If a tenant is allowed to move-in early, it is common for the tenant to pay rent, in the pro-rated amount, for the length of the stay. This is calculated by getting the monthly rent and dividing by 30 days and using the per day amount ($/day) + the number of days the tenant is moving in early.
Example: Rent is $1,500/month. Divide by 30 days which is $50/day. If the tenant is allowed to move in 5 days early, it would be $250 in pro-rated rent.