Ambiguity in Fama’s market equilibrium?

This report shows how to determine in analytic form the security prices implied by the market equilibrium model described by Fama in his book "Foundations of Finance", Chapter 8, Section III. The model assumes that all investors agree on the expected values and covariances of the random final prices at the end of an investment period. The investors interact so that the initial prices lead to an efficient portfolio with security weights proportional to the total initial value of the corresponding firm. If we assume that the expected portfolio return or the risk-free rate is specified, we find that there is a one-dimensional continuum of sets of initial prices satisfying the conditions of the model. It is unclear how to resolve this ambiguity about which model is correct.