Expected Utility

1. Caption evidence
   Observation

   The problem statement on screen includes the utility function written as $u(w)=\ln (w)$.

2. Audio
   Observation

   The speaker says Sam's utility function is given by the ln of w.

$u(w)$

Sam's utility from wealth or money made; in this example it is evaluated by the natural logarithm of wealth.

Defined for positive wealth values used in the example, namely 70,000, 40,000, and 90,000.

1. Caption evidence
   Observation

   The displayed formula uses w inside $\ln (w)$.

2. Audio
   Observation

   The speaker describes w as how much money he makes or wealth.

w

Wealth or monetary payoff associated with a job choice or outcome.

Positive dollar amounts appearing in the example: 70,000, 40,000, and 90,000.

1. Caption evidence
   Observation

   The on-screen formula shows $\ln (w)$.

2. Audio
   Observation

   The speaker repeatedly refers to the ln of 70,000, 40,000, and 90,000.

$\ln$

Natural logarithm function applied to wealth in the utility specification.

Applied to positive wealth values in the example.

1. Caption evidence
   Observation

   The slide writes $u(w)=\ln(w)=\ln(70,000)=11.156$ under Scenario 1.

2. Audio
   Observation

   The speaker says his utility from scenario one is the ln of 70,000, equal to 11.156.

11.156

Numerical utility value for keeping the current job with certain income of $70,000.

Approximate decimal value shown on screen and spoken aloud.

1. Caption evidence
   Observation

   The slide writes $u(w)$=.5\ln(40,000)+.5\ln(90,000) and then $u(w)=5.298+5.7037=11.001$.

2. Audio
   Observation

   The speaker says the new job utility is 11.001.

11.001

Numerical expected utility value for accepting the Berkshire job with uncertain income. Calculation check: the source slide says 11.001, but $0.5 \ln (40,000)+0.5 \ln (90,000)=\ln (60,000)$≈11.002; the job choice is unchanged.

Approximate decimal value shown on screen and spoken aloud.

1. Audio
   Observation

   The speaker says they must compare the utility he will receive from each option, not just the expected wealth from each option.

2. Caption evidence
   Observation

   On-screen text states: 'So we are going to compare two scenarios when deciding which decision Sam will make. We must compare the utility he will receive from each option.'

In this example, the decision rule is to compare the utility associated with each option rather than comparing only expected dollar payoffs. For a risky option, the relevant quantity is the probability-weighted average of utilities across outcomes.

1. There are multiple decision options.

2. At least one option has uncertain outcomes.

3. Preferences are represented by a utility function.

1. Logarithmic utility function

1. Caption evidence
   Observation

   The problem statement on screen gives Sam's utility function as $u(w)=\ln (w)$.

2. Audio
   Observation

   The speaker says Sam's utility function is given by the ln of w.

The video specifies Sam's preferences over wealth using the utility function $u(w)=\ln(w)$, where $w$ denotes wealth or money made. This function converts monetary outcomes into utility values before comparison.

1. w is positive wealth or income.

2. Utility comparisons are made after applying the function to each monetary outcome.

1. $u(w)$
2. w
3. $\ln$

1. Caption evidence
   Observation

   The slide labels 'Scenario 1: Keep current job' with '$70,000' and computes $u(w)=\ln(w)=\ln(70,000)=11.156$.

2. Audio
   Observation

   The speaker says that with 100% certainty he'll make $70,000, so his utility is simply the ln of 70,000.

When an option yields a single certain payoff, its utility is found by directly substituting that payoff into the utility function. Here the certain current-job income of $70,000 gives utility$$\ln(70,000)$$.

1. The option has one outcome with probability 1.

2. The utility function is known.

1. Logarithmic utility function

1. Caption evidence
   Observation

   The slide writes $u(w)$=.5\ln(40,000)+.5\ln(90,000) for Scenario 2.

2. Audio
   Observation

   The speaker explains that 50% of the time Sam gets $40,000 and 50% of the time he gets $90,000, then forms 50% times $\ln (40,000)$ plus 50% times $\ln (90,000)$.

For an option with several possible monetary outcomes, the video computes expected utility by weighting each outcome's utility by its probability and summing the results. The Berkshire job has two outcomes, $40,000 and $90,000, each with probability 0.5.

1. The risky option has discrete outcomes with known probabilities.

2. Probabilities sum to 1.

3. Each outcome is first mapped through the utility function.

1. Logarithmic utility function
2. Expected utility as the comparison criterion

1. Audio
   Observation

   The speaker says since 11.156 is a greater amount of utility, Sam will keep his current job.

2. Caption evidence
   Observation

   The final displayed numerical comparison is between 11.156 and 11.001.

After computing the utility of the certain option and the expected utility of the risky option, the decision is made by selecting the larger value. In the example, $11.156>11.002$, so the current job is preferred.

1. Both options have been evaluated on the same utility scale.

2. The decision maker prefers more utility to less.

1. Utility of a certain outcome
2. Expected utility of a risky option

1. Audio
   Observation

   The speaker concludes that because 11.156 is greater than 11.001, Sam will keep his current job.

2. Caption evidence
   Observation

   The displayed final values are 11.156 for Scenario 1 and 11.001 for Scenario 2.

Given $u(w)=\ln(w)$, the certain current job yielding $70,000 gives utility 11.156, while the risky Berkshire job gives expected utility 11.002, so Sam keeps his current job.

1. Sam's utility function is $u(w)=\ln(w)$.

2. Current job income is certain at $70,000.

3. Berkshire job pays $40,000 with probability 0.5 and $90,000 with probability 0.5.

4. No other considerations affect the decision.

For the specific decision problem stated in the video.

1. Caption evidence
   Observation

   The slide shows Scenario 1: Keep current job, $70,000, then $u(w)=\ln(w)=\ln(70,000)=11.156$.

2. Audio
   Observation

   The speaker says with 100% certainty he'll make $70,000, so plug 70 into the utility function $\ln (w)$, giving 11.156.

1. The audio shorthand 'plug in 70' corresponds to the displayed value 70,000; the on-screen formula is the clearer evidence.

1. Expression

   $$w=70,000$$
   Explanation

   Scenario 1 is the current job, which yields a certain income of $70,000.

   Justification

   Stated directly in the problem text and repeated by the speaker.

   Shown in the video
2. Expression

   $$u(w)=\ln(w)$$
   Explanation

   Apply Sam's given utility function to the certain wealth level.

   Justification

   The problem specifies the utility function $u(w)=\ln(w)$.

   Shown in the video
3. Expression

   $$u(70,000)=\ln(70,000)=11.156$$
   Explanation

   Substitute the certain payoff and evaluate the natural logarithm numerically.

   Justification

   Direct substitution into the given utility function, followed by the numerical evaluation shown on screen.

   Shown in the video

The utility from keeping the current job is 11.156.

1. Audio
   Observation

   The speaker says 50% of the time he does not get a bonus and makes $40,000; the other 50% he gets the bonus and makes $40,000+$50,000=$90,000.

2. Caption evidence
   Observation

   The problem statement says base salary $40,000 and a 0.5 chance of receiving a bonus of $50,000.

1. Expression

   $$p_1=0.5,\quad w_1=40,000$$
   Explanation

   With probability 0.5 Sam receives no bonus, so his total income is the base salary alone.

   Justification

   Given in the problem statement and explained verbally.

   Shown in the video
2. Expression

   $$p_2=0.5,\quad w_2=40,000+50,000=90,000$$
   Explanation

   With probability 0.5 Sam receives the bonus, so his total income is base salary plus bonus.

   Justification

   Given in the problem statement and computed explicitly by the speaker.

   Shown in the video

The Berkshire job has two equally likely outcomes: $40,000 and $90,000.

1. Caption evidence
   Observation

   The slide shows $u(w)$=.5\ln(40,000)+.5\ln(90,000) and then $u(w)=5.298+5.7037=11.001$.

2. Audio
   Observation

   The speaker says the utility function will look like 50% times the ln of 40,000 plus 50% times the ln of 90,000, and the result is 11.001.

1. Expression

   $$EU=0.5\ln(40,000)+0.5\ln(90,000)$$
   Explanation

   Weight each outcome's utility by its probability and add the terms.

   Justification

   This is the expected-utility calculation method used in the video for a discrete risky option.

   Shown in the video
2. Expression

   $$EU=5.298+5.7037$$
   Explanation

   Evaluate the two logarithmic terms numerically as displayed on screen.

   Justification

   Intermediate numerical values are shown on the slide.

   Shown in the video
3. Expression

   $$EU=11.002$$
   Explanation

   Add the two weighted utility terms to obtain the expected utility of the Berkshire job.

   Justification

   Arithmetic summation of the displayed intermediate values.

   Shown in the video

The expected utility from accepting the Berkshire job is 11.002. Calculation check: the source slide says 11.001, but $0.5 \ln (40,000)+0.5 \ln (90,000)=\ln (60,000)$≈11.002; the job choice is unchanged.

1. Caption evidence
   Observation

   The full problem statement is displayed on screen from about 6 seconds onward, including the two job options and $u(w)=\ln (w)$.

2. Audio
   Observation

   The speaker reads the problem and then solves it step by step.

Sam can keep his current job for a certain $70,000, or accept a Berkshire job with base salary $40,000 and a 0.5 chance of a $50,000 bonus. With utility function $u(w)=\ln(w)$, determine whether he keeps the current job, accepts the new job, or is indifferent.

1. Current job payoff is $70,000 with certainty.

2. Berkshire job base salary is $40,000.

3. Bonus amount is $50,000.

4. Probability of bonus is 0.5.

5. Utility function is $u(w)=\ln(w)$.

6. No other considerations matter.

Compare the utility of the two job options and identify Sam's choice.

1. Expression

   $$w_{\text{current}}=70,000$$
   Explanation

   Write the certain payoff for the current job.

   Justification

   Given directly in the problem statement.

   Shown in the video
2. Expression

   $$u(70,000)=\ln(70,000)=11.156$$
   Explanation

   Compute the utility of the certain option by direct substitution into the utility function.

   Justification

   Uses the given utility function and the definition of utility for a certain outcome.

   Shown in the video
3. Expression

   $$w_{\text{new},1}=40,000,\quad w_{\text{new},2}=90,000$$
   Explanation

   List the two possible payoffs of the risky job: no bonus and bonus received.

   Justification

   From the base salary and bonus information in the problem statement.

   Shown in the video
4. Expression

   $$EU_{\text{new}}=0.5\ln(40,000)+0.5\ln(90,000)$$
   Explanation

   Form the expected utility of the risky job by probability-weighting each outcome's utility.

   Justification

   Standard expected-utility computation demonstrated in the video.

   Shown in the video
5. Expression

   $$EU_{\text{new}}=5.298+5.7037=11.002$$
   Explanation

   Evaluate and sum the displayed numerical terms.

   Justification

   Intermediate arithmetic shown on screen.

   Shown in the video
6. Expression

   $$11.156>11.002$$
   Explanation

   Compare the utility of the current job with the expected utility of the new job.

   Justification

   Decision rule stated and applied in the video.

   Shown in the video

Sam keeps his current job.

The conclusion follows from the displayed numerical comparison: the certain job has utility 11.156, while the risky job has expected utility 11.002.

1. Animation
   Observation

   A white title card displays the centered text 'Expected Utility'.

1. White background

2. Centered title text 'Expected Utility'

1. Title text appears and then fades out.

1. No mathematical notation is shown yet.

This segment introduces the topic label only.

1. Diagram
   Observation

   A lecturer appears on the left and a text panel on the right shows the full word problem, including $u(w)=\ln (w)$.

2. Caption evidence
   Observation

   The right-side text states the two job options and asks whether Sam keeps his job, accepts the new job, or is indifferent.

1. Lecturer on left

2. Right-side text panel

3. Problem statement with dollar amounts and $u(w)=\ln (w)$

1. The full problem statement becomes visible and remains on screen while the speaker reads it.

1. The two alternatives and the utility function stay fixed during this interval.

The visual establishes the decision problem and the preference representation used for the calculation.

1. Caption evidence
   Observation

   Additional on-screen text appears: 'So we are going to compare two scenarios when deciding which decision Sam will make. We must compare the utility he will receive from each option.'

2. Audio
   Observation

   The speaker says they are not just comparing expected wealth but comparing utility under uncertainty versus certainty.

1. Existing problem text

2. New explanatory sentence below it

1. A new sentence appears beneath the problem statement explaining the comparison method.

1. The original problem data remain unchanged.

The slide visually shifts from stating the problem to stating the solution principle: compare utilities, not raw payoffs.

1. Diagram
   Observation

   The lower part of the slide adds 'Scenario 1: Keep current job' with '$70,000' and the equation $u(w)=\ln(w)=\ln(70,000)=11.156$.

1. Scenario 1 heading

2. Dollar amount $70,000

3. Equation chain ending in 11.156

1. Scenario 1 label and payoff appear first, then the utility equation is completed to 11.156.

1. The problem statement and method note remain above.

The animation presents the certain alternative and its direct utility evaluation.

1. Diagram
   Observation

   The slide adds 'His utility from scenario 2 is:' followed by $u(w)$=.5\ln(40,000)+.5\ln(90,000), then $u(w)=5.298+5.7037=11.001$.

1. Scenario 2 heading

2. Expected utility expression

3. Intermediate numerical terms

4. Final value 11.001

1. The expected-utility expression appears first, then the intermediate sum $5.298+5.7037$, then the final value 11.001.

1. Scenario 1 result 11.156 stays visible for comparison.

The visual sequence shows how a risky prospect is converted into a single expected utility number.

1. Diagram
   Observation

   Both final numerical results remain on screen together: 11.156 for Scenario 1 and 11.001 for Scenario 2.

2. Audio
   Observation

   The speaker states that because 11.156 is greater, Sam keeps his current job.

1. Scenario 1 utility 11.156

2. Scenario 2 expected utility 11.001

1. No new formula is added; the existing two results are used for the concluding comparison.

1. The two computed utility values remain visible simultaneously.

The final visual state supports the decision rule by placing the two utilities side by side.

1. Audio
   Observation

   The speaker explicitly says, 'we're not just going to be comparing the expected wealth from each option, we're going to be comparing his utility'.

2. Caption evidence
   Observation

   The added on-screen method note says they must compare the utility he will receive from each option.

One might think the choice should be made by comparing expected dollar payoffs directly.

The video stresses that the correct comparison is between utilities, because the decision criterion is expected utility under the given function $u(w)=\ln(w)$, not expected wealth alone.

1. Caption evidence
   Observation

   The problem first gives $u(w)=\ln(w)$, and later the slide applies ln to each monetary outcome in the expected-utility formula.

2. Audio
   Observation

   The speaker first states the utility function and then uses it to compute both scenario values.

The expected-utility computation for the risky job depends on first having the utility function $u(w)=\ln(w)$.

1. Caption evidence
   Observation

   Scenario 1 is evaluated as a single substituted value, while Scenario 2 is evaluated as a weighted sum of two utility terms.

2. Audio
   Observation

   The speaker contrasts the certainty of Scenario 1 with the uncertainty of Scenario 2.

The video contrasts direct utility evaluation for a certain payoff with probability-weighted utility evaluation for a risky payoff.

1. Audio
   Observation

   The speaker uses the inequality between 11.156 and 11.001 to conclude that Sam keeps his current job.

2. Caption evidence
   Observation

   The two final numerical values remain on screen for comparison.

The final conclusion depends on the decision rule that the option with the larger utility value is chosen.

1. Audio
   Observation

   The speaker explains that the decision compares utility from each option, including a risky one.

2. Caption evidence
   Observation

   The slide shows the expected-utility expression for Scenario 2.

1. Expected utility as the comparison criterion
2. Expected utility of a risky option

1. Audio
   Observation

   The speaker says they are not just comparing expected wealth from each option.

2. Caption evidence
   Observation

   The method note says they must compare the utility he will receive from each option.

1. Expected utility as the comparison criterion
2. Do not decide by expected wealth alone when utility is nonlinear

1. Caption evidence
   Observation

   The slide computes $u(w)=\ln(w)=\ln(70,000)=11.156$ for Scenario 1.

2. Audio
   Observation

   The speaker says to plug 70,000 into the utility function.

1. Utility of a certain outcome
2. Derivation of utility for the current job

1. Audio
   Observation

   The speaker splits the risky job into a 50% no-bonus case and a 50% bonus case.

2. Caption evidence
   Observation

   The slide shows .5\ln(40,000)+.5\ln(90,000).

1. Identify the two monetary outcomes of the risky job
2. Expected utility of a risky option

1. Audio
   Observation

   The speaker concludes Sam keeps his current job because 11.156 is greater than 11.001.

2. Caption evidence
   Observation

   The final displayed comparison is between 11.156 and 11.001.

1. Choose the option with higher utility
2. Conclusion of the worked example