Does updating a row with the same value actually update the row?
Due to the MVCC model of Postgres, and according to the rules of SQL, an UPDATE writes a new row version for every row that is not excluded in the WHERE clause.
This does have a more or less substantial impact on performance, directly and indirectly. &quot;Empty updates&quot; have the same cost per row as any other update. They fire triggers (if present) like any other update, they have to be WAL-logged and they produce dead rows bloating the table and causing more work for VACUUM later like any other update.
Index entries and TOASTed columns where none of the involved columns are changed can stay the same, but that is true for any updated row. Related:

PostgreSQL Initial Database Size
Redundant data in update statements

It's almost always a good idea to exclude such empty updates (when there is an actual chance it may happen). You did not provide a table definition in your question. We have to assume first_name can be NULL (which wouldn't be surprising for a &quot;first name&quot;), hence the query has to use NULL-safe comparison:
UPDATE users
SET    first_name = 'Michael'
WHERE  id = 123
AND    first_name IS DISTINCT FROM 'Michael';

If first_name IS NULL before the update, a test with just first_name &lt;&gt; 'Michael' would evaluate to NULL and as such exclude the row from the update. Sneaky error. If the column is defined NOT NULL, use the simple equality check, though, that's a bit cheaper.
Related:

How do I (or can I) SELECT DISTINCT on multiple columns?
Update column with data from another table