Once an app purchased from App Store has been installed, the buy button of the app on both the product page and Purchases page become grey and turn into [ INSTALLED ]
In theory, unless you remove the app from your machine, you won’t be able to re-download the app. However, the steps below will allow you to download an installed app again without prior to removing it from your computer:
1) Log in to App Store with your Apple ID
2) Go to the Purchases page
3) Press and hold the [ option ] key while clicking the app your want to re-download
4) You will be taken to the product page where the buy button changes to [ INSTALL ]
Now you can re-download the app.