Former Mayor Jane Byrne, spurned in her last three election bids, has signed with a New York agent and says she will write her autobiography. ``There are a lot of people who would pay me a million dollars not to write the book,'' Mrs. Byrne said Wednesday. ``I might as well write the book.'' Gerard McCauley, a former senior editor for Little, Brown & Co. and Alfred A. Knopf, has been hired to represent her. There is ``substantial interest'' on the part of New York publishers in a book by the former mayor, McCauley said. In it, Mrs. Byrne, 54, said she will discuss her tenure as mayor from 1979 to 1983 as well as her early life, her membership in former Mayor Richard Daley's cabinet, her association with the Kennedy family and her roles in local, state and national Democratic politics. Mrs. Byrne lost a primary race to former Mayor Harold Washington in her bid for re-election in 1983. She lost the Democratic mayoral primary again last year. Earlier this year, she lost a bid for the office of clerk of the circuit courts for Cook County.