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As mentioned in the comments this comes down to stylistic choices, since as you correctly pointed out: $$u(c_t)+\beta E_t[u(c_{t+1})]=E_t[u(c_t)+\beta u(c_{t+1})]$$ However, in principle both expressions are correct. The first expression states that the $U_t(c_t, c_{t+1})$ is a composite function of present utility of consumption $u_t(c_t)$ and expected ...


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