Download 6. External economies and learning by doing

Introduction to Economic Growth
6. External economies and learning by doing
“One for all! All for one!” [Alexandre Dumas]
Learning Goals:

Distinction between internal and external economies of scale

Acknowledge the different types of external economies related to capital
accumulation

Distinguish the implications of non-decreasing versus increasing returns
to scale for economic growth and convergence

Explain why increasing returns are a source of cumulative causation

Discuss the role of external economies in shaping comparative
advantages and the pattern of international trade
6.1 Introduction
As mentioned in Chapter 4, one possibility of accounting for a larger role of
reproducible factors in production than that implied by the corresponding shares in
national incomes is by assuming the existence of externalities. This chapter shows how
externalities associated to capital accumulation have indeed the potential to overcome
the limitation imposed by diminishing returns, leading to unceasing growth.
This avenue was first explored by Marvin Frankel, as early as in 1962. The aim
of Frankel was to conciliate the convenient properties of the Cob-Douglas production
function regarding factor allocation and income distribution, with the aim to generate
non-decreasing returns and endogenous growth, as implied by the Harrod-Domar
model. The key he proposed for such conciliation was to assume the existence of
externalities associated to physical capital. Frankel’s contribution remained, however,
unnoticed by the literature until Paul Romer came out with a similar idea: in a famous
article written in 1986, Romer built up on the theory of “learning by doing” previously
developed by the Nobel Laureate Keneth Arrow, according to which productive
experience improves the stock of knowledge, to argue that capital accumulation can led
to unceasing growth. A related work by the Nobel Laureate Robert Lucas Jr., in 1988,
emphasised the role of externalities associated to investment in human capital. The
theories of endogenous growth based on externalities on capital accumulation marked
the first wave of the so-called “new growth theory”.
This chapter reviews these theories and explores the policy implications. Section
6.2 describes the model introduced by Marvin Frankel. Section 6.3 explains why the
competitive equilibrium with externalities is not efficient and discusses the possible role
of the government in addressing the market failure. Section 6.4 addresses the specific
case with increasing returns. Section 6.5 focuses in the model of learning by doing.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
170
Introduction to Economic Growth
Section 6.6 discusses the implications of learning by doing for comparative advantages
and international trade. Section 6.7 concludes.
6.2. Externalities on capital accumulation
Modelling the externality
In his 1962 paper, Frankel first observed that the Cobb-Douglas production
function is capable of describing factor allocation and income distribution but is not
capable of generating sustained growth of per capita income. In turn, the AK production
function is capable of generating long-run growth, but it does not offer a satisfactory
theory for factor allocation and income distribution.
Frankel then proposed a method to conciliate the two production functions, so
that the desirable properties of each but none of the limitations are retained: the key was
to introduce a production externality, whereby the “overall level of development of a
region” impacts positively on the productivity of each private firm118. Frankel related
the externality to “various external effects” related to capital accumulation, such as
“improvements in the level of organization, technical change, better social overhead
facilities in the form of transport and communication networks, etc”.
To capture this idea, Frankel assumed that each individual firm faces a CobbDouglas production function,
where TFP is a positive function of the economy-wide capital stock. Formally,
let the production function for each individual firm i be given by:
Yi  BK i N i
1 
,
(6.1)
where Yi, Ki and Ni denote, respectively, for output, capital and labour employed by firm
i. The TFP parameter, B (the “development modifier”, as coined by Frankel) was
assumed to depend positively on the aggregate level of capital per worker:
BA
K
, with   1   and 0   '   ,
N ´
(6.2)
where K   K i and N   N i stand, respectively for the aggregate levels of capital
i
i
(human, physical) and labour in the economy.
According to (6.2), an increase in the aggregate stock of capital impacts
positively on the productivity of each firm. Thus, whenever a firm accumulates capital
for private reasons, it will be “indirectly” contributing to the productivity of all other
firms. Because each firm is small relative to the economy, it will ignore this external
effect. Production externalities specified in this manner are labelled “Marshallian
externalities” (see Box 6.3).
118
Frankel (1962): “Enterprises in relatively developed or advanced economies are able to produce more
with given inputs of capital and labour than enterprises in relative underdeveloped economies. This is the
essence of economic development”.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
171
Introduction to Economic Growth
The productivity term (6.2) also accounts for a negative externality on aggregate
labour, in case  '  0 . This effect captures the possibility of the positive externality
related to capital accumulation being partially or totally diluted by the size of the labour
force. When, for instance,    ' , the firm productivity will depend on the aggregate
stock of capital per worker, rather than with the aggregate stock of capital in absolute
terms. When instead  '  0 , what matters is the absolute level of capital in the
aggregate, not the capital labour ration. In the following, we’ll discuss cases in which
one or other assumption make sense.
The aggregate production function
Because of the externality, the aggregate production function differs from the
individual production functions, even if all firms are alike.
The aggregate production function is obtained substituting (6.2) in (6.1) and
summing up across all firms. This gives:
Y  AK   N 1  '
,
(6.5)
where Y  Yi .
i
The novelty of production function (6.5) is that it may deviate from the neoclassical assumptions of constant returns to scale and diminishing marginal returns. For
instance, when   1  , returns to capital are non-decreasing. As we already know,
this is the condition we need for a model to display unceasing growth thorugh capital
accumulation. On the other hand, whenever    ' , the aggregate production function
will display increasing returns to scale: that is, rising capital and labour by a given
proportion will lead to a more than proportional impact on output. As we will discuss
next, this property makes size an advantage, causing the model to display circular
causation.
Note that at the individual level, the production functions retain the neo-classical
properties of constant returns and diminishing returns to capital. The aggregate
production function departs from these properties because of the externality, which
individual firms – because they are too small - do not take into account.
Box 6.1. Externalities
The most basic type of market failure is an externality. Externalities are present
whenever an individual takes an action that directly affects the environment of others
but for which it neither pays nor is paid in compensation.
In a consumption externality, the utility of one agent is directly affected by the
consumption decisions of other agent. For instance, you may benefit by the fact that
your neighbour hires a private security: your house will be safer, even if you don’t pay
for it.
In a production externality, the utility of one agent is directly affected by the
production decisions of other agent. A textbook example is the steel mill and the
laundry: smoke emissions by a steel mill may directly affect the production of clean
clothes by a laundry. In that case, there is a negative externality on production.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
172
Introduction to Economic Growth
In the presence of externalities, the market mechanism does not deliver an
efficient allocation of resources. In the case of negative externalities, individuals do not
bear the full cost of their actions, so they will engage in socially excessive activity.
Conversely, in the case of positive externalities, individuals do not enjoy the full
benefits of their activities, so they will engage in too little activity. In both cases, a
careful use of discriminatory taxation by the government may improve, at least
theoretically, economic performance.
Box 6.2. Internal and external economies of scale
The distinction between “internal” and “external” economies of scale backs from
Scitovsky (1954). “Internal” economies of scale refer to the case in which a single firm
faces a downward sloping average cost curve when increasing its own output level. In
this case, there is a tendency for the firm to become larger and larger and to become
monopolists in the market. Internal economies of scale are inherently linked to
imperfect competition.
The concept of “external” economies of scale refers to the case in which scale
economies arise at the aggregate (spatial or industry) level. In that case, average costs
for the individual firm decline with aggregate output, but not with the individual firm
output. “External economies of scale” in the aggregate may co-exist with constant
returns to scale and declining marginal productivities at the firm level. Hence, one does
not need to abandon the assumption of perfect competition.
Factor prices in the competitive equilibrium
Irrespectively of the shape of the aggregate production function, each firm will
see its own (individual) production function (6.1) has having the standard neo-classical
properties of constant returns to scale and diminishing returns on capital. The reason is
that each firm is small relative to the economy: since the impact of a firm investment
decisions in the aggregate is negligible, each firm will take parameter B as exogenous.
Thus, profit maximization by price taker firms will lead to the usual conditions
stating that firms employ labour and capital until their marginal products equal the
respective factor prices:
r  
Yi
Y
 i ,
K i
Ki
(6.3)
and
wt 
Yi
Y
 1    i .
N i
Ni
(6.4)
Because all firms are equal, we have Yi K i  Y K and Yi N i  Y N . Hence, in
the competitive equilibrium, the shares of capital and of labour on domestic income will
be given, respectively, by  and . This is the very convenient result Frankel wanted
to stick with, for the model to be in accordance to the Kaldor stylized facts.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
173
Introduction to Economic Growth
The AK model again
Equation (6.5) is general enough to account for all types of external economies.
A particular case occurs when    ´ 1   . In this very special case (in which Frankel
focused on), the size of the positive externality in K is exactly the enough to overwhelm
the normal process of diminishing returns to capital, and – at the same time - the
negative externality on labour exactly matches the externality on capital, implying that
returns to scale remain constant. When this is so, the aggregate production function
(6.5) becomes exactly linear in K:
Yt  AK t , A > 0
(5.1)
That is, the production function takes the AK form at the aggregate level, but it
retains the neo-classical properties at the individual firm level. Each firm perceives its
production function as having diminishing returns to capital, so it will employ capital
and labour according to (6.3) and (6.4). In the aggregate, the production function will be
linear in K, so the marginal product of capital will never decline and the economy will
never stop growing.
The advantage of this model when compared to the simple AK model, is that it
does not rely on the peculiar assumption that labour plays no role in production. Like in
the Solow model, both factors are used in production. Moreover, the model accords to
the main Kaldor stylized facts: the share of capital in income is equal to ; the wage rate
and per capita income will grow steadily over time and the user cost of capital is
constant and equal to r    A (equation 6.3).
6.3. The market failure and optimal intervention
The capital’ social return
A novelty in the model with externalities is that the competitive equilibrium will
no longer be efficient. The reason is that each firm, being small relative to the economy,
will decide its capital stock taking into account the impact of that decision on its own
profits, only. The positive contribution of its investment decisions to the overall capital
stock will be considered negligible and hence ignored. Still, the investment decisions of
all firms taken together will impact positively on the profits of each individual firm.
Thus, the competitive equilibrium will deliver a suboptimal level of investment.
Formally, the marginal contribution of aggregate capital to aggregate production
(i.e, taking into account the externalities) as stated in (6.5) is:
Y
 Y 
    


K
 K  social
(6.6)
However, in its profit maximization problem, the firm considers only the narrow
private returns to capital (equation 6.3).
Hence, as long as there is an externality on capital accumulation (   0 ) the
competitive equilibrium will deliver an allocation of resources in which the private
return to capital and the marginal contribution of capital to production differ.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
174
Introduction to Economic Growth
Note that this constitutes an important novelty relative to the Solow model. By
assuming away externalities and other market failures, the Solow model implies that, in
a competitive equilibrium, factor prices are equal to their respective contributions to
production. So, a central planner concerned with efficiency would choose an allocation
of resources matching exactly the competitive equilibrium.
This model, by introducing externalities, implies that the equilibrium allocation
in the decentralized economy will not be efficient. The wedge between private returns
and social returns to capital implies that incentives are misaligned: in the decentralized
economy, investment will be too small.
Growth accounting revisited
The existence of a wedge between private returns and social returns to capital
has important implications for growth accounting. Conventional growth accounting (as
exemplified in Section 2.6) typically uses the share of capital on national income as
proxy for the contribution of capital to production. The discussion in Chapter 4
revealed, however, that this delivers an estimate that is too small to account for the
observed differences in per capita incomes across countries. In order to account for such
large differences, one would need a contribution of capital to production much larger
than that implied by the observed income shares.
This puzzle was actually solved by Frankel as early as in 1962: Frankel argued
that a much larger contribution of capital to production than that implied by the
observed shares in national incomes could be explained by externalities.
Formally, equation (6.5) reveals that, as long as the externality parameter  is
positive, the actual contribution of capital to production () is larger than that implied
by its share on income (. Log-differentiating (6.5), one obtains:
Yˆ     Kˆ  1     'n
(6.7)
In (6.7), a change in an has two effects, a direct one and an external effect. The
external effect may amplify or diminish the direct effect, depending on the sign and
magnitude of the respective parameter. For instance, when   1   , a one-percentage
point increase in the capital stock will result in a one-percentage point increase in
output, a result that conforms with the AK model (and that Frankel argued to conform
as well to the U.S. data).
Equation (6.7) suggests that conventional growth accounting, by
underestimating the effective contribution of capital to production, overestimates the
Solow residual.
Optimal intervention
The wedge between social returns and private returns to capital constitutes a
market distortion. Firms tend to under-invest in physical capital relative to what would
be considered optimal by a benevolent planner. The government, given this sort of
diagnosis, may have a role in using the policy to achieve the optimal allocation of
resources.
How might government policy be used to establish the efficient allocation? An
obvious avenue is to subsidize capital accumulation. To see this, let’s rewrite the
http://sweet.ua.pt/afreitas/growthbook/capa.htm
175
Introduction to Economic Growth
individual firm profit function, but allowing for a tax rate  K (subsidy, when negative)
on capital incomes:
 i  BK i N i1   r   1   K K i  wN i
(6.8)
In light of this specification, the cost of one unit of capital –the cost to firms - is
r   1  K . What households – the owners of capital – receive as net income is
 
r   .

From the first order conditions of profit maximization, one obtains (instead of
6.3):
Yi
Y

 r   1   K 
K i
K
(6.9)
To remove the distortion, the government should set the tax rate so that the (net)
rental price of capital, r   , reflected fully the marginal contribution of capital to


aggregate output, as given by (6.6). That is, the tax rate  K should be such that:
r   
Y K
Y
    
1   K 
K
(6.10)
Solving for  K , the optimal (first best) policy is to set:
K 

0
 
(6.11)
Thus, the optimal policy in this model involves subsidizing physical capital
accumulation119. This result is intuitive: if the contribution of capital to production is
given by (6.6) and private firms only perceive it to be equal to (6.3), then a subsidy
filling the gap will achieve the aim of getting private returns aligned with the social
interest.
In the particular case in which   1   (the AK model), the optimal tax rate
will be  K    1 (note however, that in this extreme case all income in the economy
would be devoted to capital owners and nothing would be left to raw labour; this would
be only possible if K referred to a broad concept of capital, including human capital).
Growth effects
An important implication of this model is that removing the distortion leads to a
greater efficiency and, by then, to a higher rate of economic growth. To see this,
consider again the optimal consumption rule   r   and let’s focus in the particular
case in which    ´ 1   (the AK model).
119
Of course, a question arises on how this subsidy will be financed. For the moment, just assume that
lump sum taxation is available, so that the policy will not imply further distortions. The issue of
distortionary taxation and second best decision-making will be addressed in Chapter 11.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
176
Introduction to Economic Growth
In the competitive equilibrium, the interest rate is determined according to (6.3).
Substituting r in the optimal consumption rule, one obtains the growth rate of per capita
income in the decentralized economy:
 
Y
     A    
K
(6.12)
If however the government managed to influence the interest rate so as to reflect
the social contribution of capital, the user cost of capital would become equal to (6.6).
Thus, the growth rate of the economy would be:
 *     
Y
    A  
K
(6.13)
Comparing (6.13), we see that the growth rate of this economy under central
planning will be higher than in the laissez fare.
This example illustrates how judicious government intervention might be used
to establish the “right” prices and thereby stimulate growth. Note however that such a
“perfect” intervention requires a high level of confidence on the size of the external
effect, as well as availability of non-distortionary taxation. Whenever these conditions
are not met, it may well be the case that the government may fail to do better than the
market.
6.4 The case with increasing returns to scale
So far, the analysis focused on the case with    ´ . This is however a very
special case. In that case, the positive effect arising from a larger stock of physical
capital is exactly offset by the “dilution” effect resulting from a larger number of
workers. In this version of the model the aggregate production function exhibits
constant returns to scale, even though returns to capital are non-decreasing.
A quite distinct situation occurs when    ' 120. In that case, expanding the use
of capital and labour by a given proportion has a more than proportional impact on
output: the aggregate production function exhibits increasing returns to scale.
Remember that these increasing returns do not arise at the individual firm level, but
instead at the aggregate level: it is because the productivity of each individual firm is
parametric on aggregate variables that returns to scale arise. Because of this, increasing
returns are said to be external to the firm (Box 6.2).
When the aggregate production function displays increasing returns, there will
be a tendency for the region to become larger and larger. To see this, just note that the
average product of labour in (6.5) becomes equal to:
y  Y N  Ak    N   '
(6.14)
In case    ´ 1   , the aggregate production function exhibits constant returns to scale and
diminishing returns to capital. As you may easily check, in that case the steady state growth rate of output
is equal to the growth rate of population, just like in the basic Solow model. Still, because of the
externality, private returns to capital in laissez faire are too low. The case with  ´ 0 is formally
addressed in Appendix 6.1.
120
http://sweet.ua.pt/afreitas/growthbook/capa.htm
177
Introduction to Economic Growth
This means that that, in a competitive equilibrium, the wage rate – determined
according to (6.4) - will also be an increasing function of the size of the workforce.
The implication is that a larger region will be a more attractive place to work
than a smaller region. This will generate a tendency for employment to move to the
larger region, further expanding the larger region and depressing the smaller region.
Cumulative causation
This discussion illustrates why increasing returns are a source of divergence: if
for whatever reason, a region starts out bigger, increasing returns will assure that it will
become a more attractive place to work and invest. With free factor movements, labour
and capital will tend to move from depressed areas to the more dynamic region and the
later will get bigger and richer, absorbing resources from the rest of the world.
The idea that development brings more development in a virtuous cycle is very
central in the development literature and is labelled as “cumulative causation”121.
Box 6.3 Alfred Marshall and the theory of external economies
The theory of external economies was pioneered by one of the founders of
modern economics, the British economist Alfred Marshall. In his book “Principles of
Economics” (1920, first published in 1890), Marshall was concerned with the question
as to why there is a tendency for some industries to concentrate in few areas within a
country (“industrial districts”). Examples of this at that time included the cutlery
manufactures in Sheffield and hosiery firms in Northampton. This type of spatial
concentration of industry could be explained by natural resources. In our days, similar
examples include the Silicon Valley, Hollywood and Las Vegas.
To explain the tendency for firms of the same industry to cluster together,
Marshal conjectured that the productivity of each firm in a given location may depend
positively on the general progress of the corresponding industry in the same location,
via three types of external effects:
First, the availability of specialized suppliers: in many industries, production
requires the use of specialized inputs, such as intermediate products and specific
supporting activities, that cannot be acquired at distance because of high transport costs.
For instance, the production of a motion picture requires a variety of services, such as
casting services, sound effects, costume design, choreography, catering, etc. Many of
these services are better purchased to specialized firms, because specialized firms can
split the fixed costs of their activity through different costumers. If, in a given region,
there is only one film producer, it will not pay for upstream suppliers to locate in that
region. Eventually, the later will prefer locations where there are already many
moviemakers, so that they have a market large enough to break even. By the same
token, moviemakers will find it profitable to join locations where other moviemakers
121
The term was coined by Veblen (1898). It was, however, the Nobel Laureate Gunnar Myrdal (1957)
who popularized the concept. This author contended that labour migration, capital movements and trade
may lead to cumulative expansion of the favoured regions and retard the development of backward
regions, leading to persistent or even divergent spatial differences in per capita income.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
178
Introduction to Economic Growth
are already located, because this will imply a higher market for – and hence a higher
availability of - specialized services, competing with each other122.
Second, labour market pooling: when many firms and specialized workers
located in a given region, both sides of the market will be less exposed to events
affecting a small number of firms or workers. For instance, the failure of one firm will
be less problematic for a specialized worker located in a region with many firms than if
located in a region with one employer only. The same holds for firms. By clustering
together, both firms and workers will benefit from the law of large numbers, being
therefore less exposed to specific shocks affecting particular agents.
Third, technological spillovers: Technological spillovers occur because people
have incentive to observe what the others are doing and imitate the best practices.
Arguably the process of technological diffusion takes place more effectively when
various firms of the same industry are concentrated in a given location, so that workers
belonging to different firms have the opportunity to meet together and discuss technical
problems, face-to-face. The mobility of workers across neighbouring firms is also a
process through which this process of knowledge diffusion accelerates.
All in all, these three types of external effects (often called “Marshallian
externalities”) imply that each firm will become more productive, the more firms of the
same type are located nearby. Formally, this is usually modelled assuming that the
technological parameter of an individual firm’ production function depends positively
on an variable measuring the size of the industry in that location (as done in equation
6.2). In that case, the aggregate production function may display increasing returns to
scale, creating the incentive for firms to cluster together123.
Box 6.10 Externalities on Human Capital
External economies may also show up in human capital accumulation. The main
idea is that people who get educated benefit more in a knowledge abundant society than
in a society with little knowledge.
To understand this, ask yourself why the best graduate economists prefer to
work in City or in Wall Street – where economics graduates are plentiful – rather than
in, say, Mongolia where they are in very short supply. The economist working at City
earn his high income in part because of the manner in which its own knowledge is
enhanced by those of fellow well-educated economists. This happens because
individuals benefit from interacting with each other. Exchange of ideas with other
professionals enhances individual capabilities.
122
We will address this argument more formally in Chapter 12.
123
In the real world, location decisions also depend on centrifugal (dispersion) forces. This includes
congestion effects, whereby the cost of a firm adopting a location rises with the number of adopters. For
instance, the concentration of activities in a small area leads to higher land prices, high commuting costs,
pollution and other sociological factors. Another dispersion force arises due to transport costs: to the
extent that some activities have to be undertaken in the periphery (for instance, agriculture, exploitation of
natural resources), being close to the centre implies higher transport costs in transactions with the
periphery. The allocation of economic activities across the space is therefore determined by the tension
between centripetal forces and centrifugal forces. Classical contributions accounting for these centrifugal
forces in the context of “Marshallian externalities” include Henderson (1974) and Fujita and Ogawa
(1982).
http://sweet.ua.pt/afreitas/growthbook/capa.htm
179
Introduction to Economic Growth
Thus, just like in an assembly line, where the value of each worker's effort
depends on the other worker's efforts, this creates and incentive for the best workers to
match up with each other: if the best economists are assembled together, they will have
better ideas and will get a higher payoff from their skills. If, instead, they are partnered
with lazy or incompetent economists, they will have a lower reward for any effort that
they might individually provide.
Note that this is exactly the opposite of the LDR: with diminishing returns, skills
substitute for each other, so they become more valuable where they are scarcer – in
Mongolia and not at City. Under diminishing returns, skilled labour would move from
rich countries to poor countries. By contrast when externalities are present and this
effect is strong enough to overwhelm the conventional diminishing returns, then skilled
labour will be more valuable where it is more abundant: returns to skills for each
individual are an increasing function of the existing skill level in the society.
This story explains why we see immigration of skilled labour at maximal
allowable rates and beyond from poor countries to wealthy ones and not the other way
around (remember the Lucas paradox in Section 4.3). For instance, the stock of
immigrant workers in US is, on average, better educated that the average worker in the
home countries. Moreover, for most developing countries, the highest migration rates
are observed in the group of individuals with tertiary education. That is, skilled workers
tend to move to where skilled workers are.124
Like the case with physical capital, complementarities in human capital imply
cumulative causation and vicious cycles: for instance, in a nation where skill levels are
already deep and well established, people in that nation will have strong incentives to
invest in their human skills. But in poorer economies where the skill base is thin, the
incentive of individuals to invest in human skills is low. Thus, a country will be rich if it
started out rich, a country will be poor if it started out poor125.
124
Carrington and Detragiache (1998, 1999).
125
Note that the same mechanism applies to regions, cities, families and ethic groups (Lucas, 1988).
Leaving in cities, people have more opportunity to work near the highly skilled. This helps explain why
wages for similar skills and education levels are higher in cities than in rural areas and also why people
are able to pay higher rents and property prices there. At the family level there is a tendency for literate
parents (specially literate mothers) to raise healthier and more literate children. This gives rise to vicious
circles: a low human capital generation is succeeded by another low human capital generation, while an
initially high human capital generation would give rise to another high human capital generation. As for
ethnic groups, to the extent that social segregation increases the probability of people of the same ethnic
group to match and work together, there will be a tendency for education levels to converge within each
group: people belonging to the low education ethnic group will not invest in education because working
with people with low education implies a low return to education. On the contrary, people belonging to
the highly educated ethnic group will have an incentive to invest in education, because the chances of
being matched with well-educated people are high.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
180
Introduction to Economic Growth
6.5 Learning by doing
The benefits of experience
A particular type of external economies arises from the benefits of experience.
The main idea is that firms engaged in a specific production process tend to become
more productive as time goes by, because workers, by undertaking similar actions,
perfection their routines and learn to solve minor problems. This benefit occurs through
practice - hence the label “learning by doing” - and is often summarized by a “learning
curve”, that relates the average cost of producing a given good to the cumulative
experience in producing that good126.
Kenneth Arrow used the idea to build a model of endogenous technological
change127. Arrow modelled learning-by-doing at the individual firm level, assuming that
investment in new capital impacts proportionally in the firm’ stock of knowledge. The
rationale is that, when firms buy a new capital good, they also acquire a new production
technique: learning how to use the new equipment and adapting their production
processes so as to extract full profit from the opportunities opened up takes time. As
workers become more accustomed to the new capital good, their common stock of
technical knowledge increases128.
A key assumption of the learning by doing model is that technological occurs as
a mere by-product of capital accumulation, so it does not involve deliberate economic
decisions.
Knowledge spillovers again
Another key assumption of the learning-by-doing model is that knowledge
leaks: that is, firms tend to imitate the improvements achieved by fellow firms, so they
all end up benefiting from the accumulated experience of each other.
Thus, when one firm invests in new capital, it adds to its own stock of
knowledge and at the same time to the common stock of knowledge. Formally, an
equation similar to (6.2) arises, whereby total factor productivity at the firm level is an
increasing function of the economy-wide accumulated stock of capital.
With this assumption, the model follows in an intuitive manner: each firm,
perceiving its production function as a CRS, buys new capital until the private marginal
product of capital equals the user cost of capital (eq. 6.3). Buying the state-of-the-art
126
The first person to describe the “learning curve” was a German psychologist Herman Ebbinghaus
(1850-1909), in a series of tests consisting in memorizing nonsense syllables. In economics, the concept
was first described by an aeronautical engineer called Theodore Wright. Wright (1936) observed that, as
more aircrafts of a given type are produced, the costs of production fall dramatically, and proposed a
mathematical model to describe it.
127
Arrow (1962).
128
Arrow (1962), p. 157: “each new machine produced and put into use is capable of changing the
environment in which production takes place, so that learning takes place with continuous new stimuli”.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
181
Introduction to Economic Growth
capital, the firm inadvertently increases its own stock of knowledge, but this effect is
small. Since knowledge leaks, however, the acquisition of physical capital by each one
firm adds to the common stock of knowledge, which impacts positively on the
productivity of all firms. Thus, each firm will be more productive, the higher the
productive experience (measured by the stock of capital) in the economy as a whole (eq.
6.2).
Note that the assumption of knowledge spillovers is critical for the model to be
consistent with perfect competition: if the knowledge created did not leak, the
individual firm accumulating capital would become more productive than its
competitors; its returns would be higher and higher and the conditions would exist for
this firm to grow alone and capture the entire market.
Another critical assumption of the learning-by-doing model is that there is no
negative externality associated to the number of workers. Formally, it is assumed that
 ´ 0 in (6.2). The reasoning is that knowledge is non-rival: that is, once knowledge is
acquired, many workers and firms can use it without reducing its effectiveness. Thus,
the stock of knowledge is better described as an increasing function of the total capital
stock in the economy, rather than as a function of the economy’ capital per worker. The
implication is that the learning by doing model unequivocally displays increasing
returns.
Two versions of the learning by doing model
In the literature, two main branches of the learning-by-doing model were
developed.
The first, introduced by Kenneth Arrow, retains the neo-classical assumption of
diminishing returns to capital, even after accounting for the aggregate externality. In
terms of equations (6.1), (6.2), (6.5) and (6.14), the Arrow model corresponds to the
case in which  ´ 0 and     1 . In this version of the model there are increasing
returns to capital and labour altogether, but there are diminishing returns to capital
alone. Hence, the model accounts for agglomeration effects but cannot generate
endogenous growth (details in the Appendix 6.1).
The second version of the learning-by-doing model, due to Romer (1986),
assumes non-decreasing returns in the aggregate (     1 ), so it delivers unceasing
growth. In that case, however, the interest rate becomes an increasing function of the
population size and so does the growth rate of per capita income. In other words, the
model displays a “strong scale effect”, whereby a larger economy should grow faster
than a smaller economy (details in the appendix).
Box 6.4 presents a popular illustration for the argument that learning by doing
combined with knowledge spillovers may lead to unceasing growth.
Endogenous technological change
Comparing to the Solow model, the learning-by-doing model retains the
assumption that knowledge is a public good: in this model, knowledge is non-rival and
spills over instantaneously across firms at zero cost (at least locally). Hence, like in the
Solow model, there are no economic incentives to produce new knowledge.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
182
Introduction to Economic Growth
In contrast to the Solow model, the level of technology is endogenous:
technology arises without purposeful efforts, but comes out as a by-product of capital
accumulation, which is driven by economic decisions. Thus, policies influencing the
rate of capital accumulation will also influence the level of technology and therefore
economic development.
Box 6.4. The Noorul Quader's Desh Factory
A popular illustration of how learning by doing combined with knowledge
spillovers may lead to economic growth is due to Rhee (1990).
The story backs to 1979, when Daewoo Corporation, a major world textile
producer from South Korea, was looking for a new base to evade the U.S. and European
import quotas against Korean products. Since these quotas did not cover Bangladesh,
the company created a joint venture there to produce shirts, with a former government
official called Noorul Quader. The new company, called Desh Garment Ltd, start
producing in April 1980.
Because Bangladesh had no experience in garment production, 130 Bangladeshi
workers were trained in Korea. This familiarisation with modern production and
techniques allowed the Daewoo collaborative agreement to have long lasting effects in
Bangladesh: Bangladesh became itself a strong exporter of textiles.
The mechanism through which knowledge leaked around was labour mobility.
During the 1980s, 115 workers trained by Daewoo left the company to start their own
businesses. The new firms not only produced garments, but also gloves, coats and
trousers. All in all, an entire exporting sector emerged, just through learning by doing
and knowledge spillovers.
William Easterly, in his famous book “The Elusive Quest for Growth” (2001)
concludes: "The story of the birth of the Bangladeshi garment industry illustrates the
principle that investment in knowledge does not remain with the original investor.
Knowledge leaks". ... "Why hadn't Bangladeshi already been making shirts on their
own, before Daewoo volunteered its service? The answer is that Daewoo had learned
something about how to produce shirts and how to sell them on the world market. ... and
transmitted this knowledge to Desh workers". ... "Creating Knowledge does not
necessarily mean inventing new technologies from scratch. Some aspects of garment
manufactory were probably several centuries old. The relevant technological ideas
might be floating out there in the ether, but only those who apply them can really learn
them and can teach them to others". (pp 148-150).
Localized versus global technological spillovers
A problem with the Learning-by-doing model is that it gives rise to a “scale
effect”, whereby the productivity in a given economy rises with the size of its
workforce. This is a direct implication of the non-rival nature of knowledge: since
sharing knowledge does not involve loss of its effectiveness, the larger the population
being served with that knowledge, the better. This prediction does not square well,
however, with the real world facts: in general, there is no systematic tendency for large
countries to be better of than small countries.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
183
Introduction to Economic Growth
As already argued in Chapter 1, one way of conciliating the idea that knowledge
leaks with the real world facts is to delink knowledge spillovers from country borders.
Indeed, when one looks at the effects of changes in policies, such as fiscal policy or
monetary policy, the nation is a natural unit. But from the viewpoint of knowledge,
there are no reasons to believe that France and Luxembourg are isolated countries that
grow solely based on the knowledge created by their own workers. National economies
are embedded in an interdependent global system, so firms belonging to different
economies learn from each other new production methods, models of organisation,
marketing and product design.
Taking this into account, one may interpret the model with technological
spillovers as describing, not the path of a single country along one or two decades, but
instead collections of interdependent economies in the long run. According to this
interpretation, France and Luxembourg should grow at similar rates because they share
the same body of technological knowledge.
The idea that knowledge leaks across borders linking the growth rates of
interdependent economies has to be taken seriously and shall be part of our
deliberations. However, it shall not be overstated. A well-documented fact in our days is
that technological levels are not uniform across the space. Despite all progresses in
telecommunications and the internet, we are far from the neo-classical assumption that
knowledge spills over instantaneously at any distance at no cost. In general, the
empirical evidence gives supports to the idea that proximity matters for technological
diffusion (see Box 6.5).
Thus, while keeping an eye to the idea that technology has the potential to flow
across the space – thereby promoting economic convergence, one has to recognize that
geographical distance and other factors may create barriers to technological diffusion,
thereby creating incentives for economic activities to cluster in a given territory129.
This discussion suggests that, one needs to deepen our understanding of the
mechanics of knowledge diffusion and the role of factors such as geographical distance
and economic policies in delaying that diffusion. In that discussion, one shall take into
account that knowledge is not all alike: while some knowledge travels well around the
globe, much knowledge tends to be geographically localized. This question will be
addressed in more detail in Chapter 9.
Box 6.5: Proximity and technological diffusion: empirical evidence
The question as to whether knowledge spillovers tend to be bounded in space or
not is of crucial importance for economic growth and convergence: if most knowledge
spillovers are localized, companies operating nearby benefit more from each other
innovations than companies located elsewhere. In this case, there will be an incentive
for firms to operate in the same location, giving rise to cumulative causation and
divergence. If, in contrast, knowledge spillovers are mostly global, there will be a
tendency for laggard economies to catch up and to converge.
129
This is not to say that geographical distance is the only variable influencing the pace of technological
diffusion. As we will see in Chapter 9, technology diffuses at different speeds across the space, depending
on a number of factors, including the recipient region’ economic, political and social conditions. For the
moment, just hold on to the idea that the possibility of “imperfect technological diffusion” may give rise
to asymmetries in economic development.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
184
Introduction to Economic Growth
A strand in the literature has examined technological diffusion in its
geographical dimension, and the general conclusion is that proximity indeed matters.
Jaffe et al, (1993) and Eaton and Kortun, 1999) using data on patent citations, found
that technological diffusion is stronger within countries than across countries. Keller
(2002), using intra-industry data, found that with every additional 1200 kilometres
distance there is a 50-percent drop in technological diffusion (irrespectively of country
borders). Ciccone and Hall (1996) found out that employment density increases labour
productivity, supporting the existence of knowledge spillovers across workers in the
same locations. Audretsh and Feldman (1996) found that innovation structures in US
tend to be geographically more concentrated than production structures, suggesting that
agglomeration advantages are more prevalent in R&D. Other authors pointed out that,
although proximity matters for technological diffusion, the advantage of proximity has
declined in recent years, suggesting an impact of communication technologies (see
Keller for a survey, 2004).
6.6 Learning by doing and international trade
A simple model with cumulative causation
External economies of scale and learning by doing have important implications
for international trade. These may be summarised in two main ideas:

First, external economies and learning by doing intensifies the comparative advantages that led each
country to specialize in the first place. The reason is that external economies and cumulative
experience make the home firms progressively more productive in each of the goods initially
produced at home, while foreign firms become progressively more productive in each of the goods
initially produced abroad.

Second, trade openness may involve a trade-off between static efficiency and dynamic efficiency: if
different goods differ in terms of their learning potential, the pattern of specialization of a given
country with free trade is not necessarily the one that delivers faster economic growth130.
To illustrate this, let’s return to the two goods model introduced in Section 1.6.
In that model, the home economy is small relative to the world economy, the total
labour force is equal to 1, and there are two consumption goods, agriculture (Z) and
manufactures (Y), produced using labour only (equations 1.12 and 1.13):
Y  ANY
(1.12)
Z  BNZ  B1  NY 
(1.13)
Instead of assuming that the productivity parameters A and B are exogenous,
however, lets now assume that they evolve over time as a positive function of the
country cumulative experience in the respective sectors131:
130
The idea that, in the presence of Marshallian externalities, the static gains from trade and the dynamic
gains from trade may not go along was first formulated by Graham (1923). Authors revisiting this idea
include Krugman (1981, 1987), Lucas, (1988), Young (1991), Stokey (1991) and Matsuyama (1992).
131
Krugman (1987) discussed an expanded version of this model, accounting as well for the possibility of
cross-border technological spillovers: that is, of firms learning with the productive experience of firms
located abroad. This is an important assumption, because it opens a channel through which free trade may
promote economic convergence, by increasing a country’ exposure to foreign technologies. The model
above ignores however such complication.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
185
Introduction to Economic Growth
A   Y Y   Y AN Y with  Y  0 (6.16)
B   Z Z   Z B 1  N Y  with  Z  0 (6.17)
As before, assume that learning-by-doing takes place as a pure external effect:
each producer ignores the effect of its decisions in the aggregate.
Now, let p be the relative price of the agriculture good in terms of manufactures
in the world economy. If, at the time of trade openness, p  A B (that is, if the
opportunity cost of producing the agriculture good at home is lower than the relative
price of agriculture goods in the world economy), then the home country has
comparative advantage in agriculture. If international trade is free of impediments, the
home country will specialize in agriculture.
According to (6.6) and (6.17), countries accumulate skills by doing what they
are already doing. This mechanism intensifies whatever comparative advantage
countries begin with. Thus, once a pattern of specialization is established, changes in
relative productivity will act to further lock the pattern in. In terms of the example
above, home firms will be progressively more productive in agriculture goods while
firms abroad will be progressively more productive in manufactures.
Of course, if the learning potential of both industries was the same (i.e, if
Y   Z ), then the growth rate of per capita incomes would be independent of the
specialization pattern. If however the two industries differ in terms of learning
opportunities, then growth rates will depend on which good the country specializes in.
For instance, in the extreme case in which  Z  0 (i.e, if there is no learning-bydoing in agriculture), then by specializing in agriculture the home country will achieve
no growth at all. In other words, openness to trade will lead to lower productivity
growth than the average achieved under autarky.
This model suggests that, in some cases, restricting imports may lead to faster
economic growth. By closing the economy to manufacture imports, the home country
could feed its own manufacture sector, accumulating productive experience and
eventually developing comparative advantages in manufactures in the future. Based on
this idea, some authors gave argued that laggard countries should use temporary import
protection to catch up132.
It should be noted that the argument relies on the assumption that the economy
is small relative to the world economy, so that world prices remain unchanged. If
instead one assumed that the home economy was large, the conclusion could be
different133. To se this, suppose there were two large economies, say, North and South,
132
In a model with many goods, Krugman (1987) argued that a country could improve its economic
performance by protecting an industry until thus industry gets strong enough to survive in the
international markets and then move protection to another industry. The author argued that such strategy
was followed by Japan, during its industrialization process. Young (1991) introduces a model where
learning by doing opportunities in each good are bounded up. In his model, goods are ranked
hierarchically according to their productivity (learning) potential. Hence, as “knowledge” accumulates in
a given economy, the economy becomes progressively more endowed to produce goods with higher
productivity. Trade openness impacts asymmetrically across countries, because it leads some countries to
specialize in goods in which learning by doing opportunities are exhausted, while other countries
specialize in goods in which learning by doing still proceeds apace.
133
This point was made in Lucas (1988).
http://sweet.ua.pt/afreitas/growthbook/capa.htm
186
Introduction to Economic Growth
the North being specialized in manufactures and the South being specialized in
agriculture. If learning by doing opportunities only occur in manufactures, then
manufactures production will grow over time, while agriculture production remains
constant. When both goods are normal, this implies that the world relative price of
manufactures declines over time, so the North faces an adverse terms of trade effect. In
the South, agriculture production remains constant, but its purchasing power in terms of
manufacture goods increases over time. Whether the terms of trade effect is enough to
compensate the diverging output or not, this depends on the demand conditions:

Suppose, first, that the two goods are highly substitutes (that is, the
elasticity of substitution on consumption is greater than one). In this
case, the fall in manufactures prices leads to a more than proportional
increase in the world demand for manufactures, so the South’ terms of
trade do not improve the enough. In this case, a comparative advantage
in the good with high learning potential leads to faster growth in real
income.

If however, the elasticity of substitution was equal to one (as in the case
of Cob-Douglas preferences), then the terms of trade effect would
exactly offset the differential productivity growth and countries would
grow at the same rate.

Finally, if the substitutability between the two goods was lower than one,
then the terms of trade effect dominates the learning effect and real
income in the North will grow at slower pace despite this country having
faster technological progress. That would be an (unlikely) case of
immiserizing growth.
Box 6.6. Learning by doing and the European fears of globalization
In light of the conventional theory of international trade, countries should
specialize according to their comparative advantages. In the real world, however, many
policymakers and think tanks believe that giving up a country’s manufacturing sector is
a bad thing. The reason is that the country looses productive experience.
A recent intervention by the European Commissioner, Jacques Barrot (2008),
illustrates this. Barrot contended that allowing the low skill labour intensive
components of the production chain to migrate to emerging economies, taking
opportunity of the lower labour costs there, may benefit the European consumer in the
short run, but rises the risk of Europe losing its accumulated knowledge: “it is not
possible to maintain the knowledge accumulated through learning by doing if not
supported by a production activity”, the author argues. According to Barrot, giving up
the industrial base will imply sooner or later the loss of the accumulated knowledge, so
it will not be possible to explore the potential synergies between universities, research
centres and firms, as envisaged by the European leaders.
Article for discussion: Barrot, J., 2008. Les illusions d’une Europe sans
industries”, Les Echos, 28/4/2008 (http://www.lesechos.fr/info/analyses/4697530.htm).
Box 6.7 Trade openness and convergence
http://sweet.ua.pt/afreitas/growthbook/capa.htm
187
Introduction to Economic Growth
The question as to whether trade openness is good or bad for growth has been
subject to intensive debate by economists of all times. The general case in models with
a widely accepted set of assumptions is that international trade is good for growth. Still,
one may find models stressing less common but equally realistic assumptions showing
that trade can be detrimental to growth. Models with learning by doing are typically in
the second category. Thus, the question as to whether trade openness is good or bad for
growth is to a large extent an empirical one.
Empirically, most evidence points to the case that trade openness is indeed good
for growth. A seminal contribution is from Jeffrey Sachs and Andrew Warner (1995 and
1997). The authors first constructed an “index of trade openness” according to which a
country was classified as “open” if it satisfied 5 requirements at the same time134. Using
this index, the authors found that, along the period 1970-1989, open economies
outperformed closed economies in different dimensions.
Table 6.1 summarizes some of the authors’ results. According to the table, 11
out of the 15 “open economies” in the sample expanded above 3.0% per year, while
only four of the 74 “closed economies” achieved such a fast rate of economic growth.
Controlling for other explanatory variables, the authors found that, on average, open
economies grew by 2-2.5 p.p. faster than closed economies. The authors also concluded
that open economies tend to exhibit higher investment rates than closed economies (a
similar conclusion was no found for investment in human capital).
Table 6.1
Developing countries growth and openess, 1970-1989
Growth rate
always open
not always open
Average growth > 3.0
11
4
Average growth < 3.0
4
70
Source: Sachs and Warner (1995), p. 36.
The authors then investigated how this results change with a country level of
economic development. They found that within the group of “developing countries”,
those that were considered as “open economies” expanded at 4.49% per year, while
“closed economies” expanded at 0.69%, only. Among “developed economies”, those
that are open economies expanded at 2.29%, while closed economies expanded at
0.74%. The authors also concluded that poor countries tend to grow faster than richer
countries as long as they are linked together by international trade. Closed economies,
in contrast, do not display any tendency towards convergence. This suggests that
international trade may an important channel for international technological diffusion.
134
These are: average tariff rates below 40 percent; average quota and licensing coverage of imports of
less than 40 percent; a black market exchange rate premium that averaged less than 20 percent during the
decade of the 1970s and 1980s; a non-socialist economic system; no extreme controls (taxes, quotas, state
monopolies) on exports.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
188
Introduction to Economic Growth
Finally, the authors investigated whether trade openness helps improve the
quality of economic policies 135 . The authors found that, among the 73 closed
economies, 59 experienced a severe macroeconomic crisis. In contrast, only one open
economy experienced a serious economic crises (Table 6.2).
In general, this evidence supports the general claim that trade openness is good
for economic performance.
Table 6.2
Developing countries growth and macroeconomic crisis
Growth rate
Macroeconomic crisis in 1980s
No macroeconomic crisis in 1980s
Open in 1970s Not open in 1970s
1
59
16
14
Source: Sachs and Warner (1995), p. 56.
Note: “Macroeconomic crisis” is defined by one of the following occurrences: a rescheduling of foreign
debt; arrears on external payments; an inflation rate in excess of 100 per year.
Box 6.8. “You become what you export”
An implication of learning by doing models is that some specialization patterns
are more favourable to economic growth than others. This proposition is however
difficult to test, because a measure summarizing the “quality” of a country
specialization pattern is not easy to define.
In a recent article, Hausmann et al. (2007) proposed a methodology to overcome
this limitation. In particular, they proposed an index to rank goods in terms of their
implied “income content”. This index (PRODY) is estimated as a weighted average of
the per capita GDPs of the countries exporting a product, where the weights are
proportional to each country specialization level in that product. Using the PRODY
indexes, the authors then constructed a measure of the average sophistication level of a
country export basket (EXPY).
Figure 6.1 replicates the results obtained by Hausmann et al., (2007). The figure
displays the relationship between EXPY indexes and per capita incomes, as of 2005.
The figure confirms a high correlation between EXPY and per capita incomes, giving
support to the idea that “poor countries tend to export poor country goods, while rich
countries tend to export rich country goods”. Hausmann et al., (2007) also found that
EXPY indexes are a good predictor of future growth, after controlling for the standard
covariates. The authors conclude that the “quality” of a country specialization pattern
matters for growth. “You become what you export”, they conclude.
Figure 6.1 EXPY indexes and Per capita incomes
135
Sachs and Warner (1995): “(...) the international opening of the economy is the sine qua non of the
overall reform process. Trade liberalization not only establishes direct linkages between the economy and
the world system, but also effectively forces the government to take actions on the other parts of the
reform program under the pressures of international competition”.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
189
Introduction to Economic Growth
11,0
USA
10,5
IRL
ITA
GR
10,0
PRT
SP
FRA
SA
ARG
PC GDP 05 (PPP, Logs)
9,5
CHI
IRAN
ROM
BRZ
9,0
8,5
THA
MOR
IND
8,0
y = 2,5103x - 14,706
R2 = 0,8148
7,5
MOZ
7,0
MAD
MLW
6,5
6,0
8,0
8,5
9,0
9,5
10,0
10,5
EXPY 05 (Logs)
Source: own calculations based on Hausmann et al (2007) and using
COMTRADE data.
6.7 Discussion
This chapter reviews the so-called “first wave” of endogenous growth theories,
which stressed the role of externalities related to capital accumulation. The main feature
of these models is that production externalities lead to an aggregate production function
exhibiting non-diminishing returns, even if at the firm level the production function is
perceived to have diminishing returns. This allows the model to stick with the
convenient assumption of perfect competition.
Depending on the size of the external effect, this class of models may lead to
different conclusions. If the size of the external effect is enough to overwhelm the
diminishing returns to capital, then the economy displays unceasing growth. In the
particular case in which the external effect exactly offsets the diminishing returns, the
production function assumes the AK form.
Aggregate externalities may be a source of cumulative causation. Irrespectively
as to whether returns to the reproducible factor are increasing or decreasing, whenever
the production function exhibits increasing returns in all factors, there will be a
tendency for agglomeration of economic activities and to the self-reinforcement of
economic disparities: everything else constant, richer economies will be more attractive
to new investment than poorer economies, so they will get richer while poor economies
will remain poor. In a sense, this was what happened across the World after the
Industrial Revolution.
However, externalities may also be a source of convergence. If knowledge spills
over across borders, poor economies will have the opportunity to catch up by importing
foreign technology. Such pattern has also been observed in the real world: along the last
couple of centuries, many countries managed to join the rich countries club, by
importing or adapting technologies and institutions developed abroad. This discussion
suggests that we need to improve our understanding on the factors that influence the
pattern of international technological diffusion.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
190
Introduction to Economic Growth
Appendix 6.1. Strong versus weak scale effects in endogenous growth
models
An important dimension of knowledge is its non-rival nature: once a particular
technology his discovered, many firms can make use of it without reducing its
effectiveness. To account for this important property, learning by doing models
typically rule out dilution effects through the size of population. In terms of equation
(6.2), this implies restricting attention to the case in which  ´ 0 .
With all firms identical, the aggregate production function becomes:
Y  AK   N 1
(6.18)
Thus, whenever   0 , the production function exhibits increasing returns to
scale on capital and labour altogether. Hence, this model displays a tendency for
agglomeration and cumulative causation. Note however that increasing returns on
capital and labour altogether is not a sufficient condition for endogenous growth. For
the growth rate to depend on the saving rate, you need a model without diminishing
returns on the reproducible factors, that is,     1 .
In the literature, two versions of the learning by doing model emerged. The first,
due to Arrow (1962), assumes that     1 . The second, due to Romer (1986),
assumes     1 .
In the Arrow (1962) model there are diminishing returns to capital, so the
economy converges to a steady state. But the steady state is different from the one in the
Solow model, because of increasing returns to capital and labour altogether. To see this
formally, let’s take log differences in (6.18), obtaining:
Yˆ     Kˆ  1   n
(6.19)
In the steady state, capital and output should grow at the same rate. Imposing
this restriction in the equation above, one obtains:
Yˆ 1       1   n
(6.20)
Now using   Yˆ  n , you obtain the growth rate of per capita income in the
Arrow (1962) model:
  n 1      .
(6.20)
Equation (6.20) shows that, as long as population growth is positive, per capita
income in this economy will grow over time, without the need to assume an exogenous
rate of technological progress. Still, because the growth rate of per capita income in the
steady state is determined by the growth rate of population, which is an exogenous
parameter, this model displays exogenous growth: changes in policy influencing the
saving rate or the efficiency parameter A will alter the steady state level of per capita
income, but not its growth rate (only level effects). Moreover, long run growth will only
obtain if the growth rate of the labour force is positive: whenever n=0, diminishing
returns will force the growth rate of per capita income to decline to zero, as in the neoclassical growth model.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
191
Introduction to Economic Growth
The second version of the learning by doing model assumes     1 and is due
to Romer (1986). Substituting (6.18) in (6.3), the user cost of capital in this case
becomes:
r     Ak    1 N 
(6.21)
The endogenous growth rate of per capita income is:
   Ak    1 N     
(6.22)
In the particular case in which     1 , both the interest rate and the growth
rate of per capita income become increasing functions of the capital-labour ratio, so the
model displays explosive growth. Even in case     1 , the growth rate of per capita
income will still be a positive function of the size of the labour force, N , displaying a
“strong” scale effect: if the workforce grows at a constant rate, n, there will be everaccelerating growth. This possibility raises a fundamental problem: one does not
observe in the data a general tendency for growth rates to be explosive136.
Note that the model with     1 also display a kind of “scale effect”:
according to (6.20), the growth rate of per capita income depends on how fast
population is growing. Since this “scale effect” is of a second order, as compared to the
one in (6.22), it has been dubbed as a “weak scale effect”.
136
To avoid explosive growth arising from increasing returns, Romer (1986) assumed that the growth rate
of knowledge is bounded up, due to diminishing returns in learning by doing. With this assumption, the
interest rate becomes bounded up too, and the same will happen to the growth rate of per capita income.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
192
Introduction to Economic Growth
Key ideas of Chapter 6

An avenue to account for a larger role of capital in production than that
implied by the share of capital in national incomes is to assume the
existence of externalities.

The rationale for externalities on capital accumulation can be static
(availability of specialized suppliers, labour market pooling and
technological spillovers, complementarities), or dynamic (learning by
doing).

With externalities, one may observe increasing returns without the need
to depart from the perfect competition paradigm.

In the presence of positive externalities on capital, conventional growth
accounting overestimates the Solow residual.

If externalities are significant enough to imply non-decreasing returns in
capital, then the model will display unceasing growth.

Even if externalities are not strong enough to induce unceasing growth,
they may well cause the aggregate production function to display
increasing returns on capital and labour all together, giving rise to a scale
effect. This scale effect implies that bigger economies will be more
attractive places to produce and invest, favouring economic
agglomeration.

In order to evaluate whether technological spillovers are source of
agglomeration and divergence or, instead, of convergence, it is necessary
to assess how localized spillovers tend to be. In this discussion one shall
take into account that technology is not all alike. While some
technologies and ideas diffuse well across the space, others tend to
diffuse slowly. This calls for a deeper understanding of the process of
technological diffusion, which will be specifically addressed in Chapter
9.

With externalities, the competitive equilibrium will no longer be a social
optimum. So, at least theoretically, government intervention has the
potential to improve economic performance.

With externalities, free openness to international trade may imply a trade
off between the static gains of free openness to trade and the potential
dynamic gains resulting from restricting temporarily imports so as to
achieve a critical mass in sectors with significant spillover and learning
by doing effects.

Empirically, the evidence suggests that, indeed, some industries have
higher growing potential than others. However, there is also evidence
that closing an economy to international trade tends to result in lower
permeability to technological diffusion from abroad, poorer economic
policies and slower growth.
http://sweet.ua.pt/afreitas/growthbook/capa.htm
193