1.65 Cost and Break-Even Model Definitions

Cost and Break-Even Model Definitions establishes the vocabulary for translating business-related cost and revenue situations into linear equations, describing the portion of total cost that does not depend on quantity produced, the portion that does, the linear relationship representing money earned from sales, and the specific quantity at which total cost and total revenue become equal.

Fixed Cost

Fixed cost is the portion of a total cost that remains constant regardless of how many units are produced or sold, such as rent, equipment purchases, or salaried wages that must be paid whether zero units or a thousand units are produced. In a linear cost model, fixed cost corresponds to the constant term, representing the cost incurred even at a production quantity of zero.

Unit Variable Cost

Unit variable cost is the additional cost incurred for producing one more unit, remaining constant per unit even as total variable cost grows with quantity, such as the cost of raw materials needed for a single item. In a linear cost model, unit variable cost corresponds to the coefficient multiplying the quantity variable, and multiplying it by the number of units produced gives the total variable cost, which is then added to the fixed cost to obtain the overall total cost.

In this cost model, m represents the unit variable cost, x represents the quantity produced, and b represents the fixed cost.

Linear Revenue

Linear revenue is a linear model representing the total money earned from selling a given quantity of units, expressed as the selling price per unit multiplied by the number of units sold, with no separate constant term, since revenue is zero whenever zero units are sold. A linear revenue model takes the form R(x) = px, where p is the selling price per unit and x is the number of units sold.

Break-Even Quantity

The break-even quantity is the specific number of units at which total cost and total revenue are equal, meaning neither a profit nor a loss is being made, found by setting the cost model equal to the revenue model and solving for the shared quantity variable. Setting C(x) = R(x) and solving for x yields the break-even quantity, and producing fewer units than this value results in a loss, while producing more units results in a profit, assuming the unit variable cost and selling price remain unchanged.

Together, these definitions describe the standard linear framework for modeling business cost and revenue: fixed cost and unit variable cost combine into a total cost model, linear revenue models the money earned from sales, and the break-even quantity marks the specific point where these two linear models intersect, separating a loss-producing quantity from a profit-producing one.