Econ 604 Advanced Microeconomics



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

Next 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

Px/Py = Ux/Uy, which implies

Ux/Px = Uy/Py

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, (xo, yo), (x1, y1), it is the case that

U((xo + (1-()x1, (yo + (1-()y1)> U

This condition will be satisfied if the quasi-concavity condition in 2.107 is met. It will alternatively be satisfied if d2Y/dX2.>0

Example, Suppose U* = XY. Differentiating,

dU = YdX + XdY.

On a level curve, this expression equals zero, so

dY/dX = -Y/X

Now suppose that you wanted to check the concavity of this level curve.

What if we just took the derivative of the first order condition w.r.t. X? That is,

d2Y/dX2 = Y/X2 > 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

[U22U11 -2 U1U2U12 + U12 U22]/ U22 ................
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