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lecture6 - Professor Jay Bhattacharya Spring 2001 Example...

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Professor Jay Bhattacharya Spring 2001 Econ 11--Lecture 6 1 Spring 2001 Econ 11-Lecture 6 1 Demand II • Recap: last lecture we covered: – Income Expansion Paths and Engel curves – Inferior and Normal Goods – Necessities and Luxuries – “Marshallian” Demand Curves Spring 2001 Econ 11-Lecture 6 2 Example: Calculating IEPs and Engel Curves Find the IEP and Engel Curve for a consumer with • To find the solution: – Solve for the Marshallian demand curves. This will automatically give you the Engel Curve – Solve each demand curve for income – Set these equations equal to each other to derive the IEP. () α = 1 , : Function Utility Douglass Cobb y x y x U I y p x p y = + x : Constraint Budget Spring 2001 Econ 11-Lecture 6 3 Solved Example • Set up the Lagrangian: • Calculate the first order conditions: I y p x p y x L y x + = λ 1 0 = ú û ù ê ë é = y p y x y L 0 1 = ú û ù ê ë é = x p x y x L 0 = + = I y p x p L y x Spring 2001 Econ 11-Lecture 6 4 Solved Example (II) • FindtheMarshalliandemandcurves: • These demand curves are the same as the Engel curves, since they show how the optimal levels of x and y change with income. • Note that for Cobb-Douglass utility, Engel curves are linear in income. x y x p I I p p x = , , y y x p I I p p y = 1 , , Spring 2001 Econ 11-Lecture 6 5 Solved Example (III) • Solve each demand curve for income: • Setting these equations equal to each other gives the income expansion path: • For Cobb-Douglass utility, the IEP is linear. * x p I x = = 1 * y p I y y x x y p x p y x p y p * * * * 1 1 = Þ = Spring 2001 Econ 11-Lecture 6 6 What happens to demand when price changes? x 2 x 1 2 * 1 p p slope = 2 0 1 p p slope =
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Professor Jay Bhattacharya Spring 2001 Econ 11--Lecture 6 2 Spring 2001 Econ 11-Lecture 6 7 x 1 0 1 p * 1 p Marshallian Demand Curve p 1 Spring 2001 Econ 11-Lecture 6 8 What Causes the Change in Demand?
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lecture6 - Professor Jay Bhattacharya Spring 2001 Example...

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