= Solution
Write $m_i=\mu+\alpha_i$. Apart from a constant, the <log-likelihood> is
$$
\ell=-\frac{IJ}{2}\log\sigma^2-\frac{1}{2\sigma^2}\sum_{i=1}^I\sum_{j=1}^J(Y_{ij}-m_i)^2.
$$
For each group,
$$
\sum_j(Y_{ij}-m_i)^2=\sum_j(Y_{ij}-\overline Y_i)^2+J(\overline Y_i-m_i)^2,
\qquad \overline Y_i=J^{-1}\sum_jY_{ij}.
$$
Thus <maximum likelihood estimation> sets $\widehat m_i=\overline Y_i$. The <corner-point constraint> $\alpha_1=0$ identifies $m_1=\mu$, giving
$$
\boxed{\widehat\mu=\overline Y_1,\qquad
\widehat\alpha_i=\overline Y_i-\overline Y_1\quad(i=2,\ldots,I).}
$$
The baseline mean is the first group mean, rather than the grand mean, because of the chosen <identifiability> constraint.
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