5. Consumer and Producer Surplus; Price Ceilings and Floors

Quantitative Analysis of Consumer and Producer Surplus at Equilibrium

5. Consumer and Producer Surplus; Price Ceilings and Floors

Quantitative Analysis of Consumer and Producer Surplus at Equilibrium - Video Tutorials & Practice Problems

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Let's calculate consumer and producer surplus when we are given equations for supply and demand. Algebra time!

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Quantitative Analysis of Consumer and Producer Surplus

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Alright, now let's calculate consumer and producer surplus when we're given equations and we have to use just a little bit of algebra, let's do it. All right. So um when we're calculating consumer and producer surplus, at least in this video, we're only gonna deal with equilibrium. We're gonna be at equilibrium calculating consumer and producer surplus. Ok. And if you'll notice, I've got our step by steps, I've got some formulas for you. Um But I just want to point out that first step one, this is something we've done in a previous video, this is finding equilibrium price and quantity using a little bit of algebra. And we've done that before. Um I suggest if you don't remember how to do that to watch that video again. Um But we're gonna we're gonna do it in an example here, just just to reiterate, Alright, And in step two, I want to note this term access price that I'm using, so find the access price on quantity demanded equals zero. I made up this term, that's not the real term, that's kind of how I think about it. So, let me let me just show you what I mean over here, before we get to the example, I'm gonna draw a little example uh standard supply demand curve graph. Excuse me? Graph over here. So let's say we've got our supply and demand right there, right? And we've got our equilibrium. So this is our price. Access our quantity axis. Right. Um We've got our equilibrium price right here, P star and our equilibrium quantity right here. Q. Star. Right. And notice that um to calculate our areas of our producer and consumer surplus. So if this is our consumer surplus right here, right? The area below the demand curve above the market price, this triangle right here um to calculate that we're gonna need this price way up here where the demand curve is crossing the axis and that's what I call the access price, the demand access price right there, right? And then down here we're also gonna be solving when we do our producer serpa plus we're gonna need this point where the supply curve is crossing the price axis right there, right this point right here where the supply curve touches the access, we need that number to be able to calculate um our producer surplus just to get the length of that segment. So that's gonna be our supply access price right there. Okay so let's go ahead and jump into the example and we'll see all of this in action. Alright? So the example says calculate consumer and producer surplus using the given information. And just so you know these these demand and supply curves represent the market for apartment rentals. That's kind of where I got this example. Okay so let's go ahead and on the graph I'm just gonna draw our standard price quantity. Um and just you know, get those those lines out there just to have the visual support, right? So it doesn't really matter what you draw like there's no to scale here, but one thing you want to make sure is that you cross the supply curve on the price axis here right? You don't want to draw it like this because we want to cross it on the price axis. Because in all of these examples we're gonna have that type of supply crossing the price axis. So we can calculate uh those that access price, right? So we're gonna need three prices here, we're gonna need the P star, we're gonna need the demand access price up here and we're gonna need the supply access price down here and we're gonna need Q star, right? The quantity at in an equilibrium to be able to make our calculations of consumer and producer surplus. So let's go ahead and start with step one of the algebra which you see up above this is what we've done before and it's gonna be finding the equilibrium price and quantity. Alright. So if you remember when we're at equilibrium quantity demanded equals quantity supply, right, quantity demanded equals quantity supplied at equilibrium at eq. So what we can do is set these equations equal to each other, right? Because the quantity demanded equals the quantity supplied. That means these equations are equal to each other at that point. So let's find um let's go ahead and do it. We've got three million minus 1000 P equals 1300 P minus 450,000. Alright. So what we wanna do is we want to solve for P which is our equilibrium price. So what I'm gonna do is I'm gonna try and get all the peas on one side, all the numbers on the other side. So I'm gonna go ahead and add 450,000 to both sides. That will get rid of the 400 the negative 4 50 on that side. And let's get all the peas on the same side. So I'm gonna add 1000 P. On this side and add 1000 P. On that side. Right? So these are gonna cancel out right here and this is gonna cancel out right here. So what are we left with? We've got uh 300. So the three million plus the 4 50 is three million. 450,000. And that's going to equal on the other side. The 1300 P. Plus 1000 P. 2300 P. Right? So last thing to isolate our price is to divide both sides by 2300. This is gonna cancel right here. And when we do three million for 50,000 divided by 2300. I'm gonna just mental math real quick. Just kidding. Um But the answer is gonna be 1500 P star is gonna equal 1500. Alright, so we've gotten our P. Star in this in this case. So I'm gonna go ahead and put that onto our graph. 1500 right here. That is going to be our equilibrium price. Let's go ahead and find our equilibrium quantity. I'm gonna scroll down a little bit. Alright, equilibrium quantity to do that. All we have to do is we take our equilibrium price and plug it into either equation. I'm gonna go ahead and use the demand equation. It looks a little easier to me. Um So we're gonna have quantity demanded, which is also quantity supplied at equilibrium, right, quantity demanded equals quality supply. So it's gonna be that equilibrium quantity um equals three million. Right? I'm taking our quantity demanded equation from up above this equation up here three million minus 1000. And now instead of P I'm gonna put in our equilibrium price of 1500. So let's find out what the quantity is this price. So quantity demanded. Um Well, I'm just gonna use Q Star really because this is our equilibrium quantity, right, quantity demanded equals Q star at equilibrium. So three million minus 1000 times 1500 is 1.5 million. 1,500,000. So if we do that, Q star is going to be three million minus 1,500,000. It is one million 500,000. Alright, so we've got Q star now, that's going to be this number right here. One million. 500,000 is our Q star. Alright, so let's go up here and let's see what the next step was. We just finished. Step one finding uh equilibrium price and equilibrium quantity. Now we need to find the access price when quantity demanded equals zero. And the access price and quantity supplied equal zero. Okay so let's start with the access price when quantity demanded equals zero. And this is gonna be our demand access price. So quality demanded. Right? We're gonna use this equation again except now we're gonna set quantity demanded equal to zero. So the left hand side of the equation is zero. So zero equals three million minus 1000 P. Right? And let me just label this real quick demand access price. Right? So we're calculating the demand access price here. So um let's go ahead and solve for what that is. Um 1000. We're gonna add 1000 P. To both sides. We're gonna try and isolate P. So 1000 P. Over here and we're going to get um 1000 P. Equals three million. Right? Because those cancel out over here this cancels. So all we gotta do is divide by 1000. Let me do it in a different color, divide by 1000 divide by 1000. And what's left right P. We're gonna cross out these three zeros and we're left with 3000. So three million divided by 1000 P. Equals 3000 as the demand access price. So that is gonna be our number right up here. The demand access price is the one where the demand touches the the price axis. And let's do the same thing with supply access price. So I'll do it right here under the demand access price, supply access price. And let's go ahead and find our supply equation. So that was this one over here. Right, quantity supplied equals 1300 p minus 450,000. So now quantity supplied is what we're gonna set to zero right? The left hand side of that equation is zero. So we have zero Equals 1300 P -450000. Right? So now the quantity supplied is equal to zero. Let's find out what that price is. So let's add 450,000 to both sides. Right? And we're gonna get 450,000 equals 1300 p. Now last step is just to divide by 1300 on both sides. Right? These have canceled out this cancels out. And we're left with just p. So if we do 450,000 divided by 13 1300. This I can't do in my head, I'm gonna pull up 450,000 divided by 1300 we get 3 46.15, we're just gonna round it off to 3 46. Forget about the decimals. Um just because we have such big numbers here. So the supply access price is going to be this 3 40 six. Okay and let's go ahead and put that into our graph. And with these numbers we're ready to start calculating our consumer and our producer surplus. 3 46 is our supply access price. Right? So up here, I've got the equation straight up that you can do that. You can use to calculate um consumer Producer surplus when you have this information right? You can just use that formula. But I'm gonna do it with the graph and I'm gonna use my triangles here. So I'm gonna start with consumer surplus. And that's gonna be the area above the market price. The equilibrium price of 1500, but below the demand curve. Right? So to find out that area, we need to know the base and the height. So the base will say, is that right? The base. Let me make it. So you can see it. The base here is gonna be the space between those and the height. It's gonna be this other side of the triangle, the height which is equal to the quantity, the quantity at that price. The equilibrium quantity. Alright, So we can figure out our base in our height, our base is gonna be the difference between the 4500 which is exactly what I have appeared, right. The difference between the demand access price and the equilibrium price. That is the base that I've been talking about and Q. Star that's the height that I've been talking about, right? That's the H on my graph. So there you go. That's how we're using that formula up there um in the same way as we're just calculating it on the graph. So let's go ahead and and calculate it, I'm gonna do it right here. Consumer surplus. So what's consumer surplus gonna equal? Well, we've got half times the base which was 3,000 -1500 times our height, which is just our equilibrium quantity of 1.5 million. Alright, so let's go ahead and do this math. The hard work is behind us already. So half times 1500 Right? This is gonna be half times 1,503,000 minus 1500 is 1500 times 1500. And then times 1.5 million. And we are gonna get this number right here. Um Wait, did I do it 15000.5 times 1500 times 1.5 million apartments, Yep, we got a pretty big number. But this is a big market, right? We've got a huge amount of of of apartments being supplied here. So, consumer surplus is gonna equal 1125 million. Right? That's our consumer surplus. That's a pretty big number. one million. 1125 million. Is our consumer surplus. That is the area of that purple region. Can you see it there? All right. Now let's do the same thing with producer surplus down here and we'll be done with this problem. So, producer surplus Again, we're gonna go to the graph, but I'll show you how it works on uh using our Formula two. So, producer surplus is going to be now, I'm gonna erase this basin height here. Are basin height. Now, we've got this base right here right between 1503 46 our height actually is the same quantity right there. Right? So, I'm gonna highlight it in green here. Everything below market price and above the supply curve. So that green area. Alright, so our base is gonna be that difference between 1503 46. Our height is gonna be the 1.5 million, and that's exactly what you see over here. Right? The difference between P Star and supply access prices, The base and Q Star is our height in in that green box equation. Alright, so, let's go ahead and finish this up down here at the bottom. All right, I'm gonna kind of do it. Let me get out of the way. So I'm not uh I'm not there. Okay, so let's go ahead and do it. We've got half, right, half times base times height. Our base is 1500 minus 3 46. Right? That's the base part times the height of 1.5 million apartments. And I'm gonna write the answer over here. I'm not gonna I'm just gonna go straight to the final answer, 0.5. So 1500 minus 3 46 Times 460.5 times 1.5 million. You should be getting 865,500,000. That is gonna be our final answer here. Okay, so our consumer surplus was the 1,000,125,000. Our producer surplus is right here, right? Consumer surplus up here. Producer surplus right there. So that is how we're going to calculate uh consumer and producer surplus using algebra, right? There's a lot of steps there. But you can kind of see that it's not so complicated. We're just looking for a few specific prices, our equilibrium quantity, and then we just calculate our areas of a triangle. Alright, alright, let's go ahead and practice some of this on your own. Let's do it now.

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Problem

Problem

The supply and demand curves for a product are as follows. What is consumer surplus in this market?

Q_{D} = 15 - 2P Q_{S} = -15 + P

A

6.25

B

12.5

C

20

D

22.5

E

25

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Problem

Problem

The supply and demand curves for a product are as follows. What is producer surplus in this market?