# Does the Prospect Theory value curve change according to reference wealth size?

After reading "Thinking Fast and Slow" and half a day of internet search on this topic I resort now to asking this question here.

Prospect Theory seems to only attribute value to changes in wealth. The reference point in wealth is only used to calculate the change in wealth. The change is then valued according to a value function where losses weigh heavier than gains and both have diminishing marginal utility (see figure below). It seems to me however that the absolute value of the reference point matters, but is ignored.

The diminishing marginal utility of wealth from expected utility theory is still a valid concept, as is the utility of change of wealth from prospect theory. In my view they should be combined. Losses weigh heavier than gains and the size of your reference wealth influences how you weigh the loss/gain.

In other words, this is a typical value function in Prospect Theory:

I think it needs a third axis for the reference wealth value. Higher reference wealth values lead to flatter curves while lower reference wealth values lead to taller curves. A rich person has a different value function than a poor(er) person. They value the loss (or gain) of 500 euro differently.

To illustrate this think of the following "Thinking Fast and Slow"-style problems:

• Anna had 1500 euro yesterday. Today she wakes up with 1000 euro.
• Ben had 20000 euro yesterday. Today he wakes up with 19500 euro.

It seems to me Anna and Ben would not attribute the same (negative) value to their loss. Prospect Theory seems to claim they would. They incur the same loss (-500 euro) with the same (weighted) probability and value it using the same value function. I haven't found an explanation of Prospect Theory where absolute wealth influences the value function.

In Thinking Fast and Slow, Kahneman does allure to an influence of your reference wealth (page 284):

All bets are off, of course, if the possible loss is potentially ruinous, or if your lifestyle is threatened.

But this seems a sidenote covering an exception rather than a intrinsic part of the theory.

Does prospect theory ignore the influence of reference wealth size on the value curve? And if it does, why is that valid to do?

You are certainly right that Kahneman and Tversky neglect the importance of levels in their model (even though they do acknowledge the importance of levels informally, as in the quote you reference). However, this situation has now been rectified by Kosegi and Rabin, who include both 'levels' utility (i.e.'consumption utility') and 'gain/loss' utility (they also generalise Prospect Theory to handle many goods, and endogenise the reference point).

As I understand it, the point of the Anna/Ben story is that marginal utility is a decreasing function of wealth. I would think that this is adequately captured by Kosegi and Rabin, assuming that we specify that levels utility is a strictly concave function of consumption, as is standard.

Of course, a different route would be to allow the curvature of the Prospect Theory value function to depend on wealth (as you propose). However, I'm not sure what this would buy us above and beyond the Kosegi and Rabin approach. Moreover, the theory would still be missing the rather obvious fact that people do care about absolute levels of wealth, at least to some extent.

In Kahneman and Tversky's original paper, they write the following (pdf page 16):

The emphasis on changes as the carriers of value should not be taken to imply that the value of a particular change is independent of initial position. Strictly speaking, value should be treated as a function in two arguments: the asset position that serves as reference point, and the magnitude of the change (positive or negative) from that reference point. An individual's attitude to money, say, could be described by a book, where each page presents the value function for changes at a particular asset position. Clearly, the value functions described on different pages are not identical: they are likely to become more linear with increases in assets. However, the preference order of prospects is not greatly altered by small or even moderate variations in asset position. The certainty equivalent of the prospect (1,000, .50), for example, lies between 300 and 400 for most people, in a wide range of asset positions. Consequently, the representation of value as a function in one argument generally provides a satisfactory approximation.

Thus, they realized "the asset position" is indeed a factor in defining the outcome of the value function, next to "the magnitude of change". However, generally it's not needed to include asset position as a parameter to the value function because "the preference order of prospects is not greatly altered by small or even moderate variations in asset position".

I also see that the valuations of Anna and Ben could be very different, while giving the same prospect order. So Kahneman and Tversky's argument for using only change as a parameter works when comparing options (i.e. what they do in decision theory ; )), not when evaluating and comparing raw utilities.

I wonder if more research has been done in the influence of leaving out the "asset position" parameter to the value function? Is its effect always negligible? Does anyone know of a source for typical value functions for different asset positions?