Question

In deciding to park in an illegal place, any individual knows that the probability of getting a ticket is \(p\) and that the fine for receiving the ticket is \(f\). Suppose that all individuals are risk averse (i.e., \(U^{\prime \prime}(W)<0\), where \(W\) is the individual's wealth). Will a proportional increase in the probability of being caught or a proportional increase in the fine be a more effective deterrent to illegal parking? Hint: Use the Taylor series approx- \\[ \text { imation } U(W-f)=U(W)-f U^{\prime}(W)+\left(f^{2} / 2\right) U^{\prime \prime}(W) \\]

Short answer

Answer: For risk-averse individuals, increasing the probability of being caught is a more effective deterrent to illegal parking than increasing the fine amount.

Step-by-step solution

Write down the initial utility function

The initial utility function is given by: \\[ U(W) \\]

Write down the utility function after receiving a fine

Using the Taylor series approximation provided, we can find an expression for the utility function after receiving a fine, U(W-f): \\[ U(W-f) = U(W) - fU'(W) + \frac{1}{2}f^2 U''(W) \\]

Write down the expected utility when parking illegally

The expected utility when parking illegally can be calculated by taking into account the probability of getting caught (p) and not getting caught (1-p): \\[ EU_illegal = pU(W - f) + (1-p)U(W) \\]

Substitute the expression for U(W-f)

Now, we'll substitute the Taylor series approximation for U(W-f) from Step 2 into the expected utility formula: \\[ EU_illegal = p[U(W) - fU'(W) + \frac{1}{2}f^2 U''(W)] + (1-p)U(W) \\]

Calculate the change in expected utility due to an increase in probability (p)

To analyze the impact of an increase in the probability of getting caught, we'll calculate the partial derivative of the expected utility with respect to p: \\[ \frac{\partial EU_{illegal}}{\partial p} = U(W-f) - U(W) \\]

Calculate the change in expected utility due to an increase in the fine (f)

Similarly, we'll calculate the partial derivative of the expected utility with respect to the fine (f) to analyze the impact of an increase in the fine amount: \\[ \frac{\partial EU_{illegal}}{\partial f} = p\left[-U'(W) + fU''(W)\right] \\]

Compare the effectiveness of increasing p and f

We can compare the two partial derivatives to each other qualitatively: \\[ \frac{\partial EU_{illegal}}{\partial p} = U(W-f) - U(W) \\] \\[ \frac{\partial EU_{illegal}}{\partial f} = p\left[-U'(W) + fU''(W)\right] \\] Observing both expressions, we can see that an increase in the probability of being caught (p) has a direct impact on the utility difference, U(W-f) - U(W). On the other hand, an increase in the fine amount (f) has a mixed effect on the expected utility, as it has both positive and negative impacts (through the risk aversion term fU''(W)). Given all individuals are risk-averse, it is likely that the decrease in utility function would be larger when the probability of being caught is increased compared to increasing the fine amount. Thus, increasing the probability of being caught is a more effective deterrent to illegal parking for risk-averse individuals.

Key concepts

Risk Aversion

Risk aversion is a fundamental concept in microeconomic theory that describes an individual's preference for certainty over uncertainty. An individual who is risk-averse prefers outcomes that are certain and is willing to forego a potentially higher payoff to avoid risk. This behavior is particularly relevant when making financial decisions or evaluating potential losses.

In the context of the illegal parking problem, risk-aversion translates to the individual's inclination to avoid the potential loss of wealth due to a parking fine. Mathematically, risk aversion is represented by the second derivative of the utility function with respect to wealth, \( U''(W) \), being less than zero \( (U''(W) < 0) \). A deeper concavity in the utility function points to a higher degree of risk aversion.

When individuals are risk-averse, they'll likely be more responsive to changes in the probability of incurring a fine (or any negative consequence) rather than changes in the amount of the fine itself. This propensity is because, for a risk-averse person, the increased likelihood of any loss can induce more anxiety and a stronger behavioral response, in this case, deterring from illegal parking.

Expected Utility

Expected utility is a pivotal concept in decision theory and behavioral economics. It represents the anticipated average of all possible outcomes under uncertainty, each weighted by the probability of occurrence. Expected utility combines probabilities and the utility values of outcomes to provide a single number that an individual can use to make a rational decision when faced with uncertain scenarios.

In terms of our parking problem, we calculate the expected utility of engaging in illegal parking \( EU_{illegal} \) by considering both outcomes: getting caught with a probability \( p \) and not getting caught with a probability \( 1-p \).

Due to risk aversion, individuals would consider the negative utility from a possible fine more significantly than the utility without the fine. Thus, even if the expected utility might seem favorable when considering the potential fine as a monetary loss alone, the individual's subjective weighting due to risk aversion can make the decision to park illegally less attractive.

Taylor Series Approximation

The Taylor series approximation is a technique used in mathematics to approximate complicated functions using polynomials. By expanding a function at a certain point, the Taylor series provides an approximate value of the function around that point through a finite sum of its derivatives at the point.

In our scenario, we use the Taylor series to approximate the change in utility \( U(W-f) \) after receiving a fine \( f \) based on the individual's current wealth \( W \). The approximation up to the second order is given by:
\[ U(W-f) \approx U(W) - fU'(W) + \frac{1}{2}f^2 U''(W) \]
Here, \( U(W) \) is the utility at the original wealth, \( U'(W) \) is the first derivative (representing the marginal utility of wealth), and \( U''(W) \) is the second derivative (indicating risk aversion). The Taylor series approximation enables us to simplify the calculation of the expected utility when the fine's exact impact on utility is unknown or too complex to model directly.

By applying the Taylor series approximation to the expected utility function for parking illegally, we see the comparative impact of increasing the fine \( f \) versus the probability of getting caught \( p \). This application illustrates neatly how even intuitive concepts like risk aversion and expected utility can be quantified and compared using the powerful mathematical tool of Taylor series.