The letter υ is pronounced as ee (like i in info), only when it doesn't follow a vowel. For example, υποστηρίζω (support).
When it follows a vowel like α, ε, or ο, the two vowels together (αυ, ευ, ου) become a dipththong, pronounced as av or af, ev or ef, and oo, depending on the case. (ηυ and ιυ are really are, you'll never see those).
There are some cases that people can replace υ with β, for example αυτί- αφτί (ear), or αυγό-αβγό (egg), but these cases are definitely not many. It's a grammar rule. ^.^
Have in mind that the orthography in Greek is historical. So pronunciation could be the same but for reasons of history in most cases the Modern Greek language tries to keep the Ancient Greek orthography. If there was not the Ancient Greek word or it is largely modified as the language has been evoluted since then, there is no problem to use the phonetic orthography, as the word has weak historical roots. An example: αυγό or αβγό (=egg)? The modern trend is to use the form αβγο, as there was not such a word in Ancient Greek and it came from the Ancient word "ωόν"=egg in Ancient Greek. So there is no special reason to use αυ. The same happens in foreign words. Example: τραίνο or τρένο =train. Some years ago the official orthography was τραίνο from English, but now it is phonetic: τρένο. More examples one can meet in many words.
So the verb "ξοδεύω" comes from the Ancient Greek Koene verb "εξοδεύω" and the Ancient noun "έξοδος". More here: https://el.wiktionary.org/wiki/%CE%BE%CE%BF%CE%B4%CE%B5%CF%8D%CF%89. So no way to use β in this case. It has a real sense this verb, as the word "έξοδος"=exit, that is the same in Modern Greek, means that one takes his money out his pocket :), to spend :) (joking :) ) .
Btw, the word "to spend" has a Latin origin that is the verb expendere=to lose weight ( I think, of the wallet (joking too :) ), but it is too difficult to find out its origin as much distort. Look at: http://www.etymonline.com/index.php?term=spend
Actually both mean to lose weight, to exhaust, but nobody is thinking the origin when he is using these words.
That's why etymology is so beautiful!
Woow! Thanks Stergi3 ;) . Now i can understand why ( etymology is beautiful)! :) ;) By the way i found out from your answer that the word "ωόν"=egg wich comes from the Antient Greek is the ''grand,ggrand father'' of the word ''ou''=egg in romanian language (like many other words i discovered while learning greek language). Thanks again.
Actually all Romance languages are Latin, evoluted in different countries of the old Roman Empire, a theory that I believe is valid. So if you think the origin, it is easy to compare words in different languages, including Romanian, I have a schedule to proceed to learn some. So it is easy for a Romanian to learn these languages, Italian, French, Spanish etc. Greek language is not a Romance language, it is a single one. But: All these languages (Latin and Greek and many others) come from a Proto-Indo-european (PIE) language, that is not reconstructed by linguists yet, it is very difficult, so that one can find similarities in many. I try to learn this way, to compare the words of Romance languages. English is full of Latin-origin words, but not only of course. It is more difficult to me to learn languages that do not belong to this PIE family, so that Turkish for instance is difficult to me. Since the time of Ottoman Empire though, many words from Turkish passed to Greek, so this makes things easier. Also, many Turkish words have Arabic or Farsi origin, so in way, all these languages interacted each other. It is a real game and pleasure to find the origin of the words :) for people who love languages.