Suppose a cost function is given as . C= 0.05Q^2+.0.1Q+20
Determine:
A. TFC
. TVC and ATC of producing 6 units
. MC of the 6th unit
To determine the Total Fixed Cost (TFC), we need to identify the component of the cost function that represents fixed costs.
In the given cost function, C = 0.05Q^2 + 0.1Q + 20, the constant term 20 represents the fixed costs. Therefore, TFC = $20.
To determine the Total Variable Cost (TVC), we can subtract the fixed cost from the total cost:
TVC = C - TFC = 0.05Q^2 + 0.1Q + 20 - 20 = 0.05Q^2 + 0.1Q.
To find the Average Total Cost (ATC) of producing 6 units, we divide the total cost by the quantity:
ATC = C/Q = (0.05Q^2 + 0.1Q + 20)/Q = 0.05Q + 0.1 + 20/Q.
Substituting Q = 6, we find:
ATC(6) = 0.05(6) + 0.1 + 20/6 = 0.3 + 0.1 + 20/6 ≈ 0.5 + 3.333 ≈ 3.833.
To find the Marginal Cost (MC) of the 6th unit, we take the derivative of the cost function with respect to quantity (Q) and evaluate it at Q = 6:
MC = dC/dQ = 0.1Q + 0.1.
Substituting Q = 6, we find:
MC(6) = 0.1(6) + 0.1 = 0.6 + 0.1 = 0.7.
To determine various costs and the marginal cost, we can break down the cost function into its components:
C = 0.05Q^2 + 0.1Q + 20
1. Total Fixed Cost (TFC):
Total Fixed Costs (TFC) are the costs that do not change with the level of output. In other words, these costs are incurred even when no units are produced. In this case, the TFC is the constant term in the cost function, which is 20.
So, TFC = 20.
2. Total Variable Cost (TVC):
Total Variable Costs (TVC) are the costs that vary with the level of output. To determine TVC, we need to subtract TFC from the total cost.
TVC = C - TFC
TVC = 0.05Q^2 + 0.1Q + 20 - 20
TVC = 0.05Q^2 + 0.1Q
3. Average Total Cost (ATC) of producing 6 units:
The Average Total Cost (ATC) is the total cost per unit of output. To find ATC when producing 6 units, we divide the total cost by the number of units.
ATC = C / Q
ATC = (0.05Q^2 + 0.1Q + 20) / Q
ATC = 0.05Q + 0.1 + (20 / Q)
Substituting Q = 6:
ATC = 0.05(6) + 0.1 + (20 / 6)
ATC = 0.3 + 0.1 + 3.33
ATC = 3.73
So, the Average Total Cost (ATC) of producing 6 units is 3.73.
4. Marginal Cost (MC) of the 6th unit:
The Marginal Cost (MC) is the additional cost incurred when producing one more unit. We can find MC by taking the derivative of the cost function with respect to Q.
MC = dC / dQ
MC = 0.1Q + 0.1
Substituting Q = 6:
MC = 0.1(6) + 0.1
MC = 0.6 + 0.1
MC = 0.7
So, the Marginal Cost (MC) of the 6th unit is 0.7.
To determine the different cost values and measures, we need to understand the definitions of each term:
- TFC (Total Fixed Cost): The fixed cost does not change regardless of the level of production output.
- TVC (Total Variable Cost): The variable cost changes as per the level of production output.
- ATC (Average Total Cost): It is the average cost per unit of production and is calculated by dividing the total cost by the total quantity produced.
- MC (Marginal Cost): It is the additional cost incurred by producing one additional unit of output.
Given the cost function C = 0.05Q^2 + 0.1Q + 20, let's determine each value step by step.
A. TFC (Total Fixed Cost):
Total Fixed Cost (TFC) refers to the constant cost that does not change with the level of production output. In the given cost function, the term '20' represents the fixed cost. Hence, TFC = 20.
B. TVC (Total Variable Cost):
Total Variable Cost (TVC) refers to the variable cost that changes with the level of production output. In this case, we need to find the variable cost component of the cost function.
In the given cost function C = 0.05Q^2 + 0.1Q + 20, the terms '0.05Q^2' and '0.1Q' represent the variable cost component. Hence, TVC = 0.05Q^2 + 0.1Q.
C. ATC (Average Total Cost) of producing 6 units:
Average Total Cost (ATC) is the average cost per unit of production and is calculated by dividing the total cost by the total quantity produced. We can find ATC by substituting the value of Q as 6 in the cost function C and dividing it by 6.
ATC = [0.05Q^2 + 0.1Q + 20] / Q
= [0.05(6)^2 + 0.1(6) + 20] / 6
= [0.05(36) + 0.6 + 20] / 6
= (1.8 + 0.6 + 20) / 6
= 22.4 / 6
= 3.7333... (approx.)
Hence, the ATC of producing 6 units is approximately 3.7333.
D. MC (Marginal Cost) of the 6th unit:
Marginal Cost (MC) is the additional cost incurred by producing one additional unit of output. It can be calculated by taking the derivative of the cost function with respect to Q and substituting the value of Q as 6.
MC = dC/dQ
= (d/dQ) [0.05Q^2 + 0.1Q + 20]
= 0.1Q + 0.1
= 0.1(6) + 0.1
= 0.6 + 0.1
= 0.7
Hence, the MC of the 6th unit is 0.7.