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1 Advanced Microeconoics (ES3005) Advanced Microeconoics (ES3005) Matheatics Review : The Lagrange Multiplier Outline: I. Introduction II. Duality Theory: Co Douglas Exaple III. Final Coents I. Introduction The siplest ethod of solving a constrained axiization or constrained iniization prole is y the Lagrange ultiplier ethod. In what follows we illustrate the ethod y considering, firstly, the consuer s proles of axiizing (Co-Douglas) utility suject to a udget constraint, therey yielding the consuer s Marshallian deand function. We then, secondly, consider the consuer s prole of iniizing expenditure suject to a (Co- Douglas) utility constraint, therey yielding the consuer s ( Hicks) Copensated deand function. We also allude to duality of the two proles and coent on the interpretation of the Lagrange Multiplier itself. II. Duality Theory: Co-Douglas Exaple Marshallian Deand Functions The Marshallian deand function is otained y axiizing utility suject to a udget constraint. Thus: ax x, st Ax x + () Solution: ax { x,,λ} Lg Ax x + λ( x ) () Ax x x λ 0 (3) ( ) Ax x λ 0 (4) λ x 0 (5) Dividing (3) y (4):

2 x Advanced Microeconoics (ES3005) Ax ( ) Ax x λ λ ( ) x x ( ) x (6) Rearrange (5): x (7) Sustitute (7) in (6): x x (8) Thus: x p p p (9) ( ) The consuer s Marshallian deand functions are thus: x x ( ) (0) Recall that the Marshalian deand function easures the overall change in deand when the price of a good changes and the price of other goods and consuer incoe is held constant. Optiized utility is:

3 Advanced Microeconoics (ES3005) u u x (, ) A( x ) ( ) u A p ( ) u A ( ) p () Note that the udget constraint is satisfied: x + x p + p ( ) () Copensated Deand Function The consuer s Copensated deand function is otained y iniizing expenditure suject to a utility constraint. Thus: in x, x + st Ax u (3) Solution: in { x,,λ} Lg x + + θ u Ax ( ) (4) p x θax x 0 (5) θ ( ) Ax x 0 (6) θ u Ax x 0 (7) Dividing (5) y (6) yields: 3

4 Advanced Microeconoics (ES3005) θax θ x ( ) Ax ( ) x (8) Rearrange (7): u Ax (9) Sustitute in (9) in (8): x x + x x u Ax ( ) u ( ) ( ) x u ( ) u ( ) (0) Raising every ter y the power (-) yields: x ( ) u () Sustitute equation () into (9): 4

5 Advanced Microeconoics (ES3005) u Ax ( ) u ( ) ( ) x u ( ) ( ) ( ) u A ( ) ( ) u () u u The consuer s ( Hicks) Copensated deand functions are thus: x h x h ( ) u u (3) Recall that the ( Hicks) Copensated deand functions easure the change in deand following a change in price holding the price of the other good and the consuer s real incoe (vis. His aility to enjoy a particular level of utility) constant. Thus, the (Hicks) Copensated deand function easures the own-price sustitution effect. Note that the utility constraint is satisfied: h A( x ) h ( ) A Ax u ( ) u u (4) Note that iniized expenditure is given y: h x h + h h Thus: ( ) u + p u (5) 5

6 Advanced Microeconoics (ES3005) h p u ( ) + h p u h p u h p u + (6) III. Final Coents Interpretation of the Lagrange Multiplier The Lagrange ultiplier easures the arginal effect of a relaxation in the constraint on the consuer s ojective function. Thus, the Lagrange ultiplier in the first prole, λ in the first prole easures the arginal effect on utility of a relaxation in the udget constraint and ay therefore e interpreted as the consuer s arginal utility of incoe. The Lagrange ultiplier in the second prole, θ, easures the arginal effect on expenditure on a relaxation in the utility constraint. Second-Order Conditions Note that we have not explored the second-order conditions to confir that we have indeed found the utility axiizing and cost iniizing choices for the consuer. We will generally assue that second-order conditions are satisfied in Lagrange ultiplier constrained optiization proles. Duality Note the duality of the prole. Set the constraint level of utility u in the consuer s (Hicks) Copensated deand function (3) equal to the Marshallian axiized utility u in equation (): 6

7 x Advanced Microeconoics (ES3005) ( ) p x u p A ( ) p x A ( ) p x + ( ) ( )+ ( ) ( ) ( ) ( p ) p (7) This is the level of deand deterined y the Marshallian deand function (0). A siilar result occurs if we set the udget constraint in the consuer s Marshallian deand function (0) equal to the (Hicks) Copensated iniized expenditure, h in Equation (6): x h p x x x u u u u (8) This is the level of deand deterined y the (Hicks) Copensated deand function (3). 7

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