Economics 106P
Spring 2006
UCLA
E. McDevitt
STUDY QUESTIONS-SET #3
LINEAR PROGRAMMING
1. A firm produces tires by separate processes that require different quantities of capital (K), labor (L), and
raw materials (M). Process 1 requires one unit of K, four units of L, and two units of M to produce a tire
yielding a $4 profit. Process 2 requires one unit of K, two units of
L, and four units of M to produce a tire
yielding a $6 profit. The available supply of capital is 10; of labor, 32; and of raw materials, 32.
a. Set up the objective function.
b. Give the constraints.
c. Find the profit maximizing combination T
1
and T
2
by graph. First draw the constraints, showing the
feasible region. Then draw a graph of the feasible region superimposing the objective function on it.
After doing this, confirm your answer using the slope rule.
d. Find the shadow prices of raw materials and labor. At what available quantity would the shadow price of
raw materials be zero?
2. To maintain his health, a person must fulfill certain minimum daily requirements for several kinds of
nutrients. Assume that only three kinds of nutrients need to be considered: calcium, protein and calories.
Also assume that the person's diet is to consist of only two food items, x
1
and x
2
, whose price and nutrient
contents are shown in the table below, where we have also listed the minimum daily requirement for each
nutrient. What is the cost-minimizing combination of
x
1
2
that will satisfy the daily requirements?
Provide a graphical solution. Support your graphical solution using the slope rule. What is total cost
at this optimal combination? What is cost at the other corners?
x
1
x
2
Minimum Daily Requirement .
Price
$0.60/unit
$1/unit
Calcium
10
4
20
Protein
5
5
20
Calories
2
6
12
TRANSFER PRICING
1. Using a set of graphs, answer each question below.
a. Find the profit-maximizing quantity of
the upstream product and downstream product and the optimal
transfer price assuming there is no outside market for the intermediate good.
b. Find the profit-maximizing quantity of
the upstream product used and produced by the upstream
division, quantity of downstream product, and optimal transfer price. Assume a competitive outside market
for the intermediate good and that the competitive price is below the transfer price that would exist if there
were no outside market.
c. Find the profit-maximizing quantity of the intermediate good produced,
the quantity sold to outside
buyers, the quantity "sold" to downstream division, and the optimal transfer price.
2. The House Products Division of Acme Corporation manufactures and sells digital clock radios.
A major component is supplied by the electronics division of Acme. The Cost functions for the radio and
the electronic component divisions are, respectively,
TC
r
= 30 + 2Q
r
and TC
c
= 70+6Q
c
+ Q
c
2
Note that TC
r
does not include the cost of the component. Manufacture of one radio set requires the use of
one electronic component. The firm's demand curve for the digital clock radio is P
r
= 108-Q
r
.