Questions: a) What is the function, O(q)? b) What is the revenue function, N(y)? c) What is the breakeven point? What does this point mean in terms of cost and the revenue?

a) What is the function, O(q)?
b) What is the revenue function, N(y)?
c) What is the breakeven point? What does this point mean in terms of cost and the revenue?
Transcript text: a) What is the function, $O(q)$? b) What is the revenue function, $N(y)$? c) What is the breakeven point? What does this point mean in terms of cost and the revenue?
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Solution

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To address the questions, let's analyze each part based on the given information:

a) What is the cost function, \( O(t) \)?

The cost function, \( O(q) \), is defined as the total cost of producing a quantity \( q \) of a given good. However, the question asks for \( O(t) \), which seems to be a typographical error or a misunderstanding, as the cost function is typically expressed in terms of the quantity produced, \( q \). Without additional context or information, we assume \( O(t) \) is meant to be \( O(q) \). Therefore, the cost function \( O(q) \) remains as it is, representing the total cost of producing \( q \) units.

b) What is the revenue function, \( \boldsymbol{N}(\mathbf{y}) \)?

The revenue function, typically denoted as \( R(q) \), represents the total revenue generated from selling a quantity \( q \) of goods. The question uses \( \boldsymbol{N}(\mathbf{y}) \), which might be a different notation for the revenue function. Without specific details on how revenue is calculated (such as price per unit), we cannot provide an explicit form for \( \boldsymbol{N}(\mathbf{y}) \). Generally, if the price per unit is \( p \), the revenue function would be \( R(q) = p \times q \).

c) What is the breakeven point? What does this point mean in terms of cost and revenue?

The breakeven point is the quantity of goods produced and sold at which total revenue equals total cost, resulting in zero profit. Mathematically, it is the point where:

\[ R(q) = O(q) \]

At the breakeven point, the profit \( P \) is zero because:

\[ P = R(q) - O(q) = 0 \]

This means that the business is covering all its costs with the revenue generated, but it is not making any profit. The breakeven point is crucial for businesses to understand the minimum sales required to avoid losses.

In summary:

  • The cost function \( O(q) \) represents the total cost of producing \( q \) units.
  • The revenue function \( \boldsymbol{N}(\mathbf{y}) \) is likely another notation for revenue, typically expressed as \( R(q) = p \times q \) if the price per unit is known.
  • The breakeven point is where total revenue equals total cost, resulting in zero profit, indicating the minimum sales needed to avoid losses.
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