I do not think Test Automation to be really expensive. Because automation simply helps you reduce manual labor and increase productivity.
I will give you a simple example,
Let's say you have a web application with 1k+ test cases. Each test case takes 1 min (or 60 seconds) on an average to be executed manually. So, 1k+ test cases will take 1000 minutes of your time. Now the same application has to be tested on 5 different browsers. Now it will take 5000 minutes or (300000 seconds) for you to complete the whole process.
Obviously, this is very time-consuming. And of course, you can not expect the same result every time as a human tendency to make mistakes.
Automation on another hand will execute same test case for an average of 15-20 seconds. On top of it, you can do parallel execution with the right amount of resources. So your whole execution will take 20000 seconds. So, it will reduce your release life cycle.
Since your app is being tested by the machine, chances of any error are very minimal. This will help you release the quality product with reduced time and resources.
So, the bottom line is, automation saves lots of money and time.