Quantitative Analysis of Price Ceilings and Floors: Finding Areas - Video Tutorials & Practice Problems

Graph Intuition

Alright, now, we're gonna put all of our algebra tools to work to calculate consumer surplus, producer surplus, deadweight loss in situations where we have price ceilings and price floors. So it's gonna get pretty in depth here, but I think you're gonna see that a lot of it has to do with stuff we've already been doing. All right, So let's go ahead. And first let's review um what we're gonna be looking for on the graph, right? When we're dealing with price floors and price ceilings, we're gonna be dealing with these areas looking for consumer surplus and producer surplus. Alright. So, first, let's talk about this price floor. Okay. A situation where we have a price floor and we have our equilibrium. So let's label our graph price and quantity here, right? Our downward demand, upward supply. Alright. And where's our equilibrium? We got it right here in the middle right, at a price of P star and here's Q. Star over here. Right. So we're gonna talk about an effective price floor. When is the price floor effective? Well, that's when it's above equilibrium, right? And just to remember, we've got our trick for the ceiling, right? The price ceiling below equilibrium, right? Because the ceiling we make the house there and the ceiling would be below. So price floor is gonna be above. So we're gonna pick this price right here as our price floor price. Okay, I'm gonna put P. F. For our price floor. And let's go ahead and um discuss what we're gonna be looking for in this problem, right? So um first, when we're at equilibrium, right, and we have a price floor, excuse me, when we're at equilibrium, what is our our consumer surplus when we're at equilibrium? Well, we we know that it's everything above the the price, but below the demand curve, right? So when we're at equilibrium, everything in the purple triangle above the P star is going to be um consumer surplus, so that's gonna be a plus B plus C. Right? When we're at equilibrium, and how about that producer surplus, that was everything below the price and above the supply curve? Excuse me. Alright, so that's gonna be that green area which we'll call d plus E plus F, right? That's everything that is um under the under the price, but above the supply curve. And you should remember that when we are at equilibrium, we're being efficient, and we have no dead weight loss, right? There's no dead weight loss when we're at equilibrium. So let's go ahead to this situation where we have a price floor, right? When we have a price floor, what's our consumer surplus? So we've got this high price right up here, and everything above that price is going to be consumer surplus, right? And below the demand curve. So consumer surplus is going to be just this little area right here, area of a so that's our consumer surplus with the price floor is just a all right. And remember when we're doing this, we're gonna be doing this with algebra, and we're gonna be calculating the area of A. We're gonna want to know what that area is and to find out what that area is. We're gonna need to have like we've used before our demand access price. Remember these access prices that we were solving for? So this one up here is our demand access price, right? So we're gonna need that demand access price. And we're gonna need the price floor amount. And we're also gonna need this quantity here, right? This quantity um at the price floor, I'll call it Q. L. For like a lower quantity. We're gonna need that number as well to be able to calculate that area. All right. So I'm just priming you write for the kind of numbers we're gonna be searching for and how we're gonna look for them. Alright. So how about producer surplus when we have a price floor uh in that situation? Right. So our producer surplus, if you remember from our previous discussion, is everything below the price, but it's not gonna include C. And E. Right? Because those trades don't happen, this price floor is stopping those trades from happening, right? We're only gonna trade up to quantity L. Q. L. And we don't reach Q. Star. So those trades where C. And E. Exist, those didn't happen. So those are not gonna be surplus, right? So we're gonna see that B. D. And F. The F. Is our producer surplus. So they're the producers have effectively taken some of the surplus from the consumers. But at a cost to society of this dead weight loss, right? And that's what we see here in C. And E. These trades didn't happen. Society would have been better off with them but we get nothing out of this, right? We've lost these benefits. So that is our dead weight loss C. Plus E. Right? So notice producer surplus and deadweight loss. These are all things that we could calculate. And but to calculate um producer surplus, let's talk about that one first. For producer surplus, we're gonna need definitely need this supply access price down here, right? We're gonna need this one the very bottom of the graph supply access price, right where it's crossing here and this is the demand access price here, The price floor, right here, P. Star right here and there's one more number that we're gonna need. Right, it's this last price. And I call it I call it the missing price, there's no real term for it. But I call it the missing price because there's always gonna be this one extra price that we need to solve for. And that's gonna be the last piece of the puzzle, right? So what do you see in this situation? Right? We're gonna see that we have a demand access price, the price floor, the equilibrium price, the missing price and the supply access price. There's five prices that we need to find out what they are. Um If we need to solve for all of this, remember when you see these problems on the exam, they're usually gonna only ask you for a portion, right? They're gonna say what was consumer surplus only, they're not going to say in one multiple choice, what was consumer surplus? That was producer surplus and deadweight loss? That would be really intense, Right? But you might get a few questions where you solve this in pieces, right? So we're gonna need those five prices and we're going to need these two quantities, right? Two quantities. So when we're solving for everything, we're gonna need all of these numbers to successfully figure out what consumer surplus is producer surplus and deadweight loss. Right? So let's do the same thing with a price ceiling. Okay, so now a price ceiling, right? We already set up effective price ceiling, we made the house, it's gonna be somewhere below equilibrium, right? So here's gonna be our p star equilibrium price and equilibrium quantity, right? And we're gonna say, a price ceiling, I'm gonna put pc for the price ceiling right here as a price below equilibrium. Right? So, in this situation, when we have a price below equilibrium, let's go ahead and uh find our consumer surplus, our producer surplus and deadweight loss. Let's start with producer surplus, because it's easier, that's gonna be this green area right here, right? Everything below the price. Um But above the supply curve. so that's the only producer surplus in this case and that's just f right F. Is the only producer surplus. And to calculate that again, we're gonna need that supply. Let me do it in black, that supply access price, right? Supply access price. Just like we've used in previous videos and that's that point right here where it touches the price access. Okay, so now let's do consumer surplus just like before we're going to see that it's gonna be this area A. B. And D. Right? It's not gonna include C. And E. Because those trades didn't happen and we did not get that benefit. So here we go, we've got a B. And D. Right above the price, but below the demand curve. And that is going to be our consumer surplus. So let's write it in here, A plus B plus D. And when we're calculating that area, we're gonna need um that demand access price up here, right demand access price, That's where it's touching the curve there. But we're also gonna need to because when we calculate this is a pretty weird shapes. So we would split it up into two shapes, just like we would do with producer surplus on the other side. So let me do this in blue real quick. We would split it up into this rectangle and we would solve the area of the rectangle and then add the area of the triangle up here, right? So we were treated as two shapes. So to get that area of the rectangle we would need that missing price in this case. But notice the missing price and this one is not where the missing price was in the other one right there in a different spot. So that's why it's handy to have the graph and do it with the graph because you can see and you just start putting in the numbers as you solve it, you put them where you have them on the graph. So again, the last thing we're gonna need here because to solve that we also need this length right here, right what is this quantity? And I'm gonna call it Q. L. Again right? The lower quantity because of the inefficiency. And so that yeah so there you kind of see the parallel right? With the price floor and last but not least. We've got the deadweight loss right? This C plus E trades that didn't happen. The surplus that was lost right? C plus E again is dead weight loss. So to calculate that we would need the missing price and the price is right? We would need this length right here. C. To E. So we would need the missing price and the price ceiling number there and we would need the difference between the quantities, right this height right? There is the difference between Q. Star and Q. L. So we would need all those numbers. They're pretty intense, right? So in the end you're gonna see that a lot of the steps we do is stuff that we've done before. Right again, we're dealing with this demand access price that we've solved for before supply access price. We've solved for P star we've solved for the price ceiling has been given to us, right? They're gonna have to tell us that number in the question or else we're not gonna know what the ceiling was or the price floor. So really the only new thing that's happening here is solving for that missing price, right? And potentially this lower quantity as well. So there's not so much new stuff here, we're gonna see that there's a lot of parallels to what we've been doing but then you know, it just adds that little extra layer. So let's go ahead and move on to the next video where we can start and you'll see the kind of stuff that's already similar to what we've been doing. Alright, let's go ahead and do that in the next one.

The Steps We've Done Before

All right. So, let's go ahead and do this example and follow our step by step in this video. We're going to do the steps that you should be used to already. Things that we've done previously. All right. So, it's telling us calculate consumer surplus producer surplus and deadweight loss if a price ceiling of 1000 is in effect. Alright. A price ceiling of 1000 is enough. We're going to calculate consumer producer and deadweight loss when you get this on a test. I wouldn't expect you to get everything at once. So you might not have to solve for everything, right? This is gonna take a while because we're going to do everything we're going to solve for all the prices all the quantities but on the test, and we'll see when we do some practice problems. You can skip some steps because you don't need all the information when you're not looking for everything. All right. So, look at step one, find equilibrium price and quantity boom. We do this all the time, Right? We're used to this. Let's go ahead and do that right now. So, we're gonna set these equal to each other, Right? We've got three million minus 1000 P equals 1300 P minus 450,000. Right? So, let's get all the peas on one side, all the numbers on the other side. So, we're gonna get this out of here. Put it over here. Alright? So, what's gonna cancel? This is gone. This is gone. What's left? 3,450,000 equals 2300 PPI Right? Last step is to isolate the p by dividing by 2300. Right? And we are going to get an equilibrium price here, P equal to 1500. So we figured out equilibrium price is 1500. Right now, we've done this example before, so I'm not I don't know those numbers off the top of my head. Um So what I like to do is I draw the graph and remember there were those five prices and two quantities that are going to be imported. Right? So let's go ahead and draw it. So this is kind of gonna be the standard drawing that we're gonna have when we do these is gonna be something like this, right? We're gonna have five prices. So we want to mark five different prices here. Where we're gonna have this demand access price, we're gonna have this one here, right? One of these is gonna be the missing price. One of these is gonna be the price ceiling or the price floor. We've got our equilibrium price in the middle and then our supply, uh access price on the bottom, Right? And then we're gonna have these two quantities as well. We'll have the quantity when it's low and the quantity at equilibrium. Right? All of these numbers are going to be important When we when we solve this. Right? So what have we solved so far? We just found that equilibrium price is 1500. So we've gotten that one. Let's go ahead and get our equilibrium quantity. Right? So all we gotta do is plug that into either equation, right? Because at equilibrium the quantities are the same. So let's do that. Now, I'm gonna pick the demand equation looks easier. So we've got quantity demanded. So which is Q. Star, right? Because it's gonna be the same for for demand and supply at equilibrium, it's gonna be the same. So three million minus 1000 times P at equilibrium, which was 1500. Right? So let's go ahead and solve for Q star 3,000,000,000 -15 1.5 million. That's gonna give us 1.5 million. Right? That's the answer. Let me get out of the way, 1.5 million. That is gonna be our Q star, right? So we've now gotten this number down here. Let me scroll down just a little bit and we've figured out that this is 1.5 million right? At Q star. So we've gotten two out of the seven numbers already and that stuff that we're already used to. So let's look at step to confirm that the price floor or ceiling is effective. We've done this before too, right? Price ceiling is effective. So remember that whole house thing, right? My house method. So that helps us with price ceilings, right? We're gonna have our demand and our supply and we get the the house when we're below equilibrium, right, equilibrium being right here? And we're below. So the price ceiling is when we are below equilibrium is when it's effective price floor is the other one is when we're above. Right? So we need to confirm, is this an effective price ceiling? They told us the price ceiling is 1000 right? 1000. And the quantity, excuse me, the price at equilibrium is 1500 so it's less, right, It's below equilibrium, it is going to be effective and just like that, we're ready to put this one in right here, right? This one's gonna be our price ceiling of 1000. We just put it into our graph um Yeah, I'm gonna come in right here. Um So we just took the price ceiling and we put it in our graph where it would go, right? So now we know the missing price number is gonna be this one, this is gonna be the missing price number. The price ceiling is in there. But we're not gonna solve for missing price yet. That's the new thing. Right? We're gonna do that in the next video. So in this video we're gonna do the rest that we're comfortable with which was solving for the supply access price and the demand access price. This is stuff we've done before. Right? So what we need to do is we're gonna set quantity demanded equal to zero and we're gonna set quantity supplied equal to zero and that will give us the demand access price and supply access price. So let's start here with demand. So right here demand. So when is demand equal to? What is the price? When demand is equal to zero? So quantity demanded is zero equals So I'm using our demand equation up here. Right? Three million minus 1000 P. So quiet demanded zero equals three million minus 1000 P. Right? So we're just gonna add 1000 to both sides or not 1000 1000 P. Right? So 1000 P. Is gonna equal, whoops, 1000 P. Is going to equal three million. So what is gonna be P We divide by 1000 on both sides. Whoops, I'm out of space there. Oh no, whoops, sorry about that. Let's let's get back to where we were. All right, so there we go. Three million. We're gonna divide both sides by 1000. Right? And we are going to get a price. I'm gonna go this way P equals three million divided by 1000 3000. Right? So that is gonna be our demand access price because we use the demand equation. Um So P is 3000 when quantity demanded equals zero. And let's go ahead and do the supply one right here right behind me. Alright so supply. Um what do we have quantity supply? So we're gonna use our supply equation. We're gonna set quantity supply to equal to zero. Right? And let me put these zeros in in a different color so you can see that um that's what I was doing there. Okay, so zero is gonna equal 1300 p minus 450,000. Right? That's our supply equation. And we set quality supply equal to zero. So let's add 450,000 to both sides. Right? And we are going to get Uh 450,000 equals, I'm gonna put this line right here. 450,000 equals 1300 P. And when we divide this by 1300 on both sides, we're gonna get a minimum price Or excuse me, a supply access price of I'm gonna go over here, P equals 3 46. Right? We're gonna round this off to 3 46. All right, no decimals there, we'll keep it easy. So that is our supply access price. Right? And guys, this is all stuff we've done before, right? We've done all these steps before um we've been able to solve all of these just using what we've done in previous videos. So now in the following video, we're gonna go ahead and we're going to solve for the missing price. We're gonna get that quantity and we're gonna start calculating areas. Alright, let's go ahead and do that. Now

The New Steps

Alright, let's keep it moving here. So here you're gonna see the new steps right here, we are going to calculate that missing price. We're gonna calculate that lower quantity. And then we're going to calculate the areas for consumer surplus, producer surplus and deadweight loss. Alright, so the first step is we're gonna find that quantity. Right? So if you go down here um on the graph, I've already brought over the information from the previous video, right? If you want to pause and just update this graph, so you have it all here, um I'll be here when you get back. Don't worry. So let's go ahead and do step four. Okay, So we're gonna find the floor ceiling quantity by by plugging the floor ceiling price into the correct equation. Right? So in this case we have a ceiling, right? We have a price seal. So we're gonna use the for ceilings, we're gonna use the supply equation and the floor ceiling price, right? The price, the floor ceiling price. Right? I say both. Because it can be either um is going to be 1000 in this case. Right? And we're gonna solve for that other quantity. And I'm gonna show you a trick in a second regarding this idea. So let's use our supply equation and our quantum our price of 1000. So 1300. Remember we're looking for what that quantity supplied is at this price ceiling um price quality supplied equals 1300 times 1000 right minus 450,000. So the quantity supplied is gonna equal 1.3 million minus 450,000, which equals 850,000. Right? That is going to be our lower quantity and it's gonna be this number right here. Right. So just to reiterate these are the numbers we're looking for in this video. Right? That missing price and this quantity down here, right? This quantity which we just solved for is 850,000. Now I want to give you real quick tip because it might be hard to remember oh if it's a ceiling you supply equation. If it's a floor, use the demand right? You might forget. And on the test, what I would do is I would just try one. Right? If you try to use the demand equation in this situation you're gonna end up with a um a quantity. A quantity that's bigger than equilibrium. Right? You would end up with some quantity out here, right? Some bigger number. And you would know that's wrong, right? Because we're looking for a lower quantity. We want the quantity to be less than equilibrium. So worst case scenario you you can try it in either equation, you go ahead and try the demand equation and if you get a number bigger than your equilibrium quantity, you know, you got to use the other equation. Okay, So in this case we've got our steps, it told us to use supply and we've got this lower quantity. Alright, so the last thing to find is our missing price. before we calculate areas. All right, so let's find the missing price at the floor ceiling quantity. Alright. And what does it tell us here? It says for floors use the supply equation and for ceilings, which is what we have. We're gonna use the demand equation and the floor ceiling quantity. Alright. So it's the quantity we just solved for. We're gonna plug it into the other equation. Alright. So that's kind of the flow. So if you had forgotten which was which in the first equation, Once you figured it out now, you know that you use the other equation for this one. Right? So here we're gonna use the demand equation. Um and we're gonna use the floor ceiling quantity. So the floor ceiling quantity was this 850,000. So this was step four right here, right and let's go ahead and do step five right under it. So we're gonna use that floor ceiling quantity of 850,000 and we're gonna set that equal. That's going to be what our quantity demanded is. So let's see what the price is when that is the quantity demanded three million minus 1000 P. Right? So we're gonna I'm gonna get the peas on the left hand side of the equation and the the money on the other side are just the numbers. So let me do that better. I'm gonna do plus 1000 P. Out here and minus 850,000 on both sides, right? To get the 800 numbers on one side and the p on the other side. So this is gone, this is gone. What's left? We've got 1000 P. Equals to 1 50 right? 2,150,000. Let's go ahead and divide by 1000 on both sides. And what's that left with? We're gonna get a p of 21 50 Right? So there we go, 21 50. Hopefully I wasn't in the way the whole time there. Uh But there you go. So let's go ahead and use 21 50 that is our missing price right here. So now we have all the numbers we need we're ready to start areas, right? We can figure out what the consumer surplus producer surplus and deadweight loss are at this price of 1000. So let's go ahead and start with consumer surplus, right? Actually, let's start with producer surplus, because it's gonna be the easiest one. So we've got a price of 1000 and producer surplus. We've been using green, right? At a price of 1000. Producer surplus is everything that's below the price of 1000. Right? So we'll start with this one because it's just a triangle, right? We've got a triangle in this case. Um And we just need to find the area of the triangle and we've got our producer surplus. So let's go ahead? I'm gonna write producer surplus here. And let's do the calculation, right? So what is our base and what is our height base there and height there? Right. That is gonna be the base and height of the triangle. So we're gonna go half times base and the base here, right? It's gonna be the 1000 minus the 3 46. Right? That's the length of that segment. And we're gonna multiply that by the height and the height is always just gonna be the quantity that we use there. So, the height here is gonna be 850,000. We didn't go all the way to equilibrium. We can only go to the 8 50,000. So let's go ahead and calculate what the producer surplus would be. I'm getting out my handy dandy calculator, 1000 minus 3 46 is 6 54 times a half times 850,000, Wow. So we get a producer surplus of 277 million, 950,000. All right, that is our producer surplus. We're dealing with big numbers here. Right? That's a big quantity. We're gonna get a big number. So our producer surplus, I'm gonna write it in this box 2779 50,000. Right. There we go. Almost doesn't fit. So let's go ahead and do um consumer surplus now. Right. So our Consumer surplus when we have the price floor, we're gonna have everything above the price, but below the demand curve. Right? Well let me get rid of this basin height because we're gonna have a different one now. Um But we're only gonna go up to this low quantity, right? Because the trades beyond that didn't happen because of the price floor. So it's gonna be this area right here that I've shaded in purple, Right? So how do we calculate that area? Well, it's a little tricky, but we're gonna break it up into two pieces. So I'm gonna highlight here in I'll use green to mark off the areas we're gonna calculate. So here is a rectangle that we're gonna calculate. And then we're also going to calculate this triangle, right? So when we calculate the two of them, we just add the areas together and we'll have the total area. Let's start with the rectangle. So consumer surplus, oops, consumer surplus. Alright, so let's start with the rectangle. So rectangle is just base times height. Right? The base times the height without the half part. So what is the base? Well, we're gonna have this right here right from 21 50 to 1000. And our height is gonna be this distance right here up to the quantity of 850,000. Right? So let's go ahead and do that. So our base is gonna be 21 50 minus 1000 right? That's the length of that segment times our height, which is just 850,000, right? That's the the quantity exchange. That's the length of that segment. So let's do this math. 21 50 minus 1000 is 11 50 right? Times the 850,000. That's gonna give us a consumer surplus of 977 500,000. Quite a big number there as well. Alright, so there we go. We've oh, excuse me, That's not our consumer surplus, of course, right? That's only part of our consumer surplus. We also need to find the area of the triangle and this is a huge mistake students make all the time and it almost caught me right there. So you gotta be really careful just like I was right here. Okay, so let's go ahead and find the area of the triangle. So I'm gonna write here rectangle, right, let's go ahead and do the triangle. And it might even be better actually. Now that I think about it to label this, before you start doing the math, right, then you can't forget if you're just written rectangle and then started math, you'd be like, okay, time for the triangle, right? So so it's usually good to do those headings first and I will make sure to do that in the future. So let's go ahead and calculate the area of the triangle. So we've got half times 3000 minus 21 50. That's going to be this, sorry, this is the base right here. Now, whoops, the base is going to be this length right between the 3021 50 and we've got that same height, right? The height of the quantity. So half times 3000 minus 21 50. That's the length of that segment times our height of 8 50,000. Right? So this is the other part of the producer consumer surplus. Let me get out of the way. So let's go ahead and calculate that 3000 minus 21 50. Alright. Times 500.5 times 850,000. And we're gonna get that. The triangle has an area of 3 61 250,000. Right? So now all we gotta do is add both of these together, right? And that's going to be our total consumer surplus. So, if we add that plus 9 +77 500,000, we get our total consumer surplus of 1,338,750,000. Right, huge number here. So that is our total consumer surplus. Let's go ahead and write that in the box. 3 1,000,000,038 750,000. All right, So that's our consumer surplus. Let's go ahead and finish up with the deadweight loss. Right? So this one's a little tricky. Let's go ahead and highlight the area on the graph here. It's gonna be this blue area, right? The trades that weren't made, we lost this surplus. So how do we calculate that triangle? Well, so we're gonna need this right here. Right? That is going to be our our base and this is gonna be our height right from there to there. So we can need um the difference between the quantities for the height now and the difference between those prices for our base. So let's go ahead and do that. I'm gonna do it right here at the bottom. We're kind of running out of space. So I'll do it right here. Dead weight loss. All right, So it's a triangle. Right? So we're gonna use our triangle equation half times base times height. So half times our base our base is gonna be this length, right? The length of the prices. So it's gonna be the 21 50 minus the 1000. Right? That's the those two prices there. Right. That length is the difference between those and the height is the difference between the quantities. Right? This right here, the length of the of that that size. So what's the difference between those 1.5 million minus 850,000. Right, So once we do all this math, we will have our dead weight loss, let's go ahead and do this. So it's easier to do um everything inside the parentheses first. I like to do my subtractions first. You can't mess up your order of operations. So we've got the 1.5 million minus the 850,000 is 650,000. Right? So, I'm gonna do half times and 21 50 minus 1000 is 11 51.5 million minus 850,000 is 650,000. Right? So let's just multiply it out and we're gonna get 373 million. 750,000. All right. That's our dead weight loss. And wow, that was quite a hefty problem. Right? But remember on the on the exam, they're not gonna generally ask you, um, to solve for everything. In one question, you'll probably get multiple points, right? You get points for the consumer surplus points for producer, points for deadweight loss, right? Or they could just ask you for a little peace instead of asking you for everything. So, I've got that little note here at the bottom. You might not need to do all the steps if you're only asked for a portion of it. Right? So, we're gonna do some practice problems and you'll kind of see what I mean by that. All right. So, let's go on and do practice with this topic.

The supply and demand curves for a product are as follows. What is producer surplus if a price floor of $21 is set? 

Q_D = 45 – 2P 

Q_S = -15 + P

• A. 6.25
• B. 10.5
• C. 12.5
• D. 13.5
• E. 18

The supply and demand curves for a product are as follows. What is deadweight loss if a price ceiling of $2 is set? 

Q_D = 600 – 100P 

Q_S = -150 + 150P

• A. 150
• B. 187.5
• C. 212.5
• D. 300
• E. 375