Relative Performance and R&D
Competition
Joint work with Susumu Cato (加藤晋) and
Noriaki Matsushima (松島法明)
OT2010
1
Plan of the Presentation
(1) Relative Performance and Competitiveness of the
market
(2) Stability of Collusion
(3) Relative Performance and the Stability of Collusion
(4) Strategic R&D Competition
(5) R&D Competition and Competitiveness in the
Product Market
(6) R&D Cooperation and Relative Performance
OT2010
2
Importance of Relative
Performance
OT2010
3
relative profit maximization
Consider quantity-setting competition (Cournot-type
competition). Suppose that each firm maximizes
relative profit (its own profit minus the rival’s profit)
→Firms produce (more , less) aggressively.
OT2010
4
Equilibrium
Consider quantity-setting competition (Cournot-type
competition). Suppose that each firm maximizes
relative profit (its own profit minus the rival’s profit)
Homogeneous product market, symmetric firms.
Firms 1 and 2 chooses their outputs Y1 and Y2
independently.
P(Y):inverse demand function, Ci: Firm i’s cost
function
F.O.C. P+P'Y1-C1' -P'Y2=0
→At the symmetric equilibrium (Y1=Y2), Price (>,＝,<)
marginal cost
OT2010
5
Vega-Redondo (1997)
Consider quantity-setting competition (Cournot-type
competition). Firms 1 and 2 chooses their outputs
Y1 and Y2 independently. Homogeneous product
market. No cost asymmetry.
Firm choose the same output as the rival it the rival
obtains the higher profit (Imitation)＋mutation
→The Walrasian (perfectly competitive equilibrium
outcome) is an evolutionally stable.
OT2012
6
Vega-Redondo (1997)
The Walrasian (perfectly competitive equilibrium
outcome) is an evolutionally stable.
Intuition
Relative profit is maximized at Walrasian equilibrium.
Mutation (deviation form the Walrasian equilibrium)
→Even if the profit of this mutant increases, the rival’s
profit further increases→Imitation again yields
Walrasian.
Close relationship between evolution game and
relative profit maximization.
OT2012
7
Relative profit, relative performance
approach
U1=π1-απ2
α=1 perfect competition, α=0 Cournot, α=-1 Collusion
We can analyze many situations, from perfect
competition to collusion by a single simple model
OT2010
8
Relative profit, relative performance
approach
Examples of applications
Cartel becomes more stable when α is smaller.
(Matsumura and Matsushima, forthcoming)
R&D level is non-monotone with respect to α, U-shaped
(Today’s paper)
An increase of α reduces innovation size and increases
R&D expenditure
The degree of product differentiation is decreasing in
αwhen α is positive and not too small.
When α is large or small, Multi-Store Paradox is solved.
OT2010
9
Rationalizations for relative
performance approach
(1) market evaluation for CEOs
(2) evolutionary approach
(3) envy, altruism
(4) Fershtman and Judd (1987)
(5) election, political science
(6) status, macroeconomics (relative wage, relative
consumption, relative wealth, relative income)
OT2010
10
Conjectural Variation
conjectural variation
Each firm chooses its output assuming that one unit
increase of its output yields r unit increase of the
total output.
Cournot→r=1
Conjectural Variation Model is a general model
including the Cournot model as a special case (？)
Do the cases where r≠1 make sense?
OT2010
11
Conjectural Variation Model
Suppose that r≠1.
Firm 1’s output affects firm 2’s output. Thus, firm 2
must choose its output before firm 1’s choice.
Firm 2’s output affects firm 1’s output. Thus, firm 1
must choose its output before firm 2’s choice.
→Mutually inconsistent
OT2010
12
Why is this inconsistent model
used by IO researchers?
(1) an unmodeled dynamic model
～conjectural variation model analyzes a dynamic
interaction between firms
→If so, we should formulate a dynamic model rather than
using an inconsistent static model
(2) There exists models yielding the same solution
→If so, we should use these model
（３）This model represents various market structure with
different degree of competition.
OT2010
13
Conjectural Variation Modelの解
First-order condition in CV Model P+P'rＹ1 =c
First-order condition in Cournot Model P+P'Ｙ1 =c
→corresponds to the case of ｒ＝１.
First-order condition in Bertrand Model ~ perfect
competition, P =c
→corresponds to the case of ｒ＝０
First-order condition in Joint Profit Maximization
(collusion）, P+P'(Ｙ1+Ｙ2) =c
→corresponds to the case of ｒ＝２
A larger r represents a more severe competition.
OT2010
14
advantage of relative profit
approach
(1) Consistent model.
(2) Foundation for any competitive structure between
Bertrand and Collusion.
(3) More realistic (I think).
OT2010
15
Stability of Collusion
OT2010
16
Prisoner’s Dilemma
２
C
D
C
(3,3)
(0,4)
D
(4,0)
(1,1)
１
Nash Equilibrium：(D,D)
OT2010
17
Prisoner’s Dilemma and
Cooperation
We often observe cooperative behaviors under the
situations of prisoner’s dilemma. Why？
(1) Players are irrational.
(2) Player’s payoff is different from what we
observe
(3) Long-run relationship→infinitely repeated game
OT2010
18
Idea of (2)
２
１
Ｃ
Ｄ
C
(3,3)
(0,2)
D
(2,0)
(1,1)
altruism
Nash equilibrium :(C,C) (D,D)
OT2010
19
Idea of (3)：infinitely repeated
game
Stage game is played repeatedly.
Each player maximizes the discounted sum of the
payoff obtained at the each stage game.
Payoff at period 1+δ(payoff at period 2)
+δ2 (payoff at period 3)+δ3 (payoff at period 4)+...
δ∈(0,1)：discount factor
OT2010
20
Interpretation of discount factor
(1) Interest rate δ＝1/(1＋r) r：interest rate
(2) Subjective discount rate ~ indicates how important
the future income is, indicates how patient the
player is
(3) The probability that the game continues until the
next period
⇒The probability that the game terminates within
10,000 years is almost 1. Infinitely repeated game
never implies that the game never terminates.
~Much more realistic than what it seems at the first
glance.
OT2010
21
subgame perfect Nash equilibrium
The following strategies constitute a subgame
perfect Nash equilibrium if and only if δ≧1/3.
Each player takes C at period t if and only if no
player took D before period t. Otherwise each
player takes D.
OT2010
22
The measure of the stability of
collusion
Collusion is sustainable if and only if δ ≧ δ*.
→ Collusion is more stable when δ* is smaller.
OT2010
23
Infinite Nash Reversion
infinite Nash reversion (grim trigger strategy)
If one player deviates from the collusion, then all
play the one-shot Nash equilibrium thereafter.
cf Optimal Penal Code
OT2010
24
Matsumura and Matsushima
(forthcoming)
・relative profit maximization approach
・Investigate the relationship between α and δ*.
If there is no collusion, a larger α indicates that firms
face a more severe competition in the product
market. If a larger α facilitates collusion, a larger α
indicates less severe competition.
OT2010
25
Two effects
The larger α is, the more severe competition after the
deviation is.
→more severe punishment
⇒δ* can be decreasing in α.
The larger α is, the deviation more effectively improve
its payoff because deviation reduces the rival’s
profit.
⇒δ* can be increasing in α.
OT2010
26
Results
δ* is increasing in α.
This is true if we adopt optimal penal code.
This is also true if we adopt price competition with
product differentiation.
OT2010
27
R&D and Competition
OT2010
28
Two Views on Competition and
R&D
Monopoly View ~ Monopoly stimulates innovation
・R&D investments are financed from monopoly
profits
・Monopolists internalize the spillover effects of R&D
・R&D has economy of the scale
Competition View ~ Competition stimulates innovation
・Replacement effect (Arrow ,1950)
・Competitive pressure disciplines the management
OT2010
29
Brander and Spencer (1983)
Two stage Strategic R&D game
Cost-reducing R&D
Cournot Competition
No Spillovers
OT2010
30
Cournot Model
Ｙ２
Reaction Curve of Firm 1
Reaction Curve of Firm 2
Y2C
0
OT2010
Y1
C
Y１
31
Shift of Reaction Curve of Firm 1
Ｙ２
through strategic R&D
New Reaction Curve of Firm 1
Reaction Curve of Firm 2
Y2C
0
OT2010
Y1C
Y１
32
Cost-Reducing Investments
Model
Duopoly, homogeneous goods market
First stage: Each firm i independently chooses Ii
(R&D investment level), which affect its
production costs.
Second stage: After observing firms' production
costs, firms face Cournot competition.
Payoff: Π1=P(Y1+Y2)Y1-C1(I1)Y1-I1
OT2010
33
backward induction
Second stage ~ Cournot Competition
Y1 C (I1,I2), Y2C(I2,I1)
Firm's output is increasing in its own investment and
decreasing in the rival's investment.
First stage ~ R&D Competition
F.O.C.
P'Y1 (∂Y1C/∂I1 + ∂Y2C/∂I1)+P ∂Y1C/∂I1 -C1'(I1)Y1- C1
∂Y1C/∂I1 -1=0
OT2010
34
First stage
First Stage F.O.C.
P'Y1 (∂Y1C/∂I1 + ∂Y2C/∂I1)+P∂Y1C/∂I1 -C1'(I1)Y1- C1
∂Y1C/∂I1 -1=0
P'Y1∂Y2C/∂I1 -C1'(I1)Y1-1=0 (envelope theorem)
Cost-Minimizing Level
-C1'(I1)Y1-1=0
Investment level exceeds cost minimizing level under
strategic substitutes~ strategic effect
OT2010
35
Shifts of Reaction Curves
New Reaction Curve of Firm 1
Ｙ２
New Reaction Curve of Firm 2
Y2C
0
Y1
C
Competition Reduces
Y１
profits of Both firms
OT2010
36
This paper
We investigate the relationship between the strategic
cost-reducing R&D investment and α.
Does competition accelerate R&D?
Does envy society foster or deteriorate R&D?
OT2010
37
Two effects
A larger α accelerates competition in the product
market.
→The equilibrium output is larger
⇒A larger incentive for cost-reducing (Economy of the
scale)
A larger α reduces the degree of slope of the reaction
curve at production stage.
→Strategic effect of R&D becomes weaker
OT2010
38
Proposition 1
U-shaped relationship between R&D and α.
Investment level is minimized when α＝1/3.
・Both monopoly and very competitive situation yields
intensive R&D. This result support both competition
view and monopoly view.
・Introducing a small (large) degree of `envy’ into the
standard Cournot competition reduces (increases)
R&D.
The opposite result to Aghion et al (2005).
OT2010
39
Joint Implementation
Cooperation in R&D stage.
Competition in production stage.
R&D is decreasing in α.
→If we consider joint implementation, severe
competition reduces R&D.
OT2010
40
Oligopoly
The same result is obtained if we consider the joint
implementation.
If we consider non-cooperative R&D, again similar
result is obtained. However, the range of the
decreasing part becomes wider.
→If the number of firms is larger, competition more
likely reduces R&D.
OT2010
41
Welfare
Consider cooperative R&D.
The equilibrium R&D level is too low for social welfare.
Consider non-cooperative R&D.
The equilibrium R&D level is too low for social welfare
when α is large (small)
OT2010
42

Product Diversification, Entry

シュタッケルベルグ均衡の補足