Interest rate determination
The real interest rate r will be equal to the equilibrium real interest rate
In the classical model we define equilibrium real interest rate r* as the real interest rate where savings is equal to investments, S(r*) = I(r*). As we know that S = I is a requirement for the financial market to be in equilibrium.
In the classic model, real interest rate determines the flow of funds into and from the financial market. A higher real interest rates will result in larger flows of funds into the market (savings depends positively on r) and smaller flows out from the market (investment depends negatively on r). Real interest rate will be such that the flows into market are specifically equal to the flows out of the market.
Figure: Determination of the real rate
From this graph we can determine the size of investments and savings. In equilibrium when r = r*, S = I that is what we need for GDP identity to hold. Once we know savings, we can determine household savings from SH = S - SG - SR.
In the classical model, expected inflation pe is an exogenous variable and because R = r + pe we can determine nominal interest rate from the real rate.