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When trading two goods (or currencies) it is relatively easy to say if you are making profit when you know the exchange rates. For example Person A trades Bananas for Person B's Apples. Person A gives Person B 5 Bananas and receives 10 Apples from Person B:

Trade 1:

A -> 5 Bananas -> B

A <- 10 Apples <- B

Now after a while Person B wants to have more Apples because supplies are running low. Person A now sells the Apples back for Bananas, but of course, a profit is favourable, so now they cost 7 Bananas (we can assume that Person B has sufficient supply of Bananas to trade):

Trade 2:

A -> 10 Apples -> B

B -> 7 Bananas -> A

If we only look at these two transactions you can clearly see, that Person A had 5 Bananas before and now has 7 (ignoring any Bananas obtained by means of production or other trades). There is profit for Person A. It is possible for Person A to calculate exchange rates from the information obtained by looking at Trade 1 only and then subsequently calculate if the proposed Trade 2 will be profitable.

Now it gets more complex, as Person A has developed the production of more goods and Person B did so as well. They both now produce Bananas Apples and Oranges and trade them.

So now if we have the following trade:

Trade 3:

A -> 5 Bananas -> B

A <- 10 Apples <- B

We can again calculate a exchange-rate between Bananas and Apples and check out if a future trade is profitable between these goods. But now both Persons decide to trade Oranges as well:

Trade 4:

A -> 5 Bananas -> B

A <- 20 Oranges <- B

I now have the following questions:

Let's say only Trade 3 and Trade 4 have taken place and there is no other influence on the exchange rates between the goods. Can Person A calculate the profitability of future trades of Apples or Oranges for Bananas or Bananas for Apples or Oranges? In detail, for which of the following trades can Person A calculate the profitability using only the information obtained from Trade 3 and Trade 4, and if it is possible, how would this be calculated?

Bananas -> Apples

Bananas -> Oranges

Apples -> Bananas

Oranges -> Bananas

Additionally: Again only from the information drawn from Trade 3 and Trade 4 can Person A calculate the profitability of a trade between Oranges and Apples or vice versa?

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