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Student Number:
QUIZ 4
Econ 212-Section B
Question 1. [4 marks] Consider the short-run production function
Q  50L  100K
a) [2 mark] Derive the cost-minimizing quantity of labor as a function of output and
capital. How does the cost-minimizing quantity of labor vary with Q ? with K ?
Answer:
Q  100K
50
It increases with Q and decreases with K bar.
Q  50L  100K  L 
b) [2 marks] Assume the price of labor is w and the rental rate of capital is r . Assume
that K is variable now. Derive the long-run total cost function.
Answer:
Note that we have a production function in which inputs are perfect substitutes. Thus,
given w and r , we can compare marginal product of Labour per dollar spent on labor
50 100
1 2

  ,
with marginal product of capital per dollar spent on capital. So if
w
r
w r
then we use only labor to produce Q and the Labor demand function is
Q
wQ
L
, and K=0. Thus the total cost function is 
50
50
If
50 100
1 2
Q
rQ

  K
and L  0 . And the total cost function is 
w
r
w r
100
100
Thus, in general we have a corner solution for choice of K or L. If we have interior
50 100
1 2

  , we can use any combination of
solution, it is indeterminate i.e. if
w
r
w r
Labour and capital that produces Q, including just using either capital or labor.
1 w
 =Price ratio. If this
2 r
inequality holds, then use K to produce Q. If the inequality is in the other direction, only
use L to produce Q .
Alternatively, Absolute value of slope of isoquant 
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Question 2. [4 marks] Dave and Carolyn run a landscaping company. They employ
people and rent equipment to dig holes for tree planting. They rent power auger
machines, which need two people to run each machine. The production function can be
written as Q  min 10K ,5L . Dave and Carolyn pay their workers $10 per hour. The
machines rent for $20 per hour. Dave and Carolyn need to dig 500 holes by the end of
the week.
a) [2 marks] In order to minimize their costs for this level of output, how many labor
hours should they hire? How many machine hours should they rent?
Answer:
500  min 10 K ,5 L
10 K  500
K  50
5 L  500
L  100
b) [2 marks] What is the total cost of digging the 500 holes?
Answer:
Total cost = 100 w + 50 r = 2000
Question 3. [2 marks] Identify the returns to scale (increasing, constant, or decreasing)
for the following production functions.
a) [1 mark] Q  2 * min K ,2L
Answer: It is constant returns to scale.
b) [1 mark] Q  5 K 0.25L0.5
Answer:
It is decreasing returns to scale.
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