Question

How do I calculate break-even quantities for different machines with given cost functions?

Answer 1

To calculate break-even quantities for different machines with given cost functions, you'll need to follow these steps:

### Step 1: Understand the Cost Function
First, identify the cost function for each machine. A typical cost function can be expressed as:

\[ C(x) = F + V \cdot x \]

where:
- \( C(x) \) is the total cost for producing \( x \) units,
- \( F \) is the fixed cost (costs that do not change with the level of output),
- \( V \) is the variable cost per unit (costs that change with the level of output),
- \( x \) is the quantity of units produced.

### Step 2: Identify Revenue Function
Next, specify the revenue function. The revenue can often be expressed as:

\[ R(x) = P \cdot x \]

where:
- \( R(x) \) is the total revenue from selling \( x \) units,
- \( P \) is the selling price per unit.

### Step 3: Set Up the Break-Even Condition
The break-even point occurs when total revenue equals total cost. Therefore, set the revenue function equal to the cost function:

\[ R(x) = C(x) \]

This leads to:

\[ P \cdot x = F + V \cdot x \]

### Step 4: Solve for the Break-Even Quantity
Rearranging the equation gives:

\[ P \cdot x - V \cdot x = F \]

Factoring out \( x \) gives:

\[ (P - V) \cdot x = F \]

Now, solve for \( x \):

\[ x = \frac{F}{P - V} \]

### Step 5: Calculate for Each Machine
Repeat the above procedure for each machine using its specific cost function and revenue details to calculate the break-even quantity.

### Example
Let's say you have two machines with the following numbers:

1. **Machine A:**
- Fixed cost \( F_A = 1000 \)
- Variable cost \( V_A = 5 \) per unit
- Selling price per unit \( P_A = 12 \)

2. **Machine B:**
- Fixed cost \( F_B = 2000 \)
- Variable cost \( V_B = 10 \) per unit
- Selling price per unit \( P_B = 15 \)

#### For Machine A:
\[ x_A = \frac{F_A}{P_A - V_A} = \frac{1000}{12 - 5} = \frac{1000}{7} \approx 142.86 \]

#### For Machine B:
\[ x_B = \frac{F_B}{P_B - V_B} = \frac{2000}{15 - 10} = \frac{2000}{5} = 400 \]

### Conclusion
Thus, Machine A has a break-even quantity of approximately 143 units, and Machine B has a break-even quantity of 400 units. You can compare these results to determine which machine is more cost-effective at different levels of output, considering the break-even quantities.