We can use both constructions to express purpose - to-infinitive and for+verb-ing.
In answer to the question asking "WHY", a to-infinitive is often used. On the other hand, in answer to the question asking "WHAT FOR", we use for+verb-ing or for+noun.
This is not very helpful as both the questions mean more or less the same thing.
From Practical English Usage (PEU) by Michael Swan -
213 FOR: Purpose and Cause
1. People's purposes
For can be used to talk about somebody's purpose in doing something, but only when it's followed by a noun.
He stopped at the pub for a drink.
I went to the college for an interview with Professor Taylor.
For is not used before a verb in this sense. The infinitive alone is used to express a person's purpose.
We stopped at the pub to have a drink. (NOT ...for having a drink.)
I went to the college to see Professor Taylor. (NOT ...for seeing Professor Taylor.)
2. The purposes of things: -ing forms and infinitives
For can be used before the -ing form of a verb to express the 'purpose' of a thing - what it's used for - especially when the thing
is the subject of the clause.
Is that cake for eating or just for looking at?
An altimeter is used for measuring height above sea level.
When the clause has a person as subject, it is more common to use an
infinitive to express the purpose of a thing.
We use altimeter to measure height above sea level.
3. Causes of reactions
For...ing can also be used after a description of a positive or negative reaction, to explain the behaviour that caused it.
We are grateful to you for helping us out.
I'm angry with you for waking me up.
They punished the child for lying.
He was sent to prison for stealing.
Apart from that there are verbs that collocate well with one construction over the other. It depends on individual verbs.
For example - The verb need prefers infinitive construction, and using for+verb-ing is considered incorrect.
I need to get some sleep. (NOT ...for getting some sleep)
But
I need glasses for reading.
I need time to study.
The men need courage to confront the obstacles in their lives.
I need you to help me with cooking. (you are helping me in cooking.)
I need Sam to complete the mission. (NOT ...for completing the mission) [REASON? Refer to point #2 from the quotation from PEU]
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Now coming to the sentences you quoted.
? 1. I have decided to purchase a jet for getting to NYC and back more quickly.
The verb decide takes infinitive construction. So this is correct - I have decided to purchase a jet.
Now after that the sentence adds why I have decided to purchase the jet. There is not preference of infinitive construction or for+verb-ing construction after the verb purchase, so we can use either one of them. Now checking from the PEU quotation, we can find out that infinitive construction is preferred because the purchase is carried out by a human.
I have decided to purchase a jet to get to NYC and back more quickly.
Your second sentence -
? 2. I have decided to buy a jet for having a quicker way to get to NYC and back.
The verb decide takes infinitive construction. So this is correct - I have decided to buy a jet.
Now after that the sentence adds why I have decided to purchase the jet. There is not preference of infinitive construction or for+verb-ing construction after the verb buy, so we can use either one of them. Now checking from the PEU quotation, we can find out that infinitive construction is preferred because the purchase is carried out by a human.
I have decided to buy a jet to have a quicker way to get to NYC and back.
Your third sentence -
? 3. I hate hairdryers, so I am buying a space heater for drying my hair more naturally.
As already mentioned there is no preference from these two constructions after the verb buy, we can use either one of them. But as the quotation from PEU (point #2) mentions for person's purpose infinitive construction is preferred, infinitive construction is preferred here. It's I who is buying the drier for some purpose. It's I's purpose.
I hate hairdryers, so I am buying a space heater to buy my hair more naturally.
Your fourth sentence -
? 4. For having a lasting shine in your hair, buy ShineOn Shampoo.
As already mentioned there is no preference from these two constructions after the verb buy, we can use either one of them. But as the quotation from PEU (point #2) mentions for person's purpose infinitive construction is preferred, infinitive construction is preferred here. It's you who is buying the drier for some purpose. It's you's purpose.
Your fifth sentence -
? 5. My neighbor and I have decided to cooperate more for having a better life together.
After the verb decide infinitive construction is preferred. So it's correct - My neighbour and I have decided to cooperate more.
Now after that the sentence adds the information why they decided to do that. So we have two options open. We can use either one of them. From point #2 (quotation from PEU), we came to know if it's a person's purpose infinitive construction is preferred. Therefore in this sentence infinitive construction is preferred. My neighbour and I cooperated, and we have some purpose.
My neighbor and I have decided to cooperate more to have a better life together.
I never had a problem with this, and seeing all these questions makes me reevaluate my knowledge. – laureapresa – 2015-06-16T08:49:33.940
I can't provide an explanation, but I think except sentence #4 in all the sentences it's acceptable to use both constructions - to-invinitive and for+verb-ing. I am not sure if for+verb-ing construction is incorrect in sentence #4, but I think to-infinitive is preferred there. Please correct me if I am wrong. A very interesting question, looking forward to a good answer. And I am still reading the links you provided :-) – Man_From_India – 2015-06-21T04:53:41.190
But I prefer to use to-infinitive in all these sentences. – Man_From_India – 2015-06-21T06:24:30.357
the explanation from Cambridge (as in the link) is up-to-the mark for this question. What's the problem? – Maulik V – 2015-06-23T06:14:11.993