But if new machines are becoming cheaper as well, then firms don't have charge as much against their operating income for the replacement of capital--so where's the big profitability problem?
Maybe in his Theory of Economic History the great neoclassical economist John Hicks, also inventor of the IS-LM model for which he won a Nobel Prize, had the problem right. (I am drawing from the appendix from memory, which may be a dangerous thing.)
If labor saving machines embody more labor than the machines they replace, firms may still adopt these new machines due to their beneficient effect on net revenue but if they are not going to eat into their annual surplus, the firms will only be able to replace the old machines with fewer new labor saving machines and thus potentially reduce gross output and employment. Of course this pursuit of net revenue at the expense of gross revenue and employment is the kind of infamous abstraction Marx criticized Ricardo for (and to his scientific credit, Ricardo famously revised his panglossian views towards machinery). But now the problem is less severe because new labor saving machines themselves embody less labor and thus their adoption does not come at the expense of gross revenue and employment; indeed they may embody less labor than the less powerful machines they replace! Marx's theory generalizes from the early industrial revolution at which point systematic, science-based productivity advances in the capital goods industries were just beginning. Or so John Hicks seems to think. Nathan Rosenberg does show that Marx was the first to theorize the centrality of innovation in the capital goods industries but it seems that Marx did not think through its critical implications in terms of the falling profitability theory.
best, rakesh