The fallacy here is that by neglecting quality and incurring debt, they'll eventually be increasing both the time and cost requirements many times over.
Therefore, it is the duty of the developer to inform the product manager and business owner of the unpredictable effects of incurring technical debt to give them all of the advice they need to make an informed decision. You may want to turn the tables and approach it from a positive perspective—cleaning up technical debt would allow future development of new features to be completed more quickly.
Do this to prevent the worst-case scenario where the effort required to fix the code is greater than rewriting everything from scratch.
