Market Entries and Exits and the Nonlinear Behaviour of the Exchange Rate Pass-Through into Import Prices

Abstract

This paper develops an empirical framework giving rise to a nonlinear behaviour of the exchange rate pass-through (ERPT). Rather than shifts between low and high inflation, the nonlinearity arises when large swings in the exchange rate trigger market entries and exits of foreign firms. Switching regressions are used to distinguish between low and high pass-through regimes of the exchange rate into import prices. For the case of Switzerland, the corresponding results suggest that, though inflation has been low and stable, the ERPT still doubles in value in times of a rapid appreciation of the Swiss Franc.

Appendices

Appendix A: Derivations

1.1 A.1 Derivation of the Optimal Pricing Rule

Multiplying (2) with $E_{t}/p^{*}_{t}$ and rearranging yields $\pi _{t}^{*}E_{t}/(p_{t}^{*})=[1-1/p_{t}^{*}e_{t}]E_{t}$. Since quantities are normalised to 1, we have from footnote 1 that $E_{t}/(p_{t}^{*})= 1/s_{t}^{*}$ wherefore profits expressed in market share form are given by

$$\pi_{t}^{*}=\left[1-\frac{1}{e_{t}p_{t}^{*}}\right]s_{t}^{*} (p_{t}^{*}, p_{t})E_{t}. $$

Differentiating this with respect to import prices yields

$$ \frac{\partial \pi_{t}^{*}}{\partial p_{t}^{*}}= \frac{1}{e_{t}(p_{t}^{*})^{2}}s_{t}^{*}E_{t}+\left(1-\frac{1}{e_{t}p_{t}^{*}}\right) \left[\frac{\partial s_{t}^{*}}{\partial p_{t}^{*}}+\frac{\partial s_{t}^{*}}{\partial p_{t}}\frac{\partial p_{t}}{\partial p_{t}^{*}}\right]E_{t}=0. $$

Since (1) defines the import share $s_{t}^{*}$, we have that

$$\left(1-\frac{1}{e_{t}p_{t}^{*}}\right) \left[-\frac{\gamma^{*}}{p_{t}^{*}}+\frac{\gamma^{*}{\Gamma}}{p_{t}}\frac{\partial p_{t}}{\partial p_{t}^{*}}\right]= -\frac{1}{e_{t}(p_{t}^{*})^{2}}s_{t}^{*}. $$

Multiplying this with $(p_{t}^{*})^{2}$ yields

$$(p_{t}^{*}-1/e_{t}) [-\gamma^{*}+ \gamma^{*}\nu{\Gamma}]= -s_{t}^{*}/e_{t} $$

where $\nu = \frac {\partial p_{t}}{\partial p_{t}^{*}}\frac {p_{t}^{*}}{p_{t}}$ is the conjectural elasticity. Solving for the import price yields

$$ p_{t}^{*}= \left[1 + \frac{s_{t}^{*}}{(1-\nu{\Gamma})\gamma^{*}}\right]\frac{1}{e_{t}}. $$

1.2 A.2 Derivation of the Profit Function

Inserting (1) into $\widetilde {\pi }_{t}=(s^{*}_{t}/(\gamma ^{*}(1-\nu {\Gamma }))(1/e_{t})$ yields

$$\widetilde{\pi}_{t}^{*}=\left[\frac{1}{(1-\nu{\Gamma})\gamma^{*}}\omega_{t}-\frac{1}{1-\nu{\Gamma}}\ln p_{t}^{*}+\frac{\Gamma}{1-\nu{\Gamma}}\ln p_{t}\right](1/e_{t}). $$

Taking logarithms and using the first order Taylor approximation⁷ yields

$$\ln \widetilde{\pi}_{t}^{*} \approx \frac{1}{(1-\nu{\Gamma})\gamma^{*}}\omega_{t}-1-\frac{1}{1-\nu{\Gamma}}\ln p_{t}^{*}+\frac{\Gamma}{1-\nu{\Gamma}} - \ln e_{t}. $$

1.3 A.3 Derivation of High Pass-Through Regime

Inserting $\omega _{t} = \gamma ^{*} \ln p_{t}^{*} - \gamma ^{*}{\Gamma }\ln p_{t} + (1-\nu {\Gamma })\gamma ^{*}\ln e_{t}+c$ of the first line of Eq. (6) into (4) yields

$$\begin{array}{@{}rcl@{}} \ln p_{t}^{*} &\approx& \frac{\Gamma}{(2-{\Gamma}\nu)} \ln p_{t} - \frac{1-\nu{\Gamma}}{2-\nu{\Gamma}}\ln e_{t} -\frac{1}{1-{\Gamma}\nu}\\ \ln p_{t}^{*}&+&\frac{\Gamma}{2-\nu{\Gamma}}\ln p_{t}-\frac{1-\nu{\Gamma}}{(2-\nu{\Gamma})}\ln e_{t} +constant \end{array} $$

Ignoring the constant and solving this for $\ln _{t}p_{t}^{*}$ yields.

$${\Delta} p_{t}^{*} \approx \left[(2{\Gamma})/(3-\nu{\Gamma})\right] \ln p_{t} - \left[(2-2\nu{\Gamma})/(3-\nu{\Gamma})\right]\ln e_{t} $$

B Data Appendix

Table 5 Description of the data set