Suppose the short run cost function of a firm is given by: TC=2Q3 –2Q2 + Q + 10.
Find the expression of TFC & TVC
Derive the expressions of AFC, AVC, AC and MC
Find the levels of output that minimize MC and AVC and then find the minimum values of MC and AVC
To find the expressions of TFC and TVC, we need to understand the definitions of these terms:
- TFC (Total Fixed Cost) represents the total cost that does not vary with output. In the short run, it means the cost that the firm has to pay regardless of the quantity produced.
- TVC (Total Variable Cost) represents the total cost that varies with output. In the short run, it means the cost that changes as the firm produces different quantities.
To determine TFC, we need to find the cost component that remains constant regardless of output. In the given cost function, there is no explicit term indicating fixed cost (constant with output). Therefore, we assume that the constant term, 10, represents the total fixed cost. Thus, TFC = 10.
To determine TVC, we need to find the cost component that varies with output. In the given cost function, the terms that change with output are 2Q^3, -2Q^2, and Q. Therefore, we subtract the constant term, 10, from the total cost function to isolate the variable cost component:
TVC = TC - TFC = 2Q^3 - 2Q^2 + Q
Next, we derive the expressions of AFC, AVC, AC, and MC:
- AFC (Average Fixed Cost) is the fixed cost per unit of output and is given by AFC = TFC / Q.
- AVC (Average Variable Cost) is the variable cost per unit of output and is given by AVC = TVC / Q.
- AC (Average Cost or Average Total Cost) is the total cost per unit of output and is calculated as AC = TC / Q.
- MC (Marginal Cost) is the change in total cost resulting from a one-unit change in output and can be found by differentiating the total cost function with respect to Q.
Let's calculate these expressions:
AFC = TFC / Q = 10 / Q
AVC = TVC / Q = (2Q^3 - 2Q^2 + Q) / Q = 2Q^2 - 2Q + 1
AC = TC / Q = (2Q^3 - 2Q^2 + Q + 10) / Q = 2Q^2 - 2Q + 1 + 10 / Q = AVC + AFC
MC = d(TC) / d(Q) = d(2Q^3 - 2Q^2 + Q + 10) / d(Q) = 6Q^2 - 4Q + 1
To find the levels of output that minimize MC and AVC, we need to equate the derivative of MC and AVC to zero:
d(MC) / d(Q) = 6Q^2 - 4Q + 1 = 0
d(AVC) / d(Q) = 4Q - 2 = 0
Solving the equations:
6Q^2 - 4Q + 1 = 0
Using the quadratic formula: Q = (-b ± √(b^2 - 4ac)) / (2a)
Q = (-(-4) ± √((-4)^2 - 4(6)(1))) / (2(6))
Q = (4 ± √(16 - 24)) / 12
Q = (4 ± √(-8)) / 12
As the square root of a negative number is undefined, there is no real solution for Q that minimizes MC.
4Q - 2 = 0
Q = 2
Therefore, the level of output that minimizes AVC is Q = 2.
To find the minimum values of MC and AVC:
MC(Q=2) = 6(2)^2 - 4(2) + 1 = 24 - 8 + 1 = 17
AVC(Q=2) = 2(2)^2 - 2(2) + 1 = 8 - 4 + 1 = 5
Hence, the minimum value of MC is 17 and the minimum value of AVC is 5 when the output level is Q = 2.
To find the expression of TFC (Total Fixed Cost) and TVC (Total Variable Cost):
TFC is the portion of total cost that remains constant regardless of the level of output. In this case, TFC can be represented as the vertical intercept of the cost function, which is the constant term.
TFC = 10
TVC is the portion of total cost that varies with the level of output. In this case, TVC can be represented as the difference between the total cost (TC) and TFC.
TVC = TC - TFC
= 2Q^3 – 2Q^2 + Q + 10 - 10
= 2Q^3 – 2Q^2 + Q
To derive the expressions of AFC (Average Fixed Cost), AVC (Average Variable Cost), AC (Average Cost), and MC (Marginal Cost):
AFC is the fixed cost per unit of output. It can be calculated by dividing TFC by the level of output (Q).
AFC = TFC / Q
= 10 / Q
AVC is the variable cost per unit of output. It can be calculated by dividing TVC by the level of output (Q).
AVC = TVC / Q
= (2Q^3 – 2Q^2 + Q) / Q
= 2Q^2 – 2Q + 1
AC is the total cost per unit of output. It can be calculated by dividing TC by the level of output (Q).
AC = TC / Q
= (2Q^3 – 2Q^2 + Q + 10) / Q
= 2Q^2 – 2Q + 1 + 10 / Q
MC is the change in total cost resulting from a one-unit change in output. It can be calculated by taking the derivative of the total cost function (TC) with respect to the level of output (Q).
MC = d(TC) / dQ
= d(2Q^3 – 2Q^2 + Q + 10) / dQ
= 6Q^2 – 4Q + 1
To find the levels of output that minimize MC and AVC, we need to find the minimum points of these functions. To do that, we need to find the derivative of MC and AVC and set it equal to zero.
d(MC) / dQ = 6Q^2 – 4Q + 1 = 0
Solving this quadratic equation will give us the levels of output that minimize MC and AVC. To find the minimum values, we substitute these values into the MC and AVC functions.