Effects of background risks on cautiousness with an application to a portfolio choice problem

In an exchange economy in which there is a complete set of markets for macroeconomic risks but no market for idiosyncratic risks, we consider how the efficient risk-sharing rules for the macroeconomic risk are affected by the heterogeneity in the consumers' risk attitudes and idiosyncratic risks. We provide sufficient conditions under which an idiosyncratic risk increases cautiousness (the derivative of the reciprocal of the absolute risk aversion), the determinant of the curvatures of the efficient risk-sharing rules. While the curvature of the risk-sharing rules at high consumption levels are governed by the consumers' risk attitudes, the curvature at low consumption levels depend not only on the risk attitudes but also on the lower tail distributions of the idiosyncratic risks.