2.1 Derivation of the IS curve

The IS curve is a model of the market for goods and services in an economy. We first consider a closed economy, i.e. without exports and imports.
In the upper graph we look briefly at the supply and demand or the production and use of the gross national product. A detailed description can be found in the corresponding chapters of basic textbooks on macroeconomics. In a closed economy, GDP is used for three areas: private consumption C, corporate investment I, and government consumption G. Private consumption depends positively on disposable income $Y-T$ i.e. produced GDP minus taxes. Investments depend positively on income and negatively on interest rates. The demand curve is designated ZZ and the following applies

The supply of goods corresponds obviously to the produced goods $Y$. Therefore, the supply curve is the angle bisector $Y$ = $Y$.

In equilibrium supply = demand or production = consumption, i.e.

The point of equilibrium is the point of intersection of the ZZ curve with the angle bisector. The IS curve represents the function of i and Y, which is implicitly defined by this equation.
Now, the question arises, what happens when the interest rate changes. The lower graph shows the change of the income depending on the change of the interest rate.
Let us assume that the interest rate increases from $i0$ to a higher value $i$. For each level of production, the higher interest rate leads to a decrease in investment, which in turn leads to a decrease in income. This causes a decrease in consumption and investment. Ultimately, this leads to a decrease in demand. Thus, the demand curve ZZ (green) shifts downwards. For each level of production, overall economic demand is now less. The new equilibrium is at GG, the intersection of the lower demand curve ZZ and the 45 degree line. The new equilibrium income is at $Y$.
Because of the multiplier effect, the total decline in production is greater than the original decline in investment triggered by the rise in interest rates.

In the graph, the equilibrium income Y is plotted on the horizontal axis and the interest rate i on the vertical axis. Point GG in the upper graph corresponds to point GG in the lower graph. Thus, the graph shows that, because of the equilibrium on the goods market, the higher the interest rate, the lower the equilibrium income. This relationship between interest rate and income is described by the falling curve in the graph, which is called the IS curve. The resulting IS curve is highlighted by the red dots.
When the interest rate falls, the opposite effects can be seen for all variables. Thus, a fall in the interest rate from $i0$ to $i$ leads to an increase in investment at each level of production. The increase in investment induces an increase in income. This, in turn, triggers an increase in consumption and investment, and overall economic demand increases. Therefore, the ZZ curve shifts upwards. The new, higher equilibrium income is at Y and the new, lower equilibrium interest rate is $i$.