Log-linearization of real exchange rate

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.

• Can you please show us what you attempted so far? The policy on homework questions here is that you should at least show some effort into solving them before you get help.
– 1muflon1
Mar 4, 2020 at 11:22
• Alejandro, you usually post interesting questions and I would be glad to help you, but you should avoid writing variables in formulas, without without saying what they stand for. Mar 6, 2020 at 8:51

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:

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