Electric vehicles (EVs) will play an important role in the future smart grid because of their capabilities of storing electrical energy in their batteries during off-peak hours and supplying the stored energy to the power grid during peak hours. In this paper, we consider a power system with an aggregator and multiple customers with EVs and propose a novel electricity load scheduling which, unlike previous works, jointly considers the load scheduling for appliances and the energy trading using EVs. Specifically, we allow customers to determine how much energy to purchase from or to sell to the aggregator while taking into consideration the load demands of their residential appliances and the associated electricity bill. Under the assumption of the collaborative system where the customers agree to maximize the social welfare of the power system, we develop an optimal distributed load scheduling algorithm that maximizes the social welfare. Through numerical results, we show when the energy trading leads to an increase in the social welfare in various usage scenarios.