Let's break down the problem by calculating the bakery's profits at both price points to understand which condition must hold for profits to increase.

Case 1: Selling 800 cupcakes at $3 each

Revenue = $800 \times 3 = 2400$

Let the total cost of producing 800 cupcakes be $C(800)$.

Profit at this price: $\text{Profit} = \text{Revenue} - \text{Total Cost} = 2400 - C(800)$

Case 2: Selling 1,000 cupcakes at $2.50 each

Revenue = $1000 \times 2.5 = 2500$

Let the total cost of producing 1,000 cupcakes be $C(1000)$.

Profit at this price: $\text{Profit} = 2500 - C(1000)$

For profits to increase by switching to a price of $2.50 per cupcake, the bakery's profit at this price must be higher than their current profit at $3 per cupcake. In other words, the profit from selling 1,000 cupcakes at $2.50 each should exceed the profit from selling 800 cupcakes at $3.

Now, let's analyze the conditions:

A. Their current total costs from making cupcakes are less than $2,500.

• If the bakery’s total costs for 1,000 cupcakes are below $2,500, their profit from selling 1,000 cupcakes would be positive. However, this doesn't directly indicate profit maximization compared to selling 800 cupcakes at $3.

B. The marginal costs of cupcakes are less than $0.50.

• Marginal cost is the additional cost of producing one more unit. Since the bakery increases production by 200 cupcakes (from 800 to 1,000), the marginal cost of those additional cupcakes plays a crucial role.
• The additional revenue from these extra 200 cupcakes is $200 \times 2.5 = 500$. To increase profits, the total cost of producing the extra 200 cupcakes must be less than this additional revenue. So, the marginal cost per cupcake must be less than: $\frac{500}{200} = 2.50$ Hence, B is correct because the marginal costs must be less than $0.50 to justify increasing production to 1,000 cupcakes at a lower price, ensuring higher overall profit.

C. The marginal costs of cupcakes are less than $2.50.

• If the marginal cost were exactly $2.50, the bakery would break even on the additional cupcakes. To increase profit, the marginal cost needs to be lower than $2.50, making C true but not specific enough for maximizing profit.

D. The average costs of cupcakes are less than $3.00.

• While true, this doesn’t directly indicate why profits increase with more sales at a lower price, as average cost doesn’t provide insight into the profit from the additional 200 cupcakes.

Thus, the correct condition is:

Answer: B. The marginal costs of cupcakes are less than $0.50.

This ensures that the cost of producing the additional cupcakes is low enough for the bakery to increase its profits by switching to the lower price and higher sales volume.

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Do you need any further details or clarification?

Here are 5 questions that expand on this topic:

1. What is the difference between marginal cost and average cost in production decisions?
2. How does increasing production quantity affect total costs and profits?
3. What happens if the marginal cost of production exceeds the price per unit?
4. How does the elasticity of demand impact pricing strategies in a bakery?
5. How can economies of scale benefit a bakery when increasing production?

Tip: Understanding marginal cost helps businesses determine the optimal level of production and pricing to maximize profit.