Marginal cost is the derivative of the sum of fixed costs and variable costs of production divided by the quantity produced. This measure of cost allows the producer to understand the cost required to produce each additional unit of output. Marginal costs are defined differently in the short and long term, due to the producer's inability to increase fixed costs in the short term.
Basic Definition

In the most basic sense, marginal cost is the affect on total cost caused by a unit change in production. The total cost is the summation of the variable costs (expenses that change with business activitysuch as material supplies) and fixed costs (expenses unrelated to business activitysuch as rent). You may also simply think of marginal cost as the total cost required to produce one more good. Due to the complexity of market conditions and production, marginal cost is usually examined in both the short term and long term. Other terms that may assist the producer in evaluating production costs are the averages for fixed, variable and total costs.
Mathematical Definition

You can use basic calculus to define marginal cost mathematically. First, you must take the first derivative of the total cost function. Then, you must find the derivative of the quantity function. The marginal cost is thus the derivative of the total cost relative to the derivative of the quantity.
In equation form, marginal cost is written as: MC = dTC/dQ.
Short Term

To calculate marginal cost in the short term, you must keep the fixed costs as constant; they must remained unchanged. The shortterm calculation refers to changes in variable inputs, such as labor and materials, and its affect on total cost.
When looking at a shortterm marginal cost curve on a graph, notice how the curve is steep. This is due to the affect of the law of diminishing marginal returns on the marginal cost curve. The law of diminishing marginal returns states the point at which a decline in production occurs as the factors of production increase.
A simple way of explaining this would be to describe a family attempting to cook a Thanksgiving meal together. Productivity will increase with help from each additional family member, but only to a point. Once the entire family is in the kitchen, productivity will actually begin to decrease. It becomes impossible to increase productivity without enlarging the kitchen or buying extra ovens so that the additional family members may work more efficiently. Obviously, this is not possible in the short term, as fixed costs (kitchen expansion, investment in longterm appliances) are held constant.
Long Term

In the long term, the fixed costs are not held constant. The graph of a longterm marginal cost curve is much flatter than the shortterm marginal cost curve. This is due to the fact that the longterm curve is shaped by economies of scale. The principle of economies of scale takes into account the cost benefits due to expansion that lead to an increase in production. The producer wants the average cost per unit of production to decrease as the scale, or number of inputs, increases.
Other Terms

Other cost terms, in addition to marginal cost and total cost, that are necessary to fully understand the producer's costs of production are average cost, average total cost, average fixed cost and average variable cost. These averages of cost are useful to the producer as they all directly affect the supply curve and allow the producer to see a variability in cost in both the short and long term.
References
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