Exercises: Preferences, utility and choice (‘problem set’):

Exercises: Preferences and utility

We will also have a discussion and writing exercise (time permitting) on the realism of particular assumptions and on ‘observing preferences and observing violations of transitivity’.

(Highly optional): Properties of binary relations - O-R problem 1a.

The problem begins…

Assume that \(\succsim\) is a preference relation. …

Defining ‘strictly preferred to’ “\(\succ\)”:

\(x \succ y\)

if \(x \succsim y\) and not \(y \succsim x\).

Please read O-R (p. 13) for the full problem description.

Osborne and Rubinstein have requested that we not post their problem sets (neither questions nor answers) directly on the web. I will engage with students on these directly.

Think about this and give it a try. You can skip the ‘antisymmetric’ part. I will share a suggested answer.

I assign this problem to give you a sense of:

  • ‘How the \(\succ\) operator is defined based on the \(\succsim\) operator’ (as previously asserted)

  • What proofs look like using these operators.

Consider this optional… but at least worth having a quick look at to understand the proof.

Shepard Scale and Penrose stairs (O-R problem 6)

MC Escher:  Penrose stairs

Figure 2.4: MC Escher: Penrose stairs

… See O-R chapter 1 problem 6, page 14.

(The video HERE combines these examples.)

Think about the question they ask, and write a short answer. We will discuss this. I give an answer and discussion in the video below.

Representations with additive utility [O-R 9b]

Apartment building preferences:

Follow-up point:

Exeter students: Or see recording of online tutorial (no students on screen/audio) HERE

An individual has preferences over the set of units

(Not assigned, but maybe helpful … Part A: “Show that the individual’s preference are consistent with a preference relation.”)

We want preference relations that lead to the strict preferences stated in the problem….

\[(5, 12) \succ (4, 20)\] \[(4, 5) \succ (2, 12)\] and \[(2, 20) \succ (5,5)\]

Where the first element in each pairing is ‘rooms’ and the second element is ‘floor’.

Recall the \(\succ\) meaning ‘strictly preferred’ is defined as \(A \succ B\) if \(A \succsim B\) but not \(B \succsim A\).

OK, we could merely restate the preferences above. But perhaps we want to generalise.

One preference relation (but not the only one) over all pairs \((r,l)\) that is consistent with the preferences given above is defined by

\((r,l) \succsim (r′,l′)\) if and only if either

  1. \(r \geq r′\) and \((r,l) \neq (5,5)\), or

  2. \((r′,l′) = (5,5)\).

Why the “and \((r,l) \neq (5,5)\)” in point 1?

Considering the \((2, 20) \succ (5,5)\) preference … try the above rule applied in both directions:

First, is (2, 20) weakly preferred to (5,5)?

  1. \(r ≥ r′\) and \((r,l) \neq (5,5)\) \(\rightarrow\) FALSE (because \(r<r'\)), so go to step 2 (interpret the ‘or’ as an ‘inclusive or’)

  2. \((r′,l′) = (5,5)\). \(\rightarrow\) TRUE

As “1 or 2” is TRUE, the conditions hold, so (2, 20) is weakly preferred to (5,5).

Next, is (5, 5) NOT weakly preferred to (2,20)? … now let (5,5) be (r,l) and 2,20 be (r’,l’)

  1. \(r ≥ r′\) and \((r,l) \neq (5,5)\) \(\rightarrow\) FALSE … because now (r,l) is (5,5) … so go to the second condition…

  2. \((r′,l′) = (5,5)\). \(\rightarrow\) FALSE

So (5, 5) is NOT weakly preferred to (2,20).

Thus, as we have th … (2,20) weakly preferred to (5,5) … (5,5) not weakly preferred to (2,20) We thus have (2,20) strictly preferred to (5,5).

But note how the “and \((r,l) \neq (5,5)\)” condition above was needed … if it weren’t there, we would have concluded a weak preference in both directions (implying indifference).

Try part B… here we consider whether this person’s preferences can be represented by a utility function that is additive in the number of rooms and the floor. I.e., a utility function that is the sum of two functions, one that is a function of the floor (\(\ell\)), and another that is a function of the number of rooms (r). For example \(r^2 + exp(l)\) is the sum of the functions \(f(r)=r^2\) and \(g(\ell)= exp(\ell)\).

It turns out that it cannot be represented by a utility function that is additive in this way…

(Note that the given the \(r\) and \(\ell\) functions simply identify the number of rooms and the floor of a unit, they are not ‘really’ functions.)

Think about it, try it; we will discuss the answer in some detail.

2.5.6 Some easier exercises (from Varian’s intermediate text)

  1. What does it mean to say that a function, \(u(x_1; x_2)\) represents a consumer’s preferences over bundles of \(x_1\) and \(x_2\)?
Answer: A utility function \(u\) represents a consumer’s preferences, \(\succsim\), over bundles X and Y , if \(u(X) \geq u(Y)\) if and only if \(X \succsim Y\). I.e. whenever bundle \(X\) is (weakly) preferred to bundle \(Y\), u assigns a higher number to \(X\) than it does to \(Y\) (and vice-versa) ,

ii. What assumptions do we need to make so that preferences can be represented by a utility function?

Answer: If the choice set is finite (i.e., it contains some finite number of bundles to choose from, e.g., 100 bundles) and preferences are complete, transitive, and reflexive, then there exists a utility function that can represent the consumer’s preferences.

If the choice set is infinite (e.g.,\(\{x_1,x_2\} \in \mathbb{R^2_{+}}\)) we also need to assume preferences are continuous.

Exercises: Choice

Please be sure you can do all the ‘comprehension questions’ throughout section 2.5 above. Note that some of O-R’s exercises for this are too mathematical for this module/for my taste.

2.5.7 OR chapter 2 Exercise 6 “Caring up to a limit”

2.5.8 OR chapter 2 Exercise 8 “Money pump”

2.5.9 O-R ex. 2.3

The waiter informs him … he may have either broiled salmon at 2.50 USD or steak at 4.00 USD. … he elects the salmon. Soon after the waiter returns from the kitchen … to tell him that fried snails and frog’s legs are [also available] for 4.50 USD each. … his response is “Splendid, I’ll change my order to steak”.

Explain in your own words, and using the formal language of preferences and choices, why this seemingly perverse behavior could or could not be rationalizable. Does it depend on how we define the goods in the choice set?

A discussion of an aside question HERE: what if my choices are influenced by a freind’s choice. Could this be rationalisable?

2.5.10 Exercise: Decoy effect (Attraction effect)

Attraction effect: example

Formally depict a choice function representing the ‘decoy effect’ (aka, “asymmetrically dominated choice”, aka “the attraction effect” ).

  • What is (some of the) the evidence for this? (give a very brief description, with some citations, explaining what the authors do and what they find)

  • Can these be rationalised by a preference relation? Explain why or why not.