Supply and Demand: Quantitative Analysis - Online Tutor, Practice Problems & Exam Prep

To find equilibrium in supply and demand, isolate the same variable in both equations. For demand, use $P=800-2Q$, and for supply, $P=200+Q$. Set them equal to find $Q=200$ and $P=400$. This intersection represents the equilibrium price and quantity, essential for understanding market dynamics.

Equation of a Demand Curve

Now let's learn how to find equilibrium using just a little bit of algebra, and I know I scared you there with that nasty word I just dropped, but I promise you that algebra is not going to get so intense in this segment, alright? There's just going to be a little bit of formula rearranging and solving, so it's going to be pretty basic and we'll do some reviews as we go along, so hopefully, you'll get a good hang of the kind of algebra we'll need. So let's start with the demand curve here.

Sometimes you can be given an equation for a line like you see here, p=800-2q, right? And the easiest way to take this information and be able to put it on a graph is to just pick some values for the quantity here and we'll solve for price, right? We'll pick values for quantity because it's on the right-hand side and price is already by itself, so it'll be a lot easier to just solve for price in that situation.

So let's go ahead and start with a very easy value for quantity. Why don't we start with the value of 0? What if the quantity is 0? Let's go ahead and see what happens in our equation. p=800-2×0 that's going to disappear, so we're going to end up with p=800, right? That is going to be our price. When the quantity is 0, the price is going to be 800. So if people, if the market is charging a price of 800, there will be 0 quantity demanded. Let's go ahead to our graph and let's plot this point.

So we've got our price axis over here, our quantity axis over here, alright? And let's take that point. So we have a quantity of 0, which is right here along the line and let's go ahead and mark the price of $800. So at a price of $800 up here, I'm going to use blue. Price of $800, we're going to have a quantity demanded of 0. Alright, let's go ahead and pick some other quantities.

So how do we go about picking the quantity we're going to use? Well, you want it to be something that's going to be easy to put on your graph, right? Our graph is already quantities of 100, 200, 300, so I'm going to go ahead and just pick a quantity of 200 now. Okay, because I know I'll be able to find that easily on the graph. Okay. And you can pick any number. You could pick 6 right now but I think it'd make it a lot more difficult to graph. So I'm going to pick a number 200. Let's go ahead and plug that in our equation. So quantity of 200, p=800-2×200. So that's going to be p=800-400, and p=400. So with a price of 400, quantity demanded will be 200. Let's go ahead and get that on our graph. Price 400, quantity demanded 200 will be somewhere right here. Alright, let's and by the way right now with these two points, we can make our demand curve, right. It's going to be a straight line, so we know that our curve is going to be like this, right. Something like that. It's going to pass through both of those points. I guess I could draw a little better. Let me try one more time here. Oh, I missed the other way now. One more. Alright. That was a little better. So there we go. That's what our demand curve is going to look like, but it doesn't hurt to do an extra one. We can always just confirm what we were doing here.

So let's go ahead and pick a quantity of 300, right. It looks like at 300, we're going to be at this 200 price level, right? That's where it looks like it's going to cross. p=800-2×300. p=800-600, and sure enough we got a price of 200 here. Right? So there it is. We got that answer. Feel pretty good about our line. So that is what our demand curve is going to look like here. Alright? So now I want to do a quick recap of how we can isolate different variables, and let's go ahead and do that in the next video.

Isolating Variables in the Demand Equation

Alright. So you'll notice in our equation, price is isolated by itself on the left-hand side of the equation, right? But sometimes we might want to rearrange the equation just so quantity is instead the isolated variable, right? So sometimes it could be the case that we want quantity to be just by itself on one side of the equation. So how do we do that? We're going to use just a little bit of algebra here to rearrange the equation. So let's go ahead and do that. I'm going to rewrite it.

P = 800 − 2 q d

So we want to get the quantity by itself, so the first thing we want to do is move the 800 to the other side.

P − 800 = − 2 Q D

Now, to get the

Q D

all the way by itself, we need to get rid of the coefficient, the negative two in front of the

Q D

, is right of that coefficient there for the quantity, but if we do that side by negative 2, we also have to do the other side by negative 2 as well. So this will cancel out these negatives, and we're going to be left with just quantity demanded on the right-hand side of the equation. But how do we do this P−800 over negative 2? Well, you might not remember exactly how to do that, so one thing I suggest is just take the denominator, the negative two, and just put it under each of them. So what we're going to have is

p ÷ − 2 − 800 ÷ − 2

and that's exactly what is happening here right?

P ÷ − 2 − 800 ÷ − 2

and 800 divided by negative 2 is going to give us negative 400, right? And

P ÷ − 2

, well that's the same thing as saying negative half p, right?

− P 2 − − 400

Right? So one more thing, do you guys remember from algebra is when we have two negatives, negative and a negative makes a positive, and we're going to end up with this formula right here.

− P 2 + 400 = Q D

That is how we got the quantity demanded by itself. We could also rewrite this.

400 − P 2 = Q D

right? I just rearranged the terms so that the negative term wasn't first. I think it's just easier to read like that.

Q D = 400 − P 2

right? So I just took the right hand side and I flipped it. I just put the QD on the left and the little equation stuff on the right. So that is how we switch which variable is isolated with just a little bit of algebra. There's more than one way to do this, so if you have another way you're more comfortable with, to switch the variables around, go ahead and do it your way. This is just the way I would have done it.

Alright, we're going to do the same thing now with supply and we'll have another example of isolating variables there as well. Alright. Let's move on to supply.

Equation of a Supply Curve

So just like we did with demand, let's go ahead and put a supply curve on the graph. So here we have an equation given to us: p=200+q_s. Okay. So p is by itself on the left-hand side of the equation, so we'll pick numbers for quantity, and then it will be easy to just find out what the price would be at that quantity. Alright, so let's go ahead and pick, let's start with an easy number, 0, right? So at a quantity of 0, let's solve for the price. P=200+0. Hopefully, you guys can do this math right here. P=200. Right? So at a quantity of 0, we're going to have a price of 200. Let's go ahead and put that in there. We've got our price axis, quantity axis. It's going to be right here.

Alright, let's go ahead and pick other numbers. I believe in the other video we did 200, 300, 100. Let's go ahead and just pick, we'll do 200 as well here, I guess. Why not? So p=200+q_s=200. So p=400 right there. Alright? So we'll say at a quantity of 200, price will be 400, and it will be something like that.

So let's go ahead and pick one more. Right now, we could go ahead and make our whole line because we have the two points, but just for fun, let's do one more point, and we did 300 before, so let's do it here. p=200+q_s=300 and we're going to get p=500 here. So at 300, we have a price of 500 right here, and it looks like it's going to be along the same line. Here's my attempt at the line, attempt number 1. Oh, it started so good. Alright, let's go again. Oh my god, people. Alright, that we're just going to have to settle for that. It almost goes through that last point, but there we go, that is going to be our supply curve right there, okay? So we've taken those points, we just plug them into the equation, and then we put it onto the graph. Let's do one more about isolating variables.

Isolating Variables in the Supply Equation

So just like we did with demand, we can also isolate different variables with the supply curve, right. There could be a situation where we want to get the quantity supplied by itself on one side of the equation, so we're going to have to rearrange and luckily, we can move this 200 out of here by subtracting 200. Let me do it in blue. We subtract 200 from here, subtract 200 from here, and we will get p - 200 = Q_S. So this equation was simpler than the other one; we were able to isolate Q just by moving the 200 from one side to the other. Alright, so now let's see how these work together.

Finding Equilibrium Using Algebra

Alright, so now let's see how we can find equilibrium using the equations for the supply and demand curves. Alright, so I've got the steps listed here. The first thing we're going to want to do is make sure that for our demand equation and for our supply equation, the same variable is isolated. That means we want only the price on one side of the equation or only the quantity on one side of the equation, right. We want to isolate the same variable for both. After we do that, and that's if needed, right, you could be given 2 equations where you're already isolated the same variables like you see here. Right here in both of these equations, the one for demand and supply, price is isolated in both situations, so we are good to go with step 1 in that situation, right?

So step 2 would be to take both of the curves and set them equal to each other, alright? We'll see how to do this in a second, but the idea is to remember when we're at equilibrium, there is going to be the same quantity supplied as the quantity demanded. So quantity supplied and quantity demanded are going to be the same amount, that equilibrium quantity and the price is going to be the same too, right?

The demand and then we're going to use a little bit of algebra to solve for the remaining variable, and then we're going to use a little bit of algebra to solve for the remaining variable and then once we get that answer, we'll use it in one of our original equations to get the second variable. So let's see it all in action.

So here I have the demand and supply curves that we were using earlier in this segment. On the left I have the price isolated, on the left or excuse me on the right I have the quantity isolated, and we're gonna do it both ways just so you see we get the same answer and that it doesn't matter which variable is isolated as long as it's the same one for both. Okay, so step 1 is done already in the sense that we have price isolated here and and quantity isolated here. So we're gonna do 2 examples on the left, we'll do the example where price is isolated, on the right where quantity is isolated.

Let's start with price. Scroll down a little more. So step 2 was to set the equations equal to each other, okay? So how do we do that? Remember price is going to be the same in both, okay? So if p = 802 QD and P also = 200 + QS, that means that 802 QD = 200 + QS, alright? And one more thing is gonna be that we don't need to write QD or QS anymore, we can just use Q because the quantity demanded and quantity supplied are the same at equilibrium.

800 - 2 q = 200 + q So you see what I did there? I took this side of the equation and this side of the equation and set them equal to each other.

So let's go ahead to step 3. In step 3, we're gonna take this equation we just made and we're gonna solve for q. Right? The variable that's in there. So I'm gonna rewrite it here. 800 - 2 q = 200 + q. So the first thing we want to do is get all the q's on one side of the equation, and I like dealing with positive numbers so I'm going to move them to the right. So I'm gonna take these 2 q's and I'm gonna add 2 there, and I'm gonna add 2 here. So this is gonna cancel out and we're gonna have 800 = 200 + 1 q + 2 q's = 3 q's. Alright?

So now let's go ahead and move the 200 to the other side. So we're gonna subtract 200 from each side there. We're gonna get 600 = 3q and now we want just 1q right? We've got 3q so we gotta divide by 3 on both sides. So we're gonna divide by 3, I'll do it in blue, divide by 3, divide by 3, and what do we get? 600 ÷ 3 = 200 and 3 ÷ 3 = 1, so 200 = q. We've found our equilibrium quantity of 200. All right, so that's the step 3 and now step 4 is pretty easy. We're just going to take our equilibrium quantity that we just solved for and we're going to plug it into one of our original equations.

So I'm gonna go up here and you can pick either one, you can pick the demand equation or the supply equation. I like to look at them and pick the easier one, the one that has less math involved and in this case it looks like the supply equation is a lot easier. It's just 200 + q. So I'm gonna pick that P = 200 + q. So we've got P = 200 + our equilibrium quantity of 200. So our equilibrium price is gonna be 400. We've got our equilibrium so P* = 400 and q* = 200. We just solved that using algebra. Pretty cool.

Now let's go ahead and use the other side of the equations just so you see we get the same answers here. Right? Alright. So let's go ahead and set these equal to each other right. Quantity is isolated by itself. We're gonna set this equal to this. 400 - 12p = p - 200.

Alright? So now what we want to do is solve for P , and that's going to be step 3. I'm going to rewrite the equation 400 - 12p = p - 200. So I want to get all the p's on one side of the equation So that's going to take adding 12 p here and adding 12 p over here, and let me go back to red. So this is going to cancel and we're going to be left with 400 = p + 12 p, so p is the same as 22 p. So 2 ÷ 2 p + 1 p ÷ 2 p is going to give us 32 p, right? 1 and a half p's. We had 1 p, we added another half p, we've got 32 p. Minus 200. Alright.

And now let's go ahead and get the 200 on the other side. So we're gonna add 200 to both sides and we'll have 600 = 32p, and this will cancel out. So how do we get the p by itself now, right? We've got 600 = 32p. Well if you remember from algebra, the trick here is we're going to multiply by the reciprocal. So if we multiply 32p times 23, This 23 is gonna cancel. I'm just gonna get out of the way so I'm not dodging. The 23 is gonna cancel with the 32, and we need to multiply the other side of the equation also by 23. Okay. So the 32 oh actually I'm going to write all those in blue times 23, and this side also times 23. Okay so let's go ahead and cancel stuff out. The 2's cancel, the 3's cancel, and we're left with just p on that side of the equation, and then we'll do 23 × 600. So 2 × 600 is 1200 ÷ 3 = 400. So 23 × 600 is gonna be 400, alright? So there we go. We've gotten a price of 400, which we can confirm in step 4 the other time, we got a price of 400. So it looks like we're getting the same answer.

And let's go ahead and do the last step where we solve for quantity using this price that we have. So again, I'm gonna pick the easier formula and to me it looks like the supply formula again is easier in this situation, so I'm gonna go ahead and plug actually I'm going to use the demand one just to prove that we could use either one. So I'm going to use quantity demanded equals 400 - 12 P and I'm going to plug in our P there. So Q equals 400 - 12 O f P which was 400. So we're going to get Q = 400 - 200 is 200 and that confirms what we just got on the other side, right? We got a P* = 400 and Q* = 200. So either way, either variable was isolated and we've got the same answer there. Cool?

So let's go ahead and try some practice with this stuff.

Finding Equilibrium Using Algebra

Alright. Let's go ahead and try this one here. So, it gives us a couple of curves here. We've got a supply curve and a demand curve, and it's asking us to find the equilibrium price and quantity. First thing you should notice is that different variables are isolated in each equation, right. In our demand equation, the first equation, price is isolated and in the second equation, the supply equation, quantity is isolated. And how do I know which is which? It's because the first one has quantity demanded in it, the second one has quantity supplied in it, okay? So the first thing we want to do is isolate the same variable. So I don't like the looks of the first equation, I kind of think that fraction is just messing with me. I'm going to rearrange that so that I have quantity supplied or quantity demanded by itself, but remember, you could also rearrange the other one, and we'll get to the same answer. This is the way I'm going to do it.

So we've got p = 6 − 150 QD. Right? And I want to get that Q_D by itself. So, the first thing we need to do is move that 6 from one side to the other, and we're going to get p − 6 = −150 QD. So how do we get rid of that pesky fraction? We need to multiply by the reciprocal, right? So, I'm going to multiply by 50 over 1, right? So just negative 50, so multiply by negative 50, and negative to get rid of the negative sign in the front there. So I want to do negative times a negative to get rid of the negative, and then the 50 times the 150 to cancel out that fraction.

So if I'm going to multiply this side by negative 50, I need to multiply the other side by negative 50 as well. Let's go ahead and isolate this variable. So, this negative and this negative cancel, the 50 and the 150 cancel, and we're left with just quantity demanded on this side of the equation, and let's expand this out. We've got negative 50 times p, so negative 50

and negative 50 times negative 6. Those negatives are going to cancel out. We're going to get a positive, 6 times 50 is 300. Alright. So, negative 50 times p and then negative 50 times negative 6. So there we go. That is going to be the same equation with quantity demanded isolated. Rights. Now we can go on to the next step where we set the 2 equations equal to each other based on that isolated variable.

Alright. So, I'm going to take this side of this equation right here and this side of this equation right here, our supply and our demand equation, and let's go ahead and set them equal. So − 50p + 300 = 150p − 100. Alright, so now I want to isolate my Ps, so I'm going to get them all on one side. Plus 50

plus 50

, and I will have 200

over here minus 100, and on this side the ps are canceled, and I'll have 300. 300 equals 200

minus 100. Let's go ahead and add 100 to each side. So 400 equals 200

, that cancels. We'll divide both sides by 200, and we'll get an answer of p = 2. Alright, so we've figured out what p is, Now it's just a matter of plugging this into either of our original equations, and we will get our quantity. So, this will be our equilibrium price and based on this alone, we'll know which answer to the question it is, right? It's going to be B. See, only one with an equilibrium price of 2. So on a test, you're crunched for time. Hey, this is enough information to get this right, but let's go ahead and finish it up.

So with the price of 2, I'm going to go ahead and use the supply equation. It looks easier to just plug a number in, so quantity supplied or just quantity, right, at equilibrium, they're going to be the same for demand and supply. Quantity equals 150 times our price of 2 minus 100. Right, so I just took this equation up here, QS = 150p − 100, plugged in 2 for p, and let's find out what Q equals. 300, 150 times 2 is 300, minus 100, Q = 200. Just like we see in that answer, so the answer is going to be B. Alright, let's move on.

The supply and demand curves for a product are as follows: 

Q_d = 10560 − 80P 

P = 1/40 Q_S + 6 

What is the equilibrium price and quantity of the product? 

• A. P* = $60, Q* = 5760
• B. P* = $70, Q* = 4960
• C. P* = $80, Q* = 4160
• D. P* = $90, Q* = 3360

Finding Equilibrium with Equations and a Graph

So now I want to draw our curves on the graph and find equilibrium on the graph, right. Just like we had been doing before when we were doing this without any math and we were just setting up our equation, how did we know where equilibrium was? It was where they cross, right? That was the equilibrium point. So we can do the same thing when we start plotting these lines that we're given, these equations. If we plot them on the graph and find where they intersect, that's going to be our equilibrium. We can find them on the graph as the intersection, right. They're going to be the intersection of the supply curve and the demand curve just like we saw when we weren't using math. So let's go ahead and take these equations and we are going to pick some values for price and then we will find the quantity demanded, the quantity supplied at those prices, and we'll go ahead and plot them on the graph. So let's start with these equations. I'm going to pull down the equation where quantity supplied is isolated, right, we're going to be picking prices and solving for quantity, so it's easier to do that when the quantity is already isolated in the graph, or in the equation. Let me go ahead and grab these equations, the first one being quantity demanded equals 400 minus half P, and the other one was quantity supplied equals P minus 200. So QS equals P minus 200 and you could do this with either one, you could use either equation, price isolated, quantity isolated, you just have to do a little more algebra when selecting prices here.

So let's go ahead and start with a really easy price, a price of 0. So at a price of 0 we're going to find a quantity demanded, 400 minus half times 0. We're going to get quantity demanded equals 400. Let's do the same thing with supply. Quantity supplied equals 0 minus 200. So we're actually going to get a negative number here, -200. At a price of 0, this is kind of meaningless, right? They're not going to produce anything with the price of 0. They can't produce negative supply so this we can't even really plot this on our graph because we don't have negative numbers on the graph, this isn't going to be so useful for us. So let's try another number that might give us something that we can plot. So we might want to try something a little bigger. Remember on our graph, our axes are going up 200, 400, 600, 800, it's going up pretty quickly there for price. So I think it'd be safe to pick a price, let's say around 400. Let's see what we get at a price of 400. So we're going to get \( q_d = 400 - \frac{1}{2} \) and our price in this case is 400. So we'll get 400 minus half of 400 is 200. We'll get a quantity demanded of 200. Let's do the same thing with supply. Quantity supplied equals price of 400 minus 200, and we will get a quantity supplied of 200, right. So we just plugged in 400 into that equation and we got 200. So notice what we got here. We got a quantity demanded of 200 and a quantity supplied of 200. What did we learn about equilibrium? That's when quantity demanded equals quantity supplied. We actually found it by mistake here so this is actually going to be pretty helpful once we're making our graph, but let's go ahead and find one more point so that we can make our supply curve, and I'm going to pick a price even a little higher. Let's pick a price of 600 and see what we get. So quantity demanded equals 400 minus half of our price which is 600. So quantity demanded equals 400 minus half of 600 is 300. Our quantity demanded is going to equal 100.

Alright let's do that over here with supply. Quantity supplied equals our price of 600 minus, our equation was \( P - 200 \), price minus 200, quantity supplied is going to equal 400. Alright, so now we have some points that we will be able to graph, right? So for demand, they all worked. For supply, this first one didn't really work for us, right? And that was because we got a negative number. So let's go ahead and plot these other points. So demand at a price of 0, quantity demanded was 400. So this will be our price axis, quantity axis, and I will use blue for demand here and red for supply because we're not going to be shifting or anything so I think it's safe to use 2 different colors. So we're going to be at that point there. Let's just go ahead and do all of the demand ones first. At a price of 400, they're going to demand 200. So price 400, demand 200. And last but not least, at a price of 600, demand is 100. So we'll be right there. So our demand line is going to look something like this. Oh, wow. First try. That's about as good as I'm gonna get. Alright. Let's do the same thing with supply. So we can't plot that first one. Let's go to the second one. Remember, you only need 2 points to be able to make a line. You just connect the line and keep going. So at a price of 400, quantity supplied is 200, and you can kind of see where we're going to end up here where I'm going to draw that red right on top of the blue, and at a price of 600 quantity supplied is 400. Price of 600, quantity supplied 400. We'll end up somewhere like that. So let me try and draw that. Wow. I'm improving. Okay. So what have we found? We found that they intersect on the graph right there at that point. 400 and 200, right? So that is going to be our equilibrium. Price of 400 and a quantity of 200. So we've now solved that algebraically just using algebra, we've been able to find it on the graph as well. Alright, so that's about as deep as this algebra stuff is probably going to go, so let's go ahead and move on.