= Solution
With $\alpha_1=0$, the marginal moment model gives
$$
\frac{\mathbb E(Y_{ij}\mid x_i)}{\mathbb E(Y_{i1}\mid x_i)}=e^{\alpha_j}.
$$
Thus \b[$e^{\alpha_2}$ and $e^{\alpha_3}$ are ratios of expected visits in Lent and Easter to expected visits in Michaelmas, respectively], at the same baseline covariates. These are population mean ratios, including students with no propensity to attend; they are not <odds ratios> or probabilities of any attendance.
In the hierarchical model, for a fixed positive student effect $b_i$, the conditional mean ratio is also $e^{\alpha_j}$. It describes a multiplicative within-student change in expected visits holding the student effect fixed. For the <structural-zero component> all three means are zero, so no nonzero ratio is defined there. Since the distribution of $b_i$ and the mixture weight do not vary by term, integrating out $b_i$ preserves this same marginal mean ratio. This is <mean-ratio preservation under a multiplicative random effect>: the conditional and marginal term coefficients coincide for this multiplicative log-link model. They need not coincide in a general nonlinear mixed model, such as a logistic random-effect model. A log-mean difference $\alpha_j$ is not a difference of expected logarithms of counts, which can be undefined at zero.
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