What's the difference between public debt, government debt and national debt?

With all these types of debt, there are various descriptions rather than standard definitions, and sometimes the same thing can be labelled with different names or the same name used for different things.  In addition, they can be measure gross or net, and there will be questions of how money owed by one part of the public sector to another is treated.

National debt can mean the debt of the national or central government.  This is how it was used for example with the debt the British government built up during the Napoleonic wars, which became the risk-free asset basis of 19th century banking
Government debt can mean the combined debt of different layers of government, so combines national, municipal and other levels
Public debt may include government debt and debt of other parts of the public sector, such as public corporations

None of these explicitly appear in the balance of payments statistics, especially as most public debt is owed to the domestic private sector.

What does appear is in the international investment position, where all debts (public or private) owed to foreign residents or governments appear as liabilities.  Payments of interest to foreigners count as debits in the primary income part of the current account, while borrowing or repayments of principal are flows in the financial account.


How does the companies set up utility function for its own purposes?

Most agents, including firms, do not actively set-up utility functions in their heads. A firm might however might have a worker create a profit function. A course related to labor economics will usually give you an idea of what sorts of things a firm considers.

A generic macroeconomic profit maximization problem that you'll commonly see is

$$\max_{K, L} \ \Pi = A[\delta K^{-\rho} + (1- \delta)L^{-\rho}]^{-1/\rho} - (rK + wL)$$

Where you have a constant elasticity of substitution ($\rho$ is the substitution parameter) with some level of technology ($A$) and weight of distribution ($\delta$) of inputs between capital and labor ($K, L$) which have costs per unit in the form of the rental rate of capital and wages ($r, w$). Maximize profit with respect to capital and labor.

In labor, you might see a very basic profit maximization setup where the worker maximizes

$$\max_{e} \  U = \mathbb{E}(\alpha + \beta (e + v) - C(e))$$

where $e$ is some level of effort, $\alpha + \beta q$ is pay given $q$ output and $q = e + v$, where $v$ is an i.i.d. random variable with an expected value of zero in this case.

The first order condition is

$$\frac{\partial}{\partial e} : \beta - c'(e) = 0$$

which will create the labor supply function.

The constraint is that the worker is individually rational, so $\alpha + \beta e \geq C(e)$, which can be simplified to just equality since the firm will maximize profit by giving just enough pay to elicit the effort they are looking for.

So the firm maximizes

$$\max_{\alpha, \beta} \ E(q) - (\alpha + \beta e) = e - C(e)$$

FOC:

$$\frac{\partial}{\partial \beta} : [1 - c'(e)] \frac{\partial e}{\partial \beta} = 0$$

$$\implies \beta = 1 \ (= c'(e))$$

In the above, we see that the firm makes sure that they get the full pay for their value of effort, but in real life, that may not mean the worker gets a piece rate part of their wage equal to 1; a worker like a car salesman may only make a commission of 8-20% of the value of the car because there are other labor and capital inputs that go into selling the car, so the salesperson may be contributing 8-20% of the effort needed to sell the car.

There are other things the firm needs to consider like screening costs, heterogeneous labor markets (differentiated skill levels), signalling versus human capital or firm specific capital, etc. I'll provide one more economic setup a firm might look at when seeing how their workers invest in human capital.

This setup comes from the Heckman and Killingsworth paper (1986 I want to say) on women in the labor force.

We have a dynamic labor supply with endogenous wages.

$$E(t) = E[H(t), K(t)]$$ where $E(t)$ is earnings at time $t$. $H(t)$ is hours of work for some period of time at time period $t$, and $K(t)$ is the stock of human capital.

$$\frac{\partial K(t)}{\partial t} \equiv \dot{K} = i[I(t), G(t), K(t)] - qK(t)$$

where $q$ is the depreciation of human capital, $I(t), G(t)$ are investment and goods meant to increase human capital (note that human capital is an input into its own production.)

The worker then has some utility to maximize:

$$U = \int_0^D e^{-st} u[c(t), m(t)L(t), K(t)] \ dt$$

where $c(t)$ is consumption at time $t$, $L(t)$ is leisure, and $m$ is a multiplicative factor, an underlying preference for leisure over time that changes involvement in the labor force.

Constraints:

$T = I(t) + H(t) + L(t)$

$W(t) = E(t)/(I(t) + H(t))$

$A(D) = A(0) + \int_0^D e^{-rt} [E(t) - P(t)C(t)] \ dt \geq 0$

(the last condition is just a transversality condition; cannot die in debt)

Heckman neutrality:

$E(t) = k H(t)K(t)$

$\dot{K} = i[I(t)K(t)] - qK(t)$

$U = \int_0^D e^{-st} u[c(t), m(t)L(t)K(t)] \ dt$

(the third condition is different from the above actually)

$u(t)$ will be the shadow value of cash, and $w(t)$ will be the shadow value of human capital.

Solving from the first order conditions will get you a time path for $w(t)$

Other recommended reading: "Personnel Economics in Practice", 2009; Lazear, Edward P. and Gibbs, Michael


GDP/PPP influence - Adjusted savings: carbon dioxide damage

From the definition of adjusted net savings:

  Adjusted net savings are equal to net national savings plus education
  expenditure and minus energy depletion, mineral depletion, net forest
  depletion, and carbon dioxide and particulate emissions damage.

It is a little easier to see on the World Development Indicators Table 4.11

That is:
Adjusted net savings =  Gross savings - Consumption of fixed capital + Education expenditures -Net forest depletion- Energy depletion   - Mineral depletion - Carbon dioxide damage - Local pollution damage

The idea is that ruining the environment is like depreciation in the sense that if you look at savings but ignore depreciation you overstate how much wealth has increased from that savings. Similarly, if you pollute your country that ruined environment makes you poorer than your level of savings suggests.

Update:

Despite the label, you should think of the field "Adjusted savings: carbon dioxide damage" as actually meaning "adjustment to savings from carbon dioxide damage". If you want to calculate national savings rate net of carbon dioxide damage but not net of the other variables, you would calculate it as follows:
=  Gross savings + Education expenditures - Carbon dioxide damage


A possible typo in Herbst and Schorfheide book of DSGE estimation

After asking this same question on the dynare forum, I got the answer that it's really a typo, and I should augment the state vector.

The interesting thing is that afterwards, I rerun the Gensys function of Sims(2002), and I got a solution for an indeterminate model, which is a bit strange.

Augmenting the state vector by a lag from a previously already present variable shouldn't change the model, since I just added the equation y_{t-1}=y_{t-1}to the model. However, the gensys function states that it does.

This makes me wonder if there's something wrong with the code in Sims(2002).

If anyone knows more, please feel free to share. Thanks


Interesting questions. To go through the list of questions you have...

Is there a more efficient mechanism? Will it be better to pay the lowest bidder the amount of the second lowest bid?
According to the revenue equivalence theorem, both types of auctions will result in the same expected revenue to the "cleaner," and will result in the same person doing the cleaning.
Should the amount of funding of each person depend on his/her bid?
This is the same question of choosing first-price or second-price auction. Indeed, any type of auction will depend on the bidders' bids to an extent.
Should the person cleaning also contribute to the funding?
This doesn't make a difference, as the difference would be constant at a factor of $\frac{n}{n-1}$ depending on which "rule" you apply. That is, if you switch the rule to saying that the person cleaning woulnd't have to contribute to the funding, then everyone could just increase their bids by a factor of $\frac{n}{n-1}$.

If your constraint is that someone in your dorm will have to do it, then I'd say this (along with simply keep raising the "price" until someone says, "I'll do it") would be the most efficient. If this is not a constraint, though, an outsider (cleaning company, random guy outside of your dorm) might be willing to do it for less.


MSNE and the probability of encountering each payoff

Denoting your five decks by $d_1, d_2,...,d_5$, you will get a $5 \times 5$ matrix
$$
\begin{array}{|c|c|c|c|c|}
\hline
a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
\hline
a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\
\hline
a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\
\hline
a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\
\hline
a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \\
\hline
\end{array}
$$
where $a_{ij}$ denotes the the expected payoff of deck $i$ against deck $j$. I am guessing this is actually the probability of deck $i$ beating deck $j$ but I am not familiar with Heartstone or your exact intentions. If it is in fact a probability then the payoff of deck $j$ against deck $i$ is $a_{ji} = 1-a_{ij}$.

Which desk yields the highest expected payoff?
 There is no answer this question without some assumptions about your opponent. E.g. in rock-paper-scissors there is no best move unless you specify what your opponent will do. If your opponent plays $d_j$ with probability $p_j$ then your expected payoff for playing $d_i$ is
$$
\sum_{j=1}^5 p_j \cdot a_{ij}.
$$
Whichever $i$ (or $i$'s) maximize this is (are) the best deck(s) to play.

How does one derive the probabilites $p_j$?
 There is no clear answer to this. You can use the concept of Nash-equilibrium to calculate mutual best responses. Or you could rely on experience by calculating the actual frequency of opponents playing $d_j$.


Actually, Section 3 of the paper you linked is probably one of the most comprehensive lists you could find of the movers and the shakers with regard to predictive game theory. I know that many of them have both published and working papers addressing these issues. Fudenberg even addresses those specific two topics. A few selections from the paper:

Ignacio Esponda, Philippe Jehiel, and David K. Levine are leaders in studying adaptive processes in extensive form games, and the sorts of non-Nash equilibrium outcomes that can persist even when players have a lot of experience with the game.
Michel Benaïm, Josef Hofbauer, William Sandholm, and Sylvain Sorin are making important advances in the mathematics of dynamical systems and applying them to non-equilibrium dynamics.
Chaim Fershtman and Ariel Pakes are developing estimation methods for field data that allow for incorrect off-path beliefs.
Jeff Shamma is a pioneer in bringing techniques from the feedback-control literature to the study of learning in games.

I would recommend going through these authors' websites and looking for papers that fit.


Interest rate and yield curve in a mild recession

Because interest rates have been at abnormally low levels for quite some time and the Federal Reserve bank is required to pay stockholders' dividends at 6 percent, the Federal Reserve must either issue or collect more money.

Both of these choices negatively impact the value of the US dollar as they both hurt the American economy one way or another.

A mild recession followed by a sluggish recovery may well lead to a politically initiated response that pressures the Federal Reserve to increase the money supply and inflation.

The reason the Federal Reserve concludes that interest rates rapidly increase is because they need the borrowers of this newly created money to pay back their loans sooner (to get their stockholders' 6 percent dividends).

Source:
http://www.federalreserve.gov/aboutthefed/section7.htm


SA = seasonally adjusted

WDA = working day adjusted

NSA = not seasonally adjusted


How did Pareto came up with Pareto Distribution of wealth law?

The Pareto distribution was derived based on the observation of Vilfredo Pareto that in Italy $80\%$ share of land belonged to $20\%$ of the country’s population, and he showed this held for many other countries as well (see Pareto Manual of political economy - the book is still being reprinted).
Consequently, based on this empirical observation it was argued that number of people $N$ with incomes higher than $x$ can be modeled as:
$$N = A x^{-\alpha}$$
and the Pareto distribution is based on the above.
In fact Pareto distribution is just another case of more general power law like Zipf's law (see Newman 2005). Power laws are quite common in nature for various reasons (see this article) - for example even human language follows Zipf's law (vsauce has nice video about this).
It is definitely not just due to chance wealth distributions can be approximated by pareto distribution as it is unlikely that this would be observed again and again in various countries across time if this would be just due to random chance. There are also economic models that imply that wealth distribution should follow pareto distribution. For example,  Wold & Whittle (1957) show that if you try to  model  the ratio of growth rate of wealth and mortality rate (which is important as wealth is distributed through inheritances) as a system of differential equations you can show that wealth should follow the Pareto distribution. However, this being said Pareto distribution pops out in many economic relationships (it even matters for income distribution which is log-normal with pareto tail), and I don't know of any generally accepted explanation for why this is (aside for the potential solutions offered in the sources mentioned here which still claim that this is not fully resolved puzzle).
As to whether there is any criticism of Pareto distribution in economics I am not aware of any substantial criticism. Pareto distribution is an empirical distribution so even in theoretical modeling it is mainly applied in situations where we can actually empirically verify that such distribution is present.
For example, optimal income and wealth taxation theory would work with any income/wealth distribution you can imagine. Pareto distribution was used for wealth for some time but until recently when it comes to income people used mainly log-normal distribution (e.g. Mirrlees; 1971), but recently people realized based on empirical observations that we should actually use log-normal distribution with Pareto tail for the upper incomes (e.g. Atkinson, A. B., Piketty, T., & Saez, E. (2011)). Hence if anything when it comes to research on income inequality the criticism is that it was not used enough. This being said I suppose if you would dig deep enough in  literature you could probably find examples when it was assumed/used even if it should not be. However, this is not really criticism of the Pareto distribution itself but rather its (mis)application in some specific instances.


reference request - dynamic discrete time optimization methods

As suggested by 1muflon1:
I don't really use dynamic discrete optimization (yet) in my work, but I did explore several books once. Here are some suggestions:

Economic Dynamics: Theory and Computation by John Stachurski: It's very easy to fall in love with this guy. He writes casually yet authoritatively. The book covers required real analysis and includes python codes and exercises.

If you like the above book do explore https://quantecon.org/lectures/

Recursive Methods in Economic Dynamics by Stokey, Lucas and Prescott


Why, in general, do product or service prices rise in time?

Prices increase over time generally because of inflation. Money does not have constant value over time $\\\$1$ in 2010 is not equal to $\\\$1$ in 2020. As you mentioned in your question you can observe that not only prices of products and services increase but also wages. This is precisely because money is becoming less valuable so the same value embedded in work as well as in products and services require more money.
There are various reasons why value of money might decrease over time. Following Blanchard et al. Macroeconomics an European Perspective (a standard undergraduate macro textbook) the money market equilibrium (where the value of money is determined) can be expressed with the equation of exchange as:
$$MV=PY$$
where $M$ is the money supply, $V$ velocity of money, $P$  the overal level of prices  in the economy and $Y$ real output. This model is a bit of an oversimplification in more complex versions expectations of these variables are important as well but for this question it will do. We can solve the model for prices to get:
$$P= MV/Y$$
Hence prices can change either if:

$Y$ drops. This is because if the number of goods and services in the economy decreases but the amount of money and velocity remains fixed the same amount of money will chase lower amount of goods and services which will bid up all prices across an economy. However, it is worth noting that this is not the cause of inflation that we see over long period of time as most economies are on net basis growing (i.e. $Y$ is increasing) with an occasional recession which interrupts the growth.
$M$ increases. The mechanism is similar as with drop in $Y$ but now the amount of money increases in the economy and the number of goods and velocity remains constant. In modern economies $M$ is controlled by government institutions most often by central banks which can contract or expand $M$ directly or indirectly in various ways. Over long periods of time most economists believe that inflation is caused by increase in the money supply so this is the major reason why you will see the inflation going up over long periods of time.
If $V$ increases. Velocity of money is the average number of times a bill is used to purchase goods and services. It tells us how fast are money circulating in the economy. Generally the more faster the money circulate in the economy the higher will be the inflation rate. $V$ changes over business cycle, generally dropping in recessions and increasing in expansion but over long periods of time it remains more or less constant.
Some combination of all of the above.

Why can't they stay stable?

They can be stable. This is ultimately a policy choice, there were historical periods of near zero inflation and in principle central banks/governments, which control $M$ around the world could achieve long term price stability if they wanted (even though they might not be able to do it precisely every single year due to random economic shocks they can do that on average over business cycle).
However, as pointed in the link Brian provided in his +1 comment most economists believe having certain level of inflation is desirable. This was already answered excellently by FooBar so I wont be repeating it here.

Is this inherent in the western economical model?

First of all what is 'western economical model'? Currently almost all countries around the world have a mixed economy - a system that combines free market with central planning. For example USA, South Korea, Japan, UK, Denmark, South African Republic, Russia or China are all mixed economies. Countries differ in the quality of their institutions in policies they follow and in many aspects but looking at the big picture most countries around the world follow the same economic system. Few exceptions would include countries such as North Korea.
Second, inflation is actually  much lower in 'the west' (i.e. Europe and North America and other countries that are colloquially known as 'western' such as Australia or New Zeland) then elsewhere as the data from IMF for year 2019 below show. Feel free to look at any recent year in the IMF dataset and 'western' countries will be always among those with lowest inflation rate.
This is however not due to following some fundamentally different economic system but rather due to different institutions. Most western central banks are independent from government with a mandate that either completely or partially requires them to maintain low inflation. Also just to be clear I am guessing that by 'west' you are not meaning just a western hemisphere which includes many African or South American Countries, but rather you mean nations with roots in the Western Europe (i.e. USA, UK, France, Canada, Australia etc.).
    Region                       Average Inflation Rate in % 2019
    South America                      no data
    Western Europe                      1.3
    Australia and New Zealand           1.6
    Europe                              2
    North America                       2
    Central America                     2.2
    East Asia                           2.3
    Southeast Asia                      2.3
    Caribbean                           2.8
    Pacific Islands                     3.2
    Asia and Pacific                    3.3
    Western Hemisphere (Region)         3.4
    Eastern Europe                      3.9
    South Asia                          4.8
    Middle East (Region)                7.9
    North Africa                        8.2
    Africa (Region)                     9.2
    Sub-Saharan Africa (Region)         9.7
    Central Asia and the Caucasus      11.9


Why does the bond market induce efficient allocation of capital?

This is no different from any other markets... just for money.

On the demand side of the goods, those who want money (those who sell bonds) say how much they're willing to pay in terms of bond yields. The supplier of the goods (bond buyers) come to the bond market and choose the product.

Under a developed bond market, it's guaranteed that those who get the goods (money) are the bond sellers who could pay the highest [risk-adjusted] price, which would infer that they could use that goods most productively. It's also guaranteed that those who sell the goods (bond buyers) are the ones who require the lowest price, i.e. opportunity cost is the lowest.


Take $y_1=y_2=1$. As the figure below shows, the lower left segment of the boundary of the upper contour set of the lexicographic preference is open.


Government bonds - the most simple example

Using an example from the US treasury's website you can buy a \$100 dollar face value bond that pays 3% annual interest semiannually for thirty years. That means you would get $100 \cdot 3\% / 2 = 1.5$ dollars every six months, and at the end of the thirty years you also get \$100. In total you would get $1.5 \cdot 2 \cdot 30 + 100 = 190$ dollars but as it is explained in the link in Jamzy's comment adding up dollars in this case is not very sensible, because dollars in different years have different value.  (If you disagree I would love to trade with you. For 1000 dollars today I will happily give you 1000 dollars 10 years from now.)
 The price you pay for this bond is determined by auction. Using this you can calculate the yield of the bond. (Again, please see the explanation in the link.)
 The answer would follow the same logic for €100,000. You can convert the euros to dollars or find a suitable European bond.


Finding an economic policy that worked in a developed country but failed in a developing one

In his book The Elusive Quest for Growth: Economists' Adventures and Misadventures in the Tropics, William Easterly discusses the failed efforts of states to use education, family planning, and debt forgiveness as means to grow out of poverty. You may find one policy that worked in developed countries.


General Equilibrium Involving Production

You seem to be looking for the Second Welfare Theorem.

  The Second Theorem states that out of all possible Pareto optimal outcomes one can achieve any particular one by enacting a lump-sum wealth redistribution and then letting the market take over.


What would be the income tax rate be if it were 'flattened'?

Having worked on these issues myself for quite some time, I am not aware of any study on this for New Zealand. For a back-of-the-envelope calculation of the revenue-neutral flat tax rate, you'd need the following:

The tax base $Y$, i.e. the amount of income subject to income tax. Most probably, the New Zealand Treasury has published statistics on this. Note that total taxable income is different to total earnings which are reported in national accounts.
The number of taxpayers $N$.
You might want to maintain the current exemption $E$ which remains untaxed for each individual.

With current income tax revenue $R$, the revenue neutral flat tax rate $\tau$ can be obtained from:

$ R = \tau * (Y-N*E)$

Note that is the minimal thing to do which abstracts, among other things, from behavioral effects of such a reform.


Why would a government allow currency speculation?

Speculators very rarely have large positions that come anywhere near what would count as a monopoly in other sectors. When Soros bet on the pound devaluing he only had a couple of billion dollar in his position.
The distinction between a currency trader and a currency speculator can be hard to make in practice. The difference between market making and market speculation is one reason why the Volker rule has been difficult to write and implement.
There is tremendous social value to informative prices. When prices are right, consumer and producers can rely on them to make better plans of economic activity. When prices are wrong (not reflecting the best information and most informed beliefs) those same agents are mislead by prices, leading to malinvestment and sub-optimal savings and consumption decisions.
It can be difficult to ban such speculative activity because it moves off-shore. In general, a country could ban speculation locally but not in other countries. The UK has little motive to help France ban speculation on the Euro in London. The only way to really do this effectively is with currency controls that make it difficult to bring currency out of the country. But currency controls can bring additional complications into currency markets, often a serious form of financial repression and results in exchange rates even less informative.

Faced with these issues, most well run countries, most of the time,  would take functioning currency markets with the warts and wrinkles of speculators, volatility, and the like, over the downsides of trying to control the price of their own currency.


This is an interesting question!

It would seem that the game is not fair: Player A is at a pretty serious disadvantage.

Let's imagine each player has their own probability distribution of beliefs that is a sampling distribution of some population distribution centered at the true value. Both players are aware of this, and know their own distribution but have no further information on the other's belief distribution.

The optimal strategies are for player A to guess at the center of their own distribution and for B to guess "More" if the center of B's distribution is above A's guess and "Less" if it's below.

Player A will win if their guess is between the true value and Player B's guess.

-Since there is a 50/50 shot for either guess to be above or below the true value, 50% of the time the guesses will lie on either side of the true value and player B will win, regardless of whose guess is closer.

Via symmetry:

-25% of the time their guesses will be on the same side of the true value and player B's will be closer. Player B will also win here (deservedly).

-25% of the time their guesses will be on the same side of the true value and player A's will be closer. Here player A will win (deservedly).

Hence, player A will win only 1/4 of the time.

However the game also becomes increasingly complicated once you allow each player to have first order beliefs about the other player's beliefs (ie. specific information about the shape and center of the other's belief distribution), because then there's the opportunity for say, A to exploit their belief that B's guess is below the true value by guessing slightly lower than they would otherwise. And with second-order beliefs things get even more complex, and so on and so on. I'm not sure whether player A's disadvantage is robust to these nuances, but it's still quite striking to me that a game that at first glance seems fair is actually under normal conditions pretty lopsided!


What is the "bequest condition" in a finite-horizon discrete optimization problem?

It is simply the terminal condition of the model. You do not specify the variable that x represents, but if it were, for example, capital, this would simply state that in the last period (since horizon is finite) there is a condition that states that capital must be greater or equal to some given amount (bequest --> legacy, you might find some sense in that word).
Again, you do not specify the model, but normally these types of conditions relate to either a simplifying assumption that allows you to solve models using certain methods (for example undetermined coefficient, iterative solution methods, etc.), or to some logical notion.


Why do we add Net Foreign Factor Income instead of substracting it while calculating GDP?

Seems like I got incorrect information about how to calculate NFFI.

I was told by a worker of the BEA that NFFI would be calculated as NFFI=FH-CA. In this case everything is okay.


What obstacles/economic barriers do stock exchanges face in developing countries?

Stock Exchanges are made of the investments, and serve as meeting point for company's ownership to be traded fast and legally. It's need and value increases with the the number of investors and companies in the market.

Several developing countries lack legislation and law enforcement that would incentivize the use of a stock exchange. On the other hand, in developing countries the ownership of the largest companies is typically in the hands of a few powerful elites or foreign investors, and neither benefit from using a stock exchange, as they are usually not looking to exchange their investments.


Why can two functions represent the same preferences?

Suppose I strictly prefer Apple ($A$) to Banana ($B$) and that these are the only two goods in existence.

Consider the following functions:†

The function $u$ with $u(A)=1$, $u(B)=0$.
The function $v$ with $v(A)=-99$, $v(B)=-100$.
The function $w$ with $w(A)=0.1$, $w(B)=0.01$.

Each of the above functions correctly represents my preference because in each case, the function's value at $A$ exceeds that at $B$.

A utility function (representation) is merely a (very) convenient way to numerically encode my preference. Here there are only two goods and I prefer $A$ to $B$. So, any function whose value at $A$ is greater than that at $B$ would correctly represent my preference.

In this example, we have only two goods, so a utility function isn't particularly useful or convenient.

But with more goods (and sometimes also more assumptions), utility functions can help to produce useful results that tell us something about the real world.

†Real-valued, with domain $\{A,B\}$.


Decomposition of an additive functional into a Martingale part and other

Using the given formulas (inserting the expression for the $X$'s in the sum), we arrive at

$$Y_t = (1-\beta_0)\alpha_0\cdot t + \beta_0\sum_{j=0}^{t-1}X_j +\sum_{j=1}^{t}W_j $$

We do not need a theorem to obtain that, given the description of the problem.

Manipulating,

$$Y_t= (1-\beta_0)\alpha_0\cdot t  - \beta_0(X_t-X_0) + \beta_0Y_{t}  +\sum_{j=1}^{t}W_j$$

$$\implies Y_t = \alpha_0t +\frac{1}{1-\beta_0}\sum_{j=1}^{t}W_j-\frac{\beta_0}{1-\beta_0}(X_t-X_0)$$

and we want to match this with

$$Y_t = r_1 t + M_t - r_2 (X_t - X_0)$$

We immediately get that we must have  $r_1 =  \alpha_0$, $r_2 = \beta_0/(1-\beta_0)$, and
$$M_t = \frac{1}{1-\beta_0}\sum_{j=1}^{t}W_j$$

Is $M_t$ as obtained a martingale? We have

$$E[M_t\mid \sigma(t-1,t-2,..)] = \\=\frac{1}{1-\beta_0}E[W_t\mid \sigma(t-1,t-2,..)] + \frac{1}{1-\beta_0}\sum_{j=1}^{t-1}W_j$$

$$= \frac{1}{1-\beta_0}E[W_t\mid \sigma(t-1,t-2,..)] + M_{t-1}$$

So if $E[W_t\mid \sigma(t-1,t-2,..)] = 0$, namely, if the original disturbance is mean-independent from the past (in a more fancy term, if $\{W_t\}$ is a "Martingale difference"), then $M_t$ is a martingale and we have our mapping.


Supplier offers "Price Protection Tariff" w/o exit fees, and more expensive Variable Price tariff. Does that mean it is betting on the price falling?

It is possible that the supplier is trying to segment the market:

For those customers who do not frequently review the pricing options, it has a "standard variable tariff" which is not the lowest price on the market
For those customers of theirs or of other suppliers who do review the options, it has a lower price option which it sets competitively, though it hopes that some of these will not review at the end of the 12 month period and will then be automatically switched to the higher price "standard variable tariff"

Whether you regard this as promoting competition or as an abuse in competitive markets is a complicated political and regulatory question


Suppose I define "net appliances" as dishwashers plus appliances lent to others minus appliances borrowed from others.

Suppose I build a refrigerator and lend it to you.  Then your "net appliances" have been reduced by one, my "net appliances" have increased by one, and the total of the world's "net appliances" has not changed.

On the other hand, the world does now have one more refrigerator, which suggests that "net appliances" was never a very interesting notion in the first place.


GDP per Capita in Panel Data: Must the base year be the same for all cross-sections?

Base rates account for the change in value of a currency that happens overtime because of inflation.
Not only do the base rates need to be the same, but if you are including all of these in the same panel, then the currencies need to be in the same unit as well, otherwise, the different country GDP's are not comparable.
Fortunately, most of the big international databases with GDP time series usually have multiple versions, with different base years, and converted to common currencies (usually USD) that you can use to assemble what you need.  Just a few places to start you off:
FRED
World Bank WDI
OECD Data
Note: for GDP data, these sites often present data from the same sources


What are the (immediate) effects of changing a good's price on consumer and producer surplus?

What is CS and PS out-of-equilibrium at $P=30$:
I think that answer to this question lies in misunderstanding the definition's of consumer and producer surplus.
The consumer and producer surplus are actually (following Mankiw Principles of Economics, 8th  ed, pp 135 and 140

consumer surplus:
the amount a buyer is
willing to pay for a good
minus the amount the
buyer actually pays for it

producer surplus
the amount a seller is
paid for a good minus the
seller’s cost of providing it [where Mankiw later in chapter explicitly states cost is here to be interpreted broadly as "the value of everything
a seller must give up to
produce a good", including the opportunity cost (or more broadly s wilingness to sell)].

You will note that the definitions above talk about price people pay (i.e. actual market price) not 'equilibrium price'. Some lecture notes or textbooks will include the word equilibrium in the definitions but that is only because virtually all analysis using Supply-Demand is an analysis of some equilibrium.
More rigorously, since the maximum willingness to buy is bounded by the demand curve and willingness to sell by supply curve we can define consumer surplus as the area below the demand curve and above price paid by consumers and
$$CS =  \int^{q_m}_0 D(q) dq − p_mq_m $$
where $D(q)$ is arbitrary demand curve, $p_m$ is price consumers have to pay in the market, and $q_m$ quantity sold at the market.
$$PS =  p_mq_m - \int^{q_m}_0 S(q) dq $$
wher $S(q)$ is some arbitrary supply curve (also if we would want to be even more general we could allow $p$ to be different for suppliers and consumers e.g. situations with taxation where there would be wedge between price consumers pay and price per quantity suppliers get due to the tax wedge).
Yes, in some chapters or lecture notes the above formula will be defined in terms of equilibrium price but that is only because the market price/quantity is the equilibrium price/quantity in any equilibrium analysis (i.e. in equilibrium $p_e = p_m, q_m=q_e$).
However, in an out-of-equilibrium analysis you have to use actual market prices (this would be equivalent of analyzing price floor or price celling (e.g. see Mankiw Principles of Microeconomics, 8th  ed, pp. 116).
Consequently, in your case the consumer and producer surplus can be properly visualized as shown on the tikz graph below I made:

where the area of consumer surplus (out-of-equilibrium) at price $P_m=30$ is the blue shaded region below demand and above market price bounded by market quantity, and the producer surplus (out-of-equilibrium) at price $P_m=30$ is the red trapezoid (area under the market price and above supply bounded by market quantity).
We can also calculate the area of that trapezoid, although to do that we need to know what is the price at which $Q_S=0$ (or explicitly know what the supply and demand equations are). In your question you don't state what the  price at which supply is equals zero is but if you would provide it we could calculate it (also you seem to have made some mistake when copying down the problem as your demand does not lie on a line since it is given by points $(0,40);(15,30);(20,20)$ - which simply mathematically cannot lie on a single line - I originally planned to go all the way deriving the linear demand and supply equations based on the points but this mistake in problem set up makes it impossible ).
Why is total welfare maximized in equilibrium?
Well the reason for this is that under all the background assumptions of the stylized partial equilibrium model (e.g. no market failures and other 'shenanigans' that would affect PS and CS in way not shown in the partial equilibrium analysis here), the total combined area of PS and CS is maximized at the equilibrium price and quantity.
I won't be providing a rigorous proof of this as that would require a lot of calculus and this problem is long as it is, but I encourage you to take a look at proofs provided in MWG, Microeconomic analysis pp 328 or Varian, Microeconomic Analysis pp 224 if interested.
However, the reason why the welfare  be easily seen graphically by the following thought experiment: would the total surplus be larger if $P>P_e$ or if $P<P_e$? The answer is no as you can see visualized in the two figures below. As in both figures below the total surplus area will always fit within the big triangle given by area below demand and above supply clearly the total welfare is in both situations lower than the total welfare in a situation where $P_m=P_e$.

PS: An important caveat to note is that the above applies when you want to analyze what would be the total welfare outside equilibrium vs the welfare in equilibrium. If you want to compare two situations which are both equilibrium e.g. supply shifts to the right due to technological progress, you have to compare the total welfare in the market in the old equilibrium vs the new one. You should not confuse such situations with market not being in equilibrium.


First, You may model the sale or demand as a function of price. You can use a regression model to control for controls. This regression will provide you with price elasticity of demand which show how your demand is sensitive to a change in price.
Then, you may use the above relationship to plug in for sale in the optimization step. This way you only solve for the optimum price.


As the answer says, the management fee is directly taken from the fund’s assets. The only way that anyone “pays” the fees is via lower returns. There is no cash flow to pay the fee.


Why is common stock included in bank capital?

Equity/stock/capital does absorb losses. Here is a stylized example:

Ann creates a new bank. She puts in \$100 of her own money. This \$100 is her equity/stock/capital.

She borrows another \$900 (debt). (For simplicity we'll assume that all debt/loans have an interest rate of 0%.)

She has a total of \$1,000. She lends all of it to Bob.

A little later, Bob is able to repay only \$925. So, \$75 was lost on this loan.

This \$75 loss is entirely absorbed by the equity/stock/capital, which now falls from \$100 to \$25.


You’re looking for gross output; GDP is final output.

Per the BEA:

  Economy-wide, real gross output—principally a measure of an industry's sales or receipts, which includes sales to final users in the economy (GDP) and sales to other industries (intermediate inputs)

For calculation purposes, it’s generally assumed that a firm purchasing a good (for example, a computer) is using it as an intermediate input, while a household purchasing the same good is using it as consumption (so it’s a “final sale” counted in GDP), though corrections are applied in particular cases where this assumption doesn’t hold well. Cases where households act as producers, like farms and real estate, are prime areas for these corrections.


From the Chicago Federal Reserve:

  Following a minimum wage hike, household income rises on average by about \$250 per quarter and spending by roughly \$700 per quarter for households with minimum wage workers. Most of the spending response is caused by a small number of households who purchase vehicles

http://www.chicagofed.org/digital_assets/publications/working_papers/2007/wp2007_23.pdf

Aaronson, D., S. Agarwal, and E. French  (2011): "The Spending and Debt Responses to Minimum Wage Increases" Federal Reserve Bank of Chicago Working Paper, WP 2007-23.


If I understand you correctly, all customers prefer economy class to business class (that is, they only care about the lower price).

First, the way you've specified the problem, the airline will always make every seat a business class seat, because if no economy seats are available, the airline will (a) sell the same number of tickets (since customers buy business class if economy isn't available) and (b) sell all the tickets at a higher price.

But maybe you meant something like, fewer people will buy business class seats because the price is higher, so the airline may wish to make some seats economy class so they fill every seat. That is, the airline may initially wish to offer tickets for business class, wait until they stop getting new orders, then lower the price to economy class in order to fill the rest of the seats. Depending on the underlying model of demand and purchasing you may get different results.

I'll highlight one, the 'Coase conjecture', which argues that the sort of pricing scheme I described above will break down because customers rationally anticipate that the airline will lower prices as soon as they stop getting new orders, so they wait for the price to drop.


General Equilibrium with Perfect Substitutes

To solve for competitive equilibrium, we can first find the demand :
Demand for commodity $X$ by A is $x_A = \frac{5}{p_x}$ if $p_x < 1$, $x_A \in [0,5]$ if $p_x = 1$, $x_A = 0$ otherwise.
Demand for commodity $X$ by B is $x_B = \frac{(30p_x+5)}{2p_x}$.
Now we can equate demand and supply and solve for $p_x$.
$x_A + x_B = 30$ yields $p_x = \frac{1}{2}$.


Why do commercial banks buy government bonds at negative interest rates instead of keeping cash?

For a bank , the basic choice of assets is between (a) loans to customers (b) invest in financial asset such as government bond (c) deposits with the central bank.  Note that (c) may return a negative interest rate, so (b) may be better than (c).  Holding a large amount of physical cash in the vault is not practical for a bank of any decent size, due to the cost of storage and insurance.  It would also be discouraged by the bank regulators , because it would frustrate the negative interest rate policy the central bank is trying to apply.


Does the concept of Nash-equilibrium conflict with the concept of market equilibrium in the lemon market

So this seems to be a known issue. Quoting from the Wilson article of 1980,
The Nature of Equilibrium in Markets with Adverse Selection:

  Using a variant of Akerlof's model of the used car market, we examine the equlibrium of the model under three distinct conventions: (1) an auctioneer sets the price; (2) buyers set the price; (3) sellers set the price. Only in the case of the auctioneer is the equilibrium necessarily characterized by a single price which equates supply and demand.

Thus the answer to the question is a partial yes. Given some (most) variants of the lemon market the two equilibrium concepts do not result in identical outcomes.


Intermediate Case of Bertrand and Cournot

These papers could be interesting to you. First, a classical contribution:
Singh Nirvikar and Xavier Vives, 1984. "Price and Quantity Competition in a Differentiated Duopoly," RAND Journal of Economics, vol. 15(4), pages 546-554.
And these two interesting papers, using the concept of competition toughness to reconcile Cournot and Bertrand:
d'Aspremont, Claude & Dos Santos Ferreira, Rodolphe, 2009. "Price-quantity competition with varying toughness," Games and Economic Behavior, 65, 62-82.
d'Aspremont Claude and Rodolphe Dos Santos Ferreira & Louis-André Gérard-Varet, 2007. "Competition For Market Share Or For Market Size: Oligopolistic Equilibria With Varying Competitive Toughness," International Economic Review, 48, 761-784.


How about this: For each vector $(x,y)$, there is a unique $z\in \mathbb R$ such that $(x,y) \sim (z,z)$. WLOG assume $x \geq y$. Then to see this claim, first notice by A2 that $(x,x) \succeq (x,y) \succeq (y,x)$. Then traveling along the $45^\circ$ from $(y,y)$ to $(x,x)$, A3 ensures the existence of our $z$. (Strict) Monotonicity assures uniqueness in the obvious way. Let $u: (x,y) \mapsto z$ where $z$ is defined in this way.

Now let $(x,y) \sim (z,z)$ and $(x',y') \sim (z',z')$. Then by A1 we have
\begin{align}
(x + x',y+y') &\sim (z + x',z + y') \\
(x' + z,y' +z) &\sim (z'+z,z'+z)
\end{align}
so by transitivity, $(x+x',y+y') \sim (z+z',z+z')$. Verifying that $u$ is continuous is trivial.

So $u$ is additive and continuous. As this Math SE post explains if $\mathbb R^2 \to \mathbb{R}$ is additive and continuous then it is linear.


Is the implicit function theorem needed to relate the EMP to UMP?

Fix prices at some arbitrary positive values $p_x^0,p_y^0$ and view the functions $v$ and $e$ as $v:\mathbb{R}\rightarrow \mathbb{R}$ and $e:\mathbb{R}\rightarrow \mathbb{R}$. Then they are inverses.


Implications of abolishing Fractional Reserve Banking on mortgages and interest rates

There is a long standing problem in most discussions of Fractional Reserve Banking (FRB), around the precise definition of money. Cash money (that is M0) is an asset on the banking system balance sheet, while deposit money (the liability) is the money that is created by lending.

Ever since the introduction of cheques specifically, and in general the ability to make direct transfers between two deposit accounts, deposit money has been the de facto form of money that is used in the economy. Cash money is maintained at a fractional reserve of deposits (it used to be a regulator of total deposits and lending, but in practice this isn´t the case anymore), and used primarily for inter-bank liquidity settlements, as transfers occur between bank accounts at different banks. Reserve requirements are now down to 2% or less in most countries.

So taken literally, the question is what happens to an economy when its money supply is reduced by 98%, and at the same time total lending shrinks by whatever proportion bank lending is of total lending in the economy (typically 1/3 or more).

In the short run, congratulations, you no longer have an economy. Nobody can pay their non-bank debts any more, because there isn´t enough money to pay them with. Bear in mind debt payments do not automatically adjust to changes in the money supply. That triggers a debt cascade causing wide-spread company failure. The accompanying impact on all prices causes massive disruption in the supply chain, because everything in the economy is now completely mis-priced relative to everything else. A 20% reduction in the liability money supply in the US in 1930-33 for example caused the price of food bought from farmers in some parts of the USA to be less than the cost of the fuel required to get the food to the market for sale. This event would clearly be far worse than that. Market based systems can and will adjust prices over time to compensate for money supply changes, but that´s the long term. It has been said that 'every civilisation is only 2 meals away from barbarism', and I´m afraid you just pushed yours over the edge.

In the long term, after a period of market exchanges being performed with cigarettes, and other money-substitutes, and those responsible for this mess being hunted down and dealt with appropriately, people will probably feel that fractional reserve banking wasn´t such a bad idea after all.  Some form of Germany Year 0 event will take place and the banking system will be restarted. Since all debt was effectively discharged during the "unfortunate event" the next few decades will probably be referred to as a golden age. By the survivors.

If instead we assume that rather than reducing the money supply (which we know is a bad idea), the banking system is instead converted so that there is no fractional reserve lending, asset cash is injected to replace debt, and maintain the current quantity of deposits. All deposits in the system are simply held constant from this point on, and you now have a constant money supply economy.

As a side note, everybody now has to pay for their bank accounts, because since banks are no longer lending institutions, they need a revenue source to fund their operations. Quite a large revenue source - ATM networks aren´t cheap.

The problems caused by a sudden drop in total lending are still present. In practice this will disrupt the economy in the short term, but hopefully not quite as badly as in the previous scenario.

The short answer though, is we simply don´t know how a constant money economy behaves in practice. There hasn´t been a constant money economy since fractional reserve banking was invented, and that´s at least 3-400 years now (depending where you put goldsmith banking in the banking continuum, and what country you´re in).

A lot of financial statistics would probably start looking very strange, since even correcting for CPI doesn´t adequately control for the impact monetary growth has on price measurements. It´s possible that the economy would settle down after a few years as everything adjusts around the constant money supply - that´s essentially the general equilibrium theory argument. It´s also entirely possible that the growth in the money supply that FRB provides over time makes it easier for adjustments to occur - a lot of arguments in favour of small positive inflation amounts essentially revolve around this - and that a bunch of issues we didn´t previously know about emerge and cause new problems.

While this is very much an open research question, it´s interesting to note that a side effect of some of the recent Basel 3 regulatory changes is that at least for the last couple of years, several countries including the UK have experienced stable money supplies. If this continues (it´s perfectly possible to regulate fractional reserve banking so that it doesn´t expand the money supply if you know what you´re doing - whether the central banks really know what they´re doing is also an open research question), then we may start to have some empirical data that can help shed light on this very interesting question.


As regards the first question, the "$p_t=...$" expression is conceptually and qualitatively useful because, at the optimum, it relates price with consumption and expectations. Mathematically it is an implicit function of course, so it is not a "closed-form solution" to "tell us how to find $p_t$".

Cochrane acknowledges this as he writes in p. 6

  "We have stopped short of a complete solution to the model, i.e., an expression with exogenous items on the right-hand side. We relate one
  endogenous variable, price, to two other endogenous variables,
  consumption and payoffs. One can continue to solve this model and
  derive the optimal consumption choice $c_t , c_{t+1}$ in terms of more
  fundamental givens of the model. In the model I have sketched so far,
  those givens are the income sequence $e_t , e_{t+1}$ and a specification of
  the full set of assets that the investor may buy and sell. We will in
  fact study such fuller solutions below. However,
   for many purposes one can stop short of specifying (possibly wrongly) all
  this extra structure, and obtain very useful predictions about asset prices
  from (1.2), even though consumption is an endogenous variable."

Regarding the first example, the "no-investment consumption" sequence $\{e_t\}$ is in general considered random, due to factors unmodeled at this introductory level, like an income stream that is affected by exogenous random shocks etc, so the use of the expected value relates to the existence of $e_{t+1}$  in there.

Regarding the second example, Cochrane writes p. 7 (my emphasis)

  "In this context, equation (1.4) is obviously a generalization, and it says something deep: one can incorporate all risk corrections by
  defining a single stochastic discount factor—the same one for each
  asset—and putting it inside the expectation. $m_{t+1}$ is stochastic
  or random because it is not known with certainty at time $t$. The
  correlation between the random components of the common discount
  factor $m$ and the asset-specific payoff $x_i$ generate asset-specific
  risk corrections."


You can easily think of an endowment economy with taxes, and think of the taxes as transaction costs or trading costs. I'm sure classic textbooks like Mas-Collel address this point, perhaps in an exercise. Your intuition is right, the larger "t" is, the more deadweight loss there is, and this is reflected by transactions that do not occur. A large enough "t" can prevent all transactions from occurring, just as a large enough tax can kill a market.


Expectations in Gali's classical monetary model

As for your first question, we are not just assuming $\partial U/\partial B > 0$. We simply assume monotonicity of preferences, which for the model here is a plausible condition. In simple terms this means that "more is better". Hence, the consumer will always choose to spend all of his budget for current or future consumption, given by $C$ and $B$ respectively. Hence, the constraint holds with equality. In these types of models we pretty much always assume equality of the budget constraint.

As for your second question, solving without expectations and then taking the expectation is strictly speaking incorrect. It will give you the correct result here though. What you can do here is to take the derivative in the expectation. For these models that usually works in the non-stochastic setting as here. If you have for example stochastic income with log-normal distributions you would have to be more careful. What you can do then, depending on the case, is solve the expectation before taking the derivative or use the Leibniz Rule.

Hence for your problem you don't forget the expectation in the first place, which corresponds to $\partial U / \partial C_{t+1}$. Instead for your model you can take $\partial E_0[U] / \partial C_{t+1} = E_0[\partial U / \partial C_{t+1}]$. As you can see, it will yield the same result as simply taking the expectation later on $\partial U / \partial C_{t+1}$. However, that is the wrong way and you'll probably lose points in an exam if you write it down that way.


The original problem was probably of the form $$\max_{\{W_t\}_{t=1}^\infty}\sum_{t=0}^\infty \beta^t u(W_t-W_{t+1}),$$
$$\mbox{s.t. } W_{t+1}\in[0,W_t] \ \forall \ t,  ~~ W_0 \mbox{ given}$$
When formulated that way there is no need to solve for any function $V$, but the infinite elements of the sequence ${\{W_t\}}_{t=1}^\infty$ are all choice variables. Bellman's main insight is that (under some conditions) the problem can be simplified. Instead of solving for an infinite number of variables, you could solve for a single "policy function" that tells you what to do each period as a function of the current state variables, and a "value function" that summarizes what is the value (in terms of today) of inducing each possible state in the future. Even if there is no uncertainty, it is clear that this value function is an endogenous object, since it must somehow capture the future stream of utility as a single function of the future state. In your case, that function is  $V(W')$. This is why the Bellman equation:
$$V(W)=\max_{W'\in[0,W]} u(W-W')+\beta V(W')$$
has two properties:

1) it lacks the subscripts on the $W$'s, this reflects that you will solve for $W'$ as a function of $W$, i.e. the policy function.

2) has a new object, $V$, that was not in the original problem and it is a function of $W$ (or $W'$) and just by eyeballing the two problems you can see that for them to be equivalent $V$ must be the discounted sum of future payoffs assuming future actions are also optimal. So it is not a choice variable, but definitely, an endogenous object that you need to find when solving the problem.

You can see that the problem of optimally choosing infinitely many objects is transformed into one of choosing a finite number (2) of functions, which are infinite-dimensional objects. Brilliant!


why federal fund rate vs treasury bond yield generally move in the same direction?

Federal funds rate is a rate at which banks can borrow extra money at an overnight market. This can be viewed as 'banks' cost' since when people want to lend from certain bank, the bank will often get the money not from some new deposits it itself attracts but from the overnight market.
Consequently if the Fed funds rate drops they can afford to lend money more cheaply. The reason why they also do so instead of pocketing all the difference as a higher profit is the competition amongst the bank.  If you want to completely see the mechanism laid bare you can have look at chapter 3 of Freixas and Rochet, Microeconomics of banking. Hence this is the reason why mortgage rates will fall as well. Banks simply can create those mortgages more cheaply now.
The bond yield is also affected by the above. The current bond yield ($Y$) is given as:
$$Y = \frac{C}{A}$$
where $C$ is the annual coupon payment and $A$ is the asset price. The coupons are already fixed so bond which pays ${\\\$}100$ coupon and trades for ${\\\$}1100$ will have yield of $\approx 9\%$. Now whether that $9\%$ is a lot or not depends on prevailing interest rate. If the interest rate  is $10\%$ then yield of $9\%$ is small and if interest rate is $5\%$ yield of $9\%$ is high.
If the interest rate falls relatively to the yield the bond will get more desirable (like the situation when interest rate would be $5\%$ and yield $9\%$. In such case people will simply demand more bonds and then you will get the standard supply-demand story. People will bid the bond prices until $A$ increases sufficiently enough so people are indifferent between buying bond and putting money in some deposit account bearing that $5\%$ interest (or some other instruments). As you can clearly see from the formula above, higher bond price will imply lower yield. Hence when interest rate falls yield prices fall as well.


Can you set $C / P^{\eta}$ to be the numeraire in a NK model?

I'm going to answer my own question. Consider multiplying all prices and income (i.e., the wage) by a common factor $\lambda>0$. The new ideal price index is just $P' = \lambda P$. The new wage is just $W' = \lambda W$. So real expenditure, $C' = W'/P' = W/P = C$, is invariant. The "numeraire" should now be $C' (P')^{\eta} = \lambda^{\eta}$, hence prices relative to the new "numeraire" should now be $p_j' = p_j \lambda^{1-\eta}$, not $\lambda p_j$. That's a contradiction---$CP^{\eta}$ is not a stable unit of account.

Another way to see the problem is that demand is no longer homogeneous of degree zero in prices and income under this normalization: $(c_j^{*})' = \lambda^{-\eta} p_j^{-\eta}$, so $(c_j^{*})' / c_j^{*} = \lambda^{-\eta}$.


Creation of Money - Hypothetical Situation

No, you didn’t. All money is debt, but not all debt is money.

There are two easy ways of thinking about this. The first is to think about the properties of money: unit of account, store of value, medium of exchange. Focusing on the latter two:

Is the money you owe to me a good store of value? No offense, but I’d rather have cash or keep it in an insured bank account. So no, your debt to me isn’t a good store of value.
Can I use it as a medium of exchange (i.e., can I trade your debt to me to someone else as payment for something)? Not likely, so no, it’s not a medium of exchange.

The second is to do the accounting exercise and compare it to the case where a bank makes a loan. I’ll bold all the things that count as money in the hands of people in the real economy. In this case, it’s:

  Me

  My assets: \$100 owed from you
  My liabilities: \$0

  You

  Your assets: \$0
  Your liabilities: \$100 owed to me

Now compare the case where I deposit \$100 into the bank and the bank lends out \$50 to you (I’m going to ignore bank capital to make it clearer):

  Me

  My assets: \$100 owed from the bank (bank deposit).
  My liabilities: \$0

  Bank

  Bank assets: \$50 cash + $50 owed from you
  Bank liabilities: \$100 deposit owed to me

  You

  Your assets: \$50 cash
  Your liabilities: \$50 bank loan

You’ll notice that in the first case, no money has been created, because I can’t pay anyone with your promise. In the second case, I have \$100 of money, because I can pull \$100 out of the bank (again, ignoring bank capital here) and pay someone with it, and you have \$50 of money because you have \$50 in cash (though you also have an outstanding loan for \$50), so the amount of “money” in the real economy has increased.


Why is instantaneous utility of current period discounted?

What you see as
$$
\frac{1}{1+r_{t+1}}MU_t = MU_{t+1}
$$
can also be seen as
$$
\frac{1}{1+r_{t+1}}MU_t = \frac{1}{1}MU_{t+1}
$$
where $1+r_{t+1}$ and $1$ are the respective prices of current and future money measured in future money. So basically what is happening is that you calculate how much utility an additional unit of future money would buy you in the current and in the future period. In optimum these quantities are equal. You can also rewrite the equation into the form of the familiar MRS condition:
$$
\frac{MU_t}{MU_{t+1} } = \frac{1}{\frac{1}{1+r_{t+1}}}.
$$


Testing Roberts's Law: data request on CEO pay and company characteristics

There are plenty of papers studying CEO pay and firm size empirically. Check a long list here. Some examples of data sources:

US: Execucomp, used by Gabaix, Landier, and Sauvagnat (2014).
UK: Hemscott  and Datastream, as used by Ozkan (2009) or Gregg, Jewell, and Tonks (2011). Also, BoardEx and Datastream, as used by Tarkovska (2017). You also have the Bureau van Dijk FAME dataset (used by the Bank of England, e.g. here), but it is proprietary.
Germany: handpicked from financial reports as in Heimes and Seemann (2012), or data obtained from the consulting firm
Kienbaum, as in Fabri and Marin (2015).


Question about Strict Preference Relation

You are correct that the motivation regards issues surrounding completeness. As you point out, when $\succcurlyeq$ is complete $y \not\succcurlyeq x$ implies $x \succcurlyeq x$ (hence, $x \succ y$).
If, however, we do not want to assume complete preferences then (2) does not imply (1), but strict preference can still constructed from weak using the more complex definition. (One could also take the set difference of weak preference and indifference).


Why does marginal cost slope upwards after the minimum point?

The variable unit in the quotation is the factor of production that is varied keeping the others fixed. For example it could be the amount of workers hired, while keeping the amount of machinery (capital) fixed. Or it could be the amount of machines reented, keeping the amount of workers fixed.

As for your other question diminishing returns does not refer to workers specifically, although it can.

Suppose we do consider workers as variable input and keep capital fixed. In the beginning usually adding workers will increase output a lot, so the costs of producing an additional unit are low, and might even be decreasing.

However, at some point adding extra workers will increase output only a little. After all the amount of capital is fixed so adding extra workers mean that they e.g. have to wait for another at the machines or get in another's way. So at that point whereas earlier you only needed say 1/2 a worker to produce an additional unit of output, now you need 2 extra workers to produce that one extra unit. That means that the costs of producing an additional unit (the marginal costs) have risen.


Quality of French wine in the US vs French wine in France

This is called the Alchian–Allen effect.

  The Alchian–Allen effect was described in 1964 by Armen Alchian and
  William R Allen in the book University Economics (now called Exchange
  and Production). It states that when the prices of two substitute
   goods, such as high and low grades of the same product, are both
   increased by a fixed per-unit amount such as a transportation cost or
   a lump-sum tax, consumption will shift toward the higher-grade
   product. This is true because the added per-unit amount decreases the
  relative price of the higher-grade product.

  Suppose, for example, that high-grade coffee beans are \$3/pound and
  low-grade beans \$1.50/pound; in this example, high-grade beans cost
  twice as much as low-grade beans. Now add a per-pound international
  shipping cost of \$1. The effective prices are now \$4 and \$2.50;
  high-grade beans now cost only 1.6 times as much as low-grade beans.
  This reduced ratio of difference will induce distant coffee-buyers to
  now choose a higher ratio of high-to-low grade beans than local
  coffee-buyers. (Prices are illustrative only).

  The effect has been studied as it applies to illegal drugs and it has
  been shown that the potency of marijuana increased in response to
  higher enforcement budgets, and there was a similar effect for alcohol
  in the U.S. during Prohibition.

  Another example is that Australians drink higher-quality Californian wine than Californians, and vice versa, because it is only worth the  transportation costs for the most expensive wine.

From the Alchian–Allen Wikipedia page, emphasis mine.

I learned about this paradox in the context of Why the Best Washington Apples are Shipped out of State, discussed in a chapter of a fun popular economics book Puzzles and Paradoxes in Economics which also happens to be a great source of undergraduate economics exam questions.


What are the best theoretical or empirical defenses of equilibrium analysis in economics?

There is a paper fully dedicated to this topic by Backhouse (2004).

The concept of equilibrium is often misunderstood by non-scientists and non-economists.  Following Robinson (1956: 57):

The word equilibrium, in ordinary speech, describes a relation between
bodies in space. The scales of a balance are in equilibrium when the balance is at rest. . . . If we are continually throwing coppers at random into
either scale, the balance is continually wobbling and never reaches equilibrium; but, at any moment, there is a definite equilibrium position
which it would quickly reach if, from that moment, we left it alone.

Hence, the equilibrium analysis can be readily applied to dynamic systems. Equilibrium analysis can be viewed as sort of thought experiment where to organize your thinking passage of time is held constant. Now thinking about equilibrium analysis in the way above still has few issues.
Again Robinson (1956: 59) writes:

Nor can we apply the metaphor of a balance which is seeking or tending
towards a position of equilibrium though prevented from actually reaching it through constant disturbances. In economic affairs the fact that
disturbances are known to be liable to occur makes expectations about
the future uncertain and has an important effect on any conduct (which,
in fact, is all economic conduct) directed towards future results. . . . A
belief that a particular share is going to rise in price causes people to
offer to buy it and so raises its price. . . . This element of ‘thinking makes
it so’ creates a situation where a cunning guesser who can guess what the
other guessers are going to guess is able to make a fortune. There are
then no solid weights to give us analogy with a pair of scales in balance.
The metaphor of equilibrium is treacherous. . .

However, this issue was later addressed as discipline evolved Backhouse (2004):

There is, however, an alternative way to think of equilibrium, not as the
static equilibrium analogous to that of a pendulum, but as inter-temporal
equilibrium where expectations are fulfilled (Milgate 1979, 1987, Phelps
1987). Because this idea is now so strongly associated with the New
Classical Macroeconomics and related theories, it is worth noting that Erik
Lindahl and his Swedish colleagues used the notion to attack the MisesHakek notion of neutral money. If expectations were correctly anticipated,
any rate of inflation was consistent with monetary equilibrium. The door
was opened to discretionary monetary policy. (see Laidler 1999: 58–60).

Furthermore, Backhouse (2004) also argues:

... This is the notion that equilibrium is where (a) agents’ decisions are compatible with each other and (b) no agent has any reason to
change his or her behavior. The second part of this condition (which might
be thought to encompass the first) is often represented by saying that agents
are maximizing utility subject to the constraints they face.

Consequently, in economics there are already several possible notions of equilibria:

Equilibrium as the absence of endogenous tendencies for change.
Equilibrium as balance of forces (using a mechanical analogy).
Equilibrium as correct expectations.
Equilibrium as meaning that no agent has any reason to change his
or her behavior.

We use the concepts above because they allows us to describe and analyze economy and economic relationship as well as to test theories we make about them. Following  Huw Dixon (1990: 356),

At its most general, we can say that ‘equilibrium’ is a method of solving
economic models. At a superficial level, an equilibrium is simply a
solution to a set of equations. ... However, there is more to it than that. Whilst economists rarely argue
over how to solve equations, they do argue over whether a particular
solution represents a ‘real’ equilibrium or not. What is at stake is the
economist’s view of economic agents and the market.

Next following further Backhouse (2004):

An economist who has argued the case for equilibrium analysis very forcefully is Robert Lucas (1980). He interprets equilibrium to mean individual
optimization and consistency of plans (market clearing though perfect
competition). The alternative is to use ‘free parameters’ – parameters that
are not based on individual optimizing behavior, such as those involved
in equations relating price changes to excess demand for commodities. His
reason is that we can never know whether free prarameters are reliable.
Equilibrium models, he claims, are reliable in that parameters are based on
microeconomic behavior which is observable and directly measurable in a
way in which free parameters are not. When economists had no means of
handling more complicated equilibrium models, it was necessary that they
work with free-parameter models in order to explain dynamics. Such work
was immensely productive.

... The substantive points in his argument are that
microeconomic behavior can be established by experimentation; that this
behavior can be described in terms of optimization; and (implicitly) that
aggregation is possible without the introduction of what he calls free
parameters. They define Lucas’s view of how economies work at the macroeconomic level. These are the points on which discussion should centre, not
the issue of equilibrium. This was nicely expressed by McCallum (1986:
412):

Whether or not one works with a model in which “all markets clear” is a
matter of convention, but whether this clearing pertains to auction-type
markets, or to ones with nominal contracts (perhaps implicit) that have
allocational effects, is not.
Another way to express Lucas’s support for equilibrium models and his
hostility toward free-parameter models is that he believes that it provides
explanations of why parameters are what they are. Free parameters are
unexplained, however strongly they are supported by data.

Hence to sum it up.

Equilibrium is far more wider and complex concept that commonly perceived.
It is a way how to put structure on raw data and how to understand mechanisms that generate the data we observe
It is useful concept even for empiricists as it helps us test different explanations for economic phenomena.

The arguments above respond to proper criticism of concept equilibrium in the philosophy of economics. There quoted passages from that anarchist website have far more problems and can be dismissed more easily.

[Equilibrium analysis] is essentially a static tool used to analyse a dynamic system. It assumes stability where none exists. [...] Thus the whole concept is an unreal rather than valid abstraction of reality.

This is incorrect prima facie.

You can use static tools to analyze dynamics systems if you do it at a given point in time.

For example, the whole concept of equilibrium is omnipresent in physics (see the famous Feynman Lectures on Physics). Physical systems such our solar system, earths crust etc are often analyzed using concept of static equilibrium (here is full lecture on this topics). Now I am fully aware Economics is not Physics, the point is that general statement made there is prima facie false assertion.
Static models can and are routinely applied to systems that in real life are ultimately dynamic (since time always marches forward all systems are). This is because we often on purpose want to analyze problem holding the passage of time constant as that makes problem more tractable. This is done in any field of science and there is no reason why it should not be applicable to economic problems. It would  be like arguing English can be used in one field but not another without providing any additional reasons.

The concept of equilibrium directly extend to dynamic analysis. As one can see in any modern micro or macro textbooks (e.g. see MWG Microeconomics Theory or Advanced Macroeconomics by Romer) economics often analyzes equilibria in dynamic fashion or explores dynamic properties of various equilibria.

However, the assumptions required prove to be somewhat unrealistic (to understate the point).

This was already addressed by the essay of Friedman which you cite. Models are not meant to be realistic if they would be they would cease to be models in the first place. Models simplify reality because it is beyond the capability of any human mind to comprehend whole reality at the same time. Good models will simplify reality in some aspects while be true to reality in others. One can criticize any model for being unrealistic and realistic in wrong places but arguing that 'realism' of a model per se, without any qualifications,  is relevant criterion to judge it is false.

There is almost no discussion of how scarce means are organised to yield outputs, the whole emphasis is on exchanges of ready made goods.

The above is literary not truth, as the discussion of how scarce means are organized to yield output is virtually omnipresent in economics (see sources cited above).
The criticism above is not exclusive, one could criticize sentence after sentence mentioned in that excerpt you posted in your question. Even if you would like to criticize equilibrium analysis most of that excerpt is not carefully constructed criticism but rather a discussion of some straw-man version of equilibrium/economic analysis.


What does it mean to say that deficits are financed by debt issue?

Your formula can be simply reaaranged to

$$(G+rB_t)-T=B_{t+1}-B_t.$$

The left-hand side of this equation is the deficit (the difference between public spending—including the spending needed to pay interest on existing debt—and public revenue from taxation). The right-hand side is the amount of new debt issued. Thus, this equation tells you that you must issue enough new debt to pay the deficit—in other words, deficits are financed by debt issuance.


Why Keynesians prefer short run measures despite straight long-run Phillips curve?

Actually both New Keynesians, Keynesians and Monetarists are advocating for short-run government activism, just the underlying mechanisms are different and Monetarists stress monetary over fiscal policy.  According to the New Palgrave Dictionary of Economics (Cagan 2017) monetarism is defined as:

Monetarism is the view that the quantity of money has a major influence on economic activity and the price level and that the objectives of monetary policy are best achieved by targeting the rate of growth of the money supply.

Actually Friedman (1968) and Phelps (1968) criticized the original Philips curve mainly because it posited permanent employment inflation trade-off.
The monetarist Phillips curve would  be very close to something like:
$$\pi_t = \pi_t^e + \lambda( \ln Y_t - \ln \bar{Y}_{t}) + \epsilon^s_t$$
where   $\pi_t$ is inflation, $\pi_t^e$  inflation expectation $\ln Y_t$ log of current output $\ln \bar{Y}_{t}$ log of the natural rate of output. The argument here is that if government follows inflationary policy and people are not grossly irrational then in the end $\pi_t = \pi_t^e$ and there is no inflation-output trade-off. However, even Friedman himself didn't argued against short-run trade-off. The problem in his view was that in order for there to be permanent trade-off government would always have to 'surprise' people by targeting ever increasing inflation in an unexpected way. However, the moment when the expectations would adjust $\pi_t= \pi^e_t$ and the Philips curve would be 'flat' in a sense output would be orthogonal on rate of inflation as the terms would cancel out.
Thus, if we would argue that $\pi_t= \pi^e_t$ will always hold then there would be no trade-off even in short-run - but that is not really what was ever argued by Monetarists or at least not by Friedman and Phelps. Friedman and co. were writing their research before the rational expectations revolution really took hold so their treatment of how rational expectations work was not really as rigorous as today which probably lead to some misconceptions. I think either this got somehow either not properly explained by the Froyen's book or there is some misunderstanding (I dont know the book So I can’t really comment on that).
Also from your question it’s not clear if the author of your textbook means New Keynesian Philips curve or something like  the accelerationist Philips curve.
The accelerationist Philips curve that cab be considered ‘Keynesian’ would look like this:
$$\pi_t = \pi_{t-1} + \lambda( \ln Y_t - \ln \bar{Y}_{t}) + \epsilon^s_t$$
The difference between this Philips curve and the previous one is that instead of inflation expectations $\pi^e_t$ we have there just past inflation $\pi_{t-1}$ (I am guessing that perhaps this is the assumption that was being alluded to in your textbook). This implies that there is always some short run trade-off between inflation and employment.
The New Keynesian Philips curve is yet even more different and given by:
$$ \pi _{{t}}=\beta E_{{t}}[\pi _{{t+1}}]+\kappa y_{{t}}.$$
The New Keynesian philips curve is actually based on ideas of Friedman and Phelps (see Roberts; 1995) where now the mechanism depends on current expectations on future inflation $ E_{{t}}[\pi _{{t+1}}]$. Here there is again always short-run trade-off between inflation and output with a caveat that the trade-off depends on not what past inflation was but what is the future expectation of inflation.
As you can see all three models allow for monetary policy to stimulate economy in short-run but the mechanisms are different:
In the Friedman & Phelps case short run inflation-output trade-off exists when the inflation expectation of people are not correct or if they are being surprised by government.
In the accelerationist case there is always short-run trade-off between inflation and employment because of the assumption of inflation expectations being just past inflation $\pi^e = \pi_{t-1}$.
In the New Keynesian Philips curve the mechanism depends on present expectations of future inflation and there is also always trade-off between inflation and employment with subtle difference that the trade-off depends on what the future inflation expectations are.
I think Romer Advanced Macroeconomics has a good treatment of the Philips curve so if you would like to know more about this that is the source I would recommend.


What physically happens when foreign currency reserves change?

Typically foreign reserves will just be assets denominated in a foreign currency, most importantly government debt. Another popular reserve asset are IMF special drawing rights, which are convertible into currencies of member countries. I don't think there is any amount of physical cash currency in international reserves. You can see the makeup of US reserves here: https://www.treasury.gov/resource-center/data-chart-center/IR-Position/Pages/12302016.aspx

If the US treasury would decide to increase it's Yuan holdings, it will just buy bonds that pay are traded and denominated in Yuan.


How do I calculate price elasticity of demand using historical price and quantity data?

You've fallen into a really common pitfall -- the spurious regression. The parameters you chose to include can't be chosen 'willy nilly' by throwing data into a regress command. Ultimately this can't be answered in so little words, without data, while maintaining accuracy.

That said I can try to answer your question as a reference point. Before even attempting this I couldn't emphasize enough caution. Just remember that a model that's almost correct -- for practical purposes is 100% wrong.

So it seems you've settled on a linear log-log specification. I would do a quick Google Scholar search on papers forecasting consumer demand for a product similar to the one of interest. The functional form along with parameters you choose to include must be derived from theory. You'd be able to just copy their equation and just modify it to your data. Otherwise, you're doing blind statistics. This is the main reason you're getting inconsistent signs.

What you're trying to do is specify a derived demand for the category of goods you're focussing on. Your general model is going to look like this:

$ logQ_t = \alpha + \beta_llogP_t + X'_t\gamma + \epsilon$

where:
Q sales    P selling point
  X vector of factors other than selling price
$\epsilon$ random component in demand

Have you included any economic indicator parameters? This would be things like GDP, income, population growth, unemployment, interest rates... etc. Depending on the good -- for your purpose -- there's usually theory providing necessary parameters regardless of statistical significance. I would do this first. Add some macro indicators and re-check the F-statistic for the model. Your model would then provide not only own price elasticity of the good, but you'd get an income elasticity, as well as cross-price elasticities for competing/complementing goods. If your company has spent money on advertising this would be necessary to include as well. Have you added dummy variables for the 'category' of the good itself? And trend variables? Lags of seasonal effects, their quadratic effects? Interaction effect variables?

Any cursory approach would lead to omitted variable bias over your misspecified parameters. On most circumstances you'll have to verify the multiplier effect isn't changing through time.

Have you checked for normality, autocorrelated errors?

This is as broad and general a reference I can think to give you. The functional specification here might be better calculated through an autoregressive form. But this is beyond the scope I think.

I'm not sure as to your level of econometric/demand analysis so I kept things devoid of much math. I hope this was somewhat helpful. It was great intuition to stop what you're doing when you saw incorrect signs. Keep to that rule of thumb.


How much do real-estate prices correlate with apartment rental prices?

In theory there should be a direct positive correlation (especially in countries where house prices are a part of the index which is used to define rental rates), but in reality there is a lot of variation in the results for this question depending on the country, possible effects of different laws, the business cycle, whether new housing is being built or not, etc.

For Italy, Manganelli, Morano and Tajan (2014) in "House prices and rents. The Italian experience" study response functions and conclude:

2)  Rents depend on real-estate prices, the short-term interest rate and the differences between actual and expected GDP with a one year delayed reaction.

On the other hand, Ambrose, Eichholtz and Lindenthal (2013) in House Prices and Fundamentals: 355 years of evidence, Journal of Money, Credit and Banking find that when housing prices stray from equilibrium growth, rental rates do not react the same way even if it takes decades for the prices to return to steady growth.

For a recent study on U.S. data (model built to match trends from 1960-2013), I'd check Gelain and Lansing (2014): House prices, expectations, and time-varying fundamentals, Journal of Empirical Finance


How is it cheaper to take on debt when taking on a project?

I assume EBT = Earnings before tax.

Take a \$200 project with a 6% return on assets. If 109% equity financed, EBT = \$12 (6% return).

If debt has a cost of 5%, and 50% financed (\$100 borrowed), earns \$12 - \$5 = \$7, or a 7% return on equity. So the rate of return is higher, even though total earnings are less.

However, if interest were not deductible, would need to pay taxes on the \$12 pre-interest earnings, not the \$7, and so rate of return would be lower with a high enough tax rate. (E.g., if the tax rate is 50%, there would be \$2.50 paid on the cash flow required for interest, and the rate of return would be lower than a 100% equity investment.)


Some older references, from the first wave of fascination with the subject:

1) Baumol, W. J., & Benhabib, J. (1989). Chaos: significance, mechanism, and economic applications. The journal of economic perspectives, 77-105.

A general introductory overview.

2) Lorenz, H. W. (1993). Nonlinear dynamical economics and chaotic motion (Vol. 334). Berlin: Springer.

Theoretical treatment

3) Brock, W. A., Hsieh, D. A., & LeBaron, B. (1991). Nonlinear dynamics, chaos, and instability.

Applied econometrics for the detection of chaotic behavior in economic time-series, from two of the most known researchers in the field (W. Brock and B. LeBaron)/


Without much information on your regression model it's hard to make a good suggestion. But you may have a look at the Frankel and Romer AER's paper. They construct an instrument for trade, but it could be used for FDI, based on geographic characteristics. I assume that geographic distance between two countries is a good predictor of FDI flows and that ceteris paribus a Canadian firm based in Vancouver has a higher probability to set up an affiliate in Seattle than in Dallas.


Why is deflation a likely short-term of an increase in M2?

None, you are misreading the passage. The paragraph does not say that increase in M2 will lead to deflation.
As the paragraph clearly states the deflation is most likely because of the severity of the 'shock'. Shock here is not the increase in the money supply it is the recession. Fall in real output is deflationary.
According to standard textbook equation of exchange  (see for example Blanchard et al Macroeconomics an European perspective)  price level $P$ is given by:
$$P=MV/Y$$
where $M$ is money supply, $V$ velocity and $Y$ output. This equation is an oversimplification as in more advanced models also expectations of these variables and other factors play a role (see Romer's Advanced Macroeconomics or Krugman (1998)), but for this question more complex model would arguably be an overkill.
As the equation shows price level (change of which gives you inflation) increases with $M$ but also decreases with $Y$. The article in the Financial Times simply states that although $M$ increased considerably deflation is still expected due to the fall in $Y$ (i.e. the shock).


Socio-Economic Consequences of Nation-Wide Vegan Diet

I think your questions are very interesting. I am no expert on US farming data sources. However, I have briefly pursued these questions for the UK. As currently this question has no responses, I'll post my findings below verbatim along with my data sources.

I encourage you to pursue these research questions further, perhaps using a similar framework to below.

"My response uses DEFRA data to examine the extent of the agri-food economy that must change, in equilibrium, to complete a transition to a Vegan Economy. Although in the short run Rural Unemployment may rise, once we distribute the gains from a Vegan Economy, we can raise rural incomes and reduce rural unemployment.

Output Data

DEFRA's Food Statistics in your Pocket reveals that the Agri-Food Industry made up 6.3% of Gross Value Added in 2016. However, a Vegan Economy only entails removing animal products, which in equilibrium is unlikely to influence the distribution, retail, or catering sectors negatively; people under the vegan norm must consume sufficient calories to be healthy.

Note: The Vegan Norm assumes that people spend, counterfactually, the same proportion of income and that they eat a healthy, varied Vegan Diet. We also assume that UK Welfare can distribute any economic gains from a Vegan Economy.

Only Agriculture, Fishing, and Food Manufacturing remain relevant, which has ~2% GVA contribution. This figure overestimates the contribution of Animal Products as plant-based firms can partially fill in for Food Manufacturing. Also, these figures fail to distinguish between animal and non-animal products. The ratio of the value of plant to animal products for imports or exports is 2 to 1, calculated from Observation of Economic Complexity (OEC) data, which seems a reasonable prior for domestic production too. This data also indicates that the UK balance of trade does not suffer.

Yet, the Vegan Economy also sees growth in GVA from old and new Vegan Alternatives to Animal Products. The market for alternative products is growing rapidly across the agri-food chain. We reference The Vegan Society who have collated such evidence on their website. As such, we can expect this market to fill in the anticipated output gap.

Thus, in equilibrium, the Vegan Economy may see at most a 1% reduction in Gross Value Added from eliminating Animal Products from Agriculture and Food Manufacturing. Trend GVA growth may even rise thanks to alternative products.

Employment Data

DEFRA's Pocketbook indicates that we employ 820,000 people in Agriculture, Fishing, and Food Manufacturing. Animal Products are more labour intensive but likely make up only a third of GVA. So, perhaps half of these workers (~1% of UK Labour Force) have their employment at risk under a Vegan Economy.

A sectoral shift towards crops and alternatives could fill in the output gap, suggesting that the general equilibrium effects on employment are likely negligible.

Yet, even for those who do lose their work, we can redistribute the benefits of a Vegan Economy. These benefits include increased government revenue from higher GVA growth, reduced healthcare costs and improved productivity from healthier workers. We also avoid costs associated with accelerated climate change.

Our options for helping those farmers or manufacturers out of work include but are not limited to:
incentives to switch to crop farming
start-up programs to encourage new land use such as afforestation or sanctuaries
incentives for transport, retail, accommodation, or local government to provide training or apprenticeship schemes for those out of work. They are often the main employers in rural areas."

UK Economic Data Sources:

DEFRA (October 2018), Food Statistics in your pocket 2017 Summary
url: https://www.gov.uk/government/publications/food-statistics-pocketbook-2017/food-statistics-in-your-pocket-2017-summary

Page 7 of Rural productivity and Gross Value Added (GVA) statistics, DEFRA (2019)
url: https://www.gov.uk/government/statistics/rural-productivity

Observation for Economic Complexity (2017), United Kingdom
url: https://oec.world/en/profile/country/gbr/

The Vegan Society (2019), The Vegan Society Statistics: Business/Food Sales
url: https://www.vegansociety.com/news/media/statistics

Public Health Evidence

Chen, J., Peto, R. Pan, W., Liu, B., Campbell, T.C. Mortality, Biochemistry, Diet and Lifestyle in Rural China. Geographic study of 69 counties in Mainland China and 16 areas in Taiwan. Oxford University Press, 2006
url: https://nutritionstudies.org/the-china-study/

Academy of Nutrition and Dietetics (2016), The Position of the Academy of Nutrition and Dietetics: Vegetarian Diets
url: https://www.eatrightpro.org/-/media/eatrightpro-files/practice/position-and-practice-papers/position-papers/vegetarian-diet.pdf

British Dietetic Association (August 2017), British Dietetic Association confirms well-planned vegan diets can support healthy living in people of all ages
url: https://www.bda.uk.com/news/view?id=179

Environmental Evidence

Poore and Nemecek (2018), Reducing food’s environmental impacts through producers and consumers

CH5 page 77 Figure 5.12 of
Climate Change and Land: an IPCC special report on climate change, desertification, land degradation, sustainable land management, food security, and greenhouse gas fluxes in terrestrial ecosystems (IPCC 2019)
url: https://www.ipcc.ch/srccl-report-download-page/


These assumptions actually have distinct mathematical ground, but your professor likely decided to leave this out for (of course bad) pedagogic reasons.

scarcity: society has insufficient productive resources to fulfil all human wants and needs. Scarcity is the motive behind economics. If it is cheaper for everybody to get a given good from the ground than to produce it, then they are not economic goods, they don't have a market value and are not treated by introduction classes in economics. This can be stated as: for all actor, for all (studied) goods, the cost of acquiring the good is strictly positive.
Rationality: people adopt the best actions to achieve their goals. Usually, this is translated in maths by introducing a utility function from the possession set to a measuring set (usually $\mathbb{R}$), and stating people always act in order to maximize their utility.
Stable Preferences: Using the same utility function as above, this states that this function does not depend of time. This implies that if you prefer having a kilogram of soup than a kilogram of raw potatoes today, it will still be the case tomorrow.
Equilibrium: Here I have to guess, but given your example I suppose you're studying equilibrium on the market. I guess what your teacher meant is that you study the economy at a time when the amount of goods or services sought by buyers is equal to the amount of goods or services produced by sellers.

All these definitions are not guaranteed to match with those implied by your teacher, but are consistent and widely taught in economics. When not otherwise specified, economical answers require referring to a consistent framework of axioms from which mathematics should be able to guide you to the solution you're trying to state. Different models can show different solution, but one model should give only one solution (which can be composite: "people buy more or less goods at the end of the month" is a valid solution, but "people buy more goods at the end of the month" and "people buy less goods at the end of the month" should not be obtained from the same data with the same model...).

However, economic practice does not require you to re-demonstrate any known result (mostly, the results which you know a name for can be used as starting points for your answers). If you're not confident with "explaining", you need to be able to think about a one-line mathematical justification for each explanation you give, and thus you will ensure your explanations give you the expected marks.