We study the effect of Demand-Response (DR) in dynamic real-time electricity markets. We use a two-stage market model that takes into account the dynamical aspects of generation, demand, and DR. We study the real-time market prices in two scenarios: in the former, consumers anticipate or delay their flexible loads in reaction to market prices; in the latter, the flexible loads are controlled by an independent aggregator. For both scenarios, we show that, when users are price-takers, any competitive equilibrium is efficient: the players' selfish responses to prices coincide with a socially optimal policy. Moreover, the price process is the same in all scenarios. For the numerical evaluation of the properties of the equilibrium, we develop a solution technique based on the Alternating Direction Method of Multipliers (ADMM) and trajectorial forecasts. The forecasts are computed using wind generation data from the UK. We challenge the assumption that all players have full information. If the assumption is verified, then, as expected, the social welfare increases with the amount of DR available, since DR relaxes the ramping constraints of generation. However, DR actions alter the internal states of elastic loads; if the day-ahead market cannot observe how elastic loads are affected by DR, a large quantity of DR can be detrimental and leads to a decrease in the welfare. Furthermore, the DR operator has an incentive to under-dimension the quantity of available DR. Finally, we compare DR with an actual energy storage system. We find that storage has a faster response-time and thus performs better when only a limited amount is installed. However, storage suffers from charge-discharge inefficiency: with DR, prices do concentrate on marginal cost (for storage, they do not) and provide a better welfare.