= Solution
The <complete market> makes every state-contingent <consumption> vector attainable. Its budget is $\sum_sp_sc_s=100$, and expected <logarithmic utility> under the equally likely states is $\tfrac14\sum_s\log c_s$, with $c_s>0$. Increasing <utility> makes the budget bind. For a <Lagrange multiplier> $\eta$, the <first-order conditions> are
$$
\frac1{4c_s}=\eta p_s,\qquad c_s=\frac1{4\eta p_s}.
$$
Substitution in the budget gives $1/\eta=100$, so $\eta=0.01$ and
$$
\boxed{c=(250,\ 125,\ \tfrac{250}{3},\ \tfrac{250}{3})^t.}
$$
Every state receives an expenditure $p_sc_s=25$, rather than equal <consumption>. <Strict concavity> of <logarithmic utility> makes this feasible stationary point the unique global maximizer. The physical probabilities enter the <utility> objective; the supporting prices enter the budget, and they must not be interchanged.
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