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Growth
ECON 30020: Intermediate Macroeconomics
Prof. Eric Sims
University of Notre Dame
Fall 2016
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Economic Growth
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When economists say “growth,” typically mean average rate
of growth in real GDP per capita over long horizons
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Long run: frequencies of time measured in decades
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Not period-to-period fluctuations in the growth rate
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“Once one begins to think about growth, it is difficult to think
about anything else” – Robert Lucas, 1995 Nobel Prize winner
2 / 51
US Real GDP per capita
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
50
55
60
65
70
75
80
85
Real GDP Per Capita
90
95
00
05
10
15
Linear Trend
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Summary Stats
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Average (annualized) growth rate of per capita real GDP:
1.8%
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Implies that the level of GDP doubles roughly once every 40
years
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Growing just 0.2 percentage points faster (2% growth rate):
level doubles every 35 years
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Rule of 70: number of years it takes a variable to double is
approximately 70 divided by the growth rate
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Consider two countries that start with same GDP, but country
A grows 2% per year and country B grows 1% per year. After
100 years, A will be 165% richer!
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Small differences in growth rates really matter over long
horizons
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Key Question
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What accounts for this growth?
In a mechanical sense, can only be two things:
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Growth in productivity: we produce more output given the
same inputs
Factor accumulation: more factors of production help us
produce more stuff
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Two key factors of production on which we focus are capital
and labor
Labor input per capita is roughly trendless – empirically not a
source of growth in per capita output
The key factor of production over the long run is capital:
physical stuff that must itself be produced that in turn helps
us produce more stuff (e.g. machines).
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Which is it? Productivity or capital accumulation? What are
policy implications?
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What accounts for differences in standards of living across
countries? Productivity or factor accumulation?
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Stylized Facts: Time Series
1. Output per worker grows at an approximately constant rate
over long periods of time picture
2. Capital per worker grows at an approximately constant rate
over long periods of time picture
3. The capital to output ratio is roughly constant over long
periods of time picture
4. Labor’s share of income is roughly constant over long periods
of time picture
5. The return to capital is roughly constant over long periods of
time picture
6. The real wage grows at approximately the same rate as output
per worker over long periods of time picture
6 / 51
Stylized Facts: Cross-Section
1. There are large differences in income per capita across
countries table
2. There are some examples where poor countries catch up
(growth miracles), otherwise where they do not (growth
disasters) table
3. Human capital (e.g. education) strongly correlated with
income per capita table
7 / 51
Solow Model
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Model we use to study long run growth and cross-country
income differences is the Solow model, after Solow (1956)
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Model capable of capturing stylized facts
Main implication of model: productivity is key
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Productivity key to sustained growth (not factor accumulation)
Productivity key to understanding cross-country income
differences (not level of capital)
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Important implications for policy
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Downside of model: takes productivity to be exogenous.
What is it? How to increase it?
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Model Basics
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Time runs from t (the present) onwards into infinite future
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Representative household and representative firm
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Everything real, one kind of good
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Production Function
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Production function:
Yt = At F (Kt , Nt )
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Kt : capital. Must be itself produced, used to produce other
stuff, does not get completely used up in production process
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Nt : labor
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Yt : output
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At : productivity (exogenous)
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Think about output as units of fruit. Capital is stock of fruit
trees. Labor is time spent picking from the trees
10 / 51
Properties of Production Function
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Both inputs necessary: F (0, Nt ) = F (Kt , 0) = 0
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Increasing in both inputs: FK (Kt , Nt ) > 0 and
F N ( K t , Nt ) > 0
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Concave in both inputs: FKK (Kt , Nt ) < 0 and
FNN (Kt , Nt ) < 0
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Constant returns to scale: F (qKt , qNt ) = qF (Kt , Nt )
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Capital and labor are paid marginal products:
wt = At FN (Kt , Nt )
Rt = At FK (Kt , Nt )
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Example production function: Cobb-Douglas:
F (Kt , Nt ) = Ktα Nt1−α ,
0<α<1
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Consumption, Investment, Labor Supply
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Fruit can either be eaten (consumption) or re-planted in the
ground (investment), the latter of which yields another tree
(capital) with a one period delay
Assume that a constant fraction of output is invested,
0 ≤ s ≤ 1. “Saving rate” or “investment rate”
Means 1 − s of output is consumed
Resource constraint:
Yt = Ct + It
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Abstract from endogenous labor supply – labor supplied
inelastically and constant
Current capital stock is exogenous – depends on past decisions
Capital accumulation, 0 < δ < 1 depreciation rate:
Kt +1 = It + (1 − δ)Kt
12 / 51
Equations of Model
Yt = At F (Kt , Nt )
Yt = Ct + It
Kt +1 = It + (1 − δ)Kt
Ct = (1 − s )Yt
It = sYt
wt = At FN (Kt , Nt )
Rt = At FK (Kt , Nt )
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Six endogenous variables (Yt , Ct , Kt +1 , It , wt , Rt ) and three
exogenous variables (At , Kt , Nt )
13 / 51
Central Equation
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First four equations can be combined into one:
Kt +1 = sAt F (Kt , Nt ) + (1 − δ)Kt
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Define lowercase variables as “per worker.” kt =
per-worker terms:
Kt
Nt .
In
kt +1 = sAt f (kt ) + (1 − δ)kt
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One equation describing dynamics of kt . Once you know
dynamic path of capital, you can recover everything else
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Plot of the Central Equation of the Solow Model
𝑘𝑡+1 = 𝑘𝑡
𝑘𝑡+1
𝑘𝑡+1 = 𝑠𝐴𝑡 𝑓(𝑘𝑡 ) + (1 − 𝛿)𝑘𝑡
𝑘∗
𝑘∗
𝑘𝑡
15 / 51
The Steady State
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The steady state capital stock is the value of capital at which
kt +1 = kt
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We call this k ∗
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Graphically, this is where the curve (the plot of kt +1 against
kt ) crosses the 45 degree line (a plot of kt +1 = kt )
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Via assumptions of the production function along with
auxiliary assumptions (the Inada conditions), there exists one
non-zero steady state capital stock
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The steady state is “stable” in the sense that for any initial
kt 6= 0, the capital stock will converge to this point
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“Once you get there, you sit there”
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Since capital governs everything else, all other variables go to
a steady state determined by k ∗
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Work through dynamics
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Algebraic Example
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Suppose f (kt ) = ktα . Suppose At is constant at A∗ . Then:
sA∗
k =
δ
∗
∗ ∗α
y =A k
∗
1−1 α
c ∗ = (1 − s )A∗ k ∗ α
i ∗ = sA∗ k ∗α
R ∗ = αA∗ k ∗α−1
w ∗ = (1 − α )A∗ k ∗ α
17 / 51
Dynamic Effects of Changes in Exogenous Variables
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Want to consider the following exercises:
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What happens to endogenous variables in a dynamic sense
after a permanent change in A∗ (the constant value of At )?
What happens to endogenous variables in a dynamic sense
after a permanent change in s (the saving rate)?
For these exercises:
1. Assume we start in a steady state
2. Graphically see how the steady state changes after the change
in productivity or the saving rate
3. Current capital stock cannot change (it is
predetermined/exogenous). But kt 6= k ∗ . Use dynamic
analysis of the graph to figure out how kt reacts dynamically
4. Once you have that, you can figure out what everything else is
doing
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Permanent Increase in A∗
𝑘𝑡+1 = 𝑘𝑡
𝑘𝑡+1
𝑘𝑡+1 = 𝑠𝐴1,𝑡 𝑓(𝑘𝑡 ) + (1 − 𝛿)𝑘𝑡
𝑘1∗
𝑘𝑡+2
𝑘𝑡+1
𝑘𝑡+1 = 𝑠𝐴0,𝑡 𝑓(𝑘𝑡 ) + (1 − 𝛿)𝑘𝑡
𝑘𝑡 = 𝑘0∗ 𝑘𝑡+1 𝑘1∗
𝑘𝑡
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Impulse Response Functions: Permanent Increase in A∗
𝑦𝑡
𝑘𝑡
𝑦1∗
𝑘1∗
𝑘0∗
𝑐𝑡
𝑦0∗
𝑡𝑖𝑚𝑒
𝑡
𝑖𝑡
𝑐1∗
𝑐0∗
𝑤𝑡
𝑡
𝑡𝑖𝑚𝑒
𝑖1∗
𝑖0∗
𝑡
𝑡𝑖𝑚𝑒
𝑡
𝑡𝑖𝑚𝑒
𝑅𝑡
𝑤1∗
𝑤0∗
𝑅0∗
𝑡
𝑡𝑖𝑚𝑒
𝑡
𝑡𝑖𝑚𝑒
20 / 51
Permanent Increase in s
𝑘𝑡+1 = 𝑘𝑡
𝑘𝑡+1
𝑘𝑡+1 = 𝑠1 𝐴𝑡 𝑓(𝑘𝑡 ) + (1 − 𝛿)𝑘𝑡
𝑘1∗
𝑘𝑡+2
𝑘𝑡+1
𝑘𝑡+1 = 𝑠0 𝐴𝑡 𝑓(𝑘𝑡 ) + (1 − 𝛿)𝑘𝑡
𝑘𝑡 = 𝑘0∗ 𝑘𝑡+1 𝑘1∗
𝑘𝑡
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Impulse Response Functions: Permanent Increase in s
𝑘𝑡
𝑦𝑡
𝑘1∗
𝑦1∗
𝑘0∗
𝑦0∗
𝑡𝑖𝑚𝑒
𝑡
𝑖𝑡
𝑐𝑡
𝑡
𝑡𝑖𝑚𝑒
𝑖1∗
𝑐1∗
𝑐0∗
𝑤𝑡
𝑖0∗
𝑡
𝑡𝑖𝑚𝑒
𝑡𝑖𝑚𝑒
𝑡
𝑅𝑡
𝑤1∗
𝑤0∗
𝑅0∗
𝑅1∗
𝑡
𝑡𝑖𝑚𝑒
𝑡
𝑡𝑖𝑚𝑒
22 / 51
Discussion
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Neither changes in A∗ nor s trigger sustained increases in
growth
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Each triggers faster growth for a while while the economy
accumulates more capital and transitions to a new steady state
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Aren’t we supposed to be studying growth? In the long run,
there is no growth in this model – it goes to a steady state!
We’ll fix that. You can kind of see, however, that sustained
growth must come from increases in productivity. Why?
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No limit on how high A can get – it can just keep increasing.
Upper bound on s
Repeated increases in s would trigger continual decline in Rt ,
inconsistent with stylized facts
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Bottom line: sustained growth must be due to productivity
growth, not factor accumulation. You can’t save your way to
more growth
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Key assumption: diminishing returns to capital
23 / 51
Golden Rule
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What is the “optimal” saving rate, s?
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Utility from consumption, not output
Higher s has two effects – the “size of the pie” and the
“fraction of the pie”:
I More capital → more output → more consumption (bigger
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size of the pie)
Consume a smaller fraction of output → less consumption (eat
a smaller fraction of the pie)
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Golden Rule: value of s which maximizes steady state
consumption, c ∗
I s = 0: c ∗ = 0
I s = 1: c ∗ = 0
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Implicity characterized by A∗ f 0 (k ∗ ) = δ. Graphical intuition.
24 / 51
Growth
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Wrote down a model to study growth
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But model converges to a steady state with no growth
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Isn’t that a silly model?
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It turns out, no
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Can modify it
25 / 51
Augmented Solow Model
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Production function is:
Yt = At F (Kt , Zt Nt )
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Zt : labor-augmenting productivity
Zt Nt : efficiency units of labor
Assume Zt and Nt both grow over time (initial values in
period 0 normalized to 1):
Zt = (1 + z )t
Nt = ( 1 + n ) t
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z = n = 0: case we just did
Zt not fundamentally different from At . Mathematically
convenient to use Zt to control growth while At controls level
of productivity
26 / 51
Per Efficiency Unit Variables
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Define kbt = ZKt Nt t and similarly for other variables. Lower case
variables: per-capita. Lower case variables with “hats”: per
efficiency unit variables
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Can show that modified central equation of model is:
kbt +1 =
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h
i
1
sAt f (kbt ) + (1 − δ)kbt .
(1 + z )(1 + n)
Practically the same as before
27 / 51
Plot of Modified Central Equation
𝑘𝑘�𝑡𝑡+1 = 𝑘𝑘�𝑡𝑡
𝑘𝑘�𝑡𝑡+1
𝑘𝑘� ∗
𝑘𝑘�𝑡𝑡+1 =
𝑘𝑘� ∗
1
�𝑠𝑠𝐴𝐴𝑡𝑡 𝑓𝑓�𝑘𝑘�𝑡𝑡 � + (1 − 𝛿𝛿)𝑘𝑘�𝑡𝑡 �
(1 + 𝑧𝑧)(1 + 𝑛𝑛)
𝑘𝑘�𝑡𝑡
28 / 51
Steady State Growth
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Via similar arguments to earlier, there exists a steady state kb∗
at which kbt +1 = kbt . Economy converges to this point from
any non-zero initial value of kbt
Economy converges to a steady state in which per efficiency
unit variables do not grow. What about actual and per capita
variables? If kbt +1 = kbt , then:
Kt
Kt + 1
=
Z t + 1 Nt + 1
Z t Nt
Kt + 1
Z t + 1 Nt + 1
=
= (1 + z )(1 + n)
Kt
Zt Nt
Zt + 1
kt + 1
=
= 1+z
kt
Zt
Level of capital stock grows at approximately sum of growth
rates of Zt and Nt . Per capita capital stock grows at rate of
growth in Zt
This growth is manifested in output and the real wage, but
not the return on capital
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Steady State Growth and Stylized Facts
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Once in steady state, we have:
yt +1
= 1+z
yt
kt +1
= 1+z
kt
Kt + 1
Kt
=
Yt +1
Yt
w t + 1 Nt + 1
wt Nt
=
Yt +1
Yt
Rt +1 = Rt
wt +1
= 1+z
wt
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These are the six time series stylized facts!
30 / 51
Understanding Cross-Country Income Differences
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Solow model can reproduce time series stylized facts if it is
assumed that productivity grows over time
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Let’s now use the model to think about cross-country income
differences
What explains these differences? Three hypotheses for why
cross-country income differences exist:
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1. Countries initially endowed with different levels of capital
2. Countries have different saving rates
3. Countries have different productivity levels
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Like sustained growth, most plausible explanation for
cross-country income differences is productivity
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Consider standard Solow model and two countries, 1 and 2.
Suppose that 2 is poor relative to 1
31 / 51
Convergence
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Suppose two countries are otherwise identical, and hence have
the same steady state.
But suppose that country 2 is initially endowed with less
capital – k2,t < k1,t = k ∗
𝑘𝑘𝑗𝑗,𝑡𝑡+1 = 𝑘𝑘𝑗𝑗,𝑡𝑡
𝑘𝑘𝑗𝑗,𝑡𝑡+1
𝑘𝑘𝑗𝑗,𝑡𝑡+1 = 𝑠𝑠𝐴𝐴𝑓𝑓�𝑘𝑘𝑗𝑗,𝑡𝑡 � + (1 − 𝛿𝛿)𝑘𝑘𝑗𝑗,𝑡𝑡
𝑘𝑘 ∗
𝑘𝑘2,𝑡𝑡
𝑘𝑘1,𝑡𝑡 = 𝑘𝑘 ∗
𝑘𝑘𝑗𝑗,𝑡𝑡
32 / 51
Catch Up
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If country 2 is initially endowed with less capital, it should
grow faster than country 1, eventually catching up with
country 1
Country 1
Country 2
𝑦𝑦
𝑔𝑔𝑗𝑗,𝑡𝑡+𝑠𝑠
𝑘𝑘𝑗𝑗,𝑡𝑡+𝑠𝑠
𝑦𝑦
𝑔𝑔2,𝑡𝑡
𝑘𝑘1,𝑡𝑡 = 𝑘𝑘 ∗
𝑦𝑦
𝑔𝑔1,𝑡𝑡 = 0
𝑘𝑘2,𝑡𝑡
0
𝑠𝑠
0
𝑠𝑠
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Is There Convergence in the Data?
16
14
Correlation between cumulative growth
and initial GDP = -0.18
Y2010/Y1950
12
10
8
6
4
2
0
0
2,000
6,000
10,000
14,000
Y1950
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Correlation between growth and initial GDP is weakly negative
when focusing on all countries
34 / 51
Focusing on a More Select Group of Countries
14
12
Correlation between cumulative growth
and initial GDP = -0.71
Y2010/Y1950
10
8
6
4
2
2,000 4,000 6,000 8,000 10,000
14,000
Y1950
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Focusing only on OECD countries (more similar) story looks
more promising for convergence
Still, catch up seems too slow for initial low levels of capital to
be the main story
35 / 51
Pseudo Natural Experiment: WWII
1.2
1
0.8
US
Germany
0.6
UK
Japan
0.4
0.2
0
1950
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1960
1970
1980
1990
2000
2010
WWII losers (Germany and Japan) grew faster for 20-30 years
than the winners (US and UK)
But don’t seem to be catching up all the way to the US:
conditional convergence. Countries have different steady
states
36 / 51
Differences in s and A∗
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Most countries seem to have different steady states
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For simple model with Cobb-Douglas production function,
relative outputs:
y1∗
=
y2∗
A1
A2
1−1 α s1
s2
1−α α
.
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Question: can differences in s plausibly account for large
income differences?
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Answer: no
37 / 51
Differences in s
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Suppose A∗ the same in both countries. Suppose country 1 is
y∗
US, and country 2 is Mexico: y1∗ = 4. We have:
2
s2 = 4
α −1
α
s1
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Based on data, a plausible value of α = 1/3. Means
α −1
α = −2
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Mexican saving rate would have to be 0.0625 times US saving
rate
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This would be something like a saving rate of one percent (or
less)
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Not plausible
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Becomes more plausible if α is much bigger
38 / 51
What Could It Be?
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If countries have different steady states and differences in s
cannot plausibly account for this, must be differences in
productivity
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Seems to be backed up in data: rich countries are highly
productive
160,000
GDP per worker (2011 US Dollars)
Correlation between TFP and GDP = 0.82
140,000
120,000
100,000
80,000
60,000
40,000
20,000
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
TFP (relative to US = 1)
39 / 51
Productivity is King
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Productivity is what drives everything in the Solow model
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Sustained growth must come from productivity
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Large income differences must come from productivity
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But what is productivity? Solow model doesn’t say
40 / 51
Factors Influencing Productivity
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Including but not limited to:
1.
2.
3.
4.
5.
6.
7.
Knowledge and education
Climate
Geography
Institutions
Finance
Degree of openness
Infrastructure
41 / 51
Policy Implications
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If a country wants to become richer, need to focus on policies
which promote productivity
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Example: would giving computers (capital) to people in
sub-Saharan Africa help them get rich? Not without the
infrastructure to connect to the internet, the knowledge of
how to use the computer, and the institutions to protect
property rights
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Also has implications when thinking about poverty within a
country
42 / 51
Output Per Worker over Time
11.6
Actual Series
Trend Series
11.4
log(RGDP per Worker)
11.2
11
10.8
10.6
10.4
10.2
1950
1960
1970
1980
1990
2000
2010
2020
Year
go back
43 / 51
Capital Per Worker over Time
12.6
Actual Series
Trend Series
12.4
log(Capital per Worker)
12.2
12
11.8
11.6
11.4
1950
1960
1970
1980
1990
2000
2010
2020
Year
go back
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Capital to Output Ratio over Time
3.5
3.4
K/Y
3.3
3.2
3.1
3
2.9
1950
1960
1970
1980
1990
2000
2010
2020
Year
go back
45 / 51
Labor Share over Time
0.68
0.67
Labors Share
0.66
0.65
0.64
0.63
0.62
1950
1960
1970
1980
1990
2000
2010
2020
Year
go back
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Return on Capital over Time
0.125
0.12
Return on Capital
0.115
0.11
0.105
0.1
0.095
1950
1960
1970
1980
1990
2000
2010
2020
Year
go back
47 / 51
Real Wage over Time
3.6
Actual Series
Trend Series
3.4
log(Wages)
3.2
3
2.8
2.6
2.4
2.2
1950
1960
1970
1980
1990
2000
2010
2020
Year
go back
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Income Differences
GDP per Person
High income countries
Canada
Germany
Japan
Singapore
United Kingdom
United States
$35,180
$34,383
$30,232
$59,149
$32,116
$42,426
China
Dominican Republic
Mexico
South Africa
Thailand
Uruguay
$8,640
$8,694
$12,648
$10,831
$9,567
$13,388
Cambodia
Chad
India
Kenya
Mali
Nepal
$2,607
$2,350
$3,719
$1636
$1,157
$1,281
Middle income countries
Low income countries
go back
49 / 51
Growth Miracles and Disasters
South Korea
Taiwan
China
Botswana
Madagascar
Niger
Burundi
Central African Republic
Growth Miracles
1970 Income
2011 Income
$1918
$27,870
$4,484
$33,187
$1,107
$8,851
$721
$14,787
Growth Disasters
$1,321
$937
$1,304
$651
$712
$612
$1,148
$762
% change
1353
640
700
1951
-29
-50
-14
-34
go back
50 / 51
6
7
log(RGDP per Person)
8
9
10
11
Education and Income Per Capita
1
1.5
2
2.5
Index of Human Capital
3
3.5
go back
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