# Log relative price index - calculation question

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?

• 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 May 13 '17 at 0:54
• 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 May 13 '17 at 1:07
• @EconometricsHelp Could you please add the PPP equation to your question and explain what the variables in it mean? – Giskard May 13 '17 at 6:16

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 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 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 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, $e$).

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 ?

• Any question @NewInvestor ? – keepAlive Oct 17 '19 at 11:38