Question is in the attached image. Thank you! ### Monopolist Pricing and Profit Analysis#### Problem Description1. **Consumer Demand Functions**: - Consumer 1 has the demand function $ y(p_1) = 20 - \frac{p_1}{2} $ - Consumer 2 has the demand function $ y(p_2) = 15 - \frac{p_2}{2} $ **Inverse Demand Functions**: - For Consumer 1: $ p_1 = 40 - 2y_1 $ - For Consumer 2: $ p_2 = 30 - 2y_2 $ The monopolist wants to implement price discrimination by charging different prices to each consumer.2. **Marginal Cost**: - The marginal cost (MC) of the monopolist is 10.3. **Marginal Revenue Functions**: - From Consumer 1: $ MR_1 = 40 - 4y_1 $ - From Consumer 2: $ MR_2 = 30 - 4y_2 $#### Tasksa) **Calculate the monopolist's price for each consumer**.b) **Common Pricing**: - If the monopolist cannot differentiate between consumers and must charge a common price $ p $, what would $ p $ and $ y $ be? - The inverse demand function for the common price is given by $ p = 70 - 2y $ and the marginal revenue is $ MR = 70 - 4y $.c) **Calculate the difference in profits between the two cases**.#### Detailed Steps1. **Individual Pricing**: - **For Consumer 1**: - Marginal Revenue (MR) function: $ MR_1 = 40 - 4y_1 $ - Set $ MR_1 = MC $: $ 40 - 4y_1 = 10 $ ⟹ $ 4y_1 = 30 $ ⟹ $ y_1 = 7.5 $ - Inverse demand, $ p_1 = 40 - 2y_1 = 40 - 2(7.5) = 25 $ - **For Consumer 2**: - Marginal Revenue (MR) function: \( MR_2 = 30 - 4

Answered: Image /qna-images/answer/788b4a9c-a38e-4139-9461-d087c9a81044.jpg

1) Suppose there are two consumers. Consumer 1 has the demand func- tion y(pı) 20 – P1/2, the second consumer has the demand function y(P2) = 15 – = 40 – 2yı – p2/2. Inverse demands are then given by p1 and p2 = 30 – 2y2 The monopolist wants to charge different prices for each consumer. Suppose that marginal cost of the monopolist is 10. Marginal revenue from consumer 1 is then MR 40 – 4y1 and from consumer 2, MR2 3 30 — 4у2 %3D a) Calculate the monopolists price for each consumer. b) Suppose the monopolist cannot differentiate between each consumer and much charge the common price p. What would p and y be? In this case the inverse demand is given by p = 70–2y and marginal revenue is MR=70–4y c). Calculate the difference in profits between the two cases.

1) Suppose there are two consumers. Consumer 1 has the demand func- tion y(pı) 20 – P1/2, the second consumer has the demand function y(P2) = 15 – = 40 – 2yı – p2/2. Inverse demands are then given by p1 and p2 = 30 – 2y2 The monopolist wants to charge different prices for each consumer. Suppose that marginal cost of the monopolist is 10. Marginal revenue from consumer 1 is then MR 40 – 4y1 and from consumer 2, MR2 3 30 — 4у2 %3D a) Calculate the monopolists price for each consumer. b) Suppose the monopolist cannot differentiate between each consumer and much charge the common price p. What would p and y be? In this case the inverse demand is given by p = 70–2y and marginal revenue is MR=70–4y c). Calculate the difference in profits between the two cases.

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

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Chapter1: Making Economics Decisions

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Problem 1QTC

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Question

Question is in the attached image. Thank you!

### Monopolist Pricing and Profit Analysis

#### Problem Description

1. **Consumer Demand Functions**:
- Consumer 1 has the demand function $ y(p_1) = 20 - \frac{p_1}{2} $
- Consumer 2 has the demand function $ y(p_2) = 15 - \frac{p_2}{2} $

**Inverse Demand Functions**:
- For Consumer 1: $ p_1 = 40 - 2y_1 $
- For Consumer 2: $ p_2 = 30 - 2y_2 $

The monopolist wants to implement price discrimination by charging different prices to each consumer.

2. **Marginal Cost**:
- The marginal cost (MC) of the monopolist is 10.

3. **Marginal Revenue Functions**:
- From Consumer 1: $ MR_1 = 40 - 4y_1 $
- From Consumer 2: $ MR_2 = 30 - 4y_2 $

#### Tasks

a) **Calculate the monopolist's price for each consumer**.

b) **Common Pricing**:
- If the monopolist cannot differentiate between consumers and must charge a common price $ p $, what would $ p $ and $ y $ be?
- The inverse demand function for the common price is given by $ p = 70 - 2y $ and the marginal revenue is $ MR = 70 - 4y $.

c) **Calculate the difference in profits between the two cases**.

#### Detailed Steps

1. **Individual Pricing**:
- **For Consumer 1**:
- Marginal Revenue (MR) function: $ MR_1 = 40 - 4y_1 $
- Set $ MR_1 = MC $: $ 40 - 4y_1 = 10 $ ⟹ $ 4y_1 = 30 $ ⟹ $ y_1 = 7.5 $
- Inverse demand, $ p_1 = 40 - 2y_1 = 40 - 2(7.5) = 25 $

- **For Consumer 2**:
- Marginal Revenue (MR) function: \( MR_2 = 30 - 4

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