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Learn the toughest concepts covered in your Macroeconomics class with step-by-step video tutorials and practice problems.

Deriving the Aggregate Expenditures Model

AE Model: Algebraic Approach


Algebraic Approach to the AE Model

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So here we go, we're gonna use algebra. Now we're going to use an algebraic approach to calculate our macro economic equilibrium where our aggregate expenditures equal are real GDP, let's check it out. So we're gonna use for uh some linear equations rather than a graph or just numerical numbers uh to solve for this equilibrium, where we're gonna have our aggregate expenditures equal our GDP, right, We're looking for that point where aggregate expenditures equal G. D. P. And we're gonna have to use a line for our aggregate expenditures. And we're gonna we're gonna use algebra to back into our GDP. Okay, so remember when we calculate aggregate expenditures, we're gonna have our consumption, which is gonna be some base amount of consumption A plus MPC times why write our disposable income? And we're not gonna generally we're not gonna uh differentiate between income and disposable income. It's generally just gonna be y just like this and all of these other ones are gonna be constant. Okay, they can throw some tricks at you that I'm gonna definitely show you as we go through some examples, but in general they're just constant numbers. So the macroeconomic equilibrium can be stated at the point where y R G D. P equals C plus I plus G plus N X. Which is our aggregate expenditures. Okay, so we're gonna be looking for this point where Y equals C plus I plus G plus N X. Now the trickiest part here is that when we solve for consumption. Well, remember consumption depends on GDP. So there's this kind of relationship, whereas GDP goes up, consumption goes up. So they kind of both go up together so they since consumption depends on GDP, we're gonna have this situation where we're calculating Y. D. But it's dependent on why, right? Because how did we see that the higher GDP leads to higher disposable income? Because the G. D. P. Remember we're producing more, we're hiring more people, there's more people employed making more money, which in turn leads them to consume more. So this higher disposable income leads to higher consumption. Okay, so that's the biggest trick here and we'll go through an example to see how we do this, but we're gonna have to use the the GDP number to calculate our consumption. Alright, so let's pause here and let's go through an example together and then you guys can practice

Aggregate Expenditures

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Alright, so let's go through this example. We want to use the following information to solve for macroeconomic equilibrium. And remember the macroeconomic equilibrium is where G D. P Y. Is equal to C plus I plus G plus and X. Okay, Y equals C plus I plus G plus N X. So we want to add all of those together. So let's go ahead and start here. We want Y to equal C plus I plus G plus an X. So let's start with C notice. They gave us C. I. G. And then they didn't give us N. X. They gave us exports and imports. This is the terminology they might use for that. But let's start here with consumption. So let's start with consumption as C. Which is 2000 plus 0.65 Y. So that is our consumption function right there. 2000 plus 0.65. Y. So I'm gonna go like this for C and then we're gonna add to it plus investment of 30 200. And this is i right here We're gonna add to it government purchases of 2800 which is G. And then net exports, they didn't tell us directly they gave us an amount of exports and an amount of imports. So we have to remember is that net exports is equal to exports? So exports would be exports X minus imports. Em right exports minus imports. Is our net exports. Okay, so in this case they tell us that our exports were 500 And our imports were 1500. So that means we were we were importing more than we were exporting right? There were more imports and exports. So we're gonna have a negative number for our net exports. And that's generally the case that that kind of happens a lot in these problems. Especially because currently the US is in a trade deficit where we import more than we export. So we generally are going to see a negative number for exports. So there we go. We've got C. Plus I. Plus G. Plus N. X. Right? So notice they gave us the C. As an equation and net export as an equation. So let's go ahead and simplify this first. Let's add up all of our constants together. So we had 2000 let's circle them all. 2000 plus 3200 plus 2800 plus 500 minus 1500. So let's do that math real quick. 2000 plus 3200 plus 2800 plus 500 minus 1500. That comes out to 7000. So this is generally how you're gonna wanna do these problems, add up all of your constants together and then you're gonna have your 0.65 Y piece right here plus 0.65 Y. Okay so notice we've simplified this quite a bit right now we've got just one constant and our um marginal propensity to consume times Y. So we want to do. Now is we want to solve for y we want to find that equilibrium amount of G. D. P. So what we wanna do is we're gonna subtract 0.65 Y. From both sides To get all the wise on one side and we're gonna be left with Y -0.65. Y. Do you guys remember how to do that? Well this is just like one Y. Right, so one y minus 0.65 Y. Is equal to 0.35. Y. I know a lot of you still struggle with these algebra things but this is about as tough as it's gonna get here. That that that's the toughest thing you're gonna have to do with these algebraic equations. So nothing too crazy. So here we go. 0.35 Y equals 7000. We divide both sides by 0.35 and that will get us to have why by itself. So finally we've got Y equals So our final equation here is 7000 divided by .35. And we get 20,000.20,000 is our equilibrium g. d. p. And that's where aggregate expenditures equals GDP, which is why write a equals G. D. P. When y equals 20,000, simple as that. All right, so that's all you gotta do is you just gotta add up all your consumption, investment, government and net exports together and solve for Y. Cool. Alright, let's pause here. And you guys try the next one in the practice problem. I've thrown some tricks at you, so if you get hung up, go ahead and watch the video and we'll work.

Use the following information to solve for macroeconomic equilibrium (T is a lump-sum tax): 

C = 1,500 + 0.75(Y-T) 

I = 3,400 

G = 2,600 + T 

X = 750 

M = 2,000 

T = 500

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