2. A perfectly competitive firm producing lamps has fixed costs of €1000 per week; each lamp
costs €15 to produce and is sold at €35.
(a) Calculate the break-even quantity.
(b) Does the firm make a profit or loss when: (i) 500 lamps; (ii) 1000 lamps are produced and
sold?
(c) Confirm the answers to (a) and (b) graphically

### (a) **Break-even Quantity:**

To find the break-even quantity, we first need to calculate the firm's costs and revenues.

#### Total Cost (TC):
- **Fixed Cost (FC)**: €1000 per week
- **Variable Cost (VC)**: €15 per lamp

The total cost for producing \( Q \) lamps is given by:
\[
TC = FC + VC \times Q
\]
\[
TC = 1000 + 15 \times Q
\]

#### Total Revenue (TR):
- The selling price of each lamp is €35.

The total revenue for selling \( Q \) lamps is:
\[
TR = 35 \times Q
\]

#### Break-even Condition:
The firm breaks even when total revenue equals total cost:
\[
TR = TC
\]
\[
35 \times Q = 1000 + 15 \times Q
\]
Solving for \( Q \):

\[
35Q - 15Q = 1000
\]
\[
20Q = 1000
\]
\[
Q = \frac{1000}{20} = 50
\]

So, the **break-even quantity** is **50 lamps**.

---

### (b) **Profit or Loss at Different Quantities:**

To determine whether the firm makes a profit or a loss at specific quantities, we calculate total revenue and total cost for each case.

#### (i) **500 lamps**:
1. **Total Revenue** \( TR \):
\[
TR = 35 \times 500 = 17,500 \, \text{€}
\]
2. **Total Cost** \( TC \):
\[
TC = 1000 + 15 \times 500 = 1000 + 7,500 = 8,500 \, \text{€}
\]

3. **Profit**:
\[
\text{Profit} = TR - TC = 17,500 - 8,500 = 9,000 \, \text{€}
\]

So, the firm makes a **profit of €9,000** when producing and selling 500 lamps.

---

#### (ii) **1000 lamps**:
1. **Total Revenue** \( TR \):
\[
TR = 35 \times 1000 = 35,000 \, \text{€}
\]
2. **Total Cost** \( TC \):
\[
TC = 1000 + 15 \times 1000 = 1000 + 15,000 = 16,000 \, \text{€}
\]

3. **Profit**:
\[
\text{Profit} = TR - TC = 35,000 - 16,000 = 19,000 \, \text{€}
\]

So, the firm makes a **profit of €19,000** when producing and selling 1000 lamps.

---

### (c) **Graphical Representation:**

To confirm the answers graphically, we can plot both the **Total Cost (TC)** and **Total Revenue (TR)** as functions of the quantity \( Q \).

- The **Total Revenue (TR)** line will start from the origin (because no revenue is generated if no lamps are sold) and will slope upwards with a slope of €35 per lamp.
- The **Total Cost (TC)** curve will start at €1000 (the fixed cost) and slope upwards with a slope of €15 per lamp.

The break-even point is where the two lines intersect, and the distance between the TR and TC curves beyond the break-even point shows the profit.

I can plot this graph for you if you'd like.

---

Let me know if you have questions or want details! Here are 5 related questions to deepen your understanding:

1. How would an increase in fixed costs affect the break-even point?
2. What happens to the profit if the variable cost per lamp increases to €20?
3. How does the break-even quantity change if the selling price drops to €30?
4. What is the impact on profit if the firm sells 750 lamps instead of 500 or 1000?
5. Can a firm remain operational in the long run if it's only breaking even?

**Tip:** In a perfectly competitive market, firms can only make normal profit in the long run due to the entry and exit of firms.

A perfectly competitive firm producing lamps has fixed costs of €1000 per week; each lamp costs €15 to produce and is sold at €35. (a) Calculate the break-even quantity. (b) Does the firm make a profit or loss when: (i) 500 lamps; (ii) 1000 lamps are produced and sold? (c) Confirm the answers to (a) and (b) graphically.

Learn how to calculate the break-even quantity and profit for a perfectly competitive firm producing lamps with fixed costs of €1000 and a variable cost of €15 per lamp. Explore detailed steps for finding the break-even point and analyzing profits for 500 and 1000 lamps sold. Suitable for Grades 11-12.

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