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...