/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 A consumer has an expected utili... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 10

A consumer has an expected utility function given by \(u(w)=\ln w\) He is offered the opportunity to bet on the flip of a coin that has a probability \(\pi\) of coming up heads. If he bets \(\$ x\), he will have \(w+x\) if head comes up and \(w-x\) if tails comes up. Solve for the optimal \(x\) as a function of \(\pi\) What is his optimal choice of \(x\) when \(\pi=1 / 2 ?\)

Short Answer

Expert verified
The optimal \( x \) is 0 when \( \pi = \frac{1}{2} \).

Step by step solution

01

Set Up the Problem

The consumer's utility function is given by \( u(w) = \ln w \). Let's denote the initial wealth as \( w \) and the amount bet as \( x \). The probability of winning (heads) is \( \pi \) and losing (tails) is \( 1-\pi \). When winning, the wealth becomes \( w+x \); when losing, it is \( w-x \). The expected utility \( EU \) is:\[ EU = \pi \ln(w+x) + (1-\pi) \ln(w-x) \].
02

Differentiate the Expected Utility

To find the optimal amount \( x \), we need to find the maximum expected utility. Take the derivative of the expected utility \( EU \) with respect to \( x \):\[ \frac{dEU}{dx} = \pi \cdot \frac{1}{w+x} - (1-\pi) \cdot \frac{1}{w-x} \].
03

Set the Derivative to Zero

Set the derivative equal to zero to find the critical points:\[ \pi \cdot \frac{1}{w+x} = (1-\pi) \cdot \frac{1}{w-x} \].
04

Solve for x

Solve the equation from Step 3:Multiply both sides by \( (w+x)(w-x) \):\[ \pi (w-x) = (1-\pi)(w+x) \].Expand and rearrange terms:\[ \pi w - \pi x = (1-\pi)w + (1-\pi)x \].Combine like terms:\[ \pi w - (1-\pi)w = (1-\pi)x + \pi x \].\[ (2\pi - 1)w = (2\pi - 1)x \].If \( 2\pi - 1 eq 0 \), divide both sides by \( 2\pi - 1 \):\[ x = w \].
05

Analyze Special Case \( \pi = \frac{1}{2} \)

Substitute \( \pi = \frac{1}{2} \) into the equation from Step 4: \[ (2\pi - 1) = 0 \].This implies that any \( x \) will maintain the expected utility as the utilities for winning and losing balance out precisely. Hence, the optimal bet \( x \) is 0 to play conservatively.

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Key Concepts

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

Consumer Utility
Consumer utility is a key concept in economics that describes the satisfaction or happiness gained from consuming goods or services. In our exercise, the utility is represented by a function, specifically a logarithmic function: \( u(w) = \ln w \). This reflects how the consumer's satisfaction changes with wealth \( w \). Why use a logarithmic function for utility? Because it captures diminishing marginal utility, meaning each additional dollar of wealth brings less added satisfaction than the previous one. This mirrors how in real life, the importance of each additional dollar reduces as our wealth grows.Understanding consumer utility in the context of betting requires looking at how changes in wealth (due to winning or losing a bet) affect utility. When making choices about betting, consumers aim to maximize expected utility, which involves weighing potential gains against potential losses.
Optimal Bet
The optimal bet is the amount a consumer decides to wager to maximize their expected utility. In our exercise, this involves finding the value of \( x \) (the amount bet) that maximizes the consumer's expected utility. This requires taking into account the probability \( \pi \) of winning the bet.To find the optimal \( x \), we differentiate the expected utility and set it to zero. This mathematical process helps identify the point where the utility is maximized or minimized. The resulting critical point indicates the optimal bet.In cases where the probability simplifies things, like when \( \pi = \frac{1}{2} \), the optimal bet is zero. This shows that with even odds, it鈥檚 best not to bet at all to avoid unnecessary risk.
Probability of Winning
The probability of winning directly influences the expected utility and hence the optimal betting strategy. In our problem, the probability of winning the coin flip is denoted as \( \pi \), while the probability of losing is \( 1 - \pi \).If the probability of winning is high \( (\pi > 0.5) \), betting becomes more attractive since the consumer is more likely to increase their wealth and hence, their utility. Conversely, if the probability is low \( (\pi < 0.5) \), the risk of decreasing their wealth outweighs potential gains, leading to more conservative betting behavior or none at all.Understanding this balance helps consumers make informed decisions that align with their risk preferences and desired outcomes.
Utility Function
A utility function is a mathematical representation of how a consumer derives satisfaction from wealth or consumption. In our scenario, the utility function is \( u(w) = \ln w \), which suggests diminishing returns on additional wealth.This specific utility function can be used to calculate expected utility based on different outcomes. It transforms complex decisions about possible bets into a tangible form we can analyze mathematically by considering certain probabilities.The utility function reflects how different scenarios affect overall satisfaction and helps consumers determine the best course of action to maximize their utility. By considering the expected utility, we derive decisions that are strategically beneficial in the face of uncertainty, such as flipping a coin.

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Most popular questions from this chapter

A coin has probability \(p\) of landing heads. You are offered a bet in which you will be paid \(\$ 2^{3}\) if the first head occurs on the \(j\) th flip. (a) What is the expected value of this bet when \(p=1 / 2 ?\) (b) Suppose that your expected utility function is \(u(x)=\ln x\). Express the utility of this game to you as a sum. (c) Evaluate the sum. (This requires knowledge of a few summation formulas. (d) Let \(w_{0}\) be the amount of money that would give you the same utility you would have if you played this game. Solve for \(w_{0}\)

Esperanza has been an expected utility maximizer ever since she was five years old. As a result of the strict education she received at an obscure British boarding school, her utility function \(u\) is strictly increasing and strictly concave. Now, at the age of thirty-something, Esperanza is evaluating an asset with stochastic outcome \(R\) which is normally distributed with mean \(\mu\) and variance \(\sigma^{2} .\) Thus, its density function is given by \\[ f(r)=\frac{1}{\sigma \sqrt{2 \pi}} \exp \left\\{-\frac{1}{2}\left(\frac{r-\mu}{\sigma}\right)^{2}\right\\} \\] (a) Show that Esperanza's expected utility from \(R\) is a function of \(\mu\) and \(\sigma^{2}\) alone. Thus, show that \(E[u(R)]=\phi\left(\mu, \sigma^{2}\right)\) (b) Show that \(\phi(\cdot)\) is increasing in \(\mu\) (c) Show that \(\phi(\cdot)\) is decreasing in \(\sigma^{2}\)

Suppose that a consumer faces two risks and only one of them is to be eliminated. Let \(\tilde{w}=w_{1}\) with probability \(p\) and \(\tilde{w}=w_{2}\) with probability \(1-p .\) Let \(\bar{\epsilon}=0\) if \(\bar{w}=w_{2} .\) If \(\bar{w}=w_{1}, \tilde{e}=\epsilon\) with probability \(1 / 2\) and \(\bar{e}=-c\) with probability \(1 / 2 .\) Now, define a risk premium \(\pi_{u}\) for \(\tilde{\epsilon}\) to satisfy: \\[ E\left[u\left(\bar{w}-\pi_{u}\right)\right]=E[u(\bar{w}+\bar{\epsilon})] \\] (a) Show that if \(\epsilon\) is sufficicntly small, \\[ \pi_{u} \approx \frac{-\frac{1}{2} p u^{\prime \prime}\left(w_{1}\right) e^{2}}{p u^{\prime}\left(w_{1}\right)+(1-p) u^{\prime}\left(w_{2}\right)} \\] [Hint: Take Taylor expansions of appropriate orders on both sides of \((*)\) first-order on the left and second-order on the right. (b) Let \(u(w)=-e^{-a w}\) and \(v(w)=-e^{-b w} .\) Compute the Arrow-Pratt measure for \(u\) and \(v\) (c) Suppose that \(a>b\). Show that if \(p<1\) then there exists a value \(w_{1}-w_{2}\) large enough to make \(\pi_{v}>\pi_{u} .\) What does this suggest about the usefulness of the Arrow-Pratt measure for problems where risk is only partially reduced?

Consider the case of a quadratic expected utility function. Show that at some level of wealth marginal utility is decreasing. More importantly, show that absolute risk aversion is increasing at any level of wealth.

For what form of expected utility function will the investment in a risky asset be independent of wealth?
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