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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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 ...
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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, ...
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Concave Utility function

I'm trying to solve the following question in the book Optimal Statistical Decision: Question: Consider two boxes each of which contains both red balls and green balls. It is known that one-half the ...
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Solve optimization problem using KKT method

Question: Solve the UMP $${Max} \quad u(x,y) = ln(x+1) + ln(y+1)$$ $${s.t.}\quad p_1x + p_2y \leq w$$ $$x,y \geq 0$$ Then the Lagrangean function is $$L = ln(x+1) + ln(y+1) +\lambda(w - p_1x - ...
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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 ...
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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 ...
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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 ...
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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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Find the utility of each agent whenever the social welfare is maximized.

Question: Suppose that the utility possibilities curve of 2 people economy is given by the equation $u_1^2 + Au_2^2=20$ where $A\in R_+$ and the social welfare function of the economy is $W(u_1,u_2)=...
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Meaning of Mathematical Statements in Kelly Criterion

The Wikipedia article on Kelly Criterion states, "the Kelly bet size is found by maximizing the expected value of the logarithm of wealth, which is equivalent to maximizing the expected geometric ...
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How to find the marshallian demand of utility function n goods (logit) beforehand?

Biggest issue is finding the demand beforehand for x0 as a function of (x1,...xn) ​ It should be obvious and trivial to find the demand beforehand, so I hope someone with knowledge in mathematical ...
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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 ...
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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 ...
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Where to learn about utility maximization using lagrangian method

Can you please suggest some well organized resources for learning utility function maximization problem using lagrangian multiplier method?
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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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
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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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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 ...
2 votes
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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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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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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: $\...
1 vote
1 answer
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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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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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$max - \frac{1}{x} - \frac{1}{y}$ 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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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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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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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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