Following information shows that a firm offering a good at different prices to groups of consumers with different levels of willingness to pay.
Inverse Demand for movies: P1 = 20 – 4Q1
Inverse Demand for students: P2 = 10 – Q2
MC = 4Q LKR /ticket
(a) What price and quantity and maximizes profits if the firm charges each market?
(b) Demonstrate that charging different prices for the two groups results in higher profits than charging the same price for everyone.
Solution:
a.). A firm maximizes profits at the point where: MC = MR
Market 1 - Movies:
Inverse Demand for movies: P1 = 20 – 4Q1
Derive MR1:
TR1 = P "\\times" Q1
TR1 = (20 – 4Q1) "\\times" Q1 = 20Q – 4Q12
MR1 = "\\frac{\\partial TR_{1} } {\\partial Q_{1} }" = 20 – 8Q1
MR1 = 20 – 8Q1
MC1 = 4
MR1 = MC1
4 = 20 – 8Q1
8Q1 = 20 – 4
8Q1 = 16
Q1 = 2
Market 1 quantity = 2
Substitute in the demand function to derive price:
P1 = 20 – 4Q1
P1 = 20 – 4(2) = 20 – 8 = 12
Market 1 price = 12
Market 2 - Students:
Inverse Demand for students: P2 = 10 – Q2
Derive MR2:
TR2 = P "\\times" Q2
TR2 = (10 – Q2) "\\times" Q2 = 10Q2 – Q22
MR2 = "\\frac{\\partial TR_{2} } {\\partial Q_{2} }" = 10 – 2Q2
MR2 = 10 – 2Q2
MC2 = 4
MR2 = MC2
4 = 10 – 2Q2
2Q2 = 10 – 4
2Q2 = 6
Q2 = 3
Market 2 quantity = 3
Substitute in the demand function to derive price:
P2 = 10 – Q2
P2 = 10 – 3 = 7
Market 2 price = 7
b.). Profits in terms of revenue:
TR – Market 1 = 12 "\\times" 2 = 24
TR – Market 2 = 7 "\\times" 3 = 21
This means that market 1 will generate more revenues than market 2, signifying that charging different prices will generate more revenue for the firm.
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