Merging Panel data with two different splits. Is it possible?

I would just merge them horizontally... so that the main dataset is the second one, and you include country-wide statistics into each row:

ID  Year  State  xx1  xx2  xx3   x1  x2  x3
A1  2000   AL    581  135  1.22  10  11  12
A1  2000   FL    584  51   1.5   10  11  12
A2  2000   AL    5138 84   1.7   13  25  20
...


Fundamental question on marginal utility

How is marginal utility interpreted as the additional "happiness" gained from consuming one more unit of some good?

Not sure what you mean. Utility is not interpreted as some biological measure of happiness. A bundle of goods with high utility is prefered by the consumer to bundles with lower utilities. This is all that utility describes; it is a modelling/storytelling tool.

If marginal utility is the derivative of utility, then isn't it only measuring how the gains from an incredibly small consumption of some good?

Yes, it is.
On your graph: kudos for using Desmos!

The MRS at (20,5) is -1/4 which should mean that at (20,5) this person will be ok with trading in 4 units of good x for 1 unit of good y correct?

Strictly speeking this is only true for marginal quantities (infinitesimally small quantities). Basically you are approximating the indifference curve with a linear curve (a line) at the (20,5) point. This line only coincides with the indifference curve at (20,5), and as you move farther away from this point, the line becomes a more and more imprecise approximation of the curve. (Note that 6.25 instead of 6 is not too bad as an approximation.)
The line about MRS representing the ratio at which a consumer is willing to trade is just to get the intuition across, it is not an exact definition.


Does GDP count both the investment and the house rental?

You are correct that what you describe would be an overcount, but fortunately, your description of what is counted in GDP is incorrect. When a new home is constructed and sold, the full sales price is not counted in GDP. Instead, only the value of the construction put in place is counted in GDP— when the construction is completed. Then, the value of the stream of housing services (i.e., the imputation for owners' equivalent rent) is counted annually as an output of the asset.

Quoting "How BEA Accounts for Investment in Private Structures":

  In the NIPAs, private investment in new construction
  is measured mainly as the sum of the costs of inputs
  of all construction “put in place,” that is, all
  construction activity completed in a given period. For
  individual projects, BEA’s measure of investment in
  structures includes the following:

  Cost of materials installed or erected
  Cost of labor and the cost of construction equipment
  rental for the period
  Cost of architectural and engineering work
  Miscellaneous overhead and office costs incurred
  by the project’s owners
  Interest and taxes paid during construction
  Contractors’ profits

Houses are therefore counted just as any other capital asset is. Just as a machine that makes bolts first appears in GDP when the machine is constructed then later its output is also counted, a house appears in GDP as output when it is constructed (or renovated) and its output— housing services— is also counted.


Public debt and GDP in post-socialistic countries

The statement is not generally true. If you look at gross debt, the Western countries of Finland, Denmark and Norway have lower debt ratios (53%,50%,34%) than post-socialistic Croatia (80%) and Hungary (79%). If you look at net debt, these Western countries have even better numbers.

So right now the correct answer is that your premise is not true.

Your statement is probably true in some form (e.g. on 'average', whatever that would mean in this context, or given some specific Western countries), but please make it exact and back it up with a source. (Perhaps in a new question, seeing that I put effort into answering your current one. Changing it post-answer would be somewhat unsporting.)


I'm not going to answer the questions for you, but here is some food for thought, you should be able to conclude yourself.

Ad 1

We know that in the long run equilibrium, there are zero profits: If they'd be positive, firms would enter, if they'd be negative, firms would leave. Can we conclude something about a firm's profits if it has average total costs smaller (larger) than the price?

Ad 2

What induces change in the total amount of companies in such a branch? Is having zero-profits one of these reasons?


The simplest thing is to calculate the expected cost in the two scenarios:

Scenario 1: Develop both at the same time
$$E[cost]= X + Y$$

Scenario 2: Wait and see if task two is needed
$$E[cost]= X + 0.7Z$$

However I would not say expected value calculation represents the "economics view". The general view rather depends on your preferences, often expressed as utility functions. It could be your utility is a function of expected value, but it could also not be the case. What is your risk tolerance? Some people seem risk-seeking. Are you ok with a a small possibility of ruin (bankruptcy)? Preferences can also depend on reference points and more...

For other simplistic approaches you could start with value at risk.


Why do the income and substitution effects cancel for log preferences? Trouble reconciling Slutsky decomposition

You interpreted the passage you read as essentially saying that "with log preferences, the Marshallian demand for the good does not depend on its price."
That doesn't sound very plausible, don't you think?

But the passage clearly refers to an intertemporal choice problem, it is about log-preferences over a single good and over time, it does not relate to the Slutsky decomposition of the static Marshallian demand function.

I would suggest to continue reading the notes up to page 14.


In a New Keynesian framework, expectation-augmented Philips curve is the following:

$$ \pi_t = \beta \;\mathbb{E}\pi_{t+1} + \alpha x_t$$
where $\beta$ and $\alpha$ are constant, $\pi_t$ stands for the inflation rate (actually long-linear approximation of it but doesn't matter intuitively) at time $t$, $\mathbb{E}\pi_{t+1}$ is inflation rate at time $t+1$ anticipated at time $t$,  and $x_t$ is the output gap -gap between natural level of output at time $t$ and the actual output at time $t$-.

As the equation above suggests, provided a constant $\pi_t$,  an increase in the exptected inflation at time $t+1$ leads to a decrease in the output gap at time t, in other words, total production at time goes down -so unemployment increases-. In this model to obtain a relation between expectations and the current economic decisions, we make some assumptions on technology and preferences. In particular, we assume that firms set nominal prices based on the expectations of future marginal costs, thus output gap today depends on marginal cost of production at the current period and at the following periods.


If someone goes for a haircut, does it increase GDP?

Existing answers are correct, yes, you are increasing GDP.
GDP is a crude measure of how much is spent during a year. This is used as a proxy for living standards (the thinking goes: if an economy is spending more, there's more economic activity and it's probably doing better).
But the logic doesn't always hold, for example:
Consider a scenario where you break my window and I break yours, and we both pay the glazier \$100 each to fix our windows. Our living standards are unchanged, despite the fact that our actions increased GDP by \$200!


You're making this way more complicated than it needs to be.

Edit: Okay it's a little more complicated that I thought but hey! What a cool result!

$AC = \frac{C(q)}{q} \\
MC = C'(q)$

When you minimize $AC$ with respect to $q$,

$$\frac{\partial AC}{\partial q} = \frac{C'(q) \cdot q - C(q)}{q^2} = 0$$
$$\implies C'(q) \cdot q - C(q) = 0$$
$$\implies C'(q) \cdot q = C(q)$$
$$\implies C'(q) = \frac{C(q)}{q}$$

Then marginal cost equals average cost. Which seems to be what you've gotten so far.

So now we check the second order conditions:

$$\frac{\partial AC}{\partial q} = \frac{C'(q)}{q} - \frac{C(q)}{q^2}$$
$$\frac{\partial^2 AC}{\partial q^2} = \frac{C''(q) \cdot q - C'(q)}{q^2} - \frac{C'(q) \cdot q^2 - C(q) \cdot 2q}{q^4}$$

What makes this expression greater than zero?

$$\frac{\partial^2 AC}{\partial q^2} = \frac{C''(q) \cdot q - C'(q)}{q^2} - \frac{C'(q) \cdot q^2 - C(q) \cdot 2q}{q^4} > 0$$
$$\implies \frac{C''(q) \cdot q - C'(q)}{q^2} > \frac{C'(q) \cdot q^2 - C(q) \cdot 2q}{q^4}$$
$$\implies C''(q) \cdot q - C'(q) > \frac{C'(q) \cdot q^2 - C(q) \cdot 2q}{q^2}$$
$$\implies C''(q) \cdot q - C'(q) > C'(q) - \frac{2C(q)}{q}$$
$$\implies C''(q) \cdot q > 2\left(C'(q) - \frac{C(q)}{q}\right)$$
$$\implies C''(q) > 2\left(C'(q) \cdot q - C(q)\right)$$

But recall that given the first derivative,

$$C'(q) = \frac{C(q)}{q}$$
$$\implies C'(q) \cdot q = C(q)$$

So substitute that into the above:

$$\implies C''(q) > 2\left(C(q) - C(q)\right)$$
$$\implies C''(q) > 0$$

And this is true as long as the total cost curve is convex, which is a pretty standard assumption. (I guess also positive production has to be a thing, so basically the firm has to be not shut down.) So we're done.


Would be there redistribution of wealth from actors with sticky prices to actors with more flexible prices under stagflation?

This would redistribute some resources from those who can't change their prices quickly enough, but studies (like this study here) show that the most important redistribution effect of inflation occurs along debtor/lender lines.


Unfortunately for your case all GMM estimators suffer from finite sample bias. In the presence of an endogenous explanatory variable the GMM estimator is only consistent.

There's not much you can do to remedy this issue, except collect more data. Check for consistency of sign and magnitude of estimates between OLS and GMM, as well as sensitivity to instrument choice (if your model is over-identified).


What conditions must we demand for the economy to be always on the saddle path?

To stick to the simpler deterministic case, it is perfect foresight and the Transvarsality condition (TVC)that guarantees that an optimizing agent will stay on the saddle-path, because all other paths lead to the violation of TVC (corner solutions violate the TVC).  In a deterministic environment and with perfect foresight the agent knows this and so he sticks to the saddle path. The chapter on the basic Ramsey model in Barrow and Sala-i-Martin's Growth  book has a good discussion on the matter.


What is the difference between a free market and a perfectly competitive market?

Perfectly competitive market: Numerous participants, so that no single participant has any influence.

  A perfectly competitive economic model is meant to describe a situation in which there are many participants, and the influence of each one individually is negligible. The state of the economy is thus insensitive to the actions
  of any single agent; only the aggregate behaviour matters. (Source.)

Free market: Voluntary exchanges; no coercion (in particular, no government coercion).

  "Free market” is a summary term for an array of exchanges that take place in society. Each exchange is undertaken as a voluntary agreement between two people or between groups of people represented by agents. (Source.)

Notes:

The two terms are not mutually exclusive.
Neither exists in the real world, though we can imagine examples of both.


Separating and supporting hyperplane theorems

I will sketch a solution for your question on the separating hyperplane theorem together with one economic application to producer theory (for more details please look at MWG).

Theorem: Suppose $B \subset \mathbb{R}^N$ is a convex and closed and that $x \notin B$. Then there is a $p \in \mathbb{R}^N$ with $p \neq 0 $ and a value $c \in \mathbb{R}$ such that $px>c$ and $py < c$ for every $y \in B$

Application (Producer theory): Suppose the production set $Y$ is closed and convex. Then every efficient production $y \in Y$ is a profit maximizing production for some nonzero price vector $p \geq 0 $.

Sketch of proof:
Start with an efficient $y$ and define $P_y$ as the "better than" set with respect to $y$. This set is clearly convex (pick $y', y'' \in P_y$ and their convex combination is clearly "in it"). Next realize that because $y$ is efficient $P_y \cup Y = \emptyset$. We know have all the ingredients to apply our theorem above, and in particular $\exists p \neq 0$ such that $p y' \geq py'''$ $\forall$ $y' \in P_y$ and $\forall$ $y''' \in Y$. Because $y'$ can be chosen arbitrarily close to $y$ (remember $y' \in P_y$) we can straightforwardly show that $y$ is profit maximizing for $p$

Other "famous applications" that you may want to look at are:

1) The "Second Fundamental theorem of Welfare Economics

2) The saddle point approach to dynamic programming developed by Karlin and Uzawa.


Finding labor supply function from equation of labor market

Here is another way we could solve this problem with some simple calculus. We first start by maximizing the utility function subject to the budget constraint and the time constraint.
$$\underset{L,C}{max}\;\underset{s.t.\;C=100+w(40-L)}{L^{1/2}C^{1/2}}\qquad (1)$$
$$\underset{L}{max}\;L^{1/2}[100+w(40-L)]^{1/2}\qquad (2)$$
$\textbf{FOC:}$
$$\bigg(\frac{100+w(40-L)}{L}\bigg) ^{1/2}=w\bigg(\frac{L}{100+w(40-L)}\bigg) ^{1/2}\qquad (3) $$

With some math
$$L=\frac{50+20w}{w}\qquad (4)$$

Let $N$ be the number of hours worked. If you plug 5 and 8 in for $w$, you get 10 and 13.75 for $N$ respectively.

Now we have two points $(5,10),(8,13.75)$. Solving for slope we get $\frac{13.75-10}{8-5}=1.25$.

Now we must consider the intercept. Because $w=0$ is not defined, we must use the other intercept (where labor hours equal zero).
$$40-L=0\implies 40-\bigg(\frac{50+20w}{w}\bigg)=0\qquad (4)$$

With some math, we get $w=2.5$
$$\implies\qquad N=1.25(w-2.5)\qquad (5)$$

If we did not want to impose the restriction of linearity on the relationship between labor hours and real wage, we would simply use equation (4):

$$N=40-\bigg(\frac{50+20w}{w}\bigg)\qquad (6)$$

I know your question specified a linear relationship between labor hours and real wages, however this method of solving this question does not impose this restriction, which is nice because, as we can see, the relationship between labor hours and real wage is not actually linear.


Example of Law of One Price holds but No Arbitrage Fails

Examples where this happens are always extreme and contrived. I can think of two kinds of examples. The first is where you have an asset that for some reason has a price of zero or negative but a positive payoff in some states (perhaps no other asset pays off in that state). The second is where you have two assets with positive prices that have negative payoffs in all states (or vice versa).

You can come up with other varieties, but they're always a little weird. To show you how to do this, I'll be a little more precise. To answer this question, we need to be a little more clear about what arbitrage means. A lot of books give definitions for two kinds of arbitrage (e.g. see the asset pricing book by Darrell Duffie). Let me set up some notation first, though.

Notation

Let $p_i$ be the price of security $i$ with $p = [p1,...,p_N]'$. Let $D_{ij}$ be the payoff of security $i$ in state $j$ with $D$ being an $N \times S$ matrix, where $N$ is number of assets and $S$ is the number of states. Let $\theta_i$ be the number of securities of type $i$ that are owned, with $\theta = [\theta_1, ..., \theta_N]'$.

Some definitions

"The law of of one price": For any portfolio $\theta$ that satisfies $D' \theta = \vec 0$, it must be that $q' \theta = 0$.
"Arbitrage of the First Type": There exists a portfolio $\theta$ such that $p' \theta \leq 0$ and $D' \theta > \vec 0$. This means that all elements of the vector $D' \theta$ are strictly positive.
"Arbitrage of the Second Type": There exists a portfolio $\theta$ such that $p'\theta < 0$ and $D'\theta \geq \vec 0$. This means that all elements of the vector $D' \theta$ are nonnegative.

Examples

In this first example, consider the extreme case of the existence of only one asset. Suppose that $p = 0$ and $D=1$ (one state, but we can have more). Then the law of one price holds and arbitrage of the first type exists. This is pretty contrived, though.
Consider the example of $p = [1 \quad 2]'$ and
$$
D =
\begin{bmatrix}
-2 & -1 \\
-4 & - 2
\end{bmatrix}.
$$
This satisfies the law of one price. However, when $\theta = [-1 \quad 0]'$ or $\theta = [0 \quad -1]'$, we have $p'\theta < 0$ and $D' \theta \geq 0$ and $\theta \neq 0$.


Matlab: Solow Model Convergence of capital

Since, I was urged to present an answer for didactic reasons to which I totally agree, I will provide the full set of corrections to avoid any ambiguity.
% Model parameters
alpha = 1/3;    %
s     = 0.2;    % Investment rate
delta = 0.2;    % Depreciation rate
n     = 0.02;   % Growth of labor force
g     = 0.01;   % technological progress
eps  = 1;
k(1) = 0.5;
t=1;

while  eps >= 1.0e-5
    t = t+1;
    k(t) = (1/((1+n)*(1+g)))*((s*k(t-1)^alpha+(1-delta)*k(t-1)));
    z = k(t);
    z1 = k(t-1);
    kdif = z-z1 ;
    eps = max(abs(kdif));
   if t >= 300
     break;
   end
 end


How infinite Nash equilibria are possible in a game?

Well the intuition is quite straightforward: as you have mentioned yourself, player 2 is indifferent between playing any action in pure strategies, or randomizing in any possible way. So, for a mixed strategy equilibrium to exist, player 2 needs to play L w/ probability 4/7. If this is the case, it is irrelevant what Player 1 player---player 2's outcome is the same. As a result, in pure strategies the Equilibria are L,L and R,R and, in Mixed strategies, q=4/7 and p can take any value between 0 and 1. Hence, there exist infinite possible Nash Equilibria (p just has to obey the fundamental laws of probability).


Solving an OLG model, value function iteration vs. projection (chebyshev polynomial)

You could solve for the value function ex-post from the Bellman equation. For simplicity, consider a deterministic dynamic programming problem (in OLG model, you'd have age as another state variable, but I'll abstract from that). If $x$ is state and its next-period value $x'$ is control, $U(x,x')$ is one-period utility and you have somehow obtained decision rule $x' = g(x)$, it must be the case that the value function satisfies

$$
V(x) = U(x,g(x)) + \beta V(g(x))
$$

This is a functional equation in $V()$ that could also be solved by projection method. Alternatively, you could iterate on the Bellman equation (on some grid over state space) just like in VFI, but skipping the optimization step (since you already know the optimal choice), and eventually the value function should converge.


Has this differential calculus inequality approach to optimizing the production possibility curve exist?

I don't really understand your optimization problem as it stands. Just as a few examples, when you write out

$$\min \sum_m \sum_v \left( \frac{\partial \vec f_v (\vec Q_v)}{\partial \vec f_m (\vec Q_m)} - \frac{\partial \vec f_m (\vec Q_m)}{\partial \vec f_m (\vec Q_m)} \right)$$

I have questions about why you are minimizing the difference between optimal conditions for output that has all equal opportunity costs. The justification is unclear to me. The function $\vec f_v(\vec Q_v)$ also leaves me confused looking at your work. Why is $f$ taking a vector of $n$ aggregate outputs as its argument? $f(q)$ is defined as the adjusted cost of purchasing $q \cdot n$ resources, so how does $f$ transform a vector as opposed to a scalar and what interpretation are you trying to give it?

Or in your optimization problem when you write out

$$\frac{dy}{d\sqrt{r^2 - x^2}} \rightarrow - \frac{x}{\sqrt{r^2 - x^2}} = y$$

I have no idea what you are differentiating with respect to and how you arrived at the equality on the right. In step c.) later on you substitute $x = 0$ which doesn't have a justification either. (If it does, I think you should try to be more explicit about what is happening.)

I think as is, the method of finding an optimal point on the production possibilities frontier is cumbersome, and I don't think the optimization problem is well defined.

If you wanted to refine your method, you'd have to give

conditions under where an optimum exists (the conditions you state are not sufficient)
identification strategies for an optimum
a situation where this method is more practical than the industry standard

So for that last point, why can't someone just have a production function (where they can derive the PPF) and a budget constraint from prices, regularity conditions and then solve? Since you seem to have some familiarity with higher level math, you may find Chapter 5 and 15 of Mas-Colell, Whinston, and Greene a good resources for seeing the foundation of how production problems are approached. Chapter 15 in particular more explicitly talks about production possibility sets.


CES: Production function: Elasticity of substitution $\sigma = 1/(1 + \rho)$

The production function is:
$$q = (l^\rho + k^\rho)^\frac{1}{\rho}$$
The MPL and MPK are respectively:
$$q_l = \frac{\partial q}{\partial l} = \frac{1}{\rho} \cdot (l^\rho + k^\rho)^{\frac{1}{\rho}-1} \cdot \rho\cdot l^{\rho-1}$$
$$q_k = \frac{\partial q}{\partial k} = \frac{1}{\rho} \cdot (l^\rho
+ k^\rho)^{\frac{1}{\rho}-1} \cdot \rho\cdot k^{\rho-1}$$
What is the rate that l can be substituted for k?

Where $f$ is a differentiable real-valued function of a single variable, we define the elasticity of f(x) with respect to x (at the point x) to be
$$\sigma(x) =  \frac{x f'(x)}{f(x)}\equiv \frac{\frac{df(x)}{f(x)}}{\frac{dx}{x}}$$

Do a change of variables such that $u = ln(x)$ ($\rightarrow x = e^u$) and $v=ln(f(x))$ ($\rightarrow f(x) = e^v$)
Note that $v' = f'(x) / f(x)$ and $u'=\frac{1}{x}$ so that
$$\frac{v'}{u'}=\frac{\frac{f'(x)}{f(x)}}{\frac{1}{x}} = \sigma(x)$$
Note that this is also the result you get by solving for $ \frac{d ln f(x)}{d ln(x)}$ because  $ \frac{d ln f(x)}{d ln(x)} = \frac{d v}{d u}$ which we solve via the chain rule:
$$ \frac{d v}{d u} = \frac{d v}{d x} \cdot \frac{d x}{d u} = \frac{f'(x)}{f(x)} \cdot x $$
which happens to be exactly the definition of $\sigma(x)$.

Now let's tackle your elasticity problem.

$$ ln(\frac{q_k}{q_l})= log(\frac{\frac{1}{\rho} \cdot (l^\rho + k^\rho)^{\frac{1}{\rho}-1} \cdot \rho\cdot l^{\rho-1}}{\frac{1}{\rho} \cdot (l^\rho
+ k^\rho)^{\frac{1}{\rho}-1} \cdot \rho\cdot k^{\rho-1}}) = ln (\frac{l}{k})^{\rho-1} = (\rho-1) ln (l/k) = (1 - \rho) ln (k/l)$$
$$ \Rightarrow ln (k/l) = \frac{1}{1-\rho} \cdot ln(\frac{q_k}{q_l})$$

So $\sigma = \frac{1}{1-\rho}$


I feel like I do not understand the exact meaning behind the notion of
  the Pareto optimality.

It's not you.  There are different senses of the phrase "Pareto Optimal," and you have to figure out from context which one is being used.  The dictionary definition of Pareto Optimal is something like "An allocation from which any feasible change which makes any agent better off makes at least one agent worse off."  The word to keep your eye on is "feasible."

In particular, it makes a big difference whether or not you consider side-payments feasible.  In your example, if it is impossible to make side payments from "society" (i.e. whomever is harmed externally by the production of the good) to the producer, then $Q^\pi$ is Pareto Optimal.  Of course, with no side payments, every point between $0$ and $Q^\pi$ is Pareto Optimal.

Almost always, however, we implicitly assume that side payments are possible.   In that case, the point $Q^\pi$ is not optimal.  Why not?  Well, we could move from $Q^\pi$ to $Q^*$.  This would make the producer worse off by $C$ and "society" better off by $C+D$.  Now, if it is feasible to make "society" pay the producer some amount between $C$ and $C+D$, then we can say that moving from $Q^\pi$ to $Q^*$ while compensating the producer at least $C$ at the expense of "society" makes everyone better off.  The fact that you can move off of $Q^\pi$ while making everyone better off means that $Q^\pi$ is not optimal.

This actually leads into a common criticism of Pareto Optimality.  Normally, we just assume side payments are feasible.  In fact, you get so habituated to making the assumption that it becomes invisible to you.  So, you start to think (for example) that "free trade is Pareto Optimal" is a good argument for free trade.  And that's untrue.  The necessary side-payments are probably not feasible and are not actually made in practice.


Why would banks not want to lend, under Federal Reserve's Main Street Lending Program?

I saw these two answers on Reddit or Quora, but I can't find them now. Just edit this if you do.

Because the potential losses are much bigger than 5% if you read the program. Consider the pari passu clause. In a liquidation, what's the collateral that the Main Street loan has a claim to? Collateral that is securing any existing loans you’ve granted! Thus your loss can easily outstrip the 5%, as it eats into existing collateral and increases a bank's deficiency.
Even if the bank just loses 5%, the people needing this money are going to be very risky and banks can focus their time and efforts on way less riskier programs. Contrary to popular belief, a bank doesn’t just magically get bailed out with zero pain anytime they lose some money. It’s still a 5% loss which on loans of this size are significant once they get added up. Moneys have much less risky and better ways to sell their money right now.

That is why the loan program should have been more like FAFSA that use IRS data to verify businesses and grant them an amount. FAFSA get to pick a bank to service or another servicer. This is how the student loan program works. If the loans were based on credit alone then no one would get them. Doing this through the banks was a horrible idea. If you grant businesses an amount, then they are the customer that chooses the bank that gets the servicer fees as the loans are guaranteed. Not the other way around where banks are the gatekeeper to this money.
It is amazing how bad banks screwed up in the last bailout which they used just for themselves and to shore up capitalization/leverage. We were stupid to trust the banks this time ago, and by design are helping larger companies by the cronies.
The market is efficient. You have to set the right incentives and targets though.
Banks are in self-preservation mode. If they were cut out and had to go court customers with free money in guaranteed loans with windfall service fees from the bailout they would, though since we let banks be the gatekeepers to survival again they get to keep free money and they will cut risk.


Why would a common currency such as the Euro that can't be devalued be incentive for banks to lend money to a country?

When you lend Spain 10 million Deutsche marks, the loan is denominated in a foreign currency: they have to return you 10 million Deutsche Marks (ignoring interest for the moment). But the Spanish earn their income in local currency. If in the meantime their local currency depreciates relative to the Deutsche Mark, the value of the repayment becomes greater in terms of the proportion of their income (which is in local currency) and their ability to repay becomes smaller.

If you lend them Euros and they receive their income in Euros, a devaluation doesn't affect the (local) value of the debt.


Instrumental Variable Paper for Minimum Wage

Neumark and Wascher did a lot of work on the minimum wage in the 90s and 00s, including their 1992 piece "Employment Effects of Minimum and Subminimum Wages: Panel Data on State Minimum Wage Laws". A non-paywalled version of the article can be found here. It includes use of panel data (cross sectional data that you are interested in can be considered a special case of panel data) to study the effects of the minimum wage on employment.

From page 65:

  "Another possible source of bias arises from the potential endogeneity of state minimum wage levels. If state legislators choose the timing of minimum wage increases to correspond to periods in which minimum wage increases will have the smallest (or perhaps least noticed) disemployment effects, the estimated minimum wage effects in Table 2 will be biased upward..."

They too were concerned with endogeneity between the minimum wage and employment, so their solution was to use a fixed-effects specification and put in state and year effects in the first stage of the regression, with the instrument being "the [adjusted] mean of the minimum wage level in all geographically bordering states" (p. 65). Their data was from US states. So the setup for their study could be a useful starting point for you, or those interested in a similar question to this.

As Tobias mentioned, you can also use political variables as an instrument. In Lemas (2004), Political Variables as Instruments for the Minimum Wage, lags for the endogenous variable (min. wage) can be used with serially uncorrelated errors, but if that does not hold, you can use political measures such as influence-weighted politicians' support for or against the minimum wage, among other measures that can be found in section 4.2. of the paper.


Is a loan lent with a rate under the inflation rate free money?

Inflation: 2%
Interest:  1%

This means the real rate is 3%.
So no; not free money.

Free money would be negative interest rates.
Example: Inflation: 0%.
         Interest on loan I get from the bank: -1%.
This would mean they are paying ME 1% to take the loan from the bank.

Negative interest rates are a real practice, with many historical examples that can be researched.

The banks are theoretically losing 1%(+2% inflation) on the loaned amount, however it's important to understand that deals are made among negotiations with central banks in the background that can still produce gains from this.


There are two ways to organise social security. One scheme is known as fully-funded social security. The alternative is called pay-as-you-go.

Fully funded schemes

Under a fully-funded scheme, the government collects regular payment from each member. These payments are invested until the member needs to claim (e.g. their pension), at which point the member receives payments from the invested money and the returns that it has generated. This corresponds to the situation you describe where your social security payments are sat in an investment somewhere waiting to go back to you.

Pay as you go schemes

This may be disturbing reading, so brace yourself!

Countries such as the UK and the USA do not use a fully-funded scheme. Instead, a significant portion of their social security provision operates on the basis of a so-called pay-as-you-go scheme in which the money paid by today's workers (e.g. you) is not invested at all. Instead, it is immediately spent on making social security payments to people who are currently retired. The hope is that today's workers, when they eventually retire, will likewise have their pensions paid from the social security contributions of their children (who will by then be working). And so the chain continues, each generation handing money directly to the last with little provision for self-sufficiency.

Assessment

The obvious problem with this is that the generation of 'boomers' now retiring is unusually large and had unusually few children. Moreover, that generation enjoys unprecedented levels of longevity, meaning that they are likely to claim far more out of the system than they paid in. All this means that current workers face a large burden paying the social security claims of the much larger generation that preceded them.

However, we are kind of stuck. If we want to switch to a fully-funded system then somebody has to get a very bad deal. Option 1: current employees have to pay twice (for their own fully-funded retirement and for the pay as you go claims of their parents). Option 2: current workers pay only for their fully-funded retirement and current retirees (who already paid for their parents' pensions) do not receive any pension at all. Neither alternative is very politically tractable.


Is there collaboration between cyber criminals and banks?

The vast majority of cyber crime is fraud. The criminals don't hack the bank directly; rather they hack the customers of the bank and commit fraudulent transactions.

I strongly doubt there is active collaboration. Some criminals may have insider contacts, but that's different to a symbiotic relationship.

There may be an effect somewhat like you mention. Increasing security is expensive for banks, so it is only worth doing if they are experiencing a certain level of losses. In theory it would make sense for criminals to restrict their level of activity, to avoid banks implementing additional security, which would make their job harder. In practice I don't think this happens, because their are multiple cyber gangs and I strongly doubt there is a gentleman's code between then.

It is theoretically possible for someone to hack the bank itself, and directly modify balances - effectively creating money out of thin air. In fact, it is thought that some major nation states (mostly US and China) have penetrated many banks around the word. However, if they started to extract money, this is likely to be noticed, due to internal checks and balances. It is more valuable for these state-sponsored actors to silently remain present, capturing information. And one day, perhaps they will strike, as part of a larger war strategy.


How digital currency (like Bitcoin) proponents solve the problem of its high volatility to make it a good candidate to replace fiat?

tl;dr:
fiat money can be fully digital money as noted in the +1 comment of Jasper so it does not make much sense to talk about replacing fiat money with digital money as fiat money can be fully digital - those are two separate categories.
But I understand that your question is  about bitcoin/cryptocurrency not just any digital money. There are several proposals on how to make bitcoin competitive (as there are not many economists who think it could replace fiat money). These proposals address the biggest weakness of the cryptocurrencies which is inflexible money supply and range from pegging cryptocurrency value to some other currency to inflation targeting.
Full Answer:
Part 1: Clearing out Misconceptions
I think this question is based on confusion about what fiat money and digital money is so I will clear that misconception first. By definition, following Mankiw Principles of Economics pp 606:

fiat money:
money without intrinsic
value that is used as
money by government
decree

Fiat money can be both physical (for example paper notes, coins etc.) and can also be digital (money on you debit card). Also note, not all physical paper notes would be fiat money. Under gold standard where paper notes are backed by gold there would no longer be any fiat money because then the value of money is not created by government decree anymore. So whether money is fiat or not has no relationship to what medium is used as money (paper/bits etc.).
If you want a comparable term that distinguishes bitcoin from fiat digital currency, the term would be "fiduciary money". Fiduciary money is basically a 'version' of fiat money where the value is not created by government decree but solely by trust (again fiduciary money can be both in form of physical paper and digital). In fact this is also how economists categorize bitcoin. According to Velde (2013):

Bitcoin is a fiduciary currency
Fiduciary currencies—in contrast with
commodity-based currencies (such as
gold coins or bank notes redeemable in
gold)—have no intrinsic value, and derive their value in exchange either from
government fiat or from the belief that they may be accepted by someone else.
They are inherently fragile; government
orders can be ignored or doubted, and
a currency that has value only because
of the belief that it will have value may
have no value at all (for instance, if I
believe that no one will accept it, I will
not accept it either).
The term “mining” may lead one to
think that bitcoin is not fiduciary. It
is true that real resources (computing
hardware and energy) are expended
in creating bitcoins. Since entry to mining is free, the value of these resources
will be the market value of bitcoins produced (whose number per day does
not depend on the size of the network).
But once created, the bitcoin has no
value other than in exchange, contrary
to a gold coin.

Part 2: How to Solve the Problem of Volatility
First, it is important to understand little bit about what determines the price level in terms of some money, where 'value' of money is inversely related to the price level. For example, the US-bitcoin exchange rate ($S$) should be given by $S=\frac{P_{US}}{P_{Bc}}$, where $P_{US}$ is price level (aggregate prices) in terms of USD and $P_{Bc}$ in terms of bitcoin. This needs to hold otherwise if the equality would be violated one could just buy bundle of goods in bitcoin sell it for US dollars and then exchange dollars back to bitcoin for riskless profit (or vice versa depending on how the equality is violated). If there would exist such opportunity it would quickly disappear because as people would take advantage of the exchange the demand for dollars or bitcoin would change until it would reach equilibrium where the above would hold. So there is an inverse relationship between value of bitcoin and its price level.
Broadly speaking the price level for any money is determined by money market equilibrium.  In its simplest form this can be described by equation of exchange (See Mankiw Macroeconomics pp 87) as:
$$MV=PY$$
Where $M$ is the money supply, $V$ velocity of money, $P$ price level and $Y$ output (this is oversimplification, expectations of these quantities matter as well but since you are not an economist I wanted to only use most basic models to provide structure). Solving for price level and log-linearizing we get:
$$\ln P = \ln M + \ln V - \ln Y $$
So if money supply would increase by $1\%$ you would expect price level (decrease in value) to increase by $1\%$, holding all else equal, the same for velocity and for real output you would expect decrease in price level (of course in real life there might not be 1:1 proportionality but again this is just to provide some structure for our thinking - in addition with bitcoin which is in some sense close to commodity money the production costs of mining new bitcoin would play a role as well, but lets assume that away for now for simplicity).
Here you can nicely see what the problem with cryptocurrencies vis-à-vis standard fiat money is. Most of cryptocurrencies  have a very inflexible money supply. For example, money supply of bitcoin cannot normally decrease - there is no way how you can 'un-mine' properly mined bitcoins, and there is hard cap on what maximum number of bitcoins ever will be. This means that for bitcoin $M$ will eventually become fixed. This means that its value (inverse of the price level) will be always at mercy to the shocks to velocity or real output (and in more complex models expectations of those). In fact according to the Iwamura et al (2019), the inflexibility of money supply is precisely the main reason what causes the price instability of crypto currencies.
Fiat money monetary system does not have such problem as central bank can, at least in principle, always choose what sort of $M$ it wants, and those central banks that have mandate for stable prices try to do so in a way that keeps prices stable (although usually this is done in indirect way through interest rates as opposed to directly choosing some $M$ - but in principle they could choose any $M$ if they  would want to).
Now finally we can turn to your main question:

How digital currency proponents solve the problem of its high volatility to make it a good candidate to replace fiat?

Well coincidentally there is whole paper that tries to address this question, namely:
Iwamura,  Kitamura, Matsumoto,  & Saito (2019). Can we stabilize the price of a cryptocurrency?: Understanding the design of Bitcoin and its potential to compete with Central Bank money
They basically argue that in order to make bitcoin or other cryptocurrencies competitive there must be some way how to make supply of money flexible both upward and downward. Their potential fixes include:

Currency Peg/Currency rule: The cryptocurrency could be pegged to some  real life currency, where there would be some rule that would adjust money supply in a way to always keep exchange rate between lets say USD and Bitcoin or euro and Bitcoin fixed. Alternatively, there could be some less restrictive rule where bitcoin would have to follow other currency within some band.

Monetary Policy without a Central Bank:  you can simply have some monetary rule (some argue central banks follow monetary rules such as the Taylor rule) that affects money supply by making it easier or harder to mine bitcoins that depends on some parameters in economy (for example, real output).

Inflation Target in Cryptocurrency: crypto currency money supply could be set up in a way that it always targets lets say $2\%$ inflation (or zero inflation - which would be perfect price stability, but most economists argue small positive inflation is vital for economy (see more on that here).

However, the above being said I do not know of any conventional economist who would think that cryptocurrencies should replace fiat money. Most macroeconomist believe that state should manage business cycle using, among other tools, monetary policy. In addition to that cryptocurrencies have further problems as mentioned in the paper cited above:

First, cryptocurrencies are more expensive to produce, and the production costs are hard to retrieve. Bank notes issued by the central banks require some printing and material costs. These costs are negligible compared with the face (nominal) value.
Second, bank notes are reversible between new issues and absorption because the central bank basically buys and sells securities with bank notes. A cryptocurrency cannot be absorbed, but if equipped with a built‐in value stabilization mechanism, this shortfall of irreversibility can be softened (but not eliminated) in practice.
Third, Bitcoin‐type cryptocurrencies use a delayed finality confirmation structure to avoid double spending. Consequently it typically takes hours to use obtained money. Bank notes can be used  immediately as obtained. Fourth, Bitcoin type cryptocurrencies face security risks, such as Denial of Service attacks, more widely than bank notes.

Some of the above can be aleviated further, as authors explain:

The third and fourth points are relative problems, and also intrinsic to Bitcoin‐type currencies. They are not general problems with cryptocurrency. The third point considers a problem directly compared with bank notes for direct transactions. Considering transactions with Bitcoin‐type currency may occur over a remote distance, finality confirmation may be quicker and much cheaper with a Bitcoin‐type currency than that through a bank. The fourth point is closely related to the protocol design of a Bitcoin‐type currency, and is not a general cryptocurrency issue. The instability associated with mining pools due to strategic behaviors between and within pools can be reduced substantially if the valuation system is improved in line with our suggestions.

However, the first and second point are inherent problems of cryptocurrencies, as authors further explain:

The first and second points are fundamental shortfalls of cryptocurrency. As currently described, cryptocurrency values are based on associated production costs. This mechanism is similar to commodity money, notably gold and silver coins. Historically gold and silver coins have been replaced by credit (or fiat) money basically because of the above‐mentioned first and second points.

Consequently, at best it is reasonable to aim to make crypto currencies competitive vis-à-vis fiat currency, rather than as their replacement.


Here is a fairly instructive figure showing a cross-country measure of income inequality:

It shows that inequality was increasing for most of the last 35 years, but has started to decrease again recently. (You can read more at the source for this figure: http://www.conferenceboard.ca/hcp/hot-topics/worldinequality.aspx)

Once data is weighted by population, we see a longer-term trend for decreasing inequality (due to rapid growth in China and India, which contain around a third of the world's people:

As @FooBar notes in a comment, a pattern of decreasing income inequality is consistent with economic theories of convergence.


Kelly criterion wealth allocation alternatives

Yes.  I just published a paper deriving the distributions of returns for all asset and liability classes.  It isn't one distribution or one family of distributions.  Stocks are a mixture of multiple distributions.  Some of these distributions have covariance structures, and some do not.  You may have to do a bunch of your own math though.  I have been contemplating writing a Kelly bet paper, but have not had the time.  I am about to present a distribution-free replacement for Ito calculus that does not assume the existence of first or higher moments at the Southwestern Finance Association Conference.  As such, I have been busy.  It is nearly independent of the economic model in use.  I say that because it would cover a model built around panda bears if that were the true model in nature but nobody knew it, but it may not hold if certain implicit biological assumptions became untrue in the future about humans.

What you will want to do is account for bankruptcy risk, cash merger risk, stock-for-stock merger risk and liquidity risk in addition to return risk.  There are distributions for each in the paper.  Because a Kelly bet is equivalent to having logarithmic utility, my recommendation is to solve it as a log problem.

The difficulty you will find in the paper is that for stocks, there is no equivalent to a covariance matrix.  As you add dimensions, you do not add terms for the scale parameter.  If you ignore liquidity, the limitation on liability, bankruptcy and mergers then the returns for one asset are: $$\frac{1}{\pi}\frac{\sigma}{\sigma^2+(r-\mu)^2},$$ but for two assets it becomes $$\frac{1}{2\pi}\frac{\gamma}{(\gamma^2+\sum_{i=1}^2(r_i-\mu_i)^2)^{3/2}}$$ and for three assets it becomes $$\frac{1}{\pi^2}\frac{\delta^2}{\sqrt{\delta^3}\left(\delta+\sum_{i=1}^3(r_i-\mu_i)^2\right)^2}.$$

Notice that the scale parameter changes as assets are added which I denote by simply changing the variable name $(\sigma,\gamma,\delta)$.  This is really challenging to work with because of the poor form.  Also, I didn't work out the correct constant of integration since I didn't want to sit around working out the integrals under the limitation of liability.  It is late, and I teach in the morning.  These assume up to infinitely negative returns.

There is a practical alternative as well here, and you should assume no correlation as well.  Graham and Dodd investing provide the clue.  If you assume an all-or-none utility function, that you receive a utility of one if your return exceeds a threshold and a zero otherwise, then you get a simple answer.

First calculate the Kelly bet on a binomial, which is trivial, for your threshold rate of return.  That becomes your allocation level.  Second, as price falls, the probability of meeting the level increases and so your variance falls.  So pay as little as possible for a bet.  As you are reducing your price, you are also exposing less money to each bet, so you are reducing your risk in three ways, both through the probability of loss is reduced, the variance is reduced and the exposure size being reduced per unit of risk.

You can find the paper at

  Harris, D.E. (2017) The Distribution of Returns. Journal
  of Mathematical Finance, 7, 769-804.

The paper is built around the observation that questions such as $\text{var}\left(\frac{S_{t+1}}{S_t}\right)$ are built around two observable stock prices.  This implies that stock prices and stock volumes are data, but returns are a function of data and therefore a statistic and not data itself.

Since the return is future value divided by present value minus one, it follows that returns are a ratio distribution of future prices divided by current prices.  Under mild assumptions about how stocks are traded and the number of investors involved, stock prices should be normally distributed at any static moment.  More precisely, the limit order book should be.  These distributions are just the ratios of two normal distributions for each stock.  It also nicely fits the data if you make adjustments for things like bankruptcy and so forth.

By converting your problem to a binomial you tame it and put yourself in Warren Buffett's shoes with a mathematical explanation as to why that is the case.  The downside is you now have all the terrible reading to do of annual reports as you have to restate them from their official values to their economic values for this to work.  So a ten year deferred tax liability for $100 million needs reduced to present value since this is an interest-free loan from the Treasury and the difference added to equity.  Still, once you do the analysis, you can make a good return.

Also note that these distributions lack both a mean and a sufficient statistic. In the real world they are truncated at -100% and so no non-Bayesian solution exists for any of these that is unbiased and admissible.  Don't plug in your R function. It won't work.  You have to build this from scratch if you are going to make predictions.  Nicely, Bayesian predictions are of the form:

$$\Pr(\tilde{x}|\mathbf{X})=\int_{\theta\in\Theta}\Pr(\tilde{x}|\theta)\Pr(\theta|\mathbf{X})\mathrm{d}\theta,$$ so you can form predictions that do not depend upon the value of the parameters.  Use the CDF of $\tilde{x}$ to form your binomial density.

Technically, this is not quite correct because the multi-asset form is not independent.  No two returns can ever be independent, even if there is no correlation.  Still, it is a reasonable heuristic.


How to auction shared item with a roommate?

If you think that the distribution of your value and your roommate's value are the same then I would suggest that you consider the following bidding protocol:

each bidder submits a sealed bid, $b_i$.
the highest bidder, $i$, received the console and makes a payment of $\frac{b_i+b_j}{2}$ to the loser, where $b_j$ is the loser's bid.

Cramton, Gibbons, and Klemperer (Econometrica, 1987; Section 4) show that this is an efficient way to dissolve a partnership with joint ownership stakes in an asset.

If you think that the two of you have differing value distributions (e.g. because your roommate has a new friend) then Cramton et al. show in Theorem 2 how to adjust the pricing rule to restore efficiency.

I highly recommend taking a look at the Cramton et al. paper. It's surprisingly readable for an Econometrica.

Let us know how you get on if you decide to try it out!


When do the curves touch the axes and when don't they?

Broadly, the answer to your question is it depends on the context. Generally, if you have some sort of functional form for the curves, you can tell whether they touch the axes by seeing if there is an intercept on either the P or Q axis (set P = 0 to see if there is a Q-intercept, and vice versa).
So, for example, if you're working on a monopoly problem that assumes a linear demand function of the form $Q_d = a - bP$, the demand curve will intersect the P-axis at $P = a/b$ and the Q-axis at $Q = a$.
It's pretty standard to assume that Q is never negative, so you'll generally never see any curve go to the left of the P-axis. For some curves (like supply), it doesn't make sense for the curve to exist when P is negative, so those won't go below the Q-axis, while for others (like MR), that can still make sense.
But, if you're just doing a general analysis to see what happens conceptually when, say, demand shifts to the right, or supply shifts to the left, then it's irrelevant whether the curves intersect the axes or not. If you don't know that a curve crosses an axis (like from the functional form), it's usually best not to assume either: don't assume it crosses, but don't assume it doesn't cross.


Can a market function if everyone has the same preferences

Even if everyone's preferences are exactly the same, as long as you leave diminishing marginal utility (e.g. each person would rather have nonzero proportions of apples and bananas rather than exclusively apples or exclusively bananas) and the advantage of division of labor (e.g. a farmer can produce more apples or more bananas if he focuses exclusively on growing one of those rather than trying to grow some of both), there will still be trade because people will want to maximize the value they accumulate by specializing their labor, but possess that value in the form of a wider variety of goods than their (efficiently allocated) labor produces.


Conditions for an additive value function

This is only a partial answer because it does not exactly fit your framework, but I hope it will still be helpful (and it's too long for a comment).

If you are ok with discretizing your cake into (possibly arbitrarilly small) pieces of cake, then you will find an answer in

Kraft, C. H., Pratt, J. W., & Seidenberg, A. (1959). Intuitive Probability on Finite Sets. The Annals of Mathematical Statistics, 30(2), 408–419.

the bulk of which is very well summarized in the introduction of

Fishburn, P. C. (1996). Finite linear qualitative probability. Journal of Mathematical Psychology, 40(1), 64–77.

Althought the setup of the papers is in terms of probability jugments, it can be reinterpreted from a preference point of view as follows :

A finite set of objects $S = \{1,2,\dots,n\}$ (in your problem $S$ could contain the pieces of the cake)
A preference relation $\succeq$ over $2^S$ the set of subsets of $S$.
The question : when is $\succeq$ representable by an additive utility function $U$ on $2^S$.

A classical conjecture by de Finetti's was that the following conditions should suffice (here I follows the presentation in Fishburn (1996)):

(Order) : $\succeq$ on $2^S$ is a weak order,
(Nonnegativity) : $A \succeq \emptyset$ for every $A \in 2^S$,
(Nontriviality) : $S \succ \emptyset$,
(Additivity) :  For all $A,B,C \in 2^S$, if $(A\cup B) \cap C = \emptyset$, then $[A \succ B] \Leftrightarrow [(A\cup C) \succ (B\cup C)]$.

de Finneti observed that these were necessary but could not determine whether they were sufficient. Eventually, Kraft, Pratt & Seidenberg (1959) provided a counter-example as well as an additional condition which, together with the four others implied the existence of an additive representation:

(Strong additivity) : for all $m\geq 2$ and all $A_j,B_j \in 2^S$, if $(A_1,\dots,A_M)$ and $(B_1,\dots,B_M)$ contain the same number of replicas of each elements of $S$ (i.e. if $s_1$ appears three times in all the $A_j$ sets, it also appears three times in all the $B_j$ sets, etc) and $A_j \succeq B_j$ for all $j<m$, then we do not have $[A_m \succ B_m]$.

The last condition is often referred to in the literature as the "cancelation" property. Now (Strong additivity) is not the most intuitive condition. In general, it can be hard to to check and navigate, which has spurred a large literature on alternative sufficient condition. I can send you a reading list if you are interested. Unfortunately, I don't remember of any paper directly tackling preferences over subsets of infinite sets, like your real interval.

From my experience with these kinds of problems, changing the domain over which preferences are defined makes a huge difference in terms of the results that hold and the proof techniques you can use. If a result is not already out there in the literature, it is rarely easy to derive it from apparently similar results on different domains.


Why would overemployment mean that labor is used inefficiently?

Because using efficiently does not mean that all labor is used (this holds also for any other factor of production). This is so because on the margin at some point additional inputs might not produce enough value to justify the cost.

Consider simple supply and demand on perfectly competitive labor market with no market failures and all standard background assumptions.

Supply is: $S_L=10+w$ where $w$ is wage. Also let’s suppose there are enough people so that maximum 100 labor hours ($S_L \leq 100$) could be supplied after that there simply is no more labor hours due to lack of people.

Demand for labor hours is: $D_L=100-w$.

Now given the supply and demand it’s trivial to calculate that equilibrium wage is $w=45$ and equilibrium employment will be: $S_L=D_L=55$.

This equilibrium is completely efficient yet only 55 hours of labor is demanded. Employing 60 or 70 or 100 hours of labor would just reduce total welfare as measured by total surplus, and hence it would be inefficient.

Intuition is that because given the firms demand which is derived based on their own optimal production decisions given the prices they face for their product the equilibrium wage is too high to justify employing everyone. Employing more than 55h of labor would start destroying value as those workers would have to be applied to activities where simply the value of output is not sufficient to cover the cost of that labor. Of course, there could be situations where all labors would be employed if equilibrium demand is equal to maximum number of labor hours there is or exceeds it.


How to calculate the intercept of an indifference curve in an Edgeworth box?

In an edgeworth box model the total amount of goods must be constant. In your problem this translates to

\begin{eqnarray}
x_{A_1} + x_{B_1} &=& \omega_{A_1} + \omega_{B_1} = 10 + 4 = 14 \tag{1}\\
x_{A_2} + x_{B_2} &=& \omega_{A_2} + \omega_{B_2} = 6 + 2 = 8 \tag{2}\\
\end{eqnarray}

So let's consider the first intercept

$$
(x_{A_1}, x_{A_2}) = (0, 38/5) \tag{3}
$$

which you properly derived. If you replace these values in Eq. (1) you get

$$
0 + x_{B_1} = 14 ~~~\Rightarrow~~~ x_{B_1} = 14 \tag{4}
$$

and if you replace in Eq. (2) you get

$$
\frac{38}{5} + x_{B_2} = 8 ~~~\Rightarrow~~~ x_{B_2} = \frac{2}{5} \tag{5}
$$

So there you have it

$$
(x_{A_1}, x_{A_2}) = (0,38/5), ~~~(x_{B_1}, x_{B_2}) = (14,2/5)
$$

You can do the same for the other intercept, I will leave that for you to complete


Intuition for effect of interest rate changes in New Keynesian model

Yes the 'standard story' you reference is the correct intuition here. Lower nominal interest rate encourage consumption which puts pressure on both output and prices.
Household can actually save and borrow in the model. However, because it is a representative household model in equilibrium there would not be any borrowing - the interest rate should just adjust in order to make agent's indifferent between consuming today and in the future - but the intertemporal consumption choice still depends on $i$ with higher $i$ incentivizing  future consumption at the expense of the present one.
Also, even if in the description provided in question this is not shown the Euler equation has its micro foundation in model where households are allowed to save and borrow - otherwise the interest rate would not be part of the Euler equation in the first place, and I am not sure how it would even be modeled if there would be no link between present and future. For example, see the treatments of IS-LM models in lets say Woodford (2003). Interest and Prices. You will see that in dynamic models the Euler equation is typically derived from model of household behavior where households have access to saving/borrowing.


Assuming the auction has a pure strategy equilibrium, we can set up the problem as follows:

Each bidder has a value $v_i\sim F$ on support $[\underline{v},\overline{v}]$. The symmetric equilibrium, we posit, is an increasing continuous function $b(v)$.

Now, imagine bidder $i$ bids as if his value were $\widetilde{v}$. Then he wins iff $v_j<\widetilde{v}$ and $i$'s expected payoff is therefore

$$U_i=-[1-F(\widetilde{v})]b(\widetilde{v})+\int_0^{\widetilde{v}}\! \left(v_i-b(v_j)\right)F'(v_j)\,dv_j$$.

The first term is the payoff if $i$ loses (he then pays his own bid, which is $b(\widetilde{v})$ sinc $i$ is using $\widetilde{v}$'s bidding strategy). The second term is the payoff if he wins (he pays his rival's bid, provided his rival's value is less than $\widetilde{v}$).

Now we compute the best $\widetilde{v}$ for $i$ to choose by computing a first-order condition:
$$\frac{\partial U_i}{\partial \widetilde{v}}=-[1-F(\widetilde{v})] b'(\widetilde{v})+v F'(\widetilde{v})=0.$$

We know that, in equilibrium, it must be optimal for $i$ to choose $\widetilde{v}=v_i$ (i.e., it must be a best response for $i$ to use his intended strategy rather than deviate to someone else's strategy):

$$-[1-F(v)] b'(v)+v F'(v)=0.$$

Thus,
$$b(v)=b(\underline{v})+\int_{\underline{v}}^v\!\frac{xF'(x)}{1-F(x)}\,dx.$$

We can check that $$b'(v)=\frac{vF'(v)}{1-F(v)}>0$$ as required. All we need is a boundary condition to pin-down $b(\underline{v})$. For example, we might think it natural that $b(\underline{v})=0$ because a $\underline{v}$ type knows he can never win the auction.


What textbook can I use to really understand microeconomics, macroeconomics and mathematical economics

Your question is quite broad, but here are a couple of (open) book sites you might get started with:
https://open.umn.edu/opentextbooks/SearchResults.aspx?subjectAreaId=4
http://www.freebookcentre.net/Business/Economics-Books.html

When you're familiar with the basics, you might ask the question again with a more specific topic you'd like to study.

Expertise on economics (or pretty much any academic subject) is gained through years of studying and thought. It's, at least to me, a fascinating way of thinking about the world.


Changing the base year of Real GDP from 2001 Euros to 1990 Geary–Khamis dollars

In general, real output is  calculated like this:
$$RGDP_{t, b} = NGDP_t \cdot p_{base} / p_t$$
Where $NGDP_t$ is nominal GDP at time t, $p_t$ is the price level in time $t$ and $p_{base}$ is the price level in the base period.

To covert RGDP between base periods, note that the ratio to a common year's $RGDP_t$ in two different base years is:
$$RGDP_{t, b1} / RGDP_{t, b2}  =  p_{base1} /  p_{base2}$$

You can rearrange this equality to solve for what you want:
$$RGDP_{t, b2}  =RGDP_{t, b1} \cdot p_{base1} /  p_{base2}$$


Why does supply equal demand? Problem if large increase of price, decreases demand only by a bit

"Sellers" as a whole indeed do not have a reason to decrease the price under these conditions. But sellers don't decide together collusively, acting like a monopoly seller. Instead, each of the many individual sellers decides on its own. And each does have an incentive to lower its price, because by posting a lower price it can suddenly sell all of its individual supply (as long as this is below total market demand), which always increases profit. In other words, even if the market demand curve is very inelastic, the demand facing a single seller is highly elastic. It is therefore profit maximizing for an individual seller to slightly lower its price. If all others follow suit, the market price has decreased. This continues until equilibrium is reached.


In log-linearized New Keynesian model, what do $Y$, $Y_t$, $y_t$ actually mean?

The following post explains in a somewhat easier way what exactly is happening when we log linearize a model.

http://economictheoryblog.com/2012/06/22/latexgx_t/

Going through the provided example should make it clear what the single steps are.


Estimating the cost of an assett that needs to be replenished

The retail price should include the total net present value of use of the item. At any given point in time, the value of the item is the purchase cost minus accumulated depreciation. Depreciation should include both physical wear and any non-physical types of depreciation such as obsolescence.


Usually one tries to construct a Social Welfare Function (SWF), i.e. a general rule how to aggregate individual preferences to a social preference relation. Then various axioms are formulated that a useful SWF should obey, and some such SWF is selected. A social welfare maximizing alternative would then be one which is weakly preferred to all other alternatives by society.
However, if there are at least three alternatives and the axioms include Universal Domain, the Pareto Criterion, and Independence of Irrelevant Alternatives, then Arrow's Impossibility Theorem shows that the only SWF obeying all three of these axioms is a dictatorship.


From a practical standpoint, bidding wars are effectively informal auctions. What is usually done is that the buyers inform their agents of their maximum price, and the buyers' agents tell the seller's agent "This is my client's bid, but please let us know if we're outbid." Sometimes this is formalized through what is known as an escalation clause, which is essentially an offer to bid some increment more than any other highest bid, up to some reservation price.

Some sellers accept escalation clauses, some don't, and this decision appears to be influenced partially through strategy and partially through custom in the area. Not accepting an escalation clause can force buyers into what is effectively a first-price sealed-bid auction, while accepting them turns it into more of a traditional (English) auction.

As an additional note, the point that @desnep makes is true— while proof of other bids is required, it's not always possible to determine that those bids in fact represent genuine offers to buy. My understanding is that for this reason, when bids are expected to be few, buyers are encouraged to submit fixed bids and be prepared to walk away if they lose out, as there's a stronger incentive for sellers to fabricate bids (i.e., market manipulation is more effective in less-liquid markets).


Nonparametric estimation: Interaction of two continuous variables

(I misunderstood the question at the beginning)
The model is a binary non parametric regression.
I don't know of a general model, I know of a semiparametric model by Klein and Spady
http://www.ssc.wisc.edu/~bhansen/718/NonParametrics7.pdf
The model is as follows let $x´=[x_1´ \quad x_2´]$
Now $P(y=1|x)=E[y=1|x]=g(x'\beta)=f(x)$.
The details in the link.
Another possibility is to estimate the joint distribution (that has both discrete and continuous variables). Let g(y,x) be the joint pdf, then estimate $g(y|x)=\frac{g(y,x)}{g(x)}=f(x)$. The estimation of a joint of this type needs a product kernel that you can check
Nonparametric estimation of regression functions with both categorical and continuous data
Journal of Econometrics, Volume 119, Issue 1, March 2004, Pages 99–130
Jeff Racine, Qi Li

For $y$ continuous the following answer works.
First of all is your data deterministic or random.
If deterministic you may wish to do interpolation. Splines are your best option.
If you think your dataset has noise or you have unobserved variables then you can do a non-parametric regression, I think this is what you want.
The first step is to actually make assumptions about the structure of the randomness.
The usual assumption is that
$y=f(x_1,x_2)+\epsilon$ and you let $x´=[x_1´ \quad x_2´]$ be orthogonal -indepdent- of the unobserved variable $x\perp\epsilon$. Then you can replace $f$ by some high degree polynomial on $x$ and estimate this using Non linear least squares, technically this is called series or polynomial estimators (NLS is used when you know f, but you can argue that a high degree polynomial approximates the true f).
Another way is using a ¨kernel¨ approach.
In fact  you can directly assume that f is the conditional expectation.
$E[y|x]=f(x_1,x_2)$. In that case you can estimate it using the joint distribution recall the definition of conditional expectation
$E[y|x]=\int yf(y|x)dy$, with $f(y|x)=\frac{f(y,x)}{f(x)}$. Replacing this elements by its kernel estimators you get the Nadaraya-Watson estimator, you can google it.
A third option is to use local linear regression. A more advanced option is to use Sieves estimators that use function basis to estimate the model.
If f is not the conditional expectation then you have to use non parametric regression with instrumental variables, this is more complicated there is a paper by Renault (and others) on Econometrica about this.
In summary it depends on the randomness structure that you are assuming.
The book to go if you want to learn non-parametric regression is Li & Racine http://press.princeton.edu/titles/8355.html
There is also a nice implementation of these econometric procedures in the open source statistical language "R" under the name (package) "np".


What is the name of the economics free trade "punnett square"

Perhaps you mean an Edgeworth box?
Here is an example demonstrating Ricardo's theory of comparative advantage using an Edgeworth box:

Source: Policonomics

This shows that through trade both Portugal and England can have more wine and more cloth after opening up trade between the countries. It also shows that even if Portugal is more productive than England at producing both commodities then trade is still improving (that is what makes it comparative and not absolute advantage).

Alternatively, perhaps you mean a 2x2 table like this which shows the labor needed to produce each good:

Source: KALYAN CITY LIFE

To my knowledge this table has no special name.


Why do low labour and wage flexibility forestall a single currency?

It's probably referring to the standard OCA theory of the tradeoff between symmetry and flexibility, but in the latter flexibility is usually one index:

The theory of optimal currency areas (OCA) has created a set of ideas that has an significant influence on the governance of the eurozone and on views about how this governance should be strengthened in the future. The best way to make this clear is to present the core of the OCA theory, using a well-known graphical representation of this theory (see De Grauwe (2014)). This is done in Figure 3. On the horizontal axis we set out the degree of flexibility in the labour and goods markets.
This measures the degree to which wages and prices adjust freely to shocks and the degree to which workers are mobile. We assume that these different dimensions of flexibility can be represented by one index. On the vertical axis we set out the degree of symmetry between countries, i.e. the degree of co-movement (correlation) of macroeconomic variables such as output and employment. Thus, when there are a lot of asymmetric shocks we move downwards along the vertical axis. By contrast, when shocks become less asymmetric we move upwards along the vertical axis.
The downward sloping OCA line represents the trade-off between symmetry and flexibility. Hence, when the degree of symmetry declines (there are more asymmetric shocks) countries in a monetary union need more flexibility to deal with these shocks.

The OCA-line separates the space into two zones. The OCA-zone above the OCA-line contains the collection of points at which symmetry and flexibility are high enough to guarantee that the benefits of the monetary union exceed the costs. The points below the OCA-line are the points at which symmetry and flexibility are too low, i.e. countries located in that zone will find that the costs of the monetary union exceed the benefits. The OCA-line that separates the two zones can therefore also be defined as the collection of points for which the benefits and the costs of the monetary union are equal.

So they probably mixed up usage a little and instead of wage flexibility and labor mobility they referred to labor flexibility in the 2nd term. I suppose the latter may encompass the former for some broad meaning... but then it also includes wage flexibility.
And if the quoted theory above isn't too accessible, a simple illustration is to consider Greece: its economy was quite unlike Germany's. So in order to respond to the (asymmetric) shock the Eurozone experienced as a whole via the Greek problem, Greek workers had to take lower pay or migrate to other EU countries. And they did both to some extent. Of course, the flexibility question/measure is how fast these two reactions happen in response to a shock.
Finally, labor mobility works better as an automatic stabilizer if you also have  federal taxes; without these you need some other mechanism to transfer money from the income of the workers that have migrated back to the problem area of their origin.


Are lexicographic preferences strictly monotonic?

It seems you are not confused about part (1) because that's exactly what it means.

Lexicographic preferences are monotone. Monotonicity means more is better. If I have more of every good in the bundle, then I like that bundle more. This is still true for lexicographic preferences, even though parts of the bundle may not matter. A condition for monotonicity is e.g. that (5,4) is preferred to (3,1). In other words if 5>3 and 4>1 then the first bundle is preferred. This is true for lexicographic preferences.

Your definitions of strict monotonicity and lexicographic preferences seem to deviate a bit from conventional definitions though. Here is a proof using your definitions.

Proof:

Let us compare two bundles with $n$ elements each: $x=(x_1, x_2, ..., x_n)$ and $y=(y_1, y_2, ...,y_n)$

It is best to start with the if statement when going about proving such claims. The condition and therefore starting point is $x_1>y_1$ and $x_2>y_2$, .... , and $x_n>y_n$.

In that case since  $x_1>y_1$ we have according to your preference definition that bundle $x$ is preferred to bundle $y$. Furthermore we have that bundle $y$ is not preferred to bundle $x$. So bundle $x$ is strictly preferred.

Hence, we have shown that if $x_i$>$y_i$ for each $i$, then $x$ is preferred (because in that case $x_1 > y_1$), which is your definition of monotonicity. The preferences are therefore monotonic.


Did Elinor Ostrom (or others) resolve the tragedy of the commons for common-pool resources?

Depends on your definition of 'resolve'.  In economic there were already proposed solutions for both public good and common-pool resources. However, most of those require some sort of government action, for example Pigouvian taxes/subsidies or in some cases government ownership/provision.
To be more specific before Ostrom and others, literature split goods mainly as either public or private goods (see Samuelson 1954), where it was argued that private goods can be provided by the market and public goods require 'top down' government involvement in one way or the other.
Ostrom's (and her coauthors) contribution was mainly twofold:
First, she showed that actually there is more nuance to this and that there are more dimensions to goods than simply being public or private as the matrix in your question shows (Ostrom 2005). To be fair the category of 'club goods' was already introduced by Buchanan (1965), but she added the idea of subtractability and the fourth category (Ostrom & Ostrom 1977).  Furthermore, she also showed that incentives people face between the public good and common pool resources are different and not similar as previously thought.
Second, she showed that actually once you split public goods into multiple categories, there is a category of common pool resources where 'top-down' government involvement is not necessary and the resources can be managed privately (where by privately in this case is mean to mean without government involvement) or with 'bottom-up' local government (Ostrom 1962; Weschler 1968; Warren 1966; Ostrom 1965, Ostrom 1990, Ostrom 1999, Ostrom 2005). However, her work was not really focused on developing solutions to these problems as much as describing and empirically examining how different communities manage to find solutions to these problems, and those solutions are often case and place specific (for example, the Netherlands where I live has a specific parallel local government for management of dams that is virtually separated from central government with its own local elections and separate structure called Water Boards that would likely qualify as the 'bottom-up' local solution Ostrom was advocating).
However, her work does not really contain specific proposal of resolution, rather as she herself put it in her Nobel Prize lecture:

The most important lesson for public policy analysis derived from the
intellectual journey I have outlined here is that humans have a more
complex motivational structure and more capability to solve social dilemmas
than posited in earlier rational-choice theory. Designing institutions to force
(or nudge) entirely self-interested individuals to achieve better outcomes has
been the major goal posited by policy analysts for governments to accomplish
for much of the past half century. Extensive empirical research leads me to
argue that instead, a core goal of public policy should be to facilitate the development of institutions that bring out the best in humans.

Hence, her solution was to put it simply to just create framework that allows people on local/private level  to find their own unique and specific ways of resolving the problems, which will likely be different depending on local context (e.g. culture etc).
However, I would be hesitant to say that this was a resolution of the tragedy of commons for three reasons:

If you actually take a deep look at the models that study/describe 'resolutions' that arise privately/locally from 'bottom-up' they often show that although this is substantially better than situation without any mechanism or more efficient than government 'top-down' solution they are still not always provided as efficiently as could be done if there would be no such problem (e.g. as in case of private goods). To see that you can have look at the Mueller Public Choice III - the most comprehensive textbook on the public choice (i.e. area of economics that deals with these issues). This is not to say that this would not be the best solution given alternatives, but it does not fully eliminates the problem. To give an analogy if you loose leg, having some modern prosthetic is more efficient than living without leg, but to the extent that the prosthetic is not as good/better as original leg I would be hesitant to call it a resolution of the problem.  In fact Ostrom (1999) herself often describes these solutions as 'coping with the tragedy of commons' rather than 'solving the tragedy of commons'.
Since you wanted to discuss Ostrom, most of the above focuses on her work, but while her work is immensely appreciated, and her Nobel Prize was definitely deserved, it is fair to say that most mainstream economist would argue for much greater top-down government involvement with Pigouvian taxes or cap-and-trade even in cases of common pool resources which she criticized (although here the point 1 would still apply but for different reasons - e.g. it is very hard to know what the optimal Pigouvian tax is).
Rather than developing solution to the tragedy of commons it  was an observation of how people manage to overcome it privately or on local decentralized level of government. This is by no means critique of her work, she was obviously excellent scholar, but saying she resolved the problem would be like saying that Adam Smith resolved the problem of low productivity problem by division of labor, whereas he rather provided an explanation and advocated for something he observed by visiting manufactures and the division of labor was 'invented' as a 'solution' to increase productivity by people not him.


Publishing in conferences and then journals

In my experience peer reviewed proceedings are okay but if its a partner journal its not. I actually have fairly recent experience with this because one of my papers that I just submitted to a journal was published at one conference as a part of peer reviewed conference proceedings. Since, I am young scholar myself I also first consulted this with both my mentor and I actually even send an email to my target journal asking if submitting paper to that conference counts as prior publication that would disqualify it from publishing there and they said its okay they dont count that as prior publication. However, it would not be okay if the publication would be in actual journal (if the conference has a partnership with one) - the website you linked mentions this.

But in case the conference is partnered with some journal and this is not just proceedings they wont publish there your work without your consent - at least that is my experience. Last summer I send one other paper to a conference organized in my home country and they were also having similar opportunity as the conference you linked here and actually they selected my paper to be in the journal but since its a local publication I declined and they were fine with it - at least to my best knowledge (although probably try to first send an email to the conference organizers).

Also when I was inquiring about this generally more experienced professors told me its usually positive to have your paper been already in peer reviewed proceedings. They told me that reason for that is that most journals dont have double blind peer review. Hence the reviewers can actually google you and some might think that if your paper was accepted at a good conference and already been peer reviewed once then it is of high quality, although I suppose the converse could be true with bad conference. This being said I only have two papers under my belt so maybe take this advice with a grain of salt, but I think the advice of the more experienced professors at my uni should be solid, some of them have quite impressive publication records.


STATA: Impulse Response Function after ARDL?

In order to compute IRF in stata you should check "varbasic" in Stata's manual: http://www.stata.com/manuals13/tsvarbasic.pdf


Although I'm not sure that Piketty ever directly discusses the exact definition of $r$, he does make it clear indirectly. On page 52 of the hardcover English-language edition of his book, Piketty declares his "first fundamental law of capitalism":

Piketty obtains the share of income $\alpha$ from capital in national income from the income side of the national accounts, dividing "capital income" by national income. The capital share $\alpha$ is just the inverse of the aggregate labor share of income, on which there is an extensive literature. (Of course, there are several methodological choices that must be made when defining these concepts, and some of Piketty's choices differ from choices elsewhere.)

When finding $\beta$, Piketty appears to take the aggregate value of capital from respective countries' national accounts as well (though alternative sources are needed for older values). There is lots of information on the relevant choices in the technical appendix of Piketty and Zucman (2014), which is the basis of most discussion of $\alpha$ and $\beta$ in the book.

Since Piketty makes clear that $r=\alpha/\beta$, his definition of $r$ for a given year is essentially
$$r=\frac{\alpha}{\beta}=\frac{\text{aggregate capital income taken from national accounts}}{\text{aggregate value of capital taken from national accounts}}$$
Note that this actually does not (directly) involve tax data. Tax data is the basis of much of Piketty's other work, but not this.


Did William Nordhaus exclude 87% of economic industries from his climate change analysis?

No, Nordhaus did not do that but Nordhaus did exclude $\approx 87\%$ economic activity as measured by national income and Keen either confuses economic activity with industry or does not use common terminology in economics (or given that he is known for sensationalism perhaps because $87\%$ of all sectors would be even bigger portion of the economy).
As the table below from Nordhaus (1991) shows, Nordhaus split economic activity into 12 sectors, 11 of which can be considered industries (the word industry is in economics commonly applied to private sectors not to public sector, see also usage in texts such as Mankiw Principles of Economics).
Nordhaus did tabulate effects only on sectors which belonged to category of potentially severe impact and potentially moderate impact which is 7 out of 12 listed sectors so approximately $0.58\%$ sectors were included and approximately $42\%$ sectors were not included in further tabulation.
However, those $42\%$ of not included sectors did generate $87\%$ of economic activity.
In addition, the assertion that Nordhaus excluded these $42\%$ of sectors on account of them 'being indoors' is simply wrong here.
Nordhaus references indoor activity in that section only in one paragraph where he says the following:

The most sensitive sectors are likely to be those, such as agriculture and
forestry, in which output depends in a significant way upon climatic variables.
At the other extreme are activities, such as cardiovascular surgery or
microprocessor fabrication in 'clean rooms', which are undertaken in carefully
controlled environments that will not be directly affected by climate change.

Hence he uses 'clean rooms'  as an example of an extreme case of place not directly affected by climate change. I am an economist not a biologist/climatologist, so I can be wrong, but I think stating that clean rooms - hermetically sealed and sterilized rooms where every single precaution is made that not even a speck of dust will make it in - will not be directly affected by climate change is not completely unreasonable.
Additionally, note the above was never used in the paper as a primary reason for exclusion of the $42\%$ of the sectors (many of the $52\%$ included sectors such as electricity production are also done mostly in indoor environments e.g. nuclear/coal/gas/oil plants).
The true reason why the sectors were excluded is that Nordhaus was simply not able to find evidence for significant negative effects of climate change on those industries:

But for the bulk of the
economy - manufacturing, mining, utilities, finance, trade, and most service
industries - it is difficult to find major direct impacts of the projected climate
changes over the next 50 to 75 years.

Note, Nordhaus does look at various reports like EPA (I988). U.S. Environmental Protection Agency, The Potential Effects of Global Climate Change on the United
States draft to the congress to identify which industries are at risk. One could here levy a criticism that 'absence of evidence is not evidence of absence',  but this is hardly some sort of willful ignorance or a blunder on the side of Nordhaus.


It is true that we tend to think of Investment as something firms and businesses do, while consumers practice savings or "invest" in utility-enhancing assets (like a house to live in).

One could argue that when consumers buy shares of a corporation in the stock market they participate in productive investment, even though they do not control the decision-making process.

But exactly because they do not affect the decision-making process it is better to think of such "investments" also as saving, and retain the word investment for the purposeful and decided-upon transformation of resources into productive use.


Is Soros' theory of reflexivity aligned with normative vs positive economy dichotomy?

I would say there is some degree of correspondence between the two dichotomies. However, in the economic side of things, there is in my opinion a much clearer relation between positive and normative economics (at least in the theoretical side), as it is through the use of positive economics that normative means are supposed to be achieved.

The other dichotomy however seems to be less formal in this relation. Individuals or agents can have a given understanding of the world, but this might not be correct, or precise. Thus, if they base their manipulative actions based on their perceptions of the objective reality, it does not necessarily mean they are using the correct map of reality to act upon it.

In other words, positive economics is (allegedly) science, whereas agents cognition is not necessarily (and perhaps not mainly) built upon scientific knowledge.

Now, having said this, it is also debatable that the normalite-positive dichotomy is as clear as initially theorised by Milton Friedman in his 1953 book "Essays in Positive Economics". For example, some Marxists argue that science itself (at least social sciences) is part of the tools used by the ruling/capitalist class to exert control over society.
In philosophy of science, there has been a long debate about what science is and about whether there is such thing as "scientific progress", or whether science is just ideology (for example, see here).


Nominal Growth rate above Nominal Interest Rate

I have not read the text in question (other than the quote), but I would guess that his logic runs as follows. If we are discussing the government, one common analytical concept is the primary fiscal balance - the fiscal deficit excluding interest payments. If the primary fiscal balance is zero, the change in government debt will equal interest costs. The assumption is that a primary balance of zero is in some sense neutral (that assumption could be questioned), nominal debt levels will grow at the nominal rate of interest (since the government needs to borrow to cover the interest payment).

If we then look at the debt/GDP ratio, we see that is the nominal interest rate is the same as the nominal growth rate of GDP, the ratio will be unchanged under that circumstance. However, if the nominal interest rate is less than nominal GDP, the ratio would fall.

However, it is unclear to me why growth rates being “marginally above”interest rates makes anything other than a marginal difference. That might be better explained elsewhere in his text.

There is a literature on the concept of fiscal sustainability, where the relationship between growth rates and interest rates is covered in nore depth. Trying to summarise that theory appears to be beyond the scope of this particular question, but that is a key word you could use to search for more details.


The linkage principle does not depend on risk-aversion, because risk-neutral bidders simply bid according to the expected value they assign to the auctioned object. Having private information not revealed to other bidders, will generate information rents, reduce competition among bidders and lead to lower seller revenue.

Daniel Quint (U. Wisconson, eBay) has some lectures notes where he works out the Milgrom-Weber result in detail, and also gives the following intuitive explanation:

  One way to think about this is that bidder’s expected payoff can be
  thought of as his “information  rents,” that is, the extra surplus he
  is able to get by having private information. But in auction formats
  where information is revealed which is correlated with his private
  information, his private information becomes “less private” in a
  sense, so he gets a smaller surplus, and therefore more goes to the
  seller. In the logical limit – where information revealed over the
  course of the auction fully reveals the highest type – the seller
  could simply make a take-it-or-leave-it offer to extract full surplus.


Showing utility function gives preferences that are rational and convex

I'll give a few hints to get you started. First, note that since the preference $\succsim$ is represented by the utility function $U(x_1,x_2)=x_1+\ln x_2$, it follows that
\begin{equation}
(x_1,x_2)\succsim (x_1',x_2')\quad\Leftrightarrow\quad U(x_1,x_2)\ge U(x_1',x_2') \tag{1}
\end{equation}
Keeping this equivalence in mind, consider:

Completeness: $\succsim$ is complete if for all $(x_1,x_2),(x_1',x_2')\in\mathbb R_+^2$,
\begin{equation}
\text{either }(x_1,x_2)\succsim (x_1',x_2'),
\quad\text{or }(x_1',x_2')\succsim(x_1,x_2). \tag{2}
\end{equation}
Using $(1)$, we can rewrite $(2)$ as
\begin{equation}
\text{either }U(x_1,x_2)\ge U(x_1',x_2'),
\quad\text{or }U(x_1',x_2')\ge U(x_1,x_2). \tag{2*}
\end{equation}
Now $(2^*)$ should be easy to prove using the property that $\mathbb R$ is an ordered field.

Transitivity: Use the same trick to translate preference ordering into ordering of real numbers.

Convexity: Start from the definition that $\succsim$ is convex if for any $\alpha\in[0,1]$,
\begin{multline}
(x_1,x_2)\succsim(x_1'',x_2'') \text{ and } (x_1',x_2')\succsim (x_1'',x_2'')
\\\Rightarrow \quad
\alpha(x_1,x_2)+(1-\alpha)(x_1',x_2')\succsim(x_1'',x_2'')
\end{multline}
Again, translate the preference ordering into ordering of real numbers to prove the implication. Since $U$ is quasi-linear, this way will save you some trouble of dealing with Hessians and so on.


Is there a collection of Economics PhD prelims questions?

Exams from:

Harvard
UC Davis
MIT (only problem sets; go to graduates).

Although more cumbersome, you can also google something like "advanced macroeconomics exams pdf" or so, and you can discover personal pages of lecturers/professors that put up previous exams (or problem sets). For example, resources from a lecturer at Leipzig (go to bottom of page), Oslo, Dublin (one exam only), Helsinski.


Electric Vehicle effects on family and macro-economics?

Macro
One of the big issues with EVs is that several countries collect a lot of revenue from taxes on fossil car fuels. Those revenue streams will shrink, meaning that something else will have to be taxed to make up the shortfall.
Will the West pay back its debt? Well, that's barely a meaningful question. If you're thinking of government debt (which is not equivalent to Western debt), then EVs are neither good nor bad. If revenue drops from one source, it will be taken from another source instead.
The much bigger threat to car manufacturers is self-driving vehicles. Those could reduce vehicle demand by a huge proportion, if (as expected) they trigger a growth in shared ownership of vehicles. Most cars are used for only a tiny fraction of the time. And even at rush hour, only a minority of cars are in use at any one time.
Family
EVs currently differ from fossil cars in that they have higher capital expenditure, and lower operating expenditure. That will make household cash outflow much lumpier.
However, this also encourages the move towards shared ownership of self-driving EVs, which, for those households that choose it, will remove the capital expenditure element, making their cash outflow much smoother.


Does inflation/expected inflation in a given country devalue its currency or increase its value?

I've seen both arguments. [...]

You are comparing two arguments that are not incompatible. The first one scenarizes a situation within which the CB does nothing, while the second argument implies a CB that counteracts. Note that when the CB hikes its rates, it also has some forex-unrelated domestic consequences going in the same direction.

  Which of these two views make more sense theoretically [...] ?

As explained, both. It is all about politics actually. There is no deterministic math here. Put differently, human is an unpredictable variable in the model.

  Which [...] is more supported empirically ?

Once again, both. Notorious investments such as Soros's against the British pound in 1992 perfectly illustrates the second argument. But if Britain hadn't done anything to keep the pound's value high (in relation to the German mark), it would have decreased because of inflation.


Are you referring to real life circumstances or in a hypothetical world? There is an interesting article about the correlation between wages and unemployment found here: https://www.stlouisfed.org/On-The-Economy/2014/December/Are-Wages-and-the-Unemployment-Rate-Correlated

Essentially this article states that higher levels of wages are weakly correlated with the unemployment rate based on data that was compiled and published in The Regional Economist. They also argue that when business conditions shift, it impacts both the unemployment rate and wages.


Hypothetical price of a city (e.g. Munich)

A rough estimation technic consists in (i) getting the half of the Gross Urban Product (GUP), thinking of it as a perpetual yearly net income flow (i.e. a profit flow) attributable to the entire city as such, and (ii) dividing it by the average cost of capital that is immobilized locally, say, $2\%$, minus the GUP's growth rate, say, $1\%$ (both real and assumed constant over an infinite horizon)

In the case of Munich it would be

$\frac{1}{2}\frac{GUP_{2018}}{.02 - .01} = 105\text{E}^{11}€$

$10.5$ trillions euros, actually calculated using the DD Model.

Note that public infrastructures are not excluded from the above valuation since the city is considered as an indivisible whole.


How saving and investment influence growth in the long run?

The result you are asking about follows more formally from the Solow model. It's definitely worth looking into. I won't go into the model specifics, since I believe you are after a simple example.

First, note that what economists refer to as capital is not really what most people mean when talking about capital. For us, capital is typically a production factor that is not labor. A good example is a machine. For agriculture it could also be something like fertilizer. These are all things that help you produce more and that is how you get growth from capital.

Suppose you are a farmer who can produce 10 potatoes a year currently. You sell those potatoes and then can decide to spend that money consuming (e.g. buying steaks) or you can invest it in "capital". If you spend the money on a tractor and some fertilizer, you can produce 20 potatoes next year. That is how investment leads to growth. That is also why you need to give up some consumption today in order to grow your production and have more consumption tomorrow.