Questions tagged [uncertainty]
The uncertainty tag has no usage guidance.
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How to solve for the competitive equilibrium in this very lopsided model?
Suppose there's an infinite-horizon pure-exchange economy with 2 agents. Utility function for both agents is
$$u(c)=\frac{c^{1-\sigma}}{1-\sigma}$$
The state at $t\geq 0$ is a random variable, $s_t$, ...
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Minimal assumption for a “certainty equivalence” exists
Let $R$ be the set of real number. Let $N$ be an infinite set. Let utility $u:R^N\to R$. The utility function is strictly monotonic.
My question is, does the certainty equivalence $CE$ exist? Do we ...
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What are the theoretical approaches to ambiguity?
I'm trying to understand the different approaches that economists took to investigate ambiguity. Two approaches particularly caught my eyes:
the model by Klibanoff, Marinacci and Mukerji (2009), and
...
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How do you prove NO difference in Expected Value, between gambling (probable $−$ EV) vs. investing (probable $+$ EV)?
How do you mathematically prove that "From a mathematical expected-value standpoint, there is no difference between gambling (e.g. buying a lottery ticket) and investing (e.g. buying a share of ...
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How to decide whether to buy a lottery with a too negative EV, but passable $\Pr($you win jackpot at least once│n plays)?
Daily Keno's too negative Expected Value looks scammy.
For all the different wagers of Keno's $10 PICK, the odds of winning jackpot are selfsame : 1 in 2,147181. I computed their EV.
Wager
Jackpot
EV ...
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Lotteries in an equilateral triangle
A variable can take 3 values, each represented by a vertex of an equilateral triangle. Every point in the triangle is then a lottery over the vertices. How are the elements of each lottery related to ...
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Willingness to sell a lottery ticket vs. willingness to buy a lottery ticket
I'm struggling with this question: There is a lottery which gives you D with p = 0.25 and L with p = 0.75 while initial wealth is w (w > D > L > 0). What is the minimum price the person would ...
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Does local non-satiation hold for this problem?
I am getting some confusing results solving this problem:
$max_{c_0\geq0, c_1\geq0} \bigg\{EU = R(1-c_0) [p t_1 + (1-p) c_1^{-2} t_2] \bigg\} ~ s.t. ~c_0+c_1 \leq 1$
where $p$ is the probability of $...
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CARA risk-parameter estimation for discrete data as in Holt and Laury (2002)
I have a dataset with around 40 participants who choose between a pair of lotteries A or B while the probabilities of prizes change. This setting is similar to that in Holt and Laury (2002): Risk ...
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Problem on VNM utility
Preferences are represented by the function $u(w) = √w$ .
An investor has 5000 dollars that can be invested. Investing $X ∈ [0,5000]$ he will receive $r_H X$ in addition to W with probability $P$ and ...
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Lower bound for the utility in a decision problem with uncertainty
Model
Consider a single-agent decision problem with uncertainty.
A decision maker (DM) has to choose action $y\in \mathcal{Y}$ possibly without being fully aware of the state of the world. $\mathcal{Y}...
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Arrow-Debreu Market with Uncertainty
I am reading some lecture note on stochastic macro model. Say an endowment economy with agent $i=1,2$, who receive the random endowment each period $e_{t}^{i}\left(s^{t}\right)$ where $s^{t}=\left(s_{...
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Expected utility function and Full insurance
Bob is an expected utility maximizer with utility function $u(x) = −e^{−ax}$, where $a > 0$ is a parameter. Bob has wealth $w$. There are two states of the world, a good state and a bad state. The ...
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Von-Neumann vs Bernoulli Utility? Do properties of Bernoulli translate to VNM?
This is for a HW question but I am not asking on how to solve the question, just a conceptual question.
According to my understanding,
A Von-Neumann Utility function represents preferences over ...
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Arrow debreu equilibrium or Radner equilibrium and spot prices
Suppose there are 2 states, 2 goods and 2 consumers and consumers have identical expected utility function:
$U^i (x)= \sum_{s=1,2} \pi_s (\ln x_{1s}+\ln x_{2s} )$ where $\pi=(1/3,2/3)$.
Endowments are ...
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Most utility functions under risk and uncertainty generalizes expected utility. What is deadly wrong if a model does not include EU as special case?
Why do people generalize EU instead of making an entirely new model, or create a model that is neither a special case nor an extension of EU?
To my knowledge, most utility functions under risk and ...
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Risk Aversion under Worst Case Utility Representation
The preference relations (≿A and ≿B) over lotteries is defined as:
p ≿ q iff min{v(z) : p(z) > 0} ≥ min{v(z) : q(z) > 0}
Under what conditions can you say that ≿A is more risk averse than ≿B?
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Negative certainty equivalent
Let us consider an agent of initial wealth $w_0$ whose utility function is $u(x)=\sqrt{x}$. This individual faces a risk of loss $Z$ which occurs with probability $p$.
It is assumed that $w_0=60000$, $...
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What is the intuition behind Expected Utility Theorem?
I am referring to the definition in Proposition 6.B.3 on Page 176 of Mas Colell. I follow the formal proof and the application of the Independence axiom at various steps (mathematical application of ...
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Volatility indexes
I want to use a volatility index as a measure of economic policy uncertainty. The index that I want to use is CBOE Volatility Index: VIX, but I don't know if I only can use this for US or if I can use ...
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Drawing a Probability simplex
There are 3 possible payoffs - \$4, \$9 and \$36. The utility function for these payoffs is $\sqrt x $.
I have to find all the lotteries preferable over getting $9 with probability 1 in a probability ...
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certainty equivalent and lotteries [closed]
suppose an agent has $u(z)=-e^{-bz}$ where $b>0$ as her Bernoulli utility function and faces two gambles:
G1: win 1000 dollars with probability $\frac{1}{2}$ and zero with probability $\frac{1}{2}$ ...
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comparing two lotteries
Suppose the prize space (in dollars) is $\mathbb{Z}$ = {1, 2, 3, 4, 5, 6, 7, 8} and consider choices by an agent
whose preferences (over lotteries) satisfy the von Neumann-Morgenstern axioms.
A risk ...
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Choice under uncertainty
I am practising past micro economics questions from the internet and I am not sure how to proceed with this question:
Imagine a situation where a risk averse agent has positive wealth(w) and may face ...
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Deducing beliefs from choices when the Savage Axioms are true
We know, that given a set of possible outcomes $X$, a set of states of nature $\Omega$, and the set of all acts from $\Omega$ to $X$, if a DM has rational preferences over the acts and if the Savage ...
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Proof that $U(\sum_{n=1}^{N}{p_nL_n})=\sum_{n}^{N}{p_nU(L_n)}$
I understand the expected value of a lottery is $\sum_n^N{p_nL_n}$ where there are $N$ possible outcomes, each with a probability $p_n$ with $n=1,...,N$ and $\sum_{n}p_n=1$ (that's rather trivial I ...
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Pure Nash equilibrium in bidding game?
According to the answer key for a problem set, there is no pure strategy Nash equilibrium in the following problem. Yet I can't see why not. Could it be an error in the answer key? Here's the problem:
...
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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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Choice under Uncertainty: Relation between the certainty equivalent and the coefficient of relative risk aversion
That's my question!
Propositon 6.C.4 (MWG (1995)): The following conditions for a Bernoulli utility function $u(\cdot)$ on amounts of money are equivalent:
(i) $r_R(x,u)$ is decreasing in $x$.
(iii) ...
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Uncertainty and Pareto efficient policies
There are two economic agents $i\in \{1,2\}$ with state dependent utility $u_{is}=-(x-b_{is})^2$ where $x\in R$ and $b_{is}\in R$ is bliss point of $i$ in state $s\in\{1,2\}$. Assume $b_{1s}\lt b_{2s}$...
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Epstein zin and resolution of uncertainty
I'm reading Simon Gilchrist's notes here. I understood everything until and including page 14, where it reads
$$\frac{W_h^{1-\rho} + W_l^{1-\rho}}{2} \geq \frac{c_h^{1-\rho} + c_l^{1-\rho}}{2} $$
and ...
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Proof of certainty equivalence
The questions say two lotteries are denoted L1 = (0.3,0.7,0.0) and L2= (0.9,0.0,0.1) and denote c(L1) and c(L2) the certainty equivalents of those two lotteries.
Then prove L1 ≻ L2 if and only if c(L1)...
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Can the Certainty Equivalent be negative?
I am questioning if the CE of a lottery can be negative? For me it doesn't make much sense by definition.
I encountered this problem on the following exercise:
Imagine a case where we have a lottery(...
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Lotteries = probability distribution?
Are "lotteries" in the model for choice under uncertainty not just probability distributions?
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Term for risk AND ambiguity
This question is related to References for particular definitions of risk and uncertainty, which offers an excellent description of risk and uncertainty.
Just as a recap: Knight (1921) described risk ...
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Can we compare risks of lotteries?
I understand the concepts of risk-aversion, risk neutrality and risk-attraction. I wonder if it possible to compare risks between two lotteries without giving the utility function. For instance, Let ...
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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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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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Conditions for Radner Equilibrium
This is definition of Radner Equilibrium from Microeconomic Theory (Mas-Collel, Whinston and Green - Third Edition).
I'm confused about two conditions:
$\sum_k q_k \cdot z_{ki} \le 0$ (in yellow)
The ...
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Existence of optimal strategy in a choice problem with uncertainty and information structure
Consider a decision maker choosing an action, $y$, from the finiteset $\mathcal{Y}$, possibly without having complete information about the state of the world.
More precisely, let $V$ be a ...
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Robust predictions in single-agent decision problem with uncertainty
I would like your help to better understand the possibility of using the notion of Bayes Correlated Equilibrium (BCE) in a single-agent decision problem with uncertainty to make predictions on optimal ...
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Optimal strategy in a single-agent choice problem under uncertainty
Consider the following single-agent choice problem under uncertainty.
Let $V$ be the state of the world with support $\mathcal{V}$ and probability distribution $P_V\in \Delta(\mathcal{v})$. First, ...
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Budget constraint in Radner Sequential Trade Equilibria
Suppose that $q$ is a k-tuple vector of prices for the k assets whose quantities are given by the k-tuple $\theta$. I have just read that in the Radner Sequential Trade Equilibrium (not sure if this ...
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Dominated lotteries in CPE
I have been looking into expectation-based loss aversion following Kőszegi-Rabin (2005, 2007). In particular, I find their choice-acclimating personal equilibrium (CPE) interesting, but it has a ...
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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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Archimedean but not mixture continuous
In the context of preferences on a set of lotteries on a finite set $X$, what is an example of a preference that is independent, Archimedean but not mixture continuous?
I know the mixture continuous ...
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Why Certainity Eqivalence in PIH only holds for quadratic utilities
In my current macro economics course, it has been stated that there is certainty equivalence in the random walk permanent income hypothesis ("which implies individuals act as if future consumption was ...
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Question on uncertainity
Please imagine that Nicole is uncertain of her future wealth. Her wealth in the bad state of the world is zero. Her wealth in the good state is $w>0$. Each state is initially equally likely. ...
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Indiference between two lotteries
Suppose that a binary relation satisfies only:
Independence axiom: $L≿L′⟺α\circ L+(1−α)\circ L′′≿α\circ L′+(1−α) \circ L′′$
Reduction to simple lotteries: For all $g$, $g~g'$, $g'$ is the simple ...
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Uncertainty in an unfair gamble
If a risk averse person is given the option of a certain amount of 2000 or playing a lottery game giving him 10000 with 25% probability, and 500 with 75%, then what would he do ?.
The bet here is not ...
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