BACKGROUND
In this question, it was asked why it sounds better to omit 'with' in
I have no money to buy a bed (with).
whereas 'with' sounds right in
I have no ball to play soccer with.
Indeed, the 'with' feels redundant at best in
I have no money to buy a bed with.
On the other hand, leaving out 'with' here would make it ungrammatical
I have no ball to play soccer. (??)
This is not limited to 'soccer'
I have no ball to play. (??)
Instead, it should be:
I have no ball to play with.
As one of the answers in the quoted question says, it may be okay to use 'with which' as in:
I have no money with which to buy a bed.
Given that the version without 'with which' sounds right, it might be argued that this version with 'with which' may be redundant and thus as unidiomatic as I have no money to buy a bed with. But native speakers seem to find the former grammatical.
Now, moving on to the finite relative clause, compare these:
(1) I have no money I can buy a bed. (??)
(2) I have no money I can buy a bed with.
(3) I have no money with which I can buy a bed.
Here, it seems clear to me that 'with' cannot be omitted, unlike the infinitive relative clause.
QUESTION
Therefore, I'd say that the idiomatic I have no money to buy a bed resisting with at the end is more of an exception in the sense that corresponding finite relative clauses do require 'with' either at the beginning or at the end of the clause, and that the corresponding infinitive clause with 'which' requires 'with'.
What do you think triggers this exception?
EDIT
It may be worthwhile to note that the version with "with" is not ungrammatical:
I have no money to buy a bed with.
Some native speakers might find this version more correct -- if not more common -- than the one lacking "with".
