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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Expected utility function is continuous over discrete probability distributions

I'm having trouble proving that the follwing function is continuous: Let $A$ be a non-empty set (not necessarily finite) and $$ X = \left\{ x : A \to [0, 1] \ \middle| \ \text{supp}(x) \ \text{is ...
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33 views

Why does the uniform law of large numbers hold with non-i.i.d. random variables in Bayesian experimental design?

This paper, Asymptotic theory of information-theoretic experimental design, studies Bayesian experimental design where in each round $n$, the experimenter selects a stimuli $X_n$ that maximizes mutual ...
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What is the discounted (4%) difference in health outcomes between Treatment A and Treatment B using ICER, QALY and utility value?

I'm studying health economics and have been racking my brain trying to find the right answer to this problem, but I keep getting it wrong no matter what I do. I haven't had any trouble calculating ...
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20 views

Difference between Pareto optimal redistribution and strict pareto optimal redistribution

Can someone explain the difference between Pareto optimal redistribution and strict pareto optimal redistribution? Because I know the definition but I do not understand it.
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Pareto optimal redistribution in binary exchange economy

Consider binary exchange economy with two goods and two agents, whose preferences are defined as follows: $\textbf{x}\succ \textbf{y}$ iff $x_{1}x_{2}>0 $ and $y_{1}y_{2}=0 $. In Edgeworth's box ...
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  • 137
4 votes
1 answer
70 views

Game holder is always losing money in the St. Petersberg Paradox?

The St. Petersberg Paradox is described as follows: A gambler pays an entry fee $M$ dollar to play the following game: A fair coin is tossed repeated until the first head occurs and you win $2^{n-1}$ ...
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  • 123
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13 views

Indifference preference and utility function

I met a question about indifference preference with the following conditions valid: ($\boldsymbol{\alpha}$,$x_2$)~($\boldsymbol{\beta}$,$y_2$); ($\boldsymbol{\beta}$,$x_2$)~($\boldsymbol{\gamma}$,$y_2$...
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7 views

Prove that the utility of an efficient allocation is higher than an inefficient one

An allocation is efficient if it maximizes return $$r=E[X]=\sum_{j=1}^nw_j\mu_j=w^t\mu$$ with a fixed $σ^2=w^tCw$ (C is the covariance matrix) and budget constraint $$\sum_{j=1}^nw_j=w^t1=1$$ Suppose $...
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2 votes
1 answer
43 views

Von Neumann–Morgenstern: compare coefficients in Archimedean axiom

Now we have: Axiom1: Completeness of $\succeq$. Axiom2: Transitivity of $\succeq$. Axiom3: Independence: For any $N$ and $p\in (0,1]$, if $L\succ M$, then $pL+(1-p)N\succ pM+(1-p)N$. Axiom4: ...
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39 views

Should we stay or pay the exit fee given the utility function

You have invested $10\%$ of your wealth in a hedge fund; the other $90\%$ is in cash and there is no time value of money. One year from now the hedge fund will cease operations; it will either fail ...
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What is α in a Cobb-Douglas utility function?

Sorry if this is not the place to ask, I'm new here. I'm studying economy but I'm struggling to understand the Cobb-Douglas utility function. If we've one such that xt is consumption in period t, and ...
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2 votes
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53 views

How can I differentiate the expected value of a utility function?

Suppose I have a maximisation problem $$\underset{0\le R(y)\le y} \max E\left\{u_B\left[ y-R( y)\right]\right\}$$ subject to$$E\left\{u_L\left[R( y)\right]\right\}\ge \bar U_L$$ $B$ and $L$ are simply ...
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1 answer
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How to solve this expectation problem?

I truly don't know how to name this question properly, and I will try my best to be more specific. And here is my problem Suppose a person has 2 choices. For choice A, he gets $Z$ guaranteed dollars; ...
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3 votes
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61 views

Closed form for sum of $\ln(x+2^n)/2^n$ from $n=1$ to $\infty$

Is there a closed form for the following sum, where $x>0$? $$\sum_{n=1}^{\infty} \frac{\ln(x+2^n)}{2^n}$$ The sum pops up when considering the expected utility of a game where one receives $\$2^n$ ...
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0 answers
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Some special two variable function $f(x, y)$

I'm looking for a function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ such that: $$f(a, b) \ge f(c, d) \quad\Longleftrightarrow\quad a \ge c\ \ \text{and}\ \ b \ge d$$ The origin of this problem is ...
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Eliminating weak strategy isn't order dependent

consider we have a finite one step game and a utility function u. Claim Weak strategy elimination isn't order dependent. meaning that for each iterated elimination we will get the same result prove ...
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2 votes
1 answer
138 views

How good is an optimal allocation of randomly-valued goods?

Suppose we have $n$ items to be given to $n$ people in some permutation (everyone must receive exactly one item). Each person's value for a given item is an independent draw from the uniform ...
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0 answers
32 views

Every utility function that converges pointwise converges uniformly

Let $u_n$ be a sequence of point-wise convergent utility functions on a finite actions space. I claim that $u_n$ actually converges uniformly. My try: every $u_n$ is multi-linear by definition: so it ...
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Indifference with utility function

Let u be the utility function $u(x)=-\frac{x^{-\eta}-1}{\eta},$ with $x,\eta>0$ Assume that an investor is indifferent between an investment with riskless outcome of 101.005 and a stochastic ...
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2 votes
1 answer
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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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2 answers
75 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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1 vote
1 answer
223 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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1 answer
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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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1 vote
0 answers
98 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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0 answers
19 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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0 answers
60 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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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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4 votes
1 answer
85 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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0 votes
0 answers
72 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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1 vote
1 answer
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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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-2 votes
2 answers
71 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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2 votes
1 answer
130 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 ...
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2 votes
1 answer
335 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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0 votes
1 answer
63 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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1 vote
1 answer
51 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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1 vote
0 answers
38 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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1 vote
1 answer
40 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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0 votes
1 answer
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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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1 vote
4 answers
1k 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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0 votes
1 answer
71 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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0 votes
1 answer
705 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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0 votes
1 answer
127 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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-1 votes
1 answer
35 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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1 vote
1 answer
87 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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-2 votes
1 answer
74 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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0 answers
105 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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1 vote
1 answer
48 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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0 votes
0 answers
249 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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1 vote
1 answer
51 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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1 vote
1 answer
74 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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