A MATHEMATICAL APPROACH TO REVENUE AND COST FUNCTIONS
Recall that TR = P x Q
This implies that P(AR) = TR
Q
For example, assuming that the AR function is given by:
AR = 20 - 1 Q
3
TR = P x Q
= 20Q - 1 Q^{2}
3
Marginal revenue is measure of the instantaneous rate of change of total revenue with respect to output Q. (Refer to the basic rules of differentiation in Appendix 1 of Modern Economics by Mudida)
MR = d TR
d Q
Thus, for example, given the following TR function:
TR = 2Q - 1 Q ^{2}
^{ } 2^{ }
^{ } AR = 2 - 1 Q
2
MR = d TR
d Q
= 2 - Q
The cost concepts studied earlier can also be expressed in functional form. Cubic functions are commonly used to represent cost functions. For example, a cost function may take the form:
TC = a + b Q + c Q ^{2} + d Q ^{3}
Average cost refers to the cost per unit of output.
AC = TC
Q
= a + b + cQ + dQ ^{2}
Q
Marginal cost refers to the instantaneous rate of change of the total cost function with respect to output.
MC = d TC
d Q
Given TC = a + bQ + CQ^{2} + dQ ^{3}
MC = d TC
^{ }dQ
= b + 2 cQ + 3dQ ^{2}
For example, given a total cost function
TC = Q^{3 }- 8Q^{2} + 68Q + 4
MC = d TC = 3Q^{2} - 16Q + 68
d Q