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C_DynamicLS

# C_DynamicLS - Dynamic or Life-Cycle Labor Supply 1 The...

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Dynamic, or Life-Cycle Labor Supply 1. The setup Dymo lives for T periods. Let his utility in period t be given by the function: ) , ( t t L C U (1) where C is consumption and L is leisure. We’ll assume U has the usual properties: it is increasing in both C and L , and strictly concave. Strict concavity implies negative (own) second derivatives, but (as noted in the static case) does not restrict the sign of the cross- partial derivative, U CL . Plausible stories can be told for both positive and negative U CL ’s and both are consistent with well-behaved solutions to the problem. 1 If utility is intertemporally separable and ρ is the intertemporal (subjective) discount factor, total lifetime utility is: = + = T t t t t L C U W 1 ) 1 ( ) , ( ρ (2) Note that the function U is not indexed by t ; in this baseline case we are therefore not allowing tastes for consumption or leisure to vary systematically with age (this is easy to modify). Dymo can work as many hours as he wants in each period of his life at fixed rate of pay per hour. But the wage, w , Dymo can get isn’t the same in each period: instead it is indexed by t . Suppose that Dymo knows, even at the start of his working life, what w will be in each period (this assumption is also fairly easily relaxed). If Dymo also faces a known stream of nonlabor income, G t , t = 1, … T , we can write his income in period t as: period t income = ) 1 ( t t L w - + G t (3) (Note that we are normalizing the total amount of time available in each period to equal one.) In the above definition of income, G t does not include interest income on money that was saved earlier in life (the amounts of interest income Dymo actually ends up earning in each period of his life will be endogenous outcomes of Dymo’s utility- maximizing choice of a lifetime income and consumption plan). G t as defined in (3) includes only “exogenous” items of income such as inheritances, lottery winnings, demogrants, etc. (It is quite OK to think of all or most of the G ’s as zero, but keeping track of them helps in the interpretation of some results below). 1 If you can enjoy your spending more effectively when you have more leisure (e.g. a dollar yields more marginal utils when spent on vacation in a beautiful place) then U CL >0. If your need to spend at the margin is greater when you are working (e.g. you need to have better clothes, pay for commuting, eat out more, and don’t have time to do your own home repairs), then U CL <0.

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2. Static Labor Supply in this context. Suppose there were no capital markets, i.e. no way for Dymo to use income earned in one period of life for consumption in another. Then Dymo’s lifetime consumption and work plan would have to satisfy T separate constraints, given by: T t G L w C t t t t ,... 1 , ) 1 ( = + - (4) Now, the consumption and labor supply plan, C 1 , …. C T ; (1- L 1 ), …. (1- L T ), that maximizes lifetime utility (2), consists of the C t , and L t that maximize each period’s utility (1) separately given each period’s own budget constraint (4). This just means that

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