Effect of a US Federal Reserve rate hike on the Dow Jones Industrial Average

Each component of the Dow Jones Industrial Average (DJIA) has a yield, computed as the earnings divided by the price, expressed as a percentage. For example, if a stock is priced at $100.00 and yearly earnings of $10.00 per share, then the yield for that stock is

yield = earnings / price = $10.00 / $100.00 = 10%

That yield could be paid out as a dividend, or retained and reinvested to grow the company, doesn't matter to this analysis.

The first step in the analysis is to determine the yield for each of the 30 components of the DJIA. On August 18, 2015 the DJIA closed at 17511.34, and the yields for the component companies were

    Company     Price Earnings  P/E  Yield
Visa            74.47   2.16   34.5  2.90%
Microsoft       47.27   1.46   32.4  3.09%
Nike           114.82   3.70   31.0  3.22%
Procter         75.13   2.44   30.8  3.25%
HomeDepot      122.80   4.71   26.1  3.84%
Coca-Cola       41.30   1.60   25.8  3.87%
WaltDisney     106.94   4.26   25.1  3.98%
Pfizer          35.52   1.42   25.0  4.00%
UnitedHealth   125.50   5.70   22.0  4.54%
McDonald's     100.76   4.82   20.9  4.78%
3M             148.31   7.49   19.8  5.05%
Verizon         47.46   2.42   19.6  5.10%
Boeing         144.60   7.38   19.6  5.10%
Johnson         99.37   5.70   17.4  5.74%
GE              26.07   1.50   17.4  5.75%
Cisco           28.25   1.73   16.3  6.12%
Merck           59.66   4.07   14.7  6.82%
AmExpress       81.24   5.56   14.6  6.84%
UnitedTech      99.41   6.82   14.6  6.86%
WalMart         69.48   5.05   13.8  7.27%
DuPont          53.90   3.92   13.8  7.27%
Apple          116.50   8.66   13.5  7.43%
Caterpillar     78.82   5.88   13.4  7.46%
IBM            156.01  11.90   13.1  7.63%
JPMorgan        68.21   5.29   12.9  7.76%
Intel           28.91   2.36   12.2  8.16%
Goldman        201.18  17.07   11.8  8.48%
Exxon           77.90   7.60   10.3  9.76%
Travelers      107.82  10.70   10.1  9.92%
Chevron         83.44  10.14    8.2 12.15%

The next step in the analysis is to determine the adjusted yield after the Federal Reserve increases the interest rates by 0.25%.

The expectation is that the market will adjust prices to increase yields by 0.25%. In other words, market yields represent a premium above the Federal Reserve rate.

So for example, the yield on Visa must increase from 2.90% to 3.15% to compensate for a 0.25% increase in the Federal Reserve rate. Given earnings of $2.16, the price must drop from $74.47 to $68.56 (a 7.9% drop) to compensate for the rate increase.  At the other end of the scale, the yield on Chevron must increase from 12.15% to 12.40%. Given earnings of $10.14, the price must drop from $83.44 to $81.76 (a 2% drop).

We can now augment the previous table to include the adjusted yields, adjusted prices, and the drop in price as a percentage of the original price.

                                                  Adjusted    Price
    Company     Price Earnings  P/E  Yield     Yield   Price  Change
Visa            74.47   2.16   34.5  2.90%     3.15%   68.56  -7.94%
Microsoft       47.27   1.46   32.4  3.09%     3.34%   43.73  -7.49%
Nike           114.82   3.70   31.0  3.22%     3.47%  106.55  -7.20%
Procter         75.13   2.44   30.8  3.25%     3.50%   69.76  -7.15%
HomeDepot      122.80   4.71   26.1  3.84%     4.09%  115.29  -6.12%
Coca-Cola       41.30   1.60   25.8  3.87%     4.12%   38.80  -6.06%
WaltDisney     106.94   4.26   25.1  3.98%     4.23%  100.62  -5.91%
Pfizer          35.52   1.42   25.0  4.00%     4.25%   33.43  -5.89%
UnitedHealth   125.50   5.70   22.0  4.54%     4.79%  118.95  -5.22%
McDonald's     100.76   4.82   20.9  4.78%     5.03%   95.76  -4.97%
3M             148.31   7.49   19.8  5.05%     5.30%  141.31  -4.72%
Verizon         47.46   2.42   19.6  5.10%     5.35%   45.24  -4.67%
Boeing         144.60   7.38   19.6  5.10%     5.35%  137.85  -4.67%
Johnson         99.37   5.70   17.4  5.74%     5.99%   95.22  -4.18%
GE              26.07   1.50   17.4  5.75%     6.00%   24.98  -4.16%
Cisco           28.25   1.73   16.3  6.12%     6.37%   27.14  -3.92%
Merck           59.66   4.07   14.7  6.82%     7.07%   57.55  -3.54%
AmExpress       81.24   5.56   14.6  6.84%     7.09%   78.38  -3.52%
UnitedTech      99.41   6.82   14.6  6.86%     7.11%   95.91  -3.52%
WalMart         69.48   5.05   13.8  7.27%     7.52%   67.17  -3.33%
DuPont          53.90   3.92   13.8  7.27%     7.52%   52.11  -3.32%
Apple          116.50   8.66   13.5  7.43%     7.68%  112.71  -3.25%
Caterpillar     78.82   5.88   13.4  7.46%     7.71%   76.26  -3.24%
IBM            156.01  11.90   13.1  7.63%     7.88%  151.06  -3.17%
JPMorgan        68.21   5.29   12.9  7.76%     8.01%   66.08  -3.12%
Intel           28.91   2.36   12.2  8.16%     8.41%   28.05  -2.97%
Goldman        201.18  17.07   11.8  8.48%     8.73%  195.42  -2.86%
Exxon           77.90   7.60   10.3  9.76%    10.01%   75.95  -2.50%
Travelers      107.82  10.70   10.1  9.92%    10.17%  105.17  -2.46%
Chevron         83.44  10.14    8.2 12.15%    12.40%   81.76  -2.02%

The adjusted DJIA computed from the adjusted prices is 16747.96. Thus, the DJIA was expected to drop from 17511.34 to 16747.96 (a 760 point 4.4% drop) to compensate for a 0.25% increase in the Federal Reserve rate.

What actually happened was the DJIA dropped to a low of 15666.44 low on August 25, 2015 (an 1840 point 10.5% drop) before recovering to a range between 16300 and 16500, slightly below the expectation.


What benefits does Bitcoin (i.e. cryptocurrency) offer?

I'd say that some benefits of Bitcoin and other cryptocurrencies (i.e. "altcoins"), include:

Decentralization -- which is advantageous for those who are skeptical of monetary policy, runs on banks, etc.
Anonymity, or at least the perception thereof -- with increasingly comprehensive monitoring of transactions (particularly non-cash transactions), those seeking relatively anonymous methods of exchange have traditionally been out of luck in the realm of digital transactions. Bitcoin has widely been regarded as a "digital cash", providing some measure of relative anonymity in online transactions, though this is a decreasingly accurate perception as government databases cataloging transactions grow in sophistication (and some of these databases, such as that of the FBI's, have been opened up to public developers via coding marketplaces like Kaggle). Some altcoins, such as Monero (XMR), add measures to enhance the benefit of relative anonymity.
Portability -- they can be easily stored and transported offline (paper-based hashes or QR codes), on portable media, over the Internet, with or without the aid of a "wallet" or bank-like service.
Region Neutrality -- unlike fiat currency or Eastern-only currencies such as Perfect Money, cryptocurrencies seem to have less bias based on location or language.


Concept for "when a mechanism exists which can be exploited, people will exploit it"

I think the broad principle you are looking for is: people respond to incentives.

This is an important principle (some would say the most important in economics), with many implications and corollaries. But I think the particular implication you're looking for is unintended consequences.

So in your example, welfare was intended to alleviate poverty. However, because people respond to incentives, welfare has the unintended consequence that some people have more children.

(Other similar terms you may be thinking of: Cobra effect, perverse incentive.)


@GraemeWalsh, another option is to use nonlinear least squares to estimate the elasticitiy of substitution, $\sigma$. This is what you want to estimate. The rest is either assumption or data. If you assume the technology is directed i.e. induced, you should use Box-Cox transformation of technology growth so that factor augmenting technological change grows at varying rates over time.

Values of $\gamma$ depend on the average growth rate of output and how you want the technological progress to be augmented. If you think labour-augmentation was strong, then choose a value that is close to actual average rate of output growth for $\gamma_N$, something smaller for the growth rate of the capital augmenting tech growth rate $\gamma_K$.

$\pi$s are constant share parameter.I do not know much about normaisation except that it should be chosen with reference to a common time period. You could normalise all the variables and parameters using the first period, or any other. Alternatively, use non-linear methods of estimation. You could try using micEconCES package in R.


What is the evidence that econometrics has empirical value?

To (slightly) paraphrase the OP:

  Economies (Human Bodies) are extremely complex systems with many variables, not to mention the fact that they emerge from the
  interactions of complex
  beings (factors). I agree that economies (human bodies) have certain underlying principles, but I remain skeptical of the
  overall value of econometrics (medicine) as a science.

  A truly useful econometrics (medical science) would be a
  valuable tool in predicting the future behavior of the economy
  (human body), particularly in predicting shocks like the recent financial crisis and Great Recession (severe illnesses or near
  death). But it did not happen.

  If econometrics (medical science) can't predict the future, how
  do we know it even effectively describes the past? What is the
  evidence that it has empirical value?

Comment: There is a Present that needs to be dealt with, just like with human bodies.

To (slightly) paraphrase Milton Friedman:

  I don't care how I predict, as long as I predict adequately.

Comment: But what do we do when our predictions are not adequate ?

Econometrics is based on Mathematical Statistics on the one hand, and on the assumptions made by Economic Theory on the other, which in turn are based on (imperfect) empirical observation, and are then led to their logical conclusions. In other words, Econometrics uses rigorous mathematics, induction, deduction and all the words dear to an epistemologist. Its epistemological foundations are solid as a rock.

What hangs on the balance is the observed validity of its abstractions. So the question is not whether Econometrics is a science, in the sense of whether it follows the scientific method or not. It does, fully.  The question is whether its results are useful.

But what are the criteria in order to determine "useful"? Is it just adequate predictions? That would be a matter of disagreement between humans.

And is it a matter of a "yes/no" answer?  Or is it a matter of degree to which it is useful? In which case, we have to somehow measure this degree (after we have agreed on the criteria), which brings us back to where we have to collect, analyze, assess, and debate the evidence.


I don't have a question to "correctly interpret" , but I think the way you are calculating the demand function is correct. The demand curve in an oligopoly can be kinked (bowed out), as this one is. I think it is mathematically accurate to write the function split across the domains, as such:

$Q = \begin{cases}
700 - P, & 0  \leq P\leq 500 \\
2200 - 4P, & 500< P< 550
\end{cases}$

where I have written quantity as a function of price, and assumed that the quantity is strictly positive.


Since you are stuck at just estimating AR(1) models I can show you some example code for estimating the equations you gave. Keep in mind that I did not set slopes: $a$ and $c_1$ in the random walk and  random walk with drift. Furthermore, the example code is in Python, however it should be self-explanatory and reproducible in STATA.

import numpy as np
import matplotlib.pyplot as plt

# Setting sample size
sample_size = 100

# Vectors with variables of interest
b = np.zeros(sample_size)
x = np.zeros(sample_size)
y = np.zeros(sample_size)

# IID random errors (normally distributed)
u = np.random.normal(size = sample_size)
n = np.random.normal(size = sample_size)
e = np.random.normal(size = sample_size)

# Drift
c_0 = 2

# Simulating each series
for t in range(sample_size):
    b[t] = b[t-1] + u[t]
    x[t] = c_0 + x[t-1] + n[t]
    y[t] = b[t] * x[t] + e[t]

# Plotting the results
plt.rcParams["figure.figsize"] = [10,10]

plt.subplot(321)
plt.plot(b)
plt.title('Random walk: $b_t = b_{t-1} + u_t$')

plt.subplot(322)
plt.plot(x)
plt.title('Random walk with drifft: $x_t = c_0 + x_{t-1} + n_t$')

plt.subplot(323)
plt.plot(y)
plt.title('$y_t = b_t * x_t + e_t$')

plt.show()


Accounting profit in macroeconomics as compared to microeconomics

There is no difference in definitions. In macroeconomics the economic profit is also surplus after accounting opportunity cost and all  other implicit and explicit cost.
What the equation above says is just that the accounting profit also contains return to capital. That also hold in microeconomics for any firm that owns and uses capital in its production function.
For example, typical microeconomic profit equation will be given as:
$$\Pi = PF(K, L) − wL − rK$$
where $P$ is price, $F(\cdot)$ output, $w$ wage, $r$ return, $L$ labor and $K$ capital. In the equation above the firm rents capital from some other firm. However, if If firm owns its capital rather than rent it the $rK$ will belong to the firm owner and be part of their profit (you can think of it as firm 'renting' capital from itself).
The equation above simply states that the accounting profit will be sum of economic profit (i.e. surplus after all costs are accounted for) plus any return on the capital they provide.
In fact in both microeconomics and macroeconomics this extends even further. If firm owner provides also labor/entrepreneurial labor, then the accounting profit will be actually given by $\text{accounting profit}  = \text{Economic Profit} + wL + rK$.
Again this is not something that would be different in economics. To see this just rearrange the expression for economic profit.
$ \text{Economic Profit}  = \text{accounting profit} - wL - rK$
The above literary tells you that economic profit is surplus after accounting the explicit and implicit costs of labor and capital. The wage and rent should already reflect the opportunity costs of using that labor and wage elsewhere.


I assume $t=pq$ is meant. $p$ is the price per unit.

We want to know when

$$U(q,p)=a \ln(1+q) - pq$$

is maximal, given $a>0$ and $p>0$, since this is the utility function and the quantity $q$ the customer buys is the quantity where the utility function is maximal.

The derivative with respect to $q$ is

$$\frac{d}{dq} U(q,p) = \frac{a}{1+q} - p$$

Set derivative to zero gives $q =\frac{a-p}{p}$.

Now we want to know when $(p-c)q$ is maximal, since that gives maximal profit.

Insert $q =\frac{a-p}{p}$ to get $\frac{(a-p)(p-c)}{p}$ is maximal. Rewrite this to $-p+a+c-\frac{ac}{p}$. Take the derivative with respect to $p$ and set to zero to get

$$-1+\frac{ac}{p^2}=0$$

$$-p^2+ac=0$$

$$p^2=ac$$

$$p=\sqrt{ac}$$

Thus we get $t=pq=a-p=a-\sqrt{ac}$.


How many US Dollars are lost or destroyed annually?

Are there any estimates on how many US Dollars are lost or destroyed
annually? By "lost or destroyed", I mean permanently removed from
circulation because the currency is no longer usable.

The link I offered as a comment covers this question nicely.

When Currency Is Physically Destroyed
Obviously, not all money is electronic. Just look at your wallet.
Bills and coins are destroyed every day. There are three destroyers of
money, and they're the same ones who create and regulate it.
(1) The Bureau of Engraving and Printing and (2) The U.S. Mint
The U.S. Bureau of Engraving and Printing creates all of the nation's
bills, while the U.S. mint creates its coins. But they also destroy
money.
Banks and individuals will hand over "mutilated" bills and coins to
these agencies. They then validate its authenticity and issue a
Treasury check in return. The Bureau of Engraving and Printing
receives around 25,000 mutilated currency redemption claims annually.
Each bill is shredded and sent to waste energy facilities for
disposal.

Here is a great video I found on the subject.

For instance, if I had a briefcase full of $100 bills that I either
misplace (any nobody ever finds it) or toss into a fire, that money is
no longer usable and cannot circulate within the economy. How much
money is lost or destroyed in ways like these annually?
I'm primarily interested in estimates for physical currency, but would
be interested in other means of money loss (can this happen
electronically?) as supplementary information.

By your situation previously, it is neigh-impossible to track such bills in circulation. As these bills don't transmit a signal as to their location, or who the buyer/seller is in a private party transaction. i.e. I meet up with you to buy your phone and pay in cash.
But at the same time, all US bills are serialized, meaning each are unique. A 1 dollar bill might have the same value as another $1, but each has its own unique identifier (serial number). When the US Treasury (through the Mint for coins and Bureau of Engraving and Printing for paper bills) issues new currency, the flow and supply chain of the money is tracked from its raw materials (ink, cotton, etc.) to distribution (banks, credit unions, foreign countries). Whenever a bill 're-enters' the system (i.e. deposited at a bank) the serial can be tracked from its origin point to its reentry point. However I am unaware of a system that can actively track its location in between.
At the same time, counterfeits are also taken into account the total supply of US dollars in circulation.
As for electronic loss, the current ACH System acts like a central authority that syncs all of the electronic activity that takes place. This is why when you pay, items on your bill are 'pending' until a couple of days later when they are 'posted'. At the same time, banks have several layers of backup systems and redundant systems to ensure 100% up-time and preventing data-loss.
Some reading to get started, but by no means definitive.


Is there any pressure to keep currency exchange rates around 1?

There are some counterexamples to the premise of your question:

A USD has been on the order of 100 Japanese Yen JPY for decades. By the way, 1/100 JPY used to be called "sen", and 1/1000 JPY used to be called "rin" (source: https://en.wikipedia.org/wiki/Japanese_yen#Introduction )

Bank notes:

Price tags:

A USD has been on the order of 1000 South Korean Won for some time.

Bank notes:

Price tags:

A USD is about 3500 Colombian Pesos (COP) now. Again, it's been in the thousands for decades.

Bank notes (note that they write "mil" in small letters instead of 000):

Price tags:

Note that $ denotes the peso, not U.S. dollars.

A USD is about 20,000 Vietnamese Dong (VND). Again, it's been in the thousands for decades.

Bank notes:

Price tags:

etc. These countries are not in any kind of economic crisis. The exchange rates are reasonably stable. The large numbers don't inconvenience anyone (except for some badly designed IT systems that can't handle very large or very small numbers accurately). These countries are all doing well.
Conversely, let us consider two countries at the opposite end from "doing well", whose currency is rapidly depreciating versus USD:

A USD is somewhere around 1,000,000 Venezuelan Bolivar (VES). Venezuela did redenominate its currency from "VEB" to "VEF" to "VES", over the last few years, dropping many zeroes. It did not help. The rapidly depreciating currency is one of the symptoms of their economic problems, not the cause.

A USD is somewhere around 100 Russian roubles (RUB). The rouble has been depreciating rapidly for years, especially after the West imposed painful sanctions to punish Russia for its aggression against Ukraine in 2014. Russia did already redenominate the rouble from "RUR" to "RUB" dropping 3 zeroes not long ago:

That did not help. Russia is still broke:

Russians looking for expired food.
Desperate Russian officials can't avert another imminent sovereign default, but are looking to delay the inevitable. They discuss redenominating their rouble again, dropping 2 more zeroes, and making 1 new rouble more valuable than 1 U.S. dollar, probably some variant of the Soviet "confiscatory redenomination" of 1947 (see https://en.wikipedia.org/wiki/Soviet_ruble#Fifth_Soviet_ruble,_1947%E2%80%931961 ). This won't help. Except for stroking Russian national pride, I fail to see the benefit of having a currency unit worth more than 1 USD.
Let us therefore look at some examples of currencies whose 1 unit is worth substantialy more than 1 USD.

A Kuwaiti dinars (KWD) is somewhere around 3 to 3.5 USD . This does not pose any inconvenience when a subunit of the currency is small enough. A 1/1000 fraction of KWD, worth about 1/3 of US cent, is called "fil". Here is a 1/2 KWD (500 fil) bank note:

Price tags:

This bottle of water costs 0.055 dinars, i.e. 55 fils, which is about 16 U.S. cents.

Chile widely uses an inflation-adjusted currency en.wikipedia.org/wiki/Unidad_de_Fomento - 1 unit of account is worth about 40 USD. It is not available as cash (bank notes or coins), but is primarily used to denominate real estate prices, stock prices, loans, etc.

1 Bitcoin (XBT) is worth thousands of dollars as of this writing. Conveniently, one hundred millionth of a bitcoin is called a satoshi and can be used in transactions.

Back when the U.S. dollar had a lot more purchasing power than it has today, prices commonly used "mill" (1/1000 USD). It's still perfectly legal (see https://en.wikipedia.org/wiki/Mill_(currency) ) and is often seen as the "9/10" on U.S. gasoline prices:

Contemporary gasoline prices.

Antique gasoline price sign: the gas costs 10.3 cents and the tax is 1.7 cents, i.e. 17 mills.

Also the Euro (EUR), the British pound (GBP), and the Swiss franc (CHF) are all worth more than 1 USD (but usually less than 2 USD).

A good (but not quite complete) list of recent redenominations can be found in Wikipedia https://en.wikipedia.org/wiki/Redenomination#List_of_currency_redenominations . As you see, just swapping 1,000 units of the old currency for 1 unit of the new currency changes nothing. However in the 1947 Soviet confiscatory redenomination cited above, if the state owed you 10 old roubles before, then the state now owed you only 1 new rouble. But if you owed the state 10 old roubles, then you still owed 10 new roubles to the state. This may be the kind of scheme that Russian officials are contemplating again.


What are the economic drivers of urbanisation?

This is quite broad question, in fact the full answer to your question would probably require a textbook on urban economics, so also this answer is based on one; Brueckner (2011) Lectures on Urban Economics, MIT press,  which is whole book largely dedicated to this (and other urban economics) issue(s). I will try to provide summary of main arguments found in the source cited above and also some additional ones.  According to economic theory there are several reasons for urban areas cities to develop

Specialization/division of labor - the more people live in certain area the greater scope there is for people to specialize in tasks and divide labor. This in one aspect self-explanatory since clearly when you have more people you can subdivide their tasks more but there.
In turn division of labor is extremely economically beneficial and allows us to increase production and consequently our material standards of living (as already observed by Adam Smith in the Wealth Of Nations). This drives people to aggregate in cities since thanks to specialization they can enjoy greater standards of living.

Economies of scale: Some production processes exhibit economies of scale - meaning the more you produce the more efficiently you can produce. This is technically connected to the division of labor mentioned above (Brueckner puts division of labor and economics of scale under one sub-chapter), but the division of labor is not exclusive cause of economics of scale and it also enhances production on its own so I thought they both deserve their own point.

Pecuniary agglomeration economies:  "Pecuniary agglomeration economies lead to a reduction in the cost of a firm ' s inputs without affecting the productivity of the inputs" (Brueckner 2011). This is best explained by an example. Let us assume that firm needs to hire person with some rare combination of skills and let us assume only $0.5\%$ of population has such skill. Well then in a village of 100 people you would on average expect less than one such person to exist, in town of 10,000 people you would expect 50 such people, in city with 10,000,000 you would expect 50000 of such people. Hence, it would be easier to gather the special workers you need in large city than search across many small villages.
In addition when it comes to business to business (b2b) there will be higher competition among suppliers in big cities rather than small villages. Erosion of market power of suppliers reduces costs of businesses further down supply chain.  There are other examples of these effects in the above mentioned book.

Technological agglomeration economies: "Technological agglomeration economies arise when a firm’s inputs are
more productive if it locates in a big city, amid a large concentration of
employment, than if it locates in a small city" (Brueckner 2011). This is because for example, in R&D there might be technological spillovers between firms.

Transport Costs and Firm Location: Transportation costs are obviously lower in places with high population density then when population is spread very thinly. It is also more efficient for similar firm to cluster together (e.g. silicon valley, Detroit's now infamous car manufacturing).

Retail Agglomeration: Cities historically originated around some important market and generally you will even nowadays find big shopping  centers in cities not small villages. This reduces consumer costs and also their prices as there is more competition/they can do comparison shopping etc.

Geography: This one is not given significant treatment in Brueckner, but some sources consider it important (Fujita, 2002; Krugman, 1991).  Earth surface is not homogenous. Some places might have excellent harbor, be at an important trade route and so on.

There might be also some other reason in the literature, my specialization is not Urban Economics so I do not purport to know the whole corpus of this body of research, but these seem to be the major reasons given by urban economics that are widely accepted.


Using web scraped data for consumer price indices

Take a look at Billion Prices Project:

  The Billion Prices Project is an academic initiative that uses prices
  collected from hundreds of online retailers around the world on a
  daily basis to conduct economic research.

Also take a look at PriceStats

  PriceStats collects online prices to provide daily inflation updates
  for 22 economies.

Overview of the PiceStats approach

More broadly, price index construction is a huge literature full of pitfalls so it is difficult to point you to a generic but also advanced presentation of the issues. But you might start with Consumer Prices, the Consumer Price Index, and the Cost of Living (Boskin, Dulberger, Gordon, Griliches and Jorgenson (1998)), a Journal of Economic Perspectives piece on the subject.


The question you are showing work out for where $S(w) = 0.01W - 100$ means that you will have no nurses who want to work unless you have a minimum wage of over 10,000 per week, and since the marginal benefit of every nurse is 2,000 per week, this hospital will not be in business. Your original question you typed at the very beginning is making you work with $S(w) = 0.1W - 100$. So let's try approaching that question instead. It also seems you need some help understanding how to get expenditure, so let's go through that too.

First, we find inverse supply.

$$W = 10S + 1000$$

Expenditure is the price of labor times quantity of labor.

$$E = WS = 10S^2 + 1000S$$

The derivative of this is your marginal expenditure.

$$\frac{dE}{dS} = 20S + 1000$$

Set up the profit maximization problem and take the derivative (which we set to zero).

$$\begin{align}
 \pi & = 2000S - (10S^2 + 1000S) \\
 \frac{d\pi}{dS} & = 2000 - 20S - 1000 = 0\\
 & = 1000 = 20S\\
 \implies & \boxed{S^* = 50} \\
\end{align}$$

Then to calculate wage, you just plug in the optimal supply to your inverse supply function. To find dead-weight loss, try drawing out a graph for labor supply and demand. Labor demand should be a straight line across from $W = 2000$. Labor supply was given to you. Notice where the labor supplied equals labor demand. What would $S$ equal there and what wage would be set there? It should be different from the $S^*$ we got. That should give you enough insight to the rest of the question.


Why did the Dollar weaken after interest rate increase?

It is correct that higher rates should strengthen the dollar... IF

the increase in interest rates is unexpected and
all else equals.

I haven't followed this particular rate hike, but here is a basic explanations why the dollar might weaken after an increase in interest rates:

The rate hike was widely expected and already priced in by the market, As a result, it didn't have much of an effect on the exchange rate. Keep in mind that the dollar can fluctuate quite significantly even on days without a Fed decision.  The move you refer to might be one of these fluctuations and have nothing to do with the rate hike. Or a small move due to the rate hike might have been overshadowed by a bigger move due to a different factor.

It's worth keeping in mind that it's not the events themselves that matter for exchange rates (or stock prices), but how these events turn out relative to expectations -- markets are forward-looking.


How to work out average income based on square meter of property rented

It is only possible if you also know the average fraction of income that is spent on housing in that area... and even then only if you assume either that rental costs per square foot are similar to ownership costs per square foot, or that the rent/buy decision is uncorrelated with income.

So: not really. Rental costs are correlated with income, but there's a lot of error.


How does depreciation affect the cash flow in a tax-paying company?

Depreciation is tax-deductible. The firm in question pays taxes. If depreciation increases, then taxes paid decrease. If taxes paid decrease, then cash flow increases. ceteris paribus


Economic interpretation of the concavity of the expenditure function

The expenditure function is always concave.

Suppose you consume two goods. Suppose the price of one of those goods, say $p1$, increases. Suppose also that, rather than rebalance your demand, you increase consumption of good one enough to maintain some given level of utility (let's say whatever level of utility you had whenever consuming goods 1,2 before the price of good 1 increased). Then your expenditure rises linearly.  Suppose instead that you rebalance your demand. Then your expenditures rise but not as much as whenever you do not rebalance your demand. Thus, concavity.

So, I suppose an interpretation here is that a consumer always has the ability to rebalance demands s.t. he/she maintains some desired level of utility in the cheapest way possible. Worst case, expenditures increase linearly. But linear functions are both concave and convex.

This is a very loose way to discuss this. However, I think it gives intuition, which is what I think you are looking for here.


How successful is the Islamic banking model?

In Ernst & Young 2016 Report we read (p. 17) that for the period 2010-2014 the Average Return on Equity for "Participation Banks" (this is the secular name of Banks under Sharia) was $12.6$% compared to $14.5$% of the "comparable conventional" banking. Although the ROE is lower, it is not that lower.

We also read that in the same period (p. 10), the assets of International Participation Banks grew by $16$% yearly on average.

It looks like a profitable way of doing business with strong growth.

From a philosophical point of view one should ask: how bankers in Participation banks view what they do?

Are they viewing their business model as essentially a way to cheat the religious doctrine-turned secular law, and everything they do they "translate" it into "what was the interest earned" or, after accepting the constraint that they won't do banking business in the traditional way, they explore different ways of making available funds earn a return?

The two examples offered by the OP are very useful: first, "profit sharing" makes a Participation bank more like a joint venture enterprise -and it reminds us of Ancient Greek bankers (they were the first bankers), whose bulk of business consisted of funding ships to travel and trade around the Mediterranean Sea, for a share of the profits.

Second, when a Participation Bank, instead of loaning an amount of money for a house-buy against a mortgage, buys the house and sells it to the interested party at a higher price (to be paid gradually through installments), it may appear as just a trick to avoid calling it "loan with interest", but a large construction firm that constructs houses and then sells them with a long-term payment plan, does nothing different.

So it appears that Participation banks use very standard ways to do business, only, they are ways that we usually associate with other sectors of the economy, and this is why they may appear "strange" for the banking sector.

As regards the pros and cons of Islamic Banking, there are many studies, for example

Beck, T., Demirgüç-Kunt, A., & Merrouche, O. (2010). Islamic vs. conventional banking: Business model, efficiency and stability. The World Bank.

where the abstract reads

  This paper discusses Islamic banking products and interprets them in the context of financial intermediation theory. Anecdotal evidence
  shows that many of the conventional products can be redrafted as
  Sharia compliant products, so that the differences are smaller than
  expected. Comparing conventional and Islamic banks and controlling for
  other bank and country characteristics, the authors find few
  significant differences in business orientation, efficiency, asset
  quality, or stability. While Islamic banks seem more cost-effective
  than conventional banks in a broad cross-country sample, this finding
  reverses in a sample of countries with both Islamic and conventional
  banks. However, conventional banks that operate in countries with a
  higher market share of Islamic banks are more cost-effective but less
  stable. There is also consistent evidence of higher capitalization of
  Islamic banks and this capital cushion plus higher liquidity reserves
  explains the relatively better performance of Islamic banks during the
  recent crisis.


Solow-Swan production function- What is '$L$'

$L$ is labor, which is equal to population in Solow-Swan model. Solow did not actually believe the workforce and population had the same size, but what's important is that labor is supposed to be in direct relationship with population i.e. there is a number w between 0 and 1 such that

$L = w *$ population

Direct proportionality between population and workforce is one of the assumptions of the Solow model.


There is an active body of research on network economics. See, for example,

Social and Economic Networks by Jackson
Connections by Goyal

for an introduction to the topic.

These models occasionally use weighted edges ("bandwidth") to model economic phenomena.

I'm not aware of work making use of the concept of latency, except when literally modelling latency (as in models of network congestion). An example of such a paper (though without the notion of nodes or edges) is Net neutrality and investment incentives by Choi and Kim.


How can CVS which has about 3 billion in cash on hand buy Aetna for 69 billion?

The deal is a mixture of cash and shares. Existing CVS shareholders get some cash, plus some shares in Aetna. It is likely that they would borrow in the bond market to raise the cash.

Yes, existing Aetna shareholders end up diluted. Ownership percentages in the combined firm will reflect the terms of the share conversion. The cash payment is there to reduce the dilution of Aetna shareholders, as it means that CVS shareholders get less Aetna shares.

You would only be able to take over a large firm as a small firm if the large firm’s shareholders want shares in a company that is managed by yourself. Unless you have a stellar reputation as a manager, why would they want that?


Alternative measure for market capitalization, company "value"

This is the reverse of the accounting procedure discussed in my answer to this question: link to question

What you want is the average cost basis for all holders of company.

There is no practical way that this could be calculated for a public firm. Nobody is required to report that information, and not everyone uses that accounting method for equities for taxes.

It would also be highly biased against firms where the founders hold a significant portion of shares. In an IPO, only a portion of the total number of shares are sold, sometimes only 10% of the firm. (Your example implies that 100% of the shares are sold in the IPO, which is misleading.) it is entirely possible for a founder of a company to retain control of a large percentage of the shares for decades. This method would assign almost no value to that portion of the company, which makes the valuation given meaningless for analysis.


How to intuitively understand the 'Intuitive criterion'?

A concise, completely informal way of putting it is this: The intuitive criterion rules-out any out-of-equilibrium beliefs that can only be correct if some player did something stupid.
Below is a slightly more long-winded explanation with an informal example.

In many signalling games (that is, games in which one player—the sender—can communicate information to another—the receiver), there are often a lot of implausible equilibria. This happens because the Perfect Bayesian solution concept does not specify what the receiver's beliefs must be when the sender deviates; we can therefore support a lot of equilibria simply by saying that if the sender deviates from those equilibria then he will be "punished" with very bad beliefs. Such punishment will usually be enough to make the sender play a strategy that would otherwise not be a best response.
For example, in Spence's classic job market signalling paper there is an equilibrium in which high-ability individuals invest in education (learning is easy for them) whilst low-ability individuals do not (because they find it too costly to do so). Education is then a signal of ability. We might ask: is there also an equilibrium of this game in which nobody chooses to get an education and no information is transmitted to the receiver? The answer is 'yes'. We can support such an equilibrium by saying that a deviation in which a sender is educated causes the receiver to adopt the belief that the sender is certainly low-ability. If education has the effect of signalling low-ability then, of course, everyone is happy to play along with the putative equilibrium and not get educated.
It is also clear that this equilibrium is not very plausible: the receiver knows that it is less costly for a high-ability agent to get an education than a low-ability one, so it doesn't make much sense for him to think of an education as signalling low-ability. The intuitive criterion rules out this kind of equilibrium by requiring beliefs to be "reasonable" in the following sense:

Suppose the receiver observes a deviation from the equilibrium. The receiver should not believe that the sender is of type $t_{\text{bad}}$ if both of the following are true:

the deviation would result in
type $t_{\text{bad}}$ being worse off then if he has stuck to the equilibrium for any beliefs.
there is some type $t_{\text{good}}$ who is better off by playing the deviation than by sticking to the equilibrium for some belief other than $t_{\text{bad}}$.

Returning to the education signalling model: Suppose that the equilibrium is that nobody gets an education and that the receiver believes that a deviation to getting education signals low ability. Anticipating these beliefs, a low ability worker is made worse-off by deviating because he not only incurs the cost of the education but is then thought of as a bad type as a result. Thus, condition 1. is satisfied.
Can we find some alternative belief such that the high-ability worker would like to deviate to getting education? The answer is yes: if the receiver believes that education signals high ability then this deviation is indeed profitable for the high-type. Thus, condition 2 is also satisfied.
Since both conditions are satisfied, the intuitive criterion rules-out the implausible pooling equilibrium.


Note that $u_i$ is a random residue. In the linear regression model, we assume the independence of the random residue (error term). We have two slides above for case i.i.d.: $\epsilon_i \sim (0,\sigma^2)$, thus $\forall_{i\neq j}E[\epsilon_i\epsilon_j|X]=0$ and $E[\epsilon_i\epsilon_i|X]=\sigma^2$. Later (slide 10) we assume i.n.i.d. and $\forall_{g\neq g'}E[\epsilon_{ig}\epsilon_{jg'}]=0$.

Slide 12 shows the example matrix $\Omega=diag(\Sigma_g)$ that dispels doubts.


Existing metrics for amounts of leverage used in financial sector

The first place to start is the Federal Reserve’s Flow of Funds report (Z.1) - link to Fed website.

It has aggregate balance sheets on an instrument and sector basis (including the financial sector, as well as its components). One can then look at growth rates, or scale relative to national income/GDP.

Although I have personal preferences on what data I would look at, someone else might be able to suggest studies on particular indicators.


Does it affect locals negatively when a new company moves in?

This is a very unique situation in Dallas/North Texas. This is the absolute worst economic growth possible for locals. Here are two key elements to look into:

Wages are fairly stagnant in DFW. This paired with 40-50% home/rent increases is the first warning sign.
Re-assessed tax values - expect numerous foreclosures in the next two years. Many people are purchasing homes without even thinking about the fact that many cities in North Texas have property tax rates of nearly 3%. While mortgage companies are quoting "monthly payments" based on the previous year's taxes paid on those properties (many of which are capped by homestead exemptions and have been for decades), once taxes are reassessed the following year, their payments will spike significantly. Here is an actual example from Sachse, Tx :

Total monthly payments on a 240k home for the last 6 months of 2016 were 1,536. The family accounted for this in whether or not they could afford this home. Their payment after the new tax appraisal is 1,860 per month. That is a $330 increase that they were unaware of and can no longer afford, as they purchased on the upper end of their budget. Expect to see this across the board, even as those from locations such as California will be paying more than the California state average for monthly house payments.

The average monthly payment in California was 2,542 on a 480,000 mortgage. Their annual property tax on average was 2,500. A 480,000 mortgage in Sachse, Tx with 10% down would currently run you about 3,640 per month with an annual property tax of 14,000. (Sheds a little light on the 'but we don't have a state income tax' argument, and how invalid it truly is. They pay the state income tax in the form of double/triple many other states' property tax.)

Beyond this, North Texas has room to expand, significantly. While the heart of Dallas may reach its limit and increase in price exponentially, the suburbs will likely take a large hit in value once construction begins to catch up. The home values are already overheated, and this is, by definition, a housing bubble.

My best guess for when it will pop is around 2018-2019 when we begin to see delinquent payments. Anyone purchasing a home between 2016-2017 in North Texas may be upside down for 5-10 years.


Does the inflation rate reflect only the change in quantity of money?

The growth of the amount of money in the economy is just one factor that can affect inflation. Other factors include:

changes in a nations productivity
changes in the exchange rate with other countries
changes in the velocity of money
changes in what fraction of financial exchanges are counted in inflation incidences

Given all these additional factors, its no surprise to find that in the short term the money supply does not correlate brilliantly with inflation... but in the long term the correlation is definitely there - i.e. if the money supply grows by a factor of ten over many years, then prices would probably grow by a similar factor. Milton Friedman won his Nobel prize for his work highlighting the long term correlation.


How does the US have gross national savings while running a trade deficit?

Premises of your Q are simply false:

If a country has a positive savings rate, that means that more money had to come into the country from somewhere outside of the country.

This is completely incorrect. Total savings of a country is sum of private and public saving. For an open economy output is given by $Y=C+I+G+X-M$ (where $Y$ is output $C$ consumption $I$ investment $G$ gov spending, $X$ exports and $M$ imports, the private saving is $S_p=Y-C-T$ and public saving is $S_g=T-G$ (see Blanchard et al Macroeconomics a European Perspective). Thus we can clearly see that $S_p+S_g=I+X-M$. In another words in open economy public and private saving must be equal to investment plus exports minus imports.
Let us suppose investment is 1000 exports are 0 and imports are 500 thus trivially country runs trade deficit yet has positive saving of $S_p+S_g = 1000+0-500 = 500$. Trivially saving is possible.
In addition, country having trade deficit is not equivalent to country sending out money. Generally you can only use home currency in your home country. Generally speaking one cannot use Japanese yens in the UK. If Japan wants to import cars from UK it has to sell its yens and buy pounds. Yens will be purchased by people who want to buy something in Japan because yens can be generally used to buy stuff etc only in Japan. So the total number of yens in Japanese economy will not change just because of trade deficit vis-a-vis UK, the same way as money supply of pounds in UK will not increase just because UK is running trade surplus vis-a-vis Japan.


Did previous researchers fail to detect the hot hand simply because of a statistical fallacy?

(This answer was completely rewritten for greater clarity and readability in July 2017.)
Flip a coin 100 times in a row.
Examine the flip immediately after a streak of three tails. Let $\hat{p}(H|3T)$ be the proportion of coin flips after each streak of three tails in a row that are heads. Similarly, let $\hat{p}(H|3H)$ be the proportion of coin flips after each streak of three heads in a row that are heads. (Example at bottom of this answer.)
Let $x:=\hat{p}(H|3H)-\hat{p}(H|3T)$.
If the coin-flips are i.i.d., then "obviously", across many sequences of 100 coin-flips,
(1) $x>0$ is expected to happen as often as $x<0$.
(2) $E(X)=0$.
We generate a million sequences of 100 coin-flips and get the following two results:
(I) $x>0$ happens roughly as often as as $x<0$.
(II) $\bar{x} \approx 0$ ($\bar{x}$ is the average of $x$ across the million sequences).
And so we conclude that the coin-flips are indeed i.i.d. and there is no evidence of a hot hand. This is what GVT (1985) did (but with basketball shots in place of coin-flips). And this is how they concluded that the hot hand does not exist.

Punchline: Shockingly, (1) and (2) are incorrect. If the coin-flips are i.i.d., then it should instead be that
(1-corrected) $x>0$ occurs only about 37% of the time, while $x<0$ occurs about 60% of the time. (In the remaining 3% of the time, either $x=0$ or $x$ is undefined — either because there was no streak of 3H or no streak of 3T in the 100 flips.)
(2-corrected) $E(X) \approx -0.08$.
The intuition (or counter-intuition) involved is similar to that in several other famous probability puzzles: the Monty Hall problem, the two-boys problem, and the principle of restricted choice (in the card game bridge). This answer is already long enough and so I'll skip the explanation of this intuition.
And so the very results (I) and (II) obtained by GVT (1985) are actually strong evidence in favor of the hot hand. This is what Miller and Sanjurjo (2015) showed.

Further analysis of GVT's Table 4.
Many (e.g. @scerwin below) have — without bothering to read GVT (1985) — expressed disbelief that any "trained statistician would ever" take an  average of averages in this context.
But that is exactly what GVT (1985) did in their Table 4.
See their Table 4, columns 2-4 and 5-6, bottom row. They find that averaged across the 26 players,

$\hat{p}(H|1M) \approx 0.47$ and $\hat{p}(H|1H) \approx 0.48$,
$\hat{p}(H|2M) \approx 0.47$ and $\hat{p}(H|2H) \approx 0.49$,
$\hat{p}(H|3M) \approx 0.45$ and $\hat{p}(H|3H) \approx 0.49$.

Actually it is the case that for each $k=1,2,3$, the averaged $\hat{p}(H|kH)>\hat{p}(H|kM)$. But GVT's argument seems to be that these are not statistically significant and so these are not evidence in favor of the hot hand. OK fair enough.
But if instead of taking the average of averages (a move considered unbelievably stupid by some), we redo their analysis and aggregate across the 26 players (100 shots for each, with some exceptions), we get the following table of weighted averages.
Any                     1175/2515 = 0.4672

3 misses in a row       161/400 = 0.4025
3 hits in a row         179/313 = 0.5719

2 misses in a row       315/719 = 0.4381
2 hits in a row         316/581 = 0.5439

1 miss in a row         592/1317 = 0.4495
1 hit in a row          581/1150 = 0.5052

The table says, for example, that a total of 2,515 shots were taken by the 26 players, of which 1,175 or 46.72% were made.
And of the 400 instances where a player missed 3 in a row, 161 or 40.25% were immediately followed by a hit. And of the 313 instances where a player hit 3 in a row, 179 or 57.19% were immediately followed by a hit.
The above weighted averages seem to be strong evidence in favor of the hot hand.
Bear in mind that the shooting experiment was set up so that each player was shooting from where it had been determined he/she could make roughly 50% of his/her shots.
(Note: "Strangely" enough, in Table 1 for a very similar analysis with the Sixers' in-game shooting, GVT instead present the weighted averages. So why didn't they do the same for Table 4? My guess is that they certainly did calculate the weighted averages for Table 4 — the numbers I present above, didn't like what they saw, and chose to suppress them. This sort of behavior is unfortunately par for the course in academia.)

Example: Say we have the sequence $HHHTTTHHHHH…H$ (only flips #4-#6 are tails, the remaining 97 flips are all heads). Then $\hat{p}(H|3T)=1/1=1$ because there is only 1 streak of three tails and the flip immediately after that streak is heads.
And $\hat{p}(H|3H)=91/92 \approx 0.989$ because there are 92 streaks of three heads and for 91 of those 92 streaks, the flip immediately after is heads.

P.S. GVT's (1985) Table 4 contains several errors. I spotted at least two rounding errors. And also for player 10, the parenthetical values in columns 4 and 6 do not add up to one less than that in column 5 (contrary to the note at the bottom). I contacted Gilovich (Tversky is dead and Vallone I am not sure), but unfortunately he no longer has the original sequences of hits and misses. Table 4 is all we have.


What happens in a bartering system when a resource is plentiful but cannot be paid for?

Tribe A will not necessarily sever barter with tribe B. This has to do with Production Possibility Frontier and related concepts such as specialization & utility.

Tribe A's production of bricks could certainly press tribe B to agree to a bricks/fish exchange rate that is more convenient to tribe A than it was prior to the latter's invention of tools to create bricks.

In an extreme scenario, tribe A's bricks might be of better quality and much cheaper than tribe B's bricks. That might render A unlikelier to maintain barter with tribe B. If so, tribe B might want to consider fish substitutes such as hunting, veganism, veganism's closest alternative (i.e. cannibalism), acuphagia (of A's new tools), or outright eating bricks.


How to interpret income elasticity of demand

Hints:

Looking at the definitions, if the income elasticity of demand for apples is 2, how much does apple demand increase when consumers' income increase by 10%?
Looking at the definitions, if the cross elasticity of demand between apple quantity and the orange price is 0.5, how much does apple demand decrease when the orange price falls by 15%?
How might you combine the increase with the decrease?

Multiplication might be a useful tool.


International Trade: Two country trade problem

Part (a) is correct. Your error in (b) is to implicitly assume that relative prices are equal to $1/2$.

The best way to understand the problem is using the Edgeworth box (excuse the sloppy picture; will add a custom one when had the time):

The axes represent the two goods. Their lengths are 1800 (global endowment of food, x axis) and 900 (global endowment of clothing, y axis). Point 3 represents the endowment point, which is (1200,300) for Home and (600,600) for Foreign. As the figure shows, that point is not an optimal allocation. Such optimal allocation is where the MRS for the two countries is the same (point 1), MRS which is equal to the relative price of goods (same for both countries, i.e. they are international prices).

Therefore, this allocation has the following properties:

$$ \frac{H_f^2}{2H_c} = \frac{F_f^2}{2F_c} = \frac{P_c}{P_f}$$

(I replaced $D$ with home and foreign letters, for clarity).

Notice that the ratios of consumption are the same across countries. Yet, before trade, production is such that the ratio of food versus clothing is larger in home than in foreign. Therefore, home will export food and import clothing, and vice versa for foreign. Yet, since imports of one country are the exports of the other country, you get:

$$ \frac{1200 - A}{300 + B} = \frac{600 + A}{600 - B} $$

where $A$ are exports of food and $B$ exports of clothing.

Additionally, you have that the trade balance must be in equilibrium. Therefore:

$$ P_fA = P_cB $$

Here you have all the equations you need to find the solution in terms of final consumption levels and relative prices. Let me know if this is enough. I can guide you further if you need.


Its legal to borrow and lend money, but if you become anything close to a bank, (take deposits, make loans, take risks, make a business out of it) then you will be regulated. You will be insured by the FDIC, and since you are insured, you will then be required to follow certain guidelines regarding how you asses risk and which risks you can take or not. Moreover, they will not be allowed to lose money in a large scale, so they would have to recapitalize the 'bank' after any of the charitable projects swallows up capital.

So to answer your question, they would not have more wealth, but they would manage a bigger pot of capital with which to invest. However, at some point they will lose the freedom of doing whatever they want. Since its not their money, then they can't do whatever they want.


CES First order Condition with two labour types

Simply multiply and divide one $\left(a_{F} \boldsymbol{F}\right)^{\frac{(\sigma-1)}{\sigma}}$ in the bracket and then take one outside the bracket. And by the way your FOC is incorrect in that $\frac{\sigma}{(\sigma-1)}$ should be cancelled out.


Why is the social value curve NOT the supply curve shifted downwards?

Here is how I interpret your question: "Why when modeling positive externalities do we shift the marginal benefit curve upward instead of shifting the marginal cost curve downward?"

Either way of thinking about it is fine. Positive externalities can be thought of as resulting in a social marginal benefit curve that's above the private marginal benefit curve, or as resulting in a social marginal cost curve that's below the private marginal cost curve. Which side you place it on is only a matter of accounting, since

Private Marginal Benefit + Externality - Private Marginal Cost = Private Marginal Benefit - (Private Marginal Cost - Externality).

Your question doesn't make sense in some ways though and I think it's important you understand why. You say "Because SV benefits society, the suppliers of this SV can consider the SV supplied, as follows: Suppliers' true cost = their original costs − SV, for all quantities supplied.
So the new supply curve must underlie, and be parallel to, the old supply curve."

The positive externality in this case cannot shift the producer's supply curve downward, because the producer's supply curve is his private marginal cost. Importantly, externalities only affect social marginal cost.

Remember: The supply curve is a function that tells you what quantity a supplier will produce at any given price, whereas the marginal cost curve is the cost per unit of producing an additional unit. These two things only happen to coincide in the absence of externalities.


Currency substitution after withdrawal from a currency union?

Summary:
Using a national currency gives you full control over monetary policy while using a larger foreign currency provides stability.

Detailed answer:
If a country has its own currency then the government, or to be more precise the central bank, can intervene and change exchange rates. This might be necessary if the countries products are not competitive on the international markets on the current exchange rates. An example:

Cost of a foreign laptop: 200 USD. Cost of a domestic laptop: 100 Peruvian Nuevo Sol (PEN). The domestic laptop can only be sold if the USD/PEN exchange rate is over 1/2.

In theory the government of a country using a foreign currency could launch policies that would simply reduce all salaries and rents, but this is politically and psychologically undesirable (downward rigidity of wages).

Unless big shocks (drastic events or drastic new information) occur such interventions would not be necessary as prices, wages and exchange rates slowly adjust thanks to the free market.

So a benefit of having a national currency is that it gives a country full control over its monetary policy and this makes such interventions possible. A disadvantage would be that a small nation's currency may be less stable than a large ones, for an example see this question. Stability (or at least predictability) of exchange rates is important for foreign investments and as they have a direct effect on the prices of imported goods and on the profit of export firms it is also important for the citizens of the country.

In the circumstances described there may also be a secondary effect:
A government in control of its monetary policy and facing extremely serious problems may start printing extreme amounts of cash because it turns up indirectly as income in the governments balance sheet (this is known as seigniorage). This can cause undesirable rates of inflation. Most economies avoid this problem by delegating money printing to a central bank which is independent from the government (to some extent). The mandate of the central bank is not to provide the government with income but to stabilize prices so it will probably try not to cause hyperinflation. This however is ultimately a political decision and in drastic times it may be overruled.


Why Certainity Eqivalence in PIH only holds for quadratic utilities

Certainty equivalence in the context of the Permanent Income Hypothesis implies that $u'(E_t[c_{t+1}]) = E_t[u'(c_{t+1})]$, which only holds if marginal utility $u'(\cdot)$ is linear (by linearity of expectations) and thus only if actual utility $u(\cdot)$ is quadratic.

To see where this comes from, consider the utility maximization problem of a representative household over an infinite horizon where consumption stream $\{c_t\}$ and asset holdings stream $\{a_{t+1}\}$ are chosen, taken as given the real interest rate, fixed at $r$, initial wealth $a_0$ and a stream of income $\{y_t\}$. Lastly, take utility to be quadratic over consumption: $u(c_t) = \alpha c_t - \frac{1}{2}c^2_t$ with sufficiently large $\alpha$.

So the household's maximization problem is

\begin{equation*}
\begin{aligned}
& \underset{\{c_t, a_{t+1}\}}{\text{max}}
& & E_0\sum_{t=0}^{\infty} \beta^{t}u(c_t), \text{ where } \beta = \frac{1}{1 + \rho} \text{ , } \rho > 0 \\
& \text{s.t.}
& & c_t + a_{t+1} = (1+r)a_t + y_t & \forall t.\\
\end{aligned}
\end{equation*}

$$
$$
Therefore, the Lagrangian for this maximization problem is

\begin{equation*}
\mathcal{L} = \sum_{t=0}^{\infty} \beta^{t} \{E_t[u(c_t) + \lambda_t((1+r)a_t+y_t-c_t-a_{t+1})]\}.
\end{equation*}
$$
$$

The first-order conditions of the Lagrangian are
\begin{align}
\frac{\partial \mathcal{L}}{\partial c_t} &= \beta^{t} u'(c_t) - \beta^t\lambda_t = 0 &&(1)\\
\frac{\partial \mathcal{L}}{\partial a_{t+1}} &= -\beta^{t}\lambda_t + \beta^{t+1} (1+r)E_t[\lambda_{t+1}] = 0 &&(2).
\end{align}
$$
$$

Solving for $\lambda_t$ using (1) and substituting into (2) gives
\begin{align*}
u'(c_t) = \beta (1+r)E_t[u'(c_{t+1})].
\end{align*}
$$
$$

Take $\rho = r$ in this economy for simplicity. Thus $\beta(1+r) = 1$:
\begin{align*}
u'(c_t) = E_t[u'(c_{t+1})].
\end{align*}
$$
$$

Since utility is quadratic, marginal utility is linear which implies that the expectation of marginal utility is the marginal utility of the expectation:
\begin{align*}
u'(c_t) = u'(E_t[c_{t+1}])
\end{align*}
which implies that
\begin{align*}
c_t = E_t[c_{t+1}]
\end{align*}
since $u'(\cdot)$ is injective. Thus we have certainty equivalence.


Why are trade deficits considered unfair?

You are correct.  Trade deficits are not unfair, although it is hard to define unfair in an economic sense. No one is forced to trade and trade represents a consensual agreement.

The strongest economic argument for considering trade deficits unfair is so-called dumping, which is a form of unfair competition. That is when firms sell goods abroad at below a normal price in order to drive out the local competition in the long run. As the foreign businesses would make losses doing so, it only works if the foreign governments subsidize those firms. That is why subsidies leading to unfair competition are banned according to the World Trade Organization rules. In the case of the steel tariffs imposed by the U.S., dumping by China was used as an argument, although there was no evidence of China conducting steel dumping and China isn't even one of the main exporters of steel to the U.S.

Another reason is that a trade deficit means that foreign firms are doing better than domestic ones, thereby threatening the jobs of domestic workers. However, for this to happen  consumers must be voluntarily buying those foreign products. If foreign firms offer a much better and/or cheaper product, then a trade deficit is great for consumers, but bad for domestic workers who produce those goods. In this case, trade still represents an overall net increase in welfare, but there are winners and losers. If the winners (consumers) would compensate the workers who lost their jobs, everything would be fine. If they do not it could be considered unfair, although it is not the foreign country being unfair in this case.

From my experience, there may also be illogical reasons that people consider trade unfair, which I discuss below.

When a country runs a trade deficit that means that it is consuming more than it produces. The country consumes at least as much as it produced and then a bit from what other countries produced too. When that happens, a country is essentially borrowing money from the rest of the world, because consuming more than you have is like spending more money than you have. That is what people mean when they say the U.S. owes a lot of money to China. In this sense many people who take out a loan with a bank to buy something think it is unfair that they have to pay it back. That could be one way in which deficits are considered unfair.

Another way in which deficits are considered unfair is by comparing a country to a business, which is wrong and leads to wrong policy. A business wants to maximize profits, so it would like to sell more goods than it buys. The business fails otherwise. In this sense, a business seeing other business succeeding where it fails might think it is being treated unfairly. As discussed, however, a trade surplus just means that you are lending money to the rest of the world. That may be good for some time, but only if you eventually get it back, i.e. eventually run a deficit. So for a business, perpetually making profits is a good thing. For a country, a perpetual surplus forever is not welfare enhancing.


Whats the name of the famous hostile takeover where the winner payed less than the loser

With the help of a former fellow student I was able to find out some details.
The anecdote was about the following three companies:

Macy's, Inc.
Robert Campeau
Federated (covered in the article about Macy's)

Some news coverage exists:

https://mindyourdecisions.com/blog/2011/09/20/when-game-theory-backfires-a-case-study-of-robert-campeaus-takeover-bid/
https://archive.fortune.com/magazines/fortune/fortune_archive/1990/06/18/73686/index.htm

In short, Macy's offered \$67 per share contingent on getting a majority of shares. Campeau offered \$74 per share if less than half was tendered, otherwise $\frac {0.5}{x} * \\\$74 + \frac {x - 0.5}{x} * \\\$60$ with $x$ the fraction of shares offered to Campeau. It seems as if the bidding war continued a bit after this scheme. The prices given in the blog above differ, I am sticking to these from the lecture.
I am very grateful for having had the opportunity to attend Prof. Kirste's lectures at the Universität Rostock. They were full of impressive details like this gem here.


How does investment have a negative impact on the current account (trade balance)?

Balance of payments is composed of:

the capital account (foreign assets and foreign liabilities)
and current account (includes exports and imports)

One way to define the current account is savings minus domestic investment, $S-I$. When $S<I$, a country's investing has to be financed by capital inflows, or investment from foreign countries. This is a deficit on the current account, making the country a net borrower that must seek matching capital inflows like foreign direct investment, that are a credit on the capital account (capital account surplus).

Whereas, when $S>I$, the excess savings are used to invest in foreign opportunities abroad, creating a current account surplus that makes the country a net lender, giving a capital account deficit.

In relation to the current account, a trade deficit is not identical but is a component of the current account. When there is a current account deficit, a trade deficit happens (imports exceed exports, where imports are not to be confused with investments $I$) because the country does not produce everything it needs and needs to borrow from overseas in order to pay for imports. From Greg Mankiw:

  when we write Y=C+I+G+NX, what is NX? The answer depends on how we define Y. If Y is Gross Domestic Product, then NX is the trade balance. If Y is Gross National Product, then NX is the current account.

Even though a trade deficit sounds bad, standard of living usually increases from a wider variety of imported goods and services, and prices are pushed downwards curbing inflation, but only for so long.


Economies of scale: when is it disadvantageous?

I would give you a simple answer: when the marginal cost is equal to the marginal income. This would be the equilibrium and beyond that the company would lose money.


Why's Federal Reserve buying individual corporate bonds, when corporations have been selling them successfully?

"...Its March 23 initial announcement is largely considered a watershed
  moment for the financial markets, reeling from the coronavirus threat
  spread..."

As the quoted statement implies,
people are spurred into buying corporate bonds in the current environment precisely because of the Fed's previous announcement that it would purchase such bonds. The market has been---more precisely, believes it has been--- front running the Fed, which achieves intended effect of the policy. As shown by the examples listed, corporates have been able to tap into the bond market for financing. The Fed now has to demonstrate it can/will implement its announced program to signal its credibility, and thereby maintain the effectiveness of its current and future policies.


Do current Central Banks need to target a monetary aggregate as they set interest rates for a target inflation?

The demand for liquidity is usually defined as
$$
L(Y,i)
$$
where $Y$ is income and $i$ is the nominal interest rate. In equilibrium this equals the aggregate money supply $M$ which is function of several things, like $M_0$ the reserve rate, etc. Let us assume that everything except $i$, $M_0$ and $M = f(M_0)$ is constant. Then
$$
L(Y,i) = M = f(M_0).
$$
Demand for liquidity decreases in $i$ and $M$ increases in $M_0$. Thus for any given value of $i$ there is at most one equilibrium value of $M_0$, hence the central bank cannot simoultaneously set both $i$ and $M_0$ and always expect an equilibrium outcome. Lacking equilibrium either $i$ or $M$ will shift.


Strategic form representation of extensive form games

I rely on the definitions from Chapter 2 of the Handbook of Game Theory, Volume 1, by Sergiu Hart.

If I understand you correctly, the proposition can be re-written as

Proposition. For, every finite extensive form game $\Gamma^E$, there exists a single
 strategic form representation $\Gamma^N =[I^N,\{S^N_i\},\{u^N_i(\cdot)\}]$ (up to relabeling of the agents) such that

$I^N = I^E$,

and for all $i\in I^N = I^E$,

$S^N_i = \{$ pure strategies of $i$ in $\Gamma^E$ $\}$ ,
and $u^N_i(s) = u^E_i(c(s))$, where $c(s) $ associates every profile of pure strategy in $\Gamma^E$  with a terminal node of  $\Gamma^E$ resulting from the pure strategy profile $s$.

I think 1. and 2. are obvious. There only remains to show 3, which is equivalent to proving that $c(s)$ is a function, i.e. every profile of pure strategies is associated with one and only one terminal node in $\Gamma^E$. This follows directly from the fact that the pure strategy of some player $i$ is a function selecting one and only one possible action from every information set.

Start from the root node, $r_0$.
By definition of a game in extensive form, the nodes are partitioned between the agents.
Thus $r_0$ belongs to some agent $i_0$.
By definition of the game in extensive form, the nodes of $i_0$ are partitioned into information sets.
Thus $r_0$ belongs to some information set of $i_0$, say $H_i^0$.
By definition of the pure strategy $s$, for any node in  $H_i^0$, $i_0$ always choses a single successor, among the possible successors at $H_i^0$.
Let this successor be $r_1$.
Then $r_1$ must be the successor of $r_0$, and the next node on the path.
Now consider $r_1$.
By definition of agame in extensive form, the nodes are partitioned between the agents.
Thus $r_1$ belong to some agent $i_1$ (where $i_1 = i_0$ is allowed).
$\vdots$
Repeating the argument as many time as needed, because the game is finite, we necessarily reach a single terminal node $r_* = c(s)$.


So what exactly happens to the real interest rate in the long run?

Blanchard examines a case with a single interest rate, and what happens to it "in the medium long-run" (in the sense of the economy being in equilibrium). His discussion is the most abstract.

Acemoglou discusses a more realistic case where many different interest rates exist at the same time, and analyzes the observed phenomenon that interest rates on short-term loans are different than interest rates on long-term loans, even when both contracts are entered into at the same calendar time: the time horizon of the contract matters for the interest rate itself. He then goes on to discuss how monetary policy may or may not affect expectations.

So when Acemoglou says "the long-term real interest rate tends to fall..." he does not mean "the real interest rate characterizing long-run equilibrium tends to fall". He means "the real interest rate that is currently imposed on loans with a long-term horizon, tends to fall".

These aspects are simply absent in the basic IS-LM / AS-AD analysis.


What are the differences between cryptocurrencies and digital currencies backed by CBs?

Two more differences
(1) You can pay your taxes in a government controlled digital currency, and you can't do that in cryptocurrency.  According to Modern Monetary Theory, that is an essential element to the fundamental value of a currency
(2)  even though an individual cryptocurrency may have the feature of a limited supply, there's nothing to stop multiple cryptocurrencies being created, potentially diluting each other's value.  That is not the case with the single currency of the realm.


Why is buying a put option a cash outflow, but shorting inflow?

When you buy a put option, you're taking possiession of something from someone else, and paying for it.

When you short, you're borrowing shares, and selling them, which brings cash in for you.


How are housing costs reflected in factor shares of GDP?

They enter through the output. By definition the labor share of GDP is: $w/Y$ where $w$ is factor income that is by statisticians assigned to the labor. Note  this does not necessarily correspond to the economic concept of labor as the statistical labor share of GDP calculation lumps together multiple factors of production and calls them labor - the same way as ‘capital’ share lumps together returns to land, capital, entrepreneurial labor etc.
Now the labor share of income is derived from incomes, but at the macroeconomic aggregate level all spending must be equal all income (although statistically there might be discrepancies due to measurement error). So the numerator can be replaced by total spending in the economy giving us:
$$\frac{w}{Y}=\frac{w}{C+I+G+NX}$$
Where $C$ is consumption spending, $I$ is investment spending, $G$ is government spending and $NX$ is spending on net exports. Now expenditure on rent is included in the $C$ part so that is how these expenditures enter the labor share of income calculation.
However, the back of an envelope calculation at the bottom of your question is not correct for several reasons.
First, If the labor income is 30k then that 30k is already the result of multiplying GDP by labor share i.e. it is $\frac{w}{Y }(Y)=w=30k$
Second if the labor income per year is 30k and rent is 10k and we assume that there is nothing else going on in the economy,  then it is the 10k that goes to the capital income and  calculation of its share. To be more precise here the total output by income approach would be 40k (30k wages 10k rent - this would by the way be also equal to all spending where rent would be part of $C$ and the other income also part of $C$ or other components of aggregate spending) so the labor share would be 30k/40k or $75\%$ while the capital share would be 10k/40k or $25\%$.
Lastly, using that $30\%$ statistics in this case is not really appropriate to begin with. All factors of production are provided by households, and that statistics is an average across both labor and capital providing households. Some capital providing households will just pay that income to another capital providing households (not even mentioning that large share of households empirically provides both). In addition, even though you specify that the 30k is only from a labor income of a households, I don’t think it is reasonable to assume that capital and labor providing households spend the same share of income on housing. The numbers for such household would likely be different from this average.


Why has increased productivity over the last 100 years not affected how much people work?

Let's work such a very simple model. We have a Robinson Crusoe island economy, an isolated individual that lives totally alone. In order to consume something Crusoe must work. Assume for even more simplicity that capital is not needed (say, fruit-gathering by hand). Crusoe does not like to work but he would rather sit relaxed and enjoy the good weather in his island. Apart from the biological necessity to eat something, Crusoe also likes the taste of fruits he gathers. Fruits cannot be stored.

So we have a, daily utility function of the form

$$U = U(c, \ell),  \;\; \ell + L = T$$

$$U_c >0,\;\; U_{\ell}>0,\;\; U_{cc} <0, \;\; U_{\ell \ell} <0,\;\; U_{c\ell} =U_{\ell c} \geq 0$$

The last assumption on the cross-partial seems like the more reasonable one.
$\ell$ is leisure and $L$ is amount of work, and $T$ the maximum amount of time that Crusoe can work (say, $16$ hours). We also have a  "fruit-gathering" function

$$Q = f(L) = AL^a$$

We can reasonably assume that we have decreasing marginal product of labor (due to the body getting tired going up and down all these trees), so $0<a<1$.

Since fruit cannot be stored and capital is absent, the only optimization problem that makes sense here is a static one, meaning $c=f(L)$.

Then we want to max over $L$ the function $U[f(L), T-L]$.

The f.o.c. is

$$\frac {\partial U[f(L), T-L]}{\partial L}\equiv G = U_c[f(L), T-L]\cdot f'(L) - U_{\ell}[f(L), T-L] = 0$$

and the s.o.c is

$$ \partial G/\partial L = U_{cc}\cdot f'(L) - U_{c\ell}\cdot f'(L) + U_c\cdot f''(L) - U_{\ell c}f'(L) +U_{\ell \ell} <0$$

Given our assumptions, this expression is everywhere negative, so we have our maximum.

We want to examine what happens as $A$ in the fruit-gathering function increases. Using the implicit function theorem we have, around the solution,

$$\frac {dL}{dA} = -\frac {\partial G/\partial A}{\partial G/\partial L}$$

We know that the denominator is negative, so together with the minus sign in front becomes positive. Therefore

$$\text{sign}\left\{\frac {dL}{dA}\right\} = \text{sign}\big\{\partial G/\partial A\big\}$$

$$= U_{cc}\cdot[ \partial f/\partial A ]\cdot f' +[\partial f'/\partial A]\cdot U_c  - U_{\ell c}\cdot [\partial f/\partial A] $$

This expression is of ambiguous sign, which is a first indication that things may not be that straightforward after all.

To validate the conjecture of the OP we would need to obtain,

$$\text{sign}\left\{\frac {dL}{dA}\right\} < 0 \implies U_{cc}\cdot[ \partial f/\partial A ]\cdot f' +[\partial f'/\partial A]\cdot U_c  - U_{\ell c}\cdot [\partial f/\partial A] < 0$$

$$\implies U_{\ell c}\cdot [\partial f/\partial A] >  U_{cc}\cdot[ \partial f/\partial A ]\cdot f' +[\partial f'/\partial A]\cdot U_c$$

$$\implies U_{\ell c} >  U_{cc}\cdot f' +\frac {\partial f'/\partial A}{\partial f/\partial A}\cdot U_c$$

Using the first order condition, we can substitute for $U_c = U_{\ell}/f'$, so

$$\text{sign}\left\{\frac {dL}{dA}\right\} < 0 \implies  U_{\ell c} >  U_{cc}\cdot f' +\frac {\partial f'/\partial A}{\partial f/\partial A}\cdot [U_{\ell}/f']$$

Now

$$\frac {\partial f'/\partial A}{\partial f/\partial A} \cdot \frac {1}{f'} = \frac {(a/L)L^a} { L^a} \cdot \frac {1}{(a/L)AL^a}$$

$$= \frac {1}{f} = \frac {1}{c}$$

So

$$\text{sign}\left\{\frac {dL}{dA}\right\} < 0 \implies  U_{\ell c} >  U_{cc} \cdot f'+ U_{\ell}/c$$

and re-arranging while multiplying throughout by $c/U_c$ and using the f.o.c. again, we get

$$\text{sign}\left\{\frac {dL}{dA}\right\} < 0 \implies  f'\cdot (RRA_c-1) > - U_{\ell c}\cdot c/ U_c$$

where $RRA_c$ is the coefficient of relative risk aversion related to consumption.

So we see that if $RRA_c\geq 1$, which is the usual assumption, this inequality is certain to hold and so indeed higher productivity will lead to less time spent working... so why reality does not conform with our theoretical results?

But it does. Historically, the amount of time spent working has been declining . The problem with the OP's question is that the phrase "we do as much as we can" is contaminated by framework effects: if what happens around me is people working, say, $10$ hours a day, $5$ days a week, and I work $11$ hours a day, $6$ days a week, I tend to think that, since I exceed the standard around me, "I do what I can", being oblivious to the fact that people used to work $16$ hours a day, $7$ days a week, with perhaps a few days per year off at best. So no, we do not really "do as much as we can".


I don't believe that a solution manual to Gali's book has been written. In 2015 the book saw a second edition, so maybe this time around something will appear.
These course notes may be indirectly helpful to you.


Cost function from CES production function

If you are interested in the case where $\rho \geq 1$ then look at the post CES
$\ \ \rho \geq 1$. For the standard case where $0 < \rho < 1$ you should get a result like this
$$C(w_1,w_2,y)  = \left(w_1^{\frac{\rho}{\rho -1}} +w_2^{\frac{\rho}{\rho -1}}\right)^{\frac{\rho - 1}{\rho}} y.$$
To see this you should start by setting up the cost minimization problem
$$\min_{x_1,x_2} \ \ w_1x_1 + w_2x_2 \\[8pt]
s.t. \ \ (x_1^\rho + x_2^\rho)^{1/\rho} \geq y$$
for this problem the Lagrangian function is
$$\mathcal L(x_1,x_2,\lambda) = w_1x_1 + w_2x_2 - \lambda((x_1^\rho + x_2^\rho)^{1/\rho} -y).$$
From the first order conditions of the Lagrangian you can show the constraint is binding in optimum $(x_1^\rho + x_2^\rho)^{1/\rho} = y$ and get MRS equal to relative prices
$$(1) \ \ \frac{w_1}{w_2} = \frac{x_1^{\rho - 1}}{x_2^{\rho - 1}},$$
given this information you should be able to solve for $x_1$ and $x_2$ as a function of the parameters of the problem which in this case is $\rho,y,w_1,w_2$.
Try to get $(x_1^\rho + x_2^\rho)^{1/\rho}$ to appear in the MRS equal to relative prices. So manipulate (1) to get
$$w_1^{\frac{\rho}{\rho -1}}x_2^\rho =  w_2^{\frac{\rho}{\rho -1}}x_1^\rho,$$
then add $w_2^{\frac{\rho}{\rho-1}}x_2^\rho$ to both sides of equation
$$w_1^{\frac{\rho}{\rho -1}}x_2^\rho +w_2^{\frac{\rho}{\rho -1}}x_2^\rho =  w_2^{\frac{\rho}{\rho -1}}x_1^\rho + w_2^{\frac{\rho}{\rho -1}}x_2^\rho,$$
isolate factors on both sides and exponentiate with exponent $1/\rho$ to get
$$\left(w_1^{\frac{\rho}{\rho -1}} +w_2^{\frac{\rho}{\rho -1}}\right)^{1/\rho}x_2 =   w_2^{\frac{1}{\rho -1}}(x_2^\rho + x_1^\rho)^{1/\rho} = w_2^{\frac{1}{\rho -1}} y ,$$
from here you can solve for conditional demand $x_2^\star(w_1,w_2,y)$. However, it is easier to oberserve that the factor $\left(w_1^{\frac{\rho}{\rho -1}} +w_2^{\frac{\rho}{\rho -1}}\right)^{1/\rho}$ do not change when interchanging indexes - it is symmetric. So define $a := \left(w_1^{\frac{\rho}{\rho -1}} +w_2^{\frac{\rho}{\rho -1}}\right)^{1/\rho}$ and conclude that
$$ax_1 = w_1^{\frac{1}{\rho -1}} y \\[8pt]
ax_2 = w_2^{\frac{1}{\rho -1}} y,$$
multiply first equation with $w_1$ and second with $w_2$ and add them to get
$$a(w_1x_1 + w_2x_2) = (w_1^{\frac{\rho}{\rho -1}}+w_2^{\frac{\rho}{\rho -1}})y = a^\rho y$$
solve for $(w_1x_1 + w_2x_2)$ which are the costs to get the result that
$$C(w_1,w_2,y) = a^{\rho -1} y = \left(w_1^{\frac{\rho}{\rho -1}} +w_2^{\frac{\rho}{\rho -1}}\right)^{\frac{\rho - 1}{\rho}} y$$


How to calculate equilibrium income given a certain level of unemployment?

this is how I've solved it :

(a) Y=C+I+G = 800+0.6 Y
    So Y = 2000

(b) With 400 as government expenditure, the new equilibrium income will be 2500. So the increase in the income is 500.

(c) I think this will be solved using Okun's law ut-ut-1=0.4(gyt-3%). But I'm not sure how to proceed.


Because Sunk Costs can encode information, should Sunk Costs really be ignored?

It's all about the precise idea these authors are trying to communicate:

A) Mankiew is explaining that we are sometimes too stuborn: we sometimes make a bad investment and then feel some vindication in making another bad investment that might rescue the first one, even though we know it's a bad investment on its own. In abstract terms that can be stated by saying that sunk costs should not be taken into account when evaluating whether to increase an investment in a project. Say you put in 100 dollars in a project (A) and got nothing back but you think if you only put in another 100 you will at least get you original 100 back. However, there is another project (B) that pays you 101 if you put in 100. We have the tendency to try to rescue our bad investment and put our money in project A again, but project B pays us a higher return, and we should go for project B.

B) Baliga and Eli are explaining that although, as Mankiew says, you should not use sunk costs in your calculation of whether a new investment is good or bad, it makes sense to keep track of sunk costs from an informational perspective. If you tend to forget what you did or how well it turned out, then recording your sunk costs in a notebook might help you figure out which were good investments previously and might be good in the future too, or not.


Given your equations, you can find

$$Y = C + 50 + 20 \implies Y = C + 70$$

You can then substitute $Y$ for its equal into your other equation

$$C = 100 + 0.5(C + 70 - T)$$

Finally, given a balanced buget, you know that $G = T$, which should be enough for you to solve for $C$.


How does one deal with trending variables in Linear Regression?

Depends on a trend. Trends come generally in two categories:

Deterministic trend - this can be controlled for using various methods. For example, in panel regression you could include time fixed effects that would correct for an effect of each time period on all firms and hence should control for a trend in data. This would look like this: $$y_{it} = \beta_{0} + \beta_{1}x_{it} + \beta_{2}z_{it} + \gamma_t+ u_{it}$$ Alternative approach is to just include time as separate variable $t$ which will be a series that will just increase by 1 in each time period this would look like this $$y_{it} = \beta_{0} + \beta_{1}x_{it} + \beta_{2}z_{it} + \beta_3 t+u_{it}$$This would control for any linear trend in the series, you could also make the trend quadratic but quadratic trends are quite rare.
Stochastic trends (unit root) - if there is stochastic trend in your data you generally can’t use the variable in standard regression models. You can test for presence of stochastic trend by some unit root test for example fisher test is quite popular for panel data, but it’s always context dependent. In case your variable contains unit root you can’t use it directly in most regressions but you can still transform it by taking first difference and provided the first difference does not contain stochastic trend (after you test it again) you can use the differenced variable. So in this case if all variables would contain stochastic trend you would want to run model like this: $$\Delta y_{it} = \beta_{0} + \beta_{1} \Delta x_{it} + \beta_{2}\Delta z_{it} + u_{it}$$Alternatively, sometimes variables can be cointegrated, in such case this indicates that variables tend to move together because they tend towards some long run equilibrium, you can still include them in their levels in cointegrated regression for example in panel version of fully modified OLS or other similar model, or alternatively build an error correction model where you would have both the first difference to capture short run dynamic of the model and level variables which capture the long run equilibrium.


Why not talk about utility functions on the surreal line when preferences are lexicographic, etc?

The first thing that comes to mind is that some economic models have more than one agent.

If one agent has a utility function with values in ${\mathbb R}$, then two agents have a pair of utility functions with joint values in ${\mathbb R}^2$.  If one agent has a utility function with values in $S^*$, then two agents have a pair of utility functions with joint values in ...what?  We can't just form a product of $S^*$ with itself, because $S^*$ is not a set.

There are probably ways around these difficulties, but they're going to be clumsy and they're going to require lots of mucking around with foundations that are almost surely not interesting to most economists.

Set theory lets us form unions, products, coproducts and power sets.  It lets us talk about subsets and supersets without having to worry about whether they exist.  Economists use these constructions all the time without stopping to think about them.  I doubt that we want to switch to a new foundation that forces us to think about them constantly.


$W(4) = \max\left\{4+bW(4),\frac{1}{2}\left(4 - k + bW(4)\right) + \frac{1}{2}\left(16 - k + bW(16)\right)\right\}$

$W(16) = \max\left\{16+bW(16),\frac{1}{2}\left(4 - k + bW(4)\right) + \frac{1}{2}\left(16 - k + bW(16)\right)\right\} = 16+bW(16)$

First solve for $W(16)$ to get $W(16) = \frac{16}{1-b}$. Then substitute it in $W(4)$ to solve for $W(4)$ as a function of $k$ and $b$.


First of all good question. I tried myself on that one, but if any other member of this wonderful site has additional input please also answer :)
In a monopol we know there exists a consumer who would be willing to pay a price for an additional unit of the good that is higher than the additional cost to produce that unit. Possibility of Pareto improvement: monopolist produces one additional unit and receives marginal cost from consumers. Giving away one unit for marginal costs does not make the monopolist worse off in the first case. But if the monopolist wants to sell an additional unit, he must lower the price not only for the last unit, but also for all remaining units. This is the result of one critical assumption: no perfect price discrimination. But from the perspective of a social planner one could produce an additional unit and find a trade where $MC \leq  P$ by allowing the monopolist only to change price for its last produced unit. In absence of such a planner the monopolist falls back to the old logic.
This leads us to the definition of market failure: individual rational behaviour leads to collective irrational outcomes.
Pareto optimality is also a very narrow definition, take for example the Kaldor Hicks criteria which states: An economic policy measure is welfare increasing, if in the society as a whole, increases in benefits outweighs the losses of benefits. This simply means winners could compensate losers, or in other words the sum of gains needs to be greater than the sum of losses (independent from individual changes). This holds true for the destruction of the monpoly because sum of consumer and producer surplus will increase. The pareto criterium is often criticised to favourite the status quo, while Kaldor Hicks creates more possibilities for restructuring in exchange for violating the methodological individualism.


Is the Occupy Wall Street famous 1% stable?

I've not see a strict analysis of the top one percent but I have seen an analysis of the top 400 tax payers, the very highest income taxpayers:

  Who are the rich? It depends who you ask and when. Since 1992, the IRS
  has tracked the top 400 earners in terms of adjusted gross income—the
  so called fortunate 400. ... Most importantly, this IRS report
  demonstrates there is a lot of income mobility at the top. Of all the
  filers who have made the list since 1992, 73 percent were on the list
  just once. Virtually no one remains on the list for all 18 years, but
  for privacy concerns the IRS did not report exactly how many did, if
  any. In last year’s report, just 4 people remained on the list for all
  17 years. This suggests that most top earners do not have a portfolio
  of big investments that can be cashed in year after year, but rather
  one big asset, such as a family farm or business or stock, the sale of
  which triggers a capital gain.

The Fortunate 400 By William McBride

A slightly more targeted answer:

  Considering those whose information was found in both years,
  approximately half  of taxpayers in the lowest and highest income
  quintiles remained in the same quintiles 20 years later. Nearly
  one-fourth of those in the bottom quintile moved up one quintile,
  while 4.7 percent moved to the top quintile. About one-fourth of those
  in the top 1 percent were also in the top 1 percent 20 years later,
  but nearly 70 percent remained in the top income decile. The overall
  results suggest that, while there is considerable persistence among
  observed taxpayers, there is also meaningful movement even within this
  narrow age cohort. Some taxpayers start from the bottom and move to
  the top and vice versa.

NEW PERSPECTIVES ON INCOME MOBILITY AND INEQUALITY
Gerald Auten, Geoffrey Gee, and Nicholas Turner (2013)


Solow Model with a Capital Stock Externality

There is no steady state in the Solow framework. The system is explosive because $a,b >0 \wedge a+b<1 \Rightarrow a+b + (1-a) > 1$. Thus $(zK_t)^{a+b} (L_tz)^{1-a} = z^{1+b}(K_t)^{a+b} (L_t)^{1-a}$. I.e. this production function has increasing returns to scale. Such capital stock externalities are discussed in the AK-Models.


Why does lowering interest rates transfer incomes from households to corporations?

A student of mine came up with a very plausible explanation for this statement. I think he's almost certainly on the right lines.

Low interest rates encourage consumption spending and discourage saving. (They might also encourage people to take out loans to fund consumption spending.)
Consumption spending transfers money from households (who are buying goods/services) to corporations (from whom they're buying).
These corporations are assumed to save at a higher rate than households (see the quote in the question), so increased corporate savings more than offset the decrease in household savings.

It's obvious once you think about it!


Why does a stronger currency have a tightening effect on economic conditions?

Tight monetary policy reduces the amount of money available in the economy and strengthens the currency. It is usually implemented by raising interest rates, increasing capital requirements for banks, or sometimes selling government bonds.

When there is less money and credit to go around, business activity slows down, economic growth slows down. Businesses that could have expanded by taking out loans do not do so; consumers who could have purchased more with their credit cards can't do so; investors who could have invested more money into businesses (equities) put the money into government bonds.


What is the meaning of the weights in a CES-production function?

The weights are usually called "distribution parameters" (top of page 12 of linked document), as they correspond, in the case of a Cobb-Douglas, to each factors share in total output. For example, if the two factors were labour and capital respectively, $(1-\omega)$ would be the labour share, and $\omega$ would be the capital share. Because in the case of a Cobb-Douglas ($\sigma=1$) both specifications are identical, the weights refer to the distribution parameters in both specifications.

To see this, take the second equation. First, define the share of $x_1$ in total output as

$$ \lambda_1 \equiv \frac{w_1x_1}{py} $$

where $\frac{w_1}{p}$ is the real return to factor $x_1$ (e.g. real wage).

Under competitive conditions, the firm(s) employ $x_1$ until its marginal product equal $\frac{w_1}{p}$. This marginal product is:

$$ \frac{\partial y}{\partial x_1} = (1-\omega)^{\frac{1}{\sigma}}\left(\frac{y}{x_1}\right)^{\frac{1}{\sigma}} = \frac{w_1}{p} $$

Combining the later equality with the definition of the factor share, you get that

$$ \lambda_1 = (1-\omega)^{\frac{1}{\sigma}} \left(\frac{y}{x_1}\right)^{\frac{1-\sigma}{\sigma}}  $$

In the case of a Cobb-Douglas ($\sigma=1$):

$$ \lambda_1 = (1-\omega) $$

Equivalently,

$$ \lambda_2 = \omega $$

which means that $\lambda_1 + \lambda_2 = 1$, in line with the assumption of constant returns to scale, in both specifications.

Regarding the difference between the two, according to my calculations, the second specification does not encompass the Leontieff case ($\sigma = 0$). This is because, whereas in the first specification you can solve the limit of $(1-\omega)x_i^{\frac{\sigma-1}{\sigma}}$ when $\sigma \rightarrow 0$ (using the minimum of $x_i$, see equation (23) here), you cannot do the same for $(1-\omega)^{\frac{1}{\sigma}}x_i^{\frac{\sigma-1}{\sigma}}$, because the first component goes to zero (provided the standard assumption $0<\omega<1$).

Thus, if you want to be as general as possible, I would recommend the first specification. Additionally, whereas the first specification is imo "very common", the second is not, which would require further justification as to why you choose it. If there is no extra benefit of it, then, by Occam's Razor, the first one is definitely preferred.


What are macro-economics simulations horizons and limits possible with exascale computing?

Some example of large macro models would be the the World Banks macro-fiscal model or Smets-Wouters DSGE model.

How will having access to high computing power change the models is very opinion based question I don’t think anyone can predict that but I will indulge you. With increasing computer power it will be definitely possible to make these models more detailed, perhaps even to the point where you don’t just model demand and supply in a given market based on just representative agents but also explicitly modeling interactions of heterogenous individuals and firms and deriving demand and supply from that. Also this is my two cents on the issue but I don’t think that in current macroeconomic modeling lack of computer power would be a main hurdle, I would say lack of available data is far more bigger problem but getting better. In the end no amount of computer power will help if you don’t have enough data to properly parametrize the model.


Newly discovered (by mainstream investors) asset chart/theory