Essential difference between Neo-Classical and Neo-Liberal economical philosophies

To expand on my comment. Your question is not really about economics but about philosophy or political science. The reason for this is that the Economics is usually defined following the definition of Lionel Robbins as: "Economics is the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses."

Neo-liberalism is an ideology. It is a set of values or set of ends that based on the given values are worth pursuing. For example, from Neoliberal value of freedom one could derive the end goal of free speech. But economics does not examine either ends or values. Economics treats ends and values as given by an individual. Hence there is not really anything like Neo-liberal economics (although I seen the term used as a pejorative, like trickle down economics - but those are not really economic terms that have any meaning in economics - you generally wont find them in papers or textbooks or conferences) as that entails value judgments that can only be judged based on philosophy not science.

There is a Neoclassical school of economics. However, it has nothing to do with capitalism. Capitalism is a term that’s poorly defined and not used in economics. In most economics textbook you would not even find the word, outside some inserted exempts of newspaper articles in undergraduate books. Neoclassical economics is an umbrella term so its easier to classify theories. The term for example implies that models used will assume rational agents (as opposed to lets say behavioral school that relaxes this assumption). This being said the classification is used mostly by historians of economic thought as in present day its not uncommon for people to use different models and set of assumption to fit given purpose, so there are many economists who use neoclassical model in one work but behavioral model in another.

Also, Neoclassical economic models have no relationship to idea of capitalism (many neoclassical models suggest quite large role for government intervention in economy or redistribution given certain welfare function). For example, Jozef Stiglitz's work is Neoclassical, meaning almost all his papers use neoclassical models, but he does not like the idea of 'classic capitalism'. Another example would be Greg Mankiw, who mostly works with Neo-Keynesian models in his research, but personally he is supportive of the idea of 'classic capitalism' - his ideology could be described as quite libertarian or Neo-liberal.


Whether a good is a Giffen good should be based on circumstance?

Yes, whether a good is a Giffen or inferior good depends on "the circumstances", or to be more narrow, it can depend on the income and price levels.

Claim 1. Being an inferior good is a local attribute.

Proof. Suppose a good $x$ is inferior for all income levels $I$ and all price vectors $\mathbf{p}$. Trivially demand for $x$ at $I = 0$ is 0. If $x$ is inferior for all income levels, then for any $I > 0$ demand for $x$ is also 0. This contradicts the strict definition of inferior good, which specifies that as $I$ increases $x$ should decrease.

Claim 2. All Giffen goods are inferior goods.

This is covered in some detail in textbooks, usually in a chapter that mentions the "Slutsky-equation".

Combining the two claims we have that being a Giffen good is also a local attribute; as being Giffen everywhere would imply being inferior everywhere, and we have shown that that is not possible.


Why is the yen getting stronger even with negative interest rates?

Negative interest rates are not incompatible with appreciation of the local currency. As pointed out by the economist (Why investors buy bonds with negative yields), some investors buy bonds to place bets on currencies. Thus, it does not matter so much that the interest rate is negative.

Thus, the variation of the yen reflects other trends. It is very tricky to point out at one event or idea in particular, but here are a few possible drivers of the appreciation of the Yen:

The dollar is weakening, following the release of the minutes of the Fed meeting which showed that the central bank does not intend to raise interest rates in April.
Japan is running out of options to put downward pressure on the yen. The central bank already buys everything in sight, and thus would have difficulties to increase the money supply. The feeling that central banks are loosing grip is very tangible since a few months (if not years)
G20 countries recently agreed to not proceed to competitive devaluations of their currency. Therefore, investors might expect that Yen won't go down easily, and that the most logical path is unpward.
Japan's Prime Minister Shinzo Abe seems to hesitate to proceed to planned increase in sale tax. The first increase had a desastrous short-term effect on the economy, and most firms have a pessimistic outlook (latest PMI 49.1). Therefore, it becomes more and more likely that the government will delay the increase in sale tax, and potentially also try to stimulate the economy further.

These are all guesses. It might be that one or more of the above guesses actually moved the Yen. Or maybe it didn't and something else is happening. Maybe investors aren't as rational as we think they are and they just bet on the Yen because of their gut without logical reasons.


How does the core relate to strong equilibrium?

Strong Nash equilibrium is different from core mainly because of communication. In a Strong Nash, unlimited private communication is allowed. The core is a concept that is linked to Coalition-proof Nash equilibrium rather than Strong Nash. People can freely communicate but cannot make binding commitment before deciding.

In some games, both happen to be the same, but in generality, the core is a concept derived from Coalition-proof Nash equilibrium rather than from Strong Nash Equilibrium.


Is there a difference between buying a currency and selling the currency to be converted to?

Buy and sell rates are different because the currency vendor wants to make a profit and is also taking some risk with the exchange. It is possible that she cannot unload all the CADs they buy from you immediately and then perhaps in the future they will depreciate. (Perhaps they will appreciate. It is uncertain, hence there is risk.)

The same applies to buying and selling euros.

Usually there is some middle rate $x$ EUR/CAD. The vendor will deviate 1-2% from this in the direction favorable to her, depending on the direction of the deal (buying EURs with CAD or the other way around). You can look up something close the middle rate on XE. Currently this is about 1.394 CAD/EUR. Your vendor seems to sell at a premium that seems higher than normal to me, so perhaps see if you can find another currency merchant.


$k_t$ is your capital after investment, so if you subtract the capital you carried over from last period, $k_{t-1}(1-\delta)$, you obtain the amount that must have been invested in order to have $k_t$ of capital today.

Remember that $k$ is a stock variable, while $i$ is a flow variable. This is why the flow $i_t$ is the difference between the current and the former stock: $k_t-k_{t-1}(1-\delta)$.


Why is Indian rupee is stable against the USD even though most other currencies weakening?

That is because India maintains strong economic relations with the United States. In addition, India might be looking at maintaining a fixed currency with the US. In exchange, India would have to give up Monetary Autonomy, or Free Flow of Capital.

As far as I know, India does not restrict the flow of capital to and from the country, so I assume that most policies are more government related when it comes to the US (although the new bilateral agreement is to work to reduce that). Overall, India's strong economic relations with the United States and a tied trade allows them to maintain growth.

In fact, recently with numerous trade agreements, I wouldn't say India has reached it's lowest point (that would be a few years ago), but is rather trying to secure a very high peak for the next 5-10 years.


Lessons From Successfully small island economies

I like this question, in the sense that although it is broad, I think it can be answered concisely and factually.

Singapore and China (from your previous question) are largely authoritarian countries. So they were both able to exert a lot of government influence over their development programs.  In this respect, they are not similar to Caribbean countries.

However, in other respects, Singapore is a good case analog because it is an island with very few natural resources and a relatively small population with correspondingly small domestic economic prospects.

Here are some pillars that underpinned Singapore's meteoric economic development.

Create stable and market-friendly government.  Economic development takes a long time, and it's hard to execute it if government changes frequently.  Also, governments need to be prepared -- over time -- to relinquish control over their economies as they transition to market-based.  Both Singapore and China accomplished this deliberately and in an ordered fashion (moving too quickly can cause chaos...see Russia for example).
Create stable currency and a good banking system.  Singapore, like many developing countries, decided to peg its currency to the US dollar, which gave it exchange rate stability and familarity.  Banking systems are crucial to ensuring liquidity and investment for economic growth, so installing a currency board, conservative fiscal policy, a decent bank, and a set of quasi-government commercial banks helped create financial infrastructure from scratch.
Invest in general economic infrastructure.  Singapore carefully built out physical infrastructure (roads, communications), legal infrastructure (courts and business law), and talent infrastructure (through schools and workforce training) to reduce economic friction, create economic resources, and maximize economic opportunity for development.
Seed strategic industries.  Singapore identified major industries that could help drive economic growth, and used a combination of government subsidies, pro-business laws, and economic protectionism to nurture these industries.  These investments included: making Singapore a low-cost manufacturing hub and a shipping hub (early economic development), and then subsequently a banking and communications hub (later development).
Assist trade.  The Singapore government worked closely with private and semi-public enterprises to help push through trade negotiations with countries (US, Japan, China, etc), reduce tariffs, and generally make Singapore an easy location for trade.  This was crucial to expanding the scope of the economy, because with fewer than 4 million people (at the time), the domestic economy could not be sufficient to drive growth.
Attract foreign direct investment.  As the economic infrastructure took root, Singapore became an attractive destination for foreign direct investment.  The government assisted with courting and promoting foreign direct investment into the country, which provided investment capital to build out the economy.  Tax and relocation subsidies helped a lot here.

Not all of these approaches will be applicable to Caribbean countries (e.g. Singapore's location was particularly strategic as a shipping hub), but not all of them need to be replicated in order to create fertile conditions for economic development in the Caribbean.


Germany's current account surpluses: Negative effect? How to solve them?

As requested in comments:

German banks lost a lot of money in the US sub-prime crisis and would have lost a lot more in Greece if the Eurosystem had not saved them. This is what might reasonably be expected if you run persistent trade surpluses: you gain foreign assets which may never be repaid or may be less valuable than you expected

The European Commission is supposed to investigate under its Macroeconomic Imbalance Procedure when a country's current account surplus exceeds 6% of GDP, and Germany's is now about 8%. In the end the Macroeconomic Imbalance Procedure might in theory become an Excessive Imbalance Procedure and potentially sanctions for euro-area member states of up to 0.1% of GDP. But it will not happen to Germany, for European political reasons. Wikipedia has a description and you can also read the latest Commission report

In effect Germany is making stuff and sending it to foreigners without exchanging it for a similar value of imports. Eventually people in Germany might notice that they could have a higher standard of living which their mercantilist politicians and economists have been denying them

German savers and politicians are already protesting about low interest rates and perhaps one day might think that in a period of negative interest rates it could be sensible to consume (and import) more


Why are nations engaging in austerity when bond yields are historically low?

Debt being cheaper than ever doesn't mean that it is optimal to borrow. Unless you plan to invest the borrowed money into projects that yield higher returns with certainty (we would call this arbitrage), any debt implies a tradeoff between the welfare of current versus future generations.

Perhaps the borrowing before to higher rates was non-optimal under the current government's perspective. I.e. now we place a higher value onto future generations versus current generations
Perhaps now in the slump, the government doesn't see investment opportunities that yield a sufficiently high return.
Perhaps the current government has a different prior on how high returns to debt are. Different governments (or the same government over time) can have a different prior (guess) on how important/productive government spending onto infrastructure, education, health, military etc. is. This also includes a prior on how large the Keynesian multiplier is, and how effective demand stimuli are. Then, even presented with the same investment opportunities at a
lower interest rate, a current government might chose differently
than the previous government.

This was all without political economy consideration. There are arguments on optimal spending that take into account how your current budget will affect the budget of future (different) governments. Have a look at Persson and Svensson (1989) to get a feel for this kind of political economy reasoning.

Note that the argument in the cited paper would go the wrong way (i.e., too much spending of a conservative government), but it shows you how optimal policies under a permanent ruler can differ from optimal policies in a democracy.


Why high levels of inflation doesn't leads to high levels of employment?

here are several reasons for that.
First, if we look beyond the simplified version Philips curve, to the more appropriate modern version of Phillips curve we discover that what matters is not just inflation but that inflation is higher than expected inflation. For example, following Romer Advanced Economics pp 261, the Phillips curve will look like:
$$\pi_t= \pi_t^e +\lambda (\ln Y_t -\ln \bar{Y}_t) + \epsilon^s$$
Where $\pi$ is the inflation, $\pi^e$ expected inflation, $Y$ output, $\bar{Y}$ natural level of output, and $\epsilon^S$ are supply shocks.
The expression above implies:
$$\pi_t- \pi_t^e = \lambda (\ln Y_t -\ln \bar{Y}_t) + \epsilon^s$$
The equation above implies that it’s not just inflation that increases output and thus  employment, but it is situation when inflation is higher than expected inflation. The intuition is that if people expect certain level of inflation they will advance make adjustments for that (for example unions may demand indexing wages to price level) and as a result you don’t get any of the short term positive benefits of inflation that come from decrease in real wages and increases in prices)
If people, expect 5% increase in inflation and inflation increases by 5% this will have no effect on employment. Usually in countries like Venezuela people expect the high inflation because it becomes a commonplace.
In addition, inflation does not just stimulate employment and output but also creates costs
A non-exhautive list includes:

increase in uncertainity and volatility. This holds exclusively for high levels of inflation since modest levels of inflation are not associated with increase in uncertainty and volatility. Higher uncertainity and volatility is generally bad for the economy as it leads to less investment and lower economic growth (see discussion in Mankiw Macroeconomics or Wilson (2006)).

next there are shoeleather costs of expected inflation. As explained in Mankiw Macroeconomic:

One cost is the distorting effect of the inflation tax on the amount of money people hold. As we have already discussed, a higher inflation rate leads to a higher nominal interest rate, which in turn leads to lower real money balances. If people hold lower money balances on average, they must make more frequent trips to the bank to withdraw money—for example, they might withdraw \$50 twice a week rather than \$100 once a week. The inconvenience of reducing money holding is metaphorically called the shoeleather cost of inflation, because walking to the bank more often causes one’s shoes to wear out more quickly

Of course, the above is written for undergraduates, it is not really the cost of new shoes that is biggest cost here, but opportunity cost of wasted time and general costs of adjusting to the inflation.

High inflation causes menu costs. These are costs firms incur to change prices (see Mankiw Macroeconomics).

High inflation tends to be more volatile and this complicates financial planning and distorts peoples intertemporal choices (again have look at Mankiw Macroeconomics).

The list above is not exclusive, the point is that high level of inflation has many costs associated with it. These costs increase disproportionally as inflation becomes higher, while at some point very high inflation won't even necessary have any short run benefits because the increase in uncertainty will depress economic activity more than the output get stimulated by the effect of short run effect of increase in prices
The way how you can think intuitively about this is that at unexpected and low levels of inflation the positive effects of inflation on employment and output dominates any possible negative effects.
However, at large levels of inflation the negative effects of inflation dominate and inflation has more costs for economy than benefits.


Are businesses included in economic institutions?

Unfortunately, economists don't have a standard, universally-accepted definition of what an 'institution' is.

Hence, depending on what you're trying to study and whom you ask, businesses may or may not be an institution. So, in a way, both definitions are correct.

If you're interested in different conceptual approaches to institutions, Avner Greif has written an interesting survey on it, with plenty of references that might be worth looking at.


The United States' position on the Laffer Curve

Question B in the link answers your question: most participants believe that taxable income would not rise enough to offset the tax cut, indicating that they do not believe we are on the wrong side of the Laffer Curve.


Limit of random walk auto correlation function

Hi: The expression tends to 1.0. The intuition is that, as $t$ gets larger and larger, the $h$ lags that seperate the two processes become more and more negligible and the processes begin to look the same. This happens because, as $t$ gets larger and larger, the proportion of observations common to both processes becomes larger and larger.


How does economic theory handle full automation of production resulting in full unemployment?

If this were to occur, there are three things we need to look at.

1) Will production occur?

2) Without wages, how will households make money?

3) What will be the welfare effects?

$\textbf{1)}$ To answer the first question, if we assume that capital replaced all labor, we are assuming that capital and labor are substitutes which means that the production function will have the following form:
$$F(k,l)=\alpha k+\beta l$$
What this means is that if $l=0$, $F(k,l)$ will still be greater than zero, so production will still occur.

$\textbf{2)}$ To answer the second question, we can look at how households make money. Households make money from wages, and they make money from owning a fraction of firms (so they get that fraction of the profits the firm makes). All firms are owned by households (either through stocks or private ownership). The typical income of a household looks like the following:
$$Y(\frac{}{})=wl+\theta\Pi$$
Where $w$ is wages, $l$ is labor hours, $\theta$ is the fraction of firms which the household owns, and $Pi$ is those firms' profits. If $l=0$, our income reduces to $Y(\frac{}{})=\theta\Pi$

One thing that immediately comes to mind is "without wages, won't all households' income decrease?" Well, we don't know for sure. According to economic theory, the reason the automation will have occurred is because it will cost less. What this means is that profit will increase, so $\theta\Pi$ will also increase. It really depends on whether $\Delta\theta\Pi$ is larger than, smaller than, or equal to $wl$.

$\textbf{3)}$ The welfare effects are where we run into problems. When we look at the equation for income of a household, we can conduct a thought experiment: How does the fraction of income attributed to each term change when in different income groups? In the highest income groups, most of their income will come from $\theta\Pi$. An example of this would be the show SharkTank. In this show, companies come to the very wealthy "sharks" looking for capital, and in turn, they often offer a share of their company. Now, when we think about lower income groups, where does most of their income come from? Well, most of their income comes from wages because they cannot afford to save money (and therefore invest in stocks or entrepreneurship). What would happen as a result from full automation, we would theoretically see the income inequality grow to be even larger than it currently is. This is a very undesirable result.


How to calculate Shapley Value with uncertainty?

It sounds like you may have a quadratic loss function, but I am guessing a little.  If the players have played enough to be experienced in how the game works, then I would simply use the sample mean reward as my calculation point.

You should read up on the theory behind loss functions.  The players are making estimates, certain types of estimates are the natural minimization of certain loss functions.  Most gambling type behavior results from quadratic loss, though there are particular exceptions.  Equity securities center of location is the mode because there can be no such thing as a mean return.  Equity securities behaviors should result from an all-or-nothing loss function.  Physicians are another example where there is an asymmetric loss.

If a physician fails to diagnose you with a disease, then they lost the revenue from treatment and face malpractice claims.  If they misdiagnose you as having a disease but you do not, then they still collect the medical billings and may not face malpractice claims, especially if they do not reveal it was a misdiagnosis.  A diagnosis is a form of gamble.

Take a look at the losses experienced from failing to take a particular type of behavior.


Bayesian Nash Equilibrium - Mixed Strategies

I believe that the answer given by @denesp is incorrect. The second method involves simply writing the game in strategic of "normal" form. I believe
the first method is better (easier to use), but I think that they can both be used.
In the answer given by @desesp, the following explanation is given.

  The reason why method two is flawed is that the probabilities $a$, $b$
  and $c$ are not  independent as $$ a = p \cdot q, \hskip 20pt b = p
 \cdot (1 - q), \hskip 20pt c = (1 - p) \cdot q, \hskip 20pt  1 - a - b
 - c = (1 - p) \cdot (1 - q). $$

Here, it appears that mixing is occurring over L in game 1 (with probability $p$) and L in game 2 (with probability $q$). This interpretation does make sense. I believe that @denesp is confusing conditional and unconditional probabilities. To better understand this, I'm going to start with a discussion of actions versus strategies. Then I'll discuss how the set of strategies considered in methods 1 is included in method 2. I'll note that method 2 contains a larger strategy set, which may or may not be useful. I'll conclude with an example of how both methods can produce the same answers.

Difference between actions and strategies

In the explanation given above, it may appear that mixing is occurring over actions. This is not a
correct interpretation. It is technically incorrect because the player is not mixing over actions but mixing over strategies. This is because a player chooses strategies, not actions. A strategy is a plan
that denotes that actions that a player takes in any and every contingency.
We can think of it as mapping information sets to actions. This is important because we would like player 1's actions to depend on the state of nature---we want them to depend on which game he/she is playing.

For reference, we can find definitions of actions and strategies in the first chapter of Rasmusen's book, Games and Information (4th edition). The relevant text is given here:

What are the sets of strategies considered in each method?

In the case of the game that you have given, the pure strategies available can be written succinctly (LL, LR, RL, RR), as you have already done in method 2.
Therefore, the method that you described in method two mixes over the pure strategies, with probabilities: $a$, $b$, $c$, and $1 -a-b-c$.

What strategies, then, are we mixing over in method 1? Suppose that $p$
is the probability of choosing L is game 1 and $q$ is the probability of choosing L in game 2. Then in method 1, we can see that we are choosing
the conditional probability of taking each action in each contingency.
Suppose that game 1 is denoted $G_1$ and that game 2 is denoted $G_2$.
When we specify $p$ and $q$, we are really specifying
$$
p=P(L|G_1)\\ q=P(L|G_2).
$$
Now, in order to show that these two methods are equivalent, we need to show that the sets of strategies represented by each of these sets is the same.

Method 2 contains the same strategies as method 1, plus more.

First, note that the pure strategies LL, LR, RL, and RR can be represented in method 1 by setting $p$ and $q$ to zero or 1.

Suppose that we are using method 2 and that we choose a particular $a$,$b$, and $c$, as defined above. This can be represented in method 1
with
\begin{align*}
p &= a + b \\
q &= a + c.
\end{align*}

However, suppose we choose a particular $p$ and $q$ in method 1. It can be represented in method 2, but not uniquely.
Suppose $p=1/2$ and $q=1/2$. Then two possibilities are $(a,b,c) = (1/2,0,0)$
or another is $(a,b,c)=(0,1/2,1/2)$.

Why does method 2 contain more strategies?

Method 2 contains more strategies because it allows more flexibility
to specify off-equilibrium behavior. This can end up capturing non-credible
threats. Depending on which equilibrium concept you're using, you may or may not want to include these. If you're only interested in Bayesian Nash equilibria, then you want to include these. If you're interested in sub-game perfect Nash equilibria or Bayesian sequential equilibria, then you don't want them.

Here is an example of how method 1 can miss some equilibria

The following game is again take from Rasmusen's book. Suppose that in this game
Smith moves first.

Then, Jones must choose among 4 strategies. In each of these strategies, he specifies his actions in each contingency. The 4 strategies are listed here and the game is represented in strategic or "normal" form.

There are three equilibria, denoted $E_1$, $E_2$, and $E_3$.
If we were simply interested in the Nash equilibria of this game,
we would include all of these equilbria. However, if we are interested
in only the subgame perfect equilibria, we would only want $E_2$.
That is because $E_1$ and $E_3$ involve non-credible threats.

I believe that if we were to try to solve this game using method 1, we would not be able
to identify all three of these equilibria.

Strategic, "Normal" form

In the question you've given, method 2 is essentially transforming this
into a static game in which we consider all the strategies. This means that we are considering the "normal" form of the game. This method is easy and appropriate if you're interested in finding the pure strategy equilibria. It can probably also used to find the mixed strategy BNE, but is perhaps more complicated then what is described in methods 2.
For reference,
here are some notes on the topic.
These notes give instructions on how to solve for the pure strategy Nash equilibria using the transformation that you've given. It also demonstrates how to solve the mixed strategy equilibria using method 1. (See http://www.sas.upenn.edu/~ordonez/pdfs/ECON%20201/NoteBAYES.pdf .)

Further reference

You can also use this online tool to test how the methods can give you the same answers. This is a tool to solve for the Nash equilibria of n by n games. I would recommend using this tool on the examples given in the previous section.
I found this tool referenced in this other question.


Maximum level of profit attainable when discount factor is too low

I do not want to do all the algebra, but I give you a hint.
You are correct: You are asked about $\pi^C$ in your Condition (1) and you are also right that $\pi^D$ changes.

In your parameter setting with $c=0$:
$$\pi^C := \pi (q^C, q^C) = (1- 2 q^C) q^C,$$
where $q^C$ is the quantity they coordinate on.

Then $\pi^D$ is the profit from best-responding to $q^C$ with some $q^D$:
$$ \pi^D := \pi (q^D, q^C) = (1- q^C-q^D)q^D,$$
i.e., $q^D = \arg max \quad \pi^D$. Now you solve $q^D$ as a function of $q^C$ and plug that into $\pi^D$. Having Condition (1) hold with equality gives you some $q^C$ that you can plug into $\pi^C$ again.


Calculating Nash Equilibrium prices for Bertrand duopolists

The important thing to remember here is that Bertrand duopolists compete over price, not quantity. This means that in the game each player sets a price and the quantity sold is then determined by the demand curve. In this game the firm with the lowest price sells to the entire market and if both firms have the same price they each sell to half of the market. Now, we have to think about the logic that goes into determining equilibrium. It is easy to see that neither firm will ever set a price lower than $c$ because setting a price lower than marginal cost will give the firm negative profit. So we have to start that it must be the case that $p_1, p_2 > c$. Now, suppose that $p_1 > p_2 > c$. This cannot happen because firm 1 has an incentive to set a price below that of firm 2 and above $c$ so they will take the whole market. The same logic can be applied to the case where $p_2 > p_1 > c$. Now, consider the case where $p_1 > p_2 = c$. In this case firm 1 has the incentive to set there price equal to $c$ and split the market with firm 2, so this cannot be a Nash equilibrium. Again, we can apply the same logic to the case where $p_2 > p_1 = c$. So, the only candidate left for Nash equilibrium is scenario where $p_1 = p_2 = c$. This result is very interesting considering that with one firm we get a profit maximizing monopoly result but by adding just one additional firm we get the same pricing as perfect competition as long as the firms are playing the Bertrand price competition game.


How to estimate Italian parking data starting from German parking data?

I'm not entirely sure this would be possible to do using German data, though very happy to be corrected.
It seems that the data you have is largely driven by parking search costs, therefore could you not try and find internationally comparable data on travel to work time, control for a few bits and then try and see whether you can find parking search costs as having a significant effect? Then you could just apply that effect to the data on italian travel to work time and make the assumption that's reasonable.
For data:
The Italian statistical authority (https://db.nomics.world/ISTAT) provide some brilliant survey data on citizens views on quality of life - perhaps there's something you could infer from there, or view whether data is offered on expenditure on parking enforecemnt. Similarly, the WEF's Travel and Tourism/Global competitiveness index may have indicators on travel to work time.
A very all out approach would be to collect data on council meeting minutes to see how often parking comes up as an issue.


Has the natural rate of unemployment decreased with modern travel and the Internet?

To me, it seems that it has increased, not decreased, due to the factors you mention. Yes, transportation and information networks enable workforce movement. But they also enable movement of goods and information - and because goods and information are more mobile than humans, they profit more, and the results of their portability outpaces the results of the workforce's increased mobility.

When goods were hard to move, a local demand had to be met by local suppliers. If a manufacturer lost a deal, it went to a local competitor instead, and the workforce could adapt easily. Now, if a manufacturer loses a deal, it goes to a competitor 5000 km away. Take for example the Ruhr area: it's still the most densely populated area in Germany, even though the steel industry which attracted workforce in the mid-20th century is gone. The Germans literally shipped one of their steel mills around the globe to China - but the workers stayed.

So, because the workforce cannot adapt to structural changes with the speed at which they are happening, the structural unemployment is higher than before networks which enabled such changes. But the increased speed of change also means more frictional unemployment. A large part of the adaptation which is happening requires people to change jobs. Also, your observation of an increased commute radius contributes to frictional unemployment too - when a person is unhappy with his job, he has more opportunities in his now increased commute area, and is more likely to change jobs.

There are lots of indirect effects too. The Internet and the transport networks enable changes in culture (e.g. today people move out of their birth town early, so may be more willing to relocate again later), business models (e.g. movie streaming) and market structure (today's trend towards ever larger corporations is enabled through, among other things, efficient information and people movement, which reduces overhead costs of giant companies, and global knowledge - I doubt that 50 years ago, an American chain for mediocre coffee in fancy large cups would have been able to establish itself in Europe. Nowadays, the culture export means that the trendy European teenager recognized the brand before they had one of the shops in their location). And they go both ways - a giant corporation may actually be more stable, and afford to keep workforce during short-term demand slumps which would have capsized a tiny manufacturer.

Whatever globalization (spearheaded by more exchange of information and goods, and even people on their respective networks) may bring in the future, it currently seems to create more unemployment - not because it's a bad thing in itself, but because in a more dynamic market, there is less job stability.


Is the government a producer, a consumer, or both?

The government is a producer of goods/services that are then usually not subject to market transactions.

Some goods/services that the government produce can be said to increase directly the utility of the households/individuals in the economy.

Others can be said to function as intermediate goods, entering, usually as a positive externality, the production function(s) of the private sector.

Yet others, the literature contents, can be treated as both being utility-enhnacing and private production enhancing.

An example: national roads (without full-pricing tolls). They are used by individuals to go to recreational trips (utility-enhancing) but also by businesses to transport their goods (production function externality).

Another example: a police force. It can be said that it provides a sense of security to individuals (utility enhancing) but also, it reduces uncertainty for businesses and so increases the productivity of production inputs (in the sense that one does not have to divert scarce private resources to attain the desired degree of security , when a police force operates). Likewise for military (where instead of local outlaw, the danger comes from foreign aggressors).

Etc. In fact, for most goods/services provided by the government, I would say that the dual nature prevails.

In the Political Economy sub-field, in some models the concept of government self-consumption arises, in order to capture the phenomenon of corruption which leads to part of taxes being consumed by corrupted government officials and not used to produce the aforementioned goods and services.


How do the factors of production map to the GDP identity?

They all map through production function to output $Y$. Since output is produced from these factors we have

$$Y=f(K,L,E,N,...)$$

Where, $K$ would capital, $L$ labor, $E$ would be entrepreneurship and $N$ nature - you can also include any arbitrary number of factors and also note while entrepreneurship is in some textbooks considered factor, nature is often just included under the land label.

The right hand side of the identity is all determined by output.

Consumption is determined by output as its function of disposable income based on output. A simple undergraduate example would be:

$$C=c_0+c_1(Y-T)$$

Where $c_0$ and $c_1$ are parameters $Y$ is output and $T$ taxes. You can have more complex consumption functions but they will all be functions of $Y$.

Investment $I$ depends on the $c_1$ parameter which determines what proportion of income people save. That is $(1-c_1)Y=S$ where $S$ is saving and $I=S$ (assuming balanced budget).

$G$ is again determined by the output - in the long run government can only spend maximum what the society can produce and what government can extract from society through taxes $G=T$.

$X-M$ tracks the movement of some of the home output abroad and import of some of the foreign output.

So to sum up, the factors of production map to GDP as they through production function map to $Y$ and the right hand side of the equation is just all determined by what $Y$ is.


Difference between automated and living labor pertaining to surplus value

This question is oddly phrased.

The main marxist thesis about profit is that firms can't profit from technology, but ultimately profit comes from exploitation of human labor, i.e. by extracting surplus value from workers.

In marxism, profit always comes from "living labor", in the sense that the price of a commodity both reflects the technology used to create it and also the human labor congealed in it. However,  a firm can't "pay less" machines because machines have fixed values while a living human being can be paid less than their value of their labor if the value of their labor exceeds the value the worker needs for subsistence (e.g. sufficient for their bills, calories, leisure etc).


Why Banks give more discounts/offers on Credit Card as compared to Debit Cards?

Yes of course you are right about both of your points. The fee charging the credit card POS transaction is about 2% to 3% depending on the company. The fee charging to debit card transaction is about flat 25 cents or lower.
For online transaction, the credit card fee can be as high as 4%, and the debit card fee can be negligible. Banks make more money on credit card.
So, to answer your question:

Credit department usually have more money to make promotions.
Average people (in US) have almost zero savings so they cannot use their debit card very often. But everyone has a credit card.
Of course, credit card can default, and the bank can legally charge hefty interests.

And as mentioned by Brian, competition is an important reason.


There can be market imperfections if information is not passed on immediately - historically that definitely happened all the time before telecommunication was well established.

Furthermore, if they have sufficient control of the in- and outflows countries can fix exchange rates for their currency at rates different from those of the free market. That happened in many socialist countries. The free market exchange rate for the Eastern and Western German currencies was substantially different from what the official exchange rate that was used by Eastern German banks. After unification of the countries again an exchange rate that was different from the market rate was used for the monetary union.

In Cuba a convertible peso was introduced in 1994 with a fixed exchange rate with the US dollar internally. It cannot be traded outside of Cuba.

The currency crises in Mexico and Asia in the 90s, and in Argentina in 2001-02 are associated with attempts to fix exchange rates that were in conflict with the market rates.


"The citizens know their own utility functions better than the politicians"

This idea in the title of your question sometimes falls under the name of consumer sovereignty.†
Macmillan Dictionary of Modern Economics (edited by David W. Pearce, 1992):

consumer sovereignty. The idea that the CONSUMER is the best judge of his or her own welfare. This assumption underlies the theory of consumer behaviour and through it the bulk of economic analysis including the most widely accepted optimum in WELFARE ECONOMICS, the PARETO OPTIMUM. The notion is, however, challenged in that governments do provide MERIT GOODS, while ADVERTISING is held, especially by J.K. GALBRAITH, to distort consumers' preferences.

William Vickrey (Microstatics, 1964, p. 217):

consumers are to be considered the court of final recourse as to the relative satisfactions that they derive from different situations, and that they are capable of predicting accurately the satisfaction which they will derive from the consumption of various contemplated combinations of goods.

Abba Lerner ("The Economics and Politics of Consumer Sovereignty", 1972):

The basic idea of consumer sovereignty is really very simple: arrange for everybody to have what he prefers whenever this does not involve any extra sacrifice for anybody else. … One of the deepest scars of my early youth was etched when my teacher told me, “You do not want that,” after I had told her that I did. I would not have been so upset if she had said that I could not have it, whatever it was, or that it was very wicked of me to want it. What rankled was the denial of my personality—a kind of rape of my integrity. I confess I still find a similar rising of my hackles when I hear people's preferences dismissed as not genuine, because influenced or even created by advertising, and somebody else telling them what they “really want.”
… as an economist I must be concerned with the mechanisms for getting people what they want, no matter how these wants were acquired.

Joel Waldfogel ("Does Consumer Irrationality Trump Consumer Sovereignty?", 2005):

According to economic theory, individual utility—and social welfare—are maximized when individuals make their own consumption choices. This justifies the doctrine of consumer sovereignty that underlies standard lessons of economics: for example, that lump-sum grants are more efficient than price-changing subsidies or government grants in kind. An obvious corollary is that consumption choices made by others tend to generate less satisfaction than one’s own choices

Lester Thurow ("Cash Versus In-Kind Transfers", 1974):

the received wisdom of economics seems to support transfers in cash (the 'New Federalism') rather than in kind. Governments may modify the market distribution of purchasing power, but then should stand aside and allow consumer sovereignty plus competitive markets to work. Any further intervention lowers consumer utility below what it otherwise could be. In a federal government, presumably an ideal system of cash transfers would consist of unrestricted cash transfers from the federal government to individuals and to state and local governments, who would in turn make unrestricted cash transfers to individuals. The result would be an optimum distribution of income within each area with each individual free to make his own expenditure decisions.

†I avoid the term consumer sovereignty because it is a rather muddled and vague term that has at least two distinct interpretations other than the one given above. Rather than use the term consumer sovereignty  at all, to express the ideas discussed here, I would simply say that:

"Each individual is the best judge of her own welfare"; and
"In general, cash transfers are better than in-kind transfers".


Effect in the Ramsey model of a decrease in population growth

Let $\left(\hat{c},\hat{k}\right)$ and $\left(\tilde{c},\tilde{k}\right)$ be the old and new steady-state levels of $c$ and $k$ respectively.

You make the correct observation that $\tilde{k}=\hat{k}$ and $\tilde{c}>\hat{c}$.

However, starting from the steady-state $\left(\hat{c},\hat{k}\right)$, and experiencing an unexpected negative shock to the population growth rate $n$, it is easy to see from the equations determining equilibrium dynamics that the economy does not instantaneously transition to the new steady-state $\left(\tilde{c},\tilde{k}\right)$.

It would be a good exercise for you to work out what the qualitative features of the transition path from $\left(\hat{c},\hat{k}\right)$ to $\left(\tilde{c},\tilde{k}\right)$ look like after the population growth shock.


How much is currently invested in artificial general intelligence research and development?

I agree with @1muflon1's comment that this is quite broad, because AGI can be defined in many different ways. For instance, do we consider simple learning tasks as AGI or metacognition?
This is a brief survey of AGI startups, but another point to keep in mind is that few startups/firms are actually gunning for AGI because we generally have problems in other, "easier" subfields of machine learning. For example, there are still difficulties with reinforcement learning for robotics or few-shot learning, both of which are much more specific tasks than AGI.
In sum, it's hard to provide good estimates because 1) the sum of money is likely to be small and 2) AGI is often defined broadly and used more in pop culture than by technical researchers.


Generally, price is endogenous in this set of simultaneous equations.  One strategy we can use to overcome the bias is to find a valid instrument for price—call it $Z$.  We’d need something that satisfies $Cov(P, Z) \neq 0$ and $Cov(Z, \mu_1) = 0$.

The trouble with simultaneous equations is that upon observing some $(P, Q)$ pair, all we know is that it lies at the intersection of supply and demand. As we get more pairs (more data), we’re not tracing out supply or demand curves—we’re just getting a bunch of equilibrium points and aren’t
certain whether it’s supply or demand (or both) that moved!

To estimate the demand curve, what we'd like to do conceptually is to hold it fixed while we shift the supply curve around.  In theory, these new equilibrium points would trace out the demand curve itself.

Returning to the instrument, the statement $Cov(P, Z) \neq 0$ means we'd like to find some variable $Z$ that shifts the supply curve.  Maybe this is something like subsidies to tech firms or weather in agricultural regions.  Then you'd hope $Cov(Z, \mu_1) = 0$ so $Z$ doesn't shift demand around, too (then we'd be back where we started).

If the instrument were to be valid, you could form a consistent estimator.


Why use empirical macroeconomic models when they are not policy invariant (Lucas Critique)?

The response to the Lucas Critique was the emergence of RBC and DSGE models. Using microeconomic foundations of macro models we can simulate how behavior changes when policy changes and only estimate "deep" structural paramateres that are not policy variant. Before microfoundations we were estimating models where the estimation included the actions of people. With microfounded models today we try to seperate the actions. This was not possible in older models as we actually did not consider how people act or react, whereas microfoundations tell you how people would react.

However this is a difficult task as once you introduce agents that actively think about their actions you must take into acount their future expectations, which are unknown. One way to deal with this are rational expectations.

A further issue is that such models predict reactions that are quicker than in the data. If agents are perfectly rational, have perfect foresight and all information, they react quickly and perfectly. The solution to this today is to add frictions that slow these reactions down. However old models that we used to estimate (think about IS-LM and especially AS-AD models) also have the huge issue that people are too stupid (only adaptive (backward looking) expectations, do not take information into account when forming expectations, don't think about what may come in the future) and this is partially a piece of the Lucas Critique. Now we have the problem that people are too smart or super-rational. Some models (see models where some fraction of agents are "Rule of Thumb Consumers") now also assume that some agents are fully rational (many indeed are) while some just are backward looking (as in the normal IS-LM or AS-AD model), as a balance of the two things to generate a more realistic picture.

As for the critique of rationality: In many experimental micro studies rationality fails. However this does not tell us how many small deviations from rationality aggregate, which is what macro is interested in. It may occur that from many different deviations in different directions that in aggregate rationality is still a good approximation.

Furthermore approaches have been developed now to deviate from rational expectations. These are very difficult to solve though. One interesting way is to assume people are rational, but do not have all the information, which is a reason most people make mistakes. These are models of information frictions. Key words: Rational Inattention (e.g. Sims) and Inattentiveness (e.g. Reis). Another related approach includes Learning Models.

To summarize: we try to have models that are immune to the Lucas Critique. These often require rational expectations to be solved, but other approaches are also being developed.


The answer will vary slightly depending on which notion of elasticity you're using.

Arc elasticity (or midpoint elasticity) uses the formula
\begin{equation}
\epsilon^\text{arc}=\frac{Q_1-Q_0}{P_1-P_0}\cdot\frac{\frac12(Q_0+Q_1)}{\frac12(P_0+P_1)},
\end{equation}
where $\frac12(Q_0+Q_1)$ is the midpoint between $Q_0$ and $Q_1$. Note that in your case, $7.5$ is the midpoint between $7$ and $8$.

Point elasticity uses the formula
\begin{equation}
\epsilon^\text{point}=\frac{\mathrm dQ}{\mathrm dP}\cdot\frac{P}{Q}.
\end{equation}
Here, $\frac{\mathrm dQ}{\mathrm dP}$ is the derivative of the demand function (evaluated at the point you want to calculate the elasticity, but it will be constant if demand is linear), and in your case, $P=7.5$ and $Q$ is the quantity demanded corresponding to that price.


Does every structural policy change count as structural break?

Assuming I understand the question appropriately, the answer, generally, is yes - but in its application to times series analysis/forecasting not all "structural breaks" would end up being included in a given model.  For example, if we are modeling labor supply, policy related to minimum wage would be hugely important (an increase can be seen as a structural break or a shock).  However, a small tariff placed on sugar imports, though it may effect labor supply in some minute way in some region, would not constitute a structural break in consideration of our model (though, by definition, it is a break).

To answer the second part of your question related to what conditions would necessitate inclusion in a model, I would argue that it is an intersection of theory/logic and statistical relevance.  If the influence of the break on other variables is virtually unobserved, or if its coefficients are far from significant then it need not be included (barring some other compelling case for its inclusion).  Or, we could say during the modeling process some break/shock variable added no advantage to the model fit (thinking AIC or SIC/BIC here).

Hopefully that at least gets at your question.


Allowing the reserve requirements drop to zero

I believe that the Fed is more focused on saving the US economy from collapsing because of the effects of the coronavirus situation.
When the Fed wants to loosen monetary policy and increase liquidity, it lowers the reserve requirements ratio which will make lending cheaper.

Lowering the reserve requirement increases the amount of money that banks have available to lend. Since the supply of money is higher, banks can charge less to lend it.
Thereby more people will access to have loans at a cheaper interest in order to encourage spending.
https://www.thebalance.com/reserve-requirement-3305883

With the fast-spreading coronavirus posing a dire threat to economic growth the Fed is hoping that this move will help prevent the US economy from collapsing.
https://www.cnbc.com/2020/03/15/federal-reserve-cuts-rates-to-zero-and-launches-massive-700-billion-quantitative-easing-program.html

I am sure that the Fed is concerned about bank runs but it may be the case that they only have few options available to help encourage spending to prevent our economy from collapsing.


Why is GDP equal to the sum of revenues?

It is right that the whole savings are invested.

You can use the (simplified) national income and the national output to show the equality.

The national income is $NI=S+I$

The national product is equal to the national income: $NP=NI$

The national product is equal to the sum of consumption and investment: $NP=C+I$

By using this three equations it can be shown, that $I=S$.

This presumption is made for the IS-Curve at the IS-LM-Model.


Is it possible to repay the world debt? If not, why do we care?

So there are a couple of things you want to consider:

1st: It is not true that if some country gains 1% some other country must have lost that 1% of economic growth. The economy is not a zero-sum game where the gains of one person are the losses of another one. In fact, usually, it is the opposite. Think about it this way, if the US grows by a lot it also means that they will buy more things from other countries so these countries will probably also grow faster.

2nd: Money is just the means by which we keep track of, and do transactions. At the end, what really matters is how much stuff is produced (goods or services). That is why if some country produces more things, everyone can benefit from it.

I do understand your logic that if the Bank produces 100B of coins how can they expect to get back 101B? Once you think of it in terms of stuff, the puzzle is resolved as follows:

The bank lends you the money so you can buy and plant seeds (these are the 10B), they expect to get back some of the benefits from this productive project after you produce apples (they could directly ask to pay back in apples, but this would be impractical, that's one of the reasons we use money, so instead  they ask you for the 10.1B). Of course, if the bank does not print more money where will the .1B come from? Well, indeed the bank will never ask for 1% interest if they know there is no physical way to actually pay it back.
What really happens is that the bank understands that if an economy is growing, she needs to constantly be printing money, so people can easily do and keep track of all the transactions they do. Therefore, when they ask for the 1% interest, first this is possible, because there is more money out there in the economy, and second, they choose how much interest to charge in order to reflect that they expect you to do something productive with the money (interest also depend, of course, on how risky is your project, and other factors).

I am over-looking the whole issue of inflation to keep things simple and explain the main logic of why banks (including the central bank) charges interests. Whenever you think about whether a debt can be paid or not; the relevant question is not if there is enough money out there to pay it. Rather the relevant question is if the money loaned is being used in a productive way so that it can later pay the interests. And if the answer is no, we care because that means that the money loaned is being wasted, instead.


How to derive utility possibility frontier?

Given that $u_A(x_A, y_A) = \sqrt{x_Ay_A}$, $u_B(x_B, y_B) = \sqrt{x_By_B}$ are the utility functions of A and B, and total endowment of X and Y in this pure exchange economy is $\omega_X$ and $\omega_Y$, utility possibility frontier (UPF) is the set of all utility pairs $(\mu_A, \mu_B)$ such that the there exist a Pareto optimal allocation $((x_A, y_A), (x_B, y_B))$ with the property: $\mu_A = u_A(x_A, y_A)$ and $\mu_B = u_B(x_B, y_B)$. So to find UPF, we will use the following conditions:

Optimality:

$\displaystyle \frac{y_A}{x_A} =\frac{y_B}{x_B}$

Feasibility:

$x_A + x_B = \omega_X$ and $y_A + y_B = \omega_Y$

Utility:

$\mu_A = \sqrt{x_Ay_A}$ and $\mu_B = \sqrt{x_By_B}$

Solving the above system of equations we get the UPF as:
$\mu_A + \mu_B = \sqrt{\omega_X\omega_Y}$.


Growth theory / Initial GDP: positive coefficient sign

It's really difficult to give a satisfactory answer without having the data at hand. I would do two things:

Check carefully the data and computation of growth rates. By experience, it happens that some "particular" result is the consequence of data or computation issues. You may for example observe any issue by plotting your growth rates and initial GPDs with the name of the countries.
Use your data but exactly the same period, same countries and same methodology than the literature and check if you still find a contradictory result. Then, you can mix: use the same source of data, same countries, same period but your methodology, etc...


Why hasn't JPE formally retracted Emily Oster's article "Hepatitis B and the Case of the Missing Women" (2005)?

Why hasn't JPE formally retracted Oster (2005)?

In this case there simply is no reason to retract the paper.
Academic papers are typically retracted for one of the following reasons:

Retraction for error: For example, if paper claims that if $3+x=15$ then $x=10$ and such error significantly changes conclusions from the study (usually the mistake has to be severe). However, this has to be some serious error not something like well when your regression errors have just heteroskedasticity it might be better and more appropriate to use white errors instead of HAC errors - this can also sometimes change result of a study but people are allowed to make some judgements when setting up their models.
Retraction for fraud or misconduct: For example, plagiarism, falsifying data or falsifying result. Some ethical issue (e.g. it was later uncovered data for the paper was collected unethically and so on).
Retracting Paper on Request of an Author: Sometimes authors will ask publication to retract the paper. This can happen even if the paper has no issues. For example, recently a paper on police killings was retracted by authors who did not liked how their paper was portrayed in media.
Retracting Paper Due to Public Relations: It sometimes happen readers or wider public wants the article to be retracted. For example, PLOS One retracted one biology paper after readers complained it had reference to "creator"  - although retracting paper for this reason is always controversial, and not all journals will do such thing (and the journal was criticized for this retraction heavily).

In the case above 1 and 2 does not seem to apply. As written in by Levitt in his blog (also mentioned in the comments) [emphasis mine]:

Emily wrote a paper arguing that high rates of Hepatitis B in China explained a large part of the missing women puzzle. Medical data suggested women with Hepatitis B gave birth to more sons — many women in China are infected, thus too many sons. It seemed like a crazy theory when I first heard it, but she put together extremely compelling evidence from a variety of sources to support her argument. Eventually we published it in the Journal of Political Economy, where I was an editor.

Then along came a host of other academics, including my friend and former student Ming-jen Lin, who gathered data from new sources that didn’t support Emily’s conclusions. Usually, these debates become quite acrimonious and linger on until no one cares any more. Certainly no one would admit they were wrong.

Consequently, in this case there is no evidence that there was any error in the study, or fraud or other misconduct. From what I understand study was overturned thanks to new better data.  Next, obviously the author did not requested retraction otherwise the article would not be still listed. In addition, there does not seem to be any 'mob' pressure to retract it (the point 4) and as mentioned under the point 4 many journals would not just succumb to mob pressure.

And later, Oster et al published what amounted to a full retraction: "Hepatitis B does not explain male-biased sex ratios in China" (2008, 2010).

No that was actually not retraction that was new publication with new method and new data that did not confirmed the result of original publication. Authors will usually specialize in a single area of research. It is always possible, and in fact in empirical studies its not too rare, to see that new and better data and statistical techniques change conclusions of a study.
For example, if a researcher conducts randomized control trial with 30 participants they might find positive effect that is publishable and publish it. However, even if the results are interesting having just 30 participants is a weakness (which can be criticized but might not necessary be reason for rejection of paper). Later in their career the scientist might decide to repeat the experiment  with bigger sample lets say 1000 participants and it might happen that new study will show previous study was wrong. That new study subsequently can still get published, and the old one wont get either retracted or any errata.

What I found strange is that Oster (2005) still appears on the JPE website (and also JSTOR). There is no retraction, errata, or any other clue to the uninformed reader that there might be anything wrong with the article.

Well this is actually not strange at all. This is how scientific publishing normally works. Older results are getting challenged every day in science. For example, you will still find literally hundreds of old physics, economics, biology articles discussing results or theories that were proven wrong. That is no reason to retract those old articles as they are and should still remain part of the corpus of scientific literature.
This is the reason why any scientist will tell you that if you want to understand some topic you have to do a literature review on the issue. You cannot ever in any field of science just take any single paper at face value, no matter who wrote it and in what journal it is published in. Any paper has to be properly placed in the mosaic  of wider literature to be properly understood. This is why it is not enough for a layman to simply read a paper on lets say some medication and then start prescribing it.


Why are indifference curves (often) of infinite length?

If you are using the definition of indifference curve on wikipedia then they can be of finite or infinite length. For example if you are completely indifferent between the two goods, the indiffernce curve would be of the form $x+y=const$ which is of finite length (in the positive quadrant).

A necessary criteria for infinite length would be that having $0$ of one good provides you with zero utility, so that any combination of a little bit of both has a higher utility than an arbitrary quantity of one good and zero of the other good.

Edit in response to comment: Indifference curves come in families, one can consider the family of indifference curves $x\cdot y = const.$, they satisfy the definition, are of infinite length and if either $x$ or $y$ is equal to zero, then the utility is equal to $0$ as well.


Market baskets: cross-border comparisons

It's very likely that that these major categories in the hierarchy of goods in the market basket differ somewhat by country.  The sub-categories within these major groups are even more likely to differ across countries, as the market basket is supposed to reflect the purchases of a typical consumer, and typical purchases are obtained from consumption surveys of the population.

In the U.S., the eight major categories used for calculating the rate of inflation (Consumer Price Index) are:

Food and Beverages
Housing (includes furnishings)
Apparel
Transportation
Medical
Recreation
Education and Communication
Other Goods and Services

In Canada, the major groups are:

Food and non-alcoholic beverages
Shelter
Household operations, furnishings and equipment
Clothing and footwear
Transportation
Health and personal care
Recreation, education and reading
Alcoholic beverages, tobacco products and recreational cannabis

Sources:

U.S. CPI Handbook of Methods (https://www.bls.gov/opub/hom/pdf/cpihom.pdf)
The Canadian Consumer Price Index Reference Paper (https://www150.statcan.gc.ca/n1/en/pub/62-553-x/62-553-x2019001-eng.pdf)


What is the difference between USD/EUR and EUR/USD when talking about Forex

When the forex market does a quote, it is done in currency pairs. When you sell a currency, you have to be buying something in return. That is why they are quoted in pairs.

For example when you buy an apple, you are selling your dollars to get it. Similarly in forex, when you sell EUR to get USD, you are using the EUR/USD currency pair, or USD/EUR depending on which way you want it, one is simply the inverse of the other.

However, these pairs are counter-intuitive in a such a way that they are not fractional terms. It is simply a quote. What I mean is that in the quote EUR/USD, it is expressing how much 1 EUR is worth in USD and in USD/EUR, how much 1 USD is worth in EUR. The first currency in a pair is called the "Base" currency and the second is called "Quote" or "term" currency. Think of it this way, the first currency in a pair is your "commodity" currency and the second currency is its price.

As for what "EUR/USD -0.04% -1.1208" means, I think I'll need more context to it. If by -0.04%, you mean that EUR/USD has dropped by 0.04%, in example, it simply means that if 1 EUR is 1.128 USD, it is now 1.1275 USD. Therefore, your EUR is now cheaper, USD has appreciated against the EUR or, EUR has depreciated against the USD, or a combination of both.

If you are confused, just remember the first currency in a pair is your "commodity" currency. Replace EUR with apples and you should see it clearer.

Read more about it here and here.


Can a game with a unique pure strategy Nash equilibrium also have a mixed strategy equilibria?


Does concavity of the utility function has any bite?

This post shows clearly why in the world of "standard" ordinal utility, concavity of a utility function cannot obtain an economically meaningful interpretation, although it may be useful as a mathematical property.

But "standard" ordinal utility is not compatible with Econometrics, because Econometrics deal inherently with situations where there exists uncertainty, and in a framework with uncertainty we move from "fully ordinal utility" to Expected Utility theory, where properties like concavity have economically meaningful content -they express attitude towards risk (as well as "preference intensity", as this laborious post shows).

The concavity assumption/restriction is made because there is universal consensus that the vast majority of people exhibit risk aversion in their economic behavior.


How to enhance one's comprehension of the descriptive meaning of formal economic statements?

I faced a similar issue coming from mathematics where context plays very little role in framing theorems. I was a top maths student. But Econ grad school was difficult for this. The solution I found (and successfully so since I publish well) is to read as much “soft Econ” as possible: policy discussions, business reports, newspaper analysis. Then I went back to formalized economics, and it worked. The conclusion I reached is that economics theorems are very context dependent, and typically cannot be grasped without cultural/policymaking/business context.


Are there strains of economics that are "no (political) economics"?

Government behavior/actions play a big role in economics, but economic study certainly isn't constrained by "arbitrary" political forces.  As @dismalscience pointed out in his comment, the existence of black markets points towards the drive towards a market-based system no matter the laws or political systems of a given country.  Rather than picture economics as a derivative of a set of laws, picture the laws and political systems as being constructs to help shape the underlying economic system (to correct market failures, etc.).  The goal and job of those that study economics is to help explain how all of these moving forces interact with one another.

Also, in the comments it is asked why we need economic theory to set prices if it is achievable through "common sense."  We don't need economic theory necessarily to set the price, but rather to understand why the price is set, what led to a given price, and how might the price change in the future.  The example used of grocery store prices was used for simplicity, but certainly you could imagine far more complex economic interactions where a solid economic theory would come in quite useful (think international trade, wall street, the labor market of the EU, etc.).


I am not a CS guy so this answe may be flawed in many ways. Take it more as a long comment and please give some feedback.

Economists view Ethereum as a pure Arrow-Debreu market economy. In an AD economy you can write fully contingent contracts specifying commodity trades conditional on any future event at any particular date. Therefore, an AD economy is, by definition Turing complete. In that sense, for economists Ethereum is a cool implementation of an ideal but nothing new.

The problem is that, as Arrow himself recognized (but not Debreu!), this model of the economy is not realistic in any way because there are frictions to contracting, undescribable events, unawareness, asymmetric information, uninsurable risks, etc.

At a more fundamental level, economics has another problem that seems to me to relate to the halting problem. if an economic system was decidable, then the agents in the economic model would have to either know the answer or not know the answer. In the former case, there would often be an eternal loop, since the answer to the problem would be part of the problem itself (this is the essence of the Lucas critique). If you assume they do not know the answer then you are imposing a certain rule on their actions that will not be universal and therefore would not allow for Turing completeness on the whole economy.


Difference between no perfect multicollinearity and no multicollinearity

Perfect multicolinearity means that two independent variables are perfectly correlated with their $r^2=1$ multicolinearity technically refers to any non zero correlation between two independent variables but when it is mentioned as a problem it usually implies that the correlation is high. So there is a difference between the two terms perfect multicollinearity is a special case of multicolinearity.

Also they have different implications for OLS. Under perfect multicolinearity OLS cannot be identified. Under high muliticolinearity OLS can be identified but standard errors will be inflated.

However, to make the things confusing, I noticed that sometimes people use the terms interchangeably. I only have the Wooldridge book so I can’t check the text of the other one to compare it but I think they actually refer to the same thing.


what in real terms causes one nations currency to be more valuable than another?

Exchange rates are prices.  Just like any other price, they change based on supply and demand.  Country A's currency appreciates relative to B if the citizens of country B want county A's currency.  Why does B want A's currency? Typically for the purpose of buying the goods, services, and investments produced by A.

So what would be some examples of events that would cause A's currency to appreciate relative to B?

If A starts producing some goods B wants, or producing them efficiently, then B will want A's currency in order to buy them
If A has desirable investments (savings accounts, financial assets, or real assets) and B wants a place to save for the future
If A expands their money supply more slowly than B, which reduces the available supply of A's currency relative to B's.
If A's currency is useful for some other purpose, like acting as a medium of exchange between B and another country C.


Graphics card shortage - why don't they just raise prices?

Actually, just recently (a few days ago; January 2021 - for anyone reading this later), the major retailers of these graphics cards did raise their prices (see example here for EVGA). Or more broadly Forbes reports that at the beginning of this year due to shortages and in addition also due to increased tariffs on China graphics card prices:

are going up anywhere from 7 to 25%.

It’s difficult to know how much of that is due to the China tariff and how much due to shortage, but most sites reporting on this quote the shortage of GPUs due to increased demand from Bitcoin miners as one of the main reasons for price rise.
But the question still can be reformulated in another way. Why did it take them so long to adjust their prices?
I believe the answer to that is sticky prices. Research generally finds that in the EU and the US respectively data on the frequency of price changes (Alvarez, et al., 2006).

imply average price durations close to one year in the euro area and slightly above half a year in the US.

In addition, more recent studies for US show that there is also very large heterogeneity in price stickiness as about 1/5 of prices take more than a year to change (see discussion Klenow & Malin 2010).
The average price durations above are not that far from the time it took graphics card prices to start to rise.
It is very difficult to know exactly what is the reason for price stickiness on a firm level, but research on price stickiness further shows that:

Firms usually don’t want to raise prices ahead of their competitors, hence in data we can observe a great degree of price change coordination among firms in the same industry (see Levy et al. 1998; Klenow & Malin 2010).    This seems to fit the facts of this case as the big price increase followed the tariff news (but as mentioned above, observers seems to agree that tariff could not account for all the price increase), and thus firms likely reasoned that everyone will have to increase prices now, so now is the right time to increase their prices as well.
The above is also further corroborated by Eichenbaum et al. (2011) who show that price changes usually take place when firms costs change but that the change in costs itself is not sufficient to induce the change in price (similar findings were also found by Dias et al. 2011).

Firms will change prices less frequently when ‘menu’ costs are high. Anderson et al (2015) show that menu costs can easily reduce frequency of price changes by over 13 percent which is not trivial. I would not be surprised if it would be expensive to change pricing strategy of big multinational company such as Nvidia or AMD. Possibly some of the prices might be also locked by contracts with retailers.

When manufacturers use reference prices and recommended prices, which is usual strategy for many manufactures that want to ensure uniform pricing, the prices tend to be stickier than retail prices  (Klenow & Malin, 2010 - although not all studies find this result). Indeed, in this case we can clearly see that retailers were rising prices ahead of manufacturers and especially scalpers were already taking advantage of the situation by reselling GPUs with higher prices much earlier (see news article on that here).

Empirical evidence shows that prices tend to be stickier when competition is less intense (Dias et al. 2011). GPU market is for all practical purposes a duopoly of Nvidia and AMD.

Empirical evidence also shows that goods with higher elasticity of demand adjust their prices more slowly, and this holds even in presence of demand shocks  (Dias et al. 2011). However, I could not find any empirical study on the elasticity of demand for GPUs so it is hard to know what role this plays in this case, but a priori I would expect GPU demand to be more price elastic since GPUs are not necessities and especially high-end GPUs could be considered a luxury good, and you can still play games with reasonably good graphics even on less powerful GPUs. Consequently, this could be a reason as well.

the evidence also shows that large firms are slower to adjust their prices (again see Dias et al. 2011). This also seems to fit the observable events. That is, as mentioned previously, scalpers adjusted their prices almost immediately, large retailers and manufacturers took their time.

This being said, as mentioned earlier, it is important to keep in mind that it is very hard to explain the behavior of few firms especially as the events unfold. There might further issues playing some role here. Possibly even a failure to act optimally perhaps due to some agency problems within the firms.


(The second equation for the value function of the unemployed should be
$$
v(w,U)= \max \{v(w,E); \,u[c,1]+\beta\int v(w', U) dF(w')\}. \quad (*)
$$
)

...how do you know when your problem solution require more than
one Bellman equation?

Whenever the state space of the problem contains discrete coordinates,
there would be  "multiple" value functions, indexed by the discrete coordinates.
Here the state space is $[0, \infty) \times \{E, U\}$. The second coordinate is discrete. So there are "two" value functions $v(w, E)$ and $v(w, U)$.

How do you realize that?

Any variable that determines the optimization problem faced by the agent is part of the state.
In this particular example, an unemployed agent has the option to switch to being employed. An employed agent has no choice but to stay in the job---"...to work forever".
The choice set, therefore the decision problem, faced by the agent are different depending on his employment status.
This tells you that employment status is part of the state.
(Note that, if the employed worker has the option to leave the job, the decision problem faced by the agent are still different depending on his employment status. In that case you would still have two value functions and they would be "intertwined", so to speak.)

Argue...the optimal strategy has the reservation wage property.

This is more or less immediate from the equation $(*)$.
The reservation wage $w^*$ is given by
$$
v(w^*,E) = u[c,1]+\beta\int v(w', U) dF(w').
$$
At $w = w^*$, agent is indifferent, $v(w^*, U) = v(w^*, E)$.
You would expect that for $w < w^*$.
$$
v(w,U) = u[c,1]+\beta\int v(w', U) dF(w') > v(w,E).
$$
This is a typical option-exercise problem.
The difference $v(w, U) - v(w,E)$ when $w < w^*$ is the option value. If the offered wage $w$ is too low, the agent would rather keep the option and not exercise it.
(This is an American-type option, which can be exercised anytime. Same "exercise threshold" phenomenon occurs in the continuous-time setting, which is sometimes more convenient. There the threshold is given by the smooth-pasting condition.)
General Comments
The general formulation of the Bellman equation is
$$
V(s) = \max_{c \in \mathcal{C}(s)} \int_{\mathcal{S}} V(s') dF(s'; s, c),
$$
where $\mathcal{C}(s)$ is the choice set faced by the agent at state $s$ and
$s' \mapsto dF(s'; s, c)$ is the Markov transition kernel if agent makes choice $c$ at state $s$.
(For notational simplicity, assume no period utility/discounting/etc. The discussion would not change.)
Therefore, by definition, any variable that determines the optimization problem faced by the agent is part of the state.


Check out sunk cost. Although this might seem completely counterintuitive, the investment you have made should be irrelevant to your decision to stay or quit, if this investment has already been made and cannot be recovered.

What matters is whether the benefits you are yet to incur outweigh the costs (that you are yet to incur). So the decision should be a forward-looking one.

The above reference mentions that in the real-world, people often do not behave in this way. According to sunk cost theory, this is irrational, but this is only one theory.


maximizing leontief-type utility function

You can re write $u(x, y)$ as
\begin{eqnarray*} u(x, y) = \begin{cases} ax + y & \text{ if } x > y\\ ay + x & \text{ if } x \leq y\end{cases} \end{eqnarray*}

When one plot the indifference curves, this is how they will look:

Here $\mu_1 > \mu_2 > \mu_3$.


If two variables are not cointegrated, can one still cause the other?

Causality between time-series variables does not require the two to be cointegrated.

First, cointegration requires that each series be $I(1)$. It is certainly possible for two $I(0)$ series to follow a causal relationship (or two $I(d)$ variables for that matter).

Second, cointegration implies a long-run equilibrium among the series, which is not required for a causal relationship. A long-run equilibrium between two time series variables $x$ and $y$ implies that
$$y_t-a-bx_t$$
is a stationary process. In other words, if shocks are 'turned off', the values of $x$ and $y$ will converge such that $y_t=a+bx_t$.


Can an F-test be used in a regression that is quadratic, or can it only be used in a linear regression model?

Since your model is like
$$
Y = \alpha + \beta_1 X_1 + \gamma_1 X_1^2 + \beta_2 X_2 + \gamma_2 X_2^2 + \epsilon,
$$
it is still a linear model -- because it is linear in the terms. ("Linear" refers to the terms like $X_1$ and $X_1^2$ being linearly combined.)
You can use an $F$-test for your model.


What is the difference between present value and face value?

Suppose the face value of a bond is $M$ and its interest rate is $\tau$. This means it will pay $\tau \cdot M$ interest every year (other periods are also possible) and at the end of its run (its maturity) it will also repay the face value $M$. Government bonds are usually sold in auctions. Whatever ends up being the market price is considered to be the present value of the bond. Using the cash flow you can also calculate the yield of the bond. If and only if the face value and the present value are equal the yield will be equal to the interest rate.

For an example see the US treasury's website. Click on bonds to see face value, interest rate and maturity, click on the bond serial to see its price (present value).

Further reading here. (Link copied from a comment by @Jamzy .)


How does one derive the elasticity of substitution with implicit functions?

To derive the formula for the elasticity of substition I consider a function with two arguments $f(x_1,x_2)$. I then consider a level curve $f(x_1,x_2) = c$ assumed to define $x_2$ as a function of $x_1$. From this it follows that
$$\frac{\partial }{\partial x_1} f(x_1,x_2(x_1)) = c \Leftrightarrow \\[8pt]
f_1(x_1,x_2) + f_2(x_1,x_2) \frac{\partial x_2}{\partial x_1} = 0 \Leftrightarrow \\[8pt] \frac{\partial x_2}{\partial x_1} = - \frac{f_1(x_1,x_2) }{f_2(x_1,x_2) }$$
giving the slope of the level curve in terms of $f_i := \partial f/\partial x_i$ for $i=1,2$. Given the definition of MRS I conclude that
$$(1) \ \ MRS(x_1,x_2) := \frac{f_1(x_1,x_2) }{f_2(x_1,x_2) } = -\frac{\partial x_2}{\partial x_1}.$$
Then define the ratio function
$$r(x_1) = x_2(x_1)/x_1$$
again using that $x_2$ is a function of $x_1$. This function can be inverted to get $x_1 = r^{-1}(r)$ and using again that $x_2$ is a function of $x_1$ the MRS can be written as
$$MRS(r) := \frac{f_1(r^{-1}(r),x_2(r^{-1}(r)))}{f_2(r^{-1}(r),x_2(r^{-1}(r)))} $$
and differentiated with respect to $r$ to get
$$(2)\ \ \frac{\partial MRS(r)}{\partial r} = \left[\frac{\left(f_{11} + f_{12}\frac{\partial x_2}{\partial x_1}\right) f_2 - \left(f_{21} + f_{22}\frac{\partial x_2}{\partial x_1}\right)f_1}{f_2^2} \right]\frac{\partial r^{-1}(r)}{\partial r},$$
where the expression in square brackets is $\partial MRS(x_1,x_2(x_1))/\partial x_1$.
I then use that $r^{-1}(r) = x_1$ implying that $\partial r^{-1}(r)/\partial r = 1/\partial r(x_1)/\partial x_1$ and from the definition of $r(x_1)$ it follows that
$$(3) \ \ \frac{\partial r(x_1)}{\partial x_1} = \frac{\frac{\partial x_2}{\partial x_1} x_1 - x_2}{x_1^2} = \frac{}{}-\frac{f_1}{f_2}\frac{1}{x_1} - \frac{x_2}{x^2_1} = -f_1 r\left[ \frac{1}{f_2x_2} + \frac{1}{f_1x_1}\right],$$
using that $r =x_1/x_1$ and that $\frac{\partial x_2}{\partial x_1}  = - f_1/f_2$. Inserting the reciprocal value of (3) into (2) it follows that
$$(4)\ \ \frac{\partial MRS(r)}{\partial r} = -\left[\frac{\left(f_{11} + f_{12}\frac{\partial x_2}{\partial x_1}\right) f_2 - \left(f_{21} + f_{22}\frac{\partial x_2}{\partial x_1}\right)f_1}{f_2^2} \right] \left[ \frac{1}{f_1 r\left[ \frac{1}{f_2x_2} + \frac{1}{f_1x_1}\right]}\right].$$
What remains to be done is inserting this expression into the definition of the elasticity of substitution and cleaning it up. The Elasticity of substition is defined as the elasticity of $r = x_2/x_1$ as a function of $MRS$:
$$(5) \ \  \sigma := \frac{MRS}{r} \frac{\partial r}{\partial MRS} = \frac{1}{\frac{r}{MRS} \frac{\partial MRS(r)}{\partial r}}$$
being the reciprocal of the elasticity of MRS with respect to $r=x_2/x_1$. Inserting (4) into (5) it therefore follows that
$$1/\sigma =- \left[\frac{r}{f_1/f_2}\right]\left[\frac{\left(f_{11} + f_{12}\frac{\partial x_2}{\partial x_1}\right) f_2 - \left(f_{21} + f_{22}\frac{\partial x_2}{\partial x_1}\right)f_1}{f_2^2} \right] \left[ \frac{1}{f_1 r\left[ \frac{1}{f_2x_2} + \frac{1}{f_1x_1}\right]}\right],$$
which is reduced by removing the factor $r$ and using again that $\partial x_2/\partial x_2 = -f_1/f_2$ to get
$$1/\sigma =- \left[\frac{1}{f_1/f_2}\right]\left[\frac{\left(f_{11} - f_{12}\frac{f_1}{f_2}\right) f_2 - \left(f_{21} - f_{22}\frac{f_1}{f_2}\right)f_1}{f_2^2} \right] \left[ \frac{1}{f_1 \left[ \frac{1}{f_2x_2} + \frac{1}{f_1x_1}\right]}\right] \\[8pt]
= -\left[\frac{\left(\frac{f_{11}f_2}{f_1} - f_{12}\right) f_2 - \left(f_{21} \frac{f_2}{f_1}- f_{22}\right)f_1}{f_2^2} \right] \left[ \frac{1}{f_1 \left[ \frac{1}{f_2x_2} + \frac{1}{f_1x_1}\right]}\right]\\[8pt]
= -\left[\frac{\frac{f_{11}f^2_2}{f_1} - f_{12} f_2 - f_{21} f_2+ f_{22}f_1}{f_2^2} \right] \left[ \frac{1}{f_1 \left[ \frac{1}{f_2x_2} + \frac{1}{f_1x_1}\right]}\right]\\[8pt]
= -\left[\frac{f_{11}}{f^2_1} - \frac{f_{12}}{f_1f_2}  - \frac{f_{21}}{f_1f_2} + \frac{f_{22}}{f_2^2} \right] \left[ \frac{1}{\left[ \frac{1}{f_2x_2} + \frac{1}{f_1x_1}\right]}\right].$$
Taking the reciprocal and using that $f_{12} = f_{21}$ it follows that
$$ \sigma = - \frac{\frac{1}{f_2x_2} + \frac{1}{f_1x_1}}{\frac{f_{11}}{f^2_1} - \frac{2f_{12}}{f_1f_2}   + \frac{f_{22}}{f_2^2}} ,$$
which is the formula found in for example: Daniel McFadden Constant Elasticity of Substitution Production Functions
The Review of Economic Studies , Jun., 1963, Vol. 30, No. 2 (Jun., 1963), pp. 73-
83 [stated as equation (1)] or in
Knut Sydsæter, Arne Strøm, Peter Berck Economists’ Mathematical
Manual Fourth Edition  [stated as formula 5.21].
I leave it for you to show that this formula is equivalent to the one stated in the question (I however prefer this one because the way the indeces are written makes it easier to remember).


History $(D,D)$ (Yes, this is with some abuse of notation.):

On Path: Each player gets an average payoff of $4$.

Deviating: Say player $1$ deviates and instead chooses $H$. Then the resulting sequence of actions is the alternating sequence, $(H,D)$, $(D,H)$, $(H,D)$,...

Accordingly, player $1$’s sequence of payoffs is $(8,0)$, $(0,8)$, $8,0$... Player $1$'s average payoff is $(1-\delta)\big(8 + 0 + \delta^{2}8 + 0 + \delta^{4}8 + \cdots\big)$, which is $$(1-\delta)\frac{8}{1-\delta^{2}}=(1-\delta)\frac{8}{(1-\delta)(1+\delta)}=\frac{8}{1+\delta}$$
Hence, there is no profitable deviation here iff

$$\begin{split}
4 &\geq \frac{8}{1+\delta}\\
4\delta &\geq 4\\
\delta &\geq 1
\end{split}$$

Thus, $\delta \geq 1$ is a necessary condition (not necessarily sufficient because we still need to check that following tit-for-tat is still optimal for the other histories. I'll leave that to you!)


Do actively managed ETFs help the economy in any way, and if so, how?

Yes they do in in a several ways. The main ones are:

Trading (not just of the people who actively manage the EFT but generally) helps price discovery. Price discovery is an important role of stock market. It is socially beneficial for companies to be appropriately priced given their fundamentals and all other avaiable information as that helps divert capital to the right companies. Even if traders do not buy shares from company directly the market price of a share affects how much capital company get by issuing new shares. Furthermore, ETF are also not just for stocks but also commodities or exchange rates, where they help to correctly price those commodities or exchanges - which is again socially very valuable activity. Quick price discovery is one an important characteristics of well functioning capital markets. In turn empirically well-functioning capital markets have first order impact on economic growth  (have a look at Levine, 1997 or Levine 2005).

Trading improves market liquidity. It makes sure that whenever you want to liquidate stocks etc. you hold there is some buyer. For example, it would be much harder for people to save  by investing in stock market if it would be difficult to sell stocks you own when you need to. Having low liquidity can thus deter investment. In fact liquidity is so important that it is commonly listed in textbooks as one of the major determinants of asset demand (see Mishkin & Eakins: Financial Markets and Institutions ch 4). Consequently, even ignoring anything else just the fact that there is someone trading and creating liquidity in itself creates value and is beneficial for the economy.


Why is the ECB asking EU banks to hold off on buybacks?

The reason for the ECB to limit dividend payments is to preserve bank capital, to ensure safety and soundness of the banking system at a time of stress.  Remember that when banks pay dividends , the capital base is reduced.   If that happens at the same time that banks have to make provisions against bad loans , taxpayers could be at risk from bank bailouts.


If women are paid less for the same work, why don't employers hire just women?

There could be several reasons here are just few:

Principal-agent problems. Firms are typically not managed by their owners but by managers (agents) who act on the behalf of owners/shareholders (principals). While, owners might desire to maximize profits agents can to some degree act to pursue their own goals (see discussion in Hendrikse Economics and Management of Organizations).
For example, CEO might spend more lavishly on things like their private jet, limo, having HQ built by famous architect than profit maximizing CEO would. In the same way CEO/manager with taste for discrimination might pursue that taste even at the expense of profit (to a degree of course).

Consumers could have taste for discrimination. This is in fact classic example coming from one of the earliest work on discrimination. If consumers simply have preference for seeing men instead of women at some work and are willing to pay for satisfying this preference, there will be wage gap between men and women (see Mankiw Principles of Economics pp 395).
Mankiw has a good example of this for the race wage gap, but the logic of the example directly extends to gender wage gap:

Studies of sports teams suggest that racial discrimination has, in fact, been common and that much of the blame lies with customers. One study, published in the
Journal of Labor Economics in 1988, examined the salaries of basketball players and
found that black players earned 20 percent less than white players of comparable
ability. The study also found that attendance at basketball games was larger for
teams with a greater proportion of white players. One interpretation of these facts
is that, at least at the time of the study, customer discrimination made black players
less profitable than white players for team owners. In the presence of such customer discrimination, a discriminatory wage gap can persist, even if team owners
care only about profit.

It could be result of government intervention (See discussion in Mankiw Principles of Economics pp 395).
For example, many governments around the world mandate that firms have to provide women with job guarantee during late pregnancy/mothers leave. Perks like these are costly and in competitive markets firms will just pass costs of these perks directly on their female employees (in form of lower wage - although this would not affect total compensation it is part of explanation for wage differentials).

Most labor markets are not perfect. In imperfect labor markets there are often quasi-rents to have that can be split between employee and employer. If for example women tend to be less assertive during negotiation (e.g. see Amanatullah & Morris, 2010), they might end up with less quasi-rents than their male counterparts (this argument is covered in greater detail in the Papayapap's answer).

The list above is not exclusive, but those are one of the major reasons you will commonly find in economics textbooks.
To sum it up, the argument you make in the question is valid argument but it  is only guaranteed to work under host of additional supporting assumptions. For example, assumption that customers do not care about gender might be true in some profession but not in others (e.g. perhaps male anchor appears more trustworthy to audience?).
Although careful studies that control for all relevant characteristics show that gender wage gap is much smaller than laymen commonly perceive it to be, based on senseless comparison of aggregates between male and female pay, many studies show that there still is some gender pay gap that cannot be accounted for by objective factors. See discussion of these results in this nice explainer from The Economist. For example, in 2016 the gender pay gap in the UK was estimated to be only $0.8\%$, in France$2.7\%$ and Germany $3\%$, once objective factors were accounted for. That is admittedly small, but should not be trivialized nonetheless.


While you're on the right track, minimum wage is not the best way to get at it. What you're discussing is called "real wages", and it is usually calculated based on some average wage, as many workers make more than minimum wage. So yes, the price of eggs has gone down (if minimum wage is a useful metric for average income).


Implications of differentiable demand function on the utility function properties

In the (quantity, income) space, not being everywhere differentiable would mean that we would observe gaps, discontinuous jumps, or kinks, in the corresponding graph of the Marshallian individual demand function.

GAPS: if there was a gap, it would mean that for a certain range of income, the demand function does not exist. In other words, it is not defined for a certain range of income (but it still is defined with respect to prices). This could be interpreted that for this range, income does not affect quantity demanded. So "differentiable everywhere" excludes such a phenomenon.

DISCONTINUOUS JUMPS: If there was a discontinuous jump, it would mean that a specific level of income induces a "threshold effect" on quantity demanded. So "differentiable everywhere" necessitates that no such effects exist on quantity demanded.

KINKS: If there was a kink, we would have a "second-order" threshold effect, where quantity demanded would change at a discontinuous rate with respect to income. So "differentiable everywhere" excludes such phenomena.

These should now be "mathematicized" and projected back to the properties of the utility function (if, that is, they are relevant to it).

Note: In many utility maximization problems, income does not enter the utility function, but it is part of the problem's constraints. In others, we postulate a utility function as a bivariate function of one good and income.


Labour-saving vs. Labour-augmenting technical change

This is how I'd approach the problem. Please point out any issues on this method as it is based on my own approach (I have no textbook to reference this to).

Based on the information you have, you would run a regression of log output on log-labor, log-capital and log-human capital. This would give you a model like this.

$$\ln(Y)=\beta_0+\beta_1\ln(L)+\beta_2\ln(K)+\beta_3\ln(H)+\mu$$

in terms of a more "economic looking" equation, we take the expectation of this equation and take $e$ and raise it to the power of both sides giving us our production function.

$$\mathbb{E}[\ln(Y)]=\mathbb{E[}\beta_0+\beta_1\ln(L)+\beta_2\ln(K)+\beta_3\ln(H)+\mu]$$

$$\ln(Y)=\beta_0+\beta_1\ln(L)+\beta_2\ln(K)+\beta_3\ln(H)$$

Recall that we view $\beta_0$s the co-efficient on omitted variable $\ln(A)$ as the rate of technological change1,2
$$\exp\{\ln(Y)\}=\exp\{\beta_0+\beta_1\ln(L)+\beta_2\ln(K)+\beta_3\ln(H)\}$$

$$Y=A^{\beta_0}L^{\beta_1}K^{\beta_2}H^{\beta_3}$$

Using this form you can more comfortably  calculate elasticity of subsitution between $L$ and $K$. If your elasticity of substitution is greater than or equal to 1 you have a labor saving process, however if elasticity of substitution is less than 1, we either have a process which is either a Human capital augmented process of TFP augmented process 3.

Hope this helps

1. https://en.wikipedia.org/wiki/Solow_residual#Regression_analysis_and_the_Solow_residual

2. the actual "quantity" of $A$ can be calculated by
$$A=\left(\frac{Y}{L^{\beta_1}K^{\beta_2}H^{\beta_3}}\right)^{\frac{1}{\beta_0}}$$

3. this is of course assuming that either  $\beta_3>0$ and/or $\beta_0>0$.