The Lumins Lamp Company, a producer of old-style oil lamps, estimated the following demand function for its product:
Q = 120,000 – 10,000P, where Q is the quantity demanded per year and P is the price per lamp. The firm’s fixed
costs are $12,000 and variable costs are $1.50 per lamp.
a. Write an equation for the total revenue (TR) function in terms of Q.
P = (120,000 - Q)/10,000 = 12 - Q/10,000
TR = P*Q = 12Q - Q^2/10,000
b. Specify the marginal revenue function.
MR = TR' = 12 - Q/5,000
c. Write an equation for the total cost (TC) function in terms of Q.
TC = 12,000 + 1.5Q
d. Specify the marginal cost function.
MC = TC' = 1.5
e. Write an equation for total profits (π) in terms of Q. At what level of output (Q) are total profits maximized?
What price will be charged? What are total profits at this output level?
π = TR - TC = 12Q - Q^2/10,000 - 12,000 - 1.5Q = 10.5Q - Q^2/10,000 - 12,000
MR = MC = P = 1.5, if it is perfect competitive market.
P = 12 - Q/10,000 = 1.5
Q/10,000 = 10.5
Q = 105,000
π = TR - TC = 1.5*105,000 - 12,000 - 1.5*105,000 = -12,000
So, P = AVC, it is the minimal point, where the firm can produce.
f. Check your answer in Part (e) by equating the marginal revenue and marginal
cost functions, determined in Parts (b) and (d), and solving for Q.
MR = MC
12 - Q/5,000 = 1.5
Q/5,000 = 10.5
Q = 52,500
First calculations for Q get another answer, so MR = MC not equal P. So, it is monopolistic market.
π = TR - TC = 10.5Q - Q^2/10,000 - 12,000 = $263,625
P = 12 - Q/10,000 = $6.75
g. What model of market pricing behavior has been assumed in this problem?
So, monopolistic model of pricing was assumed, because P = 6.75 is greater than MR = MC = 1.5.