Aggregation with a non-convex labor supply decision, unobservable effort, and incentive ("fair") wages

The purpose of this note is to explore the problem of a non-convex labor supply decision in an economy with unobservable effort and incentive ("fair") wages a la Danthine and Kurmann (2004), and explicitly perform the aggregation presented there without a formal proof, and thus provide - starting from micro-foundations - the derivation of the expected utility function used for the aggregate household. We show how lotteries as in Rogerson (1988) can be used to convexify consumption sets, and aggregate over individual preferences. With a discrete labor supply decisions, the elasticity of aggregate labor supply becomes a function of effort.