Balanced budget and change in taxes Consider a simple Keynesian economy in which the government expenditure ($G$) exactly equals its total tax revenue: $G = tY$ where $t$ is the tax rate and $Y$ is the national income. Suppose that the
government raises $t$. Then what happens to $Y$?.

My working: With the balanced budget multiplier in effect $Y$ becomes equal to $\frac I {(1- c) (1 - t)}$

However without the balanced budget in effect ( when taxes are increased over government expenditure), $Y$ becomes equal to $\frac {I + G} {1 - c(1-t)}$
Is the working right ?. How should I proceed in order to come to a solution ?.
Here's my take, which could be wrong.  We assume a closed economy, so net exports, imports and exports themselves are all zero.

We have Y = C + I + G
Suppose we have the simple consumption function C= k +c*(Y-T).  We are only allowed to consume out of disposable income or have autonomous consumption.

Then Y = k+c*(Y-T) + I + G
This goes to 

Y*(1-c) = k - c*T + I + G

You take T = t*Y.  So this is

Y*(1-c) = k - ctY + I + G

Subtract ctY from both sides.  Then we have

Y*(1-c*(1-t)) = k + I + G  

The solution in the absence of the balanced budget would be

Y = [k + I + G]/(1 - c*(1-t))

Now have G = t*Y - the balanced budget

Y*(1 - c*(1-t)) = k + I + t*Y  or

Y*((1-t) - c*(1-t)) = k+ I or

Y*((1-t)*(1-c)) = k + I

or 

Y = [k + I]/((1-t)*(1-c))

So my take is that the multiplier is right, in both cases, but that there is or will have to be some autonomous consumption expenditure in the economy, and that's what's left out.  You can say, "I don't need autonomous expenditure!"  But just try coming up with a Keynesian cross to settle output in nominal terms without one!