Closed-Loop Nash Equilibrium (CLNE)

I attempt to compare the difference between and through illustrated the trend chart. As revealed by figure 3, the ratio of and are increased, the trend of capital investment is a little tardy fall from to under sticky price; however, the increased investment of followed the increase of investment ratio . The value of surpassed _∞ under . On the other hand, the value of _∞surpassed under . When the value of investment ratio is the peak of price _∞ is reached under the quick adjustment price. After that, the trend of capital investment is a tardy fall.

Figure 31. Setting range are between and , the values

of and _∞are obtainment.

I attempt to compare the difference between and _∞ through illustrated the trend chart and obtain the higher value of capital investment with the sticky price.

paths in the initial game., but rather is admitted to make output yield and R&D exert strategy decisions conditional on the real development of the state of the system. CLNE is different from OLNE. CLNE, also named feedback Nash equilibrium (FNE) strategies are equilibrium strategies for any sub-game that starts from the initial time t and connected ₀ states and , although and are not on the state trajectories that begin from and . Due to restriction of LQ game with the infinite time horizon, the approach of Bellman’s value function is employed to detect linear FNE strategies. Kydland (1975), Kyland and Prescott (1977), Reynolds (1987), Reynolds (1991), and Fershtman and Kamien (1990) have been suggested this approach. The Equations of Hamilton-Jacobi-Bellman must be satisfied by using this approach in the following.

(118) where is the discounted current-value function with respect to firm ; furthermore, FNE starts at and . Because Equation (118) have to keep any allowable states and , the pair of FNE strategies are sub-game perfect Nash equilibrium strategies. The necessary conditions with respect to maximization indicated in Equation (118) are

– (119) (120) Since the right-hand side of (118) is concave in and , Equation (119) and (120) are sufficient conditions for the FNE strategies.

Substitute Equation (119) and (120) into Equation (118) produces.

– – – – –

–

–

– –

– – –

(121) Where

Equation (121) is composed of the system of two partial differential Equations (PDE).

I can acquire a pair of FNE value functions through solving this system and can be employed to obtain the FNE strategies. Because our formulation problem is the LQ structure, I suppose quadratic value functions as solution for the system of PDE in

Equation (121). As a result of the system is supposed in the following.

(122) Where are unknown parameters. It is proved by (Fershtman &

Kamien, 1987). Substitute (122) and its partial derivatives and into (121) productions the following systems of six non-linear algebraic Equations that have to hold for and . These unknown parameters must be satisfied in the following Equations.

– –

(123)

–

(124) –

(125)

(126)

(127)

(128) Abovementioned six Equations I can obtain the FNE for the firm’s R&D exertion level and production. To promise the equilibrist existence, sufficient conditions are demonstrated in the following. In practice, it is able to be illustrated that for special values of s such as or ∞ there absolutely exist FNE that are unique.

Firstly, I consider the condition that closes to zero and explain the range of price stickiness is sufficiently large.

Theorem 2. In the event of and if

, then (1) the existence of a set of FNE strategies is particular within the rank of linear, symmetric FNE strategy spaces.

(2) the steady linear FNE strategies are in the following.

–

(129)

(130) where and are value function parameters; furthermore, both and are positive.

(3) the steady linear FNE strategies produces R&D experience of Fu-kang bus transport service accumulation and price trajectories that converge conditionally locally asymptotically to the following stable state FNE R&D experience accumulation and price level for both firms.

(131)

(132) I can proof the theorem 2 from Appendix B and use these results for attempting to research relationship between the ratios of capital investment of R&D experience accumulations with Fu-kang bus transportation service and the differentiate rate of configuration service of Fu-kang bus routing under . For simplicity, I intend to use the method of numerical analysis for discussions on the numerical experiments in next subsection.

Secondly, I ponder the condition that the adjustment of speed of the market price changes into sufficiently large, i.e. that is ∞. It ponders that the market price adjusts concurrently along the demand function. Hence, the linear inverse demand function is a particular case of Equation (95). I use the illustrated procedure previously and obtain the results in the following.

Theorem 3. In the event of ∞ and if

then (1) the existence of a set of FNE strategies is particular within the rank of linear, symmetric FNE strategy spaces.

(2) the steady linear FNE strategies are in the following.

∞

–

(133)

∞ (134) where and are value function parameters; furthermore, both and are

positive. Because of ∞, is the manipulation of mathematical technique.

(3) the steady linear FNE strategies produces R&D experience of Fu-kang bus transport service accumulation and price trajectories that converge conditionally locally asymptotically to the following stable state FNE R&D experience accumulation and price level for both firms.

∞

–

(135) Furthermore, the stable state market price is

∞

(136) I can proof the theorem 3 from Appendix E and use these results for attempting to research relationship between the ratios of capital investment of R&D experience accumulations with Fu-kang bus transportation service and the differentiate rate of configuration service of Fu-kang bus routing under ∞. For simplicity, I intend to use the method of numerical analysis for discussions on the numerical experiments ∞ in next subsection.

Finally, a plausible conjecture is that the Fu-kang bus routing price should be lower when the speed of price adjustment is sufficiency. I discuss that the difference between the investment of Fu-kang bus transport service and configuration service are based on sticky price and the costs of firm’s exert. I give the numerical experiments in order to verify our model.

4.4.2.1 Numerical Experiments: .

In this subsection, I use the numerical experiments to simulate our model as Appendix F in the following; furthermore, I plot illustrations of Equation (131) and (132) under and .

Figure 32. Setting ranges are between and , the

values of and are obtainment.

I illustrate the trend chart of and under sticky price. As revealed by figure 32, the ratio of and are increased, the trend of the Fu-kang bus quantity is a tardy fall; furthermore, the trend of the firm’s R&D service experience accumulation is a tardy rise. When I more close to sticky price, the Fu-kang bus quality is declined obviously.

In addition, the value of the firm’s R&D service experience accumulation is tardy rise.

4.4.2.2 Numerical Experiments: ∞

In this subsection, I use the numerical experiments to simulate our model as Appendix G in the following; furthermore, I plot 3D illustration of Equation (135) and (136) under and

Figure 33. Setting ranges are between and

the values of _∞ and _∞ are obtainment.

0

Z

s

0

P

s





P

s





Z

s

I illustrate the trend chart of _∞ and _∞ under the quick adjustment price, i.e.

∞. As revealed by figure 33, the ratio of and are increased, the trend of the Fu-kang bus price is a little tardy fall in the initial value between _∞ and _∞ ; however, the increased price of Fu-kang bus followed the increase of investment ratio . When the value of investment ratio is , the peak of price is reached.

On the other hand, the firm’s R&D experience accumulation is remained steady ( _∞ ), although the values of and are increased.

Figure 34 . Setting ranges are between and

the values of and _∞ are obtainment.

I attempt to compare the difference between and _∞ through illustrated the trend chart. When the range of the investment ratio is , I obtain the higher value with the quick adjustment price. When the range of the investment ratio is , I obtain the higher value with sticky price. After that, as revealed by figure 34, the ratio of and are increased, the trend of the firm’s R&D service experience accumulation is a steady rise with sticky price; however, R&D service experience accumulation is a tardy fall with the quick adjustment price.





Z

s

0

Z

s

Figure 35. Setting ranges are between and , the

values of and _∞ are obtainment.

I attempt to compare the difference between and _∞ through illustrated the trend chart. When the range of the investment ratio is , I obtain the higher value with sticky price. When the range of the investment ratio is , I obtain the higher value with the quick adjustment price. After that, as revealed by figure 34, the ratio of and are increased, the trend of price is a tardy rise with the quick adjustment price and the peak of price _∞ is reached; however, the trend of price is a tardy fall with sticky price.

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