Week 4 · Chapter

Chapter 4 — Economic Growth & Productivity

Principles of Macroeconomics Free sample edition

The module's primary reading · ~35–45 minutes · Objective 4 (explain the sources of long-run economic growth and productivity, and compute and interpret growth rates, doubling times by the rule of 70, and per-capita growth), with Objective 1 (apply the tools of macroeconomic modeling — scarcity, opportunity cost, the production possibilities frontier, and the circular-flow model — and distinguish positive statements (what is) from normative statements (what ought to be)) · Read it first this week; the lecture, the tutorial, the quiz, and the Model Workshop all build on it.

In plain language

Some countries produce far more per person than others, and the gap usually comes from growth that piled up over many years. This week you measure that growth. A growth rate (how fast something gets bigger, as a percent of where it started) of 5 percent means next year's output is 5 percent larger than this year's. A shortcut called the rule of 70 (a quick way to estimate how long something takes to double) says an economy growing 7 percent a year doubles in about 10 years. You will also learn why output per person, not total output, tracks living standards, and where lasting growth comes from: better equipment, better skills, and better ideas.

Why you already care about this

Think about what changes over your lifetime. If you are 20 today, you may work until you are 65 or older. That is 45 years or more of an economy growing, or not. Over 45 years, growing 3 percent a year instead of 2 percent adds up to a very different way of life. Ask an older relative what life looked like when they were your age, and you will hear about things that are now faster, cheaper, or newly possible. Behind each of those changes is the same quiet force: an economy that makes a little more, year after year.

A little more each year turns out to matter a lot to you. In this chapter you will follow an illustrative economy that grows from 800 billion dollars of real output to 840 billion dollars in one year. You will see that 100 billion dollars growing at 2 percent reaches about 121.90 billion in ten years, while the same 100 billion growing at 7 percent reaches about 196.72 billion. The two rates are 2 percent and 7 percent. The results differ by about 75 billion dollars in only ten years.

So the student's question for this week is a big one. Why are some economies so much richer than others, and what does 2 percent versus 7 percent growth really mean over a lifetime?

By the end of this chapter you will be able to:

read long-run growth as an outward shift of the production possibilities frontier;
compute a growth rate by dividing the change by the starting value;
estimate a doubling time with the rule of 70 and say why it is an approximation;
compare two growth rates over ten years and explain how small differences compound;
compute per-capita growth as total growth minus population growth, and say what it means for living standards;
compute productivity as output per worker, and name the three sources of growth.

Carry this sentence with you: divide by where you started — and watch the rate, not the level.

Where we are

Already covered: Week 1, The Macro Perspective, gave you the production possibilities frontier. Week 2, Measuring Output, taught you to measure an economy with real GDP. Week 3, Measuring Inflation & Unemployment, taught you to turn two levels into a percentage change.
This week: how an economy's real output grows over many years, how to measure that growth and its doubling time, and where growth in output per worker comes from.
How they connect: growth is a percentage change in real GDP, the measure from Week 2, computed with the same (new − old) ÷ old × 100 move you used for inflation in Week 3, and pictured on the Week 1 frontier.
Where we're going next: Week 5, Aggregate Demand & Aggregate Supply, builds the model of the short run, where output can swing above or below the long-run path you study this week.

Three earlier ideas carry this chapter. From Week 1, the production possibilities frontier shows the most an economy can make with its resources and technology, and an outward shift of the whole frontier was your first look at growth. From Week 2, real GDP values output at fixed base-year prices, so a rise in real GDP is a rise in the amount of stuff made, not in prices. From Week 3, a rate of change is the change divided by the starting level, times 100. This week puts all three together and stretches them across years.

1. Define growth: the whole frontier moves outward

Picture a small town that, ten years ago, had one bakery, one clinic, and a bus that ran twice a day. Today it has three bakeries, a clinic with a new scanner, and buses every half hour. Nobody worked twice as hard. The town has more equipment, better-trained people, and better ways of doing things. It can now produce more of everything at once. That is what economists mean by growth.

Here is the course's definition.

Economic growth is a sustained rise over many years in an economy's real GDP, the amount of final goods and services it can produce; on a production possibilities frontier it shows up as an outward shift of the whole frontier.

Go back to the Week 1 labor-hours frontier. That illustrative economy had 24 million labor-hours. Each unit of consumer goods took 3 hours, and each unit of capital goods took 6 hours. So the frontier was the line 3x + 6y = 24, where x is the units of consumer goods and where y is the units of capital goods. Let x be counted in millions of units, and let y be counted the same way. The line runs from (8, 0) to (0, 4). The point (6, 3) sat outside it. It needed 3 × 6 + 6 × 3 = 18 + 18 = 36 million hours, and only 24 million existed. Written out in full, that is 36,000,000 hours needed against 24,000,000 hours available.

Worked example — reading growth on the frontier. Use the Week 1 frontier, which is illustrative and constructed for this course.

Today. With 24 million labor-hours, (6, 3) needs 36 million hours. It is unattainable.
After growth. Suppose, over many years, the economy gains more workers, better machines, and faster methods. Each unit of output takes fewer hours, or more hours exist. The whole frontier shifts outward.
The result. A point such as (6, 3), which today's 24 million labor-hours cannot reach, comes within reach.

In words: growth does not move the economy along its frontier; it moves the frontier itself, so combinations that were impossible become possible.

There is a second way to say this, and you will use it next week. The most an economy can produce with its labor, capital, and technology in normal, full use has its own name.

Potential output is the real output an economy can produce when its labor, capital, and technology are in normal, full use; long-run growth raises it year after year.

You can picture growth as a steady rightward march of potential output. Next week, in the AD–AS model, potential output becomes a vertical line on a graph. For now, keep the picture of a frontier that moves outward over the years.

Week 1 also hinted at a choice that feeds growth. A nation that makes more capital goods today gives up consumer goods today. Those machines and buildings help produce more later, so the frontier moves outward faster. Choosing capital goods today trades some comfort now for more capacity later.

Misconception check. Students often think any rise in output is growth. Suppose the Week 1 economy sits at the inside point (2, 2), with 6 million labor-hours idle, and then moves to (4, 2) on the frontier. Output rose, but the frontier did not move. That is idle resources going back to work, not long-run growth. The cure: ask, "Did the frontier move, or did the economy move toward it?" Only the first is growth.

Pause and reflect — choose one. Pick either prompt; they count the same.

(a) Name one change in your town or neighborhood over the last ten years that let it produce more of something without anyone working longer hours. Explain whether it came from new equipment, new skills, or a new method.
(b) Name one thing your family or a family you know can afford today that an earlier generation could not. Explain in two sentences how growth in output made it possible.

2. Compute a growth rate: divide by where you started

Suppose your rent goes from 1,000 dollars a month to 1,050 dollars. The change is 50 dollars. Is that a big change? It depends on where you started. Fifty dollars on a 1,000-dollar rent is 5 percent. Fifty dollars on a 500-dollar rent would be 10 percent. A growth rate puts every change on the same footing by dividing it by its starting value.

A growth rate is the percentage change in a quantity from one period to the next, found as the new value minus the old value, divided by the old (starting) value, times 100.

As a formula on one line: growth rate = (new − old) ÷ old × 100. This is the same move you made for inflation in Week 3. There you divided the change in the CPI by the old CPI. Here you divide the change in real GDP by the old real GDP.

Worked example — the illustrative growth economy. This economy is constructed for this course, not real data. Real GDP is in billions of dollars.

Real GDP in the illustrative growth economy, in billions of dollars
Year	Real GDP
Last year	800
This year	840

Find the change. 840 − 800 = 40.
Divide by the starting value. The starting value is last year's 800, so 40 ÷ 800 = 0.05.
Turn it into a percent. 0.05 × 100 = 5%.

In words: the economy produced 5 percent more real output than the year before. Because this is real GDP, the 5 percent is more stuff made, not higher prices.

Misconception check. Students often divide by the new value. They write 40 ÷ 840 × 100 ≈ 4.76% and report a growth rate a little under 5 percent. That divides by where the economy ended, not where it started. The wrong base makes the growth rate too small. The cure: say the formula out loud, "change divided by the starting value," and point to last year's number before you divide.

In practice (your tools). In Google Sheets, type last year's real GDP in B2 (800) and this year's in B3 (840). In C3, type =(B3-B2)/B2*100. The cell shows 5. The formula divides by B2, the cell for the starting year. If you add more years below, copy C3 down the column, and each row will divide by the year before it.

Here is a caveat for you about the word "growth." A growth rate of real GDP measures output. A growth rate of nominal GDP mixes output with prices. Before you call any rise "growth," check that the number is real, as Week 2 taught.

3. Estimate doubling time: the rule of 70

Imagine a savings account, a city's population, or a country's output, each growing at a steady rate. A natural question is, "How long until it doubles?" You could multiply by the growth factor year after year until you pass twice the starting amount. That works, but it is slow. Economists use a shortcut.

Doubling time is the number of years a quantity takes to reach twice its starting size when it grows at a steady rate.
The rule of 70 is a shortcut that estimates doubling time as about 70 divided by the growth rate in percent; it is an approximation, close for the small rates economies usually grow at.

As a formula on one line: doubling time ≈ 70 ÷ the growth rate in percent. The "≈" sign means "approximately equal to." The 70 is a rounded number. It comes from 69.3, a constant that appears when you solve the doubling problem exactly, and rounding it to 70 makes the division easy to do in your head.

Worked example — the rule-of-70 table. These rates are illustrative, constructed for this course.

Doubling times by the rule of 70 for four growth rates, in years
Growth rate	Rule of 70	Doubling time
2%	70 ÷ 2	35 years
5%	70 ÷ 5	14 years
7%	70 ÷ 7	10 years
10%	70 ÷ 10	7 years

At 2%. 70 ÷ 2 = 35 years to double.
At 5%. 70 ÷ 5 = 14 years to double. The illustrative growth economy from Section 2 grew 5 percent, so if it kept that pace, its output would double in about 14 years.
At 7%. 70 ÷ 7 = 10 years to double.
At 10%. 70 ÷ 10 = 7 years to double.

In words: the faster the growth rate, the shorter the doubling time, and the rule of 70 turns any rate into a doubling time with one division.

Does the shortcut really work? You can check it at 2 percent. Growing 2 percent a year for 35 years means multiplying by 1.02 a total of 35 times, written 1.02³⁵. That product is about 2.0. So 1.02³⁵ ≈ 2.0, and 35 years of 2 percent growth does about double a quantity. The shortcut works.

Misconception check. Students often multiply instead of divide. At 2 percent they write 70 × 2 = 140 "years." A faster growth rate should give a shorter doubling time, but multiplying makes it longer. The cure: ask, "Should a faster rate double sooner or later?" Sooner, so the rate goes on the bottom: 70 ÷ 2 = 35.

In practice (your tools). In Google Sheets, type the growth rates 2, 5, 7, and 10 in A2 through A5. In B2, type =70/A2 and copy it down to B5. The column shows 35, 14, 10, and 7. To check the 2 percent row the long way, type =1.02^35 in any cell. It shows about 2.0. In Desmos, typing y=70/x draws the whole rule as a curve, and you can read the doubling time for any rate.

4. Watch compound growth: small differences widen

Picture two friends who each start with the same amount in a savings account. One earns 2 percent a year, and the other earns 7 percent. After one year the difference is small. After ten years it is not. The reason is that each year's growth is added to the base, and next year's growth applies to that bigger base. Growth grows on itself.

Compound growth is growth in which each year's increase is added to the base for the next year, so the quantity grows on its earlier growth and small differences in rates widen into large gaps over time.

Worked example — 100 billion dollars for ten years. These figures are illustrative, constructed for this course.

At 2 percent. Each year multiplies output by 1.02. Ten years multiply it by 1.02¹⁰. So 100 × 1.02¹⁰ ≈ 121.90. The economy grows from 100 billion dollars to about 121.90 billion.
At 7 percent. Each year multiplies output by 1.07. Ten years multiply it by 1.07¹⁰. So 100 × 1.07¹⁰ ≈ 196.72. The same 100 billion dollars reaches about 196.72 billion.
Compare. From the same start, in the same ten years, the 7 percent economy ends with about 1.6-fold the output versus the 2 percent economy: 196.72 billion against 121.90 billion.
Check against the rule of 70. At 7 percent, 70 ÷ 7 = 10 years to double, so the rule predicts about 200 billion after ten years. The exact figure, about 196.72 billion, is nearly 200.

In words: a 7 percent economy almost doubles in ten years, while a 2 percent economy grows by about a fifth. The rates are 2 percent and 7 percent; the outcomes differ by about 75 billion dollars.

That is why economists, and you, should care about a point or two of growth. Over one year, 2 percent versus 3 percent is a small difference. Over a working lifetime of forty years or more, the same small difference compounds into a very different standard of living.

Misconception check. Students often treat the rule of 70 as exact. It is not. The 7 percent economy reaches about 196.72 billion in ten years, not exactly 200. The rule is a shortcut built from a rounded constant, and it is closest at the small rates economies usually grow at. The cure: write "≈," say "about," and use the rule for quick comparisons, not for keyed, exact answers.

In practice (your tools). In Desmos, type y=100*1.02^x and y=100*1.07^x on two lines. In these lines, let x be the number of years. Move along both curves to 10 years on the horizontal axis. The first reads about 121.90 and the second about 196.72. Watch how the gap between the curves keeps widening as x grows. In Google Sheets, =100*1.02^10 and =100*1.07^10 give the same two values.

5. Measure living standards: per-capita growth

Picture a pizza shared at a party. If the pizza gets 10 percent bigger but 10 percent more guests arrive, nobody gets a bigger slice. An economy is the same. Total output can grow quickly, but if the number of people grows equally fast, each person's share stays flat. To track living standards, economists divide output by population.

Real GDP per capita is real GDP divided by the population, which gives the amount of output per person and is the usual yardstick for an economy's average living standard.
Per-capita growth is the growth rate of real GDP per person, which is approximately the growth rate of total real GDP minus the growth rate of the population.

As a formula on one line: per-capita growth ≈ total growth − population growth. The "≈" is there because the subtraction is a close shortcut, not the exact answer, much like the rule of 70.

Worked example — an illustrative city. This city is constructed for this course. Its total output grows 3 percent in a year, and its population grows 1 percent.

Write the rule. Per-capita growth ≈ total growth − population growth.
Fill it in. 3% − 1% = 2%.

In words: output per person in this city rises about 2 percent a year, so the average resident's share of output is growing.

Worked example — an illustrative country. This country is also constructed for this course. Its total output grows 2 percent a year, and its population also grows 2 percent a year.

Fill in the rule. 2% − 2% = 0%.
Read it. Total output is growing, but output per person is not.

In words: total output grows but output per person stays about the same, so the average living standard is flat even though the economy is getting bigger.

Misconception check. Students often read impressive total growth as rising living standards. The illustrative country's output grew 2 percent a year, which sounds healthy, yet each person's share did not grow at all. The cure: before you say people are better off, subtract population growth from total growth and look at what is left.

In practice (your tools). In Google Sheets, put total growth rates in column B and population growth rates in column C. In D2, type =B2-C2 and copy it down. For the city, B2 holds 3 and C2 holds 1, and D2 shows 2. For the country, 2 and 2 give 0.

One limitation is worth naming. Real GDP per capita is an average. It says how much output there is per person, not how that output is shared. A country can raise its average while some groups gain a great deal and others gain little. Per-capita growth is the best single yardstick for average living standards, read with that caveat in mind.

6. Measure productivity: output per worker

Walk into a print shop where 5 workers turn out posters all day. Then imagine the shop gets a faster press, and the same 5 workers now turn out many more posters in the same day. Nothing about the workers' effort changed. What changed is how much each worker can produce. That is productivity.

Productivity is the amount of output produced per worker, or per hour of work; rising productivity is the main source of rising living standards over the long run.

As a formula on one line: productivity = output ÷ number of workers.

Worked example — an illustrative print shop. This shop is constructed for this course.

Posters printed per day at an illustrative print shop, before and after a faster press
	Workers	Posters per day	Posters per worker
Before	5	1,000	200
After a faster press	5	1,250	250

Productivity before. 1,000 ÷ 5 = 200 posters per worker per day.
Productivity after. 1,250 ÷ 5 = 250 posters per worker per day.
Productivity growth. (250 − 200) ÷ 200 × 100 = 50 ÷ 200 × 100 = 25%.

In words: each worker produces a quarter more than before, and the new press is the reason.

Notice that the growth-rate formula from Section 2 did the work in step 3. It is the same move — change divided by the starting value — applied to output per worker instead of to real GDP.

Why should you care about productivity for a whole country? Output per person can rise in only two broad ways. More of the population can work, or each worker can produce more. The first has limits; not everyone can work, and nobody can work every hour. The second has no fixed ceiling. Over the long run, rising productivity is what lifts output per person and, with it, the average way of life.

Misconception check. Students often confuse more output with higher productivity. Suppose the illustrative print shop hired 5 more workers and printed 2,000 posters a day on the old presses. Total output doubled, but 2,000 ÷ 10 = 200 posters per worker, exactly the old figure. Productivity did not change at all. The cure: divide by the number of workers before you call it a productivity gain.

In practice (your tools). In Google Sheets, put workers in B2 and B3 (5 and 5) and posters per day in C2 and C3 (1,000 and 1,250). In D2, type =C2/B2 and copy it to D3; the column shows 200 and 250. In E3, type =(D3-D2)/D2*100. It shows 25, the productivity growth rate.

7. Name the sources of growth: capital, skills, and ideas

Think about what made the print shop's workers more productive. It was a machine. Now think of two other shops. In one, the workers take a course and learn to set up jobs faster. In another, the owner finds a smarter way to schedule jobs so the presses never sit idle. Each shop raised output per worker. Each did it a different way. Those three ways are the three sources of growth.

Physical capital is the stock of equipment, machines, tools, and buildings that workers use to produce goods and services; more and better physical capital lets each worker produce more.
Human capital is the education, skills, training, and health that workers bring to their jobs and that raise how much each of them can produce.
Technology, in growth economics, is the stock of ideas and methods for turning labor and capital into output, including new products and better ways of organizing work.

So the three sources of growth in productivity are physical capital (more and better equipment), human capital (education, skills, and health), and technology (new ideas and better ways of producing).

Economists tie the three together with one framework.

An aggregate production function is a description of how an economy's total output depends on its labor, its physical and human capital, and its technology taken together.
The Solow growth model is the framework economists use to organize the sources of long-run growth with an aggregate production function, tracing how capital, labor, and technology together set output per worker over time.

The model is named for the economist R. M. Solow, whose 1956 paper set it out. This course names the model and uses its organizing idea; it does not derive the model's equations. The organizing idea is enough for now: output per worker rises when workers have more and better capital, more skills, and better technology.

Worked example — sorting three illustrative changes. Each change is constructed for this course. Name the source of growth.

A delivery company buys electric vans that carry twice as much. More and better equipment, so this is physical capital.
A hospital trains its staff on a new scheduling routine, and each nurse sees more patients safely. The staff's skills grew, so this is human capital. If the routine itself is a new method, the idea behind it is technology.
A farm adopts a new seed variety that yields more grain per acre. A new idea about how to produce, so this is technology.

In words: every gain in output per worker can be traced to more capital, more skill, better ideas, or a mix of them.

Misconception check. Students often think technology means only computers and gadgets. In growth economics it means any better way of turning labor and capital into output, including a smarter schedule, a better seed, or a new way to organize a store. The cure: ask, "Is this a new thing, a new skill, or a new method?" A new method is technology even if no machine is involved.

Pause and reflect — choose one. Pick either prompt; they count the same.

(a) Pick a job you have done or watched closely. Name one change that would raise output per worker there, and say which source of growth it uses.
(b) Pick a public service you use, such as a library, a bus line, or a clinic. Name one change that would let it serve more people with the same staff, and say which source of growth it uses.

Key terms

Economic growth — A sustained rise over many years in an economy's real GDP, the amount of final goods and services it can produce; on a production possibilities frontier it shows up as an outward shift of the whole frontier. (first used: 1. Define growth: the whole frontier moves outward)
Potential output — The real output an economy can produce when its labor, capital, and technology are in normal, full use; long-run growth raises it year after year. (first used: 1. Define growth: the whole frontier moves outward)
Growth rate — The percentage change in a quantity from one period to the next, found as the new value minus the old value, divided by the old (starting) value, times 100. (first used: 2. Compute a growth rate: divide by where you started)
Doubling time — The number of years a quantity takes to reach twice its starting size when it grows at a steady rate. (first used: 3. Estimate doubling time: the rule of 70)
Rule of 70 — A shortcut that estimates doubling time as about 70 divided by the growth rate in percent; it is an approximation, close for the small rates economies usually grow at. (first used: 3. Estimate doubling time: the rule of 70)
Compound growth — Growth in which each year's increase is added to the base for the next year, so the quantity grows on its earlier growth and small differences in rates widen into large gaps over time. (first used: 4. Watch compound growth: small differences widen)
Real GDP per capita — Real GDP divided by the population, which gives the amount of output per person and is the usual yardstick for an economy's average living standard. (first used: 5. Measure living standards: per-capita growth)
Per-capita growth — The growth rate of real GDP per person, which is approximately the growth rate of total real GDP minus the growth rate of the population. (first used: 5. Measure living standards: per-capita growth)
Productivity — The amount of output produced per worker, or per hour of work; rising productivity is the main source of rising living standards over the long run. (first used: 6. Measure productivity: output per worker)
Physical capital — The stock of equipment, machines, tools, and buildings that workers use to produce goods and services; more and better physical capital lets each worker produce more. (first used: 7. Name the sources of growth: capital, skills, and ideas)
Human capital — The education, skills, training, and health that workers bring to their jobs and that raise how much each of them can produce. (first used: 7. Name the sources of growth: capital, skills, and ideas)
Technology — In growth economics, the stock of ideas and methods for turning labor and capital into output, including new products and better ways of organizing work. (first used: 7. Name the sources of growth: capital, skills, and ideas)
Aggregate production function — A description of how an economy's total output depends on its labor, its physical and human capital, and its technology taken together. (first used: 7. Name the sources of growth: capital, skills, and ideas)
Solow growth model — The framework economists use to organize the sources of long-run growth with an aggregate production function, tracing how capital, labor, and technology together set output per worker over time. (first used: 7. Name the sources of growth: capital, skills, and ideas)

Summary

Long-run growth is an outward shift of the whole production possibilities frontier.
A growth rate is the change divided by the starting value, times 100.
Doubling time is about 70 divided by the growth rate in percent.
The rule of 70 is an approximation, so write "about."
Small differences in growth rates compound into large gaps over time.
Per-capita growth is about total growth minus population growth.
Productivity is output per worker, and it is what lifts living standards.
The three sources of growth are physical capital, human capital, and technology.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

An illustrative regional economy's real output rises from 250 billion dollars to 260 billion dollars in a year. Compute its growth rate, say what a student who divided by the wrong value would get, and use the rule of 70 to estimate how long the region takes to double at that rate.
An illustrative town's population is growing 3.5 percent a year. Use the rule of 70 to estimate how many years its population takes to double, and explain why your answer is written with "about."
An illustrative island nation's real GDP grows 4 percent a year while its population grows 2.5 percent a year. Estimate its per-capita growth, and say in words what it means for the average resident.
An illustrative bike-repair cooperative has 6 mechanics who repair 90 bikes a week. After a training course, the same 6 mechanics repair 108 bikes a week. Compute productivity before and after, the productivity growth rate, and name the source of growth.

Answers (worked reasoning — read after attempting)

The change is 260 − 250 = 10. Divide by the starting value: 10 ÷ 250 = 0.04, and 0.04 × 100 = 4%. The region produced 4 percent more real output than the year before. A student who divided by the new value would get 10 ÷ 260 × 100 ≈ 3.85%, which is too small. At 4 percent, the rule of 70 gives 70 ÷ 4 = 17.5, so the region doubles in about 17.5 years.
The rule of 70 gives 70 ÷ 3.5 = 20, so the town's population doubles in about 20 years. You write the answer with "about" because the rule of 70 is an approximation built from a rounded constant; the exact doubling time is close to 20 years, but not exactly 20.
Per-capita growth ≈ total growth − population growth = 4% − 2.5% = 1.5%. Output per person rises about 1.5 percent a year, so the average resident's share of output is growing, but much more slowly than the 4 percent headline suggests.
Before: 90 ÷ 6 = 15 bikes per mechanic per week. After: 108 ÷ 6 = 18 bikes per mechanic per week. Productivity growth: (18 − 15) ÷ 15 × 100 = 3 ÷ 15 × 100 = 20%. The mechanics' skills improved through training, so the source of growth is human capital.

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, and the quiz — and next week the AD–AS model shows what happens to output in the short run, around this long-run path.

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