The demand for item A is
P=40 -3.5Q
The production of A entails the following average variable costs:
AVC=1.5Q - 35
Fixed Costs are 24.
a) Calculate the revenue maximizing price of A
Revenue= PQ
Revenue= 40Q-3.5Q^2
Revenue' = 40-7Q
Q=40/7
P=40-3.5(40/7)
P=20
seems right?
b) Calculate the output level that minimizes the Average total cost.
-What is the average total cost equation?
do you just add the fixed cost to the AVC
ATC=1.5Q-35+24
ATC=1.5Q-11
or
is it finding variable cost first by dividing AVC by Q, then adding the fixed cost, then divide by Q
VC=1.5Q^2-3.5Q
TC=1.5Q^2-3.5Q+24
ATC=1.5Q-3.5+24/Q
a) To calculate the revenue maximizing price of item A, you correctly derived the revenue equation as Revenue = PQ, where P is the price and Q is the quantity sold. Substituting the demand equation P = 40 - 3.5Q into the revenue equation, we have:
Revenue = (40 - 3.5Q)Q = 40Q - 3.5Q^2
To find the quantity that maximizes revenue, we need to take the derivative of the revenue equation with respect to Q and set it equal to zero:
dRevenue/dQ = 40 - 7Q = 0
40 = 7Q
Q = 40/7
Now, substitute the value of Q back into the demand equation to find the corresponding price:
P = 40 - 3.5(40/7) = 20
Hence, the revenue-maximizing price for item A is $20.
b) To find the output level that minimizes the average total cost (ATC), you need to determine the equation for ATC. ATC is defined as total cost (TC) divided by quantity (Q). The total cost consists of fixed costs (FC) and variable costs (VC). Therefore, the equation for ATC can be derived as follows:
TC = FC + VC
Given that fixed costs are $24, we need to determine the equation for variable costs (VC). The average variable cost (AVC) equation you provided, AVC = 1.5Q - 35, is already in terms of variable costs. However, we need to double-check whether it includes the per-unit cost for each item.
If the equation AVC = 1.5Q - 35 represents the variable cost per item, then the total variable cost (VC) is simply AVC multiplied by quantity (Q):
VC = (1.5Q - 35)Q = 1.5Q^2 - 35Q
Now we can substitute the values for FC and VC into the equation for TC:
TC = FC + VC = 24 + (1.5Q^2 - 35Q)
Finally, to determine ATC, we divide TC by Q:
ATC = (24 + 1.5Q^2 - 35Q) / Q
Therefore, the correct equation for ATC is:
ATC = 24/Q + 1.5Q - 35
To find the output level that minimizes ATC, you can take the derivative of ATC with respect to Q, set it equal to zero, and solve for Q.