The IS curve shows all the combinations of interest rates i and outputs Y for which the goods market is in equilibrium.

To derive the IS curve, we equate the aggregate income with the aggregate expenditure:

Aggregate Income = Aggregate Expenditure

C + S + T = C + I + G + (X – M)

This can be simplified as follows:

(S – I) = (G – T) + (X – M)

This means that the excess savings minus investments goes towards funding government deficit and it also helps in supporting a trade surplus. This equation can be used to derive the IS Curve.

We will work with two graphs. 

The first graph plots Income (Y) with (S – I). When income (Y) increases, savings (S) also increase. Investments (I) also increase, however, the increase in investments is less than the increase in savings. This means that S – I goes up. This will result in an upward sloping (S - I) curve.  

The second curve plots expenditure (G – T) + (X – M) with Income Y. When income increases, the government deficit will generally go down, i.e., G – T will go down. People will also spend more money on imports, so X – M also goes down. So, when income (Y) goes up the sum (G – T) + (X – M) goes down.

The intersection of these two graphs represents the equilibrium level of income (Y*). 

Let’s say this equilibrium level of income is 100. Let’s also say that at this equilibrium, the interest rate was r = 5%. If interest rates go down to 4%, the savings will go down, but the investments will go up because of the decrease in their cost of capital. Overall, S – I will go down.   The S – I curve will shift towards the right/downwards leading to an increase in the equilibrium level of income.

From the above graph we can conclude that when real interest rates decrease, the income increases. This is the IS curve, the graph of real interest rates (r) vs. income (Y).

Note: In this context both output and income refer to the same thing (Y) and will be used interchangeably.