macro economics - Fishers Intertemporal Consumption Model...

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1 Fisher’s Intertemporal Consumption Model Formal Problem: 1 2 1 2 , 2 2 1 1 max ( , ) . . 1 1 C C U C C C Y s t C Y r r This is a constrained maximisation problem. That is, we wish to maximise utility which depends on consumption in periods 1 and 2 respectively, but we are constrained by our intertemporal budget constraint (IBC). Example: Suppose a consumer has the following utility function: 1/2 1/2 1 2 1 2 ( , ) = U C C C C . Further suppose that his income is $120 in period 1 and $100 in period 2 and the real interest rate is 25%. Calculate the consumer s optimal consumption in periods 1 and 2 and his saving in period 1. Using the method of Lagrangean Multipliers, we set up the Lagrangean as follows: 1/2 1/2 2 2 1 2 1 2 1 1 ( , , ) = 1 1 Y C L C C C C Y C r r First Order Conditions (FOC) 1/2 1/2 1 2 1 1 0 2 L C C C (1) 1/2 1/2 1 2 2 1 0 2 1 L C C C r (2) 2 2 1 1 0 1 1 Y C L Y C r r (3)
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2 Equations 1, 2 and 3 can be rearranged to show:
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