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I'm working for a multi-part question which is confusing me. Below are the questions and what I've tried so far. As well as the questions that I'm having difficulty with understanding.


According to Bernoulli, money has Diminishing Marginal Utility. We’ll set our scale for value with endpoints at 0 and 1: $V(\$100) = 1$ and $V(\$0) = 0$.

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(a) Now suppose you are offered a gamble that has probability $p$ of paying \$100, and probability $1 − p$ of paying \$0. How high would p have to be for you to be willing to trade a guaranteed \$50 for this gamble? (This is a question about your personal preferences.)

For this question, it's obvious that having $p$ at 0.5 works, but since it's personal preference, I'd rather go with 0.6 or higher.


(b) Based on your answer to part (a), how much value does an extra $50 have for you?

This question is what confuses me, what do they mean by "how much value does and extra \$50 have for you"? I'd say the \$50 is worth \$50 but I don't think that's what is expected in this question. What do they mean by this?


(c) Now consider a gamble that has probability p of paying \$100, and probability $1 − p$ of paying \$50. How high would p have to be for you to be willing to trade a guaranteed \$75 for this gamble? (This is another question about your personal preferences.)

Again, another personal question, where 0.5 works but I'd rather have 0.6 or above.


(d) Based on your previous answers, how much utility does an extra $75 have for you?

Also where I don't know what to answer with. What exactly do they mean by utility?


(e) In terms of dollars, a gain of \$75 is 1.5 times as large a gain as \$50. In terms of your utilities, how does V(\$75) compare to V(\$50)? Is it more than 1.5 times as large? Less? The same?

I know this question builds up on (b) and (d) from earlier, so I'm hoping to learn how to answer them so I can answer this final question.


What I've tried so far:

Google didn't help, but I did find a Khan academy video related to the topic at hand, but I'm afraid it wasn't exactly what I'm looking for.

I'd appreciate a complete answer here, but anything helps and I'd really like to understand how to work these kind of questions for an upcoming midterm.

There are more parts to this question (more subquestions) but these are the only ones I'm really interested in. If you'd like to know more or there's anything that will help in clearing this question up, please let me know.

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  • $\begingroup$ You are saying $1 \times V(\$50) = 0.6 \times V(\$100) + 0.4 \times V(\$0)$. The others are similar $\endgroup$ – Henry Feb 17 '17 at 10:37
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$v(0) = 0$ and $v(100) = 1$

(a) Now suppose you are offered a gamble that has probability $p$ of paying \$$100$, and probability $1 − p$ of paying \$$0$. How high would $p$ have to be for you to be willing to trade a guaranteed \$$50$ for this gamble? (This is a question about your personal preferences.)

For this question, I am taking your personal preference as given that you are indifferent between a gamble that has probability $p=0.6$ of paying \$$100$, and probability $1 − p=0.4$ of paying \$$0$, and \$$50$ for sure.


(b) Based on your answer to part (a), how much value does an extra $50 have for you?

Given your answer to (a),

$v(50) = 0.6v(100) + 0.4v(0) = 0.6$


(c) Now consider a gamble that has probability $p$ of paying \$$100$, and probability $1 − p$ of paying \$$50$. How high would $p$ have to be for you to be willing to trade a guaranteed \$$75$ for this gamble? (This is another question about your personal preferences.)

Again I am taking your personal preference that you are indifferent between a gamble that has probability $p=0.6$ of paying \$$100$, and probability $1 − p=0.4$ of paying \$$50$, and \$$75$ for sure.


(d) Based on your previous answers, how much utility does an extra \$$75$ have for you?

$v(75) = 0.6v(100) + 0.4v(50)= 0.6 + 0.4(0.6) = 0.84$


(e) In terms of dollars, a gain of \$$75$ is $1.5$ times as large a gain as \$$50$. In terms of your utilities, how does $v(75)$ compare to $v(50)$? Is it more than $1.5$ times as large? Less? The same?

$v(75) = 1.4v(50)$.

It is less because you are risk averse.

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