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Oct. 17th, 2005 10:14 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
which_chickMore fun from the brother in graduate school. I have no fucking idea what a Utility Function is. Never seen one before today.
Risky work. This problem requires you to calculate through the decision-making process of tow employees and an employer concerned with the tradeoff between risky jobs and wages.
Suppose that Naomi has a utility function Un(W,L). The means her utility depends on her wealth W in millions of dollars and on a binary variable , L, which tells if Naomi is alive (1) or dead (0). In particular suppose Naomi's utility function is as follows: Un (W, L) = L - e^-w Her initial wealth is finite, she lives a risk free life and places no value on her time.
a. Naomi is offered a job carrying a one time risk one percent risk of death. What is the minimum compensation (C) that Naomi would require to run this risk. Think of C as a lump sum increase in Naomi's wealth. Is it possible she would refuse this job for any compensation, no matter the size? Explain why or why not.
b. A second job is available, but its risk is uncertain. Naomi believes there is a one percent chance the job is very risky (50% chance of death), a 50% chance the job is moderately risky (1% chance of death), and a 49% chance the job is absolutely safe with zero risk. What is the minimum compensation she would require for this job?
c. The potential employer is also uncertain about the risks in part b. He has the same subjective distribution as Naomi. At no cost he can discover if the job is very risky, moderately risky, or safe. If he finds out he must tell Naomi. If his only goal is to minimize the expected value of hte risk premium he pays to Naomi to induce her to take the job, should he obtain the information? Why or why not.
d. Return to a and assume the answer you obtained is that Naomi will demand at least $C to take the job. OSHA frowns on paying of risk premiums for work so her employer offers her a life insurance policy of 100$C, which provides her with the same expected financial return as wage $C. Will Naomi accept the insurance in lieu of the risk premium? (note that she receives nothing from the insurance if she lives where as in A she got the wages regardless of being alive or dead)
e. John, a friend of Naomi, has the following utility function Uj( W,L) = L *1n(1+W) He is a bachelor with no dependents. His initial wealth is also W. The employer in part A is willing to pay a wage premium, pay an insurance contract, or provide some other incentive to John. The employer wants to minimize his expected monetary cost. What is the main difference between Un and Uj in terms of how John and Naomi evaluate different levels of wealth and the tradeoff beween money and the risk of dying. Discribe the general nature of the contract he should offer John.
I don't go to the lectures and I don't have the book. There are no examples for me. I assure you that my everyday life does not include anything that can be classed as a utility function. That thar's the preamble and disclaimer now that the brother has started giving me the problems he can't finish. So, y'know, don't get your hopes up.
a. Naomi is offered a job carrying a one time risk one percent risk of death. What is the minimum compensation (C) that Naomi would require to run this risk. Think of C as a lump sum increase in Naomi's wealth. Is it possible she would refuse this job for any compensation, no matter the size? Explain why or why not.
WTF do we do here? We are going to set the options to have equal worth if Naomi lives or dies. This is so that we can find the price point where it'll be worth her time. If we pay her more than that, she should be willing to take the job.
Living: .99(1-e^-x)
Dying: .01(1-e^-x)
.99(1-e^-x)=.01(0-e^-x)
.99 -.99(e^-x) = -.01(e^-x)
.99 = .98(e^-x)
1.010 = (e^-x)
ln(1.010)=.0101523
x=.0101523, which is "wealth in millions" as we read way up above. To make it a six-of-one, half dozen of the other choice, the premium C paid to Naomi has to be 10152. To make it a done deal, C has to be more than that, so call it $10153.00. No, I don't have any fucking idea where the minus sign went. I don't really understand negative exponents. (Shocking, innit?)
Is it possible that she would refuse this job for any compensation? Fuck yes. People do not always live via the cost-benefit analysis that you can see. Maybe she has a small child. Maybe the job requires too much travel. Maybe she doesn't LIKE cleaning out sewer lines. Maybe she wants more of a posh job with less, you know, filing. Perhaps she doesn't want her ass grabbed every day in your hostile workplace. People do lots of things for less than obvious reasons and expecting them to behave rationally according to YOUR definition of rational is asking to be disappointed. The comprehension questions make me think that they're not writing the books for cluebies anymore.
Checking because I feel really stupid here. Say that it's a 70live/30die split instead of 99 to 1. Naomi should want an assload more money. Does she?
.70(1-blah)=.30(-blah)
.70 -.7blah = -.3blah
.70 = .4blah
.7/.4=blah
x=.559615
Yes. She wants an assload more money for riskier work. Makes sense to me, anyway.
b. A second job is available, but its risk is uncertain. Naomi believes there is a one percent chance the job is very risky (50% chance of death), a 50% chance the job is moderately risky (1% chance of death), and a 49% chance the job is absolutely safe with zero risk. What is the minimum compensation she would require for this job?
.01 for 50% death
.50 for .01% death
.49 for 0% death
We still gotta find the tipping point for Naomi and now there's probability in there, too. Sheesh. Can't we just drug her drink and shanghai her aboard before she recovers?
Divvy it up into lives and dies.
Lives: .01[.50(1-e^-x)] + .50[.99(1-e^-x)] + .49[1(1-e^-x)]
Dies: .01[.50(0-e^-x)] + .50[.01(0-e^-x)] + .49[0(1-e^-x)]
Set 'em equal to each other and solve for tipping point.
.01(.50-.50blah) + .50(.99 -.99blah) + .49(1-blah) = .01(-.5blah) + .5(-.01blah) +.49(0)
(Doing this on the computer is a pain in the ass. I hope you appreciate the effort.)
.005 -.005blah + .495 -.495blah + .49 -.49blah = -.005blah -.005 blah
.99 -.99blah = -.01blah
.99 = +.98blah
.99/.98 = blah
x= .0101523 millions of dollars
Gee, we've seen that before, haven't we? Am I even doing this right? I'm starting to wonder... Tipping point is $10153.00
c. The potential employer is also uncertain about the risks in part b. He has the same subjective distribution as Naomi. At no cost he can discover if the job is very risky, moderately risky, or safe. If he finds out he must tell Naomi. If his only goal is to minimize the expected value of the risk premium he pays to Naomi to induce her to take the job, should he obtain the information? Why or why not.
If his only goal is to minimize the expected value of the risk premium, should he find out or not?
49% of the time he is going to pay zero death premium because no chance of death.
50% of the time he is going to pay 10153 (see part a)
1% of the time there is not enough money on the face of the earth to make her take the job, an assumption I am making because the equation I've been told to use goes irrational at that point.
(Divide by cheese error. +++Redo from start+++)
Not kidding on the divide by cheese error. When the probability of death goes to 50%, Naomi will not take the job no matter how much money she is offered, according to this shit-ass equation we are given. Where, I wonder, did the equation come from? Do I have a Utility Function tattooed on my ass? Does everyone? Or is all of this just pretend, like analyzing literature and acting like that's really a scientific and rational pursuit instead of rather complicated, insular mental masturbation? (I'm not saying literary analysis isn't fun or a great way to blow four years of one's life... but pretending that it's a science really bothers me.)
Should he find out? Well, yeah. Half the time he's not going to pay any premium at all. 49% of that is because there is zero risk of death and 1% of the time is because she cannot be paid enough to take the job. The other half of the time, he'd just be paying the premium he would ALREADY BE PAYING HER judging by the work we did in part B. I don't see how this could be seen as a loss EXCEPT if it were imperative that he hire Naomi, which is NOT what the question says. If it is mission-critical that he hire the bitch, he'd best not find out at all and pay the premium. Since the hiring of Naomi was not a condition of the question, he'd be ahead to find out and tell her the truth. Also: What's wrong with lying to the help?
d. Return to a and assume the answer you obtained is that Naomi will demand at least $C to take the job. OSHA frowns on paying of risk premiums for work so her employer offers her a life insurance policy of 100$C, which provides her with the same expected financial return as wage $C. Will Naomi accept the insurance in lieu of the risk premium? (note that she receives nothing from the insurance if she lives where as in A she got the wages regardless of being alive or dead)
Fucking A. Okay.
Expected outcome of the old way was .99(1-e^-x+c) + .01(0-e^-x+c)
Expected outcome of the new way is .99(1-e^-x) + .01(0-e^-x+100c)
These should be equal to each other if it's a fair deal and the problem TELLS US that the "expected financial return" is equal over the whole deal. (I'm glad they did that because I wasn't getting anywhere with it myself.) But... what about per case?
Assume Naomi does not die.
OLD WAY is .99(1-e^-[x+c])
NEW WAY is .99(1-e^-x)
I can't see how the fuck those are supposed to be equal assuming that C is a number other than zero. It's possible, even quite fucking likely, that Naomi's willingness to take a job depends more on the day-to-day expected result than on the wild-outlier result that she gets tragically kilt in her workplace. I don't know that Naomi would accept the insurance... it'd make her weekly pay lower and most people look at that more carefully than they look at the death benefits of their job. People aren't rational.
e. John, a friend of Naomi, has the following utility function Uj( W,L) = L *1n(1+W) He is a bachelor with no dependents. His initial wealth is also W. The employer in part A is willing to pay a wage premium, pay an insurance contract, or provide some other incentive to John. The employer wants to minimize his expected monetary cost. What is the main difference between Un and Uj in terms of how John and Naomi evaluate different levels of wealth and the tradeoff beween money and the risk of dying. Discribe the general nature of the contract he should offer John.
The main difference between Un and Uj is that Naomi is willing to consider jobs where she might die, so long as the odds of her buying the farm are solidly less than .50 whereas our man John is not at all willing to take a job where he might die. (What a big girl's blouse!) The employer needs to forget about hiring John for the job in part A because John cannot be paid enough to accept a job that includes the possiblity that he might die... or the employer can conveniently fail to disclose the dying part. That'd work, too.
Real statistics are more fun. Can we have some more of those, please?
Risky work. This problem requires you to calculate through the decision-making process of tow employees and an employer concerned with the tradeoff between risky jobs and wages.
Suppose that Naomi has a utility function Un(W,L). The means her utility depends on her wealth W in millions of dollars and on a binary variable , L, which tells if Naomi is alive (1) or dead (0). In particular suppose Naomi's utility function is as follows: Un (W, L) = L - e^-w Her initial wealth is finite, she lives a risk free life and places no value on her time.
a. Naomi is offered a job carrying a one time risk one percent risk of death. What is the minimum compensation (C) that Naomi would require to run this risk. Think of C as a lump sum increase in Naomi's wealth. Is it possible she would refuse this job for any compensation, no matter the size? Explain why or why not.
b. A second job is available, but its risk is uncertain. Naomi believes there is a one percent chance the job is very risky (50% chance of death), a 50% chance the job is moderately risky (1% chance of death), and a 49% chance the job is absolutely safe with zero risk. What is the minimum compensation she would require for this job?
c. The potential employer is also uncertain about the risks in part b. He has the same subjective distribution as Naomi. At no cost he can discover if the job is very risky, moderately risky, or safe. If he finds out he must tell Naomi. If his only goal is to minimize the expected value of hte risk premium he pays to Naomi to induce her to take the job, should he obtain the information? Why or why not.
d. Return to a and assume the answer you obtained is that Naomi will demand at least $C to take the job. OSHA frowns on paying of risk premiums for work so her employer offers her a life insurance policy of 100$C, which provides her with the same expected financial return as wage $C. Will Naomi accept the insurance in lieu of the risk premium? (note that she receives nothing from the insurance if she lives where as in A she got the wages regardless of being alive or dead)
e. John, a friend of Naomi, has the following utility function Uj( W,L) = L *1n(1+W) He is a bachelor with no dependents. His initial wealth is also W. The employer in part A is willing to pay a wage premium, pay an insurance contract, or provide some other incentive to John. The employer wants to minimize his expected monetary cost. What is the main difference between Un and Uj in terms of how John and Naomi evaluate different levels of wealth and the tradeoff beween money and the risk of dying. Discribe the general nature of the contract he should offer John.
I don't go to the lectures and I don't have the book. There are no examples for me. I assure you that my everyday life does not include anything that can be classed as a utility function. That thar's the preamble and disclaimer now that the brother has started giving me the problems he can't finish. So, y'know, don't get your hopes up.
a. Naomi is offered a job carrying a one time risk one percent risk of death. What is the minimum compensation (C) that Naomi would require to run this risk. Think of C as a lump sum increase in Naomi's wealth. Is it possible she would refuse this job for any compensation, no matter the size? Explain why or why not.
WTF do we do here? We are going to set the options to have equal worth if Naomi lives or dies. This is so that we can find the price point where it'll be worth her time. If we pay her more than that, she should be willing to take the job.
Living: .99(1-e^-x)
Dying: .01(1-e^-x)
.99(1-e^-x)=.01(0-e^-x)
.99 -.99(e^-x) = -.01(e^-x)
.99 = .98(e^-x)
1.010 = (e^-x)
ln(1.010)=.0101523
x=.0101523, which is "wealth in millions" as we read way up above. To make it a six-of-one, half dozen of the other choice, the premium C paid to Naomi has to be 10152. To make it a done deal, C has to be more than that, so call it $10153.00. No, I don't have any fucking idea where the minus sign went. I don't really understand negative exponents. (Shocking, innit?)
Is it possible that she would refuse this job for any compensation? Fuck yes. People do not always live via the cost-benefit analysis that you can see. Maybe she has a small child. Maybe the job requires too much travel. Maybe she doesn't LIKE cleaning out sewer lines. Maybe she wants more of a posh job with less, you know, filing. Perhaps she doesn't want her ass grabbed every day in your hostile workplace. People do lots of things for less than obvious reasons and expecting them to behave rationally according to YOUR definition of rational is asking to be disappointed. The comprehension questions make me think that they're not writing the books for cluebies anymore.
Checking because I feel really stupid here. Say that it's a 70live/30die split instead of 99 to 1. Naomi should want an assload more money. Does she?
.70(1-blah)=.30(-blah)
.70 -.7blah = -.3blah
.70 = .4blah
.7/.4=blah
x=.559615
Yes. She wants an assload more money for riskier work. Makes sense to me, anyway.
b. A second job is available, but its risk is uncertain. Naomi believes there is a one percent chance the job is very risky (50% chance of death), a 50% chance the job is moderately risky (1% chance of death), and a 49% chance the job is absolutely safe with zero risk. What is the minimum compensation she would require for this job?
.01 for 50% death
.50 for .01% death
.49 for 0% death
We still gotta find the tipping point for Naomi and now there's probability in there, too. Sheesh. Can't we just drug her drink and shanghai her aboard before she recovers?
Divvy it up into lives and dies.
Lives: .01[.50(1-e^-x)] + .50[.99(1-e^-x)] + .49[1(1-e^-x)]
Dies: .01[.50(0-e^-x)] + .50[.01(0-e^-x)] + .49[0(1-e^-x)]
Set 'em equal to each other and solve for tipping point.
.01(.50-.50blah) + .50(.99 -.99blah) + .49(1-blah) = .01(-.5blah) + .5(-.01blah) +.49(0)
(Doing this on the computer is a pain in the ass. I hope you appreciate the effort.)
.005 -.005blah + .495 -.495blah + .49 -.49blah = -.005blah -.005 blah
.99 -.99blah = -.01blah
.99 = +.98blah
.99/.98 = blah
x= .0101523 millions of dollars
Gee, we've seen that before, haven't we? Am I even doing this right? I'm starting to wonder... Tipping point is $10153.00
c. The potential employer is also uncertain about the risks in part b. He has the same subjective distribution as Naomi. At no cost he can discover if the job is very risky, moderately risky, or safe. If he finds out he must tell Naomi. If his only goal is to minimize the expected value of the risk premium he pays to Naomi to induce her to take the job, should he obtain the information? Why or why not.
If his only goal is to minimize the expected value of the risk premium, should he find out or not?
49% of the time he is going to pay zero death premium because no chance of death.
50% of the time he is going to pay 10153 (see part a)
1% of the time there is not enough money on the face of the earth to make her take the job, an assumption I am making because the equation I've been told to use goes irrational at that point.
(Divide by cheese error. +++Redo from start+++)
Not kidding on the divide by cheese error. When the probability of death goes to 50%, Naomi will not take the job no matter how much money she is offered, according to this shit-ass equation we are given. Where, I wonder, did the equation come from? Do I have a Utility Function tattooed on my ass? Does everyone? Or is all of this just pretend, like analyzing literature and acting like that's really a scientific and rational pursuit instead of rather complicated, insular mental masturbation? (I'm not saying literary analysis isn't fun or a great way to blow four years of one's life... but pretending that it's a science really bothers me.)
Should he find out? Well, yeah. Half the time he's not going to pay any premium at all. 49% of that is because there is zero risk of death and 1% of the time is because she cannot be paid enough to take the job. The other half of the time, he'd just be paying the premium he would ALREADY BE PAYING HER judging by the work we did in part B. I don't see how this could be seen as a loss EXCEPT if it were imperative that he hire Naomi, which is NOT what the question says. If it is mission-critical that he hire the bitch, he'd best not find out at all and pay the premium. Since the hiring of Naomi was not a condition of the question, he'd be ahead to find out and tell her the truth. Also: What's wrong with lying to the help?
d. Return to a and assume the answer you obtained is that Naomi will demand at least $C to take the job. OSHA frowns on paying of risk premiums for work so her employer offers her a life insurance policy of 100$C, which provides her with the same expected financial return as wage $C. Will Naomi accept the insurance in lieu of the risk premium? (note that she receives nothing from the insurance if she lives where as in A she got the wages regardless of being alive or dead)
Fucking A. Okay.
Expected outcome of the old way was .99(1-e^-x+c) + .01(0-e^-x+c)
Expected outcome of the new way is .99(1-e^-x) + .01(0-e^-x+100c)
These should be equal to each other if it's a fair deal and the problem TELLS US that the "expected financial return" is equal over the whole deal. (I'm glad they did that because I wasn't getting anywhere with it myself.) But... what about per case?
Assume Naomi does not die.
OLD WAY is .99(1-e^-[x+c])
NEW WAY is .99(1-e^-x)
I can't see how the fuck those are supposed to be equal assuming that C is a number other than zero. It's possible, even quite fucking likely, that Naomi's willingness to take a job depends more on the day-to-day expected result than on the wild-outlier result that she gets tragically kilt in her workplace. I don't know that Naomi would accept the insurance... it'd make her weekly pay lower and most people look at that more carefully than they look at the death benefits of their job. People aren't rational.
e. John, a friend of Naomi, has the following utility function Uj( W,L) = L *1n(1+W) He is a bachelor with no dependents. His initial wealth is also W. The employer in part A is willing to pay a wage premium, pay an insurance contract, or provide some other incentive to John. The employer wants to minimize his expected monetary cost. What is the main difference between Un and Uj in terms of how John and Naomi evaluate different levels of wealth and the tradeoff beween money and the risk of dying. Discribe the general nature of the contract he should offer John.
The main difference between Un and Uj is that Naomi is willing to consider jobs where she might die, so long as the odds of her buying the farm are solidly less than .50 whereas our man John is not at all willing to take a job where he might die. (What a big girl's blouse!) The employer needs to forget about hiring John for the job in part A because John cannot be paid enough to accept a job that includes the possiblity that he might die... or the employer can conveniently fail to disclose the dying part. That'd work, too.
Real statistics are more fun. Can we have some more of those, please?
