To address the "chicken-and-egg" dilemma in the electric vehicle (EV) market, the government intervenes by offering EV purchase subsidy to consumers and charging station construction subsidy to installers. Different from previous studies, this study investigates the effects of different types of government subsidies, including only consumer subsidies, only station subsidies, and a combination of above both subsidies, on EV demand and the charging station construction scale. In addition, we model a Stackelberg game between the government and two installers (an EV manufacturer and a charging platform). This study also differs from previous works by considering two installers instead of only one. Results indicate that the station subsidy is more conducive to EV demand than the consumer subsidy in the case of insufficient charging stations. Although the combination of both subsidies demonstrates the most significant effect on the charging station construction scale among all subsidy structures, it does not lead to an optimal ratio of EVs to charging stations (E2C ratio). Whereas only the station subsidy makes E2C ratio reach the lowest. Notably, both installers benefit more from the consumer subsidy when charging stations achieve the scale effect (beta >= 0.6), while they gain more profit from the station subsidy when charging stations are insufficient (0 < beta < 0.6). Moreover, we examine the effects of equal and different amount of station subsidies given to both installers. Interestingly, providing different amounts of station subsidies to two installers results in higher EV demand and a larger scale of charging station construction than those offering equal subsidies to both. Our outcomes also show that, compared with subsidizing the manufacturer to install charging stations, more station subsidy given to the platform can arouse its greater social responsibility and increase EV demand and the charging station scale. This study not only guides enterprise managers in decisions related to charging station installation but also offers nuanced policy recommendations for the government regarding subsidy structures.