SOLOW GROWTH MODEL WIFH MATHS

1. Economic Growth
Spring 2013
1 The Solow growth model
Basic building blocks of the model
• A production function
Yt = F (Kt, Lt, At)
– This is a hugely important concept
– Once we assume this, then we are saying that any growth has to be the result of more
capital, more people or better technology!
• Constant returns to scale
F (λKt, λLt) = λF (Kt, Lt)
• Often we’ll look at a special case: Cobb-Douglas production function with labour-augmenting
technology
F (K, L) = Kα
(AL)1−α
• Sometimes people formulate the function as
F (K, L) = AKα
L1−α
(i.e. “neutral” rather than “labour-augmenting” technical change). With a Cobb-Douglas
functional form, it doesn’t make much difference.
• Exogenous population growth
Lt = (1 + n) Lt−1
This is actually a really important assumption
• A constant rate of capital depreciation: δ
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2. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
• An exogenous savings rate
St = sYt
Ct = (1 − s) Yt
• A closed economy, so savings equals investment
It = St
• Exogenous technological progress
At = (1 + g) At−1
Mechanics of the model
• Suppose that there is no technological progress:
– How much capital will the economy accumulate?
– Will the economy grow? How much? For how long?
• Assume g = 0 for now and normalize A = 1
• Express production function in per capita terms
yt ≡
Yt
Lt
=
1
Lt
F (Kt, Lt)
= F
Kt
Lt
, 1
≡ f (kt)
where kt ≡ Kt
Lt
• Note that we use “per-capita” and “per-worker” interchangeably, but workforce
population
can vary over
time and across countries
• The capital stock evolves according to
Kt+1 = (1 − δ) Kt + It
= (1 − δ) Kt + sYt
∆Kt+1 = −δKt + sYt
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3. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
• In per-capita terms
∆kt+1 ≡ kt+1 − kt
=
(1 − δ) Kt + sYt
Lt+1
− kt
=
(1 − δ) Kt + sYt
Lt
Lt
Lt+1
− kt
= [(1 − δ) kt + syt]
1
1 + n
− kt
=
syt − (δ + n) kt
1 + n
=
sf (kt) − (δ + n) kt
1 + n
• Interpretation
• Graph: sf (kt) and (δ + n) kt
• The steady state and convergence
• No long-term growth! (GDP grows, GDP per capita does not)
• Growth during transition
• Examples:
– Increase in the savings rate
– Increase in the rate of population growth
– A one-time improvement in technology
Steady state with Cobb-Douglas
• For Cobb-Douglas case we can compute steady-state capital and output explicitly
• Production function is
Yt = Kα
t L1−α
t
yt =
Kα
t L1−α
t
Lt
= kα
t
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4. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
• The steady state is defined by
ss : ∆kt+1 = 0
so
skα
ss − (δ + n) kss = 0
s
δ + n
= k1−α
ss
kss =
s
δ + n
1
1−α
and
yss =
s
δ + n
α
1−α
so we get an expression for steady state GDP per capita in terms of parameters
The Golden Rule
Question How much should society be saving?
Answer According to one possible criterion known as the Golden Rule, society should have the
“Golden Rule” savings rate. If this sounds a bit tautological it’s because it is. It becomes more
concrete once we describe what the Golden Rule criterion is.
What do we mean by “should”? There are different possible criteria one could use to define
what should be done. The Golden Rule criterion is a very loose interpretation of the moral principle
“one should treat others as one would like others to treat oneself”. Applied to the question of the
savings rate, it can be thought to mean that societies should save in such a way as to maximize
the level of consumption in the steady state. Whether this is a good interpretation of the moral
principle is more of a literary question than an economic one, but let’s accept it for now. One
justification for this objective is that if you were going to be born into a society that is and will
remain in steady state, the Golden Rule society will be the one where you achieve the highest
utility.
Steady state consumption If the economy is at a steady state, consumption will be
css = (1 − s) yss
css depends on s in two ways:
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5. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
• Directly: the more you save, the less you consume
• Indirectly: the more you save, the higher the steady state capital stock, the higher the output
out of which you can consume
Cobb-Douglas case For the special case of a Cobb-Douglas production function
yss =
s
δ + n
α
1−α
Here we can see the indirect effect: Higher s means higher yss. Therefore steady state consumption
is
css (s) = (1 − s)
s
δ + n
α
1−α
so we can find the maximum by taking a first order condition:
−
s
δ + n
α
1−α
+ (1 − s)
α
1 − α
s
δ + n
α
1−α
−1
1
δ + n
= 0
−1 + (1 − s)
α
1 − α
s
δ + n
−1
1
δ + n
= 0
1 − s
s
α
1 − α
= 1
s = α
General Case Beyond the Cobb-Douglas case, a more general optimality condition for the
Golden-Rule-optimal savings rate comes from the following reasoning.
css (s) = (1 − s) f (kss (s)) (1)
FOC:
−f (kss) + (1 − s) f (kss (s)) ·
∂kss (s)
∂s
= 0 (2)
Now use the steady state condition:
sf (kss) = (δ + n) kss (3)
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6. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
to compute ∂kss
∂s
:
sf (kss (s)) = (δ + n) kss (s)
f (kss (s)) + sf (kss (s)) ·
∂kss (s)
∂s
= (δ + n)
∂kss (s)
∂s
∂kss (s)
∂s
=
−f (kss (s))
sf (kss (s)) − (δ + n)
(4)
Replace (4) into the FOC (2):
−f (kss) + (1 − s) f (kss (s)) ·
−f (kss (s))
sf (kss (s)) − (δ + n)
= 0
−
f (kss)
sf (kss (s)) − δ
[sf (kss (s)) − (δ + n) + (1 − s) f (kss (s))] = 0
f (kss (s)) = (δ + n) (5)
Let’s go over what this means because it’s not (just) a bunch of maths. We start from equation (1),
which says how much we consume in steady state. This depends on the savings rate directly and
indirectly through the effect of s on kss. We then take first order conditions to find an optimum
and come up with (2). This says that the direct effect, which is negative, is just proportional to
output: the higher the output level, the more we reduce consumption when we increase savings
rates. The indirect effect depends on
1. how much output would increase if we increase the capital stock (that’s why f (kss (s))
appears in the expression)
2. how much more capital we would have if we saved more (that’s why ∂kss(s)
∂s
appears in the
expression)
3. how much of the extra output would we in fact be consuming (that’s why 1 − s appears in
the expression)
This is not the end of it, because we still don’t know how much more capital we are going to have
if we increase the savings rate: we just have the expression ∂kss(s)
∂s
and we need to solve for that.
That’s where we use the fact that in steady state sf (kss) = (δ + n) kss and take derivatives on
both sides to get to (4). We then replace this in the first order conditions and get to (5).
Interpretation of the first order condition Equation (5) has a neat interpretation. Suppose
a society raised its level of savings in such a way that the steady state capital stock were higher.
Would that society have higher consumption? The answer depends on comparing f (kss) against
δ + n. Why? In a steady state, an economy will be saving/investing just enough to make up
for depreciation and population growth. That is what makes a steady state steady! In order to
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7. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
increase the steady state capital stock by a little bit (call this ∆k) , the economy would have to
increase the absolute amount of investment by just enough to make up for the depreciation and
population growth for this extra capital, every period, i.e. invest an extra (δ + n) ∆k. How much
extra output would society get out of this extra capital? f (kss) · ∆k. When is it the case that
this extra output is enough to cover the required extra investment and have a little extra left over
to consume? Whenever
f (kss) ∆k > (δ + n) ∆k
⇔ f (kss) > δ + n
Therefore it makes sense, according to the Golden Rule, to increase s (and therefore kss) if and
only if
f (kss) > δ + n
The explains condition (5)
• For the Cobb-Douglas case: compute the marginal product of capital in a steady state
f (kss) = αkα−1
ss
= α
s
δ + n
α−1
1−α
= α
δ + n
s
• In order for extra savings to increase steady state consumption, we need
MPK > δ + n
⇔ s < α
Example: economic growth in the USSR in the 1930s
Markets
• So far, “engineering” approach
• Now suppose there is a market for labour and for capital services
– Note metaphor of firms renting capital from households
– Distinction between profits and return on capital
• Questions:
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8. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
– How many workers will firms want to hire?
– How much capital will firms want to use?
– What ensures that what the firms want coincides with what is actually available?
– What will be the price of labour (the wage)?
– What will be the (rental) price of capital?
• Firms:
max
Ki,Li
F (Ki, Li) − wLi − rK
Ki
FOC:
FK (Ki, Li) − rK
= 0
FL (Ki, Li) − w = 0
• Graphical illustration
• In the Cobb-Douglas case:
αKα−1
i L1−α
i = rK
(1 − α) Kα
i L−α
i = w
• Taking a ratio
1 − α
α
Ki
Li
=
w
rK
Ki
Li
=
w
rK
α
1 − α
(6)
so all firms use the same ratio of capital and labour
• If workers are expensive relative to capital, firms use more capital per worker (and vice-versa)
• (6) implies that they all must have a capital-labour ratio that equals that aggregate, i.e.
Ki
Li
=
K
L
• This lets us find out the factor prices
rK
= αKα−1
L1−α
= αkα−1
w = (1 − α) Kα
L−α
= (1 − α) kα
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9. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
• Discuss how market-clearing makes these prices come about
• Capital deepening (i.e. increases in K
L
) increases wages and depresses the rental rate of
capital
• We can also compute the total compensation of all workers
wL = (1 − α) Kα
L−α
· L = (1 − α) Kα
L1−α
= (1 − α) Y
and the total capital-income of capital-owners
rK
K = αKα−1
L1−α
· K = αKα
L1−α
= αY
• Constant factor shares. See graph. Is this still true?
• Factor income sums up to total output
• No pure profits for firms (profits = capital income)
• Interest rates:
– If you lend to someone else, tomorrow you get
1 + rt+1
– If you build capital and rent it out, tomorrow you get the rental rate plus your depre-
ciated capital
(1 − δ) + rK
t+1
– Indifference requires
1 + rt+1 = 1 − δ + rK
t+1
rt+1 = rK
t+1 − δ
= FK − δ
• We’ll talk more about the condition
rt+1 = FK − δ
when we talk about investment
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10. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
Technological progress
• “Steady state”: no tech progress ⇒ no growth
• Now re-introduce technological progress. Focus on Cobb-Douglas case with labor-augmenting
technology
Yt = Kα
t (AtLt)1−α
At+1 = (1 + g) At
• Define “efficiency units of labour”
˜L = AL
and output and capital per “efficiency unit”
˜yt ≡
Yt
˜Lt
=
Kα
t (AtLt)1−α
AtLt
=
Kt
AtLt
α
AtLt
AtLt
1−α
=
Kt
AtLt
α
≡ ˜kα
t ≡ f ˜kt
• ˜k is the amount of capital per efficiency unit of labour.
• The growth rate of ˜L is given by
˜Lt+1
˜Lt
− 1 =
At+1Lt+1
AtLt
− 1
=
At (1 + g) Lt (1 + n)
AtLt
− 1
= (1 + g) (1 + n) − 1
= g + n + gn
≈ g + n
• Rate of “capital-per-efficiency unit of labour” accumulation computed the same way as “cap-
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11. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
ital per worker” accumulation
∆˜kt+1 =
sf ˜kt − (δ + n + g) ˜kt
1 + n + g
• Same logic for computing steady state in Cobb-Douglas case:
˜kss =
s
δ + n + g
1
1−α
and
˜yss =
s
δ + n + g
α
1−α
• But “steady state” is not so steady! There is growth!
• The key is increasing number of “efficiency units” per worker
• Since everything is constant “per efficiency unit”, everything grows at the same rate: “balanced
growth”
Putting numbers on the parameters
• Parameters:
– α, δ, s, n, g
• Data from NIPA
• Savings rate: s ∈ [0.15, 0.20]
– What about savings=investment? See graph
– Investment by government
– Set s ≈ 0.2
• Share of labour in GDP (from previous graph): 1 − α ≈ 0.65 , so α ≈ 0.35
• Rate of population growth n ≈ 0.01
• Rate of technological progress: g ≈ 0.02.
– How do we know this? According to model, this will be the rate of growth of GDP per
capita when we have balanced growth
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12. ECON 52, Winter 2012 1 THE SOLOW GROWTH MODEL
• Depreciation: δ ≈ 0.06
– This is hard to measure, because it’s different for different kinds of capital. BEA uses
– Buildings δ ≈ 0.02
– Equipment δ ≈ 0.15
– Computers δ ≈ 0.3
• Capital-output ratio in the model
K
Y
=
s
δ + n + g
≈ 2.22
• Interest rates in the model
r = FK − δ
Compute FK:
FK = αKα−1
(AL)1−α
= α
K
AL
α−1
= α
Y
K
so
r = α
Y
K
− δ
= α
δ + n + g
s
− δ ≈ 9.8%
• Higher than the rates we typically observe
– Mismeasurement?
– Role of risk?
Growth Accounting
• How much growth do we attribute to technological progress, capital accumulation and pop-
ulation growth?
• Start from production function:
Yt = F (Kt, Lt, At)
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13. ECON 52, Winter 2012 2 EMPIRICAL EVIDENCE ON GROWTH
• Use chain rule:
˙Yt = YKt
˙Kt + YLt
˙Lt + YAt
˙At
• Notation:
˙Xt ≡
dXt
dt
• Divide by Yt to express in percentage terms:
gY ≡
˙Yt
Yt
=
YKt
Yt
˙Kt +
YLt
Yt
˙Lt +
YAt
Yt
˙At
=
YKt Kt
Yt
˙Kt
Kt
+
YLt Lt
Yt
˙Lt
Lt
+
YAt At
Yt
˙At
At
(assume Cobb-Douglas production function)
= αgK + (1 − α) gL + (1 − α) gA
= capital share × % growth of capital
+ labour share × % growth of labour force
+ Solow residual
• Growth of Singapore and Hong Kong (table from Williamson textbook page 231)
2 Empirical evidence on growth
• One view:
– Technology A is the same in all countries
– Countries are not at steady states
– Some countries have more capital than others
– Growth consists of countries approaching the steady state
• This hypothesis is decisively rejected by the evidence!
Convergence
• Compute the growth rate of GDP-per-efficiency-unit of labour (see exercise)
g˜y = αg˜k
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14. ECON 52, Winter 2012 2 EMPIRICAL EVIDENCE ON GROWTH
• Recall formula for growth in capital-stock-per-effective-unit-of-labour
∆˜kt+1 =
sf ˜kt − (δ + n + g) ˜kt
1 + n + g
• Reexpress in terms of growth rate rather than aboslute change
g˜k =
s
f(˜kt)
˜kt
− (δ + n + g)
1 + n + g
• Replacing:
g˜y = α
s
f(˜kt)
˜kt
− (δ + n + g)
1 + n + g
• If production function is Cobb-Douglas
f ˜k = ˜kα
f ˜k
˜k
= ˜kα−1
= f ˜k
α−1
α
= ˜y
α−1
α
• Replacing
g˜y = α
s˜y
α−1
α − (δ + n + g)
1 + n + g
• Conclusion: countries with lower GDP-per-effective-worker should grow faster!
• If we assume technology is the same in all countries, countries with lower GDP-per-capita
should grow faster!
• By how much? We want to say something like “if country A is x% poorer than country B, it
should grow z% faster”
• Use Taylor approximation
g˜y ≈ gss +
∂g˜y
∂ log ˜y ˜y=˜yss
[log ˜y − log ˜yss]
=
∂g˜y
∂ log ˜y ˜y=˜yss
[log ˜y − log ˜yss] (7)
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15. ECON 52, Winter 2012 2 EMPIRICAL EVIDENCE ON GROWTH
• Now compute the derivative:
g˜y =
α
1 + n + g
s˜y
α−1
α − (δ + n + g)
=
α
1 + n + g
s exp
α − 1
α
log ˜y − (δ + n + g)
∂g˜y
∂ log ˜y
=
α
1 + n + g
s exp
α − 1
α
log ˜y
α − 1
α
=
α − 1
1 + n + g
s˜y
α−1
α
• In steady state we know that
s˜y
α−1
α = (δ + n + g)
so evaluating the derivative at the steady state leads to
∂g˜y
∂ log ˜y ˜y=˜yss
=
α − 1
1 + n + g
(δ + n + g)
so replacing in (7):
g˜y ≈
α − 1
1 + n + g
(δ + n + g) [log ˜y − log ˜yss]
• This formula tells us how much growth varies when a country is away from steady state
• By extension, also how much growth varies across different countries at different distances
to steady state.
• Putting numbers on parameters
α − 1
1 + n + g
(δ + n + g) ≈ −5.7%
• Countries should converge approximately 5.7% of the way to steady state every year
Evidence on Convergence
• If we plot growth rates against log(GDP per capita), we should find a slope of that magnitude
• Convergence graphs
– Full sample: slope=0
– OECD: slope= -0.11
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16. ECON 52, Winter 2012 2 EMPIRICAL EVIDENCE ON GROWTH
– W. Europe slope = -0.13
– US states
– Other regions
• Evidence on convergence is mixed at best
• “Conditional” convergence: maybe countries converge to different points
Direct measurement
• Measure the capital stock for each country
• Assume technology is the same in all countries
• Predict that GDP per capita in country i will be
yPredicted
i = kα
i
• Compare this prediction to actual measured GDP per capita
• (see graph)
• Prediction is far off and in a systematic way: based on their capital levels alone, poor countries
should not be so poor
• Maybe the problem is measuring the capital stock?
Evidence from interest rates
• Suppose we don’t trust our measurement of the capital stock
• Ask instead:
– what should k be to account for cross-country differences in GDP?
– if that were the level of k, what interest rates should we observe? are those reasonable?
y = kα
yA
yB
=
kA
kB
α
yA
yB
1
α
=
kA
kB
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17. ECON 52, Winter 20123 WHAT DOES TFP DEPEND ON? WHERE DOES GROWTH IN TFP COME FROM?
• See chart
• Implications for interest rates:
r = FK − δ
= αkα−1
− δ
= α
Y
K
− δ
• For the US,
Y
K
≈ 2.22
α ≈ 0.35
δ ≈ 0.06
⇒ r ≈ 9.8%
• Replacing the implied Y
K
levels of other countries (see chart)
• Extreme capital scarcity implies extremely high marginal product of capital!
Conclusion
• Differences in capital levels are not the full explanation of differences in GDP across countries
• Policy implications:
– Poor countries will not catch up by rising investment ONLY
– (But investment that brings in new technologies can be important)
3 What does TFP depend on? Where does growth in TFP
come from?
1. Research and development
2. Technological catch-up for countries away from technological frontier
3. “Learning by doing”. Learning by doing what exactly?
4. “Human capital”. Workers not all the same
• How much does this matter? Hall & Jones (1999)
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18. ECON 52, Winter 20123 WHAT DOES TFP DEPEND ON? WHERE DOES GROWTH IN TFP COME FROM?
• Data on wages by education levels to see how much “human capital” you get per year
of education
• Data on education levels across countries to see how much total human capital different
countries have
• Data: education differences matter, physical capital accumulation matters but a large
chunk is “unexplained” (see graph)
5. Geography: Landlocked and tropical poorer that coastal and temperate (see graph)
• Sachs (2001) proposes explanations. See also Diamond (1997): Guns, Germs and Steel.
(a) Crop yields different by climate.
• But agriculture is not a large fraction of GDP in rich countries
(b) Tropical diseases reduce productivity
• Temperate-zone diseases are/have become less burdensome
(c) Availability of energy resources (especially coal in the beginning of industrialization)
• These factors amplified by
(a) Technologies that don’t transfer well across climates
(b) Demography
(c) Political power
6. Political/social institutions
• Strong correlation between GDP and measures of political transparency, respect for
property rights, rule of law, social trust, etc.
• See graph
• Causality?
• Weaker evidence on more detailed policies
7. Misallocation
• Barriers to entry / expansion
– World bank project on costs of starting a business
• Restrictions on FDI
• Monopolies
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19. ECON 52, Winter 20123 WHAT DOES TFP DEPEND ON? WHERE DOES GROWTH IN TFP COME FROM?
• Different taxes/subsidies for different firms
• Imperfect capital markets
• Imperfect contract enforcement
– Bloom et al. (2013): the main determinant of firm size in India is the number of
male family members of owners, not the quality of management.
• Labour market regulations
• Discrimination
– Hseih et al. (2013): 1960: 94 percent of doctors and lawyers were white men. 2008:
62 percent. If the difference is due to reduced discrimination, improved allocation
of talent could account for 15-20% of US economic growth.
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