Economics: the simple multiplier (1 Viewer)

[WildestDreams]

Hey guys, so I was looking at the derivation for the simple multiplier, and I don't understand whats the role of {Co + 1o}.
I'm confused, why does it like disappear?

 

[sida1049]

Keep in mind that we're looking at the three sector Keynesian model. And like all models, some assumptions are made. I'll mention these assumptions as we go along. Also, I'm using underscores ("_") to denote subscripts. (Haven't gotten around to using LaTeX on BoS, lol.)

C_0 appears because of the following formula, which is assumed to be true:

C = C_0 + cY, where C_0 is a fixed amount of consumption and c is the marginal propensity to consume.

This plots as a straight line if you plot C against Y, with the slope being the marginal propensity to consume c. And this makes sense, because you expect that as the national income increases, consumers in the economy would generally spend more.

C_0 is a fixed amount of consumption that occurs in the economy anyway. This makes sense, as there are other things outside of c and Y which influences consumption. More technically, this term is called exogenous consumption (don't memorise this), and is assumed in the model to take into account factors influencing consumption in the economy that isn't based on the MPC c or the national income Y. (That's why it's called the exogenous; it's based on factors outside of our model.)

Now our second assumption is that the amount of investment in the economy is fixed. That is to say:

I = I_0, where I_0 is a fixed number.

In this three-sector model, we decide to include the business sector by introducing the investment term I, but since we're not looking at the relationship between I and anything else, we decide to keep it constant in this model. (In other words, the entire business sector is captured in this model exogeneously; we assumed it's only influenced by factors outside of our model, and cannot be explained by any of the other terms in our model.) In the notes you have provided, the 1_0 is a typo; it's supposed to be I_0.

I hope that answers your question.

But in general, the reason why terms like C_0 and I_0 appear in a model is not based on pure logic, but rather a subjective judgment on the part of the person who is mathematically modelling. We can choose to, say, instead of modelling investment as fixed, to model investment as a function of c, Y, and other variables, but this would make the model a lot more complicated and messy. We could also choose to ignore investment completely and get rid of the I, but this would make the model weaker (in the sense that it completely neglects an important sector). So Keynes, who developed this model, decided to strike a balance between simplicity and strength of the model by including investment, but leaving it only as a fixed constant, to suit his purpose for the model (which was to demonstrate ideas like the multiplier effect).

 

[sida1049]

If you're interested in another derivation for the simple multiplier that uses 2U maths...

Suppose in an economy, consumers receive a boost of income Y_0.

They spend an amount of Y_1 according to their marginal propensity to consume c:
Y_1 = cY_0

Now the economy receives Y_1 as income, since we assume that this economy is closed. So we spend some portion of that again:
Y_2 = cY_1 = c.cY_0 = c^2 . Y_0

And again, this process propagates:
Y_3 = cY_3 = c^3 . Y_0
.
.
.

So if we want to calculate the total amount of income received in this economy Y, we need to sum up all of Y_i's, where i = 0,1,2,... :
Y = Y_0 + Y_1 + Y_2 + Y_3 + ... = Y_0 + cY_0 + c^2 . Y_0 + c^3 . Y^0 + ... = Y_0 (1 + c + c^2 + c^3 + ...)

We can see that we have an infinite geometric series. Since c is a number between 0 and 1, we can apply the sum of an infinite geometric series formula:
1 + c + c^2 + c^3 + ... = 1/(1-c)

And so we conclude that the total amount of income the economy receives is
Y = Y_0/(1-c) = mY_0, where m = 1/(1-c) is the Keynesian multiplier, and Y_0 being the initial change in national income.

 

[WildestDreams]

Keep in mind that we're looking at the three sector Keynesian model. And like all models, some assumptions are made. I'll mention these assumptions as we go along. Also, I'm using underscores ("_") to denote subscripts. (Haven't gotten around to using LaTeX on BoS, lol.)

C_0 appears because of the following formula, which is assumed to be true:

C = C_0 + cY, where C_0 is a fixed amount of consumption and c is the marginal propensity to consume.

This plots as a straight line if you plot C against Y, with the slope being the marginal propensity to consume c. And this makes sense, because you expect that as the national income increases, consumers in the economy would generally spend more.

C_0 is a fixed amount of consumption that occurs in the economy anyway. This makes sense, as there are other things outside of c and Y which influences consumption. More technically, this term is called exogenous consumption (don't memorise this), and is assumed in the model to take into account factors influencing consumption in the economy that isn't based on the MPC c or the national income Y. (That's why it's called the exogenous; it's based on factors outside of our model.)

Now our second assumption is that the amount of investment in the economy is fixed. That is to say:

I = I_0, where I_0 is a fixed number.

In this three-sector model, we decide to include the business sector by introducing the investment term I, but since we're not looking at the relationship between I and anything else, we decide to keep it constant in this model. (In other words, the entire business sector is captured in this model exogeneously; we assumed it's only influenced by factors outside of our model, and cannot be explained by any of the other terms in our model.) In the notes you have provided, the 1_0 is a typo; it's supposed to be I_0.

I hope that answers your question.

But in general, the reason why terms like C_0 and I_0 appear in a model is not based on pure logic, but rather a subjective judgment on the part of the person who is mathematically modelling. We can choose to, say, instead of modelling investment as fixed, to model investment as a function of c, Y, and other variables, but this would make the model a lot more complicated and messy. We could also choose to ignore investment completely and get rid of the I, but this would make the model weaker (in the sense that it completely neglects an important sector). So Keynes, who developed this model, decided to strike a balance between simplicity and strength of the model by including investment, but leaving it only as a fixed constant, to suit his purpose for the model (which was to demonstrate ideas like the multiplier effect).

Hey wow! This is awesome! Thanks for clearing it up!

Just one more thing though, the derivation in the screenshot I've posted, does C_o + I_o = 1?
Because equation (2) shows it as Y = (C_o+Io_)/(1-c) but then all of a sudden in equation (3) it becomes k = 1/1-c

 

[sida1049]

Hey wow! This is awesome! Thanks for clearing it up!

Just one more thing though, the derivation in the screenshot I've posted, does C_o + I_o = 1?
Because equation (2) shows it as Y = (C_o+Io_)/(1-c) but then all of a sudden in equation (3) it becomes k = 1/1-c

No; we don't have any clear relationship between C_0 and I_0, except that they are numbers given to us.

Y = (C_0+I_0)/(1-c) = k(C_0 + I_0), where k = 1/1-c is the multiplier.

I can see how you would get confused, because the equation is typed out badly in the notes.

Hope this helps!

 

[WildestDreams]

Ohhh thanks for the quick reply! Yes, that makes a lot of sense now thank you!
Haha yes, the notes do get quite confusing...