Leibniz

7.4.1 Isoprofit curves and their slopes

A firm’s profit is the difference between its revenue (the price multiplied by quantity sold) and its total costs. If we know the firm’s cost function, $C(Q)$, we can determine its isoprofit curves—the combinations of $P$ and $Q$ that give the same amount of profit. In this Leibniz, we obtain the equation of an isoprofit curve, explain its shape and find its slope.

Economic profit is revenue minus costs. For a firm such as Language Perfection (LP), profit depends on the amount of courses sold ($Q$) and the price ($P$) at which each course can be sold. We denote profit by $\Pi,$ as before. If the firm’s cost function is $C(Q)$, then its profit can be written as a function of $P$ and $Q$:

$$\Pi = PQ - C(Q)$$

The isoprofit curves are a family of curves in the $QP$-plane, each of which corresponds to a given level of profit. The equation of a typical isoprofit curve is:

$$PQ - C(Q) = k$$

where $k$ is a constant representing the level of profit. There is a different curve for each value of $k$. We will represent the isoprofit curves in a diagram with $P$ on the vertical axis, so it is helpful to rewrite this equation in a form that expresses $P$ as a function of $Q$:

$$P = \frac{C(Q) + k}{Q}$$

This equation implies that if $k$ increases, then $P$ also increases for any given $Q$. This means that in a diagram depicting the family of isoprofit curves, higher curves correspond to higher levels of profit. You can see this in the diagram in the text for Language Perfection (Figure 7.7), reproduced here as Figure 1.

We now explain why the isoprofit curves for LP have the shapes shown in this diagram. The equation of the isoprofit curve corresponding to the level of profit $k$ may be written:

$$P = \frac{C(Q)}{Q} + \frac{k}{Q}$$

or equivalently

$$P = \text{AC} + \frac{k}{Q}$$

Focus first on the case where $k = 0$: the zero-economic-profit curve. Setting $k = 0$ in the equation above shows that the zero-economic-profit curve is the average cost (AC) curve (here it’s called unit cost curve). At all points below this curve in the diagram, the firm would be making a loss. For LP the average cost is constant: each course costs Rs. 360 to produce, whether the total quantity is large or small. So the zero-economic-profit curve is a horizontal line at $P = 360$.

Now consider the curves corresponding to positive levels of profit, $k\gt0.$ Then the equation of the isoprofit curve expresses $P$ as the sum of AC and $k/Q$. Notice that $k/Q$ is high when $Q$ is small, and

$$\frac{d}{dQ} \left( \frac{k}{Q} \right) = - \frac{k}{Q^2} \lt0, \quad \frac{d^2}{dQ^2} \left( \frac{k}{Q} \right) = \frac{2k}{Q^3} \gt0$$

So $k/Q$ is a decreasing, convex function of $Q$.

The shape of the isoprofit curves depends on the shapes of $k/Q$ and the AC curve. In the case of LP this is particularly simple. AC is a horizontal line and the equation of the isoprofit curves is $P = 2+k/Q$. So the isoprofit curves are decreasing and convex, like $k/Q$, as we see in Figure 1.

The other property is that the marginal cost curve passes through the minimum points of the isoprofit curves. In Leibniz 7.3.1 we proved that this is true for the AC curve (the zero-isoprofit-curve) by showing that $\text{MC} - \text{AC}$ always has the same sign as the slope of the AC curve. We now use the same approach for the slopes of the other isoprofit curves.

Consider the isoprofit curve corresponding to a profit of $k\gt0$. Along this curve:

$$\frac{dP}{dQ} = \frac{d} {dQ} \left( \frac{C(Q)}{Q} \right) - \frac{k}{Q^2}$$

Thus $dP/dQ$ is the difference of two terms, the first of which is the slope of the AC curve; we showed in Leibniz 7.3.1 (using the quotient rule) that this is $(\text{MC} - \text{AC)}/Q$. Also, we know from the equation of the isoprofit curve that $k/Q = P - \text{AC}$. Therefore:

$$\frac{dP}{dQ} = \frac{\text{MC} - \text{AC}}{Q} - \frac{P - \text{AC}}{Q}$$

Simplifying the right-hand side, we see that:

$$\frac{dP}{dQ} = \frac{\text{MC} - P}{Q}$$

This equation tells us the slope at any point on the isoprofit curve $P = \text{AC}+ k/Q$. When $Q$ is small $P$ is high—above marginal cost MC—and the curve slopes down. So as $Q$ increases, $P$ decreases; this continues as long as $P\lt \text{MC}$.

What about the case of LP? Since the unit cost of a language course is Rs. 360 whatever the level of production, both the marginal and average cost is Rs. 360. The zero-isoprofit-curve is not only the AC-curve, but the MC-curve as well. The equation of any isoprofit curve can be written as $P = \text{MC}+ k/Q$. So if $k \gt 0$, then $P \gt \text{MC}$, which means that the slope is always negative. As you can see in Figure 1, all the positive isoprofit curves slope downward, but never meet the MC-curve.

Read more: Chapter 8 of Malcolm Pemberton and Nicholas Rau. 2015. Mathematics for economists: An introductory textbook, 4th ed. Manchester: Manchester University Press.