Mean–field moral hazard for optimal energy demand response management

Romuald Élie, Emma Hubert, Thibaut Mastrolia, Dylan Possamaï

Research output: Contribution to journalArticlepeer-review

16 Scopus citations


We study the problem of demand response contracts in electricity markets by quantifying the impact of considering a continuum of consumers with mean–field interaction, 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 incentivizes each consumer to reduce the average and the volatility of his consumption in different usages, without observing the efforts he makes. We prove that the producer can benefit from considering the continuum 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 maximizes 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.

Original languageEnglish (US)
Pages (from-to)399-473
Number of pages75
JournalMathematical Finance
Issue number1
StatePublished - Jan 2021
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Accounting
  • Social Sciences (miscellaneous)
  • Finance
  • Economics and Econometrics
  • Applied Mathematics


  • McKean–Vlasov controlled SDEs
  • demand response models
  • electricity markets
  • mean–field games with common noise
  • moral hazard


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