Technology Changes Constraints

A thou­sand years ago, books were gen­er­ally writ­ten by hand, on parch­ment made from sheep skin. I don’t have a good source on how long it took a per­son to tran­scribe a typ­i­cal book, so for the pur­pose of this post let’s just call it 30 days. I do know that a typ­i­cal book re­quired the skins of about 12 sheep (source: Braudel).

We can rep­re­sent this via two pro­duc­tion con­straints:

… and of course we could add more con­straints to re­flect all the other in­puts to a book. We write it like this, rather than just say­ing “1 book = 12 sheep + 30 tran­scrip­tionDays”, to high­light that each in­put is an in­de­pen­dent limit on the num­ber of books pro­duced. If we only have 15 sheep on hand, then we can make at most 1 book, no mat­ter how many bored tran­scrip­tion­ists are sit­ting around.

Another rea­son why writ­ing out the con­straints is use­ful: it offers a nat­u­ral way to in­tro­duce tech­nol­ogy changes.

Let’s con­sider two pos­si­ble tech­nol­ogy changes:

  • switch­ing from parch­ment to paper

  • switch­ing from tran­scrip­tion­ists to a print­ing press

How do these mod­ify the con­straints? Well, pa­per elimi­nates the sheep con­straint and re­places it with a pa­per con­straint (of the form for some ) - yet the tran­scrip­tion con­straint re­mains ex­actly the same. Con­versely, a press elimi­nates the tran­scrip­tion con­straint—yet the sheep con­straint re­mains ex­actly the same. The con­straint rep­re­sen­ta­tion is mod­u­lar with re­spect to tech­nol­ogy changes: in­tro­duc­tion of new tech­nol­ogy re­moves/​mod­ifies some con­straints, while leav­ing most of them un­altered.

With a lit­tle cre­ativity, this rep­re­sen­ta­tion can be ex­tended to other kinds of tech­nol­ogy changes as well:

  • Be­fore the in­ven­tion of tele­vi­sion, we had the con­straint . The in­ven­tion of tele­vi­sion re­placed this con­straint with a bunch of tele­vi­sion pro­duc­tion con­straints, like

  • Fixed-cost cap­i­tal goods, e.g. a print­ing press, add a con­straint that we need at least one of the cap­i­tal good, in­de­pen­dent of the num­ber of things pro­duced:

  • A more effi­cient sheep-skin pro­cess­ing tech­nique might re­place with

… etc.

Conjugacy

One of the main les­sons of op­ti­miza­tion the­ory—be it lin­ear pro­gram­ming, con­vex anal­y­sis, what have you—is that ev­ery con­straint has a con­ju­gate “shadow price” (math­e­mat­i­cally given by the La­grange mul­ti­plier). The price in­di­cates how “taut” or “slack” the con­straint is—i.e. how much more we can pro­duce if the con­straint is re­laxed a lit­tle bit. If we’re a me­dieval book-maker with 15 sheep and a thou­sand tran­scrip­tion­ist-years on hand, then the sheep con­straint is very taut (more sheep means more books), whereas the tran­scrip­tion­ist con­straint is very slack (more tran­scrip­tion­ists does noth­ing). It’s like a rope: pul­ling on a rope won’t do any­thing un­less the rope is taut; hiring more tran­scrip­tion­ists won’t do any­thing un­less the tran­scrip­tion­ist con­straint is taut. The shadow price quan­tifies this: it tells us how much the book-maker will pay for ad­di­tional sheep ver­sus ad­di­tional tran­scrip­tion­ists. With 15 sheep and a thou­sand tran­scrip­tion­ist-years, the book-maker will hap­pily pay for more sheep, but will offer roughly zero for more tran­scrip­tion­ists.

Quick re­cap:

  • abun­dant re­source = slack con­straint = ex­tra in­put won’t pro­duce much/​any ex­tra out­put ⇔ pro­ducer won’t pay for more in­put = low shadow price of input

  • scarce re­source = taut con­straint = ex­tra in­put will pro­duce ex­tra out­put ⇔ pro­ducer will pay for more in­put = high shadow price of input

If you want to see the math, I highly recom­mend Stephen Boyd’s lec­tures & book on con­vex op­ti­miza­tion.

So what does all this tell us about tech­nol­ogy changes?

Well, new tech­nol­ogy re­moves some con­straints and re­places them with new con­straints. If the old con­straint is slack, then this doesn’t do any good. If we already have a mil­lion tran­scrip­tion­ist-hours available, and only 15 sheep, then we have no use for a print­ing press. Con­sider Pi Cheng: he in­tro­duced a mov­able-type print­ing press in China around 1045, but it mostly failed to catch on. Why?

Here’s one hy­poth­e­sis: across the board, in many differ­ent in­dus­tries, we see me­dieval/​re­nais­sance China us­ing la­bor in places where Europe used ma­chines. That sug­gests that la­bor, in gen­eral, was read­ily available in China—those con­straints were gen­er­ally slack. La­bor in China had a very low shadow price, com­pared to the shadow price of ma­chin­ery (i.e. cap­i­tal goods).

How could we test that hy­poth­e­sis?

Eco­nomic the­ory pro­vides var­i­ous con­di­tions un­der which pro­duc­ers’ shadow prices are (roughly) equal to mar­ket prices. The sim­plest such con­di­tion is com­pe­ti­tion, but that as­sump­tion usu­ally de­grades grace­fully: even if com­pe­ti­tion is less-than-perfect, mar­ket prices will still usu­ally ap­prox­i­mate shadow prices, with the ap­prox­i­ma­tion im­prov­ing as com­pe­ti­tion in­creases. So one rough mea­sure of a shadow price is the mar­ket price. (Even when the com­pe­ti­tion as­sump­tion fails com­pletely, we can probe the shadow price in other ways—e.g. by look­ing di­rectly at pro­duc­ers’ records, or by look­ing at how hard pro­duc­ers try to ob­tain var­i­ous in­puts.)

If com­pe­ti­tion among book-mak­ers is even re­motely rea­son­able as an ap­prox­i­ma­tion, then the book-mak­ers’ shadow prices will be close to mar­ket prices of the rele­vant goods. So, we could (very roughly) test our hy­poth­e­sis by com­par­ing the price of la­bor rel­a­tive to cap­i­tal in me­dieval/​re­nais­sance China to the price of la­bor rel­a­tive to cap­i­tal in me­dieval/​re­nais­sance Europe. Our hy­poth­e­sis pre­dicts that la­bor was much cheaper rel­a­tive to cap­i­tal in China.

Gen­er­al­iza­tion & Gears

Of course, there are many other pos­si­ble hy­pothe­ses about why mov­able-type print­ing didn’t catch on in China. A similar ap­proach would ap­ply to other pos­si­ble hy­pothe­ses about the adop­tion of the press.

For in­stance, in Europe at least the re­place­ment of parch­ment with pa­per is of­ten cited as a key fac­tor, sug­gest­ing that be­fore the press was adopted, the parch­ment con­straint was much more taut than the tran­scrip­tion con­straint—i.e. parch­ment was a much larger share of the book’s price than tran­scrip­tion. Only af­ter the parch­ment con­straint was re­laxed did the tran­scrip­tion con­straint be­come more taut, at which point the press caught on.

Note that, in both the cap­i­tal/​la­bor hy­poth­e­sis and the pa­per hy­poth­e­sis, we don’t have a root cause. If prices were differ­ent, then some up­stream fac­tor must have caused the price differ­ence. Rather, con­straints/​slack­ness/​prices are gears in our model of the world: each con­straint/​price pair is a gear, which can me­di­ate the causal in­fluence of a wide va­ri­ety of in­ter­ven­tions/​root causes.

Th­ese gears can in­ter­act with each other—the “out­put” in one con­straint may be an “in­put” in an­other. For in­stance, yet an­other hy­poth­e­sis for China’s non-adop­tion of mov­able-type print­ing is a rel­a­tive lack of liter­acy in China; Europe had a much larger mar­ket for books. At an eco­nomic level, the sup­ply of books was it­self a slack con­straint in China—peo­ple weren’t will­ing to pay much for more books. This con­straint would in­ter­act with both the cap­i­tal con­straint and the pa­per con­straint—e.g. if few peo­ple read books, then there would be lit­tle de­mand for more scal­able tech­nolo­gies like print­ing and pa­per. Book price/​con­straint would be a sep­a­rate gear in the model, alongside the cap­i­tal, la­bor, and pa­per prices/​con­straints.

Summary

In gen­eral, we can rea­son about the adop­tion and im­pact of new tech­nol­ogy by look­ing at prices as­so­ci­ated with con­straints. If a tech­nol­ogy re­laxes a slack con­straint, then it likely won’t be adopted at all, and won’t have much im­pact on to­tal out­put even if it is adopted. On the other hand, tech­nol­ogy which re­laxes a taut con­straint likely will be adopted, and have a large im­pact on to­tal out­put—as­sum­ing that the tech­nol­ogy doesn’t in­tro­duce an even more re­stric­tive con­straint! (Con­versely, though, gen­er­al­ized effi­cient mar­kets says that new tech­nol­ogy which re­laxes a taut con­straint will be harder to dis­cover in the first place—there was already an in­cen­tive to pick the low-hang­ing fruit.)