At first glance, with the little information we have, I dare to speculate an storm in a teapot.
For what it seems this is an small company, aside for the proper solutions already laid on here, what I see is a simple problem of communication.
What's wrong with the CEO asking the developer: How much do you think it will cost?
The developer should have some idea (doesn't have to be precise). If he/she doesn't then the CEO is completely right in not trusting him/her (then you have a bigger problem at hand, and it's not technical). By the same token, the CEO should be able to handle the estimate the developer gives. BTW, a developer shouldn't have free reign, he/she must be able to laid out what are their needs and what it should cost. If not, then again there's a bigger problem here. If the developer doesn't how much will it cost, then the CEO must find a way to know this, at the very least the developer must have an idea of needs for this task.
Given this small company, the CEO can simply put the company credit card and monitor what is being spent. If he/she can't do that, then he/she can't run the company. We are not talking about a Fortune 500 here, this is an start-up, right? So these kind of costs and actions are part of the growing process. Later in the game, maybe the company might think about having an independent credit card for certain employees.