Let’s move right along.
Actually, we only have two more to do.
Partial Y star partial I and partial Y star partial G.
And, actually, they’re going to have the same value.
What is partial Y star partial I?
Remember I is investment expenditure, exogenous investment expenditures.
[Student comment]
The same answer as before.
Because you can see a, I and G enter in a symmetrical way in this numerator.
And, in fact, the coefficient on I is 1 over 1 minus b.
So, it’s 1 over 1 minus b.
And, again, we know that b is between zero and 1, so the sign of this derivative is what?
[Student comment]
It has to be positive.
So, this means investment expenditures by businesses and national income move in the same direction.
So, another way to get us out of our current unemployment trap
or depressed economy trap is if businesses were all of a sudden to decide they wanted to invest more.
That would have a positive impact on national income.
And what about partial Y star partial G?
[Student comment]
Same answer, 1 over 1 minus b.
Again, this is positive.
And, in fact, this 1 over 1 minus b, we were calling that earlier what?
[Student comment]
The multiplier.
So, if you increase government expenditure by one dollar,
national income will increase by 1 over 1 minus b dollars.
For example, suppose the marginal propensity to consume is 0.8.
Then, 1 over 1 minus b is equal to 1 over 0.2.  Equals 5.
So, in this simple model, a one dollar increase in government expenditure
would ultimately lead to a five dollar increase in national income.
And in macroeconomics, that’s called the multiplier.
In this model, it’s also partial Y star partial G.
Multiplier, partial Y star partial G, is equal to 5.
We could also do comparative static analysis on the equilibrium value of consumption.
And I’ll leave that as an exercise for you to do.
All you would do is get C star expressed in terms of the parameters and exogenous variables
and then just do the same differentiation except you’re differentiating C star
with respect to each of the parameters and exogenous variables.
And you would find that the sign of all these partial derivatives is the same as it was in Y star.
