Mathematical Economics I

COURSE OBJECTIVE
- Acquaint participants with classic mathematical models of economic decision making developed in the second half of the twentieth century, the fundamental critique of fact-driven behavioral economics (classic anomalies) and a sketch of economic models of the future.
- The focus is on three topics: individual decision making, collective decision making (voting in groups or societies) and interdependent decision making (or game theory).
- Participants understand the purpose and the mathematical properties of each model. Participants are able to execute several strategies to calculate simple models by hand, embed such strategies in algorithms (pseudo-code for software) and being able to use freeware Gambit.
- Participants are confronted with the important difference between descriptive theory, aimed at explaining and predicting reality, and normative theory, what intervention should ideally be done.
- Economic modeling of reality, embedding economic models in software and bringing economic models to the data will also be addressed. To summarize, participants will learn, understand and reflect on important economic models, their implementation in algorithms, and experiments.

COURSE CONTENT
Society asks for evidence-driven economic theories that can be used in economic decision making in complex economic situations. This requires on the one hand descriptive theory that explains and predicts economic reality, and on the other hand normative theory that guides the decision maker what economic intervention should ideally be done. The financial crisis of 2007, and its aftermath, made clear that the classic models of economic decision making developed in the second half of the twentieth century are not up to this task. Also, these models ignored the classic anomalies (some dated early 1950s) and fundamental critique raised by fact-driven behavioral economics for too long. Since the financial crisis, (mathematical) economics is in transition, and for good reasons. This transition is reflected in this course and requires more academic reflection from participants than they are used in other EOR courses.

This course deals with individuals, companies, governments, NGOs that (need/want to) take economic decisions. Each decision maker is embedded by an economic context, e.g. you deciding how much effort to put in a team assignment. The interaction of decision makers and their economic surrounding is at the heart of this course. We distinguish three major topics: individual decision making, collective decision making (how do groups or societies reach decisions) and interdependent decision making (how to bid in an auction anticipating others’ bids).

Individual decision making (period 1)
In order to evaluate whether a decision is a good decision, economists developed the notion of preference relations that rank possible alternatives (possible choices) and utility / objective functions. In this course we introduce these concepts and investigate what mathematical structure needs to be imposed to move from preference relations to utility functions. From a descriptive perspective, this course addresses whether the mathematical structure is evidence-based. From a normative perspective, how to obtain preferences and how to compute what is best according to these preferences. This is facilitated by constructive mathematical proofs that can transformed into algorithms (and would lend itself for programming, which is outside the scope of this course). Classic economic theories about market demand of consumers, or the market supply of a product and market demand for inputs by price-taking firms are derived from objective functions. Noisy decision making, as introduced by Duncan Luce and popular in A/B testing in Data Analytics, will be introduced. Preferences for risky decisions are developed and expected utility theory derived. The famous Allais-paradox experiment that empirically rejects this theory is discussed, and Prospect theory, which can explain the paradox, will be discussed.

Collective decision making (period 1)
Individual decision makers often participate in groups or teams, and live in a society. Is it mathematically possible to derive group preferences from individual preferences? Impossible. What then? This part of the course is merely normative in analyzing classic ranking methods and voting procedures that are observed in reality. These methods and procedures will be compared with each other. One criterion employed is Pareto efficiency.

Interdependent decision making (period 2)
In many, if not all, economic situations what eventually happens depends upon decisions made by more than one individual of individuals. Whether your team assignment is evaluated with a high grade depends upon your own effort and that of your other teammates. Or, whether you win the item in an auction depends upon your own bid and the others’ bids. Predicting what others will do, how they predict what you will do, etc. becomes crucial in the mathematical analysis. Although this part can be used for normative theory (f.e. all driving on the same side of the road reduces accidents, or to develop good antitrust policy to destabilize cartels), the focus of this part of the course is mainly descriptive because of the need for evidence-based theories.

We focus on Nash equilibrium, the empirical need to refine Nash equilibrium and two of such refinements: k-level reasoning and (agent) quantal response equilibrium. The latter is a descriptive theory, Nash equilibrium is in trouble while quantal response equilibrium deals better with experimental data and is easier to bring to data.

In many economic situations some individual are better informed than others, which is called private information. For example, in Poker you know the cards you are holding while the others do not. You will be introduced to the fascinating world of interdependent decision making with private information. Because analyzing such games by hand is rather hard, you will solve such games numerically in Gambit. (Freeware Gambit is an open-source software tool programmed in Python that computes Nash equilibrium and quantal response equilibrium.) Interpretation of the computed solution and its economic implications will be addressed. Gambit will be part of an assignment that counts as part of the final grade.

This part will also focus on the economic literature during the 1980s and 1990s that were so influential that many mathematical economists became Nobel laureates in Economics. Classic theories about Cournot competition in quantities (e.g. OPEC cartel), Bertrand competition in prices, sustainable cooperation in repeated games, antitrust policy to destabilize cartels are part of the course.

TEACHING METHODS
Classes. One lecture and one practical per week. Active participation is key.
Participants may be partitioned to groups for the practical.
Participants of the practical PREPARE BEFORE coming to class and are expected TO PRESENT their answers before the Canvas in class and discuss where problems in solving questions arose.

TYPE OF ASSESSMENT
One team assignment based upon Gambit in period 2 – team assessment
Partial exams in October (covering period 1) and December (covering period 2) – individual assessment
An exam in March (covering period 1 and 2) – individual assessment Individual
Assignment (presenting before class) – individual assessment

RECOMMENDED BACKGROUND KNOWLEDGE
Knowledge of elementary mathematics and elementary probability theory. This includes differentiation, the Lagrange method, expectation, Bayes Rule. For EOR students this translates in knowledge from Analysis I and II, Linear Algebra and Probability Theory, and to a much lesser extent Finance, Statistics and Programming. Preferably, you have a sufficient mathematical background and can reason logically.