1 / 22

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: .

alika-tyler
Télécharger la présentation

Lagrange Method

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related