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Lagrange Method

Lagrange Method. Lagrange Method. 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. Let’s look at a Utility Function: U = U( ,y) Take the total derivative: .

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Lagrange Method

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  1. Lagrange Method

  2. Lagrange Method • 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.

  3. Let’s look at a Utility Function: U = U(,y) Take the total derivative: For example if MUx = 2 MUy = 3

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

  5. y x • Taking the total derivative of a B.C. yields • Px dx + Py dy = dM • Along a given B.C.dM = 0 • Px dx + Py dy = 0

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

  7. 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, MBA’s and Economics.

  8. u2 u1 y u0 Idea: Maximising U(x,y) is like climbing happiness mountain. x y But we are restricted by how high we can go since must stay on BC - (path on mountain). x

  9. u2 u1 y u0 So to move up happiness Mountain is subject to being on a budget constraint path. x Maximize U (x,y) subject to Pxx+ Pyy=M

  10. = 0 = 0 = 0 Known: Px, Py & M Unknowns: x,y,l 3 Equations: 3 Unknowns: Solve

  11. Note: Trick: U But:

  12. = 0 = 0 = 0 Known:Px, Py & MUnknowns:x,y,l 3Equations:3Unknowns:Solve

  13. Notice: U = x2 y3 <=> Slope of the Indifference Curve Recall Slope of Budget Constraint = Slope of IC = slope of BC

  14. Back to the Problem:  +  But But  + 

  15. Back to the Problem:  +  But But  + 

  16. So the Demand Curve for x when U=x2y3 If M=100:

  17. Recall that: U = x2 y3 Let: U = xa yb For Cobb - Douglas Utility Function

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

  19. Why does the demand for x not depend on py? 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 Px only effects demand for x in C.D.

  20. Constraint Objective fn So l tells us the change in U as M rises Increase from U1 to U2 Increase M  in objective fn  in constraint

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