Keynesian Cross and Balanced Budget Multiplier Hi! I am trying to understand the problem above, and was wondering if someone could help me with the last question. I think I am fine with all other questions. 

Here is my attempt: 

(i)
The fiscal multiplier is $\frac{1}{1-c} \Delta G$, as it is the infinite sum $ \Delta G + c\Delta G + c^2\Delta G + ... $. 

Depends on c because the increase in government spending first leads to an increase in Y, which leads to an increase in C, which then leads to another increase in Y, but this time the increase in only $cY$, as part of the income is saved. 

(ii) Balanced budget multiplier is 1, as $\Delta Y = \frac{1}{1-c} \Delta G + \frac{-c}{1-c} \Delta T$ with $ \Delta T = \Delta G$

This means that an increase in government spending that is matched by an identical increase in lump sum taxed increases output by the same amount as the increase in G (and T).

(iii) $$\Delta Y = \frac{\Delta X}{0.01} = 100$$ 

This means that giving me an additional unit of currency will increase total income by 100, as it feeds back into income and consumption infinitely. 

(iv) This is the question where I would like some guidance. Here is my work so far: 

We have: 
$$ C= a + c(Y-T) - d\alpha Y$$

The multiplier associated with an increase in G should now be: 
$$\frac{1}{1-c} + \frac{1}{1-d\alpha}$$
The first part is the "old multiplier, the second is the effect of the new part of the consumption function? I am not sure that this is correct, happy to receive comments!

This would mean that $c=d\alpha$.

And then, the multiplier associated with an increase in T:


1st round: Y increases by $c\Delta T$
2nd round: C increases by $c^2\Delta T + d\alpha \Delta T = 2 c^2\Delta T$
Then Y increases by $2 c^2\Delta T$
3rd round: C increases by $4 c^3\Delta T$


Does this mean that the multiplier is this sum: $$2^0C^1\Delta T + 2^1C^2\Delta T + 2^2C^2\Delta T + ... $$

And then the balanced budget multiplier would be this last result + 1 (the other multiplier, from the increase in G)? 

I'd be super grateful to anyone who could help me with question (iv)!
The fiscal multiplier in (iv) would arise from

$$ Y = C+I+G$$

$$ Y = a + c(Y-T)-d\alpha Y +G+I.$$

This can be rearranged as 

$$Y(1-c+d\alpha)=a-cT+G+I,$$

and, therefore, the fiscal multiplier would be

$$\frac{1}{1-c+d\alpha}.$$

In order for this to equal $1$, you just need $d\alpha=c.$

Note that setting $d\alpha=c$ in your "multiplier" does not yield 1:

$$\frac{1}{1-c}+\frac{1}{1-d\alpha}=\frac{1}{1-c}+\frac{1}{1-c}=\frac{2}{1-c}.$$