Handout on ‘Risky Habits’ and the Relationship Between Saving and Growth

Assume that each infinitely-lived consumer in the economy is endowed with one unit of labor which is supplied inelastically in an aggregate labor market, and that the individual consumer’s labor income is subject to multiplicative idiosyncratic transitory shocks i,t+1 such that Et[̃i,t+1] = 1. 1 Idiosyncratic ‘permanent labor income’ is defined as the level of labor income that would obtain for individual i if the transitory shock to labor income took on its mean value of one. Permanent labor income Pi,t is also subject to lognormally distributed shocks Ni,t+1 (again such that Et[Ñi,t+1] = 1) in each period; the marginal propensity to consume out of permanent income will be determined by examining the response of consumption to these shocks. Denoting by Hi,t the habit stock inherited from past consumption behavior, and designating total resources available for consumption (the sum of undepreciated past capital, current capital income, and current labor income) as Xi,t, the consumer’s optimization problem for a finite horizon beginning at the current period t and ending at period T (and dropping the i subscripts to reduce clutter) is given by