Monopoly Pricing Question help?

Question

A monopolist produces a product whose demand price and production costs vary with quality s and quantity q according to

P (s; q) = s (1 - q)

C (s; q) = s^2 q [i.e s-squared multiplied by q]

Calculate the price and quality levels that a monopolist would choose, and the corresponding quantity sold.

Answer 1

What you need to do is to solve the following maximization problem:



             max { qs(1-q) - s^2 q }

You take first order conditions with respect to q and s (i.e. take first derivatives of the above objective function with respect to s and q). You will get 2 equations with 2 unknowns. Once you solve for s and q you will get the price as well.

Answer 2

p= s (1-q) = s - qs

TC= qs²

TR=p*q= sq - sq²

Equilibrium condition: MR=MC

Profit = TR-TC = sq - sq² - qs²

Profit → MAX/MIN;  if δ(TR - TC)=0

δ(sq - sq² - qs²)/δq = 0

δ(sq - sq² - qs²)/δs = 0

s - sq - s² = 0

q - qs - q² = 0

Solving this system you will get following solutions:

|...........{q → 0 ; s → 0} . [1]

|...........{q → 0 ; s → 1} . [2]

|...........{q → 1 ; s → 0} . [3]

|......{q → 1/3 ; s → 1/3} . [4]

By testing FOC & SOC you will find that function is maximized at [4]-th solution,

thus q=1/3 ; s=1/3 ; p=1/3(1-1/3)=2/9