recursal/Fanatic-Fandom huggingface dataset:
Micro Theory

**Debreu’s Theorem**: Let

≿

{\displaystyle \succsim }

 be a preference relation on

X

{\displaystyle X}

, and suppose that

X

{\displaystyle X}

 admits a countable, order-dense subset

Z

{\displaystyle Z}

. Then,

≿

{\displaystyle \succsim }

is continuous if and only if

≿

{\displaystyle \succsim }

 admits a continuous utility representation

u
:
X
→

R

{\displaystyle u:X\to \mathbb {R} }

. (see Debreu 1954, 1964)
Homo Economicus

A model for how individuals or aggregates of individuals make choices.

As a combination of which factors is this concept defined?

* Fully selfish, no altruism
* Perfectly “rational” (what does this really mean?)
* Perfect calculator
* Perfect memory/recall
* No environmental influences that are not incentives (no nudges)
* No morality
* No other norms
* No mistakes
* No weakness of will
Lebesgue Integral

To be written when properly understood.

* Wrote an article on simple functions already. Link to there, and maybe add some graphics there (of non-equal function value steps)
* Add graphics here of several examples/different types of functions.

This page is very helpful: Lebesgue Integral - graphical concept - Mathematics Stack Exchange. The crucial difference is equally spaced partition of the range instead of the domain? Seems like the representation of horizontal vs. vertical bars is wrong. Instead, both Riemann and Lebesgue are sums of vertical bars, but in Riemann the bars are of equal width, where in Lebesgue the function values at points in a bar is equally different between all bars.
Sustainability of Collusion

I think it is generally not possible in static situations. Here, there is always an incentive to undercut the current price/competitor? Are there non-standard assumptions under which it does become possible?

In dynamic settings, it seems likely there is a rich combination of conditions under which collusion is possible.

* Differentiate between explicit and tacit collusion?
* Collusion in auctions

### Some refs:

https://ec.europa.eu/competition/mergers/studies\_reports/the\_economics\_of\_tacit\_collusion\_en.pdf
Wiki Conventions

This page lists conventions that serve towards internal consistency in this wiki.

* Set: denoted by the letter S, as

S

{\displaystyle S}
* Define mathematical concepts ideally on metric spaces. This seems general enough to include most econ purposes I run into while not unnecessarily complicating things by defining on topological spaces. [NO, NOT GENERAL ENOUGH], do all definitions on Euclidean, metric, and topological spaces? Is that a sensible way to categorize things?
* Denote events by A and B, and random variables by X and Y. Note: event is a set of outcomes of an experiment (subset of sample space), to which a probability is assigned. Denote outcomes by a, b maybe? Hence, if some event A is defined by three outcomes, a, b, c, then A = {a, b, c}.
* Denote the probability space as

Ω

{\displaystyle \Omega}

 for the sample space, something like

F

{\displaystyle \mathcal{F}}

 for the sigma algebra of events, and something for probability (maybe

P
,

P

,
Pr
,

P

{\displaystyle P,\mathbb {P} ,\Pr ,{\mathcal {P}}}

)
* Add a 'Motivation' heading at a page. Why is this important? What is its use?
Differentiation Rules

Exponential function:

* d

d
x

a

u

=

a

u

ln
⁡
(
a
)

d

d
x

u

{\displaystyle \frac{d}{dx} a^{u} = a^{u} \ln (a) \frac{d}{dx}u}
* d

d
x

a
x

=

a
x

ln
⁡
(
a
)

{\displaystyle \frac{d}{dx}a^x = a^x \ln(a) }
* d

d
x

e

u

=

e

u

d

d
x

u

{\displaystyle \frac{d}{dx} \mathrm{e}^{u} = \mathrm{e}^{u} \frac{d}{dx}u}
* d

d
x

e

x

=

e

x

{\displaystyle \frac{d}{dx} \mathrm{e}^{x} = \mathrm{e}^{x}}
Limit of a sequence of real numbers

Notation:

* a

n

→
L

{\displaystyle a\_{n}\to L}

 as

n
→
∞

{\displaystyle n\to \infty }
* lim

n
→
∞

a

n

=
L

{\textstyle \lim \_{n\to \infty }a\_{n}=L}

## Definition

Let

(

a

n

)

n

=

a

1

,

a

2

,
.
.
.

{\displaystyle (a\_{n})\_{n}=a\_{1},a\_{2},...}

 be a sequence of real numbers. If the limit of this sequence exists,

L
∈

R

{\displaystyle L\in \mathbb {R} }

 is the limit of the sequence if and only if: for every

ε
>
0

{\displaystyle \varepsilon >0}

,

∃

N

ε

{\displaystyle \exists N\_{\varepsilon }}

 such that

∀
n
≥

N

ε

{\displaystyle \forall n\geq N\_{\varepsilon }}

, we have

a

n

−
L

<
ε

{\displaystyle |a\_{n}-L|<\varepsilon }

.
Apostel's Axioms for the Real Numbers

There are many ways to construct the real numbers. I think being slightly outside the scope of Apostel 1, it takes a nonconstructive approach starting out rather far in the process:

* He takes the real numbers

R

{\displaystyle \R}

 as undefined objects
* States a list of 10 axioms that the real numbers must satisfy

All properties of

R

{\displaystyle \R}

 can be deduced from these ten axioms. I suppose that if we would consequently study a constructive approach, we could use all the results we find in Apostel by proving these ten axioms as theorems.

## Axioms

The ten axioms consist of: 6 field axioms, 3 order axioms, and a completeness axiom. Note that the six field axioms are the same as the definition of a field.
Infinitesimal

An **infinitesimal** or **infinitesimal number** is a quantity that is closer to zero than any standard real number, but that is not zero. In common speech, an infinitesimal object is smaller than any feasible measurement, but not zero in size, or so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective in mathematics, *infinitesimal* means infinitely small, smaller than any standard real number.

The term **differential** is used (in calculus) to refer to an infinitesimal ('infinitely small') change. E.g. for a variable

x

{\displaystyle x}

, a change in the value of

x

{\displaystyle x}

 is often denoted as

Δ
x

{\displaystyle \Delta x }

. The differential is an infinitely small change in

x

{\displaystyle x}

, denoted as

d
x

{\displaystyle dx}

. It is possible to relate infinitesimal changes of variables to each other using derivatives. If

Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives. If

y
(
x
)

{\displaystyle y(x)}

s a function of

x

{\displaystyle x}

, then the differential

d
y

{\displaystyle dy}

 of

y

{\displaystyle y}

 is related to the infinitesimal change in

x

{\displaystyle x}

,

d
x

{\displaystyle dx}

, by the formula

d
y
(
x
# )

d
y
(
x
)

d
x

d
x

{\displaystyle dy(x) = \frac{dy(x)}{dx}dx}

, where the fraction gives the derivative. This formula summarizes the intuitive idea that the derivative is the limit of the ratio of differences

Δ
y

/

Δ
x

{\displaystyle \Delta y / \Delta x}

 as

Δ
x

{\displaystyle \Delta x }

becomes infinitely small.

An integral can be regarded as an infinite sum of infinitesimal quantities, the area under a graph is obtained by subdividing the graph into infinitely thin strips and summing their areas. In an expression such as

∫
f
(
x
)
d
x

{\displaystyle \int f( x) dx}

, the integral sign denotes the infinite sum,

f
(
x
)

{\displaystyle f(x)}

 denotes the "height" of a thin strip, and the differential

d
x

{\displaystyle dx}

denotes its infinitely thin width.
Paper: Arrow (1950)

19 pages in total. The theorem is popularized more by the 1951 book.

## Summary

### I. Introduction (328-331)

* Par 1-2: mention 4 methods of social choice: voting, markets, dictatorship, and convention. He contrasts the latter two as not involving conflict of individual wills, with the former two as "methods of amalgamating the tastes of many individuals". Are the latter who *collective* modes of choice rational?
* **Main question**: "we ask if it is formally possible to construct a procedure for passing from a set of known individual tastes to a pattern of social decision-making, the procedure in question being required to satisfy certain natural conditions."
* Example of this failing: the paradox of voting:

A

≻

1

B
,
B

≻

1

C

⟹

A

≻

1

C

{\displaystyle A\succ \_{1}B,B\succ \_{1}C\implies A\succ \_{1}C}

, and

B

≻

2

C
,
C

≻

2

A

⟹

B

≻

2

A

{\displaystyle B\succ \_{2}C,C\succ \_{2}A\implies B\succ \_{2}A}

, and

C

≻

3

A
,
A

≻

3

B

⟹

C

≻

3

A

{\displaystyle C\succ \_{3}A,A\succ \_{3}B\implies C\succ \_{3}A}

. Then A wins from B, B wins from C, but C wins from A. So, the majority voting method here fails to produce rational collective choices.
* Can we find other methods of aggregating individual tastes which imply rational behavior of the community (and which will be satisfactory in other ways)? If we adopt the identification of rationality with maximization of some sort, then the problem of achieving a social maximum derived from individual desires is precisely the problem central to welfare economics.
* Voting and Kaldor's compensation principle are two social choice methods, but both suffer from the same problem outlined above: a paradox need not occur, but there are preference patterns which will give rise to inconsistent social choice patterns. Therefore, both must be regarded as unsatisfactory techniques for the determination of social preferences.
* "**The aim of the present paper is to show that these difficulties are general. For *any* method of deriving social choices by aggregating individual preference patterns which satisfies certain natural conditions, it is possible to find individual preference patterns which give rise to a social choice pattern which is not a linear ordering**."

### V. Some Implications for the Formation of Social Welfare Judgments (342-346)

#### V.4. Distributional Ethics Combined with Individualism

1. ↑ Makes a nice note here that earlier welfare economics was based on cardinal preferences and was plagued by difficulties of interpersonal comparisons. It is noted that emphasis shifted to a weaker definition: Pareto optimality. But, this is not sufficient to make social choices, it is only preliminary to it.
2. ↑ I'm fairly confident linear ordering here means total ordering in the mathematical sense.
Measurable set

Given a set and an associated sigma-algebra,

(
S
,
Σ
)

{\displaystyle (S,\Sigma)}

, a set

A
∈
S

{\textstyle A \in S}

 is measurable if

A
∈
Σ

{\textstyle A \in \Sigma}

.

This should be intuitive because a measure tends to be defined on a sigma-algebra??

* I think because of the sigma-additivity of a measure, it is necessary to have the countable unions/intersections also be in the set of subsets (like for a sigma-algebra).
* Do we need the closed under complementation for a measure???
Choice Function

A **choice function** is a function

C
:

2

X

→

2

X

{\displaystyle C:2^{X}\to 2^{X}}

 such that:

* C
(
A
)
⊆
A
,
∀
A
∈

2

X

{\displaystyle C(A)\subseteq A,\forall A\in 2^{X}}
* the choice function is non-empty:

C
(
A
)
≠
∅
,
∀
A
≠
∅

{\displaystyle C(A)\neq \emptyset ,\forall A\neq \emptyset }

.

where

X

{\displaystyle X}

 is a *finite* set of alternatives, and

2

X

=
{
A

A
⊆
X
}

{\displaystyle 2^{X}=\{A|A\subseteq X\}}

 is the set of all possible subsets

A

{\displaystyle A}

 in

X

{\displaystyle X}

. Note, that the second line does not imply that the consumer can not choose to do nothing, but instead doing nothing is included as an option in the choice set.

Intuitively, as the choice function is intended to describe observed behavior, an empty choice set would mean that there was no choice moment?
Calculus

**Calculus** is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.

Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing.

In mathematics education, calculus d *d*enotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits.

## Etymology

Thword *calculus*d *calcul* is Latinin for "small pebble" (the diminutive of *calx,* meaninn medicine. Because such pebbles were used for counting out distances, tallying votes, and doi(counting-frnme) ag abacus arithmetic, the word came to mean a method of computation. In this sense, it was used in English at least as early as 1672, several years prior to the publications of Leibniz and Newton.

Contents:

* Integral Calculus
* Differential Calculus
* Series
* Vector
Solution Approaches for Dynamic Optimization

* Variational Approach (from calculus of variations, which uses small changes to find minima or maxima of a function. Think of derivatives)
	+ Also, the sequential problem formulation? (so called because one chooses the entire sequence of choices).
* Optimal Control Theory
* Dynamic Programming/Recursive formulation:
	+ Transforms the dynamic problem into many static problems.

Questions:

* How is dynamic programming a solution approach from a dynamic optimization problem? I thought these were essentially acronyms.
Squaremen

The squaremen are the most common enemy type. They often move around and hurt if touched. In most games, Squaremen are ruled by Octagonman and are a metaphor for the working class. They are seen as a negative thing to The Judge who often sentences criminals to Squaremen duties which has the convict killing a group of them. Notable squaremen include zm and That One Squareman from Triangleman 3.
Expected Value

In probability theory, the **expected value** (a.k.a. (**mathematical**) **expectation**, **mean**, **average**, or **first moment**) is a generalization of the weighted average.

The expected value of a random variable with a *finite* number of outcomes is a weighted average of all possible outcomes,

E
[
X
# ]

∑

i
∈
X

x
i

p
i

{\displaystyle E[X] = \sum\_{i \in X} x\_i p\_i}

. When the set of possible outcomes is not countable, but a continuum, the expectation is defined by integration instead of summation

E
[
X
# ]

∫

−
∞

∞

x
f
(
x
)
d
x

{\displaystyle E[X] = \int\_{-\infty}^\infty x f(x) dx}

. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration.

## Notation

The expected value of a random variable X is often denoted by

E
(
X
)
,
E
[
X
]
,
E
X
,

E

[
X
]

{\displaystyle E(X), E[X], EX, \mathbb{E}[X]}

, or

μ
X

,

X
¯

{\displaystyle \mu\_X, \bar{X}}

 where the latter are more common in physics to denote the mean.

## Definition

Put in the form of Expectation under different possibilities of the set X.
Pages to add

* Pages for countries, not with google-able facts, but overviews such as the list of kings/governments/leaders over time.
* Economies of Scale
* Economies of Scope

Econ Math thms/tools

* Implicit function thm
* Envelope thm

Calculus:

* Squeeze theorem
* First fundamental theorem of calculus
* Second fundamental theorem of calculus (or combined)

Probability theory/statistics:

* Cauchy-Schwartz inequality
* Chebychev's inequality
* Kolmogorov's 0-1 law
* Boole's inequality
* Holder's inequality
* Page with common distributions (possibly linking to pages for individual distributions?)
Handbook Ch: Jantti & Jenkins (2015)

This is chapter 10 in the Handbook of Income Distribution vol 2A (2015).

"We survey the literature on income mobility, aiming to provide an integrated discussion of mobility within and between generations. We review mobility concepts, descriptive devices, measurement methods, data sources, and recent empirical evidence."
Menus

This page is a basic guide through all of the menus in the game. The *New Super Mario Bros.* menu music plays while in most menus.

## Title Screen

Upon launching the game, you will be greeted with the Title Screen. It displays the game's logo, the game version you are playing on, and says "Click to Continue...", and shows a random stage as the backdrop. Clicking or pressing any button will send you to the Main Menu.

## Main Menu

The Main Menu is the main starting point that will lead you into other menus.

 Options
 Quit
 About

### Play Game

> *Main page: Lobby Menu*
>
>

The Play Game button will take you to the Lobbies screen, where you can view and join other player's lobbies.

 Region Select
 Player's Lobby
 Join a Private Lobby
 Back Arrow
 Join Lobby
 Create Lobby

#### Create Lobby

The Create Lobby menu lets you create your own lobby, with a couple of available options.

 Private Lobby
 Cancel
 Confirm

#### Join a Private Lobby

The Join a Private Lobby menu allows you to join another player's private lobby by entering a code.

 Cancel
 Confirm

#### Lobby Menu

These are the options available while waiting in a lobby.

 Chatbox
 Send
 Character
 Colors
 Spectate Match
 Exit Lobby
 Start Game
 Copy ID

##### Game and Server Settings

These options can only be modified by the game's host.

 Coins for Item
 Time Limit
 Draw on Time Up
 Custom Powerups
 Players
 Private Lobby

### Options

> *Main page: Options*
>
>

These are the settings found in the Options menu.

#### Controls

Allows you to view and rebind your controls.

 Sprint Shoots Fireball
 Reset to Defaults
 Back Arrow

### About

> *Main page: Credits*
>
>

The About menu shows all the people who contributed to the game.

 Links

## Pause Menu

> *Main page: Pause Menu*
>
>

The Pause Menu can be opened while in a match by pressing the Pause button.

 Options
 Quit
 End Match
 Main Menu
 Cancel

## Update Available

This menu only shows up when you're playing on an outdated version. It tells you that a new version is available, along with the new version number as well as the version you're using.

 Ignore
Simple Function

In mathematics, a **simple function** is a real- or complex-valued function over a subset of the real line.

They look similar to step functions, in that they have are defined everywhere on the domain subset (really???) and consist of flat lines with jumps.

They attain only a finite number of values (note this does not mean there are a finite number of subsets of the domain).

Note: important for the definition of the Lebesgue integral, since the simple function is a function, each point in the domain only maps to one point in the image, so that when defining the 'bars to add up' in the definition, there is no overlap.

## Definition

Let

(
S
,
Σ
)

{\displaystyle (S,\Sigma )}

 be a measurable space,

(

A

i

)

# i
1

n

⊂
Σ

{\displaystyle (A\_{i})\_{i=1}^{n}\subset \Sigma }

 be a sequence of disjoint measurable set, and

(

a

i

)

# i
1

n

{\displaystyle (a\_{i})\_{i=1}^{n}}

a sequence of real or complex numbers. Then the function

f
:
S
→

C

{\displaystyle f:S\to \mathbb {C} }

 is a simple function if it is of the form:

f
(
x

# i
1

n

a

i

1

A

i

(
x
)

{\displaystyle f(x)=\sum \_{i=1}^{n}a\_{i}\mathbf {1} \_{A\_{i}}(x)}

where

1

A

i

(
x
)

{\displaystyle \mathbf {1} \_{A\_{i}}(x)}

 is the indicator function of the set

A

i

{\displaystyle A\_{i}}

.

## Integral of simple functions

If a measure

μ

{\displaystyle \mu }

 is defined on the measurable space

(
S
,
Σ
)

{\displaystyle (S,\Sigma )}

, then the integral of

f

{\displaystyle f}

 with respect to this measure is

∑

# i
1

n

a

i

μ
(

A

i

)

{\displaystyle \sum \_{i=1}^{n}a\_{i}\mu (A\_{i})}

if all terms in the sum are finite.
Bayes' Rule

**Bayes' rule** (a.k.a. **Bayes' theorem** or **Bayes law**) describes the probability of an event, based on prior knowledge of conditions related to the event.

## Statement of Theorem and Proof

From the definition of conditional probability:

P
(
A

B
# )

P
(
A
∩
B
)

P
(
B
)

{\displaystyle P(A|B)={\frac {P(A\cap B)}{P(B)}}}

P
(
B

A
# )

P
(
A
∩
B
)

P
(
A
)

{\displaystyle P(B|A)={\frac {P(A\cap B)}{P(A)}}}

Solving for the intersection event probability in the one and substituting the solution in the other gives:

P
(
A

B
# )

P
(
A
∩
B
)

P
(
B
)

=

P
(
B

A
)
P
(
A
)

P
(
B
)

{\displaystyle P(A|B)={\frac {P(A\cap B)}{P(B)}}={\frac {P(B|A)P(A)}{P(B)}}}
Abstract Algebra

**Abstract algebra** (a.k.a. **modern algebra**), is the study of algebraic structures.

## Algebraic structure

An algebraic structure consists of a:

1. Non-empty set, say

X

{\displaystyle X}

 (called the underlying set, carrier set, or domain)
2. Collection of operations defined on

X

{\displaystyle X}
3. Finite set of identities, known as axioms, that these operations must satisfy

Common axioms include: commutativity, associativity, and distributivity which are expressed as identities. E.g. an operation

⋆

{\displaystyle \star }

 is commutative if

x
⋆
# y
y
⋆
x

{\displaystyle x\star y=y\star x}

.

## Common algebraic structures

### One set with operations:

**Simple structures**: one set, no binary operation:

* set

**Group-like structures**: one set, one binary operation:

* Semigroup
* Monoid
* Group
* Abelian group a.k.a commutative group
* Magma
* Quasigroup
* Loop
* Semilattice

**Ring-like structures**: one set, two binary operations:

* Semi-ring
* Near-ring
* Ring
	+ Commutative ring
	+ Division ring
	+ Field
* Lie Ring
* Jordan Ring
* Boolean Ring
* Lattice
* Boolean Algebra

### Two sets with operations:

* Module
* Vector space
* Algebra over a field
Sampling

In statistics, **sampling** is the selection of a subset of elements from a statistical population.

## Sampling methods

* Random sampling
* Stratified sampling
* Clustered sampling
* From finite population with replacement (iid)
* From finite population without replacement (a.k.a 'simple random sampling', though its not random sampling since not iid)
Total Order

A **total order** or **linear order** is a partial order in which any two elements are comparable. That is, a total order is a binary relation on some set, which satisfies

∀
a
,
b
,
c
∈
S

{\displaystyle \forall a, b, c \in S}

:

1. **Reflexivity**:

a
≤
a

{\displaystyle a \le a}
2. **Transitivity**:

a
≤
b

{\displaystyle a \le b}

 and

b
≤
c

⟹

a
≤
c

{\displaystyle b \leq c \implies a \leq c }
3. **Anti-symmetry**:

a
≤
b

{\displaystyle a \le b}

 and

b
≤
a

⟹

# a
b

{\displaystyle b \leq a \implies a = b }
4. **Connectedness**:

a
≤
b

{\displaystyle a \le b}

 or

b
≤
a

{\displaystyle b\le a}

Note that (1) follows from (4), but is often included in the definition for comparison to that of partial orders.
Number

A **number** is a mathematical object used to count, measure and label.

## Classification of numbers

For now, excluding less practically common categories especially those that are primarily of mathematical interest.

[NEED TO MAKE INTO A BETTER FORMAT]

* Complex numbers
	+ Real numbers
		- Irrational numbers
			* Algebraic irrational numbers
			* Transcendental numbers
		- Rational numbers
			* Fractions
				+ Finite decimal
				+ Dyadic
				+ Repeating decimal
			* Integers
				+ Negative integers
				+ Natural numbers
					- 0
					- 1
					- Prime numbers
					- Composite numbers
Stochastic process

## Overview of types of stochastic processes

The flowchart shows different types of stochastic processes. Processes lower in the chart are special cases of the processes hierarchical above them. DTMC, CTMC, and SDE stands for discrete-time Markov chain, continuous-time Markov chain, and stochastic differential equations, respectively. Example of processes that are provided in this chapter are shown in *blue color*.

1. ↑ From https://www.sciencedirect.com/topics/mathematics/time-stochastic-process
A Checklist for Interpreting Statistics

Often, in response to some headline figure, summary stat, number, figure, etc, there is some qualification, adjustment or different interpretation for the headline that substantially changes the meaning.

For example, see this piece by Matthew Yglesias that provides a different look at the same data that produces a totally different interpretation.

Is it possible to write a sort of checklist of the causes of different interpretations? E.g.

* Sample selection bias
* Compositional changes by var x, where var x is correlated to the magnitude of the conclusion.
* etc?
Reasons of 'Justified' Inequality

Not all differences in economic outcomes represent *unjustified inequality*. Differences do not necessarily imply inequality.

* Working more hours
* More unpleasant work
* Working harder; more work per hour done
* Level of responsibility
* Uncertainty of income (extensive margin)
* Fluctuations of income (intensive margin)
* Skill level (talent?)
* Amount of education/training/preparation (human capital?) required or going into one's work

1. ↑ Even if a certain skill level is not required, it might have higher productivity, and possibly should still be compensated as such. (see Book: Inequality (Atkinson) 25-28: not all differences in outcomes are inequality)
Weierstrass Extreme Value Theorem

From Efe Ok: (no proof?)

If

X

{\displaystyle X}

 is a compact metric space and

f
∈

R

X

{\displaystyle f\in \mathbb {R} ^{X}}

 is a continuous function, then there exists

x
,
y
∈
X

{\displaystyle x,y\in X}

 with

f
(
x
# )
sup
f
(
X
)

{\displaystyle f(x)=\sup f(X)}

 and

f
(
y
# )
inf
f
(
X
)

{\displaystyle f(y)=\inf f(X)}

.

From micro lecture notes: (also no proof)

Let

(
X
,

d

x

)

{\displaystyle (X,d\_{x})}

 and

(
Y
,

d

Y

)

{\displaystyle (Y,d\_{Y})}

 be two metric spaces. If

f
:
X
→
Y

{\displaystyle f:X\to Y}

 is a continuous function and

S

{\displaystyle S}

 a compact set in

(
X
,

d

X

)

{\displaystyle (X,d\_{X})}

, then

f

{\displaystyle f}

 attains a maximum and a minimum in

S

{\displaystyle S}

:

arg
⁡

max

x
∈
S

f
(
x
# )
∅

{\displaystyle \arg \max \_{x\in S}f(x)=\varnothing }

 and

arg
⁡

min

x
∈
S

f
(
x
# )
∅

{\displaystyle \arg \min \_{x\in S}f(x)=\varnothing }

.
Multicollinearity

In statistics, **multicollinearity** (also **collinearity**) is a phenomenon where one predictor variable in a multiple regression model can be linearly predicted from the others with a substantial degree of accuracy.

In statements of the assumptions underlying regression analyses, "no multicollinearity" usually refers to the absence of *perfect* multicollinearity, which is an exact (non-stochastic) linear relation among the predictors. In such a case, the design matrix

X

{\displaystyle X}

 has less than full rank, and therefore the moment matrix

X

⊤

X

{\displaystyle X^{\top }X}

 cannot be inverted. Under these circumstances, for a general linear model

# y
X
β
u

{\displaystyle y=X\beta +u}

, the ordinary least squares estimator

β
^

O
L
S

=
(

X

⊤

X

)

−
1

X

⊤

y

{\displaystyle {\hat {\beta }}\_{OLS}=(X^{\top }X)^{-1}X^{\top }y}

 does not exist.

* Multicollinearity is a characteristic of the design matrix, not the underlying statistical model.
* Multicollinearity leads to parameters not being point identified. Set identification may be possible. For example, if two elements of

x

{\displaystyle x}

, say

x

1

,

x

2

{\displaystyle x\_{1},x\_{2}}

 (with coefficients

β

1

,

β

2

{\displaystyle \beta \_{1},\beta \_{2}}

) are such that

x

2

=
α

x

1

{\displaystyle x\_{2}=\alpha x\_{1}}

 then

β

1

α

β

2

{\displaystyle \beta \_{1}+\alpha \beta \_{2}}

may be point identified but

β

1

,

β

2

{\displaystyle \beta \_{1},\beta \_{2}}

 can only be set identified with

(

β

1

,

β

2

)
∈
{

b

1

,

b

2

b

1

α

b

2

=
c
}

{\displaystyle (\beta \_{1},\beta \_{2})\in \{b\_{1},b\_{2}~|~b\_{1}+\alpha b\_{2}=c\}}

 where

# c

β

1

α

β

2

{\displaystyle c=\beta \_{1}+\alpha \beta \_{2}}

, the point identified value.
Paper: Elster (1989)

Social Norms and Economic Theory

## Intro

Contrast Smith & homo economicus to Durkheim and homo sociologicus.

Define social norms as: (1) not outcome-oriented, and (2) social (not private); they must be shared by other people and partly sustained by their approval and disapproval. In addition, there are limits to the flexibility of norms (if there wasn't, of what use would they be?).
Types of Functions

See the wikipedia page, probably want to make some types of distinctions .

* Set function: input is a set
* Real function: input is real (then, what is a function with both a real input and output called? should this not be function of a real variable?)
* Real-valued function: output is real

different:

* measurable function
* correspondence
Probability/Stats Definitions

Probability facts/definitions/theorems:

* **Law of Total Probability**:

P
[
Y
# ]
P
[
Y

# X
1
]
P
[
# X
1
]
P
[
Y

# X
0
]
P
[
# X
0
]

{\displaystyle P[Y]=P[Y|X=1]P[X=1]+P[Y|X=0]P[X=0]}

. More generally for

n

{\displaystyle n}

 possible discrete values of

X

{\displaystyle X}

:

P
[
Y
# ]

∑

n

P
[
Y

X

n

]
P
[

X

n

]

{\displaystyle P[Y]=\sum \_{n}P[Y|X\_{n}]P[X\_{n}]}

Linear Algebra facts/definitions/theorems:

* **Rank of a Matrix**: the maximum number of linearly independent columns. The dimension of the vector space generated/spanned by its columns.
* **Full rank**: a matrix

X

{\displaystyle X}

 is said to have full rank if

r
a
n
k
(
X
# )
m
i
n
{
#
r
o
w
s
,
#
c
o
l
u
m
n
s
}

{\displaystyle rank(X)=min\{\#rows,\#columns\}}
* **The column rank and row rank of a matrix are equal**: for any given matrix

X

{\displaystyle X}

,

r
a
n

k

r
o
w

(
X
# )
r
a
n

k

c
o
l
u
m
n

(
X
)

{\displaystyle rank\_{row}(X)=rank\_{column}(X)}
* A non-square matrix is never invertible.
* **Invertible/Nonsingular/Nondegenerate Matrix**: for some

n

-by-

n

{\displaystyle n{\text{-by-}}n}

 square matrix

X

{\displaystyle X}

 over a field

K

{\displaystyle K}

, these statements are equivalent:
	+ There exists a matrix

	Y

	{\displaystyle Y}

	 such that

	X
	Y
	=

	I

	n

	=
	Y
	X

	{\displaystyle XY=I\_{n}=YX}
	+ X

	{\displaystyle X}

	 has **full rank**. That is,

	n
	=

	rank

	(
	X
	)

	{\displaystyle n={\text{rank}}(X)}

	.
	+ The columns of

	X

	{\displaystyle X}

	 are linearly independent
	+ The columns of

	X

	{\displaystyle X}

	 span

	K

	n

	{\displaystyle K^{n}}
	+ The columns of

	X

	{\displaystyle X}

	 form a basis for

	K

	n

	{\displaystyle K^{n}}
	+ 0

	{\displaystyle 0}

	 is not an eigenvalue of

	X

	{\displaystyle X}
	+ The transpose

	X

	⊤

	{\displaystyle X^{\top }}

	is an invertible matrix
Integration Rules/Lists of Integrals

Exponential function:

* ∫

a

u

d
x

{\displaystyle \int a^{u}dx}

: no general rule. Maybe helpful:

∫

a

b
k
x

d
# x

1
k

a

b
k
x

l
n
(
a
)

{\displaystyle \int a^{b+kx}dx={\frac {1}{k}}{\frac {a^{b+kx}}{ln(a)}}}
* ∫

a
x

d
# x

a
x

ln
⁡
(
a
)

C

{\displaystyle \int a^x dx = \frac{a^x}{\ln(a)} + C}
* ∫

e

u

d
x

{\displaystyle \int \mathrm {e} ^{u}dx}

: no general rule. Maybe helpful:

∫

e

b
k
x

d
# x

1
k

e

b
k
x

{\displaystyle \int e^{b+kx}dx={\frac {1}{k}}e^{b+kx}}
* ∫

e

x

d
# x

e

x

C

{\displaystyle \int \mathrm {e} ^{x}dx=\mathrm {e} ^{x}+C}
Euler Equation

An **Euler Equation** is a difference or differential equation that is an intertemporal first-order condition for a dynamic choice problem. It describes the evolution of economic variables along an optimal path. It is a necessary but not sufficient condition for an optimal path, and so is useful for partially characterizing the theoretical implications of a range of models for dynamic behavior.

Whatever solution approach one employs to analyze a dynamic choice problem, part of the solution is typically an Euler equation stating that the optimal plan has the property that any marginal, temporary and feasible change in behavior has marginal benefits equal to marginal costs in the present and future.
Modes of Governance/Governing Dynamics

A mode here is a way or manner in which something occurs or is done.

Together, these modes form a system of governance, which is a set of (distinguishable, but not totally distinct) modes, combined with their interactive dynamics.

The results, outcomes, empirics, actions, behavior, etc. are the things that occur. The purpose of the system is to provide a theoretical framework through which the determining dynamics of such occurrences can be thought about and analyzed rigorously.

These modes are different *dimensions or types* of mechanisms, procedures, processes, methods, or dynamics. Their particular realization at play in the world will depend on underlying data. One can think about this as a set of equations where each represents a mode and which jointly determine an outcome variable. These equations depend on parameters/state variables that determine the exact level/grade/degree (??) of their functioning, hence they need to be 'calibrated' (?) with the data. (E.g. the specific desires, objectives, habits, beliefs, norms and morals that people have)

Here what will be a parameter or state variable depends on particular question in play. Depending on the horizons of the analysis different feedback mechanisms will be become relevant or less so. For example, in the very short run norms are roughly fixed while in the very long run they might be as flexible as things can be, determined by their purpose or role in governing behavior.

## Comparison with motives

Some modes of governance probably closely correspond to the set of motives individuals have. Yet, they are not the same.

Where there are regularizable patterns of behavior that do not result from a particular motive, we have this. E.g. think about nudges: these leverage the unconscious choice heuristics or habits that people have.

## Institutions of Governance/Governance Institutions

The ways in which choices are made in practice in society.

E.g. Democracy, Dictatorship, Markets, natural social equilibrium (freedom?).

## Good sources

* Paper: Arrow (1950), especially the first two pages. I think he mischaracterizes convention as reflecting a 'common will' not involving any conflict of individual wills, which he contrasts to methods for social choice of voting and the market which amalgamate tastes of many individuals.
Market Failure

## List of (possible) market failures

Note: this is work-in-progress. This list should not be considered exhaustive, nor do I have the conceptual clarity to make these items totally distinct. My guess is some phenomena could be captured (possibly in part) by multiple of these concept.

* Rent-seeking
* Monopoly power

### Rent-seeking

Definition of this I find difficult. (see note in Book: The Worldly Philosophers | Thought Wiki | Fandom, p. 186). My attempt there was: something that grows your wealth without creating any new wealth/surplus. At best this is purely redistributional? (e.g. hedge fund buying spectrums before the US double auction). If the extraction of rent/increased price lowers consumption of the commodity, it even results in a net decrease of total surplus? (Likely also an increase in inequality?)

If e.g. buying a piece of land in expectation of its value increasing (in theory prob need to add some form of asymmetric or costly info?), a counter argument could be made by arguing that this provides an incentive for the costly information exploration to identify high-value commodities that would otherwise not be 'in the market'? Or maybe even just with asymmetric info without costly exploration, buying a commodity not in the market, sharing your info to become public and then selling it?

Maybe oil exploration is a good example of the costly info? E.g. here Saudi-Arabia and Norway are interesting examples of non-market approaches, and the Netherlands as a mixed state-market approach?

Another interesting example is appreciation resulting from zoning changes. E.g. allowing housing to be built on agricultural land next to a city causing an increase to its value, when anticipating this a speculator can buy the land pre-change and make a profit. The increase is surplus produced by the permit change ideally falls to society? This would be a rare example of a high-revenue tax channel that is non-distortionary? Spectrum allocation is another interesting example I think: no value is added when buying/selling it, and the change in its value accrues to the state by leasing it out for periods of time.

Generally, see the part of the above book from 186-onwards about Henry George about a single massive tax that absorbs all rents.
Slack Variable

In an optimization problem, a slack variable is a variable that is added to an inequality constraint to transform it into an equality. Introducing a slack variable replaces an inequality constraint with (1) an equality constraint plus (2) a non-negativity constraint on the slack variable.

For example, by introducing the slack variable

s
≥
0

{\displaystyle s \geq 0}

, the inequality

A
x
≤
b

{\displaystyle Ax \le b }

 can be converted to the equation

A
x
# s
b

{\displaystyle Ax + s = b}

. Instead of the one inequality constraint, there is now (1) an equality constraint (

A
x
# s
b

{\displaystyle Ax + s = b}

) and (2) a non-negativity constraint (

s
≥
0

{\displaystyle s \geq 0}

).

In the Lagrange multiplier method, the slack variable is usually denoted by

λ

{\displaystyle \lambda}

.
Limit of a Sequence of Sets

In sequences of sets, it seems to me that ‘size’ is thought about as the number of elements in a set, not some idea of the smallest element in the set. Consider a sequence of sets

(

A

n

)

# n
1

∞

{\displaystyle (A\_{n})\_{n=1}^{\infty }}

 where

A

n

∈
Ω

{\displaystyle A\_{n}\in \Omega }

 for all

n

{\displaystyle n}

. Note:

* ∩

# n
k

∞

A

n

{\displaystyle \cap \_{n=k}^{\infty }A\_{n}}

 consists of elements that are in *every*

A
n

{\displaystyle A\_n}

 with

n
≥
k

{\displaystyle n\geq k}

. (note: if an element is only in finitely many

A
n

{\displaystyle A\_n}

, then it is outside

∩

# n
k

∞

A

n

{\displaystyle \cap \_{n=k}^{\infty }A\_{n}}

for all

k

{\displaystyle k}

 sufficiently large.
* ∪

# n
k

∞

A

n

{\displaystyle \cup \_{n=k}^{\infty }A\_{n}}

 consists of elements that are in *some*

A
n

{\displaystyle A\_n}

 with

n
≥
k

{\displaystyle n\geq k}

. (note: if an element is only in finitely many

A
n

{\displaystyle A\_n}

, then it is outside

∪

# n
k

∞

A

n

{\displaystyle \cup \_{n=k}^{\infty }A\_{n}}

for all

k

{\displaystyle k}

 sufficiently large.
* If

x
∈

∩

# n
k

∞

A

n

{\displaystyle x\in \cap \_{n=k}^{\infty }A\_{n}}

 then *all*

A
n

{\displaystyle A\_n}

have

x

{\displaystyle x}

 in it, not just an infinite subsequence. Hence,

∩

# n
k

∞

A

n

{\displaystyle \cap \_{n=k}^{\infty }A\_{n}}

 is a lower bound (in the sense of number of elements) for all

A
n

{\displaystyle A\_n}

 for all

n
≥
k

{\displaystyle n\geq k}

.
* If

x
∈

∪

# n
k

∞

A

n

{\displaystyle x\in \cup \_{n=k}^{\infty }A\_{n}}

 if

x

{\displaystyle x}

 is in infinitely many of the

A
n

{\displaystyle A\_n}

it is in the union

∀
k

{\displaystyle \forall k}

. Hence,

∪

# n
k

∞

A

n

{\displaystyle \cup \_{n=k}^{\infty }A\_{n}}

 is an upper bound (in the sense of number of elements) for all

A
n

{\displaystyle A\_n}

 for all

n
≥
k

{\displaystyle n\geq k}

.

There are two notions for the limit of a sequence of sets: the limit infimum, and the limit supremum.

1. **Limit Infimum**:

lim
inf

A

n

:=

⋃

# k
0

∞

(

⋂

# n
k

∞

A

n

)

{\displaystyle \lim \inf A\_{n}:=\bigcup \_{k=0}^{\infty }{\Bigl (}\bigcap \_{n=k}^{\infty }A\_{n}{\Bigr )}}

 this is the collection of

x

{\displaystyle x}

‘s that are (1) in infinitely many of the

A
n

{\displaystyle A\_n}

, and (2) in all

A
n

{\displaystyle A\_n}

 for

n

{\displaystyle n}

 sufficiently large.
2. **Limit Supremum**:

lim
sup

A

n

:=

⋂

# k
0

∞

(

⋃

# n
k

∞

A

n

)

{\displaystyle \lim \sup A\_{n}:=\bigcap \_{k=0}^{\infty }{\Bigl (}\bigcup \_{n=k}^{\infty }A\_{n}{\Bigr )}}

 this is the collection of all

x

{\displaystyle x}

‘s that are in infinitely many of the

A
n

{\displaystyle A\_n}

.

As can be seen from the definitions, the limit infimum is a subset of the limit supremum,

lim
inf

A

n

⊆
lim
sup

A

n

{\displaystyle \lim \inf A\_{n}\subseteq \lim \sup A\_{n}}

. [ADD PROOF]
Books: All Chapters

## General

This page includes all the books of the Unknown World.

## Book I

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

## Characters

Sumak

Nila

Brigitta

Katya

## Trivia

● In Chapter 1, Sumak and Nila make them debut.

● In Chapter 3, Kan, Elke, Anka, Mary, Rothio, Odalys, Damisy, Yioko, Li-Xiao and Brigitta make them debut.

● In Chapter 8, Katya appears in Brigitta's flashback.
Nila

Manga

## General

Nila is one of the main characters in the Unknown World.

## Appearance

Nila is a tall, muscular boy with black hair and hazel eyes. He wears a suit that is the same as Sumak's and is white with dark blue particulars.

## Personality

Nila is shown to have pride in himself and likes to go around looking for a fight. He is described as abnormal by Teresa because he goes with the girl with the best body, he started to like Katya knowing that she is in a relationship (not necessarily knowing that she is in a relationship with Sumak) and he wanted her because of her incomparable beauty. It has been reported by him that if Katya doesn't love him he will try to sleep with her and then rape her.
Places Studying Poverty

* UChicago Harris - Comprehensive Income Dataset
* UC Berkeley - Opportunity Lab
* Harvard - Opportunity Insights
* Notre Dame = Wilson Sheehan Lab for Economic Opportunities (LEO)

### U.S. Collaborative Of Poverty Centers (CPC)

* Institute for Research on Poverty (IRP) - UWisconsin-Madison: coordinates the network
* Center for Population, Inequality, and Policy - UC Irvine
* Center for Poverty and Inequality Research - UC Davis
* Center on Poverty & Social policy - Columbia University
* Center on Race and Wealth | Howard University COAS Centers
* Hispanic Institute - Child Trends – ChildTrends
* https://poverty.umich.edu
* Stanford Center on Poverty and Inequality
* University of Kentucky Center for Poverty Research (uky.edu)
* WCPC | West Coast Poverty Center (washington.edu)
* Wilson Sheehan Lab for Economic Opportunities | University of Notre Dame (nd.edu)

## Other

* Conference on Research in Income and Wealth | NBER
* US Partnership on Mobility from Poverty (now the Urban Institute??)

## No Longer Active

* Northwestern University/University of Chicago Joint Center for Poverty Research

1. ↑ They have a summer institute with its own website here.
2. ↑ Established in 1996. Was directed by Rebecca Blank.
Kiyosho Shirayuki

## History:

In the bustling city of Shinobi, a young warrior named Kiyosho received distressing news. The ancient scrolls of the Shirayuki clan, passed down through generations, had been stolen by the nefarious Mitsurugi Nakamura and his powerful corporation, the Nakamura Empire.

Determined to restore honor to his clan and protect the sacred knowledge within the scrolls, Kiyosho decided to enter the prestigious Chi No Ken tournament. This legendary tournament brought together the world's greatest martial artists, each vying for the chance to prove their skill and claim the grand prize. For Kiyosho, it was his only hope of retrieving the stolen scrolls.

As the tournament commenced, Kiyosho showcased his lightning-fast strikes and impeccable technique, swiftly defeating opponents with ease. The crowd marveled at his agility and precision, but unknown to Kiyosho, Mitsurugi was watching closely, pleased that his plan to eliminate the ambitious warrior was well underway.

In the early stages of the tournament, Kiyosho faced off against Hachi Hinogami, a renowned and legendary martial artist. Hachi had once been Mitsurugi's long-time rival and was filled with wisdom, strength, and an unparalleled mastery of his craft. The battle was fierce, and even though Kiyosho fought with all his might, Hachi emerged victorious, leaving the young warrior with a newfound admiration for his opponent's skill.
Shepard's Lemma

* If Hicksian demand is single-valued, it coincides with the derivative of the expenditure function.
* Thm: if the utility function

u
(
⋅
)

{\displaystyle u(\cdot)}

is continuous and Hicksian demand

h
(
p
,
u
)

{\displaystyle h(p,u)}

 single-valued (function), then the expenditure function

e
(
p
,
u
)

{\displaystyle e(p,u)}

 is differentiable in price at

(
p
,
u
)

{\displaystyle (p,u)}

 with

∂
e
(
p
,
u
)

∂

p
i

=

h
i

(
p
,
u
)

{\textstyle \frac{\partial e(p,u)}{\partial p\_i} = h\_i(p,u)}

.
* Intuition: if the price of a good increases by a very (infinitesimally) small amount (

ε

{\displaystyle \varepsilon}

, say $0.01), then the unique optimal consumption bundle now costs

ε

h
i

(
p
,
u
)

{\textstyle \varepsilon h\_i(p,u)}

more.
* This is basically the envelope theorem intuition, and the proof for Shepard's lemma accordingly uses the envelope theorem.

## Proof for the Differentiable Case

The expenditure function

e
(
p
,
u
)

{\displaystyle e(p,u)}

 is the value function of the constrained optimization problem (expenditure minimization problem) characterized by the Lagrangian:

L

(

x

,
λ
;

p

,
u
# )

p

x

λ
(
u
−
U
(

x

)
)

{\displaystyle \mathcal{L} (\mathbf{x}, \lambda; \mathbf{p}, u) = \mathbf{p} \mathbf{x} + \lambda(u - U(\mathbf{x}))}

By the envelope theorem, the derivatives of the expenditure function with respect to the price of a good are:

∂
e
(

p

,
u
)

∂

p
i

=

∂

L

∂

p
i

=

x
i
h

{\displaystyle \frac{\partial e(\mathbf{p},u)}{\partial p\_i} = \frac{\partial \mathcal{L}}{\partial p\_i} = x^h\_i
}

Where

x
i
h

{\textstyle x\_i^h
}

 is the minimizer. In other words/by definition of the Hicksian demand function:

x
i
h

=

h
i

(

p

,
u
)

{\textstyle x^h\_i = h\_i(\mathbf{p},u)
}

. Hence:

∂
e
(

p

,
u
)

∂

p
i

=

∂

L

∂

p
i

=

h
i

(

p

,
u
)

{\displaystyle \frac{\partial e(\mathbf{p},u)}{\partial p\_i} = \frac{\partial \mathcal{L}}{\partial p\_i} = h\_i(\mathbf{p},u)
}