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3 Institutions, technological progress and growth
Protection from expropriation and GDP per person
There is a strong positive relation
between the degree of protection from
expropriation and the level of GDP per
Source: Daron Acemoglu, ‘Understanding institutions’, Lionel Robbins Lectures,
2004, London School of Economics, http://
Kenya’s index is 6. Kenya is below
the regression line, which means that
Kenya has lower GDP per person than
would be predicted based just on the
Log GDP per capita per person, PPP, in 1995
248 THE CORE The long run
Average protection against risk of expropriation, 1985–1995
constructed for each of these countries by an international business organisation. The positive correlation between the two is striking (the figure also plots the regression line). Low
protection is associated with a low GDP per person (at the extreme left of the figure are Zaire
and Haiti); high protection is associated with a high GDP per person (at the extreme right are
the United States, Luxembourg, Norway, Switzerland and the Netherlands).
What does ‘protection of property rights’ mean in practice? It means a good political system, in which those in charge cannot expropriate or seize the property of the citizens. It
means a good judicial system, where disagreements can be resolved efficiently, rapidly and
fairly. Looking at an even finer degree of detail, it means laws against insider trading in the
stock market, so people are willing to buy stocks and so provide financing to firms; it means
clearly written and well-enforced patent laws, so firms have an incentive to do research and
develop new products. It means good antitrust laws, so competitive markets do not turn into
monopolies with few incentives to introduce new methods of production and new products.
The list obviously goes on. (A particularly dramatic example of the role of institutions is given
in the next Focus box.)
The importance of institutions: North Korea and South Korea
Following the surrender of Japan in 1945, Korea formally
acquired its independence but became divided at the 38th
parallel into two zones of occupation, with Soviet armed
forces occupying the north and US armed forces occupying
the south. Attempts by both sides to claim jurisdiction over
all of Korea triggered the Korean War, which lasted from
1950 to 1953. At the armistice in 1953, Korea became formally divided into two countries, the Democratic People’s
Republic of North Korea in the north and the Republic of
Korea in the south.
M12 Macroeconomics 85678.indd 248
An interesting feature of Korea before separation was
its ethnic and linguistic homogeneity. The north and the
south were inhabited by essentially the same people, with
the same culture and the same religion. Economically, the
two regions were also highly similar at the time of separation. PPP GDP per person, in 1996 dollars, was roughly the
same, about $700 in both the north and south.
Yet, 50 years later, as shown in Figure 12.7, GDP per
person was 10 times higher in South Korea than in North
Korea: $12,000 versus $1,100! On the one hand, South
Chapter 12 Technological progress and growth 249
intervention but also private ownership and legal protection of private producers. North Korea relied on central
planning. Industries were quickly nationalised. Small firms
and farms were forced to join large cooperatives, so they
could be supervised by the state. There were no private
property rights for individuals. The result was the decline
of the industrial sector and the collapse of agriculture.
The lesson is sad, but transparent; institutions matter very
much for growth.
Korea had joined the OECD, the club of rich countries.
On the other, North Korea had seen its GDP per person
decrease by nearly two-thirds from its peak of $3,000 in
the mid-1970s and was facing famine on a large scale. (The
graph, taken from the work of Daron Acemoglu, stops in
1998. But, if anything, the difference between the two
Koreas has become larger since then.)
What happened? Institutions and the organisation
of the economy were dramatically different during that
period in the south and in the north. South Korea relied on
a capitalist organisation of the economy, with strong state
GDP per capita
PPP GDG per person (1996 dollars)
Source: Daron Acemoglu, ‘Understanding institutions’, Lionel Robbins Lectures,
2004, London School of Economics, http://economics.mit.edu/files/1353
PPP GDP per person: North and South Korea, 1950–98
This still leaves one essential question: Why do poor countries not adopt these good institutions? The answer is that it is hard! Good institutions are complex and difficult for poor
countries to put in place. Surely, causality runs both ways in Figure 12.5: low protection
against expropriation leads to low GDP per person. But it is also the case that low GDP per
person leads to worse protection against expropriation. Poor countries are often too poor to
afford a good judicial system and to maintain a good police force, for example. Thus, improving institutions and starting a virtuous cycle of higher GDP per person and better institutions
are often difficult. The fast-growing countries of Asia have succeeded. (The following Focus
box explores the case of China in more detail.) Some African countries appear also to be succeeding; others are still struggling.
M12 Macroeconomics 85678.indd 249
A quote from Gordon Brown, a former
UK prime minister, ‘In establishing the
rule of law, the first five centuries are
always the hardest!’
250 THE CORE The long run
What is behind Chinese growth?
From 1949 – the year in which the People’s Republic of
China was established – to the late 1970s, China’s economic system was based on central planning. Two major
politico-economic reforms, the Great Leap Forward in
1958 and the Cultural Revolution in 1966, ended up as
human and economic catastrophes. Output decreased by
20% from 1959 to 1962, and it is estimated that 25 million people died of famine during the same period. Output
again decreased by more than 10% from 1966 to 1968.
After Chairman Mao’s death in 1976, the new leaders
decided to introduce market mechanisms progressively
in the economy. In 1978, an agricultural reform was put
in place, allowing farmers, after satisfying a quota due to
the state, to sell their production in rural markets. Over
time, farmers obtained increasing rights to the land and,
today, state farms produce less than 1% of agricultural
output. Outside of agriculture, and also starting in the late
1970s, state firms were given increasing autonomy over
their production decisions, and market mechanisms and
prices were introduced for an increasing number of goods.
Private entrepreneurship was encouraged, often taking the
form of ‘Town and Village Enterprises’, collective ventures
guided by a profit motive. Tax advantages and special
agreements were used to attract foreign investors.
The economic effects of these cumulative reforms have
been dramatic. Average growth of output per worker has
increased from 2.5% between 1952 and 1977 to more than
9% since then.
Is such high growth surprising? One could argue that
it is not. Looking at the 10-fold difference in productivity between North Korea and South Korea we saw in the
previous Focus box, it is clear that central planning is a
poor economic system. Thus, it would seem that, by moving from central planning to a market economy, countries
could easily experience large increases in productivity.
The answer is not so obvious, however, when one looks at
the experience of the many countries that, since the late
1980s, have indeed moved away from central planning.
In most Central European countries, this transition was
typically associated with an initial 10% to 20% drop in
GDP, and it took five years or more for output to exceed
its pre-transition level. In Russia and in the new countries
carved out of the Soviet Union, the drop was even larger
and longer lasting. (Many transition countries now have
strong growth, although their growth rates are far below
that of China.)
M12 Macroeconomics 85678.indd 250
In Central and Eastern Europe, the initial effect of
transition was a collapse of the state sector, only partially
compensated by slow growth of the new private sector. In
China, the state sector has declined more slowly, and its
decline has been more than compensated by strong private
sector growth. This gives a proximate explanation for the
difference between China and the other transition countries. But it still begs the question: How was China able to
achieve this smoother transition?
Some observers offer a cultural explanation. They
point to the Confucian tradition, based on the teachings
of Confucius, which still dominates Chinese values and
emphasises hard work, respect of one’s commitments, and
trustworthiness among friends. All these traits, they argue,
are the foundations of institutions that allow a market
economy to perform well.
Some observers offer an historical explanation. They
point to the fact that, in contrast to Russia, central planning in China lasted only for a few decades. Thus, when
the shift back to a market economy took place, people still
knew how such an economy functioned and adapted easily
to the new economic environment.
Most observers point to the strong rule of the Communist Party in the process. They point out that, in contrast to
Central and Eastern Europe, the political system did not
change, and the government was able to control the pace
of transition. It was able to experiment along the way, to
allow state firms to continue production while the private
sector grew and to guarantee property rights to foreign
investors (in Figure 12.5, China has an index of property
rights of 7.7, not far from its value in rich countries). With
foreign investors has come the technology from rich countries, and in time the transfer of this knowledge to domestic firms. For political reasons, such a strategy was simply
not open to governments in Central and Eastern Europe.
The limits of the Chinese strategy are clear. Property
rights are still not well established. The banking system is
still inefficient. So far, however, these problems have not
stood in the way of growth.
For more on China’s economy, read Gregory Chow,
China’s Economic Transformation, 3rd edition (Chichester:
For a comparison between transition in Eastern Europe
and China, read Jan Svejnar, ‘China in light of the performance of Central and East European economies’, IZA Discussion Paper 2791, May 2007.
Chapter 12 Technological progress and growth 251
12.4 The facts of growth revisited
We can now use the theory we have developed in this and the previous chapter to interpret
some of the facts we saw previously (see Chapter 10).
Capital accumulation versus technological progress in rich
countries since 1985
Suppose we observe an economy with a high growth rate of output per worker over some
period of time. Our theory implies this fast growth may come from two sources:
It may reflect a high rate of technological progress under balanced growth.
It may reflect instead the adjustment of capital per effective worker, K/AN, to a higher
level. As we saw in Figure 12.4, such an adjustment leads to a period of higher growth,
even if the rate of technological progress has not increased.
Can we tell how much of the growth comes from one source and how much comes from
the other? Yes. If high growth reflects high balanced growth, output per worker should
be growing at a rate equal to the rate of technological progress (see Table 10.1). If high
growth reflects instead the adjustment to a higher level of capital per effective worker, this
adjustment should be reflected in a growth rate of output per worker that exceeds the rate
of technological progress.
Let’s apply this approach to interpret the facts about growth in rich countries we saw in
Table 10.1. This is done in Table 12.2, which gives the average rate of growth of output per
worker (gY - gN) for 1985 to 2014 and the average rate of technological progress, gA, for
1985 and 2013 for selected countries in Europe – including Denmark, France, Germany,
Italy, Spain, Sweden and the United Kingdom, together with Japan and the United States
we looked at in Table 10.1. Note two differences between Tables 10.1 and 12.2. First, as
suggested by the theory, Table 12.2 looks at the growth rate of output per worker, whereas
Table 10.1, which was focusing on the standard of living, looked at the growth rate of output per person; the differences, however, are rather small. Second, because of data limita- ➤ In the United States, for example, the
tions, Table 12.2 starts in 1985 rather than in 1950. The rate of technological progress, gA, ratio of employment to population
is constructed using a method introduced by Robert Solow; the method and the details of decreased slightly from 60.1% in 1985
to 59% in 2014. Thus, output per person
construction are given in the appendix to this chapter.
and output per worker grew at virtually
Table 12.2 leads to two conclusions. First, over the period 1985–2014, output per worker the same rate over this period.
has grown at rather similar rates across the selected countries. In particular, there was little or
no catching up of the United States by the other countries. This is in contrast to the numbers
in Table 10.1 which looked at the period 1950–2014 and showed substantial convergence to
the United States. Put another way, much of the convergence happened between 1950 and
1985, and since then appears to have slowed down or even stopped.
Table 12.2 Average annual rates of growth of output per worker and technological
progress in selected rich countries since 1985
Rate of growth of output per
worker (%) 1985–2014
Rate of technological
progress (%) 1985–2014
Source: Calculations from OECD Productivity Statistics.
M12 Macroeconomics 85678.indd 251
252 THE CORE The long run
Second, growth since 1985 has come from technological progress only in a few cases,
while in most cases it has come from unusually high capital accumulation. This conclusion
follows from the fact that only in some countries has the growth rate of output per worker
been roughly equal to the rate of technological progress. This is what we would expect when
countries are growing along their balanced growth path. In many European economies (with
the exception of France and Germany among the selected countries in Table 12.2), the rate
of growth of output per worker has been substantially higher than the rate of technological
progress, which means that growth in these countries has come from an unusual increase in
the ratio of capital to output, or capital accumulation.
Note what this conclusion does not say. It does not say that capital accumulation was
irrelevant in countries that have been growing along their balanced growth path. Capital
accumulation was such as to allow these countries to maintain a roughly constant ratio of
output to capital and achieve balanced growth.
Capital accumulation versus technological progress in China
Going beyond growth in OECD countries, one of the striking facts of Chapter 10 was the high
growth rates achieved by a number of Asian countries in the last three decades. This raises
again the same questions as those we just discussed: Do these high growth rates reflect rapid
technological progress, or do they reflect unusually high capital accumulation?
To answer these questions, we shall focus on China, because of its size and because of
the astonishingly high output growth rate, nearly 10% since the late 1970s. Table 12.3 gives the
average rate of growth, gY, the average rate of growth of output per worker, gY - gN, and
average rate of technological progress, gA, for two periods, 1978 to 1995 and 1996 to 2011.
Warning! Chinese data for output, ➤
employment and the capital stock (the
Table 12.3 yields two conclusions. From the late 1970s to the mid-1990s, the rate of techlatter is needed to construct gA) are
nological progress was close to the rate of growth of output per worker. China was roughly
not as reliable as similar data for OECD
on a (rapid) balanced growth path. Since 1996, however, although growth of output per
countries. Thus, the numbers in the
worker has remained high, the contribution of technological progress has decreased. Put
table should be seen as more tentative
another way, more recently, growth in China has come partly from unusually high capital
than those in Table 12.2.
accumulation – from an increase in the ratio of capital to output.
Table 12.3 Average annual rate of growth of output per worker and technological progress
in China, 1978–2011
Rate of growth of
Rate of growth of
output per worker
Rate of technological
Source: Penn World Table version 8.1.
We can look at it another way. Recall, from Table 12.1, that under balanced growth
gK = gY = gA + gN. To see what investment rate would be required if China had balanced
growth, go back to equation (12.3) and divide both sides by output, Y, to get:
= (d + gA + gN)
Let’s plug in numbers for China for the period 1996–2011. The estimate of d, the depreciation rate of capital in China, is 5% a year. As we just saw, the average value of gA for the
period was 5.9%. The average value of gN, the rate of growth of employment, was 0.9%.
The average value of the ratio of capital to output was 2.9. This implies a ratio of investment
of output required to achieve balanced growth of (5% + 5.9% + 0.9%) * 2.9 = 34.2%.
M12 Macroeconomics 85678.indd 252
Chapter 12 Technological progress and growth 253
The actual average ratio of investment to output for 1995–2011 was a much higher 47%.
Thus, both rapid technological progress and unusually high capital accumulation explain
high Chinese growth. If the rate of technological progress were to remain the same, this suggests that, as the ratio of capital to output stabilises, the Chinese growth rate will decrease,
closer to 6% than to 9.8%.
Where does technological progress in China come from? A closer look at the data suggests
two main channels. First, China has transferred labour from the countryside, where productivity is low, to industry and services in the cities, where productivity is much higher. Second,
China has imported the technology of more technologically advanced countries. It has, for
example, encouraged the development of joint ventures between Chinese firms and foreign
firms. Foreign firms have come with better technologies and, over time, Chinese firms have
learned how to use them. To relate to our discussion, growth has come largely through imitation, the importation and adaptation of modern technologies from more advanced countries.
As China catches up and gets closer to the technology frontier, it will have to shift from imitation to innovation, and thus modify its growth model.
When we think about the implications of technological
progress for growth, it is useful to think of technological progress as increasing the amount of effective labour
available in the economy (i.e. labour multiplied by the
state of technology). We can then think of output as
being produced with capital and effective labour.
In steady state, output per effective worker and capital per
effective worker are constant. Put another way, output per
worker and capital per worker grow at the rate of technological progress. Put yet another way, output and capital
grow at the same rate as effective labour, thus at a rate
equal to the growth rate of the number of workers plus
the rate of technological progress.
When the economy is in steady state, it is said to be on
a balanced growth path. Output, capital and effective
labour are all growing ‘in balance’, that is at the same rate.
The rate of output growth in steady state is independent
of the saving rate. However, the saving rate affects the
steady-state level of output per effective worker. And
increases in the saving rate will lead, for some time, to
an increase in the growth rate above the steady-state
Technological progress depends on: (1) the fertility of
R&D, how spending on R&D translates into new ideas
and new products; and (2) the appropriability of the
results of R&D, which is the extent to which firms benefit
from the results of their R&D.
When designing patent laws, governments must balance
their desire to protect future discoveries and provide incentives for firms to do R&D with their desire to make existing
discoveries available to potential users without restrictions.
Sustained technological progress requires that the right
institutions are in place. In particular, it requires wellestablished and well-protected property rights. Without
good property rights, a country is likely to remain poor.
But in turn, a poor country may find it difficult to put in
place good property rights.
France, Japan, the United Kingdom and the United States
have experienced roughly balanced growth since 1950.
Growth of output per worker has been roughly equal to
the rate of technological progress. Growth in China is a
combination of a high rate of technological progress and
unusually high investment, leading to an increase in the
ratio of capital to output.
state of technology 237
effective labour 237
labour in efficiency
balanced growth 241
M12 Macroeconomics 85678.indd 253
research and development
technology frontier 246
rate of growth of total factor
fertility of research 243
property rights 247
rate of tfp growth 257
Solow residual 257
254 THE CORE The long run
Questions and problems
All ‘Quick check’ questions and problems are available on
1.Using the information in this chapter, label each of the following statements true, false or uncertain. Explain briefly.
a. Writing the production function in terms of capital and
effective labour implies that as the level of technology
increases by 10%, the number of workers required to
achieve the same level of output decreases by 10%.
b. If the rate of technological progress increases, the investment rate (the ratio of investment to output) must increase
to keep capital per effective worker constant.
c. In steady state, output per effective worker grows at the
rate of population growth.
d.In steady state, output per worker grows at the rate of
e. A higher saving rate implies a higher level of capital per
effective worker in the steady state and thus a higher rate
of growth of output per effective worker.
f. Even if the potential returns from research and development (R&D) spending are identical to the potential
returns from investing in a new machine, R&D spending
is much riskier for firms than investing in new machines.
g.The fact that one cannot patent a theorem implies that
private firms will not engage in basic research.
h. Because eventually we will know everything, growth will
have to come to an end.
3.Sources of technological progress: leaders versus
a. Where does technological progress come from for the economic leaders of the world?
b.Do developing countries have other alternatives to the
sources of technological progress you mentioned in part
c. Do you see any reasons why developing countries may
choose to have poor patent protection? Are there any dangers in such a policy (for developing countries)?
All ‘Dig deeper’ questions and problems are available on
4.For each of the economic changes listed in (a) and (b), assess
the likely impact on the growth rate and the level of output over
the next five years and over the next five decades.
a.A permanent reduction in the rate of technological
b. A permanent reduction in the saving rate.
5.Measurement error, inflation and productivity growth
Suppose that there are only two goods produced in an economy:
haircuts and banking services. Prices, quantities and the number of workers occupied in the production of each good for year
1 and for year 2 are given in the following table:
i. Technology has not played an important part in Chinese
2.R&D and growth
a. Why is the amount of R&D spending important for growth?
How do the appropriability and fertility of research affect
the amount of R&D spending?
How do each of the policy proposals listed in (b) to (e) affect the
appropriability and fertility of research, R&D spending in the
long run, and output in the long run?
a. What is nominal GDP in each year?
b. Using year 1 prices, what is real GDP in year 2? What is the
growth rate of real GDP?
c. What is the rate of inflation using the GDP deflator?
b. An international treaty ensuring that each country’s patents are legally protected all over the world. This may be
a part of the proposed Trans-Pacific Partnership.
d. Using year 1 prices, what is real GDP per worker in year 1
and year 2? What is labour productivity growth between
year 1 and year 2 for the whole economy?
c. Tax credits for each euro of R&D spending.
Now suppose that banking services in year 2 are not the same
as banking services in year 1. Year 2 banking services include
telebanking, which year 1 banking services did not include. The
technology for telebanking was available in year 1, but the price
of banking services with telebanking in year 1 was $13, and
no one chose to purchase this package. However, in year 2, the
d.A decrease in funding of government-sponsored conferences between universities and corporations.
e. The elimination of patents on breakthrough drugs, so the
drugs can be sold at a low cost as soon as they become
M12 Macroeconomics 85678.indd 254
Chapter 12 Technological progress and growth 255
price of banking services with telebanking was $12, and everyone chose to have this package (i.e. in year 2 no one chose to
have the year 1 banking services package without telebanking).
(Hint: Assume that there are now two types of banking services:
those with telebanking and those without. Rewrite the preceding table but now with three goods: haircuts and the two types
of banking services.)
technology, K is the level of capital stock and H is the level of the
human capital stock.
e. Using year 1 prices, what is real GDP for year 2? What is
the growth rate of real GDP?
e. Low tax rates
f. What is the rate of inflation using the GDP deflator?
g. Low population growth
g. What is labour productivity growth between year 1 and
year 2 for the whole economy?
h. Consider this statement: ‘If banking services are mismeasured – for example, by not taking into account the introduction of telebanking – we will overestimate inflation
and underestimate productivity growth.’ Discuss this
statement in light of your answers to parts (a) to (g).
6.Suppose that the economy’s production function is:
Y = 2K2AN
that the saving rate, s, is equal to 16%, and that the rate of
depreciation, d, is equal to 10%. Suppose further that the number of workers grows at 2% per year and that the rate of technological progress is 4% per year.
a. Find the steady-state values of the variables listed in (i) to
i.The capital stock per effective worker
ii.Output per effective worker
iii.The growth rate of output per effective worker
iv.The growth rate of output per worker
v.The growth rate of output
b. Suppose that the rate of technological progress doubles to
8% per year. Recompute the answers to part (a). Explain.
c. Now suppose that the rate of technological progress is
still equal to 4% per year, but the number of workers now
grows at 6% per year. Recompute the answers to (a). Are
people better off in (a) or in (c)? Explain.
7.Discuss the potential role of each of the factors listed in (a) to
(g) on the steady-state level of output per worker. In each case,
indicate whether the effect is through A, through K, through
H, or through some combination of A, K and H. A is the level of
a. Geographic location
c. Protection of property rights
d. Openness to trade
f. Good public infrastructure
The appendix to this chapter shows how data on output, capital
and labour can be used to construct estimates of the rate of growth
of technological progress. We modify that approach in this problem to examine the growth of capital per worker. The function:
Y = K 1/3(AN)2/3
gives a good description of production in rich countries. Following the same steps as in the appendix, you can show that:
(2/3)gA = gY - (2/3)gN - (1/3)gK
= (gY - gN) - (1/3)(gK - gN)
where gY denotes the growth rate of Y.
a. What does the quantity gY - gN represent? What does the
quantity gK - gN represent?
b. Rearrange the preceding equation to solve for the growth
rate of capital per worker.
c. Look at Table 12.2. Using your answer to part (b), substitute in the average annual growth rate of output per
worker and the average annual rate of technological progress for the United States for the period 1985 to 2013 to
obtain a crude measure of the average annual growth of
capital per worker. (Strictly speaking, we should construct
these measures individually for every year, but we limit
ourselves to readily available data in this problem.) Do the
same for the other countries listed in Table 12.2 (where
data goes to 2014). How does the average growth of capital per worker compare across the countries in Table 12.2?
Do the results make sense to you? Explain.
Log on to MyEconLab and complete the study plan exercises for this chapter to see
how much you have learnt, and where you need to revise most.
M12 Macroeconomics 85678.indd 255
256 THE CORE The long run
For more on growth, both theory and evidence, read Charles
Jones’s Introduction to Economic Growth, 3rd edition (New
York: W.W. Norton, 2013). Jones’s web page, (http://web
.stanford.edu/~chadj/) is a useful portal to the research on
For more on patents, see The Economist’s Special Report:
Patents and Technology, 20 October 2005.
For more on growth in two large, fast-growing countries,
read the article by Barry Bosworth and Susan M. Collins,
‘Accounting for growth: comparing China and India’,
Journal of Economic Perspectives, 2008, 22(1), 45–66.
For the role of institutions in growth, read ‘Growth
theory through the lens of development economics’, by
Abhijit Banerjee and Esther Duflo, Handbook of Economic
Growth (Amsterdam: Elsevier, 2005), Chapter 7. Read
sections 1 to 4.
M12 Macroeconomics 85678.indd 256
For more on institutions and growth, look at the slides from
the 2004 Lionel Robbins Lecture ‘Understanding institutions’
given by Daron Acemoglu. These are found at http://economics.mit.edu/files/1353
On two issues we have not explored in the text:
Growth and global warming. Read the Stern Review on
the Economics of Climate Change (2006). You can find it at
www.wwf.se/source.php/1169157. (The report is long.
Read just the executive summary.)
● Growth and the environment. Read The Economist’s
Survey on the Global Environment, ‘The great race’, 4 July
2002, and the update titled ‘The Anthropocene: a manmade world’, 26 May 2011.
Chapter 12 Technological progress and growth 257
Constructing a measure of technological progress
In 1957, Robert Solow devised a way of constructing an estimate of technological progress. The method, which is still
in use today, relies on one important assumption: that each
factor of production is paid its marginal product.
Under this assumption, it is easy to compute the contribution of an increase in any factor of production to the increase
in output. For example, if a worker is paid €30,000 a year,
the assumption implies that his or her contribution to output
is equal to €30,000. Now suppose that this worker increases
the amount of hours worked by 10%. The increase in output
coming from the increase in hours will therefore be equal to
:30,000 * 10%, or €3,000.
Let’s write this more formally. Denote output by Y, labour
by N and the real wage by W/P. The symbol, ∆, means
‘change in’. Then, as we just established, the change in output
is equal to the real wage multiplied by the change in labour:
Divide both sides of the equation by Y, divide and multiply
the right side by N and rearrange:
Note that the first term on the right (WN/PY) is equal to
the share of labour in output – the total wage bill in euros
divided by the value of output in euros. Denote this share by
a. Note that ∆Y/Y the rate of growth of output, and denote
it by gY. Note similarly that ∆N/N is the rate of change of the
labour input, and denote it by gN. Then the previous relation
can be written as:
gY = agN
More generally, this reasoning implies that the part of output growth attributable to growth of the labour input is equal
to a times gN. If, for example, employment grows by 2% and
the share of labour is 0.7, then the output growth due to the
growth in employment is equal to 1.4% (0.7 times 2%).
Similarly, we can compute the part of output growth
attributable to growth of the capital stock. Because there
are only two factors of production, labour and capital, and
because the share of labour is equal to a, the share of capital
in income must be equal to (1 - a). If the growth rate of
capital is equal to gK, then the part of output growth attributable to growth of capital is equal to (1 - a) times gK. If, for
example, capital grows by 5%, and the share of capital is 0.3,
then the output growth due to the growth of the capital stock
is equal to 1.5% (0.3 times 5%).
Putting the contributions of labour and capital together,
the growth in output attributable to growth in both labour
and capital is equal to [agN + (1 - a)gK].
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We can then measure the effects of technological progress by computing what Solow called the residual, the
excess of actual growth of output gY over the growth
attributable to growth of labour and the growth of capital
[agN + (1 - a)gK]:
Residual K gY - [agN + (1 - a)gK]
This measure is called the Solow residual. It is easy to
compute. All we need to know to compute it are the growth
rate of output, gY, the growth rate of labour, gN, and the
growth rate of capital, gK, together with the shares of labour,
a, and capital, (1 - a).
To continue with our previous numerical examples,
suppose employment grows by 4%, the capital stock
grows by 5% and the share of labour is 0.7 (so the share
of capital is 0.3). Then the part of output growth attributable to growth of labour and growth of capital is equal to
2.9% (0.7 * 2% + 0.3 * 5%). If output growth is equal,
for example, to 4%, then the Solow residual is equal to
1.1% (4% - 2.9%).
The Solow residual is sometimes called the rate of growth
of total factor productivity (or the rate of TFP growth, for
short). The use of ‘total factor productivity’ is to distinguish
it from the rate of growth of labour productivity, which is
defined as (gY - gN), the rate of output growth minus the
rate of labour growth.
The Solow residual is related to the rate of technological
progress in a simple way. The residual is equal to the share of
labour times the rate of technological progress:
Residual = agA
We shall not derive this result here. But the intuition for
this relation comes from the fact that what matters in the production function Y = F(K, AN) (equation (12.1)) is the product of the state of technology and labour, AN. We saw that to
get the contribution of labour growth to output growth, we
must multiply the growth rate of labour by its share. Because
N and A enter the production function in the same way, it is
clear that to get the contribution of technological progress
to output growth, we must also multiply it by the share of
If the Solow residual is equal to zero, so is technological
progress. To construct an estimate of gA, we must construct
the Solow residual and then divide it by the share of labour.
This is how the estimates of gA presented in the text are
In the numerical example we saw previously, the Solow
residual is equal to 1.1% and the share of labour is equal to
0.7. So, the rate of technological progress is equal to 1.6%
(1.1% divided by 0.7).
258 THE CORE The long run
Remember the definitions of productivity growth you
have seen in this chapter:
(a)Labour productivity growth (equivalently, the rate of
growth of output per worker): gY - gN.
(b)The rate of technological progress: gA.
In steady state, labour productivity growth (gY - gN)
equals the rate of technological progress gA. Outside of
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steady state, they need not be equal. An increase in the
ratio of capital per effective worker, due, for example, to an
increase in the saving rate, will cause gY - gN to be higher
than gA for some time.
Source: The original presentation of the ideas discussed in this appendix is found in
Robert Solow’s article, ‘Technical change and the aggregate production function’,
Review of Economics and Statistics, 1957, 312–320.