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 approximation \(U(W-f)=U(W)-f U^{\prime}(W)+\left(f^{2} / 2\right) U^{\prime \prime}(W)\)

Let's denote the individual's wealth as W and their utility function as U(W). We are given that individuals are risk averse, meaning that the second derivative of the utility function with respect to W is negative (\(U^{\prime \prime}(W)<0\)).

The individual's decision will be based on their expected utility, so let's set up the two relevant utility functions: 1. The utility of not getting a ticket: \(U(W)\) 2. The utility of getting a ticket and paying the fine: \(U(W-f)\) Now, given the hint, we can use the Taylor series approximation to simplify \(U(W-f)\): \(U(W-f)=U(W)-f U^{\prime}(W)+\left(\frac{f^2}{2}\right) U^{\prime \prime}(W)\)

The expected utility of parking in an illegal place will be the weighted sum of the two utility outcomes. The probability of getting a ticket is p, so the probability of not getting a ticket is (1-p). Therefore, the expected utility of illegal parking (EU) can be expressed as: \(EU = p[U(W-f)] + (1-p)[U(W)]\)

To compare the deterrent effects of increasing either the probability of getting a ticket (p) or the fine (f), we will calculate their partial derivatives with respect to the expected utility: 1. Partial derivative with respect to p: \(\frac{\partial EU}{\partial p} = [U(W-f)] - [U(W)]\) 2. Partial derivative with respect to f: \(\frac{\partial EU}{\partial f} = p[- U^{\prime}(W) + f U^{\prime \prime}(W)]\)

Now we will analyze the signs of the partial derivatives: 1. \(U(W-f) < U(W)\) since individuals are risk averse, so \(\frac{\partial EU}{\partial p} < 0\). This means that an increase in the probability of getting a ticket will decrease the expected utility and act as a deterrent. 2. \(- U^{\prime}(W) + f U^{\prime \prime}(W) < 0\) since \(U^{\prime \prime}(W)<0\), so \(\frac{\partial EU}{\partial f} < 0\). This means that an increase in the fine will also decrease the expected utility and act as a deterrent. Since both their partial derivatives are negative, a proportional increase in either the probability of getting caught or the fine would both result in a deterrent effect. However, to compare their deterrent effects, we must look at their magnitudes relative to each other.

We must now examine the magnitude of the deterrent effect: 1. The effect of increasing the probability of getting a ticket by a proportion \(\alpha\) would lead to a change in expected utility: \(\Delta EU_p = \alpha \frac{\partial EU}{\partial p}\) 2. Similarly, the effect of increasing the fine by a proportion \(\alpha\) would lead to a change in expected utility: \(\Delta EU_f = \alpha p(- U^{\prime}(W) + f U^{\prime \prime}(W))\) By comparing the magnitudes of \(\Delta EU_p\) and \(\Delta EU_f\), we can determine which proportional increase leads to a greater deterrent effect. Without specific information about \(p\), \(f\), and the form of \(U(W)\), it is difficult to make a general conclusion about which proportional increase will serve as a more effective deterrent in all cases. However, these values can be analyzed on a case-by-case basis to provide more insight into the optimal strategy for deterring illegal parking.