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concept

## Quantitative Analysis of Taxes

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now let's see how we can calculate the price the buyers pay and the price the sellers receive in the case of attacks when we're given equations and use algebra. All right, so, um we're gonna do the same thing. We've got a few steps we're gonna follow, but you're gonna see that we're still just solving for equilibrium right? We're gonna figure out some numbers, it's gonna be pretty similar to what we've been doing. So we've got a couple of steps here, but the end goal is to find out what is the price the buyers pay and what is the amount of the sellers receive when we've got a tax? It's gonna be easiest to do with an example. So let's go ahead and just jump right in. It says the original supply and demand curves are as follow as follows, what is the new equilibrium price and quantity? If suppliers are tax, $1 per unit, suppliers are tax $1 per unit. Right? Um So we're gonna look for the new equilibrium price and quantity and then we're gonna look for the amount of suppliers receive and the amount of consumers pay. Right? So there's gonna be a shift to the supply curve because of the tax and then we are going to also see the new prices, Right? So let's go ahead and just start following these step by step. So we've got our two equations there, one for supply, one for demand, and it tells us to replace P with P minus tax in the supply, right? Or P with P plus tax in the demand. Well, in this case it's the suppliers that are taxed, right? So we're gonna use this first one, it's gonna be one or the other here, depending on whose taxed. But regardless we will end up at the same answer. Right? We've seen that with taxes that no matter whose tax will get the same answer. So even if we were to do the P plus tax, we'd end up at the same place here, we have a different equilibrium. I guess we would have a different equilibrium, but our prices would end up the same. So to get the correct equilibrium, we have to shift the correct graph. And that is by uh replacing P with P minus tax in the supply, Right? And this makes sense that we're gonna replace P with p minus tax because the suppliers are gonna receive less, right by the amount of the tax they're gonna have to pay a tax. So they're gonna, the end money that they end up with is gonna be the p minus the tax. Right? So let's go ahead and do that. We've got our supply equation which was quantity supplied equals to p minus six right to p minus six. But we're gonna replace that P with p minus tax. And in this case the tax was $1 right? That's the tax. So let's do it. Q quantity supplied equals two. And instead of p I'm gonna put p minus tax, right? And then I'll put in the $1 in a second minus six. Right? So there's our whole our original equation, but instead of P I've substituted substituted P minus tax. So let's continue here, quantity supply equals two and I'm still not gonna do it out, I'll go P minus and now we'll put the tax in, right, it's $1 here from the problem minus six. Okay, now we have to um what's it called, foil this out right to P Oh man, it's been around been awhile, so two times the P and then two times the $1 to p minus two. And then we've still got this minus six out here. Right? So to p minus two minus six, we're gonna get a new quantity supplied equation to p minus eight. Cool. All right. Um So now that we've got our new supply equation, this is the supply equation with the tax. This is essentially a shifted supply equation. And and we've also got our quantity demanded, right? They gave us the quantity demanded uh equation. So next all we gotta do is find equilibrium at this price. Right? Or excuse me at this with this new curve. So this was step one right here that we just did let's go ahead and do step two over here, we're gonna solve for equilibrium. We've done this before. The only difference here is that we have a new curve right? We created this curve right here by shifting with the tax, Right? So we're gonna use that quantity supplied, not the original one. So we've got two p minus eight. Right? That's our supply. And remember this quantity supplied equals quantity demanded at equilibrium. So if two p minus eight is gonna equal 10 minus P, right, that 10 minus P is our demand equation over here. So we've set quantity supplied equal to quantity demanded, which is what we want at equilibrium. So let's solve for p we're gonna get all the peas on one side and all the numbers on the other and let's see what we get. So these cancel these cancel and we're gonna get three P Oops, let me do it in red. Three P equals 18. We're gonna get P equals six. Right? So our new equilibrium price with the tax is gonna be six but equilibrium is gonna be only for one party, right? Because there's the part, the buyer and the seller, we have different prices. Right? So here is the new equilibrium price. We don't know whose price that is. It's gonna either be the seller or the buyers price. We'll get to that in the second, let's go ahead and get our our equilibrium quantity in this situation. I'm just gonna use our quantity demanded. Um Because it's it's very easy, right? 10 minus. And we put in our equilibrium price of six to find our equilibrium quantity. So the quantity, man, it's gonna be four. Right? And that is our equilibrium quantity. Right? At equilibrium quantity supplied equals quantity demanded. So that is going to be the equilibrium quantity here of four. Okay? Um So we got our equilibrium price and quantity. So that's step two and now in step three, it's easy enough. It just tells us that in step three the equilibrium price is the amount paid by the non taxed party. Well, in this case the supplier was the tax party, right? So the non tax party was the demand, so demand price. Right? This is gonna be um P. B. Is going to equal the six, right? That is um that's how we do it. The non tax party gets that equilibrium. Okay. And then last we need to solve for the remaining price. Well, that's easy enough, right? If we know that this one's $6 well, the suppliers are gonna receive $1 less, right? The supply price is gonna be $1 less because that is the amount of the tax. So the supply price, let me scroll down a little more. Um The supply price, which is P. S. Is gonna be that six minus $1 tax. They're gonna get a price of five. Right? So what is our new equilibrium? Our new equilibrium is $6 right? And a quantity of four. That's the new equilibrium after we've shifted the supply curve for the tax, the suppliers are gonna receive $5. And the commanders are gonna pay $6. Right? And we can confirm we can confirm that that's gonna all work out, right? Because we've got a dollar tax and there's a dollar in between the suppliers receive and the consumers pay. So that seems good to me. All right, let's go ahead and do a practice problem where you guys can try these steps out and solve for the prices of the buyer and the seller. Alright, let's do that in the next video.

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Problem

The supply and demand curves for a product are as follows. What is the amount suppliers receive if a $0.50 tax is imposed upon consumers?

Q_{D} = 600 – 100P

Q_{S} = -150 + 150P

A

$2.80

B

$3.00

C

$3.20

D

$3.30

E

$3.50