Are Technology and Learning joint inputs?

[More musings. I started writing this post arguing that they are joint inputs. Now I'm not so sure. So I changed the title into a question.]

We need both land and labour to produce wheat, but land and labour are not "joint inputs", because we can combine them in variable proportions. The marginal product of land is the extra wheat per extra land, holding labour constant; the marginal product of labour is the extra wheat per extra worker, holding land constant. Both are well-defined under variable proportions. If we are combining land and labour optimally (to minimise costs of producing a given amount of wheat), the envelope theorem (IIRC) tells us that the marginal cost of an extra ton of wheat will be the same whether we produce that extra ton by: adding extra land to labour; adding extra labour to land; or adding both extra land and extra labour.

But suppose we could only produce wheat by combining labour and land in fixed proportions: one acre of land plus one (full-time) worker produce one ton of wheat. Two workers and one acre produce one ton; two acres and one worker produce one ton. The marginal product of land is either: zero (if there is more land than labour); one (if there is less land than labour); or undefined (if there is equal quantities of land and labour). The same for the marginal product of labour. Land and labour would then be "joint inputs". We could only talk about the marginal product of the composite input land+labour (one acre of land plus one full-time worker have a combined marginal product of one ton of wheat); and the marginal cost of wheat would be the cost of the extra land+labour needed to produce an extra ton of wheat.

The above assumes that all land is identical and all labour is identical. It gets more complicated if there are two types of land and/or two types of labour.

The above was all just by way of introduction, to explain what "joint inputs" means, and why it matters.

Forget about land. Assume labour produces haircuts, but the amount of haircuts produced per worker per year (productivity) depends on that worker's technology (or knowledge).

It (sometimes) takes resources to invent a new technology ("R&D"). That is a fixed cost for the economy as a whole, that is independent of the number of workers that use that technology. And it's a sunk cost because that R&D only needs to be done once for each new technology (provided it's not forgotten).

It (sometimes) takes resources to learn a new technology ("learning"). That is a variable cost for the economy as a whole, that is increasing in the number of workers that learn that new technology. And it's a sunk cost for each individual worker, but not a sunk cost for the economy as a whole, if workers have finite lives, so each new generation must learn it anew.

This post is about the costs of learning, not about the costs of invention. This is empirically important, because most people spend much more time learning existing ideas than inventing new ideas. Assuming costly learning seems to be at least as important as assuming costly R&D.

Just like I assumed identical land and labour above, let's assume that all workers are identical, that each new technology is equally hard to learn, and that each new technology increases productivity by the same amount. (So if learning one new technology increases productivity by one haircut per worker per year, then learning two new technologies increases productivity by two haircuts per worker per year, though all I'm really doing here is defining units for "technology".)

Are new technologies, and learning those new technologies, joint inputs?

It seems to me that they are joint inputs. And if they are joint inputs, that can only be combined in fixed proportions, we are forced to define a composite input, and talk about the marginal product of that composite input.

What is the marginal product of adding a new technology, holding time spent learning constant? That is only well-defined if it can be learned in zero time. Otherwise, the only way workers could learn a new technology is if they either increase total learning time, or else leave some other technology unlearned. But if all technologies and workers are identical, in the sense defined above, leaving some other technology unlearned would mean zero net increase in productivity. (Just like if labour and land are joint inputs, then employing one extra acre of new land, holding employment of labour constant, simply means leaving another acre of identical land unemployed, with no change in the total output of wheat).

But maybe I'm wrong about technology and learning being joint inputs. Maybe they are complements, but not strict complements, so we can have variable proportions. The students learn one more idea, in the same total time, but learn each idea less intensively. (Just like the same number of farmers cultivate an additional acre of land, but farm less intensively.)

We have to recognise learning time as important. If one person invents a new idea it won't just appear in every producer's head by magic, even if the inventor does give them permission to use it.

To make the same point more metaphysically: if one inventor creates an idea, and 100 producers need to use their time to learn that idea, can we talk about the marginal product of one idea, or can we only talk about the marginal product of 100 ideas plus the time spent to learn them?

Or maybe we shouldn't be too persnickety about this; maybe it makes sense to model technology and learning as joint inputs, if it's a close enough approximation to reality.

[This post can be read as it is, or as a belated response to Paul Romer. Sometimes it takes time to think about things.]

Comments

[baconbacon]

Does it change anything to think of the learning and production function together? The first haircut a barber makes takes him an hour, but also teaches him how to cut hair better, so the second takes a little less than an hour, and the third a little less than that.

[Nick Rowe]

bacon: Hmmm, yes I think it does change things. Because we now have joint products (joint outputs) as well as joint inputs. The apprentice's labour is producing both haircuts and learning. Which complicates things even more. So I will ignore it ;) .

[Majromax]

> What is the marginal product of adding a new technology, holding time spent learning constant? That is only well-defined if it can be learned in zero time. Otherwise, the only way workers could learn a new technology is if they either increase total learning time, or else leave some other technology unlearned. But if all technologies and workers are identical, in the sense defined above, leaving some other technology unlearned would mean zero net increase in productivity.

I don't think this quite passes the common-sense test for what we think of as a "technology."

This does hold for fashion, where learning one style means not learning another, functionally equivalent style. But we tend to consider things new "technologies" only if they represent an improvement over existing technology.

I don't need a new hammer. So if someone invents a hammer that behaves exactly like existing hammers but has a different interface, I won't spend the time to learn it. New students can learn either the new hammer or the old hammer equally, but gain no (direct) benefit to learning both.

On the other hand, having a calculator means that I do not need to learn long division. Having Excel means I don't need to be familiar with amortization tables. Having Wolfram Alpha means I don't need to know where to look things up in a big book of integral tables, or special functions. (Or, having once learned them, I no longer need to pay the "depreciation" to keep the skill useful.)

This separation also allows us to delinate "skilled" versus "unskilled" work. Skilled professions are those where the marginal product of learning (given a fixed set of available technology and an a priori-defined fixed amount of baseline learning) is significant; unskilled professions are those where the marginal product of learning is insignificant.

[Nick Rowe]

Majro: yes. I've assumed that "more technology" means more productivity as well as more learning. But sometimes we can have a "better technology" that replaces an old technology, that either raises productivity for the same amount of learning, or else reduces learning for the same productivity.

Historically though, at the macro level, we do tend to see people needing to stay longer at school. The marginal returns to education still look pretty good I think, despite a lot more people being a lot more educated. Which suggests technology and learning have been complements.

[Jeff]

Considered as a response to Romer, I don't think this particular distinction matters, only the fact that both technology (R&D) and learning have monetary costs which cannot be paid their marginal costs as production inputs (joint or not) at the same time that workers are also paid their marginal products.

[Oliver]

Would you not have to put the incidence of new ideas in relation to lifespans? It would seem to make a difference whether it's the same person that has to relearn stuff from scratch every couple of years or whether it's the next generation that learns things in a new way. A bit like: Science advances one funeral at a time. learning =/= relearning because the latter includes unlearning.
I guess that violates your assumption that all labour is identical. Just a thought...

[Nick Rowe]

Jeff: it's not about whether the economy is competitive, or whether factors are paid their marginal products. I already dealt with that, in my first post.

[Nick Rowe]

Oliver: if people had infinite lives, then the cost of learning new ideas would be a once-and-for-all sunk cost at the macro level, as well as at the individual level. If learning one idea would make it *harder* to learn a second idea (that contradicts the first)....hmm...I think that would create an additional sunk cost, by increasing the learning time for new ideas.

[Majromax]

> Historically though, at the macro level, we do tend to see people needing to stay longer at school. The marginal returns to education still look pretty good I think, despite a lot more people being a lot more educated. Which suggests technology and learning have been complements.

Marginal returns to education in which professions?

The education premium is high, but this comes through enabling a switch to a different, high-paid, more productive occupation. With a fixed product, both learning and technology (and those together) could face sharply diminishing marginal returns. This seems to be most the case with what we consider "unskilled" work such as being a retail clerk, where ordinary high-school graduates are already overqualified in terms of general knowledge and the set of job-specific technology to learn is small.

At the economy-wide level, the returns to education are interesting because we can only push education so far without running into human-lifespan limitations, unless we see productivity increases in learning.

But to answer your metaphysical question:

> if one inventor creates an idea, and 100 producers need to use their time to learn that idea, can we talk about the marginal product of one idea, or can we only talk about the marginal product of 100 ideas plus the time spent to learn them?

I think we can talk about the marginal product of one idea as a substitute for another. That marginal product could be positive (if it is an improvement over the least-improving learned idea), zero, or even negative. To pick upon carpentry, since I can't even consistently hammer nails straight it would represent a loss of productivity if I had learned dovetailing instead.

With [one technology plus one learning] as a joint input, the marginal product will be nonnegative.

With one learning by itself, the marginal product will also be nonnegative, but it will be zero if there are no remaining unlearned technologies that complement the existing learned stock. It will always be positive if there are some technologies that can be continually mastered (with diminishing individual return) as a skill rather than as a binary "yes/no."

This division is also where newly-trained and experienced workers differ. From the perspective of an individual, tabula-rasa new worker, they can only consider the joint input of technology+learning, since they have no existing set of mastered technologies. Experienced workers face different tradeoffs, since newly-learned technologies may either complement or substitute existing ones. It's rational for a teenager to get a smartphone, but it may not be rational for that teenager's parents or grandparents.

[Majromax]

> If learning one idea would make it *harder* to learn a second idea (that contradicts the first)....hmm...I think that would create an additional sunk cost, by increasing the learning time for new ideas.

Is the field of new ideas constrained or infinite?

If ideas have joint learning benefits or penalties, then a wide enough expanse of ideas could cause the population to spontaneously develop into cliques.

In real life, I don't think we see much in the way of real un-learning penalties, but we do see non-monotonic returns to scale. The first few steps of learning to be a carpenter or economist are relatively difficult to learn and provide no real productivity gain, but the intermediate steps are tremendously productive until we finally reach diminishing returns.

The macro effect, I think, would be to promote specialization.

[Roger Sparks]

I think changes in learning and technology are components of PARADIGM changes.

To follow your land-wheat example, the move from sickle-cradle harvesting technique to combines was a huge technological change but it was much more than a knowledge change that required learning to accomplish. It also required a massive shift in the work habits and work distribution of the economy. To me, this is a paradigm change.

Another example is the discovery of the Maxwell equations that describe the electro-magnetic relationship. After 150 years, this discovery continues to evolve and allow reshaping of the economy. To me, this is paradigm change.

I don't yet see any meaningful relationship between learning and technology. The two terms co-exist. Each term is just a fuzzy description of a nebulous feature of a way of doing things.

[J.V. Dubois]

Just talking about the model you describe I do not see why inventing and learning have to be join inputs in the sense you described with labor and land. You definitely can have idea that is learned by 1 person (inventor) as well as an idea that is learned by everyone.

Imagine that we have ideas very similar to land. Each new idea improves upon the old idea in the same way each new unit of land improves upon total land area. So starting with one idea (one acre of land) adding one idea (acre of land) doubles total idea effectivity, another idea increases the total by 50% and so forth. In the same way if we can invest only so many time for learning (agricultural work) by investing 1 unit of time we can only learn 1 idea (work 1 acre of field). If labor and land are not really joint products in your model why should learning and inventing be such.

So in the end you can either have more same labor working more fields (having same people learning more ideas) or you can have two or more people working any particular field (having more people learning the next idea)

Or am I missing something?

[Nick Rowe]

JV: One person (the inventor) creates the new technology. But that new technology has no effect on productivity until workers learn the new technology. And learning takes time.

[J.V. Dubois]

Nick: Ok, I can say that one person (dutch engineer) creates the new land. But that new land has no effect on productivity until workers invest time to work the land. And working the land takes time. Does it mean that land and labor is joint labor? You don't think so at the beginning of the blog as because "we can combine them in variable proportions". Why not combine new ideas and ammount resources invested into these ideas in variable proportions? Actually we do that. Maybe hairdresser can offer better service by being able to recite Vergilius in original (at least to the customers who would enjoy such a thing). But most of them can get by with good hairdressing skills and knowledge of latest gossip.

[notsneaky]

"We could only talk about the marginal product of the composite input land+labour"

Why? (I do understand that there's a single point where MPs are undefined but that's not all that important)

[rsj]

Nick,

My own learning costs are being strained here. I could reply to some points that I take exception to -- such as the joint input assumption. But at the end of the day, I would only add to the noise because I don't know where you are trying to take your readers. Could you first clearly state your ultimate thesis and then work backwards? Is this still about the existence of a competitive equilibrium? Are you going to tie this all together somehow? Knowing that would really help me follow along.

[Nick Rowe]

rjs: "Could you first clearly state your ultimate thesis and then work backwards?"

I don't really have an ultimate thesis from which I'm trying to work backwards. I have a question, which I'm trying to answer, then trying to work forwards from that answer, to see where it leads.

But I'm pretty sure it won't lead to a competitive equilibrium (unless I rig some other assumptions to make it lead there). ***And I have no interest in leading us to competitive equilibrium anyway, because I have spent nearly all of my professional life as a macroeconomist (ever since 1985 IIRC) arguing that the economy is monopolistically competitive (or, that MC is a better macro modelling assumption than PC (except where the difference between MC and PC doesn't matter much for the point you are making, and you assume PC just to keep it simple)).***

God you lefties, always suspecting a PC conspiracy under the bed!

Here is where it *might* lead:

Empirically, we know that both technology and learning are really big and important things, that we probably shouldn't ignore in growth models. Empirically, we have seen both technology and learning increase, over the last century or two. Was it just a coincidence that both increased together? Or are they somehow related, like by being joint inputs?

If we wanted to model growth, with technology and learning, how should we write a simple aggregate production function? Should we write it as:

Y = F(min{Tech,Learning},other labour)

Is "learning" just a part of human capital, and quite distinct from technology? Or do learning and technology go together like right shoes and left shoes? Paul Romer said I was conflating technology and human capital. I think that technology and learning (a big part of human capital) might be a composite commodity, in which case it is right to treat them as a package.

A (strict) "non rival" good is one where the Marginal Cost of making an extra copy of the computer program is zero (or as close to zero that it can be ignored). But if we are talking about ideas and humans, the MC of an extra copy of the idea is the cost of teaching-and-learning that extra human, so the idea gets downloaded into his head. And that cost is not (usually) trivial.

notsneaky: how do wages W and land rents R get determined? Normally, we draw a downward-slopng MPL curve, an upward-sloping labour supply curve, and W gets determined where the 2 curves cross. Then do the same for land. But if labour and land are joint products, that won't work, because there's a great big vertical discontinuity in the MP curves. Instead we draw the MP curve for the composite input, draw the supply curve for each input, add up the two supply curves vertically to get the supply curve of the composite input, determine W+R where those two curves cross, then drop a line down to each of the two supply curves to determine W and R separately.

[Frances Woolley]

Nick - "draw the supply curve for each input" - but if one person owns all the land....?

[Nick Rowe]

Frances: we would still need to do what I said, but then we would need to draw the MR curve for the input, as well as the MP curve, to find the monopolist's equilibrium rent.

[Nick Rowe]

JV: here are two production functions (Y is wheat, L is labour, N is land):

1. Y = L^0.5 N^0.5

2. Y = min{L,N}

In both cases you need both L and N to produce any Y. But 1 has variable proportions, and 2 has fixed proportions. Take the derivative dY/dN to see what I said about marginal products.

[Roger Sparks]

Try this:

Define "technology" (T) to mean the off-site resources that an economy can bring to influence the production of any product. This would include items such as fertilizer, pesticides, irrigation, improved product storage, and improved transportation of the product. The need for "learning" would be included some place in these infrastructure improvements.

Then your formula could be modified with a term for technology (T) such as

1A. Y = L^0.5 N^0.5 T^xx

where the factor xx is the exponent empirically determined to balance the equation compared to output with no technology at all.

[Nick Rowe]

Roger: there's a lot of natural resources in fertiliser.

[Jeff]

There is definitely some conflating/equivocating going on here. Workers don't have to "learn the technology". Using new technology in production generally means workers need *less* knowledge overall. Example: computers. Workers don't need to know binary arithmetic, semiconductor science, nor computing languages. All they need to know is how to *operate* the new technology, and they can forget (or never need learn) all the skills the technology replaces (hand computation etc.). Once I have calculators at work, I no longer need to learn arithmetic, nor do I need "knows arithmetic" as a hiring requirement, and I certainly don't need to learn the internal "technology" of the calculator. The extreme example I suppose would be robots (complete automation), where the "worker learning" part is eliminated entirely (because the worker himself or herself is eliminated). Right?

[Jeff]

Anticipating the attempted fix/objection "ok just take 'learning' as 'learning to use' then", the conflation/equivocation is exposed here in the post: "What is the marginal product of adding a new technology, holding time spent learning constant? That is only well-defined if it can be learned in zero time." If the user (worker) interface to a new technology is identical to old already-known technology, then "learning to use" *is* accomplished in zero time. The fact that new technology *can* (sometimes) be introduced "under the covers" like that is an important rebuttal (to the assumptions in the post), no?

[Jeff]

Can't believe I didn't give this highly apropos example (of identical interface to highly different technologies), given that I've got three physical ones right in front of me, and that I'm right now typing on a simulated one on an iPad: QWERTY keyboard!

[notsneaky]

"notsneaky: how do wages W and land rents R get determined?"

Institutionally of course. Whatever that means; markets, politics, bargaining. But that has nothing to do with whether or not marginal products for individual factors are well defined (except at that single point which doesn't matter all that much). A "marginal product" is a purely technological concept which is independent of things called "W" and "R". You give me a production function and I can tell you all about marginal products. You don't have to tell me anything about the social organization of this economy.

This is the same argument that the Joan Robinson folks try to make against the marginal theory of distribution. Marginal products are not well defined at a point point. Presto! Wages and rents cannot be determined by markets! Of course this is wrong - that little point doesn't matter (assume that both labor and land come in arbitrarily small but non-infinitesimal chunks, problem solved. Or another way to put it, "marginal" and "infinitesimal" are not the same thing). The actual problem (and this is what "sophisticated" "Joan-Robinson-folks" recognize(d)) is that with this kind of production function the marginal product may be zero. Not undefined. Zero. And since we don't observe zero wages in the real world either of two things are true:

1) This production function is bunk and not a very useful way of thinking about real life economies (my view (asterisk))
2) Wages and rents are actually determined through non-market processes (what the argument wants to argue)

But again, this has nothing to do with marginal products being defined at one particular point (second asterisk)

(asterisk - for some purposes, like development models which deal with structural transformation in the context of surplus agricultural labor it's not that big of a deal to assume this kind of production function and assume that agricultural wages get determined by average product rather than marginal product. But that's because the focus of these models is somewhere else and it doesn't really matter)

(second asterisk - I guess if you're thinking of something like the Harrod growth model, then the economy could wind up at this "knife's edge" case via capital accumulation and that one point could matter. But again, that's only if you think that's a good way of thinking about growth to begin with...)

[notsneaky]

In the context of your model, as long as there's always some ideas which haven't been learned yet, the production function is just Y=AL, no? And that certainly has a well defined marginal product of labor which one can equate to the wage rate if one wants to.

[rsj]

Nick,

I don't think you should include learning in this discussion. Technology makes you more productive given any other input as fixed, including learning. Forget learning, we are talking about algorithms, drug patents, music and software.

But if we are going to be all free form, I was thinking about video game evolution recently.

The best video game of 1978 was Space Invaders. It earned $14 Billion (2015$) over the next 4 years. 14 Billion!

Sample of the game: https://www.youtube.com/watch?v=437Ld_rKM2s

Tomohiro Nishikado spent a year writing the game. He wrote the software, created the graphics, and composed the music.

Design goals:
"Nishikado was inspired to create Space Invaders by an early Taito electro-mechanical game called Space Monsters, and by the Atari arcade game Breakout. The game was planned to have tanks, planes, and battleships as enemies, but Nishikado was not happy with their onscreen movements. He considered making the enemies human (which would have been easier to animate), but scrapped the idea since he thought it would be immoral for players to shoot them. After he saw a magazine featuring the 1977 film Star Wars, he decided to use a space theme, and based his alien enemies on the squid-like antagonists from the 1953 movie version of The War of the Worlds."
(https://en.wikipedia.org/wiki/Space_Invaders)

35 years later, the best video game was Grand Theft Auto 5.

A core team of 360 people worked with a larger team of 1000 for over 5 years. Workers included musicians, visual artists, voice actors, art directors, writers. Development budget of about $200 million. It earned $2 Billion (2015 $) over the next year.

Sample: https://www.youtube.com/watch?v=ehEc4C6bleY

Design Goals:
"Grand Theft Auto V‍'​s central story theme is the "pursuit of the almighty dollar". Mission content is structured around the lead characters' efforts to plan and execute complicated heists to accrue wealth for themselves. The team decided to focus on money as the game's central theme in response to the 2007–08 financial crisis; the effects of the crisis on the main characters are the catalyst for them to conduct heist missions. Houser explained, "We wanted this post-crash feeling, because it works thematically in this game about bank robbers"." (https://en.wikipedia.org/wiki/Development_of_Grand_Theft_Auto_V#Design_goals)

[Nick Rowe]

notsneaky: "The actual problem (and this is what "sophisticated" "Joan-Robinson-folks" recognize(d)) is that with this kind of production function the marginal product may be zero. Not undefined. Zero."

No. It is (almost) never true that MP *determines* W, even in a standard competitive model. It is the MP *curve*, and the supply curve, that together determine MP, W, and L. But with joint inputs, the MP curve has a weird shape. It is horizontal at some positive number, then vertical, then horizontal at zero.

rsj: Space Invaders is (almost) the last video game I ever played. If I did want to play GTAV, I wouldn't know where to begin. Where do I buy it, and what equipment do I need, and how do I set it up, and how do I play it? Probably I would need to ask some kids to teach me all these things.

My mother (very old) can use a keyboard much better than me (because she learned it at secretarial school), but she never learned how to use a computer to do email.

My computer learned Excel extremely quickly, but I learn it very slowly. I use it once a year, and every year I have to relearn how to do do the couple of trivial things I need to do with it. And it's very frustrating and costly for me. And every couple of years they "improve" Excel, which only increases my learning costs.

My new computer learned the new improved version of Firefox extremely quickly, but I didn't. Now I am unable to learn how to add a link to my bookmarks. There is something called "save to pocket", and I have no idea what it is, and whether it's the same as a bookmark.

Young people tell me there are these things called "apps". Same story. I am still learning how to do the basics on the (very simple and old-fashioned) mobile phone I bought last year.

OK, so the costs of invention are large, and rising over time. But I can't forget learning. Those costs are large too.

[rsj]

The costs of many things change as you get older -- from learning a new language to learning a new technology. But we don't say that languages are getting harder to learn over time because we know that a change happening in us is not the same as a change happening to the population as a whole. There's a word for that type of projection, but now I can't remember it!

This was technology in the 1950s:

https://www.youtube.com/watch?v=bOwKh9XLO1c

You should give GTA5 a try! Or, just watch some of the trailers. This is the art form of the current age.

[Jeff]

Psychologist's fallacy.

wikipedia.org/wiki/Psychologist%27s_fallacy

[Best Beta Trading]

It might be industry specific. For farming, I learned from the Prairie Farm Report that all farmers are essentially inventors. Much of their equipment is custom-made or alterations of Deere and Buhler equipment. I suppose for the mass market, the technology is done by companies who hope enough customer will learn it at the same time. John Deere switched over to 3rd world markets and much of recent farming innovation is mapping sensor technologies that are ported from other sectors.
I guess if your "producer market" is homogenous, it will be a joint input. If it a heterogenous industry, not joint. Canada has diverse landforms, distances from port, silo and rail infrastructures, and varying markets. The availability of land is uniquely alterable due to lots of marginal land and lots of potash. But a country like Singapore might procure farm implements en bloc. Every farmer needs modern satellite/drone/air imagery, but not every farmer needs certain GMO strains or electric vehicles...
Maybe try other industries as examples? All banks went to ATMs at the same time I assume. All warehousing logistics actors used just in time computer inventory systems and RFID tags at around the same time in this globalized economy.

[Best Beta Trading]

...I'll troll a comment that private bonds might have an application for counter-cyclical policy. In a boom they can be the source of public funding for programmes that will create jobs during a bust. I assume more boom gvmt revenue eclipses the higher bond interest rates. Ideally you use them to fund training for stimulus infrastructure programmes like power metering retrofits. If you can time the business cycle.

[notsneaky]

"It is the MP *curve*, and the supply curve, that together determine MP, W, and L."

Of course. But then, what does your supply curve look like? In fact, what does the supply curve look like in the "standard competitive model"? At any point in time, K (whether you're calling this "capital" or "land" or "ideas") and L are given. Hence your supply curve is vertical. Given a vertical supply curve, it *is* the MP curve which determines W (L is exogenous). You want a non-vertical supply curve? Then you need to talk about leisure or accumulation or something else. That's not in here.

"But with joint inputs, the MP curve has a weird shape. It is horizontal at some positive number, then vertical, then horizontal at zero."

Yes. I understand this. Basic Leontief production stuff. But the conclusion that "It is (almost) never true that MP *determines* W" does not follow from this. In fact it is exactly the opposite - "It is (almost) *always* true that MP determines W". It doesn't at one particular point. How did "almost always" turn into "almost never"?

So say MP has a weird shape. So what? It's still weakly downward sloping. The supply curve either intersects it at the positive portion - MP is well defined and so is W. This is the Y=AL case where W=A and all output goes to labor (and hence ideas HAVE TO be non-excludable, which is what Romer has been saying). Or it intersect it at the "horizontal at zero" portion in which case MP is zero. 0 is still a well defined number. At that point you have to ask yourself whether this is a useful way of thinking about the world, or, alternatively argue that even when the MP is zero workers get paid positive wages for "institutional" reasons.

Or, by some miracle, the supply curve just happens to intersect/overlap the MP curve at that vertical portion. It could happen. And then indeed MP (and W) would be indeterminate. But like I've said above 1) you have to make an argument as to why exactly the economy's exogenous supply curve will wind up exactly at that one single funny point (Harrod did, for capital, although that involves a lot of other sketchy assumptions), 2) it's trivial to get rid of that one funny miracle point simply by assuming some indivisibility in factors.

(the actual problem - as I've said above - is that the vertical supply curve may intersect the MP curve at a point where MP is zero and since we don't observe zero wages this means that either the model is bunk or marginal products play no role in factor renumeration. But that's a different argument)

"It is (almost) never true that MP *determines* W, even in a standard competitive model"

Unless you're going to talk about leisure or time preference, it is actually true that MP determines W at a particular point in time. Given K and L, MP determines W. Now, if we allow for accumulation, labor/leisure choice and all that, maybe even Malthusian population growth, then (steady state) K and L are determined by some other things. Time preferences or whatever parameter determines how we split up labor and leisure. Evolution. In *that* sense W is not determined by MP, because K and L shift around until they get to where they need to be. At that point you're talking about an intersection of a vertical curve and a horizontal curve, but it's all parameters.

[Nick Rowe]

notsneaky: "Or, by some miracle, the supply curve just happens to intersect/overlap the MP curve at that vertical portion. It could happen."

Profit-maximisation says it will happen. If Y=min{L,N}, why would farmers ever hire more land than labour, or more labour than land? They wouldn't. Unless one or the other is a free good. Even if the supply of land is vertical, the supply of labour won't be (which doesn't mean labour supply is horizontal).

[notsneaky]

No, no, no. The supply of labor is what it is. Unless you want to talk labor/leisure. Otherwise it's just a vertical line which may or may not intersect the MP curve at a particular point.

Read your last sentence again - "even if .... *the supply of labor won't be (vertical)*." Why exactly? If I say the labor supply is inelastic then it's inelastic (again, other than labor/leisure, or Malthus). You are saying that BECAUSE the DEMAND for labor is funky (and let's say it is), the SUPPLY of labor cannot be something. As if the shape of the labor demand curve determined the shape of the labor supply curve. But these are two independent curves. They may not cross. They may cross at a weird point. They may cross multiple times. But one doesn't change the shape of the other.

[notsneaky]

Or let's do equilibrium. Say there's a bunch of firms (yeoman farmers) with production functions Y=Min{L,N}, where N is land per firm and L is workers hired per farmer. N is given (vertical land supply). There's only one (homogenous) good so it's only about how output gets split up and we don't worry about output prices. The payment to workers that each land owner has to make is w. And they're wage-takers so w is given. The profit function of each one, as a function of L is then a triangle with a peak at L=N. This is what you're talking about above.

Suppose that there is 10 farmers and each one has 1 unit of land (so Y=Min{L,1}). Hence if w<1 each farmer wants to hire 1 worker. For total labor demand of 10. But let's say there's only 5 workers in the economy. What happens? Well, we could say that each farmer gets 1/2 worker. But at w<1, each farmer has the incentive to offer a slightly higher wage to get 1 worker (and say "screw you" to any other ones that wish to come and work for them - yes, that's a violation of perfect competition but that's how these models work and that's where you have to start talking about rationing rules and all that other fun stuff). So wage will get bid up until w=1.

Of course if w>1 nobody wants to hire any workers. So any individual worker will have an incentive to lower their wage offer, etc., until w=1.

So with L<10, we have w=1, zero profits, and wages determined by marginal product. No indeterminacy, simply because... L<10, and 10 is where that funny point happens.

What if L>10? Say 20. Say w=1. Each farmer hires 1 worker and there's unemployment of 10 workers. Any unemployed workers has the incentive to offer their labor for 0

Of course, like I said, if your model brings you to this point, you should probably start thinking about a different model, since, Marx not withstanding, we don't actually observe 0 wage economies. Either assume a different production function or assume different concept of wage/factor price determination.

Or scratch all of the above and just assume that labor has to be hired in little indivisible chunks of 1/16th. That'll solve the problem too. In other words, this is just obsessing over an essentially meaningless, unrealistic, and local mathematical property.

[notsneaky]

There's a part in the above that got cut in the editing (my computer does this weird jumping around thing with the cursor):

"Any unemployed workers has the incentive to offer their labor for ... less, until wages reach 0. At that point farmers are willing to hire all 20 workers. Labor is free (and land is not)."

[Best Beta Trading]

You might not be able to get 1/2 a farmer but 1/2 a farm labourer is easy in many areas of Canada. One of the two planters we picked up in Garland was the guy who dumped poop outside the tent of the arrogant guy. I was the one who hit the touchy feely guy in the eye with a football. I wish I hadn't because the supervisor stole him Mom's cookies like in the Simpsons...
Medical equipment is a better industry example. It clearly is not a joint input. Doctors diagnose based upon what technologies they are familiar with. Some use more time to read cutting edge journals, some don't. MLB fan told me Zinn dreamt of a utopia he had no idea how to realize. He is missing how to weight the various gvmts he identifies against eachother, and identifying the transaction costs (often war) to realize the better gvmt/human-capital models...
I mention this because there are too many diagnostic tests on both sides of the border. But at the same time you'd want MRIs/EEGs to identify age related diseases in leaders, and perhaps fewer breast cancer tests for paranoid patients not respecting the division of labour.
A better question is what research do you want, not how to maximize the uptake of research. You want lots of brain imaging available even at high costs/person even if the costs won't come down, but not necessarily for the general industry. And the USA does not respect the Hippocratic Oath as much; they get paid for overdiagnosis while our doctors get power to compensate for being better people getting paid less. Lobbying against medical coverage makes their citizens inhumane...
So maybe you need to identify what % of a producer/provider market you'd want to have a service at a given price, and how much early adaption lowers this price? The latter questions notes that the joint input ratio is not fixed and varies among industries. If you fund networking technologies it will drop their price. So if you define them as not joint input, they won't be, if you define them as joint input, they will be.
A solution is to model what technologies you want to exist, who you want to have these technologies at a few different cost levels on the curve, how joint input (the ratio) their industries are, and the price depreciation caused by early adaption; how this lowers the cost level. Then you can raise taxes on those who got wealth/income in old/inferior paradigms (we didn't know about AGW or financial instrument meltdowns in the 80s when high bracket income tax rates were lowering) to fund whatever technologies you want some people to have. You want our soldiers, at least the gentlemanly ones, to have rifles. There are many you don't want to have rifles. In terms of AI, it is seemingly hard to figure out using economy modalities, what to invest in and what not to. Robotics would be the farming relevance. You don't want a hacker turning your tractor against your region's humans.

[Jeff]

In case anyone was following my dialog with David A. on his blog, he simply cut off my reply to his accusation that I was confused (circling the wagons again), so I'll post it here (paraphrased from memory) for consideration:

On the contrary, you've got a model where workers have to pay for their own work tools without any reimbursement (for the tools) from the employer, which means their effective wage is their MP *less* the cost of the tools -- breaking the "competitive equilibrium" requirement. (Or maybe their "human capital" is earning MP, but *they* aren't. Quintessential mathiness.)

Or another way to look at it is that the model treats workers as *both* parts and labor, but only pays for the labor piece.

David thinks he can get around an impossibility proof somehow. He can't of course.