<p>As already mentioned in the comments, the question appears to be framed in a way that you should look up <a href="https://en.wikipedia.org/wiki/Stochastic_dominance" rel="nofollow noreferrer"><strong>stochastic dominance</strong></a>, and contemplate about its relation to expected utility theory, that can lead you to an answer.</p>
<p>Two aspects of the question worth pointing out:</p>
<ol>
<li><p>The payoffs per lottery are the whole <span class="math-container">$\mathbb R$</span>, this is the meaning of &quot;the lottery has a [XXXX continuous] distribution&quot;.</p>
</li>
<li><p>The question does not specify a utility function, only that the consumer has Von-Neumann Morgenstern preferences. So apparently, the correct answer does not depend on, say, specific preferences about risk, but it should be more general.</p>
</li>
</ol>

CC BY-SA 4.0
2024-02-27T22:23:09.127Z
Then I compare the figures of CDF of exponential distribution for two different $\lambda's$. I find the $F_1$ associated with smaller $\lambda_1$ dominates the distribution $F_2$. So we should choose the first lottery. That's it?
Jonathen
CC BY-SA 4.0
2024-02-27T22:30:47.090Z
@Jonathen No. I would strike out such an answer because it does not cover preferences properly. As a comment mentioned, this is not just about higher expected value, because then we would implicitly assume that the consumer is risk neutral. But what about if they are risk-averse or risk-loving? You need to link preferences under uncertainty to first-order stochastic dominance, which is the case here.
Alecos Papadopoulos
CC BY-SA 4.0
2024-02-27T23:59:52.340Z
I am confused. first-order stochastically dominates criterion doesn't depend on what type of $u$ we have.
Jonathen
CC BY-SA 4.0
2024-02-28T00:14:18.243Z
@Jonathen Exactly. So you have to think from first principles. Be the consumer and think which lottery you would choose, and why.
Alecos Papadopoulos
CC BY-SA 4.0
2024-02-28T00:17:07.873Z
My first comment in this answer is my idea...Since $F_1$ dominates $F_2$, then the consumer should choose lottery 1. But you said this answer is wrong. So that's where I am confused.
Jonathen
CC BY-SA 4.0
2024-02-28T00:50:13.253Z
@jONATHEN It is not wrong but fatally incomplete. What does "dominate" signifies as regards preferences and utility over uncertain outcomes? You must link the two, otherwise you are using the concept mechanically. To create I debate, I say to you, "I disagree. Exactly because $F_2$ is dominated, the consumer should choose lottery 2". How would you prove that I am wrong?
Alecos Papadopoulos
CC BY-SA 4.0
2024-02-28T00:54:16.487Z
Aha, now I see your idea. My understanding is that my conclusion is correct, but I need a more rigorous proof. Am I right?
Jonathen
CC BY-SA 4.0
2024-02-28T00:59:27.167Z
@Jonathen "Rigorous" mostly as conceptual arguments, not necessarily arrogant mathematics.
Alecos Papadopoulos
CC BY-SA 4.0
2024-02-27T21:31:17.357Z
2024-02-27T21:31:17.357Z
Alecos Papadopoulos
<p>Consider two lotteries each having an exponential distribution. The function of
cumulative distribution of an exponential distribution is:</p>
<p><span class="math-container">$$F(x;\lambda)=1-e^{-\lambda x}   \forall x \in \mathbb R_{+}.$$</span></p>
<p>The expected gain given by this distribution is <span class="math-container">$E[X]=\frac 1 \lambda$</span>.</p>
<p>Suppose that the first lottery, <span class="math-container">$F_1$</span> has a parameter <span class="math-container">$\lambda_1$</span> and the second, <span class="math-container">$F_2$</span> has a parameter <span class="math-container">$\lambda_2$</span>. Suppose that <span class="math-container">$\lambda_1 &lt; \lambda_2$</span>. Which of these two lotteries, a consumer with Von-Neumann Morgenstern preferences will choose?</p>
<p>I am confused on this question.</p>

CC BY-SA 4.0
2024-02-27T18:11:07.547Z
What part of it confuses you, please elaborate!
Giskard
CC BY-SA 4.0
2024-02-27T18:16:47.273Z
@Giskard Since $\lambda_1 < \lambda_2$, then it follows that $1/\lambda_1 > 1/\lambda_2$ meaning the expected gain from the first lottery is higher than from the second. That's it?
Jonathen
CC BY-SA 4.0
2024-02-27T18:39:00.007Z
The expected value is different from the expected utility; the decision-maker need not be risk-neutral. Probably, the assumption is that the decision-maker prefers more to less money. Then, you can solve this using first-order stochastic dominance. Just compare the CDFs pointwise.
Michael Greinecker
CC BY-SA 4.0
2024-02-27T18:53:23.030Z
@MichaelGreinecker I still cannot get the solution. Could you write an answer for me? I will appreciate your help.
Jonathen
CC BY-SA 4.0
2024-02-27T20:36:30.637Z
Where are you stuck? I'm not going to simply solve what might be someone's homework.
Michael Greinecker
<microeconomics>
Which of these two lotteries, a consumer with Von-Neumann Morgenstern preferences will choose under exponential distribution?
Jonathen
# Which of these two lotteries, a consumer with Von-Neumann Morgenstern preferences will choose under exponential distribution?

(Asked by: username_0 on 2024-02-27)

Consider two lotteries each having an exponential distribution. The function of
cumulative distribution of an exponential distribution is:

$$F(x;\lambda)=1-e^{-\lambda x} \forall x \in \mathbb R\_{+}.$$

The expected gain given by this distribution is $E[X]=\frac 1 \lambda$.

Suppose that the first lottery, $F\_1$ has a parameter $\lambda\_1$ and the second, $F\_2$ has a parameter $\lambda\_2$. Suppose that $\lambda\_1 < \lambda\_2$. Which of these two lotteries, a consumer with Von-Neumann Morgenstern preferences will choose?

I am confused on this question.

**Tags:** microeconomics

> **Comments:**
> **username_1**: What part of it confuses you, please elaborate!  
> **username_0**: @username_1 Since $\lambda_1 < \lambda_2$, then it follows that $1/\lambda_1 > 1/\lambda_2$ meaning the expected gain from the first lottery is higher than from the second. That's it?  
> **username_2**: The expected value is different from the expected utility; the decision-maker need not be risk-neutral. Probably, the assumption is that the decision-maker prefers more to less money. Then, you can solve this using first-order stochastic dominance. Just compare the CDFs pointwise.  
> **username_0**: @username_2 I still cannot get the solution. Could you write an answer for me? I will appreciate your help.  
> **username_2**: Where are you stuck? I'm not going to simply solve what might be someone's homework.  

---

## 1 answers

### Answer by username_3 on 2024-02-27. Score: 1 | Accepted answer

As already mentioned in the comments, the question appears to be framed in a way that you should look up [**stochastic dominance**](https://en.wikipedia.org/wiki/Stochastic_dominance), and contemplate about its relation to expected utility theory, that can lead you to an answer.

Two aspects of the question worth pointing out:

1. The payoffs per lottery are the whole $\mathbb R$, this is the meaning of "the lottery has a [XXXX continuous] distribution".
2. The question does not specify a utility function, only that the consumer has Von-Neumann Morgenstern preferences. So apparently, the correct answer does not depend on, say, specific preferences about risk, but it should be more general.

> **Comments:**
> **username_0**: Then I compare the figures of CDF of exponential distribution for two different $\lambda's$. I find the $F_1$ associated with smaller $\lambda_1$ dominates the distribution $F_2$. So we should choose the first lottery. That's it?  
> **username_3**: @username_0 No. I would strike out such an answer because it does not cover preferences properly. As a comment mentioned, this is not just about higher expected value, because then we would implicitly assume that the consumer is risk neutral. But what about if they are risk-averse or risk-loving? You need to link preferences under uncertainty to first-order stochastic dominance, which is the case here.  
> **username_0**: I am confused. first-order stochastically dominates criterion doesn't depend on what type of $u$ we have.  
> **username_3**: @username_0 Exactly. So you have to think from first principles. Be the consumer and think which lottery you would choose, and why.  
> **username_0**: My first comment in this answer is my idea...Since $F_1$ dominates $F_2$, then the consumer should choose lottery 1. But you said this answer is wrong. So that's where I am confused.  
> **username_3**: @jONATHEN It is not wrong but fatally incomplete. What does "dominate" signifies as regards preferences and utility over uncertain outcomes? You must link the two, otherwise you are using the concept mechanically. To create I debate, I say to you, "I disagree. Exactly because $F_2$ is dominated, the consumer should choose lottery 2". How would you prove that I am wrong?  
> **username_0**: Aha, now I see your idea. My understanding is that my conclusion is correct, but I need a more rigorous proof. Am I right?  
> **username_3**: @username_0 "Rigorous" mostly as conceptual arguments, not necessarily arrogant mathematics.  

---

