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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Isoelastic utility functions, how to calculate risk boundaries.

I am trying to replicate a calculation I found in a paper that calculates the level of risk aversion for an individual to choose one option over the other, but I am struggling to solve it. Ideally, ...
anona's user avatar
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A concrete example with Arrow-Pratt coefficient of absolute risk aversion

Let $u_1$ and $g$ be increasing strictly concave functions from $\mathbb{R}$ to $\mathbb{R}$. Let $u_2:=g\circ u_1$. If we regard $u_1$ and $u_2$ as utility functions of two players, this is saying ...
No-one's user avatar
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Wealth equation for optimal consumption

Consider a market with $d = 1$ risky asset with prices $(S_n)_{n \geq 0}$ and interest rate $r$. Suppose an investor has initial wealth $X_0 > 0$, consumes $C_n$, and holds $\theta_n$ shares over ...
John David's user avatar
2 votes
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27 views

Characterization of Optimal Payoff (under Expected Utility) via Gateaux-Derivative/Fréchet Derivative

Background: Let $(\Omega, \mathcal{F}, \mathbb{P})$ model a financial market and $T>0$. Denote by $(S_t)_{t\in[0,T]}$ the price process of the risky asset in the financial market. Assume that the ...
MWilk's user avatar
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Utility function and bundles

I want to isolate $x_2$. Is this correct? Lets say I have $(x_1,x_2)$ = (2,4) and the utility function $v_1(x_1,x_2)=g(x_1^2x_2)$ $$v_1(2,4)=g(2^2\cdot4) = g(16)$$ $$g(x_1^2x_2) = g(16) \...
Yonathan's user avatar
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10 views

$ bL_h + (1-b)L_l $ is preferred over $ aL_h + (1-a)L_l $ iff $ b > a $

I am struggling to understand the proof of the second step in the Expected Utility Theorem, particularly the part that deals with preferences over weighted sums of lotteries. The statement I am trying ...
Lorena_dok's user avatar
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59 views

Optimizing the portfolio in modern portfolio theory

I am trying to understand some aspects of the modern portfolio theory, which has brought me to a point I don't fully understand. I would appreciate any hep/suggestions/references. Lets assume that the ...
Seyed Mohsen Ayyoubzadeh's user avatar
2 votes
1 answer
78 views

Analyzing a Strategic Form Game for Locating a Public Facility

I'm exploring a scenario where a public facility needs to be located along a street segment represented by the interval $[0, 1]$. In this setting, there are $n$ agents, each having their preferred ...
pacmanscuriousbloob's user avatar
2 votes
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190 views

Prices in a lottery with given utility problem

Suppose a person has a Bernoulli utility function $u(\cdot)$ and an initial wealth $w_0$. A lottery $L$ offers a payoff $A$ with probability $p$ and payoff $B$ with probability $q$, where $q = 1-p$. ...
SupremePickle's user avatar
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Lagrange Multiplier interpretation for larger changes in the value of constraint

I have two questions regarding the Lagrange Multiplier. One : Suppose we are solving this: $$\text{$\max xy$ such that $2x+y=100$}.$$ Solution is: $(x,y)= (25,50)$ and lambda is $25$. So $\text{...
Confused_intense_thoughts's user avatar
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1 answer
41 views

Real life analysis - how to break down revenue growth drivers

Real life growth driver analysis problem I am getting two Answers via two Methods. I would like to know which is Correct & why the other is Wrong. To really simplify this problem let's say there ...
Emil's user avatar
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2 answers
466 views

Understanding the "Just one more" paradox on a logarithmic scale

I got somewhat puzzled after watching this video on Kelly Criterion in economics and the associated "just one more" paradox. This question should be self-contained, so watching the video ...
Aleksejs Fomins's user avatar
1 vote
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Is Maximizing utility with compliment/substitutes NP

I was trying to code a simple economics simulator where consumers try and maximize their utilities based on a lot of parameters and I think that maximizing utilities is NP-hard but I wanted to ask you ...
SlimeyGuy123's user avatar
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1 answer
59 views

Optimal strategy that maximizes fortune

A player can bet a quantity $u_k\geq0$, at each instant $k$ if $u_k \leq x_k$, where $x_k$ is his current fortune at instant $k$. He wins the money he bets with probability $\frac{1}{2}<p <1$ or ...
Davi's user avatar
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Determine all values $\lambda$ for which $\mu \succ 0 \succ \upsilon$

Suppose an investor has a preference represented by the relation $\succ$ for which there is a von-Neumann Morgenstern representation with the utility function $u$: $$u(x)=\begin{cases} x & x\ge 0 \...
dsk62's user avatar
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Find certainty equivalent $C[x]$ with respect to the utility function $u(x)=-e^{-x}$

Let $X$ - a random variable with a Poisson distribution with parameter $\lambda >0$. Find certainty equivalent $C[x]$ with respect to the utility function $u(x)=-e^{-x}$. My try: $$u(C[X])=\mathbb ...
qerty149's user avatar
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19 views

How to take expectation of the Utility Function $\mathbb{E} (\left(y_2k+R_+B_+\right)^{-\gamma})$

How do I take the Expectations of the equation: $$\mathbb{E} (\left(y_2k+R_+B_+\right)^{-\gamma})$$
SunnyD_'s user avatar
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Expected utility theory, risk aversion, mean preserving spread

(See also short version below). Consider two lotteries $A$ and $B$ \begin{align} L_A&= ( (1/2),w_0-h;\ (1/2),w_0+h )\\ L_B&= ( (1/2),w_0-2h;\ (1/2),w_0+2h ) \end{align} where $0<h&...
Alessandro's user avatar
1 vote
1 answer
252 views

Utility in case one good is Bad and other is Good

I have a question Suppose there are $2$ goods, $X$ and $Y$ where, $X$ is Good good, $Y$ is Bad good Now Can the Utility be 1.) $U = x-y$ 2.) $U = \frac{x}{y}$ 3.) $U = \frac{\ln x}{y}$ They all are ...
Sahil Bagdi's user avatar
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24 views

Looking for an utility function for which St. Petersburgh paradox becomes unbounded

A professor has explained to my class St. Petersburgh paradox and has introduced the concept of utility function. The professor then asked us to find an utility function with a positive first ...
slow_learner's user avatar
1 vote
1 answer
239 views

Intuition behind the Compensating Variation!

In Economics, we can calculate the compensating variation (CV), which (to my understanding) is the amount of money we would need to give back to a consumer to keep them at the same level of Utility ...
CormJack's user avatar
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Define a cost function for an agent walking on a 3-parts segment.

Let us consider an agent that can walk along a one-dimensional room (i.e. along a segment). The segment is split into three equal parts and we assume the agent is in the middle piece of the segment, ...
Mark's user avatar
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-6 votes
1 answer
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Lagrange multiplier Optimization with three variables and a constraint

Ok, I'm running up against my deadline and am totally stuck on this utility maximization problem. $$U=-\frac1x-\frac1y-\frac1z$$ subject to $$I=P_xx+P_yy+P_zz$$ where $P_x$, $P_y$ and $P_z$ are the ...
mattg444's user avatar
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0 answers
26 views

Expected Utility, decision theory

I’m slightly confused how to assign utility if there are 2 pieces of information given about its value. For example, there are 2 decisions: to go to the movies or to go fishing. Provided that you get ...
David J's user avatar
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38 views

A continuous time model where Nash equilibrium is build in a dynamic programming setting or as a system of backward looking SDEs?

I am looking for a continuous time model, that builds a game among a continuum of agents who interact strategically and they have mean-variance utility function. In particular mean-variance utility ...
Oliver Queen's user avatar
-1 votes
1 answer
67 views

Generate six random numbers that come $1, 2, 3, 4, 5, 6$ with the given ${\tt PMF}$ according to a sequence $.1, .1, .2, .3, .2, .1$ using non-uniform

Problem. Generate six random numbers that come $1, 2, 3, 4, 5, 6$ with the given $\texttt{PMF}$ according to a sequence $0.1, 0.1, 0.2, 0.3, 0.2, 0.1$ using non-uniform random number generator. For ...
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How to find the marshallian demand of utility function $n$ goods (logit) beforehand?

I have a utility function (logit) that is kinda weird and hard to work with given the constraints mentioned in the problem. I’m trying to derive the marshallian demand for $x_0$, but I'm not sure how ...
Patrick Nodi's user avatar
2 votes
0 answers
1k views

Alternate formulas with NonEquivalent Averages to judge an ending quarter of one season

FA Premier League 2019/20. The season was affected by the COVID-19 Pandemic while each team had a so-called quarter of their schedule left. ("quarter" ? Since each team has 4/9 or 5/9 number ...
user avatar
1 vote
1 answer
51 views

Conditional maximization of consumer utility

I'm trying to solve the following consumer problem: Consumers: The economy is populated by an infinity of homogeneous individuals who inelastically supply an amount L of work. The individual has ...
Alien Economista's user avatar
1 vote
1 answer
59 views

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 ...
PinRod3's user avatar
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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 ...
Qcer's user avatar
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0 answers
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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 ...
mermaid45's user avatar
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0 answers
26 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.
Martin N.'s user avatar
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0 answers
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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 ...
Martin N.'s user avatar
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4 votes
1 answer
89 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 $2^{n-1}$ ...
Jerry's user avatar
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2 votes
1 answer
100 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: ...
graphitump's user avatar
0 votes
0 answers
43 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 ...
Celine's user avatar
  • 49
1 vote
1 answer
1k views

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 ...
K A's user avatar
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2 votes
0 answers
233 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 ...
Alex Wang's user avatar
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0 votes
1 answer
50 views

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; ...
Cooper's user avatar
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3 votes
0 answers
64 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 $\$2^n$ ...
BaroqueFreak's user avatar
2 votes
1 answer
147 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 ...
RavenclawPrefect's user avatar
0 votes
0 answers
53 views

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 ...
Alex's user avatar
  • 137
2 votes
1 answer
92 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 ...
Billy's user avatar
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0 votes
2 answers
259 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 ...
Mathfreak23's user avatar
1 vote
1 answer
768 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 $$I = P_xx + P_yy$$ ...
BDot35's user avatar
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0 votes
1 answer
31 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 ...
Anuj Manoj Shah's user avatar
1 vote
0 answers
228 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 ...
peppa_student's user avatar
0 votes
0 answers
54 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 ...
stevGates's user avatar
  • 107
0 votes
0 answers
95 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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