Cost cutting culture is contrary to innovation

We see all companies claiming or patting themselves to be innovators. And yet, you see them pruning budgets at all places which motivate employees: be it salary, benefits, facilities or perks.
The cost cutting measures in Indian companies have reached a limit where toilet paper I hear is sometimes in shortfall. I'm sure they ensure this is not so in the restrooms used during client visits!
Then there is the question of bandwidth: innovation as a goal as trickled down. But not the time to innovate! This leads to unplanned, half-hearted measures just to fill up appraisal forms. You can bet that the ones who come up with significant monetization worth innovation ideas are definitely spending 80% of their bandwidth on it.
If companies want to ingrain innovation, the the way to operate it is to have a focused team. Let me put it more clearly, a "swat" team of developers, architects and leads, under an " innovation" unit. This team then has to pick up one production/implementation project at a time (priority to multi year production support), and walk through their use case with the production team, and maybe even collate some potential ideas from them, but that's it. The existing team has the task to support/implement, and can't be doing justice to both the delivery at hand and the out of box innovation aspired. This should be the task of the innovation team.
There will be projects where the cost to innovate will be higher than the benefits. The innovation team can consider if the cost can be recovered in case the innovation idea can be replicated across other other projects. If not, just drop it and move on to the next project.
If any company is serious about moving beyond bits and pieces innovation, this is the model. Will be an investment or at best a break even for some time. Until reusable code base is created, and thereafter it'll be about picking and plugging the right one. The idea is to have Central control on innovation.