You may get this message trying to update drush globally via composer,
"To initialize a project, please create a composer.json file as described in the...."
so lets make it sure both are globally installed and required.
Download phar file with
curl -sS https://getcomposer.org/installer | php
then move it to /usr/local
mv composer.phar /usr/local/bin/composer
Then also add drush as required
composer global require drush/drush
and in future when you run
composer global update
it will auto update drush or if you want to manually update drush run.
composer global update drush/drush
Add this to your path file, probably with ~/.bash_profile
as this is where composer has put your drush executable, it is possible that you have it on some other place like "/usr/local/bin" remove it from there so you don't have multiple drushes (which is also probably in path and will run globally) and use the one that is not updated.