Question

You run a nail salon. Fixed monthly cost is $5,800.00 for rent and utilities, $5,909.00 is spent in salaries and $1,619.00 in insurance. Also every customer requires approximately $5.00 in supplies. You charge $63.00 on average for each service.
You are considering moving the salon to an upscale neighborhood where the rent and utilities will increase to $10,864.00, salaries to $6,428.00 and insurance to $2,383.00 per month. Cost of supplies will increase to $6.00 per service. However you can now charge $156.00 per service. At what point will you be indifferent between your current location and the new location?

Answer 1

To determine the point at which you will be indifferent between your current location and the new upscale location, we need to analyze both scenarios in terms of profit.

### Current Location:
- **Fixed Costs:**
- Rent and utilities: $5,800.00
- Salaries: $5,909.00
- Insurance: $1,619.00
- **Variable Costs:**
- Supplies per service: $5.00
- **Revenue per service:** $63.00

**Total Fixed Costs (Current Location) = Rent + Salaries + Insurance**
\[
\text{Total Fixed Costs} = 5,800 + 5,909 + 1,619 = 13,328
\]

**Total Cost per Service (Current Location):**
\[
\text{Total Cost per Service} = \text{Supply Cost} = 5.00
\]

**Profit per Service (Current Location):**
\[
\text{Profit per Service} = \text{Revenue} - \text{Total Cost per Service} = 63 - 5 = 58
\]

- To find the total profit for a given number of customers \(N\):
\[
\text{Total Profit (Current Location)} = (58 \cdot N) - 13,328
\]

### New Location:
- **Fixed Costs:**
- Rent and utilities: $10,864.00
- Salaries: $6,428.00
- Insurance: $2,383.00
- **Variable Costs:**
- Supplies per service: $6.00
- **Revenue per service:** $156.00

**Total Fixed Costs (New Location) = Rent + Salaries + Insurance**
\[
\text{Total Fixed Costs} = 10,864 + 6,428 + 2,383 = 19,675
\]

**Total Cost per Service (New Location):**
\[
\text{Total Cost per Service} = \text{Supply Cost} = 6.00
\]

**Profit per Service (New Location):**
\[
\text{Profit per Service} = \text{Revenue} - \text{Total Cost per Service} = 156 - 6 = 150
\]

- To find the total profit for a given number of customers \(N\):
\[
\text{Total Profit (New Location)} = (150 \cdot N) - 19,675
\]

### Finding the Indifference Point:
To find the indifference point, we need to set the total profits of both locations equal to each other:

\[
(58 \cdot N) - 13,328 = (150 \cdot N) - 19,675
\]

Now, solve for \(N\):

1. Rearranging the equation:
\[
58N - 150N = -19,675 + 13,328
\]
\[
-92N = -6,347
\]

2. Dividing by -92:
\[
N = \frac{6,347}{92} \approx 69
\]

### Conclusion:
You will be indifferent between your current location and the new upscale location at approximately **69 customers per month**. If you expect to serve more than this amount, the new location becomes more profitable; if fewer, the current location is preferable.