7. The Specialization Problem

Economists know that specialization is important, and they are good at enthusing about it eloquently, and briefly. Then they move on to think much harder about topics they know to be narrower and less important. It's an odd procedure, but it goes all the way back to Adam Smith, whose Wealth of Nations starts with three chapters on the division of labor, then drops the subject and goes into great detail about price determination, and never really circles back. Modern undergraduate textbooks do the same.

Not spending much time on specialization makes sense if you assume that although specialization is important, there’s no particular problem to be solved. If you believe some breezy story like “people specialize, then trade for what they need, and markets efficiently organize the trading,” then maybe economists have better things to do than talk more about it.

But there is a problem to be solved, and I plan to spend the next few posts proving that the specialization problem exists, and is very serious.

There is, in fact, a fundamental tension between specialization and competitive markets. Specialization tends to reduce competition. It reduces the number of practitioners in each increasingly narrowly-defined field. In the logical limit, it reduces it to one, as each task gets sub-divided again and again until each sub-task is being performed by just one agent. Yet markets rely on competition to set prices. The closer the economy approaches to total specialization, the less competitive markets become.

I intend to illustrate that problem in several different ways, in order to drive the important point home. My first example will follow an old tradition by traveling to an imaginary desert island, where Robinson Crusoe and Friday are stranded together. As there are only two people in this simplified model, so there are only two goods—fish, and coconuts. Each person’s utility is FC, where F is consumption of fish, and C is consumption of coconuts.

Specialization as a Prisoner’s Dilemma

Crusoe and Friday allocate their time between fishing and picking coconuts. Then, if it’s mutually agreeable to do so, they can trade. And we may further assume that if they specialize, with each one producing only one good, the total production possibilities are expanded. There are a lot of reasons why this should be so. They may have different talents, so that each is relatively more productive when doing what he does best. There may be fixed (start-up) costs to each production activity. Time spent walking to and from the fishing hole, or the coconut grove, is a cost incurred just once for the activity and not tending to accumulate with the number of fish caught or coconuts picked. Likewise, Crusoe and Friday may spend time making tools, or mastering skills.

Let’s assume throughout that if both specialize, Crusoe can catch 10 fish and Friday can pick 10 coconuts, and that this 10 fish plus 10 coconuts is the maximum the pair can produce, in the double sense that (a) no Pareto-superior joint consumption bundle is technically feasible, and (b) no feasible joint consumption bundle would push the utility possibilities frontier out further. It’s not too important to fully understand the last sentence. It basically just means specialization is the best outcome for this two-person society, but not necessarily for each of the agents involved.

If both agents specialize, so that Crusoe has 10 fish and Friday 10 coconuts, it’s pretty clear how the trade should work out. They go halves. Crusoe gives Friday 5 fish, Friday gives Crusoe 5 coconuts, and they both get utility of 25. But the concept of the Nash bargaining solution is a way to formalize this result, and at the same time, in a sense, to generalize it. Nash worked hard to figure out a bargaining outcome for any game that is (a) symmetric, and (b) unaffected by monotonic transformations of the utility function, and in that sense non-arbitrary. We'll borrow the Nash bargaining solution to predict how Crusoe and Friday will trade.

Now consider the case where one agent specializes while the other does not. What happens then?

To keep the problem symmetric, let’s assume that Crusoe, the better fisherman, can catch X fish and pick Y coconuts if he doesn’t specialize, while Friday, the better coconut picker, can pick X coconuts and catch Y fish. Further assume, for now, that X is 5 and Y is 4. If Friday specializes and Crusoe does not, then Crusoe enters trade negotiations with a consumption bundle of 5 fish and 4 coconuts that gives him a utility of 20, while Friday’s 0 fish and 10 coconuts give him a utility of 0. That puts Crusoe in a better bargaining position, and the Nash bargaining solution, by which terms of trade are set so as to maximize the product of the two agents' utility gains, provides a way of making the resulting outcome concrete.

As simple as this problem seems to be, an analytical solution turns out to be difficult (perhaps impossible, I'm not sure) to achieve. But to solve it by numeric methods via the Solver tool in Microsoft Excel is pretty easy. That's how I got the below results.

Under Nash bargaining (with X=5, Y=4, Friday specializing and Crusoe not) Crusoe the non-specialist will trade 1.48 coconuts for 5.84 of Friday’s fish. As a result, Crusoe and Friday will consume fish and coconuts in equal ratios, but Crusoe will consume 2.37 times as much as Friday. As a result, Crusoe will get utility of 34.6, while Friday gets utility of 6.2.

We’ve now arrived at the counter-intuitive heart of my argument. By specializing, Friday enriches Crusoe, but impoverishes himself. He impoverishes himself by putting himself in a weak bargaining position. He has lots of a resource, coconuts, that, thanks to his efforts, is relatively abundant, and therefore relatively low in value at the margin. Meanwhile, his complete lack of fish puts him in a wretched utility position, desperately in need of trade in order to meet his needs. Crusoe enters negotiations from a much stronger position, and can afford to walk away from them and still have a well-balanced diet. So Friday gets quite disadvantageous terms of trade, offering almost four coconuts for one fish, and ends up worse off than he would have been if he had caught his own fish instead of trying to buy them from Crusoe.

It turns out that Crusoe and Friday are playing a Prisoner’s Dilemma game. The most appealing outcome is for both to specialize, and get utility of 25. But it’s not in the interest of either to specialize. Each player thinks as follows:

• If he specializes, I can get utility of 25 by specializing, or 34.6 by not specializing (and then trading at advantageous terms).
• If he does not specialize, I can get utility of 6.2 by specializing (and getting stuck with disadvantageous terms of trade) or 20 by not specializing.

Whatever you think the other player will do, you shouldn’t specialize. Here’s the game in table form:

Table 1

		Friday
		Specialize	Don’t specialize
Crusoe	Specialize	25, 25	6.2, 34.6
	Don’t specialize	34.6, 6.2	20.3, 20.3

 

What a perverse outcome! Crusoe and Friday could increase their utility by almost 20% if only they could specialize and trade, yet it’s a dominant strategy equilibrium for Crusoe to go on picking most of his own coconuts and Friday to go on catching his own fish, and trade remains minimal. (Friday gives Crusoe half a fish in exchange for half a coconut.) Of course, it seems likely that Crusoe and Friday could solve this problem somehow, but there is a problem to be solved. 

An Institutional Fix

Now, it seems reasonable to suppose that if Crusoe or Friday specializes and the other doesn’t, the non-specialist will offer terms of trade a bit more generous than the Nash bargaining solution predicts. Some reasons for this are moral in nature, e.g., if people feel sympathy or value equality, or if the non-specialist feels gratitude to the specialist for enriching him, or is loath to seem to be “taking advantage” of his neighbor.

Yet it turns out that the moral sense does not need to be invoked. There is what I call an institutional fix, by which I here mean a set of incentive-compatible rules that can be set up which mitigate the problem.

In general, the outcome of bargaining games is rather underdetermined. The gains from trade might be almost completely captured by either party if they can somehow make a credible ultimatum, and the other prefers something to nothing. (In bargaining games, paradoxically, it can pay, in a sense, to be irrational.) The Nash bargaining solution is a way to push through the interminable bargaining puzzle with a result that’s general and non-arbitrary, if somewhat inadequately motivated. But it can be just as satisfactory to say that one party successful makes an ultimatum and captures most, or for simplicity all, of the surplus resulting from trade, if you can give any plausible account of why one player rather than the other captures the surplus. And one reason why one player might be favored to grab the surplus is that they have the "moral high ground."

To make this more concrete, suppose that Crusoe and Friday agreed that they would specialize. Friday does so, and comes home with 10 coconuts. Crusoe, however, fails to specialize, and comes home with 5 fish and 4 coconuts. Now Friday has the moral high ground, so he might think he deserves to get good terms of trade. He offers 3.5 coconuts in return for 2.3 fish, a 1.5:1 price ratio. This trade doesn’t help Crusoe at all, but it doesn’t hurt him either. It leaves him indifferent while conferring a large benefit on Friday. If Crusoe feels a bit ashamed at reneging on the deal they made earlier, a trade like this should be a price well worth paying to partially redeem himself. So Crusoe ends up with 2.7 fish, 7.5 coconuts, and utility of 20, while Friday gets 2.3 fish, 6.5 coconuts, and utility of 15.2.

While this arrangement is much better for Friday than Nash bargaining was, he’ll still regret having specialized, since even after capturing all the surplus from trading with Crusoe, he still ends up with less utility than the 20 he would have received if he hadn’t specialized in the first place. Crusoe and Friday are still stuck in a prisoner’s dilemma game. So far, the institutional fix doesn’t seem to solve the problem.

But it turns out that for some values of X and Y, the institutional fix does work. If a non-specialist can produce 5 of good A and 3 of good B, and if a sole specialist uses the moral high ground to capture all the gains from trade, then the utility of a sole non-specialist turns out to be 15, and that of the specialist, 17.6. The game then takes this form:

Table 2

		Friday
		Specialize	Don’t specialize
Crusoe	Specialize	25, 25	17.6, 15
	Don’t specialize	15, 17.6	16, 16

Now the dominant strategy is to specialize. If the other player specializes, specializing gives you utility of 25 instead of 15. If the other party does not specialize, specializing gives you 17.6 instead of 16, as you use the moral high ground to insist on better terms of trade from a reneging neighbor who, unfortunately, has impoverished you both. Either way, you’re better off specializing.

The phrase “institutions” is quite popular nowadays in development economics, and it’s used in what I regard as some quite unhelpful senses. In particular, “institutions” is often a way of invoking the government as a deus ex machina. When development economists say that the West is rich because of its superior institutions, they often mean little more than that the West is rich because its government are somehow—there is little agreement about how or why—more beneficent than governments elsewhere. (Probably they are, but I suspect the degree of difference is greatly exaggerated, and anyway, quality of governance might be—I think it probably is—more a result than a cause of the development process.)

When I say "institutions," I don't mean "government." There is no government on Crusoe and Friday’s island, nor any coercion or violence for that matter. The rule “bargain hard if you specialized and your neighbor didn’t; bargain soft in the converse case; and trade on equal terms if both or neither specialized” is self-enforcing, once it is established, in the sense that Crusoe and Friday each plays by that rule and expects his neighbor to do so, and expects his neighbor to expect  him to do so, etc.

The institutional fix is also not an ethical fix. It’s quite possible that Crusoe and Friday would specialize because, instead of narrowly maximizing self-interest, each follows the Golden Rule and loves his neighbor as himself, and specializes for the sake of the social interest regardless of what he perceives his individual interest to be, if he even bothers to think about it. But that’s quite a separate story. In this case, the individuals are acting in a strictly self-interested fashion.

To make this clear, it may help to integrate the Prisoner’s Dilemma and the Institutional Fix cases together in a single game, which is done in Table 3.

Table 3

		Friday
		Specialize, Punish	Specialize, Don’t Punish	Don’t Specialize, Fair	Don’t Specialize, Soft
Crusoe	Specialize, Punish	25, 25	25, 25	0, 15	17.6, 15
	Specialize, Don’t Punish	25, 25	25, 25	7, 29.3	7, 29.3
	DS, Fair	15, 0	29.3, 7	16, 16	16, 16
	DS, Soft	15, 17.6	29.3, 7	16, 16	16, 16

Table 3 represents a game with the following rules.

• There are two stages, production and bargaining.
• At the production stage, players can choose to specialize or not to specialize.
• At the bargaining stage, players have three options: 
• Hard: insist on capturing all the surplus;
• Fair: insist on capturing half the surplus;
• Soft: accept any terms of trade that hold you at least harmless.
• Players choose a single strategy before the game, which contains contingencies, such that play in the bargaining stage can depend on outcomes in the production stage.

Rather than show all the strategic possibilities, I simplify by assuming that at the bargaining stage, players never play Hard except as part of a Punish strategy, or Soft except to yield to an anticipated Punish strategy by the other player after failing to specialization in the production round. A Punish strategy consists in Fair bargaining when both or neither have specialized, but Hard bargaining when one has specialized and one’s neighbor has not (thereby giving the Punisher the “moral high ground”).

In this game, there are two Nash equilibria: one where both players play “Specialize, Punish,” and one where both play “Don’t Specialize, Fair.” If non-specialization will be punished at the bargaining stage, it’s not worth it, and specialization will prevail. If non-specialists refuse to accept punishment at the bargaining stage, and nix the deal instead, then punishment backfires, and non-specialization prevails.

All other strategic combinations fail to be Nash equilibria. If one of the players won’t punish, the other, rather than specializing, should be a non-specialist and gain by exploiting advantageous terms of trade. But in that case, the other player shouldn’t specialize either, leading to a non-specialized equilibrium. If the players are non-specialized but one is a soft bargainer willing to accept a hard bargain from an offended specialist, the other player should specialize, in which case the first player should specialize as well.

The institutional fix is self-enforcing, yet it still feels rather fragile, because its stability depends on some belief about what the other person would do under circumstances that, in the Nash equilibrium, never arise. Thus, I think that my neighbor would be a hard bargainer if I failed to specialize… but how do I know, if I never try it? Or, I think that my neighbor would nix a deal rather than accept punishing terms of trade… but again, how would I know?

Does society, then, depend on a whole array of such fragile institutions, which induce cooperation only through threats that aren’t ordinarily carried out? Could a wave of doubt cause such institutions to dissipate, perhaps suddenly? It’s a disquieting thought, and the quintessential worry of Burkean conservatives.

Specialization as a Stag Hunt

If we change X and Y further in the direction favoring specialization, the basic form of the game changes again, to what is called a “stag hunt.” The charming name of this type of game is an allusion to a practice of hunting a stag by first surrounding it, then closing in for the kill. This hunting technique requires multiple hunters—but for simplicity we’ll assume just two—to cooperate. Meanwhile, the hunters might be tempted to abandon the stag hunt and just hunt rabbits, which they can catch without help. The difference from the prisoner’s dilemma game is that if all the other hunters play along, there’s so much more meat on a stag than on a rabbit, that all the hunters are likely to be better off for having kept on task. But if one of the other hunters goes off chasing rabbits, the stag gets away, and the other hunters get no meat, and wish they'd hunted rabbits, too. So the stag hunt game has two Nash equilibria: one where all the hunters stay on task and catch the stag, and another where all the hunters go chase rabbits.

Consider the scenario where a non-specialized Crusoe can catch 4 fish and pick 2 coconuts, and a non-specialized Friday can pick 4 coconuts and catch 2 fish. Specialized, each can still get 10 of what he gets best. Under Nash bargaining, if Friday specializes and Crusoe doesn’t, Friday trades 5.58 coconuts for 1.47 of Crusoe’s fish.

Crusoe ends up with utility of 19.2, and Friday with 6.5. Yet though Crusoe seems to get the better end of the deal, he still has regrets. Since Friday specialized, if only he had specialized too, they could both have had utility of 25. By not specializing, he impoverished Friday, as before, but this time he impoverishes himself a little bit, too. But if Friday hadn’t specialized, he would have been glad not to have done so himself. How was he to know?

The payoff structure of the game is as shown below:

Table 4

		Friday
		Specialize	Don’t Specialize
Crusoe	Specialize	25, 25	6.5, 19.2
	Don’t Specialize	19.2, 6.5	9, 9

The lesson of Table 4 is a bit obscure. The game suggests that you always benefit others by specializing, but whether you benefit yourself depends on others. If they’re trustworthy, if they’ll hold up their end, you can all thrive together, and there might not be any sacrifice. Yet there’s a danger of being left in the lurch, stuck as a specialist in a glutted market, and regretting that you did the right thing. So whether to specialize is an easy choice for altruists but a tricky choice for egoists. The game also suggests that a society of altruists might be more prosperous than a society of egoists, because they could more easily and reliably cooperate.

Lastly, it’s worth mentioning that all the production specifications that give the game a Stag Hunt form under Nash bargaining are also amenable to the institutional fix. If specialists can credibly threaten to punish non-specialists at the bargaining stage, then specialization becomes the dominant strategy at the production stage. I don’t know whether this holds over all possible version of the Crusoe-Friday specialization game (e.g., varying the utility function), but it holds for this one (at least for integer productivities, which are the only ones I checked).

The Full Outcome Typology in the Two-Person Case

I have been developing a typology of specialization scenarios. There is a prisoner’s dilemma case; a prisoner’s dilemma with an institutional fix; and a stag hunt case. There is one more: if one’s productivity in the other good is sufficiently low, it’s not worth it, and one should just specialize. Thus, if a non-specialized Crusoe can catch 4 fish and pick only 1 coconut, his utility if he doesn’t specialize will be 14.7 if Friday specializes, and 6.25 if he doesn’t. But if Crusoe specializes, he gets 25 if Friday specializes, and 7.8 if he doesn’t. Either way, Crusoe is better off specializing, and specialization is a dominant strategy. This case might be called spontaneous specialization.

Table 5 applies the typology to all the integer productivities that can be imputed to Crusoe and Friday in their respective goods. It also shows the utilities that accrue to the non-specialist (left) and the specialist (right) under Nash bargaining.

Table 5

		Non-specialist Good B Production
		5	4	3	2	1
Non-specialist Good A Production	5	N/A
	4	34.6, 6.2	25.7, 4.9
	3	29.3, 7.0	23.5, 5.6	17.6, 4.2
	2	24.0, 8.1	19.2, 6.5	14.4, 4.9	9.6, 3.3
	1	18.4, 9.8	14.7, 7.8	11.0, 5.9	7.4, 3.9	3.7, 2.0

In Table 5, mauve cells represent pure prisoner’s dilemma cases; orange cells, prisoner’s dilemma cases where an institutional fix is available; yellow cells, stag hunt cases; and green cells, spontaneous specialization cases. I’m struck by how large a share of the cells are yellow stag hunt cases. Often, it seems, both specialization and non-specialization are equilibria. But the big, basic lesson is that there is a specialization problem, that it’s common for specialization to be socially beneficent, yet not automatically incentive-compatible for individuals.

Crusoe and Friday’s desert island may seem remote and irrelevant, but I believe the specialization problem is applicable to many or most situations where there is division of labor in a non-market context, such as a family or an organization. A common phenomenon in large organizations is the “turf war,” where different individuals or teams engage in a somewhat surreptitious struggle to annex different functions to their scope of work. A team looking for job security might start performing a bunch of tasks that lie on the margins of their work. Another team has more specialist expertise and could have done those tasks better, but it’s faced with a fait accompli. Meanwhile, the turf-grabbing team’s core functions have been relatively neglected, yet they still need to be done, and the turf-grabbing team is still best placed to do them. Management intervenes, and maybe urges the turf-grabbing team to focus on its core tasks, but it’s difficult to understand what happened, and when some compromise is reached in order to settle the issue and move on, the turf-grabbing team’s scope of work, and maybe its budget, has expanded at the expense of the other team, which now faces an increased risk of redundancy. The interests of the organization would have been better served if the turf-grabbing team had specialized in its core functions, but the turf-grabbing team benefits by becoming less specialized and grabbing more work.

That’s not to say that turf grabbing is always bad. Sometimes the motive for turf grabbing is that a team is capable of doing more than it has been tasked with, and its ambition and enterprise in expanding its scope of work benefit the organization as a whole. An organization rife with turf-grabbing might be doing well, expanding and innovating, and a lack of clarity about roles is both a cause and a consequence of its culture of creativity and change. But organizations should understand that specialists are vulnerable, and tend to face incentives to broaden their functions so as to improve their bargaining positions, and that it’s often contrary to the interests of the organization for them to do so.

Two lessons that can be drawn from the specialization problem of Robinson Crusoe and Friday are likely to become monotonous: we've seen them before, and we’ll see them again. But they are so basic and important that it’s worth insisting on them to the point where they become truisms.

First, virtue can often achieve socially beneficent outcomes that self-interested behavior makes difficult or impossible to achieve. A selfish and amoral Crusoe and Friday will face some difficulty in reaping the full gains from specialization and trade, but for a Crusoe and Friday who follow the Golden Rule and love one another as themselves, or even a Crusoe and Friday who are merely scrupulous in keeping their agreements, there is hardly a problem to solve. The altruists will always specialize for their fellow’s benefit if not their own, and the promise keepers will soon discern, and commit to, the optimal specialization pattern, then do it.

Second, “institutions”—the words customs, protocols, and habits would do almost as well here—can sometimes achieve socially beneficent outcomes even when individuals are selfish, but they tend to have a certain fragility, and can be disrupted if expectations change.

In the prisoner’s dilemma cases, where the specialization problem is most acute, the gains from specialization and trade are apparently rather small. At most, consumption increases by 20% as a result of specialization, from 4 of each good to 5 of each good. This might suggest that the specialization problem isn’t too severe and isn’t a candidate for explaining the vast income disparities between, say, Bangladesh and the United States. But the gains look small only because there are only two people and two goods. In the next post, I’ll present an example in which the gains from specialization and trade are absolutely enormous—and the specialization problem, in its acute prisoner’s-dilemma form, is fully applicable.