Chapter 3

Economic Growth

Imagine that by chance an independent observer from outer space arrived on earth and got to look aroundfor some time. At the end of the trip, the alien has the opportunity to talk to a macroeconomist and askexactly one question about the state of affairs on our planet. Of course, the question has to be related tomacroeconomics. It seems almost certain that the alien would want to know why it is that people in someareas of the world are so rich while people elsewhere are so poor. The differences in GDP per person acrosscountries are stunning. For example, income per person in the United States is about 15 times higher thanin India. Finding an explanation for these huge differences in wealth across countries remains the mostimportant challenge for macroeconomists.

If we look back in history, it becomes clear that if we want to explain differences in wealth and income acrosscountries, we need a theory of economic growth. Until the early nineteenth century, all countries were poorrelative to modern standards. Although there were substantial differences in wealth across countries, mostof the world population had an income at or slightly above the subsistence level. But in Britain, and soonthereafter in other European countries and America, a growth process of hitherto unknown proportionsstarted. Per capita incomes started to rise, and economic growth up to the present day has made millionsof people enormously wealthy by historical standards. However, only a relatively small part of the worldpopulation has participated in this economic growth. Many countries remain poor and have growth ratesmuch lower than those experienced by today’s industrialized countries.

In this chapter, I will examine the sources of economic growth. I begin with a look at the empirical facts. Itturns out that growth facts depend on which group of countries we are looking at. The experience of thenow industrialized countries has been different from the rest of the world. Economists have not succeededyet in developing a single model that explains the growth experience of industrialized countries and de-veloping countries at the same time. Therefore my search for explanations of growth will proceed in threeparts. First, I will concentrate on industrialized countries. I will introduce the Solow model (also called theneoclassical growth model) and argue that it is consistent with most facts about growth in industrializedcountries. Next, I will ask the question why the whole world was poor until about 1800, and many coun-tries are still poor today. A model due to Malthus, which emphasizes a feedback from income to populationgrowth, predicts stagnation even in the face of productivity growth. The Malthusian model therefore pro-vides an explanation for early economic history, and the economic conditions in many developing countriestoday. The chapter concludes with a discussion of how countries can escape from stagnation and go fromMalthus to Solow, and what our findings imply for economic growth around the world in the 21st century.

30

Figure 3.1: GDP Growth from 1960 to 1990 and GDP Per Person in 1960

3.1 Empirical Regularities

Figure 3.1 plots average GDP growth from 1960 to 1990 against GDP per person in 1960 for a large set ofcountries. There are huge variations both in initial wealth and subsequent growth. There is a group of richcountries with very high per capita incomes, and a large number of countries with per capita incomes farbelow $1,000 per year. Most rich countries experienced moderate growth between one and four percent.The variation in growth rates is much higher among the poorer countries. At the top is Singapore with agrowth rate in excess of seven percent a year, while a set of almost 20 countries had negative growth rates,so that the citizens are even poorer today than in 1960. Since we are looking at average growth rates overlong periods of time, by today the differences in growth have translated into large changes in per capitaGDP. Japan, Hong Kong and Singapore started near the bottom of the income scale, but today they areamong the richest countries in the world. In the figure there is no clear relationship between income andgrowth, but for low income countries the variation in growth rates is much higher.

If we look at industrialized countries only, we can identify some empirical regularities in the growth pro-cess. The British economist Kaldor summarized these regularities in a number of stylized facts. Althoughhe did that more than 50 years ago, the Kaldor facts still provide an accurate picture of growth in industri-alized countries. Here is the list:

31

Figure 3.2: Growth vs. Income for the American States

1. Capital per worker grows at a roughly constant rate

2. Output per worker grows at a roughly constant rate

3. The capital/output ratio is almost constant

4. The return to capital is almost constant

5. The capital and labor shares are almost constant

In the next section, I will present a model that accounts for all these facts. Figure 3.2 shows another reg-ularity of the growth process in the developed world: The convergence of different regions and countriesover time. The Figure plots per capita income in 1880 against the average growth rate over the next 100years for the American states. There is a clear negative relationship. The states that were poorest in 1880,in this case the Southern states which suffered from the Civil War, had the highest growth rates over the100 years. This higher growth led to a gradual convergence in income of the different states. While theNorth is still a little richer than the South today, the relative difference has diminished a lot during the last100 years. A similar picture arises if we look at different regions in Europe or prefectures in Japan. Thereis also convergence across countries: Germany and Japan lost the war and were relatively poor after 1945,but subsequently had higher growth rates than other industrialized countries. Today, the USA, Germany,and Japan have very similar per capita incomes.

There are no empirical regularities comparable to the Kaldor facts that apply both to industrialized anddeveloping countries. However, we can identify some factors that distinguish countries that went throughindustrialization and have a high income today from countries that remained relatively poor. An explana-tion of the role of such factors might be an important step towards understanding the large internationaldifferences in wealth. The fact that I am going to focus on is the relationship between growth and fertility.Each and every now industrialized country experienced a large drop in fertility rates, a process known

32

Figure 3.3: Population and Output Growth In the European World and Elsewhere

33

as the demographic transition. All industrialized countries have low population growth rates. Withoutimmigration, the population of countries like Germany and Japan would actually shrink. 200 years back,fertility rates were much higher, as they are in most developing countries today. Figure 3.3 shows pop-ulation and output for the “European World” (which is meant to include former colonies like the UnitedStates, Canada, and Australia), and the rest of the world. Until about 1850 population and output growthwere closely connected in the European World. But then the connection broke down, output started togrow much faster than population, so that today incomes per person are higher than ever before. In the restof the world, population and output moved in proportion at least until 1950, and even now the populationgrows significantly faster than in the European World. Consequently, per capita incomes are lower. I willcome back to these observations in the section on the Malthusian model, after introducing a model thataccounts for growth in industrialized countries.

3.2 The Solow Model: Explaining Growth in Developed Countries

In this section I will present a model that accounts for all stylized facts of growth in developed countries.The model is a straightforward application of the building blocks developed in Chapter 2 to the questionof economic growth. One problem that we need to deal with is that so far only one- or two-period modelswere considered, whereas the question of long-run economic growth calls for models with multiple periods.We will get around this problem by considering a model with overlapping generations. In the model world,people will live for two periods only, just as in the credit-market model in Chapter 2. However, eachperiod a new person is born, so that even though individuals die, the world lives on forever. People areborn young, and they work in their first period of life. In the second period, people are retired and liveoff their savings. At any point of time, two people are alive, one young person who was just born, andone old person who was born the period before. In this sense generations overlap in this model. Using anoverlapping-generations allows us to analyze an environment with infinitely many periods, while retainingthe relative simplicity of a two-period model for the household’s problem.

When denoting consumption values, we will use superscripts to denote the period when a person wasborn, and subscripts to denote the period when consumption takes place. That is, ctt is consumption of theperson born at time t when young, and c

tt+1 is consumption of the same person when old. Each person has

the same preferences, given by:

u(ctt; ctt+1) = ln(ctt)+� ln(ctt+1);

where � is a discount factor between zero and one. In the first period of life, the young consumer suppliesone unit of labor and receives wage wt. The budget constraint for the first period therefore is:

ctt = wt � st;

where st are the savings of the person born at time t. The consumer keeps his savings until the secondperiod of life, and at that time rents the savings to the firm as capital. In the old period, the consumer doesnot work, and consumes the returns on his savings:

ctt+1 = (1� Æ+ rt+1)st:

The return on savings has two elements in this model. The first is Æ, which is a discount factor between zeroand one. Fraction Æ of savings is used up when used as capital by the firm. As compensation, the consumerreceives rt+1 as the return on capital from the firm. The return on capital and depreciation combined playthe same role as the interest rate in the credit-market model in Chapter 2.

Apart from the sequence of two-period lived consumers, there is also a firm operating in every period inthis economy. The firm uses the following production technology:

Yt = (AtLt)�K1��t :

34

The firm hires labor Lt and capital Kt and aims to maximize profits in each period. The specific functionalform of the technology (Cobb-Douglas with constant returns) is the most important part of the model.

To close the model, we have to specify the market-clearing conditions, just as we did in the market modelin Chapter 2. In this case, we need two market-clearing conditions, one for labor and one for capital. Labordemand by the firm is given by Lt. Labor is supplied by young people only, and since we assumed thatexactly one young person is born every period, the market-clearing condition for labor is:

Lt = 1: (3.1)

Capital demanded by the firm is given by Kt. Since the young person does not own any capital, all capitalhas to be supplied by the savings of the old person. The market-clearing condition is:

Kt = st�1: (3.2)

In other words, the savings of the person born one period before (t � 1) are used as capital today (t). Anequilibrium for this economy is an allocation fc

tt; c

t�1t ;Kt; Ltg

1

t=1 and a set of prices frt; wtg1

t=1 such thatgiven prices, consumers and firms solve their problems, and all markets clear.

With the specification of the model completed, we can start to analyze its implications for economic growth.When analyzing growth, our aim will typically be to find the equilibrium law of motion for capital. A law ofmotion specifies the value of a variable tomorrow, given the value of this variable today. If we find the lawof motion for capital, we can infer how the capital stock changes over time, and since labor supply equalsone, this also pins down output in every period.

To find the law of motion, we have to solve the consumer’s problem, solve the firm’s problem, and thencombine the solutions to these problems with the market-clearing condition. After substituting the budgetconstraints into the utility function, the problem of the consumer born at time t is:

maxst

fln(wt � st) + � ln((1� Æ + rt+1)st)g :

The first-order condition for this problem is:

�

1

wt � st

+�

st

= 0;

which gives the solution:

st =�wt

1 + �: (3.3)

The firm’s problem is:

maxLt;Kt

�(AtLt)

�K1��t � wtLt � rtKt

;

and the first-order conditions give:

wt = �A�t

�Kt

Lt

�1��; (3.4)

rt = (1��)A�t

�Lt

Kt

��

: (3.5)

We now have all the conditions that are needed to derive the law of motion for capital. Using the market-clearing condition (3.2) on the left-hand side of (3.3) gives:

Kt+1 =�wt

1 + �:

35

Using (3.4) for wt and using (3.1) gives us the law of motion for capital:

Kt+1 =��A

�t

1 + �K1��t : (3.6)

To proceed with the analysis, we have to specify how the level of productivity At changes over time. I willassume that At grows at a constant rate, so that:

At+1 = (1+�)At:

The analysis of the law of motion (3.6) is complicated by the fact that At changes over time. It is oftenpossible to redefine a model in terms of variables that are constant in the long run. In the present case, it willturn out that while the capital stock Kt grows indefinitely, the ratio of the capital stock to the productivitylevel At approaches a constant. Define kt as the ratio of Kt to At:

kt =Kt

At

:

We then have Kt = Atkt and Kt+1 = At+1kt+1. Plugging this into (3.6), we get:

At+1kt+1 =��A

�t

1 + �(Atkt)

1��:

Since At+1 = (1 + �)At, this can be written as:

(1+�)Atkt+1 =��

1 + �Atk

1��t ;

or:

kt+1 =��

(1 + �)(1 + �)k1��t : (3.7)

Thus the law of motion for the redefined variable kt does not depend on At, therefore it is easier to analyzeit. Figure 3.4 shows the law of motion together with a 45-degree line. We see that the there is exactly onepositive value for kt at which the law of motion crosses the 45-degree line. At this point, we have kt+1 = kt.In other words, kt stays constant over time. If there is a value �k for kt such that kt+1 = kt = �k, we say thatkt reaches a steady state. We can also see that the steady state is stable, because if kt < �k, the law of motionis above the 45-degree line so that kt+1 > kt, and if kt > �k, we have kt+1 < kt. When we start with a low kt,kt will increase over time and approach the steady state �k, and the opposite happens if we start with a highinitial kt. We can use equation (3.7) to compute the steady-state �k analytically. �k satisfies kt+1 = kt = �k,therefore we have:

�k =��

(1 + �)(1 + �)�k1��;

which gives:

�k =

���

(1 + �)(1 + �)

� 1

�

: (3.8)

So far we have determined the long-run behavior of the variable kt, which was defined as the ratio of capitalKt to productivity At. Our ultimate interest is verify that the model is consistent with the growth facts fordeveloped countries. We now have to determine what the behavior of kt implies for the original variablesKt and Yt.

36

0

0.05

0.1

0.15

0.2

0 0.05 0.1 0.15 0.2

Figure 3.4: The Law of Motion for kt

Recall from the definition of kt that Kt = Atkt. Once kt reaches the steady-state, (3.8) gives:

Kt = At�k = At

���

(1 + �)(1 + �)

� 1

�

:

Since �k is constant in the steady-state,Kt grows at the same rate as At in the steady-state. Since we assumeda constant growth rate 1 + � for At, this implies that Kt grows at rate 1 + � as well, which verifies the firstof the growth facts on our list (since there is just one worker in our model, Kt can be interpreted both astotal capital and capital per worker).

Output (total or per worker) is given by:

Yt = (AtLt)�K1��t :

Plugging in the last equation for Kt and Lt = 1 from the market-clearing condition (3.1), we get the follow-ing expression for output per worker Yt in the steady state:

Yt = A�t A

1��t

���

(1 + �)(1 + �)

� 1��

�

or:

Yt = At

���

(1 + �)(1 + �)

� 1��

�

:

Thus output equals At multiplied by a constant. Therefore the long-run growth rate of Yt is 1 + �, whichverifies the second growth fact on our list.

The capital-output ratio is given by:

Kt

Yt

=Kt

(AtLt)�K1��t

=

�Kt

AtLt

��

:

37

Since Lt = 1 in equilibrium, this is:

Kt

Yt

=

�Kt

At

��

= k�t :

Thus the capital-output ratio is a function of kt only, and since kt is constant in the steady-state, the capital-output ratio is constant as well. This verifies the third growth fact.

From (3.5), the return on capital is given by:

rt = (1��)A�t

�Lt

Kt

��

:

With Lt = 1, this can be written as:

rt = (1��)

�At

Kt

��

= (1��)k��t :

Thus the return on capital only depends on kt as well, and is therefore constant in the steady state. Thisverifies growth fact number four.

So far, all the growth facts we checked hold only in the steady state, but not necessarily during the transitionto the steady state. The last growth fact, constant shares for labor and capital, is satisfied both in the steadystate and during the transition. The labor share is given by:

wtLt

Yt

=�A

�t L

��1t K

1��t Lt

(AtLt)�K1��t

= �;

which is constant. Similarly, for the capital share we get:

rtKt

Yt

=(1� �)(AtLt)

�K��t Kt

(AtLt)�K1��t

= 1��:

Thus the model satisfies all the growth facts on the Kaldor list. The model is also consistent with theobservation of convergence of countries and regions that start with different levels of income and capital.The model predicts convergence to the steady state, regardless what the initial conditions are. In the longrun, all countries or regions reach the same level of production per worker, as long as the technologies usedare the same.

To look at this in another way, we can compute the growth rate of the capital stock as a function of initialcapital. The law of motion (3.6) for capital Kt is given by:

Kt+1 =��A

�t

1 + �K1��t :

The growth rate of the capital stock therefore is:

Kt+1

Kt

=��A

�t

1 + �K��t :

Notice that the coefficient on Kt is negative. The higher the initial Kt, the slower is the capital stock goingto grow subsequently. Countries or regions that start with little capital will accumulate capital faster andcatch up with the leaders. The intuition for this result follows from the fact that the production functionexhibits decreasing returns to capital. The more capital a country already possesses, the smaller are thegains in output from installing even more capital. This makes it impossible for countries to grow fasterthan others indefinitely simply by saving and investing more than others.

38

Other production functions which, for example, do not feature decreasing returns to capital or do not implyconstant factor shares, produce outcomes that are hard to reconcile with the growth facts. This is the mainreason why the Cobb-Douglas production function is used as much as it is in macroeconomic studies.The Cobb-Douglas production function is often referred to as the neoclassical production function, andtherefore the growth model presented in this section is sometimes called the neoclassical growth model. Itis also called the Solow model, after its inventor Robert K. Solow.

The Solow model succeeds in explaining all stylized facts of economic growth in industrialized countries.The model explains the convergence across the American states that we saw in Figure 3.2, as well as therapid growth of Germany and Japan after World War II. Since the capital stock was low after most of itwas destroyed during the war, the return to capital was high in those countries, and the economies weregrowing fast until they reached the steady state. The model also explains why different savings rates indifferent industrialized countries do not translate into differences in the growth rate. The savings rateaffects the level of the steady state, but not the steady state growth rate.

Since the Solow model matches the facts of economic growth, it forms the basis of many more advancedmodels in macroeconomics. For example, the real-business-cycle model that will be presented in the nextchapter is a variant of the Solow model, with the addition of productivity shocks. On the other hand,the model works well only for countries that satisfy the assumption of constant population growth anda constant rate of technological progress. I will develop a model that explicitly accounts for populationgrowth below. Before that, I will introduce growth accounting, a method that allows us to decompose thegrowth rate of a country into growth in population, capital, and productivity.

Growth Accounting

Consider the neoclassical production function:

Yt = (AtLt)�K1��t :

Yt is interpreted as GDP, Lt is the number of workers, Kt the aggregate capital stock, and At measuresoverall productivity. I will be concerned with measuring the relative contributions of At, Lt, and Kt toGDP growth. I assume that data for GDP, the labor force and the aggregate capital stock is available. Thefirst thing that we need to do is compute the productivity parameter At. Solving the production functionfor At yields:

At =Y

1

�

t

LtK

1��

�

t

If � were known, we could compute the At right away. Luckily, we found out earlier that � is equal to thelabor share. Therefore we can use the average labor share as an estimate of � and compute the At.

Now that At is available, the growth rates in At, Lt and Kt can be computed by using log differences. Wecan see how the growth rates in inputs and productivity affect the GDP growth rate by taking logs of theproduction function,

lnYt = �[lnAt+lnLt]+(1��) lnKt:

Using two consecutive years, we get:

lnYt+1�lnYt = �[lnAt+1�lnAt+lnLt+1�lnLt]+(1��)[lnKt+1�lnKt]:

Thus the growth rate in output is � times the sum of growth in productivity and labor, plus 1 � � timesgrowth in capital. Using this, we can compute the relative contribution of the different factors. For example,the fraction of output growth explained by growth of the labor force is:

�[lnLt+1 � lnLt]

lnYt+1 � lnYt:

39

The fraction due to growth in capital equals:

(1� �)[lnKt+1 � lnKt]

lnYt+1 � lnYt:

Finally, the remaining fraction is due to growth in productivity and can be computed as:

�[lnAt+1 � lnAt]

lnYt+1 � lnYt:

It is hard to determine the exact cause of productivity growth. The way we compute it, it is merely a resid-ual, the fraction of economic growth that cannot be explained by growth in labor and capital. Neverthe-less, measuring productivity growth this way gives us a rough idea about the magnitude of technologicalprogress in a country.

3.3 The Malthusian Model: Explaining Stagnation

The Solow model does an excellent job of explaining the facts it is designed to explain: the growth factsfor developed countries. However, when compared to history and other countries today, the recent growthexperience of developed countries is rather special. From early human history until about 1800, there wasno sustained growth in income per capita and living standards in any country of the world. Today, in anumber of countries people are still not much better off than they were hundreds of years ago. Historically,stagnation, not growth in income per capita was the typical situation.

If we want to understand why some countries escaped stagnation and others did not, we first need toknow why there was stagnation in the first place. To answer this question, it is essential to consider theinteractions between economic growth and population growth. In the Solow model, we did not analyzefertility decisions, and simply took it as given that every generation has the same size. In this section, wewill derive the implications of letting people choose their optimal number of children.

The first economist to think in a systematic way about growth and fertility was Thomas Malthus. Backin 1798, he published the “Essay on Population”. His basic thesis was that fertility is checked only by thefood supply. As long as there was enough to eat, people would continue to produce children. Since thisleads to population growth rates in excess of the growth in food supply, people will be pushed down tothe subsistence level. According to this theory, sustained growth in per capita incomes is not possible,population growth will always catch up with increases in production and push per capita incomes down.

Stated in modern terms, Malthus thought that children are a normal good. When income goes up, morechildren are “consumed” by the parents. I assume that parents have children for their enjoyment only, thatis, I abstract from issues like child labor. As a simple example, consider a utility function over consumptionct and number of children nt of the form:

u(ct; nt) = ln(ct)+ ln(nt):

I assume that the consumer supplies one unit of labor for real wage wt, and that the cost in terms of goodsof raising a child is p. The budget constraint is then:

ct+ pnt = wt:

By substituting for consumption, we can write the utility maximization problem as:

maxnt

fln(wt � pnt) + ln(ntg):

40

The first order condition with respect to nt is:

0 = �

p

wt � pnt

+1

nt

;

which yields:

nt =wt

2p:

Thus the higher the wage, the more children are going to be produced.

If we assume that people live for one period, the number of children per adult nt determines population inthe next period Lt+1:

Lt+1 = ntLt =wt

2pLt: (3.9)

That is, the population tomorrow equals population today times number of children per person.

To close the model, we have to specify how the wage is determined. Malthus’ assumption was that thefood supply cannot be increased in proportion with population growth. In modern terms, he meant thatthere are decreasing returns to labor. The basic reason for this is that in agricultural production land playsan important role, and the amount of land cannot be increased. As an example, assume that the aggregateproduction function is:

Yt = (AtX)1��L�t ;

where 0 < � < 1, At is the level of productivity, and X is the fixed amount of land available in the country.

The wage wt could be determined in two different ways in the economy. If there is land ownership, theland owners will hire workers, and as in the Solow model the wage will equal the marginal product oflabor. Alternatively, we could assume that land is a public good that is used by all workers together. In thatcase, we would assume that each worker gets an equal share of output. Both assumptions lead to the sameconclusions. In order to avoid dealing with land ownership, I will assume that land is a public good andthe real wage therefore is:

wt =Yt

Lt

=

�AtX

Lt

�1��:

Notice that the wage decreases in the size of the population Lt. The more people, the less output there isper person. This is a basic consequence of decreasing returns to labor due to the fixed factor land. Pluggingthe wage wt into the law of motion for population (3.9) yields:

Lt+1 =(AtX)1��

2pL�t :

The result is very similar to the law of motion for capital in the Solow model. The law of motion has theconvergence property in the sense that population growth is higher when population is low:

Lt+1

Lt

=(AtX)1��

2pL��1t :

Once again, it is possible to redefine variables such that the system reaches a steady state. Define lt as theratio of population to productivity:

lt =Lt

At

:

41

Rewriting the law of motion by substituting Lt = Atlt and Lt+1 = At+1lt+1 gives:

At+1lt+1 =X1��

2pAtl

�t :

I will assume again that productivity grows at a constant rate:

At+1 = (1+�)At:

We then have:

(1+�)Atlt+1 =X1��

2pAtl

�t

or:

lt+1 =X1��

2p(1 + �)l�t :

Again, this law of motion is very similar to the law of motion for kt in the Solow model. The variable lt

reaches a steady state �l, given by:

�l =

�X1��

2p(1 + �)

� 1

1��

:

Since we have Lt = Atlt, in the steady state population grows at the same rate as productivity At.

So far, we have a developed a theory of population size as a function of productivity. Specifically, we foundthat population growth depends negatively on initial population. This implies that population growthshould be expected to rise after a sudden decrease in population, for example because of an epidemicor a war. This prediction is in line with historical observations, for example, the fast recovery of Europeanpopulation after the Black Death. The theory also implies that population growth and therefore density willbe higher in countries or areas where productivity growth � is high. Again, this prediction is confirmedby evidence. Throughout history, population density is closely related to the technological knowledge ofa society. For example, in the last 2000 years first China and later Western Europe were the leaders intechnological progress, and both these areas ended up with a very high population density.

What are the implications of the model for living standards? In the model, people’s only income is thewage wt, given by:

wt =

�AtX

Lt

�1��=

�X

lt

�1��:

The wage depends on the ratio of population to productivity lt, and since lt is constant in the long run, thesame applies to the wage. In other words, the model predicts stagnation in living standards even thoughthe level of productivity is increasing. The reason for this surprising result is that the rate of populationgrowth increases when technological progress increases. Higher productivity in itself would tend to raisewages, but higher population lowers wages. In the long-run equilibrium, these effects cancel. Populationgrowth makes up for all technological advances, and wages stagnate. The Malthusian model shows that itis easy to produce stagnation in living standards in an economic model. Only two assumptions are needed:Decreasing returns to labor, and a positive relationship between income and population growth.

The Malthusian model predicts that technological progress will affect population density only, and thatwages will stagnate. These predictions are roughly consistent with the economic history of the world before1800, and the conditions in many developing countries up to the present time. In industrialized countries,however, income per capita is growing, and birth rates have fallen tremendously from to their historicalaverages. Something must have changed in the industrialized countries that made families decide to havefewer children. The result was that economic growth no longer affected population growth. The economiesstarted to behave as predicted by the Solow model, which takes the growth rate of population as constant.Technological progress then raises wages and leads to sustained improvements in living standards, whileleaving population density unchanged.

42

3.4 Malthus to Solow: How to Escape from the Stagnation Trap

If we want to understand why some countries are rich today and others are poor, we have to find out howrich countries escaped from the Malthusian trap. In other words, why is it that people in industrializedcountries decided to have smaller families, instead of increasing the number of children as the Malthusianmodel would have predicted? Something must have changed in Europe in the nineteenth century thatmade it attractive to people to have less children, so that fertility rate fell and income per capita could startto grow. While these changes are not fully understood, we can identify an number of important factors.I will concentrate on two of them: The time cost of raising children, and a quality-quantity tradeoff indecisions on children.

In the Malthusian model, we assumed that the cost of children is fixed in terms of the consumption good. Inthe real world, however, rearing children requires both goods and time. Since the cost of time is measuredby the wage, as the economy grows and wages increase, having children becomes more expensive. Thiseffect can counteract the tendency to more children as people become richer. The quantity-quality tradeoffconcerns the observation that people tend to invest more in their children than is strictly necessary forkeeping them alive. For example, many parents incur significant expenses to pay for the education of theirchildren. We can interpret paying for education as investing in the quality of children. Parents care not onlyabout the number of children they have, but also about the quality of each child. As people become richer,they may decide to invest more in quality, instead of having more children. This effect can counteract thetendency to larger families as income rises as well.

As an example, consider the problem of a parent who has preferences:

u(ct; nt; et) = ln(ct)+ ln(nt)+ ln(et)

over consumption ct, number of children nt, and education per child et. I assume that the cost of educationis given by pe, and that raising each child takes pn units of time. The parent has an endowment of oneunit of time. Consequently, 1 � pnnt units of time are left for work after raising the children. The budgetconstraint then is:

ct+ peet = (1� pnnt)wt:

Substituting the budget constraint into the utility function results in the following maximization problem:

maxnt;et

fln((1� pnnt)wt � peet) + ln(nt) + ln(et)g :

The first-order conditions for nt and et are:

�

pnwt

(1� pnnt)wt � peet

+1

nt

= 0;

and:

�

pe

(1� pnnt)wt � peet

+1

et

= 0:

These equations can be solved for the optimal choices for child quantity nt and child quality et:

nt =1

3pn;

and:

et =wt

3pe:

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We find that the number of children nt no longer depends on the wage, but is constant instead. Thisprediction is consistent with the roughly constant fertility rates in developed countries over the last 80years. The model also predicts that investment in child quality, which can be interpreted as education,will grow with income wt. This prediction is consistent with the large increases in education levels inindustrialized countries. In summary, our simple model of a consumer deciding on child quantity andquality is in line with fertility behavior in industrialized countries. Combined with the Solow model, itprovides a unified theory of income and population in developed countries.

The model also provides a potential explanation of how industrializing countries escaped from the Malthu-sian trap. If at some point people decided to substitute child quality for quantity, fertility rates would falleven if incomes rise. There could be different reasons for such a change. The first possibility is improvedhealth and medicine, which lowers mortality rates. When people can expect to live a longer productive life,investing in education becomes more attractive. Also, technological changes that increase the demand forskilled labor can increase the returns on education. Again, this makes investing in the education of childrenas opposed to the number of children more attractive. A third possibility is a declining role of child labor.If there is no demand for child labor, the opportunity cost of sending children to school is low, which onceagain would raise the demand for child quality.

3.5 Conclusion and the 21st Century

In this chapter we developed two growth models with radically different predictions. The Solow modelpredicts sustained growth in income per capita, a stable population, and convergence across countries andregions. This model accounts for the recent growth experience of industrialized countries, but is inconsis-tent with the developing world and earlier economic history. The Malthusian model predicts stagnation inliving standards and rising population. This model is consistent with earlier economic history and devel-oping countries, but it is inconsistent with the recent economic history of developed countries.

During the 20th century, one part of the world developed as predicted by the Solow model, while anotherpart of the world developed in line with the predictions of Malthus. The result was a dramatic widening ofthe world income distribution: Some countries got rich, many stayed poor, and the gap between rich andpoor increased over time. What are our prospects for the 21st century? In the end, only history will tell, butthe models developed in this chapter allow us to make some educated guesses. We know that the Solowmodel predicts convergence across countries and regions. If all countries were to follow the Solow model,we would expect a narrowing of the world income distribution and convergence to a common growth path.How do we know whether countries are going to follow the Solow model or the Malthusian model? Again,we cannot know for sure, but in the last section we developed a theory that shows that a substitution ofchild quality for child quantity plays a major role in the transition from Malthus to Solow. This allows us toidentify some early signs that the transition from Malthus to Solow is taking place. Specifically, if a countryswitches from Malthus to Solow, we would expect to observe rising levels of education, falling fertility,falling child-labor rates, and beginning growth in income per capita. All these indicators can be observedtoday in many developing countries. Education is rising and fertility as falling throughout the world. Eventhough income levels are still low, growth rates in income per capita are picking up in the poorest countriesof the world, and often exceed the average growth rates in industrialized countries.

Observing these early signs suggests that many countries have started the transition from Malthus to Solow.In consequence, from the point of view of growth theory it seems likely that the world income distributionwill narrow in the 21st century, and that the current levels of inequality will enter the economic history ofthe world as a temporary phenomenon.

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