Questions from economics honors exam at Oberlin College

Steven Landsburg was chosen by the economics department at Oberlin College to be an outside examiner to "determine who among its top graduating seniors should receive an honors degree." He posted the written exam, which consists of 10 questions, to his blog.

I feel confident in stating that if I took the test I would get a score of 0.

Question 6. When Eve works, she produces exactly one apple per hour. Adam is completely unproductive and can produce nothing at all. Eve’s income is taxed at a flat percentage rate, with the proceeds delivered to Adam. What determines the optimal tax rate? What does “optimal” mean here, and what philosophical justification would many economists give for adopting this tax rate?
To make the problem concrete, you can assume that both Adam and Eve, if it were both possible and necessary, would be willing to work up to 1 hour for 1 apple, up to 2 hours for 4 apples, up to 3 hours for 9 apples, and up to x hours for x^2 apples. Now what is the optimal tax rate? (Your answer should be a number.)
Question 8. The five Dukes of Earl are scheduled to arrive at the royal palace on each of the first five days of May. Duke One is scheduled to arrive on the first day of May, Duke Two on the second, etc. Each Duke, upon arrival, can either kill the king or support the king. If he kills the king, he takes the king’s place, becomes the new king, and awaits the next Duke’s arrival. If he supports the king, all subsequent Dukes cancel their visits. A Duke’s first priority is to remain alive, and his second priority is to become king. Who is king on May 6?

The Honors Class, Part I | The Honors Class, Part II

Mark Frauenfelder is the founder of Boing Boing and the editor-in-chief of MAKE and Cool Tools. Twitter: @frauenfelder. Come and hear Mark speak at the ALA conference in Chicago on July 1.

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Ian70

Will someone -please- get an Oberlin Econ grad in here to answer this damn king-killing question once and for all? Please?
Anonymous

It is obvious that Duke One is the only Duke who can kill the king with impunity. Duke Two would never kill Duke One because Duke Five would, conditional on not canceling his visit, be able to kill the King with impunity.
Anonymous

Clearly the optimum tax rate is 0%. Eve is the only producer in the economy and deserves to keep all her production. Adam produces nothing and deserves to starve to death.
wakeupsilver

Eve decides to live off the land instead of working, and the dukes decide that they’d rather just hug everyone they meet and soon there are lines 10000 people long waiting to get hugged every day.
daneyul

For those who don’t understand why duke 1 is the answer—assume 3 dukes to make it simpler.

Keeping in mind each duke has first priority—stay alive; second priority—become king.

Duke 1 KNOWS that duke 2 can’t honor his “stay alive” priority by killing him, therefore duke 1 safely kills the king.

Has to be that way because:

- duke 3 WILL always honor his priorities by killing the king since the first is met whether he does or doesn’t kill so the 2nd priority takes precedence.

- duke 2 knows this so he will ALWAYS honor his priorities by NOT killing the king since that’s the only way he can stay alive

- duke 1 knows this so will always honor his priorities by killing the king since he will survive either way and the 2nd priority can be the determining factor.

Now will someone please answer the tax one????
mypalmike

Consider the additional rule: All other Dukes hate Duke 1, and would rather die than see Duke 1 become King.
Anonymous

For the taxation one.
If Eve works for 1/2 hours, she produces 1/2 apples. She is willing to work for x^2 apples so if she is left with 1/4 apples, she is happy. So she can be taxed at 50%.
If she works for 1/4 hours, she produces 1/4 apples but she is fine with being left with 1/16 apples, so she can be taxed at 75%, leaving Adam with 3/16 apples.
However, optimum taxation must deliver equal or more apples to Eve. I think the answer is 50% flat tax, as it gives Eve the most apples per minuite
Joe

The real-world application for these questions is to train students to support a right-wing, money-centric way of looking at the world that can lead to false answers.

For example, an economist must answer that if Whiplash is already charging monopoly prices for housing, so much that if he charges any more people will leave town, he therefore cannot increase the price of groceries without decreasing the price of housing. But this assumes that people can leave town freely. Since they can’t, he can raise prices even higher and throw everyone into debt that they can never escape. Classical economics would frown on any measures to help the workers, and use the official (and wrong) answer to this question to claim that the market, all by itself, checks Snidley Whiplash.

The assumption that human beings are robotic utility maximizers is the flaw in many of the questions. Consider the dukes. No duke can risk that subsequent dukes will perform the calculation correctly: an irrational visiting duke could mean death. So only the last duke can safely kill the king, therefore the original king will remain king, the first duke will support him and cancel all remaining visits. All other assumptions presume that subsequent dukes will calculate their interests in a particular way.
mechaxl

The answer?

The first duke.

The first duke kills the king, and then cancels all subsequent appointments with the other dukes, thus fulfilling his first and second objectives.
Rick York

So forget about all the dukes and kings. Didn’t we overthrow all that stuff in 1776?

Did this guy post the answers anywhere? I gave up at Question #1. But, I’m an old geezer whose brain is missing 5th gear. Does the questioner himself have answers?
swamprat

For the taxation one.
If Eve works for 1/2 hours, she produces 1/2 apples. She is willing to work for x^2 apples so if she is left with 1/4 apples, she is happy. So she can be taxed at 50%.
If she works for 1/4 hours, she produces 1/4 apples but she is fine with being left with 1/16 apples, so she can be taxed at 75%, leaving Adam with 3/16 apples.
I think the philosophical justification is a mix of communism and utilitarianism. From each according to their ability, to each according to their wants.
Eve can only produce 1 apple (as working more than this, she gets less than x^2 apples). She only wants 1/2 an apple. The utility gained from any piece of apple is inversely proportional to the number of pieces one has ie a starving person values 1/2 an apple more than someone who has already had 1/2 an apple. Assuming Adam and Eve gain the same utility for the same amount of apple, with the same inverse relationship between utility per unit and total number of units, utility is maximized through an equal division, ie 50% tax
philoponia

Assuming the King takes a 50% cut from the taxes and that 10% is distributed back to each of the two peasants, and that Eve is only willing to work for the most food/work then she will work for 30 minutes, have 0.5 apples (which makes a whole lot of sense when you are growing apples by the hour). Thus, she will give up 0.25 apples to the king, who is ready to distribute back 0.025 apples each back to Adam and Eve, but he sees that Eve is wealthy and does not need the apple back.

Now the king has 0.225 apples and Eve has 0.25 apples and Adam has 0.025 apples. Duke 1 sees this and does not like that Eve has more apples than the king, so he kills Eve and gives 10% of his earnings, or 0.025 apples, to the king. The king now has 0.25 apples, Adam has 0.025 apples and Duke 1 has 0.225 apples.

Duke 2 knows that Duke 3 was sleeping with Eve and will be mighty upset with any Duke that he thinks killed her. Since he knows that Duke 3 knows that Adam was a lazy slob that would never work, he can safely kill Adam and take his 0.025 apples without worrying about Duke 3 thinking that anybody other than Duke 1 had killed Eve.

Unfortunately at this point, Duke 3 sees that the King has 0.25 apples, Duke 1 has 0.225 apples, and Duke 2 has 0.025 apples, so he assumes that the king, having the most apples, most have killed Eve. So Duke 3 kills the king and becomes king. He then takes in a 50% cut from the peasants to bolster his new kingdom.

Duke 4 sees that Duke 3 is king with 0.375 apples, Duke 2 with 0.0125 apples, and Duke 1 with 0.1125 apples. He knows that nobody has yet sworn allegiance and that that bastard Duke, Duke 5, will sure as hell not pledge allegiance to anybody but the original king. Knowing this, Duke 4 kills Duke 1 and takes his 0.1125 apples.

Sure enough, Duke 5 sees that the original king has been killed and knows that the new king, Duke 3, must be at the bottom of this. Duke 2, however, is starting to get upset that he has so little an allotment of apples and he decides to kill Duke 4, knowing that Duke 5 will only be after Duke 3. Duke 5 does kill Duke 3 and takes his 0.375 apples and Duke 2 kills Duke 4 to increase his apples to 0.125. Duke 5, now king, takes another 50% cut from his peasants to bolster the kingdom leaving Duke 2 at 0.0625 apples and Duke 5 at 0.4375 apples.

Duke 5 (the King on May 6) and his peasant, Duke 2 are both happy with this situation, however on May 7th, Duke 5 is trying to remain alive (his first priority) and so he orders Duke 2 to apple-farm. Duke 2, realizing that it is his second priority to become king, kills Duke 5 and is now king and alive and the proud owner of 0.5 apples.

On May 8th, the new king, Duke 2, dies of starvation because as king he is not even willing to work 1 hour for 1 apple. To answer the questions, on May 6 Duke 5, then king, had a 50% tax rate and yet people were still dying, so Duke 2, now king, decides that the optimal tax rate must be higher so that he can appropriately distribute the wealth, so he raises it to 75%, realizing that that number is halfway between 50 and 100, so it must be correct!
Anonymous

Actually, given the “rules” for the Dukes (#1 to stay alive, then #2 to become king), I still support the that the original King remains. But, I agree with Ian70. I would love to hear the answer, although I doubt it will be posted. It’s a test for the students.
mypalmike

The tax one is probably 1/e.
i_prefer_yeti

replace royal palace with “first”
replace king with “who”
replace duke #1 with “what”
replace duke #2 with “I don’t know”
replace duke #3 with “why”
replace duke #4 with “because”
replace duke #5 with “I don’t care”

mix with proper timing to evoke hilarity.
Anonymous

The Lady of the Lake, her arm clad in the purest shimmering samite held aloft Excalibur from the bosom of the water, signifying by divine providence that I, Arthur, was to carry Excalibur. THAT is why I am your king.
Anonymous

The optimal tax rate is 99%. Eve is free to not work and starve. If she gets hungry, she is free to work and keep her 1% of an apple to eat. If she wants a full apple, she better move her lazy arse and work 100 hours.

Sure, Adam eats the rest, but no mater the percentage, 100% of every apple produced gets eaten – so if you are looking for “optimal” you already have 100% effeciency on the consumption side. The only lever you have left is to alter the production side. Only Eve can decide how much gets produced.

This system creates the condition for the most production.

Feel free to add decimal places to 99 as far as the sky.
DIYer

1. Adam takes video of apple production process;
2. Posts vid on intertubes;
3. ???;
4. Profit!!1!1
daneyul

>> THAT is why I am your king.

Yeah, but the answer is still duke number 1. You can’t expect to wield supreme executive power just because some watery tart threw a sword at you
ikegently

I think that the Duke question is a formulation of the chain store paradox.

I would think that any answer to this question that would satisfy an honors examiner at Oberlin would bring up the complexities of making decisions based on beliefs about the actions of future players. This is not about training students to be right-wingners. If you think that is what goes on at Oberlin, you should probably just throw in the towel now.
daneyul

Yes, of course—the answer (duke 1) is what it is due to the artifice that all the dukes will behave like automatons—-it is distinctly NOT the correct answer in a real world situation (where the answer would be that the king would remain in power).

But then, the hoary “two trains travel at the exact same speed from opposite points in the country” isn’t possible in the real world either, nor are all those word problems I did in Physics that conveniently said “assume no friction”. Problems like this are mental exercises, not real world examples.
Anonymous

As an Oberlin econ grad (though certainly not Honors), the King/Duke question reminds me of a ‘proof’ my Calc II professor showed us…

He walked into the room Monday morning and wrote “pop quiz” on the board. He announes there will be a pop quiz this week. As we write nervously in our notebooks he reasons that if Thursday night rolls around and we haven’t had the quiz we’ll know it’s on Friday and it won’t be a surprise, so the quiz can’t be Friday. He further reasons, given that the quiz can’t be Friday, when Wednesday night comes we’ll know the quiz is on Thursday and it won’t be a surprise, so the quiz can’t be Thursday either.

He uses the same logic for the remaining days of the week and finally determines that a pop quiz is impossible. He erases “pop quiz” from the board, states that there will be no pop quizes this semester, and begins class.
- SamSam
  
  That’s actually called the “Pop quiz paradox” or the “Unexpected hanging paradox.”
  
  The paradox actually goes further than your professor brought it — he only stopped half-way.
  
  After “proving” that he couldn’t logically give you the pop quiz on any day of the week (as it wouldn’t be a surprise), he could have just gone ahead and given you the pop quiz on any day of the week, and you would have been surprised. Why? Because you would have “known” that he couldn’t give you the pop quiz, and so would not have expected it.
- Anonymous
  
  Was that prof. Young? or Walsh, perhaps?
  - Anonymous
    
    It was indeed Professor Young. What a guy. He’d come into class every day (often in an undershirt) with nothing in hand and produce a small piece of paper with the homework for that night. After copying the assignment onto the board he would throw away the paper and teach the entire class from memory. I still have not forgotten the phrase “this lies very deep”, although I am not entirely sure what it means.
    
    SamSam – Thanks for the additional info. I’d assumed it was a known and studied paradox, but I’d never bothered to look it up. I think what was so delightful about my professor’s presentation was that, after initially making us nervous about the possibility of a pop quiz, he then pretended to consider it sufficient reason to never give one.
Anonymous

This is not science, or even an attempt at producing objective knowledge of the world. Instead it is either normative indoctrination or a critique of the dogma imposed upon economics students. I sincerely hope this is the point of the exercise is the latter but from my experience in econ. depts it seems to be the former.

These questions make no sense. Adam and Eve is a reduction of society to absurdity – there is no state, no commerce, no social provision of public goods, only redistribution. The conclusion to be drawn is the overly simplistic view that there are only the worthy and the parasites. This is childish and irresponsible distortion of the world.

The question on the dukes is training in rational choice game theory. It is a non-iterated “zip-back game” in which rational choice becomes unpredictable. As a game it is cute and fun to talk about, but what does it tell us about economics? About as much as it tells us about coups. Palace coups are never individual actions shaped only by an arbitrary incentive structure, yet this asks us to imagine they, and most human interactions, are.
John Mark Ockerbloom

It’s worth noting that the first-Duke-becomes-king solution is a very fragile solution; the slightest deviation from the abstract model (i.e. not everyone knows that everyone is perfectly rational, follows exactly the rules given, and has no other priorities) can make the whole solution fall apart.

Fragile solutions may be suitable for mathematics, but they’re disastrous for economics. One unexpected development can bring your whole scheme crashing down.

In a similar vein, a solution that’s fair in an abstract simplification, but fragile, can be used as propaganda to let the powerful game the system (because the solution doesn’t actually do what it claims) while preventing the less fortunate from complaining (because it *seems* to do what it claims if you don’t look too closely).

It’s not hard to see examples of both of these phenomena in recent economic history. If I were to evaluate economists, I’d pick one who could better recognize the fragility of a model over the one who could better apply it by rote.
- ikegently
  
  “If I were to evaluate economists, I’d pick one who could better recognize the fragility of a model over the one who could better apply it by rote.”
  
  Which I would bet is what the examiner was looking for. I guarantee that an answer that simply said: Duke 1. would get zero points on an exam like this.
- djn
  
  @mark, 44:
  That’s why I like the “first duke supports the king”-solution: If we add any element of doubt or uncertainty, it’s the sensible way to go.
Anonymous

What the Hell. I get on to BoingBoing to get a break from studying from studying for my Econ final at Oberlin in a week, and what do I find?

I’m absolutely serious. Stop playing with my head.
Jonathan Badger

If Eve produces apples, then clearly she is a apple tree and has no conscious control of the production process. Therefore tax rates in no way will influence her output.
- Pantograph
  
  If Eve produces apples, then clearly she is a apple tree and has no conscious control of the production process. Therefore tax rates in no way will influence her output.
  
  Either an apple tree or an underpaid Chinese factory worker assembling iPods.
Anonymous

I don’t know if fractional apple production is possible, but it appears the only way to generate any result that could be described as optimal. Given Adam and Eve’s joint valuation of x^2 apples consumed for every x hours worked, the only solutions that generate a possible surplus are those where Eve works for less than an hour. For instance, she is willing to work three quarters an hour for nine sixteenths of an apple or a third of an hour for a ninth of an apple. Optimality in my mind involves (1) encouraging the largest possible apple production and (2) distributing apples according to the value functions of all citizens. Optimality viewed in this rate suggests a tax rate of fifty percent, inducing Eve to produce half an apple and consuming half of the half (a quarter of an apple). The other quarter of the apple is given to Adam who values apples the same as Eve according to the problem statement.

There is no information about whether Adam has a disability because he fought heroically for his country or is simply a loafer who is faking an apple-producing disability, so I can not factor any teabaggery into the solution. In fact, the societal solution in my mind is to give enough apples to Adam so that his war wounds heal and he can join in the production of apples. If our population were all healthy, we could afford a lower tax rate.
Mia MacHatton

The dukes and the king are a farce of government! Rise up and destroy the oppressors!! Bring in the guillotines!

Now pardon me while I go off to knit in the corner.
Anonymous

The Duke on the first day (Duke 1) knows that if he kills the King, he will be in the same position as the King when Duke 2 arrives, and so forth.

Since no one is there to go after Duke 5 (Duke on 5th day), Duke 5 will always kill the current King. Therefore all Dukes before him will have to choose to support the King else end up dead when Duke 5 arrives.

Since Duke 1 knows this, he has to support the King, therefore none of the other Dukes arrive and the King is safe.
Anonymous

It depends on what support means. But if support does not mean defend against being killed, then it’s 5, because no matter who killed who beforehand, 5 comes in and kills the king. If support does mean defend against being killed, it’s 1, because 5 will kill 4 if 4 is king. 4 supports to avoid being killed by 5. 3 is prevented by 2 who supports because he knows 3 can kill him and be supported by 4, who is otherwise a sitting duck to 5. So 1 kills and becomes king.
Tdawwg

The king is the king, no matter who is the king: cf. “The king is dead, long live the king.” Easy.
- george57l
  
  Tdawwg, see post #27
Anonymous

Duke 5 is willing to kill Duke 4 because he has nothing to lose for doing so. Duke 4 will not kill Duke 3 because he knows that he will be killed by Duke 5. Duke 3 is therefore safe if he kills Duke 2. Duke 2 knows that he will be killed by Duke 3 if he kills Duke 1. Duke 1 kills the King. The other Dukes don’t lose any sleep over the dead King.
jere7my

No duke would ever become king with one duke left to visit after them â€” that would be suicide. The last duke would have nothing to lose, and would kill the king. So the penultimate duke (#4) must support, and thus the last duke (#5) never arrives.

Since duke #3 knows this â€” #4 must support him, no matter what â€” he can do whatever he likes. Thus he will always choose to kill the king.

Duke #2 knows this, so he has to make sure duke #3 never arrives. He must act to support duke #1.

Thus, duke #1 can kill the king with impunity, knowing that #2 is trapped into supporting him. (This is not so bad for #2 â€” he fulfills his second-best winning condition, and is safe until the next scheduled ducal visits in June.)
daneyul

>>>Duke One answer *cannot* be correct. Why? Because, at that point (Duke One is king), you have to process a recursive version of the story. You now have a king who will be visited by FOUR dukes…

Which is an even number, which ALWAYS means the original king (who -was- duke 1) survives. Duke 1 is still the correct answer, recursion or not.
jackball

Does the king have a say in what goes on? The king would probably have enough power to manipulate the outcome in his favor.

Assuming the king was a former duke he knows how the game is played. Give the first duke the option of loyalty of death, the duke will want to stay alive. Thus the king makes the decision for the dukes.

The dukes will have to come up with a better plan then one showing up each day.
Anonymous

surely optimal tax rate is 0 since adam and eve are the only people and therefore have no expensive govn’t to support
Patrick Dodds

Thank you for this post Mark – I would join you in the “zero correct” category. Mind you, it would be interesting trying to figure out why on earth Eve would want to work for the benefit of that loser Adam.
helmut_hed

Given this is an economics exam at Oberlin, the question about Eve’s “optimal” tax rate (given that Adam is “completely unproductive”) bears with it a strong whiff of academic left-wing dogma.

Nevertheless, there’s two pieces of information that can be used: Eve’s rate of production and the trade-off (for both Adam and Eve) of work vs. consumption in the form of apples. One might choose a tax rate so that Eve will maximize her output – at some point, the marginal apple enjoyment (measured in hours of labor) she got from her apples drops below that labor input. Unfortunately, it seems to me that this point arrives immediately – even for the first apple, she’d have to work an hour, and if she gave any of it away to Adam she would receive less in benefits than her labor input. The only situation in which she’d work for both of them is if she placed a value on his happiness as well. If she valued his apple consumption as much as her own, the tax rate should be set at 50%, Eve would work for two hours producing two apples, and each would receive one hours worth of consumption from her two hours of work. After that point it goes negative, although if they can split apples, Eve could work for one hour making the first apple, and they would each get 1/sqrt(2) enjoyment, or about 1.42 hours worth of pleasure from her one hour of labor.
Neuron

I thought it was Steve Landesberg. Imagine my disappointment.
Chrs

Re: Kings

Guys, either read the question thoroughly (all future visits canceled post-support), or read the comments. The correct pure-logic answer has already been posted half-a-dozen times.

The arguments about uncertainty about how rational the other Dukes are, however, are a considerable fun. We are far away from being Homo economicus, the theoretical purely rational, self-interested actor that economics so frequently assumes.
Anonymous

You are Duke #1.

Are you willing to bet your *LIFE* that Duke #2
is going to work the logical problem correctly,
and reason that he should support you?

Duke #2 may be a logician, or he may have beat up
his poor Logic tutor. Or …

I am duke #2. You are worried that I may not follow the
logic, and you meet with me before visiting the king,
and teach me that Logic dictates that I shoud support
you as King, for all the reasons laid out above. I nod
and smile.

You kill the king, and await my visit.

Meanwhile, I go have a talk with Fred (Duke #3), who I
used to play with when we were children. He agrees to
support me. So upon my arrival at your castle, my trusted
henchman slips some Iocane powder into your drink.

Now I, Duke #2, am King. I await the visit of my old
school chum Fred, who I am *sure* has not conspired
with Duke #4 …

Were I grading the paper, the only answer I would mark
as correct would be “As Duke #1 is not willing to bet his
life that his peers will act logically, Supports the
existing king, so in the end, the original King rules.”

Anyone constructing elaborate scaffolds of logic about
what each Duke would do, would get a big red scrawl
reading “We don’t want another Econ Idiot who builds
fragile houses of cards.”
Anonymous

Adam and Eve:
“if it were both possible and necessary,” they would be willing. But since it is not possible for Adam to produce any apples, Eve is not required to produce any either. Therefore, an optimal tax rate is 0% as it is the only way to optimize productivity. If Eve is taxed at any other rate, she will be demotivated to produce anything at all. If Adam dies, it minimizes consumption, so per capita productivity (the best measure of an optimized tax rate) will be highest if Eve is not taxed at all. 0 is a number, so it counts.
Optimality

Please delete if we’re not supposed to post spoilers :).

I think the answer to question 8 is Duke One.

Duke 5 will kill the king (no penalty for doing so).
Duke 4 knows this, and so will support (so as to not die).
Duke 3 knows 4 will support, so Duke 3 kills.
Duke 2 knows 3 will kill, so Duke 2 supports.
Duke 1 knows 2 will support, so Duke 1 kills and becomes the king.
- Mantari
  
  A lot of it also spins on the meaning of priority. If you say priority = requirement, then the *top* priority is *staying alive*. Once that priority is fulfilled, then the second priority can be explored. The top priority is only fulfilled for the last duke. Therefore, the first duke is going to support the king. The game is over. The starting king (Duke Zero?) is the final king.
  
  The Duke One answer is compelling, at first. However, there is one major flaw: recursion.
  
  Duke One answer *cannot* be correct. Why? Because, at that point (Duke One is king), you have to process a recursive version of the story. You now have a king who will be visited by FOUR dukes, with the same rules as above. So, by that logic, wouldn’t the next duke kill the king, also believing that nobody else would challenge him? Or does the answer change between the five and four mark?
  - Mantari
    
    CONDITION: Hard priorities “I am safe” must be fulfilled before “I want to become king.”
    
    RESULT: Duke One is not safe. Therefore, he supports. Game is over. The original king remains king.
    
    I believe in that outcome. However, what about a less rigid take on it…
    
    == RESET ==
    
    CONDITION: Weighted priorities, applied in the same manner by all dukes, with ‘conventional wisdom’.
    
    Conventional wisdom is that Duke Five will kill the current king, so unless you’re Duke Five, you don’t want to kill the king yourself, because you’ll end up dead.
    
    Duke One says, “It would be insane to kill the king, because Duke Five is going to do the same. So I am going to kill the king. Another duke would have to be insane to try to kill the king, because they’ll never make it.” Duke One becomes the king.
    
    Duke Two says, “It would be insane to kill the king, because Duke Five is going to do the same. So I am going to kill the king. Another duke would have to be insane to try to kill the king, because they’ll never make it.” Duke Two becomes the king.
    
    Duke Three says, “It would be insane to kill the king, because Duke Five is going to do the same. So I am going to kill the king. Another duke would have to be insane to try to kill the king, because they’ll never make it.” Duke Three becomes the king.
    
    Duke Four says, “The only Duke left is Duke Five. Killing the king will meet all of his priorities, without risk. Therefore, I support the current king.”
    
    So knowing that could happen, how does that affect the results?
    
    == RESET ==
    
    CONDITION: Weighted priorities, applied in the same manner by all dukes, knowledge of outcomes above.
    
    RESULT 1: Duke One knows that if he becomes the king, the Duke Two and Duke Three are going to want to kill him. So he supports the king.
    
    RESULT 2: Duke One knows that Duke Five will kill the king. He also knows that Duke Four will prevent that from happening by supporting the king. So he knows that this makes Duke Three the safest to kill the king. He knows that since all Dukes apply the logic in the same way, that Duke Two will come to the same conclusion. So Duke Two will not kill the king.
    
    Therefore, Duke One kills the king, knowing that Duke Two fears Duke Three’s immunity that is created from Duke Four’s fear of Duke Five’s unlimited power.
    
    So, depending on the base rules and knowledge (and assuming that all dukes process the logic in the same way, but based upon their position), the answer is either the starting king (absolute priorities) or Duke One (weighted priorities).
    - mypalmike
      
      It’s an interesting, but flawed analysis.
      
      CONDITION: Hard priorities “I am safe” must be fulfilled before “I want to become king.”
      
      RESULT: Duke One is not safe. Therefore, he supports. Game is over. The original king remains king.
      
      This is incorrect. Duke one is safe. Based on the rules, he can deduce with 100% certainty that he will survive, regardless of what action he takes. So he kills the king because it fulfills his second priority.
      
      RESET… Conventional wisdom is that Duke Five will kill the current king, so unless you’re Duke Five, you don’t want to kill the king yourself, because you’ll end up dead.
      
      Unfortunately this is also mistaken. Duke 1 is a rational actor, so he knows with 100% certainty that he won’t end up dead. This “conventional wisdom” is actually “flawed reasoning” and leads to the wrong answer.
      
      RESET…Therefore, Duke One kills the king, knowing that Duke Two fears Duke Three’s immunity that is created from Duke Four’s fear of Duke Five’s unlimited power.
      
      Right.
- SamSam
  
  Optimality: I hadn’t seen your reply, but we both got the same answer for the same reasoning. :)
jeaguilar

I’m going with the current king, supported by the Duke of Earl #1.
ikegently

at a tax rate of 50%, Adam and Eve will both work for 0.5 hours (though Adam is completely unproductive, he is still working). Eve makes .5 apples. .25 goes to her and .25 to Adam. This satisfies her condition for working (she will work .5 hours for .5^2 apples) and Adam’s conditions (same). At this rate, the total output of the system is maximized. At any other tax rate, they will work less than .5 hours and get less apple in order to satisfy their willingness to work at all.
Anonymous

The first one I don’t quite understand. If only one person produces, and they produce at a rate of 1 unit per hour, what is with the stipulation of being willing to work up to x hours per x^2 units? That’s utterly irrelevant, as the only way to produce x^2 units is to put in x^2 hours.

As for the second one, with the assumption of survival superceding kingship, my intuition suggests the original king remains in power. The first duke has nothing to gain by taking kingship, because he more than likely will be killed. The fifth duke has everything to gain by taking kingship, but his visit is more than likely going to be canceled, most likely by the first duke who would rather live than be king and die, and thus will submit to the king.

~D. Walker
Dave McCaig

Wouldn’t the answer to #2 just be Duke 5? Am I missing something? It seems like a no brainer to me, but maybe I’m missing something.
- Dave McCaig
  
  Whoops, I meant #1. Duh.
helmut_hed

daneyul is correct. I didn’t like the “Duke 1″ answer either at first but now it seems to me logically unassailable. If you assume:

1) all dukes are perfectly rational and know the others are too
2) they will adhere to their priorities as stated
3) any king (including former Dukes), once supported, never faces another challenge

then there is no other conclusion to be drawn. It all starts with Duke 5. If you don’t believe it, start with smaller numbers of dukes. You will see the same even-odd behavior, with either Duke 1 or the original King living, depending on the parity of the Duke count.
SamSam

I think the king must remain the king. EDIT: No! I correct myself below.

It seems to me that the only person who would ever benefit from killing the king would be the Duke Three. This is because Duke Four will never kill the king, because if he does, Duke Five will kill him. So Duke Four must support whoever is on the throne. So Duke Five will never come.

So if Duke Three were to come, he would kill the king. Both Duke One and Duke Two know this, so they would never let Duke Three come, so they would support the king.

WAIT! But if Duke Two will always support the king because he knows that Duke Three would kill him, then Duke One can safety become king!

Duke One becomes king. Duke Two knows that if HE becomes king, then Duke Three will kill him, because Duke Four will always support the king. Therefore, Duke Two will support the king.

So: Duke One.
Anonymous

Well the duke question is easy. Only the last duke could kill, as he has no danger of beeing killed. All other dukes would have to fear beeing killed. As they would support the king, the last duke doesn’t get any chance. That’s trivial.

The taxation question can be easily solved by mathematics.
Eve could produce P=t apples in t hours time.
She demands D=t^2/2 apples. (the numbers are for both people)
So P must be equal to D, so

t^2/2-tax*t=0

So this equation is satisfied if either t=0 or t=2*tax. If t=0 Eve will not want to work at all, that’s probably not optimal.
So with the taxrate (1-tax), she will want to work 2*tax.

So she more she keeps, the more she will work. Now what’s optimal? Maybe equal distribution of wealth. In this case it would be 50%. She’d work for one hour, producing one apple which they would both share.

If course if you use the total production as a measure, there shouldn’t be any tax at all, as then she’d work for 2 hours, and the economy would produce 2 apples, heavily unevenly distributed.

Of course, the model shown in that question probably has nothing to do with the real world.
Zadaz

Are economics students still being thought that all participants are rational actors?

Oh my god, we are so fxxxd.
Anonymous

Beleive jeaguilar is right. Optimality seems to have missed that the first Duke to support the King automatically cancels all subsequent visits. But would REALLY like to know the definitive answer for this…
Anonymous

Q8 Answer. Duke one is King. How? Duke one arrives knowing that if he support the King currently in power he does not have to face the other 4 Dukes. So he supports the king fulfilling his first priority. To fulfil his second priority he only needs to wait until the King in power is killed or dies of natural causes. This answer is assuming the efficient market hypothesise. Without the efficient market hypothesise the answer is either the King stays in power or the fifth Dukes become King. It all depends on the desire of the first duke. And the power of his need to stay alive. How the question is written suggests the Dukes desire to stay alive is more powerful than his desire to become King. So, who is King on May 6th? The original King. So the answer is that a co-operative game is played out.

Q6 The answer to this question is more complex in that it is an extensive form game. Where may scenarios are played out in Eves mind, being human. â€˜Optimalâ€™ means she is both â€˜happyâ€™ with her pay for her time (at work) and â€˜happyâ€™ with Admas â€˜payâ€™ as her tax rate. So the answer is not a number but a philosophy. The philosophy is what Eve values. Adam may never get paid if Eves values are such.

Since Eve and Adam are willing to work for X=X^2 the number figure becomes when she is comfortable with her lifestyle. If she is comfortable with her lifestyle (work life balance) working 3 hours she gets paid 9 apples and taxed at 30% she will be â€˜happyâ€™.
Anonymous

Further proof that economics is not a science.

Seriously, how stupid do you have to be to actually believe most of these questions have ANY real world application?
- Anonymous
  
  “Further proof that economics is not a science.
  
  Seriously, how stupid do you have to be to actually believe most of these questions have ANY real world application?”
  
  It’s called logic… And it has far-reaching real-world applications. The fact that so many people DON’T have it is why the world is as messed up as it is.
Anonymous

If there’s one duke, that duke kills the king, period.

If there’s two dukes, the first duke can’t safely kill the king as the remaining duke will kill them in turn. In that case, the first duke supports and the king is safe.

If there’s three dukes, the first duke kills the king, leaving two dukes in line. The second duke has to support to avoid being killed by the last duke.

If there’s four dukes, the first duke must support, as killing leaves only three dukes and the first duke of those three can kill with impunity.

The puzzle has five dukes. Duke One kills the king. Duke Two is first of the remaining four and must support, as being king with three dukes left is fatal. All hail, Duke One, King of May 6th!
Anonymous

But what if one of the Dukes is on an airplane trying to take off on a conveyor belt…
Anonymous

The Duke with the biggest army.
JB NicholsonOwens

The first question strikes me as more of a “review the reading material” question with no clearly defined wrong answer other avoiding self-conflicting answers or failing to convey one did not read the assigned text. This, therefore, is wholly uninteresting without knowing what was the reading material.

The second question seems exceptionally easy to me: the first duke defends the king, canceling all other duke visits, remains alive (his first priority) as do all subsequent dukes (which they agree with since they share the same priority). The king remains unchanged and gains a defender (duke #1).

I think that someone along the line got the second question wrong. The second question is more interesting if you reverse the duke’s priorities.

Some tricks in the second question arise because people don’t read closely enough: some assume the duke’s first priority is to become king, some get themselves twisted around considering what other dukes will think and then try to outthink the other dukes, one considers things backwards going against the natural flow of time.
Anonymous

Q*: It depends whether the Dukes know the rules…

So It will be either The Original King (as Duke 1 knows of the future shenanigans and chooses to support the king in return for a life of luxury lest he be killed by Duke 2).

If the Dukes don’t know the rules…. The King Is Dead! Long Live King Duke 5!
Xopher

Assuming that a) there is already a King (implied by the fact that all the Dukes can “kill the King”), b) the Dukes all know they’re all operating by the same rules, and c) they’re rational actors, the original King will be King on May 6.

Duke One’s first priority is to stay alive. He knows there is a chance that he will be killed if he kills the King, and no chance that he will be killed if he supports the King. He cannot optimize for his second priority and still maintain optimization for his first priority, so the second priority (becoming King) becomes irrelevant. Therefore he supports the King, the other Dukes stay home, and the King remains King.

Interestingly, if you remove the part about “all subsequent Dukes cancel their visits,” Duke Five is King on May 6. His first priority is automatically fulfilled, because there are no subsequent Dukes to threaten him, so he gets to his second priority, which is becoming King, and kills whomever had that role on his arrival. The actions of the others, while they may also be distorted by this change, are irrelevant.

I assume rational actors because this is a problem, and because the part about priorities implies rational actors. Real Dukes would not be so reasonable; real people do not have clear “first priorities,” in general, and are inclined to risk more important things when reaching for less important things. In the real world, it’s impossible to predict who would be King on May 6, even with such a neat scenario.
scifijazznik

I think I’ve finally figured out why our economy is in the crapper.
Anonymous

I think 5 becomes king by killing the original king.

Duke 5 will kill the king because there’s nothing to lose.
Dukes 1-4 each knows that even if he kills to become king, he will ultimately be killed by 5. So each of them supports the original king.
- Anonymous
  
  That’s no good. If any Duke supports than the remaining Dukes cancel their visit.
Anonymous

My ECON101 textbook had this story in its introduction:

A chemist, a physicist, and an economist are stranded on a never-inhabited desert island with a single can of food. The chemist wants to boil it open, the physicist wants to bash it open, and the economist says, “Let’s assume we have a can opener.”

It was at that moment I realised the futility of studying economics.
Anonymous

proof that |econ| < <<< |Math /intersect xkcd /intersect pirates|. http://forums.xkcd.com/viewtopic.php?t=330

SamSam

To extrapolate further on the Duke question, if there are an odd number of Dukes, Duke One will always kill the king, and if there are an even number, the original King will always remain.

The first statement can in fact be proven from the second statement alone.

El Mariachi

The original king is still king. Since Duke #1′s priority is to stay alive, the best guarantee of achieving that is to support the king, thereby forestalling any subsequent challenges.

More importantly, Duke Sucks.

Anonymous

“If he supports the king, all subsequent Dukes cancel their visits” Doesn’t that line ruin most of you guys’ reasonings. I think the king stays, nobody wants to die.

willy359

I never studied economics, but I did study history. Therefore I can say with some confidence that the king will have all five dukes arrested for treason sometime around mid-April. This is the problem with economics: it’s based on models that cannot possibly account for all the variables at play in the real world.

daneyul

>> Some tricks in the second question arise because people don’t read closely enough: some assume the duke’s first priority is to become king, some get themselves twisted around considering what other dukes will think and then try to outthink the other dukes, one considers things backwards going against the natural flow of time.

Uh…no. What? It’s not going backwards against the flow of time at all, it’s acting according to what you know (not guess, or think, or hope, but KNOW) the next duke will do. You know because they are bound by the same priorities you are.

Simplified yet further: a duke will always do the opposite of the next duke in line. If I know the next duke will support the king, I will become king. If I know the next duke will kill the king, I don’t kill the king. Not “may” not. WILL not, because I am bound by my 2 priorities.

Once that’s understood, no matter how long the row of dukes is, if you KNOW what the final duke will do It’s a series of toggles.

Imagine a row of A/B switches. Switching any one of them to “A” forces the one to it’s left to “B”. Switching any one of them to B force the one to it’s left to A.

So, by switching the final one to B, you know you get ?…B,A,B,A,B. Switching to A, you’ll end up with ?…A,B,A,B,A. The question mark’s place is simply determined by the number of switches. Nothing else.

Since the final DUKE will always kill the king (since priority one, stay alive, is moot and priority 2, be king, is therefore elevated) we know, all down the line, what each duke toggle is set to–kill, or not kill.

C’mon. It’s not that difficult.

Anonymous

At least some of the questions (esp. the tax rate ones) seem very much in the category where a intelligent student would try to figure out what the professor/examiner wants to hear and then rationalise the result.

Kinda why I chose engineering instead :-)

And I agree with #9.

KeithIrwin

I understand that the math works for the taxation rate being 50%, but I’m not sure why everyone seems to be stating that it’s optimal. The real question is what utility function you’re going to use. If for example, you use the sum total of all apples produced, then the optimal tax rate is 0%. Eve will work 1 hour to produce 1 apple, thus giving a total of 1 apple, which is more than the 1/2 apple total which the 50% taxation rate produces.

The proper equations are that if Eve works for as long as she’s happy to (based on her payout), then if the tax rate is t and her number of apples produced is a, then a^2 = a*(1-t), the solutions for which are a=0 and a=1-t. Presuming that Eve works at all, the higher you set the tax rate, the less work Eve does. So if you want to maximize total production, the solution isn’t 50%, it’s 0%.

Now, if you suggest an alternate utility function to maximize which gives higher value to equitable distribution, such as using the minimum number of apples which anyone receives as the overall utility, then you might get 50%. Or if you make a more complex function which is somewhere in between simple sum of all individual utilities and minimum of all individual utilities, then you would get an answer somewhere in between.

I don’t think that the questioner is dictating what you use as your utility function, just requiring you to be able to lay out the proper equations to produce an answer and justify it.

obdan

Duke 1 supports the king, remains alive and cancels further visits from dukes. On May 6th he then kills the king?

caitifty

I’m delighted to see ethical behavior remains such a strong focus of economics education.. I look forward to the new generation of Dukes^H^H^H^H corporate leaders produced by Oberlin.

daneyul

>> I’m delighted to see ethical behavior remains such a strong focus of economics education.. I look forward to the new generation of Dukes^H^H^H^H corporate leaders produced by Oberlin.

You are a very silly person.
- caitifty
  
  Well, yes, I’ve been accused of that before. What’s your point though? :)

Vaxjo Aberg

Well said, SamSam. I think that boils the logic down nicely.

Zergonapal

It seems logical to me that regicide is a bad idea because by setting such a precedent you are placing your own head on the chopping block should you wish to be king.
Better to take the long view and forge closer ties with the king and perhaps arrange a marriage between families with the long view of having a descendant inherit the throne.
Then again this is economics which is all about short term gains with little regard for the future.

Anonymous

The answer to # 6 is “Any rate from 0 to 100% will produce the same result. Eve will work enough to feed herself and maintain Adam to whatever degree she feels like. As Eve is the only woman, she will have a monopoly on sex and can charge monopoly rates. In other words, all apple taxation will flow back to her at her discretion”

Anonymous

You assume Adam is straight.

Mirza

It’s a small step from “all dukes are rational” to “all Americans pay back their mortgages.” Assuming a vacuum is a dangerous thing when you’re dealing with human behavior.

Gemma

Q8. I’m going with the original king still being in power. Working backward from Duke 5 makes no sense to me, since Duke 5 ONLY gets the opportunity to attend court/the king IF NONE of the previous dukes have supported the king.

Start from Duke 1′s visit. Duke 1 has two choices:
1) kill the king and become new-king and give duke’s 2, 3, 4 and 5 the chance to kill him
2) support the king, prevent the other bloodthirsty dukes from visiting, live a long life arranging treaties and paying apple taxes

Since his first priority is staying alive, and his choice is between 0 dukes considering killing him and 4 dukes considering killing him, he must take the option of supporting the original king.

Q6. I don’t understand the apple-tax question.

Is “producing” apples the same as having an “income” of apples?

If Eve can “produce” only 1 apple an hour, why are we told that she is willing to work up to 3 hours for 9 apples? She can only produce 1 apple an hour, but expects the square of her production value in wages? (I want to tell my boss this.)

Who owns the apple tree/factory? Who taxes Eve? Why does Adam get an income from the tax? We are told that Adam would be willing to work if possible and necessary…. am I supposed to think that it isn’t possible for him to work, or that it isn’t necessary since Eve’s taxes support him?

Utterly baffled and wondering if there’s an obvious answer that cuts out a lot of the detail.

DarthVain

Duke Leto Atreides – The Spice must flow!

jaytkay

Eveâ€™s income is taxed at a flat percentage rate, with the proceeds delivered to Adam.

Is that unsubtle teabaggery or is it just my imagination?

Anonymous

The king remains the king. It’s very logical, actually. The first Duke will support him for the simple fact that if he does not, he will be killed by the next Duke (apparently the Dukes will succeed in killing the king, since it doesn’t say “attempt to kill the king”).

Thus, by supporting the king, all the subsequent visits by Dukes is cancelled…and things remain as they are.

There is no other logical answer that I can think of.

endymion

These are great questions. This posting increases my chances of taking a serious look through Steven Landsburg’s new book. Thanks BB.

Mechalith

I’m with Xopher. Assuming the priorities given, Duke 1 will support the king and the others will stay home.

Anonymous

The comments about the unreality of the situations in these exam questions reminds me of this recent XKCD http://imgs.xkcd.com/comics/experiment.png

Which should serve to remind all the self-righteous here that it is not only economists who posit wholly unrealistic situations in order to instruct (and to create exam questions).
george57l

May I be the first to point out that on 6th May the King will be the King?
tedrock

Duke #1.
But which goblet did the Duke put the iocane powder in?
- Kimba.
  
  LOL! I loved that movie, too.
Nelson.C

Implicit in the economist’s idea of the “rational actor” is each having the complete knowledge of the very limited universe contained in the word-problem. While the solution of Duke#1 works in this reductionist scenario, and would have to be mentioned as part of the answer to question 8, I’d expect an honors grad to talk a little about the real world, and how real actors are working within a much wider world and a less limited and clear rules-set. A real Medici-style duke would probably vault over the rules for the chance to be king, turning up even when the agreement is to not do so, for example, or staying away from court until the blood has stopped flowing, or making an alliance with other dukes, turning up with an army after Duke#1′s regicide, or being ill with ague and having his less-rational son stand in his place.
- SamSam
  
  Lots of people above have said this argument, “for an honors degree I’d expect them to talk about the real world and such… &c. &c.” but that just seems dumb to me.
  
  Logic puzzles aren’t about the real world. How could anyone read those rules and think that it’s supposed to represent the real world? Should we ask “well, what would happen if Duke 1′s plane got in late?” No, that’s not part of the puzzle.
  
  In a math puzzle about trains speeding towards each other at such and such speeds at such and such a time, we don’t say “well, in an honors class I’d expect them to talk about the difficulty of keeping timetables, and maybe throw in the Mussolini is famous for making the trains run on time…”
  
  It’s a logic puzzle. The actors are all rational, unless we are otherwise told. They all know the rules. We don’t need to wonder whether Duke 2 is likely to mess up and accidentally kill Duke 1. That’s just not part of the puzzle.
  
  A whole bunch of other commentors have said things along the lines of “wow! If economists are so dumb that they think everyone is rational, no wonder the economy tanked!”
  
  While that attitude may or may not have a grain of truth in it, it can in no way be gleaned from these questions. First, the question about Dukes is a logic puzzle, nothing else. Second, the questions about taxation and such have to do with models. Models are an attempt to understand why things happen in the economic world. For example, if Coke were to randomly triple it price, what would happen? Most likely, consumption would go down. There’s a model to explain that. “Ah,” you say, “but that dumb model doesn’t account for the fact that some people might think it’s now worth more.” No, there’s a model for that too.
  
  Most economists, at least today, don’t believe that their models are 100% accurate, nor do they believe that they can model any one individual. But crowd do usually behave as the models predict, or otherwise the models get refined. How else can the Fed decide to cut its interest rates, and why else would the market generally respond in a way that’s predictable after the cut?
george57l

And what about question 2?

“Question 2: Bananas cost …”

JUST LOOK!
Kimba.

Think outside the box, folks. Duke #1 supports the king, thereby canceling the visits of all the other potentially murderous Dukes and ensuring that he himself lives. THEN Duke #1 kills the King, ensuring that he also achieves his secondary goal. Not a course of action I condone, of course, but it does successfully resolve the riddle.
daneyul

::::Well the duke question is easy. Only the last duke could kill, as he has no danger of beeing killed. All other dukes would have to fear beeing killed. As they would support the king, the last duke doesn’t get any chance. That’s trivial.

No. Trivial would be the effort expended in reading any of the preceding posts before spouting out the obvious, but very wrong, answer.
- SamSam
  
  Now go up and repeat that comment for about 12 others up the line… ;)
jmzero

Q8. I’m going with the original king still being in power. Working backward from Duke 5 makes no sense to me, since Duke 5 ONLY gets the opportunity to attend court/the king IF NONE of the previous dukes have supported the king.

Duke 1 will kill the king. Why? Because he knows Duke 2 will support him. Why will Duke 2 support him? Because Duke 2 knows Duke 3 would kill him. Why would Duke 3 kill him? Because Duke 3 knows that Duke 4 would support him. Why would Duke 4 support him? Because Duke 4 knows that if he kills, Duke 5 would kill him. Would would Duke 5 kill him? Because Duke 5 wants to be king and there’s no more dukes.

This is the same chain of reasoning from post 2 – hopefully my reasoning above makes it somewhat more clear.

And, to be clear, it’s an even-odd thing based on the number of dukes remaining. If there’s an even number of dukes after you (including zero) you kill. If there’s an odd number of dukes after you, you support.