Custom post type, taxonomies and url rewriting

After many countless hours of scouring this forum and the web I still haven’t found a solution to my problem. I set up a custom post type called ‘product’ and a taxonomy called ‘product-category’. I am trying to achieve a parent/child url structure like the following:
/term/subterm/post-name. A real world example is: /vectors/design-elements/engraved. The following is the code I have in functions.php.

add_action('init', 'post_type_product');
function post_type_product() {
	register_post_type('product', array(
		'label' => __('Products'),
		'singular_label' => __('Product'),
		'public' => true,<br />
		'show_ui' => true, // UI in admin panel
		'_edit_link' => 'post.php?post=%d',
		'capability_type' => 'post',
		'hierarchical' => false,
		'rewrite' => array("slug" => ""), // Permalinks format
		'supports' => array('title','editor','custom-fields','page-attributes')
	));
}
add_action( 'init', 'create_productcategory_taxonomy');
function create_productcategory_taxonomy() {
    $labels = array(
        'name' => _x( 'Categories', 'taxonomy general name' ),
        'singular_name' =>_x( 'Category', 'taxonomy singular name' ),
        'search_items' =>  __( 'Search Categories' ),
        'popular_items' => __( 'Popular Categories' ),
        'all_items' => __( 'All Categories' ),
        'parent_item' => null,
        'parent_item_colon' => null,
        'edit_item' => __( 'Edit Product Category' ),
        'update_item' => __( 'Update Product Category' ),
        'add_new_item' => __( 'Add New Product Category' ),
        'new_item_name' => __( 'New Product Category' ),
        'separate_items_with_commas' => __( 'Separate categories with commas' ),
        'add_or_remove_items' => __( 'Add or remove product categories' ),
        'choose_from_most_used' => __( 'Choose from the most used categories' )
        );
    register_taxonomy('product-category', array('product'), array (
        'hierarchical' => true,
        'labels' => $labels,
        'show_ui' => true,
        'query_var' => true,
        'rewrite' => array('hierarchical' => true, 'with_front' => false, 'slug' => 'categories'),
       ));
}

Apparently having ‘hierarchical’ => true, and ‘rewrite’ => array(‘hierarchical’ => true) is supposed to allow for hierarchical urls but its not working for me. I would be extremely grateful if someone could help me out.