Few companies take the time to estimate the value of a good customer (and often spend little effort to keep one) Suppose that a customer at a restaurant spends, on average R per visit and comes F times each year (for example, if a customer purchases once every two years, then F= =05 The restaurant realizes a gross profit margin of M (expressed as a fraction) on the average bill for fod and drinks In addition, the fraction of customers defecting (not returning) each year is D 2 Complete parts a and b a. Develop a mathematical model to compute V, the gross profit during a customer's lifetime in doing business with the restaurant (this is often called the economic value of a customer) V= (rF•m) • (Simplify your answer) b. If the average purchase per visit is $40, the gross profit margin is 03 (that is, 30%) customers VIsit an average of five times each year and 40% of customers defect each year what is the economic value of the customer? The economic value of the customer is $ (Round to the nearest dollar as needed)

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**Estimating the Economic Value of a Customer for a Restaurant**

Many companies strive to estimate the value of a loyal customer, though they might not always invest efforts in retaining them. Let's consider a scenario where a customer at a restaurant spends, on average, \( R \) dollars per visit and visits \( F \) times each year. For example, if a customer makes a purchase once every two years, then \( F = \frac{1}{2} = 0.5 \). The restaurant realizes a gross profit margin of \( M \) (expressed as a fraction) on the average bill for food and drinks. Additionally, a fraction \( D \) of customers do not return each year, known as the defection rate.

**Tasks:**

**a. Develop a Mathematical Model:**

To compute \( V \), the gross profit during a customer’s lifetime spent with the restaurant (often referred to as the economic value of a customer), we use the formula:

\[
V = \left(\frac{r \cdot F \cdot m}{d}\right)
\]

- **Simplify your answer.**

**b. Calculate the Economic Value:**

Given:
- The average purchase per visit is \( R = \$40 \).
- The gross profit margin is \( M = 0.3 \) (i.e., 30%).
- Customers visit on average \( F = 5 \) times each year.
- \( 40\% \) of customers defect each year, thus \( D = 0.4 \).

**Question:**
What is the economic value of the customer?

Be sure to round to the nearest dollar as needed.
Transcribed Image Text:**Estimating the Economic Value of a Customer for a Restaurant** Many companies strive to estimate the value of a loyal customer, though they might not always invest efforts in retaining them. Let's consider a scenario where a customer at a restaurant spends, on average, \( R \) dollars per visit and visits \( F \) times each year. For example, if a customer makes a purchase once every two years, then \( F = \frac{1}{2} = 0.5 \). The restaurant realizes a gross profit margin of \( M \) (expressed as a fraction) on the average bill for food and drinks. Additionally, a fraction \( D \) of customers do not return each year, known as the defection rate. **Tasks:** **a. Develop a Mathematical Model:** To compute \( V \), the gross profit during a customer’s lifetime spent with the restaurant (often referred to as the economic value of a customer), we use the formula: \[ V = \left(\frac{r \cdot F \cdot m}{d}\right) \] - **Simplify your answer.** **b. Calculate the Economic Value:** Given: - The average purchase per visit is \( R = \$40 \). - The gross profit margin is \( M = 0.3 \) (i.e., 30%). - Customers visit on average \( F = 5 \) times each year. - \( 40\% \) of customers defect each year, thus \( D = 0.4 \). **Question:** What is the economic value of the customer? Be sure to round to the nearest dollar as needed.
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