Let’s consider one more national income model.
And this time, let’s take a slightly more complicated national income model with taxes.
Remember this model right here?
Y equals C plus I plus G.
C equals a plus b times Y minus T.
Now consumption is a linear function of disposable income, where Y minus T is disposable income,
income after taxes.
Finally, let’s make the tax equation a little bit simpler.
Let’s assume there’s no intercept in the tax equation.
So, the tax equation is just capital T equals little t times Y.
Where little t in words is what?
Little t is the tax rate.
And the tax rate you could just think of as the marginal income tax rate.  It is some number between zero and 1.
As before, we will assume that the parameter a is positive.
And b, the marginal propensity to consume, is between zero and 1.
And, again, if we want to know how Y star is going to be affected by a change in one of the parameters
or one of the exogenous variables, the first step is to solve for Y star.
And how would we solve for Y star, the equilibrium value of national income?
How would we go about solving for Y star?
[Student comment]
You want to substitute the T equation into here?
[Student comment]
And then do what?
[Student comment]
Let me just do what you said in a slightly different order.
Let’s first substitute C into this equation and then we’ll substitute T in here.
But it amounts to the same thing as what you said.
If first we substitute C into that top equation, we would get Y equals a plus b times Y minus T plus I plus G.
Now, let’s substitute in for capital T in this expression right here.
So, then we get Y equals a plus b plus Y minus tY, substituting for T, plus I plus G.
Now, notice what we’ve accomplished.
We now have only one endogenous variable in that equation.
So, now we can get all the Y terms over to the left and solve for Y.
So, doing so, we get Y.
We have a bY here, we’ve got to move it over.
Minus bY.
We’re going to have a minus btY.
Bring it over to the other side.  It becomes a plus btY.
Equals a plus I plus G.
What do we do next?
[Student comment]
We factor out the Y on the left-hand side
and then we can write this left-hand as what?
[Student comment]
Y times 1 minus b plus bt.
Equals a plus I plus G.
And then, the last step is to do what?
[Student comment]
Divide by that expression in parentheses, 1 minus b plus bt.
So, now we get our equilibrium value of Y is a plus I plus G over 1 minus b plus bt.
So, now the comparative static derivatives of interest are these.
Let’s do the parameters first.  Partial Y star partial a, partial Y star partial b, and partial Y star partial little t.
And what is partial Y star partial a?
[Student comment]
The coefficient on a is just 1 over 1 minus b plus bt.
In fact, this 1 over 1 minus b plus bt is just, in macroeconomics terminology, called what?
What is this called?
[Student comment]
It’s just the multiplier.
Now, what can we say about this term right here?  About its sign?
Can we figure out the sign of that expression?
[Student comment]
It’s going to be positive.  How do you know it’s going to be positive?
[Student comment]
b itself is less than 1, so 1 minus b is positive.
This part right here we know is positive because b is less than 1 and b and t are both positive,
so you’re adding two positive things to the denominator.
So, the overall denominator has to be positive because those two terms are positive.
So, this is clearly positive.
And what does that mean in words?
See how easy this stuff is?
What does that mean in words?
[Student comment]
Y star and a move in the same direction.
If a goes up, Y star goes up.  If a goes down, Y star goes down.
