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Saving and the Golden Rule
• What determines the saving rate, s? We assumed:
It = sYt
Economic Growth II
and
Ct = (1 − s)Yt.
1. Saving and the Golden Rule
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2. Convergence
• So what is the optimal s?
• Suppose we want to maximize steady-state consumption.
3. The Solow model with labor-augmenting technological progress
4. A quick return to growth accounting
c∗ = (1 − s)y ∗
= (1 − s)Ak ∗α
⎛
⎞ α
sA ⎠ 1−α
= (1 − s)A ⎝
n+δ
• We can get the law-of-motion for Kt in per-worker (lower case terms) as well. You
can ﬁll in the intermediate steps ...
• Easier question: What steady-state level of the capital-to-labor ratio, k ∗, maximizes
steady-state consumption per worker?
Kt+1 = (1 − δ)Kt + It
...
(1 + n)kt+1 = (1 − δ)kt + it
• Deﬁne two more variables.
– ct ≡ Ct/Nt. So ct denotes consumption per worker.
– i∗ ≡ It/Nt. So it denotes investment per worker.
Yt = Ct + It
Yt
Ct
It
=
+
Nt
Nt Nt
y t = c t + it
So in steady-state:
y ∗ = c ∗ + i∗
So in steady-state:
(1 + n)k ∗ = (1 − δ)k ∗ + i∗
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3
• Then we can write the expenditure identity as:
∗
Solving for i yields:
i∗ = (n + δ)k ∗
∗
∗
• If we substitute in for y and i are per-worker expenditure identity becomes:
Ak ∗α = c∗ + (n + δ)k ∗
or
c∗ = Ak ∗α − (n + δ)k ∗
• Now remember why we did all this algebra. We wanted to ﬁnd the k ∗ that maximizes
c∗. So we want take the derivative of the above equation with respect to k ∗. So the
k ∗ that maximizes c∗ is the k ∗ that satisﬁes:
αAk ∗α−1 − δ = n
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5
or
MPK − δ = n.
∗
• Graphically, this k is the capital-to-labor ratio that maximizes the diﬀerence between
the production function and steady-state investment line in Figure 6.2.
s
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
k∗
0
54.9
147.8
263.9
398.0
547.4
710.2
885.2
1071.3
1267.6
1473.5
y∗
0
52.2
70.2
83.6
94.5
104.0
112.5
120.1
127.2
133.8
140.0
δk ∗
0
4.4
11.8
21.1
31.8
43.8
56.8
70.8
85.7
101.4
117.9
c∗
0
47.0
56.2
58.5
56.7
52.0
45.0
36.0
25.4
13.4
0
MPK MPK - δ
∞
∞
0.2850 0.2050
0.1425 0.0625
0.0950 0.0150
0.0713 -0.0087
0.0570 -0.0230
0.0475 -0.0325
0.0407 -0.0393
0.0356 -0.0444
0.0317 -0.0483
0.0285 -0.0515
Table 1: A Numerical Comparison of Steady-States
Why should we poor people make sacriﬁces for those who will in any case live in
luxury in the future?
Robert Solow
Evidence of Convergence
• Unconditional convergence: Poor countries should eventually catch up to rich countries.
• In the case of Post-WW2 Japan, Germany, and England, much capital was destroyed
in the war so reasonable to think of them being far away from steady-state so high
GDP growth in decades following war partially explained as transitioning to steadystate.
– This should occur if saving rates, population growth rates, and production functions are the same everywhere.
• For developed countries, Solow model predicts we would already be at or near steady
state.
– This really hinges on whether there is free ﬂowing international borrowing and
lending.
• If one does not account for diﬀering population growth rates and saving rates, there
is simply no evidence for convergence.
Convergence
– Is there too little capital mobility in the world?
• Conditional convergence: Living standards will converge in countries with similar
characteristics: s, n, δ, α.
– There would be convergence except diﬀerent countries have diﬀerent parameters
in the Solow model.
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7
– Why isn’t all new investment occurring in the Third World?
• If one accounts for diﬀering saving and population growth rates, convergence might
be occurring, although there is a small sample problem. The theory predicts that
countries like Japan and Germany in 1950 will have higher growth rates than US.
• US states also seem to be converging. (Poor states grow faster than rich ones.) right
now the South is growing faster than the North. Evidence of conditional convergence
since the U.S. states have similar characteristics.
• After the Civil War, average per capita income in South was 40% of the average per
capita income in North. One hundred and twenty years later is about the same.
The Solow model with labor-augmenting technological progress
We are going to take the Solow growth model with talk about last time and add a new
variable:
E ≡ eﬃciency of labor
Motivate growth in A.
• Recall that throughout this class we have worked with a Cobb-Douglas production
function:
• Why does India have a 1.4% growth rate and South Korea have a 7.0% growth rate?
Is it that India is closer to its steady state?
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• So the Solow model fails along two key dimensions
Yt = AKt1−α Ntα
Now we are going to modify this production function as follows:
– Convergence
Yt = AKt1−α (EtNt)α
– Ultimately, it is not a theory of growth – it is theory of eventually hitting a
steady-state.
where the term (EtNt) now stands for eﬀective labor.
• The eﬃciency of labor reﬂects society’s knowledge of production methods: as the
available technology improves, the eﬃciency of labor rises. The eﬃciency of labor also
rises if there are improvements in health, education, or skills of the labor force.
• We want to think about models in which we allow growth in A.
• We are going to assume that E grows each period at the rate γ. So
Et+1 = (1 + γ)Et
• Capital and labor have the following laws of motion::
Kt+1 = (1 − δ)Kt + It
• Once again we are going to assume A is ﬁxed.
Nt+1 = (1 + n)Nt
• Since we cannot measure E, when we do our growth accounting E is going to show
up in our measure of total factor productivity.
• The fraction of income that is saved each period, s is ﬁxed. So
It = sYt
• Why not just assume A itself grows?
– A steady-state exists if the growth is in E.
– No steady-state exists if we just let A grow.
• Output is split each period between consumption and investment:
Yt = Ct + It.
and
Ct = (1 − s)Yt.
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– We want to solve for a steady-state within our model.
• Deﬁne a new variable Yt
Yt ≡
So Yt is output per eﬀective labor.
Yt
EtNt
• Likewise we are going to deﬁne Kt as the capital-to-eﬀective labor ratio. So
Kt ≡
Kt
Et N t
• So let’s do a little algebra:
Yt =
=
=
=
=
Yt
E Nt ⎞
1 ⎠
⎝
AKtα (EtNt)1−α
E⎛tNt
⎞⎛
⎞
α
Kt ⎟ ⎜ (EtNt)1−α ⎟
⎜
⎠⎝
⎠
A⎝
α
1−α
(EtNt)
(EtNt)
⎛
⎞α
Kt ⎟
⎝
⎠
A⎜
(EtNt)
α
AKt
⎛ t
• We solve for the steady-state:
• Let’s get the law of motion for the capital-to-eﬀective labor ratio, Kt:
=
=
=
=
=
Kt+1
Et+1Nt+1
(1 − δ)Kt + It
(1 + γ)Et(1 + n)Nt
(1 − δ)Kt + sYt
(1 + γ)Et(1 + n)Nt
(1 − δ)Kt + sAKtα (EtNt)1−α
(1 + γ)Et(1 + n)Nt
Kt
Ktα
(1 − δ)
sA
+
(1 + γ)(1 + n) EtNt (1 + γ)(1 + n) (EtNt)α
(1 − δ)
s
Kt +
AKtα
(1 + γ)(1 + n)
(1 + γ)(1 + n)
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Kt+1 =
(1 − δ)
sA
K∗ +
K∗α
(1 + γ)(1 + n)
(1 + γ)(1 + n)
(1 + γ)(1 + n)K∗ = (1 − δ)K∗ + sAK∗α
((1 + γ)(1 + n) − (1 − δ)) K∗ = sAK∗α
(1 + γ)(1 + n) − (1 − δ)
= K∗α−1
sA
⎛
⎞ 1
(1 + γ)(1 + n) − (1 − δ) ⎟ α−1
⎝
⎠
K∗ = ⎜
sA
K∗ =
Let’s get the exponent positive
⎛
K∗ =
⎜
⎝
⎞ 1
1−α
sA
⎟
⎠
(1 + γ)(1 + n) − (1 − δ)
• So instead of getting a steady-state in the capital-to-labor ratio, k, we are getting a
steady-state in the capital-to-eﬀective-labor ratio, K. But what we really care about
is the capital to labor ratio kt and output per worker yt – not Kt and Yt.
• This looks a lot like the laws of motion for kt that we derived in the last lecture. We
can show that the model has reaches a steady state in Kt.
• So let’s get a law of motion for kt
kt+1 =
=
=
=
=
• Note that Et is growing over time, so there is no steady-state for kt. It turns out that
when Kt reaches its steady-state, kt grows at the same rate as Et; that rate is γ.
• Since kt grows at the rate γ, so does yt.
• Once again we get on the computer:
1. Select parameter values:
α = 0.3 s = 0.15
n = 0.015 δ = 0.08
γ = 0.035
2. Choose starting values
E1980 = 1.0
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=
Kt+1
Nt+1
(1 − δ)Kt + It
(1 + n)Nt
(1 − δ)Kt + sYt
(1 + n)Nt
(1 − δ)Kt + sAKtα (EtNt)1−α
(1 + n)Nt
⎛
⎞
(1 − δ) Kt
sA Ktα 1−α ⎝ Nt ⎠1−α
+
Et
α
(1 + n) Nt (1 + n) Nt
Nt
(1 − δ)
s
α 1−α
kt +
Ak E
(1 + n)
(1 + n) t t
so
K1980 = k1980 = 56.08
A = T F P1980 = 14.75
3. Compute and plot.
Punchline: In the long run, the rate of productivity improvements is the dominant
factor determining how quickly living standards rise.
capital−to−labor
output (Billions of 1992 dollars)
220
200
180
160
140
120
100
80
60
40
1980
57
56.9
56.8
56.7
56.6
56.5
56.4
56.3
56.2
56.1
56
1980
1990
1995
2005
2010
2015
Simulated time path of the Capital−to−Labor Ratio (alpha = .3, n=0.015)
1985
2000
year
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Figure 2: Simulated time path of capital-to-labor ratio: alpha=0.3
1990
1995
2000
year
2005
2010
2015
Simulated time path of capital−to−effective−labor ratio(alpha = 0.3, n=0.015)
1985
Figure 1: Simulated time path of capital-to-eﬀective-labor ratio: alpha=0.3
17
2020
2020
output (Billions of 1992 dollars)
output per worker (1992 dollars)
14000
13000
12000
11000
10000
9000
8000
7000
6000
5000
4000
1980
75
70
65
60
55
50
45
1980
1985
1995
2000
year
2005
2010
Simulated time path of real GDP (alpha = 0.3, n=0.015)
1990
1990
1995
2000
year
2005
2010
Simulated time path of real GDP per worker (alpha = 0.3, n=0.015)
20
Figure 4: Simulated time path of GDP: alpha=0.3
1985
2015
2015
2020
2020
Figure 3: Simulated time path for GDP per worker: alpha = 0.3
19
A quick return to growth accounting
Suppose the Solow model with exogenous labor-augmenting technological progress is the
way the world really works. But suppose we ran the growth accounting exercise we discussed last class. That is, we computed the percentage growth in total factor productivity
as:
∆Kt
∆Nt
∆At ∆Yt
=
−α
− (1 − α)
At
Yt
Kt
Nt
No population growth Population growth
Population growth = n
n>0
Technological progress = γ
n=0
N is constant
N grows at rate n
N grows at rate n
EN grows at the rate γ + n
K grows at rate n
K grows at rate n + γ
k is constant
k is a constant
k grows at rate γ
Well in the model we solved today:
Yt = AKtα (EtNt)1−α
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K is constant
What would we get?
Taking logs of both sides of the equation yields:
K is a constant
Y is constant
Y grows at rate n
Y grows at rate n + γ
y is a constant
y is a constant
y grows at rate γ
ln Yt = ln A + α ln Kt + (1 − α) ln Et + (1 − α) ln Nt
We go back one period to get:
ln Yt−1 = ln A + α ln Kt−1 + (1 − α) ln Et−1 + (1 − α) ln Nt−1
(2)
If we subtract equation (2) from equation (1) we get
Y is a constant
ln Yt − ln Yt−1 = α(ln Kt − ln Kt−1) + (1 − α)(ln Et − ln Et−1) + (1 − α)(ln Nt − ln Nt−1).
We can write the above equation as:
∆Kt
∆Et
∆Nt
∆Yt
=α
+ (1 − α)
+ (1 − α)
Yt
Kt
Et
Nt
(1)
(3)
So in this model we would measure the percentage change in total factor productivity
growth as
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∆Nt
∆Yt
∆Kt
∆T F P
−α
− (1 − α)
=
TFP
Yt
Kt
Nt
∆Et
= (1 − α)
Et
So in this model we get total factor productivity growth even though we did not model
growth in A explicitly.

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