Install composer and drush globally

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 | 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

Some troubleshooting

Add this to your path file, probably with ~/.bash_profile

export PATH="$HOME/.composer/vendor/bin:$PATH"

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.