Thomson Young Person’s Guide to Writing Economic Theory

Journal of Economic LiteratureVol. XXXVII (March 1999), pp. 157183

Thomson: The Young Persons Guide to Writing Economic Theory

The Young Persons Guideto Writing Economic Theory

WILLIAM THOMSON1

1. Introduction

HERE ARE MY recommendations forwriting economic theory (and, tosome extent, giving seminar presenta-tions). My intended audience is youngeconomists working on their disserta-tions or preparing their first papers forsubmission to a professional journal.

Although I discuss general issues ofpresentation, this essay is mainly con-cerned in its details with formal models.It does not cover the writing up of em-pirical work. However, since most pa-pers begin with the introduction andthe analysis of a model, I hope that itwill be useful to anyone, irrespective offield, and not just to fledgling theorists.

The principles of good writingsim-

plicity, clarity, unityare universal, butwhen it comes to putting them intopractice, multiple choices are oftenavailable, and these recommendations tofollow unavoidably reflect my personaltastes. Also, they are occasionally in-compatible. This is where judgementcomes in. Exercise yours. I make muchuse of the imperative mode, but I canwell imagine that you will come downdifferently on a number of the issues Iraise. What is important is for you tothink about them.

Good writing requires revising, revis-ing, and revising again. Undoubtedly,you will spend many months perfectingyour first papers, but this work is one ofthe wisest investments that you willever make. In your future papers, youwill face the same issues again andagain, and with the experience you willhave gained, you will be able to handlethem quickly and efficiently.

Do not think that if your ideas areinteresting, people will read your workwhether or not it is well written. Yourpapers are competing with many othersthat constantly arrive on the desks ofthe people you hope to reach, so if it isnot clear to them fairly quickly thatthey will get something out of readingyour work, they will not even start.

Finally, putting your results on paperis not subsidiary to producing them. Theprocess of writing itself will lead you to

157

1 Department of Economics, University ofRochester, Rochester, NY 14627. This is an abbre-viated version of a paper entitled WritingPapers, which is available from the author uponrequest. I encourage readers to send me theircomments at [email protected]. I thankMarcus Berliant, Youngsub Chun, JacquesCrmer, John Duggan, James Foster, Tarik Kara,Jerry Kelly, Bettina Klaus, Kin Chung Lo, LeslieMarx, Lionel McKenzie, Philip Reny, SuzanneScotchmer, and Jean-Max Thomson for their help-ful comments and James Schummer for making areality of my fantasy of letters tumbling down acliff (footnote 10) in accordance with the laws ofphysics. I also thank Toru Hokari for the figures.My greatest debt is to John McMillan, Martin Os-borne, John Pencavel, James Schummer, the edi-tors, and two anonymous referees of this journalfor their numerous and extremely useful sugges-tions.

new knowledge. Learn to write but alsowrite to learn.2

2. General Principles

Convey your message efficiently.3 Byleafing through your article, a readershould be able to easily spot the mainresults, figure out most of the notation,and locate the crucial definitionsneeded to understand the statement ofeach theorem.

Readers who have found your centralpoints interesting and want to knowmore, but have little time to invest inyour work, should then be able to getan idea of your methods of proof byvisual inspection. It is often quite infor-mative just to glance at the way anargument is structured and to identifythe central assumptions and the knowntheorems on which it is based. Thinkabout the way you read a paper. Youprobably do not proceed in a linear way.Instead, you scan it for the formal re-sults and look around them for an expla-nation of the notation and terminologythat you do not recognize or guess. Youdo not like having to spend too muchtime to find what you need. Your read-ers probably feel the same way aboutyour work.

The Components of a Paper. Your ti-tle should be as descriptive of yourtopic as possible. Devote time to yourabstract, as it is on that basis that manypotential readers will decide whether tocontinue. In your acknowledgment foot-note, be generous. Include the seminarparticipant who suggested a name for a

condition you introduced, or directedyou to a pertinent reference. Your ap-portionment of credit among the vari-ous people who helped you, however,should be commensurate with the timeand effort they spent and the usefulnessof the suggestions they made. Thereferee who sent you five pages ofcomments deserves recognition in aseparate sentence.

In your introduction, briefly placeyour work in the context of the existingliterature and describe your main find-ings. Do not start with a two- or three-page survey of the field; your readerwill want to know what your contribu-tion is sooner than that. Use plain lan-guage, and skip the technical details.Your literature review should not be amere enumeration of previous articles.In describing the work on which youbuild, give priority to the developmentof the ideas rather than to telling uswho did what, although this informationshould be included, and where youstepped in should be unambiguous. Youneed not repeat in the body of the pa-per all of the points that you made inthe introduction, although some repeti-tion is unavoidable. On the other hand,I do not generally favor relegating proofsto appendices (more on this later).

Your conclusion should not be a re-hashing of the introduction. However, acompact summary of your results and astatement of the main lesson to bedrawn from your analysis is a good leadto a list of specific open questions and ageneral discussion of promising direc-tions for future work, all of which dobelong there. In your bibliography, givethe relevant background papers. If agood survey is available, mention it. Youmay have to include papers that you didnot use, and papers that you discoveredonly after you completed yours. Checkreferences carefully, and update themas papers get published.

2 I owe this formula to William Zinssers 1989pedagogical essay. Writing to Learn, a book that Istrongly recommend.

3 This paper is longer than the average, but ex-cept in Lake Wobegon, not all papers can beshorter than the average. Actually, I do not have arecommendation on how long a paper should be,except for Make it as long as it needs to be, nolonger, and no shorter. If its structure is clear,length by itself is not a problem.

158 Journal of Economic Literature, Vol. XXXVII (March 1999)

The structure of your paper shouldbe clear, as should the structure of eachsection, each subsection, and each para-graph. To better see how your para-graphs fit together, summarize each ofthem in one sentence. Does the stringof these sentences make sense? Itshould. Perform this exercise also at thelevel of subsections, and then sections.

Show that what you did is interestingand has not been done before. To showthat your results are significant, thetemptation is great to present themwith the utmost generality, with bigwords, and in gory detail. Resist it! Tryinstead to make your argument appearsimple, and even trivial. This exercise inhumility will be good for your soul. Itwill also give referees a warm feelingabout you. Most importantly, it willhelp you prove your results at the nextlevel of generality.

Because the refereeing process andpublication constraints often have theunfortunate effect of wiping out from apaper most of what could make it easilyunderstandable, you may think that ifyours does not contain at least one re-sult that looks difficult, it is not readyfor submission. You are rightly proud ofthe sophisticated reasoning that led youto your findings. Nevertheless, workhard to make them look simple.4

To show that what you do has notbeen done before, explain how your as-sumptions differ from the assumptionsused in related literature, and whythese differences are significant, bothconceptually and technically. Demon-strate your knowledge of this literatureby citing the relevant articles and tell-ing us how they pertain to your subject.

Also, motivate your work, but do notover-motivate it, or your readers willget suspicious.

Do not forget the process by whichyou made your discovery. By the timeyour paper is finished, it will cover anarbitrary number of goods and agents,general production possibilities, uncer-tainty, and so forth, and nobody will un-derstand it. If you read it severalmonths later, you will not understand iteither. You got to your main theorem insmall steps, by first working it out fortwo agents, two goods, linear technolo-gies, and with no uncertainty, and bydrawing lots of diagrams. It is also bylooking at simple versions of yourmodel that your reader will understandthe central ideas, and it is most likelythese central ideas, not the details ofproofs, that will help her in her ownwork.

Reproducing the process of discoveryin a paper is not easy, but try. In a semi-nar, quite a bit more can be done be-cause of the informality of the occasion.Explaining how you came to the formu-lation you eventually chose and to yourresults, however, is not a license to arambling discussion in which notation,definitions, assumptions, and motiva-tion are all mixed up, like the ingredi-ents of a big salad. Even worse isadding some semi-formal algebraic ma-nipulations (tossing the salad?), andsuddenly confronting us with the sen-tence: We have therefore proved thefollowing theorem: . . . As a reader, Ifeel as if I have been mugged when Ifind myself in that situation.

Another principle that has wide valid-ity is that good exposition means goingback and forth between the general andthe particular, and I will give severalillustrations of it.

Learn from your errors. There isnothing like having misunderstoodsomething to really understand it, and

4 As a young economist, it is natural that youshould be proud of the complicated things youachieve; as you get older, you will become proudof the simple things you do. (Of course, it is notbecause you will not be able to handle the compli-cated things anymore.)

Thomson: The Young Persons Guide to Writing Economic Theory 159

there is nothing like having seriouslymisunderstood it to really, really under-stand it. Instead of being embarrassedby your errors, you should cherishthem. I will even say that you cannotclaim to have understood something un-til you have also very completely under-stood the various ways in which it canbe misunderstood. It has been said be-fore, and better: Erreur, tu nes pas unmal. (Gaston Bachelard 1938)

Your readers are likely to be victimsof the same misunderstandings that youwere. By remembering where you hadtrouble, you will anticipate where youmay lose them, and you will give betterexplanations. In a seminar, quicklyidentifying the reason why someone inthe audience is confused about some as-pect of your paper may save you from a10minute exchange that otherwisewould force you to rush through thesecond half of your presentation.

3. Notation

Choose notation that is easily recog-nizable. If you have no problem remem-bering what all of your variables desig-nate, congratulations! But you havebeen working on your paper for severalmonths now. Unfortunately, what youcall x is what your reader has beencalling m since graduate school.

The best notation is notation that canbe guessed. When you see a man walk-ing down the street with a baguette un-der his arm and a beret on his head, youdo not have to be told he is a French-man. You know he is. You can immedi-ately and legitimately invest him withall the attributes of Frenchness, andthis greatly facilitates the way you thinkand talk about him. You can guess hischildrens namesRene or Edmondand chuckle at his supposed admirationfor Jerry Lewis.

Similarly, if Z designates a set, call

its members z and z, perhaps x, y, andz, but certainly not b, or . Upon en-countering z and z, your reader will im-mediately know what space they belongto, how many components they have,and that these components are called ziand zj. If is a family of functions, re-serve the notation and ~, (perhaps or even f ) for members of the family,but certainly not or m.

If Ri is agent is preference relation,you may have to designate his most pre-ferred bundle in some choice set bybi(Ri), his demand correspondence bydi(Ri), and so on, but dropping this func-tional dependence may not create ambi-guities. For instance, you may write biand di, provided that you designateagent js most preferred element in thechoice set and his demand correspon-dence by bj and dj, and the comparableconcepts when agent is preferences arechanged to Ri by bi and di.

Designate time by t, land by , alter-natives by a, mnemonic notation by mnand so on (and make sure that no twoconcepts in your paper start with thesame letter).

Some letters of the alphabet are usedin certain ways so generally in your fieldthat their common interpretation mayget in the way of other uses that youwant to make of them. You will prob-ably be better off accepting tradition.Do not designate just any quantity by .Reserve this letter for small quantitiesor quantities that you will make go tozero.5 Call your generic individual i, hispreference relation Ri, his utility func-tion ui, and his endowment vector i.The production set is Y. Prices are p,

5 I like the fragile look of my , especially whenmy printer is running out of toner. How could onedoubt that the quantity it designates is about tofade into nothingness? However, as a referee re-minded me, in econometrics, the error term isnot necessarily a small quantity, but rather a quan-tity that one would like to be small.


quantities q. Calligraphic letters oftenrefer to families of sets; so, a is a mem-ber of the set A, which is chosen fromthe family A.

Choose mnemonic abbreviations forassumptions and properties. Do not re-fer to your assumptions and propertiesby numbers, letters, or letter-numbercombinations. Since you state your firsttheorem on page 10, it will be virtuallyimpossible for us to remember thenwhat Assumptions A1A3 and B1B4are, but the fact that Assumptions Diff,Mon, and Cont refer to differentiabil-ity, monotonicity, and continuity will beobvious to a reader starting there.Choose these abbreviations carefully: Ifyou write Con, we may not knowwhether you mean continuity or convex-ity, so write Cont or Conv. The cost toyou is one extra strike on your key-board, but your small effort will save usfrom searching through the paper tofind which property you meant. Admit-tedly, naming each assumption in a waythat suggests its content is not alwayspossible, especially in technical fields.

It is common to introduce in paren-theses an abbreviation for a condition,next to the full name of the condition atthe time it is formally stated. When theabbreviation is used later on, the paren-theses are no longer needed.6

In axiomatic analyses, many authorsrefer to axioms by numbers or abbrevia-tions, but I do not see any advantage tothat. The argument that numbers andabbreviations save space is not very con-vincing given that they will not shortena 20page paper by more than fivelines, and they certainly will not savetime for your reader. If you use differ-ent typeface for your axioms, which Istrongly recommend (for instance ital-ics, or slanted type), each axiom stands

out from the text and is perceived glob-ally, as a unit: it is not read syllable bysyllable. An alternative way to achievethis important visual separation of theaxioms from the text is to capitalize them.

Never use abbreviations in a sectionheading.

Do not bother introducing a piece ofnotation if you use it only once or twice.There is no point in defining a newpiece of notation if you hardly ever useit. How many times should a conceptbe used to deserve its own symbol?Three times? Four times? I will letyou decide. Certainly, do not botherintroducing notation that you never use.

I feel the same way about utility nota-tion when only preferences are in-volved. It is wonderful, of course, thatpreference relations satisfying certainproperties can be represented by nu-merical functions, and these repre-sentations are sometimes useful or evennecessary. But it has become a commonexcuse to use them even in situationswhere in fact they only clutter the text.Suppose, for instance, that you want towrite that the allocation rule S is strat-egy-proof. This means that for everyagent i, announcing his true preferencerelation Ri is preferable to announcingany false preference relation Ri inde-pendently of the announcements madeby the other agents. Then (here I willskip the quantifications) you can writeui(S(u)) ui(S(ui,ui)), but is such anexpression preferable to S(R) Ri S(Ri,Ri)? If your paper involves long stringsof terms of that form, as may well bethe case, utility notation will contributeto an unnecessarily messy look.

Matters are worse if you discuss cer-tain normative issues of welfare eco-nomics, social choice, or public finance,because in these fields utility functionshave cardinal significance. Even thoughyour theory may only involve the under-lying preference relations, some of your

6 When you begin a proof, write Proof: andnot (Proof:).


readers will come from a different tra-dition and be tempted to compare utili-ties, or equate them, or maximize theirsum, and so on. On the other hand, ifyou address some problem of demandtheory and you need to calculate matri-ces of partial derivatives, then of courseyou cannot avoid utility notation.

Do not define in footnotes importantnotation that is unlikely to be familiarto your reader, and that you will use inthe body of the paper. More generally,do not refer in the main text to terms,ideas, or derivations introduced in afootnote or in a remark, since thereader may have skipped it. There is ahierarchy here that you have to respect.

Save on mathematical symbols. Donot use symbols that are not necessary.For instance, try to avoid multiple sub-scripts and superscripts. If you haveonly two agents, call their consumptionbundles x and y, with generic coordi-nates xk and yk (instead of x1 and x2, withcoordinates x1k and x2k). In a text, com-binations of subscripts and superscriptslook a little better than only subscripts,but in a blackboard presentation, watchout for the sliding superscripts that endup as subscripts. If F is your generic no-tation for a solution to the bargainingproblem, you can certainly refer to theNash solution as FN, and when you ap-ply it to the problem (S,d) with feasibleset S and disagreement point d, you willget FN(S,d). But why not simply desig-nate the Nash solution by N? If you canchoose the disagreement point to be theorigin, as is almost always the casewithout loss of generality, ignore it inthe notation. Altogether, you will calcu-late N(S), a much lighter expressionthan FN(S,d). If you systematicallysearch for such notational simplications,your text will be much cleaner.

Bounds of summation or integrationare often (I agree, not always) unambi-guous. There is then no need to indi-

cate them. Do not write i = 1n xi, iN

xi,i xi, N xi, or i = 1, ,n xi when, in mostcases, xi is perfectly clear. I assureyou, upon encountering xi, your read-ers will be unanimous in assuming thatyou are summing over i when i runsover its natural domain. Similarly, andalthough the set consisting of agent ialone should be denoted by {i}, if you needto refer to it on multiple occasions, youare better off dropping the curly brack-ets. Do apologize for the abuse of nota-tion though. Similarly, if O designates alist of objects indexed by agents in theset N, you should refer to the shorterlist from which the i-th component hasbeen deleted as ON\{i}, but it has becomestandard to write O

i. I welcome theshortcut, and I used it earlier. Expres-sions can be considerably lightened byusing such tricks. Imagine that you areon a diet and that each symbol is worthone calorie. You will quickly discoverthat you can do with half as many. Youwill improve the readibility of your textand lose weight.

Do not let the reader guess or inferfrom the context what your inequalitysymbols mean. Define them the firsttime you use them. Doing that in a foot-note is acceptable.7 Alternatively, youcan give them in a preliminary sectionof notation.

4. Definitions

Be unambiguous when you define anew term. Make it immediately clearthat indeed it is new. Do not let yourreader think that you may have already

7 x >=

y means xi yi for all i; x y means x >=

yand x y; x > y means xi > yi for all i. You couldalso use x >

= y, x > y, and x >> y. It is a very common

convention to define these symbols in a footnote,and this is where most of us will look for themwhen we need them. It is therefore a good idea foryou also to define yours in a footnote. Some peo-ple have an aversion to footnotes, but personally, Ilove them. In academic writing, they are often theonly place where you will find evidence of life.


given the definition but she missed it,or that you are assuming she knows thedefinition.

Here are three possible ways of in-troducing a definition: 1. A function ismonotone if . . .; 2. A function ismonotone if . . .; 3. A function issaid to be monotone if . . . I preferthe first format and use it throughoutthis essay, because its phrasing is directand its different typeface will facilitateits retrieval, if needed. Concerning thetypeface, I recommend boldface orboldface italics over italics or plain textbetween quotation marks, neither ofwhich makes the new terms stand outsufficiently. You should probably dis-play the crucial definitions separately,and you may precede each of them bythe word Definition in boldface (seethe examples below). But do not intro-duce all definitions in this way, espe-cially if you have many of them, as itwill get tedious. Focus on the criticalones.

To avoid repeating quantificationsthat are common to several definitions,you can group these definitions andstate the quantification once: An allo-cation rule is efficient if for all prefer-ence profiles R, and all allocations zthat it selects for R, there is no otherallocation z that all agents find at leastas desirable as z and at least one agentprefers; it is weakly efficient if instead

there is no other allocation z that allagents prefer to z.

To emphasize certain aspects of yourpaper, such as important conclusions,exploit the typographical choices atyour disposal. Italics is a good one.However, if everything is emphasized,nothing is.

When introducing a novel definition,give illustrative examples. If the defini-tion is a property that an object may ormay not have, exhibit: 1. Objects thatsatisfy the definition; 2. Objects that donot satisfy the definition; 3. Objectsthat satisfy the definition but almost donot; 4. Objects that do not satisfy thedefinition but almost do. Examples inCategories 3 and 4 are particularly im-portant as they are responsible for mostof the work in the proofs. Conversely,they may be the ones that allow theproofs to go through! In a paper, givinga range of examples that are repre-sentative of all four categories is, onceagain, not easily achieved because ofspace limitations, but in seminars thiscan sometimes be done. Here are twoillustrations:

Definition. A function f: [0, 1] R isincreasing if for all t, t [0, 1] with t >t, we have f(t) > f(t).

Figures 1a and 1b are dangerous, be-cause they may plant in your readersmind the seed that you will work with

(a)

Figure 1. Examples of increasing functions and of functions that are not increasing. (a), (b), and (c): These functions are increasing. (d) and (e): These functions are not.

(b) (c) (d) (e)


functions that are linear, or perhapsconcave. Figure 1c is what you need: itrepresents an increasing function in itsfull generality, with a kink, a convexpart, a concave part, and a discontinu-ity. Figure 1d is useful too, as it showsa typical violation of the property.Figure 1e is very important becauseit makes it clear that you wantmore than that the function benondecreasing.8

Definition. The continuous preferencerelation R defined on [0,1], with asym-metric part P, is single-peaked if thereexists x [0, 1] such that for all x,x [0, 1] with either x < x x orx x < x, we have x P x.

Figure 2 presents the graphs of thenumerical representations of six prefer-ence relations. Obviously, R2 is single-peaked and R5 is not. But your viewermay not immediately think of R1 as be-

ing single-peaked because its repre-sentation achieves its maximum at acorner, or may think that R6 is admissi-ble, although its representation has aplateau and not a peak. You shouldalso make her aware of the fact that youinclude preferences that do not exhibitthe symmetry illustrated in Figure 2b.All of these examples will be very usefulto ensure that she fully perceives theboundary of your domain.

Write definitions in logical sequences.Introduce terms in such a way that thedefinition of each new one only involvesterms that are already defined, insteadof asking your readers to wait until theend of the sentence or paragraph foreverything to be clarified.

For instance, state the dimensionalityof the commodity space before you in-troduce consumers or technologies. Inthe standard model, a consumer is nomore than a preference relation definedover a subset of that space, togetherwith an endowment vector in the space;a technology is simply a subset of thespace. In each case, it is therefore natu-ral to specify the space, that is, the

8 Several readers of this essay objected to sen-tences such as this function is nondecreasing,which sounds too much like this function is nota decreasing function, but means something else.Perhaps we should speak of a nowhere-decreas-ing function.

Figure 2. Examples of single-peaked and of non-single-peaked preference relations. (a) These relations are single-peaked, with peaks at a for R1 and at b for R2. (b) These relations are single-peaked too, but they are not sufficiently representative of the whole class due to their symmetry. Readers who have not worked with such preferences often assume that symmetry is part of the definition, so you should emphasize that most single-peaked preferences do not have that property. (c) These relations are not single-peaked, since R5 has two local maxima, at a and b, and R6 is maximized at any point of the non-degenerate interval [c, d].

(c)

a b

R6

R5

c d

(b)

a b

R4

R3

(a)

a b

R2

R1


number of goods, first. Therefore, donot write: rmon is the class of increas-ing preferences R, where by increasingis meant that for all x, y R+ with x y,we have x R y, being the dimensional-ity of the commodity space. Insteadwrite: Let N be the number ofgoods. The preference relation R de-fined on R+ is increasing if for allx, y R+ with x y, we have x R y.Let Pmon be the class of increasingpreferences.

As another example, in which rn de-notes a domain of preference profiles inan n-person economy, do not write:

Definition. The social choice correspon-dence F:rn A is Maskin-monotonic iffor all R, R rn and all a F(R), if forall i N, L(a,Ri) L (a,Ri), then a F(R),where L(a,Ri) is the lower contour set ofthe preference relation Ri at a, with R andR being profiles of preference relations de-fined over A, some alternative space, andMaskin being an economist at Harvard.

Instead write:

Definition. Let Maskin be an economistat Harvard. Let A be a set of alterna-tives. Given Ri, a preference relation de-fined over A, and a, an alternative in A,let L(a,Ri) be the lower contour set of Riat a. The social choice correspondence F:rn A is Maskin-monotonic if for allR, R rn and all a F(R), if for all i N,L(a, Ri) L(a, Ri), then a F(R).

Even better, first introduce the basicnotationyou will probably use it inother definitions and in the proofsandonly then give the definition. This sepa-ration will help highlight the essentialidea of the definition.9 Begin with:

Let A be a set of alternatives. GivenRi, a preference relation defined over A,and a, an alternative in A, let L(a,Ri) bethe lower contour set of Ri at a. Let r

be a class of admissible preference rela-tions defined over A. A social choicecorrespondence associates with everyprofile of preference relations in rn anonempty subset of A.

Now, you can state the definition: Definition. The social choice correspon-dence F:rn A is Maskin-monotonicif for all R, Rrn and all a F(R), if forall i N, L(a, Ri) L(a, Ri), then a F(R).

You may also want to display thehypothesis and the conclusion:

Definition. The social choice correspon-dence F:rn A is Maskin-monotonic iffor all R, R rn and all a F(R), if

for all i N, L (a, Ri) L(a, Ri),then

a F(R).If the hypotheses and the conclusionsare simple enough, however, as they arein this example, displaying them may notbe needed.

Some will object to the double if inthe condition as I wrote it, and it doessound awkward. What about replacingthe first one with something like when-ever? Another option is to write:L(a, Ri) L(a, Ri) for all i N impliesa F(R).

I have seen the recommendation todrop the punctuation at the end of dis-played formulas (the hypothesis and theconclusion of the last statement ofMaskin-monotonicity), but there is farfrom complete agreement about this.Personally, I prefer all my sentences tobe fully punctuated. 10

9 Same thing with propositions and theorems:Do not introduce new notation in their state-ments.

10 When my daughters were in primary school, Ioccasionally went to their school to help out withthe kids writing, and my main job was to checkthat every sentence they wrote began with a capi-tal letter and ended with a period. I have learnedthis lesson well, and when I see a sentence thatdoes not end with a period, I experience the samequeasiness as when I step too close to the edge ofan open s


Make sure that the informal descrip-tions of your definitions match theirformal statements. If you write: A fea-sible allocation is Pareto efficient ifthere is no other feasible allocation thatall agents find at least as desirable andat least one agent prefers, your formaldefinition should not be (still using R todenote a preference profile and intro-ducing p for the set of Pareto efficientallocations): z p if (i) z Z and (ii)for all z Z such that for some i N,zi Pi zi, there is j N such that zj Pj zj.Instead write: z p if (i) z Z and (ii)there is no z Z such that for all i N,zi Ri zi, and for some j N, zj Pj zj.

Separate formal definitions from theirinterpretations. Formal models canoften be given several interpretations.It is, therefore, of great value to sepa-rate the formal description of yourmodel from the interpretation you in-tend in your particular application. Forexample, first write:

Definition. Let Vn be a domain of n-person coalitional games. A solution onVVn is a function that associates with everygame vVn a point x Rn such thatxi v(N).11

Then explain: If F is a solution onVn, v is a game in Vn, and i is a player inN, the number Fi(v) can be interpretedas the value to player i of being in-volved in the game v, that is, theamount that he would be willing to payto have the opportunity to play it. Alter-natively, it can be thought of as theamount that an impartial arbitratorwould recommend the player shouldreceive.

The advantage of this separation isthat it will help your reader (and even

yourself) discover the relevance of yourresults to other situations that she (andyou) had not thought about initially. Topursue the example I just gave, the the-ory of coalitional games is also the the-ory of cost allocation. Some of yourreaders are interested only in applica-tions, and not in abstract games; othersdo not care for the applications. Youcan catch the attention of all by firstgiving general definitions and thenpointing out the various possible inter-pretations of your model.

Present the basic concepts of yourtheory in their full generality. You willalmost certainly use concepts that aremeaningful much beyond the frame-work of your paper. It is preferable tointroduce them without imposing theextra assumptions that you will need toinvoke for your analysis. When you ex-plain what a Walrasian equilibrium is,do not assume convexity, monotonicity,or even continuity of preferences.12 Ofcourse, these properties are relevantwhen you turn to the issue of existenceof these equilibria, but they have noth-ing to do with the concept of a Wal-rasian equilibrium itself.

When you introduce a piece of nota-tion, tell your reader what kind ofmathematical object it designates,whether it is a point in a vector space, aset, a function, and so on. Do not writeA pair (p,x) is a Walrasian equilib-rium if . . . Instead, first define theprice simplex 1 in the -dimensionalEuclidean space and define the alloca-tion space X. Then, write A pair(p, x) 1 X is a Walrasian equilib-rium if . . . Similarly, do not writeThe function is strategy-proofif . . . k, but instead, after havingdefined the set of possible preference11 Here we have a bit of a notational problem as

the n exponent to Vn indicates the n-player case,whereas the n exponent to Rn indicates the n-foldcross-product of Rn by itself. To avoid it, you couldwrite V(n), but I do not think that the risk of confu-sion is sufficiently high to justify the parentheses.

12 Discontinuous preferences are not easy to il-lustrate graphically, so if you give a graphical illus-tration of your concept, you probably should pre-sent it for continuous preferences.


profiles rn (the cross-product of |N|copies of r indexed by the members ofN), and the allocation space X, writeThe function :rn X strategy-proofif . . .

Indicating explicitly the nature of theobjects that you introduce is especiallyimportant if the reader may not be fa-miliar with them. By writing A triple(pi, x, y) (1)n R+(ml)n R is a Lin-dahl equilibrium if . . . , you helpher realize that pi has components in-dexed by agents (these are the Lindahlindividualized prices).

By the way, a sequence of elementsof X is not a subset of X, but a functionfrom the natural numbers to X. So, youcannot write {xk}kN X. Nor can youwrite xk

X. Speak of the sequence xk

of elements of X, or of the sequence{xk} where for all k N, xk X.

When you define a concept, indicatewhat the concept depends on. Do notwrite The function f is differentiableat t if blah, blah, blah of t. Since whatfollows if depends on t, you shouldwrite The function f is differentiableat t (including at t in the expressionin italics) if blah, blah, blah of t. Then,you can continue and say The functionf is differentiable if it is differentiableat t for all t in its domain. A marginalrate of substitution is calculated at apoint, so speak of agent is marginalrate of substitution at xi. For an ex-ample taken from the theory of imple-mentation, speak of a monotonic trans-formation of agent is preferences atxi, and not just of a monotonic trans-formation.

When you define a new variable as afunction of old ones, it should appearon the left-hand side of the equality oridentity symbol. If M has already beendefined, and M is introduced next, witha value equal to M, you should writeLet M = M, and not Let M = M.

Do not assume that your readers are

necessarily familiar with the definitionsyou use. There is rarely complete agree-ment on definitions in the literature.Apparently standard terms are often un-derstood differently by different peo-ple. Therefore, define the terms youuse, even some that you can legiti-mately assume everyone has alreadyseen. Core, public goods, and in-centive compatibility are examples ofterms that are common enough, but de-fine them. The word rationality fre-quently appears in formal developmentsin game theory without a definitionbeing given. Do not make such a mis-take.

Refer to a given concept by only onename or phrase, even if you have sev-eral natural choices. Make one and stickto it. Indicate in parentheses next toyour definition, or in a footnote, theother terms that appear in the litera-ture. When you first discuss the generalidea, you may use different terms in or-der to vary language and avoid repeti-tions repetitions, which admittedly donot sound very good, but after you haveformally defined the concept and bap-tized it, only refer to it by its name.

The terms game, game form, andmechanism are used by differentauthors to designate the same concept.Choose one. For example write: Agame form13 is a pair (S,h). . . Youcan also write: a game form (alsoknown as a mechanism), thereby tell-ing us that your intention is to use thephrase game form since it is inboldface italics, but reminding us thatthe term mechanism is also used. Youwould be confusing us if you wrote amechanism (or game form) . . .

Do not populate your paper with in-dividuals, agents, persons, consumers,and players. One species is enough.

13 The terms game or mechanism are some-times used.


Universal quantifications can be writtenas for all, for any, and for every;given can also introduce some objecttaken arbitrarily from some set. I haveseen proofs in which all four ways ofquantifying were used, and that did notlook good. Be careful about for any. Ifyou write If for any x X, f(x) > a, . . .,it really is not clear whether you meanfor all x or for some x. The termspreference relation, utility, andutility function are used interchange-ably by some authors, but you shouldnot do so. There are important concep-tual distinctions here, to which I al-luded earlier. Choose language so as tohelp keep them straight.

In areas where language has not set-tled yet, you may have several, perhapsmany choices. Do not take this as alicense to go back and forth betweenseveral terms. Instead, seize theopportunity to steer terminology in thedirection you favor.

Name your concepts carefully. Whenyou introduce a definition, you need tofind a good name for it, a term or aphrase that suggests its content. If youuse a multi-word expression, do notworry too much about its length. Yourpriority is that it should be clear whichconcept you are designating. In anycase, you can also use abbreviatedforms of the expressions you chose. Agood way of preparing us for an abbre-viated expression is as follows: A feasi-ble allocation is (Pareto)-efficient ifthere is no other feasible allocation thatall agents find at least as desirable andat least one agent prefers. Later on,you can simply talk about efficient al-locations. Unless you use several no-tions of efficiency, in which case youobviously need to distinguish betweenthem by means of different phrases, theshorter expression is unambiguous andslightly easier to use.

Actually, I do not think that long ex-

pressions are much of a problem in atext, as I explained earlier. In a seminarpresentation, however, they may be. Onthese occasions, look for relatively shortones. Alternatively, you can use thelong and more descriptive expression afew times, and when you think that theconcept has been absorbed by youraudience, tell them: From here on, Iwill only use the following shorterexpression: . . .

Avoid unnecessary technical jargon.If a function is order-preserving, do notsay that it satisfies order-preserv-ingness; the name of the property isorder-preservation. I do not like thephrase one-player coalition, which weuse when discussing cooperative games;you may have to speak separately of in-dividual players and of coalitions (setsof two or more players). A theorem isproved by a person, not by a paper:this result is established by Smith(1978) is better than this result is es-tablished in Smith (1978). In commonlanguage, preferring means what ineconomese we often call strictly pre-ferring, and in our dialect we have thephrase weakly preferring, which doesviolence to standard English too. Inmost cases, we can rephrase so as toavoid these conflicts with common us-age. When you feel you cannot avoid aconflict, give priority to your statementbeing unambiguous.

Keeping in mind that a given condi-tion may have different interpretationsthat depend on the context, choose neu-tral expressions that cover the variousapplications over expressions that aretoo intimately linked to the particularset-up to which your paper mainly per-tains. The requirement that an alloca-tion rule be monotonic with respect toan agents endowment can be seen fromthe strategic viewpoint; it will make itunprofitable for the agent to destroysome of the resources he controls.


Alternatively, it may be motivated byfairness considerations; the agent shouldderive some benefit from an increasein the resources he has earned. Insteadof phrases taken from game theory orfrom the theory of fair allocation, how-ever, use a neutral expression suchas monotonicity, (or endowmentmonotonicity if you also discuss mono-tonicities with respect to other parame-ters), and let your readers decide whichinterpretation they prefer.

Designate assumptions by names thathelp keep the logical relations betweenthem in mind. Strict monotonicityshould imply monotonicity, a conditionthat in turn should imply weak mono-tonicity. In an axiomatic study, axiomsoften come in a variety of forms of dif-ferent strengths. Name them so as tomake their hierarchy clear.

Challenge dominant terminology andusage if you find them inadequate. Ifyour paper is a follow-up to someonespublished work, as it almost certainly is,do not feel compelled to use the samelanguage if it was not well chosen, evenif the writer is a prominent member ofthe profession. The same comment ap-plies to notation. For instance, whyshould the adjective fair be used todesignate allocations that are both equi-table and efficient, as it was in the earlyfairness literature? In common lan-guage, the term has no efficiencyconnotation. Refer to equitable andefficient allocations. The word endow-ment suggests (admittedly, it does notimply) resources that are owned ini-tially, prior to exchange and produc-tion, so the expression initial endow-ment is redundant. Just speak of theagents endowments.14 The condition of

independence of irrelevant alterna-tives that Nash used in his axiomaticderivation of what we now call the Nashsolution, is dangerous. I prefer a phrasesuch as contraction independence,which is suggestive of the geometric op-eration that is being performed, withoutof course allowing us to infer exactlywhat this operation is, but Nashs ex-pression is no more informative. Thereader will decide on her own whetherthese contractions are irrelevant.Maskin-monotonicity is really an in-variance condition: it states the invari-ance of the social choice under certaintransformations of preferencesthe term monotonic is appropriate todescribe these transformationsanddesignating it by a phrase such as in-variance under monotonic transfor-mations might be a better idea, espe-cially for audiences that are not familiarwith the implementation literature.(In general, naming conditions aftertheir authors is not as useful as namingthem in a way that suggests their con-tent.) If the length of this alternate ex-pression bothers you, what aboutMaskin-invariance? If you decide tointroduce a new phrase, do not forgetto also indicate the names that arecommonly used.

Of course, the English language wasnot developed to label concepts ofmathematics or economics, but thecloser the fit between the concept youhave to name and the common meaningof the word you choose, the better. Formost of your conditions, you cannothope to find a short phrase describ-ing without ambiguity hypothesis andconclusion; strike the right balancebetween compactness and precision.

14 Besides, if you have to consider changes inthe endowment of a player, to find out for in-stance whether the owner of two left gloves maygain by throwing away one of them prior to enter-ing the market, you will have to make him go from

the pleonastic initial initial endowments to theoxymoronic final initial endowments, and what-ever benefit he may derive from his clever movewill be more than cancelled by the embarrassmentof using bad English.


Use technical terms correctly. Do notuse the term vector unless you willperform vector space operations. If youhave in mind a collection of objectstaken from some set, the appropriateterms are lists, ordered lists, orprofiles. For instance, the notation(R1, , Rn) refers to an ordered list ofpreference relations (or a preferenceprofile), not a vector of preference rela-tions: you will probably not compute(R1 + R2) 2. On the other hand, it isoften appropriate to present a list(s1, , sn) of strategies as a strategy vec-tor; for instance, in a game form de-signed to implement a solution to apublic goods problem, a strategy for anagent may be a public good level, andthe outcome function may select the av-erage of the announced levels. Con-sumption bundles are usually vectors.You often compute averages of con-sumption bundles or multiply them bytwo.

Do not confuse functions with the val-ues they take. If f:R R is a function,f(x) is the value the function takes whenits argument is x. So f(x) cannot be dif-ferentiable, or concave, and so on.These are properties of f and not of itsvalues. Designate the function simplyby f (this is better than f()). By thesame token, ui(xi) is not agent is utilityfunction; ui is. Conversely, if ui is agentis utility function, it is not also the par-ticular value that this function takes fora certain choice of its argument. If F isa solution to a class of bargaining prob-lems, and S is a problem in its domainof definition, F(S) is not a solution any-more, but something like a solutionoutcome, the solution outcome ofS. Alternatively, you can call F a so-lution concept and refer to F(S) as thesolution of S.

Get a good dictionary, and, if Englishis not your first language, ask for assis-tance. To weed out from your text galli-

cisms, nipponisms, sinicisms, and so on,get the help of a native gardener.

5. Writing Proofs

Learn LATEX or Scientific Word. Oneof the first choices you have to make isthat of a typesetting software. For yourdissertation, I strongly endorse LATEX,(or TEX, or Scientific Word, whicheverone you can handle). LATEX makes plaintext look beautiful, and because it un-derstands the structure of mathematicalexpressions, its benefits for the writingof mathematics cannot be measured.Moreover, it is widely used (in mathe-matics, it has truly become the type-setters LATIN, and you will find it veryconvenient when collaborating withcoauthors dispersed throughout theworld. A reader of a previous version ofthis essay suggested that I recommendthe LATEX Graphics Companion ofGoosens, Rahtz, and Mittelbach (Ad-dison-Wesley) and PSTricks of Timo-thy van Zandt, advice that was secondedby another reader. If you do not knowhow to use these softwares, ask one ofyour younger classmates to teach you(knowledge about computers goes fromthe young to the old). Also, use a spell-check. When submitting a paper to ajournal, respect their style guidelines.

The optimal ratio of mathematics toEnglish in a proof varies from reader toreader, but there is a consensus on amiddle range. A proof written entirelyin English is often not precise enoughand is too long; a proof written entirelyin mathematics is impossible to under-stand, unless you are a digital computerof course. Modern estimation tech-niques have shown that the optimal ra-tio of mathematics to English in a prooflies in the interval (52%, 63.5%). Pickthe point in that interval that is rightfor you and stick to it. However, thetheorems themselves should be stated


in the simplest English possible. Thereader who wants to know more thanthe probably informal description of re-sults given in your introduction, butdoes not have much time, will be ableto gain a much more precise under-standing of your contribution at a verysmall cost by just reading the theorems.I admit that this is sometimes difficultto achieve, and for technical papers it isprobably impossible, but you should try.

Avoid long sentences. A good way toprevent ambiguities is to mainly writeone-clause sentences. If English is notyour native language, this will alsogreatly help you avoid grammatical er-rors. Finally, it will force you to write

sentences in logical sequences. Here isan illustration of the idea: Let (S,h) bea game form. Let rn be a class of ad-missible profiles of preference rela-tions. Given R rn, the triple (S, h, R)is a game. A Nash equilibrium of (S,h, R) is a point s S such that for all i Nand all si Si, we have hi(si, si) Ri hi(s). Ifs S is an equilibrium, h(s) Z is its cor-responding equilibrium outcome. LetE(S, h, R) Z denote the set of equilib-rium outcomes of the game (S, h, R).The game form (S, h) implements thecorrespondence :rrrrn Z if for allpreference profiles R rn, we haveE(S, h, R) = (R).

You may think that your chance for a

All I have to do is deduce, from what I know of you, theway your mind works. Are you the kind of man who wouldput the poison into his own glass, or into the glass of hisenemy? . . . Now a great fool . . . would place the winein front of his own goblet, because he would know thatonly another great fool would reach first for what he wasgiven. I am clearly not a great fool, so I will clearly notreach for your wine . . . We have now decided the poi-soned cup is most likely in front of you. But the poison ispowder made from iocane and iocane comes only fromAustralia and Australia, as everyone knows, is peopledwith criminals and criminals are used to having people nottrust them, and I dont trust you, which means that I canclearly not choose the wine in front of you . . . But again,you must have suspected I knew the origins of iocane, soyou would have known I knew about the criminals andcriminal behavior, and therefore I can clearly not choosethe wine in front of me.

Proof: This follows from the inclusion P, Part (i) Proposition 1, and Lemma 1applied to . QED

(c)

Figure 3. The ratio of mathematics to English in a proof should be in the interval [52%, 63.5%]. (a) This proofhas too much math. Due to the density of mathematical symbols, it is virtually impossible to understand. (I canonly make out that it states the existence of ducks having certain properties.) (b) This game-theoretic proof dueto William Goldman (1973) has too much English; it is not precise enough and is too long. Not surprisingly,two paragraphs down, the character who produced it is dead. (c) This proof is just right, said Goldilocks, andthat is the one she read. It is indeed pleasantly short and clean. Wouldnt you like to know what theorem itproves?

(a) (b)


Nobel prize in literature will not im-prove much by this staccato style. Yet Icould name several grammatically im-paired writers who hardly ever usedsubordinate or relative clauses and yetgot to make the trip to Stockholm! Ifyou really do not like such choppywriting, in your very last draft, recon-nect some of your shortest sentences.Similarly, break your text into para-graphs of reasonable size, keeping inmind that too much of a good thing isa bad thing: a sequence of one-sentenceparagraphs is not pleasant to read.

A certain amount of redundancy isuseful, but do not overdo it. Giving aninformal description of the main stepsof a proof in addition to the formalproof is not strictly necessary, but itmight be quite helpful. Any such expla-nation, however, should not appearwithin the proof itself, but outside andpreferably before, so as to prepare usfor it. The proof itself should be as con-cise as you can make it without hamper-ing readibility. Similarly, when youstate a difficult definition, assist us bygiving an informal explanation in addi-tion to the formal statement. Here, too,give it before the formal statement, asthis placement will prepare your read-ers for it. It will also save them15 frusta-tion: it is indeed annoying to spendtime trying to understand a complicatedconcept when it is first given, only todiscover two paragraphs down that theauthor was willing to help after all.

The same comment applies to fig-ures. If you provided a figure to illus-

trate a proof, thank you, but why didntyou say so ahead of time, so that wecould identify on it the variables as youfirst introduced them and use it to fol-low your argument? Warning us of theexistence of a figure is especially impor-tant because, if your typesetting experi-ence is as limited as mine, you will findit hard to control where the figure endsup (my computer always seems to makethose kinds of decisions), and a figureillustrating a particular proof might verywell appear on the page that follows theproof instead of next to the proof.

It is often worth explaining very sim-ple things, especially in seminars whereyou will not have the time to explain thecomplicated ones in any detail, and es-pecially at the beginning. Indeed, if youlose your audience then, you may have ahard time retrieving it.

After stating an if and only if theo-rem, do not refer to the if part andthe only if part, or the sufficiencypart and the necessity part. Mostpeople will not know for sure which di-rection you mean. I have even seensome of the greatest economists beingconfused about this, and in my personalpantheon, they are people whose ap-proach to economics cannot be de-scribed as literary. Restate the resultin each direction as you discuss it. Simi-larly, would you guess that most of yourprofessors really do not know what amarginal rate of substitution is? But itis true! To most of us, a sentence suchas Agent 1s marginal rate of substitu-tion at z0 is greater than agent 2s onlymeans that the two agents indifferencecurves through z0 have different slopesat z0. We just hope that which is steeperwill be clear when we really need toknow. Of course, we would never admitit in public, and I most certainly wouldnever put such a confession in writing,for fear of being forever shunned bymy colleagues! Instead, compare the

15 Did you notice that I sometimes refer to thereader, sometimes to your reader (in the singu-lar), sometimes to your readers (in the plural),sometimes to us, your readers? This is an exam-ple of an inconsistency of style that should beavoided. Just like this should be avoided since Ihave throughout addressed you, my reader; there-fore, I should have written, that you shouldavoid. I return to this issue at the end of thisessay.


agents marginal rates of substitution ofgood 2 for good 1 at the point z0; evenbetter, simply talk about their indiffer-ence curves being more or less steepat z0.

It is a great unsolved mystery of neu-roscience that someone can prove thefanciest theorems in the most abstractspaces and yet have trouble with somevery elementary operations. Rememberthat. After all, havent you called yourrelatives in England when it was 3 a.m.there, after having carefully calculatedthat it would be 3 p.m.? You might havefailed in such a trivial calculation,and yet brilliantly passed exams wheremuch more of your intellect was beingtested.

Use pictures. Even simple picturescan be of tremendous help in makingyour seminar presentations more vivid.Figures are also very important tolighten a paper, to provide relief fromlong verbal or algebraic developments,and to illustrate definitions and steps ofproofs. Of course, a figure is not a sub-stitute for a proof, and the proof shouldbe understandable without it, but it maygive the main idea, and thereby cut byhalf (probably much more than that, ac-tually) the time your reader will need tounderstand it. Again, remember thehundreds of little diagrams that youdrew on your way to your results.

Label your figures as completely aspossible. Label the allocations, the sup-porting prices, and the endowments. Toindicate the efficiency of an allocation,it often helps to shade the upper con-tour sets in the neigborhood of that al-location. Label a few indifferencecurves for each agent (some redundancyis useful). If you assume convexity ofpreference relations and if in fact youdraw the indifference curves strictlyconvex, who owns which indifferencecurve will be unambiguous. But if youdo not make that assumptionyou may

very well work with linear preferencerelations or non-convex onesthis own-ership will not always be so clear. Avoidunnecessary arrows. You can most oftenposition your labels close to the itemsthey designate without creating ambi-guities. Use arrows only if the figurewould get too crowded, in particular ifthe label is too long.16

Have one enumeration for each cate-gory of objects. Number definitionsseparately from propositions, theorems,and so on. Some authors use a single listfor all of their numbered items, so thatfor example, Definition 15, which is thetenth definition, is followed by Theo-rem 16, which is the third theorem, thistheorem being followed by Corollary17, which is the only corollary . . . andso on. Multiple lists are preferable, asthey help us understand the structure ofthe paper. If you have two main sec-tions, with one theorem in each, labelthe theorems Theorem 1 and Theorem2. Having a single list certainly fa-cilitates retrieving a needed item, butthis benefit is too small. Bringing outthe structure of your paper is moreimportant.17

State your assumptions in order ofdecreasing plausibility or generality.When introducing your assumptions,start with the least controversial ones,and write them in order of decreasingplausibility. For utility functions, do notwrite A1: ui is strictly concave; A2: ui isbounded; A3: ui is continuous. Instead,and here I do not attempt to give namesto the conditions, write: A1: ui is

16 Look at the map of the city where you livethere are hundreds of themand you will notethat all the streets are labeled without arrows andyet without ambiguities! You surely do not needarrows in your figures.

17 For long documents such as books, adding tothe label of a theorem the page number on whichit is stated might be useful: Theorem 3.123 is thethird theorem of the chapter and appears on page123.


continuous; A2: ui is bounded; A3: ui isstrictly concave.

Introduce your assumptions in re-lated groups. For a general equilibriummodel, Assumptions A1A5 pertain toconsumers and Assumptions B1B6 per-tain to firms. For a game, AssumptionsA1A3 pertain to the structure of thegame and Assumptions B1B2 to thebehavior of the players.

Figure out and indicate the logical re-lations between assumptions and groupsof assumptions. If you have many condi-tions, and many logical relations be-tween them, it is helpful to presentthese relations in the form of diagrams.The best way to do this is by means ofVenn diagrams, each bubble symboliz-ing the set of objects satisfying one ofthe conditions.

When you draw two partially overlap-ping bubbles associated with ConditionsA and B, it is because you have identi-fied at least one object satisfying A butnot B; at least one object satisfying Bbut not A; at least one object satisfyingboth.

You can also use a diagram of arrowsand crossed arrows. The advantage ofVenn diagrams is that by drawing thebubbles of appropriate size, you canalso convey information about the rela-tive strengths of conditions. If A ismuch stronger than B, draw a muchsmaller bubble for A. If you prove atheorem under B, whereas A was usedin previous literature, your reader willcertainly want to know how significantyour weakening is. You need to give hersome sense of it.

Another advantage of Venn diagramsis that they make it easy to indicate thejoint implications of several conditions.If A and B together imply C, the twobubbles symbolizing them intersectwithin the bubble symbolizing C. Withthe other technique, you would have tomerge two arrows emanating from A

and B and point the merged arrow at C.You will end up with a big mess. A dis-advantage of Venn diagrams is that forthem not to be misleading, you need tofigure out all of the logical relationsbetween your conditions. But this isanother advantage: you need to figureout all of the logical relations betweenyour conditions!18 You will not regretdoing the work. When you use arrows,by not linking two conditions, you un-ambiguously indicate not knowing howthey are related. That option does notexist with Venn diagrams.

When you use Venn diagrams, youcan sometimes draw the bubbles in away that suggests some of the structureof the sets they designate: if the setis convex, draw a convex bubble; ifit is defined by a system of linear

X

F

P

Wed

Bed

Figure 4. How to indicate logical relations between concepts. Key: X is the feasible set, P the set of Pareto-efficient allocations, F the set of envy-free allocations, Bed the set of allocations meeting the equal division lower bound, Wed the set of equal division Walrasian allocations. The set of feasible allocations is so large in relation to the set of Pareto-efficient allocations that its bubble does not even fit in the page. There are continua of Pareto-efficient allocations and of envy-free allocations but typically a finite number of Walrasian allocations. A small tip: breaking the boundary of a bubble to make room for its label is the best way to make unambiguous what is being labelled.

18 An effective way to do this is as follows: figureout all the illogical relations; what is left are thelogical relations.


inequalities, make its boundary polyg-onal; if it is a lattice, draw it as adiamond, and so on.

Make sure that there are objects sat-isfying all the assumptions that you areimposing. Have at least one example.After stating that you will considereconomies satisfying Assumptions 110,exhibit one that does satisfy all of theseassumptions (try CobbDouglas; it willprobably work). If the class of objectssatisfying your assumptions is empty,any statement you will make about allof them will be mathematically correct,but of limited usefulness.

Use a common format for the formalstatements of your results, and for partsof proofs that are similar. If you haveseveral results that are variants of eachother, present them in the same formatso as to make their relation to eachother immediate. If you first state:

Theorem 1. If A, B, and C, then D andE.

do not write your next theorem, whichdiffers from Theorem 1 in that C isreplaced by C and E is replaced by E

~,

as

Theorem 2. Suppose A and B. In ad-dition, consider the class of economiessatisfying C. Then D. Also, E

~ holds.

Instead, use a paralell format:19

Theorem 2. If A, B, and C, then D andE~

.

The relation between Theorems 1and 2 will then be obvious, and yourreader will discover it by simply scan-ning them. By choosing a different for-mat, you would force her to actuallyread their entire statements, and makethe comparisons, hypothesis by hy-pothesis and conclusion by conclusion,that are needed for a good under-standing of how the results are related.In some cases, it will be possible to pre-sent the two theorems as Parts 1 and 2

19 This incorrect spelling of paralell (Darn, I didit again!) is an unfortunate consequence of myhaving finally mastered that of A. Mas-Colellsname (the name for which, in my estimation, theratio of occurrences of incorrect to correct spell-ings is the highest in the profession). Do spellnames correctly. Dupont does not want to be con-fused with Dupond any more than Schultze identi-fies with Schulze. Hernandez and Fernandez aretwo different people. Thompson is very attachedto his p, and I know for a fact that Thomson hasno desire for one.


of a single theorem.20 Physical proxim-ity and common format are two impor-tant ways in which you will facilitateyour readers task.

Similarly, a proof may contain severalparts having identical or almost identi-cal structures. Present them so as tomake this obvious. Instead of writingCase 1 and Case 2 separately, writeCase 1 first, and make sure it is in per-fect shape; then copy it and make theminimal adjustments that are necessaryto cover Case 2. The similarity of phras-ing and format will unambiguously sig-nal to your reader that if she has under-stood the first part, she can skip thesecond part. Or if she decides to readCase 2, the marginal cost she will incurwill be very small.

Divide proofs into meaningful units,clearly identified. Indent and double in-dent to indicate structure. Name andnumber these units: Step 1, Step 2,Case 1, Subcase 1a, Subcase 1b, Case 2,Claim 1, Claim 2. If the proof is suffi-ciently complex, give each step or claima title indicating its content. Make surewe know whether this title is a state-ment that you will prove, or an obviousconclusion that we should reach on ourown:

Step 1: The domain of the correspon-dence is compact.

Claim 1a: The domain is bounded. Tosee this . . .

Claim 1b: The domain is closed. Thisfollows from Lemma 1.

Step 2: The correspondence is uppersemi-continuous.

If the steps are conceptual units ofindependent interest, and certainly ifthey are used in other parts of the pa-

per, as opposed to pertaining to a list ofsimilar cases that have to be checked inturn, call them lemmas (or lemmata,which is the plural form of lemma inGreek; not lemmatas, unless you reallyhave lots of them!), and present themseparately. If a proof is long, you mayhave to number the successive state-ments that it is composed of. Then, youcan refer to them by numbers. Unfortu-nately, this quickly increases the com-plexity of the proof, (I mean, how com-plex it looks). If you do this, onlynumber the essential statements. Forinstance, if you end a sentence by estab-lishing a statement that is used as a hy-pothesis in your next sentence, and ifthe statement is not used elsewhere,you need not number it.

Gather all the conditions needed for aconclusion before the conclusion insteadof distributing them on both sides. Hy-potheses come first and together. Donot write If A and B, then D since C.or If A and B, then D. This is becauseC. Instead, write If A, B, and C, thenD. Especially for long statements, ithelps to visually separate the hypothe-ses from the conclusions by then, wehave, or it follows that. If you writeSince A, B, C, and D, we will not besure whether you mean Since A, thenB, C, and D, or Since A and B, then Cand D, although technically, the for-mer interpretation has to be the correctone.

Similarly, mathematical statementsusually look better when all the quanti-fications appear together, preferably atthe beginning, instead of being dis-tributed on both sides of the predicate.For instance, instead of For all x X,we have xi > yi for all i N, write Forall x X and all i N, we have xi > yi. Bythe way, this example illustrates a con-flict between two of the recommenda-tions that I have made. I just advised toseparate mathematical expressions by

20 Capitalize the word theorem when you referto a specific theorem, as in Theorem 1 above, butnot in a sentence such as Capitalize the wordtheorem when . . . Same rule for propositions,sections, and so on.


English words: for all i N, xi > yi,does not read as well as for all i N, wehave xi > yi, but the formulation xi > yifor all i N, in which the quantificationover agents occurs after the inequality,also achieves the desired separation,and it is shorter.

Be specific about which assumptions,or which parts of assumptions, you needfor each step. Do not write The aboveassumptions imply that f is increasingif you need only some of the above as-sumptions to prove that f is increasing.Write Assumptions 3 and 4 imply that fis increasing. Even better, if you donot need Part (i) of Assumption 4, writeAssumption 3 and Part (ii) of Assump-tion 4 together imply that f is increas-ing. Similarly, if Theorem 3 followsfrom Lemmas 1 and 2, show us exactlyhow it follows. Do not write A and Bimply C and D, if in fact A implies Cand B implies D. At a very small addi-tional typing cost, you can be muchmore precise.

When you cite a theorem, be as exactas possible. Refer to a textbook thatmost of your readers are likely to ownor be familiar with. This is especiallyimportant for theorems that exist in sev-eral forms; we need to know which ver-sion you are using. Also, you shouldprobably cite the English edition of aclassic text instead of the translated ver-sion in your native language, eventhough that is the one you know well.So write: By the Brouwer fixed pointtheorem (Debreu 1959, p.26) . . . Add-ing the page number is a nice touch.

Verify the independence of your hy-potheses. For each hypothesis in eachtheorem, check whether you could pro-ceed without it. Do not write UnderAssumptions A, B, and C, then D, if Aand B together imply C, or if A and Btogether imply D.

Having put together a toy for one ofmy daughters, I discovered some parts

left in the box. Either these were re-placement parts, or I had done some-thing wrong (I will not tell you which,but as a clue, let me say that therenever are replacement parts in the box).Similarly, after QED, look in the box forstranded hypotheses. You might havemade a mistake, but you might also bepleasantly surprised to find that you canactually prove your theorem withoutdifferentiability. Wouldnt you bethrilled if your result applied to Banachlattices (which you did not even knowexisted two weeks ago), while youthought you were working in boring n-dimensional Euclidean space?

Sometimes, you will be unable toshow that a certain hypothesis is neces-sary for the proof and unable to con-clude without it either. This is an un-comfortable situation that should keepyou up late at night.

A given hypothesis may be the con-junction of several more elementaryones. Then, try to proceed without eachof its components in turn. For instance,if you have shown that Under compact-ness of the set X, conclusion C holds,do not only check that without compact-ness, C might not hold anymore. In-stead, ask whether Under boundednessof X, C holds and whether Underclosedness of X, C holds.

Explore all possible variants of yourresults. If you prove that A and B to-gether imply C, do not limit yourself tothat statement. Find out whether simi-lar statements hold with A replaced bythe closely related conditions A, A0,and A

~, or B replaced by B and B, or C

replaced by C0. Knowing statement P isnot enough. Discover as many state-ments as possible that are close to P andare also true, and statements that areclose to P but are not true. It is as use-ful to understand the multiplicities ofstatements around the one you areproving that could be true but are not,


as the statement that you are proving. Itmay even be more useful. Comment onthe main variants of your theorem butkeep to yourself the least significantones.

Do not leave (too many) steps to thereader. Give complete arguments. Somesteps in a proof may involve standardmanipulations and detract from yourmain point. Perhaps they should not bein the body of the paper, but in an ap-pendix. Do not take them out though.Your reader may not be familiar with aderivation that you have seen and per-formed hundreds of times. Just havingthe option of assessing the length of astep and recognizing the names of fa-miliar theorems on which it is basedwill be helpful to her in checking herunderstanding of the logic of your argu-ment, even if she does not actually readall the details. In general, I do not liketoo much of the work to be relegated toappendices. When I first look at a pa-per, I skip most of it anyway, and if Idecide to study it more seriously, I findit annoying to have to go back and forthbetween the body of the paper and theappendix.

If you think a step is obvious, lookagain. Do not think that your errorsnecessarily occurred in the hard parts ofyour proofs (I should say, what youthink are the hard parts of your proofs).They may very well have hidden in(what you think are) the easy parts, tak-ing advantage of your overconfidence.After completing your paper, search forthe clearlys and obviouslys andmake sure that what you claimed wasclear and obvious is, if not clear andobvious, at least true.21

Numerical examples are not alwaysuseful. It is commonly thought that nu-merical examples provide easy introduc-

tions to complicated proofs. This is trueonly if the examples are well chosen. Ageneral algebraic expression has in factthe advantage of reminding us of thelogic of an argument. If, to fix ideas,you choose x1 = 1 and x2 = 8, the number9 will refer to the sum x1 + x2 butit might be helpful to remember thisorigin: so write 1 + 8 instead, or9 (= 1 + 8). The expression x1 + x2 isoften preferable. In a three-playergame, write the number of coalitionsas 23 1; we do not care much if thatnumber is equal to 7.

Also, by using numerical examples in-stead of algebraic notation, you losetrack of units of measurement. It makesit harder to check the correctness ofexpressions.

When you vary a parameter, as a re-sult of which agent 1s income goesfrom 5 to 7 and agent 2s income from 8to 5, it will soon be difficult to remem-ber which ones are the initial incomes,which ones are the final incomes, andwhose income is 5 and when. If you usewell-chosen algebraic notation, for in-stance by calling the incomes I1 and I2before the change and I1 and I2 afterthe change, your reader cannot beconfused.

If you insist on using numbers,choose them so that whatever opera-tions you perform on them do not turnthem into monsters. If you will divide x1by 2, choose x1 even; if you will take itssquare root, do not choose x1 = 10. Actu-ally, I take this back. It depends: if theincomes are 5 and 7 initially, and theyare cut in half, they will be 5/2 and 7/2after the change and the fractions willmake it easier to remember that theyare the new ones. If they were even,you would be tempted to perform thedivision to get integers and again, thenew incomes would be hard to tell apartfrom the old ones.

In filling a payoff matrix, take all

21 Do not deduce from this, however, that sim-ply deleting the clearlys and obviouslys willnecessarily eliminate all of your errors.


payoffs to be integers between 0 and 9so that you do not need to separatethem by commas. In each cell of thepayoff matrix you can also place thepayoffs of the row player slightly higherthan that of the column player.

More useful than numerical examplesare examples with a small number ofagents, a small number of goods, and noproduction. Then you can save on sub-scripts, you can use an Edgeworth box,and in your proof you can appeal to theintermediate value theorem instead ofto a general fixed point theorem. By thesame token, general arguments aresometimes easier to understand thantheir applications to special situations:it is more transparent why a competi-tive equilibrium is Pareto efficientwhen the proof is presented in the gen-eral case than for a CobbDouglas ex-ample, say. There is indeed little tobe learned from the calculations for aspecial case.

Similarly, illustrating a general phe-nomenon by means of a perhaps incom-pletely specified geometric example ismore informative than a complete argu-ment based on a particular numericalexample. The reason is that it may behard to identify which features of thenumerical example are essential to thephenomenon. For instance, to provethat in an Edgeworth box economythere could be several Walrasian equi-libria, an example in which preferencesare suggested by means of a few indif-ference curves for each of the twoagents suffices. Of course, a few indif-ference curves do not constitute a pref-erence map, and you have to rely onyour readers experience with suchmaps for them to mentally completeyour figure, or convince themselves thatthe completion can be done. The alter-native is for you to give entire maps,which in most cases will be by means ofexplicit numerical representations for

them. These representations will oftenbe quite complicated, and although theywill prove your point beyond doubt, Istrongly believe that they will hamperthe understanding of the circumstancesunder which multiple equilibria occur.

If you want to name your agents, doit in a way that helps. When you thinknumbering your agents from 1 to 4 istoo dry in describing an example, try realnames, but choose them carefully so asto make it easy to remember who iswho. Naming them Bob, Carol, Ted,and Alice will be cute but may be coun-terproductive. Ted most certainly doesnot belong in this group. Also, theyshould be ordered alphabetically: Alice,Bob, Carol, and Dwayne are your fourconsumers.

In honor of a favorite writer, I havelong wanted to call agents 1 and 2Qfwfq and Xlthlx, but which is actuallyeasier to remember, that agent 1 isendowed with good 1 and agent 2 is en-dowed with good 2, or that Qfwfq is en-dowed with apples and Xlthlx endowedwith oranges?

By the way, in a seminar, avoid cul-tural references that are obscure to toolarge a fraction of your audience, butby all means, do not avoid culturalreferences altogether because you fearthat some of your audience may not un-derstand them. Sometimes it will not beeasy to decide. Do you think that in or-der to prevent those of my readers whodont know French from feeling ex-cluded, I should have resisted the temp-tation to quote Erreur, tu nes pas unmal, thereby depriving the others ofthis beautiful maxim? Which of the cri-teria of social choice theory is the rightone here?22

22 Once, I referred to Bob and Carol, Ted andAlice in a seminar in which I discussed matchingtheory, and a member of the audience commentedthat I was showing my age! I was unfortunately notquick enoughshowing my age once againto


Do not collapse two or three simi_lar statements into one by indicatingthe variants in parentheses. Considerthe following definition: The func-tion f:R R is decreasing (increasing;non-decreasing) if for all x, y R withx > y, f(x) < f(y) (respectively f(x) > f(y);f(x) f(y)).

The only way for us to be sure weunderstand this triple definition is toread it three times (once for decreasing,once for increasing, and once for non-decreasing), and yet it is pretty simple.More complicated statements in thatformat require a mental gymnastics thatwill unnecessarily exhaust us. Just re-state the complete sentence in thevarious forms you need. I also have a lotof trouble with and/or (or is itor/and?).

Do not start a sentence with a pieceof mathematical notation. Journal edi-tors will red-pencil you if you do, and Iagree with them that it does not lookgood, especially if the notation is lowercase. x designates an allocation is notpretty. I is the set of individuals is notas bad because I is uppercase (but whata grammatical provocation!). Let x des-ignate an allocation is what editors willprefer.

Be consistent in your writing style.Do not switch back and forth betweenfirst person singular, first person plural,and passive forms. If you write: In sec-tion 3, I show that an equilibrium ex-ists. In Section 4, we establish unique-ness. To prove these results, it isassumed that preference relations are

strictly convex and have infinitely dif-ferentiable numerical representations.For the proof of the main theorem, oneappeals to the Brouwer fixed pointtheorem. Section 5 concludes. yourreaders will think you need psychiatrichelp. Are you I or we? Is it becausethese assumptions are embarrassingthat you suddenly hide behind the pas-sive form? Believe me, we have allmade embarrassing assumptions. Andwhy do you let Section 5 conclude whenyou did all the work? The passive formis found awkward by me and our advicehere is to have it replaced. I is per-haps too personal. Between I andwe, I choose we, but if you chooseI, we will respect your choice.23

Similarly, do not travel back andforth between present and futuretenses. Do not write: First, I prove ex-istence. Then I will apply the theoremto exchange economies. I conclude withopen questions. In most cases, usingthe present tense throughout, even indescribing past literature, is just fine.

Choose the sex of your agents onceand for all. Flip a coin. If it is a boy,rejoice! If it is a girl, rejoice! And dontsubject them to sex change operationsfrom paragraph to paragraph.24 Two-person games are great for sexual equal-ity. Make one player male and the otherfemale. This will actually facilitate talk-ing about the game and help yourreader keep things straight. It will alsosave you from the awkward he or she,him or her,, his or her! Alterna-tively, you may be able to refer to your

reply that by understanding that I was showing myage, and remarking on it, he was showing his! Hewas right though. I recently asked the students inmy graduate class whether they understood theallusion. Not one of them did. And yet, Bob &Carol & Ted & Alice (its a movie) came out onlyyesterday (30 years ago, to be precise)! From nowon, I will use this example only when I givelectures in retirement homes.

23 As a reader, I rather like the I form, whichis more engaging, but I am not comfortable usingit in formal papers. I use I here only because ofthe informal style that I chose for this paper. Para-doxically, the we form is less obstrusive than theI form. We can also be interpreted as you andthe reader, whom you are taking along, but thenbe careful if you refer to our previous work.

24 For a book, alternating between male andfemale between chapters might be acceptablethough.


agents in the plural, or choose one ofthem to be a firm, and refer to it as it.

Be consistent in your choice of run-ning indices. If N = 1, , n, do not writeinterchangeably for all i N, or forall i 1, , n, or for all i = 1, , n.Pick one formula and stick to it. In mostsituations, the quantification on the setof agents (to take an example) is clear.Skip it and write for all i. Thishelps keep down the density of symbols.In general though, it is good to indi-cate membership explicitly. For in-stance, instead of There exists z forwhich . . ., write There exists zZ forwhich . . . . Therefore, for consistencyof style and esthetic reasons, wheneverything else is explicity quantified, itbothers me a little not to see member-ship indicated for the set of agents,even if it is pretty obvious that theycome from N and not from Mars. Soinstead of For all i such that . . . ,I would write For all iN suchthat . . .

Do not put quantifiers in the middleof a sentence in English. A sentencesuch as

Blah, blah, blah, x such that P(x),blah, blah, blah y such that Q(x, y) andblah, blah, blah. does not look good. Write for all andthere exists. If the mathematicalstatements introduced by the quantifi-ers are complex enough, pull them outfrom the text in English and displaythem on separate lines, as follows:

Blah, blah, . . . , blah, blah,x such that P(x), y such that Q(x, y),and blah, blah, blah.

The quantifications should always beunambiguous. Remember also that tak-ing the negation of a properly writtenmathematical statement, with no hiddenquantifications, is a trivial operation.

The only mathematical symbols thatdo not bother me in a text in English

are 0 can simply be written as:Positivity: F(S) > 0.26

Indicate the end of proofs clearly.Use QED (for quod erat demonstran-dum), or Halmos u (I suppose, forquod erat quadrandum.27). Delete

25 See the problem with starting a sentence witha piece of mathematical notation (Section 6.10)!When I wrote earlier that you should not putquantifiers in the middle of a sentence in English,I should have said: do not put them anywhere insuch a sentence.

26 Or F > 0. By the way, do not place your foot-note markers at the end of mathematical expres-sions, as they will look like exponents. Placingthem beyond the punctuation mark, as the typo-graphical convention requires, and as I have donehere, helps, although logic would sometimes dic-tate that the marker be attached to a word insidethe clause (or the sentence) that ends with thepunctuation mark. Compare the marker for thisfootnote with the marker for the previous one: theposition of that earlier marker did not create anyambiguity, as I am sure that you did not think thatmy intention was to raise the universal quantifierto any power; still it did not look pretty. The sameproblem arises with quotation marks. I just wroteF > 0. The rule is to write F > 0. This is inagreement with logic if you think of the whole sen-tence, including the period that ends it, as beingthe unit that is being discussed. In other contexts,it may be the requirement F > 0 that is underdiscussion but here, and given that quotationmarks look a little like double prime, I admit thatplacing them after a needed punctuation mark isbetter, so refer to the requirement F > 0, whichis proved in Section 2.

27 Circulus? What about a little circle to indi-cate the beginning of a proof, matching the littlesquare that closes it?


the redundant This completes theproof, which precedes u in yourcurrent draft.

6. Conclusion

If you follow all of the above recom-mendations, not only will you bepleased with yourself, your seminaraudiences enlightened, your classmatesimpressed, your parents proud of you,and you will land a job in a top-five de-partment, but most importantly, youradviser will be happy. I readily admitthat each of them does not amount tomuch. However small imperfections,when added together, will take your pa-per over the line that separates those thatcan be understood from those that can-not. An Archimedian principle is at workhere. You will lose your readers or yourseminar audiences much earlier than nec-essary. In fact, you too will be confused.

Do not fool yourself: very few of yourreaders will take the time to understandyour whole paper, and a large fractionof your seminar audience will not havethe faintest idea of what you are talkingabout when you are half-way through.So, every bit will help in keeping theattention of a few a little longer.

If you are used to certain notationalconventions, or terminology, or ways ofstructuring a proof, they almost neces-sarily seem the best to you, and perhapsthe only ones worth considering. Youhave to be open-minded and genuinelyexperiment with other formulations.Only then can you decide what is trulybest. The first few times you use a newpiece of notation or a new term or anew format, it will appear strange toyou. Give it a chance.

Let time elapse beween revisions. Ifyour paper is so familiar to you that youessentially know it by heart, you willnever discover your mistakes. You needto let it sit in a drawer for a while.

When you pick it up again, it will have afreshness that will allow you to seeimmediately where it can be improved.

Good writing requires rewriting, an

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Thomson Young Person’s Guide to Writing Economic Theory