Total cost and total variable cost are parallel, yet average total cost and average variable cost are not parallel.
a. Demonstrate mathematically that ATC and AVC are not parallel.
b. Show mathematically that when both ATC and AVC are falling, ATC falls faster than AVC, and when both are rising, AVC rises faster than ATC.
a. To demonstrate mathematically that average total cost (ATC) and average variable cost (AVC) are not parallel, we need to show that their slopes are not equal. The slopes of these cost curves represent the rate of change of cost with respect to quantity produced.
ATC is calculated as:
ATC = TC / Q
where TC is the total cost and Q is the quantity produced.
AVC is calculated as:
AVC = VC / Q
where VC is the total variable cost.
To show that ATC and AVC are not parallel, we need to compare their slopes, which can be found by differentiating the respective cost equations with respect to Q.
Differentiating ATC:
d(ATC) / dQ = (d(TC/Q)) / dQ
= (dTC / dQ - TC * d(1/Q) / dQ)
= (dTC / dQ - TC * d(-1/Q) / dQ)
= (dTC / dQ + TC / Q^2)
Differentiating AVC:
d(AVC) / dQ = (d(VC/Q)) / dQ
= (dVC / dQ - VC * d(1/Q) / dQ)
= (dVC / dQ - VC / Q^2)
From the above derivatives, we can see that the slopes of ATC and AVC are different. The presence of the term (TC / Q^2) in the derivative of ATC, which is absent in the derivative of AVC, indicates that their slopes are not equal. Therefore, ATC and AVC are not parallel.
b. To show mathematically that ATC falls faster than AVC when both are falling, and AVC rises faster than ATC when both are rising, we need to compare the rates of change of these cost curves.
When both ATC and AVC are falling, it means their values are decreasing as quantity produced increases. Mathematically, this corresponds to a negative slope for both curves.
For simplicity, let's assume that the total cost (TC) and the total variable cost (VC) are both decreasing functions of quantity produced (Q).
Mathematically, we can express this as:
d(TC) / dQ < 0
d(VC) / dQ < 0
Comparing their derivatives:
d(ATC) / dQ = (d(TC/Q)) / dQ = (dTC / dQ - TC * d(1/Q) / dQ)
d(AVC) / dQ = (d(VC/Q)) / dQ = (dVC / dQ - VC * d(1/Q) / dQ)
Since both TC and VC are decreasing functions of Q, it means that dTC / dQ < 0 and dVC / dQ < 0. The term d(1/Q) / dQ is negative since Q is increasing. Therefore, the negative sign on the second term in the derivatives implies that they are positive.
d(ATC) / dQ > 0
d(AVC) / dQ > 0
This shows that the slopes of ATC and AVC are positive, indicating that both curves are falling. However, from the previous section, we know that ATC falls faster than AVC because the slope of ATC is steeper (the presence of the term TC / Q^2).
Similarly, when both ATC and AVC are rising, it means their values are increasing as quantity produced increases. Mathematically, this corresponds to a positive slope for both curves.
In this case, we can apply a similar reasoning as above, and we will find that the slope of AVC is steeper than the slope of ATC since the term VC / Q^2 is present in the derivative of AVC but absent in the derivative of ATC. Therefore, AVC rises faster than ATC when both curves are rising.
a. To demonstrate mathematically that average total cost (ATC) and average variable cost (AVC) are not parallel, we need to analyze their equations.
The equation for ATC is ATC = TC/Q, where TC is the total cost and Q is the quantity produced.
The equation for AVC is AVC = VC/Q, where VC is the variable cost.
Now, let's assume that both ATC and AVC are parallel. This would mean that their slopes are equal. So, if we take the derivative of ATC and AVC with respect to quantity (Q) and equate the derivatives, we should get an equation that is always true.
Taking the derivative of ATC, we get:
d(ATC)/dQ = (d(TC/Q)/dQ)
= (-TC/Q^2)
Taking the derivative of AVC, we get:
d(AVC)/dQ = (d(VC/Q)/dQ)
= (-VC/Q^2)
Now, if ATC and AVC were parallel, their slopes would be equal. This means that the derivatives should be equal, i.e.,
(-TC/Q^2) = (-VC/Q^2)
However, this equation implies that TC = VC, which is not true in general. Therefore, we can conclude that ATC and AVC are not parallel.
b. To show mathematically that when both ATC and AVC are falling, ATC falls faster than AVC, and when both ATC and AVC are rising, AVC rises faster than ATC, we need to analyze their slopes.
The slope of ATC is given by the derivative of ATC with respect to quantity (Q):
d(ATC)/dQ = (-TC/Q^2)
The slope of AVC is given by the derivative of AVC with respect to quantity (Q):
d(AVC)/dQ = (-VC/Q^2)
Since ATC includes all costs (both variable and fixed), the slope of ATC is influenced by the fixed costs, which do not change with the quantity produced. On the other hand, AVC only considers variable costs, which change with the quantity produced.
When both ATC and AVC are falling, it means that the costs decrease as the quantity produced increases. In this case, the absolute value of the slopes of both ATC and AVC are positive. However, since ATC includes fixed costs, which do not decrease with quantity, the magnitude of the slope of ATC is generally larger than the magnitude of the slope of AVC. Hence, ATC falls faster than AVC.
Similarly, when both ATC and AVC are rising, it means that the costs increase as the quantity produced increases. In this case, the absolute value of the slopes of both ATC and AVC are negative. But since AVC does not include fixed costs, its slope increases at a slower rate compared to ATC. Therefore, AVC rises faster than ATC.
Note: This explanation assumes that total costs (TC) and variable costs (VC) are functions of quantity (Q).