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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