The problem is that the meaning of a good being "under-produced" is subjective in the first place. I like Chinese food, and don't like Indian food. In my opinion, the market is under-producing Chinese food and over-producing Indian. Therefore I think the government should step in and forcibly subsidise Chinese food.
For the Soviets, most luxury and consumer goods were allegedly "over-produced" by the market. For the Nazis, gas chambers for killing Jews were a public good, "under-produced" by the market. The question is, who are you to tell the free market (i.e. free people and the consensual trades they undertake) what it should be producing more of? Your fundamental misunderstanding is that you think there is a point the market should be producing at.