Let's analyze and solve each part step-by-step. --- # Part (a): Competitive Market (Price Taker) ### Given: - Cost function: \( c(q) = q^2 \) - Market price: \( P = 60 \) ### Objective: Maximize profit \( \pi(q) \): \[ \pi(q) = \text{Revenue} - \text{Cost} = P \times q - c(q) \] \[ \pi(q) = 60q - q^2 \] ### Step 1: Find the optimal quantity \( q^* \) by setting the derivative of profit to zero: \[ \frac{d\pi}{dq} = 60 - 2q = \] \[ 2q = 60 \] \[ q^* = 30 \] ### **Answer for (a):** **Maximum profit occurs at** \( \boxed{q = 30} \). --- # Part (b): Monopoly with Demand Curve \( p = 100 - q \) ### Given: - Demand curve: \( p = 100 - q \) - Cost function: \( c(q) = q^2 \) ### Objective: Find profit-maximizing price and quantity. ### Step 1: Write revenue \( R(q) \): \[ R(q) = p \times q = (100 - q) \times q = 100q - q^2 \] ### Step 2: Write profit function: \[ \pi(q) = R(q) - c(q) = (100q - q^2) - q^2 = 100q - 2q^2 \] ### Step 3: Maximize profit: \[ \frac{d\pi}{dq} = 100 - 4q = \] \[ 4q = 100 \] \[ q = 25 \] ### Step 4: Find the corresponding price: \[ p = 100 - q = 100 - 25 = 75 \] ### **Answer for (b):** - **Profit-maximizing quantity:** \( \boxed{q = 25} \) - **Profit-maximizing price:** \( \boxed{p = 75} \) --- # Part (c): Selling in Two Markets ### Given: - Domestic market demand: \( p_{dom} = 100 - q_{dom} \) - Foreign market demand: \( p_{foreign} = 60 \) (price taker) - Cost function: \( c(q) = q^2 \) ### Step 1: Domestic market - Revenue: \[ R_{dom} = p_{dom} \times q_{dom} = (100 - q_{dom}) q_{dom} = 100 q_{dom} - q_{dom}^2 \] - Profit: \[ \pi_{dom} = R_{dom} - c(q_{dom}) = 100 q_{dom} - q_{dom}^2 - q_{dom}^2 = 100 q_{dom} - 2 q_{dom}^2 \] - Maximize: \[ \frac{d \pi_{dom}}{dq_{dom}} = 100 - 4 q_{dom} = \Rightarrow q_{dom} = 25 \] - Market price: \[ p_{dom} = 100 - 25 = 75 \] ### Step 2: Foreign market (price taker) - Price: \( p_{foreign} = 60 \) - Revenue: \[ R_{foreign} = 60 \times q_{foreign} \] - Cost: \[ c(q_{foreign}) = q_{foreign}^2 \] - Profit: \[ \pi_{foreign} = 60 q_{foreign} - q_{foreign}^2 \] - Maximize: \[ \frac{d \pi_{foreign}}{dq_{foreign}} = 60 - 2 q_{foreign} = \Rightarrow q_{foreign} = 30 \] ### **Final results:** - **Domestic market:** - Quantity: \( \boxed{25} \) - Price: \( \boxed{75} \) - **Foreign market:** - Quantity: \( \boxed{30} \) - Price: \( \boxed{60} \) --- # **Summary:** | Part | Quantity | Price | |---------|--------------|--------| | (a) Competitive market | 30 | 60 | | (b) Monopoly | 25 | 75 | | (c) Two markets | Domestic: 25 at 75 | Foreign: 30 at 60 | Let me know if you'd like a detailed explanation of any step!

# Detailed Explanation of Steps Here's a breakdown of the calculations and reasoning for each part of the problem. --- ## Part (a): Competitive Market (Price Taker) ### Given: - Cost function: \( c(q) = q^2 \) - Market price: \( P = 60 \) ### Objective: To find the profit-maximizing quantity of production. ### Step-by-Step Breakdown: 1. **Profit Function**: The profit \( \pi(q) \) is defined as: \[ \pi(q) = \text{Revenue} - \text{Cost} \] Here, Revenue is \( P \times q = 60q \) and Cost is \( c(q) = q^2 \). Thus: \[ \pi(q) = 60q - q^2 \] 2. **Maximizing Profit**: To find the maximum profit, we need to take the derivative of the profit function with respect to \( q \) and set it to zero: \[ \frac{d\pi}{dq} = 60 - 2q \] Setting this equal to zero gives: \[ 60 - 2q = 0 \implies 2q = 60 \implies q^* = 30 \] ### Conclusion for (a): **The firm should produce \( \boxed{30} \) units to maximize profit.** --- ## Part (b): Monopoly with Demand Curve \( p = 100 - q \) ### Given: - Demand curve: \( p = 100 - q \) - Cost function: \( c(q) = q^2 \) ### Objective: To find the profit-maximizing price and quantity. ### Step-by-Step Breakdown: 1. **Revenue Function**: The revenue function \( R(q) \) is calculated as: \[ R(q) = p \times q = (100 - q)q = 100q - q^2 \] 2. **Profit Function**: The profit function is: \[ \pi(q) = R(q) - c(q) = (100q - q^2) - q^2 = 100q - 2q^2 \] 3. **Maximizing Profit**: We take the derivative of the profit function: \[ \frac{d\pi}{dq} = 100 - 4q \] Setting this to zero gives: \[ 100 - 4q = 0 \implies 4q = 100 \implies q = 25 \] 4. **Finding the Price**: Substitute \( q = 25 \) back into the demand equation: \[ p = 100 - q = 100 - 25 = 75 \] ### Conclusion for (b): **The profit-maximizing quantity is \( \boxed{25} \) and the price is \( \boxed{75} \).** --- ## Part (c): Selling in Two Markets ### Given: - Domestic market demand: \( p_{dom} = 100 - q_{dom} \) - Foreign market price: \( p_{foreign} = 60 \) (price taker) - Cost function: \( c(q) = q^2 \) ### Objective: To determine how much to sell in each market. ### Step-by-Step Breakdown: 1. **Domestic Market**: - **Revenue Function**: \[ R_{dom} = (100 - q_{dom})q_{dom} = 100q_{dom} - q_{dom}^2 \] - **Profit Function**: \[ \pi_{dom} = R_{dom} - c(q_{dom}) = 100q_{dom} - 2q_{dom}^2 \] - **Maximizing Profit**: \[ \frac{d\pi_{dom}}{dq_{dom}} = 100 - 4q_{dom} \implies 100 - 4q_{dom} = 0 \implies q_{dom} = 25 \] - **Finding the Price**: \[ p_{dom} = 100 - 25 = 75 \] 2. **Foreign Market**: - **Price**: \( p_{foreign} = 60 \) (price taker) - **Revenue Function**: \[ R_{foreign} = 60 \times q_{foreign} \] - **Profit Function**: \[ \pi_{foreign} = 60q_{foreign} - q_{foreign}^2 \] - **Maximizing Profit**: \[ \frac{d\pi_{foreign}}{dq_{foreign}} = 60 - 2q_{foreign} \implies 60 - 2q_{foreign} = 0 \implies q_{foreign} = 30 \] ### Conclusion for (c): - **Domestic market**: \( \boxed{25} \) units at \( \boxed{75} \). - **Foreign market**: \( \boxed{30} \) units at \( \boxed{60} \). --- ## Summary of Answers: - **Part (a)**: Quantity = \( \boxed{30} \) - **Part (b)**: Quantity = \( \boxed{25} \), Price = \( \boxed{75} \) - **Part (c)**: Domestic Quantity = \( \boxed{25} \), Domestic Price = \( \boxed{75} \); Foreign Quantity = \( \boxed{30} \), Foreign Price = \( \boxed{60} \) Let me know if you need further clarification on any specific step!