Several times in my career, I've encountered situations where the business has a technical problem, and I have the previous experience and skills necessary to resolve the issue.
Instead of being trusted to solve the issue, these companies have typically hired (expensive) external consultants to come in and "save us". Their theory being "they're paid to work in that area, so they must be experts"...
In each case, I've already presented my understanding of the problem and my proposed solution. Each and every time, the consultants produce a report that supports my conclusions (but often goes into less detail - as I've had the advantage of examining the original issue over more time).
Then, what usually happens is that I've gained some "business credibility" in subject "A", but if a problem ever occurs with topic "B" it'll be the same uphill battle all over again.
I've also seen attitudes resembling:
"We don't know, so you can't know."
If they've had an issue for months or years, then you join and suggest something they haven't tried, there's almost a willful refusal to consider the possibility.
"The consultants are employed by the vendor/OEM, so they must know."
Even when it's a relatively trivial issue, they'll assume they need an oracle on the subject (thinking along the lines of "we have a Java issue, James Gosling must be the only person able to help us").
Now, I'd hope that I have a reasonably realistic assessment of my own abilities (or lack thereof), so when there's a problem outside of my "domain" I'll just do everything possible to stay out of the way and absorb the knowledge demonstrated by whoever gets tasked with resolving it (basically, I don't pretend to know what I don't know about).
How can I convince my employer that I have the level of competence they require when it's in an area where they lack the expertise to assess competence?
A few people commenting and answering have missed a nuance to my question. The people I'm working with are not incompetent in general (quite the opposite in fact), they simply lack competence for a given skill - usually on a rare topic.
This means I do not patronize, talk down to them or speak to them with any disdain or disrespect (as some answerers have assumed). In fact, when the topic is in an area they have a degree of familiarity with, I find it very easy to get "buy-in" and support for my ideas. The only time I encounter any issues is when they lack the tools to assess my competence in a skill they lack themselves.