> Yes they are. Imagine World War II. The best person to dig a particular ditch might be Alan Turing. But that's completely different from the best job for Alan Turing being digging that ditch.
No, Alan Turing would be pretty bad at digging ditches, and there would be many people who could do the job just as well or not better than he.
So in that environment, the code breaker team needs someone, and the ditch digging team needs someone, and if it turns out that Alan is in the ditch and Joe doing an awful job breaking codes, then the code breaking team will try to fire Joe and hire Alan, and the ditch digging team wont mind switching Joe for Alan. Thus this swap is a pareto improving trade, and there are no pareto improving trades in equilibrium, therefore no equilibrium process would settle on Alan in the ditch and Joe breaking codes.
So look, this is an actual body of science and math that you can't just dismiss without understanding and thinking about. There are legitimate criticisms of welfare theorems -- the ones I outlined, namely around lack of perfect information -- does the code breaking team know how good Alan is at breaking codes? Does Alan know there is a job opening other than ditch digging? Etc. It is all about information, but to just pretend that the welfare theorems don't exist because you haven't thought of them or they don't seem intuitively "true" to you is neither a good avenue of exploration nor of debate. These are, after all, theorems. You attack theorems by attacking their hypotheses, not by "disagreeing" with them or just asserting that they are false.
> I think that optimum and equilibrium are different things and a single price cannot be optimum even if it is the equilibrium.
This is, mathematically, false. But it's also irrelevant because there is not one price, there is a single price vector. Alan and Joe don't earn the same wage. The budget of the ditch project and code breaking project are also not the same.
>No, Alan Turing would be pretty bad at digging ditches
It's a contrived example that illustrates something general. It could still be true - maybe Turing would have some insight that let him do half as much labor. But whether it's actually true for him is irrelevant.
Let's say the manager of a McDonald's started out making hamburgers and does it better than any of the other employees, but doesn't normally do it any more, unless there's a crisis and they have to fill in when nobody else is available. Or some software engineer at Google would be better at processing data for litigation purposes than anyone in that industry, yet they outsource the work.
No, Alan Turing would be pretty bad at digging ditches, and there would be many people who could do the job just as well or not better than he.
So in that environment, the code breaker team needs someone, and the ditch digging team needs someone, and if it turns out that Alan is in the ditch and Joe doing an awful job breaking codes, then the code breaking team will try to fire Joe and hire Alan, and the ditch digging team wont mind switching Joe for Alan. Thus this swap is a pareto improving trade, and there are no pareto improving trades in equilibrium, therefore no equilibrium process would settle on Alan in the ditch and Joe breaking codes.
So look, this is an actual body of science and math that you can't just dismiss without understanding and thinking about. There are legitimate criticisms of welfare theorems -- the ones I outlined, namely around lack of perfect information -- does the code breaking team know how good Alan is at breaking codes? Does Alan know there is a job opening other than ditch digging? Etc. It is all about information, but to just pretend that the welfare theorems don't exist because you haven't thought of them or they don't seem intuitively "true" to you is neither a good avenue of exploration nor of debate. These are, after all, theorems. You attack theorems by attacking their hypotheses, not by "disagreeing" with them or just asserting that they are false.
> I think that optimum and equilibrium are different things and a single price cannot be optimum even if it is the equilibrium.
This is, mathematically, false. But it's also irrelevant because there is not one price, there is a single price vector. Alan and Joe don't earn the same wage. The budget of the ditch project and code breaking project are also not the same.