...In general, hypothesis testing concerns trying to decide whether a parameter θ lies in one subset of the parameter space or in its complement...We also demonstrate an equivalence between hypothesis tests and confidence intervals...
Chapter 9, Testing Hypothesis
Risk or uncertainty?
In this textbook version of Raiffa and Schlaifer's statistical decision theory of experimentation combined with Savage's theory of statistics, the authors scratched the risk/uncertainty difference with a stroke:
...a number of economists have attempted to distinguish between risk and uncertainty, as originally proposed by Frank H. Knight. (1) Risk, Knight said, refers to situations where an individual is able to calculate probabilities on the basis of an objective classification of instances. For example, in tossing a fair die the chance of any single one of the six faces showing is exactly one-sixth. (2) Uncertainty, he contended, refers to situations where no objective classification is possible, for example, in estimating whether or not a cure for cancer will be discovered in the next decade...
...in this book, we disregard Knight's distinction. For our purpose, risk and uncertainty mean the same thing. It does not matter, we contend, whether an 'objective' classification is or is not possible. For we will be dealing throughout with a 'subjective' probability concept (as developed especially by Savage, 1954): probability is simply degree of belief. In fact, even in cases like the toss of a die where assigning 'objective' probabilities appears possible, such an appearance is really illusory. That the chance of any single face turning up is one-sixth is a valid inference only if the die is a fair one - a condition about which no one could ever be 'objectively' certain. Decision makers are therefore never in Knight's world of risk but instead always in his world of uncertainty. That this approach, assigning probabilities on the basis of subjective degree of belief, is a workable and fruitful procedure will be shown constructively throughout the book...
In a similarly direct manner, the authors defended analytic decision theory:
...the approach here does not allow for the psychological sensations of vagueness or confusion that people often suffer in facing situations with uncertain (risky) outcomes. In our model, the individual is neither vague nor confused. While recognizing that his knowledge is imperfect, so that he cannot be sure which state of the world will occur, he nevertheless can assign exact numerical probabilities representing his degree of belief as to the likelihood of each possible state. Our excuse for not picturing vagueness or confusion is that we are trying to model economics (/rational behavior), not psychology...The ultimate justification, for indifference-curve diagrams or for theories of decision under uncertainty, is the ability of such models to help us understand and predict behavior.
...a number of economists have attempted to distinguish between risk and uncertainty, as originally proposed by Frank H. Knight. (1) Risk, Knight said, refers to situations where an individual is able to calculate probabilities on the basis of an objective classification of instances. For example, in tossing a fair die the chance of any single one of the six faces showing is exactly one-sixth. (2) Uncertainty, he contended, refers to situations where no objective classification is possible, for example, in estimating whether or not a cure for cancer will be discovered in the next decade...
...in this book, we disregard Knight's distinction. For our purpose, risk and uncertainty mean the same thing. It does not matter, we contend, whether an 'objective' classification is or is not possible. For we will be dealing throughout with a 'subjective' probability concept (as developed especially by Savage, 1954): probability is simply degree of belief. In fact, even in cases like the toss of a die where assigning 'objective' probabilities appears possible, such an appearance is really illusory. That the chance of any single face turning up is one-sixth is a valid inference only if the die is a fair one - a condition about which no one could ever be 'objectively' certain. Decision makers are therefore never in Knight's world of risk but instead always in his world of uncertainty. That this approach, assigning probabilities on the basis of subjective degree of belief, is a workable and fruitful procedure will be shown constructively throughout the book...
In a similarly direct manner, the authors defended analytic decision theory:
...the approach here does not allow for the psychological sensations of vagueness or confusion that people often suffer in facing situations with uncertain (risky) outcomes. In our model, the individual is neither vague nor confused. While recognizing that his knowledge is imperfect, so that he cannot be sure which state of the world will occur, he nevertheless can assign exact numerical probabilities representing his degree of belief as to the likelihood of each possible state. Our excuse for not picturing vagueness or confusion is that we are trying to model economics (/rational behavior), not psychology...The ultimate justification, for indifference-curve diagrams or for theories of decision under uncertainty, is the ability of such models to help us understand and predict behavior.
On logic
First, the author considers the postulates he made about preference relations on acts to have both an empirical interpretation - as a 'prediction about the behavior of people' - and a normative one - as a 'logic-like criterion of consistency'.
Then some discussion of the role of logic in general:
Then some discussion of the role of logic in general:
...logic itself admits an empirical as well as a normative interpretation...to summarize (the empirical interpretation), logic can be interpreted as a crude (since people make mistakes and have limited computing power) but sometimes handy empirical psychological theory...(not very successful though)
...the principle value of logic, however, is in connection with its normative interpretation...as a set of criteria by which to detect, with sufficient trouble, any inconsistencies there may be among our beliefs...'not appropriate' here to discuss 'why and in what contexts we wish to be consistent'. We simply 'often do wish to be so'.
Implication of reducing multi-stage to single-stage
Defining acts as functions from states to consequences, reduction of many multi-stage actions to a single action follows naturally, bringing up a surprising implication for research planning though:
...the great majority of experimentalists...suppose that the function of statistics and of statisticians is to decide what conclusions to draw from data gathered in an experiment or other observational program...
...but statisticians hold it to be lacking in foresight to gather data without a view to the method of analysis to be employed...they hold that the design and analysis of an experiment should be decided upon as an articulated whole...The author then went on to propose the main theme of the discussion as the 'Look before you leap' type, as is opposite to 'You can cross that bridge when you come to it'. I wonder whether the precautionary principle amounts to 'Look and be cautious before you leap' or 'You can cross that bridge when you come to it but be cautious'. The latter would seem more widely applied.
Savage on Schlaifer
(preface to Dover edition)...the (personalistic) movement itself has other sources apart from those from which this book itself was drawn ...one with great impact on practical statistics and scientific management is a book by Robert Schlaifer...his ideas were developed wholly independently of the present book, and indeed of other personalistic literature...they are in full harmony with the ideas in this book but are more down to earth and less spellbound by tradtions...
The honorably working Bourgeois
Many essays start by quoting dictionaries and definitions, but few can be much more revealing than the dictionaries and encyclopedia themselves, as is the one done by McCloskey in Chapter 2 to start talking about the Bourgeoisie, plural of a Bourgeois whose female partner would be a Bourgeoise, not to be distinguished in spoken language from its plural form Bourgeoises, who all share ancestry with the German Bürger and Bürgerinnen, the hard-working "towns-man-ly" people, what McCloskey tries to portray as a virtuous class.
One thing peculiar about her portrait is the inauthenticity and the lack of creativity of the Bourgeois class, whose greatest drives for hard work are freedom and the ability to imitate, rather than any pure profit maximization with resolution nor logic. I wonder what economics would be like if such a homo socius becomes the protagonist.
One thing peculiar about her portrait is the inauthenticity and the lack of creativity of the Bourgeois class, whose greatest drives for hard work are freedom and the ability to imitate, rather than any pure profit maximization with resolution nor logic. I wonder what economics would be like if such a homo socius becomes the protagonist.
Sufficiency of self-interest in cooperation
The author argues that self-interested m-Cooperate strategy is unstable and easily afflicted by chance events of for example one m-Cooperator turning to Defect. Why couldn't the strategies also adapt to become stochastic and tolerant to chance events? The author admits himself that this doesn't exclude the possible existence of strategies immune to chance events. Nevertheless, such partial results are commonly used as arguments for inefficiency of self-interest and for the necessity of exogenous factors such as institutions.
The other argument using implausibility of self-interested cooperation in large groups instead of in a dyadic setting is also debatable. Individuals in large groups are not isolated from each other. There is the social network structure connecting group members and graph theory has shown that it is possible to connect most individuals in the world with one another within a very small number of links (social networks theory, 6 degrees of separation). This makes reciprocal actions not unlikely, and reciprocal altruism not unlikely, if the individuals realize this connectivity.
The other argument using implausibility of self-interested cooperation in large groups instead of in a dyadic setting is also debatable. Individuals in large groups are not isolated from each other. There is the social network structure connecting group members and graph theory has shown that it is possible to connect most individuals in the world with one another within a very small number of links (social networks theory, 6 degrees of separation). This makes reciprocal actions not unlikely, and reciprocal altruism not unlikely, if the individuals realize this connectivity.
Chapter 3
last remarks on Ethics and Reciprocity
...the origin of ethics (supposedly defining ethics as out of contractual motives, i.e. compromise between doing and suffering from injustice) tells us nothing about where it will end...
...the ultimate reason for entering into the ethical contract is...self-interest...(which is not economic-theoretically impossible but philosophically not appealing, since ethics as a philosophy is more unifying than exploratory as is specialized by sciences like economics)
Equal rights of murderers
How to spot a murderer's brain, The Guardian: In 1987, Adrian Raine, who describes himself as a neurocriminologist, moved from Britain to the US. His emigration was prompted by two things. The first was a sense of banging his head against a wall. Raine, who grew up in Darlington and is now a professor at the University of Pennsylvania, was a researcher of the biological basis for criminal behaviour, which, with its echoes of Nazi eugenics, was perhaps the most taboo of all academic disciplines.Murderers are genetically different from non-murderers. If taking this as no less scientific than the genetic difference between humans and cows, Peter Singer could have another chapter in his book of Practical Ethics on the equal rights of murderers. And perhaps another appendix On Being Silenced in Germany, for his Nazi-eugenicist ideals.
In Britain, the causes of crime were allowed to be exclusively social and environmental, the result of disturbed or impoverished nurture, rather than fated and genetic nature. To suggest otherwise, as Raine felt compelled to, having studied under Richard Dawkins and been persuaded of the "all-embracing influence of evolution on behaviour", was to doom yourself to an absence of funding. In America, there seemed more open-mindedness on the question and, as a result, more money to explore it. There was also another good reason why Raine headed initially to California: there were more murderers to study than there were at home...
Modern Utilitarianism
(Ch3)...philosophers have advocated equal considerations of interests...only a few have recognized...applications beyond our own species...one of the few being Jeremy Bentham...the founding father of modern utilitarianism...
(Ch1)...it differs from classical utilitarianism...in that 'best consequences' is understood as meaning what...furthers the interests of those affected (individual preferences incorporated)...rather than merely what increases pleasure and reduces pain...
A Normative Implication of Game Theory
Whatever game theories can tell us, a mutual understanding within the gaming body is the momumental presumption. However selfish people want to or have to be, they can at least try to understand each other's selfishness first before they try to arrive at any equilibrium solutions. Whether the solution fits some ethical ideals such as equality is a matter of interpretation, and the very pursuit for such ideals can hinder the effort to understand each other, without which games become more asymmetric and we will be farther away from whatever we were seeking.
The decision making rationale of eliminations
This topic came up with the concept of 'uncovered set' in spatial equilibriums of voting games. It makes pretty much sense to me as it coincides perfectly with my own decision making rationales. Probably out of risk aversion, putting much weight on avoiding the Type II error which rejects the less clear cut hypothesis and thus focuses on trying to reject the nulls.
Testing applicability of economic theories on issues
Whatever economics is, it is not about "I like juice and she likes tea", which says nothing about a third person's preference. To differenciate theoretical conclusions from mere facts, it is probably sufficient to see if it can predict, within a certain scope. As one possible test for presence of predictability, any issue that used to be the subject of a fortune-teller should well be said as predictable. In this aspect, what economists do are not so much different from a fortune-teller. They predict markets, wealth, and give advices on how to prevent certain things from happening. They should be able to compete well with the fortune-tellers and do whatever their competitors can do. At the same time, it would be smart if they consider themselves no more than a fortune-teller, as long as they want to compete with them in predicting, and stay away from other businesses, such as philosophy or wisdom, in which realms the predicting enterprise itself will be questioned.
Chapter 5
...the replacement of physical capital accumulation by human capital accumulation as the prime engine of economic growth has changed the qualitative impact of inequality on the process of development...
...inequality in the ownership of factors of production has generated an incentive for the better-endowed agents to block the implementation of institutional changes and policies that promote human capital accumulation...
Market Equilibrium
People have very different interpretations of the notion of an equilibrium, reflected in different formulations of the problem. Some consider it as rather a dynamic process, so formulated it as fixed point, which is more common in growth problems (also in the taxation dynamics). It is not clear yet how this is related to using duality in linear programming. According to Takayama, Arrow formulated it as games. Takayama's advocated formulation is actually by recognizing a resemblance between Pareto Optimum and the vector maximization in nonlinear programming, and as a result, the use of nonlinear programming to solve for competitive equilibrium ensures direct satisfaction of Pareto Optimum, and the existence conditions of such an equilibrium is already implicit in the requirements of nonlinear programming (such as concavity, local non-satiation, etc).
It seems that to understand the equilibrium problem is somewhat equivalent to understanding the relationship between the following few concepts:
fixed-point
duality (envelope)
nonlinear programming (saddle point)
intermediate value theorem (related to taylor expansion and asymptotic properties)
convergence of sequences
mean value theorem
central limit theorem
It seems that to understand the equilibrium problem is somewhat equivalent to understanding the relationship between the following few concepts:
fixed-point
duality (envelope)
nonlinear programming (saddle point)
intermediate value theorem (related to taylor expansion and asymptotic properties)
convergence of sequences
mean value theorem
central limit theorem
Demand as the Envelope of Optimization and Duality
Takayama arrived at the substitution properties of demand in Section 2.D directly from properties of the preference orderings (probably without transitivity). He also covered the other approach of using duality, which he derived in Section 1.F with nonlinear programming and separation theory (while MWG could only provide some intuitions of it with normal calculus). Takayama's advocated approach was centered around a 'minimum expenditure funcion', which can be just another way of expressing the supporting hyperplane or support function. Also convexity is included in the conditions of the stated demand properties, which resonate what I have wrote in the other note about the connection between linearity and convexity which supports the approach of solving optimization problems with its dual form.
Demand Theory
This is about the properties of the 'envelope' of optimization problems (solved readily with nonlinear programming), a correspondence between the solutions of optimization processes and its conditions, i.e. properties of a projection from others behavior to an individual's reactions which is determined by properties of another correspondence confined within one individual, under the optimizing principle. The behavior of this 'envelope' correspondence under some principles like equilibrium is what people want to know, and the behavior is determined by properties such as continuity, elasticities, and as related to its differentials.
These properties are traditionally derived with the preference ordering representation of behavior, elaborated in Takayama's book. Yet as emphasized in the more recent Microeconomic Theory, they can also be derived from the choice representation. This connection is essentially due to the fact that the axiom of transitivity of the preference orderings is generally irrelevant to properties of the demand function, thus can be relaxed in investigating demand, reducing the behavioral representation to something more or less equivalent to the choice approach.
About the detailed results from demand theory. All properties are derived directly from the preference representations, and the internal optimization process inside the envelop is taken as already finished (leading to equivalence with choice approach, as pointed out in Takayama's footnote 4 on Page248).
The compensated demand function is a function from the desired demand to the demand that a rational consumer adjust to so that one can still enjoy the same utility but achieved within one's budget. In a very strong sense, it is an envelope closing at an indifference curve. In this perspective, an envelope is just a tangent line that is not tangent at any specific point. Such an concept enabled the description of a tangent or sloping relationship without being restricted to one point, i.e. envelope is the tangent line of a set, and all the efforts with separation theories or minimum expenditure function are just to mathematically express an envelope. Then under certain conditions, the envelopes form a dual relationship with the demand function, and substitution properties of the demand function which is an implicit function involving optimization can be conveniently transferred to the substitution properties of its envelope.
These properties are traditionally derived with the preference ordering representation of behavior, elaborated in Takayama's book. Yet as emphasized in the more recent Microeconomic Theory, they can also be derived from the choice representation. This connection is essentially due to the fact that the axiom of transitivity of the preference orderings is generally irrelevant to properties of the demand function, thus can be relaxed in investigating demand, reducing the behavioral representation to something more or less equivalent to the choice approach.
About the detailed results from demand theory. All properties are derived directly from the preference representations, and the internal optimization process inside the envelop is taken as already finished (leading to equivalence with choice approach, as pointed out in Takayama's footnote 4 on Page248).
The compensated demand function is a function from the desired demand to the demand that a rational consumer adjust to so that one can still enjoy the same utility but achieved within one's budget. In a very strong sense, it is an envelope closing at an indifference curve. In this perspective, an envelope is just a tangent line that is not tangent at any specific point. Such an concept enabled the description of a tangent or sloping relationship without being restricted to one point, i.e. envelope is the tangent line of a set, and all the efforts with separation theories or minimum expenditure function are just to mathematically express an envelope. Then under certain conditions, the envelopes form a dual relationship with the demand function, and substitution properties of the demand function which is an implicit function involving optimization can be conveniently transferred to the substitution properties of its envelope.
The Simple-Strategy Complex-Payoff Game
This terminology is referring to the other note on Subgame Optimization. I compared Pareto optimum to Nash equilibrium there, and here there are more things to say about the Pareto optimum.
First there is still something more about the comparison. Both are about decisions. In games the individual decisions are connected and converged with each individual playing both as herself and as every other players, so that she knows where the equilibrium is and act accordingly. In competitive markets, this thinking process is replaced by simply following the price, knowing that the price contains information about the others' actions. This price-taking behavior can actually be interpreted as a solution to the game of market, where the condition of Nash equilibrium is interpreted as Pareto Optimum.
From reading Fei's lecture notes, it became clear that the Pareto Optimum (pareto efficient allocation) is nothing but a further level of constrained utility / profit maximization, which in this sense is just a pareto efficient allocation of the individual's constrained wealth / production set (?) which maximizes individual preference (utility) / production pareto efficiently. On the market exchange level, this constrained maximization becomes pareto efficient allocation of constrained total resource that maximizes group preference (utility) pareto efficiently.
pareto optimum = vector maximumdecision
theory = vector maximization
As such, the existence question of an equilibrium should be the same as the existence question of a maximum which is solved for in utility maximization with nonlinear programming conditions, e.g. KTCQ.
First there is still something more about the comparison. Both are about decisions. In games the individual decisions are connected and converged with each individual playing both as herself and as every other players, so that she knows where the equilibrium is and act accordingly. In competitive markets, this thinking process is replaced by simply following the price, knowing that the price contains information about the others' actions. This price-taking behavior can actually be interpreted as a solution to the game of market, where the condition of Nash equilibrium is interpreted as Pareto Optimum.
From reading Fei's lecture notes, it became clear that the Pareto Optimum (pareto efficient allocation) is nothing but a further level of constrained utility / profit maximization, which in this sense is just a pareto efficient allocation of the individual's constrained wealth / production set (?) which maximizes individual preference (utility) / production pareto efficiently. On the market exchange level, this constrained maximization becomes pareto efficient allocation of constrained total resource that maximizes group preference (utility) pareto efficiently.
pareto optimum = vector maximumdecision
theory = vector maximization
As such, the existence question of an equilibrium should be the same as the existence question of a maximum which is solved for in utility maximization with nonlinear programming conditions, e.g. KTCQ.
Competitive Markets
What is meant by 'competitive'?
It is nothing about ability or attitude. In economic theories, it is rather stated as the opposite to monopoly, and it is actually an 'inability' to affect something unilaterally.
In a sense, it is a decision concept, where there is an equilibrium of the decisions of various agents rather than an individual decision. A 'competitive equilibrium' is first and foremost an equilibrium, and the term 'competitive' can be actually eliminated since when talking about an equilibrium, the competitiveness or involvement of interacting agents is already inferred.
While Nash equilibrium is a kind of equilibrium, which is actually also 'Pareto Optimum' equilibrium as they both measure individual welfare states, there can also be definitions of equilibrium where aspects other than welfare is emphasized, for example goods, the equilibrium of goods, or equilibrium of demand and supply of goods, which in case can be measured cardinally and equilibrium can be equivalently translated as equality, so 'competitive equilibrium' actually refers to equality of demand and supply.
In Nash and Pareto definitions, welfare can only be measured ordinally, unlike goods, and equilibrium refers to an ordinal or relative 'equality' of welfare for all agents. Thus to ask what is the relationship between 'competitive equilibrium' and 'Pareto optimum' is actually to ask whether the cardinal goods equilibrium and ordinal welfare equilibrium can be consistent, i.e. achieved simultaneously / interchangeably.
As another way to put this, competitive equilibrium is the equality of demand and supply, a positive matter, and Pareto optimum is the 'equality' of welfare, a normative matter. And the question is whether the same mechanism that drives an equilibrium allocation of goods is automatically conforming to our ethical standard of equal welfare. The equilibrium of goods allocation is something naturally occurring in nature. What economists found is the mechanism behind, which is selfishness (individual optimization), meaning that nature achieves equilibrium with our being selfish, and it is our being selfish that helped nature allocate its resource. And the economists did something even further than reconciling selfishness with nature. They reconciled with themselves, and proved that being selfish is being ethical, and to be ethical you have to be selfish.
It is nothing about ability or attitude. In economic theories, it is rather stated as the opposite to monopoly, and it is actually an 'inability' to affect something unilaterally.
In a sense, it is a decision concept, where there is an equilibrium of the decisions of various agents rather than an individual decision. A 'competitive equilibrium' is first and foremost an equilibrium, and the term 'competitive' can be actually eliminated since when talking about an equilibrium, the competitiveness or involvement of interacting agents is already inferred.
While Nash equilibrium is a kind of equilibrium, which is actually also 'Pareto Optimum' equilibrium as they both measure individual welfare states, there can also be definitions of equilibrium where aspects other than welfare is emphasized, for example goods, the equilibrium of goods, or equilibrium of demand and supply of goods, which in case can be measured cardinally and equilibrium can be equivalently translated as equality, so 'competitive equilibrium' actually refers to equality of demand and supply.
In Nash and Pareto definitions, welfare can only be measured ordinally, unlike goods, and equilibrium refers to an ordinal or relative 'equality' of welfare for all agents. Thus to ask what is the relationship between 'competitive equilibrium' and 'Pareto optimum' is actually to ask whether the cardinal goods equilibrium and ordinal welfare equilibrium can be consistent, i.e. achieved simultaneously / interchangeably.
As another way to put this, competitive equilibrium is the equality of demand and supply, a positive matter, and Pareto optimum is the 'equality' of welfare, a normative matter. And the question is whether the same mechanism that drives an equilibrium allocation of goods is automatically conforming to our ethical standard of equal welfare. The equilibrium of goods allocation is something naturally occurring in nature. What economists found is the mechanism behind, which is selfishness (individual optimization), meaning that nature achieves equilibrium with our being selfish, and it is our being selfish that helped nature allocate its resource. And the economists did something even further than reconciling selfishness with nature. They reconciled with themselves, and proved that being selfish is being ethical, and to be ethical you have to be selfish.
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(topology)
Examples of continuous functions, with which topological properties such as compactness and therefore extremums can be preserved.
This is important because it enables the search for optimal solutions without requiring a continuity of possibilities, which is required when approaching the problem with calculus thus differentiation, so that in addition to requiring the continuity of functions, continuity of the set on which the function is defined (set of possibilities) is also required to ensure differentiability, which is not necessary, as a non-continuous set can as well achieve optimality, and as long as such properties can be transferred consistently in between possibilities (e.g. production sets/function or consumption bundle) and welfare measures (e.g. profit or utility), answering questions such as how to allocate resource (possibilities) to achieve an optimum social welfare state would become possible.
Examples of continuous functions, with which topological properties such as compactness and therefore extremums can be preserved.
This is important because it enables the search for optimal solutions without requiring a continuity of possibilities, which is required when approaching the problem with calculus thus differentiation, so that in addition to requiring the continuity of functions, continuity of the set on which the function is defined (set of possibilities) is also required to ensure differentiability, which is not necessary, as a non-continuous set can as well achieve optimality, and as long as such properties can be transferred consistently in between possibilities (e.g. production sets/function or consumption bundle) and welfare measures (e.g. profit or utility), answering questions such as how to allocate resource (possibilities) to achieve an optimum social welfare state would become possible.
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