given that the firm demand function as p=20-Q with the cost function as C=1/2q^2+2q+100 what is the firm output level that will maximize profit
revenue r(q) = q * p = q(20-q)= 20q-q^2
profit P(q) = r(q)-c(q) = (20q-q^2)-(1/2 q^2+2q+100) = -3/2 q^2 + 18q - 100
dP/dq = -3q +18
max profit is at q=6
To find the firm's output level that maximizes profit, we need to determine the quantity that maximizes the difference between total revenue and total cost.
1. Start by finding the firm's total revenue (TR):
Total Revenue (TR) = Price (P) * Quantity (Q)
TR = (20 - Q) * Q
TR = 20Q - Q²
2. Next, calculate the firm's total cost (TC) using the given cost function:
Total Cost (TC) = 1/2Q² + 2Q + 100
3. Profit (π) is calculated as the difference between total revenue and total cost:
Profit (π) = Total Revenue (TR) - Total Cost (TC)
π = 20Q - Q² - (1/2Q² + 2Q + 100)
Simplifying,
π = 20Q - Q² - 1/2Q² - 2Q - 100
π = -1.5Q²+ 18Q - 100
4. To maximize profit, we can find the level of output where the derivative of the profit function is equal to zero.
Taking the derivative of the profit function with respect to Q:
dπ/dQ = -3Q + 18 = 0
-3Q = -18
Q = 6
Therefore, the firm's output level that maximizes profit is Q = 6 units.