Please show tables!
The Case: Jing Tsong of Bayside Laundry has been providing cleaning and linen service for rental condominiums on the Gulf Coast for over
1
0
years. Currently, Jing is servicing

Question

Please show tables!
The Case: Jing Tsong of Bayside Laundry has been providing cleaning and linen service for rental condominiums on the Gulf Coast for over 
1
0
 ﻿years. Currently, Jing is servicing 
2
6
 ﻿condominium developments. Jings two major competitors are Cleanco, which currently services 
1
5
 ﻿condominium developments, and Beach Services, which performs laundry and cleaning services for 
1
1
 ﻿condominium developments.Recently, Jing contacted Bay Bank about a loan to expand business operations. To justify the loan, Jing has kept detailed records of customers as well as the customers that Bayside received from the two major competitors. During the past year, Jing was able to keep 
1
8
 ﻿of the original 
2
6
 ﻿customers. During the same period, Jing was able to get 
1
 ﻿new customer from Cleanco and 
2
 ﻿new customers from Beach Services. Unfortunately, Jing lost 
6
 ﻿of the original customers to Cleanco and 
2
 ﻿of the original customers to Beach Services during the same year. Jing has learned that Cleanco has kept 
8
0
%
 ﻿of its current customers. Jing also knows that Beach Services will keep at least 
5
0
%
 ﻿of its customers. To get the loan from Bay Bank, Jing needs

VIPSolutions · Accepted Answer

Let's break down and organize the information using tables, and then analyze the equilibrium market shares using **Markov (transition) analysis**.

---

## **Step 1: Organize the Data**

We have 3 competitors:

| Company         | Initial Customers |
|-----------------|------------------|
| Bayside Laundry | 26               |
| Cleanco         | 15               |
| Beach Services  | 11               |
| **Total**       | **52**           |

---

### **Customer Movements (Last Year)**

#### **Bayside Laundry:**
- Retained: **18** out of **26**
- Lost to Cleanco: **6**
- Lost to Beach: **2**
- Gained from Cleanco: **1**
- Gained from Beach: **2**

#### **Cleanco** (given):
- Retained: **80% of 15 = 12**
- Lost: **3**
    - Lost to Bayside: **1** (from above)
    - Lost to Beach: **2** (must deduce)

#### **Beach Services** (given):
- Retained: **at least 50% of 11 = 5.5** (but must be integer, so at least 6)
- Lost: **at most 5**
    - Lost to Bayside: **2**
    - Lost to Cleanco: **?**
    - The rest must be deduced.

---

## **Step 2: Construct the Transition Table**

Let’s denote:
- **B:** Bayside Laundry
- **C:** Cleanco
- **S:** Beach Services

Let’s fill out the transition table based on the customer movements. Each row is "from," each column is "to."

### **Transition Table: Last Year**

| From/To | Bayside (B) | Cleanco (C) | Beach (S) | Total from |
|---------|-------------|-------------|-----------|------------|
| **B**   | 18          | 6           | 2         | 26         |
| **C**   | 1           | 12          | 2         | 15         |
| **S**   | 2           | ?           | ?         | 11         |

Let’s fill in the missing entries:

- For **Beach Services**:
  - Retained: At least 6 (let's use 6 to be conservative).
  - Lost to Bayside: 2 (given)
  - Lost to Cleanco: The rest. So, 11 - (2+6) = **3** lost to Cleanco.

Thus:

| From/To | Bayside (B) | Cleanco (C) | Beach (S) | Total from |
|---------|-------------|-------------|-----------|------------|
| **B**   | 18          | 6           | 2         | 26         |
| **C**   | 1           | 12          | 2         | 15         |
| **S**   | 2           | 3           | 6         | 11         |

---

## **Step 3: Compute the Transition Probability Matrix**

To get probabilities, divide each entry by the row total:

| From/To | Bayside (B)          | Cleanco (C)         | Beach (S)         |
|---------|----------------------|---------------------|-------------------|
| **B**   | 18/26 ≈ 0.692        | 6/26 ≈ 0.231        | 2/26 ≈ 0.077      |
| **C**   | 1/15 ≈ 0.067         | 12/15 = 0.800       | 2/15 ≈ 0.133      |
| **S**   | 2/11 ≈ 0.182         | 3/11 ≈ 0.273        | 6/11 ≈ 0.545      |

---

### **Transition Probability Matrix (P):**

|         | To B     | To C    | To S    |
|---------|----------|---------|---------|
| **B**   | 0.692    | 0.231   | 0.077   |
| **C**   | 0.067    | 0.800   | 0.133   |
| **S**   | 0.182    | 0.273   | 0.545   |

---

## **Step 4: Find the Long-Run (Equilibrium) Market Shares**

Let the steady-state vector be **[x, y, z]** for Bayside, Cleanco, and Beach, respectively. In equilibrium:

$$
[x\ y\ z] \cdot P = [x\ y\ z]
$$
And,
$$
x + y + z = 1
$$

So, we solve the following system:
$$
x = 0.692x + 0.067y + 0.182z
$$
$$
y = 0.231x + 0.800y + 0.273z
$$
$$
z = 0.077x + 0.133y + 0.545z
$$
$$
x + y + z = 1
$$

Subtract left side from right:

$$
x - (0.692x + 0.067y + 0.182z) = 0 \implies 0.308x - 0.067y - 0.182z = 0
$$
$$
y - (0.231x + 0.800y + 0.273z) = 0 \implies -0.231x + 0.200y - 0.273z = 0
$$
$$
z - (0.077x + 0.133y + 0.545z) = 0 \implies -0.077x - 0.133y + 0.455z = 0
$$
$$
x + y + z = 1
$$

Let’s solve this system.

---

### **Solving for Steady-State Market Shares**

#### **Equation 1:** $0.308x - 0.067y - 0.182z = 0$  
#### **Equation 2:** $-0.231x + 0.200y - 0.273z = 0$  
#### **Equation 3:** $-0.077x - 0.133y + 0.455z = 0$  
#### **Equation 4:** $x + y + z = 1$

Let’s solve step by step.

---

#### **Step A: Express x in terms of y and z (from Eq 1):**

$0.308x = 0.067y + 0.182z$  
$x = (0.067/0.308)y + (0.182/0.308)z$  
$x ≈ 0.217y + 0.591z$

---

#### **Step B: Express y in terms of x and z (from Eq 2):**

$-0.231x + 0.200y - 0.273z = 0$  
$0.200y = 0.231x + 0.273z$  
$y = (0.231/0.200)x + (0.273/0.200)z$  
$y ≈ 1.155x + 1.365z$

---

#### **Step C: Express z in terms of x and y (from Eq 3):**

$-0.077x - 0.133y + 0.455z = 0$  
$0.455z = 0.077x + 0.133y$  
$z = (0.077/0.455)x + (0.133/0.455)y$  
$z ≈ 0.169x + 0.292y$

---

#### **Step D: Use Eq 4 (x + y + z = 1)**

Now substitute the expressions above to get everything in terms of one variable.

But it's easier to use a matrix or some algebra. Let's use Eq 1 and Eq 2 to express x and y in terms of z:

From Step A:  
$x = 0.217y + 0.591z$  
From Step B:  
$y = 1.155x + 1.365z$

Substitute x from Eq 1 into Eq 2:

$y = 1.155(0.217y + 0.591z) + 1.365z$  
$y = 0.251y + 0.683z + 1.365z$  
\(y - 0.251y = 0.683z +

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