Total Revenue Along a Linear Demand Curve

So now let's see how elasticity changes along a linear demand curve. So a lot of students trip this up and they tend to think that slope and elasticity are the same thing, although they're slightly related. Um they are not the same right slope is a ratio of changes between two variables where we're in a certain unit. But an elasticity is a ratio of percentage changes, right? We've got percentage changes and that little word percentages there. It makes a huge difference between the two definitions. So let's go over to this purple box where I can maybe show you an example of what's happening when we have a price of $1 and we increased it to $2. What was our unit change in that situation? Well, we went from 1 to 2. We increased by one, right? We increased by $1 there. But what was our percentage change? We started at one. Whoa, calm down. Okay. We started at one and now we're at two. We doubled our value. Right? We started at one and we doubled it up to two. So our percentage change in that situation was 100%. We increased it 100% from 1 to 2. And how about the second situation? We were at a price of two and now we increased it to three. Again, we have a unit change of one. Right? We just increased it by $1 there. But what about our percentage change this time? It's not 100%. Right? We didn't double it. Again, we actually only had increased it by half of the amount we went from 2 to 3. We only increased it by one which is half of two. So in that situation our percentage change was just 50% right? And that's because the units the numbers got bigger, right? So you can imagine from 3 to 4 we're still gonna have a unit change of one, but a smaller percentage change. 4 to 5% change smaller. So you're gonna see that even though the linear right? We have this change is constant of one unit at a time. We're having percentage changes every time, right? So even if you don't get everything on a really deep level, they're just at least understand that. We do have a difference here between slope and elasticity. Cool. So let's go ahead and go onto this graph where we've got a linear demand curve. Um And I just want to cut to the chase real quick. Um When we've got linear demand curve, we're gonna have sections of the line that are elastic sections of the line that are in elastic and a point on the line that's gonna be unit elastic. Alright. And that unit elastic point is going to be the point where we want to produce and it's the point where revenue is maximized. Okay, so I'm just gonna cut all through the crap here and I'm just gonna go straight to it. So right here, this section here to the left of the middle, right? So when you when you connect your demand curve from one access to the other like we have here, it's touching the price axis, it's touching the quantity axis, you just go to the middle point, right, right here is the middle and you can visualize that very easily. Where is the middle of that line? Right there? So to the left of that middle point which in that middle I'm gonna highlight in green, right to the left of that middle point, we are going to have elastic demand and to the right of that point. What do you guys think, yep I heard someone, it's any elastic on that side of the point. Now here's the real kicker. What do you guys think at that specific point? What are we dealing with, yep, Unit elastic is at that point right in the middle and at that unit elastic point, just like I said, that's where we want to produce and that's where revenue is maximized. So let's go ahead and go to this table right here where we've got the prices and quantities demanded which are shown on that line? On the graph. Okay, so I've gone ahead and taken these points and plotted them on the graph right there. Let's go ahead and calculate our total revenue in each of these cases. So total revenue. Remember it's just price times quantity. So all we gotta do is just multiply across here. So price times quantity, That's 07 times 2 14. It looks like we're doing our multiplication tables here again, take a journey back to arithmetic six. But let's go ahead and fill out this table 30 four times 8 32. Right? So I'm just multiplying across price times quantity and there we go. We've got all our total revenues, so you'll notice what's happening, right? We started with a revenue of zero when when we had a price of eight and no quantity demanded. And as we lowered the price and people started to demand quantity, our revenue went up up up to a point and then it started to fall again. Right? Once we passed that point, the price started to fall again. And what did we see at that point where it was four and eight. Right? That's that point, right in the middle that we were just talking about. So at a price of four, quantity of eight, that's our unit elastic point. That is where revenue is maximized easy enough. Right? I didn't want to over complicate it because this is kind of just the way to think about it. You're gonna connect your line to the two axes and you're gonna find that middle point on the line so easy. So let's go ahead and take this same data and make a graph here. So this first graph, right? We had our price and quantity, the graph that we're used to. But check out this second graf, I've got our total revenue rather than price on the Y axis. So we've got total revenue, revenue versus quantity here. And just like we saw in our schedule, the total revenue is going to increase up to a point and then decrease again. Right? So notice during this first section, that's where we're elastic in that section. That is where the quantity demanded, the percentage of quantity demanded is increasing faster than the prices decreasing. So that's why our revenues going up are the quantity demanded is going up faster than the price. And the opposite is happening in the in elastic part. After after it starts decreasing again there the price increases are excuse me, The price decreases are bigger than the quantity increases. So we're losing revenue again. Alright, so long story short, right. We just want to find that middle point most of the time. We just want to remember the left of that point is elastic. The right of that point is any elastic and at that point is unit elastic. And that's what I've got here at the bottom, right. This is our little summary. We've got demand is elastic to the left of the middle, demand is unit elastic at the middle. Let me do a little better. And any elastic to the right of the middle. It's as easy as that. Let's go ahead and move on

What is the elasticity of demand when the price of the good changes from $3 to $5? 

• A. 0.25
• B. 0.50
• C. 1.00
• D. 2.00

At what price is the elasticity of demand for the product equal to one? 

• A. $2
• B. $3
• C. $4
• D. $5

At what price is revenue maximized? 

• A. $2
• B. $3
• C. $4
• D. $5