  A shell-model version of Kraichnan's (1994 {\it Phys. Rev. Lett. \bf 72},
1016) passive scalar problem is introduced which is inspired from the model of
Jensen, Paladin and Vulpiani (1992 {\it Phys. Rev. A\bf 45}, 7214). As in the
original problem, the prescribed random velocity field is Gaussian,
delta-correlated in time and has a power-law spectrum $\propto k_m^{-\xi}$,
where $k_m$ is the wavenumber. Deterministic differential equations for second
and fourth-order moments are obtained and then solved numerically. The
second-order structure function of the passive scalar has normal scaling, while
the fourth-order structure function has anomalous scaling. For $\xi = 2/3$ the
anomalous scaling exponents $\zeta_p$ are determined for structure functions up
to $p=16$ by Monte Carlo simulations of the random shell model, using a
stochastic differential equation scheme, validated by comparison with the
results obtained for the second and fourth-order structure functions.
