89

American Economic Review: Papers & Proceedings 2015, 105(5): 89–93
http://dx.doi.org/10.1257/aer.p20151066

Politics does not lead to a broadly shared con-

sensus. It has to yield a decision, whether or not 

a consensus prevails. As a result, political insti-

tutions create incentives for participants to exag-

gerate disagreements between factions. Words 

that are evocative and ambiguous better serve 

factional interests than words that are analytical 

and precise.

Science is a process that does lead to a broadly 

shared consensus. It is arguably the only social 

process that does. Consensus forms around the-

oretical and empirical statements that are true. 

Tight links between words from natural lan-

guage and symbols from the formal language of 

mathematics encourage the use of words that are 

analytical and precise.

For the last two decades, growth theory has 

made no scientific progress toward a consensus. 

The challenge is how to model the scale effects 

introduced by nonrival ideas. Mobile telephony 

is the update to the pin factory, the demonstra-

tion that scale effects are too important to ignore. 

To accommodate them, many growth theorists 

have embraced monopolistic competition, but 

an influential group of traditionalists continues 

to support price taking with external increas-

ing returns. The question posed here is why the 

methods of science have failed to resolve the 

disagreement between these two groups.

Economists usually stick to science. Robert 

Solow 

(1956) was engaged in science when he 

developed his mathematical theory of growth. 

But they can get drawn into academic politics. 

Joan Robinson 

(1956) was engaged in academic 

Mathiness in the Theory of Economic Growth

By 

Paul M. Romer*

* Stern School of Business, New York University, 44 W. 

4th St, New York, NY 10012 

(e-mail: promer@stern.nyu.

edu

). An appendix with supporting materials is available 

from the author’s website, paulromer.net, and from the web-
site for this article. Support for this work was provided by 
the Rockefeller Foundation. .

† 

Go to http://dx.doi.org/10.1257/aer.p20151066 to visit 

the article page for additional materials and author disclo-
sure statement.

politics when she waged her campaign against 

capital and the aggregate production function.

Academic politics, like any other type of pol-

itics, is better served by words that are evocative 

and ambiguous, but if an argument is transpar-

ently political, economists interested in science 

will simply ignore it. The style that I am calling 

mathiness lets academic politics masquerade 

as science. Like mathematical theory, mathi-

ness uses a mixture of words and symbols, but 

instead of making tight links, it leaves ample 

room for slippage between statements in natu-

ral versus formal language and between state-

ments with theoretical as opposed to empirical 

content.

Solow’s 

(1956)  mathematical theory of 

growth mapped the word “capital” onto a vari-

able in his mathematical equations, and onto 

both data from national income accounts and 

objects like machines or structures that some-

one could observe directly. The tight connection 

between the word and the equations gave the 

word a precise meaning that facilitated equally 

tight connections between theoretical and empir-

ical claims. Gary Becker’s 

(1962) mathematical 

theory of wages gave the words “human capital” 

the same precision and established the same two 

types of tight connection—between words and 

math and between theory and evidence. In this 

case as well, the relevant evidence ranged from 

aggregate data to formal microeconomic data to 

direct observation.

In contrast, McGrattan and Prescott 

(2010) 

give a label—location—to their proposed new 

input in production, but the mathiness that they 

present does not provide the microeconomic 

foundation needed to give the label meaning. 

The authors chose a word that had already 

been given a precise meaning by mathemati-

cal theories of product differentiation and eco-

nomic geography, but their formal equations are 

 completely different, so neither of those mean-

ings carries over.

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The mathiness in their paper also offers lit-

tle guidance about the connections between its 

theoretical and empirical statements. The quan-

tity of location has no unit of measurement. The 

term does not refer to anything a person could 

observe. In a striking 

(but instructive) use of 

slippage between theoretical and the empirical 

claims, the authors assert, with no explanation, 

that the national supply of location is propor-

tional to the number of residents. This raises 

questions that the equations of the model do not 

address. If the dependency ratio and population 

increase, holding the number of working age 

adults and the supply of labor constant, what 

mechanism leads to an increase in output?

McGrattan and Prescott 

(2010) is one of sev-

eral papers by traditionalists that use mathiness 

to campaign for price-taking models of growth. 

The natural inference is that their use of mathi-

ness signals a shift from science to academic 

politics, presumably because they were losing 

the scientific debate. If so, the paralysis and 

polarization in the theory of growth is not sign 

of a problem with science. It is the expected out-

come in politics.

If mathiness were used infrequently to 

slow convergence to a new scientific consen-

sus, it would do localized, temporary damage. 

Unfortunately, the market for lemons tells us 

that as the quantity increases, mathiness could 

do permanent damage because it takes costly 

effort to distinguish mathiness from mathemat-

ical theory.

The market for mathematical theory can sur-

vive a few lemon articles filled with mathiness. 

Readers will put a small discount on any article 

with mathematical symbols, but will still find 

it worth their while to work through and verify 

that the formal arguments are correct, that the 

connection between the symbols and the words 

is tight, and that the theoretical concepts have 

implications for measurement and observation. 

But after readers have been disappointed too 

often by mathiness that wastes their time, they 

will stop taking seriously any paper that contains 

mathematical symbols. In response, authors will 

stop doing the hard work that it takes to supply 

real mathematical theory. If no one is putting in 

the work to distinguish between mathiness and 

mathematical theory, why not cut a few corners 

and take advantage of the slippage that mathi-

ness allows? The market for mathematical the-

ory will collapse. Only mathiness will be left. It 

will be worth little, but cheap to produce, so it 

might survive as entertainment.

Economists have a collective stake in flushing 

mathiness out into the open. We will make faster 

scientific progress if we can continue to rely on 

the clarity and precision that math brings to our 

shared vocabulary, and if, in our analysis of data 

and observations, we keep using and refining the 

powerful abstractions that mathematical theory 

highlights—abstractions like physical capital, 

human capital, and nonrivalry.

I.  Scale Effects

In 1970, there were zero mobile phones. 

Today, there are more than 6 billion. This is the 

kind of development that a theory of growth 

should help us understand.

Let   q  stand for individual consumption of 

mobile phone services. For  a 

∈  [0, 1],  let 

 

p

 

=  D(q)  =   q   

a

   be the inverse individ-

ual demand curve with all-other-goods as 

numeraire. Let  N  denote the number of people in 

the market. Once the design for a mobile phone 

exists, let the inverse supply curve for an aggre-

gate quantity  Q 

=  qN  take the form  p  =  S(Q)  

=   Q   

b

   for  b 

∈  [0, ∞]. 

If the price and quantity of mobile phones are 

determined by equating  D

(q)  =  m × S(Nq),  so 

that  m 

≥  1  captures any markup of price rela-

tive to marginal cost, the surplus  S  created by the 

discovery of mobile telephony takes the form

 

 S 

=  C(abm) ×  N   

  

a

(1+b)

 

_____ 

a

+b

 

 

 

where   C

(abm)  is a messy algebraic expres-

sion. Surplus scales as  N  to a power between  a  

and  1 . If  b 

=  0,  so that the supply curve for the 

devices is horizontal, surplus scales linearly in  

N

 .  If, in addition,  a 

=    

1

 

__

 

2

  ,  the expression for sur-

plus simplifies to

 

 S 

=    

2m − 1

 

_____

 

 m   

2

 

    N

With these parameters, a tax or a monopoly 

markup that increases  m  from  1  to  2  causes  S  to 

change by the factor  0.75 . An increase in  N  from 

something like   10   

2

   people in a village to   10   

10

   

people in a connected global market causes  S  to 

change by the factor   10   

8

  .

Effects this big tend to focus the mind.

VOL. 105 NO. 5

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MATHINESS IN THE THEORY OF ECONOMIC GROWTH

II.  The Fork in Growth Theory

The traditional way to include a scale effect 

was proposed by Marshall 

(1890). One writes 

the production of telephone services at each 

of a large number of firms in an industry as  

 g

((x) , where the list  x  contains the inputs 

that the firm controls and the list  X  has inputs 

for the entire industry. One obvious problem 

with this approach is that it offers no basis for 

determining the extent is of the spillover bene-

fits from the term  g

() . Do they require face-to-

face interaction? Production in the same city, the 

same country, or anywhere?

If we split  x 

=  (az)  into a nonrival input  a  

and rival inputs  z,  a standard replication argu-

ment implies that  f  must be homogeneous of 

degree   1  in the rival inputs  z . Euler’s theorem 

then implies that the value of output equals the 

compensation paid to the rival inputs  z .  In a full 

equilibrium analysis, anything that looks like 

producer surplus or “Marshallian rent” is in fact 

part of the compensation paid to the rival inputs.

It follows that there can be no nonrival input  

a

  that the firm can use yet exclude other firms 

from using. Production for an individual firm 

must take the form  A f 

(z)  where  A  is both non-

rival and fully nonexcludable, hence a public 

good.

I started by my work on growth using price 

taking and external increasing returns, but 

switched to monopolistic competition because 

it allows for the possibility that ideas can be at 

least partially excludable. Partial excludability 

offers a much more precise way to think about 

spillovers. Nonrivalry, which is logically inde-

pendent, is the defining characteristic of an idea 

and the source of the scale effects that are cen-

tral to any plausible explanation of recent expe-

rience with mobile telephony or more generally, 

of the broad sweep of human history 

(Jones and 

Romer 2010

).

In models that allow for partial excludability 

of nonrival goods, ideas need not be treated as 

pure public goods. In these models, firms have 

an incentive to discover a new idea like a mobile 

phone 

(Romer 1990) or to encourage interna-

tional diffusion of such an idea once it exists 

(Romer 1994). In such models, one can ask why 

some valuable nonrival ideas diffuse much more 

slowly than mobile telephony and how policy 

can influence the rate of diffusion by changing 

the incentives that firms face.

As many growth theorists followed trade 

theorists and explored aggregate models with 

monopolistic competition, the traditionalists 

who worked on models with a microeconomic 

foundation maintained their commitment to price 

taking and adhered to the restriction of 0 percent 

excludability of ideas required for Marshallian 

external increasing returns. Perhaps because of 

unresolved questions about the extent of spill-

overs, attention turned to models of idea flows 

that require face-to-face interaction. Because 

incentives in these models motivate neither 

discovery nor diffusion, agents exchange ideas 

in the same way that gas molecules exchange 

energy—involuntarily, through random encoun-

ters. Given the sharp limits imposed by the 

mathematics of their formal framework, it is no 

surprise that traditionalists were attracted to the 

extra degrees of freedom that come from letting 

the words slip free of the math.

III.  Examples of Mathiness

McGrattan and Prescott 

(2010) establish 

loose links between a word with no meaning 

and new mathematical results. The mathiness 

in “Perfectly Competitive Innovation” 

(Boldrin 

and Levine 2008

) takes the adjectives from 

the title of the paper, which have a well estab-

lished, tight connection to existing mathemati-

cal results, and links them to a very different set 

of mathematical results. In an initial period, the 

innovator in their model is a monopolist, the sole 

supplier of a newly developed good. The authors 

force the monopolist to take a specific price for 

its own good as given by imposing price taking 

as an assumption about behavior.

In addition to using words that do not align 

with their formal model, Boldrin and Levine 

(2008) make broad verbal claims that are discon-

nected from any formal analysis. For example, 

they claim that the argument based on Euler’s 

theorem does not apply because price equals 

marginal cost only in the absence of capacity 

constraints. Robert Lucas uses the same kind of 

untethered verbal claim to dismiss any role for 

books or blueprints in a model of ideas: “Some 

knowledge can be ‘embodied’ in books, blue-

prints, machines, and other kinds of physical 

capital, and we know how to introduce capital 

into a growth model, but we also know that 

doing so does not by itself provide an engine 

of sustained growth.” 

(Lucas 2009, p.6). In 

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each case, well-known models show that these 

verbal claims are false. Any two-sector growth 

model will show how Marshall’s style of partial 

equilibrium analysis leads Boldrin and Levine 

astray. Any endogenous growth model with an 

expanding variety of capital goods or a ladder 

of capital goods of improving quality serves as a 

counter-example to the result that Lucas claims 

that we know.

In Lucas and Moll 

(2014), the mathiness 

involves both words that are disconnected from 

the formal results and a mathematical model that 

is not well specified. The baseline model in their 

paper relies on an assumption  P  that invokes a 

distribution for the initial stock of knowledge 

across workers that is unbounded, with a fat 

Pareto tail. Given this assumption, Lucas and 

Moll show that the diffusion of knowledge from 

random encounters between workers generates 

a growth rate  g

[P](t)  that converges to  γ  >  0  as  

t

  goes to infinity.

Assumption   P  is hard to justify because it 

requires that at time zero, someone is already 

using every productive technology that will ever 

be used at any future date. So the authors offer 

“an alternative interpretation that we argue is 

observationally equivalent: knowledge at time 0 

is bounded but new knowledge arrives at arbi-

trarily low frequency.” 

(Lucas and Moll 2014, 

p.11

). In this alternative, there is a collection of 

economies that all start with an assumption  B  

(for bounded initial knowledge.) By itself, this 

assumption implies that the growth rate goes 

to zero as everyone learns all there is to know. 

However, new knowledge, drawn from a distri-

bution with a Pareto tail, is injected at the rate  

β  , 

so a  B  economy eventually turns into a  P  econ-

omy. As the arrival rate  

β  gets arbitrarily low, 

an arbitrarily long period of time has to elapse 

before the switch from  B  to  P  takes place. 

(See 

the online Appendix for details.

)

For a given value of  

β  >  0,  let  β : B  ⇒  P  

denote a specific economy from this collection. 

Any observation on the growth rate has to take 

place at a finite date  T.  If  T  is large enough,  

 g

[P]()  will be close to  γ,  but  g[β : B  ⇒  P]()  

will be arbitrarily close to 0 for an arbitrarily 

low arrival rate  

β.  This means that any set of 

observations on growth rates will show that the  

P

  economy is observably different from any 

economy   

β : B  ⇒  P  with a low enough value 

of  

β.  They are not observationally equivalent in 

any conventional sense.

The mathiness here involves more than a 

nonstandard interpretation of the phrase “obser-

vationally equivalent.” The underlying formal 

result is that calculating the double limit in one 

order   lim  

β→0

  

 

   

( lim  

T

→∞

  

 

    g

[β : B  ⇒  P])  yields one 

answer,  

γ  , which is also the limiting growth rate 

in the  P  economy. However, calculating it in 

the other order,   lim  

T

→∞

  

 

   

( lim  

β→0

  

 

    g

[β : B  ⇒  P]),  

gives a different answer,  0.  Lucas and Moll 

(2014) use the first calculation to justify their 

claim about observational equivalence. An argu-

ment that takes the math seriously would note 

that the double limit does not exist and would 

caution against trying to give an interpretation to 

the value calculated using one order or the other.

IV.  A New Equilibrium in the Market for 

Mathematical Economics

As is noted in an addendum, Lucas 

(2009) 

contains a flaw in a proof. The proof requires that 
a fraction    

α

 

__

 

γ

    be less than  1.  The same page has an 

expression for  

γ,   γ  =  α   

γ 

____ 

γ + δ 

   , and because  α, γ,  

and  

δ  are all positive, it implies that    

α 

__ 

γ 

   is greater 

than  1.  Anyone who does math knows that it is 

distressingly easy to make an oversight like this. 

It is not a sign of mathiness by the author. But 

the fact that this oversight was not picked up at 

the working paper stage or in the process leading 

up to publication may tell us something about 

the new equilibrium in economics. Neither col-

leagues who read working papers, nor review-

ers, nor journal editors, are paying attention to 

the math.

After reading their working paper, I told 

Lucas and Moll about the discontinuity in the 

limit and the problem it posed for their claim 

about observational equivalence. They left their 

limit argument in the paper without noting 

the discontinuity and the Journal of Political 

Economy

 published it this way. This may reflect 

a judgment by the authors and the editors that at 

least in the theory of growth, we are already in a 

new equilibrium in which readers expect mathi-

ness and accept it.

One final bit of evidence comes from Piketty 

and Zucman 

(2014), who cite a result from a 

growth model: with a fixed saving rate, when the 

growth rate falls by one-half, the ratio of wealth 

to income doubles. They note that their formula  

W

/Y  =  s/g  assumes that national income 

and the saving rate  s  are both measured net of 

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MATHINESS IN THE THEORY OF ECONOMIC GROWTH

 depreciation. They observe that the formula has 

to be modified to  W

/Y  =  s/(g + δ),  with a 

depreciation rate  

δ,  when it is stated in terms of 

the gross saving rate and gross national income.

From Krusell and Smith 

(2014), I learned 

more about this calculation. If the growth rate 

falls and the net saving rate remains constant, 

the gross saving rate has to increase. For exam-

ple, with a fixed net saving rate of  10 percent  

and a depreciation rate of  3 percent,  a reduction 

in the growth rate from  3 percent  to  1.5 percent  

implies an increase in the gross saving rate from  

18 percent  to  25 percent.  This means that the 

expression  s

/(g + δ)  increases by a factor  1.33  

because of the direct effect of the fall in  g  and by 

a factor  1.38  because of the induced change in  s .  

A third factor, equal to  1.09  , arises because the 

fall in  g  also increases the ratio of gross income 

to net income. These three factors, which when 

multiplied equal  2,  decompose the change in  
W

/Y  calculated in net terms into equivalent 

changes for a model with variables measured in 

gross terms.

Piketty and Zucman 

(2014) present their data 

and empirical analysis with admirable clarity 

and precision. In choosing to present the theory 

in less detail, they too may have responded to 

the expectations in the new equilibrium: empir-

ical work is science; theory is entertainment. 

Presenting a model is like doing a card trick. 

Everybody knows that there will be some sleight 

of hand. There is no intent to deceive because 

no one takes it seriously. Perhaps our norms will 

soon be like those in professional magic; it will 

be impolite, perhaps even an ethical breach, to 

reveal how someone’s trick works.

When I learned mathematical economics, a 

different equilibrium prevailed. Not universally, 

but much more so than today, when economic 

theorists used math to explore abstractions, 

it was a point of pride to do so with clarity, 

precision, and rigor. Then too, a faction like 

Robinson’s that risked losing a battle might 

resort to mathiness as a last-ditch defense, but 

doing so carried a risk. Reputations suffered.

If we have already reached the lemons market 

equilibrium where only mathiness is on offer, 

future generations of economists will suffer. 

After all, how would Piketty and Zucman have 

organized their look at history without access 

to the abstraction we know as capital? Where 

would we be now if Robert Solow’s math had 

been swamped by Joan Robinson’s mathiness?

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