Connecting the keynesian cross to the IS curve

In the last video we hopefully got the intuition between how real interest rates might impact planned investment. We saw that if real interest rates went up, then planned investment went down. If real interest rates went down, then planned investment went up. What we want to do in this video is take this conclusion right over here, this hopefully fairly intuitive conclusion right over here and apply it to our Keynesian Cross and think about how real interest rates would effect overall planned expenditure and what that would do in a model like the Keynesian Cross, what that would do to our equilibrium real GDPs. Just as a reminder, let's just draw our Keynesian Cross first, or parts of it. On this axis right over here, we have expenditures. This axis right over here, we have income. We know, from many videos now, that an economy is a equilibrium when income is equal, when aggregate real income is equal to aggregate real expenditures. Circular flow of GDP. Let's draw … Let me make a line that's all the points where Y is equal to expenditures. Along this 45 degree line right over here. This is our expenditures. At this point right over here, that should be the same value as what our aggregate income is. That's part of the Keynesian Cross. The other part is to actually plot planned expenditures relative to this and then see where they intersect. What the equilibrium for that planned expenditure line? I'll write it here as ... I've written it in the past as planned. I just wrote out the word. Planned expenditures. We could write it as expenditures planned, like that. It's equal to our aggregate consumption. Our aggregate consumption, we can write it as a function of disposable income. Y - T is disposable income. Aggregat income minus aggregate taxes. I want to be very clear here. This is not saying C x Y - T. This is saying C is a function of Y - T. Give my a Y - T and I will give you a C. For the sake of our Keynesian Cross analysis, and this is kind of kind of what you would see in a traditional intro class, we assume that we have a linear consumption function. We assume that our consumption functions. C as a function of disposable income. It might be something like our autonomous consumption plus our marginal propensity to consume times our aggregate income, minus taxes. This right over here really is multiplication. We could distribute this C 1. This is just saying C as a function of Y - T. That's only one part of planned expenditures. Above and beyond that, we have planned investment. We're talking about the planned side of things. Now we know that planned investment ... In the past we viewed it as a constant, but now we know it can actually be a function of real interest rates. Above and beyond that, we have government expenditures and then net exports. For some given real interest rate, we can plot this line. The consumption function right over here is just a line with a positive slope that intersects the vertical axis at some place up here. It has a positive intersect. All of these, for given interest rate, these are all going to be constant. Our planned expenditures would look something like this. It might look something like that. This is YP. Let's call this YP_1. This is the YP we get when we pick … I'll just write ... I'll just rewrite the whole thing over again. We have our consumption, which is a function of Y - T, plus the level of planned investment at ... Let's say interest rate R1, so at some given interest rate, plus government spending, plus net exports. We see ... We've done this Keynesian Cross analysis several times now, already. This is our equilibrium level of GDP. This is where along our planned expenditure line, where income is equal to expenditures, or output is equal to expenditures. We are equilibrium right over here. We're not eating into inventories in an unplanned way and we're not building excessive inventory above and beyond what we had planned. Now, what I want to think about, what happens if interest rates go from R1 to R2? What happens if interest rates go from R1 to R2 and in particular let's assume that R2? Now, we're going have planned investment at R2 and we're going to assume that R2 is less than R1. We're essentially saying, what happens when interest rates go down. We already know. When interest rates go down, planned investment goes up. Everything else equal, if this thing shifts up, if this term right over here goes from R ... if the input into it, if the real interest rate goes down, then this whole expression is going to go up and so you're going to have an increase. You're going to have a shifting up of your planned expenditure for any level of income. It might look something like this. It would look something like this. This delta right over here, this ... Let me do it right over here. This distance right over here is going to be your change in planned investment. It went up because interest rates went down. We saw that in the last video. We saw that we got to a new level, or we see now that when you shift that up, that investment goes up. Because real interest rate went down, you get to a new equilibrium point. That equilibrium point is a higher level ... it's a higher level of GDP or income. We know from previous videos as well, that this distance right over here is the same as our multiplier times the amount that things got bumped up. The amount that things got bumped up was the change in planned investment. Then, we multiply that times our multiplier. Our multiplier is 1 over the marginal propensity to save, or 1 over 1- the marginal propensity to consume. The marginal propensity to consume ... We assume it's going to be constant in order to even be able to do this map. That's this piece right over there. That is equal to our C1. The main theme here, the real big picture here as we go on our way to constructing our ISLM model, is really that all we're seeing ... when real interest rates go up, planned investment goes down. When interest rates go down ... which is what we saw in this example right over here. Actually, let me write this down. Y, planned expenditures 2 at C as a function of Y - D +. Our new planned investment, at this lower interest rate, + G + net exports. This is our Y2 right over here, our planned expenditures. We saw in this example, when real interest rates went down, planned expenditures ... When real interest rates went down, planned investment went up. That made total planned expenditures go up. That made total GDP go up. Now we can have another relationship, which is really very analogous to this. Really, by changing this, we're just shifting this curve. Then, you have the multiplier effect on our equilibrium output. The big takeaway from here is, if real interest rates go up, not only does planned investment go down, that would shift this entire curve down. Then, that would also cause our equilibrium real GDP to go down. It would go down by some multiplier, by the multiplier of how much this goes down. If real interest rates go down, then planned investment, because of what we saw in the last video, goes up. Then, that would cause ... That would cause this whole ... That's what we did in this video. This curve would shift up. If this curve shifts up, our equilibrium GDP is going to be however much this shifted, times the multiplier, so your equilibrium GDP is going to go up. You really have a very similar relationship in terms of just how things move. We can plot this. Economist are famous for not always plotting the independent variable the way you would want to. As we construct our ... What we're going to see is our IS curve. It stands for investment savings. What we're going to do and we'll talk more about that in the future. We plot the convention is to put real interest rates on the vertical axis and to put real GDP right over here. If you want to look at this relationship, when we have a high real interest rate, we're going to have a low real GDP. When we have a low real interest rate, we're going to have a high GDP. It's going to make spending go up. If spending goes up, you have a multiplier effect. It makes our equilibrium output go up. Low interest rate, high real GDP, so you have a curve that relates. If you want to relate real GDP to real interest rates you get a curve like this, and it's called the IS curve. IS comes for investment savings. We're really more focused on the I part of it, the way we analyzed here. The whole reason, based on the logic in this video and the last one as well, the whole reason why we have this relationship is due to real interest rates impact on investment. When you have high real interest rates, you don't have much investment. Also, you'll be sapping out of GDP. If you lower interest rates, then that makes you end up having a lot more investment, like we saw in the last video. That will expand GDP by the multiplier that we see right over there.