# Derive IS-Curve (Y)

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

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.