University:
University of Michigan
Course:
ECON 613 | Advanced Topic in Mathematics for Economists
Academic year:

2024
Views:

126

Pages:

1
Author:

Alvinska H.

Extremum Points in a Power of Three Function: Determining Critical Points in a Company Total Cost Model Name: Alvinska Hebat Grade: 910-Math Triple Power Functions (Third Degree Polynomials) • • • • A power of three function describes the dependency relationship between variables where the independent variable is a power of three. The general form of a power-of-three function: Y = f(X) = aX3 + bX2 + cX + d where a ≠ 0. When depicted this function is as shown in the following figure. If you look at the graph of the function, there are 2 (two) turning points, namely at point A and point B. Point A is known as the maximum turning point (extreme point) while point B is called the minimum extreme point. To find the position of these points is easier if we have learned the concept of Differential. Application of Power of Three Functions in Economic Concepts • • • The form of a power function in economic concepts is often used in the total cost function (TC) as follows TC = f(Q) = aQ3 + bQ2 + cQ + d; where TC = Total Cost; Q = total output; a,b,c = coefficients; d = constant. So from these equations can be known function fixed cost (FC) and variable cost (Variable Cost), average cost or total cost per unit (Average Cost, AC), average variable cost or variable cost per unit (Average Variable Cost, AVC), and average fixed cost or fixed cost per unit (Average Fixed Cost, AFC) as follows: FC = d; d = constant; VC = f(Q) = aQ3 + bQ2 + cQ AC = f(Q) = TC/Q = aQ2 + bQ + c + d/Q AVC = f(Q) = VC/Q = aQ2 + bQ + c AFC = f(Q) = FC/Q = d/Q Example: Determine the equation of the fixed cost function (Fixed Cost, FC); variable cost (VC), average cost or total cost per unit (Average Cost, AC), average variable cost or variable cost per unit (Average Variable Cost, AVC), and average fixed cost or fixed cost per unit (Average Fixed Cost, AFC) from the total cost equation (TC) as follows: TC = 4Q3 - 240 Q2 + 5200Q + 20,000 Answer: FC = 20,000 VC = 4Q3 - 240Q2 + 5200Q AC = 4Q2 - 240Q + 5200 + 20,000/Q AVC = 4Q2 - 240Q + 5200 AFC = 20,000/.