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.
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
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