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.
179
questions
2
votes
1answer
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 ...
0
votes
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 ...
1
vote
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 $$I = P_xx + P_yy$$ ...
0
votes
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 ...
0
votes
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 ...
0
votes
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+...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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( { (...
4
votes
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 ...
0
votes
0answers
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 ...
1
vote
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, ...
-2
votes
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 ...
0
votes
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 ...
2
votes
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
votes
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 ...
0
votes
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:
$\...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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
$$\operatorname{Max}𝑢(𝑥, 𝑦, 𝑧, 𝑤) = \min\{𝑥, 2𝑦\} ...
1
vote
3answers
451 views
Proving a function is quasi-concave
This is from economics, but I think there's a lot of math involved and I want to make sure I didn't mess anything up. There is a utility function U = $x_1$ + $\ln(x_2)$ such that $x_1$ and $x_2$ are ...
0
votes
0answers
8 views
Utility Representation and Continuous preferences
I have a question regarding the monotonic transformation of utility functions. If U(x) and V(x) represent a preference relation R defined on X such as U: x-->R and V: X-->R, is there always a ...
0
votes
1answer
65 views
Compute a definite integral that involves exponentials and trigonometric functions.
Let $ r > 0 $ and let $\vec{\gamma} = (\gamma_i)_{i=1}^2 $ such that $\gamma_{1} > 0 $, $\gamma_2 > 0 $ and such that $ \gamma_1 + \gamma_2 < 1$. We consider the following integral:
\begin{...
0
votes
1answer
144 views
Convexity of CES functions
I'm asked to confirm if the CES utility function is convex, and I know it is, I just don't understand why :(
My function is: $$U(x_1,x_2)=(αx_1^ρ+(1-α)x_2^ρ)^{1/ρ}$$ pictured here
I've seen ...
0
votes
1answer
44 views
Existence of Maximum and Minimum (Utility Functions)
I'm given the following question:
Show that any utility function on a finite set of alternatives attains maximum and minimum values by using Bolzano-Weierstrass Thm.
Well, if it were the case that we ...
-1
votes
1answer
27 views
maximizing using lagrangian
So I have a question from my quiz. I don't want a specific answer but a help or guidence.
My objective function is F(x,y)=x+4y and my subject is I-Pxx-Pyy=0 where I,Px and Py are both positive ...
1
vote
2answers
77 views
The Arroyo Game versus The Pasadena Game
This choice problem builds on St. Petersburg’s Paradox. Part 7 of this SEP article indicates the problem.
Question
For all integers $n\geq1$, the Arroyo game pays $X$ where
$$\text{P}\left(X=(-1)^{...
-2
votes
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 $[a,b]$?
0
votes
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....
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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, ...
0
votes
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 ...
0
votes
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 ...
0
votes
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}...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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....
0
votes
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 ...
1
vote
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:...
