# Is this logical contradiction in explanation of Big Mac Index calculation or am I wrong?

Let we have currency A and currency B.

In the first country 1A=50B and Big Mac costs 2A. In the second country Big Mac costs 200B.

Old exchange rate: 1A=50B

New exchange rate must be: 200B/2A=100(B/A). Therefore: 1A=100B.

It follows that currency A is overrated by 100%.

It's explanation of Big Mac Index calculation.

But... if currency A was equal to 100B then Big Mac couldn't cost 2A in the first country because Big Mac consist of components that depend on exchange rate.

Is it logical contradiction indeed or am I wrong?

Thanks.

You did not obtained any logical contradiction.

Logical contradiction by definition is a situation where two statements cannot be true at the same time. Example of logical contradiction would be I am in Amsterdam, and I am in Utrecht at the same time. Since it is impossible both first part of the statement and second part of the statement is true we arrived at a logical contradiction.

In your case your statements are; Big Mac Index is a measure of whether exchange rate is overvalued, and Big Mac Index is based on value of an item that also depends on an exchange rate. Both of these things can be true at the same time.

Consider simpler example, suppose we want to measure if building in Utrecht is taller than expected. A simple measure could be to calculate measure such as follow;

$$T= T_i - E[T]$$

Where $$T$$ is the measure, $$T_i$$ is height of building $$i$$, $$E[T]$$ is the expected height of the building in Utrecht.

In turn the expectation operator can be written as; $$E[T] = \sum_j^n p_jT_j$$ where $$p$$ is probability of selecting some building $$j$$ and $$T$$ is the height of building $$j$$. Next $$j$$ is the set all buildings in Utrecht that includes the building $$i$$. Hence we have here also measure that determines whether building $$i$$, is too tall based on comparing building $$i$$ to expectation that is again based partially on building $$i$$ since the value $$E[T]$$ depends on inclusion of building $$i$$.

Yet is there any contradiction? Statement building $$i$$ is taller then expected if $$T>0$$, and statement building $$i$$ is included in the set $$j$$ used to calculate $$E[T]$$ are completely logically consistent because both statements can be true at the same time.

Similarly one could simply devise a measure of whether currency is overvalued purely by comparing current exchange to its average which would include current exchange rate. It might not be good measure to make investment decisions on but it is not logically inconsistent measure.

In case of Big Mac Index it is based on assumption of law of one price so the Big Mac Index is given by;

$$BM = P_x/P_y$$

where $$P$$ is price of Big Mac in $$x$$ and $$y$$ respectively. Then suppose these prices have some relationship to exchange rate, so $$P(S)$$ where $$S$$ is an exchange rate.

You determine whether exchange rate is over or undervalued based on comparison;

$$V=\frac{BM - S}{S} =\frac{P_x(S)/P_y(S) - S}{S}$$

Where $$V>0$$ means its overvalued, $$V<0$$ it is undervalued. Now the claim that for some particular country pair;

$$\frac{P_x(S)/P_y(S) - S}{S}> 0$$

i.e. that the currency is overvalued by this measure, is completely consistent with claim that BMC is a function, (i.e. depends on) of $$S$$.

Both claims that $$V>0$$ and $$BM$$ is composite function of $$S$$ can be true at the same time. There is no logical contradiction.

• @Mike_bb you are making some leaps here. First, I wont use your numbers because they are not easy for calculation and if you want to use functions you cannot use just random numbers, but there is no issue with BMI being based indirectly on exchange rate. Suppose the price of big mac depends on exchange rate in x as $P_x = 2S$ and in y as $P_y = 0.5S$, then suppose exchange rate between x and y is 1 to simplify calculations. That would mean BMI is 4, according to the big mac index currency is overvalued by $(4-1)/1=3$ so by 300%, yet all these numbers are completely consistent with each other
– 1muflon1
Commented Jun 20 at 9:16
• Exchange rate is consistent with big mac prices 2 and 0.5 and also BMI being 4 and being based on exchange rate S=1 and also with claiming there is 300% overvaluation, all these statements are simultaneously true. No logical contradiction exists. Even for your numerical example I could show the same but I would have to find functions $P_x(S)$ and $P_y(S)$ that are consistent with your numbers and that would be too much work. Still even in your case it would produce consistent system. When I said you can't use random numbers I meant it in a sense that numbers have to be consistent with P(S)
– 1muflon1
Commented Jun 20 at 9:18
• @Mike_bb no, Big Mac Index is not measure of "true" exchange rate, whatever that even means. The exchange rate is actually the rate you get somewhere at a window of a bank. Big Mac Index measures whether the actual exchange rate deviates from exchange rate under PPP/law of one price. Idea is that people believe in long run exchange rate should approximately converge to theoretical exchange rate derived from law of one price. Just because BMI shows overvaluation does not imply central bank has to somehow change exchange rate
– 1muflon1
Commented Jun 20 at 11:00
• if we talk about fixed/managed exchange rate (since you are invoking central bank). When BMI shows overvaluation, it shows overvaluation relative to theoretical exchange rate under law of one price. Also, BMI is itself a more humorous take on PPP/law of one price, its not necessarily the most rigorous measure. Furthermore, there are more complex models of exchange rate where PPP violations are possible.
– 1muflon1
Commented Jun 20 at 11:03
• @Mike_bb what exactly you mean that it is needed? 1. BMI is not needed, again its an humorous take on PPP. 2. BMI can be useful if you believe the that exchange rate should follow law of one price in long term, then if BMI shows some currency is overvalued you can sell it and if undervalued you can buy it or make some other economic decisions like buying abroad/home based on it. Its not state of the art forecasting model, but its a simple model to give you more info than saying future exchange rate could be anything between $(0, \infty)$. Any model that can be used to get some info
– 1muflon1
Commented Jun 20 at 11:21