Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 6

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

Expert verified
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

01

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 \(U(W)\).
02

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.
03

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: \[E[U(W)]=kp[U(W)-f U^{\prime}(W)+\frac{f^{2}}{2} U^{\prime \prime}(W)]+(1-kp)U(W)\]
04

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.
05

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose Molly Jock wishes to purchase a high-definition television to watch the Olympic Greco-Roman wrestling competition. Her current income is \(\$ 20,000,\) and she knows where she can buy the television she wants for \(\$ 2,000\). She has heard the rumor that the same set can be bought at Crazy Eddie's (recently out of bankruptcy) for \(\$ 1,700,\) but is unsure if the rumor is true. Suppose this individual's utility is given by \\[\text { utility }=\ln (Y).\\] where Fis her income after buying the television. a. What is Molly's utility if she buys from the location she knows? b. What is Molly's utility if Crazy Eddie's really does offer the lower price? c. Suppose Molly believes there is a \(50-50\) chance that Crazy Eddie does offer the lowerpriced television, but it will cost her \(\$ 100\) to drive to the discount store to find out for sure (the store is far away and has had its phone disconnected). Is it worth it to her to invest the money in the trip?

An individual purchases a dozen eggs and must take them home. Although making trips home is costless, there is a 50 percent chance that all of the eggs carried on any one trip will be broken during the trip. The individual considers two strategies: Strategy 1: Take all 12 eggs in one trip. Strategy 2: Take two trips with 6 in each trip. a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that on the average, 6 eggs will remain unbroken after the trip home under either strategy. b. Develop a graph to show the utility obtainable under each strategy. Which strategy will be preferable? c. Could utility be improved further by taking more than two trips? How would this possi bility be affected if additional trips were costly?

In some cases individuals may care about the date at which the uncertainty they face is resolved. Suppose, for example, that an individual knows that his or her consumption will be 10 units today \((Q)\) but that tomorrow's consumption \(\left(\mathrm{C}_{2}\right)\) will be either 10 or \(2.5,\) depending on whether a coin comes up heads or tails. Suppose also that the individual's utility function has the simple Cobb-Douglas form \\[U\left(Q, C_{2}\right)=V^{\wedge} Q.\\] a. If an individual cares only about the expected value of utility, will it matter whether the coin is flipped just before day 1 or just before day \(2 ?\) Explain. More generally, suppose that the individual's expected utility depends on the timing of the coin flip. Specifically, assume that \\[\text { expected utility }=\mathbf{f}^{\wedge}\left[\left(\mathrm{fi}\left\\{17\left(\mathrm{C}, \mathrm{C}_{2}\right)\right\\}\right) \text { ) } \text { ? } |\right.\\] where \(E_{x}\) represents expectations taken at the start of day \(1, E_{2}\) represents expectations at the start of day \(2,\) and \(a\) represents a parameter that indicates timing preferences. Show that if \(a=1,\) the individual is indifferent about when the coin is flipped. Show that if \(a=2,\) the individual will prefer early resolution of the uncertainty - that is, flipping the coin at the start of day 1 Show that if \(a=.5,\) the individual will prefer later resolution of the uncertainty (flipping at the start of day 2 ). Explain your results intuitively and indicate their relevance for information theory. (Note: This problem is an illustration of "resolution seeking" and "resolution-averse" behavior. See D. M. Kreps and E. L. Porteus, "Temporal Resolution of Uncertainty and Dynamic Choice Theory," Econometrica [January \(1978]: 185-200\).

Investment in risky assets can be examined in the state-preference framework by assuming that \(W^{*}\) dollars invested in an asset with a certain return, \(r,\) will yield \(\mathrm{W}^{*}(\mathrm{l}+r)\) in both states of the world, whereas investment in a risky asset will yield \(\mathrm{W}^{*}\left(1+r_{g}\right)\) in good times and \(\left.\mathrm{W}^{*}\left(\mathrm{l}+r_{b}\right) \text { in bad times (where } r_{g}>r>r_{b}\right)\) a. Graph the outcomes from the two investments. b. Show how a "mxied portfolio" containing both risk-free and risky assets could be illustrated in your graph. How would you show the fraction of wealth invested in the risky asset? c. Show how individuals' attitudes toward risk will determine the mix of risk- free and risky assets they will hold. In what case would a person hold no risky assets? d. If an individual's utility takes the constant relative risk aversion form (Equation 8.62 ), ex plain why this person will not change the fraction of risky asset held as his or her wealth increases.

Ms. Fogg is planning an around-the-world trip on which she plans to spend \(\$ 10,000\). The utility from the trip is a function of how much she actually spends on it \((Y),\) given by \\[ U(Y)=\ln Y \\] a. If there is a 25 percent probability that Ms. Fogg will lose \(\$ 1000\) of her cash on the trip, what is the trip's expected utility? b. Suppose that Ms. Fogg can buy insurance against losing the \(\$ 1000\) (say, by purchasing traveler's checks) at an "actuarially fair" premium of \(\$ 250 .\) Show that her expected utility is higher if she purchases this insurance than if she faces the chance of losing the \(\$ 1000\) without insurance. c. What is the maximum amount that Ms. Fogg would be willing to pay to insure her \(\$ 1000 ?\)
See all solutions

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free