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 [s-squared multiplied by q]
Calculate the price and quality levels that a monopolist would choose, and the
corresponding quantity sold.
A monopolist will produce where marginal cost = marginal revenue.
Total revenue is P*Q = s(1-Q)*Q = sQ - sQ^2
Total cost is s^2Q
Take the first derivitive (with respect to Q):
MR = s - 2sQ
MC = s^2
Take it from here, Solve for Q.
To find the price and quality levels that a monopolist would choose, we need to determine the values of s and q that maximize the monopolist's profit.
The monopolist's profit can be calculated as the difference between the total revenue and total cost:
Profit = Total Revenue - Total Cost
Total Revenue is given by the product of the price (P) and the quantity sold (q):
Total Revenue = P(s, q) * q = s(1 - q) * q = sq - sq^2
Total Cost is given by the function C(s, q) = s^2q:
Total Cost = C(s, q) = s^2 * q
Substituting the expressions for Total Revenue and Total Cost into the profit equation, we get:
Profit = sq - sq^2 - s^2q
To find the price and quality levels that maximize profit, we take the derivative of profit with respect to both s and q, and set them equal to zero:
∂Profit/∂s = q - 2sq = 0
∂Profit/∂q = s - 2sq = 0
Solving these equations simultaneously gives us the values of s and q. Let's solve each equation:
1. ∂Profit/∂s = q - 2sq = 0
This equation implies that q = 2sq. Substitute this value of q into the second equation.
2. ∂Profit/∂q = s - 2sq = 0
Now substitute q = 2sq into this equation and solve for s.
s - 2s(2s) = 0
s - 4s^2 = 0
s(1 - 4s) = 0
This equation gives us two possible solutions: s = 0 or s = 1/4.
If s = 0, then the quality level would be zero, which is not realistic. So, we discard this solution.
If s = 1/4, then the quality level would be 1/4.
Substitute the value of s = 1/4 into the equation q = 2sq:
q = 2(1/4)q
1 = 1q
q = 1
Therefore, the monopolist would choose a quality level (s) of 1/4, a quantity sold (q) of 1, and the corresponding price (P) can be calculated using the demand price equation:
P(s, q) = s (1 - q)
P(1/4, 1) = (1/4) (1 - 1)
P(1/4, 1) = 1/4 * 0
P(1/4, 1) = 0
Hence, the monopolist would choose a price level of 0, a quality level of 1/4, and would sell a quantity of 1.