3. intertemporal utility maximization - consumption and saving

3. intertemporal utility maximization - consumption and saving

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Intertemporal utility maximization — consumption and saving 10 September 2009 1 Reading Appendix 4A of Abel/Bernanke/Croushore 2 The Intertemporal Budget Constraint The household’s initial holding of real asset (nominal asset divided by the price level) is a 0 . Then in period 1, it faces a budget constraint, c 1 + a 1 = y 1 +(1+ r 0 ) a 0 . (1) Similarly for period 2, c 2 + a 2 = y 2 +(1+ r 1 ) a 1 . (2) Solve (1) for a 1 and substitute the result into (2) , c 2 + a 2 = y 2 +(1+ r 1 )( y 1 +(1+ r 0 ) a 0 c 1 ) c 1 + c 2 1 1+ r 1 + 1 1+ r 1 a 2 = y 1 + 1 1+ r 1 y 2 +(1+ r 0 ) a 0 . (3) The sum c 1 + c 2 / (1 + r 1 ) denotes the PV of consumption, while the sum y 1 + y 2 / (1 + r 1 ) represents the PV of labor income. There is also the principal plus interest (1 + r 0 ) a 0 from the initial asset holding. Finally the amount a 2 / (1+ r 1 ) is the PV of asset to carry over to period 3. In all, the PV of labor and investment income is to be divided among consumption over the two periods and the asset to leave aside at the end of the planning period. We call this constraint the Intertemporal Budget Constraint. 1
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Figure 1: Diminishing marginal utility of consumption 3 The Intertemporal Utility Maximization If we f x the amount on the RHS of (3) and the end of planning period asset, the amount a 2 / (1 + r 1 ) , the household’s decision is over how best to allocate a given amount of resource between periods 1 and 2 consumption. Assume that the decision is made to maximize the sum of utility over the two periods, U = u ( c 1 )+ β u ( c 2 ) , where u is a strictly increasing and concave function that exhibits diminishing marginal utility, and that β 1 is known as the subjective discount factor that measures the degree of impatience–the extent to which future utility is discounted relative to present utility. The utility maximization problem is max c 1 ,c 2 u ( c 1 )+ β u ( c 2 ) , subject to c 1 + c 2 1 1+ r 1 = y 1 + 1 1+ r 1 y 2 +(1+ r 0 ) a 0 1 1+ r 1 a 2 . Consumption smoothing —Suppose β =1 ; i.e., the utility of future consump- tion is not discounted at all, and that r 1 =0 ; i.e., delaying consumption and accumulating asset earns no net return. In this case, the utility—maximizing 2
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Figure 2: The Marginal Utility of Consumption { c 1 ,c 2 } pair must be such that c 1 = c 2 . Suppose we have c 1 >c 2 .T h e n MU 1 = u 0 ( c 1 ) <u 0 ( c 2 )= MU 2 ,g iventha t u is strictly concave. But then a small decrease in c 1 for an amount say dc , which would help allow for the same increase in c 2 ,g iventheIBC : c 1 + c 2 = y 1 + y 2 +(1+ r 0 ) a 0 a 2 , must result in a higher U = u ( c 1 )+ u ( c 2 ) since MU 1 dc
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3. intertemporal utility maximization - consumption and saving

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