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# More on total revenue and elasticity

In this video, take a deeper dive into the total revenue rule and the relationship between total revenue and elasticity. Created by Sal Khan.

## Want to join the conversation?

• In this example the Total Revenue is maximum when the elasticity of demand is unity, is this example specific or does it happen in any scenario? I mean, does the total revenue peak whenever the elasticity of demand is one and vice versa? •   Good question!

Short answer: Yes! When the elasticity of demand equals 1, the Total Revenue is ALWAYS at a maximum.

Long answer: If you're familiar with Differential Calculus, this fact is easy to prove because Total Revenue = Price x Quantity Demanded (which is the same as saying Price x Amount sold) and the maximum amount of Revenue occurs at the point where the derivative of Total Revenue with respect to Price is zero. Then, rearranging the terms gives us a definition of elasticity of demand.
Here is a proof:

d/dP(TR) = d/dP(QP)
= P*(dQ/dP) + Q
Setting equal to zero:
0 = P*(dQ/dP) + Q
Subtracting Q from both sides and dividing by P:
-Q/P = dQ/dP
then dividing both sides by Q/P gives us:
-1 = (P/Q)(dQ/dP)

Now, taking the absolute value of the left side (as we normally do for Elasticity of Demand) we have:
(P/Q)(dQ/dP) = abs(-1) = 1

where the left side is the Point-Price Elasticity definition of the Elasticity of Demand.
If this sounds confusing (either because you are unfamiliar with Calculus or with Point-Price Elasticity or both), don’t worry: the important thing is that Total Revenue is ALWAYS at a maximum when the elasticity of Demand equals 1 (It’s just that sometimes I find a proof more satisfying than a simple “yes” or “no” answer).

Hope this helps!
• You are wrong about the change in TR being not equal to one at unity elasticity.
If you were allowed go from P3 to P3'=0.99*P3 AND go from Q3 to Q3'=1.01*Q3
Then the Elasticity would not be unity:
[0.01/1.005]/[0.01/0.995] ~ 0.99
Truth is, if your price drops by 1%=1/100, then your QD will go up by 1/99, in which case you will get elasticity one and TR the same as before. • In this example the Total Revenue is maximum when the elasticity of demand is unity. But how about the cost of revenue (salaries, etc.)? The firms real objective is to maximise profit, not revenue. But higher quantity at a lower price increases Total Revenue but will also increase Total expense. Surely that would be taken into account when setting up prices? • Awesome videos I am really enjoying them. I noticed all the examples given have a negative slope or a decrease in price. My question is what if the price increased (positive slope), is the area of elasticity still furthest to the left on the graph and in-elasticity toward the right (highest ascending point) or vice-versa. Thanks for any and all advise. • The calculations would be exactly the same. However, the scenario itself would be bizarre.
If there was a positive slope, that would would mean that the more expensive a commodity is, the more people are willing to buy it! This weird situation might happen a bit in some circumstances (e.g. someone might believe that an expensive item must be of higher quality), but overall, not really. Eventually it would be too expensive for anyone at all to buy.
Suppose that it happens anyway, and we have a demand curve that has a positive slope everywhere. Then there would be no "maximum" revenue, since whatever price you charge for the items, you could increase your revenue further by charging even more.
• but how does the knowledge of Price Elasticity be of any use to a producer? • • Is unit elasticity the maximied possible Total Revenue in the linear graph? I would assume so since a square maximizes area. • Yes. Try taking a graph or table of price and quantity and multiplying each point to get corresponding total revenue for a given quantity.

Now create a graph of TR on the y-axis and Q on the x-axis. When you graph each point, you'll get a parabola. At the point where the slope of the parabola = 0 is the same point where unit elasticity occurs, it's also where you get the greatest total revenue.
• I know this is kind of late in the video or the chapter but, Can someone explain to me how to understand the graph that sal makes at the beginning of the video?
(1 vote) • Hello, Nightmare252! If you mean the demand curve, this is pretty simple. The demand curve basically shows how much product is demanded at a given price by consumers. It slopes downwards since the higher the price the less the consumers will demand.

You can not sell more than demanded that is why the revenue determined by the price you sell your product.

The elasticity of demand is a term related to price. And it is basically indicating whether for a given sacrificed unit of price (price reduction) you would get equal, more, or fewer units of quantity demanded.

It helps to decide whether it is worth reducing or increasing the price.

The elasticity of demand answers the question - if I reduce the price, would people buy much more of the product than the reduction of price or if I increase the price, would people still buy the same or nearly the same?

Sometimes your profits would be much higher with a price increase (inelastic), the same (unit elastic) or less (elastic).

Dear Nightmare252, please let me know if that what you meant.

Sincerely, Anatoly.
• I am confused with slope and elasticity of demand. So, if we have a horizontal straight line, then the slope is 0, elasticity of demand is 0 and therefore it is perfectly inelastic. Now, if we have a vertical straight line, then the slope is infinity, elasticity of demand is infinity and therefore it is perfectly elastic. Now, when the slope is constant the elasticity of demand isn't constant and the graph would have 3 regions of elasticity: inelastic, elastic and unit constant. When the graph has elasticity of demand constant, the slope wouldn't be constant. So, the slope and elasticity of demand coincide only at extreme cases. Please verify what I just said is accurate.  