Utility assignment (Assignment 3), Psych/PPE 153, Spring 2010

Here were the questions for the utility measurement part, in the order in which they are in the data. (Some got them in reverse order, and at the beginning; others got them at the end. This was actually a mistake, which also caused the questionnaire not to load properly half the time. As a result, we will use the class data from last term, along with your individual data.)

Difference question

Which would have a greater effect on what is important to you about your life as a retiree or senior?
* The difference betwen a household annual (pre-tax) income of \$40,000 and \$X, or
* The difference between \$X and \$120,000.

Here enter (by cutting and pasting or retyping) the single line for your ten difference judgments from the SECOND email you received with your results from the first part:

If you did not do the first part in time, you can still answer most of the questions here, so you can get some credit for this assignment; just leave this blank.

Gamble question

Supposed you had a choice of two investments for retirement. Each would provide your sole income during your entire retirement at the given rate (the same for all years). Which would you choose?
* This one would pay \$X per year (in current dollars) throughout your retirement.
* This one has a 50% chance of paying \$40,000 per year and a 50% chance of paying \$120,000.

Here enter the single line for your gamble judgments:

Explanation of data for Gamble and Difference questions

The values of X were 50,000, 60,000, 70,000, 80,000, and 90,000. The sequence was repeated with everything doubled, so that, for example, the range was \$80,000-\$240,000 instead of \$40,000-\$120,000. This was mainly just to get more data.

In the data, 1 represents a choice that implies a less concave utility function and 0 represents a choice that implies a more concave utility function. That is, 1 represents a choice of the riskier investment, or a judgment that the higher interval is larger than the lower one (e.g., the difference between \$X and \$120,000 is greater than the difference between \$40,000 and \$X). For each subject, the score was simply the mean judgment. For example, if the judgments were "1 1 1 0 0 1 1 1 0 0" the score would be .6

If a subject got a score of 1.0, what would that imply about the utility function; is it convex, concave, or linear?

What scores would be consistent with a linear utility function (no concavity and no convexity)? Note that one of the items would yield indifference in this case, but you were forced to answer, so there could more than one score consistent with linear utility.

Note that this scoring method assumes that subjects were consistent. For example, a rejection of a gamble for a given value of X should imply rejection of the gamble when the sure thing (X) is even higher. Some responses were not consistent, but I just assumed that these were random errors and the mean response is still the best estimate we could get.

Were all your responses consistent? If not, was the incosistency in the gambles, the differences, or both?

Here are summary statistics for the two measures of the utility function:

```Gambles
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
0.0000  0.3000  0.4000  0.4113  0.6000  0.7000
Differences
Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
0.000   0.400   0.500   0.529   0.700   1.000
```
The means were significantly different (t61=2.91, p=0.0050). How would you explain the difference?

Are utility functions inferred from your risk choices more concave (more risk averse) than those inferred from your difference judgments?

Luxury questions

How does this affect what is important to you about your life as a retiree or senior?
1 This would be nice, but it would have little effect.
2 This would have a noticeable effect.
3 This would have a large effect.
4 This is absolutely essential.

The most essential items were:
* owning one inexpensive car (vs. no car)
* owning a television (as opposed to none)
* having a high-speed (fiber optic or cable) Internet connection (as opposed to something that uses the phone)
* flying to see relatives (including children) or friends once a year
* being able to buy appropriate presents for friends and relatives on holidays, birthdays, etc.

The least essential, hence most luxurious, items were:
* having an extra two bedrooms in your home, for visitors, as opposed to one
* regularly buying wine that costs about \$10 for an ordinary bottle
* regularly buying wine that costs over \$20 for an ordinary bottle
* hiring a chauffeur or cook
* hiring someone to maintain a garden or lawn

To assess the taste for luxury, I computed a measure called Lux. For each subject, I found the slope of the best-fitting regression line for predicting that subject's responses to these items from the mean responses of all subjects across the 34 items. The following two plots show the two members of the class with the highest and lowest slopes. The lines are the best fitting predictors of the subject's ratings from the class mean ratings of the 34 items.

The idea here is that a high slope means that the subject cares relatively little for the things that most others care about less; these are the luxuries. A low slope means that the subject cares just as much about these things as the things that other subjects regard as essential. Thus, a high slope means LESS of a taste for luxuries. Those with a high slope should this have a more concave utility function. It would be more important for such people to have a safe but basic retirement rather than to take any risk in hopes of becoming really wealthy, because they would not benefit that much from the extra wealth.

Here is the summary for Lux:

```   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
0.1525  0.7698  1.0550  1.0010  1.2130  1.8500
```

The results of greatest interest are, first, that Lux did not correlate at all with the gamble measure. The correlation was .05, which is slightly positive and thus in the wrong direction. Lux did correlate with the difference measure, however. The correlation was -.30 (negative, as predicted) and significant (t60 = -2.4222, p = 0.0185), although small. A plot of this correlation is this:

How would you explain this difference in correlations?

Why wasn't the correlation for differences greater (more negative) than -.30?