(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.

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(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.

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(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.

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(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.

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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.