Motivated by a desire to calculate the optimal credit limit for a credit card account in terms of the card-holder’s purchasing behaviour, we consider a single-period inventory model in which the total value of all attempted purchases increases in jumps throughout the period. If a purchase does not cause the limit to be exceeded, then that purchase is approved and the total value of approved purchases is increased by its value. On the other hand, if the limit is exceeded, then the purchase is rejected and the total value of approved purchases remains at its previous level.

We derive an equation for the Laplace–Stieltjes transform of the expected total value of approved purchases at the end of the period. Unfortunately this equation appears to be intractable, a situation that we address by providing upper and lower bounds. We provide numerical examples that show that these upper and lower bounds can be close, which leads to good upper and lower approximations for the optimal limit.