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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.

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.

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    I don't understand. What specifically is the text element that's confusing you? And what exactly is the question you're asking about it?
    – FumbleFingers
    Commented Dec 28, 2023 at 11:35
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    "How expensive we are" is not a question - that would be "How expensive are we?"
    – Kate Bunting
    Commented Dec 28, 2023 at 12:02
  • Wouldn't that quote you added be "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 YOU know how expensive we are." Also, it is unclear to me what question you are asking. "We are aware of our high cost." seems like a statement not a question.
    – Ellie K
    Commented Dec 28, 2023 at 12:20
  • Here, "how" defines a degree of cost. Assuming that how defines a question is like thinking the word question forms a question. I don't know how, why, or when they would necessarily form a question.
    – Yosef Baskin
    Commented Dec 28, 2023 at 14:53
  • Wikipedia would call this an interrogative content clause; it gives the example 'I can't guess how he managed it.' [ICC italicised] It also says that another term used here is 'indirect question': expanding, the direct question appears: 'we know the answer to the question 'How expensive are you/we?' // This has been covered many times before on ELU.
    – Edwin Ashworth
    Commented Dec 28, 2023 at 15:10

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So let's consider this simpler example:

I know how expensive it is.

This sentence is ambiguous. First, it could mean "I know that it is remarkably expensive." On that reading, it's an exclamative content clause; in other words, it's the subordinate clause version of "How expensive they are!" Alternatively, it could mean "I know how much it costs"; in that case, it's an (open) interrogative content clause--in your terminology, an "embedded question"; in other words, it's the subordinate clause version of "How expensive are they?" Similarly ambiguous sentences include "She forgot how old they were" and "I know what games they play" (Huddleston & Pullum (2002), pp. 991-992).

You mention that you think it can't be an embedded question (in your terminology) for semantic reasons. But what's relevant here is not its meaning but its syntactic structure. You can tell that it can be an interrogative clause since you can put any interrogative clause there. For example:

  1. I know who committed the crime.
  2. I know which of them committed the crime.
  3. I know whose car they used.
  4. I know why/when/where/how they committed the crime.

In this sense, the verb know means (essentially) that you know the answer to the question that the subordinate clause asks.

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