How would profit from labor exploitation be possible in a relatively competitive market?

In competitive industry it is not possible as wages will be paid their marginal product of labor, and 'accounting' profit will arise either from marginal product of capital or entrepreneurial labor.
In economics, it is important to distinguish between economic and accounting profit, the economic profit in perfectly competitive market is zero, but you cannot observe that just by looking at the accounting profit firms report as that does not take into account various costs such as opportunity costs (generally accounting serves practical purposes not to scientifically describe true underlaying economic reality - for example in many countries sole traders/self employed people will be by accounting systems allowed to write of let's say $30\%$ of revenue as a cost - this may make sense from accounting perspective but economically it is completely absurd to state that all self-employed people have such costs).
In addition, just because there are imperfections on goods market (allowing firm to earn economic profit) that does not mean there will be imperfections on labor market (meaning employees will still get their marginal product).
Lastly, even when labor market is imperfect, and these imperfections result in labor getting less then its marginal product (there could also be imperfections that would result in more) there is no exploitation going on in the Marxian sense of exploitation.
The Marx's theory of labor exploitation can be summed as (see Stanford Encyclopedia of Philosophy):

To understand Marx’s charge of exploitation, it is first necessary to understand Marx’s analysis of market prices, which he largely inherited from earlier classical economists such as Adam Smith and David Ricardo. Under capitalism, Marx argued, workers’ labor power is treated as a commodity. And because Marx subscribed to a labor theory of value, this means that just like any other commodity such as butter or corn, the price (or wage) of labor power is determined by its cost of production—specifically, by the quantity of socially necessary labor required to produce it. The cost of producing labor power is the value or labor-cost required for the conservation and reproduction of a worker’s labor power. In other words, Marx thought that workers under capitalism will therefore be paid just enough to cover the bare necessities of living. ...

But while labor power is just like any other commodity in terms of how its price is determined, it is unique in one very importance respect. Labor, and labor alone, according to Marx, has the capacity to produce value beyond that which is necessary for its own reproduction. In other words, the value that goes into the commodities that sustain a worker for a twelve-hour work day is less than the value of the commodities that worker can produce during those twelve hours. ...

According to Marx, then, it is as though the worker’s day is split into two parts. During the first part, the laborer works for himself, producing commodities the value of which is equal to the value of the wages he receives. During the second part, the laborer works for the capitalist, producing surplus value for the capitalist for which he receives no equivalent wages. ...

Capitalist exploitation thus consists in the forced appropriation by capitalists of the surplus value produced by workers.

As you can plainly see above the idea of exploitation rests on the idea of labor theory of value (LTV). Labor theory of value is no longer accepted theory since it leads to empirical inconsistencies and it is generally considered incorrect (the current accepted theory of value is subjective theory of value). Under subjective theory of value anything can create value, including capital and indeed as we are seeing more and more automation that eliminates the need for labor this is becoming more painfully clear than ever.
In addition, the Marx's analysis implies that industries which are more labor intensive should have higher rates of profit (since according to Marx profit is earned through exploitation) and this empirically does not hold (Böhm-Bawerk 1898).
Next, the whole theory on which Marx build contained analytical mistakes that were not resolved by Marx (see Grant & Brue The History of Economic Thought 7th ed).
Lastly, the Marx's theory of exploitation also relied on an argument that workers are forced to work for capitalist as they don't have access to capital. That could seem reasonable in context of early 19th century England, when capital was very expensive and incomes low so for most part only richer people could afford to invest in it, and thus they were 'forced' by circumstance to work for capitalist.  However, this is not applicable when workers have access to out option (welfare benefit) as they often do in modern countries. In addition,  with modern financial systems the access to capital if you have good idea is arguably better than in 19th century where capital markets were still developing.
Hence as pointed by KennyLJ in the comments the Marx theory of exploitation is discredited in the profession. There were some scholars who tried to 'resurrect' the Marx idea of exploitation by dropping the labor theory of value and arguing only some of the 'worker surplus' is being exploited, and also trying to argue it applies to situations where workers are not forced to work  (see Kymlicka 2002, Contemporary Political Philosophy: An Introduction), these ideas are not generally accepted in economics and rather live on in some corners of philosophy/sociology.


Relevance of government liabilities to GDP

The government debt represents the total liabilities outstanding that the government must (at some point) repay to investors. Generally to do so, governments tax their populace, either now, or at some future point.

The often cited Debt to GDP ratio figure is one shorthand for understanding how difficult it will be for any given government to raise the revenue necessary to cover its debts, and/or decrease government spending. If country "A" had \$100 billion in outstanding debt, and country "B" had \$400 billion in public debt, which would be in worse shape (that is, which would likely have a harder time raising the necessary funds)? Ultimately, it depends on a variety of factors, one of which being "how much can the country realistically raise in revenue?" If country "B"'s GDP is ten times the size of country "A," then even with a higher debt load, it represents a smaller relative amount owed, and might then be easier to pay back.

Of course, this is just one common, convenient shorthand statistic often used to quickly assess how sustainable a country's debt is. There are many other factors which must be considered if you're trying to comprehensively measure "default risk." But Debt/GDP is a pretty easy, useful one that has become popular over time.


Why offer curves of a power plant are stepwise increasing?

One minor point- it could be that the factory schedules labor and other factors of production (coal deliveries, etc) in daily increments, and therefore pricing is fixed within each increment.


Why do some phones depreciate faster than other?

I think it might have something to do with how much a phone model is valued in certain markets. For example, the amount of utility you get from a certain phone model is higher than another phone - they are not perfect substitutes for each other. Hence, the discounted utility of the phone will still be higher, because the user still values it higher.

There is no strict formula that I know of for this comparison, but by simple logic, if you feel proud of having an iphone, it's demand will be high, will be more plastic, and hence, whether the phone is used or unused, it will sell at a higher price, so will depreciate lesser.

Must note, though, that the depreciation is not constant over time. It reduces much more in the beginning, and then sort of flattens out towards the tail end.


Does dollar cost averaging actually have any advantage over one time investment?

The seminal academic criticism of dollar cost averaging on many specifications of economic conditions is A Note on the Suboptimality of Dollar-Cost Averaging as an Investment Policy (Constantinides (1979)). You might also be interested in these papers:

  Dollar Cost Averaging is an investment system that is widely advocated
  by brokerage firms and mutual funds. In its best known form, an
  investor seeking to put a lump sum into risky assets is counseled to
  invest the money over a period of time in equal installments in order
  to avoid the devastating effect of a market fall immediately after a
  single, lump-sum investment. Using graphical analysis, historical
  stock market returns, and Monte Carlo simulations, this article
  demonstrates that no such benefit accrues to a Dollar Cost Averaging
  Strategy. Two alternative strategies, optimal rebalancing and buy and
  hold achieve better performance in all three analyses.

Nobody gains from dollar cost averaging analytical, numerical and empirical results (Knight and Mandell (1993))

  Some studies find the dollar-cost averaging investment strategy to be
  sub-optimal using a traditional Sharpe ratio performance ranking
  metric. Using both the Sortino ratio and the Upside Potential ratio,
  we empirically test four investment strategies for alternative asset
  investments. We find the relative ranking of dollar-cost averaging
  remains inferior to alternative investment strategies. (JEL G1, G11,
  N2)

An empirical examination of the effectiveness of dollar-cost averaging using downside risk performance measures (Leggio and Lien (2003))

  The widespread practice of dollar-cost averaging (DCA) amongst the
  investing public, has puzzled most financial economists, ever since
  Constantinides demonstrated the dynamic inefficiency of this
  strategy under very general conditions. This enduring phenomena has
  forced researchers, such as Statman , to suggest behavioral
  explanations for DCA's popularity, predicated on the prospect theory
  of Kahneman and Tversky .

  In this paper we reexamine the payoff structure of DCA via
  continuous-time financial mathematics and then ask the question: Is it
  possible to reconcile the theory and practice of dollar-cost
  averaging?

  To answer this question, we take a slightly different approach to the
  issue by using the tools of stochastic calculus and Brownian bridges.
  We demonstrate that engaging in a dollar-cost averaging strategy is
  akin to purchasing a zero strike arithmetic Asian option on the
  underlying security. In other words, people who engage in dollar-cost
  averaging are implicitly purchasing a path-dependent contingent claim.
  We then prove that the expected return from this exotic option — i.e.
  the DCA strategy — conditional on knowing the final value of the
  security will uniformly exceed the return from the underlying security
  for all sufficiently large volatilities.

  This leads us to argue that investors may be dollar-cost averaging
  because they have "target prices" for the underlying asset price. The
  strategy of dollar-cost averaging would then exceed the returns from
  lump-sum investing, based on their subjective conditional expectation.
  In fact, the more volatile the underlying security, the greater is the
  benefit to dollar-cost averaging — conditional on knowing the final
  value — which is consistent with common practice.

A CONTINUOUS-TIME REEXAMINATION OF DOLLAR-COST AVERAGING (MILEVSKY and POSNER (2003))

  Dollar Cost Averaging is a strategy for purchasing equity securities
  that is widely recommended by professional investment advisors and
  commentators, but which has been virtually ignored by academic
  theorists and textbook writers. In this paper we explore whether the
  strategy is but another instance of irrational behavior by individual
  investors, or whether it is an investment heuristic that has survival
  value in an environment in which security prices exhibit mean
  reversion behavior that has only belatedly been recognized by academic
  theorists. Our evidence supports the view that the uninformed
  individual investors who follow this strategy in purchasing individual
  stocks to add to an existing portfolio are better off than if they
  followed the ‘rational’ strategies traditionally recommended by
  academics.

Dollar Cost Averaging (Brennan, Li and Torous  (2005))


About studying econometrics in undergraduate

If you are looking to understand how Econometrics are and should be grounded in Statistics, read Aris Spanos "Probability Theory and Statistical Inference: Econometric Modeling with Observational Data". Be prepared for a bit of a shock.

As for Mathematical Statistics, they are a difficult subject, and there is no magical textbook that has ever succeeded in making them "accessible".


Real wage stagnation and union membership

My understanding is that theories and empirical methodologies demonstrating the relationship between declining union membership and wages risk overestimating the causality of the relationship.

Anyone who argues that businesses do NOT, in a relevant share of cases, endeavour to suppress union formation, activities or recruitment, whether individually or collectively, is either ignorant or a political shill. (Things are not nice 100% of the time in the other direction either.) This should be recognized, in order to proceed with rational discussion of the economics of the issue.

However, factors like increased international competition in lower-skilled jobs from a very large international labour pool (which is much more accessible to producers than in 1950 or 1980) can be one reason that the causality of declining union membership can be overestimated.

Also, mechanisms of tacit collusion between businesses can act to hold down wages as well, for example with various agreements which prevent an employee from working for a competitor for some number of years after they leave (e.g., Google and Apple could both force employees to sign such an agreement as a condition of employment, thus holding down wages for both firms, which both account for a substantial share of employment in their specific areas of activity). This could be another explanation for some portion of wage stagnation that would be difficult to frame as resulting from declining union membership per se.

Between 1) international competition and 2) a proliferation of methods to suppress wages unrelated to union formation or membership per se, it should suffice to say that significant effort for robustness must be demonstrated before any empirical work can be taken seriously when discussing causality between declining union membership and 50 years of stagnant average wages.

The risk, then, is that an inadequate methodology which involves a) under-specification of a model, and/or which b) includes inadequate robustness checks, could indicate that declining union membership caused stagnant wages, when theoretically this could be a false positive when other factors explain both the declining union membership and also stagnant average wages.


Papers on time it takes to reach market equilibrium?

There is the famous Cobweb model. Starting from a situation of equilibrium, it analyses the effect of a shock, and how long it takes to reach the equilibrium, which depends entirely on the elasticities (or slopes) of demand and supply. Thus, by estimating empirically such elasticities for a given period (e.g. year), you can predict/forecast how many periods (e.g. years) it will take the market to settle after a shock. (Strictly speaking, it is infinity, but you just need to select a threshold or margin of error to consider the market in equilibrium)

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(Source of image here)

A huge literature exploded after the original paper, so if you google Cobweb models you get tons of hits. A very good treatment of this model can be found in Waugh (1964). He extends the simple model to multiple markets.


Are there any real world examples of potential Pareto improvements?

By "potential Pareto improvement", what Fallis means is a "Kaldor-Hicks improvement" or more simply an overall increase in wealth.

The argument by Kaldor (1939) and Hicks (1939) was that if there is an overall increase in wealth, then we could potentially compensate the losers, so that there is potentially a Pareto improvement. Therefore, a "potential Pareto or Kaldor-Hicks improvement" (i.e. an increase in wealth) is "just like" an "actual Pareto improvement".†

So, a free-trade agreement is a "potential Pareto" or "Kaldor-Hicks" improvement so long as it boosts our country's GDP. However, it is probably not an "actual Pareto improvement" because there will likely be at least one person in our country who suffers a loss due to the free-trade agreement.

And so, as you can tell and as Fallis says, most policy recommendations by economists are "potential Pareto" or "Kaldor-Hicks" improvements, but not "actual Pareto improvements".

†Note that this argument by Kaldor and Hicks is controversial, but is also one that most economists have gleefully latched on to, precisely because it is nearly impossible to ever have an actual Pareto improvement. Further reading: Feldman, 1998, "Kaldor-Hicks compensation".


How can I prove that a utility function does (or does not) satisfy diminishing MRS?

If you remember, in a two-dimensional curve, its concavity or convexity (the slope of its slope) is given by the second derivative. For a three-dimensional function, you want to look at the Hessian table (the table of all second derivatives).
If the Hessian is negative definite for all values then the function is strictly concave, and if the Hessian is positive definite for all values then the function is strictly convex. If the Hessian is not negative semidefinite for all values then the function is not concave, and hence of course is not strictly concave. So on, so forth.
This particular function will be a bit more tricky to find second derivatives, I presume.
There are various examples shown in a very nice website here.
There is even a Cobb-Douglas example, which I'm certain you'll find valuable.


Do countries with centrally-planned economy have Entrepreneurship as a factor of production?

This is hard to answer because it is disputable if entrepreneurship is a factor of production just to begin with. There are economists who would argue that any entrepreneurial activity already falls under labor/human capital.
Setting the caveat above aside there are two major views on how entrepreneurship as a factor should be treated (see here). In one view entrepreneurship cannot be divorced from ownership of assets since as argument goes people can take entrepreneurial risk only if they actually have 'skin in the game'. In centrally planned economy asset are collectively owned state and all market risks are socialized. Under this view there can be no entrepreneurship in centrally planed economy. Moreover, managers would not count under such view either.
A second view would treat entrepreneurship as purely an intellectual activity of combining different factors of production (to be clear the previous view also agrees on this part but they add assets as additional requirement). For example, in Fed St. Louis series for students the entrepreneurship as a factor of production is defined as:

The fourth factor of production is entrepreneurship. An entrepreneur is a person who combines the other factors of production - land, labor, and capital - to earn a profit.

The above definition does not have asset ownership as a part of requirement and manager's work here could be considered to be an entrepreneurial factor. The additional important requirement is that this is done to earn profit but as far as I understand here it does not matter if the payment is considered profit or wage in accounting manner but rather from economic perspective. Furthermore, generally speaking profits can exist in centrally planned economies, to the extent these are shared with the managers in some explicit or implicit way, you could argue that under this definition entrepreneurship can exist in centrally planed economy.
This being said, centrally planned economies most likely don't have much of it. According to highly cited work of Audretsch (2007):

Measurement of entrepreneurship capital is no less complicated than is measuring
the traditional factors of production. Just as measurement of physical capital, labour,
and knowledge invokes numerous assumptions and simplifications, creating a metric for
entrepreneurship capital has also presented a challenge. Many of the elements that constitute
entrepreneurship capital defy quantification. In any case, entrepreneurship capital, like all
of the other types of capital, is multifaceted and heterogeneous. However, entrepreneurship
capital manifests itself in a singular way—the start-up of new enterprises.

The emphasis was added by me. Hence an entrepreneurship can be measured by the rate at which new enterprises form (and in fact this is the way in which most literature measures it). It goes without saying that historically speaking centrally planned economies are not exactly known for high rates of new enterprise formation.


Autocorrelation test for AR(p) (Breusch-Godfrey LM test)

Yes you can use  Breusch–Godfrey  (BG) test for autocorrelation also in AR(p) models and dynamic models in general (see Verbeek Guide to Modern Econometrics where BG is applied to dynamic models in some examples - one of such examples is on page 142 in the 4th ed). As a matter of fact BG test is, generally speaking, the preferred test for autocorrelation in AR models and dynamic models  (see Maddala Introduction to Econometrics).
Furthermore, actually Ljung-Box's Q test should not be used for testing AR processes or other dynamic models with lagged dependent variable. The reason for that is that this test is in the models with lagged dependent variable biased towards the null hypothesis. This is because in such models the Ljung-Box's Q test no longer has asymptotically $\chi^2 $ distribution (see the Maddala textbook above for further discussion of this issue).


Economic Profit and the Return on Invested Capital

Both of them are consistent. The economic profit is the total revenue $TR$ minus total cost $TC$ but in economics costs must include also opportunity costs not just accounting ones.

However, for all standard market structures  $TR>TC$ happens only if the marginal revenue or price  $P$ is above marginal costs $MC$.

Also if there is positive economic profit then also return on the capital will be higher than opportunity cost of that capital, because by definition such profit must be higher than the minimum necessary to compensate for the opportunity cost.

So it’s not that the definitions are inconsistent they are just different results based on the same principle. However, if you would try to calculate the true economic profit by different estimation methods you could get different answers because some opportunity cost might be not easy to observe. For example, weighted average costs of capital (WACC) is a crude way how to estimate opportunity cost it’s not the definition of it. Also many inputs in WACC are only proxies not actually directly observable variables.

Now turning attention to your example it’s simply not true the demand for oil is elastic or that oil market is perfectly competitive market. Due to organizations as OPEC oil market was for long time almost monopolized and even though the strength of OPEC decreased in recent years I would still put it somewhere more close to oligopoly. It’s definitely not even close to perfectly competitive market which requires many sellers not just few key players. Also perfect competition requires (virtually) no barriers to entry and in oil industry the entry barriers can be high.

Moreover, demand for oil is also very inelastic. Unfortunately even now in 21st century our civilization crucially depends on fossil fuels to run. Oil is still virtually a necessity and demand for it is highly inelastic. In fact if you try to survey the literature on google scholar by checking for elasticity of oil demand you find that most mean estimates report elasticity less than 1 and many close to 0.1 which is very small. In such setting in most market structures you get a room for significant economic profit as market power varies inversely with elasticity of demand.


[Graduate Macro]Land in the Economic Growth Model: CES Function

Suppose that the total amount of land ($\Lambda$) is fixed but the population measure increases exogenously ($\dot{L}>0$).

The only way the per capita product ($y(t)$) remains constant over time is when you can fully substitute capital against land. In formulas, you need to set $\psi=1$ to get:
$$ y(t) = A[a k(t)^\alpha + (1-a) \lambda(t)] $$
and, as you can see, you can always trade-off declining land per capita ($\lambda(t) =\frac{\Lambda}{L(t)}$, then $\dot{\lambda}<0$) with more investment in capital.

On the other hand, as soon as $\psi \neq 1$, then $Y/L$ will steadily decrease over time as a consequence of population growth in conjunction with decrease land per capita not being fully substituted by increases in capital.

This would be my intuition, hope it helps, I'm sure one can make the argument more rigorous, especially for the case of $\psi\neq 1$.


tl;dr: In the hypothetical you set in the body of your question redistribution cannot help poor. However, this is not because redistribution could not significantly raise the welfare of the poor but rather because in your question 'the rich' actually dont have any resources to share with the rest. In fact the way how you set up your hypothetical example what you call 'redistribution' is equivalent to monetary expansion.
However, in real life redistribution, even redistribution of money, is not equivalent to monetary expansion and can help to raise material welfare of people especially of the poor and to the extent quality of life depends on material welfare their quality of life as well.

Full Answer:
The full answer will be structured in three parts. I will first try to answer your question in body of the text, then explain the problems with your assumptions and finally try to answer question in your title with small changes that will be still true to the spirit of the question and make it actually answerable in objective fashion (I do this because I think that most of the attention this question is driving comes from the question in its title).
PART I: Answer to the Question in the Body:
In order to answer your Q lets try to model it. In your example you talk about hypothetical economy when nobody works there are just people with some given wealth so we are in some endowment economy where people are endowed with some output $Y$ which constitutes their 'wealth', but rich people have only cash/money $M_r$. What even more you assume that the cash was not used in anyway so for all practical purposes it simply is not even part of the economy. Moreover, as per the question I will assume the money are equally distributed among everyone.
Well taking assumptions above seriously we can just analyze the effect of such money transfer the same way as just an expansion of money supply using simple equation of exchange (this is not normally appropriate model for evaluating redistribution but in your case redistribution is just monetary expansion in disguise and this model is a simple textbook example of a model that can be used to analyse effect of monetary expansion):
$$MV=PY$$
where, $V$ and $P$ which were not yet introduced  are velocity of money and price level respectively. In this case $Y$ is fixed as there is no production just everyone has some certain level of wealth. Hence when $M$ increases by adding the $M_r$ either $P$ increases or $V$ must drop (which would happen if the money people get from the rich end up 'buried' and not used), or some combination of thereof.
In either case whether prices or velocity changes people cant be made better off by the transfer of cash in your hypothetical scenario as in your scenario the rich $1\%$ really has nothing of value to give to the $99\%$. Their endowments $Y$ remain fixed. Again, I would normally never recommend using this model for analyzing redistribution but in your case you are not really talking about redistribution.
PART II: Problems With Your Question:
The main problem to your question is that if really taken to its logical conclusion it boils down to the following. Does creating more money make people any richer? Well answer to that is no. What makes people rich is the amounts of goods and services they have access to. In fact from an economic perspective the $1\%$ of population in your question must be miserable bunch because by your assumption they only have money while the other $99\%$ actually have some other 'wealth' which presumably in your question implies some goods and services to enjoy.
However, the above situation is clearly absurd and does not correspond to reality. Rich people do not sit upon piles of sterile cash that is not used in economy. If their money circulates in the economy either because the rich spend it, or put it in bank into their accounts or invest it into some assets they will already be part of the $M$ and redistribution, even redistribution of money, will not expand $M$.
Redistribution might decrease output due to the fact that it distorts incentives, however at the same time there are arguments that it might boost output as high levels of inequality can make harder for poor people to become entrepreneurs or realize their full potential in different ways. Empirical studies actually show that the relationship between economic growth and intensity of redistribution is mixed, suggesting that on net it has no impact on economic growth save for cases of extreme redistribution that would be well above what we typically see in developed nations nowadays (see Ostry, Berg, and Tsangarides 2014 and sources cited therein). When it comes to velocity redistribution can slightly increase it as some research shows that people with lower savings rate contribute to higher velocity (see Wang & Ding 2005), however the effects are nowhere near strong enough so any transfer will have no net effect on welfare of recipients.  As a matter of fact it is generally agreed by the profession that over long time periods it is an increase in $M$ which leads to inflation (see pretty much any conventional textbook for example Blanchard et al Macroeconomcis an European Perspective or Mankiw Principles of Economics).
As a consequence of the above, redistribution can be more or less viewed not just as a transfer of money but as a transfer of resources and in fact this is how redistribution is treated in wast majority of the literature in public economics (see models presented in any public economics textbook). Many redistribution models will not even explicitly include money at all and this is not due to lack of oversight but deliberate simplification (the same way as physicists often may assume that some part of space contains perfect vacuum even if it has some particles) reflecting the view that redistribution is not simply some money creation especially when financed by taxes (even though relative prices might change which can still affect welfare but this is accounted for in the models).
Part III: Answer to the Question in the Title:
The question in your title:

Would a fair distribution of wealth from the super rich increase the purchasing power/life quality of the average person?

is actually far more sensible but I will still make 3 changes that I think are true to the spirit of what you are actually interested in.
First I am going to change the word fair distribution to redistribution to get rid of the word 'fair'. Well what is fair or fair distribution? That's a question that human race pondered ever since civilization emerged from the fertile crescent and perhaps even in prehistory without any generally agreed upon answer and discussion of fairness belongs to moral philosophy not economics.
Second I will change wealth to wealth/income. The reason for this is that I feel that like many other non-economist you do not distinguish between wealth and income and treat them jointly. For example, who is in your opinion a richer person? A old retiree whose net assets are worth $\\\$1,000,000$ because his house happened to be in gentrified area and went up in value but with measly salary of $\\\$500$ working as shop greeter, or a superstar which might have no net assets (i.e. living in fancy hotel instead of owning house) but with monthly paycheck of $\\\$70,000$? The retired person has more wealth but I think many would actually consider person in that situation poor whereas the second person has no wealth but I think most reasonable people would consider such person rich.
Third I will replace average person with low income people. The reason for this is that in any right skewed income distribution (an income distributions are world are right skewed) an average person will actually be better off than more than half of the population and optimal redistribution systems are designed to help people at the bottom not people who are better off than most (even though unfortunately in practice governments often engage in such 'perverse' redistribution as well).
Hence I will try to answer the following question:
Would a redistribution of wealth/income from the super rich increase the purchasing power/life quality of the low income people?
the answer here when it comes to income is resounding yes, when it comes to wealth the answer is not clear or rather the answer would be maybe.
Redistribution via Income Taxes
The literature on optimal income tax shows that optimal top marginal tax rates for the richest can be somewhere in rage of $50-75\%$ in the US (see Saez 2001) an generally similar estimates are also found for other developed countries.
The simulations for these tax rates are already made with an explicit goal of maximizing actual welfare of the recipients and society overall  - not to maximize any monetary transfer per se - but to maximize their underlying utility. They are based on the following optimal non-linear tax function which comes from the seminal works of Mirrlees (1971) - who in fact got Nobel Prize in Economics for this contribution, Diamond (1998) and Saez (2001).
Furthermore, the optimal tax rate even takes into account all the labor supply response and other factors in fact the formula is given by:
$$ \frac{T'(z_n)}{1-T'(z_n)} = \left( 1 + \frac{1}{\epsilon_{lT}} \right)\frac{\int (1-b_m)f(z_m)dz_m}{1-F(z_n)} \frac{1-F(z_n)}{z_nf(z_n0)}$$,
with $b_n \equiv \frac{\Psi'(u_n)u_c}{\eta}+ nT'(z_n) \frac{\partial l_n}{\partial \rho} $.
I wont go over every single term in the formula as this would turn this answer into a book but broadly speaking the first part $\left( 1 + \frac{1}{\epsilon_{lT}^*} \right)$ is given by elasticity of labor supply to income taxes and you can think of it as an 'efficiency' parameter, the second part $\frac{\int (1-b_m)f(z_m)dzm}{1-F(z_n)}$ tells us what the marginal benefit of redistribution is and this marginal benefit factors in underlying actual welfare which is captured by $b_n$ which depends on both utility of consumers and the societal utility function,  and finally $\frac{1-F(z_n)}{z_nf(z_n0)}$ is the part that captures the relative magnitude of distortions created by this taxation. Again since we are talking about improvements in actual underlying utility what the aggregate price level is does not matter.
If redistribution would not be able to improve welfare in poor this formula would give you zero marginal tax rates. Since simulations based on real world parameters (Saez 2001) show that top marginal tax rates could be as high as almost $80\%$ clearly redistribution can improve the welfare of poor.
An important caveat is that this optimal taxing formula does not take into account general equilibrium effects and general equilibrium effects generally (no pun intended) result in lower taxes but no reasonable estimate of the magnitude of general equilibrium effects would push the top marginal tax rates to zero. Even if we would say that in general equilibrium these taxes would be half of what partial equilibrium analysis suggests they would still result in redistribution that would be able to significantly help the low income individuals.
Redistribution via Wealth Taxes
When it comes to wealth taxes the literature is much smaller as studying wealth taxes is exponentially more difficult than income taxes due to the fact that they are rare and we also don't have good data. As a consequence the discussion on wealth taxes to some extent takes a form of twitter fights between various economists (I am referring to Summers-Saez & Zucman infamous twitter battle) rather than in some proper research.  This being said some academic work was already done on this topic but it remains highly inconclusive. I will try to present both views fairly starting with the no wealth tax view and then presenting the pro wealth tax view.
Anti wealth tax view:
Some economists argue that optimal wealth tax is simply $0$. If this is so then obviously wealth tax cannot help improve welfare of the poor. There are several reasons why this would be so which are all summed nicely in this article from Larry Summers. Here is the digest:

Wealth taxation also raises practical concerns — for example, issues of valuation: Is a partnership in a law firm wealth? How will illiquid assets — like football teams or newspapers — be valued? And issues of liquidity: If someone owns 1 percent of Uber — still a private company — she will owe roughly $20 million in taxes each year, but it’s unclear where she can get this money. She can’t sell shares and, if involved with the operation of the company, is likely to be barred from borrowing against the value of her stock.

There are also family unit issues: If a couple files separately or gets divorced, do they get two $50 million exemptions? And issues of gaming: There will be incentives to use legal structures to relinquish direct ownership of assets while maintaining control of them. For example, owning assets in a trust or a nonprofit to benefit from wealth while avoiding tax liability. Granting that capital income should be taxed more heavily than it now is, and that unrealized capital gains going untaxed is a serious problem, there is also a question of just how a punitive tax is appropriate.

It is important not to be misled by the 2 percent annual rate: A 50-year old who has accumulated a substantial fortune can expect to pay more than half of it in taxes before she dies.

Imagine that a wealthy person invests in 10-year treasury bonds, with a 2.4 percent return. The wealth tax would extract 2 of the 2.4 percent return. Combined with income taxes levied at a 40 percent rate, the wealth tax could make the effective tax rate on capital income well over 100 percent. And then at the end of life would come the estate tax. While we are not aware of formal estimates of the loss in economic efficiency from wealth taxes, we suspect that if levied without concomitant reductions in income tax rates or estate tax rates, the ratio of burden on the economy to revenue raised would be far higher than with the base-broadening measures we advocate.

....

The Organization for Economic Cooperation and Development recently assessed wealth taxation and concluded that “from both an efficiency and equity perspective, there are limited arguments for having a net wealth tax.” Of the three countries with a wealth tax, two — Norway and Spain — raise an average of 0.305 percent of GDP. These taxes generate less than one-third of what the wealth tax estimates despite having a much broader base: While precise data are hard to come by, we suspect that less than 10 percent of this revenue — or 0.03 percent of GDP — comes from those in the top 0.1 percent of the wealth distribution.

Pro wealth tax view
However, Saez and Zucman the main proponents of wealth tax  would argue that the worries mentioned above are overstated (see here).
According to Saez and Zucman the main reason why the European wealth taxes failed is that Europe tolerates (and arguably even encourages) tax competition. In this article they argue that:

The specific form of wealth taxation applied in a number of European countries had three main weaknesses. First, European
countries were exposed to tax competition and tax evasion through offshore accounts, in a
context where until recently there was no cross-border information sharing. Second, European
wealth taxes had low exemption thresholds, creating liquidity problems for some moderately
wealthy taxpayers with few liquid assets and limited cash incomes. Third, European wealth
taxes, many of which had been designed in the early 20th century, had not been modernized,
perhaps reflecting ideological and political opposition to wealth taxation in recent decades.
These wealth taxes relied on self-assessments rather than systematic information reporting.
These three weaknesses led to reforms that gradually undermined the integrity of the wealth
tax: the exemption of some asset classes such as business assets, a preferential treatment of
others such as real estate, or a repeal of wealth taxation altogether.
A modern wealth tax can overcome these three weaknesses. First, offshore tax evasion can
be fought more effectively today than in the past, thanks to recent breakthrough in cross-border
information exchange, and wealth taxes could be applied to expatriates (for at least some years),
mitigating concerns about tax competition. The United States, moreoever, has a citizenshipbased tax system, making it much less vulnerable than other countries to mobility threats.
Second, a comprehensive wealth tax base with a high exemption threshold and no preferential
treatment for any asset classes can dramatically reduce avoidance possibilities. Third, leveraging
modern information technology, it is possible for tax authorities to collect data on the marketvalue of most forms of household wealth and use this information to pre-populate wealth tax
returns, reducing evasion possibilities to a minimum. We also discuss how missing market values
could be obtained by creating markets. In brief, the specific way in which wealth was taxed in a
number of European countries is not the only possible way and it is possible to do much better
today

This being said even the most extreme serious wealth tax proposals are somewhere in ballpark of $5\%$ - this is much less then the top marginal taxes that are optimal for income redistribution. It can still be argued to be somewhat well-being enhancing but clearly it is debatable of how significant would this effect be.
Nonetheless as correctly pointed by Brian in his +1 answer/comment the argument can be made that wealth tax will help to erode the political power of the rich which will ultimately help the poor. There are also counterarguments, in fact Summers argues that wealth tax might actually increase the political power of the rich but this debate is something that is outside the field of economics so feel free to read their papers and articles I linked here in full to make up your mind about this. I personally am agnostic about this power argument as there is not much research done on this issue. I suspect that even Summers and Saez and Zucman would agree that on both sides of this argument evidence is mixed.
Hence when it comes to wealth taxes it is hard to say if they can improve the lives of low income people significantly, but this is not because of a high inflationary effect redistribution via these taxes would have but rather due to the fact that optimal wealth taxes are either zero or low.
The only exception of the above is land or similar type of property which can still be counted as wealth and which is actually easy to be taxed even at high rates without generating some of the above issues (although the liquidity and difficulty of valuation argument would remain) . However, this being said most of the wealth around the world does not really consist of land so this is more or less side note.
PS: The above is just a tip of an iceberg. I did not discuss how factoring in the effect on incentives for education change the result, what is the role of capital, inheritance, consumption taxes in all of this, I also did not discussed in great lengths general equilibrium analysis, or fully factored in all dynamic effects, but all I have written still holds in spirit even if the issues mentioned here could change the actual numbers presented. If your intent is to actually learn about the field then in addition to sources already presented here and sources cited therein I wholeheartedly recommend reading the Mirrlees review which is probably the best literature review on optimal taxation there is and also the book by Inequality Atkinson who before his death was one of the top experts in the area of inequality and redistribution.


Is there any empirical evidence for the so-called "law" of diminishing returns

The above chart is from Agroecosystems, nitrogen-use efficiency, and nitrogen management by Cassman KG, Dobermann A, Walters DT, and it deals with this sort of question quite extensively for those interested in all the gory details.


Articles about police complaints for a lit review?

I found these articles about measuring the quality of service of a police department:

  Chen, Chien Min et al. 2013. The police service quality in rural
  Taiwan. Policing: An International Journal of Police Strategies &
  Management 37 no. 3:521-542.

  Editorial Desk. 1983. A Better Report Card for the Police. The New
  York Times (Late City Final Edition), February 11.

  Slothower, Mary, Peter Neyroud, Lawrence W. Sherman. 2015. Tracking
  Quality of Police Actions in a Victim Contact Program: A Case Study of
  Training, Tracking, and Feedback (TTF) in Evidence-Based Policing.
  International Criminal Justice Review 25 no. 1:98-116.

Then I found this about productivity and trust.Sort of has to do with supply side, and a little about the quality of service.

  Brown, Sarah et al. 2015. Employee trust and workplace performance.
  Journal of Economic Behavior & Organization 116 (August): 361-378.


The main economic journals are slowly starting to require authors to make their data and the code of their analysis available as part of the online appendix. When this is the case, it is easy to figure out which software was used.

One example are recent publications in the American Economic Review. For instance,

Calsamiglia, Caterina, Guillaume Haeringer, and Flip Klijn. 2010. "Constrained School Choice: An Experimental Study." American Economic Review, 100(4): 1860-74. https://www.aeaweb.org/articles.php?doi=10.1257/aer.100.4.1860

the online appendix of which contains the complete STATA code of the statistical analysis.

If the code is not available at the journal's webpage, other alternatives involve :

Checking the website of the authors.
Contacting the authors themselves to ask if they could send you their code.

None of these methods are great if you're trying to gather a sizable or representative dataset of software used in econ papers, but for anecdotal data, they should do the trick.


Mortgage loans from foreign banks at lower interest rates

The low interest rate will be in a different currency. If your domestic currency falls in value, the value of the mortgage in terms of the domestic currency goes up.

Entities borrowing in a foreign currency and then running into difficulties is a standard features of financial crises over history.


Is Quantitative Easing an asset-swapping or money-printing exercise by the Federal Reserve?

Seignorage is harmful if it is abused to cause hyperinflation. Without quantitative easing, we probably would've had mass deflation instead (which is arguably worse), and makes the Fed look not credible in striving for its target inflation rate.

There are different ways of implementing quantitative easing in other areas I think, but more or less, it is buying massive amounts of financial assets from banks and other financial institutions in order to lower their yield, which raises the price and increases the money supply. It is supposed to be inflationary, but QE is usually in response to when the short term interest rate approaches zero, and it becomes hard to do that.


Log-linearization of Euler equation with an expectation term

Let's ignore for the moment the existence of the expected value. If this was a deterministic set-up, linearization through taking logs would be straightforward, and without the tricks of the links the OP provided. Taking natural logs on both sides of the first equation we obtain:

$$0 = \theta \ln \delta-\frac {\theta}{\psi}\ln \left (\frac{C_{t+1}}{C_t} \right )-(1-\theta) \ln(1 + R_{m,t+1}) + \ln (1 + R_{i, t+1}) \tag{1}$$

Set

$$\hat c_{t+1} = \frac{C_{t+1}-C_t}{C_t} \Rightarrow \frac{C_{t+1}}{C_t} = 1+\hat c_{t+1} \tag{2}$$

Also, note that it is standard approximation to write $\ln (1+a) \approx a$ at least for $|a|<0.1$. Usually this is the case with growth rates and financial rates so we obtain

$$0 = \theta \ln \delta-\frac {\theta}{\psi}\hat c_{t+1} -(1-\theta) R_{m,t+1} + R_{i, t+1} \tag{3}$$

which is a clear dynamic relation that links the three variables present.
If in the model, the steady-state is characterized by constant consumption and constant returns, then at it we will have $\hat c_{t+1} =0$ and so the steady-state relation will be

$$R_{i} = - \theta \ln \delta + (1-\theta) R_{m} \tag{4}$$

But we did all these ignoring the expected value. Our expression is $E_t\left[f\left(C_t, C_{t+1},R_{m,t+1},R_{i,t+1} \right)\right]$, not just  $f\left(C_t, C_{t+1},R_{m,t+1},R_{i,t+1} \right)$. Enter first-order Taylor expansion of $f()$. We need a center of expansion. Represent the four variables simply by $\mathbf z_{t+1}$ (it doesn't hurt that a variable with $t$-index is present in $\mathbf z_{t+1}$). We choose to expand the function around $E_t(\mathbf z_{t+1})$. So

$$f\left(\mathbf z_{t+1}\right) \approx  f\left(E_t[\mathbf z_{t+1}]\right) + \nabla f\left(E_t[\mathbf z_{t+1}]\right)\cdot \big(\mathbf z_{t+1}-E_t[\mathbf z_{t+1}]\big) \tag{5}$$

Then

$$E_t\left[f\left(\mathbf z_{t+1}\right)\right] \approx f\left(E_t[\mathbf z_{t+1}]\right) \tag{6}$$

Obviously this is an approximation, i.e. it has error, even if only because of Jensen's inequality. But it is standard practice. Then we see that all the previous work we did on the deterministic version, can be applied in the stochastic version inserting conditional expected values in place of the variables. So eq. $(3)$ is written

$$0 = \theta \ln \delta-\frac {\theta}{\psi}E_t[\hat c_{t+1}] -(1-\theta) E_t[R_{m,t+1}] + E_t[R_{i, t+1}] \tag{7}$$

But where are the steady-state values? Well, steady state values in a stochastic context are a bit tricky -are we arguing that our variables (which are now treated as random variables) become constants? Or is there another way to define a steady-state in a stochastic context?

There are more than one ways. One of them, is the "perfect foresight steady state", where we forecast perfectly a not-necessarily constant value (this is the concept of "equilibrium as fulfilled expectations"). This is for example used in Jordi Gali's book mentioned in a comment. "Perfect-foresight steady state" is defined by
$$E_t(x_{t+1}) = x_{t+1} \tag{8}$$

Under this concept, eq. $(7)$ becomes eq. $(3)$ which is now the "perfect-foresight stochastic steady state" equation of the economy.

If we want a stronger condition, saying that variables become constant in the steady-state, then it is also reasonable to argue that, again, their forecast will eventually be perfect. In that case, the steady-state of the stochastic economy is the same as that of the deterministic economy, i.e. eq. $(4)$.


Value flow in economy - redefinition credit and money?

Your question is very broad so it is hard to answer but I will try to do it

Can there be sustainable growth without debt? Yes! Most models of economic growth don't even have debt in them. Most economists believe that growth comes form increase in factors of production like capital accumulation and from increases in avaiable technology. Debt can also play role in promoting growth by extending credit to entrepreneurs who are poor, but in most macro models it does not play role in long term growth.
The vignette that you described holds only in the short run. In the short run it is possible to boost economic activity by 'supporting' loans by lowering interest rate, and in short run savings can exaggerate recession, but this does not hold in the long run (at least not in standard textbook macro models). Regardless, you should not confuse the short run dynamics with long run economic growth. Unless there is some hysteresis effect, which is still controversial, business cycle does not necessary affect the economic growth it just happens around it. Even in presence of hysteresis economic growth would still occur but business cycle could made it permanently lower than it could have been.
Value does not come from labor - that is an archaic classical economics theory that was since disproved countless times and does not square well with empirical observations. Value is subjective. This being said boosting labor supply definitely increases GDP because going back to the point 1. it increases the avaiable stock of factors of production, although unless you can do it consistently it will lead to level increase in GDP not really to sustainable growth.
Every single advanced country I can think of has some programs that are trying to improve labor force participation. For example, many countries have earned tax credit, unemployment offices that help people find job, vocational training programs, lifelong learning programs etc. So the "boosting" of labor supply is definitely happening in most countries - in fact the only exception I can think about are some defunct states like Somalia which are in state of anarchy. Of course some states do this better than others, but that's completely different topic.
It is not a job of central bank to conduct or even suggest labor market policies. Central banks job is to conduct monetary policy and bank/financial supervision in accordance with its mandate. This mandate differs across the world but most central banks have mandate that is similar either to the Fed's or ECB's mandate. The Fed has dual mandate of price stability and full employment, but full employment here means of resources in the economy not just labor and its done through the monetary policy. ECB has only single mandate of price stability so it is not its job to even care about labor supply, except to the extend it can have some influence on price stability. In short central banks talk about quantitative easing and debts because that is their job. It is simply not in their competence to conduct some policies on labor market even if they would want to - that is job of labor department or ministry of finance/economics depending on what country we are talking about.


In a bi-weekly payroll cycle, there are only ever 26 pay dates, correct?

You mean 26 pay dates per year, right?

Then no, because a year is 365 days long and a 26 biweeks are $26 \cdot 14 = 364$ days long. Therefore any year that starts with a Friday (which is usually a pay date) should have 27 pay dates. However if Jan 1 is a holiday (depends on where you live), the paydate is probably moved to Dec 31 so the preceding year would have 27 pay dates.


Why is increased productivity infinitely good

Productivity gains either allow to produce more with constant inputs, or to produce the same amount as before with fewer inputs. It is to individuals and firms to decide what to do do with productivity gains. In a market economy, the price and wage
adjustments induced by productivity changes help individuals to decide whether to work less or more, to invest and hire more employees, etc.

Usually productivity gains do not fall from heaven, firms and individuals have to pay for it (R&D, organizational change, etc) so it is not necessarily good, it is however welfare
improving if the price paid to achieve productivity gains does not increase the benefits (private or public). How costs and benefits from productivity changes are shared
in order to promote both equity and further productivity gains is a challenging societal and political issue.

Contrary to the claim of your citation, productivity gains do not systematically increase wages, it depends on the technology and the structure of the commodity and labour market.


What is the producer surplus given the price the consumer paid + price they are willing to pay, and the marginal cost of producing the product?

Yes in this case producer surplus is $\\\$1$ and consumer surplus is $\\\$2$. Your calculations are thus correct.
Generally, producer and consumer surplus are not just willingness to pay minus market price times quantity, and market price minus marginal cost times quantity, but in this case they are.
Generally producer and consumer surplus are defined as area under the demand curve and above market price (for consumer surplus) and area under market price but above supply curve for producer surplus (see Mankiw Principles of Economics for undergraduate explanation or Varian Microeconomic Analysis for more rigorous info on the definition). More rigorously they are given as:
$$CS =  \int^{q_m}_0 D(q) dq − p_mq_m $$
where $D(q)$ is arbitrary demand curve, $p_m$ is price consumers have to pay in the market, and $q_m$ quantity sold at the market.
$$PS =  p_mq_m - \int^{q_m}_0 S(q) dq $$
where $S(q)$ is arbitrary supply curve.
However, in your special case you can use formulas $CS=(W-p_m)\cdot q_m$ and $PS=(p_m-MC)\cdot q_m$, because in your special case both CS and PS are rectangles and area of an rectangle is given by $ab$ and here the expressions in the brackets give you the hight of the rectangle ($a$) and quantity gives you the base of rectangle $b$ (you could still use the integrals that would give you same results but it would be overkill and waste time in this simple case).  Graphically your problem looks like:


Bayesian Nash Equilibria in Battle of Sexes

Yes, you are correct. All types $t_{1}$ choose O (B) and all types $t_{2}$
choose O (B) are both Bayesian equilibria.
Note that there are other Bayesian equilibrium in this game, if you are interested this is explained in detail here (p. 10, see reference below) for this particular battle of the sexes with two-sided incomplete information. The basic idea is to note that in this game, each player has a continuum of types, and so the set of types is infinite. You can look for a Bayesian equilibrium in which player 1 goes to the $Opera$ if $t_{1}$ exceeds some critical value $x_{1}$ and chooses $Fight$ otherwise, and player 2 chooses to $Fight$ if $t_{2}$ exceeds some critical value $x_{2}$ and goes to the $Opera$ otherwise. To find the values $x_{1}$, $x_{2}$ that make these strategies a Bayesian equilibrium you can calculate each player's expected payoffs given the other player's strategy and find the optimal values based on this.
Game Theory: Static and Dynamic Games of Incomplete Information
Branislav L. Slantchev Department of Political Science, University of California – San Diego


Remove Linear Good From Quasi-linear Utility Function

This is one possible interpretation. Good 2 being removed from the market can simply be interpreted as $x_2 = 0$. In an economic interpretation the good does not simply disappear from the utility function in the sense that preferences do not change, it is just the availability of the good that changes. This is an external condition, so you can simply think of this as a market constraint $x_2 = 0$.

Now, looking at indifference curves as the different bundles for which the consumer obtains the same level of utility, and defining this level as $k$. It is clear that for any $k$ when there is only one good, each "indifference curve" will consist of only one point (in particular $x_{1}|u(x_1,0) = k$). In a 2-D graph this will simply correspond some point ($x_1$,0) for each $k$ level.

The demand function should be quite straightforward.


Morten O. Ravn and Harald Uhlig (2002)

  This paper complements these insights using two different analytical
  approaches. The first approach uses the time domain  and  focuses  on
  the  ratio  of  the  variance  of  the cyclical component to the
  variance of the second difference of the trend component: this ratio
  is often used for calculating the smoothing parameter. For a
  particular benchmark stochastic process, it is shown that time
  aggregation changes this ratio by the fourth power of the observation
  frequency. The second approach uses the frequency domain and
  investigates the transfer function of the HP filter, thereby obtaining
  a general result. Again, a change-of-variable argument shows that one
  should adjust the HP parameter with approximately  the  fourth  power
  of  the  frequency  change.  Both approaches therefore yield a value
  of approximately $1600 / 4^4 = 6.25$ for annual data, which is close
  to the value of 10 given by Baxter and King (1999).

Source: Notes On Adjusting the HP-Filter for the Frequency of Observations

Mueller's result is not obvious to me given this rule. By that rule of thumb, a monthly parameter  given an quarterly parameter of 900,000 is $900,000 / 3^4 \approx 11,111$ (because there 3 months in a quarter). It also gives a yearly parameter of $900,000 / 12^4 \approx 43$ (12 months in a year).

The following quote seems to support Mueller.

  On suboptimality of the Hodrick–Prescott filter at time series
  endpoints Hodrick and Prescott (1997) proposed, on somewhat subjective
  grounds, a value λ = 1600 for quarterly data. However, it is desirable
  to adjust this value when observations of different frequencies are
  subject to the filter. Backus and Kehoe (1992) suggested an adjustment
  of the value by multiplying the standard value of 1600 with the square
  of the frequency of observations relative to quarterly data. For
  example, the relative frequency is 3 for monthly data and 1/4 for
  annual data. Hence, the corresponding values of the smoothing
  parameter is $\lambda$ = 100 and 14,400 for annual data and monthly data,
  respectively. This suggestion has been also used in commercial
  packages such as EVIEW. We shall use these values throughout the
  paper. With regard to the choice of the smoothing parameter, it is
  worth noting that, in research that has gone largely unnoticed in this
  field, Akaike (1980), while further allowing a seasonal component in
  the decomposition, proposed precisely the HP approach together with a
  data-dependent Bayesian procedure for the choice of $\lambda$.

So just as $1,600 (quarterly) \rightarrow 3^2 \cdot 1,600 =  14,400 (monthly)$ so to would $100,000 (quarterly) \rightarrow 3^2 \cdot 100,000 =  900,000 (monthly)$

Misea, Kimb, and Newboldc (2005)

But the cited Backus and Kehoe (1992) result seems to be superseded by the Ravn and Uhlig (2002) result, so this may not reflect the state of the art thinking on this matter.


Career change to economics from software engineering

If you are looking for a job with a title "economist," I doubt that anyone would look at you if you do not have at least an undergraduate degree in economics (in North America, anyway).

However, you could end up working for a firm that does economics, or financial research. I had a background in control systems engineering, with summer job experience as a programmer, and then ended up working as a quantitative analyst for fixed income. The programming experience ended up being probably the training that I ended up using the most. However, there is a lot of competition to get a job in finance.

Outside of finance, I believe that there is a lot of work in "big data," which involves a lot of programming.

You could end up working with economists, but you would probably not be considered an "economist."

You ask:

  Lastly, do economics ever serve in the private sector. I'm asking this
  because it seems that a lot of econ jobs would be as advisory roles in
  government institutions.

There's a lot of market economists in finance. A few large corporations still have people with the title "economist," but my guess is that most people with economics training have a different title when working elsewhere in the private sector. (As noted by denesp, Google famously has someone wih a Chief Economist title.) Public sector jobs are different in that respect.


Could low interest rates encourage savings?

tl;dr Yes it is possible, in fact it is possible even under the most basic macroeconomic consumption models (see chapter 8 in Romer's Advanced Macroeconomics textbook). This is because price change at any market (and interest rate is just price for savings) will always cause two effects. A substitution  effect and income effect.
Lower interest rate creates an intertemporal  substitution effect which diverts consumption from future to present causing people to save less.
The income effect depends on whether a person is net lender (saver) or net borrower. For net lender lower interest rate has negative income effect which indeed causes them to  consume less and save more. Opposite holds for net borrower though.
This being said empirically in most cases substitution effect dominates the income effect (see discussion in Romer Advanced Macroeconomics) and as mentioned in the above paragraph the income effect will lead to increase in savings only for net lenders. However, in those rare cases where income effect dominates it is possible to observe increase in savings after interest rate decreases.
Full answer:
The intuition mentioned above  can be seen from any standard consumption model. For example, consider simple two period model with bond market. In first period consumer budget constraint is given as:
$$C_1 + S_1 = Y_1 \tag{1}$$
where $C_1$, $S_1$ and $Y_1$ are consumption, savings and income respectively in period 1. Saving is done by purchasing a simple bond that carriers real interest $r$. Consequently in second period we will have budget constraint given by:
$$ C_2 = (1+r)S_1+ Y_2 \tag{2}$$
We can merge the two budget constraints through rearranging the first period budget constraint for savings $S_1=Y_1 - C_1$, which gives us the intertemporal budget constraint:
$$C_1 + \frac{C_2}{1-r} = Y_1 + \frac{Y_2}{1-r} \tag{3}$$
The equation 3 already showcases the income effect since if we can see that if interest rate decreases the right hand side of an equation 3 $Y_1 + \frac{Y_2}{1-r} $ which is the person's lifetime wealth also becomes lower.
Next the person will try to maximize their lifetime utility $U(C_1,C_2)$ and hence we have the following problem:
$$ \max U(C_1,C_2) \quad s.t. C_1 + \frac{C_2}{1-r} = Y_1 + \frac{Y_2}{1-r} \tag{4}$$
Now I will skip some derivation steps but we can solve the above problem with Lagrangian where from FOCs we will be able to find that marginal rate of substitution between future and present consumption is equal to:
$$ \frac{U'_{C_2}}{U'_{C_1}} = 1+r \tag{5}$$
And hence the marginal rate of substitution depends on interest rate $r$ where higher $r$ leads us to consume more in the future. Since saving $S_1=Y_1-C_1$, lower consumption today means we save more.
Now the optimum consumption will be found at the point where marginal rate of substitution is tangent to the budget constraint. Now in this can be seen from figure 8.2 in Romer's macroeconomics posted below where

Panel (a), the individual is initially at the point ($Y_1,Y_2$); that is, saving is
initially zero. In this case the increase in r has no income effect—the individual’s initial consumption bundle continues to be on the budget constraint.
Thus first-period consumption necessarily falls, and so saving necessarily
rises.
In Panel (b), $C_1$ is initially less than $Y_1$, and thus saving is positive. In
this case the increase in r has a positive income effect—the individual can now afford strictly more than his or her initial bundle. The income effect
acts to decrease saving, whereas the substitution effect acts to increase it.
The overall effect is ambiguous; in the case shown in the figure, saving does
not change.
Finally, in Panel (c) the individual is initially borrowing. In this case both
the substitution and income effects reduce first-period consumption, and
so saving necessarily rises.

Consequently it is theoretically possible that decrease in real interest will reduce the amount of saving provided that person is a net lender in their first period and provided that income effect dominates the substitution effect.  The case (b) is also the most relevant one since on net the wealth in any real life economy is positive so people must be on average savers rather than lenders, however even despite of that empirically substitution effect usually dominates the income effect on savings market. This why its very hard to find any evidence for lower interest rates leading to more savings with Japan, Korea and Taiwan perhaps being exceptions as argued in the source you provided.


printing money after making the value by setting prices by law

Why cant America set prices by law; and then print money and give it to those who qualify

    This is the main question

So the Fed along with the US government declares that it will start controlling the prices within the US territory. There will be some simple consequences to this decision,

$1)$ Investors who invest in enterprises and corporations, big and small will run away, literally run away. This will drive the investments at an all time low. Ask yourself, why would anyone invest their hard earned cash in the US if they would not be allowed to enjoy the profits. After all, profits revenues and other benefits are a result of price flexibility.

$2)$ No price means no profit, means no investment means soon many private entities will either stop functioning or just find some other country to move to. If there are no other entities to tax on, whom will the government tax on.

$3)$ If prices remain fixed, then who will the government give its money? Who will even want the money. What good use is the money if an individual cannot buy or sell, in short have any purchasing power.

Here you go, http://www.socialstudieshelp.com/economics_basic_concepts.htm


The government just consumes within this model, as is similar to other standard DSGE models. Therefore, its consumption (G) never has an effect on productivity. Imagine a wartime situation where the government is buying goods that are used to provision the army; the goods are consumed, but will not help develop the domestic economy. If you wanted to simulate government infrastructure spending raising productivity within these models, the government infrastructure build would have to be added to business investment (raising the capital stock).

The "real government debt" constraint says that although the nominal level of debt is fixed, if future taxation policy is expected to be "too low" relative to the nominal stock of debt, the price level will rise now. This the Fiscal Theory of the Price Level. The governmental budget constraint is used to determine the formal relationship between the expected primary surpluses and the real value of current government debt (effectively, the current price level, since the initial nominal debt level is fixed).

With regards to your examples, even if debt due was greater than GDP, the situation can be sustained. (We have seen debt-GDP ratios greater than 100% in a number of countries, including Japan at present.) Most of the debt will be rolled over into new debt. That is, in year 2 of your example, the government could just borrow the nominal amount equivalent to two tons of corn, and roll over the debt from year one. (This is possible since these are models of monetary economies, in which nominal claims - money and bonds - are used in exchange, and not just goods themselves, see below.) The issue with sustainability is whether the debt/GDP ratio will tend to infinity.

If transactions were barter transactions with real goods, you could then have "real debts" which are payable by goods (corn, say). What the limits are for paying back debt depends upon your assumptions about the payment system. Normally these models assume simultaneously clearing of all transactions in a period. However, if we assume that there are multiple transactions possible within a period (and we are only simulating the aggregate), the government in your example could borrow corn from some members of the private sector, and then use that to pay off other lenders, and then repeat the process until the entire debt stock is rolled over. If that is assumed not be posisble, the government would have to lengthen maturities, so that only part of its debt stock needs to be rolled over, and the previously described procedure is again possible. Having debt with multiple maturities is avoided in DSGE models due to complexity concerns, but it would be necessary if the payments system was restricted like this.


Are principal payments on government bonds included in normal budget spending?

Normally, they aren’t, but there are exceptions. For example. The Japanese Ministry of Finance includes principal repayments in its definition of “debt service” - see budget summary for 2014, page 4.

(The inclusion of principal payments in debt service seems to be a convention for external debt: how much foreign currency debt that needs to be paid back can matter for credit risk. Floating currency sovereigns like the U.K. have negligible credit risk for their local currency borrowings.)

The reason why it is ignored principal repayment is a fairly meaningless number in government finance. If the government issues \$1 billion in short-term debt, in one year it will have:

(roughly) \$12 billion in principal repayments if it rolls over 1-month bills,
(roughly) \$4 billion in principal repayments if it rolls over 3-month bills, or
(roughly) \$1 billion in principal repayment if it issues a 12-month bill.
(Numbers are approximations since Treasury bills are discount instruments.)

The same principle works for longer-dated debt: you have twice the principal repayments if you roll over a 5-year bond instead of issuing a 10-year bond.

So all principal payments tell us about is the debt management strategy of the government.

If you want to know how government debt is out there, you just look at the debt stock.


Quickest way to learn about yield management and revenue maximization?

You should seek for topics under Industrial Economics then. I have these books has reference (see below list). Keep in mind that Industrial Economics is highly focus on competition and market power.

Industrial Organization: Markets and Strategies.
Microeconomic Theory: Basic Principles and Extensions

I find them both easy to read and understand concepts, mainly the first one. When I first had to work with it, my common problem was with the math, but if your background is in Physics, I'm sure you'll have no problems understanding the functions transformations.

Coursera also has some courses that slightly approach Industrial Economics, like this one, but I guess this will not suffice to you.


Who really benefits from economic growth?

I suppose that it depends on the time scale you are considering. Over the very long run the poor seem to have benefited far more than the rich from growth. See for example Robert J. Samuelson in the Washington Post:

  By 1915, the United States was the world’s richest nation — and yet,
  most Americans were dirt poor by today’s standards. Adjusted for
  inflation, men’s average wages were about a third of what full-time
  workers now earn. The average workweek in manufacturing hovered around
  50 hours, and many employees worked a half day on Saturday. Less than
  a third of homes had electric lights. Less than a fifth of the adult
  population were high school graduates.

A hundred years ago leisure and electric lighting were luxuries. Now they are near ubiquitous in the west. Higher education used to be rare, now it is common. Heath-care was once dangerous and of limited efficacy for poor and rich alike, and now we use lasers, chemicals, and radiation to great curative effect.

Billionaires All (Or Practically So) by Don Boudreaux

  In 1916 even Rockefeller had little access to air conditioning; the
  range of cuisines from which he could choose his meals was drearily
  narrow; he was attached by wire to a wall or a floor whenever he spoke
  on the telephone; he could entertain and inform himself neither with
  radio nor television; forget about computers or tablets.  Every movie
  he watched was silent and in black and white.  Unlike us today, in
  1916 it took Rockefeller days, rather than hours, to get from New York
  to Europe – or even to California.  Although he was chauffeured in his
  (un-air-conditioned) limousine, the ride he experienced was bumpier
  and more perilous than is any ride enjoyed today by an ordinary
  American.  Rockefeller’s fine Swiss watch was far less accurate than
  is today’s cheapest Timex.  Were he to need the services of a fire and
  rescue team, it would have taken that team longer to reach
  Rockefeller, and the effectiveness of their efforts to save his life
  or his property were much less than is the effectiveness of such
  teams’ efforts today.

  If in 1916 Rockefeller or any of his children or grandchildren caught
  pneumonia or cut a finger on a rusty nail, the risk of death was much
  higher than it is today because antibiotics were unavailable.  Of
  course, Rockefeller also had no access to statins, hip- or
  knee-replacement surgeries, Lasik procedures, contact lenses,
  antihistamines, or even sunglasses that protected his eyes from
  ultraviolet rays.  Yet he was then one of the world’s richest men –
  deprived as he nevertheless was of many of the miraculous goods and
  services that we ordinary Americans today take for granted.


What resources are available for learning Dynare?

Eric Sims at Notre Dame has an excellent set of self contained notes.

He uses a small neoclassical model as an example and goes over the model derivation, how to construct each block of Dynare code, different simulation options, interpreting output, and how to do higher order approximations. Full code is included for each step in the notes.


Monetary Approach to a depreciating currency

Let us call the country with the trade deficit, country A, which is our domestic country. If a country builds up a trade deficit, it is selling less goods to other countries than it buys from them. To do so, A must exchange its currency for foreign currency. This means more demand for the foreign currency and more supply for the domestic currency. More supply reduces the price of domestic currency relative to other currencies. More demand for foreign currency increases their price relaive to domestic currency. All this causes a depreciation of the domestic currency. For more detail on this see, for example, the answer to this question.
Currency values are fundamentally determined by supply and demand for currency (see answer above). Both of those are driven by both the goods and financial markets. So trade deficits have an effect here, as well as investors flocking to higher rates.
Foreign investors both affect and are affected by the exchange rates. If these investors want to invest in a foreign country with higher interest rates, they need the currency of that country in one way or another. So they sell their currency to buy the foreign currency, which like in the answer to 1) will have an effect on the value. Even if foreign investors have the foreign currency to invest, they will need to exchange it at some point in order to consume their earnings at home.
Solid empirical evidence is rather difficult when it comes to long term effects. It is true that the US has had a trade deficit for a long time. However, it is unclear how long a time is a long-run. Furthermore, before having a deficit, the US has had a surplus for a very long time, so it is natural it would eventually have a deficit. Whether this deficit has exceeded past built up surpluses is not necessarily clear.


Relationship between forward rate, future spot rate, and liquidity preference

In addition to comment given by @dismalscience, here you may find partial answer
(hope I got everything right below). Since many similar terms refer to concepts
that are close to each other, I'm also regularly fighting to get these somehow
in order. For example, there are forward and futures contracts that use similar
terms and are related to the yield curve concepts.

Yield curve premiums

On liquidity: lenders like to prefer short periods, because they are more easy
to convert to cash, while borrowers prefer long periods, since roll-over
terms involve risk. So lender wants compensation for which the borrower is ready
to pay. Longer you go, the premium is likely to increase but at decreasing rate.
This is different to the liquidity concept when the market for a bond is very
thin.

It is healthy to consider also other aspects with respect to risk-free bonds.
E.g. how good vs. bad times affect premiums. And why term premium? There is some
evidence of negative correlation between short bond return and economic growth,
and especially this correlation has larger magnitude than the correlation the
long term bonds have with economic growth.  This in turn, would mean that short
term bonds work out as better hedges against economic downturn leading to larger
premium requirement for the longer bonds.

It is possible, that we have positive liquidity premium at the same time when
other premiums are not, even so that the yield curve inverts, because the
spot rate
is geometric mean of forward rates. In this situation, the forward rate curve
would be below the spot yield curve. (This is not shown in the equations
below.)

Contracts

Forward rate (i.e. forward price) is the price agreed on the contract initiation
date. Pricing means a method to find out the forward rate "today" for the
contract.

Let us denote the value of the underlying asset $S_T$. Then, the value of
contract is $S_T - F_{0,T}$ at the "end" (time $T$) and we could write it as
$V_{0,T}(T) = S_T - F_{0,T}$. To get the value today, we discount with risk-free
curve, and is common to set the forward price $F_{0,T}$ such that neither party
does not have to pay to the other: $V_{0,T}(0) = S_0 - F_{0,T}/(1+r)^T = 0$,
where $r$ is the risk-free interest rate. So, $F_{0,T} = S_0(1+r)^T$.

Futures rate (i.e. futures price) is priced similarly, $f_{0,T} = S_0(1+r)^T$,
and it is said that "the futures price is the spot price compounded at the
risk-free rate". For the value, the marking-to-market makes the value zero in
each day, but just before that, the value is $f_{0,T}(t) - f_{0,T}(t-1)$.
If there is a carrying cost $c$, the price is  $f_{0,T} = S_0(1+r)^T +c_T$

Future rate and expected spot rate: consider carrying costs $c$ and risk $k$,
for which you don't want to pay, and that we don't know, what the spot price
will be at $T$, thus using expectation:
$$ S_0 = \frac{E(S_T) - c_T - k}{(1+r)^T}. $$

Put the future price into that and you will obtain $f_{0,T} = E(S_T) - k$, that
is, the futures price equals expected spot price less risk premium.

Yield curve

Forward rate $f$ is an interest rate today for a loan to be made in future.

The spot rate $r_T$ is the geometric mean of the forward rates, that is,
$$ (1+r_T)^T = (1+r_1) \Pi_{i=1}^{T-1} (1+f_{i,1}^i) $$
or
$$ r_T = ((1+r_1) \Pi_{i=1}^{T-1} (1+f_{i,1}^i))^{1/T} - 1.$$

Do forward rates imply future spot rate movements? Not necessarely, but find out
more about the trade called "rolling down the yield curve". However, the forward
rates do provide an estimate of the expected spot rate that is biased by
liquidity premium (premium is not shown in the above equation).


Is there a theory of financial regulation?

Jean Tirole just won the Nobel Prize for his studies in regulation in different markets, including the banking market.

Check:

Balancing the Banks:
Global Lessons from the Financial Crisis and
The Prudential Regulation of Banks


King-Plosser-Rebello Preferences: Scale leisure

If you specify

$$ U(c, l) =\left(c\cdot [(v(l))^A]\right)^{1-\sigma}$$

then

$$\ln U(c,l) = (1-\sigma)\ln c + A(1-\sigma)\ln[v(l)]$$

...and $A$ can be seen as regulating the relative weight of leisure, for any $v(l)$.


Are there "Grossman/Hart/Moore" models without efficient negotiation outcomes?

Yes, there are models that reconcile Hart’s property rights theory with Williamson’s emphasis on haggling and ex post inefficiencies.

Hart and Moore have developed a behavioral theory of contracts as reference points. They argue that contractual parties engage in ex post inefficient shading activities when they are aggrieved (which happens when they do not get what they feel entitled to given the reference point that was set by a contract). In this sense, the theory is based on non-standard preferences: Hart, O. and Moore, J. (2008). Contracts as Reference Points, Quarterly Journal of Economics, Vol. 123, pp. 1–48. See also Hart (2009) for an extension of this approach.

Another possibility to formalize ex post inefficiencies in a Grossman/Hart/Moore framework is to allow for asymmetric information: Schmitz, P.W. (2006). Information Gathering, Transaction Costs, and the Property Rights Approach, American Economic Review, Vol. 96, pp. 422–434. This approach maintains the standard assumption of profit-maximizing behavior and studies the trade-off between investment incentives and ex post inefficiencies (see also the more recent contribution by Schmitz (2017) and the literature mentioned there).

Finally, while less related to the Grossman/Hart/Moore framework, there are also models that study property rights in adverse selection models without investments (building on the prominent paper by Myerson and Satterthwaite, 1983), where the optimal ownership allocation minimizes ex post inefficiencies: McKelvey, R.D. and Page, T. (2002). Status quo bias in bargaining: An extension of the Myerson–Satterthwaite theorem with an application to the Coase theorem, Journal of Economic Theory, Vol. 107, pp.336–355. A literature review with particular emphasis on models building on the Myerson/Satterthwaite framework: Segal, I. and Whinston, M.D. (2012), Property rights. Handbook of Organizational Economics, pp. 100–158.


Understanding the Components of Optimization

I will give a general answer (for the specification of utility maximization), and I will give a more specific example of Cobb-Douglas utility:
$$U(x_i,y_i)=x_i^{\alpha_i}y_i^{1-\alpha_i}$$

$\textbf{Objective function:}$ In the utility maximization problem, the objective function is going to be the utility function. This is the function which you are trying to maximize. Therefore the objective function for agent $i$ in our example is:
$$U(x_i,y_i)=x_i^{\alpha_i}y_i^{1-\alpha_i}$$

$\textbf{Constraint function:}$ The constraint function is the function that limits the agent's ability to consume. If these functions were not in place, any agent with a strictly increasing utility function would choose to consume infinite goods. The two most common types of constraints are resource constraints and budget constraints. Resource constraints are just a maximum amount of a resource available (which makes sense because we know that resources are finite). A budget constraint is basically a constraint that says that you cannot consume more goods than you can buy. Examples of the two are below:

Resource constraint:
$$\sum_{i=1}^nx_i=A,\quad \sum_{i=1}^ny_i=B$$
where $A,B<\infty$

Budget constraint:
$$P_xx_i+P_yy_i= Z_i$$
where $Z_i$ is income of agent $i$ and $Z_i<\infty$

$\textbf{Choice variables:}$ These are the variables that each agent chooses to maximize utility. In our example the choice variables for agent $i$ are $x_i$ and $y_i$.

$\textbf{Parameters:}$ Parameters are also called state variables. These are the variables which the agent has no control over. In our example the parameters are $\alpha_i,B,A,Z_i, P_y,P_x$.

$\textbf{Policy function:}$ The policy function is also called the decision rule. This is what you get from solving the maximization problem. In the context of utility maximization problem, the policy function is the Marshallian demand (how much you want of each good). The policy function gives you your choice variables as functions of your state variables/parameters.


Why are pricing charts treated as continuous series?

Inherently discrete variables (like "count data") have special properties, and it matters when it comes to econometric estimation.

The usual criterion to treat a variable as continuous or discrete, is I believe not what we observe but what could be conceivably observable.
Example: if one counts number of people, the variable is inherently discrete. In large samples, the continuous approximation may not matter -in numerical analysis, starting the search for a solution by "pretending" that the data is continuous is done all the time. But in small samples it may matter. Also, there are variables that take only a few specific values (usually integers). Here, taking into account the discreteness is crucial.

Prices have no reason to be treated as discrete. They are continuous because in principle nothing constrains a price to not take any value on the half-real line.

As for the data sets: data is always discrete -simply because between any two real numbers there is an "infinite" number of other real numbers (with the appropriate mathematical definition of infinity). So since the sample is always of finite size, it will always be discrete. The issue is whether the Data Generating Mechanism can be conceived as being continuous or not.

Finally, for certain purposes continuity is a great analytical helper bar one-dimension: the time-dimension, especially in stochastic setups. Continuous stochastic calculus notoriously carries a mathematical baggage that many times (but not always) one feels that is not worth the trouble.


Why does competition work for capitalism?

When two companies produce commodities and/or services that are substitutes then they are simply producing for a market that allows the presence of two or more substitutes.

  "...If two (or more) companies produce similar products that compete against other, these companies are in a competition"

Two firms do not necessarily compete with each other when they are supplying substitutes in the market.

What needs to be clear is that qualifying a market as competitive amounts to making a series of statements about the way the participants in the market behave, their numbers, how potential new participants can interact with the market, the relevant time scale etc.

Narrowing the focus on two members of a market and claiming that they are in competition is not warranted by the assumptions. Nothing has been said about the structure of the market they are embedded in. They could eg be simply trying to flood the market with alternatives to a third commodity so as to narrow the market share of a third producer; the possibilities are really unbounded.

  "...Because the objective of a company is to earn as much money as possible, the company will try to maximise the selling of their product."

The objectives of firms are really as diverse as is the number of industries and producers immense. "Making money" is not always a single operative constraint. Firms have a plurality of goals and obtaining liquidity is one of them. How binding that goal is depends on a host of issues like the market they are embedded in, the firm's size and relative position in the market, probably its access on financing etc.

Profit maximization is an assumption about the behavior of firms which is historically relevant and analytically useful but it is not always the case that firms are maximizing profits. They may be going for market share, or they may be simply optimizing some other metric. And, of course it could very well be the case that they are just winging it!

  "...However, because normally there are more than a single company, every company has to sell their best possible product for the lowest price without making enough profit..."

Again, this claim is not universally true. The number of firms in the market is not always a guarantee of 'quality' and 'affordability'. What matters is the cost structure of the industry, the technology and the assumptions about the market structure. Also, profits that vanish to zero is, again, a feature of the competitive model; not what 'really' obtains.

Now, as far as the 'Prisoner's dilemma' is concerned: In an one-shot game the best thing to do is to 'defect'. In a repetitive setting, that is also open-ended (no predetermined end date) I think that anything goes. You would have to make more assumptions about the number of players and their strategies over time (and in response to others') in order to arrive at definitive conclusions.

The short and non-definitive answer to the question of why don't firms collude when involved into price-competition with each other would be that there are laws that prevent them from that ie the fundamental cost structure of the game is different than what the 'dilemma' posits.


You interview one employer per week, and your discount rate is $\delta = 1/(1+.07)$.  Suppose $x$ is the lowest quality job you would accept: anything worse than $x$, and you reject and keep searching. Let $J$ be the value of searching. Then we have the functional equation:
$$
J = \max_{x} \int_{x}^{1}z dF(z) + \int_{0}^x \delta J dF(z)
$$
The first term is the jobs you accept, since they are above your cutoff, and the second are the scenarios where you keep searching and forego the bad offer, so that you end up where you started tomorrow with an expected payoff of $J$. You have a uniform distribution so $dF(z) = dz$ on $[0,1]$.

Maximizing over $x$ gives the optimization condition
$$
-x*f(x*) + \delta f(x*) \delta J = 0
$$
so that $x^* = \delta J$.  Only if the offer is better than $\delta J$, the discounted value of continuing, do you accept; indeed, $\delta J$ is the opportunity cost of accepting, so I assume that is your $c$.

Now, let's solve for $J$. We have the equation
$$
J = \int_{\delta J}^1 zdz + \int_0^{\delta J}dz \delta J
$$
$$
J = \dfrac{1}{2}(1-(\delta J)^2) + (\delta J)^2
$$
$$
J = \dfrac{1+(\delta J)^2}{2}
$$
$$
0 = 1+\delta^2 J^2 - 2J
$$
This is a quadratic equation. The solution is
$$
J^* = \dfrac{1}{\delta^2} - \dfrac{\sqrt{4 - 4\delta^2}}{2\delta^2}
$$
$$
J^* = \dfrac{1-\sqrt{1 - \delta^2}}{\delta^2}
$$
and your search strategy is to accept if the job is better than
$$
x^* = \dfrac{1-\sqrt{1 - \delta^2}}{\delta}
$$
and reject otherwise.


Convex games: equivalence of definitions

Define the marginal contribution of $i \in N$ to any $C \subseteq N \setminus \{i\}$ by
\begin{align}
m_i(C) = v(C \cup \{i\}) - v(C).
\end{align}

We are going to show C.1 $\Rightarrow$ C.2 $\Rightarrow$ C.3 $\Rightarrow$ C.1.
Note that C.1 $\Rightarrow$ C.2 is trivially true.
If C.1 holds for all $S,T \subseteq N$, then C.2 must hold for all $S,T \subseteq N$ with the restriction $|S \setminus T| = |T \setminus S| = 1$.
In order to show C.2 $\Rightarrow$ C.3 consider any $S \subseteq T \subseteq N \setminus \{i\}$.
Let $T \setminus S = \{j_1, \ldots, j_k\}$ and define $S_\ell := T_{\ell-1} \cup \{i\}$ and $T_\ell := T_{\ell-1} \cup \{j_\ell\}$ for $\ell \in \{1,\ldots,k\}$ with $T_0 = S$.
Note that
\begin{align}
\begin{split}
T_{k-1} &~= T_{k-2} \cup \{j_{k-1}\} \\
&~= T_{k-3} \cup \{j_{k-2}\} \cup \{j_{k-1}\} = T_{k-3} \cup \{j_{k-2},j_{k-1}\}\\
& ~ \vdots\\
&~ =  T_0 \cup \{j_1,\ldots,j_{k-1}\}\\
&~ = S \cup \{j_1,\ldots,j_{k-1}\}\\
&~ = T \setminus \{j_k\}
\end{split}
\end{align}
Since $|S_\ell \setminus T_\ell| = |T_\ell \setminus S_\ell| = 1$ we can apply C.2, i.e.
\begin{align}
\begin{split}
&v(S_\ell \cup T_\ell) + v(S_\ell \cap T_\ell) \geq v(S_\ell) + v(T_\ell)\\
\Longleftrightarrow \quad &v(T_{\ell-1} \cup\{i, j_\ell\}) + v(T_{\ell-1}) \geq v(T_{\ell-1} \cup \{i\}) + v(T_{\ell-1} \cup \{j_\ell\})\\
\Longleftrightarrow \quad &v(T_{\ell-1} \cup\{i, j_\ell\}) - v(T_{\ell-1} \cup \{j_\ell\}) \geq v(T_{\ell-1} \cup \{i\}) - v(T_{\ell-1})\\
\Longleftrightarrow \quad &m_i(T_{\ell-1} \cup\{j_\ell\}) \geq m_i(T_{\ell-1})\\
\Longleftrightarrow \quad &\sum_{\ell=1}^k{m_i(T_{\ell-1} \cup\{j_\ell\})} \geq \sum_{\ell=1}^k{m_i(T_{\ell-1})}\\[2mm]
\Longleftrightarrow \quad &\sum_{\ell=1}^{k-1}{m_i(T_{\ell-1} \cup\{j_\ell\})} + m_i(T_{k-1} \cup\{j_k\}) \geq m_i(T_0) + \sum_{\ell=2}^k{m_i(T_{\ell-1})}\\[2mm]
\Longleftrightarrow \quad &\sum_{\tau=2}^{k}{m_i(T_{\tau-2} \cup\{j_{\tau-1}\})} + m_i((T \setminus \{j_k\}) \cup\{j_k\}) \geq m_i(S) + \sum_{\ell=2}^k{m_i(T_{\ell-1})}\\[2mm]
\Longleftrightarrow \quad &\sum_{\tau=2}^{k}{m_i(T_{\tau-1})} + m_i(T) \geq m_i(S) + \sum_{\ell=2}^k{m_i(T_{\ell-1})}\\
\Longleftrightarrow \quad & m_i(T) \geq m_i(S)
\end{split}
\label{eq:pvex}
\end{align}
The bottom line corresponds to C.3.
The final step C.3 $\Rightarrow$ C.1 is provided in Moulin (1988, p. 113).


Why is national debt bad if the central bank can keep buying bonds?

Yes.

The central bank (say the Fed in U.S.) can purchase bonds, but notice:

The Fed is not going purchase t-bonds and t-bills directly from the Treasury. It will only purchase or sell bonds in the market. ("Open market purchase" or "open market sell.")
Open market purchase is not only the buying of bonds, but also the creation of money. (You sell bonds to the Fed, and the Fed will pay you.) Increase in money supply means inflation. Obviously, the Fed can't buy too much bonds, or the U.S. will have to deal with disastrous hyperinflation.
Most central banks have some degree of independence in determining what to do. The Fed won't purchase more than they want, and the government is not able to force them.


Must a market-based economy constantly expand?

No idea who said that first, but I have seen similar assertions in economic commentary many times. The logic was never fully explained, but revolved around the notion that we need compunding income to pay compund interest. Authors treated the idea as so obvious or well known that they never saw the need to explain the details, or cite earlier authors. (It’s been awhile since I’ve seen the assertion, so I cannot give examples.)

In theory, the belief that the economy must expand appears incorrect. One could look at the various stock-flow consistent (SFC) models in “Monetary Economics” by Wynne Godley and Marc Lavoie, and one sees that steady solutions are easy to find. Although debt compounds, borrowers just pay down debt.

As a practical matter, the tendency is to prefer growth. However, in a services based economy, it is unclear whether real GDP growth gas to correspond to greater natural resource use, which creates a debate whether it is “real growth” or “paper growth.”