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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%."?

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The real exchange rate is the exchange rate adjusted for inflation. Try a numerical example. imagine 10p=1USD. –  Jamzy Jul 28 at 5:27
    
added self-study tag, if it's not self study, feel free to delete it. –  Jamzy Jul 28 at 5:28
    
What exchange rate are you using? "Dollars per One peso" or "Pesos per one dollar"? –  Alecos Papadopoulos Jul 28 at 12:29
    
The example doesn't clearly say any of them, but I don't think it would cause any trouble. –  SiXUlm Jul 28 at 14:41

3 Answers 3

up vote 3 down vote accepted

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

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

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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 Jul 28 at 14:46
    
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 Jul 28 at 14:49
    
@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. –  bilbo_pingouin Jul 28 at 15:10
    
I understand your argument now. Based on it, I present my understanding below. I think your whole reasoning is correct. –  SiXUlm Jul 28 at 15:12

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!

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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.

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