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 use these curves to find the optimal point of production, where they make the most profits. There are a few different types of cost curves, each relevant to a different area of economics.

The short-run average total cost curve (SATC or SAC)

Typical short run average cost curve

The average total cost curve is constructed to capture the relation between cost per unit and the level of output, ceteris paribus. A productively efficient firm organizes its factors of production in such a way that the average cost of production is at lowest point and intersects Marginal Cost. In the short run, when at least one factor of production is fixed, this occurs at the optimum capacity where it has enjoyed all the possible benefits of specialization and no further opportunities for decreasing costs exist. This is usually not U shaped, it is a checkmark shaped curve. This is at the minimum point in the diagram on the right.Example: Q=2K^.5L^.5 STC=Pk(K)+Pw(Q²/4K) SATC or SAC= (Pk(K)/Q)+Pw(Q/4K)Short run average cost equals average fixed costs plus average variable costs. Average fixed cost continuously falls as production increases. The shape of the average variable cost curve is directly determined by diminishing marginal returns to the variable input (conventionally labor). ^[1] Average variable cost eqauls w/AP_L or the wage rate divided by the average product of labor.^[2]

The long-run average cost curve (LRAC)

Typical long run average cost curve

Essentially, the long-run average cost curve depicts what the minimum per-unit cost of producing a certain number of units would be if all productive inputs could be varied. Given that LRAC is an average quantity, one must not confuse it with the long-run marginal cost curve, which is the cost of one more unit. The LRAC curve is created as an envelope of an infinite number of short-run average total cost curves. The typical LRAC curve is U-shaped, reflecting economies of scale when negatively-sloped and diseconomies of scale when positively sloped. Contrary to Viner, the envelope is not created by the minimum point of each short-run average cost curve. This mistake is recognized as Viner's Error.

In a long-run perfectly competitive environment, the equilibrium level of output corresponds to the minimum efficient scale, marked as Q2 in the diagram. This is due to the zero-profit requirement of a perfectly competitive equilibrium. This result, which implies production is at a level corresponding to the lowest possible average cost, does not imply that other production levels are not efficient. All points along the LRAC are productively efficient, by definition, but are not equilibrium points in a long-run perfectly competitive environment.

In some industries, the LRAC is always declining (economies of scale exist indefinitely). This means that the largest firm tends to have a cost advantage, and the industry tends naturally to become a monopoly, and hence is called a natural monopoly. Natural monopolies tend to exist in industries with high capital costs in relation to variable costs, such as water supply and electricity supply.

The average cost is the total cost divided by the number of units produced.

The marginal cost curve (MC)

Typical marginal cost curve

A marginal cost that graphically represents the relation between marginal cost incurred by a firm in the short-run product 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 U-shaped. Marginal cost is relatively high at small quantities of output, then as production increases, declines, reaches a minimum value, then rises. The marginal cost is shown in relation to marginal revenue, the incremental amount of sales that an additional 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 - Diminishing returns). Marginal cost equal w/MP_L.^[3] For most production processes the marginal product of labor initially rises, reaches a maximum value and then continuously falls as production increases. Thus marginal cost initially falls, reaches a minimum value and then increases. ^[4]

The long run marginal cost curve is the minimum cost incurred per unit change in output when all factors of production are variable. The long run marginal cost curve is shaped by economies of scale rather than the law of diminishing marginal returns. The long run marginal cost curve tends to be flatter than its short run counterpart due to increased input flexibility.

Combining cost curves

Cost curves in perfect competition compared to marginal revenue

Cost curves can be combined to provide information about firms. In this diagram for example, firms are assumed to be in a perfectly competitive market. The marginal cost curve will cut the average cost curve at its lowest point. In a perfectly competitive market a firm's profit maximising price would be at or above the price at which the average cost curve cuts the marginal cost curve. If the marginal revenue is above the average total cost price the firm is deriving an economic profit.

Cost curves and production functions

Assuming that factor prices are constant, the production function determines all cost functions. ^[5]The variable cost curve is the inverted short run production function or total product curve and its behavior and properties are determined by the production function. ^[6] Because the production function determines the variable cost function it necessarily determines the shape and properties of marginal cost curve and the average cost functions. ^[7]

Cost functions

• Total Cost = Fixed Costs (FC) + Variable Costs (VC)

C = 420 + 60Q + Q²

FC = 420

VC = 60Q + Q²

• Marginal Costs (MC) = ∂C/∂Q

MC = 60 +2Q

• MC equals slope of the total cost function and the variable cost function.

VC' = 60 +2Q = MC

C' = 60 + 2Q = MC

• Average Total Cost (ATC) = Total Cost/Q

ATC = (420 + 60Q + Q²)/Q

ATC = 420/Q + 60 + Q

• Average Fixed Cost (AFC) = FC/Q

AFC = 420/Q

• Average Variable Costs = VC/Q

AVC = (60Q + Q²)/Q

AVC = 60 + Q

• MC curve determines the shape of the ATC and AVC functions.^[8]

If MC curve is above average cost or average variable cost curves, then curves are rising.^[9]

If MC is below average cost curve or average variable cost curve, then curves are falling.^[10]

If MC equals average cost, then average cost is at its minimum value.^[11]

if MC equals average variable cost, then average variable cost is at its minimum value.^[12]

Relationship of functions – Example: MC = ATC at minimum ATC

• To find minimum ATC take the first derivative of the ATC function and set it equal to zero and solve for Q.

ATC = 420/Q + 60 + Q

ATC = 420Q-1 + 60 + Q

ATC’ = -420Q-2 +1

ATC’ = (-420/Q2) + 1

ATC’ = 0

(420/Q2) + 1 = 0

-420/Q2 = -1

-420 = -Q2

Q = √420

Q = 20.494

MC = 60 +2Q

MC = 60 +2(20.494)

MC = 60 + 40.988

MC = 100.988

ATC = 420/Q + 60 + Q

ATC = 420/(20.494) + 60 + 20.494

ATC = 20.494 + 60 + 20.494

ATC = 100.988

Relationship between short run and long run cost functions

1. The SRTC can be tangent to the LRTC at only one point. The SRTC cannot intersect the curve. ^[13][14]The SRTC can lie wholly “above” the curve with no tangency point.^[15]

2. The SRTC curve is tangent to LRTC at long run cost minimizing level of production. At the point of tangency LRTC = SRTC. At all other levels of production SRTC will exceed LRTC.^[16]

3. Average cost functions are the total cost function divided by the level of output. Therefore the LRATC is also tangent to the SRATC at cost minimizing level of output. At the point of tangency LRATC = SRATC. At all other levels of production SRATC > LRATC^[17]

4. The slope of the total cost curves equals marginal cost. Therefore when LRTC is tangent to SRTC, SRMC = LRMC.^[18]

5. At the long run cost minimizing level of output LRTC = SRTC; SRATC = LRATC and SRMC = LRMC.^[19]

6. The long run cost minimizing level of output may be different from minimum SRATC. ^[20][21]

7. If constant returns to scale then (min) SRATC = LRATC = SRMC = LRMC.^[22]

8. If increasing returns to scale (min) SRATC will occur at higher level of production than long run cost minimizing level of production.^[23] While LRTC = SRTC; SRATC = LRATC and SRMC = LRMC. SRMC does not equal LRMC and LRMC does not equal LRAC.

9. With decreasing returns (min) SRATC will be tangent to the LRAC curve at a lower level of production lower than long run cost minimizing level of production.^[24] While LRTC = SRTC; SRATC = LRATC and SRMC = LRMC. SRMC does not equal LRMC and LRMC does not equal LRAC

10. A firm that is experiencing increasing (decreasing) returns to scale and is producing at min SRAC can always reduce average cost by expanding (reducing) the use of the fixed input.^[25][26]

U-shaped curves

Both the SRAC and LRAC curves are typically expressed as U-shaped.^[27] However, the shape of the curves are not due to the same factors. For the short run curve the initial downward slope is largely due to declining average fixed costs^[28] while the upward slope is due to diminishing marginal returns to the variable input. ^[29]With the long run curve the shape is due to economies of scale.^[30] At low levels of production long run production functions generally exhibit increasing returns to scale which means that the long run average costs is falling.^[31] The upward slope of the long run average cost function is due to decreasing returns to scale which set in at relatively high levels of production.^[32]

See also

• Cost
• Economic cost
• Point of total assumption

Footnotes

Reference

1. Binger, B & Hoffman, E,. 1998 Microeconomics with Calculus, 2nd ed. Addison-Wesley. ISBN 0321012259

2. Kreps, D.: 1990 A Course in Microeconomic Theory Princeton. ISBN 0691042640

3. Melvin & Boyes, 2002 Microeconomics 5th ed. Houghton Mifflin.

4. Perloff, J. 2009 Microeconomics 5th ed. Pearson. ISBN 0321564391

5. Perloff, J: 2008 Microeconomics Theory & Applications with Calculus Pearson. ISBN 9780321277947

6. Pindyck, R & Rubinfeld, D: 2001. Microeconomics 5th ed. Prentice-Hall. ISBN 0130196738