Game theory analysis of technology adoption timing and pricing decision in supply chain system under asymmetric nash equilibrium

Abstract

The technology adoption of a firm is endowed with complexity if the production technology upgrades continually and the price competition become intensified. This paper establishes a supply chain involving one technology supplier and two manufacturing enterprises. The supplier provides a kind of production technology and offers its upgraded versions for enterprises to produce products. On this basis, this paper studies the problem that whether or not to buy this kind of production technology in the perspective of each enterprise, explores the problem that when to buy an upgraded version of production technology in the perspective of each enterprise, and analyzes the price competition between two kinds of products. The results show that the asymmetric Nash equilibrium can effective avoid the prisoner’s dilemma of technology investment, and help to maximize the interest of the social welfare system that includes the technology supplier, manufacturing enterprises and consumers.

Keywords

Asymmetric nash equilibrium duopoly technology adoption bertrand game social welfare

1 Introduction

The introduction of advanced production technology from universities, research institutes and other economic and social knowledge centers not only helps themselves (especially small and medium-sized enterprises) achieve leaping development, but also makes the value of technological R & D input realized. However, in reality, the transfer of technology and the commercialization of scientific and technological achievements are still facing many challenges. First, the fluctuation of the technology purchase price is getting bigger and bigger, and the cycle of technology upgrading is getting shorter and shorter. When an enterprise is preparing to purchase a production technology, it will consider whether to wait for a period of time to purchase the technology at a low price or wait for some time to purchase the upgraded production technology. Then, the cost of introducing new technologies is often transferred to the consumer in the form of increasing the price of the end product. When the enterprise faces the competition of the competitor’s product price, the enterprise will be very concerned about the necessity of upgrading the production technology. Therefore, considering the speed of technological innovation, the degree of innovation, the cost of introduction and the competition of peers, it is of great reference value for enterprises to introduce the production technology and participate in the price competition of products market.

In view of the low conversion rate of technology, some scholars have studied the key factors that affect the transformation of technology and put forward the corresponding solutions. Anderson thinks that scientific and technological achievements have quasi public attributes, and whether the University’s running nature (public or private) affects the efficiency of University’s research and technology transformation is studied [1]. Wang Bangjun and Zhao Leiying will be the key factors of technology transformation effect is attributed to environmental factors (market demand level, the protection of intellectual property rights and legal support) and process factors, as Science and technology intermediary service, collaborative innovation service platform, information communication network construction and cultural value integration [2]. Huang Jingjing pointed out that the efficiency of scientific and technological cooperative development of small and medium-sized enterprises is better than that of large enterprises, but the conversion efficiency of scientific and technological achievements is not as good as that of large enterprises [3]. Liu Jun and others studied the impact of urbanization on R&D and transfer [4]. It is considered that the market of the urban area is developed and the transaction efficiency is high, which can improve the efficiency of technology transformation. In order to promote the transformation of technology, governments and scholars from various countries have proposed their own solutions from different angles. For example, the governments of China, the United States, Britain, Japan, Germany and other countries have introduced laws that are conducive to the transformation of scientific and technological achievements, and provide legal protection for the transformation of scientific and technological achievements [5]. Wang Kai and Zou Xiaodong suggested that Chinese universities establish a new organization operation mode, PoCCs, to solve the funding gap of technological innovation projects in the seed stage. It is believed that the innovation of organization operation mode can improve the success rate and conversion rate of scientific and technological achievements in universities [6]. Qi Yong pointed out scientific and technological achievements will be the basis for the classification of public welfare, common technology and proprietary technology three categories, establishing principal-agent model, analyze the transformation of scientific and technological achievements of the market, information technology that the transferor in the market weakness in the transformation is the key factor hindering technology transformation, relying on science and technology intermediary service institutions the information advantage of efficiency can facilitate the transformation of technological achievements [7]. From the above research, we know that the government and scholars have gradually realized the importance of technology transformation, and gave a way to promote the transformation of technology from the macro level. However, in order to better guide the practice of technology transformation between universities, scientific research institutions and enterprises, we need to further study the technology import strategy of enterprises under the market competition from a micro perspective.

Considering the competitive environment of the enterprise, some scholars have actively explored the technology introduction strategy of the enterprise under the market competition. For example, Reinganum considers market competition and sets up a game model between enterprises, and analyzes the technology investment behavior of enterprises under the competition of product market [8]. Although Reinganum research does not consider the difference between manufacturing enterprises and ignores the continuity of technological innovation, his pioneering research successfully guided scholars to conduct in-depth research on technology investment game of enterprises based on product competition. Hendricks takes the technology innovation type in Reinganum as the case where the cost of technology purchase falls [9], while Nair describes the technological innovation as the decline of marginal production cost, and analyzes the timing of enterprise purchasing technology [10]. In the market entry theoretical framework, Li and Jin have studied the timing of product entry into the market and the timing of product upgrading [11]. Because the market competition of products often has an impact on the technology purchase behavior of enterprises, the research ideas and analysis methods of technology purchase competition and product competition have gradually attracted the attention of researchers [12]. For example, Rahman and Loulou believe that the renewal of production technology has no effect on the quality and price of products [13], but it has an impact on the marginal production cost of products, so the two enterprises compete on the number of products. In reality, even if two enterprises purchase the same production technology at the same time, there are differences between the products produced by the two enterprises due to the different factors such as brand, sales market, and employee skill level [14, 15]. Therefore, the analysis of the price competition between the two enterprises is more in line with the actual situation of the operation of the enterprise. In this context, it has become a very practical and theoretical issue to study whether or not the enterprises buy or buy new technologies at any time.

In this paper, the price game model of Bertrand is introduced to examine the product price competition in the duopoly market. Under the circumstance that the expected technology providers will provide two stages of future innovative production technology, the technology purchase strategy of duopoly enterprises consisting of single technology providers and duopoly enterprises is studied [16]. First calculate the duopoly enterprises are willing to pay for future technology supplier first and second generation technology critical purchase price, then the sales price and the enterprise technology supplier is willing to pay the purchase price of technology were compared. The effect analysis for supplier sales price of buying the game results. The results show that the different technology will make the sales price between duopoly firms buying the game results into a symmetric Nash equilibrium or non symmetric Nash equilibrium, non symmetric Nash equilibrium can effectively avoid the prisoner’s dilemma, technology investment, and to maximize the technology suppliers, technology buyers and consumers of the whole the interests of the social welfare system.

2 Model design

2.1 Basic assumptions

The BertRand model is a competition model proposed by the French economist Joseph Bertrand in 1883. It is the model of the analysis of the price competition of the enterprises in the oligopoly market. It is assumed that there are only two companies on the market:Enterprise 1 and enterprise 2 are priced at the same time, the products they produce are exactly the same (i.e. homogeneous), and the cost function of oligarch enterprises is also the same: the marginal cost of production is equal to the unit cost, assuming that there is no fixed cost. The market demand function D(p) is a linear function without any formal or informal collusion between each other.

Because the products produced by the two oligopoly enterprises are homogeneous, so they are completely substitutable, so the two enterprises will get all the needs with low price, while the high priced ones will lose the whole market. If the two enterprises have the same price, they will share the market equally.

When the quality of the two enterprises is the same and the marginal cost is constant, the reaction function and its Nash equilibrium level are shown in Fig. 1.

Fig. 1.

Bert Rand competition model.

In Fig. 1, $P_{1}^{*} (P_{2})$ , $P_{2}^{*} (P_{1})$ is the optimal response function of the enterprise 1 and the enterprise 2, and the two coordinate axes represent the strategic choices of two enterprises respectively.

Because enterprise 2 and enterprise 1 have the same marginal cost, their reaction function curves are the same shape and about 45 degree line symmetry. The intersection point of the two reaction curves represents the Nash equilibrium point of the game.

Here, Nash equilibrium is a combination of price strategies, and no business can make a profit by unilaterally changing the price. The equilibrium point $P_{1} = P_{1}^{*} (P_{2})$ is not only the optimal pricing of enterprise 1, but also the optimal pricing of enterprise 2 as $P_{2} = P_{2}^{*} (P_{1})$ . In addition, the optimal pricing of the two enterprises is equal to the marginal cost, that is $P_{1}^{*} = P_{2}^{*} = MC$ .

Through the above analysis, we can see that under the condition of the same quality and marginal cost of the two enterprises, the Bert Nash model has the unique equilibrium. When the prices of the two enterprises are the same, they are all equal to the marginal cost, and the profit is equal to zero (but they still get the normal profit).

The Nash equilibrium under the Bert Rand model of the two enterprise product homogeneity but the marginal cost change is analyzed.

Suppose that a major innovation technology (T₀) has been developed successfully and has been successfully applied to production by two duopoly enterprises. At the same time, it is expected that the technology providers will provide the two generation of improvement and innovation of the technology (T₁, T₂) in the future, and the selling price of T₁ and T₂ is X and Y respectively. The technology T₀ used by duopoly enterprises is the major innovation technology, the marginal production cost is C, and the two enterprises launch the Bertrand price game on the product market. When buying T₁ and T₂ in the future, the marginal production cost of enterprises will be reduced to C₁ and C₁¹ due to the different learning effects formed by the process of T0, and the marginal production cost of enterprises 2 will be reduced to C₂ and C₂¹. The goal of a duopoly is to maximize the discount rate when the discountrate is R.

At the next stage, two enterprises need to decide whether to buy new technologies at the same time (B = “buy”, “W = does not buy, wait for better technology”), and determine the price of their products to participate in the price competition of the product market. Each enterprise has four kinds of production plan: (1) has been used for production technology of T₀ (2) has been used in the next stage of a purchase T₁ replacement T₀ and (3) wait for the future stage two to buy T₂ to replace the T₀ (4) and has been used in the next stage of a purchase T₁ replacement T0 until stage two to buy technology T₂ replaced T₁.

The analysis process is not simplified. In this analysis, the profit calculation of enterprise 1 is based on the assumption that enterprise 2 is a competitor. At the same time, for analysis convenience, this paper assumes that the new technology purchased by enterprises can be immediately put into production and completely replace the previous generation technology, and will not take into account the new technology competition in the technology sales market when the last generation technology discount is involved in the new technology competition. It is assumed that the reverse demand (demand decision price) function of the product is as follows: $q_{1} = α - p_{1} {+ β p}_{2}$ (1) $q_{2} = α - p_{2} {+ β p}_{1}$ (2)

It can be seen that each profit function is a one price two times function of price. With the increase of price, the profit will increase first and then decrease. When a certain p value is reached, the profit will reach the maximum. $\frac{\partial π_{1}}{\partial p_{1}} = α - 2 p_{1} + β p_{2} + c_{1}^{i} = 0$ (3) $\frac{\partial π_{2}}{\partial p_{2}} = α - 2 p_{2} + β p_{1} + c_{2}^{i} = 0$ (4)

To do the corresponding transformation, find out respectively, as follows: $p_{1} = \frac{2 α + α β + β c_{2}^{i} + 2 c_{1}^{i}}{4 - β^{2}}$ (5) $p_{2} = \frac{2 α + α β + β c_{1}^{i} + 2 c_{2}^{i}}{4 - β^{2}}$ (6)

Assuming that enterprise 2 is a competitor, The profit of enterprise 1 is π₁ =π_X1X2/Y1Y2, (X1X2) the two phase of the 1 representative enterprises in future technology purchasing decisions and Xi∈(B, W), (Y1Y2) on behalf of the decision-making of enterprise 2, Yi ∈(B, W). Enterprise 1 and enterprise 2 carry out the purchase game of the future two stages of technology, and the result of the game is shown in Table 1.

Table 1

Game results for the future two stages of technology purchase

Ent. 1	BB	WB	BW	WW
Ent. 2
BB	πBB/BB	πWB/BB	πBW/BB	πWW/BB
WB	πBB/WB	πWB/WB	πBW/WB	πWW/WB
BW	πBB/BW	πWB/BW	πBW/BW	πWW/BW
WW	πBB/WW	πWB/WW	πBW/WW	πWW/WW

2.2 The critical technology purchase price that a duopoly is willing to pay

In the process of the above analysis, the Bert Rand price game between two enterprise products has introduced the general form of the profit function of the enterprise. In the two stage of the technology purchase game, considering the purchase cost of technology T₁ and T₂, the 1 profit of enterprises can be calculated by the general form of profit function 1. For example, in the two stage game, enterprise 1 adopts the second production plan mentioned above, that is, buying T₁ does not buy T₂, while enterprise 2 uses third kinds of production plan, that is, do not buy T₁ to buy T₂, the profit of enterprise 1 is:

$\begin{matrix} π_{B W / W B} = \frac{{[2 α + α β + β c + (β^{2} - 2) c_{1}]}^{2}}{{(4 - β^{2})}^{2} (1 + r)} \\ + \frac{{[2 α + α β + β c_{2}^{1} + (β^{2} - 2) c_{1}]}^{2}}{{(4 - β^{2})}^{2} {(1 + r)}^{2}} - X \end{matrix}$ (7)

In the same way, when two enterprises adopt fourth production schemes at the same time, the profit of enterprise 1 is:

$\begin{matrix} π_{BB / BB} = \frac{[2 α + α β + β c_{2} + (β^{2} - 2) c_{1}]^{2}}{(4 - β^{2})^{2} (1 + r)} \\ + \frac{[2 α + α β + β c_{2}^{1} + (β^{2} - 2) c_{1}^{1}]^{2}}{(4 - β^{2})^{2} (1 + r)^{2}} - X - Y \end{matrix}$ (8)

For enterprise 1, the result of any game in Table 1 is to be Nash equilibrium, and the profit of the result must be greater than the other three profit results.

When π_BB/BB> π_WB/BB $\begin{matrix} \begin{matrix} \frac{[2 α + α β + β c_{2} + (β^{2} - 2) c_{1}]^{2}}{(4 - β^{2})^{2} (1 + r)} + \frac{[2 α + α β + β c_{2}^{1} + (β^{2} - 2) c_{1}^{1}]^{2}}{(4 - β^{2})^{2} (1 + r)^{2}} \\ \begin{matrix} - X - Y \geq \end{matrix} \frac{[2 α + α β + β c_{2} + (β^{2} - 2) c]^{2}}{(4 - β^{2})^{2} (1 + r)} \\ + \frac{[2 α + α β + β c_{2}^{1} + (β^{2} - 2) c_{1}^{1}]^{2}}{(4 - β^{2})^{2} (1 + r)^{2}} - Y \end{matrix} \end{matrix}$ $\begin{matrix} \Rightarrow X \leq \\ \frac{(2 - b^{2}) (c - c_{1}) [4 α + 2 α β + 2 β c_{2} + (β^{2} - 2) (c + c_{1})]}{(4 - β^{2})^{2} (1 + r)} = x_{1} \end{matrix}$ (9)

Similarly, in Table 1, when other game results become Nash equilibrium, we can solve the critical technology purchase price Xi and Yi that the enterprises are willing to pay to the technology providers. $x_{2} = \frac{(2 - b^{2}) (c - c_{1}) [4 α + 2 α β + 2 β c + (β^{2} - 2) (c + c_{1})]}{(4 - β^{2})^{2} (1 + r)}$ $x_{3} = \frac{(2 - b^{2}) (c - c_{1}) [4 α + 2 α β + 2 β c_{2} + (β^{2} - 2) (c + c_{1})] (2 + r)}{(4 - β^{2})^{2} (1 + r)^{2}}$ $x_{4} = \frac{(2 - b^{2}) (c - c_{1}) {(2 + r) + 2 b (c + cr + c_{2}^{1}}}{(4 - β^{2})^{2} (1 + r)^{2}}$ $x_{5} = \frac{(2 - b^{2}) (c - c_{1}) [4 α + 2 α β + 2 β c + (β^{2} - 2) (c + c_{1})] (2 + r)}{(4 - β^{2})^{2} (1 + r)^{2}}$ $y_{1} = \frac{(2 - b^{2}) (c - c_{1}^{1}) [4 α + 2 α β + 2 β c_{2}^{1} + (β^{2} - 2) (c + c_{1}^{1})]}{{(4 - β^{2})}^{2} {(1 + r)}^{2}}$ $y_{2} = \frac{(2 - b^{2}) (c - c_{1}^{1}) [4 α + 2 α β + 2 β c + (β^{2} - 2) (c_{1} + c_{1}^{1})]}{(4 - β^{2})^{2} (1 + r)^{2}}$ $y_{3} = \frac{(2 - b^{2}) (c - c_{1}^{1}) [4 α + 2 α β + 2 β c + (β^{2} - 2) (c + c_{1}^{1})]}{{(4 - β^{2})}^{2} {(1 + r)}^{2}}$ $y_{4} = \frac{(2 - b^{2}) (c - c_{1}^{1}) [4 α + 2 α β + 2 β c_{2} + (β^{2} - 2) (c + c_{1}^{1})]}{{(4 - β^{2})}^{2} {(1 + r)}^{2}}$ $y_{5} = \frac{(2 - b^{2}) (c - c_{1}^{1}) [4 α + 2 α β + 2 β c + (β^{2} - 2) (c + c_{1}^{1})]}{(4 - β^{2})^{2} (1 + r)^{2}}$

After solving the critical technology purchase price Xi and Yi, which is willing to be paid to the technology provider, the result of technology purchase game between duopoly enterprises depends on the selling price of technical suppliers. Here is an example of the relationship between the game results and the price of the sale of the technology. When we set the parameters of alpha = 10, beta = 0.7, c = 12, c1 = 6, c2 = 5, c = 2, C₂¹ = 1.5, r = 0.1, the Xi, Yi and tagged in Fig. 2 are solved in turn. As shown in Fig. 1, the game result is affected by the selling price of X and Y of future technology T₁ and T₂. When Y is fixed, the purchase quantity of technology T1 is less and less with X increasing. When X >X₅, no enterprise will buy T₁ again. When X is fixed, as Y increases, the number of purchases of T₂ is less and less, and no enterprise will buy T₂ again when Y >Y₅. When X and Y are increasing at the same time, the best strategy for enterprise 1 and enterprise 2 is to buy one of the technologies each.

Fig. 2.

the critical purchase price of technology T1 and T2 that enterprises are willing to pay.

3 Model analysis

In the above models, the value of critical purchase price that enterprises 1 are willing to pay to technology providers is influenced by market size, substitution coefficient between products, marginal production cost of enterprises and marginal production cost of competitive enterprises. On the one hand, the different values of these parameters make the enterprises willing to pay the critical purchase price of the technology providers differently, and on the other hand, the game results of the enterprises change under the same technology selling price. In this paper, the learning ability is different, even in the future to buy the same innovative technology, the marginal cost of production enterprises and products will decline between different c–c1≠c-c2. These differences not only make the enterprise different from other enterprises, but also bring different competitive advantage and market profit to the enterprise [14]. Next, we can examine the learning effect of the enterprise and the influence of Nash equilibrium on the strategy and profit of enterprise’s technologypurchase.

3.1 The learning effect of enterprise

Generally speaking, when a company first purchases technology T0 for production, it will gradually form the learning effect of the technology, so that the marginal production cost will be significantly reduced when the technology T1 is subsequently purchased. However, the learning effect of the enterprise wills gradually diminishing. With the continuous purchase of this series of technologies, the marginal production cost will gradually become stable. Based on the principle of marginal decline in learning effect of technology purchase, literature [15] divides technological innovation into major innovation and improvement innovation, and analyzes the optimal technology purchase strategy of enterprises in these two situations. Through the purchase of major innovative technologies, enterprises can gain great learning effects in future production. But with great innovation and characteristics of the time interval of strong randomness, so as to improve the innovation of technology innovation based on the major is also very important, at the same time, improve the learning effect of innovative technology leads to a certain extent of marginal decline will give birth to great innovation.

On the other hand, the ability of the competitor’s learning ability at the same price of the same technology will also affect the purchase behavior of the enterprise and the corresponding profit. As shown in Fig. 2, When the T2 price of the technology supplier is Y = y₂–ɛ, ɛ>0, and when the price of T1 is X <X1, T2<Y2 and the game result is (BB/BW). At this point, if the competitor’s C2 decreases so that the Y2 moves down to y/₂ = y₂-θ, and θ> ɛ, the T2>Y2 game result is (BW/BW). Therefore, if the margin of difference between the marginal production cost of competitors C, C₂ and C₂¹ is bigger, that is, the stronger the learning ability of enterprise 2, the less the number of technology purchased by enterprises 1, and the less profits theywill get.

3.2 Advantages of asymmetric Nash equilibrium

In this paper, according to the definition of literature [15], the game results in Table 1 are divided into symmetric Nash equilibrium and asymmetric Nash equilibrium. Game two enterprises at the same time to buy or not to buy at some stage or two stage technology as the result of symmetric Nash equilibrium, the enterprise faces the prisoner’s dilemma, such as the technology of the future purchase price is too low or too high to cause the result of the game for the symmetric equilibrium (BB/BB), (BW/BW), (WB/WB), (WW/WW) when compared to the current situation of the two enterprises will not buy because of better future technology innovation. While the result of the game is asymmetric Nash equilibrium (BB/BW), (BB/WB), (BW/WW), (WB/WW), the more profits the more frequent the purchase of technology is obtained, such as:

$\begin{matrix} π_{BB / BW} - π_{BW / BB} = \frac{{[2 a + ab + {bc}_{2} - (2 - b^{2}) c_{1}]}^{2}}{(4 - b^{2})^{2} (1 + r)} + \frac{{[2 a + ab + {bc}_{2} - (2 - b^{2}) c_{1}^{1}]}^{2}}{(4 - b^{2})^{2} (1 + r)^{2}} \\ - \frac{{[2 a + ab + {bc}_{2} - (2 - b^{2}) c_{1}]}^{2}}{(4 - b^{2})^{2} (1 + r)} - \frac{{[2 a + ab + {bc}_{2}^{1} - (2 - b^{2}) c_{1}]}^{2}}{(4 - b^{2})^{2} (1 + r)^{2}} \\ = \frac{[2 b (c_{2} - c_{2}^{1}) + (2 - b^{2}) (c_{1} - c_{1}^{1})] \times [4 a + 2 ab + 2 b (c_{2} + c_{2}^{1}) - (2 - b^{2}) (c_{1} + c_{1}^{1})]}{(4 - b^{2})^{2} (1 + r)^{2}} > 0 \end{matrix}$ (10)

Therefore, how to realize asymmetric Nash equilibrium and to buy or upgrade new technology more actively in the game have become a hot spot of research. On the other hand, to give up the purchase of technology corporate profits will not necessarily lose a lot, because it no longer pay the purchase cost of the technology but also by “leapfrog” strategy for the purchase of higher efficiency technology, forming technology purchase the advantage of [16]. The above examples also point out the drawbacks of scale economy. Nowadays, with the rapid development of economy, many enterprises are pursuing economies of scale unilaterally. For example, production enterprises have the motivation to increase investment in equipment, technology and labor. In this paper, when the scale of the enterprise technology investment has made the game equilibrium become non symmetric Nash equilibrium (e.g. BW/BB), if we continue to increase the investment will make the game become a symmetric Nash equilibrium (e.g. BB/BB) form the prisoner’s dilemma of technology investment.

4 The influence of technology purchase strategy on social welfare

In the above models, a duopoly monopolizes enterprises to compete in the product market under the circumstance of price competition, so as to maximize their profit and make technological buying game. This part will further analyze the impact of different game results on the whole social welfare [17, 18]. In the general study, the social welfare is divided into two parts of the consumer surplus and the producer’s surplus. But in this article, the product manufacturer’s technology is acquired through the purchase, so the social welfare analysis should also include the welfare of the technical supplier.

In the previous analysis, if the duopoly monopolized the manufacturer to buy the future each stage technology would form the game result (BB/BB). At this time, the quality of products is greatly improved, and because of the price competition between enterprises, the price of products is very low, and consumers get the maximum welfare [19]. Technology providers get the most benefits because they sell more technology [20]. But because the producers are facing a huge loss in the prisoner’s dilemma, the welfare of the whole society may fall. In this paper, the Nash equilibrium results of manufacturer’s game are decided by the price of future technology T₁ and T₂, so the change of price of technology T₁ and T₂ will cause the change of Nash equilibrium. There are also differences in social welfare under different game results, so it is very important to study the influence of X and Y on social welfare in a small range of changes.

For example, in Fig. 2, when the T1 price as X = x5+ɛ, ɛ>0, And the price of T2 is Y >Y5, Nash equilibrium (WW/WW) given some incentive technology vendors will T1 the price drop for X = x5–θ, θ>0, Nash equilibrium will transfer from (WW/WW) to (BW/WW), at least one companies will choose to buy the future first generation technology of T1, and the operating status of the company will buy alone because the technology has improved. Because at least one technology has been sold, the benefits of technical suppliers have been improved. At the same time, the total social welfare has been improved as the technology improves the quality of the product and makes the consumer surplus. Similarly, when Nash equilibrium is (BB/BB), if the technology supplier is given some incentive, if the price of T2 is constant, the price of T1 will be increased in a small range, and the equilibrium will be changed to (WB/BB). At this time although the technology suppliers, consumer welfare fell slightly, but the only buyers of the T1 welfare has been greatly improved, and the other one to give up to buy T1 enterprises due to the purchase cost and expenditure can be purchased through alternative technology or development way of diversification to improve the return on investment. Compared with the situation of the prisoner’s dilemma in the technology purchase, the total social welfare will be greatly improved. Conversely, if the game result is asymmetric Nash equilibrium (WB/BB), if the technology supplier continues to reduce the technology price, the game result will turn to the symmetric Nash equilibrium (BB/BB), and the whole social welfare will decrease.

The government’s goal is to maximize the social welfare, so if the excitation control of the government through price subsidies and other ways to give technical supplier a future technology T1, T2 price makes the game results change from the symmetric Nash equilibrium for non symmetric Nash equilibrium, the social welfare will be improved.

5 Productivity equilibrium game

In order to reflect the merge as well as reorganization within industries in the duopoly monopolized model, we applied such concept of cross ownership. Cross owner is different from cross shareholding as well as the equity investment without the limitation. Cross owner is the conduct that an enterprise hold the share of another enterprise in a same or similar market constructed with multiple main bodies [21]. The studies showed that, through the cross ownership, an enterprise can obtain the equity of competitive opponent, although they might not own the decision making right, they can obtain profit and inhibit the competition of market, and acquire the specialized knowledge of competitive opponent, further to fulfill the identical effect and diverse investment.

Assumption 1. The arrange of cross-ownership of enterprise 1 to enterprise 2, i.e. holding the other party’s equity in δ proportion 0 ≤ δ ≤ 1. If $0 \leq δ \leq \frac{1}{2}$ and then the right of proprietary will be controlled by the original shareholder of enterprise 2; if $\frac{1}{2} < δ \leq 1$ and then the right of proprietary will be controlled by the shareholder of enterprise 2.

Assumption 2. Two enterprises implemented two-stage dynamic sequential game. At first stage, two enterprises define the capacity according to the efficiency maximum or social welfare maximum. If the government does conduct the management, the enterprise will make decision on the capacity according to efficiency maximum; otherwise, decision can be made on the social welfare according to social welfare. At second stage, two enterprises conduct Cournot competition around capacity according to efficiency maximum to define the capacity.

5.1 When $0 \leq δ \leq \frac{1}{2}$

If $0 \leq δ \leq \frac{1}{2}$ , and then it implies that enterprise 2 decision making will be controlled by original shareholders. The efficiency function of twoenterprise is $u_{1} = π_{1} + δ π_{2}, u_{2} = (1 - δ) π_{2}$

Firstly, to view the game at first stage, the operators of two enterprises will make decision on the capacity according to efficiency maximum, it needs to meet: $\frac{\partial u_{1}}{\partial q_{1}} = 0 and \frac{\partial u_{2}}{\partial q_{2}} = 0,$ it can be concluded: $q_{1} = \frac{(3 - δ) (a - m) + 8 x_{1} - 2 (1 + δ) x_{2}}{15 - δ}$ (11) $q_{2} = \frac{3 (a - m) - 2 x_{1} + 8 x_{2}}{15 - δ}$ (12)

And then view the game at secondary stage, it can be divided into two cases. In order to realize the efficiency maximum, two enterprises’ capacities need to meet: $\frac{\partial u_{1}}{\partial x_{1}} = 0 and \frac{\partial u_{2}}{\partial x_{2}} = 0$ put the Equation (11), (12) into the enterprise efficiency function, it can be concluded that: $x_{1} = \frac{(δ^{3} - 34 δ^{2} + 301 δ - 208) (a - m)}{δ^{3} - 37 δ^{2} + 363 δ - 559}$ (13) $x_{2} = \frac{16 (δ - 13) (a - m)}{δ^{3} - 37 δ^{2} + 363 δ - 559}$ (14)

Put Equation (13), (14) into $x_{1} - q_{1} = \frac{(δ - 3) (a - m) + (7 - δ) x_{1} + 2 (1 + δ) x_{2}}{15 - δ}$ $x_{2} - q_{2} = \frac{- 3 (a - m) + 2 x_{1} + (7 - δ) x_{2}}{15 - δ}$ it can be concluded that: $x_{1} - q_{1} = \frac{(- δ^{2} + 18 δ - 13) (a - m)}{δ^{3} - 37 δ^{2} + 363 δ - 559} > 0$ (15) $x_{2} - q_{2} = \frac{(δ^{2} - 12 δ - 13) (a - m)}{δ^{3} - 37 δ^{2} + 363 δ - 559} > 0$ (16)

If the government strictly regulates the capacity, they need make decision according to social welfare maximum, i.e. W maximum

$\frac{\partial W}{\partial x_{1}} = 0$ and $\frac{\partial W}{\partial x_{2}} = 0$ , it can be concluded that: $x_{1} = \frac{(- 3 δ^{4} + 96 δ^{3} - 861 δ^{2} + 1530 δ - 1350) (a - m)}{- 3 δ^{4} + 100 δ^{3} - 988 δ^{2} + 2640 δ - 2925}$ $x_{2} = \frac{(4 δ^{3} - 126 δ^{2} + 1080 δ - 1350) (a - m)}{- 3 δ^{4} + 100 δ^{3} - 988 δ^{2} + 2640 δ - 2925}$

Put Equation (17);(18)into $x_{1} - q_{1} = \frac{(δ - 3) (a - m) + (7 - δ) x_{1} + 2 (1 + δ) x_{2}}{15 - δ}$ (17) $x_{2} - q_{2} = \frac{- 3 (a - m) + 2 x_{1} + (7 - δ) x_{2}}{15 - δ}$ (18) it can be concluded that: $\begin{matrix} x_{1} - q_{1} \\ = \frac{(δ^{3} - 45 δ^{2} + 675 δ - 3375) (a - m)}{(15 - δ) (- 3 δ^{4} + 100 δ^{3} - 988 δ^{2} + 2640 δ - 2925)} > 0 \end{matrix}$ (19) $\begin{matrix} x_{2} - q_{2} \\ = \frac{(- δ^{4} + 46 δ^{3} - 720 δ^{2} + 4050 δ - 3375) (a - m)}{(15 - δ) (- 3 δ^{4} + 100 δ^{3} - 988 δ^{2} + 2640 δ - 2925)} > 0 \end{matrix}$ (20)

Proposition 1. When enterprise 1 does not execute the cross ownership to enterprise 2 or execute low level of cross ownership and proprietary right is controlled by the original shareholders of enterprise 2, excess capacity phenomenon can occur regardless whether the government executes strict capacity regulation.

5.2 When

\frac{1}{2} < δ < 1

enterprise’s decision making right will be held by the shareholders of enterprise. The efficiency functions of two enterprises turn to be u₁ = π₁ + δπ₂, u₂ = δ (π₁ + δπ₂).

In the game at secondary stage, the decision making state for the capacity of two parties needs to meet $\frac{\partial u_{1}}{\partial q_{1}} = 0, \frac{\partial u_{2}}{\partial q_{2}} = 0$ $q_{1} = \frac{(δ^{2} - 3 δ) (a - m) - 8 δ x_{1} + (2 δ^{2} + 2 δ) x_{2}}{δ^{2} - 14 δ + 1}$ (21) $q_{2} = \frac{(1 - 3 δ) (a - m) + 2 (1 + δ) x_{1} - 8 δ x_{2}}{δ^{2} - 14 δ + 1}$ (22)

If the enterprise holds the right of capacity decision making by itself, in the case that capacity is guaranteed to be positives, the balanced capacitylevel is: $x_{1} = 0$ (23) $x_{2} = \frac{(1 - 3 δ) (a - m)}{δ^{2} - 6 δ + 1}$ (24)

Put(23), (24) into $\begin{matrix} x_{1} - q_{1} \\ = \frac{(- δ^{2} + 3 δ) (a - m) + (δ^{2} - 6 δ + 1) x_{1} - 2 (δ^{2} + δ) x_{2}}{δ^{2} - 14 δ + 1} \\ x_{2} - q_{2} \\ = \frac{(3 δ - 1) (a - m) - 2 (1 + δ) x_{1} + (δ^{2} - 6 δ + 1) x_{2}}{δ^{2} - 14 δ + 1} \end{matrix}$ it can be concluded that: $x_{1} - q_{1} = \frac{(- δ^{4} + 15 δ^{3} - 15 δ^{2} + δ)}{(δ^{2} - 14 δ + 1) ((δ^{2} - 6 δ + 1)} < 0$ (25) $x_{2} - q_{2} = 0$ (26)

Proposition 2. In the case that enterprise 1 holds enterprise 2’s decision making right (not completely merged),if the enterprise determines the capacity by itself, enterprise 1 will not generate excess capacity but generate capacity shortage. Enterprise 2’s capacity is equal, realizing the balance of capacity.

In the case that the government executes strict governance to the capacity and holds the decision making right, it can be concluded that: $x_{1} = 0$ (27) $x_{2} = \frac{(- 4 δ^{4} - 14 δ^{3} + 98 δ^{2} - 18 δ + 2) (a - m)}{14 δ^{4} - 32 δ^{3} + 120 δ^{2} - 24 δ + 2}$ (28)

Put the Equation (27),(28)into $\begin{matrix} x_{1} - q_{1} \\ = \frac{(- δ^{2} + 3 δ) (a - m) + (δ^{2} - 6 δ + 1) x_{1} - 2 (δ^{2} + δ) x_{2}}{δ^{2} - 14 δ + 1} \\ x_{2} - q_{2} \\ = \frac{(3 δ - 1) (a - m) - 2 (1 + δ) x_{1} + (δ^{2} - 6 δ + 1) x_{2}}{δ^{2} - 14 δ + 1} \end{matrix}$ it can be concluded that: $\begin{matrix} x_{1} - q_{1} \\ = \frac{(- 6 δ^{6} + 110 δ^{5} - 384 δ^{4} + 224 δ^{3} - 42 δ^{2} + 2 δ) (a - m)}{(δ^{2} - 14 δ + 1) (14 δ^{4} - 32 δ^{2} + 120 δ^{2} - 24 δ + 2)} < 0 \end{matrix}$ (29) $\begin{matrix} x_{2} - q_{2} \\ = \frac{(- 4 δ^{6} + 52 δ^{5} + 68 δ^{4} - 228 δ^{3} + 16 δ^{2}) (a - m)}{(δ^{2} - 14 δ + 1) (14 δ^{4} - 32 δ^{2} + 120 δ^{2} - 24 δ + 2)} > 0 \end{matrix}$ (30)

Proposition 3. In the case that enterprise 1 holds enterprise 2’s decision making right (not completely merged),if the government hold the decision making right of the enterprise, enterprise 1 will not generate capacity shortage, enterprise 2 will generate excess capacity.

5.3 When δ = 1

if the enterprise 1 completely merges enterprise 2, the efficiency function of these two enterprises will be same, u₁ = π₁ + π₂u₂ = π₁ + π₂.

Two parties’ balanced capacity at first stage will be: $q_{1} = \frac{2 (a - m) + 8 x_{1} - 4 x_{2}}{12}$ (31) $q_{2} = \frac{2 (a - m) - 4 x_{1} + 8 x_{2}}{12}$ (32)

In the first stage game, if the enterprise can determine the capacity by itself, the balanced capacity will be: $\begin{matrix} x_{1} = x_{2} = q_{1} = q_{2} = \frac{a - m}{4}, x_{1} - q_{1} \\ = x_{2} - q_{2} = 0 \end{matrix}$

If the government hold the decision making of capacity, and then $\begin{matrix} x_{1} = x_{2} = \frac{2 (a - m)}{5}, q_{1} = q_{2} = \frac{3 (a - m)}{10} \\ x_{1} - q_{1} = x_{2} - q_{2} = \frac{a - m}{10} > 0 \end{matrix}$

Proposition 4. When enterprise 1 completely merges the enterprise 2, if the enterprise holds the capacity decision making right, two enterprises’ capacity will be equal; if the government holds the capacity decision making right, two enterprises will generate excess capacity.

6 Conclusion

Based on game theory, this paper studies the strategy of purchasing technology and upgrading from a technology supplier in a duopoly market, and analyzes the pricing of products under different technology procurement strategies. The research shows that, under the competition of product price, the game results of the enterprise’s purchase of technology will be divided into symmetrical Nash equilibrium or asymmetric Nash equilibrium. On the one hand, the symmetric Nash equilibrium makes the enterprise face the prisoner’s dilemma, which reduces the profit growth of the two enterprises. On the other hand, the asymmetric Nash equilibrium makes the technology enterprises acquire high profits, and at the same time, it will help to maximize the benefits of the whole social welfare system made up of technology providers, technology buyers and consumers. If the government gives certain incentive technology suppliers through price subsidies, the technology supplier price within a certain range, so that the game between enterprises to buy results from the symmetric Nash equilibrium results into non symmetric Nash equilibrium, the welfare of the whole society will be improved.

However, this paper only studies the general mode of new technology purchased by duopoly enterprises. Further research can examine the impact of the uncertainty of the marginal production cost of products on the game of technology import. We can also study how to play the role of government in the diffusion of technology under the goal of maximizing social welfare. Government’s strict capacity regulation might not solve the problem of excess capacity. It is found that in the study, in the case that low level cross-ownership, although strict government’s capacity can guarantee the capacity decision making conform to the requirements of social welfare maximum. In the case that enterprise 1 holds enterprise 2’s decision-making right (not completely merged), if the government holds the enterprise’s capacity decision making right, enterprise 1 will have capacity shortage, enterprise 2 will have excess capacity. But if the enterprise makes the decision by itself, excess capacity phenomenon will not occur. When enterprise 1 completely merges enterprise 2, the effect of strict capacity governance policy is not as good as that when the enterprise makes decision by itself, it will cause excess capacity instead. Its economic implication is that, government’s strict capacity governance policy must have sufficient scientific demonstration. Market’s decisive role must be upheld, otherwise government’s dysfunction will be caused, which is not helpful for the settlement of excess capacity problem.

Footnotes

Acknowledgments

This paper is supported by National Natural Science Foundation of China (71372139).

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