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.

Quantity Discount or Not? (This Article Could Save You Money.)

At a certain chain drugstore (which shall not be named), about a year ago, I came across an amazing quantity discount: ten containers of oatmeal for $10. I was immediately sure that I wanted to buy these ten items, but not sure that I could carry them all home, as I had travelled there on foot. But the temptation of cheap oatmeal was too great, and I chose to lug all of them to the cash register and brave the walk home, overloaded with oats.

But at the cash register, something shocked me. As the cashier scanned each container of oatmeal, the green text on the monitor facing me read: "oatmeal - sale $1.00", implying that, in order to obtain the discount, I didn't have to buy ten items in the first place! I could have bought three of them for three dollars, or five of them for five dollars. This quantity discount had not been a quantity discount at all!

I felt cheated and betrayed. But there is a silver lining here for me: now whenever I shop there, I find whatever deep "quantity" discounts they offer, and only buy one item. This saves significant amounts of money on the reasonable (and easier to carry) quantities I buy.

I did not, and still do not understand why this retailer (and others, as I later discovered) would offer a false quantity discount.

I have several theories. But for now let's just break the situation down so we can understand it better. Here I want to logically demonstrate why under all circumstances, the offering of a false quantity discount provides less revenue for a vendor than offering a real quantity discount. Let's say that a hypothetical retailer offers a 10 for $10 dollar oatmeal deal, but in reality the retailer has simply lowered the price of oatmeal from $2 to $1. In this situation there are five distinct groups of consumers (identified by number):

1. There are consumers who would buy 10+ items, and won't notice at the cash register, or on their receipts, that they didn't really need to buy the 10 items to get a discount.

2. There are consumers who would buy 10+ items, and will notice that they didn't really need to buy the 10 items to get a discount.

3. There are consumers who would buy fewer than 10 items, who won't notice that they didn't really need to buy 10 items to get a discount.

4. There are consumers who would buy fewer than 10 items, who will notice that they didn't really need to buy 10 items to get a discount.

5. There are consumers (like me, now that I have learned) who already know that the quantity discount is false, and will thus buy with full knowledge of what the price will be at the cash register.

Groups 2 and 4, the crafty consumers who take notice of what they have been charged, will each be given a surprise at the cash register. For group 4 it will be a pleasant surprise, and for group 2 it will be an unpleasant one.

Let's say there is someone named Steve, from group 4, who comes to the store, sees the oatmeal, is not willing to buy 10 units, and instead buys 3 units. Steve expects to be paying a total of $6 for his oatmeal, but at the cash register, he will notice the unit price is only $1. Steve will thus receive a pleasant surprise of $3.

Now what if someone named Jane from group 2 comes to the store. She sees the quantity discount and is blown away. She excitedly grabs 10 containers, and rushes to the cash register. Jane expects to be saving $10 as a result of her thrifty bulk-buying behavior, but at the cash register she will find that she did not need to buy in bulk to get the discount. This will be an unpleasant surprise.

If this had been a real quantity discount, the firm could be capturing an extra $3 from Steve, because he would have bought the oatmeal for $6 anyway. They would not be capturing anything more from Jane, but she would not have had that unpleasant surprise at the cash register, a surprise that may affect her opinion of the retailer in the future. Also, keep in mind that any customer from group 2 or 4 automatically transfers to group 5 when they find out that the firm's quantity discounts are bogus. And it is very important to note that the retailer could also be capturing extra money from each consumer from group 3, just as it could have captured surplus from Steve and the rest of group 4. Just like Steve, the consumers in group three weren't going to buy 10+ items in the first place and were happy with the price without the discount. But when they get to the cash register, they will be charged less than they were willing to pay. Simply put, the retailer is giving away money on every transaction with a group 3 or 4 person!

Discounts are only useful if consumers know about them. If prices fall in the woods, and there's no one around to hear them, they do not make a sound. A discount not known by the customer causes the firm to lose revenue on every transaction. At best, such a discount works as a pleasant surprise to the customer, who may then want to go back to the same store. The best case scenario for the firm seems to be when a consumer is from group 1. But in this scenario, the same result would be achieved if the firm had offered a real quantity discount.

So, in the face of all of this lost revenue and possible consumer frustration, why doesn't this retailer offer real quantity discounts? Maybe the firm wishes to create a subgroup of customers who know that the quantity discounts are false (group 5), thereby receiving the loyalty of this group of insiders (this is the far-fetched conspiracy theory explanation). Maybe there would be high costs involved in changing the computer system of the company's cash registers to be able to process quantity discounts (a more realistic explanation). Another possible explanation is managerial incompetence. But I try not to be quick to jump to that conclusion, however tempting it may be to make myself feel smarter than successful business managers. I'd like to think there is a reason for this policy, even a bad reason. Nonetheless, this company is losing money every time a customer from group 3 or 4 comes to the cash register. If any readers have theories or inside information on why this is, please let me know, I am very eager to solve this mystery. While its perhaps not rare that a company would mislead its customers, it is rare that a company would choose not to take money from its customers. I'm sure you can understand why I am confused.

(Follow Up to this story 1/6/11 by Alex Trenchard-Smith.

In the years since I wrote this article, I have noticed several more grocery and drugstore chains that follow the same policy of false quantity discounts. So as a shopper, definitely check your receipt to see if you really need to buy the recommended quantity. Since then I've also gained some knowledge of computer programming and reached the conclusion that companies probably do this because it's too complex to deal with Point Of Sale software that needs to be updated all the time with required quantities. They find it is more cost effective to mislead their customers than to add/and or utilize some simple if-else statements in their POS software.)

The Debits and Credits of Cigarettes and Food

(This article contains a highly simplified model of the economy, ignoring such things as the time value of money, and the unpredictability of health costs. What is important here is the broad concept.) (Also, the format of the accounting entries in this article is getting messed up by the website. Just remember, Debits on the left, Credits on the right, even if it's a bit uneven.)
Double entry bookkeeping is a beautiful thing. It is the yin and yang of the economic world. It demonstrates that there are always two sides to everything.
In this article, using simple double entry bookkeeping, I shall demonstrate that not all economic activities are created equal. In particular, the production and sale of cigarettes is not equally valuable to the economy as the production and sale of food. To demonstrate this, I shall go through some very simple journal entries, similar to anything that could be found in an accounting textbook. But unlike most accounting textbooks, these shall be journal entries for several entities listed in the same journal. The three entities involved here, which shall be denoted by letters in parentheses, are the Tobacco User (U), the Tobacco Retailer (R), and The Rest of Society (S). Remember, Assets = Liabilities + Equity, and as you shall see, the key difference between food and tobacco consumption is in the liabilities. Our first journal entry happens when a retailer purchases a pack of cigarettes for resale.

Tobacco Inventory (R) $x

Cash (R) $x


Next, the smoker purchases from the retailer. (m = markup for retailer).

Tobacco inventory (U) $x+m
Cash (R) $x+m
Cost of goods sold (R) $x
Cash (U) $x+m
Tobacco inventory (R) $x
Sales Revenue (R) $x+m


Now, the smoker consumes the tobacco. We all know that smoking causes health problems, and that these health problems are a burden to the smoker, and the rest of society. So to take this into account, a liability shall be part of the journal entry. This liability will be equal to the total cost to the smoker, and the rest of society of bad health brought about by the consumption of the tobacco in question. We will divide up the bad health expense and liability into proportions paid by the User and by the Rest of Society by using the variables Bu and Bs. (We shall lump the retailer in with the rest of society for simplicity's sake). Here, is what happens when the smoker smokes:



Tobacco expensed (U) $x+m
Deferred bad health expense (U) $Bu
Deferred bad health expense (S) $Bs
Tobacco inventory (U) $x+m
Long term bad health liability (U) $Bu
Long term bad health liability (S) $Bs


Over the course of the smoker's lifetime this expense will be realized by both the smoker and the rest of society. This is the journal entry for that.

Bad health expense (U) $Bu
Long term bad health liability (U) $Bu
Bad health expense (S) $Bs

Long term bad health liability (S) $Bs
Deferred bad health expense (U) $Bu
Cash (U) $Bu
Deferred bad health expense (S) $Bs
Cash (S) $Bs


Society's net income or loss from this entire transaction is equal to the tobacco retailer's profit from the transaction, m (x + m - x = m) minus the sum of Bs and Bu.
Here's the net income or loss for society:


Society's net income or loss = m - (Bs + Bu)


Now, what if the product in question, instead of cigarettes, was a healthy food item that caused no bad health liability or expense? Run through the same simulation and you will see that society's net income (and not a loss unless there is negative markup) will simply be equal to m.


Double entry bookkeeping exposes both sides of everything. In this case it has exposed the seedy economic underbelly of tobacco consumption. Though food and cigarettes may be accounted for in GDP the same way, using the wisdom of double entry bookkeeping, we can see through this. Clearly the two activities are not equally healthy, for people, or for the economy.

Investors or Inventors

There was an infomercial on T.V. the other day selling a groundbreaking system of stock market analysis, that when put into practice, would guarantee great returns; "We'll teach you how to get in when the stocks are low, and get out when they're high" etc. This was obviously a ridiculous scam. But it may be successful in drawing in victims. The stock market tantalizes people with the possibility of free money. But free money does not, and will never exist. All money gained from investments, (outside of that which arises from pure luck), comes from one of two sources, and both involve hard work:

1. Hard work by an investor in finding profitable investments.

2. Hard work by the invested-in company in gaining profit through its operations.

Nonetheless, in the American mind lies the fantasy of being able to sit at a computer and generate cash with the click of a mouse. People hope that somewhere there lies a perfect investment strategy. But the search for such a strategy is the modern equivalent of alchemy, and just like alchemy, it is a waste of time.

There can never be a fail-safe strategy of investing, because, no matter how sophisticated one's analysis of the stock market is, stock prices depend on an infinite number of unknown events that occur in the actual physical world. Only a piece of omniscient software could predict the future. Secondly, if a trader got his/her hands on this omniscient software, it would become useless if other traders had it too. The best that investors can settle for is to be sensible and manage their risk as much as possible.

Investment strategies are merely ways of siphoning money from the physical or intellectual assets of an economy. And if the productive capabilities of assets in an economy are impaired, it will become harder to make money as an investor. This is a law of financial physics. Even successful short selling depends on an initial price that reflects a perception of high productive capabilities. Securities' prices are merely shadows of the real economy.

Investment is the lifeblood of the economy, but careers as investment professionals are not what I hope the majority of kids of the next generation will strive to attain. I hope they will want to become inventors, not investors. (By inventor I mean someone who creates something new that is valuable to society). How useless would investors be if there were no inventors? There would be nothing to invest in. Inventors also need investors, to fund their endeavors. But good ideas will tend to attract investment. It does not take a degree in Finance to recognize a good idea. I hope the intelligence and creativity of the next generation, instead of going towards the alchemical quest of beating the stock market through technical analysis, will go towards innovations in the actual physical world. Imagine if the world's great geniuses decided to become stock traders instead of chemists, surgeons, authors, engineers etc. These geniuses might think of some great trading strategies, but there would be a lot fewer assets to invest in.

Pure speculation on the rising and falling of stock prices is a zero sum game. But sensible investment in good ideas benefits investors, inventors, and the economy as a whole.

The Joey and Chandler Theorem Pt. 2

Here is an application of the "Joey and Chandler Theorem" of my last post.

In the example of my previous post, there were two assets involved: pizza and cash. But assets and other pieces of economic reality, come in many forms. And theoretically, human behavior should manifest itself in a similar, or even identical way regardless of what assets are involved. With that being said, here's a new example involving other assets.

The first asset we shall consider is leisure. In this example I am speaking of leisure as the leisurely pace taken by an employee in completing a task. This is not to be confused with leisure as a measure of hours spent outside of work. To quantify leisure, lets assume that management has an idea of a standard time for an employee to complete a certain task. Using time (T) as a variable, lets call the standard time:

Ts

But, employees will often finish the task in more or less time than the standard. Lets call the actual time:

Ta

Let's assign the variable leisure the letter L. Using these variables, and assuming no external delays in the completion of the task, the function of leisure taken in completing a task can be measured as:

L = (Ta-Ts)/Ts

Thus, if a task that management believes should be completed in 1 hr gets completed in 1.5 hrs, L would be 0.5. If the same task were completed in 3 hrs, L would be 2. We can also measure leisure as a negative number if tasks are completed in less than standard time.

Leisure is something valued by everyone. (Even the craziest workaholics can't move at light-speed). And, like all things of value, Leisure has a price. In the case of leisure, the price an employee pays for being slower than the standard time is the loss of regard by his/her employer. This price will be different for whatever employer one has, depending on the amount of such things as firings, demotions, or just plain nagging that the employer will respond with if an employee is works too slowly. These things are hard to quantify, so lets just give the price of leisure a dollar value. We can do this by simply multiplying L by the hourly wage of an employee (this example doesn't work for employees paid flat salaries). It seems reasonable that the negative consequences of being slow would be directly related to the amount of money in wages being wasted by an employee's leisurely pace. This amount of wasted money (in standardized, not absolute terms, because of the division by Ts), would be equal to L multiplied by the wage (W). Lets call the price of leisure P. So:

P = W*(Ta-Ts)/Ts

But an employee does not automatically pay a price for leisure. This will only happen if management is aware of the time each employee takes. Now, what if a task is completed by a team rather than an individual? It will then be harder for management to trace the performance of the team back to its individual members. Lets assume that two workers, (I'll call them Joey and Chandler just to emphasize the point that the same economics of my last post is at work in this situation) work together to complete a task. Management has a standard time for the task of 1 hr. J and C both get paid $10 per hr. Assuming that management can only evaluate the team by measuring the time it takes them to finish, J and C will be sharing the cost of any leisureliness each one of them might take in completing the task. The free-rider problem will take effect. If J works lazily, and C works extra hard or vice versa, the team might still finish in standard time. And just as in the pizza eating example, the marginal cost of more leisure is cut in half with the introduction of another participant. As a team the only L that matters is that of the team, which is dependent upon the L of each participant. An individual's L is based upon the amount of time it would take that person to complete the whole task in the absence of other people. (To make it simple, lets assume that this is a task that theoretically could be completed by only one person, and that neither one will be waiting for the other for any reason).

So the price each person pays for being leisurely is based on an average:

(I'm sorry if the subscripts are confusing, let me explain: Taj = time, actual, Joey. Et cetera.)

Pj = ($10*((Taj - Tsj)/Tsj) + $10*((Tac - Tsc)/Tsc))/2

Here's an example of how it could work out. Let's say that Joey works at a pace that would have taken 1.5 hours if he had worked alone, and that Chandler works at a pace that would have taken 1 hour if he had worked alone. Then:

Pj = Pc = ($10*(0.5) + $10*(0))/2 = $2.50

Anyway, if we take the partial derivative for either P with two participants, we see that it will be half of what it would be if there had been no one else on the team. Lets cut to the chase here. Just as in the pizza example, the marginal cost of L will be cut down as more people join the team. The marginal cost, if there were 1000 people on the team would be $10/1000 = $0.01, which isn't much to worry about. Assuming individual demand functions for leisure of negative slope, individuals on teams will choose more leisure. So, the conclusion I have reached here is:

Ceteris paribus, the free-rider problem causes teams, being measured as teams, to be lazier than individuals.

But does this mean that individual performance must be emphasized above that of the team? I have my doubts, for reasons I shall discuss in a future post.

The Joey and Chandler Theorem Pt. 1

   

(The economic ideas contained in this article were inspired by a certain episode of a certain sitcom. Economic wisdom can be acquired anywhere.) 

   Let's say a group of friends are sitting down for a meal at a restaurant. To make this simple at first, let's assume that there are only two friends, Joey and Chandler. Now lets consider two ways that Joey and Chandler, (J and C for short) could pay for the meal. The ordinary way (scenario 1) would be for J and C to pay only for what they each ordered. Another way (scenario 2) would be for J and C to each pay a proportionate amount of the total cost of the meal. 

Let's look at these scenarios mathematically. Using subscripts J and C to indicate which friend is involved with the variable, Let P = the price of food, and let Q = the quantity of food. To simplify, we shall assume that the restaurant only serves one item in discrete units, slices of pizza for $2 each. 

   Now let's look at how the costs will change under the different scenarios. 

Under scenario 1 the total cost for Joey will be: 

P*Qj

And for Chandler:

P*Qc 

This is because J and C will only pay for their own orders. The marginal cost of a slice of pizza will simply be $2. Under scenario 2 things get a little more interesting. With each friend paying a proportional amount of the total, the total cost (TC) for Joey will be:

TCj = P*(Qj + Qc)/2 

and for Chandler, it will be the same:

TCc = P*(Qj + Qc)/2

To generalize, with N people the function would be 

TC = (P*Q1 +P* Q2 +... + P*Qn)/N. 

Under scenario 2 everyone just pays the mean order for the group. So how does scenario 2 change marginal cost (MC) for Joey and Chandler? Taking the partial derivative of Joey's function, 

TCj = $2*(Qj + Qc)/2 

gives us:

MC = d(Pj)/d(Qj) = $2/2 = $1. 

Thus, with the introduction of Chandler under scenario 2, Joey's marginal cost of a slice of pizza has been cut in half. Now what if another friend, Monica, joined in with scenario 2? Joey's new marginal cost for pizza would be cut to a third: $2/3 = $0.67. If six friends total joined in the meal, Joey's marginal cost would be $0.33. This would be the marginal cost for all other participants as well. With a lower marginal cost, each participant will consume more. But as will be shown, some will benefit more than others from this decrease.

   While marginal cost decreases under scenario 2, the total cost of the meal between J and C will not change. It will still be: 

TCj = TCc = $2*(Qj + Qc)/2    

(The following part of this post is in the process of retooling, though the major points will be the same, I think, unless I've been totally wrong)

Now let's say that Joey has the following linear demand function, (in terms of Marginal Cost) for pizza: 

MC = 15 - 2Qj

And Chandler has:

MC = 7 - 2Qc 

We can see from these demand functions that Joey likes pizza more than Chandler. Joey will thus benefit from scenario 2 more than Chandler will. Finding the point where Marginal Benefit = Marginal Cost, by substituting the $1 for MC, and solving, we see that Joey will consume 7 slices of pizza, and Chandler will only consume 3. They both will however, cover the same cost of $10 per person. Through the agreement on proportional payment Joey, in essence, gets Chandler to pay for two of his pizza slices. So under scenario 2, Chandler gets screwed, because he is not a pizza eating machine like Joey is. 

Two lessons can be learned from this example:

1. When the cost of pizza (or any asset) is to be split proportionally among a group, all members of the group will consume more than they would have otherwise. 
2. Under a proportional cost sharing agreement, those who consume more than the mean are able to shift their costs onto those who consume less than the mean.

Due to the cost sharing agreement, a simple meal among friends becomes like a pizza eating contest, and the more participants there are, the more of a contest it becomes. Imagine, instead of a small group, that 2000 people all decided to eat this same $2 pizza, with the agreement that each person would pay for a proportional amount. For all participants, the marginal cost of pizza would be $2/2000 = $0.001 per slice, which is practically free. Lets say that the mean quantity of pizza consumed by these 2000 people was 5 slices, thus a total of $10 would be paid by each participant. But what if, out of these 2000 people, one person happened to be the crowned world champion pizza eater, who consumed 30 slices? This person would end up getting a major bargain for pizza (but would probably pay for it later with digestive problems).  

   I have only recently begun thinking about these properties of cost sharing, and will be looking for other real-life instances of this behavior. I hope to soon expand upon these ideas in more depth, in future postings that involve assets other than Pizza. 

(See my graph at the top of the page, displaying the increase in quantity of pizza consumed when Marginal Cost decreases due to another participant in scenario two)

Welcome to my blog!

Hello, 

I just created a blog, but I then realized I had misspelled "economics" on the web address (I had spelled it "econonomics"). Anyway, here's my blog, for real this time. I hope to be posting substantial articles here soon, on the subjects of economics and management.