Capital Accumulation Game Under Spatial Competition

The Optimal Competitive Policy

Abstract

The purpose of this paper is two fold. One is to investigate the dynamic behaviors of the duopolistic firms. The other is to derive the government optimal policy to maximize the discounted sum of the future economic surplus given the behavior of the firms. The control variable of the government is the public transportation fare, which can be considered as the proxy of the degree of the competitiveness in such a spatial duopolistic economy. When the fare is high, the profit of the firms would be low and the speed of the capital accumulation is slow since the investment is financed by the current profit. In this way, the government can control the speed of the capital accumulation of the economy.
Two firms act as Hotelling-type competitors in the short run. In the long run, they aim to maximize the discounted sum of the future profits. The control variable of each firm is the level of the investment to improve the efficiency of the production. We assume that the dynamic competition is also a Nash type as well as in the short run, where each firm considers the time path of the opponent's control variable as given.
The equilibrium path for this dynamic game is characterized as follows: The solution path is the bang-bang control, since we assume that the profit function is linear in investment, which is bounded. The investment should be nonnegative and should not exceed the current profit. In ordinary optimal growth theory, the firm accumulates the capital until the marginal profit equals the disconut rate plus the depreciation rate. But in our dynamic game, there exists the case where the firm with the larger discount rate ceases to accumulate its capital before it reaches this point and decreases capital by means of depreciation. If not, the firm can not reach the optimal stationary state because of the behavior of the opponent, who has the smaller disconunt rate and is still accumulating the capital. At the stationary state, the firm with the smaller discount rate has the greater capital.
Given the optimal behaviors of the firms, the government plays the role of the Stackelberg's leader in this dynamic game. We assume that the public transportation fare has the upper and the lower limits. The optimal path of the government control variable is characterized as follows: In the early stage, the transportation fare is high and lies at the upper limit. This implies that the degree of competition is sufficiently low, and the firms earn the monopoly profits and the speed of the capital accumulation is high at the sacrifice of the consumer's surplus. But in the later stage, where the capital is sufficiently accumulated, the fare drops to the lower limit. That is, the optimal solution for the government is bang-bang control.
The larger the government's discount rate of the future surplus, the earlier emerges the switching of the transportation fare from the upper limit to the lower one to enjoy the greater economic surplus in the early stage at the cost of the future surplus.
We show that the imperfect competititon is desirable for some periods if not all.