Why isn't a definite article used before "a labyrinth"?

Solution 1:

The reason that a is not used with a labyrinth here is because although the phrase has the form of X of Y it is not a partitive genitive construction. Not every X of Y construction is a genitive in the sense that Y owns X. To illustrate ...

• the dinner of the dog

... equates to

• the dog's dinner

However, the box of wood does not equate to:

• the wood's box.

In the example given, the labyrinth doesn't really belong to the subsidies and tax credits. It's more a case of the labyrinth being made of the subsidies and tax credits. So it is not a phrasal genitive.

Having said this, the Original Poster is absolutely correct in thinking that the Prepositional Phrase of subsidies and tax credits does not narrow down which labirynth we are talking about - and that this is why the author uses a instead of the. It is not working like a restrictive relative clause. It is merely attributing some characteristics to the labyrinth. If the description did explain which labyrinth we were talking about, then the author would have used the - but it doesn't.

It should be noted here that a phrasal genitive will not always show us exactly which person or thing is being talked about. For example, if we are talking about a long period in French history, then a particular king of France is just one of many kings. In this situation we could still talk about a king of France because of France is not enough to tell us which king of France we mean.

Solution 2:

The decline is the result of ... a labyrinth of [things].

vs

The decline is the result of ... the labyrinth of [things].

Here, a means something that is proposed to exist that is labyrinthine in structure or concept.

the would mean something that is arguably well-known to exist, and the only or major example.

Solution 3:

"a labyrinth" here means a sort of labyrinth, it's not a known, not a definite labirinth, therefore they used "a" in the sentence.