Given a total cost function: C= Q3 - 3Q2 + 8Q + 48 ,then find,
a. TFC, TVC , b)AFC,AVC, AC c) MC, d)the minimum point of MC and the
minimum point of AVC e) Determine the level of output for which MC=AVC?
Using the more usual notation,
c(q) = q^3 - 3q^2 + 8q + 48
as I understand it,
TFC = c(0)
TVC = c(q)-c(0)
AFC = TFC/q
AVC = TVC/q
AC = c(q)/q
MC = dc/dq
use derivatives to find minima
answer
use derivatives to find minima
a. To find TFC (Total Fixed Cost) and TVC (Total Variable Cost), we need to look at the equation C = Q^3 - 3Q^2 + 8Q + 48.
TFC is the constant term, which in this case is 48.
TFC = 48
TVC is the rest of the equation since it varies with the level of output (Q).
TVC = Q^3 - 3Q^2 + 8Q
b. To find AFC (Average Fixed Cost), AVC (Average Variable Cost), and AC (Average Cost), we divide the respective costs by the level of output (Q).
AFC = TFC / Q
AVC = TVC / Q
AC = C / Q
For this particular case:
AFC = 48 / Q
AVC = (Q^3 - 3Q^2 + 8Q) / Q
AC = (Q^3 - 3Q^2 + 8Q + 48) / Q
c. To find MC (Marginal Cost), we take the derivative of the Total Cost function with respect to Q:
MC = 3Q^2 - 6Q + 8
d. To find the minimum points of MC and AVC, we set the derivative (MC = 3Q^2 - 6Q + 8) equal to zero and solve for Q:
3Q^2 - 6Q + 8 = 0
Using the quadratic formula, we find that the minimum point of MC is at Q = 1.04 (approximately).
To find the minimum point of AVC, we need to find the value of Q that makes AVC a minimum. This occurs when AVC is at its minimum point when Q > 1.04.
e. To determine the level of output for which MC = AVC, we set MC equal to AVC and solve for Q:
3Q^2 - 6Q + 8 = (Q^3 - 3Q^2 + 8Q) / Q
Simplifying and solving this equation will give us the level of output for which MC equals AVC.
To find the total fixed cost (TFC), total variable cost (TVC), average fixed cost (AFC), average variable cost (AVC), average total cost (AC), marginal cost (MC), and the minimum points of MC and AVC, we need to first understand the definitions and calculations for each of these cost concepts.
a) TFC (Total Fixed Cost):
Total Fixed Cost refers to the fixed expenses that a company incurs regardless of the level of output. In the given cost function C = Q^3 - 3Q^2 + 8Q + 48, the fixed cost portion is the constant term 48. So, TFC = 48.
b) TVC (Total Variable Cost):
Total Variable Cost represents the cost that changes with the level of output. In the given cost function C = Q^3 - 3Q^2 + 8Q + 48, the variable cost portion is Q^3 - 3Q^2 + 8Q. So, TVC = Q^3 - 3Q^2 + 8Q.
c) AFC (Average Fixed Cost):
Average Fixed Cost is the fixed cost per unit of output. It is calculated by dividing the TFC by the level of output. In this case, AFC = TFC/Q = 48/Q.
d) AVC (Average Variable Cost):
Average Variable Cost is the variable cost per unit of output. It is calculated by dividing the TVC by the level of output. In this case, AVC = TVC/Q = (Q^3 - 3Q^2 + 8Q)/Q = Q^2 - 3Q + 8.
e) AC (Average Total Cost):
Average Total Cost is the total cost per unit of output. It is calculated by dividing the TC by the level of output. In this case, AC = C/Q = (Q^3 - 3Q^2 + 8Q + 48)/Q = Q^2 - 3Q + 8 + 48/Q.
f) MC (Marginal Cost):
Marginal Cost is the additional cost incurred by producing an additional unit of output. It is calculated by taking the derivative of the total cost function with respect to the level of output. In this case, MC = dC/dQ = 3Q^2 - 6Q + 8.
To find the minimum points of MC and AVC, we need to find where the derivative of the respective cost functions is equal to zero.
To find the minimum point of MC:
1. Set MC = 0: 3Q^2 - 6Q + 8 = 0.
2. Solve the quadratic equation to find the values of Q at which MC = 0.
(Using the quadratic formula, Q = (-(-6) ± √((-6)^2 - 4(3)(8)))/(2(3)))
Q = (6 ± √(36 - 96))/6
Q = (6 ± √(-60))/6 (Discriminant is negative, so no real roots exist)
Therefore, there is no minimum point for MC.
To find the minimum point of AVC:
1. Set AVC = 0: Q^2 - 3Q + 8 = 0.
2. Solve the quadratic equation to find the values of Q at which AVC = 0.
(Using the quadratic formula, Q = (-(-3) ± √((-3)^2 - 4(1)(8)))/(2(1)))
Q = (3 ± √(9 - 32))/2
Q = (3 ± √(-23))/2 (Discriminant is negative, so no real roots exist)
Therefore, there is no minimum point for AVC.
e) To determine the level of output for which MC = AVC:
We need to set MC equal to AVC and solve for Q.
3Q^2 - 6Q + 8 = Q^2 - 3Q + 8.
2Q^2 - 3Q = 0.
Q(2Q - 3) = 0.
Q = 0 (one solution)
or
2Q - 3 = 0.
Q = 3/2 (another solution).
Therefore, the level of output for which MC equals AVC is Q = 0 and Q = 3/2.