<p>I think you should calculate the log of the ratio of two price series. When googling "log of relative price index", I did not find many references, but I found three which seem sufficient to me:</p>

<ul>
<li>Footnote 23 in page 14 in <a href="https://www.imf.org/external/pubs/ft/wp/2001/wp01194.pdf" rel="nofollow noreferrer">this</a> IMF research paper states:</li>
</ul>

<blockquote>
  <p>The RPI is defined as the ratio of domestic CPI for a country to trade weighted averages of the CPI's of partner countries.</p>
</blockquote>

<ul>
<li>Table D2 in online appendix of <a href="https://assets.aeaweb.org/assets/production/articles-attachments/aer/app/10408/20111536_app.pdf" rel="nofollow noreferrer">this</a> American Economic Review paper, which defines the "Log relative price index" as $\ln(P_i)$, where:</li>
</ul>

<blockquote>
  <p>$P_i$ [is] the price of good $i$ relative to an aggregate price index. (p.21)</p>
</blockquote>

<ul>
<li>Equation 1 in <a href="https://www.csae.ox.ac.uk/workingpapers/pdfs/2007-11text.pdf" rel="nofollow noreferrer">this</a> working paper, which uses the log of the ratio of import prices versus domestic (manufacture) prices.</li>
</ul>

<p>On top of this, I think using the ratio of two log prices is also quite counterintuitive. Notice that the following equality is true:</p>

<p>$$ \frac{\ln P_{it}}{\ln P_{jt}} = \log_{P_{jt}}P_{it} $$</p>

<p>Thus, by defining your log RPI as the ratio of two logs, you actually have a (non-natural) logarithm, where one price index is used as a base. Even more, this base is changing over time. This is not the case when you use the log of a ratio, as both have a common base (normally, $e$).</p>

CC BY-SA 3.0
2017-05-15T07:13:30.410Z
2017-05-15T07:13:30.410Z
luchonacho
<p>Actually, as @Henry mentions in comments</p>

<blockquote>
  <p>[...] the logarithm of the ratio [...] might make some sort of sense.</p>
</blockquote>

<p>And given that you are using US CPI as base, you are very likely to want the logarithm of the ratio. Why ?</p>

<p>What would you do If I were asking you to compute the growth rate (relative difference) of two aggregates, say, productions (e.g. of cars), denoted by <span class="math-container">$Q$</span>, between two consecutive years <span class="math-container">$t$</span> and <span class="math-container">$t+1$</span> ? Actually, you would use the following formulae:</p>

<p><span class="math-container">$g_{t}^{d} = \frac{Q_{t+1}-Q_{t}}{Q_{t}} = -1 + \frac{Q_{t+1}}{Q_{t}}$</span></p>

<p>where <span class="math-container">$^{d}$</span> stands for <em>discrete</em>. Indeed, note that when one computes <span class="math-container">$g_{t}^{d}$</span> like so, there is an implicit assumption. This assumption is that we reasonably suppose our productions to be discrete values. If we were considering those as continuous values (e.g. liters of water), we would actually use an other formulae, as follows</p>

<p><span class="math-container">$g_{t}^{c} = \ln{\left(\frac{Q_{t+1}}{Q_{t}}\right)}$</span></p>

<p>Note that <span class="math-container">$g_{t} \rightarrow g_{t}^{c}$</span> when <span class="math-container">$Q_{t+1}\rightarrow Q_{t}$</span>.</p>

<p>Thus, we just compared two types of production over time, conditioning the formulae that is used to what these productions are (discrete <em>versus</em> continuous). </p>

<p><em>What if I were asking you to compare CPIs over two nations ?</em></p>

CC BY-SA 4.0
2019-10-17T11:38:38.557Z
Any question @NewInvestor ?
keepAlive
CC BY-SA 4.0
2017-05-13T18:16:16.983Z
2019-10-25T11:03:11.167Z
2019-10-25T11:03:11.167Z
keepAlive
<p>I have US CPI prices &amp; Canada CPI prices (where US is my base). I am trying to calculate the 'log relative price index'.</p>

<p>Do I log the individual CPI prices first then calculate log Canada CPI/log US CPI or do I calculate the ratio and then log the answer?</p>

CC BY-SA 3.0
2017-05-13T00:54:02.513Z
It depends on what you are trying to do. But the logarithm of the ratio (equal to the difference of the logarithms) might make some sort of sense.  You might also consider what *base* means
Henry
CC BY-SA 3.0
2017-05-13T01:07:05.170Z
I want to chart the log relative price index and the nominal exchange rate to examine purchasing power parity theory. Base in my question means the nominal exchange rate is in US dollars.
NewInvestor
CC BY-SA 3.0
2017-05-13T06:16:27.663Z
@EconometricsHelp Could you please add the PPP equation to your question and explain what the variables in it mean?
Giskard
<self-study><price-index>
Log relative price index - calculation question
NewInvestor
# Log relative price index - calculation question

(Asked by: username_0 on 2017-05-13)

I have US CPI prices & Canada CPI prices (where US is my base). I am trying to calculate the 'log relative price index'.

Do I log the individual CPI prices first then calculate log Canada CPI/log US CPI or do I calculate the ratio and then log the answer?

**Tags:** self-study, price-index
> **Comments:**
> **username_1**: It depends on what you are trying to do. But the logarithm of the ratio (equal to the difference of the logarithms) might make some sort of sense.  You might also consider what *base* means  
> **username_0**: I want to chart the log relative price index and the nominal exchange rate to examine purchasing power parity theory. Base in my question means the nominal exchange rate is in US dollars.  
> **username_2**: @EconometricsHelp Could you please add the PPP equation to your question and explain what the variables in it mean?  

---

## 2 answers

### Answer by username_3 on 2017-05-15. Score: 1

I think you should calculate the log of the ratio of two price series. When googling "log of relative price index", I did not find many references, but I found three which seem sufficient to me:

* Footnote 23 in page 14 in [this](https://www.imf.org/external/pubs/ft/wp/2001/wp01194.pdf) IMF research paper states:

> The RPI is defined as the ratio of domestic CPI for a country to trade weighted averages of the CPI's of partner countries.

* Table D2 in online appendix of [this](https://assets.aeaweb.org/assets/production/articles-attachments/aer/app/10408/20111536_app.pdf) American Economic Review paper, which defines the "Log relative price index" as $\ln(P\_i)$, where:

> $P\_i$ [is] the price of good $i$ relative to an aggregate price index. (p.21)

* Equation 1 in [this](https://www.csae.ox.ac.uk/workingpapers/pdfs/2007-11text.pdf) working paper, which uses the log of the ratio of import prices versus domestic (manufacture) prices.

On top of this, I think using the ratio of two log prices is also quite counterintuitive. Notice that the following equality is true:

$$ \frac{\ln P\_{it}}{\ln P\_{jt}} = \log\_{P\_{jt}}P\_{it} $$

Thus, by defining your log RPI as the ratio of two logs, you actually have a (non-natural) logarithm, where one price index is used as a base. Even more, this base is changing over time. This is not the case when you use the log of a ratio, as both have a common base (normally, $e$).


---

### Answer by username_4 on 2017-05-13. Score: 0

Actually, as @Henry mentions in comments

> [...] the logarithm of the ratio [...] might make some sort of sense.

And given that you are using US CPI as base, you are very likely to want the logarithm of the ratio. Why ?

What would you do If I were asking you to compute the growth rate (relative difference) of two aggregates, say, productions (e.g. of cars), denoted by $Q$, between two consecutive years $t$ and $t+1$ ? Actually, you would use the following formulae:

$g\_{t}^{d} = \frac{Q\_{t+1}-Q\_{t}}{Q\_{t}} = -1 + \frac{Q\_{t+1}}{Q\_{t}}$

where $^{d}$ stands for *discrete*. Indeed, note that when one computes $g\_{t}^{d}$ like so, there is an implicit assumption. This assumption is that we reasonably suppose our productions to be discrete values. If we were considering those as continuous values (e.g. liters of water), we would actually use an other formulae, as follows

$g\_{t}^{c} = \ln{\left(\frac{Q\_{t+1}}{Q\_{t}}\right)}$

Note that $g\_{t} \rightarrow g\_{t}^{c}$ when $Q\_{t+1}\rightarrow Q\_{t}$.

Thus, we just compared two types of production over time, conditioning the formulae that is used to what these productions are (discrete *versus* continuous).

*What if I were asking you to compare CPIs over two nations ?*

> **Comments:**
> **username_4**: Any question @NewInvestor ?  

---

