Post on 17-Dec-2015
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Economics of Innovation
Patent Races:Dasgupta- Stiglitz Model
Manuel Trajtenberg2005
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Patent Races
Dasgupta, P. and J. Stiglitz, "Uncertainty, Industrial Structure, and the Speed of R&D." The Bell Journal of Economics, 1980 (11), pp. 1‑28.
1. No uncertainty, R&D determines time of discovery; baseline, useful for further analysis.
2. Uncertainty as to time of discovery, Poisson process
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Dasgupta – Stiglitz (DS) 1
0)('' ,0)('
0)( ,)0(
:)(
xTxT
TT
xT
cmsjer
V trjtj ,, ),1( If finite patent:
R&D: x, determines the timing of invention, T(x):
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R&D: x
Discovery Time T(x)
R&D and time to discovery
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DS 2
1)(' :
, ),(
)(
)(
xrTj
xrTj
x
eVxrTFOC
msjxeVMax
Therefore,
)()( ****msms xTxTxx
6
45
*mx *
sx x
)( xrTjeV
m
s
Optimal R&D: “s” versus “m”
sV
mV
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DS 3
If competition, game not well defined. Clearly, in eq. just one firm (recall no uncertainty); free entry => zero profits:
*)(* *
: cxrT
cc xeVx c
*
*
0
cI
cIIf
xxif
xxifxx
Stackelberg eq. (I: incumbent, f: follower):
8
45
*mx *
cx*sx x
)( xrTjeV
m
c
s
Optimal R&D for each case
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DS 4
**
****
?
,
sc
mcms
xx
xxxx
DS show that for constant-elasticity demand functions,
The more so the more elastic the demand.
From social point of view, dx just moves forward invention date by a bit; for individual firm, it may bring the full reward.
**sc xx
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DS 5 • Without uncertainty, in competition just one firm does R&D, but makes zero profits (it may license to others, or do limit pricing) => anti-trust implications…
• It may overinvest in R&D…(but still, recall spillovers); partial appropriability (smaller V) versus pressure of competition.
• In a real patent race: many do R&D, one gets the full “prize”. Basic notion: to actually compete you must stand a chance ex ante…or be able to split the rewards (often happens)
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Introducing uncertainty: a reminder
)()(
!)(
YVarYE
y
eyYPr
y
The Poisson distribution for a discrete random variable Y (such as the number of innovations or patents) is:
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The Poisson Process
Consider the random variable Y, the number of occurrences over a fixed time interval, and assume that,
(i) The probability of an occurrence during a short time interval t+t is: t.
(ii) The probability of more than one occurrence during such time interval is negligible.
(iii) The probability of an occurrence during such time interval does not depend on what happened prior to time t.
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The Poisson Process – cont .
If those conditions hold, then the number of occurrences over a time interval of length t has a Poisson distribution with mean: t
Thus is the expected number of occurrences per unit time
Examples:
• Number of telephone calls received at a switchboard during a given time interval.
• Number of atomic particles emitted from a radioactive source that strike a target during a given time interval.
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DS w/uncertainty 1
If the occurrence of innovations over time follows a Poisson process, then the probability distribution of the waiting time until innovation, T, is:
tt
ete
g tions durin0 innovatPtTP
1!0
)(1
)(1)(0
!)(
y
eyYPr
y
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DS w/uncertainty 2
Now assume that the parameter is determined by the amount of R&D that firm i invests, xi , i.e .
)( ii x Then the probability that firm i will innovate until time t is given by
txi
tx
i
i
extf
etFtTP)(
)(
)()(
1)()(
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)(x
x
The “R&D lab technology”
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DS w/uncertainty 3
The expected innovation time for firm i is then,
)(
1)|(
ii x
xTE
)(
1
)(1)()(
0
)(
0 0
i
tx
xdte
dttFtdttfTE
i
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DS w/uncertainty 4
Assume that the R&D programs of different firms are (statistically) independent. Then,
ni in x1 )(
Assuming symmetry,
)(xnn
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DS w/uncertainty 5
Then the probability that some firm will invent until t is,
tn
nexxtTP 1),...,|( 1
and,
)(
11),...,|( 1 xn
xxTEn
n
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Maximizing social surplus
),;()(
o
rts nxdtexntfVSE
nxrxn
xnVSE
exnxntf
s
txn
)(
)( )(
)(),;( )(
Notice twice discounting…Assuming symmetry :
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Maximizing social surplus 2
1])([
)(' 0 )2(
])([
)( 0 )1(
:
)(
)( )(
2
2
rxn
rxV
x
E(S)
xrxn
rxV
n
E(S)
FOC
nxrxn
xnVSEMax
s
s
sn,x
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Maximizing social surplus 3
*)('*
*)(
)('
)(x
x
xx
x
x
Divide (1) by (2),
That is, the marginal equals the average, hence exhaust “scale economies” in R&D, then replicate R&D labs.
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)(x
xx*
Optimal amount of R&D in each lab
If concave replicate many tiny labs (stat independence…), if convex just a huge one…
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Comparing social optimum to monopoly
A monopoly firm maximizes same objective function, except that Vm<Vs, and therefore,
**** , smsm nnxx
That is, the monopoly will set up a smaller number of labs, in each same amount of R&D (x*: just a technical issue); total R&D for monopoly less than
socially optimal .
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1st FOC for “s”
snmn
x*
1st FOC for “m”
Optimal n : s versus m
xrxn
rxV
n
E(j)j
2])([
)( 0 )1(
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The competitive case: free entry
txii
iexxtf )()()|(
ij j txij extF
)()|(1
ij j txij extF
)(1)|(
The unconditional probability of firm i inventing at t:
The probability that some other firm invents up until t:
Hence the probability that nobody else invents up until t:
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competitive case – cont. 1
n
j j txi
iji
ex
xtFxtf
1)(
)(
)|(1)|(
The probability that firm i invents up until t, conditional on nobody else having invented till then:
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competitive case – cont. 2
0
)(1)()( i
rttxicci
xxdteexVPVEMax
n
j j
i
xrxn
xVPVEMax cc
x
)(
)( )(
The objective function of a competitive firm,
Assuming symmetry,
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competitive case – cont. 3
)(')(
)(
)( :
1)(
)(' :
xx
x
xrxn
xVentryfree
rxn
xVFOC
c
c
xrxn
xVPVEMax cc
x
)(
)( )(
Assuming that each firm is small enough so that there are no strategic interactions, i.e. each takes (xj)=n(x) as
given (like taking as given the expected date of discovery), the free entry equilibrium is:
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Compare social optimum to free entry equilibrium
xrxn
xVPVEMax
ximandPrivate ma
nxrxn
xnVPVEMax
MaximandSocial
ccx
ssnx
)(
)( )(
:
)(
)( )(
:
,
Social: maximand multiplied by n, and cs VV
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Social optimum vs. free entry
xrxn
xVxE
Private op
xrxn
rxV
n
E
op.Social
c
s
)(
)( )(
: .
)(
)(0
)(
:
2
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Social optimum vs. free entry 2
• If Vc = Vs then nc > ns;
• If Vc < Vs then can go either way.
• If Vc < Vs and too little R&D (i.e. too few firms),
could in principle increase patent life so as to increase Vc and bring nc = ns ; but it could be
that even an infinite patent will not do.
• What if Vc < Vs and too much R&D? In any case, optimal x
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Social optimum vs. free entry 3
(i.e. how “drastic” is the innovation)
Demand elasticity Too little
R&D
Too much R&D
oldnew cc /
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Qualifications
• Limitation of Poisson: cannot change riskiness (the variance) without changing the mean.
• The “V” does not include spillovers
• The labs not statistically independent
• The “lab technology” may vary across firms
• Other ?