Existence, uniqueness and determinacy of equilibrium in C.A.P.M. with a riskless asset

In this note, we study the problem of existence, uniqueness and determinacy of equilibrium in the two period mean-variance C.A.P.M. with a riskless asset and possibly an infinite number of assets. The existence, uniqueness and determinacy problem is brought down to a two-dimensional problem. We construct a reduced two-dimensional economy which has the same equilibria as the original economy. In particular, we provide a very elementary proof of existence of equilibrium. We then show that when utilities are additively separable in mean and variance, sufficient conditions for uniqueness of equilibrium may be given in terms of ‘risk aversion'. Lastly, we show that generically equilibria are determinate.