Intermediate Microeconomics Econ 301 UBC Professor Sergei ...

Now substitute (4) into (3) and solve for Y. Once you have solved for Y, you can
substitute Y back into (4) and solve for X. Note that algebraically there are several
ways to solve this type of problem; it does not have to be done exactly as shown here.
The demand functions are:
Y
X
P XI
2
PY PY PX PYI 2
PX PY PX or Y I
12
or X 3I
4 .
b. Assume that her income I = $100. How many candy bars and how many espressos will Sharon
consume?
Substitute the values for the two prices and income into the demand functions to find
that she consumes X = 75 candy bars and Y = 8.33 espressos.
c. What is the marginal utility of income?
As shown in the appendix, the marginal utility of income equals . From part a,
1 1
. Substitute into either part of the equation to get
0.
5
2
Y
0.
5
PX X 2PY
This is how much Sharon’s utility would increase if she had one more dollar to spend.
9. Exercise 5, Appendix to Ch 4, p 157
Maurice has the following utility function: U(X,Y) 20X 80Y X 2 2Y 2 , where X is his
consumption of CDs, with a price of $1, and Y is his consumption of movie videos, with a rental price
of $2. He plans to spend $41 on both forms of entertainment. Determine the number of CDs and
video rentals that will maximize Maurice’s utility.
Using X as the number of CDs and Y as the number of video rentals, the Lagrangian
equation is
20X 80Y X 2 2Y 2 (X 2Y 41).
To find the optimal consumption of each good, maximize the Lagrangian equation with
respect to X, Y and
constraint. The necessary conditions for a maximum are
(1)
20 2X 0
X
(2)
80 4Y 2 0
Y
(3)
X 2Y 41 0.
Note that in condition (3), both sides have been multiplied by –1. Combining conditions (1)
and (2) results in