The question in the title is actually a bit broader than the question in the body, I will try to answer both questions starting with the one in the title.
Answer to Question in the Title

Can banks 'create' money on their own or do they need help from other banks?

Yes, banks actually can crate deposits on their own without help of other banks, thanks to getting excess reserves in the interbank market or from central bank. For example, you can have look at McLeay (2014) Money creation in the modern economy which explains that in greater detail, and it will likely be covered in later chapters in the textbook unless it is old one (or you can have look at other textbooks such as Blanchard et al Macroeconomics a European Perspective).
Answer to the Question in the Body
In the body your question is about whether single bank can increase money supply within the money multiplier model. There the answer is no.
The multiplier model is one model of money creation that applies under certain circumstances, such as when central banks decide to exogenously control money supply by changing quantity of reserves (although the money multiplier exists in multiple models usually when people talk about money multiplier they talk about the simplest exogenous supply of money model). But the multiplier model won't hold (save for special cases) when the central bank conducts its policy through setting interest rate and supplying amount of reserves that is demanded.
This is because if there is only single bank it won't be able to create more reserves out of new deposits For example, let us assume that single bank starts with 100 reserves (R), which are assets  and 100 deposits (D) which are liabilities. The ballance sheet of such bank will look like:
  Bank A
   A | L
 -----------
100R | 100D

Now if bank issues a loan that means that it is converting some of the reserves to a loan. If a single bank would issue a loan and the person would kept the loan with the same bank then the balance sheet would look like this:
  Bank A
   A | L
 -----------
 10R | 100D
 90L |

And bank would not be able to lent any more money as with 10% reserve ratio it would already issue the maximum amount of loans it can make as it has no more excess reserves to lent out. However, the 90 loan that exists the bank can then become a reserve for a new bank, so for bank B the balance sheet would look like:
   Bank B
   A | L
 -----------
 90R | 90D

Hence now that original 90 loan became new 90 reserve and that would not happen if the original bank would just keep the money. Now the new bank can further break the 90R into 10R and 80L and so on.
Now as mentioned in the first paragraph the above won't hold in practice as even single bank nowadays can support more lending by getting excess reserves from other banks  or when central bank is willing to supply additional reserves instead of just setting the quantity reserves from the get go. Nowadays central banks mainly change the price for reserves (i.e. the interest) not amount of reserves. In that case that single bank can just increase the amount of R to create new loans (and matching deposits). However, even that is currently no longer the case because reserve requirements were dropped by Fed in 2020 so now banks can lend as much as they can (at least in US and other countries).
The multiplier model is a simplification that is mostly used as an introduction to money creation since it is an exogenous money supply model and those are easier to solve and learning the more nuanced endogenous money supply model is easier after learning the exogenous one. I am not sure to which Mishkin textbook you refer to, but I would be surprised if in later chapters it would not cover endogenous money supply models (money multiplier is an exogenous money supply model). For example, textbook by Blanchard et al. covers endogenous money supply model right after money multiplier, unless the Mishkin textbook you refer to is very old (prior 2009) it will likely cover it as well in subsequent section.


Does it matter if your decision might not matter?

Interesting question. I don't have any empirical evidence to offer, so I can't answer your question if that's what you're asking for.

But it's easy to provide "theoretical evidence", as it were, that your concerns may be well-founded, even without leaving the EU model.

You simply need to add a decision cost --- or a cost to figure out your true WTP --- to the decision-theoretic model of your participants.
Then, as the EU stakes decrease (because $p$ decreases), participants are less likely to invest in the introspection cost of discovering their true WTP.
As a consequence, participants' behavior become a much noisier indication of their WTP:

Some of your participants will decide to buy the good although their true WTP is lower than the price. This could happen if the participant decided not to invest in introspection and therefore got a sufficiently upward-biased signal of their true WTP.
Other participants will decide not to buy the good although their true WTP is above  the price. This could happen if the participant decided not to invest in introspection and as a consequence got a sufficiently downward-biased signal of their true WTP.

Assuming the cost of introspection does not change with $p$, participants become less likely to invest in introspection as $p$ decreases (because decreasing $p$ decreases the stakes, and therefore the benefits of introspection), and you'll get more of this noise as $p$ decreases.


Commercial bank balance sheet after Fed purchases $10 million in Treasury Bonds

The sale of the bonds (assets) will only affect the assets-side of the balance sheet.

If Acme bank did not lend out the money, your “bonds” would go 50-10 = 40. And your cash would go 0+10 = 10.

If they lend out the money, bonds still go to 40 and lending goes up by 10, so 50+10 = 60.


Specification form for household utility function

In growth models, in order to obtain a steady-state in growth rates (i.e. a constant growth rate in long-run equilibrium), the CRRA utility function when labor-leisure choice is present must have a specific functional form.
The mathematical proof can be found in Barro & Sala-i-Martin book (2nd ed) , Appendix 9.4, pp 427-428.
It is shown that the CRRA utility function must take the form ($\ell$ represents labor here)
$$u(c,\ell) = \frac {c^{1-\theta}\cdot \exp\{(1-\theta)\cdot \omega(\ell)\}-1}{1-\theta}$$
where $\omega(\ell)$ is some function of labor, and $\omega'(\ell)<0$.
For $\theta=1$ this simplifies to $u(c,\ell)=\ln(c) + \omega(\ell)$.

After we impose additive separability, the exact functional form of the component related to leisure/labor is sometimes formed so as to have a constant Frisch elasticity of labor supply, For example
$$u(c,\ell) = \ln(c) +\alpha \frac {(1-\ell)^{1+1/v}}{1+1/v}$$
has Frisch elasticity equal to $v$, free of $\ell$. See this thread for more details on this.
The Frisch elasticity of labor supply is defined as the elasticity of labor while keeping the marginal utility of wealth constant. Its expression is derived given the solution to the utility maximization problem (i.e. by using the relations emerging from the optimaztion conditions).  After some manipulations this expression is
$$\eta_F = \frac {U_{\ell}\cdot U_{cc}}{\ell\cdot [U_{cc}\cdot U_{\ell \ell}-U^2_{c \ell}]}$$
Under separability between consumption and labor/leisure,
$$U^2_{c \ell} = 0 \implies \eta_F = \frac {U_{\ell}}{\ell\cdot U_{\ell \ell}}$$
To use one of the functional forms of the OP
$$U(c,N)=\dfrac{c^{1-θ}}{1-θ}-\alpha \dfrac{N^{1-γ}}{1-γ}$$
we have
$$U_{N} = -\alpha N^{-\gamma},\;\;\; U_{NN} = \alpha \gamma N^{-\gamma-1}$$
so
$$\eta_F = \frac {-\alpha N^{-\gamma}}{N\cdot \alpha \gamma N^{-\gamma-1}} = -\frac 1 {\gamma}$$
So functional forms that lead to a constant Frisch elasticity, treat it as a "deep" preference parameter. Functional forms that lead to a Frisch elasticity that includes (the optimal) level of labor (as is the first functional form of the OP), treat it as a derived measure, dependent on the optimization framework also.


A question about the property of quasi-linear preference

This seems like a homework question, so I'll just give hints.

By definition, a quasilinear utility has the form $u(x, y) = x + v(y)$ where $y$ is a vector of all other goods and $v(\cdot)$ is strictly concave. In this case, $x$ is called the numeraire.

From utility maximization, what's the first order condition that relates $MU_x$, $MU_y$, $p_x$, and $p_y$? Write down the equation.
What's the utility gain from consuming one more unit of $x$?

You should be able to work things out from there.

Edit

$MU_x$ will be constant... more precisely, $MU_x$ is always one (by definition).

You can then plug this into FOC and get that $MU_y = 1/z$. where $z = p_x/p_y$ is simply how many $y$'s you could buy with "additional amount of wealth equal to the cost of one unit of good 1 on all other goods".

Since one more unit of good $y$ will get you $1/z$, you need (approximately) $z$ more of good $y$ to get one more unit of utility increase (which is the same as what you get when you increase $x$ by one unit).


The Pareto optimal set should always pass through the origin because it's one of the point that makes it impossible to better off a consumer without harming the other.


How can trailing and forward P/E ratio of main stock markets show such huge difference

Am I right concluding that:

Stock markets are probably over-evaluated, considering the earnings in 2016?
Stock markets are over-optimistic regarding the earnings in 2017?

No, you cannot draw that conclusion a priori. You can conclude that the market expects a big increase in profits in 2017. You can have an opinion about whether the market is over-optimistic, but unless you got divine insight, or have better models than they do, you cannot conclude that their forecast is wrong. Remember that the market is a collection of people with access to roughly the same information that you have access to.


Finding Optimal Bundle and Change in Satisfaction after Changes in Budget Constraint

I understand what you though, the old indifference curve sure can touch but cannot conclude to be the optimal one.

As the evidence of the money he can pay 1050 of the last year budget but for this year he could only use just 1020 to buy the same combination of those goods with the same utility so that means he still can buy and consume more with the relative purchasing power.

I will just use 2 assumptions that are "more is better", "no bliss point". It is said that the budget which has shown or given here is not significant to the level that the consumer will keep and not spend so that as he can spend he will and with "no bliss point" that means this consumption is too small and yet not to the level that he will have negative marginal utility.

Therefore, as he can buy the old combination with the lesser price so he can allocate the leftover budget for either of two goods.


Any information on a toy economy with 3 types of agents in it

Becker has a very widely cited paper about crime that generally defines the literature, here.

To examine your proposal, increases in productivity generally lead to higher wages in the legal sector, but could easily increase crime rates if they disproportionately increased the returns to crime.

No agent modeling or network theory is needed for this to an accepted idea in the literature, but you're welcome to explore it if you think the consequences are somehow divergent from what I've outlined here! Furthermore, the actual task to -verify- his theory is very difficult. To search "the literature", I would search within the body of papers citing this particular Becker paper.


What is the relevance of economic growth?

World population currently is estimated at $7.3$ billion people, while world GDP at $74.5$ trillion USD (nominal). So the global average of per capita GDP currently stands at USD $10,205$, again nominal, not adjusted for purchasing-power-parity. But in any case, this number, although not negligible, it is small enough so that, even if we had absolute global equidistribution, we would hardly be in an "Affluent Society".

This alone could justify why Growth is still the main economic target of societies all over the world: even if inequality was out of the picture, we will still be far from living really "comfortably".

Add inequality to the picture, and one realizes that Growth is also seen as something that can help mitigate possible social tensions due to inequality: one accepts the existence of inequality easier, if ones sees his own fortunes getting better (this requires of course that everybody gets something out of Growth, even in unequal shares).


After posting a bad solution yesterday I believe I got a better one:

The strategy of the buyer consists of two functions, $(f_1(v,p_1),f_2(v,p_1,p_2))$ where both functions map to $\left\{A,R\right\}$ (where $A$ stands for Accept, $R$ for Reject).
The strategy of the seller is $(p_1,p_2(f_1(v,p_1)))$. You get the solution via backward induction. In PBE $f_2(v,p_1,p_2)$ maps to $A$ if and only if $v \geq p_2$. (There is inconsequential leeway at equality.) In PBE the seller believes that there is a set $H$ of types for which the buyer refused her offer $p_1$. Then
$$
p_2^* = \arg\max_{p_2} p_2 \cdot Prob(f_2(v,p_1,p_2) = A | f_1(v,p_1) = R).
$$
The buyer will accept offer $p_1$ if and only if
$$
v - p_1 \geq \delta \cdot (v - p_2).
$$
From this you get
$$
v \cdot (1 - \delta) \geq p_1 - \delta \cdot p_2.
$$
The left hand side of this equation is increasing in $v$, so types with high valuation will Accept. This means that in PBE the set $H$ is such that
$$
H = [0, \bar{v}).
$$
From this we get the optimal $p_2$ given $\bar{v}$:
$$
p_2^* = \arg\max_{p_2} p_2 \cdot Prob(v \geq p_2 | v \in [0, \bar{v})) = \frac{\bar{v}}{2}.
$$
In PBE $\bar{v}$ is a function of $p_1$:
$$
\bar{v} \cdot (1 - \delta) = p_1 - \delta \cdot \frac{\bar{v}}{2},
$$
so
$$
\bar{v} = \frac{p_1}{1 - \frac{\delta}{2}}.
$$
We have determined all the PBE strategies but $p_1$.
The expected payoff of the seller is
$$
p_1 \cdot \left( 1 - \frac{p_1 - \delta \cdot p_2(\bar{v}(p_1))}{1 - \delta} \right) + \frac{1}{2} \cdot p_2(\bar{v}(p_1)) \cdot \left( \frac{p_1 - \delta \cdot p_2(\bar{v}(p_1))}{1 - \delta} - p_2(\bar{v}(p_1)) \right),
$$
where
$$
p_2(\bar{v}(p_1)) = \frac{\bar{v}(p_1)}{2} = \frac{\frac{p_1}{1 - \frac{\delta}{2}}}{2} = \frac{p_1}{2 - \delta}.
$$
Substituting this we get
$$
p_1 \cdot \left( 1 - \frac{p_1 - \delta \cdot \frac{p_1}{2 - \delta}}{1 - \delta} \right) + \frac{1}{2} \cdot \frac{p_1}{2 - \delta} \cdot \left( \frac{p_1 - \delta \cdot \frac{p_1}{2 - \delta}}{1 - \delta} - \frac{p_1}{2 - \delta} \right),
$$

You have to maximize this w.r.t. $p_1$. With $\delta = 0.5$ I got
$$
p_1^* = \frac{9}{20}, \hskip 20pt \bar{v} = \frac{3}{5}, \hskip 20pt p_2^* = \frac{3}{10}.
$$


A firm's contribution to GDP isn't revenues. A firm's contribution to GDP is roughly the sum of the wage bill and the money earned by capital (debt (interest) and equity (economic and not accounting profits)). That is, profits + interest payments + wages (look elsewhere for a more precise definition). Your example doesn't really have enough to go on to estimate the GDP impact.
The contribution could be zero. This can and did happen in the dot com bubble when companies would agree to buy advertisements on each other's sites of equal amounts to boost sales figures. This increases revenues and expenses by the same amount with no changes in wages, profits, or interest expense (maybe minimal changes in practice).
The contribution could also be arbitrarily large. Imagine a firm that makes a buys a piece of software for \$2 million that allows them to make \$10 billion in sales (no other expenses and sales and no profit without this expense). That would make a GDP contribution $9.998 billion. Now imagine splitting that company in half so that two different firms each make \$5 billion in additional sales after buying \$1m in software.
In summary, the impact on GDP depends and there isn't enough detail here to be more specific.


Why do anti-trust regulators in China need to approve NXP's takeover?

Generally, antitrust regulators have authority over any company doing business inside the zone of its control, hence why the EU antitrust regulators hit Google in 2017, or the EU's recent examination of US company Apple's attempted acquisition of another US based technology firm, Shazam.

Of course, for these companies, there's always the option of simply pulling out of those markets (not selling any goods or services, or holding any capital inside their borders) which dodges their authority. But given the size of these markets (US, China and the EU particularly), that's simply not a feasible option.


Found my mistake. I guess I'm not allowed to move the thetas to the other side.

Define $q^{-1}$ be the inverse of $q$. Then,

$$ \theta = q^{-1} \left( \frac{c}{J} \right)$$

$$ \frac{W_1}{W_0} = \frac{f(\theta_0)}{f(\theta_1)} = \frac{f \left( q^{-1} \left(\frac{c}{J_1}\right)\right)}{f \left( q^{-1} \left(\frac{c}{J_0}\right)\right)}$$

$J_1 > J_0$, hence $\theta_1 > \theta_0$ and $W_1 < W_0$.


Reason for marginal revenue is less than price in monopoly

Most internet providers rarely consider to decrease their price to win market shares. One possible reason could be because the additional revenue P(Y) from a new costumer joining the provider, would not compensate for the total loss of revenue P'(Y)Y from the group of regular subscribers (Y = several millions?). Instead, most providers invest a lot of effort to discriminate between these two groups and try to propose interesting prices only for new subscribers.
In "real life", you may not observe such a foolish thing, and will find only few concrete examples of monopolies having reduced their price in the hope of winning market shares. It is not because you do not observe such a behavior, that monopoly theory is useless. Quite the contrary, the chapter explains why you rarely observe such a foolish behavior.


China - US Trade War: Holes in this analogy?

A very interesting question

The analogy is wrong in a few aspects

1st of all, the technology. The us produces patents and knowledge in general which china doesn't posess
Furthermore the us government supports innovation and the development of new technologies
Since it intangible, its hard to dismiss, but when you think about it, the US most valuable companies main assets are tech patents

The ingredient of creating technology is highly skilled workers, which in turn, make the us low tech industry  unsustainable, which, in its turn, gives china its advantage

China doesn't have a functioning innovation model, on the contrary, stealing patents are almost a Chinese specialty...

Another thing to consider is the game that china is playing:
The stronger the demand for its goods, the stronger the demand for labor, which increases local salaries, then the product cost itself
And along with the rise of its foreign currency balance that pressures the Yuan's rate to rise,
there is a huge pressure on the Chinese products' price in dollars

China holds it currency artificially low, essentially subsidizing its products at a large cost, and in way, hurting the us economy

So the analogy is not really holding...
The us is not the free rider in this story


Difference between two ways of calculating real return

First, it's better to cancel $P$ in your first definition. Then
$$
R=\frac{N-I}{1+I}
$$
And the second expression is
$$
R=N-I
$$
Actually, it is the famous $\textit{Fisher Equation}$.

The difference that there is a $1+I$ on denominator. Of course they are different. But economists sometimes use both since they are approximately equal when $R$ and $I$ are small. If you look at the data, both real interest rate and inflation rate are just a little above 2%.

Instead of using your first definition, let's start from an alternative form
$$
1+N = (1+R)(1+I) = 1+R+I+R\cdot I
$$
When $R,I$ are both small, their product can be ignored. Say $2\%\times 2\%=0.0004$. Hence, you can say
$$
N= R+I \Leftrightarrow R= N-I
$$

Here is the Wikipedia Link.
https://en.wikipedia.org/wiki/Fisher_equation


The hint is to use the recursive Euler equation to express all future consumptions $c_{t+j}$ as a function of current consumption $c_t$, and to play with the indexes in the sum of the left.


Why are the same set of controls used for both OLS and 2SLS?

The reason for that is that 2SLS does not solve omitted variable bias without its cost.

First, for every omitted variable you would have to find suitable instrument. Finding instruments is incredibly hard so why would you purposely create more omitted variable bias? Unless you are trying to challenge yourself for fun or doing it as a practice exercise for students it does not make sense.

Second, actually 2SLS still has bias - the advantage of 2SLS is that bias decreases as strength of the instrument increases, so with good instruments 2SLS is better than OLS in terms of bias. But in practical applications you rarely have perfect instruments- so even in case you could feasibility instrument the other variables you omit on purpose you will be just adding to the bias of your model.

Hence you should always include all relevant controls that you can include


Why is government debt level presented as ratio to GDP, not against government budget or revenue size?

The ratio of government debt to GDP is a useful indicator because, broadly, the larger a country's GDP, the larger is the tax revenue the government could potentially raise to service the debt while maintaining other government expenditure at an acceptable level.  Thus a very high debt / GDP ratio would raise doubts about the government's ability to service the debt.

One reason why a larger GDP implies a larger potential tax revenue is that most of the transactions making up GDP from an income perspective, eg wages and salaries, company profits, could be taxed.  However, taxes could also be levied on items that are not part of GDP, eg capital values of property.  The more fundamental point, therefore, is that a larger GDP, again from an income perspective, tends to imply a larger aggregate ability to pay, out of that income, such taxes as may be levied by the government.

The government's ability to tax is also influenced by political factors and by the effectiveness of its tax collection, so the debt / GDP ratio can only be a crude indicator of ability to service debt.


NA, in my opinion, is the lack of a trading strategy to generate excess returns, that means returns greater than those of assets in the same risk category. Note that it doesn't mean no risk.

A market is efficient, again in my own understanding, when all the contents of an information set are reflected in the market's assets prices. There are 3 forms of market efficiency weak, semi-strong and strong.

Weak form efficiency is given when the information set contains historical data about the assets, for example past prices, returns, past news about the companies, etc. An example of weak form efficiency would be when past growth data of a company is incorporated into its equity prices.
Semi-strong form efficiency occurs when the market is weak form efficient and the information set also includes all currently available public data about the assets in the market. For example press releases, macroeconomic indicators, etc. An example of semi-strong efficiency would be when due to an increase in the interest rate, the cost of capital of a company increases and the equity prices fall.
Strong form efficiency takes place when a market is semi-strong form efficient and the information set also includes all available private information about assets in the market. For example, plans yet not publicly available about a new product release or private events in the lives of the members of the management team. An example of strong form efficiency would be if, due to the CEO of the company going through a divorce, the company's shares prices fall.


What strategies can the president employ to deflate the dollar

Direct intervention by the U.S. Treasury in the foreign exchange market offers one option for the president.    The Treasury would use dollars to buy foreign currency bonds, bidding up the relative prices of foreign currencies and weakening the dollar.

Another option is  imposing taxes on the foreign purchase of U.S. assets as a way of weakening the dollar and reducing the trade deficit.


Are state owned enterprises really inefficient?

SOE's don't have to perform worse that private enterprises, but they often do.  The reasons are manifold:

Some SOE's are not set up for profit motive, but rather to seed strategic industries for a nation (e.g. Port of Singapore, or Huawei Telecom).
Frequently, state influence at market-oriented is counterproductive because politicians placed in leadership positions are not good business managers (different skillsets), and political contacts can have a detrimental effect on competitiveness (overreliance on government contracts, etc).
Even if the SOE is run by market-selected managers and employees, states often have different shareholder incentives and objectives as owners, so they may not exercise market-driven ownership which leads private firms to attempt maximizing equity value and earnings.

Note that it's not true that SOE's are not funded directly by government.  Many SOE's receive partial funding through government, and many others receive heavily subsidized contracts from governments.

There is quite a lot of research on SOE performance.  Do a search.  For example, McKinsey did a study on Chinese SOE's and concluded that on average they deliver less than half the return on assets as their public counterparts:

http://www.forbes.com/2004/11/04/cx_1104mckinseychina6.html


Does working abroad cause less unemployment? Or is it actually harmful (directly or opportunity cost)?

On remittances:
It is not unusual to hear about income being sent home by, say, US immigrants. However, the product of their labor and the side effects of their intellectual capital, sometimes called positive spillover, do not occur in the home country.  As such, they are not as beneficial to the home country as some counterfactual world where an equally profitable job existed at home. The immigrants are then beneficial to the host country.  The remittances still benefit the home country, however, and are particularly important when no employment is available at home.
On unemployment:
Broadly, immigration does not appear to have much of an impact on the unemployment in the host country. Or at least, something happens which masks that effect and makes it tremendously difficult to measure.  There has been a lot of controversy on this study but it helps highlight a lot of the issues if you take the chain of papers and responses seriously: https://www.jstor.org/stable/2523702?seq=1#metadata_info_tab_contents
On the type of people immigrating:
"Are the ones going abroad necessarily like money-hungry people who do not love their country or something."  This isn't really the type of question economists answer. Everyone is money-hungry and equally self-interested, by default, and that is fine.  This helps (in some ways) avoid stereotyping immigrants. You may want to open another question about the types of people who like to immigrate.


Why are there estate taxes after the owner passes away?

From an equality of opportunity perspective, it is desirable to have some taxes on inheritance. This levels the playing field for each generation to some extent. If real estates are exempt from such a tax, individuals would prefer real estate over other assets, creating a distortion in the asset price market.

For a recent contribution to optimal inheritance taxation, see:
Piketty, Thomas, and Emmanuel Saez. "A theory of optimal inheritance taxation." Econometrica 81.5 (2013): 1851-1886.

For an overview over the equality of opportunity literature, see:
Corak, Miles. "Income inequality, equality of opportunity, and intergenerational mobility." The Journal of Economic Perspectives (2013): 79-102.


Is it considered acceptable or unacceptable to use currency as a measure of utility?

Utility functions as ordinarily used are not a measure of well-being comparable among people, but a representation of preferences. Moreover, preferences could principally be elicited from choice experiments.
A utility function assigns real numbers to alternatives so that one alternative is preferred to the other if and only if it is assigned a higher number.
Now in many situations, one can use so-called "money metric utility functions." Start by holding prices fixed (this should already tell you a fundamental limitation of the idea.) For each commodity bundle $x$, let $u(x)$ be the price of the cheapest commodity bundle at least as good as $x$ according to the preferences. Under some assumptions, this gives you a genuine utility function $u$ representing preferences and it gives you actual dollar values. But the natural interpretation depends crucially on prices being fixed.


Test: Part-time Workers Effect on Unemployment Rate

This is mostly related to 2):

Not only is there reversed causality, but also potential for omitted variable bias. Identifying causality instead of correlation is something that no econometric method can do for you, you will have to work for that yourself.

For example, you can

Find some proxy that is going to move part-time employment, but not any other variable that could be somehow linked to the unemployment rate
Find exogenous (random?) variation in part-time employment. The most common tool here are (quasi-) natural experiments.


Doubt about Mixed Strategy Nash Equilibrium

Suppose player $i$ plays the mixed strategy $\mathbb{P}_i(B)= p_i$, and assume for now that the support of $\mathbb{P}_i$ is $\{B,F\}$ (i.e. player 1 plays a fully mixed strategy). For both $B$ and $F$ to be in 1's support, he must obtain the same expected payoff from either strategy (otherwise, he would put all the weight on the strategy with the higher payoff!).
Now, the expected utility of player 1 from playing $B$ is:
$\mathbb{E}[u_1(B,.)] = p_2u_1(B,B)+(1-p_2)u_1(B,F) = 2p_2$
Similarly, the expected utility of player 1 from player in $F$ is:
$\mathbb{E}[u_1(F,.)] = p_2u_1(F,B)+(1-p_2)u_1(F,F) = 1-p_2$
The important point to note here is that the expectation is over the actions of player 2 - since player 1 knows the distribution used by player 2 (in equilibrium), but not the realised action.
You can write the best response function of player 1 as follows:
$BR_1(p_2) = \begin{cases}
0 & \text{ if } 2p_2 < 1-p_2 \\
(0,1) & \text{ if } 2p_2 = 1-p_2\\
1 & \text{ if } 2p_2 > 1-p_2 \\
\end{cases}$
Since we assumed that 1 uses a fully mixed strategy, the $BR_1$ function dicates that this can happen only when $2p^*_2 = 1-p^*_2 \implies p^*_2 = \frac{1}{3}$.
In other words, $p^*_2$ is the unique probability that is consistent with player 1 mixing over both her strategies. Whether this forms an equilibrium is still not clear - for that you need to calculate $BR_2$ (using the same steps) and see if $p^*_1\in(0,1)$. In that case, both players are best responding to each other - and hence playing a Nash Equilibrium.
P.S - for instance, if you found out that $p^*_1 = 1$ (i.e. 1 would like to play pure strategy $B$), then our starting assumption is wrong! So we need to redo the calculation for $p^*_2$.


Labor demand and supply curves in terms of elasticities

This result is presented on page 14 of Foundations of Modern Macroeconomics. The key is to total differentiate both expressions.
Start with the supply of labor:
\begin{gather}
W/P^e = g(N^S) \\
d(W/P^e) = d\Big( g(N^S) \Big) \\
\frac{1}{P^e} dW - \frac{W}{(P^e)^2}dP^e = g_NdN^S \\
dN^S = \frac{1}{g_N}\frac{dW}{P^e} - \frac{1}{g_N}\frac{W}{P^e}\frac{dP^e}{P^e} \\
dN^S = \frac{1}{g_N}\frac{dW}{P^e} - \frac{g(N^S)}{g_N}\frac{dP^e}{P^e}, \quad \text{where} \quad W/P^e = g(N^S) \\
dN^S = \frac{1}{g_N}\frac{g(N^S)}{W}dW - \frac{g(N^S)}{g_N}\frac{dP^e}{P^e} \\
dN^S = \frac{g(N^S)}{g_N} \bigg[ \frac{dW}{W} - \frac{dP^e}{P^e} \bigg] \\
\frac{dN^S}{N^S} = \frac{g(N^S)}{N^Sg_N} \bigg[ \frac{dW}{W} - \frac{dP^e}{P^e} \bigg]
\end{gather}
To derive the expression for the demand of labor, one can follow a similar process (I will ommit the superscript $N^D$ for simplicity, but we know we are talking about labor demand). Recall that the FOC of the firm's problem is given by:
\begin{gather}
PF_N(N,\bar{K}) = W \\
F_N(N,\bar{K}) = \frac{W}{P} \\
d\Big( F_N(N,\bar{K}) \Big) = d\Big( W/P \Big) \\
F_{NN}dN + F_{NK}d\bar{K} = \frac{1}{P}dW - \frac{W}{P^2}dP \\
F_{NN}dN + F_{NK}d\bar{K} = F_N\frac{dW}{W} - F_N\frac{dP}{P} \\
F_{NN}dN = F_N \bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg] - F_{NK}d\bar{K} \\
dN = -\frac{F_{NK}}{F_{NN}}d\bar{K} + \frac{F_N}{F_{NN}}\bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg] \\
\frac{dN^D}{N^D} = -\frac{F_{NK}}{N^DF_{NN}}d\bar{K} + \frac{F_N}{N^DF_{NN}}\bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg] \\
\frac{dN^D}{N^D} = \frac{d\bar{K}}{\bar{K}} + \frac{F_N}{N^DF_{NN}}\bigg[ \frac{dW}{W} - \frac{dP}{P} \bigg], \quad \text{where} \quad KF_{NK} = -NF_{NN}
\end{gather}


Definition: What Is The "Trend Growth Rate"?

Any time series can be decomposed in four terms:

trend: the underlying change in the variable if no other factor is present. In the simplest case, think of an upward slopping line. The growth rate of this trend is it's slope.
cycle: the cyclical component of the series, defined by ups and downs like a wave, with a certain "frequency" and "amplitude". Think of this like a wave.
seasonal component: usually a within year cycle. The standard textbook example is that of ice cream sales, which pick up in summer. This is different from the longer cycle described above, and perhaps less smooth (e.g. as ice cream sales are low 3/4 of the year)
noise: natural randomness due to measurement errors or other unobserved yet minor factors, including shocks (e.g. a very hot weather in a week, increasing ice cream sales)

For a series $y_t$, this is something like

$$ y_t = d_t + c_t + s_t + e_t $$

Importantly, all the variables are defined in the same units by construction. E.g. if this is GDP, they are all in value added units (real, nominal, whatever).

A general case of a decomposition is (assume just trend and cycle):

$$ f(y_t) = d^{'}_{t} + c^{'}_{t} $$

In this case, the trend growth of $f(y_t)$ is the growth rate of $d^{'}_t$. But this is not the same as the trend growth rate of $y_t$.

Consider an example. Let us construct a series $y_t$, where $d_t=d_0(1+g)^t$ and $c_t=sin(t)$, and $y_t=d_t+c_t$. This is a trend-cycle decomposition where the growth rate of the trend is $g$:

$$ \frac{d_{t+1}-d_t}{d_t} = \frac{d_0(1+g)^{t+1}-d_0(1+g)^t}{d_0(1+g)^t} = \cdots = g $$

Now, transform the process into log, and let us postulate a new decomposition:

$$ \log y_t = D_t + C_t $$

where $D_t = D_0(1+G)^t$, and $C_t$ is an unknown function. This is by definition correct, but $C_t$ might be an irregular, perhaps not algebraically defined function (like $sin(t)$). In this new case, it is correct, by definition, that the trend growth rate of the log of $y_t$ is $G$, computed with the standard formula.

The central question is: can we find a general method to calculate the trend growth rate of $y_t$ ($g$) using only some transformation of $D_t$? (as in your attempt using the exponential). In other words, can we use only $D_0(1+G)^t$ to obtain $g$?

(Here I highlight the point of why would you ever go down this route, if you can decompose $y_t$ in levels and solve your problem immediately; but anyway, let us continue).

The answer seems to be negative. It is trivial to see that

$$ e^{D_0(1+G)t} e^{C_t} = d_0(1+g)^t + sin(t) $$

where you cannot find a direct function between $G$ and $g$ which does not involve the use of $C_t$ and $sin(t)$. In other words, you need to use the information of both decompositions to solve the problem. But this is circular. To make the link between logs and levels, you need to make the decomposition in levels. But if you do the latter, the answer is there already. No need to go further.

Finally, consider the very special case where the series $y_t$ only has a constant growth rate (trend):

$$ y_t = d_t = d_0(1+g)^t $$

Then,

$$ \log y_t = D_t = D_0(1+G)^t = \log \left(d_0(1+g)^t\right) $$

The following formula recovers the original growth rate of $y_t$ in terms of the trend from the series in log:

$$ \dfrac{e^{D_{t+1}} - e^{D_t}}{e^{D_t}} = \frac{d_{t+1} - d_t}{d_t} = \cdots = g $$

The result can be generalised to any $f(y_t)$, where the transformation of the growth rate calculation has to be $f^{-1}(\cdot)$.


Finding Pareto efficiency with negative externalities

Your statement "Isn’t any level of noise Pareto-Efficient?" would be true only if the only good traded was "levels of noise". Indeed, given a single good, Pareto-Efficient means that you simply do not waste anything.

I understand that here you have two exchangeable goods, Decibels and Money, and you need to evaluate possible combinations of the two. You'll see that, your task is to find the combinations of decibels and money, which is done by finding the contract curve.

The common method, as I'm sure you have learnt, is the comparison of the two persons' Marginal Rates of Substitution, in your example: $ MRS_F = MRS_S$, as given by the Edgeworth Box:
Notice how all Pareto-Efficient contracts lie in the contract curve? You can use this equation to input values for decibels and get the requested "Pareto-Efficient" level. The hint on the budgets of the two students also works in favor of you finding a proper budget constraint and finding precise levels for the two commodities.
This would be my process.


What does "writes a check to Congress in exchange for them" mean in this context?

The US Government has $\\\$X$ in spending and raises $\\\$Y$ in revenue.
Usually, $X>Y$. So, the US Government (or more specifically the US Treasury) has to cover this $\\\$(X-Y)$ difference (or funding requirement/shortfall).
It does so by issuing bonds (or notes or bills) worth $\\\$(X-Y)$. Pretty much anyone is free to buy these bonds, including the public and the Fed.
So "writes a check to Congress in exchange for them" probably simply describes the act of the Fed buying some of these bonds.

The buying and selling of US Government bonds by the Fed is called open market operations. See e.g. Investopedia or this Fed webpage for how this works.

Three clarifications:

The US Congress doesn't have any direct official/technical/legal role in any of this, except insofar as it determines spending ($X$) and revenue ($Y$). So, it's not really correct to say that the Fed "writes a check" to Congress. We should instead say that Fed "writes a check" to Treasury. (Of course, we could quibble and say that Congress and the Treasury are all part of the US Government, so "writing a check" to the Treasury is "really the same thing" as to Congress.)
This writer writes, "the Fed takes all the government bonds which the public does not buy." This statement may suggest incorrectly that the government is free to sell as many bonds at whatever interest rate it likes. But this is incorrect. Instead, what happens is that the Fed targets certain interest rates. The government must then adjust its interest rates—so as to meet its $\\\$(X-Y)$ funding requirement and, at the same time, the Fed's targets.
"Writes a check" is not strictly speaking correct, since nowadays this is all done electronically with keystrokes rather than any physical writing of checks.

By the way, I've never heard of this writer, but Wikipedia calls him a conspiracy theorist and states that The Creature from Jekyll Island (1994) "promotes false theories about the motives behind the creation of the Federal Reserve System." So I wouldn't necessarily believe anything written by this writer.


:History: Industry threatening to leave if social change takes effect

What's going on when a for-profit company makes a threat?

The same thing that is happening when does anything. It is trying to increase its profits, relative to what would happen if it didn't do the thing.

In this case, it's trying to influence policy to protect its profits. It has decided that issuing the threat gives their profits a better chance than not issuing the threat. That probably means that they believe that their profits will be lower with the new policy than without it.

The thing it's threatening is, in a sense, neither here nor there: the threat itself is the action.

Should the policy that a company opposes, get implemented, then the company then has a choice, to stay or to go, and the decision criterion will be: will them staying result in higher or in lower profits than them staying. That's a completely different issue to the thing they issued the threat over, and the threat doesn't tell us anything useful about it.

And companies are often wrong about the impacts of policy anyway.  In summary, treat their statements on policy with a lot of scepticism.


What are some database like websites about lower levels of economics (micro/macro) for a research project?

Fama-French library
has good data on which you can test your first Fama-French regression model.

I have found this video on Youtube that might help you with this first project.

Ps.: It would help a lot if you were more specific about what you want to learn. I can give other suggestions, but I really don't know what you are looking for.


What is preventing the United States from printing USD to buy up prized assets from foreign countries given that USD is the world's reserve currency?

To a certain extent it can.  But not too much, else it will create a run on the dollar.  If speculators no longer trusted the dollar, they would sell it for other stores of value (like Euros, gold, etc...).  Foreign dollars would have to come home and be redeemed by real wealth which means less real wealth for the US domestic economy and inflation.  This can feed on itself and cause a self-reinforcing inflationary fire that would be difficult to put out.


How are real income and utility the same thing?

Real income and utility are not the same thing but utility can be expressed as a function of income because income is what allows you to consume goods and services.
For example, let us assume there are only two goods $x_1$ an $x_2$ and consumer is given budget $p_1 x_1 + p_2 x_2 = m$, where $p_i$ are prices for good 1 and 2 respectively and $m$ is an income.  Let us assume Cobb-Douglas utility function:
$$ u(x_1, x_2) = x_1^\alpha x_2^{1-\alpha}$$
Here after solving the consumer optimization problem we would get optimum consumption of $x_1$ and $x_2$ as (I skipped the steps but you can easily derive it by setting up an Lagrangian):
$$x_1^*= \alpha \frac{m}{p_1};x_2^* = (1 -\alpha) \frac{m}{p_2}$$
Consequently, given optimal consumption of $x_1^*$ and $x_2^*$ is:
$$u(x_1^*, x_2^*) = \left( \alpha \frac{m}{p_1} \right)^\alpha \left((1 -\alpha) \frac{m}{p_2} \right)^{1-\alpha} $$
So the utility will end up being function of nothing else than a function of income $m$ an prices $p_1$ ($\alpha$ is just parameter). A utility function expressed in terms of prices and income even has its own special name 'indirect utility' typically denoted as $v$ (e.g. $v(\textbf{p},m) =\max u(\textbf{x})$ such that $\textbf{px} = m$ see Varian Microeconomic Analysis 3rd ed pp 99.)
Consequently as shown above utility does depend on income. Utility is generally not equal to real income, and I don't have the copy of the textbook you mention and you did not gave us the page for that claim, but they might have just used a utility function where utility would be proportional to real income. For a didactic purposes even assuming $v(m,\textbf{p})=\frac{m}{\textbf{p}}$ can be reasonable depending on setting even if it is not realistic because it simplifies calculations for novices (e.g. supply and demand are also typically not linear but your textbook likely draws them as linear functions).


Equity can be negative when a firm accumulates losses and liabilities exceed the number of assets. In that case debt ratio can be bigger than 1.


Infinitely repeated game: prisoners' dilemma

the payoff from play the trigger strategy will be:
$$ \sum_{i=0}^{\infty}(x+1) \delta^i=\frac{x+1}{1-\delta}$$

if I deviate and I play $l$ or $d$ the payoff will be
$$ 2x + \sum_{i=1}^{\infty}x \delta^i = 2x+ x\frac{\delta}{1-\delta}=\frac{2x(1-\delta) + x\delta}{1-\delta}= \frac{x(2-\delta)}{1-\delta}$$

then, the condition is $$ \frac{x+1}{1-\delta} \geq \frac{x(2-\delta)}{1-\delta}$$

I hope that this will help you ;)


The MA terms are lagged errors (you don’t need to fetch them manually - for example in R you can use Arima function which does this for you and any program/language will have this basic function as mentioned +1 answer of @RichardHardy in +1 comment of @Dayne).
For example, following Verbeek (2008) A Guide to Modern Econometrics 4th ed pp 261 ARMA(p,q) in general form can be written as:

$$y_t=\theta_1 y_{t−1}+ … + \theta_p y_{t−p} + \epsilon_t+ \alpha_1 \epsilon_{t−1}+…+ \alpha_q \epsilon_{t−q}$$

The distinction between the effect of AR coefficients and MA coefficients is that they model fundamentally different forms of relationships with past realizations. AR terms model autocorrelation of the dependent variable - that is relationship with its own past realizations. MA terms model dependence on the errors that model made in the past. These have different implications for how persistent the series is.

I think it is best to visualize this. Here are two pictures I took from the above mentioned Verbeek from pages pp 260-261.
On this first picture you can see effect of autocorrelation in AR(1) process when $\theta=0.5$. You can see that even though we have only single AR term it’s effect is persistent across several periods.

However, let’s consider now case of MA(1) process where $\alpha=0.5$. As you can see clearly below the effect is seen only for one period.

The advantage of including MA terms (provided that they actually should be there and you are not adding them just for heck of it), generally speaking, are that you will more accurately model the persistence in your model, you will have better fit and more accurate forecasts than otherwise. It is always important to get the right functional form functional form as wrong functional form can even bias the results. I mean on a very basic level (and I am oversimplifying here) this is like asking why we sometimes use $\ln y = \alpha + \beta \ln x + e$ instead of $y = \alpha + \beta x +e$ it is because we want to get the functional form right so there is no misspecification in the model.


It is possible for firms to have constant marginal costs in monopolistic competition in theory. Nevertheless, they must also have fixed costs. The fixed costs prevent firms from entering in sufficient numbers such that you would have perfect competition.

As to whether constant marginal costs are realistic in you scenario, that depends on what you believe about the industry in practice. That would require detailed knowledge and expertise of this specific industry, which you are unlikely to find on this site.

Note, however, that just because the company has already bought the materials does not neessarily mean that these costs do not matter. For example, the company may still want to use less material per production output, because it could resell the unused materials. So the company in that case still cares about the costs associated with the use of each unit of material.

It is a bit unclear in your example what this company actually produces. However, if you think the costs of producing one extra unit are the same, regardless of how many units you have already produced, then you are justified in assuming constant marginal costs.


The equality in question follows from the expression for expectation of a log-normal distribution.
For example, if $X \stackrel{d}{\sim} N(\mu, \sigma^2)$, then
$$
E[e^{a X}] = e^{a \mu + \frac12 a^2 \sigma^2}.
$$
This is the reason that CARA/normal or CRRA/log-normal (agent utility/asset return distribution pair) set-ups reduce to the mean-variance case, up to certainty equivalence.


I am not sure about a way to look at it from an economics perspective since I am no economist. However, I am someone interested in this question and looking into doing academic research into this. Here is a few way I would look at this:

Economic Perspective: One of the things I would be interested in is looking at the correlations between the severity of the outbreak (cases and deaths) and the economic prospects of a country. I have already seen one article that discusses this in greater detail through examining the analysis of vintage data. Check: [Growth forecasts and the
Covid-19 recession they convey:
End-2020 update][1]: https://cepr.org/content/covid-economics-vetted-and-real-time-papers-0
In terms of GDP, I would look at the GDP (any of the multiple versions of GDP) growth rates and the growth rates of the outbreak. After all, I would expect their to be a correlation between the two since greater outbreak usually corresponds to a lower supply of labor (due deaths and absenteeism). Also a greater outbreak should be signaling a greater prospect for a government intervention through lockdowns and hence more economic pessimism over the outlook of the economy.

That being said, this is not a thorough nor a comprehensive answer to your question. Looking forward to seeing more answers on this very interesting question.
For my own purposes, I am looking into government effectiveness through indicators through moving averages of the number of cases and deaths. This bears some resemblence to technical analysis of stocks, although, this techniques is not widely respected among the academic and economic circles.


Elasticity of substitution between leisure in two periods

I think there are two problems with your calculations. First, as $l_i$ is the time devoted to work, you should care about $\frac{d\ln\frac{1-l_2}{1-l_1}}{d\ln (U_1/U_2)},$ rahter than the $\frac{d\ln\frac{l_2}{l_1}}{d\ln (U_1/U_2)}$ you consider. Second, I don't think your $U_i$ are correct (perhaps it's caused by first). I get

$$\begin{matrix} U_1=\frac{b}{1-l_1} & U_2=\frac{be^{-\rho}}{1-l_1}\end{matrix},$$

that results in the marginal rate of substitution being

$$MRS=\frac{U_1}{U_2}=\frac{1-l_2}{1-l_1}\frac{1}{e^{-\rho}}.$$

If you take the logarithm, you get

$$\ln MRS = \ln \frac{1-l_2}{1-l_1}+\ln\frac{1}{e^{-\rho}},$$

where it is useful to observe that the last term of the RHS is a constant. Now, the total differential is

$$ d\ln MRS = d\ln \frac{1-l_2}{1-l_1}+0,$$

and you get the desired result.


There are a bunch of examples for incomplete markets in the finance literature.  The oldest (that I know of) is Hart (1975).  In finance, the problem is that if you have two different assets they have different prices that make the market complete.  But then equilibrium considerations make them have the same price, which forces the market to no longer be complete.  (If the market was complete, then they would have the same price.)


Difference between unemployment rate between ILO and bureau of labor statistics

With regards to the concept of 'unemployment' and the 'unemployment rate' it is important to note that, while the general definition (number of unemployed /number labor force * 100) is pretty standard, the exact parameters or "operational criteria", in ILO-speak, for how some of these concepts are defined can vary from place to place, often with large impacts on the final statistic.

Specifically, it is important to consider the criteria what is used to classify people as "employed" and "unemployed", with the latter concept having many different criteria to define--e.g. working age, paid or not, what constitutes "actively looking for work", what constitutes "available for work", etc.

With this in mind, the ILO has a standard definition with specific operational criteria for all of the concepts used to define employement, unemployment, and the labor force, so that calculations of the unemployment rate using this ILO definition are sure to be comparable from place to place.

Now with all of that said, the US BLS uses the Current Population Survey to calculate unemployment rate according to the ILO definition, so the stat should be the same. If you go the the ILO-STAT dashboard for the United States, you can see that the data-source for ILO stats on the U.S. is in fact, the BLS and the US Current Population Survey.

On a more general note, the ILO recently published this very useful, concise guide that explains specifically l their definition, but also, more generally, how to interpret uemployment rates and all of operational criteria and how they can impact rate calculations. You can find the guide to  unemployment rate on the ILO site.


New Keynesian IS curve: question about time dependence

As you already mentioned this equation can be derived from the households consumption rule.

the equation is derived based on consumption C rather than income Y, by considering the marginal utility of comsuming right now vs saving and delaying consumption. Investment/net export is not a part of this model, so C=Y

So lets take a look at the Euler condition for the household (where $\pi$ represents inflation, $\beta$ the discount factor and $R_t$ the noninal interest rate)  and remember the market clearing condition $C=Y$:
$$\begin{align}C_t^\sigma = \beta^{-1}  E \left[\frac{\pi_{t+1}}{R_t} C_{t+1}^\sigma \right]\end{align}$$
You can directely see, that if the household expects future consumption to increase, current consumption will also increase (assuming no changes in inflation and interest rates). This is called consumption smoothing. The households tries to balance consumption over different periods. A rise in nominal interest rate makes saving more attractive and thus households shifts some of the current consumption into future periods.


Is there some kind of consensus on the determinant factors of education achievement?

There is no solid consensus on the inputs that go into educational achievement or attainment. The list of variables is vast and varied.

One of the most common applications of this is attempting to calculate the returns to education - i.e. the average increase in income that results from one additional year of schooling.

I myself have performed research on this topic in grad school, and a good reference is:

Bronars, Stephen G., and Gerald S. Oettinger. “Estimates of the return to schooling and ability: Evidence from sibling data.” Labour Economics 13.1 (2006) : 19-34. http://www.sciencedirect.com/science/article/pii/S0927537104000983

In my research there were many variables that were statistically significant:

age
gender
race
marital status
union membership
year at time of measurement
number of siblings
religious beliefs
opinion on sex ed in schools
opinion on national educational spending
opinion on education as catalyst for racial/class divides

Most researchers in this area agree that parental involvement is one of the main determining factors for a child's educational outcome. The difficulty is in quantifying this and other subjective data such as teacher quality and peer effects. This is one of the main issues in the field now - figuring out how to statistically measure non-numerical variables. The opinion variables I mentioned above are from national social surveys and are an attempt to quantify these subjective things. Specifically, my data came from the General Social Survey (GSS) available here: http://www3.norc.org/GSS+Website/


Government expenditure multiplier in the new-Keynesian model?

The "standard" New Keynesian model could be many things, but suppose that we're dealing with the basic log-linearized 3-equation model (intertemporal Euler equation, New Keynesian Phillips curve, and monetary policy rule) exhibited, for instance, in Gali's textbook.

In most variants of this simple model, a temporary increase in government expenditure will almost always increase output, but the intuition is different from the Old Keynesian IS/LM model. The extent of the increase depends on the monetary policy rule.

If the monetary policy rule targets inflation of exactly zero (assuming trend inflation is also zero), then it turns out that there is no difference between the effects of government expenditure in the NK model and the effects in a purely real model; by successfully targeting zero inflation, we effectively negate the effect of the NK model's nominal frictions. Since the underlying real model produces an increase in output when government spending increases (ignoring effects of possible distortionary taxation used to fund this spending), we get this in the NK model with inflation targeting as well, but it's not very exciting. It's a purely classical story: all else equal, consumers work harder when their consumption is lower, and thus consumption does not drop one-for-one with increases in government spending.

If the monetary policy rule happens to target a constant real interest rate, then in the basic NK model the "multiplier" is exactly 1: the constant path for the real interest rate leaves consumption unchanged, and the increase in output exactly equals the increase in spending. If the monetary policy rule targets a constant nominal interest rate during the stimulus, then there is an additional effect because future stimulus spending produces expected inflation that lowers the real interest rate and boosts consumption. This is the source of "multipliers" >1 in the basic NK framework. See Woodford's 2011 AEJ macro paper for more details on this.

None of this comports with the traditional Keynesian 1/(1-MPC) multiplier. This is for a couple reasons, but most important is the fact that the basic NK model features fully Ricardian consumers who intertemporally smooth consumption and anticipate future taxes levied to pay for current spending. Gali, Lopez-Salido and Valles's 2007 JEEA comes much closer to the traditional story by adding ad-hoc "non-Ricardian" households.


I see two ways to produce histories with multi-state deviations (not including the degenerate case in which all states defect):

Multiple states defect in different periods.  This possibility is ruled out so long as Equation 3 (the necessary and sufficient condition for unilateral defection at time $t-1$ to be punished by all other states at time $t$) is satisfied - in that case, only one state can ever defect at a time because all other states have credibly committed to Penance.
Multiple states defect in the same period. There are two obvious ways this can happen: 1) regional agreements are in effect and one of the regions is punishing a defection.  Equation 7 gives the conditions required for both regions to revert to Regional Penance in the following period, so this case isn't interesting.

The other way to have multiple states defect in the same period is the interesting case. It implies collusion between a subset of states. This reduces to the Regional Penance case if the subset is equal to half the states.  If it isn't, Footnote 10 explains that such an arrangement is not weakly renegotiation-proof. This makes sense, as collusion implies that (Regional) Penance is not, in fact, subgame perfect.


Why are capital gains not counted in a poverty status calculation?

Most Exchange Traded Funds of stocks or bonds have an associated income (even those which reinvest or accumulate this income)

For example Vanguard Total Stock Market ETF has a dividend yield of $1.81\%$.  So multiplying by $\$10,000,000,000$, this would give an annual income of about $\$18,100,000$, a little above income poverty levels

On the other hand, people with a large amount of physical cash or gold coins and no income, and using some of this for living expenses, would count as being in income poverty, no matter how high their wealth.  But there are relatively few people like this


What is the returns to scale of the production function q = min {K, L^(1/2)}?

You have a Leontief production function and in optimum you will always have $K=\sqrt{L}=q_1$.  Now increase both inputs by factor $k>1$ and you arrive at $k*K > \sqrt{kL}=q_2$ where the first inequality follows from $k>\sqrt{k}$ for $k>1$ and the second ineqality from the fact that only the minimum matters in your production function.
Therefore you have $F(kK,kL)= \sqrt{kL}<kK=k\sqrt{L} =k*F(K,L)$ with any optimal input combination $(K,L)$ and therefore decreasing returns to scale.


Why do so many people die in hospitals? Aren't doctors there supoosed to save lives?

The IMF is the trauma surgeon of government finances. When a country asks the IMF for help, it is because its finances are in very bad shape. They no longer have enough money to pay for public services, infrastructure, pensions, salaries. Without some kind of external help, the government has to fix the problem by some combination of reduced spending, increased taxes, or by printing money (which will increase inflation).

What this shows is that austerity is not imposed by the IMF. Austerity is "imposed" by the inability of the government to finance its own spending.

The IMF, by providing loans when no one else will, actually helps to soften the adjustment. With the extra funding, the government is able to cut spending and raise taxes less than they otherwise would.

Of course, one thing to keep in mind is that the IMF is not a charity. It expects its money back. If the IMF provided grants instead of loans, there would be no need for austerity. But then the IMF wouldn't exist, because it would have run out of money long ago.


Volatility of investment: Joint-Volatility of mutually independent variables

The variances of the sum of independent random variables are additive. Therefore, if X and Y are independent, Var(X+Y)= Var(X) + Var(Y). The standard deviation is defined as $\sigma_z = \sqrt{Var(z)}$. Therefore, if X and Y are independent, $\sigma_{X+Y} = \sqrt{Var(X+Y)} = \sqrt{Var(X) + Var(Y)}$. This gives the result above.


Are there economic policies that help countries fight corruption?

Kaushik Basu has proposed (Why, for a Class of Bribes, the Act of Giving a Bribe should  be Treated as Legal) legalizing paying bribes to separate the ties of criminality between the bribe payer and receiver. This allows the bribe payer to pay the bribe and report the bribe taker. If bribe paying is illegal then the shared criminality prevents this.
Dina Pomeranz empirically examines the widely held belief that VAT taxes discourage tax evasion (No Taxation without Information Deterrence and Self-Enforcement in the Value Added Tax) and finds that they do indeed discourage tax evasion:

Claims that the VAT facilitates tax enforcement by generating paper
trails on transactions between  firms contributed to widespread VAT
adoption worldwide, but there is  surprisingly  little  evidence. This  paper  analyzes  the  role  of  third  party  information for
VAT enforcement through two randomized experiments among over 400,000
Chilean  firms.  Announcing additional monitoring has less impact on
transactions that are subject to a paper trail, indicating the paper
trail's preventive deterrence effect. This leads to strong enforcement
spillovers up the VAT chain.  These  findings confirm that when taking
evasion into account, significant differences emerge between otherwise
equivalent forms of taxation.
Other forms of taxation that are harder to monitor and enforce provide more opportunities for corruption and so switching to a broad VAT may reduce corruption.

Another possibility is outsourcing corruption enforcement:

The International Commission against Impunity in Guatemala (Spanish:
Comisión Internacional contra la Impunidad en Guatemala, CICIG) is an
international body charged with investigating and prosecuting serious
crime in Guatemala. It was created on December 12, 2006, when the
United Nations and Guatemala, under the government of Óscar Berger,
signed a treaty-level agreement setting up CICIG as an independent
body to support the Public Prosecutor's Office (Procuraduría General
de la Nación), the National Civilian Police (Policía Nacional Civil)
and other state institutions in the investigation of a limited number
of sensitive and difficult cases. The ultimate goal is that through
CICIG's work, national judicial sector institutions will be
strengthened to continue to confront illegal groups and organized
crime in the future.

Wikipedia: International Commission Against Impunity in Guatemala (CICIG)
Civil service exams (as opposed to the spoils system which still has defenders) were, among other reasons, adopted as an anti-corruption mechanism. Perhaps these countries could also adopt such exams.

This article traces a number of cases of graft in Zambia to show the
importance of this practice within the political system. Graft is
treated as one element of a spoils system through which clientelism
operates and through which, more generally, the state is used as a
resource for private ends. Graft and patronage are shown to have
negative consequences for the state through undermining efficiency and
legitimacy and displacing policy ends. But perhaps most importantly,
it is argued that graft ultimately involves a transfer of wealth
between classes and more specifically is an important factor in the
growth of an indigenous owning class.

Political graft and the spoils system in Zambia — the state as a resource in itself
Well defined property rights provide fewer opportunities for regulatory discretion and therefore fewer opportunities for bribe takers to exert influence. Easier said that done.
Restrict financial privacy of and gift taking, side-employment, and post-employment by government employees. May be hard to enforce in low-development societies.


I think I understand the problem here: you're thinking of exchange rates as "units", i.e you're thinking that looking at GDP numbers in dollars tells you the same thing as looking at the same numbers in the local currency after adjusting for inflation. Unfortunately, exchange rates need not reflect what you think they reflect at all. In a world without capital flows between countries and with floating exchange rates, exchange rates would adjust in order to ensure that the current account position of any given country is balanced. This adjustment means that exchange rates primarily reflect relative prices on the international market of tradable goods, like commodities, for instance. Services such as medicine, education, finance (the capital account is closed) are not easily traded, therefore the exchange rate would carry little information about their prices, which are relevant to any person living in the country.

However, the real situation is even worse, because in reality there are capital flows between countries; there's borrowing, there's portfolio investment, there's direct investment, etc. Therefore, the condition I mentioned above fails: instead, the exchange rate adjusts so that the balance of payments balances, i.e such that

$$ \textrm{Current account} + \textrm{Capital account} = 0 $$

This means that if investors in the US decide to cut exposure to Turkish assets in the face of increasing political risks, the lira would depreciate and the nominal GDP measured in dollars would fall. However, such a devaluation would likely be expansionary for exports, and if the country has no problems of inflation (which is not true for Turkey), and there is no currency mismatch on domestic balance sheets (which is, again, not true for Turkey) their central bank could maintain monetary policy at low interest rates, causing the depreciation to become expansionary. Turkey would be producing more goods and services, it would just be that their value measured in local currency and then converted to USD would be less than it was before.


Why can't a government print money to pay its debt?

Your get the basic intuition, but real-life situation is a  bit different.

First, I didn't really understand what you meant by "try to catch some 'fool'", but what you are calling foreign exchange bank is actually called the Foreign Exchange (market), and consist of buyers and sellers that are basically ready to buy and sell any currencies (almost all currencies in fact) provided that the price aligns with their expectations. Classic laws of market apply to this market, and if X's government (its central bank to be more precise) prints a lot of money, money supply increases. The result is a decrease in the price of money relative to other money, which is the exchange rate.

Money may be hard to understand, but it is actually a right to future cash flow of a given country. It is a sort of perpetual debt with zero coupon issued by a given country. So, if the country issues more money, it means that it issues more debt, so as a dilution mechanism it yields less cash flow to each unit of money, and since the price of money (as the price of any debt) is the expectation of future cash flow, the price will decrease.

Then, this will indeed result in more inflation. I expose two ways of seeing that. First, you can think about the price of goods as the price of goods relative to the price of money. Indeed, you're going to pay $x$ for a particular good because the seller knows that it will receive $x$ and that $x$ will give him the right to a certain amount of GDP (cash flow of country). But if the amount of GDP that the money represents is lower, he will ask for a higher price. Second, traditional monetary research has derived, a long time ago, the following formula which represents what I explained in my first point, but more formally. The following is always true, it is an accounting relation :
\begin{equation}
M\cdot V = P\cdot Q
\end{equation}

$M$ is the amount of money, $V$ is the speed of money, $P$ is the price level and $Q$ is the output or quantity of goods exchanged.

Without entering into too much details on this equation (there are extended information on Internet), you can see that increasing $M$, holding $V$ and $Q$ constant, results in inflation.

To come back to your question, countries don't (only) print money to repay their debt, because they are afraid of inflation. There is a big debate on inflation (You can read it on Internet), but loosely speaking inflation seems hard to contain. So to avoid hyper-inflation situation, they try to maintain inflation (in general below 2%).

Your story does not need another company investing in  X because, as you said,  people will end-up with more money, but X's production  will effectively increase. But it is true that companies buy currencies for investment purposes in their own country. Even local company may end-up buying their own currency, if for instance they sell product in USA (exportations), and sell the USD they receive against buying their own currency. This is why import and export (worldwide) are good indicators of exchange rate fluctuations.

Printing money to pay debt is a complicated question, and I only tried to sketch some insight here. For more information, read more on money creation and inflation (either in the academic literature or in economic textbook for instance).


Does risk aversion of utility function cause the existence of positive interest rate?

Yes and no; it depends on which interest rate you look at.

You are right in that risk aversion affects interest rates, but the direction can go both ways. In what follows I look at an economy with risky (stochastic) and non risky assets and risk averse agents. For the thought experiment, we increase the volatility of the risky asset, "increasing its riskiness".

Risky Asset

Despite having the same expected return, due to Jensen's inequality, the risky asset gives less utility to the agents. The risky asset will need to pay a higher interest, a risk premium, to attract investors.

Safe Asset

We have increased the aggregate risk of the economy by making the stochastic asset more volatile. Agent's added utility from insurance (through safe assets) has increased. Demand for safe assets at the old interest increases. Holding the supply of safe assets fixed, this means that the safe asset's interest rate will go down.

To understand this fully, you might want to write down a two period model with two different assets, one standard normal with standard deviation $\sigma$ and one deterministic. Examine how interest rates for both assets change if you increase $\sigma$.


Here's a quick and dirty trick:

  DWL is the triangle that points (horizontally) towards the efficient quantity.

If equilibrium quantity is lower than the efficient quantity, DWL should point rightward, and it represents the amount of unrealized gains from trade.

If equilibrium quantity is higher than the efficient quantity, DWL should point leftward, and it represents the amount of social loss due to over-production/consumption (when social marginal cost is greater than social marginal benefit).


What does Piketty mean by the C90/C10 ratio?

My understanding is that Piketty uses the term "P90/P10" only to measure wage inequality  and the term "C90/C10" for wages plus social charges. The underlying distributions are different, but they both measure the ratio of the 90th percentile to the 10th percentile.


It is not war itself that helps to counteract economic crises, but rather the increased spending needed to support the war. In terms of the evidence to support this claim, Lee Hudson Teslik from the Council on Foreign Relations writes:

  There is an ongoing debate about the extent to which war spending affects a country’s economy. Experts disagree on the most fundamental point—whether war helps or hurts national economic prospects. Massive U.S. national defense spending during World War II is sometimes credited with rejuvenating U.S. economic prospects following the Great Depression. The journalist Robert J. Samuelson, in a primer on the topic, says there can be little doubt that military spending and mobilization during World War II reduced U.S. unemployment rates and revitalized the economy. A recent paper from the National Bureau of Economic Research concludes that countries with high military expenditures during World War II showed strong economic growth following the war, but says this growth can be credited more to population growth than war spending. The paper finds that war spending had only minimal effects on per-capita economic activity. The effects of more recent wars are equally disputed. A historical survey of the U.S. economy from the U.S. State Department reports the Vietnam War had a mixed economic impact. The first Gulf War typically meets criticism for having pushed the United States toward a 1991 recession. In a 2003 op-ed in the Guardian, economist [Joseph] Stiglitz wrote that the aftermath of the Gulf War exposes the “myth of the war economy.” Indeed, he argues that increased military spending is “unambiguously bad” for the living standards of normal citizens. Other economists argue the opposite.

I would argue the consensus generally leans towards wartime spending as beneficial to a nation's economy.


19th Century American Economist Henry George argued that the real conflict wasn't between Labor and Capital, but rather between the owners of Land and everybody else.  Neoclassical reformulation of the factors of production essentially merged Capital with Land, erasing the distinction and setting the stage for the current Socialism/Communism vs. Capitalism conflict we've had ever since.

Georgists today support a return to the earlier model, which points to a middle way between the two extremes.  They argue that by taxing economic rent (the logical extension of payments made for the use of Land) governments can make infrastructure projects self-financing, eliminate the root causes of financial speculation (and the boom/bust cycles they trigger) and reduce inequality — all while maintaining (or increasing) the freedom of markets.


What can explain increasing exports after default episodes?

at periods of default of Argentina, Ukraine and Panama, the data tells
  exactly the opposite: exports increased in the wake of the sovereign
  default. What can explain this?

Sovereign default may trigger a depreciation of the domestic currency. When that happens, it has a two-fold effect on the domestic economy: On the one hand, foreign immediate inputs (that is, imports) become more expensive. On the other hand, exports become cheaper (that is, more attractive) to foreign markets.

The net effect depends on the particularities of the economy being analyzed. For instance, a depreciation of the domestic currency might prompt firms to resort to domestic substitutes of the foreign intermediate inputs, in which case the drawback of sovereign default is reduced or mitigated.

An exporter is more immune to the detrimental effect of sovereign default if (1) its need for foreign inputs is not too high, (2) it hedged against a depreciation of the domestic currency, and/or (3) it does not compete for or need subsidies from the government because these could be harder to obtain in the context of sovereign default.


Available code for computing solutions to matching algorithms?

While answering a comment, I realized I had a post-worth response. R has become the "default language" for a lot of computational research statistics (for a number of reasons; nice NYT article here). It's high level, free and open-source, and has a closely-related journal for publishing statistical algorithms. Citations and peer review are key for academia, so you get a lot of well-described code posted to the R archives (CRAN) with descriptions posted to JStat. This spills over into a lot of blogs and quick demonstration code posts.
That is to say, there's an enormous user-create code base for R. When I need to find an algorithm online, I'll often first look to the massive R codebase. A quick search for R code turned up the following:
From an R blogger, with code (see the gist link):

The Deferred Acceptance Algorithm (DAA) goes back to Gale and Shapley (1962). They introduce a rather simple algorithm that finds a stable matching for example for college admissions or in a marriage market. ... Variations of this algorithm are used in Hospital assignments in the USA, whereby recently graduated doctors submit preferences over hospitals, and hospitals submit preferences over graduates. ... Here I'm going to use R to make a little simulation of this

From an install-able github repository for matching markets:

R package matchingMarkets comes with two estimators:

stabit: Implements a Bayes estimator that estimates agents' preferences and corrects for sample selection in matching markets when the selection process is a one-sided matching game (i.e. group formation).

stabit2: Implements the Bayes estimator for a two-sided matching game (i.e. the college admissions and stable marriage problems).

and three algorithms that can be used to simulate matching data:

hri: Constraint model for the hospital/residents problem. Finds all stable matchings in two-sided matching markets. Implemented for both the stable marriage problem (one-to-one matching) and the hospital/residents problem, a.k.a. college admissions problem (many-to-one matching).

sri: Constraint model for the stable roommates problem. Finds all stable matchings in the roommates problem (one-sided matching market).

ttc: Top-Trading-Cycles Algorithm. Finds stable matchings in the housing market problem.

Functions hri and sri allow for incomplete preference lists (some agents find certain agents unacceptable) and unbalanced instances (unequal number of agents on both sides).

Hopefully one of these can help. The second one in particular looks extremely useful, particularly if it provides an empirical estimator.


What was the reason for a boom on Wallstreet in 1938?

There might even be some quantitative analyses on this (which would make a [much] better answer), but the "word on the street" (i.e. popsci explanation) is that after the 1937 recession (which actually dragged into the first half of 1938), the rebound was due to FDR's massive spending/stimulus program.
However, looking at the Wikipedia page on the 1937 recession, the discussion on the 1938 rebound probably cannot be easily disentangled from the disputes on what caused the 1937 recession.
The uncontroversial facts are that

According to the National Bureau of Economic Research, the 1937 contraction, which lasted from May 1937 until June 1938, was America’s third-worst recession of the twentieth century, paling in comparison to the 1920 and 1929 downturns.
A few statistics reveal the severity of the 1937 recession: Real GDP fell 10 percent. Unemployment, which had declined considerably after 1933, hit 20 percent. Finally, industrial production fell 32 percent (Bordo and Haubrich 2012).

The causes for the 1938 rebound are thought to be among the observed associations:

The recession ended after the Fed rolled back reserve requirements, the Treasury stopped sterilizing gold inflows and desterilized all remaining gold that had been sterilized since December 1936, and the Roosevelt administration began pursuing expansionary fiscal policies. The recovery from 1938 to 1942 was spectacular: Output grew by 49 percent, fueled by gold inflows from Europe and a major defense buildup.

Actually disentangling the fiscal from the monetary contributors is probably quite difficult, hence the various theories advanced by various macro schools. It's still an active area of research.
For the (mostly) monetarist view in a quantitative perspective see Irwin (2012). But even he acknowledges that:

Stocks began to recover in April 1938 when the sterilization program was officially terminated but two months before the end of the recession. But here too, it is not clear if the stock market recovery was driven by the end of sterilization itself or by the expectation that the economy would recover.

Somewhat on the other side of the fence, Park and Van Horn (2015):

After implementing the difference-in-difference estimators, we find that the
increases in reserve requirements did not create financing constraints for member banks and lead
them to reduce lending. Therefore, the actions of the Federal Reserve concerning the required
reserve ratios cannot be blamed for instigating the economic downturn of 1937-38.

I haven't read this latter paper in detail, but on a quick scan they note in a footnote:

Many scholars have blamed the premature tightening of monetary policy, fiscal policy, or both
(Fridman and Schwartz 1963, Romer 1992, Eggertson 2008, Velde 2009, Irwin 2012). On the monetary
side, the Federal Reserve doubled reserve requirements, and the Treasury sterilized gold inflows from
Europe to the United States. On the fiscal side, the Roosevelt administration attempted to achieve a
balanced budget by reducing the growth in government spending and increasing taxes. Other scholars have
proposed alternative explanations, such as labor policies and a shift in expectations. Cole and Ohanian
(2001, 2009) and Hausman (2014) argue that New Deal industrial and labor policies raised wages in the
manufacturing sector and stunted economic recovery. Eggertsson (2008) argues that a shift in beliefs about
future inflation and income caused the recession. Mathy (2014) finds that uncertainty regarding fiscal and
monetary policy by the Roosevelt administration, not reserve requirements, induced the recession of 1937.


How are historic economic statistics calculated?

The site has actually multiple estimates for historical UK/England GDP (as the Maddison data include data for UK as well). But based on your question I will assume you are referring to this chart:

As the chart itself states the sources for the data are Campbell, Klein, Overton & van Leeuwen(2015) and Bank of England (2020).
Summarizing a methodology and all sources of project that requires whole book to describe is beyond scope of what can be done reasonably within format of this site which holds for all sources as well.
In short, they used output based approach - that is they try to estimate GDP by looking at various historic sources that measure different kinds of output and extrapolate from that. They use very large amount of different sources so I will just give you some examples of historic sources not all (which you can view in their book I referenced above):
Sources on Agricultural Production

Medieval Accounts Database assembled by Campbell (2000; 2007)
Overton’s Early Modern Probate Inventories Database (Overton, Whittle,
Dean and Hann, 2004)
Modern Farm Accounts Database constructed by Turner,
Beckett and Afton (2001)

Sources on Industrial Production:

Hatcher (1973). English Tin Production and Trade before 1550, Oxford: Clarendon
Press.

Mitchell (1988). British Historical Statistics, Cambridge: Cambridge University
Press.

King (2005).  “The Production and Consumption of Bar Iron in Early Modern England
and Wales”, Economic History Review, 58, 1-33.

Sources on Services

Deane and Cole (1967). British Economic Growth, 1688-1959: Trends and
Structure, 2nd edition, Cambridge: Cambridge University Press

Carus-Wilson and
Coleman (1963) England’s Export Trade, 1275-1547,
Oxford: Clarendon.

Fisher (1940) “Commercial Trends and Policy in Sixteenth-Century England”,
Economic History Review, 10, 95-117

ibid (1950). “London’s Export Trade in the Early Seventeenth Century”,
Economic History Review, New Series 3, 151-161.

Davis (1954). “English Foreign Trade, 1660-1700”, Economic History Review, 7,
150-166.

ibid (1962). The Rise of the English Shipping Industry in the Seventeenth and
Eighteenth Centuries, London: Macmillan.

All of the above sources are just examples of course (in fact even calling them a tip of an iceberg would be understating how many more sources are used for these estimates), the lists are not exclusive. For full list consult their book. Furthermore, naturally some of the above sources are not primary but secondary themselves so you will have to dig deeper if you want to know all the primary sources.

Finally, is these any metric to determine how accurate they may be?

There is no straightforward way of measuring the degree of accuracy of this kind of historic data. One way would be to compare it to other historic estimates which use different methods (for example income approach for historic reconstructions is also used) and then see how well they match together. However, a problem to begin with is that even the historic sources themselves might not be accurate as standards of record-keeping/bookkeeping were compared by present day standards low.
Furthermore, as the authors of the data themselves note these are rough estimates and frankly just being able to provide that for such a long span of time is incredible intellectual achievement.

PS: I managed to found an un-gated 2011 paper of the aforementioned authors which serves as the basis for the book. You should still consult the book for full list of sources and description of methodology (as they might have changed something) but if you want to get quick summary of what they done the paper will probably serve you well. In addition in that 2011 paper they also make comparisons of their approach with some of the aforementioned income based approaches.


Can a commercial bank lend money from transaction/checking account as it would from a deposit account?

Per fractional reserve system, banks lend deposited money as credit to clients provided they keep a reserve requirement.

This is a misleading statement. A bank loan creates a deposit, which can be transferred to other forms of deposits.

  Does this apply to both transaction accounts and deposit accounts?

As the above explained, this is not applicable.

  If yes, is the reserve requirement percent the same for both?

Each country has its own rules. Countries like Canada abolished reserve requirements in the 1990s. The United States was the main countries with reserve requirements, but I believe those were suspended during the 2020 Crisis. Before the suspension, reserves depended upon the type of account, and so were somewhat incoherent (e.g., banks would “sweep” deposits between accounts at the end of the day to reduce reserve requirements).


The notation $q_{i}(t_{0})$ does not necessarily mean number of goods sold in instant $t_0$. It can also mean the number of goods sold in the period denoted by $t_0$. As you point out the first one would be somewhat difficult to measure.

In fact I have not yet seen the notation you use. The notation I am familiar with is usually something like $q_{i,t_0}$, which better indicates that time is treated as a discrete variable. The unit of time could be anything, but it is usually a year or quarter.


A partial answer: convex analysis is extensively used in axiomatic decision theory, at least in its recent developments. Most of these papers focus on individual behavior. You can have a look for instance at the following papers on ambiguity-averse preferences:

"Maxmin Expected Utility with Non-Unique Prior" (Gilboa & Schmeidler)
"Ambiguity Aversion, Robustness, and the Variational Representation of Preferences" (Maccheroni, Marinacci & Rustichini)
"A Smooth Model of Decision-Making under Ambiguity" (Klibanoff, Marinacci & Mukerji)
"Ambiguity in the Small and in the Large" (Ghirardato and Siniscalchi)

Here is a paper that applies convex analysis to a model of trade under ambiguity aversion: "Subjective Beliefs and Ex-Ante Trade" (Rigotti, Shannon & Strzalecki).

Beyond models of ambiguity aversion, virtually all recent work in axiomatic decision theory makes use of convex analysis, and applies its tools to study various phenomena: regret aversion (Sarver, Ergin), cost of thinking (Ortoleva), random choice (Gul, Pesendorfer)... Please tell me if you want more precise suggestions.

For the mathematical part, a very good reference is Convex Analysis by Rockafellar (1970). It is cited by most of the papers above ;-).


well, LIFO also lowers the value of held inventory too...IF one special thing is occurring:  Inflation.

When there is inflation, LIFO makes things look, well, a little bit better.  During deflation (or flat) FIFO makes more sense.


What are different ways of specifying utility and decision making?

This won't get at individual choice, but how about evolutionary approaches? Perhaps this isn't what you are looking for, but one way to model decisions is to wander from the rational paradigm entirely. All changes in behavior are driven by natural selection, and so an equilibrium is based on stability.

  In a symmetric normal form game, an evolutionarily
  stable strategy is a (possibly mixed) strategy with the following property: a population in
  which all members play this strategy is resistant to invasion by a small group of mutants
  who play an alternative mixed strategy.

Text reference: Sandholm (2010)

Edit: Wanted to add some more exotic models, inspired by @CompEcon. These are interesting models for temptation, inconsistency, etc because you can get at internal conflict by seeing a DM as the unification of multiple parts.

Dual-Self Model of Impulse Control (Fudenberg and Levine 2006)

The Brain as a Hierarchical Organization (Brocas and Carrillo 2008)


what can be the proxy variable for measure of product innovation?

UN COMTRADE has export data by country-product at the HS 6-digit level. What you could do to find out when the product was first exported is to get data for a country's export for product, say, with HS code 123456, from as far back as possible, and see when was the first year that the export of this product is not zero.

I've seen number of patents filed used as a proxy for product innovation in some papers. That might work for your paper as well?


Are prepaid cards, and gift cards considered a form of money?

Prepaid cards are definitely money by any definition— they are essentially demand deposits. An article from the SF Fed states this clearly:

  In contrast to purchases with credit cards, debit card purchases transfer money electronically from individual and business bank accounts to the sellers’ accounts. Spending with a debit card would affect demand deposits and the money supply in the same way that purchases with a check or cash does. Because a debit card transfers your existing financial assets—the financial assets that you may access with a debit card are included in the money supply.

Gift cards, in contrast, aren't generally considered to be part of the measured money supply, but they are in some sense also a form of money. They can be used as a medium of exchange; they are tied to a broader unit of account, and they're (in a manner that depends significantly on the terms of the gift card) a store of value. Rather than being liabilities of banks, as deposits are, they're liabilities of non-financial firms. As a result, gift cards are not quite as fungible as currency is; as you note, they typically trade at a discount to face value. This isn't terribly unusual for money— money during the free banking period often traded at discounts to its face value, particularly as the distance from the issuing bank increased. One of many discussions of this phenomenon can be found in Gorton (1999).


Theories relevant to institutional quality and economic development

The following researchers have written a number of papers and books on this topic:

Doran Acemoglu
James Robinson

You may want to read their papers for a good start.


Question regarding Carlstrom and Feurst (1997)

What you address (and what is done in your referred paper) is called
Model Calibration
Many economic models use this method to determine one or more parameters:

Calibration is a strategy for finding numerical values for the parameters of artificial economic worlds.

As the widespread use of this technique, let me explain it with a universal example and order your statements to the correct steps here:
The Model

In the paper the authors derive 9 equations that uniquely derive an equilibrium in their model.

Let us construct a simple model to explain some dependent variable $y$. The only independent variable to explain $y$ is $x$. The only restriction in our model is, that $\frac{dy}{dx}=1$ must hold for any observation.
The steady state
Without any further requirements, it is simple to derive the whole model:
$$y=x + \mathbf{t}$$
What can be seen, is the appearance of an unknown variable t. As our model contained a differential equation, such additional variables occur. How to deal with them?
Calibration

Then in the next section they run empirical tests by starting at a steady state and then hitting the system with productivity and wage shocks to see the dynamics.

Make use of your economic model and calibrate t to your data. Thats exactly what the authors do in order to see the dynamics. Lets assume you have the following three observations:

|    y    |    x    |
---------------------
|    3    |    1    |
|    4    |    2    |
|    5    |    3    |

Calibration finds an optimized solution for our yet unknown parameter t. For the three observations above, t would have a value of $2$. Now, you have your full model given with the formula:
$$y=x+2$$
Further information
I recommend to read this paper, which discusses the use of calibration techniques in economic models. The authors discuss concrete various macroeconomic models on their critical use of calibration and offer more details and best practices on this technique.


Are there Utility Monsters in Economics?

There is a type of protection called a liability rule, where I, $A$, can take something from $B$, if I pay the damages $c$ which are court-ordered preemptively. Copyright law is all about liability rules. If the damages are correlated with $B$'s valuation appropriately, then efficiency holds. You are interested in the opposite case. If IP law doesn't get it right, this would be an example matching your hypothetical as I understand it. And some might say this is justified. A company might want to keep some research under wraps to prevent competitors from also using it and making it even more valuable to society through investment. So the expiration of a patent might fit this scenario in a roundabout way too if we start the model where the company already owns the patent and a court is deciding when to let it expire.

Reference: Exchange Efficiency With Weak Property Rights.


Inflation without Increase in Money Supply?

It all depends on what you mean by inflation and by money supply. Technical questions and answers need specific definitions, otherwise everyone ends up talking at cross-purposes.

Is it possible to have an increase in general price levels without any changes to the amount of money in circulation? Yes: if the velocity of circulation of money increases, and the amount of goods and services available to buy does not increase by as much.

Is it possible to have an increase in general price levels without any changes to the amount of money in circulation or the velocity of circulation? Yes: if the amount of goods and services available to buy, decreases, so that there's more money chasing fewer goods.

Is it possible to have an increase in general price levels without any changes to the amount of money in circulation or the velocity of circulation, and with no decrease in the amount of goods and services available to buy? Yes, if the demand curve changes so that the same amount of money is now used to buy a smaller quantity of stuff at higher prices.

Is it possible to have an increase in general price levels without any changes to the amount of money in circulation or the velocity of circulation, and with no change in the amount of goods and services bought? No, because the velocity of circulation is by definition total transaction value divided by the amount of money in circulation, so if velocity, quantity and money supply are constant, then prices must be too, because total transaction value equals prices times quantity.