  We revisit the classical topic of quadratic and linear mean-variance
equilibria with both financial and real assets. The novelty of our results is
that they are the first allowing for equilibrium prices driven by general
semimartingales and hold in discrete as well as continuous time. For agents
with quadratic utility functions, we provide necessary and sufficient
conditions for the existence and uniqueness of equilibria. We complement our
analysis by providing explicit examples showing non-uniqueness or non-existence
of equilibria. We then study the more difficult case of linear mean-variance
preferences. We first show that under mild assumptions, a linear mean-variance
equilibrium corresponds to a quadratic equilibrium (for different preference
parameters). We then use this link to study a fixed-point problem that
establishes existence (and uniqueness in a suitable class) of linear
mean-variance equilibria. Our results rely on fine properties of dynamic
mean-variance hedging in general semimartingale markets.
