What about this one?
To get the partial derivative with respect to b, what rule are we going to use?
[Student comment]
We’re going to use the quotient rule.
And notice there’s no b in the numerator.
So, if we take the derivative of the numerator with respect to b, that derivative is what?
[Student comment]
Not 1.
We’re taking the derivative of the numerator with respect to b.
There’s no b in the numerator.
So that means, when we differentiate that numerator with respect to b,
it’s going to be zero because there’s no b up there.
So, it's zero.
[Student comment]
So, the derivative of the numerator with respect to b is zero times the denominator minus –
now, what is the derivative of the denominator with respect to b?
[Student comment]
Negative 1.
Because the coefficient of b is actually minus 1.
So, the derivative of the denominator with respect to b is negative 1.
So, it’s minus the derivative.
Minus a minus 1.
Times the numerator, a plus I plus G.
All over what?
[Student comment]
One minus b squared.
So, we can simplify this equation.
Notice you’ve got a zero here, so that drops out.
You’ve got a minus times a minus 1, so it becomes a plus 1.
So, the numerator just becomes a plus I plus G divided by 1 minus b squared.
So, what is the sign of partial Y star partial b?
[Student comment]
Because both the numerator and the denominator are positive, it will be positive.
1 minus b itself is positive, then you’re squaring it.
So, it will still be positive.
This is positive, so that is going to be positive.
And, in fact, in terms of Y star, what is partial Y star partial b?
Let’s look at the magnitude a little bit.
It’s clear that the sign of this positive, but we could get some other information.
If you use Y star, you could write partial Y star partial b is equal to what?
Notice Y star is a plus I plus G over 1 minus b.
So, this expression right here is the same thing as what?
[Student comment]
This is the same thing as Y star over 1 minus b.
That’s kind of interesting because this partial
derivative, Y star is the equilibrium value of national income.
That’s going to be a large number.
And it’s divided by 1 minus b, so it’s divided by some fraction.
So, this partial derivative is not only going to be positive, it’s going to be huge in magnitude.
So, that suggests that a small change in marginal propensity to consume, b,
is going to have a large impact on national income.
Not only is this partial derivative positive, but it’s huge.
It’s going to have a very powerful impact on national income.
So, if people start spending a greater fraction of their income on consumption, that will have a
powerful impact on national income.
So, one of the ways of trying to get out of this current recession that we’re in. . .
I just actually heard this morning on the radio that unemployment in North Carolina is still over ten percent.
So, it’s very, very high.
So, certainly, our economy is still in recession.
So, what would be one way of trying to get us out of that recession?
Getting us out of this situation where unemployment is so high?
[Student comment]
Let’s relate it to today’s discussion a little more directly.
[Student comment]
What would be one way of increasing consumption?
Let’s tie a little more closely to what we’re talking about here.
This is not a complete tangent I’m going on.
If there was some way of raising b, the marginal propensity to consume,
that would have a big impact on national income.
So, one way would be to have the president come out and say,
“Everybody, spend more.
Go out and spend.
Don’t save.
Go out and spend.”
And that would have the effect of doing what to b if people listened?
[Student comment]
It would increase b.
And that would have a big impact on national income.
So, that would be one way of trying to stimulate the economy.
Would be to have people save less.
Now, this is interesting.
This is called the paradox of thrift.
You know, you’re taught as a child.
Or maybe you weren’t. I was.
It’s a good thing to save. You should be thrifty.
You should save for a rainy day.
And while that’s good at the individual level, at the collective level, it can sometimes be a bad
thing.
If everybody’s saving too much, it could cause the economy to be depressed.
If everybody spends more, it can be a boost to the economy.
[Student comment]
We are talking about the short run.
You're right.
In the longer run, maybe it’s not such a good
thing.
Especially for the country as a whole. Not to save.
But in the short run, it could be a big boost to the economy.
So, that’s partial Y star partial b.
It’s positive, large.
Meaning that the equilibrium value of national income and b move in the same direction.
