Is there a way to link Berge's theorem of maximum to Envelope theorem?

They are related and usually fall into the same discussion, but as @Alecos mentions in the comments, the two theorems show different things.

I suppose the connection that you're after is the fact that if
the derivative
$$
\left .\frac{\partial f(x, a)}{\partial a} \right |_{x=x(a)}
$$
exists, then because differentiability implies continuity, you might be able
to get part of theorem of the maximum out of it. However, to compare and contrast two theorems you mustn't just look at the results. You need to look at the assumptions as well. For example, the theorem of the maximum doesn't assume any sort of differentiability. The envelope theorem does (at least some forms of it). In any case, the assumptions that go into each are different (some stronger, some weaker).

Also, there is this. The envelope theorem
doesn't tell you that anything about the control function. Therefore, you definitely wont be able to get the result that $C^*$ is upper hemicontinuous.


If a central bank stopped printing money then deflation would eventually occur meaning the value of $1 would increase over time.

There is a simple intuition as to why this would happen. Imagine there is an economy where the only product either produced or sold is widgets. Lets assume that the central bank fixes the total amount of currency in the economy at $1,000. Lets also assume to keep the math simple that every widget produced every year is bought and consumed and that people in this economy don't ever save money and spend 100% of their income every year on widgets (this is a dubious assumption in real life but makes the math easy, it is a fair assumption in real life that citizens will spend a certain % of their income on goods and services every year that are consumable).

If the economy produced 100 widgets per year based on its level of production and those widgets were bought using the available currency then each widget could sell for $100.

Lets now say that due to population growth and increases in technology the economy can now produce 200 widgets. There is only $1,000 to buy the 200 widgets so each widget now sells for $50.

The price level has halved. In order to buy the same product (1 widget) you have to spend half as much money. If the central bank wished to keep price levels the same they would have to increase the money supply to $2,000 to keep up with he increase in productivity. This is why you expect a central bank to increase the money supply over time.


I need something like “how to calculate equilibrium price for complete idiots”

First, the equilibrium price is, at a very basic level, just the market price for livestock. If you have a perfectly competitive market (a lot of suppliers) and all livestock is about the same, then the market price should be about the best your grandfather can do.

That being said, there are lots of reasons why perfect competition breaks down. This can come from number of sellers (just a few for a buyer to pick from), unobserved product heterogeneity (some livestock is better than others, but can't always see it just by looking), and more. Preferences over different types of products also interact with these other issues (if there's only one person selling and he knows how much a buyer is willing to pay, he can charge exactly that amount, etc.).

This is where the profit-maximizing price issue comes in to play. Your grandfather wants to charge the price that maximizes his profit

$\pi = p*q(p) - c(q)$

with $q$ the demand function (how much he can sell at price $p$) and $c$ the cost function (how much it costs, including what is being given up, to "produce" -- whatever that means in this market -- $q$ units of the good). (Quick note: I'm abusing notation a little in the profit function, but I'm doing it to make the point). The profit maximizing price is just the price level at which the first order condition of $\pi$ is equal to zero (assuming $\pi$ satisfies some regularity conditions). This will end up in the solution that he should produce/charge such that the marginal revenue is equal to marginal cost.

Taking advantage of this in practice requires having a really good measure of both $q$ and $c$. Although difficult for economists, $c$ is usually straight forward for firms themselves to get because they can actually calculate their own costs. The difficult part for everyone is estimating $q$. It takes a lot of data and a really good model of the market, taking into account all the issues previously discussed.

This answer discusses how this might be done by really big firms. Note that one of the main methods they use is experimentation. Large firms can do this easily and often. For smaller firms -- like your grandfather's, I'd expect -- some of the best data your grandfather might have is, like @denesp pointed out in a comment, his experience. If he's been at it for a while, the data that he has gathered over time selling in this market has likely given him a pretty good feel for what he can charge for what and to whom.

To give an example of this type of experience in practice, an economist friend of mine as an undergrad used his father's hardware business as the motivation behind a paper he was writing. He decided that he would use the store's data to estimate preferences and improve profits. He ended up pitting his estimates against his father's experience-based pricing and, try as he might, could not outperform his father's method.

This is not to say that you can't use models and estimation to improve profits. It does indicate, however, that this is not a simple problem and can require a lot of pretty sophisticated methods to outperform the natural market evolution that would be in many ways already internalized by your grandfather.


Why are my 2SLS estimates much larger than my OLS estimates?

Let me take an example based on the estimation of returns to education, which has been a well-studied problem. The usual result is that researchers find the 2SLS estimate to be larger than the OLS estimate by approximately 25%-50%, e.g. Card (1999, 2001).

Three reasons :

An omitted variable that could be negatively correlated with the amount of education. This omitted variable would lead to an downward bias in the OLS estimate of the schooling coefficient.
Errors in variables that arise because years of schooling are a noisy measure of education that leads to higher earnings. This measurement error in education biases the OLS estimate of the treatment effect toward zero. OLS estimates are thus too small. Since the IV estimate is unaffected by the measurement error, they tend to be larger than the OLS estimates.
It's possible that the IV estimate to be larger than the OLS estimate because IV is estimating the local average treatment effect (ATE). OLS is estimating the ATE over the entire population. By contrast, IV is estimating the local ATE: the instrument shifts the behavior of a subgroup of individuals for whom the returns to education are larger than average. In other words, the IV estimate is the effect of increasing education only for the population whose choice of the treatment was affected by the instrument, while the OLS estimate describes the average difference in earnings for those whose education differs by one year. Then IV estimates will be larger than OLS estimates because of heterogeneity in the studied population.

References

Card, D. (1999), The causal effect of education on earnings. In: Ashenfelter, O. C., and D. Card (eds). Handbook of Labor Economics Vol. 3A. Amsterdam: Elsevier, 1801–1863.
Card, D. (2001), Estimating the Return to Schooling, Econometrica, 1127-1152.


What has caused the recent 25% unemployment rate in Spain?

Here's an explanation from Paul Krugman. You can read more about this in Krugman's book End This Depression Now!.

Since joining the Euro, Spain has experience large capital inflows—money flowing into Spain, mostly from Northern Europe. These inflows caused a boom in investment, coupled with an increase in prices of virtually everything (including labor) relative to other Eurozone countries.

One consequence of this is that the recession in Spain has been exacerbated by the fact that high costs of production (especially high labor costs) made the Spanish economy less competitive so that the country wasn't able to rely on exports to substitute for the reduced domestic demand.

In order to remedy this situation, Spain needs to become more competitive (i.e. for labor costs to fall relative to those in other countries). Normally, this would happen automatically: as a country exports less, demand for its currency from importers (and hence the currency's value) falls so that its products become cheaper for foreigners. However, the fact that Spain is in the Eurozone means that it can't devalue its currency—it doesn't have one of its own!

Instead, Spain must rely on 'internal devaluation', i.e. reducing the wages of its workers relative to those elsewhere in the Eurozone. This is problematic because workers are typically reluctant to accept a pay cut (so-called downward nominal rigidities). Thus, the way that the economy adjusts is to have a sufficiently large share of the workforce unemployed that people are prepared to accept jobs on significantly lower wage than they might have expected in the pre-recession years.

It should be noted that this line of reasoning is not without controversy. For one, a significant share of macroeconomists do not believe the nominal rigidities story.


Does the Marshallian demand function always include prices and income?

These are proper Marshallian demand functions, even though Income does not appear in them. This is due to specific form of the utility function (and the candidate solution of all goods being purchased at strictly positive quantities). It emerges that there is no income effect for goods $x_1$ and $x_2$ - optimal uncompensated demand does not depend after all on the level of income, but only on the relative prices.

The reason why Marshallian demand is defined as it is, is to make clear that it does not include any kind of "income compensation" as Hicksian demand does. But this does not preclude a case like yours, which again, depends on the form of the utility function.

Thinking economically, what goods can you think of whose demand may realistically not depend on the level of income but only on relative prices? Answering this question would be useful in order to map the mathematical expression of the utility function to real-world economic phenomena.

Also, note that the specific utility function points to "quasi-linear" frameworks, where the good that enters additively, essentially functions as Income itself.


Textbooks claim that the difference between GDP and GNP (or GNI) is about geography vs citizenship--is this correct?

Residency is a complex definition. For example, the Council for Science and Technology of my country supports many graduate students abroad every year. Those guys stay abroad for periods of 1-5 years, yet statistics compilators still count them as residents of the National economy. Why? Because their economic interests are based at home, not the other country.
But there's more to statistics compilation than just that. Typically, the Council for Sci&Tech will provide the statistics office with data on those scholarships. But what do they do with all the folks coming and going in the corporate world? They won't report that, at least not voluntarily. So there are a few rules of thumb that the guys doing the compilation will follow:

Companies working on defined projects abroad are still residents of their home economy.
For instance, a construction company from Mexico building a bridge in Guatemala, a project that will take them 3 years, will be recorded as a resident of the Mexican economy;
Employees working for 1 year or less in another country will still be considered residents of their normal country of residence;
People whose residence moved (declared intention of moving and remaining in the new country) will now be considered residents of their new country.
Companies establishing a branch abroad will be recorded as Direct Investors, and the new branch, a resident of that economy where they are located.

Beware: this may or may not coincide with the definitions by fiscal authorities. This is strictly from a statistics compilation viewpoint. Also, these are rather guidelines, but the more information a statistical office has, the better the compilation. This is why offices of different countries should seek to cooperate with others.


There is the market segmentation theory.

The demand on the longer end of the yield curve in theory comes from institutional investors, pension funds, insurance companies, foreign central banks etc. who prefer longer durations.


Is the global Automobile Industry changing from an oligopoly to monopolistic competition?

The automotive industry is being disrupted by new trends such as electric vehicles and self-driving cars. These, still, represent a minor part of the market.

So the answer would be no, for the moment besides media attention, there weren't gigantic shifts in the industry as to have monopolies. Looking at the last years data it seems to remain an oligopoly with 4 very large manufacturers.

If you need further info this is usually not free - look for the automotive industry in business databases such as IBIS world, or get a report from a consulting firm such as this one.


Producers of corn may not be able to afford a decrease in prices.

Remember that farmers are (usually) not speculating and not “investing”. They are producing a product that they need to offload in order to make money. They have zero interest in uncertainty (again, assuming that most are not speculating). The future price they can receive today may be better than the spot price in 6 months, or it may be worse - you really have no idea, but at least you can lock it in, so that you know what to plan for.

Remember that farmers have two uncertainties about their product when laying the first seed: 1) quantity and 2) price. By hedging the price on some quantify you can reduce your risk significantly (and hence also your upside) which may be far more appealing then the risk of losing everything.


Deriving a demand curve from a Cobb-Douglas utility

If you take the general class of CES utility functions, of which Cobb-Douglas is a special case, you do indeed get a demand function that depends on other prices. Specifically, the CES utility function (over $n$ goods, $x_1,\dots,x_n$) takes the form
\begin{equation}
  u(x_1,\dots,x_n)=\bigl[\alpha_1x_1^\rho+\cdots+\alpha_nx_n^\rho\bigr]^{1/\rho},
\end{equation}
where $\rho\in(-\infty,1]\setminus\{0\}$, $\alpha_i\in[0,1]$ and $\sum_i\alpha_i=1$. We interpret $\alpha_i$ as the consumption share of good $i$ and $\sigma\equiv\frac{1}{1-\rho}$ as the constant elasticity of substitution. Note also that when $\sigma=1$ (or $\rho\to0$), we get the Cobb-Douglas utility form.

Solving utility maximization subject to the usual budget constraint, we get the demand for good $i$ as
\begin{equation}
  x_i(p_1,\dots,p_n,M)=\frac{M(\alpha_i/p_i)^\sigma}{\sum_{j=1}^n\alpha_j^\sigma p_j^{1-\sigma}},\quad i=1,\dots,n.
\end{equation}
Again, observe that when $\sigma=1$ we get the demand associated with Cobb-Douglas utility.

The elasticity of substitution governs how relative expenditures on different goods change as relative prices change. Take a two-good example. An increase in the relative price $p_1/p_2$, i.e. good 1 becoming relatively more expensive, causes two effects simultaneously:

per unit expenditure on good 1 rises, as good 1 now costs more in relative terms; and
quantity demanded for good 1 decreases due to law of demand.

These are opposing effects on the expenditure on good 1 relative to that on good 2. It turns out that the elasticity of substitution determines which effect dominates. If $\sigma>1$, the second effect dominates, and if $\sigma<1$, the first effect dominates. When $\sigma=1$, which is the case of Cobb-Douglas, the two effects cancel each other exactly, so that relative expenditure is independent of the relative prices, and depends only on preference parameters (the $\alpha_i$'s).

Of course, CES utility is not the only class of utility functions that generate demands dependent on prices of other goods. Another common form of utility function, the quasi-linear utility function,
\begin{equation}
u(x_1,\dots,x_n)=x_1+v(x_2,\dots,x_n),
\end{equation}
where $v(\cdot)$ is strictly increasing and strictly concave, also generates demand functions that depend on other prices. A common example is $u(x_1,x_2)=x_1+2\sqrt{x_2}$. I trust that you can verify that both $x_1$ and $x_2$ depend on the two prices $p_1$ and $p_2$.


How does raising interest rates calm inflation?

I will give you some simplified thoughts in three brief sections:

The traditional view is that higher rates increase the cost of investment and increase the opportunity cost of spending (rather than saving). This places downward pressure on prices. Another way to think about this (opportunity cost part) is considering that higher rates reduce liquidity, which in turn reduces the demand for goods and services.

Asset prices typically move in the opposite direction of interest rates. So, raising rates to cool inflation would actually, according to the common understanding, place downward pressure on asset prices. However, if one looks into the data, one can see that there are co-movements in asset prices and rates. These co-movements are the basis of Karadi's newest paper that is currently R&R'd. However, Gali does have a paper suggesting a link between rates and asset bubbles wherein increasing interest rates can impact the short-run behavior of rational bubbles.

Higher rates can encourage capital inflows. This can cause currency appreciation. This generates its own set of issues.


What was the Summers memo of 1991 sarcastic about?

The context of this memo was an emerging debate about structural adjustment and its environmental implications. The World Bank has attached a wide range of liberalization policies to many of its loans. These policies were meant to increase trade and foreign investment and to decrease fiscal deficits, all of which should help countries to pay back their loans. Critics argue that imposing these forms of liberalization on LDCs forces them to prioritize repayment over other policy goals, including environmental protection.

Here is a 2003 review on environmental impacts of structural adjustment by Bank researchers which briefly mentions and dismisses the "pollution haven... hypothesis that 'dirty' industries migrate to low-income countries after trade liberalization because of differences in costs of pollution abatement and looser environmental standards (p. 27)."


Is it viable to put a cap on personal wealth in the U.S.?

Wealth cap that would confiscate and redistribute wealth above certain limit is in an essence just different term for wealth tax (as opposed to situations where there would be just a ban on having more wealth which would be proper cap). Consequently, to see whether such policy is optimal one has to look in literature on wealth taxes. A wealth cap that takes and redistributes wealth tantamount to 100% marginal tax rate on wealth above the 'cap' set by the policy. This is definitely not optimal (in fact one of this is the first rule that you will learn in class on optimal taxation).
First, generally speaking any top marginal tax of 100% and above is suboptimal because it removes any incentive for further economic activity, be it working when talking about income taxes or accumulating wealth. This is undesirable if the objective is to maximize the tax revenue, for lets say redistribution, because if the tax would be even just 99.9% the individual being taxed would still have incentive to be economically active and have incentive to create more wealth that then can be taxed.
Second, the literature shows that optimal top marginal wealth taxes, even in case of Rawlsian social utility function (the most redistributive social function there is), are somewhere in range 0-10% (see Kocherlakota 2005; Fama 2019; Zucman and Saez 2019) or discussion in the famous Mirrlees Review: Dimensions of Tax Design).  Also, note the consensus seems to still be closer to the zero result and only very few scholars would go as high as 10%.
The reason why the optimal wealth taxes are so low even with the most redistributive social preferences is that they create very large behavioral responses that distort economy (e.g. see Jakobsen, Jakobsen, Kleven, & Zucman, 2020), they are very hard to enforce (e.g. large portion of wealth is difficult to price) and they are to a large extent implicit taxes on capital incomes (since company valuation counts as a wealth so they are further implicit taxes on wealth invested in business) and per famous Chamley-Judd result (Chamley, 1986; Judd, 1985) the burden of capital income tax is in long run shifted to labor (although the results does not hold perfectly in practice).
Such high wealth taxes could be justified only if the objective is not maximizing welfare of the poor  (e.g. for example with Ralwsin max-min principle) through redistribution, but just lowering wealth inequality no matter the cost, but I don't think there are many if any economists that would endorse such course of action.
In addition, the literature cited above might be too dense for non-economist, especially if you don't have solid math background. An accessible book discussing inequality and taxation is Inequality by Atkinson. Most of the book does not focus on wealth taxes since many economists consider zero wealth taxes to be optimal, but they are mentioned in some chapters.


Have there been instances where economists have advocated a trade embargo?

Free trade is, on the whole, one of the few otherwise controversial policy topics on which economists have near-perfect consensus. Historically, this consensus has long been strong in the English tradition (Hume, Smith, Ricardo, Mill), albeit less strong elsewhere. Famously, 1028 American economists signed an unsuccessful petition in 1930 begging Herbert Hoover not to approve the Smoot-Hawley tariff. If the IGM Economic Experts panel is any guide, consensus remains firm today.

That said, off the top of my head, various cases in which some modern economists have departed from advising free trade include:

Import substitution and related protectionist development philosophies in the early postwar era. These were never (as far as I know) advocated by many economists in the neoclassical tradition, but they certainly had support among influential other figures, notably Raúl Prebisch; enough that they were put into action in many parts of the developing world, especially Latin America. The mainstream verdict on import substitution is that it was a costly failure, although there are notable heterodox dissenters like Ha-Joon Chang. Dani Rodrik also has a slightly less heterodox record of free trade skepticism.
Cases of market power. Here, economists do not necessarily advocate departing from free trade in practice; but they do recognize that (in principle) it can be individually optimal for countries with either some monopoly or monopsony power to try to manipulate the terms of trade in their favor via trade restrictions (potentially either import or export tariffs). Many large, developed countries probably have some market power of this kind, and specialized commodity suppliers do as well. This observation is the basis of some economic theories of trade agreements, which are modeled as devices for countries to coordinate on a Pareto optimal free trade regime and overcome their individual desire to manipulate the terms of trade.
Aggregate demand management when monetary tools are limited (due to the zero lower bound). Paul Krugman has discussed how protectionist clauses in stimulus plans could in principle be globally optimal, by allowing countries to retain more of the benefits of their own stimulus and thereby encouraging them to do more. Another case is where countries facing the zero lower bound could impose tariffs on unconstrained countries, in an attempt to redirect expenditure to demand-constrained economies. That said, Krugman still (mostly) favors free trade and doubts that the benefits of such a policy would overcome the cost to the global free trade regime.
Distributional consequences. This is a common refrain in popular critiques of free trade: many pundits argue that even if trade is beneficial in some aggregate sense, its adverse distributional impact (e.g. hurting the already-suffering manufacturing workforce) negates the overall benefit. Indeed, in models where gains from trade arise from differing factor endowments, the whole point is that some factor (the domestically scarce but internationally not-so-scarce one) will lose; this is the idea behind the Stolper-Samuelson theorem.

Traditionally, most economists have argued that it is better to have free trade and address any distributional or insurance objectives through the overall tax-and-transfer system. Whether not this conclusion holds in a formal model, however, depends on exactly what instruments are available to the government; it's conceivable that trade barriers would be an optimal second or third-best policy in some cases. The left-wing heterodox economist Dean Baker has strenuously argued along these lines (though he has certainly not offered a formal model). More in the mainstream, an early version of Autor, Dorn, Hanson (AER 2013) made a suggestive stab in this direction with a back-of-the-envelope calculation showing that the deadweight loss from transfers induced by Chinese trade was a substantial fraction of the theoretical gains from trade - though this calculation was rough and evidently removed from the published version. Notably, Autor was one of the few IGM panelists with an "uncertain" reply about the benefits from trade.


How can Aggregate Supply = National Income

Aggregate supply is the total supply of goods and services in an economy given the price level, over a period of time. So, according to this your first statement is correct.

The second statement emphasized on the potential output when all factors of production are fully utilized. It is a long term statement because only in the long run does an economy fully utilizes all the resources.

Here, second is a particular case of the general aggregate supply function definition.


What happens with corporate deposits when a bank goes bankrupt?

In simplified terms a bank or shadow bank (non-bank financial intermediary operating a balance sheet similar to a bank balance sheet) holds a mix of cash, marketable securities, and investment portfolio on the asset side of the balance sheet. The bank or FI issues a mix of equity, insured liabilities, and uninsured liabilities on the liabilities and equity side of the balance sheet. If the bank or FI is forced to write-down the value of assets in the investment portfolio then the creditors holding equity are forced to take the first loss and the creditors holding uninsured liabilities take the remainder of the total loss. Sometimes uninsured liabilities convert to equity in a new venture carrying the good assets or there is a bailout via government recapitalization effort in a systemic crisis.
These are FAQs describing the resolution of IndyMac Bank via the FDIC:
https://www.fdic.gov/resources/resolutions/bank-failures/failed-bank-list/indymac-q-and-a.html#Uninsured

If it is determined that you have some uninsured funds, the FDIC will mail you a Receiver Certificate.  This certificate entitles you to share proportionately in any funds recovered through the sale of the assets of IndyMac Bank.  You will eventually recover some of your uninsured funds.  Of course, you will receive immediate full payment for your insured amount by transfer to IndyMac Federal Bank or from your broker.  In addition, the FDIC will pay uninsured depositors an advance dividend of 50% of your uninsured deposit.

Large cash investors seek secure liquid investments and cannot rely only on insured deposits for this purpose. See Institutional Cash Pools and the Triffin Dilemma of the U.S. Banking System by Zoltan Pozsar (2011):
https://www.imf.org/external/pubs/ft/wp/2011/wp11190.pdf

Institutional cash pools prefer to avoid too much unsecured exposure to banks even through insured deposits. Short-term government guaranteed securities are the next best choice, but their supply is insufficient. The shadow banking system arose to fill this vacuum. One way to manage the size of the shadow banking system is by adopting the supply management of Treasury bills as a macroprudential tool.

Warren Buffet testified to the Financial Crisis Inquiry Commission (FCIC). There is an audio copy online. As I recall he describes having to make an $8 billion payment in the fall of 2008 during the financial crisis. He said Treasuries were the best option at the time to ensure the cash would be available to make the contractual payment obligation.


Meaning of $dF(z)$ in expected utility framework

Not all cdf’s have a density function, (for example if $F$ is not differentiable). However, when they do have a density, the notation $dF(z)$ is equivalent to $f(z)dz$. When performing integrals. However, even if the density does not exists, you can still write the expectation using the notation $dF(z)$. The details of what it actually means and the subtle differences can be learned in any measure theory course. But hopefully this short explanation helps you not freak out every time you see that notation.


Yes, there's daily data available on the St. Louis Fed's FRED.


How the Federal Reserve Manages Money Supply?

Federal reserve manages money supply in various different ways (see Blanchard et al Macroeconomics ch4). The main ones are:

By managing interest rates on a reserves: lower interest rate reserves allow banks to lend more and expand money supply and vice versa.
By open market operations: where purchase Fed purchases debt in exchange for new money (which eventually increases reserves as that money finds it way to banks). Issuing new debt increases money supply and repayment of the debt decreases it.
By changing reserve requirement. Currently they are abolished by Fed, but nothing prevents Fed from bringing them back if they desire so. Here higher reserve requirement contracts money supply and lower one expands it.
Other: Fed can also affect money supply through other means such as banking regulation, although that is too broad of a topic to discuss in detail in one SE post. You can find some explanation of that in McLeay et al (2014) and sources cited therein.

But when the issuer pays off the bond, wouldn't that reduce bank reserves, thus reducing lending, thus reducing the money supply?

When the bond is payed off it does lead to reduction in money supply unless the money are lent again. However, money supply is reduced just to the original point. If Fed injected 100 USD to the economy through OMOs and if through fractional reserve banking system that injection multiplied up to 1000 USD then repaying that debt only destroys the 1000 USD not any additional money (e.g. if M before injection was 1 billion USD after repayment it will again be just 1 billion USD).

It does not reduce money supply further than it was before it was expanded. The interest rate is paid to the bank. In case of commercial banks they keep the money as a profit (after paying for all costs of course) so it will still be circulating in the economy. In case of US government paying interest to Fed, again money supply is not additionally destroyed because Fed is government institution and it sends all profits from interest back to the treasury (e.g. see WSJ article about profits they send back in 2020) so the money used to pay interest stays circulating in the economy.


In the UK, government spending is 38.5 per cent of GDP.  To oversimplify, that means that for every £1 spent on private goods, 63p needs to be raised in taxes.  At present, the UK raises a little over a quarter of that from income tax.  One of the reasons income tax is a significant component is that it is progressive, unlike other major sources such as VAT and excise duties.

It is also worth mentioning that the categories of government expenditure you mention are relatively small.  Half of all government spending goes on pensions, healthcare and education.  The categories you mention are marginal by comparison.


A country borrowing from the rest of the world

The idea is that the world economy is closed and value only comes from production. It is maybe best to forget about money for a moment here. Since money derives its value directly from production, anything we say about production applies to money as well.
Since the world economy is closed, a country that is producing less than it consumes must "borrow goods" from other countries to consume (otherwise it cannot consume more than it produces by definition). We can't have the whole world consuming more than it produces, so someone else must be consuming less, i.e. "lends". This is the case of a trade deficit, i.e. positive net imports. If a country "borrow goods" it effectively also borrows money.
A currency's value (purchasing power) is proportionate to the number of goods in that economy, since all money must be able to buy all goods by definition and legally. (If only 1 loaf of bread and only 1 dollar exist, the dollar has a purchasing power of 1. If (only) 2 dollars exist then each dollar has a purchasing power of 0.5).
How this may work is that the exporting country lends money to the importing country in order to be able to buy more than it sells. In a barter economy this happens directly without money, the extra goods given are loans (as goods in equal value have not been receieved). Since money is not "real" in an economic sense, the same must happen with money. This happens, since each currency's value is related to production and the importing country is producing less (as it doesn't have enough to spare), while the exporting country is producing more (it has more than enough), therefore the respective currency values reflect this.
Very often an importing company will take out a credit from the exporting company and this reflects itself at the country level. Even if money isn't actually lent, it happens through adjustment in exchange rates and interest rates.
Since money's value depends on production value, this is also the  reason the current acount + the capital account must be equal to 0. For any goods bought or sold, money must be paid. Its like saying the value of goods you receive at the store must equal the value of money you give to the cashier. If you get more goods than you have money for you'll simply owe the cashier. Internationally (but alson domestically) you only have money in value of what you produce. Therefore if you produce less than you consume, you don't have enough money (production=money) to pay the cashier for the goods you consume.
Further, the current account is constructed similarly as an account from accounting. It sums up to 0 by (that) construction. Goods leave, money enters, they must have the same value. For more on this you can also consult any introductory accounting book.


This is a very poorly written question.

The gist of the question is this: should a company take a lump-sum payment from a customer of \$8,525 or a \$12k payment that is spread uniformly over 12 years. The thing the question wants you to consider between 1a and 1b is that in 1a the company cannot earn interest (which the answer just assumes is compounding) and in 1b the company can earn 7% interest.

For the given answer, part 1a is saying that, since the company cannot earn interest on investments, it is better to just take the \$12k. However, one ought to know something about how this company discounts to actually answer this question. That is, one should make this comparison by comparing \$8,525 to the discounted present value of the \$12k, taking into account that the \$12k is received in payments spread uniformly over 12 years. The real answer to this question as written is, ' IT DEPENDS'.

Part 1b is saying that $8,525(1+.07)^{12} \equiv 19,199.93 > 12,000$ and so the company should take the lump-sum payment and invest it. HOWEVER, this is again the wrong way to answer this question. Instead, one should compare
$8,525(1+.07)^{12}$ with the money the firm will have if it chooses the \$12k and subsequently invests each annual payment. This value is \$19,140.64 (link to calculate). The answer here is correct but for the wrong reason.

Really, this is a very bad question and a very bad answer.


As a result we have $$ z\{p(t)\}\beta(t) = F\{K(t),L(t)\}. $$ Solving
  for the output price we obtain $$ p(t) = p[F\{K(t), L(t)\}/\beta(t)]$$

I am pretty sure that what is meant here is that there is a $p(t)$ trajectory for which
$$
z\{p(t)\} = F\{K(t),L(t)\} / \beta(t)
$$
(see first equation)

Given the function $z$, this depends only on the right hand side of the equation, hence let us denote the trajectory $p(t)$ as a function of this, i.e.
$$ p(t) = p[F\{K(t), L(t)\}/\beta(t)]$$
A different notation would be
$$
p(t) = z^{-1}\left( F\{K(t), L(t)\}/\beta(t) \right).
$$
This second notation is useful because all the partial derivatives follow from this notation and the chain rule.


Floating exchange rate and cost-push inflation?

I would imagine that a falling exchange rate causes the price of imports to rise as the foreign currency becomes stronger against the local currency.

With high inflation exports become more expensive because foreign buyers have to exchange more of their currency to account for the increase in price. This means they cost more per unit in terms of the foreign currency and consequently become less competitive.

The subsequent fall in demand for exports causes the fall in the exchange rate as the demand for exports and local currency are complementary. A weaker local currency means that one unit of local currency is now worth fewer units of foreign currency.

Thus the result is that imports become more expensive as any given amount of imports now costs more in terms of the local currency. Imagine that the exporting nation, say England, sells teddy bears for £5.00 a piece. If the dollar weakened against the pound from \$1.50 to a pound to \$1.56, then as far as a buyer from the United States is concerned, the price of a teddy bear increases from \$7.5 (i.e. 1.50 * 5.00) to \$7.8 (i.e. 1.56 * 5.00).


I haven't read the paper in full detail (I should, it's interesting) but let me give you my interpretation of the paper's title.

Blanchard uses a constant probability of instanteneous death for each agent. As mentioned in footnote one of the paper this is most realistic as assumption for people between 20 and 40. Agents in the model therefore are like young people for their entire life, hence the title.

As for your question 2, the way I read the paper yes there are many people alive of all ages. After all at each instant in time a new cohort of size p is born (and starts dieing off immediately at rate 1/p). So eventually at time is infinity that cohort will have completely passed away.


How to determine price per unit and quantity sold of product from financial statement

What you are trying to do here is estimate the demand curve of a firm. Broadly speaking, even if you got data on price and quantity, you would not be able to estimate this because your right-hand-side variable (quantity) is also determined by the supply curve. This is what people call the 'simultaneity' form of the problem of endogeneity.

The way to resolve this is to find a credible instrument that allows you to model movements in quantity that arise from changes in supply. For example if you found data on changes in the oil price, and found that they affect the quantity of goods purchased in the market you are estimating the equation for, you would essentially be able to 'trace out' the demand curve.

However, reading between the lines of your query, I suspect you are not particularly familiar with econometrics so the above may not be useful to you. If this is a school project (as opposed to university or other higher level of education), I think you could take the approach you are suggesting and then get additional points if you can explain what I just did above.

Company annual reports should contain the information you need, so long as you focus on companies that sell 'real goods' as opposed to services (think automobile manufacturers rather than tech companies or law firms). For example, page 71 of the Daimler annual report (see here) contains data on number of cars sold, and page 228 contains total revenues. If you want to break this down by type of vehicle sold, see page 253 for revenue by type of vehicle.

In general (i.e. when you look at other annual reports) you will find that:

You will find volumes sold somewhere in the text of the annual report (before the financial statements start)
You will find revenues at the start of the financial statements (p.228 in the above report example)
Further breakdown of revenues can be found in the notes to the financial statements (i.e. some point beyond p.228 - the note I refer to on p.253 is the note on revenue in that annual report, so naturally breaks the revenue out. You can find out which notes refer to which items in the financial statements by looking at the financial statements on p.228 - there you will see a '4' next to revenue which means you want to look at note 4)

I would recommend of doing a mix of looking through annual reports for a given company over time, and also multiple different companies' reports.


Will printing more money during COVID cause hyperinflation?

Likely not for several reasons. First, the article from economicshelp.org you mention is oversimplifying the economics, which is understandable as it is article written for non-economists, but to understand this issue we need to go little bit further.
Inflation, is simply just positive change in price level and price level of an economy is determined by the equilibrium at money market. The money market equilibrium, in its simplified form (more complex models include expectations of the quantities as well but this should suffice for this answer) is given by equation of exchange (See Mankiw Macroeconomics pp 87) as:
$$MV=PY.$$
Where $M$ is the money supply, $V$ velocity of money, $P$ price level and $Y$ output.
Solving for price level and log-linearizing (so % changes in right hand side variables give us the % change in P) we get:
$$\ln P=\ln M+\ln V−\ln Y.$$
Hence, it is true that inflation depends on increase in money supply but it also depend on changes in velocity of money and real output. If money supply increases by $30\%$ and real output decreases (due to pandemic) by $10\%$ and velocity drops by $40\%$ these effects cancel out and even $30\%$ increase in money supply would not lead to any inflation.
In fact one of the main reasons why we did not see much inflation recently is that velocity of money dropped significantly as you can see from Fred data. It is not clear if velocity will pick up in near future and as long as it remains low it will be offsetting considerable amount of increase in money supply despite of significant drop of real output.
Next, it is important to understand what hyperinflation is. Following Cagan The Monetary Dynamics of Hyperinflation, hyperinflation can be defined as an increase in price level that exceeds $50\%$ per month.
This means that in terms of the simple model $\Delta M + \Delta V - \Delta Y  = 50\%$ per month.  There is simply not enough money creation going on  in most countries to hit that $50\%$ per month value to classify as hyperinflation. For example, in the US the money stock as measured by $M2$ increased during last year by $\approx 19\%$ (see Fred data). While, that is without doubt large increase relative to previous years, it is far cry from increase in money supply that would lead to hyperinflation, even taking into account drop in output due to covid. Hyperinflation would not be caused by changes in velocity either because even though it is possible it would pick up later velocity usually does not increase fast month by month.
Lastly, if there would be any signs of hyperinflation independent central banks would start acting. In countries like Zimbabwe central banks are not independent and often they are forced to expand money supply rapidly by politicians. In most developed countries central banks are independent and for most part run by technocrats. Central banks have plethora of tools to sharply reduce money supply.
Even though people often just label all money creation money printing this is really just analogy as it is easier to explain money supply increases to people that way, but in reality most money are created by banking system via lending and central banks control the amount of money that can be created by changes in interest rates on its reserves (see for example McLeay, Radia, & Thomas; 2014). Next money are also created when Fed borrows money to the government via its open market operations. Only very small fraction of money are actually printed each year and similar picture holds for most developed nations.
Consequently, if Fed or any other modern independent central bank would spot signs of not just hyperinflation but even high inflation (like $10\%$ per year) they would quickly start hiking interest rates which would slow money creation and eventually even destroy part of new money, they would stop buying government bonds or even sell them again destroying part of money supply and so on. Independent central banks are quite good 'inflation fighting machines' and empirical literature, for the most part, shows that there is negative relationship between central bank independence and inflation (e.g. see Jácome & Vázquez; 2008).
Of course, it is not literally impossible that the situation will be mishandled, and heads of central banks are still typically appointed by politicians in most countries  so there is possibility of political capture of central banks, but as the things look like now I would say that hyperinflation in most developing countries is astronomically unlikely in near future.


mixted strategy that assigns positive weight to a pure startegy that is dominated

The short answer is that a mixed strategy $\sigma_i$ that uses with positive probability a dominated pure strategy $s_i$ can always be improved by excluding $s_i$ from its mixing support and redistributing its "probability weight" to one of its dominating strategies.

More formally, suppose that strategy $s_i'$ dominates $s_i$ (you can think of $s_i'$ as a pure or mixed strategy itself, this does not make any difference for the argument; for simplicity I suppose $s_i'$ is a pure strategy), that is $\forall s_{-i}\in S_{-i}: \pi(s_i',s_{-i})>\pi(s_i,s_{-i})$.

Also, suppose that strategy $\sigma_i$ plays strategy $s_i$ with positive probability $p_{s_i}\in(0,1)$. Build strategy $\sigma_i'$ as follows: identical to $\sigma_i$ but replace $s_i$ with $s_i'$ , that is play $s_i$ with probability 0 and play $s_i'$ with (additional) probability $p_{s_i}$.

By linearity of the payoff function you have, $\forall s_{-i}\in S_{-i}$:
$$
\pi(\sigma_i',s_{-i})=p_{s_i}\pi(s_i',s_{-i}) + (1-p_{s_i})\pi(\Lambda_i,s_{-i}) > p_{s_i}\pi(s_i,s_{-i}) + (1-p_{s_i})\pi(\Lambda_i,s_{-i}) = \pi(\sigma_i,s_{-i})
$$
where $\Lambda_i$, loosely speaking, denotes the residual combination of strategies played in $\sigma_i$, apart from $s_i$, inherited by $\sigma_i'$ too.


Expanding on jmbejara's answer and your question about integration, there are really three steps here:

Solve the HJB equation $\rho V(k) = \max_c g(c)+V'(k)(z-c)$. (You're already done this.)
Obtain the policy function $c(k)=\text{argmax}_c g(c)+V'(k)(z-c)$. (You presumably already did this while solving the HJB.)
Substitute $c(k)$ into the law of motion for $k$ and numerically solve that differential equation for $k(t)$:$$\dot{k}=z-c(k)$$

It sounds like you already had (1) and (2) and were just unsure about what exactly to do in (3). If you're working in a deterministic environment (and from the HJB it looks like you are), no discretization on your part is necessary: you can just use a canned ODE solver from Matlab or any other software package to numerically solve the differential equation above for $k(t)$.

To be a little more specific:

If the HJB is simple and you have an analytical solution for $V(k)$, and thus can easily compute $c(k)$ as well, that's great. You can just plug $c(k)$ into the law of motion for $k$ and tell your software package of choice to solve it and obtain the trajectory of $k(t)$.
If you had to numerically solve the HJB, then that's great because it means that you probably have already used some kind of numerical ODE solution technique and will be familiar with it. On the minus side, it means that when you initially obtain $V(k)$ and $c(k)$, you'll only get it for a grid of $k$, rather than having some kind of formula you can apply everywhere - which is a problem, because you'll need to be able to determine $c(k)$ at any point $k$ in a continuum if you're going to numerically solve the $\dot{k}$ differential equation for $k(t)$. The simplest way around this, to my knowledge, is to fit a spline to (or use some other approximation method on) the discrete grid of $(k_{grid},c(k_{grid}))$ values that you have. Then you can obtain an approximate value of $c(k)$ anywhere that's covered by the spline, and it will be extremely accurate assuming that your ODE is well-behaved and you have a dense grid.

Things get more complicated in a stochastic environment, but your case looks deterministic. (And to be clear, I'm just a dabbler here: numerical methods are not my area of expertise. But this is what I know.)


Prove that if a bond trades at a discount, its yield to maturity will exceed its coupon rate

If a bond trades at a discount, $P<FV$.

$\displaystyle\implies CPN\times\frac{1}{YTM_n}\left(1-\frac{1}{(1+YTM_n)^n}\right)+\frac{FV}{(1+YTM_n)^n}<FV$

$\displaystyle\implies CPN\times\frac{1}{YTM_n}\left(1-\frac{1}{(1+YTM_n)^n}\right)<FV-\frac{FV}{(1+YTM_n)^n}$

$\displaystyle\implies CPN\times\frac{1}{YTM_n}\left(1-\frac{1}{(1+YTM_n)^n}\right)<FV\left(1-\frac{1}{(1+YTM_n)^n}\right)$ --------------(1)

$\displaystyle YTM_n>0\implies 1+YTM_n>1\implies(1+YTM_n)^n>1\implies\frac{1}{(1+YTM_n)^n}<1\implies 1-\frac{1}{(1+YTM_n)^n}>0$

So, we divide both sides of (1) by $\displaystyle 1-\frac{1}{(1+YTM_n)^n}$ to get

$\displaystyle CPN\times\frac{1}{YTM_n}<FV$

$\displaystyle\implies\frac{CPN}{FV}<YTM_n$

$\displaystyle\implies r<YTM_n$

$\displaystyle\implies YTM_n>r$

QED


General equilibrium - exchange economy with time and perishable goods

Of course, note that in general, consumer B is willing to increase her consumption in period 1. it is a matter of simply solving the problem in the regular way. i.e, MRS=p and find the $p$ which "clears" the market.

Normalize the price of the current period good to $1$ and let $p$ be the price of the second period good, then the optimum for player $A$ is:
$\max_{x_{A1},x_{A2}} \sqrt{x_{A1}}+\sqrt{x_{A2}}$ s.t $x_{A1}+px_{A2}\le 1$
Clearly the constraint is binding and thus the problem becomes:
$\max_{x_{A1}} \sqrt{x_{A1}}+\sqrt{\frac{1-x_{A1}}{p}}$
Calculate the first order condition, repeat the process for consumer B etc...


Event Study vs. Difference-in-Differences

The statement is reasonably accurate. When we talk about the 'standard event study' in economics and financial literature (e.g. MacKinlay, 1997),  which based on your text  you are, they are not appropriate for analyzing single effects that affect whole market. There are several reasons for this:

A key assumption of event study is cross-sectional independence. A cross-sectional independence will be violated when in your sample multiple events happen at the same time. A textbook example would be stock listings occurring at the same day (See Brooks Introductory Econometrics for Finance). Such event clustering makes your estimates of test statistics and consequently $p$-values biased and as a result any inference from such event study will be most likely incorrect. If you are analyzing an impact of a single event you will have the most extreme case of event clustering as you are modeling the regulation as a separate event for every firm that happens exactly at the same period of time.
There are some solutions to event clustering. For example, you could solve it by not aggregating across the firms and just examining event impact on firm level do a summary analysis of all individual firm level studies.  An alternative would be to create a portfolio of firms  that experience the event at the same time and analyze the portfolio as a single firm (see again Brooks). However, in the first case the solution is not very elegant and it has its own problems that are beyond scope of this answer, but you could in principle do it. The second solution is not appropriate because in your case you would end up with a one big portfolio of all firms so you would end up with one aggregate observation.

Usually when you are interested in evaluating an effect of policy you do not want to just know if the policy had an effect on one day but if there are also effects that persist. For example, if you evaluate a policy like deworming pills in developed countries on student performance, you dont want to just know if there was one-off effect of this policy on student performance which lasted for few days or months, you would be interesting in knowing what was the effect long term. In such cases using event study is problematic as that would require you to use very long event window. However, this introduces another issues as event study is very sensitive to even small misspecifications if the event window is long (see Brooks again). This is why longer event studies use buy-and-hold abnormal returns (BHAR) instead of just cumulative abnormal returns (CAR) but if your event window is too long you will still have problems.

There are also further issues but the two I mentioned above are in my opinion the major ones when compared to DiD. Also yes there are other alternatives but I think that mentioning them all would be too broad. For example, you could model single event as a structural break or even as regime switch and analyze it that way. It will always depends on the specifics of the problem, for that it is always best to do very careful literature review on the topic and see how other people approach the problem and which issues they consider most important.


I apologize for my previous comment; it doesn't make much sense.
Upon inspecting your data, I've found that there are IDs for which you only have observations for two years. The problematic IDs are: 9231, 24873, 24907, 78341, 78998, 90246, 92673, 92787, and 93846. You can see the full frequency table here.
By looking at the source code for the package, we can see what causes that error message to appear:
  if((use.mc.diff | use.mc.lev) && (length(unique(dat[, varname.t])) <
 3)){

stop("Insufficient number of time periods to derive linear
     moment conditions.")
}

which I understand as: if the length of the vector containing the unique dates is less than 3, then the estimation cannot be computed.
Looking at the package help, we can run the example code provided:
## Load data from plm package

    if(!requireNamespace("plm", quietly = TRUE)){

     stop("Dataset from package \"plm\" needed for this example.
         Please install the package.", call. = FALSE)
    }

    else{

    data(EmplUK,
         package = "plm")  dat <- EmplUK  dat[,c(4:7)] <- log(dat[,c(4:7)])

         ## Arellano and Bond (1991) estimation in Table 4, column (a1)

     m1 <- pdynmc(dat = dat, varname.i = "firm", varname.t = "year",
                  use.mc.diff = TRUE, use.mc.lev = FALSE, use.mc.nonlin = FALSE,
                  include.y = TRUE, varname.y = "emp", lagTerms.y = 2,
                  fur.con = TRUE, fur.con.diff = TRUE, fur.con.lev = FALSE,
                  varname.reg.fur = c("wage", "capital", "output"), lagTerms.reg.fur = c(1,2,2),
                  include.dum = TRUE, dum.diff = TRUE, dum.lev = FALSE, varname.dum = "year",
                  w.mat = "iid.err", std.err = "corrected", estimation = "onestep",
                  opt.meth = "none")
  summary(m1)
    }

which works just fine. Furthermore, if we run the following code:
length(unique(dat[, "year"]))

we get 9 as the output, which is clearly greater than 3. However, if we do the same using the data you have provided:
length(unique(datacheck[, "Year"]))

the result is 1, which is smaller than 3, just as the error message pointed out.
I can't figure out why that is happening. My guess is that perhaps you need at least 3 unique sequences of years or something else, because you do have enough observations with more than 3 years available.
While this is not a definitive answer, it's too long to post as a comment. Hope it helps anyway.
EDIT:
Source code
Vignette


UPDATE 19-9-2017

I just received word from the author that the book is now freely downloadable in English, at this link.

Got it. I remember browsing the pages of this book. But it was in Greek, and it does not appear to have been published also in English:

Euclidean Economics: A Mathematical Method for Construction and Application of Economics Models

  "The book establishes that economics can be studied and utilised on
  the basis of a minimum of fundamental hypotheses and the general laws
  of mathematics. Starting from 12 major propositions, it proves that
  closed exchange networks are replicated by systems of 2n first-order
  non-homogeneous linear differential equations, which simulate
  household, business, bank and government receipts and payments. Such
  networks are defined in the (R2n, t) vector space, where n is the
  number of sectors or markets and countries or currencies, while t is
  the elapsed time. Uniqueness, stability and controllability of the
  steady state as well as takings and outgoings are functions of
  behaviour coefficients and policy parameters. Annual sales of
  services, products and securities divided by average salaries, prices
  and values yield the man hours, output items and liquidity units
  traded. Demanded quantities do not coincide with available quantities.
  Potential inflation or deflation is proportional to the weighted
  deficit or surplus of market goods. Digital animation of the
  endogenous variables is possible on the basis of historical, projected
  or hypothetical data about the exogenous variables."

Author: Sophocles Michaelides
Language: Greek
Publisher: Mesogios Publications – Ellinika Grammata
ISBN: 9963-9176-0-7
Year: 2006

The summary does not state it clearly, but the "12 major propositions" are Euclidean, as I recall.


There is no inconsistency here. Following Blanchard et al. Macroeconomics an European Perspective (which is by the way same textbook where you got that picture from) the goods market can be described as:
$$ Y = C(Y − T) + I(Y, i) + G   \quad (1)$$
where, $Y$ is the output, $C$ consumption, $T$ taxes, $I$ investment $i$ interest and $G$ gov. spending. Then if we assume that consumption follows:
$$C = c_0 + c_1(Y − T) \quad (2)$$
and that investment function is given by:
$$I = \bar{I} + d_1Y − d_2i \quad (3) $$
then the goods market equilibrium will be given as:
$$ Y = \frac{1}{1-c_1-d_1}A - \frac{d_2}{1-c_1-d_1}i \quad(4)$$
where $A=[c_0 + \bar{I} + G − c_1T]$. Finally the IS curve is derived just by solving the above equation for $i$:
$$i = \frac{1}{d_2}A - \frac{1-c_1-d_1}{d_2} Y \quad (5)$$
Now this last expression clearly corresponds to the IS from the graph. As the equation lay bare the relationship between interest rate and output is still the same - lower interest rate is associated with higher output. The intuition behind negative relationship between
investment and the interest rate is determined by the fact that the higher the interest rate, the lower the incentive for a company to borrow to make new investment. Higher investment means also higher output $Y$ so the lower the interest rate the higher output will be.
Really the upper graph just plots the equation (4) although rearranged in different way and the bottom graph is the equation (5).  So the relationship was already negative and same in both upper and lower graph.  The graphs also might not be drawn to scale - they dont look to me like they are generated by specifying the underlying functions in lets say tikz picture in LaTex.


If a business charges more for a product than the cost to make it, how is our economy not inevitably doomed for debt?

To continue your example:
Resources (including labor) cost $x$, product sells for 4$x$. The fat cat business owner makes 3$x$. What does he do with it? If he spends it, for example plays golf or gets his hair done then the people providing those services get the money. Thus there need not be debt, the hairdresser can use these earnings to buy the product the fat cat's business was selling. The usual water analogy is of a circulatory system: water (money) flows from one vessel to the other.


The yield to maturity is the internal rate of return on the bond assuming no default. Given a series of promised cashflows in the future, to quote Prof. John Cochrane, "... the yield is just a convenient way to quote the price."

In the case of a zero-coupon bond with price $p_0$ and face value $c_t$ due $t$ periods in the future, the yield to maturity is the fictional, constant interest rate $y$ that solves the equation:

$$ p_0 = \frac{c_t}{(1 + y)^t} $$

More generally in the case of a non-negative series of payments $(c_1, \ldots, c_T)$, the yield to maturity is the fictional, constant interest rate $y$ that solves:

$$ p_0 = \sum_{t=1}^T \frac{c_t}{\left(1 + y\right)^t}$$

Given a series of promised coupon payments and the bond's face value, listing either a bond price or bond yield conveys the same information.

The yield to maturity does not tell you the expected return because you don't know the probability of default, recovery rate, etc....

Problems/errors with Investopedia's definition

The Investopedia definition you quoted is flawed because it's missing the critical phrase, "assuming no default." I've seen other sloppy, incomplete, or confused definitions on Investopedia and prefer Wikipedia.

Problems with your statements on the cost of capital

The yield to maturity has an unambiguous, widely agreed upon definition and is easily computable. On the other hand, the cost of capital is a more nebulous concept. Different practitioners or academics often use somewhat different definitions for the cost of capital and estimate these somewhat different notions of the cost of capital in different ways.

If you define the cost of capital as the return expected to be earned by investors, then yield to maturity is not the cost of capital over the period because the yield to maturity ignores the probability of default. (That said, it appears common practice in corporate finance to just ignore default probabilities and use yields as the cost of debt.)


Although this question will be flagged and closed I assume, I would recommend getting a variety of texts. Mankiw has a good intermediate macro book, that can get you from zero to a foundation, however his introductory book isn't as good.

For econometrics, Woolridge gives a good bang for your buck, but I like Gujarati's basic econometrics as well. For time series, Enders' "Applied econometric time series" is my bible.

For micro I'm not as sure, I didn't have any great experiences with introductory texts. I can't really give a good opinion on the intermediate level either, as I've only used Varian's.

Once you get an idea of the concepts the best thing to do is to delve right into academic journals, so that you can start learning how to read them. Just that skill will help you be able to catch up fast.

I assume you're already strong in linear algebra and your other math, and some of your engineering will carry over to economics, especially your ability to think about minimizing and maximizing.

Good luck!


Why do firms losing money almost always reduce labor costs via layoffs instead of pay cuts?

Here are four possible reasons:

Reason #1. Fixed costs per employee.

Due to taxes, insurance, 401k plans, etc., the cost of an employee (to the employer) is significantly greater than the employee's take-home pay.

For simplicity, assume an employer must bear an additional lump sum cost per employee that is equal to 20% of each employee's current take-home pay. Assume also that all employees are identical. Then to cut payroll costs by 30%, the employer can either:

A. Fire 30% of the employees; OR

B. Cut each employee's take-home pay by 36%.

Many firms will prefer A to B.

Reason #2. Parkinson's Law

Parkinson's Law is the humorous adage that "work expands so as to fill the time available for its completion".

This may apply also to workers in a firm. Laying off say 5% of employees may make no difference to total production -- especially if a firm was already bloated and overstaffed to begin with.

Reason #3. Keeping employees on their toes.

By occasionally laying off a significant fraction of employees during downturns, one credibly demonstrates that employees should work hard and well or risk being fired.

[There are here two separate questions: (a) Does this tactic really work (in motivating employees)?; and (b) Do there exist employers who believe in this tactic? An affirmative answer to (b) suffices to serve as an explanation to your question.]

Reason #4. Worker morale.

Suppose we ignore Reason #1 and happily, we can save 30% in payroll costs by either:

C. Firing 30% of the employees; OR

D. Cutting each employee's take-home pay by 30%.

Under C, the 30% who are fired will indeed be very heavily demoralized, but they are no longer your problem. As for the remaining 70%, they might be slightly demoralized about losing some of their work buddies. But they might even be reinvigorated after discovering they survive the cut.

In contrast, under D, every employee is rather demoralized about the 30% pay cut.

Many firms will prefer C to D.


In fisher equation, Should I use "expected inflation" not "actual inflation"?

The Fisher equation as $$r=i-\pi$$ is little more than an approximation to the definition of the real interest rate $$r=\dfrac{1+i}{1+\pi} -1$$ where $r$ is the real interest rate, $i$ is the nominal interest rate and $\pi$ is inflation (the proportionate change in prices). You can complicate things by "coninuous compounding" but it makes little actual impact.

Looking backwards you should look at actual interest and actual inflation at the time of the loan. Looking forward, if you do not know what inflation or interest rates will be then you can only use estimated values and forecaster often prefer to estimate inflation and real interest rates, turning the Fisher equation round to give $$i = r+ \pi$$


Calculate optimal discount for product bundling

Do you have price elasticity figures at your disposal from other sources or were you planning on trying to estimate elasticities yourself using the data? If you are planning on estimating elasticities using the data, you need to be very careful. Running a simple log-log OLS regression will give you poor estimates because your data points are all market equilibrium points and are not a representative sample of points along a static demand curve. You have serious endogeneity issues there.

I have tried undertaking similar projects so I know the pitfalls well. If you still want to back out sales using elasticities, you could try an instrumental variables approach using some of your other variables as instruments to try and circumvent the endogeneity problem via supply-side identification. Another option is to use a simultaneous equations model where you posit functional forms for supply and demand separately and then estimate the reduced form using, say, two-stage least squares (2SLS).

If you are strictly interested in forecasting sales well and not really concerned with understanding causal channels, you can look into machine learning algorithms. Depending on how many data points and how many variables you have at your disposal, there are various models that may suit your needs. You should understand these models well before running them, however, because most of them are not like the models in your standard econometrics toolkit. For instance, the more "flexible" models (such as unpruned decision trees, multilayer perceptrons, etc.) are very susceptible to overfitting the data by essentially modeling noise, leading to higher mean squared errors than anticipated due to high variance.

If you're interested in learning more, I found this to be an outstanding, easy-to-understand introduction into statistical learning.


What does Eugene Fama mean "the amount of debt that's financed with equity isn't put aside"?

Fama's idea is to look at the level of equity that would have been necessary in the last crisis and multiply by 1.1 and call that a starting point for a level of equity that makes the company safer from insolvency. He phrased it as adding 10 percent.
In Fama's view there will still be risk of insolvency. "You have to watch them because they are very clever at getting around these things."
The actual quote from the other person is "locking up that kind of money...".
Then Fama talks about the balance sheet but he's just making the point that even with regulation the assets as a whole still bear investment risk.  I would add, although he does not say it explicitly, "because they are very clever" he is suggesting it is not practical to consider assets as being locked up, frozen, or set aside.


Are symmetric equilibria continuous with respect to the payoff matrix?

If I understand your question correctly then the answer is no. Consider

$$
A(h) =  \left( \begin{array}{ccc}
1 & 1 & 0 \\
0 & 2 & 0 \\
2+2h & 0 & 3 \end{array} \right)
$$
For the game defined by $A(0)$ the vector $a^t(0) = \left( \frac{1}{2}, \frac{1}{2}, 0\right)$ yields an equilibrium. For any $h>0$ however
$$
A(h) \cdot a(0) = \left( \begin{array}{ccc}
1 \\
1 \\
1+h \end{array} \right),
$$
so $a(0)$ is no longer an equilibrium. If play around with the probabilities you will see that nearby strategies do not constitute an equilibrium either. Hence if you have a positive sequence $h_1, h_2, ...$ such that $\lim_{n\to \infty} h_n = 0$ then you cannot select a sequence of vectors $a(h_1), a(h_2), ...$ such that they are equilibrium strategies of their respective games $A(h_1), A(h_2), ...$ and
$$
\lim_{n\to \infty} a(h_n) = a(0).
$$

In this counterexample I used that an action that was never played in $a(0)$ yielded the same payoff as actions with positive supports. If this is not the case then I think the answer to your question would be affirmative. Here is a proof for the case where $a>0$.

Since $a^t$ is a best response to $a$ we have
$$
A\cdot a = \underline{1} \cdot c
$$
where $\underline{1}$ is a vector with $1$ at all coordinates and $c \in \mathbb{R}$.
If $A$ is non-singular then
$$
a = A^{-1} \cdot \underline{1} \cdot c.
$$
As $a$ is a probability vector, $c$ is uniquely determined by $A$:
$$
1 = \underline{1}^t \cdot a = \underline{1}^t \cdot A^{-1} \cdot \underline{1} \cdot c
$$
I will use $A(h)$ to denote the perturbed matrix, and $a(h)$, $c(h)$ to denote the associated vectors. The original matrix and vectors occur at $h=0$. Since matrix inversion is continuous
$$
\lim_{h \to 0} c(h) = \lim_{h \to 0} \frac{1}{\underline{1}^t \cdot A(h)^{-1} \cdot \underline{1}} = \frac{1}{\underline{1}^t \cdot A(0)^{-1} \cdot \underline{1}} = c(0).
$$
Similarly
$$
\lim_{h \to 0} a(h) = \lim_{h \to 0} A(h)^{-1} \cdot \underline{1} \cdot c(h) = A(0)^{-1} \cdot \underline{1} \cdot c(0) = a(0).
$$
This is not an exact proof. To be exact you would have to specify the norm of a matrix, and then to prove that the above limits exist you would have to have a condition like
$$
\lim_{h \to 0} ||A(h)|| = ||A(0)|| \neq 0.
$$


Why does an increasing demand for dollars lead to the depreciation of a currency?

I am by no means an expert in monetary policy, but this is how I understand it. An increasing demand for dollars implies that more people want to trade the local currency for dollars. The supply of the local currency then increases while demand for it decreases, implying that the currency will depreciate.


What's the next largest company after Facebook of which the majority of the votes is controlled by a single person?

This question is likely to go stale pretty fast, for two reasons. First, ownership and control of companies is always subject to change; and second, market capitalization can also show substantial swings from quarter to quarter (see Apple's valuation at \$835 billion in Q1 2019 vs \$1,305 billion in Q4 2019 for just one example). I'm writing this in June 2020 and will try to indicate dates on my sources, but it won't be constant.
However, I do think there is value to answering this, to the extent that it explains not-commonly-understood aspects of how corporations work in practice.
The crucial distinction being made in this question is between voting rights and share ownership. You have to go pretty far down on the list to find a company for which stock constituting majority ownership is still in the hands of a single individual; however, it is not uncommon for companies to have dual voting structures which allocate more votes to the share types owned by insiders--a control structure which is considered harmful by knowledgeable commentators. In the example given, Zuckerberg does not own 58% of Facebook; he just has a majority of the insider shares, which get 10 votes each (as opposed to 1 vote for the publicly traded shares).
So ultimately the question is about control of the companies' decision-making--leadership and policies--and the extent to which outside investors can direct that. To answer that we have to look in quite some detail at their voting structures.
The following list is companies by largest market capitalization as of March 31, 2020:

Microsoft: no dual structure

Apple, Inc: one-share one-vote, no absolute majority shareholder. Has made moves to allow large investors more of a say in nominating board members.

Amazon.com: one-share one-vote, no absolute majority shareholder (Bezos remains plurality controller with roughly 16% of the vote)

Alphabet has majority control in the hands of two people (Larry Page and Sergey Brin)

Alibaba Group: Up until late last year, when founder Jack Ma stepped down and relinquished a lot of control, this would probably have been the winner. It's tough to say, because due to restrictions on foreign ownership, Chinese business structures are very complicated--lots of shell corporations. The company was originally listed in the US because its dual-share structure was illegal in China. Also, Alibaba is worth more than Facebook, so it wouldn't be the next one after Facebook...

Facebook, Inc -- already discussed

Tencent -- fuzzy. I haven't gone to the Chinese sources on this, and there's a lot of chaff in the search results; news about Tencent's control of online gaming and music sales is drowning out information about who controls Tencent. This is a possible candidate.

Berkshire Hathaway -- Warren Buffett controls about a third of the votes as of 2018

Visa -- charter appears to prevent anyone from owning more than 15% of the vote

Johnson & Johnson -- does not appear to have dual voting structure

I suspect it's going to be tough to find a similarly high-placed firm with sole-individual voting control, because even among firms with dual-class stock structures, most of the time those shares are split among cofounders, as with Page & Brin, or the cofounders of Lyft (who share something like 49% of the voting rights). Also, I'm having trouble finding a good summary of firms by current market capitalization; the easily-structured lists I'm seeing are for end 2019 (e.g.) or are from dodgy-looking sites.
However, even looking at majority vote control doesn't resolve the issue; you still really have to dig into the details on a company-by-company basis. For instance, Elon Musk effectively controls Tesla, despite having only around a fifth of the vote, because the charter requires nigh-impossible supermajorities for investors to challenge management decisions.
We can at least set a floor on the query: EchoStar Communications, market cap as of this writing \$3 billion, has a "substantial majority" of votes controlled by the founder. So any other candidate would have to have a market capitalization between \$3 billion and \$265 billion.
The short answer is it's difficult to find any single individual who has the level of influence and direct control that Zuckerberg has, but there are plenty of firms--especially in the tech sector--with structures that give founders a long-term controlling interest. Facebook is unique in having one founder and being so blatant, but the trend is quite common.


The independence of the central bank means that the bank is independent of the government. A mandate of the central bank is to ensure price stability. However any excess cash the bank prints is the property of the government. The government enjoys having extra income and were it allowed to exercise direct control it may push the bank to print more money. This would undermine the price stability mandate of the bank. Hence in most countries they give the central bank leadership that is independent of the government.

As Bitcoin is not issued by any government that I know of it does not in any way affect this. Perhaps if the government were to bribe the leaders of the bank in Bitcoin...


Utility of expected income or expected utility of income?

Your problem has problems with it.  There is no disutility of studying and there is no interest rate or it is zero percent.  However, let's consider the three possible cases.

If $y_2>F$ then for all cases you should study.  If $y_1<F$ then all cases you should not study.  The difficulty happens when $y_2<F$ and $y_1>F$.  Then you should study if $E(U(s))\gg{U}(F)$ and not if $E(U(S))\ll{U}(F)$.  The difficulty with the phrasing of the problem arises when $E(U(s))\approx{U}(F)$.

From the form of your problem, indifference should happen at: $$(y_1^\alpha-y_2^\alpha)p+y_2^\alpha=F^\alpha.$$  It should have been written as: $$(y_1^\alpha-y_2^\alpha)p+y_2^\alpha-U(s=study)=F^\alpha(1+\bar{r})^\alpha.$$  You cannot differentiate over $s$ because it is a binary choice.  In fact, $y_1,y_2$ are really $y_1(s=study)$ and $y_2(s=study)$ and $F$ is really $F(s=\text{don't study})$.

For the form of your problem, you should be indifferent if: $$p=\frac{F^\alpha-y_2^\alpha}{y_1^\alpha-y_2^\alpha}.$$  This permits an objective solution that does not depend upon the person.  For the more expansive problem at the indifference point, the solution depends upon the market rate of interest and the subjective cost of studying.

This implies two important things.  The first is that the value of an education depends on market rates of interest.  Because of this, there should be a decrease in the rate of education when rates are high.  The second is that the indifference point for the probability depends on the specific person.  This implies that probability has a subjective component, even when the probability is external to the person.  Indeed, if the probability is a function of all the choices of all the people, then the probability actually depends upon the marginal rate of disutility experienced in studying.


How do deal with negative values when finding log difference?

In excel use the LN function to perform the log.  The difference in logs should be done with the log of the most recent value minus the log of the past value
LN(Present) -LN(PAST) = Growth%

Take the excel handle and pull it down so that you can copy fill the formula to do all the
quarterly periods.

At this point you may also want to chose which decimal place you want to set your value to make the data look presentable by using the decrease/increase decimal value in the number section of the home tab within the menu bar.

I am not sure about your last statement about negative values.  If you mean that what to do when you get a negative value after the log transformation then that value is normal and be part of your regression analysis because you are seeing a decrease in change over time.
However, if the numeric value that you are log transforming is a negative value (which should not show up in the aggregate Real GDP values time series)  then you should pick a method to adjust the value before you use the log.
There are some sites that does offer suggested options ( https://blogs.sas.com/content/iml/2011/04/27/log-transformations-how-to-handle-negative-data-values.html  ) ; however, just like in data science, pick a method and stick to it and report what you did.


Question on shifts in Supply and Demand Curves

Write $i=I(y,a)$ for the IS function (expressed as some function of $y$). Likewise, let $i=L(y)$ be the LM function. Assume the following properties:

$\frac{\partial I(y,a)}{\partial a}>0$ : $a$ is a parameter that causes the IS curve to shift up.
$\frac{\partial I(y,a)}{\partial y}<0$: the IS function slopes downwards
$\frac{\partial L(y)}{\partial y}>0$ : the LM function slopes upwards

For any $a$, we find the equilibrium $y^*(a)$ at the intersection of the two functions:
$$I(y^*(a),a)- L(y^*(a))=0.$$
Applying the implicit function theorem:
$$\frac{\partial I}{\partial a}+\frac{\partial y^*}{\partial a}\left(\frac{\partial I}{\partial y^*}-\frac{\partial L}{\partial y^*}\right)=0,$$
which implies
$$\frac{\partial y^*}{\partial a}=-\frac{\frac{\partial I}{\partial a}}{\frac{\partial I}{\partial y^*}-\frac{\partial L}{\partial y^*}}.$$

The numerator is positive by hypothesis. Thus, for $y^*$ to increase with $a$ we require that the denominator be negative: $\frac{\partial I}{\partial y^*}<\frac{\partial L}{\partial y^*}$. This is clearly true because $I$ slopes downward and $L$ slopes upward. But $y^*$ would also increase even if $I$ sloped upward—provided itdoes so more gradually than $L$.


In Alecos Papadopoulos's answer, both the conditional mean and the conditional median are linear in $X$. In the following example, the conditional mean is linear in $X$ while the conditional median is quadratic in $X$. This example is a constructed toy but it touches the heart of the issue.

Example: Suppose that $y = \beta_0 + \beta_1 x + u$ and $u = x^2 (w - 1)$, where $w \sim \chi_1^2$ and $x$ and $w$ are mutually independent. Then, as the R command qchisq(0.5,1) gives, $med(w) \doteq 0.455$ so that $med(u|x) = x^2 (-0.545)$, while $E(u|x) = 0$.  We thus have
\begin{equation}
  E(y|x) = \beta_0 + \beta_1 x, ~~\text{ while }~~ med(y|x) = \beta_0 + \beta_1 x - 0.545 x^2.
\end{equation}

Note: Asymmetry of the error distribution and the dependence of $u$ on $x$ are both important. If the error distribution is symmetric, then the conditional mean function and the conditional median function coincide. Also, if the distribution of $u$ is independent of $x$, then $med(u|x)$ is nonzero but constant and thus $med(y|x)$ is again linear in $x$.


@Afreelunch gave you the right intuition. Non-regular economies will exhibit an infinite amount of equilibrium points. Intuitively since there is no local uniqueness, the continuum can be expressed as an interval on the real line so there is an infinite number of such points.


Probability of the event knowing that I received no informations

The right denominator should be $P(\bar C)$, and the right numerator should be $P(A\cap \bar C)$ since $\bar C$ is the event of no receiving information, either because the guy is not there or the guy is there but gives no information. The denominator should not depend on the true state of his "coolness".

An easy way to picture this problem is by doing a diagram. Think of a square and divide it in half, say vertically where the mass (or probability) in the left side represents $P(A)$ and complementary probability in the right. Then divide the square in another half, say horizontally, so that the mass in the top half represents $P(B)$. each of the resulting smaller squares contains the mass of each of the possible intersections between these two events. Finally, you can divide the top half (created in the second step) in half (say horizontally again) where that top part represents $P(C)$, and the bottom part (from this last line up to the bottom of the square) represents $P(\bar C)$. Clearly, $C$ does not intersect with $\bar B$ and $\bar C$ contains $\bar B$, as desired. And you can clearly visualize the probability you are trying to calculate.


I have the 2008 hardcover edition, and in that edition, Chapter 10 starts on page 249. I could not spot that quote - what is the section title?

In any event, in the 2008 edition, there is a section “Prudence and Surveillance” within Chapter 10. In it, he describes “customer and collegiate surveillance.” The argument is that when banks were largely allowed to fail, the private sector had no choice but to monitor bank risk taking.

Customer surveillance refers to surveillance depositors; they will pull their deposits out if they view the bank as especially risky.
Collegiate surveillance refers to surveillance by banks and other institutions that lend to a bank in the wholesale money markets (e.g., in the Fed Funds market to make up reserve deficiencies).

As for why this matters, this shows up in the rest of Chapter 10. He describes how bank lending determines the level and type of investment.

  Financial market and banking affect investment because the current value of capital assets and thus the demand price for investment outout are determined in financial markets, because the amount of investment that will be financed depends upon banking processes, and because the supply price of investment depends upon the cost of finance. (Page 254 of the 2008 edition; first section of Chapter 10).

In his section “Coda” he then argues that “there are no effective barriers to bank expansion and thus to the destabilising impact of banks on demand.”

In other words, he is worried more about the instability created by banking as a result of the lack of regulation of their portfolio growth, rather than whether people lose money.


Printing money causes inflation because someone (or some institution) will get the money and will spend it somehow. This increases demand for goods, but the number of goods, the supply was not affected by printing money. Hence demand at the old price level is larger than supply at the old price level. Prices will rise until a new equilibrium is reached.

The problem is that you seem to think that the N dollars you give to China would be gone. But it wouldn't be, the people who got it in China can now spend it in the US. They can import things, come as tourists, make investment purchases etc. This increases demand and results in inflation again.


Should central bank raise interest rates during a condition of liquidity trap?

In rough terms the central bank wants the nominal interest rates low so as a) to incentivize people to consume their income rather than save it and b) to make loans more attractive as they are linked with lower interest, thus bolstering consumption (again). During a crisis however, consumption remains low as expectations of economic growth deteriorate. Moreover, as interest rates are low, there is no point in saving money into the banks. Thus, people choose to hoard money, essentially removing it from the economy. With less money in the banking system and a generally pessimistic economic environment, banks become more skeptical towards providing loans as they doubt that they will get their investment back. All these processes empower the endurance of the vicious circle of the liquidity trap.

In theory, such prolonged periods of near zero interest rates would mean higher inflation. However, the recent crisis has proven that zero interest rates and stable inflation can co-exist. Therefore, the central bank has, up until now, been reluctant towards increasing the interest rates and in the past years it has made sure to inform the public that it is going to keep them quite low for some time (forward guidance). There is a whole debate of whether low interest rates are or should be linked with high inflation during financial crises and recently we see a drift, especially in the US, towards an interest rate increase even though it has not yet materialized.

Bonds are not considered a "go to" solution under a liquidity trap as their demand is really low. This stems from the fact that when a government faces a recession, its bonds are not considered a safe investment as they are paired with a high interest rate. Thus, the public awaits that they are unlikely to be honoured in the future. And if the recent financial crisis has taught us anything is that even bond holders are not safe.


Economic interpretation: IS curve contra GDP in equilibrium

So graphically this is pretty straight forward, as you may have already understood: Since the LM curve has a positive slope the entire shift of IS curve ($\frac{\partial Y}{\partial G})$ is not fully translated into final output. It would happen when LM curve is flat (which is usually the case at very low interest rates in liquidity trap situation).
Interpretation: As govt's consumption/expenditure increases at a given interest rate, it increases output. But higher output requires more money circulating in the economy to sustain the higher output. Since no additional money is injected into the economy this additional money demand is met from the speculative demand for money. This means that speculative demand should come down.
The decrease in speculative demand comes from increase in interest rates. (Mechanically, this happens because as govt floats more bonds into the market (for expanding its borrowing), it triggers a speculation of increase in interest rates which, in turn, triggers the money held by speculators to invest back in the economy - increasing the money available for transactions). Further there are more bonds in the market too causing the bond prices to fall and interest rates to increase (wealth effect channel).
But as the interest rates goes up, the private investment demand comes down (called - crowding out of private investment) ultimately making final increase in demand ($\frac{\partial Y^*}{\partial G}$) to be lower at equilibrium as compared to $\frac{\partial Y}{\partial G}$.


Are there examples of strategies where a combination of third degree pricing and two-part tariffs are used together?

Here's an easy example: movie theaters.

If you view theater pricing models as a two-part tariff (ticket + snacks) then any time there's a "seniors discount" or something like it, this would fit your strategy.  It's a simple exercise to determine the conditions for profitability of such a campaign, but the fact that they've been around forever suggests that on net, they are.


$p_i (t_j) $ is the PDF of $i$'s belief over $j$'s types.

Recall that if $t_j$ is uniformly distributed on $[0,x]$, then the probability that $t_j \in [a,b]$, where $0 \leq a \leq b \leq x$ is given by

$$ \Pr ( a \leq t_j \leq b) = \int_a^b p_i (t_j) d t_j $$

In particular,

$$ \Pr (t_j = a) = 0 $$

for any $a$.


Gini index vs Herfindahl index: one increases while the other decreases

They measure different things.  In particular the Gini index measures inequality and is strongly affected by the number of individuals with almost nothing, while the Herfindahl index measures the diversity of the shares (for example the choice in a market) and is almost unaffected by those with almost no share

If $n$ people had equal shares then the Gini index would be $0$, not changing as $n$ increased since inequality would not change, while the Herfindahl index would be $\frac1n$, reducing as $n$ increased to reflect greater diversity

As an an illustration, the following $36$ values give roughly the same results as your two examples

V2 <- c(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
        0.05, 0.19, 0.19, 0.19, 0.19, 0.19)
Gini(V2)
# [1] 0.8527778
Herfindahl(V2)
# [1] 0.183

But the zeros have no impact on the market shares and the Herfindahl index: having somebody extra with no share is as irrelevant as having nobody extra, and somebody with a tiny share is almost as irrelevant.  Removing those zeros would lead to

V3 <- c(0.05, 0.19, 0.19, 0.19, 0.19, 0.19)
Gini(V3)
# [1] 0.1166667
Herfindahl(V3)
# [1] 0.183

with the Gini index being much smaller (there is less inequality when those with nothing are excluded from consideration) but the Herfindahl index staying the same, corresponding to somewhere between $5$ and $6$ equal shares

You could looking at the Lorenz curves for your examples with something from the package like

plot(Lc(V0), col="blue")
lines(Lc(V1), col="red")

or something homemade like

plot(c(0,1), c(0,1), type="l")
points((0:length(V0))/length(V0),cumsum(c(0,sort(V0)))/sum(V0),
    type="b", col="blue")
points((0:length(V1))/length(V1),cumsum(c(0,sort(V1)))/sum(V1),
    type="b", col="red")

gives a picture like this with V0 in blue and V1 in red

So you can see that there is a greater area between the diagonal and the red curve (V1) than between the diagonal and the blue curve (V0), explaing why V1 gives a higher Gini index.  This is largely due to V1 having more extremely low values than V0 does

But on the right hand side, the top two values are greater for blue (the cumulative sums of the others are lower), so shares are more concentrated for V0 leading to a higher Herfindahl index


What does "(Earnings Per Share) sen" mean in a shareholders' context?

Perhaps "sen" is no more than a 1/100 sub-unit of the (e.g.) currency Malaysian ringgit? Such a sub-unit is called "sen", and is etymologically related to the English word "cent".

It seems even more likely that "sen" in  "Earnings Per Share (sen)" refers to a 1/100 sub-unit of a currency, as the google query "Earnings Per Share (cents)" yields many similar results.

But even if that would be the case, I think there can be more elaborate answers than mine, which also explains a bit more about the meaning of the words "Earnings Per Share", in relation to numbers expressed in either "cents" or "sen".


What type of graph can be hepful to explain the type of errors that should be used?

You are saying: $y$ is regressed on $x_1$ and $x_2$, say, and you think it would be better to use cluster errors since you expect $y$ is correlated with abilities for each person.

First, $y$ is naturally correlated with the error term $u$ (if you mean the error term by "abilities") because $u$ is a part of $y$. You have no problems with that; $y$ is always strongly correlated with $u$. It also has nothing to do with cluster errors.

If you happen to mean cluster standard errors (SE) by "cluster errors", and if you are saying that you would use cluster SE because $x_1$ is correlated with $u$, I would say that the regressor-error correlation (endogeneity) has nothing to do with cluster SE, what so ever. Cluster SE provides valid inference when the the error term show within-cluster (auto)correlation.

Please enlighten me if I misinterpreted your question.


Post Brexit Referendum employment theory

Is there anything in existing economic theory that could explain a
  fall in immigration correlating and/or causing higher difficulty in
  finding employment?

In addition to the lagged effect you outline, there are at least two possible causes for the paradox you describe. These theoretical possibilities are independent of which interpretation of unemployment is adopted. I'm unaware of (and not even searching for) statistical evidence of whether this actually occurs.

A reduction of immigration could also entail a reduction of demand of goods and services. If there's no demand, some businesses are likely to reduce their staff, leading to an oversupply of labor (i.e., unemployment).
High-skilled immigrants might boost the industry in a different country to an extent that renders that foreign economy more competitive than the domestic economy. A significant trend in that direction could prompt domestic companies to slash jobs or even to relocate to one such country where supply of high-skilled labor is stronger. That would lead to the unemployment of low-skilled workers in the domestic labor market.

As for 2., it is reasonable to say that --almost by definition-- the low-skilled workers outnumber high-skilled ones.


Crude oil benchmarks and production places

This article states:

  Brent, WTI and Dubai-Oman are the main crude oil benchmarks of
  the current oil pricing system. Nearly all oil traded outside America and the Far East is priced using Brent as a benchmark. WTI is the main benchmark used for pricing oil imports  into  the  USA.  Dubai-Oman  is  used  as  a  benchmark  for Gulf  crudes (Saudi Arabia, Iran, Iraq, the UAE, Qatar and Kuwait) sold in the Asia-Pacific market.

Regarding where the oil is produced, according to Investopedia:

  “Brent” actually refers to oil from four different fields in the North Sea: Brent, Forties, Oseberg and Ekofisk. [...] WTI refers to oil extracted from wells in the U.S. and sent via pipeline to Cushing, Oklahoma. [...] [Dubai-Oman is a] “basket” product consisting of crude from Dubai, Oman or Abu Dhabi.

The above is sometimes visualised in the following map: