But as most people know, robots [...] are expensive
If they are ubiquitous, one could assume the cost of robot-cops (totally not a trademark) is a solved problem. If you want to make a fair comparison, you must account for the price of training a human, their equipment, their retirement, their hospital bills or funeral, and maybe the checks sent to their families.
Why robots over humans?
You can train a robot much faster. If the robot dies, you can transfer its databanks to a new one. They are typically superior to humans in terms of strength, speed and reflexes. Intellectually, they can process more information, and access any data in the system on the spot, which is an interesting trait for a detective. They can also operate 24/7 without having to pay overtime.
Lack of morality is an interesting point, one that can also be found in humans. On one hand, there's the cold logic of a robot's programming, which drives them to make the "right" choice, depending on their programming. On the other hand, human can blatantly disregard human life. In a number of countries, police corruption is a big problem. You can't corrupt a robot.
Related to the previous point, robots have no second thought. If there is a danger, they are more likely to dive into it if it means saving a life, whereas a human would rightfully so fear for their own lives. In case of an injury, a robot can get repaired and get back in active duty much faster as well.
And because they have no strong feeling one way or the other, they might not kill a kid on sight because he's black.
Why not robots over human?
In the present, humans aren't really ready to trust robots with their lives. Likely, that doesn't change much in the future, so first contact would probably be cold. Human-cyborg relations would probably get better over time though.
Security would be a major concern, the possibility of your police force switching off or turning against you is a frightening one. However, as the hacking capabilities of bad guys increase, so does cybersecurity. You can take steps to mitigate the risks of hostile takeover. Ultimately, the common criminals will likely lack the proper skill to hotwire high-tech machinery so the real threat is organized crime, which is already known to bribe/threaten police officers. There is still no such thing as zero risk, so it comes down to how acceptable the risk is versus the risk of humans losing control of themselves.
Another thing to consider is that if they'll be chasing human criminals, then a human mind might be a better frame of reference. Humans are most likely to better understand how a human criminal think than a robot would.
A favorable environment I think would be high crime rates and/or high probability of getting your skin shot. In that case, a replaceable police force would make sense.
Another possible scenario would be lack of trust in human police. If your police force is known for its corruption, then a robot might be a better sight than a cop.
What I would recommend is to have humans and robots doing the job together, as opposed to one or the other. They both have strengths and weaknesses, so going one way may solve some problems while creating other.
A last point I'd like to make is that police is supposed to protect and serve the public. In an ideal world, financial considerations shouldn't matter as long as the public service is effective. But you know, that's ideally speaking.