MODULE 4: Risk and Reward: Individual Decision

Overview and Course Learning Objectives

Individual decision-making is a foundational area of study within risk management and economics. Understanding how individuals weigh risks against potential rewards provides critical insights into personal financial behavior, market demands, and public policy formulation. Course learning objectives for individual decision-making (CLO 1 and CLO 2) mandate a comprehensive understanding across several key domains:

• Recognizing the fundamental importance of understanding individual decision-making mechanisms.

• Identifying the critical theoretical and operational differences between organizational decision-making models and individual decision-making frameworks.

• Applying Utility Theory to concrete decision scenarios under conditions of uncertainty.

• Recalling and evaluating the Cost/Benefit approach to individual decision-making, specifically incorporating opportunity costs, marginal costs, marginal benefits, sunk costs, and externalities.

Organizational vs. Individual Decision Making

In organizational contexts, risk management decisions are predominantly modeled using expected monetary value ($EV$). Expected monetary value provides a linear, rational benchmark that works remarkably well for explaining business and corporate choices. The mathematical formula for expected value of strategy $j$ is expressed as:

$EV_j = \sum_{i=1}^{n} p_i a_{ij}$

Where $p_i$ represents the probability of outcome $i$ occurring, and $a_{ij}$ represents the payoff or monetary value of outcome $i$ under strategy $j$.

While expected monetary value functions effectively as a normative and predictive model for organizations, it performs poorly when applied to individual human behavior. Individuals do not make choices purely to maximize expected monetary payout. Unlike corporations, which often operate with substantial capital pools and diversified portfolios, individual decisions are profoundly shaped by risk preferences, psychological biases, and starting levels of personal wealth.

Empirical Survey Findings on Decision Preferences

Empirical surveys conducted among university students enrolled in Risk in Business (RMI 2302) consistently demonstrate the failure of expected value as a direct descriptor of individual choice. When presented with controlled gambling scenarios, individual preferences deviate systematically from expected monetary calculations across three primary dimensions.

In Question 1, individuals are asked to choose between two options starting from an initial wealth of $\$0$:

• Option A: A fair flip of a coin. If it lands heads, receive $\$1000$; if tails, receive $\$0$.

• Option B: Receive $\$500$ for sure.

Under expected value calculations, Option A yields 0.50 \times \1000 + 0.50 \times \$0 = \$500$, while Option B yields $1.00 \times \$500 = \$500$. Because both options share the exact expected value of $\$500$, a decision maker choosing purely on expected value would be completely indifferent. If an entire section of over 500 students operated on expected value, choices would split approximately $50\%$ for Option A and $50\%$ for Option B. However, empirical polling shows that nearly $85\%$ of students select Option B (the sure thing). This demonstrates a foundational trait of individual decision-making: all else being equal, individuals tend to prefer certainty to uncertainty.

In Question 2, the parameters are adjusted to test risk tolerance further, starting from an initial wealth of $\$0$:

• Option A: A fair flip of a coin ($\$1000$ for heads, $\$0$ for tails; $EV = \$500$).

• Option B: Receive $\$400$ for sure ($EV = \$400$).

Pure expected value maximization dictates that $100\%$ of participants should choose Option A due to its $\$100$ expected gain over Option B. Nevertheless, empirical results reveal that nearly $75\%$ of class section respondents actively choose Option B. This establishes a second critical principle: individuals are routinely willing to sacrifice expected monetary value in exchange for increased certainty. The threshold at which an individual shifts back to Option A depends on how low the cash-certain amount drops (e.g., $\$300$, $\$200$, or $\$100$).

In Question 3, the base financial condition is altered by shifting starting wealth:

• Initial Condition: You have $\$50,000$ in your pocket.

• Option A: Fair coin flip ($\$1000$ for heads, $\$0$ for tails).

• Option B: Receive $\$500$ for sure.

When starting wealth increases from $\$0$ to $\$50,000$, response patterns invert dramatically. While $\sim 85\%$ of respondents selected Option B when broke, nearly $85\%$ select Option A when holding $\$50,000$. This establishes a third core insight: an individual's existing wealth level directly influences how much risk they are willing to assume. In expected value models for organizational decisions, initial balance sheet wealth is omitted because it does not alter expected monetary payoffs; for individuals, initial wealth is paramount.

Decision Maker Classifications and Utility Theory

To resolve the discrepancies between standard expected value calculations and observed human behavior, economists John von Neumann and Oskar Morgenstern developed Expected Utility Theory in 1947. Utility theory does not attempt to train individuals to alter their choices; rather, it provides a rigorous framework designed to model individual risk preferences accurately.

Decision makers are categorized into three distinct classifications based on their risk posture:

• Risk-Averse: An individual who is always willing to accept a smaller cash-certain amount than the expected monetary value of a gamble. The majority of the human population is risk-averse. Risk-averse mindsets are graphically characterized by concave utility functions.

• Risk-Neutral: An individual who is indifferent between a cash-certain amount and a gamble whose expected monetary value equals that cash-certain amount. Decisions made by risk-neutral individuals align perfectly with expected monetary value calculations. Risk-neutrality is graphically represented as a linear $45^{\circ}$ function.

• Risk-Seeker (Risk-Lover): An individual who demands a cash-certain payout in excess of a gamble's expected monetary value to forfeit the opportunity to gamble. Risk-seeking mindsets are characterized by convex utility functions that curve upward at an increasing rate.

Mathematical Formulation of Utility and Function Properties

Expected Utility Theory evaluates decision strategies by transforming monetary payouts into "utils"—an abstract unit of measurement quantifying satisfaction or benefit derived from wealth—via a specialized utility function $U(X)$. The expected utility equation for strategy $j$ is expressed as:

$EU_j = \sum_{i=1}^{n} p_i U(a_{ij})$

Where $p_i$ is the probability of outcome $i$, and $U(a_{ij})$ represents the utility output generated by the monetary wealth resulting from outcome $i$.

Standard risk-averse utility functions exhibit specific mathematical properties:

1. Wealth Basis: Utility functions are formulated primarily as a function of total wealth $X$.

2. Positive First Derivative ($U'(X) > 0$): Utility is always strictly increasing in wealth. Individuals continuously prefer more wealth to less wealth (more is better).

3. Negative Second Derivative ($U''(X) < 0$): Utility increases at a decreasing rate. This property models diminishing marginal utility of wealth—meaning an additional $\$100$ produces substantially higher incremental utility for a impoverished student than for a multi-millionaire.

Common continuous mathematical functions meeting these conditions include logarithmic functions (such as the natural log $U(X) = \ln(X)$) and root functions (such as the square root function $U(X) = \sqrt{X}$).

Graphical Representation of Utility, Expected Utility, and Certainty Equivalent

Graphing wealth ($X$) on the horizontal axis against utility ($U(X)$) or wealth value on the vertical axis visually demonstrates how risk aversion influences choices under uncertainty.

For a risk-neutral decision maker, the utility line is a straight $45^{\circ}$ line. If an individual possesses $\$50$ in initial wealth and enters a gamble yielding a $50\%$ chance of $\$50$ and a $50\%$ chance of $\$0$ (expected gamble value $EV = \$25$), expected total wealth reaches $\$75$. On the linear risk-neutral function, the value of entering the gamble and the value of receiving $\$75$ cash-certain lie on the exact same point.

For a risk-averse decision maker, the utility curve bows upward (concave). Comparing a $\$50$ wealth gain from $\$50$ to $\$100$ against a $\$50$ wealth gain from $\$250$ to $\$300$, the height jump $U(100) - U(50)$ is visibly larger than $U(300) - U(250)$, illustrating diminishing marginal utility.

When evaluating a gamble between outcome $X_1$ and outcome $X_2$, a straight chord (dashed line) is drawn connecting point $(X_1, U(X_1))$ and point $(X_2, U(X_2))$. The expected utility $E[U(X)]$ corresponds to the vertical coordinate on this linear chord evaluated at expected wealth $E(X)$. Because the utility curve is strictly concave, the direct utility of expected wealth $U[E(X)]$ lies higher than the expected utility of the gamble $E[U(X)]$:

$U[E(X)] > E[U(X)]$

The Certainty Equivalent ($CE$) is the exact dollar amount of guaranteed wealth that provides the identical utility level as the uncertain gamble. Graphically, tracing horizontally from the expected utility coordinate $E[U(X)]$ on the dashed chord over to the concave utility curve identifies the Certainty Equivalent on the horizontal wealth axis. Because the curve is concave, $CE < E(X)$. The difference $E(X) - CE$ represents the risk premium—the dollar amount of expected value an individual is willing to forfeit to eliminate risk completely.

For an individual with initial wealth $\$0$ facing a fair flip yielding $\$1000$ or $\$0$ under square root utility $U(W) = \sqrt{W}$:

• Utility of $\$500$ for sure: $U(500) = \sqrt{500} \approx 22.361\text{ utils}$.

• Expected utility of gamble: $0.50 \times \sqrt{0} + 0.50 \times \sqrt{1000} = 0.50 \times 0 + 0.50 \times 31.6227 = 15.811\text{ utils}$.

• Because $22.361 > 15.811$, the individual chooses $\$500$ for sure.

• Calculating the Certainty Equivalent: Solve $\sqrt{CE} = 15.81137\implies CE = (15.81137)^2 \approx \$249.96$.

• Surrendered Expected Value: $\$500.00 - \$249.96 = \$250.04$. The individual is indifferent between receiving $\$249.96$ for sure and taking the gamble.

Quantitative Applications of Expected Utility

Expected Utility Theory enables precise mathematical modeling across varied individual risk scenarios.

Example 1: Lottery Ticket Purchase

An individual possesses an initial wealth of $\$500$ and operates under a square root utility function $U(W) = \sqrt{W}$. They are offered a lottery ticket for a purchase price of $\$50$. The ticket features a $50\%$ chance of winning $\$125$ and a $50\%$ chance of winning nothing.

• Option 1 (Do Not Buy Ticket): Retain guaranteed initial wealth of $\$500$.   $EU_{\text{No Buy}} = 1.00 \times U(500) = 1.00 \times \sqrt{500} \approx 22.361\text{ utils}$

• Option 2 (Buy Ticket): Spend $\$50$, reducing immediate wealth to $\$450$. If the ticket loses ($50\%$ probability), final wealth remains $\$450$. If the ticket wins ($50\%$ probability), final wealth becomes $\$450 + \$125 = \$575$.   $EU_{\text{Buy}} = 0.50 \times U(450) + 0.50 \times U(575)$   $EU_{\text{Buy}} = 0.50 \times \sqrt{450} + 0.50 \times \sqrt{575}$   $EU_{\text{Buy}} = 0.50 \times (21.2132) + 0.50 \times (23.9792) = 10.6066 + 11.9896 = 22.596\text{ utils}$

Decision Rule: Compare expected utilities. Because $22.596\text{ utils} > 22.361\text{ utils}$, the individual should purchase the lottery ticket.

Example 2: High-Risk Startup Investment

An investor holds an initial wealth of $\$10,000$ and has a utility function $U(W) = \sqrt{W}$. They are presented with a start-up investment requiring $\$5,000$. The venture has an $80\%$ probability of complete failure (losing the entire $\$5,000$ investment) and a $20\%$ probability of success (returning the original $\$5,000$ plus an additional gain of $\$20,000$).

• Option 1 (Do Not Invest / Keep Under Pillow): Retain initial guaranteed wealth of $\$10,000$.   $EU_{\text{Pillow}} = 1.00 \times U(10000) = 1.00 \times \sqrt{10000} = 100.00\text{ utils}$

• Option 2 (Invest): Liquidate $\$5,000$, leaving liquid cash at $\$5,000$. If the startup fails ($80\%$ probability), final wealth is $\$5,000$. If the startup succeeds ($20\%$ probability), final wealth expands to $\$10,000 + \$20,000 = \$30,000$.   $EU_{\text{Invest}} = 0.80 \times U(5000) + 0.20 \times U(30000)$   $EU_{\text{Invest}} = 0.80 \times \sqrt{5000} + 0.20 \times \sqrt{30000}$   $EU_{\text{Invest}} = 0.80 \times (70.7107) + 0.20 \times (173.2051) = 56.5685 + 34.6410 = 91.21\text{ utils}$

Decision Rule: Compare expected utilities. Because $100.00\text{ utils} > 91.21\text{ utils}$, the investor should decline the investment and keep the money under the pillow.

The Cost/Benefit Approach to Individual Decision Making

Beyond pure formal utility models, economists evaluate choice through the Cost/Benefit approach. The underlying decision rule is straightforward: define activity $x$, compute its total costs $C(x)$ and total benefits $B(x)$, and undertake activity $x$ if and only if:

$B(x) > C(x)$

While simple in theory, implementing this decision rule requires identifying and accurately quantifying all implicit costs and non-monetary benefits.

Opportunity Cost and Utility Equivalents

When engaging in activity $x$ prevents an individual from performing activity $y$, the value surrendered by foregoing activity $y$ constitutes the opportunity cost of activity $x$.

Consider an individual scheduled to work a 6-hour shift (10 AM to 4 PM) at Chuck's Chicken Palace. Pay is $\$20\,\text{per hour}$. The individual dislikes working; their reservation price—the absolute minimum monetary compensation required to induce them to work one hour—is $\$15\,\text{per hour}$. A friend invites them to go waterskiing all day Saturday. The individual's direct cost share for boat rental, fuel, and supplies is $\$100$. The individual enjoys waterskiing immensely and values the experience at $\$200$ (their willingness to pay for 6 hours of waterskiing).

Evaluating the decision to go waterskiing:

• Total Costs of Waterskiing $C(\text{skiing})$:

  1. Direct Out-of-Pocket Expense: $\$100$.

  2. Foregone Earned Income (Opportunity Cost): 6\,\text{hours} \times \20/\text{hour} = \$120$.   $C(\text{skiing}) = \$100 + \$120 = \$220$

• Total Benefits of Waterskiing $B(\text{skiing})$:

  1. Direct Enjoyment Value (Utility equivalent of fun): $\$200$.

  2. Avoided Labor Disutility: 6\,\text{hours} \times \15/\text{hour} = \$90$. (Avoiding an unpleasant task provides a real economic benefit equal to the labor reservation price).   $B(\text{skiing}) = \$200 + \$90 = \$290$

Decision Analysis: Since B(\text{skiing}) = \290 > C(\text{skiing}) = \$220$, the net benefit of waterskiing is $\$70$. Alternatively, calculating net monetary benefit of work gives Gross Income ($\$120$) minus Disutility Cost ($\$90$) = $\$30$. Net benefit of waterskiing ($\$200 - \$100 = \$100$) exceeds net benefit of working ($\$30$) by $\$70$. The individual should go waterskiing.

Dollar figures assigned to non-monetary experiences (such as $\$200$ for fun or $\$90$ for disutility of labor) represent utility equivalents, allowing non-financial satisfaction to be directly compared alongside cash transactions.

Marginal Cost and Marginal Benefit Analysis

Decisions frequently involve variable quantities rather than binary choices. In these cases, decision makers evaluate marginal benefit ($MB$)—the additional benefit obtained from undertaking one more unit of an activity—against marginal cost ($MC$)—the additional cost incurred from that extra unit.

Marginal Decision Rule: Continue increasing the level of an activity as long as marginal benefit exceeds marginal cost ($MB > MC$). Stop or reduce the activity the moment $MC > MB$.

Example 1: Overtime Work vs. Social Activity

An employee works at Chuck's Chicken Palace for $\$20\,\text{per hour}$ ($MB = \$20$ fixed). Scheduled work ends at 7 PM, but the employee's fraternity is throwing the biggest party of the year starting at 9 PM. Starting at 7 PM, the manager asks the worker to stay extra hours on an hour-by-hour basis. The marginal costs of working increase over time due to missing the party:

At 10 PM, the marginal cost of working ($\$21$) exceeds the marginal benefit ($\$20$). The employee tells the boss no and goes to the party.

Failure to Distinguish Average from Marginal Metrics: If the worker incorrectly computed average cost across the 4-hour block from 7 PM to 11 PM, Average Benefit equals $\$20/\text{hour}$, while Average Cost equals $\frac{15 + 15 + 18 + 21}{4} = \$17.25/\text{hour}$. Because average benefit ($\$20$) exceeds average cost ($\$17.25$), averaging leads to the flawed choice of working through 11 PM or midnight. Marginal evaluation prevents this error.

Example 2: Study Hours vs. Exam Score

A student prepares for the RMI 2302 Final Exam. Additional study time increases exam performance, but exhibits diminishing marginal returns:

Optimal study time cannot be determined without specifying the marginal cost of studying. If the student establishes that their marginal cost of study time is fixed at equivalent to $6\,\text{exam points per hour}$:

• Hour 1: $MB = 18 > MC = 6$ (Study)

• Hour 2: $MB = 10 > MC = 6$ (Study)

• Hour 3: $MB = 7 > MC = 6$ (Study)

• Hour 4: $MB = 5 < MC = 6$ (Stop Studying)

The student stops studying after 3 hours, achieving an expected score of 80.

Sunk Costs and Their Exclusion from Decision Making

Sunk costs are expenses that have already been incurred or committed to permanently and cannot be recovered regardless of future actions. Sunk costs must be completely excluded from cost/benefit evaluations of future decisions.

Consider an FSU student deciding whether to travel home to Orlando for the weekend (a 500-mile round trip) via private car or via commercial bus (Red Coach / Flix). A round-trip bus ticket costs $\$120$. The student drives $\sim 20,000\,\text{miles per year}$ and tallies annual auto expenses:

Flawed Calculation (Including Sunk Costs): Calculating overall average cost per mile driven:

\text{Average Cost} = \frac{\14,400}{20,000\,\text{miles}} = \$0.72\,\text{per mile}$ \text{Driving Cost for 500 Miles} = 500\,\text{miles} \times \0.72/\text{mile} = \$360.00$

Comparing $\$360.00$ for driving against $\$120.00$ for the bus ticket suggests taking the bus.

Correct Calculation (Excluding Sunk Costs): The annual lease ($\$8,400$) and insurance ($\$2,400$) payments are sunk costs that remain identical whether the vehicle stays parked in Tallahassee or drives to Orlando. The only variable marginal cost incurred by taking the trip is gas and vehicle wear/maintenance ($\$3,600$):

\text{Marginal Variable Cost} = \frac{\3,600}{20,000\,\text{miles}} = \$0.18\,\text{per mile}$ \text{Actual Driving Cost for 500 Miles} = 500\,\text{miles} \times \0.18/\text{mile} = \$90.00$

Because driving costs $\$90.00$ compared to $\$120.00$ for the bus ticket, the student should drive home.

Externalities, Positive Economics, and Normative Economics

Individual decision-making often creates spillover impacts on third parties not directly involved in the transaction. These spillovers are termed externalities:

• Negative Externalities: Actions that generate private benefits for the decision maker but impose uncompensated costs on society or third parties (e.g., choosing to drive a private car adds traffic congestion and environmental pollution for others).

• Positive Externalities: Actions that cost the decision maker directly but create uncompensated benefits for third parties.

Ignoring externalities can lead to socially suboptimal choices. Economic analysis evaluates these choices through two distinct analytical perspectives:

• Normative Economics: Addresses subjective values and asks "what should be." (e.g., "Should government mandate public transportation usage to protect the natural environment?").

• Positive Economics: Addresses objective, testable consequences and asks "what is / what will happen." (e.g., "If private driving is banned, what specific economic impacts will occur in the automotive industry and university enrollment decisions?").

Interactive Concept Checks and Review Questions

Question 1

You have absolutely no money ($\$0$ initial wealth) and are offered a choice:

• Option A: Fair coin flip ($\$1000$ if heads, $\$0$ if tails).

• Option B: Receive $\$500$ for sure.

Result & Rationale: No single correct answer exists for all people, but $\sim 85\%$ pick B due to risk aversion and preference for certainty.

Question 2

You have absolutely no money ($\$0$ initial wealth) and are offered a choice:

• Option A: Fair coin flip ($\$1000$ if heads, $\$0$ if tails; $EV = \$500$).

• Option B: Receive $\$400$ for sure ($EV = \$400$).

Result & Rationale: $\sim 75\%$ pick B, demonstrating willingness to forfeit $\$100$ in expected value to secure certainty.

Question 3

You have $\$50,000$ in your pocket and are offered a choice:

• Option A: Fair coin flip ($\$1000$ if heads, $\$0$ if tails).

• Option B: Receive $\$500$ for sure.

Result & Rationale: $\sim 85\%$ pick A, proving that higher base wealth increases willingness to accept risk.

Question 4

Which of the following statements is/are correct?

• a) Most people are risk neutral.

• b) A risk neutral mindset is graphically represented as a 45-degree linear function.

• c) Both A and B are correct.

• d) Neither A nor B is correct.

Correct Answer: b. Most people are risk-averse, not risk-neutral. A risk-neutral mindset is correctly represented as a linear $45^{\circ}$ line.

Question 5

The UTIL is a function of:

• a) Income

• b) Savings

• c) Wealth

• d) Spending

Correct Answer: c. Utility functions within standard expected utility theory are evaluated as a function of total Wealth.

Question 6

At what point do you tell the boss no, and go party!?

• a) 8 PM

• b) 9 PM

• c) 10 PM

• d) Unable to determine

Correct Answer: c. At 10 PM, marginal cost ($\$21$) exceeds marginal benefit ($\$20$).

Question 7

Given the study score chart, at what point do you stop studying?

• a) After 1 hour of studying

• b) After 3 hours of studying

• c) After 5 hours of studying

• d) Unable to determine

Correct Answer: d (if marginal cost is unknown); or b (if marginal cost is explicitly defined as $6\,\text{points/hour}$).