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Cost of Living in Denmark
Having in your backhead that Denmark is one of the most expensive places in the world to love, 40 euros seems reasonable, but its not overwhelming... Dont know if its negotiable, but try and ask for 50... That way you will be able to afford to go out in the weekends...
I have a couple of questions about your stay in Denmark:
In what part of Denmark will you be staying? Noting that Copenhagen is more expensive, other parts are quite obviously less.
Does the accommodation provided for you include a kitchen, refrigerator, etc. and laundry facilities? The reason I ask is because if not, eating out (as normal) will be quite expensive and laundry isn't cheep either. If a kitchen is provided, I would suggest humbly asking one of your Danish counterparts take a lunch or some time after work to take you shopping. It might seem a little odd to be led around shopping for simple things, but it will help immensely, and it will help you feel not so lost doing such a simple thing.
Do you speak any Danish? I have looked through many English-Danish dictionaries, and quite frankly, the translations there are quite bad or very formal and not practical. It is great, however, that Danes begin learning English very young. This will help a lot.
On the note of your per diem, I did some conversions on the internet and it appears that 40 Euros will equate to about 300 Kroner each day (at yesterday's exchange rates). It has been a long time since I was there for a visit, but that seems like a decent amount. However, I agree with Asanka87 that you could try asking for 50. Worst that can happen is that your company says no.
I would like to make a simple suggestion also if I could: When I lived there (in Vejle) it was an old fashioned custom to bring small gifts if invited to someone's house as to say thank you (which, if working there for 6 months, I am sure you will be invited a few times). I think more currently people resort to bottles of wine and things of that nature. But if it is possible for you to find things that are nice, quaint, and inexpensive to take with you that are distinctly American, this might make a very nice gift to your hosts. I'm sure some of the Danes that read this will be able to make better suggestions, but I am remembering from many years ago when I lived there.