opportunity cost when inside production possibility frontier

This would only work if the resources have literally no alternative uses (e.g. think of mana that according to the Judeo-Christian mythology could be only just used for immediate nourishment or it would just quickly spoil and bred worms, hence it had no other uses and could not even be saved or traded).
However, this does not hold in traditional PPF analysis. Usually PPF will look like the picture below (picture taken from Wikipedia). In such case there is always an opportunity cost. If you start at point A and move to point B on the PPF there is an opportunity cost, because instead of moving to B you could move to D. The opportunity cost of choosing allocation B is the allocation D that we forgone.
In order to get rid of opportunity cost you would have to get rid of all other options and have just  one single use to which all resources can be allocated (which would be just a straight line overlapping either $y$-axis or $x$-axis). But such PPFs are not used because resources virtually always have alternative uses. In fact a proper PPF that goes beyond didactic purposes would even have more than just two dimensions, one for each good, as factors (i.e. labor, capital etc.) can be alternatively used in various production processes not just two.


Arguing uniqueness of Bayes-Nash equilibrium in an auction setting

Let $BR=(BR_1,BR_2,BR_3,...)$ denote the best response mapping. This gives a mapping given message vector $m$. (For any $BR_i$ only the messages of non $i$ players are used to derive the best response mapping.)

Let $m^* = (m_1^*,m_2^*,m_3^*,...)$ denote the symmetric equilibrium. As it is an equilibrium, for any $i$ you have $BR_i(m^*) = m_i^*$, so
$$
BR(m^*) = m^*.
$$
Suppose we have a non truth-telling equilibrium $m'$. As this is an equilibrium
$$
BR(m') = m'
$$
so
$$
\forall i\in N: m_i' = \theta_i + \gamma \sum_{j \ne i} (\theta_j-m_j').
$$
Rearranging this
$$
\forall i\in N: 0 = \theta_i - m_i' + \gamma \sum_{j \ne i} (\theta_j-m_j').
$$
Adding up all the equations yields
\begin{eqnarray*}
0 & = & \sum_i (\theta_i - m_i') + \gamma \sum_i \sum_{j \ne i} (\theta_j-m_j')
\\
0 & = & \sum_i (\theta_i - m_i') + \gamma \cdot (n-1) \cdot  \sum_i (\theta_i - m_i') \\
0 & = & (1 + \gamma \cdot (n-1)) \cdot \sum_i (\theta_i - m_i')  .
\end{eqnarray*}
There are two ways this can hold. Either $\sum_i (\theta_i - m_i') = 0$ or $\gamma = -\frac{1}{n-1}$.

Case 1. Suppose $\sum_i (\theta_i - m_i') = 0$. Then from
\begin{eqnarray*}
m_i' & = & \theta_i + \gamma \sum_{j \ne i} (\theta_j-m_j') \\
m_i' - \theta_i & = & \gamma \sum_{j \ne i} (\theta_j-m_j') \\
(m_i' - \theta_i) - \gamma \cdot (m_i' - \theta_i)  & = &  \gamma \cdot (\theta_i - m_i') + \gamma \sum_{j \ne i} (\theta_j-m_j') \\
(1 - \gamma) \cdot (m_i' - \theta_i) & = & \gamma \sum_{j} (\theta_j-m_j') = 0.
\end{eqnarray*}
In the truth-telling equilibrium this is true for all $i \in N$. If $\gamma = 1$ non truth-telling equilibria are also possible for any number of players, e.g. $n-1$ players reporting their type $+1$ and one player reporting her type $-n+1$. If $\gamma < 1$ and the equilibrium is non truth-telling then this is not possible.

Case 2.
Suppose $\gamma = -\frac{1}{n-1}$.   Let $x_i$ denote $\theta_i - m_i'$, so the equation system
$$
\forall i\in N: 0 = \theta_i - m_i' + \gamma \sum_{j \ne i} (\theta_j-m_j')
$$
becomes
$$
\forall i\in N: 0 = x_i + \gamma \sum_{j \ne i} x_j.
$$
For $\gamma = -\frac{1}{n-1}$ the matrix representation of this equation system is invertible so there is only a single solution, and $\forall i: x_i = 0$ is clearly a solution. This means that the only equilibrium is the truth-telling equilibrium.


Where can I find a complete list of goods affected by NAFTA?

As highlighted by @Bkay, these linked documents are likely to rot. For future users, the keyword I was missing when searching for these documents was 'tariff schedule'. The below links to 2019 tariff schedules. I suggest that in the future a user search:

'YEAR tariff schedule COUNTRYOFINTEREST'

It may also be useful to specify the trade agreement of interest.

For USA: Harmonized Tariff Schedule 2019 Revision 2

Link: https://hts.usitc.gov/current

For Canada: Customs Tariff by Chapter (Customs Tariff 2019-1)

Link: https://www.cbsa-asfc.gc.ca/trade-commerce/tariff-tarif/2019/html/tblmod-1-eng.html

For Mexico (and all three): Mexico Import Export Codes

Link:http://www.siicex-caaarem.org.mx/

Chapters of interest will vary by personal use.


R or Python for private sector economist/strategist

If you already know how to code in MATLAB then python is more similar to it than R so I would say you will have easier time to transitioning there.

Otherwise, both R and Python are programming languages so you will be able to do all those things in both of them as you can always program your own functions.

The strength of R is that the language has a large following in the research community, which then in turn means that people write a lot of packages that focus on exactly the models that are commonly used in research. For more than two decades academics and statisticians basically helped to develop R so you can now find library for anything there save for really the most cutting edge econometric techniques - and you can bet that the first place where those will get their packages will be R. For most of the things you describe in your post you will have some excellent libraries and they are relatively more easier to estimate with R than with Python.

Python is more of an general purpose programming language which strength is very intuitive syntax. Writing in Python can sometimes feel like writing in pseudo code - its very intuitive and natural. However, since its more general purpose language its not as tailored to statistical analysis as R. You will definitely be able to do all you want to do there but more routine statistical analysis will be a bit more time consuming to code.

I actually prefer using R for any standard statistical analysis and Python for numerical analysis & machine learning & mining data through web-scraping.


Axiom of Minimal Liberalism & Sen's Theorem of Paretial Liberal

Your question is a bit confused, because it mixes together several different things.  For example, in the title, you mention Sen's Minimal Liberalism, but in the actual question, you don't mention Sen at all --- instead you talk in more general terms about "rights that cannot be taken ... in any situation".  You mention "Harsanyi's axioms", but it isn't clear what you mean by this ---or what you mean by "compatible".  So I will try to answer a couple of different interpretations of your question.
Presumably, by "Harsanyi's axioms", you mean the axioms in Harsanyi's Social Aggregation Theorem.  (These are: (1) All individuals and the social planner have preferences over lotteries that satisfy the von Neumann-Morgenstern axioms, and hence admit expected utility representations; and (2) The social preferences over lotteries satisfy the Pareto axiom with respect to individual preferences over lotteries.)  Harsanyi's Social Aggregation Theorem is an axiomatic characterization of utilitarianism. (Or at least, that is how Harsanyi interpreted it.)  So I will take your question to really be asking:  "Is utilitarianism compatible with an absolute, non-negotiable respect for individual rights?"
The answer is "No".  And this has nothing to do with Harsanyi, or even modern  axiomatic social welfare theory in general.  The incompatibility of utilitarianism with absolute individual rights has been understood for two hundred years --indeed, this is one of the major philosophical objections to utilitarianism.  Of course, utilitarianism generally promotes human rights, because utilitarianism wants to maximize sum-total utility, and violating people's rights tends to cause a massive loss of utility, so it is something you generally want to avoid, if possible. However, because it aggregates over the preferences of many people, utilitarianism has the feature that the desires of a very large group of people can override the desires ---or even the "rights" ---of a small group of people, in some situations.  The moral philosophy literature is full of thought experiments where utilitarianism leads to some egregious violation of individual rights.  (Of course, the same is true for almost every other social welfare function.)
But it is important to nuance this point.  We want to maximize social welfare over a "feasible set" of possible social outcomes.  Whether or not this will involve violating individual rights depends upon what is contained in this feasible set.  One way to prevent utilitarianism from violating rights is to just exclude rights-violating outcomes from the feasible set.  In the terminology of Robert Nozick, we can impose "side constraints" on the social optimization problem, so that certain policies or social outcomes are just off-limits.
Most moral philosophers who endorse utilitarianism are not fans of using "side constraints" to protect individual rights, because it ruins the beautiful conceptual simplicity and unity of pure utilitarianism.  Most utilitarians don't actually believe that "rights" are part of the fundamental structure of morality ---except in the limited, instrumentalist sense that I described two paragraphs back.  Jeremy Bentham (often seen as the originator of utilitarianism) once described the idea of "natural rights" as "nonsense upon stilts".
However,  I am raising this issue because it has a bearing on whether or not Harsanyi's Theorem is "compatible" with some notion of rights.  The reason is that Harsanyi's Theorem does not make any assumptions about the feasible set.  It simply supposes that there is some set $X$ of "social alternatives", and every agent (i.e. the social planner and every individual) has preferences over the space $\Delta(X)$ of lotteries over $X$.  So the question of whether Harsanyi's theorem is compatible with rights is simply the question: what is contained in the feasible set $X$?  If  $X$ contains outcomes that violate rights, then under some conditions, a maximization of utilitarian social welfare will choose these outcomes, so Harsanyi's Theorem is incompatible with rights.  But if $X$ does not contain outcomes that violate rights, then  Harsanyi's Theorem is incompatible with rights.
Turning to Sen:  Sen's Minimal Liberalism axiom can in fact be seen as an assertion about what is in $X$.  It stipulates that $X$ contains a pair of outcomes $x$ and $y$ such that (on the grounds of "liberty" or "rights") the social preference between $x$ and $y$ should be entirely determined be the preferences of some individual $i$.  For example, it might be that $x$ and $y$ are two outcomes which are identical in every way except for what colour shirt $i$ is wearing.  If we believe (plausibly) that people should have the "right" to chose their own clothing, then the social preference between $x$ and $y$ should be determined by $i$'s preferences between $x$ and $y$.  This is what Sen's Minimal Liberalism Axiom is saying.  Sen's Impossibility Theorem says that even this minimal liberalism condition is incompatible with the Pareto axiom.
However, notice that this is only an issue because somehow the set $X$ of social outcomes included information about what colour of clothing people can wear. So Sen's Minimal Liberalism is really saying, "$X$ contains outcomes which bear upon individual liberties, so that it is possible to violate individual rights by choosing the wrong element of $X$".  In other words, Minimal Liberalism is saying something about what is contained in $X$.  If we simply restrict $X$ to social outcomes where rights violations are not possible (i.e. impose "side constraints"), then the problem goes away.


Marx's theory of product-value is often called a "labour theory of value" though he never used it himself to describe his "law of value". He also had a theory of exchange-value to try to explain market phenomena.

Marx did have a theory of the materialisation of labour in gold. But fiat or credit money has no intrinsic value (if it did then it would not be fiat), though it does have an exchange value, broadly speaking the value that can be acquired by spending it.  Marx argued that the use of money obscured the real value ("congealed labour-time") involved in transactions and exchanges.

That is not really testable, but what may be testable is the extent that relative prices of different commodities reflect the labour involved in their production, and if so whether these reflections are of average or marginal labour, and what causes changes in relative prices over time.  Whether you would want to is another question: it is all a bit 19th century.


Example in which revealed preference doesn't imply preference

The text basically explains that consumer is assumed to be rational. This means the consumer can identify the best bundle and picks that as her choice.

If the consumer was unable to identify the bundle that is best for her or was simply choosing randomly rather than based on her preference then her choice would not reveal anything about her preferences.


How to visually present a simultaneous game with continuous strategies

This thread has some game trees. In case you want put them on your documents, you can do like this (using LaTeX with istgame package):

\documentclass{standalone}

\usepackage{istgame}

\begin{document}

\begin{istgame}[font=\scriptsize]
\setistgrowdirection{east}
\cntmdistance{20mm}{20mm}
\cntmAistb{q_1=0}[at end,below]{q_1=1,000}[at end,above]
\istrootcntmA(0){1}
  \istbA[draw=none]
  \endist
\cntmAistb{q_2=0}[at end,below]{q_2=1,000}[at end,above]
\istrootcntmA(1)(0-1)<[xshift=3pt]90>{2}
  \istbA[draw=none]{}{\pi_1,\pi_2}
  \endist
\end{istgame}

\end{document}

\documentclass{standalone}

\usepackage{amsmath}
\usepackage{istgame}

\begin{document}

\NewDocumentCommand\vpay{m}
{ \begin{matrix} #1 \end{matrix} }

\begin{istgame}[scale=1.5,font=\scriptsize]
\cntmApreset{.9}
\cntmdistance{15mm}{20mm}
\istrootcntmA(0)[null node]{1}
  \istbA(.977)[->,thick]{q_1}[r]
  \istbm
  \endist
\cntmAInfosetO[thick,dashed](0)[-11](2em)
\istrootcntmA(1)(0-1)[null node]<180>{2}
  \istbm
  \istbA(.977)[->,thick]{q_2}[r]
    {\vpay{\pi_1(q_1,q_2)\\\pi_2(q_1,q_2)}}
  \endist
\end{istgame}

\end{document}

\documentclass{standalone}

\usepackage{amsmath}
\usepackage{istgame}

\begin{document}

\NewDocumentCommand\vpay{m}
{ \begin{matrix} #1 \end{matrix} }

\begin{istgame}
\xtdistance{15mm}{40mm}
\cntmdistance{15mm}{30mm}
\cntmApreset[draw=none]{1}[red!50,opacity=.2]
\istroot(0){Challenger}
  \istb{In}[al]  \istb{Out}[ar]  \endist
\cntmAistb[draw=none]
\istrootcntmA(1)(0-1)<135>{Incumbent}
  \istbA{q_I}[left]  \endist
\cntmAInfosetO(1)
\cntmAistb[draw=none]
\istrootcntmA(2)(0-2)<45>{Incumbent}
  \istbA*{q_I}[left]
    {\vpay{\pi_C(0,q_I)\\\pi_I(0,q_I)}}
  \endist
\cntmApreset[draw=none]{1}[blue!50,opacity=.2]
\cntmAistb[draw=none]
\istrootcntmA(3)(1-1)<180>{Challenger}
  \istbA*{q_C}[midway,left]
    {\vpay{\pi_C(q_C,q_I)\\ \pi_I(q_C,q_I)}}
  \endist
\end{istgame}

\end{document}


This is a static Bayesian game. The two players share a common prior over the type space which is characterized by the uniform distribution. For Bayesian games, you have to specify the belief of each player given his/her type, and the belief of each player determines their action. Basically, the strategy is a mapping from one's type to an action through belief specification.

What is player $i$'s belief about $v_j$? It is independently drawn from the uniform distribution over $[0,1]$, so $E(v_j) = 0.5$ for now. And he will exchange only when $v_i \leqslant \frac{3}{2}v_j$, but he does not know exactly the value of $v_j$, so he takes an expectation over $v_j$, then as long as $v_i \leqslant \frac{3}{2} \times 0.5 = \frac{3}{4}$, he will choose to exchange.

The same goes for player $j$ because they are symmetric, so player $j$ will only exchange when $v_j \leqslant \frac{3}{4}$.

Given this, $E(v_j)$ now decreases to $\frac{3}{8}$, player $i$'s expected payoff from exchange decreases to $\frac{3}{2}\times \frac{3}{8} = (\frac{3}{4})^2$. Therefore, player $i$ will only choose exchange when $v_i \leqslant (\frac{3}{4})^2$.

Since $\frac{3}{4} < 1$, this process will go on and on such that no one will choose exchange in the end which is the only equilibrium.


How did the Exchange Rate Mechanism work?

How the system operated changed over time.

This Bank of England paper does a review of the literature. Rather than give an inaccurate summary, I will let you look through it.


Testing for heteroskedasticity in panel data vs time series?

You can regress residual squares (from RE or FE depending on your estimation) on $X_{it} \hat\beta$ and its square using the clustered standard errors (the vce(cl id) option), and read the F statistic and the associated p value. This is basically the same as Het test for cross sectional models (White's simplified test).

xtreg y x1 x2, re
predict uhat, ue
predict xb, xb
gen uhatsq = uhat^2
reg uhatsq c.xb##c.xb, vce(cl id)
testparm c.xb##c.xb

Another test is for groupwise heteroskedasticity proposed by Greene (2000). This test assumes that the variance of the error term is $\sigma_i^2$ (no heteroskedasticity over $t$) and then test whether $\sigma_i^2$ is the same for all $i$. You can do it using the xttest3 module. The xttest3 manual says:

  In terms of small sample properties, simulations of the test statistic have shown that its power is very low in the context of fixed effects with "large N, small T" panels. In that circumstance, the test should be used with caution.


Here is my suggestion:

I would say the optimal level of consumption is $\frac{m}{p_1}$ if $p_1 < p_2$ and $\frac{m}{p_2}$ if $p_2 < p_1$. So the individual consumes only whichever good is cheaper. Since utility is just the maximum of whichever two amounts you could consume, you would maximize that by consuming the highest amount possible.
Good 1 is a normal good if it is cheaper than Good 2, then the amount consumed rises with income.
I would say these goods are perfect substitutes, since you consume as much of the cheaper good as possible with no further constraints.
EDIT: This wrong, they are not perfect substitutes, since they are not substitutable - if you consume only Good 1, you wouldn't want to exchange any for Good 2, except if you then had more of Good 2 than of Good 1.


Why will the holder of a broad-based index fund, by definition, achieve the average return for the market?

1. The Red Sentence

Because it is market-weighted, the market portfolio is the average of what's going on. It is the default scenario, in which you don't need to trade to maintain your asset allocation.

As soon as someone trades and diverge from this default scenario, one of the two parties will lose, relative to the market (they can both be better off, but one of them will underperform).

In addition, your setup with two funds cannot work. The smart beta index is going to trade. With whom? Not with the market-weighted fund: he does not need to trade (the weights basically adjust all by themselves with the price changes, he only needs to reinvest dividends). Therefore if there are two funds only in your universe, and one of them holds the market portfolio, then the second either holds nothing, or he holds the market portfolio as well. If he holds any other security, then the market portfolio funds cannot hold all securities with market-weights.

2. The Bold Sentence

I think this sentence is wrong. It confuses return and performance. Performance is the efficiency of the process which generates return from risk. Basically, you input risk, and you output return. That's what investing is all about.

The claim behind smart betas is that the market portfolio is not correctly diversified. Basically, you get exposure for some risk without being paid for it. This can be shown quite intuitively : equally-weighted portfolios overperform market-cap weighted portfolios (Platen and Rendek, 2010, Platen and Rendek, 2012). That's because it gives you exposure non non-rewarded risks!

To sum up: the market-weighted portfolio does give you the average return. It can be outperformed by other strategies. Your example is too simple to work.


Who is working right now? (coronoavirus)

Data on employment usually comes out a month or two (sometimes a year) after the reference period.   The Bureau of Labor Statistics (https://www.bls.gov/) puts out data on employment (who is working right now)  but to see the effects of our current crisis will take time to show.

Here are some links to the employment data that the BLS produces:

National employment (monthly):  https://www.bls.gov/ces/

State and local employment (monthly): https://www.bls.gov/sae/

State and county quarterly employment:  https://www.bls.gov/cew/

JOLTS data or Unemployment claims data can show you jobs that are hiring, laying off, separations,  and how many are collecting unemployment claims.  This could give you an idea about how many people are out of a job.

Job Openings and Labor Turnover Survey (monthly): https://www.bls.gov/jlt/data.htm

Unemployment Insurance Weekly Claims Data (weekly) : https://oui.doleta.gov/unemploy/claims.asp


What are the differences between "Crawling peg" and "Crawl-like arrangement"?

The answer is already in your question. As the IMF paper says:

Classification as a crawling peg involves the confirmation of the country authorities’ de jure exchange rate arrangement.

So the difference between crawling peg and crawling-like is that in the first case country officially states that it is doing crawling peg, while with the crawling-like arrangement the country does not officially follow crawling peg, but behaves in a way as if it would be following it.


conditional factor demand functions for capital and labor

$$(\frac{r}{w} \frac{1-\alpha}{\alpha})*K$$
$$\equiv (\frac{r}{w} \frac{1-\alpha}{\alpha})[\frac{y}{a}(\frac{w}{r} \frac{\alpha}{1-\alpha})^{1-\alpha}]$$
$$\equiv (\frac{r}{w} \frac{1-\alpha}{\alpha})^{\alpha-1+1} \frac{y}{a}$$

$$\equiv (\frac{r}{w} \frac{1-\alpha}{\alpha})^{\alpha} \frac{y}{a}$$

Is that what you're looking for?  Just notice that between steps two and three, I inverted the part of K in parentheses and changed the exponent to account for the inversion.


Are there any examples where fractal geometry is applied in economics or finance?

Fractals applied to finance was a research area of Benoît Mandelbrot. He wrote books and research articles in the matter. Looking at his work could be a good way to start with.


Money supply M0 and M1 data for students

I think one of the best websites for this rather general economic data is

https://fred.stlouisfed.org/

On the first look I found money supply data for many countries. Additionally, you can check the websites of national central banks, which usually have comprehensive data, especially on monetary time series.


Can inflation show up in assets without necessarily being reflected in classic inflation indexes?

Yes inflating asset prices does not count as an inflation generally speaking although your friend's explanation is incorrect.
Fed is not an authority that decides how inflation is defined. Inflation is generally in academic literature simply defined as (Lebow & Rudd, 2016)

Inflation measurement is the process whereby changes in the prices of individual goods and services are combined to yield a measure of general price change.

Assets are neither goods and services so they should not be considered at all if we talk about inflation generally.
This being said economists also sometimes use the inflation concept to measure change in price level of things other than goods and services. Price indexes that track asset prices and concept of asset price inflation does exist, but if an economic text, book, article etc talks about inflation they will mean inflation as defined above. When they are using other meanings of inflation economists will always add clarification like  asset price inflation etc. However, again if the discussion is just about inflation, without any other caveats it is about general goods and price inflation. Also this is not because Fed would decree so, this is just general economic terminology (Fed has power over methodology they use for their measure of inflation but they try to capture inflation as accurately as possible).


Is it possible to tell anything about market structure if I have annual price data of the industries?

That that we have linear supply and demand functions, pretty much the simplest case such that quantity supplied of good $i$ is:
$$ Q_{s,i,t} = \alpha_{0,i,t} + \alpha_{1,i,t} \cdot P_{i,t}$$
and quantity demanded is:
$$ Q_{d,i,t} = \beta_{0,i,t} + \beta_{1,i,t} \cdot P_{i,t}$$

In equilibrium  $Q_{i,t}=Q_{d,i,t}=Q_{s,i,t}$ so substitute and solve:
$$ Q_{i,t} = \alpha_{0,i,t} + \alpha_{1,i,t} \cdot P_{i,t} = \beta_{0,i,t} + \beta_{1,i,t} \cdot P_{i,t}$$
$$ P_{i,t} = \frac{\alpha_{0,i,t}-\beta_{0,i,t}}{\beta_{1,i,t} - \alpha_{1,i,t}}$$

As you can see, the price depends on supply and demand. Unfortunately, shifts to the level or slope of either the supply or demand functions both affect the price. Therefore, in  general, you can't tell anything about the supply function from the behavior of the price. And if you can't tell anything about the supply function I don't see how you'd know anything about the IO of the producer industries.


Why is it said that Sweden has a regressive income tax system?

Excellent question. You can understand this by reading the original study referred in the article. I will try to provide a short explanation here.

Basically the study argues that if you take into account all sorts of taxes like taxes on properties, consumption, dividends, etc. the European countries tax their rich residents relatively less than the poor residents, while the US does the opposite. This is possible because there are other significant taxes besides the income tax. As the study states:

  The main reason for this is the relatively low reliance of the
  American fiscal system on sales taxes, which constitute a much higher
  share of revenues in other countries (even when including state and
  local taxes in the US case, as the figures above do). General sales
  taxes place a disproportionate share of the tax burden on the lower
  end of the income distribution because expenditures are higher as a
  share of income as we move down the distribution.

To gain an exact understanding of the key claim in the WP article we have to understand both the Lorenz Curve and the Kakwani index. The Lorenz Curve has a pretty good Wikipedia page, I recommend browsing it. The Kakwani index is chiefly used to calculate payments for health services.

  The Kakwani index is defined as twice the area
  between a payments’ concentration curve and the Lorenz curve and is
  calculated as, $\pi_K = C - G$ , where $C$ is the health payments’
  concentration index and $G$ is the Gini coefficient of the ATP variable.
  The value of $\pi_K$ ranges from –2 to 1. A negative number indicates
  regressivity; $L_H(p)$ lies inside $L(p)$. A positive number
  indicates progressivity; $L_H(p)$ lies outside $L(p)$. In the case of
  proportionality, the concentration lies on top of the Lorenz curve and
  the index is zero.

Source

The difference in the Kakwani index is small though, the range of the graph in the article is in the hundredths.


Why would Switzerland, with its export-driven economy, worry about weakening against the dollar?

Its not totally clear, but it seems that what they are saying is that, as long as the euro was a strong currency, they did not want to have the frank much stronger than the euro. Later, on, when the euro lost value, they no longer wanted to insure that the franc wasn't much stronger than the euro.

In other words they don't want a very strong currency, but they are ok with having a stronger currency than their neighbors.

The question includes just one of the concerns for central bankers. It says that the Swiss export to their neighbors, so they don't want their products to be uncompetitive with those of other Europeans. That would mean less purchases, lower GDP etc. This was the logic used when they imposed the cap on the franc price relative to euros.

However, the central bank has another main concern, which is that it wants to make sure the currency remains a good asset for investors. It has to make sure the currency doesn't lose value, otherwise, it hurts all those investors that trusted in it. So, for the world wide capital markets to demand the Swiss franc, the central banker needs to maintain a reasonably high value for it. This is the core concern when they are allowing the franc to increase in value relative to a (falling) euro.


Are debt/GDP ratios calculated with real or nominal GDP as the denominator?

Usually, the debt/gdp ratio has no unit. For instance, if the debt is equal to 60% of the gdp, the debt/gdp ratio is 0.6. If you use real gdp as denominator and nominal debt as numerator, you end up with a number which is less clear to interpret. It is more relevant to have either both variables in nominal terms or both in real terms. In these two cases, the result is the same.

If you denote D and GDP respectively nominal debt and nominal GDP. As long as you take the same price index P, the ratio in nominal variables, D/GDP, and the ratio in real variables, (D/P)/(GDP/P) are mathematically identical.


How purchasing power reduces when currency is commodity money(say gold)

Here is a chart from The Economist in July 2010 showing the nominal and real (i.e. inflation adjusted) price of gold in US dollars back to 1970.  The nominal price later continued to rise until 2012 and then fell, and it is currently at about its 2010 level; the real price also rose and then fell and is currently below its 2010 level.  The lines would look different in other currencies

So while the purchasing power of most currencies including the US dollar tends to fall over ten-year periods due to inflation, the purchasing power of gold (shown by the real line) may rise or fall depending on precisely which ten-year period you choose.  For example it rose from 2000 to 2010 and from 1970 to 1980 but fell from 1980 to 1990 and from 1990 to 2000; indeed from 1980 to 2000 the purchasing power of gold fell faster than the purchasing power of the dollar

An additional consideration is that you can store your currency in a bank by depositing it, and you can then receive interest on it, which may offset the decline in the purchasing power. If you store gold in a bank, the bank is likely to charge you a storage fee and you will not receive interest


How to deal with misspecification of models in Macroeconomics?

Thomas Sargent and Lars Peter Hansen have a series of work on "robustness", a theory (and technique) that deals directly with potential model misspecifications in the context of macro. [The term "robustness" derives from robust control theory].
Sargent's website has a page dedicated to this topic. The two also published a book, Robustness, that contains the main results of their work to date. Below is the publisher's description:

The standard theory of decision making under uncertainty advises the decision maker to form a statistical model linking outcomes to decisions and then to choose the optimal distribution of outcomes. This assumes that the decision maker trusts the model completely. But what should a decision maker do if the model cannot be trusted?
Lars Hansen and Thomas Sargent, two leading macroeconomists, push the field forward as they set about answering this question. They adapt robust control techniques and apply them to economics. By using this theory to let decision makers acknowledge misspecification in economic modeling, the authors develop applications to a variety of problems in dynamic macroeconomics.
Technical, rigorous, and self-contained, this book will be useful for macroeconomists who seek to improve the robustness of decision-making processes.


Intrinsic value of non-voting shares which don't pay dividends

Just because a share doesn't presently pay dividends, doesn't mean it never will. The gamble is that eventually, it will pay dividends. So the share's value is the market's estimate of the net present value of a future cashflow discounted by two factors: one, the discount rate to reflect that this is a future cashflow not a present one; and two, the probability the cashflow will start later, or never.


Software for field work: collecting survey data (multi-platform, on-line and off-line)

I've worked in 2015 on a survey collecting ~300 answers from 130 different countries. We had similar requirements to what you described, with multilanguage functionality on top. it was not a RCT (Randomized Controlled Trial), we were surveying on conditions of work.

We ended up using SurveyMonkey which costed 900€ and had only the online functionality, which is a mobile optimized web page survey (compatible with any system), very fast to set up and with the ability to skip questions and add pictures. We could not capture GPS data (never heard of surveys doing that), but we could get the IP address of respondents, which can be a proxy for their location. Pictures were also possible to add, but we used it only to brand the survey.
We used Word-files via email for "offline answers", which we then manually inserted back on the website. We didn't have much field work, but I know it would have worked great to fill the survey in a phone or tablet easily, as long as there was internet access.

The main thing that made us take SurveyMonkey vs using self-hosted open-source software like LimeSurvey was:

The ability to set up quickly (your survey is up as soon as you insert the questions, no IT configuration time needed),
Speed of connection for respondents: our server in Europe would mean long loading times for the respondents in other continents, and SurveyMonkey has powerful servers everywhere
Text mining features for open-answers and pre-made dashboards to share quick and high-level results

If you want to look at other providers, most of the surveys I've seen recently used Qualtrics, Typeform, SurveyMonkey or GoogleDocs.


Why does lowering the value of yuan against the USD disadvantage the US and benefit China

The article reflects a consensus view of the trade situation between US and China.  Personally I see it differently :  China has been willing to make and ship trillions of dollars’ worth of goods to US in return for entries in a computer (Treasury bonds).  This is undoubtedly good for US consumers, as you say.


To get back to the original production function just multiply both sides with capital. Here:
$$Y/K = A (N/K)^{1.1} = F(N/K,1) \implies Y = AN^{1.1}K^{-0.1}  =F(N,K)$$
Also, the marginal product of labor here will be:
$$F_N'(N,K) =  1.1 A N^{0.1}K^{-0.1} = 1.1 A (N/K)^{0.1}$$


What separates "rich" developing nations from "poor" developed nations?

All country classifications are conventions, which depend on the rule used by the particular institution. There is no hard (ontological) rule on any classification. There is no even consensus on the definition of categories. Advanced/rich/developed/industrialized economies are sometimes used interchangeably, but who is in each of them depend on who is talking. Let see an example:

International Labor Organisation (ILO)

(I had to cut it because it is too long. See the full list in page 67 of this report).

Citigroup

(Report here).

Conclusion

As you can see, ILO puts Czech Republic, Greece, Argentina, and Chile as developed countries, whereas Citi puts them as emerging countries (Argentina as emerging can be seen in other reports). (ILO even distinguishes between emerging and developing).

This shows that there is no hard-coded rule about classification of countries. It all depends on the institution.


Asymptotic variance vs. conditional variance in calculating regression estimator standard errors

So the asymptotic variance in the first expression that you give is the variance under heteroskedasticity as you note. The definition you gave of $V(b|X)$ is under homoskedasticity. As one of the comments states, we could use $V(b|X)$ (as you define it) under this assumption. It seems that Hayashi uses this notation because he wants to consider heteroskedastic errors.


As of April 2018, the tariffs are targetted at products in which Chinese companies dominate.  They would apply to products made by American companies in China and imported to the US, but the exact products covered have been chosen to minimise this effect.  Nevertheless,

  it will most likely increase costs for American manufacturers that depend on imported parts because it concentrates heavily on machinery and high-tech components.

In other words, US firms that use components made in China by Chinese companies as inputs in creating more complex products will be harmed.  For instance, Harley Davidson have said that

  Import tariffs on steel and aluminum will drive up costs for all products made with these raw materials, regardless of their origin.


How does Infrastructure impact on Aggregate Supply?

Aggregate supply is typically modelled as

$$ Y = F(A, K, L) $$

where

$Y$ = total production

$L$ = labour input

$K$ = capital input

$A$ = total factor productivity

Investment in infrastructure can increase both capital input $K$ and total factor productivity $A$. In a standard $AS - AD$ model, the aggregate supply curve would shift to the right, as a result, and the equilibrium output of the economy would increase.

Investment in infrastructure can increase capital input, because if there are more roads in the country, the economy can produce more goods and services (think about a truck company, for example, which can do more jobs per day if its trucks travel on more direct roads or faster highways).

It can also increase total factor productivity, because the proportional increase in output is potentially higher than the proportional increase in capital input. If with a short road I link a town to a national highway network, the benefits can be disproportionally high, as a short road link now provides access to a huge existing highway network. Or think about the 50 mile Panama Canal. It provides ships an 8,000 mile shortcut.


Endogenous Gentrification and Rates of Return to Education

@denesp comment is right to the point as regards the scope of this question. In street-talk "Gentrification" just means "things get more expensive" ("rightly" or "wrongly", that's another matter). The displacement of poor people will happen, for those who remain poor, where here "poor" is a relative concept "poor enough not to be able to afford the new cost levels".

What I believe is interesting here is to ask :"Assume that the amenities of an area increase in value, and so rents and "cost of living" goes up. If this value-increase of amenities is also increasing the productivity/efficiency/earnings of the local population, which effect will dominate?"

Will the income-increase of local residents be enough to cover the increased costs of living?

Instinctively speaking, I would say "no", if the productivity-increase comes through education, because the phenomenon of price-increases in such areas happens much faster than the time needed for education to bear income fruits.


Is it sufficient to conclude by looking the following weekly demand schedule that the good is Perfectly Elastic?

No, you do not have sufficient information to conclude that the demand is perfectly elastic.

The following figures show two ways that demand curves that are not perfectly elastics could have generated the first two data points in your series:

It should be easy to see how to add other demand curves to generate the rest of the data points.

This is actually a common and more general problem: price and quantity are jointly determined by the interaction of supply and demand. Given two data points, it is impossible to determine whether they were caused by a shift in supply or demand or both. There are two solutions to this dilemma:

estimate a simulatenous equation model that accounts for both supply and demand. You can read about simultaneous equation models in a good econometrics textbook or, for example, in Dan McFadden's lecture notes.
work in a context where you know either the supply or the demand curve are fixed. If you know the supply curve didn't move then any change in price or quantity must have come from a shift in demand and you know that all of your data points lie on (and characterise) the supply curve. The converse is true if you that the demand curve did not move.


If Bitcoin becomes a globally accepted store of value, would it be liable to the same problems that mired the gold standard?

It would not just be plagued by the same problems it would create some new ones.
Following the Weber (2016) who actually written research exploring exactly the question that you are asking here:

The scope of monetary policy would be more limited under the Bitcoin standard than
under the gold standard. The ability to issue fiduciary currency would give central banks
limited ability to act as lenders of last resort. However, virtually costless arbitrage of Bitcoin across countries would prevent central banks from implementing interest rate policies to affect their domestic economies.

An empirical examination of countries’ experience with the gold standard leads to the
following conjectures about how the Bitcoin standard might perform:

In the long run, there would be moderate deflation that would increase over time until
reaching a rate of deflation equal to the negative of the rate of growth of world output
around 2026.
Price levels of the various countries would be highly, but not perfectly, correlated, much
as they were under the gold standard.
Exchange rates among the fiduciary currencies of various countries would be fixed at
par, because the cost of Bitcoin arbitrage is essentially zero.
There would still be financial crises, because they can occur under any fractional reserve
financial system.

The paper concludes by speculating that even if the Bitcoin standard were to come into
existence, it would not last long, for two reasons: (1) The payments world is changing so
rapidly that there will be a technological innovation that provides a potential medium of
exchange with the same or greater benefits of Bitcoin or with lower costs. Such an innovation
could come either from the private sector or from the government. (2) There would be
pressure to return to a fiat money system so that a more activist monetary policy could be
pursued.

In a nutshell, it would be pretty similar to gold standard but with some extra strings attached that would just additionally constrain monetary policy making it even slightly worse. As a consequence, if Bitcoin standard would ever become adopted it would likely perform even worse than Gold standard did (which is saying something), and would likely go the same way.  You can find more detail on the workings of such standard in the paper itself.

Response to Edit:
The above still applies but let me also respond to the new infographic that was added to the question.

Scarcity - This is exactly why Bitcoin would perform worse, and it is connected to what was written above.
Durability - Given that gold has very long half-life and under gold standard it was mostly stored in vaults not actually used (hence no wear & tear) this never was issue with gold standard.
Portability - Same as above, under gold standard people mostly moved claims to gold, actual gold moved from vault to vault only occasionally. Also note in both Bitcoin and Gold standard people would not pay with Bitcoins or Gold respectively, they pay with banknotes that are backed by either Bitcoin or Gold respectively (and which would be on fixed exchange with banknotes e.g. 1USD=0.1 Bitcoin for example).
Divisibility - This is moot point. You can have 0.000001 troy ounce. In fact gold is for all practical purposes infinitely divisible (up to the scale of individual atoms of course).
Storage - this was never significant issue with gold standard, in addition people still store investment gold in vaults its just not tied to money.
Counterfeit - Again completely moot point. Under gold standard gold would mostly just sit in vaults. People would not pay with gold coins. Banknotes backed by either Bitcoin or gold would be equally difficult to counterfeit as regular fiat banknotes since that is what people would be using as money under Bitcoin standard.
Adoption - Again not relevant, if we talk about Bitcoin standard adoption requires fixed exchange between a currency and Bitcoin - to my best knowledge no country did this to the date. Also, I never heard of use of market capitalization as measure of adoption when it comes to monetary system so the metric does not provide any argument either in favor of gold or Bitcoin.


Example of a utility maximization problem with a non-binding budget constraint

Assuming positive prices, any utility function that doesn't satisfy strong monotonicity of at least one good would have a non-binding budget constraint. Since you asked for some examples:

$U(x, y) = 0$,
$U(x, y) = -x$.

Re: Giskard's comment.

I'm not sure I used the correct term here when I said "strong monotonicity of at least one good." To be precise, what I meant to say was:

Let $U(x_1, x_2)$ be the utility function. The function would have non-binding budget constraint if there are some $(x_1, x_2)$ such that $dU/dx_1 \leq 0$ AND $dU/dx_2 \leq 0$.


Was there a Role of Technology in Macroeconomic Models before Solow?

The Solow model was indeed the first model to account for technological progress.

While models like the the  Harrod-Domar model and the Neo-Marxian  Feldman–Mahalanobis model conciser investment and saving as contributors to economic growth, they don't consider any measure of technology like total factor productivity to contribute to economic growth, like the Solow Residual.


Why is economic growth correlated with long-term interest rates?

Very simply, economic growth is correlated with long term interest rates in the sense that interest rates are set according to the growth of the overall economy and GDP. If economic growth is weak, interest rates are set very low as a means of economic stimulus. If economic growth is too strong due to demand and the economy is overheating, interest rates are slashed.

The Federal Reserve alone determines interest rates, as well as the federal funds rate and the discount rate, through the Federal Open Market Commitee. Visit Federal Reserve System


So let's decided that decision A is taking the certain sum and decision B is taking the risky sum, where $\bar{w}$ is current wealth and $\mathcal{U}$ is a utility function where $\mathcal{U}'>0$ and $\mathcal{U}''<0$, then you have two potential outcomes in utility.  The first is $$\mathcal{U}(A)=\mathcal{U}(\bar{w}+2000)$$ and the second is $$\mathcal{U}(B)=\frac{\mathcal{U}(\bar{w}+10000)+3\mathcal{U}(\bar{w}+500)}{4}.$$

Given only this information, the decision depends upon $\bar{w}$, the shape of $\mathcal{U}$, the payoff outcomes, and the probabilities involved.  If you wanted to determine how sensitive a decision would be to change, you could treat the currently fixed factors as variable factors one at a time.  To provide an example, you could create a new variable $\pi$ so that option A would become $$\mathcal{U}(\bar{w}+\pi).$$

You could also do this to substitute for $\bar{w}$, the other payoffs, or the odds.  If you added the assumption that the actors were indifferent, then you could also use the Envelope Theorem to look at local sensitivity.


The word "distortion", (as used in e.g. the Ramsey–Boiteux pricing), is not intrinsically negative.

It simply means that people change their consumption behaviour because of changes in price(s).

For example, a tax on oxygen would not (and cannot)  be distortive because people have no choice but consuming oxygen.

A contrario, a tax on tabacco is distortive, and is actually precisely implemented because of this very trait: it will change the consumption behaviour of smokers.

Put differently, the notion of distortion is strongly related to the notion of price elasticity.

Also, constraining markets equilibrium can generate deadweight losses, which is likely the reason why people think of the word "distortion" as negative. Is changing behaviour via price signals negative per se? Is trying to make people quit smoking negative per se ? No.

The word "distortion" is probably one of the most misconceived/misused notion in economic theory, especially when it is used in the political sphere. But originally it has, I repeat, no negative connotation. And I really think that this is how Mankiw uses/perceives it.

  Why is price dispersion from Inflation called relative price “distortion”?

It is called so because consumption behaviors do not change if all increases of prices in the economy are of the same relative magnitude $\iff$ the dispersion is null. E.g. a tax on, say, tobacco, has to be distortive to be useful, i.e. it has to be greater than inflation, even if such intervention can actually generate deadweight losses.


By what steps and contracts does new money enter the economic system?

The phrasing of your question (the same is true for your description of your understanding) suggests to me that what you need to understand about money creation in banking is much more basic than "steps and contracts". As a full treatment of money creation would not be appropriate on a Q&A site (it'd be really long!), I will instead recommend that you first read about the different types of money (specifically, the difference between base money and broad money), and then that you read "Money Creation in the Modern Economy", which will answer your question about how banks create money.

The short answer to your question is that banks don't "receive" money to loan out as a part of any formal arrangement, rather, they are allowed to lend out a large fraction of the money that is deposited with them. Importantly, when they give someone a loan, the money that is loaned will tend to be deposited at a bank (because banking is a mostly closed system), which gives the bank that receives the deposit the option to loan out more money. That's how money creation happens— loans that, through the magic of double-entry accounting, result in the bank swapping one kind of asset (reserves) for another (an outstanding loan), while the borrower suddenly has a new liability (the loan they owe to the bank), and a new asset (cash).

When we talk about money creation, we are not saying that banks literally create money in the sense of asking that some be printed up for them, it's that the banking system acts in a way that generates total liabilities (or claims on others) that are much greater than the monetary base. Because all money is really just claims on others, this is, in effect, money creation.


If I don't pay a debt, then the creditor takes my goods. Why, then, do Greek creditors not take Greece?

In general, there are three kinds of debt:

Secured debt, like a mortgage or a repurchase agreement. With a mortgage, for example, the debt is secured by a lien on the home, and if the debtor does not pay, the creditor can seize the home.
Unsecured debt, like a credit card or corporate bond. Governments will generally allow creditors to liquidate many of the assets of an insolvent individual or firm to meet unsecured obligations, subject to (often significant) limits. For individuals, things like retirement accounts and primary residences are often off-limits for debt collection.
Sovereign debt. Sovereigns typically issue debt under their own law, and countries generally don't set up their legal systems in such a way as to allow other countries to take their things. However, when sovereigns issue debt under foreign countries' law, as Greece has done with some UK-law bonds, payouts tend to be higher, and overseas property is in fact occasionally seized. However, seizing the domestic property of a defaulted sovereign is effectively impossible, as one of the benefits of being a sovereign is that you control your own territory— attempting to seize another nation's "structures, cities, industries, lands, etc." (setting aside the fact that most structures, cities, industries, and land in Greece are not owned by the government) is what is known as an act of war.


Profit for business owners vs. incomes of the self employed

I'm guessing you may be confused why we add both wages of self-employed people who run businesses, and profits of businesses, as they should be the same given that profits of businesses go to their owners.

The important fact is that when we calculate the profit of each business, we deduct wages (even to their owners) from their revenues and the left-over is what we add as profit.

Therefore, the wages deducted is added back into the GDP via wages of self-employed people.

Hence, there is no double-counting in the Income Approach to GDP calculations.


Is the marginal cost the same for every firm in a perfectly competitive market?

No, the marginal cost curves are not necessarily the same for each firm in the market. However the values of marginal costs are.

To disprove the general claim that "The marginal cost curve of each firm in a competitive market is the same" we simply need to find one counter-example, such as the one given below:

Suppose there are two firms in the market and the market inverse demand function is given by $P=7 - q^m = 7 - (q_1 + q_2)$

Here $P$ represents the price, $q^m$ the market supply and $q_1, q_2$ represent the respective supplies of the firms 1 and 2.

Suppose the marginal cost curves for each firm are given by:

$MC_1 = 2q_1$,

$MC_2 = 4q_2$

Note that these are not the same! Since we are looking for a competitive equilibrium, the firms behave competitively and hence we must assume that they do not realize they have market power. Hence they each set:

$P = MC_i, \; i\in\{1,2\}$

From this, solving for quantity we have the respective Supply functions of each firm:

$S_1 = q_1 = 0.5P$

$S_2 = q_2 = 0.25P$

Inserting these two supply functions into the inverse demand function we have:

$P=4$, $q_1=2$, $q_2=1$.

Is this an equilibrium with perfect competition? Well we found it by combining Supply and Demand assuming perfectly competitive firms, so it must be. To verify:

$Market \; Demand = 7 - p = 3$

$Market \; Suppy=q_1 + q_2 = 3$

So the market is indeed in equilibrium. Are both firms setting $P=MC$ as they should in perfect competition?

$MC_1 = 2*q_1 = 4 = P$

$MC_2 = 4*q_2 = 4 = P$

So indeed we have a market equilibrium with perfect competition and different marginal cost curves. QED

Note however that the values of the marginal cost curves at the equilibrium quantities are the same for each firm (and must be the same for each firm in equilibrium).


Studying growth in US and Sweden, what variables should I watch out for in a regression?

"Solow's Growth Accounting"

A first pass idea is what MIT's Bob Solow proposed many decades ago: to do a growth accounting exercise. You'd like to know how much physical capital changed and how much human capital changed and then with some assumptions or estimation of the relative importance of each of these then estimate how much total factor productivity changes. One way to do this is by running a regression of (log) output on (log) human capital, and (log) physical capital, for data from both the US and Sweden. The errors are the "total factor productivty" component or "Solow residual"....


How exactly is a Bertrand equilibrium defined?

What do we mean by "makes the best decision based on what the other player has decided to do"

Your question touches a little bit on the philosophical foundations for Nash Equilibrium. As you know, a Nash Equilibrium occurs when each player chooses the best response to the strategies chosen by others. In other, words, they act as if they know what the strategies of the other players will be, and play the best response accordingly—even if all strategies are chosen simultaneously.

Why does it make sense to model people behaving as if they knew everyone else's strategy? The whole point of looking for an equilibrium of a game is to try to find a way to 'solve' the game such that the solution gives a plausible prediction about how people might play the game. Suppose we have identified a way to compute the solution. If we can compute the solution then the players can compute the solution too (using the same method), and figure out what everyone's equilibrium strategy will be. If the players compute the solution and discover that their equilibrium behavior in the solution does not maximise their own payoff then they will refuse to play the solution! So any solution concept that (a) players can use to solve the game and (b) has one or more players behaving sub-optimally will be self-defeating. Flipped on its head, any solution concept that players are actually willing to go along with must necessarily involve them playing a best response—i.e must be a Nash Equilibrium.

One concise way to phrase this is that a player's strategy can be accurately predicted by the other players (using the same solution concept we are using the solve the game)! Note that this is slightly different to what you wrote in a comment ("the competitor's action can be accurately predicted"). We can use the equilibrium to predict strategies, but might not be able to predict actions (because the strategy might be a mixed strategy, meaning the action is random).

What is a Bertrand Equilibrium

Recall that to set up a static game of complete information we need to know (1) the players, (2) the actions, and (3) payoffs. To turn to your question, a Bertrand Equilibrium is just another name for the Nash Equilibrium of the following particular static game of complete information:

There are N firms (players)
Each firm's action is to choose a price, $p_i$
The payoffs are that the lowest price firm serves all consumers and earns $p_i-c$ per-consumer (where $c$ is the marginal cost). If $m$ firms tie at the lowest price then each of them serves $1/m$ of the consumers. firms that serve no consumers earn zero profit.

So, in a Bertrand equilibrium, using the principle of Nash, each firm acts as if it knows the others' price and chooses the best price in response.

Characterising the Bertrand Equilibrium?

To find the Nash ("Bertrand") equilibrium of this game, we have to find a set of strategies (prices) such that each firm is playing a best response (i.e., such that no firm could profit by changing its price).

Claim There is no equilibrium in which the lowest firm's price is bigger than $c$.

To prove this claim, suppose on the contrary that $\min p_i>c$. We will show that at least one firm can profitably change its strategy (and is therefore not playing a best response). Order the firms so that $p_1\leq p_2\leq p_3\leq\ldots\leq p_n$. There are two possibilities:

All firms charge the same price (so $n$ ties at the lowest price). Then $n$'s profit is $(p_n-c)/m$. But suppose $n$ reduced its price by some amount $\Delta$. Then it would have a price strictly below anyone else and earn profit $p_n-\Delta-c)$. If $\Delta$ is small enough, this is a strict increase in $n's$ profit.
$p_n>p_1$ so $n$ does not attract any consumers. $n$'s profit is zero. But $n$ could deviate to choosing a price of $p_1-\Delta$, yielding a profit of $(p_1-\Delta-c)$. Provided $\Delta$ is small enough, this is positive so $n$'s profit has increased.

Claim At least two firms must set $p=c$ in equilibrium. All firms earn zero profit.

To prove this claim, start by noticing that if some firm sets $p_i<c$ then at least one firm will make negative profits and would prefer to deviate to $p=c$ (thus guaranteeing zero profit).

Suppose that fewer than two firms set $p=c$. By the previous claim, we know at least one firm sets $p=c$. This means precisely one firm sets $p=c$, while all others set $p>c$. But then the firm that sets $p=c$ earns profit $(p-c)=0$, but could get positive profit by increasing $p$ (just a little so it is still the cheapest firm). Because there is a profitable deviation, we can't have an equilibrium with fewer than two firms setting $p=c$.

Lastly, we know the firms that set $p=c$ earn zero profits. We also know that no firm sets $p<c$ so any firm not setting $p=c$ must set $p>c$, meaning it attracts no consumers and also makes zero profit.


Forgive me in advance if this is already familiar. I'll be talking about stability of equilibrium the way I (briefly) learned it.

Let $\xi^j(p)$ denote the excess demand for good $j$ given price $p$.

$$\xi^j(p) = \sum^m_{h=1}\left( x^j_h(p) - e^j_h \right) = x^j(p) - r^j$$

For $m$ households/consumers, $e$ being an endowment to a household, and $r^j$ being the total resource available of good $j$. We say the rate of price adjustment over time $\left(\frac{\partial p^j}{\partial t}\right)$ is proportional to excess demand.

For either of your market clearing prices $p^*$, we then know that $x^j(p^*) = r^j \ \forall j \implies \xi^j(p) = 0$
Otherwise,

$\xi^j(p)> 0 \implies \text{excess demand} \implies \frac{\partial p^j}{\partial t} > 0$

$\xi^j(p)< 0 \implies \text{excess supply} \implies \frac{\partial p^j}{\partial t} < 0$

Out of the three equilibrium you have up there, we'd probably want to find one that was locally stable and say that's where the market is more likely to realistically go.

Steady state $p^*$ is locally stable if $\exists \epsilon > 0 \ \text{s.t.} \ \forall \ p(0) \in B_\epsilon (p^*)$, an open neighborhood with radius $\epsilon$ around $p^*$, then we have $p(t, p(0)) \rightarrow p^*$

I don't want to fuddle this too much with math (particularly when you linearize the system and have to deal with eigenvalues/eigenvectors), so let's just take a look at your supply and demand curves above.  We can peruse the math on our own time.

Case A: Let's say you are at point A, your nice equilibrium. Suddenly there is a downwards quantity shock and you are now to the left of $q_A$. Since there is excess demand, the change in price over time should be positive, so the price will rise, and you will arrive back at point A.

What if you are shocked to the right of $q_A$ in the short run? Now there is excess supply and price over time should fall. If you were still above $p^*_A$ at this point, you will arrive back at point A. Otherwise price will fall all the way until you reach point B.

Case B: We know what happens with a shock to the left of point B now. What about to the right? If point C looks like a global minimum of the supply curve from what it looks like, then any shock that gets you before point C gets you excess demand, which means the price will rise and you will arrive back at point B.

Case C: To the right of point C, there is excess supply, and prices will fall back down until you reach C again.

None of these equilibria seem globally stable. It seems C is not locally stable on the left, so I wouldn't say the market would hang out there for very long. Points A and B are locally stable, but it seems B has greater room to get shocked and still move back towards B.

So for this scenario, I personally like B as your long run equilibrium.

If you had extra information about how shocks in the market happen (how often, variance, etc.), it may allow you to describe what percentage of the time you stick around at each equilibria, and you could describe an aggregate equilibrium like that as well.


I think your notation makes this more confusing that is necessary since $A$ and $B$ represent sets of alternatives in one setting and alternatives in the other setting.
So write IIA as:
If $x\in C\big(\{x,y\}\big)$ and $y\notin C\big(\{x,y\}\big)$, then $y\notin C\big(x,y,z\big)$.
To prove this from choice coherence, simply choose $A$ and $B$ appropriately.


What impact did major fiscal policy actions taken during the Great Recession (2007-09) have on U.S. economic performance?

The fiscal policies to fight the great recession in the United States were not administered in a way that would allow us to assess this a natural experiment. However, there have been many situations in the past where there was such a natural-experiment setting, and that allows us to approximate the effect of these fiscal policies. For example, if a dollar of additional government spending, when such variation is experimental in nature creates two dollars in private consumption and investment (a multiplier of two), then when the government spends a trillion dollars in a way that may be endogenous, you could guess that the effect on private consumption would be something like two trillion.

The CBO used this methodology in their report Estimated Impact of the American Recovery and Reinvestment Act on Employment and Economic Output from October 2010 Through December 2010 (2011):

  The American Recovery and Reinvestment Act of 2009 (ARRA) contains
  provisions that are intended to boost economic activity and employment
  in the United States. Section 1512(e) of the law requires the
  Congressional Budget Office (CBO) to comment on reports filed by
  recipients of ARRA funding that detail the number of jobs funded
  through their activities. This CBO report fulfills that requirement.
  It also provides CBO's estimates of ARRA's overall impact on
  employment and economic output in the fourth quarter of calendar year
  2010. Those estimates—which CBO considers more comprehensive than the recipients' reports—are based on evidence from similar policies
  enacted in the past and on the results of various economic models.

Key findings:

  On that basis, CBO estimates that ARRA’s policies had the following
  effects in the fourth quarter of calendar year 2010:

  They raised real (inflation-adjusted) gross domestic product (GDP)
  by between 1.1 percent and 3.5 percent,
  Lowered the unemployment rate by between 0.7 percentage points and
  1.9 percentage points, Increased the number of people employed by between 1.3 million and 3.5 million, and
  Increased the number of full-time-equivalent jobs by 1.8 million to
  5.0 million compared with what would have occurred otherwise, as shown in Table 1. (Increases in FTE jobs include shifts from
  part-time to full-time work or overtime and are thus generally
  larger than increases in the number of employed workers.)

Another CBO paper, The Fiscal Multiplier and Economic Policy Analysis in the United States (Whalen and Reichling (2015), discusses an economic parameter at the heart of these estimates, the fiscal multiplier.

  The recession from 2007 to 2009 sparked wide interest in the economic
  effects of fiscal policy.  That interest is reflected in an ongoing
  debate over the size of the fiscal multiplier. This  working paper
  addresses three questions: What models do economists use to estimate
  that multiplier?  Why do estimates of it vary widely? How can
  economists use those estimates to judiciously analyze  U.S. economic
  policy?

Tables 1 and 2 are of particular interest, they can help you understand the estimated contributions of various parts of the stimulus to the recovery by time and project type.


Let $v_i$ be the valuation that the $i$-th consumer has for the product, and $v^*$ the common valuation of the product, where the $v$'s are measured in monetary units, and are assumed continuous.  Consumers are here assumed binary -they either purchase $0$ or $1$ (i.e. the product is indivisible). We assume that if the valuation is equal to the price, they do want to purchase the product.

Write the indicator function $I\{v_i\geq p\}$, that takes the value $1$ if the valuation is greater or equal than the price (and so the consumer will want to buy the one unit of product. Let there be $N$ consumers. Then, demand can be expressed as

$$Q^{d} = \sum_{i=1}^NI\{v_i\geq p\}$$

and since the valuation is assumed common to all,

$$Q^{d} = N\cdot I\{v^*\geq p\}$$

We see that  if $p\leq v^*$, quantity demanded will be equal to $N$, while if $p> v^*$ quantity demanded will be equal to $0$.

Diagrammatically, this gives


Should we expect more structural technological unemployment if growth becomes more limited by natural resources and less by labour?

I understand that you are asking the following

  Up to now, we have avoided structural unemployment because we increased
  production by more than what the technological efficiency gains
  implied (and so eventually re-employed the labor that had become
  initially obsolete, usually in other industries). But if finite resources put constraints on how
  much we can increase production, won't a time come that structural
  unemployment will emerge?

As @Foobar notes in a comment, technically speaking this appears to have to do with the rate of increase of efficiency in resource use compared to the rate of decrease of resources available.

But one could argue that, efficiency in resource use has its limits, and the constraint will eventually set in. In such a technological unemployment scenario (which I don't believe anyone can really put a reliable time-line on), there is one resource that won't decrease but rather increase in availability: labor. So, and if nothing truly life-changing happens (like devastation of human population, massive colonizations of other planets, or the equivalent of Amalthea's Horn of Plenty), production will tend to utilize this available resource (which we should expect to command a relatively lower reward being in high supply). Meaning, a tendency for production to grow in sectors where services delivered by humans remain central, by their nature (or by the prevailing consumer preferences). Servants, personal assistants, waiters, nurses, sex workers, trainers, teachers, baby sitters, security guards, body guards, arts and sports entertainers... even if only as a relative "luxury", rather than strictly needed in the amounts employed. And we may even see innovative ways of using labor (i.e. once more creating new sectors) -but I leave that to anyone's imagination...
Of course this labor should get paid something for its services, so those employing it should have the income/wealth needed to pay this something... which indirectly brings in the issue of wealth and income inequality, and whether it plays (or not) a part in all these.


Why is the Market-Clearing Price also an Equilibrium Price?

(This is referring both to the initial question and to the comment. I would respond to each individually, but I don't think I have the reputation to address the comment directly.)

Paraphrasing a bit from your question, you're right- an equilibrium price is one where no firm has an incentive to deviate in their pricing or production decisions, and demand is perfectly satisfied (again, given the price). While the exact "mechanism" that would push price down might differ depending on what market type you're considering, the common feature will be that there is always a marginal incentive to deviate down to the lower price.

In terms of perfect competition- as you mentioned, it can be seen as a race for different firms to undercut their competitors, until they hit the no profit condition. That pushes the market clearing price down to marginal cost, which is the equilibrium price.

For monopolists, they're looking to maximize profit, not necessarily revenue. But, as every firm does, they choose a price p to satisfy:

$$
max\{q(p-c)\} = \pi
$$

They will choose a price higher than the one found under a perfectly competitive framework; hence why the market clearing price, and equilibrium price, is higher under monopoly than perfect competition.

But if the monopolist were to set a price different from the one that satisfies that maximization problem, they are pricing, as you described, too high. That tells them when to produce more units at a lower price; until their profit maximization price is hit. That becomes the clearing price, and the equilibrium price for a monopolist.

I'm sorry if that was too repetitive or elementary. You're question is an excellent one, and a somewhat subtle point has to be made. If it wasn't clear, or you'd like it re-framed, let me know!


Have you seen http://faculty.econ.ucdavis.edu/faculty/dlmiller/statafiles/ ?

I see some entries there such as Multi-way clustering with OLS and Code for “Robust inference with Multi-way Clustering”.

EDIT: At least we can calculate the two-way clustered covariance matrix (note the nonest option), I think, though I can't verify it for now. See the following.

set more off
clear all
local n 50
local T 50

set obs `=`n'*`T''

gen id = floor((_n-1)/`T')+1
by id, sort: gen year = 1970+_n
xtset id year

set seed 1

gen x = rnormal()
gen y = 1+0.5*x+rnormal()

xtreg y x, re vce(cl id)
mat b = e(b)
mat V1 = e(V)
xtreg y x, re vce(cl year) nonest
mat V2 = e(V)
xtreg y x, re vce(cl id year) nonest
mat V12 = e(V)
mat V = V1+V2-V12
mat l V
ereturn post b V
ereturn display

set more on

This code has yet to be verified manually.

EDIT 2: Sorry for keeping editing. Just found that Stata's reg (for pooled OLS) does not allow for clustering by multiple variables such as vce(cluster id year). We should use vce(r) or just r. However, it seems that xtreg does (usually requiring nonest), though I counldn't find documentation.


Given that this is a macroeconomic study, the cost to write a script to average your data is simple and effectively costless. If you obtain your variables based off of monthly formats, then it would be wise to also obtain your other variables also in the same format.

Each time you simplify, you may potentially lose small but significant outliers in your data that are averaged out as part of your calculations. But if you have more granular data at hand, you can revisit your figures and revise your analysis as needed.

In other words, if you can do your analysis month-by-month I would recommend that you do so, it is relatively simple to scale up and analyze it by quarter as needed; it is however difficult to 'scale' down, don't mix and match two variables measured by different time segments.


Converse in supermodularity and single-crossing

I'm not sure there are enough assumptions here to move forward, in particular I would think you would need some conditions on your function $h$, at the very least. It's pretty easy to show that $g(\cdot,\cdot)$ will have increasing differences, which implies the single-crossing property for $g$ (this can be quickly seen by letting $h$ be identically zero). But I don't see how this is enough, since single-crossing doesn't imply supermodularity. Maybe I'm missing something, though.


The first-order condition would be satisfied under the assumption $c=G(l)$ if $U'(0)=0$. But this latter requirement is incompatible with the assumptions: indeed, $U'$ is nonincreasing (by the concavity of $U$) which yields $U'(x) \leq 0$ for any $x \geq 0$, which contradicts the fact that $U$ is increasing.

If $U'(x)>0$ for all $x$, the only possibility to meet the first-order condition is indeed $\dfrac{dc}{dl}=G'(l)$.


Non Collusive Cournot Duopoly model with two firms, zero costs and linear demand curve

The temporal constraints (i.e. what happens in a period) are not very clear in this question. It is likely that the good sold is not a durable good and hence there is no "leftover demand" between periods, demand is simply 'reset'.

In period 2 leftover demand appears because firm B assumes firm A will not change its production from period 1.

Then in period 3 firm A will best respond to the unchanging production of firm B from period 2.


With interest rates and inflation in the U.S. low and growth moderate, where is the money going?

The money is not going anywhere. In fact, it never even "left the building". The banks simply increased their reserve balances with the Federal Reserve, mainly because the Fed began paying interest on reserves in October 2008.


Printing Money vs. Issuing International Bonds

Bonds are backed by real resources. Money is backed by nothing.

Lets say I'm a British investor. The price of 1 kilo of cotton is 100 British pounds. If I buy a bond for 1 kilo of cotton redeemable in 1 year for 90 pounds then in one year when I redeem the bond and the 1 kilo of cotton is sent to me I can sell it for 100 pounds for a profit of 10 pounds or 10% (100 - 90). The number of pounds in the south has nothing to do with the value of the pound since no new pounds are being created. A southerner with payment in pounds can go to Britain and use the pounds and if anything the transfer of pounds from Britain to the South will cause deflation as the supply of pounds in Britain decreases since they are now in the South.

This is different from the South printing money which actually increases the money supply. If the south has 100K total confederate dollars in circulation which can be used to purchase 1000 arbitrary goods and services then each good can be purchased for 100 confederate dollars. If the south wants to purchase an additional 1000 arbitrary goods and tries to print an additional 100K this leads to inflation as now there are 200K total confederate dollars. The money supply in this instance has expanded whereas it has not in the bonds example.


Finding pure-strategy subgame-perfect Nash equilibria

Because you are looking for subgame-perfect equilibria, it is the correct approach to solve this game backwards. There are 2 proper subgames, and you identified the correct Nash equilibria there. However, note that you specified the payoffs, not the strategies. The correct formulation would be $(a_1,b_1)$ instead of $(2,4)$.
Next, you substitute the subgame perfect payoffs for the subgame and check for the optimal selection of A in the beginning. It is important that an equilibrium is always a full strategy profile. That is, it needs to specify an action for every information set, regardless of whether it is reached on equilibrium path. A has three actions, one at the beginning, one in the left and one in the right subgame. B has two actions, one for each subgame. So a strategy profile would be of this form [(first,left,right),(left,right)].
You have to check for all of the combinations. If you consider payoff (2,4) on the left and (1,4) on the right, you have a SPNE [(don't reduce,a1,a1),(b1,b1)]. Next, consider (2,4) and (3,2), you have a SPNE [(reduce,a1,a2),(b1,b2)]. You also have two more SPNE: [(don't,a2,a1),(b2,b1)] and [(don't,a2,a2),(b2,b2)]. That is, there are four SPNE.
If you do not specify what happens off path, you cannot evaluate what happens when player A deviates in the first stage and thus cannot determine whether no profitable deviations exist.


Net Positions vs Gross Positions as they Relate to the Current Account

The assets and liabilities are what make up what is called the International Investment Position, often broken down into three main parts:

Direct investment: subsidiaries abroad or domestic companies with foreign owners, where their full profits or losses count as overseas income for the balance of payments
Portfolio investment: foreign shares and bonds or domestic shares and bonds owned by foreigners, where the dividends and coupons count as overseas income for the balance of payments
Other investment: including foreign bank accounts and loans, where the interest counts as overseas income for the balance of payments; there are also derivatives and reserves

When these are added up, the amounts involved are big league (to channel President Trump).  At the end of 2016 the US had gross assets abroad worth about USD 24 trillion and gross liabilities to foreigners worth about USD 32 trillion, each more than US annual GDP. (The UK gross positions are even more extreme)

The net International Investment Position is simply the difference between gross assets and gross liabilities, so with the US having a net liability of about USD 8 trillion at the end of 2016

One might naively think

having a persistent current account deficit might lead to steadily widening net liabilities as financial flows need to balanced current account flows
having a net liability could lead to a deficit on income flows (if you owe more you might pay more), making the current account deficit difficult to close even if trade in goods and services moved into balance

That was not what was observed in the paper and so two rationalisations were offered

US assets abroad seemed to be paid a higher rate of return than US liabilities to foreigners for specific reasons
US assets overseas had been revalued over time (for reasons such as exchange rate changes or market price changes) so the net position had not exploded

The rest of the paper speculated that this sort of process could lead to a smoother transition to a current account balance than one might otherwise naively think, especially if accompanied by currency depreciation.  It combined this with some mathification and a toy numerical illustration


My suggestion would be to take the habit of fully writing down and then solving formally even the simplest model. Yours is a static one,

$$\max_{C,L} U(C,L) = CL,\;\;\;\; s.t. C = (24-L)w$$

since we silently assume that the individual cannot consume more than his income. Here $L$ is leisure, not work.

Inserting the constraint into the objective function,

$$\max_L U[C(L), L] = (24-L)w\cdot L = 24wL - wL^2 = w(24L - L^2)$$

This permits you to see that the wage doesn't matter for the optimal allocation (it will matter, for the level of utility attained). This may sound counter-intuitive... It is, and it has to do with the category of functions to which the specific utility function belongs to. In other words, it is an unrealistic aspect of the model introduced by a mathematical choice we made.

The above has a straightforward solution for a maximum.

(PS: Has Walras been notified that his name is used in such a manner?)


Why is the martingale factor a martingale in Hansen's 2012 Dynamic Valuation Decomposition?

The idea is that after the Perron-Frobenius problem is solved, you have $e$ such that
\begin{align}
E \left[\frac{M_{t+1}}{M_t} e(X_{t+1}) \middle | X_t = x \right]
 &= e(x) \exp(\eta) \\
E \left[\frac{M_{t+1}}{M_t} \frac{e(X_{t+1})}{e(X_t)} \exp(-\eta) \middle | X_t = x \right]
 &= 1 \\
E \left[\frac{\widetilde M_{t+1}}{\widetilde M_t} \middle | \mathcal F_t \right]
 &= 1 \\
E \left[\widetilde M_{t+1} \middle | \mathcal F_t \right]
 &= \widetilde M_t.
\end{align}

Thus we see that $\widetilde M_t$ is a martingale.


Which is the origin of the "economic" approach to production?

The book from Alfred Marshall you are talking about is "Principles of Economics"

Principles of Economics PDF

As much I know the idea for Economic Surplus (Consumer Surplus and Producer Surplus) originated from Alfred Marshall, he is considered the person who "put" mathematics in the Economics theory.


I am unsure how economists deal with the transition from continuous time to discrete time. My background is in systems engineering/applied mathematics, and the economics literature I am familiar with is either discrete time or continuous time. Unfortunately, I have no good references on the conversions from discrete to continuous time from the systems engineering literature (got rid of the textbooks).

The key in this case is that the “a” coefficient in these systems do not necessarily correspond to the same number. We need to adjust it so that behaviour lines up.

If we are transitioning from continuous time to discrete time, we effectively are sampling the continuous time system. The frequency we sample at matters.

This is easy to see. Imagine that a continuous variable grows by 1% in six months. If the discrete time step is six months, the variable grows by 1.01 each period. However, if we our sample is annual, we get a transition growth step of $(1.01)^2$ per period. Essentially, the formula to convert the growth coefficient is similar to changing interest rate quote conventions from continuous compounding to a bond yield equivalent.

Although a continuously growing variable (like a compunding deposit) is relatively straightfoward to deal with, something like consumption series is more difficult to specify. If we want the discrete time to correspond to the sampled values of the continuous time version, we need to pin down the continuous time dynamics in some fashion. Is it constant during the time interval, or is it also growing in an exponential fashion (which would be the case if it were an endogenous variable inside a linear model)?

Going from discrete time to continuous time is more awkward. A continuous time system contains much more information than a discrete time system. There is an infinite number of continuous time systems that can generate the same discrete time system. We need a machanism to pin down the dynamics of the continuous time version, but the decision can be viewed as arbitrary. This is a topic that is not heavily studied in systems engineering, as physical systems are already truly operating in continuous time.

A simple economic example of the loss of information is as follows. Imagine that we have a quarterly model frequency, but a household receive a cash flow at some point during the quarter. Interest compounds daily (like a bank deposit). The final amount held at the end of the quarter depends upon the length of the holding period, which could be from 0-3 months in length. Since the discrete time model has no notion of times in between the quarterly sample periods, it has no way of modelling this timing effect that is a property of the continuous time model. This means that a discrete time model is likely only going to approximate a continuous time model.


Could massive derivatives exposures at banks lead to disaster?

Its possible, but, in principle, most non-financial companies use derivatives to "hedge risk", rather than to "bet". Hedging risk means that the firm will use derivatives to lower its chances of going bankrupt, for  example an oil company will buy a put option on its oil output, so that if the price of oil falls after the firm invested in producing it,  the firm can still sell the oil at a high price, and avoid bankruptcy. Other times, the bank is an intermediary in futures. futures allow a firm to sell its output forward in time, and another firm to buy the output forward in time. this reduces the risk for both of the firms. Of course, selling or buying  forward reduces the  chance of making  it big if the prices improve in your favor.

Also, financial regulation forces banks to consider multiple possible future scenarios and their effects on their balance sheet. These exercises result in a measure of the riskiness of the bank, and in turn let authorities ask the bank to hold more capital or unwind some of its positions if has too much risk of some sort. The exercises include dramatic swings in commodity prices, unemployment, or the failure of large corporations, or large financial counter-parties.

There are certainly some financial agents out there that hold tons of risk. They might be indeed betting on one view or another, but because of the regulatory environment, these can't be "banks". This is very important, because if they are not banks, then their collapse does not then create contagion. A big investor looses his money, but if its not a deposit taking, (or similarly, a highly leveraged institution), then the losses, or gains, are in sense expected.

Therefore, although you are right in principle when you say that if banks stand in the middle between two firms and one of them goes bankrupt then the bank gets hit, regulation is meant to push banks away from risk. Also, the possible losses to the banks owners move them towards avoiding risk. And "risk" here includes the one that comes from their clients on one leg or another of a deal collapsing...

Here's a more detailed analysis: Congressional Research Service article on Financial Regulation.
You could also do worse than by taking Nobel Prize winner Bob Shiller's  online course.


Is it constant returns to scale if the output of a production function is purely a function of one variable?

Returns to scale is a concept that we use to think about how output changes as we continually add more inputs. It does not matter if your production function takes one input or $N$ inputs. All that matters is how inputs behave within the function itself. Another way to think about this is to consider that a production function with a single input can exhibit decreasing, constant, or increasing returns to scale.

For example:

$f(L) = L^2$ exhibits increasing returns to scale.
$f(L) = L$ exhibits constant returns to scale
$f(L)= L^{\frac{1}{2}}$ exhibits decreasing returns to scale

Your production function is linear in its one input [case number 2 above] and so it exhibits constant returns to scale. That is, the answer to your question is yes.


Estimation of the future growth of real GDP per capita of the world

I don't think there is a consensus about this. In 1980s and 1990s, growth theorists have tried to endogeneize the technological progress in growth models, that shows a "balanced growth path" (in which economy grows at a constant rate in long distance.) So, we can say that economists don't want a decrease in GDP.

But note that there are some people who support the idea of "degrowth". You can find a definition on Wikipedia.


Why goods prices are normalized in each period in growth models?

In fact, in growth models, people usually assume that there exists a one generic good for which the price is equal to one and not an infinite number of goods. Then, it is possible to set the price equal to one.

You can look at Acemoglu's Introduction to Modern Economic Growth textbook. In this book, you can see in a decentralized economy (where there are different sectors, final good producer, intermediate good producer etc.) you can see that the price of the final good is normalized to one and there exists a price $p$ for the intermediate good.

Therefore, to sum up, in simple canonical solow model with only one sector, there exists a one generic good and for more complicated growth models (endogenous growth models), where there are different sectors, there are different goods, then this implies different prices.

Hope that is a clear answer for you !


Your Lagrangian would be
$$L = (ax+by)+\lambda (I−p_x x−p_y y) +\mu_x(x−0)+\mu_y(y-0),$$
where the final two terms represent the restriction that $x,y\geq0$.
You then arrive at conditions
$$\frac{\partial L}{\partial x}= a -\lambda p_x +\mu_x=0$$
$$\frac{\partial L}{\partial y}= b -\lambda p_y +\mu_y=0$$
$$I=p_x x+p_y y$$
and complementary slackness conditions
$$\mu_x x =0 , \quad \mu_y y =0.$$
You then consider many different cases: $(\mu_x>0,x=0,\mu_y=0,y>0),(\mu_x=0,x>0,\mu_y=0,y>0), (\mu_x=0,x=0,\mu_y=0,y>0), ....$
and for each of them you look at your conditions:
Suppose $(\mu_x>0,x=0,\mu_y=0,y>0)$. Then,
$$a -\lambda p_x +\mu_x=0 \Rightarrow a -\lambda p_x <0$$
$$b -\lambda p_y +\mu_y=b -\lambda p_y=0$$
$$\Rightarrow a -\frac{b}{p_y} p_x <0 \iff \frac{a}{p_x} <\frac{b}{p_y}$$
This condition can be checked as $a,b,p_x,p_y$ are given. If it does not hold, $x=0,y>0$ cannot be the solution. If it does hold, it means that the utility per dollar is higher for good $y$ and all money should be spend on $y$ (which $x=0,y>0$ and the budget constraint imply).
You have to check all the cases, but you will arrive at a solution of the above form.


Has anybody seen anything this expression before?

$p\cdot z(p)$ is total revenue (price times quantity), and so its derivative $\frac{\mathrm d}{\mathrm dp}pz(p)=pz'(p)+z(p)$ is marginal revenue.


Such a function is called homogeneous of degree 1.


Is it better for low-income people to live in the countryside where it's cheap or in the city where there are more services?

It depends not just on the cost of living, but on the available means of income (wages) relative to each location.  Rural areas may have cheaper rent, but they often have fewer employment opportunities.  The right balance will often depend on the skills/qualifications of the individual.


I just taught my students that signalling is useless in Matching Pennies and Rock, Paper, Scissors. You would just lie and be random anyway so why would the other player ever believe any signal at all? That renders signals useless. Perhaps the people you overheard meant just that.


Why is devaluing your currency good for exports?

The problem lies with your assumptions. Practical world trade doesn't usually work that way. No one ever devalues their currency by half.

This would be a practical assumption:

US buys toys from China - say x units. It also buy toys from Japan - say y units.

Chinese toys cost xx, and Japanese toys cost yy.

Now, the US would start buying more from China (say x + a), by substituting Japanese toys ( now the US buys y - b) with Chinese toys. (The US would never import more without substituting because it won't import more than what it needs, as pointed out by someone else here)

If production is China had slowed down before devaluation, this would result in more production. More jobs, more income for employees. A devaluation is done to boost exports, resulting in more production, resulting in more employment, resulting in an overall increase in economic activity, whether or not it results in a direct increase in the income received through exports (although it usually does)


Is the income derived from owning a live animal profit or rent?

Live animals held and raised in captivity (and any products created) are counted as production. In particular, they're counted as "livestock value-added". From World Bank (2014):

Outputs from the livestock sector include the increase in the number of animals and the production of livestock products.
The increase in number of animals is represented by both fixed capital formation — i.e. animals that are inputs into the production process, such as breeding animals and adult males for breeding or animal traction — and by so-called ‘work-in progress’ animals, namely those reared for slaughter and young animals reared to become fixed assets.
Livestock products include meat, milk, eggs, hides and skins and other by-products, such as manure, hides and skins, fat, offal, honey, transport services, etc.

You also ask:

Does it make a difference whether the tree or animal is wild or has been planted on a plantation or bred in captivity respectively?

Yes. From UN System of National Accounts (2008, PDF, emphasis added):

Certain natural processes may or may not be counted as production depending upon the circumstances in which they occur. A necessary condition for an activity to be treated as productive is that it must be carried out under the instigation, control and responsibility of some institutional unit that exercises ownership rights over whatever is produced. For example, the natural growth of stocks of fish in the high seas not subject to international quotas is not counted as production: the process is not managed by any institutional unit and the fish do not belong to any institutional unit. On the other hand, the growth of fish in fish farms is treated as a process of production in much the same way that rearing livestock is a process of production. Similarly, the natural growth of wild, uncultivated forests or wild fruits or berries is not counted as production, whereas the cultivation of crop-bearing trees, or trees grown for timber or other uses, is counted in the same way as the growing of annual crops. However, the deliberate felling of trees in wild forests, and the gathering of wild fruit or berries, and also firewood, counts as production. Similarly, rainfall and the flow of water down natural watercourses are not processes of production, whereas storing water in reservoirs or dams and the piping, or carrying, of water from one location to another all constitute production.


Why is South Korea classified as an Emerging Market economy?

Economists already consider South Korea to be developed economy. In fact some scholars already argued that Korea was a developed economy already sometime in late 90s - 2000s (see Kim, 1988; Kim, 2011). Opinions might differ because there is no precise definition of what developed country is but quite often in literature this is proxied with whether economy reached high income status and South Korea already did that by World Bank definition some time ago.
MSCI index has its own classification that does not necessary reflect what economists think about the country. MSCI is a private company and they create their own classification in order to help people who invest to decide how to invest. The MSCI classification framework can be seen here. The classification is based on economic performance but really the numbers and thresholds are all arbitrary and do not even refer to economic literature or literature on economic development. For example, other indexes such as  Financial Times Stock Exchange
(FTSE) index already reclassified South Korea as a developed country as you can read here.
You should not take classification by different indexes  too seriously, or give them too much weight, they do not necessarily reflect economic literature. If you would start your own private company doing the same business as MSCI you could decide to classify even Maldives as developed country or USA as a developing one. This being said the indexes do this to help to guide their investors so they will try not to do it completely without any thought (and go to such extreme as I suggested in the previous sentence) but in the end it is their private decision, and different indexes will decide differently.