Lecture 14

Malthusian Model

Economics Growth

Solow model explains all growth facts for industrialized countries

Why are many poor countries not growing?

Why did all countries stagnate before 1800?

The Malthus Model

1798:"Essay on Population" Explains stagnation as resulting from

limited supply of land and population growth

Outline

Production function uses land and labor People decide on number of children Population growth offsets productivity

growth

Differences to Solow Model

Solow model does not consider land as an input

Solow model takes population growth as constant

Structure of the Model

Consumers work on their own farm (no firms in the model) and decide to consumption and fertility

Each consumer owns an equal share of land

Law of motion for population is determined by fertility decisions

The Consumer Consumer cares about consumption and

number of children Consumer works one unit of time, owns

equal share of land, and keeps farm output

Each child eats units of consumption

tc

tn

tx

ty

b

The Maximization Problem

Consumer takes income as given and chooses and

ty

tc tn

ttt

tt

bnyc

ts

ncMax

..

)log()log(

The Solution

Plugging in the constraint:

First-order condition:

)log()log( ttt nbnyMax

0/1)/( ttt nbnyb

)2/( byn tt

Result

Number of children depends positively on farm income

The richer the consumer is, the more children she will have

Land and Population

Total amount of land X is fixed; for simplicity, assume X=1

Population size is denoted by Amount of land per person:

tP

ttt PPXx /1/

The Farm

Farm technology is Cobb-Douglas in labor and land (z is productivity):

One unit of labor is supplied ( ) and land per person is

tl

tx

1tltt Px /1

at

att lzxy 1

at

att zPPzy )/1(

The Law of Motion for Population

Population tomorrow is population today times children per person:

Plugging in actual

Plugging in actual

ttt nPP 1

tn)2/(1 byPP ttt

ty

)2/()2/( 11 bzPbzPPP a

ta

ttt

Convergence in Population

Computing the Steady State Find steady-state population :

Find steady-state income :

Income is independent of z!

aSS

aSSSS

bzP

bzPP/1

1

))2/((

)2/(

SSP

SSy

bzPy aSSSS 2

The Malthusian Mechanism

Fertility in the steady state:

Income adjusts such that each person has one child (zero population growth)

Therefore income adjusts to the same level regardless of productivity

12/ byn SSSS

Explanation for the Result

When income in high, people have many children and population growth is high

High population growth lowers income until each parent has just one child

What Happens if Productivity Increases?

Increase in z will lead to higher income and higher population growth

Higher population growth pushes incomes down

Ultimately, a new steady state is reached with higher population, but same income per person

A jump in productivity

Long-Run Predictions Income per capita will be roughly constant

over time Increase in productivity will increase world

population without raising living standards Inventive countries and regions will have

higher population growth

Why is the Solow Model Different? The Solow model does not have

endogenous fertility-population growth assumed to be constant

The Solow technology does not use land-population increases do not lower income per capital

Why Do We Use Both Models?

The Malthus model is an accurate description of the world until 1800

--Agriculture and land were important --Income and fertility were positively related The Solow model is an accurate description of

industrialized countries --Agriculture is small sector today --Fertility no longer closely related to income

How Did We Get From Malthus to Solow?

Two promising approaches: --Technological change replaced land-

based technology by capital-based technology

--People started to invest in human capital instead of having many children

Technological Changes and Growth Industrialization replaced land-based

technology by capital-based technology Since the importance of land diminishes,

returns to population size no longer decreasing

Growth in income per capita is possible even as population is growing

The Malthus Model with a Constant-Returns Technology Consumer:

Firm:

ttt

tt

bnyc

ts

ncMax

..

)log()log(

tt zly

Results

Fertility Choice:

Output per person:

population growth as a function of z:

)2/( byn tt

tt zy

)2/( bzn tt

Implication

Productivity growth (increase in z) is not offset by population growth

Income per capita increases with z, as in the Solow model

However, population growth ever increasing

Can get transition in theory, not in practice

The Demographic Transition

The income-fertility relationship also changed

In late 19th century, mortality rates and fertility rates fell steeply in England, U.S., and Germany

Today richer people have fewer children All industrialized countries experienced

demographic transition

The Malthus Model with Fixed Fertility Exogenous limit on fertility:

Decreasing-returns technology:

1tn

at

att lzxy 1

Results

Constant population:

Constant labor ( ) and land per person( )

Resulting output per person:

01 PPnPP tttt

1tl0/1 Pxt

aat zPPzy 00 )/1(

Implication

Once again, income per capita increases with z, as in the Solow model

However, share of land stays constant In data, share of land in output declining

over time True explanation is probably a

combination

Why Did Fertility Fall?

Increased cost of children (education, no more child labor)

Low mortality Social security Increased female labor-force participation

The Current Situation in Developing Countries

Many developing countries are still in between Malthus and Solow

Agriculture plays a more important role Less education, less social security, less

female labor force participation, and more child labor result in higher fertility and population growth

Income per Worker vs. Population Growth

Income per Capita vs. Share of Agriculture

Additional Problems in Developing Countries

Lack of protection of property rights Inefficient government and corruption Wars and civil conflict Tropical diseases and AIDS Lack of human capital

Economics for the 21th Century

What are the prospects for world economy in the next 100 years?

Key questions: --Are there reasons why growth

might stop in developed countries? --Will developing countries start to

be more like industrialized countries?

Educated Guesses

No reason why Solow model should cease to work-continued growth in rich countries is likely

Fertility is falling fast, and the role of agriculture is declining-more developing countries should start to grow

Expect convergence around the world

Birth Rates in a Few Developing Countries