Marcie DeGiovine
258-10-0153
February 1, 2013 – May 20, 2013
BA635
Lesson #8, Assignment #3:
P
= $75 - $0.003Q
MR
= Δ TR/ Δ Q = $75 - $0.006Q
TC
MC
= Δ TC/ Δ Q = $25 + $0.004Q
The company has assets of $100,000 used for call waiting services and the utility commission has authorized a 12% return on investment.
A. Calculate Black Hills' profit-maximizing price (monthly and annually), output, and rate-of-return levels.
B. What monthly price should the commission grant to limit Black Hills to an 12% rate of return?
(Demand for shelled almonds)
(Marginal revenue from shelled almonds)
(Demand for shell by-product)
(Marginal revenue from shell by-product)
TC
= $3,000,000 + $6.25Q
(Total cost)
MC
= $6.25
(Marginal cost)
Answers:
A.
To find the profit-maximizing level of output, we must set MR = MC where:
MR = MC
$75 - $0.006Q = $25 + $0.004Q
0.01Q = 50
Q = 5,000
P = $6.25 - $0.00025(5,000)
= $5
(Monthly price)
P = $75 - $0.003(5,000)
= $60
(Annual price)
p = TR - TC
= $60(5,000) - $108,000 - $25(5,000) - $0.002(5,0002)
= $17,000
If the company has $100,000 invested in plant and equipment, its optimal rate of return on investment is:
Return on investment =
$17,000
$100,000
= 0.17 or 17%
(Note: Profit is falling Q > 5,000)
B.
With a 12% return on total assets, Black Hills would earn profits of:
p = Allowed return Total assets
= 0.120($100,000)
= $12,000
To determine the level of output that would be consistent with this level of total profits, consider the profit relation:
p = TR - TC
$12,000 = $75Q - $0.003Q2 - $108,000 - $25Q - $0.002Q2
12,000 = -0.005Q2 + 50Q - 108,000
0 = -0.005Q2 + 50Q - 120,000
which is a function of the form aQ2 + bQ + c = 0 where a = -0.005, b = 50 and c = -120,000 and can be solved using
the quadratic equation:
Q =
-b
√b² + 4ac
2a
=
-50
√50² + 4(-0.005)(-120,000)
2(-0.005)
=
-0.01
= 4,000 or 6,000 customers
Because public utility commissions generally want utilities to provide service to the greatest possible number of
customers at the lowest possible price, the "upper" Q = 6,000 is the appropriate output level. This output level will result in a
monthly service price of:
P = $6.25 - $0.00025(6,000)
= $4.75
This $4.75 per month price for call waiting service will provide Black Hills with a fair rate of return on total investment,
while ensuring service to a broad customer base.