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Create an on-budget account called Cash in YNAB. When you take money out of the bank transfer the money to cash. When you spend it record the payment out of the cash account.
I give each of us a Petty Cash account. Since I am quite anal about tracking expenses, I pretty much track most of my cash to the penny. But, I don't even try to track my husband's cash. I give him a certain amount each month and as far as I'm concerned, it's gone. So, when I update his balance from time to time, I just categorize the difference as going to "Spouse Petty Cash" category. So, I have a Petty Cash account for each of us and a Petty Cash category for each of us. It's basically just like Dave Ramsey's "Mad Money" category. I really wouldn't need to have an actual account for my husband's, but he likes me to occasionally update what he has as he tends to stash his $ in various places and forget about it. So, once in a while we "round it all up", count it, and update the balance! His way of saving - and, he does save every penny of his petty cash he can -- he puts it towards vacation each year! It encourages him to spend less when eating lunch out, etc.