Microeconomics: Uncertainty P . v. Mouche Spring 2020 Wageningen University
Utility maximisation Three main types for setting of utility maximisation: standard (we studied in the first part of the course); inter temporal (Chapter 9; we did not study); with uncertainty (we now are going to study). Understanding the standard setting is very efficient for dealing with the other two.
Motivation It is not true that a (‘right-thinking’) consumer always prefers outcomes with a higher expected money value. In order to see this, we consider two examples. Example 1 Getting 100 . 000 e with chance 1 / 80 has expected value 1250 e . Getting 1000 e for sure has expected value 1000 e . What would You choose?
Motivation (ctd.) Example 2 Sint-Petersburg paradox (Daniel Bernouilli, 1738). Suppose a fair coin (with a heads and tails side) is tossed until it comes up heads. Your payoff depends on the number of tosses before heads appears for the first time. When this number is k , You receive 2 k e . Note that the expected money value of this game = 1 22 + 1 44 + 1 88 + · · · = + ∞ . What is the maximum amount You will pay in order to play this game?
Motivation (ctd.) The Sint-Petersburg paradox is a classical situation where a naive decision criterion (which takes only the expected money value into account) would recommend an action that no (real) ‘right-thinking’ person would be willing to take. The above two examples make clear that we need some concept other than expected (money) value to analyse how people make decisions in risky situations. Expected utility is such a concept. This chapter provides the basics for dealing with uncertainty. It has many applications, for instance in insurance (see Exercise 2 below).
The model Consider a decision maker with uncertainty about the outcomes of given choices he can made. The outcomes might well be consumption bundles, amounts of money, or anything at all. The choices the decision maker can make concern gambles. There are two types of gambles: simple gambles and compound gambles. Interpretation of a (simple) gamble: a lottery ticket with various possible outcomes that occur with a certain probability. Interpretation of a compound gamble: outcomes are themselves (simple or not simple) gambles.
The model (ctd.] Fundamental assumption: the decision maker has a preference relation � (first part of the course!) over the set of gambles and makes choices according to this relation. In particular the decision maker has preferences about the for sure outcomes. SIMPLIFYING ASSUMPTIONS: 1. We only consider simple gambles. 2. The outcomes are amounts of money; a negative amount means a loss.
The model (ctd.] Notation for a gamble with n outcomes c 1 , c 2 . . . , c n that occur with probabilities π 1 , π 2 , . . . , π n respectively: (( c 1 , π 1 ) , ( c 2 , π 2 )) , . . . , ( c n , π n )) . So π 1 + π 2 + · · · + π n = 1. If π i = 1 for some i , then this gamble leads to outcome c i for sure. We denote such a gamble also by ( c i , 1 )) or just by ( c i ) . For our purposes it is sufficient to consider further only gambles with two possible outcomes c 1 , c 2 .
Preferences Suppose Mariko has 1000 yen and can participate in a gamble that gives her a 50 percent probability of winning 500 yen and a 50 percent probability of loosing 500 yen. Not participating concerns the gamble g np = ( 1000 yen , 1 ) . Participating concerns the gamble g p = (( 1500 yen , 1 / 2 ) , ( 500 yen , 1 / 2 )) . It depends on Mariko’s preferences whether she will participate or not. As in the first part of the course we are going to represent preferences (but now for the specific case of lotteries) by utility functions, so-called Von-Neumann-Morgenstern utility functions.
Von-Neumann-Morgenstern utility Theorem: under reasonable conditions (including completeness, transitivity and monotonicity of the preference relation � ) there exists a utility function U defined on the set of gambles representing � . There even exists a utility function U with the expected utility property, i.e. it assigns to each gamble the expected value of the utilities that might result: U (( c 1 , π 1 ) , ( c 2 , π 2 )) = π 1 U ( c 1 ) + π 2 U ( c 2 ) . U is unique op to positive affine transformations. Thus utility here is cardinal (and not ordinal)! Remark: some people say ‘linear’ instead of ‘affine’.
Von-Neumann-Morgenstern utility (ctd.) Such an utility function U is called a ‘Von-Neumann-Morgenstern utility function’. Important observation: when we know the value of von-Neumann-Morgenstern function on all sure outcomes, i.e. when we know all U ( c ) , then we know it on all gambles. The distinction between ordinal and cardinal utility has been explained in the first part of the course. Concerning this remember: A positive affine transformation of a utility function U ( c ) is a utility function of the form aU ( c ) + b with a > 0. � U ( c ) is not a positive affine transformation of U ( c ) , but it is an ordinal transformation.
Risk Given a Von-Neumann-Morgenstern utility function U and a gamble g = (( c 1 , π 1 ) , ( c 2 , π 2 )) , the following three notions are very fundamental: U ( g ) : expected utility of g; E ( g ) = π 1 c 1 + π 2 c 2 : expected money value of g ; U ( E ( g )) : utility of expected money value of g .
Types of risk Suppose the decision maker is given a choice between accepting the gamble g on the one hand or receiving with certainty the expected money value E ( g ) of g on the other. He is risk loving at g if U ( E ( g )) < U ( g ) ; i.e. the gamble is preferred above its expected money value risk neutral at g if U ( E ( g )) = U ( g ) ; i.e. indifferent between the expected money value and the gamble. risk averse at g if U ( E ( g )) > U ( g ) ; i.e. the expected money value is preferred above the gamble.
Types of risk (ctd.) This definition together with the expected utility property of U imply that for g = (( c 1 , π 1 ) , ( c 2 , π 2 )) we have risk loving at g if U ( π 1 c 1 + π 2 c 2 ) < π 1 U ( c 1 ) + π 2 U ( c 2 ) ; risk neutral at g if U ( π 1 c 1 + π 2 c 2 ) = π 1 U ( c 1 ) + π 2 U ( c 2 ) ; risk averse at g if U ( π 1 c 1 + π 2 c 2 ) > π 1 U ( c 1 ) + π 2 U ( c 2 ) .
Types of risk (ctd.) The decision maker is risk loving if he is risk loving at all gambles; Mathematically this means that the function U is strictly convex (increasing marginal utility). risk neutral if he is risk neutral at all gambles; Mathematically this means that the function U is affine (constant marginal utility). risk averse if he is risk averse at all gambles; Mathematically this means that the function U is strictly concave (decreasing marginal utility). Examples: U ( c ) = c 2 is strictly convex, U ( c ) = c is affine, √ U ( c ) = c is striclty concave.
Exercise 1 Heinrich has Von-Neumann-Morgenstern utility function which for a sure outcome c is given by U ( c ) = √ c . So √ c 1 + π 2 √ c 2 . U (( c 1 , π 1 ) , ( c 2 , π 2 )) = π 1 Suppose Heinrich has 28 e and he can participate in a card game (with standard cards). If he gets a red card, then he obtains 36 e . Otherwise he has to pay 12 e . Is Heinrich risk loving? Will Heinrich participate in the gamble?
Exercise 1 (ctd.) SOLUTION As the square root function is strictly concave, Heinrich is not risk loving, but risk averse. Not participating corresponds to the gamble g np = ( 28 , 1 ) and participating to the gamble g p = (( 64 , 1 / 2 ) , ( 16 , 1 / 2 )) . We have √ √ U ( g np ) = U ( 28 , 1 ) = 1 · 28 = 28 = 5 . 29 .., U ( g p ) = U (( 64 , 1 / 2 ) , ( 16 , 1 / 2 )) = 1 64 + 1 √ √ 16 = 6 . 2 2 As U ( g p ) > U ( g np ) , Heinrich will participate (although he is risk averse).
Exercise 2 Socrates is an expected utility maximiser with von Neumann-Morgenstern utility function U ( c ) = c . Socrates owns just one ship. The ship is worth 200 million e . If the ship sinks, Socrates loses 200 million e . The probability that it will sink is 2 / 100. Socrates’ total wealth, including the value of the ship is 225 million e . What is the maximum amount x that Socrates would be willing to pay in order to be fully insured against the risk of losing his ship?
Exercise 2 (ctd.) SOLUTION No insurance is the gamble g ni = (( 225 , 98 2 100 ) , ( 25 , 100 )) and insurance with price x is the gamble g i = ( 225 − x , 1 ) . The maximum amount x satisfies U ( g ni ) = U ( g i ) . As U ( g ni ) = 98 100 U ( 225 )+ 2 100 U ( 25 ) = 98 100 · 225 + 2 100 · 25 = 221 , U ( g i ) = 1 · U ( 225 − x ) = 1 · ( 225 − x ) = 225 − x , we find x = 4 million e .
John von Neumann John von Neumann (1903 – 1957). Enjoy (if You like) the following videos about the genius von Neumann: https://www.youtube.com/watch?v=vqzkyWIfe0I https://www.youtube.com/watch?v=X4cbFtbfm-8 (in French)
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