Solow Growth Model IISolow Growth Model IIEconomic Development & Construction 0008Economic Development & Construction 0008
Dr. Kumar AniketDr. Kumar Aniket
Lecture 2Lecture 2
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Some Basic MathSome Basic Math
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Growth Rates Primer IChange in the value of between times period and is given by
Growth rate of is given by
x Change Growth
2 NA NA
3 1 0.50
6 3 1.00
7 1 0.17
13 6 0.86
15 2 0.15
x t − 1 t
Δx = xt − xt−1
x
gx =Δx
x
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Growth Rates Primer IIIf variable are multiplied
the growth rates get added up
x y z Growth_x Growth_y Growth_z
3 2 6 NA NA NA
6 5 30 1.00 1.50 4.00
7 6 42 0.17 0.20 0.40
13 11 143 0.86 0.83 2.40
15 14 210 0.15 0.27 0.47
z = x ⋅ y
= +Δz
z
Δx
x
Δy
y
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Growth Rates Primer IIICapital per-effective labour ratio is given by
where is marginal productivity of labour. The growth rate of is given by
K A L k Growth_K Growth_A Growth_L Growth_k
100.00 1.00 10.00 10.00 NA NA NA NA
120.00 1.05 10.30 11.10 0.2 0.05 0.03 0.10957
144.00 1.10 10.61 12.31 0.2 0.05 0.03 0.10957
172.80 1.16 10.93 13.66 0.2 0.05 0.03 0.10957
207.36 1.22 11.26 15.16 0.2 0.05 0.03 0.10957
k =K
AL
A k
= − −Δk
k
ΔK
K
ΔA
A
ΔL
L
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Properties of Production FunctionProduction function
If the production function has diminishing returns to capital output-capital
ratio falls as more capital is employed,
i.e., falls as increases
intuition: each subsequent addition of capital produces a smaller increase
in output at the margin
Y = F(K, AL)
YK
K
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Output-capital ratio
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Solow Growth ModelSolow Growth Model
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Goods Market EquilibriumPeople consume proportion of their income
And save the rest
Economy's output is either consumed or invested by its denizens
It implies that saving equals investment in the economy
non-consumed output becomes investment1
(1 − s)
C = (1 − s)Y
S = Y − C = sY
Y ≡ C + I
S ≡ I
1. Output cannot easily be transformed into investment good. This implies thatresources that produce output can be redirected towards investment goods.
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where proportion of current
capital stock needs to replaced
every year
InvestmentInvestment gets divided up between depreciation and adding to capital
stock in the economy
Change in capital stock is given by
I
I = ΔK + δK
δ
K
ΔK = I − δK
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Simplifying AssumptionsThere is no population growth:
Marginal Product of labour is constant:
= 0ΔL
L
= 0ΔA
A
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the closer the value of gets to ,
the smaller the growth rate of
capital
Fundamental equationCapital accumulation is saving that is in excess of depreciation
dividing through by
If the economy's production function has diminishing returns to capital, the
economy heads to convergence
sY δK
ΔK = sY − δK
K
= s( ) − δΔK
K
Y
K
s YK
δ
gK = ΔKK
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Convergence to Steady State
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where
Steady StateCapital accumulation stops when . This implies that
Steady-state condition: the output-capita ratio equals a constant
(depreciation saving rate ratio)
Higher the saving rate , the richer the country
Lower the depreciation rate , the richer the country
Δ ( ) = 0KL
s( ) = δ( )Y
L
K
L
=Y
K
δ
s
= /YK
YL
KL
s
δ
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New AssumptionsPopulation grows at the rate
Marginal Product of labour grows at the rate
Since , we can write this as
n > 0
= nΔL
L
g > 0
= gΔA
A
k = K
ALK = k ⋅ A ⋅ L
= + g + nΔK
K
Δk
k
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Fundamental equation Growth rate of capital
Revisiting the Fundamental equation
Growth rate of capital per-effective worker is
Further the economy from the steady state, the faster it grows. The closer its
gets, the smaller the growth rate.
= s( ) − δΔK
K
Y
K
K
= + g + nΔK
K
Δk
k
k = KAL
= s( ) − (δ + n + g)Δk
k
Y
K
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Convergence to Steady State
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For to be constant, growth rate of
numerator should be equal growth
rate of the denominator
Steady State Growth rateIn steady state
This implies growth rate of capital-labour ratio in steady state is equal to
the growth rate of marginal productivity of labour.
The faster marginal productivity of labour grows, the more capital each
worker has, the richer the worker becomes.
= 0Δk
k
KL
k =( )K
L
A
k
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Infrastructure has
complementarities with capital and
thus plays a crucial role in ensuring
that returns to capital are non-
diminishing
Erie canal
Railway Construction by the British
Raj in India
Taking StockConvergence driven by diminishing returns to capital
What if returns to capital did not diminish?
No convergence to steady state
Perpetual growth
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Opened on October 26, 1825 and ran
584 km from Hudson River to Lake
Erie
It was faster than carts pulled by
draft animals and cut transport costs
by about 95%
Erie canal
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Non-diminishing returns to Capital
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Economic Growth (1760-1990)
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Economic Growth (1760-1990)
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Economic Growth (1760-1990)
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Infrastructure like plays a crucial
role in ensuring returns to capital
are non-diminishing
e.g., roads, railways, canals,
electricity, education and health
infrastructure
ConclusionsCapital accumulation crucial for economic development
The more capital workers have, the more they can produce and consume
Diminishing returns to capital constrains the growth
Factors that makes returns to capital non-diminishing crucial for growth
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