I can get the same for cheaper somewhere else.
Heard this before? Ever heard yourself say it?
It's impossible. You can never get exactly the same solution from someone else for less. There's always aspects of a solution that are unique. If someone says this to you, then they probably have a relationship with you which stems back to when they first had an idea or desire to look at possible solutions. The fact that they say this to you is out of respect - because they will feel some tinge of guilt if they buy the products you proposed off someone else. The better you've done your job, the greater their emotional tie to you, and the bigger the guilt.
As a bare minimum they are giving you the opportunity to retender your offer - maybe restructured to appear better value - you know the deal, get 5% off if you pay a 25% deposit. Stuff like that.

Let's not miss the point though – it's very rare that 2 solutions are exactly the same.
I often buy the cheaper and less functional item where for me the benefits don't justify the expense. Paying less for a lesser product is different from paying cheaper for the same. However, if it comes down to two companies offering the same product it's rare that I'll buy from the supplier who has invested the least in me. Out of respect for how one supplier helped me I'll help them and award them the sale. It just seems fair. The older I get the greater value I place on good advice.
You simply can't get the same for less. It's impossible.