  Despite the success of demand response programs in retail electricity markets
in reducing average consumption, the random responsiveness of consumers to
price event makes their efficiency questionable to achieve the flexibility
needed for electric systems with a large share of renewable energy. The
variance of consumers' responses depreciates the value of these mechanisms and
makes them weakly reliable. This paper aims at designing demand response
contracts which allow to act on both the average consumption and its variance.
The interaction between a risk--averse producer and a risk--averse consumer is
modelled through a Principal--Agent problem, thus accounting for the moral
hazard underlying demand response contracts. We provide closed--form solution
for the optimal contract in the case of constant marginal costs of energy and
volatility for the producer and constant marginal value of energy for the
consumer. We show that the optimal contract has a rebate form where the initial
condition of the consumption serves as a baseline. Further, the consumer cannot
manipulate the baseline at his own advantage. The second--best price for energy
and volatility are non--constant and non--increasing in time. The price for
energy is lower (resp. higher) than the marginal cost of energy during
peak--load (resp. off--peak) periods. We illustrate the potential benefit
issued from the implementation of an incentive mechanism on the responsiveness
of the consumer by calibrating our model with publicly available data. We
predict a significant increase of responsiveness under our optimal contract and
a significant increase of the producer satisfaction.
