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.

(d) An advertising campaign can change consumers preferences so that the inverse demand curve
becomes

P = 10 − 1/2Q.
The cost of this campaign is 7. Should the monopoly run the advertising campaign? (4 marks)

We need to determine whether the advertising campaign will increase the monopoly's profit, taking into account both the change in demand and the cost of the campaign.

### Step 1: Profit Without the Advertising Campaign
The original demand curve is \( P = 10 - Q \). The monopoly maximizes profit where marginal revenue (MR) equals marginal cost (MC). 

First, derive the total revenue \( TR(Q) \):
\[
TR(Q) = P \cdot Q = (10 - Q)Q = 10Q - Q^2
\]
Now, calculate the marginal revenue (MR) by taking the derivative of \( TR(Q) \):
\[
MR = \frac{d(TR)}{dQ} = 10 - 2Q
\]
The marginal cost (MC) is constant and equal to the derivative of \( C(Q) = 4Q \), which is:
\[
MC = \frac{dC(Q)}{dQ} = 4
\]
Set \( MR = MC \) to find the profit-maximizing quantity:
\[
10 - 2Q = 4 \quad \Rightarrow \quad 2Q = 6 \quad \Rightarrow \quad Q = 3
\]
Substitute \( Q = 3 \) back into the demand curve to find the price:
\[
P = 10 - 3 = 7
\]
The profit without the advertising campaign is the total revenue minus the total cost:
\[
\text{Profit} = TR - TC = (P \cdot Q) - C(Q) = (7 \cdot 3) - (4 \cdot 3) = 21 - 12 = 9
\]

### Step 2: Profit With the Advertising Campaign
The new demand curve is \( P = 10 - \frac{1}{2}Q \).

The total revenue with the new demand curve is:
\[
TR(Q) = P \cdot Q = \left(10 - \frac{1}{2}Q\right) Q = 10Q - \frac{1}{2}Q^2
\]
Now, calculate the marginal revenue (MR):
\[
MR = \frac{d(TR)}{dQ} = 10 - Q
\]
The marginal cost remains the same, \( MC = 4 \). Set \( MR = MC \) to find the new profit-maximizing quantity:
\[
10 - Q = 4 \quad \Rightarrow \quad Q = 6
\]
Substitute \( Q = 6 \) back into the new demand curve to find the price:
\[
P = 10 - \frac{1}{2} \cdot 6 = 10 - 3 = 7
\]
The profit with the advertising campaign is:
\[
\text{Profit} = TR - TC - \text{Advertising Cost} = (7 \cdot 6) - (4 \cdot 6) - 7 = 42 - 24 - 7 = 11
\]

### Step 3: Decision
The profit without the advertising campaign is 9, and the profit with the advertising campaign is 11. Since 11 > 9, the monopoly should run the advertising campaign because it increases the profit by 2 units.

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Would you like more details on any specific part of this? Here are five related questions you might find interesting:

1. How would a change in the cost structure (e.g., increasing marginal costs) affect the decision?
2. What happens if the monopoly faces a price cap, limiting the price it can charge?
3. How would the analysis change if the cost of the campaign were higher than 7?
4. What role does elasticity of demand play in the decision to advertise?
5. How would price discrimination alter the monopolist's decision?

**Tip**: In profit maximization problems, always remember to compare the marginal revenue to the marginal cost for optimal output determination.

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. An advertising campaign can change consumer preferences so that the inverse demand curve becomes P = 10 − 1/2Q. The cost of this campaign is 7. Should the monopoly run the advertising campaign?

Analyze whether Gamerix, a video game console monopolist, should run an advertising campaign based on profit maximization. The problem covers key concepts like monopoly pricing, marginal revenue, marginal cost, and demand curve shifts. Suitable for undergraduate economics or advanced high school students.

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