In all likelihood, you can't. Sorry, there it is.
The problem is departmentalization: beancounter 1 is responsible for office arrangement, beancounter 2 for machines, beancounter 3 for programmer's productivity and project.
BC 1 and 2 do not care the least bit about BC 3's interests.
Read good discussion how noise and open space ruin team productivity in Peopleware (deMarco) IIRC and McConnell's books. Both are top of the line experts on sw development processes, not to mention other people like Paul Graham, all oppose noisy environments like open space and then... nothing happens. It all stays the same. The problem is that assessing programmer productivity is loaded with lots of difficulties or risks, while beancounting on $100 on hard drive or office arrangement is easy, so beancounters go with the latter option. Remember: people do not go for profitable options, but for EASY ones!
I'm typing this on laptop with mechanical drive even though this kills productivity given our sw dev environment. Trying to phrase this in economics of entire company falls on deaf ears.
OK scratch the 3 beancounters, roll them all into 1, namely the boss in question. That still does not change the basic problem I described: assessing impact of old machines on programmer productivity is hard, bc assessing productivity of programmers as such is hard. The losses in employee time given the old machines in all likelihood are much bigger than the cost of the new equipment. But evaluating the cost of the former is easy vs evaluating the latter is hard, so your boss does the former.
All you could do I think is simple calculation like:
compilation done n times on average each day * number of employees doing compilation * employee wage per compilation instance time = $X.
If you tried to put this in numerical terms, maybe it will work - but even then I think it won't, as such people typically have vain hope that it will work well enough as it is and that those other losses are not real since evaluating those losses is debatable and hard.
That's your best chance although I'm afraid the boss response will be "don't make programming errors and reduce the number of compilations!".