left(\frac{\bar{z}\bar{\ell}N_t}{\bar{n}}\right)^{\gamma} \qquad\Longrightarrow\qquad g_y^* = \gamma\bar{n}\]

• More people → more researchers → more ideas → higher income for everyone.
• The only term that changes over time is \(N_t^{\gamma}\), which grows at \(\gamma\bar{n}\).
• With \(\bar{n} = 0\) there is no long-run growth at all.

This is the engine that the Solow model was missing.

Step 5 substitutes A* into y = A^γ(1 − ℓ̄). Step 6 uses the growth rule for powers: g_y = γ g_A, and g_A = n̄ on the balanced growth path.

Two textbook cases, from the notebook: γ = 2 and n̄ = 1 percent give 2 percent growth; γ = 1/2 and n̄ = 4 percent also give 2 percent.

Why ideas grow with researchers rather than faster: Romer’s original version let the growth rate of ideas rise with the number of researchers. Then growth itself would keep speeding up as population grows. U.S. growth has stayed close to 2 percent a year for more than a century, with no such acceleration, so the version here fixes that.

Drawing the balanced growth path

With \(\gamma = 0.7\) and the population growing 5% a year, what does the path of output per person look like on a ratio scale? And on a linear scale?

Remind students of the ratio scale from Chapter 3: equal distances mean equal percentage changes, so constant growth is a straight line. Ask them to sketch both versions on paper before the figure appears.

On a ratio scale, the Romer economy grows along a straight line

Model figure: the Romer model’s balanced growth path with \(\bar{n} = 0.05\), output per person relative to 2025. Adapted from Fig. 6.2 in Jones (2024), Macroeconomics, 6th ed.

How to read it: output per person from 2025 to 2100, relative to its 2025 level. The slider sets γ from 0.2 to 1; the buttons switch between a ratio and a linear vertical scale. The title reports the simulated growth rate, γ·n̄, and output in 2100 relative to 2025.

Do live:

1. At γ = 0.7: growth of 3.47 percent a year (γn̄ = 3.50 percent; the small gap is the yearly step, 1.05^0.7). By 2100 output per person is 13.0 times its 2025 level.
2. Switch to the linear scale: the same path now curves upward, because constant percentage growth adds more dollars every year.
3. Move γ down to 0.4: about 2 percent a year, close to U.S. history.

Small differences in \(\gamma\bar{n}\) compound into large differences in income

Importance of ideas \(\gamma\)	0.2	0.4	0.6	0.7	1.0
Growth a year (\(\bar{n} = 5\%\))	0.98%	1.97%	2.97%	3.47%	5.00%
Output in 2100, times 2025	2.1	4.3	9.0	13.0	38.8

Growth is constant forever, as long as the number of researchers keeps growing.

The numbers are the slider titles of Fig. 2.

Use the rule of 70 from Chapter 3 as a check: at 3.5 percent a year, income doubles every 20 years, so in 75 years it doubles almost four times, about 13 times its starting level.

Ask: the United States has grown at about 2 percent a year for more than 150 years. Which combinations of γ and n̄ give that? (Any with γ × n̄ ≈ 0.02, for example γ = 0.5 and n̄ = 4 percent.)

More people do not make us poorer in the Romer model

A growing population means more mouths to feed and less of everything to go around, so it must lower income per person.

For the world’s ideas the opposite holds: more people means more researchers, and each new idea serves everyone at once. Income per person grows at \(\gamma\bar{n}\), so a faster-growing population means faster growth.

The myth is Malthusian: with fixed resources, each extra person leaves less for the others. That is true of rival goods, of objects. Ideas are nonrival, so an extra inventor’s idea does not have to be shared out.

Be fair to the myth: in the Solow model of Chapter 5, faster population growth lowered steady-state income, because capital had to be spread over more workers. Both forces are real. The Romer force works through the world’s stock of ideas, so it is about the number of people who can create ideas anywhere, not the population of one country.

Kremer (1993, Quarterly Journal of Economics) finds that over very long periods of history, regions and eras with larger populations had faster technological progress.

Romer’s Nobel lecture puts it as a choice between “us versus them” and “us and them”.

Some changes move the growth path up; only population growth tilts it

A permanent rise in …	Long-run growth \(g_y^*\)	Level of the path \(y_t^*\)	Output on impact
research productivity \(\bar{z}\)	unchanged	rises	unchanged
research share \(\bar{\ell}\)	unchanged	rises, if \(\bar{\ell}\) is not too large	falls
population growth \(\bar{n}\)	rises by \(\gamma\Delta\bar{n}\)		unchanged

A level effect shifts the path up or down; a growth effect changes its slope. Only \(\gamma\) and \(\bar{n}\) have growth effects.

Take logs of the BGP level: ln y* = ln(1 − ℓ̄) + γ ln ℓ̄ + γ ln z̄ + γ ln N − γ ln n̄. Each coefficient is an elasticity. A 1 percent rise in z̄ raises the BGP by γ percent (the notebook checks it: +0.599 percent with γ = 0.6).

The research share appears twice, once with a positive sign (more researchers, more ideas) and once with a negative one (fewer people making goods), so its elasticity γ − ℓ̄/(1 − ℓ̄) is positive only when ℓ̄ is small.

On impact the stock of ideas is what past research left, so output per person jumps only when the number of goods workers changes, that is, when ℓ̄ changes.

The next two experiments are the first two rows.

Experiment: research becomes eight times as productive

In 2050, artificial intelligence makes every researcher eight times as productive (\(\bar{z}\) rises from 1 to 8). With \(\gamma = 0.6\) and \(\bar{n} = 5\%\), the economy grows 3% a year before. Does long-run growth rise? What happens to growth in 2051?

Collect predictions on two separate points: the growth rate in the long run, and the growth rate right after the change. Many students expect long-run growth to rise eightfold, or at least to rise. Some will remember that z̄ does not appear in g* = γn̄.

Ask them to sketch output per person on a ratio scale, with the old balanced growth path as a straight line.

A jump in research productivity lifts the path, not its long-run slope

Model figure: simulation of the Romer model with \(\gamma = 0.6\) and \(\bar{n} = 0.05\); \(\bar{z}\) rises in 2050. Adapted from Fig. 6.3 in Jones (2024), Macroeconomics, 6th ed.

How to read it: output per person relative to 2025, on a ratio scale. The straight lines are the old and the new balanced growth paths; the solid line is the economy. The slider sets the new research productivity from 0.5 to 10. The title reports the level effect and the growth rates in 2051 and 2100.

Do live:

1. At z̄′ = 8: the new path is 3.48 times the old. Growth is 22.4 percent in 2051, falls back to 3.2 percent by 2100, and the balanced growth rate is still 3.0 percent.
2. Point at the moment of the change: output does not jump, because the stock of ideas is what it was. It is the flow of new ideas that jumps.
3. Move the slider to 0.5: research becomes less productive, growth slows to 1.5 percent in 2051, and the economy settles on a path 0.66 times the old one.

A level effect with a long transition

New \(\bar{z}'\)	New path, times old	Growth in 2051
0.5	0.66	1.5%
2	1.52	5.9%
4	2.30	11.6%
8	3.48	22.4%
10	3.98	27.5%

\[g_{At} = \bar{n}\,\frac{A_t^*}{A_t}\]

In 2050, \(A_t^*\) becomes 8 times \(A_t\), so ideas grow by \(0.05 \times 8 = 40\%\), and output per person by \(1.40^{0.6} - 1 \approx 22\%\).

Principle of transition dynamics: the farther below its balanced growth path, the faster an economy grows.

The numbers are the slider titles of Fig. 3 and the notebook’s Check 3 (22.37 percent simulated and from the formula; level effect 3.482 = 8^0.6).

The formula on the right holds in every year: ideas grow at n̄ times the ratio of the BGP stock to the actual stock. When the stock is below its BGP value, ideas grow faster than n̄, and the gap closes.

Why only a level effect? Long-run growth of ideas is pinned down by growth in the number of researchers. More productive researchers produce more ideas each year, a bigger stock, but the stock still grows at the rate of the research workforce in the long run.

Experiment: more people go into research

In 2050, the share of people doing research rises from 3% to 35% and stays there. What happens to output per person in 2050 itself? And in the long run?

Ask students to think about the two terms in y = A^γ(1 − ℓ̄). In 2050 the stock of ideas has not changed yet, but the number of people making goods has. Ask for a number: how much does output per person fall on impact? (0.65/0.97, a fall of about a third.)

More researchers cost output today and raise it later

Model figure: simulation of the Romer model with \(\gamma = 0.6\) and \(\bar{n} = 0.05\); \(\bar{\ell}\) rises from 0.03 in 2050. Adapted from Fig. 6.4 in Jones (2024), Macroeconomics, 6th ed.

How to read it: as in Fig. 3, with the old and new balanced growth paths and the economy’s path, on a ratio scale. The slider sets the new research share from 0.05 to 0.75. The title reports the impact in 2050, the level effect and growth in 2051.

Do live:

1. At 0.35: output falls by 33.0 percent in 2050, grows 31.7 percent in 2051, and ends on a path 2.93 times the old one.
2. Point at the crossing: years pass before the economy is back above where it would have been. That is the cost of investing in ideas.
3. Move the slider to 0.65 and 0.75: the drop is deeper and the level effect smaller (2.28, then 1.78).

The research share has a hump: the best share is \(\gamma/(1+\gamma)\)

New share \(\bar{\ell}'\)	On impact	New path, times old
0.10	−7.2%	1.91
0.20	−17.5%	2.57
0.35	−33.0%	2.93
0.50	−48.5%	2.79
0.65	−63.9%	2.28
0.75	−74.2%	1.78

\[y_t^* \propto (1-\bar{\ell})\,\bar{\ell}^{\gamma}\]

More researchers make more ideas (\(\bar{\ell}^{\gamma}\)), but leave fewer people to make goods, the factor \(1-\bar{\ell}\). The path is highest at \(\bar{\ell} = \gamma/(1+\gamma) =\) 0.375.

Even the best research share costs output for years before it pays off.

The numbers are the slider titles of Fig. 4 and the notebook’s key-numbers table (starting from ℓ̄ = 0.03 with γ = 0.6: 1.91 at 0.10, 2.93 at 0.35 and 0.40, 1.48 at 0.80).

The calculus: maximize ln(1 − ℓ̄) + γ ln ℓ̄. The derivative −1/(1 − ℓ̄) + γ/ℓ̄ is zero where ℓ̄/(1 − ℓ̄) = γ, so ℓ̄ = γ/(1 + γ) = 0.6/1.6 = 0.375. The notebook finds the same maximum numerically.

The impact effect is simply 1 − (1 − ℓ̄′)/(1 − ℓ̄): it depends only on how many people leave goods production.

For the policy students: a policy that moves people into research is an investment. It lowers output now and raises it later; whether it is worth it depends on how much we value the future.

More research is not always better

If ideas drive growth, a country should put as many people as possible into research.

In the model, the path peaks at a research share of 0.375. Raising the share from 0.03 to 0.80 leaves a path only 1.48 times the old one, far below the 2.93 reached at 0.35, because too few people are left to use the ideas.

The myth takes a true statement (ideas drive growth) one step too far. Ideas are only valuable if someone uses them to make goods.

Two cautions for the policy debate. First, real economies are far below the peak of the hump: researchers are a small share of the workforce, and in Romer’s original model markets choose too little research, because inventors cannot capture all the benefits of their ideas. So the practical question is usually how to raise research, not how to cap it. Second, even a share below the peak lowers output for years, as Fig. 4 showed.

The number 1.48 is from the notebook’s table of level effects.

Ideas cross borders: the Romer model describes the world

• Read literally, \(y_t^*\) rises with a country’s own population, so large countries should be richer. They are not.
• Ideas travel through trade, multinational firms, licenses and migration. A country can use ideas invented anywhere.
• So \(N_t\) counts the people anywhere who create ideas, and \(\gamma\bar{n}\) is the growth of the world frontier.

Why countries grow at different rates is a question for the combined model, next.

This slide heads off the most common misreading of equation (4). China and India have the largest populations but are not the richest countries per person.

The resolution is that ideas are nonrival across borders as well as across firms, as long as countries can access them. Oral rehydration therapy and mobile phones are ideas that spread to poor countries quickly; patents granted in the United States go in large part to foreign inventors.

So the Romer model is best read as a model of the world’s stock of ideas. Each country then sits somewhere relative to that frontier, depending on its investment, its institutions and its access to ideas.

Combining Solow and Romer: capital amplifies the growth of ideas

\[Y_t = A_t^{\gamma}K_t^{1/3}L_{yt}^{2/3}, \qquad \Delta K_{t+1} = \bar{s}Y_t - \bar{d}K_t\]

\[g_y^* = \gamma\bar{n} + \tfrac{1}{3}g_y^* \quad\Longrightarrow\quad g_y^* = \tfrac{3}{2}\gamma\bar{n}\]

• Ideas raise output; part of it is invested; the extra capital raises output again: \(1 + \tfrac{1}{3} + \tfrac{1}{9} + \dots = \tfrac{3}{2}\).
• With \(\gamma = 2/3\) and \(\bar{n} = 3\%\): growth of 3% a year, against 2% in the Romer model alone.
• On the path, capital per person grows as fast as output per person, and \(K^*/Y^* = \bar{s}/(g_Y^* + \bar{d}) \approx 0.62\).

The combined model has six equations: the five of the Romer model, with capital added to production, and the capital-accumulation equation of Chapter 5.

The derivation: output per person is y = A^γ k^(1/3)(1 − ℓ̄)^(2/3), so g_y = γ g_A + (1/3) g_k. Ideas still grow at n̄. On a balanced growth path, K/Y is constant (from the capital equation), so g_k = g_y. Substituting gives g_y = γn̄ + (1/3)g_y, and solving gives (3/2)γn̄.

The capital-output ratio: the Solow ratio s̄/d̄, with depreciation replaced by g_Y + d̄, because investment now must also keep capital growing with output. With s̄ = 0.05, d̄ = 0.02 and g_Y = 6.09 percent, it is 0.62. The notebook’s Check 5 simulates 300 years from very little capital and lands on 0.618.

Capital cannot drive growth on its own, but it multiplies whatever ideas do.

Saving sets the level of the path; ideas set its slope

\[y_t^* = \left(\frac{\bar{s}}{g_Y^*+\bar{d}}\right)^{1/2}(A_t^*)^{3\gamma/2}\,(1-\bar{\ell})\]

A permanent rise in …	Long-run growth	Elasticity of the level
investment rate \(\bar{s}\)	unchanged	+0.5
research productivity \(\bar{z}\)	unchanged	+1
research share \(\bar{\ell}\)	unchanged	+0.89
population growth \(\bar{n}\)	rises

The level comes from substituting K* = [s̄/(g_Y + d̄)] Y* into the production function and solving for Y. The textbook prints the exponent on A as 3/2, which is right only when γ = 1; for any γ it is 3γ/2. The notebook flags this.

The elasticities use the values of Fig. 8 (γ = 2/3, n̄ = 3%, d̄ = 2%, ℓ̄ = 0.1). Every elasticity linked to ideas is 3/2 times its Romer value, for the same reason as the growth rate: capital amplifies ideas. Depreciation has an elasticity of −0.12.

The headline: every parameter except γ and n̄ has only a level effect. Saving behaves as it did in Solow; ideas set the long-run growth rate.

Experiment: the investment rate jumps

In the combined model, the investment rate jumps from 5% to 60% of output in 2050. Does output per person jump? Does long-run growth rise above 3% a year?

This mirrors the Solow experiment of Chapter 5, now on top of growing ideas. Ask students to sketch the path on a ratio scale. Two things to predict: whether there is a jump on impact (no: capital takes time to build), and whether the long-run slope changes (no).

Saving more lifts the path, not its long-run slope

Model figure: simulation of the combined Solow–Romer model with \(\gamma = 2/3\), \(\bar{n} = 0.03\) and \(\bar{d} = 0.02\); \(\bar{s}\) rises from 0.05 in 2050. Adapted from Fig. 6.8 in Jones (2024), Macroeconomics, 6th ed.

How to read it: output per person on a ratio scale, with the old and new balanced growth paths. The slider sets the new investment rate from 0.05 to 0.8. The title reports the level effect and growth in 2051 and 2100.

Do live:

1. Before 2050, growth is 3 percent a year, the combined model’s (3/2)γn̄.
2. At s̄′ = 0.6: no jump in 2050, growth of 26.2 percent in 2051, 3.2 percent by 2100, and a new path 3.46 times the old one.
3. Move the slider to 0.1: the path rises by 1.41 = √2, with growth of 5.6 percent in 2051.

A twelvefold rise in saving lifts income only \(\sqrt{12}\) times

New investment rate \(\bar{s}'\)	0.1	0.2	0.3	0.4	0.6	0.8
New path, times old	1.41	2.00	2.45	2.83	3.46	4.00
Growth in 2051	5.6%	10.3%	14.7%	18.8%	26.2%	32.8%

In the first year capital grows by 95% and output per person by 26%; then growth slows back to 3%. The level effect is \((0.6/0.05)^{1/2} \approx 3.46\).

The numbers are the slider titles of Fig. 8 and the notebook’s Check 6 (capital grows 95.1 percent in the first year, simulated and from the transition formula; output per person 26.2 percent).

The transition formula: g_K = (g_Y + d̄)(K/Y)/(K/Y) − d̄. Raising s̄ from 0.05 to 0.6 multiplies the BGP capital-output ratio by 12 overnight, while the actual ratio has not moved: 0.0809 × 12 − 0.02 ≈ 0.95.

The comparison with ideas: the level elasticity of saving is only 1/2, as in Solow.

Why growth rates differ across countries for decades

• In the long run, every country that can use the world’s ideas grows at the same rate: \(\tfrac{3}{2}\gamma\bar{n}\).
• Over any decade, countries differ in how far they sit below their own growth paths.
• A reform that raises \(\bar{s}\), \(\bar{\ell}\) or access to ideas opens a gap, and the transition brings decades of fast growth: Japan and South Korea.

Think of each country as a Solow economy riding on the world’s growing stock of ideas.

This is the full theory of long-run growth that the textbook’s three growth chapters build: the Romer model explains the trend of the frontier; Solow’s transition dynamics explain why some countries grow much faster or slower than the frontier for decades.

South Korea, from Chapter 5: its investment rate rose steeply, and its access to technology improved, so its growth path moved up and it grew fast during the transition. Countries whose investment or access to ideas deteriorated fall behind instead.

What the model does not say is why some countries change their parameters and others do not. That is the question of institutions from Chapter 4.

Ideas in the data

Act III

Experiment: a century of industrialization

Over the twentieth century, the world’s industry used vastly more aluminum, copper, coal and iron ore, and the supply of each is finite. What happened to their real prices between 1900 and 2000?

Ask for a show of hands: rose a lot, rose a little, stayed flat, fell. Most students choose “rose”, following simple supply and demand with growing demand and fixed supply. Ask those who choose “fell” to explain why.

Real commodity prices fell for most of the century

Data: Jacks (2019), Cliometrica 13(2), real commodity prices 1850–2020, extended to 2025 with the World Bank Pink Sheet and deflated by the U.S. CPI via FRED; geometric mean of six industrial commodities, 1900 = 100. Chart inspired by Fig. 6.5 in Jones (2024), Macroeconomics, 6th ed.

How to read it: an index of the real prices (divided by U.S. consumer prices) of aluminum, coal, copper, lead, iron ore and zinc, combined with equal weights in logs and set to 100 in 1900. Hover over the line for the values.

Do live:

1. Hover over 1916: the World War I spike, 124.1.
2. Follow the line down to the trough in 2002: 30.1.
3. Point at the rise after 2002, to 84 in 2011, which the textbook links to the fast growth of China and India, and the fall since.

Data note: the textbook uses proprietary data and reports a fall of “almost five” times between 1900 and 2000. With open data the fall is a factor of about 3. The levels differ; the ups and downs are the same.

Supply grew faster than demand, because of new ideas

Real commodity price index (1900 = 100)
Peak during World War I, 1916	124.1
Lowest value, 2002	30.1
Fall from 1900 to 2000	factor 2.9
Rise from 2000 to 2011	factor 2.43
Latest year, 2025	58.9

Demand rose enormously, so supply must have risen even more: new deposits, new ways to extract them, and new ways to use less.

The numbers are the notebook’s key numbers for Fig. 5.

The supply-and-demand reasoning: if prices fell while demand grew, the supply curve must have shifted out by more. New ideas did that: geological exploration, new mining and refining techniques, and products that use less material for the same job.

Aluminum is the clearest example (next figure): a new way to refine it and cheap electricity turned a precious metal into foil.

The index is a geometric mean, the average of the logs. A simple average would be dominated by one commodity whose price shoots up: in 2022 it would be 190.8 instead of 83.8, because of coal.

Beyond the textbook’s six

The index covers six industrial commodities. Across 40 commodities, metals, energy, grains, coffee and cotton, meat and minerals, which do you expect cost more in real terms today than in 1900?

Let students make a short list. Common guesses: oil, gold, coffee, meat. Write them on the board and check them in the next figure.

Most commodities got cheaper; energy and precious metals did not

Data: Jacks (2019), Cliometrica 13(2), 40 real commodity prices 1850–2020; extended to 2025 with the World Bank Pink Sheet for 30 of them; deflated by the U.S. CPI via FRED; 1900 = 100. Chart inspired by Fig. 6.5 in Jones (2024), Macroeconomics, 6th ed.

How to read it: real prices on a ratio scale, each set to 100 in 1900. The textbook’s six are shown; click a name in the legend to add a commodity, or double-click to show it alone. The buttons switch between the six and all lines.

Do live:

1. With the six: aluminum fell to 8 percent of its 1900 price; coal is the only one above 100, with a spike in 2022 during the energy crisis that followed Russia’s invasion of Ukraine.
2. Add rubber (down to 5 percent of its 1900 price) and gold (more than four times its 1900 price).
3. Add petroleum and natural gas: energy is the main exception.

Data caveat: ten commodities have no World Bank match and stop in 2020, so check where each line ends before comparing latest values.

28 of 40 commodities cost less than in 1900

Fell the most	Latest, 1900 = 100
Rubber	5
Aluminum	8

Rose the most	Latest, 1900 = 100
Gold	421
Petroleum	297
Beef	232
Coal	227

The Simon–Ehrlich bet

In 1980 the economist Julian Simon bet the biologist Paul Ehrlich that metal prices would fall over the next ten years. Four of the five metals are in the data, and all fell from 1980 to 1990:

chromium −12%, copper −23%, nickel −14%, tin −77%.

The numbers are the notebook’s key numbers for the 40-commodity chart (real prices in the latest year, 1900 = 100, and the changes from 1980 to 1990).

The exceptions are mostly energy and precious metals, plus beef, lamb, tin, chromium, manganese and coffee. Ask: why might energy be different? (Demand for energy grew even faster than for materials, extraction faces depletion, and prices are affected by cartels and wars.) Why gold? (It is held as an asset, not used up.)

The bet: Simon let Ehrlich pick five metals and a ten-year window. Ehrlich, who had predicted mass shortages, paid up in 1990. The policy slide comes back to whether Simon was skillful or lucky.

Finite resources do not have to mean rising prices

The planet’s supply of raw materials is fixed, and demand keeps growing, so their prices must rise over time.

Over the twentieth century, the real price of the six industrial commodities fell by a factor of about 3, and 28 of 40 commodities are cheaper than in 1900. New ideas kept expanding the usable supply faster than demand grew.

The myth applies a correct static argument (fixed supply plus rising demand) to a world where supply is not fixed. What is fixed is the number of atoms; what is not fixed is our knowledge of where to find them, how to extract them and how to use fewer of them.

Be careful not to overclaim. Prices rose steeply after 2002, and a few commodities, especially energy, are more expensive than in 1900. The point is that rising prices are not inevitable: whether ideas keep up with demand is an empirical question.

Ask: what would rising prices do to the incentive to find new ideas? (Raise it, which is part of why the long-run trend has been down.)

Limits to growth: can ideas keep beating scarcity?

Can new ideas keep growth going on a finite planet, or will resources and pollution set the limit?

Why ideas can win

• Most raw materials got cheaper as use soared.
• Higher energy prices brought more energy-saving inventions.
• Simon won his bet against scarcity.

Why it is contested

• In many other decades, Simon would have lost.
• Market prices leave out pollution and carbon.
• Ideas must be steered toward clean technology.

The debate goes back to Meadows and others, The Limits to Growth (1972), which projected that resource depletion and pollution would end growth within a century.

Left column: the commodity figures above. Popp (2002, American Economic Review) shows that energy-saving patents in the United States rose with energy prices: scarcity directs research toward saving the scarce input. The Simon–Ehrlich bet, from the notebook.

Right column: Kiel, Matheson and Golden (2010, Ecological Economics) repeat the bet for every ten-year window of the twentieth century and find that Ehrlich would often have won: the result depends on the window. Prices measure scarcity of the commodity to buyers, not the damage from burning it: carbon emissions are an externality. Acemoglu, Aghion, Bursztyn and Hémous (2012, American Economic Review) show that innovation follows profits, so without policy it keeps improving dirty technologies; a carbon price plus research subsidies can redirect it toward clean ones.

The 2018 Nobel Prize went jointly to Romer, for ideas and growth, and William Nordhaus, for climate economics: the two questions belong together. Ask the policy students: which policy would you use to steer ideas, a carbon tax, research subsidies, or both?

Has U.S. growth slowed down?

Since 1960, U.S. GDP per person has grown about 2% a year on average. Has the pace changed since the Great Recession of 2007–2009? Guess the 10-year average growth rate before and after.

Ask students for two numbers. Explain why we average over ten years: a single year’s growth jumps around with booms and recessions, and the trend is what the growth model is about.

Growth was steady for half a century, then fell after the Great Recession

Data: U.S. Bureau of Economic Analysis, real GDP (GDPCA) and population (B230RC0A052NBEA), via FRED; recessions from the NBER based Recession Indicators (USREC). Chart inspired by Fig. 6.6 in Jones (2024), Macroeconomics, 6th ed.

How to read it: in each year, the average growth of real GDP per person over the preceding ten years, 100 × (ln y_t − ln y_(t−10))/10. Shaded bands are recessions. The slider sets the window to 5, 10, 15 or 20 years. The title reports the means for 1960–2006 and since 2009.

Do live:

1. At 10 years: a mean of 2.2 percent from 1960 to 2006, and 1.2 percent since 2009.
2. Hover over 2016, the low point: 0.70 percent.
3. Hover over 1983 and 1985: 1.24 percent, then 2.43, the quick rebound after the 1980–82 recessions. Ask why this time was different.
4. Move the window to 5 and 20: shorter windows show more ups and downs.

The 10-year average has recovered only partly

10-year average growth of U.S. GDP per person
Mean, 1960–2006 (range 1.24 to 3.36)	2.18%
Mean, 2009–2018	0.87%
Lowest, 2016	0.70%
2019 → 2025	1.64 → 1.82%

Even with seven more years of data than the textbook, growth is still below its pre-2007 mean. What is behind the slowdown?

The numbers are the notebook’s key numbers for Fig. 6. The 10-year averages from 2019 to 2025 are 1.64, 1.2, 1.72, 1.75, 1.82, 1.86 and 1.82 percent.

The textbook’s data end in 2018, at the bottom of the slump. With data to 2025, the average has risen back to about 1.8 percent, but not to 2.2.

Possible explanations come from both the short run and the long run: a legacy of the financial crisis on investment and research, or a slowdown in the arrival of new ideas. Growth accounting, next, tells us which part of growth slowed.

Growth accounting: TFP is what is left over

\[\underset{\text{output per hour}}{g_Y - g_L} \;=\; \underset{\text{capital per hour}}{\tfrac{1}{3}(g_K - g_L)} \;+\; \underset{\text{labor composition}}{\tfrac{2}{3}(g_{L_y} - g_L)} \;+\; \underset{\text{TFP}}{g_A}\]

• Output, capital, hours and the skills of workers can be measured; TFP cannot. So TFP growth is the residual: the Solow residual.
• Example: output per hour +2.0%, capital per hour +2.4%, composition +0.2: TFP \(= 2.0 - 0.8 - 0.2 =\) 1.0 point, half of all growth.

The derivation: take logs of Y = A K^(1/3) L_y^(2/3) and the change from one year to the next: g_Y = g_A + (1/3)g_K + (2/3)g_Ly. Subtract the growth of hours, g_L, from both sides, splitting it as (1/3)g_L + (2/3)g_L.

L is total hours worked; L_y is the effective labor input, which also rises when the workforce becomes more educated or experienced. That is the “labor composition” term.

The worked example is the notebook’s Check 7. Capital per hour contributes a third of its growth, 0.8 points; labor composition 0.2; the rest, 1.0 point, is TFP.

This is a decomposition, not a causal model. Keep that in mind for the myth slide.

Both U.S. slowdowns were mostly slowdowns in TFP growth

Average growth, % a year	1948–73	1973–95	1995–2003	2003–21	2003–25
Output per hour	3.3	1.5	3.2	1.9	1.8
Contribution of capital per hour	1.0	0.8	1.4	0.8	0.7
Labor composition	0.2	0.2	0.3	0.3	0.3
TFP	2.1	0.5	1.5	0.8	0.8

TFP is the part of growth that the Romer model links to new ideas.

The numbers are the notebook’s growth-accounting table, computed from the Bureau of Labor Statistics indexes. The last column extends the textbook’s table to 2025. Our numbers match the textbook’s to 0.1 point; three cells differ by 0.1 because the BLS has revised its data.

Read the TFP row: 2.1 percent a year before 1973, 0.5 from 1973 to 1995, a burst of 1.5 during the information-technology boom of 1995–2003, and 0.8 since. Capital per hour and labor composition change much less.

Ask: what happened in the mid-1990s? (Computers and the internet spread through the economy; the productivity gains came largely in the sectors that made or used information technology.)

Growth accounting does not tell us what caused growth

Capital per hour accounts for about a third of U.S. growth, so about a third of growth was caused by investment in machines.

In the combined model with \(\gamma = 1\), output per hour grows 1.5% if TFP grows 1.0%. Accounting credits 0.5 points to capital, yet all of the growth is caused by ideas: the capital is there only because new ideas made it worth building.

The numbers are the notebook’s Check 7: on the balanced growth path with γ = 1, g_y = (3/2)g_A, and capital per hour grows as fast as output per hour, so equation (10) credits (1/3)(1.5) = 0.5 points to capital and the remaining 1.0 to TFP.

The lesson: accounting splits growth into proximate sources, not ultimate causes. Without TFP growth, capital would stop growing too, as in the Solow model. So the true contribution of ideas is larger than the TFP row of the table.

A second test from the model: on a balanced growth path, output per hour and capital per hour grow at the same rate. An economy whose capital per hour grows much faster is still in transition.

Was the slowdown only American?

Did other rich countries slow down after 2007 too? What about large, poorer economies such as China and India?

Ask students to predict separately for the rich countries near the frontier and for countries far below it. Recall the combined model: frontier countries grow with world ideas; countries below their paths grow faster while they catch up.

Growth slowed after 2007 in every G7 country, but rose in China and India

Data: Penn World Table 11.0 (rgdpna / pop; Feenstra, Inklaar and Timmer 2015), 185 countries; regions: World Bank country groups. Chart inspired by Fig. 6.6 in Jones (2024), Macroeconomics, 6th ed.

How to read it: the same 10-year average growth of GDP per person, for any of 185 countries. The menu picks a region; clicking a country in the legend adds it. It starts with the United States, China and India.

Do live:

1. Point at China and India rising above the United States after 2009.
2. Pick East Asia & Pacific and add Japan: 8.6 percent a year in 1970, 1.0 in 2000. Japan’s slowdown came decades earlier and was deeper.
3. Pick Europe & Central Asia and add Italy: negative growth after 2009.

Data note: many series start in 1960, 1970 or 1990, so a country’s 10-year average appears only ten years after its data begin. For the United States the PWT numbers differ from the BEA’s by at most 0.2 points.

The frontier slowed; countries catching up sped up

GDP per person, growth a year	1960–2006	2009–2023
United States	2.2%	1.0%
Japan	4.5%	0.6%
Germany	3.0%	1.1%
Italy	3.2%	−0.3%
China	5.4%	7.8%
India	2.3%	4.9%

A slowdown at the frontier is a slowdown in world ideas; fast growth far below the frontier is catch-up, the transition dynamics of the combined model.

The numbers are the notebook’s key numbers for the country chart (Penn World Table 11.0).

Every G7 country grew more slowly in 2009–2023 than in 1960–2006. Japan’s growth fell earlier: its 1960–2006 average includes the fast catch-up of the 1960s.

China and India grew faster after 2009 than before: they are far below the frontier, and their reforms moved their growth paths up. In the combined model, they are in a long transition.

Ask: what will happen to China’s growth as it gets closer to the frontier? (It should slow toward the frontier’s rate, as Japan’s did.)

Is growth slowing down for good?

Is slow productivity growth a passing phase, or the new normal? And could artificial intelligence change the answer?

The pessimists

• The great inventions are behind us.
• Ideas are getting harder to find.
• Mismeasured free apps cannot explain it.

The optimists

• AI can transform how goods are made.
• AI may also speed up research itself.
• In the model, a higher \(\bar{z}\) means decades of fast growth.

Left column. Gordon (2016, The Rise and Fall of American Growth) argues that electricity, sanitation, antibiotics and the internal combustion engine transformed life in a way that email and smartphones cannot repeat. Bloom, Jones, Van Reenen and Webb (2020, American Economic Review) show that the number of researchers has risen steeply while productivity growth has not, so each researcher produces less growth over time: research productivity is falling, in the economy and in specific fields such as computer chips and crop yields. Syverson (2017, Journal of Economic Perspectives) finds that the gains from free digital goods are too small to explain the missing growth.

Right column. Brynjolfsson and McAfee (2014, The Second Machine Age) argue that machine learning and robots will raise productivity sharply. Agrawal, McHale and Oettl (2019, in The Economics of Artificial Intelligence) model AI as a tool that helps researchers find useful combinations of existing ideas, which is a rise in z̄. Fig. 3 showed what that does: a level effect and decades of fast growth, though not a permanent rise in the growth rate.

For the policy students: what would you do to raise z̄ or ℓ̄? Research funding, education in science and engineering, immigration of researchers, competition policy. Each has costs and uncertain effects.

“If growth needs a growing population, does growth end when population stops growing?”

A skeptic says: “In the Romer model, \(g_y^* = \gamma\bar{n}\). World population growth is slowing and may turn negative this century. So the model predicts that growth will stop, or that a shrinking world will get poorer.”

Reply: the skeptic reads the model correctly, and economists take the warning seriously. What matters is the number of people creating ideas, not population itself, and that number can keep growing for a long time through education, a larger research share and billions of people in poorer countries joining the frontier.

Take the objection seriously: in the model, ideas grow with researchers, and researchers are a share of the population. If the population stops growing and the share is fixed, the stock of ideas grows ever more slowly in percentage terms, and so does income.

Jones (2022, American Economic Review, “The End of Economic Growth? Unintended Consequences of a Declining Population”) makes exactly this point: if population declines, an economy can end up with living standards that stagnate, as the stock of ideas stops growing. He calls it the “empty planet” result.

The reply has three parts. First, the research share can rise: in the past century the number of researchers in rich countries grew far faster than population, because of mass education. Second, most of the world’s people live in countries far from the frontier; as they become richer and better educated, many more become idea creators. Third, research productivity z̄ can rise too, for example with AI, though in the model that gives a level effect, not lasting growth.

Ask: which policies would you consider if the number of researchers stopped growing? (Education, immigration of researchers, research subsidies, family policy.) Each one is a choice about the parameters ℓ̄ and n̄.

What to remember from this chapter

• Ideas are nonrival, so returns to objects and ideas together are increasing, and prices at marginal cost cannot pay for research.
• In the Romer model, income depends on the total stock of ideas, and long-run growth is \(g_y^* = \gamma\bar{n}\).
• Research productivity, the research share and saving have level effects; only \(\gamma\) and \(\bar{n}\) change long-run growth.
• With capital added, growth is \(\tfrac{3}{2}\gamma\bar{n}\), and transition dynamics explain decades of catch-up.
• In the data, raw materials got cheaper, and the U.S. slowdowns were mostly TFP slowdowns.

Ask each student to write one sentence answering the opening question: can new ideas keep income growing forever? A good answer: yes, as long as the number of people creating ideas keeps growing; saving and research policies change the level of the path, not its long-run slope.

This closes the textbook’s study of long-run growth (Chapters 3 to 6). The open question is why some countries have the parameters, investment, research and access to ideas, that put them on high paths, and others do not.

Walk the three branches of the mind map in about a minute each: why ideas are different (cyan), the Romer model (lavender), ideas in the data (green). Click the image to zoom in.

Sustained growth comes from nonrival ideas; saving and research policies move the level of the path, not its slope

Next: Chapter 7, The Labor Market, Wages and Unemployment

The growth chapters end here. Chapter 4 explained income levels with a production function; Chapter 5 made capital endogenous and found its limits; this chapter made ideas endogenous and found the engine of growth.

Chapter 7 turns to the labor market: why wages rise with productivity, and why some people who want a job cannot find one.

Sources

• Figs. 1, 2, 3, 4, 7 and 8, the economics of ideas and the Romer and Solow–Romer models. Model figures adapted from Figs. 6.1 to 6.4, 6.7 and 6.8 in Jones (2024).
• Fig. 5, commodity prices. Jacks, D. (2019), “From Boom to Bust: A Typology of Real Commodity Prices in the Long Run,” Cliometrica 13(2), doi:10.1007/s11698-018-0173-5; World Bank, Commodity Price Data (Pink Sheet); U.S. BLS, CPI (CPIAUCSL), via FRED. Inspired by Fig. 6.5 in Jones (2024).
• Fig. 6, U.S. and world growth. U.S. BEA, GDPCA and B230RC0A052NBEA, and NBER recessions (USREC), via FRED; Penn World Table 11.0 (Feenstra, R., R. Inklaar and M. Timmer 2015, American Economic Review 105(10), doi:10.34894/FABVLR); regions: World Bank country groups. Growth accounting: U.S. Bureau of Labor Statistics, total factor productivity indexes. Inspired by Fig. 6.6 and Table 6.2 in Jones (2024).
• Paying for ideas. Williams, H. (2013), Journal of Political Economy 121(1); Sampat, B. and H. Williams (2019), American Economic Review 109(1); Budish, E., B. Roin and H. Williams (2015), American Economic Review 105(7); Moser, P. (2005), American Economic Review 95(4); Azoulay, P., J. Graff Zivin, D. Li and B. Sampat (2019), Review of Economic Studies 86(1); Kremer, M. (1998), Quarterly Journal of Economics 113(4); Kremer, M., J. Levin and C. Snyder (2020), AEA Papers and Proceedings 110.
• Limits to growth. Meadows, D. H., D. L. Meadows, J. Randers and W. Behrens (1972), The Limits to Growth, Universe Books; Popp, D. (2002), American Economic Review 92(1); Kiel, K., V. Matheson and K. Golden (2010), Ecological Economics 69(7); Acemoglu, D., P. Aghion, L. Bursztyn and D. Hémous (2012), American Economic Review 102(1).
• The slowdown. Gordon, R. (2016), The Rise and Fall of American Growth, Princeton University Press; Bloom, N., C. Jones, J. Van Reenen and M. Webb (2020), American Economic Review 110(4); Syverson, C. (2017), Journal of Economic Perspectives 31(2); Brynjolfsson, E. and A. McAfee (2014), The Second Machine Age, W. W. Norton; Agrawal, A., J. McHale and A. Oettl (2019), in The Economics of Artificial Intelligence, University of Chicago Press.
• Population and ideas. Kremer, M. (1993), Quarterly Journal of Economics 108(3); Jones, C. I. (2022), American Economic Review 112(11).
• The model. Romer, P. (1990), Journal of Political Economy 98(5).
• 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 6.