Theory of and studies on the long-term effect of governmental redistribution

This is a very broad question and no one except the respondents in question know exactly what they believe and why.

However, I suspect one important result in this context is the Second Welfare Theorem.

In simple language and subject to some assumptions:

The First Welfare Theorem tells us that any competitive equilibrium leads to a Pareto efficient allocation of resources.

The Second Welfare Theorem tells us that almost any Pareto optimal equilibrium can be achieved via the competitive mechanism, provided appropriate lump-sum taxes and transfers are imposed on individuals and firms.

The main idea here is that markets lead to efficiency. Thus, no intervention of the government is required. However, a situation where someone holds every good and the rest of the population holds none, is also a Pareto efficient distribution. This distribution can hardly be considered as perfect under any welfare definition. The second theorem helps us here.

The two main issues economists are concerned about in this context are efficiency and distribution. The first is all about not wasting resources and maximizing the total economic pie. The second issue is about distributing that pie. The first theorem tells us that a free market achieves an efficient allocation. The second tells us that we can achieve any distribution we want through transfers without harming efficiency.

It is unlikely that the one particular efficient allocation which results from a free market (out of the endless possibilities) also is optimal from a distributional point of view (and optimal distribution will depend on a society's preferences). Luckily, according to the Second Theorem, we can use transfers to achieve both optimal efficiency and distribution. Hence, it is unsurprising that many economists would be in favor of that.

The controversies start with discussion how exactly to design those transfers. Any economist would agree that lump sum taxes and distribution are optimal. However, these taxes are considered impossible politically, which means we often have to rely on transfer mechanisms that trade-off efficiency and distribution.

About your question "is there substantial evidence showing that the policy of governmental redistribution improves absolute well-being of poor people?": The answer must be yes by definition. If we give poor people more resources it can be safe to assume that their economic well-being is improved.


There are several explanations for this in the literature (the order does not necessarily reflect importance of each explanation).

Different Price Levels: Non-trivial portion of the wage differential is due to different price levels/cost of living. Once you adjust for different price levels and compare real wages gap narrows. This is obvious but since you do not mention in your question whether you are talking about real or nominal wages I thought it is useful to mention that.
Regional Institutional Differences: For example Blackaby and Murphy (1991) argue that inter regional real wage differentials are at least partially due to labour market institution heterogeneity (unionisation and the bargaining system). This applies especially to unions/federations like US where there can be large heterogeneity in regional laws & regulations.
Regional Heterogeneity/compensating differentials: Even within one country the geography is usually heterogenous which creates opportunity for compensating differentials  (Farber and Newman, 1989). If you are offered two jobs that are same in all aspects but one job requires you to relocate somewhere to Alaskan wilderness and the other job requires you to relocate to some pleasant state with nice beaches you will likely be willing to accept lower salary from the job in nicer place because you are compensated by the fact that you are living in a nice place. Some regional differences could be also explained by agglomeration economics (e.g. urban vs rural areas have different economic implications), see for example Combes & Gobillon (2015).
Worker Heterogeneity: The wages would only equalize among workers that are homogenous across all work relevant characteristics. However, people are different from each other and often people who are similar tend to cluster together. For example, would it be surprising to find that average wage in silicon valley is higher than average wage in some farming community? This would be just comparison of apples to oranges. Hence looking at average wages can be deceptive and research that takes into account worker heterogeneity find that the wage differentials shrink substantially (see Gerking and Weirick, 1983; Dickie and Gerking, 1987;  Farber and Newman, 1989 and Combes, Duranton and Gobillon, 2008).
Regional Differences in Human Capital: Workers are not just compensated for their labor but also get return for their human capital. Human capital endowments might differ across regions (perhaps due to regional differences in quality of education) leading to wage differentials (see Duranton and Monastiriotis, 2002; Motellón, López‐Bazo & El‐Attar, 2011).

I think the above summarizes all the major factors but there might be some further ones as well for that you can see the sources above and ones cited therein.


Why did the Swiss National Bank abandon its currency peg so suddenly?

Just to synthesize the other two answers together a bit and "join-the-dots".

Setting the Scene

Since 2011 SNB had decided that it wanted to prevent the CHF from strengthening past 1.2 CHF per EUR.  In order to do this it had to do either or both of:

Discourage people from buying CHF.  One way to try do this is to keep rates low.
Sell CHF by the bucket load.

Keeping rates low at the short end is easy enough - that's the base rate.  But that was scuppered by the low EUR short rates.  Keeping rates low at the long end is trickier, but can be done by not issuing too many bonds.

Selling CHF is fine, but that means buying other currencies.  By the end of December the SNB had bought 85% of GDP worth of foreign currencies.

At this point the SNB decided that it did not want to keep expanding the balance sheet.  Why it decided this is a different question (as the question points out).  The question is how to end a cap once you hold a huge Euro position.

The Sudden Act

Once the SNB had decided it could not defend the cap, it had three ways it could act.

Stop supporting the cap, but not tell anyone
Tell people at the same time that it stops supporting the cap
Tell people before it stops supporting the cap (or as in your example, schedule a reducing cap).

The third case would have almost certainly bankrupted the SNB.  If they had, as you suggest, let everyone know in advance what was going to happen, then everyone in the world would just arbitrage the situation into the ground.  Everyone would just buy CHF from the SNB at the cheap rate while they support the cap, and sell as soon as the cap is dropped.

The first case is a little dishonest, and would almost certainly get found out (the worst kind of dishonesty!).  As soon as they stopped supporting the cap, the FX rate would drift slowly down from 1.2.  Once people realised that this was not "supposed to happen" and therefore the SNB must have stopped defending it, then it would probably snap anyway.  There are two dubious plus sides to this approach I see though

The SNB could have started offloading some of its 85% holdings before the market realised.  This is called front running and market abuse, and would land a stock trader in jail, but for a central bank trading FX, it would not technically come under any legal challenge (at the moment...).
The reputational damage to the SNB would be huge, which would go some way to undermine investment in the country, causing a weakening in the CHF!

As you can see this would be a rather dubious strategy, but I have often wondered if Thomas Jordan (chairman of SNB) could simply address the world with underpants on his head and his willy hanging out as a way to depress the CHF.

The strategy they opted for was the "ripping off a bandaid" style.  The sudden shock is estimated to have cost the SNB about 13% of GDP due to all those foreign holdings.  Since the SNB made a massive profit in 2014 (see: http://www.bloomberg.com/news/2015-01-09/snb-sees-2014-profit-of-38-billion-francs-resumes-dividend.html) it probably saw this as an opportunity to get out before things got any worse.  Hold out longer and have 100+% GDP in foreign currency, and the bank might not have survived the jolt itself.


Whats the difference between local non-satiation and monotonicity?

Monotonicity of preferences is a stronger condition than local nonsatiation. Monotonicity implies local nonsatiation, but not the other way around.

To see this:

Claim: Let $\succsim$ be a monotonic preference relation over $\mathbb{R}^n_{+}$. Therefore, $\succsim$ is locally nonsatiated.

Proof: Fix some $\varepsilon > 0$. Let there be an arbitrary $x \in \mathbb{R}^n_{+}$ and let $\mathbf{1} \in \mathbb{R}^n_{+}$ be the unit vector. For any $\lambda > 0$, we also have $x + \lambda \mathbf{1} \in \mathbb{R}^n_{+}$. Since clearly $x + \lambda \mathbf{1} \gg x,$ $x + \lambda \mathbf{1} \succ x$ by monotonicity. Consider the following metric over $\mathbb{R}^n_{+}$:

$$
d(x+\lambda \mathbf{1}, x) = ||x+\lambda \mathbf{1} -x|| = \lambda||\mathbf{1}|| = \lambda \sqrt{n}.
$$

Thus for $\lambda < \frac{\varepsilon}{\sqrt{n}}$, $d(x+\lambda \mathbf{1}, x) < \varepsilon$ yet $x + \lambda \mathbf{1} \succ x$. Since $x$ was arbitrary, the existence of such a point implies that $\succsim$ is locally nonsatiated. $\blacksquare$

To show that locally nonsatiated preferences do not imply monotonic preferences, you can come up with a utility function $u(\cdot)$ over various goods that strictly increases with respect to some of the goods but reaches a satiation point for at least one of the others. For example, in $\mathbb{R}^2_{+}$:

$$
u(x_1,x_2) = x_1 - |1-x_2|.
$$

A consumer with such preferences is satiated with respect to $x_2$ at $x_2=1$.


How can the share price of a company be less than its (theoretical) liquidation value?

A company can have a positive book value and still be in trouble because of insolvency.  This is very common with banks.  Banks operate by borrowing short term to buy long term assets.  This is inherently risky and vulnerable to a liquidity crisis.  A bank can show a profit and positive book value and still be in a world of trouble.  If investors feel that a bank will not be able to rollover their short term debt, then the bank will be in trouble.  Selling their long term assets will not be the answer because long term assets are by their very natural not very liquidable.  You can never trust a bank's balance sheet nor their income statement.  A more accurate assessment of a bank would be of their cashflow, maturity mismatching and credit rating.


Is it easier to profit from selling, than buying, options?

The logic given in the question is inadequate, since it misses the point that upside potential in long option positions is much higher than the upside potential for a short option position.
If one were buying/selling options at fair value on instruments that followed the random process used to price the options, a long or short position should not have an advantage over the other. The higher chance of a short position being profitable at maturity is offset by lower profits.
However, in the real world, options can be mis-priced, such as under-valuing extreme events. Furthermore, implied volatility is quite often higher than realised, which is equivalent to there being a risk premium. Those real world pricing discrepancies need to be looked at to discuss the relative merits.


Bolzano-Weierstrass Theorem and Pareto Efficient Allocation

What one needs are two facts:

Scalarization/Sufficiency for Pareto Optimality
Suppose the economy has finitely many agents $u_i$, $i = 1, \cdots, I$, with feasible allocations given by some $X \subset \mathbb{R}^I$. If
$x = (x_1, \cdots, x_I) \in X$ solves the problem
$$
\max_{x \in X} \sum_{i=1}^I \lambda_i u_i(x_i),
$$
where $\lambda_i > 0$ for all $i$, then $x$ is Pareto optimal. This maximization problem is called the social planner's problem with social weights $\lambda_i$.
(The converse is not true. In other words, not all Pareto allocations arise as such, not even when $u_i$'s are concave.)
Continuous functions map compact sets to compact sets. (The Bolzano–Weierstrass gives a characterization of compactness on $\mathbb{R}^n$---closed and bounded.)

So if the preferences are continuous and feasible set $X$ is compact, existence of Pareto optimal allocation follows immediately. Choose any social weights $\lambda_i > 0$. A solution of the resulting social planner's problem is a Pareto optimal allocation.


when a=0, would economy accumulate more physical capital? Why?

From the looks of this question, it seems like a typical consumption-saving problem with possibly endogenous leisure.
I am assuming the production function is constant returns to scale. I am assuming that $a\in[0,1]$
The problem is:

$\underset{\{c_t,k_{t+1},l_t\}} {Max} \sum_{t=0}^{\infty} \beta^t U[c_t,l_t] = (1-a)ln[c_t]+aln[l_t]$

subject to $k_{t+1} = f[k_t,l_t]+(1-\delta)k_t-c_t$

The Bellman Equation:

$V[k_t] = U[c_t,l_t]+\beta V[k_{t+1}] $

Differentiating w.r.t $k_{t+1}$ and using the envelope condition you will find the Euler as:

$\frac{c_{t+1}}{c_t} = \beta (f_{k_{t+1}}+(1-\delta))$

Which at steady state gives you:
$k^* =(1-l^*)(\frac{e}{\frac{1}{\beta}-(1-\delta)})^{\frac{1}{1-e}}$

If a = 0, there is no labor selection problem and $l^*=0$ leading to full use of available time for labor. Clearly, there is more capital in the steady state equilibrium.


Is it possible, that the global warming will affect the growth of the economy?

Global warming doesn't impact energy production in a major way (nuclear plants getting shut down because the cooling water is too hot is a thing, but it's not a major thing). If anything we're building far more energy production (solar and wind) before we ordinarily would have to (plants reaching end of life). If energy consumption went up we would just not close a power station early....

All of the above assumes you're talking about europe/AUS/US. In other continents demand still outpaces production in many places. But that's not a global warming problem, they will happily build coal plants all day long.


As long as trading faster is not explicitly forbidden, there will be people who want to trade faster if they can take advantage of it.

If you are able to trade fast enough, you can take advantage of arbitrage situations.
For instance, if you can detect a fall in the price of soy on the New-York exchange, buy soy on the New-York Exchange, and sell it on the Chicago exchange before the price in Chicago adapts, you will profit from it.

So, on the positive side, the answer to your question in the title "Why do stock exchanges not operate at a fixed frequency?" seems to be "because it's not impossible/forbidden to operate faster and people benefit from it".

Now, on the normative side, there are certainly a fair amount of people who believe that stock exchanges would perform ``better" if they operated at a fixed frequency.

Roughly, increasing the frequency of trades is good if it allows prices to adapt faster to new economic information. Some people argue, however, that past a certain threshold, increasing the speed of trade cannot foster more effective price adjustments. The argument is that relevant economic information only arrives every so often, and there is no point in trading faster than the fastest stream of relevant information.
Past this threshold -- so the argument goes, increasing the speed of trade only allows for increased high-frequency speculation in which traders benefits from arbitrage situations that are unrelated with ``actual" economic fundamentals.

I don't think they are the first to propose it, but Budish, Crampton and John have a recent pair of papers in which they advocate for a cap on the speed of transactions for similar reasons : The High-Frequency Trading Arms Race: Frequent Batch Auctions as a Market Design Response, and Implementation Details for Frequent Batch Auctions: Slowing Down Markets to the Blink of an Eye.


Advice for finding literature for thesis

Your supervisor might want for you to come up with a research question yourself, maybe because that's part of the assignment. At the thesis, you should be able to demonstrate independent thinking rather than being told what to do for an assignment, and your supervisor might want you to do some more research on your own. However, it is also part of his/her job to give you guidance on the relevant fields where s/he considers your skills fit best.

Since you want to research on macro, check which papers were part of the recommended readings for that subject and ask your supervisor if s/he would recommend you to start there.

Another way to get ideas is to scout the papers in economics search engines. The IDEAS/RePEc can be a good starting point. You can look for topics and sort by citations.
Elsevier also has a list of most cited work in economics since 2013. Keep in mind that importance in academic terms is usually often measured by number of citations.

If you want a more mainstream reading, I'm sure a google search will also help you find articles like this list of 2017's most important researches.


In a production function, the technology can vary across the firms and times. But the "power" must be same for the same industry?

You can estimate it in multiple ways. If you have panel data then you could first log linearize the expression giving you:

$$ln(F_{it})=ln(A_{it}) + \alpha ln(K_{ti}) + (1-\alpha) ln(L_{it}) $$

Which could be estimated in multiple ways. You could estimate it as a pooled cross-section where the technology would be assumed constant across the time and same for all firms (lower case variables are in logs, and betas here give you the estimates of exponents from original specification):

$$f_{it}=a+\beta_1 k_{it} + \beta_2 l_{it} +e_{it}$$

You could run firm fixed effects specification where you allow every firm to have its own technology but technology is constant across the time:

$$f_{it}=a_i+\beta_1 k_{it} + \beta_2 l_{it} +e_{it}$$

You could assume that it’s the same for all firms but different across time:

$$f_{it}=a_t+\beta_1 k_{it} + \beta_2 l_{it} +e_{it}$$

Or you could also have a specification where you would assume that technology can be different across firms and also change in time (but the change in time has to be the same for all firms)

$$f_{it}=a_t+a_i+\beta_1 k_{it} + \beta_2 l_{it} +e_{it}$$

If you would also want to allow for technology to change differently across firms you could employ some multi equation estimation where you estimate the parameters for each firm separately.

So really depending on what you think is the most appropriate for your dataset you could make different assumptions on this. For example if it’s an industry that relies on basic research so there are no intellectual property rights - it might be appropriate to assume all firms have the same technology which can change in time. If it’s some industry where technological progress is unlikely - let’s say some services like barbers then the best assumption might be to have constant technology for all firms and times etc.


What is "indirect mechanism" in the context of adverse selectoin?

In direct mechanism agents directly report their preferences (preferences are observable).

In indirect mechanism agents don’t report their preferences directly. Preferences can be observed only indirectly through signals or behavior.

By Revelation Principle if some outcomes can be implemented in indirect mechanism they must be also implementable in the direct mechanism where everything is observable - that’s what the quote is trying to say.


What you are missing is the concept of GDP deflator .
Which is there while studying National Income Accounting.

When you are comparing change in Valve Added (GDP) of 2016 and 2017 , the change could have been mainly due to two reasons :-

Increase in production
or Change in prices

While comparing Value Added by the company in two years , second reason i.e change in prices would give us false result as while comparing GDP we want to see how much the production has increased/decreased between the years , but the change in prices might show an exaggerated value of GDP if we do not take into account the change in prices .

For getting the correct answer what we do is that we take one year as base year(generally the first year given to us i.e 2016 in your case) and calculate the GDP on the basis of the prices prevalent on that year , so that we get a true picture of real change in GDP .

$$
\text{GDP Deflator} = \frac{\text {Nominal  GDP}}{\text{Real GDP}} \cdot {100}
$$

Where Nominal GDP is the GDP of 2017 at current year i.e 2017 prices
And Real GDP is the GDP of 2017 calculated at base year prices i.e 2016

Also remember the formulae $\text{GDP = Units produced * Price per unit} $

Do refer to the following hyperlink

GDP deflator

Hope you find my answer useful. :)


Relationship between default risk and value of bond

[...] underestimating the risk of default is tantamount to overestimating the value of the bond.

The reason behind this is simply due to how bonds are priced.

$$
P = \sum_{n=1}^N \frac{C + M[{I_{n=N}}]}{(1+i)^n}
$$

Where $i$ is the contractual interest rate, $C$ is the coupon payment, $N$ is the number of payments, $I_{n=N}$ is an indicator function taking the value of $1$ when $n=N$ is $true$ or $0$ otherwise, $P$ is the market price of the bond, $M$ is the value of the bond at maturity.

When $i$ goes up, $P$ goes down and vice versa, see

$\rightarrow$ if $i$ is underestimated, $P$ will be overestimated. Indeed, it is $i$ which, via a premium, internalizes the default risk consideration and is an increasing quantity of it.


Are the goods in additively separable utility functions normal goods?

Yes if you assume that the sub-utility functions are concave. Notice that this is a standard assumption as otherwise, the utility function $u = \sum_i f_i$ is not guaranteed to be concave (nor quasi-concave).
Let denote by $u_i = \dfrac{\partial u}{\partial x_i}$ and by $u_{i,j} = \dfrac{\partial^2 u}{\partial x_i \partial x_j}$. By additivity $u_{i,j} = 0$ if $i \ne j$.
The first order conditions for the utility maximisation problem give:
$$
\begin{align*}
u_i = \lambda p_i, \tag{1}\\
\sum_i p_i x_i = m \tag{2}
\end{align*}
$$
Differentiating both with respect to income $m$ (and using $u_{i,j} = 0$) gives:
$$
\begin{align*}
&u_{ii} \frac{\partial x_i}{\partial m} = p_i \frac{\partial \lambda}{\partial m} \tag{3}\\
&\sum_i p_i \frac{\partial x_i}{\partial m} = 1 \tag{4}
\end{align*}
$$
Substitute $(3$) into $(4)$:
$$
\begin{align*}
&\sum_i (p_i)^2 \frac{\partial \lambda}{\partial m} \frac{1}{u_{ii}} = 1,\\
\to &\frac{\partial \lambda}{\partial m} = \frac{1}{\sum_i \frac{(p_i)^2}{u_{ii}}} \tag{5}
\end{align*}
$$
Notice that $u_{ii} < 0$ by concavity of the sub-utility functions. As such, $\frac{\partial \lambda}{\partial m} < 0$ and also, by $(3)$:
$$
\frac{\partial x_i}{\partial m} = p_i \frac{1}{u_{ii}} \frac{\partial \lambda}{\partial m} > 0
$$
as the right hand side is the product of two negative numbers.
This shows that $x_i$ is a normal good. By symmetry, this holds for all goods.
An alternative quicker way to notice this is to see that

by $(3)$ the sign of $\dfrac{\partial x_i}{\partial m}$ will be the reverse of the sign of $\dfrac{\partial \lambda}{\partial m}$.
This has to be true for all goods, which means that either all goods are normal, or all goods are inferior.
From $(4)$, it follows that at least one good should be normal (as otherwise the sum cannot be equal to 1 which is greater than zero).
Conclude that all goods have to be normal.


World bank data - GDP per capita current US\$ vs constant 2005 US\$

The two charts are consistent in the basic sense that the 2005 values are the same in both, with Croatia about $\$10200$ and Russia about $\$5300$. What happens in other years depends on exchange rates and inflation in the three countries concerned (Croatia, Russia and the United States).

The current USD-based figures are calculated much as you would expect: take the GDP per capita in local currency (kuna in Croatia, rubles in Russia) and covert these to dollars at the exchange rates applying in the different years.

Your question would make sense if the constant USD-based figures were calculated by taking the current USD-based figures and deflating them for price changes in the US since 2005. So the Croatian and Russian numbers current USD-based figures would have been deflated by the same ratios, and if they were to meet in a particular year in one series then they would meet in the same year in the other series.

But they were not calculated that way.

Instead the constant USD-based figures were calculated by taking the GDP per capita in local currency each year, and were deflated by price changes locally since 2005 to give GDP per capita in constant local currency units, and then converted to dollars using the 2005 exchange rates. Because exchange rates did not match varying relative local price changes in Croatia and Russia compared with the United States, this produces a different result.

There is a way to approach this question to reduce the problem, but that would involve purchasing power parities (PPP) looking at what money can actually buy in different countries rather than market exchange rates, and the statistical invention of the international dollar (not an actual currency) to handle price changes year-to-year. It results in different data again and can lead to different rankings between countries (famously, China would appear to be relatively much richer). Here it would suggest Russia having significantly higher GDP per capita than Croatia since 2011 on a PPP basis, the opposite of your second chart due to the the ruble seeming to have been undervalued in the market exchange rates compared with the kuna in terms of what it can apparently buy locally.


Is there existing research on predicting the number of house listings based on exogenous data?

It looks like what you're really after is determinants of housing mobility or migration. You get that, you can then just apply it to a specific city and aggregate to predict. One of the earliest and most seminal works in this field was Rossi's Why Families Move, originally published in '56 and redone in '80. The book is here and a review is here.

Another giant paper in the literature is Massey "Social Structure, Household Strategies, and the Cumulative Causation of Migration". It is also a little older (1990), but brings together a lot of the previous literature (great paper for lit review) and formalizes the decision making process of migration.

Newer papers in the literature look more at specific issues related to relocation, like eviction, fertility and housing decisions, and connection with life-course events.


Why are CAD, AUD known as commodity currencies

From wikipedia: A commodity currency is a name given to currencies of countries which depend heavily on the export of certain raw materials for income. These countries are typically developing countries, e.g. countries like Burundi, Tanzania, Papua New Guinea; but also include developed countries like Canada and Australia.

In a more practical sense, if the value of a currency is heavily dependent on the value of the natural resources exported, it can be considered a commodity currency. For example, look at the appreciation of the AUD during the recent mining boom. This appreciation was driven primarily by increased Chinese demand raising prices of commodities Australia was endowed with.


Why don't most of the highly educated people in poorer part of Europe migrate to the Western part?

There are numerous reasons why one would want to stay in a particular country even though earning power is lower.  These are usually summarized as "frictions" in the labor market.

Culture, familiarity with an area, language, family are all non-negligible "frictions".

I note that while there will be a tendency to vote this closed as not economic, I think the reminder that there are meaningful human incentives other than money and tangible goods may be worth remembering.


My question is that isn't the central bank only responsible for determining the interest rates?

No, central banks do not even have directly responsibility to adjust interest rate. Depending on which country we talk about they will have responsibility to keep prices stable, high employment and bank regulation and often combination of the above. In order to follow their mandates they will often fiddle with interest rates but typical central bank has many instruments at its disposal. For example, they can also increase money supply or even adjust reserve requirements that also affect money supply (Fed recently set them to zero in their effort to increase inflation).
As mentioned above  most of the time contemporary central banks adjust interest rates on their reserves that influences interest rate in the economy and let the new money supply be determined by what the interest rate is but I don't know of any central bank that would not be allowed to directly increase $M$ if they would want to in some way. Also, note even if central bank sets itself some goal achieving some interest rate without stating any goals in terms of money supply, since interest rates depend on what $M$ is, they in principle could fully control interest rate just by varying $M$. They might want to use more tools because relying only on $M$ would be more difficult but it is possible.
In fact the open market operations that you mentioned are one way how central bank can directly increase money supply in the economy and the open market operations are not that scarce. These are becoming increasingly common, for example you can see from the graph below what an effect QE (which is not fundamentally much different from OMO expect in its scale and that it is part of holistic strategy that includes other tools) had on increase in monetary base in the US (first round of QE started in 2009 you can easily spot it by the shar increase in monetary base), so that is non-trivial  change in $M$ (technically $M0$ but $M$ consist of base money as well) that wasn't created with interest rates.


Why would a low wage country affect negatively the high wages of a country with trading?

They mention low-high wages countries due to the following common narrative on trade and inequality:

Globalization allowed low-wage countries to export more labor intensive goods to high-wage countries.
Less skilled workers in those high-wage countries faced import competition and lost their jobs.
Bottoming out of the middle class and rise in inequality.
Leading to a backlash against globalization.

This common narrative suggests development of low-wage countries has come at the expense of low skilled workers in high-wage countries. This narrative has been investigated in many papers and surely in your book!


Does the Efficient Market Hypothesis Solve the Tragedy of the Commons?

Your description of the Efficient Market Hypothesis (EMH) is not correct.

The EMH simply states that asset prices fully reflect all publicly-available information. It does not make the wild claim that:

  the market will find the best solution both in the short and in the long term.

You write:

  It seems that market players are not investing in ensuring the effects of global warming are minimized (i.e. market players are not ensuring the continuing existence of a market place). Does this mean that the Efficient Market Hypothesis is a model that does not describe our actual reality?

Market players do not invest with the goal of minimizing the effects of global warming. Instead, their goal is to maximize their returns.

To the extent that their interests and those of society's (or the world's) are not aligned, we have a problem of externalities. But this says nothing whatsoever about the EMH.

Example. At a horse-racing track, individuals place bets with the goal of maximizing their winnings. They do not place bets with the goal of raising the horses' welfare or of sustaining the horse-racing business. Their goal may not align with that of the horses or the business owner. But this says nothing whatsoever about the EMH.

(What the EMH would say in this example is that the bets reflect the expected performances of the horses. The bets do not say anything about the welfare of the horses of the viability of the horse-racing business.)


Real world evidence that voluntary exchange is mutually beneficial?

First of all, there is no need to believe any economic dogma. The real world is usually more complicated than these stories. If anyone can convince me of something with a two minute anecdote, that was probably not an important aspect of my world view, and I should probably not engage in setting such policy. (E.g. via voting for the person who tells the same anecdote.)

The statement that voluntary exchange is mutually beneficial relies on the assumption that both parties are rational and well-informed. If the exchange was not mutually beneficial, the one who is losing via the exchange would just say "no, thank you". Hence literally all voluntary exchange is mutually beneficial (so long as the parties are rational and well-informed). This does not mean that the situation voluntary exchange leads to is optimal, just that it is better than having no market at all.

Some further remarks on taxation:

Deadweight loss is indeed bad. (By the way what you are describing is comparative advantage, not deadweight gain, as you yourself noted correctly in the tags.) However taxation is not the only thing that can cause  market failure, which is when the free market leads to a suboptimal outcome. For example market power (monopoly, oligopoly, monopsony, etc.) may also result in deadweight loss.

Unregulated externalities (e.g. pollution) may cause harm, and in this case taxation (or a proper market for the externalities) can actually improve everyone's situation.

Underfinanced public goods (e.g. roads) may also mean market failure. Voluntary financing usually leads to underfinancing, hence taxes (or some revenue mechanism) are needed.

So taxation, while it does have downsides, is not the devil. It may do more harm than good and it may do more good than harm, it all boils down to the details.


Welfare status when a positive externality is subsidised

Yes. While producer and consumer surplus rise, the total cost to government exceeded the increase, leaving deadweight loss.


Why is the nature of graph of utility function different from indifference curve?

You seem to be confusing some things.

  The indifference curve and utility function both have the same
  equation

The utility function has a formula, not an equation.

$U(x_1,x_2)$ is a utility function.

The points $(x_1,x_2)$ for which $U(x_1,x_2) = 13$ form an indifference curve. (13 was picked as a random number)

The graph of a utility function has one more dimension than the graph of an indifference curve: in addition to the space of goods, this graph would also include the utility value, so its points would be something like $(x_1,x_2,u)$. Chances are you have not seen a graph of any such function, as 3D drawings are complicated. This is why maps of 2D indifference curves are used to get a feeling for the shapes of utility functions.


Significance of Y-axis in Aggregate Supply curve (AD-AS model)

I didn't quite well understood your question, but from what I could make of it I have answered the relation behind the formation of the Short run aggregate supply

In the short run, nominal rigidities , and/or imperfect information, make the AS
curve positively sloped (SRAS - Short run aggregate supply)

The position of the SRAS curve is determined by the factors
contributing to equilibrium in the labour market. In particular, an increase
in the expected price level makes workers bargain for higher wages (for
any given unemployment rate). The increase in the wage raises the actual
price that firms set. As a result, an increase in the expected price level
causes the AS curve to shift upwards

Diagrammatically , the SRAS curve should shift upwards ;

This equation states that the deviation of the actual price level from the
expected price level, P – Pe, is proportional to the deviation of output
from its natural level, Y – Yn

If you are confused between Workers negotiation and AS fall or rise - you should read about the WS-PS model. The WS stands for wage setting, PS for price setting.


I'm guessing the author has split the series between a trend component and noise component, for example with the HP-filter, and studies the smooth trend component.


Basic question on calculating unemployment rates

No, it's not correct. You are supposed to exclude people in FTE from labour force. Unemployment rate is a proportion of labour force which is unemployed. Labour force is defined as a sum of those employed and those looking for jobs (so it also excludes pensioners, parents on parental leave etc.). In your example the labour force should consist of only the people 16-24 years old.

I.e. $$12.2\% = \frac{\text{unemployed who are 16-24 years old}}{\text{labour force who are 16-24 years old (employed + unemployed)}}$$

Edit: I added that in this example labour force is restricted to specified age group.


I found some references that might help you explore this topic. Other topics like game theory can also be applied to border carbon adjustments.
LSE: https://www.lse.ac.uk/granthaminstitute/publication/the-global-consumer-incidence-of-carbon-pricing/
This is an excerpt from the website: "the author investigates complementing the EU ETS with a carbon price on traded goods, through so-called Border Carbon Adjustments (BCA), and finds this would most affect the poorest as well as the richest consumers in the EU. This is the first estimate of the EU-wide consumer incidence of BCA to complement the EU ETS." They have a working paper you can download.
LSE: https://www.lse.ac.uk/granthaminstitute/wp-content/uploads/2012/04/Trade-climate-change-game-theory-border-carbon-adjustments.pdf
Here is an excerpt from the abstract: "This paper
advances two arguments. First, it argues that the conventional view of Border Carbon Adjustments (BCAs) as a “dirty” trade barrier should be turned on its head. Rather, the absence of a carbon price comprises an implicit subsidy to dirtier production in non-regulated markets. Second, BCAs could act as a gamechanger when climate policy negotiations move at a glacial pace, if at all. Materially stronger progress could be achieved indirectly from the threat of unilateral trade policies. The paper shows how this could come about, using a simple political game theory model."
Vivid Economics: https://www.vivideconomics.com/casestudy/border-carbon-adjustments-and-industrial-competitiveness-in-a-european-green-deal/
Here is an excerpt from the website: "Driving industrial decarbonisation while safeguarding international competitiveness is a major policy challenge. As part of the European Green Deal, the EU is considering the introduction of a Border Carbon Adjustment (BCA) to ensure that the price of imports into the EU more accurately reflects the environmental costs of their carbon content. BCAs could be an alternative to free allocation to emissions-intensive trade-exposed sectors as a measure to address the risk of carbon leakage in the EU’s ETS." They have a report you can download too.


Lifecycle Consumption Models and Dispersion in Consumption

There are two major "qualifiers" to the life-cycle hypothesis (LCH). Both put forth by John Maynard Keynes in "The General Theory of Employment, Interest, and Money."

The first is the "precautionary effect," that people save more than the LCH would predict because they are uncertain about their future health, life span, medical bills, etc.

A second qualifier is that old people would want to leave a bequest, or legacy to their descendants, and therefore not spend all their savings.


Does an instant settlement system (such as blockchains) eliminate the possibility for short selling?

Imagine:

I borrow 1 bitcoin from you and agree to pay you back $(1+r)$ bitcoin in a month.
I immediately sell the bitcoin you lent me.
One month from now, I buy $(1+r)$ bitcoin and pay you back.

If I can conduct those transactions, I can short sell bitcoin.


Macroeconomics nowadays tend to require microfoundations. As a result, general equilibrium are more of a topic for macro than micro analysis. The dynamic stochastic general equilibrium (DSGE) model and its variants underlie much of macroeconomic analysis nowadays.

Standard texts:

Stokey and Lucas (1989) Recursive Methods in Economic Dynamics
Ljungqvist and Sargent (2004) Recursive Macroeconomic Theory


Why do different countries have different currency ? Why not there are different currencies for each 10-20 kilometers?

If the division is true why not there are different currencies for
  each 10-20 kilometers(or where there is more rice growth,or where
  there is no agriculture but more tourism difference based on these
  differences)?

If I'm grasping your question correctly, that type of hyper-fragmentation of currency areas would largely wipe out the standardization advantage that money has over barter (By standardization I am not referring to the ISO classification of currencies, but to their feature of being a mutually-acknowledged unit for trading).

Historically, fragmentation of currency areas has occurred during wars and/or early postwar. Two examples come to my mind. One is the Mexican Revolution (1910-1921), where various groups of rebels and armies got in the habit of inventing their own currency and deploying them in territories they occupied. It was jokingly said that Francisco "Pancho" Villa used to print his bills while on horseback.

Another, more recent example dates from the mid 1990s, during the precarious period once the Balkan war was brought to an end (see Dayton Agreement). I recall a Polish reporter of name Kasia Wyderko mentioning during a TV roundtable from late 1990s that there were three currencies in Bosnia-Herzegovina, their areas being delimited by the ethnic profile of each territory (croats, serbs, and muslims, accordingly). Unfortunately, the roundtable doesn't appear to be uploaded, but some reference to the multiple currencies in Bosnia-Herzegovina in the mid 1990s is this Wikipedia site.

Other than the answers to the "duplicate" question, the aforementioned instances reflect that fragmentation of currency areas is somehow a "last resort" where marked distrust, instability, or resentment precludes economic cohesion and the adoption of a common currency. The isolation and barter-like limitations that ensue from that fragmentation in a small economy most often hinders its dynamism or adaptability.


a risk lover agent preferences and the preference of risk natural agent may be the same

Don't commit the cardinal mistake of equating preferences with choices.

In the context of Expected Utility Theory, the fact that a risk-averse agent ($RA$) would choose $N$ over $M$ implies that

$$E[u_{RA}(N)] > E[u_{RA}(M)]$$

The fact that a risk-neutral agent ($RN$) could choose $M$ over $N$ implies that

$$E[u_{RN}(N)] < E[u_{RN}(M)] \implies E(N) < E(M)$$

A risk-lover is a person that could choose $M$ over $N$ even if $E(N) > E(M)$. So here that the opposite inequality holds, it is certain that it will also choose $M$ over $N$, given also the fact that there appears to be no other lotteries to choose from.

But never say that the risk-neutral and the risk-lover have "same preferences". They don't -it just happens, given the available lotteries, that they choose the same lottery, even though they have different preferences.


If protectionism is injurious to a state, is the success of mercantilist China in spite of its protectionist measures?

I am not an expert on this issue, but pending an expert's answer on this matter, here are my two cents.

1. It is not true that there is a consensus that protectionism is almost always a bad thing for a country.

Many slightly non-orthodox economists (e.g. Dani Rodrik, Chang Ha-Joon) have pointed out that protectionism is a perfectly good strategy that many developing/industrializing countries have adopted, including the US in the 19th century, Japan and Korea in the 20th, and now China in the 21st.

I believe Rodrik and Chang's main argument for protectionism in this context is the infant industry argument. Another argument is the second-best argument. See this other answer of mine.

2. China's protectionism has been selective.

In my vague impression as a non-expert, in most industries in which China exports heavily (except perhaps steel), there has been little protection and great competition. Think for example of clothing and the electronics manufacturing base centered around the Pearl River Delta and Shenzhen. These are industries in which every cost advantage and innovation are ruthlessly and instantly seized upon, to the advantage of both producers and consumers (including consumers around the world).

I believe the IT industry is where China engages most heavily in protectionism. As is well known to the man on the street, many technology platforms like Google, Facebook, and YouTube do not operate in China either because they have been outright banned or because of the onerous censorship requirements. The result is that local substitutes have flourished (Baidu/Sogou for search, WeChat/Weibo/QQ for social media, Tudou for video, Taobao/Alibaba for ecommerce) and Tencent and Alibaba are among the worlds largest firms, even though most people outside China may know little of them.

But perhaps the greater bone of contention is China's "theft" of intellectual property (IP). As an explicit/implicit condition of entry, tech firms trying to enter China are usually forced to hand over their IP.

Again, these are merely my vague impressions and hopefully an expert can back these up or correct me with more concrete facts and statistics.

3. One argument against protectionism is that it provokes retaliation, which can be damaging.

However, for the past many decades, the many countries to which China exports have shown great restraint and retaliated little. These are for reasons that I am not sure about and which I hope an expert will explain. (Perhaps this is simply the same restraint that has been somewhat shown -- not always consistently and often arbitrarily -- to other developing countries in the past.)

This is now changing with the aggressive trade war launched by Trump and there are now indeed fears and perhaps even signs that China may be suffering as a result of this trade war. Huawei and ZTE are two Chinese companies that have suffered heavily from the US's uncertain on-and-off bans in the past year.

This however is to not to say that Trump's trade war is the best approach. See for example this recent piece by the right-wing, pro-trade Cato Institute ("Disciplining China’s Trade Practices at the WTO - How WTO Complaints Can Help Make China More Market-Oriented", Nov 2018). I think many economists would agree that if China does indeed engage in protectionism and one wishes to retaliate, then the means of retaliation should be through multilateral, rules-based organizations like the WTO, and not unilateral, arbitrary trade wars such as those currently being undertaken by Trump.

4. Your example of the EU Common Agricultural Policy.

I think economists are almost universally agreed that protection for agriculture in rich countries is bad for most parties (the only beneficiaries being the farmers who receive the subsidies). I have never come across any modern economist defending any such policies.

Besides, I am not aware that China engage in much agricultural protectionism and indeed, I don't think this is what people think of when talking of China's protectionism. (Recently China has imposed heavier tariffs on pork and soybeans imports from the US, but these I believe are merely strategic retaliatory measures against Trump's trade war.)


Does it ever make sense to underproduce as a firm?

An interesting case is: should a firm ever produce less than the profit-maximizing output? This could make sense in a dynamic setting: by underproducing in the short run, you might increase future long-run demand. For example, when popular toys like Cabbage Patch Kids and Power Rangers were first released, they were in short supply. Parents went nuts trying to find them, and there was lots of media coverage. Eventually the supply increased; to be honest, I don't know if they raised the price but they could have.


I agree with @BBKing, just to put it here as an answer: if we go back to your two definitions you can find quiet easily some counter examples. Let's take a firm, which uses a machine as capital. Let's assume also that this machine can only be used by one man at a time. Do you think this machine is a non-excludable / non-rival good ?


In an intertemporal (2-period) consumption model, why is the investment rate independent of discount factor?

This idea is known as the Fisher separation theorem.
Without the investment opportunity to transfer $h$ units of present day value into $w(h)$ units of future value, the perfect credit market gives us the intertemporal budget constraint of
$$
c_1 + \frac{c_2}{1+r} = y,
$$
which can be represented by a straight line.

Without knowledge of the consumer's preferences, it is not possible to say what the optimal $(c_1,c_2)$ is here. But we can say with certainty, that the higher the budget line is, the more options the consumer has, and assuming monotonic preferences she will be better off.
Now the yield curve $w$ of the investment opportunity allows us to shift the basic credit market budget line. Starting from the lower right point of $(y,0)$ (i.e. not putting money into the credit market at all) we can reach cash flows $(y-h, w(h))$.

But by also putting some money into the credit market (or borrowing), one can reach other cash flows as well. These always transfer $x$ units of present value into $x(1+r)$ units of future value (in case of borrowing $x$ is negative), so they are always paralel to the original budget line, but instead of starting from $(y,0)$, they start from the point $(y-h, w(h))$ where $h$ is the size of the investment.
The optimal investment size is the one that reaches the highest budget line:

The idea is that given perfect credit markets one can separate consumption behavior and investment behavior. The higher the net present value of my cash flow, the better of I will be once I adjust my cash flow via the credit market to my intertemporal preferences (this is where $\beta$ matters). Thus I need to make investment decisions that maximize the present value of my cash flow, and I do not have to take anything else into consideration, e.g., the timing of the payments.

A mathematical tidbit:
In the above example, given the optimal $h$ (assuming it is an interior point) the budget line is tangential to the orange curve depicting the possible investments, so we have
$$
w'(h) = 1 + r,
$$
the marginal yield from the investment and the credit market are equal. This is equivalent to the final equation in your question.


When using consumables consider the purchase or market price?

Assuming you plan to keep both oil and wood in stock, and they both give you the same quality of heating, presumably you should burn whichever will cost less for you to replace. That would be the fuel that will cost less to buy in the future when you buy it.

If you don't have any special knowledge about what prices will do in the future, I think the most neutral position would be to assume they will stay where they are now, so you should burn whichever currently has the lowest market price - oil in your scenario.

Having done that you could if you wanted to sell the wood that you didn't burn, which you say is worth more than the oil, buy the exact quantity of oil that you did burn, and have money left over. That must be better than burning the wood.

I've assumed for simplicity that there are no transaction costs incurred in the act of buying and selling.


Proof coefficient in log-log model is equal to coefficient of elasticity

Because $\Bbb E[\varepsilon \mid x]= 0$ is one of the key assumptions for the estimation.


How would increase labor productivity lead to improved living standards of people in an economy?

In economics we measure living standards by the amounts of goods and services people can consume. Increasing labor productivity allows us to produce more goods and services and thus also consume more goods and services, hence it improves the living standards.

For example consider classic Cobb-Douglas production function:

$$ Y = K^{\alpha} (AL)^{1-\alpha}$$

in this production function $A$ can be viewed as a productivity of labor (or technology that makes labor more effective). In the equation above $Y$ is income or output (i.e. amount of goods and services) $K$ capital and $L$ labor supply.

Even if everything stays constant (meaning economy does not have more capital or people dont work more hours and so L is also constant), but the labor productivity increases ($A$ goes up) the output of economy goes up as well. Higher output means that there are more goods and services for people to enjoy and in economics we would count that as an increase in (material) living standards.


Is money received from my mother every month an income?

I will simply repeat what I've said in the comments, just so that there's a concentrated version of the argument.

What you are describing is not 'income' but a kind of 'transfer' payment. Your mother transfers an amount of money from her account to yours.

It is possible to consider it an 'investment' on her part; she would be investing in the increase of your human capital stock; such an approach would imply that she would have a claim on a part of your future income stream.

Since, usually, parents don't enter into contractual agreements with their children concerning their future income possibilities (that I know of) I would argue that from the viewpoint of you both, what you are receiving and what she is delivering is a cash transfer.


Subgame perfect equilibrium and expected profit

After obtaining the loan, entrepreneur has two possible strategies to choose from :

Strategy $a$ : Expected payoff from strategy $a$ is $\frac{3}{4} (12000-R)$
Strategy $b$ : Expected payoff from strategy $b$ is $\frac{1}{4} (16000-R)$

He'll choose strategy $a$ if $\frac{3}{4} (12000-R) \geq \frac{1}{4} (16000-R)$, and strategy $b$ if $\frac{3}{4} (12000-R) < \frac{1}{4} (16000-R)$. Or simply, he'll choose strategy $a$ if $R \leq 10000$, and strategy $b$ if $10000 < R \leq 16000$.

Bank will maximize it's expected profits by choosing $R$ taking as given the entrepreneur's strategy. Therefore, Bank's expected profits equal $\frac{3}{4}R - 1000$ if it chooses $R \leq 10000$, and $\frac{1}{4}R - 1000$ if it chooses $10000 < R \leq 16000$. Clearly, expected profits of the bank are maximum at $R^*=10000$. Observe that entrepreneur is willing to pay more for the loan than $R^*=10000$ since his expected payoff is $\frac{3}{4}(12000 -10000) = 1500 > 0$, but the bank  will not charge more than 10000 because that would change the entrepreneur's choice of strategy to $b$ which is a more risky choice and yields lower expected returns to the bank.


Why don't we draw National Income Curve(45 deg. line) simultaneously with graph for Saving Function?

The role of 45 degree line while showing a consumption function is that the 45 degree line translates the values on x-axis to equal values on y-axis. The consumption function is drawn on an income and aggregate expenditure plane. Hence the 45 degree line shows consumption + savings which is aggregate output or aggregate supply or income.

The other way to look at it is when you add up consumption function and saving function on the income- expenditure plane you will get a 45 degree line.

You can show the 45 degree line along with the saving function and the difference between 45 degree line and the saving function will reflect the consumption expenditure.

Another intuition that you will get is that the saving function will never intersect the 45 degree line which is representative of the fact that there is some consumption at the 0 level of income and rate of increase in savings is less than the rate of increase in income which is very obvious.


A thing that bothers me here is the following: the Incentive Compatibility constraint is

$$IC: w'p(a') + w(1-p(a')) - 1 \geq w'p(a) + w(1-p(a))$$

$$\implies w'-w \geq \frac {1}{p(a')-p(a)}  \tag{1}$$

...since by assumption $p(a')-p(a) >0$. We are told that we should find that at the optimum,
$$x'-w' = x-w \implies x'-x = w'-w \tag{2}$$

Combining $(1)$ and $(2)$, if indeed this is the optimum under the given constraints, we must also have

$$x'-x \geq \frac {1}{p(a')-p(a)} \tag{3}$$

But this is an additional, necessary constraint on a priori magnitudes, that must hold if the postulated optimal solution is to be admissible. Even if indeed such a constraint is assumed, in any case, it visibly reduces the generality of the problem (which purports to show something general, i.e. how the risk-neutrality of the agent affects the solution).

Nevertheless, let's work this a bit more formally. I will assume that $w, w'$ can be zero, but not negative. This is a maximization problem in normal form with inequality constraints, non-negative decision variables and non-negative multipliers. The full Lagrangean of the problem therefore is (I will compact notation in an obvious way),

$$\Lambda =  u(x'-w')p' + u(x-w)(1-p') + \lambda\cdot [w'p' + w(1-p') - 1 ]\\ + \mu \cdot [ w'p' + w(1-p') - 1 - w'p - w(1-p)] + \xi w + \xi' w'$$

The essential first order conditions are

$$\frac {\partial \Lambda}{\partial w} \leq 0, \;\;\frac {\partial \Lambda}{\partial w} \cdot w = 0$$

and analogously for $w'$. These result in

$$\frac {\partial \Lambda}{\partial w} = -u'(x-w)(1-p') +\lambda (1-p') - \mu (p'-p) + \xi \leq 0$$

$$\implies u'(x-w)(1-p') \geq \lambda (1-p') - \mu (p'-p) + \xi $$

$$\implies u'(x-w) \geq \lambda - \mu \frac {p'-p}{1-p'} + \frac {\xi}{1-p'} \tag{4}$$

$$\frac {\partial \Lambda}{\partial w'} = -u'(x'-w')p' +\lambda p' + \mu (p'-p) + \xi' \leq 0$$

$$\implies u'(x'-w') \geq \lambda + \mu\frac {p'-p}{1-p'} + \frac {\xi'}{p'} \tag{5}$$

First note that not both wages can be zero, because the constraints would be violated. Given this, consider the possibility that the $IR$ is binding (so $\lambda >0$). If it is binding, then with not both wages zero, the $IC$ constraint will necessarily be violated. So we conclude that

$$\lambda_* = 0$$

and the first-order conditions now become

$$u'(x-w) \geq  - \mu \frac {p'-p}{1-p'} + \frac {\xi}{1-p'} \tag {4a}$$

$$ u'(x'-w') \geq  \mu \frac{p'-p}{1-p'} + \frac {\xi'}{p'} \tag{5a}$$

Now note that if $\xi =0$ (i.e. $w >0$) then $(4a)$ should hold as an equality and with the last term on the right equal to zero. But this would require negative marginal utility which is inadmissible. We also know that not both wages can be zero. So we conclude that we must have

$$\xi_* > 0 , w_*=0,\;\;\; \xi'_* = 0,\; w'_* >0$$

and the conditions now become

$$u'(x) \geq  - \mu \frac {p'-p}{1-p'} + \frac {\xi_*}{1-p'} \tag {4b}$$

$$ u'(x'-w') =  \mu \frac{p'-p}{1-p'}  \tag{5b}$$

Eq. $(5b)$ implies that $\mu_*>0$ , under a usual utility function specification, which does not give zero marginal utility except at infinity. This in turn means that the $IC$ constraint should hold as an equality. Given that $w_*=0$ this gives

$$IC:  w'p'  - 1 - w'p =0 \implies = w'_* = \frac 1{p'-p} \tag{6}$$

This should ring a bell, because the right-hand-side of $(6)$ is the same as the right-hand-sides of $(1)$ and $(3)$.

Namely, if we are assuming a priori that $x'-x = \frac 1{p'-p}$, then the solution we have arrived at validates the claim $x'-w'_* = x-w_*$

Under this additional assumption, we also obtain

$$u'(x) \geq  - \mu_* \frac {p'-p}{1-p'} + \frac {\xi_*}{1-p'} \tag {4c}$$

$$ u'(x) =  \mu_* \frac{p'-p}{1-p'}  \tag{5c}$$

Combining, we obtain

$$ \mu \frac{p'-p}{1-p'} \geq - \mu \frac {p'-p}{1-p'} + \frac {\xi_*}{1-p'}$$

$$\implies \mu_* \geq \frac {\xi_*}{2(p'-p)} \tag{7}$$

This is admissible. So under $x'-x = \frac 1{p'-p}$, we obtain the solution

$$\left \{w'_* = x'-x = 1/(p'-p), w_* =0, \lambda_*=0, \mu_* \geq \frac {\xi_*}{2(p'-p)}, \xi_* >0, \xi'_* =0\right\}$$


This is accountancy and double-entry bookkeeping, but intuitively there are two ideas underlying this:

The net difference between domestic investments and domestic savings has to be financed from abroad (or leads to financing foreigners if domestic savings exceed domestic investments) and this is reflected in the financial or capital accounts of the balance of payments.  Put more simply, if you want to build new buildings or buy new machinery or just put stuff into warehouses to sell next year, and your domestic economy is not saving enough to pay for this, then you need to get the extra stuff (probably represented in money terms, but really it is goods and services) from foreigners, creating new liabilities to them.
The financial or capital accounts should be balanced by the current account of the balance of payments (note the accounting use of the word balance). The net stuff you need for the excess investment over savings needs to bought from abroad (or if savings exceed investment then this needs to represent net stuff sold abroad).

There is a subtle difference between the balance of trade and the current account balance, with the difference representing income flows on existing international investments, but this is often relatively small and in accountancy terms is taken into account in the calculation of domestic savings.


Why do game theorists use a discounted payoff of this form?

In my experience, it's mainly just for cleanliness for results.

Consider an infinite horizon repeated game, with discounted payoff representation (where I use $\delta = (1-\lambda)$ in your notation)
$$
(1-\delta)\sum_{t=0}^{\infty}\delta^t R_t
$$
where $0 < \delta < 1$.

Suppose I play a strategy that gives me the same payoff, say $a$, for each period $t$. Then,
$$
(1-\delta)\sum_{t=0}^{\infty}\delta^t a = (1-\delta)\frac{a}{1-\delta} = a
$$
which is cleaner than
$$
\frac{a}{1-\delta}
$$
As a side note, multiplying the utility by a constant does not change preferences, so we keep the inherent preferences the same.


Printing money to put into a Superannuation or UBI fund

Here is an interesting article on the link between helicopter money (“printing money”) and UBI:
Helicopter Money and Basic Income: Friends or Foes?
First, helicopter money (or funding a SWF) is arguably a more straightforward way than QE (asset purchases) to increase inflation to the target:

Instead of QE, the ECB could start a helicopter money scheme by giving
200 euros per adult citizen for one year – no strings attached, no
taxes involved, simply courtesy of the ECB’s (digital) printing
presses. This would involve about one-third of the money printing of
that under QE and yet would be more likely to fulfil the ECB’s
objective. … The ECB’s temporary scheme would allow some time for EU
policymakers to create the institutional and fiscal infrastructure for
such a eurodividend to be functional.

Instead of paying a UBI directly to citizens, or fostering economic activity by buying government debt like in QE, the central bank could also use newly created money to set up SWF to make consistent UBI payments over time.
The UCL IIPP is proposing the wider use of SWF:
Public wealth funds: Supporting economic recovery and sustainable growth
November 2020

Previous pandemics have, according to some studies, depressed growth
for decades (Jordà, Singh and Taylor 2020). One explanation of this is
that state-backed loans given out during the pandemic build a
corporate debt mountain that stifles many companies’ growth for years
to come. ... Under such conditions, public sector equity investments
or debt-for-equity swaps become an attractive option. By taking stakes
in threatened, strategically important firms, governments can not only
maintain jobs and provide a financial cushion to allow for firms to
recover and invest, but also ensure the taxpayer earns some reward
when the economy does emerge on the other side.

While you would likely want to use public leverage to impact investment towards public policy objectives you would arguably not want to end up with the government actively managing the economy.
The most neutral investment would be for the government to invest in land given that who owns land does not affect supply, which is fixed, or in practical terms governed by public planning laws.


Do the partial derivatives of the compensated demand have an interpretation?

To the question on whether the same classification can be achieved through the hicksian demand function: in general I believe the answer is no. To see this, relate the classifications to the Slutsky decomposition.

(Changes in) Hicksian demand is related to the substitution effect from the Slutsky decomposition. Any standard textbook should provide a reference to this.

There are two broad classifications:

1.Ordinary and Giffen goods: These refer to marshallian demand changes when prices change. This is what we call the Total Effect in the Slutsky
decomposition.

Normal and inferior goods: These refer to marshallian demand changes when income changes. This is by definition the Income Effect in the Slutsky decomposition.

As you see, the substitution effect, which is represented by changes in hicksian demand, is not a part of any of these classifications you mentioned.


What were M.F. Chamans's calculations about how much trade would gain by the burning of Paris?

Auguste Louis Philippe de Saint-Chamans (1777-1860), also known as Vicomte de Saint-Chamans, was a French politician who served as the master of requests to the Council of State in 1820, Councilor of State in 1827, and the deputy for the Marne from 1824 to 1827. He also wrote a number of economic treatises. (The "M.F." is, I believe, incorrect and actually just "M." for Monsieur in the original.)
Unfortunately, Bastiat remembered Saint-Chamans's quote incorrectly -- though his critique of the idea is just as valid.
Saint-Chamans argued that the Great Fire of London (1666) in its destruction led to a net economic gain (due to spending to rebuild) of 1 million pounds (or 25 million French francs).
You can read more about this fallacy here or read Saint-Chamans's work in its original French: M. le vicomte de Saint-Chamans, Traité d’économie politique suivi d’un apercu sur les finances de la France, vol. 1 (1852).


Why is there a structural deficit/surplus?

In terms of rule's logic, they are the same. The logic of structural budget is to provide stability and long-term planning to government spending. This is implemented by first finding the "long-run" revenue levels. Long run here means the economy in its potential. Additionally, for countries which rely a lot on some particular industries like natural resources (e.g. Copper in Chile), an estimation of the "long-run" price of such resources is required. These estimates are usually conducted by an independent, non-partisan, expert-based body, not affiliated with the government.

Once these revenues are defined, then a rule is chosen. It could be a balanced budget (all long-run revenues are spent), which is what you are calling Cyclical Budget, or a deficit/surplus, defined as % of total revenues, which is what you seem to refer to as Structural budget deficit/surplus. Either of these methods provide stability to spending, as the committee evaluating the long-term revenues of the government does this for several years in advance.

Why to chose a structural deficit/surplus? A structural surplus can be chosen to accumulate savings over time, and have a cushion in time of a crisis. In the case of Chile, the aim was to build a cushion in case the long-run price of Copper fell sharply, and long-term government revenues shrank considerably. Others might chose a deficit in order to provide a constant stimulus to the economy. Naturally, the size of the public debt is something to have in consideration regarding which rule to choose.

You can read a bit more about this rule, for the case of Chile, here or here.


Help with Monopolistic Competition Proof, Prove Love for Variety

You have

$$ \begin{align} u(\lambda a + (1 - \lambda) b, \lambda b + (1 - \lambda)a) &= \sqrt{\lambda a + (1 - \lambda)b} + \sqrt{\lambda b + (1 - \lambda)a} \\
&\geq \lambda\sqrt{a} + (1 - \lambda)\sqrt{b} + \lambda\sqrt{b} + (1 - \lambda)\sqrt{a} \\ &=  \sqrt{a} + \sqrt{b} \\ &= u(a, b) \end{align}$$

The inequality in the second line follows from Jensen's inequality for concave functions. I don't think you need to prove Jensen but you can even do that moderately easy with the AM-GM inequality if you want to.


Interpreting the paper you linked:

  a European family of four would see their annual
  disposable income increase by an average of €545
  per year as a result of the agreement. This figure
  takes account not only of increases in wages and
  other household income but also price reductions.

This means that price changes have been taken into consideration and so this income change is already corrected for any inflation. It is possible that a trade deal results in more disposable income, for a basic example see Ricardo's theory of comparative advantages.

I can't say how this figure was arrived at by the paper because no explanation is given.

The following has very little to do with your question, it is more of a reaction to what I perceive to be your anti-TTIP stance. If I am mistaken, apologies.
I find that people who don't like TTIP focus on strange things even though there are some valid critiques. The paper you linked says that the gains are a result of, among other causes, a

  more aligned regulatory systems agreed
  under the TTIP

Some critiques claim that the regulatory system will be closer to US regulation which favours companies over consumers. In terms of patent rights it may also not be optimal, though the effect of patent systems on long term growth is unclear anyway. I find that these are valid worries. Even if TTIP does decrease prices (which I think it will) there may be hidden costs. I don't know if there are, but I think it should be this that is scrutinized about TTIP.


Does treasury pay interest+ face value to the fed?

Yes, Fed sends all its profit including the ones it earns on government debt back to the treasury (this was already answered on this site here). Fed can create an unlimited amount of money by buying treasuries but not because of the interest payments but because of the debt itself.
The interest payments themselves do not generate new money in this relationship when they are paid back to the treasury, that is just juggling money around. Government, has to pay interest on its bonds and through the transfer of the profits from the Fed to the treasury any interest government paid is sent back. So for example, if the government pays ${\\\$}100$ interest payment to Fed, Fed's interest revenue will be ${\\\$}100$ higher. Fed has mostly fixed costs, variable costs associated with processing extra interest payments are small so we can reasonably assume that in this case their net profit also increases by ${\\\$}100$ and hence this money will just make a full circle from the treasury through fed back to the treasury. So the interest itself does not generate new money.
However, when Fed purchases a bond from the government it actually can do that with newly created money. That money will exist until government decides to repay the face value of the bond and Fed could just let that money exist by simply 'scrapping' the bond itself.
Through creation of new money, this way Fed and government in general is able to generate some real revenue - this is called 'seigniorage'. However,  only limited amount of real revenue can be created this way. Empirical studies show that most countries including US have seigniorage revenues around $1-2\%$ of GDP (see Haslag's Dalas Fed brief on this). That is not entirely non-trivial but also not much compared to other revenue sources.
The reason why government cannot just create unlimited ‘profits’ for itself this way is that, in economics, there is a strong consensus among top economists that only a limited amount of real spending can be financed through money creation. This is  because creating more money (aside from special circumstances as when economy is at zero lower bound) is generally inflationary which means that although you created new money they are also worth less - have lower real purchasing power (see textbooks like Blanchard et al. Macroeconomics; Mankiw Macroeconomics etc.). The government can still earn little bit of extra revenue from this, but even if it can in principle create an unlimited amount of money that does not mean it can also purchase an unlimited amount of goods and services.


Measuring asymmetry between countries GDP per capita

To improve the explanation of asymmetries you could add:

You can look at the standard deviation, and see how the GDP of a certain country is compared to the average +/- standard deviation
You can make subgroups of the countries analysed and compare their averages to the target country. One example of subgroup is the country total population: could be the main explanation of asymmetries for Lichtenstein and Luxemburg to have such high GDPs per capita
Compare the specific elements of the GDP, if the data is available (C vs C + G vs G...)

The best measure depends on what specifically do you want to measure.

If you want to measure the asymmetry per se, then you can focus on alternative indicators and descriptive statistics, as you suggested.


Is the relation between expenditure and indirect utility in the Mascolell's diagram wrong?

We can say that $v(p,w)=e(p,v(p,w))$ (or, equivalently, that $e(p,u)=v(p,e(p,u))$ if we are using the "money metric" indirect utility function. You can read about this in section 3.I of MWG. I guess that is what the authors had in mind when they drew figure 3.G.3.

More generally, the relationship between $e$ and $v$ is one of inverses: $e(p,u)$ tells us the mapping $$\text{target utility}->\text{money spent}$$ while $v(p,w)$ tells us the mapping  $$\text{money available}->\text{attainable utility}.$$
So $e$ maps from money to utility and $v$ maps from utility back into money.

In words: if we take a given $w$ and feed it into $v$, we will get an answer to the question "what is the most utility I could ever achieve, $\bar{u}$?" Now, if I take that $\bar{u}$ and feed it into the expenditure function I get an answer to the question "what do I have to spend to achieve utility level $\bar{u}$. That answer will be exactly $w$ (so $e$ 'undoes' the transformation applied by $v$, and vice-versa).

That's just a long-winded way of saying that you can obtain the expenditure function by inverting $v$ in its second argument, or obtain $v$ by inverting $e$. This is made explicit in equation 3.E.1 of MWG.


Are businesses and individuals buying clean / green / renewable energy contribute to slowing down the climate change?

Your intuition is correct, actually switching from dirty good to clean good will also increase demand for clean good so price there might increase as well. However, this does not mean doing some voluntary change could not help as it also depends on how the quantity demanded changes with price (elasticity of demand). So switching is not necessarily unhelpful.

However, at the same time it is also only a partial solution. In order to fully  internalize the externality it’s necessary to apply Piguvian tax to the dirty good at a level that fully internalizes the externality to people consuming the good.


Having trouble solving system of equations through matrix approach

You may find better help in the Math SE, but let's see if we can help here.

So we can arrange your system as such:

\begin{align}
 s_1 - \frac{1}{3} s_2 & = 2 \\
 -\frac{1}{8} s_1 + s_2& = 7 \\
\end{align}

So we form the matrix setup:

$$\begin{bmatrix}
        1 & -1/3  \\
        -1/8 & 1  \\
        \end{bmatrix}
\begin{bmatrix}
        s_1 \\
        s_2 \\
        \end{bmatrix}
=
\begin{bmatrix}
        2 \\
        7 \\
        \end{bmatrix}
$$

You can multiply the matrices out to see that you get the same system.

The first step of solving any system, regardless of method, is to always check for linear dependence of the coefficient matrix.

$$
\det
\begin{bmatrix}
        1 & -1/3  \\
        -1/8 & 1  \\
        \end{bmatrix}
=
(1 \cdot 1) - (-1/3 \cdot -1/8) = \frac{23}{24} \neq 0
$$

So Cramer's rule is indeed useable here. To solve for $s_1$. You replace the column for $s_1$ with the answer matrix and divide the determinant of that by the determinant of the original coefficient matrix. Use a similar process for $s_2$

$$
s_1^* =
\frac
{\det
\begin{bmatrix}
        2 & -1/3  \\
        7 & 1  \\
        \end{bmatrix}}
{\det
\begin{bmatrix}
        1 & -1/3  \\
        -1/8 & 1  \\
        \end{bmatrix}}
= \frac{\frac{13}{3}}{\frac{23}{24}} = \frac{104}{23}
$$

$$
s_2^* =
\frac
{\det
\begin{bmatrix}
        1 & 2  \\
        -1/8 & 7  \\
        \end{bmatrix}}
{\det
\begin{bmatrix}
        1 & -1/3  \\
        -1/8 & 1  \\
        \end{bmatrix}}
= \frac{\frac{29}{4}}{\frac{23}{24}} = \frac{174}{23}
$$

And then we're done. Plug in the answers and make sure they work.


Difference between Income-Expenditure Identity and the Goods Market Equilibrium condition

The output defined with spending approach: $Y\equiv C+I+G+NX$ is an identity because it holds by definition. By definition output must be equal to sum of spending, investment government spending and net exports. However, this is actually not the equilibrium condition.
The equilibrium condition is that goods market demand $Z$ must be equal to output in the equilibrium. Hence the equilibrium condition is that $Z=Y$ (this is really just macro version of saying that demand = supply, since output $Y$ is the supply). This is the equilibrium condition not the identity $Y\equiv C+I+G+NX$. However, given that the equilibrium condition is that $Z=Y$ we can from that deduce the following: $Z=Y \implies Z= C+ I + G + NX$ in equilibrium.
This might be easier to explain with graph. Consider the following graph from Blanchard et al. Macroeconomics a European Perspective pp 47. In the graph below the demand is the purple line while the identity $Y \equiv C+I+G$ (they omit net exports for simplicity) is the $45°$ line that just definitionally says output is equal to output.


Second order partial derivative and cross second-order partial derivative

Is there anyone who can help me with this?

Here it is. Equations 1-3, and 5-6 are obtained in preparation for the 2nd derivatives of V with respect to L and K.

Let me know if you have any questions.


How can foreign protectionism affect local FDI

A protectionist policy in other countries could take the form of providing more incentives for companies to invest domestically instead of abroad. For example, U.S. companies face penalties if they invest abroad compared to investing in the U.S. or they have more benefits from investing in the U.S. now compared to before. Anyhow, companies outside of Turkey could have more inventives to invest in their own countries, which in turn means they have less incentives to invest abroad, which in turn means they have less incentives to invest in Turkey, which in turn means they invest less in Turkey.


I'm studying the effect of a law on a sector: how do I select the control group?

Ideally, but not always feasible, the first option would be to select a similar region where the law is not in place and compare them (e.g. different country / State / city)

If it is not possible to compare with another region, then chose a similar industry. I would try to justify it well though, and comparability would be harder to prove. For example, if you want to study subsidised education, you might want to compare it with subsidized healthcare - the argument could be they're both services, and provide basic needs at some point, with standardised outcomes expected.


Search terms to find a menu of currency exchange mechanisms?

The area of research you are looking for is market microstructure, the study of how markets work and the process of price formation. Naturally, this means studying liquidity, information diffusion, uncertainty, and dynamic games. There is not a lot of mechanism design in market microstructure -- though work along those lines would certainly be welcomed.
For an introduction to the area, a good first read might be Foucault, Pagano, and Röell's book. (Sadly, the review articles which come to mind are old.) The Journal of Financial Markets is chock full of microstructure research and you would do well to start following it.
FX markets have been changing recently with more retail flow and trading on ECNs while the inter-dealer network has become less segregated and more affected by trading on those ECNs. Thus the old Olsen & Associates work like the book by Gençay, Dacorogna, Muller, Pictet, and Olsen (2001) might help explain the interbank market but won't be up-to-date.
More relevant (and shorter) would be to check into Evans and Rime (2019).


Preferences exhibiting local nonsatiation

Since we are working in $\mathbb{R}_+$ you can't go below (0,0). So consider the point (0,0). Are we guaranteed a point around it that is preferred? So using the notation you used, let $(x_1,x_2)=(0,0)$, then any other point $(y_1,y_2)$ where $y_1$ or $y_2$ is positive will be worse than $(0,0)$. So we have a satiation point at $(0,0)$.

Now consider anywhere else. We can also go lower by reducing either coordinate so there is always a better point until we reach both points =0.


Can a risk-averse agent's Certainty Equivalent be lower than the lowest possible outcome of a gamble?

"Since agent is risk averse, we would expect that $U(E[g]) < U(CE)$ ,
  where $CE$ is the certainty equivalent."

This is wrong. I presume the Expected Utility Property holds here, so, if we denote the gamble by $G$, a discrete uniform random variable taking three values according to the setup, we have

$$U(CE) \equiv \sum_{i=1}^3p_iU(g_i) = E[U(G)] < U[E(G)]$$

the inequality to the right due to Jensen's Inequality and the assumption that $U()$ is concave. This also gives us

$$ CE < E(G)$$

which should be intuitive: a risk-neutral person would demand $E(G)$, the expected value of the gamble, in order not to take it. A risk-averse person would require less, to leave the gamble.

Having cleared this, the OP asks: Is it possible that $CE < \min G$?

The answer is : No. Assume that the gamble outcomes are ordered, so $\min G = g_1$.

Ad absurdum, assume that $CE < g_1$ holds. Then we will have

$$U(CE) < U(g_1)$$

Using the definition of $CE$ we replace the left-hand side

$$\sum_{i=1}^3p_iU(g_i) < U(g_1) \implies p_2U(g_2) + p_3U(g_3) < (1-p_1) U(g_1)$$

$$\implies p_2U(g_2) + p_3U(g_3) < p_2U(g_1) + p_3U(g_1)$$

$$\implies p_2[U(g_2)-U(g_1)] + p_3[U(g_3)-U(g_1)] < 0$$

But this is impossible since $g_1 = \min\{g_1,g_2, g_3\}$ and so

$U(g_2)-U(g_1) >0$ and $U(g_3)-U(g_1) >0$.

So assuming $U(CE) < U(g_1)$ led us to an impossible situation, and therefore  it cannot hold.

Intuitively, the worst outcome of being in the gamble is to receive the minimum payoff -so for a rational agent, even if it is risk-averse, it would be irrational to accept less than the worst outcome, since then it would certainly do worse than being in the gamble. Note that "aversion to risk" does not mean "take away all risk at all costs".


According to the most basic theory of economics, the consumer surplus
  is maximized if and only if the price is equal to the marginal
  opportunity cost.

You are mixing agents in using these concepts. Consumer's surplus (or utility) is independent of the producer's marginal cost. Paraphrasing Investopedia's explanation, the consumer surplus or utility is maximized when the purchase of an additional unit equals the marginal benefit the consumer obtains from that additional unit.

  Should publishers of scientific papers such as Elsevier be regulated
  by the government?

No. Doing so could prompt the publisher to reduce the quality of its product or --in an extreme scenario-- close operations altogether. The existence substitute publications which are available for free further weakens the need for governmental intrusion in a private entity's decision-making.


Modern descendants of Frank Ramsey's paper "A Mathematical Theory of Saving"?

If I understand correctly you are not interested just in saving but to mathematical approach to normative questions in general. This approach is actually quite common in the whole literature that uses normative economics.
For example, the question of redistribution is predominantly normative question,  because as opposed to questions of social efficiency, how much our society should redistribute depends on normative judgment. For example, optimal redistribution and marginal tax rates under Rawlsian social welfare function (based on Rawls MaxMin principle and his moral philosophy) will look different than tax system if one adopts utilitarian moral philosophy.
You could have a look at Saez  (2001) or Diamond (1998). Here is an example from Saez which shows how different marginal tax rates look under different normative judgements:

The simulations above are based on:
$$ \frac{T'(z_n)}{1-T'(z_n)} = \left( 1 + \frac{1}{\epsilon_{lT}} \right)\frac{\int (1-b_m)f(z_m)dz_m}{1-F(z_n)} \frac{1-F(z_n)}{z_nf(z_n0)}$$,
with $b_n \equiv \frac{\Psi'(u_n)u_c}{\eta}+ nT'(z_n) \frac{\partial l_n}{\partial \rho} $.
I wont go over every single term in the formula as this would turn this answer into a book but broadly speaking the first part $\left( 1 + \frac{1}{\epsilon_{lT}^*} \right)$ is given by elasticity of labor supply to income taxes and you can think of it as an 'efficiency' parameter, the second part $\frac{\int (1-b_m)f(z_m)dzm}{1-F(z_n)}$ tells us what the marginal benefit of redistribution is and this marginal benefit factors in underlying actual welfare which is captured by $b_n$ which depends on social welfare function,  and finally $\frac{1-F(z_n)}{z_nf(z_n0)}$ s the part that captures the relative magnitude of distortions created by this taxation.
In turn the social welfare function is nothing more than just mathematized version of some moral philosophy, the most common ones I seen in literature are utilitarianism, Rawls MaxMin, libertarianism or sometimes called conservativism - especially in US and some variations of those (although sometimes you will see more exotic ones as well).
Generally, I think it is fair to say that most economic papers that discus normative judgements mathematize these. You should look at various graduate textbooks or notes on public economics/economics of public sector or on economics of inequality or public choice (this is because these are the sub-fields that deal with normative judgments most frequently as opposed to let's say monetary theory).


What is the implicaiton of agent risk neutrality in moral hazard?

It is true that when both principal and agent are risk neutral, the first best can be obtained despite asymmetric information. You should refer to a textbook, such as MWG (ch.14), for the technical details of such models. I'll give an intuitive explanation here. The intuition of the result lies in optimal risk sharing between the principal and the agent.

With a risk neutral principal and risk averse agent, optimal risk sharing requires that the principal bear all the risk and the agent bear none. Hence, first best is generally not attainable in such a setting because, with asymmetric info, the principal cannot condition pay on effort but on outcome instead. As a result, the agent would have to bear some risk in equilibrium. To satisfy the agent's participation (or individual rationality) constraint, the principal needs to compensate the agent for his risk taking, thereby reducing the total gains from the contract.

When both parties are risk neutral, there is no cost for the agent to bear risk. Therefore, optimal risk sharing between the principal and the agent is attained even if pay is based on outcome, not effort. This results in the same outcome as the first best.

Let's view the problem of contracting with moral hazard from a slightly different perspective. Suppose, instead of trying to monitor the agent's effort, the principal simply sells the shop to the agent for a fixed fee, and let the agent retain whatever profit it can make from the business. As the de facto owner, the agent would thus have incentive to exert the optimal level of effort. [This is the quote from Bolton and Dewatripont mentioned in @Hal_Incandenza's answer.] But how much of the fixed fee can the principal charge for transferring ownership?

Well, to respect the participation constraint, the fee can be at most the certainty equivalent of the agent. If the agent is risk averse, his certainty equivalent is less than the expected value of the business, hence the efficiency loss in the second best outcome. If the agent is risk neutral, his certainty equivalent is the same as the expected value of the business; thus there is no efficiency loss in the second best outcome.


Why doesn't national savings, in aggregate, always equal to zero in monetary terms?

I think this boils down to misunderstanding what saving in macroeconomics is. Saving does not cancel each other out. In closed economy, private saving is difference between income (which is by definition equivalent to output so I will be using output and income interchangeably) and consumption $S=Y-C$ and public saving is difference between government spending and taxes $T-G$. In your case there are only private individuals so let us assume there is no government so that $T=G=0$ and consequently national identity will be given just by $Y=C+I$ (also note this shows that investment and saving must be equivalent since $Y=C+I \implies Y-C=I \implies S=I$). See Blanchard et al Macroeconomics a European Perspective for further explanation of those identities.

You you can't just save without having income. The money you saved in previous time periods would be part of your net wealth. For example, if in 2010 nothing is produced and sold both individuals have no income  and thus cannot form any new saving. In this example saving for 2010 would be 0. The money they hold from previous time period would be their net wealth.   However, note in this case there cannot be any consumption either and consumption will be zero. Why? If there is no income and production there cannot be any consumption as well. If something would be produced in 2009 and it was not consumed in 2009 so that it is still avaiable for purchase in 2010 it would be recorded as an inventory investment. So in your example, no matter what those people do $Y, C$ and $I$ are all zero $Y=C=I=0$.

If they actually produce something saving is totally possible. Consider the following trivial  example, suppose the person A produces in 2010 $\\\$75$ worth of widgets and person B produces in 2010 $\\\$50$ worth of food. This means that the national income in 2010 will be $Y=50+75={\\\$}125$. Now let us suppose person A buys all food from person B for $\\\$50$ and person B buys $\\\$50$ of widgets from person A. In this case consumption for B is $C={\\\$}50$ B has zero savings since for $B$ $Y_B-C_B=0$. However, A will have some saving since for A income was $Y_A={\\\$}75$ and consumption was $C_A={\\\$}50$ so there is $S_A=Y_A-C_A={\\\$}75-{\\\$}50={\\\$}25$ which on national accounts will turn up as and inventory investment. So everything balances since $Y=C+I$ holds as we have $Y_A+Y_B ={\\\$}125$

Note it makes no difference whether everything is expressed in purely monetary units or real terms. In above I assumed all those quantities are expressed in nominal, so monetary terms, but I could as well just make them real by deflating those prices by some deflator. The equality would still hold equally well.


Where's a good place to find Information on Non-OCED countries?

The IMF Data, the UN Country Data or the World Bank Open Data should be good data sources covering a wide range of countries.


How to estimate a model by pooled OLS when we DO NOT have panel data?

You don't need panel data for pooled  OLS. You can use pooled cross section data which is similar to panel data but it is different data type. In panel data you follow the same units (e.g. individuals, firms, countries, stocks, etc.) consistently over time, whereas in pooled cross-section data you will have 'a time series of cross-sections' where the same units are not necessarily followed.
If you don't have some sort of data collected on multiple units over time you can't run pooled OLS. However, in description you mention the data contain years and locations so this should not be issue as you should have multiple time periods and units and thus at very least the data structure should be pooled cross-section.
In R you can can use the lm function or plm package to run pooled OLS.
PS: You mention that most of the data are binary - if your dependent variable is binary as well, you should probably consider pooled logit/probit model instead of OLS.


How should I evaluate my startup?(specific question)

First of all, valuation, especially for startups, is a topic without much consensus on the single way to do it. You can take several approaches, such as DCF, multiplayer, prepetuity and EVA (or residual income).

One frequently used way to make a valuation of projects and businesses is to use the Discounted Cash-Flow model, where you estimate the profits and losses for each year, including the financing and the depreciation you would have from the initial investments. This will give you the expected net result of the project, discounted to today (the net present value).

As some startups have difficulties in projecting costs and revenues, alternatives such as the multiplier method are used by some entrepreneurs & investors. I do not consider this method very accurate, as it is easily manipulated by the choice of multiplayer.

Go beyond the numbers

If you're looking for investors to fund your large initial investment, you usually need to provide proof that you can actually sell to your customers. If you could get customers to sign already a future commitment to buy this would help investors trust you and fund you.

You can also try to resort to debt or public grants for starting a business, or try to anticipate some sales revenue to finance your startup. Get legal advice  before signing with an investor, this is critical to ensure you still get a fair share of the deal.


Constructing a Production Possibility Curve (PPC)

$ \textbf{Answer:}  $
Simply put, it does not matter which good goes on which axis as long as the relationship between the two goods is adjusted for correctly.

$\textbf{Explanation:}$

This can be done as long as the relationship (i.e. the PPC curve) is correctly maintained, that is the function is inverted when you change between the x and y axis.

$\textbf{For Example:} $

Suppose you were given the ppf which was linearly defined $good_1$ as a function of $good_2$  by:

$$good_1(good_2)= -a\times good_2 +b$$
$$\implies good_1= -a\times good_2 +b$$

graphically this can be seen as;

By inverting the function, we now make $good_2$ a function of $good_1$ (i.e. $good_2(good_1)$), which simplistically means solving $good_1= -a\times good_2 +b$ for $good_2 $;

$$ good_2(good_1)=-\frac{1}{a}\times good_1 +\frac{b}{a}$$

i.e. From this
$$good_1= -a\times good_2 +b$$
to this
$$\implies good_2=-\frac{1}{a}\times good_1 +\frac{b}{a}$$

and graphically:

We can see that everything is the same, except that our axes are different but the relationship between the two good is exactly the same. The relationship is now inverted to accommodate the change of "perspective" between having $good_1$ or $ good_2$ in the axis, which is graphically seen by the change in the intercepts.

$*$ I apologise if this is not mathematically rigours, and would appreciate any feedback!


$a=0$ means implies the household does not enjoy leisure. In other words, the trade-off between consumption and leisure is switched off. It's not generally true that with $a=0$, utility is higher for a given value of $c_t$, as this depends on the previous value of $l$. Whether or not consumption rises as a consequence depends also on the parameter $b$, i.e. the utility discount factor.
If the household does not value leisure, he might work 80 hours per week and obtain a higher income. If he is very forward-looking however, he might also save a lot for later consumption.


Does the linear probability model require the regressand to be zero/one-valued?

The "LPM" label refers to the structure of the equation, not to the estimator. LPM models can be estimated not only by least-squares methods but also by maximum-likelihood for example.

As regards the nature  of the dependent variable, we are talking about an affine transformation here.  Let a model be with a binary dependent variable and a single regressor for simplicity,

$$y_i = \beta_0 +\beta_1x_i +e_i$$

with $y_i \in \{0,1\}$. We are then told that the original variable was

$$z_i = a+(b-a)y_i$$

so $y_i=0\implies z_i =a$ and  $y_i=1 \implies z_i =b$.

Then the model for $z_i$ is

$$z_i = a+(b-a)[\beta_0 +\beta_1x_i +e_i]$$

or

$$z_i = \gamma_0 +\gamma_1x_i + u_i$$

$$\gamma_0 = a+(b-a)\beta_0,\;\;\; \gamma_1 = (b-a)\beta_1,\;\;\; u_i = (b-a)e_i$$

The structure of the model remains the same, an equation linear in parameters, so it is an LPM model still.

Since $(a,b)$ are presumably known values, after estimating the $z$-model, and obtaining estimates for the gammas, we can recover the estimates for the betas also, if they are of interest.