Such metrics exist. I've seen the evidence in the past with regards to software for automotive control (running the engine). Basically if you have to recall 300000 cars to be fixed at your cost (even just an ECU software upgrade) you have at the very least say 300000 hours of technician time and all the rest. Plus possibly if it was bad a few lawsuits to burn cash defending or paying out on.
I think full life cycle metrics in automotive put bug fixing once a vehicle was out the door at something like 50 - 200 times the cost of finding it early on during design reviews, code reviews and ultimately testing of unit, module and system as different stages.
But in that instance you are talking scale of units in the millions ultimately with many years of life so many millions of operational years in which bugs can bite! Plus the goods are high value. If you are talking low value goods and a few thousand copies well loose it under GPL with suitable disclaimers of suitability of anything is denied:-)
You get the picture, you might not be able to prove what you want because it is a time money equation with varied parameters.
In short you need the metrics for what you are focussed on and gathering those metrics may cost more than it's worth. The collection cost itself is a similar time money equation to consider in advance. Reality for contractor style companies is they don't want to bother testing because they get to fix the things they break in adding features for you or fixing other bugs. Testing is a path to being redundant.