ECONOMIC GROWTH

DEBRAJ RAY

OVERVIEW ON GROWTH THEORY

• How to study growth?

• Observations on growth.

• Questions asked in growth theories.

• Short history of modern growth theory.

• Basic concepts in growth models.

• Harrod- Domar model.

• Solow model.

OBSERVATIONS ON GROWTH

• 1. GDP/capita varies a lot from country to country.

• About 50% of the world population live in countries with GDP/capita less than 10% of that of the richest countries.

• Growth rates vary a lot, but there is no huge difference between the average growth rates of developing and developed countries

• Average per capita growth rate in 16 today's developed countries • (Europe, USA, Canada, Australia) • period growth rate, % per year • 1870-1890 - 1.2 • 1890-1910 - 1.5 • 1910-1930 - 1.3 • 1930-1950 - 1.4 • 1950-1970 - 3.7 • 1970-1990 - 2.2 • Average per capita growth rate in 15 developing countries in Asia

and • South America. • 1900-1913 - 1.2 • 1913-1950 - 0.4 • 1950-1973 - 2.6 • 1973-1987- 2.4

• Growth rates are not necessarily constant over time

• Ex. India: 1960-97 average growth rate was 2.3%, but

• 1960-80: 1.3%

• 1980-1997: 3.5%

• China:

• 1960-1978: 1.9%

• 1978- 1997: 5%

• Country’s relative position in the world distribution of per capita incomes can change.

• Countries can move from being ‘poor’ to being ‘rich’:

• Korea, Taiwan, Singapore, Japan, Hong Kong.

• Ex. Korean GDP/capita 7.4 times higher in 1990 than in 1960 (880 -> 6580 USD, 1985 prices).

• Or from being ‘rich’ to being ‘poor’:

• Ex. Iraq GDP/capita fell from 3300 to 1780 USD from 1960 to 1990 in 1985 prices).

• Growth in output and growth in the volume of international trade are closely related.

• Both skilled and unskilled workers tend to migrate from poor to rich countries or regions.

QUESTIONS ASKED

• Why are some countries poor and others rich?

• Why are some countries growing and others not?

• Where does growth come from?

Neoclassical growth

• The neoclassical growth model, also known as the Solow–Swan growth model or exogenous growth model, is a class of economic models of long-run economic growth set within the framework of neo classical economics.

• Neoclassical growth models attempt to explain long run economic growth by looking at productivity, capital accumulation, population growth and technology.

• The neo-classical model was an extension to the 1946 Harrod - Domar model that included a new term: productivity growth.

• Important contributions to the model came from the work done by Robert Solow and T. W. Swan who independently developed relatively simple growth models.

SHORT HISTORY OF MODERN GROWTH THEORY

• Modern growth theory originates from 1950s (by Robert Solow)

role of physical capital and technological progress central.

perfect competition as a starting point.

technology assumed to grow exogenously in time as ‘manna from heaven’

Harrod- Domar Model

• Developed by Sir Roy Harrod and Evsey Domar in the 40s .

• The Harrod–Domar model is used in development economics to explain an economy's growth rate in terms of the level of saving and productivity of capital.

The Harrod-Domar Model

R. F. Harrod, The Economic Journal, Vol. 49, No. 193. (Mar., 1939), pp. 14-33.

Econometrica, Vol. 14, No.

2. (Apr., 1946), pp.

137-147

“ Economic Growth is the result of abstention from current consumption”.

• Two type of commodities:

1. Consumption Goods: produced to satisfy human wants and preferences.

2. Capital Goods: Commodities that are produced to produce other commodities.

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The Harrod-Domar Growth Model (continued)

Firms

Households

Wages, Profits, Rents

Consumption Expenditure

Outflow

Inflow

Inflow

Outflow

Investment

Savings

BASIC CONCEPTS

• Capital (K) and Labor (L) used as inputs to produce the output (Y).

• Fixed factor proportions assumed.

• The state of technology is given and requires that inputs to be used in fixed proportion.

The state of technology is given and requires that inputs to be used in fixed proportion. Thus production function is of fixed coefficient type:

• α = labour output ratio;

• β = capital output ratio

})(

,)(

min{)(

tKtLtY

• It is also assumed that the economy is closed and is producing a single commodity, which is partly consumed and partly invested.

• Labor forces grows at an exogenous determined constant exponential rate.

• Investment is proportional to change in output. It is also assumed that the capital stock does not depreciate and there is no technical progress.

• The society is inclined to save a constant proportion of its output all the time.

• Entrepreneurs are profit maximizers.

Use of aggregate production

• Y(t)= C(t)+ I(t)……………..1

• Y(t)= real GDP in year t.

• C(t)= Consumption in period t.

• I(t)= Investment in period t.

Use of aggregate income

• Y(t)= C(t)+ S(t)………….2

• C(t)= Purchase of consumption in period t.

• S(t)= Household saving in period t.

• Using the above equations==

• S(t)=I(t)….savings =investment.

• How are capital stock (K) and investment flow (I) related?

• Investment augments the national capital stock K and replaces that part of it which is wearing out.

• I(t)= K(t+1)-K(t) + δ K(t)

• Or, K(t+1) = (1-δ) K(t) + I(t)

• This tells us how capital stock must change over time.

• δ = rate of depreciation of the capital stock

• t = index for time

In equilibrium

• C +S = C+ I ( from 1 and 2)

• <=> S = I……………..(3)

• => S(t) = K(t+1) –(1- δ) K(t)

• Investment augments the national capital stock K and replaces that part of it which is wearing out.

• Let Savings rate:

……………..(4)

• Share of income that can be allocated to investment to increase the growth rate.

)(

)()(

tY

tSts

• Define Capital Output ratio= θ

• The amount of capital required to produce a single unit of output in the economy.

• θ = …………….(5)

• θ is assumed to be a technologically given constant.

• Combining equation 4 and 5, we get

= g+ δ

H-D Equations • K(t+1) = (1-δ) K(t) + I(t)

• S(t)=I(t)

• K(t+1) = (1-δ) K(t) +S(t)

• S(t)=s Y(t)

• K(t)=θ Y(t)

• θY(t+1)= (1-δ) θY(t)+ s Y(t)

• Or, θ[ Y(t+1)- Y(t)]= Y(t) (s-δθ)

• or,

s

tY

tYtY

)(

)()1(

• g= overall rate of growth

• g=[Y(t+1)-Y(t)]/ Y(t) .

• This is the Harrod Domar Equation.

sg

sg

• is called the warranted rate of growth.

• Under the assumption of constant θ , g increases proportionally with s.

• Because s is considered to increase proportionally with income per capita, s is bound to be low.

• Hence, g will be low in low-income economies if savings and investment are left to private decision in the free market.

• The model implies, that promotion of investment is needed to accelerate economic growth in low-income economies.

• Infact, the Harrod - Domar model provided a framework for economic planning in developing economies, such as India's Five Year Plan.

• Suppose θ= 4 and s = .02 (20%).

• The growth rate would then be 20/5 = 5%. These numbers in fact roughly describes the Indian economy in the 1980s. Policy makers in India argued how India needed to increase its savings rate or make capital more productive (i.e. lower θ)

• H-D model links growth rate of the economy to two fundamental variables:

Ability of the economy to save.

Capital output ratio.

Higher saving rate would push up the economy.

Increasing the rate at which capital produces output (a lower θ ), growth would be enhanced.

What causes growth in this model?

How does the Harrod-Domar model conceive of growth?

• Expanding yields the approximate equation:

• Ability to save and invest (s)

• Ability to convert capital into output (θ)

• Rate of capital depreciation (δ)

• Rate of population growth (n)

ngs */

• For developing countries, the key to successful development is increasing the rate of savings.

• Capital created by investment is the main determinant of growth.

• Saving makes investment possible.

• The ‘tricks’ of economic growth, according to this model, are simply a matter of increasing savings and investment.

• The main obstacle to or constraint on development then is the relatively low level of new capital formation or investment in most LDCs.

The Harrod-Domar Model • Consequences

– Saving as crucial for growth

– The preceding result is valid as long as there is no labor shortage. If n = s/c population, capital and income will grow at the same rate.

– Knife-edge dynamics • If n>g , then chronic unemployment

• If n<g , then chronic labor shortages, capital becomes idle.

– No endogenous process to bring the economy to equilibrium

What are the problems with the Harrod-Domar model?

• Model assumes economy grows forever.

• Saving as sufficient.

• No diminishing returns; no factor substitution.

• No technological change.

• All three factors (s, n and θ) are given facts of nature .

• Constancy of capital output ratio .

• Thus, only capital and not labour contributes to production.

• Labour and capital are not substitutable in Production.

• Output does not increase by applying more labour for a given stock of capital.

• Harrod Domar Model is a neutral theory of economic growth.

39

Beyond Harrod- Domar model

• Endogeneity of savings – Savings are influenced by per capita incomes and

distribution of income in an economy – Both of these are influenced by economic growth – Economic growth mirrors the movement of savings

with income

• Endogeneity of population growth – Relationship between demographic transition and

per capita income – External policy can prevent an economy from sliding

in to a “trap” (process of demographic transition)

• Endogeneity of capital-output ratio – Captured in Solow’s model

Solow Model

• Solow altered the Harrod - Domar story by making the capital-output ratio endogenous.

• Solow’s model is based on the diminishing returns to individual factors of production.

• Capital and Labor (L) are both needed to produce output.

• K/Y ratio is no longer fixed but depends e.g. on the economy with relative endowments of capital and labor.

Assumptions:

• Savings rate (ratio of savings to income), s, is constant. Savings will be channelled into investment as previously.

• Population growth rate is constant:•L/L = n, where L denotes population or labor.

• There is perfect competition: the firm takes the market wages on labor and rents on capital as given.

• Constant returns to scale. If labor and capital inputs are doubled, the output gets doubled as well.

• The Solow model is built around two equations, a production function and a capital accumulation equation.

• The production function is assumed to have the Cobb-Douglas form and is given by

• Y=F(K,L)= KαL1-α

• where a is some number between 0 and 1

• Production function exhibits constant returns to scale: if all of the inputs are doubled, output will exactly double.

• Perfect competition prevails and the firms are price-takers.

• w=workers wage

• r= rent payment to capital.

• Profit maximizing condition implies:

• max F(K, L) - rK - wL.

• First-order conditions implies:

• Firms will hire labor until the marginal product of labor is equal to the wage and will rent capital until the marginal product of capital is equal to the rental price:

• w = = (1-α)

• r= = α

• wL + rK = Y; payments to the inputs ("factor payments") completely exhaust the value of output produced so that there are no economic profits to be earned.

• share of output paid to labor is:

• wL/Y = 1 - α

• the share paid to capital is:

• rK/Y = α

• These factor shares are constant over time.

• Rewrite the production function as

• output per worker, y = Y/L,

• capital per worker, k = K/L:

ky

• Equation on Capital Accumulation:

• • • =Kt+l – Kt

= sY = gross investment; dK = depreciation.

dKsYK .

• we assume that workers/consumers save a constant fraction, s, of their combined wage and rental income, Y = wL + rK.

• a constant fraction, d, of the capital stock depreciates every period (regardless of how much output is produced).

• Rewrite the capital accumulation equation in terms of capital per person.

• The production function in equation will tell us the amount of output per person produced for whatever capital stock per person is present in the economy.

• k= K/L= log k= log K-log L

• Also, , so that

• Or,

• Capital accumulation equation in per worker terms:

• This equation says that the change in capital per worker each period is determined by three terms:

Investment per worker, sy.

depreciation per worker, dk.

population growth n.

SOLVING THE BASIC SOLOW MODEL

1.......................ky

2...................)(.

kdnsyk

• The first equation shows how output is produced from capital and labour.

• The 2nd equation shows how capital is accumulated over time.

• Y and K are endogenous variables, so is y and k .

• Solving a model means obtaining the values of each endogenous variable when given values for the exogenous variables and parameters.

The Solow Diagram:

(n+d)k

Investment: sy

Investment, depreciation

Capital, k k*

Net investment

k0

Steady state.

• The first curve is the amount of investment per person, sy = skα. This curve has the same shape as the production function.

• The second curve is the line (n + d)k, which represents the amount of new investment per person required to keep the amount of capital per worker constant .

• The difference between these two curves is the change in the amount of capital per worker.

• When this change is positive and the economy is increasing its capital per worker, we say that capital deepening is occurring.

• When this per worker change is zero but the actual capital stock K is growing (because of population growth), we say that only capital widening is occurring.

• At ko, the amount of investment per worker exceeds the amount needed to keep capital per worker constant, so that capital deepening

• occurs-that is, k increases over time.

• This capital deepening will continue until k = k*, at which point sy = (n + d)k, so that = 0.

• At this point, the amount of capital per worker remains constant, and we call such a point a steady state.

k

• At points to the right of k* the amount of investment per worker provided by the economy is less than the amount needed to keep the capital-labor ratio constant.

• The term k is negative, and therefore the amount of capital per worker begins to decline in this economy.

• This decline occurs until the amount of capital per worker falls to k*.

• The Solow diagram determines the steady-state value of capital per worker.

• The production function of equation then determines the steady-state value of output per worker, y*, as a function of k*.

The Solow Diagram and the Production Function

Investment, depreciation,

and output

Capital, k

Y0

k0

y*

k*

Consumption

sy

Output: Y

(n+d)k

• Long run capital output ratio must be constant.

• Per capita capital stock settles to some steady state.

• Per capita income also settles to some steady state.

• Thus, no long run growth of per capita output.

• Total output grows at the rate of growth of population.

• Savings do no have any long run effect on the rate of growth rate of per capita income. But affects the L-R level of income.

• Diminishing returns to capital creates endogenous changes in capital output ratio.

• This chokes off growth.

How Parameters affect the steady state

• What happens to per capita income in an economy that begins in steady state but then experiences a "shock."

• Increase in the saving rate, s.

• Increase in the population growth rate, n.

Investment, depreciation

Capital, k

New investment

exceeds depreciation

Depreciation: d K

k** k*

Old investment: sy

An Increase in the Saving Rate

s’y

A Rise in the Population Growth Rate Investment, depreciation

Capital, k k** k*

sy

(n+d)k

(n’+d)k

PROPERTIES OF THE STEADY STATE

1

1

*

.

.

)(

0

)(

dn

sk

k

kdnskk

In steady state =0 k

• Substituting the above in the production function:

• Thus, we have a solution for the model at steady state.

1* )(

dn

sy

• This equation reveals the Solow model's answer :

• "Why are we so rich and they so poor?"

• Countries that have high investment rates will tend to be richer, ceteris paribus.

• Countries that have population growth rates, in contrast, will tend to be poorer.

• A higher fraction of savings in these economies must go simply to keep the capital-labor ratio constant in the face of a growing population

• This capital-widening requirement makes capital deepening more difficult, and these economies tend to accumulate less capital per worker.

Empirically, countries with higher investment rates have higher capital to output ratios:

ECONOMIC GROWTH IN THE SIMPLE MODEL

k/k

1 skk

sy

n+d

k* k

• What does economic growth look like in the steady state of this simple version of the Solow model?

• There is no per capita growth in this version of the model!

• Output per worker is constant in the steady state.

• Output itself, Y, is growing, of course, but only at the rate of population growth.

• there is no long-run economic growth in the Solow model.

• in the steady state: output, capital, output per person, and consumption per person are all constant and growth stops.

• An economy that begins with a stock of capital per worker below its steady-state value will experience growth in k and y along the transition path to the steady state.

• Over time, however, growth slows down as the economy approaches its steady state, and eventually growth stops altogether.

• The further an economy is below its steady-state value of k, the faster the economy grows.

• Also, the further an economy is above its steady-state value of k, the faster k declines.

• empirically, economies appear to continue to grow over time

– thus capital accumulation is not the engine of long-run economic growth

– saving and investment are beneficial in the short-run, but diminishing returns to capital do not sustain long-run growth

– in other words, after we reach the steady state, there is no long-run growth in Yt (unless Lt or A increases)

Level effects and Growth effects.

C

D F

B

A

E

Time

(Lo

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• A growth effect : Changes the rate of growth of a variable.

• A level effect leaves the rate of growth unchanged.

• Savings rate has a level effect only in the Solow model.

• Savings rate, s, and population growth ,n, only has level effects.

Solow Model with Technology

• To generate sustained growth in per capita income , we must introduce technological progress to the model.

• Y = F(K, AL) = Kα(AL)1-α

• Technology variable: A

• The technology variable A is said to be "labor augmenting“. Technological progress occurs when A increases over time - a unit of labor, for example, is more productive when the level of technology is higher.

exogenous technical progress

• Consider the labour-augmenting production function:

• Technical progress occurs when A rises over time, with labour becoming more productive when the level of technology is higher.

• Let,

Y F(K,AL)

gA

A

• Capital accumulation equation:

• Production function in terms of output per worker:

• y= kαA1-α

• Taking log and differentiating:

dK

Ys

K

K

A

A

k

k

y

y )1(

• From the capital accumulation equation we can see that the growth rate of K will be constant if and only if Y / K is constant.

• If Y / K is constant, y/k is also constant.

• y and k will be growing at the same rate.

• A situation in which capital, output, consumption, and population are growing at constant rates is called a balanced growth path.

• Let, g, to denote the growth rate of some variable x along a balanced growth path.

• Then, along a balanced growth path, gy = gk

• Recalling that,

• gy=gk =g

gA

A

• Along a balanced growth path , output per worker and capital per worker both grow at the rate of exogenous technological change.

• No technological progress, and therefore there was no long-run growth in output per worker or capital per worker; gy = gk = g = 0.

• Technological progress is the source of sustained per capita growth.

THE SOLOW DIAGRAM WITH TECHNOLOGY

• k is no longer constant in the long run.

• The new state variable will be:

• = ratio of capital per worker to technology or capital technology ratio and is equal to k/A and constant along the balanced growth path.

ky~~

k~

AL

Kk ~

A

y

AL

Yy ~

y~Output technology ratio

• Rewriting the capital accumulation equations in terms of

• Combining it with capital accumulation equation:

k~

L

L

A

A

K

K

k

k ~

~

kdgnysk~

)(~~

the Solow diagram with technical progress

f (k)

k

(d n g)k

ys~

0

~k

*

~k

• Capital techology ratio converges to a stationary steady state.

• Long –run increase in per capita income takes place at the rate of technical progress.

Steady-state income and growth • Setting

• Substituting this into the production function gives

1/(1 )

* sk

n d g

/(1 )

* sy

n d g

0~k

• Recall the capital accumulation equation :

• This can be re-written as

k sy (n g d)k

k ys (n g d)

k k

An Increase in Investment

ys~

ys ~'kdgn~

)(

**~k

*~k

k~

a rise in the saving rate

k

n+g+d

1s 'k

*k **k

1sk

k / k

THE EFFECT OF AN INCREASE IN INVESTMENT ON GROWTH

y

y

Time

g

*t

THE EFFECT OF AN INCREASE IN INVESTMENT ON y

Level Effect

*t Time

Log y

EVALUATING THE SOLOW MODEL

• How does the Solow model answer the key questions of growth and development?

• First, the Solow model appeals to differences in investment rates and population growth rates and (perhaps) to exogenous differences in technology to explain differences in per capita incomes.

• Why are we so rich and they so poor?

• Invest more and have lower population growth rates, both of which allow us to accumulate more capital per worker and thus increase labor productivity.

• Second, why do economies exhibit sustained growth in the Solow model?

• Technological progress.

• Without technological progress, per capita growth will eventually cease as diminishing returns to capital set in.

• Technological progress, can offset the tendency for the marginal product of capital to fall.

• In the long run, countries exhibit per capita growth at the rate of technological progress.

• How, then, does the Solow model account for differences in growth rates across countries?

• Transition dynamics can allow countries to grow at rates different from their long-run growth rates.

• An economy with a capital-technology ratio below its long-run level will grow rapidly until the capital-technology ratio reaches its steady-state level.

• The principle of transition dynamics says that the farther below its steady state an economy is, in percentage terms, the faster the economy will grow

• Similarly, the farther above its steady state, in percentage terms, the slower the economy will grow

• This principle allows us to understand why economies may grow at different rates at the same time

How do we understand the discrepancy between Harrod-

Domar and Solow models?

• In the former the savings rate, for example, affected the long run growth rate, while in the Solow model savings rate does not affect the growth rate.

• There is no sustained growth to begin with in this version of the Solow model!

• This is because there are diminishing returns to capital, which create endogenous changes in capital-output ratio.

• This chokes off growth in the Solow model. If capital were to grow faster than labor, each unit of capital had less labor to work with it, and the output per unit of capital would be reduced.

• There can be no steady state growth.

Unconditional Convergence

(Lo

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Time

B

A

AB plots the time path of per capita income at the steady state.

• If Countries in the long run have the same rates of technical progress, savings, population growth and capital depreciation .

• Solow model predicts that all countries , capital per efficiency unit of labour converges to the common vales of k*.

• If a country starts below the steady state level per efficiency unit , the country will initially display a rate of growth that exceeds the steady state level and over time growth will decelerate to steady state level. Vice versa.

• Convergence shows a strong relationship between growth rate of per capita income and the initial value of per capita income.

• Poor countries grow faster than the rich ones.

• A country which is poor initially (with low per capita income and capital stock) will then grow faster than the rich country.

• Relative income differences between countries must die away in the LR.

Evidence

• Option 1. Small number of countries, long horizon.

• Option 2. Large number of countries, short horizon.

Empirical evidence

• The growth rates of homogenous countries do converge more clearly than the growth rates of non-homogenous countries (the US states, OECD vs. the world).

• Homogenous countries are more likely to have ’the same’ steady state.

Conditional convergence: • Unconditional convergence assumes all

parameters are the same.

• Parameters, population, capital depreciation, savings differ.

• The steady state could be different from country to country.

• Countries need not converge to other.

• Weaker hypothesis called conditional convergence.

• Assume that knowledge flows freely across countries.

• Technical progress determines the growth rate of per capita income in the long run.

• This leads to the prediction of convergence in growth rates.

• Long run per capita incomes vary from country to country.

• Long run per capita growth rates off all countries are predicted to be the same.

(Lo

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Time

• Conditional convergence implies convergence in growth rates.

• Growth rate convergence implies that a country that is below its own steady state grows faster than its steady state growth rate.