# Macro economics

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Macroeconomics ECO 403
VU
LESSON 21
ECONOMIC GROWTH (Continued...)
The Golden Rule: introduction
·
How do we know which is the "best" steady state?
·
Economic well-being depends on consumption, so the "best" steady state has the highest
possible value of consumption per person:
c* = (1­s) f(k*)
·
An increase in s
· leads to higher k and y , which may raise c
*
*
*
· reduces consumption's share of income (1­s),
which may lower c*
·
So, how do we find the s and k* that maximize c*?
The Golden Rule Capital Stock
K*gold = the Golden Rule level of capital, the steady state value of k that maximizes
consumption.
To find it, first express c* in terms of k*:
=  y* - i*
C*
- i*
= f (k*)
- δk*
= f (k*)
In general:
i = Δk + δk
In the steady state:  i* = δk* because Δk = 0.
Then, graph f(k*) and δk*, and look for the point where the gap between them is biggest.
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Macroeconomics ECO 403
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δ k*
output and
depreciation
f(k*)
C*gold
i*gold = δk*gold
k*gold
Y*gold = f(k*gold)
capital per
worker, k*
c* = f(k*) - δk*is biggest where the slope of the production function equals the slope of the
depreciation line: MPK = δ
δ k*
output and
depreciation
f(k*)
*
C gold
k*gold
capital per
worker, k*
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Macroeconomics ECO 403
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The transition to the Golden Rule Steady State
·
The economy does NOT have a tendency to move toward the Golden Rule steady state.
·
Achieving the Golden Rule requires that Policymakers adjust s.
·
·
But what happens to consumption during the transition to the Golden Rule?
Starting with too much capital
If k  * > k  gold
*
then increasing c* requires a fall in s.
In the transition to the Golden Rule, consumption is higher at all points in time.
y
c
i
t0
Time
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Macroeconomics ECO 403
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Starting with too little capital
If k  * < k  gold
*
then increasing c* requires an increase in s.
Future generations enjoy higher consumption, but the current one experiences an initial drop
in consumption.
y
c
i
Time
t0
·
The basic Solow model cannot explain sustained economic growth. It simply says that high
rates of saving lead to high growth temporarily, but the economy eventually approaches a
·
We need to incorporate two sources of growth to explain sustained economic growth:
population and technological progress.
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Macroeconomics ECO 403
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Population Growth
·
Assume that the population--and labor force-- grow at rate n. (n is exogenous)
ΔL
= n
L
EX: Suppose L = 1000 in year 1 and the population is growing at 2%/year (n = 0.02).
·
Then ΔL = n L = 0.02 × 1000 = 20,
so L = 1020 in year 2.
Break-even investment
(δ + n)k = break-even investment, the amount of investment necessary to keep k constant.
Break-even investment includes:
·
δ k to replace capital as it wears out
·
n k to equip new workers with capital
(otherwise, k would fall as the existing capital stock would be spread more thinly over a
larger population of workers)
The equation of motion for k
·
With population growth, the equation of motion for k is
Δk = s f(k) - (δ + n) k
Where
S f(k)= actual investment
(δ + n) k = breakeven investment
The impact of population growth
(δ +n2) k
Investment,
(δ +n1) k
break-even
investment
sf(k)
k2*
k1*
Capital per
worker, k
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Macroeconomics ECO 403
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Prediction:
Higher n lower k*.
·
And since y = f(k) ,
·
lower k* lower y* .
Thus, the Solow model predicts that countries with higher population growth rates will
·
have lower levels of capital and income per worker in the long run.
The Golden Rule with Population Growth
To find the Golden Rule capital stock, we again express c* in terms of k*:
=  y* - i*
c*
= f (k* ) - (δ + n) k*
c* is maximized when
MPK = δ + n
or equivalently,
MPK - δ = n
In the Golden Rule Steady State, the marginal product of capital net of depreciation equals the
population growth rate.
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