You are the manager of a monopoly, and your demand and cost functions are given by P = 300 – 2.5Q and C(Q)=1000 + 2.5Q2.
a.At what price and quantity are firm’s profits maximized?
revenue = price * quantity
profit = revenue - cost
so, the profit is
Q(300-2.5Q) - (1000 + 2.5Q^2)
= 300Q - 2.5Q^2 - 1000 - 2.5Q^2
= -5Q^2 + 300Q - 1000
= -5(Q^2 - 60Q + 200)
max profit at Q=30, P(30)=225
To find the price and quantity at which the firm's profits are maximized, we need to find the profit maximizing output level (quantity) and then determine the corresponding price.
Step 1: Calculate the total revenue (TR) and total cost (TC) functions:
Total Revenue (TR) = Price (P) x Quantity (Q)
TR = (300 - 2.5Q) x Q = 300Q - 2.5Q^2
Total Cost (TC) = Cost function (C(Q)) = 1000 + 2.5Q^2
Step 2: Calculate the profit function (π) as the difference between total revenue and total cost:
Profit (π) = TR - TC = (300Q - 2.5Q^2) - (1000 + 2.5Q^2) = 300Q - 2.5Q^2 - 1000 - 2.5Q^2 = -5Q^2 + 300Q - 1000
Step 3: Determine the derivative of the profit function with respect to quantity (Q):
dπ/dQ = 0 when the profit is maximized.
Taking the derivative of the profit function:
dπ/dQ = -10Q + 300
Step 4: Set the derivative equal to zero and solve for Q:
-10Q + 300 = 0
10Q = 300
Q = 30
Step 5: Substitute the value of Q into the demand function to find the corresponding price (P):
P = 300 - 2.5Q
P = 300 - 2.5(30)
P = 300 - 75
P = 225
Therefore, the firm's profits are maximized at a price of $225 and a quantity of 30 units.