Choice under uncertainty questions

I have some questions concerning the problem set and the notes of Choice under Uncertainty.

First, for problem #3, I was confused by the question “state necessary and sufficient conditions on u(.), ensuring the consumer will underinsure/fully insure/overinsure”. In fact, I have solved the FOC, and find that the property of u'(.) rather than u(.) do matter in this problem; However, the problem doesn’t provide any information about u'(.). Can I make any additional assumption about continuity, convexity/concavity, differentiability… of u'(.)?

When you solve through the FOC, you should get that the result in question (over vs. perfect vs. underinsurance) depends on an expression that looks like E[u'(x)] <=> u'(E[x]). The problem doesn’t actually give you enough info to really state a necessary condition, I agree. But try for a pretty loose sufficient condition (i.e., u(x) = x^(5/4) may be sufficient for overinsurance, but that’s not what I’m looking for).

Doesn’t that E[u'(x)] <=> u'(E[x]) look familiar? (Perhaps something to do with Jensen’s inequality…)

Second, a question about the definition of “continuous preference over lotteries P”: if p>p’>p”, there exists some lamda, such that lamda*p+(1-lamda)*p”~p’, then whether such lamda is unique? I also found in the proof of Theorem 1 (Representation Theorem, in step 3), the author assumes that such lamda is unique. But how can we prove the uniqueness from the definition?

A good question… how can we prove that the lambda in question is unique. I won’t lay out a complete proof, but here’s a sketch. Suppose as you do that p > p’ > p” (where those are strict preference, not greater than signs). Further suppose that there were more than one lambda (i.e., L1 != L2) such that Li p + (1-Li) p” ~ p’ for both i=1 and i=2.

  • By transitivity of indifference, we get L1 p + (1-L1) p” ~ L2 p + (1-L2) p”
  • Since p > p”, I know p != p”. Thus since L1 != L2, I know that L1 p + (1-L1) p” != L2 p + (1-L2) p”.
  • Notice that I now have four points that are colinear: p, p”,  L1 p + (1-L1) p”, and L2 p + (1-L2) p”. Further, I know the first two are nonequal and the second two are nonequal. The second two are not equal.
  • Using a slight extension of our that independence gives linear indifference curves, we can show that since I am indifferent between the second two points above, I must be indifferent among all four (since they are colinear). This is the slightly tricky step, but give it a think.
  • Contradiction: p ~ p” but p > p”

Finally, a small question: which one is correct: Shepard’s Lemma, or Shephard’s Lemma?…

Wikipedia likes “Shephard” http://en.wikipedia.org/wiki/Shephard%27s_lemma
Google search count’s also way outnumber for this spelling compared to others.

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