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Lagrange Method

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Why do we want the axioms 1 – 7 of consumer theory? Answer: We like an easy life! By that we mean that we want well behaved demand curves.

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Lets look at a Utility Function: U = U(,y) Take the total derivative: For example if MU x = 2 MU y = 3

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Look at the special case of the total derivative along a given indifference curve: dy dx

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Taking the total derivative of a B.C. yields P x dx + P y dy = dM Along a given B.C. dM = 0 P x dx + P y dy = 0 y x

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=> Slope of the Indifference Curve = Slope of the Budget Constraint Equilibrium x y

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We have a general method for finding a point of tangency between an Indifference Curve and the Budget Constraint: The Lagrange Method Widely used in Commerce, MBAs and Economics.

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y x y x u0u0 u1u1 u2u2 Idea: Maximising U(x,y) is like climbing happiness mountain. But we are restricted by how high we can go since must stay on BC - (path on mountain).

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y x u0u0 u1u1 u2u2 So to move up happiness Mountain is subject to being on a budget constraint path. Maximize U (x,y) subject to P x x+ P y y=M

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Known: P x, P y & MUnknowns: x,y,l 3 Equations: 3 Unknowns: Solve = 0

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Trick: But: Note: U

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Known: P x, P y & MUnknowns: x,y, 3 Equations: 3 Unknowns: Solve = 0

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Notice: U = x 2 y 3 Recall Slope of Budget Constraint = Slope of the Indifference Curve Slope of IC = slope of BC

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Back to the Problem: + But But +

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Back to the Problem: + But But +

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So the Demand Curve for x when U=x 2 y 3 If M=100: PxPx xDxD 104 85 58 220

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Recallthat: U = x 2 y 3 Let: U = x a y b For Cobb - Douglas Utility Function

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Note that: Cobb-Douglas is a special result In general: For Cobb - Douglas:

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Why does the demand for x not depend on p y ? Share of x in income = In this example: Constant Similarly share of y in income is constant: So if the share of x and y in income is constant => change in P x only effects demand for x in C.D.

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So l tells us the change in U as M rises Increase M Increase from U 1 to U 2 in constraint Constraint Objective fn in objective fn

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