21.2 The equilibrium and its graphical presentation

We now summarize all assumptions and close the model. The central equation is the equilibrium of the balance of payments.

$$F\left(y,S,r\right)=0$$

This implicit function of three variables can be represented as an area in the $\mathit{ySr}$ coordinate system. However, in the three-dimensional illustration a graphical analysis of the equilibrium, shocks and reactions is not possible. In particular, if the graphs of the IS-LM model are added for the illustration of policy measures, the many surfaces would obscure the essential points from any viewing direction. Therefore, we depict the equilibrium with three connected 2d graphs. In the figurative sense, we unfold the $\mathit{ySr}$-cube. We look at the $\mathit{yr}$-plane while keeping $S$ constant, the $\mathit{yS}$-plane while keeping $r$ constant and the $\mathit{rS}$-plane while keeping $y$ constant. In each of the graphs, we show the relationship between the two variables, which results from $F\left(y,S,r\right)=0$, if the third variable is kept constant at the value resulting from the other two graphs. For example, the $\mathit{rS}$-graph plots the implicitly defined curve $F\left(y_0,S,r\right)=0$ where $y_0$ is the equilibrium resulting from the goods market.

If, for example, $y$ changes in the $\mathit{yr}$-graph, this change is transferred to the $\mathit{yS}$-graph and the $\mathit{rS}$-graph, whereby a shift of the curve $F\left(y_0,S,r\right)=0$ occurs in the $\mathit{rS}$-graph, since the (locally fixed) value for $y_0$ changes.

Each of the three graphs has a well-defined interpretation:

• The $\mathit{yr}$-plane represents the domestic goods market. In this market, the IS-LM curves are also plotted, which can be used to illustrate shocks and policy measures.
• The $\mathit{yS}$-plane represents the international goods market.
• Die $\mathit{rS}$-plane represents the international capital market.

Using the IS-LM model, monetary or fiscal shocks and policy measures are illustrated and the transmission channels in the model can be traced.