Functions in Business — Cost, Revenue, Demand and Profit

Cost function

A firm's total cost function expresses total cost $C$ as a function of output $x$:

where $F$ is the fixed cost (cost incurred even at zero output — rent, salaries, depreciation) and $v$ is the variable cost per unit (cost that varies directly with output — raw material, labour per unit). This is a linear function of $x$, with $F$ as the intercept and $v$ as the slope.

Average cost at output $x$ is $\overline{C}(x) = \dfrac{C(x)}{x}$.

Marginal cost — the extra cost of producing one more unit — can be approximated at this stage (before differentiation is studied later in this syllabus) as the difference between two consecutive total costs:

Revenue function

If a firm sells $x$ units at a constant price $p$ per unit, total revenue is

— a linear function of $x$. If, instead, price itself must fall to sell more (a common real-world situation), $p$ is itself expressed as a linear function of $x$ (from the demand function below), and substituting it into $R(x) = px$ produces a quadratic revenue function in $x$.

Demand and supply functions

The demand function expresses the quantity demanded $Q_d$ as a function of price $P$:

— a linear function with a negative slope, reflecting the Law of Demand (quantity demanded falls as price rises). The supply function is typically

with a positive slope (quantity supplied rises with price). Market equilibrium is the price and quantity at which $Q_d = Q_s$ — found exactly the way any two linear functions are set equal and solved.

Profit function

Profit is the function

The break-even point is the output level at which $P(x) = 0$, i.e. where $R(x) = C(x)$ — the firm neither makes a profit nor a loss. Beyond the break-even output, $P(x) > 0$ (profit); below it, $P(x) < 0$ (loss). …