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