Makro

3 The Fundamentals of Economic Growth 53

4 Explaining Economic Growth in the Long Run 80

5 Labour Markets and Unemployment 105

6 Money, Prices, and Exchange Rates in the Long Run 137

PART II

The Macroeconomy inthe Long Run

Part II studies the long run. The long run is what economistsmean when they talk about the behaviour of an economy over aperiod of decades, rather than over short time spans of quartersor a few years. It describes attainable and sustainable aspects of the national economy, and goes far beyond the short-termperspective of the business cycle fluctuations described inChapter 1. Most important, it represents the basis of sustainableevolution of standards of living.

We begin with economic growth, the most fundamental of alllong-run macroeconomic phenomena. Economic growth is therate at which the real output of a nation or a region increasesover time. As the ultimate determinant of the poverty or wealthof nations, sustained economic growth is a central aspect of thelong run. Because this is such an important topic, two chaptersare dedicated to studying it.

Next, we look at the labour market, one of the most importantmarkets in modern economies. In the labour market, householdstrade time at work for the ability to purchase goods and servicesin the goods market. We will see how labour is allocated: where it comes from, who demands it, and how to think aboutunemployment.

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© Oxdord University Press 2009. Michael Burda and Charles Wyplosz. Macroeconomics A European Text 5e

PART II THE MACROECONOMY IN THE LONG RUN52

The last chapter in Part II introduces the long-run role of monetary and financialvariables: money, interest rates, and the nominal exchange rate, which are generallydenoted in nominal terms—in pounds or euros or dollars. Nominal variables determinethe real terms of exchange between goods within a country, between countries, or overtime—the command of resources represented by one type of goods and services overothers.

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The Fundamentals ofEconomic Growth

3.1 Overview 54

3.2 Thinking about Economic Growth: Facts and Stylized Facts 54

3.2.1 The Economic Growth Phenomenon 54

3.2.2 The Sources of Growth: The Aggregate Production Function 55

3.2.3 Kaldor’s Five Stylized Facts of Economic Growth 59

3.2.4 The Steady State 60

3.3 Capital Accumulation and Economic Growth 61

3.3.1 Savings, Investment, and Capital Accumulation 61

3.3.2 Capital Accumulation and Depreciation 61

3.3.3 Characterizing the Steady State 62

3.3.4 The Role of Savings for Growth 63

3.3.5 The Golden Rule 65

3.4 Population Growth and Economic Growth 68

3.5 Technological Progress and Economic Growth 71

3.6 Growth Accounting 73

3.6.1 Solow’s Decomposition 73

3.6.2 Capital Accumulation 75

3.6.3 Employment Growth 75

3.6.4 The Contribution of Technological Change 76

Summary 77

3

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The consequences for human welfare involved in ques-tions like these are simply staggering: Once one starts tothink about them, it is hard to think about anything else.

R. E. Lucas, Jr1

3.1 Overview

The output of economies, as measured by the grossdomestic product at constant prices, tends to growin most countries over time. Is economic growth a universal phenomenon? Why are national growthrates of the richest economies so similar? Why do some countries exhibit periods of spectaculargrowth, such as Japan in 1950–1973, the USA in1820–1870, Europe after the Second World War, or China and India more recently? Why do otherssometimes experience long periods of stagnation,as China did until the last two decades of the twen-tieth century? Do growth rates tend to converge, sothat periods of above-average growth compensatefor periods of below-average growth? What doesthis imply for levels of GDP per capita? These questions are among the most important ones ineconomics, for sustained growth determines thewealth and poverty of nations.

This chapter will teach us how to think system-atically about growth and its determinants. The production function is the tool that will help us

identify the most important regularities of eco-nomic growth among nations around the world.These stylized facts serve to point economic theoryin a sensible direction. First, investment can add tothe capital stock, and a greater capital stock enablesworkers to produce more. Second, the working population or labour force can grow, which meansthat more workers are potentially available for market production. This growth can arise for manyreasons—increases in births two or three decadesago, immigration now, or increased labour forceparticipation by people of all ages, especially bywomen. The third reason is technological progress.As knowledge accumulates and techniques im-prove, workers and the machines they work withbecome more productive. For both theoretical andempirical reasons, technological progress turns out to be the ultimate driver of economic growth.Because it is such an important topic, a detailed discussion of technological progress will be post-poned to Chapter 4.

1 Robert E. Lucas, Jr (1937–), Chicago economist and NobelPrize Laureate in 1995, is generally regarded as one of themost influential contemporary macroeconomists. Among his many fundamental contributions to the field, he hasresearched extensively the determinants of economic growth.

3.2 Thinking about Economic Growth: Facts and Stylized Facts

3.2.1 The Economic GrowthPhenomenonDespite setbacks arising from wars, natural disas-ters, or epidemics, economic growth seems like animmutable economic law of nature. Over the cen-turies, it has been responsible for significant, long-run material improvements in the way the world

lives. Table 3.1 displays the annual rate of increasein real GDP—the standard measure of economic

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CHAPTER 3 THE FUNDAMENTALS OF ECONOMIC GROWTH 55

output of a geographic entity—for various periods in a number of currently wealthy countries since1820. (The early data are clearly rough estimates.)Over almost two centuries, GDP has increased by60- to 100-fold or more, while per capita GDP hasincreased by 12- to 30-fold. Our grandparents areright when they say that we are much better o3than they were.

The table also reveals that the growth process is notvery smooth. We will see that this variation reflectsthe e3ect of wars, colonial expansion and annexa-tion, and dramatic changes in population as well as political, cultural, and scientific revolutions.Despite these swings, it is striking that the overallaverage growth of GDP per capita is remarkablysimilar across these countries, regardless of wherethey come from.

Small average annual changes displayed in Table 3.1cumulate surprisingly fast. The advanced econom-ies of the world grow by roughly 2–4% per year. A growth rate di3erence of 2% per annum com-pounds into 49% after 20 years, and 170% after halfa century. The recent phenomenal growth successes

of China and India and the troubling slowdowns inGermany and Japan show that growth is by no meansan automatic birthright. Moreover, fortunes canchange: as Box 3.1 shows, China was a leadingworld economy in the fourteenth century, only to fall into a half-millenium of decline and stagnation.For this reason, politicians and policy-makers areconcerned about persistent di3erences in growthrates between countries.

3.2.2 The Sources of Growth: The Aggregate Production FunctionIt is common and useful for economists to reasonabstractly about economic growth. To do so, theyusually think of an economy producing a single output—real GDP—using various inputs, or ingredi-ents. We discussed these inputs, the factors of pro-duction, in Chapter 2. To recap, these are:

(1) labour;

(2) physical capital, which is equipment and structures;

(3) land and other measurable factors of production.

Average rates of growth in GDP (% per annum) Av. growth GDP per capita1820–2006 1820–1870 1870–1913 1913–1950 1950–1973 1973–2001 1973–20061820–2006(% per annum)

Austria 2.1 1.4 2.4 0.2 5.2 2.5 2.4 1.6

Belgium 2.1 2.2 2.0 1.0 4.0 2.1 2.1 1.5

Denmark 2.4 1.9 2.6 2.5 3.7 2.0 2.0 1.6

Finland 2.6 1.6 2.7 2.7 4.8 2.4 2.6 1.8

France 2.0 1.4 1.6 1.1 4.9 2.3 2.1 1.6

Germany 2.2 2.0 2.8 0.3 5.5 1.8 1.7 1.6

Italy 2.1 1.2 1.9 1.5 5.5 2.3 2.0 1.5

Netherlands 2.4 1.7 2.1 2.4 4.6 2.4 2.3 1.4

Norway 2.7 2.2 2.2 2.9 4.0 3.4 3.2 1.9

Sweden 2.3 1.6 2.1 2.7 3.7 1.9 2.0 1.6

Switzerland 2.4 1.9 2.5 2.6 4.4 1.2 1.3 1.7

United Kingdom 2.0 2.0 1.9 1.2 2.9 2.1 2.2 1.4

Japan 2.7 0.1 2.4 2.2 8.9 2.7 2.6 1.9

United States 3.6 4.1 3.9 2.8 3.9 3.0 2.9 1.7

Source: Maddison (2007).

Table 3.1 The Growth Phenomenon

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Growth theory asks how sustained economicgrowth across nations and over time is possible. Dowe produce more because we employ more inputs,or because the inputs themselves become more productive over time, or both? What is the con-tribution of each factor? To think abstractly aboutgrowth, we will need a number of tools. The mostimportant tool we will use is the production func-

tion. The production function relates the output of an economy—its GDP—to productive inputs. The two most important productive inputs are thephysical capital stock, represented by K, and labouremployed, represented by L. The capital stockincludes factories, buildings, and machinery as wellas roads and railroads, electricity, and telephone

networks. Employment or labour is the total numberof hours worked in a given period of time. Thelabour measure L is the product of the average num-ber of workers employed (N) during a period (usuallya year) and the average hours (h) that they work dur-ing that period (L = Nh). We speak of person-hours oflabour input.3 Symbolically, the production func-tion is written:

(3.1) Y = F (K, L).+ +

Box 3.1 China and the Chinese Puzzle of Economic Growth

Most scholars agree that, at the end of the fourteenth century, China was the world’s most advanced eco-nomy. While Europe was just beginning to recover from centuries of inward-looking backwardness and relativedecline, Chinese society had reached a high degree ofadministrative, scientific, and economic sophistication.Innovations such as accounting, gunpowder, the mari-time compass, moveable type, and porcelain manufac-ture are just a few attributable to the Middle Empire.Marco Polo was one of many famous European traderswho tried to break into the Chinese market. Accord-ing to crude estimates by economic historian AngusMaddison, fourteenth-century Western Europe andChina were on roughly equal footing in terms of marketoutput—and many experts claim the Chinese weretechnically more advanced.2 Yet over the next six centuries,standards of living increased 25-fold in Western Europecompared with only sevenfold in China.

Most of that sevenfold increase in GDP per capita hasoccurred in China over the last 25 years. This makes the Chinese story a growth phenomenon without com-parison. After adopting far-reaching market economyreforms in the 1980s, economic growth has averaged a phenomenal 10.2% per annum since 1990. At thisrate, the economy will double in size every seven years.

If this growth continues, China will easily reach the standard of living of poorer EU countries by 2025.

The Chinese growth phenomenon raises a host ofintriguing questions. Why did China stagnate for cen-turies, while Europe flourished? Why did China literallyexplode in the 1990s? While there are many theories, itis widely agreed that the Chinese success story wouldhave been impossible without China’s recent policy ofopenness to international trade and foreign directinvestment. Almost as a converse proposition, some his-torians associate the economic stagnation of China afterthe fifteenth century with the grounding of 3,500 greatsailing ships of the Ming dynasty in 1433, the world’slargest naval expeditionary fleet under the command ofAdmiral Zheng He. A policy of ‘inward perfection’, fear of Mongol threats, lack of government funding, and adeep mistrust of merchant classes which benefitedmost from the international excursions of the Imperial‘Treasure Fleet’, all led China to close itself off from foreign influences, with disastrous consequences. Formany economists, this is a warning shot about potentialrisks of unbridled anti-globalization. In Chapter 4, werevisit the theme of international trade and economicgrowth in more detail.

2 Maddison (1991: 10).

3 Since output and labour inputs are flows, they could also bemeasured per quarter or per month, but should be measuredover the same time interval. Note that capital is a stock,usually measured at the beginning of the current or end ofthe last period. We discussed the important distinctionbetween stocks and flows in Chapter 1.

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The plus (‘+’) signs beneath the two inputs signify that output rises with either more capital or morelabour.5

The production function is a useful, powerful,and widely-used short-cut. It reduces many andcomplex types of physical capital and labour inputto two. In microeconomics, the production functionhelps economists study the output of individualfirms. In macroeconomics, it is used to think aboutthe output of an entire economy. Box 3.2 presents anddiscusses the characteristics of a widely-used pro-duction function, the Cobb–Douglas productionfunction.

The production function is a technological relationship.It does not reflect the profitability of production,

and it has nothing to do with the quality of life or thedesirability of work. It is meant to capture the factthat goods and services are produced using factors ofproduction: here, equipment and hours of labour. In the following, we describe some basic propertiesthat are typically assumed for production functions.

Marginal productivity

One central property of the production functiondescribes how output reacts to a small increase in one of its inputs, holding other inputs constant.Consider an economy producing output with work-ers and a stock of capital equipment. Then imaginethat a new unit of capital—a new machine—isadded to the capital stock, raising it by the amountΔK, while holding labour input constant.6 Outputwill also rise, by ΔY. The ratio ΔY/ΔK, the amount ofnew output per unit of incremental capital, is calledthe economy’s marginal productivity. Now imagine

4 To see this, note that the elasticity of output with respect to capital is defined as (dY/dK)(K/Y) and is given by (αKα−1L1−α)(K1−αLα−1) = α. Similarly, 1 − α is the elasticity of output with respect to the labour input.

5 Formally, this means that the two first partial derivativesFK(K, L) ≡ ∂F/∂K and FL(K, L) ≡ ∂F/∂L are positive.

6 Throughout this book, the symbol ‘Δ’ is used to denote a stepchange in a variable over some period of time.

Box 3.2 For the Mathematically Minded: The Cobb–Douglas Production Function

The use of mathematics in economics can bring clarityand precision to the discussion of economic relation-ships. An illustration of this is the notion of a productionfunction, which formalizes the relationship betweeninputs (capital and labour) and output (GDP). One par-ticularly well-known and widely-used example is theCobb–Douglas production function:

(3.2) Y = KaL1−a,

where a is a parameter which lies between 0 and 1, andis called the elasticity of output with respect to capital: a1% increase in the capital input results in an a increasein output.4 Similarly 1 − a is the elasticity of output with respect to labour input. It is easy to see that theCobb–Douglas production function possesses all theproperties described in the text.

Diminishing marginal productivity

The marginal productivity of capital is given by thederivative of output with respect to capital K: ∂Y/∂K =aKa−1L1−a = a(L/K)1−a. Since a < 1, the marginal product

of capital is a decreasing function of K and an increasingfunction of L. Similarly, the marginal productivity oflabour is given by ∂Y/∂L = (1 − a)(K/L)a, which is increas-ing in K and decreasing in L.

Constant returns to scale

The Cobb–Douglas function has constant returns toscale: for a positive number t, which can be thought ofas a scaling factor,

(3.3) (tK )a(tL)1−a = ta t1−aKaL1−a = tKaL1−a = tY.

Intensive form

The intensive form of the Cobb–Douglas production func-tion is obtained by dividing both sides of (3.2) by L, whichis the same as setting t = 1/L in equation (3.3), to obtain:

(3.4) Y/L = y = (KaL1−a )/L = KaL−a = (K/L)a = ka,

where k = K/L and y = Y/L are the intensive form measuresof input and output defined in the text. Since a < 1, theintensive form production is indeed well represented byFigure 3.2.

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repeating the experiment, adding capital again andagain to the production process, always holdinglabour input constant. Should we expect output toincrease by the same amount for each additionalincrement of capital?

Generally, the answer is no. As more and morecapital is brought into the production process, itworks with less and less of the given labour input,and the increases in output become smaller andsmaller. This is the principle of diminishing mar-

ginal productivity. It is represented in Figure 3.1,which shows how output rises with capital, holdinglabour unchanged. The flattening of the curve illus-trates the assumption. In fact, the slope of the curveis equal to the economy’s marginal productivity.

It turns out that the principle of diminishingmarginal productivity also applies to the labourinput. Increasing the employment of person-hourswill raise output; but output from additional per-son-hours declines as more and more labour isbeing applied to a fixed stock of capital.

Returns to scale

Output increases when either inputs of capital orlabour increases. But what happens if both capital andlabour increase in the same proportion? Suppose,for example, that the inputs of capital and labour were both doubled—increased by 100%. If outputdoubles as a result, the production function is said

to have constant returns to scale. If a doubling ofinputs leads to more than a doubling of output, we observe increasing returns to scale. Decreasing

returns is the case when output increases by lessthan 100%. It is believed that decreasing returns toscale are unlikely. Increasing returns, in contrast,cannot be ruled out, but we will ignore this possibilityuntil Chapter 4. In fact, the bulk of the evidencepoints in the direction of constant returns to scale.

With constant returns we can think of the linkbetween inputs and output—the production func-tion—as a zoom lens: as long as we scale up theinputs, so does the output. In this case, an attractiveproperty of constant returns production functionsemerges: output per hour of work—the output–

labour ratio (Y/L)—depends only on capital per hourof work—the capital–labour ratio (K/L). This sim-plification allows us to write the production functionin the following intensive form:7

(3.5) y = f(k),

where y = Y/L and k = K/L. The output–labour ratio Y/Lis also called the average productivity of labour: it sayshow much, on average, is being produced with oneunit (one hour) of work.8 The capital–labour ratioK/L measures the capital intensity of production.

The intensive-form production function is de-picted in Figure 3.2. Because of diminishing mar-ginal productivity, the curve becomes flatter as the capital–labour ratio increases. The intensive-form representation of the production function isconvenient because it expresses the average pro-ductivity of labour in an economy as a function of theaverage stock of capital with which that labour isemployed. If average hours worked per capita areheld constant, the intensive form production func-tion is a good indicator of standards of living (Y/N).

7 The constant returns property implies that if we scale up Kand L by a factor t, Y is scaled up by the exactly same factor—for all positive numbers t, it is true that tY = F(tK, tL). In thetext we use the case t = 2; we double all inputs and producetwice as much. If we choose t = 1/L, we have y = Y/L = F(k, 1).Rename this f (k) because F (k, 1) depends on k only. Theintensive production function f (k) expresses output producedper unit of labour (y) as a function of the capital intensity ofproduction (k).

8 It is important to recall the distinction between averageproductivity (Y/L) and marginal productivity (ΔY/ΔL).

Production functionY = F(K, L)

Ou

tpu

t (Y

)

Capital (K )

Fig. 3.1 The Production FunctionHolding labour input L (the number of hours worked)

unchanged, adding to the capital stock K (available

productive equipment) allows an economy to produce

more, but in smaller and smaller increments.

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It follows from (3.5) that in a world of constantreturns, the absolute size of an economy does not matter for its economic performance. Indeed, Ire-land, Singapore, and Switzerland have matched orexceeded the per capita GDP of the USA, the UK, or Germany.

3.2.3 Kaldor’s Five Stylized Facts ofEconomic Growth At this point it will prove helpful to look at the data: How have inputs and outputs in real-worldeconomies changed over time? In 1961, the Britisheconomist Nicholas Kaldor (1908–1986) studied eco-nomic growth in many countries over long periodsof time and isolated several stylized facts about eco-nomic growth which remain valid to this day.Stylized facts are empirical regularities found in the data. Kaldor’s stylized facts will organize ourdiscussion of economic growth and restrict ourattention to theories which help us to think about it,just as a police detective uses clues to limit the num-ber of possible suspects in a criminal investigation.

The first of Kaldor’s stylized facts concerns thebehaviour of output per person-hour and capitalper person-hour.

Stylized Fact No. 1: output per capita and capitalintensity keep increasing

The most remarkable aspect of the growth phe-nomenon is that real GDP seems to grow withoutbound. Yet labour input, measured in person-hoursof work (L), grows much more slowly than both cap-ital (K) and output (Y). Put di3erently, average pro-ductivity (Y/L) and capital intensity (K/L) keep rising.Because income per capita is closely related to aver-age productivity or output per hour of work, eco-nomic growth implies a continuing increase inmaterial standards of living. Figure 3.3 presents theevolution of the output–labour and capital–labourratios in three important industrial economies.

Stylized Fact No. 2: the capital–output ratioexhibits little or no trend

As they grow in a seemingly unbounded fashion,the capital stock and output tend to track eachother. As a consequence, the ratio of capital to out-put (K/Y ) shows little or no systematic trend. This isapparent from Figure 3.3, but Table 3.2 shows thatit is only approximately true. For example, whileoutput per hour in the USA has increased byroughly 600% since 1913, the ratio of capital to out-put actually fell slightly over the same period. Atany rate, the capital–output ratio may not be exactlyconstant, but it is far from exhibiting the steady,unrelenting increases in average productivity andcapital intensity described in Stylized Fact No. 1.

Stylized Fact No. 3: hourly wages keep rising

The long-run increases in the ratios of output and cap-ital to labour (Y/L and K/L) mean that, over time, an

Intensive-formproduction function y = f (k)

Ou

tpu

t–la

bou

r ra

tio

(y =

Y/L

)

Capital–labour ratio (k = K/L)

Fig. 3.2 The Production Function inIntensive FormThe production function shows that the output–labour

ratio y grows with the capital–labour ratio k. Its slope is

the marginal productivity of labour since with constant

returns to scale ΔY/ΔK = Δy/Δk. The principle of declining

marginal productivity implies that the curve becomes

flatter as k increases.

1913 1950 1973 1992 2008*

France n.a. 1.6 1.6 2.3 2.7

Germany n.a. 1.8 1.9 2.3 2.5

Japan 0.9 1.8 1.7 3.0 3.7

UK 0.8 0.8 1.3 1.8 2.1

USA 3.3 2.5 2.1 2.4 3.0

* Estimates

Sources: Maddison (1995); OECD; authors’ calculations.

Table 3.2 Capital–OutputRatios (K/Y), 1913–2008

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hour of work produces ever more output. Simplyput, workers become more productive. It stands toreason, then, that their wages per hour also rise(this link will be shown more formally in Chapter 5).Growth delivers ever-increasing living standards for workers.

Stylized Fact No. 4: the rate of profit is trendless

Note that the capital–output ratio (K/Y) is just theinverse of the average productivity of capital (Y/K).

The absence of a clear trend for the capital–outputratio (K/Y) implies that average productivity too istrendless: over time, the same amount of equip-ment delivers about the same amount of output. It is to be expected therefore that the rate of profitdoes not exhibit a trend either. This stands in sharpcontrast with labour productivity, whose secularincrease allows a continuing rise in real wages. Yetincome flowing to owners of capital has increased,but only because the stock of capital itself hasincreased. Indeed, with a stable rate of profit, in-come from capital increases proportionately to thecapital stock.

Stylized Fact No. 5: the relative income shares ofGDP paid to labour and capital are trendless

We just saw that incomes from labour and capitalincrease secularly. Surprisingly perhaps, it turnsout that they also tend to increase at about the samerate, so that the distribution of total income (GDP)between capital and labour has been relatively sta-ble. In other words, the labour and capital shareshave no long-run trend. We will have to explain thisremarkable fact.

3.2.4 The Steady State

Stylized facts are not meant to be literally true at alltimes, certainly not from one year to the other.Instead, they highlight central tendencies in thedata. As we study growth, we are tracking moving targets, variables that keep increasing all the time,apparently without any upper limits. Thinkingabout moving targets is easier if we can identify sta-ble relationships among them. This is why Kaldor’sstylized facts will prove helpful. Another example ofthis approach is given by the evolution of GDP: itseems to be growing without bounds, but could itsgrowth rate be roughly constant? The answer is yes,but only on average, over five or ten years or more.In Chapter 1, we noted the important phenomenonof business cycles, periods of fast growth followed by periods of slow growth or even declining output.As we look at secular economic growth, we are notinterested in business cycles. We ignore shorter-term fluctuations—compare Figures 1.4 and 1.5 inChapter 1—and focus on the long run.

40.00

35.00

30.00

25.00

20.00

15.00

10.00

5.00

0.001820 1840 1860 1880 1900 1920

Output per hour

1940 1960 1980 2000

1820 1840 1860 1880 1900 1920 1940 1960 1980 2000

1990

$

120.00

100.00

80.00

60.00

40.00

20.00

0.00

1990

$

USA

UK

Japan

USA

UK

Japan

Capital per hour

Fig. 3.3 The Output–Labour andCapital–Labour Ratios in Three CountriesOutput–labour and capital–labour ratios are

continuously increasing. Growth accelerated in the

USA in the early twentieth century, and after 1950 in

Japan and the UK.

Sources: Maddison (1995); Groningen Total Economy Database,

available at <www.ggdc.net>, OECD, Economic Outlook, chained.

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This is why it is convenient to imagine howthings would look if there were no business cycles at all. Such a situation is called a steady state. Its characteristic is that some variables, like theGDP growth ratios, or the ratios described in thestylized facts—the capital–output ratio or the lab-our income share—are constant. Just as the styl-ized facts are not to be taken literally, think of the steady state as the long-run average behaviourthat we never reach, but move around in real time.From the perspective of 10 years ago, we thought

of today as the long run, but now we can see all the details that were unknown back then. Giventhat modern GDPs double every 10–30 years, a temporary boom or recession which shifts today’sGDP by one or two percentage points amounts to little in the greater order of things, the power-ful phenomenon of continuous long-run growth.Steady states—and stylized facts—are not just con-venient ways of making our lives simpler; they areessential tools for distinguishing the forest from the trees.

3.3 Capital Accumulation and Economic Growth9

3.3.1 Savings, Investment, and CapitalAccumulationKaldor’s first stylized fact highlights a relationshipbetween output per hour and capital per hour. This link is in fact predicted by the production function in its intensive form. It suggests that agood place to start if we want to explain economicgrowth is to understand why and how the capitalstock rises over time. We will thus study how the savings of households—foregone consumption—is transformed in an economy into investment incapital goods, which causes the capital stock togrow.

The central insight is delivered by the familiar circular flow diagram in Figure 2.2. GDP representsincome to households, either directly to workers or to the owners of firms. Households and firmssave part of their income. These savings flow into the financial system—banks, stock markets, pen-sion funds, etc. The financial system channels theseresources to borrowers: firms, households, and thegovernment. In particular, firms borrow—includ-ing from their own savings—to purchase capital

goods used in production. This expansion of pro-ductive capacity, in turn, raises output, which thenraises future savings and investment, and so on.

We now examine this process in more detail. Tokeep things simple, we first assume that the size ofthe population, the labour force, and the numbers of hours worked are all constant. At this stage, we ask some fundamental questions: can capital accu-

mulation proceed without bound? Does more sav-ing mean faster growth? And since saving meanspostponing consumption, is it always a good idea tosave more?

3.3.2 Capital Accumulation andDepreciationLet us start from the national accounts of Chapter 2.Identity (2.7) shows that investment (I) can befinanced either by private savings by firms orhouseholds (S), by government savings (the consoli-dated budget surplus, or T − G), or the net savings offoreigners (the current account deficit, Z − X):

(3.6) I = S + (T − G) + (Z − X ).

As a description of the long-run or a steady state,suppose that the government budget is in balance (T = G), and the current account surplus equals zero (Z = X ). In this case, the economy’s capitalstock is ultimately financed by savings of resident

9 This section presents the Solow growth model, in referenceto Nobel Prize Laureate Robert Solow of the MassachusettsInstitute of Technology.

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households.10 More precisely, we reach the con-clusion that, in the steady state, I = S. Investmentexpenditures are financed entirely by domestic savings. This is a first explanation of the growthphenomenon: we save, we invest, we grow. As afirst approximation, let s be the fraction of GDPwhich households save to finance investment. Thatinvestment equals saving implies:

(3.7) I = sY and therefore I/L = sY/L = sy = sf (k).

This relationship is shown in Figure 3.4 as the saving schedule. It expresses national savings as afunction of national output and income. The sav-ing schedule lies below the production functionbecause we assume that national saving is a con-stant fraction of GDP.

We next distinguish between gross investment,the amount of money spent on new capital, and net

investment, the increase in the capital stock. Grossinvestment represents new additions to the physicalcapital stock, but it does not represent the netchange of the capital stock because, over time, pre-viously installed equipment depreciates—it wearsout, loses some of its economic value, or becomesobsolescent. Some fraction of the capital stock isroutinely lost. It is called depreciation and the pro-portion lost each period δ is called the depreciation

rate. The depreciation rate for the overall economyis fairly stable and will be taken as constant: themore capital is in place, the more depreciation willoccur. Depreciation is represented in Figure 3.4 by a ray from the origin, the depreciation line, with aslope δ.

If gross investment exceeds depreciation, netinvestment is positive and the capital stock rises. If gross investment is less than depreciation, thecapital stock falls. While it may seem odd to imag-ine a shrinking capital stock, it is a phenomenonnot uncommon in declining industries or regions. Net investment is therefore:

(3.8) ΔK = sY − δK

or equivalently, written in intensive form:

Δk = sy − δk.

We see that the net accumulation of capital per unitof labour is positively related to the savings rate s andnegatively related to the depreciation rate δ. Therole of capital intensity k is ambiguous: on the one hand, it increases income (y = f (k)) and there-fore savings and investment but, on the other hand, it increases the amount of depreciation. This ambiguity is a central issue in the study of economic growth and will be addressed in the following sections.

3.3.3 Characterizing the Steady State Let us summarize what we have done up to now.The production function (3.5) relates an economy’soutput to inputs of capital and labour. Its intensiveform, presented in Figure 3.3 and Figure 3.4, relatesthe output–labour ratio to the capital–labour ratio.According to equation (3.8), capital accumulation is

10 This need not be true for a region within a nation: the capitalstock of southern Italy, eastern Germany, or NorthernIreland may well be financed by residents of other parts oftheir countries. Yet even these financing imbalances areunlikely to be sustainable for the indefinite future.

Production functiony = f (k)

Ou

tpu

t–la

bou

r ra

tio

(y =

Y/L

)


Saving sf (k)

Depreciation (dk)B

A

D

C

k1 k2Q

Fig. 3.4 The Steady StateThe capital–labour ratio stops changing when

investment is equal to depreciation. This occurs at

point A, the intersection between the saving schedule

sf (k) and the depreciation line dk. The corresponding

output–labour ratio is determined by the production

function f(k) at point B. When away from point A, the

economy moves towards its steady state. Starting below

the steady state at k1, investment (point C) exceeds

depreciation (point D) and the capital–output ratio will

increase until it reaches its steady-state level Q.

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also driven by the output–labour ratio. Putting allthese pieces together, we find that capital accumu-lation (Δk) is determined by previously accumulatedstock of capital (k):

(3.9) Δk = sf(k) − δk.

In Figure 3.4, Δk is the vertical distance between thesavings schedule sf(k) and the depreciation line δk. It represents the net change in the capital stock perunit of labour input in the economy. The sign of Δk tells us where the economy is heading. When Δk > 0, the capital stock per capita is rising and the economy is growing, since more output can beproduced. When Δk < 0, the capital stock per capitaand output per capita are both declining. At theintersection (point A) of the saving schedule and the depreciation line, gross investment and depre-ciation are equal, so the capital–labour ratio (pointB) no longer changes. The capital stock is thus sim-ilar to the level of water in a bathtub when thedrain is slightly open: gross investment is like thewater running through the tap, while depreciationrepresents the loss of water through the drain. Thenewly accumulated capital exactly compensatesthat lost to depreciation—the water flows into thebathtub at the same speed as it leaks out. This is thesteady state.

Capital formation process is not a perpetualmotion machine. Wherever it starts, the economy will gravitate to the steady state and stay there.Suppose, for instance, that the economy is to theleft of the steady-state capital–output ratio Q, say atthe level k1.11 Figure 3.4 shows that gross invest-ment sf(k1) at point C exceeds depreciation δk1 atpoint D. According to (3.9), the distance CD repre-sents net investment, the increase in the capital–labour ratio k, which rises towards its steady-state level Q.

Can the capital stock proceed beyond Q, going allway to say, k2? It turns out that it cannot. As theeconomy gets closer to point A, net investmentbecomes smaller and smaller and nil preciselywhen the steady state is reached. To see how theeconomy behaves when capital is above its steadystate, consider k2 > Q. Gross investment sf (k2) is less

than depreciation δk2, the capital–labour ratio de-clines, and we move leftward towards Q, the eco-nomy’s stable resting point. Later we shall see thatthe stability of capital and output per capita carriesover when we account for population growth.

3.3.4 The Role of Savings for Growth We now show that the more a country saves, themore it invests; the more it invests, the higher is itssteady–state capital–output ratio; and the larger its capital–output ratio, the higher its output–labourratio in the steady state. Thus, as a long-run pro-position, we should expect to find that countrieswith high savings and investment rates have high per capita incomes. Is this true? Figure 3.5(a) looks at the whole world and indeed detects such a link.The poor countries of Africa typically invest little, in contrast to richer countries of Europe and Asia.

Yet, the link is not strong. In addition, Figure3.5(b) shows that the investment rate fails to ac-count for di3erences in economic growth betweencountries. Obviously, our story is too simple and wewill soon put more flesh on the bare bones that we have just assembled. Still, at this stage, we canexplain why savings and investment only a3ect the steady-state level of output, and not its growthrate. This means that nations which save moreshould have higher standards of living in the steadystate, not that they will not indefinitely grow faster.This is an important and slightly counter-intuitiveresult.

To see this, consider Figure 3.6, which illustratesthe e3ect of an increase in the savings rate from s tos′. The savings–investment schedule shifts upwardswhile the production function schedule remainsunchanged. As announced, the new steady-stateoutput–labour and capital–labour ratios are bothhigher at point B than they were at point A before-hand. It will take time for the economy to reach the new steady state. Now that the saving state has increased, at point A, the initial steady-stateposition, gross investment has risen, depreciation is the same, so net investment is positive. The capital–labour ratio starts rising, which raises the output–labour ratio. This will go on until the newsteady state is reached at point B. During thisinterim period, therefore, growth is higher, which can

11 In general, steady-state values of variables will be indicatedhere with an upper bar, e.g. Q, Y, etc.

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60,000

50,000

40,000

30,000

20,000

10,000

00 5 10

Average investment rate (% of GDP)

15 20 25 30 35 40

0 5 10

Average investment rate (% of GDP)

15 20 25 30 35 40

Leve

l of

real

GD

P pe

r ca

pita

in 2

004

(in U

S$)

Gro

wth

in r

eal G

DP

per

capi

ta (%

per

an

nu

m)

EuropeAmericaAsiaAfrica

EuropeAmericaAsiaAfrica

9.0

6.0

3.0

0.0

–3.0

(b) Investment Rate and Real Growth in GDP per Capita (% per annum)

(a) Investment Rate and Real GDP per Capita (level)

Fig. 3.5 Investment, GDP per Capita, and Real GDP Growth For a sample of 174 countries over the period of 1950–2004, the correlation coefficient between the investment rate (the

ratio of investment to GDP) and the average per capita GDP over the period is high and positive (0.51). The correlation of

the investment rate in the countries with real GDP growth is also positive but less striking (0.31).

Source: Penn World Table Version 6.2 September 2006.

9780199236824_000_000_CH03.qxd 2/2/09 11:06 AM Page 64



give the impression that higher investment ratescause higher economic growth. The boost is onlytemporary: once the steady state has been reached,no further growth e3ect can be expected from ahigher savings rate. We still need a story to explaingrowth in output per capita. This is the story told inSections 3.4 and 3.5.

It may be surprising that increased savings does not a3ect long-run growth. The reason that highersavings cannot cause capital and output to grow forever is the assumption of diminishing returns.An increase in savings causes the capital stock torise, but as more capital is put into place, more capital depreciates and thus needs to be replaced.Increasing amounts of gross investment are neededjust to keep the capital stock constant at its higherlevel. Yet the resources for that increased invest-ment are not forthcoming, because the marginalproductivity of capital decreases. Further addi-tions to the capital–labour ratio yield smaller and smaller increases in income, and therefore in sav-ings. Depreciation, however, rises with the capitalstock proportionately. Put simply, the decreasingmarginal productivity principle implies that, atsome point, saving more is simply not worth it.12

3.3.5 The Golden RuleFigure 3.6 contains an important message: tobecome richer, you need to save and invest more. But is being richer—in the narrow sense of accu-mulating capital goods—always necessarily better?Saving requires the sacrifice of giving up some con-sumption today against the promise of higherincome tomorrow, but does saving more todayalways mean more consumption tomorrow? Theanswer is not necessarily positive. To see why, notethat in the steady state, when the capital stock per capita is Q, savings equal depreciation and thesteady-state level of consumption N (the part ofincome that is not saved) is given by:

(3.10) N = Y − sY = f (Q) − δQ.

In Figure 3.7, consumption per capita is given by the vertical distance between the productionfunction and the depreciation line.13 If we couldchoose the saving rate, we could e3ectively pickany point of intersection of the savings schedulewith the depreciation line, and therefore any level of consumption we so desired. Figure 3.7 shows that consumption is highest at the capital stock forwhich the slope of the production function is paral-lel to the depreciation line.14 The correspondingoptimal steady-state capital–labour ratio is indic-ated as Q ′. Now remember that the slope of the production function is the marginal productivity ofcapital (MPK) while the slope of the depreciationschedule is the rate of depreciation δ. We have justshown that the maximal level of consumption isachieved when

(3.11) MPK = δ.

This condition is called the golden rule, and can bethought of as a recipe for achieving the best use ofexisting technological capabilities. In this case, withno population growth and no technical progress,

Ou

tpu

t–la

bou

r ra

tio

O Capital–labour ratio

f (k)

Depreciation

A

B

s ′f (k)

sf (k)

Fig. 3.6 An Increase in the Savings RateAn increase in the savings rate raises capital intensity (k)

and the output–labour ratio (y).

12 In Chapter 4, we will see that the outcome is very di3erentwhen the marginal productivity of capital is not declining.

13 Note that everything, including consumption and saving, is measured as a ratio to the labour input, person-hours. As already noted, if the number of hours worked does notchange, the ratios move exactly as per capita consumption,saving, output, etc.

14 An exercise asks you to prove this assertion.

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the golden rule states that the economy maximizessteady-state consumption when the marginal gainfrom an additional unit of GDP saved and investedin capital (MPK) equals the depreciation rate.

What are the consequences of ‘disobeying’ thegolden rule? If the capital–labour ratio exceeds Q ′,

too much capital has been accumulated, and theMPK is lower than the depreciation rate δ. By reduc-ing savings today, an economy can actually increaseper capita consumption, both today and in thefuture. This looks like a free lunch, and indeed, it is one. We say that the economy su3ers fromdynamic inefficiency. Dynamically ine5cient eco-nomies simply save and invest too much and consumetoo little.

A di3erent situation arises if the economy is tothe left of Q ′. Here, steady-state income and con-sumption per capita may be raised by saving more,but not immediately; consumption only can beincreased in the long run after the adjustment has occurred. No free lunch is immediately avail-able, but must be ‘earned’ by increased saving and reduced consumption at the outset. Movingtowards Q ′ from a position on the left requires cur-rent generations to sacrifice so future generations can enjoy more consumption which will resultfrom more capital and income in the steady state. Aneconomy in such a situation is called dynamically

efficient because it is not possible to do better without paying the price for it. The di3erencebetween dynamically e5cient and ine5cient sav-ings rates is illustrated in Figure 3.8, which shows how we move from one steady state to another onewith higher consumption.

In the dynamically ine5cient case (a), it is possibleto permanently raise consumption by consuming

Con

sum

ptio

n

Con

sum

ptio

n

Time Time

(a) Dynamically inefficient case (b) Dynamically efficient case

Fig. 3.8 Raising Steady-State Consumption In a dynamically inefficient economy (a), it is possible to permanently raise consumption by reducing saving. In a

dynamically efficient economy (b), higher future consumption requires early sacrifices.

Ou

tpu

t–la

bou

r ra

tio

Y ′

Q ′

Capital–labour ratio

A

y = f (k) ′

Consumption

Investment

Depreciation

Fig. 3.7 The Golden Rule Steady-state consumption N (as a ratio to labour) is the

vertical distance between the production function and

the depreciation line Q . It is at a maximum at point A

corresponding to Q , where the slope of the production

function, the marginal productivity of capital, is equal

to d, the slope of the depreciation line.

9780199236824_000_000_CH03.qxd 2/2/09 11:06 AM Page 66



more now and during the transition to the newsteady state. In the dynamically e5cient case (b), ahigher steady-state level consumption is not freeand implies a transitory period of sacrifice.

Dynamic ine5ciency arises when excessive sav-ings have led to too high a stock of capital. Sav-ing must remain forever high merely to replace depreciating capital. Dynamic ine5ciency mayhave characterized some of the centrally plannedeconomies of Central and Eastern Europe. We say‘may’ because the proof that an economy is ine5-cient lies in showing that its marginal productiv-ity of capital is lower than the depreciation rate,and neither of these is easily measurable. What we do know is that Communist leaders oftenboasted about their economies’ high investmentrates, which were in fact considerably higher thanin the capitalist West. Yet overall standards of living were considerably lower than in marketeconomies, and consumer goods were in notori-

ously short supply. Box 3.3 presents the case ofPoland.

In dynamically e5cient economies, future genera-tions would benefit from raising saving today, butthose currently alive would lose. Should govern-ments do something about it? Since it would rep-resent a transfer of revenues from current to futuregenerations, there is no simple answer. It is truly adeep political choice with no solution since futuregenerations don’t vote today. A number of factorsinfluence savings, such as taxation, health andretirement systems, cultural norms, and social cus-tom. Importantly, too, saving and investment areinfluenced by political conditions. Political instabil-ity and especially wars, civil or otherwise, can leadto destruction and theft of capital, and hardlyencourage thrifty behaviour. As we discuss inChapter 4, in many of the world’s poorest coun-tries, property rights are under constant threat or non-existent.

Box 3.3 Dynamic Inefficiency in Poland?

From the period following the Second World War until theearly 1990s, Poland was a centrally planned economy.Savings and investment decisions for the Polish eco-nomy were taken by the ruling Communist party. The panels of Figure 3.9 compare Poland with Italy, a coun-try with one of the highest saving rates in Europe. The firstgraph shows the increase in GDP per capita between1980 and 1990 (the GDP measure is adjusted for pur-chasing power to take into account different price systems). While Italy’s income grew by 25%, Poland’sactually shrank by about 5%. The second graph shows theaverage proportion of GDP dedicated to saving over thesame period. Clearly, Poland saved a lot, but receivednothing for it in terms of income growth.

As the third and fourth panels of Figure 3.9 show, thesituation was reversed after 1991, when Poland intro-duced free markets and abandoned central planning.From 1991 to 2004, per capita GDP increased by 68%, witha lower investment rate than Italy’s (which grew by 17%).However, our theory predicts that savings affect the

steady-state level of GDP per capita, not its growth rate.In 1980, Poland invested 21.6% of its GDP. By 1990 thisrate had fallen to 18.3%. In the period 1990–2004 percapita consumption rose in Poland from $2,908 to $7,037,an increase of 142%, compared with 61% in Italy over thesame period. Is this proof of dynamic inefficiency, i.e.that a significant part of savings was used merely to keep up an excessively large stock of capital? Anecdotalevidence would suggest so. Stories of wasted resourceswere common in centrally planned economies: unin-stalled equipment rusting in backyards, new machineryprematurely discarded for lack of spare parts, tools ill-adapted to factory needs, etc. One important cause ofwastage was a reward system for factory managers.These were often based on spending plans, and not onactual output. An alternative interpretation is that theinvestment was in poor quality equipment, which couldnot match western technology. No matter how we lookat it, savings were not put to their best use in centrallyplanned Poland.

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3.4 Population Growth and Economic Growth

A major shortcoming of the previous section is that it does not explain permanent, sustainedgrowth, our first stylized fact. Capital accumula-tion, we saw, can explain high living standards andgrowth during the transition to the steady state but the law of diminishing returns ultimately kicksin. Clearly, some crucial ingredients are missing.One of them is population growth, more precisely,growth in the employed labour force. This sectionshows that sustainable long-run growth of both output and the capital stock is possible once weintroduce population growth.

Recall that labour input (person-hours) growseither if the number of people at work increases, orif workers work more hours on average. Later on inthis chapter and Chapter 5, we will see that thenumber of hours worked per person has declinedsteadily over the past century and a half. Figure 3.10shows that, despite this fact, employment has beenrising, either because of natural demographicforces (the balance between births and deaths) or

immigration. Overall, more people are at work butthey work shorter hours, so the balance of e3ects isambiguous. Because the number of hours workedper person cannot and does not rise without bound,we will treat it as constant. Then any change in person-hours is due to exogenous changes in thepopulation and employment, and output per person-hour changes at the same rate as output per capita.

Even though population and employment aregrowing, the fundamental reasoning of Section 3.3remains valid: the economy gravitates to a steadystate at which the capital–labour and output–labour ratios (k = K/L and y = Y/L) stabilize. With L grow-ing at the exogenous rate n, output Y and capital Kwill also grow at rate n. The relentless increase in the labour input is the driver of growth in this case.Quite simply, if income per capita is to remainunchanged in the steady state, income must grow atthe same rate as the number of people.

The role of saving and capital accumulationremains the same as in the previous section, with only

70.00

60.00

50.00

40.00

30.00

20.00

10.00

−10.00

0.00

ItalyPoland

GDP growth(1990 relative to 1980, %)

30.00

25.00

20.00

15.00

10.00

5.00

0.00ItalyPoland

Investment/GDP(average 1980–1990, %)

70.00

50.00

60.00

40.00

30.00

20.00

10.00

0.00ItalyPoland

GDP growth(2004 relative to 1991, %)

30.00

25.00

20.00

15.00

10.00

5.00

0.00ItalyPoland

Investment/GDP(average 1991–2004, %)

Fig. 3.9 Was Centrally Planned Poland Dynamically Inefficient?Despite a high investment and savings rate, Polish per capita GDP shrank during the period 1980–1990 while Italy’s grew.

During the transition period, Poland grew much faster, with a lower investment rate than in Italy.

Source: Heston, Summers, and Aten (2006).

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a small change of detail. The capital accumulationcondition (3.9) now becomes:15

(3.12) Δk = sf(k) − (δ + n)k.

The di3erence is that, for the capital–labour ratio to increase, gross investment must not just com-pensate for depreciation, it must also provide newworkers with the same equipment as those alreadyemployed. This process is called capital-widening

and it explains the last term (n).The situation is presented in Figure 3.11. The only

di3erence with Figure 3.4 is that the depreciation line δk has been replaced by the steeper capital-

widening line (δ + n)k. The fact that the capital-widening line is steeper than the depreciation line captures the greater need to save when moreworkers are being equipped with productive capital.The steady state occurs at point A1, the intersection

of the saving schedule and the capital-wideningline. At this intersection Q1, savings are just enoughto cover the depreciation and the needs of newworkers, so Δk = 0.

The role of population growth can be seen bystudying the e3ect of an increase in the rate of population growth, from n1 to n2. In Figure 3.11 thecapital-widening line becomes steeper and the new steady state at point A2 is characterized by alower capital–labour ratio Q2. This makes sense. Weassume that the savings behaviour has not changedand yet we need more gross investment to equipnew workers. The solution is to provide eachworker with less capital. Of course, a lower Q im-plies a lower output–labour ratio f (Q). Thus we findthat, all other things being equal, countries with arapidly growing population will tend to be poorerthan countries with lower population growth. Box 3.4 examines whether it is indeed the case thathigh population growth lowers GDP per capita.

At what level of investment does an economywith population growth maximize consumptionper capita? Because the number of people who are able to consume is growing continuously, the

Euro area

United States

Euro area

United States

(b) Employment

1960 1965 1970 1975 1980 1985 1990 1995 20052000

(a) Working age population

240,000,000

220,000,000

200,000,000

180,000,000

160,000,000

140,000,000

120,000,000

100,000,000

150,000,000

140,000,000

130,000,000

120,000,000

110,000,000

100,000,000

90,000,000

80,000,000

70,000,000

60,000,0001960 1965 1970 1975 1980 1985 1990 1995 20052000

Fig. 3.10 Population GrowthPopulation of working age (between 15 and 64) has been growing both in the USA and the euro area, the part of the

European Union that uses the euro. Employment has also been growing, albeit less fast in Europe. Note the jump in the

euro area in 1991, the year after German unification.

Source: OECD, Economic Outlook.

15 The proof requires some calculus based on the principlespresented in Box 6.3. The change in capital per capita is Δk/k = (ΔK/L) − (ΔL/L). After substituting ΔK = I − δK and ΔL/L = n and setting I = sY, the equation can be rearranged to yield Δk = sy − δk − nk = sf (k) − δk − nk.

9780199236824_000_000_CH03.qxd 2/2/09 11:06 AM Page 69



golden rule must be modified accordingly. Follow-ing the same reasoning as in Section 3.3, we note that steady-state investment per person-hour is (δ + n)Q, so consumption per person-hour N is givenby f (Q ) − (δ + n)Q. Proceeding as before, it is easy to see that consumption is at a maximum when

(3.13) MPK = δ + n.

The ‘modified’ golden rule equates the marginalproductivity of capital with the sum of the depre-ciation rate δ and the population growth rate n. The intuition developed above continues to apply: the marginal product of an additional unit of capital (per capita) is set to its marginal cost, whichnow includes not only depreciation, but also thecapital-widening investment necessary to equipfuture generations with the same capital per head as the current generation. A growing populationwill necessitate a higher marginal product of capitalat the steady state. The principle of diminishingmarginal productivity implies that the capital–labour ratio must be lower. Consequently, outputper head will also be lower.

Ou

tpu

t–la

bou

r ra

tio

(y =

Y/L

)


Capital-widening(d + n1)k

(d + n2)k

Saving sf (k)

Q2 Q1

A2

A1

Fig. 3.11 The Steady State with PopulationGrowth The capital–labour ratio remains unchanged when

investment is equal to (d + n1)k.

This occurs at point A1, the intersection between the

saving schedule sf(k) and the capital-widening line

(d + n1)k. An increase in the rate of growth of the

population from n1 to n2 is shown as a counter-clockwise

rotation of the capital-widening line. The new steady-

state capital–labour ratio declines from Q1 to Q2.

Box 3.4 Population Growth and GDP per Capita

Figure 3.12 plots GDP per capita in 2003 and the averagerate of population growth over the period 1960–2004. Thefigure could be seen as confirming the negative rela-tionship predicted by the Solow growth model. Taken atface value, this result might be interpreted as supportfor the hypothesis that population growth impoverishesnations. Thomas Malthus, a famous nineteenth-centuryEnglish economist and philosopher, also claimed thatpopulation growth causes poverty. He argued that afixed supply of arable land could not feed a constantlyincreasing population and that population growth wouldultimately result in starvation. He ignored technolo-gical change, in this case the green revolution which significantly raised agricultural output in the last half of the twentieth century. As we confirm in Section 3.5,

technological change can radically alter the outlook forgrowth and prosperity.

Yet the pseudo-Malthusian view has been taken seri-ously in a number of less-developed countries, whichhave attempted to limit demographic growth. The mostspectacular example is China, which has pursued a one-child-only policy for decades. At the same time, we needto be careful with simple diagrams depicting relation-ships between two variables. Not only do other factorsbesides population growth influence economic growth,but it may well be that population growth is not exogen-ous. Figure 3.12 could also be read as saying that as people become richer, they have fewer children. Thereexists a great deal of evidence in favour of this alternativeinterpretation.

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0

10.000

20.000

30.000

40.000

60.000

50.000

–2.0 0.0 2.0 4.0

Kuwait

Qatar

UAE

Oman

Bahrain

Brunei

Saudi Arabia

6.0 8.0

Average population growth rate, 1960–2004 (%)

GD

P pe

r ca

pita

in 2

004

(in U

S$, 2

000)

Fig. 3.12 Population Growth and GDP per Capita, 1960–2000The figure reports data on real GDP per capita and average population growth for 182 countries over almost a half century.

The plot indicates a discernible negative association between GDP per capita and population growth, especially when the

rich oil-producing countries (United Arab Emirates, Qatar, Kuwait, Bahrain, Oman, Brunei, and Saudi Arabia) are excluded.

The sharp population growth observed in these countries is largely to due to immigration.

Source: Heston, Summers, and Aten (2006).

3.5 Technological Progress and Economic Growth

Taking population growth into account gives onegood reason why output and the capital stock cangrow permanently, and at the same rate. While thissatisfies Kaldor’s second stylized fact, the pictureremains incomplete: in our growth model, cap-ital–labour and output–labour ratios were constant.Standards of living are not rising in this economy, andthis is still grossly inconsistent with Kaldor’s firststylized fact and the data reported in Table 3.1.Under what conditions can per capita income and capital stock grow, and grow at the same rate?

So far, we have ignored technological or tech-

nical progress. It stands to reason that, over time,increased knowledge and better, more sophistic-ated techniques make workers and the equipment

they work with more productive. With a slightalteration, our framework readily shows how tech-nological progress works. To do so, once more, wereformulate the aggregate production functionintroduced in (3.1). Technological progress meansthat more output can be produced with the samequantity of equipment and labour. The most con-venient way to do this is to introduce a measure of the state of technology, A, that raises output atgiven levels of capital stock and employment:

(3.14) Y = F(A, K, L).+ + +

When A increases, Y rises, even if K and L remainunchanged. For this reason, A is frequently called

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total factor productivity. It should be emphasizedthat A is not a factor of production. No firm pays for it, and each firm just benefits from it. It is bestthought of as ‘best practice’ and is assumed to be available freely to all. At this point, it will be con-venient to assume that A increases at a constant rate a, without trying to explain how and why.Technological progress, which is the increase in A,is therefore considered as exogenous.

It turns out that it is possible to relate our analysisto previous results in this chapter in a straight-forward way. First, we modify (3.14) to incorporatetechnical progress in the following particular way:

(3.15) Y = F(K, AL).

In this formulation, technological progress actsdirectly on the e3ectiveness of labour. (For this reason it is sometimes called labour-augmenting technical progress). An increase in A of, say, 10% hasthe same impact as a 10% increase in employment,even though the number of hours worked hasn’tchanged. The term AL is known as effective labour tocapture the idea that, with the same equipment,one hour of work today produces more output than before because A is higher. E3ective labour ALgrows for two reasons: (1) more labour L, and (2)greater e3ectiveness A. For this reason, the rate ofgrowth of AL is now given by a + n.

Now we change the notation a little bit. Weredefine y and k as ratios of output and capital relat-ive to e4ective labour: y = Y/AL, k = K/AL. Once this is done,it is possible to recover the now-familiar productionfunction in intensive form, y = f (k).16 Not surpris-ingly, the ratio of capital to e3ective labour evolvesas before, with a slight modification:

(3.16) Δk = sf (k) − (δ + a + n)k.

The reasoning is the same as when we introducedpopulation growth. There we noted that, to keepthe capital–labour ratio K/L constant, the capitalstock K must rise to make up for depreciation (δ) andpopulation growth (n). Now we find that, to keepthe capital–e3ective labour ratio k = K/AL constant, the capital stock K must also rise to keep up with

workers’ enhanced e3ectiveness (a). So k will in-crease if saving sf (k), and hence gross investment,exceeds the capital accumulation needed to make up for depreciation δ, population growth n, andincreased e3ectiveness a. From there on, it is a simple matter to modify Figure 3.11 to Figure 3.13.The steady state is now characterized by constantratios of capital and output to e3ective labour (y = Y/ALand k = K/AL).

Constancy of these ratios in the steady state is a very important result. Indeed, if Y/AL is constant,it means that Y/L grows at the same rate as A. If the average number of hours remains unchanged,then income per capita must grow at the rate oftechnological progress, a. In other words, we havefinally uncovered the explanation of Kaldor’s firststylized fact: the continuous increase in standards ofliving is due to technological progress. Since K/AL isalso constant, we know that the capital stock per

16 Constant returns to scale implies that y = F(K, AL)/AL= F(K/AL, 1).

Ou

tpu

t–ef

fect

ive

labo

ur

rati

o (y

= Y

/AL)

Capital–effective labour ratio (k = K/AL)

Capital-widening (d + a + n)K

Saving sf (k)

Q

A

Fig. 3.13 The Steady State with PopulationGrowth and Technological Progress In an economy with both population growth and

technological progress, inputs and output are measured

in units per effective labour input. The intensive form

production function inherits this property. The slope of

the capital accumulation line is now d + a + n, where a

is the rate of technological progress. The steady state

occurs when investment is equal to (d + a + n)k (point A),

which is the intersection of the saving schedule sf(k) with

the capital-widening line (d + a + n)k. At the steady-state

Q, output and capital increase at the rate a + n, while

GDP per capita increases at the rate a.

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capita also grows secularly at the same rate, i.e.Kaldor’s second stylized fact. Figure 3.14 illustratesthese results. Because of diminishing marginal pro-ductivity, capital accumulation alone cannot sustaingrowth. Population growth explains GDP growth,but not the sustained increase of standards of living

over the centuries. Technological progress is essen-tial for explaining economic growth in the longrun. Rather than creating misery in the world, itturns out to be central to improvements in stand-ards of living.

Note that an increase in the rate of technologicalprogress, a, makes the capital-widening line steeperthan before. In Figure 3.13 this would imply lowersteady-state ratios of capital and output to e3ectivelabour. This does not mean that more rapid techno-logical progress is a bad thing. On the contrary, in fact,when Y/AL is lower, Y/L grows faster so that stand-ards of living are secularly rising at higher speed.

The discussion can be extended in a natural wayto address the issue of the golden rule. Redefining cas the ratio of aggregate consumption (C) to e3ect-ive labour (AL), the following modified version of(3.10) will hold in the steady state:

N = f (Q) − (δ + a + n)Q.

The modified golden rule now requires that themarginal productivity of capital be the sum of therates of depreciation, of population growth, and oftechnological change:

(3.17) MPK = δ + a + n.

Maximizing consumption per capita is equivalentto making consumption per unit of e3ective labouras large as possible. To do this, an economy nowneeds to invest capital per e3ective unit of labour to the point at which its marginal product ‘covers’ the investment requirements given by technicalprogress (a), population growth (n), and capitaldepreciation (δ).

Time

Growth rate = a + n

Growth rate = a

Growth rate = 0y = Y/AL or k = K/AL

Y/L or K/L

Y or K

Fig. 3.14 Growth Rates along the SteadyStateWhile output and capital measured in effective labour

units (Y/AL and K/AL) are constant in the steady state,

output–labour and capital–labour ratios (Y/L and K/L)

grow at the rate of technological progress a, and output

and the capital stock (Y and K) grow at the rate a + n, the

sum of the rates of population growth and technological

progress.

3.6 Growth Accounting

3.6.1 Solow’s Decomposition

As shown in Figure 3.14, we have now identifiedthree sources of GDP growth: (1) capital accumulation,(2) population growth, and (3) technological progress.It is natural to ask how large the contributions of

these factors are to the total growth of a nation or a region. Unfortunately, it is di5cult to meas-ure technological progress. Computers, for instance,probably raise standards of living and growth, but by how much? Some people believe that the ‘neweconomy’, brought on by the information technology

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revolution, will push standards of living faster thanever. Others are less optimistic that the e3ect is anylarger than other great discoveries which mark eco-nomic history. Box 3.5 provides some details on thisexciting debate.

Robert Solow, who developed the theory pre-sented in the previous sections, devised an ingeni-ous method of quantifying the extent to whichtechnological progress accounts for growth. Hisidea was to start with the things we can measure: GDP growth, capital accumulation, and man-hoursworked. Going back to the general form of the pro-duction function (3.14), we can measure output Yand two inputs, capital K and labour L. Once weknow how much GDP has increased, and how muchof this increase is explained by capital and hoursworked, we can interpret what is left, called theSolow residual, as due to the increase in A, i.e. a = ΔA/A:

Solow residual = − output growth due to growth in cap-

ital and hours worked.18

We now track down the Solow decomposition.

ΔYY

Box 3.5 The New Economy: Another Industrial Revolution?

The striking changes brought about by the ICT (informa-tion and communications technologies) revolution,which include the internet, wireless telecommunica-tions, MP3 players, and the conspicuous use of elec-tronic equipment, have led many observers to concludethat a new industrial revolution is upon us. Figure 3.15reports estimates of overall increases in multifactor pro-ductivity in the USA, computed as annual averages overfour periods. A difference of 1% per year cumulates to 28%after 25 years. The figure shows a formidable accelerationin the period 1913–1972, and again over 1995–1999;hence the case for a second industrial revolution.

While initially there was much scepticism about the trueimpact of the ICT revolution—Robert Solow himselfsaid early on that ‘computers can be found everywhere

except in the productivity statistics’—there is com-pelling evidence that ICT have indeed deeply impacted theway we work and produce goods and services, and haveultimately increased our standards of living signific-antly. These total factor productivity gains, according to recent research by Kevin Stiroh, Dale Jorgenson, andothers, can be found both in ICT-producing as well asICT-using sectors.17 Strong gains in total factor produc-tivity have been observed in the organization of retailtrade, as well as in manufacturing and business services.Even more interesting is the fact that not all economiesaround the world have benefited equally from productivityimprovements measured in the USA. In particular, someEU countries continue to lag behind in ICT adoption aswell as innovation.

17 See the references at the end of the book.

18 Formally, the Solow residual is = − (1 − sL) + sL ,

where sL is the labour share, defined as the share of nationalincome paid to labour in the form of wages and non-wagecompensation, and 1 − sL is the income share of capital. TheWebAppendix shows how this formula can be derived fromthe production function.

JL

ΔLL

ΔKK

GI

ΔYY

ΔAA

1.4

1.2

1

0.8

0.6

0.4

0.2

01870–1913 1913–1972 1972–1995 1995–1999

Fig. 3.15 Multifactor Productivity in theUSA (average annual growth, %)The average annual increase in A (multifactor

productivity) accelerated sharply after 1913, came

to a near stop over 1972–1995, and seems to have

vigorously bounced back at the end of the 1990s.

Source: Gordon (2000).

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3.6.2 Capital AccumulationTable 3.3 shows that, typically, capital has beengrowing at about 3–5% per year over most of thetwentieth century in the developed countries. Cap-ital accumulation accelerated sharply in the 1950s and 1960s as part of the post-war reconstruction.Many European countries accumulated capital con-siderably faster than the USA and the UK up until the mid-1970s, the reason being that continentalEurope was poorer at the end of the Second WorldWar. These sustained periods of rapid capital accu-mulation fit well the description of catch-up, whenthe capital stock is below its steady-state level.

3.6.3 Employment Growth The most appropriate measure of labour input istotal number of hours worked. For several reasons,however, growth in population or the number of employees does not necessarily translate intoincreased person-hours. To understand this, we canrewrite the total number of hours in the followingway:

total hours worked = (hours/employee) ×(employee/population) × population.

The total number of hours worked can increase forthree reasons:

1890–1913 1913–1950 1950–1973 1973–1987 1987–2008

France n.a. 1.2 5.1 4.5 3.2

Germany 3.1 1.1 6.6 3.5 2.2*

Netherlands n.a. 2.4 5.8 3.3 2.8

UK 2.0 1.5 5.1 2.9 3.9

USA 5.4 2.1 3.2 3.3 4.2

Japan 3.0 3.9 9.1 7.6 3.2

* 1991–2008

Sources: Maddison (1991); OECD, Economic Outlook.

Table 3.3 Growth of Real Gross Fixed Capital Stock, 1890–2008 (% per annum)

Population Employment Growth in Hours worked Hours worked growth growth hours worked per person per person (% per annum) (% per annum) per person in 1913 in 2006

(% per annum)

France 0.3 0.3 −0.5 2,588 1,529

Germany 0.5 0.6 −0.5 2,584 1,437

Netherlands 1.1 1.3 −0.5 2,605 1,413

United Kingdom 0.5 0.6 −0.4 2,624 1,624

United States 1.5 1.7 −0.4 2,605 1,791

Japan 1.0 0.9 −0.4 2,588 1,775

Sources: Maddison (2006); Groningen Growth and Development Centre and the Conference Board, Total Economy Database, January 2007.

Table 3.4 Population, Employment, and Hours Worked, 1870–2006

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◆ Obviously, population growth. Everything elseunchanged, more people provide more workinghours. But many things change.

◆ The proportion of people who work. Some working-age people are unemployed and othersvoluntarily stay out of work for various reasons. In addition, people live longer, study longer, and retire earlier. Furthermore, women have in-creased their labour force participation over thepast 30 years.19

◆ Hours worked per person. Over time, those whowork tend to work fewer hours per day and fewerdays per year.

Table 3.4 shows that these e3ects have roughly can-celled each other out so that, in the end, employ-ment and population size have increased by similaramounts in our sample of developed countries.

Table 3.4 also documents the sharp seculardecline in the number of hours worked per per-son in the developed world. The long-run trend is aconsequence of shorter days, shorter workweeks,fewer weeks per year, and fewer years worked per person. This is why the number of person-hours has declined across the industrial world. Overall,European labour input has increased between niland 0.3%, while immigration lifted it well above 1%in the USA and Australia. The dramatic decline inhours worked per person is a central feature of thegrowth process; an average annual reduction of0.5% per year means a total decline of 40% over acentury. As societies become richer, demand forleisure increases. The last two columns of the tablereveal a massive jump in leisure time available,which is as important a source of improvement ofhuman welfare as increases in material wealth.

3.6.4 The Contribution of TechnologicalChange Table 3.5 presents the Solow decomposition in twodi3erent contexts. The first employs historical datafor the period 1913–1987. The second examines thesame countries over the last two decades. We seethat growth in inputs of labour and capital account

(a) 1913–1987*

Country GDP Contribution Residualof inputs

France 2.6 1.1 1.0

Germany 2.8 1.4 0.8

Netherlands 3.0 2.0 0.4

UK 1.9 1.2 0.5

USA 3.0 2.0 0.7

Japan 4.7 3.0 0.5

* An adjustment is made to account for the modernization of

productive capital

Source: Maddison (1991: 158).

(b) 1987–1997

GDP Contribution Residualof inputs

France 2.0 1.1 1.0

Germany* 1.4 0.2 1.2


United Kingdom 2.2 1.4 0.7

United States 3.0 2.5 0.5

Japan 2.7 1.4 1.3

* 1991–1997

(c) 1997–2006

GDP Contribution Residualof inputs

France 2.2 1.3 1.0

Germany 1.4 0.6 0.8


United Kingdom 2.7 1.7 1.0

United States 3.0 1.9 1.1

Japan 1.2 0.1 1.1

Sources: Maddison (1991); authors’ calculation based on

Groningen Growth and Development Centre and the

Conference Board, Total Economy Database, January 2007,

available at <www.ggdc.net>; OECD, Economic Outlook.

Table 3.5 The SolowDecomposition (average annual growth rates)

19 Chapter 4 explores these and related issues in more detail.

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for only one-half to two-thirds of total economicgrowth in these economies. The rest is the Solowresidual, and confirms the importance of tech-nological progress. A puzzling observation is theapparent slowdown in technological change duringthe 1970s and the 1980s, which contrasts with

reports of an acceleration since the late 1990s. In the table, only the USA and Germany exhibithigher growth in technical progress (the Solowresidual) in the second decade, which is generallythought to be a period associated with technologicalacceleration.

1 Summary

1 Economic growth refers to the steady expansion of GDP over a period of a decade or longer.Growth theory is concerned with the study of eco-nomic growth in the steady state, a situation inwhich output and capital grow at the same pace andremain in constant proportion to labour ine3ective terms. This approach reflects key styl-ized facts.

2 The aggregate production function shows thatoutput grows when more inputs (capital andlabour) are used, and when technology improvesthe e3ectiveness of those inputs.

3 The capital stock, the sum of productive equip-ment and structures, is accumulated throughinvestment, and investment is financed by sav-ings. When savings are a stable proportion of out-put, the steady-state capital stock is determinedby savings and capital depreciation.

4 The assumption that the marginal productivity of capital is declining implies that output, andtherefore savings, grow less proportionately tothe stock of capital, in contrast to depreciation,which is proportional to capital. This implies thatcapital accumulation eventually exhausts thepotential of savings and comes to an end. At thispoint, the steady state is reached.

5 In the absence of population growth and techno-logical change, the steady state is characterizedby zero output and capital growth. Adding population growth provides a first explanation ofsecular output growth, but standards of living—

measured as output per capita—still do notincrease. It is only when we allow for technolog-ical progress that permanent growth in per capitaoutput and capital is possible.

6 The golden rule describes a steady state in whichconsumption is as high as possible. It occurswhere the marginal productivity of capital equalsthe rate of depreciation (so as to replace worn-out capital) plus the rate of population growth (to provide new workers with adequate equip-ment) plus the rate of technological change(adjusting capital to enhanced labour e3ective-ness): MPK = δ + n + a.

7 An economy is dynamically e5cient whensteady-state consumption can be raised tomor-row only at the expense of lower consumptiontoday. An economy is dynamically ine5cientwhen both current and future steady-state con-sumption can be increased. In the former case,the capital stock is lower than the golden-rulelevel. In the latter case, the capital stock is abovethe golden-rule level.

8 Saving does not a3ect the steady-state growthrate, but only the level of output per capita.

9 The Solow decomposition is a method of account-ing for the sources of economic growth. It breaksdown growth in GDP into the sum of growthattributable to changes in the factors of productionand growth due to improved production. The latter is called the Solow residual, and is usuallyinterpreted as technological change.

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3 Key Concepts

◆ economic growth

◆ technological progress

◆ Solow growth model

◆ stylized facts

◆ steady state

◆ capital accumulation

◆ aggregate production function

◆ diminishing marginal productivity

◆ returns to scale (constant, increasing, decreasing)

◆ output–labour ratio

◆ capital–labour ratio

◆ saving schedule

◆ gross investment

◆ depreciate, depreciation rate

◆ depreciation line

◆ capital-widening line

◆ golden rule

◆ dynamic inefficiency, dynamically efficient

◆ effective labour

◆ Solow decomposition

◆ Solow residual

◆ total factor or multifactor productivity

2 Exercises

1 Draw intensive-form production functions f (k)with decreasing, constant, and increasing returnsto scale.

2 Use graph paper (or a computer spreadsheet) to plot the following intensive-form productionfunctions (expressing y as a function of k) from theinterval [0, 100]:

(a) f (k) = 2k;(b) f (k) = 10 + k0.5;(c) f (k) = k1.1;(d) f (k) = 10 + 2k − 0.5k2;(e) f (k) = max(0, − 10 + 2k).

Which of these functions has constant returns?Decreasing or increasing returns? Are youranswers always unambiguous, i.e. do they hold for all k ∈ [0, 100]?

3 Define the steady state. How is a steady stateimportant in the context of the Solow growthmodel? Explain why stylized facts are necessaryto organize the discussion of economic growth.

4 Suppose K/Y is constant at 2. (a) Assume first that there is no population growth and no technological progress. What is the steady-statesaving–output ratio consistent with a rate ofdepreciation of 5%? (b) Now allow for positiveeconomic growth, due to either populationgrowth or technological progress. What is thesteady-state saving–output ratio consistent witha rate of depreciation of 5% and 3% real growth?

5 Consider a country with zero technological pro-gress and K/L = 3. Its population grows at the rateof 2% per year. What is the steady-state rate ofgrowth of GDP per capita if the saving rate is20%? If it is 30%? How do your answers change if depreciation occurs at a rate of 6% per year?

6 Suppose the aggregate production function is given by Y = KL. Does it have increasing,decreasing, or constant returns to scale? Showthat the marginal products of capital and labourare declining. Show that they are increasing in the input of the other factor.

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7 Define the concept of a golden rule. Explain whyit is achieved at Q′ in Figure 3.7. To establish thisresult, imagine that you start to the left of Q′ andexplain why moving to the right increases con-sumption. Similarly show that consumptiondecreases when moving rightwards from a posi-tion to the right of Q′.

8 The golden rule (3.17) is MPK = δ + a + n. In com-parison with the no-technological change case, we now require a higher marginal productiv-ity of capital. With diminishing marginal pro-ductivity, this means a lower capital stock pere3ective unit of labour. Is that not surprising?How can you explain this apparent paradox?

9 Draw a graph showing the evolution of Y/L inthe catch-up phase and then in the ensuingsteady state when the economy starts from acapital–e3ective labour ratio below its steady-state level. Draw a picture showing investment(not the capital stock).

10 Explain, formally or informally, the di3erencebetween average labour productivity and outputper capita.

11 In the steady state, output per e3ective labour Y/AL is constant. What happens to output percapita when the average number of hoursworked declines, holding all else constant?

5 Essay Questions

1 Japan in the 1960s, Korea in the 1980s, China in the1990s, and India in the current decade have expe-rienced periods of rapid growth, in e3ect catchingup on the richer and more developed countries.How can you explain this phenomenon, and whydid it not happen earlier?

2 In earlier centuries, colonial expeditions werelaunched to increase a country’s land and popula-tion. How might such activities make a countryricher?

3 ‘Globalization is bad for growth since it meansthat countries invest abroad and expand the capi-tal stocks of other countries instead of their own.The Solow model would consider this a decrease

in the savings rate, thus leading to lower GDP percapita.’ Comment.

4 Examining the panels of Figure 3.5, one immedi-ately notices that African countries are bunched in the lower left-hand part of the diagram, whilethe European countries are clustered in the upperright-hand region. What explanation of this fact is o3ered by the Solow growth model? Do youthink this is a su5cient explanation? Explain.

5 It is often claimed that the defeated nations afterthe Second World War grew faster than the victornations. Is this hypothesis consistent with theSolow growth model?

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