Illustration: "What's so Funny About Science?" by Sidney Harris (1977)

The updated version of this document and all the links are available at:
http://toomas-marit.hinnosaar.net/pages/suvekool

The material discusses Math requirements for graduate Economics studies and suggests some useful Math courses from Mathematics department in Tartu University. The first part of the material describes results from a small survey: seven persons, who (have studied) study in foreign universities in Europe, with undergraduate degrees from Tartu University or from Tallinn Technical University, gave their opinion about the need of Mathematics in their graduate studies. Mathematics requirements in US universities are described in the second part. The final part gives suggestions about some of the useful Math courses in University of Tartu.

The material is compiled for the Doctoral Summer School in Economics in 2006 (organized by University of Tartu and Tallinn University of Technology).

We asked 10 persons, who (have studied) study in foreign universities in Europe, with undergraduate degrees from Tartu University or from Tallinn Technical University, to complete a small survey. We received responses from 7:

• Kalle Ahi, CERGE-EI, Praha, Karl University (PhD student)
• Reimo Juks, Stockholm School of Economics (PhD student), Central European University (MA)
• Alvar Kangur, University of Oxford (PhD student)
• Dmitry Kulikov, University of Aarhus (MSc, PhD)
• Egle Tafenau, Christian-Albrechts-Universität zu Kiel (Univ. of Kiel) (PhD student)
• Ott Toomet, University of Aarhus (PhD)
• Lenno Uusküla, European University Institute (PhD student)

Main textbooks used in graduate courses in the foreign university (only those, which were mentioned by more than one person):

• Macroeconomics
  • Ljungqvist and Sargent, Recusive Macroeconomic Theory (5)
  • Stokey and Lucas, Recursive Methods in economic Dynamics (4)
  • Blanchard and Fischer, Lectures on Macroeconomics (2)
  • Walsh, Monetary Theory and Policy (2)
• Microeconomics:
  • MasColell, Whinston, Green: Microeconomic Theory (4)
  • Osborne and Rubinstein, A Course in Game Theory (2)
• Econometrics:
  • Greene, econometric Analysis (4)
  • Wooldridge, econometric Analysis of Cross Section and Panel Data (3)
  • Amemiya, Advanced Econometrics (2)

We asked whether survey participants had insufficient preparation in any of the main fields of Mathematics.

Summary of comments to this question:

• Everybody agreed that the Math preparation was sufficient for the program.
  • But only one person answered that he did not need to learn more Mathematics during his graduate studies.
  • The rest of the survey participants answered that it was possible to fill the holes in the preparation either independently or during the Mathematics courses offered by the Economics department or Mathematics department.
• Still, most of the participants agreed, that they would have benefited from better preparation in the following areas:
  • Real Analysis (matemaatiline analüüs): 4 persons
  • Functional Analysis (funktsionaalanalüüs): 3 persons
  • Theory of Probability (tõenäosusteooria): 3 persons
  • Measure Theory (mõõduteooria): 3 persons
  • Stochastic Differential Equations (stohhastilised diferentsiaalvõrrandid): 2 persons
  • Topology (topoloogia): 2 persons
  • Numerical Methods (numbrilised meetodid): 2 persons
  • Optimization Theory (optimeerimismeetodid): 1 person
  • Other topics (muud teemad):
    • Dynamic Programming
    • how to construct proofs
• All the participants answered that preparation in the following areas was appropriate:
  • Linear Algebra (lineaaralgebra)
  • Functions of Complex Variable (kompleksmuutujate funktsioonid)
  • Differential Equations (diferentsiaalvõrrandid)

Survey participants estimated their preparation in Mathematics compared to their coursemates in the foreign university as follows:

• one of the best (2)
• over average (2)
• average (2)

Mathematics books that survey participants have used during their studies in the foreign university:

• Simon and Blume, Mathematics for economists (3)
• Rudin, Principles of Mathematical Analysis (1)
• Apostol, Mathematical Analysis (1)
• Royden, Real Analysis (1)
• Hoffmann-Jorgensen, Probability With a View Toward Statistics
• Fuerst
• Sydsaeter, Knut, Peter Hammond, Atle Seierstad and Arne Strom (2005), Further Mathematics for economic Analysis, Prentice Hall.
• Chiang
• Mathematical appendixes of the Economics textbooks

Math courses that survey participants have taken during their studies in the foreign university:

• Math for econ PhD or Math Camp (5)
• Real Analysis
• Functional Analysis
• Probability Theory

Compulsory or non-compulsory prerequisite Mathematics courses for the admission process to the PhD program in the foreign university:

• No one said that there was any compulsory Mathematics requirements.
  • But one person mentioned that the admission committee was interested in Mathematics background.
  • One person mentioned that Math camp was used to select students to the program.

Suggestions to the Economics graduate students who plan to take courses in abroad:

• Different persons from different universities had many quite different suggestions
  1. One answer suggested, that Mathematics preparation from compulsory undergraduate courses in the Faculty of Economics in Tartu University was sufficient preparation and no further Mathematics studies were needed.
  2. One answer suggested that in order to do research in Microeconomics or Econometrics, one needs to have undergraduate degree in Mathematics (not Economics). The situation is not as hopeless if you choose Macroeconomics, but even then, those who have studied Mathematics and no Economics, are strictly better off, compared to those who have studied only Economics in whichever university.
  3. You can find a list of excellent suggestions in the end of this document.

Could Steve Levitt get into a top Economics Ph.D. program today?

At the PhD level, the study of Economics requires an absolute minimum of one year of college calculus and a quarter (or semester) each of both matrix algebra and Mathematical statistics (that is, statistics using calculus, as distinct from introductory statistics for social science).

Beyond these basic prerequisites, many of our applicants have taken other advanced Mathematics courses, such as real analysis, have completed some graduate-level classes in Economics or related fields, or have had some other significant exposure to research in Economics.

FAQ: What can I do over the next few years to better prepare myself for a top PhD program in Economics?

… We would strongly encourage you to take some advanced courses in Mathematics, such as real analysis, to develop your ability to read and write rigorous Mathematical arguments.

Proficiency in Mathematics is crucial for successful completion of the first year of study in the Department of Economics at the University of Pennsylvania. A minimal level of preparation consists of two years of Mathematics courses in college, including courses in:

• multivariable calculus
• probability theory and statistics
• linear algebra.

An excellent level of preparation consists of additional courses in:

• real analysis
• point-set topology
• measure-theoretic probability theory

How important is a background in Mathematics?

It is imperative that you have had exposure to Mathematics. A degree in Mathematics is not required, though we look especially carefully at applicants with a joint degree in Economics and Mathematics.

What sort of Mathematics courses should I have taken?

You should definitely have taken single-variable and multivariate calculus. It is expected that you would have a background in linear algebra, and an exposure to probability and statistics. Many of our applicants also have a background in difference or differential equations, and they have been exposed to rigorous thinking in limits, continuity, and basic topological concepts (openness, compactness, etc.). There are a good number of applicants who have more than this: e.g., some measure theory and exposure to rigorous probability theory and stochastic processes.

Math Camp is compulsory (or sometimes just strongly recommended) course before the start of the PhD program (2 to 5 weeks). Sometimes there is additional Math course in the Fall semester.

Math camps usually use 2 or 3 of the following textbooks:

• Rudin, Principles of Mathematical Analysis;
• Simon and Blume, Mathematics for economists;
• Sundaram, A First Course in Optimization Theory.

Example: Math camp in University of Pennsylvania 2004:

Thomas J. Sargent recommends: If you take one course a quarter from the second year on, you will have at least nine quarters under your belt by the time you graduate. While it may be painful at first, like jogging, after a while the pain will vanish and you will acquire a facility in writing and reading Economics.

Math Department:

• Math 103, 104, Linear algebra
• Math 113, 114 Linear algebra and matrix theory
• Math 106, Introduction to functions of a complex variable
• Math 124, Introduction to stochastic processes
• Math 130, Ordinary differential equations
• Math 131, Partial differential equations
• Math 175, Functional analysis
• Math 205A, B, C, Real analysis and functional analysis
• Math 230A, B, C, Theory of Probability
• Math 236, Introduction to stochastic differential equations

Engineering economic Systems and Operations Research

• EESOR 313, Vector Space Optimization.
• EESOR 322, Stochastic calculus and control

Statistics

• Stat 215-217, Stochastic processes
• Stat 218, Modern Markov chains
• Stat 310 A, B, Theory of probability

These are very useful courses for applied work in Econometrics, Macroeconomic theory, and applied industrial organization. They describe the foundations of methods used to specify and estimate dynamic competitive models.

• These courses are recommended in addition to ALL Mathematical Economics, Statistics and Econometrics courses you can take in Economics Department.
• Most of the courses suggested below require Algebra I and/or Calculus II as prerequisites. As you can see below, we believe that Economics graduate students are probably familiar with the topics covered in Calculus II (but if differentiation and integration is not easy for you, take it anyway), but Algebra I is not covered in Economics program and thus highly recommended.
• Keep in mind that if you have a plan to apply to top Economics graduate programs in Europe or US, then good grades in these Mathematics courses are essential (much more important than the grades in undergraduate Economics courses). Thus, it is recommended to take the courses seriously and not too many in one semester. (Read more here and here).
• All the information below is from 2006, and may change in the future.
• All the courses here are lectured in Estonian.

• Simple introduction to set theoric notation and proofs. The course is compulsory to all first-year undergraduate Mathematics and physics students and we think it should be compulsory to all graduate Economics students. We strongly recommended to take it in order to understand the notation used in graduate Economics textbooks and articles. For example: ,
• Covers: Notion of a sets, operations with sets, functions relations, relations of equivalence and partitions to the classes, capacity of a set, ordered sets. Main notions of propositional calculus. Logically equivalent formulae. Disjunctive normal form of formulae.
• Material: Peeter Oja, «Hulgateooria», 2002.
• NB! Star-exercises1) are perfect training of Mathematical proofs.
• More information in ÕIS

• The course gives basic knowledge in abstract algebra2). For Economics students the knowledge of advanced matrix algebra and vector spaces is very important (extensively used in Econometrics and economic theory), but also for taking other Mathematics courses, as they often use results from algebra.
• Covers: Basic algebraic structures (group, ring, field, vector space) on the level of definitions and examples. Linear algebra (base of vector space, Laplace's theorem on determinants, rank of matrix, systems of linear equations, linear mappings, eigenvectors and eigenvalues of linear transformations, inner product spaces). Elementary theory of complex numbers and polynomials in one variable.
• Material: Mati Kilp, Algebra I, 1998.
• Compulsory Prerequisites: MTPM.02.022 Algebra and Geometry (this is probably covered in undergraduate courses in Economics faculty), MTRM.01.020 Set Theory and Mathematical Logic
• More information in ÕIS

• The course, which every graduate Economics student should take. It is useful for two things: 1) it introduces simple concepts from matematics, which are needed in graduate Economics courses, 2) it teaches how to construct proofs. The course covers the first half of the Rudin's «Principles of Mathematical Analysis». After Set Theory and Mathematical Logic this is the first real Math course to take.
• Covers: theory of real numbers, continuous functions on closed intervals, number and functional series, the Riemann integral.
• Materials:
  1. http://Math.ut.ee/pmi/kursused/ma3/
  2. Kangro, Matemaatiline analüüs I and II
  3. Rudin's «Principles of Mathematical Analysis»
• Compulsory Prerequisites: Calculus I and II. But undergraduate training from Economics department should cover most of the topics from these courses.
• More information in ÕIS

• Continues with the material from Calculus III by doing more or less the same in multivariate case.
• Covers: multivariable differential and integral calculus and also the theory of Fourier series.
• Materials:
  1. Kangro, Matemaatiline analüüs I and II
  2. http://Math.ut.ee/pmi/kursused/ma4/
• Compulsory Prerequisite: Calculus III.
• More information in ÕIS

• Functional Analysis I is almost as important as Calculus III for all the students who intend to take graduate level economics courses. It covers important topics in the material, which is needed for graduate Economics courses (for example, fixed point theorems are used everywhere to prove the existence of equilibrium).
• Covers: metric spaces (convergence, examples (including normed linear spaces), balls, open sets, closed sets, closure, interior), complete metric spaces (examples (including Banach spaces), main theorems), continuous operators (continuity criteria, Banach fixed point theorem), compactness (extensions of Weierstrass theorems, compactness criteria), introduction to topological spaces, finite-dimensional normed spaces, series in normed spaces, the space of continuous linear operators, basic principles of classical functional analysis
• Compulsory Prerequisites: Set Theory and Mathematical Logic and Calculus II are official prerequisites for FAI. But students often take FAI after Calculus III, therefore it might be useful to take the courses in the same order as other students.
• Material:
  1. http://Math.ut.ee/pmi/kursused/fa1/
• NB! Again, star-exercises are challenging and we suggest to take them seriously.
• More information in ÕIS

• Continues FA I in more advanced level. Probably not as important to take, but covers topics useful for economists.
• Covers: continuous linear functionals (dual spaces, Hahn-Banach theorem, reflexivity), differential calculus of operators in normed spaces, Hilbert spaces (main theorems, Fourier series), operators in Hilbert spaces.
• Compulsory Prerequisite: Functional Analysis I.
• Material:
  1. http://Math.ut.ee/pmi/kursused/fa2/
• More information in ÕIS

• General Topology is certainly not required for Economics, but may be really useful. It discusses the concepts on a general level, which is helpful when trying to understand these concepts in specific cases (as discussed in Calculus or Functional analysis). Thus it is often recommended for Economics graduate students.
• Covers: Definitions and properties of topological spaces. Closed and open sets, closure and boundary. Basis for a topology, local basis. Factor topology and product topology. Continuous, open and closed functions, homeomorfisms. Families of sets, convergence of families. Filters and ultrafilters, convergence, continousness and closures in filters notation. Separation axioms, Hausdorff space. Regular and completely regular spaces. Normal spaces. Compact spaces, locally compact spaces.
• Compulsory Prerequisites: Set Theory and Mathematical Logic, Calculus II.
• Materials:
  1. http://Math.ut.ee/~zolki/Math/topo.ps
  2. http://Math.ut.ee/~zolki/Math/topo_pr.ps
• More information in ÕIS

• Numerical methods are often used in Macroeconomics and Econometrics to approximate non-linear systems that would be impossible to handle otherwise. Therefore it is recommended to learn some numerical methods or read for example Judd's "Numerical Methods in Economics".
• Covers: Calculation with errors, numerical solution of equations and systems of equations, approximation of functions, numerical differentiation and integration.
• More information in ÕIS

• Highly recommended for the students interested in Econometrics.
• Covers: Banach-Tarski paradox, algebras and sigma-algebras, measure spaces, Lebesgue measure, measurable functions, Lebesgue integral and convergence theorems, modes of convergence of measurable functions, product measures.
• Materials: http://Math.ut.ee/pmi/kursused/lebesgue/index.html
• Compulsory Prerequisite: Calculus III
• More information in ÕIS

• Good preparation for Econometrics in graduate level.
• Covers: covergence of random variables, the laws of large numbers and central limit theorems. Also, some basic consetration inequalities and the elements of weak covergence of probability measures are itroduced.
• Materials: http://www.ms.ut.ee/ained/Tnt2/TNT2-05.pdf
• Compulsory Prerequisite is Probability I, but the preparation in Economics department should be enough. Not compulsory prerequisite is Measure and Lebesgue Integral.
• More information in ÕIS

• The course should be taken, if you are seriously interested in Econometrics.
• Covers: signed measures (the Radon-Nikodym Theorem, differentiation of measures in R^n, functions of bounded variation), Radon measures (regularity properties, the Riesz Representation Theorem), basic theory of L^p-spaces (including the dual spaces of L^p-spaces).
• Materials: http://Math.ut.ee/pmi/kursused/mooduteooria/index.html
• Compulsory Prerequisites: Measure and Lebesgue Integral, Topology I, Functional Analysis I and II.
• More information in ÕIS

• Useful in time-series analysis
• Covers: Starting with discrete time martingales, it also covers some basics of continuous time martingales. Ito calculus as well as Black-Scholes formula are introduced.
• Material: http://www.ms.ut.ee/ained/Martingaalid/Martingaalid2006.pdf
• Compulsory Prerequisite: Probability Theory II
• More information in ÕIS

• Especially useful in time-series analysis
• Covers: Differentiation and integration of functions of complex variable. Cauchy theorem and Cauchy formula are proved. Taylor and Laurent series expansions of differentiable functions are investigated. Theory of residuals and conformal mappings of domains are presented.
• Materials: E. Jürimäe, Kompleksmuutuja funktsioonide teooria lühikursus, Tallinn, 1983.
• Compulsory Prerequisites: Algebra I, Calculus II.
• More information in ÕIS

We believe, that it is much more efficient to learn graduate level Economics after learning the language of Set Theory3), training Mathematical proofs4) and knowing the main results from Linear Algebra5), Calculus6), Functional Analysis7) and Measure Theory8). If you are seriously interested in theoretical research (in Microeconomics, Macroeconomics, or Econometrics), you should continue with courses useful for your research. Thus, if you feel that your Mathematics preparation for Economics graduate studies is not sufficient, we would recommend something like the following:

1. Start in Fall with
   1. Set Theory and Mathematical Logic
   2. Calculus III (if Calculus II seems easy enough, otherwise take it before)
   3. Measure and Lebesgue Integral (if it does not seem too mysterious)
2. Suggestions for spring
   1. Algebra I
   2. Functional Analysis I
   3. Calculus IV (if you survived Calculus III in Fall)
3. Continue and take (as much as time permits) the courses useful for your research

If you are uncertain about your preparation in Mathematics, look at the textbooks and materials refered above. If they all seem too easy, you are well equipped. But otherwise you will probably need some additional Mathematics studies and taking courses is usually easier than studing large amount of new Mathematics material yourself.

Some suggestions by the Estonians studying in Europe:

• «Vaja on korraliku matemaatika tausta + põhiaineid – mikro, makro, matemaatika. Lisaks on vaja põhilisi meetodeid, nagu Nashi tasakaal, dünaamiline programeerimine, optimaalne kontroll. Tõenäosusteooria (mõõduteooriat pole vaja): jaotused, maatriksalgebra, Taylori rida.»
• «Valmistage ennast piisavalt ette matemaatika ja tõenäosusteooria alal.»
• «(Välis)ülikool katab oma kursustega kõik vajaliku ära, nii et kui millestki puudu jääb, siis ole aga mees ja õpi.»
• «Kokkuvõtteks ütleksin, et mulle on seni piisanud enam-vähem sellest, mida meile õpetati esimestel kursustel kõrgema matemaatika, matemaatilise statistika ja tõenäosusteooria ning J. Vehviläineni kursuste raames.»
• «Üht ja ainsat head soovitust ei ole - kõik sõltub sellest, mida tahad teha. Esiteks, kui tahta uurida mikrot või ökonomeetriat, siis tuleb Tartus mitte õppida majandust, vaid matemaatikat. Makroga on lood paremad, kuid kergem on neil, kes teavad matemaatikat, kuid mitte majandust võrreldes majandust õppinutega ükskõik millisest ülikoolist inimesed siis ka pärit poleks.»
• «Majanduses (vähemalt vanasti) olid täiesti korralikud kohustuslikud matemaatika kursused (Sakkov, Vehviläinen), kuid kolmandal-neljandal aastal ei järgnenud majanduses kursuseid, mis oleks nende peale 'ehitanud'. Mitmed teadmised jäid rakenduseta. Palju keskenduti ka 'keerulistele' matemaatilistele probleemidele nagu mitmendat järku differentsvõrrandid jne, kuid välisülikooli õppima minnes oleks eelkõige osata lahendada esimest järku dif. võrrandit, kuid seda ka unepealt.»
• «Kui erilist huvi matemaatika vastu ei ole, siis tuleb suhteliselt varakult otsustada, millise valdkonnaga ennast siduda. Siis on võimalik täpselt valida, millist osa matemaatikast õppida, sest hilja alustanutel (st peale bakalaureust majanduses) on raske teada matemaatikast kõike, kuid võimalik on ennast hästi kurssi viia kas arvutuslike meetodite, topoloogia, matemaatilise statistika vms. See oleks hea doktoritöö kirjutamiseks juba piisav.»
• «Matemaatika õpetamise osas oleks mul Tartu Ülikoolile selline palve, et seda ei tehtaks eraldiseisvana teistest ainetest. Isikliku kogemuse põhjal võin öelda, et mul tekkis huvi matemaatikast rohkem teada siis, kui sellega oli seotud konkreetne majanduslik küsimus. Väga hästi on võimalik ühte kursusesse mahutada nii matemaatika kui majanduse õpetamine-õppimine, nii on varakult ka selge, millist osa matemaatikast kõige rohkem tulevikus tarvis võib minna.»
• «Väga palju oleneb uuritavast valdkonnast. Iga valdkonna tarbeks on üldiselt ka erinevad tehnikad. Selgita välja omale vajaminevad tehnikad ja püüa need endale selgeks teha. Ülejäänud tehnikatega tegele siis, kui aega ja vajadust on.»
• «Püüa hoida majanduse ja matemaatika õppimine lahus, kuid
  1. matemaatika õppimisel püüa mõelda, kuidas saaks rakendada õpitut majandusprobleemide lahendamisel. Püüa konstrueerida omatehtud majandusmaigulisi ülesandeid ja lahenda need.
  2. majandusmudelite õppimisel püüa formuleerida ülesanne matemaatiliselt korrektselt, ja seejärel lahenda see nagu oleks tegemist ainult matemaatilise ülesandega.»
• «Tavaliselt lähevad inimesed välismaale majanduskursuseid võtma. Mina soovitaksin võtta ka matemaatika kursuseid, mida pakutakse majandusteaduskonnas. Olen kindel, et Te ei kahetse.»
• «Ma arvan, et Eesti ülikoolid võiksid majandusmatemaatika õpetamisel pöörata rohkem tähelepanu korrektsele matemaatilisele tähistusele. Minu esimeses matemaatikatunnis välismaal oli kõige suuremaks probleemiks arusaamine matemaatilistest terminitest ja tähistustest. Nt: või .»
• «Muidugi tuleb uurida põhjalikult võetava aine programmi ning selgitada välja eeldusained. Vajadusel viia end sellega kurssi. Kui aine osutub oodatust raskemaks, ei tohi ära ehmatada vaid püüda süstemaatiliselt oma nõrgad kohad järele aidata. Veel, teaching assistants ongi peamiselt selleks et neile 'pinda käia' :).»