Abstract: With the increasing popularity of cross-shareholding among firms, the problems of profit distribution after cross-shareholding, how cross-shareholding affects the economic behavior and market performance of firms, and what the optimal shareholding should be have always been hot issues that people pay more attention to. Some literatures have studied the problem of profit distribution of cross-shareholding among firms. However, the profit formulations in literature usually emphasize only one firm's returns acquired by holding shares of other firms, while neglecting the rival firms' returns demanded by holding the firm's stock, thus overstating the profit; or only consider the direct shareholding between firms, but ignore the indirect shareholding, resulting in profit omission. Clayton and Jorgensen (2005) analyzed the externality and strategic interactivity when Cournot duopoly market products were substitutes and complements respectively, and put forward the optimal cross-shareholding. They argued that, in an imperfect competitive market, the equilibrium equity of cross-shareholding is positive if the products are complements, the profits of the firms will increase, and the equilibrium equity is negative if the products are substitutes, the profits of the firms will fall. In both cases, consumer surplus will always increase. However, we think their approach and conclusion to be open to question for the substitutes market. First, in general, cross-shareholding equity should be non-negative, the purpose of cross-shareholding is to increase profitability, but their research on substitutes market had an opposite result. Second, their study claimed that cross-shareholding always increase consumer surplus in both substitutes and complements markets, while restrictions and regulations on cross-shareholding always reduce consumer surplus. This conflicts with general perception in substitutes market.
In this paper, the benefit distribution of cross-shareholding among n firms is studied, a profit formulation of cross-shareholding is presented which revises previous literatures. It takes into account not only the returns obtained by one firm holding shares in other firms, but also the returns claimed by the latter holding shares in the former, as well as the returns among firms through direct and indirect shareholdings, avoiding the exaggeration or omission of the profit or the red tape of the formula form. Secondly, on this basis, the general expressions of the price decision response function and equilibrium price of firms with cross-shareholding in Bertrand oligopoly market in the form of vector and matrix are derived, which avoids the complicated mathematical operation of finding partial derivative of profit formulations and solving n dimensional linear equations. Thirdly, by constructing a two-stage dynamic game, the equilibrium strategy for cross-shareholding of two firms in Bertrand duopoly market is focused on. The equilibrium pricing and optimal shareholding among firms are obtained by backward induction method. Finally, the effect of the equilibrium strategy on firm's economic behavior and market performance is analyzed.
Our results show that cross-shareholding is anti-competitive for substitutes market. The lower the degree of the product differentiation, the stronger the anti-competition; the more the percentage of the cross-shareholding, the more obvious the anti-competition. Each firm's profit is a monotone increasing function of its equity holding in the other firm, and a monotone decreasing function of the other firm's equity holding in the firm. Both firms have the incentive to increase their own profits by holding as much equity in the other as possible until they reach equilibrium. Compared with non-cross-shareholding, the equilibrium cross-shareholding increases the equilibrium price level, lowers the output level of each firm, and enables each of them to gain higher profit at lower production level, so as to realize a win-win situation between them. However, the product market equilibrium characterized by higher price and less output leads to decrease of the consumer surpluses and economic welfare. This result supports the common view that limiting cross-shareholding increases consumer surplus for substitutes market. That is in stark contrast to the conclusion of Clayton and Jorgensen (2005).
The research results and conclusions of this paper provide theoretical basis for the economic behavior of cross-shareholding among enterprises and the regulation of cross-shareholding by the government.

Key words: Bertrand duopoly market, cross-shareholding, profit formulation, pricing strategy, optimal shareholding