Pay is not determined by the difficulty or unpleasantness of the job, or at least, not directly. It is determined by the balance of supply and demand. How many people are ready and willing to be police officers at some level of pay, and how many police officers are needed?
Policing isn't trivial, but you know what? Neither are most other jobs. There are many, many people with the capability of being police officers; supply is high. Demand is not actually all that high compared to supply. There's even more supply than may initially meet the eye because many people not physically strong enough to be police officers right this instant could bring themselves up to spec if needed. The difficulty or unpleasantness of the job factors into the supply but I think you'd find a lot of people who would take that job; it's hard, but there's a lot of hard jobs about that people do for much less than that amount of money. (After all, first responding is hard from one point of view, but it is uniquely rewarding too; how many lives have you saved in the course of your work? It's not all good but it's not all bad either.)
You argue that they are valuable, and this is true, but the value of an employee is not what determines their pay... it is what caps it; long term, anyhow. You can't be paid more than what you are worth, you can't even be paid exactly what you are worth, you in fact inevitably must be paid less than what you are "worth" for the whole arrangement to work. Governments aren't immune to this. They must run at a net profit or they'll bring their society down. Measurement of profit is somewhat different than a private company, but profit they must; they must be extracting more value from their employees than they are paying their employees or the society is running the government at a net loss, which can only be tolerated to a finite degree as determined by what other surpluses the society is running elsewhere. What governments can do that private industry can't is put off the pain until much later before the fact they are paying people more than the value they are actually bringing bites them.
Pay is not determined by the difficulty or unpleasantness of the job, or at least, not directly. It is determined by the balance of supply and demand.
That's true, but the market is ridiculously deformed on both sides of that equation. On the supply side, there are many who would like to be cops but the requirements bar them entry (most often for legitimate reasons, but I suspect not always). On the demand side, America's constant quest for new crimes to be defined and new police powers creates an artificial demand. I mean, most people I know don't see a demand for policing those smoking marijuana, but legislation forces it.
You can't be paid more than what you are worth
That depends on the definition of "worth". If you determine it rigorously as an economist would -- how much less money would the organization make without this employee -- then you're right. But if you define it in terms of the employee's productive contribution, it's a different story. Again, regulations cause some of that different (the need to have a figurehead owner who is a state resident, in some jurisdictions, is just deadweight but a requirement for doing business); in other cases it's contractual obligations (i.e., to be certified to sell some Cisco product you must have on staff one Cisco-certified engineer, even though you're not using his services as such).
I think you misunderstood when I said you can't be paid more than you are worth; I don't mean that you can't overpay someone, ever, I'm saying that nobody, government or otherwise, can persistently over the long haul pay its employees more than they are worth. I'd say any definition of "value" or "worth" that does permit that is so broken as to be useless. It's simply impossible; a system simply can not have more value flowing out of it than the sum total of the value it generates, the value it takes in, and the value it has on hand, and imbalances in income vs. outflow tend to be able to rapidly deplete the stored value.
That the market is "ridiculously deformed" here is sort of my point, along with the fact that such deformations can not be sustained.
I don't think this pay is rooted in the laws of economics. The number of people applying for police officer jobs is huge - but that doesn't mean the pay gets lowered for anyone. The contracts are negotiated by the veterans in the force, and the people with criminal justice degrees waiting in the wings who would gladly take less money are not stake holders in contract negotiations.
Nothing stops people from being paid more than they are worth. Doctors in the US are a classic example of an industry that captures regulations which enables them to artificially limit supply and over charge for their services.
It is determined by the balance of supply and demand
That's not actually true - here in the UK they are turning away applicants for the Met, even cancelling start dates. The supply of police officers far outstrips the demand. The same is true of the fire brigade.
Policing isn't trivial, but you know what? Neither are most other jobs. There are many, many people with the capability of being police officers; supply is high. Demand is not actually all that high compared to supply. There's even more supply than may initially meet the eye because many people not physically strong enough to be police officers right this instant could bring themselves up to spec if needed. The difficulty or unpleasantness of the job factors into the supply but I think you'd find a lot of people who would take that job; it's hard, but there's a lot of hard jobs about that people do for much less than that amount of money. (After all, first responding is hard from one point of view, but it is uniquely rewarding too; how many lives have you saved in the course of your work? It's not all good but it's not all bad either.)
You argue that they are valuable, and this is true, but the value of an employee is not what determines their pay... it is what caps it; long term, anyhow. You can't be paid more than what you are worth, you can't even be paid exactly what you are worth, you in fact inevitably must be paid less than what you are "worth" for the whole arrangement to work. Governments aren't immune to this. They must run at a net profit or they'll bring their society down. Measurement of profit is somewhat different than a private company, but profit they must; they must be extracting more value from their employees than they are paying their employees or the society is running the government at a net loss, which can only be tolerated to a finite degree as determined by what other surpluses the society is running elsewhere. What governments can do that private industry can't is put off the pain until much later before the fact they are paying people more than the value they are actually bringing bites them.
Later's pretty much here, by the way.