How do I programmatically change the body text format on the page content type to full HTML? I have a custom install profile and I must programmatically change the body text format on the page content type to full HTML. However, I didn't manage to find how to do it.

How can I do it?
You can do it with hook_element_info_alter, here is a snippet.

&lt;?php
/**
 * Implements hook_element_info_alter().
 *
 * Sets the text format processor to a custom callback function.
 * This code is taken from the Better Formats module.
 */
function MODULENAME_element_info_alter(&amp;$type) {
  if (isset($type['text_format']['#process'])) {
    foreach ($type['text_format']['#process'] as &amp;$callback) {
      if ($callback === 'filter_process_format') {
        $callback = 'MODULENAME_filter_process_format';
      }
    }
  }
}

/**
 * Callback for MODULENAME_element_info_alter().
 */
function MODULENAME_filter_process_format($element) {
  $element = filter_process_format($element);
  // Change the default text format of the 'field_company_spotlight' field to
  // 'Media HTML'. 
  if ($element['#bundle'] == 'company' &amp;&amp; $element['#field_name'] == 'field_company_spotlight') {
    $element['format']['format']['#default_value'] = 'media_html';
  }
  return $element;
}
?&gt;


As THIS post suggests you could try 

$form['field_name'][LANGUAGE_NONE][0]['#format'] = 'full_html';
$form['field_name'][LANGUAGE_NONE][0]['#format'] = 'filtered_html';


in your hook_form_alter or in hook_FORM_ID_alter

Also there is Better Formats module 


  Better formats is a module to add more flexibility to Drupal's core
  input format system.