A tag for all questions involving a type of utility function.

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1answer
34 views

Scaling in utility maximisation

If I have the wealth process $$dw_t=rw_tdt+n_tS_t(\sigma dB_t+(\mu-r)dt)-c_tdt,$$ where $n$ is number of $S_t$ and $B_t$ is Brownian motion. If we define the admissible set $A$ as follows: ...
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48 views

Optimal Insurance Coverage - risk neutral and risk loving consumers

I'm struggling with understanding a problem in finance: We have 2 states: $S_1$: bad state $Y-K = 2000$; probability $\pi$ = 10% $S_2$: normal state $Y = 5000$; probability $1-\pi$ = 90 ...
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20 views

Is it a Risk-averse utility function?

I'm a little unsure whether this utility function represents a risk-averse attitude, while it's not wholly concave: Would you define it as both risk-averse and risk-neutral as it seems to have ...
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0answers
8 views

Quasiconcavity of a function

can you help me check whether the following function is quasi-concave? $u(x,y) = x^2y$ if $x \geq y$ and $xy^2$ if $y>x$ FYI, a function $u(x)$ is quasiconcave if for any $x,y \in X$ and $\alpha ...
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1answer
39 views

Stochastic dominance characterization

Consider two probability measures on $\Bbb R$ given by $\mu$ and $\nu$. We write $\mu\leq \nu$ if there exists a joint distribution $P$ with the latter marginals such that $P(x\leq y) = 1$. In ...
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13 views

A question concerning pareto optimal allocations

If both agents have utility functions of the form U= log x + y, having endowments e=(1,1), what would be the set of pareto optimal allocations? After equalizing the MRS x=1, so should p.o allocations ...
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15 views

Differentiability of a process containing a brownian motion

I have trouble understanding a statement of the paper of Constantinides'1990 "Habit Formation: A resolution to the equity premium puzzle" (for example here ...
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0answers
10 views

Comparing cost of two alternative given their distribution

I have distribution for cost of two alternative through Monte Carlo simulation. The distributions are not normal. Given the benefit of the two alternatives is the same but ungiven, I want to choose ...
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0answers
32 views

Expected utility for cost

I want to compare two alternatives under uncertainty. I know if I had the values and the probability of their occurrence, I would be able to calculate the expected utility of each alternative. ...
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0answers
65 views

Ramsey-Cass-Koopmans model/Economic Growth/Phase Diagrams

I haven't had to ask for your service since my second year (I'm in my third now) of University. However, after today's seminar, which was filled with equations, terminology, and more equations, I'm ...
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0answers
18 views

Specific utility (error) function for machine learning

I need a differentiable analog of following piecewise-defined function for machine learning application: $E=E(x,y)$ when $y=1$, $E=1/(x+1)$ when $y=-1$, $E=-1/(x-1)$ $y\in \{-1,1\}$ (two values, ...
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1answer
56 views

Derivatives defined on a discrete state space

Ive been looking at certain economic papers, and optimal control papers. They define a state variable, $\omega$, which follows a discrete time Markov Chain. Then they define a utility function ...
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1answer
70 views

Deriving demand from quadratic utility function

How do you derive the demand for utility $u(x_1, x_2) = x_1^2 + x_2^2 $ and initial endowment is $\omega = (2,2) $? I believe this demand has three cases. Thank you
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52 views

Utility Theory: Risk Averse, which should I choose?

Question: If I am slightly risk averse which do I prefer. (Give a mathematical justification for your conclusion): [0.5, \$450; 0.5, \$400] or [0.1, \$4375; 0.9, \$0] Okay so I get that the risk ...
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1answer
33 views

Finding Preto Optimal allocation if utlities are of the form $u_1=x_{11}x_{12}$ and $u_2=2x_{21}+x_{22}$

There are two persons and two goods in an exchange economy. Initial endowment is $$ \omega = (\omega_1,\omega_2) =\left((1,0),(0,1)\right)$$ The utilities of two agents are given by: ...
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3answers
119 views

Economics, numeraire, utility, demand, marginal rate of substitution

I typed my question in Microsoft Word and printscreen it instead of typing it, this is because I don't know how to type mathematical questions here, sorry for the inconvenience caused.
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1answer
55 views

Intemporal budget by lagrange

Assume that a representative agent lives forever and receives an endowment, denoted yt, in each period. The entire endowment sequence is known with certainty on date 0. The representative agent ...
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21 views

Aggregated demand function for several similar offers?

I want to generate a realistic demand function for a service, depending on the price and properties of offers. The service is passenger travel, for whatever purpose. There are several companies that ...
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1answer
52 views

Making a matrix full rank through affine transformations

If I have (finite) $k$ vectors, $u_1,...,u_k\in\mathbb{R}^N$ that are in general linearly dependent is it possible to take positive affine transformations of the form: $$u'_i=\alpha_i u_i ...
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1answer
85 views

Looking for resources for understanding derivation of demand from utility

I am struggling with my homework and would very much appreciate a rundown of the math or pointers to where I can find help otherwise. Quoting from the assigment: There are $n$ sectors in the ...
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2answers
32 views

Can a game with negative expectation still have a positive utility?

Intuitively, I think not. But I can't clearly prove why. Specifically, I've been thinking about lottery games, where the expectation is obviously negative. But can the utility of hitting the jackpot ...
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1answer
251 views

How to derive demand function from a utility function without any knowledge of Lagrange Multipliers?

How do I derive the demand function for a utility function of, say, $U(x,y)=\sqrt{11x+11y}$ for goods X and Y in terms of $P_x$, $P_y$, and income $I$, with basic mathematics (basic calculus, but no ...
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1answer
285 views

Utility function, plotting indifference curves

I know how to plot indifference curves; simply take the utility function and plot some level curves in $2D$. But how to plot a specific indifference curve, so all bundles on it are indifferent to a ...
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1answer
68 views

Solving first order constraints; lagrangian function and utility maximisation

I am supposed to find the demand curve if the following is given; $U(x,y) = xy$ price of $x * x$ + price of $y * y = m$ (so a general case, and I will be adding certain prices and income levels ...
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26 views

Maximizing utility function equivalence

Let's denote our expected utility $U_{\pi} = \int ((1-e^{-kx}) \frac{1}{\sqrt{2\pi}\sigma}e^{-(x-z)^2/2\sigma^2} dx$ I'd like to show that maximizing this is equivalent to maximizing $E[X] - ...
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1answer
46 views

Utility Max Problem

I have a utility function $U(x,y)=\frac{xy}{x+y}$ and a budget of $200=2x+2y, P_x=P_y=2$. But for the first 50 units of product 1 sell for 2 dollars but for "$x>50$" the price of product 1 ...
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1answer
28 views

Multivariable Calculus Application Question: Utility and MCRS

If a student has a utility function given by $$U(x_1, x_2) = −x_1 + > 10x_ 2^2 − 2x_1x_2$$ where $X_1 = 5$ and $X_2 = 20$. If this student was to eat $5$ less hot meals per month, estimate the ...
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0answers
40 views

Infinite fourth moment and maximum entropy

Alright, I expect this is a silly question, but I don't actually know, so. Suppose there is some random variable that's distributed on the reals, and all I know about the distribution is its mean ...
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123 views

preference relation.

In the exercise below I need to check whether the relation below is a preference relation ( need to be transitive (if $x>y$ and $y>z$ then $x>z$) and connected ). But I cannot find an ...
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1answer
75 views

Counterexample in axioms of expected utility theory

This is an exercise problem. Suppose $X_1=200$ with probability $1/3$, $0$ with probability $2/3$. $X_2=200$ with probability $p$, $0$ with probability $1-p$. $X_3=200$ with probability $1-p$, ...
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1answer
49 views

Expected utility representation

I am stuck on some question on utility theory. The question is as follow: Consider $A=[0,+\infty)$, and $Q=${F-cumulative distribution function on $A: \int^{+\infty}_0 x dF(x)<\infty$}, the set of ...
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0answers
65 views

Is the expected utility function linear?

Given the definition of the mixed extension of a finite game as in the link below (only first 7 lines): How to find perfect equilibria in a finite game? We define the expected utility function in the ...
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2answers
73 views

A little question about payoff functions being continuous.

In the mixed extension of a finite game $G$, why are the payoff functions of players continuous? Does it has something to do with being von Neumann and Morgenstern utility functions? Is there other ...
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1answer
71 views

utility and uncertainty of economics

United Petroleum is operating a deep water oil rig in the Gulf of Calexico. Management have been informed that the drilling riser may be susceptible to methane build up and hence at risk of an ...
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1answer
76 views

Finding the value of $p$

I need some help with this question: Consider an individual who possesses the Bernoulli utility function of $u(x)=\dfrac{x^{1-\gamma }}{1-\gamma }$ where $\gamma>0$, $\gamma \neq 1$. Who maintains ...
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1answer
63 views

utility function for nominal attributes

I'm working in a decision making topic where a product (e.g., a hotel) is described by some attributes, that is: $p=(p_1,\ldots,p_n)$. An attribute $p_i$ can either be numeric (e.g., the room average ...
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1answer
602 views

Curvature and the Arrow Pratt Absolute Risk Coefficient

So I'm in my first year of grad school, and I'm taking a decision analysis course. One of the topics we're covering is risk aversion, and with that comes discussion of the Arrow Pratt Absolute Risk ...
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1answer
105 views

How do I optimize a function subject to a two-part constraint?

I would like to maximize the following function $$\max\; U= log(xT_o + (1-x)T_s) + log(Y)$$ by choosing levels of $T_o$, $T_s$, and $Y$, and where $x\in[0:1]$ subject to $$N = \binom{P_sT_s+Y ...
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1answer
339 views

Expected utility and St. Petersburg paradox

Can someone explain to me how they get the $10.94$ at the Expected utility theory section of the solutions to the St. Petersburg paradox? My problem is that they use a formula to calculate the ...
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1answer
29 views

Markov Decision Process - Optimal policy invariance to scaling in the Utility Function

The title says it all. If i use a discounted Utility Function, why is the optimal policy invariant with respect tot the scaling of the Utility Function by a positive Factor?
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1answer
69 views

Optimal Price for Monopolist

I want to find the total demand for good r and the optimal price p for a monopolist using the following information: Marginal cost per good = $c$ (constant) All consumers have the following utility ...
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0answers
39 views

How to define a utility of an information source?

This is a more specific (and, hopefully, clearer version of a previous question). The utility of discovering the value of a random variable $X$ can be defined to be its information content. When ...
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1answer
160 views

Mistake in Wikipedia article on St Petersburg paradox?

I suspect that there is a mistake in the Wikipedia article on the St Petersburg paradox, and I would like to see if I am right before modifying the article. In the section "Solving the paradox", the ...
3
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1answer
266 views

Maximum amount willing to gamble given utility function $U(W)=\ln(W)$ and $W=1000000$ in the game referred to in St. Petersberg's Paradox?

The game works as such: I flip a fair coin until it lands on tails. $h$ is the number of heads obtained until the first tail occurs and the game stops. My payoff from this game is: $\hat G=2^{h}$ I ...
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0answers
55 views

How to derive recursive equation for expected discounted utility function?

I am trying to mathematically represent expected discounted utilities when the length of a lifetime is uncertain. $V_0$ is the expected discounted utility at $t=0$ and can be represented as such: ...
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1answer
54 views

What type of utility function is this?

I'm a little unsure about what kind of utility function this is? u(x1, x2)=x1*x2+x1 Is it correct that this is a Cobb-Douglas function?
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1answer
80 views

What causes the change in the expected value of the product of random variables?

The following question is part of a homework exercise on portfolio theory that I have to do. Suppose that $Y$ is a random variable representing the returns on an investment. Now, let $f$ be a ...
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1answer
146 views

Functional equation for scale invariant utility functions

Two utility functions $u,v:\mathbb{R}_{>0}\rightarrow\mathbb{R}$ (giving the utility of, say, an amount of money) are considered equivalent if $u(x)$ is given by $m\,v(x)+c$, for some constants $c$ ...
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2answers
333 views

What does Arrow's theorem say about Kaldor-Hicks social welfare functions with von Neumann-Morgenstern utility?

Let $A$ be the set of all possible states of the world, let $G(A)$ be the set of all "lotteries" or "gambles", i.e. the set of all probability distributions over $A$. Now consider an individual with ...
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1answer
280 views

How can a social welfare function be a linear combination of von Neumann-Morgenstern utility functions?

The von Neumann-Morgenstern axioms were an attempt to characterize rational decision-making in the presence of risk. The von Neumann-Morgenstern utility theorem says that if someone is vNM-rational, ...