The five richest countries produce 82 times as much per person as the five poorest

82× GDP per person in 2023: Switzerland, Luxembourg, Norway, Singapore and Ireland against Burundi, the D.R. of the Congo, the Central African Republic, Mozambique and Madagascar

Is the gap about how many machines each worker has, or about how well a country uses its machines and workers?

Open with the number. It is the notebook’s key number for Fig. 7 (Penn World Table 11.0, 2023): the average income of the five richest countries divided by the average of the five poorest. The textbook finds 75 for 2019.

Make it concrete: a worker in the richest group produces in about four and a half days what a worker in the poorest group produces in a year.

Ask the room for explanations and write them on the board: more machines, better schools, better technology, less corruption, natural resources, geography. By the end of the lecture we will have a way to split the gap into two parts: one due to capital and one due to everything else.

Today: three questions

• The production model. How do machines and workers turn into output, and who gets paid what?
• The model meets the data. If countries differed only in capital per person, how rich would each one be?
• TFP and institutions. How much of the gap is productivity, and what could explain it?

These are the three acts and the three branches of the mind map on the title slide. Act I builds and solves the model: one good, a fixed number of machines and workers, competitive firms. Act II takes its per-person prediction to data on 110 countries. Act III measures productivity as what the model cannot explain, and ends with four policy questions: capital flows, institutions, misallocation and the speed of reform.

This is the first behavioral model of the course. The growth facts of Chapter 3 are what it, and the Solow model of Chapter 5, try to explain.

The production model

Act I

A model is a small economy we can solve

• One good, measured in tons a year. Its price is 1, so every price is in units of the good.
• Given inputs: \(\bar{L}\) workers and \(\bar{K}\) machines. A bar marks a parameter, set outside the model.
• Technology: a production function \(Y = F(K, L)\).
• Competitive firms take the wage \(w\) and the rental price \(r\) as given and maximize profit.
• Markets clear: \(w\) and \(r\) adjust until firms want exactly the workers and machines that exist.

Five assumptions, five equations, five unknowns: \(Y\), \(K\), \(L\), \(r\) and \(w\).

The textbook’s good is ice cream; any single good works. One good means no relative prices between goods: the good is the numéraire, so the wage and the rental price are measured in tons of the good.

Why so simple? A model is a laboratory: we decide what the toy economy contains, then switch it on and see what it does. If the toy world reproduces some facts, we have learned which ingredients matter for them. If it fails, the way it fails is informative too, which is exactly what happens in Act II.

Notation: letters with a bar are parameters (given), letters without a bar are endogenous (solved for), and an asterisk marks equilibrium values.

Cobb-Douglas: double both inputs and output doubles

\[Y = F(K, L) = \bar{A}\,K^{1/3}L^{2/3}\]

\[F(\lambda K, \lambda L) = \bar{A}(\lambda K)^{1/3}(\lambda L)^{2/3} = \lambda^{1/3+2/3}\,F(K, L) = \lambda\,F(K, L)\]

Constant returns to scale: the exponents add up to one. A firm can always build a copy of itself.

Ā is productivity: with the same machines and workers, a larger Ā gives more output. It is the parameter Act III is about.

The derivation uses two rules for exponents: (ab)^c = a^c b^c and a^c a^d = a^(c+d). With exponents that sum to more than one we would have increasing returns, below one decreasing returns.

The replication argument: a second plant with the same staff and equipment makes the same output again, so doubling everything doubles output. Ask: why might it fail in practice? (Land and natural resources cannot be copied; coordination in a very large firm.) We set those aside here.

Why 1/3 on capital? Act I ends with the answer: it is the share of income that capital earns.

Experiment: add machines, keep the workers

A plant has 64 workers. With one machine it makes 16 tons of ice cream a year. How much does it make with eight machines and the same 64 workers? Write down a number.

Collect guesses before revealing anything. Many students say 128 tons (“eight times the machines”). Some will say “less than 128” without a number.

The numbers: with Ā = 1 and L = 64, one machine gives 1^(1/3) × 64^(2/3) = 16 tons. We come back to the guesses on the interpretation slide.

Each extra machine adds less than the one before

Model figure: \(Y = \bar{A}K^{1/3}L^{2/3}\) with \(\bar{A} = 1\). Adapted from Fig. 4.1 in Jones (2024), Macroeconomics, 6th ed.

How to read it: output in tons against the number of machines K, with the workforce fixed. The dots mark K = 1, 2, …, 10. The slider sets the number of workers L, from 8 to 112; it starts at 64. The title reports the output gained from the second machine and from the eighth.

Do live:

1. At L = 64, read the title: the second machine adds 4.16 tons, the eighth only 1.39 tons. Point at the dots getting closer together vertically.
2. Hover over K = 8: 32 tons. That answers the prediction.
3. Move the slider to 112 workers: the whole curve shifts up, and the second machine now adds 6.04 tons. Machines are more useful when more people run them.
4. Move to 8 workers: the curve is almost flat beyond a few machines.

The notebook’s own version also has a slider for the exponent: at an exponent of 1 the curve becomes a straight line and the marginal product stops falling.

Eight times the machines, only twice the output

With 64 workers	Output	Added by the last machine
1 machine	16 tons
2 machines	20.16 tons	4.16 tons
8 machines	32 tons	1.39 tons

\[MPK = \frac{\partial F}{\partial K} = \frac{1}{3}\,\bar{A}\left(\frac{L}{K}\right)^{2/3} = \frac{1}{3}\cdot\frac{Y}{K}\]

Diminishing returns: with the workforce fixed, the marginal product of capital falls as machines are added.

Reveal: 8^(1/3) = 2, so eight machines make 2 × 16 = 32 tons, not 128. The gains are the title of Fig. 1.

The formula: the marginal product of capital is the slope of the curve, the extra output from one more machine with L fixed. Apply the power rule to K^(1/3): the derivative is (1/3) K^(−2/3). Rewriting K^(−2/3) L^(2/3) as (L/K)^(2/3) shows the diminishing returns: more K lowers L/K and so the MPK. Rewriting it as Y/K shows a Cobb-Douglas property: each marginal product is the exponent times the average product.

By the same steps, MPL = (2/3) Y/L, which falls as workers are added and rises with more machines.

Doubling the machines does not double output

If a firm doubles its machines, it doubles its output.

Only if it doubles its workers too. Capital alone multiplies output by \(2^{1/3} \approx\) 1.26: each new machine has fewer hands to run it. Constant returns apply to all inputs at once.

The two properties sit side by side in the same function: constant returns to scale (all inputs) and diminishing returns to each input alone. Students often mix them up.

Ask: a company buys twice as many delivery vans but hires no new drivers. What happens to deliveries? (Little, once every driver has a van.)

This is the most important property for Act II: large differences in capital per person will turn into much smaller differences in income.

Firms hire each input until its marginal product equals its price

\[\max_{K,L}\; F(K, L) - rK - wL \quad\Longrightarrow\quad MPK = r, \qquad MPL = w\]

\[r = \frac{1}{3}\cdot\frac{Y}{K} \qquad\qquad w = \frac{2}{3}\cdot\frac{Y}{L}\]

Because the MPK falls as \(K\) rises, the hiring rule for capital is a downward-sloping demand curve. The same holds for labor.

The logic without calculus: if one more machine adds more output than its rent (MPK > r), renting it raises profit, so the firm keeps renting and the MPK falls. If MPK < r, returning a machine raises profit. Only at MPK = r is there nothing left to gain. Same for workers and the wage.

The supplies are fixed at K̄ and L̄, so the supply curves are vertical lines. With the two hiring rules, the production function and the two market-clearing conditions, we have five equations for Y, K, L, r and w.

Because both markets clear at once, this is a general equilibrium, not a single supply-and-demand diagram.

Experiment: the economy gets more machines

The economy doubles its stock of machines; the number of workers does not change. What happens to the rental price of capital, \(r\)? And to the wage, \(w\)? Up, down or unchanged?

Ask for a show of hands for each price. Most students get r right (more supply, lower price) and are split on w. Some think the wage falls because “machines replace workers”.

We will start from K̄ = L̄ = Ā = 1 and move to K̄ = 2.

More machines lower their price and raise the wage

Model figure: the hiring rules and fixed supplies, with \(\bar{A} = \bar{L} = 1\). Adapted from Fig. 4.2 in Jones (2024), Macroeconomics, 6th ed.

How to read it: left panel, the market for capital (rental price r against K); right panel, the market for labor (real wage w against L). The downward curves are the demand curves (MPK and MPL); the vertical lines are the fixed supplies; the dots are the equilibria. The slider sets the supply of capital K̄ from 0.5 to 2, with L̄ = Ā = 1. The title reports r, w and the factor shares.

Do live:

1. Start at K̄ = 1: r* = 0.333 and w* = 0.667.
2. Move to K̄ = 2: the capital supply line shifts right and r* falls to 0.210. In the right panel the labor demand curve shifts up, and w* rises to 0.840.
3. Move down to K̄ = 0.5: r* = w* = 0.529.
4. Watch the end of the title as you move: capital always gets 33 percent of output and labor 67 percent.

Each input is worth more when the other one is plentiful

	\(\bar{K} = 1\)	\(\bar{K} = 2\)
Rental price \(r^*\)	0.333	0.210
Wage \(w^*\)	0.667	0.840
Capital’s share of output	1/3	1/3

More machines move the economy down the capital demand curve, but shift the labor demand curve up: each worker now has more machines to work with.

Reveal the predictions. The rental price falls by about 37 percent: (1/2)^(2/3) ≈ 0.63. The wage rises by about 26 percent: 2^(1/3) ≈ 1.26. Both numbers are in the notebook’s Model overview.

The “machines replace workers” intuition fails here because there is only one good and every worker is employed. In this model, capital and labor are complements: more of one raises the marginal product of the other. Whether new kinds of machines (robots, AI) substitute for some workers is a question for a richer model with different kinds of labor.

A policy reading: in this model, investment raises wages. That is one reason capital per person matters for living standards.

The solution: every input is employed, and each is paid its marginal product

\[K^* = \bar{K}, \quad L^* = \bar{L}, \quad Y^* = \bar{A}\bar{K}^{1/3}\bar{L}^{2/3}\]

\[r^* = \frac{1}{3}\,\bar{A}\left(\frac{\bar{L}}{\bar{K}}\right)^{2/3} \qquad w^* = \frac{2}{3}\,\bar{A}\left(\frac{\bar{K}}{\bar{L}}\right)^{1/3}\]

Each unknown on the left, only parameters on the right: the model is solved.

The steps: market clearing gives K* and L* directly. Substitute them into the production function for Y. Substitute Y into the hiring rules: divide by K̄ for r* (K̄^(1/3)/K̄ = K̄^(−2/3)), divide by L̄ for w* (L̄^(2/3)/L̄ = L̄^(−1/3)).

Read the prices: r* depends on L̄/K̄, the number of workers per machine; w* on K̄/L̄, machines per worker. Neither depends on the size of the economy, only on the ratio. We return to this in Act II.

With the values of Fig. 2, K̄ = L̄ = Ā = 1, the solution is Y* = 1, r* = 1/3 and w* = 2/3.

Labor gets two‑thirds, capital one‑third, and nothing is left over

\[\frac{w^*L^*}{Y^*} = \frac{2}{3} \qquad \frac{r^*K^*}{Y^*} = \frac{1}{3} \qquad w^*L^* + r^*K^* = Y^*\]

With \(\bar{K} = \bar{L} = \bar{A} = 1\)
Output \(Y^*\)	1.000
Capital income \(r^*K^*\)	0.333 (33%)
Labor income \(w^*L^*\)	0.667 (67%)
Economic profit	0.000

The table is the notebook’s key numbers for Fig. 2. The shares come straight from the hiring rules: w* L/Y = (2/3)(Y/L)(L/Y) = 2/3, whatever Ā, K̄ and L̄ are. Only the exponent can change the split.

This is why the exponent on capital is 1/3: in U.S. data, capital receives roughly a third of GDP and labor roughly two-thirds, and the split has been fairly stable (Chapter 2 showed the labor share). The model is built to match that fact.

The last equation says income equals production, one of the national-accounts identities of Chapter 2. Profit is zero, which is what perfect competition leads us to expect.

Zero profit does not mean that firms earn nothing

If economic profit is zero, firm owners and shareholders get nothing.

Economic profit is what is left after paying all inputs, including capital. Accounting profit is revenue minus labor costs: the payment to capital, \(r^*K^*\), which is one-third of output. Shareholders own claims to that flow.

The two words “profit” mean different things. In the model, firms rent machines from their owners; in reality, corporations own their machines, and the owners of the corporations (shareholders) receive the capital income as accounting profit, dividends and retained earnings.

So the stock market puts a price on capital income: a firm’s market value is the value of its current and future accounting profits. That is the next figure.

Ask: if accounting profit is a third of output, why would economic profit be zero? (Because investors could have earned the same return elsewhere: r is the opportunity cost of capital.)

Capital income in the data

What does a stock price index measure, and how much has the real S&P 500 grown since 1920?

Ask students to guess a factor: is the real S&P 500 today 2 times, 10 times or 50 times its 1920 value? Then ask which crashes they can name.

The S&P 500 covers 500 large U.S. companies. “Real” means divided by the consumer price index, so the index is in dollars of the same purchasing power (here, September 2026 dollars).

The real S&P 500: about 50 times its 1920 value, with deep falls

Data: Robert Shiller, U.S. stock market data (S&P composite), extended with S&P 500 (SP500) via FRED; deflated by the BLS CPI (CPIAUCNS). Chart inspired by Fig. 4.3 in Jones (2024), Macroeconomics, 6th ed.

How to read it: the real S&P 500 index, monthly, 1920 to September 2026, in September 2026 dollars. The buttons switch between a ratio scale (the default) and a linear scale; the range slider under the chart picks a period. The title reports the factor and the growth rate a year: a factor of 49.8, or 3.7 percent a year.

Do live:

1. On the ratio scale, point at the big swings: the 1929 peak and the 1932 crash, the long real fall from the late 1960s to 1982, the 2000 peak and the 2009 trough.
2. Switch to the linear scale: the early decades flatten near zero and only recent swings look large. For a series that grows this much, the ratio scale is the honest picture (Chapter 3).
3. The textbook’s question: narrow the range slider to 1965–1995 and read the change. Then pick your own birth year to today.

Data caveat: the index is in 2026 dollars, so it is about 10 percent higher than the textbook’s 2023 dollars. It is a price index; dividends are not included, so it understates the total return to shareholders.

Share prices value future capital income, so they swing with expectations

• Over 1920–2026 the real index rose by a factor of 49.8: about 3.7% a year.
• The falls are large and can last a decade or more: 1929–32, the late 1960s to 1982, 2000–09.
• A share is a claim to a firm’s future accounting profits, its part of \(r^*K^*\). Its price moves with expected profits and with the rate used to discount them.

The stock market puts a value on capital income, the third of output that the model pays to capital.

The factor and the growth rate are the title of Fig. 3. The swings are the notebook’s “What to notice”.

Why is the index so volatile when GDP is not? Because the price reflects expectations about the whole future stream of profits. A small change in expected growth, or in the interest rate used to discount the future, changes the value of the whole stream. The production model is static and says nothing about expectations; Chapter 17 (investment) returns to asset prices.

The notebook also compares real share prices in 40 countries: Japan’s rose about 11-fold from 1959 to its 1989 bubble peak, then lost about three-quarters of their value by 2011. Not every stock market goes up.

The model meets the data

Act II

Per person, income depends on productivity and capital per person

\[y^* \equiv \frac{Y^*}{L^*} = \bar{A}\,\bar{k}^{1/3}, \qquad \bar{k} \equiv \frac{\bar{K}}{\bar{L}}\]

• Productivity \(\bar{A}\) raises income one for one.
• Capital per person raises it with strong diminishing returns: doubling \(\bar{k}\) multiplies \(y^*\) by \(2^{1/3} \approx 1.26\).
• The wage moves with income per person: \(w^* = \frac{2}{3}\,y^*\).

Divide Y* by L̄ and use L̄^(2/3)/L̄ = L̄^(−1/3). Everyone works in this model, so “per person” and “per worker” are the same thing.

The second bullet is the key prediction to take to the data: large differences in capital per person should show up as much smaller differences in income.

The wage formula will matter in Act III: if schooling raises wages by some percentage, it raises income per person by the same percentage, which lets us size the role of human capital.

Productivity is the only parameter that raises every outcome

A 1% rise in …	\(Y^*\)	\(y^*\)	\(w^*\)	\(r^*\)
productivity \(\bar{A}\)	+1%	+1%	+1%	+1%
capital \(\bar{K}\)	+⅓%	+⅓%	+⅓%	−⅔%
labor \(\bar{L}\)	+⅔%	−⅓%	−⅓%	+⅔%
both \(\bar{K}\) and \(\bar{L}\)	+1%	0	0	0

Each entry is an elasticity: the exponent of the parameter in the log of the solution.

Take logs of the solution: ln Y* = ln Ā + (1/3) ln K̄ + (2/3) ln L̄, and so on. The coefficient on each parameter is the percent change in the outcome when the parameter rises 1 percent.

Read the rows:

• Productivity shifts both demand curves up, so every outcome rises 1 for 1.
• More capital lowers its own price (down the demand curve) and raises the wage (labor demand shifts up). More labor does the opposite.
• For y* the elasticity with respect to L̄ is 2/3 − 1 = −1/3: the same output shared by more people.
• The last row is next slide’s misconception.

The factor shares are 1/3 and 2/3 in every row.

A bigger economy is not a richer one, per person

Countries with more workers and more machines are richer per person.

Scale both \(\bar{K}\) and \(\bar{L}\) up by the same percentage: total output rises in proportion, but \(y^*\), \(w^*\) and \(r^*\) do not change. Only the ratio \(\bar{K}/\bar{L}\), and productivity, matter for income per person.

Constant returns to scale is what makes size irrelevant per person. Small countries are among the richest: the five richest countries of the opening slide include Luxembourg, Singapore and Ireland.

Size can matter in richer models: through the number of ideas a large population produces (Chapter 6), or the size of the market for new products. In this model, it does not.

Ask: India and the United States have very different populations. Which number would you compare to say who is richer? (GDP per person, not total GDP.)

Development accounting: first, assume equal productivity

\[y^* = \bar{k}^{1/3} \qquad (\bar{A} = 1;\ y \text{ and } \bar{k} \text{ relative to the U.S.})\]

Capital per person (U.S. = 1)	Predicted income (U.S. = 1)
1/8	1/2
1/27	1/3

Diminishing returns compress income gaps: in logs, an income gap is one-third of the capital gap.

The assumptions: every country produces with the same Cobb-Douglas function, with exponent 1/3; y is real GDP per person and k̄ is the stock of buildings, machinery and equipment per person; both are measured relative to the United States, so the U.S. values are 1. And, for this first pass, the same productivity everywhere.

The data: Penn World Table 11.0. Capital per person is our own estimate, built up from each country’s past investment (the perpetual-inventory method).

Ask before showing the data: if this model is right, how poor can a country with one-hundredth of U.S. capital be? (1/100^(1/3) ≈ 0.22, about a fifth of U.S. income.)

Capital alone: how rich should each country be?

If countries differed only in capital per person, how much of U.S. income would China, India or Burundi have?

Let students place three countries on a sketch of the curve y = k^(1/3): where are they along the k axis, and where does the curve put them on the y axis?

The next figure places the textbook’s eight countries on the curve, for any year from 1980 to 2023.

Huge gaps in capital become modest gaps in predicted income

Data: Penn World Table 11.0 (rgdpo, pop, csh_i; Feenstra, Inklaar and Timmer 2015); capital per person is our perpetual-inventory estimate. The curve is \(y^* = k^{1/3}\). Chart inspired by Fig. 4.4 in Jones (2024), Macroeconomics, 6th ed.

How to read it: capital per person on the horizontal axis, the model’s predicted income on the vertical axis, both relative to the United States. Every dot sits on the curve by construction: these are predictions, not actual incomes. The slider picks the year (1980, 1990, 2000, 2010, 2019, 2023). Click “Other countries” in the legend to add the rest of the textbook’s sample as gray dots.

Do live:

1. In 2023, Burundi has less than 1 percent of U.S. capital per person and Switzerland about 60 percent more. Hover over both.
2. Point at the United Kingdom and Switzerland: Switzerland has more than twice the capital per person, but is predicted to be only about 30 percent richer.
3. Move the slider back to 1980: China slides down the curve, from 35 percent of U.S. capital per person in 2023 to about 3 percent. Its predicted income goes from 71 percent to 30 percent (the title).

Data caveat: capital per person in the early years depends on a starting guess, which is why the figure begins in 1980.

The curve is steep at the bottom and flat at the top

• China, 2023: 35% of U.S. capital per person → predicted 71% of U.S. income.
• India, 2023: about 9% of U.S. capital per person → predicted about 45%.
• Switzerland has more than twice the United Kingdom’s capital per person, yet is predicted to be only about 30% richer.

With equal productivity, the model predicts that poor countries should be much less poor than their capital stocks suggest.

The numbers are the title and the “What to notice” of Fig. 4.

The steep part of the curve at the bottom is also where the marginal product of capital is highest. Keep that in mind for the capital-flows question in Act III: with equal productivity, a machine in Burundi would add far more output than a machine in Switzerland.

Transition: these are predictions. Now we compare them with the incomes countries actually have.

Experiment: is the model right?

Plot each country’s predicted income against its actual income. If the model were right, every country would lie on the 45-degree line. Where do you expect the poor countries: above the line, on it or below it?

Ask for a show of hands. Remind students what “above” means: the model predicts more income than the country actually has.

Hint if needed: on the previous figure, India’s predicted income was about 45 percent of the U.S. level. Do students think India actually has 45 percent of U.S. income per person?

The model predicts poor countries should be far richer than they are

Data: Penn World Table 11.0 (rgdpo, pop, csh_i; Feenstra, Inklaar and Timmer 2015), 110 countries in 2023; capital per person is our perpetual-inventory estimate. Chart inspired by Fig. 4.5 in Jones (2024), Macroeconomics, 6th ed.

How to read it: actual GDP per person in 2023 on the horizontal axis, the model’s prediction k^α on the vertical axis, both relative to the United States and on ratio scales. The dashed line is the 45-degree line, where prediction equals data. The slider changes the exponent α on capital, from 1/4 to 1; it starts at 1/3. The title reports the share of countries above the line and the typical gap (the median of predicted over actual income).

Do live:

1. At α = 1/3: 95 percent of countries lie above the line, and the typical country is predicted to be 2.8 times richer than it is.
2. Hover over Burundi, at the far left: predicted about 1/6 of U.S. income, actual about 1/74.
3. Point at the top right: the richest countries lie close to the line. The miss shrinks as income rises.
4. Move α to 1: the poor countries move toward the line, but now many rich ones fall below it (next slides).

Data lesson from the notebook: Venezuela appeared at 0.2 percent of U.S. income, about 160 dollars a year, because its price data broke down during hyperinflation. We dropped it, leaving 110 countries. Always look at your data first.

Almost every country is poorer than its capital predicts

2019, U.S. = 1	Capital per person \(\bar{k}\)	Predicted \(\bar{k}^{1/3}\)	Actual \(y\)
Switzerland	1.557	1.159	1.252
Japan	0.985	0.995	0.645
Italy	0.825	0.938	0.693
China	0.287	0.659	0.265
India	0.084	0.437	0.111
Burundi	0.005	0.169	0.015

The model gets the ranking right but the gaps far too small. Something other than capital makes poor countries poor.

The table is the notebook’s version of the textbook’s Table 4.3, for 2019, from a newer Penn World Table release. The textbook’s numbers differ a little; the pattern is the same.

Reveal the predictions: the poor countries lie far above the line. Italy has about 82 percent of U.S. capital per person, so the model predicts 94 percent of U.S. income; Italy actually has 69 percent. Burundi is predicted at about a sixth of U.S. income and has about 1.5 percent.

Switzerland is one of the few countries richer than predicted. Ask: what could make a country richer than its capital? (Higher productivity: better skills, technology or institutions.)

A bigger exponent on capital is no fix

Exponent \(\alpha\)	Countries above the line	Typical gap
1/4	95%	3.3 ×
1/3	95%	2.8 ×
1/2	95%	2.0 ×
2/3	94%	1.4 ×
1	34%	0.8 ×

With \(\alpha = 1\) the poorest countries fit better, but the rich ones overshoot, and capital’s share of income is about one-third, not one.

The table is the slider of Fig. 5: each row is the title at that exponent. The typical gap is the median of predicted over actual income; below 1 means the model now predicts too little income for the typical country.

The argument: the exponent is not free. The model says it equals capital’s share of income (Act I), and that share is about one-third in the data. Raising the exponent to fit the income gaps would contradict the factor shares.

So we keep α = 1/3 and drop the other assumption: equal productivity. That is Act III.

TFP and institutions

Act III

TFP is the productivity that makes the model fit exactly

\[y^* = \bar{A}\,\bar{k}^{1/3} \quad\Longrightarrow\quad \bar{A} = \frac{y^*}{\bar{k}^{1/3}}\]

A country with 1/8 of U.S. capital per person is predicted to have 1/2 of U.S. income. If it actually has 1/8, its implied TFP is \(\frac{1/8}{1/2} = \frac{1}{4}\).

Total factor productivity (TFP) is a residual: everything that raises income other than physical capital.

We observe y and k̄ for every country, but there is no direct way to count Ā: no census of efficiency. So we assume the model is right and solve it for the one term we cannot observe.

Because it is chosen to make the model fit, TFP measures how much the model leaves unexplained. That is both its strength (it is a precise number) and its weakness (it is a name for what we do not explain), the subject of the Devil’s advocate at the end.

The example is the notebook’s: with income 1/8 and capital 1/8, capital explains a factor of 2 and TFP a factor of 4.

China: productivity versus capital

China has 35% of U.S. capital per person. What would its income be with U.S. productivity, and how much does it actually have?

Use the curve from Act II: 0.35^(1/3) ≈ 0.71. Ask students to guess China’s actual income relative to the U.S.; most will remember the “about a quarter to a third” from Chapter 3.

Lower TFP shifts the whole production function down

Data: Penn World Table 11.0 (rgdpo, pop, csh_i; Feenstra, Inklaar and Timmer 2015), 2023; capital per person is our perpetual-inventory estimate, and TFP is computed from it. Chart adapted from Fig. 4.6 in Jones (2024), Macroeconomics, 6th ed.

How to read it: output per person against capital per person, both relative to the U.S. The upper curve is the U.S. production function (A = 1); the lower curve is the compared country’s, with its own implied TFP. Each country has two points: where it would be with U.S. TFP (on the upper curve) and where it actually is (on the lower one). The slider picks the country: China, India, Brazil, Mexico, Nigeria, Japan.

Do live:

1. China: 71 percent of U.S. income with U.S. TFP, 30 percent with its own; implied TFP 0.42.
2. India: 45 percent versus 12 percent, TFP 0.27. The vertical distance between the two points is the effect of lower TFP.
3. Nigeria: 37 percent versus 6 percent, TFP 0.18.
4. Japan: almost as much capital per person as the United States (98 percent of U.S. income with U.S. TFP) but 61 percent of U.S. income, TFP 0.62.

Data caveat: the textbook, with 2019 data from an older Penn World Table release, finds a TFP of 0.35 for China. The newer release revises China’s income up, and China has kept growing since 2019.

Two countries with similar capital can be far apart in income

2023, U.S. = 1	Income with U.S. TFP	Actual income	Implied TFP
Japan	0.98	0.61	0.62
China	0.71	0.30	0.42
Brazil	0.59	0.26	0.44
India	0.45	0.12	0.27
Nigeria	0.37	0.06	0.18

Low TFP means the same machines and workers produce less: the curve itself is lower.

The numbers are the slider titles of Fig. 6. The first column is k̄^(1/3), the second y, and their ratio is TFP.

Point out Brazil, Mexico and China: very similar TFP (0.44, 0.44 and 0.42) and similar incomes, a quarter to a third of the U.S. level. In each, lower TFP costs more than half of the income the country would have with U.S. productivity.

Ask: what kinds of things would make the same machine and the same worker produce less in one country than in another? Collect answers for the last slides: skills, technology, management, corruption, roads and power cuts, regulations.

Is low TFP the exception or the rule?

Is low productivity a feature of a few poor countries, or a pattern across the whole world? And has it changed since 1980?

Ask students to sketch what a plot of TFP against income should look like if TFP explained most of the income gaps: a tight upward cloud. If capital explained most of them: a flat cloud near TFP = 1.

Rich countries have high TFP; the poorest have less than a tenth of U.S. TFP

Data: Penn World Table 11.0 (rgdpo, pop, csh_i; Feenstra, Inklaar and Timmer 2015), 110 countries, 1980–2023; capital per person is our perpetual-inventory estimate, and TFP is computed from it. Chart inspired by Fig. 4.7 in Jones (2024), Macroeconomics, 6th ed.

How to read it: implied TFP against GDP per person, both relative to the U.S. and on ratio scales. Each dot is a country. Press the play button to move through 1980, 1990, 2000, 2010, 2019 and 2023, or drag the year slider. The axes stay fixed, so the dots move.

Do live:

1. Point at the tight upward cloud: rich countries have high TFP, poor countries low TFP.
2. Hover over the bottom left: in the poorest countries TFP is below 1/10 of the U.S. level.
3. Hover over the top right: Ireland, Norway, Singapore and Switzerland have higher TFP than the United States.
4. A few countries, such as Egypt and Syria, have high TFP for their income level. Ask why (natural resources and measurement problems are candidates).
5. Press play and follow China: it moves up and to the right.

Most of the gap between rich and poor is TFP

\[\underbrace{\frac{y_{rich}}{y_{poor}}}_{82} = \underbrace{\frac{\bar{A}_{rich}}{\bar{A}_{poor}}}_{15.9\ (\text{TFP})} \times \underbrace{\left(\frac{\bar{k}_{rich}}{\bar{k}_{poor}}\right)^{1/3}}_{5.2\ (\text{capital})}\]

• Five richest against five poorest in 2023: TFP explains about two-thirds of the gap in logs, capital about one-third (\(\ln 5.2/\ln 82 \approx 0.37\)).
• Burundi against the U.S.: actual income 1/74; with capital differences only, 1/6.0; with TFP differences only, 1/12.2. Capital explains 42% of the log gap.

The numbers are the notebook’s key numbers “Capital or TFP?”. The textbook, for 2019, finds 75 = 5 × 15; our 2023 split is very close.

Why logs? The decomposition is a product, so the parts add up only in logs: ln 82 = ln 15.9 + ln 5.2 (up to rounding). “TFP explains two-thirds” means ln 15.9 / ln 82 ≈ 0.63.

Policy reading: even if a poor country matched rich-country capital per person, it would close only a third or so of the log gap. Raising productivity is where most of the gain lies, which is why the rest of the lecture asks what TFP is.

TFP is more than technology

TFP measures technology: poor countries are poor because they use older machines and methods.

TFP is a residual. It collects everything except physical capital: skills and schooling, technology, management, institutions, how inputs are spread across firms, natural resources, and errors in measuring capital and output.

Technology is one candidate, and Chapter 6 develops it. But the residual does not label itself.

Human capital can be sized with w* = (2/3) y*: if extra schooling raises wages by some percentage, it raises income per person by the same percentage. The textbook’s rough calculation: if U.S. adults have about 9 more years of school than adults in the poorest countries, at about 10 percent more wages a year of school, education alone would make U.S. income about 1.9 times as high. That is a large effect, but far from the factor of 15.9 that TFP accounts for in the rich-poor comparison.

Measurement: our capital stock is built from past investment, with a guess for the starting value. Errors in that guess, or in the price data, end up in TFP. Venezuela in Act II is an extreme example.

Why doesn’t capital flow from rich to poor countries?

With equal productivity, a country with 1/8 of U.S. capital per person would have a marginal product of capital \(8^{2/3} = 4\) times as high. Why don’t investors move their capital there?

What the evidence suggests

• With TFP of 1/4, the two effects cancel: capital earns the same return.
• Measured carefully, returns to capital differ little across countries.
• Weak institutions raise the risk of losing an investment.

Why it is contested

• Capital often flows out of fast-growing poor countries.
• Default risk and thin financial markets also matter.
• Measured returns depend on how capital is priced.

The question is the title of Lucas (1990, American Economic Review): with the production model and equal productivity, the return to capital in India should be many times that in the United States, and capital should rush there. It does not.

Left column: per person, MPK = (1/3) Ā k̄^(−2/3). With 1/8 of the capital, k̄^(−2/3) is 4 times higher; with TFP of 1/4, the two cancel exactly (the notebook’s Model overview). Caselli and Feyrer (2007, Quarterly Journal of Economics) measure marginal products directly and find them similar across rich and poor countries, once differences in natural resources and in the price of capital goods are accounted for. Alfaro, Kalemli-Ozcan and Volosovych (2008, Review of Economics and Statistics) find that institutional quality is the main explanation for the paradox.

Right column: Gourinchas and Jeanne (2013, Review of Economic Studies) document an “allocation puzzle”: among developing countries, those with faster productivity growth attract less foreign capital, not more. China, a fast grower, exports capital to the United States (Chapter 2’s trade balances).

Ask the policy students: what would make you invest your own savings in a factory abroad?

Institutions: what natural experiments suggest

When one country is split in two under different rules, what happens to the two incomes over a few decades?

What the evidence suggests

• Korea and Germany were split in two: same history and culture, different rules, very different incomes.
• At night, South Korea is full of light; the North is dark except its capital.
• Secure property and contracts make investment pay.

Why it is contested

• Which institutions matter, and how much, is hard to pin down.
• Schooling and income may improve institutions, not only the reverse.
• Rules that work in one country often fail elsewhere.

Olson (1996, Journal of Economic Perspectives, “Big Bills Left on the Sidewalk”) uses the divided countries as natural experiments: the pairs shared geography, culture and history until the split, so the large income gaps that opened afterwards point to institutions and policies. The textbook’s Fig. 4.8 is a night-time satellite photo of the Korean peninsula; there are no reliable income data for North Korea, so we cannot rebuild it from data. NASA’s “The Koreas at Night” shows a similar image (link in the book chapter).

Acemoglu, Johnson and Robinson (2001, American Economic Review) use the mortality of European settlers to isolate the effect of institutions on income, and find it large. Hall and Jones (1999, Quarterly Journal of Economics) link output per worker, and TFP, to “social infrastructure”.

Contested: Glaeser, La Porta, Lopez-de-Silanes and Shleifer (2004, Journal of Economic Growth) argue that human capital is the more basic cause, and that poor countries often grow first and improve institutions later.

Ask: name one institution you would change first in a poor country, and how you would know it worked.

Misallocation: the same inputs, used in the wrong firms

Can a country raise its TFP without new machines, more workers or new technology, just by moving inputs between firms?

What the evidence suggests

• Well-allocated inputs have equal marginal products in every firm.
• In China and India they differ a lot: some firms get cheap credit or favors.
• Moving inputs to the most productive firms would raise TFP substantially.

Why it is contested

• Part of the measured gaps is error in firm data.
• Some gaps reflect adjustment costs and risk, not bad policy.
• Removing distortions creates losers who resist it.

The idea follows from the hiring rule MPK = r: in an efficient economy, every firm faces the same prices, so marginal products are equalized. Taxes, subsidies, cheap credit for state firms, or favors for a politically connected firm break that rule. A connected but unproductive firm keeps workers and machines that a more productive rival would use better, and aggregate TFP falls.

Evidence: Hsieh and Klenow (2009, Quarterly Journal of Economics) measure marginal products across manufacturing plants in China and India and estimate large TFP gains from equalizing them. Restuccia and Rogerson (2008, Review of Economic Dynamics) show in a model that policies favoring some firms over others can lower TFP considerably.

Contested: Bils, Klenow and Ruane (2021, Journal of Monetary Economics) argue that measurement error inflates the measured dispersion. Asker, Collard-Wexler and De Loecker (2014, Journal of Political Economy) show that adjustment costs alone can create large measured gaps.

For the policy students: misallocation is attractive because it does not need more saving; it needs reforms of credit, competition and regulation.

Reform after 1990: two paths

After 1990, Russia replaced its institutions quickly, in a “big bang”, while China reformed step by step. Which path raised income per person more over the next three decades?

Ask students to sketch both paths, Russia and China, from 1990 to today. Many expect the fast reformer to do better, as many economists did at the time.

Russia’s income halved in the 1990s; China’s grew steadily

Data: Penn World Table 11.0 (rgdpo / pop, 2021 dollars; Feenstra, Inklaar and Timmer 2015); regions: World Bank country groups. Chart inspired by Fig. 4.9 in Jones (2024), Macroeconomics, 6th ed.

How to read it: GDP per person in 2021 dollars since 1990, for China and Russia. The buttons switch between a linear and a ratio scale; the menu picks a region, and clicking a country in the legend adds it. The title reports average growth a year since 1990: China +7.6 percent, Russia +2.3 percent.

Do live:

1. On the linear scale, follow Russia: it falls for most of the 1990s and passes its 1990 level only in 2006.
2. Switch to the ratio scale: China’s line is close to straight, a similar growth rate for three decades.
3. The notebook’s exercise: choose Europe & Central Asia and add Poland and Ukraine. Both left communism around 1990. Which looks more like Russia, and which more like China?

Russia lost half its income in a decade; China grew about 8% a year

• Russia: $18,735 per person in 1990, $9,617 at the low point in 1999, a fall of 49%. Back above the 1990 level only in 2006.
• China: $1,923 in 1990 and $17,973 in 2019, growth of 8.0% a year.
• Since 1990, average growth a year: China +7.6%, Russia +2.3%.

The same model, very different outcomes: the way institutions change can matter as much as which institutions a country ends up with.

The numbers are the notebook’s key numbers for Fig. 8; the growth rates since 1990 are the figure’s title.

Data note: the textbook says Russia’s income fell by 40 percent, but its own data show about 48 percent from 1990 to 1999, and ours about 49 percent. For China the textbook reports about 6 percent a year; the newer Penn World Table revises China’s 1990 income down and its 2019 income up, which raises the growth rate to about 8 percent.

In the production model, a fall in income with the same machines and workers is a fall in TFP: the old way of organizing production broke down before a new one worked. That is one way to read the Russian 1990s.

Big bang or gradualism?

Should a country leaving central planning reform everything at once, or step by step?

The case for gradual reform

• China kept the planned sector while letting a market sector grow beside it, so few people lost at first.
• Reforms in sequence let new institutions form before old ones disappear.

Why it is contested

• China started as a poor, rural economy; Russia was industrial. Starting points differ.
• Part of Russia’s measured fall was output nobody wanted.
• Some fast reformers, such as Poland, did much better than Russia.

The case for gradualism: Lau, Qian and Roland (2000, Journal of Political Economy) explain China’s “dual-track” reform: firms delivered planned quotas at plan prices and sold anything extra at market prices, so reform created winners without immediate losers. Roland (2000, “Transition and Economics”, MIT Press) argues that reform sequencing matters when institutions take time to build.

Contested: Sachs and Woo (1994, Economic Policy) argue that China’s success came from its starting point (a mostly rural labor force moving into industry), not from gradualism. Shleifer and Treisman (2005, Journal of Economic Perspectives) argue that Russia’s fall is overstated, because much Soviet output had little value, and that Russia became a fairly normal middle-income country. Stiglitz (2002, “Globalization and Its Discontents”) criticizes the speed of Russia’s privatization.

Ask the policy students: with Fig. 8 open, add Poland and Ukraine. What would you conclude about the speed of reform, and what else differs between the countries?

“TFP is just a measure of our ignorance”

A skeptic says: “You explained a third of the gap with capital and called the rest TFP. That is a name for what you cannot explain. The model has not explained why poor countries are poor.”

Reply: the size of the residual is the finding. It rules out the idea that poor countries only need more machines, and it tells us where to look: skills, technology, institutions and misallocation. Each of these can then be measured and tested, as the last slides showed.

Take the objection seriously. Abramovitz (1956, American Economic Review) called the growth residual “a measure of our ignorance”, and the label stuck.

The reply has three parts. First, a negative result is valuable: before this kind of accounting, many development policies were built on raising investment. The model says that equal capital would close only about a third of the log gap. Second, the residual can be broken down: human capital through wages (w* = (2/3) y*), misallocation through firm data, institutions through natural experiments. Third, the residual is a target for theory: Chapter 6 models technology as ideas.

But the skeptic has a point: TFP also absorbs measurement errors, and the model does not say why TFP differs. Ask: what would convince you that a particular cause, say institutions, explains a large part of TFP?

What to remember from this chapter

• \(Y = \bar{A}K^{1/3}L^{2/3}\): constant returns to scale, but diminishing returns to each input alone.
• Firms hire until \(MPK = r\) and \(MPL = w\). Capital earns one-third of output and labor two-thirds; profit is zero.
• Per person, \(y^* = \bar{A}\bar{k}^{1/3}\). With equal productivity, the model predicts income gaps far smaller than in the data.
• TFP, \(\bar{A} = y^*/\bar{k}^{1/3}\), is the residual. It explains about two-thirds of the log income gap between rich and poor.
• Institutions, misallocation, skills and technology are candidates for TFP, and how to change them is contested.

Ask each student to write one sentence answering the opening question: are poor countries poor because of fewer machines or because of lower productivity? Then ask what the model cannot tell us: where capital per person comes from. That is the next chapter.

Walk the three branches of the mind map in about a minute each: the production model (cyan), the model meets the data (lavender), TFP and institutions (green). Click the image to zoom in.

Capital per person explains only a small part of the gap between rich and poor; most of it is how well countries use what they have

Next: Chapter 5, The Solow Growth Model

The production model takes capital as given. The Solow model of Chapter 5 explains where capital per person comes from (saving and depreciation), and asks whether capital accumulation can sustain growth. The answer, by diminishing returns again, is no: it is the beginning of the case for ideas in Chapter 6.

Sources

• Figs. 1 and 2, the production model. Model figures adapted from Figs. 4.1 and 4.2 in Jones (2024).
• Fig. 3, the S&P 500. Robert Shiller, U.S. stock market data (S&P composite index), extended with S&P Dow Jones Indices, SP500, via FRED; deflated by the BLS CPI, CPIAUCNS, via FRED. Inspired by Fig. 4.3 in Jones (2024).
• Figs. 4 to 8, capital, TFP and income. Penn World Table 11.0 (rgdpo, pop, csh_i; Feenstra, R., R. Inklaar and M. Timmer 2015, American Economic Review 105(10), doi:10.34894/FABVLR); capital per person is our perpetual-inventory estimate, and TFP is computed from it. Regions: World Bank country groups. Inspired by Figs. 4.4 to 4.7 and 4.9 in Jones (2024).
• Capital flows. Lucas, R. E. (1990), American Economic Review 80(2); Caselli, F. and J. Feyrer (2007), Quarterly Journal of Economics 122(2); Alfaro, L., S. Kalemli-Ozcan and V. Volosovych (2008), Review of Economics and Statistics 90(2); Gourinchas, P.-O. and O. Jeanne (2013), Review of Economic Studies 80(4).
• Institutions. Olson, M. (1996), Journal of Economic Perspectives 10(2); Acemoglu, D., S. Johnson and J. Robinson (2001), American Economic Review 91(5); Hall, R. and C. Jones (1999), Quarterly Journal of Economics 114(1); Glaeser, E., R. La Porta, F. Lopez-de-Silanes and A. Shleifer (2004), Journal of Economic Growth 9(3).
• Misallocation. Hsieh, C.-T. and P. Klenow (2009), Quarterly Journal of Economics 124(4); Restuccia, D. and R. Rogerson (2008), Review of Economic Dynamics 11(4); Bils, M., P. Klenow and C. Ruane (2021), Journal of Monetary Economics 124; Asker, J., A. Collard-Wexler and J. De Loecker (2014), Journal of Political Economy 122(5).
• Reform. Lau, L., Y. Qian and G. Roland (2000), Journal of Political Economy 108(1); Roland, G. (2000), Transition and Economics, MIT Press; Sachs, J. and W. T. Woo (1994), Economic Policy 9(18); Shleifer, A. and D. Treisman (2005), Journal of Economic Perspectives 19(1); Stiglitz, J. (2002), Globalization and Its Discontents, Norton.
• The residual. Abramovitz, M. (1956), American Economic Review 46(2).
• 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 4.