11

GROWTH AND ACCUMULATION

MD Siyam HossainMD Siyam Hossain Bangladesh Institute of Business & Technology.Bangladesh Institute of Business & Technology. Narayangonj,DhakaNarayangonj,Dhaka Dhaka,BangladeshDhaka,Bangladesh www.facebook.com/mdsiyamhossain

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Economic Condition: Developed and less developed countries

Western has enormously higher incomes than did their great grandparents

People in industrialized nations are far wealthier than people less developed countries

Americans and Europeans had higher incomes a century ago than people in poor countries now

Graph -1 shows GDP for four countries over more than 150 years

33

Graph has four striking characteristics:

a) It shows that in average in USA income increased more than sixteen fold over this period

b) Japan was poor country before 2nd World War. Now it has a standard of living rough equal to USA

c) Norwegian income has spurted in the last 25 years

d) Bangladesh was poor 150 years ago, grew very slowly and remains that way today

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Figure 3-1: GDP Growth of USA, Japan, Norway and Bangladesh

GDP USA

Japan

Norway

Bangladesh

1860 2000 (Year)

55

Question is: What accounts for these differences? And What will determine our standard of living in future? The answer of these questions are given by: (i) Growth accounting and (ii) Growth theory Growth Accounting Growth accounting explains what part of growth in

total output is due to growth in different factors of production (capital, labour, technology, etc.)

Growth Theory Growth theory explains how economic decisions

control the accumulation of factors of production Or, how rate of saving today affects stock of capital in

future

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1. GROWTH ACCOUNTING

General Explanation of GrowthOutput grows through increases in:

(i) Inputs (K and N) and(ii) Productivity due to (a) improved technology and (b) a more able work force

Production Function Production function (I) provides a quantitative link

between inputs and outputs It relates level of output to level of inputs and level of

technology

Y = A(K, N) (1)

Equation (1) is the functional equation of GDP Equation (1) is also called production function

77

Where: N = Labour Force K = Capital (K) and A = Level of TechnologyEquation (I) shows that output (Y) is a function of:

(i) Inputs (K and N) and (ii) Level of technology (A)

Inputs More input means more output Increase in labour force (one input) increases output

(marginal product of labour/MPN) Increase in capital, the other input increases output

(marginal product of capital/MPK)

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Level of technology

A in production function is called ‘productivity’

It is a more neutral term than ‘technology’ Higher level of A means more output It means more output is produced for a given

level of inputs Production function (1) can be transformed

into a growth of output accounting equation

Y/Y = [(1 - ) N/N] + ( K/K) + A/A (2)

It is called growth accounting equation

99

(Dividing both side by Y)

ANKFKMPKNMPNY ),(

ANKFK

KKMPK

N

NNMPNY

),()()(

Y

ANKF

K

K

Y

KMPK

N

N

Y

NMPN

Y

Y

),()()(

),(

),()1(

NKAF

ANKF

K

K

N

N

Y

Y

A

A

K

K

N

N

Y

Y )1(

ANKFK

KKMPK

N

NNMPNY

),(

)()(

1010

Where: (1 - ) is the weight equal to labour’s share of income is weight equal to capital’s share of income

In words the equation can be written as Growth of Output=[(weight of labour share)(labour

growth)]+[(weight of capital share)(capital growth)] + technical progress

Equation 2 summarises the contribution of growth of input and productivity to growth of output

It also says There is growth in total factor productivity, when we

get more output from same factors of production

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Example The capital share of income is 25% and that of labour

is 75%, which correspond to the value of US economy Labour force growth is 1.2%, growth of the capital stock is 3% and total factor productivity is 1.5% annually What is the growth of the output?

SolutionApplying equation 2, we obtain a growth rate of: Y/Y = (.75 1.2%) + (.25 3%) + 1.5% = 3.15%.

ConclusionGrowth of Output depends on:

(i) Kapital(ii) Labour, and (iii) Level of over all Technological Progress

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2. ACCOUNTING FOR GROWTH IN PER CAPITA OUTPUT

Equation (2) helps calculating growth in total output (absolute growth)

But we are not really interested in total national output and its growth

Because it expresses little about development status and living standard of a country

To know about real development level and compare it with other countries:

Per capita GDP is to be known As for instance, Norway is a rich country and India is a poor

country However, aggregate Indian GDP is higher than that of

Norway Standard of living refers to individual well-being Per capita GDP is the ratio of GDP to population

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Equation (2) helps deducing equation for accounting per capita growth in output:

Y/Y = [(1 - ) N/N] + ( K/K) + A/A Y/Y = [N/N - N/N] + (K/K) + A/A Y/Y = N/N - N/N + (K/K) + A/A Y/Y - N/N= K/K - N/N + + A/A Y/Y- N/N = [K/K - N/N] + A/A y/y = k/k + A/A y = k/k + A/A y = (k) + A/A (3)

Where: Y/Y - N/N = y/y = per capita out put growth, and [Growth rate of per capita GDP equals the growth rate of

GDP minus growth rate of population] K/K = k/k + N/N = k/k = k = per capita capital growth [Growth rate of capital equals growth rate of per capita plus

growth rate of the population]

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Growth accounting equation (3) expresses that per capita growth rate of GDP depends on:

(i) Per capita capital growth (k) and (ii) Technological development (A/A)

k is the number of machines per worker It is also called the capital-labour ratio It is a key determinant of the amount of output a

worker can produce Since is around .25, equation (3) suggests that:

1% increase in amount of labour increases only a quarter of 1% of per capita output

In other word, to increase 1% of per capita output (y), GDP, 4% capital has to be employed

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From growth accounting equations, technological change could be calculatedIf growth of GDP, Population (Labour) and Capital are knownWe know that:

Hence:

Again we know that:

(3)

Hence:(4)

A

A

K

K

N

N

Y

Y

)1(

K

K

N

N

Y

Y

A

A

)1(

A

A

k

k

y

y

k

k

y

y

A

A

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Technological progress A is also called total factor productivity

Technological progress also cause increase of labour productivity

Such technological progress is called labour augmenting technological progress

For labour augmenting technological progress equation (3) is written as:

Hence:

(5)

A

A

k

k

y

y

)1(

K

K

Y

Y

A

A 1

1

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There is another production function of GDPIt is the Cobb-Douglas production function of GDP:

Here is the capital share to the GDP [For USA = .20]Here 1- is the labour share to the GDPCobb-Douglas production function is at same time GDP accounting equationIf K and N are known the Y could be calculatedCobb-Douglas production accounting equation could be transferred into per capita form:

In functional form this equation could be written as:y = (k)

1NAKY

AkN

KA

N

KANAK

N

NAK

N

Yy

1

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For labour augmenting technological progress equation Cobb-Douglas production accounting equation is written as:

1)(ANKY

1NAKY

11 NAKY

11

NK

YA

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3. EMPIRICAL ESTIMATE OF GROWTH

For growth capital accumulation is important, but technical progress is more important

In this regard Robert Solow studied period 1909-1949 in USA He concluded that over 80% of the growth was due to technical

progress Important determinants of per capita growth are technical

progress and capital accumulation Increased population means decrease GDP per capita even

though GDP increases More workers means more output, but output increases

proportionately slowly Each percentage growth of labour force leads to about .75-

percentage growth in output As output grows less quickly than numbers of workers, per

capita output falls

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4. FACTORS OTHER THAN CAPITAL AND LABOR

Besides capital and labour other important inputs for growth are:(i) Natural resources and (ii) Human capital

Natural Resource Early prosperity of the US was due to abundant fertile From 1820 to 1870 arable land of USA grew at 1.41% annually Recent example of importance of natural resources is sharp

increase in Norwegian GDP From 1970-1990 Norway's per capita GDP rose from 61% to

77% of US per capita GDP This was due to the discovery and development of massive oil

reserves However, some empirical evidence suggests that: Countries with more natural resources generally do worse Explanation is: Some such countries squander their wealth

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Human Capital

High investment leads to high income Question is, is there any relationship between human

capital and output In industrialized countries, qualified workers is more

important than unqualified labour Stock of unqualified workers could be increased by

investment in human capital

Adding human capital (H), production function could be written as:

Y = AF(K,H,N) (4) Human capital is difficult to measure precisely However, length of schooling can serve as a proxy

for human capital

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Impact of Immigration Immigration boosts per capital output when skilled

workers enter the country From immigration USA has frequently benefited In USA qualified workers immigrated

Impact of Refugees Immigration consisting of refugees depresses per

capita output in the short run A factor of production adds to output growth only

when supply of this factor grows

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5. GROWTH THEORY - THE NEOCLASSICAL MODEL

There have been two periods of intense work on growth theory:

(i) Late 1950s and 1960s and (ii) Late 1980s and early 1990s

Research in the first period produced neoclassical growth theory

This theory focuses on capital accumulation and its link to savings

The other theory that focuses on the determinants of technological progress is called:

Endogenous growth theoryThis theory is studied in the next chapter

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Neo-classical growth theory

Neo-classical growth theory begins with a simplification of assumption

It predicts that there is no technological progress in the long run

This implies that economy reaches a long-run level of output and capital

It is called the steady-state equilibrium

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Steady-state equilibrium

Steady-state equilibrium is the combination of per capita GDP and per capita capital where economy remains at rest

It means in the steady state: Per capita economic variables are no longer changing

That is in steady state equilibrium: Per capita GDP does not grow [y = 0] Per capita capital does not grow [y = 0]

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Scope of Neo-classical growth theory

Neo-classical growth theory studies transition economy from its current position to steady state

It assumes that ultimately:

There will no technological progress and Per capital GDP will remain unchanged Hence it develops first the growth model for steady

state and As a final step adds technological progress to the

model

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We know that

Production function in per capita terms for classical model is written as:

y = (k) + A/A If technological progress A/A = 0 and = 0, then:

y = (k) (5) Here k is the capital-labour ratio

Considering technological progress; production function can be expressed as:

y = ak or Y = aK

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Let us graph per capita GDP against capital-labour ratio (Figure-2)

The graph shows that: As capital rises, output rises (marginal productivity of

capital is positive) However, output increases less at higher levels of

capital than at low levels [Following law of diminishing marginal productivity

of capital] Each additional machine adds to production, but adds

less than previous machine Law of diminishing marginal productivity is the key

explanation why the economy reaches a steady state rather than growing endlessly

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Figure-2: GDP per capita graphed against capital-labour ratio

Y

y* y = f(k)

k* K

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5.1 Steady State

An economy is in a steady state:

When per capita income and per capita capital are constant (Figure-2)

In steady state per capita income cannot be increased more

Steady-state value of per capita income = y* Steady-state value of per capita capital = k* At steady-state capital required for new workers and

machines worn out is just equal to saving generated by economy

At steady state levels of output saving and re-investment balance

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Explaining the situation before steady state

Before steady state: Per capita output, y, could be increased Per capita capital, k, could be increased If saving is greater than investment requirement,

capital per worker and output rise If saving is less than investment requirement, capital

per worker and hence output fall

ConclusionHave we: Steady-state value of per capita income, y*, and Steady-state value of per capita capital, k* We can examine transition path of economy from any

arbitrary point to steady state

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5.2 GROWTH IN NEOCLASSICAL MODEL

Investment required maintaining a certain level of growth (capital per capita) depends on:

(i) Population growth and (ii) Depreciation rate Let per capita capital required to maintain present

growth is k Let population grows at constant rate n (n = N/N) Let the rate of depreciation is d Capital dk is required for new machinery to keep

present growth So, investment required to maintain level of growth

is (n + d) k

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Let us now examine the link between saving and growth in capital

[Let there is no government sector and no foreign trade or capital flows]

Let the rate of saving is constant and s of income So, per capita saving is sy Since income equals production So, we can also write sy (and sy = sf (k)

Change in per capita capital (k) is excess of saving over required investment:

k = sy - (n + d)k (6)

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We know that:Steady state is defined by k = O

So, steady state occurs at:

0 = sy - (n + d)k

sy = (n + d)k

At steady state the values of y and k are:

y* and k*

Hence

sy* = (n + d)k* (7)

sy* = sf(k*) (8)

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Figure-3: Shows a graphical solution for the steady state

y

D y = f(k)

C (n+d)k

A sy

B

k0 k* k

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Curve sy: Curve sy presents a constant proportion of output [Here s is the rate of saving of per capita capital and y is the

per capital output] Curve sy shows the level of saving at each capital-labour ratio Straight-line (n + d)k Straight-line (n + d)k shows amount of investment needed to

keep capital-labour ratio constant Intersection of the curve sy and the straight-line (n + d)k at

point C means balance of saving and required investment at steady state

Here per capita capital is k*, that is steady-state per capita capital (k*)

Here per capita capital is k*, that is steady-state saving rate (k*)

Steady-state income (output) is read off the production function at point D

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5.3 THE GROWTH PROCESS

Let us study the adjustment process that leads the economy:

From a initial capital-labour ratio to steady state (Figure-3)

Critical element in this transition process is:

Rate of saving (s),

Investment (k),

Rate of depreciation (d), and

Population growth (n)

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Key to the neoclassical growth model is the saving, sy When sy exceeds the investment requirement (to

maintain running growth), k increases [When sy exceeds (n+d)k, k increases] So, over time the economy grows (Figure-3)

Let economy the starts at capital-output ratio ko

(Figure -3)

At capital-output ratio ko investment at B is needed to

maintain current growth rate However, more is saved at point A,

So, capital-output ratio increases and greater than ko

Hence, the economy growths and shifts left

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End of the Growth

Adjustment process comes to a halt at point C Here the capital-labour ratio, k*, match the

investment requirement Beyond this point capital-labour ratio cannot be

more increased by saving And per capita output cannot also be increased more At this point so is saved as the growth rate demand If population increases, investment increases, but per

capita capital labour ratio remain constant At steady state, both k and y are constant At steady state growth rate is not affected by the

savings rate It is one of key results of neoclassical growth theory

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5.4 AN INCREASE IN THE SAVING RATE

Let explain how increase in savings affects growth In short run, increase in savings rate raises growth

rate of output Let the economy is in equilibrium, at which saving

precisely matches investment requirement Let now people want to save a larger fraction of

income This causes an upward shift of savings schedule to

dashed schedule (Figure-4) Let at point C (Figure-4) we have initially

equilibrium Let now more is saved than is required to maintain

capital per head constant

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Figure-3: Shows a graphical solution for the steady state

y y1 = f(k1)

D y = f(k)

C1 (n+d)k

C sy

B

k0 k* k

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That means, enough is saved to allow capital stock per head to increase

Capital per head, k, rises until it reaches higher amount

At point C1 (Figure-4) both capital per head and output per

head have risen

However, economy has reached to its steady-state growth rate

Now: Savings rate does not affect growth rate

[Long run Neoclassical growth theory]

In transition, higher savings rate increases growth rate of output and per capita output

It means, capital-labour ratio rises from k* to k**, steady state

Only way to achieve an increase in capital-labour ratio is:

Capital stock has to grow faster than the labour force and depreciation