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So i have this question:enter image description here

I go along and get enter image description here

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

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

enter image description here

From my knowledge i know that enter image description here

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?

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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

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

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  • $\begingroup$ 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)?? $\endgroup$ – Sara Saletti Jul 9 '19 at 22:18
  • $\begingroup$ 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. $\endgroup$ – callculus Jul 10 '19 at 4:25

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