Productivity and wages aren’t intrinsically linked.
Say you’ve got someone digging a trench with a manual shovel, and then the Bobcat is invented. Let’s say that the Bobcat lets someone do five times as much digging. The wage paid isn’t going to be five times the shoveler.
The wage will be set by whatever it takes to get ahold of someone who can operate the Bobcat. That’ll depend on how many people are out there who can operate a Bobcat, and what else they might be doing.
The only guarantee is that it won’t be more than five times the manual shoveler, because then (setting aside, for a moment, the non-labor costs) digging the thing with the Bobcat would be less-efficient than having it manually-shoveled.
In fact, productivity and wages can be inversely-correlated.
Let’s say that instead of a Bobcat operator and a manual shoveler, where the skillset is different and the pool of people who can do each may differ, you have some technological improvement that doesn’t change the pool of labor at all. Let’s say that someone suddenly realizes that the Bobcat shovel could be twice as large and it can scoop twice as much. Ignoring, for simplicity, things like setup time, suddenly every Bobcat operator is twice as productive.
Now, usually there’s some level of price elasticity of demand. If you can make something more-cheaply, then more people will buy it – some people wanted a trench but it just didn’t make financial sense, but now suddenly it does. But let’s assume that demand is entirely inelastic. There is still the same amount of demand then, even if the trench can be dug more-cheaply, and the same amount of trench will be dug.
In that case, one only needs half the number of Bobcat operators. The market allocates workers based on their wage – pay more, more people will be willing to do a job, pay less, and fewer will. What will happen is that Bobcat operator wages will drop until about the required number of Bobcat operators are willing to do the work. Those who were already on the edge will exit the field, do something else.
Wages can also change when productivity doesn’t. North Dakota had an oil boom about twenty years back. There weren’t nearly enough people to work the fields. Wages skyrocketed, and people entered the field or moved to the area. They weren’t more-productive than the previous workers. They were paid more because the supply was limited; paying more resulted in the needed number of workers showing up.
Wages can closely track productivity in some situations. Suppose you have zero price elasticity of supply – that is, no new workers are able/willing to enter a field, no matter what wage is being offered. And there is infinite price elasticity of demand – in practice, immense demand for the thing at the particular price, but not above that. An example – maybe a bit contrived – would be if a number of people with identical cars all locked their keys in their uninsured cars prior to a flood, and a lone locksmith is available. They can break a window to get their keys and rescue their car, or have the locksmith open the car. Anyone who can will pay the locksmith to open the car for up to the cost of replacing the window, but not more than that. There is no time for more locksmiths to show up – supply is inelastic. In that case, if the locksmith could manage to open a car in half the time, he’s be paid twice as much.
But normally, wage serves the role in a market of allocating workers to a given field. It isn’t directly bound to productivity. And you wouldn’t want it to do that, because that’d kill its use to do that labor allocation, which is how the market moves workers where they’re needed. Let’s set aside practical difficulties and imagine that we could pass a law to lock productivity and wage. Suppose it resulted in a lower wage than market rate – as it would with the North Dakota oil workers above – then you wouldn’t have enough workers, and oil that should be extracted would go unextracted. Suppose it resulted in a higher wage than market rate, as it would with the Bobcat operator above. Then you’d have a line of capable-of-using-a-Bobcat people, all of whom want the Bobcat operator’s job. In practice, because the wage is locked, non-wage compensation probably changes – that is, the conditions of the job get worse. The Bobcat operator has to be on-site the instant the job starts, any mistake on his part and he’s immediately replaced, etc. And you have the crowd of people trying to get his job probably trying to offer bribes and the like to get him ejected and themselves put in place.