Editing Marginal cost (section)

== 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 cost]]s 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]] (ATC), which is the total cost (including fixed costs, denoted C<sub>0</sub>) divided by the number of units produced:

:<math>ATC=\frac{C_0 + \Delta C}{Q}.</math>

For discrete calculation without [[calculus]], marginal cost equals the change in total (or variable) 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 ($40 – $30 = $10). In another example, when a fixed cost is associated, the marginal cost can be calculated as presented in the table below.

{| class="wikitable"
|+ Marginal Cost Example
|-
! Output (units) !! Total Cost !! Average Cost !! Marginal Cost
|-
| 0|| 10 (Fixed Cost)|| ∞|| –
|-
| 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.<ref>Silberberg & Suen, The Structure of Economics, A Mathematical Analysis 3rd ed. (McGraw-Hill 2001) at 181.</ref> 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 &mdash; conventionally assumed to be labor. Adding more labor to a fixed capital stock reduces the marginal product of labor because of the [[diminishing returns|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.<ref>See http://ocw.mit.edu/courses/economics/14-01-principles-of-microeconomics-fall-2007/lecture-notes/14_01_lec13.pdf.</ref> Denoting variable cost as VC, the constant wage rate as w, and labor usage as L, we have
:<math> MC = \frac{\Delta VC}{\Delta Q}</math>
:<math> \Delta VC = {w \Delta L}</math>

:<math> MC = \frac{w \Delta L}{\Delta Q}=\frac{w}{MPL}.</math>

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 <math> \frac{\Delta L}{\Delta Q}</math> is the change in quantity of labor that brings about a one-unit change in output.<ref>Chia-Hui Chen, course materials for 14.01 Principles of Microeconomics, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [12 Sept 2009].</ref> Since the wage rate is assumed constant, marginal cost and marginal product of labor have an inverse relationship&mdash;if the marginal product of labor is decreasing (or, increasing), then marginal cost is increasing (decreasing), and AVC = VC/Q=wL/Q = w/(Q/L) = w/AP<sub>L</sub>