7.4.1 ISOPROFIT CURVES AND THEIR SLOPESA firm’s profit is the difference between its revenue (the pricemultiplied by quantity sold) and its total costs. If we know thefirm’s cost function, , we can determine its isoprofit curves—the combinations of and 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 manufacturing firm such asBeautiful Cars, profit depends on the quantity of output produced ( ) andthe price ( ) at which each unit of output can be sold. We denote profit by

as before. If the firm’s cost function is , then its profit can bewritten as a function of and :

The isoprofit curves are a family of curves in the -plane, each of whichcorresponds to a given level of profit. The equation of a typical isoprofitcurve is:

where is a constant representing the level of profit. There is a differentcurve for each value of . We will represent the isoprofit curves in a diagramwith on the vertical axis, so it is helpful to rewrite this equation in a formthat expresses as a function of :

This equation implies that if increases, then also increases for any given. 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 thediagrams in the text for Apple Cinnamon Cheerios (Figure 7.4) andBeautiful Cars (Figure 7.10) of the text, reproduced here as Figures 1 and 2.

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Figure 1 Isoprofit curves for Apple Cinnamon Cheerios.

Figure 2 Isoprofit curves for Beautiful Cars.

We now explain why the isoprofit curves for these two firms have theshapes shown in this diagram. The equation of the isoprofit curvecorresponding to the level of profit may be written:

or equivalently

Focus first on the case where : the zero-economic-profit curve. Settingin the equation above shows that the zero-economic-profit curve is

the average cost (AC) curve. At all points below this curve in the diagram,the firm would be making a loss. For Apple Cinnamon Cheerios the averagecost is constant: each pound costs $2 to produce, whether the total quantityis large or small. So the zero-economic-profit curve is a horizontal line at

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. Beautiful Cars has a U-shaped average cost curve and hence a U-shaped zero-economic-profit curve.

Now consider the curves corresponding to positive levels of profit,Then the equation of the isoprofit curve expresses as the sum of AC

and . Notice that is high when is small, and

So is a decreasing, convex function of .The shape of the isoprofit curves depends on the shapes of and the

AC curve. In the case of Apple Cinnamon Cheerios this is particularlysimple. AC is a horizontal line and the equation of the isoprofit curves is

. So the isoprofit curves are decreasing and convex, like , aswe see in Figure 1.

For Beautiful Cars the AC curve is U-shaped, and therefore convex, witha minimum point at (point B). The isoprofit curve corresponding toa level of profit must then be convex too, since the sum of two convexfunctions is always convex (the second derivative of is

, which is positive if and are positive).If , both and are decreasing functions of , so

the isoprofit curve slopes downward. If is large, the derivative of isclose to zero, so the slope of the isoprofit curve is almost the same as theslope of —the isoprofit curve slopes upward (as does the AC curve).Hence the isoprofit curve for , like the AC curve, is U-shaped, with aminimum point at some positive value of .

Let be the value of where the minimum occurs. Notice thatdepends on . We know that all the isoprofit curves slope downward until

, so : the minimum point on an isoprofit curve with isto the right of the minimum point of the zero-profit curve. A similarargument shows that as we increase , also increases: isoprofit curvescorresponding to higher levels of profit have their minimum point furtherto the right (Figure 2).

We have now explained why the isoprofit curves for Beautiful Cars areU-shaped. The other property you can see in Figure 2 is that the marginalcost curve passes through the minimum points of the isoprofit curves. InLeibniz 7.3.1 we proved that this is true for the AC curve (the zero-iso-profit-curve) by showing that always has the same sign as theslope of the AC curve. We now use the same approach for the slopes of theother isoprofit curves.

Consider the isoprofit curve corresponding to a profit of . Alongthis curve:

Thus is the difference of two terms, the first of which is the slope ofthe AC curve; we showed in Leibniz 7.3.1 (using the quotient rule) that thisis . Also, we know from the equation of the isoprofit curvethat . Therefore:

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Simplifying the right-hand side, we see that:

This equation tells us the slope at any point on the isoprofit curve. When is small is high—above marginal cost MC—and

the curve slopes down. So as increases, decreases; this continues as longas . In the case of Beautiful Cars we eventually reach a point where

and at that point, the equation tells us that the slope is zero: wehave reached a minimum point of the isoprofit curve. The MC-curve slopesupward through this point. And so beyond this point, and the iso-profit curve slopes upward too.

What about the case of Apple Cinnamon Cheerios? Since the unit cost ofa pound of Cheerios is $2 whatever the level of production, both the mar-ginal and average cost is $2. The zero-isoprofit-curve is not only the AC-curve, but the MC-curve as well. The equation of any isoprofit curve can bewritten as . So if , then , which means that theslope is always negative. As you can see in Figure 1, all the positive isoprofitcurves 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.

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