Advanced Microeconomics (ES30025)


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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 (CoDouglas) 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: CoDouglas 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 ownprice 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. SecondOrder Conditions Note that we have not explored the secondorder conditions to confir that we have indeed found the utility axiizing and cost iniizing choices for the consuer. We will generally assue that secondorder 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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