Managing peak energy demand is a critical problem for energy utilities. The energy costs for the peak periods form a major component of their overall costs. Real-time pricing mechanisms have been explored as a means of flattening the demand curve and reducing the energy costs. In this paper, we examine a model of ex-post real-time pricing mechanism that can be used by the utilities for this purpose. In particular, we study a convex piece-wise linear cost function that modulates the price of energy based on the aggregate demand of the utility. We provide a game-theoretic analysis of the mechanism by constructing a non-cooperative game among the consumers of a utility wherein the cost to each consumer is decided by the pricing mechanism. We formally characterize the Nash equilibrium and other properties for two settings: (i) consumers have full flexibility in shifting their demand, and (ii) consumers can shift only a fraction of their demand at any time to another time.