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Leibniz

7.3.1 Average and marginal cost functions

The total costs of production for a firm such as Language Perfection (LP) include the rent on the building where the courses are taught, the price of raw materials (including utilities), and the wages of all its employees (teachers). The cost function, \(C(Q)\), describes how the firm’s total costs vary with its output—the number of language courses, \(Q\). In this Leibniz we show how the firm’s average and marginal cost functions are related to \(C(Q)\).

In general, total costs will increase with the quantity of output produced. In what follows, we treat \(Q\) as a continuous variable, a common and useful approximation when dealing with large numbers. It then makes sense to assume that the function \(C\) is differentiable, and describe the fact that it is an increasing function by the inequality:

\[C'(Q)\gt0\]

The average cost (AC) of producing the language courses is defined as the total cost divided by the number of courses sold. Thus if \(Q\) courses are sold:

\[\text{AC} = \frac{C(Q)}{Q}\]

The average cost of producing \(Q\) courses is the slope of the line from the point \((Q, C(Q))\) to the origin. The slope varies with \(Q\): the average cost AC is itself a function of \(Q\).

The marginal cost (MC) is the rate at which costs increase if \(Q\) increases. Thus if \(Q\) courses are sold:

\[\text{MC} =C'(Q)\]

You can interpret MC, as in the text, as the cost of producing one more course: but remember that this is only an approximation. Geometrically, MC is the slope of the curve \(C(Q)\). As mentioned earlier, this cost function has the property that the slope increases as \(Q\) increases: we are assuming that for Language Perfection the cost of producing an additional course is an increasing function of the number of courses already being produced. This means that the marginal cost is an increasing function of \(Q\). However, in the unit, we assume that the marginal cost is constant, i.e. \(C'(Q)=0\).

Now suppose we want see how the average cost function changes with output. By the rule for differentiating a quotient:

\[\frac{d}{dQ} \left( \frac{C(Q)}{Q} \right) = \frac{QC'(Q)- C(Q)}{Q^2}\]

Now \(C'(Q)= \text{MC}\) and \(C(Q) = Q \times \text{AC}\). Therefore:

\[\frac{d}{dQ} (\text{AC} ) = \frac{\text{MC}-\text{AC}}{Q}\]

Since \(Q\gt0\), it follows that the slope of the AC curve at each value of \(Q\) has the same sign as \(\text{MC} - \text{AC}\).

Read more: Sections 6.4 and 8.1 of Malcolm Pemberton and Nicholas Rau. 2015. Mathematics for economists: An introductory textbook, 4th ed. Manchester: Manchester University Press.