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325_Lecture4_Feb2

325_Lecture4_Feb2 - CONSUMPTION-LEISURE FRAMEWORK(CONTINUED...

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1 C ONSUMPTION- L EISURE F RAMEWORK (C ONTINUED ) F EBRUARY 2, 2011 February 2, 2011 2 C ONSUMER O PTIMIZATION The Graphics of the Consumption-Leisure Model Consumer’s decision problem: maximize utility subject to budget constraint – bring together both cost side and benefit side Choose c and l subject to Plot budget line Superimpose indifference map At the optimal choice (1 ) 168(1 ) Pc t Wl t W ± ² ² leisure c slope = -(1- t ) W / P 168 optimal choice ( c*,l* ) ( *, *) (1 ) ( *, *) l c u c l t W u c l P ² CONSUMPTION-LEISURE OPTIMALITY CONDITION - key building block of modern macro models MRS (between consumption and leisure) After-tax real wage Derive consumption-leisure optimality condition using Lagrange analysis IMPORTANT: the larger is (1- t ) W / P , the steeper is the budget line
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2 February 2, 2011 3 L AGRANGE A NALYSIS The Mathematics of the Consumption-Leisure Model Apply Lagrange tools to consumption-leisure optimization Objective function: u ( c , l ) Constraint: g(c,l ) = 168(1- t ) W – Pc – (1- t ) Wl = 0 Step 1: Construct Lagrange function Step 2: Compute first-order conditions with respect to c , l , NJ > @ ( , , ) ( , ) 168(1 ) (1 ) L c l u c l t W Pc t Wl O O ± ² ² ² ² February 2, 2011 4 L AGRANGE A NALYSIS The Mathematics of the Consumption-Leisure Model
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