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 / Suppose that all individuals are risk averse (that is, \(U^{\prime \prime}(W)<0\), where Wis 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? \([\text {Hint}\) : Use the Taylor \(\text { series approximation }\left.U(W-f)=U(W)-f U^{\prime}(W)+-U^{\prime \prime}(W) .\right]\)

Short answer

Answer: A proportional increase in the fine amount is a more effective deterrent to illegal parking as it results in a larger reduction in expected utility from parking illegally.

Step-by-step solution

Expected utility for legal parking

We know that the utility when parking legally is simply the utility of the current wealth, which is given by \(U(W)\).

Expected utility for illegal parking

Expected utility when parking illegally can be found using the hint provided. When parking illegally, there's a probability \(p\) of getting a ticket and having a wealth of \((W-f)\), and a probability \((1-p)\) of not getting a ticket and having a wealth of \(W\). Therefore, the expected utility of parking illegally is: \[E[U(W)]=pU(W-f)+(1-p)U(W)\] Now, using the Taylor series approximation provided: \[U(W-f)=U(W)-f U^{\prime}(W)+\frac{f^{2}}{2} U^{\prime \prime}(W)\] Replace \(U(W-f)\) in \(E[U(W)]\) by the approximation: \[E[U(W)]=p[U(W)-f U^{\prime}(W)+\frac{f^{2}}{2} U^{\prime \prime}(W)]+(1-p)U(W)\] Now, let's examine the effect of a proportional increase in probability and fine on the expected utility of illegal parking.

Effect of a proportional increase in probability:

Let the new probability be \(p'=kp\), where \(k>1\). Expected utility with the increased probability is: \[E[U(W)]=p'[U(W)-f U^{\prime}(W)+\frac{f^{2}}{2} U^{\prime \prime}(W)]+(1-p')U(W)\] Substitute \(p'=kp\) and analyze the impact on expected utility: \[E[U(W)]=kp[U(W)-f U^{\prime}(W)+\frac{f^{2}}{2} U^{\prime \prime}(W)]+(1-kp)U(W)\]

Effect of a proportional increase in fine:

Let the new fine be \(f'=kf\), where \(k>1\). Expected utility with the increased fine is: \[E[U(W)]=p[U(W)-(kf) U^{\prime}(W)+\frac{(kf)^{2}}{2} U^{\prime \prime}(W)]+(1-p)U(W)\] Now, compare the impacts of both increases on the expected utility of illegal parking.

Comparison of increased probability and increased fine:

In both cases, the deterrent effect comes from the reduction in the expected utility of illegal parking. The higher the reduction in expected utility, the more effective the deterrent is. After comparing the expected utility functions for the increased probability and increased fine, it can be seen that the reduction in utility is greater when the fine is increased compared to when probability is increased. This is because the term involving the fine (both linear and quadratic) has a greater effect on the overall expected utility compared to the term involving probability. Thus, a proportional increase in the fine will be a more effective deterrent to illegal parking as it results in a larger reduction in expected utility from parking illegally.

Key concepts

Risk Aversion

Understanding risk aversion is essential when evaluating individual choices under uncertainty. Risk-averse individuals prefer a sure thing to a gamble with the same expected value. For example, given the option between receiving a guaranteed \(50 or having a 50% chance of receiving \)100, a risk-averse person would choose the guaranteed $50.

The textbook exercise alludes to a utility function 'U' where a second derivative that is less than zero (U''(W) < 0) indicates risk aversion. This characteristic of their utility function means that as wealth increases, the additional satisfaction from an extra dollar decreases. In the context of the parking problem, risk-averse individuals would value the certain outcome of legal parking over the risk of receiving a fine through illegal parking.

Probability

At the heart of making decisions under uncertainty is the concept of probability. It quantifies the likelihood of an event occurring. In the textbook exercise, p represents the probability of an event—receiving a parking ticket if one parks illegally. When parking illegally, there is a 'p' chance of incurring a fine, and a '(1-p)' chance of avoiding it.

Greater probability or increased fines both negatively affect a risk-averse individual’s expected utility, but it is vital to quantify these changes to understand their deterrent effects. Understanding how small changes in probability influence decisions can illuminate the nuances of behavioral economics and decision-making processes under risk.

Taylor Series Approximation

Tackling complex problems in economics often involves simplifying assumptions or approximations, such as the Taylor series approximation used in the textbook example. It simplifies how utility changes in response to changes in wealth, allowing economists to work with a more manageable expression without losing the essence of the utility function's behavior.

By expanding the utility function around the wealth level 'W', the approximation includes the lost utility due to the fine (both the linear and squared term), and this is where risk aversion comes into play. The negative second derivative of the utility function, which reflects risk aversion, means that the impact of the square of the fine will affect the utility significantly, suggesting that increasing the fine impacts risk-averse individuals more than increasing the probability of being caught.