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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41 views

Question regarding Allais Paradox (I know title unspecific; Sorry!; I don't know how to specify the question other than “what am I getting wrong?”)

Reading about Decision Theory I have come about Allais Paradox to be an argument against expected utility theory. One faces the following lotteries each with 100 tickest and the following payoffs per ...
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2answers
35 views

Is the utility function $u(x)=x_1 x_2 + \gamma x_2$ concave or quasi-concave?

I want to prove, given $\gamma>0$ and $x\in \mathbb{R}^2_+$, if the utility function: $$u(x)=x_1 x_2 + \gamma x_2$$ is concave, strictly concave, quasi-concave or strictly quasi-concave. I have ...
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1answer
33 views

Finding Marshallian Demand without Lagrange?

I need to find Marshallian demand for goods x and y (in terms of $P_x, P_y,$ and $I$) with the following utility function: $$U(x,y) = x + 10y - y^2$$ and general budget constraint $$I = P_xx + P_yy$$ ...
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0answers
8 views

Bayesian estimation for partial observation

Let $B$ be the competitor's hidden bid which follows some stationary distribution $F_B$ parameterized by $\theta$, and let our bid price be denoted by $x$. We win the auction when we bid higher than ...
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1answer
24 views

How to Quantify Utility/Pleasure/Pain using the Positive Real Numbers?

I am studying about Cardinal Utility in Economics (or more generally, how to quantify pleasure and pain!) Intuitively, I assign a positive number to pleasurable experiences, and a negative number to ...
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0answers
13 views

Derive demand function from a specific quadratic utility function

Suppose that the following function is the utility function of a representative consumer: U(x1,x2,y)=a/(b−c)∗(x1+x2)−b/(2∗(b−c)2)∗(x21+x22)−c/(b2−c2)∗x1∗x2+y The budget restriction is given by p1∗x1+...
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0answers
32 views

Please help: Derive the demand function from a specific quadratic utility function

Suppose that the following function is the utility function of a representative consumer: $$U(x_1,x_2,y)=a/(b-c) *(x_1+x_2)-b/(2*(b-c)^2)*(x_1^2+x_2^2)-c/(b^2-c^2)*x_1*x_2+y$$ The budget restriction ...
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0answers
15 views

How can I implement a sigmoid function composed from three distinct phases?

I want to divide a sigmoid function to three functions, I found this function: Where The five parameters in this model have the following meaning: alpha is the initial size of y, beta represents the ...
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0answers
39 views

How can I decompose sigmoid function on three parts?

I want to create a function F(x, x1, x2, x3, c1, c2, c3), where the function F represent a segmoid function. This function defined by 3 parts [0,x1], [x1,x2] and finally [x2,x3]. Can you please help ...
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0answers
15 views

Calculate premium for claim, the principle of zero utility

$$\begin{array}{|c|c|c|c|c|} \hline X&0&200&400&800&1000&2000\\ \hline \mathbb{P}&0.4&0.2&0.1&0.1&0.1&0.1\\ \hline \end{array}$$ The principle of ...
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0answers
48 views

Finding the core of a 4 player game

How do you find the core of this game? $V({0}) = V({1}) = \dots = V({4}) = 0 , V({1,2,3}) = V({1,2,4}) = 6, V({1,3,4}) = 5,V({2,3,4}) = 3, V({N}) = 11$ This is supposed to be the core: $C(v)=conv( { (...
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1answer
68 views

Philosophically motivated risk metrics

During a random process over a sample space $\Omega$, an agent incurs a cost (say, in money) given by a random variable $Z:\Omega \rightarrow R$, which is determined by the agent's action. The agent ...
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69 views

Assume the marginal utility of $u(w)$ is $\alpha e^{-\alpha w}$, $\alpha > 0$ i.e. $u'(w)=\alpha e^{-\alpha w}$, $\alpha>0$.

Assume the marginal utility of $u(w)$ is $\alpha e^{-\alpha w}$, $\alpha > 0$ i.e. $u'(w)=\alpha e^{-\alpha w}$, $\alpha>0$. a) Compute the utility function $u(w).$ b) Let $X_1$ and $X_2$ be two ...
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1answer
39 views

Find the optimal weighting distribution for a die given a set prize values

Say you have $B$ boxes, each of which contains a prize of value $v_b$. You have a $B$-sided die, and you win the value of box $b$ by rolling $b$. You get $R$ rolls, and so can collect multiple prizes, ...
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2answers
70 views

What is the probability of being selected for execution? [closed]

Thirty Arkton hostages in a Brumton prison in occupied Arkland need to select three of them to be executed by their captives in retribution to the killing of three Brums by the Ark army. They tear ...
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0answers
15 views

General approach to solving Pareto-Optimal allocations with Leontief Utility

Problem Consider the following case in a two-person, two-good economy $U_{1}(x_1,y_1)=U_{1}(x_1,x_2)$ $U_2(x_2,y_2) = min(\alpha x_2, \beta y_2)$ Where $Q_1=x_1+x_2$ and $Q_2 = y_1+y_2$, with ...
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1answer
68 views

Problem using Lagrange Multipliers in Utility function

I am to optimize utility given the utility function $$ u(c,l):=c-\frac{\eta}{\eta+1}(24-l)^{\frac{\eta+1}{\eta}},$$ where $c$ represents consumption and $l$ represents leisure. The budget constraint ...
2
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1answer
74 views

The indifference curve of $U(x,y)=\min(x,y)^2+\max(x,y)$

I am trying to draw the indifference curves for $U(x,y)=\min(x,y)^2+\max(x,y)$. It should not come as a straight line, right? I tried to calculate it by setting one variable $= 0$ and the other ...
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1answer
55 views

Pareto allocation problem when the utility is decreasing in one good

I have to characterize the Pareto optimal allocation of the following problem: Consider two-agents-two-goods economy. The preferences of the agents are given by the following utility functions: $\...
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0answers
19 views

Utility maximization: two-period portfolio

Question: Utility function $U(w) = aw-bw^2$, where $a>0$, $b>0$ and wealth $w>0$ Two-period portfolio. Second-period wealth is determined with the portfolio’s return, $R^\sim$ which is a ...
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0answers
36 views

Economics-utility function

We let $g(z)$ be a strictly monotonous function so: $$\frac{dg(z)}{dz}>0$$ Consumer 1 has preferences given by the utility function $u(x_1,x_2)=ln(x_1)+2ln(x_2)$, while consumer 2 has preferences ...
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0answers
19 views

Cobb-Douglas taxes advantage mathematically

Let a utility function for a consumer be defined as $u(x_{1},x_{2})=x_{1}^{1/2} x_{2}^{1/2}$. With the budget $x_{1}p_{1}+x_{2}p_{2}=m$. With values $p_1=p_2=1, m=32$. The state now adds a tax of unit ...
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0answers
6 views

Taking First Order Condition of a max Function

$max \int_i G[u^i((1-\tau)z^i+\tau Z(1-\tau)-E dv_i$ where Z (capital z) denotes a function. In my notes, I am told this is equal to $\int \frac{\partial G}{\partial u^i} \frac{\partial u^i}{\partial ...
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0answers
28 views

Total Derivative of a Max Function

I'm studying public economics but my question here is purely mathematical in nature. I have a function: $$ V(1-\tau, R) = \max_zu((1-\tau)z+R,z) $$ I need to take the total derivative of this, in my ...
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1answer
38 views

$max - \frac{1}{x} - \frac{1}{y}$ s.t. $2x + y ≤ 10, x ≥ 0, y ≥ 0$

$\max - \frac{1}{x} - \frac{1}{y}$ s.t. $2x + y ≤ 10,\quad x ≥ 0,\quad y ≥ 0$ I set up the lagrangian and take FOC. $ \frac{\frac{1}{x^2}}{\frac{1}{y^2}}$=$\frac{2}{1}$ $y=\sqrt2$ x Substitute in ...
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1answer
52 views

Optimization question on a function $𝑢(𝑥, 𝑦, 𝑧, 𝑤) = \min\{𝑥, 2𝑦\} + \max \{3𝑧, 4𝑤\}$

I have the following utility function $$𝑢(𝑥, 𝑦, 𝑧, 𝑤) = \min\{𝑥, 2𝑦\} + \max \{3𝑧, 4𝑤\}$$ I want to find its demand function. For that $$\operatorname{Max}𝑢(𝑥, 𝑦, 𝑧, 𝑤) = \min\{𝑥, 2𝑦\} ...
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3answers
451 views

Proving a function is quasi-concave

This is from economics, but I think there's a lot of math involved and I want to make sure I didn't mess anything up. There is a utility function U = $x_1$ + $\ln(x_2)$ such that $x_1$ and $x_2$ are ...
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0answers
8 views

Utility Representation and Continuous preferences

I have a question regarding the monotonic transformation of utility functions. If U(x) and V(x) represent a preference relation R defined on X such as U: x-->R and V: X-->R, is there always a ...
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1answer
65 views

Compute a definite integral that involves exponentials and trigonometric functions.

Let $ r > 0 $ and let $\vec{\gamma} = (\gamma_i)_{i=1}^2 $ such that $\gamma_{1} > 0 $, $\gamma_2 > 0 $ and such that $ \gamma_1 + \gamma_2 < 1$. We consider the following integral: \begin{...
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1answer
144 views

Convexity of CES functions

I'm asked to confirm if the CES utility function is convex, and I know it is, I just don't understand why :( My function is: $$U(x_1,x_2)=(αx_1^ρ+(1-α)x_2^ρ)^{1/ρ}$$ pictured here I've seen ...
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1answer
44 views

Existence of Maximum and Minimum (Utility Functions)

I'm given the following question: Show that any utility function on a finite set of alternatives attains maximum and minimum values by using Bolzano-Weierstrass Thm. Well, if it were the case that we ...
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1answer
27 views

maximizing using lagrangian

So I have a question from my quiz. I don't want a specific answer but a help or guidence. My objective function is F(x,y)=x+4y and my subject is I-Pxx-Pyy=0 where I,Px and Py are both positive ...
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2answers
77 views

The Arroyo Game versus The Pasadena Game

This choice problem builds on St. Petersburg’s Paradox. Part 7 of this SEP article indicates the problem. Question For all integers $n\geq1$, the Arroyo game pays $X$ where $$\text{P}\left(X=(-1)^{...
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1answer
34 views

Utility function square root [closed]

Could somebody tell my how to calculate $EU(X+2)$, where $U(X)={\sqrt X}$ and X is distributed evenly over the interval $[a,b]$?
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0answers
65 views

Maximizing the probability of choosing a ball from two boxes

I am new here but I have a question that I would like to ask. If any body is in the know, kindly assist. The problem is from Berger (1985) statistical decision theory and Bayesian Analysis Exercise No....
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1answer
42 views

Total Utility Value Composition of Different Utility Functions

Let's suppose we have a variable $x$ with a domain $X \in [0,1000]$ and two utility functions $uf_1(x)$ and $uf_2(x)$ that describe the utility of $x$ with respect to two different properties. We ...
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0answers
190 views

Under which conditions is the expected value of a function a (strictly-quasi) convex function?

I have an indirect utility function - $P(a,b ;\theta)$ - where $a$ and $b$ are positive, deterministic parameters and $\theta$ is a random variable. I would like to study the properties of the ...
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1answer
47 views

Intuition for complete monotne functions

While reading papers about utility theory, I've stumbled upon a definition of a completely monotone function (AKA proper), which is a function with $u'>0$, $u''<0$, $u'''>0$ and so on. See ...
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1answer
52 views

How to find a utility function

The choices are of the form $(x; y)$ where $x$ represents the amount of time you have left to live, say anywhere from $0$ to $50$ years, and $y$ represents the amount of time you have left to work, ...
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1answer
28 views

Why can utility functions be continuous, and what does this imply for marginal utility?

I am studying microeconomics at the introductory undergraduate level and two related but distinct math-related questions are puzzling me. First, my textbooks express utility functions as continuous ...
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2answers
42 views

Strictly monotonic utility function

Given a utility function of $U = x - 3y^2$ for $x>0$ and $y>0$ Are the preferences strictly monotonic for all $x>0$ and $y>0$? what happens to the marginal utility as each good is being ...
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1answer
38 views

How to draw the graph of this utility function?

This is in connection to a problem in Economics. I am trying to draw the graph of $U(x,y)=\min(x+y,2\sqrt{xy})$. In my attempt I tried break the definition into different cases. Now , $x+y=2\sqrt{xy}...
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1answer
48 views

Deconvolution with respect to a particular function

Let $\mathcal L, \mathcal L^*: \Theta \times \mathcal A \to \mathbb R$ be functions. When can $\mathcal L$ be expressed as the convolution of $\mathcal L^*$ with some third function $U$? That is, when ...
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0answers
39 views

Finding FONC and Maximising a utility function

So we were discussing a utility function in class today and I'm not sure how my teacher arrived at the First Order Necessary Condition that he did (using substitution) We want to maximise the ...
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1answer
55 views

Interpretation of this Lagrange Multiplier

I have the following utility maximization problem with inequality constraints: Objective function given by $U(x_1,x_2)=\ln(x_1)+\beta \ln(x_2)$ where $0<\beta<1$, and the constraints are given ...
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0answers
88 views

How to solve a VCG auction game?

So i have this question: And these are the 5 highest marginal valuations according to the answer sheet: This is how the answer sheet defines the vcg mechanism How is the answer sheet chosing these ...
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1answer
57 views

Finding the Nash equilibria of games

So i have this question I have some learning disabilities and have no clue whatsoever how the best reply of each player is what it is or how all players will demand the values shown in the answer ...
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1answer
109 views

Utility function and Insurance premium

A policy maker has utility function $u(w)=b^2-(b-w)^2$ where $w>10$ (wealth ) and $b>0$ constant such as $b \geq 3w$. The policy maker is exposed to risk of loss $X$. $X=1$ with probability $0....
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2answers
69 views

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
35 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:...