Deriving the IS Curve

To determine how income changes when the interest rate changes, we can combine the investment function with the Keynesian cross diagram. Let's consider the effect of an increase of the interest rate $r$:

As investment is inversely related to the interest rate, the increase in the interest rate reduces the quantity of investment $I$. The reduction in planned investment, in turn, shifts the planned-expenditure function downward in the Keynesian cross. The shift in the planned expenditure function causes the level of equilibrium income to fall. Hence, an increase in the interest rate lowers income. By varying the interest rate $r$, we obtain the relationship between the interest rate and the level of income, which we will call the IS curve.

In essence, the IS curve combines the interaction between $r$ and $I$ expressed by the investment function and the interaction between $I$ and $Y$ demonstrated by the Keynesian cross. Each point on the IS curve represents equilibrium in the goods market, and the curve illustrates how the equilibrium level of income depends on the interest rate. Because an increase in the interest rate causes planned investment to fall, which in turn causes equilibrium income to fall, the IS curve slopes downward.