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Mean-field moral hazard for optimal energy demand response management

We study the problem of demand response contracts in electricity markets by quantifying the impact of considering a mean-field of consumers, whose consumption is impacted by a common noise. We formulate the problem as a Principal-Agent problem with moral hazard in which the Principal - she - is an electricity producer who observes continuously the consumption of a continuum of risk-averse consumers, and designs contracts in order to reduce her production costs. More precisely, the producer incentivises the consumers to reduce the average and the volatility of their consumption in different usages, without observing the efforts they make. We prove that the producer can benefit from considering the mean-field of consumers by indexing contracts on the consumption of one Agent and aggregate consumption statistics from the distribution of the entire population of consumers. In the case of linear energy valuation, we provide closed-form expression for this new type of optimal contracts that maximises the utility of the producer. In most cases, we show that this new type of contracts allows the Principal to choose the risks she wants to bear, and to reduce the problem at hand to an uncorrelated one.