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CHAPTER 2
THE MATHEMATICS OF OPTIMIZATION
The problems in this chapter are primarily mathematical. They are intended to give students some practice
with taking derivatives and using the Lagrangian techniques, but the problems in themselves offer few
economic insights. Consequently, no commentary is provided. All of the problems are relatively simple and
instructors might choose from among them on the basis of how they wish to approach the teaching of the
optimization methods in class.
Solutions
2.1 U( x, y) 4 x2 3 y2
U U
a. = 8 x , = 6 y
x y
b. 8, 12
U U
c. dU dx + dy = 8 x dx + 6 y dy
x y
dy
d. for dU 0 8x dx 6y dy 0
dx
dy 8x 4x
= =
dx 6y 3y
e. x 1, y 2 U 4 1 3 4 16
dy 4(1)
f. 2 / 3
dx 3(2)
g. U = 16 contour line is an ellipse centered at the origin. With equation
2 2 dy 4x
4x 3y 16 , slope of the line at (x, y) is .
dx 3y
2.2 a. Profits are given by R C 2q 2 40q 100
d *
4q 40 q 10
dq
* 2
2(10) 40(10) 100 100
2
d
b. 2 4 so profits are maximized
dq
dR
c. MR 70 2q
dq
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