A single price setting monopolist faces the demand : P = 4000-5Q, TC = 0 + 400Q. For the single price-setting monopolist, tell me profit maximizing quantity, price,total revenue, total cost, profit and consumer surplus.
To find the profit-maximizing quantity, price, total revenue, total cost, profit, and consumer surplus for a single price-setting monopolist, we need to analyze the given information:
Demand Function: P = 4000 - 5Q
Total Cost Function: TC = 0 + 400Q (where the fixed cost is 0, and the variable cost per unit is 400)
1. Profit-Maximizing Quantity:
To determine the profit-maximizing quantity, we need to find the quantity at which marginal revenue (MR) equals marginal cost (MC).
MR is the derivative of the total revenue (TR) function with respect to quantity (Q). From the demand function, we can find TR by multiplying the price (P) with the quantity (Q):
TR = P * Q = (4000 - 5Q) * Q = 4000Q - 5Q^2
Using calculus, we differentiate TR with respect to Q to find MR:
MR = d(TR)/d(Q) = 4000 - 10Q
MC is the derivative of the total cost (TC) function with respect to quantity (Q):
MC = d(TC)/d(Q) = 400
Setting MR equal to MC:
4000 - 10Q = 400
Solving for Q:
10Q = 3600
Q = 360
So, the profit-maximizing quantity is 360 units.
2. Price:
To find the price, substitute the profit-maximizing quantity (Q = 360) into the demand function:
P = 4000 - 5Q = 4000 - 5(360) = 4000 - 1800 = 2200
So, the price at the profit-maximizing quantity is 2200 dollars.
3. Total Revenue:
To find total revenue, multiply the price (P) with the quantity (Q):
TR = P * Q = 2200 * 360 = 792,000
So, the total revenue is $792,000.
4. Total Cost:
Use the total cost function to find the total cost (TC) at the profit-maximizing quantity:
TC = 0 + 400Q = 0 + 400(360) = 0 + 144,000 = 144,000
So, the total cost is $144,000.
5. Profit:
To find the profit, subtract the total cost from the total revenue:
Profit = TR - TC = 792,000 - 144,000 = 648,000
So, the profit is $648,000.
6. Consumer Surplus:
Consumer surplus is the difference between the value that consumers are willing to pay and the price they actually pay. To find consumer surplus, we need to calculate the area under the demand curve (P = 4000 - 5Q) from Q = 0 to Q = 360 and above the price (P = 2200).
Consumer Surplus = 0.5 * base * height = 0.5 * (360 - 0) * (4000 - 2200) = 180 * 1800 = 324,000
So, the consumer surplus is $324,000.