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# Optional calculus proof to show that MR has twice slope of demand

Using some basic calculus to show that marginal revenue has twice the slope of the demand curve for a monopolist. Created by Sal Khan.

## Want to join the conversation?

• why is demand curve a straight line for monopoly?
• It can be, but doesn't have to be. It could be nearly any shape.
• Why is calculus so important?
• Like anything in mathematics you can go a long way with just arithmetic. Since math is built from more basic operations into higher more elaborate concepts (Khan Academy is based on this fact) you can do much with just addition.

So why learn multiplication if it is just addition repeated? 5x4 = 5+5+5+5. Awesome! I don't have to learn my multiplication tables. However if we do learn them we have the advantage of a short cut we know that 5x4 = 20.

Not only does multiplication allow a shortcut it also creates a new concept to think about the operation. Calculus is similar, it allows us to have short cuts and concepts that relate to rates of change, maximums of curves, areas under curves, and thinking about very small (infinitesimal) or very large (infinite) numbers.

In the case of this video knowledge of the derivative allowed me to understand the concept of marginal revenue right away because MR is a derivative of the TR curve.
• are there any activities or skill problems that go along with these microeconomics videos? if not does anyone know of a good place I can get some extra practice? (i'm an active learner so I need to do it to get it)
• wait why is MR the tangent of TR?
(1 vote)
• MR is the change of TR at a given point which can be viewed as the slope of the tangent of the TR curve at that point. So MR is not the tangent of TR but rather the slope of TR.
• So what if the demand curve is not linear but exponential? Is it possible and will the slope of Marginal Revenue have twice the demand slope?
• No, this is only true for linear demand. Take this simple example:

Demand P = (Q - 6)^2 (exponential)
TR = Q*(Q - 6)^2
TR = Q*(Q^2 - 12Q + 36)
TR = Q^3 - 12Q^2 + 36Q
(calculus)
MR = 3Q^2 - 24Q + 36 (not twice the slope)
(1 vote)
• In the starting of the economics playlist, we say that the quantity is a function of the price, then how can we compare P = Q + k (constant) to the general form of linear equation y = mx +c, because over here y is a function of x, but the price is not the function of quantity, its the other way round. Please clear my doubt, thanks :)
(1 vote)
• We are not looking at quantity as a function of price, but rather marginal revenue as a function of quantity. Quantity is clearly the independent variable here: the firm chooses to produce a certain quantity and takes the point at which marginal revenue is equal to marginal cost.
• What can be inferred from this derivation? I mean what does it tell us in economics terms? Thanks in advance.
(1 vote)
• It tells us that monopolies have the incentive to cut supply to below the equilibrium in order to get a higher price.
(1 vote)
• Is this the case for Monopolistic competition and Oligopolies too? :)
(1 vote)
• At , (TR)'/(Q)' what happens to Q? Isn't the equation wrote out just the deriveded of (TR)'?
(1 vote)
• It's the derivative of TR with respect to Q.
(1 vote)
• Why marginal revenue is half of the demand curve ?
(1 vote)
• That is just how calculus works. Of course, this rule is only true for linear demand curves, not more complex ones.