
Question 1 (12 marks). Gamerix is a monopolist in the industry of video game consoles. Its market demand is given by
P = 10 − Q,
where P and Q are, respectively, the market price and quantity of consoles. The total cost of producing Q units is given by the cost function C(Q) = 4Q. Gamerix practises simple monopoly pricing.
(a) Find the Marginal Revenue function and the profit-maximising quantity. (2 marks)

To solve this, we need to follow a few steps to find the Marginal Revenue (MR) function and the profit-maximizing quantity for Gamerix. Let's break it down.

### Step 1: Demand function
The demand function is given as:
\[
P = 10 - Q
\]
This represents the relationship between price \(P\) and quantity \(Q\).

### Step 2: Total Revenue (TR)
The total revenue is given by the product of price \(P\) and quantity \(Q\):
\[
TR = P \times Q = (10 - Q) \times Q
\]
\[
TR = 10Q - Q^2
\]
This is the total revenue function.

### Step 3: Marginal Revenue (MR)
Marginal Revenue is the derivative of Total Revenue with respect to \(Q\):
\[
MR = \frac{d(TR)}{dQ}
\]
Differentiate the total revenue function:
\[
MR = \frac{d}{dQ}(10Q - Q^2) = 10 - 2Q
\]

### Step 4: Marginal Cost (MC)
The total cost function is given as \(C(Q) = 4Q\). The Marginal Cost (MC) is the derivative of the total cost with respect to \(Q\):
\[
MC = \frac{dC(Q)}{dQ} = 4
\]

### Step 5: Profit Maximization
A profit-maximizing monopolist produces where Marginal Revenue equals Marginal Cost (MR = MC):
\[
10 - 2Q = 4
\]
Solve for \(Q\):
\[
10 - 4 = 2Q
\]
\[
Q = 3
\]

### Final Answer:
- **Marginal Revenue function**: \(MR = 10 - 2Q\)
- **Profit-maximizing quantity**: \(Q = 3\)

Would you like a detailed explanation of how we derived the profit-maximizing condition?

---

Here are 5 related questions:
1. How do we calculate the price Gamerix charges at the profit-maximizing quantity?
2. What is the total revenue Gamerix earns at the profit-maximizing quantity?
3. How does Gamerix's cost function influence its pricing strategy?
4. What is the relationship between marginal cost and total cost?
5. How would a shift in demand affect the profit-maximizing quantity?

**Tip**: In monopoly pricing, the marginal revenue curve always lies below the demand curve because the monopolist must lower the price to sell more units.

Gamerix is a monopolist in the industry of video game consoles. Its market demand is given by P = 10 − Q, where P and Q are, respectively, the market price and quantity of consoles. The total cost of producing Q units is given by the cost function C(Q) = 4Q. Gamerix practises simple monopoly pricing. (a) Find the Marginal Revenue function and the profit-maximising quantity.

Learn how to find the marginal revenue function and profit-maximizing quantity for Gamerix, a monopolist in the video game console market. This problem explores key economic concepts such as demand, total revenue, and the profit-maximizing condition MR = MC. Suitable for university-level students studying economics or business mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation