On Aggregate Precautionary Saving: When is the Third Derivative Irrelevant?

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Abstract
When does idiosyncratic earnings uncertainty increase aggregate saving? We address this question in the context of a general equilibrium model where infinitely-lived agents receive idiosyncratic labor endowment shocks, hold a risk-free asset to smooth consumption and face a liquidity constraint. We prove that the steady-state capital stock is always larger in any equilibrium with idiosyncratic shocks and a liquidity constraint than without idiosyncratic shocks (i.e. there is aggregate precautionary saving) as long as utility functions are strictly concave. We also prove that aggregate precautionary saving occurs if and only if the liquidity constraint binds for some agents.
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