Answer to Question #275307 in Microeconomics for Ima

Question

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 = 4

(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.

(c) Graph the demand curves, the marginal revenue curves, the marginal cost

curve and highlight the equilibrium.

Answer 1

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.

c.)

from the graph

the blue represents Inverse Demand for movies: P1 = 20 – 4Q1

the red represents Inverse Demand for students: P2 = 10 – Q2

the red represents mc = 4