This one is a little trickier.
Maybe we need a little more space for this one.
Partial Y star, partial b.  That’s the one we’re doing next.
So, who wants to tell me what partial Y star partial b is?
[Student comment]
Any volunteers?
What rule are we going to use?
[Student comment]
We’re going to use the quotient rule.
And what is the derivative of the numerator with respect to b?
[Student comment]
Zero.  Here’s the numerator right here.
There’s no b up here.
So, the derivative of that numerator with respect to b is zero, so this is zero times the denominator,
1 minus b plus bt.
Minus – what is the derivative of the denominator with respect to b?
[Student comment]
Negative 1 plus t.
Times the numerator, a plus I plus G.
Over the denominator squared.
So, this zero means this first term vanishes.
Notice, if I multiply minus 1 times this, this term over here becomes 1 minus b, so this just becomes 1 minus b.
[Student comment]
Oh, it’s t.
In this expression right here, we were taking the derivative of the denominator with respect to b
and you’re absolutely right.
And I wrote down the wrong thing even though you told me the right thing.
It’s t.  Thank you.
So, this should be a 1 minus t.
You people are on top of things.
1 minus t times a plus I plus G.
Over 1 minus b plus bt squared.
And what do we know about the sign of this partial derivative?
It’s got to be positive.
Because 1 minus t has to be positive.  t is the tax rate.
All this is positive.  And this is positive and it’s squared.
So, this has to be positive.
In fact, not only is it positive, but how could we rewrite this in terms of Y star?
Get a feeling for the magnitude of this.
If we look at the expression for Y star.
[Student comment]
1 minus t times Y star.
Notice this over one of these is Y star.
But then you’ve got one of these left.
Again, not only is this positive but it’s going to be quite large.  Because national income is quite large.
This is a fraction.
So this is going to be quite large.
Positive and quite large.
So, again, a rise in the marginal propensity to consume in this more complicated model
would cause the equilibrium value of national income to rise.
Again, the paradox of thrift.
If we want to stimulate the economy in the short run, just have people spend more of their income, save less.
