Why didn't Mathilde admit what happened with the lost necklace in "The Necklace"?

Mathilde does not tell Madame Forestier that she lost the necklace because she is embarrassed and proud.

Mathilde Loisel believes that she was born below her proper station in life.  Because of this, she pines for a life of riches and beautiful things.  Her husband is aware of this, but he is only a lowly clerk.  One day he surprises her with the gift of tickets to a fancy ball, and she gets upset because she doesn’t think she has anything to wear.

Monsieur Loisel gives his wife everything he has saved, and suggests that she borrow a proper jewel from her wealthy friend, Madame Forestier.  She agrees, excited to have a pretty dress and an expensive jewel to wear to the fancy ball.

Madame Forestier makes some suggestions, but Mathilde sees the one she likes.

Suddenly she discovered, in a black satin case, a superb diamond necklace; her heart began to beat covetously. Her hands trembled as she lifted it. She fastened it round her neck, upon her high dress, and remained in ecstasy at sight of herself.

What Mathilde thinks is a very expensive diamond is actually fake.  She has no idea. She is on top of the world at the ball.  Everyone wants to dance with her and thinks she is beautiful.  Unfortunately, as she is leaving she realizes she has lost the jewel.

Monsieur Loisel does not want to tell Madame Forestier what happened at first because he thinks he can still find the necklace. He probably feels partially responsible, since borrowing it was his idea.

"You must write to your friend," he said, "and tell her that you've broken the clasp of her necklace and are getting it mended. That will give us time to look about us."

When they can’t find it, he finally decides they have to replace it.  It takes some time to find one that looks like it, and they go into great debt to do it.  For years they have to pay off the debt, until one day Madame Forestier sees Mathilde and does not recognize her.  That is when they learn that the necklace was a paste.  It was fake.

If Monsieur and Madame Loisel had told the truth, they would have saved themselves great distress and suffering.  Unfortunately, they were too proud and never owned up to what happened.  Because of that, it was years until they learned of their mistake and by then they had destroyed themselves.