In Hedda Gabler, towards the end of Act Four, when Brack proposes to Hedda that "You could always say he'd stolen it.", why does Hedda respond "I'd rather die!"
You are quite correct in your conclusion that the character of Hedda is far from moral. Moreover, she is also quite careless about people, as she lacks the depth of character that would define an otherwise wiser and less selfish woman.
Everything Hedda does is for the sole benefit of Hedda. Therefore, in Act Fourth, when she said to Judge Brack that she'd rather die than saying that Loevborg had stolen the gun is a statement that she really means: Hedda would much rather, literally, die than to be an accomplice in a case that could bring her disgrace.
Most importantly, she would rather die rather than to have to depend on Judge Brack’s knowledge of the matter. Would he ever tell? Will he ever go to the police and declare against her? These are questions that would gnaw at Hedda’s conscience and will make her forever fear for her life.
[...] I saw the pistol found in Lovborg's pocket, and I knew it at once as the one I had seen yesterday--and before, too.
What will the police do with it?
Search till they find the owner.
Do you think they will succeed?
[Bends over her and whispers.] No, Hedda Gabler--not so long as I say nothing.
This is one of the few times when we see Hedda being put in a desperate situation. She understands the tone with which Brack is speaking, and he makes the situation worse later in the conversation. This is the final catalyst to the end of Hedda
[Looks up at him.] So I am in your power, Judge Brack. You have me at your beck and call, from this time forward.
[Whispers softly.] Dearest Hedda--believe me--I shall not abuse my advantage.
I am in your power none the less. Subject to your will and your demands. A slave, a slave then! [Rises impetuously.] No, I cannot endure the thought of that! Never!
[Looks half-mockingly at her.] People generally get used to the inevitable.
[Returns his look.] Yes, perhaps.
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