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When individuals face uncertainty, such as the decision to park illegally and risk a fine, they weigh the possible outcomes using the concept of expected utility. Expected utility helps them to make rational choices by considering the weighted sum of all possible utilities based on their probabilities.

To determine the expected utility of a risky choice, such as illegal parking, two possible outcomes are typically evaluated:

• Getting a ticket, which may involve a financial penalty, impacting wealth and causing a utility loss.
• Not getting a ticket, meaning no penalty and the utility remains unchanged.

The formula for expected utility in this context is:

\[ EU = p[U(W-f)] + (1-p)[U(W)] \]

Here, \(U(W)\) is the utility when no ticket is issued, \(U(W-f)\) is the utility after paying the fine \(f\), and \(p\) is the probability of being caught. Risk-averse individuals, who dislike losses, experience a decrease in expected utility when there is a higher chance or severity of fines.

To simplify complex functions and make them easier to analyze, a Taylor series approximation can be used. This is particularly helpful when examining the utility of wealth after facing a potential penalty.

In this scenario, using Taylor series expansion, we approximate \(U(W-f)\), the utility after a fine, around \(W\):

\[ U(W-f) = U(W) - f U^{\prime}(W) + \left( \frac{f^2}{2} \right) U^{\prime \prime}(W) \]

This approximation breaks down the change in utility into simple terms:

• The first term, \(U(W)\), is the initial utility.
• The second term, \(-f U^{\prime}(W)\), represents the linear loss due to the fine.
• The third term, \(\left( \frac{f^2}{2} \right) U^{\prime \prime}(W)\), accounts for the curvature and risk aversion, making this correction more significant as the fine size increases.

By understanding these terms, one can better comprehend how financial penalties affect individuals’ decision-making.

Partial derivatives are a useful mathematical tool when analyzing how changes in variables, like the probability of getting caught \(p\), or the fine \(f\), affect expected utility. In assessing the deterrent impact of penalties for illegal parking, these derivatives help to determine the sensitivity of the expected utility to changes in \(p\) and \(f\).

For the probability \(p\):

\[ \frac{\partial EU}{\partial p} = [U(W-f)] - [U(W)] \]

This derivative tells us that changes in probability affect utility negatively because, generally, \(U(W-f) < U(W)\) due to risk aversion; hence, increasing \(p\) reduces \(EU\).

For the fine \(f\):

\[ \frac{\partial EU}{\partial f} = p[-U^{\prime}(W) + f U^{\prime\prime}(W)] \]

This indicates that increasing fines also decrease expected utility since \(U^{\prime \prime}(W)<0\) for risk-averse individuals. Understanding how these derivatives work helps us assess which factor, probability or fine, has a larger deterrent effect on illegal activities.