There are numerous directions to go which start moving you beyond ordinary least squares (OLS), linear regression. The universe of statistical methods is large!

Two books that I particularly enjoyed are Econometrics by Hayashi and Elements of Statistical Learning by Hastie et. al. Looking back at your question, these books may be too advanced. But maybe not. An easier version of the latter is An Introduction to Statistical Learning (and may be interesting from the perspective of a broader exposure to data science than just econometrics).

Hayashi's Econometrics introduces a variety of methods through the lens of GMM and with an eye towards time-series econometrics.
Elements of Statistical Learning is a modern classic of the statistics, machine learning literature. It's great for opening your eyes to methods outside of traditional econometrics.

Some examples beyond ordinary least squares...

Maximum likelihood estimation (MLE)

You must know this if you continue doing statistics. It's a ubiquitous workhorse.

If you can specify the likelihood function then parameters of the likelihood function can be estimated by maximizing the likelihood function. In certain special cases, (eg. linear regression with conditionally normal error terms) the OLS estimator is the MLE estimator. You've undoubtedly encountered MLE estimation before if you've estimated a logit model. MLE is all over physics, engineering, and the sciences.

There are issues with applying MLE to economic models though. Often we know that an overall economic model is false. A model will imply certain facts that are entirely fallacious. Forcing the model to match the data in a maximum likelihood sense may not induce a useful choice of parameters. GMM can be used as an alternative approach to selectively test certain predictions of a model.
GMM is another broad method for estimating parameters based upon moment conditions that in expectation should be zero. Hayashi's book Econometrics develops ordinary least squares regression, instrumental variables, maximum likelihood, and other methods as special cases of GMM with different moment conditions. OLS can be thought of as GMM using the orthogonality condition of the regressors and the error terms. MLE can be derived as GMM on the score.

A John Cochrane ode to GMM is here.
Matching methods for estimating causal effects are common in certain areas of economics.

The idea is to match a treated entity with an untreated entity based upon observable characteristics. A widely used technique for example is propensity score matching
There are all kinds of variations on classic linear methods:

The idea here is to start with ordinary least squares but then to bias coefficient estimates towards zero to reduce overfitting and improve out of sample prediction.

Ridge regression
Lasso

References

Fumio, Hayashi, 2000, Econometrics

Hastie, Trevor, Robert Tibshirani, Jerome Friedman, 2009, Elements of Statistical Learning

James, Gareth, Daniela Witten, Trevor Hastie, and Robert Tibshirani, 2017, An Introduction to Statistical Learning


Utility function and homogenous of degree zero

Consider the utility maximisation problem:
$$
\max_{x_1, x_2} u(x_1, x_2) \text{ s.t. } p_1 x_1 + p_2 x_2 \le m.
$$
If we multiply all prices and income by the same number $t > 0$, we obtain the problem:
$$
\max_{x_1, x_2} u(x_1, x_2) \text{ s.t.} t p_1 x_1 + t p_2 x_2 \le t m
$$
However if we divide the left and right hand side of the constraint by $t$ we get the original problem back.
In other words, the two problems are identical. This also means that the two problems will also give exactly the same solution.
Summarizing, all demand functions are homogeneous of degree zero whatever the utility function because multiplying all prices and income by the same positive number does not change the problem: everything is $t$-times more expensive but you also have $t$-times more income.


What would happen if China called in its debt?

You have to understand how international debt works. These are not loans, but bonds. China buys a US bond for, e.g., 98 USD. This bond is a promise by the US Treasury to pay 100 USD one year from now. China owns a lot of this type of bonds. Once the bond hits maturity, China is paid 100 USD and the thing it typically does with these 100 USD is it buys the same type of bond again. If it decides it no longer wants to hold US debt, it can do one of two things:

First, it could decide to sell all its bonds at market prices to someone who's willing to buy them. This would be a terrible idea for the Chinese, since they'll lose a lot of money in this kind of fire sale. Second, it could just stop rolling over the debt and invest its foreign reserves in other kinds of bonds.

In any case what will not happen is that the US suddenly has to pay back all its debt at once. It will become more expensive for the US to refinance its debt, since the demand for treasuries will go down, but this is more of a long run problem. Also, it is not trivial to find an alternative asset market that can absorb an investment of 1200 billion dollars without prices going through the roof.


What are some further readings in Econometrics you recommend?

I might be wrong but from what you write, it seems you've been given a "classical" introduction to econometrics: You've covered IVs and Diff-in-diff but apparently only in passing, and causal inference does not look like it was the core of the classes you've taken.

If that's correct, then I would recommend reading:

Mastering 'Metrics (https://www.amazon.com/Mastering-Metrics-Path-Cause-Effect/dp/0691152845),
Mostly Harmless Econometrics (https://www.mostlyharmlesseconometrics.com/)

before reading anything else (e.g., Wooldridge or Greene, as referenced in E.Sommer's answer).


From OECD website,

  Definition of Tax wedge

  Tax wedge is defined as the ratio between the amount of taxes paid by
  an average single worker (a single person at 100% of average earnings)
  without children and the corresponding total labour cost for the
  employer. The average tax wedge measures the extent to which tax on
  labour income discourages employment. This indicator is measured in
  percentage of labour cost.

So here, it is a ratio, but not between wages, but rather of the direct tax burden (including social security contributions), as a percentage of total labor cost (i.e of total reward of labor). But it does not include indirect taxes.

As you can see, there are different ways to define the concept (extending it to include indirect taxes makes sense -the problem would be to measure it at that level).But it appears that in general we are talking about a ratio rather than a difference.

To reconcile, it is the "difference (in absolute terms maybe) between the real consumption wage and the real product wage" as a percentage of real product wage.


preference relations and monotonic transformations of utillity functions

Depending onhow one reads the question, the converse holds either always or basically never. Let $u:X\to\mathbb{R}$ and $v:X\to\mathbb{R}$ be arbitrary functions with ranges $u(X)$ and $v(X)$, respectively. It is not hard to show that $u$ and $v$ represent the same preferences if and only if there exists a strictly increasing function $h:v(X)\to\mathbb{R}$ such that $u=h\circ v$. So this is the "always"-answer.

Now one might require $h$ to be defined and strictly increasing on all of $\mathbb{R}$. In that case, one can basically never guarantee the existence of such an $h$, at least for standard economic applications. Let $X=[0,1]$, $u$ be given by $u(x)=x$ and $v$ be given by $v(x)=x$ for $x\leq 1/2$ and $v(x)=x+1$ for $x>1/2$. Then $v(X)=[0, 1/2]\cup (3/2,2]$. If $h:\mathbb{R}\to\mathbb{R}$ is a nondecreasing function such that $u=h\circ v$, all $r\in(1/2,3/2]$ must be mapped to the same point by $h$, so $h$ cannot be strictly increasing.

Such a problem occurs whenever $X$ is a connected topological space (say, a convex subset of some $\mathbb{R}^n$) and some continuous utility representation exists that is not constant.

The problem can even occur if both $v$ and $u$ are continuous. Let $X=\mathbb{R}$, $u$ be given by $u(x)=x$ and $v$ be given by $v(x)=e^x$. Then $v(X)=\mathbb{R}_{++}$, the set of strictly positive numbers. Suppose there were some strictly increasing function $h:\mathbb{R}\to\mathbb{R}$ such that $u=h\circ v$. Then $h$ would be a strictly increasing extension of the logarithm function to all of $\mathbb{R}$, which is impossible.

Finally, some positive result: If $X$ is a connected topological space, and $u:X\to\mathbb{R}$ and $v:X\to\mathbb{R}$ are continuous functions that represent the same preferences and are either both bounded above or unbounded above and are either both bounded below or unbounded below, then there exists a strictly increasing continuous function $h:\mathbb{R}\to\mathbb{R}$ such that $u=h\circ v$.


How is software treated in Gross Domestic Product calculations?

The National Income and Product Accounts (NIPA) are used to calculate GDP. The Bureau of Economic Analysis (BEA) prepares methodology papers discussing how they calculate various contributions to the NIPA. They happen to have one on software for business software and this paper should be of use:

  This section describes the methodologies used to prepare (1) annual
  estimates of business and government purchases of software
  (prepackaged and custom), (2) annual estimates of own-account
  production of software, and (3) price indexes that are needed to
  prepare the real estimates of capitalized software. The description is
  limited to the period beginning with 1992.

Estimation of Software in the U.S. National Accounts: New Developments (Moylan (2003))

The methodology for household consumption of software is in the personal consumption expenditures document.

Open-source software is a non-market transaction and not in GDP directly though it may boost the value of goods (like computers) and services (like consulting):

  But by the 1950s, critics of GDP as a metric were emerging, critics
  such as the economist Moses Abramovitz. Indeed, Kuznets himself had
  written that ‘the welfare of a nation can scarcely he inferred from a
  measure of national income. One criticism of the use of GDP focuses on
  the effectiveness of GDP as an economic measure per se. As a number of
  critics have pointed out, GDP can he considered flawed. It does not,
  for example, measure volunteer or non-market transactions. These types
  of transactions are both numerous and wide ranging and can cover
  anything from the care of sick relatives to the creation of
  open-source software. Furthermore, GDP is either a sum total or per
  capita measure. As such, it does not reveal the distributions of
  wealth behind the per capita figure. Indeed, the GOP per capita
  figures may hide important and potentially divisive wealth
  inequalities within a country.

Government for a new age: The transformation agenda by Khoury and Abouchakra (p. 190)


Find high resolution market data for a single ticker?

Many universities have access through WRDS to the NYSE Trade and Quote (TAQ) data. Despite the name, TAQ has " intraday transactions data (trades and quotes) for all securities listed on the New York Stock Exchange (NYSE) and American Stock Exchange (AMEX), as well as Nasdaq National Market System (NMS) and SmallCap issues."

Quandl has intraday data, but only at a minute level frequency. Tesla is a member of the NASDAQ 100 and so is one of the minute by minute firms available through Quandl via AlgoSeek. The pricing for an individual user for the whole year is \$800 (as of 2/2/17).


Calculating the growth rate of a stock over a year with imperfect information

I don't think that on yearly returns, a day more or less matters too much. In the unlikely case it actually does, you could interpolate the final day.

A first approximation would be the average over June 12th and June 11.


Find equilibrium price using excess demand function

Let $m_i$ be the market value of the endowment of player $i\in\{1,2\}$.
In optimum player $i$ spends everything such that $p_x x_i + p_y y_i =m_i$.
And because of the preferences in optimum also $x_1 = 4 y_1$ and $x_2=y_2$ holds.
You use that to get 2's demand as $x^*_2 = \frac{m_2}{p_1+p_2}=y^*_2$ and a similar bur slightly different formula for $4 y^*_1 = x^*_1$.
The excess demand would be this demand minus the endowment. The sum of excess demands is zero in equilibrium.
In equilibrium, $x_1^*+x_2^* =2+4$ and $y_1^*+y_2^* =2+4$, which you can both use to calculate an equilibrium price ratio. Any price combination with that ratio clears the market.


What does liberal economic theory say about monopolies?

Monopoly and competition are one of the central topics in microeconomics, so there many theories on the source of monopoly, and they are not universally applicable.  Government intervention is certainly one possible source of monopoly, but there are many others.

I don't think I could give a general answer that does the question justice, so I will just give links to some of the topics are related:

Cournot competition
Economies of scale
Network effect
Diamond search paradox
Monopolistic competition
Market contestability

The general area is known as industrial organization.  The classic text is Jean Tirole's "The Theory of Industrial Organization", though I think this only covers a small fraction of the topic.


Data Set/Database listing NAICS codes for U.S. firms?

If NAICS are not provided in the dataset, you need a second variable, related to NAICS, to do the match. Two options are ISIC and SIC, related to industry classification. You can find the official link between these two and NAICS here. These are Excel files with the concordances, which will not be perfect (sometimes you have m-to-1 or 1-to-m matches). There are more crosswalks between classifications here, here, and here.

In terms of Stata files (if you need that), check file [C1] here. There is also this very good NBER crosswalks (under Concordances heading), where you get SIC-NAICS crosswalks in Stata, SAS, Excel and CSV formats.

For reference, here you can find a mapping between SOC (occupations) and NAICS, but I am do not sure how it works, as I cannot see a 1-to-1 mapping between occupation and industry.

Finally, here there is a document describing some technical aspects of the crosswalks.


We have an allocation $(x^*,y^*)$ and a price system $p$ forming a price equilibrium with transfers.The assignment of wealth levels $(w_1,\ldots,w_I)$ has to satisfy $$\sum_i w_i=p\cdot\bar{\omega}+\sum_j p\cdot y_j^*.$$
One way to interpret the wealth levels is that the planner hands out coupons that can be traded at existing prices for commodities and chooses the production plans of the economy. But one can interpret the wealth distribution also as a distribution of the initial endowments and of firm shares. Let $w=\sum_i w_i$ and for each consumer $i$, let $m_i=w_i/w$, which is just the share consumer $i$ has in the aggregate wealth. Now give each consumer an endowment $\omega_i$ of $m_i \bar{\omega}$ and for each firm $j$ a share $\theta_{ij}$ equal to $m_i$. For the resulting private ownership economy, $(x^*,y^*)$ and $p$ constitute a Walrasian equilibrium, so one has reduced redistributing wealth to redistributing endowments and shares in firms. To see that everything works out, note that
$$w_i=m_i w=m_i\bigg(p\cdot\bar{\omega}+\sum_j p\cdot y^*_j\bigg)=p\cdot m_i\bar{w}+\sum_j m_i p\cdot y_j^*=p\cdot \omega_i+\sum_j\theta_{ij} p\cdot y_j^*.$$


How does a country devalue its currency?

Typically, a devaluation is achieved by selling the domestic currency in the foreign exchange market and buying other currencies. Suppose China sells one trillion Renminbi and buys 157 billion US dollars. From the point of view of the market, it is as if the supply of Renminbi just increased. As in any competitive market, an increase in supply will cause the price (i.e. the exchange rate) to fall: one Yuan will be worth less than before.

Devaluations are good for a country's balance of trade. Companies based in China ultimately care about how many Yuan they end up with. Suppose that the value of the Renminbi is 10 Yuan = \$1US. That means that a Chinese product priced at 10 Yuan would cost an American \$1 to buy. Now suppose that the value of the Renmimbi falls by half: 10 Yuan = \$0.50. Now the same product, still priced at 10 Yuan, will only cost an American 50 cents. It's as if everything China exports just got cheaper! This fall in the apparent price of Chinese exports will make people in other countries want to buy more Chinese products so that China will experience an increase in its exports. The argument also works in reverse: to a Chinese person, the devaluation makes it look as if American products got more expensive, so Chinese will demand fewer American (and British, and German, etc.) products and China will import less. Together, these two effects mean that China's balance of trade (i.e. the difference between exports and imports) will improve as a consequence of the devaluation.


Consider the following production function $Q=min \left(\frac{L}{2a}, \frac{K}{4b}\right)$. Let $w$ and $r$ be the wage and rental rate respectively

The steps to optimally go from the Production function to the Cost function is to solve the cost-minimization problem for the firm, which determines the input mix given prices and some level of output, and in turn allows us to express one input with respect to the other (in this simplest two-inputs case).

When the production function is of the Leontief variety, optimal input mix is indeed determined by the equality stated by the OP, since by simple reasoning, any amount of one input above the amount satisfying the equality will incur the cost to  buy it but not any extra output.

Inserting the equality

$$\frac{L^*}{2a}=\frac{K^*}{4b} \implies L^* = \frac {a}{2b}K^*$$

into the cost function we get Cost expressed as a function of one of the inputs only, $C^* = g(K^*)$

In order to eliminate it too, we turn again to the production function which can also be expressed as a function of one input at the optimal solution,

$$Q = h(K^*) \implies K^* = h^{-1}(Q)$$

Inserting $h^{-1}(Q)$ into the Cost function we obtain

$$C = g\left[h^{-1}(Q)\right]$$

where also the various exogenous constants and prices will be present. In fact, this is easier done when the production function is Leontief, than when it is Cobb-Douglas for example, and so it should be easy for the OP to arrive at the correct solution.


market equilibrium quantity $\ne$ firm profit maximising quantity?

In intro micro you usually assume that every firm in a perfectly competetive market has the same costs and production function etc. Hence, the MCs are the same for every firm. Under this assumptions, every firm will produce the same quantity in equilibrium.


Is the time value of money natural or artificial?

Let's first write down two realistic reasons as to why the "time value of wealth" (since here "money" is considered in its function as store of value), is "natural and not artificial", to use the OP's words.

1) The phenomenon of inflation. If we expect to see price inflation, then a given amount of nominal "wealth tomorrow" will buy for us less goods and services than the same amount of "wealth today". So "wealth today" is preferable than "wealth tomorrow", since what we care about, is goods and services which are the things generating utility for us.

But note that inflation is not some always-present phenomenon in all economic systems, economies, countries, eras: for example the 19th century has been largely a deflationary period, with prices tending to fall during the century, in the main capitalistic economies.

2) The uncertainty of the future. This is always present, even if it has to do with the possibility of unexpected death. So "wealth tomorrow" may not be enjoyed because I will have die suddenly and unexpectedly till then. So "wealth today" is preferable, and so more valuable.

In a lending and borrowing situation, we can also map to the "uncertainty of the future" the risk of not-repayment etc. and so view the time value of wealth as being also a compensation for undertaking these risks.

Also, in a long-term approach, eventually we all die, so the very finiteness of our horizon creates a situation where "wealth today" is more preferable and so more valuable than "wealth tomorrow", if this "tomorrow" is far in the future.

The thing is, economists argue that "wealth today" is always preferable to "wealth tomorrow"
a) even if there is no inflation, or alternatively, if we examine wealth in "real terms" (i.e. in terms of goods and services),
b) even if there is no uncertainty at all,
c) even if we have an "infinite" time horizon, i.e. even in the hypothetical situation that we are certain that we will never die, not even from "natural causes."

This is why in intertemporal models of consumer choice and utility optimization that are deterministic /incorporate no uncertainty, and assume an infinite time horizon, still, a "discount factor" exists that discounts future certain utility in real terms, which indirectly reflects that "wealth today" is preferable to "wealth tomorrow".

What is the argument behind that? Well, sometimes there is no philosophical argument only a methodological argument: "a deterministic economic model with infinite time horizon and in real terms is an approximation to reality, which is uncertain and has finite horizon for economic agents, and so we include the discount factor to keep it a little closer to reality". That's fine, that's what economic models do (models in general actually).

But could we come up with an argument that even an immortal living in a perfectly deterministic world, would prefer "wealth today" to "wealth tomorrow"?

We can. It's called "opportunity cost". It can stand on its own without invoking uncertainty or the possibility of death. It depends on the following very simple axiom, which is supported by timeless experience related to human behavior: "more is better". Under this axiom, having wealth today will permit me, if I choose so to have more wealth tomorrow and for all tomorrows. So it potentially expands my "choice set" for all eternity.  If more is better, then I want a larger choice set, so I would prefer to have all my future wealth today rather than tomorrow.

...one could of course argue that the "more is better" postulate is not an independent trait of humans, but is one more incarnation of the "survival instinct" and so it is in reality a response to uncertainty and risk of death. Certainly, but everything can be eventually attributed to the "survival instinct", and what explains everything explains nothing.


What would happen to US economy before US government lose control over its national debt?

It depends.  Mathematically, you just need GDP to grow at the same rate as the debt grows, and then you can sustain deficit spending indefinitely. But this simply shifts the line of questioning to "can GDP grow indefinitely", and for most people the obvious answer is "no".

Ironically, the band-aid solution that most developing nations have taken is to grow the population through immigration (ironically because this approach is seeing significant geopolitical pushback).  It's well-known that more affluent countries have lower birth rates, and so immigration is the "obvious" solution to this.  But it clearly carries its own challenges.


How does the Federal Reserve remove money from circulation?

One of the Federal Reserve's (the Fed) primary policy tools is the Federal Open Market Committee (FOMC). Through open market operations the Fed can buy or sell securities on a secondary market. By buying securities they bring new money into circulation, by selling securities they take money out of circulation.
Thus the Fed has a direct influence on the so called monetary base (M0, consisting of currency in circulation and reserves held by banks at the Fed).

It is true that the Fed cannot take the $-bills out of your pocket, but they don't have to. The Fed trades in securities, and every security has a price. Hence, if the Fed wants to take money out of circulation they "buy" dollars, by selling securities. At the market price there will by definition be people who are willing to give their money to the Fed in return for securities.

The value of money is determined by numerous factors beyond the monetary base (e.g. the money supply M1,M2,M3..., exchange rates) and is a concept harder to grasp, because value can be defined in various ways (as opposed to the "price" of money). But I think that goes beyond your question.


you can do this using comparative statics and envelope theorem. The profit function specification was correct given the assumptions, let me just rewrite it as:
$$\pi (q,t) = p q - C(q) - tq$$
However, given the above we also know that the optimal quantity will depend also on a tax rate, so we will have $q^* = q(t)$.
Now by envelope theorem:
$$\frac{\pi^*(t)}{dt} = \pi'_2(q^*(t),t) = - Q^*(t) $$
So the optimal level of output given the tax $t$ will depend inversely  on change in profit when the tax rate changes or  $-\pi'_2(q^*(t),t) = Q^*(t)$.


What is Production Possibility Frontier (PPF) diagram drawn on consumption-leisure?

Edited because wrnog answer, did not took the time to read the page.

Without maths

So your PPF would just be the link between leisure and consumption set by technology : the amount of leisure you choose give the time you can spend working, so the wage you earn and therefore the consumption you have (since there is no saving, all you earn is what you consume).

With a bit of math

You have a fixed amount of time, which we normalize to $1$. You can spend it either in leisure F (fun) or in work L (labour) so that
$$1=L+F$$
Then you get a wage $w$, that is related to $L$ in some way : $w=\psi(L)$, and you consume $C=w=\psi(L)$, so that $L=\psi^{-1}(C)$

Then your PPF is
$$F+\psi^{-1}(C)=1 $$
It is a relation between leisure $F$ and consumption $C$ set by technology (the way wages are fixed).

Then the program is to maximize the agent utility $U(C,F)$ subject to this contraint. You can visualize it by the common tangent between iso-utility curves and your PPD.


Deloitte managed a semi-intelligible explanation. It's called "super" when an income tax deduction is applied twice for the same item (e.g. for R&D)...

  In early 2015, Slovakia introduced a new type of tax incentive aimed at supporting companies conducting corporate R&D (most common is development of new or substantially improved products, services, technologies and processes). [...] Similar regimes have been in use for years in the world's most advanced economies.

  In practice, eligible expenses can be deducted from the tax base twice. First as a normal tax-deductible expense and second as an additional super-deduction. [...]

  Super-deductions can be utilized by any company, which in the relevant tax period incurs expenses directly related to R&D projects. In case of tax losses, the super-deduction may be transferred to the next tax period (up to maximum of four subsequent years).

KPMG has a more concise definition and some examples/figures:

  Income tax is reduced by deducting
  R&D expenses from the tax assessment basis by
  more than 100% (“super deduction”) or from the
  tax liability (“tax credit”). [...]

  Tax-free amounts are applied to obtain a discount
  of more than 100% of the qualifying R&D costs.
  From a practical perspective, this is usually
  achieved by deducting these costs against tax
  more than once. The majority of multipliers in
  other jurisdictions are between 130% and 210%.
  This leads to the fact that these increased R&D
  expenses can be also claimed as expenses
  for tax purposes and reduce the taxable profit
  accordingly.


It should be pointed out that just because one encounters complex numbers does not mean one is doing "complex analysis", e.g. complex eigenvalues, complex Borel measures, Fourier transforms, etc. where trivial properties of complex numbers come up.

Complex analysis is a very focused subject unlike, say, real analysis, which is eclectic by comparison. At its core are holomorphic functions of one or more complex variables.

This paper

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=932693

is a specific instance of an economic model where complex analysis is used. The model solution technique used there is the identification between holomorphic functions on the unit disk and their continuation on the boundary. (The resulting function space is called the Hardy space, which contains the players' strategy spaces in the game being played in the paper.)


Elucidation needed on textbook's description of the effect of increased Interest rates on the money supply curve?

The graph is a little misleading.

Imagine the interest rates increase without $MS$ shifting to $MS1$. Here the demand for money is less than the supply, so there is a surplus. If the MS does not shift, eventually the interest rate will fall again until the shortage is no longer (we are back in equilibrium). If the Fed wants to maintain the higher interest rate, $r_1$, it also must shift $MS$ to $MS1$.


Why should I get a bond with negative interest instead of having a bank deposit account either zero interest or positive interest

It makes little sense to me either, but here are some possible reasons for buying a bond with negative interest rates rather than depositing the same amount in a bank:

The deposit-taking bank may go bankrupt during the lifetime of the bond and your deposit would be too large to be guaranteed, with the cost of default insurance being higher than the the negative interest
The bank may start charging you for holding your deposit at some time during the lifetime of the bond, at a higher cost to you than the negative interest
You may be legally obliged to hold bonds, for example as a pension or insurance company
You run an Exchange Traded Fund or similar investment tracking a bond index and some of the bonds in the index have negative interest rates
You are a foreign investor who only cares about the combination of the current exchange rate, future exchange rate and interest rate, not any individual component of this triangle, or you believe currency appreciation will more than compensate for negative interest
You are a derivatives trader, and the market prices of derivatives lead to an arbitrage opportunity where you can only guarantee a profit by buying the bond
You believe that at some time in the future you may be able to sell the bond for more than you paid for it (i.e. an even more negative yield), perhaps to a central bank's quantitative easing programme


How do debt-free governments control the money supply?

There are several ways how central banks can still control money supply even if we assume the government does not issue any bonds.

First, there is the discount rate at which other banks can borrow money form the central bank. The nominal interest rate throughout the whole economy depend on the discount rate as the rate at which banks are willing to borrow to households and businesses also depend on the rate which they can borrow from the central bank.

This is what the Norway (a country with negative net debt) is doing. According to the Norges Bank

  Instruments and implementation
  The main monetary policy instrument is the policy rate, which is the interest rate on banks' deposits in Norges Bank up to a specified quota.

  The policy rate is set by Norges Bank's Executive Board. Normally, the Executive Board makes monetary policy decisions eight times a year. The Executive Board meetings where monetary policy decisions are made are called monetary policy meetings.

Second, there is the reserve requirement. In principle central banks could increase money supply by decreasing the reserve requirement which would increase the size of money multiplier  $1/RR$ where $RR$ is the reserve ratio. This is quite crude way ho to conduct monetary policy though and its quite uncommon.

Furthermore, as was correctly pointed out in the comments even government with no net debt might still issue bonds, so the assumption that they do not is not necessarily correct. Hence in practice that option is still there at least in principle - although as correctly pointed by @BrianRomanchuck your question excludes it, its worth while to keep this in mind.

Hypothetically, there could also be other ways how this could be done but they are not used too much in practice. For example. there are also precedents for central banks buying up private bonds but its not common. Central banks could in principle also directly change the monetary base with something like helicopter money (depositing money directly to private accounts and hence increasing the monetary base - or in cases like ECB where central bank also has power to directly coordinate printing of the money by literary printing them and throwing them out of helicopter), but since your question is more about how its actually done not how hypothetically it could be done I wont go into them.


What's is the purpose of "conditional preferences" as specified in Savage's framework?

The statement $f\succcurlyeq f^\prime$ and for every $s$ not in $E$, $f(s)=f'(s)$ says that $f$ and $f^\prime$ have the consequences unless $E$ occurs, and $f$ is preferred to $f'$. Loosely, the sure thing principle says that if two acts differ only in their consequence should $E$ occur, their ordering should depend only on their consequences in $E$ and not on states outside of $E$. What the sure thing principle gets you is that if $f\succcurlyeq_E f^\prime$ then $f$ must be conditionally preferred to $f^\prime$ because of the consequences of $f^\prime$ in $E$. Otherwise this would not be known.

Here's a plain English interpretation of conditional preference. Imagine I put together two portfolios, $A$ and $B$. I start both portfolios off with the same asset, let's just say some money in a mutual fund. In portfolio $A$ I put an additional asset that pays a dollar in all states of the world $E$ where a democrat wins the 2016 election. I give you a choice between $A$ and $B$. In action $f$ you choose $A$ and in action $f'$ you choose $B$. Presumably you prefer $f$ to $f'$. But the payoffs of $f$ and $f'$ differ only should a democrat win the 2016 election. So by the definition Karni has given you $f$ is preferred to $f'$ given $E$. With the sure thing principle,  I can separate your preference in $E$ states from your preferences in not $E$ states. So I now know your preference for $f$ must stem from $f$ being in some sense better than $f'$ in $E$ states.

Technically (read: obnoxiously) preferences are defined over the functional space $F$, so the abstruse definitions come from the fact that the notation $f(s)\succcurlyeq f'(s)$ or $f(s)\sim f'(s)$ is not yet defined. Instead Karni has to work with the $f_E$ notation. In this notation you'd write your idea for conditional preference as $f\succcurlyeq_E f'$ iff $f_Eh\succcurlyeq f'_Eh$ (the first part of the sure thing principle). I agree that this is more in line with my notion of what conditional preference should be. Savage mentions both senses in his discussion:

  What technical interpretation can be attached to the idea that $f$ would be preferred to $g$, if $B$ were known to obtain? Under any reasonable interpretation, the matter would seem not to depend on the values $f$ and $g$ assume at states outside of $B$. There is, then, no loss of generality in supposing that $f$ and $g$ agree with each other at states outside of $B$, that is, that $f(s)=g(s)$ for all $s \in \sim B$. Under this unrestrictive assumption, $f$ and $g$ are surely to be regarded as equivalent given $\sim B$, that is, they would be considered equivalent, if it were known that $B$ did not obtain. The first part of the sure-thing principle can now be interpreted thus: If, after being modified so as to agree with one another outside of $B$, $f$ is not preferred to $g$; then $f$ would not be preferred to $g$, if $B$ were known. The notion will be expressed formally by saying that $f\leq g$ given $B$.

(Here $\sim$ is set notation instead of preference ordering.)

The sense that Karni involves two acts that agree with each other outside of $B$. However Savage's definition of conditional preference is exactly $f\succcurlyeq_E f'$ iff $f_Eh\succcurlyeq f'_Eh$.  If there really is no loss of generality in the rest of the proofs it doesn't matter which definition you pick. However Savage's definition is in line with both yours and mine, and the definition of conditional expectation. I'll look to see if that has any real consequences for Karni.

ADDENDUM Using equality instead of $\sim$ is plenty general, simply because if the two acts only differ in consequences that do not have an effect on the preference ordering, you could just redefine the consequence space such that the equality was still maintained. Which is a cop out, but also true.

I believe I've figure it out. The only place Karni seems to use his conditional preference definition is in P7. Savage uses an analogous P7 but with his own definition of conditional preference. P7 says (Savage's wording):

  If $f \leq g(s)$ given $B$ for every $s\in B$, then $f \leq g$ given $B$.

This is a statement of conditional dominance: "a certain consequence $g(s)$ is better than an (uncertain) $f$ for every possible $s$ in $B$, so $g$ should be preferred to $f$ given $B$." Because we have already accepted the sure-thing principle, P7 from Savage and P7 from Karni are equivalent. If we had not accepted the sure-thing principle, Savage's would be more restrictive. So Karni is actually being less restrictive with his definition of conditional preference, and that is presumably the purpose.


Declaring bankruptcy is a way to prevent creditors from seizing assets and pursuing lawsuits to recover what they are owed in a piecemeal fashion. It forces all parties to wait for a judgement that determines in what fashion various classes of creditors will be repaid.

In the United States, there are two main ways a private firm goes bankrupt - known as Chapter 7 and Chapter 11 (chapters of the bankruptcy act).

In a Chapter 11 bankruptcy, the firm aims to reorganise its debts. The typical outcome is that the original equity holders lose almost all control of the firm as subordinated debtors have their claims converted into equity claims. Other debtors may get paid only a percentage of their claim, or they are given new debt. What happens is determined by negotiation among the major creditors and firm management, overseen by the bankruptcy judge. By lowering the debt burden of the firm (and possibly eliminating pension obligations that cannot be sustained), the firm is hoped to then be profitable, and able to pay back the remaining debts.

Chapter 7 bankruptcy is a liquididation. All assets are sold, and then the proceeds are used to pay off debtors. However, the liquidation value of most assets is far below their carrying cost on the balance sheet, and so the recovery values are extremely low. The exceptions might be things like real estate, which another firm might be able to use in a more profitable fashion. From the prospective of creditors, the recovery value is typically much lower in a liquidation (barring special cases) - look at the prices for clothing in liquidation sales versus normal retail price. Industrial equipment is specialised, and the buyers know that this is a liquidation, and so they will only bid if prices are extremely cheap versus new equipment. However, if a firm tried Chapter 11 and fails to recover, it is more likely to give up and liquidate the next time.


Is the S&P 500 now in a boom-bust cycle?

Here's the Efficient Market Hypothesis (EMH) explanation for why one cannot say that a correction is overdue:

Suppose that it were widely known that, by looking at the above figure, one could ascertain that the market was "due for a drop" one week from now. Financial analysts and investors would look at the graph and come to this conclusion. How would people respond to this knowledge? They would go and sell their stocks before those stocks loose a substantial portion of their value. However, this flurry of selling activity would cause the price of the stocks (and hence the stock index) to fall today—one week earlier than forecast.

This example is obviously a highly simplified and stylised exposition of how markets function, but it should serve to make clear the basic point: any hypothesised means by which one can look at a chart like this and forecast the future movement of prices is automatically self-defeating and any information about the future of the market is instantly built in to the current price of securities. The drop you think might be due will come, says the EMH, if and only if new information about the economy becomes available that suggests the economic outlook is less promising that people had thought.


Is there consensus on why inflation is not rising?

If your question is about Europe, yes there is. Inflation took time to rise because commercial banks have used injected liquidities to consolidate their balance sheet instead of stimulating demands by creating credits (that did not previously exist when they make loans). This is called the money multiplier. The multiplier was somehow blocked (to ~$1$) because of their precaution behavior.

On March 10, 2016, the ECB has even considered the possibility of giving 200€ per month and per person to stimulate inflation. And this with the main idea of circumventing the commercial-bank system. This unconventional monetary policy is called monetary helicopter.


Sharing my answer out there, correct me if I'm wrong.
$u(c,l)=u(rk+w(1-l), l)$
$U$ is strictly concave and differentiable. Let $max$ u attained at $(c^*, l^*)$ i.e $(k^*, l^*)$.
Then $v(k^*)=u(k^*)$ and $u(k) ≤ v(k)$
Then by Theorem 4.10 (Benveniste & Scheinkman), $v$ is differentiable at $k^*$.
$V_k^*=U_k(rk^* + w(1-l^*, l*)
      = U_c(rk^* + w(1-l^*, l*)$


Calculating price elasticity by asking a "this" or "that" question

The elasticity of demand measures how changes in prices affect changes in quantities demanded So if you have the total quantity demanded at the original prices and reduced them by, say, $10\%$, then observe the total quantity demanded of both goods, you can calculate elasticity by simply using the arc formula,  or the midpoint formula, etc. In fact, by just varying one good at a time, or reducing one good more than the other, you could also estimate cross-price elasticities.

In your example, however, it sounds like you have only information on a single consumer that buys either one good or the other. So you only know if they were buying at the original prices and which good and if they start buying after prices decrease, or if they decide to change the good they buy.

This exercise cannot really measure any elasticity because there is no sensitivity to measure. Best case scenario, if the person was not buying a good and after the decrease in price starts buying it, it can give you a lower bound on their valuation for the good.

In contrast, if the question was changed to how many units of each good would the consumer buy? then you could estimate demand elasticity and cross-price elasticities with careful exercises of lowering each price at a time.


Under what conditions does market failure occur?

Any situation where markets fail to clear leads to market failure. Some common reasons for this are:

Externalities, or agents not being responsible for costs or benefits of their actions
Information assymetry, or agents operating with different information
Irrationality, or agents doing random things for no reason
Principal agent problems, where someone entrusted with others funds uses them wastefully
Productive inefficiency, or firms failing to produce goods effectively given the allocated capital (think using Gatorade to water plants)
Allocative inefficiency, where firms are not provided the appropriate capital
Economies of scale, or the problem of capitalist firms failing to grow large enough to establish a service
Destructive competition- eg, private militaries blowing each other up
Shortsighted management of natural resources
Monopoly
Moral hazard, adverse selection, and other problems in contract theory


Remittance: Migrant workers sending money to their families

There are actually more pros then cons for the "home countries" in my opinion, and I have official/credible sources to back this opinion up.

Financial remittances have been recognized as an important developmental vehicle associated with migration. Financial remittance flows have steadily increased in volume from the 1990s to the present day. In 2017, migrants sent an estimated $466 billion to families in developing countries. Money sent home from abroad is shown to be more stable than both private debt and portfolio equity flows, and several times larger than international development aid.

An estimated 800 million people worldwide are directly supported by remittances from relatives and loved ones abroad, according to the International Fund for Agricultural Development (IFAD). Remittances lift families out of poverty, improve health and nutrition conditions, increase education opportunities for children, improve housing and sanitation, promote entrepreneurship and reduce inequality.

However, apart from financial remittances, transnational communities also contribute by way of 'social remittances' - the flow of skills, knowledge, ideas and values that migrants transmit home. The impact of social remittances was most strongly felt in areas such as education, health, employment, business and aspects of governance, found a study conducted by IOM in Tanzania in 2014.

In short, there is a lot work to be done. IOM, UN Migration Agency, in partnership with specialized organizations, civil society, academia and the private sector, have engaged in several concrete projects and initiatives aimed at improving knowledge of social and financial remittance usage and corridors. They have conducted studies on the impact of remittances on families and communities, and worked to enhance the development impact of remittances, financial inclusion and diaspora engagement.

EDIT to address cons:
There are indeed cons; let's look at the ones you pointed out:             inflation: I think most people would say this form of inflation is insignificant when you compare it the global economic perspective. There is a literal currency war between most if not all of the worlds nations, going on for a while now, where advanced economies are technically importing inflation to emerging market economies. These amounts are huge and when you factor in our topic here, it makes our topic a bit insignificant in comparison. See: https://en.wikipedia.org/wiki/Currency_war
James Rickards also has a great book on the subject.

As for social conflicts, this is a pretty micro oriented con, and a bit political as well. I'm afraid I don't have much knowledge on this one.

For accelerated emigration, this is perhaps the biggest con, but is easily countered in the long run. Here's why:

I think it is commonly accepted that most of the skilled workers in the long run wish to return to their home country/their families. When these workers leave their home countries, the country loses the potential GDP input of said workers. What do they gain in return? Cash inflow. Which we already accepted doesn't have much of a significant effect on inflation. What that cash inflow does do is increase consumption and investment a little bit, which promotes the prosperity potential of the country in the long run. Assuming this consumption and investment is aimed at long term growth for the home country, the workers would eventually come back(their original goal in the first place) to a stable economic system(that they inadvertently helped create) and provide further.


You can show this concerning the optimization problem with the objective function $U_0 = f(x_1) + x_2$ and the budget restriction $M - p_1 x_1 - p_2 x_2 = 0$. Using the Lagrangian, this leads you to
$$
f'(x_1) = \frac{p_1}{p_2} \quad \text{or} \quad f'^{-1}(\frac{p_1}{p_2}) = x_1^{*} = D_1(p)
$$
You can see that in this special case the optimum quantity of $x_1^{*}$ (Marshallian demand function) does not depend on the income $M$
$$
\frac{\partial D_1}{\partial M} = 0,
$$
The income effect is therefore zero, and you will not consume a different amount of $x_1^{*}$ if the income $M$ varies.

Some further considerations: Based on the Marshallian $D_i(p, M) = x_i^{*}$ and Hicksian $H_i(p, u) = x_i^{*}$ demand function, you can show some interesting properties of this particular utility function using the Slutsky equation:
$$
\frac{\partial D_i}{\partial p_i} = \frac{\partial H_i}{\partial p_i} - x_i^{*} \frac{\partial D_i}{\partial M}
$$
This shows that the derivative of the Marshallian demand function with respect to price equals the derivative of the Hicksian demand function with respect to price minus the optimal $x_i^{*}$ times the derivative of the Marshallian demand function with respect to income. In this special case, the Marshallian demand function equals the Hicksian demand function, as $\frac{\partial D_i}{\partial M} = 0$.


How does government spending on imported goods affect GDP?

Recall the definition of GDP that it is the aggregate value of all goods and services produced within the domestic territory.

The concept of domestic territory is central here. Only those government expenditures that creates a domestic income is counted in GDP.

For instance, if US government gives economic aid to a poor African country it will not be a part of US GDP, although it is a government expenditure.

In the same way the government expenditure on imports does not create any domestic income hence the value of imports will be subtracted from the GDP without any complementary addition.


The main point is that the principal's expected utility from a payoff $z$ conditional on a certain level of effort $e$ can be written as

$$ \text{E}[{z}|e] - \frac{r_p}{2}\text{Var}(z|e).$$

In other words, since wealth is normally distributed, exponential utility has a simple `mean-variance' representation. For a derivation, see here.

I take it that the principal's payoff $z$ equals $x - w(x) = (1 - b)x - a$. It is then straightforward to compute the (conditional) mean and variance of $z$:

$$ \text{E}[z|e] = (1 - b)\text{E}[x|e] - \text{E}[a] = (1 - b)\mu(e) - a,$$

$$\text{Var}[z|e] = (1 - b)^2 \text{Var}(x|e) - \text{Var}(a)= (1 - b)^2\sigma^2. $$

It follows that the principal's expected utility can be written as

$$ (1 - b)\mu(e) - a - \frac{r_p}{2}(1 - b)^2\sigma^2.$$


Stock return predictability. Why do papers continue to use Granger Causality tests

Because the papers which use these methods are not properly refereed. That's why you should read papers published in high impact factor journals.


A question about savings and investments

It is a statement about national accounts that has to hold (an accounting identity). Unless you understand all the terms in the national accounts, the accounting identity may not make sense.

One non-obvious property of the $S=I$ accounting identity is that increases in inventories are considered investment. That is, it is not just fixed investment (machinery, buildings) which people normally associate with “investment.”

Take the simple example of a firm that sells goods with a zero profit margin. Its workers are its customers. It pays its workers \$100, and they save \$10. This gives it sales of \$90, and a build up of inventory of \$10. The savings of the workers did not go to fixed investment, it resulted in a possibly undesired build up of unsold goods at the firm.

One thing to note that we are discussing flows within a single accounting period; in my example, the \$10 build up of inventories happens in the same accounting period as the savings (the same day, for example). As a result, when you write in the question that the \$500 savings today can be used for investment tomorrow is incorrect if we our accounting period is one day at a time - the savings today has to match investment today. (If we had a longer accounting period, such as quarterly, today and tomorrow could end up in the same accounting period.)

As a result, in order to understand the identity, you need to understand the definitions of the various components of the national accounts. If you added an example of a discussion that you had difficulties with to your question, someone might be able to give you more guidance.

My trick to looking at this is to note that the accounting identity holds for any accounting period, including extremely short ones. In particular, any group of transactions satisfies the identity. I then try to see how the low level transactions would end up if they were the only transactions in the national accounts (like the example above).


The corner solutions are where $q_1=0$ or $q_2=0$. So just take the first-order conditions, plug in zero for the value, and solve for income. For incomes less than that amount, the quantity demanded is zero.


Daron Acemoglu's papers on institution could be a very good way to enter in this kind of issues.


Is it possible for shareholders to get a companies assets in place of currency?

The short answer is no.

Even on a liquidation scenario it’s unlikely that a common stock shareholder would be distributed physical assets.

Assuming this is a C-Corporation you are asking about, shareholders do not have rights to ‘money’ or assets held inside the company- If the shares are of a preferred class with dividend rights you may receive a portion of retained earnings.


Is there a discrepancy between the ratio and total expenditure methods of measuring elasticity?

If something rises by $25\%$ then you then need a $20\%$ reduction in the new larger number to get back where you started: $1.25 \times 0.8=1$. This asymmetry means comparing large percentage changes can produced oddities

Another approach, closer to the idea of approaching the derivative as a limit, is to use logarithms and calculate the elasticity as a ratio of logarithms, something like $$\frac{\log\left(\frac{Q_2}{Q_1}\right)}{\log\left(\frac{P_2}{P_1}\right)}= \frac{\log(Q_2)-\log(Q_1)}{\log(P_2)-\log(P_1)}$$

and with your example this would give $\frac{\log(15)-\log(12)}{\log(8)-\log(10)} = \frac{\log(1.25)}{\log(0.8)} =-1$ as expected


How is money destroyed when banks issue debt?

Generally, people do not pay for bonds with currency (e.g. pound notes). They pay with bank deposits. This means that Y has lost a deposit.

If the deposit was with X, the deposit is just destroyed; X does not hold deposits against itself.
If the deposit was with another bank Z, bank Z loses the deposit, and it has to transfer the 100 pounds to X via the payments system. X will not hold a deposit at Z.

In either case, bank deposits are destroyed, and they are a component of the money supply.


What is the interpration of results if adding or subtracting a control variable effects the significance of explanotary variables?

When you run a regression you are making the assumptions of the Gauss Markov Theorem, one of which is that the error is uncorrelated with all of the independent variables (exogeneity).

If you remove from your regression a variable that is correlated with other explanatory variables then the error term and those explanatory variables will be correlated and the Gauss-Markov Theorem will then be violated. The consequence of this is that the estimates produced by the model will be biased. This particular form of bias is known as Omitted Variable Bias.

A consequence of omitted variable bias is that spurious variables can often be made significant be deleting important variables from the model. In these instances, the fact that they become statistically significant doesn't signify that they suddenly became important, it just means that you are estimating a mis-specified model that violates your own assumptions. So we can't just blindly interpret the hypothesis tests associated with a regression model, we must pay attention to whether the model is well-specified—including the question of whether all of the relevant variables are included.

Last point: your question is "What does this tell about whether the explanatory variables influence economic growth?" The answer is nothing because a regression in itself will not tell you anything about causality, only whether your variables are correlated. To make a causal connection between variables you have to think about how to build causal identification into your empirical strategy from the beginning.


Show that this income tax is effectively a lump sum tax

A lump sum tax is a tax which only has an income effect, it has the equivalent effect of reducing the agent's wealth. In particular, it has no substitution effect, it does not incentivise the agent to switch between $c$ and $l$.

In your case the agent's optimal choice of $c$ and $l$ is given by:
$(c^*,l^*)=\left(\frac{w(1-t)}{1+w(1-t)},\frac{w(1-t)}{1+w(1-t)}\right)$
(show this)

It is easy to see that increasing $t$ is equivalent to decreasing $w$.


I will assume that you are talking about the Solow, R. M. (1956). A contribution to the theory of economic growth. The quarterly journal of economics, 70(1), 65-94. as to my best knowledge Solow did not published paper entitled the "Theory of Economic Growth" in 1956 and the first equation you use in your question is from the paper.

Moreover, I also think that you either made a mistake or your textbook assumed whole output is saved (which would be most unorthodox) because normally the equation (1) should be looking like this:

$$\dot{K} = s F(K,L) - \delta K$$

See for example the Romer's Advanced Macroeconomics or Blanchard's et al Macroeconomics an European Perspective (both authors use slightly different notation but the point is both textbooks show that change in capital depends on the fraction of saved output not the whole output).

In the 1956 paper the closest equivalent of equation (1) in your question would be equation (3) in Solow's paper:

$$ \dot{K} = sF(K,L)$$

The reason why the depreciation is not explicitly there as a separate term is that Solow assumes that "Output is to be understood as net output after making good the depreciation of capital".


Where can I find the data for income, equity, assets and sales of a company that is presented in terms of U.S. and outside of US.?

Have you tried the Mergent database?  If you're part of a University system, Mergent is a typical offering through University libraries.  Your University librarian should be able to help you find and navigate it if necessary.


Why are we comparing the ratios of MPL to price of labor and MPK to price of capital not ratios of MPL and MPK to price of labor and price of capital

You can do both. Effectively you say "let me move one dollar from labor to capital expenditure and see what happens". By doing so you have reduced labor by 0.01 and thus reduced production by 0.15 chairs, but increased capital by 0.02 and thus production by 0.2 chairs. Net effect is 0.05 chairs more at the same cost.

Equivalently, MPL/MPK = 1.5< w/r =2, or MPL/w= 0.15 < MPK/ r = 2. Second is just rearrangement of the first, but with both you answer where the dollar of you cost is more efficient.


How does GNU software development sustain economically?

I would like to start by saying that I'm not a programmer and I have never contributed to any open source project. However, I have been interested in open source for a long time and I believe that I understand the general concepts of open source and how it works.

To start of, I would like to say that open source does not mean that you cannot make money on the software. It just means that the code has to be publicly available. Companies like Red Hat and Canonical make money not by selling the software, but by selling their expertise. If i wan't my company to run a Linux server, I can get the software for free. But I need somebody to install it, set it up and give support. This is where specialist from e.g. Red Hat comes in and makes money. For the company it makes sense, because hiring their own specialist would probably be much more expensive. This also gives these companies an incentive to contribute the the code. They want their product to be good so people will use it and by their services.

But lets talk about your points about scalability.

The cool thing about open source is that you do not have to develop everything from scratch. An operating system like Ubuntu has not been build by a single person. Instead a lot of people has contributed to different parts of the system (actually I think it would be hard to find one person with all the skills to make and effective operating system). For example, the Ubuntu people does not develop the Linux kernel. They just use one developed by others. So what was without open source probably impossible, is now possible,because you can build on other peoples work.
Reading and understanding others code it not more time consuming than writing your own. At least not in many cases. Beyond that, you do not have to understand all the code you use. If I want to write a program for Linux, I do not have to understand how all the parts in that program works in detail. I just have to know what they do. I can then take these parts and put them together with other parts to create my program. Or I can take an existing program and modify it for my needs.
tools like git and github makes it incredibly easy to collaborate. You just get the code and make modifications. You then submit them to the person in charge of the project. If it is good, it will be accepted.
people go in and out of projects all the time. But if the project is popular, enough will be working on it.

Here are some reasons why open source works.

I think the main reason that open source software has become so good is that the  large number of people working on the a project, insures a level of expertise that i hard to archive in a small team of developers. While it might seem weird, this single fact, seems to outweigh all the negative problems that can arise in open source.
In commercial programming, the project dies with the firm. Lets say you  by some software from a company that then closes. Then your screwed, as you will not receive updates and bug fixes, and you will need to by new software to keep up. With open source you can just find another company to support you software or develop it yourself.

If you are still interested, I suggest you read
The Cathedral and the Bazaar


Competitive equilibrium in Leontief economies

Strict convexity of preferences is not needed in existence results for competitive equilibria. Leontief preferences are quite well-behaved. They are continuous, convex, and strongly monotonic.   If all endowments are strictly positive, the existence of a competitive equilibrium in an exchange economy (or a production economy satisfying standard conditions) exists by the first result of the original Arrow-Debreu paper.

Arrow-Debreu actually do not just require convexity, they make, as pointed out by denesp in a comment, the convexity assumption (III.c) on utility functions that $u(x)>u(x')$ and $0<t<1$ implies $u(tx+(1-t)x')>u(x')$. Plain convexity suffices for existence, but Leontief preferences do also satisfy condition (III.c).:  Assume $\min\{\alpha_i x_i\}>\min\{\alpha_i x_i'\}$. Then
$$\min\big\{\alpha_i (tx_i+(1-t)x_i')\big\}>\min\big\{\alpha_i tx_i\big\}+\min\big\{\alpha_i(1-t) x_i'\big\}$$ $$=t\min\{\alpha_i x_i\}+(1-t)\min\{\alpha_i x_i'\}>\min\{\alpha_i x_i'\}.$$


Dynamic Market Equilibrium: motivation for differential equations

As the link you provided mentions:

  "Differential equations can be used to include the dynamic aspects to
  economics into a mathematical framework which takes into account the
  volatility present in economics."

So the author does not derive the differential equations as the result of a theoretical model -the author just wants to obtain the movement through time of equilibrium price (the author assumes market-clearing), as a second order differential equation.

So it is a purely descriptive/forecasting exercise, devoid of any economic theory background. The author obtains the solution, which includes a number of coefficients to be determined, for which the author mentions

  "These values can be estimated using statistics and econometric
  methods"

Some constraints on the parameters may come from observing the past price process (is it oscillating? around a trend or around an horizontal line? etc), and in the end, the hope is that we may obtain a good predictor for the price movements -again, without basing it on economic theory.


When can one safely talk about decreasing marginal utility?

The concept of "marginal utility" (and therefore of decreasing such) has meaning only in the context of cardinal utility.

Assume we have an ordinal utility index $u()$, on a single good, and three quantities of this good, $q_1<q_2<q_3$, with $q_2-q_1 = q_3-q_2$.
Preferences are well behaved and satisfy the benchmark regularity conditions, so

$$u(q_1)< u(q_2) < u(q_3)$$

This is ordinal utility. Only the ranking is meaningful, not the distances. So the distances $u(q_2) - u(q_1)$ and $u(q_3) - u(q_2)$ have no behavioral/economic interpretation. If they don't, neither do the ratios

$$\frac {u(q_2) - u(q_1)}{q_2-q_1},\;\; \frac {u(q_3) - u(q_2)}{q_3-q_2}$$

But the limits of these ratios as the denominator goes to zero would be the definition of the derivative of the function $u()$. So the derivative is devoid of economic/behavioral interpretation, and so comparing two instances of the derivative function would not produce any meaningful content.

Of course this does not mean that the derivatives of $u()$ do not exist as mathematical concepts. They can exist, if $u()$ satisfies the conditions needed for differentiability. So one can ask the purely mathematical question "under which condition the  function representing ordinal utility has strictly negative second derivative" (or negative definite Hessian for the multivariate case), trying not to interpret it as "decreasing marginal utility" with economic/behavioral content, but as just a mathematical property that may play some role in the model he examines.

In such a case, we know that:
1) If preferences are convex, the utility index is a quasi-concave function
2) If preferences are strictly convex, the utility index is strictly quasi-concave

But quasi-concavity is a different kind of property than concavity: quasi-concavity is an "ordinal" property in the sense that it is preserved under an increasing transformation of the function.

On the other hand, concavity is a "cardinal" property, in the sense that it won't necessarily be preserved under an increasing transformation.
Consider what this implies: assume that we find a characterization of preferences such that they can be represented by a utility index which is concave as a function. Then we can find and implement some increasing transformation of this utility index, that will eliminate the concavity property.


Does more variable input cause marginal product to decline?

The two statements are similar.

For the website example, you could say that the marginal product of the first 5 workers is 50 units. For the next 5 workers, however, you could deduce that the marginal product is less than 50 units (otherwise "the next 50 units" would require exactly 5 workers, not 10 workers).

The website is merely stating that, in order to get the same number of outputs (in this case 50 units) you need to hire proportionally more inputs. To judge whether this is counter productive requires knowledge about the price you could sell these products, etc.


Difference between a 3-months UK nominal spot rate and a 3-months UK treasury bill discount rate?

They are based on completely different assets.

Gilts:

As this document states:

the government liability nominal yield curves are derived from UK gilt prices and General Collateral (GC) repo rates.

Later on, the same document defines:

A conventional gilt is a guarantee by the Government to pay the holder of the gilt a fixed cash payment (coupon) every six months until the maturity date, at which point the holder receives the final coupon payment and the principal.

These assets are usually long term. As this page states:

In recent years the Government has concentrated issuance of conventional gilts around the 5-, 10- and 30-year maturity areas, but in May 2005 the DMO issued a new 50-year maturity conventional gilt. In June 2013, following market consultation the DMO issued a new 55 year maturity conventional gilt.

Meanwhile, the document referred earlier defines "General Collateral (GC) repo rates" as:

General collateral (GC) repo rates refer to the rates for repurchase a
greements in which any gilt may be used as
collateral.  Hence, GC repo rates should in principle be close to true risk-
free rates.  Repo contracts are actively traded for maturities out to one year;  the rates prevailing on these contracts are very similar to
the yields on comparable maturity conventional gilts.

Notice the phrase in bold. The rates are similar to comparable (e.g. 3-month T-Bills), but not identical. Unfortunately, they do not state why.

3-months T-Bills:

According to Bank of England definitions, the "Treasury bill tender 3 month (91 days) bills" is:

Treasury Bills are bearer Government Securities representing a charge on the Consolidated Fund of the UK issued in minimum denominations of £5,000 at a discount to their face value for any period not exceeding one year. Although they are usually issued for 3 month (91 days), on occasion they have been issued for 28 days, 63 days and 182 days. They are issued:

by allotment to the highest bidder at a weekly (Friday) tender to a range of counterparties;
in response to an invitation from the Debt Management Office to a range of counterparties;
at any time to Government departments (non-marketable bills only).

Regarding why they have less data available, the same article points out that:

The secondary market in Treasury bills has in recent years become illiquid and representative rates are no longer obtainable other than those for the most recently issued 91 day bills. The rates shown are the average rates of discount at the weekly tender for 91 day bills.

Finally, regarding to which one to use, I would be definitively inclined to use the data related to the yield curve (gilts). This, as indicated above, are traded daily in the secondary market. Thus, data is more extensive, and perhaps more "reliable".


What do supply-demand curves really look like?

You sometimes find textbooks drawing the supply and demand curves as concave upwards, as such:

The straight-line supply and demand curves can be thought of as a magnification of this graph, where the two intersect. Thus, the units on the axes would give you a clue as to how high up the graph is being drawn, or how far to the right (if the units start at a number other than 0 or skip an interval). This shows you why they are indeed called 'curves', even when they are sometimes straight lines.
Supply and demand curves are drawn using straight lines for simplicity. For example, two straight-line equations may be given, from which it is relatively simple to calculate the point of intersection.
In the real world
Supply and demand curves are an approximation of what happens in real life. The curves are a simplified model which show the general trend in the two functions. In reality, supply and demand curves are approximated using data that is collected over many years, with many short-term variables affecting the results, if the curves are even drawn up at all (the curves are more a theoretical model than a calculative method).
Economic factors which affect supply and demand

Price elasticity at different points of output affect the gradient of the curve at those points. A higher price elasticity results in a shallower gradient and vice versa. A unit price elasticity at all points on a demand curve will result in a hyperbola. In that case, the asymptotes of the demand curve are the x and y axis. FYI, the asymptotes of a demand curve will always be the x and y axis if the graph is drawn using this definition of price elasticities, but different elasticities will change the compression or expansion of the graph either vertically or horizontally so that it becomes not a hyperbola, but a hyperbolic graph.
The supply curve for a firm (with any level of competition in the market) is the marginal cost curve faced by that firm. The marginal cost curve looks like this:

However, due to the firm's variable costs, output is always greater than that at the turning point of the marginal cost curve.

For interest, in the special case of an oligopoly, there exists a kinked demand curve faced by an individual firm, as shown:


Historical returns for popular American funds over 20-30 years?

There are a few important things you need to consider when planning your retirement.

1. Risk tolerance.

You cannot invest only in stocks, whether indices or individual stocks. Even the most risk loving investors will get stressed out when their lifetime savings drop 10% over a week amidst some market volatility. It is usually best to put a portion of the money into bonds or other fixed income assets. They usually have much lower yields than stocks so it will bring down the yield of the entire portfolio. How much to put into bonds and how much into stocks depends on your risk preference and how old you are. The older you get the more you should allocate into fixed income assets.

2. Inflation.

Inflation will eat up a lot of your savings over the course of your life. Not much can be done about it. Stocks are usually more resistant to inflation so having a decent position in stocks helps.

3. Hidden costs of index funds/ETFs

ETF and index providers sometimes try to hide some costs like annual fees, commissions for good performance, dividends etc. It is very important to look thoroughly into the costs of the fund you are going to invest in.

So if you build a balanced portfolio of stocks (index funds) and bonds, and the broker charges are either very low or free, then your annual returns on average should be around 3-5% depending on inflation. 4% is the rough estimate of returns you can get on a portfolio that is mostly composed of index funds. There are tons of literature out there, but most legitimate sources converge on the number 4%.

Here is an article from a good practical blog about retirement that I came across recently: http://www.mrmoneymustache.com/2012/05/29/how-much-do-i-need-for-retirement/. It has been featured on the Wall Street Journal so it seems quite credible.


Why was the term 'Economic Rent' chosen? How does it relate to the ordinary meaning of 'rent'?

Note that the Etymoline page you link traces the origin of the word rent as meaning, more generally, "income revenue". Here's their definition (my emphasis):

  rent (n.1) "payment for use of property," mid-12c., a legal sense, originally "income, revenue" (late Old English), from Old French rente "payment due; profit, income," from Vulgar Latin *rendita, noun use of fem. past participle of *rendere "to render" (see render (v.)).

This meaning (income revenue) is quite close to the use of the term rent by economists—i.e., an income arising from private control of some scarce factor of production.

Indeed, if we consult the Oxford English Dictionary (which I usually take as the definitive reference on the language), here's what we find in the first entry:

  1. b. Revenue, income. Also fig. Obsolete.

The OED has examples of the word being used in this sense as late as 1783. Meanwhile, the first recorded usage of "rent" in the sense meant by economists is 1662. So economists didn't simply invent a new meaning for the word rent; they adopted the word because, at the time early economists were writing, it meant exactly what they wanted to say.

The more modern meaning of "payment for use of property" is essentially a special case of the older, more general meaning of the word rent. Indeed, such a payment is an expense for the payer but an income for the recipient.

If a recipient owns an apartment and earns an income ("rent") from letting it out to a tenant then s/he is earning rent both in the common modern sense of the word, and in the sense described by Pindyck and Rubinfeld. So the usage is clearly related. It's just that this situation (someone earning an income because they own a scarce factor of production) is somewhat more general than the simple contexts in which the word rent finds every day usage.


Is there an alternative to currency to facilitate exchange of goods and services?

First, there are many misconceptions and mistakes in your question.

You state:

Instead, I have just caused inflation, since theres more stuff in relation to money.

No, inflation is inversely related to output. If the amount of money is fixed and you create more valuable products that (ceteris paribus) creates deflation not inflation (see Blanchard et al. Macroeconomics: An European Perspective).

You state:

Whats weird about this is, that even though I did all the work, tom became 10\$ poorer yet both me and the bank became \$10 richer.

No this does not make any sense from economic perspective. Tom purchased those seashells from you for \$10 which clearly shows that Tom values seashells more than \$10 and you sold those seashells for \$10 which clearly shows that you value the money more than seashells. Hence after trade you both have higher value than before and you are both richer.
In addition, if Fed loans bank \$10 the bank did not became richer by \$10. Now bank has \$10 liability so their net wealth is zero (since \$10 in cash assets - \$10 in liability =0). So asserting that banks becomes richer just by borrowing from Fed is absurd. Banks earn money on later lending out the funds on intermediation margins and those intermediation margins are their payment for providing financial intermediation services (e.g. screening clients, having accessible branches, etc.).

You state

money can only be used to exchange a fixed supply of things back and forth and can't actually account for creation of new wealth without "hacks" like the federal reserve

No this again makes little sense. Money existed long before Fed. If you have fixed supply of money and amount of products and services in economy increases then prices across economy adjust (decrease via deflation).  That means that even with fixed money supply output can increase indefinitely. As a matter of fact even when Fed prints money to keep prices constant, in real terms value of output is still allowed to increase as in economics what matters is the value of goods and services (e.g. real output) what matters not how price change.
Now to your question:

Is there a system of exchange that would be similarly easy to money, but that accounts for the creation of wealth in a natural and , if possible, fair way?

No, alternative to exchange with money is barter, but barter is much less convenient and inefficient due to the problem of double coincidence of wants. With barter you always have to search for a person that both sells what you want and want what you sell.
At the fundamental level the only difference between barter and exchange with money is that the problem of double coincidence of wants is eliminated. Instead of searching for someone who sells what you want as well you just sell your seashells and then buy what you want from some another person. In the end you are still exchanging just goods and services for other goods and services just with extra step that makes everything more convenient (but clearly less obvious to people without economic training).
In addition I can't see how money is less 'natural' (whatever that is supposed to mean) than barter. Going with dictionary definition of natural as something not man made neither barter or money is natural.
Lastly, what is fair or not fair is not for economists to say (although I have hard time seeing any unfairness with concept of money per se) and requires moral judgements - for that you need to visit philosophy stack.


How can the stock market keep growing indefinitely?

As mentioned in the other answer, the amount of money does not have to correspond to the total value of all assets in the economy. However, there is some correspondence that is not mentioned in the other answer so I will focus on that.
First, there should always be enough money in the economy so people can carry all the transactions they want. If that is not true the money market would not be in equilibrium. This can be formalized using the equation of exchange (see Mankiw, Macroeconomics 7th ed pp 86):
$$MV=P_TT$$
where $M$ is the stock of money, $V$ is the velocity of money (how many times is on average one bill used), $P_T$ are the prices and $T$ is the amount of transactions (which for the purposes of this answer can include asset transactions as well although normally this sort of model would be used only for goods and services).
We can learn several things from this formalization.
First, as the previous answer pointed out amount of money $M$ does not need to be equal to the value of assets, since the equation is $MV=P_TT$ not $M=P_AA$ (where $A$ would be assets). So the velocity of money matters as well. If you buy from me 1 stock of Apple for ${\\\$}100$ I can use the same ${\\\$}100$ dollar bill to buy other assets - so even though there is only single ${\\\$}100$ dollar bill multiple things can be purchased with it if it circulates fast enough. Moreover, because not all assets $(A)$ are always transacted $(T)$ there does not need to be correspondence between value of all assets and money as well.
Second the formula $MV=P_TT$ can be rearranged by decomposing the prices into aggregate price level $P$ and relative prices $P_r$ of the 'things' that are being sold on the market where the following equality would hold $P_TT=P P_r Q$  where $P_r Q$ would be the real value (i.e. inflation adjusted value) of $Q$'s  that are being sold (which we could also be in principle some asset). Lets also denote the real value of things being sold as $T_r$. So now the model would look like:
$$MV = PT_r$$
The equation above also shows us another thing. Even when $M$ and $V$ are fixed the real value of transactions $T_r$ can grow because prices adjust.
Ultimately what matters are real returns not nominal returns. When someone says that stock market returns beat inflation by $7\%$ that means that they real value increased by $7\%$ not that their nominal value is by $7%$ higher. For example, imagine that economy is experiencing deflation (negative inflation of $10\%$ and stock increases in value in real terms by $7\%$ per year. In such such situation if you invest ${\\\$}100$ dollars for one year in nominal terms at the end of next year your stock will be worth only ${\\\$}97$. However, real value of your asset (correcting for inflation or in this case deflation) is ${\\\$}107$ because deflation means everything in economy is getting cheaper and so you can purchase much more with ${\\\$}97$ than with ${\\\$}100$ year before.
The real value of asset  such as stock depends, very broadly speaking, on peoples' valuation of the underlaying companies. Constant improvement in technology and productivity on average leads to companies being more valuable on average over long time periods (of course, technological progress destroys some companies but it also allows other to flourish or completely new industries to rise up and we are talking about whole market here). Consequently, in real terms the only limit for how much stock market can grow are technological & resource constraints avaiable to humanity (with emphasis on the technology).
In addition, it is worth noting there is no limit on how many money there can be in the world. Nowadays most of money is not even printed it is just created at a keystroke virtually. In principle if central banks would be willing to they could always find a way to expand money supply to any size they desire, so in theory $M$ can be any non-negative real number you can think of.
Hence in conclusion, the premise in your question ignores the velocity of money and the fact that not all assets are always transacted so there does not need to be enough money to purchase them all. And secondly even if velocity of money and and money supply is fixed the prices can still adjust in a way that allows for the market to grow in real terms.


If you are looking for data on costs for production plants you might consider the BLS productivity data https://www.bls.gov/mfp/mprdload.htm.

Under the combined sector and industry multifactor productivity excel file you can find cost measures organized by NAICS (North American Industry Classification System) at the three digit level [https://www.census.gov/cgi-bin/sssd/naics/naicsrch?chart=2012] .

Depending on what type of production plant you are looking for there is data by energy usage, rent, and other types of costs.


Show a compound function is increasing and supermodular given its components

I don't think this proposition holds without assuming that $g$ is also increasing.
Take $f(x_1,x_2)=x_1x_2$ (actually any supermodular function) and $g(z)=-z$ (decreasing and convex). Then, $g(f(x_1,x_2))=-x_1x_2$ which is neither increasing in its arguments or supermodular.

If you assume also that $g$ is increasing, you get $g'(f(x))\geq 0$ straightforwardly.

You can easily show supermodularity using the conddition on the second derivatives. Denote $x=(x_1,...,x_n)$. Supermodularity is equivalent for function f to $$\frac{\partial^2 f}{\partial x_i \partial x_j}(x)\geq 0$$ for any $i\ne j$.
If you twice-differentiate $g \otimes f$ ($\otimes$ is the compound operator),
$$
\frac{\partial^2 g \otimes f}{\partial x_i \partial x_j}(x)=\frac{\partial^2 f}{\partial x_i \partial x_j}(x) . g'(f(x)) + \frac{\partial f}{\partial x_i}(x) .\frac{\partial f}{\partial x_j}(x) .g''(f(x))
$$
You can check that $\frac{\partial^2 g \otimes f}{\partial x_i \partial x_j}(x)\geq 0$ so that $g \otimes f$ is supermodular since $f$ is supermodular, $g$ is increasing and $g$ is convex.


It is perfectly consistent for the marginal revenue to increase in $q$, even if the demand curve decreases.

Marginal revenue is
$$p(q)+ q p'(q).$$

The first term says "if I sell one extra unit then I will receive an extra $p$ in revenue". The larger is this effect, the higher is the MR.
The second term says "in order to sell one extra unit, I will have to reduce my price by $p'(q)$, which means that much less revenue on each of the $q$ units I would have sold anyway". The larger is this effect, the lower is the MR.

If $p''(q)>0$ it means the price cut needed to sell an extra unit (the second effect above) gets smaller as we move down the demand curve. So the negative part of the marginal revenue is diminished, which can cause the marginal revenue overall to increase.

Note 1: although the marginal revenue function can be increasing, it can never rise above the demand curve. Indeed, from the expression above we see that the marginal revenue is strictly below the demand curve whenever $p'(q)<0$.

Note 2 (slightly more technical): the possibility of an increasing marginal revenue implies a potential non-uniqueness of the solution to the profit-maximisation problem (even when marginal costs are constant). This is because there may be more than one point at which MR=MC, as the below figure shows (this figure was generated with the demand function $p(q)=1-q+3q^2-3q^3$):

We only have to think about points where MR=MC and MR is decreasing (the one in the figure where MR is increasing is a local minimum for profits rather than a local maximum). The other two are both local maxima, but typically only one will be the global profit-maximising point.

Becuase dealing with cases where there are multiple potential solutions is messy, people often make an assumption to guarantee a unique solution. One such assumption is that "demand is log-cocave", meaning $ln(p(q))$ is concave. Let's see why this works. Demand is log concave if

$$\frac{d^2\ln(p(q))}{dq^2}<0$$
$$\frac{p''(q)}{p(q)}-\frac{p'(q)^2}{p(q)^2}<0$$
So if demand is log-concave we know
$$p''(q)<\frac{p'(q)^2}{p(q)}.$$

We can use this information to see how large we can make the derivative of the MR curve that you calculated:
$$\frac{d MR}{dq}=2p'(q)+qp''(q)\leq 2p'(q)+q\frac{p'(q)^2}{p(q)}.$$
Can this be positive? The answer is yes if
$$2p'(q)+q\frac{p'(q)^2}{p(q)}>0$$
i.e. if
$$2p(q)+qp'(q)=MR+p(q)<0.$$
(in the last line I changed the direction of the inequality because I divided by $p'$, which is negative).

We find that, given log-concave demand, $\frac{d MR}{dq}$ can be positive only if $MR<0$. So the marginal revenue curve will never slope up in the region the firm would consider choosing.


For annual individual-level earnings data:

From 1962: try Current Population Survey (CPS) https://cps.ipums.org/cps/intro.shtml
From 2000, try American Community Survey (ACS) https://usa.ipums.org/usa/acs.shtml

For decadal individual-level census data:

From 1850: IPUMS-USA https://usa.ipums.org/usa-action/faq#ques0


what does historical volatility explain?

No. Volatility by itself is a meaningless number.

Volatility becomes relevant when used in a context, such as when comparing between assets, or to identify how much the stability of a financial instrument has changed over time. In these two examples, dividing volatility by the mean rate of return gives a more useful, dimensionless number known as coefficient of variation.

Historical volatility may also serve as input parameter for simulation of an asset price, provided that the analyst already has in mind a probabilistic model. The example you outline is akin to estimating a confidence interval. Your example reflects an assumption that the asset's behavior is modeled with a probability density function which is symmetrical with respect to the latest price.

There are other uses of historical volatility, but the ones above are among the most typical applications of that concept.


The statement is not true.
Let $x_A + x_B = 1$, $y_A + y_B = 1$.
Let $U_A(x_A,y_A) = x_A + \ln(y_A)$, $U_B(x_B,y_B) = x_B + \ln(y_B + 1)$.
$f \neq g$ are both strictly increasing & concave. For all $z \in [0,1]$ the distributions
$$
(x_A,y_A) = (z,1), \ (x_B,y_B) = (1-z,0).
$$
are Pareto-efficient. This follows from $MRS_A(x_A,y_A) \geq MRS_B(x_B,y_B)$.
 In these cases $y_A \neq y_B$.


The Effect of Two income Households on Housing Prices

Todd Zywicki has tried to directly evaluate this claim. He finds that it housing prices and other costs are much lower than the increase in the tax burden.

  Thus, taxes increase in the example by \$13,086. By contrast, annual
  mortgage obligations increased by only \$3690 and automobile
  obligations by \$2860 and health insurance \$620. Those increases are
  not trivial, but they are swamped by the increase in tax obligations.
  Too put this in perspective, the increase in tax obligations is over
  three times as large as the increase in the mortgage (the supposed
  driver of the "two income trap") and about double the increase in the
  combined obligations of mortgage and automobile payments. This also
  leaves aside the peculiarity that the 2000s family is paying \$9670 in
  new child care and \$2860 in new automobile expenses supposedly to meet
  a \$3690 increase in mortgage expenses, the supposed driver of the
  model.

Evaluating The Two-Income Trap Hypothesis

Arnold Kling (Taxes and the Two-Income Trap) notes that Zywicki's estimate doesn't include payroll taxes and therefore underestimates the change in tax burden.


Statistical models which represent the impact of strict measures vs no strict measures (and less economic impact), enforced due to Covid-19

"The Benefits and Costs of Using Social Distancing to Flatten the Curve for COVID-19", Thunstrom et al.

  We examine the net benefits of social distancing to slow the spread of COVID-19 in the United States. Social distancing saves lives but imposes large costs on society due to reduced economic activity. We use epidemiological and economic forecasting to perform a rapid benefit-cost analysis of controlling the COVID-19 outbreak. Assuming that social distancing measures can substantially reduce contacts among individuals, we find net benefits of about $5.2 trillion in our benchmark case. We examine the magnitude of the critical parameters that might imply negative net benefits, including the value of statistical life and the discount rate. A key unknown factor is the speed of economic recovery with and without social distancing measures in place. A series of robustness checks also highlight the key role of the value of mortality risk reductions and discounting in the analysis and point to a need for effective economic stimulus when the outbreak has passed.

"A Simple Planning Problem for COVID-19 Lockdown" (NBER) Alvarez, Argente, and Lippi

  We study the optimal lockdown policy for a planner who wants to control the fatalities of a pandemic while minimizing the output costs of the lockdown. We use the SIR epidemiology model and a linear economy to formalize the planner's dynamic control problem. The optimal policy depends on the fraction of infected and susceptible in the population. We parametrize the model using data on the COVID19 pandemic and the economic breadth of the lockdown. The quantitative analysis identifies the features that shape the intensity and duration of the optimal lockdown policy. Our baseline parametrization is conditional on a 1% of infected agents at the outbreak, no cure for the disease, and the possibility of testing. The optimal policy prescribes a severe lockdown beginning two weeks after the outbreak, covers 60% of the population after a month, and is gradually withdrawn covering 20% of the population after 3 months. The intensity of the lockdown depends on the gradient of the fatality rate as a function of the infected, and on the assumed value of a statistical life. The absence of testing increases the economic costs of the lockdown, and shortens the duration of the optimal lockdown which ends more abruptly. Welfare under the optimal policy with testing is higher, equivalent to a one-time payment of 2% of GDP.

Also

"The Macroeconomics of Epidemics" by Eichenbaum, Rebelo, and Trabandt. Published in NBER in March when it got NYT coverage; there's an "April update" now. The abstract of that paper isn't very revealing (as it only describes one scenario), but they actually consider several scenarios in the paper, with various exogenous constraints like treatments being found effective or not, vaccines being discovered etc. And then they consider a lockdown sequence optimization problem [in each scenario] as a Ramsey problem; see section 5 in the paper... and then some "exit strategies" from each. (Even that can be formulated as an optimization problem, which they call "smart containment" but it requires knowing the immunity status of everyone--which is a bit unrealistic as they admit.) Teaser figure from the paper (I won't try to explain it here):

"Macroeconomic Dynamics and Reallocation in an Epidemic", (NBER, also in CEPR with somewhat more copyediting) Krueger, Uhlig, Xie

  In this paper we argue that endogenous shifts in private consumption behavior across sectors of the economy can act as a potent mitigation mechanism during an epidemic or when the economy is re-opened after a temporary lockdown. Extending the theoretical framework proposed by Eichenbaum-Rebelo-Trabandt (2020), we distinguish goods by their degree to which they can be consumed at home rather than in a social (and thus possibly contagious) context. We demonstrate that, within the model the "Swedish solution" of letting the epidemic play out without government intervention and allowing agents to shift their sectoral behavior on their own can lead to a substantial mitigation of the economic and human costs of the COVID-19 crisis, avoiding more than 80 [percent] of the decline in output and of number of deaths within one year, compared to a model in which sectors are assumed to be homogeneous. For different parameter configurations that capture the additional social distancing and hygiene activities individuals might engage in voluntarily, we show that infections may decline entirely on their own, simply due to the individually rational re-allocation of economic activity: the curve not only just flattens, it gets reversed.

"Internal and External Effects of Social Distancing in a Pandemic" (NBER) Farboodi, Jarosch, Shimer

  We use a conventional dynamic economic model to integrate individual optimization, equilibrium interactions, and policy analysis into the canonical epidemiological model. Our tractable framework allows us to represent both equilibrium and optimal allocations as a set of differential equations that can jointly be solved with the epidemiological model in a unified fashion. Quantitatively, the laissez-faire equilibrium accounts for the decline in social activity we measure in US micro-data from SafeGraph. Relative to that, we highlight three key features of the optimal policy: it imposes immediate, discontinuous social distancing; it keeps social distancing in place for a long time or until treatment is found; and it is never extremely restrictive, keeping the effective reproduction number mildly above the share of the population susceptible to the disease.

Not strictly about Covid-19, but also recent paper from FRB members (and one MIT faculty):

Correia, Luck and Verner. "Pandemics Depress the Economy, Public Health Interventions Do Not: Evidence from the 1918 Flu",

  what are the economic costs and benefits of non-pharmaceutical interventions (NPI)? Using geographic variation in mortality during the 1918 Flu Pandemic in the U.S., we find that more exposed areas experience a sharp and persistent decline in economic activity. The estimates imply that the pandemic reduced manufacturing output by 18%. The downturn is driven by both supply and demand-side channels. Further, building on findings from the epidemiology literature establishing that NPIs decrease influenza mortality, we use variation in the timing and intensity of NPIs across U.S. cities to study their economic effects. We find that cities that intervened earlier and more aggressively do not perform worse and, if anything, grow faster after the pandemic is over. Our findings thus indicate that NPIs not only lower mortality; they may also mitigate the adverse economic consequences of a pandemic.

A 2nd (recent) paper from a FRB author on the 1918 pandemic, albeit a bit more descriptive: "What Happened to the US Economy During the 1918 Influenza Pandemic? A View Through High-Frequency Data" (and the part most relevant to the question here is left in teaser form in the abstract...)

  Interventions to hinder the contagion were brief (typically a month) and there is some evidence that interventions made a difference for economic outcomes.