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The cost, also called cost, is the amount of money needed for a certain economic activity that involves the production of a good, a service or the development of an activity with social value. Seven parameters are involved in determining cost: marginal cost , total cost , fixed cost , total variable cost , average total cost , average fixed cost , and average variable cost .
In turn, the information that must be available to calculate each of these parameters is usually obtained in three formats, which record the relationship between the production parameters, for example the total cost (TC parameter), and the quantity produced (variable Q), which is information associated with the economic activity on which the cost is being analysed. A table of values or a graph relating the production parameter to the variable Q is one of the possible formats. Another format can be to present this information as a linear equation that relates the production parameter to the variable Q, while the third format can be a non-linear equation.
Definition of the parameters associated with a cost evaluation
Marginal cost is the cost a firm incurs in producing a good in addition to the amount it is producing. Suppose the company is producing two goods and the company’s managers would like to know how much costs would increase if production were increased to three goods. The difference of going from producing two goods to three is the marginal cost, and it is calculated as follows.
Marginal cost = Total cost of producing 3 goods – Total cost of producing 2 goods
For example, if the cost of producing three goods is $600 and $390 is the cost of producing two goods, the difference is $210, so the marginal cost is $210.
Total cost is simply the sum of all the costs associated with the production of a certain number of goods. The fixed cost is the cost of production that does not depend on the quantity of goods that are produced; It is, then, the cost incurred by the production system even when no good is produced.
The total variable cost is the cost incurred by the production system when a certain quantity of products is produced. It is the difference between the total cost and the fixed cost. For example, the total variable cost of producing four units is calculated as follows.
Total variable cost of producing 4 units = Total cost of producing 4 units – Total cost of producing 0 units
Assigning values to this example, if the total cost of producing four units is $840 and $130 is the fixed cost, that is, the cost of the production system when no product is produced, the total variable cost is $710, that is Say, the difference $840 – $130 = $710.
Average total cost is the total cost of producing a certain number of units divided by the number of units. For example, if five units are produced, the average total cost is calculated as:
Average total cost of production of 5 units = Total cost of production of 5 units / 5
If the total cost of producing five units is $1,200, the average total cost of producing five units is $240, that is, $1,200 / 5 = $240.
Average total cost is often also called average cost per unit or average cost per unit.
Similarly, average fixed cost (also average fixed cost per unit or unit fixed cost) is the fixed cost divided by the number of units produced. The average fixed cost is determined with the following formula:
Average fixed cost = Total fixed cost / Number of units produced
Following the same criteria, the average variable cost (with equivalent denominations) of producing a certain number of units is the total variable cost divided by the number of units produced. The average variable cost is determined with the following formula:
Average variable cost = Total variable cost / Number of units produced
Calculation of the parameters of a cost evaluation
Tables and graphs
As explained, the information for calculating costs relates some of the parameters to the quantity produced (variable Q) and is usually obtained in three formats. One possibility is that the available information is presented in a table or graph. The figure below shows an example of a chart describing total cost, fixed cost, and variable cost, and the relationship to their respective average values, in particular average total cost.
Another possibility is that from a table the relationship between the marginal cost and the variable Q is obtained, and the total cost should be calculated from this information. To calculate the total cost of producing two goods, the following expression can be used:
Total cost of production of 2 goods = Total cost of production of 1 good + marginal cost of production of 2 goods
From the table it will be possible to obtain the cost of producing one good, the marginal cost of producing two goods and the fixed costs. If the cost of producing one good is $250, and the marginal cost of producing one additional good is $140, then the total cost of producing two goods will be $390, or $250 + $140 = $390.
Linear equations
It is possible that to calculate the 7 cost parameters there is a linear equation that represents the relationship between the total cost TC and the quantity produced (variable Q). Linear or first-order equations are those that relate the dependent variable to the independent variable in a polynomial expression with the independent variable raised only to the exponent one, and that do not involve any other function such as logarithms or exponentials. Linear equations are represented on a graph as lines, as shown in the figure above. An example of a linear equation that relates the total cost parameter TC with the variable Q would be:
TC = 50 + 6 × Q
If we wanted to calculate the total cost for a specific quantity Q, all we have to do is substitute the variable Q for the quantity of units that we want to produce. Therefore, the total cost of producing 10 units is:
50 + 6 × 10 = 110.
This expression means that total cost increases by 6 for each additional good added: there is a constant marginal cost of $6 per additional unit produced. In addition, a cost of $50 is added even when Q is 0, when no good is produced; thus, the fixed cost of this production system is $50.
To calculate the average variable cost, divide the variable cost by the quantity of goods produced, the variable Q. Since the addend of variable cost in this total cost equation is 6 × Q, the average variable cost will be the constant value 6. In the case where the total cost is represented by a linear equation, the average variable cost does not depend on the quantity produced, just like the marginal cost. Generalizing the example, when there is a linear relationship between the total cost and the quantity of products, the total cost is expressed as:
CT = CF + CM × Q
being CF the fixed cost and CM the marginal cost, which in this case is a constant value and does not depend on the quantity of products that are generated.
nonlinear equations
There are production systems in which the relationship between the total cost TC and the quantity of goods produced is represented by non-linear equations.That is, equations that relate the dependent variable to the independent by a polynomial expression with the independent variable raised to exponents greater than one or with non-polynomial functions. Let’s look at two examples of nonlinear equations; in the first case, a polynomial equation of degree 3, and in the second an equation that combines a polynomial function of degree 1 and a logarithmic function.
TC = 34 × Q 3 – 24 × Q + 9
CT = Q + log(Q + 2)
When there are non-linear equations, the appropriate way to obtain the expression of the marginal cost is through mathematical calculation. The marginal cost is the variation in the total cost associated with the variation in the quantity of products; therefore, the expression of the marginal cost will be the derivative of the expression of total cost with respect to the variable Q. Let’s see what expressions of marginal cost CM are obtained in the two previous examples.
TC = 34 × Q 3 – 24 × Q + 9
MC = 102 × Q 2 – 24
CT = Q + log(Q + 2)
MC = 1 + 1/(Q + 2)
As we have seen previously, if you want to obtain the total cost or the marginal cost for the production of a certain amount of goods, you have to substitute the value of Q in the previous expressions.
The case of the linear relationship that was seen in the previous section, this relationship is a particular case of the nonlinear equations that is seen here. If the expression of the total cost were linear, with the form CT = CF + CM × Q, the derivative of this expression with respect to Q would be CM, coinciding with the previous result.
Let’s see how to obtain the other parameters involved in a cost evaluation from the non-linear relationships that are presented as examples.
The fixed cost CF is determined when Q = 0. In the first example:
TC = 34 × Q 3 – 24 × Q + 9
If Q = 0, then CF = $9.
In the second example:
CT = Q + log(Q + 2)
If Q = 0 then CF = 0 + ln(0 + 2) and CF = log(2) = $0.30.
The total variable cost TVC is determined as:
CVT = CT – CF
In the first example:
CT = 34 × Q 3 – 24 × Q + 9 and CF = 9
Therefore:
CVT = 34 × Q 3 – 24 × Q
In the second example:
CT = Q + log(Q + 2) and CF = log(2)
Therefore:
TVC = Q + log(Q + 2) – log(2)
The average total cost CTP is determined by dividing the total cost by the variable Q. Therefore, in the first example the expression for CTP is:
CTP = 34 × Q 2 – 24 + 9 / Q
In the second case, the CTP expression is:
CTP = 1 + log(Q + 2) / Q
In the same way, the average fixed cost CFP is determined by dividing the fixed cost by the variable Q. In the first case, the expression of the CFP is:
PFC = 9 / Q
In the second example, the CFP expression is:
CF = log(2) / Q
Lastly, the average variable cost CVP, as in the two previous cases, is determined by dividing the total variable cost CVT by the variable Q. The expression for CVP in the first case is:
CVP = 34 × Q 2 – 24
The expression of CVP in the second case is:
CVP = 1 + log(Q + 2) / Q – log(2) / Q
Sources
E. Bueno Campos E., Cruz Roche I., Durán Herrera JJ Business economics. Analysis of business decisions . Pyramid, Madrid, Spain, 2002. ISBN 84-368-0207-1.
Omar Alejandro Martínez Torres, OA Economic analysis . Astra editions, Mexico, 1984.
