"Group" animation events in jQuery
The way I see it, you have 2 options; either use Deferreds, or store your delay in a variable:

var delay = 0,
    $left_box = $(".balance_left_box");

$("#bgd_balance").animate({
    backgroundPositionY: "0px",
    backgroundPositionX: "0px",
    'background-size':'100%'
}, 800, "swing");

$left_box.delay(delay += 2000).slideDown(200, "easeInCirc");

$left_box.find("p.first-line").delay(delay += 2400).slideDown(600, "easeInCirc");

$left_box.delay(delay += 1000).animate({
    height:"270px",
    top:"64px"  
}, 100, "easeInCirc");