AE Model: Algebraic Approach: Videos & Practice Problems

To find macroeconomic equilibrium, where aggregate expenditures equal real GDP, use the equation $Y=C+I+G+\mathrm{NX}$. Aggregate expenditures consist of consumption, investment, government spending, and net exports. Consumption is influenced by GDP, as higher GDP leads to increased disposable income and consumption. Understanding this relationship is crucial for analyzing economic stability and fluctuations, particularly in the context of fiscal policy and aggregate demand management.

Algebraic Approach to the AE Model

So here we go. We're going to use algebra now. We're going to use an algebraic approach to calculate our macroeconomic equilibrium where our aggregate expenditures equal our real GDP. Let's check it out. So we're going to use some linear equations rather than a graph or just numerical numbers, to solve for this equilibrium, where we're going to have our aggregate expenditures equal our GDP. Right? We're looking for that point where aggregate expenditures equal GDP and we're going to have to use a line for our aggregate expenditures and we're going to use algebra to back into our GDP. Okay? So remember, when we calculate aggregate expenditures, we're going to have our consumption which is going to be some base amount of consumption \( a \) plus MPC times \( Y \), right? Our disposable income and we're not going to, generally, we're not going to differentiate between income and disposable income. It's generally just going to be \( Y \) just like this and all of these other variables are going to be constant. Okay? They can throw some tricks at you that I'm going to 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 \( Y \), our GDP equals \( C+I+G+NX \) which is our aggregate expenditures. Okay. So, we're going to be looking for this point where \( Y = C + I + G + NX \). 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 wherein as GDP goes up, consumption goes up, so they kind of both go up together. Since consumption depends on GDP, we're going to have this situation where we're calculating \( Y_d \), but it's dependent on \( Y \). Right? Because how do we see that? The higher GDP leads to higher disposable income because the GDP, 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 going to have to use 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

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 GDP \( y \) is equal to \( c + g + nx \). Okay? \( y = c + g + nx \). So we want to add all of those together. So let's go ahead and start here. We want \( y \) to equal \( c + g + nx \). So let's start with \( c \). Notice they gave us \( c \), \( I \), \( g \), and then they didn't give us \( nx \). 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 \( 2000 + 0.65y \). So that is our consumption function right there. \( 2000 + 0.65y \). So I'm going to go like this for \( C \) and then we're going to add to it plus investment of 32,100, and this is \( I \) right here. We're gonna add to it government purchases of 28,100 which is \( g \). And then net exports, they didn't tell us directly. They gave us an amount of exports and amount of imports. So we have to remember that net exports is equal to exports. So exports would be \( x - m \), 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 1,500. So that means we were importing more than we were exporting, right? There were more imports than 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 gonna see a negative number for net exports. So there we go. We've got \( C + I + g + nx \), 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 2,000, let's circle them all, 2,000 plus 32,100 plus 28,100 plus 500 minus 1,500. So let's do that math real quick. 2,000 + 32,100 plus 28,100 plus 500 minus 1,500, that comes out to 61300. So this is generally how you're gonna want to do these problems. Add up all of your constants together, and then you're going to have your 0.65y piece right here, plus \( 0.65y \). Okay? So notice, we've simplified this quite a bit, right? Now, we've got just one constant and our marginal propensity to consume times \( Y \). So what we want to do now is we want to solve for \( Y \). We want to find that equilibrium amount of GDP. So what we want to do is we're going to subtract \( 0.65y \) from both sides to get all the \( y \)'s on one side. And we're gonna be left with \( y - 0.65y \). Do you guys remember how to do that? Well, this is just like one \( y \), right? So one \( y - 0.65y \) is equal to \( 0.35y \). 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's the toughest thing you're gonna have to do with these algebraic equations. So nothing too crazy. So here we go, \( 0.35y = 61300 \). We divide both sides by 0.35 and that'll get us to have \( y \) by itself. So finally, we've got \( y \) equals. So our final equation here is \( 61300 / 0.35 \) and we get 175,143. \( 175,143 \) is our equilibrium GDP, and that's where aggregate expenditures equals GDP, which is \( y \). Right? \( A = GDP \) when \( y = 175,143 \). Simple as that. Alright? 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.

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

Alright, let's see how you did with this one. I threw in a trick here. I included taxes, right? I included taxes that are going to affect our consumption equation and they're going to affect our government purchases, right? Because they're going to receive the taxes. It comes out of consumption. But remember, all you need to do, look, taxes are a lump sum amount, right? There are 500 in taxes. We just need to plug that in. They gave us the number. We need to plug it in and we just need to solve for Y just like before. So let's go ahead and set up our equation just like we did before. We have Y=C+G+NX. Now, let's go ahead and plug in everything we know. So we're going to have Y= and C=1500+0.75Y-t, right? Y-t and then our investment is equal to 3,400, easy enough. Government purchases is 2600+t and net exports, remember exports, net exports are equal to exports minus imports, right? Everything we export minus everything we import is the net amount of exports. In this case, exports were 750, imports were 2,000. So we subtract the 2,000. So we've put everything in place except we still have this taxes number, right? We've got t=500. So let's go ahead and plug in 500 everywhere that we see t. So let's go ahead and do that. Y=1500+0.75Y-500+34100+26100+500+750-2000. All right? Let's go ahead and add up all of our constants together now. We've got a bunch of constants here to add up. Let's go ahead and circle them so we don't miss any. 1500, that's all inside of that, plus 34100, plus 26100, 500, 750, minus 2,000. All of those numbers need to be added together now. So we've got 1500+34100+2600+500+750-2000. So this comes out to 6,750. 6750+0.75Y-500. Alright. So you're going to have to remember one more little trick from algebra to get that 500 out. And that's how we're going to expand this equation by multiplying 0.75 by Y and multiplying 0.75 by -500. Okay? So that's what we're going to do. We're going to expand the equation by multiplying that by both of those numbers. That's what we would do in algebra and we're going to get Y=6750+. So the first term would be 0.75Y which is just 0.75Y and the second term is going to be 0.75*(-500. So it's going to be a negative number and we're going to have to do 0.75*500. So it's a negative 375. Okay? So there we go. That 500, basically what that's telling us here when we have this Y minus T is it's telling us that some of the income that we get is being taxed. So they're taking that out of our disposable income. That's why they've expanded it a little bit to make it a little trickier, a more little more realistic in that case. So we've got a new constant here of 375. So we've got the 6,750 and the 375 we've got to bring together now. So we're almost done here. Y=6750-375,6375+0.75Y. So all I did was I took the 6750 and I subtracted this 375 from it, alright? So now, we look very similar to what we've done in a lot of problems, right? We've got a constant plus our MPC times Y, right? So all we got to do is solve for Y now. Let's go ahead and subtract 0.75 Y from both sides. So Y-0.75Y=, I'm going to move over here now. And we're going to have on the left-hand side, Y-0.75Y=0.25Y, right? 1Y-3/4Y leaves us with 0.25Y, this cancels out right here and we're left with 0.25Y=6375. We divide both sides by 0.25 and lo and behold, this will be our final answer. So 6375/0.25,Y=25500. Alright. This is about as tough as these questions are going to get and I would expect most of you, your professors probably not even going to go this far when it comes to doing these types of questions. I wanted to take you there because I knew you could figure it out and just in case you have that challenging professor that really wants to give you that tough question. Well, now you're ready for it. Okay? So all this question added was that we had this variable t that affected our consumption function a little bit, but they're always just going to give you the number. So you're just going to have to do a little bit of extra algebra to solve for Y. The goal is always we're just trying to find out what Y is, they gave us all the other numbers and we're just reducing our equation until we find Y, okay? So in this case, Y equals 25,500. Let's go ahead and move on to the next.

Here’s what students ask on this topic:

What is the algebraic approach to finding macroeconomic equilibrium?

The algebraic approach to finding macroeconomic equilibrium involves using the equation Y = C + I + G + NX, where Y is the real GDP, C is consumption, I is investment, G is government spending, and NX is net exports. Consumption (C) is typically expressed as C = a + MPC * Y, where 'a' is the base level of consumption and MPC is the marginal propensity to consume. By solving these linear equations, we can find the point where aggregate expenditures equal real GDP, indicating macroeconomic equilibrium.

How does consumption depend on GDP in the AE model?

In the AE model, consumption (C) depends on GDP (Y) because higher GDP leads to higher disposable income, which in turn increases consumption. This relationship is expressed as C = a + MPC * Y, where 'a' is the base level of consumption and MPC is the marginal propensity to consume. As GDP increases, more people are employed and earning income, leading to higher disposable income and thus higher consumption.

What is the significance of the equation Y = C + I + G + NX in macroeconomics?

The equation Y = C + I + G + NX is significant in macroeconomics because it represents the aggregate expenditures model, which is used to determine the macroeconomic equilibrium. This equation shows that real GDP (Y) is the sum of consumption (C), investment (I), government spending (G), and net exports (NX). Understanding this relationship helps in analyzing economic stability, fiscal policy, and aggregate demand management.

How do you calculate consumption in the AE model?

In the AE model, consumption (C) is calculated using the equation C = a + MPC * Y, where 'a' is the base level of consumption and MPC is the marginal propensity to consume. 'Y' represents the real GDP. This equation shows that consumption increases as GDP increases, reflecting the relationship between higher income and higher consumption.

What role does the marginal propensity to consume (MPC) play in the AE model?

The marginal propensity to consume (MPC) plays a crucial role in the AE model as it determines the change in consumption resulting from a change in disposable income. It is a coefficient in the consumption function C = a + MPC * Y, where 'a' is the base level of consumption and 'Y' is the real GDP. A higher MPC means that a larger portion of additional income will be spent on consumption, influencing the overall aggregate expenditures and economic equilibrium.