Low period in deficit (8)
I believe the answer is:
shortage
'deficit' is the definition.
('shortage' can be a synonym of 'deficit')
'low period' is the wordplay.
'low' becomes 'short' ('short' is associated in meaning with 'low'**).
'period' becomes 'age' (I've seen this before).
'short'+'age'='SHORTAGE'
'in' is the link.
(Other definitions for shortage that I've seen before include "Lack, deficiency" , "Deficit" , "may put price up" , "Horse tag (anag.)" , "famine" .)