He said it was discussed at the committee a month ago whether Elder should appear in order to put make his side of the story public, while giving the committee and the public the chance to ask the hard questions.
"In response to the public interest I felt that it would be good to invite him sooner rather than later.
"I think a simple invitation extended to him is appropriate.
"I would assume that he would want to appear, because he has received a lot of criticism - I don't think it will be an easy appearance for him by any means. I think he deserves the right to put his side of the story," said Young.
Elder is still receiving his $1.3 million salary until April while acting in an advisory capacity from home.
Cosgrove said he believed Elder's exit in April would be punctuated with a payment of $1 million.
Cosgrove has also called for former chairman John Palmer to appear to explain.
In the last financial year Solid Energy lost $40 million and subsequently axed 450 jobs nationwide plus around 240 jobs at Westport's Stockton mine.
Speaking from Chile last week Prime Minister John Key said he had no problem with Elder appearing at the committee to answer questions about the problems at Solid Energy.