ECON5102 Macroeconomics 2
Lecture 3
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Growth and Ideas
Overview/background
▶ Our model so far for output per person: y = Ak1/3
▶ In the basic production model we assumed both technology A
and capital per-worker were exogenous.
▶ The Solow-Swan model relaxed the assumption of exogenous
capital, but still assumed A was exogenously determined.
▶ In the Solow-Swan model technological innovations play a
very important role.
▶ Now, we are going to allow A, technology, to be endogenous.
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Growth and Ideas
Readings — Jones, chapter 6
Framework is based on a model due to Paul Romer (1990).
Romer interview: www.youtube.com/watch?v=iCdr86rkVBc
Nobel Prize: https://www.youtube.com/watch?v=vZmgZGIZtiM
Overview:
▶ Introducing new ideas are the key to sustained long-run
growth
▶ A new model of economic growth: the Romer Model
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Introduction
Today class:
▶ Talk about economics of ideas
▶ We will develop a number of key insights in this process
▶ We will build a simple model of idea-based economic growth
▶ We will combine the Solow-Swan model with the model of
ideas → A rich theory of long run economic performance.
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The economics of ideas
Romer proposed the division of economic goods into:
▶ Objects: physical goods
▶ Ideas: instructions, recipes
Idea Diagram
Ideas → Non-rival → Increasing Returns
→ Failure of Perfect Competition
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1. Ideas
Ideas→ Non-rival→ Increasing Returns→ Failure of Perfect Comp.
Ideas
▶ Objects are finite, limited by the raw materials we have.
▶ Ideas, the way we can arrange and organize raw materials, seem to
be unlimited.
▶ Sustained economic growth occurs because we discover new ideas.
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An example
▶ Glass-making
sand︸︷︷︸
an object
+ heat︸︷︷︸
an object
+ glass-making knowledge︸ ︷︷ ︸
an idea
= glass
▶ Corrective eyeglasses
sand︸︷︷︸
an object
→ glass︸︷︷︸
an object
+ knowledge of optics︸ ︷︷ ︸
an idea
= corrective eyeglasses
▶ Silicon computer chips
sand (silica)︸ ︷︷ ︸
an object
+ lots of technical knowledge︸ ︷︷ ︸
an idea
= silicon chips
Production of these objects is limited by sand but not by the
required knowledge.
Unless the knowledge is somehow restricted, say through patents.
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An example (cont)
Further effects:
▶ When introduced, simple lenses to correct nearsightedness
doubled the working life of skilled European craftsmen.1
▶ With corrective lenses, the incentive to invest in education
and skills training are now much greater; human capital
increases, leading to further gains in production.
So ideas are clearly important; and they have important properties
that make them different from objects.
Another useful way to think of ideas — as recipes.
1David Landen, The Wealth and Poverty of Nations, (New York: W. W.
Norton, 1999).
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Non-rival
Ideas→ Non-rival→ Increasing Returns→ Failure of Perfect Comp.
Objects are generally rival and this gives rise to scarcity (and economics,
the science of choice). How about ideas?:
▶ Ideas are non-rival. One person’s use of an idea does not reduce the
(potential) availability of that idea to other people.
▶ For example, the quadratic formula is not scarce. Any number of
people can simultaneously use the formula to solve a quadratic
equation.
▶ Difference between the blueprints for the design of a car and the car
itself.
▶ Note that new ideas are scarce (we don’t know how to produce
cheap carbon-free electricity). But existing ideas are not scarce.
Can an idea depreciate?
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Excludability
▶ This refers to the extent to which someone has property rights
to a good (or an idea) and can legally restrict the use of the
good.
▶ Non-rivalry says it is feasible for an idea to be used by
everyone simultaneously — it does not imply that this is an
economically desirable outcome.
▶ Societies often grant property rights to people over ideas —
patents.
rivalry ̸= excludability
So, we can take an non-rival idea, grant patent protection, and
make it excludable.
Example: Microsoft Office (an idea? an object?) — definitely
excludable
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Increasing Returns to Scale — Production Function
Perspective
Non-rival ideas lead to increasing returns to production.
The principle is straightforward:
▶ Consider a firm producing some good or service using various
inputs and a particular production process (idea).
▶ By replication (doubling the inputs) we can double the output
— CRS.
▶ If we double inputs and the stock of ideas, we must more than
double output — increasing returns to scale, IRS.
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IRS
But what does it mean to ‘double the stock of ideas’?
▶ Think of two production methods - A1 and A2.
▶ Using A1, labour and capital (bundled together as X ) produce
an amount of output, say Y1.
▶ Using A2, labour and capital (X ) produce an amount of
output, say Y2 = 2× Y1.
▶ An implication: this means average product (Y over X ) is
increasing.
▶ For constant returns to scale, average product is constant.
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IRS
A bit more detail (‘scale’ is a bit confusing here). We have
Y = AX ;
▶ if CRS, then scaling means just X , so Y /X = A is constant
(since A is unchanged).
▶ if IRS, then scaling means A and X , so Y /X = A is increasing
(since A is changing).
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IRS
Mathematically, we can use the Cobb-Douglas production function
to model IRS.
Production: Y = F (Kt , Lt ,At) = AtK
1/3
t L
2/3
t
Note: At is technology or ideas. No longer exogenous and
constant (recall A.) Constant returns to scale in Lt and Kt since
F (2Kt , 2Lt ,At) = 2Yt . (Or any other scale factor.) Note: we can
call Xt = K
1/3
t L
2/3
t our input ‘bundle’ as we did on previous slide.
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IRS
But, we have increasing returns to scale in At , Lt , and Kt :
F (2Kt , 2Lt , 2At) = (2At)(2Kt)
1/3(2Lt)
2/3 = 4AtK
1/3
t L
2/3
t = 4Yt
Output Y is more than doubled.
Put it another way,
▶ If we double inputs K and L, we double output;
▶ If we double inputs K and L and move to better
production methods (At → 2At), we more than double
output.
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Graphically
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Graphically
▶ For the CRS production process, average product Y /X is constant.
▶ For the fixed cost (IRS) production process, average product Y /X
is increasing.
Average product at points a and b is the slope of the dashed
line.
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Problems with Perfect Competition with IRS
Under certain conditions perfectly competitive markets result in an
optimal or efficient allocation of resources.
One condition is that:
Price = Marginal Cost.
The price of a good is equal to its marginal cost of production.
With increasing returns this condition is unlikely to hold.
In our example of the new drug, the marginal cost of a tablet was
$10, so it might be thought that competitive forces would push its
price to $10.
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Problems with Perfect Competition with IRS - cont.
But this ignores the initial fixed cost ($2.5 bn) of developing the
drug.
If Price = Marginal Cost, then no firm would be willing to
undertake the costly research needed to invent new ideas.
For the inventor to be compensated we require (expected) Price >
MC. So firms will earn monopoly profits.
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In Summary
▶ Ideas are non-rival, which means there are IRS to inputs
and ideas.
▶ Perfect competition model is not going to work with the
market for ideas & innovation.
▶ We need a model that emphasizes the production & use
of non-rival ideas — Romer model.
▶ Unfortunately, fully modelling the production of ideas is
very difficult so this part is greatly simplified.
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2. The Romer Growth Model
To understand sustained growth:
We need a model that distinguishes between goods and ideas
We need increasing return to scale (IRS) because ideas are
non-rival
The Romer Model
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2. The Romer Growth Model
A model of economic growth that makes endogenous the
production of new ideas.
To focus on the accumulation of ideas we will omit physical
capital: fix capital stock at K = 1.
Labour can be used, along with knowledge to produce output
(physical goods) or it can be used to produce new ideas
(knowledge).
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2. The Romer Growth Model
Production of goods
Yt = AtLyt
This equation describes the production of goods. It requires labour
and also the stock of ideas.
For a given At , there are constant returns to scale in labour. There
are increasing returns to scale for ideas and labour.
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Production of new ideas
∆At+1 = zAtLat
This models how the stock of ideas At changes over time.
The production of new ideas depends on:
▶ the stock of existing ideas, At .
▶ the amount of labour used in the ‘technology sector’, Lat .
▶ a productivity parameter for the technology sector, z .
The economy is assumed to begin with a initial stock of ideas A0.
The same stock of ideas can be used to produce both goods and
new ideas — ideas are non-rival between these sectors.
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Labour Supply and Allocation of Labour Across Sectors
Labour is not non-rival; either in one sector or the other and
subject to availability of labour supply:
Lyt + Lat = L
Labour has two uses: output and research; assume a simple rule
for allocation. Similar to how investment is determined in the
Solow-Swan model.
Lat = ℓ · L 0 ≤ ℓ ≤ 1
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Complete Model — 4 equations in 4 unknowns
Yt = AtLyt
∆At+1 = zAtLat
Lyt + Lat = L
Lat = ℓ · L 0 ≤ ℓ ≤ 1
Endogenous variables: Yt , At , Lyt , Lat .
Exogenous variables: z , ℓ, L, A0.
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Solution: solving the Romer model
Lat = ℓ · L Lyt = (1− ℓ)L
yt ≡ Yt/Lt = At(1− ℓ)
▶ Output per person depends on the total stock of knowledge
From the idea production function
∆At+1
At
= zLat = zℓL
▶ The growth rate of knowledge is −−−−−−−− over time
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Solution: solving the Romer model
We can assume that the stock of knowledge at any point in time is
given by:
At = A0(1 + g)
t
Where g is the growth rate of ideas (from previous slide)
▶ The growth rate of ideas is proportional to the number of
researchers in the economy (proportional to L also)
▶
g ≡ z · ℓ · L
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Solution
Lat = ℓ · L Lyt = (1− ℓ)L
yt ≡ Yt/Lt = At(1− ℓ)
At = A0(1 + g)
t see over
yt = A0(1− ℓ)(1 + g)t
Notice that output per person grows at a constant rate g (a theory
of sustained economic growth).
This growth rate depends upon,
▶ how much total labour supply there is, L;
▶ how much labour is in the technology sector, ℓ;
▶ how productive is labour in the technology sector, z .
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Solution Detail:
First, we need to solve ∆At+1 = z · ℓ · LAt . Here’s a reminder.
Re-write
∆At+1 = At+1 − At = z · ℓ · LAt
as,
At+1 = z · ℓ · LAt + At = (1 + z · ℓ · L)At
= (1 + g)At
Now, start at A1, given A0, and iterate a few times:
A1 = (1 + g)A0
A2 = (1 + g)A1 = (1 + g)
2A0
...
At = (1 + g)
tA0
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Solution Detail: (cont.)
Second, we need to solve for output per person, yt ≡ Yt/L.
From the production function,
Yt = AtLyt = At(1− ℓ)L
So,
yt = Yt/L = (1− ℓ)At
Using our solution for At ,
yt = (1− ℓ)(1 + g)tA0
Or,
yt = (1 + g)
ty0
Since y0 = (1− ℓ)A0.
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cont.
The source of sustained economic growth in this model is the
non-rivalry of ideas (At) and the sustained production of new ideas.
No diminishing returns to ideas:
∆At+1 = zAtLat
(Exponent on At is equal to one. The tutorial has a different
version with decreasing returns to ideas.)
In the Solow-Swan model, economic growth comes from
accumulating capital (until we reach steady state).
Capital is a rival good and has diminishing returns — hence no
long run sustained economic growth.
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cont.
We have a number of exogenous variables. We can ask what
happens when one of these changes.
Experiment No. 1: An increase in the population
Here we will treat the model as one of world economic growth.∗
yt = A0(1− ℓ)(1 + g)t
g = z · ℓ · L
An increase in L¯ at time t will have no direct effect on the level of
output yt since L¯ only influences the growth rate g¯ . (The level of
output at t occurs because the economy was previously growing at
the old growth rate.)
An increase in L¯ at time t will increase the growth rate for the
future path of output. dg
dL
= z · ℓ > 0
* Why is this not a good model on a country by country basis?
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cont.
* Why is this not a good model on a country by country basis?
▶ It predicts that more populous countries will grow faster
than less populous countries, which isn’t true empirically.
See Case Study — A Model of World Knowledge.
▶ More importantly, it is not a sensible model for a single
country. Applying it to a single country assumes that the
stock of knowledge is country specific and that all ideas
must come from within a country.
▶ But ideas and knowledge are more universal than this,
especially over time.
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Graphically
log yt
year2000 2020
g = z · ℓ · LOLD
g = z · ℓ · LNEW
An increase in L at 2020
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Experiment No. 2: An increase in the share of labour force
in technology sector, ℓ
yt = A0(1− ℓ)(1 + g)t
g = z · ℓ · L
Key points:
▶ A growth effect through g and a level effect on yt .
▶ Growth effect raises the path of output over time.
▶ Level effect lowers current and near future levels of output
since less labour to produce goods.
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Graphically
log yt
year2000 2020
g = z · ℓ · L
g = z · ℓ ′ · L
An increase in ℓ at 2020 to ℓ
′
.
Jump down (level effect): current and near future levels of output
fall since less labour to produce goods.
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3. Growth Accounting
The general production function we have for output is:
Yt = At︸ ︷︷ ︸
technology/ideas
× K 1/3t︸︷︷︸
capital
× L2/3yt︸︷︷︸
output specific labour
Notice that this now has a changing capital stock (as in
Solow-Swan) and output specific labour (as in Romer).
In growth rates, the above is
gYt = gAt +
(
1
3
)
gKt +
(
2
3
)
gLyt
Let gLt be the growth rate of the total hours worked in the
economy (production and technology sector).
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cont.
gYt − gLt = gAt +
(
1
3
)
(gKt − gLt) +
(
2
3
)
(gLyt − gLt)
The growth rate of output per hour worked (or employee)
comprises:
▶ Growth rate of ideas, or more generally Total Factor
Productivity (TFP)
▶ Growth rate of capital per hours worked (K/L)
▶ Labour composition. Here, gLyt − gLt ; more generally, labour
quality.
In practice, everything but gAt is measurable; so we construct the
growth rate of TFP as a residual.
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Australian Bureau of Statistics
5260.0.55.002 Estimates of Industry Multifactor Productivity,
Australia
Table 20. Labour productivity, growth accounting - Market
sector
2010-11 2011-12 2012-13
Industry labour productivity growth -1.28 1.10 3.28
Contribution to labour productivity growth from:
Information technology capital per hour 0.18 0.14 0.32
Non-information technology capital per hour -0.46 0.28 2.41
Labour composition 0.20 0.20 0.20
Multi-factor productivity -1.19 0.47 0.35
Negative multifactor productivity — increase in inputs not matched
by outputs. For example, mining often sees large capital invest-
ments that take many years to see increased production.
gyt − gLt = gAt +
(
1
3
)
(gKt − gLt) +
(
2
3
)
(gLyt − gLt)
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Example
As an example, consider the US between 1995–2007.
Output per hour 2.8 %
Capital per hours worked 1.1 %
Labour composition 0.2 %
TFP Growth ? %
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4. Summary
Solow-Swan model (based purely on accumulation of objects, i.e.
capital) is not able to provide a model for persistent growth in
output per person.
Romer model focuses on the accumulation of ideas (which are
assumed to be non-rival) and this generates increasing returns to
scale.
Sustained grow in the total stock of knowledge is able to produce
sustained growth in output per person.
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4. Summary
Jones (Appendix) shows how to combine the Solow-Swan and
Romer models:
▶ Romer model provides the underlying source of long-run
growth
▶ Solow-Swan model gives rise to transition dynamics —
explains the process by which countries who start-out
relatively poor are able grow fast in the short-run and so
catch-up to the rich countries.
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