Innovation Mode of an Asymmetric Duopoly with Semi-collusion

Systems Engineering — Theory & Practice Volume 29, Issue 3, March 2009 Online English edition of the Chinese language journal Cite this article as: SETP, 2009, 29(3): 21–27

Innovation Mode of an Asymmetric Duopoly with Semi-collusion SUN Cai-hong1 ,∗ , QI Jian-guo2 , YU Hui3 1. Applied Technology College, Chongqing Technology and Business University, Chongqing 400067, China 2. Institute Quantitative & Technical Economics, Chinese Academy of Social Sciences, Beijing 100732, China 3. College of Economics and Business Administration, Chongqing University, Chongqing 400030, China

Abstract: Asymmetric mode is proposed to analyze the effects of asymmetries on R&D investment, output, profit, and social welfare in a duopoly with semi-collusion. Simulations show that the agent with lower initial cost has higher R&D expenditures, output, and profit; the agent with a higher innovation ability has higher R&D expenditure and output, while the profit depends on its spillover; under the condition of asymmetric spillover, the agent with lower spillover has lower R&D expenditure higher output and profit; when the difference of initial cost and innovation ability is lower and the two agents’ spillover is close to 0 or 1, the welfare reaches high. Key Words: asymmetric duopoly; semi-collusion; game theory; numerical simulation

1 Introduction Cooperative innovation can internalize spillovers effectively, achieve economies of scale, bring about technological complementarities and synergies, promote mutual knowledge flow, and avoid duplication of research. Therefore, cooperative innovation as opposed to independent R&D tends to have higher innovation performance. This is the reason why cooperative innovation has become a hotspot in the fields of industrial organization in recent years. The study on R&D cooperation of the duopoly can be dated back to Katz[1] , while the two-stage duopoly game model with technology spillover (referred to as AJ model) established by Apremont & Jacqumin[2] has laid the foundation for study on collaborative innovation game model. In AJ model, the two agents decide R&D investment in the first stage and the productivity in the second stage. According to whether the two agents cooperating in the two stages, cooperative innovation would be divided into three models: totally non-cooperative innovation (agents act non-cooperatively in both output and R&D), semi-collusion (cooperation in R&D and noncooperation in productivity), total cooperation (cooperation in both stages). In terms of joint R&D investment, joint production, joint profits, and social welfare, several pros and cons of the models are compared. There are a number of further researches on AJ model, such as Fershtman & Gandal[3] , Petit & Tolwinski[4] . As the collusion in the production and sales phase will reduce the degree of competition of the product in the market, and even lead to the loss of social welfare, it is usually intervened directly or prohibited by the governments, for example, the“National Cooperative Research Act” (NCRA) was adopted in the United States in 1984, which allows cooperation only in the prototype production and technology development phase to avoid conspiracy in

the product market; China’s Anti-monopoly Law also came into effect on August 1, 2008. In view of this, the second innovation model (that is semi-cooperative innovation model) relative to the whole collaborative innovation is more realistic. It is significant in practice and theory to analyze and research the innovation. Therefore, the semi-collusion innovation model was addressed. Brod & Shivakumar[5] studied a semi-collusion mode , which referred that agents act cooperatively in production and non-cooperatively in R&D. In this article, the semicollusion refers to cooperation in R&D phase and noncooperation in the production phase. In addition, the former studied the benefits of the consumer and producer as compared to noncooperation in both R&D and production stage, while the latter emphasized how the asymmetry of initial conditions had impacted contrasts of production and R&D. AJ Model and the vast majority of AJ-based models usually assumed that the agents are identical. Dynamic game with perfect information under the condition of symmetry laid the foundation for the cooperative innovation. However, as it placed excessive emphasis on the symmetry, the scope of application of the results had been greatly hampered. In reality, more asymmetric agents had been involved in cooperative innovation, such as asymmetric initial costs, asymmetric innovative ability (the same R&D investment does not mean the same output, etc.), asymmetric technology spillover (the leakage of technical information is different, as well as the absorptive capacity of external technology and knowledge differences), so that recent research studies have started to pay attention to this phenomenon by introducing asymmetry to the AJ model. Lukatch and Plasmans[6] expanded the hypothesis that the initial cost is symmetry and analyzed the pros and cons of different innovation models with asymmetric initial cost and innovation ability. Lahiri &

Received date: November 27, 2008 ∗ Corresponding author: E-mail: [email protected]

Foundation item: Supported by the National Nature Science Foundation of China (Nos.70871126; 70501015; 90924009); New Century Excellent Talents

in University, Ministry of Education (2009) c 2009, Systems Engineering Society of China. Published by Elsevier BV. All rights reserved. Copyright °

SUN Cai-hong, et al./Systems Engineering — Theory & Practice, 2009, 29(3): 21–27

Ono[7] studied optimal R&D subsidies of two-stage duopoly with the asymmetric initial cost and concluded that the agents with lower initial marginal cost should be subsidized, whereas the agents with higher initial marginal cost should be taxed. Atallah[8] and De Bondt[9] expanded AJ model by introducing asymmetric spillover. Atallah[8] analyzed the effect of spillover on the motive of cooperation and joint profit. De Bondt[9] and Henriques[10] pointed out the elements that caused asymmetric strategic investment are initial marginal cost, innovation ability spillover, etc.; and for the quadratic payment function, the asymmetric spillover is important to determine R&D investment successively or at the same time. Asymmetry is introduced into AJ model, but it differs from the studies mentioned above; First, from the standpoint of the construction of the model, we introduce asymmetric initial cost, asymmetric innovation ability, and asymmetric spillover into the model simultaneously and provide a generalized analysis. Lukatch & Plasmans[6] , Atallah[8] , and Lahiri & Ono[7] only introduced one asymmetric factor, which is equivalent to a special case of this article. Second, from the points of the research contents, Lukatch & Plasmans[6] and Atallah[8] compared the pros and cons of R&D and production of the entire industry in different models of cooperation after the introduction of asymmetry, which is the same as AJ model. De Bondt & Henriques[9] studied the effects of asymmetric spillover on non-cooperative R&D investment. Lahiri & Ono[8] studied the effect of asymmetry of initial costs on the R&D subsidies. This paper analyzes how the asymmetries have affected the contrast of R&D investments, output, and social welfare. Overall, the focus of this paper is to construct two– stage game including R&D and production in the case of three asymmetric factors (that is initial cost, innovation rate, spillover) to expand AJ model. The paper aims to find the equilibrium solution of R&D investments, production, and profit under the condition of asymmetry and then compare the equilibrium solution to determine whether the asymmetries will affect corporate behavior and social welfare, and explore the law that the contrasts of agents’ behavior change with the asymmetries.

yi = 21 γi x2i , i = 1, 2 where γi > 0 is the per unit marginal cost of research, yi is the cost of innovation of agent i’s. According to the parameters, the agent with low initial marginal production cost will advantage in cost. The lower the per unit marginal cost of R&D is, the higher the ability of innovation is. Consider the following two-stage game, in the first stage (R&D phase), two agents act cooperatively to coordinate R&D investment, reduce cost of production, and maximize profits; in the second stage (production stage), two agents compete in production, and choose qi to maximize the profit after the R&D investments are given in the first stage. The agents will expect the effects of profits caused by R&D investments. Therefore, the whole game is a two-stage dynamic game with perfect information and the result is a refining sub-game perfect Nash-equilibrium, which can be solved through backward induction. Agent i’s payment function is the profits of the second stage deducting the expenses of the first stage, which can be written as πi

2.1

= (p − ci )qi − yi = [(a − qi − qj ) − (Ai − xi − (1) βj xj )]qi − 21 γi x2i (i, j = 1, 2; i 6= j) Competition in production stage

Consider the equilibrium solution of production in the second stage first. In the production stage, agents determine the output independently and compete in the product market. Therefore, there is a non-zero- sum- game in the second stage, and the solution sets of Nash Equilibrium exist. In this stage each chooses its profit-maximizing output depending on the R&D expenditures (conditional on xi and xj ). i The first order condition is ∂π ∂qi = 0, i = 1, 2, so that NashCournot equilibrium can be computed to be qi

=

2 Two-stage game model of asymmetric duopoly Consider an industry with two agents producing homogeneous goods facing an inverse demand function p = a − (q1 + q2 ), where a > 0, and a is the parameter of demand function, p is the price, qi is the output of agent i (i = 1, 2). Assume the initial cost of each agent is Ai , and the agents can reduce the cost by process innovation, where 0 < Ai < a (i = 1, 2). The R&D externalities or spillovers imply that some benefits of each agent’s R&D flow without payment to other agents. The effective reduction of cost is Xi = xi + βj xj , where 0 ≤ βj ≤ 1(the interval of the spillover effect levels), and βj xj indicates that the degree of cost reduction agent j caused by agent i’s R&D investment. Therefore, agent i’s unit cost is ci = Ai − Xi = Ai − xi − βj xj (i, j = 1, 2; i 6= j) Assume that agent i’s R&D investment is quadratic, reflecting the existence of diminishing returns to R&D expenditures:

(a − 2Ai + Aj ) + (2 − βi ) + (2βj − 1)xj 3 (i, j = 1, 2; i 6= j)

(2)

Equilibrium profit (the profits of the second stage deducting the expenses of the first stage) is: πi

=

1 1 qi2 − γi x2i = [(a − 2Ai + Aj ) + (2 − βi ) + 2 9 1 (3) (2βj − 1)xj ] − γi x2i (i, j = 1, 2; i 6= j) 2

2.2 Cooperation in R&D stage We introduce cooperation into R&D, while the production remains non-cooperative. At the first stage, the agents maximize the joint profits, as a function of x1 and x2 . max (π1 + π2 ) =

{x1 ,x2 }

1 [(a − 2A1 + A2 ) + (2 − β1 )x1 + 9 1 (2β2 − 1)x2 ]2 + [(a + A1 − 2A2 ) + 9 (2β 1 − 1)x1 + (2 − β2 )x2 ]2 − 1 1 γ1 x21 − γ2 x22 (4) 2 2

SUN Cai-hong, et al./Systems Engineering — Theory & Practice, 2009, 29(3): 21–27

We obtain the following unique solutions for the equilibrium: Fi Dj − Fj C x ˆi = Di Dj − C 2

(i, j = 1, 2; i 6= j)

=

∧

∧

(a − 2Ai + Aj ) + (2 − 3 (i, j = 1, 2; i 6= j) ∧

yi =

1 ∧2 γi x i 2

w

=

∧ 1) xj

(6)

(i = 1, 2)

∧ 1 ∧ ∧ πi = qi2 − γi x2i 2 ∧

+(2βj −

(i = 1, 2)

∧ + q2 )

2.3 Limitations of equilibrium We maximize profits as a function of two variables x1 and x2 .The first-order condition is expressed as a 2×2 matrix equation. Maximization requires the first principal minor of the coefficient matrix (Hessua matrix) to be negative and the second principal minor to be positive. The first principal minor is H1 = 2(2 − β1 )2 + 2(2β1 − 1)2 − 9γ1 < 0, which implies γi >

10 9

β2 )2 + 2(2β2 − 1)2 − 9γ2 ] − 4[(2 − β1 )(2β2 − 1) + (2β1 − 1)(2 − β2 )]2 > 0

(i = 1, 2)

(7)

(8)

In addition to the limitations on parameters implied by second-order conditions, the model must also be calibrated with respect to the basic rationales. There is an important condition in the problem setup that must be satisfied. The condition is q1 + q2 < a

3

(9)

Simulation

The asymmetry makes the model more realistic but also makes it more difficult to analyze and solve the model. As so many parameters and asymmetries are involved, it is difficult to draw conclusions through direct analysis. So simulation was used to analyze the asymmetry in R&D and production. 3.1

1 ∧ ∧ ∧ ∧ ∧ − (q1 + q2 )2 − (c − x1 −β2 x2 ) q1 2 1 ∧ 1 ∧ ∧ ∧ ∧ −(c − x2 −β1 x1 ) q2 − γ1 x21 − γ2 x22 2 2 ∧ a(q1

[2(2 − β1 )2 + 2(2β1 − 1)2 − 9γ1 ][2(2 −

[10 − 9γ1 ][10 − 9γ2 ] − 64 > 0

where xi satisfies the first-order condition. The symmetric cooperative equilibrium in production, R&D investments, profits, and social welfare corresponding to the following unique solution are qi =

=

We have

−2(2 − βi )(a − 2Ai + Aj ) − 2(2βi − 1)(a + Ai − 2Aj ) (i, j = 1, 2; i 6= j)

∧ βi ) xi

H2

(5)

where Di = 2(2 − βi )2 + 2(2βi − 1) − 9γi (i = 1, 2) C = 2(2 − β1 )(2β2 − 1) + 2(2β1 − 1)(2 − β2 ) Fi

The second principal minor:

Setup of parameters, the simulation of model

We propose the following two-step procedure of calibrating the model for simulation. First, we set the values of the parameters in such a way that second-order conditions and simpler model conditions are satisfied. After that we run the series of simulations and sensitivity tests to figure out which set of parameters will satisfy more complicated conditions. Because we are interested in the testing of solutions for a wide range of asymmetry levels, we must be sure that selected parameters also satisfy the above conditions on both upper and lower limits of the selected interval. In this paper, we use that parameter set, which gives a valid range of asymA2 γ 2 , ≤ 5, 0 ≤ ββ12 ≤ 10. Table 1 contains 14 metry: 1 ≤ A 1 γ1 sets of values of parameters corresponding to each asymmetric position simulated. The parameters can be classified into 3 categories: the values of parameters from date1 to date5 are

Table 1. Values of Parameters

Data 1 2 3 4 5 6 7 8 9 10 11 12 13 14

β1 0.0 0.1 0.5 0.9 1.0 0.0 0.1 0.5 0.9 1.0 0.1 0.5 0.9 1.0

β2 0.0 0.1 0.5 0.9 1.0 0.0 0.1 0.5 0.9 1.0 From 0 to 1 successively From 0 to 1 successively From 0 to 1 successively From 0 to 1 successively

A1 10 10 10 10 10 10 10 10 10 10 10 10 10 10

A2 From 10 to 50 successively From 10 to 50 successively From 10 to 50 successively From 10 to 50 successively From 10 to 50 successively 10 10 10 10 10 10 10 10 10

γ1 15 15 15 15 15 15 15 15 15 15 15 15 15 15

γ2 15 15 15 15 15 From 15 to 75 successively From 15 to 75 successively From 15 to 75 successively From 15 to 75 successively From 15 to 75 successively 15 15 15 15

a 250 250 250 250 250 250 250 250 250 250 250 250 250 250

SUN Cai-hong, et al./Systems Engineering — Theory & Practice, 2009, 29(3): 21–27

on behalf of asymmetric initial cost; the values of parameter from date6 to date10 are on behalf of asymmetric innovation ability; the values of parameter from date11 to date14 are on behalf of asymmetric spillover. Second, carry out the simulation with the determined parameters. Figure 1(a) and Figure 1(d) respectively simulate

∧ y2 ∧ y1

(R&D investment comparison),

∧ q2 ∧ q1

(Production

∧

∧

comparison), π∧2 (profit comparison), and w(social welfare π1

level)varying with

A2 A1 (cost

2(d) respectively simulate γ2 γ1 (innovation

∧

∧

∧ y1

,∧,

Figure 2. Variation diagram of

∧ ∧ ∧ y2 q2 π2 ∧

,∧,

∧

,w along

y1 q1 π1

∧ ∧ ∧ y2 q2 π2 ∧ ∧

,∧,

∧

y1 q1 π1

∧

y2 q2 π2 q1

asymmetry).

∧

∧ y1

∧

,∧,

∧

q1

∧ π1

∧

and w varying with

rate asymmetry); Figure 3(a) to 3(d) respec-

tively simulate

Figure 1. Variation diagram of

asymmetry); Figure 2(a) to ∧

y 2 q2 π 2

,w along

A2 A1

γ2 γ1

∧ π1

∧

and w varying with

β2 β1

(spillover

SUN Cai-hong, et al./Systems Engineering — Theory & Practice, 2009, 29(3): 21–27

Figure 3. Variation diagram of

∧ ∧ ∧ y2 q2 π2 ∧ ∧

,∧,

∧

,w along

y1 q1 π1

4 Simulations analysis Numerical simulation indicates that asymmetries affect R&D investments and production in semi-collusion, which can be grouped. First, under the conditions of asymmetric initial cost, the lower the cost is, the higher R&D investment, output, and profit are. The gap of R&D investment, output, and profit expands with initial cost conditional on technology spillover. While the gap is narrowed, the gap of initial cost tends to be small as spillover increases. The change of spillover has little effects on the gap of output and profit. (Figure1(a) to Figure1(c)) Second, under the conditions of asymmetric initial cost, the wider the initial cost is, the lower social welfare is. However, social welfare is not monotonic with spillover conditional on initial cost. Third, the higher the innovation ability is, the more R&D investment and output are. The gap of the innovation ability increases with that of R&D investments and output conditional on spillover. When spillover increases, the gap of two agents’ R&D investment and output tends to be narrowed conditional on the gap of innovation ability, however, the change of spillover has little effects on R&D investments. (Figure 2(a), Figure 2(b)). Fourth, as for profit, when spillover is close to 0, the agent with lower innovation ability has little profits; moreover, the gap of profit will increase with the gap of innovation ability. When the spillover is close to 1, the agents with higher innovation ability has higher profits. Moreover, the gap of profit will widen with the gap of increasing innovation ability. (Figure 2(c)). Fifth, the wider the gap of innovation ability is, the lower social welfare is. However, social welfare is not monotonic with spillover. (Figure 2(d)) Sixth, conditional on the asymmetric spillover, the

β2 β1

smaller spillover is, the less R&D investment is. The gap of spillover is monotonic with that of R&D investment. (Figure 3(a)) Seventh, the lower the spillover is, the higher the equilibrium output is and vice verse. The narrower the gap of spillover is, the less the equilibrium output is. The gap of output increases with that of spillover. We can draw similar conclusions from the analysis of profits. (Figure 3(b), Figure 3(c)) Eighth, as the spillover takes different values, the gap of social welfares varies with the spillover. When the spillovers of two agents are very high or very low, social welfare is high. If one agent’s spillover is very low while that of other agent is very high, social welfare is low. If one agent’s spillover is close to 0.5, the narrower the gap of spillover is, the higher social welfare is. (Figure 3(d))

5

Conclusions

The simulation shows that the introduction of asymmetry into the semi-collusion model will affect R&D and productivity, which can be grouped. From the view point of R&D investment, the agent who advantages in initial cost and innovation ability or high spillover will be active in R&D investment. Moreover, with the asymmetry intensified, the gap of R&D investments sharpened. With spillover increased, the gap of R&D investment tends to be narrowed. In terms of the contrast of production, the agent who advantages in initial cost and innovation ability or low spillover will have a higher output. With the asymmetry intensified, the gap of output will be expanded. However, Figure 2(b) and Figure 2(d) shows that the asymmetries of innovation ability and spillover have little effects on the contrast of output. If market concentration is to be reduced, the government should encourage the agents with asymmetric innovation ability and spillover to cooperate.

SUN Cai-hong, et al./Systems Engineering — Theory & Practice, 2009, 29(3): 21–27

As for the contrast of the profit, the agent who advantages in initial cost always has higher profit. The gap of profit tends to be narrowed with the increase of spillover; however, the gap is not very high. The gap of innovation ability has little effects on spillover coefficient. When spillover is close to 0, the agent with higher innovation ability has more profits, and the gap narrowed with the increase of spillover. When spillover is close to 1, the agent with low innovation ability has higher profits, and the gap narrowed with the increase of spillover. When spillovers are asymmetric, the agent with low spillover has higher profits. The gap of spillover increases with the gap of profits. Due to the effects of the asymmetric profits, the government should encourage semicollusion, and subside the agent with lower profits, including higher initial costs, higher spillovers, higher innovation ability (when the two agents’ spillovers are close to 1) to cooperate. As to social welfare, the higher the initial cost or innovation ability is, the lower the social welfare is. However, social welfare is not monotonic with the gap. When the two agents’ spillovers are very low or high, social welfare is higher. When one agent’s spillover is very low and the other’s is very high, social welfare is very low. When one agent’s spillover is close to 0.5, the narrower the gap of spillover is, the higher social welfare is. From the point of social welfare, the government should encourage the agents with close initial cost and innovation ability or the agents

whose spillover is close at a low or high level to cooperate.

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