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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