MACROECONOMICS. Chapter 7 Economic Growth I: Capital Accumulation and Population Growth. Solow Growth Model. Real GDP in US is 5X its level 50 years ago; per capita real GDP is 3X. In some poor countries, real GDP per person is only 2-5% of US.
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Economic Growth I: Capital Accumulation and Population Growth
The production function is the familiar one with
constant returns to scale. The little trick of
defining z allows us to show the output as per
worker real GDP and the input as capital per
worker (also called capital-labor ratio).
The lower case depiction of the production
function, therefore, says that per worker output
depends on capital per worker.
When x changes by one unit, by how many units will Z change?
Negative exponent means reciprocal.
Using an arbitrary Cobb-Douglas
function, we can see how the
production function can be
presented in terms of GDP per
The MPK will be decreasing as
capital increases because as K
goes up the denominator
Likewise, when capital-labor ratio
(k) increases, the marginal
product of k decreases.
The exercise here shows that
MPK=MPk and it doesn’t matter
If one uses Y or y.
The total output (GDP) is divided between C, I, G, and NX. For
simplicity, pretend that G=0 and NX=0. Then, Y = C + I
Let’s show this equation as per worker:
Output per worker (y) is determined by capital
per worker (k). Given k, we know what y will be.
The output per worker (y) will be divided between
consumption per worker and investment per
worker according to the size of savings rate (s).
The higher the savings rate, more of the output
will be used for investment and less for
y=f(k) and i=sy
which is the
same as i=sf(k)
per worker, c
per worker, i
What happens if s rises?
k (Capital per worker)
If y = 10k0.25, and δ = 0.15, what is the golden rule k?
MPK = 2.5k-0.75
MPK = δ
2.5k-0.75 = 0.15
k0.75 = 16.6667
k = 16.66671.333
k ≈ 42.6
Compare with slide # 15!
Δk = i – δk – nk
Δk = i – (δ + n)k
It has to be nk because growth rate of L
has to match growth rate of K.
0 = sf(k) - (δ+n)k
i = (δ+n)k
n2 < n1