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"Sometimes it's good to build really weird models."

That may possibly be true. But could you explain why you think "it's good" to build specifically this weird model? What specific real world problem does it illuminate?

Sandwichman: many goods in the real world are non-rival, or close to non-rival. Maybe an increasing share of total GDP?? It can help explain why productivity falls in recessions. It may also shed light (see the last bit) on the short run Phillips Curve, and why inflation failed to fall as much as we expected in the recent recession.

Plus it's fun, and good for the brain, to see how are results depend on assumptions we don't even see.

Prescription drugs are (mostly) non-rival. First pill costs billions, subsequent pills are cheap.

Nick: "Maybe an increasing share of total GDP??"

absolutely definitely an increasing share of total GDP - any entertainment or information good is non-rival, anything that's patented has an information content (the other phrase that's sometimes used is collective consumption good). One could even argue that traditional public goods e.g. national defence are more collective consumption goods than they once were. An old-style fort protected a much much smaller area than modern-day missiles.

"absolutely definitely an increasing share of total GDP - any entertainment or information good is non-rival, anything that's patented"

Whoa! Perhaps you could explain, Frances, why a good that by its "nature" may indeed be non-rival still remains so even after the state has intervened with the intention of making it what might well be called a pseudo-rival good? Just because the turnstiles are artificial barriers doesn't make them non-barriers.

Sandwichman - that's the difference between rivalry and excludability. Rival is about the technological nature of production. A good is non-rival if the marginal cost TO THE PRODUCER if another person consuming the good is zero (or as close to zero as makes no difference). This web site, which you can access 10000 per month without paying, is no more or less rival than the NY Times website, with 20 free views per month.

That's why Nick said "non-rival" as opposed to "non-excludable" or "pure public" goods.

Michael: good example. I had forgotten that one.

Frances: so maybe this model is becoming increasingly less weird over time? neat. Because, apart from the weird assumption, it's got rather weird results!

Sandwichman: remember the difference between non-rival and non-excludable. Turnstiles make a good excludable, not non-rival.

Nick: "so maybe this model is becoming increasingly less weird over time?"

It's not that you're wrong, Nick, it's just that the world hasn't quite caught up to you yet ;-)

"Turnstiles make a good excludable, not non-rival..."

Nick, you're giving me definitions. I know what the definitions are. My question has to do with the effects of juridical modifications. Turnstiles are not free. I mean they are not free to install. Patent lawyers are costs, too. One does not invest a billion dollars making a "first pill" without also spending several tens of millions insuring that the subsequent pills will not be cranked out by a competitor (and continuing to spend tens of millions to keep it that way).

Put my question this way: doesn't your assumed distinction between rival and non-rival goods rest upon the assumption that there are no transaction costs involved in making the hitherto non-rival goods excludable?

Sandwichman - as long as the cost of the turnstile is fixed, e.g. tens of millions regardless of the # of pills produced, Nick's argument should go through.

"as long as the cost of the turnstile is fixed..."

Frances, that is a theological argument. There is no such thing as "fixed." My ceteris paribus is your fallacy. Elinor Ostrom wrote, "When the world we are trying to explain and improve is not well described by a simple model, we must continue to improve our frameworks and theories so as to be able to understand complexity and not simply reject it."

I don't think it is unimaginable that the cost of the turnstile may be a function of the intrinsic non-rivalousness of the goods. I also don't think it is unimaginable that a considerable portion of that cost may be shifted from the beneficiary of the turnstile to society (AKA "externalities"). My question then is, does Nick's model take into account the possibility of such transaction costs and cost shifting or does it finesse these questions with simplistic simplifying assumptions? If it is the latter, what real world problem does the model solve?

I should point out that I spend a lot of time wrestling with the implications of the rivalous/non-rivalous, excludable/non-excludable goods spectrum. Thus if Nick's model were indeed useful, it would be useful TO ME. But as soon as I get to the expression "strictly non-rival" an alarm goes off that says, "there is no such thing!" I mean there is no such thing because rival and non-rival are only meaningful as comparatives, like larger or smaller. There is no such word as "smallerist."

Incresing share of GDP?.
We live in a high-fixed-low-marginal cost world. Planes that cost $ 200 Millions to build and are cheaper to operate by seat than a compact car. Our manpower spend 15 years at school then consume only 2000 calories to think.
We live in the empty Samuelson core. In "Brave New World", people made the sign of the T. Maybe we should now make the sign of the S.

I should also mention that the pharmaceutical example is a particularly egregious one if the assumption is that the cost of the turnstile is fixed. After a drug has been developed, pharma companies spend a lot of money promoting it. Much new drug development has more to do with protecting or carving out a market niche than with any substantive health benefit. Research shows that pharma sales reps often (usually?) fail to mention the harmful side effects in their presentations to prescribing physicians.

This looks like a multiple equilibrium model, really. In the initial equilibrium, firms are producing very little software and charging high average costs, so they have high 'real' costs of production (or rather, 'real' costs as a fraction of the economy). Then a demand shock occurs, so the firms distribute the sofware more widely. Total revenue at each firm (as a fraction of cost) increases. Whether each individual firm chooses to decrease their price depends on whether the demand curve they face becomes more or less elastic, as in your model. But if they do decrease prices, then we transition to a "good" equilibrium where more software is distributed, at a lower average cost, so each firm faces a low 'real' cost of production. The positive demand shock has obviated a coordination problem.

Interesting stuff. This might help make sense of some post-Keynesian literature, which stresses the importance of large-scale monopolistic producers, and posits that demand shocks do have long-run effects.

anon: we are on the same page. In this model, depending on preferences, it would easily be possible to have (say) three LRAS curves, with the outer two locally stable, and the one in between unstable. (I would just need to assume preferences such that elasticity of demand first decreases below one, then increases above one, then decreases below one again, as Y increases). So if we are initially stuck in the bad equilibrium (on the low LRAS curve), a big enough and sudden enough increase in AD, plus a little bit of purely temporary price stickiness, could switch the economy permanently (well, strictly, for an indefinitely long time, or until there's a sudden decrease in AD) to the good equilibrium.


Sorry if I’m missing your point, but I don’t really see why the non-rivalry assumption is needed here. Suppose N monopolists each supplying a distinct good. Suppose technology is such that to produce a unit of a good “i” requires using up a fraction v of a unit of output obtained from another supplier “j” (doesn’t matter which). Then the profit-maximization condition for supplier “i” is

Pi*dQi/dPi + Qi - v*Pj*dQi/dPi = 0

(where Pj = the price charged by the “other” supplier) or, in elasticity form

e(Pi/Pj,Y) + 1 - v*(Pj/Pi)*e(Pi/Pj,Y) = 0

(where e( ) = own-price elasticity of demand). Assuming a symmetric equilibrium (and using the fact that real income Y = (1 - v)*N*C), supplier-specific equilibrium output C is defined by

-e[1,(1 - v)*N*C] = 1/(1 - v) (*).

In the case described in your post v = 0, so (*) reduces to

-e(1,N*C) = 1 (**).

But (*) and (**) are formally identical: any results you can get from (**) for a particular e( ) - including multiple equilibria and “crazy” dynamics - will also be implied by (*) for a somewhat different version of e( ).

Your analysis is perfectly correct but, as far as I can tell, all your results go through even if there's a positive marginal cost of production. It’s your monopoly pricing assumption that’s the key theoretical ingredient here.

Giovanni: "Suppose technology is such that to produce a unit of a good “i” requires using up a fraction v of a unit of output obtained from another supplier “j” (doesn’t matter which)."

Your technology assumes that there is no labour (or land) needed to produce goods.

Giovanni: but yes. If marginal costs are positive, *but do not increase when all firms expand output together* (which they would if you needed extra labour or land to increase aggregate output) then you would get much the same sort of model as mine. And you would need monopolistic pricing to prevent the model exploding.

If you take the pictures in this old post, and change them slightly, you can see a digrammatic representation of this model. Make the mc curve in the first picture, and the MC curve in the second picture, perfectly flat at zero. Then make the MR curve in the second picture downward-sloping, to get a unique stable equilibrium where that MR curve cuts the MC curve.


I think what you mean is that even if you need land or labour to produce goods, it is not a binding constraint on the quantity of goods that can be produced. Even a software developer needs a place to sit down and someone to write the code.

Even monopoly producers of semi-durable goods (like software) face the constraint of market saturation. I would be happy to pay monopoly pricing for one copy of a software package, but I would not purchase a second copy.

One final thought would be that much as the producer is not resource constrained, the buyer may not be resource constrained as well. Can an unconstrained buyer put upward pressure on prices despite the monopoly presence of the producer?

Frank: "Even monopoly producers of semi-durable goods (like software) face the constraint of market saturation. I would be happy to pay monopoly pricing for one copy of a software package, but I would not purchase a second copy."

What that means is that elasticity of demand falls as the number of copies of all goods increases. In which case the Long Run equilibrium is unique and stable.


But I am resource constrained as well (money, time, and ingenuity). The only reason I don't buy two copies is that I don't have the ability (on my own) to maximize the utility of having the extra copy. I recognize that other buyers may not face the same constraints that I have.


Update: The Short Run Phillips Curve slopes the wrong way. Prices start falling in a boom, and start increasing in a recession.

That is not wrong at all for the types of goods that we are talking about. The best way to think about it is that firms do not lower their prices, but invest more in the next version of the good during the boom -- the software has more features, or the song is better produced, or more artists are signed to the record label. The basket of varieties produces more utility per dollar charged during the boom, so prices fall. During the bust, firms cut back on R&D and also produce fewer varieties, so that the basket provides less utility per dollar.

As a simple example, consider broadcast television. When there is a lot of ad revenue to be had, stations compete with each other by investing a lot in programming content to lure viewers. When ad revenue plummets, viewers are less valuable, so stations put on a lot of cheaply-produced reality tv shows. Newspapers would be another example. On the other hand, video games, which have been a booming industry, see ballooning production costs as the games become more realistic, with bespoke soundtracks and custom art. The video game industry is hiring their own composers now. It is a great comparison to see what is happening, in terms of production quality, with broadcast television versus video games.

Nick, now I see a difference. The way I think about SW is that the firm invests more into the product in order to add more features to fight for customers. This has diminishing returns, not in terms of number of units produced, but in terms of the utility provided by each additional feature. The firm keeps investing until it earns no excess profits.

At first, I thought your model was effectively the same as this, because you would keep adding more varieties, but now I see this is not the case, because your price per variety is the same, which does not reflect the diminishing returns. Your production function is linear, and I wonder how much of the weirdness arises from this linearity, when a bit more realistic production function would have diminishing returns in terms of product quality, even if all copies of the product are free.

E.g. MS OFFICE version 1 has 100 features that everyone uses. Say this takes 100 units of labor. MS OFFICE version 2 has 1000 features, requiring 1000 units of labor, but most people don't value those extra features that much. The price of MS OFFICE version 2 is not 10 times the price of MS OFFICE version 1, but Microsoft's labor demand is very sensitive to perceived increases in demand for its products. If that demand shifts back a bit, then Microsoft may want to lay off 99% of its labor force and only produce 100 features again.

Unfortunately, I don't know how to add this wrinkle to your example.

Anon nailed it, and I also immediatelly remembered this post which is kind of a follow-up for a post that Nick linked above. Everyone should go read both of them as they are excelent (the first post was actually a result of my random google search and since then WWI blog made it into my regular reading list)

And yes, Scott Sumner put this as a comment "I believe this is why a monetary shock that pushes us to the left of the natural rate of output seems to be more destructive than a monetary shock that pushes us to the right of the natural rate, even though it really isn't more destructive in the long run." I strongly feel that there is more to sticky prices than it seems and I think this is an important direction. And it needs to be repeated.

Thanks JV! The first post you link in your comment is the most clear. But it's only partial equilibrium/micro. The second post is harder to follow, but it's general equilibrium/macro.

rsj: With my assumption of exogenous labour supply, full employment, and the simple linear technology of one good per worker, the investment in new goods, and the number of different goods, stays the same over the business cycle. All the action is in the number of copies. But yes, I think it would not be too hard to relax those assumptions a bit, so that investment in existing and/or new goods would vary over the business cycle. I need to think of a simple and clear way to do it.

In the real world, some goods are rival, and other goods are non-rival. There's a mix. But *all* (AFAIK) macro models assume that *all* goods are rival. (Hmmm, except maybe models with investment in R&D? And some models with monopolistic competition assume fixed costs, which is partway there?) So here I'm taking the exact opposite assumption. The truth will lie somewhere in between.

Real world problem. Model this.

Suppose the "good" is the capacity to work. There are two kinds of work to be done -- paid market work and unpaid household work. Women perform three-quarters of the household work and men perform two-thirds of the market work. Total annual hours of paid work equal total annual hours of unpaid work. Thirteen times as much energy is consumed during each hour of paid work as during each hour of unpaid work. How do you improve the economic status of women without penalizing men while decreasing energy consumption?

I think you have to model a multi-agent ecology. Agents with different cost structures will try to fill different niches.

For instance, a large company with a large income won't try to enter markets where the total demand is small. So they will avoid markets where people only need one copy, unless the number of such people is large, and they can sell a continuing stream of upgrades. Charging high prices while selling small numbers of copies tends not to be a stable strategy unless the monopoly is airtight, and there are no reasonable substitutes. Otherwise someone leaner and meaner will eat their lunch.

You can make money even if your product is free if you sell ancillaries. Bands don't make money from selling records anymore, they make it from tours and T-shirts.

There will be a substantial presence of truly free and open sourced goods. You can base products on these with certain restrictions, but they are hard to compete with in the long term. Open source development is often slow, but its developers are very persistent. Eventually the tide will reach you.

The IP monopoly system is inherently unstable. As the average number of IP components needed for a high-tech product increases, the licensing becomes some combination of unaffordable and unmanageable. In a major software product, it would cost more to do a patent clearance on the source code than it cost to produce the code - quite a lot more. Also the system will be captured by nonpracticing entities, who, since they don't produce a product can't be counter-sued for infringement. This negates the mutual assured destruction detente arising from patent portfolios and cross-licensing. Also the MAD is failing as the patent tangles get thicker. The more slices to the pie, the thinner they are. The recent Cell phone patent war is a good example.

Our patent system has also been captured by patent lawyers, and patent quality is very low. We're seeing the consequences of this now.

So it's very hard to model a non-rivalrous monopoly system in the limit of a vast number of monopolies with overlapping clams and non-practicing entities.

I'm afraid that without these details, you won't get a model that is useful for interpreting any real current or future economy.

You might be better off starting from game theory.

Your Captcha system doesn't seem to work with my configuration of Firefox. I'm using Chrome for this.

Sandwichman: your comment is totally off-topic. If you want that modelled, then you model it.

Peter N: my model above is in fact game-theoretic. What I am describing is a Bertrand-Nash equilibrium between firms. (Or a Cournot-Nash equilibrium with a large number of firms selling differentiated products, because the two equilibria are the same in the limit as the number of firms gets large.)

Totally off-topic? You say in your model that these things have these properties:
goods -- non-rival
wages -- flexible
prices -- sticky
employment -- full
labour supply -- fixed
My "off-topic" point is that "labour" is your transmission gear. The teeth (properties) of your gears have to mesh somehow, not only for your model to be realistic but for it to even function as a model. I suppose you could say that in your model "labour" is not an activity performed by humans for the purpose of obtaining provisions for their subsistence. I do hope you pay your words extra when you make them do so much work!

I'm not an expert in these equilibrium models, but assuming this is correct, the restrictions are much too severe for a useful model.

"The model rests on very specific assumptions. There are at least two firms producing a homogeneous (undifferentiated) product and can not cooperate in any way. Firms compete by setting prices simultaneously and consumers want to buy everything from a firm with a lower price (since the product is homogeneous and there are no consumer search costs). If two firms charge the same price, consumers demand is split evenly between them. It is simplest to concentrate on the case of Duopoly where there are just two firms, although the results hold for any number of firms greater than 1.

A crucial assumption about the technology is that both firms have the same constant unit cost of production, so that marginal and average costs are the same and equal to the competitive price. This means that as long as the price it sets is above unit cost, the firm is willing to supply any amount that is demanded (it earns profit on each unit sold). If price is equal to unit cost, then it is indifferent to how much it sells, since it earns no profit). Obviously, the firm will never want to set a price below unit cost, but if it did it would not want to sell anything since it would lose money on each unit sold."

1) homogeneous (undifferentiated) product
But product differentiation real and purported is a fundamental principle of product marketing -

SOS with a rust arrester
Ivory 99 44/100th percent pure....

2) Firms compete by setting prices simultaneously
Usually there is a leader who tests the waters and various followers. The leader takes the risk of having to retract the price increase if there is consumer resistance, and other firms don't follow suit.

3) firms... can not cooperate in any way.

4) consumers want to buy everything from a firm with a lower price

This would make non-simultaneous price changes impossible since increases would result in 0 sales, and simultaneous changes violate the non-cooperation condition (and, of course, antitrust laws)

5) A crucial assumption about the technology is that both firms have the same constant unit cost of production

Differences in company size and strategy produce different unit costs, which can never be 0, though in free software models, they can be very small.

6) marginal and average costs are the same and equal to the competitive price

But in most cases the marginal production costs (as opposed to cost of sales) are very small compared to the cost of making the first unit. This results in the first basic marketing choice - whether to aim to sell a small number of units for a high price or a large number for a small price.

7) This means that as long as the price it sets is above unit cost, the firm is willing to supply any amount that is demanded

Not necessarily. Customers will often buy more if the product is more expensive, since low price is equated with low quality. A sure way to destroy a luxury brand is to discount it, and mass distribution conflicts with the image of exclusivity. Why do people buy Swiss watches for $10,000? They don't keep better time. Why don't the manufacturers cut prices to increase sales. There's no real barrier to entry into the luxury watch business.

Would Tide (the most expensive brand of soap) make more money if it cut prices?

It seems economists usually ignore most of the business school curriculum. If what it teaches is wrong, you have to wonder why so many people pay so much for it.

I meant a game theoretic model with different agents with different characteristics using different strategies - more of an ecological model.

Also, it matters whether a given customer will buy more than one copy of a particular non-rivalrous good.

And there is competition within niches. For instance consider the concept of the entertainment dollar. Assuming that people budget a fixed percentage of disposable income in this category, they can spend it on concerts, movies, CDs, DVDs, gaming, books, theme parks...

For example, the record companies claim that the fall in CD sales is due to piracy, but what has really fallen is the CD's share of the entertainment dollar. The number of entertainment dollars has been increasing.

Peter: I'm not sure what your source is there, but it's rather misleading in this case. It refers to a sort of Mark 1 Bertrand-Nash model. Bertrand's original 19th century model. We have generalised it since then.

I'm explicitly dropping assumptions 1,4,5, and 6 in my model. I'm keeping 2 and 3. And 7 is a conclusion, rather than an assumption (I am assuming demand curves slope down).

But yes, the results you get from a model depend on the assumptions you put in. And I'm getting quite enough action from my (false in some cases) "MC=0 for an extra copy" assumption.

Sandwichman: I'm afraid that made even less sense to me than your previous comment. Just stop. There are comments that add to the conversation. But yours don't. So stop. This is not your blog.


That clarifies things quite a bit.

The first question then is, what counts as a marginal cost? Royalties, advertising costs, shipping, support, pay to play are all unit costs. So, while it may not cost more to produce another copy, it may cost more to sell one.

Something close to an exception to this is the open source model, but if your model forces this form of distribution, it would seem a bit unreasonable.

Second, what sorts of violations of condition 1 do you allow. That is, by what methods can a vendor differentiate a product. If you look at software, you'll see vendors try all sorts of methods to differentiate their products. When they don't, one vendor will achieve a near monopoly (like IBM in the 1970s and Microsoft 1985-2000 or so). Eventually someone will find a way to differentiate, and competition is restored.

once someone can create non-rival consumers of goods, we're all set.

Peter N: Almost anything that makes firms products different from other firms' products (or just makes at least some actual or potential purchasers think they are different), so that the individual firm faces a downward sloping demand curve (rather than taking price as given as in perfect competition) is OK for my model (I think).

Shipping costs would be included in marginal costs, if there is a cost of shipping each copy. Maybe support too. Pay to play, and royalties, are part of the price, not the cost. Advertising costs could go either way, it depends. If there are positive marginal costs (like in all other macro models) my model won't work exactly. If they were small, it wouldn't change the results much. It would simply make it more complicated.

But what you maybe don't get is that I am deliberately making this model extreme, and so "unrealistic". I am deliberately going to the exact opposite extreme to all other macro models, just so I can see what difference it makes.

It's your blog, Nick, and you're welcome to delete this comment (as I expect you will). But I find your exaggerated "incomprehension" of my banal criticisms to be rather disingenuous and histrionic.

There's an old saying, sometimes attributed to Abraham Lincoln (but predating him), about how many legs a cow has if you call its tail a leg. It still has four legs; calling a tail a leg doesn't make it one.

You called an element in your model "labour": "each good requires one unit of labour input." Sorry, Nick, there is no labour in your model. Calling a tail a leg doesn't make it one. Calling an input "labour" doesn't make it labour. What part of the word "no" do you not comprehend? I'll stop now. Thanks for the gratuitous insults.

"Because there is no upper limit on the number of copies that can be made, even with finite resources. And firms will produce and sell as many copies as are demanded, at existing prices."

But are there limits on the number of copies one person wishes to own? Say you have 3 goods - CDs, movies in DVD format and video games. Each person has a number of entertainment dollars that is a function of disposable income. Let's say this budget is 100% of earnings (this immediately creates a problem, but let's ignore it for now). Each of the three categories comes in titles and nobody wants more than one copy of a given title.

So with N workers the maximum sales per/title is N. Demand for a given title depends the amount of money spent on promotion, the number of copies sold, the percentage of income spent in the category and the price

That is, you can never get all DVDs with no CDs or games, because the utility functions have gradients like a diffusion that increase as the population ratio becomes more extreme. Furthermore, if you have a lower promotional budget, you'll have a lower unit sales cost which decreases one factor of demand and increases the other. You'll have a mix of highly promoted more expensive titles and cheaper less promoted titles. People get utility both from having more titles and from having more popular titles.

The model has time steps, since a static equilibrium model is too unrealistic. You never see such a thing, and it would make monopolies permanent, which they never are. A company would start out with a certain amount of capital, and subsequently it would have to fund itself from retained earnings (later you might add borrowing). A poor company would be forced into a low budget strategy. However a small company would also have lower fixed costs. This combination should keep the model from ending up with a static monopoly.

Likewise the gradients should prevent video games from driving CDs and DVDs out of the market. French fries may be cheap and popular, but they don't reduce the sales of asparagus to 0, even if they are free.

I don't think you can get much simpler and get realistic behavior, but you can certainly start with only CDs. You do have to look at how your fixed costs, variable costs and retained earnings interact. Do you get a monopoly or can you model a situation where one strategy can never completely dominate the market? There have to be disadvantages to being big as well as advantages.

There is another problem in closing the S=I loop. It's quite a different thing to model the entire economy as being of this form as opposed to saying that non-rivalrous goods are a certain percentage of the economy. It might be better to make them a certain percentage and then investigate behavior as that percentage goes to 0. In math there's an art to deciding when to take the limit. Later is usually better unless you can't make further progress without doing it.

Finally, you can model free as very very low unit costs and very low fixed costs. People don't want to produce free software that nobody wants to use, and you always have some costs. There's at least a web page with usage charges. Large free software products can have fairly large budgets, but then you might need to expand the model, because they have a different ways of funding themselves (the FSF, Ubuntu, Red Hat, the GIMP).

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