# 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.

138 questions
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...