<p>You did not obtained any logical contradiction.</p>
<p>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.</p>
<p>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.</p>
<p>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;</p>
<p><span class="math-container">$$T= T_i - E[T]$$</span></p>
<p>Where <span class="math-container">$T$</span> is the measure, <span class="math-container">$T_i$</span> is height of building <span class="math-container">$i$</span>, <span class="math-container">$E[T]$</span> is the expected height of the building in Utrecht.</p>
<p>In turn the expectation operator can be written as; <span class="math-container">$E[T] = \sum_j^n p_jT_j$</span> where <span class="math-container">$p$</span> is probability of selecting some building <span class="math-container">$j$</span> and <span class="math-container">$T$</span> is the height of building <span class="math-container">$j$</span>. Next <span class="math-container">$j$</span> is the set all buildings in Utrecht that includes the building <span class="math-container">$i$</span>. Hence we have here also measure that determines whether building <span class="math-container">$i$</span>, is too tall based on comparing building <span class="math-container">$i$</span> to expectation that is again based <em>partially</em> on building <span class="math-container">$i$</span> since the value <span class="math-container">$E[T]$</span> depends on inclusion of building <span class="math-container">$i$</span>.</p>
<p>Yet is there any contradiction? Statement building <span class="math-container">$i$</span> is taller then expected if <span class="math-container">$T&gt;0$</span>, and statement building <span class="math-container">$i$</span> is included in the set <span class="math-container">$j$</span> used to calculate <span class="math-container">$E[T]$</span> are completely logically consistent because both statements can be true at the same time.</p>
<p>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.</p>
<p>In case of Big Mac Index it is based on  assumption of law of one price so the Big Mac Index is given by;</p>
<p><span class="math-container">$$BM = P_x/P_y$$</span></p>
<p>where <span class="math-container">$P$</span> is price of Big Mac in <span class="math-container">$x$</span> and <span class="math-container">$y$</span> respectively. Then suppose these prices have some relationship to exchange rate, so <span class="math-container">$P(S)$</span> where <span class="math-container">$S$</span> is an exchange rate.</p>
<p>You determine whether exchange rate is over or undervalued based on comparison;</p>
<p><span class="math-container">$$V=\frac{BM - S}{S} =\frac{P_x(S)/P_y(S) - S}{S} $$</span></p>
<p>Where <span class="math-container">$V&gt;0$</span> means its overvalued, <span class="math-container">$V&lt;0$</span> it is undervalued. Now the claim that for some particular country pair;</p>
<p><span class="math-container">$$\frac{P_x(S)/P_y(S) - S}{S}&gt; 0 $$</span></p>
<p>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 <span class="math-container">$S$</span>.</p>
<p>Both claims that <span class="math-container">$V&gt;0$</span> and <span class="math-container">$BM$</span> is composite function of <span class="math-container">$S$</span> can be true at the same time. There is no logical contradiction.</p>

CC BY-SA 4.0
2024-06-20T05:58:44.777Z
I understand. But my question is about dependence of Big Mac cost of exchange rate. Can you explain following example? In Russia Big Mac costs 150RUB, in USA Big Mac costs 5USD. Official exchange rate 1USD=90RUB.  According to Big Mac Index calculation true exchange rate must be 1USD=30RUB. But if it was so then Big Mac in Russia couldn't cost 150RUB because different exchange rate lead to different Big Mac costs. Thanks. @1muflon1@
Mike_bb
CC BY-SA 4.0
2024-06-20T09:16:40.263Z
@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
CC BY-SA 4.0
2024-06-20T09:18:48.990Z
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
CC BY-SA 4.0
2024-06-20T10:32:59.927Z
I asked about another thing namely: when you find $V$ you know which new exchange rate is. But your Big Mac cost is depend on it and must be changes if exchange rate is another than official exchange rate. It's impossible that if Big Mac costs 150RUB then Big Mac must also costs 150RUB always even  exchange rate is calculated as you wrote above. If we make in country new exchange rate that was calculated then Big Mac cost must be another than it was before our calculation (not 150RUB). @1muflon1
Mike_bb
CC BY-SA 4.0
2024-06-20T10:39:26.017Z
For example: imagine that we are Central Bank of Russia and official exchange rate is 1USD=50RUB. Big Mac costs 150RUB. According to calculation of Big Mac Index we know that really exchange rate is 1USD=30RUB. We sold dollars to change exchange rate and make it 1USD=30RUB. Now Big Mac cost must be changed because we changed exchange rate. Is it true? @1muflon1
Mike_bb
CC BY-SA 4.0
2024-06-20T11:00:30.570Z
@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
CC BY-SA 4.0
2024-06-20T11:03:44.273Z
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
CC BY-SA 4.0
2024-06-20T11:15:59.287Z
But then why is Big Mac Index needed if no one can't influence on the exchange rate?@1muflon1
Mike_bb
CC BY-SA 4.0
2024-06-20T11:21:39.283Z
@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
CC BY-SA 4.0
2024-06-20T11:24:53.513Z
about what the long term trend in exchange rate will be, even with very low accuracy can be potentially be useful for *some* decision making. For example, people in project management often use the concept of NPV to evaluate different projects, which is also crude concept with unrealistic assumptions (e.g. discount rate is constant, assumes you know future flows etc). Clearly there is some utility to this metric if people use it even if its accuracy might be very low.
1muflon1
CC BY-SA 4.0
2024-06-20T11:34:06.550Z
Ok. If many people will buy many undervalued currency then will Big Mac cost be changed?@1muflon1
Mike_bb
CC BY-SA 4.0
2024-06-20T11:48:30.253Z
@Mike_bb what do you mean by costs? You mean the price of burgers? Under your assumptions that the prices depend on exchange rate yes
1muflon1
CC BY-SA 4.0
2024-06-20T11:51:58.583Z
Yes, I mean price of burgers. It's my assumption but I don't know if this assumption is true or not. I ask you about it. @1muflon1
Mike_bb
CC BY-SA 4.0
2024-06-20T12:00:35.087Z
@Mike_bb I don't know either, I would have to empirically investigate McDonalds in particular location. Its definitely not something that you can claim is axiomatically true nor its something that has to be the case based on standard economic theories. It could be the case in some special local cases, that would have to be empirically estimated. Regardless, whether these prices depend on exchange rate or not its not really relevant for your question. In both cases (when prices depend on S and when they are independent) there isnt any logical contradiction (as my math above demonstrates)
1muflon1
CC BY-SA 4.0
2024-07-03T09:39:52.920Z
Hello. You wrote that "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". For what purposes do we buy or sell currency? Thx. @1muflon1
Mike_bb
CC BY-SA 4.0
2024-07-03T10:55:31.437Z
@Mike_bb usually people buy/sell currency to get another currency so they can buy something with that another currency.
1muflon1
CC BY-SA 4.0
2024-07-03T11:03:34.260Z
Yes. But why do people buy undervalued currency and sell overvalued currency? I mean in this context. @1muflon1
Mike_bb
CC BY-SA 4.0
2024-07-03T11:05:41.337Z
if a currency is overvalued people expect the price to drop in future so they would sell now while price is high if undervalued the opposite
1muflon1
CC BY-SA 4.0
2024-06-19T20:19:14.147Z
2024-06-19T20:19:14.147Z
1muflon1
<p>Let we have currency A and currency B.</p>
<p>In the first country 1A=50B and Big Mac costs 2A. In the second country Big Mac costs 200B.</p>
<p>Old exchange rate: 1A=50B</p>
<p>New exchange rate must be: 200B/2A=100(B/A). Therefore: 1A=100B.</p>
<p>It follows that currency A is overrated by 100%.</p>
<p>It's explanation of Big Mac Index calculation.</p>
<blockquote>
<p>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.</p>
</blockquote>
<p>Thus we obtained logical contradiction.</p>
<p>Is it logical contradiction indeed or am I wrong?</p>
<p>Thanks.</p>

<currency><exchange-rates><purchasing-power-partity>
Is this logical contradiction in explanation of Big Mac Index calculation or am I wrong?
Mike_bb
# Is this logical contradiction in explanation of Big Mac Index calculation or am I wrong?

(Asked by: username_0 on 2024-06-19)

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.

Thus we obtained logical contradiction.

Is it logical contradiction indeed or am I wrong?

Thanks.

**Tags:** currency, exchange-rates, purchasing-power-partity


---

## 1 answers

### Answer by username_1 on 2024-06-19. Score: 1 | Accepted answer

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.

> **Comments:**
> **username_0**: I understand. But my question is about dependence of Big Mac cost of exchange rate. Can you explain following example? In Russia Big Mac costs 150RUB, in USA Big Mac costs 5USD. Official exchange rate 1USD=90RUB.  According to Big Mac Index calculation true exchange rate must be 1USD=30RUB. But if it was so then Big Mac in Russia couldn't cost 150RUB because different exchange rate lead to different Big Mac costs. Thanks. @username_1@  
> **username_1**: @username_0 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  
> **username_1**: 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)  
> **username_0**: I asked about another thing namely: when you find $V$ you know which new exchange rate is. But your Big Mac cost is depend on it and must be changes if exchange rate is another than official exchange rate. It's impossible that if Big Mac costs 150RUB then Big Mac must also costs 150RUB always even  exchange rate is calculated as you wrote above. If we make in country new exchange rate that was calculated then Big Mac cost must be another than it was before our calculation (not 150RUB). @username_1  
> **username_0**: For example: imagine that we are Central Bank of Russia and official exchange rate is 1USD=50RUB. Big Mac costs 150RUB. According to calculation of Big Mac Index we know that really exchange rate is 1USD=30RUB. We sold dollars to change exchange rate and make it 1USD=30RUB. Now Big Mac cost must be changed because we changed exchange rate. Is it true? @username_1  
> **username_1**: @username_0 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  
> **username_1**: 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.  
> **username_0**: But then why is Big Mac Index needed if no one can't influence on the exchange rate?@username_1  
> **username_1**: @username_0 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  
> **username_1**: about what the long term trend in exchange rate will be, even with very low accuracy can be potentially be useful for *some* decision making. For example, people in project management often use the concept of NPV to evaluate different projects, which is also crude concept with unrealistic assumptions (e.g. discount rate is constant, assumes you know future flows etc). Clearly there is some utility to this metric if people use it even if its accuracy might be very low.  
> **username_0**: Ok. If many people will buy many undervalued currency then will Big Mac cost be changed?@username_1  
> **username_1**: @username_0 what do you mean by costs? You mean the price of burgers? Under your assumptions that the prices depend on exchange rate yes  
> **username_0**: Yes, I mean price of burgers. It's my assumption but I don't know if this assumption is true or not. I ask you about it. @username_1  
> **username_1**: @username_0 I don't know either, I would have to empirically investigate McDonalds in particular location. Its definitely not something that you can claim is axiomatically true nor its something that has to be the case based on standard economic theories. It could be the case in some special local cases, that would have to be empirically estimated. Regardless, whether these prices depend on exchange rate or not its not really relevant for your question. In both cases (when prices depend on S and when they are independent) there isnt any logical contradiction (as my math above demonstrates)  
> **username_0**: Hello. You wrote that "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". For what purposes do we buy or sell currency? Thx. @username_1  
> **username_1**: @username_0 usually people buy/sell currency to get another currency so they can buy something with that another currency.  
> **username_0**: Yes. But why do people buy undervalued currency and sell overvalued currency? I mean in this context. @username_1  
> **username_1**: if a currency is overvalued people expect the price to drop in future so they would sell now while price is high if undervalued the opposite  

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