II' Endogenous Growth

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II. Endogenous Growth

• Part I
• i. The AK model
• ii. A Simple Model of Human Capital

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Recap Solow and Ramsey

• Solow and Ramsey models share commons
• DR to K holding AL constant
• Economy eventually settles down to a s.s. growth
  path in which kK/AL is constant and where the
  only source of growth in output per capita is
  exogenous technical progress
• Convergence hypothesis

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How to amend their models?

• Redefine the capital stock
• increase the capital share to eliminate the
  counterfactual implications for fast convergence
  and high real interest rates in developing
  countries
• Make technology endogenous by
• incorporating RD
• studying technology adoption

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i. The AK growth model

• AK model YAK
• A reflects the level of technology
• CR to K the simplest version of production
  function without diminishing returns
• A model of perpetual growth through capital
  deepening rather than innovation
• Good introduction to the macro. mechanics of new
  growth models.

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Assumptions and simple math

• Closed economy
• No technical progress
• Population size is constant or ngt0
• Infinitely-lived representative agent

To show that per capita growth can occur in the
long run even without exog tech change
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sA
gk gt 0
nd
k
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No absolute or conditional convergence

• Countries with identical saving and depreciation
  rate, population growth rate, and technology
  level will grow at the same rate but there level
  or GDP or capital per capita is based on their
  initial levels, k0.

k
k0,H
k0,L
time
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No convergence in the AK model

• Speed of convergence
• For a Cobb-Douglas
• The speed of convergence
• Now the AK model

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AK Ramsey

• Level of tech is constant
• Utility function
• Assuming ndgA0 no externality central
  planner problem

s.t.
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Optimal growth of consumption
The current-valued Hamiltonian

FOC
Euler equation of consumption
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Under the restrictions on parameters, the s.s.
has perpetual consumption growth. Such a s.s. is
feasible provided the k and y grow at the rate gc
as well.
Balanced growth
In s.s., c/k, must be constant thus c, k and y
grow at the same rate. In s.s. gcgkgy
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In steady state gc gk

• Higher g and s when (s q-1(1-r/A))
• The higher the MPk(A) relative to r, the more it
  pays to depress current c to enjoy higher future
  c
• the more willingness to substitute
  intertemporally (small ?)
• Higher A
• The economy reaches its s.s. growth path
  immediately, i.e., there is no transitional
  period (since production function is linear in K,
  so tie down the interest rate, MPk, independently
  of K)
• Market outcome CE PO (since no externality) or
  private MPk social MPk

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A Simple Model of Human Capital

• AK model can be thought as a reduced-form
  representation. An example with physical and
  human capital.
• Assume Y AKaH1-a A (H/K)1-aKAK
• Output can be used on a one-for-one basis for
  consumption, investment in physical capital and
  investment in human capital.
• For simplicity, let depreciation be the same
  (rate d) for both types of capital.

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ii. A Simple Model of Human Capital

• In equilibrium RK RH r d.
• Firms optimization
• RK aA(H/K)1-a (1 -a)A(H/K)-a RH
• Thus, in equilibrium,

All variables in the economy grow at the constant
rate
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From AK to Lucass model

• NGT ? K flows from rich to poor countries
• But in fact, it is not the case! Why?
• Physical K is complementary to skills (human K)
  and people in poor countries dont have the
  required skills.
• Model the endogenous skill accumulation by LBD
  (skills grow as a function of past production or
  investment) or LBS (spend time in receiving
  education)

schooling
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Simplified Lucass model

• Suppose people live forever and that a person's
  human capital evolves according to
• where u(t) is the fraction of time spent not
  producing. Notice the linearity constant returns
  in the accumulation technology.
• Suppose each individual rents capital and has her
  own backyard technology. Her output is

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Simplified Lucass model

• There are no externalities here, so the
  equilibrium coincide with the allocation chosen
  by a social planner the second welfare theorem
  applies.
• So we proceed to solve the planner's problem. It
  can be written as follows.
• subject to

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Simplified Lucass model

• H(0) and K(0) are given set d 0 to further
  simplify things. The Hamilatonian is
• The optimality conditions are

MPL
MPH
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• As Lucas says, time should have the same return
  in both uses working and studying, and the
  consumption good should have the same return in
  both uses eating and accumulating.
• Introducing w(t) to denote the MPL (where L
  (1-U)H) and r(t) to denote the MPK, we get
• From (7), we get

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Simplified Lucass model

• From (5) we can deduce
• Combining these two equations, we get
• Interpretation consumption should grow when
  return to waiting is greater than the (utility)
  cost of waiting.

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• Now let's focus on characterizing a balanced
  growth path, where U(t) U (constant). Then
• and, by definition of a balanced growth path,
• On such a balanced growth path, w(t) and r(t)
  will be constant. Meanwhile, equation (6) says
  that
• But then equation (8) becomes
• and, when w is constant,

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Simplified Lucass model

• Meanwhile, since C and H are growing at the same
  rate,
• The long-run growth rate is

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Important Implications

• 1. Growth is increasing in the productivity of
  the human capital production sector and
  decreasing in the rate of patience.
• 2. Notice how we assumed that existing human
  capital is an input in the production of new
  human capital, and that returns are constant.
  This is the engine of sustained growth.
• 3. Be consistent with the fact that capital does
  not seem to flow from rich to poor countries.

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Important Implications

• We also get a further implication the rate of
  growth should be positively related to the
  fraction of time spent in school. The evidence is
  mixed but broadly supportive of this view. Hong
  Kong versus Mali. 
• A problem with this human capital model is that
  it is inconsistent with the obvious migration
  pressures that we observe. Lucas deals with this
  by assuming that returns to human capital are to
  some extent external. But why should the scope of
  this externality have any regard to national
  borders?

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Still confused

• But what determine the level of technology?
• Spillover / externality
• No explicit production of A
• A is generated indirectly by investment and thus
  accumulation of capital
• RD
• Explicit production of A
• A is produced by the RD sector