lecture4 - Professor Jay Bhattacharya Spring 2001 Two...

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Professor Jay Bhattacharya Spring 2001 Econ 11--Lecture 4 1 Spring 2001 Econ 11--Lecture 4 1 Two Example Problems and One Policy Application Two solved example problems One policy applications using budget constraints and indifference curves It is only the 4th class, but we already know enough to tackle an important policy question. – Housing Subsides Spring 2001 Econ 11--Lecture 4 2 Problem #1 from Nicholson “One young connoisseur has $300 to spend to build a small wine cellar. She enjoys two vintages: an ‘expensive’ 1981 French Bordeaux ( W f ) at $20 per bottle and a less expensive 1983 California wine ( W c ) at $4 per bottle. How many bottles of each wine should she buy if her utility function is given by: ( ) 3 1 3 2 , c f c f W W W W U = Spring 2001 Econ 11--Lecture 4 3 Wine Connoisseur’s Budget Constraint where, I W P W P c c f f + , 20 $ = f P , 4 $ = c P 300 $ = I The connoisseur should spend her whole budget on wine! g, simplifyin after or 300 4 20 = + c f W W 0 75 5 = + c f W W Spring 2001 Econ 11--Lecture 4 4 Wine Connoisseur’s Problem 0 75 5 : subject to = + c f W W 3 1 3 2 , max c f c f W W W W ( ) 75 5 , , max : Lagrangian the up set 3 1 3 2 + = c f c f c f W W W W L W W Spring 2001 Econ 11--Lecture 4 5 First Order Conditions 0 5 3 2 3 1 = = λ f c f W W W L (1) 0 3 1 3 2 = = λ c f c W W W L (2) (3) 0 75 5 = + = c f W W L λ Spring 2001 Econ 11--Lecture 4 6 Solving the Problem 5 2 5 3 1 3 2 : (2) by (1) Divide 3 2 3 1 3 2 3 1 = = f c f c c f f c W W W W W W W W λ λ 5 2 f c W W =
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Professor Jay Bhattacharya Spring 2001 Econ 11--Lecture 4 2 Spring 2001 Econ 11--Lecture 4 7 Plug Into Budget Constraint 0 75 5 into 5 2 : Plug = + = c f f c W W W W 0 75 2 = + c c W W bottles 25 = c W bottles 10 = f W Spring 2001 Econ 11--Lecture 4 8 Check Second Order Conditions In a linearly constrained optimization problem with two variables, the second order conditions are given by: • Here, 0 2 2 1 22 2 1 12 2 2 11 < + f f f f f f f ( ) 3 1 3 2 , c f c f W W W W U f = = Spring 2001 Econ 11--Lecture 4 9 Second Order Conditions (II) 0 10 25 3 2 3 2 3 1 3 1 1 > = = = f c f
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