1. prove mathematically that MC does not depend on TFC
2. if the average cost of producing 8 units of output in $80 and that of prodcuing 9 units of output is $90, find the marginal cost of the 9th unit of output and the total cost of the last unit
1. To prove mathematically that marginal cost (MC) does not depend on total fixed cost (TFC), we need to understand the relationship between these two concepts.
Marginal cost is the cost of producing an additional unit of output. It is calculated as the change in total cost (TC) when producing an additional unit of output. Mathematically, MC = ΔTC / ΔQ, where ΔTC is the change in total cost and ΔQ is the change in quantity.
On the other hand, total fixed cost (TFC) is the cost that does not change with the level of output. It includes expenses like rent, utilities, and salaries, which remain constant regardless of the quantity produced.
Since TFC does not change with the level of output, it does not affect the change in total cost (ΔTC) when producing an additional unit. As a result, the ΔTC in the MC formula becomes independent of TFC.
Therefore, mathematically proving that MC does not depend on TFC can be done by observing that MC is calculated solely based on the change in total cost divided by the change in quantity, while TFC remains constant.
2. To find the marginal cost of the 9th unit of output and the total cost of the last unit, we can use the given information about average cost and apply some basic principles.
Average cost (AC) is calculated by dividing the total cost (TC) by the quantity of output (Q). Mathematically, AC = TC / Q.
We are given that the average cost of producing 8 units of output is $80 and the average cost of producing 9 units of output is $90. From this information, we can calculate the total cost for each level of output.
For 8 units of output:
AC = $80
TC = AC * Q = $80 * 8 = $640
Now, to find the marginal cost (MC) of the 9th unit of output, we need to compare the total cost of producing 9 units to that of producing 8 units.
For 9 units of output:
AC = $90 (given)
TC = AC * Q = $90 * 9 = $810
To calculate the marginal cost of the 9th unit:
MC = TC(9) - TC(8) = $810 - $640 = $170
Therefore, the marginal cost of the 9th unit of output is $170.
To find the total cost of the last unit, we need to subtract the total cost of producing 8 units from the total cost of producing 9 units.
Total cost of the last unit = TC(9) - TC(8) = $810 - $640 = $170
Therefore, the total cost of the last unit is also $170.