1. Consider a monopolistic market with market demand function q = 8 2p. Also sup
ID: 1218082 • Letter: 1
Question
1. Consider a monopolistic market with market demand function q = 8 2p. Also suppose marginal cost of production is constant and equal to 2. Part I In this part we examine the fact that profit maximization should yield the same result regardless of the fact that profit is maximized over price or quantity. a. Write down the profit function both in terms of price and quantity. b. Find the quantity and price the the monopolist charges by maximizing the profit function over quantity. c. Find the quantity and price the the monopolist charges by maximizing the profit function over price. d. Compare your answers in parts (b),(c). Are they the same? d.Since we know that profit is also maximized by using MR(q) = MC(q), check if you get the same answer with this approach.
Explanation / Answer
a) A monopolist operates with an aim of earning maximum profit. Profit is the difference of total revenue and total cost. Total revenue is the product of price and quantity. In terms of price, total revenue is p*q which is equal to 8p – 2p2and total cost is marginal cost multiplied by the quantity which becomes 2*(8 – 2p) or 16 – 4p.
So the profit as a function of price is given by = 8p – 2p2– 16 + 4p.
In terms of quantity, demand function is p = 4 – (q/2). Total revenue is p*q which is equal to 4q - q2/2. Total cost is marginal cost multiplied by the quantity which becomes 2q. Profit as a function of quantity is given by
= 4q - q2/2 – 2q
b) Profit as a function of quantity is given by
= 4q - q2/2 – 2q
= 2q - q2/2
Profit is maximum at a price level at which its first derivate is zero
d()/dp = 0
2 – q = 0
q = 2
Substitute q = 2 in the demand function as q = 4 – (2/2) = 3. So profit maximizing price p* is 3 and profit maximizing quantity q* is 2
c) Profit function is
= 8p – 2p2 – 16 + 4p
= 12p – 2p2 – 16
Profit is maximum at a price level at which its first derivate is zero
d()/dp = 0
d(12p – 2p2 – 16)/dp= 0
12 – 4p = 0
p = 3
Substitute p = 3 in the demand function as q = 8 – 2(3) = 2. So profit maximizing price p* is 3 and profit maximizing quantity q* is 2
d) Yes. From both methods, the answer is same. MR is given by 4 - q (derivative of TR). MC is given as 2. So MR = MC implies 4 - q = 2. This gives profit maximizing quantity q* as 2. And so profit maximizing price p* is 4 - 2/2 = 3.
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