IV' THE SCHUMPETERIAN APPROACH

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IV. THE SCHUMPETERIAN APPROACH

• Aghion and Howitt (Econometrica, 1992) A second
  strand of EGT models a different pattern of
  innovation in which innovation takes the form of
  improvements in existing products.
• Innovation thus creates new products or
  technologies, as well as destroying the value of
  old products or technologies by making them
  redundant. These models are referred to as
  vertical innovation or quality ladder models.

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THE SCHUMPETERIAN APPROACH

• The approach is much closer in spirit to the
  process of creative destruction, which is how
  Schumpeter famously characterized technical
  progress
• The fundamental impulse that keeps the capitalist
  engine in motion comes from the new consumers
  goods, the new methods of production or
  transportation, the new markets, the new forms of
  industrial organisation that capitalist
  enterprise creates. The process incessantly
  revolutionizes from within, incessantly
  destroying the old one, incessantly creating a
  new one. The process of Creative Destruction is
  the essential fact about capitalism.-- Schumpeter
  (1947), pp. 823.

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THE SCHUMPETERIAN APPROACH

• Aghion and Howitt (1992) introduced the seminal
  model in this vein, which they also summarize in
  Aghion and Howitt (1998, Chapter 2).
• Unlike the model in Romer (1990), the Aghion and
  Howitt (1998) version of this model abstracts
  completely from capital accumulation.
• Romer horizontal innovation, technical progress
  takes the form of introducing new inputs. But
  input i introduced at t is no more productive
  than input ik introduced at ts.

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THE SCHUMPETERIAN APPROACH

• Two new issues
• Creative destruction (Schumpeter). Technical
  progress create loss (destruction of rents) as
  well as gains.
• Critical role of forward-looking expectations.
  The amount of research carried on today depends
  on the expected duration of the rent generated by
  the innovation. If there is expectation that much
  research activities will be carried on in the
  future the expected benefit from todays
  innovation is low, and little research activity
  is financed today.
• (RD)t VE((RD)t1), Vlt0

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The Model

• Productive Inputs
• Unskilled labor (or any land-type factor) L
• Skilled labor H
• Knowledge, or an index of the level of
  technology A
• Intermediate inputs of different quality, each
  indexed by an integer saying the progressive of
  its invention x(i), i1,2,

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The Model 3 sectors

• (i) Research sector using H to produce DESIGN
  for better intermediate goods. The designs are
  patented, and the patents sold to firms in the
  intermediate sector
• (ii) Intermediate good sector a firm buys a
  design (i) and acquires the exclusive right to
  produce the corresponding input (i) forever. The
  technology to produce each intermediate input is
  identical, CRS to the only variable input H.
  However, once a new design is introduced the
  producer goods previously manufactured become
  obsolete.
• (iii) Final good sector using unskilled labor
  (L) and the most productive intermediate input
  (x(i)) to produce the unique consumption good.

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The Model Technology

• Final good sector
• (L1)
• T is a continuous time index i0,1,2 indexes
  inventions by the order of introduction.
• A(i) is a parameter indicating the productivity
  of the last intermediate input invented before
  time t.
• Intermediate good sector
• Firm producing x(i) must have bought the design
  (i) before starting manufacturing the
  corresponding product

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The Model Invention

• Research sector
• The invention process is modeled as a stochastic
  process, i.e., when resources are invested by a
  firm into RD, there will be a positive
  possibility of success (inventing a new design)
  and a positive possibility of failure (no
  invention).
• The possibility of success increases with the
  number of skilled workers involved in the
  activity. A firm employing h2 workers in RD at
  time t will discover a new design with
  probability lh2dt and will discover nothing with
  probability (1- lh2)dt. There are no spillovers
  across firms in research new discoveries
  randomly occur with a Poisson arrival rate of
  lH2t.

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Assumptions Invention

• A Poisson process with arrival rate q(lH2) has
  the following properties
• Let s be the length of a time interval (t, ts).
  Then
• The interval from 0 up to the first event, and
  thereafter the interval between successive
  events, are independently distributed with the
  exponential probability density function qe-qt
  The exponential distribution
• The expected waiting time between two successive
  events is 1/q, which is a function of the amount
  of skilled workers employed in the Res sector.
• When a new invention arrives, A(i) jumps to a
  higher level, meaning productivity growth occurs
  at discrete intervals. The law of motion of A is
  A(i) A0gi A(i) g A(i-1) where ggt1.

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Solution Final Good

• Final goods (competitive) producers choose
  intermediates to maximize profits.
• The derived inversed demand for intermediate i is
• Like in Romer (1990) a constant elasticity demand
  function for the intermediate input x(i)

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Solution Intermediate Good

• IGS is a monopoly. Given the demand constraint,
  the monopolist chooses production so as to max

The value of wH and H1 will change only when an
invention is introduced successfully
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Solution Research

• The objective of a (risk-neutral) firm in RD is
  to choose H2 to max the expected profit, define
  as
• where V(i1) is the value of the (i1)st
  innovation, and lH2 is the prob of discovering a
  new design by employing H2 workers. For
  maximization, the Kuhn-Tucker conditions are
• Now we need to determine V(i1).

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Solution Research

• V(i1) expected present discounted value of the
  flow of monopoly profit pI generated by the
  (i1)st innovation over an interval whose length
  is random.
• 1/lH2(i1) expected duration of the monopoly
• Why? See the example.

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An example

• EXAMPLE A risk-neutral agent is offered at t a
  bond which yields 10 dollars at t1, then at t2
  it participates to a lottery where
• with prob. p 0.15 it expires ( and becomes
  worthless )
• with prob. (1-p) 0.85 it pays 10 dollars and the
  right to participate to a new round of the
  lottery at t3.
• The story repeats itself thereafter until when a
  negative draws is incurred.
• The yearly interest rate is constant, r 0.1.
• What is the value of this bond to our agent?

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An example
SOLUTION. The value is So, the value is 40
Dollars. The problem in continuous time (where
p is the arrival rate of a negative draw and r as
the interest rate ) is analogous Now,
remember that the probability that a design
becomes worthless is precisely the probability
that a new invention comes along, i.e.
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Intertemporal Spillover

• There is an important spillover in the model.
• An innovation raises productivity forever.
• It follows each subsequent innovation to raise
  At, by the same multiple g.

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Equilibrium

• Combining those FOCs in the Res sector and
  substituting away V(i1), we get
• 
  
• LHS MC the cost of a RD worker divided by her
  MP,
• RHS MB the present discounted value of the
  rent(profit) generated by an innovation, where
  discounting considers the obsolescence rate of
  the innovation

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Equilibrium

• Combining FOCs in the other two sectors and the
  resource constraint, H1H2 H, thus,
• Note
• MC is increasing with current employment in Res
  sector
• MB is decreasing with future employment in RD
• This is the original point about creative
  destruction.

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Equilibrium

• Two reasons why current research depends
  negatively on future research (f lt0)
• It shortens expected lifetime of the monopoly
  enjoyed by the innovator
• It raises future wages of skilled workers - also
  employed by the IGS firm- hence reducing expected
  pI(i1).

See Fig 1, p. 332 H2(i) f(H2(i1))
MC, MB
MC
MB
H2
H2,0
H2,2
H2,1
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Two interesting cases

• Two-period cycle a pair (H20, H21) s.t. H20 f
  (H21) and H21 f (H20)
• Zero growth trap a two-period cycle s.t. either
  H20 or H21 is zero. The prospect of high research
  in odd intervals discourages research in even
  intervals.

MC, MB
MC
MB
H2,0
H2,1
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Steady State

• A unique stationary equilibrium exists,
• Thus, steady state H2
• decrease with r
• increase with g (size of each innovation)
• increase with l (efficiency of RD)

• increase with H
• decrease with a.

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Market power and steady state H2

• For s.s. H2 to be positive we need
• H2 increases with the degree of market power
  (measured by 1-a(p-mc)/p in IG sector, the share
  of the equilibrium revenue accruing to monopoly),
  implying monopoly is good for growth
• A min degree of monopoly power ( )
• If the degree of market power falls short of this
  minimal value, the flow of monopoly profit from
  the next innovation would not be sufficient to
  induce positive research.

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Balanced Growth

• In the interval during which the ith innovation
  is adopted, real output is
• Y remain constant through all the period in which
  technology I is adopted. Thus,
• The time path will be a random step function
  starting at lnY(0).
• The size of each step equal to the constant ln(g)
• The time between each step (D1, D2, ) is a
  sequence of iid variables exponentially
  distributed with .

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Balanced Growth

• Output is a nonstationary stochastic process
• .

Average growth rate and its variance are
increasing functions of the size of innovation
and the proportion of skilled labor force
employed in research.
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Welfare analysis

• CE vs. PO? No K ? Ct Yt
• The social planner simply choose the sequence
  H2tt08 to max

U(0) is an initial flow of output discounted by
the social discount rate, lower than r since
output will be growing over time.
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Welfare analysis

• PO-FOC
• Market-FOC

• Appropriability effect monopolist only cares
  about his surplus, not the total welfare effect
  of innovation

• Business stealing effect the planer realizes
  that innovation also causes a destruction

• Intertemporal spillover (discount rate) social
  planner considers that the benefit to the next
  innovation will continue forever, whereas private
  research firm ignore it.

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Welfare analysis

• Which growth rate is higher? Market or PO?
• Ambiguous since
• gm lt go if appropriability and intertemporal
  spillover effects dominate.
• gm gt go if business-stealing and monopoly
  distortion effects (wage to skilled labor is
  less) dominate.
• The solution

Economic intuitions?
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Welfare analysis

• For large innovation sizes growth tend to be
  inferior in the decentralized than in the optimal
  economy
• When a is very small (high monopoly power) and
  innovation size is small the laissez-faire
  economy can generate too much growth.

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The exponential distribution

• P(T1gtt) P(N(t) 0) eqt
• T1 the waiting time of the first occurrence
• P(T1ltt) 1- eqt
• ?P(T1t) dP(T1ltt)/dt qeqt Assumptions
  Invention