Cost, Revenue and Marginal Analysis

The most direct commercial use of the derivative is marginal analysis — measuring the effect on cost, revenue or profit of producing or selling one more unit. Because a derivative is precisely the rate of change of one quantity with respect to another, the marginal quantities in economics are simply derivatives of the corresponding total functions.

Let $x$ denote the number of units produced and sold. The basic functions are:

• Total cost $C = C(x)$ — the cost of producing $x$ units. It usually has a fixed part (independent of $x$) and a variable part.
• Total revenue $R = R(x)$ — the money received from selling $x$ units. If each unit sells at price $p$ (which may itself depend on $x$ through the demand relation), then $R = p \cdot x$.
• Profit $\pi = R(x) - C(x)$ — revenue minus cost.

Average and marginal functions. From the total cost we form:

Average cost is the cost per unit over all $x$ units; marginal cost is the (approximate) cost of producing the next unit — the instantaneous rate at which total cost rises as output rises. Likewise,

Marginal revenue is the additional revenue from selling one more unit. (Note that average revenue always equals the price $p$, since $R = px$.) …