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This video will help you figure out or compute the consumer surplus, the producer surplus,
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and the social gain given a demand function and a supply function that are functions of
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the quantity, not of the price.
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So if you find yourself in a situation where you have demand as a function of price, you're
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going to have to find the inverse function. One that has quantities and input in order
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to do this.
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So if you want to find the consumer surplus, producer surplus, and total social gain at
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the equilibrium quantity in price, first you're going to have to find the equilibrium quantity
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in price.
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So to do that, as you learned in a previous class, math 140, you need to find the place where
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both these functions are the same. In other words, where they're graphs cross, where they're
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equal, their outputs are equal.
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So there's a couple of ways to do that. You can do a graph and I've put both functions
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here. I've put both functions here in the calculator and I've already set up a window
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for them.
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I know my window is going to be at a Y max a little bit bigger than 80, right?
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Because this is going down and this one's going up from 34. I'm not quite sure about
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what X should be, so I made it big enough and you can always adjust the X max to whatever
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you need. So if I go ahead and graph it, I can see that I was okay. My window is good.
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I find I can see the intersection. So now I can do a second, calc, and intersect, which
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is 5. And I can do enter, enter, enter in order to find the intersection.
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Okay, so here's my equilibrium quantity at 15 and the other one, the equilibrium price
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is 54, 84. Alright, so that's the way to get it graphically and you get both answers at
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once. There's another way you can actually find the solution to the equilibrium quantity
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and that's using solver. So if, for example, you create another one called Y, 1 minus Y
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2, whenever that function is 0, then these two must have the same output, right? So I
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could solve for X, I eat the quantity in this case. So if I fire up a solver with Y, 4,
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then, and I go ahead and do alpha, solve. I'm the only thing I see there, which is X. I
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find out it's 14.98. Then I can go back, for example, and go out to the home screen.
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I can evaluate either one of these functions. It doesn't matter which one I evaluate and
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find out the corresponding price. So here I'm going to evaluate Y 1 at 15 and get the
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other part of this, the equilibrium price. Okay, so you just saw two ways to get the answer
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to the first part, which was the equilibrium price and quantity here. So yes, we got
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those correct. 15 and 54, 84. Alright, so now how do you do the last three problems? Well,
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here I have a little description written out. So you should already know that the producer
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surplus and the consumer surplus represent areas. For example, the consumer surplus when
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you're at the equilibrium quantity is actually this area right here. And that area is the
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integral under the curve, under the demand curve, all the way out here, minus the area
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of this rectangle right here, this area of this rectangle right here. And if you do that,
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you end up with the shaded area that I just made up here, which is the consumer surplus.
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If you do it in the opposite sense, you do it in the opposite sense, then you end up with
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the consumer, excuse me, the supplier, producer surplus, which is this area right here. That's
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the producer surplus. And that's the area underneath the supplier curve. And then you
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got to subtract that from the rectangle, that is, P star, Q star. And that'll give you
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this area right in here. And that's the producer surplus. The producer surplus is right here.
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Right? Let me just get this out here and I can just write it in. It'll be really easy.
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So this is the producer surplus. And this is the consumer surplus in here. When you put
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those two areas together, you get what's called the social gain. Okay? So I need to integrate
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zero to the equilibrium quantity of the demand function and subtract off the rectangle to
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get the one. And the other one I have to integrate the supply function of zero to the equilibrium
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quantity and subtract that from the rectangle. And I get the producer surplus. And if I
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add those two together, I can get the social gain. So that's how I do the computation.
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Now I just got to do it in the calculator. So in my calculator, in Y zero is my integrator
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is my formula for integration. So in other videos, I showed you how to do this, how to
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have this in here. And there's a sheet that reminds you of how to put these things in.
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So mine always refers to Y four. So I need to make sure that whatever I'm integrating
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is in Y four or is referenced to Y four. So I don't want to integrate the difference.
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That's not what I'm trying to do here. I'm going to integrate first the demand function
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which is Y one. So since my integrator refers to Y four, I got to edit this now and take
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out this part right here. So now I'm only going to integrate Y one. So if I fire up
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solver and I switch it over to looking at Y four instead, which is Y zero, my integrator.
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So I'm going to go over here, type in a zero. Okay, now I got Y zero. So here's my integrator.
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Remember we don't care about X. We ignore X. We're going to integrate from zero to where.
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So 15, right? That's my equilibrium quantity. And now I do alpha enter on the I for integrate
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for that integral. And then I wait. And there's my integration, but then to do the consumer
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surplus, I have to subtract off the rectangle whose dimensions are what? Subtract off the
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rectangle whose dimension is 15 by what? Let's see. What the other one was here? By 54,
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84. So by 54.84. If I do that, just press downward. Then I can check my answer. And
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I see I got it right. 183.5. All right. And how do I do the consumer? Excuse me, producer
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surplus. I just got to integrate the other function. So I got to integrate Y two. So that
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means I got to put Y two here in my schema things. And now I can go back to solver. And
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everything set. I'm just integrating a different function. I'm still integrating it on the
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same limits. So all I got to do is go ahead and integrate. So that's what I'm going to
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do. Now, according to what I showed you earlier, I need to subtract this from the box. From
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the box. So I need to go second insert. And then I need to multiply 15 times. And once
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again, what was the corresponding equilibrium price? 54, 84, 54, 84, 54.84. And I need
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to subtract my area under the curve that I just found from that. And I get 164. So yes,
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I got that right. And then if you notice, the last answer is just adding these two together.
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To get 384.2. Okay, so that's how that actually do the computations of these things with the
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assistance of your calculator. Consumer surplus, producer surplus, total social gain.