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7 Thomson: The Young Person s Guide to Writing Economic Theory 163 (a) (b) (c) (d) (e) 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. given the definition but she missed it, or that you are assuming she knows the definition. Here are three possible ways of introducing a definition: 1. A function is monotone if... ; 2. A function is monotone if... ; 3. A function is said to be monotone if... I prefer the first format and use it throughout this essay, because its phrasing is direct and its different typeface will facilitate its retrieval, if needed. Concerning the typeface, I recommend boldface or boldface italics over italics or plain text between quotation marks, neither of which makes the new terms stand out sufficiently. You should probably display the crucial definitions separately, and you may precede each of them by the word Definition in boldface (see the examples below). But do not introduce all definitions in this way, especially if you have many of them, as it will get tedious. Focus on the critical ones. To avoid repeating quantifications that are common to several definitions, you can group these definitions and state the quantification once: An allocation rule is efficient if for all preference profiles R, and all allocations z that it selects for R, there is no other allocation z that all agents find at least as desirable as z and at least one agent prefers; it is weakly efficient if instead there is no other allocation z that all agents prefer to z. To emphasize certain aspects of your paper, such as important conclusions, exploit the typographical choices at your disposal. Italics is a good one. However, if everything is emphasized, nothing is. When introducing a novel definition, give illustrative examples. If the definition is a property that an object may or may not have, exhibit: 1. Objects that satisfy the definition; 2. Objects that do not satisfy the definition; 3. Objects that satisfy the definition but almost do not; 4. Objects that do not satisfy the definition but almost do. Examples in Categories 3 and 4 are particularly important as they are responsible for most of the work in the proofs. Conversely, they may be the ones that allow the proofs to go through! In a paper, giving a range of examples that are representative of all four categories is, once again, not easily achieved because of space limitations, but in seminars this can sometimes be done. Here are two illustrations: Definition. A function f: [0, 1] is increasing if for all t, t [0, 1] with t > t, we have f(t) > f(t ). Figures 1a and 1b are dangerous, because they may plant in your reader s mind the seed that you will work with

8 164 Journal of Economic Literature, Vol. XXXVII (March 1999) R 4 R 6 R 2 R 3 R 5 R 1 a b a b a c d b (a) (b) (c) Figure 2. Examples of single-peaked and of non-single-peaked preference relations. (a) These relations are singlepeaked, with peaks at a for R 1 and at b for R 2. (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 R 5 has two local maxima, at a and b, and R 6 is maximized at any point of the non-degenerate interval [c, d]. 8 Several readers of this essay objected to sentences such as this function is nondecreasing, which sounds too much like this function is not a decreasing function, but means something else. Perhaps we should speak of a nowhere-decreasing function. functions that are linear, or perhaps concave. Figure 1c is what you need: it represents an increasing function in its full generality, with a kink, a convex part, a concave part, and a discontinuity. Figure 1d is useful too, as it shows a typical violation of the property. Figure 1e is very important because it makes it clear that you want more than that the function be nondecreasing. 8 Definition. The continuous preference relation R defined on [0,1], with asymmetric part P, is single-peaked if there exists x [0, 1] such that for all x, x [0, 1] with either x < x x or x x < x, we have x P x. Figure 2 presents the graphs of the numerical representations of six preference relations. Obviously, R 2 is singlepeaked and R 5 is not. But your viewer may not immediately think of R 1 as being single-peaked because its representation achieves its maximum at a corner, or may think that R 6 is admissible, although its representation has a plateau and not a peak. You should also make her aware of the fact that you include preferences that do not exhibit the symmetry illustrated in Figure 2b. All of these examples will be very useful to ensure that she fully perceives the boundary of your domain. Write definitions in logical sequences. Introduce terms in such a way that the definition of each new one only involves terms that are already defined, instead of asking your readers to wait until the end of the sentence or paragraph for everything to be clarified. For instance, state the dimensionality of the commodity space before you introduce consumers or technologies. In the standard model, a consumer is no more than a preference relation defined over a subset of that space, together with an endowment vector in the space; a technology is simply a subset of the space. In each case, it is therefore natural to specify the space, that is, the

11 Thomson: The Young Person s Guide to Writing Economic Theory 167 profiles r n (the cross-product of N copies of r indexed by the members of N), and the allocation space X, write The function ϕ:r n X strategy-proof if... Indicating explicitly the nature of the objects that you introduce is especially important if the reader may not be familiar with them. By writing A triple (π, x, y) ( 1)n + (m l)n is a Lindahl equilibrium if..., you help her realize that π has components indexed by agents (these are the Lindahl individualized prices). By the way, a sequence of elements of X is not a subset of X, but a function from the natural numbers to X. So, you cannot write {x k } k X. Nor can you write x k X. Speak of the sequence x k of elements of X, or of the sequence {x k } where for all k, x k X. When you define a concept, indicate what the concept depends on. Do not write The function f is differentiable at t if blah, blah, blah of t. Since what follows if depends on t, you should write The function f is differentiable at t (including at t in the expression in italics) if blah, blah, blah of t. Then, you can continue and say The function f is differentiable if it is differentiable at t for all t in its domain. A marginal rate of substitution is calculated at a point, so speak of agent i s marginal rate of substitution at x i. For an example taken from the theory of implementation, speak of a monotonic transformation of agent i s preferences at x i, and not just of a monotonic transformation. When you define a new variable as a function of old ones, it should appear on the left-hand side of the equality or identity symbol. If M has already been defined, and M is introduced next, with a value equal to M, you should write Let M = M, and not Let M = M. Do not assume that your readers are necessarily familiar with the definitions you use. There is rarely complete agreement on definitions in the literature. Apparently standard terms are often understood differently by different people. Therefore, define the terms you use, even some that you can legitimately assume everyone has already seen. Core, public goods, and incentive compatibility are examples of terms that are common enough, but define them. The word rationality frequently appears in formal developments in game theory without a definition being given. Do not make such a mistake. Refer to a given concept by only one name or phrase, even if you have several natural choices. Make one and stick to it. Indicate in parentheses next to your definition, or in a footnote, the other terms that appear in the literature. When you first discuss the general idea, you may use different terms in order to vary language and avoid repetitions repetitions, which admittedly do not sound very good, but after you have formally defined the concept and baptized it, only refer to it by its name. The terms game, game form, and mechanism are used by different authors to designate the same concept. Choose one. For example write: A game form 13 is a pair (S,h)... You can also write: a game form (also known as a mechanism), thereby telling us that your intention is to use the phrase game form since it is in boldface italics, but reminding us that the term mechanism is also used. You would be confusing us if you wrote a mechanism (or game form)... Do not populate your paper with individuals, agents, persons, consumers, and players. One species is enough. 13 The terms game or mechanism are sometimes used.

19 Thomson: The Young Person s Guide to Writing Economic Theory 175 inequalities, make its boundary polygonal; if it is a lattice, draw it as a diamond, and so on. Make sure that there are objects satisfying all the assumptions that you are imposing. Have at least one example. After stating that you will consider economies satisfying Assumptions 1 10, exhibit one that does satisfy all of these assumptions (try Cobb Douglas; it will probably work). If the class of objects satisfying your assumptions is empty, any statement you will make about all of them will be mathematically correct, but of limited usefulness. Use a common format for the formal statements of your results, and for parts of proofs that are similar. If you have several results that are variants of each other, present them in the same format so as to make their relation to each other immediate. If you first state: Theorem 1. If A, B, and C, then D and E. do not write your next theorem, which differs from Theorem 1 in that C is replaced by C and E is replaced by E ~, as Theorem 2. Suppose A and B. In addition, consider the class of economies satisfying C. Then D. Also, E ~ holds. Instead, use a paralell format: 19 Theorem 2. If A, B, and C, then D and E ~. The relation between Theorems 1 and 2 will then be obvious, and your reader will discover it by simply scanning them. By choosing a different format, you would force her to actually read their entire statements, and make the comparisons, hypothesis by hypothesis and conclusion by conclusion, that are needed for a good understanding of how the results are related. In some cases, it will be possible to present the two theorems as Parts 1 and 2 19 This incorrect spelling of paralell (Darn, I did it again!) is an unfortunate consequence of my having finally mastered that of A. Mas-Colell s name (the name for which, in my estimation, the ratio of occurrences of incorrect to correct spellings is the highest in the profession). Do spell names correctly. Dupont does not want to be confused with Dupond any more than Schultze identifies with Schulze. Hernandez and Fernandez are two different people. Thompson is very attached to his p, and I know for a fact that Thomson has no desire for one.

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