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21. Determine consumers' surplus (CS) and producers' surplus (PS) under market: equilibrium 0 Demand: p = 400-92.1 Supplv: p = 20q + 100...

Question

21. Determine consumers' surplus (CS) and producers' surplus (PS) under market: equilibrium 0 Demand: p = 400-92.1 Supplv: p = 20q + 100

21. Determine consumers' surplus (CS) and producers' surplus (PS) under market: equilibrium 0 Demand: p = 400-92.1 Supplv: p = 20q + 100



Answers

Consumer Surplus and Producer Surplus In Exercises $61-64,$ (a) graph the systems representing the consumer surplus and producer surplus for the supply and demand equations and (b) find the consumer surplus and producer surplus.
$$\begin{array}{ll}{\text { Demand }} & {\text { Supply }} \\ {p=100-0.5 x} & {p=25+0.1 x}\end{array}$$

Okay, So we are given demand and supply curves and we essentially want to find the consumer and producer surplus. But to do that, the first thing we're gonna have to do is find the point of intersection. And the reason I say find the point of intersection instead of just graphing them is because these are actually really difficult to graph. And the reason isn't because it's challenging, right, this still has a slope on a Y intercept. It's just that if the y intercept of this one is 140 the rate of change is that that is so close to zero that this is almost going to look like a horizontal line, Which means this is going to start at 80, Wherever 80 is and it's going to increase at such a small rate. It's also going to look like a horizontal line. Okay, But they aren't horizontal and eventually at some very, very large number they're going to intercept. So there's no point in even trying to draw it. We're just going to set the equation is equal to each other to find the point of intersection. Okay, so I'm gonna set 140 .002 x equals 80 plus 800.1 X. So I'm going to subtract 80 from both sides so that this side becomes 60. I'm going to add This to both sides that I get .03 X. Right, okay. So now I'm gonna plug that into my calculator because I don't feel like thinking today. Um And I get a very large number, I get two million. So X is going to be two million now to figure out the price at which this occurs, I plug it into either one of these equations. It honestly doesn't matter which one you do. I'm actually going to plug it into this one because it's easier to do mentally. Okay, so the price is going to be 80 plus when you multiply by this decimal, It's equivalent to moving the decimal .5 places to the left. So I'm going to take two million and move the decimal five places to the left. That's going to be one 234 five, which is 20 An 80 plus, 20 is 100. Okay. Or you could just plug that into a calculator and substitute the correct value of X. It really honestly doesn't matter. You'll get the same if you plug it in either one by the way and you should check that. Okay, so the ordered pair or the point of intersection is going to simply be uh two million, which I'm going to. Well I guess I shouldn't abbreviate that. two million. Common. 100. Okay, so I don't know why it's being all weird and stuff. Okay, sorry about that shouldn't be erasing all of this. But basically these two points are going to intersect. So for the sake of this question, I'm actually just going to exaggerate this. So the demand curve is always going to be negative And the supply curve is always going to be positive. Okay, so they're going to intersect at some point and this is not a correct visualization of this graph, because this point happens at two million. Right, I'm going to abbreviate to with a capital at two million and this point is going to be 100. But I'm just doing that so that we can understand how to find consumer and producer surplus. Now, the first thing you do is you take this equilibrium point, you trace it back to the y axis, and then you're going to have two right triangles, this right triangle, the area of this right triangle that's going to represent your consumer surplus, consumer surplus, and this right triangle is going to represent your producer surplus. So all we need to do is find the area of those two triangles. And then we're good. Well, that's not terribly difficult. Okay, the first one is going to have a base of two million Because that's what this distance is, it's going to have a height of 40 because 140 is over here, and 100 is over here. To find the distance between those two points, you subtract 140 and 100 to get a value of 40. Okay, so this is going to be uh formula for the area of a triangle is one half base times height. So it's gonna be one half Times two million times What did I say, 40? So it's going to be a million times, Which is 40 million. This capital M just means million. That's not a math thing. That's just abbreviate. Okay, so the second one, the producer surplus in this case the bases two million. And the height is this distance? Well, the distance between 180 is just a positive 20. So we do the same thing. You take one half, one half of two million and 20 And you'll get 20 million.

This problem we are asked to find the consumer and producer surpluses for a demand function of 300 minus X. They supply function of 100 plus X. So the first thing that we're going to need to do is solve for the X value where we have equilibrium between supply and demand. So we said, we'll see daisy, we said supply 300- X equal to demand or opposite way around, but no difference here significantly. Uh So we can add X to both sides, subtract 100 from both sides, so we'll get to X equals 200 therefore X is going to equal 100. Then we can plug that back into one of our price functions and figure out that the price at 100 The equilibrium price is going to be 200. So now that we have that the consumer surplus, which I'll call sc going to be the integral from zero up to 100 of the demand function minus the price. That's 300 minus x minus 200. Which will be just 100 minus X. D X. So that will be 100 X minus X squared over two, Evaluated from zero up to 100. So that will be uh one second here Will be 100,000 or sorry, 10,000 rather 10,000 -10000 over two. So 10,000 -5000. So the result there should be 5000 and then the producer surplus Is going to be the integral from 0 to 100 Of the price. 100 minus the supply function. There will be 100 minus one. Yeah. Excuse me. One second here. All right. So yeah, that will turn out to be 100 -100 -1. So that will be just negative x dx evaluated from 0 to 100. So that will turn out to be negative X squared over two I evaluated from 0 to 100. So the result there Is going to be negative 5000. Excuse me, That should be 200 minus X. So 200 X. So 20,000 minus 5000. So the result there should be 15,000, I believe. One second. Yeah, no, I take that back. It was only in the producer surplus that I screwed that up. So the actually this should be down here 200 minus X. So that's going to it would be 200 minus 100 minus x. So that will turn out to be 100 minus X there. So the result that we arrive at, when we actually do everything properly here should be the same thing as what we had above. Essentially we should end up having At the end of the day we should end up having 5000 for the producer surplus

For this problem we are asked to find the consumer and producer surpluses or the demand function is 975 minus 23 X. And the supply function is 42 X. So to begin, we need to find the equilibrium price value so we can do that by setting the two price is equal to each other. So we have 9 75 minus 23 X equals 42 X. After doing a little bit of rearranging, we should find that X is going to equal uh 15 Yeah, 15. Then having that the consumer surplus, it's going to equal the integral from 0 to 15 of 975 minus 23 X minus the price at X equals 15, which I should have said earlier, the price at X equals 15. That equilibrium value is going to be 630. So we'd have 975 -23 x -630. So one second here. Okay, that will be the equivalent to Just 3 45 -23 x. So integrating will have yes, 304. Come on. 345 X minus 23 X squared over two, evaluated from 0 to 15 And plugging in the values there. We should find that the result is going to be 2587 0.5. Then For the next part, the producer surplus, it's going to be equal to the integral from zero up to 15 of 6 30 minus 42 X dx. Which will be equal to 6 30 x minus 42 X squared over two or minus 21 X squared, evaluated from 0 to 15 and plugging in the values and calculated that we should get a result of 4725.

For this question, we are given both the price demand equation as well as the price supply equation. So we're asked to find the consumer surplus and the producer surplus at the equal living in price level. So equilibrium price, that is where these two curves interesting. So we can find it by equating them to each other. So we'll let 50 -0.1 x. Be equal to 11 plus 0.5 X. Okay. And then This gives us 39 is equal to 0.15. And then now we can divide both sides by 0.1 by When we get access to six. So this is the equilibrium quantity or otherwise known as as known as X bar. Okay, so now that we have the quantity, we can go and find the price by substituting in this X value to either the demand curve or the supply curve. So let's say we pick the supply curve and we get he is equal to P bar is equal to 11 plus 0.5 times +26 Okay. And then that should equal to 24. So this is our equilibrium price now that we have X. Bar and P. Bar. We can go ahead and find the consumer surplus and the producer surplus. So let's first find the consumer surplus. That is going to be equal to Um the integral from 0 to X. Bar Of the demand curve. So that's 50 -0.1 x. And now we have to subtract explore times people. Okay, so this is equal to anti derivative is 50 x minus 0.05 x squared, Evaluated from 0 to 60. And now we subtract to 60 times 24. Okay, so let's go ahead and put this into a calculator. So this is going to be 50 times to 60 -15 times to 60 squared. And then we subtract 260 times 24 mm. So this ends up being 3380. So this is our consumer surplus. Now let's find a producer surplus. Producer surplus is X Bar times P bar minus The integral from 0 to X. Bar of the Supply equation, which is 11 plus 0.05 banks. Okay, so this is equal to the same thing that's to 60 times 24 minus. And the integral where the anti drood over that is 11 X plus 0.25 X squared Evaluated from 0 to X. Bar, which is 26. And the same thing, let's go ahead and put the values in. So that's 260 times 24 minus. And we have 11 times to 60 plus .025 times to 66. So this ends up being 1690. Okay, so there you go. This is our consumer surplus and this here is our producer surplus. Now the last thing we need to do is um is to include a graph that identifies um those two surpluses. Okay, so I have that right here in the middle. Is this, so that's right over here. So you can see this red curve is the demand curve, and then this purple curve right here is the supply curve um X. Bar P bar is the equilibrium price and quantity, which is 24 2 60. So now this purple area is our supply, start producers surplus. And then this green area here is our consumer service.


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