"No model can have a competitive equilibrium with price-taking behavior and partially excludable nonrival goods.
If you are not an economist, this would be a model in which someone who has a monopoly on an idea can charge for its use, but somehow is unable to influence the price that users have to pay, which should sound implausible at least. If you are an economist, you know that there is a very simple argument based on Euler’s theorem that proves this type of model is impossible."
That seems wrong to me. Maybe I don't understand it; or maybe it is wrong.
Here is a very simple growth model:
Initially, one acre of land produces one ton of wheat per year. No labour, all land identical, fixed supply of land, constant returns, perfect competition, no funny stuff.
Then I come up with an idea that lets one acre of land produce two tons of wheat per year. My idea is non-rival (just because one landowner uses my idea doesn't mean another landowner can't use it too). My idea is excludable (I patent my idea, so nobody can use it without my permission). If you like, we can assume my idea is only partially excludable, because my patent only works in Canada, and I can't stop non-Canadians using my idea.
How much can I charge landowners for using my idea? The demand curve for the use of my idea is perfectly elastic at a price of one ton of wheat per acre per year, then goes vertical at the total number of acres in the world (or in Canada, if it's only partially excludable). If I set the price at more than one ton, nobody will use my idea; if I set the price at less than one ton, everyone will use my idea. So I will set the price for using my idea at one ton of wheat per acre per year (or maybe a smidgen less, if you want to be picky).
Then David Andolfatto comes up with a new idea, that is better than mine. David's idea lets you grow three tons of wheat per acre per year. David can also charge one ton of wheat per acre per year. If he charges more than one ton, people will use my idea instead (I won't charge more than one ton, even if David charges more than one ton, because nobody would use my idea if I charged more than one ton); if he charges less than one ton, everyone will use his idea. So David charges one ton (or maybe a smidgen less), and everyone uses David's idea and stops using mine.
[Update: I'm assuming Bertrand competition between me and David (where we each set price taking the other's price as given). If instead we collude, or play Cournot (each setting quantity, taking the other's quantity as given), we can earn higher profits than this.]
Then Glenn MacDonald comes up with an even better idea, that lets you grow four tons per acre.
And so on. Productivity grows at one ton per acre per new idea. The person with the newest idea earns one ton of wheat per year for every acre in the world (or every acre in Canada, if it's only partially excludable).
That still looks like a competitive equilibrium to me. The person with the newest idea faces a demand curve that is perfectly elastic up to the quantity where he captures the whole market. Just like the demand curve facing an individual wheat farmer in a perfectly competitive market for wheat. The individual farmer can set a price for his wheat above the market-clearing price P* if he wants, but he will sell no wheat if he does this; and he can set a price for his wheat below the market-clearing price P* if he wants, and everyone in the whole world will want to buy only his wheat if he does this, but he won't maximise his profits if he does this.
Here (I think) is where Euler's theorem kicks in. Sure, landowners earn the marginal product of land under the second-newest idea, which is less than the marginal product of land under the newest idea. But that's just an integer problem (is that the right word, in math-speak?). In the limit, as ideas get smaller and smaller, or productivity gets larger and larger, that difference matters less an less. It doesn't matter at all if there's a continuous flow of tiny new ideas. And it doesn't make any difference to the efficiency of equilibrium anyway, if land is in perfectly inelastic supply (Henry George and all that). And it doesn't affect what I said about the demand curve for the newest idea being perfectly elastic.
[Update: see jonathan's comments below.]
What is logically wrong with my counterexample to what Paul Romer said?
I'm crap at math (it took me some time before I vaguely remembered reading about Euler's theorem in Mark Blaug's book on history of thought). I don't do growth theory (unless I have a co-author to help). I drove across Minnesota once (Typepad says that's how you spell it). But me and Minnesota economics … have issues.
Maybe I'm just signalling my membership of the Western club?
No. I'm pissed because somebody said something (in words) on the internet that seems to me to be wrong, and backed it up with mathy theory that I don't understand. And either I'm wrong, and I learn something new when people explain it to me, or I'm right, and I further my personal agenda of seeking fame and fortune on the internet.
Worthwhile Canadian Initiative 
Perhaps I’m misunderstanding, but I’m not sure why a landowner would earn any more than 1 ton of wheat per year (assuming there is no labor input), regardless of how many new ideas there are. How could a landowner not pay for use of the second newest idea? Can he use this patent for free just because it’s not the latest one?
are you and Romer using the same definition of competitive equilibrium?
Did anybody else read the related post on Volrath’s (excellent) blog? I need somebody to put me out of my misery, when it comes to idea that any aggregate production function must be CRS in rival inputs, because of replication arguments. It seems to me that if you allow IRS in rival inputs at the firm level, then what happens at the aggregate depends on whether you are giving each firm more inputs or whether you are duplicating firms without changing their scale. I am sure I have missed the point, can anybody help?
FYI, the blog referenced by Romer here:
makes a very clear and easy to understand argument with almost no math
ah rsj you look like the person to help me
Put your model in continuous time. Suppose there is a unit interval of land, with productivity A(t) (using state of the art tech). At time t there is an invention that raises productivity to A(t) + dA, and this tech stays best for a time interval dt, when it is supplanted by a better invention.
Then the land-owners earn A(t) for time interval dt, while the inventor earns dA for this time interval. Thus total income to land-owners in this interval is A(t)dt, and total income of inventors is dAdt. Tech growth rate g = (dA/dt)/A is constant.
As you point out, for dt large this is not a competitive equilibrium, because land-owners are not earning their marginal product.
What happens if we take the limit as dt -> 0? Now we have continuous innovation. But the problem is that the innovators have zero income in the limit!
To see this, note that total income of land-owners is the integral of A(t)dt over time, which integrates to a positive quantity (average A). The total income of innovators, though, is the integral of dA(t)dt over time. This integrates to zero!
This is because dAdt is the product of TWO things that are converging to zero. i.e. you can write it as:
dAdt = (dA/dt) dt^2 = gA (dt)^2
where g is the constant growth rate.
So the limiting economy doesn’t violate Romer’s comment. The process converges to a competitive equilibrium with no profits to inventors.
Luis,
The assumption is that you have an aggregate production function in two inputs, F(P, X). X is the rival goods (land, people), and P is the excludable non-rival goods (patents).
Now duplicate the earth, by putting an exact copy of the old earth next to it. The production function aggregates across both earths, doubling the rival inputs but leaving the patents the same. But the output of the two duplicate earths is the 2the output of the single earth.
F(P, 2X) = 2F(P, X).
So the production function is CRS in X but must be increasing returns to scale overall. Because the production is constant returns to scale, you can take the partial with respect to X and, by euler’s theorem, get
F(P,X) = partialF_xX + “profits”, where profits is just the residual (but is positive).
Therefore there doesn’t exist a competitive equilibrium.
In Nick’s argument, I take it that the “mathiness” he is employing is to assume, for some reason, that all patents except the newest one are free (receive no profits), and then assume that the latest patent receives a payment “close” to the other patents (close to zero), because it’s innovation is very small.
rsj:
The new idea fully supplants the old one.
A farmer has the choice of using old tech with productivity A, or new tech with productivity A + dA. Then if the new tech is priced at any level above dA, say dA + eps, the inventor of the old tech can undercut it by offering a price in the interval (0,eps). This is worthwhile because the idea is already invented, and it doesn’t cost anything to sell someone the right to use the idea.
The highest price that won’t be undercut in this way is dA, so this is the equilibrium price.
Of course, we can look around and see different technologies and observe that the second best technology is not free, it is worth whatever gain is obtained from adopting that technology over just using land. The landowner, faced with the choice of technology A or B or C, that each allow him to earn an extra 2, 3, or 4 tons of wheat, is willing to pay up to an extra 2, 3, or 4 tons of wheat for using each technology. That’s the indifference point of the landowner when deciding which technology to use. Just because a technology isn’t the best doesn’t mean that it’s worthless and available to be used for free.
Jonathan,
It does not. See above. You are playing fast and loose with differentials here — translate the problem to integers. Or, if you want differentials, tech A gives an advantage of dA, and that is the price. Tech B gives an advantage of dB and that is the price. Whether dA > DB is beside the point. Saying that, in the limit, the differential is “zero” and therefore the technology is free is the height of mathiness.
Here’s another way of thinking about why the price of new tech is the difference in productivity from the old tech, which hopefully will clarify things.
Consider a 3-person game. The players are farmer, inventor1, and inventor2.
The farmer’s choice is who to buy from. The inventors choices are to set a price for their technology.
Then the Nash equilibrium is for the inventor with the less productive tech to set P = 0, and the more productive inventor to set a price equal to the difference in productivity. Any choice by the farmer is a NE, but it makes sense to assume he buys the more productive technology (by a limiting argument).
Why is this the Nash Eqm? Well, clearly no inventor will set a negative price. If the inventor with the better tech sets a lower price, the farmer will buy from him, but he can do better by raising his price. If the inventor with the better tech sets a higher price, the other inventor will undercut him. So this is a NE.
This clearly isn’t competitive. But if we let the productivity difference go to zero (which is what you do when you take the limit as dt -> 0), then it converges to a competitive eqm with zero profits for inventors.
jonathan: OK. But the demand curve facing an inventor is still perfectly elastic, in either discrete or continuous time, which does violate Romer’s comment.
And so the question is: do inventors get paid their marginal product? And as far as I can see, that depends on whether David would have had his 2003 idea (3 tons per acre) if I hadn’t had my 2002 idea (2 tons per acre). If David’s 2003 idea depends on my 2002 idea, and if I can’t charge David for using my idea to develop his own, then David rips off my marginal product. But if David would have come up with his idea anyway, in 2003, then David gets paid his marginal product.
Is that right?
rsj: If David charges 1 ton to use his idea, then if I charge more than zero, everybody will use David’s idea. In Bertrand equilibrium between me and David, David charges 1 ton, and I charge 0. David and I would need to collude, to share the market between us, to get anything better.
Jonathan,
We’re talking about competitive equilibria here, not Nash Equilibria. You can find Nash equilibria with monopolistic competition, which is the point of the example.
But yes, that’s a good point — if we solve this as a Nash equilibrium game, we get a (non-competitive) equilibrium in which only the latest tech is used. If we try to solve this by determining what is the marginal gain by putting a unit of land into production with a given tech, then there is a different price vector. That alone should be a proof that the competitive equilibrium doesn’t exist.
rsj: see my update. We need to distinguish between Bertrand-Nash and Cournot-Nash equilibria.
I think that this can be considered a particular case of a general rule:
– A monopolist is in the same situation as a perfect competitor if: a) the utility of his product is not subjected to decreasing marginal utility (or if the only decrease is from “have utility” to “no utility”); AND b) the product has the same utility (or the same reserve price) to all potential customers.
Miguel: that sounds right to me.
jonathan @10.41 That sounds right to me. You are talking about Bertrand-Nash.
Nick,
I’m not sure how a perfectly elastic demand curve violates Romer’s comment. This still isn’t a competitive eqm, because farmers aren’t earning their marginal product.
As to your question — I think you’re right about why inventors are not paid their marginal product.
Suppose there was some cost of invention, say you pay a cost C(p) for a probability p of an invention that is an advance dA above the existing tech. Then it’s pretty clear that you don’t get the optimality properties of a competitive equilibrium. This is because inventors don’t capture the benefits of advancing the technology frontier and therefore allowing FUTURE innovations to raise productivity still higher.
An inventor WOULD take this into account if there was just one inventor who owned all the patents. Then he wouldn’t let his past patents compete against the current state of the art, and would capture all of the accumulated benefits of tech progress. Thus he would take the future benefits into account when deciding current expenditures on invention.
No model can have a competitive equilibrium with price-taking behavior and partially excludable nonrival goods
Then I come up with an idea that lets one acre of land produce two tons of wheat per year. My idea is non-rival (just because one landowner uses my idea doesn’t mean another landowner can’t use it too). My idea is excludable (I patent my idea, so nobody can use it without my permission). If you like, we can assume my idea is only partially excludable, because my patent only works in Canada, and I can’t stop non-Canadians using my idea.
Maybe silly, but worth a shot: By introducing land you now have a (excludable) rival good?
Nick,
Thinking about it a little more, I think I understand how to fit it into your model.
If inventors were paid their marginal benefit it would look like this:
year 1: Nick earns 1
year 2: David earns 1 AND Nick earns 1
year 3: Glenn earns 1, David earns 1, Nick earns 1
etc. You don’t just capture the marginal benefit of your innovation in the first year, but also the benefit in every subsequent year from the fact that your innovation enabled future progress.
Been a while since I did micro, but price-taking firms make 0 ‘super-normal’ profits, but normal profits still exist as a minimum to justify the firms existence (basically a salary for the owner/manager, we could restrict this to being his marginal product), if we make some assumption about simply being an owner of a firm provides utility in itself (it’s nice not being a wage-slave) then why would this not provide some incentive to innovate & form a new firm? If this is true, then price-taking and at least some degree of innovation is compatible, what am I missing here?
Nick,
Yes, I agree re: Bertrand-Nash equilibria. But I still accuse you of mathiness using games with differentials.
Ask yourself, if you insist on making a limit-based argument, what would the final economy look like as you take the limit of an infinite number of technologies? E.g. Your production function is now F( P, t, X), where t is a (positive) real number indexing the productivity gains of version t of the technology.
And your function would look something like this:
F(P, t, X) = tPX
It looks to me like you’ve got an increasing returns to scale function!
How could you get rid of the increasing returns to scale function and make it constant returns to scale? Well, you want F(P, t, X) = kX.
So on the one hand, you want new technologies to be more valuable, so F(P, t, * ) is an increasing function of t, but on the other hand, you want it to be a constant. It’s only by playing fast and loose with differentials that you achieve this (the technology gets better, but really slowly).
So I think the fundamental result holds, and the multiple tech scenario is just a distraction. The core issue, whether techs come in many flavors or just one flavor, is that the function is going to be increasing returns to scale.
Britonomist,
You’re talking about another (rival) input into firm production, like entrepreneurial capital or something.
Romer is talking about a non-rival good, like ideas that anyone can use.
Jonathan, I don’t see the problem though. I come up with a new idea that will improve the productivity of producing X; I have no means of protecting this idea however & my production process will be completely transparent, meaning all other firms will be able to immediately replicate it. Does that remove my incentive to go into business? Of course not! I’ll still get to run a business, and be heralded as a great innovator, this gives me utility and an incentive in the absence of super-normal profits.
For the record, I definitely agree that a lack of super-normal profit incentive will significantly reduce innovation. But reduce it to absolutely nothing? No way.
jonathan @11.27.
Agreed. You are assuming there that Glenn’s idea depends on David’s, which depends on mine. And if so, and if ideas are excludable, I could charge David and Glenn for using my idea.
And then the problem is that landowners only get paid 1, while the MP of land under the newest idea is 4. And if new land could be produced (contra my assumption) that would mean there is underinvestment in new land, unless Glenn cuts the producers of new land a special deal.
Right?
I think it’s very sneaky to talk about land being fixed, since the idea of marginal product is to add one more unit, at least conceptually. But perhaps it was Nick Van Rowe who was making this post.
But a general principle I wish could be kept in mind is that if your argument is going to depend on taking an infinite limit of things, then see what is the final object that your limits converge to. Then see if your limits of equilibria converge to an equilibria on the final object. Then see if that final equilibria is a competitive equilibria. It’s a very long road to travel one, when we have a beautiful 2 earth argument in front of us.
This talk of invention is getting me thinking about tech companies.
Nick,
Right.
The basic point is that, no matter the assumptions, this economy is not competitive. Either farmers earn less than the marginal product of their land, or inventors earn less than the marginal product of their research.
The only exception would be if research were continuous and not costly at all (up to an upper bound). Then the latest tech would be free, and this economy would basically be a model with exogenous tech progress.
jonathan: OK. But if Henry George puts a tax of one ton of wheat per acre of land, is that economy not “competitive”? Individual landowners still face a perfectly elastic demand curve for their land, but it’s below the Present Value of MP.
But yes, we are now arguing semantics.
rsj: Marginal product is still defined, regardless of whether the input is in perfectly inelastic supply. Ricardo and all that.
jonathan: “I’m not sure how a perfectly elastic demand curve violates Romer’s comment.”
Paul Romer said: “… this would be a model in which someone who has a monopoly on an idea can charge for its use, but somehow is unable to influence the price that users have to pay,…”
Is Paul Romer going to say that an individual wheat farmer who faces a perfectly elastic demand curve for his wheat is able to influence the price that buyers of his wheat have to pay? I don’t think so.
Nick,
A competitive economy is one in which every agent is a price-taker. That means they take a price P as given, and can buy or sell any quantity at that price.
In your example, inventors are NOT price takers. They face a demand curve that is elastic up to the quantity Q=1, but they can’t sell more than that. So the demand curve is not perfectly elastic. With a perfectly elastic demand curve, if you set P = P* – epsilon, you get infinite demand. Here inventors see demand Q = 1.
This is important. If inventors were price takers, they would want to raise their production above Q=1 at any price P>0, because the marginal cost is zero. The only solution would be P = 0.
Instead of an elastic demand curve, the inventors face a kinked demand curve. Viewed in the standard (P,Q) space, it’s flat (locally elastic) at P = A’ – A up to Q=1, and then drops vertically (inelastic) to zero at Q=1.
The solution to their profit-maximization problem is to set P = A’-A and Q=1.
Nick,
I think the key here is “idea” — e.g. there is a non-rival good, and the claim is that you can’t both have monopolies on non-rival goods while at the same time being price takers for them. The non-rival good is important to the “two earth” argument, because if you duplicate the earth, you are duplicating all the rival goods (twice as much wheat), but are leaving the ideas the same. There is no production function specific effect to “duplicating” a non-rival good, since it’s already ubiquitously available. Mathematically, this is crucial to arguing that the resulting production function has increasing returns to scale. If it were a rival good, such as labor, then the resulting production function would be constant returns to scale and the argument would not follow through.
Or what Jonathan said.
I guess I’m the only one who is utterly in love with the simple and beautiful “two earth” thought experiment.
jonathan: under that definition, nobody is ever a price taker. If an individual firm in a competitive market cuts its price below the market equilibrium it can only sell a finite quantity, which is much bigger than it wants to sell, true, but still finite. Or take standard Bertrand duopoly, with identical goods. There’s a kink where the individual demand curve hits the market demand curve.
There are two ways to define “quantity” for a non-rival good: number of different songs; number of copies of each song (assuming copies have zero MC). In the equilibrium of my model, everyone buys a copy of the newest and best song, which is efficient. And we also get the efficient number of different songs. (But if land wasn’t in perfectly inelastic supply, we wouldn’t get the efficient quantity of land, true.)
rsj: I thought the 2 earth thing was neat too.
Nick,
I know you don’t like the Walrasian Auctioneer, but the standard definition is as I describe. The Auctioneer names a price vector P, and every agent takes this as given and acts as though it could buy or sell any quantity at this price.
By the standard definition then, this economy is not competitive, since the inventors are not price-takers.
(When I talk about “quantity” in the above, I mean the sale of the right to use the patent. This is what has zero marginal cost.)
Also, rsj makes a good point above. I would phrase it like this:
The aggregate production function is
Y = F(A,L) = AL
where L is land and A is the productivity of the technology being used.
Now in a competitive equilibrium, the rental price of land will be r = F_L = A, and there’s no income left over to pay the inventors.
rsj
it’s just the bit about any aggregate production function necessarily being CRS in rival inputs that I do not understand.
IRS in rival inputs can exist at the firm level, right? An an economy of N such firms has an aggregate production function? Or maybe it doesn’t. Because in such an economy when scaling inputs it matters whether you are scaling firms or duplicating firms. Which is my problem with the duplicating earth argument. Because I think you can have IRS in rival inputs, and still only double outputs when duplicating units of production without changing their scale. So when you write “the production function aggregates across both earths” that’s the bit I am objecting to – supposing doubling earths is like doubling the size of a firm. Maybe I am getting muddled because I have in mind an economy, made up of IRS firms, for which an aggregate production function does not exist?
Or maybe I am talking rubbish. Is Sum over i for i=1 to N f(X^2_i) an aggregate production function?
jonathan: yep, I think I must be going a bit Marshallian? Individual firms know they can’t sell more than total market demand at that price.
I get the bit about factor payments adding up to total product under CRS and W=MPL. But look, if the government puts a tax on something, then factor payments will be less than MPL and less than total product under CRS. Do taxes suddenly eliminate price-taking competitive behaviour? No.
Luis: as I understand it, the replication argument is an argument against decreasing returns to scale, but it doesn’t prevent increasing returns to scale. But under IRS, sum MPX.X > Y, so you can’t have all inputs being paid their Marginal Products.
ah, well that would explain my confusion!
[Sure I get the point inputs can’t be paid marginal product under IRS – and a host of other production functions (O-ring etc.) it was just that one bit about duplicate earth and CRS that I was caught on]
Nick: Are you assuming that farmers have to pay inventors PER UNIT of output produced using their technology, or just pay a one-time fee for the technology?
I was thinking about the latter. But then it makes sense for one farmer to pay the cost, and then rent everyone’s land and farm it using the best technology. In this case, since Y = AL, the rental rate will be r = A, and we get the adding up problem.
But a per-unit fee, like a per-unit tax, would affect the rental rate, and so everything would add up. Is that what you have in mind?
jonathan: I was assuming each farmer paid a fee per acre per year. So a farmer with 200 acres pays double a farmer with 100 acres. (That would be the same as a fee per ton of wheat, in my simple model.)
It wouldn’t be profit-maximising for me to charge all farmers the same fixed fee, regardless of how many acres they farm. Charging a fee per acre maximises my profits.
Nick: Interesting. Unless I’m mistaken, I think that your per-unit pricing assumption violates the conditions under which Romer proves his statement.
Reading the original mathiness paper, it looks like the argument in question is in the 2nd paragraph on page 3. It goes like this:
Production is F(a,x), where x are rival inputs. F is HOD 1 in x. Therefore, by Euler’s theorem, the entire output is spent paying for the rival inputs (x), and so nothing is left over to pay the nonrival inputs (a).
This argument assumes that the firm maximizes:
profits = F(a,x) – wa – px
and takes p and w as given.
In your setup, the firm problem is:
profits = (A+zdA)*L – rL – zpL
where z is the decision whether to buy the new technology, which raises productivity by dA.
This is clearly a different setup than assumed by Romer. In particular, you’re assuming that the total amount paid for the new technology dA DEPENDS ON the amount of land L. Romer’s formulation seems not to allow for this.
jonathan: Ah, that is interesting. Yep, I can see that if the inventor charged a fixed price per farmer, regardless of acreage, you would end up with only one monopoly farmer (or maybe a small number of Cournot oligopoly farmers). All the other farmers would lend sell him their land and live off the interest from the loan they gave him to buy their land.
But that wouldn’t be profit-maximising for the inventor.
But maybe that’s the root of the disagreement? (I haven’t read David Andolfatto’s paper to see what he assumes; scared to even look at it, because it’s probably got lotsa math.)
The way I teach “Euler’s theorem” in ECON 1000: IRS means downward-sloping LRATC curve, which means MC < ATC, which means you can’t have both P=MC and P=ATC. Each firm is gonna want to get bigger, or else exit, so you can’t have a LR equilibrium with lots of firms.
But I confess, I did have a frustrating time once trying to explain this to PhD economists, so maybe Paul Romer does have a point.
Wouldn’t the rest of the world be flooding Canada with wheat (or at least their customers), or are you assuming they do have a way to exclude this?
Lord: no. The cost of wheat would be the same. Canadian farmers would pay 1 ton rent on the land, plus 1 ton payment for my idea. ROW farmers would pay 2 tons rent.
I think Romer view’s P as the quantity of technologies or patents. Because the rival goods are CRS, they can all be aggregated into a single firm, which then licenses some quantity of P. A patent to do X better, a patent to do Y better, etc. P doesn’t measure how many licenses are sold for a single patent, but how many different patents are sold.
You need at least this to put P into the production function as a variable which would cause IRS, otherwise P is just a constant. IRS is the key here, and it comes by being able to increase the number of patents (invest in research), not the number of patent licenses.
As I read all this, I have a nagging question. Who is going to eat all that wheat?
And, even assuming no foreign competition, don’t diminishing returns kick in?
Luis,
The replication argument proves increasing returns to scale under the assumption that increasing P also increases total output. You have, by euler’s theorem,
f(P, X) = Residual(X,P) + const(P)*X
All you need to get IRS is to show that the derivative of f with respect to P is always > 0. I.e. if you increase P you get more output.
“mathiness” comes in when you assume that P is not a variable for purpose of scale determination (e.g. it is a constant), but at the same time you want to assume that P is somehow a choice variable for purposes of modeling behavior.
Romer is interested in models of endogenous technical change, so P is an input into the production process, and whenever that happens, unless there is a ceiling on technology so that further investment yields no gains, you must have IRS and therefore cannot have a competitive equilibrium.
Nick: I have a hard time calling per-acre pricing “competitive”.
The usual way to set up a firm problem with two inputs would be something like:
max F(K,L) – rK – wL
But what you’re doing is making the price of one input (technology) DEPEND ON the amount of the other input used. So it’s like you have:
max F(K,L) – r(L)*K – wL
This doesn’t seem competitive, because you’re letting a price depend on a choice variable.