I spoke with Jacob in the hallway about problem 5 and this is what he had to say:
1) Its not necessarily the case that CRRA utility plus lognormal gives you mean variance optimization. This is true if you can continually rebalance or if you are not perturbed by small shocks. (I'm not entirely clear on what this means). The crux of this arguement relies on the fact that a linear combination of log normal assets is not lognormal. In continuous time, the mean and variance dominate over the sqrt(delta t) term so its better there (Again unclear on details here).
2) Having one investor find it optimal to hold the market portfolio implies that you can use the market portfilio to do pricing (maybe not necessarily Beta pricing). This is because optimality gives us a ratio of utilities and thus a SDF. This SDF need not be unique. Its likely not in problem 5.
3) Now its also possible that every investor might want to hold the market portfolio, but its not necessary that the pfolio be MVE. In fact, if you have 2 fun seperation through utility restrictions, unless you have MV utility, the funds may not be MVE. IF you have distributional constraints leading to 2 fund sep, then all utility maximizers (including MV optimizers) will want to hold the funds, so they will have to be MVE.
He recommended reading Dybvig and Ross (?) 1980(?) paper in Econometrica. It has something to do with the fact that investors might want to hold specific assets because its optimal for them to do so but the linear combination of these assets is SSD (dominated) by something else in the economy.
I know this probably did not clear up many of your questions, but perhaps you can be more prepared when you speak with him.
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Here's a link to the Dybvig and Ross paper. They show that in general, the efficient set is not convex. The mean-variance case is just a special case where convexity holds.
So I was looking of Beta pricing in HL and I think I have some inuition as to why Beta pricing fails to hold for the market portfolio for problem 5 of Jacob.
We can always get something like
R_q = \beta_1 R_m + \beta_2 R_zM + junk
The issue is that in this case we do not get \beta_1 = 1-\beta_2, so we can't beta price, and this has to do with the fact that in this case the minimu variance zero covariance portfolio of the zero covariance portfolio of the market is not equal to the market. I am not sure if this helps, but I think it is a start towards and inuition.
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