The IS Curve
There are several more or less equivalent ways to arrive at an IS Curve.
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EconModel (Loanable Funds) Graphical
The EconModel presentation for the IS/LM Model constructs an IS Curve graphically from equilibrium movements in the supply and demand for loanable funds. The IS Curve shows those points that are consistent with equilibrium in the supply and demand for loanable funds.
An alternative, more common view is that the IS Curve shows those points that are consistent with C + I + G = C + S + T. These two views are operationally equivalent if you take the supply of loanable funds to be S and the demand to be I + G - T. That, is real capital investment is financed by borrowing.
An algebraic approach is to let savings be a function S(Y) of income and let investment be a function I(R) of the interest rate. An increase in income Y causes more savings. This forces down the interest rate R so that investment increases and the identity I(R) +G - T = S(Y) is maintained. The IS Curve is thus downward sloping. The increase in income Y causes increased savings, which drives down the interest rate and increases investment.
Simple Keynesian Model
The EconModel presentation for the Simple Keynesian Model derives an IS Curve under the simplifying assumptions of that model.
Link: Keynesian Models