Robert Barro: Sketch of a Model of Microsoft’s Social Value

One thing many people don't realize is that there is often a lot more behind the opinion pieces written by people like Paul Krugman, Greg Mankiw, Hal Varian, Dani Rodrik, Tyler Cowen, George Borjas, Robert Barro, Martin Feldstein, and others than it appears on the surface. That's not true in every case, but it is generally true when reputable economists weigh in on an issue.

For example, I recently had questions about calculations by Robert Barro in this editorial in the WSJ on Microsoft's social value. In response, he sent me this model that, though he does not regard it as definitive, explains the basis of his arguments.

In the model, production of goods requires intermediate products like software. The level of output, as explained below, depends upon the variety of these intermediate goods, i.e. how many different types are available. Thus, as we discover more "idea-type goods" such as Windows, we are able to produce more output. Therefore, in this model, the invention of Windows and other products adds to the variety of intermediate goods, the increased variety allows more goods to be produced, and the increased output adds social value by allowing higher consumption levels. Here's the analysis:

Sketch of a Model of Microsoft’s Social Value, by Robert Barro, June 2007 [pdf version]:  Goods are produced by competitive firms using the freely accessible production function:

where A>0, L is labor input, x_j is the quantity of intermediate input of type j, and N is the number of varieties of intermediates that exist.  The quantity L is in fixed aggregate supply.  Although L is called labor, it really represents all of the usual rival inputs to production (unskilled labor, skilled labor, capital—all treated here as in fixed aggregate supply).  Software and other idea-type goods are modeled as the intermediates.  These goods are treated, for simplicity, as non-durables.  The parameter α (0<α<1) will be the income share for intermediates.  The parameter σ (0<σ<1) measures substitutability among types of intermediates.  The presence of the last term in Eq. (1) will imply that total gross output, Y, is proportional to N, and this property will allow for endogenous growth in dynamic models where N grows due to R&D activity.  The present analysis considers only one-time shifts in N.

Suppose that an intermediate of type j is priced at P_j>0.  Competitive, profit-maximizing producers of final output equate the marginal product of x_j to P_j.  This condition yields the demand function:

Hence, if N is large, the elasticity of demand for x_j is approximately constant and equal to ‑1/(1-σ), which exceeds one in magnitude.  (Competitive producers of final goods hire labor at a given wage rate, w.  In equilibrium, w equals the marginal product of labor, and each producer of final goods earns zero profit.)

Each type of intermediate, x_j, is produced at constant marginal (and average) cost, c>0.  Without loss of generality, assume c=1.  Thus, physically, a unit of x_j is “produced” by taking a unit of final output and placing a j-type label on it.  This labeling is assumed to be the exclusive province of intermediate firm j, which owns the rights to produce that intermediate.  (This exclusive holder may be the inventor or developer.)  The perpetual profit flow for intermediate firm j is:

Intermediate firm j chooses P_j (at each point in time) to maximize π_j, subject to Eq. (2).  This condition yields the monopoly price, (P_j)*:

Hence, the monopoly price is the markup, 1/σ, of marginal cost, 1. 

We can generalize from pure monopoly to assume that each firm j actually prices as the fraction λ of the monopoly price:

where σ ≤ λ ≤ 1.  The first part of the inequality ensures that profit is non-negative.  The monopoly case corresponds to λ=1.

Since the model is fully symmetric across types of intermediates, the values of P_j, x_j, and π_j are the same for all j.  Denote these values by P, x, and π.  We can use the results for x to determine total output (gross of production of intermediates) from Eq. (1) to be

Total output goes to aggregate consumption, C, and aggregate intermediate production, Nx.  (This model excludes investment, including R&D outlays that might lead to changes in N over time.)  Total profit is Nπ.  Consumption is divided among wage earners and owners of intermediate firms.  The part of consumption that goes to the wage earners is C- Nπ. 

We can readily work out formulas for all of these variables.  It is convenient to express the results as ratios to Y, given by Eq. (6).  The various ratios turn out to be:

The variable NPx is the total revenue of intermediate firms.  The ratio of wage-earner consumption to this revenue follows from Eqs. (11) and (8) as

Note that the last ratio depends only on α (the share of intermediate factor income in total income) and not on σ (substitutability among intermediates) or λ (markup ratio relative to the monopoly markup).

If N increases, Y rises in accordance with Eq. (6).  The other variables (Nx, NPx, C, Nπ, C-Nπ) rise in the same proportion—that is, the ratios given in Eqs. (7)-(11) are constants.  We can think of the creation of Microsoft as raising N (adding a variety of intermediate product, corresponding to Windows and other software).  We can think of Microsoft’s observed gross revenue (say $44 billion per year) as the addition to NPx.  Therefore, Eq. (12) implies that the addition to wage-earner consumption (that is, consumption beyond that enjoyed by owners of Microsoft) is $44 billion multiplied by (1-α)/α. 

The parameter α represents the share of total income going to intermediate production—that is, inputs that have an idea-type character.  It seems that much of national income would flow to standard, rival-type factors of production, so that α would be well below one-half.  Hence, (1-α)/α tends to be well above one.  My “conservative” calculation assumed that (1-α)/α equaled one. 

This calculation gives no weight to the added consumption of Microsoft owners (including Bill Gates).  This additional consumption corresponds to the rise in Nπ.  The additional term follows from Eqs. (10) and (8) as 1 – σ/λ (which has to be non-negative).  That is, this term adds to (1-α)/α to incorporate the added consumption of Microsoft owners.  (Note that this analysis treats the increase in N as coming without cost.  In a dynamic analysis, changes in N could be related to costly R&D outlays.)

In comments, Brad DeLong says:

Mark--

I'm confused.

In most models like this that I am used to, the limit of α=0 is the case in which "ideas" are unimportant, and in which the social benefits from an increase in the number of ideas N are zero. This is how it should be as α approaches zero: rival factors of production need less and less supplement from new ideas to avoid diminishing returns to scale. But in this model the social benefits to an increase in the number of ideas N in the case of α=0 are greater than for larger values of α.

In most models like this that I am used to, the limit of σ=1 is the case in which "ideas" are perfect substitutes--intermediate goods are (for positive alpha) important to have, but it doesn't matter how many varieties you have, and so once again the social benefits to an increase in the number of ideas N are zero. This is how it should be as σ approaches one: an invention that does something completely new have a bigger impact than an invention that is a close substitute for already existing technologies? But in this model the social benefits to an increase in the number of ideas N in the case σ=1 are strongly positive.

If I can still do math, using C for net final output and S for sales of the new intermediate goods variety, I get:

dC/dS = ((1/α) - σ), which is definitely not zero for σ=1, and definitely infinity for α=0.

The key is the last "N" term in the production function. A new variety not only allows the economy to use its intermediate-goods spending more efficiently, but also shifts the whole production function upward by boosting total factor productivity by this factor N^(1 - α/σ). Even if you zero out production of the new variety completely via regulation , the economy's production rises because of the mere fact of its invention.

And I don't understand why one should assume that σ = (1/α) - 1 in the first place--which is what is needed to get the increase in net output equal to intermediate-goods sales...

Comments

[Esben]

Let me admit that these formula go over my head. Even so, it appears to me (please correct me if I am wrong) that Barro STILL operates under the assumption that in the case of Gates never being born, no one else would have created something similar to Microsoft and Windows. We are supposed to think that those meekly alternatives we have today are the same exact options we would have had in a Gates-less world where Microsoft did not monopolise the market for OS?

[Bruce Wilder]

It doesn't go over my head, but, oh my lord! What mad tripe!

[Brad DeLong]

Mark--

I'm confused.

In most models like this that I am used to, the limit of alpha=0 is the case in which "ideas" are unimportant, and in which the social benefits from an increase in the number of ideas N are zero. This is how it should be as alpha approaches zero: rival factors of production need less and less supplement from new ideas to avoid diminishing returns to scale. But in this model the social benefits to an increase in the number of ideas N in the case of alpha=0 are greater than for larger values of alpha.

In most models like this that I am used to, the limit of sigma=1 is the case in which "ideas" are perfect substitutes--intermediate goods are (for positive alpha) important to have, but it doesn't matter how many varieties you have, and so once again the social benefits to an increase in the number of ideas N are zero. This is how it should be as sigma approaches one: an invention that does something completely new have a bigger impact than an invention that is a close substitute for already existing technologies? But in this model the social benefits to an increase in the number of ideas N in the case sigma=1 are strongly positive.

If I can still do math, using C for net final output and S for sales of the new intermediate goods variety, I get:

dC/dS = ((1/alpha) - sigma), which is definitely not zero for sigma=1, and definitely infinity for alpha=0.

The key is the last "N" term in the production function. A new variety not only allows the economy to use its intermediate-goods spending more efficiently, but also shifts the whole production function upward by boosting total factor productivity by this factor N^(1 - alpha/sigma). Even if you zero out production of the new variety completely via regulation , the economy's production rises because of the mere fact of its invention.

And I don't understand why one should assume that sigma = (1/alpha) - 1 in the first place--which is what is needed to get the increase in net output equal to intermediate-goods sales...

[cm]

I don't think I can derive value from delving into the bombast of the formulae. The bottom line appears to be that once you manage to claim a majority share in an "enabling factor" (in this case technology products) in your name, economists will assign to you all benefits derived from other people's work supposedly using (or perhaps, fighting with) your enabling factor.

Kind of the same principle as assigning achievements made by workers in any successful company to the CEO, or sometimes the founders.

[baileyman]

"invention of Windows" may be a stretch. Perhaps "lifting the idea whole cloth from Apple" would be better. Again, it's really hard to ascribe the CREATION of social value to Microsoft. There is plenty of reason to ascribe the CAPTURE of social value created by others. I'll happily listen to softies who can list the good original things the company has done. It appears that the basic business model, extend monopoly by attaching others' ideas to the original operating system network effect through copyright, dominates this discussion.

[Lafayette]

MT about Barro: The level of output ... depends upon the variety of these intermediate goods, i.e. how many different types are available. Thus, as we discover more "idea-type goods" such as Windows, we are able to produce more output. Therefore, in this model, the invention of Windows and other products adds to the variety of intermediate goods, the increased variety allows more goods to be produced, and the increased output adds social value by allowing higher consumption levels.

The best example of what Barro is talking about, at least as regards the computer industry, is the OEM market. These market players take computer systems and add a purpose-built software application, which they then sell to end-users.

What he calls idea-type products (such as a computer operating system) are technological step-functions. They bring dramatic and sudden change. They are therefor revolutionary and not evolutionary.

What this means, however, is that they are quite extraordinary. They are almost impossible to predict and putting them into any econometric model, to my mind, is hazardous. Their frequency has been measured often in centuries.

Consider the example of the latest, truly revolutionary idea-product - the Internet. Historically, the dissemination of information, other than word of mouth, since the Gutenburg Bible has employed print media. It is some six centuries that it has existed and been employed by mankind.

The Internet has come to replace print media with electronic media and has enhanced its potential distribution. Written works no longer collect in libraries, but in server farms, available to a far larger population than before.

It is the Internet, a revolutionary step-function in the access of information, that changed our ability to collate information, analyze and disseminate it. But, the sheer volume of information has imposed an altogether different problem. That of finding the nugget in a mass of earth.

For the moment search engines display pages of information within which key words are found. The next evolution of the Internet, which will bring it to its fullest potential is to add to the search engine a logical purpose. For the moment, a Boolean search of indexed words has no logical purpose. It is our own mind that lends purpose by sifting through the pages for the information we want.

Barro is suggesting that “idea-products” is the precursor of other products that add social value. He may not be wrong. But, I think it is important to differentiate between revolutionary products and those that simply enhance the original idea’s potential. Both are necessary to realize a full potential.

Maybe Barro is doing precisely this, but it is not apparent (to me) by what is written and even less so when cast into a mathematical formulation. I don’t see the sense of putting such a simple idea into a mathematical model the purpose of which is to measure the “social value” created. Such idea-products are rare and spaced in time. It is the evolution of revolutionary products that is creating the new jobs and therefore economic value. (But, that is a personal opinion.)

Often, the creation of products is at the heart of destruction of existing products, since it sponsors the replacement of the latter. It took less than a century for the jet engine to create air travel that largely replaced rail travel.

It will likely not take another six hundred years for the next revolutionary “idea-product” to happen upon us. But, neither is it the sort of economic agent that one can count upon continually. Truly revolutionary ideas are comparatively random events. Evolutionary products are simply putting one’s mind to a concept towards enhancing its benefit. These are far more industrially prevalent.

Finally, “idea-products” are fine. If they are digital, they are also sexy. But, they are not the only sort of revolutionary products that humanity needs. Humanity clearly requires a revolutionary product that will unhook the production of energy, so necessary to economic growth, from the carbon molecule – thereby assuring that his planet remains habitable.

So, as much as idea-products ostensibly generate enormous follow-on wealth, they are by no means the only kind that a society may need to develop properly.

[reason]

The problem with Microsoft (as before it IBM) of course is that it owns infrastructure (or perhaps better put a standard). This means that the evolution of public resources is controlled by a private monopoly - nobody gets to vote on what happens next. And of course it receives monopoly rent.

Standards are in general a problem - they both reduce costs AND are an impediment to innovation. Eventually the need for innovation will override the cost advantage but it can take a very long time.

My personal opinion is that such standards (where identified) must be fully specified and the copywrite of the standards separated from their implementation to allow the magic of the market to work properly. The standards would still be copywrited and a fee could be charged for using them but other people then would know the inner workings in advance and could program on the basis of published information. In fact I see this everywhere in capitalists economies - the tendency to create monopolies by using secret information to facilitate vertical integration. Of course it is inevitably less secure - security by ignorance is no security at all. Intellectual property needs rethinking from top to bottom - economists could definitely help here. But first they need to recognise that there is a problem.

[Lafayette]

reason: This means that the evolution of public resources is controlled by a private monopoly - nobody gets to vote on what happens next. And of course it receives monopoly rent.

I thought of this, but in the context of what I wanted to say regarding revolutionary and evolutionary ideas, I thought it less important.

Besides, I suspect that MS is the last giant that will be made in software. Proprietary software will be supplanted by open source products in the future. At least, let's hope so. We are seeing the birth of this trend by products not only like Linux, but Wikipedia and a few others that are looking for their niches. There are many areas in which it is applicable, particularly in teaching the arts and sciences.

Still, this trend does not really matter. What Barro is saying remains regardless pretty much true. Idea-products are money spinners because they enable the development of products across a spectrum of customer needs, both consumer and industrial.

(Windows gave MS the platform on which to launch its Office Products suite, which was hugely adopted. It enabled an entire set of office products to be sold.)

We need more thinking in this area, because it is at the heart of a modern society that is increasingly taken away by new and advanced product ideas. Most of them are trashable but some will go on to spin lots and lots of economic value.

[Anonymous]

There are many ways in which a standard provides a social benefit. Posit a world with no Windows, where there are five competing OSes each with 20% market share. Sure, each of the five companies will be worse off than a monopoly provider, but will they be better off?

A monopoly platform saves software developers the costs of cross-platform development. (This also has a minus -- it saves virus and malware writers the costs of cross-platform development too.) It also saves users the costs of learning different platforms. I once watched someone try to figure out Mac OS 9. She just couldn't understand the switching paradigm (what's this menu on the top-right), wheras the Windows 95 taskbar was super-intuitive. Guess what Mac OS X has, a taskbar at the bottom of the screen!

Xerox invented the GUI, and foolishly gave it away to Apple. Did Xerox create the social value or did Apple? Apple brought the GUI to 15% of the market, Microsoft did to everyone. Did Apple create the social value or did Microsoft? Going in the other direction, Digital Research developed CP/M, Microsoft bought a clone, and IBM purchased the clone because Digital's chairman was out flying his private plane (depending on the version of the story you hear). Did Digital, or Microsoft, or IBM create the social value? IBM invented the relational database and published it in a paper. Oracle seized the opportunity and grew to be the biggest database maker. Did IBM create the social value by developing the concept, or did Oracle by making databases available to corporations while IBM dropped the ball? Or IBM, which later responded and provided an alternative to the dominant power (Oracle)? Were they copying Oracle or themselves?

Invention does not equal innovation. Responding to competitors' ideas does not equal lack of innovation. Social value is created both by invention and by exploitation of that invention.

[Lafayette]

anon: Posit a world with no Windows, where there are five competing OSes each with 20% market share. Sure, each of the five companies will be worse off than a monopoly provider, but will they be better off?

May I suggest that it is the customer who should be better off, if competition leads to lower prices? Monopoly profits please only corporations, not customers.

MS/DOS was a kluge of a product, and the Mac's more elegant design should have won out over Windows - but it didn't.

Besides, we are getting away from the subject. Barro's idea, as expressed in the title article of this thread is interesting in that it conceives "precursor" products, which of themselves derive a wider range of products responding to a customer need. This is what computers did, regardless of which OS is employed. Which is why they were originally called "general purpose computers".

It reminds me of Shumpeter's creative destruction. There are numerous examples of where revolutionary product design made obsolete entire industries. For example, air transport replacing rail transport. Flight replaced overland bulk transport for long distances. But, it did not go on to enable other derived services. It should have, but didn't. It did not replace the car, for instance. Nor commuter rail lines.

Lasers are another example of a revolutionary product that evolved into multiple uses: communications, digital recording, medical, etc.

Barro focused upon "idea products" because they have far more ability to innovate further derivative products. It is these latter which provoke the larger economic value. (He should drop the usage of the phrase "social value", which is an altogether different matter.)

[Lafayette]

What is social value and why we should not confuse it

Anytime an author wants to distinguish an idea, they employ a marketing technique called product differentiation. That is, they respond to the challenge, "How do I make my product distinguish itself from others in the pack?"

Journalists will try to distinguish their work from some one else's with eye-catcher phrases. Barro, in this thread's title article, employs the phrase "social value". What does this mean?

It means "of value to society" and no one can doubt that hi-tech brings value to society. But, it is confusing as well. Is it economic value or social value that is predominant?

Social value has numerous connotations. A characteristic of social value is "amplitude", meaning it encompasses all elements of a society. Sorry, Windows is not of permanent value to just everyone, even though it is the most ubiquitous graphic-user-interface on the planet.

Health care and education are social values, I submit, of far greater consequence/importance to a society than, er, Windows Vista (for instance). Or Linux. Or Apple. Or, whatever gadget one happens to fancy. I can do without Windows, but can I do without adequate health care? Durable employment? Or the skills necessary for a decent job?

Economic value, to my mind, is the more appropriate phrase when talking about agents/factors that stimulate economic development. Here, high-tech is clearly such a candidate.

If I insist on this, there is a reason. Barro is confusing the word, largely because the phrase “social value” has few connotations in American English. This is not the case in Europe, not even in British English. Why? Because, contrary to Europe, America never struggled for "social equity". It fought to be free, but never a class struggle as has often, in the course of history, been the case in Europe.

Social values are human rights that are considered as birthrights. Meaning, one is born intrinsically with the right, and it is not bestowed either by a Constitution or by laws.

Liberty is just such a right. No constitution gives a nation liberty, it simply guarantees its presence and endurance. Liberty is a birthright, as intrinsic as life itself.

Fairness is another social value. Fairness is the precept that whilst we may not all be equal in competence, we should all be equal in opportunity/treatment. Which translates into: If I try to succeed, my only impediments to success are my own inabilities - not the colour of my skin, not my religious beliefs (or the lack of them), not where I come from, not the school I didn't go to ... and, certainly, not the poverty I inherited from my parents.

It's an important distinction, I think. Particularly in America, which does not yet appreciate how much our penchant for exaggerated wealth, in fact, limits opportunity of the many quite simply because it concentrates it in the few.

[paine]

barro ...i love you

magic formula
for
IP- monop
love potion #13

beware cone hatted figures
juggling college math
and calling
innocent universal forms and
a few greek letters
a model of
this that and the other thing too

[Anonymous]

Lafayette: I think it is reasonably clear that I meant to say "will consumers be better off?", as the next paragraph discusses ways in which a dominant platform provides benefits. Maybe consumers will benefit from lower prices on the OS, and pay for it with higher prices on other software. Or maybe they'll make up for those higher prices with less incidence of viruses. Or maybe the money saved from viruses is more than made up for by money lost in interoperability problems (let's not forget, standards bodies move a lot more slowly than de-facto standards set in the marketplace).

There are too many variables in a network-effect model to be satisfied with a simplistic analysis that blindly assumes more competition is better than less. (e.g. look at the XM/Sirius merger.)

[paine]

key trick move caught on camera by brad

"The presence of the last term in Eq. (1)
will imply that total gross output, Y, is proportional to N, "

sounds innocent to the rubery
especially with this cover story

" this property will allow for endogenous growth in dynamic models where N grows due to R&D activity"
but why add a dynamic feature if
"The present analysis considers
only one-time shifts in N " ???

because it allows the fudge result
Barro and his backers want

in Barronia
even duplicate ideas
like big pharma produces and
thru market muscle forces onto the scene
have an " exchange value "
at least equal to their cost

[paine]

sorry to go on and on
since this thread has been left in the wake of all first class minds

but the reward for innovation should be kept apparent from the reward of enterprise

the reward of enterprise is a function
of dynamic factors
in first -> surplus profits
in this case until entering mimics
lower relative price to average returns
but a monopoly granted by buying a patent
that requires a new player to also get around the patent
and thus fortifying
market position and then maybe adding on
your own me toos to your installed network ....

obviously this is a capture move
not creation comp
as nicely and concisely
said above by baileyman

using one of a dozen "innovation" varients
to found a monopoly based in market share scaling effects
makes the soft touch a monopolu gig not an innovation reward
yes much as
IBM HAD FOR 60 OR SO YEARS

[Lafayette]

anon: Maybe consumers will benefit from lower prices on the OS, and pay for it with higher prices on other software.

I don't follow you. Or, you don't understand me.

Competition works in every direction. Why shouldn't add-on software also benefit from the competition and thus lower prices?

A dominant market player has never benefited customers, which is why, for instance, Microsoft was found guilty of abuse of a dominant market position.

No one went to jail, the management was slapped on the wrist, and the company placed under temporary surveillance - it was all swept under the rug ... in full view of everyone.

Before the law, some people seem to be more equal than others.