5

Let $U:\mathbb R^2\to\mathbb R$ be a utility function.

If $U$ is strictly increasing and continuous, then it is well known that for any $(x_1,x_2)$ there exists a certainty $(c,c)$ such that $$U(x_1,x_2)=U(c,c).$$

If we assume that $U$ is strictly increasing and upper semi continuous, can we still find a certainty equivalence $(c,c)$ or not?

Clarification:

$U(x_1,x_2)$ is not necessarily expected utility (EU). It could be other utility models under uncertainty, such as the max-min expected utility, choquet expected utility, and others.

For example, see: http://www.columbia.edu/~md3405/BE_Risk_4_17.pdf

My try:

Of course one important example of $U$ is the additive representation (subjective expected utility): $U=\sum_ip_iu(x_i)$.

Consider a special case: let $p_1=p_2=0.5$; let $u$ be upper-semi continuous, for example $u(a)=a$ when $a<1$ and $u(a)=a+3$ when $a\geq 1$.

In this case, $U(0,1)=2$.

$U(c,c)=c$ when $c<1$; $U(c,c)=c+3$ when $c\geq 1$. None of the certainty has a utility of 2.

So the answer seems to be "NO"?

High GPA
  • 1,906
  • 9
  • 21
  • Did you mean certainty equivalent, as in a $c\in\mathbb R$ such that $u(c)=\sum_ip_ix_i$? – Herr K. Nov 23 '20 at 17:18
  • 1
    @HerrK. I updated my answer based on your comment. I hope it is now clearer. Many thanks. – High GPA Nov 24 '20 at 00:09
  • Looks like you've answered your own question. I guess you're also implicitly requiring that $U$ is not simultaneously lower semicontinuous. – Herr K. Nov 24 '20 at 01:13
  • @HerrK. Does my answer make sense? – High GPA Nov 24 '20 at 03:21
  • 1
    Yes it does make sense. – Herr K. Nov 24 '20 at 04:43
  • 1
    I am sorry why are you defining certainty like this. It seems like $x_1,x_2$ are different goods in the utility function. Shouldn't it be $c$ such that $E(u(x))=u(c)$, where $c, x$ are lotteries? – Dayne Nov 24 '20 at 05:22
  • 1
    @Dayne: That threw me off at first, but I realized after OP's clarification that $U(x_1,x_2)$ is an expected utility where $x_i$ is the payoff in state $i$. So $U(c,c)$ is the EU where one gets the same payoff $c$ in both states. I guess maybe OP wanted to allow for the possibility that the probability of a state somehow depends on the amount of payoff one gets in it, e.g. $U(x_1,x_2)=p(x_1)u(x_1)+(1-p(x_1))u(x_2)$, where $p(\cdot)$ is a payoff-dependent probability function. But ultimately, it's an incarnation of $E(u(x))=u(c)$ as you pointed out. – Herr K. Nov 24 '20 at 05:50
  • @HerrK.: Yeah! Perhaps we can request the OP to edit the question to further clarify this? – Dayne Nov 24 '20 at 07:22
  • @Dayne $U$ is not necessarily expected utility. It could be other utility functions, as clarified in the question. – High GPA Nov 24 '20 at 08:19
  • Okay. So just for further clarification, $x_i$ represents a particular good $i$ or payoff from state $i$? – Dayne Nov 24 '20 at 08:43
  • @Dayne Those two make no difference mathematically I guess; for example, Arrow Debreu use the same framework for those two cases: consumer choice and Arrow-Debreu state-contingent claims – High GPA Nov 24 '20 at 13:02
  • I guess utility function is allowed much more freedom in choice of functions than E(U) as a function of states. In any case, mathematical equivalence, I think, is not enough. Unless certainty equivalence is defined for utility functions (as opposed to expected utility), this is confusing. – Dayne Nov 24 '20 at 15:22
  • Like Dayne says, there is a significant difference. An EU utility is defined as a linear function on $U: \Delta(X) \rightarrow \mathbb{R}$. What does strictly increasing mean in such a context? If $X = {0,1}$, then each lottery is of the form $(p,1-p)$. How can we even compare them with respect to a set order?

    On the other hand if we think of the consumption space as $\mathbb{R}^2$, then a utility function $U: \mathbb{R}^2 \rightarrow \mathbb{R}$ makes sense. Please clarify your question to the exact setting you are interested in.

     – Walrasian Auctioneer Nov 24 '20 at 17:45
  • 2
    Your counterexample works; the answer is, indeed, no. – Michael Greinecker Nov 24 '20 at 23:01
  • @WalrasianAuctioneer There is no mathematical difference between: "$x_i$ represents a particular good i or payoff from state i", as I replied to the "Okay" comment. Of course there is a difference between SEU and other utility functions. Dayne's "Okay" comment did not mentioned SEU or EU. – High GPA Nov 25 '20 at 01:28

0 Answers0