Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Macro Notes
Alan G. Isaac
American University
Malthusian Model
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
Every generation has perceived the limits to growth
that finite resources and undesirable side effects
would pose if no new recipes or ideas were
discovered. And every generation has underestimated
the potential for finding new recipes and ideas. We
consistently fail to grasp how many ideas remain to
be discovered.
– Paul Romer (1993), as cited in Jones (2011, p.130)
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
Within thirty years, we will have the technological
means to create superhuman intelligence. Shortly
after, the human era will be ended.
– Vinge (1993)
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Ideas
Paul Romer (1990 JPE) proposes a fundamental distinction
between objects and ideas, especially ideas about how to make
objects.
instructions
recipes
designs
blueprints
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Nonrivalry
Ideas are
costly to generate
nonrivalrous in use
Nonrivalry: my use of an idea does not limit your use.
Rivalry is the basis of scarcity. Objects are generally highly
rivalrous: one person’s use of an object reduces its usefulness
to another person.
In contrast, if I implement a production design, you can
equally well implement the design. In this sense, the design is
nonrivalrous.
(But, any particular implementation will be rivalrous.)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Nonrivalry vs. Nonexcludability
We are interested here in the idea, not the medium of
transmission. (We might transmit instructions or a recipe or a
design or a blueprint on paper or on a flash drive.)
Nonrivalry does not imply nonexcludability.
Excludability: the extent to which there are enforceable
property rights in a good, allowing the restriction of access or
use.
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Scarcity
Economics is
“the science which studies human behaviour as a
relationship between ends and scarce means which
have alternative uses.”
Lionel Robbins (1932)
The scarcity of “means” forces us to make trade-offs between
their alternative uses.
This notion of scarcity invokes rivalry in use.
For economists, the term scarcity is used to indicate a
requirement to make trade-offs.
Even so, the term has important ambiguities, and economists
are more likely to use more precise terms, such as rivalry or
excludability.
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Scarcity
Many ordinary economic goods are inherently scarce in the
sense of being rivalrous.
If I consume a peach, you cannot also consume it.
Some economic goods are less scarce in this sense.
If I watch a TV, you might also be able to watch it. But 50 of
us cannot watch it at the same time.
In a “knowledge-based economy”, some economic goods are
barely scarce in this sense. If I watch a film on cable TV, you
can too, and so can many others.
Of course such goods are still produced with scarce resources.
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Excludability
A good that is not rivalrous may be excludable. One may see
exclusion as a way to create scarcity of a nonrivalrous good.
E.g., cable TV.
Naturally the incentives of the cable TV company to provide
programming are affected by its ability to exclude viewers who
do not pay.
Ideas are perfectly nonrivalrous. In this sense, any existing idea
is not inherently scarce. However it may be rendered “scarce”
(i.e., exculdable) through patent or copyright.
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Scarcity of New Ideas
What about new ideas?
These are hard to come by: to produce new ideas we give up
other production.
In this sense, new ideas are scarce.
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
Ideas and Returns to Scale
Ideas → nonrivalry → increasing returns → problems with
pure competition
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Sunk Costs and Increasing Returns
Malthusian Model
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
1/3 2/3
Yt = At Kt Lt
We still have constant returns to scale in capital and labor.
Think about the technological parameter as another input to
production: if we double all inputs (A, K , L), then we
quadruple output.
That is, we have increasing returns to scale in all inputs.
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
Let’s work for a moment with a simplified version: a single
input, which we will call labor.
Yt = At Lyt
So we have constant returns to scale in labor alone, but
increasing returns to scale in all inputs (A, L).
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Addition to our theory: tell a story about A.
The growth rate of technology (gA ) depends on the amount of
labor we allocate to the production of new ideas.
Yt = At Lyt
∆At /At = z̄Lαt
Lyt + Lαt = N̄
Lαt = `¯N̄
(Here, ∆At ≡ At+1 − At .)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
The Romer model on the material standard of living, Y /N.
Y
Lyt
= At
N
N
¯
= At (1 − `)
(1)
(2)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
So, at point in time, allocating labor to the production of
knowledge is costly. But over time, it pays off in new knowlege
and higher levels of production:
∆At
= z̄Lαt
At
= z̄ `¯N̄
= ḡ
(3)
At = (1 + ḡ )At−1 = (1 + ḡ )t Ā0
(4)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Source: ERP (2010, Figure 10-3)
Malthusian Model
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
The growth of technology determines the growth rate of our
material standard of living:
Yt
Nt
¯
= At (1 − `)
yt =
¯
= (1 + ḡ ) Ā0 (1 − `)
¯ + ḡ )t
= Ā0 (1 − `)(1
t
(5)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Output per Person over Time
Malthusian Model
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Increase in Population
Malthusian Model
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Increase in R& D
Malthusian Model
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Word Processor Production Function
Malthusian Model
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Growth Accounting
Consider the production function.
1/3 2/3
Yt = At Kt Lyt
(6)
We can express this in rates of growth:
2
1
gYt = gAt + gKt + gLyt
3
3
So to understand the growth rate of output, we need to
understand the three growth rates on the right.
(7)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Similarly
2
1
gYt − gLt = gAt + (gKt − gLt ) + (gLyt − gLt )
| {z }
3 | {z } 3 | {z }
growth of Y /L
change in K /L
labor composition
(8)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Note the productivity slowdown after the first oil price shock.
(A decline is R&D spending has also been blamed.)
The productivity recovery in the 1990s is sometimes referred
to as the new economy. This change has been linked to IT
investment.
BUT: We cannot observe gA . We compute it as a residual.
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Our simple endogenous (Romer) growth model implies a
growth rate for technology:
gAt =
∆At
= z̄Lαt = z̄ `¯N̄ = ḡ
At
(9)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Our simplest neoclassical (Solow) growth model implies a
growth rate for capital:
gKt = ∆Kt /Kt = s̄Yt /Kt − d̄
(10)
which tells us that if we are on a balanced growth path, such
that Yt /Kt constant, then gKt is constant over time.
Along a balanced growth path we have
gK∗ = gY∗
Let us return to this in a bit.
(11)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Kaldor (1958) offered six famous “stylized facts” about the
growth of advanced industrialized economies:
[
1
Y
/N > 0 relatively constant in LR (exponential growth)
2
3
4
5
6
[
K
/N > 0 and K̂ roughly constant
[
Y
/K ≈ 0 in LR: capital output ratio fairly constant
d ≈ 0 in LR, so the rate of return to capital is
π/K
trendless. For example, the real interest rate on
government debt in the U.S. is trendless.
[
Ŷ and Y
/N vary a lot across countries
high π/Y associated with high I /Y
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
The number of production workers does not change in our
simple growth models. (Production workers are a constant
fraction of a constant population.)
gLyt = 0
=⇒
1
2
gY = ḡ + gK + 0
3
3
(12)
or, along a blanced growth path,
1
2
gY∗ = ḡ + gY∗ + 0
3
3
(13)
or
3
3
gY∗ = ḡ = z̄ `¯N̄
2
2
This is the growth rate along a balanced growth path.
(14)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Contrast this with our simplest Romer model, where gY∗ = ḡ .
The difference arises from our reintroduction of capital
accumulation. Now, in addition to the direct effect of higher
productivity, we have an indirect effect: higher productivity
leads to higher capital stock, which also raises output.
It is still the case that capital is not the engine of growth.
But capital accumulation amplifies the effect of productivity
growth.
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
Solow II: Combing Romer and Solow I
Econ 301 students are not responsible for the Solow-Romer
algebra that follows.
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Two equations from our simplest neoclassical (Solow) growth
model:
1/3 2/3
Yt = At Kt Lyt
(15)
∆Kt = s̄Yt − d̄Kt
(16)
Changes: At can vary over time; Lyt instead of all labor.
Three equations from Romer model:
∆At /At = z̄Lαt
Lyt + Lαt = N̄
Lαt = `¯N̄
(17)
(18)
(19)
At any time t, the five endogenous variables (“unknowns”)
are: Yt , Kt+1 , At+1 , Lyt , and Lαt .
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Output per Person
yt = Yt /L
= Yt (1 − `)/Lyt
=
where k = K /Ly .
1/3
At kt (1
− `)
(20)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
The capital output ratio along a balanced growth path is
constant.
gY∗ = gK∗
gY∗ = s(Yt∗ /Kt∗ ) − d̄
(21)
Solve for (K /Y )∗ :
(K /Y )∗ = s̄/(gY∗ + d̄)
(22)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Next let us solve for income per worker. Start from the
production relation
1/3 2/3
Yt = At Kt Lyt
we can rewrite this as
2/3
1/3
Yt
Kt
= At
Lyt
Yt
or
1/2
Yt
Kt
3/2
= At
Lyt
Yt
or
1/2
Kt
Yt
3/2
= (1 − `)At
Yt
N̄
Along a balanced growth path we therefore have
1/2
∗
s̄
3/2
Yt /N̄ = (1 − `)At
gY∗ + d̄
(23)
(24)
(25)
(26)
(27)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Transition Dynamics: Increase in s
Source: Jones (2011)
Malthusian Model
References
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Saving and Growth
So we see:
raising the saving rate can only temporarily raise growth.
But now we have another way to save.
As in the simplest Romer model, we can allocate more labor to
research and development. This still raises the growth rate,
even in the long run.
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Delong offers a simple Malthusian variant. http://delong.
typepad.com/sdj/2008/02/econ-101b-feb-1.html
Resources R are fixed in quantity but augmented by
technological progress.
c = Ê = gR
ER
(28)
Resources are an input into the production function:
Y = K α (ER)β L1−α−β
(29)
Y /L = (K /L)α (ER/L)β
(30)
or
Steady state factor use per “capita” will be constant, so Y /L
will too.
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
This suggests
L̂ss = gR
Population growth must move to a rate justified by
technological progress. What is the mechanism?
(31)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Adopt a Malthusian mechanism: population grows faster
above subsistence level y ∗ .
L̂ = γ(Y /L − y ∗ )
(32)
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
In the steady state, the population growth rate is constant.
L̂ss = γ(Y /L − y ∗ )
(33)
(Y /L)ss = y ∗ + L̂ss /γ
(34)
so
is also constant. This means
Ŷss = L̂ss
(35)
(Y /L)ss = y ∗ + gR /γ
(36)
Furthermore
We live about subsistence only to the extent that technical
change allows.
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Now from the production function
ˆ + (1 − α − β)L̂
Ŷ = αK̂ + β ER
(37)
[
0 = αKd
/L + β ER/L
(38)
so
which holds as the factor ratios are constant in the steady
state.
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Capital is accumulated in standard fashion:
K̂ = sY /K − δ
(39)
\
Since (K
/L)ss = 0, we must have
L̂ss = sY /K − δ
(40)
γ((Y /L)ss − y ∗ ) = s(Y /K )ss − δ
(41)
or
Jones Chapter 6: Romer Model
We can solve for
Combining the simple Solow and Romer Models
Malthusian Model
References
Y
= y ∗ + gR /γ
(42)
L
http://delong.typepad.com/delongslides/2008/02/
the-malthusian.html Only technological change holds us
above the subsistence level.
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
How do we escape from the Malthusian trap? DeLong says
the most obvious way is to raise y ∗ , but he wonders about the
effectiveness of this strategy. http://delong.typepad.com/
delongslides/2008/02/the-malthusian.html
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
References
Escape from the Trap: Permanent or Temporary
Source:
http://delong.typepad.com/delongslides/2008/02/
greg-clarks-wor.html
Jones Chapter 6: Romer Model
Combining the simple Solow and Romer Models
Malthusian Model
Romer, Paul (1990, October). “Endogenous Technological
Change.” Journal of Political Economy 98(part 2),
S71–S102.
Vinge, Vernor (1993, Winter). “The Technological
Singularity.” Whole Earth Review 81.
References