Chapter 16
Relation of two countries

Now we look at the possibility of comparing domestic and foreign growth rates, i.e. answering questions such as: What happens when the domestic economy grows, but the foreign economy grows faster?

For this purpose we look at the domestic and foreign economy symmetrically. The following applies

$$\begin{array}{rcll}{M}^d& =& \mathit{kPy}\&M^{d\ast }=k^{\ast }{P}^{\ast }{y}^{\ast }& \\ M^s& =& M^d\&M^{s\ast }=M^{d\ast }& \end{array}$$

In money market equilibrium, the respective supply of money ($M^s$ and $M^{s\ast }$) determines the quantity of money. We replace this in the money demand equations and by dividing, we obtain

$$\frac{M_0^s}{M_0^{s\ast }}=\frac{\mathit{kPy}}{k^{\ast }{P}^{\ast }{y}^{\ast }}$$

Here, we insert the purchasing power parity equation and obtain:

$$\stackrel{P=S{P}^{\ast }}{\Rightarrow }\frac{M_0^s}{M_0^{s\ast }}=S\frac{\mathit{ky}}{k^{\ast }{y}^{\ast }}$$

Solved for the exchange rate $S$, we get

$$\Rightarrow S=\frac{\frac{M}{M^{\ast }}}{\frac{\mathit{ky}}{k^{\ast }{y}^{\ast }}}=\frac{\tilde{M}}{\tilde{k}ỹ},$$

where the variables marked with $\stackrel{̃}{}$ indicate the ratio of the respective domestic to the foreign variables. This equation can be interpreted quite clearly, since a change of x% in a tilde variable $\stackrel{̃}{}$ means that the domestic variable has changed by x% more than the foreign variable. We show this with the example of money supply using the approximations $\frac{1+x}{1+y}\approx 1+x-y$ and $\left(1+x\right)\left(1+y\right)\approx 1+x+y$ which are valid for small $x$ and $y$, respectively.

$$\begin{array}{rcll}\tilde{M}& =& \left(1+\frac{x}{100}\right){\tilde{M}}_0& \\ M& =& \left(1+\frac{{\varphi }_M}{100}\right)M_0& \\ M^{\ast }& =& \left(1+\frac{{\varphi }_M^{\ast }}{100}\right)M_0^{\ast }& \end{array}$$

Now we replace $\tilde{M}=\frac{M}{M^{\ast }}$

$$\left(1+\frac{x}{100}\right){\tilde{M}}_0=\tilde{M}=\frac{M}{M^{\ast }}=\frac{\left(1+\frac{{\varphi }_M}{100}\right)M_0}{\left(1+\frac{{\varphi }_M^{\ast }}{100}\right)M_0^{\ast }}=\frac{1+\frac{{\varphi }_M}{100}}{1+\frac{{\varphi }_M^{\ast }}{100}}{\tilde{M}}_0$$

By reducing ${\tilde{M}}_0$ on both sides, we obtain

$$\begin{array}{rcll}1+\frac{x}{100}& =& \frac{1+\frac{{\varphi }_M}{100}}{1+\frac{{\varphi }_M^{\ast }}{100}}\approx 1+\frac{{\varphi }_M}{100}-\frac{{\varphi }_M^{\ast }}{100}\text{ oder }& \\ {\varphi }_M-{\varphi }_M^{\ast }& =& x.& \end{array}$$

The interpretation of the above equation can also be formally derived by "loglinearizing", i.e. logarithmizing both sides of the equation and then calculating the difference to the previous period. The result is the same as above.

$$\begin{array}{rcll}S& =& \frac{\tilde{M}}{\tilde{k}ỹ}& \\ & \Rightarrow & \log <!--\; FUNCTION\; APPLICATION\; -->S=\log <!--\; FUNCTION\; APPLICATION\; -->\left(\frac{\tilde{M}}{\tilde{k}ỹ}\right)=\log <!--\; FUNCTION\; APPLICATION\; -->\tilde{M}-\log <!--\; FUNCTION\; APPLICATION\; -->\tilde{k}-\log <!--\; FUNCTION\; APPLICATION\; -->ỹ=\log <!--\; FUNCTION\; APPLICATION\; -->M-\log <!--\; FUNCTION\; APPLICATION\; -->M^{\ast }-\left(\log <!--\; FUNCTION\; APPLICATION\; -->k-\log <!--\; FUNCTION\; APPLICATION\; -->k^{\ast }\right)-\left(\log <!--\; FUNCTION\; APPLICATION\; -->y-\log <!--\; FUNCTION\; APPLICATION\; -->y^{\ast }\right)& \\ & \Rightarrow & \log <!--\; FUNCTION\; APPLICATION\; -->S_0=\log <!--\; FUNCTION\; APPLICATION\; -->M_0-\log <!--\; FUNCTION\; APPLICATION\; -->M_0^{\ast }-\left(\log <!--\; FUNCTION\; APPLICATION\; -->k_0-\log <!--\; FUNCTION\; APPLICATION\; -->k_0^{\ast }\right)-\left(\log <!--\; FUNCTION\; APPLICATION\; -->y_0-\log <!--\; FUNCTION\; APPLICATION\; -->y_0^{\ast }\right)& \\ & \Rightarrow & \Delta \log <!--\; FUNCTION\; APPLICATION\; -->S_0=\Delta \log <!--\; FUNCTION\; APPLICATION\; -->M_0-\Delta \log <!--\; FUNCTION\; APPLICATION\; -->M_0^{\ast }-\left(\Delta \log <!--\; FUNCTION\; APPLICATION\; -->k_0-\Delta \log <!--\; FUNCTION\; APPLICATION\; -->k_0^{\ast }\right)-\left(\Delta \log <!--\; FUNCTION\; APPLICATION\; -->y_0-\Delta \log <!--\; FUNCTION\; APPLICATION\; -->y_0^{\ast }\right)& \\ & \Rightarrow & {\varphi }_S={\varphi }_M-{\varphi }_{M^{\ast }}-\left({\varphi }_k-{\varphi }_{k^{\ast }}\right)-\left({\varphi }_y-{\varphi }_{y^{\ast }}\right)& \end{array}$$

Therefore, the following applies:

1. 1. An increase of x% in the relative supply of money $\tilde{M}$ 1. causes the currency to devalue by x%.
2. If the domestic supply of money $M$ increases by x% more than the foreign supply $M^{\ast },$ the currency devalues by x%.
3. An increase of the relative real GDP $ỹ$ by x% causes the currency to appreciate by x%.
4. If the domestic GDP $y$ by x% more than the foreign GDP $y^{\ast },$, the currency appreciates by x%.
5. Ein Anstieg des relativen Kassenhaltungskoeffizienten (Stabilität der Transmissionsmechanismen am Finanzmarkt) $\tilde{k}$ Preisniveaus um x% lässt die Währung um x% aufwerten.

Herein, the individual changes are additive.