In this study, we use a game-theory-based framework to model power in a supply chain with random and price-dependent demand and examine how power structure and demand models (expected demand and demand shock) affect supply chain members' performance. We demonstrate that whether a firm benefits from its power depends on the expected demand model but not on demand shock model. A firm benefits from its power only for linear but not for constant elasticity expected demand. The impact of power structure on supply chain efficiency depends on the models of both expected demand and demand shock. With additive shock, supply chain efficiency is highest (lowest) when neither firm dominates for linear (constant elasticity) expected demand. With multiplicative shock, the supply chain efficiency is highest with a power retailer (manufacturer) for linear (constant elasticity) expected demand. The manufacturer always benefits from a reduction in demand uncertainty. However, the retailer loses (benefits) from demand uncertainty reduction for linear (constant elasticity) expected demand. With a power retailer, the retail price is always on the higher end for linear expected demand, and the customer service level is the lowest for constant elasticity expected demand. Consequently, consumers do not necessarily benefit from a power retailer.