Call Us demand function is Q=410-P and MC=10+5Q. Given that TFC=$0
a)derive an equation for TC
b)calculate the profit maximizing level
Good luck
Q = 410 – P and MC = 10 +5Q
a) TC = FC + TVC (FC = $0, given)
To derive TC (or TVC) from MC, integrate
TC = 10/1Q + 5/2Q1 +1 = 10Q + 2.5Q2
b) Profit is maximized when MR = MC
P = 410/1 – Q/1 = 410 – Q
MR = 410 – (2)Q = 410 – 2Q
MC = 10 + 5Q
410 – 2Q = 10 + 5Q
Q = 400/7 = 57.14 (profit maximizing quantity)
P = 410 – Q = 410 – (57.14) = $352.86 (profit maximizing price).
a) To derive an equation for TC (Total Cost), we need to consider the fixed costs (TFC) and the variable costs (VC).
As given, TFC = $0, which means there are no fixed costs to consider.
Variable costs (VC) are dependent on the quantity produced (Q) and the Marginal Cost (MC) equation.
MC = 10 + 5Q
To calculate the Variable Costs (VC) at a certain quantity (Q), we can integrate the Marginal Cost equation:
VC = ∫(10 + 5Q)dQ
VC = 10Q + (5Q^2 / 2) + C1
Where C1 is the constant of integration.
Since there are no fixed costs (TFC = $0), the Total Cost (TC) is equal to the Variable Cost (VC):
TC = VC
TC = 10Q + (5Q^2 / 2) + C1
b) To find the profit-maximizing level, we need to find the quantity (Q) that maximizes profit. Profit is calculated by subtracting total cost (TC) from total revenue (TR).
Total Revenue (TR) is the product of price (P) and quantity (Q):
TR = P * Q
Given the demand function as Q = 410 - P, we can substitute this value for Q in the total revenue equation:
TR = P * (410 - P)
TR = 410P - P^2
Profit (π) = Total Revenue (TR) - Total Cost (TC):
π = TR - TC
π = (410P - P^2) - (10Q + (5Q^2 / 2) + C1)
To find the profit-maximizing level, we need to find the quantity (Q) that maximizes profit. This can be done by differentiating the profit equation with respect to Q and solving for Q:
dπ / dQ = 0
-10 - 5Q = 0
-5Q = 10
Q = -2
However, a negative quantity doesn't make sense in this context. Therefore, there is no profit-maximizing level in this case.
Alternatively, to find the profit-maximizing level, we could differentiate the profit equation with respect to Q and solve for Q when the derivative equals zero. However, since we obtained a negative value for Q, it indicates that there is no profit-maximizing level.