Logarithmic Utility Function Calculator

Calculator Inputs

Scenario Name

Initial Wealth

Outcome A Gain or Loss

Probability A

Outcome B Gain or Loss

Probability B

Utility Shift

Utility Scale

Vertical Shift

Log Base

Custom Log Base

Certain Gain or Loss

Decimal Places

Normalize Probabilities

Scenario Notes

Formula Used

The calculator uses a shifted logarithmic utility function:

U(x) = a × log_b(W + x + s) + c

Here, W is initial wealth. x is the outcome gain or loss. s is the utility shift. a is the scale. c is the vertical shift. b is the selected log base.

Expected Utility = pA × U(A) + pB × U(B)

Certainty Equivalent = b^{((Expected Utility - c) / a)} - s

Risk Premium = Expected Terminal Wealth - Certainty Equivalent

How to Use This Calculator

Enter the starting wealth for the decision maker.
Enter gains as positive values and losses as negative values.
Add probabilities for both outcomes.
Use the shift field when terminal wealth plus outcome may be zero or negative.
Select the log base and utility scale.
Enter a certain gain or loss for comparison.
Press Calculate to show results above the form.
Use CSV or PDF export for records and reports.

Example Data Table

Scenario	Initial Wealth	Outcome A	Probability A	Outcome B	Probability B	Shift
Investment Case	100000	20000	0.60	-10000	0.40	0
Insurance Case	50000	0	0.80	-15000	0.20	1000
Project Case	250000	70000	0.35	-30000	0.65	0

Logarithmic Utility in Statistics

A logarithmic utility function helps compare uncertain wealth choices. It is common in decision theory, economics, and applied statistics. The curve rises as wealth rises. Yet each extra unit gives less added utility. This pattern is called diminishing marginal utility. It matches many risk averse decisions. A person may prefer a sure moderate gain over a risky larger gain.

Why This Calculator Matters

This calculator estimates utility for two possible outcomes. It also returns expected utility, expected terminal wealth, certainty equivalent, and risk premium. These measures help explain choices under uncertainty. They can support portfolio reviews, insurance examples, business decisions, and classroom models. The shift value lets you keep every log input positive. The scale and vertical shift let you match a custom utility model.

Understanding the Outputs

Expected utility is the probability weighted average of outcome utilities. It is not the same as utility from expected wealth. That difference shows risk aversion. The certainty equivalent is the guaranteed wealth giving the same utility as the risky choice. When expected wealth is higher than the certainty equivalent, the gap is the risk premium. A larger gap means the risky option demands more compensation.

Advanced Interpretation

Marginal utility shows how much utility changes near the expected wealth point. For log utility, relative risk aversion is one. Absolute risk aversion falls as shifted wealth grows. This means the same cash risk matters less to someone with higher wealth. Base selection changes the utility scale, not the ranking, when scale is positive.

Practical Use

Use realistic probabilities and outcome values. Check that probabilities represent the same period. Enter losses as negative values. Add a shift only when wealth plus outcomes can reach zero or below. Review the certainty equivalent before making a final decision. The model is powerful, but it is still an assumption. Actual behavior may include taxes, liquidity needs, goals, and emotions.

Best Practice

Compare several cases. Change one input at a time. Watch how the risk premium moves. Save CSV results for audit trails. Use the report option for quick sharing. Keep notes beside each scenario. This makes future reviews easier and improves statistical judgement. Label each trial clearly before comparing final decisions across scenarios later.

FAQs

What is a logarithmic utility function?

It is a utility model where satisfaction rises with wealth, but at a decreasing rate. This makes it useful for risk averse decisions.

Why must wealth plus shift be positive?

Logarithms are only defined for positive inputs. The shift helps handle cases where losses could make terminal wealth zero or negative.

What does expected utility mean?

Expected utility is the probability weighted average of utility values. It evaluates risky choices using preferences, not only cash amounts.

What is certainty equivalent?

Certainty equivalent is the guaranteed terminal wealth that gives the same utility as the risky choice. It converts utility back into money terms.

What does risk premium show?

Risk premium is the gap between expected terminal wealth and certainty equivalent. It estimates compensation needed for accepting risk.

Can I use negative outcomes?

Yes. Enter losses as negative values. Make sure terminal wealth plus the shift remains greater than zero for each outcome.

Does log base change the final choice?

With a positive scale, changing the base changes the utility scale. It usually does not change the preference ranking.

When should I normalize probabilities?

Use normalization when entered probabilities are weights rather than exact probabilities. The calculator rescales them so their sum equals one.