The Microeconomics of Complex Economies Evolutionary, Institutional, Neoclassical, and Complexity Pe

SOLUTION MAMUAL

The

Microeconomics of Complex Economies.

Evolutionary, Institutional, Neoclassical, and Complexity Perspectives

by Wolfram Elsner, Torsten Heinrich, and Henning Schwardt

Amsterdam, Boston, Heidelberg, London, New York, Oxford, Paris, San Diego, San Francisco, Singapore, Sydney, Tokyo: Academic Press, 2015

Instructor’s Manual

2

CONTENTS

1. The Structure of the Textbook / The Ideal Sequence of the Parts and Chapters

p. 3

2. Sample Syllabi / Roadmaps

p. 5

3. Instructor Powerpoint Presentations

p. 6

4. Solution Keys for Chapter Exercises

p. 7



Solutions Chapter 2

p. 8



Solutions Chapter 3

p. 11



Solutions Chapter 4

p. 17



Solutions Chapter 5

p. 27



Solutions Chapter 6

p. 34



Solutions Chapter 7

p. 39



Solutions Chapter 8

p. 49



Solutions Chapter 9

p. 62



Solutions Chapter 10

p. 67



Solutions Chapter 11

p. 71



Solutions Chapter 12

p. 77



Solutions Chapter 15

p. 87

5. A Note on Didactics for Teaching Computer Simulation (Chapter 9)

p. 93

6. Competing Textbooks – A Brief Comparison

p. 95

7. The Math Required – A Short Overview of Mathematical Prerequisites of Working with this Textbook

p. 99

3

1.

THE STRUCTURE OF THE TEXTBOOK / THE IDEAL SEQUENCE OFTHE PARTS AND CHAPTERS

The structure of the textbook is given in Figure X.1 most importantly indicating which chapters build heavily on which preceding chapters. See the book‘s Preface for more detail.

4

PART I

Ch 1: Basic Ideas Ch 2: Game Theory I Ch 3: Problem Structures Ch 4: Real World

PART II

Ch 5: Neoclassical GET Ch 6: Critique of GET Ch 7: Real World Markets

PART III Ch 8: Game Theory II

PART IV

Ch 9: Simulation Ch 10: Complexity I

Ch 11: Complexity II

Ch 15: Open Source

Ch 12: History of Thought Ch 13: Core Models Ch 14: Meso-Economics

Ch 16: Innovation Systems Ch 17: Policy

PART V

Ch 18: Knowing, Acting, Governance-Epistemology Figure X.1 The structure of the textbook and the interdependence of parts and chapters.

Figure X.1 The structure of the textbook and the interdependence of parts and chapters.

5

2.

SAMPLE SYLLABI / ROADMAPS

Sample Syllabi are given in Table X.1. See the book‘s Didactics part for more detail.

This textbook can be used – as a required primary or a complementary reading – for standard Microeconomics courses at the intermediate undergraduate level, for both a one-semester and a two-semester course. It can also – in parts – be presented to graduate/advanced audiences and even – in its introductory part – used at the introductory level. Furthermore, it can be used for elective courses in Game Theory and/or Industrial Organization. For all this, see some sample syllabi below. Finally, it could be assigned as a companion volume to provide more theoretical, more methodological or more applied chapter material in certain specialized intermediate or advanced economics courses with some complexity perspective.

Course Level:

Introductory

Undergraduate/Intermediate

Graduate/Advanced

Chapters to be used (1st semester):

1 2 3 4 (12)

One-semester course, focus on theory and methods:

6 (9) (13) 10 11 (14)

5 7 8 9 Two-semester course: Second semester, focus on History of Thought, important models, and applications:

Chapters to be used (2nd semester):

12 13 14 15 16 17 18 Elective Course in Game Theory and/or Industrial Economics

Chapters to be used:

2, 7, 8, (9), (11), (13), 15, 16.

Elective Course in Complexity Economics (Advanced Level) Chapters to be used:

(3), 6, 9, 10, 11, (13), (14), (15)

Table X.1: Sample syllabi / roadmaps. (Note: Chapter numbers in brackets indicate non-core chapters that may be skipped.)

6

3.

INSTRUCTOR POWERPOINT PRESENTATIONS

Lecture slides corresponding to several chapters of the textbook are available from the textbook companion website http://booksite.elsevier.com/9780124115859 in pdf format. An editable MS Powerpoint version – which allows both modifications on the part of the instructor and the inclusion of notes for her lecture – is available on the instructor‘s page.

It is the philosophy on which the presentations made available here are based that instructors should be able to change their lecture slides according to their needs and specific focuses.

Note that the presentations are rather minimalistic renderings of the content presented in the textbook as instructors will always find it easier to add material to a presentation than to cut material away.

7

4.

SOLUTION KEYS FOR CHAPTER EXERCISES

Note:There are no exercises for Chapters 1, 13, 14, 16, 17, and 18.

Solutions Chapter 1 (No exercises for Chapter 1.)

8 Solutions Chapter 2

1.: Identify: a) the Pareto optima in the anti-coordination game (Figure 2.1) and in the social optimum game (Figure 2.2) b) the best answer strategies and Nash equilibria in the prisoners‘ dilemma (Figure 2.3) and the social optimum game (Figure 2.2) c) whether there are strictly dominant strategies in the social optimum game (Figure 2.2)?

Solution key: a) Pareto Optima in the (Anti-Coordination) game in Figure 2.1 are Wait-Pass (lower left) and Pass-Wait (upper right). The (Social Optimum) game in Figure 2.2 has its only Pareto Optimum in the upper left (Strategy 1 – Strategy 1). b) In the Prisoners‘ Dilemcma (Figure 2.3), Strategy 2 is the best answer to both Strategy 1 and Strategy 2 for both players. The combination of mutual best answer strategies is by definition a Nash Equilibrium. The only Nash Equilibrium in this game is thus Strategy 2 – Strategy 2 (lower right).

In the social optimum game (Figure 2.2), Strategy 1 is the best answer to both Strategy 1 and Strategy 2 for both players. The combination of mutual best answer strategies is by definition a Nash equilibrium. The only Nash equilibrium in this game is thus Strategy 1 – Strategy 1 (upper left).

c) Strategy 1 is strictly dominant in the (Social Optimum) game in Figure 2.2 for both players.

9 2.: Consider the following game and identify Pareto optima, best answer strategies, and Nash equilibria. Are there strictly dominant strategies?

Player A

Strategy 1 Strategy 2

Player B Strategy Strategy 1 2 4 3 2 3 1 2 1 4

Solution key: The outcome Strategy 2 – Strategy 1 (lower left, crossed in the matrix below) is Pareto dominated by each of the other three possible outcomes; it is not a Pareto Optimum. The other three outcomes (Strategy 1 – Strategy 1, Strategy 1 – Strategy 2, Strategy 2 – Strategy 2) are not Pareto dominated; each of them is a Pareto Optimum.

Strategy 1 is a best answer to Strategy 1 for both players while Strategy 2 is also a best answer to Strategy 1 for both players (underlined in the matrix below). Neither strategy is, however, a best answer to the respective other strategy. The combinations of mutual best answer strategies are Nash Equilibria, here Strategy 1 – Strategy 1 (upper left) and Strategy 2 – Strategy 2 (lower right).

From its structure it follows that the game is a coordination game:

Player A

Strategy 1 Strategy 2

Player B Strategy Strategy 1 2 4 3 2 3 1 2 1 4

10 3.: How many Nash equilibria, how many best answer strategies, and how many Pareto optima does a 2-person 2-strategy normal-form game have at least? How many (of any of those concepts) does it have at most? Consider as an example the following game.

Player A

Strategy 1 Strategy 2

Player B Strategy Strategy 1 2 0 0 0 0 0 0 0 0

Solution key: In this game, all strategies are best answers to all strategies for all players (since all outcomes are equivalent); thus, each possible outcome is a Nash Equilibrium. None of the four possible outcomes is Pareto dominated. There are consequently at most 4 Nash Equilibria and at most 4 Pareto Optima in a 2-person-2strategy normal form game (and all strategies, i.e. 2+2, 2 for each player, may be best answers; they may also be best answer strategies to each of the opponent‘s strategy choices).

Since every outcome that is not a Pareto Optimum must be Pareto dominated by at least one other possible outcome, there must be at least one outcome in each normal-form game that is not Pareto dominated, thus a Pareto Optimum. Consequently, a 2-person-2strategy normal form game has at least one Pareto Optimum (examples include the prisoners‘ dilemma in Figure 2.3 of the chapter and the social optimum game in Figure 2.2 of the chapter).

In a 2-person normal form game there is for every strategy of every player at least one best answer (4 if both players have 2 strategies). Strategies may, however be best answers to multiple strategies of the opponent (consequently at least 2 strategies, one for each player, are indeed best answer strategies). However, there may be games where none of the best-answer strategies are mutual best answers and where therefore is no Nash Equilibrium in pure strategies (an example being the matching pennies game in Figure 8.7 in Chapter 8). (As shown in Chapter 8, there is, however, at least one Nash Equilibrium in pure or mixed strategies for every 2-person-2-strategy normal form game.)

11 Solutions Chapter 3

1.:

Briefly describe the range of problem structures that agents can face in interdependent decision situations.

Solution key: Problem structures in interdependent decision situations can be made accessible by use of simple games, in a first approximation, at least.These range between two cases limiting the spectrum of problem structures: the social optimum game and the social dilemma game. In both, agents have a unique optimal strategy at their disposition. In the social optimum game, choice of the strictly dominant strategy by every agent results in the sole Pareto-optimal outcome of the situation; in the dilemma game, choice of the strictly dominant strategy results in the sole non-Pareto-optimal outcome.

A number of coordination problems can be positioned in between these two.

Coordination problems permit the realization of Pareto-optimal outcomes if agents coordinate their strategy choices. We can distinguish these problems by what we have called a ‗degree of conflict‘ between the agents. Even though Pareto-optimal outcomes are in reach, it may be that agents prefer the realization of different outcomes as their individual results differ between them.

2.:

What distinguishes social optimum and social dilemma situations from coordination problems?

Solution key: In social optimum situations as well as in social dilemma situations, agents have one strictly dominant course of action to follow. In the social dilemma situation,recourse to the strictly dominated strategy would allow a Pareto-improvement, but the presumed rationality of agents, and the set-up within which they are placed, prevents them from making that strategy choice. Hence, in both games, given the basic assumptions formulated, the outcome (barring mistakes) is clear.

In coordination problems, this is not the case; there are no strictly dominant strategies for the respective agents. There are a number of Nash Equilibria, and which one constitutes the eventual outcome cannot be predicted given the basic game-theoretic assumptions used for setting the structure up.

3.:

What problem that agents may face beyond a necessity for assurance does a stag-hunt game illustrate?

Solution key: There is also the issue of risk aversion to deal with. Even though some external agent assures them that the strategy leading to the Pareto-optimal outcome will be chosen by all others, agents may still have doubts regarding the others. This can refer to others‘ ability to avoid mistakes, for instance, or a possible change of heart by other agents,

12 based on an assessment of their possible expectations regarding others‘ potential errors.

Hence, caution can still be a noticeable influence factor influencing decisions.

4.:

What game can you use for describing a coordination problem with asymmetric interests between the agents? Also think of an example of a situation where this may be helpful for understanding a real-world problem.

Solution key: One possibility is the ‗battle-of-the-sexes‘ game. Each player prefers another of the Nash Equilibria, as only one of them realizes her best result in one specific Nash Equilibrium.

Real world examples may include issues such as the independent development but joint use of technology standards, where coordination on some standard increases the outcome for all involved, but making it preferable for each agent that the other(s) would switch to the standard she developed.

5.:

Which are considered typical examples of social dilemma situations? Where do the specific problems arise from in these?

Solution key: Typical examples referred to earlier in the textbook include a) public good problems andb) common-pool resource problems.

In public good problems, the problem structure results from the fact that whereas contributing to the generation of the good requires private resources to be dedicated to this, their use cannot be prevented once they exist. This leads to two subsets of the problem-structure, the joint production of public goods in a group of agents and the production of public goods by a private company (and hence for sale).

As a dilemma problem, for members of groups of agents, non-contribution in the production of public goods is a strictly dominant strategy for each involved agent.

Private goods (time, resources …) have to be dedicated to the production of something you can use for free once it has been produced. Letting others pick up the production, using your private resources for private benefits, and then profit from the public good‘s existence is individually rational, when rationality is understood as it is in mainstream economics. For companies that means that they cannot internalize the social benefits provided by the availability of the good through profits, as agents cannot be excluded from using the good once in existence.

In common-pool resource problems, the good or resource in question exists without a need for production (its exploitability may well depend on the technological standards in a society, though). The problem that results is a potential for the overexploitation of this good or resource, as it may be advantageous for each individual agent to exceed the prudent use-levels for their private benefit (in fact, when the problem takes a dilemma structure, this is precisely the case). Then, if enough agents follow this course of action,

13 the resource will be depleted at a rate that is higher than its natural reproduction can sustain.

6.:

How can agents, who take their decisions exclusively based on their payoffs, realize a Pareto-optimal result in a social dilemma problem?

Solution key: As long as the social dilemma persists, such agents cannot achieve a Pareto-optimal resultin a social dilemma situation (as long as all other agents are following the same kind of decision rules). What permits improvements in the results are changes in the overall problem structure, such as those that may be attained through the repetition of one game in a series of such games between the same agents, a super-game.

7.:

What condition has to be fulfilled when a social dilemma problem is changed in a repeated game? Derive this condition from the payoffs of the agents.

Solution key: This condition is called the single-shot condition.For a change of the problem structure (given the strategies used in the chapter), TFT has to be a Best Answer to TFT. Rearranging this gives the single-shot condition, which applies to the value the discount factor has to take in order for the transformation to a coordination problem to succeed: 𝑎 𝑐 >𝑏−𝑐+ 1−𝛿 1−𝛿 𝛿>

8.:

𝑏−𝑎 . 𝑏−𝑐

What can you learn about the possibilities for Pareto-improvements in dilemma situations from the single-shot condition?

Solution key: Two aspects have to hold for the individual agent for the possibility for improvements to become feasible, both are related to future interactions, and the agents‘ attitudes and expectations with regard to these. These attitudes determine the value of the discount factor δ,δ=p/(1+r). In this,δ, r is the discount rate agents apply to future payoffs p gives the probability that the supergame the agents play will continue.

The first aspect thus concerns the agent herself, and the importance she gives future payoffs– the higher the weight given to future payoffs, the lower the discount rate r, and c.p., the more likely it is that the single-shot condition is met.

The second aspect concerns the expectations of the agents regarding the continuation of the interactions with a specific other player, the continuation of the super-game. The

14 more likely players assume such continuation and common future to be, the more likely it is that the single-shot condition is met.

9.:

In a dilemma game as shown in Figure 3.8, let a=4 and c=2. For calculating payoffs in the related supergame, let δ=0.9 (δ=0.5). What is the maximum value of b for which the single-shot condition still holds?

Solution key: 𝛿>

𝑏−𝑎 𝑏−𝑐

𝑎 − 𝛿𝑐 >𝑏 .

1−𝛿 For δ=0.9 =>b< 22; for δ=0.5 =>b< 6.

10.: In a dilemma game as shown in Figure 3.8, let b=10 and c=2. For calculating payoffs in the related supergame, let δ=0.8 (δ=0.5). What is the minimum value of a for which the single-shot condition holds?

Solution key: 𝛿>

𝑏−𝑎 𝑏−𝑐

𝑎 >𝑏−𝛿 𝑏−𝑐 .

For δ=0.8 =>a>3.6;for δ=0.5 =>a>5.

11.: Assume a population perspective on a dilemma-based supergame. Calculate the value for (k/n)crit (the intersection of the payoff schedules as shown in Figure 3.10).

Solution key: From 𝑒 𝜋𝑇𝐹𝑇 =

𝑘 𝑎 𝑛−𝑘 𝑐 + 𝑑−𝑐+ 𝑛1−𝛿 𝑛 1−𝛿

and 𝑒 𝜋𝐴𝑙𝑙 −𝐷 =

𝑘 𝑐 𝑛−𝑘 𝑐 𝑏−𝑐+ + 𝑛 1−𝛿 𝑛 1−𝛿

we get 𝑘 𝑎 𝑛−𝑘 𝑐 𝑘 𝑐 𝑛−𝑘 𝑐 + 𝑑−𝑐+ = 𝑏−𝑐+ + 𝑛1−𝛿 𝑛 1−𝛿 𝑛 1−𝛿 𝑛 1−𝛿

15 𝑘 𝑎 𝑘 𝑐 𝑘 𝑐 𝑘 𝑐 − 𝑑−𝑐+ − 𝑏−𝑐+ + 𝑛1−𝛿 𝑛 1−𝛿 𝑛 1−𝛿 𝑛1−𝛿 𝑐 𝑐 = − 𝑑−𝑐+ 1−𝛿 1−𝛿 𝑘 𝑎−𝑐 − 𝑑 − 𝑏 + 2𝑐 = 𝑐 − 𝑑 𝑛 1−𝛿 𝑘 𝑘 𝑐𝑟𝑖𝑡 𝑐−𝑑 = = 𝑎−𝑐 . 𝑛 𝑛 − 𝑑 − 𝑏 + 2𝑐 1−𝛿

12.: How can institutions support agents when it comes to realizing improved results in dilemma situations? Why does a simple rule not suffice in such a situation?

Solution key: Institutions, as social rules plus sanction mechanism (Chapter 1), that establish accepted behavior under which agents abstain from trying to gain a short-term advantage over others are the tool enabling agents in groups to show behavior allowing the realization of a cooperative outcome in dilemma situations.

A rule would suffice under single-shot conditions, as it enables agents to solve the coordination problem they face there, but, when transferring our results to real-world situations, we have to be more cautious. The at least latent dilemma structure in the background requires additional mechanisms enabling a stable cooperative environment.

Institutions that rely on a built-in sanction mechanism (positive as well as negative) offer such mechanisms.

A social institution directs agents to play cooperatively even in a dilemma situation. It provides a limit to the temptation of giving in to trying to exploit other players. Once it is in place, it may eventually be accepted without the connection to the agents who may have developed it, as a general norm in situations where the dilemma structure is in place and agency mechanisms play no important role. The institution has in that case become detached from any specific relations, and constitutes a generally accepted code of behavior. Agents accept these behavioral rules independently of the exchange relation they are about to enter as a generally applicable norm.An institution that is reaffirmed in semiconscious behaviors, in a habitual application, can thus lead agents to show cooperative behavior, even where the basic conditions would lead us to suspect that no such behavior should be observed.

More stability results if cooperation is understood as the result of habitual behavior that is enabled by and embodied in social institutions. Limited cognitive capacities in complex situations make such behavioral rules necessary, because agents cannot make conscious decisions in every moment and situation they face. Furthermore, such institutions likewise serve to reduce the degree of strategic complexity and uncertainty that agents face by informing them about socially acceptable behavior patterns and hence about what generally to expect from counterparts in a given situation.

(See also the section E. Ostrom in Section 13.13, especially with regard to the willingness of real-world agents to punish others for what is perceived as deviating or

16 defecting behavior, even at a cost to them; which more obviously is at play in the maintenance of stable structures in reality than a rational reconstruction can generally identify.)

17 Solutions Chapter 4

1.:

Key Terms What have you learned so far? Reconsider the following key conceptions from Chapters 1-3: Direct interdependence – Complex decision structures – (Initial strong strategic) Uncertainty – (Bounded) rationality – Complexity – Social optimum game – Competing economic perspectives: Invisible hand and fallacy of aggregation – Coordination problems and social dilemmas – Collective goods and externalities – Opportunity costs – Normal-form games – Imperfect/incomplete information – Rationality in the short-run and in the long-run – Recurrence and sequentiality – Supergames – One-shot result and single-shot solution – Social rules and institutions – Coordination and cooperation – Sacrificing and sanctioning – Common and collective problem-solving – Pathdependent, cumulative, and non-ergodic process – Increasing returns – Learning – Expectations and trust – Instrumental and ceremonial dimensions of social institutions – Strategic games – Anti-coordination games – Nash equilibrium – Pareto optimum.

You may use the index of the book to find the text passages in the first three chapters, where these key words are defined or conceptions described.

Please define/explain the following key terms/key conceptions of the present chapter: • • • • • • • • • • • • • •

Lock-in (technological lock-in) Global institutional disequilibrium VAC fragmentation Inappropriability TIC Interoperability and technological standardization Behavioral standardization/institutionalization Network-technology, network-externality/network-effect Strategic triangle of the economy Spatial industrial cluster Industrial network Functional vs. strategic interrelations among firms Hub&Spoke network structure OS.

Solution key: The student will have read Chapters 1 - 4 so far and will be able to give first, rough verbal definitions of the key terms or descriptions of the basic conceptions listed. This will be done on the basis of an understanding of  

direct interdependence among agents, intricate decision structures, and lasting tensions and ambiguity among agents in the process of their recurrent interaction (Chapter 1), basic conceptions and tools of game theory (Chapter 2),

18  

a set of basic game structures in normal form and potential processes resulting under some repetition (Chapter 3), some critical basic technological and organizational phenomena in today‘s global economy under the conditions of ever more complex products, functionally and spatially fragmented value-added chains, and networktechnologies:

•

Lock-In (technological lock-in): A simplified version of the Arthur-model of a random coordination problem in a population in the case of technology choice is given in Chapter 4 (the full model is given in Chapter 13). It should be understood that in a fragmented value-added chain agents have to decide separately, or individualistically, but need to be coordinated in terms of technological standardization; further, under network technologies, the use value (utility) of a technology grows for every individual user with a growing net for a particular technology, so that the standardization in terms of one technology out of potentially several competing technologies is required also from the technology side (equivalent with the coordination on one out of several Nash equilibria in a coordination game). The student should see that there is a tendency to lock the whole system in on one of the competing technologies, that this is necessary and beneficial for all in the short run but may be detrimental in the longer run, when a continuing innovation is required. In the pure form, such standardization correlates with some monopolization on the supplier side, and the system would not find a way out of the technological standard gained – the technological lock in, comparable, as said, with a coordination game, when an equilibrium has been found; then no one has an incentive to deviate from it. For an institutional lock-in, where mutual behavioral expectations make the system lock in certain behaviors, you may refer to the QWERTY story of P. David, as given in Chapter 13 …

•

Global institutional disequilibrium This conception referenced from the literature refers to our description of the global functional and spatial fragmentation of value-added chains and to collective goods of all kinds involved, including complex products of a value-added chain, a high degree of innovation in a value-added chain, and particularly the global commons. The contention of this thesis is that there is a growing demand for a collective production, or management, of global common goods, but the growth of individualistic fragmented structures would relatively reduce the collective decision capacity, so that a proper collective/coordinated management namely of the global commons, such as natural environment, climate, stock of resources, food, water,or landscape beauty, can be supplied to an ever lesser degree. The thesis contends a growing demand and a shrinking supply of global commons. Note that you can ask students to relate this to prisoners‘ dilemma structures, which often are played among the same agents besides an obvious coordination game.

Examples given refer to (1) the behavior in the booms and depressions of the business cycle (a dampened business cycle as a collective good), (2) the opportunity of individual agents to wait with own investment, if the standard to come out of competing technologies is not known yet, or (3) if, in an innovation chain, an agent can free ride on the innovation effort of

19 others without spending effort and cost on own innovation. These examples are given in the chapter …

•

VAC fragmentation Consider functional and technological fragmentation; ―outsourcing‖; ―global sourcing‖; the spin-off waves and firm foundations … Consider that this causes basic coordination problems, and under certain conditions also social dilemmas. Consider the tension between such fragmentation and the ever larger complexity of products ...

•

Inappropriability An important theoretical and practical phenomenon, which causes a system of externalities, where a firm may receive externalities from one firm and has to let externalities drop out to another firm. A multi-agent coordination problem or social dilemma may result. The presupposition here is, confirmed by a rich empirical industrial research, that any firm has to give knowledge away through the innovation content of its product, that this fact is common knowledge, so that the cost for the knowledge content will not be fully covered by the buyers in the selling prices. Successful and innovative production or innovation systems, value-added chains or regional systems have turned out to be systems with a successful common or collective management of externalities or knowledge involved ...

•

TIC We argue that the usual term ICT—information and communication technology is not fully appropriate, but that it is important that the relevant technologies a tele-technologies, technologies for remote, in fact global, information and communication and that it goes far beyond just face-to-face communication. With digital microelectronic tele communication, the coordination requirements have considerably increased into a qualitatively new dimension. It is not that the screw-wrench must have the same inch size as the screw; it is about the fact that a whole service industry exists working on codes, interfaces, and protocols, for billions of people to be able to communicate with each other worldwide ...

•

Interoperability and technological standardization Interoperability is the technical term of standardization in microelectronic, digital technologies, ICT/TICs (see above) …

•

Behavioral standardization/institutionalization Consider that with obvious coordination problems (modeled as coordination games) in, say, a value-added chain, a social dilemma (game) may be connected under certain conditions (waiting with investing in a new technology, free-riding on innovation, see above). The technological standard has as its other side of the coin a behavioral standard, a behavioral rule coordinating agents (without which the technological standard would not come about). In case of a social dilemma (game) the problem-solving behavioral standard must be a social institution, rather than just a social rule, according to the terminology introduced with solving coordination and dilemma games.

20 •

Network-technology, network-externality/network-effect The basic idea that a first- or second-year undergraduate student (with or without an economics major) should be familiar with is that in fragmented production or innovation processes there must be a standardization, to be yielded in a common or collective process, both across multiple agents and across related technologies. Any such technology forms a net (or network) across users (consumers). Consider that with competing technologies, consumers might be reluctant to purchase the new technology at all, when the final standard is not known yet (the classical case was video tapes/recorders/stores). The cumulative economic process entailed is equivalent with increasing returns (again, see Arthur‘s technology choice model, Chapter 13). The cumulative process tending towards one standard is based on the fact that the larger the net, the more attractive it is c.p. for any user (higher utility for those already in and those joining), and that, similarly, any new, joining user increases the utility for himself and all who already are in that technology‘s net. Again, a simplified version of the famous Arthur-model is given in Chapter 4 and may be used to require a first formal modeling as a multi-agent coordination game.

•

Strategic triangle of the economy/Figure 4.3 What is described in the text as a ―strategic triangle of the ‗new‘ economy‖, a lasting, structural tension among the angles ―complex product‖ – ―fragmented VAC‖ – ―network technologies‖, has been illustrated in Figure 4.3 as a tension (trade-off) between fragmentation and product complexity and an advance in technological (and behavioral) coordination, where better standards of one technology (―technological progress‖) allow for realizing both higher levels of product complexity and fragmentation, while different technologies entail different combination of both, at which level ever. You may want to let the students explain the tensions here and make them aware that successful coordination/cooperation and related technological and behavioral progress are not givens, obvious or self-evident but always fragile solutions from fragile processes.

•

Spatial industrial cluster Note that we have verbally explained a picture of a two-staged real-world process of institutionalization of coordinating/cooperating behaviors (always depending on whether we envisage a particular coordination or a dilemma problem). In this process, we start from an ideal ―market‖ situation (a decentralized spontaneous mechanism with prices playing some role), with increasing interdependencies, strategic uncertainties, and market failure. One spontaneous reaction of the global corporate economy is to overcome the shortcomings of just arm‘s-length and volatile market relations and to build regional proximity and some relational stability, faceto-face interactions, etc., i.e. building regional clusters worldwide, where human and social capital, knowledge exchange, and common infrastructure are critical. Note that these become somewhat resistant against price changes, as other factors (certainty, behavioral coordination/cooperation, shared knowledge, etc.) compensate low-price offers from outside (within smaller or wider limits).

•

Industrial network

21 The two-staged story of institutionalized coordination/cooperation may then lead to an outgrowing of a ―strategic‖ network out of a cluster, i.e., logically, the formation of a network of formal or informal, usually projectbased and temporary multi-agent cooperation, a more intense or stronger form of coordination/cooperation among a partial set of the cluster agents.

Note that the problems solved remain the same. So always try to lead the students back to the basic problems, usually coordination or dilemma games. This is what they should try to apply to the real-world info provided in Chapter 4. The cluster is a softer form of coordination, the network will typically be formed, when there is more at stake, e.g., when dilemmas come to the fore, which, in turn, require stronger forms (formal or informal) of coordination, i.e., cooperation.

•

Functional vs. strategic interrelations among firms This terminology is more everyday language, and ―strategic‖ here does not strictly refer to a game-theoretic reasoning. We call the cluster emergence ―spontaneous‖, ―historical‖, and ―out of the (failing) market‖; so we assume relatively less deliberate and planning action – in contrast to the formation of a ―strategic‖ network, where agents make a handshake and more or less deliberately build on the stable relations that have emerged in the cluster.

Note that a more formal game-theoretic and evolutionary modeling of the emergence of networks would in no way require any deliberate hand-shake.

So this is more our real-world story here.

•

Hub&Spoke network structure We discuss real-world networks also in this case as an organized, deliberate decision, in a world of net(work)s and hierarchies, led and dominated by large hub-type hierarchies, again a more deliberately formed network. Note that we discuss networks more generally in Chapter 13 (Sections 10 and 11), where we show that many emerged network structures in the real world in fact are ―scale free‖, i.e., with a certain distribution of firm sizes, and should have the properties of effective diffusion such as small-world networks. If students wish to discuss things deeper than just at the level of Chapter 4, you can always refer to the contemporary anchor models of complexity economics as given in Chapter 13.

•

OS We introduce open-source networks as a very strong, ideal solution to complexity problems. The idea here is to try to generalize the principles of open source as a potential more general economic form and principle of organization (knowledge sharing and common/collective information management, strong institutionalization of reciprocity, etc.). More detail on this, including a discussion of ―proprietary‖ motives coming along with voluntary contributions to the collective good, are given in Chapter 15 on open source in the information and software industries, and in Chapter 16 on the Organizational Triangle, where open source networks are considered an ideal solution to the complexity problem of firms beyond the simplistic ―market‖—―hierarchy‖ dimension. Students should comprehend here that there is a variety of organizational principles and forms, ways to deal with uncertainty and information-sharing, and institutionalized cooperation in a wide sense.

22

2.:

New Properties of the Economy: Complex Products, Fragmented Value Chains, and Network-Technologies a)

Explain the coordination problems (i) in the fragmented VAC and (ii) in technology choice among competing (non-interoperable) network-technologies.

b)

Explain the potential social dilemmas additionally implied (i) in the innovation process in the fragmented VAC and (ii) in technology choice between noninteroperable network-technologies. Explain a potential mutual blockage of (Pareto-superior) action under conditions of a social dilemma in technology choice. Also explain potential regret in technology choice.

c)

Explain potential inappropriabilityin the VAC and the system of both providing and receiving positive externalities.

d)

Explain how interoperable network-technologies may contribute to solving the coordination and dilemma problems for fragmented agents in the VAC, and to producing highly integrated complex products through a fragmented VAC.

Solution key: a) Basically, we would expect here the same story as requested in exercise (1) above, but this time verbally told, plus perhaps slightly formalized in a three-agent payoff matrix as provided, from the angle of the game-theoretic structure. b) In (i), you may let the student tell the story as in exercise (1) above, but focus more on the conditions, under which firm agents may be able (or not) to adopt the free-rider position in an innovation system in the value-added chain (transparency about strategies, actions, and payoffs; recognized interdependence; learned and habitually acquired long-term perspective; no hierarchy given, though).In (ii), you might want to let the students provide the first algebraic approximation, together with a numerical example for a payoff matrix for technology choice. Finally, regret refers to sequential decision making among agents, basically a path-dependent process, where the agent (because of the complexity of the process, or ―bounded rationality‖) cannot know in advance the final standardization outcome. Avoiding later regret may lead users to abstain from adopting one of the competing new technologies, with a collective refusal of users to adopt any of the new technologies (again, see the classical case of video technology, also, more recently, but nor that severe, the 2nd-genration DVD between HD and Blu-ray, and many other cases). c) You might want to let the same story of exercise (1) be told form a more theoretical angle, with a focus on information being public in the value-added chain (knowledge

23 being shared, if agents like it or not), thus no full reimbursement of own effort to generate knowledge (innovation) from the buyers/users, rather the firm taking some information for free from the knowledge generation of others, and giving some knowledge away for free. You may want to introduce a positive externality here. You also might let the stories of the ―Silicon Valley‖ or the ―Third Italy‖ or other innovative interaction systems (regional clusters, industrial agglomerations) be explored and told.

Another theoretical upgrading of the verbal responses might include a reference to Alfred Marshall. d) You may ask students to connect the technical term of interoperability in microelectronics and software production, as a systematically generated interoperability among competing technologies – often undermined by the power strategies of leading suppliers who want make their technology the standard –, with the conception of open source, publicly shared knowledge, and break-out from lock-in. This is meant more as an outlook, and more detail on such opportunities are given in Chapters 15 and 17 (in the latter in terms of policy options).

3.:

Technology Choice: Relative Number of Choices and Related Payoffs (1) Assume the following numerical payoff functionsfor technologies T1 and T2, depending on the number of agents i = 1, …,n having chosen the respective technology: [See Table Exercise 3-1] Determine the resulting payoffs for agents A, B, and C, PA, PB, PC, all depending on T1,2 and n, in particular depending on the following choice sequences given (indicated by X), after all agents have chosen (fill in the bottom right cells in the format: PA, PB, PC): [See Table Exercise 3-2] You may determine the other six possibilities of choice sequenceson your own.

Solution key: This exercise is a didactical attempt at inducing the students to some finger exercise with (and somewhat beyond) the normal form games (three agents in this case), to apply the matrix to different sequences of choice of three agents between two different ―strategies‖. The main learning effect should be to realize that net size is a critical factor and may relativize a one-for-all optimality. It is important for them to realize that a larger number coordinated (for what reason ever) on an inferior technology may each gain at least as much as a smaller number coordinated on the ―optimal‖ technology.

Surprise from a paradox would be the didactical objective here.

4.:

Technology Choice: Relative Numbers of Choices and Related Payoffs (2) Further consider an indefinite series of agents with random technology choices.Determine their changing payoffs, depending on the choices made by each following agent, given the following numerical payoff functions:

24

[See Table Exercise 4-1] See the following example of some sequence: [See Table Exercise 4-2] Build different choice sequences among up to ten choosing agents in a row, and determine the different results for each.

In particular, give an example for a choice sequence that yields a higher payoff for those who have chosen the inferior technology T2.

Solution key: This is just another simple finger exercise to help generating an understanding with the students of a basically open-ended process that may display different results, if it ended at different points in time. So no unique, predetermined, clear-cut equilibrium the system may tend to, but rather two potential equilibria. The two potential equilibria become more obvious, if, after the xth agent has chosen, you introduce a switch of all following choosers to the technology with the higher average payoff (imitation). From here on, you may easily connect this little introductory exercise with the more formal Arthurian technology choice model as given in Section 13.6 and/or the illustration of a process that, after the second agent has chosen one of the technologies, immediately drifts to one of the two corner solutions of coordination (attractors), as given in Figure 13.11. You may also easily introduce the Arthurian ―absorbing barriers‖, by letting, say, the first ten agents chose independently of the size-related payoffs (as suggested above) and only thereafter make then switch to imitate the choices of those, whose technology net provides the higher payoffs. Then you approach the way Arthur had put his model, i.e., at the absorbing barriers. A final step then might be discussing Arthur‘s graphical display of his computer simulation of that process. How far you can take undergraduates on this way, you have to check against the comprehension and interest they display after they understood the basic principle of random process, open process, and final lock in.

25 5.:

New Empirical Phenomena to Cope With Increased Complexity a)

Explain why and how firms in the global corporate economy tend to organize (or being organized) in • • •

b)

spatial industrial clusters industrial networks hub&spoke networks.

Explain why OS-type networks may be considered superior in collectively governing/managing information and expectations (trust), and to accelerate information and innovation flows. Consider the single-shot solution to explain institutionalized information exchange and knowledge sharing in OS-type networks.

Solution key: a) You may let the students again describe the basic situation of firms in fragmented valueadded chains, under conditions of ever more complex products and competing (and basically non-interoperable) network technologies. You may want to let them describe coordination and dilemma problems in production, investment, and innovation systems recognized by firms and their learned forms of coordination and cooperation as spontaneous individualist solutions to reduce uncertainty (and volatility or turbulence) by reducing the complexity of their decision situation. Again you may draw their attention to applying theoretical knowledge to real-world organizational phenomena, whether they star with the real-world phenomena (clustering, networking) or the theoretical model structures. Finally, their attention should be drawn to the fact that the real-world deviates from the simple initial game-theoretic forms (where equal, equally independent and equally powerless firms are supposed) on the issue of power.

Combining power and hub-functions of large and central firms with fragmented production and innovation systems and, thus, with net-structures, hub&spoke networks are a basic result for many real-world supplier systems. Later on, theoretical and empirical social network analysis might qualify the students‘ knowledge on centrality, hub functions, and size distributions combined with clustering in larger global networks, which interconnect different clusters of firms and where a clustered part of that larger network may be considered an individual value-added chain. For more detail, you may already refer again to Sections 13.10 and 11. b) If OS-type networks have come into existence and work on the principles of institutionalized voluntary contribution to the common good and reciprocity, they may indeed be highly effective, as certain aspects of ―hackerdom‖ (the ―Linux paradigm‖) have proven during the 1990s and 2000s. the then represent a strong form of institutionalization, with high levels of trust and favorable expectations (―to meet …‖), and an effective governance of information flow and information sharing. The singleshot solution of the PD-SG indicates that, given a certain incentive structure, expectations are critical and the learned and culturally acquired long-run perspective –

26 the logical conditions highlighted by the single-shot solution. With a view on more realworld aspects, you might want to discuss with students (or have students discuss) the size dimension of the relevant population or group (―meso‖-sized groups), but also the critical role of hierarchy (how flat ever), i.e., of smaller or larger hubs organizing and coordinating the contributions of the individual agents, in the OS software production networks. In this sense, you might want to take up some results from Chapter 14 here already.

27 Solutions Chapter 5

1.: Assume that there are three goods, apples, peaches, and pineapples. Construct a preference relation that violates the transitivity axiom.

Solution key: Denote Apples 𝐴𝑝, Peaches 𝑃𝑒, Pineapples 𝑃𝑖.

2 𝐴𝑝 ≻ 2 𝑃𝑒, 2 𝑃𝑒 ≻ 2 𝑃𝑖,2 𝑃𝑖 ≻ 2 𝐴𝑝 is, if this holds simultaneously, not transitive. There are numerous other possibilities for constructing intransitive preference relations.

2.: Draw the indifference curves and compute the marginal rates of substitution for the following two utility functions: u( x1 , x2 ) = x1  2 x2 u( x1 , x2 ) = x10.3  x20.7 .

Solution key: The indifference curves for both functions are shown in Figures X.2 and X.3.

First function: 𝜕𝑢 𝜕𝑥1 𝜕𝑢 𝜕𝑥2

1

= 2.

Second function: 𝜕𝑢 𝜕𝑥1 𝜕𝑢 𝜕𝑥2

0.3𝑥 0.3

= 0.7𝑥 20.7 .

1

28

Figure X.2 Indifference curves corresponding to the first utility function, exercise 2, Chapter 5.

3.: A consumer spends her entire budget on two goods, bread and beer. How do the quantities consumed change if the price of beer increases (and the price of bread stays constant)?

You might use an indifference curve to visualize the changes.

Solution key: The utility function remains unchanged, but the budget restriction is changed (turned around the axis intercept of the good bread, 𝑥2 ) effectively the budget decreases (as the same income enables the agent to consume only a smaller quantity of goods). Either both consumption quantities may decrease or a part of the consumption of the now considerably more expensive good (beer) is substituted by bread such that the consumption of this good increases (see Figure X.4).

29

Figure X.3 Indifference curves corresponding to the secind utility function, exercise 2, Chapter 5.

4.: When is the assumption of firms and households taking prices as given reasonable?

Explain.

Solution key: This may be justified when there are many small firms (technically infinitely many infinitely small firms) and households so that decisions made by any individual firm (and households) do not have a measurable impact on the equilibrium.

5.: There is a single market with supply given by X s = 10  3 p and demand given by 70  5 p . Compute the equilibrium price and quantity.

Solution key: 𝑋𝑆 = 𝑋𝐷 70 − 5𝑝 = −10 + 3𝑝 80 = 8𝑝 𝑝 = 10 𝑋 = 70 − 5𝑝 = 70 − 5 ∗ 10 = 20 (or, equivalently 𝑋 = −10 + 3𝑝 = 20).

30

Figure X.4 Possible effect of a price increase on consumption corresponding to exercise 3, Chapter 5.

6.: Explain the law of diminishing returns in words and mathematically.

Solution key: From some production quantity on, the marginal costs rise faster than the marginal revenue. This means that while the producer will be able to make profits and up to this critical point also to increase her profits by increasing the production quantity, she will not have an incentive to increase the quantity beyond this critical point as it would decrease her profit. The marginal cost and revenue functions are continuous and the difference between them (marginal profit) is monotonic, i.e. the largest profit margin exists for the first unit produced and the marginal profit (the difference) decreases monotonically.

Mathematically (𝐹 being the production function, 𝑥 the quantity, 𝑐 a constant factor) 𝐹 𝑐𝑥 < 𝑐𝐹 𝑥 for 𝑐 > 1 .

31 7.: Consider an exchange economy consisting of two agents. The agents have the utility function ui =  i ln x1i  (1   i ) ln x2i with 1 = 0.3 ,  2 = 0.7 and initial endowments

1 = (30,15) ,  2 = (10,30) . Compute equilibrium allocations and prices.

Solution key: Endowments and utility function are given. The budget restriction is by definition 0 = 𝑝1 𝑥1 + 𝑝2 𝑥2 − 𝑊𝑖 .

For both agents 𝑖 = 1,2 where 𝑊1 = 30 𝑝1 + 15 𝑝2 𝑊2 = 10 𝑝1 + 30 𝑝2 .

Construct the Lagrangian (identical for both agents) as ℒ 𝑥1 , 𝑥2 , 𝜆 = 0.3 ln 𝑥1 + 0.3 ln 𝑥2 − 𝜆(𝑝1 𝑥1 + 𝑝2 𝑥2 − 𝑊) and derive the first order conditions, the budget restriction (derivative with respect to 𝜆) as well as 𝜕ℒ 0.3 = − 𝜆𝑝1 = 0 𝜕𝑥1 𝑥1 𝜕ℒ 0.3 = − 𝜆𝑝2 = 0. 𝜕𝑥2 𝑥2 From the latter two, it follows that 𝑝1 𝑥1 = 𝑝2 𝑥2 , which, substituted into the budget restriction, gives the demand functions 𝑥1𝐷 =

𝑊 2𝑝1

𝑥2𝐷 =

𝑊 .

2𝑝2

Substituting the budgets 𝑊𝑖 into the demand functions gives for the first agent

and for the second

𝐷 𝑥1,1 =

30𝑝1 + 15𝑝2 𝑝2 = 15 + 7.5 2𝑝1 𝑝1

𝐷 𝑥1,2 =

30𝑝1 + 15𝑝2 𝑝1 = 15 + 7.5 , 2𝑝2 𝑝2

32

𝐷 𝑥1,1 =

10𝑝1 + 30𝑝2 𝑝2 = 5 + 15 2𝑝1 𝑝1

𝐷 𝑥1,2 =

10𝑝1 + 30𝑝2 𝑝1 = 5 + 15 , 2𝑝2 𝑝2

and aggregated demand functions 𝑥1𝐴𝐷 = 15 + 7.5 𝑥1𝐴𝐷 = 15

𝑝2 𝑝2 𝑝2 + 5 + 15 = 20 + 22.5 𝑝1 𝑝1 𝑝1

𝑝1 𝑝1 𝑝1 + 7.5 + 5 + 15 = 20 + 22.5 . 𝑝2 𝑝2 𝑝2

Aggregated supply is (from the initial endowments) 𝑥1𝐴𝑆 = 30 + 10 = 40 𝑥2𝐴𝑆 = 15 + 30 = 45 and must in equilibrium equal aggregated demand, hence 20 + 22.5

𝑝2 = 40 𝑝1

𝑝2 8 = 𝑝1 9 9 𝑝1 = 𝑝2 8 𝑝2 =

8 𝑝 .

9 1

(Setting demand and supply for 𝑥2 equal would give the same result.)

Now choose one of the goods as numéraire, say the second one. Thus, the prices are 𝑝1 = 1.125 𝑝2 = 1 .

This allows computing the equilibrium quantities from the demand curves as for the first agent 𝑥1 = 15 + 7.5 𝑥2 = 15

𝑝2 2 = 21 𝑝1 3

𝑝1 3 + 7.5 = 24 , 𝑝2 8

33 and for the second agent 𝑥1 = 5 + 15 𝑥2 = 5

𝑝2 1 = 18 𝑝1 3

𝑝1 5 + 15 = 20 . 𝑝2 8

34 Solutions Chapter 6

1.:

Name five main criticismsof the neoclassical general equilibrium theory and explain briefly.

Solution key: Main criticisms are: a)

the mistaken energy-value (Mirowski‘s argument),

b)

the non-existence of a unique equilibrium except under very restrictive special conditions (following from the SMD theorem),

c)

the impossibility of horizontal demand curves,

d)

the impossibility to aggregate capital (independently from the price system; Sraffian critique),

e)

the potential existence of increasing returns,

f)

neoclassical models focus on equilibria which may not necessarily exist or be stable or – even if this is the case – may coexist with other complex attactors (see Chapter 11). Further, neoclassical general equilibrium models generally neglect a number of aspects shown to be present in the economy, e.g.,

g)

transportation and transaction costs,

h)

heuristic decision making instead of optimization, and many more.

Exercise 1 requires 5 of these criticisms to be named and briefly explained.

2.:

What is an economic conservation law?

Solution key: An economic conservation law is a mathematical condition following from the use of Lagrangian/Hamiltonian optimization in economic problems, specifically in neoclassical models that fulfill the requirements necessary for applying such optimization techniques (differentiable utility functions, such that utility and prices can be modeled as a vector field in which change in utility and expenditure act as forces and counterforces). The condition requires the value of the Hamiltonian, ℋ, be constant (over time and over modifications within the set of optimal solutions). For static problems, this yields that the value of the Lagrangian, and thus the utility be a constant identical value for all optima (which is trivial). For dynamic problems (such as Ramsey optimizations as introduced in Chapter 5), the conservation law is different and it may be impossible to interpret this condition in a useful way. The question of economic conservation laws is generally avoided in neoclassical models.

35 3.:

Why are negative inclined market-demand curves and horizontal demand curves for individual companies logically incompatible given the assumptions of the perfect market model? (Show this formally.)

Solution key: There are two basic assumptions behind the perfect competition model that matter for us here: The decision to increase output 𝑞𝑖 that one firm 𝑖 may take does not lead to reactions by 𝑑𝑞 any other producer 𝑘 in the market, so that 𝑑𝑞𝑘 = 0. As a consequence, overall output 𝑖

changes in the market, 𝑑𝑄, and changes in output by one individual firm are the same, 𝑑𝑄 = 1. 𝑑𝑞 𝑖

Additionally, we have the negatively sloped demand function, or the corresponding 𝑑𝑝 inverse demand function 𝑝 = 𝑝(𝑄). The slope of this function is 𝑑𝑄 . We can multiply this by 1 and use the above relation to arrive at: 𝑑𝑝 𝑑𝑝 𝑑𝑄 𝑑𝑝 = = . 𝑑𝑄 𝑑𝑄 𝑑𝑞𝑖 𝑑𝑞𝑖 =1

In short, we have assumptions and a result we can derive if we follow the implications of these assumptions through, which shows the assumptions to be incompatible: 𝑑𝑝 𝑑𝑝 = . 𝑑𝑄 𝑑𝑞𝑖 <0

4.:

=0

Name the principal problems around the assumption of positively sloped supply curves.

Solution key: Increasing supply increases costs due to the increased need for a scarce input. As the supply-curve is given by the marginal cost curve this would lead to its upward slope.

Sraffa has approached this aspect from two perspectives on what may constitute a market with a view on factor requirements in production, a narrow definition of a market and a broad definition of a market. (Note that the upward sloping supply curve can only be conceptualized within the framework of perfect competition, as companies‘ output will be determined by demand condition as well as cost conditions as soon as they are assumed to have market power.)

The argument leads to the recognition that the assumption of scarce inputs used for arguing for increasing marginal costs is problematic. If you take a narrow view on what constitutes a market (producers of nails, for instance), the inputs used in production will be used in other markets as well. That means they are more widely available in which case an increase in their price in case of expanding output is not necessarily justified.

This is in fact covered by empirical observationsthat show companies operating under

36 constant or even increasing returns to scale, and consequently horizontal or even negatively sloped supply curves. As a consequence, demand conditions alone will determine the quantity of output, not the interplay of demand- and supply-side.

If you take a broad view on markets (such as everyone utilizing iron ore in production), the independence of supply and demand can no longer be assumed. In the broadly conceived markets, changes in factor employment have effects on income and thereby on demand. In this case, every point on the supply curve corresponds to a different demand schedule and the uniqueness of equilibrium cannot be taken as given. This is a general dilemma for partial market analyses as the desired determination of a factor‘s or good‘s unique scarcity value leads to potentially very wrong conclusions then. Think of the labor market, where partial market analyses simply ignore the fact that, in order for there to be a demand, people have to have the means to purchase something – the simplistic formula ‗higher wages lead to lower employment‘ cannot be sustained once the labor market is recognized to be embedded in the overall economy. (As Keynes already observed, Say‘s law does not hold.)

5.:

From what angle and how has the ‗revealed preferences‘ approach been criticized?

Solution key: The objective of the ‗revealed preferences‘ approach was to remove unobservable preferences from explanations of consumer behavior (Samuelson 1938). Choice is to be explained based on observed choice. Based on the assumption of rational behavior by agents, under certain circumstances, observed choices could be translated into constructs that are equivalent to mappings of indifference schedules. Hence, they could be transferred into utility functions, in fact showing the equivalence of choice sets by rational agents and utility functions constructed for describing and explaining choices by those rational agents under certain conditions.

The assumption that explanations of behavior can be possible on the basis of observed behavior alone hinges on the consistency condition: If a person chooses y when z is also available, this is taken to mean that y is preferred to z. This is translated into the weak axiom of revealed preferences stating that when y is revealed to be preferred to z through agents‘ choices, z cannot be preferred to y. When this is fulfilled by agents, behavior is consistent and a theory of consumer behavior can be built on its foundation.

Now, as Sen (1973) stresses, the choices of y at sometimes and z at others, even though both are available, are not necessarily inconsistent. They can be seen as inconsistent when we attribute specific motives to the person making these choices. This relies on a step that was precisely the one which Samuelson and those adopting the concept aimed at avoiding.

You cannot explain choice based on observation of choice alone but always have to refer to some notion external to choices for interpreting them, such as objectives and expectations.

To exemplify the impossibility of understanding preferences and beliefs based on choice alone, we can refer to the fact that individual choices can be context-dependent.

The overall situation in which people make choices matters for the choices themselves and a decision taken can differ depending on those overall circumstances. In contrast,

37 one core aspect in operationalizing and formalizing individual choice is the independence of a decision from additional available alternatives and hence contextdependence.

That is to say, if in situation A an individual chooses x over y, she should not choose y over x when a third alternative z becomes available as well in situation B. We can, however, easily construct situations where this is the case. Let x be not taking an apple from a fruit bowl and y be taking an apple. Now, if in situation A there is only one apple in that bowl, people may opt for x. If situation B differs in that there are more apples in the bowl (let z be consuming two apples), an agent may opt for y. The motivation in that case would be not to take the last apple even if you prefer consuming an apple to not consuming one. Such settings can be extended for including additional factors in individual decision-making, such as rules and institutions as norms, more explicitly, which introduce further aspects of motive and preference that are withdrawn from observation but are accessible only through introspection or contextualization, and interpretation.

In interdependent situations this is further interesting, because individual behavior depends on expectations regarding others‘ behavior, but also depends on the interpretations of others‘ motives: trusting behavior is more likely to be met with a trustworthy response then distrustful behavior. Something that is freely given, say option A, cooperation, when defection, option B, is possible, is more likely to be met with a trustworthy response than the same behavior in the absence an option that is not chosen and thereby allows to send a signal of trust (e.g. Hausman 2000). Overall, an interpretation of motivation based on behavior, choice, alone is not possible – we see ‗chooses‘, we infer ‗prefers‘ and can do so only on the basis of something external to choice, namely, preferences in one guise or another.

Sen points out that maybe the axiom could be seen as a hypothesis to be empirically validated, but that, even for small numbers of goods, as soon as they are divisible, the amount of possible combinations becomes extremely large due to their exponential growth – and actually, infinite in the usual calculus-based approach to consumer theory.

Even if numerous observations were available these could not hope to consist of sufficient numbers for confidently not rejecting the hypothesis. That the collection of such data will have to be undertaken in a timeframe in which it may reasonably be assumed that preferences have not changed further underlines the practical impossibility of the notion.

6.:

Explain the basic equation of a Sraffian one commodity-model.

Solution key: The basic equation in a Sraffian one commodity-model is based on the following considerations: For the production of one unit of corn we need a units of input (seed corn) and l units of labor. Let p denote the price of corn, w the wage rate (the price of one unit of labor) and r the profit rate. Assuming that wages are paid at the end of the production period, costs of production are given by 𝑝 = 1 + 𝑟 𝑎𝑝 + 𝑤𝑙 .

(6.1)

38 Since we assume that production is feasible we assume 𝑎 < 1, meaning that less than one unit of seed corn is needed to produce one unit of corn. We can normalize the price of our single commodity corn by setting𝑝 = 1, and rewrite Equation6.1 to see that the net output (1 − 𝑎) is distributed between profits and wages: 1 − 𝑎 = 𝑟𝑎 + 𝑤𝑙 .

(6.2)

Another way to see this would be to solve Equation6.1 for the wage as a function of the profit rate: 𝑤 𝑟 =

1−(1+𝑟)𝑎 𝑙

.

(6.3)

We see that there is an inverse relation between the wage rate and the profit rate, reflecting and illustrating the class struggle that is inherent in capitalist production.

7.:

What are mark-ups and administered prices?

Solution key: At the center stands pricing power of the firms. Prices are not cost- or demanddetermined but set according to strategic considerations by firms. They are operationalized as markup prices. The markup is added onto the average variable costs (AVC) in a period. The mark-up permits an integration of a wide set of possible influences, including the industry structure, influences on demand through, for instance, advertising, and wage setting rules and labor union influence, among others. Depending on your needs in a model, you can capture the markup as a percentage m of AVC or as a fixed amount M added to ACV: 𝑝 = 1 + 𝑚 𝐴𝑉𝐶 or 𝑝 = 𝐴𝑉𝐶 + 𝑀 .

Another aspect to be mentioned here isadministered pricesintroduced by the institutionalist economist Gardiner C. Means in the 1930s (Means 1939;compare Lee 2004). Companies enter into repeated interactions and contractual relations. Prices are thereby set for a period of time. These are called administered prices; they can be maintained if companies have some degree of market power. The rationale for the company can be understood from a long-term perspective, allowing strategic considerations regarding growth and market share to become important factors in their considerations and price policies.

39 Solutions Chapter 7

1.:

For the inverse demand function 𝑝 𝑦 = 𝑎 − 𝑏𝑦 and the cost function 𝐶 𝑦 = 𝑐𝑦 calculate the profit-maximizing price-quantity combination for a monopolist. a) b)

For a = 200, b = 1, c = 20.

For the placeholders a, b, and c for a general result in this setting.

Solution key: a) 𝑝 𝑦 = 200 − 𝑦 𝐶 𝑦 = 20𝑦 𝜋 = 𝑝 𝑦 𝑦 − 𝑐 𝑦 = 200 − 𝑦 𝑦 − 20𝑦 𝑑𝜋 = 200 − 2𝑦 − 20 = ! 0 𝑑𝑦 𝑦 = 90, 𝑝 = 110 . b) 𝑝 𝑦 = 𝑎 − 𝑏𝑦 𝐶 𝑦 = 𝑐𝑦 𝜋 = 𝑝 𝑦 𝑦 − 𝐶 𝑦 = 𝑎 − 𝑏𝑦 𝑦 − 𝑐𝑦 𝑑𝜋 = 𝑎 − 2𝑏𝑦 − 𝑐 = ! 0 𝑑𝑦 𝑎−𝑐

𝑦 = 2𝑏 .

2.:

Depict the functions from exercise 1) as well as marginal revenue and marginal cost curve and show the Cournot point in the resulting graph.

Solution key: See Figure X.5.

40

p Marginal revenue (with inclination -2b) Cournot point Demand schedule (with inclination -b) Marginalcost c(with inclination 0)

y Figure X.5 Functions from exercises 1 and 2, Chapter 7.

3.:

Modify the cost function in 1) to 𝐶 𝑦𝑖 = 𝑐𝑦𝑖 . For i = 1, 2 calculate the best-response functions of two identical companies and determine the quantity that each oligopolist chooses (no storage, all produced units are sold)(𝑦 = 𝑖 𝑦𝑖 ). Which is the resulting market price? a) b) c)

For a = 200, b = 2, c = 50, when companies decide on their output quantity simultaneously.

For a = 200, b = 2, c = 50, when company 1 decides first and company 2, knowing about y1, deciding later.

For the placeholders a, b, and c for a general result in these two settings.

Solution key: a) 𝑝 𝑦 = 200 − 2𝑦 𝐶 𝑦𝑖 = 50𝑦𝑖 𝜋𝑖 = 𝑝 𝑦 𝑦𝑖 − 𝐶 𝑦𝑖 = 200 − 2𝑦 𝑦𝑖 − 50𝑦𝑖

41 𝑑𝜋1 = 200 − 4𝑦1 − 2𝑦2 − 50 = ! 0 𝑑𝑦1 𝑦1 =

150−2𝑦 2 4

.

As the firm‘s cost functions are identical, the best response functions take the same form as well, hence: 𝑦2 =

150−2𝑦1 4

.

Substitute y2 in y1 𝑦1 =

150−2𝑦 1

150 − 2

4

4

1 = 37,5 − 18,75 + 𝑦1 4

3 𝑦 = 18,75 4 1 𝑦1 = 25(= 𝑦2 ) 𝑦 = 50, 𝑝 = 100 . b) The decision structure changes slightly. Assume firm 1 decides first, then the best response function for firm 2 takes the same for as in exercise a). The decision structure for firm 1, knowing this, changes, however. It maximizes its profits given the amount to be expected from firm 2 (it does not need a best response function): 𝜋1 = 𝑝 𝑦 𝑦1 − 𝐶 𝑦1 = 200 − 2𝑦 𝑦1 − 50𝑦1 𝜋1 = 𝑝 𝑦 𝑦1 − 𝐶 𝑦1 = 200 − 2(𝑦1 +

150 − 2𝑦1 𝑦1 − 50𝑦1 4

𝜋1 = 𝑝 𝑦 𝑦1 − 𝐶 𝑦1 = 200 − 2𝑦1 − 75 + 𝑦1 𝑦1 − 50𝑦1 𝜋1 = 𝑝 𝑦 𝑦1 − 𝐶 𝑦1 = 125 − 𝑦1 𝑦1 − 50𝑦1 𝑑𝜋1 = 125 − 2𝑦1 − 50 = ! 0 𝑑𝑦1 𝑦1 = 37,5 𝑦2 = 18,75 𝑦 = 56,25, 𝑝 = 87,5 .

42

c) For (a): 𝑝 𝑦 = 𝑎 − 𝑏𝑦 𝐶 𝑦𝑖 = 𝑐𝑦𝑖 𝜋𝑖 = 𝑝 𝑦 𝑦𝑖 − 𝐶 𝑦𝑖 = 𝑎 − 𝑏𝑦 𝑦𝑖 − 𝑐𝑦𝑖 𝑑𝜋1 = 𝑎 − 2𝑏𝑦1 − 𝑏𝑦2 − 𝑐 = ! 0 𝑑𝑦1 𝑦1 =

𝑎−𝑐−𝑏𝑦 2 2𝑏

.

As the firm‘s cost functions are identical, the best response functions take the same form as well, hence: 𝑦2 =

𝑎−𝑐−𝑏𝑦 1 2𝑏

.

Substitute y2 in y1 𝑦1 =

𝑎−𝑐−𝑏𝑦 1

𝑎−𝑐−𝑏

2𝑏

2𝑏 𝑦1 = 𝑦1 =

=

𝑎 − 𝑐 𝑎 − 𝑐 − 𝑏𝑦1 − 2𝑏 4𝑏

𝑎 − 𝑐 𝑦1 + 4𝑏 4

𝑎−𝑐 = 𝑦2 3𝑏 2 𝑎−𝑐

𝑦=3 𝑏 .

Alternatively, due to symmetry between firms (y1=y2): 𝑦1 =

𝑎 − 𝑐 − 𝑏𝑦2 𝑎 − 𝑐 − 𝑏𝑦1 = 2𝑏 2𝑏 𝑦1 =

𝑎−𝑐 1 − 𝑦1 2𝑏 2 𝑎−𝑐

𝑦1 = 3𝑏 .

For (b): 𝜋1 = 𝑝 𝑦 𝑦1 − 𝐶 𝑦1 = 𝑎 − 𝑏𝑦 𝑦1 − 𝑐𝑦1

43 𝜋1 = 𝑝 𝑦 𝑦1 − 𝐶 𝑦1 = 𝑎 − 𝑏(𝑦1 +

𝑎 − 𝑐 − 𝑏𝑦1 𝑦1 − 𝑐𝑦1 2𝑏

𝜋1 = 𝑝 𝑦 𝑦1 − 𝐶 𝑦1 = 𝑎 − 𝑏𝑦1 −

𝑎 − 𝑐 − 𝑏𝑦1 𝑦1 − 𝑐𝑦1 2

𝜋1 =

𝑎−𝑐 𝑏 𝑦1 − 𝑦12 2 2

𝑑𝜋1 𝑎 − 𝑐 = − 𝑏𝑦1 = ! 0 𝑑𝑦1 2 𝑎−𝑐

𝑦1 = 2𝑏 .

Plug into best response function: 𝑦2 =

𝑎 − 𝑐 − 𝑏𝑦1 𝑎 − 𝑐 𝑎 − 𝑐 𝑎 − 𝑐 = − = 2𝑏 2𝑏 4𝑏 4𝑏 3 𝑎−𝑐

𝑦=4 𝑏 .

Similarly for (c).

4.:

Assume n identical companies in a market described by the above functions. Show that with an increasing number n, the market price converges to the constant marginal cost c.

Solution key: 𝑝 𝑦 = 𝑎 − 𝑏𝑦 𝐶 𝑦𝑖 = 𝑐𝑦𝑖 𝑦=

𝑛 𝑖=1 𝑦𝑖 .

For every firm i: 𝜋𝑖 = 𝑝 𝑦 𝑦𝑖 − 𝐶 𝑦𝑖 = 𝑎 − 𝑏𝑦 𝑦𝑖 − 𝑐𝑦𝑖 .

For the exercise, look at firm n: 𝑑𝜋𝑛 = 𝑎 − 𝑐 − 2𝑏𝑦𝑛 − 𝑑𝑦𝑛

𝑦𝑛 =

𝑎−𝑐−𝑏(𝑛−1)𝑦 𝑖 2𝑏

𝑛−1

𝑦𝑖 = ! 0 𝑖=1 (𝑛 −1)𝑦 𝑖

𝑎−𝑐

= 2𝑏 −

𝑛−1 2

𝑦𝑖 .

As the firm‘s cost functions are identical, the best response functions take the same form as well, hence:

44 𝑦𝑛 =

𝑎−𝑐 𝑛−1 𝑎−𝑐 𝑛−1 − − 𝑦𝑛 2𝑏 2 2𝑏 2

2−𝑛+1 𝑛−1 2 𝑦𝑛 = 𝑎−𝑐 + 𝑦𝑛 4𝑏 2 𝑛−1 2 2−𝑛+1 1− 𝑦𝑛 = 𝑎−𝑐 2 4𝑏 4 𝑛2 − 2𝑛 + 1 2−𝑛+1 − 𝑦𝑛 = 𝑎−𝑐 4 4 4𝑏 3 − 𝑛2 + 2𝑛 𝑦𝑛 = 3 − 𝑛

𝑎−𝑐 𝑏

3 − 𝑛 𝑛 + 1 𝑦𝑛 = 3 − 𝑛

𝑎−𝑐 𝑏

𝑦𝑛 =

1 𝑎−𝑐 𝑛+1 𝑏 𝑛 𝑎−𝑐

𝑦 = 𝑛+1 𝑏 .

Alternatively, due to symmetry between firms (y1=y2): 𝑦𝑛 =

𝑎 − 𝑐 − 𝑏(𝑛 − 1)𝑦𝑛 2𝑏

𝑦𝑛 =

𝑎−𝑐 𝑛−1 − 𝑦1 2𝑏 2 1 𝑎−𝑐

𝑦𝑛 = 𝑛+1 𝑏 .

5.:

Explain the reasoning behind a kinked demand curve.

Solution key: Sweezy (1939) addressed the question of sticky prices in markets. He asked how companies might behave in an oligopoly market in equilibrium. The point was to find reasons for observed stickiness of prices in markets, especially in markets where you might expect price competition to drive prices down, if you applied a more standard model formulation in the analysis. In order to illustrate the point, he assumed that competitors react in different ways to price changes by other companies, depending on whether these would increase or lower their prices. These different reactions introduce a kink in the demand curve of companies. This means that in situations characterized by strategic interdependence, a standard demand function would not be useful.

The basic premise here is that companies seek the protection of their market shares. In that case, Sweezy reasoned that it was unlikely that price increases by a company would

45 be met by increases by their competitors. The result is a relatively elastic demandschedule for a firm for prices above the currently charged one. Market share would be lost and total revenue would be reduced if a firm single-handedly raised the price for its product. On the other hand, he assumed that reductions of prices would be met by competitors. Therefore, the demand schedule for the individual firm for prices below the current one is relatively inelastic. Market shares would not change much, and total revenue would decrease as a consequence.Once firms find themselves in equilibrium in an oligopoly market, the approach can therefore help to understand why market conditions, and particularly the price,may not change.

6.:

When might it be considered preferable to have a market serviced by a monopolist?

Explain why.

Solution key: One particular set of stable monopolies are so-called naturalmonopolies. A natural monopoly is characterized by asubadditive cost function which indicates that it is cheaper to produce a given amount of output for a single firm than for many smaller firms.A cost function is subadditive if 𝑚

𝑛

𝐶 𝑞𝑖 < 𝑖=1

𝑚

𝐶 𝑞𝑗 ∀ 𝑚 < 𝑛 , 𝑤𝑖𝑡ℎ 𝑗 =1

𝑛

𝑞𝑖 = 𝑖=1

𝑞𝑗 = 𝑞 . 𝑗 =1

Given the conditions for a natural monopoly, it is more efficient to have only one firm serving the market.

Subadditivity may arisedue to economies of scale, for instance. Since, in this case average cost falls as output increases (strictsubadditivity), the company with the largest production volume in the market could offer the output at the lowest price and displace its competitors.To avoid the cost of parallel investments, it may be useful to grant monopoly rights to a company and regulate it.

7.:

Explain the situation when weak subadditivity is a factor in a) b)

contestable markets, markets with entry barriers.

Solution key: a) In a contestable market a monopoly price above average cost attracts competitors (entrants) who have free access to the same technology as the establishedfirm (incumbent). The entrants can set a price below the existing monopolist‘s price and take consumers away. The incumbent firm cannot deter entry if it cannot adjust its prices downwards quickly enough. Then, the entrant can engage in a hit and run strategy, entering the market whenthe price is high, making short-term profits and leaving it

46 without any costs (a situation that depends on specific assumptions regarding investments, in turn), when the incumbent lowers its price.

If the incumbent firm has a U-shaped average cost curve (Figure 7.6) and the demand level allows the subadditivity condition to hold, the threat of entry of a competitor would also lead the incumbent to charge the second-best welfare-optimal price ps. But the resulting price output combination (ps, qs) is not sustainable, although the incumbent gains no profits. The firm is not protected from market entry by rival firms, since there is a profitable entry strategy for an entrant. An entrant could challenge the established monopoly by producing a level of output at minimum efficient scale (qe) and charging a price below ps. This may cause a price competitionthat has an end at price pe. At that point, the demand q*-qe is not met. This outcome can be improvedby regulating the market through anentry barrier or through a guaranteed minimum price. b) In real-world markets there are market entry andexit barriers. Thus, the monopoly is not necessarily threatened by an entry and has no incentive to set a social-optimum price.

On the other hand, subadditivity given, the cheapest way to produce any given level of output is to have one firm producing for the whole market. This raises thequestion of how to reach a socially optimal price and socially optimal costs simultaneously.

In the case of weak subadditivity the incumbent sets the Cournot monopoly price, since it does not have to fear the market entry of a competitor. Hence the result is inefficient and the monopoly position is sustainable. But the government has the opportunity to set the first-best price regulating the market, because at pf the monopoly makes positive profits.

8.:

What options do companies have for strategic behavior a) b) c)

amongst themselves and with regard to consumers?

Identify critical assumptions about agents‘ behavior, options, and objectives for arriving at these results.

Solution key: a) There are factors that tend to threaten companies‘ profits in real-world oligopoly markets: Reinforced size growth by others through economies of scale in production technology, network effects on the user side, and the other factors mentioned in Section 7.2. Thus, on the one hand we observe an incentive for companies to try and create a stable environment, in which profits are not threatened, whilst on the other hand each individual company‘s measures to try and increase its protection against uncertain events may lead to problems for others. Still, ways have been found for reducing the threats emerging from this situation. For instance, to avoid price wars and related turbulence in the oligopoly, oligopolists have chosen early in history to calm their direct competition down and reduce their direct interdependence through dissolving an

47 important condition of direct dilemma-prone interdependence – which also is an important common condition of all oligopoly models:Product homogeneity.

Oligopolists may, of course, avoid price competition through „meet the competition― clauses (used also as an entry deterrence), i.e., a credible threat to meet a price-cut by another oligopolist (or potential entrant) through the same price cut on their sides. What may look like fierce price competition in fact is a device to enforce price stability as a coordination form. If a price cut can be expected to be met by price cuts by competitors, oligopolists have every incentive to maintain high-price strategy coordination. The longer a relatively stable environment can be conserved, the better the opportunities to learn about each other and find even more subtle ways for coordinating their policies (tacit collusion). b) Product and price differentiation, through branding, advertising, or quality differences, allowing market segmentation, have occurred as a result of the growing action capabilities of large oligopolistic corporations vis-à-vis their customers, users, consumers,suppliers, and even politics and public administrations. Large oligopolistic firms divide the market and generate their own market segments, where they can behave similarly to monopolists. Consumers are induced into spending on goods and services because of the name and its image, for instance, rather than because of rational examinations of functional or instrumental quality. In institutional-economic terms, we may call this ceremonial belief, ceremonial consumption institutions, or ceremonial institutionalized trust. The set of phenomena that emerged this way in real-world markets from the second half of the 19th century onwards was analyzed as monopolistic competition. Monopolistic competition nowadays applies to virtually all important goods and services such as cars, computers, software, telecommunication, and internet services, private water and energy supplies, fuel, etc. Given differentiated products, the customer/user/consumer has to incur higher costs to collect and process information on the different brands that, in turn, are deliberately made non-transparent, incomparable or incommensurable to customers by the monopolistic competitors.

Binding consumers/customers/users to the oligopolist‘s brand name by advertising, brand management, and reputation building would be a prime strategy for an oligopolist in monopolistic competition. The more she succeeds in steepening her sales curve (the demand curve), the more the sales price p that she can realize will tend to be above her marginal revenue (see the Amoroso-Robinson Relation above). Changes in sales can be met through changes in marketing and other measures for securing demand. In contrast to a competition through prices for a given demand, this changes the focus of competition: Instead of competing in markets, companies compete for markets, as the Schumpeterian notion would take it. c) Critical assumptions concerning the agents as customers regard the autonomy they face with respect to companies. A number of the points we have referred to are based on the assumption that demand can be created and managed. For the agents that means, we assume that they do not show exogenously given preferences but rather, that they are influenced by their social environment when making consumption decisions.

48 There are various influence channels that can be taken for exerting this influence, or for fostering social learning dynamics. Related to these, the motivations behindthe agent‘s decision differ as well. One large field, in Western consumer societies, will be consumption spending as a means for status-signaling (conspicuous consumption).

Creating an image for a product as a potential single-carrier is one aspect, the willingness of the group (who has to interpret the signal sent out) to accept certain signals its counterpart.

49 Solutions Chapter 8

1.:

Consider the following 2-person normal-form games.

a) Can these games be solved using SESDS? b) Which strategy configurations are Nash equilibria in pure strategies in these games? c) Are there further Nash equilibria in mixed strategies? If so, compute the Nash equilibrium strategy for both players for each game. d) Assess the evolutionary stability for each of the pure and mixed strategy Nash equilibria. e) Assess, whether or not each of the equilibria in pure strategies is trembling hand perfect. f) Discuss for each of these games whether or not there are trigger strategies of a supergame consisting of an infinite sequence of repetitions of the respective normal-form game.

I

Player B Strategy Strategy 𝑠𝐵1

2

Strategy Player A

𝑠𝐴1

2

II

3 0

0

Strategy 𝑠𝐴2

𝑠𝐵2

3

1 1

Player B Strategy Strategy 𝑠𝐵1

3

Strategy Player A

𝑠𝐴1

3

III

2 0

0

Strategy 𝑠𝐴2

𝑠𝐵2

2

1 1

Player B Strategy Strategy 𝑠𝐵1

3

Strategy Player A

𝑠𝐴1

3

0 2

2

Strategy 𝑠𝐴2

𝑠𝐵2

0

1 1

50 IV

Player B Strategy Strategy 𝑠𝐵1

0

Strategy Player A

𝑠𝐴1

0

3 2

2

Strategy 𝑠𝐴2

𝑠𝐵2

3

1 1

Solution key: a) Game I: Strategy 2 strictly dominates strategy 1 for both players, strategy 1 can thus be eliminated, resulting in a solution (𝑠𝐴2 − 𝑠𝐵2 , lower right).

Game II: No strategy is strictly dominated; hence the game cannot be solved using SESDS. Game III: Strategy 1 strictly dominates strategy 2 for both players, strategy 2 can thus be eliminated, resulting in a solution (𝑠𝐴1 − 𝑠𝐵1 , upper left).

Game IV: No strategy is strictly dominated; hence the game cannot be solved using SESDS. b) Game I: 𝑠𝐴2 − 𝑠𝐵2 .

Game II: 𝑠𝐴1 − 𝑠𝐵1 , 𝑠𝐴2 − 𝑠𝐵2 .

Game III: 𝑠𝐴1 − 𝑠𝐵1 .

Game IV:𝑠𝐴2 − 𝑠𝐵1 , 𝑠𝐴1 − 𝑠𝐵2 . c) Game I: no further Nash equilibria in mixed strategies.

Game II: Π1 = 3𝑝𝑞 + 2𝑞 1 − 𝑝 + 1 − 𝑝 1 − 𝑞 = 2𝑝𝑞 − 𝑝 + 𝑞 + 1 𝜕Π1 = 2𝑞 − 1 = ! 0 𝜕𝑝 𝑞 ∗ = 0.5 .

Since the game is symmetric, the mixed strategy equilibrium (𝑀) is also symmetric, 𝑝∗ = 𝑞 ∗ = 0.5. The mixed strategy equilibrium is

51 0.5 0.5 − .

0.5 0.5 Game III: no further Nash equilibria in mixed strategies.

Game IV: Π1 = 2𝑝 1 − 𝑞 + 3𝑞 1 − 𝑝 + 1 − 𝑝 1 − 𝑞 = −4𝑝𝑞 + 𝑝 + 2𝑞 + 1 𝜕Π1 = −4𝑞 + 1 = ! 0 𝜕𝑝 𝑞 ∗ = 0.25 .

Since the game is symmetric, the mixed strategy equilibrium (𝑀) is also symmetric, 𝑝∗ = 𝑞 ∗ = 0.25. The mixed strategy equilibrium is 0.25 0.25 − .

0.75 0.75 d) Game I: Test if 𝑠𝐴2 is ESS (against 𝑠𝐴1 ): 𝑠𝐴2 𝑇 𝒜𝑠𝐴2 > 𝑠𝐴1 𝑇 𝒜𝑠𝐴2 0 1

2 0 3 1

0 > 1 0 1

2 0 3 1

0 1

1>0.

Hence, the first condition is strictly fulfilled (against the only other pure strategies and therefore also against all possible mixed strategies between the two), 𝑠𝐴2 is ESS. Game II: Test if 𝑠𝐴1 is ESS (against 𝑠𝐴2 ): 𝑠𝐴1 𝑇 𝒜𝑠𝐴1 > 𝑠𝐴2 𝑇 𝒜𝑠𝐴1 1 0

3 0 2 1

1 > 0 1 0

3 0 2 1

1 0

3 > 2.

Hence, the first condition is strictly fulfilled (against the only other pure strategies and therefore also against all possible mixed strategies between the two), 𝑠𝐴1 is ESS. Test if 𝑠𝐴2 is ESS (against 𝑠𝐴1 ):

0 1

𝑠𝐴2 𝑇 𝒜𝑠𝐴2 > 𝑠𝐴1 𝑇 𝒜𝑠𝐴2 3 0 2 1

0 > 1 0 1

3 0 2 1

0 1

52 1>0.

Hence, the first condition is strictly fulfilled (against the only other pure strategies and therefore also against all possible mixed strategies between the two), 𝑠𝐴2 is ESS. Test if 𝑀 is ESS (against 𝑠𝐴1 and 𝑠𝐴2 , 𝑠𝐴2 first): 𝑀𝑇 𝒜𝑀 = 𝑠𝐴2 𝑇 𝒜𝑀 0.5 0.5

3 2

0 1

0.5 = 0 1 0.5

3 0 2 1

0.5 0.5

1.5 = 1.5 .

Hence, the first condition is fulfilled with equality, thus the second condition must be evaluated 𝑀𝑇 𝒜𝑠𝐴2 < 𝑠𝐴2 𝑇 𝒜𝑠𝐴2 0.5 0.5

3 0 2 1

0 < 0 1 1

3 0 2 1

0 1

0.5 < 1 .

The condition is not fulfilled, 𝑀 is invadable by 𝑠𝐴2 and can thus not be an ESS. Testing against 𝑠𝐴1 is thus not necessary any more. Starting by testing against 𝑠𝐴1 would have yielded the same result (𝑀 invadable by 𝑠𝐴1 ).

Game III: Test if 𝑠𝐴1 is ESS (against 𝑠𝐴2 ): 𝑠𝐴1 𝑇 𝒜𝑠𝐴1 > 𝑠𝐴2 𝑇 𝒜𝑠𝐴1 1 0

3 2 0 1

1 > 0 1 0

3 2 0 1

1 0

3> 1.

Hence, the first condition is strictly fulfilled (against the only other pure strategies and therefore also against all possible mixed strategies between the two), 𝑠𝐴1 is ESS. Game IV: Test if 𝑠𝐴1 is ESS (against 𝑠𝐴2 ): 𝑠𝐴1 𝑇 𝒜𝑠𝐴1 < 𝑠𝐴2 𝑇 𝒜𝑠𝐴1 1 0

0 2 3 1

1 < 0 1 0

0 2 3 1

1 0

0<3.

Hence, the first condition is not fulfilled (𝑠𝐴1 is invadable by 𝑠𝐴2 ), 𝑠𝐴1 is not an ESS. Test if 𝑠𝐴2 is ESS (against 𝑠𝐴1 ):

53 𝑠𝐴2 𝑇 𝒜𝑠𝐴2 < 𝑠𝐴1 𝑇 𝒜𝑠𝐴2 0 1

0 2 3 1

0 < 1 0 1

0 2 3 1

0 1

1<2.

Hence, the first condition is not fulfilled (𝑠𝐴2 is invadable by 𝑠𝐴1 ), 𝑠𝐴2 is not an ESS. Test if 𝑀 is ESS (against 𝑠𝐴1 and 𝑠𝐴2 , 𝑠𝐴2 first): 𝑀𝑇 𝒜𝑀 = 𝑠𝐴2 𝑇 𝒜𝑀 0.25 0.75

0 3

2 1

0.25 = 0 1 0.75

0 2 3 1

0.25 0.75

1.5 = 1.5 .

Hence, the first condition is fulfilled with equality, thus the second condition must be evaluated 𝑀𝑇 𝒜𝑠𝐴2 > 𝑠𝐴2 𝑇 𝒜𝑠𝐴2 0 3

0.25 0.75

2 1

0 > 0 1 1

0 2 3 1

0 1

1.25 > 1 .

The condition is fulfilled, 𝑀 is not invadable by 𝑠𝐴2 , now test against 𝑠𝐴1 : 𝑀𝑇 𝒜𝑀 = 𝑠𝐴1 𝑇 𝒜𝑀 0.25 0.75

0 3

2 1

0.25 = 1 0 0.75

0 2 3 1

0.25 0.75

1.5 = 1.5 .

Hence, the first condition is fulfilled with equality, thus the second condition must be evaluated: 𝑀𝑇 𝒜𝑠𝐴1 > 𝑠𝐴1 𝑇 𝒜𝑠𝐴1 0.25 0.75

0 3

2 1

1 > 1 0 0

0 2 3 1

1 0

2.25 > 0 .

Hence, the conditions are fulfilled against both 𝑠𝐴2 and 𝑠𝐴1 , 𝑀 is thus not invadable by either of the two nor by any mixed strategy between the two (except for 𝑀 itself). 𝑀is consequently an ESS.

54 e) For all four games it can easily be seen that in the Selten-perturbed game, the bestanswer profile and the Nash equilibria remain the same. (For Myerson perturbation, i.e. the assessment of proper equilibria instead of trembling-hand equilibria, the result would be the same.)

Player 2 𝑠𝐵1

I Player 1

𝑠𝐴1

𝑠𝐵1 …3-2𝜀

.2-2𝜀 2-2𝜀

2𝜀

𝑠𝐴2 2𝜀

1+2𝜀

3-2𝜀

Player 2 𝑠𝐵1

II Player 1

1+2𝜀

𝑠𝐴1

3-3𝜀 3-3𝜀

𝑠𝐵1 2-𝜀 3𝜀

𝑠𝐴2 3𝜀

1+𝜀 1+𝜀

2-𝜀

Player 2 𝑠𝐵1

III Player 1

𝑠𝐴1

𝑠𝐵1 𝜀

3-𝜀 3-𝜀

2+𝜀

𝑠𝐴2 2+𝜀

1-𝜀

𝜀 Player 2 𝑠𝐵1

IV Player 1

1-𝜀

𝑠𝐴1

𝑠𝐵1

2𝜀 2𝜀

3-2𝜀 2-2𝜀

2-2𝜀 3-2𝜀

1+2𝜀 1+2𝜀

𝑠𝐴2

55 f) Game I: The game is a standard prisoners‘ dilemma with a Pareto dominated Nash equilibrium. That is, there is a strategy (mutual playing of 𝑠𝐴1 ) which would result in higher payoffs for both players. This strategy is only feasible if there is a credible threat of punishment, i.e. a trigger strategy. However, once the other player deviated from the cooperative arrangement (mutual playing of 𝑠𝐴1 ) 𝑠𝐴2 would be the best answer in the short term and (since𝑠𝐴2 ensures the agents‘ maximin payoff) a credible threat in the long term. 𝑠𝐴2 is also the agents‘ minmax strategy; playing this strategy they can guarantee that the opponent will never get more than 1.

Game II: 𝑠𝐴2 guarantees the agents‘ maximin payoff. While mutual playing of 𝑠𝐴1 would lead to a Pareto superior result, 𝑠𝐴2 is a best answer to deviating from this arrangement (to playing 𝑠𝐴2 instead of 𝑠𝐴1 ); it could therefore serve as a trigger strategy to motivate the opponent to stick to the Pareto superior Nash equilibrium.

Game III: There is a strictly dominant strategy (𝑠𝐴1 ) mutual playing of which also yields the single highest payoff for both players and is thus the only Pareto optimum in this game. Deviating from this strategy is never a rational option; there is no trigger strategy.

Game IV: There are two Pareto optimal Nash equilibria in pure strategies (and one more in mixed strategies) in the underlying normal form game; any of these the Nash equilibria may – depending on the beliefs of the agents – result in the supergame.

However, both the agents‘ minimax strategy (guaranteeing a minimal payoff for the opponent) and the strategy guaranteeing their maximin payoff is 𝑠𝐴2 . Should there be an arrangement involving that an agent plays a strategy other than purely 𝑠𝐴2 then 𝑠𝐴2 could serve as her trigger strategy punishing the opponent for deviation from the arrangement.

56 2.: In the vicinity of an illegal rave, the police arrests two persons, T.H. Cunning and L.S. Deedful; they are accused of having been involved in organizing the party. Both are presented with the following deal: Confess and testify against the other and you will be released immediately if the other person denies having been involved (0 months in prison), while the other is convicted of creating public disorder (12 months in prison); if the other person confesses as well, the punishment for both is reduced (10 months in prison). Should both refuse to testify, the prosecutor lacks convincing evidence and both are released from prison after 1 month detention for questioning.

Write this game in matrix notation and discuss its properties. Which of the solution concepts introduced in this chapter provide are appropriate to derive a solution? To which equilibrium do the respective solution concepts lead? What changes if this game is repeatedly played (say the rave takes place annually and the two above mentioned persons are always the unlucky ones that get arrested)?

Solution key: The resulting game Player 2 Refuse Refuse Player 1

-1 -1

Testify

Testify 0 -12

12 0

-10 -10

is the standard prisoners‘ dilemma. The strategy Testify is strictly dominant, the strategy Refuse can be eliminated in SESDS leading to a solution (Testify-Testify, lower right).

The same outcome is the game‘s single Nash-equilibrium; it is trembling-and perfect, proper, and constitutes an ESS. To see this, note that the best answer profile is identical to that of Game I in Exercise 1. If the game is played repeatedly, the strategy Testify can be used as a trigger strategy and is a credible threat with which the two players can punish deviation from a more cooperative and more mutually beneficial outcome, specifically mutual refusal to testify against each other.

57 3.: Kim, a first year economics undergraduate, while eating a big tub of ice cream at the beach during her/his summer vacation, is approached by a party of extraterrestrial invaders who demand a share of the ice cream. As both Kim and the extraterrestrials are both reasonable and strictly rational beings, they engage in a bargaining game with the extraterrestrials starting to propose a division of the remaining amount of ice cream. If Kim rejects, it is Kim‘s turn to propose a split. The bargaining process takes 1 minute per offer (and rejection). As the sun is burning hot, the ice cream melts slowly but steadily 1/10 of the original amount per minute (i.e. nothing is left after 10 minutes). a) Which outcome of the bargaining process is to be expected? b) Suppose, the extraterrestrials – being unfamiliar with earth food – are unaware of the fact that the ice cream melt. How does the bargaining process and its outcome change? c)

Suppose, the encounter occurs not during summer vacation but during Christmas break when it is cold and the ice cream does not melt. How does the bargaining process and its outcome change?

Solution key: a) The game consists of 9 successive periods in which Kim and the extraterrestrials alternatingly propose a division of the ice cream. The 10th period will not take place any more since no ice cream is left to be shared after this period. The amount of ice cream in period 𝑥 < 10 is exactly (10 − 𝑥)/10. Each proposal is made in such a way that the other player will have an advantage not to reject it (i.e. will obtain marginally more than she could allocate to herself in the successive period. Specifically the proposals in periods x are (𝜀 is an arbitrarily small amount): 𝑥 = 9: (Extraterrestrials proposing) Extraterrestrials:(1 − 𝜀)/10, Kim:𝜀/10 𝑥 = 8: (Kim proposing) Extraterrestrials: 1/10, Kim:1/10 𝑥 = 7: (Extraterrestrials proposing) Extraterrestrials: (2 − 𝜀)/10, Kim:(1 + 𝜀)/10 𝑥 = 6: (Kim proposing) Extraterrestrials:2/10, Kim:2/10 𝑥 = 5: (Extraterrestrials proposing) Extraterrestrials:(3 − 𝜀)/10, Kim:(2 + 𝜀)/10 𝑥 = 4: (Kim proposing) Extraterrestrials:3/10, Kim:3/10 𝑥 = 3: (Extraterrestrials proposing) Extraterrestrials:(4 − 𝜀)/10, Kim:(3 + 𝜀)/10 𝑥 = 2: (Kim proposing) Extraterrestrials:4/10, Kim:4/10 𝑥 = 1: (Extraterrestrials proposing) Extraterrestrials:(5 − 𝜀)/10, Kim:(4 + 𝜀)/10 .

Kim will accept the extraterrestrials‘ suggestion in period 1. b) Consider solution 3.c) (since this is the game the extraterrestrials believe themselves to be in). The extraterrestrials will propose an equal sharing of the ice cream in period 1, i.e. (0.45,0.45) believing this to be identical to the initial amount and also believing that

58 this amount would not decrease in future periods. Kim will accept since this exceeds his/her reservation offer ((4 + 𝜀)/10). c) The amount to be shared does not change which means the decision problem with which each proposing player is faced (each time she proposes a division) is identical. Assume she proposes a sharing (𝑥, 𝑦) with 𝑥 being the amount she allocates to herself (with 𝑥 + 𝑦 = 1). Divisions (𝑥’, 𝑦’) proposed by the opponent will only be acceptable to her if she receives at least as much as she would allocate to herself, 𝑥’ ≥ 𝑥. The same is true for the opponent, 𝑦 ≥ 𝑦’ and their positions are exchangeable, thus 𝑦’ = 𝑥 and 𝑥’ = 𝑦.

The only division for which this is fulfilled and that could be both proposed and accepted by rational players is (0.5, 0.5).

The extraterrestrials will propose an equal sharing of the ice cream in period 1 and Kim will accept.

4.: Consider the game matrices in exercise 1 again. Suppose, these games are played in an evolutionary population setting. a) Which evolutionary stable strategies are there in each of the games? b) Which equilibria are derived in a replicator dynamic model constructed from these normal-form games using the same approach used for the hawk-dove game in section 8.7.3?

Solution key: a) [See 1d).] b) Call the population share playing the first strategy 𝜃1 and the one playing the second strategy 𝜃2 = 1 − 𝜃1 ; call the fitness of the first and second population share 𝑓1 and 𝑓2 respectively.

Game I: We obtain individual fitness terms of the two population shares by computing: 𝑓1 = Π1 = 1 𝑓2 = Π2 = 0 1

2 0 3 1

0

2 3

0 1

𝜃1 = 2 1 − 𝜃1

𝜃1 = 3 1 1 − 𝜃1

as 𝑓1 = 2𝜃1

0

𝜃1 = 2𝜃1 1 − 𝜃1

𝜃1 = 3𝜃1 + 1 − 𝜃1 = 2𝜃1 + 1 1 − 𝜃1

59

𝑓2 = 2𝜃1 + 1 .

This yields the replicator equation 𝑑𝜃1 𝑡 = 𝜃1 1 − 𝜃1 −1 = 𝜃12 − 𝜃1 𝑑𝑡 with fixed points

∗ 𝜃1,1 =0 ∗ 𝜃1,2 = 1

and the eigenvalue of the Jacobian 𝜆=

𝜕 𝑑𝜃 1 𝑡 /𝑑𝑡 𝜕𝜃1

= 2𝜃1 − 1 ,

which for the fixed points assumes the values ∗ 𝜆 𝜃1,1 = 𝜆 0 = −1 ∗ 𝜆 𝜃1,2 =𝜆 1 = 1.

Hence, there is a stable fixed point (the entire population plays 𝑠𝐴2 ) and an unstable one (everyone playing 𝑠𝐴1 ).

Game II:We obtain individual fitness terms of the two population shares by computing: 𝑓1 = Π1 = 1 𝑓2 = Π2 = 0 1

3 0 2 1

0

3 2

0 1

𝜃1 = 3 1 − 𝜃1

𝜃1 = 2 1 1 − 𝜃1

0

𝜃1 = 3𝜃1 1 − 𝜃1

𝜃1 = 2𝜃1 + 1 − 𝜃1 = 𝜃1 + 1 1 − 𝜃1

as 𝑓1 = 3𝜃1 𝑓2 = 𝜃1 + 1 .

This yields the replicator equation 𝑑𝜃1 𝑡 = 𝜃1 1 − 𝜃1 2𝜃1 − 1 = −2𝜃13 + 3𝜃12 − 𝜃1 𝑑𝑡 with fixed points

and the eigenvalue of the Jacobian

∗ 𝜃1,1 =0 ∗ 𝜃1,2 = 0.5 ∗ 𝜃1,3 =1

60 𝜆=

𝜕 𝑑𝜃 1 𝑡 /𝑑𝑡 𝜕𝜃1

= −6𝜃12 + 6𝜃1 − 1 ,

which for the fixed points assumes the values ∗ 𝜆 𝜃1,1 = 𝜆 0 = −1 ∗ 𝜆 𝜃1,2 = 𝜆 0.5 = 0.5 ∗ 𝜆 𝜃1,3 = 𝜆 1 = −1 .

Hence, there are two stable fixed points (the entire population playing 𝑠𝐴1 and 𝑠𝐴2 respectively) and an unstable one (with the population equally divided between 𝑠𝐴1 and 𝑠𝐴2 ).

Game III: We obtain individual fitness terms of the two population shares by computing: 𝑓1 = Π1 = 1 0

3 2 0 1

𝑓2 = Π2 = 0 1

𝜃1 = 3 2 1 − 𝜃1 3 2 0 1

𝜃1 = 3𝜃1 + 2 − 2𝜃1 = 𝜃1 + 2 1 − 𝜃1

𝜃1 = 0 1 1 − 𝜃1

𝜃1 = 1 − 𝜃1 1 − 𝜃1

as 𝑓1 = 𝜃1 + 2 𝑓2 = 1 − 𝜃1 .

This yields the replicator equation 𝑑𝜃1 𝑡 = 𝜃1 1 − 𝜃1 2𝜃1 + 1 = −2𝜃13 + 𝜃12 + 𝜃1 𝑑𝑡 with fixed points

∗ 𝜃1,1 =0 ∗ 𝜃1,2 = 1

∗ as well as a theoretical (but invalid) third fixed point (𝜃1,3 = −0.5) and the eigenvalue of the Jacobian

𝜆=

𝜕 𝑑𝜃 1 𝑡 /𝑑𝑡 𝜕𝜃1

= −6𝜃12 + 2𝜃1 + 1 ,

which for the fixed points assumes the values ∗ 𝜆 𝜃1,1 = 𝜆 0 = +1 ∗ 𝜆 𝜃1,2 = 𝜆 1 = −3.

Hence, there is a stable fixed point (the entire population plays 𝑠𝐴1 ) and an unstable one (everyone playing 𝑠𝐴2 ).

Game IV:We obtain individual fitness terms of the two population shares by computing:

61

𝑓1 = Π1 = 1 0 𝑓2 = Π2 = 0 1

0 2 3 1

0 2 3 1

𝜃1 = 0 1 − 𝜃1

𝜃1 = 3 1 1 − 𝜃1

2

𝜃1 = 2 − 2𝜃1 1 − 𝜃1

𝜃1 = 3𝜃1 + 1 − 𝜃1 = 2𝜃1 + 1 1 − 𝜃1

as 𝑓1 = 2 − 2𝜃1 𝑓2 = 2𝜃1 + 1 .

This yields the replicator equation 𝑑𝜃1 𝑡 = 𝜃1 1 − 𝜃1 1 − 4𝜃1 = 4𝜃13 − 5𝜃12 + 𝜃1 𝑑𝑡 with fixed points

∗ 𝜃1,1 =0 ∗ 𝜃1,2 = 0.25 ∗ 𝜃1,3 =1

and the eigenvalue of the Jacobian 𝜆=

𝜕 𝑑𝜃 1 𝑡 /𝑑𝑡 𝜕𝜃1

= 12𝜃12 − 10𝜃1 + 1 ,

which for the fixed points assumes the values ∗ 𝜆 𝜃1,1 =𝜆 0 =1 ∗ 𝜆 𝜃1,2 = 𝜆 0.25 = −0.75 ∗ 𝜆 𝜃1,3 =𝜆 1 = 3.

Hence, there are two unstable fixed points (the entire population playing 𝑠𝐴1 and 𝑠𝐴2 respectively) as well as a stable one with the population equally divided between 𝑠𝐴1 and 𝑠𝐴2 .

62 Solutions Chapter 9

1.: Obtain the Python virtual machine and the integrated development environment Spyder from the websites given above in Section 9.6.1. Download the Scripts given in this chapter from the textbook‘s companion website. Run each of the short scripts explained in Section 9.6.2. Feel free to try combinations of the concepts that are illustrated by these scripts.

Solution key: This exercise concerned download and installation of programs required to run Python scripts; hence no solution key is necessary for this exercise; the scripts from the textbook website should give the results as shown in the chapter.

2.: Run the longer script discussed in Section 9.6.3. Verify the result given in Section 9.5.2.

Solution key: The script creates a 10x10 grid of cooperating (1) or defecting (0) agents; it prints out the grid for each of the 10 iterations. In the first few iterations, many of the agents remain defectors but are clustered (because the first output occurs only after the first iteration). From about iteration 5 the entire population consists of cooperators: Iteration 5 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111

3.: Append the script from exercise 2 to give additional output which would allow the observer to verify that the neighborhood structure has been arranged correctly. (Hint: The code block given for this purpose in Section 9.6.3 may be used.)

Solution key: The code block should be inserted before the line for t in range(0,iterations): The resulting additional output (which is printed before the iterations commence) is of the structure agent 0 grid position 0 0 has the following neighbors: 1 10 11

63 for all 100 agents; it verifies that the grid and neighborhood structure are arranged correctly.

4.: The simulation conducted using the script in exercise 2 contains stochastic components. As seen earlier in this chapter, it is necessary to verify simulation results, especially if the simulation makes use of random numbers. Suggest a way to do this in the case of the script from exercise 2.

Solution key: In order to verify the simulation results it must be shown that the simulation results are reproducible. An easy way to do this would be to put the entire simulation (from the line agentnumber=0 to right before the line input()) into a for-loop which is repeated for a large number of times (perhaps 100, 1000, or 10000) to see if the simulation result is always the same, i.e. that the entire population switches to cooperation.

Doing this creates another problem: the output becomes incredibly long and reading it becomes time consuming. It is therefore necessary to aggregate the results. This can be done by evaluating at the end of each simulation if all agents are cooperators, e.g. like this: all_cooperators=1 for i in range (0,grid_size): for j in range (0,grid_size): ifagent_arr[i][j]==0: all_cooperators=0 .

The variable all_cooperators then indicates if there are still defectors left (if it equals 0) after 10 iterations (of the current simulation run) or not (if it equals 1). Then the runs with defectors left after 10 iterations are recorded and printed after all repetitions are completed.

This would result in the following modification of the script:

64 #import statements and class and function definitions … runs_with_defectors=0 for v in range (10000): agentnumber=0 … all_cooperators=1 for i in range (0,grid_size): for j in range (0,grid_size): ifagent_arr[i][j]==0: all_cooperators=0 ifall_cooperators==0: runs_with_defectors+=1 print(‘Out of 10000 simulation runs, in ’) print(1-runs_with_defectors) print(‘the entire population switched to cooperation within 10 iterations. In ’) print(runs_with_defectors) print(‘runs there were defectors left’) input() This should usually result in populations with only cooperators for all 10000 runs, i.e. the simulation should pass the verification test.

5.: The number of agents considered in the script in exercise 2 is very low compared to realistic models of economic reality. Increase the number of agents and rerun the script. Do the results change? (Hint: The number of agents in the script is defined by the size of the grid.)

Solution key: The line grid_size=10 has to be changed, e.g. to grid_size=100 for a simulation with 100𝑥100 = 10000 agents. Depending on the size of the population (of the grid), the emergence of the homogenous cooperating population may take longer (e.g. for 100𝑥100 agents, the last defectors commonly last to about iteration 9) but the dynamics and the final result of an emergence of cooperation is not changed.

65 6.: The script from problem 2 models only the second setting discussed in Section 9.5 (Moore neighborhood), not the first one (total connectivity, every agent being the direct neighbor of every other agent). Make the appropriate changes to the script to model the first setting. The following code block is the one that has to be altered: for i in range (0,grid_size): for j in range (0,grid_size): if not i==0: if not j==0: mn(agent_arr[i-1][j-1],agent_arr[i][j]) mn(agent_arr[i-1][j],agent_arr[i][j]) if not j+1==grid_size: mn(agent_arr[i-1][j+1],agent_arr[i][j]) if not j==0: mn(agent_arr[i][j-1],agent_arr[i][j])

Solution key: There are multiple options; one of them is to modify the code block in question to for i in range (0,grid_size): for j in range (0,grid_size): for i2 in range (0,grid_size): for j in range (0,grid_size): if not (i==i2 and j==j2): mn(agent_arr[i][j],agent_arr[i2][j2])

7.: Make further modifications and study the behavior of the simulation program. For instance, try other neighborhood structures or run the simulation with other underlying games (e.g. a coordination game instead of a prisoners‘ dilemma). Feel free to write other simple simulation programs and to conduct your own simulation studies.

Solution key: Since this exercise is an ―open‖ exercise there is no defined and exhaustive solution.

Here are a few hints: For different neighborhood structures, the code block mentioned in Exercise 6 has to be modified again. For instance to create a von Neumann neighborhood (vertically and horizontally adjacent agents belong to the neighborhood, diagonally adjacent agents do not) use the following code

66 for i in range (0,grid_size): for j in range (0,grid_size): if not i==0: mn(agent_arr[i-1][j],agent_arr[i][j]) if not j==0: mn(agent_arr[i][j-1],agent_arr[i][j]) if not i+1==grid_size: mn(agent_arr[i][j],agent_arr[i+1][j]) if not j+1==grid_size: mn(agent_arr[i][j],agent_arr[i][j+1]) .

In order to study different game types exchange the payoff distribution in the play() function. The function goes through all possible outcomes in the order: upper left, upper right, lower left, lower right. agent1 is the row player, agent2 the column player. When considering other game structures, the strategy TFT does for most of those other game structures not make sense. Therefore the examination of the agent‘s memory (the statements and agent1.memors[idx2]==1 and and agent2.memory[idx1]==1) should also be removed from the play() function.

67 Solutions Chapter 10 1.: Consider a difference equation model 𝑥𝑡+1 = 𝑎𝑥𝑡 (1 − 𝑥𝑡2 ) (with 0 ≤ 𝑥𝑡 ≤ 1) a) Depending on parameter a, which fixed points do exist? Are they stable? b) Does the model undergo any bifurcations or other phase transitions? If so, at which point or parameter setting? c) Based on the analysis of the logistic map in this chapter, would you assume that the current system shows chaotic behavior?

Solution key: a) There are two fixed points derived by applying the equilibrium condition 𝑥𝑡+1 = 𝑥𝑡 , 1

∗ 𝑥1∗ = 0, and 𝑥2/3 = ± 1 − 𝑎 , where only the positive form of the square root 𝑥2∗ = 1

+ 1 − 𝑎 is within the boundaries of the valid area of the system.

The stability is assessed by using the eigenvalue of the Jacobian (in this case identical to the single element of the Jacobian) 𝜆 = 𝑎 − 3𝑎𝑥 2 . For the two fixed points the eigenvalue assumes the values𝜆 0 = 𝑎, hence 𝑥1∗ is stable (only) if 𝑎 < 1. In this area, the term under the square root in the formula for the other valid fixed point is negative, i.e. 𝑥2∗ is imaginary in this area. 𝜆

1

1 − 𝑎 = 𝑎 − 2𝑎

1

1−𝑎

2

= 3 − 2𝑎, hence 𝑥2∗ is stable if 1 < 𝑎 < 2 (if𝑎 < 1,

𝜆 > 1, if𝑎 > 2, 𝜆 < −1).(See Figure X.6.)

68

Figure X.6 Bifurcation diagram corresponding to the difference equation in exercise 1, Chapter 10. b) ∗ Yes, the system undergoes the first phase transition when the fixed points 𝑥2/3 become ∗ ∗ real while 𝑥1 becomes unstable at 𝑎 = 1. It has another phase transition when also 𝑥2/3

leave the stable area (the upper and lower boundaries of which are 𝐿𝐶1 = 𝐿𝐶2 =

𝑎−1 3𝑎

𝑎+1 3𝑎

and

respectively) at 𝑎 = 2; here a bifurcation occurs and the dynamic enters a

cycle of length 2 followed by further bifurcations and longer cycle lengths. c) The model is fairly similar to the discrete logistic map model analyzed in the chapter. It also undergoes a series of bifurcations (though starting at different parameter values); it is likely that chaotic regimes occur when 𝑎 becomes larger.

69 2.: Consider the following difference equation models. Determine if there are fixed points and (if so) which of them are stable. a) 𝑥𝑡+1 = 𝑥𝑡 b) 𝑥𝑡+1 = 𝑥𝑡 + 1 c) 𝑥𝑡+1 = 𝑥𝑡 𝑦𝑡+1 = 𝑦𝑡 d) 𝑥𝑡+1 = 𝑦𝑡 𝑦𝑡+1 = 𝑥𝑡 e) 𝑥𝑡+1 = 𝑎𝑥𝑡 f) 𝑥𝑡+1 = 𝑥𝑡2 Solution key: a) Fixed points: The system‘s definition is identical to the equilibrium condition for difference equations; every point 𝑥𝑡 is a fixed point.

Stability:𝜆 = 1. From the eigenvalue it follows that the fixed points (all of them) are marginally stable. b) Fixed points: none (system divergent to +∞). c) Fixed points: The system‘s definition is identical to the equilibrium condition for difference equations; every combination of values 𝑥𝑡 and 𝑦𝑡 is a fixed point.

Stability:0 = det

1 0 1 0 −𝜆 0 1 0 1

= 1−𝜆 2 𝜆1/2 = 1 .

Hence all the fixed points are marginally stable. d) Fixed points: Every combination of values 𝑥𝑡 and 𝑦𝑡 with 𝑥𝑡 = 𝑦𝑡 .

Stability:0 = det

0 1 1 0 −𝜆 1 0 0 1

= −𝜆 2 − 1 𝜆1/2 = ±1 .

70

Hence all the fixed points are marginally stable. e) Fixed points:𝑥1∗ = 0, 𝑥2∗ : 𝑎 = 1 .

Stability:𝜆 = 𝑎. Fixed points are stable if 𝑎 < 1, marginally stable for 𝑎 = 1, unstable otherwise. Each of this may be true for 𝑥1∗ , depending on 𝑎. 𝑥2∗ is always marginally stable. f) Fixed points:𝑥1∗ = 0, 𝑥2∗ = 1.

Stability: 𝜆 = 2𝑥𝑡 . 𝜆(0) = 0first fixed point is stable. 𝜆 = 2second fixed point is unstable.

3.: What is a Taylor series? What can it beused for in economic models?

Solution key: The Taylor series is a mathematical concept used to approximate functions (specifically functions that are complex or for which no simple functional representation exists). An infinite sum over the (1st, 2nd, 3rd, …,ith, …) derivatives of the function in the vicinity of point 𝑐 weighted with 𝑥 − 𝑐 𝑖 /𝑖! is constructed, hence ∞

𝐹 𝑥 = 𝑖=0

𝐹𝑖 𝑐

𝑥−𝑐 𝑖 𝑖!

.

Economically, this may be useful in order to approximate time series of functions derived from more complex models or empirical observations in order to capture then in a functional and computable form (at least in the vicinity of certain values) in spite of their complexity.

71 Solutions Chapter 11 1.: Consider a dynamic system with a single repeller (say 𝑥𝑡+1 = 𝑥𝑡 + 𝑦𝑡 ; 𝑦𝑡+1 = −𝑥𝑡 + 𝑦𝑡 ). a) Could such a system be characterized as a complex system? b) Are complexity and divergence separate phenomena?

Solution key: a) Generally, no. The system behaves in a rather regular way; two equations are obviously sufficient to describe the system. It diverges towards ±∞ while circulating clockwise around the origin. There is no emergent behavior in the system, no specific behavioral changes under any special conditions or in any specific areas of the phase space. On the other hand, the system does not possess any attractors and the only thing we can summarize about it is that it diverges in a circular pattern. Predictions about the state of the system at any specific point in time require exact knowledge of the initial conditions (positive Lyapunov exponents).

Using syntactic measures the system‘s complexity could be assigned a value which would for most measures be larger than that of a converging system but much lower than those of systems with different characteristics in different areas of their phase space.

Using semantic criteria, the system would hardly be considered complex. b) Divergence does contribute to complexity (as convergence towards attractors allows predictions about the system without exact knowledge of its initial state as long as it is in the attractor‘s basin of attraction) but hardly constitutes a sufficient condition of complexity.

72 2.: Consider the following model of the formation of innovative clusters in the economy Triangulum. Triangulum is located on an island shaped like an equilateral triangle; the first and largest cluster (the capital) encompasses a smaller triangle the three corners of which are exactly in the middle of the three sides of the island. This splits the island into 4 smaller equilateral triangles: one developed and 3 undeveloped regions. In the undeveloped regions, smaller clusters are formed in the same way as before on the whole island, a process that then continues infinitely. The prime minister of Triangulum claims that the undeveloped region in that will remain in his country is a complex structure.

a) How can the prime minister‘s statement be verified? Which measures of complexity do you know? Are they well-suited to assess the complexity of the undeveloped region? b) Compute the fractal dimension (Hausdorff dimension) of the undeveloped region as it will remain after an infinite number of steps of the cluster-formation process. c) What is the area and what is the perimeter of the undeveloped region? d) Assume that the process ceases after x iterations. What is the fractal dimension (Hausdorff dimension) of the undeveloped region in this case? What is the difference to the situation in b?

Solution key: a) Complexity of a structure could be measured in different ways including the fractal dimension of the structure (see 2.b, if it has a fractal dimension it may be seen as more complex than non-fractal objects; if it has a higher fractal dimension it may be seen as more complex), its information complexity (how much information does it require to describe the structure), as well as the computational complexity (e.g. the Kolmogorov complexity; how long would a program be that computes the structure), and potentially many more. All of these would be feasible though potentially complicated to compute (e.g. evaluating the Kolmogorov complexity would require finding the shortest possible program able to reproduce the structure).

Alternatively there are also a number of semantic measures of complexity some of which could be applied here (e.g. composition out of a huge number of individual shapes: yes) while others could not (e.g. adaptiveness, emergence, since the structure is not dynamic). b) In this model, the base unit is split in 2 with each iteration; the triangle is thereby split in 4 smaller triangles, 3 of which are part of the structure. Therefore we find corresponding to the progression 𝜁 = 1,2,3,4, … (and hence an increasing sequence of triangles 1,4,16,64) a progression 𝒩 = 1,3,9,27, … the formula for which is 𝒩 = 3log 2 𝜁 . This yields the Hausdorff dimension (fractal dimension)

73

log 3log 2 𝜁 log 3 log 2 𝜁 log 3 log 𝜁 log 3 = lim = lim = lim = 1.58496 . 𝜁 →∞ 𝜁 →∞ 𝜁 →∞ log 2 log 𝜁 𝜁→∞ log 2 log 𝜁 log 𝜁

𝐷𝐻 = lim c)

The limit (for 𝜁 → ∞) of the area of this region is 0, the limit of its perimeter is ∞; it is a fractal structure as seen from the fractal dimension computed in 2.b. d) In this case, from iteration 𝜁′ on (in which the process stops), 𝒩 would not follow the sequence above any more but would be a constant share 𝛼 of the total number of triangles. The total number of triangles develops according to 𝜁 2 , therefore 𝒩 = 𝛼𝜁 2 with the Hausdorff dimension now being log 𝛼𝜁 2 log 𝛼 + 2 log 𝜁 log 𝜁 log 𝛼 = lim = 2 lim + lim =2. 𝜁→∞ log 𝜁 𝜁→∞ 𝜁 →∞ log 𝜁 𝜁 →∞ log 𝜁 log 𝜁

𝐷𝐻 = lim

The dimension of the structure is not fractal any more.

3.: Consider an industry with many small and no large firms (think of bakeries) using two competing types of technology. The firms ‗reproduce‘ by former employees founding startups using the same technology at a constant rate. Demand is, however, limited and if too many firms are in operation, not all of them can cover their production costs and some are forced out of business. Let the system be represented by the following system of equations (demand limit is normalized to 1) 𝑑𝑥1 = 𝑥1 1 − 𝑥1 − 𝑥2 𝑑𝑡 𝑑𝑥2 = 𝑥2 1 − 𝑥1 − 𝑥2 . 𝑑𝑡 a) Derive the fixed points of the system and assess their stability. b) How would the dynamics change if one of the technologies would allow to serve a larger customer base, i.e. the limit of demand is increased to, say, 2 with one of the equations now consequently being (in this case the second one)

𝑑𝑥 2 𝑑𝑡

=

𝑥2 2 − 𝑥1 − 𝑥2 .

Solution key: a) 𝑑𝑥

𝑑𝑥

The equilibrium conditions are 𝑑𝑡1 = 0 and 𝑑𝑡2 = 0 which must be fulfilled simultaneously. It is derived by starting with one condition and substituting the result into the other one. Starting with the first one, it follows that there are two cases for

74 which the condition holds, 𝑥1 = 0 and 1 − 𝑥1 − 𝑥2 = 0 (and thus 𝑥2 = 1 − 𝑥1 ). The first case (substituted into the second condition) yields 𝑥2 − 𝑥22 = 𝑥2 (1 − 𝑥2 ) = 0 which is fulfilled if either 𝑥2 = 0or 𝑥2 = 1. Substituting the second case into the second condition leads to 0 = 0, which is always true. ∗ ∗ Consequently, there are two equilibria, the first one being 𝑥1,1 = 0, 𝑥2,1 = 0 (i.e. none ∗ ∗ of the two technologies is actually used) while the second one is 𝑥2,1 = 1 − 𝑥1,2 (and every combination fulfilling this condition is a fixed point).

b) To evaluate the stability, derive the eigenvalues from the Jacobian of the system. The Jacobian is 𝒥=

1 − 2𝑥1 − 𝑥2 −𝑥2

−𝑥1 .

1 − 𝑥1 − 2𝑥2

Now, either the Jacobian is linearized at the fixed points or an equation which depends on 𝑥1 and 𝑥2 is derived which can be linearized afterwards. Taking the latter option the eigenvalues follow from 0 = 1 − 2𝑥1 − 𝑥2 − 𝜆 1 − 𝑥1 − 2𝑥2 − 𝜆 − 𝑥1 𝑥2 as 2 − 3𝑥1 − 3𝑥2 𝑥12 + 𝑥22 + 2𝑥1 𝑥2 𝜆1/2 = ± , 2 4 which, linearized for the fixed points, gives 𝜆1/2 (0,0) = 1 ± 0 𝜆1/2 𝑥1 ,1 − 𝑥1 =

2−3𝑥 1 −3(1−𝑥 1 ) 2

±

𝑥 12 + 1−𝑥 1 2 +2𝑥 1 (1−𝑥 1 ) 4

1

= −2 ±

1 4

.

Thus, for the first equilibrium, both eigenvalues are 1 and do not fulfill the equilibrium condition (𝜆 < 0), the fixed point is unstable; for the second, the eigenvalues are 𝜆1 = 0and 𝜆2 = −1, thus the (set of) fixed point(s) is (are all) marginally stable.

75 4.: Consider again the MKG model example given in Eq. (11.17). a) What would change if the tipping point level (from which on the dynamic is growing) were increased in both equations? I.e. i.

𝑑𝑥 1 𝑑𝑡 𝑑𝑥 2

= 𝑥1 𝑥2 − 2𝑥1 = 𝑥1 (𝑥2 − 2)

ii. 𝑑𝑡 = −𝑥1 𝑥2 + 2𝑥2 = 𝑥2 −𝑥1 + 2 . b) Explain how the system and the solution would change if the system was a system of difference equations instead of a system of differential equations (note that for this system, the original economic interpretation does not apply any more), i.e. i. 𝑥1,𝑡+1 = 𝑥1,𝑡 𝑥2,𝑡 − 𝑥1,𝑡 = 𝑥1,𝑡 (𝑥2,𝑡 − 1) ii. 𝑥2,𝑡+1 = −𝑥1,𝑡 𝑥2,𝑡 + 𝑥2,𝑡 = 𝑥2,𝑡 −𝑥1,𝑡 + 1 .

Solution key: a) 𝑑𝑥

𝑑𝑥

As in Exercise 3, the equilibrium condition is 𝑑𝑡1 = 0and 𝑑𝑡2 = 0. It gives the fixed ∗ ∗ ∗ ∗ points 𝑥1,1 = 0, 𝑥2,1 = 0and 𝑥1,2 = 2, 𝑥2,2 = 2. 𝑥2 − 2 𝑥1 . Now, either the Jacobian is linearized at the −𝑥2 2 − 𝑥1 fixed points or an equation which depends on 𝑥1 and 𝑥2 is derived which can be linearized afterwards. Taking the latter option, the eigenvalues follow from 0 = 𝑥2 − 2 − 𝜆 2 − 𝑥1 − 𝜆 + 𝑥1 𝑥2 The Jacobian is 𝒥 =

as 𝜆1/2 =

𝑥2 − 𝑥1 ± 2

𝑥1 − 𝑥2 2 − 8(𝑥1 + 𝑥2 ) + 16 .

4

Linearization for the two fixed points yields 𝜆1/2 0,0 = 0 ± 4 = ±2 𝜆1/2 2,2 = 0 ± −4 = ±2𝑖 .

Hence the first fixed point (0,0) is unstable (a saddle point) while the second one is marginally stable with a cyclical dynamic around it. b) The system is a difference equation system with the same Jacobian as the differential equation system analyzed in Section 11.3.5. What changes is the equilibrium condition and the interpretation of the eigenvalues (though the eigenvalues remain the same). The equilibrium condition is now 𝑥1,𝑡+1 = 𝑥1,𝑡 , 𝑥2,𝑡+1 = 𝑥2,𝑡 . This yields (substituted into the system‘s equations) that 𝑥1∗ = 0 for all fixed points while 𝑥2 may take any value (and the resulting state remains a fixed point).

76

To assess the stability, the eigenvalues of the Jacobian – linearized for the fixed points – must be computed. Since the computed set of fixed points has variable 𝑥2 instead of a scalar value, the eigenvalues should be derived from the Jacobian 𝑥 −1 𝑥1 𝒥= 2 as functions of 𝑥2 , specifically −𝑥2 1 − 𝑥1 𝜆1/2 =

𝑥 2 −𝑥 1 2

±

𝑥 1 −𝑥 2 2 −4(𝑥 1 +𝑥 2 )+4 4

.

And for the fixed point (with 𝑥1 = 0 linearized) 𝜆1/2 =

𝑥2 𝑥22 − 4𝑥2 + 4 𝑥2 ± = ± 2 4 2

𝑥2 − 2 2 𝑥2 𝑥2 = ± −1 .

4 2 2

Hence 𝜆1 = 𝑥2 − 1and 𝜆2 = −1. The fixed points (𝑥1∗ = 0, 𝑥2∗ ∈ ℜ) are marginally stable for 0 ≤ 𝑥2∗ ≤ 2, and unstable otherwise.

77 Solutions Chapter 12

Early Scottish Classical Economics and Adam Smith

1.:

Please discuss the relations among the conceptions of ‗unintended consequences‘, ‗invisible hand‘, and ‗fallacy of aggregation‘. Refer to the historical optimism of the Scottish Enlightenment. Also refer to the vulgar apologetic version of Bernard Mandeville.

Solution key: The distinction, and potential divergence, between the individual, its intentions and actions, and the system of individuals, a society, population or group and its common or collective outcome (at the system level) is ascribed to B. de Mandeville‘s ―Fable of the Bees‖ (1705/1714/1723), although ancient and medieval philosophers had dealt with the issue as well. The idea has a positive variant, established, in fact, by Mandeville: the idea that eventhe most selfish and asocial individual ambitions (of the rich and powerful) may lead to the betterment of the general condition (―private vices, public benefits‖). The idea was considerably modified and elaborated by the Scottish Enlightenment (F. Hutcheson, D. Hume, A. Ferguson, A. Smith, among others) and found its most elaborated formulation in the works of Adam Smith, popularly known as the idea of an ―invisible hand‖. Here, a decentralized competitive system – under the condition of well-established social institutions moderating ―selfishness‖ into ―prudence‖(his TMS)!–may aggregate individual motivations and actions into an optimal economic result. The conception of unintended consequences, however, although it was always present in social philosophy and political economy, has experienced a considerable theoretical revival in the social sciences in the 20th century, and, in more recent times, also in complexity economics,reflecting increasing collective good problems, systems of mutual (positive or negative) externalities, and social dilemmas of all kinds in a non-―perfect‖ capitalist market economy. While Samuelson/Nordhaus have coined the idea a ―fallacy of composition‖, we have preferred the term ―fallacy of aggregation‖. The most simple and obvious little structure reflecting and illustrating it, is a PD (the career of which in economics has, in fact, promoted the revival of the understanding of unintended consequences in economics), but also a stag hunt game (assurance game – A. Sen), played under distrust, implies some unintended consequences, perhaps to be discussed with students. You also might wish to discuss with students, how far even an anti-coordination game (chicken/hawkdove) may have aspects of unintended consequences (for hawk). The latter may be discussed sequentially in a static normal-form one shot perspective, with the ambiguities of pure strategies in a population, and, lastly, in an evolutionary game theory perspective (with a mixed strategy being some answer to the problem of unintended consequences).

78 2.:

How may complex direct interdependence lead to unintended consequences in the sense of a fallacy of aggregation?

Solution key: You may have the different basic normal-form game structures discussed and checked in detail for unintended consequences and resulting fallacies of aggregation, as indicated above for Exercise 1.

3.:

Please explain how Adam Smith, in his Theory of Moral Sentiments, explained the emergence of basic social institutions through a consideration of others by the individual. Refer to his dual motivations and the mechanism of ‗changing places in fancy‘.

Solution key: You may have the content of the TMS reproduced by the students, using the modern conceptions of ―recognized interdependence‖, ―empathy‖, ―futurity‖ (expectations ―to meet …‖), reputation chains, reciprocity, and habituation. You may stress the fact that psychologists have qualified the approach as fully consistent with the knowledge and perspective of modern psychology. Note that Smith tends to end up with some normative content of the institutions rather than a strict long-run calculation (such as in the single-shot approach), emerging in a multi-personal process of ―changing places in the fancy‖. Also note, again, that, in Smith‘s view, such institutionalization / ―normativation‖ makes a capitalist market society workable, rather than destructive, by moderating individualism from selfishness to prudence. In this way, Smith was already aware that capitalist markets may destroy themselves, if not embedded in a proper institutional setting that civilizes and mitigates individualism, and that, in contrast, there easily may institutional conditions come to the fore, where social rules let selfishness, greed, and predation dominate.

4.:

Give a (non-formal) game-theoretic interpretation of the mechanism of ―changing places in fancy‖, using the one-dimensional approach of (long-run) maximizing agents.

Solution key: As indicated above, the logic of a one-shot normal-form consideration of individuals deciding simultaneously requires a first step of recognizing interdependence. Although the logic here still is focused on strict own maximization, it is easy to see, that in recurrent interactions, an agent might come to take a second step and ask, how the other one might perceive his action and how he would react in the future, and whether this is what he really would want in the long run … So students might learn that Smith‘s TMS provides a convincing mechanism and story for agents acquiring a more long-run perspective and solve dilemma problems by a different ―rationality‖. The difference still remains that game-theoretic reasoning is still focused on calculation, while Smith‘s focus is on building qualitative norms and a ―moral‖ of reciprocity. You might have the students consider whether this an important difference, or whether the main difference is taking the second step of recognized interdependence, i.e., ―changing places in

79 fancy‖, equivalent with the transition from one-shot to a supergame logic and singleshot solution.

Early Evolutionary Understandings in Post-Smithian and ‘Darwinian’ Economics

5.:

Please explain Malthus‘ conception of a periodically increasing selective pressure on humankind.

Solution key: This appears to be a quite simple logic, if the two different algebraic growth mechanisms and paths of the ―population law‖ and the ―law of agricultural production‖ apply. You may want to discuss with the students the historically diverse experience in this respect: While it appeared common knowledge up to the 1980s that the developed countries did overcome both laws, the global picture in the 2010s appears quite different, with renewed limits to resources and agricultural production, repeatedly exploding food prices, and dozens of millions of people dying from hunger. So while Malthus in many respects appears sarcastic and inhuman nowadays, isn‘t he right, and back on the scene with his view?

6.:

Please explain Malthus‘ understanding of a ‗principle of natural selection‘. Why does it tend to be too ‗biologistic‘?

Solution key: Of course, Malthus‘ merit is in his first provision of some evolutionary, cumulative, but also cyclical, process.On the other hand, both biology (Darwin) and social science (Marx) transcended his simple mechanisms quickly. Viewed from today‘s state of the art of evolutionary sciences (physics, biology, anthropology, sociology, psychology, economics …), in many respects a general, global, optimizing ―natural selection‖ mechanism does not exist, but is fragmented in manifold ways, and also interlocked in complex ways with diversification mechanisms. In this way, in the historical process of the development of modern evolutionary and complexity sciences, Malthus has mostly been found on the side of those misinterpreting the Darwinian insights (the Veblenian ones in economics) in simplistic, and often dangerous (―social-darwinistic‖) ways.

7.:

Please explain Marx‘ understanding of the historical evolution of the capitalist system.

Why is capitalism a (over-) complex system in his perspective?

Solution key: Marx methodological approach was to elaborate systemic contradictions, particularly in the capitalist system. A basic dynamic stems from the individualist rationality of the capitalist to cut costs and press wages down, profiting from an oversupply of labor, which in total then may lead to an adverse macroeconomic income distribution, which, in turn, dampens growth, causes imbalanced inter-sectoral growth (in a two-sector growth model) and makes the economy cycle, perhaps on a longer-run trend. Regulation

80 and institutionalization of some macroeconomic planning, and, in the last instance, the dissolution of the core mechanism of making individualist maximum profit from the competing individual capitals (socialization of the means of production) would, in this view, also reduce the (―over-―) complexity (or ―unnecessary‖ complexity), turbulence, and (in his wording) ―anarchy‖ of the rivalry among the capitals on the one hand and the capitals vs. the workers on the other (both are just permanent redistribution fights for the surplus) ... In all, as Marx was much interested in algebraic issues, and systemic issues, he was on the way towards applying to capitalism what developed shortly later as complexity sciences in physics, chemistry, biology, etc. Finally, note that you may discuss the dynamics of conflicting interests more formally either referring to the profitrate formula as given in the text or to a simple two-sector growth model (consumptiongoods sector and production-goods sector), and the resulting equilibrium condition of extended reproduction (where wages from both absorb the first sector‘s product, and surplus and reproduction of constant capital absorb the second sector‘s product).

8.:

Please explain Menger‘s ‗organic‘ analogy to the evolution of the market system. Why is it not Darwinian evolution?

Solution key: As we have tried to explain in the text, Menger contributed to the contemporary understanding of evolutionary processes. However, he did not embed his considerations in a Darwinian population thinking (―phylogenetics‖ or ―phylogeny‖) but in an organic conceptual frame (―ontogenetics‖ or ―ontogeny‖), where the elements have their roles to play to form a collective unity. ―Organicism‖ has been much misused in connection with ―Social Darwinism‖ since then. But admittedly, the differences between phylogenetic and ontogenetic mechanisms (and, with this, respective analogies in the social sciences) had not been clarified that much (not even in the contemporary interpretation of Darwin himself), as it is today.

9.:

Please explain Menger‘s approach to the evolution of money.

Solution key: Take as an instance of Menger‘s approach his theory of the evolution of money, a theoretical milestone of ―emergence‖ (although money nowadays is mostly claimed to stem from bureaucracy and state power) ...

10.: Why did Marshall aspire for an ‗organic‘ analogy and consider biology as the ‗Mecca of the economist‘?

Solution key: When working through the details of market supply and demand emergence and interactions, Marshall recognized ever more complexity and complex process, so that he realized that a static optimization approach might not be appropriate in many respect.

Comprehending the interaction of many agents led him to realize that many conclusions drawn from static optimization would reverse under more realistic assumptions …

81 11.: What did Marshall learn for economic evolution from his industrial analysis?

Solution key: You may take the industrial agglomeration case as an example of the above and perhaps refer to parts of Chapter 4 (spatial clusters) and Chapter 16 (the networking firm; regional innovation systems). Ideally, in a (at Marshall‘s time) strictly neoclassical perspective, firms were not expected to agglomerate in space. So Marshall had to extend the perspective to a different systemic view with externalities of knowledge and infrastructures as common goods, in order to explain such unexpected behavior. It is, of course, not that this cannot be modeled in a neoclassical frame, considering costs of bridging distance and other cost and profit factors for an extended marginal cost-benefit analysis (see models from J.H. von Thuenen, through W. Christaller, A. Weber, to O. Williamson‘s theory of network forms as hybrids of ―market‖ and ―hierarchy‖). But our argument, both in Chapter 4 and Chapter 16 (on the networking firm), seems more consistent with what Marshall felt would be the ―Mecca‖: Agglomerations typically do develop clusters and networks as systemic forms to deal with complexity problems …

Veblen, Keynes, Schumpeter 12.: Explain Veblen‘s critique of neoclassical economics.

Solution key: As students, who are taught history of (complexity) economic(s) thought, have studied principles of neoclassical microeconomics already, we suggest reading and discussion Veblen‘s 1898 piece ―Why is economics not an evolutionary science?‖ in original.

While his style and wordings in general are somewhat strange, students will enjoy the fresh and often ironic, but always consequently dissecting approach. It is not at all a piece of today‘s state-of-the-art criticism of mainstream economics (neither from the side of the state of the mainstream criticized, which was relatively crude then, nor from the side of the criticism, which was relatively crude as well), but a central piece in the history of thought, which founded the evolutionary-institutional school in economics (the so-called old or original institutional economics—OIE), which, in turn, had its better and worse times throughout the 20th century. Students should read this being aware of the central role of this article and the kind of argument put forward against the then new neoclassical economics and in favor of an evolutionary perspective. Also note that Veblen was more in favor of stressing the evolutionary alternative than being aware of the potential role of institutions as social coordination devices. In fact, he was quite critical towards institutions, as being always and inevitably past-bound and devices of predation.

82 13.: Please explain Veblen‘s understanding of the evolution of social institutions. Why is it Darwinian and non-teleological?

Solution key: In addition to the article mentioned, you might want to read a chapter, a more theoretical one or a more applied, story-telling one, from his Theory of the Leisure Class (1899). You may try to make the students aware that the method is not strictly theoretical but largely historical, with some evolutionary principles, such as cumulative causation, with no beginning and no end, only process. Nevertheless, different from Malthus, he was able to refer to Darwin‘s insights and to try to apply it to economics, in fact, the first economist, who did this on a broad scale. Note his fierce criticism of ―teleology‖, where he (misleadingly) subsumed classical political economy, including Smith and Marx, and neoclassical modeling. And in proceeding to a more systematic, theoretical teaching, rather than just teaching HET, you may bring in the sections on later institutionalists, such as C.A. Ayres and the P.D. Bush formalization of the Veblenian institutional dichotomy in Section 13.12, in order to demonstrate how the somewhat one-sided and overly critical view of Veblen was rectified by later institutionalists.

14.: Why does Keynes‘ understanding of aggregation and ‗macro‘ belong to the ‗fallacy of aggregation‘ side of economic theory, rather than the ‗invisible hand‘ side?

Solution key: The basic approach here is to be after the Keynesian microfoundation, i.e., his approach to strong uncertainty, often wrong expectations, and the role of ―animal spirits‖, on the side of entrepreneurs, investors, and speculators, his approach to imitating and herd behavior, hoarding rather than investing in typical situations, which in all appear individually rational, although often just helpless, often lead to mutual blockage of appropriate action, and generate an imbalanced macroeconomic outcome. It is obvious that those instructors inclined try to draw parallels of the Keynesian ―liquidity trap‖, hoarding, and equilibrium at any level of unemployment with the 2008ff. financial and economic crisis, should do so.

15.: Please explain Keynes‘ understanding of fundamental uncertainty, the investor‘s recourse to ‗animal spirits‘, and the agent‘s general reference to habits.

Solution key: As modern ―Post-Keynesians‖ argue, Keynes‘ microfoundations were overlooked in its radical consequences for microeconomics most of the time of post-WWII. So the didactical objective here might be to show that Keynes‘ theory is not just macro, but does have a substantial microfoundation, which, in turn, appears very modern and consistent with complexity economics in its use of uncertainty (you might wish to refer to Keynes‘ earlier work on probability here) and habits/‖spirits‖/institutions/social rules ...

83 16.: Why are money and prices complex phenomena in Keynes‘ theory?

Solution key: Basically, we intend here to refer to J. Kregel’s approach to money supply and demand and (his ―money theory‖ of) price formation (see Section 12.8.3).

17.: What is the specific Schumpeterian approach to evolution?

Solution key: Refer here to Schumpeter‘s motivational theory (e.g., ―energetic egoism‖), his conceptions of complexity, uncertainty, search, experimentation, all centered on a broad approach to innovation of what he terms the agent type and function of an ―entrepreneur‖. You may also refer to his broader understanding ofcompetition, including temporary monopoly and changing conditions (of competition) in the life cycles (longer and shorter waves) of products and industries. While much of resulting evolutionary process is endogenously explained, some elements remain a black box, such as the emergence of fundamental inventions (as the basis for long waves) or of entrepreneurs. Note that the early Schumpeter (Schumpeter I) developed a much more optimistic approach then the later Schumpeter (Schumpeter II), who dealt more with corporate than with entrepreneurial capitalism. Neo-Schumpeterians today represent a wide range of interpretations of the Schumpeterian system of thought between enthusiastic disciples of entrepreneurship and new, young and innovative business startups and a critical stance towards late capitalism as a power system and its systemic limits to entrepreneurial innovation.

18.: How is Schumpeterian entrepreneurial innovation subject to a diversification and a selection mechanism?

Solution key: This exercise is just to retell the Schumpeterian entrepreneur/innovation story from the specific angle of the two basic evolutionary principles. Note that innovation then has to do with (1) search, experiment, and learning under complex conditions, i.e., with bounded rationality and imperfect/incomplete information and (2) monopolistic competition in the sense that there is sufficient expectation to yield temporary monopolistic profit by establishing a niche, being different. Successful innovation and positive selection then requires a certain combination of properties of the product, production organization, knowledge and strategy, and certain market conditions (including, among others, relative costs and prices). Note that Schumpeter came closer to the idea of a working competitive market with the idea of sorting certain successful innovations out later, or generalizing them and equalizing them with the production of many suppliers.This occurs through a number of competing and imitating further suppliers. But also note that Schumpeter was aware of the evolutionary character of technology and evolution, including cumulative aspects of technology development, depending on brain, effort, and capital successively invested in it. This, finally, also relates to the possibility of economies of scale in production, a decreasing learning curve, the tacit or codifiable character of the knowledge involved, etc., which in all

84 leaves it open, whether and for how long the innovator will be able to maintain his monopoly position

Post-Veblenian Evolutionary-Institutional and Ecological Economics 19.: Please explain K. Polanyi‘s conception of a ‗Great Transformation‘.

Solution key: Make sure that the students explain (1) the ―marketization‖ movement and the counter movement of protection of certain societal areas against commodification (the ―double movement‖), (2) the social/institutional ―embedding/disembedding‖ idea of markets, and (3) the limits of commodification of certain ―goods‖ (labor, land, and money), the ―fictitious commodities‖ issue.

20.: Why are land (nature), labor, and money not easily marketable commodities but complex economic factors?

Solution key: You may wish to make the students aware of the multidimensional characters of the three ―fictitious commodities‖, where ―land‖ includes resources and ecology, landscape and space to move … These multiple properties and functions of land may not become subject to any price movement or any market concentration. Similarly, you might try to make students aware of the multiple characters and functions of labor and money.

21.: Please explain Myrdal‘s understanding of circular cumulative causations and processes.

Solution key: Students should be aware of the different particular meanings of the theoretical conceptions of ―cumulativity‖ (positive feedback) and ―circularity‖ (repeated ―rounds‖ of causation among the same critical factors, at ever higher or lower levels), suggesting the theoretical figure of a spiral, upwards or downwards. The examples might be taken from longer-run spatial up- or downgrading processes at local, regional, or national levels. You may make the students note that so-called backwash effects may at certain critical phases of a long-wave process be accompanied and eventually even overcompensated by so-called spread-effects, be these fully equalizing or not. Spread effects might reverse the character of a CCC process.

22.: Please explain Simon‘s conception of ‗bounded rationality‘.

Solution key: You may have introduced the specific definition of bounded rationality, according to Simon, and compared to other forms of imperfect or incomplete information and

85 definitions of uncertainty. There are a number of schematic overviews over these related conceptions available in the literature. You may make the students aware that bounded rationality in general refers to a complex economic environment, with large amounts of information bits potentially available to the individual (or individual organization), but, in spite of huge amounts of information, nevertheless, no way to gain full information in the sense of complete and perfect information on the global status of the whole relevant system. Bounded rationality, in particular, then refers to the relatively limited capacity of the human ―brain‖ (including perhaps also a related psychological capacity) to process all available information into disposable knowledge. Note that this is not just an issue of two different quantities (larger supply than can be processed), but also a question of quality of information: Under complexity conditions, the individual may not be able make clear distinctions on, e.g., the global or local character of a certain bit of information. You might also make the students aware of the fact that the computer, although inferior in many respects to the human brain and comprehensive decision capacity (context-building, framing, expectations-building, imagination, etc.), is superior to the human brain with respect of calculating interrelated functions (equation systems) with numbers, provided it is given enough processing time. The computer is definitely superior to the human brain in calculating and simulating a given complex system with feedback loops, a recursive structure, and among many agents – a condition, where the human brain would be unable to calculate even some few iterations (you may refer in this context to Chapter 9). Also note that when information needs to be sorted out after qualities and quantitatively filtered, then providing an appropriate information basis for decision-taking is costly.

23.: Please explain how ‗bounded rationality‘ relates to complexity and causes ‗satisficing‘ behavior rather than maximizing.

Solution key: The first remarks to Exercise 22 above apply here, and a reference might be made to particular parts of Chapter 11 (e.g., the definitions of complexity of a system and/or of an individual decision structure, and the position of the individual therein). In addition to that, you may want to relate the imperfect state of information and knowledge of the economic agent with its imperfect decision-making, its reliance on rules of thumb, to be developed from recurrent experience with decision-making, and its reduction of its aspiration level – i.e., satisficing strategies rather than any abstract maximizing, or local improvement rather than global ―peaking out‖.

24.: Please explain the complexity stemming from the economy as a sub-system open to, and exploiting, another sub-system (nature), in Georgescu-Roegen‘s theory.

Solution key: You may want to teach and discuss this more in real-world terms, in verbal form, promoting the qualitative understanding of the students, or relate this to the formal forms of mathematical systems mapping the ―real‖ economic and natural systems – and either made a-priori closed in the sense of (in the extreme neoclassical case) a unique (and stable, predetermined) equilibrium, or designed in an ―open‖ way in the sense of open (undetermined) and open-ended results, which need to be found out, on the basis

86 of complex modeling, through (ideally) computer simulations. Note that in the latter case, there remains a large requirement of partial closing and calibration. Nevertheless, the ―research program‖ is basically different from the former in that the specific dynamic properties and paths of the system are to be explored rather than determined by its input. Note that the complexity view on the economic system alone is a major issue, but that Georgescu-Roegen introduced the openness and interrelation of such system with another complex system. Both systems and their interrelation then need to be modeled, which should increase the complexity of the whole (combined) system.

25.: Please explain the tendency towards entropy of the economic system in GeorgescuRoegen‘s theory.

Solution key: This interrelation is what Georgescu-Roegen has explicitly conceptualized and modeled according to the entropy law. It is about maintaining or increasing one system‘s complexity through its metabolism with another one, as known from biological and physico-chemical systems, i.e., absorbing available and disposing bound (unavailable) energy. A major challenge with this exercise may lie in teaching the basics of the entropy law (and thermodynamics) (which is not provided in this textbook), but alternatively, this exercise might also be discussed in applied forms, i.e., in terms of real-world ecological developments, depending on the general level of study the students have reached.

26.: Please explain Kapp‘s additional institutional analysis for the (capitalist) economy‘s tendency to exploit the other two sub-systems, nature and society, and towards entropy of the total system ‗earth‘.

Solution key: Against the background of the Exercises 24 and 25, Kapp adds to the application of the entropy law to the economic system an institutional analysis of the particular costfiltering system of the capitalist firm, which explains – going beyond GeorgescuRoegen – that the capitalist market economy not just generally is a system that is open towards the natural system, but that it, in its microeconomic units, largely is designed to generate profit (which possibly is only feasible) through exploiting free energy from the second system and disposing waste (bound energy) and heat, without paying the full costs (tbd) in either case (sourcing and polluting). You may then show how institutional analysis may add to a general complexity analysis applying biological/physical knowledge. Note that Kapp insisted that he was not just using the biological/physical analogy, but applying such transdisciplinary knowledge directly to economics.

Solutions Chapter 13 (No exercises for Chapter 13.)

Solutions Chapter 14 (No exercises for Chapter 14.)

87 Solutions Chapter 15

1.: Section 15.2.1 describes an n-person game between a producer of a good with zero variable cost and a number of consumers, their respective strategies being to allow (other) consumers to access the good for free. a) Write the game as a simplified 2-person game between one producer and one consumer.

Do Nash equilibria and Pareto optima coincide? Are there Nash equilibria in mixed strategies? b) Write the game as an n-person game. Is the structure (Nash equilibria, Pareto optima, and coincidence between these) retained?

Solution key: a) As stated in the text, it is assumed that customers have no dominant incentive to prevent or not to prevent other customers from using certainsoftware; producers, however, do.

Both positions have the option to share the software with other customers for free. If either of them does, the producer loses her profits (e.g., 4). The consumer obtains utility (e.g., 1) from using the software, and more utility if the producer provides it for free (e.g. 2). Obtaining the software from other customers is not covered in this 2-person game: Consumer Share for Free

Producer

Share for Free Not for Free

Not for Free

2 0

2 0

1 0

1 4

There are consequently 3 Nash equilibria (upper and lower left as well as lower right), 3 Pareto optima (upper left as well as upper and lower right). Since the consumer‘s strategy choice has no direct effect on her payoff every mixed strategy is for her a best answer to everything. However, the producer‘s best answer to any non-pure mixed strategy of the consumer is strategy 2 (not to share for free). Computation of the mixed strategies yields for the producer Π𝑃 = 4 1 − 𝑝 1 − 𝑞 = 4𝑝𝑞 − 4𝑝 − 4𝑞 + 4 𝜕Π𝑃 = 4𝑞 − 4 = ! 0 𝜕𝑝

88 𝑞∗ = 1 (hence only if the consumer plays her first pure strategy the producer is indifferent).

For the consumer Π𝐶 = 2𝑝𝑞 + 2𝑝 1 − 𝑞 + 𝑞 1 − 𝑝 + 1 − 𝑝 1 − 𝑞 = 𝑝 + 1 𝜕Π𝐶 = 0 = !0 𝜕𝑞 𝑝∗ ∈ 0,1 . b) There are different ways to do this. One of them is to define a quantity 𝑋 =

1 𝑖𝑓 𝑡ℎ𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑟 𝑜𝑟 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 1 𝑜𝑡ℎ𝑒𝑟 𝑐𝑜𝑛𝑠𝑢𝑚𝑒𝑟 𝑠ℎ𝑎𝑟𝑒𝑠 𝑡ℎ𝑒 𝑠𝑜𝑓𝑡𝑤𝑎𝑟𝑒 𝑓𝑜𝑟 𝑓𝑟𝑒𝑒 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 .

Then the payoff of the consumers is (no matter their strategy) Π𝐶 = 2 − 𝑋.

The producers payoff from their strategy to share for free (Π𝑃,𝑆 ) or not to share for free (Π𝑃,𝑁 ) are Π𝑃,𝑆 = 0 Π𝑃,𝑁 = 4𝑋.

The profile of best answers, incentives, Pareto optima, and equilibria is identical to the 2-person case.

2.: Consider the replicator dynamic model given in Section 15.2.3. ∗ ∗ a) Show that 𝑝𝑃𝐷,2 is stable and 𝑝𝑃𝐷,1 is not. b) Could the model also be analyzed using an agent-based simulation? If so, how? Would this yield the same result?

Solution key: a) 1.2𝑝

The system is defined as 𝑝𝑃𝐷,𝑡+1 = 1+1.4𝑝𝑃𝐷 ,𝑡 .As the system is 1-dimensional, the only 𝑃𝐷 .𝑡

1.2

eigenvalue is identical to the only element of the Jacobian, 𝜆 = 1+1.4𝑝 Linearization for the fixed points leads to

𝑃𝐷 .𝑡

2

.

89

1

5

𝜆 0 = 1.2and 𝜆 7 = 6.

The first fixed point is thus unstable, the second is stable. b) Yes, the system could also be investigated as an agent-based model. Though there are a number of possibilities, a simple one would be: define a (large) number of agents each playing one of the two strategies would be defined. They contribute according to their strategy and receive future income according to the equations of the model in Section 15.2.3. After a number of periods, the simulation is terminated and the final market shares are evaluated. The simulation should be re-run with different initial conditions (i.e. different initial market shares), preferably covering an area as large as possible of the state space. Since the state variables are not constrained, the state space is infinite and cannot be exhaustively investigated; however, the relative population shares 𝑝 are constrained between 0 and 1. This range should be covered.

Stochastic perturbation of the system could be investigated in this framework as well.

The simulation could also be used to amend the model to include some of the aspects mentioned at the end of the section (thereby loosening some of the simplifying assumptions).

3.: Consider the replicator dynamic model in Section 15.2.4 in the part on compatibility and incompatibility (on the basis of the game matrix in figure 15.3). a) Show that it leads to the same dynamic system as given in the first part of the section in Equation 15.1. b) Show that it yields the 3 equilibria and their respective stability properties as stated in section 15.2.4. c) How does the system change if the parameters are constrained to 0 < 𝛼 < 1 and 0 < 𝛽 < 1? Is the equilibrium structure retained?

Solution key: a) The expected payoffs are obtained from Π𝐴 𝑡 = 1 0

1 1−𝛼

Π𝐵 𝑡 = 0 1

1−𝛽 1

1 1−𝛼

𝑝𝐴 (𝑡) = 1 − 𝛽 1 − 𝑝𝐴 (𝑡) 1 − 𝑝𝐴 (𝑡)

1−𝛽 1

Substituting into the replicator equation gives

𝑝𝐴 (𝑡) = 1 − 𝛼𝑝 𝑡 .

1 − 𝑝𝐴 (𝑡)

90 𝑑𝑝𝐴 = 𝑝𝐴 1 − 𝑝𝐴 Π𝐴 𝑡 − Π𝐵 (𝑡) = 𝑝𝐴 1 − 𝑝𝐴 𝑑𝑡

𝛼 + 𝛽 𝑝𝐴 − 𝛽 ,

i.e., an equation identical to Equation 15.1. b) There are three cases for which the equilibrium condition (Equation 15.1 being 0) would be fulfilled; namely, if any of the three factors is 0, the entire equation must be 0 as well. Consequently, there are three fixed points ∗ 𝑝𝐴,1 = 0, ∗ 1 − 𝑝𝐴 = 0 , thus𝑝𝐴,2 = 1, and

𝛼 + 𝛽 𝑝𝐴 − 𝛽 = 0,

∗ thus𝑝𝐴,3 =

𝛽 . 𝛼+𝛽

The fixed points‘ stability follows from the eigenvalue which is the (linearization of the) only element of the Jacobian, 𝜆 = −3 𝛼 + 𝛽 𝑝𝐴2 + 2 𝛼 + 2𝛽 𝑝𝐴 − 𝛽 .

Linearized for the three fixed points, this yields 𝜆 0 = −𝛽 𝜆 1 = −3𝛼 − 3𝛽 + 2𝛼 + 4𝛽 − 𝛽 = −𝛼 𝛽 𝛽 2 𝛽 𝛽 𝜆 = −3 𝛼 + 𝛽 +2 𝛼+𝛽 + 2𝛽 −𝛽 𝛼+𝛽 𝛼+𝛽 𝛼+𝛽 𝛼+𝛽 𝛽2 𝛽2 𝛽2 𝛽 𝛼 + 𝛽 − 𝛽2 = −3 + 2𝛽 + 2 −𝛽 =𝛽− = 𝛼+𝛽 𝛼+𝛽 𝛼+𝛽 𝛼+𝛽 𝛼𝛽 + 𝛽 2 − 𝛽 2 𝛼𝛽 = = . 𝛼+𝛽 𝛼+𝛽 While for the first two fixed points, the eigenvalues are always negative (fixed points stable), the eigenvalue for the third fixed point is always positive (fixed point unstable). c) No, excluding values larger than 1 for 𝛼 and 𝛽 (i.e. values for which the incompatibility does not just reduce benefits but actively inflicts damage) does not change the dynamics of the system; neither the fixed points nor their stability conditions depend on this.

91 4.: Concept of open source a) What is open source software? How does it differ from freeware? b) Open source programmers do not receive a monetary revenue from selling their products. Why do they engage in open source programming nevertheless? Give 9 possible approaches to explain this. c) Why did open source communities first emerge in the software industry and not, say, in the manufacturing sector?

Solution key: a) Open source software is software for which the source code is shared publically and for which it is consequently possible for everyone to examine and modify that source code, though there may be restrictions (from the terms under which the code is licensed) about how the modified or unmodified code may further be used, sold, or otherwise distributed.

Freeware is software that is available for free. It is not necessarily open source software (if the source code is not shared and only the compiled binary version of the program is publically shared). b) -

-

-

The programmer needed the program anyway and had no desire to keep it from others once it was completed.

The programmer just enjoys writing software, fixing problems, etc.

The programmer dislikes commercial licenses and writes open source software for political reasons (or rather, licenses her software open source for political reasons).

The programmer wants to build reputation in order to increase her chances of professional success.

The programmer uses her writing of open source code as a signaling strategy (i.e., less professional programmers cannot afford to use their time for pursuits that do not directly pay for them).

The programmer expects reciprocity (that others will also share their programs and/or source code for free if she does).

The programmer fears to be punished (shunned, excluded …) from the software community if she does not license her code under open source terms.

The programmer seeks recognition and social approval in response to her writing of open source software.

A company pays programmers to write open source software – for strategic reasons which may for instance be rooted in network externalities (i.e. the company might be trying to build a user base for standards it controls by providing free and/or open source software which makes use of these standards).

92 c) There may be a number of reasons for this, but one of the central causes is most likely that software products may be reproduced with close to zero marginal costs (and that in recent times the necessary equipment, i.e. a computer, is available to large numbers of people). Not only were the necessary requirements for the development and production of open source software thus available to many people including skilled programmers, it did also not require any substantial further investment to reproduce and distribute the software once it was written. This is different in the manufacturing sector: development cost may be comparable to the software industry but the marginal costs of producing a manufactured good are also substantial, mostly far beyond what small private groups of developers or engineers (or even small firms) could afford. A further reason may lie in the greater incidence of network technologies and network externalities in the software industry – which may make it necessary for successful goods and standards to be introduced as free goods in order to build an installed base (which obviously favors open source software over expensive commercial software).

Solutions Chapter 16 (No exercises for Chapter 16.)

Solutions Chapter 17 (No exercises for Chapter 17.)

Solutions Chapter 18 (No exercises for Chapter 18.)

93

5.

A NOTE ON DIDACTICS FOR TEACHING COMPUTER SIMULATION (CHAPTER 9)

In order to use the method of computer simulation successfully, students should not only be aware of the purpose and usual procedure of the technique, they should also be familiar with the tool, i.e. the programming language used for programming the simulation. They should also develop a feeling for how a particular simulation is computed which can later help to assess whether specific programming techniques are well-suited for a particular problem or not.

We would therefore like to give a few suggestions on how to introduce students to the technique: 1.

One might start by letting the students compute a (simple) simulation literally with pencil and paper instead of using a computer. This helps to see that the computer is not a magical device that enables us to conduct simulations at all but rather only a machine that allows us to do them faster with fewer errors.

2.

One might even start by linking simulations to scientific experiments. For instance: draw a square on a sheet of paper; draw the largest possible circle into this square. If the square covers an area on 1, the circle covers an area of 𝜋/4.

Explain to the students that the area of the circle (relative to that of the square) is to be estimated without making use of 𝜋. (Make the story more graphical by explaining that there was a time in antiquity when humans did not yet have textbooks on mathematics and consequently did not know 𝜋.) Drop a number of small objects (e.g. paper clips) on the paper and count the number of those within the square and those within the circle. Given that the number of objects is large enough, the relation of the two numbers is a fair approximation of the relation between the areas of the two geometrical figures.

3.

It may be instructive to confront the students with a simple economic example which exhibits far-form-equilibrium dynamics. This makes clear why it may be necessary to use simulation even if other methods (like computing the fixed points) might be available. To give one example: Consider a firm with a capital 𝑘 (in million $), a production function 𝑞 = 5𝑘; the capital is consumed in the production process and has to be replaced from the firm‘s revenues. Assume that the firm has a monopoly; the price thus depends only on the quantity the firm 𝑞 produces; it follows the equation = 1 − 20 . The firm‘s revenue is consequently Π = 𝑝𝑞. The firm reinvests a certain share 𝜎 of the revenues, thus the capital in the next period is 𝑘𝑡+1 = 𝜎Π. Set a couple of possible starting values 𝑘0 between 0 and 4 (recall that k=4 means $4m) and ask the students to assess the dynamics of the process for different 𝜎 (with 0 < 𝜎 < 0.8, otherwise it yields negative values). The problem is mathematically equivalent to the logistic equation as discussed in Chapter 10 of the textbook. It has a stable equilibrium for 𝜎 < 0.6, then undergoes a sequence of bifurcation and shows at first oscillations and finally (for 𝜎 close to 0.8) chaotic behavior.

4.

If the students are already familiar with a programming language other than Python, consider using this programming language instead of Python.

94

5.

Allow the students to familiarize themselves with the programming language; give them for instance plenty of time for Chapter 9‘s Exercise 1 (running the short scripts given in this chapter, try a few things, etc.).

6.

Ask the students to familiarize themselves with the programming language‘s (online) documentation, for Python; this is to be found at http://docs.python.org.

Most alternative programming languages also have an online documentation. This facilitates becoming able to solve technical problems without the help of the instructor.

7.

Before introducing the students to agent-based simulation (and object-oriented programming, classes and functions, in general) have them solve a few simple mathematical tasks with the programming language. Examples include: sorting lists of numbers, computing sums or products over lists of numbers, computing the faculty of a number …

8.

If (most of) the students have taken a class of statistics before, consider to include statistical simulation problems (such as: Consider a car park being built in a modern inner city; assume the times between the arrivals of each two successive cars are Poisson distributed with parameters x and y and the time each car stays is distributed according to a normal distribution with parameters z and s; which size would you recommend for the car park?).

9.

Several of the models considered in Chapter 13 make good examples for simulation problems. Consider using for instance Schelling‘s segregation model, Arthur‘s technology choice model or his El Farol problem as well as simulation experiments with repeated prisoners‘ dilemmas in the style of Axelrod or Ostrom.

If the students are familiar with basic network theory consider having them investigate the properties of Watts-Strogatz and/or Barabási-Albert networks with simulation models.

95

6.

COMPETING TEXTBOOKS – A BRIEF COMPARISON

H. Varian, Intermediate Microeconomics, 1984ff.

From the number of well-established top-selling standard microeconomics textbooks (among them also Pindyck/Rubinfeld), we pick Varian. Although Varian has a good name in research on the ‗new information economy‘ and has taken on many of the new applied themes of complexity economics such as uncertainty, ‗futurity‘, information/net-technologies, net externalities, lock-in, collective goods/‗commons‘, and asymmetric information, the whole book still remains rooted in conventional supply and demand microeconomics. Uncertainty issues turn out to be calculable risk rather than ‗strong‘ (‗true‘) uncertainty, and like most ‗older‘ textbooks, the new issues in the newer editions are just added on top of the old ones, without any repercussion on the core model. So the basic world view is not thoroughly reconsidered and redesigned from scratch, and the students, in the end, are left without a red thread, but with the confused view rather that there is a nicely working market – and a whole lot of disturbing new things on top. The classic representative of the mentioned ‗perfectmarket-plus-some-exemptions‘ textbook agenda.Keen (2011 [2001], see below) has thoroughly criticized Varian‘s prototypical way to deal with some internal logical contradictions of the representative agent and the aggregate demand curve. We return to this in detail in our Chapters 5 and 6.

A. Mas-Colell et al., Microeconomic Theory, 1995ff.

The ‗modern replacement for Varian‘ (Keen, op. cit., p. 58) is this book. It is ‗hypermathematical – but utterly non-empirical‘ (ibid.), a fairly neoclassical, advanced microeconomics textbook. It discusses some of the shortcomings and critiques of neoclassical economics (first and foremost the Sonnenschein-Mantel-Debreu theorems – for details and context, see Chapter 5 in our textbook), but fails to feature any alternatives to the mainstream view. Thus, no consequence drawn from the fundamental critique put forward. This might confuse potential readers. Decision theory and game theory are presented in detail but this does not lead the authors to consider any ‗heterodox‘ interpretations, usage or embedding of it in other traditions of the economics literature. But today the ‗mainstream‘ authoritative source in the field, as once was Samuelson‘s ‗Foundations‘ (1948) (see Chapters 5, 6, and 12 in our textbook). For our textbook, on the other hand, diverse modern core theories and models converge to some new core.

C. Holt, Markets, Games & Strategic Behavior, 2007 A closer competitor to this textbook may be Holt‘s book. It is also largely advanced. It somewhat overlaps with ours in terms of issues, models, game theory, and policy implications, but it is much more confined to the market and business management perspectives, so more reduced in perspective. It is somehow complementary to ours, discussing foundations and broader applications of topical market-related and management issues.

96 A. Schotter, Microeconomics. A Modern Approach, 2001 This is a micro-textbook completely based on game theory. It has similarities with Holt (and ours) in terms of issues and perspectives, but is much more formal (confined to game theory).

It is largely dominated by game theory rather than by economic problems, so little ‗embedding‘ of the formalisms.

J. Lesourne et al., Evolutionary Microeconomics, 2006 Also Lesourne et al. (3 authors and 6 collaborators mentioned) is a textbook in a complexity perspective: It is also based on game theory, with evolutionary conceptions and implications of irreversibility, replicator mechanism, institutional emergence, and all that is required for a full-fledged evolutionary process resulting from complex structures over time. It covers an array of themes and issues that is only part of our textbook, particularly parts of our Chapters 3, 8, 11, 12, 13 and 14. But this book appears (to us) to be very abstract and general, little applied, without stories or cases from the real world, and thus, not very appealing.

J. Potts, New Evolutionary Microeconomics, 2000 This book is not designed as a textbook, but a research monograph with some textbook-like systematization of the material. It is most advanced and in research style, though quite successful book in terms of citations. It follows a neo-Schumpeterian perspective (see our Chapter 12 for more detail on Schumpeterianism).

S. Keen, Debunking Economics, 2001ff.

Keen‘s book has been mentioned and cited already, and will be referred to in more detail with respect to certain of his criticisms of the neoclassical mainstream in Chapters 5 and 6 below.

It is a most critical and radical, but well-informed textbook, nicely composed form storytelling, analytical formalisms, and some striking new research results. It had a second edition ‗post-2008‘ in 2011, and Keen has indeed been a rather profiled analyst and critic of the financial crisis. Much more radical and political than ours. Keen is excellent in applying complex methods in easily comprehensible ways, with concrete ‗proofs of the pudding‘.

S. Bowles, Microeconomics, 2004 Closest to our textbook is Bowles‘ book. It resembles ours in terms of the complexity perspective in general, the stories from the real-world, the breadth of the perspectives adopted, evolutionary and institutional perspectives, and the more ‘embedded’ use of game theory (as ‗embedded‘ in proper evolutionary-institutional ‗story telling‘). It also is a book rebuilding real-world micro from scratch. Bowles is famous for having continuously written important and often path-breaking papers for more than 40 years. The book is in parts above the level of ours, i.e., it is mostly advanced, particularly in terms of a required deeper understanding of the history of economic thought, and with less introductory chapters. So it requires more from the potential student. Also, is not really a full-fledged textbook but something in between a textbook and a research monograph. With its new perspective, it has

97 been already a big-seller – which confirms that there indeed is a need for a textbook like ours ‗out there‘.

A new microeconomics textbook-like book based on Bowles‘ book has been announced by the Santa Fé Institute in 2012: Samuel Bowles and Duncan Foley, Coordination, Conflict and Competition: A Text in Intermediate Microeconomics (no publisher given). The table of content can be accessed at: http://tuvalu.santafe.edu/~bowles/books.htm. It is more textbooklike but also consequently advanced.

S. Himmelweit et al., Microeconomics. Neoclassical and Institutionalist Perspectives on Economic Behaviour, 2001ff.

Himmelweit et al. (a group from the Open University, UK), with reprints in 2002 and 2005, also comes close to our book in terms of general perspectives, although it is much more introductory and very little game theory is deployed, i.e., largely non formal, with much story-telling, but in parts rather simple and sometimes also somewhat naïve vis-á-vis the neoclassical ‗mainstream‘ material. We have worked for many years with this book in introductory micro classes. But it seems to have received no further editions.

N. Goodwin et al., Microeconomics in context, 2005ff.

Goodwin et al. (4 co-authors, a famous group from the center for development and environment at Tufts University) also comes close to our book with their evolutionary perspectives. Although still largely conventional in its structure, it is much engaged in environment and gender issues. It is also fully verbal and non-formal, thus introductory level.

Not really complexity economics and not cutting-edge. We have used this book, too, for many years, at least in parts, in second-semester micro.

D. Fudenberg, J. Tirole, Game Theory, 1991 Game-theoretic textbooks also come close to our textbook, although they are generally more reduced in perspective and method. We pick four from the competition. Fudenberg/Tirole appears as rather technical, focusing onBayesian games. Therefore, although somewhat complementary (and substitutive) to ours, it is more advanced – and more limited.

A. Rubinstein, M. Osborne, A Course in Game Theory, 1994 This is a pure game-theory textbook. It provides examples from economics and discusses the various links between game theory and economics as well as some history of game theory in the earlier economic literature on utility and preferences. The text is very mathematical, thus requiring without a background in advanced math, so definitely advanced level.

Rubinstein/Osborne and our textbook may be considered complementary in parts.

98 H. Gintis, Game Theory Evolving, 2000ff.

We might further mention Gintis‘ book (2nd ed. 2009) and S. Hargreaves Heap, Y. Varoufakis, Game Theory, 1995ff.

(2nd ed. 2004) as two most exciting and deeply substantial textbooks in game theory, far beyond standard interpretations and applications. Gintis‘ is more on complexity in general, advanced, also far beyond economics and going deeply into philosophy and more researchmonograph-like, Hargreaves Heap more intermediate, institutionalist, and textbook-style.

Both good supplements for our textbook (Hargreaves Heap more than Gintis) and similar in using substantial economic problems (but nevertheless: game theory textbooks, not microeconomics per se). We have worked with both in graduate classes.

And more …

The would be more different types of textbooks worth mentioning, such as Fireside et al., Real-World Micro 1989ff., a radical-political reader of cases, oriented towards the structure of the standard curriculum, (in order to criticize it neatly), or Fred Lee’s Microeconomic Theory: A Heterodox Approach, 2014, largely focused on alternative price theories, theories of the firm, and of market competition, in Marxian, Post-Keynesian, and Institutionalist perspectives, with a larger social-science and methodological embedding. Not to mention the many more or less critical general principles textbooks and ‘anti-textbooks’ …

99

7.

THE MATH REQUIRED – A SHORT OVERVIEW OF MATHEMATICAL PREREQUISITES OF WORKING WITH THIS TEXTBOOK

Our approach is to show in Part I of the book that even a general introduction to economics is possible from a complexity perspective. However, we recognize that there are currently very few universities, where such a ―heterodox‖ microeconomics is taught at the introductory level. Parts II-V, therefore, take into account that the great majority of readers are familiar only with neoclassical introductory microeconomics. The more formal Parts III and IV might be somewhat more demanding in their mathematical prerequisites than an average intermediate neoclassical course (see Table X.2).

Part

Prerequisites

Part I: Basics of the Interactive Economy and Its Processes

None.

Part II: Markets: General Equilibrium Theory and RealWorld Market Structures

Introductory (neoclassical) microeconomics; basic analysis, basic algebra.

Part III: Further Tools and the Analysis of Complex Economies

Introductory Microeconomics; basic algebra, basic analysis; dynamical systems / basic differential equations (intermediate analysis).

Part IV: History of Thought and Contemporary Models in Complexity Economics

Introductory Microeconomics; basic algebra and analysis; basic understanding of game theory, evolutionary game theory, complexity theory, dynamical systems, computer simulations (Parts II and III provide these prerequisites).

Part V: Further Applications and Interactive Economic Policy

Basic understanding of the methods and models used in ―heterodox‖ economics (Parts II through IV provide these prerequisites).

Table X.2 Overview of part-wise mathematical prerequisites.

‘Another Math’ for Complexity Economics?

Many heterodox economists have rightfully deplored the model Platonism of much of the neoclassical theory. It is, however, a common mistake to forego as a consequence from this

100 the use of formal models and mathematics completely. As Keen, one of the harshest critics of neoclassical economics, put it, ―to eschew mathematical analysis in the development of heterodox economics ... would be a classic case of throwing the baby out with the bathwater.‖ (Keen, 2009, 164) Mathematics is one technique to frame ideas, hypotheses, and models; verbal explanation is another. While it is advisable to keep in mind that models are always only approximations of the reality and that the model should be made to follow reality (not vice versa), mathematics is a powerful, exact, and transparent tool that has many advantages.

It can be less ambiguous than verbal explanations, it offers clarity, it enables us to take the analysis further than otherwise possible, and it has the power to illustrate rather complex concepts.

Nevertheless, there is arguably a pressing need for the inclusion of different fields of mathematics in the education in (heterodox) economics. Heterodox models have long made use of fields of mathematics that remain completely alien to the average student of neoclassical economics, fields that include dynamical systems, differential equations, chaos theory, game theory, network and graph theory, as well as potentially the theory of computation, recursion and complexity theory. As a consequence, most students are faced only with the question of whether to use mathematics or not, but never consider whether the problem at hand should be approached using real analysis or recursion theory or whether a model should be built using constructive mathematics only or not. This often results in a misleading view on many important questions, as is exemplified throughout the textbook, e.g. when discussing the relation between rationality and bounded rationality. Many students consider the latter to be a qualification of the former, but if they were aware of some basic recursion theoretic results, they would know that this is not the case and the rational actor is indeed the special case. Similar examples could be given for network structures: For many students modelling a particular network structure explicitly means a further specification of a model, not being aware that a model not modelling a network structure explicitly generally assumes a special form (usually a complete network) implicitly. The resulting ambiguity is regrettable and should be an anathema to the scientific community, especially to the economists such as Gerard Debreu advocating the use of mathematics because of the clarity and the rigor attached to its use (see e.g. his presidential address as president of the AEA, Debreu 1991).

Further – and this is also true for the education in neoclassical economics – much more value could be placed on teaching the mathematical foundations of formal methods. Students who should be able to conduct a critical evaluation of their approach need to understand how, why, and for what the method works as well as which alternatives exist. Regrettably, the mathematics taught in (micro-)economics courses often focuses strongly on the application of methods such as linear and nonlinear optimization while neglecting the mathematical background. Similarly, although many economics students use mathematics, only few know about the academic debates in the field of mathematics. For most of them, mathematics is just one big unique theory and only few know about different schools and controversies in the field (for details, see, e.g. Velupillai, 2009, Zambelli, 2011, or Velupillai et al., 2011). This limits their view about mathematical argumentation. A recent debate in economics is about the adequacy of the use of unconstructive mathematics in economic theory. Only few students could follow such a discussion as the distinction between constructive and unconstructive mathematics is not taught to economics students. If economists want to make use of the rigor and clarity of mathematics, it would be desirable that they know more about this field than just being able to apply some fundamental tools. A better understanding of mathematics as a science in general would lead to a more pertinent use of mathematical methods and the rigor of using the right mathematical tool and giving it the right interpretation would also train the economic reasoning and clarity of the students.

101 That being said, it should not go unmentioned that much has been accomplished in the improvement of teaching a broader field of mathematical methods in economics. Many universities have included courses on game theory, network theory, and economic dynamics into their curricula at least at the intermediate or advanced level in the recent years and a wide and expanding variety of very good textbooks on these topics exist (some of which were mentioned and recommended in the appropriate chapters of the current book). A few universities even include courses on less commonly accepted fields such as chaos theory and rather unorthodox applications such as agent-based modelling and simulation.

Some of those courses do not even place any particular emphasis on the application to complexity economics, though the methods would fit such applications very well. However, there is only one mathematics (though it is multi-faceted), hence standard courses in mathematical economics do not become invalid in heterodox approaches and no ‗other kind of math‘ is required for complexity economics, merely a different – possibly broader – approach to it.

References Debreu, G., 1991. The Mathematization of Economic Theory. American Economic Review, Vol. 81(1), pp.1-7.

Keen, S., 2009. Mathematics for pluralist economics. In: Reardon, J., The Handbook of Pluralist Economics Education, Routledge, Abingdon, UK/New York, NY, pp. 150-167.

Velupillai, K.V. 2009. Uncomputability and Undecidability in Economic Theory, Applied Mathematics and Computation, Vol. 215(4), pp. 1404-1416.

Velupillai, K.V, Zambelli, S. and Kinsella, S. (eds.), 2011. Computable economics, Edward Elgar, Cheltenham, UK. Zambelli, S., 2011. Introduction to the JOES Special Issue on Nonlinearity, Randomness and Complexity. Journal of Economic Surveys 25(3), pp. 395-400.