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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.

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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鈥攔eceiving 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鈥檚 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.

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Most popular questions from this chapter

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 \(\$ 1000 ?\)

Suppose an individual knows that the prices of a particular color TV have a uniform distribution between \(\$ 300\) and \(\$ 400\). The individual sets out to obtain price quotes by phone. a. Calculate the expected minimum price paid if this individual calls \(n\) stores for price quotes. b. Show that the expected price paid declines with \(n\), but at a diminishing rate. c. Suppose phone calls cost \(\$ 2\) in terms of time and effort. How many calls should this individual make in order to maximize his or her gain from search?

For the constant relative risk aversion utility function (Equation 8.62 ) we showed that the degree of risk aversion is measured by \((1-R)\). In Chapter 3 we showed that the elasticity of substitution for the same function is given by \(1 /(1-R) .\) Hence, the measures are reciprocals of each other. Using this result, discuss the following questions: a. Why is risk aversion related to an individual's willingness to substitute wealth between states of the world? What phenomenon is being captured by both concepts? b. How would you interpret the polar cases \(R=1\) and \(R--^{\circ}\) in both the risk-aversion and substitution frameworks? c. A rise in the price of contingent claims in "bad" times \(\left(P_{b}\right)\) will induce substitution and income effects into the demands for \(W_{g}\) and \(W_{h}\). If the individual has a fixed budget to devote to these two goods, how will choices among them be affected? Why might \(W_{g}\) rise or fall depending on the degree of risk aversion exhibited by the individual? d. Suppose that empirical data suggest an individual requires an average return of 0.5 per cent if he or she is to be tempted to invest in an investment that has a \(50-50\) chance of gaining or losing 5 percent. That is, this person gets the same utility from \(W_{o}\) as from an even bet on \(1.055 W_{o}\) and \(0.955 W_{o}\) i. What value of \(R\) is consistent with this behavior? ii. How much average return would this person require to accept a \(50-50\) chance of gaining or losing 10 percent? Note: This part requires solving nonlinear equations, so approximate solutions will suffice. The comparison of the risk/reward trade-off illustrates what is called the "equity premium puzzle," in that risky investments seem to actually earn much more than is consistent with the degree of risk-aversion suggested by other data. See N. R. Kocherlakota, "The Equity Premium: It's Still a Puzzle" Journal of Economic Literature (March 1996 ): \(42-71\)

In Problem \(8.5,\) Ms. Fogg was quite willing to buy insurance against a 25 percent chance of losing \(\$ 1,000\) of her cash on her around-the-world trip. Suppose that people who buy such insurance tend to become more careless with their cash and that their probability of losing \(\$ 1,000\) rises to 30 percent. What is the actuarially fair insurance premium in this situation? Will Ms. Fogg buy insurance now? (Note: This problem and Problem 9.3 illustrate moral hazard.)

A farmer's tomato crop is wilting, and he must decide whether to water it. If he waters the tomatoes, or if it rains, the crop will yield \(\$ 1,000\) in profits; but if the tomatoes get no water, they will yield only \(\$ 500 .\) Operation of the farmer's irrigation system costs \(\$ 100 .\) The farmer seeks to maximize expected profits from tomato sales. a. If the farmer believes there is a 50 percent chance of rain, should he water? b. What is the maximum amount the farmer would pay to get information from an itiner ant weather forecaster who can predict rain with 100 percent accuracy? c. How would your answer to part (b) change if the forecaster were only 75 percent accurate?
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