var date = new Date();
var firstDay = new Date(date.getFullYear(), date.getMonth(), 1);
var lastDay = new Date(date.getFullYear(), date.getMonth() + 1, 0);

or you might prefer:

var date = new Date(), y = date.getFullYear(), m = date.getMonth();
var firstDay = new Date(y, m, 1);
var lastDay = new Date(y, m + 1, 0);