BUSINESS ECONOMICS

BUSINESS ECONOMICS

PAPER NO. : 1, MICRO ECONOMICS ANALYSIS MODULE NO. : 9, CHOICE UNDER UNCERTAINITY

Subject Business Economics

Paper No and Title 1, Micro Economics Analysis Module No and Title 9, Choice under Uncertainty

Module Tag BSE_P1_M9

BUSINESS ECONOMICS

PAPER NO. : 1, MICRO ECONOMICS ANALYSIS MODULE NO. : 9, CHOICE UNDER UNCERTAINITY

TABLE OF CONTENTS

1

Learning Outcomes 2. Introduction

3. Lottery

3.1 Compound Lottery 3.2 Reduced Lottery 3.3 Preferences Axioms 4. Expected Utility Theorem

5. Violation of Expected Utility Theorem 5.1 Allais Paradox

5.2 Ellsberg Paradox 6. Risk Aversion

7. Measuring Risk Aversion

8. Summary

BUSINESS ECONOMICS

PAPER NO. : 1, MICRO ECONOMICS ANALYSIS MODULE NO. : 9, CHOICE UNDER UNCERTAINITY 1. Learning Outcomes

After studying this module, you shall be able to

 Analyze how decision makers take decisions under uncertain situations

 Learn what is lottery, and about both compound lottery and reduced lottery

 Learn what is the expected utility theorem and how it works

 Understand when a person is called a ‘risk lover’, or ‘risk averse’ or ‘risk neutral’

2. Introduction

In general, all the decisions made by a firm or an individual are assumed to be taken under a certain known situation. An individual must have knowledge of his own budget and utility before he can maximize satisfaction. A firm knows what input mix is required to maximize its fixed output. But what happens when there is an uncertain situation? How can a firm or an individual make choices under uncertain situations? This is a big question.

There are lots of uncertain situations where decision making is not easy. Firms are uncertain about their output, e.g., bad weather can affect their output, machines can breakdown, often market conditions are also unpredictable. Consumers may not be able to decide whether or not to make a purchase decision, when they face uncertainty regarding their incomes or about future prospects.

Every individual behaves differently in uncertain and risky situations. Some people try to avoid risk, while some love it. A risk averse person tries to reduce its risk by adopting certain strategies like, purchasing insurance, via a diversified portfolio of financial investments and so on. This module will provide insights into about how agents (individual consumers or firms) behave in an uncertain situation and what differentiates a risk lover from a risk averse entity.

3. Lottery

Suppose there are N states (1,…..,n) and the probability attached with each state is 𝑃_𝑖 ( where i=1,…..,n). We have 𝑃_𝑖 ≥ 0 and ∑ 𝑃^𝑛₁ 𝑖 = 1, satisfying the basic axioms of proability.

Let 𝑥_𝑖 (where i=1,…..,n) represent a prize received when a particular state (1,…..,n) has occurred. Then the expected value of the payoff received or ‘lottery’ is denoted as :

𝐿 = ∑ 𝑃_𝑖𝑥_𝑖

𝑛

It may also be expressed as (𝑃₁, … . . , 𝑃_𝑛) where the 𝑃1 ₁ represents the probability associated with the prize 𝑥𝑖.

The prizes which an individual receives may be in cash or kind. For example: if there are two states 𝑥 and 𝑦 which occur with probability 𝑃₁ and 𝑃₂ respectively then the lottery is denoted as :

𝑃1𝑥 + 𝑃2𝑦

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PAPER NO. : 1, MICRO ECONOMICS ANALYSIS MODULE NO. : 9, CHOICE UNDER UNCERTAINITY

A lottery can be graphically represented as a tree like structure as follows :

Figure 1: Representation of a simple lottery

We can also represent the lottery as a vector in a probability simplex (i.e. as a vector of probabilities). Suppose a lottery is represented by a vector of probabilities given by P*=(1/2,1/2,0) where P1=1/2 is the probability of outcome x1, P2=1/2 is the probability of outcome x2 and finally P3=0 is the probability of outcome x3. Note, that P1+P2+P3=1.

Figure 2: Representation of a simple lottery in probability simplex

Moving up in this diagram means the likelihood of the first outcome will increase (as P1

increases), similarly moving down, left and down, right will increase the likelihood of occurrence of the second and third outcomes respectively.

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The above example of lotteries can be generalized to n number of outcomes where the sum of their probabilities of occurrence is one.

3.1 Compound lottery

Suppose there are two lotteries say P* and P**; the convex combination these two lotteries is also a lottery. A convex combination of lotteries is known as a compound lottery.

In other words a compound lottery is a lottery of lotteries. It is a type of lottery whose prize is itself a lottery. Given S simple lotteries with probabilities 𝐴_𝑖 (𝑤ℎ𝑒𝑟𝑒 𝑖 = 1, … 𝑠) attached to each simple lottery then the compound lottery can be denoted as:

𝐿_𝑆= (𝐿₁, … . . , 𝐿_𝑆; 𝐴₁, … … . 𝐴_𝑠) where 𝐴_𝑠≥ 0 and ∑ 𝐴^𝑆_{1 𝑠}= 1

Figure 3: Representation of a compound lottery

We can also represent the Compound lottery as a vector in a probability simplex: Suppose there are two lotteries P* and P**. The lottery P* occurs with a probability of α and the lottery P**

occurs with a probability of (1-α). Then the compound lottery can be represented as follows:

C = α P* + (1-α) P**.

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Figure 4: Representation of a compound lottery in probability simplex

3.2 Reduced Lottery

A simple lottery which is obtained from a compound lottery is known as a reduced lottery.

A reduced lottery R, can be obtained by using the following formula:

𝑅 = 𝐴1𝐿1+ ⋯ + 𝐴𝑠𝐿𝑆

Example 1:

Figure 5: Representation of a reduced lottery

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Outcome 1: ¹

3׳

9+²

3ײ

6= ⁹

27

Outcome 2: ¹

3×⁴

9+²

3׳

6=¹³

27

Outcome 3: ¹

3ײ

9+²

3×¹

6= ⁵

27

We can also represent the Reduced lottery in a probability simplex: Suppose there is a compound lottery say C* given as follows:

Figure 6: Reduced lottery

The above reduced lottery can be represented in probability simplex as follows:

Figure 7: Representation of a reduced lottery in probability simplex

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PAPER NO. : 1, MICRO ECONOMICS ANALYSIS MODULE NO. : 9, CHOICE UNDER UNCERTAINITY

3.3 Preference axioms

There is a possibility that two or more compound lotteries would produce the same reduced lottery. In such a situation a person would be indifferent between the two (or more) compound lotteries. There is a hypothesis known as 'Consequentialist Hypothesis' which says only the consequence and probabilities attach to it matter, and not the way that is followed in order to obtain the consequence.

Therefore we assume that a decision maker who is rational will only care about the consequences and not about the way in which they are achieved. So, a rational decision maker will obviously have preferences over a cluster of lotteries. In what follows we will discuss a set of axioms that such a preference ordering must satisfy.

Let 𝑳̂ be the set of all the simple lotteries and the relation ≿ means strict preference over lotteries.

It is assumed the rational decision maker’s preference ≿ is continuous and transitive.

Axiom 1: Completeness: If there are any two simple lotteries say L* and L**:

either, L* is strictly preferred to L** (L*≿L**);

or, L** is strictly preferred to L* (L**≿L*);

or, both (L*~L**) i.e. the rational decision maker is indifferent between the two lotteries.

Axiom 2: Transitivity: Suppose there are three lotteries L*,L** and L*** . If the preference are transitive then, L*≿L** and L**≿L*** => L*≿L***

Axiom3: Continuity: A preference relation ≿ in lottery space 𝑳̂ is said to be continuous if for any lotteries L*, L** and L*** which belong to 𝑳̂, such that L* is strictly preferred to L** and L** is strictly preferred to L*** , (i.e., L*≿ L** ≿ L***), there is some β which lies between 0 and 1 i.e. 𝜷 ∈ [𝟎, 𝟏], such that,

𝜷𝑳^∗+ (𝟏 − 𝜷)𝑳^∗∗∗~𝑳^∗∗

The continuity axiom implies that if a lottery L* is preferred over another lottery L** then any lottery that is close to the lottery L* will also be preferred over lottery L**. This axiom can be understood as follows : if there are two simple lotteries and a strict preference relation defined over these two, then a small change in probabilities will not affect the order of preference between these lotteries.

We can understand Axiom 3 further with the help of a probability simplex diagram (Figure 8).

Suppose there are two lotteries L* and L** where L*≿ L** and there is a neighborhood of L*

and of L**, which are denoted by N (L*) and N (L**). It follows that, L1* ≿ L2**, where L1*∈

N(L*) & L2**∈ N(L**).

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Figure 8: Continuity Axiom

Axiom 4: Monotonicity: Suppose there are two lotteries, L* and L**. Out of these two lotteries the one that assigns a higher probability of getting a higher outcome will be preferred.

Axiom 5: Independence: A preference relation ≿ satisfies this axiom if we assume that there are three lotteries L*,L** and L***, and there is some β lying between 0 and 1, i.e. 𝛽 ∈ [0,1], then we have, L* ≿L** iff

𝛽𝐿^∗+ (1 − 𝛽)𝐿^∗∗∗≿ 𝛽𝐿^∗∗+ (1 − 𝛽)𝐿^∗∗∗

In simple words this axiom can be explained as follows: Suppose we combine two lotteries with a third lottery, the order of preference over the first two lotteries will not depend on the particular third lottery that is being used.

This axiom can be shown with the help of the following probability simplex diagrams:

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Figure 9: Independence Axiom – (i)

L* ≿L** Iff 𝛽𝐿^∗+ (1 − 𝛽)𝐿^∗∗∗≿ 𝛽𝐿^∗∗+ (1 − 𝛽)𝐿^∗∗∗

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Figure 10: Independence Axiom – (ii)

If the preference relation of a decision maker satisfies the above axioms over a set of simple lotteries 𝑳̂ then we can assume that there is a utility function U such that this utility function will represent these preferences.

4. Expected Utility Theorem

Expected Utility Theorem: An Utility function U() has an expected utility form if numbers (a1,a2,a3,……..,aT) can be assigned to a total number of T outcomes such that, for every simple lottery L*=(P1,P2,….PT) ∈ 𝑳̂, we can express,

U(L*)= a1P1+ a2 P2 +………..+ aT PT

This kind of utility function which takes the expected utility form is also known as Von- Neumann-Morgenstern (vNM) expected utility function.

The utility function which we have mentioned above is linear in probabilities. In fact a utility function will have expected utility form iff (if and only if) it is linear in probabilities.

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Suppose there are a total number of T lotteries; for any lottery Lt∈ 𝑳̂ where t=(1,2,3,…..T) if the following is true then it is said to be linear in probabilities ,where the probabilities are given by (β1, β2, β3..., βt) ≥0 and the sum of these probabilities is1.We can write,

𝑼 (∑ 𝜷_𝒕𝑳_𝒕

𝑻

𝒕=𝟏

) = ∑ 𝑩_𝒕. 𝑼(𝑳_𝒕)

𝑻

𝒕=𝟏

The utility of the expected value of the T lotteries i.e. 𝑼(∑^𝑻_𝒕=𝟏𝜷_𝒕𝑳_𝒕) is equal to the expected utility of T lotteries. i.e. ∑^𝑻_𝒕=𝟏𝑩_𝒕. 𝑼(𝑳_𝒕).

Note:

1. This expected utility property is cardinal because the rank and the number assigned by this utility function, both matter.

2. The Utility function which is in the Expected Utility Form will preserve any positive affine (linear) transformation. If U() and Z() are two utility functions representing preference ≿. If U () has the expected utility form, then Z () will also have the expected utility form if and only it is the positive affine transformation of U(); i.e. suppose there are numbers a > 0 and b such that:

Z (L*) = a U(L*) + b Where L*∈ 𝑳̂

Remember any other transformation except a positive affine transformation of U () will not preserve the same preference ≿.

5. Violation of Expected Utility Theorem

There are a number of experiments which violate the independence axiom discussed in Section 3.3. above. Among these experiments there is a famous one by Maurice Allais (1953).

5.1 Allais Paradox

Suppose there are two situations and 3 prizes in each situation. The amounts of the prizes are Rs.

20, Rs. 10 and Rs. 0 respectively. The two situations are illustrated below:

Situation 1:

Prize Rs.20 Rs. 10 Rs. 0

Lotteries Probabilities

L1 0 1 0

L2 0.10 0.89 0.01

Situation 2:

Prize Rs.20 Rs. 10 Rs. 0

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Lotteries Probabilities

L1' 0 0.11 0.89

L2' 0.10 0 0.90

It is seen that in general rational people give more value to those things which are sure to happen.

Thus in Allais’s experiment majority of the respondents chose lotteries L1over L2 in situation 1 and L2' over L1' in situation 2. Therefore, the experimental outcome can be written as follows:

The preference in situation 1: L1≻L2

By choosing L1 over the L2 the respondent is 100 % sure that he will get Rs. 10 as the prize money. But if he tries to choose L2 over L1 then probability attached to winning Rs. 20 is only 0.10 and probability of winning Rs. 10 is 0.89. Also he may not get anything at all because the probability of getting prize money Rs. 0 is 0.01. So he would choose L1 over L2.

The preference in situation 2: L2'≻L1'

As we can see in lottery L1' people will get prize of Rs. 10 with probability 0.11 and in lottery L2' people will get prize of Rs. 20 with probability 0.10. There is not much of difference in the probabilities and majority of the people have chosen L2' over L1'.

Now consider the following. According to expected Utility theorem, in situation 1, the following can be written:

𝟎 × 𝑼(𝟐𝟎) + 𝟏 × 𝑼(𝟏𝟎) + 𝟎 × 𝑼(𝟎) > 𝟎. 𝟏𝟎 × 𝑼(𝟐𝟎) + 𝟎. 𝟖𝟗 × 𝑼(𝟏𝟎) + 𝟎. 𝟎𝟏 × 𝑼(𝟎)

or, 𝟏 × 𝑼(𝟏𝟎) > 𝟎. 𝟏𝟎 × 𝑼(𝟐𝟎) + 𝟎. 𝟖𝟗 × 𝑼(𝟏𝟎) Subtracting 𝟎. 𝟖𝟗 × 𝑼(𝟏𝟎) from both sides we get:

𝟎. 𝟏𝟏 × 𝑼(𝟏𝟎) > 𝟎. 𝟏𝟎 × 𝑼(𝟐𝟎)

Clearly, if the above equation holds true then L1'≻L2'. But this is a contradiction of the fact that L2'≻L1', which was observed via Allais’s experiments. In this case, theory contradicts what was observed in reality via Allais’s experiments.

So Maurice Allais (1953) tried to show how the independence axiom gets violated in the real world.

5.2 Ellsberg Paradox

Ellsberg Paradox is based on subjective probability theory. In this experiment the respondents were informed that a box contains 300 balls, out of which 100 are red and 200 are either blue or green.

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There are two situations:

Situation 1: Choose between L1 and L2

Lottery L1: The person will get Rs. 500 if the ball is red.

Lottery L2: The person will get Rs. 500 if the ball is blue.

Situation 2 Choose between L1' and L2'

Lottery L1': The person will get Rs. 500 if the ball is not red.

Lottery L2': The person will get Rs. 500 if the ball is not blue.

It was observed, that majority of the respondents chose L1 over L2 in Situation 1, and L1' over L2' in Situation 2. This is a very common outcome because in Situation 1 it is clear that out of 300 balls there are 100 red balls, but is not clear exactly how many blue balls are there out of the remaining 200 balls; so the respondents chose L1 over L2. In Situation 2, again the respondents are clear how many balls out of 300 are not red i.e. 200, but they did not have a clear idea exactly how many balls are not blue out of 300 balls.

So here the experimental outcome can be written as:

The preference in situation 1: L1≻L2 The preference in situation 2: L1'≻L2'

But this kind of preference violates subjective probability theory. Let us see how this happens:

First we define the events R, B, R* and B* as follows:

The ball is red: R The ball is blue: B The ball is not red : R*

The ball is not blue: B*

According to basic axioms of Probability we know:

p(R)=1-p(R*) (Eq.1)

p(B)=1-p(B*)

In light of this now consider the experimental outcomes that were observed.

In Situation 1, we have L1 is preferred over L2, so : p(R).U(500) > p(B).U(500) which gives

p(R) > p(B) (Eq. 2)

In Situation 2 we have L1' is preferred over L2', so : p(R*).U(500) > p(B*).U(500) which gives p(R*) > p(B*) (Eq. 3)

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Studying Eq.1, Eq.2 and Eq.3, we can easily make out that they are inconsistent.

The Ellsberg paradox seems to hold true because people in this experiment tend to think that betting for or against R (RED) is safer than betting for or against B (BLUE).

6. Risk Aversion

Now let us consider the outcome of the lotteries in monetary terms, i.e., now we can will say that the prizes associated with the lotteries are in monetary form. We can assume here if the decision maker’s preferences satisfy Axioms 1to 5 (discussed in Section 3.3 above) then we can make use of an expected utility function to represent his preferences over the lotteries. This way we can figure out the decision maker’s actions over the money gambles, using the expected utility function.

To calculate the expected utility for a money gamble we can make use of the following formula:

𝑬(𝑳^∗) = 𝒑 𝑼(𝒙) + (𝟏 − 𝒑)𝑼(𝒚)

Where x is the prize money which a person gets with probability p and y is the prize money he gets with the probability (1-p).

Let’s use an example to understand the attitude of people towards risk. Note that risk refers to a situation where there is more than a single possible outcome of a decision and the probability of each possible outcome is given (or can be estimated).

Consider a gamble that gives me Rs. 200 with probability 1/2 and Rs. 0 with probability 1/2; and there is another offer which gives me a sure amount of Rs. A. Now the question is whether I should accept the second offer, which is certain, or I should play the gamble.

The expected value which I get by playing the gamble is:

𝑬𝑽 =^𝟏

𝟐𝟐𝟎𝟎 +^𝟏

𝟐𝟎 = 𝑹𝒔.100

If I do not want to take the risk then I will accept Rs. A, which is somewhat close to Rs. 100, but is strictly lower than it. Suppose Rs. A =Rs. 80, then in this case, my expected utility from the lottery is less than the utility of the expected value of the gamble. This means I would prefer a given sum of money with certainty, to a risky asset with a slightly higher expected value.

Here Rs. 80, which is minimum amount which I will accept for not playing the gamble, is known as the ‘certainty equivalent’. It equals the certain amount of money that offers the same utility as the lottery.

We use this concept to define risk premium as follows:

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Risk premium = Expected Value of the lottery - Certainty Equivalent

Here the risk premium is Rs. 20. This much amount I can pay to someone else to take the risk for me. This is the maximum amount that I am willing to pay to avoid the risk.

There are three types of decision makers depending upon their behavior toward risk. Consider the following example. Suppose a person has to choose either one of the following two situations:

Situation 1 : Get Rs. 100 with certainty.

Situation 2 : Get a lottery which yields Rs. 50 with probability ½ and Rs. 150 with probability ½, so that the expected value of the lottery is Rs. 100 (since 100 = ½*50 + ½*150).

A risk averse person would prefer Situation 1 to Situation 2. A risk loving person would prefer Situation 2 to Situation 1; while a risk neutral person would be indifferent between the two situations.

In what follows we discuss the differences between risk averse, risk loving and risk neutral agents in more detail.

1. Risk Averse: If the expected utility of any lottery, L* (𝒑𝑼(𝑿) + (𝟏 − 𝒑)𝑼(𝒀)), is not more than the utility of getting the expected value (𝑼(𝒑𝑿 + (𝟏 − 𝒑)𝒀)) of the lottery for sure, then we can say that the decision maker is a risk averse person.

Figure 11: Risk Averse

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If the decision maker is risk averse then:

𝑼(𝒑𝑿 + (𝟏 − 𝒑)𝒀) ≥ 𝒑𝑼(𝑿) + (𝟏 − 𝒑)𝑼(𝒀)

If ≥ is replace by strict inequality i.e. > then we can say the decision maker is strictly risk averse.

A risk averse person has a concave utility function. The more concave the curve, the more the person will avoid risk.

2. Risk Lover: The decision maker is a risk loving person if the expected utility of any lottery is more than the utility of getting the expected value of the lottery with certainty.

If the decision maker is a risk lover then:

Figure 12: Risk Lover

If ≤ is replaced by a strict inequality i.e. < , then we can say the decision maker is strictly a risk lover.

A risk lover person has a convex utility function. The more convex the curve is, the more the person will love to take risk.

3. Risk Neutral: A decision maker is said to be risk neutral if the expected utility of any lottery is the same as the utility of getting the expected value of the lottery for sure:

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Figure 13: Risk Neutral If the decision maker is risk neutral then:

𝑼(𝒑𝑿 + (𝟏 − 𝒑)𝒀) = 𝒑𝑼(𝑿) + (𝟏 − 𝒑)𝑼(𝒀)

7. Measuring Risk Aversion

The most commonly used method to measure absolute risk aversion was developed by the economists Arrow and Pratt. It is as follows:

𝑹_{𝑨𝒗𝒆𝒓𝒔𝒊𝒐𝒏}(𝒙) = −𝑼^′′(𝒙) 𝑼^′(𝒙)

Where 𝒙= consumer current wealth Note:

 If 𝑹_{𝑨𝒗𝒆𝒓𝒔𝒊𝒐𝒏}(𝒙) > 𝟎 then the person is a risk averse person

 If 𝑹_{𝑨𝒗𝒆𝒓𝒔𝒊𝒐𝒏}(𝒙) = 𝟎 then the person is a risk neutral person

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 If 𝑹_{𝑨𝒗𝒆𝒓𝒔𝒊𝒐𝒏}(𝒙) < 𝟎 then the person is a risk lover person

We can deduce that the risk aversion of a person can change with a change in his wealth. As 𝑹_{𝑨𝒗𝒆𝒓𝒔𝒊𝒐𝒏} is a function of wealth (𝒙).

The following examples demonstrate the above :

Example 1 : If the utility function has a quadratic form such as : 𝑼(𝒙) = 𝒄 + 𝒅𝒙^𝟐+ 𝒆𝒙 where d<0 and e>0

𝑹_{𝑨𝒗𝒆𝒓𝒔𝒊𝒐𝒏}(𝒙) = −𝑼^′′(𝒙)

𝑼^′(𝒙) = − 𝟐𝒅

𝟐𝒅𝒙 + 𝒆< 𝟎

In this case risk aversion increases with an increase in 𝒙 Example 2: If the utility function has a logarithmic form such as : 𝑼(𝒙) = 𝐥𝐧 (𝒙) where x>0

𝑹_{𝑨𝒗𝒆𝒓𝒔𝒊𝒐𝒏}(𝒙) = −𝑼^′′(𝒙) 𝑼^′(𝒙) =𝟏

𝒙> 𝟎

In this case risk aversion decreases with an increase in 𝒙 Example 3: If the utility function has an exponential form such as : 𝑼(𝒙) = −𝒆^−𝜶𝒙 where 𝜶>0

𝑹_{𝑨𝒗𝒆𝒓𝒔𝒊𝒐𝒏}(𝒙) = −𝑼^′′(𝒙) 𝑼^′(𝒙) = 𝜶

In this case risk aversion is not affected with a change in 𝒙

The formula given by Arrow-Pratt is used to calculate only absolute risk aversion. It is used to measure how a person thinks about lotteries, whose prizes are defined in terms of an absolute monetary value. There can be gambles that offer to increase or decrease your wealth by a certain percentage and are known to be relative lotteries. Its prizes are defined relative to the current wealth of a person. So relative risk aversion can be measured by:

𝑹𝑹_{𝑨𝒗𝒆𝒓𝒔𝒊𝒐𝒏}(𝒙) = 𝒙 × 𝑹_{𝑨𝒗𝒆𝒓𝒔𝒊𝒐𝒏}(𝒙) = −𝒙𝑼^′′(𝒙) 𝑼^′(𝒙)

BUSINESS ECONOMICS

PAPER NO. : 1, MICRO ECONOMICS ANALYSIS MODULE NO. : 9, CHOICE UNDER UNCERTAINITY 8. Summary

 Under certain preference axioms, the expected utility theorem is a useful tool to express the utility of a decision maker under uncertain situations.

 The Independence axiom is an important axiom for the expected utility theorem. If this gets violated then the expected utility theorem may not work efficiently.

 The concept of risk aversion can be applied to many situations. It helps in studying attitude towards insurance, as the decision of a person changes with the change in his attitude towards risk.