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Deriving the Aggregate Expenditures Model

Deriving the Multiplier Algebraically


Deriving the Multiplier Algebraically

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So let's see the derivation behind the multiplier equation that we use, we're gonna use a little algebra here to find out how we came up. This 1/1 minus MPC for our multiplier. Okay, so we're gonna do is we're gonna use a simplified model of aggregate expenditures um to calculate 1/1 minus MPC. Okay, so remember that this multiplier, it tells us that it signifies the multiple increase in GDP based on an initial increase in spending. So if we spend a little bit more on investment, well, that's gonna have a multiple effect on the level of GDP in the economy. Okay, so let's start with a private closed economy where there's no government and no international trade, just to make this equation look a little simpler, but it stands just the same when we add those in. Okay, so in a private closed economy, our aggregate expenditures is just consumption and investment, right? There's no government and no international trade. So we just got a C plus I for our aggregate expenditures so we can expand our C. For what we've been talking about consumption as the consumption function so far there's the autonomous level of consumption, as we say, there's that amount that's gonna be there no matter what, right? If there was no income in the economy no production, well, there would still be some consumption, we still have to eat, we still need shelter and clothes, right? So there would still be some consumption but as we increase our disposable income. Well we're going to be increasing consumption based on that marginal propensity to consume. right? So there we go. That's our consumption function. All we did was expand our C. Plus I equation to have the consumption function plus I there. Okay. So since there's no taxes or government transfers in this model, well then all of the income that earned that's earned is disposable income, right? There's no taxes, right? There's no taxes. So every income that's earned is going to be um part of our disposable income. So we could just say that our aggregate expenditures equals A plus MPC times Y plus investment. Okay, so all income is disposable income. We're not going to differentiate between the two and when we're in equilibrium. Well, remember an equilibrium are aggregate expenditures equals R G. D. P. So for GDP we're gonna use the term Y. For G. D. P. So um remember this is GDP right here, right, are all the income that's earned. All the everything that's produced is earned by somebody and all of that is going to be involved in this consumption function. So we can say that our aggregate expenditures are equal to G. D. P. Which is equal to Y. So Y equals A plus MPC. All we did was uh substitute E. R aggregate expenditures. We just put in G. D. P. For that because they're the same at equilibrium NPC times Y plus investment. Okay. So what we're gonna do is we're gonna rearrange this formula to solve for GDP we're gonna rearrange the formula to solve for GDP and um all that takes is moving some of these factors around. So let's go ahead and move the MPC y to put the G d p s on the same side of the equation and let's continue here. So we're gonna have y minus MPC times Y is equal to, well this is gone and we're left with a plus, I write y minus MPC times Y is equal to a plus I Now this next step takes a little bit of algebra but it should um it's it's just one little trick that we learned in algebra when we have y minus MPC times Y, we can factor out the Y. So we could have Y times one minus M P C. And you can double check that, right? If we were to factor this back in, if we were to multiply it back in like this, we would have Y times one is y minus MPC times Y. Right, so all we did was we factored out the Y from the equation which left us with one minus MPC, so y times one minus MPC is equal to A plus I. All right, so we're almost done. We just need to get the Y by itself by dividing dividing both sides by one minus MPC. So this will go away and we're left with Y equals A plus I over one minus M P C. Now I want to do one more thing to separate this out I want to say y equals so I want to say 1/1 minus M. P. C. We can do this we're gonna separate the denominator from the numerator times A plus I. Okay so what does this tell us? Notice what this first bit here is? That's our multiplier. Right? So what it tells us is that for any change if there's an increase let me get out of the way. Sorry I didn't realize I was in the way I was. So in the zone with this algebra so what I did is I separated the 1/1 minus MPC. Notice if you were to multiply these together you would just have a plus I divided by one minus MPC. Because the one times the A. Plus I would put a plus i in the numerator just like we have over here so I expanded it out so that we can see the multiplier by itself times A plus I. So what it tells us is that if there's any increase over here there's an increase in a plus in either A. Or I over there. Let me make it clear that these are eyes right here. Those are all A plus I the investment. So if there's any increase say an investment, well there's gonna be a multiple effect on GDP. Right? GDP is gonna be increased by not only the investment amount but times the multiplier. Right? 1/1 minus MPC. So any increase from a change in the base amount of consumption or the amount of investment or say government purchases or net exports in a open economy? Well, it will result in a 1/1 minus MPC increase in equilibrium G. D P. So that's the multiplier effect. That's how we derive that multiplier. Is that any increase in that in those constants are going to increase our GDP in a multiple amount and that is the multiplier there. That's the big thing about the multiplier, is that by increasing the constant, we're multiplying the amount of GDP. Okay, so that has great effects in the economy when we're trying to boost GDP in the economy, maybe during a recession? Well, if we boost our investment spending, it's gonna have this multiple effect on our GDP. Okay, so that's about it. Nothing like I wouldn't expect if if you don't understand this completely, I wouldn't expect the professor to like ask you to derive the multiplier, Right? But some of them like to talk about it um and maybe just have you understand on that level of how the investment increase will affect our GDP. Okay, so that's about it here, let's go ahead and move on to the next