Questions tagged [uncertainty]

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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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2 votes
1 answer
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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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1 vote
1 answer
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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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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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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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2 votes
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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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1 answer
613 views

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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1 answer
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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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3 answers
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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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3 votes
1 answer
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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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-1 votes
1 answer
203 views

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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1 vote
1 answer
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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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0 votes
1 answer
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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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0 votes
1 answer
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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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1 vote
1 answer
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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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2 votes
0 answers
42 views

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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2 votes
1 answer
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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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1 vote
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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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2 votes
1 answer
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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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-1 votes
1 answer
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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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0 votes
2 answers
329 views

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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3 votes
1 answer
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Profit maximization under uncertainity

I have a seller say S and I have a buyer say B. Buyer’s willing to pay is equal to x which is private information. But Seller believe that it falls in the range [0,x1]. Seller’s belief distribution is ...
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6 votes
1 answer
1k views

What exactly is certainty equivalence in the context of DSGE models?

I keep reading about certainty equivalence in the context of DSGE models. I understand that it has something to do with "Getting rid of the expectation operator", but I'm not entirely sure? ...
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5 votes
1 answer
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A growth model with regime switch

I have a very general question. I am reading this paper : http://www.webmeets.com/files/papers/eaere/2015/177/Discounting-HelsinkiBlind.pdf There is a catastrophic event probability and after the ...
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9 votes
5 answers
990 views

References for particular definitions of risk and uncertainty

I have some doubts about risk vs. uncertainty. I have read the thread "What is the difference between risk, uncertainty and ambiguity" and have skimmed through Knight's "Risk, Uncertainty, and Profit" ...
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0 votes
2 answers
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Help with a choice under uncertainty exercise [closed]

Let $u$ and $v$ be utility functions (not necessarily VNM) representing $\succsim$ on $\mathcal{G}$. Show that $v$ is a positive affine transformation of $u$ if and only if for all gambles $g^1, g^2, ...
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3 votes
1 answer
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von-Neumann-Morgenstern v. Bernoulli Utility Function

A great deal of time is spent distinguishing the big $U$ (von-Neumann-Morgenstern)v. small $u$ (Bernoulli Utility Function). The v.NM function maps from the space of lotteries to real number as it ...
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2 votes
1 answer
400 views

Inc Linear Transformation of Bernoulli Utility

According to MWG Proposition 6.B.2, it illustrates that the expected utility form is preserved only by increasing linear transformation. What is the significance of this proposition? The part I ...
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1 vote
1 answer
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Consequentialist View of Risk

In MWG, the authors introduce the consequentialist view of risk by assuming for any risky alternative, only the reduced lottery over final outcome matters to decision maker. From philosophical view, ...
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7 votes
8 answers
3k views

Would insurance plan be necessary if we had instant access to credit?

Assuming one could instantly have access to an loan to cover his recent loss/accident at an affordable interest, would it make sense to pay premium for an insurance plan? I was thinking that it would ...
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2 votes
1 answer
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Complete markets: security market equlibrium same as contingent commodity equilibrium?

"For complete markets, the security market equilibrium is the same as the contingent commodity equilibrium but with far fewer spot markets." I wrote this down in my notes today on what my professor ...
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