How can I obtain State level Investment in the US?

This is a great question that I think you should bring up to the US Census bureau.
I believe at one point they did explore how to come up with capital measures by state; however, nothing ever happened.    At this point and at the time of this posting the measurement of capital measures by state does not exist.  It would appear this would be the "Holy Grail" of economic measurement.
There may be some capital data on equipment and structures at the state level from the Survey of Manufactures data (ASM) from the census bureau but that would be for only one sector in the economy.
There have been some studies about state level investment but the data used for the analysis are mostly state shares from  aggregate US totals.  The ratio used to come up with these types shares usually comes from the researchers assumptions, which is the best practice available.
Below is an example of a report:
https://www.cbpp.org/research/state-budget-and-tax/its-time-for-states-to-invest-in-infrastructure
If there is capital data at the state level someone please post the source here.


Risks asscoiated short term stimulus as interest rates approach zero

The potential risk of borrowing too much now is that interest rates will be higher when the debt has to be rolled over. Even though it is completely correct to say that most government securities have fixed interest rate, most of these securities are not perpetual, meaning they expire at some point.
For example, as mentioned on the UK Debt Management Office website:

Conventional gilts also have a specific maturity date. In the case of 1½% Treasury Gilt 2047 the principal will be repaid to investors on 22 July 2047. In recent years the Government has concentrated issuance of conventional gilts around the 5-, 10- and 30-year maturity areas, but in May 2005 the DMO issued a new 50-year maturity conventional gilt. In June 2013, following market consultation the DMO issued a new 55 year maturity conventional gilt.

This carries some potential risk. For example, suppose for the sake of argument government could borrow now $£ 1000$ with coupon rate $0\%$ (coupon rate is essentially the interest paid on face value of fixed income security), for 5 years. That is great deal now but in 5 years government will face the following problem, it will either have to repay the debt (which could be difficult to do for country with high level of debt) or instead roll it over meaning the government issues another $£ 1000$, but this time if interest rates are higher and government is forced to issue it with, let us say $20\%$ coupon rate that would be a problem because suddenly the cost of servicing debt to the government would skyrocket.
This being said, it is worth noting that even though the above is potential risk that should be taken seriously, most economists believe that governments should borrow more when the interest rates are low, especially if they do so long term (e.g. 10, 30 or 50 years) or even better some argue they could start issuing perpetuities i.e. debt instruments where principal never has to be repaid (see discussion in Corsetti, Erce, Pascual 14 May 2020).  This is because issuing long term debt or perpetuities locks the low interest rates on that particular debt for very long time or indefinitely meaning there would be plenty of time to consolidate public finances if interest rates were rising sharply. In addition, government has also other venues how to handle debt such as monetization of the debt although that can have some serious disadvantages as well but discussing that is beyond scope of this answer. Consequently, while it is definitely real risk as you pointed out in the question itself most economists would argue governments should take advantage of these low rates while they can.


what alternatives can possibly back a new currency

The value of money is derived entirely from others' willingness to accept it in exchange for things of value.

Money has in fact emerged— Rai stones on Yap is the classic example— without the existence of either a "trusted authority" mandating its use (I assume that by "trusted authority" you mean a government-mandated fiat currency) or any recourse to a scare resource (e.g., notes redeemable for silver or gold).

While the Yap stones themselves (the monetary base) were in fact scarce, and some degree of scarcity is a necessary condition for money to act as an effective store of value, it's not a sufficient condition, as FooBar noted in his comment. Even if a monetary base is finite, what matters is that people are willing to accept that form of money as payment, which is in turn generally determined by whether other people will be willing to accept that form of money as payment. It's collective agreement that money has value that makes it valuable. If people some day find no reason to continue using the Rai stones as money (they've been replaced by US dollars for day-to-day use, and they're only used for certain traditional exchanges now), they will lose their value as currency, despite their continued scarcity.

Specifically regarding Bitcoin, there has been significant skepticism in the economics community about whether it is actually a currency, with Yermack (2013) noting that it functions poorly as a store of value and unit of account, and that its transaction volumes have been quite low.


what are the categorization by industry in economics?

Several countries have it defined in their tax code, or by the national statistics institute, or the central bank. Stock exchhanges also define industry categories.

For general purposes or international comparisons you can use the categories defined by the world bank (see also in word version), or from the United Nations.


Would you like to buy a pack of sugar flavored amazonian soil for 500 dollars?

You can probably answer this question even though you have never heard of this product before. Being able to answer this question, being able to place the newly available consumption alternatives within your ranking is the assumption of completeness.

The same is true if a type of good is replaced. The set of consumption alternatives is changed, but given any two you can still tell which one you prefer.


The result is not the Frisch–Waugh–Lovell Theorem. Rather, the result shows the extent of Omitted variable bias.

The Frisch-Waugh-Lovell Theorem states that the residuals of the regression of $\boldsymbol y$ on $\boldsymbol Z$, regressed on the residuals of the regression of $\boldsymbol X$ on $\boldsymbol Z$, gives $\boldsymbol{{\hat\beta}}$. To put it another way, let $\boldsymbol{M}$ be the residual maker. Then the regression of $\boldsymbol{M_Z y}$ on $\boldsymbol{M_Z X}$ gives $\boldsymbol{{\hat\beta}}$. The equation is:
$$\boldsymbol{\hat\beta} = \left(\boldsymbol{X}'\boldsymbol{M_Z}'\boldsymbol{M_Z X}\right)^{-1}\boldsymbol X'\boldsymbol{M_Z}' \boldsymbol{M_Z y}$$
Note that the residual maker matrix is symmetric and idempotent.

An analogous result holds for $\boldsymbol{\hat\gamma}$.


What did Karl Marx mean by "the movement of this antithesis"?

Town represents manufacturing and it's where a lot exchanges taking place. Country represents farming. The movement of antithesis symbolizes the urbanization of civilizations as economy progresses.

As an economy progresses, people will move from rural area into the cities. Labour associated with agriculture will decrease, at the same time, workforce in manufacturing and services industries will increase.


Preparation for the rigors of an economics graduate programme

I'd work my way through Microeconomic Analysis by Hal Varian (the PhD level one) and Mathematics for Economists by Simon and Bloom. You'll need both the micro intuition  of the former and mastery of the math bits of the the latter. You'll need thornier and more technical books at some point but these authors will hold your hand enough to work through them solo.


Does National Savings equal zero in a closed economy? Please give an economic intuition

In a closed economy, savings equal investment. Your equation actually shows this. We have:

$Y−(C+G) - I = 0$.

Note that savings, by definition, is just equal to the production $Y$ that is not consumed. Here we have consumption by private citizens $C$ and consumption by the Government $G$. Therefore, savings $S$ is:

$S = Y - (C+G)$

and therefore:

$S=I$

So national savings can only be zero if investment is zero.

Here is a concrete example. You don't spend all your income and save money in a bank. People who want to invest (e.g. to start a business or buy a tractor) go and borrow that money, which is then invested.

In an open economy, there is no reason for savings to equal investment. Nevertheless, the two are often strongly correlated in reality, which is known as the Feldstein-Horioka Puzzle. This is/was one of the major puzzles in international economics and a large literature has been devoted to explaing it.


Count panel models for endogenous regressors

Windmeijer (2000, Economics Letters) presents a treatment of the estimation of count-data models with fixed effects and endogeneity.

http://www.sciencedirect.com/science/article/pii/S0165176500002287

See this slideshow by Wooldridge for a pedagogic and progressive presentation of panel data models with endogeneity. Count-data models are introduced slide 54. He clearly explains the possible solutions and relate the count-data case to other nonlinear models.

http://www.ifs.org.uk/docs/wooldridge%20session%204.pdf


What is the probability of an unemployed worker receiving no job offer during a time period?

Unless I misunderstand the question these seem to be complementing events with probabilities $p$ and $1-p$.


How does everyone not become poor over time?

In short because economy is not a zero-sum game and because economic production does not need to decline but  can grow over time.
First of all, forget about money. Money in economics is just extra tool that solves issue of double coincidence of wants, allows you to store value over time and to do accounting. In fact, traditionally money consisted of commodities such as stone, or precious metals or even products, coffee, cigarettes or even alcohol etc at some point in time and place served as money.
As a non-economist this might come as a surprise to you but most microeconomic models do not even have money in them. Money can play an important role in macroeconomic perspective because in the short-run it can affect the real economy, but from micro perspective and even long-term macro perspective money is neutral and not important.
Next, as you pointed out a world's economy, taken as whole, is a closed system but that does not mean people should get poorer over time. This is not an equivalent of thermodynamically closed system where entropy has to dissipate energy overt time. A closed economy can be described as:
$$Y  = C + I + G$$
which says that output/income (they are economically equivalent) $Y$ has to be equal to consumption $C$, investment $I$ and government spending $G$. In turn, $Y$ is a result of production. A common specification for production process would be to use Cobb-Douglas production function, so I will do that here as well, although in principle there could be multiple functional forms that production process could have. Moreover, for simplicity I will restrict myself to two factors of production (labor and capital) even though in principle you could put into Cobb-Douglas also human capital (education), land and other factors, but I omit them for sake of brevity. Given these assumptions the production process would be given by:
$$Y = A K^{\alpha} L^{\beta}$$
where $A$ is the available technology (broadly defined - in economics better production strategies count as a technology as opposed to just 'gadgets' such as PC), $K$ is the stock of capital and $L$ is the stock of labor. Alpha and beta are parameters of the production function that determine other characteristics of production such as what sort of economies of scale production exhibits (e.g. increasing, decreasing or constant).
Thus, we can say that the consumption, investment and government spending is equal to this production process:
$$ A K^{\alpha} L^{\beta} = Y  = C + I + G$$
So as long as technology $A$ or capital $K$ or labor grows, output $Y$ will grow as well and even completely closed of economy can consume more, invest more or have more government spending.
Furthermore, what we really care about is usually output/income (in economics output and income are equivalent and interchangeable) per head. So if we for simplicity assume constant returns to scale ($\beta = 1-\alpha; 0 <  \alpha < 1$)  we can divide the last expression by $L$ to get whole expression per capita as:
$$A \left(\frac{K}{L}\right)^{\alpha} = \frac{Y}{L} = \frac{C}{L} + \frac{I}{L} + \frac{G}{L} $$
Here, if we keep the population growth constant then we can see that per capita incomes/output and consequently per capita consumption, investment and government spending will grow when our capital stock and technology stock grows. You get growth in capital stock from investment (investment is part of production that is not consumed but saved for future, possibly for further production), and growth in technology from human ingenuity or also depending on which economic growth theory you buy to also investment.
In addition, all the above is up till now just manipulation of definitional expressions. We could further continue by adding on some growth theory which could yield further insights. For example, the most popular growth model presently is the Solow-Swan growth model where it can be shown that what really matters is growth in $A$ because increase in investment and capital stock can only increase level of output not growth. If you buy into endogenous growth theories investment can even increase the growth rate further beyond that (you can read more on this in Barro & Sala-i-Martin: Economic Growth 2nd ed).
In order for people to get poorer over time (in per capita terms) an economy would have to experience either destruction of technology (for example ancient Romans, Greeks, Carthaginians etc had plumbing but this knowledge was lost in many parts of World - but not all - during dark ages, or another example would be concrete) or by destruction of capital (e.g. war, violence, natural disasters etc. but also just wear and tear i.e. depreciation). Empirically, and let me add thankfully, people historically turned out to be better at producing new technologies and adding new capital through investment, than at destroying either of them.
Consequently, to sum up people do not get poorer over time and in fact get richer because of growth in economic production (which in turn determines people's income), and economic growth occurs primarily thanks to growth in technology and investment. In fact note because the economic growth is to large extent driven by technology, it is possible for economic production to grow indefinitely (assuming technology has no limit - e.g. in real life it is possible you can't get at an end of technology tree like in some strategy game as it might have no end). In addition, more rich models would also include other important factors such as human capital (education) but the explanation above would not conceptually change (education macroeconomically functions in same way as capital - hence why economists call it human capital).


Does the contract curve always have to connect the initial points on an edgeworth box? Why or why not?

The contract curve is the locus of Pareto optimal points in an Edgeworth box. What we get from that:

To be P.O., an allocation must be feasible. So, the contract curve does not extend beyond the edges of the box (opposed to indifference curves, which we can draw as extending beyond the edges of the box) because points outside the Edgeworth box are not feasible.
Because the Edgeworth box does not allow for destruction/disposal, we know that the initial allocation point will account for all resources in the economy. This may make you think that the initial endowment should be part of the contract curve. However, this is not true. That is, the initial endowment point is not necessarily part of the contract curve because there are easily imagined cases where trading happens without making either agent worse off and makes at least one agent strictly better off.
Yes, there are certain circumstances that cause the edges of the box to form part or all of the contract curve. If you want to explore the idea more I suggest you spend some time reading about what happens when one agent has (or both agents have) linear indifference curves.


Is an increase in the money supply (by buying government bonds) recorded in bank deposits or bank reserves?

No.

Think of it as the central bank is buying the bond from the commercial bank itself, so that the bank was holding one type of asset (a bond) and now it's holding another type of asset(cash or its equivalent, reserves at the central bank). At that point deposits have not increased, it's just, that the bank now is holding more cash and less bonds than before. Because cash holdings or reserves at the central bank are called 'reserves', the commercial bank's reserves have increased.

The next step is that this bank sees no benefit in holding so much cash, more than the regulation forces it to, so ti goes and lends this cash out. At that point deposits start increasing, but still, the whole of the banking system is going to have more reserves than before because somebody is going to be holding that cash.


Central Bank liabilities and assets - Money and Bonds

My question is which bonds are these? Do these bonds belong to the same country that the CB belongs to?

This will vary a lot from central bank to central bank. Most of the assets of the US Federal Reserve are indeed government securities, but the European Central Bank is prohibited from buying such securities (because of the crisis, it now buys limited amounts of them, but the legal grounds are shaky).

Since 2009, central bank balance sheets have changed quite a bit. In 2008 the Federal Reserve did not hold any mortgage backed securities. Now, they comprise about 40% of the Fed's total assets.

Take a look at the Fed's balance sheet in Table 1 of the August 2015 report.

  Because if so, it seems that we're just exchanging one liability (bond) for another (money). Or are these bonds from any entities other than the CB itself?

Yes, they are from other entities. For example, US Treasury bonds are about half of the Fed's balance sheet. These bonds are issued by the Department of the Treasury and must be pay back to the Fed on maturity, using money collected from taxes or the proceeds from new bond issues.


Canonical set of books covering Bachelors and Masters in Economics?

Econ isn't as sequential as physics. The two core undergrad classes that everyone takes are Intermediate Micro and Intermediate Macro. Varian is the standard for Intermediate Micro.

Undergrads also take some sort of econometrics class, which can widely vary depending on the target audience (i.e., business majors vs double-majors). It's not uncommon for there to be an Econ degree geared towards double majoring.

After these courses, students generally pick about 5 electives. The classic books largely depend on the subjects at the undergrad levels. For Game Theory, Osborne is a good text as is Fudenberg and Tirole. The latter is a solid grad textbook as well. A good Auction Theory text is Krishna's.

At the grad level, folks generally go for PhDs over Masters degrees. While Masters programs exist, they aren't as common as in math, physics, or CS. A good math for economists text is Simon and Blume. Mas-Collel is the classic for the grad micro sequence, which highlights game theory, IO, and general equilibrium. Jehl and Reyni is another good reference for micro, and it's a friendlier read.

Folks at the grad level also take sequences in macro and metrics. You may wish to check out Aris Spanos' text for metrics.


Pure exchange economy with free and non-free disposal

Consider a pure exchange economy with two players - A and B.

A's utility function is

\begin{eqnarray*} u_A = x_A^{\frac{1}{2}}y_A^{\frac{1}{2}}\end{eqnarray*}

B's utility function is

\begin{eqnarray*} u_B = \frac{x_B}{1 + y_B}\end{eqnarray*}

Endowments of A and B are respectively

\begin{eqnarray*} \omega_A = (2,5), \ \omega_B =(5,2)\end{eqnarray*}

  Find the set of Pareto efficient allocations in this economy.

Set of Pareto efficient allocations consists of those feasible allocations in which B consumes $0$ unit of commodity $Y$.

$$\text{Pareto Set} = \left\{\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right) \in \mathbb{R}_+^2 \times \mathbb{R}_+^2 | x_A + x_B = y_A + y_B = 7, y_B = 0\right\}$$

  Assume there is no free disposal. Player A can make a take-it-or-leave-it offer to B. Player B can either accept or reject. If player B rejects, then each player stays with his or her current endowment. If player B accepts, A's proposal is implemented. Find the equilibrium allocation outcome $\left(\left(x_A^*, y_A^*\right), \left(x_B^*, y_B^*\right)\right)$.

In a subgame perfect equilibrium, B's strategy will be to accept an allocation proposal $\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)$ if $u_B(x_B, y_B) \geq u_B(5, 2) = \frac{5}{3}$, and reject otherwise. Given B's strategy, A will choose a proposal by solving the following maximization problem :
\begin{eqnarray*} \max_{\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)\in \mathbb{R}_+^2 \times \mathbb{R}_+^2} & \ x_A^\frac{1}{2}y_A^\frac{1}{2} \\ \text{s.t.} & \ x_A+ x_B = 7 \\ & \ y_A + y_B = 7 \\  & \ \frac{x_B}{1+y_B} \geq \frac{5}{3}\end{eqnarray*}

Solving this problem we get the following proposal :
$\left(\left(x_A^*, y_A^*\right), \left(x_B^*, y_B^*\right)\right) =  \left(\left(\frac{16}{3}, 7\right), \left(\frac{5}{3}, 0\right)\right)$

  Assume there is now free disposal. Redo the last part allowing for free disposal.

Allowing for free disposal, in a subgame perfect equilibrium, B's strategy will be to accept an allocation proposal $\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)$ if $u_B(x_B, y_B) \geq u_B(5, 0) = 5$, and reject otherwise. Given B's strategy, A will choose a proposal by solving the following maximization problem :
\begin{eqnarray*} \max_{\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)\in \mathbb{R}_+^2 \times \mathbb{R}_+^2} & \ x_A^\frac{1}{2}y_A^\frac{1}{2} \\ \text{s.t.} & \ x_A+ x_B = 7 \\ & \ y_A + y_B = 7 \\  & \ x_B \geq 5\end{eqnarray*}

Solving this problem we get the following proposal :
$\left(\left(x_A^*, y_A^*\right), \left(x_B^*, y_B^*\right)\right) =  \left(\left(2, 7\right), \left(5, 0\right)\right)$

  Find competitive equilibrium (if it exists) in both the cases : (a) no free disposal, (b) free disposal

Competitive equilibrium consists of an allocation $\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)$ and prices $(p_x, p_y = 1)$ such that

$x_A = \dfrac{2p_x + 5}{2p_x}$, $y_A = \dfrac{2p_x + 5}{2}$
$x_B = \dfrac{5p_x + 2}{p_x}$, $y_B = 0$
$x_A+x_B = 7$, $y_A+y_B=7$

Solving it we get, $p_x = \dfrac{9}{2}$.
Therefore, competitive equilibrium allocation is $\left(\left(x_A, y_A\right), \left(x_B, y_B\right)\right)=\left(\left(\dfrac{14}{9}, 7\right), \left(\dfrac{49}{9}, 0\right)\right)$ and supporting prices are $(p_x, p_y) = \left(\dfrac{9}{2}, 1\right)$. Equilibrium is the same in both cases - with or without free disposal.


The intensive form of the production function is derived from the following. Let us assume a production function, F, with inputs of capital, K, and labor and technology, AL. Thus, output, $Y = F(K,AL)$. Assuming constant returns to scale, i.e. $F(cK,cAL) = cF(K,AL)$, we can say $F(K,AL) = ALF(K/AL, 1)$. The intensive form of the production function, f, takes the argument k = K/AL. k is ratio of capital to labour, or how much capital is there per effective unit of labour. $f(k) = Y/AL = F(K/AL, 1)$, or the output per effective unit of labour.

The intensive form is relevant because although the production function may have constant returns to scale, each individual input may exhibit diminishing returns. In other words, if you increase both labour and capital, this will increase your output proportionally, but if you increase only capital, your output will rise proportionally less. It is also easier to express interest rates and wages in the intensive form. For more, you can read any graduate level macroeconomic textbook. (I've used Romer's Macroeconomics.) Hope this helps!


When should a company's revenue and share price grow faster or slower than the economy?

GDP is the sum of wages and capital income. In most developed economies, 2/3rds of GDP is wages (income and other compensation) and the remainder is capital income (owner occupied rents, interest and dividends mostly). This ratio is difficult to measure precisely but the general consensus is that it has been mostly but not perfectly stable over time.

Why Labor’s Share of Income Is Falling By Jared Bernstein
Therefore, if the economy $Y$ grows at a rate $g$ and the capital income is roughly $Y/3$ then it too must be growing at rate $g$. If discount rates $r$ are also constant this implies that wealth is $\frac{Y}{3(g-r)}$ which also grows at rate $g$. Now it turns out that discounts rates are not constant and have been in long term decline, and as long as this is the case the value of the capital and land assets of the economy can indeed grow faster than output does. The following figure, which depicts financial assets relative to GDP, doesn't fully capture this issue (it omits non-financial assets like much of real estate) but it is a decent approximation.

All of which implies that capital income like other quantities roughly proportional to output, must grow at the same speed as output. Assets can grow at a different rate in the short to medium term than output, but that only if there are long term changes in the discount rate. But most firms are very small and even the largest ones are small relative to the whole economy. Firms are being founded and failing all the time, so there is plenty of room for individual firm to grow faster or slower than the economy as a whole.


What is the correct economic term for beneficial "transaction frequency"?

The only actual question I could glean from this seems to be:

How do transactions benefit the economy?

In economics, the value of goods and services is their utility, an abstract measure of how useful they are. The food a restaurant or grocery store sells provides utility by providing nutrition and the pleasure of eating. A car or public transportation provides the utility of ease of movement. A doctor gives the utility of medical care, etc.

In economies with money, money as a medium of exchange provides potential utility. Your money gives you the ability to perform trades with other people, gaining the utility of their goods and/or services, in exchange for them being able to do the same with someone else later.

Whenever a transaction occurs, the party providing the good or service earns a profit by charging more money than they spent on labor, raw materials, etc. The consumer, meanwhile, gains a sort of utility surplus which is the difference between the amount of money spent and the maximum amount they would be willing to spend.

If a fast food restaurant spends \$2.50 to make a burger that they sell to me for \$5, but which I would be willing to pay a maximum of \$7.50 for, after the transaction the restaurant has \$2.50 more than before they began the process to make the burger, while I have gained an amount of utility I would have been willing to spend \$7.50 to gain having only spent \$5. Both parties are enriched.

Not 100% of transactions are mutually beneficial. Sometimes sellers erroneously set their prices too low to earn a profit, sometimes buyers misjudge the value of something and spend more than they should have, sometimes a party is tricked or coerced into making bad decisions. But the vast majority of them are, which is why they keep happening.

As for your P.S., there is the concept of the velocity of money, essentially the rate that money is exchanged for goods and services. When most transactions are beneficial, the more that occur the better as a general rule. If I have $1 and do nothing with it, no one benefits because money is merely potential utility unless I use it for emergency toilet paper or something. If I spend it, both I and the person I give it to (probably) benefit and when they spend it they and the third party benefit and so on. This is one of the reasons certain individuals support government-mandated transfers of money between different segments of society, arguing that the group the money is taken from doesn't use it enough.


Evidence that open source production processes increase efficiency and/or consumer surplus?

There is a large economic literature on intellectual property rights. However, the issue seems far from settled on what even the optimal duration for patents are. Note that open source is even a step further than a 0 day patent duration. A strong case for your view would probably be found in Boldrin/Levine:
http://levine.sscnet.ucla.edu/general/intellectual/againstnew.htm

Here are several further starting points in the literature:

Scotchmer, Suzanne. "Standing on the shoulders of giants: cumulative research and the patent law." The Journal of Economic Perspectives (1991): 29-41.

Besen, Stanley M., and Leo J. Raskind. "An introduction to the law and economics of intellectual property." The Journal of Economic Perspectives (1991): 3-27.

Posner, Richard A. "Intellectual property: The law and economics approach." Journal of Economic Perspectives (2005): 57-73.


Derivation of Equilibrium Strategy in 1st-price Auction?

I think you have a typo -- the equilibrium bidding strategy in the first-price auction you specify should be $$ \beta(x) = \frac{N-1}{N}x $$

Here's a hint that might help. The CDF of the order statistic, $Y_1$, is $$G(x) = x^{N-1}$$

To see why this is the case, notice that this is exactly the probability that some given quantity $x$ is greater than or equal to $N-1$ independently drawn random variables that are uniformly distributed on $[0,1]$.

I'll write something in more detail when I have more time tomorrow, but I think this should tide you over for now. (At the very least, it'll help you evaluate the integral.)


Services are just as much VATable supplies as pyhsical products are

It is a mistake to think of the increase in price along the supply chain as just each person wanting their cut. For them to achieve this they need to be increasing the value that can be delivered to end customers, even if - for example - it is simply bringing a pallet of product to a convenient location and being there so that customers can buy single items at a time and place that suits them. That adds value to the pallet of goods in a warehouse.

The value added is the differemce between the VATable outputs I can make and the VATable inputs I use.

If my  business uses a consultant's time I pay input VAT on the money they charge me. I can offset this against the VAT I had to collect from my customers and send to HMRC.

If I hire a employee instead I pay their salary without VAT (so save input VAT) but then can't offset that amount against my output VAT, meaning I have to write a cheque to HMRC that is larger by the same amount as the input VAT I didn't have to pay (and which the consultant would have had to pay to HMRC).

It ends up a lot like a sales-tax but collected across the supply chain.

The problem with exemptions and zero rating

(I don't want to go into it, but making something VAT-exempt and making it zero-rated are very different things.)

One major problem with zero-rating activities from VAT is that it creates arbitrage opportuntiies if you can shift some or all of the supplies you make into the zero-rated category.

As most supplies have in some sense a service and a product component, general zero-rating of services would create huge incentives to artificially split out product and service compoments to save on VAT, leading to huge tax bureaucracy and litigation.

That this is a real issue can be seen in the hard-fought court cases over, for example, whether a jaffa cake is a cake or a biscuit or whether shops can save the VAT on a small transactino processing cost component of every transaction.

Ultimately VAT is very similar to a retail sales tax but with collection distributed across the supply chain.


Are means-tested or universal programs more effective at reducing the poverty rate?

TL;DR:
There is generally not much support for universal welfare programs (as narrowly defined - see below) in the literature. However, it is for different reasons that what is mentioned in the question.
The main proposed advantage of universal programs is that they do not distort peoples incentives since you are not loosing your benefits as you are moving up an income ladder (less administrative costs is in the literature only auxiliary argument). However, at the same time this means that state has to redistribute resources even to billionaires, so such programs are more expensive. These expenses ultimately require higher taxes and taxes distort incentives as well. It turns out that on net having means tested welfare transfers produces less distortions to the economy.
The bureaucracy issue is very commonly cited in the press, because it likely captures better peoples imagination and attention as compared to more serious but less known deadweight loss caused by taxation/redistribution.  However, bureaucracy costs are usually trivial for either system. In addition, there is no support in a literature that welfare for rich has much stimulative effect for the economy.
Full Answer:
Narrowing the Scope:
First, in order to actually answer this question, we have to properly define the scope of subject matter. 'Programs that reduce poverty' is broad and vague term that could be applied to virtually any government program. However, given what is written in the question I suspect what you are most interested in are income redistribution taxes and transfers such universal basic income (UBI) vs means tested transfer as opposed to some other government schemes (e.g. compulsory state provided elementary and high-school education could be viewed as anti-poverty program but we have to narrow the scope of the question as providing overview of all government programs that can reduce poverty in one way or another would require several books and be well beyond scope of SE answer).
Bureaucracy Issue
Next let me address the bureaucracy issue as this is very common argument that non-economists use but you won't even see it mentioned in the literature much. Simply put bureaucracy costs are trivial. For example, consider the administrative spending for main US welfare programs below:

The administrative spending ranges from $9\%$ to $>1\%$ of total spending for the program. Moreover, you will note that the highest administrative costs are for food stamps and housing vouchers which are in kind/conditional cash transfers that are, generally speaking,  considered by most economists relatively inefficient for various reasons relative to unconditional (but still means tested - conditionality here refers to additional restrictions on spending of the transfer) cash transfers (e.g. see Currie & Gahvari, 2008; Skoufias et al., 2013; Handa et al., 2018; Mishra & Kar, 2015; Gentilini 2016 or Caldés et al., 2006).
As you can see the earned income tax credit (which is still means tested) but purely cash based transfer has less than $1\%$ administrative cost. This just goes to show that in principle means tested programs are not expensive from point of view of bureaucracy. Even if we would be generous and assume that switching to universal transfer could cut $50\%$ of all administrative costs (which I think is overtly generous given that significant part of administrative costs are fixed costs) it would be penny wise but pound foolish to make argument that such trivial savings are justifiable taking into account large costs that extending transfer to anyone entails (although as will be explained further down the road a proper efficiency argument could be made on the account of distortions they create).
Stimulus Argument
I must say I did not heard the argument that transfers to the rich provides additional stimulus to the economy (I heard the one about tax cuts but not about transfers), but it does not have any basis in economics. First, from macroeconomic perspective spending is spending whether done by rich or poor. There is not much if any evidence that spending would stimulate economy more if done by rich instead of poor.
Second, richer people tend to save higher portion of their income (see explanation in this older SE answer why), this does not matter outside recession since saving becomes investment spending which increases output equivalently to consumer spending in normal times. However, during recessions economies can experience the 'paradox of thrift' - a situation where extra saving is offset by drop in output and thus won’t help the economy  (I wont go into details on this concept as it is tangential to this question, you can see further explanation in Blanchard et al Macroeconomics: A European Perspective). But even in such an event consumer spending still stimulates the economy. Consequently, if anything if the goal is to stimulate economic activity the transfers should especially target poor who have higher propensity to consume not higher income groups who tend to save more.
Economic Efficiency Issue
When economists discuss economic efficiency of redistribution schemes they mainly focus on the economic distortions that these schemes create (see Stiglitz Economics of the Public Sector). These efficiency costs are created because taxes and redistribution distort optimal behavior of people. For example, an income tax will distort peoples incentives to work, consumption tax to consume etc. At the same getting welfare benefit conditional on low income distorts peoples incentives to move up income ladder or put more hours and so on.
The reason why economists focus on these effects is that even though these are almost unknown among general public their economic costs are massive. For example, as mentioned in Jacobs (2009) on the margin it is often believed that excess burden of taxation is approximately $50\%$ (although there is some uncertainty about this value and it could be as low as $30\%$ or as high as over $100\%$ depending on exact behavioral responses (elasticities) to taxes and country we are talking about). The commonly accepted $50\%$ figure implies that for government to raise raise $1$ dollar of tax revenue it has to reduce aggregate income of an economy by $1.5$ dollars on the margin. This is purely cost of damage to the economy, the administrative costs would have to be added on top of that, but as mentioned above they are more or less trivial (e.g.  in Netherlands where I live they would be additional 6 cents on average (see Allers, 1994) and in most developed countries it should be comparable).
Now when it comes to universal redistribution schemes their big advantage is that they eliminate distortions on the receiving side. For example, with UBI you get your welfare transfer whether you work, are poor or rich and so on. So there is absolutely no reason for you to change your behavior/economic activity. However, at the same time since the benefit is extended to everyone they are more expensive, requiring increases in taxes which as mentioned above also have serious effects on distorting the economy.
When it comes to means tested redistribution schemes they create relatively large economic disincentives at the receiving end but an advantage is that only a portion of the population receives the benefit making them much cheaper. Consequently, government needs to raise much less revenue through distortionary taxation to pay for it.
Hence, a priori it is not clear which one would create less net distortions as one creates more distortions on receiving end and the another one more on the taxing end. Consequently, the main economic debate of universal redistribution schemes vs means tested schemes is about whether the economic distortions of one are larger than other, not about administrative costs that are relatively trivial at best mentioned as an auxiliary argument/issue in the literature.
Here the literature shows that means tested schemes are preferable as they cause less distortions both from theoretical and empirical perspective. For example, the state of the art optimum income taxation simulations (that includes also redistribution which is treated as a negative tax in the literature) show that it is optimal to have means tested income redistribution, even when government adopts Rawlsian social welfare preferences (i.e. the max-min criterion, which implies that government only cares about the welfare of the poorest anyone else except for the poor has zero welfare weight in government social utility function). This is despite of the fact that such means tested programs cause distortions primarily on the receiving end and thus create a 'poverty trap' - a situation where poor have less incentive to climb up the income ladder (see Diamond 1998;  Saez 2001).  The case for broader benefits is bit more stronger under utilitarian or charitable conservative/libertarian social welfare function, but I don't know of any standard social welfare function that would result in universal benefit. This can be seen on the picture below which shows simulations of optimal marginal tax rate. The tail is so high on the lower end of income distribution precisely because that represents people loosing their benefits (withdrawal of benefit is considered tax since benefit itself is a negative tax).

If you are not convinced by theoretical optimal tax simulations then when we turn to empirical evidence on the most popular universal redistribution scheme: universal basic income (UBI), the evidence shows that the measure is simply too expensive and thus inefficient (even if it reduces the above mentioned poverty trap and has some other positive effect). For overview of literature on this you can see Hoynes and Rothstein (2019) or Martinelli (2017) and sources cited therein. The literature can be summarized nicely by the following quote form Martinelli:

an affordable UBI would be inadequate, and an adequate UBI would be unaffordable.

The issue is that there are quite a lot of people in upper income brackets. For example, in 2019 in the US over $35\%$ of households earns above $100,000$ dollars per year and about $20\%$ above $150,000$ dollars per year. These are income groups where there is little to none rationale for welfare support (at least from the point of view of welfare/public economics - politically it might be useful to support such groups for political reasons), but are still very broad groups of people that would waste enormous number of resources unless the universal benefit is set so low that it will become insufficient for those at the bottom of income distribution.
Few Caveats
However, there are few caveats that need to be mentioned.

As seen in the previous chapter taxation and redistribution cannot be divorced from each other - they are two sides of the same coin. However, if you only talk about transfers you could make an argument that you can have universal transfer while you cancel it with extra taxes on higher income groups. For example, you could give everyone $1000$ dollars per month of universal income and then tax everyone else except the poor by $1000$ dollars more essentially making the universal basic income means transfer program but in more hidden way. This is really just semantics argument hinging on definition of the words in the name of program, but some economists such as Mankiw made an argument for UBI on these grounds. However, this is to the large extent a semantic issue as economically if you cancel that $1000$ by higher taxes for better of groups you create the same disincentives as if the benefit would only exist for poor. So really this is just a semantic argument without economically affecting the discussion in sections above.
As mentioned at the beginning this analysis might not extend to every single government program. Arguments for universal education or healthcare could be made on partially redistribution grounds but address additional issues of externalities or market imperfections that make their universality necessary or more efficient, but this goes beyond any purely redistribution/poverty-alleviation concerns.


Why does Fed buying mortgage backed security help lower the interest rate for mortgages?

Addressing your question directly, yes, the rate will necessarily decrease. The Fed's purchases of MBS were entirely MBS that were securitized by the government-sponsored enterprises (GSEs). GSE MBS are not subject to credit risk, as the credit risk is guaranteed by the GSEs.

Further, banks underwriting mortgages that are securitized through this channel aren't holding any credit risk— the only real risk they face is that if a loan that defaults is found to have underwriting defects, the originating bank can be forced to repurchase the loan from the GSE through which it was securitized. This repurchase risk isn't trivial, but it can be controlled in a number of ways (proper underwriting being foremost among them). As a result, the channel that you posit— bank perceptions of mortgage risk increasing, offsetting rate decreases brought about by QE— is not possible.

Empirical evidence supports the above assertion— rates did in fact decrease. Krishnamurthy and Vissing-Jorgensen say:

  We document that the Federal Reserve’s purchase of long-term Treasuries and other long-term
  bonds ("QE1" in 2008-2009 and "QE2" in 2010-2011) significantly lowered nominal interest
  rates on Treasuries, Agencies, corporate bonds, and mortgage-backed securities, but with
  magnitudes that differed across bonds, across maturities, and across QE1 and QE2.


Why do you typically have to pay for use of the washing machines while other amenities are free?

As you said, you would prefer to have free washing machines instead of free swimming pool. What does this mean? That washing your clothes is more important than having a bath in a swimming pool. Usually, it applies to almost anyone. So you could stop using swimming pool, but not washing machines.

Also, it does not seem probable that many people will use that swimming pool at the same time. But even if they do, people will not complain about massification unless it is too much and permanent (both conditions are rare to be filled).

So:

1) They know that washing clothes is more important than having a bath in a swimming pool. And even more, they know that you cannot avoid to wash your clothes (but you can avoid having a bath).
2) They do not expect the swimming pool to be always full of people, so they do not expect many complaining about it.
3) Having a swimming pool is cool. Having a free swimming pool is very cool. Marketing.

In conclusion: they use the swimming pool as part of their marketing, while they make money from washing machines since you have to wash your clothes with no potential alternative (while having a bath in a swimming pool has several alternatives, so if you had to pay for it, maybe you would not use it).

I hope it helps.

EDIT:

For those who do not know what I am talking about, take a look of these two terms: "utility" and "elasticity" ;)


Difference between "Level" and "Flow" in the Z.1 Fed data

Say you have a balance sheet that measures a country's total stock of financial assets and liabilities for a given year. If you have a bunch of these balance sheets for several years, those data points are the actual levels of total assets/liabilities.

Now suppose you decide to subtract the levels that correspond to year $t-1$ from the levels that correspond to year $t$. That's what the Fed is doing with these data series -- the flows are the first-difference of the levels.


Why hasn't Quantitative Easing surged inflation, or caused hyperinflation?

High-powered money is another term for the monetary base or MB.  MB represents all money created by the government.  However, that is not all the only source of money.  Banks are allowed to create money too.  Which is why higher aggregates like M2 and M3 are greater than MB.

Because bank money is greater than government money, it (and not MB) is more of a primary determinant to monetary inflation.  Pretend there is an economy of blue dollars and yellow dollars.  The blue are created by the government and the yellow by banks.  There are many more yellow dollars than blue dollars.  So when the supply of blue dollars goes up it doesn't matter as much because of the number of yellow dollars.

Bank money was being destroyed in the crisis which was actually creating deflationary pressures.  The collapse in higher monetary aggregates was offset somewhat by the surge in the monetary base.

Now typically base money is used to create and leverage higher aggregates, but this was not the case during the crisis.  Because of liquidity concerns, banks curtailed monetary creation so this is why inflation did not happen.


Is there always a pure Nash equilibrium in a resource selection game?

Yes, there is always a pure Nash equilibrium. See:

I Milchtaich (1996). Congestion games with player-specific payoff functions.
Games and economic behavior 13 (1), 111-124.

You are interested in the special case of singleton congestion games with player-specific payoff functions.

And yes, they can be computed in polynomial time. See Corollary 7 in:

Heiner Ackermann, Heiko Röglin, Berthold Vöcking (2009). Pure Nash equilibria in player-specific and weighted congestion games. Theoretical Computer Science 410 (17), 1552-1563.


What is the impact of fracking on oil prices?

over the past few months, the oil harvested from fracking had come onto the market reducing the price of gas.

Obviously, this makes no sense insofar as the oil harvested from fracking displaces oil harvested by other means.

Absent changes in supply or demand, the price of oil has to rise at the rate of interest.

The development of a new extraction technology is a one-shot increase in supply, which should lead to a one-shot fall in price, followed by a resumption of the ongoing increase at the rate of interest.  It can't (by itself) explain an ongoing fall in price.


General Equilibrium - Mas Colell Problem 17.D.2

Your answer is essentially the same as Mas-Colell's.

$z_{11}$ is excess demand for good $1$ by consumer $1$. $z_{12}$ is excess demand for good $1$ by consumer $2$. They sum up to zero at the price vector $(1,1)$, which means that the market for good $1$ clears. By Walras' law, that implies that the market for good $2$ clears as well, hence $(1,1)$ is an equilibrium price vector.


What would you call a real market that is close to perfect competition?

I do not have a special term, but I give this analogy:

We know what a triangle is. But in the real world, it is impossible to draw a perfect triangle and there do not exist examples of perfect triangles.
Similarly, we have just learnt what perfect competition is. But in the real world, there do not exist examples of perfect competition. Nonetheless, here are some examples of real-world markets that come "close to" perfect competition ...
We stress that these markets are not actual examples of perfect competition. (To repeat, such examples do not exist.) Instead, these are merely examples that come "close to" perfect competition.


Policy relevance of the optimal taxation given uncertainty about social welfare specification

Here is an answer based on the following interpretation of the SWF : there is no "true" SWF, but SWFs are observable in principle, they simply represent the preferences of the policy decision maker.

Under this interpretation, the policy recommendation are relevant despite depending on the specification of the SWF, precisely because different decision makers have different preferences. So there is no problem with studying different configuration of the SWF and getting different policy recommendations. In fact this is rather desirable as it makes the theory relevant to a larger number of decision makers.

Now, whether you are aiming at policy recommendations for the average decision maker, or whether you are giving policy recommendation to a particular decision maker,  figuring out an exact social preferences might be complicated. Data may be insufficient to pin down particular preferences without much uncertainty. And if you ask the decision makers to state their preferences, they might be uncertain about their own preferences.

As is to be expected, the more uncertainty about the decision maker's preferences, the less precise the policy recommendations will be. But there is still a whole lot you can say even under relatively large uncertainty. As a matter of fact, most of the literature on optimal taxation has focused on providing general policy recommendations which hold for a wide class of SWF.

A very good example is Roëll, A. A., 1985. A note on the marginal tax rate in a finite economy.
Journal of Public Economics 28, 267–272's very weak redistribution assumption (VWR), which was informally presented earlier in Guesnerie, R., Seade, J., 1982. Nonlinear pricing in a finite economy. Journal
of Public Economics 17, 157–179. The VWR is a property of the solution of the taxation problem which formalizes the idea idea that the decision maker is egalitarian in at least a very mild way. To give a taste of the weakness of the assumption, notice for instance that

  If the utility function U is strictly concave, leisure is a normal good, single-crossing is satisfied, the social welfare function $W$ is differentiable, strictly concave, increasing, and symmetric, and at a solution of the taxation problem, the lower level of income is positive, then VWR is satisfied (Brett and Weymark, forthcoming)

It turns out that the VWR is sufficient to derive many important policy implications. For instance, under the VWR (and single crossing), marginal tax rates must be non-negative at a solution of the taxation problem. Also under the same assumptions, the marginal tax rate of the highest type should be zero, and any type different from the highest type whose consumption and income are strictly positive must face a strictly positive marginal tax rate. Provided the non-normative assumptions of the model hold (e.g. single crossing, but also the fact that decision maker's preferences are representable by a SWF, more on this below), these are policy recommendation you can safely make to any policy decision maker with some form of egalitarian concerns, even if you do not know the exact degree to which they are egalitarian.

Another question you are raising is whether policy decision makers' preference can be represented by a SWF at all. To say the least, I am not a big fan of the cardinal utility framework of SWF. This is mostly because hidden in the cardinality assumption is the idea that one can somehow "objectively" compare utilities. But it is also partly because some decision maker's preferences might be not representable through SWF. However, there are alternatives to both these problems, with applications to optimal taxation. See for instance the last chapters of A theory of fairness and social welfare, by Fleurbaey and Maniquet.

The approach described in A theory of fairness and social welfare, by Fleurbaey and Maniquet is not only ordinal, it is also purely axiomatic. Decision makers may have a hard time figuring out which SWF represent their preferences. With an axiomatic approach, they simply have to decide on the axioms they like the best to "discover" their preferences. This is often much simpler than trying to figuring and spelling out your full preference relation.


What is the example of inaccurate impact due to endogeneity?

Endogeneity can arise from several reasons and in each case the explanation will be slightly different. I won't show full review of all possible reasons but just two important examples:
Simultaneity
For example, following Verbeek a guide to modern econometrics pp 146, suppose that true model is given by system of equations:
$$y = \beta_1 + \beta_2 x_{2t}  + \epsilon_t \tag{1} $$
$$x_{2t} = y_t + z_{2t} \tag{2}$$
where $z$ will be exogenous $(cov(z,\epsilon)  = 0)$  but clearly $x$ is endogenous. We can see from the equations above that if $x$ increases not only does $y$ increase but $x$ increases again through effect of $y$ on $x$ given by $2$ which then again increases $x$.
Solving the 1 and 2 for $x$ and $y$ respectively we get:
$$ x_{2t} = \frac{\beta_1}{1-\beta_2} + \frac{1}{1-\beta_2} z_{2t} +  \frac{1}{1-\beta_2} \epsilon_t \tag{3} $$
$$ y_t = \frac{\beta_1}{1-\beta_2} + \frac{\beta_2}{1-\beta_2} z_{2t} + \frac{1}{1-\beta_2} \epsilon_t \tag{4}$$
It follows from 3:
$$ cov(x_{2t} \epsilon_t ) = \frac{1}{1-\beta_2} cov(z_{2t} \epsilon_t) + \frac{1}{1-\beta_2} V(\epsilon) = \frac{\sigma^2}{1-\beta_2} \tag{5} $$
The 5 just shows that $x$ will be correlated with $\epsilon$, but a direct consequence of this correlation is that:
$$ \text{plim } b_2 = \beta_2 + \frac{cov(x_{2t}, \epsilon_t}{V(x_{2t})}$$
So what you are estimating is not just your beta coefficient you want $\beta_2$ but you are estimating sum of the true $\beta$ coefficient together with ratio of covariance between $x$ and $z$ to variance in $x$ or  $\frac{cov(x_{2t}, \epsilon_t}{V(x_{2t})}$.  This indeed can even force coefficients to flip. For example, suppose true effect $\beta_2=5$ but $cov(x,z)=-20$ and $V(x)=1$ then if you just naively run model  $y = b_1 + b_2 x_{2t}  + \epsilon_t $ your estimated $\hat{b_2}$ will be $\hat{b_2} = 2 -20 = -18$ which has opposite sign and the magnitude of $b$ will be distorted. So you will be getting unreliable results.
Ommited Variable Bias
This would be example, where we do not observe (or fail to include) variable that should be in the model. Again following Verbeek pp 145 consider individual wage equation:
$$ y_ = x_{1i}' \beta_1 + x_{2i} \beta_2 +  u_i \gamma + v_i$$
here $y$ would be wage, $x_{1i}$ is vector of individual characteristics (e.g. gender, age, location etc), $x_{2i}$ years of schooling and $u_i$ some unobserved innate ability. We expect that $cov(x_{2i},u_i)>0$ because it makes sense that if you have higher innate ability you will have higher education as people with high innate ability should have easier time at the school/university.
Now let us suppose that because innate ability is not observable you are forced to estimate:
$$y x_i'\beta + \epsilon_i$$
with $x_i'=(x_{1i}', x_{2i}) $ and $\beta'= (\beta_1', \beta_2), $ and naturally since we omitted ability  $\epsilon_i = u_i\gamma + v_i$.
Now the $b$ estimates of $\beta$ are given by:
$$b= \beta +  \left( \sum_{i=1}^N x_i x_i' \right)^{-1} \sum_{i=1}^N x_i u_i \gamma + \left( \sum_{i=1}^N x_i x_i' \right)^{-1} \sum_{i=1}^N x_i v_i$$
Even assuming $E(x_i, v) = 0$ (i.e. there is no additional simultaneity or other endogeneity issues in the true model), we get:
$$\text{plim } b = \beta + \sum_{xx}^{-1}E[x_i u_i] \gamma$$
So as long either $\gamma \neq 0 $ (in which case it should not be in the first model to begin with as that would mean ability does not affect wage), or $E[x_i u_i] \neq 0$ which in our case wont hold because we assumed ability does affect education, your estimated beta is not going to be actual effect of the independent variable on the dependent variable but the actual effect plus term determined by correlation between omitted and included independent variable and $\gamma$. Again this can change both magnitude and sign of $\beta$ so you simply can't trust it at all.
Lastly, there could be other ways in which endogeneity arises in econometric setting, but exploring all of them is beyond scope of SE, and the two above mentioned ones are probably the most common ones and relevant.


What happens to the stock market if no one wants to buy?

Yes, price will decrease. If nobody wants to buy at lower prices, sell  orders will push stock prices toward zero.
This is possible in periods of financial turbulence, especially for companies that are very exposed to uncertainty and have do not have positive financial results to show to investors.
If you are interested in exploring this topic, I suggest that you start by reading  about the stock market crash of October 1987.


Best resource to learn Microeconomics fast?

It would help a lot to know at what level this class is exactly. You mention calculus, however most classes titled like yours ("Introductory Microeconomics") tend to be very light on math. I'm going to assume that your class is of the learning-introductory-concepts variety and not of the taking-partial-derivatives-to-maximize-utility variety. If I'm wrong please correct me (or ignore this answer).

You have two options:

1) Obtain a virtual copy of an intro micro textbook (I can vouch for Mankiw's Principles of Microeconomics) and read it cover to cover.

2) Use a (probably) less-reliable resource such as this or the link you posted in your answer.

Of course, I choose these options with the assumption that your textbook is as bad as you say it is. I would still recommend you go over the course material to determine what your instructor considers important.

Important note: at this level of economics, knowing how to do things (calculate elasticity, do simple math in simple word problems about total cost and average cost, etc) is not nearly as important as understanding the concepts being taught. You say you've taken Calculus 3; expect basically the opposite of your Calc 3 final for this final. Of course, you should know the equation for, say, the elasticity of demand. However, the much more important thing is understanding what elasticity means and how to apply the concept to novel real-world problems.

Here is my off-the-top-of-my-head list of concepts you need to master.

Competition. How does profit get driven to 0 in a competitive market? What happens when demand increases in the short run? What then happens as time goes on?
Monopolies. Understand why it is in a monopolist's best interest to produce less than the efficient amount. Understand why a monopolist faces a downward sloping MR curve while a competitive firm faces a flat MR curve. Be able to show your understanding on a graph.
Understand the difference between an increase in demand (ie, an outward shift in the demand function) and an increase in quantity demanded. Same for supply. Be able to show it on a graph.
Understand what happens to quantity supplied and quantity demanded when a binding price floor is in place. Be able to show it on a graph.
Firm cost and revenue graphs. How are ATC, MC, MR, P, AVC, and AFC related? Does the answer to this depend on the structure of the market? Why are supply curves upward sloping while demand curves are downward sloping? What are sunk costs?
Opportunity cost. what is opportunity cost? If firms in a competitive market see 0 economic profit, how is it that real-world firms achieve positive accounting profit?
Specialization and comparative advantage.
Price discrimination.
Deadweight loss. What is it and why do taxes create it. Be able to show it on a graph.
Externalities, public goods, and Pigovian taxes.
Elasticity. Bonus points for understanding in terms of calculus. Understand perfect elasticity and perfect inelasticity. Show it on a graph.


The issue is that the invention of the steamship is not exogenous to economic development. Assuming that trade exerts positive effects on development, it could be that past trade favored economic development and as result the invention of the steamship. By contrast, wind flows are exogenous.


Current Value Hamiltonian VS Present Value Hamiltonian in Economics

There is no clear right and wrong about this, it's just a matter of convenience. The current-value Hamiltonian is likely to be more convenient when the objective function includes a discount factor. Following Chiang (1), suppose the problem is:
$\qquad$Maximise $V = \int_0^T G(t,y,u)e^{-\rho t}$
$\qquad$subject to $\dot y=f(t,y,u)$
$\qquad$and boundary conditions
The standard (present value) Hamiltonian is:
$\qquad H=G(t,y,u)e^{-\rho t} + \lambda f(t,y,u)$
If we proceed from this Hamiltonian, the co-state equation (one of the first-order conditions) is:
$\qquad \dot \lambda = -\frac{\partial H}{\partial y}= -\frac{\partial [G(.)e^{-\rho t}]}{\partial y}-\lambda\frac{\partial f}{\partial y}$
While it is possible to obtain a solution this way, the discount factor complicates the derivatives and can make interpretation more challenging.
Suppose instead we use the current-value Hamiltonian:
$\qquad H_c = G(t,y,u) + mf(t,y,u)$
where $m$ is a current-value Lagrange multiplier defined by $m=\lambda e^{\rho t}$.  The co-state equation is then:
$\qquad \dot m -\rho m = -\frac{\partial H_c}{\partial y} = -\frac{\partial G}{\partial y} - m\frac{\partial f}{\partial y}$
This is simpler because it contains no discount term.
Reference

Chiang A C (1992) Elements of Dynamic Optimization pp 210 ff


Is there a statistic that shows which countries would be owed money if all external public debts were payed immediately?

The description given implies to me that you want to net the debt holdings between governments only. For example, we would subtract the Chinese government's holdings of U.S. Treasury securities from Chinese debt (and so forth).

I am unaware of any such statistics. For the developed countries, the amount of cross-governmental debt holdings is minimal (excluding the ambiguous situation of the European Central Bank). Therefore, such an adjustment would only matter for some developed countries. (The ratio of governmental debt to reserves - which are holdings of developed country debt - is probably looked at by ratings agencies and analysts of developing countries.)

A more usual statistic is the net international position, which compares national holdings of foreign assets versus foreign holding of domestic assets. However, this mixes all form of financial assets (debt and equity) and includes both the private sector. The IMF has a standard for reporting the international investment position - link to page

The usefulness of such statistics enters the realm of opinion. The main usage of the net international position seems to be the assessment of the odds of a currency crisis; I am unaware of the literature on that subject.

As for debt statistics, things like the debt-to-GDP ratio have been greatly discussed. However, I am in the camp that argues that the main "use" of debt statistics has been to generate incorrect forecasts (such as the sustainability of Japanese debt). In general, the link between debt levels and bond yields for the developed countries is questionable, which indicates that bond investors are not greatly impressed by the importance of debt statistics.


Housing econometrics: interpretation of a quadratic variable

When both the level and the square affect the dependent variable (and if the coefficient on the square is negative), we have a case where the variable in question initially affects positively the dependent variable, but after a point it affects it negatively.

When only the square is deemed to affect the dependent variable, then the effect is monotonic and accelerating, while its direction is indicated by the sign of the coefficient.

Real world interpretation: can you rationalize why latitude could have a monotonic effect, while longitude has not? Can you match this to the geographical areas you use data on? Is there anything special along latitude as regards the weather, socioeconomic conditions etc? And also, why longitude has an up-and-down effect? What could be the possible reason here?

These are the cases when empirical studies become thrilling: when unexpected econometric results provide new insights into the real world.


Does economic theory support the notion that the wealth of the wealthy is based on the poverty of the poor?

Let me preface this answer with a word of caution: your question is a very good and important one but it is also one that depends tremendously on the definitions of the terms it uses. I'm going to attempt to answer it in the most unassuming and non-technical way. You could pose it in technical terms and get a more precise answer.

The wealth of the rich can be attributed to the poverty of the poor. In North Korea, one of the poorest countries in terms of per-capita GDP, there are few rich people and many poor people. It is well-known that almost all of the rich people work either in the highest levels of government or the military. The poor people who are lucky enough to make any income largely do so at the behest of the rich people, and those who refuse are imprisoned or worse. Since almost all economic activity in North Korea is centrally planned via force, it follows that the rich derive their wealth from the suffering of the poor.

However, in societies with more private ownership of production and capital - and less centralization - force is still useful in general but less of a factor in determining who is wealthy and who is poor.

Regardless of our relative lot in life, if you and I freely trade with each other, we only do so when both of us feel like we are "better off" as as a result. So, if you are a wealthy industrialist and I am a poor banana farmer, and I freely decide to trade 100 of my bananas for 100 of your dollars, you necessarily think a banana is worth at least 1 dollar to you, and I necessarily think a banana is worth at most 1 dollar to me.

Excluding outright fraud (misrepresentation either of your dollars or my bananas) we must conclude that each of us either thinks this is a fair trade or that the other guy is a dope. For instance, you may know something about the value of bananas that I don't. Maybe you've just discovered that bananas have special healing powers or that my bananas are particularly good. On the other hand, maybe I know something you don't. Maybe I know that there are about to be many more banana producers in the area or that I can buy two bananas for 1 dollar somewhere else.

Usually, however, it's just a matter of what each party in the trade is able to do with the traded goods that determines what he/she thinks they are worth.

If you get rich from selling my bananas at a higher price, you are not doing so "at my expense" as long as you are not preventing me from doing the same thing. There may be reasons you have the ability to sell bananas at a higher price that have nothing to do with force - maybe you own a distribution system or retail outlets, and I don't - but my choice isn't between selling retail bananas and selling wholesale bananas, but rather selling bananas at the highest possible price vs not selling bananas at all.


At the HS6 level, there are thousands of product codes (I believe around 5,000) so if you're just computing the HHI using product codes, then actually it's pretty surprising you've received such a large number! However, for the interpretation of your result you should keep in mind that you've computed the export HHI of Chile using HS6 product codes, which definitely do not correspond to a more common, yet harder to define, notion of "commodities".

For example, there are many product codes that can be said to be copper related, and refer to various forms of smelted and refined copper. For instance, the HS6 code (in the HS1992 nomenclature) for copper ore is 260300, but there are dozens of product codes for refined copper and related products (i.e. 740110, 740120, 740200, etc.). Do you want to treat all these as separate products? It depends on the question you want to ask.

For instance, if you wanted to ask what the export HHI of Chile is for commodities, then you would need to define what a commodity is, and which HS codes you want to aggregate into commodities such as "nickel", "copper", etc.. and then compute the HHI based on the export shares of these groupings. There are pre-existing groupings of commodities, but the "right" aggregation will depend on your question.


What does the Federal Reserve intend to do with bonds it has purchased under QE?

The leverage ratio, as always, is a ratio between liabilities and equity. It has really no significance, as the liabilities of the Fed, which are mostly dollars, need not to ever be paid back.

Only in the case of inflation would the Fed need to reduce its balance sheet (i.e., sell the public debt or MBS). Do note that the profit of the Fed goes back to the government, thus the debt that is sitting there is an asset and liability of the government at the same time, canceling out. That is, the net debt decreases as the Fed buys bonds from the market. You may also view it as inflation or a money tax that pays back the debt.


Financialization is just a broad term for when financial markets become bigger and have more influence on the market economy.

I'd say that it wasn't so much markets becoming bigger that directly caused the 2008 crisis, but more that the shadow banking financial market became bigger (intermediaries not subject to the same regulation as normal banks). Financialization arguably caused the need for more niche ways of getting money into the market in this way, and as financial markets got bigger, particualrly in regards to housing, more banks ended up securitizing subprime mortgages, and that was a big factor in causing the 2008 recession.


Where can I retrieve a complete history of macro economic data annoucements?

Federal Reserve Economic Data (FRED) system may have some of the data you are seeking. For example the links below may be helpful directly or indirectly to inform your further search efforts.
FRED  recent release calendar:
https://fred.stlouisfed.org/releases/calendar
FRED web services release dates:
https://fred.stlouisfed.org/docs/api/fred/releases_dates.html


Definition. The opportunity cost (OC) of any alternative is the value you place on the best of the forgone alternatives.

If we adopt the above definition (and I do), the OC of $B$ is the value of $A$, which is simply $\$100$.

With the above definition, the alternative you choose ($B$ in this case) is completely irrelevant when calculating its OC. All that matters is the best of the forgone alternatives.

To illustrate this point, consider this pair of examples:

  Example #1. $A$ and $B$ are the only two alternatives. $A$: "Pay $\$500$ to get back $\$600$ in return." $B$: "Pay $\$500$ to get back $\$450$ in return." The OC of $B$ is the value of $A$, which is $\$100$.

  Example #2. Everything is exactly the same as before, except that now $B$ is changed to "Pay $\$500$ to get back $\$1,000$ in return." The OC of $B$ is again the value of $A$, which is $\$100$.

In the above examples, the value of $B$ changes (from $-\$50$ to $\$500$) but that of $A$ doesn't. And so, the OC of $B$ is $\$100$ in both examples.

An alternative's value is completely irrelevant to its OC. What's relevant is the value of the best alternative forgone.

Notes. "What exactly is OC?"

This seemingly-simple question was discussed in the past (see e.g. Alchian, 1968). Then like many of the most important questions in economics, it was then forgotten/glossed over until Ferraro & Taylor (2005) and the susbequent literature spawned (see in particular the 2016 JEE Symposium).

It all boils down to which definition you use. My view (and I believe also that of Alchian and Buchanan, 1987) is that the above definition is the "most correct" one that "everyone should use". Of course, whoever's marking your homework may disagree.


As you note in your question the (inner points of the) Pareto set are defined by
$$
x_1^1 = \frac{\alpha_1 r^1 x_1^2}{\alpha_1 x_1^2 - \alpha_2 x_1^2 + \alpha_2 r^2}
$$
In order to better examine this curve, let us treat it as a function. Let
$$
f(x) = \frac{\alpha_1 r^1 x}{\alpha_1 x - \alpha_2 x + \alpha_2 r^2}.
$$
As I stated in my comment
$$
f(0) = 0 \mbox{ and } f(r^2) = r^1,
$$
so the set indeed goes from one corner to the other. In between you have
$$
\frac{d \ f(x)}{d x} = \frac{\alpha_1\alpha_2r^1r^2}{\left(\alpha_1 x - \alpha_2 x + \alpha_2 r^2\right)^2}
$$
and
$$
\frac{d^2 f(x)}{dx^2} = 2 \cdot \frac{(\alpha_1\alpha_2r^1r^2) \cdot (\alpha_2-\alpha_1)}{\left(\alpha_1 x - \alpha_2 x + \alpha_2 r^2\right)^3}.
$$
The first derivative is always positive because $r^2 > x$ and all other parameters are positive. For the same reason, the sign of the second derivative depends on the sign of $\alpha_2-\alpha_1$.  Thus the Pareto set will be a strictly increasing curve, which is convex if agent 2 values good 1 'relatively more', concave if agent 1 values good 1 'relatively more', and a straight line if they value it 'relatively equally'.


How is monetary policy fair in the current economy?

The description you're providing of how interest works is based on a couple of fairly common misunderstandings about how the system actually works, so lets clear that up first.

Fractional reserve banking and the gold smith banking systems preceding it, are a result of the development of an accounting technology called double entry book keeping in the 13th century.

Double entry book keeping grossly simplified. Each transaction must consist of a debit and a credit tuple, occurring in two separate books. Asset books are traditionally written on the left, and liability and equity on the right. On the right hand side (liability and equity side), credit adds, debit subtracts, and its the opposite on the left hand side, debit adds, credit subtracts. Just remember the right hand side is what you think it should be, the left hand side is the opposite.

When a bank makes a loan, it performs the book keeping operation:

[debit loan, credit customer account].

This creates the loan, and it creates money in the form of the matching deposit. When  loan capital is repaid, the operation is:

[credit loan, debit customer account]

and this removes the deposit money that was created.

So how does interest get paid? Well in your example, Roger gets a loan for 100, lets say it's a 10 month, simple interest loan (easier math.)

The first month, Roger repays 10 capital, and 5 in interest. That's:

[credit loan 10,        debit Roger account 10]

[                       debit Roger account 5,
                        credit bank interest income account 5]

The bank's interest income account is also classified as a liability, although not strictly as a deposit account. Once the bank has recognised that income, it can then use it to pay expenses. Let's assume Roger works for the bank, and the Bank pays Roger a monthly salary of 5. That just reverses the operation above, ie.

[                       debit bank interest income account 5,
                        credit Roger account]

The next month, Roger pays another 10 in capital, and pays 5 interest, and gets the interest back in salary. This continues until the loan is repaid.

The key concept is that in an economy, money is exchanged and individual units of money are constantly circulating through the economy and being re-used, as in the example above. It's a flow system essentially, and this applies to interest payments just as much as payments for food, etc.

Money in a fractional reserve monetary system today also predominantly takes the form of liability bank accounts, rather than physical currency, and most banking operations these days operate as transfers from bank account to account. The writers who concentrate on physical currency in describing this system seem to miss this point, and consequently get key points incorrect.

This doesn't mean that there isn't a very challenging accompanying inflation problem. Generally over time, banking systems tend to create more new loans at a slightly higher rate than they are repaid, and so the system is in a more or less continuous state of growth. There's no reason why it couldn't be regulated to be stable - but the problem is nobody at this time knows what that would do to the economy. The best guess at the moment is that a low but positive rate of expansion is optimal.


How are shell companies used for tax evasion?

Another key feature of those shell companies is that they hide the ultimate beneficiary of the transactions. Banks, insurance companies and most financial services firm must make enquiries as part of the "Know Your Customer" (KYC) regulations: they should be able to find out who will benefit ultimately from the transactions, or in the name of whom they are carried out. THeoretically (ate least in the EU), if they fail to identify an ultimate beneficiary, they should refuse to carry out the transaction AND warn the responsible authority (typically the local equivalent or representative of the FATF - Financial Action Task Force)

But a succession of shell companies in different countries, in particular in countries which apply a strict confidentiality of the investors of a company, make it very hard to be sure that the beneficiary identified is indeed the ultimate beneficiary.

Thus, those shell companies also serve the purpose to hide their owner to their counterparts. This feature is most appreciated for tax evasion and money laundering (obvisouly).


Simple Keynesian model of income determination

I cannot see any way to have negative autonomous consumption. I believe autonomous consumption's extreme is zero if there's a kind of superheroes who don't need water, electricity and food to live and they save everything they earn. But in a case like this, there's  no reason for having aggregate supply at all as nobody will buy the goods in first place.


How to calculate the ideal maintaining time of a machine?

Suppose the probability of an error in the $n$th cycle since the most recent maintenance is
$a+bn$, and let the number of tasks (cycles) between maintenance be $x$. I'm assuming here that the tasks can be treated as discrete.
The way I would approach the optimization is to calculate both the total maintenance cost and the expected total error cost over some fairly large number of cycles - say 1,000.  What number is chosen here is arbitrary, but I find it more intuitive to think of total costs in that context than averages per cycle, although each should lead to the same result.
Total maintenance costs are:
$$\frac{1000}{x}\times 1000=\frac{1000000}{x}$$
Expected error costs between two successive maintenance events, using the triangle number formula to sum the $b$'s, are:
$$\sum_{n=1}^x 15(a+bn)=15\Big(ax+b\Big(\frac{x(x+1)}{2}\Big)\Big)$$
Hence total expected error costs are:
$$\frac{1000}{x}\times 15\Big(ax+b\Big(\frac{x(x+1)}{2}\Big)\Big)=15000\Big(a+\Big(b\frac{x+1}{2}\Big)\Big)$$
Total costs $TC$ (including both maintenance and expected error costs) are therefore:
$$TC = \frac{1000000}{x} + 15000\Big(a+\Big(b\frac{x+1}{2}\Big)\Big)$$
Setting the first derivative equal to zero to find the minimum:
$$\frac{dTC}{dx}=\frac{-1000000}{x^2}+15000\Big(\frac{b}{2}\Big)=0$$
$$-2000000+15000bx^2=0$$
$$-400+3bx^2=0$$
$$x=\sqrt{\frac{400}{3b}}$$
To confirm this is a minimum:
$$\frac{d^2TC}{dx^2}=\frac{2000000}{x^3}>0$$
Putting (as approximately suggested by the chart) $a=0.18, b=0.001$, this yields:
$$x=\sqrt{\frac{400}{0.003}}\approx365$$
Note that in this case $a+bx=0.18+(0.001\times365)=0.545 < 1$.  The formula would need modifying should the values imply a probability of error exceeding 1 before reaching the next maintenance event.
Addendum 3/11/2020
In response to your comment re integration, if the tasks are taken to be discrete then integration does not give an exactly correct formula for the total expected error costs.  The integral needed is:
$$\int_0^x 15(a+bn)dn=15\Big(ax+\frac{bx^2}{2}\Big)$$
This has $x^2$ where the calculation above has $x(x+1)$.  However, this difference vanishes on differentiating TC since:
$$\frac{d(bx)}{dx}=\frac{d(b(x+1))}{dx}=b$$
Thus this approach leads to the same optimum value for $x$.


Explaining the net capital outflow - trade balance identity

I believe this is the intuition:

What do people use to buy foreign goods?

Either they have to offer domestic goods or they have to offer domestic assets.

What do people use to make foreign investments?

Either they have to offer domestic goods or they have to offer domestic assets.

So if a country sells more exports than it buys imports then it must be that it is buying more foreign assets than they are selling domestic assets. Otherwise whey would be exchanging both less domestic assets and less domestic goods for both more foreign assets and more foreign assets. But since that's the entire set of things usable for trade, that's not possible.


A reference for most used utility functions in macroeconomic problems of intertemporal optimization

I think that this article might be helpful:

“Exotic Preferences for Macroeconomists”

http://www.nber.org/chapters/c6672.pdf

They give a thorough explanation of many preference functionals and show their usefulness with macroeconomic applications.

Im not a specialist in macroeconomics, so I dont know the literature that well, but I think that the most used preferences nowadays are the Epstein-Zin preferences. The article I cite discusses thoroughly their properties.

A nice thing this paper has is that it contains detailed comments  from Peter Lars Hansen and Ivan Werning among others at the back.

Hope it helps!


When can I say that a utility function has constant marginal utility?

Marginal utility (of $x$) in your case is $U_x(x,y)=2xy^2$. You use the sign of the derivative of MU, namely $U_{xx}$, to tell whether MU is increasing, constant, or decreasing.

Specifically, you have

increasing MU if $U_{xx}>0$,
constant MU if $U_{xx}=0$, and
decreasing MU if $U_{xx}<0$.

In your case, assuming that $y>0$, you'd have $U_{xx}>0$, hence increasing MU.


Using the income approach, GDP is the sum of compensation of employees, gross operating surplus, gross mixed income, and taxes less subsidies on production and imports. Gross operating surplus includes not only corporate profits but also increases in inventory valuation.

In the above example, using the income approach, the 15 sold cars will reflect under corporate profits and the value of the 5 unsold cars will reflect in an increased inventory valuation. Using the expenditure approach, the 15 sold cars will reflect under consumption expenditures and the 5 unsold cars will reflect changes in inventory valuation under investment. In practice, you cannot determine the value of a car until it is sold, so most accounting standards will conservatively use the cost to produce the car for inventory valuation. Since the cost of production in the example is unrealistically zero, the change in inventory valuation would also be zero in both approaches. Note that even if a less conservative accounting standard was used, both approaches would lead to the same result as it would affect the valuation of the 5 unsold cars in the same way.


Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

As you say (how did $\sin t$ in the initial problem become $\cos t$?), $\mathbb{Q}$ is a measure under which $W_t$ becomes $\tilde{W}_t - \sin t$, where $\tilde{W}_t$ is a $\mathbb{Q}$-Brownian motion. So Girsanov's theorem would say

$$
L_t = E[\frac{ d \mathbb{Q} }{ d \mathbb{P} }|\mathcal{F}_t] = e^{ - \int_0^t \sin s dW_s - \frac{1}{2} \int_0^t \sin^2 s ds}
$$

is one such measure.

As for uniqueness, if we take $\Omega$ to be the canonical path space $C[0, T]$ with $\mathcal{F}$ generated by the cylinder sets. Then the Wiener measure would be unique, simply by the fact that the Brownian law says any two such measures must agree on the cylinder sets and therefore all of $\mathcal{F}$.

In general, the Girsanov $\mathbb{Q}$ is unique up to the Brownian filtration $\mathcal{F}_{W} \subset \mathcal{F}$, since it is constructed $W$-pathwise.


Does the Labor Theory of Value hold in the long term in competitive markets?

No it does not for several reasons. Labor theory of value (LTV) central tenant is that there is some intrinsic component to value determined objectively by labor  (see Brue and Grant History of Economic Thought  7ed). This implies that there exist some unit of labor in which every value could be measured. In addition, it means that the value ($V$) is some sort of function labor $L$ i.e. $V=f(L)$. As stated by Adam Smith in Wealth of Nations who first developed the LTV (and as mentioned in the comments as well):

"The value of any commodity ... is equal to the quantity of labour which it enables him to purchase or command. Labour, therefore, is the real measure of the exchangeable value of all commodities.

Hence, at the core of labor theory of value is proposition that all value can be somehow objectively measured by labor, and that it should depend on labor.
Well there are several problems with that proposition:

What about things that require great labor inputs but are worthless (which is situation that can and empirically does arise even in competitive markets as preferences and tastes change and something that was once desired might be trash today)?

The LTV actually has no good answer for that. Adam Smith tried to
answer this objection by distinguishing between value in use and
value in exchange. As Smith further writes in his Wealth of Nations:

The word value, it is to be observed, has two different meanings, and sometimes expresses the utility of some particular object, and sometimes the power of purchasing other goods which the possession of that object conveys. The one may be called "value in use"; the other, "value in exchange." The things which have the greatest value in use have frequently little or no value in exchange; and, on the contrary, those which have the greatest value in exchange have frequently little or no value in use. Nothing is more useful than water: but it will purchase scarce anything; scarce anything can be had in exchange for it. A diamond, on the contrary, has scarce any value in use; but a very great quantity of other goods may frequently be had in exchange for it.

This is basically the argument that Marx repeats (although he had different name for it), as mentioned above (indeed I am always surprised that Marx is often quoted regarding LTV given he did not do virtually any original work on the LTV and just took it from Ricardo and Smith).
The problem with this is that it is already quite scientifically undesirable as it is essentially a cop-out. Labor is the source of value, well, except in cases where it isn't because the object is not useful. So already LTV from the get go admits it is not a theory that can explain all value, which is scientifically undesirable as we want to generalize as much as possible. Now, if we would not have alternative theories of value this could be forgiven but given we do have them and they can explain more this in itself is argument for ditching LTV (e.g. we can safely ditch Newtons theory of gravity in favor of relativity as the latter is more general, but even though theory of relativity can't explain everything we cannot ditch because there is no other competing theory that can).
With regards to your main question, already this point 1 proves that in competitive markets LTV does not need to hold as it is completely valid outcome under perfect competition that value of produced good will be 0 (or indeed even negative like with oil futures).

How to measure value in labor (the problem of transformation)?

One of the central tenets of labor theory of value as mentioned
above is that labor must be some sort of real measure of value of
all commodities. However, due to the problem mentioned in point 1 it is also necessary to argue that value of labor will not correspond exactly to the  price.
Hence, this gives raise to the question how to actually establish some correspondence between prices and labor which is also known as the problem of transformation (see Samuelson; 1971). This was in fact a problem that Ricardo devoted considerable amount of time to solve, although he was ultimately unsuccessful.
Samuelson argues there is no general solution to the problem, and although there are some scholars claiming to solve it  there are no generally accepted solutions.
In regards to your question, assuming competitive market does not provide solution to the above problem and without the above being possible it is very hard to sustain claim that value can be somehow objectively measured by labor.

Empirically value seems to be orthogonal on labor inputs and this hold regardless of how competitive industry is.

There simply does not seem to be any relationship between value and labor inputs. Relative prices fluctuate widely even when labor inputs stay same. This can occur both in theoretical models of perfect competition (e.g. consider two perfectly sectors two goods model with some idiosyncratic demand shocks), and frequently occurs in reality.
Now even if there would be a discrepancy between price and the value of labor that went into producing the product, if there is any objective value determined by labor then you would at the very least expect relative prices to depend on relative labor inputs but they do not and there is no indication that they start doing so in competitive industries. Hence $V \perp L$.
This also connects to the previous point, but for LTV to hold it is simply not enough to just show that something that was produced with labor is happens to have value - the value of it should depend on labor input and ultimately be measurable.  What even more it should be the  measuring stick of the value. But this is clearly inconsistent with observation that relative prices change and fluctuate widely even when labor inputs do not.


Will the services rendered by a worker from an unorganized sector be counted in GDP?

If by ‘un-organized’ sector you mean gray and black market then these activities are not directly recorded in GDP. For example, if you pay 10e to a neighborhood kid to mow your lawn it won’t show on GDP as a spending. Although, you could say those activities still indirectly can increase GDP, as if then that kid goes and buys a lego at a store that would otherwise not be bought that purchase would raise GDP. But I don’t know what exactly you mean by unorganized if. If the spending/incomes in a sector are tracked by statistical office it will show up on GDP.


Do minimum wage laws bring more bad than good?

There is ample evidence that increases in the minimum wage in the magnitudes that we've seen in the last 20-30 years have caused no or very small decreases of low-skilled employment. In that sense, the predictions of competitive labor market models like the one Friedman probably had in mind are clearly wrong.

A classic reference here is Card and Kruger 1993 (http://davidcard.berkeley.edu/papers/njmin-aer.pdf). A more recent contribution is Allegretto et. al. (http://ftp.iza.org/dp7638.pdf). There are many more.

Some recent minimum wage increases have been much bigger than the ones discussed in these papers (for example the 15$ minimum wage in California) and the effects of these increases may be more substantial.

The argument for minimum wages is mostly about redistribution, and modest minimum wages probably achieve this goal.


Why are things usually cheaper in absolute terms the poorer a country is?

It does seem kind of puzzling. However, there is one simple explanation. It´s a theory of why this happens, called the Balassa-Samuelson theory.

It takes about three steps to see it:

The idea is, first, that the productivity in the case of tradable goods is different in the two countries. It takes 100 hours of work in Switzerland to produce a computer. It takes 1000 hours of work in Venezuela to produce a computer. (This is easy to believe, you could think this is the result of Switzerland having more capital, infrastructure, etc.)  Because these computers are tradable, they should have the same price everywhere, once expressed in dollars. So suppose the computer is worth 1000.
The second step is to realize what this means for wages. It means that in Switzerland, people that make computers are paid 1000/100=10 dollars an hour. (Assume the workers get paid the whole price of the computer...), and in Venezuela they get paid 1000/1000 = 1 dollar per hour (!).

The third step is to realize a) that there are many goods that are non-tradable, like restaurant meals. They are usually services. These goods don't have to have the same price everywhere; and b) that there is not such as big a difference in the productivity of these goods across countries.  So it takes one hour of work to make a nice meal in Switzerland and the same in Venezuela. (One way to rationalize this is to see that if you want to invest a lot, you want to invest in producing something that can be sold worldwide, a tradable good. Therefore, rich countries have invested in increasing the productivity of tradable goods.) But now you have the result, because the price of a meal is 10 in Switzerland (1 hour of work) and 1 in Venezuela (1 hour of work) (!!).

In sum, what the Big Mac index reflects is that the price of non-tradeables is different in a way that reflects the productivity differences in tradable goods.

Magic!


The best resource I've found so far is Pitchbook's quarterly summary. It includes some degree of historical data, historical data by funding stage, and historical data by sector.


Firstly, the formula you have used for FV is equivalent to that shown here. What it calculates, with your numbers, is the value after 7 years of investing 200,000 annually at a compound rate of 7% (as a rough confirmation, 1,851,960 is somewhat more than 7 x 200,000).  Instead, you need the value after 7 years (or 6 years if the man is already at the end of his 24th year) of a one-off investment of 200,000 (call that V).

Secondly, the simple division by 20 takes no account of the interest earned during years 31-50. To allow for this, you need the present value of an annuity formula (source):

$$P\Bigg[\frac{1-(1+i)^{-n}}{i}\Bigg]$$

Set this formula equal to V and treat the annual (periodic) payment P as an unknown to be found.