SQL subquery to determine whether or not a particular row is the first item that matches the subquery
There is no natural order.  With no order by the order of the rows is not guaranteed.

What you are looking for is not totally clear to me.  But I think this is what you are looking for.  Pretty sure window function is what you are looking and it is efficient.

  select tt.AuctionID, tt.StartingPrice
       , tt.rn 
       , case when rn = 1 then 'first' else 'not first' end as [verb]
  from ( SELECT [AuctionID]
              , [Price]
              , ROW_NUMBER() over (partition [AuctionID] by order by [Price]) as rn
         FROM [Test].[dbo].[Auction]
       ) tt

select *
     , case when rn = 1 then 'entered rate of' else 'edited rate to' end as [verb]
from  ( select n.PayRate, n.CreatedOn, n.CreatedByUserName
            ,  row_number() over (partiton by RateID order by TrackRateID) as rn
          from Rates, TrackRates n
         where ClientID = @id 
      ) tt
 order by RateID, TrackRateID