Deriving the Multiplier Algebraically: Study with Video Lessons, Practice Problems & Examples

The multiplier effect illustrates how an initial increase in spending, such as investment, leads to a greater overall increase in GDP. In a private closed economy, aggregate expenditures consist of consumption (C) and investment (I). The equation for GDP (Y) can be derived as Y=1-MPC(A+I), where MPC is the marginal propensity to consume. This demonstrates that any increase in consumption or investment will multiply the effect on GDP, crucial for economic stabilization during downturns.

Deriving the Multiplier Algebraically

So let's see the derivation behind the multiplier equation that we use. We're going to use a little algebra here to find out how we came up with this one over one minus MPC for our multiplier. Okay? So what we're going to do is we're going to use a simplified model of aggregate expenditures, to calculate one over 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 going to have a multiple effect on the level of GDP in the economy. Okay? So let's start with the 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 are just consumption and investment, right? There's no government and no international trade. So we've just got C+ 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 going to 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+ I equation to have the consumption function plus I there. Okay? So since there are no taxes or government transfers in this model, well then, all of the income that earned that's earned is disposable income, right? There are no taxes. Right? There are no taxes, so every income that's earned is going to be, part of our disposable income. So we could just say that our aggregate expenditures equal a+mpc∙y+i. Okay? So all income is disposable. Income, we're not going to differentiate between the 2. And when we're in equilibrium, well, remember in equilibrium, our aggregate expenditures equal our GDP. So for GDP, we're going to use the term y for GDP. So, remember, this is GDP right here. Right? 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 GDP which is equal to Y. So Y equals A+MPC∙Y+I. All we did was, substitute a e. Our aggregate expenditures, we just put in GDP for that because they're the same at equilibrium. MPC times Y plus investment. Okay? So what we're going to do is we're going to rearrange this formula to solve for GDP. We're going to rearrange the formula to solve for GDP and, all that takes is moving some of these factors around. So let's go ahead and move the MPC y to put the GDPs on the same side of the equation, And let's continue here. So we're going to have y-MPC∙y=a+i. Right? y-MPC∙y=a+i. Now this next step takes a little bit of algebra, but it should, it's just one little trick that we learned in algebra. When we have Y - MPC times y, we can factor out the Y. So we could have Y∙(1-MPC) 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✕1 is y minus MPC times y. Right? So all we did was we factored out the y from the equation which left us with 1-MPC. So Y∙(1-MPC)=a+i. Alright. So we're almost done. We just need to get the y by itself by dividing both sides by 1 minus MPC. So this will go away and we're left with y=a+mpc. Now I want to do one more thing to separate this out. I want to say y equals so I want to say 11-MPC. We can do this. We're going to separate the denominator from the numerator times a+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 if there's any increase over here, there's an increase in a+i in either a or I over there, and let me make it clear that these are i's right here. Those are all a+i, the investment. So if there's any increase, say, in investment, well, there's going to be a multiple effect on GDP. Right? GDP is going to be increased by not only the investment amount, but times the multiplier. Right? 11-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 an open economy, well, it'll result in a one over one minus MPC increase in equilibrium GDP. So that's the multiplier effect. That's how we derive that multiplier is that any increase 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 going to have this multiple effect on our GDP. Okay? So that's about it. Nothing like I wouldn't expect 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 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.

Here’s what students ask on this topic:

What is the formula for the multiplier in macroeconomics?

The formula for the multiplier in macroeconomics is derived as follows: In a private closed economy, aggregate expenditures (AE) consist of consumption (C) and investment (I). The consumption function can be expanded to include autonomous consumption (A) and the marginal propensity to consume (MPC) times income (Y). Thus, AE = A + MPC * Y + I. At equilibrium, AE equals GDP (Y), so Y = A + MPC * Y + I. Rearranging this equation to solve for Y, we get Y = (A + I) / (1 - MPC). The multiplier (k) is therefore 1 / (1 - MPC).

How does the multiplier effect impact GDP?

The multiplier effect impacts GDP by amplifying the initial increase in spending. For example, if there is an increase in investment (I), this initial spending leads to increased income and consumption, which further increases GDP. The formula Y = (A + I) / (1 - MPC) shows that any increase in autonomous consumption (A) or investment (I) will result in a multiplied increase in GDP. This is crucial for economic stabilization, especially during downturns, as it shows how government spending or investment can significantly boost overall economic activity.

What is the significance of the marginal propensity to consume (MPC) in the multiplier formula?

The marginal propensity to consume (MPC) is a key component in the multiplier formula. It represents the fraction of additional income that households spend on consumption. The multiplier is calculated as 1 / (1 - MPC), indicating that a higher MPC leads to a larger multiplier. This means that when households spend a greater portion of their additional income, the initial increase in spending has a more significant impact on GDP. Understanding MPC helps policymakers predict the effectiveness of fiscal policies aimed at stimulating economic growth.

How do you derive the multiplier algebraically?

To derive the multiplier algebraically, start with the aggregate expenditures (AE) in a private closed economy: AE = C + I. Expand the consumption function to include autonomous consumption (A) and the marginal propensity to consume (MPC) times income (Y): AE = A + MPC * Y + I. At equilibrium, AE equals GDP (Y), so Y = A + MPC * Y + I. Rearrange to solve for Y: Y - MPC * Y = A + I. Factor out Y: Y * (1 - MPC) = A + I. Finally, solve for Y: Y = (A + I) / (1 - MPC). The multiplier (k) is thus 1 / (1 - MPC).

Why is the multiplier important in macroeconomics?

The multiplier is important in macroeconomics because it quantifies the impact of an initial change in spending on the overall economy. It shows how an increase in autonomous spending, such as investment or government expenditure, can lead to a larger increase in GDP. This concept is crucial for understanding how fiscal policies can be used to stabilize the economy, especially during recessions. By knowing the multiplier, policymakers can estimate the potential effects of their actions on economic output and make informed decisions to promote growth and stability.