Derive IS-Curve (Y)

Question

An Economy has a GDP described by the following:
$Z=C(Y −T)+G+I(r)$
$C(Y −T)=C_0 +C_1(Y −T)$
$I(r) = I_0 − I_1r$
where Z is planned expenditure, Y is GDP, T is tax, G is public consumption, I is investment, r is interest. $C_0$, $C_1$, $I_0$, $I_1> 0$ are all parameters and $C_1 <1$. T and G are exogenous variables. r is also exogenous .
How does one derive the IS-Curve Y as a function from r,G,T?
Im not sure what is meant by this because my understanding is that the IS-Curve is just $Y = C(Y-T)+ I(r) + G$
Any help is appreciated.
I have now solved this

Oscar · Accepted Answer

So, the IS curve is a set of equilibria: all combinations of income and interest rate that achieve macroeconomic equilibrium are represented by the IS curve. After all, that's why they call it IS: Investment = Savings!!!
Hence, let's take off from here:
Investment = Savings = Public Sector Savings + Private Sector Savings
What's Public savings? Remember: savings is all income left after expenses. Thus:
T - G = Public Savings: Government's income (taxes) minus expenses (gov' purchases, G).
What's Private savings? Well, the people earned an income (Y), paid taxes on it (T), and spent on their consumption (C): Y - T - C
There we have it: Savings = S = (Y-T-C) + (T-G) = Private + Public savings
Question: how do we know S = I?
Put together your first 3 equations, and let's do one bold assumption: Y = Z (i.e., actual expenditure = planned expenditure. Not that bold, is it?)
Y = C_0 + C_1*(Y−T) + I_0 - I_1*r + G

Solving for this equation (beware: Y appears on both sides of it, which means you need to to solve for it), goes like this:
Y - C_1*Y = C_0 - C_1*T + I_0 - I_1*r + G
Y*[1 - C_1] = C_0 - C_1*T + I_0 - I_1*r + G

Y = [C_0 - C_1T + I_0 - I_1r + G]/[1 - C_1]
And that's your IS curve, exhibiting a negative relationship between your two key variables, income and interest rate.

Derive IS-Curve (Y)

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