The expense of beginning to sleep in bed (4)
I believe the answer is:
cost
'the expense' is the definition.
(cost is a kind of expense)
'beginning to sleep in bed' is the wordplay.
'beginning to' says to take the initial letters.
'in' means one lot of letters goes inside another.
'bed' becomes 'cot' (cot is a kind of bed).
The first letter of 'sleep' is 's'.
's' placed within 'cot' is 'COST'.
'of' acts as a link.
(Other definitions for cost that I've seen before include "estimate price" , "Be priced at" , "Estimate the expense of doing" , "Suffering" , "Total spent for goods or services" .)