# How to calculate the multiplier from 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?

Let´s denote $$Y$$ as national income. Then the consumption depends on national income like $$C=cY^d+C^{\textrm{aut}}$$, where c is the marginal rate of consumption, $$Y^d$$ is the disposable income and $$C^{\textrm{aut}}$$ is the autonomous consumption. Next we have to regard the taxes, with tax rate $$t$$. The disposable income $$Y^d=(1-t)\cdot Y$$. In general the expenditure approach says that $$Y=C+I+G$$. Putting all together we get

$$Y=c\cdot (1-t)\cdot Y+C^{\textrm{aut}}+I+G$$

Putting all terms with $$Y$$ on the LHS

$$Y(1-c\cdot (1-t))=C^{\textrm{aut}}+I+G$$

Dividing the equation by $$(1-c\cdot (1-t))$$

$$Y=\frac{C^{\textrm{aut}}+I}{1-c\cdot (1-t)}+\frac{G}{1-c\cdot (1-t)}$$

Differentiating Y w.r.t. $$G$$

$$\frac{\Delta Y}{\Delta G}=\frac1{1-c\cdot (1-t)}$$

With $$c=0.8$$ and $$t=0.25$$ we obtain

$$\frac{\Delta Y}{\Delta G}=\frac1{1-0.8\cdot (1-0.25)}=\frac1{1-0.8\cdot 0.75}=\frac1{1-0.6}=\frac1{0.4}$$

• Last thing.... can't i take c1 just from the C equation? Like 0.8(1-t)... but in that case wouldn't c1 just be 0.8 and not 0.8(1-t)?? – Sara Saletti Jul 9 '19 at 22:18
• Yes, you can do that if $C=c_0+c_1\cdot (Y-T)=c_0+c_1\cdot Y\cdot (1-t)$. Here $T=t\cdot y$ are the taxes where $t$ denotes the tax rate. – callculus Jul 10 '19 at 4:25