marginal cost The effect on totalcost of producing one additionalunit of output. It corresponds to theslope of the total cost function ateach point.marginal revenue The increase inrevenue obtained by increasing thequantity from Q to Q + 1.

7.6.1 MARGINAL REVENUE AND MARGINAL COSTOne way to determine the price and quantity that maximize theprofits of a firm such as Beautiful Cars is to find the point wherethe demand curve is tangent to an isoprofit curve. This Leibnizintroduces an alternative method using the firm’s marginalrevenue and marginal cost.

Remember that Beautiful Cars’ profit, , is equal to its total revenue fromselling cars, minus its total cost of producing them:

The inverse demand curve, , tells us the maximum price at whichcars can be sold, so we can write revenue as a function of alone, which

we call the revenue function and denote by . Thus:

Revenue at any point on the demand curve can be represented graphicallyas the red rectangle below the curve, as shown in Figure 7.12a of the text,reproduced as Figure 1.

The expression for profits, above, can be written as a function of output, as the difference between the total revenue function and total cost

:

To find the value of that maximizes profit, we differentiate with respectto , to obtain the first-order condition , which implies that:

The term on the right-hand side of the equation is the firm’s mar-ginal cost (MC)—the rate at which cost increases as output rises. Similarly,

the derivative of the revenue function, is the rate at which revenuerises with output, and it is known as marginal revenue (MR). Thus, thefirst-order condition for profit maximization may be written as:

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Figure 1 Calculating marginal revenue.

Figure 2 Marginal revenue and marginal cost.

So the first-order condition tells us that, when is at its profit-maximizinglevel, the marginal revenue is equal to the marginal cost.

The marginal cost curve (that is, the function ) shows how mar-ginal cost changes as output changes. In the case of Beautiful Cars, we knowthat marginal cost increases with output, so the MC curve is upward-sloping. Similarly, the function is the marginal revenue curve,showing how marginal revenue changes with output. In the text we drewthe MR curve as downward-sloping. Figure 1 reproduces the middle panelof Figure 7.12b from the text, showing both curves.

The profit-maximizing quantity lies at the point where the two curvescross—at point E in Figure 2, where . Since Beautiful Cars hasan upward-sloping MC and downward-sloping MR, there is just one pointof intersection.

At point E, the company produces 32 cars. As explained in theinteractive for Figure 7.12b in the text, we can see that this is profit-maximizing by noting that the marginal cost of producing more than 32

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cars would be greater than the marginal revenue generated (MC > MR),while the opposite would be true if fewer than 32 were produced.

But note that if the curves had sloped differently this argument mightnot have worked. If MC sloped downward (which can happen, if the firmhas economies of scale) and MR sloped upward (which would be unusual,but can happen for some demand functions), the point of intersectionwould be a profit-minimizing point (try drawing the curves and explainingto yourself why this must be true).

In general, if we can find a solution to the first-order conditionMC = MR, we can say that it is the profit-maximizing quantity if MC < MRwhen and MC > MR when .

The relationship between the two methodsWe now show that the first-order condition for profit maximizationderived above, , is equivalent to the first-order condition forprofit maximization given in Leibniz 7.5.1. Using the rule for dif-ferentiating a product to differentiate , we see that:

Thus the first-order condition may be written:

Rearranging,

which is the first-order condition of Leibniz 7.5.1. Remember that it can beinterpreted as saying that the slope of the demand curve is equal to theslope of the isoprofit curve.

We drew very different diagrams to illustrate the two forms of the first-order condition. MC = MR is illustrated by drawing the MC and MRcurves and finding the intersection. The other form can be illustrated bydrawing the demand and isoprofit curves, and showing the tangency point.

This MC = MR method is often useful in the analysis of firms’behaviour. In empirical work it is sometimes easier to estimate a revenuefunction than a demand function. In Leibniz 7.8.1, when we introduce theconcept of the elasticity of demand, we shall see yet another useful way ofwriting the first-order condition. But whichever method is used, the first-order conditions are equivalent, and the solution for the profit-maximizingquantity is therefore the same.

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.

7.6.1 MARGINAL REVENUE AND MARGINAL COST

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