  We consider the Nonlinear-Cost Random Walk model in discrete time introduced
in [Phys. Rev. Lett. 130, 237102 (2023)], where a fee is charged for each jump
of the walker. The nonlinear cost function is such that slow/short jumps incur
a flat fee, while for fast/long jumps the cost is proportional to the distance
covered. In this paper we compute analytically the average and variance of the
distance covered in $n$ steps when the total budget $C$ is fixed, as well as
the statistics of the number of long/short jumps in a trajectory of length $n$,
for the exponential jump distribution. These observables exhibit a very rich
and non-monotonic scaling behavior as a function of the variable $C/n$, which
is traced back to the makeup of a typical trajectory in terms of long/short
jumps, and the resulting "entropy" thereof. As a byproduct, we compute the
asymptotic behavior of ratios of Kummer hypergeometric functions when both the
first and last arguments are large. All our analytical results are corroborated
by numerical simulations.
