Managerial & Business Economics - Decision-making under uncertainty

Risky Alternatives

  • Lottery Definition: A lottery is any choice or alternative with uncertain consequences. In microeconomics, lotteries are used to model uncertain prospects like investment returns.

  • Describing a Lottery:

    • List all possible outcomes, ensuring they are mutually exclusive.
    • Determine the probability of each outcome (between 0 and 1), ensuring the sum of probabilities equals 1.
    • Determine the monetary consequences (payoffs) of each outcome for the decision-maker.

  • Example: Job Offers

    • Alpha Corp.: Salary of 50,000 with probability 1 (sure thing).
    • Beta Inc.: Salary of 30,000 with 0.5 probability, salary + bonus of 70,000 with 0.5 probability.

  • Expected Value: Calculated by multiplying each payoff by its probability and summing the results. Represents the average payoff if the lottery is repeated many times.

    • Alpha Corp. Expected Value: EVA = pA × IA = 1 × $50,000 = $50,000
    • Beta Inc. Expected Value: EVB = pB1 × IB1 + pB2 × IB2 = 0.5 × $30,000 + 0.5 × $70,000 = $50,000

  • Risk: Measures how spread out a lottery's payoffs are. Beta Inc.'s job offer is riskier than Alpha Corp's.

    • Variance of payoffs is a common measure of a lottery's risk.

Expected Utility

  • Preferences Over Lottery Outcomes: A decision-maker’s preferences over outcomes aren't sufficient to determine preferences over lotteries.

  • Independence Axiom: States a decision-maker’s preferences over lotteries should depend only on how the lotteries differ. This is a requirement of rationality under uncertainty. Positive (describes rationality) and normative (preferences must satisfy the axiom to be rational).

  • Job Offers with Recession Risk (Example)

    • Alpha Corp.: 50,000 with 0.9 probability, $0 with 0.1 probability.
    • Beta Inc.: 30,000 with 0.45 probability, $70,000 with 0.45 probability, $0 with 0.1 probability.

  • Applying the Independence Axiom: Choice of job should be independent of recession risk. Since both jobs have a 0.1 probability of resulting in $0 income, this factor should not influence the decision.

  • Expected Utility Definition: The sum of the utilities from all possible outcomes of a lottery, weighted by their probabilities. If preferences satisfy the independence axiom, they can be represented by expected utility.

  • Example: Harry's Preferences

    • Utility function: U(I) = \sqrt{I}
    • Alpha Corp. Expected Utility: EUA = pA × \sqrt{IA} = 1 × \sqrt{50,000} ≈ 224
    • Beta Inc. Expected Utility: EUB = pB1 × \sqrt{IB1} + pB2 × \sqrt{IB2} = 0.5 × \sqrt{30,000} + 0.5 × \sqrt{70,000} ≈ 219
    • Harry prefers the job with Alpha Corp.

  • Exercise: Expected utility example with recession

    • Alpha Corp.: IA1 = $50,000withwithpA1 = 0.9,and, andIA2 = $0withwithpA2 = 0.1

    • Beta Inc.: IB1 = $30,000withwithpB1 = 0.45,,IB2 = $70,000withwithpB2 = 0.45,and, andIB3 = $0withwithpB3 = 0.1

    • Solution:

      • EUA = pA1 × \sqrt{IA1} + pA2 × \sqrt{IA2} = 0.9 × \sqrt{50,000} + 0.1 × \sqrt{0} ≈ 0.9 × 224 + 0.1 × 0 = 201.6
      • EUB = pB1 × \sqrt{IB1} + pB2 × \sqrt{IB2} + pB3 × \sqrt{IB3} = 0.45 × \sqrt{30,000} + 0.45 × \sqrt{70,000} + 0.1 × \sqrt{0} ≈ 0.45 × 173 + 0.45 × 265 + 0.1 × 0 = 197.1
      • Harry chooses Alpha Corp.

Risk Preferences

  • Stock Options vs. Bonus Check: Choosing between risky stock options and a sure cash bonus.

    • Stock options: 1000 with 0.6 probability, $16,000 with 0.4 probability.
    • Cash Bonus: 7000 (sure thing).
    • Both options have the same expected value.

  • Utilities of Outcomes: Utility function U(I) = \sqrt{I/1000}.Includescalculationsforutilityof. Includes calculations for utility of1000, $16,000, and $7000.

  • Expected Utilities of Alternatives

    • Expected utility of stock options: EU=0.6×1+0.4×4=2.2EU = 0.6 × 1 + 0.4 × 4 = 2.2

  • Risk Aversion and Diminishing Marginal Utility: Risk-averse individuals prefer a sure thing over a lottery with the same expected value. This results from diminishing marginal utility.

  • Risk Loving: A decision-maker is risk-loving if they prefer a lottery to a sure thing with the same expected value; results from increasing marginal utility.

  • Risk Neutrality: A decision-maker is risk-neutral if indifferent between a lottery and a sure thing with the same expected value. characterized by linear utility functions, expressed as U(I)=α+βIU(I) = α + βI, where α is a constant, and β is a positive constant.

  • When Risk-Averse Take Risks: Risk-averse individuals may take risks if the potential reward is high enough.

  • Risk Premium: The difference between the expected value of a lottery and the sure-thing payoff with the same expected utility. To find risk premium: EU=U(EV−RP)EU = U(EV − RP)

  • Risk Premium Exercise

    • Lola's utility function: U(I)=2IU(I) = 2\sqrt{I}
    • Lottery: 2500 with 0.5 probability, $1600 with 0.4 probability, $400 with 0.1 probability.
    • Solution:
      1. Expected value: EV = p1 × I1 + p2 × I2 + p3 × I3 = 0.5 × 2500 + 0.4 × 1600 + 0.1 × 400 = $1930
      2. Expected utility: EU = p1 × 2\sqrt{I1} + p2 × 2\sqrt{I2} + p3 × 2\sqrt{I3} = 0.5 × 2\sqrt{2500} + 0.4 × 2\sqrt{1600} + 0.1 × 2\sqrt{400} = 0.5 × 2 × 50 + 0.4 × 2 × 40 + 0.1 × 2 × 20 = 86
      3. Risk premium: solving equation EU = U(EV − RP),weget, we get86 = 2\sqrt{1930 − RP},then, thenRP = $81.

Insuring Against Risk

  • The Risk of an Accident: Scenario where Sena faces a 24% chance of a car accident costing 40,000. Sena's income is 62,500andutilityfunctionis62,500 and utility function isU(I) = \sqrt{I}.

    • Lottery: 62,500 with 0.76 probability, $22,500 with 0.24 probability.
    • Expected Value: EV = 0.76 × 62,500 + 0.24 × 22,500 = $52,900

  • Insurance: Sena can purchase insurance to compensate for accident damages. The premium is a fee, F, regardless of accidents. The policy pays out 40,000incaseofanaccident;sure−thingpayoffof40,000 in case of an accident; sure-thing payoff of62,500 - F.

  • Fairly Priced Insurance Policy: Premium equals the expected value of the insurance payment and provides a sure-thing payoff equal to the lottery’s expected value without insurance.

  • Fair Price for Insurance: Fair price is the sum of payments weighted by probability. For Sena: 0.24 × $40,000 = $9600. The sure-thing payoff is 62,500 - 9600 = $52,900.

  • Sena's Preference for Insurance: Without insurance: expected utility is EU=p1×I1+p2×I2=0.76×62,500+0.24×22,500=0.76×250+0.24×150=226EU = p1 × \sqrt{I1} + p2 × \sqrt{I2} = 0.76 × \sqrt{62,500} + 0.24 × \sqrt{22,500} = 0.76 × 250 + 0.24 × 150 = 226. With insurance: U(52,900)=52,900=230U(52,900) = \sqrt{52,900} = 230.

  • Problems with Fairly Priced Insurance: The 'fair price' doesn't account for the insurer's operational costs or capital reserves for correlated risks; premiums will typically exceed the fair price.

  • Moral Hazard: Purchasing a fair insurance policy may affect the way you drive and maintain your car. Introducing a deductible can mitigate this issue.

  • Sena's Willingness-to-Pay for Insurance: The maximum price Sena would pay is the fair price plus his risk premium. To find Sena's risk premium we need to solve the equation EU=U(EV−RP)EU = U(EV − RP), then RP = $1824. Maximum price = 9600 + 1824 = $11,424.