One problem a managerial economist is faced with is that of choosing the type of equation or curve which fits the cost data best. Three types of functions linear, quadratic and cubic especially the first two have been most commonly used in fitting statistical total cost functions.
Linear Cost Functions.
Suppose a firm is producing a certain good under the following conditions. It has fixed costs which must be met irrespective of the quantity of output produced. These fixed costs are represented by a. In order to produce X units of the good, it must buy a proportional amount of raw materials, labor and other necessary inputs, the cost of which is the variable amount bx. If the total cost of the fixed and variable quantities is denoted by Y, the type of equation representing the total cost of production for the firm is the linear function:
Y = a + bX
The shape of the curve is given below:
Certain important economic and mathematical properties of this function are given below:
At zero output, total fixed cost, such as rent, property taxes, insurance, and depreciation due to time and obsolescence, equals total cost. Increase in total cost due to increase in output are represented by total variable costs.
The average or unit cost function can be obtained by dividing the total cost function by output X.
Average Total Cost = Y/X = a/X + b
ATC = a/X + b
Since Average flied Cost in the above formula is a/X subtracting this from the equation leaves average variables cost b.
The marginal cost can be obtained by elementary differential calculus MC = b.
Quadratic Cost Functions.
This type of function which has been widely used in empirical studies is represented by the equation:
Y = a + bX+ cX2
The shape of the curve will be as
TC (Y = a + bX+ cX2)
The implications of this equation may be noted:
- When X = 0 Y = a. In other words, total costs equals fixed cost when output is zero.
- The equation is quadratic. Therefore, it has only one bend as against linear total cost function Y = a + bX, which has no bends. The number of bends is always one less than the highest exponent of X.
- The average cost equation can be derived by dividing the total cost function by output X. Average Total Cost = —Y/X = a/X+ b + cX.
- Since average fixed cost in the above a equation equals a/X subtracting this out gives average variable cost, b + cX.
- The equation for marginal cost can be obtained by differentiating the total cost function. Thus, MC = b + 2cX.
- MC = AVC = b, when X = 0.
Cubic Cost Functions.
The typical cost function of economic text books is not usually of a linear or quadratic form but rather of the cubic type:
Y= a + bX – cX2 + dX3
Its example would be:
Y = 18 + 30X – 10X22 + X3
The curve shall have two bends one less than the highest exponent of X. The shape of curve will be as:
This function combines the phases of form increasing and decreasing productivity or returns. In the above mentioned Figure, we see that increasing returns occurs at all levels to the left of the vertical dashed line as the total cost rises at a decreasing rate. Thereafter, on the right hand side of the dashed line, total cost is rising at an increasing rate denoting diminishing returns.
Cubic cost. functions have not often been fitted in actual studies. An important reason for this is that cubic functions do not usually result in significant improvement over the less complicated quadratic function.