# Question on real exchange rate

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%."?

• The real exchange rate is the exchange rate adjusted for inflation. Try a numerical example. imagine 10p=1USD. – Jamzy Jul 28 '15 at 5:27
• added self-study tag, if it's not self study, feel free to delete it. – Jamzy Jul 28 '15 at 5:28
• What exchange rate are you using? "Dollars per One peso" or "Pesos per one dollar"? – Alecos Papadopoulos Jul 28 '15 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 '15 at 14:41

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

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

• 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 '15 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 '15 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. – clem steredenn Jul 28 '15 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 '15 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! 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.