Character set conversion and database size

IT management is pushing us to migrate our databases to the Unicode character set ( NLS_CHARACTERSET= WE8MSWIN1252 to AL32UTF8). One of my internal customers is concerned that conversion to multibyte characters will dramatically increase the size of the database. I know that for VARCHAR2s Oracle only allocates storage for the actual characters stored in the string regardless maximum defined size of the column. Does the same idea apply a VARCHAR2 column whose contents can be coded as single bytes or does "Hello world!" actually take twice as much space in Unicode as it does in the default character set?

Thanks,
Peter

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