```python
def tier_cost(units: int, allowance: int, first_rate: int, later_rate: int) -> int:
    if units < 0:
        raise ValueError("Units cannot be below zero.")
    
    if allowance > 0 and units <= allowance:
        return units * first_rate
    
    remaining_units = units - allowance
    return allowance * first_rate + remaining_units * later_rate
```