325_Lecture7_Feb14

# 325_Lecture7_Feb14 - CONSUMPTION-SAVINGS...

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1 C ONSUMPTION- S AVINGS F RAMEWORK (C ONTINUED ) F EBRUARY 14, 2011 February 14, 2011 2 C ONSUMER O PTIMIZATION The Graphics of the Consumption-Savings Model ± Consumer’s decision problem: maximize lifetime utility subject to lifetime budget constraint – bring together both cost side and benefit side ± Choose c 1 and c 2 subject to ± Plot budget line ± Superimpose indifference map ± At the optimal choice 22 2 11 1 P cY Pc Y ii ± ± ± ± c 1 c 2 slope = -(1+ i )/(1+ Ⱥ 2 ) optimal choice ** 11 2 21 2 2 (,) 1 ucc i S ± ± CONSUMPTION-SAVINGS OPTIMALITY CONDITION -A key building block of modern macro models MRS (between consumption in consecutive time periods) price ratio (across consecutive time periods)

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2 February 14, 2011 3 L AGRANGE A NALYSIS: L IFETIME A PPROACH Consumption-Savings Model: Lifetime Formulation ± Apply Lagrange tools to consumption-savings optimization ± Objective function: u ( c 1 , c 2 ) ± Constraint (assuming A 0 = 0): ± Step 1: Construct Lagrange function ± Step 2: Compute first-order conditions with respect to c 1 , c 2 , Ǌ ± Step 3: Solve (with focus on eliminating multiplier) 22 2 12 1 1 1 (, ) 0 11 YP c gc c Y Pc ii ±² ² ± ± 2 1 1 1 (, ,) (, ) c Lc c uc c Y OO ª º ± ± ² ² « » ± ± ¬ ¼ ** 11 2 21 2 2 (,) 1 1 ucc i r S ± ± ± CONSUMPTION-SAVINGS OPTIMALITY CONDITION MRS (between consumption in consecutive time periods) price ratio (across consecutive time periods) using Fisher equation February 14, 2011 4 T WO- P ERIOD F RAMEWORK IN R EAL T ERMS From Nominal to Real ± Depending on application, may be useful to work with framework (independent of lifetime vs. sequential approach) in nominal terms or in real terms 2 1 P cY Y ± ± ± ± LBC in nominal terms (assuming A 0 = 0) 21 2 1 (1 ) ) PY Y cc P iP P i §· ± ¨¸ ± ± ©¹ Divide by P 1 Multiply and divide last term on right-hand-side by P 2 2 2 1 2 ) ) P Y P P i P ± ±± Use definitions: y 1 = Y 1 / P 1 , y 2 = Y 2 /P 2 , and 1+ Ⱥ 2 = P 2 / P 1 1 2 yy SS ± Use Fisher equation: (1+ Ⱥ 2 )/(1 +i ) = 1/(1+ r ) cy rr ± ± ± LBC in real terms (assuming A 0 = 0) Maximize u ( c 1 , c 2 ) subject to the real LBC Æ identical consumption-savings optimality condition (details in recitations)
3 February 14, 2011 5 L AGRANGE A NALYSIS: S EQUENTIAL A PPROACH Consumption-Savings Model: Sequential Formulation ± Sequential formulation highlights the role of net wealth ( A 1 ) between period 1 and period 2 ± Accords better with the explicit timing of economic events than the lifetime approach… ± …but yields the same result ± Advantage: allows us to think about interaction between asset prices and macroeconomic events (intersection of finance theory and macro theory in Chapter 8) ± Apply Lagrange tools to consumption savings optimization ± Objective function: u ( c 1 , c 2 ) ± Constraint s : ± Period 1 budget constraint: ± Period 2 budget constraint: ± Sequential Lagrange formulation requires two multipliers ± See Math Refresher, Chapter -1 ± Could pursue sequential approach in either nominal or real terms 10 1 1 1 (1 ) 0 Yi A P c A ± ±²² 21 2 2 2 ) 0 A P c A ±

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## This document was uploaded on 11/01/2011 for the course ECON 325 at Maryland.

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325_Lecture7_Feb14 - CONSUMPTION-SAVINGS...

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