Average and Marginal Cost
Average and Marginal Cost
Total (or global) cost, represented mathematically as CT, represents the total expenses needed to produce a given output. It is the sum of two kinds of costs: Fixed costs, represented as CF , are independent of the volume of output; they include expenses such as rents, insurance, maintenance of equipment, interest payments, and that portion of overhead and labor costs that is independent of the level of activity.
Variable costs, represented as CV , increase with activity. In the short run, modifications in the level of output are realized by modifying employment and raw material spending, with fixed equipment and plant costs. Some costs may be strictly proportional to production (for instance, raw material consumption); in other cases, cost variation may be more complicated, either for technical reasons (e.g. fuel consumption is not strictly proportional to speed), or for financial reasons (e.g. paying overtime hours). Let CV = ψ (Q ). A frequent assumption is that the rate of increase is always positive (ψ' > 0) but variable: In a first phase of activity it decreases (ψ' < 0), and increases in a second phase (ψ' > 0); consequently, and obviously, it goes through a minimum (ψ' = 0) between the two. In some cases, variable cost is considered as a constant, independent of the activity level for some production ranges.
Average total cost, or CA, is unit cost, that is, total cost divided by number of units produced: . In a similar way, one can define average fixed cost as CAF = CF/Q and average variable cost as C AV = C V /Q.
Marginal cost (Cm ) is the additional cost of producing one more output unit. It depends on cost in the following
way: An additional production of δQ units means an additional cost of δCT euros (or dollars), thereby marginal cost is represented as Cm = δCT/δQ. Given infinitely small increases of output, marginal cost is defined as derivative of total cost function: Cm = δCT/δQ = φ(Q). Marginal cost is independent of fixed cost and depends only on variable cost.
The following two cases will illustrate these principles: one using variable average costs, another using (total) variable costs strictly proportional to production, and, consequently, linear variable cost function.
When production starts, the average cost curve decreases because fixed costs are recouped through increasing sales of production units. With further production increases, however, variable costs increase more than proportionally to the output level because of increasing production difficulties; eventually this factor prevails and the cost curve increases. Consequently, at a given production level average cost reaches a minimum.
The marginal cost curve in such cases follows the course indicated in Figure 1 decreasing at first when production starts and then increasing with output level. When production starts, any increase in production level implies a decrease of average cost; consequently marginal cost is inferior to average cost, and the marginal cost curve is under the average cost curve. With high production levels, marginal cost is superior to average cost (since it is very costly to produce additional output). Consequently, marginal cost curve intersects average cost curve on the minimum of this one.
This can be demonstrated mathematically as follows:
By derivation of CA to Q (using standard rules of functions derivation), we arrive at the following:
When costs are considered in strict proportion to production, the variable cost function is linear, and with a as a proportionality factor, Cv = aQ and CT = CF + CV = CF + aQ.
Marginal cost is here constant, independent of production level
and equal to average variable cost defined as .
Both functions are identical and represented by one parallel to the quantity axis.
Average total cost CA is a constantly decreasing curve with two asymptotes, the vertical axis on one hand and the marginal cost line, or variable cost line, on the other (Figure 2).
This representation may be considered characteristic of cost functions for new products and especially for new technologies in information and communication. Development of new exploitation systems, new software, new aircraft, new medicines, cars, or drugs can cost millions of dollars. Reproduction may be quite cheap: a few dollars to manufacture a software program, more to produce a car; but the representation is also relevant since variable cost may in some cases amount to only 10 or 15 percent of total cost; the difference represents the costs of research and development, marketing, patent registration, and of course corporate profit. All these examples are characterized by very high fixed costs and constant marginal cost at very low level for large production ranges.
In the long run (which is not necessarily the same as a long time) changes in equipment and physical plant must be taken into account. The distinction between fixed and variable costs is no longer relevant because all production factors can be considered variable. With an increase in output, the average cost of production may decrease because of economies of scale due either to technical factors (improvement in task specialization, better division of labor, utilization of bigger and/or more specialized equipment) or for financial reasons (larger firms have more bargaining power with banks and suppliers).
Inversely, an excessively large production scale may create expenses: “diseconomies of scale” caused by administrative complexity, bureaucratic red tape, or organizational difficulties that may lead to increases in production costs. Such increases or decreases in costs are typical of internal economies (or diseconomies) of scale because they are the result of a firm’s own decisions. But costs may increase or decrease because of factors external to the firm that may modify its cost functions. Better organization of national production factors (raw material, labor), improvements to infrastructure such as the construction of roads and highways, or reductions in input prices or taxes qualify as external economies because they can improve a firm’s production costs regardless of the firm’s decisions. Conversely, external diseconomies may have a negative effect on production cost, such as deterioration in infrastructures, traffic jams, pollution, and tsunamis.
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