Comparing Advantages

I was having a conversation recently with a co-worker who was lamenting that the lead on his project didn't find much value in my friend’s contributions.  It seems that the team lead is better on all the various technologies they employ.  I pointed out that even if that were true, which I doubt, my friend had an invaluable resource that no one could ever exceed – his time.  Every human on this earth has, in a normal day (i.e., one in which they are neither first conceived nor in which they die) the same number of seconds as everyone else.  This immutable fact of human existence leads to the idea of what economists call comparative advantage.

What exactly is comparative advantage?  It is best explained by examining a simplified version of my friend’s situation.  Suppose that we name my friend ‘F’ and his team lead ‘L’.  Also, let’s focus on two tasks that they each can perform and label them ‘A’ and ‘G’ (based on the actual tasks performed on my friend's project).

To understand comparative advantage, we have to say a little about how each of the persons working on this project employs their skills to perform these tasks.  When L says he is better than F, he can only mean one of two closely-related things.  Either he produces a better result in a fixed amount of time or he produces the same result in a shorter amount of time.

This balance between quality and time-to-complete is present in every daily activity where expertise is involved.  Do-it-yourself home projects are excellent examples where the weekend warrior spends two days trying to repair a sink or toilet when the same job could be done in a half-an-hour by a master plumber.

So why doesn't L just get rid of F?  After all, if L’s skill is analogous to the ‘master plumber’ and F’s to the ‘weekend warrior’ why even employ F?  This is where the time comes in.  Since L has the same amount of time available in a day as F, L can maximize the use of his expertise by relying on F to ‘take work off of his plate’.

Let’s make this more understandable by using concrete numbers.  Assume that L can perform task A in 0.5 hours and with a profit of $50, and task G in 0.3 hours and results in a profit of $20. Next assume that F can perform task A in 0.7 hours with a profit of $50 and task G in 0.4 hours with a profit of $20.  Note that in this example, I am imagining a situation where tasks A and G have to be done to a certain level of quality or no profit will result.  This is the usual situation when purchasing a good like a computer where the whole thing has to work (would you buy a computer that was mostly completed but was missing a hard drive?).   There are two final ingredients that have to be considered.  First, how many of tasks A and G must the firm of L & F complete?  Let’s assume that they need to complete at least 90 of task A and 80 of task G to live up to demand.  Finally assume that L and F are working a 40-hour work week.   I will only be examining a single week so I will not require that L or F work exactly 40 hours, but I will require that they stay within 0.1 hour (39.9 - 40.1) and I allow fractions of tasks completed. In a more realistic situation the output would be averaged many weeks and the fractions would disappear.  Let’s summarize the ground rules of our game in a little table.

Task A (90/week) Task G (80/week)
time profit time profit
L 0.5 $50 0.3 $20
F 0.7 $50 0.4 $20

Comparing side-by-side we can see that L is indeed better than F at both tasks.  That is to say that L completes both tasks faster than F for the same profit per task.

If the firm of L & F wants to maximize their profit (and who doesn't) the operative question is then:  what is the proper work load for L and F?

Let’s try a couple of examples.  Suppose L & F decide that since L is better at everything they should let him try to maximize the number of tasks completed.  It shouldn't be hard to convince yourself that the maximum number of tasks L can complete is 103 distributed between A and G as shown below (if he tries to do more tasks he has to do more on G leaving too little time for F to produce the remaining units to complete all 90 A tasks and 80 G tasks). The resulting work division is shown in the table below netting the firm a profit of $6,100.

Number of tasks profit/task hours/task total profit total hours
L does A 46 $50 0.5 $2300 23
L does G 57 $20 0.3 $1140 17.1
$3440 40.1
F does A 44 $50 0.7 $2200 30.8
F does G 23 $20 0.4 $460 9.2
$2660 40

Can their firm actually do better?  The answer is yes and it involves having L do fewer tasks, focusing solely on performing A tasks, which have the higher profit margin.  The maximum profit the firm can earn is with the following work load

Number of tasks profit/task hours/task total profit total hours
L does A 80 $50 0.5 $4000 40
L does G 0 $20 0.3 $0 0
$4000 40
F does A 11 $50 0.7 $550 7.7
F does G 80.5 $20 0.4 $1610 32.2
$2160 39.9

which results in a net profit of $6,160 dollars.   Observe that in raising the profit of the firm, L actually drops his output from 103 tasks to 80 while F drops his profit from $2660 to $2160. In addition, L & F combined work 0.2 hours fewer. By allowing L to specialize, both L & F stand to make a larger profit than can be made otherwise.  And even though F is seemingly inferior to L in all regards, his ability to provide the invaluable resource that is his time allows the firm as a whole to earn more profit.  That is comparative advantage.

At this point there are two additional observations to make.

The first is that the example given above is particularly simple in that not only was L better than F at performing both tasks, he has an advantage over F in performing Task A that is greater than his advantage over F in performing Task G.  This advantage is measured as the ratio of L’s profit per hour to F’s profit per hour on a given task.  L earns $100/hr on Task A compared to F’s $71/hr giving an advantage of 100/71 = 1.40.  Likewise, L earns $67/hr on Task G compared to F’s $50/hr giving an advantage of 67/50 = 1.34.  The fact that L has the greatest advantage on the task with the greatest profit margin ($100/hr) is what makes the analysis of the maximum profit easy to do.

The situation becomes more nuanced if F can improve in his performance of Task A, completing it in 0.6 hours instead of 0.7.  In this case, L still has an advantage on Task A at a value of 1.2 but it is now lower than his advantage on Task G.  With F’s new found skill, the firm can not only earn more money, but they can do so by having L spread his efforts between G and A tasks.  A possible (not neccessarily optimal) work load is

number profit hours total profit total hours
L does A 38 $50 0.5 $1900 19
L does G 70 $20 0.3 $1400 21
$3300 40
F does A 60 $50 0.6 $3000 36
F does G 10 $20 0.4 $200 4
$3200 40

which nets a profit of $6500 for the firm.  Note that L is still better than F in all regards, but L is not only doing more G tasks than before but his total number of tasks rises to 108 while F drops his task from 91.5 to 70.  Imagine now a real business with many employees and many tasks and you can see how complex it is to actually determine how to maximize business performance.

The second observation is that there is a human element to the problem of L & F that hasn't been addressed.  I posed a parenthetical question above when I asked who wouldn't want to maximize their profit.  Of course, in that context the profit was defined as dollars.  In fact, people define profit in all sorts of different ways  - job satisfaction, time off from work, lower stress, etc. – that make maximizing the outcome nearly impossible because of competing agendas and goals .  For example, it is possible that the firm of L & F is willing to forego some dollar earnings so that L & F can each share in the tasks being performed.   It’s for this reason that when this logic is applied in the broader context where L and F are human institutions (business, governments, or countries) there is a lot of friction in deciding what to do.

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