Mathiness in the Theory of Economic Growth

By PAUL M.ROMER∗

Politics does not lead to a broadly shared con- machines or structures that we could observe. sensus. It has to yield a decision whether or not a The tight connection between the word and the consensus prevails. As a result, political institu- equations gave the word a precise meaning that tions create incentives for participants to exag- facilitated equally tight connection between its gerate disagreements between factions. Words theoretical and empirical claims. Gary Becker’s that are evocative and ambiguous better serve mathematical theory of wages (1962) gave the factional interests than words that are analytical words "human capital" the same precision and and precise. established the same two types of tight con- Science is a process that does lead to a broadly nections – between words and math and be- shared consensus. It is arguably the only social tween theory and evidence, and the same micro- process that does. Consensus forms around the- economic foundation in data and observations. oretical and empirical statements that are true. In contrast, McGrattan and Prescott (2010) In making these statements, a combination of give a label – "location" – to their proposed new words from natural language and tightly linked input in production, but their mathiness leaves symbols from the formal language of mathemat- so much slippage that the word could mean any- ics encourages the use of words that are analyti- thing. The authors choose a word that had al- cal and precise. ready been given a precise meaning by math- Economists mostly stick to science. Robert ematical theories of product differentiation and Solow (1956) was engaged in science when he economic geography, perhaps to give an impres- developed his mathematical theory of growth. sion of meaning by association. However, the But they can get drawn into academic politics. formal equations are completely different, so Joan Robinson (1956) was engaged in academic neither of those meanings carry over. politics when she waged her campaign against Their mathiness also offers no connection be- capital and the aggregate production function. tween its theoretical and empirical statements. Academic politics, like any politics, is better The quantity of location has no unit of mea- served by words that are evocative and ambigu- surement. The term does not refer to anything ous, but if an argument is transparently political, a person could observe. In a striking (but in- economists interested in science will simply ig- structive) use of slippage between the theoreti- nore it. The style that I am calling mathiness lets cal and the empirical, the authors assert, with no academic politics masquerade as science. Like explanation, that the national supply of location mathematical theory, mathiness uses a mixture is proportional to the number of residents. This of words and symbols, but instead of making raises questions that the equations of the model tight links, it leaves ample room for slippage do not address. If the dependency ratio and pop- between statements in the languages of words ulation increase, holding the number of work- as opposed to symbols, and between statements ing age adults and the supply of labor constant, with theoretical as opposed to empirical content. what mechanism leads to an increase in output? Solow’s mathematical theory of growth Does an additional 70 year-old retiree increase mapped the word "capital" onto a variable in location by the same amount as an additional 5 the mathematical equations, and onto both data year-old? from national income accounts and things like Their paper is one of several that introduce ∗ Romer: Stern School of Business, New York University, 44 mathiness into growth theory to support price- W. 4th St, New York, NY 10012, [email protected]. An ap- taking and oppose monopolistic competition. In pendix with computations and other supporting detail is available one sign that this campaign is political (in the from the author’s website and from the jounral website. The author acknowledges the generous support of the Rockefellar Foun- sense of academic politics) rather than scien- dation. tific, proponents offer neither theory nor ev- 1 2 PAPERS AND PROCEEDINGS MONTH YEAR idence to support the trade-offs that this re- fining the powerful abstractions that mathemati- quires. For example, McGrattan and Prescott cal theory highlights – abstractions like physical do not explain why price taking with no micro- capital, human capital, and nonrivalry. foundation is better than market power with a micro-foundation. I. Scale Effects For roughly two decades, growth theory has In 1970, there were zero mobile phones. To- made no progress toward a consensus about the day, there are more than 6 billion. This is the foundations for an economics of ideas. A stale- kind of development that a theory of growth mate prevails between Marshallian external in- should address, but to be able to do so, the the- creasing returns and monopolistic competition. ory must accomodate scale effects. One unfortunate side effect has been slow adop- Let q stand for individual consumption of mo- tion in aggregate theory of a powerful abstrac- bile phone services. For a ∈ [0, 1], let p = tion from public finance – nonrivalry – which is D(q) = q−a be the inverse individual demand far more consequential than the relatively minor curve with all-other-goods as numeraire. Let differences implied by price-taking versus price- N denote the number of people in the market. setting. Once the design for a mobile phone exists, let If mathiness is used infrequently to slow con- the inverse supply curve for an aggregate quan- vergence to a new scientific consensus, it will tity Q = q N take the form p = S(Q) = Qb for do localized, temporary damage. Unfortunately, b ∈ [0, ∞]. the market for lemons tells us that as the quan- If the price and quantity of mobile phones are tity of mathiness increases, it could do perma- determined by equating D(q) = m ∗ S(Nq), nent, pervasive damage in economics. It is hard so that m ≥ 1 captures any mark-up of price for readers to distinguish mathiness from math- relative to marginal cost, the surplus S created ematical theory. by the discovery of mobile telephony takes the The market for mathematical theory can sur- form vive a few lemon articles filled with mathiness. a(1+b) S = C(a, b, m) ∗ N a+b , Readers will put a small discount on any article with math, but will still find it worth their where C(a, b, m) is a messy algebraic expres- while to work through and verify that the for- sion. Surplus scales as N to a power between mal arguments are correct, that the connection a and 1. If b = 0, so that the supply curve for between the symbols and the words is tight, the devices is horizontal, surplus scales linearly 1 and that the theoretical concepts have implica- in N. If, in addition, a = 2 , the expression for tions for measurement and observation. But af- surplus simplifies to ter readers have been burned too often by mathi- 2m − 1 ness that wastes their time, they will stop taking S = N. seriously any paper that contains mathematical m2 symbols. In response, authors will stop doing With these parameters, a tax or a monopoly the hard work that it takes to supply real mathe- markup that increases m from 1 to 2 causes S matical theory. If no one is putting in the work to change by the factor 0.75. An increase in N to distinguish between mathiness and mathemat- from something like 102 people in a village to ical theory, why not cut a few corners and take 1010 people in a connected global market causes advantage of the slippage that mathiness allows? S to change by the factor 108. The market for mathematical theory will col- Effects this big tend to focus the mind. lapse. Only mathiness will be left. It will be worth little, but cheap to produce and might sur- II. The Fork in Growth Theory vive as entertainment. Economists have a collective stake in flush- The traditional way to include a scale effect ing mathiness out into the open. We will make in a growth model was proposed by Marshall faster scientific progress if we can continue to (1980). One writes the production of telephone rely on the clarity and precision that math brings services at each of a large number of firms in to our shared vocabulary, and if, in our analysis an industry as g(X) f (x) where the list x con- of data and observations, we keep using and re- tains the inputs that the firm controls and X is VOL. VOL NO. ISSUE MATHINESS 3 the list of inputs for the entire industry. One ob- turns, even when this forces mechanical mod- vious problem with this approach is that it of- els in which agents have no incentive either to fers no basis for determining the extent is of the discover ideas or encourage their diffusion. To spillover benefits from the term g(X). Do they defend this approach, they too resorted to math- require face-to-face interaction? Production in iness. the same city, the same country, or anywhere? If we split x = (a, z) into a nonrival input a III. Examples of Mathiness and rival inputs z and write output at an individual firm as Af (a, z), a standard replication The McGrattan and Prescott article links a argument implies that f must be homogeneous word with no meaning to new mathematical re- of degree 1 in the rival goods z. (If the aggre- sults. The mathiness in "Perfectly Competitive gate supply curve of goods is upward sloping, Innovation," (Boldrin and Levine 2008) takes as in the example above, this is merely a sign the adjectives from their title, which have a well that some of the factors in the list z are in fixed established, tight connection to existing mathe- supply. Anything that looks like a "Marshallian matical results, and links them to a diametrically rent" in a partial equilibrium analysis is in fact opposed set of mathematical results. In an ini- compensation to one of these fixed factors.) In a tial period, the innovator in their model is a mo- competitive equilibrium, for each firm, the value nopolist, the sole supplier of a newly developed of output equals the compensation paid to the ri- good. Nevertheless, the authors force the mo- val inputs z so there can be no nonrival inputs a nopolist to take a specific price for its own good that an individual firm can use whilst excluding as given by imposing price-taking as an assump- other firms from using them. (For an elaboration tion about behavior. of this argument, see Romer 1994b.) Hence, the Boldrin and Levine (2008) also make free- nonrival inputs A must be 0% excludable and standing verbal assertions (e.g. concerning Mar- output must take the form A f (z). No firm or shallian rents) that seem to invoke known math- person can keep a nonrival idea secret. No firm ematical results but which are in fact false. If or person will have any incentive to take advan- they had written down a formal mathematical tage of the surplus noted above by encouraging argument that was tightly linked to their words, the diffusion of an idea like mobile telephony they would have caught their error. In "Ideas and throughout the world. Discovery and diffusion Growth," Robert Lucas makes a similar verbal have to happen by accident. claim that seems to invoke known results and I started with the framework of external in- likewise turns out to be false: "Some knowl- creasing returns in my Ph.D. thesis, but soon edge can be ’embodied’ in books, blueprints, switched to a model with monopolistic compe- machines, or other kinds of physical capital, and tition that allows for nonrival inputs in produc- we know how to introduce capital into a growth tion that could be at least partially excludable. In model, but we also know that doing so does not models with partially excludable nonrival ideas, by itself provide an engine of sustained growth" it is logically possible for a firm to have an in- (Lucas 2009, p. 6). Any model of growth with centive to discover new ideas (Romer 1990) or to a growing variety of capital goods or a quality encourage the international diffusion of an exist- ladder of capital goods is a counter-example. ing idea like mobile telephony (Romer 1994a). In Lucas and Moll (2014), the mathiness in- In this framework, excludability offers a much volves both words that misrepresent the mathe- more precise way to think about spillovers. Non- matical analysis and a mathematical model that rivalry, which is logically independent, is the is not well specified. The authors develop a defining characteristic of an idea and the source model based on an assumption that I’ll call P of the scale effects that are central to any plau- (for their use of "Pareto.") They show that given sible explanation of the broad sweep of human P, the growth rate g[P](t) converges to γ > 0 history (Jones and Romer 2010). as t goes to infinity. Because P is hard to jus- As part of the campaign of academic politics tify, the authors offer "an alternative interpreta- noted above, economists commitment to price- tion that we argue is observationally equivalent: taking persevered with the assumption of 0% ex- knowledge at time 0 is bounded but new knowl- cludability required for external increasing re- edge arrives at arbitrarily low frequency." (Lucas 4 PAPERS AND PROCEEDINGS MONTH YEAR and Moll 2014, p. 11.) not picked up at the working paper stage or in Let β denote this arrival rate for new knowl- the process leading up to publication may tell edge. In their alternative interpretation, the au- us something about the new equilibrium in eco- thors consider a collection of economies that nomics. Neither colleagues who read working start under assumption B, but eventually switch papers, nor reviewers, nor journal editors, are to an economy where assumption P applies in- paying attention to the math. stead. As β gets arbitrarily low, the time it takes I, and others, told Lucas and Moll about the to switch to P goes to infinity. (See the online discontinuity in the limit in their joint paper and appendix for details.) Let β : B ⇒ P denote the problem this posed for their claim about ob- one economy from this collection specified by servational equivalence. I thought that publish- the value β. Because any β : B ⇒ P economy ing the paper in this form would be embarrass- will eventually switch to being a P economy, the ing for them. They kept this analysis in the pa- limit of the growth rate in every β : B ⇒ P per and the Journal of Political Economy pub- economy is equal to the limit of the growth rate lished it. This may reflect a judgment by the au- in the P economy. thors and the editors that at least in the theory of Yet at any date T that is large enough so that growth, we are already in a new equilibrium in we can ignore transitory effects, g[β : B ⇒ economics in which readers have come to expect P](T ) approaches 0 as β becomes arbitrarily and accept mathiness. low yet g[P](t) is close to γ . Because any obser- One final bit of evidence comes from Piketty vation on the growth rate has to be taken at some and Zucman (2014), which cites a result from specific date, all observations show that instead a growth model: with a fixed saving rate, when of being equivalent, the collection of economies the growth rate falls by half, the ratio of wealth {β : B ⇒ P} differs markedly from the P econ- to income doubles. They note that their formula omy. W/Y = s/g assumes that national income and The mathiness here involves more than a non- the saving rate s are both measured net of depre- standard interpretation of the phrase "observa- ciation. They observe that the formula has to be tionally equivalent." The underlying formal re- modified to W/Y = s/(g + δ), with a deprecia- sult is that calculating the double limit in one tion rate δ, when it is stated in terms of the gross order limβ→0(limT →∞ g[β : B ⇒ P]) yields saving rate and gross national income. one answer, γ , but calculating it the other, From Krusell and Smith (2014), I learned limT →∞(limβ→0 g[β : B ⇒ P]), gives a dif- more about this calculation. If the growth rate ferent answer, 0 < γ . The mathiness here in- falls and the net saving rate remains constant, volves picking the calculation that is convenient the gross saving rate has to increase. For exam- and acting as if the double limit exists. An ar- ple, with a fixed net saving rate of 10% and a de- gument that takes the math seriously would note preciation rate of 3%, a reduction in the growth that the double limit does not exist and would rate from 3% to 1.5% implies an increase in the caution against trying to give an interpretation gross saving rate from 18% to 25%. This means to value calculated using one order as opposed that the expression s/(g + δ) increases by a fac- to the other. tor 1.33 because of the direct effect of the fall in g and by a factor 1.38 because of the induced IV. A New Equilibrium in the Market for change in s. A third factor, equal to 1.09, arises Mathematical Economics because the fall in g also increases the ratio of gross income to net income. These three fac- As is noted in an addendum, Lucas (2009) tors, which when multiplied equal 2, decompose contains a flaw in a proof. The proof requires the change in W/Y calculated in net terms into that a fraction α be less than 1. The same page γ equivalent changes for a model with variables γ has an expression for γ , γ = α γ +δ , and because measured in gross terms. α α, γ , and δ are all positive, it implies that γ is Piketty and Zucman (2014) present their data greater than 1. Anyone who does math knows and empirical analysis with admirable clarity that it is distressingly easy to make an over- and precision. In choosing to present the the- sight like this. It is not a sign of mathiness by ory in less detail, they too may have been re- the author. But the fact that this oversight was sponding to the expectations in the new equilib- VOL. VOL NO. ISSUE MATHINESS 5 rium. Empirical work is science; theory is en- Marshall, Alfred. 1890. Principles of eco- tertainment. Presenting a model is like doing a nomics. London:Macmillan and Co. card trick. Everybody knows that there is will be some sleight of hand. There is no intent to de- McGrattan, Ellen R., and Edward C. ceive because no one takes it seriously. Perhaps Prescott. 2010. “Technology Capital and the the norm will soon be like the one in profes- US Current Account.” The American Eco- sional magic; it will be impolite, perhaps even nomic Review, 100(4): 1493–1522. an ethical breach, to reveal how someone’s trick Piketty, Thomas, and Gabriel Zucman. 2014. works. “Capital is Back: Wealth-Income Ratios in When I learned mathematical economics, a Rich Countries 1700-2010.” The Quarterly different equilibrium prevailed. Not universally, Journal of Economics, 129(3): 1255–1310. but much more so than today, when economic theorists used math to explore abstractions, it Robinson, Joan. 1956. The Accumulation of was a point of pride to do so with clarity, pre- Capital. Homewood, Illinois:Richard D. Ir- cision, and rigor. Then too, a faction like Robin- win. son’s that risked losing a battle might resort to Romer, Paul M. 1990. “Endogenous Techno- mathiness as a last-ditch defense, but doing so logical Change.” Journal of Political Econ- carried a risk. Reputations suffered. omy, 98(5): S71–S102. If we have already reached the lemons market equilibrium where only mathiness is on of- Romer, Paul M. 1994a. “New Goods, Old The- fer, future generations of economists will suffer. ory, and the Welfare Costs of Trade Restric- After all, how would Piketty and Zucman have tions.” Journal of Development Economics, organized their look at history without access 43: 5–38. to the abstraction we know as capital? Where would we be now if Robert Solow’s math had Romer, Paul M. 1994b. “The Origins of En- been swamped by Joan Robinson’s mathiness? dogenous Growth.” Journal of Economic Per- spectives, 8: 3–22. 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