At&t is the dominant firm in the local telecommunication industry, which has a total market demand given by Q
=

1
0
0

-

2
P
.
AT&T has competition from a fringe of four small firms that produ

Question

At&t is the dominant firm in the local telecommunication industry, which has a total market demand given by Q 
=
 
1
0
0
 
-
 
2
P
.
 ﻿AT&T has competition from a fringe of four small firms that produce where their individual marginal costs equal the market price. The fringe firms each have total costs given by TCi 
=
 
1
0
 ﻿Qi 
+
 ﻿Qi
^
2
.
 ﻿If AT&T
'
s total costs are given by TCa 
=
 
1
0
 
+
 
1
0
 ﻿Qa
,
 ﻿how much does the industry as a whole produce at this price?
6
0
 ﻿Units

VIPSolutions · Accepted Answer

Let's break the problem down step by step.

### **Given Data**

- **Market Demand:** $ Q = 100 - 2P $
- **AT&T (Dominant firm):**
  - Total Cost: $ TC_A = 10 + 10Q_A $
  - Marginal Cost: $ MC_A = \frac{d(TC_A)}{dQ_A} = 10 $
- **Fringe (4 small firms):**
  - Each firm's total cost: $ TC_i = 10Q_i + Q_i^2 $
  - Each firm's marginal cost: $ MC_i = \frac{d(TC_i)}{dQ_i} = 10 + 2Q_i $

---

### **Step 1: Find Fringe Supply at Price $ P $**

Each fringe firm sets $ MC_i = P $:

$$
10 + 2Q_i = P \implies Q_i = \frac{P - 10}{2}
$$

With 4 firms:

$$
Q_f = 4Q_i = 4 \left( \frac{P - 10}{2} \right) = 2(P - 10)
$$

---

### **Step 2: Residual Demand for AT&T**

$$
Q = 100 - 2P \implies Q_A = Q - Q_f = 100 - 2P - 2(P - 10)
$$
$$
Q_A = 100 - 2P - 2P + 20 = 120 - 4P
$$
So, the residual demand for AT&T is:
$$
Q_A = 120 - 4P
$$

---

### **Step 3: AT&T's Profit Maximization**

AT&T faces residual demand $ Q_A = 120 - 4P $. Inverting:
$$
Q_A = 120 - 4P \implies 4P = 120 - Q_A \implies P = 30 - \frac{Q_A}{4}
$$

Profit:
$$
\pi_A = P Q_A - TC_A = P Q_A - (10 + 10Q_A)
$$
$$
= (30 - \frac{Q_A}{4})Q_A - 10 - 10Q_A
$$
$$
= 30Q_A - \frac{Q_A^2}{4} - 10Q_A - 10
$$
$$
= 20Q_A - \frac{Q_A^2}{4} - 10
$$

Maximize with respect to $ Q_A $:

$$
\frac{d\pi_A}{dQ_A} = 20 - \frac{Q_A}{2} = 0 \implies Q_A = 40
$$

So, **AT&T produces $ Q_A = 40 $**.

---

### **Step 4: Find Market Price**

Plug $ Q_A = 40 $ into the residual demand:

$$
Q_A = 120 - 4P
$$
$$
40 = 120 - 4P \implies 4P = 80 \implies P = 20
$$

---

### **Step 5: Fringe Output at $ P = 20 $**

Each fringe firm:

$$
Q_i = \frac{P - 10}{2} = \frac{20 - 10}{2} = 5
$$

Total for all 4 fringe firms:

$$
Q_f = 4 \times 5 = 20
$$

---

### **Step 6: **Total Industry Output**

$$
Q = Q_A + Q_f = 40 + 20 = 60
$$

---

## **Final Answer**

**The industry as a whole produces 60 units at this price.**

This matches your answer. Well done!

VIPSolutions · Answer

1. Introduction:

Understanding the interaction between a dominant firm and fringe competitors in a market requires familiarity with concepts of market demand, marginal cost, residual demand, and profit maximization strategies. Market demand describes the total quantity consumers are willing to purchase at various prices, often represented by a demand curve. The dominant firm, which holds a significant market share, typically considers how its production influences market price and how it can maximize profits given the behavior of smaller, price-taking fringe firms.

Fringe firms are considered small or competitive firms that produce where their marginal costs equal the market price, effectively acting as price takers. Their individual and collective supply responses influence the residual demand faced by the dominant firm. The residual demand is the portion of the total market demand remaining after accounting for the supply provided by fringe firms. To determine total industry output at a given market price, it is essential to analyze the fringe firms’ supply behavior, the dominant firm’s profit-maximizing output, and how these combine to satisfy the overall market demand.

This approach involves calculating the fringe supply at various prices, establishing the residual demand for the dominant firm, and determining the equilibrium outputs and prices in the market. Understanding these core concepts provides a foundation for analyzing the complex interaction between the dominant firm and fringe competitors, ultimately enabling the computation of total industry production at equilibrium.

**Explanation:**  
The introduction establishes the fundamental concepts necessary to analyze a mixed market structure involving a dominant firm and fringe competitors. It clarifies how market demand, marginal costs, residual demand, and profit maximization strategies interrelate in determining total industry output. Recognizing the roles of these concepts is vital for understanding the step-by-step solution process that follows, ensuring a comprehensive grasp of the problem setting.

2. Presentation of Relevant Formulas Required To Solve The Question:

a) **Market Demand Equation:**  
$$ Q = 100 - 2P $$  
This formula expresses the relationship between the total quantity demanded $Q$ and the market price $P$. It allows for the calculation of total market quantity at any given price.

b) **Fringe Firms' Supply Condition:**  
$$ MC_i = P \implies 10 + 2Q_i = P $$  
This formula states that each fringe firm's marginal cost equals the market price, which determines their individual supply quantity $Q_i$.

c) **Total Fringe Supply:**  
$$ Q_f = n \times Q_i $$  
Where $n$ is the number of fringe firms; here, $n=4$. This calculates the total supply from all fringe firms at a given price.

d) **Residual Demand for the Dominant Firm:**  
$$ Q_A = Q - Q_f $$  
This determines the quantity the dominant firm faces after subtracting fringe supply from total demand.

e) **Inverse Residual Demand for the Dominant Firm:**  
$$ P = a - b Q_A $$  
Derived from the demand equation, expressed in terms of $Q_A$, to facilitate profit maximization.

f) **Profit Maximization Condition for the Dominant Firm:**  
$$ \frac{d\pi_A}{dQ_A} = 0 $$  
Where profit $\pi_A = P Q_A - TC_A$. This condition finds the optimal output for the dominant firm.

g) **Cost and Marginal Cost for the Dominant Firm:**  
$$ TC_A = 10 + 10 Q_A \implies MC_A = 10 $$  
Since the marginal cost is constant, the dominant firm’s supply decision is based on residual demand and profit maximization.

---

**Explanation:**  
These formulas are crucial for translating the market environment into mathematical relationships that enable calculation of equilibrium outputs and prices. The demand equation links quantity and price directly, while the marginal cost and supply conditions guide the fringe firm's behavior. The residual demand framework isolates the dominant firm's decision-making environment, and the profit maximization condition ensures that the dominant firm’s output aligns with profit maximization principles under the given costs and residual demand.

3. A Detailed Step-by-Step Solution:

**Step 1: Determine Fringe Firms’ Supply at a Given Price $ P $**

- Each fringe firm sets marginal cost equal to the price:
  
  $$ MC_i = 10 + 2Q_i = P \implies Q_i = \frac{P - 10}{2} $$
  
- Since there are 4 fringe firms, total fringe supply:

$$ Q_f = 4 \times Q_i = 4 \times \frac{P - 10}{2} = 2(P - 10) $$
  
**Explanation:**  
This step calculates individual and total fringe supply based on the assumption that fringe firms produce where their marginal costs equal the market price, reflecting competitive behavior.

---

**Step 2: Express the Residual Demand for the Dominant Firm**

- Total market demand:

$$ Q = 100 - 2P $$
  
- Residual demand faced by AT&T:

$$ Q_A = Q - Q_f = 100 - 2P - 2(P - 10) $$
  $$ Q_A = 100 - 2P - 2P + 20 = 120 - 4P $$
  
- Inverted to

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