¹It is surprising to note that for a consumer with logarithmic utility, a mean-preserving spread in risk has no eﬀect on the level of consumption (this can be seen by substituting $ρ = 1$ into (4), which causes the term involving risk $σ2r$ to disappear from the equation). The reason this is surprising is that intuition suggests that if the consumer’s consumption (and therefore current saving) are unchanged, the increase in uncertainty must constitute a mean-preserving spread in future consumption, which by Jensen’s inequality should yield higher expected marginal utility. The place where this argument goes wrong is that it forgets that the expectation in the Euler equation $u′(ct) = β 𝔼t[ℜℜℜt+1u ′(ct+1)]$ is also aﬀected by a covariance between $ℜℜℜt+1$ and $u′(ct+1 )$ ; the case of log utility is the special case where this boils down to a constant times $𝔼t[ℜℜℜt+1∕ℜℜℜt+1] = 1$ , which is why the expected marginal utility is unaﬀected by the unavoidable increase in risk. This is yet another reason (if any more were needed) to conclude that logarithmic utility does not exhibit suﬃcient curvature to plausibly represent attitudes toward risk. ( $ρ ≥ 2$ seems a plausible lower bound).