You may not need to develop a full-blown business case for computer upgrade.
Post an open-ended question to your boss/supervisor “What are the plans for computer system upgrades?”
Large companies typically plan equipment needs on an annual basis - smaller companies on a case by case basis - but most are budgeted. If your job responsibilities are same as other designers, then parity is a reasonable request. If budget is not available, at least you can get your request in the queue.