Regression Optimization problem under constraints

The situation is given in the following picture

The black line is the true conditional mean $E(y|x)$. If we truncate the data, all observations above the truncation $Y^A$ are not observed.
For low values of $x$, we will observe (on average) lower values of $y$ than we would without the truncation. As such, when $x$ is low, the observed conditional mean function (in blue) will be lower than the true one. If you fit a linear function through the observed points (say in red), you will therefore fit a linear function approximating the blue one. This will be a function with a flatter slope (compared to the true conditional mean in black).
The following is not exact, but only gives an idea how I would approach the problem analytically.
Assume that the true data generating process be given by:
$$
y_i = \beta_0 - \beta_1 x_i + \varepsilon_i.
$$
Where, as usual, $E(\varepsilon_i) = 0$ and $E(\varepsilon|x_i) = 0$. The conditional mean function is then equal to:
$$
E(y_i|x_i) = \beta_0 - \beta_1 x_i + E(\varepsilon_i|x_i) = \beta_0 - \beta_1 x_i.
$$
Now let $z_i$ be the random variable equal to 1 iff observation $i$ is not truncated, i.e. when
$$
 \beta_0 - \beta_1 x_i + \varepsilon_i < Y^A.
$$
If $i$ is not observed, i.e. truncated, we have $z_i = 0$.
Then the observed conditional mean equals:
$$
E(y_i|x_i,z_i=1) = \beta_0 - \beta_1 x_i + E(\varepsilon_i|x_i, z_i = 1).
$$
The last term is given by:
$$
E(\varepsilon_i|x_i, z_i = 1) = E(\varepsilon_i|x_i, \varepsilon_i < Y^A - \beta_0 +  \beta_1 x_i)
$$
This last term, which is a function of $x_i$ is negative (as $\varepsilon_i$ is truncated from above and the mean of the untrunctated $\varepsilon$ is zero). Define:
$$
E(\varepsilon_i|x_i,\varepsilon_i < Y^A - \alpha_0 +  \alpha_1 x_i)  = g(x_i).
$$
we have that $g(x_i) < 0$ and in general, we would expect that $g'(x_i) > 0$, because the restriction is less binding if $x_i$ becomes bigger (This however is probably not always the case, as $g$ depends on the shape of the joint distribution of $\varepsilon_i$ and $x_i$.)
Then we have:
$$
E(y_i|x_i, z_i = 1) = \alpha_0 + \beta_1 x_i + g(x_i),
$$
This is the blue curve in the picture.
The slope of this line equals:
$$
\frac{\partial E(y_i|x_i, z_i = 1)}{\partial x_i} = -\beta_1 + g'(x_i)
$$
The right hand side of the first equation is bigger than $-\beta_1$ (so the slope is flatter).
Notice that in general, the conditional mean will alsono longer be a linear function of $x_i$, so there will also be a specification bias. This also makes it not trivial to determine what exactly the bias will be from fitting a linear function.
Lowering the value of $Y^A$ will lead to a larger bias (more negative $g(x_i)$) and probably also to a flatter slope of the estimated regression. I hope this is intuitive from the picture above.


How do we know a business is allocating resources correctly? If the answer is 'profitability,' why is monetary profitability the barometer?

A firm has a business plan. From its perspective, resources are allocated “correctly” if they achieve its business plan. However, it is entirely possible that the business plan makes no sense, e.g., it is supplying a product far in excess of what customers are willing to buy. For public companies, equity market analysts attempt to project the effectiveness of those business plans. The volatility of equity prices is an indication that this is not easy.
A single project is typically evaluated with discounted cash flow analysis. (Similar to this question answer: government analysis) This gives us a project-level profitability measure, but we really need to evaluate the global business plan.
The issue is that “profitability” is somewhat vague: on what horizon? It is entirely possible for a firm to invest in projects to build capacity but those initial projects have very low profits (or even losses) during its growth years. However, the hope is that the firm is very profitable when it has reached its target size. Unless we wait, we cannot be sure whether this strategy will generate future profits or not.


Why is imputed rental income for owner-occupied housing included in the measurement of household income?

The household owns an asset, the house, from which an income can be derived. The household could decide to let the house and with the income rent another house. Alternatively, the household could decide to live in the house that they own. In the latter case household income would be lower, unless imputed rental income is considered.
So calculating and including imputed rental income serves to control household income for households' decision to let and rent or live in their own house. If it wasn't included, household income would swing considerably based on how many households decide to rent or buy their home in any given period.


Difference between yield curve and term structure of interest rate

As far as I understand these terms are used almost interchangeably with the former term being more wide while the later applies more specifically to visualization of the term structure.
According to Mishkin and Eakins, Financial Markets and Institutions. 8ed, in chapter 5 pp137 on Term Structure of Interest Rates:

A plot of the yields on bonds with differing terms to maturity but the same risk, liquidity, and tax considerations is called a yield curve, and it describes the term structure of interest rates...

Also the whole chapter on term structure of interest rates focuses almost solely just on the yield curve.
However, as I understand it, the term structure of interest rates can be also applied also more widely, while yield curve specifically refers to the above mentioned plot. However, since the yield curve is almost always used to visualize the relationship the terms are colloquially often used as synonyms.


It can be shown that under the assumption of strict concavity of utility function the investor will choose to split her wealth equally in the two assets. This happens because the variance on returns decreases by investing in two assets (assuming that the returns on two assets are independent).
We can consider two scenarios: first when the individual invests only in $X$ and the second where she splits her investment between $X$ and $Y$. It can be shown that in second case the variance of return goes down.
Let $r \equiv E(R_x) = E(R_y)$ and $\sigma^2 = Var(R_x) = Var(R_y)$
Assume that in second case, she invests $\alpha$ fraction of wealth in $X$ and $(1-\alpha) $ fraction in asset $Y$. Clearly $E(R) = r$. However:
\begin{align}
Var(R) &= Var(\alpha R_x + (1-\alpha)R_y)\\
&= (\alpha^2+(1-\alpha^2))\sigma^2 \tag{from independence} \\
&=(2\alpha^2+1-2\alpha) \sigma^2
\end{align}
The investor will choose $\alpha$ to minimize the term in braces which happens at $\alpha=0.5$ and so $Var(R) = \sigma^2/2$
Now it is a well know result that a portfolio with mean preserving spread is less preferred by risk-averse (concave utility) individuals. So the individual will choose to split her wealth equally among the two assets.


How can I get total home values on Zillow?

To answer your question of whether you can get a number for 'total home value,' the answer is no–at least not easily.

Zillow (somewhat) recently made available a data set similar to what you are searching for. I suspect, however, that they do not want the public to have access to aggregated value because it would showcase the volatility of Zillow home value estimates, i.e. their algorithm-based estimate fluctuates much more than the estimates coming out of research institutions, etc.

Sorry for the dissatisfying answer!


What is the effect on society of me choosing to make my own shoes, food, or clothing?

The consequences "on society" of you changing any consumption/production choice you make is zero. Assuming that all the markets you interact with (shoes, clothes, etc) are competitive, your contribution to the demand and supply of a good/service is neglectible. Therefore, market equilibrium (price/quantity) will not change.

The story is of course different if everyone, or a "sizable" amount of people does change their decisions. Demand and supply of goods and services will be altered, just as the supply and demand of all the inputs used to produce such goods. Since a lot of goods and services are related in one way or another, either as complements or substitutes, the consequences could be felt accross the whole economy. How far this goes, depend on the number of people changing their actions, and the relation between goods, services and inputs. In short, your actions will have a general equilibrium effect.


Foundational equations or concepts of Finance

Considering this is an Economics Stack Exchange site, I’m going to answer in the spectre of Financial Economics. These are the most foundational equations and ideas of Financial Economics to understand more complex applied or academic research.

1. Gross yield

The gross yield is the yield on an investment before the deduction of taxes and expenses.
$$1+R_{t+1}=\frac{P_{t+1}+D_{t+1}}{P_t}$$
$R_{t+1}$: net yield
$P_{t+1}$: price of the financial asset
$D_{t+1}$: dividend

2. Present Value

Present value ($PV$) is the current worth of a future sum of money or stream of cash flows given a specified rate of return.
$$PV=\frac{FV_n}{(1+R)^n}$$
$FV$: future value $R$: interest rate

3. Gordon’s growth model

The Gordon growth model is used to determine the intrinsic value ($V_0$) of a stock based on a future series of dividends ($D_0$) that grow at a constant rate ($g$).
$$V_0=D_0 \frac{1+g}{R-g}$$
$R$: interest rate

4. Bond pricing
The price of a bond is the sum of the present values of all expected coupon payments plus the present value of the par value at maturity.

$$P_{bond}= \sum_{t=1}^T \frac{\text{Interest}}{(1+r_{it})^2}+\frac{\text{Par Value}}{(1+r_{iT})^2}$$

5. Duration

A measure of the sensitivity of the price (the value of principal) of a fixed-income investment to a change in interest rates. Given the relation between present value, future cash-flow and yield to maturity:$$PV=\frac{\sum CF_t}{(1+YTM)^t}$$
The second order Taylor polynomial is the following:
$$dPV=\underbrace{
                \left(
                      \frac{dPV}{dYTM}
                \right)dYTM}_{\textbf{Duration}}+
\underbrace{\left(
                      \frac{1}{2}
                \right)
                \left(
                      \frac{d^2 PV}{dYTM^2}
                \right)dYTM^2}_{\text{Convexity}}$$

$YTM$: yield to maturity

6. Convexity

For any given bond, a graph of the relationship between price and yield is convex. This means that the graph forms a curve rather than a straight-line (linear). The degree to which the graph is curved shows how much a bond's yield changes in response to a change in price.

$$dPV=\underbrace{
                \left(
                      \frac{dPV}{dYTM}
                \right)dYTM}_{\text{Duration}}+
\underbrace{\left(
                      \frac{1}{2}
                \right)
                \left(
                      \frac{d^2 PV}{dYTM^2}
                \right)dYTM^2}_{\textbf{Convexity}}$$

7. Liquidity premium

A premium that investors will demand when any given security cannot be easily converted into cash, and converted at the fair market value.

8. CAPM (Capital Asset Pricing Model)

A model that describes the relationship between risk and expected return and that is used in the pricing of risky securities. It can be estimated by simple OLS:
$$ER_i=R_f + \beta_i(ER_m-R_f)$$

$ER_i$: expected return of portfolio $i$, $ER_m$: expected return of the market portfolio, $R_f$: risk-free portfolio,  $\beta_i$:CAPM-beta of portfolio $i$


After searching for a while, this is the best answer I can so far come up with.

1) A formalized argument for why identification could break down under BBL is from Srisuma ('13). He gives two specific examples in the online appendix  where identification is lost because of using additive rather than multiplicative perturbations to construct off-equilibrium value functions (as was suggested in the original BBL paper). This is indicative of a broader issue with BBL that there might be off-equilibrium parameters that satisfy the BBL minimum-distance estimator.

2) The two examples given in the appendix are both quite basic and standard (single-agent and Cournot). This suggests that the phenomenon could be a problem in many/most applications.

3) Be creative with policy perturbations. Although Srisuma doesn't show the benefit of multiplicative over additive perturbations in general, the couple examples given show that multiplicative perturbations could improve the estimator. Formalizing optimal perturbations seems a good place for further research.


Areas of Europe with more women than men have higher GDP and lower unemployment. Why might this be?

I believe this is generally true in the US, and I'll offer at a few reasons that are sometimes suggested here.

Women are more likely to go to college than men, and colleges tend to be located in higher productivity areas.
Women are more likely to live and work in cities (a majority in 81 of the 100 largest cities but a majority in only 58 percent of small metropolitan areas ), and wages are higher in cities.
Depending on the sample period, for much of the time since 2007 female unemployment rates have been lower than the male rate, so you might expect places with more women to have have lower unemployment rates.


What is the logic behind that one can "grow out of debt" via increase in GDP?

If they invest in productivity-enhancing stuff, GDP could grow at a faster pace than the [increase in] debt. Then the debt to GDP would decrease.


Is profit margin the same as net profit margin?

Profit margin is profit after the cost of sales. If you review an income statement, you will see revenue, and then typically a data field called “cost of selling goods”. Revenue - Cost of sales is profit margin.

Net profit margin is Revenue - all expenses. So this includes taxes, interest expense, administrative expenses, and ALL expenses. An easy way to remember this is that profit margin is simply revenue - the cost of selling products, and net profit margin is revenue - ALL costs.


Where can I find economics theories to learn about?

You should read the fundamentals. I strongly suggest two textbooks for microeconomics:

Hal Varian - Microeconomics
Snyder, Nicholson - Microeconomic Theory

If your read any of the two and find it confusing, I suggest you go for:

Mankiw, Taylor - Economics


Can I make this assumption about the utility function?

No.

In general, based on the information you have, there are no results you can infer for the partial derivatives $U_{cl}$ / $U_{lc}$. In this context they could be anything.

As to what you are allowed to assume is a different story. You can assume anything in general, of course. Nevertheless, there is no way to check if your assumptions would be correct in this case.


What is structural estimation compared to reduced form estimation?

Structural estimation is a term coined by the Cowles commission which at the time seems to have been dominated by Haavelmo, Koopmans and a few others. The motto of the Cowles commission (after 1965) was: "Theory and Measurement". The phrase represents the underlying rationale of structural modelling, that measurement cannot be done without some kind of theory. To my knowledge, the phrase was first used by Koopmans in "Identification Problems in Economic Model Construction":

  Systems of structural equations may be composed entirely on the basis
  of economic "theory." By this term we shall understand the combination of (a) principles of economic of behavior derived from general observation--partly introspective, partly through interview or experience--of the motives of economic decisions,(b) knowledge of legal and institutional rules restricting individual behavior (tax schedules, price controls, reserve requirements, etc.), (c) technological knowledge, and (d) carefully constructed definitions of variables.

Structural equations are then equations that come from an underlying economic (or physical, or legal) model. Structural estimation is precisely estimation which uses these equations to identify parameters of interest, and inform counter-factuals. Importantly, these parameters are usually taken to be invariant, and therefore counter-factuals taken from their estimates will be completely "correct". Counter-factuals were the main unit of interest to the Cowles commission.

Koopmans also discusses reduced form estimation:

  By the reduced form of a complete set of linear structural equations... we mean the form obtained by solving for each of the dependent (i.e., nonlagged endogenous) variables, and in terms of transformed disturbances (which are linear functions of the disturbances in the original structural equations).

The linearity is an artifact of the times (this was published in 1949!) but the point is that reduced-form equations are equations written in terms of economic variables which do not have a structural interpretation as defined above. So, a linear regression will be a reduced-form of some true structural model, because linear regression usually does not have a true economic interpretation. This does not mean that reduced form equations cannot be used to identify parameters in structural equations - in fact this is precisely how indirect inference works - just that they do not represent a deeper model of the data generating process. Reduced forms can (in principle) be used to identify structural parameters, in which cased you are still performing structural estimation, just through using the reduced form.

Another way to look at this is that structural models are generally deductive, whereas reduced forms tend to be used as part of some greater inductive reasoning.

For a comparison of this kind of Cowles commission structural modelling with Rubin causal modelling, check out this awesome set of slides by Heckman.

For other resources I'd check out more of what Koopmans wrote, the book Structural Macroeconomics by DeJong and Dave, these lecture notes by Whited, this paper by Wolpin (written for the Cowles Foundation, in honour of Koopmans) and a response by Rust.

Addendum: A simple example of reduced form and structural models.

Suppose we were looking at data on the prices, $p_t$ and quantities, $q_t$ produced by a monopolist. The monopolist faces a series of unknown costs in the future, and a linear demand curve (this would really have to be justified). Let's say the $\hat q_t$ and $\hat p_t$ we observe are measured with some kinds of mean-zero error, $e_t$, and $v_t$

Noting that both price and quantity seem to be associated with changes in cost, a reduced form equation for this model might be:
\begin{align}
\hat q_t &= \gamma - \lambda c_t + \epsilon_t\\
\hat p_t &= \alpha + \beta c_t + \nu_t
\end{align}
Because this is a reduced form model, it needs no justification other than that it might work empirically.

On the other hand, a structural model would start by specifying the demand curve (again to be strict this should start at the level of individual utility), and the monopolist's problem:

\begin{align}
\text{Demand curve: }&p_t=a-bq_t\\
\text{Producer's problem: }&\max E\left[\sum_{t=0}^\infty\delta^t (p_t-c_t)q_t(p_t)\right]\\
\text{Measurement equations: }&\hat q_t = q_t + e_t\\
&\hat p_t = p_t + v_t
\end{align}

From this further structural equations could be derived (structural because they are still representative of principles of economic behavior):

\begin{align}
\hat q_t&=\frac{a-c_t}{2b} +e_t\\
\hat p_t&=\frac{a+c_t}{2} + v_t\\
\end{align}

This is a case where a reduced form equation will have a meaningful structural interpretation, as consistent estimates $\hat a$ and $\hat b$ can be formed:

\begin{align}
\hat a&= 2\hat\alpha \\
\hat b&= \frac{1}{2\hat \lambda}
\end{align}

Another case of identification of structural parameters from reduced forms is the logit model in the case of valuations with extreme value errors (see McFadden (1974)). In general it is unlikely a given reduced form model will have a structural interpretation.


Convex rationalization when the budget sets are segments?

Take a dataset $D = (B^t, x^t)_{t \in T}$ such that for all $t$, $x^t \in B^t$. I'll say that $D$ is rationalisable by the utility function $u$ if for all $t$ and all $x \in B$: $u(x^t) \ge u(x)$.
If you only impose convexity on $u$, then any dataset $D$ is rationalisable by the constant (convex) utility function $u(x) = k$. So this has not testable implications.
Assume now that you impose strict monotonicity on $u$ in the sense that $x > y$ implies $u(x) > u(y)$.
Consider a budget set $\overline{y^t z^t}$ and take the comprehensive hull:
$$
CH(\overline{y^t z^t}) = \left\{x \in \mathbb{R}^n: x \le w \text{ for some } w \in \overline{y^t z^t}\right\}.
$$
Then we can show that the dataset $D = (\overline{y^t z^t}, x^t)_{t \in T}$ is rationalisable by a montone utility function $u$ if and only if the dataset $D' = (CH(\overline{y^t z^t}), x^t)_{t \in T}$ is rationalisable by the same monontone utility function $u$.

For the proof: $(\to)$ Let $D$ be rationalisable by the montone function $u$. We want to show that for all $t$ and all $x \in CH(\overline{y^t z^t})$ we have $u(x^t) \ge u(x)$. Indeed, if $x \in CH(\overline{y^t z^t})$, we know that $x \le w$ for some $w \in \overline{y^t z^t}$. As such:
$$
u(x) \le u(w) \le u(x^t),
$$
where the first inequality comes from the monotonicity conditionon $u$ and the last one from rationalisability of D.

$(\leftarrow)$ Let $D'$ be rationalisable by $u$. Let us first show that for all $t$, $x^t \in \overline{y^t z^t}$. If not, then as $x^t \in CH(\overline{y^t z^t})$ there must exist a $w \in \overline{y^t z^t}$ such that $x^t < w$; Then, by monotonicity of $u$ and rationalisability of $D'$:
$$
u(x^t) < u(w) \le u(x^t),
$$
a contradiction. Now for any $z \in \overline{y^t z^t}$ it then follows from rationalisability of $D'$ that $u(z) \le u(x^t)$, so $D$ is also rationalisable by $u$.

This shows the following equivalence:

The dataset $D = (\overline{y^t, z^t}, x^t)_{t \in T}$ is rationalisable by a convex and montone utility function $u$.
The dataset $D = (CH(\overline{y^t, z^t}), x^t)_{t \in T}$ is rationalisable by a convex and monotone utility function $u$.

As the sets $CH(\overline{y^t z^t})$ are comprehensive, the revealed preference conditions for answering the condition 2) can be found in the JET paper that you refer to (Revealed preference analysis for convex rationalizations on nonlinear budget sets)


Economic Order Quantity w/ no set up costs

The total ordering cost are

$TC(Q)=P\cdot a+\frac{a\cdot K}{Q}+\frac{h\cdot Q}{2}$

If K=0 TC becomes

$TC(Q)=P\cdot a+\frac{h\cdot Q}{2}$

P\cdot a is a constant. Let denote it as c.

$TC(Q)=c+\frac{h}{2}Q$

You can see that TC is a straight line with a slope of $\frac{h}{2}$ and a domain of $Q \in \mathbb R^+$. The marginal cost is

$\frac{dTC}{dQ}=\frac{h}{2}$

It is a constant. Therefore there is no single optimum order quantity.

The store has a capacity. This is the upper limit of the order. If you have still some goods in the store the maximum order is the capacity (c) minus the goods in the store (x).

Therfore you can order bewtween $0$ (exclusive) and $c-x$ (inclusive) goods. The total cost of inventory will be always the same for a given demand in a period.


Is conditional mean independence, $E(e_i | x_i)=0$, a different assumption from $E(x e) = 0$? (from Hansen)

$E(\mathbf{x}e)=0$ does not imply $E(e|\mathbf{x})=0$, to see that, suppose that x is a variable that always takes the value of zero, but $e$ is independent of $\mathbf x$ and has a mean of, say, $3$. Then the expected value of the product is necessarily $0$, but the conditional expectation will be equal to 3 (given independence, the conditional mean is equal to the unconditional mean).


Regression approximation for the rate of change in occupancy rate of residential market with respect to price

If the goal of the project is to model housing demand, I would suggest using the actual number of occupied housing units. Census data may calculate occupancy as an algebraic function of supply, that is, occupancy is derived using the supply variable. This may show up as a strong correlation in the model, but it doesn't capture the stochastic properties of housing demand.

If you are interested in the price elasticity of housing, just use a log-log regression. Here is a good post about equilibrium price.

What is the level of analysis? If you have panel data (historical data for $n$ neighborhoods), I would suggest looking into spatial regression.

For a given neighborhood $x$, the occupancy rate of the surrounding neighborhoods may affect demand for housing (the negative affects of vacancy "spill over" into $x$). Another example, if rent is high in the neighborhoods surrounding $x$,a shift in the demand for housing will occur, because people are making a spatial decision regarding where they will live and take into account the housing characteristics of the neighborhoods of a given area.  Therefore, to model housing demand, the occupancy rate and independent variables of surrounding neighborhoods may capture the process of housing demand in $x$.
This is refereed to as spatial dependence and it can affect the model a lot. I would recommend the software Geoda and GeodaSpace to analyze and run spatial regression models.


What does Marx mean by "the division between capital and labor"?

Two main changes happened when capitalism came into existence these being the ability to buy and sell property among the peasantry.

Under the feudal system each household was endowed land which was used to produce food/ agricultural products for the household and the lord which provided the land.

When feudalism was abolished the possibilities of production outside the land allotted to the former peasantry was an option. However the nature of the work changed in the sense that workers were still productive but did not own their means of production in the same way that they did under the feudal system. they worked with the machines and materials provided by their employer but did not take the product back, but rather settled for a working wage which is systematically less than his own work. This is exactly the split between land and labor Marx is talking about- the inability of the worker to have access to the true product of labor (or value of it).

The land provided by the lord was a place where the peasants lived so they were always one with their means of production and benefited from it. In the new capitalist system this was no longer possible.


After you get 265 you should take out -10 from Net Foreign Factor Income. This is in there implicity through disposable income etc, but was not produced in the country. Since you want domestic production you must take that out.


Proximity-concentration hypothesis: homogeneous firms and symmetric countries


The 40-week comes started with the movement of the 8h working day. The wikipedia article makes a quick summary of the historical progress of thinking from 12h work days in Britain during the industrial revolution to the nowadays 8h/day.
The quick summary is that after the first world war, the International Labour Organization (ILO) was formed and its first convention was defining limits for the work day:

Having decided (...) the "application of the principle of the 8-hours day or of the 48-hours week" which is the first item in the agenda

From Convention Limiting the Hours of Work in Industrial Undertakings to Eight in the Day and Forty-eight in the Week (ILO, 1919)
This meant that generally workers would work 8h/day as a maximum, with 1 day of rest per week. 52 countries have ratified this convention, although not immediately.
The change for the 40h week came in 1935, with another ILO convention, which explains the motivations in its preamble:

Considering that unemployment has become so widespread and long continued that there are at the present time many millions of workers throughout the world suffering hardship and privation for which they are not themselves responsible and from which they are justly entitled to be relieved;
Considering that it is desirable that workers should as far as practicable be enabled to share in the benefits of the rapid technical progress which is a characteristic of modern industry; and
Considering that in pursuance of the Resolutions adopted by the Eighteenth and Nineteenth Sessions of the International Labour Conference it is necessary that a continuous effort should be made to reduce hours of work in all forms of employment to such extent as is possible;
adopts (...) the following Convention, which may be cited as the Forty-Hour Week Convention, 1935:

From: ILO Forty-Hour Week Convention, 1935 (No. 47)
This later convention has been ratified by 15 countries only.
Further recommendations on the reduction of hours were made in 1962 (source ILO). The ILO also has a page summarizing the work done in the last 100 years on working time, which is a great resource to learn more.


Are convex preferences needed for the first welfare theorem?

No, convexity of preferences is imposed for other reasons. A general sufficient condition is local non-satiation, which says the agent can be made better off by an arbitrary small perturbation of his consumption bundle.
This can hold without the preference being convex.

  It would see so. For example...

As already pointed out by @Shomak, the example of an allocation you suggest is not an equilibrium. It also has nothing to do with convexity. Crossing of indifference curves at an allocation can occur for convex or non-convex preferences. Therefore it does not speak to the role of convexity in the theorem one way or another.

Assuming preferences are represented by differentiable utility functions (as per usual), standard marginalist reasoning tells you the allocation you describe is not an equilibrium.

Regardless of whether utility function is quasi-concave (i.e. whether the underlying preference is convex), the first order condition is always a necessary condition for optimality. Therefore agents must have the same marginal rate of substitution between commodities in equilibrium. When indifference curves cross, they don't have the same MRS.


Why does the inverted yield curve occur with US Treasury securities?

In general a longer maturity rate is practically speaking as equally profitable as a series of short maturity rates compounded. So for example, investing for 1 year thrice is as equally profitable as investing for 3 years. Investing for 6 months 16 times is as equally profitable as investing for 8 years. There is no certainty of equality and you can make money if you guess correctly. If there are many guessers then near equality will be realized.

The U.S. Treasury short maturity yield is higher than some of the long maturity yields (today), not because the short maturity investments are riskier. It is because the very short term rate (overnight) set by the Federal Reserve is high enough that the 3 month rate is higher than the 10 year rate. In other words, the expected mean rate for the next 3 months is higher than the expected mean for the next 10 years. Sometime in the future there might be a recession and the Federal Reserve might want to lower the overnight rate to help the economy. Sometime in the future the Federal Reserve might increase the overnight rate to reduce inflation. Both these considerations form expectations.

In addition to overnight rate setting, the Federal Reserve has engaged in other activity that influences Treasury rates. The Federal Reserve increased its holdings of Treasury bonds in the period from 2008 to 2014. These economists (Gagnon and others) in a paper titled "The Financial Market Effects of the Federal Reserve’s Large-Scale Asset Purchases" estimated that these purchases have been a downward force on long term rates.

The graph shows that those holdings have been decreasing since early 2018 so there might be reason to think that any downward force on long term rates is now not what it used to be so if we see a rise in those rates there is reason to think of causes that are apart from a change in risk perceptions.

Addendum: @Michael rightly points out that market participants bidding up long term bond prices is another force that lowers long term bond yields.


If the economy is not always on the Price setting curve, can the Central Bank target unemployment?

The role of the central bank is solely to enact monetary policy, not government (fiscal) policy. Monetary policy of the central bank is independent (in the legal and political sense) from government.


Macroeconometrics:How to measure capital depreciation?

I tried to empirically estimate the rate of depreciation a couple of years ago. I gave up! Following a large literature on growth, I assumed a constant rate of depreciation eventually. I did not have data for all the variables needed to estimate the model.

Take a look at Nadir and Prucha's modelling approach.
In chapter 7 of this book, the authors develop an accelerator model of investment. You may then calculate the depreciation rate using the parameter estimates of the model.


Relationship between elasticity and interpretation of approximate percentage change

The derivative of $log(x)$ is $1/x$, which implies that $dlog(x)=(dx)/x$. Therefore, $e=dlog(y)/dlog(x)$ is the same as $e=(dy)/y/(dx)/x$. This ratio $e$ is known as the elasticity of $y$ with respect to $x$. We usually express derivatives $f′(x)$ as the change in $f$ when $x$ moves one unit, but $f(x)+1∗f′(x)$ is just an approximation (a Taylor approximation) of $f(x+1)$. Therefore, saying that a 1% change in $x$ will lead to an $e$ change in $y$ is just an approximation. It's a rate of change that is exact only at the $x$ point, but not exact as you move away from it to $x+0.01∗x$.


I am asking whether or not it fully expounds some first cause of human
  action from which we can deduce answers to other questions about human
  economic behavior?

What it is getting at:

Praxeology [the science of human action] says that all economic propositions which claim to be true must be shown to be deducible by means of formal logic from the incontestably true material knowledge regarding the meaning of action. Specifically, all economic reasoning consists of the following:

an understanding of the categories of action and the meaning of a change occurring in such things as values, preferences, knowledge, means, costs, etc;
a description of a world in which the categories of action assume concrete meaning, where definite people are identified as actors with definite objects specified as their means of action, with some definite goals identified as values and definite things specified as costs. Such description could be one of a Robinson Crusoe world, or a world with more than one actor in which interpersonal relationships are possible; of a world of barter exchange or of money and exchanges that make use of money as a common medium of exchange; of a world of only land, labor, and time as factors of production, or a world with capital products; of a world with perfectly divisible or indivisible, specific or unspecific factors of production; or of a world with diverse social institutions, treating diverse actions as aggression and threatening them with physical punishment, etc; and
a logical deduction of the consequences which result from the performance of some specified action within this world, or of the consequences which result for a specific actor if this situation is changed in a specified way.

Provided there is no flaw in the process of deduction, the conclusions that such reasoning yield must be valid a priori because their validity would ultimately go back to nothing but the indisputable axiom of action. If the situation and the changes introduced into it are fictional or assumptional (a Robinson Crusoe world, or a world with only indivisible or only completely specific factors of production), then the conclusions are, of course, a priori true only of such a “possible world.” If, on the other hand, the situation and changes can be identified as real, perceived and conceptualized as such by real actors, then the conclusions are a priori true propositions about the world as it really is. [19]

Such is the idea of economics as praxeology. And such then is the ultimate disagreement that Austrians have with their colleagues: Their pronouncements cannot be deduced from the axiom of action or even stand in clear-cut contradiction to propositions that can be deduced from the axiom of action.
— Hans-Hermann Hoppe, Economic Science and the Austrian Method, pg 25.

  What are the known refutations of Mises' Axiom of Human Action?

There aren't any that have been successful...

"All of these categories which we know to be the very heart of economics—values, ends, means, choice, preference, cost, profit and loss—are implied in the axiom of action. Like the axiom itself, they are not derived from observation. Rather, that one is able to interpret observations in terms of such categories requires that one already knows what it means to act.

No one who is not an actor could ever understand them, as they are not “given,” ready to be observed, but observational experience is cast in these terms as it is construed by an actor. And while they and their interrelations were not obviously implied in the action axiom, once it has been made explicit that they are implied, and how, one no longer has any difficulty recognizing them as being a priori true in the same sense as the axiom itself is.

For any attempt to disprove the validity of what Mises has reconstructed as implied in the very concept of action would have to be aimed at a _goal_, requiring _means_, excluding other courses of action, incurring _costs_, subjecting the actor to the possibility of achieving or not achieving the desired goal and so leading to a _profit_ or a _loss_.

Thus, it is manifestly impossible to ever dispute or falsify the validity of Mises’s insights. In fact, a situation in which the categories of action would cease to have a real existence could itself never be observed or spoken of, since to make an observation and to speak are themselves actions.
— Hans-Hermann Hoppe, ESAM


Which of the following explains that the price increase of oil cause the price of natural gas to increase?

I think your answer is OK.

A doesn't connect prices in any way. In a two-goods economy and given that wealth doesn't change... you may give it some more thought but you need to add extra assumptions.
B tells us nothing on why such change would take effect. It may just as well be that oil is more valued.
C is the right answer
You need more assumptions for D, e.g. why the oil price increased. At least in the really short term, though, supply isn't a factor.


If $X$ and $Y$ are complements, an increase in the quantity demanded of $X$ also implies an increase in the demand for $Y.$ This is why both the price and the quantity of $Y$ will also have to increase.


Suppose I have a preference satisfy all basic assumptions, how to show consumer problem solution is unique

In that case you would still have a problem showing the existence of a solution. Without existence, it makes little sense to think about uniqueness. The reason a continuous utility function (implied by some of those preference assumptions you gave) has a maximum at all is because a continuous function achieves a maximum somewhere on a compact set, which in this case is B.

Luckily, we don't really have to imagine scenarios in which we have no information about the budget set. This is not some abstract set, but rather something very concretely defined as:

$B(p,w)=\{x \in \Bbb R_{+}^{n}: px \leq w \}$,

where $p$ is the price vector, $x$ is the demand, $w$ is the budget and $n$ is the dimension of the vector.

Luckily, this is always compact.

So, no, you cannot show there is a unique solution, because in that case you can't show there is a solution in the first place (using standard tools).

However,  you are correct in thinking that uniqueness itself does not hinge upon the compactness of the budget set.


Why is everyone suggested to specialize their education?

Note: I did not vote down on this question, and it is not clear why anyone would do so.

  Why is so common to suggest university students to specialize in order
  to get a better paid job?

Because the industry evolves and encounters increasingly harder, diverse challenges. Specialized knowledge is essential for coping with those challenges. At the same time, it is impossible for everybody to achieve expertise in all the diverse disciplines or branches of knowledge. In the labor market, that implies that the supply of workforce is more constrained as we move along the "specialization axis".

It is noteworthy, though, that specializing in order to get a higher compensation is not always profitable. For specialization to be profitable, there also needs to exist enough demand for a skills set. A specialist in semantics of Etruscan dialects spoken in the Po Valley will have a much harder time to find a job in his field (let alone to earn a satisfactory salary) than a specialist in 4th generation nuclear reactors.

  students are driven towards specialization, investing their most
  valuable asset, time, in a single high risk endeavor.

Not really. A certain extent of diversification is inherent to the individual's university background. In the case of the nuclear reactor specialist, his background includes calculus, probability, numerical analysis, thermodynamics, heat transfer, electricity, etc. That background makes it easier for him to adapt to drastic changes such as switching careers.

By contrast, a company is much more static and, thus, much unlikelier to cope with drastic changes successfully. Here are some factors that impair a company's adaptability:

Managers, first and foremost, aim at retaining their power;
it is harder for various stakeholders to agree on what the new
direction needs to be;
synchronizing an update of skills set of all its members is expensive
and possibly out of reach;
some members of the organization might leave during the transition;
new processes might entail new redundancies, frictions, and
additional overhead.

A shareholder has little-to-no control on how the company will fare, or whether it will adopt a unified, effective approach at all. Things are different with the individual, provided that he is determined, not erratic, and has a rational/realistic vision of what he wants.


Aggregation theorems hold if endowments are endogenous?

The answer is it depends on more assumptions on preferences. The short answer is no. To provide as much intuition we will look at the optimal conditions in each of these problems.

First, for the descentralized problem you have that individuals choose $(c_{0i}, c_{1i}, \eta_i)$ to maximize:

$\max \qquad \sum_t \beta^t u(c_{ti}) $

$s.t \qquad c_{0i} = e_{0i}(\eta_i) - \eta_i p$

$ \quad \qquad c_{1i} = e_{1i}(\eta_i) + \eta_i X$

The solution, for each individual $i$ will be optimal if they follow the Euler equation derived from the first order conditions and that markets clear:

$\frac{1}{\beta} \frac{u'(c_{0i}^*)}{u'({c_{i}^*)}} = \frac{p - e_{i}'(\eta_i^*)}{e_{1i}'(\eta_i^*) + X}$

Now look at the representative agent problem for whom you say the endowment is given by $\bar e(\eta) = \frac{1}{N}\sum_{i} e^i(\eta)$. I will drop the $N$ because scaling does not matter (you are assuming the mean endowment, I am using the total endowment so all of my solutions are $N$ times what you want). This agent's problem is to choose $(c_0, c_1, \eta)$ to maximize:

$\max \qquad \sum_t \beta^t u(c_{ti}) $

$s.t \qquad c_{0} = \sum_{i} e_{i}(\eta) - \eta p$

$ \quad \qquad c_{1} = \sum_{i} e_{i}(\eta) + \eta X$

Now the first order condition says:

$\frac{1}{\beta} \frac{u'(c_{0}^*)}{u'({c_{1}^*)}} = \frac{p - \sum_{i}e_{i}'(\eta^*)}{\sum_i e_{i}'(\eta^*) + X}$

These two need not be equal. Do you have more assumption on the preferences/endowment functions?


How to analyze small human activities from the economical standpoint?

A useful framework for analysing personal activities involving both monetary and time costs is a household production model. A simple example of such a model (adapted from Chiappori & Lewbel (2015) pp 411-2) is below. Note that a "household" could be one person (the assumption that members of a multi-person household cooperate to maximise their joint utility can be regarded as an optional extra of the model).

The model distinguishes between commodities, which directly affect utility, and goods, which affect utility only indirectly.  The taxi ride, for example, would be a good, not a commodity, because it is just a means of getting from A to B. A household is assumed to maximise utility which is a function of commodities:

$$max\quad U(Z_1,\dots,Z_m)$$

For each commodity there is a production function:

$$Z_i=f_i(\mathbf{x}_i,T_i)$$

Here $\mathbf{x}$ is a vector of goods and $T$ is time taken by a household member in producing the commodity using the goods. Production here should be understood broadly: it would include, for example, using a combination of public transport and time to "produce" a journey from A to B. Leisure activities such as visiting an entertainment can also be produced in this sense.

The goods used in producing $Z_i$ can be bought at prices $\mathbf{p}_i$, and income can be earned, over any chosen time period, at a wage $w$ (this is a simplification, eg because working hours are often fixed by an employer). Considering a period of, say, a week (and simplifying again by ignoring saving and storage of goods between weeks), the monetary constraint is ($T_w$ is time worked):

$$\Sigma_{i=1}^m \mathbf{p}_i\mathbf{x}_i\leq wT_w$$

The time constraint is ($T_{tot}$ is total time available):

$$T_w + \Sigma_{i=1}^m T_i \leq T_{tot}$$

The two constraints can be combined (by substituting for $T_w$) as:

$$\Sigma_{i=1}^m \mathbf{p}_i\mathbf{x}_i \leq w[T_{tot} - \Sigma_{i=1}^m T_i]$$

This formalises the trade-off between money and time: more time used as an input to commodities means less time available for wage-earning work and so less income with which to purchase goods inputs to commodities.

One way to bring personal energy and motivation within the model would be via the value assigned to $T_{tot}$.  Starting from 168 hours in a week, deductions might be made for sleeping time, quiet relaxation, etc to arrive at what might be termed effective time available for activities, with smaller deductions for a more energetic and motivated person.


Thanks to denesp, I realised that I was only considering outcomes resulting from the following subgame:

If you consider all outcomes within the extensive form, then it is clear that $(2,6)$ pareto dominates $(0,5)$.


How exactly does the black market affect a country's economy?

I think the question is extremely broad. The effects of drug dealing on the whole economy depend on many things, for example, on how strict is the country at enforcing the ban, how violent the drug dealers are, and what is done with the profits. The experiences in Mexico and Colombia, for example, are substantially different. While many drug lords have developed complete cities from the gains from trafficking, others have simply wasted it all. In most cases, the government is involved in the trafficking (via corruption) so it can have some positive effects in terms of govt budget, but if it becomes violent, the costs are clear.

If there is any generality I would say that the combo of low cost and huge demand makes it really hard to prohibit. There are a few studies trying to measure the economic effect of the violence of drug dealing, but any empirical work is limited due to the lack of reliable data on basically every aspect of illegal markets.


Doesn't the concept of marginal utility speak to a cardinal utility function?

To the pure ordinalist who believes that preferences are purely ordinal, the concept of marginal utility (MU) has no meaning. (And a fortiori, the concept of diminishing MU also has no meaning.)
However, the concept of the marginal rate of substitution (MRS) does have meaning.
In the course of our work, we may compute something that we call MU. But to the pure ordinalist, any such number found has no meaning in and of itself.

Example. Say an individual's preferences $\succsim$ over two goods $A$ and $B$ can be represented by the utility function $U:(\mathbb{R}^+_0)^2\rightarrow\mathbb{R}$ defined by $$U(A,B)=AB.$$
The intermediate microeconomics student may then carry out these computations:
$$MU_A=\frac{\partial U}{\partial A} = B.$$
$$MU_B=\frac{\partial U}{\partial B} = A.$$
$$MRS = \frac{MU_A}{MU_B}=\frac{B}{A}.$$
The above says that if, for example, my current bundle is $(A,B)=(200,1000)$, then  $$MU_A=B=1000\text{ and }MU_B=A=200.$$ However, these two numbers have no meaning whatsoever.
The only number that has meaning is the MRS: To get another unit of $A$, I'm willing to give up (approximately) $$MRS=\frac{B}{A}=\frac{1000}{200}=5\text{ units of }B.$$
To the pure ordinalist, the above reasoning is completely legitimate, so long as one assigns meaning only to MRS. What's illegitimate is to assign any meaning to $MU_A=B$ or $MU_B=A$.
The pure ordinalist knows that if $\hat U$ is a strictly increasing transformation of $U$, then $\hat U$ is also a utility representation of $\succsim$. So, for example, if $\hat U:(\mathbb{R}^+_0)^2\rightarrow\mathbb{R}$ is defined by $$\hat U(A,B)=2AB,$$ then $\hat U$ also represents $\succsim$.
However, with $\hat U$, our computations seem to differ slightly from before:
$$M\hat U_A=\frac{\partial \hat U}{\partial A} = 2B.$$
$$M\hat U_B=\frac{\partial \hat U}{\partial B} = 2A.$$
$$\hat{MRS} = \frac{M\hat U_A}{M\hat U_B}=\frac{2B}{2A}=\frac{B}{A}.$$
Any conclusions we arrive at with the new utility representation $\hat U$ are the same as before.
If again my current bundle is $(A,B)=(200,1000)$, then  $$M\hat U_A=2B=2000\text{ and }M\hat U_B=2A=400.$$ However and again, these two numbers have no meaning whatsoever.
The only number that has meaning is the MRS: To get another unit of $A$, I'm willing to give up (approximately) $$\hat{MRS}=\frac{2B}{2A}=\frac{2000}{400}=5\text{ units of }B.$$
The quantity MU by itself has no meaning. Confusion arises only when one attaches meaning to MU and wonders how for example it is that $$M\hat U_A = 2MU_A,$$ and what the above equation means. (Answer: It means nothing.)
Out of convenience, the intermediate microeconomics student will often compute something called $MU_A$ and $MU_B$ and these can often be evaluated as actual numbers. But on their own, these numbers have no meaning (to the pure ordinalist). Only the ratio of the two quantities has any meaning: $$MRS=\frac{MU_A}{MU_B}.$$

Some quotes. Hicks (1939):

We have now to undertake a purge, rejecting all concepts which are tainted by quantitative utility, and replacing them, so far as they need to be replaced, by concepts which have no such implication.
The first victim must evidently be marginal utility itself. If total utility is arbitrary, so is marginal utility. ...
The second victim (a more serious one this time) must be the principle of Diminishing Marginal Utility. If marginal utility has no exact sense, diminishing marginal utility can have no exact sense either.

Dittmer (2005, emphasis added):

Many introductory microeconomics textbook authors derive the law of  demand from the assumption of diminishing marginal utility. Authors of intermediate and graduate textbooks derive demand from diminishing marginal rate of  substitution and ordinal preferences. These approaches are not interchangeable;  diminishing marginal utility for all goods is neither a necessary nor sufficient condition for diminishing marginal rate of substitution, and the assumption of diminishing marginal utility is inconsistent with the assumption of ordinal preferences.


Does the break even analysis contradict the law of supply?

The short answer is no.

The law of supply only states that increases in price result in higher quantities supplied, ceteris paribus. Break even is the point where Revenue = Costs.

Your confusion might come from this sentence:

  a firm producing at break even level of output would have to increase the quantity supplied if the price of the commodity goes down in order to break even.

If the price lowers, the firm will reduce its quantity supplied. That's it. The specific quantity depends on the variable and fixed costs.
It's not mentioned that the company has to break even - think of a monopolist for example.


The easiest way to model short-term but risky debt in continuous time is to have your $\psi$ be the increment of a compound Poisson process.

Jumps in this process correspond to events that might or might not cause default; the size of the jump can enter together with the state $X(t)$ into a function like your $g(X(t),\psi)$ to determine whether or not default actually occurs. Using this formulation, one can circumvent your concern in the comment that "debt is risk-less, if it is almost immediately paid back".

If, for instance (in a simple case), default occurs whenever there is a nonzero increment in the compound Poisson, regardless of the size of this increment, then if the arrival rate of these jumps is $\mu$, the interest rate will be $r=r^*+\mu$, the risk-free rate plus the flow default rate. If (in a more complicated case) default only occurs for some increment sizes, according to a threshold that depends on $X(t)$, then we can get $r$ varying with $X$.

In this environment, your observation that $f(t)$ does not affect $r(t)$ is completely correct. But this is because $f(t)$ does not contemporaneously affect $X(t)$ and hence does not affect whether or not there will be a default if the Poisson shock occurs at time $t$. $f$ does affect future $X$, and thus will potentially affect future default probabilities and interest rates.

The property we see here - where your policy today doesn't affect your interest cost today, because short-term creditors only care about the current probability of default and your policy can't affect that right away - is generically a strange feature of short-term debt models, and is brought into starkest relief in the continuous-time environment. This is why models of long-term debt are often more realistic, because the price of long-term debt incorporates future default decisions and therefore the effects of policy on future values of the state variable.


First, note that GDP is not the quantities of all goods and services produced in that year multiplied by their prices in some base year, but the quantities of all final goods and services produced in that year multiplied by their prices. Intermediate goods and services are not included.

1) GDP is about value created in the economy. So, if there are new services or products, we should simply include them. For the price, we take the price in that year and deflate it using a factor that accounts for inflation. Note also that new services and products typically come at the expense of other products produced. People that would otherwise be, say, working at a factory, are now programming apps.

2) As you would with oranges and apples. Lawyers typically charge a price per hour. So, it would be number of hours times the price. For more general services, like, say, the police force, you would just calculate the costs of the police force for a given year.


Is the Trans Pacific Partnership good or bad for the U.S?

These estimates are almost certainly bs.
The TPP would benefit consumers because easier imports may make consumer goods cheaper directly and through more competition on product markets.
The TPP would hurt some workers, because increased import
competition would force some American companies out of business.
The TPP would benefit other workers, because their companies would
export to Asia and grow.

I think economists consensus is that on average and in the medium run, free trade will benefit people in both countries. However, it has also become pretty clear that free trade hurts some people quite badly in the short run because they'll lose their jobs and possibly won't find new ones. It has also become pretty clear, that even though in theory you could put policies in place that help people and places that are hurt by trade at the expense of those who benefit from it, this is almost never done in practice. If you are for more or less free trade depends largely on what weights you apply to this trade-off.


Has the "dismal theorem" practical implications?

My conclusion based on reading his paper is that the utility function of an individual or society can't be of the CRRA form presented in the paper. That would indeed lead to scenarios where you could not get out of the bed in the morning, as minimizing the tiniest probability of an enormous risk would warrant infinite sum of money.

I will attempt to explain the mathematics of the paper. First the utility as a function of consumption is of the form:

$$ U(c) = -c^{1-a}, \space a>1$$

This is also called a constant relative risk aversion utility function. Constant relative risk aversion with regards to consumption means that a person prefers a bundle of 1 utils over uncertain bundles with expected value of 1 utils. The same applies to 2 utils to the same degree proportionally. Note that the utility is $-\infty$ when consumption reaches 0.

Now, if you want to calculate the exact utility that a person receives you simply multiply the probability of each bundle with the utility it grants:

$$P_1*U(C_1)+P_2*U(C_2)... = \sum_{n=1}^{\ b}P_n U(C_n)$$

Where P denotes a probability of a given bundle and U(c) is the utility.

Finally, consider a case where there is a possibility of a bundle that has consumption value of 0, with probability greater than zero:

$$ P_x U(c_x) = -\infty $$

It does not matter what the probabilities and utilities of other bundles are as you can't recover from minus infinite utility.

The same principle applies to continuous probability distributions, so just replace the sum sign with an integral and consider case where it doesn't reach 0 probability when c=0.

You may be interested in readin Nordhaus's response:
http://aida.wss.yale.edu/~nordhaus/homepage/documents/weitz_011609.pdf (By the way, he uses the exact same case scenario as I did).


Optimal pricing for Crypto-Currency Exchanges

You are missing the key element, transfer fees.  If the transfer fees are large enough, then the marginal cost to move between exchanges just needs to stay inside the gross bid-ask spread.  It can probably move outside that spread because there is an additional risk for one who moves between exchanges.  Not all exchanges move back and forth between real world bank accounts equally well, nor do all have equal trading depth.  Additionally, the time value of money between the exchanges is probably sufficient to eat up any arbitrage.  It may be worth giving up arbitrage profits if they are traded for the time value of money profits.  If the arbitrage is small and the cost is great, you can carry a wider bid-ask spread and get away with it as a market maker.

It sounds like you need decision theory and not game theory unless you are considering the behaviors of the other actors as part of the game.  You should start with de Finetti's Coherence Principle.  It is a weaker principle than the "no arbitrage" principle.  In its weakest form, it only states that odds are coherent if a bookie cannot take a sure loss in all states of nature through the clever combination of bets either by a single or by multiple actors.

What makes it weaker is that it never says that no arbitrage opportunity exists, but it would imply that if it did exist, then the market makers would capture all or nearly all the profits.  I would be surprised to find that the exchanges did not profit greatly from the arbitrage opportunities on each other's exchanges.  They are perfectly placed and have deep inventory that they could borrow from their customers.  They probably allow some to go by to get speculators into their markets.  They are profit maximizers.


Why are the concepts of short-run and long-run equilibrium exclusive to aggregate supply and demand and macroeconomics?

The concepts of short-run and long-run equilibria certainly feature in microeconomics.  Indeed, they have done so for over a century: see Marshall's Principles of Economics (1890).


Why are we using exp(.) as the functional form in the household maximization problem inside the RCK model?

I guess you already know that $\exp(-\rho t)$ is the the discount factor. In particular, it is obtained by continuously compounding the discount factor

$$\exp(-\rho t)=\lim_{m \to \infty} \left(1+ \frac{\rho}{m}\right)^{mt}$$


Does there always exist a consumption bundle at which the indirect utility function is the inverse of the expenditure function?

Regarding question 1:

From the assumption that the consumer has a single income value $m$, there can only be one value for $m$. So by the notion that $e$ and $v$ are inverses, there can only be one value $e$ for which $m$ corresponds to and thus only one $u$. So there is only one point.

Regarding Question 2:

If we have (1) a utility function that is continuous and locally non-satiated and (2) if $m > 0$ and (3) if both UMP and EMP exist, then they are equivalent. And $e$ and $v$ are inverses.

Claim 1: Solving UMP solves EMP.

Proof:

Suppose bundle $c^*$ solves the UMP but not EMP. Let $c'$ solve EMP. Then (1) amount spent on $c^*$ greater then that on $c'$ and so $u(c') \geq u(c^*) $ because obviously spending more means worse off or same utility. But (2) by local nonsatiation a consumption there exists a bundle $c''$ close enough to $c'$ such that amount spent on $c''$ is less than amount spent on $c^*$ and $u(c'') > u(c^*)$. (3) Contradiction because we assumed $c^*$ solves UMP.

Claim 2: Solving EMP solves UMP.

Proof:

Suppose bundle $c^*$ solves EMP but not UMP. Let $c'$ solve UMP. Then (1) $u(c') > u(c^*)$ even though we spend the same amount on each. That is, $c^*$ doesn't solve UMP since less utility for same $m$.   Because amount spent positive, we can find $t$ where $0<t<1$ such that amount spent on $tc'$ is lower than that spent on $c^*$ yet $u(tc') > u(c^*)$ because $c^*$ does not solve the UMP. (3) Contradiction because we said $c^*$ solved EMP.

Summary:

If solutions exist either to UMP or EMP, then solutions exist for the other and they are inverse functions.
If an inverse exists, it must be a point.


Why N>T is required on system GMM estimations?

Asymptotics are done as N goes to infinity and not T. If T is larger than N, this assumption is odd. True for most of what is commonly called panel data estimators.


Will banks in the US continue to offer CDs?

Demand deposits might be cheaper, but they can disappear due to withdrawals. Banks also need stable sources of longer-term funding to match against their illiquid assets. Holding reserves is only of limited economic importance relative to the need to widen funding sources.
CD’s are cheap relative to other sources of term funding. It would be very hard to imagine that they will be phased out.


Why (neoclassical) economic models have no profits, unlike the real world?

It seems economic models do not explicitly include ownership of the firm (not of the physical capital, an input used to produce).

Many economic model do have profits. We're just more careful about what we call it. See the definition of "economic profit." When you think of profits, you need to ask yourself if the money you are making is simply a fair exchange for the costs you incur---explicit and implicit costs---or whether some portion of that profit represent money gained above and beyond the (true) costs of production (see "economic rent"). Indeed, this is the only sensible way to think about the world.

  It seems economic models do not explicitly include ownership of the firm

Many economic models do in fact include ownership of the firm. However, most economic models are built for demonstrating some particular idea in relative isolation, as opposed to trying to model every single detail. So if someone wants to demonstrate some effect in a world without economic rents, then a lot of the time modeling the ownership of the firm is less important than modeling the ownership of the capital---and thus it might be excluded.

If you want to see models that dig deep into the issues of economic rents and the ownership of a firm, I would recommend taking a look at the subfields called "industrial organization" and "corporate finance." (For example, take a look at Jean Tirole's book on corporate finance.

  Meanwhile, in real life, even in competitive markets, firms do make profits on the long run.

This is a bold claim. If you want to be scientific, you're going to have to be precise about how you (1) define profits and (2) define competitive markets, and (3) show (with data and sound analysis) that firms very commonly do make profits as you've defined them. This is how an economist would operate.

  Firms are just an ethereal "thing" that take inputs and produce output with a technology freely available.

Hidden in this comment is an very interesting object of study. What is a firm, really?
This was an important question (first?) and most famously raised by Ronald Coase. How do we define a firm? Why do they exists? etc. See the "theory of the firm."


Does Game theory consider opportunity cost?

Recall that Oppurtunity Cost is referred to the benefit forgone by choosing the other option.

In your example of a single player game, opportunity cost exists since the architect is completely in control of the benefit he receives. In this case the opportunity cost for signing the contract is 0 (and conversely -40 for not signing the contract).

However this is only applicable in a single person game.

In a case where the business owner and Architect were actively negotiating, it would appear that there would not be any practical ramifications for considering opportunity cost because the payoffs change depending on the other player and can only be seen after the fact.

for example consider a single period 2x2 game such that:
$$\left[\begin{matrix}(10,5)&(2,2)\\(2,2)&(5,10)\end{matrix} \right]$$

From the horizontal/vertical player's position his payoffs range from 10,5 and 2. He does not choose with respect to costs, he chooses based on what he thinks the payoffs will be.

To answer the broader question which  you are probably wondering about.
Opportunity cost is only considered when a buy in for a specific game is necessary

You'd have to consider expected value of a game compared to the cost. (however that only provides a fair price when the game is played infinity, you would also have to consider how many times you'd have to play a game and end up with a positive profit) using this formula:

$$n=\left(\frac{1.96\sigma}{p-\mu}\right)^2$$

I suggest reading up on Martingales and the St. Petersberg Paradox for more info.


Interpretation of FE: $\delta_{st}$  is the mean rating in each sector in each time period.

Contrived Example:  Over time, public opinion of workers in the library sector improved and their rating r improved. On the other hand, those in the used-car sector became seen as less reliable and their rating r declined.  We capture this with $\delta_{st}$ .

Identification of B (intuition):

x must vary within individuals i of the same sector s every time period. Otherwise $\delta_{st}$ will be co-linear.
x must vary within individuals i along time periods t. Otherwise $\gamma_i$ will be co-linear.

Basically, $\gamma_i$ can be identified separately from $\beta$ because individuals change over time.  And $\delta_{st}$ can be identified separately from $\beta$ because they individuals are heterogeneous within each time period and sector.


How does demand curve for labor change when there is a proportional payroll tax on employer

Your guess that somewhat is happening with the slope was a good one. The slope will indeed change. If you want to draw that new Demand line ($D_1$) it has to start at the exact same point of the old demand line (if wages are zero the percantage tax will be zero too). The correct intersection with the Y-Axis would be: $(1-t) W_0$ (this simply means one minus the tax rate ($t$) times the old intersection point ($W_0$). I also made a little graph:


How to tell if F-Stat is significant? And at which significance level?

You get:

F-statistic          24.60231
Prob(F-statistic)    0.000000

You observe that the p-value is 0.00, so this means that the F-statistic is significant at at 99+%, and also obviously at 95% and 90%.


What are the Economic reasons and explanations to make a referendum?

Polling is expensive so only issues that a lot of people feel strongly about will get polled.

You can imagine that every citizen $i$ would pay $r_i$ monetary units to have a certain policy changed. This number may also be negative to reflect that some people like the current policy better than the proposed change.
Ideally the government would like to know $\sum r_i$, if its positive or negative. But I am sure there are also a lot of practical political considerations. However polling itself has a significant cost. Let us denote this by $C$. The government probably has some vague idea about $\sum r_i$. E.g. most people don't care about the color of chewing gum wraps, so the $\sum r_i$ for banning silver coloring is likely negligable. In this case a poll to determine the exact value would be wasteful as
$$
\sum r_i - C < 0.
$$
So when you pick something to poll it is probably a salient issue, something that a lot of people know about and have strong opinions about. As I said, I am sure that these considerations are usually secondary to political ones.


How does elasticity impact market efficiency?

Good question

If we examine edge cases of full elasticity, then we get:
supply or demand with zero impact of quantity on price: no matter the quantity, price will remain the same

Looking at elastic demand, that mean that for suppliers the MC function will arrive to the particular demand price. No special change in efficiency

Looking at elastic supply, that means that the consumer surplus will derive from the fixed price

Based on theory, there will be no loss of efficiency (a certain quantity that will be lost) when adding taxes or subsidies

The common real case of full elasticity, is for imported or exported goods, where the world market's equilibrium for the product imported /exported is unaffected by local market
In this case, either consumers get cheaper prices for products, comparing to a similar product manufactured locally
or get extra prices for local goods sold abroad


GDP per industry and per region in Europe

Eurostat is the way to go.

Website: http://ec.europa.eu/eurostat (advised that you start from here to get where you want to be because then you see what is actually available)

GDP specifically (I think this is not what you need exactly - GDP per country over time - but its a starting point.): http://ec.europa.eu/eurostat/web/national-accounts/data/main-tables

Keep it in mind. Need Europe+numbers? Eurostat.


Consequences of a particular loss function

Is it the case that since you can rearrange this expression to $\Sigma (\frac{1}{y^2})(\hat{y}-y)^2$, that you are really weighting the standard L2 norm with  weights that are higher for lower absolute values of y?


What are the differences between the older gold standard and the current fiat money standard?

The gold standard in the US was actually kind of complicated.

You had the government issuing notes off of gold, and private banks issuing bank notes off of gold...but also banks issuing notes/deposits off governments notes as well.  A pyramid on a pyramid.

Neither was 100% backed and both were vulnerable to runs.

In 1933, FDR no longer let the public redeem gold for dollars.  A default of sorts...but we were never on a true gold standard!  We always had more promises than gold.  The same with the banks.  Given that gold used to be equivalent to dollars (as the Fed used to exchange the two), banks would use dollars as reserves for convenience...but soon they became the monetary base.

Now the US still maintained an international gold standard.  But this too was fractional and had more promises than assets.  Countries (most notably) the French did not like how "overbooked" the dollar was to gold and started redeeming their gold in mass.  There was a run by international central banks on the dollar and Nixon had no choice but to take us off the international gold standard as well.

So it is confusing, but we went off the gold standard twice.  Once to the public (FDR) and once to international central banks (Nixon).

So how did this work before 1972?  Gold was the international reserve currency, but on top of this the dollar was the means by which gold would be commonly traded between countries.  It was much more difficult to clear transactions in gold than paper (or accounting) dollars, which led to its popularity as a reserve currency.  On occasion of course individuals would demand gold for dollars and you would have physical transfers take place (as in load the gold onto the ship and send it over the high seas).

If you are interested in the history of the American gold standard, I highly recommend the book A History of Money and Banking in the United States:
The Colonial Era to World War II by Murray N. Rothbard:

mises.org

So what is the difference between the old system and the current system?  Not much as we were never on a true gold standard.  But in a nutshell, banks used to be able to redeem dollar bills or central bank deposits for gold.  Now dollars are not redeemdable for anything.  You can see on the Fed balance sheet a holdover for this as gold is still considered an asset and dollars (paper and electronic) as liabilities.

en.wikipedia.org

So how would trade occur pre-Bretton Woods?  Mostly through banks.  The receiver bank would obtain credit for gold, or a national currency that was convertible to gold (dollars or say pounds).  If one party wanted to convert currencies they could but it mostly took place as a transfer from say "pound deposits" to "dollar deposits".  If the new party demanded their payment in gold they could obtain this but it rarely happened.

After Bretton Woods, international trade was mostly facilitated by dollar deposits with the promise of gold redemption of a foreign central bank demanded it (which Nixon ended).

Now when gold reserves were physically demanded, did this cause problems?  Yes!  Remember, all countries and banks issued more notes/deposits for dollars than they had (somewhat dishonest but that is another matter).  Say the pound (during the standard) lost a bunch of reserves...so they go from a ratio of say 1000 central bank deposits to 200 gold to 900 central bank deposits and 100 gold.  A serious blow to credibility which could lead to a run on the currency. Do they have to print or destroy more notes/deposits...no because they do not 100% back gold.

So why do countries hold reserve currencies today?  They kind of don't need to, but there is a reason.  Basically private banks operate by maturity mismatching (balancing short term debt against long term assets).  When a currency is not stable (like from a small nation) short term interest rates are too volatile for banking and in fact bank runs can be initiated from clever predator foreign inverters (like George Soros).  Politicians like banks...and for some reason they think they are important to the economy, so go to great ends to prop them up.  For a small nation this means their currency must be somewhat stable for their banking system to survive...and the nation achieves this by maintaining a peg.  You can only maintain a peg if you can manipulate the market, so countries hoard foreign assets like dollars so they can buy and sell their home currency (like say the Peso) so it achieve its peg rate.

As for clearing of international transactions this is mostly done with bank money, but dollars are a common facilitating base for these transactions (although certainly many other currencies are being used now).


Explosiveness

The paper contains an error, which causes the explosive dynamics in your simulation (although presumably the underlying computations in the paper were correct). The equilibrium condition derived from eigenvalue decomposition is contained in the third row of matrix $Q^{-1}$ on page 12 of the paper, with variables ordered as $(c,k,h,z)$ (I'll drop tildas, so all lowercase variables are to be understood as log-deviations). Comparing with eqn. (16) on p. 13, we see that coefficients for $k$ and $h$ are switched, and so the correct condition is

$$
c_t = 0.54 k_t + 0.02 h_t + 0.44 z_t
$$

Simulation

First, we can express consumption and labor as linear function of state variables (no need to solve the system at each step of the simulation). The intertemporal and intratemporal equilibrium conditions can be written as

$$
\begin{bmatrix}1 & -0.02 \\ 2.78 & 1 \end{bmatrix} \begin{bmatrix} c_t \\ h_t\end{bmatrix} = \begin{bmatrix} 0.54 & 0.44 \\ 1 & 2.78 \end{bmatrix} \begin{bmatrix} k_t \\ z_t\end{bmatrix}
$$

so after multiplying by an inverse we get

$$
\begin{bmatrix} c_t \\ h_t\end{bmatrix} = \begin{bmatrix} 0.53 & 0.47 \\
-0.47 & 1.47 \end{bmatrix} \begin{bmatrix} k_t \\ z_t\end{bmatrix}
$$

Next, transition for states can be written as

$$
\begin{bmatrix} k_{t+1} \\ z_{t+1} \end{bmatrix} = \begin{bmatrix} -0.07 & 0.06 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} c_t \\ h_t\end{bmatrix} + \begin{bmatrix} 1.01 & 0.1 \\ 0 & 0.95 \end{bmatrix} \begin{bmatrix} k_t \\ z_t\end{bmatrix} + \begin{bmatrix} 0 \\ \epsilon_{t+1}\end{bmatrix}
$$

which can be reduced by substuting for control variables to

$$
\begin{bmatrix} k_{t+1} \\ z_{t+1} \end{bmatrix} = \begin{bmatrix} 0.94 & 0.16 \\ 0 & 0.95 \end{bmatrix} \begin{bmatrix} k_t \\ z_t\end{bmatrix} + \begin{bmatrix} 0 \\ \epsilon_{t+1}\end{bmatrix}
$$

Now the simulation should be trivial, here's a Matlab/Octave example:

T = 200;
X = zeros(2,T);
for i=2:T
    X(:,i) = [0.94 0.16; 0 0.95] * X(:,i-1) + [0; 0.007*randn()];
end
Y = [0.53 0.47; -0.47 1.47] * X;
figure
plot(1:T, [X; Y])
legend('k','z','c','h')

Of course in practice, you should probably recompute the whole solution, including the eigenvalue decomposition, so that you would be able to change parameters, etc.


Why do rich and poor countries trade in different goods and why do rich countries trade in similar goods with other rich countries?

The key is production capabilities, not relative specialization. In words of Hausmann and Rodrik (2006):

  The idea that rich (poor) countries tend to export goods exported by other rich
  (poor) countries is quite obvious and is compatible with many possible theories. For
  example, in conventional trade theory a la Heckscher-Ohlin, countries export products
  that are more intensive in the factors of production that are relatively abundant at home.
  As development takes place, physical, human and institutional capital is accumulated and
  the products countries export become more intensive in these factors. ...

  Labor-abundant countries trade with land- or capital-abundant countries in order to
  acquire more efficiently the goods that are intensive in the factors they don’t have. The
  transition between goods is of little significance: they are the passive consequence of
  changing factor endowments.


How does paying off debt to the bank work?

The total nominal income of the domestic economy is equal to nominal GDP growth. This chart shows nominal GDP growth for the U.S.: link to FRED graph.
If you dig into the numbers, we see that nominal GDP growth is often higher than most interest rates. So total income is growing faster than interest costs.
You cannot look at one loan in isolation and hope to make sense of total flows of income. At any given time, new loans are continuously made, while old loans are paid down. Mortgages are typically the largest household sector debt in aggregate. New buyers get large mortgages, while existing homeowners slowly pay down debt. The net change is normally positive, which effectively finances construction. Construction in turn adds to GDP and hence incomes. This story is repeated in different forms across all types of debt.


A bilateral currency swap line allows two central banks to immediately swap currencies directly between each other up to a specified amount. The swap allows Central Bank A to immediately buy Central Bank B's currency at the official exchange rate. The currency never leaves each Central Bank; it is just reassigned to a new owner.

Crucially, the two banks also agree that Central Bank A has to buy back its currency at the same rate plus interest at a specified future date. Central Bank B never assumes any risk for Central Bank A's newly purchased reserves and is no party to any agreements which make use of the funds.


Do banks make profits by selling bonds to the central bank?

No, banks don't directly profit from selling bonds to the central bank. These bonds are sold at face value, there is no discount or premium, this is not part of the system.

My best guess of what the author means is that by selling bonds to the central bank, commercial banks get cash, which they loan out and make a profit on the loans. So commercial banks profit from selling bonds to the central bank in an indirect way. Attributing banks' profits from lending to the central bank is a bit of stretch though and not a very appropriate statement in my view, so I fear I might have misunderstood the author.

What the article seems to promote is already a reality in communist China, where state-owned banks lend at favourable rates. There can be significant problems with this approach of cutting out private banks, as the Chinese bad loan problem shows (although private banks can also accumulate bad loans, but maybe less so due to competition), but this is a different discussion.


Derivation of Arrow-Pratt risk aversion measure

Please find below the pages which may interest you.

Arrow, K. J. Essays in the Theory of Risk-Bearing, North-Holland Publishing Company, 1971

I let the admin delete this post if the few extracts are not allowed...


Do mechanisms always result in a series of strategic games being played out?

I'm not sure what you mean by "results in a game".

When doing mechanism design, after you've fixed a mechanism, the resulting mathematical structure can be identified with a game which we solve using the tools of game theory. That is, the choice of a mechanism induces a game. I suppose you could take this to mean that it results in a game.

Just because that game has dominant strategies does not mean the game doesn't "take place". In fact, I don't even know how you can speak of dominant strategies when you don't (even if only implicitly) have a game in mind when doing so.


Sources of free European national income and output data?

The annual macro-economic database of the European Commission (AMECO) provides the data you need.

The data is available for free, in formats that are easy to work with.


How can difference equations with an infinite summation be represented in matrix form?

Maybe the differencing approach should work for you. Basically the idea is to reduce this series to a finite expression, using the lag operator. Let me explain with an example from Costa (2016, p.81) (If you're not familiar with the lag operator I suggest you to read the first section of the page linked above.):
We have the optimal price level equation (equivalent to your NKPC but less developed):
$$\tilde{P_{t}}=\theta\tilde{P_{t-1}}+(1-\theta)(1-\beta\theta)E_t\sum_{i=0}^{\infty}(\beta\theta)^i\tilde{MC}_{t+i}$$
Now, we can see that the difference between one period of time might be represented by:
$$(\beta\theta)^k\tilde{MC}_{t+k}-(\beta\theta)^{k+1}\tilde{MC}_{t+k+1}=(\beta\theta)^k(\tilde{MC}_{t+k}-\beta\theta\tilde{MC}_{t+k+1})$$
In terms of lag operator we could factor $\tilde{MC_t}$:
$$(\beta\theta)^k\tilde{MC}_{t+k}-(\beta\theta)^{k+1}\tilde{MC}_{t+k+1}=(\beta\theta)^k\tilde{MC_t}(1-\beta\theta L^{-1})$$
What this tells us is that we can use the same series lagged by $-1$ (same as forward by 1) to eliminate future periods from the original one. Let $S_t=E_t\sum_{i=0}^{\infty}(\beta\theta)^i\tilde{MC}_{t+i}$, then (omitting expectations operator):
$$S_t-S_{t+1}=
$$
\begin{bmatrix}
\tilde{MC}_{t}+(\beta\theta)\tilde{MC}_{t+1}+(\beta\theta)^2\tilde{MC}_{t+2}+(\beta\theta)^3\tilde{MC}_{t+3}+...\\
-\beta\theta\tilde{MC}_{t+1}-(\beta\theta)^2\tilde{MC}_{t+2}-(\beta\theta)^3\tilde{MC}_{t+3}-(\beta\theta)^4\tilde{MC}_{t+4}-...
\end{bmatrix}
$$=\tilde{MC}_t$$
Look that we managed to reduce the infinite sum into a single expression, now to replicate this in our prices equation, we need to get something similar to $S_t-S_{t+1}$ from $S_t$ only, taking advantage of lags because if we just sum $-S_{t+1}$ in both sides of the equation we would've done nothing. Then you can check that:
$$S_t-S_{t+1}=(1-\beta\theta L^{-1})S_t=\tilde{MC}_t$$
With that in mind, we just need to multiply both sides of our prices equation by $(1-\beta\theta L^{-1})$, and develop the corresponding algebra for arriving to:
$$\tilde{P_t}-\beta\theta E_t\tilde{P_{t+1}}=(1-\theta)(1-\beta\theta)\tilde{MC}_t-\theta^2\beta\tilde{P}_t+\theta\tilde{P}_{t-1}$$
Which allows us to make better analysis by getting rid of the series, and therefore being able to write in terms of matrices and vectors that are more "readable". I'm pretty sure you can do this same process with your equation that is in term of inflation (instead of prices). Hope this works.


Serial correlation and clustered correlation in FE and FD estimators

Yes. If the idiosyncratic error is iid, xtreg (FE) with no options (ordinary se) is valid, but the ordinary se for FD, reg d.(y x), is invalid because the differenced error is serially correlated.
Yes, it is. It's cluster-robust, which means OK  regardless of the presence of within-group correlation.


Microeconomics theory and integration by parts (proof).

Integrate the $f(t)$ (a primitive of which is $F(t)$) and differentiate the $t$. This yields

\begin{align*}
x(z) F(z) & = \int_{0}^{z}{t f(t)dt} \\
& = \big[ t F(t) \big ]_{t=0}^{t=z}-\int_{0}^{z}{F(t)dt} \\
& = z F(z)-\int_{0}^{z}{F(t)dt}.
\end{align*}

Dividing by $F(z)$ on both sides yields the result.


finding the returns to scale of a production function

What you have proven is that $$f(tx_1, tx_2) > tg(x_1,x_2)$$

which translates "scaled $f$ is higher than some other, homogeneous function $g$ scaled by the same factor". This does not prove anything about the returns to scale related to $f$, although I can see why it may appear otherwise.

You can go in reverse, starting from

$$tf(x_1,x_2) = t^{0.5}t^{0.5}f(x_1,x_2)=...$$

and compare to $f(tx_1, tx_2)$.

Also beware about the following: what are the permissible values for $t$ when returns are not constant? Are you sure we should obtain the result for $t>0$, or maybe we examine the property for a smaller interval?


Hint:
Note that
\begin{equation}
k_{t+1}=gk_t^a \quad\Rightarrow\quad g=\frac{k_{t+1}}{k_t^a}=\frac{k_{t+2}}{k_{t+1}^a}.
\end{equation}
Rearrange your Euler condition into a form such that you can use $g$ to sub-out the above ratios. Then it should be straightforward to solve for $g$.


Necessary indifference conditions in mixed equilibrium

First, because action $M$ is weakly dominated for Player 3, does everyone assume that $r_2=0$?

Without seeing the game you're considering, it's hard to tell. But the general answer is No. Weakly dominated strategies can be played in a Nash equilibrium (either mixed or pure). In the following game, both $B$ and $C$ are weakly dominated strategies for both players. But each player randomizing between them in any proportion is still a NE.

\begin{array}{|c|c|c|c|}
\hline & A & B&C\\\hline
A&1,1&0,0&0,0\\\hline
B&0,0&0,0&0,0\\\hline
C&0,0&0,0&0,0\\\hline
\end{array}

  Does Player 3 need to be indifferent between $L$, $M$, and $R$, or just between $L$ and $R$, since $M$ is weakly dominated?

The indifference condition for MSNE applies to pure strategies in the support of the equilibrium mixed strategy. If $M$ is played with positive probability in equilibrium, then you'd need indifference across all three. If $M$ is played with zero probability, then you'd need $u_3(\sigma_1,\sigma_2,L)=u_3(\sigma_1,\sigma_2,R)\ge u_3(\sigma_1,\sigma_2,M)$

More generally, the necessary and sufficient conditions for a Nash equilibrium (either pure or mixed) are summarized in the following theorem:

  A strategy profile $(\sigma_1,\dots,\sigma_n)$ is a Nash equilibrium if and only if for all $i=1,\dots,n$,
  \begin{equation}
u_i(s_i,\sigma_{-i})= u_i(s_i',\sigma_{-i}), \qquad \forall s_i,s_i'\in\mathrm{supp}(\sigma_i)
\end{equation}
  and
  \begin{equation}
u_i(s_i,\sigma_{-i})\ge u_i(s_i',\sigma_{-i}), \qquad \forall s_i\in\mathrm{supp}(\sigma_i)\text{ and }\forall s_i'\notin\mathrm{supp}(\sigma_i).
\end{equation}

Hence, we see that the indifference condition is only required for pure strategies in the support of the equilibrium mixed strategy but not necessarily outside it.


Why did the hybrid Phillips Curve become flatter (as empirical data shows) after the crisis?

(1) The Phillips curve is not a structural thing, it is not surprising that the relationship between unemployment and inflation is not constant over time.

(2) There's a nice AER paper by Simon Gilchrist and coauthors that argues that inflation did not go down as much as in earlier recessions because financially constrained firms needed to keep their cash-flows up and increased prices.


What is the economic incentive to cheat? How does an experiment capture exogenous deviations?

A couple of comments to your question below:

  When an economic experiment is designed to have predetermined outcomes of ethical and unethical decisions, how does it address the extreme levels of deviation that is present within historical record?

1) External validity --- whether results obtained in the lab can be generalized to settings outside the lab --- has long been one of the main criticisms of experimental economics, and practitioners in this field are well aware of this. In light of this concern, one must therefore exercise caution in interpreting experimental results. Lab results should be extrapolated to an external environment only if the settings of the experiment sufficiently resemble those in that environment. (To some extent, the same can be said about generalizations of results obtained using field data.)

In the experiment that you participated, decisions were labeled as "cooperate" and "cheat". However, the labels could very well be "A" and "B", or "Left" and "Right", or "Aspiring" and "Cautious". The point is that the labeling of the actions is arbitrary, and you shouldn't (over)interpret the results just because the experimenter happen to choose a non-neutral frame.

2) Judging by your very limited description of the experiment --- e.g. you didn't say what the control treatment was like, or whether the subject moving at stage 2 can observe what was chosen in stage 1 --- I suspect that the experiment was more for a pedagogical purpose than a research one. Generally speaking, experiments for research purposes should avoid recruiting subjects from the experimenter's own class, since there are strong outside incentives that may affect/dominate the salience of the payoff from within the experiment. Moreover, experiments should generally adopt neutral framing, since non-neutral frames tend to bias the results in a certain direction.

Therefore, I think you are over-interpreting the results in an experiment that is not very rigorously conducted.


What is the optimal forecast for the following moving average model?

Your equation $y_{T+1} = E(y_{T+1}|I_T)$ clearly does not make sense.

In any case,
$$
E(y_{T+1}|I_T)
$$ is the best one step ahead forecast conditional on $Y_1, \cdots, Y_T$, whereas you're asked to find the best linear one step ahead forecast given $Y_T$. They are in general not the same.
The best linear forecast for $Y_{T+1}$ given $Y_T$ is
$$
\mu + \frac{\gamma(1)}{\gamma(0)} (Y_T - \mu),
$$
where $\mu$ is the unconditional mean of the series and $\gamma$ is the autocovariance function. In your case, $\mu = 0$ and $\gamma$ is that of a MA(1) series ($\frac{\gamma(1)}{\gamma(0)} = \frac{\theta}{1+\theta^2}, \; \theta = \frac12$).

In general, the best linear forecast for covariance-stationary time series can be obtained by solving the Yule-Walker equations.

For best linear forecast, the assumption that innovations are i.i.d. is extraneous.


Demand for minimum of $4$ different goods

You can solve it sequentially by noting the nesting structure of the utility function $U$.
So first note that the utility function combines utility functions you are probably already familiar with $U=\min\{u_1,u_2\}$ of complements and $u_1=\sqrt{x+y}$ and $u_2 = z+w$ both of which are perfect substitutes. Where $u_1$ and $u_2$ are nested within $U$.
The solution procedure is based on the following argument: However much money $m_{u_1}$ the agent decides to spend on $u_1 =\sqrt{x+y}$ she must be minimizing the cost of getting $u_1$ utility points. Similarly, however much money $m_{u_2}$ the agent spends on $u_2= z + w$ it must be the case that the agent minimize cost. So first find the expenditure functions for these inner problem.
Secondly the agent must the maximize the outer utility $\min\{u_1,u_2\}$ subject to the expenditures of the inner problems adding up to total income.
Solve the second inner problem:
The second inner problem is
$$max_{z,w} \{z+w \lvert p_zz+p_ww\leq m_{u_2}\},$$
where $m_{u_2}$ is money spend on $u_2$.
How do you get the most utility for the buck when utility is $u_2 = z + w$? You simply buy the cheapest good so the price of one point of $u_2$ utility is $p_{u_2} := \min\{p_z,p_w\}$. The demand for $z$ and $w$ depends on the amount of money you decide to spend on $u_2$ assumed for now to be $m_{u_2}$. Given this amount demand for $z$ is
$$z = \frac{m_{u_2}}{p_z} 1[p_z <p_w]$$
and
$$w = \frac{m_{u_2}}{p_w} 1[p_w <p_z].$$
I leave it to you to handle the threshold case where $p_z=p_w$. How much utility $u_2$ do you get? Well this must be $u_2 = m_{u_2}/\min\{p_z,p_w\}$ - this is the value function for this inner utility maximization problem - hence it must be the case that
$$m_{u_2} = \min\{p_z,p_w\} u_2,$$
which is the expenditure function for this inner problem.
Solve the first inner problem:
The first inner problem is
$$max_{x,y} \{\sqrt{x+y} \lvert p_xx+p_yy\leq m_{u_1}\},$$
where $m_{u_1}$ is money spend on $u_1$.
How do you get the most utility for the buck when utility is $u_1 = \sqrt{x+y}$? It is perfect substitutes so you buy the cheapest good so demand is
$$x = \frac{m_{u_1}}{p_x} 1[p_x <p_y]$$
and
$$y = \frac{m_{u_1}}{p_y} 1[p_y <p_x],$$
where $m_{u_1}$ is amount of money spend on $u_1$ assumed known. How much utility $u_1$ do you get? That must be $u_1 = \sqrt{m_{u_1}/\min\{p_x,p_y\}}$ such that
$$m_{u_1} = \min\{p_x,p_y\} u_1^2$$
Solve outer problem:
The outer problem is
$$\max_{u_1,u_2}: \ \ \  \min\{u_1,u_2\}  \\[8pt]
s.t. \ \ \ \ m_{u_1} + m_{u_2} = m $$
for which we rewrite the budget constraint to
$$\min\{p_x,p_y\} u_1^2 + \min\{p_z,p_w\} u_2 = m$$
and then use that $u_1=u_2$ because the problem is perfect complements. You then have two equations in two unknowns $u_1$ and $u_2$. Solve for $u_1$ and $u_2$ and plug solutions back into expenditure functions for inner problems to get $m_{u_1}$ and $m_{u_2}$ which you then plug into the equations for the demand.


Interpretation of $\frac{\partial }{\partial p_1}Q_1(p_1, p_2)/\frac{\partial}{\partial p_2} Q_1(p_1, p_2)$

One interpretation I can offer. The demand function can be expressed as:

$$Q_1 = Q_1(p_1,p_2)$$

Let us take the total differential:

$$dQ_1 = \frac{\partial Q_1(p_1,p_2)}{\partial p_1}dp_1+\frac{\partial Q_1(p_1,p_2)}{\partial p_2}dp_2$$

Assume that $Q_1$ remains unchanged with respect to a change in prices. This implies that $dQ_1=0$. Solving the equation:

$$\frac{d p_2}{d p_1} = - \frac{\frac{\partial Q_1(p_1,p_2)}{\partial p_1}}{\frac{\partial Q_1(p_1,p_2)}{\partial p_2}}$$

The above expression implies the following (since both partial derivatives are negative):

$$t = \left|\frac{dp_2}{dp_1}\right| = \frac{\frac{\partial Q_1(p_1,p_2)}{\partial p_1}}{\frac{\partial Q_1(p_1,p_2)}{\partial p_2}}$$

$t$ can be interpreted as follows: if good 1 became more expensive, how much would the price of good 2 need to change (decrease) such that the demand for good 1 remains unchanged.

Mathematically, $t$ represents the magnitude of the slope at ($p_1,p_2$) on the level curve when we fix $Q_1$.


Regarding the group formation in an oil-price experiment

Without knowing more about your specific project, you probably want to look at difference-in-difference methodologies. It's the bread and butter of this sort of psudo-experiment.

I see you use the phrases "train" and "test".  Be very careful, typically machine learning approaches are designed to get very good $\hat{Y}$s.  Economists in general don't care about $\hat{Y}$s much at all. We care about having good $\hat{\beta}$'s. It will take some time to get used to that idea.

Naive use of traditional ML techniques will lead to highly biased estimators and very misleading estimates of $\hat{Y}$ that often lead to bad policy decisions.