Log-linearization of real exchange rate

Question

I have a doubt on how to log linearize an expression for the real exchange rate. I have read in several papers that they express it using substitutions. I haven't been able to find out what substitutions they use, so that is why I come here to ask for help.

For example, suppose that the law of one price holds such that:

\begin{gather} P_{H,t}=S_tP^*_{H,t} \quad \quad \text{and} \quad \quad P^*_{F,t}=\frac{P_{F,t}}{S_t} \end{gather}

Let $Q_t=\frac{S_tP^*_t}{P_t}$ be the real exchange rate and $T_t=\frac{S_tP^*_{F,t}}{P_{H,t}}$. Moreover, let the CPI be

\begin{gather} P_t=[aP^{1-\theta}_{H,t}+(1-a)P^{1-\theta}P_{F,t}]^{\frac{1}{1-\theta}} \end{gather}

I know that the log-linear version of $Q_t$ is given by

$$\hat{Q_t}=(2a-1)\hat{T}_t$$

Can someone please tell me how to obtain this latter expression?

Thank you very much.

Answer 1

Let $P$ be the CPI of the Home country, $P^*$ the CPI of the Foreign country, $P_H$ and $P_F$ the price of Home and Foreign intermediate goods expressed in domestic currency, and $P^*_H$ and $P^*_F$ be the price of Home and Foreign intermediate goods expressed in foreign currency.

The log-linear version of the real exchange rate is given by

\begin{gather} \hat{Q}_t = \hat{S}_t + \hat{P}^*_t - \hat{P}_t \end{gather}

We know that the CPI of home and foreign country (log-linear versions) are given by

\begin{gather} \hat{P}_t = a\hat{P}_{H,t} + (1-a)\hat{P}_{F,t} \quad \quad \hat{P}^*_t = a\hat{P}^*_{F,t} + (1-a)\hat{P}^*_{H,t} \end{gather}

We only need to substitute this in the log-linear version of the real exchange rate:

\begin{gather} \hat{Q}_t = \hat{S}_t + \hat{P}^*_t - \hat{P}_t \\ = \hat{S}_t + a\hat{P}^*_{F,t} + (1-a)\hat{P}^*_{H,t} - a\hat{P}_{H,t} - (1-a)\hat{P}_{F,t} \\ = \hat{S}_t - a(\hat{P}^*_{H,t} - \hat{P}^*_{F,t}) + \hat{P}^*_{H,t} + a(\hat{P}_{F,t}-\hat{P}_{H,t}) - \hat{P}_{F,t} \\ = \hat{S}_t - a\hat{T}^*_t + \hat{P}^*_{H,t} + a\hat{T}_t - \hat{P}_{F,t} \\ = \hat{S}_t + \hat{P}^*_{H,t} - \hat{P}_{F,t} + a(\hat{T}_t - \hat{T}^*_t)\\ = \hat{P}_{H,t} - \hat{P}_{F,t} + a(\hat{T}_t - \hat{T}^*_t) \quad \text{given that the law of one price holds}\\ = -\hat{T}_t + a(\hat{T}_t - \hat{T}^*_t)\\ = -\hat{T}_t + 2a\hat{T}_t \quad \text{given that the law of one price holds, $\hat{T}^*_t = -\hat{T}_t$}\\ \hat{Q}_t = (2a-1)\hat{T}_t \end{gather}