Dynamically Stable Cost-Saving Joint Venture

Abstract

In this chapter, we consider a common economic activity involving cooperative optimization—joint venture. However, it is often observed that after a certain time of cooperation some firms in a joint venture may gain sufficient skills and technology that they would do better by breaking away from the joint operation. Analysis on time (optimal-trajectory subgame) consistent joint ventures are presented in the following sections.

Appendix: Proof of Proposition 5.2

To prove Proposition 5.2 we first use $\hat{x}^{j(L)}$, for j∈L, to denote the optimal trajectory of the optimal control problem $\varpi [L;\tau,x_{\tau} ^{L}]$, which maximizes

subject to

$$\dot{x}^{j}(s) = f^{j}\bigl[s,x^{j}(s),u_{j}(s)\bigr],\quad x^{j}(\tau) = x_{\tau} ^{j}, \mbox{ for }j \in L.$$

Note that

(5.98)

Similarly, for the optimal control problem $\varpi [K\backslash L;\tau,x_{\tau} ^{K\backslash L}]$, we have

(5.99)

Now consider the optimal control problem $\varpi [K;\tau,x_{\tau} ^{K}]$ that maximizes

subject to

$$\dot{x}^{j}(s) = f^{j}\bigl[s,x^{j}(s),u_{j}(s)\bigr],\quad x^{j}(\tau) = x_{\tau} ^{j}, \mbox{ for }j \in K.$$

Since $\psi_{j}^{(\tau )K*}(s,\hat{x}^{K(K)}(s))$ and $\hat{x}^{K(K)}(s)$ are, respectively, the optimal control and optimal state trajectory of the control problem $\varpi [K;\tau,x_{\tau} ^{K}]$,

(5.100)

Invoking (5.98), (5.99), and (5.100), we have

$$W^{(\tau )K}\bigl(\tau,x_{\tau} ^{K}\bigr)\geq W^{(\tau )L}\bigl(\tau,x_{\tau} ^{L}\bigr)+W^{(\tau )K\backslash L}\bigl(\tau,x_{\tau} ^{K\backslash L}\bigr).$$

Hence Proposition 5.2 follows.