Setting the Cash Envelope: A Quantile Rule for Weekly Discretionary Caps That Bounds Overspend Without Starving Ordinary Weeks

Households that budget with cash envelopes must pick a weekly cap for discretionary spending, and the cap is usually picked by feel. This paper tests caps against a stated spending distribution — weekly desired spending with mean USD 180 and standard deviation USD 55, simulated over 200,000 weeks — and proposes a simple rule: set the cap at the 90th percentile of desired weekly spending, here USD 250.99. At that cap, 23.6% of weeks still press against the envelope in the nearest tabulated case, but the mean excess in those weeks is only USD 32.31, and the probability that a month's accumulated excess overwhelms a USD 150 buffer is 1.2%. Tighter caps fail more often than intuition suggests: a cap at the mean is exceeded in 50.4% of weeks. The study is simulated, not measured. SIMULATED, NOT MEASURED.