QUESTION 2 (CLO 1)
The individual demand function for three firms selling storybooks are as follows:
Firm 1: P = 240-0.5Q
Firm 2: P = 150-2Q
Firm 3: P = 90-0.4Q
a. What is the market demand equation for the storybook?
(9)
b. Write the total revenue (TR) and marginal revenue (MR) equation.
(8)
c. What is the quantity and price equilibrium that maximise the total
revenue? (9)
[Total: 30 marks

To solve the given problem, we need to address each part step by step.

### a. Market Demand Equation

The market demand equation can be found by summing up the individual quantities demanded by all firms at a given price. This involves expressing each individual demand equation in terms of \( Q \), and then summing these quantities.

For Firm 1:  
\[ P = 240 - 0.5Q_1 \]  
\[ Q_1 = \frac{240 - P}{0.5} = 480 - 2P \]

For Firm 2:  
\[ P = 150 - 2Q_2 \]  
\[ Q_2 = \frac{150 - P}{2} = 75 - 0.5P \]

For Firm 3:  
\[ P = 90 - 0.4Q_3 \]  
\[ Q_3 = \frac{90 - P}{0.4} = 225 - 2.5P \]

Now, sum up the quantities \( Q_1 \), \( Q_2 \), and \( Q_3 \) to get the market demand \( Q \):

\[ Q = Q_1 + Q_2 + Q_3 \]  
\[ Q = (480 - 2P) + (75 - 0.5P) + (225 - 2.5P) \]  
\[ Q = 480 + 75 + 225 - (2 + 0.5 + 2.5)P \]  
\[ Q = 780 - 5P \]

Thus, the market demand equation is:
\[ Q = 780 - 5P \]

We can also express this in terms of \( P \):
\[ P = \frac{780 - Q}{5} \]

### b. Total Revenue (TR) and Marginal Revenue (MR) Equation

Total Revenue (TR) is given by the price (P) times the quantity (Q):
\[ TR = P \cdot Q \]

Substitute the market demand equation \( P = \frac{780 - Q}{5} \) into the TR equation:
\[ TR = \left(\frac{780 - Q}{5}\right) Q \]
\[ TR = \frac{780Q - Q^2}{5} \]

To find the Marginal Revenue (MR), we need to take the derivative of the TR function with respect to \( Q \):
\[ MR = \frac{d(TR)}{dQ} \]
\[ TR = \frac{780Q - Q^2}{5} \]
\[ TR = 156Q - \frac{Q^2}{5} \]
\[ MR = \frac{d}{dQ} \left( 156Q - \frac{Q^2}{5} \right) \]
\[ MR = 156 - \frac{2Q}{5} \]
\[ MR = 156 - 0.4Q \]

### c. Quantity and Price Equilibrium that Maximize Total Revenue

To maximize total revenue, we set marginal revenue (MR) equal to zero:
\[ MR = 0 \]
\[ 156 - 0.4Q = 0 \]
\[ 0.4Q = 156 \]
\[ Q = \frac{156}{0.4} \]
\[ Q = 390 \]

Substitute \( Q = 390 \) back into the market demand equation to find the price \( P \):
\[ P = \frac{780 - Q}{5} \]
\[ P = \frac{780 - 390}{5} \]
\[ P = \frac{390}{5} \]
\[ P = 78 \]

Thus, the quantity and price equilibrium that maximize total revenue are:
\[ Q = 390 \]
\[ P = 78 \]

### Summary:
1. **Market Demand Equation:** \( Q = 780 - 5P \) or \( P = \frac{780 - Q}{5} \)
2. **Total Revenue Equation:** \( TR = \frac{780Q - Q^2}{5} \)
3. **Marginal Revenue Equation:** \( MR = 156 - 0.4Q \)
4. **Equilibrium Quantity and Price:** \( Q = 390 \) and \( P = 78 \)

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Would you like further details or explanations on any part? Here are some related questions that expand on this information:

1. How do changes in the individual demand functions affect the market demand?
2. What happens to the market equilibrium if a new firm enters the market?
3. How is the concept of marginal revenue used in other economic analyses?
4. What are the implications of maximizing total revenue for a firm's profitability?
5. How can firms use the knowledge of market demand to make pricing decisions?
6. What role does elasticity play in determining the optimal price and quantity?
7. How would a tax on storybooks affect the market demand and equilibrium?
8. How can changes in consumer preferences impact the market demand function?

**Tip:** Always verify the correctness of algebraic manipulations, especially when dealing with multiple functions and their summations.

Explore how to solve market demand and revenue equations for storybook firms. Understand the concepts of market demand, total revenue, and marginal revenue with step-by-step explanations and examples. Suitable for undergraduate economics studies.

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