# How to calculate the spending multiplier from a given set of equations?

So i have this question:

I go along and get

Then i need to calculate the effect on the optimal output is G increases by 80:

And on the answer sheet it states that the spending multiplier is:

From my knowledge i know that

Now how come that the spending multiplier is 1/0.4? Where are they getting the 0.4, which should be 1-c1-d1 from the original equations?

A friend told me that the 1/0.4 is derived from 1/1-c1(1-t)

then 1/1-0.8(0.75)= 1/0.4. He said that since there is no d1 in the equations ( in the investment equation) we don't use it.

Can somebody explain the meaning behind this 1/0.4) Thanks!

Your friend is right. If you look at the initial investment equation, and comparing it to your definition, you have: $$I=900+(0)Y-50i$$ So $$d_{1}=0$$ and $$c_{1}=0.8(0.75)$$ and thats why you get the spending multiplier as $$\frac{1}{0.4}$$
• $C=0.8(1-t)Y$ where $t$ is exogenously given. Further, if you look at your IS curve you derive, $c_1$ is that chunk that appears before $Y$. I believe it isn’t just the $0.8$ – Brennan Jul 9 '19 at 14:23
• It just means that it is computed outside of the model and should be taken as given. Its an assumption that simplifies the model. It is like assuming government expenditures, $G=800$ so it is easier to compute. Im sure you could dig deep into where $t$ comes from in reality. – Brennan Jul 9 '19 at 17:24