Cost curve

In economics, a cost curve is a graph of the costs of production as a function of total quantity produced. In a free market economy, productively efficient firms optimize their production process by minimizing cost consistent with each possible level of production, and the result is a cost curve. Profit-maximizing firms use cost curves to decide output quantities. There are various types of cost curves, all related to each other, including total and average cost curves; marginal ("for each additional unit") cost curves, which are equal to the differential of the total cost curves; and variable cost curves. Some are applicable to the short run, others to the long run.

Notation

There are standard acronyms for each cost concept, expressed in terms of the following descriptors:

SR = short run (costs spent on non-reusable materials e.g raw materials)
LR = long-run (cost spent on renewable materials e.g equipment)
A = average (per unit of output)
M = marginal (for an additional unit of output)
F = fixed (unadjustable)
V = variable (adjustable)
T = total (fixed plus variable)

C = cost

These can be combined in various ways to express different cost concepts (with SR and LR often omitted when the context is clear): one from the first group (SR or LR); none or one from the second group (A, M, or none (meaning “level”)); none or one from the third group (F, V, or T); and the fourth item (C).

From the various combinations we have the following short-run cost curves:

Short-run average fixed cost (SRAFC)
Short-run average total cost (SRAC or SRATC)
Short-run average variable cost (AVC or SRAVC)

Short-run marginal cost (SRMC)
Short-run fixed cost (FC or SRFC)
Short-run total cost (SRTC)
Short-run variable cost (VC or SRVC)

and the following long-run cost curves:

Long-run average total cost (LRAC or LRATC)
Long-run marginal cost (LRMC)
Long-run total cost (LRTC)

Short-run total cost (SRTC) and long-run total cost (LRTC) curves

thumb|right|240px|class=skin-invert-image|The total cost curve, if non-linear, can represent increasing and [[diminishing marginal returns.]]

The short-run total cost (SRTC) and long-run total cost (LRTC) curves are increasing in the quantity of output produced because producing more output requires more labor usage in both the short and long runs, and because in the long run producing more output involves using more of the physical capital input; and using more of either input involves incurring more input costs.

With only one variable input (labor usage) in the short run, each possible quantity of output requires a specific quantity of usage of labor, and the short–run total cost as a function of the output level is this unique quantity of labor times the unit cost of labor. But in the long run, with the quantities of both labor and physical capital able to be chosen, the total cost of producing a particular output level is the result of an optimization problem: The sum of expenditures on labor (the wage rate times the chosen level of labor usage) and expenditures on capital (the unit cost of capital times the chosen level of physical capital usage) is minimized with respect to labor usage and capital usage, subject to the production function equality relating output to both input usages; then the (minimal) level of total cost is the total cost of producing the given quantity of output.

Short-run variable and fixed cost curves (SRVC and SRFC or VC and FC)

thumb|left|240px|One can decompose total costs as the sum of [[fixed costs and variable costs. Here output is measured along the horizontal axis. In the Cost-Volume-Profit Analysis model, total costs are linear in volume.]]

Since short-run fixed cost (FC/SRFC) does not vary with the level of output, its curve is horizontal as shown here. Short-run variable costs (VC/SRVC) increase with the level of output, since the more output is produced, the more of the variable input(s) needs to be used and paid for.

Short-run average variable cost curve (AVC or SRAVC)

thumb|400px|right|class=skin-invert-image| A U-shaped short-run Average Cost (AC) curve. AVC is the Average Variable Cost, AFC the Average Fixed Cost, and MC the marginal cost curve crossing the minimum of both the Average Variable Cost curve and the Average Cost curve.

Average variable cost (AVC/SRAVC) (which is a short-run concept) is the variable cost (typically labor cost) per unit of output: SRAVC = wL / Q where w is the wage rate, L is the quantity of labor used, and Q is the quantity of output produced. The SRAVC curve plots the short-run average variable cost against the level of output and is typically drawn as U-shaped. However, whilst this is convenient for economic theory, it has been argued that it bears little relationship to the real world. Some estimates show that, at least for manufacturing, the proportion of firms reporting a U-shaped cost curve is in the range of 5 to 11 percent.

Short-run average fixed cost curve (SRAFC)

Since fixed cost by definition does not vary with output, short-run average fixed cost (SRAFC) (that is, short-run fixed cost per unit of output) is lower when output is higher, giving rise to the downward-sloped curve shown.

Short-run and long-run average total cost curves (SRATC or SRAC and LRATC or LRAC)

The average total cost curve is constructed to capture the relation between cost per unit of output and the level of output, ceteris paribus. A perfectly competitive and productively efficient firm organizes its factors of production in such a way that the usage of the factors of production is as low as possible consistent with the given level of output to be produced. In the short run, when at least one factor of production is fixed, this occurs at the output level where it has enjoyed all possible average cost gains from increasing production. This is at the minimum point in the above diagram.

Short-run total cost is given by

:<math>STC = P_{K} \cdot K + P_{L} \cdot L</math>,

where PK is the unit price of using physical capital per unit time, PL is the unit price of labor per unit time (the wage rate), K is the quantity of physical capital used, and L is the quantity of labor used. From this we obtain short-run average cost, denoted either SATC or SRAC, as STC / Q:

:<math>SRATC\ or\ SRAC = \frac{P_{K} \cdot K}{Q} + \frac{P_{L} \cdot L}{Q} = \frac{P_{K} \cdot P_{K{A} + \frac{P_{L} \cdot P_{L{A} </math>,

where <math display="inline">AP_{K} = \frac{Q}{K}</math> is the average product of capital and <math display="inline">AP_{L} = \frac{Q}{L}</math> is the average product of labor.

Within the graph above, the Average Fixed Cost curve and Average Variable Cost curve cannot start with zero, as at quantity zero these values are not defined since they would involve dividing by zero.

Short-run average cost (SRATC/SRAC) equals average fixed costs plus average variable costs. Average fixed cost continuously falls as production increases in the short run, because K is fixed in the short run. The shape of the average variable cost curve is directly determined by increasing and then diminishing marginal returns to the variable input (conventionally labor).

The long-run average cost (LRATC/LRAC) curve looks similar to the short-run curve, but it allows the usage of physical capital to vary.

Short-run marginal cost curve (SRMC)

thumb|right|class=skin-invert-image|Typical marginal cost curve

A short-run marginal cost (SRMC) curve graphically represents the relation between marginal (i.e., incremental) cost incurred by a firm in the short-run production of a good or service and the quantity of output produced. This curve is constructed to capture the relation between marginal cost and the level of output, holding other variables, like technology and resource prices, constant. The marginal cost curve is usually U-shaped. Marginal cost is relatively high at small quantities of output; then as production increases, marginal cost declines, reaches a minimum value, then rises. The marginal cost is shown in relation to marginal revenue (MR), the incremental amount of sales revenue that an additional unit of the product or service will bring to the firm. This shape of the marginal cost curve is directly attributable to increasing, then decreasing marginal returns (and the law of diminishing marginal returns). Marginal cost equals w/MPL.

Long-run marginal cost curve (LRMC)

The long-run marginal cost (LRMC) curve shows for each unit of output the added total cost incurred in the long run, that is, the conceptual period when all factors of production are variable. Stated otherwise, LRMC is the minimum increase in total cost associated with an increase of one unit of output when all inputs are variable.

The long-run marginal cost curve is shaped by returns to scale, a long-run concept, rather than the law of diminishing marginal returns, which is a short-run concept. The long-run marginal cost curve tends to be flatter than its short-run counterpart due to increased input flexibility. The long-run marginal cost curve intersects the long-run average cost curve at the minimum point of the latter. Because the production function determines the variable cost function it necessarily determines the shape and properties of marginal cost curve and the average cost curves. Likewise, it has diseconomies of scale (is operating in an upward sloping region of the long-run average cost curve) if and only if it has decreasing returns to scale, and has neither economies nor diseconomies of scale if it has constant returns to scale. In this case, with perfect competition in the output market the long-run market equilibrium will involve all firms operating at the minimum point of their long-run average cost curves (i.e., at the borderline between economies and diseconomies of scale).

If, however, the firm is not a perfect competitor in the input markets, then the above conclusions are modified. For example, if there are increasing returns to scale in some range of output levels, but the firm is so big in one or more input markets that increasing its purchases of an input drives up the input's per-unit cost, then the firm could have diseconomies of scale in that range of output levels. On the other hand, if the firm is able to get bulk discounts of an input, then it could have economies of scale in some range of output levels even if it has decreasing returns in production in that output range.

Relationship between different curves

Total Cost = Fixed Costs (FC) + Variable Costs (VC) = Average Total Cost (ATC) x Quantity (Q)
Marginal Cost (MC) = dC/dQ; MC equals the slope of the total cost function and of the variable cost function
Average Total Cost (ATC) = Total Cost/Q
Average Fixed Cost (AFC) = FC/Q
Average Variable Cost (AVC) = VC/Q.
ATC = AFC + AVC
At a level of Q at which the MC curve is above the average total cost or average variable cost curve, the latter curve is rising.
If MC is below average total cost or average variable cost, then the latter curve is falling.
If MC equals average total cost, then average total cost is at its minimum value.
If MC equals average variable cost, then average variable cost is at its minimum value.

Relationship between short-run and long-run cost curves

For each quantity of output there is one cost–minimizing level of capital and a unique short–run average cost curve associated with producing the given quantity. The following statements assume that the firm is using the optimal level of capital for the quantity produced. If not, then the SRAC curve would lie "wholly above" the LRAC and would not be tangent at any point.

Each STC curve can be tangent to the LRTC curve at only one point. The STC curve cannot cross (intersect) the LRTC curve.
One STC curve is tangent to LRTC at the long–run cost–minimizing level of production. At the point of tangency LRTC = STC. At all other levels of production STC will exceed LRTC.
Average cost functions are the total cost function divided by the level of output. Therefore, the SATC curve is also tangent to the LRATC curve at the cost-minimizing level of output. At the point of tangency LRATC = SATC. At all other levels of production SATC > LRATC
The slope of the total cost curves equals marginal cost. Therefore, when STC is tangent to LTC, SMC = LRMC.
At the long–run cost–minimizing level of output LRTC = STC; LRATC = SATC and LRMC = SMC,.
With fixed unit costs of inputs, if the production function has constant returns to scale, then at the minimal level of the SATC curve we have SATC = LRATC = SMC = LRMC.
With decreasing returns, minimum SRAC occurs at a lower production level than minimum LRAC because a firm could reduce average costs by simply decreasing the size or its operations.
The minimum of a SRAC occurs when the slope is zero. Thus the points of tangency between the U-shaped LRAC curve and the minimum of the SRAC curve would coincide only with that portion of the LRAC curve exhibiting constant economies of scale. For increasing returns to scale the point of tangency between the LRAC and the SRAC would have to occur at a level of output below level associated with the minimum of the SRAC curve.

U-shaped curves

Both the SRAC and LRAC curves are typically expressed as U-shaped. while the upward slope is due to diminishing marginal returns to the variable input.