Universal bounds for asset prices in heterogeneous economies

We establish universal bounds for asset prices in heterogeneous complete market economies with scale invariant preferences. Namely, for each agent in the economy we consider an artificial homogeneous economy populated solely by this agent, and calculate the “homogeneous” price of an asset in each of these economies. Dumas (Rev. Financ. Stud. 2, 157–188, [1989]) conjectured that the risk free rate in a heterogeneous economy must lie in the interval determined by the minimal and maximal of the “homogeneous” risk free rates. We show that the answer depends on the risk aversions of the agents in the economy: the upper bound holds when all risk aversions are smaller than one, and the lower bound holds when all risk aversions are larger than one. The bounds almost never hold simultaneously. Furthermore, we prove these bounds for arbitrary assets.