Consider two agents, \(\mathrm{A}\) and \(\mathrm{B}\), who live in a one-period economy defined by two points in time. They are given endowments at both points in time. Both agents have the possibility of making productive investments. However, there is uncertainty with respect to the return of this investment in the future. With probability \(p\) it is \(F_{1}(\epsilon)\) and with probability \(1-p\) it is \(F_{2}(\epsilon)\). As a consequence, an agent who makes a productive investment cannot be certain about the future consumption. With probability \(p\) agent \(i\) can consume \(y_{i}^{1}\) and with probability \(1-p\) can consume \(y_{i}^{2}\). Their preferences are described by a utility function \(V_{i}=E U_{i}(x, y)\) where \(U_{i}(x, y)\) has the properties described in section 1.1. Find sufficient conditions for the level of investment to be independent of the preferences and of the endowments of the agents.