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.