Marginal Cost - Cost Functions and Relationship To Average Cost

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.

Since (by definition) fixed costs do not vary with production quantity, it drops out of the equation when it is differentiated. The important conclusion is that marginal cost is not related to fixed costs. This can be compared with average total cost or ATC, which is the total cost divided by the number of units produced and does include fixed costs.

For discrete calculation without calculus, marginal cost equals the change in total (or variable) cost that comes with each additional unit produced. In contrast, incremental cost is the composition of total cost from the surrogate of contributions, where any increment is determined by the contribution of the cost factors, not necessarily by single units.

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 the second shoe is $40 - $30 = $10.

Marginal cost is not the cost of producing the "next" or "last" unit. As Silberberg and Suen note, 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 costs 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 (∂AC/∂q)q. The first component is the per unit or average cost. The second unit is the small increase in costs due to the law of diminishing marginal returns which increases the costs of all units of sold. Therefore, the precise formula is: MC = AC + (∂AC/∂q)q.

Marginal costs can also be expressed as the cost per unit of labor divided by the marginal product of labor.

MC = ∆VC∕∆q;

∆VC = w∆L;

MC = w∆L;/∆q;

∆L∕∆q the change in quantity of labor to affect a one unit change in output = 1∕MPL.

Therefore MC = w/MPL Since the wage rate is assumed constant marginal cost and marginal product of labor have an inverse relationship - if marginal cost is increasing (decreasing) the marginal product of labor is decreasing (increasing).