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What you have there are the preferences under an arbitrary policy -- what some call the prevalue function. The only thing missing is the max operator. Written with maximization (and making the state and choice explicit), the Bellman equation is \begin{align} U(K_t, \epsilon_t) =& \max_{C_t} \{ (1-\beta) C_t^{1 - \frac{1}{\eta}} + \beta [E_t(U(K_{t+1}, \...


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