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asset assume assumption capital choice commodities competitive equilibrium concept consequences consider constraint consumer consumption plan consumption set continuous converges convex Debreu decision problems decision-maker decisions defined definition denoted Dreze elementary events exists expected utility hypothesis expected utility index expected value feasible firm follows future given implies individually rational initial endowment input K. J. Arrow L. J. Savage lemma market value markets for contingent maximisation metric space nature non-negative obtain optimal allocation optimum output paper Pareto efficiency Pareto optimal period portfolio preferences preordering present price expectations price system private ownership economy probability measure problem production plans production set Proof properties proposition pt+1 public investment random result risk aversion risk margins Sandmo satisfies sequence shares stochastic stock exchange stock market stock ownership program stockholders equilibrium subset sumer tatonnement theorem theory tion trader uncertainty utility function vector zero