Resource Allocation in Electricity Markets with Budget Constrained Customers

In electricity markets, customers are increasingly constrained by their budgets. A budget constraint for a user is an upper bound on the price multiplied by the quantity. However, since prices are determined by the market equilibrium, the budget constrained welfare maximization problem is difficult to define rigorously and to work with. In this letter, we show that a natural dual-ascent algorithm converges to a unique competitive equilibrium under budget constraints. Further, this budget-constrained equilibrium is exactly the solution of a convex welfare maximization problem in which each user's utility is replaced by a modified utility that splices the original utility with a logarithmic function where the budget binds. We also provide an explicit piecewise construction of this modified utility and demonstrate the results on examples with quadratic and square root utility functions.