There are two desirable brands of speakers – Boss (a budget speaker) and Zanadoo (a highend speaker).
Consider the following key assumptions:
• The price of Boss speakers = 𝑃𝑃𝐵𝐵
• The quantity of Boss Speakers demanded = 𝑄𝑄𝐵𝐵
𝑑𝑑
• The quantity of Boss Speakers supplied = 𝑄𝑄𝐵𝐵
𝑠𝑠
• The price of Zanadoo speakers = 𝑃𝑃𝑍𝑍 = $50
• Income = I = $100
• Demand function for Boss speakers = 𝑄𝑄𝐵𝐵
𝑑𝑑 = 50 – 2𝑃𝑃𝐵𝐵 + 𝑃𝑃𝑍𝑍 + I
• Supply function for Boss speakers = 𝑄𝑄𝐵𝐵
𝑠𝑠 = -20 + 20𝑃𝑃� Calculate the equilibrium price and quantity of Boss speakers in this market. Show
workings for each value (2 x 1 mark = 2 marks).

To calculate the equilibrium price and quantity of Boss speakers in the market, we need to set the quantity demanded equal to the quantity supplied, and then solve for the price $ P_B $.

### Given:
- Price of Zanadoo speakers $ P_Z = 50 $
- Income $ I = 100 $
- Demand function for Boss speakers: $ Q_B^d = 50 - 2P_B + P_Z + I $
- Supply function for Boss speakers: $ Q_B^s = -20 + 20P_B $

### Step 1: Express the demand function
Substitute the values of $ P_Z $ and $ I $ into the demand function:
\[
Q_B^d = 50 - 2P_B + 50 + 100
\]
\[
Q_B^d = 200 - 2P_B
\]

### Step 2: Express the supply function
The supply function is already given as:
\[
Q_B^s = -20 + 20P_B
\]

### Step 3: Set quantity demanded equal to quantity supplied to find the equilibrium price
\[
Q_B^d = Q_B^s
\]
\[
200 - 2P_B = -20 + 20P_B
\]

### Step 4: Solve for $ P_B $
Add $ 2P_B $ to both sides:
\[
200 = -20 + 22P_B
\]
Add 20 to both sides:
\[
220 = 22P_B
\]
Divide both sides by 22:
\[
P_B = \frac{220}{22} = 10
\]

So, the equilibrium price $ P_B $ is $10.

### Step 5: Calculate the equilibrium quantity
Substitute $ P_B = 10 $ back into either the demand or supply function. Using the supply function:
\[
Q_B^s = -20 + 20(10) = -20 + 200 = 180
\]

So, the equilibrium quantity $ Q_B $ is 180 units.

### Final Answer:
- **Equilibrium Price $ P_B $: $10**
- **Equilibrium Quantity $ Q_B $: 180 units**

Would you like more details or have any questions?

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**Related Questions:**
1. How would an increase in income affect the equilibrium price and quantity of Boss speakers?
2. What happens to the equilibrium if the price of Zanadoo speakers increases?
3. How does the demand function change if the price of Zanadoo speakers decreases to $40?
4. What is the impact on the equilibrium if the supply function changes to $ Q_B^s = -10 + 15P_B $?
5. How would a government tax on Boss speakers affect the equilibrium price and quantity?

**Tip:** Always ensure you correctly substitute the known values into the demand and supply functions to accurately calculate equilibrium conditions.

Learn how to calculate the equilibrium price and quantity of Boss speakers in a market scenario involving demand and supply functions. Detailed step-by-step solution provided with specific data points including price of Zanadoo speakers, income, and functions for both demand and supply.

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