Solow Growth Model II - Aniket

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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