Abstract: This paper introduces and investigates an extension of the price dynamics in serial monopoly blockchain described in Nisan, tailored to accommodate \emph{quasi-patient} users. Our model reflects users' diminishing interest in having their transactions added to the ledger over time, resulting in only a fraction $\delta$ of the current demand persisting in the subsequent round. The framework presented by Lavi et al., where users are impatient and derive utility only from immediate transaction inclusion in the next block, corresponds to $\delta=0$. Fully patient users who wait forever as in , correspond to $\delta=1$ in our model. This work provides new bounds on the price dynamics for the more interesting case $\delta\in(0,1)$, showing somewhat unexpected effects on the dynamics itself. While the dynamics for the fully patient case is essentially "oblivious'' of the structure of the daily demand curve, this is no longer true for finite $\delta < 1$. Moreover, the dynamics undergoes a "transition phase'' where for some $\delta$ it behaves as in the fully patient setting ($\delta=1$), and for some smaller values $\delta'<\delta$ it stops ``oscillating'' and stays at the highest ("monopolist'') price. We provide quantitative bounds and analytical results that apply to different demand functions showing that the bounds for $\delta=1$ are not tight in general, for $\delta<1$. These provide guarantees on the minimum ("admission'') price such that transaction willing to pay that price are eventually included (and those who do not want are never included).