We know how expensive we are

Question

We know how expensive we are.

I cannot for the life of me decide if this is supposed to be interpreted as a complement clause or an embedded question or what.

My thought process so far is that it couldn't be a reported question (for semantic reasons) or fused relative ("how" here doesn't work as a relative pronoun) at all. Sources I've checked so far have been ambiguous about how small or vast a category "complement clause" can encompass, for example, with some including reported questions in the category of verbal complements (Mark Newson et al., 2006) and some not doing so.

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Edit: I'll add the context in which the sentence originally appeared, after talking with my student. I believe this doesn't fundamentally change anything, but perhaps pushes it towards an interrogative interpretation:

You may complain about our high prices, but unlike a certain competitor, we have informed you that the increase would be 15%, so at least we know how expensive we are.

Answer 1

Short answer (tl:dr):

In the Original Poster's example, the subordinate finite clause how expensive we are is syntactically ambiguous and could be either interrogative or exclamative.

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Full answer:

1. We know how expensive we are.

The fact that the analysis of this sentence is vexing the Original Poster is not surprising because it is syntactically and semantically ambiguous. And furthermore, as they note, the term complement clause seems to be widely used but poorly defined.

Let's do away with this pernicious term here.

Setting aside relative clauses and comparative clauses, both of which have their own peculiar structures, we can analyse finite subordinate clauses as belonging to one of four syntactic clause types:

• declarative
• closed interrogative
• open interrogative
• exclamative

These are the subordinate equivalents of the five main clause types, apart from that there is no subordinate form for imperatives. We can, of course, use different constructions to convey a similar sort of meaning to imperatives:

• She demanded that we leave
• It was imperative we should leave

Note that despite denoting commands or indicating a compulsion to act, the examples above use straightforward declarative subordinate clauses in terms of their syntax. And syntax is what we're looking at today.

These four types of finite subordinate clause can partake in all kinds of grammatical relations. For example, they can be the Subjects of main clauses. However, as the term subordinate may convey, they most often appear as the Complements of other words or phrases:

2. We know [that] you stole the cookies. ^declarative
3. We know whether you stole the cookies. ^{closed interrogative}
4. We know which cookies you stole. ^{open interrogative}
5. We know what a cookie thief you are. ^exclamative

In (2-5) above, you can see these four clause types all occurring as the Complement of the verb know. In this particular case we are interested in subordinate open interrogatives like (4) and subordinate exclamative as in (5).

Both subordinate interrogatives and subordinate exclamatives share some features. They both, for example, have a wh-phrase which must occur in clause initial position. And if that wh-phrase is not the subject of the clause, or part of the subject of the clause, they have a gap inside the nucleus of the clause which we identify with that wh-phrase. We could represent them like this:

6. We know which cookies₁ [you stole ___₁]
7. We know what a cookie thief₁ [you are ___₁]

The small subscript '1' there shows that the wh-phrase and the gap refer to the same things. We could alternatively represent the sentences like this:

8. We know which cookies [you stole which cookies]

9. We know what a cookie thief [you are such a cookie thief]

The wh-words that occur in subordinate interrogatives are interrogative words, and they can represent all types of word categories and correspond to phrases carrying out many different types of syntactic functions. They have many forms: who, which, what, where, when, why, how and so forth. In contrast, the wh-words in subordinate exclamatives are not interrogative words, and are limited to the words how and what. And far from having a wide range of meanings, both of these words indicate degree or extent.

These two exclamative wh-words are homophonous with the interrogative words what and how. Luckily in the case of what, the meaning is clearly different. Exclamative what indicates the degree or extent of something whilst interrogative what substitutes for a determiner or a noun phrase and is the unconstrained version of which:

• What books did you want?
• Which books did you want? (limited selection available)

Also the grammar can help us disambiguate in the case of what. Exclamative what modifies a whole noun phrase, but when interrogative what occurs within an NP, it functions as Determiner. For this reason, if the head of the noun phrase is a singular, countable noun, we will see an article or other determiner between an exclamative what and the noun. However, in the case of the interrogative, the word what is the Determiner:

10. I know what tyrant you deposed. (interrogative)
11. I know what a tyrant you deposed. (exclamative)

Like exclamative what, exclamative how is homophonous with interrogative how. However, interrogative and exclamative how are similar in meaning and also carry out a similar range of syntactic functions. This means that interrogative and exclamative how are sometimes difficult or impossible to disambiguate without any special intonation. Thus the following sentence is ambiguous:

12. My wife noted how much I had spent.

In the sentence above, the clause in bold could be either interrogative or exclamative. In the first case the sentence would mean:

My wife made a mental note of the answer to the question 'How much did my husband spend?'

And in this case the answer might be one euro, or a million euros. In the exclamative case, however, the meaning would be:

My wife made a mental note: 'How much my husband has spent!'

And in this case we don't know whether this is a good or a bad thing, but we do know that the amount of money was quite a lot and that it was surprising or noteworthy.

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The Original Poster's question

• We know how expensive we are.

The sentence above is syntactically ambiguous. The finite subordinate clause functioning as the Complement of the verb know could be either an open interrogative clause or an exclamative clause. The interrogative version means something like:

13. We know the answer to the question "How expensive are we?"

The exclamative version means something like:

14. We appreciate to what a considerable degree we are expensive.

Answer 2

The Cambridge Grammar of the English language p978 has the following in a section titled 'Survey of constructions containing subordinate interrogatives'.

(d) Internal complement licensed by the matrix verb

[23]

i We’ll [establish what caused the malfunction].

ii We [investigated whether the contract is valid].

[24]

i ascertain care check consider decide determine disclose discover establish estimate forget guess indicate inform judge know lean mind notice observe predict prove realise recall remember say see show tell think

ii ask concern2 inquire investigate ponder

Concern2 means “be about”: The debate concerned how best to contain inflation.

The verbs in [i] license declaratives as well as interrogatives, while those in [ii] do not: compare We’ll establish that the contract is valid and ∗We investigated that the contract is valid. The verbs that exclude declaratives include the verbs of asking (interpreted broadly enough to include wonder), but they are not limited to these.

How can head a fused relative as in

That's not [how I wanted it done]. [the way...]

but here we could not insert the way..., so it's definitely an interrogative if the choice is between a fused relative or an interrogative. However, as pointed out by Araucaria, it could also be interpreted as an excalmative - the heads allowing these being generally the same as those allowing interrogatives, and the form of the subordinate clause also being the same for the clause in question.