How much would it take to end global extreme poverty?

The Economist (Oct 8th, 2016, online, PDF) attempts to answer this question.

It defines the global poverty gap as the amount we'd have to transfer to the world's extreme poor (defined as those living below $\$1.90$ a day, at 2011 PPP), in order to lift them to the $\$1.90$ a day threshold.

The evolution of this global poverty gap from 1990-2013 is graphed below. As should be evident, at the current rate of decline in global extreme poverty, the world is on track to hit the World Bank's goal of completely eliminating extreme poverty by 2030.

Nonetheless, even if there were no further progress and global extreme poverty stayed frozen at 2013 levels,

a transfer of US\$78b a year would suffice to completely eliminate extreme poverty.

The Economist adds an important caveat:

  In reality, of course, money cannot be directed so precisely
  to the poor, nor transferred cost-free. In some countries, the infusion
  of money might also push up prices and currencies, making
  the endeavour more expensive. Nonetheless, this thought experiment
  illuminates the diminishing size of the problem. The world
  can afford to end poverty. Indeed, it might end poverty before it
  figures out how to measure it accurately.

$$$$

P.S. Note though that the reduction in global extreme poverty has been largely due not to transfers or aid of the sort envisioned by Jeffrey Sachs, but to economic progress, most notably in China.


Why does a decrease in interest rate reduces the velocity of money circulation?

You are absolutely right. Since higher interest rates increase the opportunity cost, you wan't to get rid of it faster (at the margin).

The core of the argument is that the cost stems not only from the amount of money holdings, but also from the duration. Say you are constrained about the amount of money holdings you need in your daily life, but you can still make sure the money is not idle for too long.

For the second part of your question, you are mixing up concepts. The interest rate influences the Consumption/Savings Decision, but it is the same decision problem as in a (model) world without money, i.e. it is a choice between real variables, whereas money velocity is a monetary question.


Model of economic cycles consistent with efficient market hypothesis

The EMH applies to assets, not just stocks, and its implications are more relevant for investors who own part of the market - not the entire thing.  This is important, because it's the difference between looking at a closed system versus an open one, and between populations versus samples.  People make money all the time by cycling between stocks, bonds, precious metals, real estate, etc.  Flights to "safe havens" occur all the time, rightly and wrongly.  Someone buying SNC Lavalin stock three months ago and selling it a few weeks after the Canadian election would have made some fairly predictable money.

The apparent issue you're running into here stems from the erroneous assumption that non-stock assets are "safe".  In fact, they have their own risks and cycles, which are (anti-)correlated with the stock market in various ways.  The "Risk-free rate" in the Fama-French model isn't constant, and is fundamentally no more predictable than any individual stock price.


(Ricardian model)Is it correct that complete specialization in comparative advantage CAN lead to DECREASE in total production?

Its not correct to say that production will be lower in the solution to the Ricardian model (in your case full specialization of US is not equilibrium outcome). In general in Ricardian model the production after specialization will always be either the same (if the relative opportunity costs are exactly same) or production will be higher. The reason why you got it lower in your case is that your model omits demands for the two goods and by your assumption on production technology US is economically big country while France is economically small (meaning the production possibility frontier (PPF) of US severely outstrips the PPF of France).

In situations like these you will find that only small country fully specializes while the big country specializes only partially. The reason for this is that even if the small country produces on full capacity it simply wont be able to satisfy the world demand, so once the US produces enough wine to fill up the word demand it will be also allocating the rest of the resources to cheese.

Fully deriving the optimum solution with adding utilities would take too much space so instead of full proof consider the following numerical simulation with your own example.

As you have shown in autarky we will have the following production:

US Cheese: 500 US Wine:  1000
France Ch: 250 France W:  333
World  Ch: 750 World  W: 1333

Now consider possible scenario with full specialization France (all in cheese) and partial US (1500 labor in wine and 500 in cheese):

US Cheese: 250 US wine: 1500
France Ch: 500 France W:   0
world Ch:  750 world W: 1500

Now you can see the production of cheese is exactly the same as before plus world has more wine. With just one extra unit of labor in cheese in US you would still get more cheese and also more wine compared to autarky.

Now to get an exact optimal solution you need to explicitly include consumer utility and demand, and you would see that in this case there would be excess demand that France could not satisfy which would be allocated to US. For the full treatment see Feenstra, R. C. (2015). Advanced international trade: theory and evidence. Princeton university press.

The undergraduate treatment of Ricardian model does not include demand functions explicitly because as long as none of the two countries have production frontiers that do not strictly outstrips the PPF of the other at all points you will always get full and never partial specialization so the exact nature of demand function does not matter (notice that in undergraduate Ricardian model you wont even find the final distribution of the goods unless the final demands are assumed in the question - this is to simplify the model to make it easier to solve but it does not represent the full model in its modern specification)

PS: of course on off equilibrium path any combination of outputs that satisfies the resource constraint is possible


The social welfare of a unique general competitive equilibrium in an exchange economy

Using the 'utility monster' critique of aggregate utility maximization it is pretty easy to show that gen. comp. eq. does not always maximize aggregate utility.

Imagine a pure exchange economy with two agents, $A$ and $B$ and two types of goods, $x_1$ and $x_2$. Agent $A$ has nothing whereas agent $B$ has one unit of each good. The agents respective utility functions are
$$
U_A(x_1^A,x_2^A) = 2x_1^A + 2x_2^A \hskip 20pt U_B(x_1^B,x_2^B) = x_1^B + x_2^B.
$$
Trivially the initial distribution is a competitive equilibrium and also trivially it does not maximize aggregate utility.

EDIT:

I am not sure I understand @spinkus's comment, as obviously not every solution maximizes the aggregate utility, but I will address the other point: Not trading. That is not at all important for me, I just wanted to make the example trivial.

Suppose given some strictly monotonic utility functions $U_A,U_B$ and some initial endowments $w_A,w_B$ some allocation $a^*$ is a competitive equilibrium. If this allocation does not maximize aggregate utility we are done. If it does, consider the following monotonic transformation of $U_B$:
$$
\hat{U}_B \triangleq c \cdot U_B
$$ where $c$ is a positive number. In a pure exchange economy with $U_A,\hat{U}_B$, initial endowments $w_A,w_B$ the competitive equilibrium will again be $a^*$ as the preferences are unchanged. Yet even if $a^*$ maximized $U_A+U_B$, it clearly will not maximize $U_A + \hat{U}_B$ if $c$ is small enough.


(Microeconomics) Why must fims always lower prices when there is excess supply?

You are absolutely right. In many markets, especially consumer markets, which people often think of, firms are not forced to lower prices. If a bakery produces to much bread one day, it might make a bigger profit if it keeps the price and throws away the breads they have not sold at the end of the day. That is because its limited competition.

What many forget to tell you when they teach microeconomics, is which markets microeconomic models is designed to model. It is market with perfectly competition (infinitely many suppliers and consumers) and where everyone have perfect knowledge about prices. Think if big international markets of standard products, like crude oil, flower, rice and stocks.

If you are a farmer wanting to sell rice on an international exchange (normally you will just sell directly to a distributer), you are to small to affect the market. Your only option is ether to sell a tiny amount below the spot price (the last transaction), or choose not to sell at all (something you probably don't want to do). If there are more sellers than buyers, the price is pressed downwards. This downward price pressure will continue until prices are either so small that buyers want to buy more, or the price is so low that some producers don't want to sell anymore.

If you did not understand the last paragraph you should probably read up on how stock/commodity market auctions work.


Why are readily available productivity data relative, not absolute?

The data you ask for is in the OECD stats website under

  Productivity > Productivity and ULC – Annual, Total Economy > Level of GDP per capita and productivity

This is the graph (in 2010 PPP US dollars):

Aternatively, you can compute it manually:

Select a series of constant GDP of your interest.
Divide it by number of hours for a group of interest.

You can find both series in the above website. GDP is under

  National Accounts > Annual National Accounts > Main Aggregates > Gross domestic product

Hours are under

  Labour > Labour Force Statistics > Hours worked


Walrasian Equilibrium under Leontief Preferences

You do not need money exchanged to establish a budget constraint, most of these standard micro questions work in cashless economies.

Your Px/Py is just the exchange rate between the goods. If it's 3:1, it means that you exchange 3 $y$ for one $x$ - no cash involved.

Not being able to differentiate sucks, but actually requires you to know what you're doing. The point of differentiation is finding a place at which all Pareto improving trade has been exhausted.

Here, that means that, indexing the two players by i:

$$ x_i, y_i \in \arg\max_{x,y} U^i(x, y) s.t. p_x x + p_y y = p_x x^i + p_y y^i \, , \forall i $$

Without first-order conditions, this problem is potentially hard to solve. However, most non-differentiable utility functions give you a hint on how the outcome must look like.

Here, aggregate resources add up: $A=x_1 + x_2 = y_1 + y_2$. Hence, the planner would chose some allocation where x=y for both players, as for any positive $a$ and $\epsilon$ it is inefficient to give one player $(a + \epsilon, a)$ and the other one $(A - a - \epsilon, A - a)$.

However, that leaves a continuum of distributions as potential outcomes: Those where we give players one and two $(a, a)$ and $(A-a, A-a)$ for any $0 \leq a \leq A$.

Which equilibrium we actually get depends on the relative value of the initial endowments. I will leave you with this: What does $p_x/p_y$ necessarily have to be given total endowments and preferences? What does this imply for the unique equilibrium distribution of resources among the players?


Question regarding demand schedule and equilibrium consumption

Yes, he is referring to a movement along the demand line. If the individual purchases less than $Q_0$, than the demand line would be higher than the price for quantities for $Q_0$, this means that the individual values consumption more than its cost (its price). Graphically, you just need to show that for a quantity less than $Q_0$, the demand line is higher than the price, i.e. the individual values consumption more than its cost.


Lecture notes competitive labor market with minimum wage

In the Varian textbook Intermediate Microeconomics (I am guessing in most micro textbooks), the chapter on Equilibrium discusses price controls in the presence of a demand and supply function. You can apply that discussion directly to the (competitve) labor market, were the price of labor is of course the wage.


Are Cobb-Douglas preferences monotone according to the marginal utility condition?

They do not satisfy either condition. An increase in the quantity of one good need not increase utility; $U(0,1)=0^a1^b=0=0^a2^b=U(0,2)$.
The condition that all marginal utilities must be positive is inherently problematic because it does depend not just on the underlying preferences. For one, not every utility representation needs to be differentiable. But even that is not enough. Suppose there is a single good and more is better than less. You can represent these preferences by the utility function given by $U(x)=x$. Now $U'(x)=1$ for all $x$, so these preferences are supposedly strictly monotone. But now consider the utility function given by $V(x)=(x-1)^3$. If $x'>x$, then $V(x')>V(x)$, more is better and $V$ represents the same preferences as $U$. But $V'(1)=0$, so the preferences are not strictly monotone according to the definition in terms of marginal utilities.
The problem just mentioned is actually even worse. Let $U$ be a differentiable utility function on $\mathbb{R}_+^2$ with strictly positive partial derivatives everywhere. Let $(x^*,y^*)\gg0$be any commodity bundle. Define $V$ by $$V(x,y)=\big(U(x,y)-U(x^*,y^*)\big)^3.$$ $V$ represents the same preferences as $U$ but the partial derivatives at $(x^*,y^*)$ are zero.
Long story short: Every partial derivative being strictly positive is a sufficient but not necessary condition for a differentiable function to be increasing in every coordinate.
Yet another issue is that it is not entirely clear how one defines the derivative at the boundary of the commodity space. There are different notions of differentiability which need not be equivalent.


Because they are different markets, which trade at different hours, leading to different results. There is also the effect of variations of variations in the USD/EUR rates.


How to assign a set of parameters to a macroeconomic model in Matlab?

Here is my approach to assign a set of parameters to a macroeconcomic model:
function para=assign_parameters(v)

if length(v)==3
   para.alpha=v(1);
   para.beta=v(2);
   para.gamma=v(3);
else
   error('parameter input has not correct format of length 3!');
end

The advantage is that the parameter set can be changed on the fly. That is, one can assign to the same underlying macroeconomic model a different set of numerical parameters without observing any conflict among the analyzed models. This would allow a quick comparison of model outcomes and ensure a clear as well as consistent model environment.
For instance, the example from the post can be assigned by
pm=assign_parameters(v)

and if one wants to study a second model like
v2=[2/3 1/100 1/2]

one should assign this set of parameters to a different structure name like
pm2=assign_parameters(v2)

Then both sets remain separated and the first parameter set will not be overwritten by the second.


How to calculate the degree of risk aversion of a consumer in a lottery?

The gamble:

$$g = \left(\pi \circ 98, (1-\pi)\circ100\right)$$

The expected value of the gamble:

$$\mathbb{E}[g]= \pi\cdot 98 + (1-\pi)\cdot 100$$

Expected utility:

$$u(g) = \pi \cdot \ln (98) + (1-\pi)\cdot \ln (100)$$

Risk premium is such that:

$$\ln \left(\mathbb{E}[g]-P\right) = \pi \cdot \ln (98) + (1-\pi)\cdot \ln (100)$$

Where $P$ is the risk premium. Solve the above equation for $P$. You can measure the degree of absolute risk aversion as follows:

$$ARA = -\frac{u''(c_h)}{u'(c_h)}$$

And the degree of relative risk aversion as:

$$RRA = - \frac{c_hu''(c_h)}{u'(c_h)}$$


Here is a reason why growth of a firm's sales could be too fast.  Suppose the volume of sales of the firm's product is $N$ in period 0 and $N+\Delta$ in period 1. Suppose also that each unit of product sells for $P$, and the cost of the inputs for one unit is $cP$ where $0 < c < 1$.  Thus the profit per unit is $(1-c)P$, and total profit is $N(1-c)P$ in period 0 and $(N+\Delta)(1-c)P$ in period 1.  So far that looks satisfactory.

But suppose further that (as is quite possible) there is a lag of one period between purchase of the inputs to produce a unit and sale of that unit.  To keep things simple, assume there is no credit allowed on either purchases or sales, so that the cash outflow for the purchases occurs one period before the cash inflow from the associated sale.  Then in period 0 the cash inflow for period 0 sales is $NP$ while the cash outflow for the inputs for the period 1 sales is $(N+\Delta)cP$.  Thus net cash inflow in period 0 ($NCI_0$) is given by:

$$NCI_0 = NP - (N+\Delta)cP = [N - (N+\Delta)c]P$$

The condition for this to be positive is:

$$N > (N+\Delta)c$$

which is equivalent to:

$$ \Delta < \frac{N(1-c)}{c}\qquad(A)$$

If sales growth is sufficiently fast that relation $A$ does not hold, then, despite the firm being profitable, there will be a net cash outflow in period 0, and if such sales growth continues over many periods, there will be a continuing net cash outflow.  The question then is whether the firm can finance the cash outflow for long enough to see it through until a time when sales become more stable so that its profits begin to be realised in cashflow terms.

Finance could in principle be via either equity or debt.  However, a growing firm is unlikely in practice to be able to obtain debt finance unless the lender is satisfied that the firm's owners are risking their own equity to a sufficient degree.  It is possible therefore for a firm's sales to be growing at a faster rate than it is likely to be able to finance given its level of equity.


In theory there is nothing that is preventing the real interest rate from becoming negative. Making the real interest rate negative by keeping the nominal rate at zero and creating inflation has been the goal of the ECB in the recent past. Once you hit the ZLB there is no other way to further loosen monetary policy. With regard to your work it is probably important to carefully select the inflation rate you're using, but you dont describe this in detail. Depending on your question you might want to consider forward looking measures (i.e.Blue Chip or Greenbook forecasts if you're talking about the US).


If $x \succsim_i x_i^*$ then $p \cdot x_i\ge w_i.$ (MWG 16.C.2)

Hint: Prove by contradiction. That is, suppose $x_i\succsim_ix_i^*$ but $p\cdot x_i<w_i$, show that this would lead to a contradiction of $x_i^*$ being maximal for the feasible set.


For stability, is there a limit to the sovereign debt that a Government owes to itself against issued fiat currency?

Does this mean that Governments can continue to print a limited amount of fiat currency (say 5% of GDP) annually forever?

Yes government can 'print' (I use quotation marks because most money is created electronically nowadays) any amount of fiat currency it wants to.

In other words, is there a problem even if this debt that the Government owes itself, grows to, say, 1000 times the GDP?

Depends on your definition of a 'problem'. In terms of affecting probability of default it is not any problem at all. Central bank can scrap all the government debt it owns at a stroke of keyboard.
However, this being said, it is agreed by virtually all conventional economists that increasing amount of money (ceteris paribus) leads to inflation (see Mankiw Macroeconomics 8th ed.). So eventually if government would try to create arbitrary high amount of money it would eventually lead to inflation and at some point even hyperinflation. For this reason it is generally agreed by economists that governments can fund arbitrary amount of spending by creating new money (see this poll among the most prestigious policy economists). This being said there would be some scope for generating seigniorage revenue that way, so monetary financing might not be bad if it is done carefully, but having debt to GDP ratio of 1000:1, completely financed through central bank's high powered money, would likely lead to disastrously high inflation.


How was the year 1925 chosen as a dividing point in the history of economic thought?

tl;dr:
It is most likely an arbitrary date because they had to set some year. In fact the year 1925 is not even a strict guideline but only loose criterion. As such the year was not set due to some special significance.
The JEL classification also responds not just to the needs of academicians but also grant agencies, government, university HR departments etc. and so classifications might be arbitrarily set in ways that do not necessary always correspond to what economists believe. However, there are some subtle hints it could have something to do with the advent of Neoclassical Economics.
Full Answer:
The JEL codes are set by American Economic Association (AEA) as they were originally developed for their Journal of Economic Literature. As such AEA is really the only authoritative source on this issue.
Normally classification codes come with guidelines but in this case JEL guidelines for "History of Economic Thought through 1925" just read:

B1 History of Economic Thought through 1925:
Guideline: Not Specified.
Keywords: Economic Thought.

However, B1 is just general category that is further subdivided into smaller sub-categories B10-B19. When you look at each of the B10-B19 guidelines individually most of them include the following caveat:

Caveats: The demarcation year of 1925 is used as a loose criterion.

When we look at the classification guidelines for History of Economic Thought since 1925" we find the following guidelines:

B2 History of Economic Thought since 1925:
Guideline: Studies that are also relevant to contemporary economics or
economies should be cross-classified here and under the other
appropriate categories.
Keywords: Economic Thought, Neo Classical.

Again quite vague but the keyword 'Neo Classical', is suggestive and suspiciously inserted there especially since neoclassical economics (although technically possible to trace prior to 1900s) first gained on prominence in early 1900s (see the History of Economic Thought by Brue and Grant). Although this is really not enough to be certain.
Also as in the previous case B2 is split into multiple sub-categories ranging from B20-B29 and most of them again include the following caveat:

Caveats: The demarcation year of 1925 is used as a loose criterion.

Furthermore, there is actually paper published by JEL about history of it's classification system and how it developed to it's present form. According to Classifying Economics: A History of the JEL Codes by Beatrice Cherrier (2017):

... Yet the AEA classification
does not provide a pure image of the discipline. Instead, it is a compromise between
looking forward and looking backward,
between AEA officials’ sometimes conflicting visions of their science and the multiple
and contradictory demands they face: editors
needed a way to select reviewers and referees; recruitment committees needed a way
to classify job candidates and their output;
the government wanted a system to draft
economists into the war effort, and later to
recruit specialists into the various bureaus
concerned with monitoring and managing
economic affairs; librarians needed help in
indexing papers and books; and the National
Science Foundation (NSF) needed a classification to quantify and evaluate national
scientific expertise ...

Author further states:

JEL codes reflected changes in
the ways theory and applied work interacted.
Second, the codes point to the transformation of the subject matters of the discipline
and the rise and fall of different approaches
to economics. Third, they reflect changes in
the external pressures on the discipline and
information technology

Furthermore, as the article explains the classification codes undergo 4 revisions and were originally set up 1911. Consequently, the year 1925 was not originally even part of classification and might very well change in the future.
The article also tells a fascinating story of how these classifications came to be and most of them were results of compromise between various stakeholders and economists holding different opinions on how to classify economics thinking. It is a result of a messy process rather than some unanimous academic consensus.
Lastly, I actually written an email to AEA requesting clarification. They confirmed that the year 1925 is not strict criterion and the person I corresponded with simply stated they do not know how the criterion was set. I am not publishing the e-mail as I don't know what would legal ramifications of such act be, and my e-mail address contains my real name, but I expect that anyone making the same inquiry would receive the same answer.
So what can we conclude from the above?

I think the evidence above points clearly to the fact that 1925 is just a loose guideline. The year itself has no special significance.
The keyword Neo Classical for B2 category is suggestive that development of Neoclassical economics could have something to do with it as it occurred around the period set by the 'loose criterion'.
The classification has to be set not only for needs of economists but also bureaucratic needs more widely and as such it might not always reflect purely academic opinions.
AEA employees (or at least one of them) themselves do not know why 1925 was chosen. Given that the most recent revision of JEL codes is over three decades old it is possible that such knowledge is lost to us.


The main problem is that wealth does not equal cash. Those ten rich people probably have most of their wealth in real estate and stocks. Imagine all you own is a Lamborghini, then the government prints a lot of money, does that impact you?

Depending on the situation the money printing may be progressive or regressive. And the cost are high: There will not just be one inflationary moment, there will be inflation because the initial inflation breeds an expectation of even more inflation leading to hyperinflation.

Finally when you've done this once, nobody will ever want to hold your currency again because they distrust you. Leading to a number of other problems.


How to compute the utility function when risk aversion is equal to 1?

Take the limit as $\sigma \rightarrow 1$,
\begin{align*}
\lim_{\sigma \rightarrow 1} \frac{C^{1-\sigma}}{1-\sigma}.
\end{align*}
This will then results in log utility. Calculating the limit goes as follows.
Since plugging in $\sigma = 0$ would give $\frac 0 0$, we use L'Hopital's rule.
This gives us
\begin{align*}
\lim_{\sigma \rightarrow 1} \frac{C^{1-\sigma}}{1-\sigma}
  &= \lim_{\sigma \rightarrow 1} \frac{- C^{1-\sigma} \log C}{-1} \\
  &= \log C.
\end{align*}


Why lower the deposit rate if it is already negative?

You are assuming that the supply of deposits is zero when the price (the rate) is zero but it definitely is not. There are several reasons for this.

While you can withdraw cash from the bank it is unwieldy, costly, and unsafe to have large amounts of cash lying around. Therefore, if there were no other alternatives, heavy spenders would still want some wealth in deposit system even if rates were quite negative. Consider, for example, that it is common for depositors to pay account fees that greatly exceed interest earnings.
These costs are actually very low as it goes. A rate of -0.1 percent a year is ten cents per \$100 of average balance, and since these costs are proportional to average balances, which themselves are small relative to total spending, the value of transaction services provided by these accounts is probably large relative to total negative interest rate costs.
Assuming non-bank storage costs are high relative to negative rates, it isn't the nominal negative rates (the advertised rate) that matters but rather the real rates (relative to inflation). Under these circumstances, you should be indifferent between 2 percent deposit rates under 4 percent inflation and -1 percent deposit rates under 1 percent inflation. So a rate can drop and the supply of funds can be unchanged if it is accompanied by a drop in inflation.

And, in the case of the EMU / ECB / Euro, that may well be what happened. Inflation fell for the five months after June 2014, which may explain why nominal rates had to fall: to keep the tightness of monetary policy unchanged or at least to keep real rates unchanged.


It is implicit in the interpretation:

Mas-Collel: the amount that must be given (+)  to compensate for a reduction (-).

Reny: The rate at which good j  can be exchanged (+ & -) for good i.

The derivation from total differentiation only requires the utility to be constant, so the derivative must be negative to express that if the quantity of good $i$ increases, the quantity of good $j$ should decrease.

Another reason why some authors omit the negative sign is that in equilibrium  $|MRS_{ij}|=|\frac{P_i}{P_j}|$ So either you use the derivation from total differentiation and also the negative ratio of prices, or leave both sides as positive and focus on interpretation. Most people prefer the latter.


What will giffen goods and Veblen goods peform in general equilibrium model?

In general, what you are referring to is what we call numeraire pricing. This means that the prices, as you mentioned, are relative to one of the goods (you can pick any good as a numeraire, which is how we measure prices through the whole economy).
The reason for this is Walras' Law states (basically) that since we have a bunch (say $N$) goods in the economy, and say we have a bunch (again, $N$) markets in the economy, then solving all of these will leave one redundant equation. This applies in particular to general equilibrium because there are essentially more equations than variables, and hence you have one (or more) redundant variable(s) at the end.
To get a little more technical if you want: if money is something that we can consume, then we would insert cash as an additional good and make that our numeraire. Thus we can have "money," in the sense that it has a value as a bubble, and we can include it as a good in our utility.
To answer your questions about Veblen/Giffen effects, we can still observe these effects, they have nothing to do with money. These effects in their essence are talking about goods that we buy more of when the price increases. Even if this price is in terms of "money" or a numeraire, the point is that we buy it when it's more expensive to us, so we push other things that we might need out of the way in order to buy this.
A quick example (don't think too deeply about it, it's just an illustration) would be college in America, using, say, $1 McDonald's McChicken sandwiches as a numeraire. An argument could be made that each year more students attend college yet the price in terms of McChickens goes up each year ((i.e., consumption increases, as the price goes up). This could be described by the Veblen effect, where we maybe don't want to be perceived as someone with "only" an associate's degree or high school degree (not that those are bad, but it's just the way things are in America). If I measured in dollars or McChickens, the Giffen or Veblen effect doesn't change.


What fraction of new businesses die out within a year?

The Business Dynamics Statistics (BDS) of the Census provides annual measures of business dynamics (such as job creation and destruction, establishment births and deaths, and firm startups and shutdowns) for the economy and aggregated by establishment and firm characteristics. The use of the LBD as its source data permits tracking establishments and firms over time. In particular, BDS provides information on Establishment openings and closings and Number of startups and firm shutdowns.

The following figure is based on BDS and from Haltiwanger, John, Ron Jarmin, and Javier Miranda, (2009) "Business Dynamics Statistics: An Overview".

Note: SUSB means Statistics of U.S. Businesses from Census firms as well.


Uncertainty and Pareto efficient policies

Pareto efficient allocations can be found by maximizing a weighted average of the utilities of the agents. Let $\lambda$ be the Pareto weight for agent 1 and $1 - \lambda$ the weight for agent 2 (where $\lambda \in [0,1]$). This then gives the following problem:
$$
\max_{x_1, x_2} -\lambda \left[\pi_1(x_1 - b_{11})^2 + \pi_2(x_2 - b_{12})^2\right] - (1-\lambda)\left[ \pi_1 (x_1 - b_{21})^2 + \pi_2 (x_2 - b_{22})^2\right].
$$
The first order conditions with respect to $x_1$ and $x_2$ give:
$$
\lambda \pi_1 2(x_1 - b_{11}) + (1-\lambda) \pi_1 2(x_1 - b_{21}) = 0\\
\lambda \pi_2 2(x_2 - b_{21}) + (1-\lambda) \pi_2 2(x_2 - b_{22}) = 0.
$$
Simplifying gives:
$$
x_1 = \lambda b_{11} + (1-\lambda) b_{21},\\
x_2 = \lambda b_{21} + (1-\lambda) b_{22}.
$$
So $x_1$ and $x_2$ are weighted averages of the bliss points.


How and when did the public learn of monetary policy changes before FOMC announcements and press conferences?

The operating procedures for the Federal Reserve have changed drastically over the years. For example, interest rates were pegged during the early post-war era. Later on, the Volcker Fed attempted to target money supply growth, and "Fed Watchers" scoured the money data. (I do not have any references for this earlier history.)

Once money supply targeting was abandoned, operating procedures were roughly similar to the situation after 1994. The Dallas Fed has discussions of Fed Communications (written by Mark A. wynne) that has a brief summary of this period.

During that era, reserves paid no interest. The Fed did open market operationa to add/subtract reserves to match the demand for required reserves; the Fed Funds rate is the "price" banks were willing to pay to borrow reserves. If they supplied too much reserves, the Fed Funds reserves would drop towards 0%; supply too little, and banks would pay a lot to meet reserve requirements. The open market desk attempted to estimate the demand for reserves, and calibrate the size of the operations based on the market fed funds rate.

From the other side, banks could see the operations undertaken by the open market desk. They tried to reverse engineer the Fed's reserve supply function, so that they could tell what the target was (and of course, they could observe the market rate on Fed Funds).

As a result, banks roughly could tell what the target was. They would then adjust the prime rate on loans. I believe that "the public" would find out as a result of banks announcing changes to their prime rates, although presumably investors with a particular interest in interest rates would be watching what was happening in the money markets (which would adjust more rapidly).


First off, just because economists say that the trade deficit is not a negative thing, does not mean it is positive.

Yes, trade is balanced in the long run. In and of itself, a trade deficit is therefore not necessarily an issue.

When a country runs a trade deficit that means that it is consuming more than it produces. The country consumes at least as much as it produced and then a bit from what other countries produced too. That in itself is not a bad thing. In fact, consuming more than you produce seems rather nice.

However, a trade deficit may also be a symptom of other economic weaknesses. A perpetual deficit may also be the precursor to a crisis. So whether a deficit in a certain case is bad or good depends on the circumstances. A good source is this IMF report.

That being said, it is incorrect and a bit naive to simply see every trade deficit as a problem.


How did reducing the employment gender gap impact the US economy?

I'm eyeballing that the unemployment rate of man rose by 3% while the rate for woman fall by 13. Clearly not all of those newly working woman replaced man.

You choose a rather long period with incredible changes to the US economy; The one I think responsible for the effect you saw being the shift from manufacturing to service.

The Situation in 1977 can be seen on Page 10 (4 in the original numbering) of the 1977 Census.
It shows 44% of all employees working in manufacturing and 39% in service (Retail + "Selected services industries".

Contrast that to the 1997 Economics Census (Page 20 or 14) where "Retail Trade" alone employed more people than manufacturing - and Services almost double compared to manufacturing.

That shift favored woman (and probably was only possible due to more woman choosing not be stay-at-home moms) which explains the diminishing employment gap.

(The methodology of those reports has changed considerably so the exact percentage rates can certainly not be compared -- but the shift is so big that the conclusion stands unchallenged)


Interpreting Disequilibrium: Inelastic Supply and Demand at different quantities

I am not sure what you mean by looking, but there is no equilibrium in this market, so any equilibrium analysis is impossible.
Situations like this are possible. Consider the Titanic. Supply of lifeboat spaces are limited and do not change depending on the price. People are willing to pay any price.

There are limitations on what an individual can pay by money though. One might argue that there were a lot of people who were willing to do anything to get a space (push in line, etc.). As long as the number of these people is larger than the supply and the price is measured in effort your graph would describe the market for places.


A market correction for an industry which has had long-run negative externalities

An aspect of the matter could be described as follows:
We want

  prompt replacement of (existing) fixed capital

because, I guess, it creates currently "unacceptable" levels of negative externalities, and we know better than to think that through the pricing of the externalities we will be able to reverse the damages, and all swell.

From this point of view, we don't care about distribution between suppliers: it should be equivalent to us whether the existing companies change their capital base, or whether there appear new entrants with new cleaner technologies which will somehow drive out of business (or marginalize) the incumbents.

Let's describe the situation with very simple relations:

Prior to any pricing of externalities, the typical incumbent (symbolized by $h$ where needed) has a after-income-tax profit function at current time $t$

$$(1-\tau)\pi_{t} = (1-\tau)\big[p_{t}q_{t} -c(q_t)\big] \tag{1}$$

The OP writes that installed capital is "fully written off by now". This means that there is no income tax offset from accounting depreciation, Of course there is a capital base here that operates, needs maintenance, some marginal replacements etc. These costs are captured in the operational cost function (i.e. we assume 100% depreciation rate for them).

Assume now that we declare that anyone who continues to use the existing technology will pay an enviromental tax that is linear in quantity, $T_t = \xi\cdot q_t$ (naturally, this tax does not decrease taxable profits). Alternatively, he will be exempt from the tax if he completely renews his capital base, with new technology.

The incumbent now has to compare two discounted cash-flow scenaria:

$$\text {OLD} : NPV_{Oh} = \sum_{t=0}^{T} \left(\frac 1{1+r}\right)^t\big[(1-\tau)\pi_{ht}-\xi q_t\big]$$

and

$$\text {NEW} : NPV_{Nh} = -C_R-I_N+\sum_{t=0}^{T} \left(\frac 1{1+r}\right)^t\big[(1-\tau)\pi_{ht}+\tau\cdot\frac 1T(C_R+I_N)\big] \tag{2}$$

where $C_R$ measure the (capitalized) side-costs of renewal (disposal, disruption of operations etc), while $I_N$ is the needed new investment (including perhaps adjustment costs). $r$ is some average measure of opportunity cost (and not interest rate). We have assumed that the operational costs of the new technology are the same. $\tau\cdot\delta(C_R+I_N)$ is the tax benefit from accounting depreciation of the new investments. $\delta$ is the depreciation rate. We use the Fixed percentage method of Accounting depreciation here.

What the situation is for a prospective new entrant? It is easily seen that if he attempts to install Old Technology, he will make less profits than the incumbent (due to the necessary investments). So $NPV_{Oe} < NPV_{Oh}$. If he opts for the new technology for production capacity equal to that of the incumbent described above, we have

$$\text {NEW} : NPV_{Ne} = -I_N+\sum_{t=0}^{T} \left(\frac 1{1+r}\right)^t\big[(1-\tau)\pi_{ht}+\tau\cdot\frac 1TI_N\big] \tag{3}$$

It is easy again to determine that, here, $NPV_{Ne} > NPV_{Nh}$ i.e. the entrant is in a better position from the point of view of investment evaluation, because he does not have to bear the side-costs of renewal The tax benefits of the incumbent for depreciation of $C_R$ do not offset in NPV terms $C_R$.

Why do we care whether $NPV_{Ne} > NPV_{Nh}$? Because we want, conditional on both using the new technology, the new entrant to have an advantage over the incumbent in terms of financial strength, in order to have the resources to sustain a possible war from the part of the latter.

If we want that everybody has an incentive to use the new technology, then we must have

$$NPV_{Oe} < NPV_{Oh}< NPV_{Nh} < NPV_{Ne}$$
The first and third inequalities already we already have. For the middle one, which will connect the chain, we need to set the tax rate accordingly:

$$NPV_{Oh}- NPV_{Nh} <0 \Rightarrow -\sum_{t=0}^{T} \left(\frac 1{1+r}\right)^t\xi q_t +C_R+I_N-\sum_{t=0}^{T} \left(\frac 1{1+r}\right)^t\tau\cdot\frac 1T(C_R+I_N)\big]$$

Using an average period quantity produced $\bar q$, and denoting

$$R\equiv  \frac {(1+r)^{T+1}-1}{r(1+r)}$$

this becomes

$$NPV_{Oh}- NPV_{Nh} <0 \Rightarrow - R \cdot\xi\cdot \bar q +C_R+I_N-R\cdot\tau\cdot\frac 1T(C_R+I_N) < 0 $$

$$\Rightarrow \left(1-R\cdot\tau\cdot\frac 1T\right)(C_R+I_N) < R \cdot\xi\cdot \bar q $$

$$\Rightarrow \xi > \left(R^{-1}-\frac {\tau}T\right)\frac {C_R+I_N}{\bar q} \tag{4}$$

Inequality $(4)$ determines the minimum level of the environmental tax rate from the point of view of financial incentives in order for the incumbents to switch technologies.

So, does it matter whether this minimum level is consistent with the tax rate that would reflect the proper pricing of negative externalities, since by setting it above it, we expect that the old technology will be wiped out? It doesn't seem so. This is important: it says that we can see the environmental tax rate as a threat that it won't be necessary to realize.
From a "political point of view", is this minimum tax rate feasible in view of possible reaction, lobbying etc? This is a different chapter.

Note that it does not affect our goal what will happen in an oligopolistic framework, where possibly, the market demand cannot sustain both the incumbent and the new entrant (i.e. double output): whatever happens, the one, the other, or both, the appropriately set tax rate ensures that we will see only new technology in operation.


Sequential Game Rollback Equilibria Question

You're correct on the definition of the strategies: Albus' strategy set is $S_{A} = \big\{N,E,S\big\}$, and Minerva's is $S_{M} = \big\{a, b\big\} \times \big\{a, b\big\} \times \big\{a, b\big\}$.

Remember, a strategy is a contingent plan that prescribes an action at every possible node at which a player plays.

Then, the unique equilibrium (via Backward Induction) is $(N, b, a, b)$. The others you mention are not equilibria, because they involve incredible threats, e.g. if Minerva found herself at the decision node following a choice of $E$ by Albus, she would never choose $b$.


Do conceptual ideas that become standardised ever make money?

This question is hard to answer because it can take so many perspectives. The only way I find to approach it is using examples.

Could John Nash profit from his paper in game theory? Could Einstein profit from general relativity? Yes, with prizes and recognition, but not everytime the concept is used. Not in a way analog to the copyrights of music or publishers.

If you talk about standard methodologies or standardized concepts, in the business world, several organisations strive for developing and enforcing standards, notably ISO and GS1. They do have revenue, mostly from consulting or memberships, but they are not-for-profit organisations, as having profit can drive standardisation away.

The history of Thunderbolt demonstrates how trying to make money of a standard (through royalties) can sometimes drive consumers to use another standard which is free, even if inferior.


Does a strong dollar achieve the same tightening effect on the economy as rising interest rates?

ceteris paribus, you are absolutely right. The price for international borrowing and lending is determined by both the interest and exchange rate.

Three remarks:

Above; I wrote ceteris paribus, meaning nothing else changes. In reality, however, both (long-term) interest rates and exchange rates are heavily impacted by expectations on how the economy is going to evolve.
Note that the exchange rate only impacts international transactions, on both financial and goods markets, whereas the interest rate impacts all borrowing and lending. The impact of a change in the exchange rate is therefore mainly a quantitative question. Without going into too much detail, the effect is likely going to be a lot bigger for a small open economy, compared to the US.
Usually, causation is assumed to go the other way. That is, the Federal reserve sets interest rates (arguably on the backdrop of domestic economic conditions), and then the exchange rate reacts on changes in interest rates. However, this is conventional wisdom, not an economic fact.


Definition of "Structural Equation" In Economics

First, structural equations assume causality. The following quotation is from Cameron and Trivedi's textbook Microeconometrics: Methods and Applications:

  At one end of the spectrum can be found very highly structured models,
  derived from detailed specification of the underlying economic
  behavior, that analyze causal (behavioral) or structural relationships
  for interdependent microeconomic variables. At the other end are
  reduced form studies that aim to uncover correlations and associations
  among variables, without necessarily relying on a detailed
  specification of all relevant interdependencies.

Second, identifiability does not exactly define whether an equation is structural or not. I would rather say that identifiability is the reason why we are forced to work with nonstructural equations. Parameters in structural equations cannot be identified most of the time. What can be estimated instead are reduced-form (or nonstructural) parameters. For instance, take the following simulatenous equation model
\begin{align*}
 x_i =\alpha_0 y_i +\alpha_1 z_i +\epsilon_i\\
 y_i = \beta_0 x_i+ \beta_1 z_i +\mu_i
\end{align*}
with two endogenous variables $x$ and $y$, an exogenous variable $z$, and $\epsilon$ and $\mu$ as error terms (suppose also $\alpha_0\beta_0\neq 1$).
These two equations are structural. They may correspond to demand and supply equations of a given good, with $x$ the quantity and $y$ the price for instance. Without more assumptions, the coefficients are not identified and cannot be estimated. We can however estimate the parameters $\zeta$ and $\gamma$ of the reduced-form equation:
\begin{align*}
 x_i = \zeta z_i + e_i\\
 y_i = \gamma z_i  +u_i
\end{align*}
with $\zeta=\frac{\alpha_1+\alpha_0\beta_1}{1-\alpha_0\beta_0}$ and $\gamma=\frac{\beta_1+\alpha_1\beta_0}{1-\alpha_0\beta_0}$.

In the second paragraph of the extract, the point is that identifiability (of the structural equations) is not necessary "as long as the structure remains unaltered". For instance, you can well predict the values of $x$ given the knowledge of $z$ and a good estimate of the nonstructural parameters $\zeta$. The last sentence illustrates the fact that, in our case,  i) we do not need to know the causal structure to make a good prediction, ii) it does not help to distinguish between structural and nonstructural equations, and iii) we do not care about identifiability of the structural model.


Log-normality assumption in consumption based asset pricing

The typical two-period Lagrangian is

$$\Lambda = \beta^t\cdot \Big(\frac{c_t^{1-\gamma} -1}{1-\gamma} + \lambda_t\cdot \big[(d_t+p_t) \pi_{t-1} + \pi_{t-1}^0- c_t - \pi_t p_t - \pi_t^0 p_t^0\big]\Big) \\
+ \beta^{t+1}\cdot \Big(\frac{c_{t+1}^{1-\gamma} -1}{1-\gamma} + \lambda_{t+1}\cdot \big[(d_{t+1}+p_{t+1}) \pi_{t} + \pi_{t}^0- c_{t+1} - \pi_{t+1} p_{t+1} - \pi_{t+1}^0 p_{t+1}^0\big]\Big)$$

The first order conditions with respect to $c_t, \pi_t$ are

$$c_t^{-\gamma} = \lambda_t \implies ... \gamma\ln \frac {c_{t+1}}{c_t} = \ln \frac {\lambda_{t}}{\lambda_{t+1}} \tag{1}$$

$$-\beta^t\lambda_tp_t + \beta^{t+1}\lambda_{t+1}(d_{t+1}+p_{t+1})=0 \implies  \frac {\lambda_{t}}{\lambda_{t+1}} = \beta \frac{p_{t+1} + d_{t+1}}{p_t} \tag{2}$$

and so, using also the definition of the gross return,

$$\ln \frac {\lambda_{t}}{\lambda_{t+1}} = \ln \beta + \ln R_{t+1} \tag{3}$$

Combining $(1)$ and $(3)$ we get

$$\ln \frac {c_{t+1}}{c_t} = \frac 1 {\gamma}\ln \beta + \frac 1 {\gamma}\ln R_{t+1} \tag{4}$$

So we see that at the optimal path, consumption growth is a direct affine function of the log-risk returns. This among other things implies that their correlation coefficient is equal to unity.

The normal distribution is closed under affine transformations (alternatively, under scaling and shifting), so if we assume that log-risky returns are normally distributed, then consumption growth is also normally distributed (with different mean and variance of course).

Note that although in general, the joint normality assumption is an additional one to be made when two normal random variables are not-independent, here, the fact that the one is an affine function of the other guarantees joint normality. By Cramer's condition for bivariate normality, it must be the case that all linear combinations of two normal random variables have a univariate normal distribution. In our case we have (generic notation) the random vavriable $Y$ and the random variable $X = a+bY$. Consider

$$\delta_1X + \delta_2 Y = \delta_1(a+bY) + \delta_2 Y = \delta_1a + (\delta_1b+\delta_2)Y$$

So for any $(\delta_1, \delta_2)$ (except the zero vector which is excluded a priori), $\delta_1X + \delta_2 Y$ follows a normal distribution if $Y$ does. So it is sufficient to assume that log-risk returns follow a normal distribution to obtain joint normality also.


Proof: Risk averse; Certainty Equivalent smaller than expected value

You basically have the proof, just remember that for any function $f(x)$ strictly increasing, then $f(a)\leq f(b)$ if and only if $a\leq b$. It is also true that for any strictly increasing function its inverse exists, but there is probably no need for that.


Why can utility functions be continuous, and what does this imply for marginal utility?

Utility functions can be continuous because quantitites can be continuous. (Think liters instead of bottles of wine or kilos instead of loafs of bread.) But even if quantities are discrete; whenever a unit is reasonably small (grains of salt: yes; cars: no) it is just way more convenient to work with smooth functions than with discrete ones, since the former allow you to calculate derivatives.

Introductory textbooks often use discrete quantities ("additional utility of consuming the next unit") to define marginal utility, since this is more intuitive. However, as soon as you have smooth utility functions, you better use the derivative of utility instead. Think of the point measure as the limit of the arc measure as the increase in quantity goes to zero. (That's more or less how the derivative is defined.) If quantities are discrete but very small and your utility function is reasonable, then the two measures are almost identical anyway. As an example, if your utility function is defined over $x$ = the number of grains of salt on your chips, then, using e.g. $u(x)=1-(x/100-1)^2$, your $MU$ at $x=100$ is $0$ if you use the derivative and $-0.0001$ if you just calculate $u(101)-u(100)$. To find the optimal quantity, setting $MU=0$ is of course easier than listing 200 discrete values and searching for the maximum.


What is the average percent of value an employee makes which he gets on salary (called)?

You are most likely looking for labor or wage share defined as the ratio of labor costs to nominal GDP. It basically expresses how much share of nominal GDP gets captured by labor.

You can find the data for the labor share in the US here, under that bit of text (for some reason I could not find direct link to pure dataset). The data are from the US bureau of labor statistics.


Some thoughts:

accurate prediction seems ambitious without empirical data;
many people have negative or zero net wealth;
a continuous distribution could lead to oddities;
the world may have a finite total wealth and so a finite average wealth thus excluding some distribution shapes such as Pareto distribution with $\alpha \le 1$

But you could certainly develop a model, even if it would be wrong.  For example, let's try:

A Pareto distribution with shape parameter $\alpha$ and scaling parameter $x_{\min}$
The application of the Pareto 80-20 principle, so $\alpha=\log_4(5) \approx 1.160964$
A total global wealth of perhaps of the order of $\$ 250$ trillion and a global population of about $7.4$ billion, giving an average wealth $\mu$ of about $\$33784$ and $x_{\min}=\mu\frac{\alpha-1}{\alpha}$ would be about $\$4684$ (looks rather high to me, but $x_{\min}$ is really a scaling factor and should not be taken too seriously)
A Gini coefficient of $\frac{1}{2\alpha-1}$ would be about $0.76$
The top $1\%$ (so $74$ million people) would have over $x_{\min} / 0.01^{1/\alpha}$ each, so here about $\$250,000$
a person with wealth $w$ would be at the $\left(\frac{x_{\min}}{w}\right)^{\alpha}$ quantile measured from the top

Since this probably does not fit reality, you can then start adjusting it if you want: for example you can get closer to a Gini coefficient of $1$ if you reduce $\alpha$ towards $1$ from above, and this would also lower $x_{\min}$.   If you use a Gini coefficient of $G$, that would make $\alpha=\frac{G+1}{2G}$ and $x_{\min}=\mu\frac{1-G}{1+G}$ so the person with wealth $w$ could be at the $\left(\frac{\mu(1-G)}{w(1+G)}\right)^{(G+1)/(2G)}$ quantile, measured from the top.

Or you could decide to give a certain share of the population a wealth of zero.  If the rest keep the same $\alpha$ on a (now zero-inflated) Pareto distribution this will increase $x_{\min}$ for the rest while also increasing the Gini coefficient overall.

Or you could look for empirical information on the actual wealth distribution and then try to fit a curve to that.  Or perhaps look at global income distribution instead, which has not had a particularly simple function in the past.


Hep with total differentiation of an AD function

For a variable $X$, let $dX$ denote its total differential. Let $k$ be a constant, and $X$ and $Y$ variables. You'll need the following rules:

$$dk = 0$$

(constant rule),

$$d(X + Y) = dX + dY$$

(sum rule),

$$d(XY) = Y \cdot dX + X \cdot dY$$

(product rule) and

$$d\left(\frac{X}{Y}\right) = \frac{ Y \cdot dX - X \cdot dY }{ Y^2 }$$

(quotient rule). These rules follow immediately from the definition of the total differential. (For more on total differentials, see e. g. chapter 8 of Chiang and Wainwright, Fundamental Methods of Mathematical Economics, 4th ed., McGraw-Hill 2005.)

Now let's start with your equation. By the product rule we have:

$$\begin{align}
dx & = d\left( \mu^{-1} (g + i + e) \right) \\
& = d\left(\mu^{-1}\right) (g + i + e) + \mu^{-1} d(g + i + e)
\end{align}$$

What is $d(\mu^{-1})$? Because $\mu^{-1} = \frac1\mu$, by the quotient rule,

$$\begin{align}
d\left(\mu^{-1}\right) & = d\left(\frac1\mu\right) \\[4pt]
& = \frac{ \mu \cdot d1 - 1 \cdot d\mu }{ \mu^2 } \\
& = -\frac{ d\mu }{ \mu^2 }
\end{align}$$

where we also made use of the fact that by the constant rule, $d1 = 0$. Plug this into the above intermediate result and also apply the sum rule:

$$\begin{align}
dx & = d\left(\mu^{-1}\right) (g + i + e) + \mu^{-1} d(g + i + e) \\
& = -\frac{ d\mu }{ \mu^2 } (g + i + e) + \mu^{-1} \left( dg + di + de \right) \\
& = -\frac{ d\mu }{ \mu } \cdot \mu^{-1} (g + i + e) + \mu^{-1} \left( dg + di + de \right) \\
& = -\dot\mu \cdot x + \mu^{-1} \left( dg + di + de \right)
\end{align}$$

where use was made of the definition of $x$, and the fact that $\frac{ d\mu }\mu = \dot\mu$.

Now we're almost there. Note that since $\frac{ dX }X = \dot{X}$ for any variable $X$, we also have $dX = X \cdot \dot X$; applying this to $g$, $i$ and $e$ and dividing the entire equation by $x$ then yields

$$\begin{align}
\dot x = \frac{ dx }x & = -\dot\mu + \mu^{-1} \cdot \frac{ dg + di + de } x \\
& = -\dot\mu + \mu^{-1} \cdot \frac{ dg }x + \mu^{-1} \cdot \frac{ di }x + \mu^{-1} \cdot \frac{ de }x \\
& = -\dot\mu + \frac{ \mu^{-1} g }x \cdot \dot g + \frac{ \mu^{-1} i }x \cdot \dot i + \frac{ \mu^{-1} e }x \cdot \dot e
\end{align}$$

which is the desired result.


I don't know the name of the term you are looking for but the ratio you are looking for can be calculated as the volume of trade data is usually available for each stock. A similar metric that you may like is Volume-Weighted Average Price which takes into account the volume of trade.

Is this suspicion accurate?

It's very subjective but your suspicion seems to be based on your hypothetical construct which needs further discussion. You say 'I really want to buy a share..'. Now why would that be? For a rational investor, it will be when she sees it's value to be \$11 rather than market value of \$10.
When enough investors are there in the market that think that the value of the share should be high then the price of the stock goes up (and there's nothing wrong with it), which reflects, in a way, the collective sentiment about the stock.
In the above reasoning, we consider the market as a non-cooperative non-atomic game, in which single player's decision has no influence on market outcome. Therefore, an individual is not pushing the price of the stock but rather the general consensus that the stock is going to giver higher returns.
So your suspicion is correct when these at least one of the following assumptions are not be valid: (a) investors are not rational (domain of behavioral finance); (b) game cannot be considered non-atomic (there are big players that can alone move the market - almost always true); and (c) there is collusion among investors to bid the price of a stock up.
So yes, for some stocks, to some extent it is perfectly possible that some investors bid the price up to the level that it doesn't reflect the real valuation. This is more common with penny stocks which is why there are more restrictions/regulations around them.


Could a tax on rental income that depends on price per square meter induce more housing with lower rents?

Why do you think that additional taxes on more expensive apartments (progressive taxes) would lead to lowering the prices and offering them cheaper than market dictates?

Adding taxes on top would just make these apartments even more expensive for the consumer, since the owner will push down the additional cost to the consumer rather than take a haircut. Especially if these apartments are laying around empty anyway.

If I am willing to sell product A for say 1.000$ but there are no buyers - will an increase in the tax help me sell it?
If you on the other hand gave me a tax discount, I might want to sell it cheaper, since I may be left with the same after-tax profit.


Rosen's Diagonal Strict Concavity condition

So you want to find a maximum of $\sigma(s,z)$. If $\sigma$ is diagonally strictly concave you can do so by starting at any point and just following the gradient $g(s,z)$ until you find the maximum and no matter where you start, you will always end up at the same point (Start at the lower black points and follow the direction of the gradient (the direction of the steepest ascent).).

However, if $\sigma$ isn't diagonally strictly concave, you could end up at different maxima by starting at an arbitrary point and following the gradient (Follow the direction of the steepest ascent starting from the two lower black dots; you'll end up at two different points.).


Could price floors improve worker safety in the meat cutting industry?

A priori this is difficult to say without any empirical experiment but it is unlikely due to:

Price floors do not necessarily lead to more competition on quality. Theoretical models do show that price floors can (depending on parameters) lead to an increase in quality (Bilotkach, 2012 and sources cited therein), but they can also just result in intensifying competition along different dimensions (such as location, advertising, and other dimensions without necessarily more differentiation of products (see Bhaskar (2003) and Bilotkach (2012) and sources cited therein).
So already an issue is that price floors cannot be guaranteed to lead to higher completion on quality. This would first have to be tested in the specific case.

High-quality products do not necessarily imply better working conditions and high safety standards. Apple products could be argued to be of high quality (although I am an economist, not an engineer), yet apple is getting regularly lambasted for poor working conditions in its factories (see this article from the Guardian).
I was not able to find on google scholar any empirical or theoretical work that would support the idea that price floors on products would lead to better working conditions. Also higher firm profitability is not necessarily associated with better working conditions. Firms can as well just keep working conditions the same and pocket higher profits.

Even if it would work in public economics we are not interested in solutions that just work, but the solutions that achieve goals that we set in an efficient manner. As mentioned by BKay in their excellent comment there are many different ways how to raise work safety, arguably most if not all ideas mentioned in the comment would be more efficient than price floor which would likely have to change with market conditions and be set in a careful manner which might be too difficult to implement.


In former times, seignorage was a de facto fee charged by a government authorized mint for converting precious (or semi-precious) metals into de facto trademarked hard currency.  Typically a mint would charge about a ten percent fee for this service.  Anyone in possession of the hard currency could "redeem" the currency by melting it down back into the original metals, and thereby recover the remainder of the currency's value.

In countries that do not rely on hard currency for most daily transactions, seignorage occurs in two stages:

A government borrows purchasing power, and issues a coin or note in exchange.  The government promises to repay the purchasing power upon return of the coin or note.  An accountant would say that this is the "cash flow" part of the seignorage transaction.
The government defaults on the promise to repay the purchasing power.  This can happen explicitly (like when the U.S. refused to repay silver certificates in silver, or when the U.S. "closed the gold window") or implicitly via inflation of price levels stated in government notes.  An accountant would say that this is the "revenue" part of the seignorage transaction.

According to the U.S. Federal Reserve, about 1.5 trillion (1.5 million million) U.S. dollars were in circulation in 2017.  This amount has been increasing at the rate of about 90 billion (90,000 million) U.S. dollars per year.

So the rate of the U.S.' seignorage incremental borrowing per year is about 90 billion dollars per year.

The official U.S. inflation rate has been in the range of 0 to 3 percent per year during the last few years.  If the average is 1.6 percent per year, then the amount of seignorage revenue (i.e. de facto default on the purchasing power of U.S. dollar notes in circulation) is about 25 billion (25,000 million) U.S. dollars per year.

In addition, the U.S. Federal Reserve purchased about 4 trillion (4 million million) U.S. dollars worth of mortgages and other securities during and shortly after the 2008 financial crisis.  It "increased the monetary base" when it purchased these securities.  In other words, it used seignorage to provide the electronic money used to purchase these securities.  At an average inflation rate of 1.6 percent per year, this is about 65 billion (65,000 million) U.S. dollars of seignorage revenue per year.

The U.S. Federal Reserve uses its net income from (after being laundered from seignorage revenues into net interest on securities and capital gains on securities sales) to make dividend payments to the United States Treasury.  From 2010 through 2013, these dividend payments were about 75 billion (75,000 million) U.S. dollars per year.


Is optimal labour zero when (i) capital fixed and (ii) elasticity of substitution less than 1?

Ok, rather embarrassing, but the problem was in the formula for $Y_0$. In effect, for $\rho>0$, it is true that

$$ Y_{_{L=0}} = AK(1-\alpha)^{\frac{1}{\rho}} $$

However, for the case of $\rho<0$, the limit of $Y$ is zero. The proof is trivial. Rewrite the CES as:

$$ Y = AL\left(\alpha + (1-\alpha)\left(\frac{L}{K}\right)^{-\rho} \right)^{\frac{1}{\rho}} $$

Then, it is trivial to see that:

$$ \lim_{L\rightarrow 0} Y = A0 \: \alpha^{\frac{1}{\rho}} = 0 $$

With this, we can see that $\Delta Y = Y > 0$. This solves the inconsistency.

PS: as it turns out, the marginal product of labour at $L=0$ is not infinity. Apparently, the Inada conditions do not hold for the CES with $\rho<0. Actually, the marginal product of labour seem to be zero! (check slide 15 here). This is a major revelation to me.


Which currency is preferred when countries with different currencies purchase imports/sell exports?

There isn't a typical answer.

The vendor may want paying in whatever currency represents the highest proportion of their costs, as this reduces their currency risk.

But there may also be restrictions on what currency they can trade in; how much of some particular currency they're allowed to hold; there may be capital controls in place; some currencies might be easier to hedge than others; trade in some currencies might attract that country's tax authorities, if significant movements have to be registered; ... and so on.

They might want to do business across various buyers in a basket of currencies, as a way to minimise currency risk through diversification.

They may prefer to do business in currencies with a higher liquidity (USD, GBP, EUR, CHF, AUD, JPY, CAD).

There are lots of variable factors in play. And it's a negotiation between vendor and buyer; the above is written as if the vendor holds all the power, but similar issues apply to the buyer too. For example, the buyer may want to pay in whatever currency represents the highest proportion of their income.


Is Canada a country of high taxation in international comparison?

Because they are high in comparison. Take a look at the data from the OECD here, for 2015. I have plotted them below:

(The red is the personal income tax set by law, and the blue is the one adding tax credits, as far as I understand).

As you can see, rates in Canada are quite high in comparison with other OECD countries like Estonia or Poland, and also higher than the UK.

Also, in international comparison, Canada is quite good in redistributing income. You can see that by comparing the Gini index before and after taxation:

Property taxes are quite high too:

The final evidence comes from this index of progressivity of both personal income tax and social security:

Conclusion

Canada is quite a progressive country in terms of taxation, redistribution quite a lot, and lowering market inequality considerably. Even more, it seems to be better than the UK in almost every comparison.

Source: OECD (2012).


How to deal with a singular Leontiev inverted matrix?

I am not sure exactly what you divided by what, but suppose your input-output table looks like this:
$$
A =
\left[
\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}
\right]
$$
If you proceed to divide the first column by $\sum\limits_i a_{i,1}$ the second by $\sum\limits_i a_{i,2}$ and the third by $\sum\limits_i a_{i,3}$ then your new matrix will have 1 as an eigenvalue. (You can show this by showing that the norm of any vector is unchanging upon multiplication with the matrix by straightforward calculation.) This is problematic, because if $s$ is an eigenvector then
$$
A \cdot s = s = I \cdot s
$$
which implies
$$
(I - A) \cdot s = 0
$$
which in turn implies that $I - A$ is not invertible (singular). So my guess would be that the problem is the division.

I thought (incorrectly, see comment below) that input-output models usually deal in physical goods, not the value of physical goods, so the unit of measurement for $a_{11}$ would not be euros by say liters (of milk). In this case you would not have to divide the value of individual inputs by the sum of the value, but the number of goods used in the individual inputs by the total output of a good. This may make your problem go away.


Splittet Value Function and Hamilton-Jacobi-Bellman equation

I don't think I've seen cases with a kink in continuous time that are solved numerically. Aguiar & Amador (2014) have a kink in their HJB, but they solve their HJB analytically.

Validation

I don't know to what extent Moll et al's algorithm is going to get that right. You will most likely get weird behavior around the kink, but whether it's the real functional form is to be doubted, and you'll have to argue ex-post that "it looks okay alright?".

If you want to get a better grip into whether the kink looks alright, make it more extreme. Without going through your equations, it seems that the first part is convex and the second part is linear. Increase the curvature for the first part so you can "see" more easily whether it still looks correct around the kink.

Also, you definitively should look at the numerical counterparts to the first-order and second-order derivative, the first-order and second-order difference of the approximation divided by the step size. Do they have the correct signs around the kink?

One thing I'd emphasize is having many grid points around the kink. However, since the algorithm gets tougher when you have a nonlinear grid (because the grid step size is now not a constant, but a vector), it might just not be worth it and you could rather jack up overall grid points and keep it linear.

Speed

Finally, you really should look into having sparse matrices for the A and B matrices. 10 iterations per minute is really slow. It takes you perhaps 30 minutes to understand and implement (if you've never worked with sparse systems before), but already after the second time you run the code you've made up that time.

If the problem persists, have a look at profilers for your coding language. For Python, this is a nice solution. Even if you have everything sparse, perhaps you forgot some matrix that you don't even need in the iteration loop. That was the case for me, and with large grid sizes that single reinitialization took up a lot of time. Profiler will tell you what's what.


Textbooks addressing Markov decision processes in microeconomics/game theory/decision theory?

I guess you learned about MDP from Reinforcement Learning since RL tries to solve MDP without parameters.

For MDP without considering Microeconomics, (Indeed MDP is a decision-making process.)

"Markov Decision Processes: Discrete Stochastic Dynamic Programming" by Martin Puterman

If you want an Economics based book,

"Recursive Methods in Economic Dynamics" by  Nancy L. Stokey and Edward C. Prescott

For multi-agent MDP, (there are not many books for this topic specifically  although some Economics books mention the topic)

"Competitive Markov Decision Processes" by Jerzy Filar and Koos Vrieze


Intuition behind Engel Aggregation and Cournot Aggregation

These expressions can be easily derived formally by taking the derivatives of Walras' law w.r.t $I$ (Engel aggregation) and $p_j$ (Cournot aggregation).
The Engel aggregation means that not all goods are luxuries ($\eta_i>1$), or necessities ($\eta_i<1$), or inferior goods ($\eta_i<0$). The Cournot aggregation says that the budget share $s_j$ of the good $j$  is small when the good $j$ has many substitutes ($\epsilon_{ij}>0$), and big when it has many complements ($\epsilon_{ij}<0$), or their respective budget shares are large.


I would say that we don't generally speaking assume that stock betas are constant. For example the paper Explanations for the Instability of Equity Beta: Risk-Free Rate Changes and Leverage Effects, from 1985 (!), cites at least six other earlier papers that the market risk of securities is not stable.
Many papers use a rolling window CAPM estimation approach (a popular way of allowing betas to vary) exactly because CAPM and more generally factor model betas do not seem to be constant. See for example the JFE paper CAPM for estimating the cost of equity capital: Interpreting the empirical evidence.


Marshallian demand with Leontif preferences

So, I have not really worked out the maths behind solving this. I prefer to take a shortcut and just compute it numerically. Here is what I have.

These are the contour surfaces and the budget constraint. Notice how the solution will be at the "kinked" curve in the middle of each surface.

The levels 1 and 2 are only for example. The third level, u = 3.92762 is related to the next part of my answer. Also, BC is the budget curve with income = 300. I have chosen $\alpha = 0.5$ and $P_1 = 2, P_2=3, P_3=1$.

Now, using the parameters mentioned in the last paragraph, the maximum is found to be at $q_1 = 65.8896, q_2 = 39.1448, q_3 = 50.7863$ and the maximized utility is $u = 3.92762$. I assumed that all goods are strictly consumed at positive amounts.

I am attaching my Mathematica code below, so that you can replicate and modify these results as you want

  u = Min[a*Log[q1] + ((1 - a)*Log[q2]), Log[q3]]

  BC = p1*q1 + p2*q2 + p3*q3
    a = 0.5; p1 = 2; p2 = 3; p3 = 1;

  ContourPlot3D[{u == 1, u == 2, u == 3.92762, BC == 300}, {q1, 0,
  100}, {q2, 0, 100} , {q3, 0, 100}, AxesLabel -> Automatic,
  ContourStyle -> {Opacity[0.5], Opacity[0.5], Opacity[0.5],     Opacity
  [1]}, Mesh -> None, PlotLegends -> "Expressions"]

  NMaximize[{u, BC <= 300, q1 > 0, q2 > 0, q3 > 0}, {q1, q2, q3}, Reals]


This paper may be of interest:

This paper examines the impact of regulatory intervention to cut termination rates of calls from fixed lines to mobile phones. Under quite general conditions of competition, theory suggests that lower termination charges will result in higher prices for mobile subscribers, a phenomenon known as the waterbed effect. The waterbed effect has long been hypothesized as a feature of many two-sided markets and especially the mobile telephony industry. Using a uniquely constructed panel of mobile operators’ prices and profit margins across more than twenty countries over six years, we document empirically the existence and magnitude of this effect. Our results suggest that although regulation reduced termination rates by about 10%, this also led to a 5% increase in mobile retail prices. We also provide evidence that both competition and market saturation, and most importantly their interaction, affect the overall impact of the waterbed effect on prices.

TESTING THE “WATERBED” EFFECT IN MOBILE TELEPHONY (Genakos and Valletti (2011))


The answer to 2. is no.
One way to see this is from MWG's Property 6.D.2: $F$ SOSD $G$ if and only if
\begin{equation}
\int_0^xF(t)\mathrm dt \le \int_0^xG(t)\mathrm dt \quad\text{for all }x.
\end{equation}
Dixit calls the two integrals super-cumulative functions of $F$ and $G$, respectively. Hence, a characterization of SOSD is that the super-cumulative of the dominating distribution always lies below the super-cumulative of the dominated distribution. (This is reminiscent of the characterization of FOSD as the cdf of the dominating distribution always lying below the cdf of the dominated one.)
For a counterexample, we only need to come up with distributions $F$ and $G$ such that their super-cumulatives cross. Here's one:
\begin{align}
f(x)&=\begin{cases}
0.1 & x\in\{0,6\}\\
0.4 & x\in\{2,4\}\\
0 & \text{elsewhere}
\end{cases}\\
g(x)&=\begin{cases}
\frac13 & x\in\{1,3,5\}\\
0 & \text{elsewhere}
\end{cases}
\end{align}
Both distributions have the same mean of $3$, but as the following figure shows, their super-cumulatives ($S_F$ and $S_G$) cross at several points, and so neither distribution SOSD the other.

Here, $S_F(x)=\int_0^x F(t)\mathrm dt = \int_0^x\int_0^tf(s)\mathrm ds\mathrm dt$, and $S_G$ is similarly defined. Since $f$ and $g$ are probability mass functions, the cdf's $F$ and $G$ are step-functions (shown below). Integrating the step-functions yields the continuous and piece-wise linear $S_F$ and $S_G$ above.

Edit
As OP noted in a comment, "ALL risk averse people with iso-elastic utility ($u=x^\alpha$, $\alpha\in(0,1)$) prefer gamble $G$ to gamble $F$". The negative answer above suggests that there must be a concave function with which $F$ is preferred to $G$. Here is an example:
\begin{equation}
u(x)=\begin{cases} 2x& x\le 2\\ 4& x>2 \end{cases}
\end{equation}
This function is concave, and $\mathbb E_F(u)=3.6>3.\overline{33}=\mathbb E_G(u)$.


What is the difference between "aggregation" and a "representative agent?"

(I cannot say if my answer will respond to your questions, which indeed, are a bit unclear).

If one browses through many-many economic papers, one will get the impression that "representative" just means identical. Indeed in large chunks of literature this is the case, for historical reasons.

The drive behind the adoption of the "representative consumer" modelling framework came from the Lucas critique on the previous-generation macro-models, and the requirement that macroeconomic models are "micro-founded". But true theoretical aggregation (with heterogeneity present) requires some skill and knowledge and the bulk of the discipline appeared to have quickly settled for the "representative means identical" framework.

The problem is that in such a case, you don't really have a macro-model, just a blown-up version of a micro-model (note: here the words micro/macro are not to be mapped to the partial/general equilibrium concepts). There is nothing to "aggregate" here: the whole point of aggregation is to see whether the behavior of the collective differs from the behavior of the individual. And in the "representative means identical" approach, no such thing can happen, by construction: Instead of micro-founded macro-models, we ended up having blown-up micro-models posing as macro-ones (this is not opinion, I am just describing).

There are some models where the term "representative" obtains some intuition -especially in models with more than one classes of agents (say, labor owners and capital owners). Here we model two agents, and each is "representative" of its class. In-class are all identical, but here it sounds more appropriate to call the two individuals "representative".

The funny thing is, the concept "representative consumer" (RC) does have a special meaning: the representative consumer represents all consumers as regards basic structure, not measurement or quantities. E.g. "all individual maximize utility from consumption ("same structure"), but their utility parameters may differ("different measure"). All consumers have wealth, but the level of wealth may differ. Etc. RC is still a modelling abstraction, but it does leave room for heterogeneity.

A good source on the matter is

Caselli, F., & Ventura, J. (2000). A representative consumer theory of distribution. American Economic Review, 909-926.

Apart from their focus on developing a theory of distribution in the context of an RC model, they make a good job in presenting what can be done in an RC framework and what not. An excerpt:

  The RC is a fictional consumer whose utility maximization problem when
  facing aggregate resource constraints generates the economy's
  aggregate demand functions. The RC assumption does not rule out
  consumer heterogeneity, but only requires that potential sources of
  consumer heterogeneity have sufficient structure to ensure that the
  sum of all consumers behaves as if it were a single consumer.


Not necessarily. One has to distinguish between (output - I just assume that you mean output, i. e. gdp) growth and per capita growth.

In (very) simple terms: Growth is the result of either population growth, capital accumulation (more investment) or technological progress.

So one reason for the low US growth could be limited population growth and looking at the data this is a possibility (Population Growth Rate US)
Another possibility might be the "unusability" of technical progress. While a lot of research is done and we discover lots of new things not everything can be used in the production progress and is therefore not reflected in gdp growth.

One way to check this possibility is to development a new measure of technical progress - I think Daron Acemoglu and some coauthors are working on a new index. Another possibility is that this technical knowledge is just worthless. Consider the internet rush around 2000 which was at that time considered as technical progress which would increase the productivity of the whole economy... Well most of this was wrong and this technical progress did not lead to an increase in productivity - an argument often used by Robert Solow.

So the question: "Why is a nation not growing" is pretty difficult and not easy to answer or even understand.


Why we need to control for the interation of year and industry fixed effects?

When you control for not just year fixed effects but instead year-region or year-industry it adds flexibility.
The year fixed effects controls in a flexible manner for the time-trend and is more flexible - less restrictive - than for example assuming that the time trend is for example linear $a \cdot t$, second order polynomial $at + bt^2$, exponential $exp(at)$ ect.
Nevertheless, year fixed effects are still restrictive compared to year-region or year-industry because the assume that the time trend is the same for all regions or for all industries instead of allowing different regions to follow different trends or different industries.
Consider for example this plot here

and assume that it show the average wage of women (red) and the average wage of men (black) over time.
We then conclude that the gender wage-gap is closing. However, someone then points out this is not really the case because although female workers on average over time get the same as male workers this is simply because female workers have changed from predominantly working in the low-wage industry to be working in the high-wage industry.
Offcourse you are sharp and you run a regression controlling for industry
$$\log y_{it} = \beta_0 + \alpha \cdot \text{male}_{it} + \lambda \cdot  \text{industry}_{it} + \delta_t + \epsilon_{it}$$
which is a good idea and for the simulated data used to generate the plot above you then get an estimate $\hat \alpha = 3.05$ clearly showing that male workers are given more in wage than female workers.
However, the true $\alpha$ is actually even larger and equal to $4$ so you are still underestimating the degree to which female workers are discriminated on the labor market.
In this example this happens because:
Over time the wage in the high wage sector increases faster than the wage in the low wage sector. So while women over time changed to the high wage sector you are undercontrolling for the sector effect because the early year with a small difference between the sectors bias the sector effect downwards.
In this case what you should have been doing to get an unbiased estimate was to run the regression
$$\log y_{it} = \beta_0 + \alpha \cdot \text{male}_{it}  + \lambda_{st} + \epsilon_{it}$$
where $\lambda_{st}$ is the sector-time fixed effect.
You can simulate this example in R using the following code
library(data.table)
library(lfe)
N1 <- 1000
N2 <- 1000
N <- N1+N2
T <- 10
b0 <- 1

# a1 is the true gender effect
a1 <- 4

gender <- rep(c(rep(0,N1),rep(1,N2)),T)
a2 <- rep(1:T * 2,N)
dt <- data.table(id=rep(1:N,each=T),time=rep(1:T,N),gender=gender)
setkey(dt,time,gender)
industry <- c()
for (t in 1:T)
    {
        temp1 <- sample(c(0,1),N1,replace=TRUE,prob=c(0.5,t/T))
        temp2 <- sample(c(0,1),N1,replace=TRUE,prob=c(0.5,0.5))
        industry <- c(industry,temp1,temp2)
    }
dt$industry <- industry
setkey(dt,id,time)
dt$log_y <- b0 + a1*dt$gender + a2*dt$industry + 0.3*dt$time + rnorm(N*T)
foo <- dt[,.(m_wage=mean(log_y)),by=.(time,gender)]
plot(1:T,foo[gender==0,]$m_wage,ylim=c(0,20),col="red",type="l")
points(1:T,foo[gender==1,]$m_wage,type="l")

dt$i_time <- interaction(dt$time,dt$industry)
# Regression with industry dummy

summary(felm(log_y ~ gender+industry|time,data=dt))
#          Estimate Std. Error t value Pr(>|t|)
# gender    3.05645    0.04164   73.41   <2e-16 ***
# industry 11.13346    0.04215  264.12   <2e-16 ***

# Regression with industry time fixed effects
summary(felm(log_y ~ gender |i_time,data=dt))

#        Estimate Std. Error t value Pr(>|t|)
# gender  4.00873    0.01443   277.7   <2e-16 ***

savePlot("male_female.jpg",type="jpg")


Renewal of multiple notions of value in contemporary (green, public good) economy and are there general game theories?

The current mainstream theory of value is the subjective theory of value: goods or services have value that people subjectively believe that they have. Rembrandt paintings cost millions because someone subjectively thinks they are worth that much.

If there are no market failures the market price captures the subjective value through supply demand interactions.

However, in case of environment there are market failures. For example, due to lack of property rights to clean air firms use it for free and hence the subjective value of clean air is not reflected in market price. This would be example of negative externality. Externalities can be also positive, for example basic research can’t be patented so it’s value is more than scientist can get out of it by supplying their ideas on market.

If there is a negative or positive externality the subjective value is still there but it’s incredibly hard to find out. An example of how economists try to find this would be to use some econometric techniques like for example hedonic pricing which tries to estimate the value not captured in market price by observing how other market values change when availability of the good suffering the externality changes (for example by looking how pollution affect housing prices). Other techniques exists I would recommend starting with the hedonic pricing as it’s the most simple ones and you will see that many articles discussing it will point out to more complex techniques.

Also note that in many cases government might already correct for these externalities by taxes and subsidies or by issuing property rights over clean air by tradable polluting permits (cap and trade). Assuming, the government where you live already corrects for externalities and does it properly the market price would reflect also the subjective valuation of environmental goods etc.


There are two possible answers here, depending on whether you want to take a partial or general equilibrium setting. Note that the partial equilibrium analysis of tax incidence is only taught in undergraduate micro textbooks but is not considered a sensible approach in current research.

Partial Equilibrium: Suppose your supply is perfectly inelastic (for example a non-rival but excludable good which you produced last period) and demand is perfectly elastic (let's say another firm produces a perfect substitute with perfectly elastic supply sold at some price p). Then the producer will bear the full burden of the tax.

General Equilibrium: In a general equilibrium setting, anything can happen. In response to a tax, both supply and demand curves may shift. However, in this case the standard definition of tax incidence is no longer meaningful. In such cases, researchers either calculate some loss function for each consumer or simply look at price changes to analyze tax incidence.


How to calculate Pareto Optimal outcome in a game with a Nash Equilibrium

For Pareto optimality, you can ignore the timing. Also, anything you know about Nash equilibria in this game is irrelevant.

Let $v_F(w, L)$ be the payoff of the firm and $v_U(w, L)$ be the payoff of the union given $(w, L)$. What you are looking for are pairs $(w, L)$ such that neither the firm nor the union can have a higher payoff without the other side having a lower payoff. Equivalently, you want to the payoff of the firm to be as large as possible under the constraint that the union cannot have a lower payoff and you want the payoff of the union to be as large as possible under the constraint that the firm cannot have a lower payoff.

This means that $(w,L)$ is Pareto optimal if and only if it solves the following two maximization problems:

$$\max_{(w',L')} v_F~\text{ s.t.}~v_U(w',L')\geq v_U(w,L).$$
$$\max_{(w',L')} v_U~\text{ s.t.}~v_F(w',L')\geq v_F(w,L).$$

Now if you just have to find one Pareto optimum and not all Pareto optima, you can just maximize $v_F+v_U$.


There is no possibility to model a

  "single" player [that] can simultaneously choose different options.

Neither in the normal nor in the extensive game form. Different types of a player can be only be used if they are mutually exclusive. So suppose the citizens (one player!) can either be poor or rich and this may affect their pay-offs and thus actions. There is no possibility for the citizens to be poor and rich at the same time (while represented by the same player).

Your only option is to follow Sub-Optimal's solution and model different types of citizens as different players.

As Bayesian pointed out, you can then model simultaneous decisions by citizen 1, 2 and so forth by setting their information sets accordingly.


Understanding the impact of Subprime Mortgages crisis on investment banking

Simply put, you can't tell exactly. It's just not possible to tell these entities' subprime losses apart from other losses, with the possible exception of Lehman Brothers, due to the detail made available through its bankruptcy proceedings. Even then, it would be an incredibly labor-intensive task.

There are a number of reasons for this. First, most subprime mortgage exposures of major banks weren't held as on-balance-sheet loans, but in the form of (usually higher-rated) tranches of private-label mortgage-backed securities, which has a number of implications, including: the exposures appeared on the balance sheet as securities, not as mortgage loans; there's no way to know what fraction of those securities contained subprime loans; and due to the structures of the securities, there's not even a linear relationship between the fraction of subprime loans in a given security and the bank's exposure to losses from those loans, among other things. Second, they often had off-balance-sheet exposures in the form of support obligations (whether explicit or implicit) on structured investment vehicles. Third, their exposures varied significantly due to differences in the amount of leverage they used in funding their assets.

As an aside: pre-crisis, none of the four entities you've named were actually banks. Though often referred to as "investment banks," all of those were nonbank broker-dealers.


Is it possible for Gross value added be negative on a company level?

Yes, it can happen if a company is selling things for less than it cost it to produce them, though for obvious reasons this doesn't happen very often (or at least not over any great length of time) as it usually doesn't make sense to continue producing in those circumstances.

You might see it occasionally, for example, in small farm businesses.  These can have high subsidies on production to begin with (which get subtracted in the calculation of GVA, narrowing the loss needed to bring it below zero), have few or no employees (just a self-employed farmer, so no or low fixed wages/CoE to be paid out), and a bad harvest or loss of livestock could result in the business not producing enough output to make GVA positive for a particular year.


Constrainted optimization: merge two constraints into one

$$0\leq u \leq \bar u  \implies -\frac {\bar u}{2} \leq u - \frac {\bar u}{2} \leq \frac {\bar u}{2}$$

$$\implies \left | u - \frac {\bar u}{2}\right| \leq \frac {\bar u}{2}$$

$$\implies \frac {\bar u}{2} - \left | u - \frac {\bar u}{2}\right| \geq 0$$

ADDENDUM
In a comment it was proposed that we could instead use the squared expression to achieve differentiability everywhere,

$$\frac {\bar{u}^2}{4} - \left ( u - \frac {\bar u}{2}\right)^2 \geq 0$$

Let's see: we then are allowed to decompose the square and write

$$\frac {\bar{u}^2}{4} - u^2 + u\bar u - \frac {\bar{u}^2}{4} \geq 0$$

$$\implies -u^2 + \bar u u \geq 0 \implies u(\bar u -u) \geq 0$$

which is nothing more than the multiplication of the two separate constraints.

ADDENDUM II
If we have $u \in [a,b]$ for $a<b$ arbitrary reals, then the general expression is

$$-u^2+(a+b)u-ab\geq 0$$


Intepretation of $dC_1=-\theta(dT_1+\frac{dT_2}{1+r})=0\Longleftrightarrow dT_1=-\frac{dT_2}{1+r}$?

Basic intro to total differentiation

Suppose we have a function $$f(x_1,x_2,\ldots,x_n).$$ The total derivative is given by $$\frac{\partial f}{\partial x_1}dx_1+\frac{\partial f}{\partial x_2}dx_2+\ldots+\frac{\partial f}{\partial x_n}dx_n,$$
where the $\partial$ terms are partial derivatives and the $dx$ terms are so-called "differentials".

Conceptually, what is going on is the following: we ask "if I change every variable $x_i$ by a small amount $dx_i$, what is the effect on the overall value of $f$?"

To answer this question, we can think about the variables one-by-one. Start with $x_1$. A unit increase in $x_1$ causes $f$ to change by $\partial f/\partial x_1$. We are making a small $dx_1$ unit change in $x_1$. Thus, the total change in $f$ due to the change in $x_1$ is $\frac{\partial f}{\partial x_1}dx_1$. At the same time, we change $x_2$ by $dx_2$, resulting in a $\frac{\partial f}{\partial x_2}dx_2$ unit change in $f$ and so on.

If you want to read more on the topic, I'd recommend the relevant chapter of "Fundamental Methods of Mathematical Economics" by Alpha Chiang.

Aside on why this might be useful (can be skipped without loss)

Let's look at an example of how this could be useful. Suppose we have a game where players $i=1,2$ each choose an action $x_i$. Player $i$'s payoff is

$$u_i=f(x_i,a)+g(x_i,x_j,a)$$

where $a$ is a parameter and $x_j$ is the other player's action. Thus, a player's best action is given by the first-order condition

$$f_1(x_i,a)+g_1(x_i,x_j,a)=0$$
where $f_i$ means the partial derivative of $f$ wrt its $i^{th}$ argument.

It is hard to know how a change in the parameter $a$ will affect $i$'s choice of $x_i$ because the change in $a$ will affect $x_i$ not only directly, but also by changing $x_j$. When all of these changes have settled back to equilibrium, what happens? It seems like we can't go any further because we need to know the functional form of $f$ and $g$ to be able to explicitly solve for $x_i$ and see how they depend on each other and on $a$.

But total differentiation can help us understand the properties of the solution here. Let's totally differentiate the first order condition, allowing all of $x_i$, $x_j$, and $a$ to change at once (I omit the arguments of the functions for brevity):

$$[f_{11}+g_{11}]dx_i+g_{12}dx_j+[f_{12}+g_{13}]da=0.$$

Now I can solve for $\frac{dx_i}{da}$:

$$\frac{dx_i}{da}=-\frac{f_{12}+g_{13}}{f_{11}+g_{11}}.$$
So long as I know something about the individual properties of $f$ and $g$, I can use this equation to tell me what the sign of $fx_i/da$ will be!

Coming back to your example

For notational convenience, I give your function a name ($f$):

$$f(C_1,T_1,T_2)\equiv\theta(V_1+Y^L_1-T_1+\frac{Y^L_2-T_2}{1+r})-C_1=0,$$

we apply the basic rules of total differentiation:

$$f_1 dC_1+f_2dT_1+f_3dT_2=0.$$

We compute $f_1=\frac{\partial f}{\partial C_1}=-1$,$f_2=\frac{\partial f}{\partial T_1}=-\theta$, $f_3=\frac{\partial f}{\partial T_2}=-\theta\frac{1}{1+r}$. This therefore gives $$dC_1=-\theta\left(dT_1+\frac{dT_2}{1+r}\right).$$

In words, this equation tells us that the change in $C_1$ must be equal to $-\theta\left(dT_1+\frac{dT_2}{1+r}\right)$, where $dT_1$ and $dT_2$ are the changes in $T_1$ and $T_2$. Thus, if we want to say that $C_1$ doesn't change then it must be that $dT_1=-\frac{dT_2}{1+r}$ so that the changes in $T_1$ and $T_2$ 'cancel-out'.


Converting current to constant USD values for trade data (UN Comtrade)

Use the World Bank data on US inflation to deflate the UN Comtrade dataset, so that it is also in constant prices.

Select Download, at the right side of the page, and you can get the dataset in csv or other formats.

To deflate the trade data, simply divide the values of bilateral trade by the value of each years CPI index, and then multiply them by the value of CPI in 2010 (the base year, which is 100 in this case). Alternatively, rebase the CPI index, by dividing all values by 100, so that CPI in 2010 is equal to 1. Then, divide the trade data by this new index. Both approaches are equivalent.


Why are some goods without close substitutes not sold in some countries?

One way to look at this is through the idea of "fixed costs". It only makes sense to pay the fixed cost if the market is big enough.

There is very likely a large a fixed cost of entering a market. You have to establish a network of retailers or at least distribution channels, a way for the customers to return the defective units, partners that will also supply the covers, the software, the adapters, etc. Often you have to register the devices with the local regulators, they have to approve them for sale in that country, etc.

From the marketing perspective there are also fixed costs. You make a single advertisement campaign per country, and you use it everywhere in that country. I suppose there is a minimum word of mouth effect that needs to take place for any marketing to be available, so I suppose a small marketing campaign is unlikely to be efficient.

Also, sometimes products are sold or presented basically as flagship products, or loss-leaders, to call attention to the technological prowess of a firm, but not because they are all profitable. So, if a brand does not face a lot of competition in a country, it might not introduce these products as selling them, might actually represent a marketing expenditure.