Consumer_Problem

# Consumer_Problem - The Math Behind Section 16-2 The...

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The Math Behind Section 16-2 The consumer’s problem is to maximize utility over two periods where utility in both periods is “time separable:” 2 1 ( ) ( ) ; ' 0; " 0(positive marginal utility; diminishing MU) 1 "time separable" because the utilty in the second period does not depend on consumption in the first period. is the "pure rate of t U c V U c U U δ = + < + 2 1 2 1 ime preference" and measures impatience. s.t. (IBC) , ( )(1 ) 1 second period consumption is first period saving plus interest. c W c c W c r r = + = - + + Substituting the intertemporal budget constraint (IBC) into V and maximizing yields: 1 1 1 1 1 1 1 2 2 (( )(1 )) ( ) 1 1 '( * ) '(( * )(1 )) 0 1 '( * ) 1 1 '( * ) '( ); 1 '( * ) 1 U W c r V U c dV r U c U W c r dc U c r r U c U c U c - + = + + + = - - + = + + + - = + + The last line above is very important (and should make sense). The first version says that at the optimum, the marginal utility of consuming today must equal (1+r) times the marginal utility (from today’s perspective) of consuming tomorrow. This should make

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## This note was uploaded on 03/02/2010 for the course ECON 57 taught by Professor Woglom during the Spring '08 term at UMass (Amherst).

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Consumer_Problem - The Math Behind Section 16-2 The...

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