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Questions tagged [utility]

For all questions involving utility functions as used in economics and decision theory, including study of their properties or how they can be used to represent preferences.

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Why is $u(z) = \frac {z^{1-p}}{1-p}$ taken as $log(z)$ when $p=1$?

We want our function $u(z)$ to have constant $-\frac{zu''(z)}{u'(z)}$. Let $u(z) = \frac {z^{1-p}}{1-p}$ when $p$ is not 1, and $u(z) = log(z)$ when $p=1$. Why do we take it as $log(z)$? How does ...
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1answer
24 views

What is $dF(z)$ in the expected utility framework?

Background: from a Microeconomics course, $F$ is a cdf. In other words, if $F$ has a density function $f$, then $$F(z)={\int_{-\infty}^z f(x) dx} $$ Write the Bernoulli utility function $u:...
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9 views

Intuition of TailVaR

As per the actuarial guide I have called the CMP - from Acted - tailVaR is the expected loss in excess of the benchmark value L. I don't really get that, so I tried splitting the equation into: $...
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1answer
78 views

St Petersburg paradox

I am wondering what it means to say: "The gamble is worth $x$ dollars." Does it mean the expected outcome\value is $x$ dollars? In the photo (last sentence of paragraph) it says that "the gamble is ...
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1answer
27 views

Why must risk averse be correlated with a concave utility function?

Let my utility function $U: \mathbb{R}\to\mathbb{R}$ be arbitrary and suppose I am a risk averse person. Now suppose there was no risk involved with the attainment of money. Should should my utility ...
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4 views

Difference and usefulness between performance measure and utility

I would like to know if my understanding is correct. Here is how I understand the difference between performance measure and utility in terms of agents. It seems that performance measure is a binary ...
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0answers
23 views

Nominals, prices, utility

An investor is considering two possible assets, a three month one A with a yield of $4\%$ convertible quarterly and some three month one B. a) For a nominal of 100, determine the price of A. b) ...
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1answer
19 views

Market prediction, utility and information

A country is home to several million citizens; all citizens are risk averse, and each citizen is exposed to an independent risky loss each year of 1,000 dollars with a probability of $\frac{1}{40}$. ...
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22 views

What is the relationship between number of hours and the hourly wage rate?

Cindy gets utility from consumption, C, and leisure, L. Her utility function is $U=C+LC$. The price of consumption is $£1$ per unit and the hourly wage rate is $w$. Cindy has no non-labour income. She ...
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24 views

Sow that one probability distribution can't be preferred to others.

Let $R= ${ $r_1, r_2, r_3, ...$} be a countable set of rewards, and let $U$ be a utility function on R. Let $P_1, P_2, P_3, ...$ be a sequence of probability distributions on R. For each distribution ...
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1answer
15 views

Can two utility representations always be connected by a strictly monotonic function?

The Microeconomics Lecture notes by Rubinstein has the following question in Problem set two. Let $U, V: X \to \mathbb{R}$ be two utility representations of the preference relation $P$ (preference ...
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1answer
21 views

Maximizing utility function

Consider a world where people live for up to three periods. In the first period each person earns 10. In the second period, each person earns 20. Earnings in the third period are 0. Assume that utility ...
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55 views

Consider an experiment in which a person first specifies a number of dollars

Consider an experiment in which a person first specifies the number of dollars x and then observes the value of a random variable Y. Suppose that if Y >= x, he receives Y dollars as his reward. If Y &...
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1answer
31 views

Can we use the Shannon Entropy as Utility Function?

Shannon entropy can explicitly be written as $$\mathrm {H} (X)=-\sum _{i=1}^{n}{\mathrm {P} (x_{i})\log _{2}\mathrm {P} (x_{i})}$$ where the discrete random variable ${\textstyle X}$ has possible ...
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1answer
37 views

Utility function and preference relations

If a utility function represents a consumers preference relation if it assigns 'higher numbers' to preferred bundles, how do we know the properties of this preference relation (i.e. complete, ...
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1answer
21 views

How to calculate marginal utility?

I am having some difficulties understanding the calculations of marginal utility. On this problem 𝑈(𝑃,𝑀)= square root of PM I know that MRS is equal to MUp/MUm but i do not understand how its ...
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18 views

Utility premium

For the customer, the utility premium $H^{\star}$ is defined as the premium making the homeowner indifferent to insure or not, e.g. $H^{\star}$ solves $$u(x-H^{\star})=\mathbb{E}u(x-X).$$ How does it ...
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79 views

What is the difference between $\min(x_1,x_2)$ and $\min\{x_1,x_2\}$?

This question refers to microeconomic utility optimization where the utility function is Leontief. Are $\min(x_1,x_2)$ and $\min\{x_1,x_2\}$ interchangeable, or is there a difference? edit: I ...
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2answers
66 views

Does the von Neumann-Morgenstern utility theorem work for infinitely many outcomes?

The von Neumann-Morgenstern utility theorem is easy to prove for a finite number of outcomes. Is it still true for an infinite number of outcomes? With infinite outcomes, a lottery can now be any ...
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0answers
34 views

Separability of a function, preservation only under linear transformation

i have a doubt about the following problem: Lets define a multivariable function $u(x):\mathbb{R}^{n} \rightarrow \mathbb{R}$, which has the "additive separability" property, meaning we can writte $\...
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27 views

Can power utility function be defined for risk-seeking agents?

Based on Wikipedia's definition for power utility functions (i.e. isoelastic utility functions), eta taking the value of 0 corresponds to risk neutrality while taking a positive constant corresponds ...
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1answer
37 views

Mean-Variance Criterion Derivation

Can you help me understand this derivation? This is finance related so $E[X]$ means the expected outcome (of a gamble) and $u(x)$ is the utility function. With any well-behaved utility function, ...
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0answers
29 views

How should I create a matrix of costs based on a set of variable?

I'm trying to make a matrix that depicts costs for a set of tasks. These tasks are the rows of the matrix. The cost matrix is to be based on a multi-attribute weightage of the tasks based on their ...
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1answer
229 views

What is log-utility?

I came across this problem today: Calculate the log-utility optimal fraction of your capital to bet on a fair coin flip where you win $x$ on heads and lose $y$ on tails. What is the meaning of log-...
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17 views

Interpretation of a game on public good contribution

"Consider 2 identical players (i.e. i = 1, 2) with utility function: πi = b(qi + q-i) - cqi. Where qi is equal to one if player i contributes to the provision of a public good and zero if she does not,...
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1answer
19 views

Expectation of A Coin Toss with Utility

Example 7: Suppose that we flip a coin until a tail first appears and if the number of tosses equals $k$, then we are paid $2^k$ dollars. What is $E[U(2^k)]$, where $ U(p)= \frac{p^{1-\gamma} -1}{1- \...
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1answer
73 views

Power Utility Function Inverse

If power utility is $p = \frac{x^{1- \gamma} -1}{1 - \gamma}$ then is the inverse of the power utility function just $\frac{1 - \gamma}{p^{1- \gamma} -1}$?
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1answer
28 views

Utility Expectation

Example 6.3 Suppose that we flip a coin until a tail first appears and if the number of tosses equals $k$, then we are paid $2^k$ dollars. What is $E[2^k]$? I believe $E[2^k] = \sum_{k=1}^{\infty} ...
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1answer
27 views

Expectation of a Utility Function

I'm trying to understand how to calculate the expectation of a utility function, but not sure if I'm understanding it correctly... If your given a utility function $U(w)$ is $E[U(w)] = U(w)$?
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1answer
18 views

Countable Sets in utility theory

What does it mean for a quotient space to be "at most countable"? I read that in a Theorem and I didn't understand the terminology. Thank you in advance!
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1answer
60 views

Solve Lagrangian function for utility

I have the following Utility function: \begin{align} U = w^\prime\mu \end{align} and Langrangian function subject to constraint: \begin{align} F (w, \lambda)= w^\prime\mu - \lambda(w^\prime i - 1) ...
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1answer
27 views

How do I maximize utility subject to contraint via Langrangian function

I have the following Utility function: \begin{align} U = w^\prime\mu - \frac{c}{2}w^\prime\Sigma w \end{align} The Langrangian function subject to the constraint that the weights need to sum up to ...
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0answers
85 views

Using kuhn tucker to solve non negativity constraints

George likes breakfast tacos and milk. His utility is given by $u(t, m) = t + 5 \ln(m)$. Suppose tacos and milk both cost 1 dollar a piece. How should George spend the $\$4$ he has set aside for ...
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1answer
433 views

demand function from utility

thanks in advance. I’m looking to find a way (apart from Lagrange and MUx/MUy=Px/Py) to derive demand function for x, not y. You see, whenever I try solving the problem, I get y’s Demand instead of x....
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66 views

expected utility, St Petersburg Paradox, effect of initial wealth on limit price

I have a question about the St-Petersburg paradox. In the case of expected utility with log utility function, how can we show analytically that, for $w > 2$, $c$ (limit price) is increasing in $w$ (...
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0answers
26 views

Utility Maximization on a finite Probability Space. Possible mistakes in a paper?

I am currently reading this paper on utility maximization in a financial market model. On page 5 the author starts with the case of a finite probability space and on page 19 he considers the ...
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1answer
39 views

Certainty Equivalent of a Poisson distributed function.

I am struggling with a calculation for my thesis and was wondering, whether you could assist me. I have the following payoff function: $$X_i=N_i R-c, N_i \sim \textrm{Poisson} (\lambda) $$ And the ...
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12 views

Understanding Recursive Utilities

I am trying to learn about recursive utilities and aggregators at the moment, but am having a lot of trouble understanding the subject. I was wondering whether anyone here could suggest a good ...
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0answers
47 views

Suggestions on how to solve maximization problem subject to capacity constraints which are indirectly inferred

I'm interested in knowing practical optimization methods which resemble what done in NUM (network utility maximization). Basically, I envision a network in which sources' sending rate is regulated by ...
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38 views

Maximal Utility Principle

There are 2 possible scenarios, A and B. Scenario A has a 50% chance of being successful. Scenario B has two stages and a 80% chance of being successful at the first stage. If Scenario B is successful ...
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1answer
14 views

Solving for $Y$, using a utility function

The question asks to solve for $Y$, below are the steps I tried to use: $U = \ln X + 3 \ln Y \iff U = \ln X + \ln Y^3 \iff \ln Y^3 = U - \ln X$ How can I move on from here if I need to isolate, why?...
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1answer
49 views

Defining a probability of success in a lottery

Suppose that I have the following lottery: $p(x)W-(1-p(x))L$ Where $p$ is the probability of winning W and x is a variable that influences the probability of success. The expected utility for the ...
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1answer
42 views

My solution to optimizing a utility function does not agree with the textbook answer.

I have a problem from an economics course. I worked out a utility function to be given by $$ U (F, C_1) = \ln C_1 + \beta \ln F + \frac{\ln ((1+r)\left( V_1 + w(1-\tau)(1-F) - C_1\right))}{1+\phi}$$ ...
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0answers
150 views

Known concave down increasing function that can be used to estimate a utility function for a dataset?

I am trying to estimate a utility function on a given dataset. The data is normalized and ranges between 0 & 1 on both the $X$ & $Y$ axes. The estimation will find the function that best fits ...
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0answers
64 views

Utility Function Specification with Integration over a continuous function

Suppose a group of people (size=N), each with an endowment $y_i$ can increase or decrease their initial endowment by gambling. Make the simplifying assumption that everybody gambles in this group and ...
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123 views

Kelly betting alternatives [closed]

Thanks for your patience with answering my question. I am interested in building an optimal wealth allocation strategy across multiple betting opportunities, correlated or uncorrelated and with ...
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1answer
47 views

Finding consumption function which maximizes utility

I can across this question in my applied real analysis textbook that I'm having trouble with. It asks us to consider the utility function $U(C) = \sqrt{e^{-rt}C}$. I'm supposed to find the consumption ...
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1answer
143 views

How does lexicographic ordering break order separability?

I have a very specific question on existence of ordinal utility functions. In many sources (e.g. Krantz et al. 1971: Foundations of measurement: additive and polynomial representations), it is stated ...
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1answer
92 views

Which route is better for a Neutral-risk person?

There are two routes, 1 and 2. It is known that route 1 takes 60 minutes and route 2 takes 41 minutes. If there is problems in the traffic: Route 1 will increase to 70 minutes; Route 2 will ...
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1answer
838 views

Continuous utility function

A rational preferences (that is, complete and transitive) are continuous. Then how can I show that there exist a continuous function u(x) that represents there preferences. Continuity of preferences ...