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I think you may substitute it directly as mentioned? $$v(A,k) = \max log(c) + \beta E[v(A',k')|A]$$ Applying the value function expression (note period change): $v(A',k') = \frac{log(k')}{1-\beta} + v(A',1)$ Substituting: $$v(A,k) = \max log(c) + \beta E\left[\frac{log(k')}{1-\beta} + v(A',1)|A\right]$$ By linearity: v(A,k) = \max log(c) + \beta E\left[\...