(Modified comparative advantage model)Can trade with a country that only has one industry be mutually beneficial?
I did not encounter this kind of example before but I think comparative advantage would still apply. In Ricardian model opportunity cost can be expressed as a ratio of outputs $a$ for food $f$ and bronze mirrors $b$ as $a_f/a_b$. If Japan can create A units of food but 0 bronze mirrors the opportunity cost would be given by $A/0$ now this is undefined in mathematics, because limit of the expression would not give you the same value from both sides, but since we are working only with positive quantities (presumably output cant be negative) we could evaluate the limit only from positive side which will give you $\infty$ meaning that no matter how much you sacrifice food you will get no mirrors.

The rest of the problem could be solved in this way, the equilibrium price that should be $a_{f}^{'}/ a_b^{'} &lt; P_f /P_b &lt; a_f/a_b$  would just have the upper bound equal to $\infty$. The exact quantities demanded would be given by the assumptions on individual demands in countries etc. Even in case this would not be mathematically rigorous I think you could carry out virtually the same analysis with just assuming some small infinitesimal production possibility for Japan and examining how the solution behaves as it goes to 0.