Hi, we are now ready for our first real growth model.

This is actually going to be a simple, [laugh] it's a Simple Growth Model.

But it's still going to be far more complicated than any model we have seen so far.It's got all sorts of moving parts.

That said, still by economists standards, by the models that economists use today, it is still relatively simple.

Alright, so hang in there!

So, here's how it's going to work.

Model's going to have a group of Workers and there's going be these Coconut trees.

And the Workers could pick the Coconuts

When they pick the Coconuts, they can do two things with them.

They can eat them, because they are so good and drink them up.

So that they can live off these Coconuts.

And they can also, instead of eating the Coconuts, use these Coconuts to build Machines.

Right, and these Machines can help them pick Coconuts even faster.

Now there's one other assumption that I'm going to add in. Right.

The machines will wear out over time, because they are made out of coconuts, right?

So, they sort of degrade over time and will need to be replaced with new machines.

So, that's it - very simple economy: Coconuts, Workers, Machines, Machines depreciate.

What we want to do is construct a Model of that.

And we're going to use that Model to explain the role that investment in capital.

In this case the Machines - produces growth, and the limits of that.

Then from that we are going to see why innovation is so important.

For now, we just want to focus on this very simple Model.

Let's get started. Lots of moving parts.

Tons of them. [laughter] So hang in there!

There's going to be Workers time t , L of t for Laborers.

Then there is going to be Machines at time t. We'll call that M of t for Machines ("Mt" = Machines at time t)

The reason we put those 't's down there, those subscript 't' , is because over time there is change.

Workers at Time 1, Workers at Time 2

Machines at Time 1, Machines at Time 2, and so on.

The Workers of Machine are going to combine to form Output.

We are going to call that O of time t. (Ot = Output of Coconuts at time t)

Then there's going to Coconuts that can be eaten, that's E of t (Et = Number consumed at time t)

Or they can be Invested. That' s I of t, and when I say Invested, they can be turned into more Machines (It = Number invested at time t)

Now, to figure out how many can be eaten and invested, there's going to be a Savings Rate (s = savings rate)

The Savings Rate will determine the percentage of savings which we will go into Investment,

and the amount we don't save that we will just eat, and that's "Et".

The last thing in the model, I realize this is a lot [laughter],

is the Depreciation rate (d = Depreciation rate), and that is the rate at which these machines wear out.

We are going to assume there is some fixed rate percentage of the machines wear out each period.

It's a simplification, but we are going to use it, All right.

So, lot's of stuff. Workers, Machines, Coconuts. The Coconuts get eaten or turned into more machines, and there's Depreciation (percentage of the machines wear out each period).

Got to make some assumptions.

The first assumption we're going to make is that the production of these Coconuts is increasing, but Concave (Ot=(√Lt)(√Mt)) .

Remember we have Concave Functions going up, but falling off - in both workers and and in machines.

That means - the more machines, the more coconuts. The more workers, the more coconuts.

But those things sort of fall off.

Right. We're going to use a specific functional form that says:

'The square root of the Laborers times the square root of the Number of Machines.' ((√Laborers)(√Machines))

Second Assumption: Output is either Consumed or turned into Machines.

The Coconuts are either eaten or turned into Coconut Picking Machines. There is no waste.

So Basically that means Output O is just equal to E plus I (Eaten + Invested). ("Ot"="Et"+"It").

Another way to write this is (I=s(O)).

Because S is our Savings Rate and O is our Total Output . So, the amount Invest is going to be equal to our (Savings Rate)(Total Output).

Assumption 3: The last thing is that these Machines Depreciate. Right.

So, the Machines we have at "Mt" plus 1 are going to be the Machines we have at "Mt" plus Our Investment minus how many ever Depreciate. (Mt+1 = Mt + It - d(Mt)).

Lot's going on. We have all these variables, and these are the equations that help us make sense of the variables.

Let's step back to tha First Assumption - Concave.

So, Concave means that like the first workers, you know gives you more Coconuts.

The second give you more coconuts, but he gives you fewer Coconuts than the first one did. It sort of 'falls off'.

Economists call this Diminishing Returns to Scale.

Here's a picture - This is the number of Workers, and this is the number of Coconuts

What we can see is this first worker gives you quite a bit. The second gives you fewer. The thirds gives you sort of even less.

So what you get, is you add more workers.

Yes you get more coconuts, but the workers become less and less valuable.

The same is going to be true of machines.

That's all Concave means.

Okay. Alot going on here. Let's simplify things a bit.

Let's assume that we just got a hundred workers.

So, remember before Output was equal to the ((√Lt) (√Mt)).

Were going to assume we have a hundred workers. That's going to mean it's going to be (10√Mt) . That will make things simpler.

In a more realistic model. We would have workers deciding to goto work depending on what the wage is.

So we would have to create a market for wages as a function of output, as a function of how much they like the Coconuts, and that would get really complicated.

[laughter] So, we are just going to skip all that stuff.

We're going to skip the entire labor market, and just assume everybody goes to work everyday, and then see what happens.

Let's do an example. Now, were going to do some math. We'll try and do it slowly.

So what we're going to do is assume the Depreciation rate is one forth (0.25), and the Savings rate is 30% (d=0.25, s=0.3), (Year 1: 4 machines).

Let's suppose we start out with four machines. The the output is ten times the square root of four ((10)(√4)=20).

So how much do they invest?

Well, remember they invest 20 times 30%, so they invest in 6 machines. So, Investment = 6 ((20 30%)=6).

How much Depreciation is there?

Well, Depreciation is on the old machines.

There were four machines in the past. 25% or one forth of those get worn out. That represents 1 machine.

We subtract those two, and get a net of +5 machines ((20(0.3)) - (4(0.25)) = 5 or (Investment) - (Depreciation) = +5 machines.

We're going to invest in 6 new ones. We loose one to Depreciation, so that gives us 5.

We started out with 4 machines. So that means in Year 2 we have 9 machines .

Here's how our economy works.

We've got four machines, we produce twenty coconuts.

We ate fourteen. We invested six.

We lost one machine to depreciation.

So now we have five new machines (6-1)

That gives us a total of nine (4+(6-1)). We started out with four machines, now we got nine.

We have a nice GDP of 20, and now we've added more machines so we should do better.

So, let's look at the next year.

The next year (Year 3) we've got nine machines. Output is ten times the square root of nine.

So that's going to be thirty ((10)(√9) = 30).

And let's think about how many new machines do we get.

Well, we're (30)(0.3), that is how much we are going to save.

So that gives us 9 new machines we're going to buy

But the questions is 'how many do we loose to depreciation'?

We had 9 machines and we multiply that times 0.25.

That's like nine over four (9/4 = 2.25), so that's like 2 1/4.

Let's just simplify this, and suppose it's 2. (fudge factor)

The Depreciation is 2. So we had 9 machines to start. We get 7 new ones (9-2=7) for a total of 16 machines. With that little fudge of 1/4 to keep the math simpler.

Let's take a look at our GDP.

The previous period, our GDP was 20.

Now our GDP is 30.

So, we have this nice sustained growth.

Let's look at Year 3. Year three began with 16 machines.

If we have 16 machines, that means our Total Output equals 100 time square root of 16 which is 40 (Output=100√16=40)

So, what's happened to our growth? We started out with 20, then 30, then 40.

Actually a little less with that fudge factor (2 instead of 2.25). But, we get this nice sustained growth.

We can ask, is this going to continue?

Are we going to go from 10 to 20, 30, 40, 50, 60, or is it going to fall off.

But, we see a hint that it is falling off a little bit. This number should not be quite 40.

It should be a little bit less than 40. Instead of 20 to 30 to less than 40.

We should be asking 'are there limits to this growth'?

To try and understand is growth is going to stop or not, let's do the problem.

Let's assume we have a big number of machines.

The number of machines continues to grow unless something different happens. Let's look at 400 machines.

If we have 400 machines, our Output is going to be ten times the square root of four hundred.

So that is going to be ten times twenty which is two hundred ((10)(√400)) or (10)(20)=200 .

So that's great, that's huge GDP, huge output!

What's our Investment going to be?

Well, we had 200 as Output (0.3)(200). Our Savings Rate is 0.3 .

So that means we are going to Invest in 60 machines.

But, Depreciation is going to be (400)(0.25) which is 100 machines.

Investment = 60, Depreciation = 100 (60-100=-40)

So if we started off with 400 machines, we would fall off to 360 machines.

[laughter] So wait a minute!

Somehow this economy is going to grow, but it can't grow this big.

If it grew to 400 machines it would shrink back down to 360 machines.

Is there some number we could reach. We started out with 4. We'd go from 4 to 9 to 16 and so on. It looks like it is never going to stop.

Then we say we have 400 (machines) would continue to grow.

We find out, No, it's not going to grow, it's going to stop.

In fact it's going to shrink down.

There must be some place where it's going to stop.

There must be some natural limit to the growth.

In fact that is what economists like to call "The Equalibrium"

Right. If we look at this growth it is going up, up, up, and it is going to flatten out.

This flat line here is going to be the Equilibrium level.

What we want to understand - what is that Equilibrium level?

Let's think about it. What's going to happen in Equalibrium?

With Equilibrium the number of machines stays fixed. We stop growing.

But, what effects the number of machines? Two things.

One, Investment. You buy new machines based on your Savings Rate.

What else? Depreciation. You loose machines due to Depreciation.

So, the Equilibrium occurs when Investment = Depreciation.

Okay. Well Guess what, that why this is so easy to solve.

That is why models are so great!

Let's do this formally.

So, think about it. What's our Output (Output equals ten times square root of M (Output=(10√M).

What is our Investment? Well, that's easy.

Investment is 0.3 ten times square root of M which equals three times square root of M. ((0.3)(√M)) = ((3)(√M)).

What's our Depreciation?

That's just (M/4). In Equilibrium, (Depreciation) must equal (Investment). So again, easy ((3)(√M) = (M/4)

So that means that ((12√M)=(M)), 12=√M, M=144

So, if my total number of machines are 144, the depreciation is going to be exactly the same as the Investment.

Again, here's the math, Investment = Depreciation (0.25M = 3√M. M=12√M, M=144

Let's go ahead and check.

d=0.25, s=0.3, machines=144

Output=10√144=120 Savings= (0.3)(120)=36 machines. Depreciation is 1/4(144)=36. So, Investment=36 and Depreciation is 36.

The total number of Machines remains at 144. We have reached and Equilibrium.

The number of new machines and the number of old machines cancel out.

That's exactly where that curve finishes.

Output is going to be at 120.

Right? So what we get is a Long Run Equilibrium rather than a Small Run of exactly 120 units of output total.

There's something ironic here. I call this a Growth Model.

Well, what's ironic about this?

What's ironic about this, is that eventually there is no growth.

We're starting with 4 machines, then 9, then 16 then 23, and so on, eventually there are 144 machines, and we stop.

Growth stops.

What's going on? Well, let's think about it.

Depreciation is linear.

So Depreciation is just a nice linear function.

But, our Output is a function of our machines is Concave, thats falling off.

At some point, the amount of more Output that we're getting is falling

to match the slope of depreciation, and those things exactly balance out.

And that's why growth stops.

Well, if growth stops, how do we get more growth?

Well, the answer is Innovation.

Innovation allows us to continue to grow.

That's why people focus so much on Innovation.

So, to get a real model of Economic Growth,

we've got to move beyond this Simple Growth Model and

actually include a 'Parameter' that takes into account technological growth.

So let's step back. What have we learned?

We've learned, that if we write down a Simple Model of Growth -

Economic Growth that involves investing money in new machines.

That there are limits to growth. That the model is going to max out at this point,

when the number of machines lost to depreciation is exactly offset by the number of machines that we invested in the previous period.

If we start with no machines, growth is going to happen really really fast initially, but then it's going to fall off when it reaches this equilibrium level.