Suppose the short-run cost function of the firm is by: TC = 100 + 60Q + 4Q^2. Find:
A. The expression of AVC, AFC and MC
B. The levels of output, that minimize AVC and MC
C. The minimum values of MC and AVC
A. To find the expressions for AVC, AFC, and MC, we need to first find the expressions for average variable cost (AVC), average fixed cost (AFC), and marginal cost (MC).
1. Average variable cost (AVC):
AVC = TVC / Q
Where TVC is the total variable cost.
In this case, the total variable cost can be found by subtracting the total fixed cost (TFC) from the total cost (TC).
TVC = TC - TFC = 100 + 60Q + 4Q^2 - TFC
2. Average fixed cost (AFC):
AFC = TFC / Q
3. Marginal cost (MC):
MC = ΔTC / ΔQ
Where ΔTC is the change in total cost and ΔQ is the change in quantity.
B. To find the levels of output that minimize AVC and MC, we need to take the derivative of both expressions with respect to Q and set it equal to zero.
1. To minimize AVC:
Take the derivative of AVC with respect to Q and set it equal to zero:
d(AVC)/dQ = (d(TVC)/dQ) / Q - TVC / Q^2 = 0
Solve for Q that satisfies this equation.
2. To minimize MC:
Take the derivative of MC with respect to Q and set it equal to zero:
d(MC)/dQ = (d(TC)/dQ) / Q - TC / Q^2 = 0
Solve for Q that satisfies this equation.
C. To find the minimum values of MC and AVC, substitute the values of Q obtained from part B into the expressions for MC and AVC, and calculate their respective values.
A. To find the expressions for AVC, AFC, and MC, we need to know the definitions of these average and marginal cost measures:
1. Average Variable Cost (AVC): AVC is the cost per unit of output that can be attributed to variable costs (i.e., costs that change with the level of output). It is calculated by dividing the total variable cost by the level of output.
AVC = TVC / Q
2. Average Fixed Cost (AFC): AFC is the cost per unit of output that can be attributed to fixed costs (i.e., costs that do not change with the level of output). It is calculated by dividing the total fixed cost by the level of output.
AFC = TFC / Q
3. Marginal Cost (MC): MC is the additional cost incurred from producing one more unit of output. It is calculated by taking the derivative of the total cost function with respect to the level of output.
MC = dTC / dQ
Given the cost function TC = 100 + 60Q + 4Q^2, we need to find the expressions for AVC, AFC, and MC:
- Total Variable Cost (TVC) is obtained by subtracting the total fixed cost (TFC) from the total cost (TC).
TVC = TC - TFC
= 100 + 60Q + 4Q^2 - TFC
- Total Fixed Cost (TFC) can be calculated by plugging in a specific value of Q into the cost function and solving for TFC. However, since the cost function does not contain any specific values for TFC, we cannot determine the exact value of TFC.
B. To find the levels of output that minimize AVC and MC, we need to find the values of Q where AVC and MC reach their minimum points.
1. To minimize AVC, we need to find the value of Q where the derivative of AVC with respect to Q equals zero.
dAVC / dQ = 0
2. To minimize MC, we need to find the value of Q where the derivative of MC with respect to Q equals zero.
dMC / dQ = 0
To find these values, we need the expressions of AVC and MC, but unfortunately, we do not have enough information to determine these expressions.
C. Similarly, without knowing the expressions for MC and AVC, we cannot find their specific minimum values.
A. To find the expressions for Average Variable Cost (AVC), Average Fixed Cost (AFC), and Marginal Cost (MC), we need to start by understanding the definitions of these cost measures.
1. Average Variable Cost (AVC): AVC is the cost per unit of output produced, and it represents the variable costs incurred to produce each unit of output. The formula for AVC is AVC = VC/Q, where VC is the variable cost and Q is the quantity of output.
2. Average Fixed Cost (AFC): AFC is the cost per unit of output that represents fixed costs incurred to produce each unit of output. The formula for AFC is AFC = FC/Q, where FC is the fixed cost.
3. Marginal Cost (MC): MC represents the additional cost incurred by producing one more unit of output. The formula for MC is MC = dTC/dQ, where TC is the total cost and Q is the quantity of output.
To find the expressions for AVC, AFC, and MC, we need to differentiate the total cost function (TC) with respect to Q.
Given TC = 100 + 60Q + 4Q^2,
a. AVC = VC/Q = (TC - FC)/Q
= (100 + 60Q + 4Q^2 - FC)/Q
= (100/Q) + 60 + 4Q
b. AFC = FC/Q = (100 + 60Q + 4Q^2 - VC)/Q
= (100 + 60Q + 4Q^2 - (100/Q) - 60 - 4Q)/Q
= (100 - (100/Q))/Q + 4Q
c. MC = dTC/dQ = d(100 + 60Q + 4Q^2)/dQ
= 60 + 8Q
B. To find the levels of output that minimize AVC and MC, we need to find the critical points of these functions. In calculus, critical points occur where the derivative of a function is equal to zero or undefined.
1. Minimize AVC:
To minimize AVC, we set its derivative equal to zero and solve for Q:
dAVC/dQ = 0
60 + 4Q = 0
Q = -15
However, since quantity cannot be negative, this result is not meaningful. Therefore, there is no level of output that minimizes AVC in this case.
2. Minimize MC:
To minimize MC, we set its derivative equal to zero and solve for Q:
dMC/dQ = 0
8Q + 60 = 0
Q = -7.5
Again, since quantity cannot be negative, this result is not meaningful. Therefore, there is no level of output that minimizes MC in this case.
C. To find the minimum values of MC and AVC, we need to evaluate these functions at their respective critical points.
1. Minimum MC:
Evaluating MC at Q = -7.5:
MC = 60 + 8(-7.5)
= 60 - 60
= 0
So, the minimum value of MC is 0.
2. Minimum AVC:
Since we found no meaningful level of output that minimizes AVC, it is not possible to find a minimum value of AVC in this case.