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 (rates of change) are different.
The formula for ATC is:
ATC = Total Cost / Quantity
The formula for AVC is:
AVC = Total Variable Cost / Quantity
To find the slopes of these cost curves, we differentiate them with respect to quantity (Q).
Let's assume that Total Cost (TC) and Total Variable Cost (TVC) are both functions of quantity (TC = f(Q) and TVC = g(Q)). Differentiating both of these functions with respect to Q will give us the slopes of ATC and AVC, respectively.
ATC = TC / Q
Differentiating both sides with respect to Q:
d(ATC) / dQ = (d(TC) / dQ) / Q
AVC = TVC / Q
Differentiating both sides with respect to Q:
d(AVC) / dQ = (d(TVC) / dQ) / Q
As you can see, the slopes of ATC and AVC depend on the derivatives of Total Cost and Total Variable Cost, respectively.
If the rates of change of TC and TVC are different, the slopes of ATC and AVC will also be different. Therefore, 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 are rising, AVC rises faster than ATC, we need to compare the rates of change of ATC and AVC.
Let's assume that both ATC and AVC are functions of quantity (ATC = f(Q) and AVC = g(Q)). Differentiating these functions with respect to Q will give us the rates of change of ATC and AVC, respectively.
Rate of change of ATC:
d(ATC) / dQ
Rate of change of AVC:
d(AVC) / dQ
If both ATC and AVC are falling, it means their rates of change should be negative. In this case, if d(ATC) / dQ is more negative than d(AVC) / dQ, ATC falls faster than AVC.
Similarly, if both ATC and AVC are rising, it means their rates of change should be positive. In this case, if d(ATC) / dQ is less positive than d(AVC) / dQ, AVC rises faster than ATC.
Therefore, to mathematically show 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 d(ATC) / dQ and d(AVC) / dQ.