The Engine

1
Norton Media Library
Chapter 5
The Engine of Growth
Charles I. Jones
2
Where Does Technological Progress Come From?

• Technological progress plays a crucial role in
  the Solow-type growth models
• However, so far technological progress and its
  growth were given exogenously
• We are now asking this question where does
  technological progress come from?
• In other words, we are seeking to endogenize
  technological progress
• Improvements in technology are understood in
  terms of the firms incentives to maximize their
  profits

3
Romer versus Schumpeterian Models

• Romer models look at technological progress from
  the point of view of increasing the variety of
  products
• Laptops in addition to desktops
• Blue coke in addition to original red-colored
  coke
• iPhones in addition to just handheld sets
• Schumpeterian models focus on improving the
  quality of existing products
• Faster computers with more memory
• Safer cars
• Creative destruction
• It turns out the implications for growth are
  similar irrespectively of whether we focus on
  more variety or better quality

4
Romer Model Production Function
Romer models explain why and how the advanced
countries experience sustained growth. The focus
is on the technological frontier, i.e. the best
practice. Technology diffusion to the less
developed countries is left aside. Main
equations are similar to the Solow-type
models. The production function is a little bit
different

• A is now the endogenous stock of ideas
• is labor input (index Y will be explained
  later)
• There are three factors of production now since A
  is endogenously determined?increasing returns to
  scale

5
Romer Model Capital Accumulation
The capital accumulation equation looks very
similar to that of the Solow models
Labor grows exponentially, as in Solow
6
Romer Model Production of New Ideas
The key addition of Romer model to the Solow
framework is the introduction of the production
function of new ideas. More researchers can
produce more ideas, recall the discussion in
chapter 4 about the benefits of population growth
in terms of generating more ideas.
is the number of people attempting to
discover new ideas.
is the rate at which new ideas are
discovered.
Labor L is allocated between work and innovation.
7
Rate of Discovery of the New Ideas
The rate at which new ideas are discovered is
modeled as follows
Several cases are possible with respect to
parameter Productivity of
research increases with the stock of ideas A
Laser, calculus, electricity etc The
fishing out case most obvious ideas discovered
first, subsequent ideas are increasingly
difficult to discover Implies
constant rate at which ideas are discovered
Productivity of research is independent of the
stock of A
8
Rate of Discovery of the New Ideas
Rate of discovery of the new ideas may also
depend on , the number of innovators
(inventors). Consider
The term rather than just reflects
the fact that with many researcher thinking about
innovations, duplication of effort is possible.
9
Intuition Behind Innovation Equation
It follows that
(why?)
An individual engaged in research creates
new ideas. Rate of discovering new ideas
will depend on the aggregate research effort.
reflects the stepping on
toes, or congestion, effect too many cooks will
spoil the broth (duplication effort)
is the positive externality effect
benefits of the discovery spill over to the other
researchers (calculus, Newtons laws). This is
the standing on shoulders effect
10
Growth in the Romer Model
Initial endowments we assume an economy starts
with initial endowments of and
.
As in the Solow models, all per capita growth is
due to technological progress. Along the
balanced growth path, all variables are growing
at the same rate
11
Rate of Technological Progress
What is the rate of technological progress along
a balanced growth path?
Taking logs and differentiating the last equation
implies
The growth rate of the number of researchers must
be equal the rate of growth of the population,
otherwise researchers will outnumber the whole
population
12
Long-Run Growth Rate
Along the balanced growth path, the economy is
growing at the rate of technological progress
Suppose there is no duplication of research
effort, and that research productivity is
independent of the stock of ideas, i.e.
In this case ideas get produced according to If
the number of researchers stays the same,
ideas do get generated since
, but the rate of growth of
ideas approaches zero with time because A is
increasing
However, if the growth rate in ideas
is constant, too.
13
Population Growth and Long-Run Growth Rate
Higher population growth rates strains resources
(Solow-type models), but it also increases the
number of researchers , which increases the
number of non-rivalrous ideas. These ideas
benefit the whole economy, so the overall effect
of the population growth is beneficial! Long-run
growth is impossible without the ongoing growth
of the population since that would mean constant
number of ideas relative to the ever growing
stock of ideas. In this way, assuming
results in a rather sour prediction.

14
Population Growth and Long-Run Growth Rate
Let us assume now that there is no duplication
problem (as before), but that research
productivity depends on the stock of existing
ideas, that is, assume that
In this case, the productivity of researchers
grows over time even if the number of researchers
is constant.
Caveat If , that would imply ever
increasing growth rates of the economy since
has been growing rapidly. However, the U.S.
growth rates have been rather modest at 1.8 for
the last century. Hence, we assume that
15
Romer and Solow Models
In Solow (neoclassical) models, government
policies produce level, but not growth
effects. Romer model produces the same result
even if technological progress is endogenous,
rather than exogenous, as it is in the Solow
models. Even increasing the share of researchers
will not change the growth in Romer model
since
16
Comparative Statics RD Share Increase
Suppose the government extends subsidy to the RD
institutions so that the share of researchers
goes up. Assume for simplicity that
Along the balanced growth path, If the share of
researchers increases relative to L, an increased
number of new ideas is produced, which increases
the growth rate of technology. Rate of
technological progress will exceed population
growth n, so the fraction of researchers will
start declining again.
17
Comparative Statics RD Share Increase
Initially, the rate of technological progress
exceeds population growth However, as time goes
by starts declining back to the
initial level
18
Comparative Statics RD Share Increase
A permanent increase in the share of researchers
increases the rate of technological progress only
temporarily
19
Comparative Statics RD Share Increase
However, the level of technology increases
permanently as a result of a permanent increase
in
Note the level and growth effects in the Romer
model produced by an increase in
are qualitatively similar to the effects produced
by an increase in the investment share in the
Solow models.
20
Solving Romer Model
The only difference with Solow is the
term Technology level increases with labor force
as
Substituting into the first equation results in
A larger economy is also a richer economy scale
effect.
21
Economics of Romer Model
We saw in Chapter 4 how RD can be the result of
profit-maximizing behavior. What is the
microeconomics behind this link? Consider a
three-sector economy final goods, intermediate
goods, and the research sector. Research sector
produces ideas that take the form of the new
variety of capital goods new computer chips,
industrial robots, or printing presses. Researche
rs sell the exclusive right to produce a specific
capital good to an intermediate goods firm, a
monopolist, that sells intermediate good to the
final-goods sector that produces consumer output.
22
Final-Goods Sector
A large number of perfectly competitive firms
combine labor and capital to produce a
homogeneous final output good, Y. The production
function in the final goods sector is
Output Y is produced using labor and a
number of intermediate goods Notice that the
level of technological progress A in this
specification is exactly the number of
intermediate goods!
For any given level of A, the production function
exhibits constant returns to scale.
23
Final Goods Sector Profit Maximization Problem
Replace for convenience the summation sign with
an integral
Normalizing the final good price to unity, we
come up with the following profit-maximization
problem
where w and are wages and capital rental
prices, respectively.
24
Final Goods Sector First-Order Conditions
Firms hire labor until the marginal product of
labor equals the wage Firms rent capital goods
(intermediate goods) until the marginal product
of each kind of capital equals the rental price
25
Intermediate Goods Sector
Intermediate goods are produced by monopolists.
Intermediate goods are capital goods that are
sold to the final-goods sector. Intermediate
goods producers are monopolists because they
purchase the design of their capital goods from
the research sector. Patent protection ensures
the intermediate goods producers monopolistic
power. To produce one unit of the capital good,
the intermediate good producer needs exactly one
unit of raw capital that costs r per unit.
26
Intermediate Goods Sector Profit Maximization
Problem
is the demand function for the capital
good. Remember it is equal to First-order
conditions
is the inverse price elasticity of
demand which is equal to
27
Intermediate Goods Sector Producers Profits
Since the demand function for the intermediate
capital goods is the same for all varieties, all
varieties are employed in identical quantities,
or As a result, each capital-goods firm earns
the same profit (why?)
28
Intermediate Goods Sector Demand for Capital and
Aggregate Production Function
Total demand for capital from the
intermediate-goods sector must equal the total
capital stock in the economy All capital
goods are used in the same amount, and the number
of varieties is equal to the level of
technological progress A. As a result, The
final-goods production function then becomes the
familiar aggregate production function
29
Research Sector
New designs (varieties) of capital goods are
discovered according to On discovery of a new
design, an inventor receives a patent from the
government, the patent lasts forever. Inventors
sell patents to the intermediate-goods sector and
use the proceeds to consume and save. What is
the price of a patent? The price of a patent
must be equal to the present discounted value of
the future profits by the intermediate-goods
firm. (why?)
30
Patent Price and the Arbitrage
Arbitrage is the method of profiting by
exploiting price differences across different
places or investment opportunities. The practice
of arbitrage leads to the equalization of prices
among different places or investment
opportunities. (why?) Suppose the patent costs
. If you invest it in the bank for one period
(year), your gain will be equal to . If
you use the same amount of money to buy a patent,
your gain will be The arbitrage condition
requires that Rewriting the above equation
results in Along a balanced growth path, r is
constant, and
31
Features of Romers Model

• The aggregate production function exhibits
  increasing returns
• Increasing returns require imperfect competition
• Firms in the intermediate goods sector are
  monopolists
• Capital goods sell at a price exceeding marginal
  cost
• Profits earned by the intermediate producers are
  extracted by the inventors who get compensated
  for the time they spend looking for new ideas
• This framework is called monopolistic competition
• There are no economic profits in the model since
  all rents are compensating some factor input
• Markets no longer result in the most desirable
  outcome due to the presence of imperfect
  competition

32
Solving for the Share of Researchers
What is the fraction of total population L that
decides to engage in research to develop new
designs of capital goods? Remember that
and Labor working in the final
goods sector receives a wage w that is equal to
its marginal product in this sector Labor in
the research sector earns the wage equal to the
value of its marginal product in that sector as
well By the principle of arbitrage, wages in
the two sectors must be the same
It follows that
33
Growth and Share of Researchers

• The faster the economy grows, the greater the
  share of researchers in the economy
• The higher the interest rate obtainable from the
  bank, the lower the fraction of researchers

Interest rate can be shown to be equal to
, which is lower than the marginal
product of capital. In the Solow model, perfect
competition ensures that all factors are paid
their marginal products. In the Romer model,
imperfect competition requires to pay factors
less in order to free funds for research.
34
Creative Destruction
In the Romer model, technological level A is
equal to the number of the varieties of the
intermediate capital goods. Technological
progress according to Romer is about the increase
in the number of existing capital goods
varieties. This paradigm implies older
inventions never give way to the newer ones.
However, we dont see typewriters or steam
engines used anymore. Joseph Schumpeter in the
1940s described capitalism as a process of
creative destruction existing technologies are
replaced by new ones. Productivity grows thanks
to the replacement of old designs with the new
inventions.
35
Schumpeterian Model Discrete Innovations
Aggregate production function Innovation occurs
in steps since Think of as walking
technology, then will be a horse cart,
a steam engine, and a modern
car. Each time we innovate, we get more
productive if igtj. New innovations
occur according to where is the size of the
innovation. The growth rate of A is then equal
to . The length of the time
period between any two innovations is uncertain.

36
Schumpeterian Model Probability of Innovation
Denote to be the probability for any
researcher to make a discovery at any given point
in time. Similarly to the Romer model, we have
the effects of standing on shoulders and
stepping on toes, but this time these effects
work on the probability of innovation by one
researcher, not on its size For the economy
as a whole, the probability of innovation equals
Standing on shoulders makes it easier to
innovate, but harder to find a new design
37
Schumpeterian Model Remaining Parts
The remaining parts of the Schumpeterian model
are identical to the Romer model Capital
accumulation Labor force growth Labor
division Initial endowments
38
Schumpeterian Growth
Since innovations occur randomly, growth is not
regular. There can be periods of one or more
years when innovations do not occur at all.
Due to the random nature of innovations in the
Schumpeterian model, we can only make statements
about growth in terms of averages, or
alternatively, in terms of mathematical
expectations. The expected growth rate of A over
time is equal to the probability of innovation
times the size of the innovation
Along the balanced growth path,
39
Schumpeterian Expected Growth Rate of Technology
Taking logs and differentiating, we obtain the
following
Since the number of researchers cannot exceed
total population, , which implies
that The average long-run growth rate in the
Schumpeterian model is identical to the growth
rate in Romerian model!
40
Growth Path in the Schumpeterian Model
The average growth is governed by the growth of
population n, the duplication of effort ,
and the spillovers parameter . Flat sections
in the graph indicate those time periods when no
innovation occurs Log income per capita jumps by
the value of each time an innovation occurs
41
Innovation Size and Growth Rate
The innovation size does not affect the
growth rate While the larger innovation size
increases the absolute size of technological
level A, increasing productivity, it also reduces
the probability of finding next innovation since
. Such reduction of
probability occurs because we are assuming The
productivity effect of a larger innovation size
gets offset by the longer period of time that
has to pass before the next innovation arrives.
42
Economics of Schumpeterian Model
Imperfect competition is again necessary to
justify monopolistic profits needed to compensate
researchers for their work. Differences are in
how intermediate goods are used, and in the
nature of innovation. Three sectors final
goods, intermediate goods, and research. Only
one single intermediate goods produced by a
monopolist who owns a patent.
43
Creative Destruction
Researchers work to produce a new version of the
capital good that is more productive for the
final good. Intermediate good producers are
eventually replaced when the new patent is sold
by the researchers to another capital goods
producer. The intermediate-goods producer is
always in danger of being replaced (destruction)
by a new, more productive, supplier (creation).
The understanding of possibility of such
replacement will be reflected in the value that
the intermediate goods producer will be willing
to pay for a patent.
44
Final Goods
Unlike Romer, in the Schumpeterian model there is
only one single intermediate good, and is
the level of technological progress that changes
discretely over time, i1,2,3,.. The production
function for final goods is Output Y is
produced using labor , and a single capital
good , the intermediate good at step i. A
large number of firms are competing in the
final-goods sector. The production function is
constant returns to scale since doubling and
labor will double the output. Example
i1 is mainframe computers, i2 is modern
servers. One server is more productive than one
old mainframe, i.e.
45
Final Goods Profit Maximization
It can be shown that only the latest version of a
capital good will be used at any given time since
the prices of all versions are the same at any
given time. Final goods producers are maximizing
their profits as follows where w is wages, and
is the rental price per unit of capital
good First-order conditions
Compare to Romer FOC
In both models, the economic reasoning is
similar hire factors until what you pay per unit
is equal to the marginal product.
46
Intermediate Goods
A single intermediate good is produced by a
single intermediate-goods firm that bought a
design from the research sector. Monopolistic
position is ensured by the patent
protection. Similarly to Romer, one unit of raw
capital will produce one unit of the capital
good, so that the profit maximization and the
resulting price of the intermediate good is the
same in the Schumpeterian model
The first-order conditions imply that Similarly
to Romer model, the intermediate firm
charges Again we have a markup over the
production cost r.
47
Intermediate Goods Why Only One Version?
Why is it that the final goods firms only
purchase one version of the capital good, and why
that version is always the latest
version? Since, as we have shown,
is the same irrespectively of the particular
version, i.e. since doesnt really depend on
i, the version number, the final goods producers
will naturally buy only the latest, most
productive, version of the capital good. For
that reason, at any moment in time, only one
single firm producing intermediate goods will be
operating.
48
Intermediate Goods Profits and Aggregate Output
The intermediate goods producers profits are
given by which is very similar to Romers
. Technically,
its Romers profits if A1. However, in the
Schumpeterian model profits are not divided over
many intermediate goods producers, rather all
going to just one firm. Capital stock in the
economy must equal the stock of the single
intermediate good Aggregate output will then
be
49
Research Sector
New versions of the capital good arrive with
constant probability A successful inventor
receives a patent from the government. The patent
lasts forever. The patent is sold to the
intermediate goods producer. A patent number
i1 will not be sold to an intermediate good
producer that uses patent number i, so the
existing producer will have to go (destruction)
to give way to the new firm (creation).
50
Arbitrage and Patent Value
The amount of money equal to , the value
of the patent, can be deposited at the bank. In
this case the increase in value will be equal to
. The same amount of money can be paid to
obtain the patent. In this case, the increase in
value will be profits from producing an
intermediate good, plus the change in the value
of the patent, minus the expected value of the
loss of due to somebody producing an
innovation with probability . The
arbitrage equation
Rearranging,
Denote to be the aggregate
probability of a new innovation occurring.
51
Patent Value and Balanced Growth Path
Along a balanced growth path, rental rate of
capital r is constant. The aggregate probability
of innovation occurring is constant
as well. For the ratio to be constant,
profits and patent value have to grow at the same
rate. Since ,
profits are growing at rate From prior
analysis, we know that The arbitrage equation
implies then
52
Patent Values in Romer and Schumpeterian Models
The patent value in Schumpeterian model is lower
compared to the patent value in Romer model, In
the Schumpeterian model, since sooner or later
the existing producer of intermediate goods must
exit the market, the patents value is declining
with the increase in probability of a new
innovation . As the size of the
innovations increases, the value of a patent
increases, too.
53
Why is Replacement Necessary?
Why is it that a new innovation always results in
a replacement (destruction) of the existing
producer of an intermediate good? Arrow
replacement effect due to Kenneth Arrow (1962)
the new firm will always bid more for a patent
since acquiring a new patent will not involve
loss in value due to the obsolescence of a
previous patent. For the existing firm, buying a
new patent will obliterate the value of the
existing patent.
54
Allocation of Labor to Research
Individuals working in the final goods sector are
earning Individuals can also work as
researchers and earn The arbitrage condition
(i.e. the requirement that the two types of wage
be the same in both sectors) implies It can
be shown that as a result,
55
Allocation of Labor to Research
The Schumpeterian fraction of researchers
is similar to the Romer fraction The term r-n
appears in both formulae, saying that the higher
the discount rate that applies to profits, the
lower the fraction of researchers. Two effects
of (1) as the chance of innovation
increases, the value of patent declines, in
(2) innovations are more lucrative
if they are more probable
56
Allocation of Labor to Research
Since the aggregate probability of innovation
exerts two opposite effects on , the
combined effect can be seen by taking a
derivative of with respect to .
The combined effect is positive since the gains
from innovation are large relative to the losses
from eventual replacement (show it).
57
Comparing Romer and Schumpeter
In both models, the endogenous growth rate is
given by
The innovation-increasing effect of population
growth works in both models to the same extent
irrespectively of the exact type of the
innovation-producing process. Schumpeterian
approach reflects the dynamics of firm behavior
by exploiting the concept of creative
destruction. The Schumpeterian model will have a
higher if , or if the
discount rate applied to profits is relatively
large people do more research if the future
destruction carries little weight.
In both Schumpeterian and Romer models, since
growth only depends on the population growth rate
n, all policy changes produce only level effects,
and no growth effects.
58
Optimal RD Knowledge Spillover
In Romer and Schumpeterian models, there are
three distortions to research lack of accounting
for the spillovers to the future research, the
stepping on toes effect that allows duplication
of effort, and the consumer surplus effect since
monopolists value innovation only according to
their profits.
The knowledge spillover effect Market values
research according to profits from new
design. However, means productivity of
research increases with the stock of ideas. Since
researchers are not compensated for increasing
the productivity of the future researchers, the
market produces too little innovation Newton,
Maxwell, Einstein This is a classic positive
externality problem
59
Optimal RD Stepping on Toes
Stepping on toes effect
means that researchers lower
their research productivity because they
duplicate effort. Duplication of effort happens
when two or more researchers are working to solve
the same problem, or to produce the same
design. In this case, the externality is
negative, and the market is producing too much
research.
60
Optimal RD Consumer Surplus
Consumer surplus effect
An inventor only captures the monopolistic
profit, which is smaller than the consumer
surplus representing social gains from inventing
the good. Too little innovation is generated.
61
Importance of Basic Research

• The discrepancy between private gains due to
  innovation, and the social gains can be very
  large in case of fundamental, or basic, research
  calculus, vaccination, the Internet etc
• In this case the government should fund basic
  research since otherwise researchers will not
  find it profitable to engage in fundamental
  science
• Non-rivalrous ideas
• No direct short-term profits
• Griliches (1991) finds rates of return in the
  fundamental inventions area to be around 40 and
  60, which is much higher than the private
  returns.

62
Too Little Research
Among the three effects associated with research,
only one (stepping on toes) results in too much
research because of the duplication of effort.
The other two (knowledge spillovers and consumer
surplus) work to result in too little
research. The combined result is, too little
research is produced.
63
The Importance of Monopolies
Classical economics views monopolies as
undesirable since monopolies charge above
marginal cost, and since they result in
deadweight losses to the society. However, in
the economics of ideas monopolies are crucial
since it is monopolistic profits that motivate
researchers to produce innovations. If
regulators destroy imperfect competition and
monopolies in the market for innovations, there
will be no gains to productivity, and as a
result, no growth.