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# Simple model to understand r and gÂ relationship

## Video transcript

What I want to do in this video is to create a simple spreadsheet to help us understand why
if R is greater than G, why that might lead to more and more of national income going
to the owners of capital as opposed to labor. So let's just say R is 3%, and we can change that assumption later, so that's the return on
capital that we're assuming. We're assuming it's just going to be fixed at that constant rate. And let's say that economic growth is 2%. So we're assuming a scenario
where R is greater than G. Now this column, let me write the year. So this is year one, year
two, and then maybe we can go, let's see, maybe we can go up to year, let's go to year 15 just for good measure. Now we can think about what
our national capital is, or should I say total capital. So the capital in our economy. Now we'd also want to think
about the national income. So we could do this as
the output of the economy, national income. And now we can think about
the split of the income between capital and labor. So income to capital, and income to labor. And now let's just think
about percentage of total. So capital as percent of total. So let's come up with some
assumptions right over here. So let's say that our aggregate capital and I'm just going to throw
out a random number here, let's say it's 4000. And if we're talking
about millions of dollars then this would be 4 trillion dollars. But I'm being currency
agnostic right over here. So let's just say it's 4000. If this was in millions
it would be 4 trillion. But let's just say 4000 for now. And let's say our national income is 1000, and we've seen charts already, at least for the U.S., that national capital as a
percentage of national income has been about 400 or 500%. It's kind of oscillated in that range. So it's been about four or
five times national income. So this is kind of a
not unreasonable ratio. And now what's the income
to the owners of capital? Well, we're saying that the
return on capital is 3%. So this is going to be equal to this 3%. I'm going to press F4 to
put those dollar signs and we'll see why that's valuable to make sure that we stay
referenced to that cell as we drag this down later on. It's going to be that
times the amount of capital that we have in the economy. So it's going to be 120. And notice, I had the
dollar signs in the B1, because I always want to refer to this, but I didn't put the
dollar signs in the B5, because as I drag this down, I always want this cell, say
when I drag it down here, I want it to refer to that same 3%. That's why I kind of anchored
it there with the dollar sign. But I want it to be times
the capital in that row. So we'll see how that happens in a second. Now what's the income to labor? Well, it's going to be what's left over. National income minus
the income to capital, and then capital is a percentage of total, the income to capital is a
percentage of total income. Well that's just going to be equal to, I can select income to capital, D5, divided by national income. And so it's 12%. And we're also going to
assume that every year that income to capital all
gets re-invested as capital. So it doesn't get consumed in some way. So the year two, the capital
that we start off with, is going to be capital from last year plus the new income to capital. That income to capital is going to get re-invested as capital. That's just going to
be my assumption there. And national income, well we know it's growth. It's going to be the previous
year's national income plus I guess we could say
times one plus this number, plus our economic growth. So there, and I'll press F4 because I want to stay referencing that. And so notice we grew by 2%. And then these two over here, actually all three of these over here, we can just drag down. And now hopefully you see
the value of what I did when I put the dollar signs here. Because now when I dragged it down, this is still referring to B1, because it has the dollar signs there. It's just taking the 3% times B6. So B6 is this. It's looking at the B
column, but in its row now. And that's one of the really useful things about spreadsheets. And actually this column, let me make this a percentage. Just so it becomes a little bit cleaner. Okay, there you go. And then we can just keep on going. So let's just keep on going down. And what do we get? And actually let me get rid of some of the decimal places here. It's making it a little bit messy. So there you go. That actually makes it
a little bit cleaner. And so what do we have going on over here? Well we see that when R
is a fixed rate of return on capital that is greater than growth, we see that capital, the
income going to capital as a percentage of the
total national income is increasing. Now this could be used
as a proxy for inequality because as we've said before, capital does tend to be concentrated amongst the upper percentile,
decile, qauartile, however you want to define it. But once again, this is just a proxy. And the other thing you
have to keep in mind is inequality is a natural byproduct of a capitalistic market economy. And even though more
and more of the income is going to the owners of capital, the labor, the income going to labor is also increasing. Now that by itself doesn't necessarily say it's a good thing. For example, if the population were increasing faster than this, then the income to labor
divided by the population which would be kind of a
per capita income to labor or kind of a proxy for how much the individual person working is making, then even if this is going up and the population is
growing faster than that, that might not be a good thing. Because that means per capita income might not be where it needs to be. But if we assume the population is stable or it's not growing as fast as this, even though we see inequality, or at least this measure
of inequality going up, more and more of the
income is going to capital, these people still might be
better off in this reality. But to kind of test the sensitivity of this model that we've created, we can actually try different things. So if the return on
capital is much larger, say it's 5%, we see the disparity becomes
much, much more significant. And let's say if it was 10%. Now you see a situation that
could get not so pleasant. Because in this situation, this is a fairly extreme
situation we have right over here, now all of a sudden the income to labor, and not even on per capita terms, is actually decreasing. So it really does matter a lot where these two numbers are. But of course if economic
growth was also pretty robust, now all of a sudden this
is still decreasing, but if economic growth was 9%, now all of a sudden this could be a pretty good scenario
for everyone involved. You do have, at least this is kind of a proxy for increased income to capital which could be a proxy for inequality, but maybe the economy
is growing fast enough that everyone is benefiting. So I encourage you to either
build a model like this or I'll give you the link
to this model as well, and play with these numbers and just try to get a
better understanding for if we assumed R and G were constant, and we made some assumptions about starting capital
and national income, how that ends up breaking down as we go further and
further into the future.