Growth Models including trade offs in Population and Human Capital:

1
Growth Models including trade offs in Population
and Human Capital
2
Overview

• Stylized facts
• Malthus
• Human capital, fertility and economic growth
  (Becker, Murphy, Tamura, JPE 1990)
• Population, technology growth from Malthusian
  stagnation to the demographic transition and
  beyond (Galor and Weil, AER 2000 and Galor,
  Handbook of EG 2005)

3
Stylized Facts

• Malthusian regime continued until the beginning
  of the industrial revolution (i.e. 1760).
• Great divergence some countries are still
  comparatively poor.

Source Galor 2004
4
Stylized Facts

• Estimated growth rate of GDP between 500-1500 is
  zero. (Angus Maddison 1982)
• Real wage in England 1800 was the same as in
  1300. (Ronald D. Lee 1980)
• Real wages in China were lower at the end of
  eighteenth century than at the beginning of the
  first century. (Kang Chao, 1986)
• Sustained growth in living standard is an anomaly
  even in the richest countries. (Mokyr 1997,
  Pritchett 1997, Lucas 1999)

Source Galor and Weil 2000
5
Stylized Facts
Source Galor 2004
6
Stylized Facts

• Demographic transition

Source Galor 2004
7
Stylized Facts
Source Galor 2004
Transition is not immediate Europes share in
world population grew during the 19th century,
but decreased later.
8
Stylized facts

• 0.1 per year population growth from the years
  500 to 1500
• .064 per year growth of world population from
  the year 1 to 1750 (Massimo Livi-Bacci 1997)

Source Galor and Weil 2000
9
Stylized Facts

• Low fertility is associated with high per-capita
  growth rates.
• Low fertility is associated with higher human
  capital.
• Demographic transition Initial increase in
  population growth with higher income, and then
  decrease in fertility throughout development
  process.

10
Stylized Facts

• Demographic transition is accompanied by other
  changes - health

Source Galor 2004
11
Stylized Facts

• Demographic transition is accompanied by other
  changes - health

Source Galor 2004
12
Stylized Facts

• Demographic transition is accompanied by other
  changes health

Source Galor 2004
13
Stylized Facts

• Demographic transition is accompanied by other
  changes education

Source Galor 2004
14
Stylized Facts

• Demographic transition is accompanied by other
  changes education

Source Galor 2004
15
Stylized Facts

• Payments to factors of production

Source Galor 2004
16
Malthus Basics of the model

• Important input of production which is fixed(e.g.
  Land)
• Thus there is decreasing returns to scale for all
  other inputs
• Increases in living standards positively increase
  population growth.

Source Galor 2000
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Malthus Innovation Shock
Consumption per worker, c
Land per worker, l
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Malthus N Growth Rate
Population Growth
Consumption Per Worker, c
19
Malthus Population
Population, N
N2
N1
T Time
Williamson, Macroeconomics 2nd ed. Page 183
20
Malthus per capita consumption
Consumption per worker, c
T Time
Williamson, Macroeconomics 2nd ed. Page 183
21
Malthus Good for its time

• Prior to the industrial revolution (approx.
  1800) the Malthusian model explain economic
  growth fairly well.

22
Evidence for Malthus
Malthus to Solow page 1207
23
Malthus More Evidence
Malthus to Solow page 1208
24
Malthus Breakdown of correlation
Malthus to Solow page 1208
25
Malthus Summing it up

• According to Malthus, when population size is
  small, the standard of living will be high, and
  population will grow as a natural result of
  passion between the sexes. When population is
  large, the standard of living will be low, and
  population will be reduced by either the
  preventive check (intentional reduction of
  fertility) or by the positive check
  (malnutrition, disease, and famine).
  (Source Galor 2000)

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Malthus result

• Stagnant growth constant per capita income
• Technological improvement leads to higher
  population density but not living standards

Source Galor and Weil 2000
27
Please explain

• We observe two types of countries
• Countries with low fertility and high output and
  HC,
• Countries with high fertility and low output and
  HC.
• Implying multiple equilibria
• Children cost money, but yetexistwhy?
• Utility from children (endogenous fertility)
• Cost of children (time\money)
• Capital doesnt flow to poor countries HC gaps.
• Many children low investment in HC of each.

Slide Source Aharonovitz
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Human Capital, Fertility Economic Growth

• Becker, Murphy, Tamura, JPE 1990 (hereafter BMT)
• Analyze the link between endogenous fertility
  (which is missing from the classical models) and
  HC (which is missing from Malthus).
• Explain the existence of multiple steady states
• High fertility, low HC
• Low fertility, high HC

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BMT Basic Properties

• This is and overlapping generations model.
• People live for two periods Childhood and
  Adulthood.
• Childhood is spent investing in HC
• At the beginning of Adulthood, the parent chooses
  to have children. And pays hours and
  units of consumption per child.

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BMT Basic properties

• Children cost time and money (i.e. consumption)
• There are substitution and income effects that
  influence the parents fertility decision.
• Substitution effect Raising children cost time
  and money, increased wage influences parents to
  substitute away from children to work.

31
BMT Basic Properties

• Income effect Children are a good in the utility
  function. So higher income makes parents
  consume more of them (i.e. increase fertility)
  
• These income vs. substitution can be thought of
  as being analogous to the those in the
  labor-leisure models that we all know and love.
• Take out the leisure and replace with children.

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BMT Basic Properties

• Human capital has increasing returns to scale
• Not like physical capital (DRS)
• We see this on an individual basis, we must learn
  to do arithmetic and algebra before we learn
  about calculus. Step by step.
• We have increasing returns over the initial
  concepts learned.

33
BMT Basic Properties

• As concepts become increasingly more difficult
  learn we begin to see decreasing returns to HC.

34
BMT Basic Properties
35
BMT Equations

• Utility
• The parents utility from current period
  consumption
• The parent gains utility from each childs future
  consumption via

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BMT Equations

• The parent discounts the future utility of
  children at a rate of
• This discount rate has the property
• Further we have the property

37
BMT Equations

• There are two production functions
• One for production of HC which is HC intensive
• One for production of consumption goods
• There is a time budget constraint

38
BMT Equations

• For convenience bdb1
• Our equations give us the following Euler
  equation
• If we collapse the three constraints we can
  substitute into the value/utility function and
  maximize in the usual fashion.

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BMT Equations

• From the maximization process we can find the
  following
• The rate of return form investing in human
  capital
• The first order condition
• Utility from an additional childCost of an add.
  child

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BMT Concluding Remarks

• Our analysis indicates that Malthusians have a
  myopic view that is inappropriate when economies
  manage to diverge enough from the undevelopment
  trap. Economies would continue to develop and
  diverge from that steady state if technological
  and other shocks either raise the policy
  functions above the steady-state or raise the
  stock of human and physical capital sufficiently

Source BMT 1990 page S32
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BMT Concluding Remarks

• Can get equilibria without diminishing returns or
  without unstable steady state

42
BMT Concluding Remarks

• Considerable luck is needed in the timing and
  magnitude of shocks to give a sufficiently big
  push to investment in human and physical capital.
  But very unlikely configurations of events do
  occur in the course of thousands of years of
  history. We believe that the West's primacy,
  which began in the seventeenth century, was
  partly due to a lucky timing

Source BMT 1990 page S33
43
BMT The Problem

• B.M.T. multiple equilibria
• Two different situations stagnation vs. growth
• But the developed world stepped out of one
  steady state to the other.
• There was no external shock, stepping out was
  gradual.
• Need for a unified theory.

44
Unified growth theory

• These theories attempt to incorporate into one
  model the epoch of Malthusian stagnation, the
  era of modern growth, and the principal factors
  that brought about the transition between these
  regimes.

Source Galor Handbook of EG 2005 page 54
45
Unified growth theory Solutions

• There is a latent variable which is growing in
  the back ground.
• This variable is growing but is not contributing
  to per capita growth during the Malthusian
  period.
• After a certain level of stock in the variable is
  achieved then the variable begins to contribute
  to per capita growth.

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Unified growth theory Basics

• Building Blocks
• Malthusian Elements
• The Engines of Technological Growth
• Origins of Human Capital Formation
• Determinants of parental choice regarding
  quantity and quality of offspring

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Unified growth theory Basics

• Malthusian Elements
• There is a subsistence consumption constraint
• When binding, increases in income increase the
  quantity of children. (i.e. the income effect
  dominates)
• Improvements in technology which increase income
  thus trigger population growth and reduce
  completely any increase in income per capita.

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Unified growth theory Basics

• The Engines of Technological Growth
• Population size The bigger your population the
  more likely one of them will come up with a good
  idea.
• Closer to Tech Frontier
• Human capital
• More important when far from the frontier

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Unified growth theory basics

• Origins of human capital formation
• Skill biased increase the returns to HC
• Long run skill biased or skill saving

50
Unified growth theory basics

• Parental choice regarding quantity and quality
• Parent gain utility from the quantity and quality
  of children
• Quality and quantity of children chosen while
  considering time constrains
• Time is devoted to labor market activity and
  child rearing
• Rises in the demand for human capital induce
  parent to choose quality over quantity

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Technology-Human Capital Cycle
Growing population
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Unified Growth theory Basics

• Overlapping generations model
• Infinite and discrete time
• A single homogeneous good produced every period.
• Inputs land and efficiency units of labor.
• Lands is fixed over time
• Number and Human capital of children determined
  in previous period by the parent.

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Unified growth theory Production

• Constant-returns-to-scale technology production
  function
• Xt is land
• Ht is effective labor
• This production function is subject to endogenous
  technology growth,

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Unified growth theory Production

• Output per capita
• Where
• And

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Unified growth theory Production

• The Wage
• Which calculated by making the assumption returns
  to land are zero, and there is no capital in the
  model. This allows for tractability.
• Then we have the zero profit condition.

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Preferences

• Utility is determined by the function
• Parents gain utility from consumption, ct , which
  needs to be above subsistence,
• And from the aggregate future income of their
  children,
• The parent chooses nt and ht1 . Then supplies
  the remaining time in the labor market for
  consumption

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Budget constrain

• Cost of a child
• is the faction of time to raise a child
  regardless of the education/quality of the child
• et1 is the additional time fraction to raise a
  child of quality level et1
• Thus we have
• Where zt is defined to be the maximum amount of
  consumption if nt 0

58
Production of Human Capital

• We have an implicit function
• This function is a increasing strictly concave
  function of the parent investment in the childs
  education, et1
• It is a decreasing strictly convex function of
  technology growth,

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Production of Human Capital

• This gives two interesting properties
• The first tells that as g increases that h
  actually falls. This means that human capital
  becomes obsolete as technology advances
• The second tells us that the affect of increasing
  technology growth can be mitigated by education.
  

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Production of Human Capital

• It should also be noted that each person has an
  innate level of human capital such that

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Optimization

• We substitute the budget constrain and the
  equation of motion for effective labor into the
  utility function, and we have
• Subject to

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Optimization

• The solution for nt
• is the minimum amount of income necessary to
  achieve subsistence income.

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Optimization
64
optimization

• Our solution for nt and the optimization for et1
  gives us a relationship between et1 and gt , the
  parents choose of et1 . Here it is stated
  implicitly as
• Note that is a threshold level of technology.

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Optimization

• Further note that the derivatives of previous
  function imply that parents will choose a higher
  level of time educating children as the rate of
  technological growth increases. This in turn
  reduce the number of children. There is also the
  additional property of diminishing marginal
  returns

66
Opitmization

• We can substitute this solution for et1 back
  into the solution for nt and we have

67
Optimization

• And so we have the following three propositions

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Technological progress

• Recall that technological progress, or gt1
  increases with et and with Lt

Rearranging we get
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The latent variable Tech growth

• That neither gt1 or et1 are dependent upon the
  subsistence level of consumption.
• This is important because it allows them to grow
  independent of the constraint.
• Eventually increasing output and quality of
  children
• And explaining the decrease in nt

70
Population growth
We use the above equation and apply our newest
solution for nt and find the below equation
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Effective resources (Land)

• Recall our definition
• To this we can apply our equations for Lt , At ,
  and nt and arrive at the following equation

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A System of Dynamic Equations
73
Low L
74
Moderate L
75
High L
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The Dynamical System summing up

• The economy shifts from one system to the other.
• Initially low investment in HC, constant
  resources per capita, but population increases.
• As population increases technological change
  speeds up.
• Eventually, when g is fast enough, parents invest
  in HC and resources per capita increases.
• When subsistence is not binding HC and
  technology grow even faster.

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Conclusions

• Explained the transition.
• HC and technological change are circular.
• Help speed up the transition in other places?
• Fertility quantity vs quality
• Quality (HC) affects growth
• Problem 1 why do we find differences in
  technology?
• Problem 2 too many technical assumptions (HC
  decreases with g).

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Critiquing their own model Culture

• Differences between countries in the
  determination of population growth or in the
  process of technological change (as a result of
  institutions and cultural factors, for example)

79
Critiquing their own model Public Schooling

• Similarly, differences in policies such as the
  public provision of education, would change the
  dynamics of the model.

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Critiquing their own model Ghost Acres

• Through the 1800s
• Large out flow of migrants from Europe
• Large in flow of migrants to the U.S.
• Large in flow of grain from the U.S. to Europe