strategic r&d in cournot markets - fields institute · 2013-08-27 · 4 strategic r&d:...
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1
Strategic R&D: Motivation Dynamic R&D Control Extensions
Strategic R&D in Cournot Markets
Mike Ludkovski1 & Ronnie Sircar2
1UC Santa Barbara
2Princeton University
Fields InstituteAug 27, 2013
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions
Outline
1 Strategic R&D: MotivationCournot Games
2 Dynamic R&D ControlGame ModelSeparable ControlsCoupled Controls
3 Extensions
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Cournot Games
Resource and Commodities Markets
Long-term prices largely driven by production levels among several largeproducers
Lots of evidence of strategic behavior by participants
Noncooperative dynamic gameMany sources of uncertainty
◮ changing production costs◮ fluctuating demand◮ policy changes◮ technological advances
Fertile application area for stochastic games
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Cournot Games
Role of R&D
Recently, attention has focused on exhaustibility: running out ofresources (peak oil)
Conversely, there has been a lot of technological breakthroughs:
Shale natural gas
Moore’s law for solar powerplants
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Cournot Games
Role of R&D
Recently, attention has focused on exhaustibility: running out ofresources (peak oil)
Conversely, there has been a lot of technological breakthroughs:
Shale natural gas
Moore’s law for solar powerplants
Technological change is abrupt, ongoing, and uncertain
Aim to endogenize investment in R&D
Embed within a dynamic stochastic game framework
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Cournot Games
Role of R&D
Recently, attention has focused on exhaustibility: running out ofresources (peak oil)
Conversely, there has been a lot of technological breakthroughs:
Shale natural gas
Moore’s law for solar powerplants
Technological change is abrupt, ongoing, and uncertain
Aim to endogenize investment in R&D
Embed within a dynamic stochastic game frameworkMotivation:
◮ Green (solar, biofuel) vs. fossil fuel energy production◮ R&D is key for switching to inexhaustible technologies◮ Game-theoretic effects can be significant
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Cournot Games
Game Model
Cournot market: players control supply
Production levels qit ; production costs c i
t
Price is given by inverse demand curve P based on aggregate supply~q =
∑
i qi
Profit from production is qit · (P(~qt )− c i
t )
Each producer i looks at her total discounted revenue:
E
[∫ ∞
0e−rt {(P(~qt )− c i
t )qit − C(ai
t)}
dt]
.
ait is R&D effort
Look for closed-loop Markov Nash equilibrium
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Cournot Games
Game Model
Cournot market: players control supply
Production levels qit ; production costs c i
t
Price is given by inverse demand curve P based on aggregate supply~q =
∑
i qi
Profit from production is qit · (P(~qt )− c i
t )
Each producer i looks at her total discounted revenue:
E
[∫ ∞
0e−rt {(P(~qt )− c i
t )qit − C(ai
t)}
dt]
.
ait is R&D effort
Look for closed-loop Markov Nash equilibrium
State : c it is stochastic, can be lowered through R&D
Everything else is deterministic
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Cournot Games
Intuition
Recall the static Cournot duopoly with production costs c1 and c2
For simplicity, will focus on linear demand P(~q) = 1 −∑
i qi
The respective revenue is R1 := q1(1 − q1 − q2 − c1) andR2 := q2(1 − q1 − q2 − c2)
Interior eqm solution is qi,∗ = 1+c i−2c i
3 yielding revenue rate (qi,∗)2
Game value vi =(1+c i−2c i )2
9r
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Cournot Games
Blockading
Production rate must benon-negative
If c i > 1+c i
2 , producer i isblockaded and does notproduce at all, q∗
i = 0. Inthat case have monopolywith q∗
i= (1 − c i)/2
Blockading when c i islarge (close to 1) relativeto c i . No blockading ifc i < 0.5
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0.5
1
0.5
Player 1Blockaded
Blockaded
0
Player 2
1 c1
c2
R&D
Figure : Fixed-cost Cournot game.
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Cournot Games
Existing Literature
Industrial Organization: optimizing R&D investment by a monopolistfacing uncertainty
Within exhaustible resource context: Kamien and Schwartz (1978),Lafforgue (2008), numerous papers addressing climate changemitigation.
But: Only single agent – no game effects
Game Theory: impact of technological change on strategic competition
Fudenberg & Tirole (1985), Weeds (2002)
But: One-shot games – focus on coordination/preemption, no dynamiceffects
Cournot Games: Hotelling (1931), Sircar et al. (2010–)
But: no R&D – production costs are fixed
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
R&D Control
Model technology as a discrete ladder: c(1) > c(2) > ... ≥ 0
If currently at n-th stage, a breakthrough moves the producer to n + 1-ststage of technology
c(n) = exp(−bn): efficiency improves proportionally by b%
c(n) = (1 − bn)+: absolute improvements in efficiency – eventually willreach “zero” costs
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
R&D Control
Model technology as a discrete ladder: c(1) > c(2) > ... ≥ 0
If currently at n-th stage, a breakthrough moves the producer to n + 1-ststage of technology
c(n) = exp(−bn): efficiency improves proportionally by b%
c(n) = (1 − bn)+: absolute improvements in efficiency – eventually willreach “zero” costs
Expend effort at ⇒ breakthroughs occur at rate λat
N it : point process for the technology advances of player i. Given (ai
t ), (Nit )
is a Poisson process with controlled intensity λiait
Dynamic production costs are c it = c(N i
t )
R&D incurs costs C(at) per unit time; convex + increasing
For convenience assume finite number of stages Ni for player i
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Dynamic R&D
Only uncertainty is from (N it ). Between R&D advances the game is
deterministic. Can be viewed as a sequence of coupled static Cournotgames
There is dynamic interaction between R&D and production. Players maybe blockaded, and may also choose to expend no effort (making somestates (c1, c2) absorbing)
Continuous-time strategies: qit , a
it
Strategies assumed to be in feedback form for (N1t ,N
2t )
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Finding Nash Equilibrium
Given initial technology stages (n1, n2), game values are denoted byvi(n1, n2)
τ i is the time of first R&D success by player i – controlled by effort (ait )
Given (qi , ai ), vi ’s satisfy the recursions
v1(n1, n2) = E
[
∫ τ1∧τ
2
0e−rs{q1
s (P(~qs)− c1(n1))− C(a1s)} ds
+ e−rτ 1∧τ2[1{τ 1<τ 2} · v1(n1 + 1, n2) + 1{τ 1>τ 2} · v1(n1, n2 + 1)]
]
By the piecewise deterministic property, under every Markov Nashequilibrium, qi
t ≡ qi , ait ≡ ai are constant for t ∈ [0, τ1 ∧ τ2]
So τ1 ∧ τ2 ∼ Exp(λ1a1 + λ2a2)
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Duopoly Game Values
Using properties of Poisson arrival times, Nash equilibria arecharacterized by
v1(n1, n2) = supq,a
q(1− q − q2,∗ − c1(n1))− C(a) + λ1av1(n1 + 1,n2) + λ2a2,∗v1(n1, n2 + 1)λ1a + λ2a2,∗ + r
Similar equation for v2(n1, n2)
q1,∗ is obtained directly as 1+c1(n1)−2c2(n2)3
Differentiating wrt a’s, obtain a system of two nonlinear equations ina1,∗, a2,∗ characterizing the Nash equilibrium
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Recursive Static Games
v1(n1, n2 + 1)
yλ2a2
v1(n1, n2)λ1a1
←−−−−− v1(n1 + 1, n2)
Can solve recursively on a lattice
Boundary condition is vi(N1,N2) =(1+c1(N1)−2c2(N2))
2
9r ; also when n1 = N1,no further R&D is possible for P1 (1-dim optim by P2)
a1,∗ depends on v1(n1 + 1, n2)− v1(n1, n2) > 0 andv1(n1, n2 + 1)− v1(n1, n2) < 0
C(a) = a2/2 + κa:◮ Have a system of two coupled quadratic equations for ai,∗
◮ if κ > 0 then R&D may be unprofitable, so a∗ = 0 is possible◮ Analytic expressions to determine whether R&D is zero
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Unilateral R&D
0.00
0.05
0.10
0.15
0.20
0.25
P1 Costs c_1
R&D
Effo
rt
0.75 0.49 0.38 0.30 0.23 0.17 0.11 0.06 0.02
c_2 = 0.5c_2 = 0.25c_2 = 0
Figure : Effort Curves for Unilateral R&D in a Cournot Duopoly. Quadratic effort costC(a) = a2/2 + 0.2a with λ = 5, r = 0.1. Here c1(n) = 0.75− 1.5
√n (q1(n) is linear)
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Unilateral R&D
0.00
0.02
0.04
0.06
0.08
0.10
0.12
P1 Costs c_1
R&
D E
ffort
1.00 0.88 0.76 0.64 0.52 0.40 0.28 0.16 0.04
P2
Blo
ckad
ed
P1
Blo
ckad
ed
Levels of R&D over time may have different shapes
Affected by: expectation of future profits (less future gains as get closeto c1 = 0), shape of the cost curve n 7→ c1(n) (marginal efficiency ofR&D), current revenue levels
If initial c1 is too high relative to c2, become too discouraged and donothing (never enter the market)
Monopoly is more conducive to R&D than duopoly
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Bilateral R&D
R&D Effort
Costs P1
Cos
ts P
2
0.78 0.37 0.17 0.08
0.78
0.37
0.17
0.08
Production
Costs P1
0.78 0.37 0.17 0.08
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Figure : Right panel shows the effort a1(n1, n2) and left panel the production rateq1(n1, n2). Quadratic costs C(a) = a2/2 + 0.2a with λ = 5, r = 0.1. c i(n) = e−n/8
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Bilateral R&D
R&D effort levels are asymmetric: put most effort when slightly ahead ofcompetitor
Therefore, player with lower costs tends to extend her advantage("mean-aversion")
Expectations of future profits can spur R&D even if currently blockadedout of the market
Zero R&D can happen even without blockading
Conversely, if c i is very low compared to competitor, may becomecomplacent and stop R&D
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Bilateral R&D
R&D effort levels are asymmetric: put most effort when slightly ahead ofcompetitor
Therefore, player with lower costs tends to extend her advantage("mean-aversion")
Expectations of future profits can spur R&D even if currently blockadedout of the market
Zero R&D can happen even without blockading
Conversely, if c i is very low compared to competitor, may becomecomplacent and stop R&D
Outside input (subsidies) can spur endogenous advances both for veryinefficient technologies and for efficient monopolies
Competition is dynamically unstable (tends to collapse into a monopoly)
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Sample Path of (N1t ,N
2t )
0 10 20 30 40 50
510
1520
25
Time
Tech
nolo
gy S
tage
P1P2
Figure : Sample path of (N1t ,N
2t ). Costs are c i(n) = e−n/8. Quadratic effort curve
C(a) = a2/2 + 0.2a with λ = 5, r = 0.1.
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Distribution of (N1t ,N
2t )
t = 2
5 10 15 20 25
510
1520
25
P1 Technology Stage
P2
Tech
nolo
gy S
tage
Figure : Distribution of (N1t ,N
2t ). Costs are c i(n) = e−n/8. Quadratic effort curve
C(a) = a2/2 + 0.2a with λ = 5, r = 0.1.
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Distribution of (N1t ,N
2t )
t = 4
5 10 15 20 25
510
1520
25
P1 Technology Stage
P2
Tech
nolo
gy S
tage
Figure : Distribution of (N1t ,N
2t ). Costs are c i(n) = e−n/8. Quadratic effort curve
C(a) = a2/2 + 0.2a with λ = 5, r = 0.1.
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Distribution of (N1t ,N
2t )
t = 8
5 10 15 20 25
510
1520
25
P1 Technology Stage
P2
Tech
nolo
gy S
tage
Figure : Distribution of (N1t ,N
2t ). Costs are c i(n) = e−n/8. Quadratic effort curve
C(a) = a2/2 + 0.2a with λ = 5, r = 0.1.
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Distribution of (N1t ,N
2t )
t = 15
5 10 15 20 25
510
1520
25
P1 Technology Stage
P2
Tech
nolo
gy S
tage
Figure : Distribution of (N1t ,N
2t ). Costs are c i(n) = e−n/8. Quadratic effort curve
C(a) = a2/2 + 0.2a with λ = 5, r = 0.1.
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Distribution of (N1t ,N
2t )
t = 25
5 10 15 20 25
510
1520
25
P1 Technology Stage
P2
Tech
nolo
gy S
tage
Figure : Distribution of (N1t ,N
2t ). Costs are c i(n) = e−n/8. Quadratic effort curve
C(a) = a2/2 + 0.2a with λ = 5, r = 0.1.
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Distribution of (N1t ,N
2t )
t = 40
5 10 15 20 25
510
1520
25
P1 Technology Stage
P2
Tech
nolo
gy S
tage
Figure : Distribution of (N1t ,N
2t ). Costs are c i(n) = e−n/8. Quadratic effort curve
C(a) = a2/2 + 0.2a with λ = 5, r = 0.1.
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
Impact of Uncertainty
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
Game value
P1 Costs c^1
v_1
(c^1
,c^2
)
c^1(n) = 1−n/10c^1(n) = 1−n/20c^1(n) = 1−n/50
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
R&D Effort
P1 Costs c^1
a^*_
t
Figure : Comparison of game values v1(·, c2) and effort levels a1(·, c2). Lineartechnology progress c1(n) = 1− n/M, λ = 0.4M with c2 = 0.7 and κ = 0.2.
Can study effect of uncertainty by linearly scaling the R&D ladder c(n)and rate of progress λ
Take c(n) = f (n/M), λ = λM where c 7→ f (c) is cont R&D curve on [0, 1]As M → ∞, R&D success becomes deterministic: dct = λat dtImpact is ambiguous: more uncertainty can spur/deter R&D investment!
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions Game Model Separable Controls Coupled Controls
R&D Complementary to Production
Firm has fixed labor supply L. Allocate L between production and R&D:ai
t + qit = Li
Sharpens the trade-off between immediate revenue and future higherprofits
Cost of R&D is now implicit (quadratic if assume linear demand P(~q))
Will tend to decrease R&D over time
May be optimal to voluntarily lower/suspend production to advancetechnology (e.g. to lock-in monopoly)
May allow a high-cost competitor to operate by temporarily focusing onR&D (i.e. strategic non-blockading)
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions
Extensions: Exhaustible Resources
When considering competition between old and new energy (fossil fuelsvs. renewables), exhaustible reserves play a crucial role
Xt – level of reserves at date t ; dXt = −qtdt lowered through production
Oil industry (P1): low production costs c1, but also marginal cost ofexhaustibility
Renewables industry (P2): high current production costs c2(0); potentialfor R&D
P1 chooses (q1t ); P2 chooses (q2
t ) and (a2t ). State is (x , n)
Leads to a system of nonlinear ODEs in x , coupled through n
Can allow P1 to also explore for new reserves (L. & Sircar 2012)
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions
Extensions: Switching Technologies
Consider two integrated producers who can each use either cheap fossilfuels, or expensive backstops (oil sands)
Resources allocated between production and R&D (advancing backstoptechnology)
Uncertainty in advances will spur earlier R&D investments as marginalvalue of cheap reserves rises
Related to the model of Harris, Howison and Sircar (2010)
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions
Conclusion
Ongoing project: stochastic framework for natural resource oligopolies
Effect of exhaustibility
Potential of R&D
Strategic interactions (blockading, uncertainty, policy impact) lead tonumerous non-trivial phenomena
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions
Conclusion
Ongoing project: stochastic framework for natural resource oligopolies
Effect of exhaustibility
Potential of R&D
Strategic interactions (blockading, uncertainty, policy impact) lead tonumerous non-trivial phenomena
THANK YOU!
Ludkovski Strategic R&D in Cournot Markets
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Strategic R&D: Motivation Dynamic R&D Control Extensions
References
C. Harris, S. Howison, and R. SircarGames with exhaustible resourcesSIAM J. Applied Mathematics 70 (2010), 2556-2581.
G. LafforgueStochastic technical change, exhaustible resource and optimal sustainable growthResource and Energy Economics 30 (2008), no. 4, 540–554.
J. WeedsStrategic delay in a real options model of R&D competitionReview of Economic Studies, 69 (2002), 729–747.
M. Ludkovski and R. SircarExploration and Exhaustibility in Dynamic Cournot GamesEuropean Journal of Applied Mathematics, 23 (2012), no. 3, 343–372.
Ludkovski Strategic R&D in Cournot Markets