e604-lect5S06 - Econ 604 Advanced Microeconomics Davis...

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Econ 604 Advanced Microeconomics Davis Spring 2005 16 February 2006 Reading. Chapter 5 (pp. 116-130) for today Chapter 5 (pp. 130-144) for next time Problems: To collect: Ch. 4: 4.2, 4.4, 4.6, 4.7 4.9 N ext time: Ch. 5 5.1 5.2 5.4 Observe: 5.1 and 5.2 don’t fit into the standard Lagrangian setup, because, in each case, the products are perfect substitutes. Below a given price level, the consumer will devote all income to only one product 5.1 is a case of perfect substitutes. 5.2 pertains to perfect complements. Lecture #5 REVIEW IV. Utility Maximization and Choice A. An Introductory Illustration. The two good case. (This is largely a graphical representation) 1. The Budget Constraint 2. First Order Conditions for a Maximum P x /P y = U x /U y , which implies U x /P x = U y /P y Intuitively: Purchase goods until the MU per dollar spent on each good is the same. 3. Second Order Conditions for a Maximum. The intersection of first order conditions will be a maximum if the quasi-convexity conditions are satisfied (that is, for any order pairs on an indifference curve U , (x o , y o ), (x 1 , y 1 ), it is the case that U( α x o + (1- )x 1, y o + (1- )y 1 )> U This condition will be satisfied if the quasi-concavity condition in 2.107 is met. It will alternatively be satisfied if d 2 Y/dX 2 . >0 Example, Suppose U* = XY. Differentiating, dU = YdX + XdY. On a level curve, this expression equals zero, so dY/dX = -Y/X 1
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What if we just took the derivative of the first order condition w.r.t. X? That is, d 2 Y/dX 2 = Y/X 2 > 0. Does this imply convexity in XY space? No! We must recognize that Y is implicitly a function of X. Rather, we have two (ultimately equivalent ways) to check for the concavity of the indifference curve. One is to totally differentiate U twice, subject to the constraint that utility is constant. We developed the sufficient condition for concavity of the constrained utility function in equation 2.107 [U 2 2 U 11 -2 U 1 U 2 U 12 + U 1 2 U 22 ]/ U 2 2 <0 For U = XY, U 1 = Y, U 2 = X,U 12 = 1 U 11 = 0, U 22 = 0, Inserting, -2YX/X 3 = -2Y/X 2 <0 (Note the sign is reversed relative to the standard second order condition. This is due to the way 2.107 is set up) The alternative approach is to solve the expression YX = U directly for Y and then differentiate twice. For example, let U = U* a constant. Then Y = U*/X dY/dX = -U*/X 2 (with U* =XY, dY/dX = -Y/X) d 2 Y/dX 2 = 2U*/X 3 = 2Y/X 2 <0 4. Corner Solutions B. The n-Good Case 1. First Order Conditions. For n goods, an optimizing consumer purchases all products so that the MU per dollar spent is equal for all goods. 2. Implications of First Order Conditions 3. Interpreting the LaGrangian Multiplier C. Indirect Utility Function We can express U(x 1 , x 2 ,….x n ) + λ (I –p 1 x 1 - …- p n x n ) as U(x 1 (p 1 …p n ,I),……. x n (p 1 …p n ,I) or V((p 1 …p n ,I) The advantage of this is that indirect utility is expressed in terms of observables. Thus, given, of course our assumptions about the functional form of utility) we can talk
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This note was uploaded on 02/23/2011 for the course ECON 604 taught by Professor Littleton during the Spring '10 term at Harvard.

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e604-lect5S06 - Econ 604 Advanced Microeconomics Davis...

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