Edit Google calendar events from Google service account: 403

I want to create new event at google calendar from Service Account. I can access correctly and print a list all my calendars. But when I want to create a new event the response is 403 Forbidden.

My code:

require_once 'src/Google_Client.php';
require_once 'src/contrib/Google_CalendarService.php';

session_start();

const CLIENT_ID = 'XXXXXXXXXX.apps.googleusercontent.com';
const SERVICE_ACCOUNT_NAME = 'XXXXXXXXXXXX@developer.gserviceaccount.com';

const KEY_FILE = 'google_src/XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX-privatekey.p12';

$client = new Google_Client();
$client->setApplicationName("APP_NAME");

if (isset($_SESSION['token'])) {
 $client->setAccessToken($_SESSION['token']);
}

$key = file_get_contents(KEY_FILE);
$client->setClientId(CLIENT_ID);
$client->setAssertionCredentials(new Google_AssertionCredentials(
  SERVICE_ACCOUNT_NAME,
  'https://www.google.com/calendar/feeds/MY_CALENDAR_ID/private/full/',
  $key)
);

$client->setClientId(CLIENT_ID);
$cal = new Google_CalendarService($client);
$event = new Google_Event();
$event->setSummary('Bon dia pel matí!');
$event->setLocation('Somewhere');
$start = new Google_EventDateTime();
$start->setDateTime('2012-08-06T10:00:00.000-07:00');
$event->setStart($start);
$end = new Google_EventDateTime();
$end->setDateTime('2012-09-06T10:25:00.000-07:00');
$event->setEnd($end);

$createdEvent = $cal->events->insert(CALENDAR_NAME, $event);

And server response:

(403) Forbidden

Any ideas? Maybe other scope, or allow edit events (I dont know where I can configure this and I have searched it, promise)

Of course, my calendar is a public calendar.

Any help will be welcome :)

Thanks!