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

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.

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.

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.

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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.

Consumption Set and Preference Ordering

As discussed in the note for competitive equilibrium, the maximizing behavior of a consumer can only be modeled in relative ordinal measures, i.e. through preference orderings. A convenient way of storing this ordinal information would be to map it into a cardinal utility measure, although such a transforming function, if to be more efficient than remembering the orders directly, thus usually preferred to be continuous, can only serve a certain type of preference assigning scheme. An exception would be the lexicographic ordering, and the essence is that in such an ordering scheme, the economic agent assigns preferences at disjoint steps, and while the orders can still be stored in an arbitrary utility function, the irregular discontinuity points makes such a transformation saving no much in storing the information.

In a sense, a very strong sense, the preference ordering itself is a function.

Most of the specifications or assumptions about the preference orderings, such as local nonsatiation or convexity, really should be understood as normal assumptions about functions.

Difficulty in Modeling Social Welfare

Takayama started discussion on this issue from the optimizing behavior of producers in a price-taking, thus competitive economy, and in his opinion, the problem of social prosperity (optimality) only arises when the behavior of consumers is 'introduced', and the critical concern is that their optimizing behavior cannot be measured as explicitly as in the case of profits for the optimizing producers. Following his logic, if it were not for this difficulty, social welfare would have been evaluated by just adding up individuals' utilities, and optimized given resource constraints, so basically just an ordinary optimization problem, as an enlarged version of the production problem. Then in his opinion, since such a 'cardinal' evaluation is not possible, people resorted to relative measures, i.e. the 'ordinal', and evaluated not the optimal amount (the exact utility, which is considered unmeasurable), but the optimal behavior when the relative orders of the optimal amount is stabilized at an equilibrium state.

In this sense, the trick of analyzing a resource constrained exchange behavior becomes switching the focus between absolute (for producers) and relative (for consumers) values.

Optimization

Linear programming (simplex/...) is a special kind of optimization in the sense that the problem can be simplified to a few vertices due to the convexity/regularity/predictability of linearity.

Separation theory translates the spacial relationship of keeping a certain distance from something and only to one side, to the algebraic expression using a notion of the linear algebraic vectors (depicting the side) and a notion of order (depicting the distance). In other words, optimization can be directly translated to spacial limit searching if ever optimization is possible (somewhat convex).

Nonlinear programming (KT/...) comes from the the concept of simplification of optimization problems in linear programming according to geometrically intuitive properties. It is in contrast to the second-order calculus approach which also come from geometrical concepts but used a more complicated representation of convexity thus making the solution process less preferred. Such complication also imposed unnecessary restrictions on the types of solvable problems, especially in terms of the effectiveness of constraints, where in settings such as the Lagrangian problem constraints have to be in equality in order for the first order conditions to be 'conveniently' fulfilled.

The envelope theorem states that the total effect on equilibrium state of the change of a parameter in the optimization process is the same as if no optimization process is gone through. Varian explained it more intuitively in his Microeconomic Analysis, page 45 (profit function).

Separation Theory

...(as compared to the use of tangents in calculus) the power of the separation theorem is that the boundary (hyper-) curve does not have to be smooth (differential)...

A Little Topology

Convex sets are closed (in a certain sense) under linear operations.
Closed sets are closed under limit operations.
Compact sets are closed and finite (in a certain sense) under limit operations.

convex set ~ concave function ~ linear operation
closed set ~ continuous function ~ limit operation

Closed sets contain all the limits of its convergent sequences. With compact sets the limits not only are contained within the set but also are insured of existence for at least one subsequence of every sequence, i.e. sequences are bounded within compact sets, while closed within closed sets.Building on the spacial property of distance in metric spaces and extending beyond to general non-metric spaces.

...open sets satisfy the axioms of topological space... 

Concept of limit. Sensible only in the context where a 'topology' e.g.distance is defined.

...a sequence is not a set of points...rather, it is a function... 

A limit is something that can be approached by a sequence.

...limit points are limits of sequences...

...a closed set contains all its limit points...

...a closed set is a set which is closed under the limit operation... 

Continuity of a function is the continuity of limit operation before and after the function.
Metric functions are continuous by definition because of triangular inequality.
Closed and bounded = compact

Linear Functions

...the set of matrices is an isomorphic representation of the set of linear functions on a finite dimensional linear space...

other than being a mere array of numbers.

These linear functions are defined on a fixed basis, while the matrices are representations of when the functions are applied to arbitrary vectors constructed from the basis vectors. The matrices cannot be defined directly as functions since they are just lists of numbers without variables, i.e. they are just the 'constants' in the functions.

Duality in Optimization

Duality in the optimization context basically means the existence of an equivalent representation of the optimization problem in linear form, coming from the connection between linearity and convexity. Basically if a person's decision behavior is more rational (convexity of consumption set), it will be more regular and predictable (convertable to linear optimization of the dual problem). A deeper root of such a property lies in the property of binary ordering.

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Relationship between preference relations and choice rules is investigated in this section, but in fact, it is the rational preference relations and the consistent choice rules that is being compared. What about the question of whether every choice rule, consistent or not, can be explained by a preference relation, and whether every preference relation, rational or not, can be realized in a choice rule? The second question obviously has an answer yes, and with multiple possibilities of realization. 举一反三. The answer to the first question is more like the mathematical / philosophical question of whether there is a solution / truth to a particular problem or not. For example, can the second case in Example 1.C.1 be explained by a preference relation while the choice rule itself is not consistent? Can it be explained by a rational preference relation? If it can then there are cases not consistent but rational, which is not impossible intuitively, suggesting that the choice rule is not exactly a generalization of preference relations and we cannot say that one assumption is weaker or stronger than the other. This should be a more fundamental question related to the structures behind these two different ways of storing behavior, of decision making or of choice.

The authors in the book actually touched on this point while introducing on Page 14 a different way of realizing choice from a rational preference relation, i.e. a different definition 1.D.1., which allowed for choosing less than one's optimal choices and leaving behind some of those one is indifferent to, suggesting that, in an extreme case, one can be rationally indifferent among all alternatives which trivially serves as an explanation for any choice behavior, including those inconsistent ones, and in this sense, consistency is actually an additional restriction rather than a relaxation. In a sense those difficulties in preference theory actually also arise from this more restrictive view of how behavior is connected to rationality, and the Condorcet paradox on Page 8 can be readily solved using the indifferent preference relation.

Preference and Choice

Takayama oriented his Mathematical Economics stressing the set-theoretic approach to behavioral modeling, in contrast to the traditional calculus approach. In this Oxford book however, it is the rationality assumption (completeness and reflexibility of preference relations), present in earlier times including Takayama's for ease of modeling, that the authors now try to substitute with the new concept of choice rule, which basically weakened the assumption of complete and transitive rationality to instantaneous and pairwise rationality (WARP). How would such a simplification change the mathematical formulations? Maybe it changed nothing, but just extended the application of these formulations to less rational conditions, like the extension of KTCQ of optimization which is calculus based thus requiring differentiability, to non-differentiable cases using non-linear programming.

Like differentiability, completeness and transitivity are also properties of functions / correspondences from X, to >() if not a function (not continuous), and to R if represented by utility. The structure (X, >()) is a correspondence commonly restricted by the axiom of rationality, while the structure (B, C()) is a correspondence commonly restricted by the axiom of WERP (consistency in behavior rather than in the intellect, i.e. people are, assumed, to be more motivated or even forced to behave consistently in a social context, which reflects some rationality of mind in the decision making process, but such a process can also be skipped to arrive at the same behavior.).

However, less assumptions usually means more information to be considered and more complex analysis. Preference relations relaxed differentiability but generally went back to utility functions (assuming continuity and very often even back to differentiability) in analyzing practical problems. The choice approach will also do the same. 

Theories are simplifications of the world, and relaxing assumptions generally brings theories down by introducing a more complex world, which however is also a way of testing how powerful and brilliant the theories are. Everyone can have their own theories, just some theories are based on relatively less information and more assumptions in the view point of those who have more information, and these theories are usually just called prejudice, presumption, or cliche if information come with time. In this sense those economic theories that do not pass the test of 'choice rule' might then be considered as prejudice, presumption, or cliche, but can still be true and influential within their circles or within their time.

What then, do the assumptions of continuity and differentiability mean exactly, if preference relations means rationality and choice rule means consistent behavior? It seems the two calculus related concepts have something to do with the comparative statics and thus the thermodynamical equilibrium concept.

Public Goods and Externalities

Are they not essentially the same? Is public good not just a positive externality where the good is shared so that consumption by one person benefits the utility of everyone else?

Elasticity and Equilibrium

Samuelson introduced physics into economics. It has become ever clearer that it is into the concepts of elasticity and equilibrium that analogy with physics was introduced. There has to be a relationship between the change in elasticity (yet not elasticity itself) of a rubber band as you stretch it and whether it springs back (whether it can stay equilibrium at the stretched state) after you release it.
From P8/Ch4 of Fei's lecture notes, it is suggested that there is a relationship between the convexity of production set and the elasticity of scale.

Duality

In a sense duality is just something related to the hyperplanes in separation theory. While duality used calculus and differentials to represent the tangent price lines, separation theory used inner product of vectors to represent the line. Varian's book is really looking more like a devolution from Takayama's, in terms of the use of mathematics, which was introduced into economics to clarify things better, and now people are going back to complicating them.
However this might be rather a necessary situation. Whenever a new existence comes in, nature has to integrate it into its system by enriching it in two directions, complicating and simplifying, so that it gets connection to everything else, and so that the connections are binded together under a single rule. Both simplicity and complexity are natural. Humans do best if they follow the natural way. And for the intellect as an existence in the world, both complexity and simplicity is needed for it to integrate into the world. In this sense, the popular Chinese system of the complicating Confucious together with the simplifying Buddhism, or rather Zenism, is quite a working combination.

Subgame Optimization

The analogy suggested in this title is also mentioned in the book's chapter on game theory, where the author said that game theory is just an extension of decision theories,

....a generalization of standard, one-person decision theory...

In a sense, the optimization part (of profit, cost on the firm's side and of utility, demand on the consumer's side) of Microeconomics is just about obtaining the payoff tables in games, and they constitute a subgame where only one agent is involved, i.e. subgames (improper) with only the terminal nodes in an extensive form game tree. And the same idea of an equilibrium found by state-wise examination (living in heaven without knowing the path to it) is probably also present in the less 'gamely' equilibrium analysis, where the strategy-payoff connection is the more complicated part to calculate, while in game theory such calculation is assumed to have already been informed for the various players but the strategy-strategy connection is more investigated to deal with complications in such ways.

How does profit / utility maximization and cost /expanse minimization genuinely become different, while appearing to be much the same? If I am faced with some decisions about how much production to make, or how much to consume (which is slightly different since utility is not generally considered measurable as in production), I can generally adjust my input amount accordingly, and the important decision is not so much on choosing the cheapest ones (although should be adjusted for), but rather finding the satiating level, before which spending more can actually bring more profits, and the actual level is determined by a kind of efficiency of production technology (or greediness in consuming). Yet if I am trying to choose between alternative inputs for production so that cost is minimized, the relative efficiencies of respective inputs (efficiency of efficiencies / substitution rate) seem to become more relevant than their aggregated efficiency, and the gaming tinge (level of satiating) is not yet there. In a sense cost minimization is more conservative than profit maximization, and while profit maximizers generally do minimize cost, they can also make profits being luxurious. The traditional (or obsolete?) profit seekers either emphasize too much on minimizing cost or they believe that the more production the better (which could be that they don't really have alternative ways to produce, for example a field tending farmer, as opposed to a modern investor). Maybe that's how we accumulated enough for the modern consumers to think exactly the opposite, and since utilities are not really measurable, they just take them as unlimited.

The satiating level mentioned above should be present at all levels of optimization, and actually is very likely to be the one thing that links the simple (optimization) and complex (game) decision theories. At an optimal / equilibrium decision, any unilateral move away from the decision would lead astray the optimum. In applying the game theory, it is fitting this equilibrium concept into a specific setting, what kind of moves that can lead you astray, is it jumping upwards or swaying sideways that take you off the balance beam, that is to be found out, and theory just gives a criteria for you to judge whether you are right or not. In applying optimization theories and deciding for optimal actions however, these game settings are already part of how people found the theory, and it is already confirmed that swaying sideways can take you down from the balance beam, so you start from this point and find some other signs that can tell you whether you are swaying sideways or not, or more precisely, whether you are standing straight, touching the beam, or not, which is more or less equivalent statements / properties / definitions of swaying sideways, and in mathematical terms they are along the necessary and sufficient line of derivation, including equations. 

Some common types of such signs of equilibrium / optimum.

(it might be actually understood as a kind of order-preserving transformation from different equilibrium states to different values of a certain property)

(1) Tangency conditions. P27. P42. P51. P73. P89. P101. P107. P109. P110. (application diagrams not included)

Marginal rates of the decision factors that prevail at optimums is connected to some external factors such as price.