In this talk, we introduce two problems of contract theory, in continuous-time, with a multitude of agents. First, we will study a model of optimal contracting in a hierarchy, which generalises the one-period framework of Sung (2015). The hierarchy is modelled by a series of interlinked principal-agent problems, leading to a sequence of Stackelberg equilibria. More precisely, the principal (she) can contract with a manager (he), to incentivise him to act in her best interest, despite only observing the net benefits of the total hierarchy. The manager in turn subcontracts the agents below him. We will see through a simple example that, while the agents only control the drift of their outcome, the manager controls the volatility of the Agents’ continuation utility. Therefore, even this relatively simple introductory example justifies the use of recent results on optimal contracting for drift and volatility control, and therefore the theory on 2BSDEs.

This will lead us to introduce the second problem, namely optimal contracting for demand-response management, which consists in extending the model by Aid, Possamai, and Touzi (2022) to a mean-field of consumers. More precisely, the principal (an electricity producer, or provider) contracts with a continuum of agents (the consumers), to incentivise them to decrease the mean and the volatility of their energy consumption during high peak demand. In addition, we introduce a common noise, impacting all consumption processes, to take into account the impact of weather conditions on the agents’ electricity consumption. This mean-field framework with common noise leads us to consider a more extensive class of contracts. In particular, we prove that the results of [1] can be improved by indexing the contracts on the consumption of one agent and aggregate consumption statistics from the distribution of the entire population of consumers.

*Talk based on Hubert (2020) and Elie, Hubert, Mastrolia, and Possamai (2021).*