How Does Dostoevsky Provide a Convincing Rebuttal of Chernyshevsky's Proposed Solutions to Russia's Social Problems?

How does Dostoevsky provide a convincing rebuttal of Chernyshevsky’s proposed solutions to Russia’s social problems?

Nikolai Chernyshevsky’s novel What is to be Done? was published in 1863. Through the actions of two of his characters, Dmitry Lopukhov and Alexander Kirsanov, he advocates the philosophy of rational egoism, while acknowledging the natural human conflict between this impulse to uphold the interests of the individual and the competing desire to promote communal prosperity. This is most simply observed in the novel when Lopukhov perceives that his wife is in love with their friend Kirsanov, and that her love is reciprocated. Because of this, he realises that the most expedient course to take is to fake his own death and move to America to begin a new life. This is explained to his wife by the “superior man” Rakhmetov as a result of the “incompatibility of your two natures.” Chernyshevsky’s rational egoism was developed from the writing of the British utilitarian John Stuart Mill. In this way he justified the importance of the individual alongside the improvement of society as a whole. This seems to resemble a mathematical equation, as Chernyshevsky explained human behaviour as a constant subconscious calculation designed to result in the least amount of pain and the most pleasure for the individual. “New men”, as he described Kirsanov and Lopukhov, would be able to recognise that their immediate needs were secondary to those of the wider society, and that this would benefit them individually later.
Fyodor Dostoevsky seems to parody Chernyshevsky himself in Crime and Punishment, published in 1866, with the character of Andrey   Lebezyatnikov, who “really was rather stupid” and who was one of “the obstinate fools who... infallibly attach themselves to the most fashionable current idea”, in this case nihilistic utilitarianism. Lebezyatnikov claims that something that is“useful” is honourable. Dostoevsky pushes his satire to absurdity when the incensed...