The Efficient Allocation of the Right of Way

This may not be representative of society in general, but lately when passing by crosswalks, I have noticed two things:

1. Cars being less willing to stop for waiting pedestrians.

2. Pedestrians being more willing to wait for a break in the traffic before starting to cross, instead of demanding that cars stop for them.

This got me thinking about the traffic rules for crosswalks (giving pedestrians the right to cross whenever they want to), and has led me to some interesting insights into why the rules are the way they are. I do not know, however, why I see so many drivers and pedestrians acting contrarily to the rules. What follows is a meditation on proper crosswalk behavior, and how economic efficiency dictates what customs we follow (or rather should follow).

To begin with, let's make some assumptions that will allow this thought experiment to take place.

Let's assume that drivers and pedestrians can all be categorized into four groups, and that people in these groups behave the same each time they approach a crosswalk. The four groups are as follows:

1. Pedestrians who wait for the cars to pass before they walk, denoted by Pw (pedestrians who wait)

2. Pedestrians who do not wait for the cars to pass before they start walking, denoted by Pnw (pedestrians not waiting)

3. Drivers who wait for pedestrians to cross, following the law as it currently stands, denoted by Dw (drivers who wait)

4. Drivers who do not wait for pedestrians to cross, just driving through and forcing the pedestrian to wait or get run-over, denoted by Dnw (Drivers not waiting)

Thus there are the following four scenarios that could happen at the crosswalk:

Pnw meets Dw (pedestrian crosses easily)

Pnw meets Dnw (pedestrian and driver both try to go, resulting in a dangerous face-off)

Pw meets Dw (pedestrian and driver both wait around like idiots, resulting in a delay until they sort out who should go)

Pw meets Dnw (Driver goes and pedestrian waits for break in traffic)

Now lets consider the costs involved in each scenario.

(denoting "meets at the crosswalk" with a /)

Pnw/Dw: the cost of the driver having to waste time and gas to stop.

Pnw/Dnw: the cost to both pedestrian and driver of a possible harmful or fatal accident.

Pw/Dw: the cost of the driver's time and gas, as well as the pedestrian's time.

Pw/Dnw: the cost of the pedestrian's time.

Clearly Pnw/Dnw entails the highest cost.

We'll return to this discussion of cost shortly, for now we have more assumptions to consider.

Let's assume that 80% of pedestrians don't wait for a break in traffic, and the remaining 20% wait for the cars to pass. This gives us probabilities that a random pedestrian will be of each group. Here are the probabilities:

("probability of an event" is here denoted by P(event))

P(Pnw)=0.8

and

P(Pw)=0.2

Secondly, lets assume that 80% of drivers are those who stop for people at crosswalks, and the remaining 20% are the jerks who just plow through. This gives us probabilities that a random driver will be of each group. Here are the probabilities:

P(Dw)=0.8

and

P(Dnw)=0.2

From these probabilities we can create a probability distribution for each of the possible scenarios at a crosswalk.

Because a certain type of pedestrian coming to a crosswalk, and a certain type of driver coming to a crosswalk are totally unrelated, independent events, we can find the probabilities for each situation by multiplying the driver and pedestrians' probabilities together.

So this gives us the following probability distribution:

P(Pnw/Dw) = 0.8*0.8 = 0.64

P(Pnw/Dnw) = 0.8*0.2 = 0.16

P(Pw/Dw): = 0.2*0.8 = 0.16

P(Pw/Dnw): = 0.2*0.2 = 0.04

So in this imaginary world I have created, 16% of all crosswalk encounters create a possibly dangerous showdown of pedestrian versus car, the most costly of the scenarios. 4% of the time there will be the boring situation of both driver and pedestrian wasting their time and/or gas. And the remaining 32% are efficient situations where either the driver or the pedestrian waste time/gas, but not both.

The first conclusion to be drawn from this hypothetical situation is that fewer costs will be incurred if all drivers and pedestrians knew what the rule was and followed it consistently. Let's imagine another world with different laws, where 100% of pedestrians were Pws and 100% of drivers were Dnws. Pedestrians would spend more time waiting than they do under the current rules, but the dangerous Pnw/Dnw scenario, and the extra time-and-gas-wasting Pw/Dw scenario would both be eliminated. This shows that there are clearly efficiencies to be gained from people behaving consistently as the result of clear property rights, regardless of who is given the rights in the first place. I believe this is the essence of the famous "Coase Theorem" in economics. In this case it would be cars that "own" the right of way. If pedestrians uniformly respected this property right, the result would be better than if pedestrians and cars didn't know or care about who has the right of way, resulting in accidents and delays.

With that being said, cars clearly should not have the right of way. The "transaction costs" (also a key element of the Coase Theorem) pedestrians face in crossing the street (e.g. the chance of getting run-over) are obviously much higher than those that cars face (e.g. wasting some gas). Thus to create a more efficient society, the law allocates the property right in a way that minimizes costs.

If drivers and pedestrians would just act like they understand who owns the crosswalk, it might be safer out there.

Tobacco: Optimal Fines for Enabling Adults

The other day I was walking past a liquor store and saw a sign in the window, explaining that adults who buy cigarettes for minors will face a $200 fine. This made me think of the concept of the optimal punishment for a crime, which I was first introduced to by Donald Wittman's Economic Foundations of Law and Organization, ( a great book I highly recommend.)
To determine the optimal punishment for a crime, societies take into account both the harm caused by the crime and the probability of catching the criminal, which tends to conform to the following model:
P*F=H,
with P being the probability of getting caught if you commit a crime, and H being the harm the crime causes to society. This leaves F, the appropriate fine to be levied out so that, in the aggregate, it will make criminals pay for their behavior, thus efficiently deterring crime.
So if smashing someone's window costs the victim $100 dollars in damage, and there is a one in ten chance of being caught for the crime (P=0.1), the appropriate fine for smashing a window would be $1000. If suddenly, (maybe due to new window smashing technology), it became twice as hard to catch a window smasher (P= 0.05), the punishment should double to $2000. The punishment should also double if the cost of window repair were to double, (H=$200). So with optimal punishments, crimes that cause more damage, as well as those that are harder to catch, are met with proportionally harsher fines.
So is the $200 fine for buying tobacco for minors an optimal punishment? My initial reaction is "heck no." But sound policy is not based on reactions, here I shall try to provide quantitative analysis to help answer this question. Though I do not have all the data needed to answer it, I will set up a framework into which data could be plugged, to lead us closer to the truth.
To begin with, lets try to get an idea of H (the cost to society of buying a pack of cigarettes for a kid).
To find H, we must isolate the smoking (both present and future) that would happen directly as a result of an adult buying a pack of cigarettes for a minor. It must be differentiated from smoking that would happen otherwise. Obviously when an adult buys a pack of cigarettes for a kid, this is increasing smoking by at least one pack, and possibly more than that because that one pack could lead to more smoking in the future. While some kids will get hooked for life because of that one pack, and possibly die of lung disease, others may give it up after a single puff. The probabilities involved in this game of slow motion Russian roulette are very hard to quantify. But to arrive at something close to H, one could start by taking a large sample of people who had been given cigarettes by adults when they were minors. The next (very challenging) step would be to use regression analysis to try to isolate the effect of each illicit tobacco purchase on the minor's cigarette consumption over a lifetime. Lets assume that an amazing statistical study determines that each purchase of a pack of cigarettes for a minor leads to 1.1 more packs to be smoked in total than would happen in absence of the crime, (the extra 0.1 being because of kids led to further smoking as a result of the one pack that was bought for them). The next step would be to find the cost to society (to the smoker and everyone else) incurred because of that one pack. Searching around the internet, I've found a group of scholars who say the total cost to society from one pack of cigarettes is $40. For our purposes, lets assume this is the cost. Under these assumptions, we have found H.
H = $40*1.1= $44
So the total cost to society from the crime is $44, the societal cost times the expected additional quantity of tobacco consumed.
Now let's try to think of a way to find P, the probability of getting caught. (As an editorial note, this seems to me to be a pretty easy crime to get away with. All an adult has to do is find a discreet way of passing the cigarettes to the youth, end of story.) It would be difficult, but one could find the probability of getting caught for the crime, by taking the total number of convictions for the crime, and dividing this into an estimate of the total number of times cigarettes were purchased for youths, which could be estimated through surveys of young smokers.
That is beyond my means, so lets just assume that one out of every hundred of these crimes is discovered and prosecuted (which I would guess to be a very generous assumption.) Now we have P.
P = 0.01
And with P and H, we can find the optimal punishment, F.
Plugging P and H into the formula gives us:
0.01*F=$44
F=$4400
So using these assumptions, the appropriate fine for buying a pack of cigarettes for a minor should be $4400. This is 22 times the actual fine for my jurisdiction.
However, this is not a real study and largely based upon numbers I pulled out of my imagination, and from an academic paper that, to be honest I only read the abstract of. But I wouldn't be surprised if the probability of getting caught for this crime is a lot lower than 1 out of 100 (thereby increasing F), and that $40 is a good estimate of the total societal cost of a pack of cigarettes.
So for the sake of argument let's now assume that this estimate of F is close to reality. What could be a reason for the big difference between F and the actual $200 fine. Perhaps because of the legal and social acceptance of smoking as an adult, the law only considers the damages to society incurred while these smokers are minors, while the actual costs of smoking (e.g. addiction, lung disease) are heavily back-loaded to times long after the young smokers have grown up, and their habit has long been accepted by the society it damages.
So who would be harmed by increased fines for this crime? Just the enabling adults and the tobacco industry.


Sources:


Donald Wittman, Economic Foundations of Law and Organization, Cambridge University Press, 2006


http://mitpress.mit.edu/catalog/item/default.asp?tid=10298&ttype=2

Innovation and Value Creation

Let's say that a person A is living in an apartment she finds sufficient at the rate she is paying, and that there are other identical apartments available for the same price. Now what would she do if her lease was up, and the rent at her current apartment was going to increase? The answer to this question depends largely on one very important factor: the transaction costs of moving. These could be the costs of van rentals, moving supply purchases, and the opportunity costs of person A's spending time searching for other housing. (To make things simple let's ignore the time value of money.) Assuming she expects to face $1000 in total costs to find and move into sufficient replacement, what would happen if her rent were to increase by $50 per month? This would amount to a $600 increase for the duration of the next yearlong lease. Clearly she would rather face the cost of higher rent ($600) than bear the costs of moving ($1000). But if the proposed rent increase were $100, for a total of $1200 she would rather face the costs of moving than the higher rent.

The transaction costs of uprooting yourself to obtain new housing are particularly high, relative to other goods. This is not the same as making a choice among buying from different vendors of apples, or electronics.

But in the situation like the one Person A faces, with high transaction costs, there is the opportunity for an innovator to create value for Person A, and profit from it. Lets assume that someone else, Person B, were able to invent an amazing new moving van that could reduce Person A's transaction costs of moving to $500. Then even with the smaller, $50 rent increase, it would be worthwhile for Person A to move. Person B could then charge person A a certain price, (less than the $100 difference between the new transaction costs and the rent increase,) and both Person A and B would benefit.

The facilitation of transactions that would otherwise not happen is one important way that innovators can create value for society. For example websites make it easier to shop around for housing, which reduces the opportunity costs for housing seekers. (The same goes for Star Trek collectibles.)

Increasing benefits and reducing costs is the essence of innovation. And in the end, innovation is the only reliable way for individuals to get abnormally rich. They must find ways to abnormally reduce the costs or increase the benefits for others, and demand a price for it. Finding ways to allow mutually beneficial trades to happen, that would not otherwise happen due to large transaction costs, is one way to achieve this goal and make a buck.

The Economics of Perception (wow, I don't often get this polemical)

"Lisa, I'd like to buy your rock."
-Homer Simpson



Imagine two people, Person A and Person B, in a world where there are only two goods: pumpkins for eating, and yo-yos for entertainment.

Person A is really good at growing pumpkins. His pumpkin growing skills are such that he can grow and harvest a pumpkin for a marginal cost of $1 each. He is less skilled at manufacturing yo-yos, however, and if he were to try, would incur a marginal cost of $5 per yo-yo.

Person B is not so good at growing pumpkins. If he were to try to grow a pumpkin, he would incur a marginal cost of $5. But he's a yo-yo manufacturing maniac, incurring only $1 of marginal cost per yo-yo.

Person B is hungry one day, so he goes to Person A and says, "I'd like to buy a pumpkin." Person A, says, "that'll be $6" and person B responds, saying: "Six Dollars! are you kidding, I can grow a pumpkin myself for cheaper than that!" Before person B can storm off, person A says, "OK I'll give it to you for $4," and B says: "you've got yourself a deal." Person B gets his pumpkin, and person A gets a margin of $3 to use to buy more yo-yos, and produce more pumpkins in the future.

Person A is bored one day, and wants to add a new yo-yo to his collection, so he goes to person B and says "I'd like to buy a yo-yo." Person B, says, "that'll be $6" and Person A responds, saying: "Six Dollars! are you kidding, I can manufacture a yo-yo myself for cheaper than that!" Before Person A can storm off, Person B says, "OK I'll give it to you for $4," and A says: "you've got yourself a deal." Person A gets his yo-yo, and person B gets a margin of $3 to use to buy more pumpkins, and produce more yo-yos in the future.

As in this example, in market economies a buyer will pay a seller a price to do something that is, (taking all costs into account, including opportunity costs) too costly for the buyer to do himself. The price will be greater than or equal to the seller's marginal cost of production (or procurement), and less than or equal to the buyer's marginal cost of obtaining that object through any other means, either through self-production or buying from a different seller. Often, because of the complexity of products, self-production is not an option, so in this case the upper limit on the price a seller can charge will be the price offered by competing producers.

Another factor in determining the price is harder, if not impossible to quantify. This factor is the purely psychological valuation of different goods and services. This value varies from person to person, and within the mind of each individual person, can vary from time to time. If there were a magical brain measuring device to give readings of exactly what people are willing to pay for different items, we could find a measurement of the mean dollar value that a group of people are willing to pay for a good or service. Let's denote this unknowable value with the letter V.

So, if the upper limit of prices is determined by the prices of competitors, what can an individual seller do to make abnormal profits? One method is to try to increase the unknowable value, V, to a level higher than the price of competitors. This can be done through marketing efforts that improve people's perception of your product, without actually changing your product. On an individual level, this can take the form of person to person salesmanship, or on a wider level through mass media advertising. The more the collective perception of value can be shifted upwards through marketing efforts, the higher the price can go.

This is how companies can sell junk for high profits, like the snake-oil diet drugs and herbal "neuroboosters" that can be seen advertised on TV. You'll probably see some zero value junk advertised at the sidebar of this very blog as you are reading this article.

Prohibitions against false advertising are a precondition for a well functioning market. Purely deception based industries are at best zero-sum activities, and at worst they can be truly harmful scams that drag people into economically dangerous situations, as in cases where deceptive contracts tie people to huge outlays of money for nothing valuable in return.

I strongly believe that all purely deceptive industries need to be shut down by regulators, not because I feel a sense of outrage on behalf of people who get duped, but because there are opportunity costs society faces from the existence of such industries. If people can make a living through deception, they will turn away from, (and turn their victims' dollars away from) useful industries. To give a functional definition of "useful", I say that a product is useful to people, if knowing all necessary information about the product, they would still want to use it.

What if Person B comes to Person A looking to buy a pumpkin, and person A says: "I don't grow pumpkins anymore, I produce and sell Neurobooster Pills" and with great salesmanship, sells person B the pills, but they're really just tic-tacs. Is this economic efficiency? In an economy where regulators allow this to happen, resources will be wasted as consumers engage in trial and error to verify the truth. And the capabilities of the information age do not make things better. Today, there is an unprecedented access to information, both true and false. This is the age of tremendously useful online encyclopedias and journals, as well as the age of fake Yelp reviews and SEO tricks. The internet can be both a font of wisdom, and an echo chamber of lies.

Regulators must shut down scam artists, not just to protect their potential victims, but also to lead business people down the path to the real, not just perceived, creation of value, not wasteful industries of subterfuge, salesmanship and fine print.

Remember, if you get scammed, don't just sit there, report it to the Federal Trade Commission.

Check out the FTC website:

http://ftc.gov/index.shtml

There's lots of information on different scams, from job scams (a truly booming industry) to phony diet pills, and using the website it's really easy to report a scam to the FTC. There's even a little cartoon to explain how to report. Don't keep quiet, it's your civic duty to our economy!

The Saw Movies, and Diminishing Marginal Returns, Pt 2

Well, my econometric predictions were way off. Saw 6 only grossed $27,693,292. However, this still proves the broader point that the marginal returns are diminishing. But it looks like they are diminishing at a faster rate than my simplistic model could predict

: (

But this was far from a box office failure. The production budget for Saw 6 was 11 mil, so that's a pretty good gross margin. There will continue to be new Saw movies until the expected marginal revenue falls below the necessary cost of production and distribution. Eventually, they might start releasing them direct to DVD (or internet), which would be a lower cost/lower revenue alternative.

The Saw Movies, and Diminishing Marginal Returns

Every year, around halloween, horror movie fans are treated to the release of the next Saw movie. For those who aren't familiar, this is a series of films about diabolical killers and mechanical deathtraps. It's a gory formula that has proven successful in attracting viewers so far. With five films made already, and a sixth coming out this October, it seems that there will be a new Saw movie released every halloween, forever. But economic wisdom tells us that this can't be true. There are bound to be diminishing marginal returns for the Saw movies, right?

Let's take a look at the data. The following are the domestic box office results for all five of the Saw movies so far:

Saw 1 - $55,185,045

Saw 2 - $87,039,965

Saw 3 - $80,238,724

Saw 4 - $63,300,095

Saw 5 - $56,746,769

After the initial jump from the first film to the first sequel, we do see diminishing box office with every film. My theory is that sequels to successful films already have a level of built in publicity, almost like a brand name. So, what will the next Saw movie gross (no pun intended) at the box office? Using an awesome piece of free online statistical software, I have built a multiple regression model to predict the box office results of the next Saw movie. Here it is.

(This is a very simple model, based upon on a sample of only 5 occurrences, and not taking many important variables into consideration. I don't seriously believe it will predict the box office of Saw VI, But it sure would be cool if it did.)

I used two variables to explain the box office of these movies:

1. The order of their release, meaning a value of 1 - 5

2. A dummy variable (1 or 0) applied if the film is a sequel. This is used to explain the "Sequel Bump" I have seen here.

Thanks to the miracle of statistical software, we have the following model:

SawBoxOffice =

$65966866.70 + $43600897.50*(sequel dummy) - $10781821.70*(film#) + e[t]

So, using the model, lets predict the box office for Saw VI:

Saw6BoxOffice = $65966866.70 + $43600897.50*(1) - $10781821.70*(6)

= $44,876,834

So, I predict that Saw 6 will make $44,876,834 at the box office. We'll find out if my prediction is close at all when the halloween spooky movie season is over.

Sources:

BoxOfficeMojo.com,

Wessa, P. (2009), Free Statistics Software, Office for Research Development and Education,
version 1.1.23-r4, URL http://www.wessa.net/

Why The Rich Get Richer, Part 2: The Danger Zone

Just as for any other asset, the marginal utility of money is a decreasing function. But the marginal utility of money inevitably decreases at a slower rate than any other asset because money can be used to purchase all other assets. Compare the marginal utility of gaining money to the marginal utility of gaining carrots. At a certain point, one can have too many carrots, especially since, as perishable goods, carrots cannot even be converted to cash once they go rotten. But money never rots, and can easily be converted into whatever one desires. So the marginal utility of money will only approach zero once one has literally satisfied all desires, including the desire for unconventional things like giving to charity. Nonetheless, there are important conclusions to be drawn from the fact that the marginal utility of money decreases.

To understand the concept of the individual's decreasing marginal utility of money, consider the difference between Person A with $4000 in the bank, with $1000 rent payable due in a week, and Person B with $500 in the bank, and $1000 rent payable due in a week. If Person A suddenly received $600 in income, he would have many choices of what to do with his money, including saving it. If person B received $600, $500 of this would most likely go towards rent, and the remaining $100 would go towards other essential items such as food.

My subjective judgment is at work here, but I am sure that many will agree with me when I say that the opportunity costs of not getting the extra $600 would be greater for Person B than for Person A. There are extra costs for Person B that would be incurred by not receiving the $600: in not making rent, or in the health costs of malnutrition. Also, at very low levels of wealth, the loss of money adversely affects one's ability to make money in the future. For example, if someone loses his front teeth because he can't afford to see a dentist, he will probably make less of a good impression at a job interview. This is a hidden opportunity cost that people with low levels of wealth must face.

This helps to better illustrate the same phenomenon presented in part 1. It is clear to me that the net benefit of making an investment is not a linear function in relation to one's level of wealth. This is because there are extra costs that naturally arise when one's wealth is at a certain low level. There are unavoidable items, such as basic food, shelter, clothing that everyone must obtain to survive, live with some dignity, and be able to produce wealth in the future. Then there are avoidable items, like MP3 players, which are biologically of a second priority, and do not affect someone's ability to produce wealth in the future. At a certain level, one's wealth will be used exclusively for unavoidable items. (Though people suffering from addiction and mental illnesses may avoid these items in favor of others.) It is apparent that the cost of being without food for a week would be greater than the cost of being without an MP3 player.

All types of investments can be a problem for people at lower wealth levels. As we have seen, if one accepts that the marginal benefit of having money is a decreasing function, one must accept that those with lower wealth will face greater marginal costs involved in any given investment. The marginal benefits of investments will also be greater at low wealth levels, but in smaller proportion than the costs. For everyone, there lies an economic danger zone, at the point where basic needs cannot be sufficiently satisfied by one's funds. All wealth from $0 rising up to the edge of this danger zone must be in a liquid form to satisfy current essential needs. Even risk free opportunities for profit would not be taken by rational people who simply cannot afford them because of the need to maintain a certain balance of liquid wealth. To escape the perils of the danger zone, and make profitable investments, such as those in vocational training, or a new car to help a job-search, one may resort to borrowing, but clearly there are extra marginal costs involved with this, e.g. interest due. Those with sufficient wealth can better avoid borrowing.

Those with enough money to avoid the danger zone are free to take greater risks and reap rewards for doing so. Thus, the rich get richer.

Why the Rich Get Richer, Part 1

Let's say you had a chance to play a game, only once, where the rules were as follows: Flip a coin. Heads, you get $100. Tails, you lose $50.
Would you want to play?
The answer to this question will come from your individual risk preference.
The expected value from the game is:
E(x) = ($100 - $50)/2 = $25
But what does the measure of expected value mean to an individual?
If a sizable sample of people play this game, the mean return will approach $25. But to each person playing this game there are only two possible results; a gain of $100 or a loss of $50. This is the origin of risk aversion. If the player could play again, there would be a chance to reverse any losses, and in the long run this would happen. But with this not being the case, people behave differently, usually in a way that is risk averse.
Now, are there any social conditions that would affect one's behavior in a game like this? It is apparent that a person's wealth would be a determinant. Let's say Person A and Person B are walking home from work. Both A and B plan to pick up milk from the grocery store. They both run into a street vendor who offers them the chance to play the aforementioned game. Person A has $100 in cash in his wallet, and Person B has $50 in his. While the monetary payoff or loss from playing the game is the same for both individuals, Person B may very well feel less willing to play. This is not, I repeat, not because Person B is more risk averse. To say that person B is more risk averse would imply that Person B is making decisions using exactly the same costs and benefits as Person A, when really this is not the case. In reality, the payoffs are different. If Person A were to lose $50, he could still pick up the milk from the grocery store using the $50 he has left. But if Person B were to lose $50, he would lose all the money he has, and not be able to get the milk he had desired. This is an added opportunity cost for Person B, so taking this cost into account, it is only natural that he would be less willing to play the game. And by not playing the game, Person B misses out on an expected payoff of $25.
As it goes for Person B, the same goes for real people with lower wealth throughout the world. And though street vendors offering games with favorable outcomes are not common, equivalent risk/reward situations exist everywhere, and for those with low enough wealth it is far more costly to play these games. Thus the rich have greater freedom to play risky games, and, as the saying goes, get richer. (I shall explore these other "games", such as higher education and changing careers, in a future post.)

Marginal Analysis of Senior-itis

We have all probably heard of "senior-itis", the phenomenon of High School or College Seniors losing interest in the outcome of their classes. It's likely that the phenomenon is largely caused by simple boredom, but marginal analysis seems to shed some light on this phenomenon. Early in a scholastic career, each grade affects GPA greatly. With the completion of each class, however, the effect of each grade on GPA decreases. This will continue to the point that in the senior year, GPA may not even budge in response to one's grades. In economic language this is a decrease in the marginal benefit of getting a higher grade in a class. Thus the return from an investment in studying time decreases, and students may find it reasonable to decrease their studying to a new equilibrium. Obviously, students will not want to incur the cost of repeating a class due to failing, and this is a separate issue. Thus the new equilibrium will always be above a failing grade, no matter how small the marginal change to GPA will be, but it will decrease to lower levels as more courses are completed. This little bit of economic analysis may partially explain why bored high school and college seniors might aim for C's in their classes.

More Economics-Based Pizza Shenanigans!

Warning! There might be major flaws in my reasoning. If so, please point them out in the comments.
What would happen if a group of people were to contribute money to a fund to buy pizza, and there was an agreement to split the pizza equally? (People eating pizza together can obviously divide it up according to how much money each person contributed, or by how hungry each person is, but we shall use the simplification of equal distribution, for illustrative purposes.) This is the flip-side of my Joey and Chandler Theorem, where the costs of each participant were the mean of the group. In this case, the benefits for each participant (slices of pizza) will be the mean of the group.

The main point of this article is to show that if a group of friends are pitching in for pizza, there will be certain effects on everybody's demand functions. Lets say there are two friends, Rachel and Monica, who pitch in for pizza together. (I will be using subscripts r and m for the two participants, and P and Q for price and quantity). The total cost for Rachel will be P*Qr. To avoid confusion, remember that there will be a difference here between quantity ordered and quantity consumed, because of the agreement to share. The total cost for Monica will be P*Qm. However the total quantity consumed for each participant will be equal to:
(Qr+Qm)/2
Let's give Rachel the following demand function for pizza:
P = 5 - Q
Since her Q will now be the mean of both her and Monica's quantity of order, her demand function will now be:
P = 5 - (Qr + Qm)/2
Notice two things about Rachel's new demand function:
First, her demand now partially depends upon Monica's order, the constant Qm/2. Secondly, the slope of Rachel's demand function has changed. It was =-1, now it is:

DP/DQr = -1/2
Notice that Rachel's demand has just become more elastic. To understand this intuitively, lets think of what would happen if the price of a slice of pizza were to increase by $0.50. Rational consumers would consume less. Now what if the price of half a slice of pizza were to increase by $0.50? Rational consumers would consume even less than in the first situation. What if the price of a slice of pizza were to decrease by $0.50? Rational consumers would consume more. What if the price of half a slice of pizza were to decrease by $0.50? Rational consumers would consume even more. This is what is happening with Rachel's demand function. For every slice of pizza she orders, she will pay the full price, but only get to consume half of it. Thus she will be more sensitive to changes in price.

Let's say that pizza is sold by the slice for the price (P) of $1. Let's also say that Rachel will contribute money to buy pizza before Monica does.
Solving for Qr, Rachel's order will be:
Qr = 8 - Qm
But this function only applies if Rachel knows how much Monica will order. If Rachel puts her money down first, she will have to use her expected value of what Monica's order will be: E(Qm)r. The person who puts his/her money down first clearly sets the tone. Let's say Monica has an identical demand function to Rachel, and that Rachel expects Monica to order 3 slices of pizza. Let's see what will happen.
Rachel will order:
Qr = 8 - E(Qm)r
Qr = 8 - 3 = 5
So, Rachel will order 5 slices of pizza, but what will Monica do now that she knows what Rachel ordered, and does not have to depend on an expected value? Monica, with her identical demand function to Rachel, will order:
Qm = 8 - 5 = 3
So, Monica will order 3 slices of pizza. But both Monica and Rachel will be splitting the pizza equally, eating 4 slices each. So, Rachel ends up paying for one of Monica's slices.

What if Rachel realizes that she got ripped off? The next time the two order pizza she has a different strategy. She immediately tells Monica that she can only afford 3 slices. What would happen? Monica, knowing that Rachel will only be contributing 3 slices to the pizza pool, to maximize her utility from the transaction, will now be forced to order...
Qm = 8 - Qr
Qm = 8 - 3
Qm = 5
...5 slices of pizza, just as Rachel had ordered the time before. Because she gave a low-balled contribution to the pizza pool, Rachel has cheated Monica out of $1. This method of low-balling has a risk, though. If Rachel misjudges how hungry Monica is, she might end up with less pizza than would maximize her utility.
In a situation like this, where costs are borne independently, but benefits are shared, the expected or actual value of what one's friend, or friends will order does something to one's demand curve. If friend A knows, or is otherwise convinced that friend B will order a higher quantity of pizza than friend A would order independently, to take advantage of this, friend A will shift her demand curve inward, and order less pizza than if she had simply been ordering by herself. And if friend A knows or is otherwise convinced that friend B will order less than friend A would order independently, to make the most of this unfortunate situation, friend A's demand curve will shift outward. For some reason the latter rule, to me seems less intuitive. But, the demand curves, with their shifting intercepts, do not lie. Or do they?
Let's get away from this algebra, and into some real life. No, the demand curves do not lie, but they are incomplete. For one thing, we must realize that these demand curves only represent the behavior of people who would choose to enter into a benefit sharing agreement. In reality, there are many other alternatives to splitting pizza with people, like getting pizza on your own, and I can imagine that many would find that arrangement more desirable than sharing. Also the price of pizza is not the only cost one pays for when pizza is ordered, and the quantity of pizza is not the only benefit. There are psychological costs and benefits as well. For example, if one does not pitch in enough money for the pizza, one might be stigmatized as a cheapskate, jerk or loser. If accountants could look into the minds of people ordering pizza in this way, they could make debits for "being called a cheapskate expense", "guilt over cheating one's friends expense" or credits for "perceived generosity revenue". But the human mind is hard to penetrate. Nonetheless we should understand that forces other than price and quantity are at work here, which is why people do not necessarily act in the ways shown on the graphs. Economic models (much like fashion models) do not have wrinkles, but real life is full of them.