Separability and Memory: Micro Causes, Macro Consequences

In a multi-period Mirrlees (1971) setting individuals face privately observed shocks to their ability. Each individual's productivity is the product of this idiosyncratic ability and an aggregate component which is also stochastic but publicly observed. Using a separable iso-elastic specification for preferences, we investigate optimal risk sharing and its macro consequences in this environment. We show that efficient allocations display memory with respect to past aggregate shocks for all levels of risk aversion different from one. This dependence on past aggregate shocks decreases with the persistence of idiosyncratic shocks. At the limit, when the economy becomes a pure heterogeneity one, the constrained efficient allocation is independent of past aggregate shocks and displays a strong form of separability with respect to aggregate shocks. Indeed, these two properties, separability and lack of memory, are shown to be connected: an allocation is memoryless only if it separable. Non-separability is interesting in itself for it is associated with state-varying labor wedges at the micro level. Using numerical examples we show that this non-separability induces efficient counter-cyclical labor wedges at the macro level as well, when risk aversion is greater than one.