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Topic 1: Utility function.

ECON30010 Microeconomics

3 March 2016

ECON30010 Topic 1: Utility function. 3 March 2016 1 / 31

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A bit of history and motivation

Classical Economics

Adam Smith, David Ricardo, Karl Marx, John Mill.

Attempted to uncover the laws of the economy, which were thoughtto be similar to the laws of physics (e.g. Newton’s laws).

Example

Kuznets curve: an inverted U-shaped relationship betweenincome per capita and inequality;

Kondratiev’s wave: a 45-60 year business cycle These ideas are derived from observing aggregate variables, not

individual behaviour.


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Neoclassical Economics

Leon Walras, Francis Edgeworth, Vilfredo Pareto, Alfred Marshall.

Decision making of agents is the main focus.

Nowadays even macroeconomics, who, as the name implies, look ataggregate variables, are concerned with “microfoundations”: is mymacroeconomic model consistent with the behaviour of theindividuals?


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Utility function: Motivation

In this course, we will be concerned with the decision making of individuals.

When we talk about individuals, we immediately start to talk about a“utility function”, which summarizes what individuals like and dislike.

Since we are going to use utility function all the time, we need tohave clear understanding what it is and how we could deal with it.


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Plan

Utility function is a model of an individual. As any model, it has its

setup and assumptions. ⇒ We start by looking at the assumptions we want to impose on

individual preferences. This is a bit of ECON20002 recap, but we willbe more careful.

As any model, it makes predictions. I will argue that these predictionsare very weak.

However, in practice we use these weak predictions to make furtherstrong assumptions (not implied by our original assumptions) and usea specific form of a utility function.

This specific form is another instance of a model, which does notalways conform to the reality (and, in fact, any model should not always conform to reality)

We then talk about testing different models.

We finish by throwing in few more assumptions on preferences.


M d l A i

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Model: Assumptions

Modeling individuals: the choice set

First, we need to agree what individuals choose. In most of the subject,they would choose two very dry commodities: good 1 (the quantity of good 1 consumed will be denoted by a letter with subindex 1, such as q 1,

x 1, or y 1) and good 2 (quantities q 2, x 2, or y 2). As often as life allows uswe will deal with only two goods, for simplicity.

When we model individuals, we need to be very precise and careful aboutthe choice set. In many models the choice set is introduced in a single

sentence, but we will see today that this choice is important.


M d l A ti

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Model: Assumptions

Modeling individuals: preferencesNote: this mostly reviews the concepts you have learned in ECON20002 last year.

Once we have settled on the choice set, we need to make someassumptions on how agents choose from this choice set.

In most economic problems, the following assumptions seem reasonable:

Completeness: any two bundles x = (x 1, x 2) and y = (y 1, y 2) can becompared.1

There may be situations where you would say “I cannot

compare these two objects” – what do you like better,

singing or a cylinder hat? – but if you cannot compare,

do you really ever need to choose between the two?

1In math notation, bold letters, such as x and y, will always stand for a vector, as itis here. Thus, x = (1, 3) should be understood as a bundle that consists of 1 unit of

good 1 and 3 units of good 2.ECON30010 Topic 1: Utility function. 3 March 2016 8 / 31

Model: Assumptions

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Model: Assumptions

Modeling individuals: preferences


Transitivity: if bundle x is at least as good as y and y is at least asgood as bundle z, then x is at least as good as z.

One example of transitivity are real numbers and greater than relation(>). It is transitive: if a > b and b > c then a > c .

What happens if transitivity does not hold? Suppose thatyou prefer ECON20001 to ECON30010 (ECON20001 ECON30010),

and you prefer ECON20002 to ECON20001 (ECON20002 ECON20001),but you prefer ECON30010 to ECON20002 (ECON30010 ECON20002).

These are not great preferences to have as a manipulator would be able tomake you whatever choice he or she likes (including taking ECON30010!).

I would hope that whenever you observe yourself making such choices

(which is, of course, possible), you would regard them as a mistake.ECON30010 Topic 1: Utility function. 3 March 2016 9 / 31

Model: Assumptions

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Model: Assumptions

A note on the choice of the choice set.

Consider a dinner where you are to choose the wine (red or white) and themain dish (fish or beef).

Suppose: red wine white wine fish beef.

Shall we conclude from this that you would choose red wine and fish for

your dinner? If not, what’s wrong?

The choice set is incorrectly defined. If you want to choose a drink and amain dish, then the choice set should be: (red wine, beef), (white wine,fish), (red wine, fish), (white wine, beef).

⇒ The choice of the choice set should be appropriate for the question youwant to ask.


Model: Assumptions

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Model: Assumptions



Continuity (not an exact definition): if bundle x is better than bundley and bundle z is sufficiently close to y, then x is better than z.

If you prefer (x =) 1 of red wine and 1 kg of beef to (y =) 1 of whitewine and 1 kg of fish, then you would prefer x to 1.01y (that is, to 1.01of white wine and 1.01 kg of fish).


Model: Assumptions

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Model: Assumptions



Completeness: any two bundles x = (x 1, x 2) and y = (y 1, y 2) can becompared.

Transitivity: if bundle x is at least as good as y and y is at least asgood as bundle z, then x is at least as good as z.

Continuity (not an exact definition): if bundle x is better than bundley and bundle z is sufficiently close to y, then x is better than z.

What do we need it for? With these assumptions, we can start to use very

powerful math tools.


Model: Results

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Utility Function Representation

Even though people simply make a choice from the choice set, we can

pretend that they have a utility function.

Specifically, we will assign to each bundle (such as 1 of red wine and 1 kgof beef) a (real) number and then compare numbers instead of bundles.

Formally: We say that a utility function u (x) represents the preferences if,whenever x better than y, we have u (x) > u (y).

This leads to a very important theorem:

Theorem

If preferences over bundles of goods x satisfy completeness, transitivity,and continuity, then there exists a continuous utility function u (x) that represents these preferences.

Why is this theorem important? Because mathematicians are very good atworking with functions, and we can use a lot of their tools.


Model: Results

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Remarks on Utility Function Representation

Some preferences cannot be represented by a continuous utility

function (but they would not satisfy our assumptions). If preferences are represented by a utility function, this utility function

is not unique. Any positive monotonic transformation of the utilityfunction will represent the same preferences.

Why: The only important thing about a utility function is how two

numbers compare (e.g., u (x) > u (y)). But the very definition of apositive monotonic transformation is that this inequality is preserved!

Example: u 1(x) = x 1x 2

u 2(x) = ln(x 1) + ln(x 2)

u 3(x) =√

x 1x 2

u 4(x) = x 21 x 22

all represent the same preferences (e.g. all four of these utility

functions could be utility functions of the same individual).How could you check it? Take any two consumption bundles, e.g.(1, 3) and (2, 1), and calculate u 1, u 2, u 3, u 4 for these two bundles.2

2Note: this is not a proof. For a proof, you need to show that the same holds for anytwo bundles, (x 1, x 2) and (y 1, y 2); that is, if for given x, y, u

i (x) > u i (y) for one of these

functions, the same holds for all three others.ECON30010 Topic 1: Utility function. 3 March 2016 14 / 31

Model: Results

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SummaryA utility function is a fake . When I say “consider an agent with utilityfunction u = F · J ”, this is short for: consider an agent who, when facedwith the choice between food and junk food, make a choice as if his or herutility function is u = F · J .

If you ever hear someone say:

Economists are silly, they assume people have utility function,but if I ask them to write their own utility function, they cannot

do even that

then you would need to say that economists do not assume that people

have utility function. Economists assume that people’s preferences are (1)complete; (2) transitive; (3) monotone. Which one do you have a problemwith?

This is, of course, not to say that utility function cannot be incorrectly

specified. We will look at it next.ECON30010 Topic 1: Utility function. 3 March 2016 15 / 31

Model: Results

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A tutorial question: Examples of utility functions

Do the preferences described by the following utility functions3

satisfy(1) completeness;(3) continuity;(2) transitivity?

1. u

(x

) = −x

for x ∈ [0, 1]

2. u (x ) = 1 for x ∈ [0, 1), u (x ) = 2 for x ∈ [1, 2].

3. u (x ) = 1/x for x ∈ (0, 1], u (x ) = 0 for x = 0.

4. u (x ) =

x 2, for x ∈ [0, 1],

10, for x ∈ (1, 2),x 2, for x ∈ [2, 3].

3Note: here x is not bold, so it is just a number.ECON30010 Topic 1: Utility function. 3 March 2016 16 / 31

An application: altruistic behaviour

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Selfish behaviour: Motivation

Next we will talk about selfish behaviour (or the lack of thereof). Why dowe want to talk about it?

Specific knowledge:

1. We have made assumptions on preferences; I want to show you that

these assumptions still allow for a variety of different utility functions.

Generic skills:

2. I want a simple environment that I can use to talk about differentmodels.

3. I want to drive home a point that models are not fixed, receivedknowledge. They are subject to testing and modifications and may besuitable in some cases and unsuitable in some other cases.



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Selfish behaviour

Sometimes you can hear thatEconomists assume selfish behaviour, which is silly as we see

many instances of altruistic behaviour, such as when people

share their money with others.

This description is a bit too generic, so let us think about a concretesituation: suppose that agent 1 is given $5 that agent 1 can allocatebetween herself and agent 2. Suppose that we observe that agent 1 givesy > 0 to agent 2. I will call this situation “Situation 1”.4

Is it a contradiction to anything we have assumed so far?

4Sometimes I will call an invented situation, such as Situation 1, an “experiment”. Itmay sometimes be confusing, but in many cases we will indeed look at experiments from

Experimental Economics literature.ECON30010 Topic 1: Utility function. 3 March 2016 18 / 31


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A “standard” model

Choice set is (x , y ), x ≥ 0, y ,≥ 0. I interpret x as the amount of moneykept to oneself and y as an amount of money given to the other person.

Utility function u (x , y ) defined over this choice set. The utility functionthat is consistent with “selfish” behaviour is u (x , y ) = x .

In Situation 1, there is an additional requirement, x = 5 − y .

What “standard” model predicts? Since agent 1’s utility function isu (x , y ) = x , then her optimal choice is y = 0.

I assumed that agent 1 splits the money so that y > 0. This isinconsistent with the prediction of the “standard” model.



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Alternative utility functionsThere is a number of theories that can rationalise “giving”. Let us look atfew of these theories (which, in this instance, take a form of anassumption on utility function – but they do not have to!).

1. u FS (x , y ) = x − α |x − y |, 0 < α 0: a person cares about own payoff,but also cares about the difference in payoffs (inspired by FS);

3. u BO (x , y ) = x − α

x x +y −

12

2, α > 0: a person cares about own

payoff and about her share of the payoff (inspired by Bolton andOckenfels, 2000);

4. u LL(x ) = x − α(x − x e )2, α > 0: x e is an expected decision (e.g.dictated by social norms) and (x − x e )2 is the moral cost of deviatingfrom an expected decision (inspired by Levitt and List, 2007).



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Prediction for u FS

in Situation 1.Suppose that u FS

is the true model of agent 1. What agent 1 will do inSituation 1?

We can re-write u FS

as (I write it as a function of x for convenience):

u FS

(x , 5 − x ) = x − α (x − (5 − x ))2 = x − α (5 − 2x )2

What choice does it predict in our environment?

d

dx U FS

(x , 5 − x ) = 1 + 4α(5 − 2x ) = 0

⇓

5 − 2x = − 14α

⇒ x = 52

+ 18α

Note that α is not known and may vary among individuals; the only thingwe assume about α is that α > 0. Thus, the prediction is that agent 1 will

keep more than 1/2 of $5 (nothing more can be said).ECON30010 Topic 1: Utility function. 3 March 2016 21 / 31


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Predictions for u LL in Situation 1.

Given our situation, we conjecture that x e = 5/2 (wait to ask me why),

hence we can re-write u LL as

u LL(x ) = x − α (x − 5/2)2

What choice does it predict in our environment?

d

dx u LL(x ) = 1 − 2α(x − 5/2) = 0

⇓

x = 5/2 + 12α

Recall that α is not known; so the prediction here is that agent 1 will keepmore than 1/2 of $5 to herself.



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Summary: Comparing predictions for u FS

and u LL

First, to avoid confusion, I will denote αs from previous slides as αFS

andαLL.

x = 5/2 + 1

8αFS

x = 5/2 + 1

2αLL

Given that we do not know αFS

and αLL, there is no difference inprediction in our Situation 1: we cannot distinguish between two models.



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Data for Situation 1

This lack of different predictions is bad news for us: if we want to test

whether u FS

or u LL

better describe “giving”, we could not say. We may,however, try to make a less precise statement whether they “seem to fitwell” the reality or not. Let us look at the data:

$ %

0 .29

0.5 .171 .152 .07

2.5 .253 .035 .04

The right column is a % of people who give the $ amount inthe left column. For example, $0 is given by 29% of the

subjects and $2.5 is given by 25% of the subjects.Recall (previous slide) that if an agent has FS and LLpreferences, she would always give less then 2.5 to agent 2.5

Hence, because we observe so much of $2.5 choices, we mayconclude that neither model does a good job explaining all thedata; yet we may still want to know which one does better.

Situation 1 does not allow us to distinguish two models, we move toSituation 2.

5A bit of a problem for my interpretation is that subjects could give in $.5

increments; we will discuss this separately.ECON30010 Topic 1: Utility function. 3 March 2016 24 / 31


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Situation 2

Now, we change the situation a little: suppose that agent 1 can not only

give some y ≤ 5 to agent 2, but also can take up to $1 from agent 2. So,in Situation 1 0 ≤ y ≤ 5 and in Situation 2 −1 ≤ y ≤ 5.

What are the predictions for FS and LL?

FS : For these who gave $0 in Situation 1, they could either give $0 or -$1.

Indeed, if α is very small, then such an agent would have loved to takeeverything for agent 2, but was more constrained in Situation 1 (couldtake at most $0) than in Situation 2 (could take at most -$1).

For these who gave more than $0 in Situation 1, their choice shouldnot change.

For them, optimal choice has been x

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Data for Situations 1 and 2

$ % in S1 % in S2-1 0 .210 .29 .44

0.5 .17 .091 .15 .07

1.5 0 .042 .07 .04

2.5 .25 .093 .03 .03

5 .04 0

This table combines the data for bothsituations and has the same structure as theprevious table. We see that in Situation 244% give $0 to the other agent.This is inconsistent with FS prediction (“forthese who gave more than $0 in Situation 1, their

choice should not change”), but consistentwith LL prediction (which allows for change).6

6The Econometricians among us may like to see a formal test of this claim, but we

will rely on a “visual” test (and a claim that it is “obvious” that LL fits the data better).ECON30010 Topic 1: Utility function. 3 March 2016 26 / 31


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Is LL better than FS ?

LL explains our Situation 2 data better. Does it mean that LL is a bettertheory?

Note x e : there is an extra parameter in this model and there is no

expectation that this parameter should stay the same in differentsituations. So, without a theory how x e is formed, LL has very lowpredictive power: by suitably choosing parameter x e it is possible to explainalmost any data. In the form I presented this theory, it is not falsifiable .7

7The theory is falsifiable, or refutable, if it is possible to design a test which proves

the theory false.ECON30010 Topic 1: Utility function. 3 March 2016 27 / 31


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Selfish behaviour revisited (a teaser), 1

FS

may not explain the data very well, but selfish model (with u (x ) = x )clearly explain it even worse. Why would we routinely assume selfishbehaviour?

It appears that in many situations of our interest – that is, in a natural

environment – they are OK. They also give extremely sharp (no α and x e

)and simple predictions.

John List run the following experiment. Let us consider a slightly morecomplex game, called “gift exchange” game, in which agent 1 first pays

agent 2 some amount of money and then agent 2 return some money backto agent 1. Note that the second part is similar to our Situation 1 (called“dictator” game).



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Selfish behaviour revisited (a teaser), 2

For added realism, in List’s experiment, agent 2 does not return money.

Agent 2 return baseball cards of different quality. An altruistic agent 2would return a high-quality, valuable baseball card after receiving highpayment and a low-quality, not valuable card after low payment. Selfishagent 2 would return a low-quality card no matter what.

As an interesting twist, List uses actual baseball card dealers.He first puts them in the lab. The results are in line with other labexperiments: baseball card dealers are “altruistic”.

He then run field experiment (that is, the same rules of trade but in real

life; dealers are unaware of the experiment). The dealers are much lessaltruistic than in the lab experiment.

As a side note, LL model explain that as well, because x e could change.


Sources

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Textbooks

Varian:

Preferences: Ch. 3

Utility: Ch. 4

Behavioural Economics: Ch. 31.4

Serrano-Feldman:

Ch. 2


Sources

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Papers FS and FS preferences: Fehr, Ernst and Klaus M. Schmidt. 1999. A

theory of fairness, competition and cooperation. Quarterly Journal of Economics , 114, 817–68.

BO preferences: Bolton, Gary E., and Axel Ockenfels. 2000. ERC: ATheory of Equity, Reciprocity, and Competition. American Economic Review , 90 (3): 166–93.

LL preferences: Levitt, Steven and John List. 2007. What DoLaboratory Experiments Measuring Social Preferences Reveal Aboutthe Real World? Journal of Economic Perspectives , 21(2), 153–74.

List’s experiment with -$1: List, John A. 2007. On the interpretationof giving in dictator games. Journal of Political Economy , 115,

482–493. List’s experiment with baseball card dealers: List, John A. 2006. The

Behavioralist Meets the Market: Measuring Social Preferences andReputation Effects in Actual Transactions. Journal of Political Economy , 114, 1–37.


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