Lecture 1 Uncertainty 1

7/29/2019 Lecture 1 Uncertainty 1

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UncertaintyLecture 1 Tom Holden

Intermediate Microeconomics Semester 2

http://micro2.tholden.org/

ECO2051 Intermediate Microeconomics
http://micro2.tholden.org/http://micro2.tholden.org/


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Introduction to Semester 2

In the second half of the module we relax some of the assumptionsof the perfectly competitive model.

This leads us to consider the role of government intervention.

And consider some interesting ways that individuals and firmsrespond to uncertainty and imperfect information.

As last semester, you will need to be able to solve problems but I alsoexpect you to be able to discuss the implications of the theory for

real life behaviour and policy.

ECO2051 Intermediate Microeconomics 2


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Key info (1/4): Contact details

Course website: http://micro2.tholden.org/

Email Me: [email protected]

Your class tutor, Tina Tse: [email protected]

Office hours: Thursday, 2 4.

E-mailing in advance is always helpful.

Other times may be possible by (e-mail) arrangement.

Classes: Thursdays (10-11 and 11-12)

http://micro2.tholden.org/mailto:[email protected]:[email protected]:[email protected]:[email protected]://micro2.tholden.org/


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Key info (2/4): Readings

Main text:

V = Varian, H. (2006) Intermediate Microeconomics: A Modern Approach,Seventh edition, Norton.

Other editions fine, just use the contents to find the right chapter.

Alternative:

MKR = Morgan, Katz and Rosen Microeconomics (2nd European edition)McGraw-Hill, 2009.

Additional exercises:

Bergstrom and Varian, Workouts in Intermediate Economics



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Key info (3/4): Draft timetable


Week Date w/c Topic Basic Readings Problem set

1 4/2 Uncertainty 1 V 12, MKR 6

2 11/2 Uncertainty 2 + Welfare 1

3 18/2 Welfare 2 V 33 MKR 12 (esp.

p.446 onwards)

1. Uncertainty

4 25/2 Externalities V 34, MKR 18

5 4/3 Public goods V 36 MKR 18 2. Welfare and

externalities

6 11/3 Inter-temporal choice V 10, MKR 5 (from

p.154)

7 18/3 Multiple Choice Test (10%)

Asymmetric Information

V 37 MKR 17 3. Public goods and

inter-temporal choice

8 22/4 Asymmetric Information 2

9 29/4 Auctions V 17 MKR 17 4. Asymmetric

information

10 6/5 Short answer test (20%)

11 13/5 Revision and feedback on test 5. Auctions and

Revision


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Key info (4/4): Draft assessment plan

Test 1: 20 question multiple choice.

60 minutes.

Worth 10%.

On material from weeks 1-6.

Sample test given in week 6.

Actual test in week 7.

Test 2: Answer 3 short-answer questions from 6.

Up to 2 hours. (Should only take around 1.)

Worth 20%.

On material from all weeks, but with a focus on the second half of the course.

Sample test given in week 9.

Actual test in week 10.

Exam: 3 sections, equal weights. Section A is multiple choice. (Similar style questions to test 1.)

Section B is compulsory short answer questions. (Similar style questions to test 2.)

Section C asks you to answer 1 longer question from 3.

2 hours.

Worth 70%.

On material from all weeks.

Mock exam given out in week 11.



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Expectations

As well as mastering the concepts discussed in the lectures, you needto practice the techniques we cover.

I will cover one or two problems in each lecture as an introduction,but you will need to work on the problem sets before the class.

Bergstrom and Varians Workouts provide many additional exercises.

Please let me know if there are any problems, or if anything can bedone to improve the running of the course. Do come to me in thefirst instance.



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Introduction

Uncertainty is pervasive.

Many of the choices we make include a element of risk.

Economic theory can help us to understand the choices made insituations where there is risk.

Examples:

Career choices

Insurance markets

Investments

Gambling



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Topic objectives

To learn the language of expected utility theory.

To understand risk aversion and risk-seeking.

To consider examples where EUT provides insights

Insurance

Investment

To consider the Ellsberg, Allais and St. Petersburg paradoxes.



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Reading

Varian, Chapter 12

Morgan, Katz and Rosen, Chapter 6



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The St. Petersburg Paradox (1/3)

How much would you pay to play the following lottery?

I will repeatedly toss a coin.

If the first toss comes up tails, you win 2, and the game ends. If the first toss, is heads, we toss again.

If the second toss is tails you win 4 and the game ends. Otherwise we toss again.

If the third toss is tails you win 8. And so on (16, 32, 64)



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You have a probability of of getting 2.

You have a probability of

=

of getting 4.

You have a probability of

=

of getting

8.

Etc.

So your expected return in pounds is

2 +

4 +

8 + = 1 + 1 + 1 + =



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So, lets play this game for real.

One slight modification:

The most I will pay out is twice the amount you pay me for the right to play.

So, if you pay me 10 for the right to play, the most I will pay out is 20 (soyou double your money).

If I make a profit, the money will be donated to the Surrey EconomicSociety.



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The foundations of expected utility theory

The basis for EUT is the assumption that individuals are just ascapable of making choices between uncertain consumption bundlesas they are between certain bundles of goods.

This means that we can apply the tools of straight-forward consumertheory (budget constraints and indifference curves) to situations thatinvolve uncertainty.



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The language of uncertainty

Generally uncertainty is discussed in terms of money as proxy for compositeconsumption.

The consumption received depends on the outcome of a random eventthis is referred to as the STATE OF NATURE. We attach some probably to this.

The CONTINGENT CONSUMPTION PLAN is a statement of the consumptionthat occurs in each state of nature.

For example: The choice over whether or not to buy a lottery ticket.

If I do not buy a lottery ticket my wealth will remain the same.

If I buy a lottery ticket then I will lose 1 if none of my numbers come up and gain1000000 if it is the winning ticket.



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Drawing the budget constraint

In straightforward consumer theory the budget constraint shows howmuch of good 1 you can trade for good 2.

In EUT the budget constraint shows how much consumption in stateof the world 1 you can trade for consumption in state of the world 2.

A 1 bet on the turn of a card gives 40 if its not a heart. You keep the 1 if you win and lose it if you lose.

Need to decide how much to bet.



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The budget constraint for a gamble


Consumption if heart

Consumption

ifnotheart

1

0.4 Endowment

Budget constraint could be extended if the

individual could offer the same bet to others

Slope is 0.4 which is the cost of gaining 1 whena heart does appear in terms of loss when a heart

does not appear.


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But so we far we havent mentioned

probability

We know that the chance of an individual winning the bet is 3 in 4.

In which case we can work out the expected net value of the gamble:

0.4 0.75 + 1 0.25 = 0.30 0.25 = 0.05 The expected net value of this gamble is positive.

Would need to lower the prize slightly to generate a fair gamble.

If the prize was 33 the expected value would be: 0.33 0.75 + 1 0.25 = 0.25 0.25 = 0 The budget constraint for a fair gamble gives the fair-odds line. Draws a line

between all the possible expected values.



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The fair-odds line


Consumption if heart

1

0.33 Endowment

We showed that the expected value of the gamble is

zero when the payoff to not heart is 0.33.

Consumption

ifnotheart


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Another example

In this case were gambling on the throw of a dice.

If its a six receive 4 times the stake (and keep the stake). Otherwise the stake is lost.

What would the budget constraint look like?

Whats the expected value of the gamble?

How would we need to change the game to make the gambleactuarially fair?



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The fair-odds line for the die game


Consumption if not six

1

5

Endowment

Consumptionifsix

= 561 = 1

6Slope= 5


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In the general case

The fair odds line gives the slope of the budget constraint when theexpected value is zero.

The odds of an outcome occurring is the probability it occurs divided by theprobability it does not occur.

I.e.

where

is the probability of the event occurring.

The odds of heart is =

.

The odds of not six is = 5.

The fair odds line has slope given by minus the odds of the outcome on thehorizontal axis.

Why?!?

We generally put the worse outcome on the horizontal axis, but we neednot.

Notice that the actual budget constraint for the gamble and the fair-oddsline need not be the same.



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Next: preferences

Choices are made over contingent commodities bundles of goods asrepresented by points on the budget line.

The individual is not making a choice about the state of the world, ofcourse they would prefer to win the gamble, but that choice isnttheirs to make.

In the examples we have been considering the choice is over howmuch to gamble.



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The expected utility function

The utility function for consumption in states of the world 1 and 2 is , ,

where is consumption in state 1, is consumption in state 2, is theprobability of state 1, 1 is the probability of state 2.

The utility of a contingent consumption plan depends on consumption in thetwo states of the world and the probability of each state of the worldoccurring.

The expected utility form is:

, , = + 1 Expected utility is a weighted average of the utility of the two possible

consumptions with the weighting dependent on the probabilities.



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More on expected utility

The expected utility function is unique up to an affinetransformation which means you can add a constant and multiply bya positive number and the preferences which result wont change.

Contrast with normal utility functions.

An expected utility function is also known as von-Neumann-Morgenstern (vNM) utility function.

Cobb-Douglas utility CD , , =

is a vNM utilityfunction, despite appearances, since: logCD , , = log + 1 log



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Why does it make sense?

The additive nature of the expected utility function means that theutility derived from the consumption in each state of the world isindependent of consumption in the other state of the world.

Once the state of the world is known you will only care aboutconsumption in the SoW that prevails, the other will be irrelevant.

This is known as the independence axiom.



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Illustration of independence

Suppose utility is given by:

, , = + + where + + = 1.

Consider the MRS between consumption in state 1 and state 2: MRS =

=

The MRS for consumption in SoW 1 and consumption in SoW 2 doesnot depend on what consumption is in SoW 3.



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Do you have vNM preferences?The Allais Paradox

Experiment 1 Experiment 2

Gamble 1A Gamble 1B Gamble 2A Gamble 2B

Winnings Chance Winnings Chance Winnings Chance Winnings Chance

1 100% 1 89% 0 89% 0 90%

0 1% 1 11%5 10% 5 10%


Evidence for Prospect Theory???

Derived from: https://en.wikipedia.org/wiki/Allais_paradox
https://en.wikipedia.org/wiki/Allais_paradoxhttps://en.wikipedia.org/wiki/Allais_paradoxhttps://en.wikipedia.org/wiki/Allais_paradox


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Formal definition of the independence

axiom

Suppose you prefer taking part in one gamble, A (e.g. the heartsone) to taking part in another, B (e.g. the dice one).

And suppose C is a third gamble.

Then if I offer you a choice between the following two new gambles:

Gamble D: With probability , I run gamble A, and with probability 1 Irun gamble C.

Gamble E: With probability , I run gamble B, and with probability 1 Irun gamble C.

Which do you prefer?

The independence axiom is the assumption that you will always prefer Dhere.



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Do you have vNM preferences?The Ellsberg Paradox

Suppose there is an urn containing:

33 red balls 66 balls that are a mix of black and yellow.

You do not know how many are black or how many are yellow.

Consider the following four gambles:

Gamble 1A: You receive 1 if you draw a red ball. Gamble 1B: You receive 1 if you draw a black ball. Gamble 2A: You receive 1 if you draw a red or a yellow ball.

Gamble 2B: You receive 1 if you draw a black or a yellow ball.



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More on vMN preferences:An important distinction

Back to first example. Assume 10 bet. 0.75 chance of having 14, 0.25 chance of having 0. Therefore the expected value of the gamble is 10.50.

The expected utility from the gamble is: 0.75 14 + 0.25 0

E.g. if = , this is 0.75 14 + 0.25 0 2.81

The utility from the expected value of a gamble is

10.5 E,g, if = , this is 10.5 3.24 Risk averse agents prefer to get the expected value of a gamble with

certainty.



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Risk aversion


140

0

consumption

Utility

14

0.75 14 +0.25 0

10.5Notice that the expected utility of

the gamble is lower than the

utility of the expected value.

I.e. + > + This is the definition of risk

aversion.

10.5


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Risk loving


140

0

consumption

Utility

14

0.75 14 +0.25 0

10.5

It is also possible to be risk

loving, the opposite of risk

aversion.

This is preferring gambles over

getting their expected value for

certain.

I.e. + < + What would risk neutral utility

look like?

10.5


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Risk aversion and marginal utility

Notice that a risk averse utility function is concave.

There is a declining marginal utility of income.

Each extra 1 is worth less the richer you become.

As a consequence income lost has more impact than income gained.

This is why a risk averse person would not want to take an actuarially fairgamble.

Empirical work suggests

= log is a reasonable approximation

to most peoples utility.



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Indifference curves with risk aversion


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Certainty line

Fair odds line (slope= )

We would expect ICs to be convex: prefer a mix of consumption in both periods.

This captures risk aversion.

Consumption if heart appears

Consumptionifnotheart


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45

Certainty line

Fair odds line (slope= )

Risk aversion rules out indifference curves like the following one. Why?

A

BConsumptionifnotheart

Consumption if heart appears


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Point A and point B have the same expected return.

But point A is a certainty.

Therefore by the definition of risk aversion, A must be preferred.

Indifference curves must be tangential to the fair odds line.

A risk averse person always chooses equal consumption in bothstates of the world if offered fair odds.



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In equilibrium

Equilibrium follows exactly as it would in all other situations.

The individual chooses the contingent consumption bundle whichmaximises his utility given the budget constraint he faces.

So a risk averse consumer will not accept an actuarially fair gamble.

What will happen in the case of a risk lover?



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Risk premia

140

0

consumption

Utility

14

0.75 14 +0.25 0

10.5

10.5

Risk premium

Risk premium is what youd need to bepaid to be compensated for taking the

risk.

Its calculated as the difference

between what youd be willing to pay

for the gamble and its expected value.

0.75 14 +0.25 0


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Or with indifference curves


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Certainty line

Fair odds line

Fair odds line shows expected values.

Would need to be compensated by the length of AB for being at A rather than Beven though expected value is the same.

A

B

Risk premiumA


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Example of optimisation(For next week perhaps)

Odds on Arsenal winning the premier league 10 to 1. You think theirprobability is .

You have 1000 to spend on consumption if you dont bet. Your von-Neumann-Morgenstern utility function is

+1 How much should you bet?

Suppose you bet . Consumption if Arsenal win: = 1000 + 10. Consumption if Arsenal lose:

= 1000 .



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Theoretical optimisation example(Certainly for next week)

Suppose that there are two possible future states of nature, 1 and 2,with probabilities and 1 respectively.

And suppose that there are two risky assets you can purchase.

Asset 1 pays out 1 in state 1 and 0 in state 2 and costs . Asset 2 pays out 1 in state 2 and 0 in state 1 and costs .

Expected (vNM) utility is + 1 You have a wealth of to spend on and together.



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Coming up next time

More concrete applications of expected utility theory

Insurance

Diversification in investment

Decisions over criminal actions

Application of expected utility theory to moral philosophy!

Lecture 1 Uncertainty 1

Documents

Transcript of Lecture 1 Uncertainty 1