Consider a monopolist facing a demand curve given by P = 20 – q, where P is the market price and q is the quantity sold. The monopolist's marginal costs are MC = 2 per unit and a fixed cost of $20. What is the monopolist's profit it is charges a uniform price?
To calculate the monopolist's profit when it charges a uniform price, we need to find the monopolist's optimal quantity and the corresponding price. We can then calculate the total cost and total revenue to determine the profit.
1. Determine the monopolist's optimal quantity:
The monopolist maximizes its profit by setting marginal cost equal to marginal revenue. Since the marginal cost (MC) is given as $2 per unit, we need to find the marginal revenue (MR) by taking the derivative of the demand function with respect to quantity:
MR = d(P*q) / dq = d(20q - q^2) / dq = 20 - 2q.
Setting MR = MC, we get:
20 - 2q = 2.
Solving for q:
2q = 20 - 2,
2q = 18,
q = 18 / 2,
q = 9.
So, the monopolist's optimal quantity is q = 9 units.
2. Determine the corresponding price:
To find the price, we substitute the quantity (q = 9) into the demand function:
P = 20 - q = 20 - 9 = 11.
So, the monopolist's price is P = $11 per unit.
3. Calculate the total cost:
The total cost is the sum of the fixed cost and the variable cost. The variable cost is the product of the marginal cost (MC) and the quantity (q):
Variable cost = MC * q = 2 * 9 = $18.
Total cost = fixed cost + variable cost = $20 + $18 = $38.
4. Calculate the total revenue:
Total revenue is the product of the price (P) and the quantity (q):
Total revenue = P * q = $11 * 9 = $99.
5. Calculate the profit:
Profit is the difference between total revenue and total cost:
Profit = Total revenue - Total cost = $99 - $38 = $61.
Therefore, the monopolist's profit when it charges a uniform price is $61.
To find the monopolist's profit when it charges a uniform price, we need to determine the optimal quantity and price that maximizes its profit. Here's the step-by-step explanation of how to calculate it:
1. Start by calculating the monopolist's total revenue:
Revenue = Price * Quantity
We need to express price in terms of quantity, so rewrite the demand function as q = 20 - P. Then substitute this into the revenue equation:
Revenue = P * (20 - P)
2. Calculate the monopolist's total cost, which consists of both the variable costs (marginal cost per unit) and the fixed cost:
Total Cost = Variable Cost + Fixed Cost
Given that the marginal cost (MC) is $2 per unit and the fixed cost is $20, the total cost is:
Total Cost = 2q + 20
3. Determine the monopolist's profit by subtracting total cost from revenue:
Profit = Revenue - Total Cost
Let's substitute the revenue and total cost equations:
Profit = P * (20 - P) - (2q + 20)
4. Simplify the profit equation by expanding and rearranging terms:
Profit = 20P - P^2 - 2q - 20
5. To find the monopolist's optimal quantity and price, we need to find the derivative of the profit function with respect to quantity (q) and set it equal to zero. This will give us the monopolist's profit-maximizing quantity:
dProfit/dq = -2 - 0 = -2
6. Set the derivative equal to zero:
-2 = 0
7. Since -2 is never equal to zero, we conclude that there is no profit-maximizing quantity. In this case, the monopolist will aim to produce and sell the quantity that maximizes its revenue, which occurs when the marginal revenue (MR) is equal to zero.
To further analyze the problem, we would need additional information about the marginal revenue or further constraints on the monopolist's behavior.