Questions tagged [expected-utility]

The expected utility theory deals with the analysis of choices among risky projects with multiple possible outcomes.

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Repeated betting game with positive expected value

Consider the following basic repeated betting game: A player can enter the game with an amount of money x. The game consists of multiple rounds. In each round a ...
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Expected utility theory (Lottery notation)

A wheel of fortune has outcomes $S=\left \{ 1000,100,50,20,0 \right \}$ as money prices. A consumer has the preferences $$20\sim \left ( \frac{2}{100}\cdot1000 \oplus \frac{98}{100} \cdot 0 \right )$$ ...
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How to Represent as a Payoff Matrix

I'm trying to represent the following as a pay-off matrix. I have 100 dollars to invest in one agricultural stocks with a choice of apples, pears or grapes. Return on investment relies on whether ...
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210 views

Comparing & contrasting decision problems and normal games

I am trying to compare and contrast between decision problems and normal games. Are there any key concepts I should know? Any help would be greatly appreciated.
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Negotiations under expected utility maximization

A buyer is negotiating with a used car salesperson. The value of the car to the seller is uniformly distributed between 0 and 5000. Value to the buyer is 50 percent more than that of the seller (i.e. ...
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Why utility rather than expected utility in Cochrane's “Asset Pricing”?

Cochrane "Asset Pricing" Chapter 1 p. 6 says We model investors by a utility function defined over current and future values of consumption, $$ U(c_t,c_{t+1}) = u(c_t) + \beta \mathbb{E_t}[...
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Is the expected utility the inverse of the utility function?

Can somebody explain to me if that it's true and also graphically explain it?
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67 views

The efficient frontier in mean variance criterion

The efficient frontier is the portfolios with the minimum of variance ($V$) at a given mean ($E$) or a maximum of mean at a given variance,Why do the optimal portfolios in the effcient frontier, is ...
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Need help with Wakker (2010) on arbitrage

In Prospect Theory (2010; Cambridge UP), Peter P. Wakker has an exercise assignment 3.3.6 without solution in the book and I'm really unsure about this one. The exercise states on pages 76-77: ...
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Calculating risk interest rate within a two period model

I am trying to calculate how to determine the interest rate ( = risk free rate + premium) within the following model where a consumer decides to invest in a safe asset or in a risky asset. The utility ...
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99 views

Should the “value function” be “utility function” in prospect theory?

I have a background in mathematics rather than economics, and currently reading Choices, Values, and Frames[1]. The paper defines a "hypothetical value function" (the s-shape that is concave ...
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Fair value that a risk averse individual would pay to enter a gamble

Introduction Assuming an individual (or corporation) with risk aversion and a von Neumann-Morgenstern utility curve and given a gamble g with E(g) > 0. From what I researched, certainty equivalent is ...
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How is the defintion of the mean preserving spread (MPS) not too general?

The mean preserving spread is defined as follows: Consider two lotteries g and h. Let $x_g$ und $x_h$ denote the corresponding random variables. Then h is a mean preserving spread (MPS) of g, if: $...
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Utility Theory/Marginal Rate of Substitution: Can the marginal rate of substitution be calculated for a point of the budget line?

This a person's budget line with various points, and their consumption, C*, and their endowment e, which is worth $5000 (unimportant). Also shows is their initial indifference curve. The difference ...
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45 views

Understanding Rabin's Diminishing Marginal Utility of Wealth Cannot Explain Risk Aversion

I am trying to understand Rabin's Diminishing Marginal Utility of Wealth Cannot Explain Risk Aversion. I am struggling to completely understand the following: Suppose you have initial wealth of $W$...
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116 views

Diminishing mariginal utility and risk preferences

Diminishing marginal utility is a concept only in cardinal utility theory rather than ordinal utility theory. As diminishing marginal utility implies a concave shape of the utility function, does it ...
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If the marginal cost is equal to 1, how does that imply marginal cost is equal to marginal benefit?

The function below is a utility function simplified after subject to an implied participation constraint. $$ E\left(\pi_{n}\right)=e^{*}-E\left(s^{*}\right)=e^{*}-c\left(e^{*}\right) $$ where $ \pi_n ...
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Who is the first one to equate “rational” with “complete and transitive preference”?

MWG taught that, suppose that the menu is finite, "rational" is the same as "complete and transitive". But it seems that it does not cite any sources. Who said this first? vNM said ...
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329 views

Why is the risk premium always positive for risk averse individuals?

I think this has to do with the definition of concavity and the fact that a risk averse person has a concave utility function, but I'm not sure how that helps.
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Intertemporal choice with possibility of death

Here is the setup: Suppose that there is an individual who lives up to two periods. He lives with absolute certainty during period $1$, and during this period his sub-utility function is given by: $$...
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Why can we write any lottery as a convex combination of the degenerate lotteries?

I know that a degenerate lottery is a lottery that yields outcome $n$ with probability $1$ and I also know the definition of convex combination: given $x_{1},x_{2}, \cdots ,x_{n} \in \mathbb{R}$, a ...
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How to prove the relationship between the expected value of a lottery and its certainty equivalent?

Utility function $u(x)$ is monotonic. I want to prove that $u(x)$ exhibits risk aversion if and only if for all lottery $F$: $E(x) \geq CE(F,u)$ (CE is certainty equivalent). (Definition of $CE$: the ...
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129 views

Expected Utility and Jensen's Inequality

Consider two random variables (costs and valuations) distributed $v\backsim G(.)$ and $c \backsim F(.)$ with pdfs $g(.)$ and $f(.)$. Let the supports of $c$ and $v$ be $[x,y]$. Let $x<a=E(v)<b&...
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Relationship between expected utility and independence axiom

Jonathan Levin in "Choice under Uncertainty" wrote in Theorem 1 " A complete and transitive preference relation on a set of lotteries P satisfies continuity and independence if and only if it admits ...
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712 views

Check if a utility function represents a monotone preference

Given a function $u(x_1, x_2) = x_1 +x_2 + \min(2x_1, x_2)$, how do we mathematically prove that it monotonic or not? Is there is a general algebraic technique to show monotonicity of suchlike ...
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Algebraic approach towards convexity

I have a function: $ u(x) = x_{1} + x_{2} + \min\{x_{1}, x_{2}\}$. How do we algebraically show if it's convex or not? Also, what would be the general way to show if any given function is convex.
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Local non-satiation in economics

I am having trouble completely understanding the mathematical definition of non-satiation. I have stated the definition from Wikipedia below. It would be great if someone can graphically explain. ...
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derive value function from utility function

We have the utility function. $$U_{t} = \ln{c_{t}} + E_{t}\sum_{s=1}^{\infty}(\beta^{s}\ln{c_{t+s}})$$ And I am trying to find the value function. $U$ is utility function. $c_t$ is consumption at ...
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359 views

How does expected utility theory treat losses?

I've been reading about prospect theory lately and have read often that prospect theory predicts people will be risk averse in gains and risk seeking in losses. This statement is typically ...
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Savage's subjective probabilities applied to Allais paradox

I've been reading up on the von-Neumann and Savage proofs for the existence of an expected utility representation. I've also been reading critiques of the expected utility hypothesis, especially the ...
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187 views

Proof: Risk averse; Certainty Equivalent smaller than expected value

I would like to show for a randomly distributed variable $x$ with CDF $F(\cdot)$ , given a Bernoulli utility function $u(x)$ the following property holds: The certainty equivalent, $CE(\cdot)$, is ...
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Expected Income Question

An urn contains equal number of green and red balls. Suppose you are playing the following game. You draw one ball at random from the urn and note its colour. The ball is then placed back in the urn, ...
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Independence and Reduction Axioms

I have read that the Independence of Irrelevant Alternatives axiom in expected utility theory implies the fact that compound lotteries are equally preferred to their reduced form simple lotteries. ...
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First Order Stochastic Domination and lottery preferences

Let $L$,$L'$ be two lotteries over the real numbers. Let $u$ be an increasing Bernoulli utility function. Let $F_L$, $F_{L'}$ be the CDFs of the two lotteries. We wish to show that $$L \succ_{FOSD} L'...
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Meaning of $dF(z)$ in 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:...
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Expected Utility with expected value and variance

I'm having trouble with a question from Ariel Rubinstein's book, Lecture Notes in Microeconomic Theory. It's the problem 2 from Problem Set 7. Here's the question: Show that the utility function $u(...
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Can I recreate an experiment on Allais paradox using student grades as payoffs?

For a project in experimental economics, I thought of doing something related to expected utility theory/prospect theory, but using grades instead of money. Is this reformulation of the Allais ...
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Does Pascal's Wager fail archimedean property?

I assume most people have heard of Pascal's Wager, in case you have not: https://en.wikipedia.org/wiki/Pascal%27s_Wager By the Stanford encyclopedia of philosphy: "We have a decision under risk,...
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281 views

Existence of 'best' and 'worst' lottery

How can 'the best and worst lotteries exist when the set of outcome is finite and the rational preference relation satisfies independence axiom' be proven?
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Proof of Expected utility theorem with three outcomes

I am trying to prove the expected utility theorem with three outcomes. The expected utility with $n$ outcomes is rather cumbersome and long in the economics textbook Mas-Colell. But I was hoping that ...
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Theories that complement/contradict prospect theory?

Kahneman and Tversky's prospect theory, which they developed to contradict expected utility theory, is obviously an interesting result. But, after their experiments has anyone tried to completement ...
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Anscombe and Auman Expected Utility

I would like to hopefully get some insights on the Anscombe and Aumann Expected utility. I've read some proofs and understood the Expected Utility Theorem (VNM) which allows us to approach consumers ...
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advanced mathematical treatement of revealed preference and utility theory?

I am looking for a textbook that treats revealed preference and utility theory much more thoroughly than does Mas-Collel. What would be suggestions for this? Specifically I'm interested in conditions ...
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Construct utility function for a risk-averse agent

I am trying to construct utility function for an agent who can be risk-seeking or risk-averse. We have an agent $i$ who has an ideal point $x$ in a policy space $X = [0,1]$. There is a policy (option) ...
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Does the Prospect Theory value curve change according to reference wealth size?

After reading "Thinking Fast and Slow" and half a day of internet search on this topic I resort now to asking this question here. Prospect Theory seems to only attribute value to changes in wealth. ...
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Can number of children in rich country is small due to this simple economic factor?

In rich countries, people are expensive. Even taxi driver in Singapore earns $6k per month. Now, the "market" tend to correct it self. If something is expensive, and it's marginal productivity is ...
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Lotteries and expected utility

Suppose we have the following four lotteries: $L_{1}=[(1,\$1)]$ $L_{2}=[(0.01,\$0),(0.89,\$1),(0.1,\$5)]$ $L_{3}=[(0.9,\$0),(0.1,\$5)]$ $L_{4}=[(0.89,\$0),(0.11,\$1)]$ If our agent says that he ...
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Expected values vs uncertainty

Most models I have seen use expected values. Why is this a better economic model than uncertainty and economic agents thus having to make 'best guesses', with the result of 'animal spirits' playing a ...
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Literature on Recursive Preferences and Time-Additive Expected Utility

In Chapter 20 of the book Economic Dynamics in Discrete Time, named "Recursive Utility", the author asserts that the Time-Additive Expected Utility Model (TAEU) has some shortcomings when applied to ...
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Is there a dutch book argument for the “independence of irrelevant alternatives” axiom?

There is a dutch book argument to show that nontransitive preferences are in a sense "unreasonable", which justifies why we pose the axiom of transitivity in the definition of "rational preferences", ...