Monday, March 30, 2009

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

Wednesday, March 25, 2009

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: 
And for Chandler:
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!

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.