Long-Run Economic Growth

If God had meant there to be more than two factors of production, He would have made it easier for us to draw three-dimensional diagrams.
— Robert Solow

5.1 The Production Function: How Economies Produce Goods and Services

In Chapter 4, we introduced gross domestic product as a measure of the value of goods and services produced in an economy. In Chapter 5, we move from measuring economic growth to explaining it.

One of the most important questions in macroeconomics is:

Why are some economies able to produce more goods and services than others?

The answer begins with understanding how economies transform resources into output.

Economists describe this relationship using a production function. A production function shows how an economy combines inputs to produce goods and services. The inputs used in production are often called factors of production.

The Solow Model focuses on three primary factors:

• Physical capital,

• Labor,

• Productivity.

These factors determine how much output an economy can produce.

Physical Capital

The variable $K$ represents physical capital. Physical capital includes the tools, machines, buildings, infrastructure, and equipment used to produce goods and services. Examples of physical capital include:

• factory equipment,

• computers,

• transportation networks,

• agricultural machinery,

• office buildings,

• power plants.

Physical capital is different from financial capital. Financial capital refers to money, stocks, bonds, and other financial assets. Financial capital can help businesses purchase resources, but it does not directly produce goods and services. Physical capital, by contrast, directly contributes to production.

A factory worker with advanced machinery can generally produce more output than the same worker using outdated tools. A farmer using a tractor can produce more food than a farmer using only hand tools. A software engineer with a powerful computer can often complete more work than one using limited technology.

Capital increases productivity because it allows workers to produce more output with the same amount of labor.

Labor

The variable $L$ represents labor, or the human effort used in production.

Labor includes:

• the number of workers,

• the hours workers spend producing goods and services,

• the skills and effort workers contribute.

An economy with more workers generally has the ability to produce more output than an economy with fewer workers. However, the number of workers alone does not determine economic success. The quality of labor also matters. A highly educated and skilled workforce can often produce more output than a workforce with fewer skills.

This is why economists distinguish between labor and human capital. Human capital refers to the knowledge, skills, education, and experience that make workers more productive.

Although the Solow Model can include human capital, this chapter will focus primarily on physical capital accumulation. We will return to the deeper causes of productivity and economic success in Chapter 8.

Productivity and the $A$ Term

The variable $A$ represents productivity, sometimes called technology or total factor productivity. Productivity measures how efficiently an economy transforms inputs into output.

Two economies may have the same amount of capital and labor but produce very different amounts of output if one economy uses those resources more effectively. For example, imagine two factories with identical numbers of workers and identical machines. If one factory uses better production methods, better organization, improved software, or superior management practices, it may produce more output than the other factory.

The difference is captured by the $A$ term.

Productivity includes many factors:

• technological knowledge,

• production techniques,

• organization and management,

• education and worker skills,

• institutions that encourage productive activity.

In this chapter, we treat $A$ as a given feature of an economy. Our focus is understanding how capital accumulation affects long-run growth. However, an important question remains:

Why do some economies have higher productivity than others?

The answer depends heavily on the institutions that shape incentives, investment, innovation, and cooperation. We will examine the role of institutions in creating productivity and economic growth in Chapter 8.

The Importance of Capital Accumulation

Capital does not appear automatically. Economies must devote resources toward producing new capital.

For example:

• A business must invest profits into new equipment.

• A government must invest in infrastructure.

• A society must save resources that can be used for future production.

When an economy accumulates capital, workers become more productive, and output increases. However, the relationship between capital and output is not unlimited. More capital increases production, but each additional unit of capital usually provides a smaller increase in output than the previous unit. This idea is known as diminishing marginal productivity of capital.

Diminishing Marginal Productivity of Capital

Suppose a farmer begins with no tractors. Adding the first tractor may dramatically increase production because the farmer can complete tasks that were previously impossible. Adding a second tractor increases production further, but the increase may be smaller because many important tasks are already being completed. Adding a third tractor helps even less. Eventually, the farmer may have more tractors than can be effectively used.

This does not mean additional capital is harmful. It means that the additional benefit of each new unit of capital becomes smaller as the amount of capital increases.

Diminishing marginal productivity is one of the most important ideas in the Solow Model because it explains why capital accumulation alone cannot generate unlimited economic growth. A country with very little capital can often grow rapidly by investing in additional capital. A country that already has a large amount of capital will experience smaller gains from additional investment. This idea will become central when we analyze why economies eventually approach a steady state.

Putting the Production Function Together

The production function tells us that output depends on three fundamental factors:

$$Y=AF(K,L).$$

An economy can increase output by:

1. increasing capital,

2. increasing labor,

3. improving productivity.

However, these sources of growth do not have identical effects. Increasing capital can raise output, but diminishing marginal productivity means the benefits become smaller over time. Increasing productivity is different. Improvements in technology, knowledge, and institutions can allow an economy to produce more output with the same amount of capital and labor. This distinction will be essential throughout this chapter.

5.2 Capital Accumulation: Savings and Depreciation

In the previous section, we introduced the production function and explained why physical capital is important for economic growth. Machines, factories, infrastructure, and equipment allow workers to produce more goods and services.

However, capital does not remain constant. Machines wear out, buildings deteriorate, and equipment becomes outdated. At the same time, economies invest in new capital that increases productive capacity.

The key question is therefore:

How does an economy’s capital stock change over time?

The Solow Model answers this question by focusing on two forces:

1. Investment, which increases the capital stock.

2. Depreciation, which reduces the capital stock.

The change in an economy’s capital stock depends on the difference between these two forces.

Savings Creates Investment

Where does investment come from?

In the Solow Model, investment comes from savings.

When households save part of their income, those resources become available for investment in physical capital. Businesses can use these resources to purchase machines, construct buildings, and expand production.

The model assumes that a constant fraction of output is saved and invested.

This fraction is represented by the savings rate:

$$s.$$

If an economy produces $Y$ units of output and saves a fraction $s$, then investment is:

$$Investment=sY.$$

For example, if an economy produces $1 trillion in output and saves 20% of that output, investment equals:

$$Investment=0.20(\$1\text{ trillion})$$

$$Investment=\$200\text{ billion}.$$

The remaining 80% of output is consumed.

The savings rate is important because it determines how quickly an economy accumulates capital. A higher savings rate means more resources are devoted to investment and capital accumulation.

However, as we learned in Section 5.1, capital has diminishing marginal productivity. Therefore, more investment does not produce constant increases in output forever.

Depreciation Reduces the Capital Stock

Capital is not permanent. Machines break down. Buildings require maintenance. Computers become outdated. Roads deteriorate. Economists refer to this process as depreciation. Depreciation reduces the economy’s capital stock over time. The amount of depreciation depends on:

• the amount of existing capital,

• the depreciation rate.

The depreciation rate is represented by:

$$\delta.$$

If an economy has capital stock $K$, depreciation is:

$$Depreciation=\delta K.$$

For example, suppose an economy has $500 billion worth of capital and a depreciation rate of 5%.

Then:

$$Depreciation=0.05(\$500\text{ billion})$$

$$Depreciation=\$25\text{ billion}.$$

This means the economy must invest $25 billion just to replace capital that wore out.

The Capital Accumulation Equation

Combining investment and depreciation gives us the central equation of capital accumulation:

$$\Delta K=sY-\delta K.$$

This equation tells us how the capital stock changes.

If:

$$sY>\delta K,$$

investment is greater than depreciation, and capital increases.

If:

$$sY<\delta K,$$

depreciation is greater than investment, and capital decreases.

If:

$$sY=\delta K,$$

investment exactly replaces depreciated capital, and the capital stock remains constant.

Worked Example: Capital Growth

Consider an economy with:

$$K=100$$

$$Y=200$$

$$s=0.25$$

and

$$\delta=0.10.$$

First, calculate investment.

$$Investment=sY$$

$$Investment=0.25(200)$$

$$Investment=50.$$

The economy creates 50 units of new capital.

Next, calculate depreciation.

$$Depreciation=\delta K$$

$$Depreciation=0.10(100)$$

$$Depreciation=10.$$

Therefore, the change in capital is:

$$\Delta K=50-10$$

$$\Delta K=40.$$

The capital stock increases by 40 units.

Why Growth Slows as Capital Increases

Now consider what happens when the economy already has a large amount of capital.

Suppose the same economy now has:

$$K=500$$

and

$$Y=600.$$

The savings rate and depreciation rate remain unchanged:

$$s=0.25$$

$$\delta=0.10.$$

Investment is:

$$Investment=0.25(600)$$

$$Investment=150.$$

Depreciation is:

$$Depreciation=0.10(500)$$

$$Depreciation=50.$$

Capital growth is:

$$\Delta K=150-50$$

$$\Delta K=100.$$

Capital still grows, but the economy is closer to a point where depreciation becomes increasingly important.

As capital increases, two things happen:

1. More capital must be replaced because there is more capital to maintain.

2. Each additional unit of capital produces less additional output because of diminishing marginal productivity.

These forces eventually push the economy toward a point where investment only replaces depreciation. That point is called the steady state, which we will analyze in section 7.3.

The Role of Savings

The savings rate plays an important role in determining how much capital an economy accumulates.

A higher savings rate means:

$$s\uparrow$$

which causes:

$$Investment\uparrow$$

and therefore:

$$K\uparrow.$$

However, a higher savings rate does not create unlimited economic growth. Because capital has diminishing marginal productivity, each additional unit of capital creates smaller gains in output. Eventually, the economy reaches a new steady state with a higher level of output but no permanent increase in the growth rate.

This distinction is crucial:

A higher savings rate can increase the level of output, but it cannot create permanent growth by itself.

5.3 The Steady State: Why Economic Growth Slows

In the previous section, we learned that capital accumulation depends on two forces:

1. Investment, which adds new capital.

2. Depreciation, which removes existing capital.

An economy grows when investment exceeds depreciation. However, the Solow Model makes an important prediction:

An economy cannot grow forever simply by accumulating more capital.

The reason is diminishing marginal productivity. As an economy accumulates more capital, each additional unit of capital produces a smaller increase in output. Eventually, investment is only large enough to replace capital that wears out. At this point, the economy reaches its steady state.

The steady state does not mean the economy is poor or inactive. It means that, without technological progress or productivity improvements, capital accumulation alone no longer increases output per worker.

The Simplified Solow Model

To understand the steady state, we simplify the production function introduced in Section 5.1.

Recall that the production function is:

$$Y=AF(K,L).$$

For this section, we assume:

• productivity is constant,

• the number of workers is constant,

• capital is the only changing input.

We therefore write the production function as:

$$y=\sqrt{k}$$

where:

• $y$ = output per worker,

• $k$ = capital per worker.

This function captures diminishing marginal productivity. As capital per worker increases, output per worker increases, but at a decreasing rate.

For example:

$$k=25$$

produces:

$$y=\sqrt{25}=5.$$

Increasing capital to:

$$k=100$$

produces:

$$y=\sqrt{100}=10.$$

Capital increased by 75 units, but output only increased by 5 units. This is diminishing marginal productivity: more capital increases output, but each additional unit of capital provides a smaller benefit.

Investment in the Solow Model

Investment depends on savings.

If the savings rate is $s$, then investment is:

$$i=sy.$$

Since:

$$y=\sqrt{k},$$

we can substitute:

$$i=s\sqrt{k}.$$

Investment increases as capital increases because more capital produces more output, creating more savings and investment. However, because output increases at a decreasing rate, investment also increases at a decreasing rate.

Depreciation in the Solow Model

Depreciation works differently from investment. While investment increases at a decreasing rate, depreciation increases at a constant rate. If the depreciation rate is:

$$\delta,$$

then:

$$d=\delta k.$$

As an economy has more capital, there is simply more capital that must be maintained and replaced.

For example, if the depreciation rate is 10%:

$$k=100$$

creates:

$$d=10.$$

But:

$$k=500$$

creates:

$$d=50.$$

More capital means more depreciation.

Finding the Steady State

The steady state occurs when investment exactly equals depreciation: $$Investment=Depreciation.$$ Using our equations: $$s\sqrt{k}=\delta k.$$ At this point, the economy is replacing exactly the amount of capital that wears out. The following model illustrates this relationship.

Graphical Intuition

The Solow Model can be understood by comparing investment and depreciation.

• Investment begins above depreciation when capital is low.

• Depreciation eventually catches up as capital increases.

• The intersection represents the steady state.

$$k<k^*$$ When capital is below the steady state: $$Investment>Depreciation.$$ The economy accumulates capital: $$k\uparrow$$ and output rises. $$k>k^*$$ When capital is above the steady state: $$Depreciation>Investment.$$ The capital stock declines: $$k\downarrow$$ until the economy returns to the steady state.

Examine the model presented in Figure 5.1. The figure displays the output (black), savings (green), and depreciation (red) for a simplified economy defined by $y = 5\sqrt{k}, s = 0.2, \delta=0.15$. You can visually see the diminishing marginal productivity of capital because the slope of the output and savings curves becomes flatter as capital is added to the economy (as seen on the horizontal axis). However, depreciation increases linearly with capital. Therefore, as capital increases, the amount capital increases by (as shown by the difference between the savings and depreciation lines) decreases. Therefore, as the economy approaches the steady state (shown by the intersection of the savings and depreciation lines) the economy grows at a slower rate, driven by the decline in additions to capital and the diminishing marginal productivity of the increases to capital.

Why Growth Slows as Economies Develop

The Solow Model explains an important pattern observed around the world:

Poor economies often grow faster than rich economies.

This does not mean poor countries automatically become rich. Instead, it means that economies with very little capital often have large opportunities for productive investment. A worker in a poor economy may become dramatically more productive after receiving basic tools, machinery, or infrastructure.

However, as capital accumulates:

• additional machines become less valuable,

• investment produces smaller gains,

• growth slows.

This is called convergence or catching-up growth. Countries that are below their steady state tend to grow rapidly as they accumulate capital. Countries near their steady state grow more slowly.

However, convergence is not automatic. A poor country with weak institutions, limited property rights, corruption, or poor incentives may fail to accumulate capital or improve productivity. This is why Chapter 8 will examine the institutions that influence productivity and long-run economic success.

5.4 The Limits of Capital Accumulation: Why Foreign Aid Alone Cannot Create Growth

In the previous section, we used the Solow Model to explain why economies grow rapidly when they have little capital and why growth slows as economies approach their steady state.

This insight leads to an important question:

If poor countries lack capital, why not simply give them more capital?

This question is central to debates about foreign aid and economic development. Many people argue that poor countries remain poor because they lack machines, factories, infrastructure, and other forms of capital. Therefore, providing additional capital should allow these economies to become richer.

The Solow Model provides an important insight:

Capital accumulation can increase output, but it cannot create permanent economic growth by itself.

Foreign aid can help an economy move closer to a higher level of output, but it does not automatically increase productivity or permanently shift an economy’s growth path.

Capital Accumulation Moves an Economy Toward a New Steady State

Recall the simplified Solow Model:

$$y=\sqrt{k}.$$

Output per worker depends on capital per worker. If an economy receives additional capital, then: $$k\uparrow$$ which causes: $$y\uparrow.$$ In the short run, additional capital increases production.

For example, imagine a poor country receives new agricultural equipment. Farmers can produce more food. Factories receive new machinery. Workers become more productive. Output increases. However, because capital has diminishing marginal productivity, each additional unit of capital produces smaller gains than the previous unit. The economy moves along the same production function. It does not create a new production function.

Understanding Foreign Aid Through the Solow Model

Suppose a country begins with a low level of capital per worker: $$k_1.$$ Because the country has little capital, the marginal benefit of additional capital is large. New investment generates significant increases in output. Now suppose the country receives foreign aid in the form of machines, infrastructure, or equipment. The capital stock increases: $$k_1 \rightarrow k_2.$$ As shown in Figure 5.1, the economy moves to the right along the production function. Output increases: $$y_1 \rightarrow y_2.$$ The country becomes richer. However, the economy has not changed the underlying relationship between capital and output. The production function remains the same: $$y=\sqrt{k}.$$ Because of diminishing marginal productivity, the additional output created by future capital accumulation becomes smaller. Eventually, the economy reaches the steady state. At that point: $$Investment=Depreciation.$$ The economy is richer than before, but growth slows.

Why Foreign Aid May Fail to Create Long-Run Growth

The Solow Model does not imply that foreign aid is always ineffective. Capital can clearly improve living standards, especially in the short-run. The problem is that capital alone does not determine long-run economic success.

For foreign aid to create lasting improvements, it must be combined with factors that increase productivity, including:

• effective institutions,

• secure property rights,

• incentives for investment,

• education and human capital,

• technological progress,

• efficient markets.

Without these conditions, additional capital may not be maintained or used effectively.

For example, a country may receive advanced agricultural equipment. If farmers lack secure property rights, they may have little incentive to invest in maintaining the equipment. If infrastructure is poorly managed, roads and electricity systems may deteriorate. If corruption prevents resources from reaching productive uses, the benefits of aid may be limited. The issue is not that capital is unimportant. The issue is that capital requires a productive environment. We will discuss the ideas presented here in detail in Chapter 8.

How Foreign Aid Can Reduce Long-Run Growth

In some cases, foreign aid can unintentionally reduce economic growth. This may occur if aid:

• reduces incentives for domestic saving,

• encourages dependence on external resources,

• supports inefficient industries,

• increases corruption,

• weakens market institutions.

For example, suppose a government receives enough foreign assistance that it no longer needs to develop a strong domestic tax system. Over time, this may weaken the government’s ability to provide effective public services or create incentives for economic activity. Similarly, if foreign aid replaces domestic investment rather than adding to it, the savings rate may decline.

Recall the capital accumulation equation: $$\Delta K=sY-\delta K.$$ If foreign aid reduces domestic savings: $$s\downarrow,$$ then investment falls: $$sY\downarrow.$$ If investment becomes smaller than depreciation: $$sY<\delta K,$$ the capital stock can actually decline. In this situation, an economy can move backward rather than forward.

The Difference Between Capital and Productivity

The central lesson of the Solow Model is that capital accumulation and productivity growth are different sources of economic improvement. Capital accumulation: $$K\uparrow$$ moves an economy along the existing production function. Productivity growth: $$A\uparrow$$ shifts the entire production function upward.

A country can become richer by accumulating capital, but sustained increases in living standards require continual improvements in productivity. This explains why some countries have achieved extraordinary long-run growth while others have struggled despite receiving large amounts of capital.

The question becomes:

What determines productivity?

The answer involves institutions, incentives, property rights, education, innovation, and economic organization. These ideas will be the focus of Chapter 8.

Chapter Summary

Economic growth is one of the most important topics in macroeconomics because it determines how living standards change over time. Small differences in economic growth rates can create enormous differences in prosperity across generations. This chapter introduced the Solow Model, one of the most influential frameworks economists use to understand why economies grow and why growth eventually slows.

Section 5.1 introduced the production function, which describes how economies transform inputs into output. The Solow Model begins with:

$$Y=AF(K,L)$$

where output depends on physical capital, labor, and productivity.

The chapter emphasized three important factors of production. Physical capital includes machines, buildings, infrastructure, and equipment used to produce goods and services. Capital increases worker productivity by allowing workers to produce more output. Labor represents the human effort used in production. The number of workers matters, but the skills, education, and experience of workers also influence productivity. Productivity ($A$) measures how efficiently an economy transforms inputs into output. Productivity captures factors such as technology, production methods, knowledge, and organizational efficiency.

A central insight of the chapter is that capital accumulation alone cannot explain unlimited economic growth. Physical capital has diminishing marginal productivity. As an economy accumulates more capital, each additional unit of capital produces a smaller increase in output than previous units.

Section 5.2 examined how an economy’s capital stock changes over time. Capital accumulation depends on two forces:

1. Investment, which creates new capital.

2. Depreciation, which reduces existing capital.

In the Solow Model, savings creates investment: $$Investment=sY$$ where $s$ represents the savings rate. Depreciation reduces the capital stock: $$Depreciation=\delta K$$ where $\delta$ represents the depreciation rate. Combining these two forces gives the capital accumulation equation: $$\Delta K=sY-\delta K.$$

Capital increases when investment exceeds depreciation. Capital decreases when depreciation exceeds investment.

The section emphasized that a higher savings rate increases capital accumulation and raises output. However, because capital has diminishing marginal productivity, higher savings cannot create permanent increases in the growth rate. Instead, it raises the level of output that an economy can sustain.

Section 5.3 introduced the steady state, the central concept of the Solow Model. To simplify the analysis, the chapter used the per-worker production function: $$y=\sqrt{k}.$$ In this model:

• $y$ represents output per worker,

• $k$ represents capital per worker.

Investment is: $$i=s\sqrt{k}$$ and depreciation is: $$d=\delta k.$$ The steady state occurs where investment exactly equals depreciation: $$Investment=Depreciation.$$ At this point, the economy is replacing exactly the capital that wears out. Capital per worker remains constant, and without productivity growth, output per worker also stops increasing.

The Solow Model explains why economies often experience rapid growth when they are poor but slower growth when they become wealthy. Economies with little capital have large opportunities for productive investment. As capital accumulates, diminishing marginal productivity reduces the gains from additional investment. This process is known as convergence.

However, convergence is not automatic. Poor countries do not necessarily catch up to rich countries. Differences in institutions, incentives, technology, education, and productivity can cause countries to follow very different growth paths.

Section 5.4 examined the limits of capital accumulation by analyzing foreign aid. A common argument is that poor countries remain poor because they lack capital, so providing additional capital should generate economic development.

The Solow Model provides a more nuanced explanation. Additional capital increases output, but it does not permanently increase economic growth. Foreign aid can move an economy toward a higher level of output by increasing capital per worker, but because of diminishing marginal productivity, the effects eventually decline.

The chapter emphasized the difference between capital accumulation and productivity growth.

Capital accumulation: $$K\uparrow$$ moves an economy along the existing production function. Productivity growth: $$A\uparrow$$ shifts the production function upward and allows sustained increases in living standards.

This distinction explains why some countries achieve long-run prosperity while others struggle despite receiving large amounts of capital. The institutions that encourage productivity growth—including property rights, reliable legal systems, incentives for innovation, and effective governance—will be the focus of Chapter 6.

The central lesson of this chapter is:

Capital accumulation can make an economy richer, but productivity growth is what allows an economy to continue becoming richer over time.

Key Terms

Capital accumulation

The process of increasing the economy’s stock of physical capital through investment.

Capital per worker ($k$)

The amount of physical capital available for each worker.

Convergence

The idea that poorer economies can grow faster than richer economies when they have similar savings rates, technology, and institutions.

Depreciation

The decline in the capital stock caused by wear, damage, or obsolescence.

Depreciation rate ($\delta$)

The percentage of the capital stock that wears out during a period.

Diminishing marginal productivity of capital

The idea that each additional unit of capital produces a smaller increase in output than the previous unit.

Foreign aid

Resources transferred from one country or organization to another to support economic development or humanitarian goals.

Human capital

The knowledge, skills, education, and experience that increase worker productivity.

Investment

Spending that creates new physical capital.

Labor ($L$)

Human effort used in the production of goods and services.

Physical capital ($K$)

Machines, buildings, infrastructure, and equipment used to produce goods and services.

Production function

A relationship describing how inputs are transformed into output.

$$Y=AF(K,L)$$

Productivity ($A$)

The efficiency with which an economy uses inputs to produce output.

Savings rate ($s$)

The fraction of output saved and invested rather than consumed.

Solow Model

A model of economic growth that explains output changes through capital accumulation, depreciation, labor, and productivity.

Stable state

The level of capital where investment equals depreciation and the capital stock remains constant.

Steady state

The level of capital where investment exactly replaces depreciation.

$$Investment=Depreciation$$

Technology

The knowledge, methods, and techniques used to produce goods and services.

Output per worker ($y$)

The amount of production generated by each worker.

Concept Check

Answer the following questions in your own words.

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