<blockquote>
<p>If the Phillipine peso falls in value against the USD by 5% in a year, but the domestic inflation rate in the Phillipines is 10%, compared to 2% in the USA, the nominal exchange rate has fallen (by 5%), but the real exchange rate has risen by 3%.</p>
<p> Could anyone help me explain why &quot;the real exchange rate has risen by 3%.&quot;?</p>
</blockquote>
<p>A Word of caution: it is not the RER that rises or falls. It is currencies. A currency (in this case the PH peso) either appreciates or depreciates relative to another. The PH peso appreciates in real terms when the cost of a PH basket of godos falls relative to the cost of the same basket in the US, when both baskets are expressed in the same currency.  Let P be the Price index for PH and P* be the Price index for US. If E is defines as the number of dollars that must be given up in Exchange for one PH peso, then the RER can be defined as:</p>
<p>RER=P/EP*</p>
<p>This indicates the number of PH baskets that must be given up in exchange to obtain a similar basket of US godos. If this number raises then the PH peso has appreciated in real terms vis-avis the US dollar.</p>
<p>Taking percentage changes and ignoring second order terms:</p>
<p>%CH_RER=%CH_P-(%CH_E+%CH_P*)</p>
<p>Thus,</p>
<p>%CH_RER=10%-(5%+2%)=3%   Voila!</p>

CC BY-SA 3.0
2015-07-29T18:51:41.883Z
2015-08-16T11:31:47.580Z
2020-06-17T09:40:54.643Z
Abraham Vela
<p>Mainly based on the idea of @bilbo_pingouin, this is my understanding:</p>

<p>Year 0: </p>

<ul>
<li><p>Price of product X in the US is: 1 dollar</p></li>
<li><p>Exchange rate is $x$ pesos for 1 USD</p></li>
<li><p>So, price of X in the P is $x$ pesos</p></li>
</ul>

<p>Year 1:</p>

<ul>
<li><p>Price of product X in the US is: $(1+\pi_{US})$ dollars, due to inflation</p></li>
<li><p>Exchange rate is $(1+\alpha)*x$ pesos for 1 USD, due to depreciation of peso.</p></li>
<li><p>So price of X in the P is $(1+\pi_{US})(1+\alpha)*x$ pesos.</p></li>
</ul>

<p>But this price has NOT been adjusted for inflation in the P, so its real price in the P should be: $\frac{(1+\pi_{US})(1+\alpha)}{1+\pi_P}*x$ pesos</p>

<p>So the real change in exchange rate should be $\frac{(1+\pi_{US})(1+\alpha)}{1+\pi_P}-1 = -2.4\%$, which corresponds to an appreciation in peso.</p>

CC BY-SA 3.0
2015-07-28T15:10:59.163Z
2015-07-29T03:36:09.867Z
2015-07-29T03:36:09.867Z
SiXUlm
<p>Note that I am not 100% sure. But in my understanding, we have </p>

<p><strong>Year 1</strong></p>

<ul>
<li>Price for a product in the US :  $p_{US}=v$ \$</li>
<li>Exchange rate: $x$ pesos for $1$\$</li>
<li>Price of the product in the Philipines: $p_{Ph})=v.x$ pesos</li>
</ul>

<p><strong>Year 2</strong></p>

<ul>
<li>Price for the same product in the US : $p^\prime_{US} = (1+\alpha_v)v$ \$. The price increased due to the inflation $\alpha_v$.</li>
<li>Nominal exchange rate: $(1+\alpha_x)x$ pesos for $1$\$. A drop in value means you need more pesos for one USD.</li>
<li>Inflation in the Philipines: $p^\prime_{Ph} = \frac{P_{Ph}}{1+\alpha_p}$. Due to the inflation, the acquisitive power of the pesos is reduced.</li>
<li>Price of that product : $p^\prime_{Ph}=[(1+\alpha_v)v].[(1+\alpha_x)x].[\frac{1}{1+\alpha_p}] = \frac{(1+\alpha_v).(1+\alpha_x)}{1+\alpha_p}.v.x$</li>
</ul>

<p><strong>Variation</strong></p>

<ul>
<li>The effective variation compared to the previous year is thus, $\frac{(1+\alpha_v).(1+\alpha_x)}{1+\alpha_p}$, which corresponds to a <em>rise</em> in effective exchange rate of </li>
</ul>

<p>$$\frac{1+\alpha_p}{(1+\alpha_v).(1+\alpha_x)}-1=2.7\%$$</p>

CC BY-SA 3.0
2015-07-28T14:46:22.397Z
Generally,I get your point and I think it is correct. Only one thing I don't understand: in year 2, in the US, you adjusted the price by MULTIPLYING with (1+inflation in US). But then in the Phillipines, you adjusted by DIVIDING by (1+inflation in P), which I find not consistent.
SiXUlm
CC BY-SA 3.0
2015-07-28T14:49:29.107Z
Also, the coefficient $(1+\alpha_v)(1+\alpha_x)/(1+\alpha_p)$ is actually not belong to any of $v$ or $x$. So it's not clear to me why it corresponds to the exchange rate.
SiXUlm
CC BY-SA 3.0
2015-07-28T15:10:43.287Z
@SiXUlm this is why I said I wasn't sure. You want an exchange rate normalised for 1\$, which is why $v$ disappears. Plus you want the evolution of the exchange rate. This is why $x$ disappears. I divided by $1+\alpha_p$ because you get less for 1 peso in year 2 compared to year 1. But the price in dollar increased.
clem steredenn
CC BY-SA 3.0
2015-07-28T15:12:38.787Z
I understand your argument now. Based on it, I present my understanding below. I think your whole reasoning is correct.
SiXUlm
CC BY-SA 3.0
2015-07-28T12:44:01.090Z
2015-07-28T12:44:01.090Z
clem steredenn
<p>If the Phillipine peso falls in value against the USD by 5% in a year, but the domestic inflation rate in the Phillipines is 10%, compared to 2% in the USA, the nominal exchange rate has fallen (by 5%), but the real exchange rate has risen by 3%.</p>

<p>Could anyone help me explain why "the real exchange rate has risen by 3%."?</p>

CC BY-SA 3.0
2015-07-28T05:27:15.597Z
The real exchange rate is the exchange rate adjusted for inflation. Try a numerical example. imagine 10p=1USD.
Jamzy
CC BY-SA 3.0
2015-07-28T05:28:03.910Z
added self-study tag, if it's not self study, feel free to delete it.
Jamzy
CC BY-SA 3.0
2015-07-28T12:29:09.970Z
What exchange rate are you using? "Dollars per One peso" or "Pesos per one dollar"?
Alecos Papadopoulos
CC BY-SA 3.0
2015-07-28T14:41:57.657Z
The example doesn't clearly say any of them, but I don't think it would cause any trouble.
SiXUlm
<macroeconomics><inflation><exchange-rates><self-study>
Question on real exchange rate
SiXUlm
# Question on real exchange rate

(Asked by: username_0 on 2015-07-28)

If the Phillipine peso falls in value against the USD by 5% in a year, but the domestic inflation rate in the Phillipines is 10%, compared to 2% in the USA, the nominal exchange rate has fallen (by 5%), but the real exchange rate has risen by 3%.

Could anyone help me explain why "the real exchange rate has risen by 3%."?

**Tags:** macroeconomics, inflation, exchange-rates, self-study

> **Comments:**
> **username_1**: The real exchange rate is the exchange rate adjusted for inflation. Try a numerical example. imagine 10p=1USD.  
> **username_1**: added self-study tag, if it's not self study, feel free to delete it.  
> **username_2**: What exchange rate are you using? "Dollars per One peso" or "Pesos per one dollar"?  
> **username_0**: The example doesn't clearly say any of them, but I don't think it would cause any trouble.  

---

## 3 answers

### Answer by username_4 on 2015-07-28. Score: 4 | Accepted answer

Note that I am not 100% sure. But in my understanding, we have

**Year 1**

* Price for a product in the US : $p\_{US}=v$ \$
* Exchange rate: $x$ pesos for $1$\$
* Price of the product in the Philipines: $p\_{Ph})=v.x$ pesos

**Year 2**

* Price for the same product in the US : $p^\prime\_{US} = (1+\alpha\_v)v$ \$. The price increased due to the inflation $\alpha\_v$.
* Nominal exchange rate: $(1+\alpha\_x)x$ pesos for $1$\$. A drop in value means you need more pesos for one USD.
* Inflation in the Philipines: $p^\prime\_{Ph} = \frac{P\_{Ph}}{1+\alpha\_p}$. Due to the inflation, the acquisitive power of the pesos is reduced.
* Price of that product : $p^\prime\_{Ph}=[(1+\alpha\_v)v].[(1+\alpha\_x)x].[\frac{1}{1+\alpha\_p}] = \frac{(1+\alpha\_v).(1+\alpha\_x)}{1+\alpha\_p}.v.x$

**Variation**

* The effective variation compared to the previous year is thus, $\frac{(1+\alpha\_v).(1+\alpha\_x)}{1+\alpha\_p}$, which corresponds to a *rise* in effective exchange rate of

$$\frac{1+\alpha\_p}{(1+\alpha\_v).(1+\alpha\_x)}-1=2.7\%$$

> **Comments:**
> **username_0**: Generally,I get your point and I think it is correct. Only one thing I don't understand: in year 2, in the US, you adjusted the price by MULTIPLYING with (1+inflation in US). But then in the Phillipines, you adjusted by DIVIDING by (1+inflation in P), which I find not consistent.  
> **username_0**: Also, the coefficient $(1+\alpha_v)(1+\alpha_x)/(1+\alpha_p)$ is actually not belong to any of $v$ or $x$. So it's not clear to me why it corresponds to the exchange rate.  
> **username_4**: @username_0 this is why I said I wasn't sure. You want an exchange rate normalised for 1\$, which is why $v$ disappears. Plus you want the evolution of the exchange rate. This is why $x$ disappears. I divided by $1+\alpha_p$ because you get less for 1 peso in year 2 compared to year 1. But the price in dollar increased.  
> **username_0**: I understand your argument now. Based on it, I present my understanding below. I think your whole reasoning is correct.  

---

### Answer by username_3 on 2015-07-29. Score: 3

> If the Phillipine peso falls in value against the USD by 5% in a year, but the domestic inflation rate in the Phillipines is 10%, compared to 2% in the USA, the nominal exchange rate has fallen (by 5%), but the real exchange rate has risen by 3%.
>
>  Could anyone help me explain why "the real exchange rate has risen by 3%."?

A Word of caution: it is not the RER that rises or falls. It is currencies. A currency (in this case the PH peso) either appreciates or depreciates relative to another. The PH peso appreciates in real terms when the cost of a PH basket of godos falls relative to the cost of the same basket in the US, when both baskets are expressed in the same currency. Let P be the Price index for PH and P\* be the Price index for US. If E is defines as the number of dollars that must be given up in Exchange for one PH peso, then the RER can be defined as:

RER=P/EP\*

This indicates the number of PH baskets that must be given up in exchange to obtain a similar basket of US godos. If this number raises then the PH peso has appreciated in real terms vis-avis the US dollar.

Taking percentage changes and ignoring second order terms:

%CH\_RER=%CH\_P-(%CH\_E+%CH\_P\*)

Thus,

%CH\_RER=10%-(5%+2%)=3% Voila!


---

### Answer by username_0 on 2015-07-28. Score: 0

Mainly based on the idea of @bilbo\_pingouin, this is my understanding:

Year 0:

* Price of product X in the US is: 1 dollar
* Exchange rate is $x$ pesos for 1 USD
* So, price of X in the P is $x$ pesos

Year 1:

* Price of product X in the US is: $(1+\pi\_{US})$ dollars, due to inflation
* Exchange rate is $(1+\alpha)\*x$ pesos for 1 USD, due to depreciation of peso.
* So price of X in the P is $(1+\pi\_{US})(1+\alpha)\*x$ pesos.

But this price has NOT been adjusted for inflation in the P, so its real price in the P should be: $\frac{(1+\pi\_{US})(1+\alpha)}{1+\pi\_P}\*x$ pesos

So the real change in exchange rate should be $\frac{(1+\pi\_{US})(1+\alpha)}{1+\pi\_P}-1 = -2.4\%$, which corresponds to an appreciation in peso.


---

