Econometrics: Is elasticity meaningful in my, or any, regression?

The answer to the question is yes, it is indeed meaningful (at least mathematically speaking). If you estimate the linear equation

$$ W = \beta_0 + \beta_1 PTR, $$

then $\beta_1=\frac{\partial W }{\partial PTR}$, meaning that $\beta_1$ represents the marginal change of $PTR$ over $W$. Now, if you estimate

$$ log(W) = \beta_0 + \beta_1 log(PTR), $$

then $\beta_1=\frac{\partial W}{\partial PTR}\cdot\frac{PTR}{W}$, which is the very definition of elasticity.

Generally speaking, linear transformations only affect the interpretation given to the coefficients, but the validity of the regression itself (in broad economic terms) is given by the model's assumptions and the economic phenomena being analyzed.


I think you've misunderstood how the (UK) income tax brackets work. They work differently to how stamp duty used to work.

Stamp duty brackets used to be absolute rates: so when the purchase price crossed over the bracket threshold by £1, the marginal rate would be huge, as the whole purchase price was taxed in the higher rate. Hence the clustering.

That's not what happens with income tax. (and it's no longer what happens with stamp duty, either)

With income tax, only the amount of income that's above the threshold, gets taxed at the higher rate. So, the marginal rate is always equal to the band rate. i.e. if the rates are as in your example, and your income rises from 10,000 to 10,001 then you would pay 10% on the first 10,000 and 20% on the extra 1. You would not pay 20% on all 10,001.


Why is a company allowed to change the number of shares it’s divided into after its IPO?

If it were impossible to raise equity capital once a corporation has gone public, it may be impossible for it to expand if lenders have reached the prudential limits for lending to the corporation. This is highly constraining, and could lead to the company’s bankruptcy.
Does raising capital lower share prices? If we make the assumption that the share price is always equal to book value (this obviously is often violated in the real world), when it sells new shares, it gets cash equal to the number of shares. The result is that book value and market capitalisation increase by the same amount. However, this is somewhat short-sighted way of looking at it. The corporation may only be able to survive in the long term via issuing new equity.


Value added GDP measures the total market value of all final goods and services produced in the same country. Intermediate goods (goods or services that are input in the production of other goods) are not included in Value added GDP to avoid double counting.

The making of bread( raw materials) would be the cost of the initial inputs to include transportation services.  In the Transportation statistics annual report of 1996 transportation services are components of both final demand and production.  However, a large share of transportation activities, like freight, is identified as intermediate inputs.

https://books.google.com/books?id=r3sgAQAAMAAJ&pg=PA42&lpg=PA42&dq=intermediate+inputs++trucking+service&source=bl&ots=2OtHjJ5qhp&sig=ACfU3U31CxEpI6jMe3hQiNIZzbhXGg1nbQ&hl=en&sa=X&ved=2ahUKEwi35tbZ07npAhX-mHIEHf0bCqEQ6AEwA3oECAUQAQ#v=onepage&q=intermediate%20inputs%20%20trucking%20service&f=false

components of Value Added (VA): Labor Income (LI)[cost of workers], Other Property Income (OPI),[capital costs] and Taxes on Production and Imports (TOPI).
So I listed labor separately to be accounted for value added.

Valued Added would omits the following:

the cost of raw material (13 dollar)

1 dollar to transport it to someplace


Why is so much of macroeconomic theory dedicated to economic growth?

Have you read:

Barro and Sala-i-Martin (2004, 2e), Introduction
Acemoğlu (2009), Introduction
Paul Romer (2016), "The Deep Structure of Economic Growth"
Robert Lucas (1988), "On the mechanics of economic development", Introduction:

  I do not see how one can look at figures like these without seeing them as representing possibilities. Is there some action a government of India could take that would lead the Indian economy to grow like Indonesia's or Egypt's? If so, what, exactly? If not, what is it about the 'nature of India' that makes it so? The consequences for human welfare involved in questions like these are simply staggering: Once one starts to think about them, it is hard to think about anything else.

Ultimately what one believes is the most important topic is a subjective/normative question, but I think the above pieces make a strong case that economic growth should be it.


Why is the derivative of this SWF non-positive?

This looks like a simple first order condition from constrained optimization. If the maximum is interior, i.e. if $t_i>0$, then the first derivative must be zero. If the maximum is on the boundary, i.e. if $t_i=0$, then the first derivative must be $\leq 0$. If it were $>0$, then we can't have an optimum, because then the value of $U$ could be raised by increasing $t_i$. It can be negative, though, assuming that $t_i$ cannot be negative.


Is there any fundamental reason that output elasticity of capital $<1$ in general?

The elasticity of output with respect to capital will be less than 1 due to the diminishing marginal returns of capital - this is both realistic on macroeconomic scale and also one of the central assumptions of the model.

According to Romer’s advanced macroeconomics, pp 12 section 1.2 assumptions:

  “The intensive-form production function, $f(k)$, is assumed to satisfy $f(0)=0$, $f’(k)>0, f’’(k)<0$ ... Thus the assumptions that $f’(k)$ is positive and $f’’(k)$ negative imply that the marginal product of capital is positive, but that it declines as capital (per unit of effective labor) rises.”

This is one of the standard assumptions of Solow-Swan models so $a_K$ (or in Romers book $\alpha_K$) will be less than 1 just by assumptions of the model.

Side note: this is actually also implied by the Inada conditions themselves (which as Romer points out are stronger than needed for the model's central result) since $\lim_{k\rightarrow 0 } f’(k)=\infty$ and $\lim_{k\rightarrow \infty } f’(k)=0$ together with other assumptions of the model imply that $a_k<1$ - and hence the model should always converge.


Research on the existence of utility functions?

The most famous one is probably Afriat's 1967 paper titled "The construction of a utility function from demand data".

You can also find the basic idea in Hal Varian's Intermediate Microeconomics: Revealed Preferences Chapter or you can read the generalized version of the theorem in the paper.


why is labor $h_t$ often average working time percentage of non-sleeping time in RBC?

Specifications of this form treat labor $h_t$ and leisure $1-h_t$ as two alternative ways of spending a total time endowment.  Labor can be used as an input to the production function, while leisure enters directly into the utility function as a desirable end in its own right.

In this environment, there are many equivalent normalizations. There is no need to say that the total time endowment is 1; instead, we could just as easily say it is 100, and by modifying the production function and utility function accordingly we'd get an isomorphic model.  There's nothing special about any particular normalization.

That said, given a particular functional form for utility $u$, it often does matter how we define the time endowment, and this issue has played an important and somewhat embarrassing role in the history of RBC. In particular, if we normalize the total "time endowment" to 1 and say that $h_t$ is the labor fraction of the time endowment, then it can matter greatly whether we define the time endowment to be 24 hours every day for all individuals, limit to non-sleeping time, limit to non-sleeping time of working age adults, etc.

Why does this matter? The basic RBC model has typically assumed that intratemporal utility depends on a constant-returns-to-scale composite of consumption $c_t$ and leisure $l_t$. Given this, we can infer that this composite must have a constant elasticity of substitution of 1 from the fact that long-run labor supply seems roughly stable even as productivity and real wages have a strong upward trend. (See, for instance, pages 6 and 7 of Prescott 1986.) An elasticity of 1 means that the composite is Cobb-Douglas in consumption and leisure, taking the form $c_t^{1-\phi}l_t^\phi$ for some $\phi$.

RBC models then generally assume a constant intertemporal elasticity of substitution $1/\gamma$ for this composite, giving us the period utility function found on page 6 of Prescott (1986):
$$u(c_t,l_t) = \frac{[c_t^{1-\phi}l_t^\phi]^{1-\gamma}-1}{1-\gamma}$$
In the common special case with unitary intertemporal elasticity of substitution $1/\gamma=1$, the above reduces to simply
$$u(c_t,l_t) = (1-\phi)\log c_t + \phi \log l_t\tag{1}$$
Now, one important feature of a dynamic macroeconomic model is the Frisch elasticity of labor supply, which is the elasticity of labor supply to real wages holding marginal utility of consumption constant. This is important because it determines how the household will intertemporally substitute labor in response to the intertemporal pattern of real wages and real interest rates.

At an optimum the marginal utility of leisure must equal the real wage times the marginal utility of consumption, so for constant marginal utility of consumption $\lambda_t$ we have the condition
$$\frac{\phi}{l_t} = w_t\lambda_t$$
It follows that the Frisch elasticity of leisure $l_t$ with respect to $w_t$ is $-1$; this is a more-or-less direct consequence of the assumption that consumption and leisure form a Cobb-Douglas aggregate. But since $l_t = 1-h_t$, the local Frisch elasticity of labor $h_t$ is
$$\text{Frisch elasticity of labor}=\frac{dh_t/h_t}{dw_t/w_t}=\frac{-dl_t/h_t}{dw_t/w_t}\\
=-\frac{l_t}{h_t}\times \underbrace{\frac{dl_t/l_t}{dw_t/w_t}}_{\text{Frisch elasticity of leisure}} = \frac{1-h_t}{h_t}$$
From this, we can see that the definition of "time endowment" 1 matters greatly. For instance, from Table 1 of the most recent American Time Use Survey we see that average working hours per day are 3.14. If we take the time endowment to be all 24 hours, then $h=3.14/24 \approx .13$, and the implied local Frisch elasticity is about 6.6. If, on the other hand, we take out sleeping and other personal care activities, the remaining time endowment is 14.5 hours, in which case $h\approx.22$ and the implied Frisch elasticity is about 3.5.

Both these values are far higher than most Frisch elasticities estimated using micro data, as discussed by Chetty et al. (2012). Moreover, it seems bizarre that something as deep and important as the Frisch elasticity should depend on a choice as arbitrary as the definition of "time endowment". This connection is ultimately attributable to the assumption that $c_t$ and $l_t$ form a constant-returns-to-scale composite; King, Plosser, and Rebelo (1988) show that if we discard this assumption, a more general "KPR" functional form is consistent with the stylized fact of stable long-term labor supply, and we can potentially calibrate this more general form to the (much lower) elasticities estimated in the labor supply literature rather than imposing an unrealistically high Frisch elasticity by fiat.

For reasons that I do not completely understand, much of the RBC literature has followed Prescott (1986) and used the "log-log" utility parameterization (1) above, rather than calibrating the KPR functional form to match some externally estimated elasticity. This is probably due to a combination of inertia and convenience: the high Frisch elasticity guaranteed by the log-log parameterization helps to make labor volatile enough to match the macro data. And this is all connected to a lengthy and convoluted debate about the possible difference between "micro" and "macro" elasticities - see Chetty's slides or paper to read more about this.


Linear Regression Assumptions of Homoskedasticity

For the same reason you're taught the very simple supply and demand model of perfect competition in econ 101. It's not that it's wrong, it's a simplification of the real world and a good place from where to start. Once you've mastered the basics, you can learn more advanced topics. Heteroskedasticity isn't that advanced of a topic tho, most undergraduate students who take a second econometrics course get to learn about robust tests and all that.


Why are high/rising property values considered a good thing?

Considered a good thing by whom?

Financial institutions and speculators: yes, since houses for them are just another asset. Higher asset prices is a key mechanism of profit making for financial institutions (capital gains). In the recent financial crisis house prices collapsed and people defaulted on their mortgage payments. Thus,  the fall in the value of houses repossessed by banks meant the return on financial assets associated to housing translated into massive losses for financial institutions and government agencies associated with the housing market like Fannie Mae and Freddie Mac in the US.
Home owners: partly yes. A price increase is good, particularly if you have a mortgage to pay (so if you sell the house you can pay the loan back). Rising house prices are also good for pensioners, which main source of wealth is their house. Rising prices also give people the feeling they are richer (and hence spend more). Some argue that the positive appreciation of the media of rising house prices is partly due to the fact that newspapers are written and read mainly by older people, not by youngsters i.e. renters.
Renters: no, because rents are associated with house prices. If incomes are not growing at the pace of house prices, renters will be squeezed, to the benefit of landlords and homeowners. This is true in the UK (and in many other advanced economies, see here):

Central banks, and perhaps the overall economy and population: not necessarily, because rising house prices fueled by speculation and indebtedness (e.g. speculators and investors entering into the buy-to-let market) mean possibilities of a bubble bursting and a subsequent financial crisis. For example, in the UK, Mark Carney has expressed such concerns in the past.
Estate Agents: yes, as their business as intermediaries depends partly on fees from sales. People is more willing to sell when the market is on the up, fostering the property chain. If prices are falling, people prefer to wait instead of selling their house, breaking the property chain.


What is the purpose of monetary expansion during an economic lockdown?

What is the purpose of monetary expansion in this environment?

Avoiding (and preventing future) supply and demand shocks

The purpose is to keep as much as possible markets (labor related, of goods and services, etc...) in their pre-crisis configuration -- imperfectly admittedly -- with the intention of avoiding and preventing future post-crisis supply and demand shocks, which empirically translate into (out of control) inflation (positive or negative: deflation).

Doing it well

But when I say "supply" or "demand", I am not only talking about final demands and final supplies. I am talking about supply chains in their entire verticality, i.e. which is related to any intermediate consumption (and related market). And you do not want these intermediary markets to be strongly shocked either, since any shock in those would then propagate to final consumers. As outlined previously, the labor market is a intermediary market like any other from this perspective.

This is where decision makers actually have their role to play, because more than supporting the economy, they have to do it well. Imagine that:

They maintain final demands without adequatly maintaining intermediate markets $\implies$ inflation (which is what happened the most historically), potentially hyper depending on how strong and badly they do it
The net effect of their stimulus boils down to supporting more supplies than demands $\implies$ deflation...

In coordination

Moreover, in a globalized and highly interconnected economy, you may have control on your domestic product, but clearly not on foreign providers whose economic health depends on foreign monetary policies. Hence the need for a coordinated central bank action.

In conclusion and to say it simplistically, what would happen after the crisis if agents use their savings to buy consumer goods to the last company that didn't go bankrupt ? $\implies$ hyperinflation + social tensions.

Targeting a V-shape recovery

  [...] with consumers so constrained, can it work in a meaningful way?

In Germany or in France, partial unemployment -- financed by governments at virtually no (short-term) cost thanks to QE -- is a way to maintain both final demands and intermediary consumptions of productive factors (such as labor force) while favouring a V-shape activity recovery:

As of April 30 2020, 11 million French workers are currently temporarily or partially unemployed. Also, almost 650,000 businesses had applied for Germany's reduced-hours work scheme by April 6th. In the USA, Trump's helicopter money is a manner to protect final demands' purchasing power and limit the number of bankruptcies.

Idem for China, who cut interest rates to be more powerful in supporting the economy, announced on February 5, 2020, 16 preferential policies to sustain the production and operation of SMEs including policies on taxation, finance, social security, subsidies, and rent reduction). China's V-shape recovery follows:

  Is the impact in financial markets sufficient?

Sufficient to what ? As Krugman says:

  I'm a bit disappointed that so many people — including financial reporters! — still assume that the stock market is telling us something about real economic prospects [...]

Not related to Economic Science as such

Then, something you could ask is "why not rely on private capital?" Economic theory stops here. And it all depends on your ideology.


Solow Model: Steady State v Balanced Growth Path

This is when the attempt at accuracy creates confusion and misunderstanding.

Back in the day, growth models were not incorporating technological progress, and led to a long-run equilibrium characterized by constant per capita magnitudes. Verbally, the term "steady-state" seemed appropriate to describe such a situation.

Then Romer and endogenous growth models came along, which also pushed the older models to start including as a routine feature exogenous growth factors (apart from population). And "suddenly", per capita terms were not constant in the long-run equilibrium, but growing at a constant rate. Initially the literature described such a situation as "steady state in growth rates".

Then it appears the profession thought something like "it is inaccurate to use the word "steady" here because per capita magnitudes are growing. What happens is that all magnitudes grow at a balanced rate (i.e at the same rate, and so their ratios remain constant). And since they grow, they follow a path..." Eureka!: the term "balanced growth path" was born.

...To the frustration of students (at least), which have now to remember that for example, the "saddle path" is indeed a path in the Phase diagram, but the "balanced growth path" is only a point! (because in order to actually draw a Phase diagram and obtain a good old long-run equilibrium, we express magnitudes per effective worker, and these magnitudes do have a traditional steady-state. But we continue to call it "balanced growth path", because per capita magnitudes, which is what we are interested in, in our individualistic approach), continue to grow).

So "balanced growth path" = "steady state of magnitudes per efficiency unit of labor", and I guess you can figure out the rest for your phase diagram.


What to do for missing points in CPI time series?

Statistical Analysis with Missing Data by Little and Rubin is the go-to reference for working with missing data, at least if nothing state of the art is required. In general, this is a complex problem that remains an area of active research. The relatively easy cases are when the data is missing-completely-at-random or missing-at-random. Even among the most basic single imputation methods you have a lot of choices (list from Little and Rubin):

Mean imputation (replace with mean values)
Regression imputation
Stochastic regression imputation
Hot deck imputation (substitute individual values drawn from "similar" response units)
Substitution (not relevant in your context)
Cold deck imputation (replace missing value with constant value from external source like last value)
Composites of the above methods

However, if all the the general price level is what's of interest and not the dynamics of the price level (e.g. because you want a deflator instead of studying inflation dynamics) linear interpolation / extrapolation may be just fine. Fundamentally, since deflation is rare, if prices are 100 at time t and 110 at t+2, realistically prices at t+1 are going to be somewhere in $[100,110]$ and lots of models can get you there.

You can check the within and out of sample prediction quality to asses if your method is a good predictor of the missing prices. Within sample testing could be as simple as asking if the $R^2$ is high of predictive model. Obviously you can do much richer analysis than that. For out of sample testing, consider splitting the sample and calibrating the model only on the first half of the data, then evaluating prediction quality on the second half of the data.


How to find the market equilibrium by surplus maximisation when there is a consumer subside?

never work on weekends :-)

What you really do here is to find the total surplus in terms of $Q*$ and then maximise it setting the derivative to $Q*$ equal to zero.
So, you just maximise
$$TotalSurplus = - \frac{3}{16}Q^{*2}+\frac{9}{2}Q^{*}  + 2 * Q^{*}$$
for $Q*$ and you obtain the expected results.


Classical Theory of Inflation - Money Supply, Money Demand and Prices

If the demand curve SHIFTS to the right (due to a thriving economy and an increase in transactional demand for money) and the central bank keeps the money supply the same, the result would be deflation?

Yes, this would result in deflation ceteris paribus. In fact Mankiw Macroeconomics 8ed also mentions that in passing on pp 335, but you are right it does not explain intution.
The intuition is that if money demand shifts to the right and money supply is fixed that means that each individual unit of money will have higher value (e.g. this is equivalent to standard supply and demand model - fix supply of houses, increase demand for houses and you get that they have both higher value and price in equilibrium). Increase in value of money means that purchasing power of money increases which decreases the price level. New equilibrium interest rate will be higher because interest rate is essentially a price at which you can obtain money from banking sector (I am oversimplifying a bit to give you an intuition). Consequently, shift in money demand to right will increase interest rate, value of money and lowers a price level.
Conversely, if demand shifts to the left and money supply is fixed the value of each unit of money will drop, the interest rate drops as well and price level increases.
In addition, it is not true that no source discusses this. It is true Mankiw focuses on money supply in his Macroeconomics, but in Blanchard et al Macroeconomics an European Perspective there is whole sub-chapter that focuses just on change in money demand (see chapter 4.2 pp 62).


Evidence on economic impact of 2015 European refugee/migrant influx

I didn't see the evidences on the recent European immigration, but the event is similar to the Mariel Boatlift story in Card (1990):

  The Mariel immigrants increased the population and labor force of the Miami metropolitan area by 7 percent. Most of the immigrants were relatively unskilled: as a result, the proportional increase in labor supply to less-skilled occupations and industries was much greater. Nevertheless, an analysis of wages of non-Cuban workers over the 1979-85 period reveals virtually no effect of the Mariel influx. Likewise, there is no indication that the Boatlift lead to an increase in the unemployment rates of less-skilled blacks or other non-Cuban workers. Even among the Cuban population wages and unemployment rates of earlier immigrants were not substantially effected by the arrival of the Mariels.

That's for a 7% shock. In contrast, Europe experience a 0.1% shock in labor supply. Maybe the only difference we'd find will be in the areas where immigrants settle densely.

For more reading:

Card, “Immigration and Inequality”;
Card, “Is the New Immigration Really so Bad?”

In general, economists suggest that low-skilled natives suffer a bit from an influx of low-skilled immigrants (but again, we're talking about large numbers, >1% of population):

http://www.igmchicago.org/igm-economic-experts-panel/poll-results?SurveyID=SV_5vuNnqkBeAMAfHv


The appeal of low-dividend stocks when trading with government debt

In the paper cited at the end of this answer, the incentives for all three parties involved are discussed. Not only did the creditors suffer from a lower return after the conversion, they also faced longer maturities and, given the strong heterogeneity in lending terms across bonds, also a destruction of bond specific information.

Nevertheless, conversions in line with your description were beneficial for the owners of the converted bonds since the conversion increased liquidity (the liquidity of the shares highly contrasted the illiquidity of government bonds) and reduced the probability of government default. In addition, the shares represented claims on rents of the company. Though, for the specific case of the South Sea Company, there have never been any rents involved.

  Stephen Quinn, 2008. "Securitization of Sovereign Debt: Corporations as a Sovereign Debt Restructuring Mechanism in Britain, 1694-1750,"
  Working Papers 200701, Texas Christian University, Department of Economics. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=991941


AR(1) in Discrete Version of Stochastic Growth Model

I find it difficult to just say government spending follows an AR(1) process this is because you will end up with violating the "No Ponzi Game Condition" on most models including government.
To model changes in government spending you would simply have to concider different ways of shocking your model.
If you are thinking about optimality in your model and if there is a place for government, formulate your environment in the context of a planners problem and check how optimality looks and if in fact there is space for government.
I hope this helps


An overview of vouchers that is a bit outdated but a good place to start building an understanding of past work:

http://www.nber.org/papers/w7092

On school vouchers:

https://www.aeaweb.org/articles?id=10.1257/aer.97.3.789

http://faculty.smu.edu/millimet/classes/eco7321/papers/rouse.pdf

https://www.aeaweb.org/articles?id=10.1257/aer.96.3.847

https://www.aeaweb.org/articles?id=10.1257/aer.90.1.130

https://jswinton.gcsu.edu/ECON%203640/Expenditure/voucher%20potential.pdf

housing vouchers:

https://www.aeaweb.org/articles?id=10.1257/000282804322970788

https://www.aeaweb.org/articles?id=10.1257/pol.20130064

https://economics.nd.edu/assets/153486/carr_jillian_jmp.pdf

http://www.nber.org/mtopublic/482.pdf

^^^ Those are a few of the papers I've read over the last two years involving vouchers.


Need Advice Debugging MLE Code to Estimate an ARMA(p,q)

If anyone per chance runs into the same issue,

The reason the MLE is exploding is that a system with explosive roots (eigenvalues outside the unit circle), might make the Kalman-filter predictions explode at a point, and thus creating a few very large MLE values.

The solution is to rig the MLE function s.t. if any of the eigenvalues lie outside the unit circle, return an MLE >=0. Thus the optimization algorithm will be forced to find a stationary local minimum.


When does the "stationary" problem become and issue with panel data?

The data sample is so small that formal testing for stationarity would be essentially worthless. Inspect visually your individual series for any obvious trend. This would be the case where even with a short sample non-stationarity would be a problem.


Is there a good mechanism to incentivize leveraged firms to take less risks?

You could augment an externality mechanism, where people are punished for the effect they have on other people, and use a convex function or variance of outcomes. You could also transform their payoffs with a concave function to induce risk aversion. Look into the VCG mechanism and extensions for more info.


I think I understand now. If we look at example 8.1 on the previous page, the $y_{tm}$ variables may be the same, which would allow for some of the $\mathbf{z}_{tm}$ variables to be endogenous. And we pool all those endogenous variables (both the $y_{tm}$ and of the $\mathbf{z}_{tm}$) in one vector,  $\mathbf{y}_{t}$, and all the remaining ones (exogenous) in the $\mathbf{x}_{t}$ vector.


What is the percentage of announced Merger & Acquistion deals actaully completed?

If I'm reading it correctly, Table X (page 2633) of Schwert (2000) (Journal of Finance, Hostility in Takeovers: In the Eyes of the Beholder?) says that about 78 percent of deals in 1975-1996 were successful.

However, this measure is constructed based on the acquisition of the firm, not the bid of the acquirer, so that if there are multiple bidders this is coded as a 1 if any succeed and a 0 only if all of them fail. So it isn't what percentage of bids succeed but rather what percentage of targets are ultimately acquired. I couldn't find any measure of the probability of a bid succeeding but it must be significantly lower.

One tricky issue is that while successful bids must be revealed unsuccessful one need not be, so there is selection bias in the sample of bids. Therefore, a narrower way of characterizing this result is to say that of the potential deals where at least one bid was made public 78 percent were ultimately acquired by some firm.


Which are the different Schools of Economic Thought?

Post-Keynesian

Post-Keynesianism (PK) is based on the criticism of the so called "Keynesianism", which according to PKs is not loyal to core Keynes ideas. As such, this school of thought aims to be called the "true" Keynesians.

The criticism starts with the workhorse model of Keynesianism, the IS-LM model, developed by Hick in a 1937's article, right after Keynes magnum opus. According to Minsky (a prominent PK), this is

  an article which ... misses Keynes’ point completely’ (Minsky, 1969, p. 225)

Later in life Hicks acknowledge this, by stating that his model

  was Walrasian rather than Keynesian in origin (Hicks, 1981, p. 142)

Given this background, PK has the following core features in its method:

emphasis on disequilibrium rather than equilibrium (as in IS-LM).
rejection of rational expectations. Agents can make inaccurate estimates of future based on present information without need to be either fully rational or incredibly naive.
against microfoundations. PKs argue that "emergent properties" of a system, arising from the interaction of individuals, means a system cannot be understood from a simple extrapolation of the properties of a few agents (e.g. representative firm and household).
the use of Leontieff (fixed proportion) production functions. PKs reject the marginalist theory, and argue that the latter is empirically inconsistent (e.g. Blinder, 1998).
Money is not neutral. This is a straightforward Keynesian argument, also present in IS-LM. Modern approaches highlight the crucial role of banks in endogenous money creation, which is not dependent on reserves (e.g. Moore, 1979).
Government has an important role in stimulating aggregate demand. Also in line with IS-LM, PKs argue that effective demand is what matters for economic activity. Say's law is therefore rejected.

Some early authors of this discipline are Michael Kalecki, Joan Robinson, Nicholas Kaldor, Luigi Pasinetti, and Piero Sraffa. More recent authors include Wynne Godley, Steve Keen, Frederic S. Lee, and Marc Lavoie.

A recent introduction to PK can be found in Godley and Lavoie (2007). There is also a very comprehensive two-volume Oxford Handbook of Post-Keynesian Economics.

Sources: own research, and Keen (2013).


Open offer made by companies to take over

An "Open Offer" is not necessarily the same as a "corporate takeover."

An "open offer" (as defined by the link above) is

  the opportunity to purchase stock at a price that is lower than the current market price

An "open offer" is primarily used to raise funds for the company.

However, it is also important to note that, in an "open offer,"

  a company allows existing shareholders to purchase additional shares directly from the company in proportion to what they currently own. This is to prevent dilution to existing shareholders

Since the existing shareholders own the same proportion of stocks both before and and after the "open offer," the leadership of the company does not necessarily change.


I recommend the Comp-Net database. It is a firm-level data for European countries. It includes a large range of variables, including mark-ups:

Check the documentation for more details.

Many researchers have also used the Amadeus database, also for Europe. A nice summary about it is here.

For the US, take a look at this paper. The authors use three different sources of markups: Compustat, NBER Census of Manufacturing, and IRS (Inland Revenue Service). The first two are micro-data whereas the latter is aggregate at industry level.


There are many such models. A prime example would be the IS-LM AS-AD model where money supply is controlled by central bank and money can be created at will from nothing (I suppose that is what you mean by 'ex nihilo').
This model is one of the 'workhorse' models of modern macroeconomics so you can find its exposition in any textbook. If you want an introductory treatment then Blanchard et al Macroeconomics: a European Perspective offers one. If you want graduate treatment Romer Advanced Macroeconomics is good source. Or if you specifically want a paper not a book then you can see the Hicks, J. (1980). IS-LM: an explanation. Journal of Post Keynesian Economics, 3(2), 139-154.


How do Gini coefficients correlate with the cost of higher education?

I think this gives an answer reasonably close to what you're looking for (from the NYTimes article, "In climbing the income ladder, location matters," featuring research by Raj Chetty, Emmanuel Saez, and others).

The team of researchers initially analyzed an enormous database of earnings records to study tax policy, hypothesizing that different local and state tax breaks might affect intergenerational mobility.
What they found surprised them, said Raj Chetty, one of the authors and the most recent winner of the John Bates Clark Medal, which the American Economic Association awards to the country’s best academic economist under the age of 40. The researchers concluded that larger tax credits for the poor and higher taxes on the affluent seemed to improve income mobility only slightly. The economists also found only modest or no correlation between mobility and the number of local colleges and their tuition rates or between mobility and the amount of extreme wealth in a region.

(emphasis added)
Note that the study is primarily about income and social mobility, but they also mention that tey find modest or no correlation between tuition rates and amounts of extreme wealth in a region.


How to compute the Marshallian demand for this specific utility function

It means that the level surfaces of $u$ don't touch the surface $x_1p_1+x_2p_2+x_3p_3=1$ unless $p_2=p_3$. But, since $x_i\ge0$, it follows that the admissible domain of $x_i$ is compact and the maximum is still achieved somewhere. Namely, in some boundary point of the  admissible domain $-$ for $x_2=0$ or $x_3=0$.


Do you get economies of scope from horizontal or vertical integration?

We'll you've pretty much answered it yourself...
Assuming any integration happens, it'll be horizontal

Vertical integration is buying your customer/ supplier up/down the value chain

Horizontal will be to buy your competitor, and in the general case, its added value and the reason of purchase will be the specific products it creates, not its finance dept. for that matter

Regarding an example, most M&As I can think of right now are of vertical integration
If I remember any I'll add them as comments


Its important to keep in mind that the exchange rate is a "price for currency" and just like any other price it is determined by supply and demand. The main question now is what determines supply and demand for currency?
There are two main models that tell us how exchange rates behave based on the two main forces driving demand (and supply) for currency.
One factor is trade of goods (the goods market). If foreigners buy our goods they need our currency, so they demand our currency and higher demand ceteris paribus (all else staying constant) leads to a higher price and the value of the domestic currency increases. Based on such considerations in the goods markets we have the model of Purchasing Power Parity (PPP). However empirically it does not always hold and in reality it is mostly expected to hold in the long run.
Answer to Q1: Beyond the goods market discussed above, the second main factor is the capital market and that is what Investopedia is referring to. This gives rise to the model of Uncovered Interest Rate Parity, called UIP (there is also a "Covered Interest Rate Parity Model"). This says that if interest rates are higher in the domestic country compared to the foreign country, then foreign investors would like to invest in our country to get the higher returns. To do so they need our currency. So they buy it (demand it) and therefore as long as supply of currency doesn't increase (the central bank printing more money) the price and value of a currency must increase. Also note that there's no reason to expect the central bank to change the supply in a floating exchange rate regime, which most countries have (i.e. there is no attempt to intervene in the exchange rate to keep it fixed).
Answer to Q2: It is not necessarily true that a higher interest rate should increase inflation in general. However increasing the interest rate can decrease inflation. Your understanding of inflation is correct. The second part of the second quote you gave, that higher inflation decreases the value of the currency, however is correct. This is due to the first model we discussed, the PPP. The idea is that inflation makes goods more expensive and therefore our goods have less foreign demand, which leads foreigners to have less demand for our currency (they buy less of our goods so they need less of our money to buy our goods) and less demand for the currency reduces the value of the currency.
Answer to Q3: I believe this follows from the general discussion above.


Continuous rational and monotone preference relation implies $x\succsim0$?

If we take the definition of monotonicity to be if $x\geqq y$  then $x \succeq y$, you can simplify the proof (though it looks right).

Note $\mathbf{0}\leq x$ for all $x\in \mathbb{R}_+^l$. So by the definition of monotonicity (essentially replacing $y$ with $\mathbf{0}$ above), $x\succeq \mathbf{0}$. I don't think continuity is required (check lexicographic preferences to see it is not necessary for the stated result).


Is there a dutch book argument for the "independence of irrelevant alternatives" axiom?

Yes, and it's actually quite simple. Start with alternatives {A, B}, where you prefer A to B. If there exists some C such that introducing it causes you to prefer B to A instead, then the following holds:
I start by offering {A, B}, and you take A. Then I offer C as well, and also let you trade B for A so long as you give up some arbitrarily small amount of utility denoted dx. This means that your options are {A, (B-dx), C}, where dx is some arbitrarily small amount of utility that you will lose which is chosen so as to not change your opinion that B-dx is better than A. As a result you choose B-dx instead of A. The result is that you have given up utility for no benefit -- you could originally have chosen B originally, but instead took B-dx -- i.e. that you have been Dutch booked.


Why does the stock market tend to grow over time

Actually you are asking little bit different question in your title and then in your question. There is difference between asking why stock market always grows and why individual companies’ stock grow. In fact in case of individual companies you can see that many companies fail and their stock price gets wiped out despite the stock market always growing over time so those are fundamentally different questions but I will try to respond to both of them.

What determines individual stock price? There are multiple stock pricing models but they more or less boil down to estimating what will be the future discounted return on your investment in one way or another. An example of a stock pricing model would be the Gordon’s growth model, where the stock price is given by:

$$ P=\frac{D_0(1+g)}{r-g}$$

Where $P$ is the price $D_0$ is the current dividend, g is the dividend growth rate and r is the required rate of return.

So the individual stock price can increase either due to higher dividend payments, higher expected growth rate of these dividends or lower required rate of return.

However, as mentioned before there is no reason to think or even expect for any individual company’s stock price to always increase in a long term. In fact you have a lot of companies that simply fail and their stock price becomes zero.

Also note the explanation above is little bit oversimplification because there are firms that don’t pay out dividends but they are still being valued and their stock price can grow, but I on purpose did not wanted to make my answer overtly technical and include some more complex models that can handle that as well, and generally the explanation is still valid one.

How come that stock market tends to grow over time? This is because even though  individual companies can and often do fail they are usually replaced by new and more efficient companies. In general thanks to economic growth, which depends on technological growth, growth of population, or even due to endogenous factors like human capital accumulation, firms are constantly getting more efficient and more productive meaning they can generate larger profits with less inputs and consequently also payout more of this profit as dividends. Firms that can’t adjust to technological change and deliver better products more efficiently will fail and be replaced by those companies that can. Recent famous examples would be Kodak or Nokia.

This being said if the economic progresses of humanity would be halted for whatever reason the stock market could start failing over time. For example in case of some apocalyptic scenario such as nuclear war. It might also happen that eventually human race would end up in some stagnant state when there would be no more future growth because we would reach a point where we can’t get any new technological progress or accumulate more human capital - but that does not seem to be probable any time soon.


Perfect competition: Finding short run equilibrium price?

These kinds of questions depend heavily on the market structure that you assume. Most importantly, when firms have (some degree of) market power, they usually take the demand function as given and try to solve something of the kind

$$ \max_p D(p)\cdot (p-c(D(p)))$$

However here, when there is no market power, firms take the price as given and solve for the optimal quantity of production $q$:

$$ \max_q q\cdot (p-c(Q))$$

Then, we get the supply of any single firm, given the price $S(p)$ from solving this problem. To clear the perfectly competitive market, we remember that there are $30$ firms and solve:

$$p: D(p) - 30\cdot S(p) = 0$$


I am not sure that I am able to fully answer your problem, but I can give you a short answer to your questions. "Aren't these games simultaneous games? Why would the players consider the choices made in the first game to play the second game?"
I would refer to collusion and repeated games. Every stage game is simultaneous, but the fact that they are played sequentially makes them a super-game. In super-games the outcome of the first stage game influences the second, because in this way every player can introduce a punishment at the second stage if the other player did not cooperate. So pretty much like in collusion games, what matters are the threats one player can pose to the other through the second stage to obtain collaboration at the first stage.
Having said that, I'd suggest some easy readings:
Introduction: Cabral "introduction to industrial organization" chapter on game theory and collusion"
Game theory: Chapter 14 and 15 Osborne "Introduction to game theory"
Regards.


regression method for data that contain a number of observation for several years

Hang on - so, what is the unit of analysis in your data set?  Is it companies, or is it managers?  Think carefully about this.  How are the "top 5" ranked?  Does the cohort change frequently?  What variables are you interested in with respect to policy impact?

It's true that from a company view, you've got a panel, but if there's enough churn in the management roster you could justify a pooled approach.  In addition, if the missing data is non-random (e.g., cyclic, confined to certain firms/sectors/etc), then it would bias a panel approach, and a pooled regression may be the best you can do.

It's down to the particulars of your data, which only you have access to.


You have a log-log model, so the coefficients on performance can be interpreted as the elasticity of your outcome (performnce) with respect to the regressor of interest (level in the corporate heirarchy).
This table only shows the coefficient of interest for four separate regressions. For each of these job levels (CEO, other board, white-collar, all other), a 1% increase in the 'total expected ex ante pay' leads to a x-% increase in 'total shareholder returns'.
For example, based on the results reported in this table, a 1% increase in total ex-ante CEO pay is associated with a 0.22% increase in total shareholder returns, a 1% increase in total ex-ante 'other board executive' pay is associated with a 0.21% increase in total shareholder returns, etc.
All four coeficients are statistically significant at the 1% level (which you can confirm by looking up the values of each coefficient's t-statistic, which is just the coefficient value divided by the standard error).


Why does the Fed control interest rates?

The fed controls interest rates via supply and demand.

The fed directly controls only one rate, the rate at which banks can borrow from the government.

Lowering this rate permits banks to lower the rates they charge while making the same profit. They don't have to do so, but supply and demand says that those banks who do lower their rate can take business away from those which don't, so the market pressures them to more-or-less follow the fed rate's shifts.


Continuing on the comments exchange, the conclusion that the overall effect is to change the discount factor, is similar to the one reached in the original Overlapping Generations Model of Blanchard, where individuals are faced with a "probability of death" (surely a "catastrophic event" I believe).

See the Blanchard & Fischer book "Lectures in Macroeconomics", p. 117. The authors note that the result has been originally obtained by Cass and Yaari (1967) "Individual Saving, Aggregate Capital Accumulation, and Efficient Growth", a paper that is part of the book Essays on The Theory of of Optimal Economic Growth


Yes. Suppose the consumer has the following utility function:

$$u(x) = - \sum_{i=1}^n x_i$$

Here's a different example, where commodities are goods instead of bads:

$$v(x,y) = -(x-a)^2 - (y-b)^2$$

When the consumer's budget $m$ is strictly greater than $p_x a + p_y b$, then the consumer will not consume all of their budget.


Retaliating against a foreign country's import tariffs

If you're using a simple standard model of trade, like the Ricardian model, trade is good and all tariffs, price floors, price ceilings, etc are bad. Retaliatory tariffs will still be problematic for the country imposing them. I think your reasoning is fine in this regard.

However, trade occurs over many many years. Responding to the initiation of new tariffs by imposing temporary tariffs will reduce the short-term welfare for everyone, including the tariff initiator.  The country which began this "trade war" will then have lower welfare from both sets of tariffs.  Hopefully, the initiating country will remove their tariffs and return to normal.

The retaliating country has some short-run cost, but if it leads to a final low-tariff state, retaliation may be worth it in the long run.

Imagine the game is a repeated version of the following:

\begin{array}
\\ & Free Trade & Tariff
\\ Free Trade&(5,5)&(3,4)
\\ Tariff & (4,3)&(2,2)
\end{array}

Where for some reason a party has ended on Tariff, perhaps because they have a "trembling hand" with regard to policy.  Note this is different then the prisoner's dilemma because the payoff does not increase as a result of imposing tariffs.


Why do different product-markets record different rates of return if the assumption of competitive markets hold?

Edit: after OP made changes to his question and clarification this answer became less relevant. I am still keeping it here because OP said some parts of it and references are still useful.
There are several reasons why even if all markets were perfect rates of return would not equalize. I will just focus on some major ones.

Many countries have some capital controls in place. This means that even if markets were perfect the rate of return would be prevented to equalize by the lack of capital mobility. In fact assumption of perfect capital mobility is necessary assumption for rate of return equalization between different markets.

If capital is heterogeneous it might not necessary be easily transferable between countries which mean that capital poor countries will get less capital inflows. For example, some capital such as some computer for some production processes such as programming might require skilled workforce. If some country does not have skilled workforce then not all capital from capital rich country will be allowed to flow from capital rich country to capital poor country.

If places differ with aspects such as infrastructure or geography in general rates of returns might be different just due to differences in these factors as they would again prevent equalization of rates of returns and capital flows (for example country with no roads and harbor cannot really sustain any modern industrial complexes).

Note that you have to also take broader view of risk which encompasses also risks of nationalizations, some institutional risks (countries might be expected to adopt worse institutions in future) and so on. Risk should be not viewed as simply just the entrepreneurial risk.

In fact the reasons above is why you get Lucas paradox (see this question). Lucas paradox is the paradox that nowadays we observe that instead of capital flowing from capital rich countries to capital poor countries (which would equalize rates of return), the capital actually flows from capital poor countries to capital rich countries. The arguments above are more formally presented in Montiel (2006), Schularick & Steger(2008), Azémar & Desbordes (2013), Akhtaruzzaman, Hajzler & Owen (2018).
Also note that if you are wondering why the rates of return are different between different places you have to venture outside pure industrial organization to international macroeconomics or international economics. Industrial organization focuses mainly on explaining how firms compete on given market and while many models from IO are heavily used in international economics or international macroeconomics pure IO will not really discuss much differences between rates of returns in different countries. Hence if you want to pursue this topic further you should also have look at some international macro or international economics textbook. Most of these will discuss international capital flows (on which the equalization of rate of return depends) in decent detail.
Furthermore, of course you are right in assuming that many markets are imperfect and that is also why rates of returns are not equalized.


Local vs. foreign currency denomination for Emerging Market Bonds

An EME's government collects taxes in domestic currency. If it issues debt in foreign currency and the domestic currency depreciates, the value of foreign currency-denominated debt increases. For example 1 000 000 000 USD is due at some date. If the exchange rate is 1 peso per dollar the government has to collect 1 000 000 000 pesos to pay it off. If the exchange rate becomes 1.25 pesos per dollar the domestc government has to collect 1 250 000 000 pesos. If the debt was denominated in pesos all along the government would still need only 1 000 000 000 pesos.

If debt to GDP is high such a depreciation may cause the EME government to default.


Why does deflation cause banks to increase their interest rates?

I suspect there is a small mistake in your notes.

Deflation does not cause banks to increase their interest rates. However it is true that a deflationary spiral (or plain deflation for that matter) causes real interest rates to increase. These are the interest rates that matter for the economy anyway, which is why in macroeconomics we often refer to the real interest rate even if it's not explicitly stated so.

The nominal interest rate, set by banks in part, is the interest rate in terms of money. It determines how much money we have to pay back for credit.
The real interest rate  is the rate in terms of goods and services. It determines how many goods and services we have to pay back for credit.

The real interest rate is defined as:

$ (1+r) = (1 + i) / (1 + \pi) $, which can be approximated by:
$r = i - \pi$, where r is the real interest rate, i is the nominal interest rate set by banks and $\pi$ is the inflation rate.

When we have deflation we get a negative $\pi$, which, as can be seen from the equations above, increases the real interest rate r. The smaller $\pi$ is, the bigger r gets. When $\pi$ is negative, even if we set $i$ to its lowest possible value, which is 0, the real interest paid on it will be positive. This means the amount paid back in goods and services is larger than the amount borrowed and increasingly so the bigger the deflation (smaller $\pi$) is.


What is the resolution down to which the annual inflation value can be controlled?

They can target arbitrary number. For example, they can target $1.9999\%$ or $3.1415\%$.

Could it be set, for the sake of the example, to 1.9% instead? Or to 1.95%?

Yes, it could. However, the target inflation rate is more or less aspirational and it is the value Fed aims to hit on average over the business cycle rather than maintain it at all times (in fact the target is not just $2\%$ but $2\%$ over the long run (see official Fed explanation here). Fed has dual mandate of keeping price stability and full employment. Following both of these objectives requires that inflation is sometimes above $2\%$ and sometimes below. In fact recently Fed chairman announced that in the near term they want to target inflation moderately above $2\%$.
In addition, note that Fed is not expected to hit its target with surgical precision. Consequently, there is not much point in setting it to $1.95\%$. For example, if Fed would keep the $2\%$ but inflation over long-run would happen to average at $1.95\%$ they would likely consider it a success since it is close enough to their target. Even though Fed can control inflation, price level (change in which gives you inflation) depends also on random macroeconomic perturbances/shocks. Consequently, most economists would not consider it a failure if average inflation deviates from its long run target by just $0.5\%$.
Regarding the means Fed has to control inflation there are several ones. Following, Blanchard et al. Macroeconomics central banks can control inflation via:

It can do it via changes in the interest rates (this is one of the most common ways).
Direct changes to the money supply.
Changes to the reserve requirement (currently abolished in the US).

An important caveat is that Fed is also institutionally constrained in the use of the above. For example, helicopter money is one way how central bank could expand money supply from economic perspective, but it might not be legal in the US (it is highly questionable). Fed can still affect money supply in other ways such as open market operations but these constraints mean that sometimes Fed cannot go as far as it would want with expansion of money supply.
Some other instruments are constrained by economic reality. For example, Fed cannot set interest rates too low as there is effective lower bound on how low the Fed funds rate can go. Typically models show it is at zero interest rate (although practically it is possible to go bit below) because cash has implicit $0\%$ interest rate and thus if Fed would set its rate significantly below zero (which would make commercial banks to go below zero as well) people would just refuse to keep their money in their banks and just keep cash (the interest rate can go bit negative as there are storage costs for keeping large sums of cash at home).
However, going over every minutiae detail of how Fed is constrained by law or economics would be beyond the scope of SE answer, but you should keep in mind that there are asterisks to the above.


What is the relation among "welfare economics", "institutional economics", "environmental economics" and "neoclassical economics"

As dismalscience said, none of these terms has a truly rigorous and concrete definition. But here is a broad outline:

Environmental economics is the branch of economics that deals with environmental issues, such as providing incentives for pollution abatement, dealling with climate change, managing shared environmental/natural resources, etc.

Neoclassical economics is the dominant paradigm for economic analysis across virtually all fields of economics (including environmental econ). It is cheifly characterised by the assumptions that all actors behave rationally in the pursuit of some well-defined objective. In modern economic work, neoclassical analysis typically manifests in the presentation of a mathematical model of behaviour that embodies this rationality assumption.

Welfare economics is the branch of economics that deals with questions about how the well being of individuals/groups in society (as well as of society in aggregate) can be definied/measured. It also deals with studying the effects of particular policies/economic phenomena on welfare. Such analysis is usually conducted within the framework of neoclassical economics.

Welfare economics provides an important set of tools used by environmental economists to understand who wins and loses under different regimes. It also provides the tools necessary to define what a "good" outcome might be, when all outcomes involve some winners and some losers.

Institutional economics studies how and why particular forms of institutions emerge and how they affect economic outcomes. One particular focus of institutional economics is how society deals with situations where markets would otherwise fail. For example, the environment is a common resource shared by everyone, meaning that individual actors have an incentive to exploit the resource for private gains and hope that others will take care of it instead. A key question addressed by institutional economics is how institutions have evolved to deal with these kinds of problems. Such analysis sometimes, but not always, occurs within the framework of neoclassical economics.


Why doesn't Costco simply raise prices and nix the membership fee?

Because there is not a single successful model of actually doing business. Costco's model is pretty text-book level, but not economics textbook level, but rather marketing textbook level:

Differentiate ("In Costco we do things differently").
Make the customer feel special (hey, here is "members only"!).
Induce loyalty or at least "loyalty" ("now that I paid the executive fee I might as well buy from Costco to get part of it back")

Whether it will indeed be successful or not, depends on time, place, culture it operates in, as well as a few boring economic facts like, whether Costco's prices are indeed lower, etc.

Economics analyzes firm behavior at a much more abstract level, "losing sight" of all these different business strategies because Economics is ultimately interested in market outcomes (and then economy-wide outcomes), not individual-firm outcomes.


One can distinguish between probabilistic randomness (which can be quantified) and uncertainty. This article by a Fed Researcher discusses this: “The Stock Market: Beyond Risk Lies Uncertainty” (Frank Schmid). As the tag to the question suggests, this is associated with Keynes, and post-Keynesian economics at present. Therefore, if you look at the post-Keynesian literature, you will see more discussion of uncertainty.

“Mainstream” economics and finance focus on quantifiable models. As a result, they rely on probability theory, and expected values are very common. However, option pricing theory does take into account the whole probability distribution, and so “uncertainty” might be thought of as the properties of the tail of the distribution. Rather than describing the price of bitcoin as uncertain, the tendency would be to find a model (such as a “rational bubble”) that might fit observed data. That is, one can argue that it is possible to analyse bitcoin using a standard probabilistic framework.

Whether the probabilistic approach or the more qualitative post-Keynesian approach is “better” is going to be opinion-based, and so I do not want to pursue that. (As a disclaimer, I am in the post-Keynesian camp.)


Why is the cash in circulation on the rise in rich countries?

For starters, I've confirmed your results for the United States using data from federalreserve.gov and alfred.stlouisfed.org:

If we take a look at the breakdown of bills in circulation, we can see that the value of $100 bill has drastically increased relative to the total value of bills in circulation but the others bills have not.

So people are indeed using cash less often for everyday purchases that most likely involve lower denomination bills. In short, people are most likely hoarding more cash in the form of higher denomination bills as a result of recent interest rate trends. According to a blog post by the St.Louis Fed, "the nominal interest rate on short-term bonds has declined essentially to zero, and, in this case, the best form of risk-free liquid asset is no longer the short-term government bonds, but money."

You might ask why people wouldn't just hold more money in electronic currency versus paper cash? If we compare paper cash to MZM (which includes various forms of electronic currency storage), we see that the % of the value in paper cash have actually gone down.

That said, if there was more confidence in technology and banking system, the % in paper cash could decline further. There also seems to be some arguments for a larger underground economy in the US since 2008. I couldn't find any long-term trends on financial literacy, but I also don't see how it would affect people to use paper cash over other forms of credit.


The question is related to topic consumer surplus

Demand function in your question is given as

Q=12-3P

In order to find consumer surplus,we first find inverse demand function from it

Q=12-3P
12-Q=3P
12/3-Q/3=P
4-Q/3=P or
P=4-Q/3

At P = 3 ,from inverse demand function, we get Q= 3

Since intercept in the inverse demand function is 4 and base is 3,we can calculate consumer surplus by using the formula for area of a triangle as ab/2 i.e 4x3/2=6

Similarly,when price falls to 1,we get value of Q=9

Here,area of triangle becomes 9x4/2=18,which is the new consumer surplus

Subtracting 6 from 18, we get the change in consumer surplus as 12

So the answer to your question is 12.


converting sum of consumption basket to integral

This is because there’s a relationship between summation and integrals. An integral is basically a way how to calculate an area under the curve. With right sum you can do the same thing. For example, mathematically

$$ \sum_{i=0}^n f(i) \approx \int_0^n f(x)dx $$

In fact under the Reimann’s definition of integral, integrals are literary defined as a limit of a sum of function output multiplied by change of x or:

$$\int_a^b f(x)dx = \lim \sum_{i=0}^{n-1} f(\xi_i) \Delta x_i$$

Now to go back to your question you can imagine $p_i c_i$ as an area of rectangle of height $p_i$ and base $c_i$. Consequently $\sum p_i c_i$ can be considered an adding of of all areas of these rectangles and $\int p_i c_i di $ is just continuous approximation of this area.


Is there inverse relationship betweet prices of interest-bearing assets and the interest rate?

The shortest possible answer to the question in the title (as stated now): “Yes.”
However, based on the comments, there are questions about definitions. I will attempt to answer them.
A fixed income instrument is anything that pays the owner cashflows on a fixed legal schedule, and the owner is not obligated to make any payments (other than the initial payment). This covers bonds, loans, and other forms of debt.
If there is no optionality in the structure (not prepayable, etc.), the internal rate of return of the security is unique and has an inverse relationship with the initial price. (Excluding corner cases such as zero cost, etc.)
This internal rate of return is typically referred to as the rate of interest or yield of the instrument. (Bond and money markets have specific calculation conventions, which end up being a more complex version of the internal rate of return.)
When trading fixed income securities, they can be traded in terms of yield or price interchangeably (although markets follow conventions for which to use in a quote). As such, prices and “interest rates” are mechanically always moving inversely.
The question seems to be focussed more on the policy rate, which is often “the” interest rate in simplistic economic models. Modern central banks force a short-term nominal instrument rate (often overnight, but can be two weeks or even three months) to some target level, that is changed over time as a policy tool.
Despite what simplistic models say, there is no relationship between the “money supply” and this rate. (People discussing this relationship are typically using Economics 101 textbooks that are decades out of date.)
The yields on other instruments move around on a daily basis. Many factors can influence prices, but there are bedrock valuation principles.

Instruments that have no perceived default risk (e.g. most non-Euro area developed country sovereigns) have a fair value that matches the expected path of short-term rates. The set out of outstanding bonds defines a risk-free yield curve.
Other instruments will incorporate a default risk premium and/or a liquidity premium that is added to the risk-free yield curve.


on economic growth and Second Welfare theorem

The two statement say two completely different things. So, no, there is no contradiction.

The second welfare theorem essentially says that a system of transfers that results in an efficient allocation can be supported by a competitive outcome. The statement about replacing income taxes with a consumption tax will increase peoples' incentives to save. These are two completely different statements.

Your mistake is that you suppose that the second welfare theorem prescribes a system of taxation to achieve the hypothetical transfers. It doesn't. The second welfare theorem just supposes that these transfers can be made---nothing more. Importantly, the alleged social planner who makes the transfers has complete information about the agents in the model. When information about the agents is hidden from the social planner (information such as the skills and true abilities of each individual), such transfers become difficult and costly. That is, they create distortions. This was a point that was famously made by James Mirrlees in his seminal paper "An Exploration in the Theory of Optimum Income Taxation" (1971). The point of this paper is that there is a trade-off between equity and efficiency. Once we consider the fact that transfers are costly and create distortions, the question is, how do we create a tax system that minimizes the amount of distortions and still is able to make desired transfers of wealth. This serves as a motivation for why people might study the incentives created by an income tax or consumption tax, for example.


Why some countries are wealthier than others?

One argument brought forward (it is a generalization/argument for lack of infrastructures) is the Acemoglu-Robinson hypothesis of extractive versus inclusive institutions.

Long-story short: some institutions are good for (long term) growth ("inclusive"), and some are good for short-term extraction of resources ("extractive"). Depending on whether the elites / rulers of a country (colony) had long-term plans of "staying in a country" or whether they wanted to just "quickly get everything they can" out of the country, they installed these different type of institutions.

See also their slides and book.


Is an income tax always more favourable for consumers compared to ad valorem/quantity tax?

You question can be answered using a revealed preference argument.
Let $B = \{q \in \mathbb{R}^n_+| p' q \le m\}$ be some budget set of a consumer (i.e. $B$ gives all possible bundles that the consumer can choose).
Let $q^\ast$ be the optimal choice from $B$, i.e. the bundle that optimizes the utility. Then for any other bundle $q \in B$, it must be that $u(q^\ast) \ge u(q)$.

proof: The proof is simple, if, towards a contradiction, we would have that $q \in B$ and  $u(q) > u(q^\ast)$, then this would contradict the assumption that $q^\ast$ was optimal in $B$.

Now, let's apply this to the two budget sets for your question.
The first budget set has a lump-sum tax equal to $R$.
$$
B^1 = \{q \in \mathbb{R}^n_+ | p' q \le m - R\}.
$$
The second budget set has an ad-valorem tax equal to $t$ for some good, say $i$.
$$
B^2 = \{q \in \mathbb{R}^n_+| p' q + t q_i \le m\}.
$$
Let $q^{1}$ be utility maximising, bundle in $B^1$ and let $q^2$ be the optimal, utility maximising, bundle in  $B^2$.
In addition we assume that the revenue of the ad-valorem tax equals the revenue for the lump-sum tax:
$$
t q_i^2 = R.
$$
We would like to show that $u(q^1) \ge u(q^2)$, so the lump sum tax always gives at least as much utility as the ad-valorem tax.
Using the above revealed preference argument, it suffices to show that $q^2 \in B^1$, which means that $q^2$ was a feasible option when $q^1$ was chosen.
Indeed, we have:
$$
q^2 \in B^2,\\
\iff p' q^2 + t q^2_i \le m,\\
\iff p'q^2 \le m - t q^2_i,\\
\iff p' q^2 \le m - R.\\
$$
So $q^2 \in B^1$.
Remark: notice that we did not impose any restrictions on the utility function. The only thing that we assumed was that the consumer was able (from any budget set) to pick an optimal bundle.


US Student debt volume and default rate data source

With regard to your title, I assume you're looking for data for the US. This report from the U.S. Department of Education presents data and selected findings about the repayment of federal student loans using data from the 2015 Federal Student Aid Supplements to two Beginning Postsecondary Students Longitudinal Studies (BPS).

The report focuses on two cohorts of borrowers: those who began their postsecondary education in the 1995–96 academic year (BPS:96/01) and those who began in the 2003–04 academic year (BPS:04/09).

The report presents estimates of both cohorts' cumulative borrowing, repayment, and default statuses as of June 30, 2015, some 20 years after the 1995–96 cohort and 12 years after the 2003–04 cohort began postsecondary education.

Interesting key findings

Some 55% of students in the 1995–96 cohort and 63% of the 2003–04 cohorttook out federal education loans within 12 years of entering postsecondary education.
Overall, 25% of borrowers in the 1995–96 cohort had defaulted on at least one loan within 20 years of beginning postsecondary education, and 14% defaulted on their most recent loan.
Among borrowers in the 2003–04 cohort, 45% of those who had never attained a credential and were not enrolled in 2009 had defaulted on a loan within 12 years. Among those who attained undergraduate certificates, the rate of defaulting within 12 years was 44%; for associate's degree recipients, it was 22%; and for bachelor's degree recipients, it was 8%.


Does Visitor Export Include Airline Tickets?

The UN SNA (2008, p. 42) has this to say:

  planes belonging to a domestic airline are always assets of the domestic economy regardless of where in the world they happen to be.

So, conceptually,* we should include in our country's GDP any services provided by any planes belonging to domestic airlines. And so for example, the fare paid in a Ryanair (based in Ireland) one-way ticket from Paris (France) to Berlin (Germany) would be counted in Ireland's GDP.

Conceptually, an export is any good or service sold by residents to non-residents. So, if the above one-way ticket were purchased by an Irish resident, then it should be counted as consumption in Ireland's GDP. If it were instead purchased by an Irish non-resident, then it should instead be counted as exports in Ireland's GDP. (I don't know though if in practice they actually make this distinction.)

*In practice, I am not sure how exactly this is done by national statistical agencies. My suspicion is that they probably don't consider each ticket sold and instead just look at aggregate figures provided by domestic airlines, e.g. their profits.


How does the Fed measure the shadow banking sector in the US?

Well before going to measurement I think it’s appropriate to start by definition. The term shadow bank is not always 100% consistently applied but most often the a shadow bank is defined as follows:

  Shadow banks are financial intermediaries that conduct maturity, credit, and liquidity transformation without explicit access to central bank liquidity or public sector credit guarantees.

This is how you see shadow banks defined a lot in research (and also other definitions come close). Moreover, this is definition used in research coming from Fed such as  this paper from NY Fed or this paper from St Luis Fed for example.

To my best knowledge there is no unified measure/indicator that’s being recorded (like let’s say GDP/inflation). Rather like this literature review from NY Fed mentions people take the definition of shadow bank and then calculate the proportion of for example financial liabilities hold by shadow banks relative to ordinary banks or other types of financial institutions. Maturity and credit transformation calculated as proportion per type of bank is also used. In wider research I have seen other measures as well.


Should selling price depend on product quality or on work to produce the product if both not in positive correlation?

No one is "right" here, there is rarely a "should" in economics, it is not about morals.
When making a purchase, you are usually willing to pay an amount $x$ larger than the price $p$ which is larger than the cost $c$ it took to make the product/provide the service. You are arguing that price should only depend on cost. The store is arguing that if you are willing to pay more, they can charge more. Unless they have competition who undercuts them, they are right.
In the theoretical model of perfect competition, where there is a single type of product and firms face very strong competition the price would equal the average cost in the long run. I would argue the model does not apply here, competition is not perfect (if I am wrong, go to the competition). In case of imperfect competition it is normal to charge people whose demand is less elastic w.r.t. price more.

What you can do: pay them or find a better offer. If you are convinced that you are right, found a rival printing firm, undercut them and put them out of business.


Total cost function and (dis)economies of scale

First of all, knowing the cost function is not enough to know whether we have decreasing, increasing or constant returns to scale. Returns to scale are a property of the firm's production function -- and the relationship of the production function to the cost function is not as straightforward as you might think. However, if we assume that the firm takes factor prices as given, then we can apply a (rather intuitive) result from Sandmo (1970):

  Average cost is decreasing, constant or increasing according as returns to scale are increasing, constant or decreasing.

In light of this, we simply need to consider the average cost curve:

$$ AC(Q) =100+20Q+3Q^2 $$

Clearly, this is strictly increasing in $Q$. Therefore, if we are willing to assume price taking in the factor markets, we know that the firm's underlying production function exhibits decreasing returns to scale.


What you're failing to consider here is that in buying a large amount of shares, the buyer also makes the price high for himself. And if he sells a large amount of shares at the same time, he's likely to lower the price of the actions while selling them. Poor boy has no chance.


Given the level you say you are at I would recommend you to go through this book
Simon and Blume: Mathematics for Economists.
it will give you the core of math skills you need for undergraduate and graduate economics at most universities I would guess. In graduate courses you will probably have to learn more depending on the course and university but it is standard to learn some of the math on the fly when you do economics.
I could recommend other books but they would then be more specialized in certain topics and anyway Simon and Blume is the book that I have most often seen references to on various university courses.
Another virtue of the book is that the examples are to a large extent taken from economics. So you do not "waste your time" on nice mathematical examples that illustrate the mathematical methods but will never appear in economics.
For some graduate courses of course there will be special math skills that are required. Econ departments of some universities specialize in different areas so what you need can also vary from university to university. However, as an economist I would dare say you are very often learning the mathematics as you learn new areas of economics.
Choose a programming language
Although you did not as for a recommendation of this kind, I would strongly encourage you to start programming. Very often being able to simulate some problem in optimization, probability theory, dynamic programming etc. is an invaluable resource to get a deeper understanding of complicated math/economic models.
Currently Matlab is used a lot especially for structural economics. R is very versatile better than Matlab when it comes to econometrics. Python is strong for Machine Learning which is also finding its way into econometrics/economics these days.


Why, in such a debt-heavy era, is inflation still such a Bad Thing(TM)?

While many points can be made about the pros and cons of inflation, the following is one of the most fundamental.

Suppose A borrows \$100 from B and inflation is 5% per year. To keep things simple, assume the terms of the loan require repayment after one year. When the repayment falls due, A pays B \$100, but because of the inflation it is only worth (at the original price level) $100/1.05 \approx \$95.2$.  So A's debt has been somewhat eased by the inflation (and if the term of the loan were much longer the easing could be much greater).  But what about B? In real terms B has lost the same amount that A has gained.

Other things being equal, therefore, inflation benefits borrowers and costs lenders. And in total, amounts lent must equal amounts borrowed. It may be that the the borrowers (government, struggling households) are more visible than the lenders (better-off households, foreigners).  Nevertheless, the effects on lenders, and on incentives for future lenders, cannot be ignored.  Where inflation is expected, lenders will as suggested in Henry's comment demand a higher interest rate to offset their expected loss on the principal repayment.


How to calculate inter-annual variation of inflation for a 3 months period?

Without further details, what I'd do is what you said. Take an 3/6/9/12-month average of the price index and calculate growth based on that.


List of commercial banks by country in Europe

I think i have answers for your question.

In "thebanks.eu" you can find banks list by country and there is a filter so you can find commercial banks in Europe.
https://thebanks.eu/banks-by-country

There is an other answer for your question; "stockopedia.com". This time you can find banks list and data. But there is no filter so you can not find only commercial banks and this service isn't free.

https://www.stockopedia.com/sectors/banking-services-551010//


Plamen Nikolov of Binghamton University presents a brief treatment of this in "Writing Tips For Economics Research Papers". He notes that there are two basic rules of economic usage:

  "Long run" (without a hyphen) is a noun. "Long-run" (with a hyphen) is
  an adjective. Same with short(-)run. and "Saving" (without a terminal
  s) is a flow. "Savings" (with a terminal s) is a stock.


How do I calculate the price and risk of a senior tranche?

Let $D_1 = 1$ be the event that the first loan defaults and let $D_2 = 1$ be the event that the second loan defaults. Assume that we know the correlation $\rho$ between $D_1$ and $D_2$ and we know the marginals $p_1$ and $p_2$.
The aim is to compute the probability that one of the two loans defaults (for the junior tranche) and the probability that both default (senior tranche).
First, notice that:
$$
\begin{align*}
&\mathbb{E}[D_1] = \Pr(D_1) = p_1\\
&\mathbb{E}[D_2] = \Pr(D_2) = p_2.
\end{align*}
$$
First, let us try to compute the probability that both loans default.
$$
p_{12} = \Pr(D_1 = 1 \text{ and } D_2 = 1)
$$
The covariance between $D_1$ and $D_2$ is given by:
$$
cov(D_1, D_2) = \mathbb{E}\left[(D_1 - p_1)(D_2 - p_2)\right] = p_{12} - p_1 p_2
$$
Next, the variance of $D_1$ and $D_2$ equals:
$$
\begin{align*}
var(D_1) &= \mathbb{E}\left[(D_1 - p_1)^2\right] = p_1 - (p_1)^2 = p_1(1-p_1),\\
var(D_2) &= p_2(1- p_2).
\end{align*}
$$
So the correlation is given by:
$$
corr(D_1, D_2) = \rho = \frac{cov(D_1, D_2)}{var(D_1)} = \frac{p_{12} - p_1 p_2}{\sqrt{p_1(1-p_1)} \sqrt{p_2(1-p_2)}}
$$
As such, the probability that both default is given by:
$$
p_{12} = \rho \sqrt{p_1(1-p_1)} \sqrt{p_2(1-p_2)} + p_1 p_2.
$$
Then also:
$$
\begin{align*}
\Pr(D_1= 1 \text{ and } D_2 = 0) &= \Pr(D_1 = 1)- \Pr(D_1 = 1 \text{ and } D_2 = 1),\\
&= p_1 - \rho \sqrt{p_1 p_2(1-p_1)(1-p_2)} - p_1 p_2\\
\Pr(D_2 = 1 \text{ and } D_2 = 0) &= p_2 - \rho \sqrt{p_1 p_2(1-p_1)(1-p_2)} - p_1 p_2
\end{align*}
$$
Then the probability that at least one defaults is given by:
$$
\begin{align*}
\Pr(D_1 = 1 \text{ or } D_2 = 1) &= \Pr(D_1 = 1 \text{ and } D_2 = 0) + \Pr(D_1 = 0 \text{ and } D_2 = 1) + \Pr(D_1 = 1 \text{ and } D_2 = 1),\\
&= p_1 + p_2 -  \rho\sqrt{p_1 p_2(1-p_1) (1-p_2)} - p_1 p_2,\\
\end{align*}
$$


Building a time-series demand forecasting model

If it's just knowing about the models, I'd suggest starting with Hamilton's "Time Series Analysis" but any book like it will do. You can also explore Google Scholar for inspiration (people likely had built similar models). It seems you have a VAR model, check this wikipedia link to see if it's what you have in mind: https://en.wikipedia.org/wiki/Vector_autoregression
If you are looking for tools to obtain these forecasts, I'd suggest EViews or R. They are not useful on their own: you must have to have a model to do any kind of forecasting.


How is Singapore able to provide first-world public services, given its low income tax rate?

I'm not an expert on this, but as a Singaporean, here are some factors off the top of my head (plus some Googling), explaining why Singapore is different from other first-world countries (in terms of revenue and expenditures).

In 2014, the personal income tax contributed only about 14.6% of government revenues (Source). In contrast, in Canada for example, it was nearly 48% of government revenues (C\$135.7b out of C\$282.3b in 2015).

One unusual revenue source deserves commentary: car taxes, including the so-called Certificate of Entitlement (COE). The COE is simply a piece of paper that entitles you to own a car. This recent news story mentions how the COE price hit a 5-year low, at "only" S\$45,002 (≈ US\$31,600) for small cars.

These COEs ("Vehicle Quota Premiums") make up 6.0% of government revenue.

Besides this very expensive piece of paper, a car-owner still has to pay all the other usual taxes (GST, road taxes) and insurance, plus a lot of unusual fees (e.g. Electronic Road Pricing - an idea which London borrowed). A lot of these flow to government coffers. Together, these make Singapore easily the most expensive place in the world to own cars.

"Public" housing is not cheap, heavily-subsidized housing.

It is true that over 80% of Singapore residents live in government housing (HDB flats). But these are not the sort of heavily-subsidized public housing you have in some Western countries. Indeed, it is a frequent gripe amongst Singaporeans that these are incredibly expensive and there are often conspiracy theories that the government sometimes actively works to push up prices.

Given the lack of transparency and also the fact that Singaporeans are actively encouraged to use their CPF (retirement) savings to pay for these homes, these conspiracy theories are not without merit.

Government housing is generally priced at market rates. For example, recently a 1,001 square foot flat went for S\$900,000 (≈ US\$632,000) - this was in the news because it broke a local (Clementi) record, but it doesn't get much cheaper than that.

There are some subsidies for first-time Singaporean home-buyers: If both you and your spouse are first-time buyers, then you get a total subsidy of S\$30,000. But this is not a large subsidy.

Some exceptional hardship cases (which involves an elaborate application process and onerous requirements) get to rent a small one-room flat for extremely cheap (as little as S\$26 a month), but these cases are so few in number as to be negligible.

Public transport is not a perennial loss-maker, but rather a profit-maker!

Unlike in say some US cities, where public transport is always losing money, public transport in Singapore ALWAYS makes very healthy profits. So these are not drains on but rather contributors to the government coffers.

If you've only been in US cities where there's often only a few people on a bus, you may find it hard to imagine. But come to any Asian city where public transport is always jam-packed and you can see why public transport is so profitable.

The CPF

This is a forced retirement savings scheme. In Sep 2015, the CPF had S\$293.9b in total members' balances (source). That's about 75% of GDP.

This is another frequent gripe of Singaporeans because these savings are all locked up (with very draconian rules on how you can gradually withdraw small amounts after you're 55).

Moreover, these funds are "managed" by the various Singapore sovereign wealth funds (GIC, Temasek). GIC in particular is not well-ranked in terms of transparency (source). So it is unclear as to how and how much exactly these help to contribute to Singapore government coffers.

Health care is very efficient (at least if compared to say the USA)

Healthcare spending is 4.6% of GDP (World Bank).

I have heard some of my friends who work at the Ministry of Health (Singapore) complaining about how inefficient the Singapore health care system is.

But as someone who has lived in both Singapore and the US for a good number of years, I would say (based purely on my own anecdotal experience) that the Singapore healthcare system is extremely efficient, especially if compared to the US. But the US is perhaps the worst-case scenario. I'm not sure how Singapore would compare to other first-world countries.


Is the international economic system too chaotic to "work towards" carbon emission reduction

There are already precedents that show the international capacity is there, technically, and economically to deliver the carbon reduction necessary, in time. And there are indications that the political will to do so is growing to sufficient levels, too.

Firstly, with the number of participating nations being 195 (the UNFCCC nation parties), that's about the right size to create a management group that prevents the destruction of the commons, despite the short-run market incentives being ones that would otherwise lead to such destruction: see the Nobel-winning work by Elinor Ostrom for lots of information on this - I highly recommend it, it's a very good read. This addresses the aspect of your question about whether the market can accommodate information about the common good and the future.

Secondly, there are already examples of individual nations putting primary legislation in place, and acting on it: the UK's Climate Change Act was the first such law in a major industrialised nation; other countries have done similar.

Thirdly, there are already examples of nations collaborating on global environmental problems successfully: the international action to reduce emissions of ozone-destroying chemicals, the Montreal Protocol, is working.

And fourthly, the COP21 Paris Agreement showed unanimity among the 195 nations plus the institution of the EU in the need to address this problem. Pause for a moment to consider the enormity of that. The Paris Agreement wasn't just approved by majority consensus, but unanimously by all 195 countries.

All of these things happened within the context of the international economic system. So it's certainly not a barrier.

And we know that the economic incentives to prevent catastrophic climate change far outweigh the costs of doing so, at any sane discount rate. However, there is a principal-agent problem, and decades of lag between cause and full outcome; therefore market mechanisms alone cannot be sufficient. But there are no pure market economies - it's all a blend of private and public agents - so that's only a hindrance, but not a barrier.


Why is the equilibrium price not anything between \$0 and \$200?

Varian is indeed wrong here in that there is a critical omission: He should have specified whether there in a 27th person and what this person's reservation price is.

If there is no 27th person, then the set of possible equilibrium prices is $[\$0,\$200]$, as you correctly point out. Explanation:

At each $p\in(\$0,\$200)$, we have $S(p)=26$ and $D(p)=26$, so that there is a possible equilibrium.

At $p=\$200$, we have $S(p)=26$ and $D(p)=\{25,26\}$, so that there is a possible equilibrium.

At $p=\$0$, we have $S(p)=\{0,1,\dots,26\}$ and $D(p)=26$, so that there is a possible equilibrium.

At any $p>\$200$, we have $S(p)=26$ and $D(p)\leq25$, so that there is no possible equilibrium.

If there is a 27th person and her reservation price is $k<\$200$, then the set of possible equilibrium prices is $[k,\$200]$. Explanation:

At each $p\in(\$k,\$200)$, we have $S(p)=26$ and $D(p)=26$, so that there is a possible equilibrium.

At $p=\$k$, we have $S(p)=26$ and $D(p)=\{26,27\}$, so that there is a possible equilibrium.

At $0<p<k$, we have $S(p)=26$ and $D(p)=27$, so that there is no possible equilibrium.

At $p=\$0$, we have $S(p)=\{0,1,\dots,26\}$ and $D(p)=27$, so that there is a possible equilibrium.

At any $p>\$200$, we have $S(p)=26$ and $D(p)\leq25$, so that there is no possible equilibrium.

Notes.

I assume throughout, as I believe Varian has, that each apartment-owner's reservation price or willingness-to-accept is \$0. I also assume that negative prices are impossible.
Following usual practice in economics, I've been a little sloppy about whether the elements of the codomains of the supply and demand functions $S$ and $D$ are numbers or sets of numbers. (Technically and most generally, they should be sets. But so as not to confuse students, we usually and more simply say they are numbers.)


What happens when government bonds purchased as part of central bank quantitative easing (QE) reach maturity?

The bonds mature and pay the principal.
However, central banks tend to reinvest the principal, keeping their holding amounts relatively stable over time. This can be validated by looking at central bank balance sheet statistics.


Derivation long run cost function of three inputs with Leontief-like characteristics

Your reasoning that $L_1=L_2^{1/3}$ is valid for any $K$. Indeed if this equality does not hold you can lower the cost by reducing the excess input of either skilled or unskilled labor. Thus you can rewrite the cost minimization problem in two dimensions instead of three

\begin{equation}
\min_{K,L_1}{200 K + 5 L_1 + 6 L_1^3}
\end{equation}
subject to
\begin{equation}
K^2 L_1 = q
\end{equation}

From the constraint you can replace $L_1$ by $q/K^2$ in the objective function, which becomes a minimization problem with one variable $K$ and a parameter $q$. Solving it will give you the cost function as the value function of the problem.


How does buying assets change net capital outflow?

Net capital outflows from the US would increase. Net capital inflows for the foreign country (investment) would increase. This keeps the accounting equation equal. For the US S-I would increase, thus creating a balance of trade surpluse (even though these are financial assets). For the country receiving the inflow S-I (savings-investment) would decrease, increasing their deficit (assuming the balance of trade began equal). This balances both sides of the trade. Our net capital outflows grow by an equal amount to their net capital inflows in this situation.


Question on returns of scale in isoquants

I've computed this in WolframAlpha and I think it'll help you understand just what exactly a production function is. Remember, the production function $Y=f(K,L)$ is defined by $f:\mathbb{R}^2 \to \mathbb{R}$. When you see a graph with axis in $K$ and $L$, you`re seeing a part of a 3D object, the surface that a function with domain in $\mathbb{R}^2$ generates. The isoquants are just contour lines of it.

The change in the isoquant's steepness is a consequence of the shape a Cobb-Douglas Produtcion Function's surface. The non-linear nature of Cobb-Douglas model makes its contour lines change steepness as $f(K,L)$ gets bigger, as production goes up. There's an interesting economic intuition behind it. The more inputs a firm employs in its production process, the smaller the marginal rate of technical substitution of the optimal plant is because of diminishing marginal returns. As a firm employs more of an input, the smaller is that input`s marginal productvity. As a firms employs more and more of two inputs, capital and labor, both the marginal productivity of capital and labor get smaller, meaning a smaller marginal rate of technical substitution.


Why we use revealed preferred condition in GARP instead of directly revealed preferred?

We need indirect comparisons because looking at direct comparisons is not sufficient to detect all types of inconsistencies. Suppose for instance that we observe $C(x,z)=\{x\},C(z,y)=\{z\}$ and $C(x,y)=\{y\}$. $C(x,y)=\{y\}$ implies that $y$ is "directly" revealed preferred to $x$. The reason why GARP is violated is that $C(x,z)=\{x\}$ and $C(z,y)=\{z\}$ imply that $x$ is "indirectly" revealed preferred to $y$, an inference that wouldn't be possible by looking at direct comparisons.


Secular Stagnation: A Plain-English Explanation

Quoting from Coen Teulings C and
Richard Baldwin R (ed.) (2014) Secular Stagnation:
Facts, Causes and Cures

  "(...) A workable definition of secular stagnation is that negative real interest rates are needed to equate saving and investment with
  full employment. (p.2)"

Later on they write (p.10)

  "Low real interest rates play a linchpin role in the secular stagnation debate for two reasons.
  • First, if real rates are
  low in normal times, adverse macroeconomic shocks are more likely to
  require negative real rates to restore a full-employment
  investment-savings balance. In today’s low-inflation environment, this
  tends to undermine the effectiveness of monetary policy.
  •
  Second, low nominal and real interest rates undermine financial stability.   Summers (i.e. Larry Summers, the economist that brought
  back the concept of Secular Stagnation) lists three channels through
  which low rates may foster instability:
  (i) they increase risk-taking as investors reach for yield;
  (ii) they promote irresponsible lending as coupon obligations become very low and easy to meet; and
  (iii) they make Ponzi financial structures more attractive as interest rates look low relative to expected growth rates."

So if interest rates tend to be low, the need to push them down even more in order to deal with macroeconomic shocks, may lead to financially-born crises.

So why interest rates have historically fallen? From the supply side (i.e. savings behavior/supply of loans) factors like,
a) increase in life expectancy
b) lower retirement age, and
c) the decline in the growth rate of the population,
tend to increase required life time savings - required in order to smooth life-time consumption (at the individual level this goes like this: projecting that we will live more, and stay in pension for more years, we tend to save more during our working years).
I will add that supply of loans will tend to increase with wealth inequality.

The effect from the demand side (i.e. investment behavior/demand for loans) is much less agreed upon, but it appears that the revolution of the Information Technology (IT) sector had a negative effect on this aspect, since it reduced the value of the investments needed. The authors write

  "The key is that the development of high value-added services by Google, Microsoft, Amazon, Facebook and the like require relatively
  little investment.

So the loan supply schedule shifted outwards, while the loan demand scheduled shifted inwards. Schematically.

which creates the needed environment for Secular Stagnation, if it goes too far down.

Of course this is not the whole story. The above e-book by CEPR is freely downloadable.