Marginal cost

In economics, marginal cost is the change in the total cost that arises when the quantity produced is increased, i.e. the cost of producing additional quantity. In some contexts, it refers to an increment of one unit of output, and in others it refers to the rate of change of total cost as output is increased by an infinitesimal amount. As Figure 1 shows, the marginal cost is measured in dollars per unit, whereas total cost is in dollars, and the marginal cost is the slope of the total cost, the rate at which it increases with output. Marginal cost is different from average cost, which is the total cost divided by the number of units produced.
At each level of production and time period being considered, marginal cost includes all costs that vary with the level of production, whereas costs that do not vary with production are fixed. For example, the marginal cost of producing an automobile will include the costs of labor and parts needed for the additional automobile but not the fixed cost of the factory building, which does not change with output. The marginal cost can be either short-run or long-run marginal cost, depending on what costs vary with output, since in the long run even building size is chosen to fit the desired output.
If the cost function is continuous and differentiable, the marginal cost is the first derivative of the cost function with respect to the output quantity :
If the cost function is not differentiable, the marginal cost can be expressed as follows:
where denotes an incremental change of one unit.

Short run marginal cost

Short run marginal cost is the change in total cost when an additional output is produced in the short run and some costs are fixed. On the right side of the page, the short-run marginal cost forms a U-shape, with quantity on the x-axis and cost per unit on the y-axis.
On the short run, the firm has some costs that are fixed independently of the quantity of output. Other costs such as labor and materials vary with output, and thus show up in marginal cost. The marginal cost may first decline, as in the diagram, if the additional cost per unit is high, if the firm operates at too low a level of output, or it may start flat or rise immediately. At some point, the marginal cost rises as increases in the variable inputs such as labor put increasing pressure on the fixed assets such as the size of the building. In the long run, the firm would increase its fixed assets to correspond to the desired output; the short run is defined as the period in which those assets cannot be changed.

Long run marginal cost

is defined as the length of time in which no input is fixed. Everything, including building size and machinery, can be chosen optimally for the quantity of output that is desired. As a result, even if short-run marginal cost rises because of capacity constraints, long-run marginal cost can be constant. Or, there may be increasing or decreasing returns to scale if technological or management productivity changes with the quantity. Or, there may be both, as in the diagram at the right, in which the marginal cost first falls and then rises.

Cost functions and relationship to average cost

In the simplest case, the total cost function and its derivative are expressed as follows, where Q represents the production quantity, VC represents variable costs, FC represents fixed costs and TC represents total costs.
Fixed costs represent the costs that do not change as the production quantity changes. Fixed costs are costs incurred by things like rent, building space, machines, etc. Variable costs change as the production quantity changes, and are often associated with labor or materials. The derivative of fixed cost is zero, and this term drops out of the marginal cost equation: that is, marginal cost does not depend on fixed costs. This can be compared with average total cost, which is the total cost divided by the number of units produced:
For discrete calculation without calculus, marginal cost equals the change in total cost that comes with each additional unit produced. Since fixed cost does not change in the short run, it has no effect on marginal cost.
For instance, suppose the total cost of making 1 shoe is $30 and the total cost of making 2 shoes is $40. The marginal cost of producing shoes decreases from $30 to $10 with the production of the second shoe. In another example, when a fixed cost is associated, the marginal cost can be calculated as presented in the table below.

Output	Total Cost	Average Cost	Marginal Cost
0	10	∞	–
1	30	30	20
2	40	20	10
3	48	16	8

Marginal cost is not the cost of producing the "next" or "last" unit. The cost of the last unit is the same as the cost of the first unit and every other unit. In the short run, increasing production requires using more of the variable input — conventionally assumed to be labor. Adding more labor to a fixed capital stock reduces the marginal product of labor because of the diminishing marginal returns. This reduction in productivity is not limited to the additional labor needed to produce the marginal unit – the productivity of every unit of labor is reduced. Thus the cost of producing the marginal unit of output has two components: the cost associated with producing the marginal unit and the increase in average costs for all units produced due to the "damage" to the entire productive process. The first component is the per-unit or average cost. The second component is the small increase in cost due to the law of diminishing marginal returns which increases the costs of all units sold.
Marginal costs can also be expressed as the cost per unit of labor divided by the marginal product of labor. Denoting variable cost as VC, the constant wage rate as w, and labor usage as L, we have
Here MPL is the ratio of increase in the quantity produced per unit increase in labour: i.e. ΔQ/ΔL, the marginal product of labor. The last equality holds because is the change in quantity of labor that brings about a one-unit change in output. Since the wage rate is assumed constant, marginal cost and marginal product of labor have an inverse relationship—if the marginal product of labor is decreasing, then marginal cost is increasing, and AVC = VC/Q=wL/Q = w/ = w/AP_L

Empirical data on marginal cost

While neoclassical models broadly assume that marginal cost will increase as production increases, several empirical studies conducted throughout the 20th century have concluded that the marginal cost is either constant or falling for the vast majority of firms. Most recently, former Federal Reserve Vice-Chair Alan Blinder and colleagues conducted a survey of 200 executives of corporations with sales exceeding $10 million, in which they were asked, among other questions, about the structure of their marginal cost curves. Strikingly, just 11% of respondents answered that their marginal costs increased as production increased, while 48% answered that they were constant, and 41% answered that they were decreasing. Summing up the results, they wrote:
Many Post-Keynesian economists have pointed to these results as evidence in favor of their own heterodox theories of the firm, which generally assume that marginal cost is constant as production increases.

Economies of scale

apply to the long run, a span of time in which all inputs can be varied by the firm so that there are no fixed inputs or fixed costs. Production may be subject to economies of scale. Economies of scale are said to exist if an additional unit of output can be produced for less than the average of all previous units – that is, if long-run marginal cost is below long-run average cost, so the latter is falling. Conversely, there may be levels of production where marginal cost is higher than average cost, and the average cost is an increasing function of output. Where there are economies of scale, prices set at marginal cost will fail to cover total costs, thus requiring a subsidy. For this generic case, minimum average cost occurs at the point where average cost and marginal cost are equal.

Perfectly competitive supply curve

The portion of the marginal cost curve above its intersection with the average variable cost curve is the supply curve for a firm operating in a perfectly competitive market. This is not true for firms operating in other market structures. For example, while a monopoly has an MC curve, it does not have a supply curve. In a perfectly competitive market, a supply curve shows the quantity a seller is willing and able to supply at each price – for each price, there is a unique quantity that would be supplied.

Decisions taken based on marginal costs

In perfectly competitive markets, firms decide the quantity to be produced based on marginal costs and sale price. If the sale price is higher than the marginal cost, then they produce the unit and supply it. If the marginal cost is higher than the price, it would not be profitable to produce it. So the production will be carried out until the marginal cost is equal to the sale price.

Relationship to fixed costs

Marginal costs are not affected by the level of fixed cost. Marginal costs can be expressed as ∆C/∆Q. Since fixed costs do not vary with changes in quantity, MC is ∆VC/∆Q. Thus if fixed cost were to double, the marginal cost MC would not be affected, and consequently, the profit-maximizing quantity and price would not change. This can be illustrated by graphing the short run total cost curve and the short-run variable cost curve. The shapes of the curves are identical. Each curve initially increases at a decreasing rate, reaches an inflection point, then increases at an increasing rate. The only difference between the curves is that the SRVC curve begins from the origin while the SRTC curve originates on the positive part of the vertical axis. The distance of the beginning point of the SRTC above the origin represents the fixed cost – the vertical distance between the curves. This distance remains constant as the quantity produced, Q, increases. MC is the slope of the SRVC curve. A change in fixed cost would be reflected by a change in the vertical distance between the SRTC and SRVC curve. Any such change would have no effect on the shape of the SRVC curve and therefore its slope MC at any point. The changing law of marginal cost is similar to the changing law of average cost. They are both decrease at first with the increase of output, then start to increase after reaching a certain scale. While the output when marginal cost reaches its minimum is smaller than the average total cost and average variable cost. When the average total cost and the average variable cost reach their lowest point, the marginal cost is equal to the average cost.