LECTURE NOTES The Microeconomics of Complex Economies Evolutionary, Institutional, Neoclassical, and

LECTURE NOTES COMPLETE

Prof. Wolfram Elsner Faculty of Business Studies and Economics iino – Institute of Institutional and Innovation Economics

Introductory Microeconomics An Introduction to the Microeconomics of Complex Economies

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

22.02.2026

Readings for this lecture  Mandatory reading this time: An Introduction to the Microeconomics of Complex Economies, in: Elsner/Heinrich/Schwardt (2014): The Microeconomics of Complex Economies, Academic Press, pp. 323.

 The lecture and the slides are complements, not substitutes  An additional reading list can be found at the companion website 2

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

22.02.2026

Economic Situations Economic situations: A much broader and more fundamental range of circumstances than the exchange of goods and services for money, e.g.:  Research and development (R&D)  Competitive set-ups  Acquiring knowledge in structured ways  Increase future earning potential Results depend on decisions made by other agents as well

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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Economic Situations Multi-personal (social) decision situations  Direct interdependence  Behavior of one influences results of others  Of interest is the choice of behavior – individuals have to have at least two options  Individually optimal decision may have negative effects on results of other agents  Overall suboptimal outcomes can result

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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Social Situations Social situations with interdependent agents can be differentiated by the degree of conflict – individual interests are fully or partly aligned, or in conflict Individually optimal behavior leads to optimal social, collective outcome

Individually optimal behavior leads to socially suboptimal outcome Comparatively worse individual outcomes Might allow for superior results

How to deal with such situations?

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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Pareto Criterion Pareto Criterion: Criterion for evaluating and comparing situations  Pareto Optimum: no agent´s situation can be improved without reducing the payoff of someone else at the same time  Criterion does not include a broader judgment regarding the desirability of an outcome

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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Social Optima and Social Dilemmas Social Optima: Rational and individualistic agents will reach a Pareto-optimal result Social Dilemmas: Rational and individualistic agents deciding will not reach a Pareto-optimal result  Individually optimal choice does not generate optimal outcome  No one has individualistic incentives to change this choice  No possibility to solve this situation (improve the outcome without outside support, or some change in the situation) 7

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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Coordination Problems Coordination Problems: No clearly optimal behavior, different options become more or less attractive, depending on what other agents in a group opt for  Individually optimal behavior cannot be defined without knowing the choices of other agents  Coordinated on a type of behavior, there is no reason to change, at least not individually  There is no guarantee that the coordinated situation would be the best possible result, but it is superior to uncoordinated outcomes

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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The Invisible Hand The Invisible-Hand World View  Individually rational behavior would automatically result in a social optimum  Concept of self-organizing social and economic system leading to desirable outcomes without need for supervision

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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The Fallacy of Aggregation The Fallacy-of-Aggregation concept contains individually optimal actions that may result in Pareto-inferior outcomes  Individually rational behavior does not result in a social optimum  Importance of expectations  Stabilizing and supporting functions for third agents  Emergence of behavior that can improve results among the agents

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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Continuum of Complex Decision Structures Coordination problems represent a continuum of problem structures beyond the easy case of rational decisions resulting in a Pareto Optimum.  Degree of conflict differs depending on the overall problem structure (Degree of conflict: Can a Pareto-optimal outcome be reached? Are individual interests aligned in the outcome?)  Different situations require different responses to ensure a Pareto optimal outcome  But what are the measures appropriate to ensure a Pareto optimal outcome in a given situation?

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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Important aspects of real world situations Beyond direct interdependence, real-world situations are commonly characterized by:  Uncertainty  Path Dependence  Boundedly rational agents  Different behavior patterns, and social rules and institutions guiding them  External effects

 In the remainder we introduce these aspects

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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(Strategic) Uncertainty Uncertainty describes a situation in which agents do not know about future states of a system  Do the agents know all possible states?  Do they know their probabilities?

Uncertainty is not risk!

In case of direct interdependence: concept of strategic uncertainty

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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Path Dependence vs. Ergodicity AA

A

AB

B

BA BB

t

Ergodicity means that the properties and the constitution of the system do not change over space and time  Several endpoints in non-ergodic systems  A dynamic view is required

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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Bounded Rationality Agents are boundedly rational – they can’t process all information given in one moment (and under uncertainty lack certain information in any case) What is helpful in such situations for the agent?  Social Rules and Social Institutions reduce complexity for the agents and stabilize behavior pattern and expectations

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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How Rules and Institutions Can Solve Coordination and Dilemma Problems Rules and Institutions are able to solve coordination and dilemma problems:  Social Rules help achieve coordination (in the best case on a Pareto-optimal outcome)  Social Institutions help achieve (common) cooperation  Cooperation is defined as coordination plus sacrifice (giving up a short-term advantage for longer term gain)  Social institution is a social rule plus sanction mechanism (should cooperation be refused)

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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Instrumental and Ceremonial Dimension of Social Institutions  The Instrumental dimension of an institution refers to its ability to solve common and collective problems  The Ceremonial dimension of an institution refers to it as an instrument to maintain differential power and status  Live-Cycle of a rule: Emergence as a problem solving device

Introduction of a normative dimension

Emergence of a ceremonial dimension

t

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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External Effects and Institutions An external effect is an uncompensated consequence of an economic decision to noninvolved agents Institutions bring about such external effects  By reducing uncertainty and strengthening expectations

Economies of scale describes the falling average costs as a result of increasing production volumes or numbers  repeated use of rules and institutions results in this effect for them as well (which can make changing them very challenging)

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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Different Types of Goods Rivalry

Nonrivalry

Excludability

Private goods

Club goods

Nonexcludability

Commons

Public good

 Positive external effects of contributing to social rules and institutions  Negative external effects of not contributing to the rules and institutions Similarities to network effects 19

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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Markets as Sets of Rules and Institutions

A market is a set of social rules and institutions, serving as a system of allocation of resources and of decision making involving prices and quantities – and one mechanism for addressing economic problems  Different markets may exhibit extremely different sets of rules and institutions  Markets are to be embedded in proper sets of rules and institutions for ensuring positive outcome  Markets are continuously changing over time – a static model of a market is likely miss important aspects of its operation, then 20

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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Summary Microeconomics thus is “the” science of complex coordination, cooperation and organization  Directed Interdependence, in different problem structures  Uncertainty  Path Dependence  Bounded Rationality  Emergent Behavioral Patterns: Social Rules and Institutions  External Effects Only if these aspects are considered, can Microeconomics help to design measures adequate to support a Pareto optimal outcome in situations that do not obey the Invisible Hand but are subject to Fallacy of Aggregation-type issues 21

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

22.02.2026

Readings for the next lecture  Compulsory reading: Introduction to Game Theory, in: Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies, pp. 25-32.

 For further readings visit the companion website

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Prof. Wolfram Elsner Faculty of Business Studies and Economics iino – Institute of Institutional and Innovation Economics

Introduction to Game Theory

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Readings for this lecture  Mandatory reading this time: An Introduction to Game Theory, in: Elsner/Heinrich/Schwardt (2014): The Microeconomics of Complex Economies, Academic Press, pp. 2532.  The lecture and the slides are complements, not substitutes  An additional reading list can be found at the companion website 2

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Game Theory  Social Interactions may be modeled as strategic games, i.e.

Game Theory models.  The use of Game Theory requires a framework of assumptions translating the social setting into a simplified mathematical model.

 Due to its abstract nature, underlying mechanisms valid for many situations can be identified.  Useful for prediction, explanation, and modeling in a wide range of fields (economics, sociology, …). Examples are given in later Chapters.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Strategic Games In Game Theory, models are specified as strategic games, which are characterized by  A set of participants (agents)  A set of behavioral options (strategies)  A set of rules (including the sequence of decision making)  The information sets available to the agents.  Agents are assumed to be rational (pursuing maximum payoff, not envious).

Now: Begin with an easy, but almost universally applicable form of a strategic game

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Simultaneous 2-Person-2-Strategy Normal-Form Games  Characterized by  2 agents

 Each agent has two alternative strategies  Simultaneous choice of the strategies  Complete information (about game, strategies, payoffs, but not the opponent’s choice)

 No collusion Let us consider some examples:

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Social Optimum Game Player B

Strategy 1 P l a y e r

Strategy 2

Strategy 1

A

Strategy 2

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Social Optimum Game Payoffs given in the columns Example / Interpretation: Player B

P l a y e r A

Strategy 1

Strategy 1

Strategy 2

4

2

4

2 2

Strategy 2 2

Payoffs given in the rows

0 0

• Invisible hand concept • The option an egoistic individual chooses is the same option chosen by a hypothetical social planner seeking to maximize the outcome for the entire society • Game will arrive at the social optimum • This point is Pareto-optimal • Model for conflict-free social interactions.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Anti-Coordination Game Player B

Pass P l a y e r

Wait

Pass

A

Wait

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Anti-Coordination Game Example / Interpretation:

Player B

Pass P l a y e r A

Wait 0

Pass

0

1 1

1

Wait

1

0 0

• Encounter in a narrow doorway • Agents cannot pass both at the same time • Pareto-efficient solution is that one waits and the other passes •Model for potential social conflict without actually conflicting agendas (necessity to coordinate) 9

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Prisoners´ Dilemma Game Example / Interpretation:

• Two burglars arrested after a robbery • They could give a guilty plea or be silent • If both remain silent they can only be held for a short time • If one defects she gets a shorter and the other a longer jail time •If both confess they are both jailed for a long time •Model for conflicting incentives (agendas) in social interactions 10

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Prisoners´ Dilemma Game Player B

Be silent P l a y e r A

Interpretation/Example:

Confess 1

2

Be silent 1

-1

-1

0

Confess 2

0

• What are the two individuals thinking?

•If they choose to confess, they “do not make a mistake, no matter what the other one does” •If they choose to be silent, there is the huge risk of being exploited if the other one confesses •To confess is therefore strictly dominant 11

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Pareto Optimal Outcomes  A comparison of different outcomes is not straightforward  In the Prisoners’ Dilemma, the agents prefer different outcomes (conflicting agendas, necessity to coordinate and to cooperate)  Also: Payoffs of different agents cannot be directly compared.

 Solution: Concept of Pareto optimality  A situation is to be preferred over another if no player is worse of but at least one player is better of.

 The concept avoids any direct comparison of payoffs of different agents and treats them as different dimensions to the problem.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Solving Games: Pareto Dominance  In the above Prisoners’ Dilemma there are 4 possible outcomes (W, V, X, and Y, see next slide): X is Pareto dominated by Y while W, V, and Y are Pareto optima.  Pareto dominance and likely outcomes of games?

 Although Pareto optimal are preferable to all agents, this is not necessarily the likely outcome of social interactions.  Consider the Prisoners’ Dilemma: Agents try to avoid being exploited (Player B wants to avoid the lower left field, W, player A wants to avoid the upper right field, V) leading to a Pareto inferior outcome (lower right field, X).  The Pareto criterion evaluates outcomes, it does not predict them.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Solving Games: Pareto Dominance

B1

B2 1

A1

2

Y

V

1

-1 -1

A2

W 2

0

X

0

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Solving Games: Dominant Strategies  If a strategy gives higher (or equal) payoffs than another, no matter what strategy the other player chooses, this strategy is said to dominate the other one (which is said to be dominated).  If a strategy always gives higher payoffs than another, it is said to strictly dominate the other (then called strictly dominated).  A strategy that (strictly) dominates all alternatives is called the (strictly) dominant strategy in this game.  Rational players would never choose a strictly dominated strategy.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Solving Games: Dominant Strategies Identifying dominant Be silent

Confess 1

Be silent

1

2

-1 -1

Confess

2

strategies

0

0

2 > 0

0 > -1 Strategy „Confess“ therefore strictly dominates strategy „Be Silent“.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Solving Games: Nash Equilibrium  Best answer strategy

 A strategy that gives the highest possible payoff given the opponent’s choice is called the best answer strategy (corresponding too the opponent’s choice).

 Nash equilibrium

 An outcome that is achieved as a result of every player playing her best answer strategy.  In a Nash equilibrium no player has incentive to deviate from her choice (given that the opponent also sticks to her choice).

 Nash equilibria (as well as strategic dominance) can therefore indeed serve as predictions of the outcome of the game.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Solving Games: Nash Equilibrium

Be silent

Confess 1

Be silent

1

2

-1 -1

Confess

2

How to find a Nash equilibrium:

0

0

 Underline the best answer payoffs given the strategy of the opponent (for both players)  Mutual best responses are Nash equilibria

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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From 2-Person Games to N-Person Games  Extending 2-person normal-form games to n-person games is straightforward and usually does not change the basic results  The agents have to choose between the same two strategies  Payoffs depend on the choice of all other agents  Consider two examples where n1 players choose the first and n2 players choose the second strategy. Resulting payoffs are:

Social Optimum Game

Social Dilemma Game

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Social Optimum vs. Prisoners Dilemma Social Optimum

Prisoners Dilemma

Nash equilibrium is the only Pareto-optimal outcome

Nash equilibrium is the only outcome, that is not Paretooptimal

Resulting payoffs:

Resulting payoffs:

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Other Types of Games  Sequential games  Players make their choices one after the other  Second player knows the decision of the first one when making her decision

 Repeated games

 Agents remember choices and outcomes from previous periods  Allows memorizing others behavior and learning

 Supergames

 Indefinitely repeated games are called supergames

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

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Conclusion  This Chapter covered:  …an introduction to the theory of strategic games  …an introduction to matrix representation of normal-form games  …an introduction to common game structures  …an introduction to simple solution concepts for strategic games.

 More will follow in Chapter 8.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 02: Introduction to Game Theory

22.02.2026

Readings for the next lecture  Compulsory reading: Problem Structures and Processes in Complex Economies, in: Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies, pp. 33-55.  For further readings visit the companion website

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Prof. Wolfram Elsner Faculty of Business Studies and Economics iino – Institute of Institutional and Innovation Economics

Problem Structures and Processes in Complex Economies

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

22.02.2026

Readings for this lecture  Mandatory reading this time: Problem Structures and Processes in Complex Economies, in: Elsner/Heinrich/Schwardt (2014): The Microeconomics of Complex Economies, Academic Press, pp. 3355.  The lecture and the slides are complements, not substitutes

 An additional reading list can be found at the companion website 2

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Introduction  Direct interdependence of agents leads to various problem structures for them to deal with

 These range from ‘easy to solve’ to ‘impossible to solve’ (solution = Pareto-optimal result)  Use of game theory for modeling these problem structures and an identification of underlying differences therein  But: Modeling a situation with the wrong game can lead to very misleading results!

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Social Optimum by Invisible Hand Player B

Strategy 1 P l a y e r

Strategy 2

Strategy 1

A

Strategy 2

4

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Social Optimum by Invisible Hand Player B

P l a y e r

Strategy 1

Strategy 1

Strategy 2

4

2

4

A

Strategy 2

2 2

2

• Strategy 1 is strictly dominant • No conflict between agents • Pareto-optimal outcome is reached if agents are rational

0 0

5

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Coordination Problems  There is no single individually optimal (strictly dominant) strategy – preferable option depends on what others do

 Several Nash Equilibria (NE) exist  At least one NE is also Pareto-optimal  Different cases  Are all NE Pareto-optimal or only some, or one?  Are the payoffs in one NE equal or different for the agents? (Are interests asymmetric?)

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Simple Coordination Problem Player B

Strategy 1 P l a y e r

Strategy 2

Strategy 1

A

Strategy 2

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Simple Coordination Problem Player B

P l a y e r A

Strategy 1

Strategy 2

Strategy 1

Strategy 2

4

2

4

2 2

2

• Two Nash equilibria • No preferred strategy • Pareto-optimal outcome as long as they can coordinate

4 4

8

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Assurance Game Player B

Strategy 1 P l a y e r

Strategy 2

Strategy 1

A

Strategy 2

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Assurance Game Player B

P l a y e r A

Strategy 1

Strategy 2

Strategy 1

Strategy 2

4

2

4

2 2

2

3

• Two Nash equilibria • Only one Pareto-optimal NE • Assurance regarding behavior of other agents can support Pareto-optimal outcome

3

10

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Stag Hunt – Problems of Trust and Signaling Player B

Strategy 1 P l a y e r

Strategy 2

Strategy 1

A

Strategy 2

11

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Stag Hunt – Problems of Trust and Signaling Player B

P l a y e r A

Strategy 1

Strategy 2

Strategy 1

Strategy 2

4

2

4

1 1

2

2

• Two Nash equilibria • One is Pareto-optimal • But: Strategy 2 as the “safe strategy” • The safe option introduces potential additional problems for the agents when deciding – expectations about others even more important

2

12

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Battle of the Sexes – Asymmetric Interests Man

Soccer

W o m a n

Soccer

Dancing

Dancing 3

4

2 2

1 1

• Two NE • Both are Pareto-optimal • But: Agents prefer different outcomes • Interests asymmetric

4 3

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Hawk Dove – Anti-Coordination Problem Driver B

pass D r i v e r A

pass

wait

wait -6

-6

4 -2

-2 4

• Two NE • Three Pareto optima • Asymmetry of interests more pronounced than in battle of the sexes

0 0

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Social Dilemma Problem Player B

Strategy 1 P l a y e r

Strategy 2

Strategy 1

A

Strategy 2

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Social Dilemma Problem Player B

P l a y e r A

Strategy 1

Strategy 2

Strategy 1

Strategy 2

3

4

3

1 1

4

• Problem as it is cannot be solved • Strictly dominant strategies do not lead to Pareto-optimal outcome • Nash equilibrium lower right • Pareto-optima in the other three

2 2

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Solutions to Social Dilemma Problems

1.

2.

3.

4.

Transformation of the problem structure Population perspective Agency mechanisms Social Institutions

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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From the Single Game to the Supergame  Consider the repeated dilemma instead of the one-shot dilemma

 The supergame of a game G, is G played finitely or infinitely many times by the agents  Among other things, strategies can include provisions to react to others’ behavior (e.g., sanction certain behavior)  Given some conditions, the dilemma situation can be transformed into a stag hunt-type coordination game

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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From the Single Game to the Supergame  Same one-shot game is repeated: Payoffs can be summed up  Future payoffs are discounted using a discount parameter 1 δ= 1+𝑟  If the game stops with probability p: 𝛿 =

𝑝 1+𝑟

 If the game is repeated n times: Π = 𝑎 + δa + 𝛿 2 𝑎 + ⋯ + 𝛿 𝑛−1 𝑎

 If the game is repeated indefinitely: Π =

𝑎 1−𝛿

 Consider the game in general structure: 19

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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General Structure of the Dilemma Game

Strategy 1

Strategy 2

a Strategy 1

a

b

Strategy 2

b

(𝑏 + 𝑑) <𝑎 2

d d

𝑏>𝑎>𝑐>𝑑

c c 20

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Strategies in the Supergame  For the supergame, agents’ strategy is more like a game plan  This overall strategy includes provisions for strategy choices for every possible situation throughout the supergame and cannot be changed having started  A simple representative of a very successful class of strategies in this context is Tit-for-Tat (TFT):  Players start with cooperation  Then do what their opponent did in the previous round  Can thereby sanction behavior in the previous round

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Tit-for-Tat Player B

𝑐

Player B

TFT P l a y e r

TFT

𝑎 1−𝛿

𝑑−𝑐+

A

All-D

All-D

𝑐 𝑑−𝑐+ 1−𝛿 𝑐 𝑏−𝑐+ 1−𝛿

𝑎 1−𝛿 𝑐 1−𝛿

𝑐 𝑏−𝑐+ 1−𝛿

𝑐 1−𝛿 𝑐 1−𝛿

b>a>c>d

𝑎

•As long as 𝑏 − 𝑐 + > All1−𝛿 1−𝛿 D strictly dominates TFT 𝑐

𝑎

• When 𝑏 − 𝑐 + < , we have 1−𝛿 1−𝛿 a second Nash equilibrium, in the upper left, and hence a coordination game • Whether the transformation takes place depends on 𝛿 •The single-shot condition: 𝑏−𝑎 𝛿> 𝑏−𝑐

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Folk Theorem Any result from individually rational decisions can be sustained as an equilibrium in a supergame with a finite number of players and a finite number of strategies.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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The Population Perspective  Agents from a larger group are randomly matched  Single-shot fulfilled – both strategies are viable, payoffs are depending on the shares of agents that play the strategies

 If a fraction of k plays TFT and the rest n-k plays ALL-D, then the expected payoffs become: 𝑘 𝑎 𝑛−𝑘 𝑐 𝑒 𝜋 𝑇𝐹𝑇 = + (𝑑 − 𝑐 + ) 𝑛1−𝛿 𝑛 1−𝛿 𝑒 𝜋𝐴𝐿𝐿−𝐷 =

𝑘 𝑐 𝑛−𝑘 𝑐 (𝑏 − 𝑐 + )+ 𝑛 1−𝛿 𝑛 1−𝛿 24

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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The Minimum Critical Mass  To the right of the intersection TFT with higher expected payoff  Intersection gives the minimum critical mass

 A population of only ALL-D players is stable, as is one of only TFT players

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Agency Mechanisms  If single-shot condition not fulfilled (dilemma maintained), there are still options:

 Allow additional capabilities for the agents – ‘agency mechanisms’:  Monitor other agents  Have a memory period  Being able to gain and use information  Build a reputation  Reject interactions with others (on this foundation)

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Agency Mechanisms 𝛼 represents agency mechanisms (0 < 𝛼 < 1) − for TFT players, likelihood for interactions with TFT players increases beyond their share in the population.

The expected payoffs are given by: 𝑘 𝑒 𝜋 𝑇𝐹𝑇 = 𝑛

𝑒 𝜋𝐴𝐿𝐿−𝐷 =

𝑘 𝑛

𝛼

𝑎 𝑘 + 1− 1−𝛿 𝑛

1 𝛼

𝛼

𝑐 𝑑−𝑐+ 1−𝛿

𝑐 𝑘 𝑏−𝑐+ + 1− 1−𝛿 𝑛

1 𝛼

𝑐 1−𝛿 27

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Agency Mechanisms  If the single shot condition is not fulfilled (as in the figure), two stable outcomes:  A situation in which both strategies are played

 Alternatively, all players choose ALL-D as their strategy

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Institutional Structure in the Real World  The considerations so far cannot explain cooperative behavior between two persons that have never met before and probably will not meet again  In the real world, institutions matter, and guide individual behavior as agents are interacting within an institutional structure

 This institutional structure can lead to the stabilization of cooperation in situations where cooperation otherwise cannot be expected

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Institutional Structure in the Real World  Social institutions reduce the complexity faced by boundedly rational agents

 Because of habituation agents are not forced to make conscious decisions every time they act  Institutions are semi-consciously applied, habitually followed – they stabilize behavior as well as expectations and thereby enable interactions

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Summary  We have learned about a variety of different problem structures that can be represented by different games Social Optimum

Coordination Problems

Dilemma Problems

One Pareto-optimal NE

At least one Pareto-optimal No Pareto-optimal NE NE Further classification according to degree of conflict / asymmetry of interests 31

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Summary  Situations characterized by a social optimum game are of minor importance as there is no problem

 The examples of coordination problems considered so far differed in the degree of asymmetric interests of the agents (which outcome is preferred by the different agents)  Dilemma problems cannot be solved by rational behavior alone  The transformation of the problem structure, agency mechanisms and an institutional structure for boundedly rational agents can lead to cooperation 32

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 03: Problem Structures and Processes in Complex Economies

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Readings for the next lecture  Compulsory reading: Approaching Real-World Interdependence and Complexity: Empirical Phenomena in the Global Economy, in: Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies, pp. 57-93.  For further readings visit the companion website

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Prof. Wolfram Elsner Faculty of Business Studies and Economics iino – Institute of Institutional and Innovation Economics

Approaching Real-World Interdependence and Complexity “ […] Reducing transaction costs is the main argument for financial deregulation, which is the root of the current financial crisis.”

Ping Chen

“There trend to be a lack of communication and co-operation in the supply chain, and the process is not marked by a great deal of trust.”

Oliver Loebel

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 01: Introduction to Microeconomics of Complex Economies

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Readings for this lecture  Mandatory reading this time: Approaching Real-World Interdependence and Complexity: Empirical Phenomena in the Global Economy, in: Elsner/Heinrich/Schwardt (2014): The Microeconomics of Complex Economies, Academic Press, pp. 5793.  The lecture and the slides are complements, not substitutes  An additional reading list can be found at the companion website 2

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Aims of this lecture  Characterize real world economic phenomena  Apply the tools acquired so far to study these phenomena  Identifying needs for further analyses and related tools, to be developed in the forthcoming parts of the lecture.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Topics  The complexity of the real world economy and its implications  How do economics deal with their complex environment?  How should policy makers deal with complex economies?

 Fragmentation of value added chains (VAC)  The increasing role of network-based technologies  Spatial clusters and networks as dominating organizational structures

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Fragmentation of VACs  Outsourcing, the delegation of parts of a company’s production process (and potentially also employees or assets) to a third party, typically the less productive elements of the VAC, leading to  Functional fragmentation  Spatial fragmentation along VACs

 Inappropriability of information

 Inevitable information and knowledge outflow  Thus impossibility to maintain a long-term monopoly on technological knowledge or to appropriate potential rents  Stimulating in cooperative environments  Continuous exchange of positive externalities  Harmful in defective environments  Free riding and exploitation 5

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Resulting coordination and dilemma problems  Two firms, A and B, in the same VAC  What does this mean for their technology choices?  The technologies must be interoperable  Coordination problem

 Continuous exchange of information through product parts  Collective good characteristics of information

 What does this mean for the firm‘s decisions to invest into R&D?  There is an incentive to free-ride on the other‘s expense  Dilemma Problem 6

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Tacit Knowledge  Inappropriability of information in business processes affects  Formal technological knowledge, including current research, trade secrets etc. (to a lesser degree)

 Knowledge on the organizational structure, etc.  Procedural knowledge on implementation of technologies, work experience, etc. (tacit knowledge)  Tacit knowledge cannot (or hardly) be formally taught or catalogued

 It is part of the firm’s employees human capital, the firm will lose it when laying off the employees (typically when closing the firm’s presence in the region)  It will spread as employees collaborate with others from other parts of the VAC. 7

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

 Manufacturing processes and  Service provision

 Direct interdependence

 between the two resulting units  with other units in the VAC and potential costumers  Coordination and Cooperation is required to the ends of  Developing tacit knowledge  Joint development  Building trust (including but not only on the management level)

 Implicit assumption: No power difference among the firms  What if there are some big firms, and many small ones?

Example: Service Providers and Customers

 Outsourcing may lead to the division of

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8

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Network Technologies  Different agents in an interconnected system  VAC, Joint Venture, Communication network

 Compatibility and interoperability required  Software, supplier parts

 The users the network has, the bigger its usefulness for the users  New users generate positive external effects for the incumbent users  New users generate negative external effects for the users of competing networks

 Choice as a path-dependent process involving many actors  Individualistic maximization becomes nearly irrelevant 9

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Network Technologies  Technologies that gain their value to the user predominantly from their user base, e.g. telephones, communication equipment, technological and social standards’  New users generate positive external effects for the incumbent users  Technology choice as a path-dependent process involving many actors  Multiple Equilibria; Individualistic maximization has low predictive power as to which equilibrium the system will converge.

 In the context of modern businesses:  Different agents in an interconnected system that requires compatibility and interoperability, e.g. VAC, Joint Venture, Communication network  E.g. software, communication infrastructure, supplier parts 10

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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A Game Theory Example  Three agents: A, B, C  Choice among two non-interoperable technologies, 𝑇1 and 𝑇2 with numbers of users 𝑛(𝑇1 ) and 𝑛(𝑇2 ) and expected individual payoffs per user Π1 and Π2  At the beginning, agents do not know which technology is superior  Suppose, in fact, 𝑇1 is superior to 𝑇2 Technology Agent

T1 A

T2 B

C

 Coordination problem  If a player makes her choice, she produces positive externalities for the other players 11

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Game Theory Example  Consider Π1 = 𝑛(𝑇1 ) + 1 and Π2 = 𝑛(𝑇2 )  Coordinated situation at the inferior technology is superior to all non-coordinated (isolated) situations  Path dependent process  The underlying mechanism may easily lead to latent collective blockage of action

12

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Incentive to wait vs. first mover advantage  To make the fist move in the technological choice model is risky  One might bet on the wrong technology  One might set the standard and receive increased profits

 Minimum critical mass required

 To wait can be profitable as well  Exploitation of the information revealed by prior movers

13

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Information as a collective good  Fragmented VACs imply a certain distribution of information

 The single firm receives less profit from their R&D expenses...  The information inherent to the project cannot be privatized entirely

 ...but it might also receive benefits from the products of others

 Potential for mutual exchange of information  But individualistic incentive to just exploit the others  Need for an institutional structure solving the underlying dilemma

14

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Path Dependence The development of an economic system depends on past events and is time-dependent  External/environmental random events  Internal random effects  Systemic structural changes  Indirect reactions of the system to both exogenous and internal stochastic effects  Transitions in the behavior of the system as result of any of the above

For optimizing individuals it is impossible to take all these aspects into account 15

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

systems and  Microsoft was hired to develop a suitable operation system.  Microsoft bought Q-DOS, made some minor changes, called it MS-DOS and sold it to IBM  The rights on the “new” operation system stayed with Microsoft

 The program became in spite of a poor performance quite successful.

The rise of Microsoft

 IBM planned to enter the market for personal computer

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 Microsoft went on to dominate the PC sector ever since 16

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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 IT startups at the time faced strong uncertainty and had several options to deal with it, the most important being

 First-mover advantage  Long-term monopoly revenues  Very risky

 Incremental innovation  Developing applied solutions

 “Wait” for dominating system to emerge  Less risky

 Sell company to major competitor  Opportunity of becoming part of the new mother company  Without bearing much risk

The rise of Microsoft

 Radical innovation:  Developing a new generation of programs/ operating systems/ etc.

 What would be the right choice for a maximizing individual?

17

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

 Microsoft realized a mixture of the three alternatives presented earlier  They benefited heavily from the tacit knowledge then available to them  After becoming the standard setter, Microsoft turned to more individualistic strategies  Ending collaboration with some major partners (those who were also competitors) removing compatibility from their products  Aggressively pursuing a strategy of gaining first-mover advantages at the cost of e.g. quality and also revenue (tolerating piracy)

The rise of Microsoft

 Microsoft for instance did not choose a single path and to be prepared for the development of the sector.  Acquiring (and investment in) in DOS  Joint venture with IBM  Negotiations with companies working with UNIX  Collaboration with Apple  Development of a next generation operating system (what would become Windows)

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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The complexity challenge  The economic environment of most economic agents is characterized by  Strong dependence  Uncertainty about the actions of others

 Need for cooperation and coordination

 How do agents react to this complexity challenge?  Individualistic strategies  Collective strategies

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

22.02.2026

Hierarchy, Power, Collusion  Network effects favor the growth of big firms  The individual form or agent cannot change the systemic properties of the economy

 Big firms often rely on hierarchy to solve the problem of coordination and cooperation under complexity  Helps them to sustain their position  Not favorable for the rest of the economy, i.e. an individualistic response

 But it is a system of institutionalized behavior  Includes barriers of entry, mergers and acquisitions, standard setting, lobbying and the use of hub & spoke networks 20

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Hub & Spoke Networks  Hub cooperation controls layered sourcing systems  Main suppliers as spokes  Less profitable steps get outsourced  Hub retains processes with highest value-added, i.e. the most profitable

 Cheap labor can be exploited  Hub retains control over the network  Standard setting 21

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Hub & Spoke Cluster  Hub & spoke structures established as  global sourcing/supplier networks (often with system suppliers supplying several assembler hubs);  regional forms of organization in spatial (regional, local) clusters of firms (i.e. firm agglomerations consisting of competitors/ cooperators on a horizontal level and suppliers, customers, service suppliers, infrastructure providers, …).

 Highly fragile structures:  High dependency of firms, employees, and regions worldwide from the central hub and its development; risky and precarious.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Industrial Clusters  Local and regional clustering as alternatives to the individualistic responses  A cooperation- and qualification-oriented system of independent and even-based small and medium-sized enterprises benefiting from  Spatial proximity (countermeasure against too much fragmentation)  Regional open information flows (tacit knowledge)

 Characteristics  Interaction density of stakeholders  Proximity of stakeholders

 Silicon-Valley or Third-Italy type systems 23

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Industrial Clusters  Quasi reintegration of parts of the VAC  Spatial institutional re-embedding  Mixtures of the two reactions are common  Firms may be clustered around Silicon Valley and still outsource parts of their VAC to other parts of the world

 Determinants for the choice  Recognized interdependence

 Degree of uncertainty  ...

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Emergence of regional clusters Stable relation

Spontaneous market action

Resistance against conventional market signals

Complex Environment

Stable expectations

Spatial proximity Relation outgrows the market

Long term reciprocity t

Initial interaction

Stable institutional setting

25

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Industrial clusters and firm networks  The emergence of industrial clusters often proceeds without formal central planning  Inside an industrial clusters, more intense relations may be built consciously among a subset of firms  Strategic networks  Special case: Open-Source Networks

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Summary  Modern Information Economies are characterized by  Fragmented VAC  Network technologies

 This leads to  Need for technological interoperability  Direct interdependence among actors  Ubiquitous coordination problems and underlying dilemmas

 As a result: A complexity challenge

27

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Preliminary conclusions  There are different reactions for the agents to cope with the complexity of their environment  Big corporations favor to preserve existing power differentials  Individualistic reactions (Hub&Spoke networks, ...)

 From a societal viewpoint, low power differentials and a cooperative institutional setting would be preferable  Industrial and strategic clusters

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Preliminary conclusions for policy  Critical time windows for intervention in technological development (standardization and monopolization processes) should be identified  Policy should foster non-individualistic reactions to complexity  To elaborate the exact functioning and to design adequate policies more concretely, more advanced tools are required  Mere deregulation is not sufficient or even harmful  Therefore: Understand the theories underlying deregulation and elaborate better alternatives 29

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 04: Real-World Interdependence and Complexity

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Readings for the next lecture  Compulsory reading: The Ideal Neoclassical Market and General Equilibrium, in: Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies, pp. 97-128.  For further readings visit the companion website

30

Prof. Wolfram Elsner Faculty of Business Studies and Economics iino – Institute of Institutional and Innovation Economics

Introductory Microeconomics The Ideal Neoclassical Market and General Equilibrium

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 05: Ideal Neoclassical Market and General Equilibrium

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Readings for this lecture  Mandatory reading this time: The Ideal Neoclassical Market and General Equilibrium, in: Elsner/Heinrich/Schwardt (2014): The Microeconomics of Complex Economies, Academic Press, pp. 97128.  The lecture and the slides are complements, not substitutes

 An additional reading list can be found at the companion website 2

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 05: Ideal Neoclassical Market and General Equilibrium

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Introduction – neoclassical method What is a neoclassical method?  To develop a “pure theory”  Axiomatic approach  Taking as little as possible as given  To construct an optimal equilibrium model  In a price-quantity world

Neoclassical economics typically ends up in price-quantity spaces employing the mathematics of maximization under restrictions (Lagrange algorithm). Economism in that sense is the idea of a “pure” economy, isolated from the rest of society and from the natural environment.

3

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 05: Ideal Neoclassical Market and General Equilibrium

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Introduction - model What is a model?  Imaginary  Analogy  Metaphor  Relating variables to one another  Diverging in the number of variables, agents and relations  Perfect competition and perfect information are assumed at least as a benchmark to which more specific models with imperfect competition or imperfect information are compared

4

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 05: Ideal Neoclassical Market and General Equilibrium

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General Equilibrium Theory (GET)…

… as the study of all interdependent ideal partial markets  Comprehensive formulation of neoclassical economics  Simultaneous equilibrium in all partial markets  Neoclassical paradigm cannot be tested and rejected straightforwardly  “data” are always selected  Stylized facts

5

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 05: Ideal Neoclassical Market and General Equilibrium

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GET – Walrasian Economics  Fictitious auctioneer (computes market-clearing prices)  No direct interdependence (only indirect via the ‘market’)

Walras was the first to emphasize the interdependence of individual partial markets in his Elements of pure economics (1874).

Neoclassical or marginalist revolution  Marginal utility (and marginal productivity in production)  Scarcity (focus on the allocation of scarce resources)

6

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 05: Ideal Neoclassical Market and General Equilibrium

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Overview  Optimization problems  Consumer theory  Production theory

 Utility/ profit maximization problems  Cost minimization problems  Partial and general equilibrium  Limitiations  Theory of the second best,  Markets of lemons 7

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 05: Ideal Neoclassical Market and General Equilibrium

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Perfect Information and Perfect Competition  All agents (firms and households) have all relevant information  Past, present and future  Reflected in prices  At no costs

 Preferences and production technologies are exogenous given

 Technology can be accessed and employed by everyone without problems

8

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 05: Ideal Neoclassical Market and General Equilibrium

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Competitive Equilibrium  No endogenous mechanism that causes a change in prices  Exogenous change in preferences or technology will result in a shift of the equilibrium  Such an equilibrium situation, as the “first welfare theorem” shows, is Pareto-optimal

9

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 05: Ideal Neoclassical Market and General Equilibrium

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Preferences Descartes division of the world into an inner subjective domain and an outer objective domain  The mind as having values and being subjective  The individual with reference to reason and consciousness

Cogito ergo sum – I think, therefore I am

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 05: Ideal Neoclassical Market and General Equilibrium

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Cartesian dualism The Cartesian dualism is the root of the neoclassical conception of the individual  Focus shifted from individual toward her individualistic choice  No explicit consideration of interaction among individuals  No explicit consideration of the institutional environment

 Individual decision making

 Completely detached from her social structure

11

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 05: Ideal Neoclassical Market and General Equilibrium

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A rational preference relation We assume that the individual has a rational preference relation over the set of all possible choices. Let 𝑥, 𝑦, 𝑧 be mutually exclusive alternatives in the set of possible choices 𝑋 A rational preference relation is characterized by

 Completeness: for 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≽ 𝑦 ∨ (read: or) 𝑦 ≽ 𝑥 or both  Transitivity: for 𝑥, 𝑦, 𝑧 є 𝑋, 𝑥 ≽ 𝑦 ∧ (read: and) 𝑦 ≽ 𝑧 ⇒ 𝑥 ≽ 𝑧  (Reflexivity: 𝑥 ~ 𝑥 ∀ (read: for all) 𝑥 є 𝑋)

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Lexicographic preferences An agent with lexicographic preferences will choose the bundle that offers the largest amount of the first good x1 no matter how much of the other good is in the bundle. (𝑥1, 𝑥2) ≻ (𝑥´1, 𝑥´2) if 𝑥1 ≻ 𝑥´1  Rational, but cannot be represented by a continuous function

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Notation I Symbol

Meaning

𝑝 = (𝑝1, 𝑝2, … , 𝑝𝑁)

Price vector

Real numbers, positive real numbers (size of commodity space and consumption set)

ℝ𝑁 +

ℝ𝑁 ,

𝑥𝑖´ = (𝑥1𝑖 , 𝑥2𝑖 , … , 𝑥𝑁𝑖 ) є ℝ𝑁 +

Commodity bundle or demand individual 𝐼

𝑥´ = (𝑥1 , 𝑥 2 , … , 𝑥 𝐼 ) є ℝ𝐼𝑁 +

Allocation

𝑖 𝜔𝑖 ´ = (𝜔1𝑖 , 𝜔2𝑖 , … , 𝜔𝑁 )

Endowment individual 𝐼 Aggregate endowment, wealth

𝐼

𝜔 ഥ

= ෍ 𝜔𝑖 ,

𝜔 = 𝑝𝜔 ഥ

𝑖=1 14

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Notation II Symbol 𝑥,ҧ

Meaning

𝑥𝑛𝑖 ,

𝑝ҧ

Equilibrium allocation, equilibrium demand agent 𝑖 good 𝑛, equlibrium price vector

𝑢 … ,

𝑣(… )

Direct and indirect utility functions

𝑥 … ,

ℎ(… )

Marshallian and Hicksian demand

e(… )

𝜆,

Expenditure function

Λ(… )

𝐾, 𝐿, 𝑦(𝐾, 𝐿)

𝑟, 𝑤

Lagrange multiplier, Lagrangian function Production factors (capital, labor), Factors productivities (unit interest and unit wage) Production function 15

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Lagrangian method  Unrestricted utility/profit-functions of this type have no maxima but individuals usually are restricted at least by a limited budget  Hence, neoclassical microeconomics needs a method for optimization under restrictions

 The Lagrangian method is a method to find local maxima and minima (originally from classical mechanics, developed by JosephLouis Lagrange in correspondence with Leonhard Euler, 1755)  Restrictions are exogenously imposed  Note that the method requires the solution to be on every one of the restrictions (i.e. all restrictions are binding) 16

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Utility function  Continuous  Strictly increasing in every argument

𝜕𝑢/𝜕𝑥𝑛 > 0

 At least differentiable twice on the interior of ℝ𝑁 +  Strictly concave 𝜕 2 𝑢/𝜕 2 𝑥𝑛 < 0  Satisfies

lim 𝜕𝑢/𝜕𝑥𝑛 = +∞

𝒙𝒏 →𝟎

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Marginal utility Marginal utility is positive, but decreasing. Quantities and prices will be strictly positive in equilibrium.

This leads to a budget set 𝑖 𝑖 𝐵 𝑖 = 𝑥 𝑖 ∈ ℝ𝑁 + : 𝑝𝑥 ≤ 𝑝𝜔

… that typically takes the form of a linear decreasing budget restriction

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The Lagrange Multiplier Approach The Lagrangian function is constructed as follows: max Λ = 𝑢 𝑥1 , 𝑥2 − 𝜆(𝑝1 𝑥1 + 𝑝2 𝑥2 − 𝜔)

𝑥1 ,𝑥2 ,𝜆

Assuming the utility function is concave, first order conditions are necessary and sufficient conditions – a system of three equations: (1)

𝜕Λ 𝜕𝑢 = − 𝜆𝑝1 = 0; 𝜕𝑥1 𝜕𝑥1

(2)

𝜕Λ 𝜕𝑢 = − 𝜆𝑝2 = 0; 𝜕𝑥2 𝜕𝑥2

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The Lagrange Multiplier Approach (3)

𝜕Λ = − 𝑝1 𝑥1 − 𝑝2 𝑥2 + 𝜔 = 0 𝜕𝜆

Dividing the first and the second equation – At the optimum, the ratio of prices equals the ratio of marginal costs:

𝜕𝑢/𝜕𝑥1 𝜆𝑝1 𝑝1 = = 𝜕𝑢/𝜕𝑥2 𝜆𝑝2 𝑝2

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Walrasian or Marshallian demand function The solution to the UMP (utility maximization problem) is a bundle of commodities as a function of prices and wealth:

𝑥 𝑖 = 𝑥 𝑖 (𝑝, 𝜔) Using the Lagrange method, we maximize: Λ = 𝑢 𝑥 − 𝜆(𝑝𝑥 − 𝜔) First order conditions (FOC): 𝜕𝑢 = 𝜆𝑝𝑛 ∀𝑛 ∈ 𝑁; 𝜕𝑥𝑛

𝑝𝑥 = 𝜔 21

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Equilibrium conditions for consumers Marginal utilities are equalized formally: 𝜕𝑢/𝜕𝑥𝑛1 𝜕𝑢/𝜕𝑥𝑛2 = 𝑝𝑛1 𝑝𝑛2

𝜕𝑢/𝜕𝑥𝑛1 𝑝𝑛1 − =− = 𝑀𝑅𝑆 𝜕𝑢/𝜕𝑥𝑛2 𝑝𝑛2 The FOCs of the UMP give us 𝑁 + 1 variables (𝑥1 , 𝑥2 , … , 𝑥𝑁 and 𝜆), a system that is solvable in principle.

The relations of marginal utilities (and good prices) is called the marginal rate of substitution (MRS) 22

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The utility function is given by:

subject to 𝑝1 𝑥1 + 𝑝2 𝑥2 ≤ 𝜔.

We set up the Lagrange function

max 𝛼 ln 𝑥1 + 1 − 𝛼 ln 𝑥2 + 𝜆(𝜔 − 𝑝1 𝑥1 − 𝑝2 𝑥2 )

𝑥1 ,𝑥2 ,𝜆

An Example

𝑢 𝑥1 , 𝑥2 = 𝛼 ln 𝑥1 + (1 − 𝛼) ln 𝑥2

and derive the FOCs: (1)

𝛼 = 𝜆𝑝1 𝑥1

(2)

1−𝛼 = 𝜆𝑝2 𝑥2

(3) 𝑝1 𝑥1 + 𝑝2 𝑥2 = 𝜔

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We combine the first two to get:

And substitute into the third, 𝑝1 𝑥1 = 𝜔 − 𝑝2 𝑥2 We obtain: 𝛼 𝑝2 𝑥2 = 𝜔 − 𝑝2 𝑥2 1−𝛼

An Example

𝛼 𝑝1 𝑥1 = 𝑝 𝑥 1−𝛼 2 2

This equations can be solved for 𝑥2 as a function of prices and wealth: 𝜔 𝑥2 𝑝1 , 𝑝2 , 𝜔 = 1 − 𝛼 𝑝2 24

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 05: Ideal Neoclassical Market and General Equilibrium

𝑥1 𝑝1 , 𝑝2 , 𝜔 = 𝛼

𝜔 𝑝1

The last two equations are the individual´s Walrasian or Marshallian demand functions, telling us, how much of each good the individual will demand as a function of prices and wealth.

An Example

Inserting and solving for demand of 𝑥1 , we get:

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𝑣 𝑝, 𝜔 = 𝑢 𝑥1 , 𝑥2 = 𝛼 ln 𝛼

𝜔 𝑝1

+ (1 − 𝛼) ln (1 − 𝛼)

𝜔 𝑝2

An indirect utility function gives utility as a function of prices and income.

An Example

The consumer´s utility at the optimum is given by the indirect utility function:

The direct utility function gives utility as a function of goods consumed

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Marshallian demand

x1 Indifference curves

Tangential point – optimum consumption bundle x1, x2 budget line

x2

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Marshallian Demand, Hicksian Demand The dual problem to utility maximization is a minimization of expenditure for reaching a specific utility level u.

The general form: min 𝑒(𝑝𝑖 , 𝑥𝑖 ) subject to 𝑢 𝑥 ≥ 𝑢ത Given an expenditure function 𝑒(𝑝, 𝑢) that shows the minimum expenditure required for reaching a certain utility level, the relation between Hicksian and Marshallian demand is given by: ℎ 𝑝, 𝑢 = 𝑥 𝑝, 𝑒 𝑝, 𝑢

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Slutsky Equation Given the indirect utility function just derived before, we can also state: 𝑥 𝑝, 𝜔 = ℎ(𝑝, 𝑣(𝑝, 𝜔)) The two kinds of demand can be related by what is termed the Slutsky equation.  Split changes in uncompensated demand  Substitution of the (after the price changes) relatively cheaper goods for relatively more expensive ones, the substitution effect  Change in the consumer´s purchasing power following the price changes, the income effect 29

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Slutsky Equation 𝜕𝑥𝑖 𝑝, 𝜔 = 𝜕𝑝𝑗

𝜕ℎ 𝑝, 𝑢 𝜕𝑝𝑗

𝜕𝑥𝑖 𝑝, 𝜔 − 𝑥𝑗 (𝑝, 𝜔) 𝜕𝜔

𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝐸𝑓𝑓𝑒𝑐𝑡

𝐼𝑛𝑐𝑜𝑚𝑒 𝐸𝑓𝑓𝑒𝑐𝑡

Using Shephard’s Lemma

𝜕𝑒 𝑝,𝑢 𝜕𝑝𝑗

= ℎ𝑗 (𝑝, 𝑢) the derivation follows with

the definition of the indirect utility above (𝑥𝑗 𝑝, 𝜔 = ℎ𝑗 (𝑝, 𝑣(𝑝, 𝜔))) from the total differential of the Hicksian demand function (with 𝑑𝑢 = 0):

𝑑ℎ𝑖 𝑝, 𝑢 =

𝜕𝑥𝑖 (𝑝,𝑒 𝑝,𝑢 ) 𝜕𝑥 (𝑝,𝑒 𝑝,𝑢 ) 𝜕𝑒 𝑝,𝑢 𝑑𝑝𝑗 + 𝑖 𝜕𝑝𝑗 𝜕𝑒(𝑝,𝑢) 𝜕𝑝𝑗

𝑑𝑝𝑗 +

𝜕𝑥𝑖 (… ) 𝜕𝑒 … 𝜕𝑒(… ) 𝜕𝑢

𝑑𝑢

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x1

IE and SE in demand changes

Hicksian demand (compensated)

New optimum

x2 IE

SE

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The Production Function As an example for firm´s production possibilities is often used the Cobb-Douglas technology, which is defined by the following production function for 𝛼 ∈ (0,1) and two inputs (capital 𝐾, labor 𝐿): 𝑦 = 𝐹 𝐾, 𝐿 = 𝐴𝐾 𝛼 𝐿1−𝛼

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Cobb-Douglas production function 33

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Marginal rate of technical substitution (MRTS) If we assume, that the production function is differentiable, the MRTS tells us at which rate one input can be exchanged for another one without altering the quantity of output.

Setting the equal to zero and sticking to the Cobb-Douglas production function for the derivation, with 𝐴 = 1:

𝐿 1−𝛼 𝑑𝑦 = 𝛼 𝑑𝐾 + 1 − 𝛼 𝐾 𝑑𝐿 Resolving for gives 𝑑𝐾

𝑑𝐿 =− 𝑑𝐾

𝐿 1−𝛼 𝐾 𝐾 𝛼 1−𝛼 𝐿

𝛼

𝐾 𝛼 𝑑𝐿 = 0 𝐿 𝛼 =− 1−𝛼

𝐿 1−𝛼+𝛼 𝐾 34

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MRTS The absolute value of the scope of the slope of the isoquant: ෫𝐾𝐿 = 𝑀𝑅𝑇𝑆

𝑑𝐿 𝛼 𝐿 =− 𝑑𝐾 1−𝛼 𝐾

… or sometimes also given in terms of the absolute value,

𝑀𝑅𝑇𝑆𝐾𝐿 = −

𝑑𝐿 𝑑𝐾

=−

𝛼 𝐿 1−𝛼 𝐾

Formally, isoquant 𝑆(𝑦) and input requirement set 𝐼(𝑦) are defined as: 𝑆 𝑦 = { 𝐿, 𝐾 : 𝐹 𝐾, 𝐿 = 𝑦} 𝐼 𝑦 = { 𝐿, 𝐾 : 𝐹 𝐾, 𝐿 ≥ 𝑦} 35

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Isoquants and input sets

α=0,5

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Law of diminishing returns The law of diminishing returns states, that if all but one input is fixed, the increase in output from an increase in the variable input decline. For the production function, this implies:  Strictly increasing in both dimensions

 Concave in both dimensions

𝜕𝐹 𝜕𝐹 > 0 and >0 𝜕𝐾 𝜕𝐿

𝜕2𝐹 𝜕2𝐹 < 0 and 2 < 0 𝜕2𝐾 𝜕 𝐿

(Note that as the function is also twice statically differentiable and its slope becomes infinite near zero (𝐾 = 0 or 𝐿 = 0) it mathematically resembles the neoclassical utility function) 37

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Cost Minimization and Cost Functions Starting with the firm´s cost-minimization problem: min 𝑟𝐾 + 𝑤𝐿 𝐾,𝐿

subject to

𝐹 𝐾, 𝐿 ≥ 𝑦ത

we construct the Lagrangian Λ 𝐾, 𝐿, 𝜆 = 𝑟𝐾 + 𝑤𝐿 + 𝜆(𝐹 𝐾, 𝐿 − 𝑦) ത To figure out which point on the isoquant is cost minimizing, we take the FOCs and set them to zero: 𝜕Λ

𝜕𝐹

(1) =𝑟+𝜆 =0 𝜕𝐾 𝜕𝐾 (3) 𝐹 𝐾, 𝐿 − 𝑦ത = 0

(2)

𝜕Λ 𝜕𝐹 =𝑤+𝜆 𝜕𝐿 𝜕𝐿

=0 38

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Cost Minimization and Cost Functions The first two FOCs can conveniently be rearranged to 𝑟=𝜆

𝜕𝐹 𝜕𝐾

and

w=𝜆

𝜕𝐹 𝜕𝐿

Combining the two equations we get the firm´s optimality condition for the relative amounts of inputs used: 𝜕𝐹 𝑟 ෫𝐾𝐿 = 𝜕𝐾 = −𝑀𝑅𝑇𝑆 𝜕𝐹 𝑤 𝜕𝐿 At the optimum, the relative price of inputs has to equal the MRTS. 39

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Input requirement set, isoquant, relative prices 40

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Cost Minimization and Cost Functions Deriving the firm´s cost function with using the Cobb-Douglas function (𝛼 = 0.5):

𝐶 𝑟, 𝑤, 𝑦ത = min 𝑟𝐾 + 𝑤𝐿 𝐾,𝐿

subject to 𝑦ത ≥ 𝐾 0.5 𝐿0.5 = 𝐾𝐿

The constraint must be binding, 𝑦ത ≥ 𝐾 0.5 𝐿0.5 . Solving for 𝐾 yields 𝑦ത 2 𝐾 = which can be substituted into the cost function: 𝐿

𝑦ത 2 𝐶 𝑟, 𝑤, 𝑦ത = min 𝑟 + 𝑤𝐿 𝐾,𝐿 𝐿

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Cost Minimization and Cost Functions Taking the FOC for labor we have

𝑟 0.5 𝐿 𝑟, 𝑤, 𝑦ത = 𝑦. ത 𝑤 The conditional demand for capital is derived in the same way; we obtain 𝑤 0.5 𝐾 𝑟, 𝑤, 𝑦ത = 𝑦. ത 𝑟 and can rewrite the cost function as 𝐶 𝑟, 𝑤, 𝑦ത =w𝐿 𝑟, 𝑤, 𝑦ത + 𝐾 𝑟, 𝑤, 𝑦ത = 2𝑟 0.5 𝑤 0.5 𝑦. ത 42

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Profit Maximization Using the cost function, we now turn to the firm´s profit maximization problem. A firm´s profits are defined as revenue minus cost: Π 𝑦, 𝐿, 𝐾, 𝑤, 𝑟, 𝑝0 = 𝑝0 𝑦 − (𝑤𝐿 + 𝑟𝐾)

The firm has now the task to choose the level of output.

Substituting the cost function for (𝑤𝐿 + 𝑟𝐾) and taking input prices (𝑤 and 𝑟) as given: max Π = 𝑝0 𝑦 − 𝐶(𝑟, 𝑤, 𝑦) 𝑦≥0

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Profit Maximization The Cobb-Douglas production function exhibits constant returns to scale since:

𝐹 𝑐𝐾, 𝑐𝐿 = 𝐴 𝑐𝐾 0.5 𝑐𝐿 0.5 = 𝑐𝐹(𝐾, 𝐿) This means that the cost function is linear and average costs as well as marginal costs are constant. 𝐶 𝑟, 𝑤, 𝑦 𝐴𝐶 = 𝑦 𝜕𝐶 𝑟, 𝑤, 𝑦 𝑀𝐶 = 𝜕𝑦 44

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Existence of Equilibrium: Constant Returns to Scale

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Existence of Equilibrium: Decreasing Returns to Scale

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Existence of Equilibrium: Increasing Returns to Scale

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Partial Equilibrium: Stylized Demand and Supply Functions

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Partial Equilibrium

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General Equilibrium Instead of just looking at one market in isolation, as we did before, we look at the complete set of markets now.

In equilibrium, marginal rates of substitution (MRS) and technical substitution (MRTS) are equal: 𝜕𝑦 𝜕𝑢ൗ ൗ𝜕𝑥 𝑝1 𝜕𝑥1 1 ෫12 𝑀𝑅𝑆12 = − =− =− = 𝑀𝑅𝑇𝑆 𝜕𝑢ൗ 𝜕𝑦 𝑝2 ൗ𝜕𝑥 𝜕𝑥2 2

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Walras´ Law The combination of equilibrium prices and allocations (𝑝,ҧ 𝑥)ҧ as Arrow-Debreu equilibrium.

We assume that all markets are in equilibrium, i.e. excess demand 𝑧 (demand minus supply) is equal to zero: 𝐼

𝐼

𝑧𝑛 𝑝 = ෍ 𝑥ҧ𝑛𝑖 (𝑝) − ෍ 𝜔𝑛𝑖 = 0 𝑖=1

∀𝑛 ∈ {1, … , 𝑁}

𝑖=1

A corollary of this is Walras´ Law, stating that if 𝑁 − 1 markets are in equilibrium, the 𝑁th market must also be in equilibrium 51

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Walras´ Law If 𝑁 – 1 markets are in equilibrium, the Nth market must also be in equilibrium. This follows from simple accounting. If we sum up all individual budgets constraints, we see that total expenditure has to equal total receipts, i.e., 𝐼

𝐼

𝑝ҧ ෍ 𝑥 𝑖 𝑝 = 𝑝ҧ ෍ 𝜔𝑖 𝑖=1

𝑖=1

For excess demand functions, Walras´ Law implies that they sum up to zero: 𝑁

෍ 𝑧𝑛 𝑝 = 0 𝑛=1 52

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Walras´ Law For the analysis of general equilibrium, this result proves useful since it implies that if all market but one are in equilibrium, the last market also has to be in equilibrium: 𝑧𝑛 𝑝ҧ = 0

∀𝑛 ∈ 1, … , 𝑁 − 1

⇒ zN 𝑝ҧ = 0

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Welfare Theorems For now, we just assume that there exists an equilibrium with positive prices and state some properties of this equilibrium:

Theorem 1 (first welfare theorem): A competitive equilibrium allocation is an efficient allocation.

Theorem 2 (second welfare theorem): Every efficient allocation can be transformed into a competitive equilibrium allocation by appropriate transfers

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Conditions for Walras´ common equilibrium of Sonnensch./Mantel/Deb. Existence  Convexity of preferences (concave utility function)  Convex production set Uniqueness  Either the initial allocation is already Pareto-efficient  Or all goods are substitutes (positive cross price elasticities)  Then the excess demand curve crosses the x-axis only once Stability  Conditions for uniqueness are sufficient for global stability  If excess demand becomes zero only once, this equilibrium will be stable 55

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Sonnenschein-Mantel-Debreu The conditions that have to be fulfilled for an existing, unique, and stable equilibrium are called SMD conditions.

The GET is a special exception, not the rule for the condition of a perfect rational agent. For all other cases the system could react differently, i.e., multiple equilibria, no equilibrium, or instable equilibria etc.

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The General Theory of the Second Best To which degree the general equilibrium model can serve to inform relevant policy decisions in the real world?

 Some distortion is present in the setup  Optimal conditions in all other relevant areas will not guarantee a second best result (i.e. there will not be an efficient allocation even between the undistorted elements) as the theory of the second best shows  Some first-best conditions are violated, there may be numerous second best results that can be reached 57

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The General Theory of the Second Best We have a function 𝐹(𝑥1 , … , 𝑥𝑛 ) of 𝑛 variables 𝑥𝑖 that is to be maximized (minimized) and a constraint 𝐿 𝑥1 , … 𝑥𝑛 = 0 that has to be taken into account in that operation.

The optimization gives the FOCs (with 𝐹𝑖 as 𝐹𝑖 − 𝜆𝐿𝑖 = 0

𝜕𝐹𝑖 etc.) 𝜕𝑥𝑖

𝑖 = 1, … , 𝑛

and, from these, the relative conditions for optimality, 𝐹𝑖 𝐿𝑖 = 𝐹𝑛 𝐿𝑛

i = 1, … , n.

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The General Theory of the Second Best In Lipsey and Lancaster’s theory of the second best, this first best solution is excluded by an additional constraint of the form: 𝐹1 𝐿1 =𝑘 with k ≠ 1. 𝐹𝑛 𝐿𝑛

This additional constraint changes the optimization problem to: min 𝐹 − 𝜆1 𝐿 − 𝜆2

𝐹1 𝐿1 −𝑘 𝐹𝑛 𝐿𝑛

(or equivalently for maximization problems). 𝜆1 and 𝜆1 will generally be different from λ.

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The General Theory of the Second Best The FOCs here are: 𝐹𝑖 − 𝜆1 𝐿𝑖 − 𝜆2 𝐹 𝐹 −𝐹 𝐹 𝐹𝑛

𝐹𝑛 𝐹1𝑖 − 𝐹1 𝐹𝑛𝑖 𝐿𝑛 𝐿1𝑖 − 𝐿1 𝐿𝑛𝑖 −𝑘 =0 2 2 𝐹𝑛 𝐿𝑛 𝐿 𝐿 −𝐿 𝐿 𝐿𝑛

Denoting 𝑛 1𝑖 2 1 𝑛𝑖 = Q i and 𝑛 1𝑖 2 1 𝑛𝑖 = 𝑅𝑖 we can write: 𝜆2 𝐹𝑖 𝜆1 𝑄𝑖 − 𝑘𝑅𝑖 = 𝜆2 𝐹𝑛 𝐿 + 𝑛 𝜆1 𝑄𝑛 − 𝑘𝑅𝑛

𝐿𝑖 +

The Pareto optimum conditions (achieved in the first best result above) can therefore only be attained if 𝜆2 = 0 (i.e. if the additional condition is irrelevant).

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Asymmetric Information – The Markets for Lemons Differences in the information (on quality) available to sellers and potential buyers of a good.

 Sellers will exploit their information advantage and sell lowquality goods at market price  Buyers use some statistics from observed frequencies of good and bad products (lemons) to form an idea of expected quality The example will show that these information sets directly lead to the collapse of the market (and without a market, no efficient or Pareto optimal allocation of goods can be established).

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Asymmetric Information – The Markets for Lemons  Quality 𝑞 of goods is distributed uniformly; sellers will not sell under value, i.e. the seller price is 𝑝𝑠 ≥ 𝑞 for each seller.

 Buyers are willing to pay a higher price 𝑘𝑞 (with 1 < 𝑘 < 2) over their expected quality; since the average value of the goods 1 𝑘 is σ𝑖 𝑞𝑖 , the buyer’s reservation prices will be 𝑝𝑑 ≤ σ𝑖 𝑞𝑖 2

2

𝑘

 Since 𝑘 < 2 there are sellers with quality 𝑞 > σ𝑖 𝑞𝑖 who will 2 exit the market which leads to a further fall in average quality and sends the market into a downward spiral

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Asymmetric Information – The Markets for Lemons Assume the following situation:  Two groups of agents, 1 and 2.  Distinguished by the utility they gain from consumption of a specific good  𝑞𝑖 is the indicator of the quality of a particular unit of the good  𝑀 being a bundle of the rest of goods

𝑈1 = 𝑀 + σ𝑛𝑖=1 𝑞𝑖

and

3 2

𝑈2 = 𝑀 + σ𝑛𝑖=1 𝑞𝑖

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Asymmetric Information – The Markets for Lemons Spending one more unit on bundle 𝑀 increases utility by one unit 𝑑𝑈𝑖 =1 𝑑𝑀 Spending on the second good, the utility effect depends on the quality

𝑞>𝑝

: The purchase is worthwhile of type 1 agents

3𝑞 >𝑝 2

: worthwhile for type 2 agents

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Asymmetric Information – The Markets for Lemons Let 𝑌1 and 𝑌2 denote the income of all types 1 and 2 agents, respectively. Then, demand 𝐷 for the good from type 1 agents is of the form: 𝑌 𝐷1 = 1 𝑖𝑓 𝑞 ≥ 𝑝 𝑝

𝐷1 = 0

otherwise

For type 2 agents, we have the following analogous expressions 𝑌 𝑝

3 𝑞≥𝑝 2

𝐷2 = 2

𝑖𝑓

𝐷2 = 0

otherwise

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Asymmetric Information – The Markets for Lemons Accordingly, total demand 𝐷 is: 𝑌 +𝑌2 𝑝 𝑌 𝐷= 2 𝑝

𝐷= 1

𝑖𝑓 𝑞 ≥ 𝑝 𝑖𝑓

3 𝑞≥𝑝≥𝑞 2

3 2

𝐷=0

otherwise (i. e. if 𝑝 > 𝑞)

With uniform quality distribution, the average quality at any price 𝑝 𝑝 would be 𝑞 = ; therefore the demand would be 𝐷 = 0.

2

No trade will never take place, even though at any price there would be someone willing to pay the asked price if quality could be assured 66

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Conclusion & Limitations of GET  This chapter provided a short introduction to neoclassical microeconomics, optimization methods, and GET

 Some critique (SMD) and two specific limitations (second best and markets for lemons) were included  Further extensions (intertemporal optimization, growth models) and limitations (imperfect information) are introduced in the chapter (but are not included in this presentation)  Chapters 6 and 7 of the textbook extend this and go more into detail on the critique of GET and on oligopoly models.

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Readings for the next lecture  Compulsory reading: Critiques of the Neoclassical ‘Perfect Market’ Economy and Alternative Price Theories, in: Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies, pp. 129-155.  For further readings visit the companion website

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Prof. Wolfram Elsner Faculty of Business Studies and Economics iino – Institute of Institutional and Innovation Economics

Real-World Markets

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Readings for this lecture  Required reading this time: Geben Sie hier eine Formel ein.

Real-World Markets: …, in: Elsner/Heinrich/Schwardt (2014): The Microeconomics of Complex Economies, Academic Press, pp.

157-190.

 The lecture and the slides are complements, not substitutes  An additional reading list can be found at the companion website 2

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Introduction  Real-world markets are not simple, clear-cut, unique, or transparent.

 They are networks of directly interdependent agents, facing information problems, different problem structures, and uncertainty.  Such situations are complex.

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Introduction  To protect themselves in such an environment, firms have various options to take strategic actions, e.g.:

 Size growth (technological factors can support this)  Influencing customers and/or suppliers in production networks  Avoidance of competition (through signals among competitors, or market segmentation, for instance)

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Introduction  Decentralized market systems thus tend to develop into power-based systems.

 Oligopolistic structures are the rule rather than the exception in real-world markets.  Due to direct interdependence, turbulence can nevertheless remain a factor in the environment, though.

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Factors leading to size and power  Real-world markets exist in limited space, they are networks of (relatively few) directly interdependent agents.

 Technological factors in production tend to favor size growth.  Informal collusion or formal cartelization help to avoid competition.  It pays to try and deter new entrants.  Network effects in consumption may favor structures with few suppliers.

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Factors leading to size and power  First-mover advantages favor few suppliers.  Learning curves in production leading to cost reductions reinforce these advantages.

 Technological knowledge may be difficult to transfer, tacit knowledge can play an important part in production and distribution.

Factors’ effects are cumulative, they can combine their influences.

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A model of a monopoly market  One firm produces the total supply in a market for one good.

Its goal is to maximize its profit.

 Numerous buyers purchase the product.  A demand function has a negative slope – the higher the price, the lower the quantity that can be sold.  The monopolist has market power: It can select its preferred point on the demand schedule (Cournot point) through its choice of output produced.

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A model of a monopoly market  Maximize profit π – the difference between revenue R and cost C, all as function of quantity q: max 𝜋 𝑞 = 𝑅 𝑞 − 𝐶(𝑞) 𝑞

 From the first order condition we can derive the general rule for profit maximization (marginal revenue = marginal cost): 𝑑𝑅 𝑑𝐶 𝑀𝑅 = = = 𝑀𝐶 𝑑𝑞 𝑑𝑞

 In contrast to the perfect market, the market-price changes when the monopolist changes output.

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A model of a monopoly market  The demand function is 𝑞 = 𝑞(𝑝), the inverse demand function hence 𝑝 = 𝑝(𝑞).

 Revenue is 𝑅(𝑞) = 𝑝(𝑞)𝑞, …  … and marginal revenue (using the product rule of differentiation): 𝑑𝑅 𝑑 𝑝 𝑞 𝑞 𝑑𝑝 𝑀𝑅 𝑞 = = =𝑝+ 𝑞 = 𝑀𝐶 𝑑𝑞 𝑑𝑞 𝑑𝑞

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A model of a monopoly market  We can rearrange this expression as (ε is the price elasticity of demand): 𝑑𝑝 𝑑𝑝 𝑞 1 𝑝+ 𝑞 =𝑝 1+ =𝑝 1+ 𝑑𝑞 𝑑𝑞 𝑝 𝑑𝑝Τ𝑑𝑞 𝑝Τ𝑞 1 𝑝 1+ 𝑑𝑝Τ𝑑𝑞 𝑝Τ𝑞

1 = 𝑝 1+ = 𝑀𝐶 𝜀

 The monopolist will offer a quantity for which marginal cost ist below the price in the market (as 𝜀 < 0).

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A model of a monopoly market  The expression can be rearranged to give the AmorosoRobinson relation: 1 1 𝑝 1+ =𝑝 1− 𝜀 𝜀

𝑀𝐶 = 𝑀𝐶 ↔ 𝑝 = 1 − 1Τ 𝜀

 In the perfect market-benchmark case, price is equal to marginal cost. The higher price in the monopoly case, is inversely related to the elasticity of demand a company faces.

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A model of a monopoly market  We can analyze the welfare effects of monopoly markets as well, the benchmark case is the perfectly competitive market.

 Consumer surplus (CS): the difference between the price consumers are willing to pay and the price they have to pay  Producer surplus (PS): the difference between unit cost and the price units are sold for (zero in a competitive market)

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A model of a monopoly market  Perfect market CS: ACF; Monopoly market CS: EDF; Monopoly market PS: ABDE; Welfare loss monopoly: BCD

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Oligopoly models  There are three standard oligopoly models, Cournot, Bertrand, and Stackelberg.

 In all models, few companies provide the supply of a homogenous good to a market with many buyers. Their individual decisions have an impact on the overall quantity supplied.  Companies take their decisions at the beginning of a period.

These cannot be changed during the period.

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Oligopoly models  In the Cournot model, companies decide simultaneously, how much to produce. Their combined output determines the market price.  In the Bertrand model, companies set their prices. The one with the lowest price serves the entire demand. If more than one company (say, n) set an equally low price, each serves 1/n of total demand.  In the Stackelberg model, companies decide sequentially, how much to produce. Earlier decisions are known to those who choose later. Combined output determines the market price.

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Oligopoly models – Cournot  2 companies, A and B.  Let the inverse demand function be p(q)=a-bq, with positive parameters a and b, and 𝑞 = 𝑞𝐴 + 𝑞𝐵 . Constant unit costs for both are c.  The problem for A is (taking the decision of B as given): max 𝑎 − 𝑏 𝑞𝐴 + 𝑞𝐵 𝑞𝐴 − 𝑐𝑞𝐴 𝑞𝐴 ≥0

 The resulting first order condition is (for B equivalently): 𝑎 − 2𝑏𝑞𝐴 − 𝑏𝑞𝐵 = 𝑐

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Oligopoly models – Cournot  A firm‘s best response function (BR) shows its optimal amount of output given the output of the rest.

 Rearranging the FOC gives: 1 𝑎−𝑐 − 𝑞𝐵 2 𝑏 1 𝑎−𝑐 = − 𝑞𝐴 2 𝑏

𝑞𝐴𝐵𝑅 𝑞𝐵 = 𝑞𝐵𝐵𝑅 𝑞𝐴

 Substituting one into the other gives the mutual best response: 1 𝑎−𝑐 𝑞𝐴 = 𝑞𝐵 = 3 𝑏 18

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Oligopoly models – Cournot  We can compare this outcome to monopoly (M) and perfectly competitive (PC) market.

 As price is equal to MC (𝑝 = 𝑐) in a PC, the inverse demand schedule gives 𝑞 𝑃𝐶 = (𝑎 − 𝑐)/𝑏  Solving for the monopoly solution above, gives 𝑞 𝑀 =

𝑎−𝑐 2𝑏

 Overall, thus: 1 𝑎−𝑐 2 𝑎−𝑐 𝑎−𝑐 < < 2 𝑏 3 𝑏 𝑏 𝑞𝑀

𝑞𝐶

𝑞 𝑃𝐶

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Oligopoly models – Cournot

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Oligopoly models – Bertrand  In the Bertrand model, companies compete through price setting. For two companies, demand takes the form:

𝑞𝐴 𝑝𝐴 , 𝑝𝐵

𝑞 𝑝𝐴 𝑖𝑓 𝑝𝐴 < 𝑝𝐵 = ൞0.5𝑞 𝑝𝐴 𝑖𝑓 𝑝𝐴 = 𝑝𝐵 0 𝑖𝑓 𝑝𝐴 > 𝑝𝐵

 In the sole Nash equilibrium in this situation, price is equal to marginal cost and the companies split the market.

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Oligopoly models – Bertrand  If there was a chance for one company to undercut the other, it would do so and attract the entire demand in the market to itself.  If 𝑝𝐴 > 𝑐, it is advantageous to charge an amount just below that price for B. The limit to this dynamic is provided by marginal cost c, as negative profits result below.  If cost structures differ, the cheaper producer can undercut the other slightly and service the entire market at the resulting price.

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Oligopoly models – Stackelberg  The sequential decisions in the Stackelberg model allow the leader to ask how the follower will react to its decision.

 The reaction functions are the same as in the Cournot case (the basic maximization problem stays the same): 1 𝑎−𝑐 𝑞𝐵𝐵𝑅 𝑞𝐴 = − 𝑞𝐴 2 𝑏  Instead of substituting in the BR, we substitute this in A’s profit function.

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Oligopoly models – Stackelberg  This gives: 1 𝑎−𝑐 max 𝑎 − 𝑏 𝑞𝐴 + − 𝑞𝐴 𝑞𝐴 ≥0 2 𝑏

𝑞𝐴 − 𝑐𝑞𝐴

1 max 𝑎 + 𝑐 − 𝑏𝑞𝐴 𝑞𝐴 − 𝑐𝑞𝐴 𝑞𝐴 ≥0 2

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Oligopoly models – Stackelberg  Solving for the optimal quantities (from FOC for A, and BR for B) gives: 𝑎−𝑐 𝑞𝐴 = 2𝑏 𝑎−𝑐 𝑞𝐵 = 4𝑏  The first mover advantage for the leader allows it higher profits.

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Natural monopolies  We find monopolies in a number of areas in the economic sphere. – Why?

 A particularly stable set are the natural monopolies.  Characterized by subadditive costs in production: 𝑚

𝑛

෍ 𝐶 𝑞𝑖 < ෍ 𝐶𝑞𝑗 ∀𝑚 < 𝑛 𝑖=1

𝑚

𝑗=1

𝑛

𝑤𝑖𝑡ℎ ෍ 𝑞𝑖 = ෍ 𝑞𝑗 = 𝑞 𝑖=1

𝑗=1 26

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Natural monopolies  Given such subadditivity, average costs fall with increased production.

 To avoid multiple investments, let one supplier provide the product in question and put regulations in place.  Examples include all the classic public utilities – products (gas, water, electricity, landline telephone, …) that rely on extensive networks for their delivery.

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Natural monopolies

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Natural monopolies

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Natural monopolies

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Sticky prices in oligopoly markets  Prices tend to be sticky over time.  Instead of asking how an equilibrium in one period may look, we can ask how companies behave in a stable market.

 Assume that competitors react differently to changes in prices, depending on whether another company raises or lowers its prices.  The objective is to at least maintain market shares.

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Sticky prices in oligopoly markets  Then, price increases may not be met. Companies raising prices lose market share.

 Price decreases may be met. Market shares remain the same, revenue is reduced.  Price competition does not make sense.

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Sticky prices in oligopoly markets  The demand function is kinked as a consequence. Standard demand schedules may not be useful under strategic interdependence.

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Heterogenization and monopolistic competition  Avoid price competition and bind customers more tightly – competition through non-price variables.

 Creation of a niche -> market power therein.  Niches occupied by imperfect substitutes.

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Heterogenization and monopolistic competition  The above refers to the monopolistic aspect. For the competitive character, free entry as long as profits are realized.

 Eventually, enough niches are occupied to drive prices down to average costs and profits down to zero.  If entry barriers exist, companies may be able to protect their profits, an oligopoly with heterogeneous products results.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 07: Real-World Markets

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Opportunities for strategic behavior  Companies have various non price-based options for (further) increasing control over their environment and strengthening their position. Some rooted in the character of production technology (e.g., increasing returns to scale). Others include:

 Brand strategies, niche creation  Horizontal integration  Deter entry  Vertical integration.

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Opportunities for strategic behavior  These measures tend to increase the turbulence for all other companies in a market. This remains a noticeable factor, attempts to the contrary notwithstanding.  Over time, though, some ways have been found for reducing direct interdependence among companies, e.g.:  avoidance of product homogeneity.  circumvention of price competition through various ways of instituting informal arrangements

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Opportunities for strategic behavior  Increasing action capabilities of companies has facilitated market segmentation.

 For consumers, this typically goes hand in hand with increased search and information costs.  Competition shifts to competition for markets, as opposed to competition in markets.

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Readings for the next lecture  Required reading: Chapter 8: Game theory II, in: Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies, pp. 193-226.

 For further readings visit the companion website

39

Prof. Wolfram Elsner Faculty of Business Studies and Economics iino – Institute of Institutional and Innovation Economics

Introductory Microeconomics More Formal Concepts of Game Theory and Evolutionary Game Theory

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Readings for this lecture  Mandatory reading this time: More Formal Concepts of Game Theory and Evolutionary Game Theory, in: Elsner/Heinrich/Schwardt (2014): The Microeconomics of Complex Economies, Academic Press, pp.

193-226.  The lecture and the slides are complements, not substitutes

 An additional reading list can be found at the companion website 2

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Outline  Basic games of classical GT was introduced in Chapter 2  Now: Understand the formal structure of classical decision theory and GT

 Classical GT relies on (boundedly) rational agents  Evolutionary GT allows to relax such assumptions and focuses on the dynamic performance of strategies

 First: Understand most important formal concepts of classical GT  Then: Move to evolutionary GT 3

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Basic Concepts of Game Theory  A strategic game is described by  The rules of the game  The agents of the game (here: finite number)  The strategies of the agents (here: finite number)  The information available to the agents

 Normal form game  Agents make decisions simultaneously  They do not know about the decision of the others

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Notation I Symbol

Explanation

𝑠𝑖

A pure strategy of the ith agent

𝑆𝑖

Set of all pure strategies of agent i

𝑆 = 𝑆𝑖 , 𝑖 = 1, … , 𝑛 𝑠−𝑖 = 𝑠𝑗

𝑖≠𝑗

𝑠 = 𝑠𝑖 , 𝑠−𝑖

Set of strategies of all players Strategies of all other agents than agent i Feasible configuration of strategies

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Notation II Symbol Π𝑖 𝑆 = Π𝑖 𝑠

= Π𝑖 𝑠𝑖 , 𝑠−𝑖 ∀𝑠

Explanation ∀𝑠𝑖∀𝑠−𝑖

Set of payoffs for all possible combinations of strategies

𝐺 = 𝑆𝑖 ; Π𝑖 𝑆 ; 𝐼𝑖 , 𝑖 = 1, … , 𝑛

General description for a normal form game

𝐺 = 𝑆1 , 𝑆2 ; Π1 𝑆 , Π2 𝑆 ; 𝐼1 , 𝐼2

Description for a normal form game with two players

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Matrix notation 𝐺 = 𝑆1 , 𝑆2 ; Π1 𝑆 , Π2 𝑆 ; 𝐼1 , 𝐼2 Strategy 1

Strategy 2

Π𝐵 (𝑠𝐴1 , 𝑠𝐵1 )

Π𝐵 (𝑠𝐴1 , 𝑠𝐵2 )

Strategy 1 Π𝐴 (𝑠𝐴1 , 𝑠𝐵1 )

Π𝐴 (𝑠𝐴1 , 𝑠𝐵2 )

Π𝐵 (𝑠𝐴2 , 𝑠𝐵1 )

Strategy 2

Π𝐴 (𝑠𝐴2 , 𝑠𝐵1 )

Π𝐵 (𝑠𝐴2 , 𝑠𝐵2 ) Π𝐴 (𝑠𝐴2 , 𝑠𝐵2 )

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Assumptions regarding the agents  In order to predict the outcome of an interaction, assumptions regarding the agents’ behavior must be made

 Utility maximizing agents with well-defined preference orderings, i.e. for any outcomes 𝑎 and 𝑏 the following holds:  Completeness:

𝑎 ≻ 𝑏 𝑜𝑟 𝑏 ≻ 𝑎 𝑜𝑟 𝑎 ~ 𝑏

 Reflexivity:

𝑎≻𝑏 ⟺𝑏≺𝑎

 Transitivity:

𝑎≻𝑏 ‫𝑐≻𝑎⟹ 𝑐≻𝑏 ٿ‬

 Common Knowledge of Rationality  Every agent knows that all agents are rational, that all all other agents also know that all agents are rational, that they also are aware that all agents know that all are rational, etc.

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A Preliminary Consideration: NonInteractive Concepts, Decision Theory  Consider a non-interactive decision situation (to bring or not to bring an umbrella) under risk (the state of the world is unknown, it may or may not rain).

S t a t e of the W o r l d

Rain P l a y e r

Bring Umbrella

No Rain

4

5

No Umbrella -10

10

 How will the player decide?  The obvious difference to game theory is that the State of the World is unknown and unconstrained by rationality 9

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Non-Interactive Concepts, Decision Theory  An optimistic concept: Maximax

 An pessimistic concept: Minimax

 Find the best possible payoff for every strategy and maximize

 Find the worst possible payoff for every strategy and maximize

S t a t e of the W o r l d

R P l a y e r

U

4

No U -10

No R 5 10

S t a t e of the W o r l d

R

No R

Min.

4

5

4

No U -10

10

-10

Max. 5 10

P l a y e r

U

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Non-Interactive Concepts, Decision Theory contd.  An opportunity cost based concept: Savage’s Minimax Regret  Construct the regret matrix (how much would the player regret this decision in this state of the world compared to the other decision she could have taken), find the highest regret for every strategy and minimize S t a t e of the W o r l d

R P l a y e r

U

4

No U -10

Regret

No R 5 10

S t r a t .

U No U

R

No R

Max.

0

5

5

6

0

6 11

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Non-Interactive Concepts, Decision Theory contd.  There are also parametric decision criteria (i.e. criteria that assign a priori probabilities to the states of the world) including, e.g. the Laplace and the Hurwicz criterion  Of course, all decision theory concepts may also be used to make predictions for strategic games (these predictions would be valid even without CKR, i.e. even if agents assume their opponents to be irrational)  However, for many problem structures, these concepts fail  GT allows for more advanced prediction methods by taking the agents’ capability to make consistent, systematic, and rational decisions into account 12

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How to solve the game  Players are rational, i.e. payoff maximizing and neither benevolent nor envious

 They know that all other players think the same way  Expectations about the behavior of the other players can be formed  The players then play the strategy giving them the best outcome (highest payoff) given their expectations about the reasoning of the other players

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Dominance of strategies  A strategy 𝑠𝑖∗ ∈ 𝑆𝑖 dominates a strategy 𝑠𝑖~ ∈ 𝑆𝑖 iff Π𝑖 𝑠𝑖∗ , 𝑠−𝑖 ≥ Π𝑖 𝑠𝑖~ , 𝑠−𝑖 ∀𝑠−𝑖 and ∃𝑠−𝑖 : Π𝑖 𝑠𝑖∗ , 𝑠−𝑖 > Π𝑖 (𝑠𝑖~ , 𝑠−𝑖 )

 The latter strategy 𝑠𝑖~ is said to be dominated by the first

 If a strategy gives strictly higher payoffs regardless of the choice of the opponent, it is said to strictly dominate the other strategy  Rational players never play a strictly dominated strategy

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SESDS  One way to predict the outcome of a game is therefore the successive elimination of strictly dominated strategies (SESDS)  SESDS does not require any assumptions about opponent’s behavior Strat. 1

Strategy 1 4

4

Strategy 2 2

2

Strat. 2

2

Strategy 1 4

4

0

Strategy 2 2

2

2

0

Strat. 1

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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SESDS  SESDS does not always yield a solution: Player B

Strategy 1 P l a y e r A

Strategy 2

2

Strategy 1

2

0

0 0

Strategy 2 0

 There are no strictly dominated strategies to remove

2 2

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Nash Equilibria  SESDS does not always yield a solution:  But: Thanks to CKR, agents can form expectations about the choices made by other agents

 They can choose the best possible response to the expected choice of their opponents  Due to CKR, the others will do so as well

 The resulting situation is a combination of mutual best responses  This situation is called Nash Equilibrium (NE) 17

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Nash Equilibria  The formal definition is as follows:

 If a game is solvable via SESDS the solution is also a NE  But: Not all NE can be explored via SESDS  Also, There are games that do not have a NE in pure strategies at all

 Therefore: Introduce the distinction between pure and mixed strategies 18

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Mixed Strategies Player B

 How do you play RockPaper-Scissors?  To always play the same strategy is a bad idea  Mixed strategies capture the idea of playing different pure strategies with some probability

Rock P l a y e r

Paper 0

Rock 0

-1

1

A

Scissors

1

-1

Paper

-1 1

0 0

1 -1

Scissors

1 -1

-1 1

0 0 19

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Nash Equilibrium and Mixed Strategies  A mixed strategy σi for player i is a vector in which every pure strategy is associated with a probability

𝑝  For the two strategy case: 𝜎𝑖 = 1 − 𝑝  The Nash Equilibrium is defined as a configuration of mixed strategies for the n players such that

∗ ∗ Π𝑖 𝜎𝑖∗ , 𝜎−𝑖 ≥ Π𝑖 𝜎𝑖 , 𝜎−𝑖

∀𝜎𝑖 ∀𝑖 20

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Nash Equilibrium and Mixed Strategies  Every finite n-person normal-form game with a finite number of strategies for each player has at least one NE in pure or mixed strategies (proven by John Nash in 1950)  Thanks to the NE and the concept of mixed strategies, all finite n-person normal-form games with a finite number of strategies can be solved in theory  How can we compute the NE in mixed strategies?

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Computation of NE in Mixed Strategies Player B

P l a y e r A

Dove

Hawk 2

Dove 2

3 1

1

Hawk

3

0 0

 Define the 𝑥 × 𝑦–matrix 𝒜 with the payoffs for the first player: 2 1 𝒜= 3 0  Expected payoffs are given by: Π1 = 𝜎1𝑡 × 𝒜 × 𝜎2

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Computation of NE in Mixed Strategies Π1 = 𝜎1𝑡 × 𝒜 × 𝜎2 Π1 = 𝑝

1−𝑝

2 3

Π1 = −2𝑝𝑞 + 3𝑞 + 𝑝 𝜕Π1 = −2𝑞 +1 𝜕𝑝

𝜕Π1 = −2𝑞 + 1 = 0 𝜕𝑝

q = 0.5

→ 𝑝 = 0.5

1 0

𝑞 1−𝑞

 This procedure can be employed for every (symmetric) game  Always take the first derivative of the expected payoff function with respect to the strategy parameter of the same player (𝑝 for 1, 𝑞 for 2) and solve it for 0  For symmetric games, the equations are identical for both (with 𝑝 and 𝑞 exchanged) and thus have to be solved only once 23

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Reaction Functions  𝑝 and 𝑞 can also be shown as the reaction functions p(𝑞) and 𝑞(𝑝) of the players’ reaction to the opponent’s choice

 Intersections denote NE (the current Hawk-Dove game has 3) Note: The caption in Figure 8.12 in Chapter 8 is incorrect (same Figure as on this slide; the Figure depicts Hawk-Dove, not Matching Pennies) 24

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Reaction Functions  Mixed strategy NE are where players are indifferent between

both pure (and mixed) strategies (i.e. where

𝜕Π1 𝜕Π =0, 2 = 0) 𝜕𝑝 𝜕𝑞

 Mixed strategy NE may seem oddly unstable but this is not necessarily the case (see Evolutionary Game Theory below).  Also, under CKR it is rational to deliberately choose mixed NE strategies in order to facilitate the emergence of the equilibrium and to avoid being exploited (Aumann’s defense)

 This is especially true if there is no pure strategy NE; consider the Matching Pennies game (or the Rock-Paper-Scissors game above) 25

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Mixed Strategy NE Player 2

P l a y e r

1

Heads

Tails

1

Heads -1

-1 1

-1

Tails 1

1 -1

 In zero-sum games (like this Matching Pennies game), unequal payoff denotes one player being exploited by the other 26

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Extensions of the Nash Equilibrium  Many extensions to the Nash Equilibrium have been developed

 Some of these, the ones for extensive or repeated (subgame perfect Nash Equilibrium) as well as evolutionary games (ESS, … ), will be presented below  There are also refinements that ensure that the equilibrium is still valid under stochastic perturbations (the ‘trembling hand’), e.g. Selten’s Trembling Hand Perfect Equilibrium, Myerson’s Proper Equilibrium (see textbook), or Harsanyi’s Bayesian Nash Equilibrium.

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Extensive Form Games  In normal-form games, the players make their decisions simultaneously but this is not the case for other types of games

 For games in sequential form we use a new notation, the extensive-form notation  Note that this is necessary only if the agents have full information about the decisions made by the previous agents  Otherwise the game is equivalent to a normal-form game

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Extensive-Form Notation Player 1

Player 2

Result 1

Result 2

 Depending on the choice of the first player, the second player faces a different decision situation Player 2

Result 3

Result 4

 It is therefore convenient to define complete strategies for the players  A complete strategy gives each player an instruction for all possible situations

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Complete Strategies Player 1

 For an extensive game 𝐺 𝐸 let 𝑉 be a set containing all possible states of 𝐺 𝐸

𝑉1 C

Player 2

D

𝑉2

 𝑉𝐴 will contain all possible situations in which player A can possibly make a decision

𝑉3

C

D

C

D

𝑉4

𝑉5

𝑉6

𝑉7

(1,1)

(−1,2)

(2, −1)

(0,0)

 A complete strategy for player A gives a an instruction for any element in 𝑉𝐴

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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How to solve extensive games  Analytical derivation of NE cumbersome  Thanks to CKR we can rely on backward induction

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Backward Induction Player 1

 The reasoning for player 1 is as follows:

𝑉1 C

 What would player 2 do in situation 𝑉2 ?

D

 Choose D

Player 2

𝑉2 C

 Result is would be 𝑉5

𝑉3 D

C

D

 What would player 2 do in situation 𝑉3 ?

 Choose D  Result would be 𝑉7

𝑉4

𝑉5

𝑉6

𝑉7

(1,1)

(−1,2)

(2, −1)

(0,0)

 Since 1 prefers 𝑉7 (via 𝑉3 ) to 𝑉5 (via 𝑉2 ), she chooses D (which leads to 𝑉3 and then 𝑉7 ) 32

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How to solve extensive games  Analytical derivation of NE cumbersome  Thanks to CKR we can rely on backward induction  Advantage: All NE found with backward induction are also subgame perfect  A NE is subgame perfect if it is also a NE of all the subgames that contain the NE

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Repeated Games and Supergames  Up to this point every game we have investigated was played only once

 The solutions we have obtained are called one-shot solutions  Now we consider repeated games  A supergame 𝒢 is a sequence of repetitions of the normal-form game 𝐺 that is infinite from the players’ point of view, i.e. either really infinite or indefinite (with stochastic probability to end in each period)

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Solving Supergames  If 𝒢 is finite it can be expressed as an extensive game and solved via backward induction  If 𝒢 is infinite or indefinite, there is no last period, so backward induction will not work  Still, strategies can be characterized as complete (if they contain instructions for every possible situation) or not complete and equilibria can be characterized as subgame perfect or not  If both players follow complete strategies, all choices are predictable and the payoffs are thus known in advance (to rational players and observers); this allows to calculate a present value to be used in deciding between strategies in advance 35

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Present Values  To obtain the present value for a future payoff 𝑎 one uses a discount parameter 𝛿 (with 0 ≤ 𝛿 ≤ 1) which denotes the player’s valuation of future payoffs (the present value in time 0 of a payoff 𝑎 in time 0 is 𝑎, … of the same payoff in time 1 it is 𝛿𝑎, in time 2, 𝛿 2 𝑎, etc.)  The present value for an infinite sequence of payoff 𝑎 is 𝑎 therefore given as Π = 𝑎 + 𝛿𝑎 + 𝛿 2 𝑎 + 𝛿 3 𝑎 +... = 1−𝛿

 Agents make the decisions about their strategy plan based on the present values of the expected payoffs from the different strategy plans 36

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Strategy Plans  Strategy plans can be very simple, e.g. always to play one strategy  In the PD, ALL-D is a strategy plan according to which the player always defects

 Strategy plans can also be more complex, esp. when the choice of the player depends on the decisions made by other players in the past  In supergames, the players usually have a memory that allows expectation formation (which is known to the player so that the strategy plan can make use of the fact)

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Folk Theorem and Trigger Strategies  Consider a supergame based on a social dilemma (a generalized PD), i.e.  a cooperative solution exists which constitutes the social optimum and a Pareto optimum of the underlying game  but the cooperative strategy is exploitable (i.e. the cooperative solution is not a NE)

 The folk theorem states that it is possible to reach the cooperative solution in the supergame  This is achieved by using trigger strategies, positing a credible thread of punishment if the opponent should deviate from the cooperative strategy (Rubinstein’s proof).

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Trigger Strategies and Rubinstein’s Proof of the Folk Theorem  Define a strategy that guarantees a minimum of payoff for the opponent as the minmax strategy (analogous to maximin, only that her payoff is minimized by her opponents choice not maximized by her own choice)  If the opponent’s payoff resulting from using the minmax strategy on her is less then her payoff from cooperation, the minmax strategy can serve as a threat of punishment for deviating from cooperation  The threat is said to be credible if the punishing player receives a higher or equal payoff from playing the minmax strategy than from allowing herself to be exploited (more generally: if the minmax strategy is not strictly dominated) 𝒢

 Credible threats can be used to construct a trigger strategy 𝑠𝑡𝑟𝑖𝑔𝑔𝑒𝑟 39

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Trigger Strategies and Rubinstein’s Proof of the Folk Theorem  A strategy profile can include the instruction to employ 𝑠𝑚𝑖𝑛𝑚𝑎𝑥 always if a specific expectation is not met  By doing so, mutually beneficial agreements that do not constitute a NE in the one-shot case can be enforced Example: The Prisoner’s Dilemma and tit-for-tat 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑒, 𝑖𝑓 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑝𝑙𝑎𝑦𝑒𝑟 ℎ𝑎𝑠 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑒𝑑 𝑙𝑎𝑠𝑡 𝑟𝑜𝑢𝑛𝑑 𝒢 𝒢 𝑠𝑡𝑟𝑖𝑔𝑔𝑒𝑟 = 𝑠𝑇𝐹𝑇 = ቊ 𝑑𝑒𝑓𝑒𝑐𝑡, 𝑖𝑓 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑝𝑙𝑎𝑦𝑒𝑟 ℎ𝑎𝑠 𝑛𝑜𝑡 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑒𝑑 𝑙𝑎𝑠𝑡 𝑟𝑜𝑢𝑛𝑑

To defect is the minmax strategy that serves as credible threat, all players playing this trigger strategy is a NE in the repeated PD  Trigger strategies can be constructed in different ways; they could for instance also – less forgivingly – punish forever after one defection 40

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Evolutionary Game Theory  Allows to conveniently construct microfounded economic models, thus closing gaps between micro, meso, and macro level  In Evolutionary Game Theory (EGT) we consider populations of agents, all continuously playing one and the same strategy according to their type  Agents are matched randomly to play again and again the same underlying game (which must be symmetric so that the positions of row player and column player are exchangable)  The population shares of these types (strategies) develop according to their performance in those games 41

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Evolutionary Game Theory  The changing share of types may be seen as reproduction with the offspring continuing the same strategies or as poorly performing agents consciously changing their strategy (type)  The most important solution techniques are 1. Analysis of evolutionary stability (evolutionary stable strategies, ESS) 2. Replicator dynamics 3. Simulation (see textbook Chapter 9)

 The players are matched randomly

 When they meet, they play the underlying game using their predetermined strategy  Composition of the population changes according to the agent’s performance in playing the underlying game 42

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Evolutionary Stability  NE: Combination of mutual best answers  In the population context we consider only strategies:  A strategy is evolutionary stable if a population dominated by it is not invadable by any other strategy  If a population is dominated by an ESS, the situation will remain stable

 What does “not invadable” mean?

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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The principle of evolutionary stability Symbol

Explanation

𝒫𝒢𝒱

An evolutionary population setting

𝐺𝒢𝒱

The underlying (symmetric) one-shot normal form game in this setting

𝒜

The payoff matrix

Π𝜎1 /𝜎2 = 𝜎1𝑇 𝒜 𝜎2

Expected payoff of the first strategy against the second

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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The principle of evolutionary stability  Consider a population of agents playing 𝜎 ∗  Now consider a very small group of players entering the group and playing 𝜎 ~ ≠ 𝜎 ∗

 If the new strategy yields better payoffs than the old one, the share of players playing 𝜎 ~ will increase  In this case the 𝜎 ~ has invaded 𝜎 ∗ and 𝜎 ∗ cannot be said to be an ESS

 We will now formalize the concept of evolutionary stability 45

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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How to test for evolutionary stability  Let 𝜀 be the (arbitrarily small) share of the invading group playing 𝜎 ~  A share of (1 − 𝜀) is therefore playing strategy 𝜎 ∗  𝜎 ∗ is an ESS if it yields a higher expected payoff than 𝜎 ~

 Formally: 𝜎 ∗𝑇 𝒜(1 − 𝜀)𝜎 ∗ + 𝜎 ∗𝑇 𝒜𝜀𝜎 ~ > 𝜎 ~𝑇 𝒜(1 − 𝜀)𝜎 ∗ + 𝜎 ~𝑇 𝒜𝜀𝜎 ~  This rather complicated formula can conveniently be tested using two simple conditions,  The first is constructed from letting 𝜀=0 for which the above inequality must hold at least weakly, with ≥ (1st condition)  The second results additionally for the case that the first condition holds with equality (2nd condition).

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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How to test for evolutionary stability (Weak) First condition of evolutionary stability:

∗𝑇

∗

𝜎 𝒜𝜎 ≥ 𝜎

~𝑇

𝒜𝜎

∗ Does not hold

𝝈∗ is not evolutionary stable

Holds

Strict first condition of evolutionary stability: 𝜎 ∗𝑇 𝒜𝜎 ∗ > 𝜎 ~𝑇 𝒜𝜎 ∗

Does not hold

𝜎 ∗𝑇 𝒜𝜎 ~ > 𝜎 ~𝑇 𝒜𝜎 ~ Second condition of evolutionary stability

Holds Holds

𝝈∗ is evolutionary stable 47

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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How to test for evolutionary stability  Test considering which candidate strategies: pure and mixed NE strategies, since non-NE-strategies are by definition not best answers to themselves (and this is required by the 1 st condition of ESS).  Test against which invading strategies: Against all competing pure strategies (in the 2-strategy case – since mixed strategies are then only linear combinations combining properties of only two pure strategies)

 In cases with more than 2 strategies, evolutionary stability must be tested against mixed strategies as well.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Evolutionary Stability: Example  Consider as an example the Hawk Dove game above.

2 1  We have game matrix 𝒜 = and three NE strategies, 3 0 0 1 0.5 H= , D= , and M = 1 0 0.5  Test H against D, 1st condition: 2 1 0 ≥ 1 0 1 3 0 1 0≱1

0

2 1 3 0

0 1

 The condition does not hold, H is therefore not an ESS because it can be invaded by D-players 49

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Evolutionary Stability: Example contd.  Test D against H, 1st condition: 2 1 1 ≥ 0 1 0 3 0 0 2≱3

1

2 1 3 0

1 0

 The condition does not hold, D is therefore not an ESS because it can be invaded by H-players

 M must be tested against both D and H; Test M against H, 1st condition: 2 1 0.5 2 1 0.5 ≥ 0 1 0.5 0.5 3 0 0.5 3 0 0.5 1.5 ≥ 1.5  Condition holds with equality, test of 2nd condition necessary 50

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Evolutionary Stability: Example contd.  Test M against H, 2nd condition: 2 1 0 > 0 1 0.5 0.5 3 0 1 0.5 > 0

2 3

1 0

0 1

2 3

1 0

0.5 0.5

 Condition holds, H cannot invade M  Test M against D, 1st condition: 2 1 0.5 ≥ 1 0 0.5 0.5 3 0 0.5 1.5 ≥ 1.5

 Condition holds with equality, test of 2nd condition necessary

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Evolutionary Stability: Example contd.  Test M against D, 2nd condition: 2 1 1 > 1 0 0.5 0.5 3 0 0 2.5 > 2

2 3

1 0

1 0

 Condition holds, D cannot invade M  Since all mixed strategies are linear combinations of D and H, none of those (except M itself) can be able to invade M  M is therefore this game’s only ESS

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Replicator Dynamics  The focus is not on single strategies but on stable compositions of populations  The population is modeled as a dynamical system  A dynamical system describes how state variables change over time  Consider a system with z state variables:

𝜃𝑡 =

𝜃1,𝑡 𝜃2,𝑡 ⋮ 𝜃𝑧,𝑡

= 𝜃𝑖,𝑡 𝑖=1,…,𝑧

The development path depends on the initial values and the development equations 53

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Replicator Dynamics  Here the state variables represent the shares of specific agents playing a certain strategy (except for the last share which is the difference to 100%, thus determined by the others)  The shares might change according to a difference or differential equation:

𝜃𝑡+1 = 𝐹𝐷~ (𝜃𝑡 )

𝑑𝜃(𝑡) ~ = 𝐹𝑑 (𝜃𝑡 ) 𝑑𝑡 54

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Replicator Dynamics Symbol

Explanation

𝑓𝑖,𝑡

The evolutionary fitness of agent i at time t

𝜙 = ෍ 𝜃𝑖,𝑡 𝑓𝑖,𝑡

Average fitness in the population

𝑖

 For the evolutionary performance of the agents it is important how their fitness compares to the average fitness

𝜃𝑖,𝑡+1 = 𝐹𝐷~ (𝜃𝑖,𝑡 , 𝑓𝑖,𝑡 , 𝜙𝑡 )

𝑑𝜃𝑖 (𝑡) = 𝐹𝑑~ (𝜃𝑖 𝑡 , 𝑓𝑖 𝑡 , 𝜙𝑖 (𝑡)) 𝑑𝑡

 One can now analyze the dynamical system and search for stable equilibria 55

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Replicator Dynamics  Replicator Dynamics requires an explicit assumption on the form and speed of the replicator (this is implicit in ESS)

 Typical forms set the dynamic as proportional to the relation of individual and average fitness with the most common canonical forms being

𝜃𝑖,𝑡+1 = 𝜃𝑖,𝑡 (𝑓𝑖,𝑡 /𝜙𝑡 )

𝑑𝜃𝑖 (𝑡) = 𝜃𝑖 𝑡 (𝑓𝑖 𝑡 − 𝜙(𝑡)) 𝑑𝑡

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How to determine (stable) equilibria  The conditions for equilibria are: ∗ 𝜃𝑖,𝑡+1 = 𝐹𝐷~ 𝜃𝑖𝑡 , 𝑓𝑖,𝑡 , 𝜙𝑡 = 𝜃𝑡∗

𝑑𝜃𝑖 (𝑡) = 𝐹𝑑~ 𝜃𝑖 𝑡 , 𝑓𝑖 𝑡 , 𝜙𝑖 𝑡 𝑑𝑡

=0

 In order to test for the stability of the equilibria one calculates the eigenvalues of the development equation for the equilibria (see Chapters 10 and 11 of the textbook for details)  For difference equations, the equilibrium is stable if all eigenvalues have an absolute value smaller than unity

 For differential equations, the equilibrium is stable if all eigenvalues are negative

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Replicator Dynamics: Example  Consider again the Hawk-Dove game above with the canonical replicator function

𝑑𝜃𝑖 (𝑡) = 𝜃𝑖 𝑡 (𝑓𝑖 𝑡 𝑑𝑡

− 𝜙(𝑡)), call

the pure strategies 𝑥𝑖 and assume the 𝑓𝑖 = Π𝑖 = 𝑥𝑖𝑇 𝒜 𝜃 (at any point in time, leaving out the (𝑡) for convenience)  We further have 𝒜 =

and therefore also 𝑓1 = 1 𝑓2 = 0

2 1 1 0 , 𝑥1 = 𝐷 = , 𝑥2 = 𝐻 = , 3 0 0 1 2 3

0 1

1 0 2 3

𝜃1 = 1 + 𝜃1 1 − 𝜃1 𝜃1 1 = 3𝜃1 0 1 − 𝜃1 58

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Replicator Dynamics: Example contd.  Since 𝜙 = σ𝑖 𝜃𝑖 𝑓𝑖 =𝜃1 𝑓1 +(1 − 𝜃1 )𝑓2 , we can rearrange the replicator equation 𝑑𝜃1 = 𝜃1 (𝑓1 − 𝜙) 𝑑𝑡 𝑑𝜃1 = 𝜃1 (𝑓1 − 𝜃1 𝑓1 − (1 − 𝜃1 )𝑓2 )=𝜃1 ((1 − 𝜃1 )𝑓1 − (1 − 𝜃1 )𝑓2 ) 𝑑𝑡 𝑑𝜃1 = 𝜃1 (1 − 𝜃1 )(𝑓1 − 𝑓2 ) 𝑑𝑡

 Substituting 𝑓1 and 𝑓2 yields 𝑑𝜃1 = 𝜃1 1 − 𝜃1 1 + 𝜃1 − 3𝜃1 = 𝜃1 1 − 𝜃1 1 − 2𝜃1 𝑑𝑡 59

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Replicator Dynamics: Example contd. 𝑑𝜃

 Applying the equilibrium condition 1 = 0 yields three fixed 𝑑𝑡 points 𝜃1,1 = 0, 𝜃1,2 = 1, and 𝜃1,3 = 0.5  To assess the stability, we bring the replicator equation into polynomial form 𝑑𝜃1 = 2𝜃13 − 3𝜃12 + 𝜃1 𝑑𝑡

and obtain the only element of the system’s Jacobian 𝜕(𝑑𝜃𝑖 /𝑑𝑡) = 6𝜃12 − 6𝜃1 + 1 𝜕𝜃𝑖 the linearization of which is the (dominant) eigenvalue.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Replicator Dynamics: Example contd.  We obtain the three eigenvalues for the three equilibria 𝜆 𝜃1,1 = 1 (i.e. 𝜃1,1 is unstable), 𝜆 𝜃1,2 = 1 (i.e. 𝜃1,2 is also unstable), 𝜆 𝜃1,3 = −0.5 (i.e. 𝜃1,3 is stable)  The result is identical to that obtained from analysis of ESS: a composition of the population (or equivalently of each individual strategy) of both strategies with equal shares is the only stable equilibrium or fixed point. The pure strategy equilibria / fixed points exist but are unstable.

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Summary  This chapter  gave a comprehensive introduction to formal game theory including different notations (matrix, formal, extensive)

 introduced advanced solution concepts of decision theory and game theory (including SESDS and Nash Equilibria in pure and mixed strategies, backward induction, …)  covered non-normal-form games (including extensive games, repeated games)  gave a formal introduction to Evolutionary Game Theory (ESS and replicator dynamics)

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Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Readings for the next lecture  Compulsory reading: Introduction to Simulation and Agent-Based Modeling, in: Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies, pp. 227-247.  For further readings visit the companion website

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