Mathiness in the Theory of Economic Growth
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American Economic Review: Papers & Proceedings 2015, 105(5): 89–93
http://dx.doi.org/10.1257/aer.p20151066
Mathiness in the Theory of Economic Growth†
By Paul M. Romer*
Politics does not lead to a broadly shared con- politics when she waged her campaign against
sensus. It has to yield a decision, whether or not capital and the aggregate production function.
a consensus prevails. As a result, political insti- Academic politics, like any other type of pol-
tutions create incentives for participants to exag- itics, is better served by words that are evocative
gerate disagreements between factions. Words and ambiguous, but if an argument is transpar-
that are evocative and ambiguous better serve ently political, economists interested in science
factional interests than words that are analytical will simply ignore it. The style that I am calling
and precise. mathiness lets academic politics masquerade
Science is a process that does lead to a broadly as science. Like mathematical theory, mathi-
shared consensus. It is arguably the only social ness uses a mixture of words and symbols, but
process that does. Consensus forms around the- instead of making tight links, it leaves ample
oretical and empirical statements that are true. room for slippage between statements in natu-
Tight links between words from natural lan- ral versus formal language and between state-
guage and symbols from the formal language of ments with theoretical as opposed to empirical
mathematics encourage the use of words that are content.
analytical and precise. Solow’s (1956) mathematical theory of
For the last two decades, growth theory has growth mapped the word “capital” onto a vari-
made no scientific progress toward a consensus. able in his mathematical equations, and onto
The challenge is how to model the scale effects both data from national income accounts and
introduced by nonrival ideas. Mobile telephony objects like machines or structures that some-
is the update to the pin factory, the demonstra- one could observe directly. The tight connection
tion that scale effects are too important to ignore. between the word and the equations gave the
To accommodate them, many growth theorists word a precise meaning that facilitated equally
have embraced monopolistic competition, but tight connections between theoretical and empir-
an influential group of traditionalists continues ical claims. Gary Becker’s (1962) mathematical
to support price taking with external increas- theory of wages gave the words “human capital”
ing returns. The question posed here is why the the same precision and established the same two
methods of science have failed to resolve the types of tight connection—between words and
disagreement between these two groups. math and between theory and evidence. In this
Economists usually stick to science. Robert case as well, the relevant evidence ranged from
Solow (1956) was engaged in science when he aggregate data to formal microeconomic data to
developed his mathematical theory of growth. direct observation.
But they can get drawn into academic politics. In contrast, McGrattan and Prescott (2010)
Joan Robinson (1956) was engaged in academic give a label—location—to their proposed new
input in production, but the mathiness that they
* Stern School of Business, New York University, 44 W.
present does not provide the microeconomic
4th St, New York, NY 10012 (e-mail: promer@stern.nyu. foundation needed to give the label meaning.
edu). An appendix with supporting materials is available The authors chose a word that had already
from the author’s website, paulromer.net, and from the web- been given a precise meaning by mathemati-
site for this article. Support for this work was provided by cal theories of product differentiation and eco-
the Rockefeller Foundation. .
†
Go to http://dx.doi.org/10.1257/aer.p20151066 to visit nomic geography, but their formal equations are
the article page for additional materials and author disclo- completely different, so neither of those mean-
sure statement. ings carries over.
8990 AEA PAPERS AND PROCEEDINGS MAY 2015
The mathiness in their paper also offers lit- will be worth little, but cheap to produce, so it
tle guidance about the connections between its might survive as entertainment.
theoretical and empirical statements. The quan- Economists have a collective stake in flushing
tity of location has no unit of measurement. The mathiness out into the open. We will make faster
term does not refer to anything a person could scientific progress if we can continue to rely on
observe. In a striking (but instructive) use of the clarity and precision that math brings to our
slippage between theoretical and the empirical shared vocabulary, and if, in our analysis of data
claims, the authors assert, with no explanation, and observations, we keep using and refining the
that the national supply of location is propor- powerful abstractions that mathematical theory
tional to the number of residents. This raises highlights—abstractions like physical capital,
questions that the equations of the model do not human capital, and nonrivalry.
address. If the dependency ratio and population
increase, holding the number of working age I. Scale Effects
adults and the supply of labor constant, what
mechanism leads to an increase in output? In 1970, there were zero mobile phones.
McGrattan and Prescott (2010) is one of sev- Today, there are more than 6 billion. This is the
eral papers by traditionalists that use mathiness kind of development that a theory of growth
to campaign for price-taking models of growth. should help us understand.
The natural inference is that their use of mathi- Let q stand for individual consumption of
ness signals a shift from science to academic mobile phone services. For a ∈ [0, 1], let
politics, presumably because they were losing p = D(q) = q−a be the inverse individ-
the scientific debate. If so, the paralysis and ual demand curve with all-other-goods as
polarization in the theory of growth is not sign numeraire. Let Ndenote the number of people in
of a problem with science. It is the expected out- the market. Once the design for a mobile phone
come in politics. exists, let the inverse supply curve for an aggre-
If mathiness were used infrequently to gate quantity Q = qNtake the form p = S(Q)
slow convergence to a new scientific consen- = Qbfor b ∈ [0, ∞].
sus, it would do localized, temporary damage. If the price and quantity of mobile phones are
Unfortunately, the market for lemons tells us determined by equating D(q) = m × S(Nq),so
that as the quantity increases, mathiness could that m ≥ 1 captures any markup of price rela-
do permanent damage because it takes costly tive to marginal cost, the surplus S created by the
effort to distinguish mathiness from mathemat- discovery of mobile telephony takes the form
ical theory.
The market for mathematical theory can sur- a(1+b)
S = C(a, b, m) × N a+b ,
_____
vive a few lemon articles filled with mathiness.
Readers will put a small discount on any article
with mathematical symbols, but will still find where C(a, b, m) is a messy algebraic expres-
it worth their while to work through and verify sion. Surplus scales as N to a power between a
that the formal arguments are correct, that the and 1. If b = 0,so that the supply curve for the
connection between the symbols and the words devices is horizontal, surplus scales linearly in
is tight, and that the theoretical concepts have N .If, in addition, a = __
12 ,the expression for sur-
implications for measurement and observation. plus simplifies to
But after readers have been disappointed too
often by mathiness that wastes their time, they S = _____
2m − 1
N.
will stop taking seriously any paper that contains 2
m
mathematical symbols. In response, authors will
stop doing the hard work that it takes to supply With these parameters, a tax or a monopoly
real mathematical theory. If no one is putting in markup that increases m from 1to 2causes Sto
the work to distinguish between mathiness and change by the factor 0.75. An increase in Nfrom
mathematical theory, why not cut a few corners something like 102 people in a village to 1010
and take advantage of the slippage that mathi- people in a connected global market causes S to
ness allows? The market for mathematical the- change by the factor 108.
ory will collapse. Only mathiness will be left. It Effects this big tend to focus the mind.VOL. 105 NO. 5 MATHINESS IN THE THEORY OF ECONOMIC GROWTH 91
II. The Fork in Growth Theory As many growth theorists followed trade
theorists and explored aggregate models with
The traditional way to include a scale effect monopolistic competition, the traditionalists
was proposed by Marshall (1890). One writes who worked on models with a microeconomic
the production of telephone services at each foundation maintained their commitment to price
of a large number of firms in an industry as taking and adhered to the restriction of 0 percent
g(X ) f (x), where the list x contains the inputs excludability of ideas required for Marshallian
that the firm controls and the list X has inputs external increasing returns. Perhaps because of
for the entire industry. One obvious problem unresolved questions about the extent of spill-
with this approach is that it offers no basis for overs, attention turned to models of idea flows
determining the extent is of the spillover bene- that require face-to-face interaction. Because
fits from the term g(X ). Do they require face-to- incentives in these models motivate neither
face interaction? Production in the same city, the discovery nor diffusion, agents exchange ideas
same country, or anywhere? in the same way that gas molecules exchange
If we split x = (a, z) into a nonrival input a energy—involuntarily, through random encoun-
and rival inputs z , a standard replication argu- ters. Given the sharp limits imposed by the
ment implies that f must be homogeneous of mathematics of their formal framework, it is no
degree 1 in the rival inputs z. Euler’s theorem surprise that traditionalists were attracted to the
then implies that the value of output equals the extra degrees of freedom that come from letting
compensation paid to the rival inputs z .In a full the words slip free of the math.
equilibrium analysis, anything that looks like
producer surplus or “Marshallian rent” is in fact III. Examples of Mathiness
part of the compensation paid to the rival inputs.
It follows that there can be no nonrival input McGrattan and Prescott (2010) establish
a that the firm can use yet exclude other firms loose links between a word with no meaning
from using. Production for an individual firm and new mathematical results. The mathiness
must take the form A f (z) where A is both non- in “Perfectly Competitive Innovation” (Boldrin
rival and fully nonexcludable, hence a public and Levine 2008) takes the adjectives from
good. the title of the paper, which have a well estab-
I started by my work on growth using price lished, tight connection to existing mathemati-
taking and external increasing returns, but cal results, and links them to a very different set
switched to monopolistic competition because of mathematical results. In an initial period, the
it allows for the possibility that ideas can be at innovator in their model is a monopolist, the sole
least partially excludable. Partial excludability supplier of a newly developed good. The authors
offers a much more precise way to think about force the monopolist to take a specific price for
spillovers. Nonrivalry, which is logically inde- its own good as given by imposing price taking
pendent, is the defining characteristic of an idea as an assumption about behavior.
and the source of the scale effects that are cen- In addition to using words that do not align
tral to any plausible explanation of recent expe- with their formal model, Boldrin and Levine
rience with mobile telephony or more generally, (2008) make broad verbal claims that are discon-
of the broad sweep of human history (Jones and nected from any formal analysis. For example,
Romer 2010). they claim that the argument based on Euler’s
In models that allow for partial excludability theorem does not apply because price equals
of nonrival goods, ideas need not be treated as marginal cost only in the absence of capacity
pure public goods. In these models, firms have constraints. Robert Lucas uses the same kind of
an incentive to discover a new idea like a mobile untethered verbal claim to dismiss any role for
phone (Romer 1990) or to encourage interna- books or blueprints in a model of ideas: “Some
tional diffusion of such an idea once it exists knowledge can be ‘embodied’ in books, blue-
(Romer 1994). In such models, one can ask why prints, machines, and other kinds of physical
some valuable nonrival ideas diffuse much more capital, and we know how to introduce capital
slowly than mobile telephony and how policy into a growth model, but we also know that
can influence the rate of diffusion by changing doing so does not by itself provide an engine
the incentives that firms face. of sustained growth.” (Lucas 2009, p.6). In92 AEA PAPERS AND PROCEEDINGS MAY 2015
each case, well-known models show that these The mathiness here involves more than a
verbal claims are false. Any two-sector growth nonstandard interpretation of the phrase “obser-
model will show how Marshall’s style of partial vationally equivalent.” The underlying formal
equilibrium analysis leads Boldrin and Levine result is that calculating the double limit in one
astray. Any endogenous growth model with an
order limβ→0 (limT→∞
g[β : B ⇒ P]) yields one
expanding variety of capital goods or a ladder answer, γ, which is also the limiting growth rate
of capital goods of improving quality serves as a in the P economy. However, calculating it in
counter-example to the result that Lucas claims the other order, limT→∞
(limβ→0
g[β : B ⇒ P]),
that we know. gives a different answer, 0. Lucas and Moll
In Lucas and Moll (2014), the mathiness (2014) use the first calculation to justify their
involves both words that are disconnected from claim about observational equivalence. An argu-
the formal results and a mathematical model that ment that takes the math seriously would note
is not well specified. The baseline model in their that the double limit does not exist and would
paper relies on an assumption P that invokes a caution against trying to give an interpretation to
distribution for the initial stock of knowledge the value calculated using one order or the other.
across workers that is unbounded, with a fat
Pareto tail. Given this assumption, Lucas and IV. A New Equilibrium in the Market for
Moll show that the diffusion of knowledge from Mathematical Economics
random encounters between workers generates
a growth rate g[P](t)that converges to γ > 0as As is noted in an addendum, Lucas (2009)
tgoes to infinity. contains a flaw in a proof. The proof requires that
α
Assumption P is hard to justify because it γ be less than 1
a fraction __ .The same page has an
γ
requires that at time zero, someone is already expression for γ,γ = α ____, and because α, γ,
using every productive technology that will ever γ+δ
be used at any future date. So the authors offer αγ is greater
and δare all positive, it implies that __
“an alternative interpretation that we argue is than 1.Anyone who does math knows that it is
observationally equivalent: knowledge at time 0 distressingly easy to make an oversight like this.
is bounded but new knowledge arrives at arbi- It is not a sign of mathiness by the author. But
trarily low frequency.” (Lucas and Moll 2014, the fact that this oversight was not picked up at
p.11). In this alternative, there is a collection of the working paper stage or in the process leading
economies that all start with an assumption B up to publication may tell us something about
(for bounded initial knowledge.) By itself, this the new equilibrium in economics. Neither col-
assumption implies that the growth rate goes leagues who read working papers, nor review-
to zero as everyone learns all there is to know. ers, nor journal editors, are paying attention to
However, new knowledge, drawn from a distri- the math.
bution with a Pareto tail, is injected at the rate β, After reading their working paper, I told
so a B economy eventually turns into a P econ- Lucas and Moll about the discontinuity in the
omy. As the arrival rate β gets arbitrarily low, limit and the problem it posed for their claim
an arbitrarily long period of time has to elapse about observational equivalence. They left their
before the switch from B to P takes place. (See limit argument in the paper without noting
the online Appendix for details.) the discontinuity and the Journal of Political
For a given value of β > 0, let β : B ⇒ P Economy published it this way. This may reflect
denote a specific economy from this collection. a judgment by the authors and the editors that at
Any observation on the growth rate has to take least in the theory of growth, we are already in a
place at a finite date T. If T is large enough, new equilibrium in which readers expect mathi-
g[P](T )will be close to γ
,but g[β : B ⇒ P](T ) ness and accept it.
will be arbitrarily close to 0 for an arbitrarily One final bit of evidence comes from Piketty
low arrival rate β. This means that any set of and Zucman (2014), who cite a result from a
observations on growth rates will show that the growth model: with a fixed saving rate, when the
P economy is observably different from any growth rate falls by one-half, the ratio of wealth
economy β : B ⇒ P with a low enough value to income doubles. They note that their formula
of β.They are not observationally equivalent in W/Y = s/g assumes that national income
any conventional sense. and the saving rate s are both measured net ofVOL. 105 NO. 5 MATHINESS IN THE THEORY OF ECONOMIC GROWTH 93
d epreciation. They observe that the formula has organized their look at history without access
to be modified to W /Y = s/(g + δ), with a to the abstraction we know as capital? Where
depreciation rate δ,when it is stated in terms of would we be now if Robert Solow’s math had
the gross saving rate and gross national income. been swamped by Joan Robinson’s mathiness?
From Krusell and Smith (2014), I learned
more about this calculation. If the growth rate
falls and the net saving rate remains constant,
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