Decisions Under Risk and Uncertainty

Uncertainty in Pervasive What is uncertain in economic systems? Tomorrows prices Future wealth Future availability of commodities Present and future actions of other people What are rational responses to uncertainty? Buying insurance (health, life, auto) A portfolio of contingent consumption goods

States of Nature Possible sates of nature: car accident (a) no car accident (na) accident occurs with probability a, doesnt with prob na a + na = 1 accident causes loss $L

Contingencies a contract implemented only when a particular state of Nature occurs in state-contingentex. insurer pays only if there is an accident aleatory contract: an ex. = insurance contract performance is required only in specific states of nature (an insured accident)ex. warranty repair/replacement has to occur ONLY if product fails (might/might not) state-contingent consumption plan: implanted only when a particular state of Nature occurs what will be consumed in each different states nature each different outcome of the random processex. take a vacay only if there is no accident discussed after expected utility for decision making

Ex. $100 now and contemplating buying a lottery ticket number 13, if 13 drawn holder will be paid $200, ticket costs $5 2 outcomes: ticket drawn, it isnt drawn original endowment $100 if didnt purchase, $100 id it isnt drawn buy for $5, wealth of $295 if ticket is winner and $95 is it isnt

Uncertainty vs. Risk A Digression Uncertainty: situation where an action has more that one possible result and the probability of each result is unknown Risk: situation where an action has more that one possible result and the probability of each result in known Decision-making under risk can be modeled similar method can be used to model decisions under uncertainty if decision maker assigns subjective probabilities to each outcome

Expected Utility as Basis for Decisions Decisions under risk are modeled using utility of wealth (von Neumann-Morgenstern)& expected UTILITY of a risky prospect, as opposed to expected monetary value of the risky venture before von Neumann and Morgenstern, there was Bernoullis St. Petersburg Paradox

Bernoullis St. Petersburg Paradox he considered decision-making under risk noted that decisions are NOT made basis of expected monetary value, but on other St. Petersburg Paper 1738 example is extreme but makes the point he reasons like this: imagine a gambling opp based on tossing a fair coin (prob of heads = 0.5) if coin lands head on 1st toss gambler receives 2 ducats if coin lands tails on 1st toss, heads on 2nd toss gambler receives 4 ducats if coin lands tails first 2 tosses, heads on 3rd toss gambler receives 8 ducats if coin lands heads on 4th toss gambler receives 16 ducats each time coin comes up as tails amount won if coin comes up head on next toss is doubles

possible winningsComputing Expected Value = Sum of (Winnings)*(Prob) - sum of last column is infinite

Expected Money Value of the Game is Infinite But: no one would pay an infinite amount to play this game no one (conjectured Bernoulli) would pay everything he/she had the amount of people will pay to play the game is strictly finites, while expected payoff is infinite concludes Bernoulli is cannot be true that people make decisions under risk on basis of expected monetary value

Decisions Based on Expected Utility people differ in their attitudes towards risk some more frightened of loss than others less likely to take risk than others less afraid of loss 2 people might agree on prob of gain/loss, and on expected monetary value of a risky prospect one will take the risk and other will not people differ in their degree of risk aversion: how mush risk is preferred

von Neumann & Morgenstern Bernoulli stated utility of wealth as: utility resulting from an small increase in wealth will be inversely proportional to the quantity of goods previously possessed Neumann & Morgenstern idea more mathematically precise with utility of wealth function that is concave downward Utility of wealth increases, but at a decreasing rate

von Neumann & Morgenstern Utility Functions Expected Utlity: given VNM utility function, the weighted average of gambles utility, where each utility is weighted by its probability von Neumann-Morgenstern (VNM) Utility Function: utility function expressed as the expected value of utility of an uncertain event such that, if U(ci) = utility associated with event ci; pi = prob of event ci & n possible eventsVNM (expected) utility:

Preferences under Uncertainty (Risk) gamble using fair coin win $90 with prob 0.5 and win $0 with prob 0.5 expected money value of gamble isEM = 0.5(90) + (0.5)(0) = $45

Preferences under Uncertainty gamble using fair coin win $90 with prob 0.5 and win $0 with prob 0.5 U($90) = 12, U($0) = 2 givenExpected utility (EU) = (0.5)*U($90) + (0.5)*U($0) = 7

Attitudes Toward Risk Risk-averse: individual always preferring a sure thing to a risky gamble with the same expected outcome as the sure thing Risk-loving (risk-affine): individual always preferring a risky gamble to a sure thing is the expected outcome of the gamble = outcome of the sure thing Risk-neutral: individual who is indifferent between a sure thing and a risky gamble with the same expected outcome as the sure things

Preferences Under Uncertainty EM = $45 assumer different VNM utility of wealth functions such that EU = 7 for all of them U($45) > 7 $45 for sure is preferred to the lottery risk-aversion U($45) < 7 the lottery is preferred to $45 for sure risk-loving U($45) = 7 lottery is preferred equally to $45 for sure risk-neutral

VNM utility Risk-AverseVNM utility Risk-lovingVNM utility Risk-Neutral

Expected VNM Utility of a Risky Gamble vs. VNM Utility of Expected Value of a Risky Gamble Careful not to incorrectly calculate VNM Utility associated with the expected income of the consumer instead of the expected utility If incorrectly calculated VNM utility associated with the expected income of the consumer You will have calculated the VNM utility for a guaranteed income equal to the expected income

Example: Job Search Jason earns $49,000 annually (after tax) in his current bookkeeping job Thinking of changing jobs, to work in software industry Believes he has change of getting programmers job which will pay him $64,000 (after tax) annually Is he looks for new job lose current Employer regards job search as sign if disloyalty so job will be lost forever If he doesnt get the programmers job can get entry-level job at software firm with pays $25,000 (after tax)

Job Search Decision Utility of income (proxy for utility of wealth) U(Y) = = ((0.1)Y)1/2 Utility of 3 possible incomes: U(64000) = 80 (programmers job) U(25000) = 50 (entry-level job) U(49000) = 70 (current) Believes his prob of getting $64k job 50% so prob of entry-level job ($25k) = 50% (1st belief set) If he quits his expected income from software firm : 0.5*(64) + (0.5)*(25) = 44.5 less that he is currently earning (49)

if these are his beliefs easy to see that he is better off at current job even if we evaluate decision based off WRONG data expected utility (right criteria): 0.5(80) + 0.5(50) = 65 less than his current utility of income (70) better of staying at new job

instead believes prob of getting $64k job = 65%, so prob $25k = 35% (2nd belief set) if he quits current job: expected income from software firm: 0.65(64) + 0.35(25) = 50.35 more than current income (49) will he want to take the risk of quitting and applying to software firm? Expected utility (right criteria) if he applies: 0.65(80) + 0.35(50) = 69.5 less than current utility of income (70) should stay at current even though his expected income from risky prospect is higher his expected utility is lower so he will not quit his job

believes prob of getting $64k job = 90%, so for $25k = 10% (3rd belief set) if he quits, expected income from software firm: 0.9(64) + 0.1(25) = 60.1 more that current income expected utility 0.9(80) + 0.1(50) = 77 more than current utility better off quitting and taking his chances with the software firm

Expected Income, Expected Utility Combinations expected income: (2nd set) expected utility

= prob of high-income outcome expected income: linear combination of income in the high state and income in the low state expected utility: linear combination of utility in the high state and utility in the low state exact value of expected income/utility depends on value assigned to line combining all the income/utility all resulted so far calculated must be on the line as prob of success increased combination of income/utility moves up & right along the line practical use: can illustrate (and help determining) the certainty equiv) of a risky prospect which is used to determine the individuals risk premium risk premium: indicators of the individuals degree of risk aversion

Certainty Equivalent & Risk Premium certainty equiv: fixed income an individual would accept instead of taking a gamble; so that indiv would be indifferent in taking the fixed income and taking the gamble; VNM utility of fixed income = expected VNM utility of gamble risk premium: difference between expected value of a gamble and the gambles certainty equiv

Certainty Equivalent (CE) find such ath expected utility of the uncertain prospect (applying for a job with software firm) = utility of the no-risk prospect income 49000 is the CE of what gamble? Need to solve:U(YNO RISK) = [U(64)] + (1-)[U(25)] 70 = [80] + (1-)[50] 70 = 80 + 50 50 20 = 30 = 2/3 Next step: sub computed prob into expression for expected monetary value (expected income) E[income] = (2/3)64 + (1/3) 25 = 51 EMV of gamble = $51,000 Compare to certain income that yields utility of 70 CE ($49,000) risk prem in this ex = $2000

Application? Does tax of gains/losses from risky investments increase the amount of venture capital risked? Decrease it? leave it the same? If gains are taxed, but losses can be deducted, we find that taxation (of a type of capital gain) can INCREASE amount of risky investment

Risk Spreading: each consumer spreads his risk over all of the other consumers and thereby reduces the amount of risk he bears Asymmetric Info: each person knows something that the other people do no, try to use it to their adv

TV game show: Choice between walking away with a fixed sum of money $< or staking it on a game called Risk Decides to play Risk host will toss coin H = get $200, T = $0