3.1 Derivation of the LM curve

The starting point for the analysis of the money and financial market is the equation

which relates the nominal demand for money (right side) to the nominal supply of money (left side). The other variables adjust according to the kind of shock that occurs. This can be seen in this and the next graph. In essence, the above equation already represents the LM curve. However, the presented graph explains in more detail how the relationships between the individual variables are established.
In order to understand the relationship between interest rates and income, we will first look at the left-hand part of the graph. The interest rate $i$ is plotted on the vertical axis and the real money stock $M∕P$ on the horizontal axis. The real money supply is given by the vertical line and is designated $\mathit{Ms}∕P$. For a given real income $Y$ , the real money demand is a falling function of the interest rate. It is designated $\mathit{Md}$.It is represented by the downward sloping curve in the left part of the graph. The initial equilibrium is at point $\mathit{Eq}0$. Thus, the money supply corresponds to the money demand at an interest rate of $i0$.
Now let´s see what happens when the income increases. At a given interest rate $i$, all economic actors increase their money demand. As the money demand increases, it shifts to the right, to the new demand curve. The new equilibrium is in the point $\mathit{Eq}$, with the higher interest rate $i$ .
Thus, a higher income leads to an increase in the interest rate from $i0$ to $i$, because with rising income the money demand increases, whereas the money supply is fixed by the vertical line Ms. Therefore, the interest rate must rise until two opposing influences on the money demand cancel each other out:
Because of the higher income, economic actors want to hold more money. Therefore, they try to sell bonds to get more cash. If the money supply remains unchanged, this causes the price of bonds to fall and so the interest rate to rise. Because economic actors want to hold less money when interest rates rise, this effect causes the money demand to decrease. Therefore, the interest rate must rise just enough to ensure that the money demand again corresponds to the unchanged money supply. Only then, the money market returns to the equilibrium at point $\mathit{Eq}$.
At the same time, the bond market will return to an equilibrium. For the new equilibrium interest rate $i$ there is no longer any reason to sell bonds. Hence, the LM-equation describes a simultaneous equilibrium on the money and financial markets.
Corresponding to the left-hand graph, in which income $Y0$, the associated money demand $\mathit{Md}0$ and the equilibrium interest rate $i0$ are plotted, now the right-hand graph will be included, where the income is plotted on the horizontal axis and the associated equilibrium interest rate on the vertical axis. Point $\mathit{Eq}$ in the left graph corresponds to point $\mathit{Eq}$ in the right sub-graph. If the income $Y$ increases, $\mathit{Md}$ is the corresponding money demand and the equilibrium interest rate $i0$ increases to $i$. The point $\mathit{Eq}$ in the right sub-graph, which in turn corresponds to the point $\mathit{Eq}$ in the left sub-graph, represents the new equilibrium on the money and financial market. The higher the income, the higher the equilibrium interest rate. This relationship between income and interest rate is described by the rising curve in the right-hand sub-graph. This curve is called the LM curve.
Analogously, it can be shown that the equilibrium interest rate falls in the case of decreasing income.

For the current illustration we use P=1 and $L\left(i\right)=\mathit{exp}\left(-i∕3\right)$. This means that the demand for liquidity is equal to 1 for zero interest rates and decreasing for rising interest rates. The curves of the graphics can thus be represented by $i=-3\mathit{ln}\left(\frac{M}{Y}\right)$, in one case as a function of M and in the other case as a function of Y. The graph of the shift of the LM curve uses a different specification.