Consider the simplest monopoly case, where there is linear demand, p(q) = a - bq
ID: 1220298 • Letter: C
Question
Consider the simplest monopoly case, where there is linear demand, p(q) = a - bq and constant costs c(y) = c. Write out the monopolist profit maximization problem with both y and p as choice variables. Use the substitution method to find the optimal p* the monopolist should charge. At this price p*, how many output, y*, will the monopolist sell? How does this optimal solution (p*, y*) compare to the other optimal solution when the monopolist maximizes profit while choosing output? (The method you used for Uncle Rico.) IF you do not know this answer (or don't believe your answer), you can quickly derive the other optimal solution by setting MR(y) = MC(y) and solving for y.Explanation / Answer
(a) Profit, Z = Revenue - Costs = (Price x Quantity) - Costs
= [q x (a - bq)] - c
= aq - bq2 - c
So, profit maximization problem is:
Maximize Z = aq - bq2 - c
(b) Monopolist will maximize profits when dZ / dq = 0
a - 2bq = 0
2bq = a
q = a / 2b
p* = a - bq = a - b x (a / 2b) = a - (a / 2) = a / 2
(c) When p* = a / 2,
q* = a / 2b
(d) Question does not mention which method was employed for the "Uncle Rico" problem.