Equilibrium

We summarize all the assumptions and conclude the model using the money market equilibrium, i.e. the supply of money ${M}^{s}$ corresponds to the demand for money ${M}^{d}$. The idea behind this last assumption is that in the event of a temporary imbalance, e.g. due to a monetary policy measure, the economic parameters are adjusted so that the economy moves back towards equilibrium.

$$\begin{array}{llll}\hfill {M}^{s}& ={M}^{d}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {M}^{s}& =\mathit{kPy}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {M}^{s}& =\mathit{kS}{P}^{\ast}y\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \Rightarrow S& =\frac{{M}^{s}}{k{P}^{\ast}y}\phantom{\rule{2em}{0ex}}& \hfill \text{(14.1)}\phantom{\rule{0.33em}{0ex}}\end{array}$$

The exchange rate is thus determined by the domestic money supply, the domestic GDP, the cash coefficient, and the foreign price level. If one of these variables changes, the equilibrium adjusts accordingly.

- An increase in domestic money supply by x% causes the currency to depreciate by x% .
- An increase in domestic real GDP by x% causes the currency to appreciate by x% .
- An increase in the foreign price level by x% causes the currency to appreciate by x% .

The identical rates of change measured in percent are a consequence of the multiplicative structure of equation 14.1.An increase of x% means a multiplication of the corresponding value by $1+\frac{x}{100}$. So if c.p. the money supply increases by x% (${M}^{s}\mapsto {M}^{s}\cdot \left(1+\frac{x}{100}\right)$), then on the left side $S$ must also be multiplied by the factor $1+\frac{x}{100}$, i.e. it also increases by x%. If, on the other hand, there is economic growth of x%, then the factor $1+\frac{x}{100}$appears in the denominator and the left side must be multiplied by $\frac{1}{1+\frac{x}{100}}$. Because $\frac{1}{1+\frac{x}{100}}\approx 1-\frac{x}{100}$, this corresponds to a decrease of $S$ by x%, i.e. an appreciation.

This can also be illustrated by a logarithmic approximation. Here ${\varphi}_{x}$denotes the growth rate of variable $x$ in %.

$$\begin{array}{rcll}\mathrm{log}S& =& \mathrm{log}\left(\frac{{M}^{s}}{k{P}^{\ast}y}\right)=\mathrm{log}{M}^{s}-\mathrm{log}k-\mathrm{log}{P}^{\ast}-\mathrm{log}y& \text{}\\ \mathrm{log}S\left(1+\frac{{\varphi}_{S}}{100}\right)& =& \mathrm{log}{M}^{s}\left(1+\frac{{\varphi}_{{M}^{S}}}{100}\right)-\mathrm{log}k\left(1+\frac{{\varphi}_{k}}{100}\right)-\mathrm{log}{P}^{\ast}\left(1+\frac{{\varphi}_{{P}^{\ast}}}{100}\right)-\mathrm{log}y\left(1+\frac{{\varphi}_{y}}{100}\right)& \text{}\\ \mathrm{log}S+\mathrm{log}\left(1+\frac{{\varphi}_{S}}{100}\right)& =& \mathrm{log}{M}^{s}+\mathrm{log}\left(1+\frac{{\varphi}_{{M}^{S}}}{100}\right)-\left(\mathrm{log}k+\mathrm{log}\left(1+\frac{{\varphi}_{k}}{100}\right)\right)-\left(\mathrm{log}{P}^{\ast}+\mathrm{log}\left(1+\frac{{\varphi}_{{P}^{\ast}}}{100}\right)\right)-\left(\mathrm{log}y+\mathrm{log}\left(1+\frac{{\varphi}_{y}}{100}\right)\right)& \text{}\\ & =& \underset{\mathrm{log}S}{\underbrace{\mathrm{log}{M}^{s}-\mathrm{log}k-\mathrm{log}{P}^{\ast}-\mathrm{log}y}}+\mathrm{log}\left(1+\frac{{\varphi}_{{M}^{S}}}{100}\right)-\mathrm{log}\left(1+\frac{{\varphi}_{k}}{100}\right)-\mathrm{log}\left(1+\frac{{\varphi}_{{P}^{\ast}}}{100}\right)-\mathrm{log}\left(1+\frac{{\varphi}_{y}}{100}\right)& \text{}\\ \mathrm{log}\left(1+\frac{{\varphi}_{S}}{100}\right)& =& \mathrm{log}\left(1+\frac{{\varphi}_{{M}^{S}}}{100}\right)-\mathrm{log}\left(1+\frac{{\varphi}_{k}}{100}\right)-\mathrm{log}\left(1+\frac{{\varphi}_{{P}^{\ast}}}{100}\right)-\mathrm{log}\left(1+\frac{{\varphi}_{y}}{100}\right)& \text{}\end{array}$$

Because $\mathrm{log}\left(1+\frac{x}{100}\right)\approx \frac{x}{100}$, this simplifies to

$${\varphi}_{S}={\varphi}_{{M}^{S}}-{\varphi}_{k}-{\varphi}_{{P}^{\ast}}-{\varphi}_{y}.$$ |

Thus, the devaluation rate of the exchange rate ${\varphi}_{S}$ corresponds exactly to the money growth rate minus the rate of change of the cash coefficient (changes in the monetary process), the foreign inflation rate and economic growth.

(c) by Christian Bauer

Prof. Dr. Christian Bauer

Chair of monetary economics

Trier University

D-54296 Trier

Tel.: +49 (0)651/201-2743

E-mail: Bauer@uni-trier.de

URL: https://www.cbauer.de