Capitalism is defined as an economic and political system in which a country's trade and industry are controlled by private owners for profit, rather than by the state.

It functions as an allocation mechanism which generally yields the most optimal results under the Fundamental theorems of welfare economics and prevents issues such as Tragedy of the commons which a communist system would fail to deal with (unless all resources were heavily policed and regulated with knowledge of every consumers utility functions).

Under a pure capitalist system, government intervention in the market place is viewed as harmful.

However according to mainstream economic theory, there is some benefit to have from government intervention when dealing with  externalities, market saturation and responding to Business Cycles.


Straub and Werning, 2014, on zero capital taxation

The non-mathematical answer I believe is well described in Straub & Werning's last paragraph:

  "In quantitative evaluations it may well be the case that one finds a
  zero long-run tax on capital, e.g. for the model in Judd (1985) one
  may set $\sigma  < 1$ ($\sigma$ is the reciprocal of the intertemporal elasticity of substitution), and in Chamley (1986) the bounds may not bind forever,
  depending on parameters. In this paper we stay away from making any
  such claim, one way or another. We confined attention to the original
  theoretical results, widely perceived as delivering zero long-run
  taxation as an ironclad conclusion, independent of parameter values.
  Based on our analysis, we find little basis for such an
  interpretation."

Just above their Proposition 2, there is also another illuminating passage: S&W write:

  "(...)This shows that the solution cannot converge to the zero-tax
  steady state. Indeed, it actually proves the solution cannot converge
  to any interior steady state, since, we argued, the only possible
  interior steady state is the zero tax steady state."

In other words, it appears that Judd and  Chamley did not fully solved their models, but provided results conditional on the parameters being such that the steady state will be an interior one. S&W argue (I have not checked mathematical correctness), that, depending on the parameter values, the optimal solution may lead to a corner steady state in some aspect (see below for an example) -and in a corner steady state, the capital tax-rate will be positive.  This needs checking because Judd explicitly considers different values for $\sigma$ (so a plain mistake may be involved after all).

Now if you ask me, this mostly indicates that the tools used may not be well suited after all (or we have somehow "misused" them) to solve the specific theoretical problem, and so they are unreliable to affect policy one direction or the other (which, by the way, reminds me of my question...)

Because, exactly of what value is (for either theory or policy) to, say, empirically determine that the elasticity of intertemporal substitution is lower than unity, and then declare "the optimal taxation solution for a government that cares only about the workers entails positive tax on capital and zero consumption of workers" at the steady state?  (see Proposition 3, p. 11 of S& W). Who is going to contemplate seriously such a proposition for purposes of real-world policy?

ADDENDUM

$\sigma >1 \Rightarrow$ the intertemporal elasticity of substitution ($1/\sigma)$ is lower than unity. $\gamma =0\Rightarrow$ the planner puts $0$ weight on capitalists' utility. $c_t$ is the workers' consumption, $C_t$ is capitalists' consumption, and $g$ is governments own consumption (i.e. it is not the part of government budget that goes to the workers as transfers).

Again, I have not checked the mathematics here. Apart from what kind of economic content and relevance one could conceivably provide for workers consumption going to zero, another issue here is that, if the limiting tax rate is going to unity as government own consumption goes to zero, then what happens to the Tax revenues (which are not given to the workers as transfers because then their consumption could not go to zero?)


What have been some trade-offs that the U.S. has made?

It might be easier to grasp the idea of a tradeoff by framing the concept at a less-than-national scale.

At a personal level, you might think of a tradeoff as giving up a little bit of one thing so you can have a little bit more of another. So, maybe you save your lunch money for a few weeks so you can get new shoes. The thing that is scarce here is money. You save your money (you accumulate your scarce resource) by forgoing lunch (the sacrifice) and then buy new shoes (the gain). The tradeoff is the lunch/shoes. You were a bit hungry so that could dress better and impress the ladies.

This is a bit naive but it illustrates the idea here: a tradeoff is just achieving a balance between desirable but incompatible/mutually exclusive things.

A better example that actually answers your question (all nations face this) is the unemployment/inflation tradeoff mapped by the Phillips curve. Here, the tradeoff is the rate of unemployment a nation must endure relative to the amount of inflation a nation must endure.

As you can see in the image, reducing inflation comes at the cost of increasing unemployment and reducing unemployment bears the cost of increased inflation. Thus, a nation must optimize its outcome by striking a balance between the two.

That should be enough for you to think of what tradeoffs America faces. A good place to look might be at national budgeting decisions. The recent financial crisis certainly forced some big decisions. If you understand the concept it is relatively easy to identify tradeoffs.


Is there an "ideal" ratio of tax revenue to GDP? Or, at least, can I assume a higher ratio is better?

There is no ideal ratio as the “ideal” (I choose to interpret the ideal as optimal/efficient) level of tax to GDP depends on the moral philosophy. Generally Ralwsians prefer more taxation in order to help poor people, through redistribution, than libertarians/classical liberals, who would prefer private solutions such as charity.
Higher is not always better. First, there is a Laffer curve - at a 100% tax rate the collected tax is zero because nobody is going to willingly become a slave by working only so that government takes all of their money. So there is always a maximum tax rate and implied tax to GDP ratio that is the maximum feasible tax any rational government can implement. Where exactly is the peak (gov revenue maximizing) tax rate changes from country to country, although in advanced countries it is usually around 60-70% (see for example this study) of all income meaning tax to GDP ratio would be 0.6-0.7.
Second, actually the peak of Laffer curve is higher than a welfare maximizing top marginal rate even under Ralwsian preferences (which are the most commonly used redistribute preferences) so if you care about welfare of population instead of just the revenue of government you would usually keep the tax to GDP ratio lower.
Country can be financially stable at any tax to GDP ratio. In order to be financially stable you just need to have debt to GDP ratio be on non-explosive path. The explosive path is a one where debt grows faster than the overall economy in long run. As long as the debt grows at the same rate as economy or at smaller rate the country’s public finances are stable.
Having high tax to GDP ratio does not mean your tax system is efficient. Algeria is not really known for its tax efficiency.


No, you can not really say that. Here is the reason:

Utility is generally used to describe preferences. Let's say my happiness is described by this utility of chocolates: $Y(X) = X$. So, two chocolates will give me two units of happiness. However, by definition of preferences, I am allowed to apply monotone transformations to them. Hence, you can just transform that function to $Y(X) = X^2$. Now two chocolates will give me 4 units of happiness.

Again, by definition, this is a completely valid transformation. However, now I am getting 4 dollars instead of $2 because I simply transformed the function. For obvious reasons, you can not do that.

Hence, utility is unitless.


Why does any treasury / central bank hold gold?

It's a holdover from the old Gold Standards. Gold standard regulation required all banks, including the central bank to hold gold as a regulatory asset. In the last gold standard, the Bretton Woods regime, the US in particular had to hold gold to back the dollar. The requirement went away with the collapse of the Bretton Woods agreement in 1973, but the gold didn't.

These days there isn't any requirement to hold gold, and indeed some countries - notably the UK in 1999 and 2002 have sold off significant amounts of their bank holdings. (Note though, it's been suggested there were stability issues behind this sale.) Gold on the Federal Reserve's book isn't even held at market prices, it's marked to a notional statutory value of ~$42.

By the same token, there isn't any requirement not to hold it, so why sell it? There tends to be a lot of inertia in these kinds of systems.

Interestingly, an examination of central bank annual reports shows a fair amount of variation in what they use for their assets. While government debt in the form of treasuries is usually present, the Norwegian central bank holds market securities - presumably due to a lack of government debt. The Federal Reserve added securitized loans in the wake of the 2007 crash in order to stabilise their financial system.


Is there a term for a game whose pareto optimal solutions and nash equilibria are disjoint?

The term I often come across is 'inefficient Nash equilibrium' or 'Pareto-inefficient Nash equilibrium'. Here is an older well cited article discussing inefficient NEs: http://www.jstor.org/stable/3690047?seq=1#page_scan_tab_contents

Here is the frequency of the term 'inefficient Nash equilibrium' used in scientific papers:
https://scholar.google.com/scholar?q=%22Inefficient+Nash+Equilibrium%22&btnG=&hl=en&as_sdt=0%2C5

'Inefficient outcome' is also frequently used to describe NEs that are not Pareto-optimal: https://scholar.google.com/scholar?q=%22Inefficient+outcome%22&btnG=&hl=en&as_sdt=0%2C5
But it has to be put into game-theoretical context and specified that it is NE.


Why are elasticities defined as logarithmic derivatives?

If I understand your question, first the elasticity haven't units. The problem with $\partial Y/\partial X$ is that if you change the measure units the result is different. Is less problematic to express it in percentages:
$$ El_X(Y)=\frac{\Delta Y/Y}{\Delta X/X}$$
if $\Delta Y \to 0$ when $\Delta X \to 0$ then you obtain $$El_X(Y)=\frac{dY}{dX}\frac{X}{Y}$$ if $Y=Y(X,W,...,Z)$ you change $d$ for $\partial$.

Second, like you know $d \log(x)=\frac{dx}{x}$ thus $$ El_X(Y)=\frac{d Y}{d X}\frac{X}{Y}=\frac{d\log(Y)}{d\log(X)}.$$


I agree that this question is too broad, but here's a perspective: much of the criticism of modern capitalism is from people who've been in the system.  What about people who haven't been in the system, like in Vietnam or China, where they haven't been under capitalism in the past?  Do they tell us what we're missing out on?

One study I've seen all over:

  A world-topping 95 percent of Vietnamese say that most people are
  better off in free-market economies, and 76 percent of Chinese agree,
  according to a Pew survey of nearly 49,000 people worldwide that might
  have astonished Ho Chi Minh and Mao Zedong. Seventy percent of
  Americans and 65 percent of those in the UK expressed support for a
  free-market economic system.

This is part of the reason why this question is too broad.  We could say that while some systems abandon it, others will use it.  I also don't know if a capitalist-like system didn't exist before Adam Smith, as from reading some Roman history, there seems to be times where there was a merchant class featuring open markets with entrepreneurs where they would meet the needs of those they traded with and while they certainly didn't call it capitalism (I assume), like we do, seems very similar.

I believe Plato pointed out that civilizations often move from one system to another, like Oligarchy to Democracy, before reverting, so the very nature of a Meritocracy may be a transitional stage in a civilization's history.


That comment said, I can provide an answer here.

Take an average cost function:

$$f(x) = mx + b$$ where x is output, b is the intercept, and m is the gradient.

So total cost is:

$$C(x) = x(mx + b) = mx^2 + bx$$

And that makes marginal cost the derivative of this function:

$$C'(x) = 2mx + b$$

Best of luck with your homework.


Mixed Strategies in Bayes Nash Equilibrium (Bayesian Battle of the Sexes). Shouldn't it depend on $p$?

Suppose player 1 chooses a mixed strategy such that the left type of player 2 is indifferent between C and H. There is a unique mixed strategy that does this. The same mixed strategy, however, does NOT make the right type of player 2 indifferent. Loosely speaking, this is simply because the preferences of the left type are exactly those of the right type switched around. Thus it is impossible for player 1 to make both types indifferent simultaneously. Therefore, the probability $p$ plays no role when choosing a strategy such that at least one of them is indifferent.
This also indicates why there are exactly two equilibria.


What are some good graduate-level econometrics books for someone with a strong mathematics background?

"Adult" Wooldridge is great intro to various microeconometrics topics.

For time series, Hamilton's Time Series and Lutkepohl's Introduction to Multiple Time Series Analysis are both nice, though Hamilton is a bit dated and Lutkepohl is more focused.

As far as more foundational, rigorous material, Herman Bierens has a short Introduction to the Mathematical and Statistical Foundations of Econometrics. Gourieroux and Monfort have a whole flock of graduate econometrics texts, but to quote an anonymous reviewer, they are "in the French tradition of excellent precision and terrible pedagogics," though they have their champions.


What is the difference between marginal cost and average cost?

Suppose initially there are no fixed costs.

What does it mean to take an average?

Consider a cost function $C(y)$. What does it mean to take the “average” of this function? Mathematically, it is just $$A(y) = \frac{c(y)}{y}$$

Let’s suppose we are considering $C(y) = y^3$. Suppose we now consider $y = 5$. Then $$A(5) = \frac{5^3}{5} = 25 $$
This is just saying that, for each unit I buy, I am buying them at $25$ each on average. So I could have paid
$$ \frac{15 + 39 + 46 + 14 + 11}{5}$$
or
$$ \frac{17 + 3 + 78 + 23 + 4}{5}$$
What information does the average cost give us?

The 'cost per unit' given by an average cost function isn't like taking an average by adding up the cost of each unit we bought. When we are given a cost function $C(y)$, this just tells me the total cost. I don't know how much my first TV cost me from this equation alone. And the average doesn't tell me that either. Note how above we have two sets of 5 TVs that yielded the same average cost. Thus, the average cost function doesn't tell me how much I paid for each specific TV.

What does marginal cost tell me?

This is exactly what marginal cost provides. Marginal cost provides the specific cost of each successive infinitesimal amount of good. Consider again $C(y) = y^3$.
$$MC(y) = \frac{d C(y)}{d y}= 3y^2$$

Suppose I have purchased $3.5$ units of TV. Then the MC equation thus says, for $c(y) = y^3$, each additional infinitesimal amount of TV costs me $$3(3.5)^2 = 36.75$$ at that point. If I purchase an additional $0.1$ amount of TV, then my MC changes and now I am at $$3(3.6)^2 = 38.88$$

Thus, this concept is a bit trickier because it involves a continuous amount of TV and the cost per infinitesimal unit changes as you buy more. So you really can't just consider how much $MC(1)$ is and $MC(2)$ is. You are considering it for some infinitesimal additional amount $dy$ at a given point (e.g. $y=2$). The trend is easier to think through if you don't restrict yourself to integers. Note, we are assuming you can have noninteger quantities of goods, otherwise we wouldn't be working in $\Bbb{R}^n$ and the integration later might be trickier.

Example to Clarify Marginal Cost

So suppose I want to buy $5$ TVs. For the $k$th TV purchased, the $$MC(k)= 3k^2$$ Going back to our example above, let's now suppose $k \in [0,5]$. To find the average cost, we simply do the addition formula used above for the 5 TV example, except now summed over each infinitesimal amount (of which there are an infinite number). This gives us

$$A(5) = \frac{3(0)^2+\cdots + 3k^2+ \cdots + 3(5)^2}{5} = \frac{\int_{0}^{5} 3y^2 dy}{5}= \frac{(125-0)}{5}=\frac{c(5)}{5}=25$$

$$A(5) =25$$
the same as we calculated earlier.

Summary

Marginal Cost is
$$MC(y) = \frac{d C(y)}{d y}$$
Average variable cost is

$$A(y) = \frac{\int_{0}^{y} MC(y) dy}{y}$$

Note, since we assumed $FC = 0$, this formula also thus defines AC but would not be true for $FC \neq 0$.

AVERAGE COST VS AVERAGE VARIABLE COST

I have been sloppy about the distinction between AVC and AC up until now. I have avoided this distinction by assuming $FC = 0$. Now I will try to clarify this point by assuming $FC$ can be anything.

Fixed costs ($FC$)are costs the firm pays that do not vary with the amount produced. For example, suppose I work for Uber and I buy a car. That money is spent and doesn't change with the amount I drive. But the amount of gasoline I consume does change as with the amount I drive. Costs that scale with production are known as variable costs ($VC(y)$).

$$AC(y) = \frac{C(y)}{y}=\frac{VC(y) +FC}{y}= \frac{VC(y)}{y} + \frac{FC}{y}=AVC + AFC $$

Hyperbolic Behavior of AFC with y >0

As production goes up ($y\rightarrow \infty$), AFC goes down ($AFC \rightarrow 0$) in an inversely proportional fashion. But note, as production goes down ($y\rightarrow 0$), AFC goes to infinity ($AFC \rightarrow \infty$). Thus, the plot of AFC will always be a hyperbola unless $FC = 0$ in which case AFC is just 0.

How does AFC affect AC?

Without specifying $VC(y)$, we cannot know the behavior of $AC(y)$ as $y$ moves away from $0$. There will be some $y_{0}$ such that, for $y > y_{0}$, $AC(y)$ can essentially be anything. Of course, as $y \rightarrow 0$, $AC(y) \rightarrow \infty$. This is because AVC cannot be negative and so we are guaranteed any variable costs will not lower the average cost below AFC. Therefore, $AVC \geq 0$, so since $AFC \rightarrow \infty$ as $y\rightarrow 0$ (remember it is a hyperbola), thus AC must go to infinity as well.

Thus, since the behavior of fixed costs are always known, AVC is the missing ingredient needed to specify the behavior of AC.

AC and AVC can't be exactly equal for $FC>0$

Since $AFC \rightarrow 0$ but does not ever equal 0 (for $FC >0$), we know that $$AC \neq AVC$$ for any $y$ and $FC >0$. But since $AFC$ approaches 0 for large enough $y$, AC approaches AVC asymptotically.

AC and AVC can't be exactly parallel for $FC>0$

If $AVC$ and $AC$ are parallel, then their derivatives should be equal. But notice that $$\frac{dAFC}{dy} = -\frac{FC}{y^2}$$
so $$\frac{dAC}{dy} = \frac{dAVC}{dy} + \frac{dAFC}{dy} = \frac{dAVC}{dy} - \frac{FC}{y^2} \neq \frac{dAVC}{dy}$$
So they aren't parallel because their derivatives aren't equal. That said, for large enough $y$, the derivatives will be very close to each other so they may appear nearly parallel over some portion of the curves.

MC intersects AC at minimum point of AC curve

See Alecos's answer.

MC intersects AVC at minimum point of AVC curve

Consider the average variable cost curve. Find $y^*$ that solves

$$\min_{y} AVC(y)$$

So at this point,
$$\frac{dAVC(y^*)}{dy^*} = 0$$

This means by quotient rule

$$\frac{VC'(y^*)y^*-VC(y^*)}{(y^*)^2} = 0$$
and since $y^*$ can't be 0, this implies $$VC'(y^*)y^*-VC(y^*)=0$$ which rearranged gives
$$VC'(y^*)=\frac{VC(y^*)}{y^*}$$
Recall,
$$C(y) = FC + VC(y)$$
Note, since $FC$ is a constant,
$$C'(y) = VC'(y)$$
Therefore,

$$C'(y^*) = \frac{VC(y^*)}{y^*}$$

EXTRA STUFF

Why do we define the firm's longrun shutdown point in terms of average cost not marginal cost?

Recall a firm's profit function is $$\pi = py - C(y)$$

I am not defining the behavior of $p$ here or who controls $p$. I am just considering for what $p$ will the firm shut down, and ignoring everything else because it's irrelevant.

So we can easily see that since $$AC(y)= \frac{C(y)}{y}$$ for $p=AC(y)$, this yields $$\pi = \left(\frac{C(y)}{y}\right)y - C(y) = C(y)- C(y) = 0$$
So the firm will shut down if $p <AC(y)$ because then $\pi < 0$.

So consider the function $C(y) = y^3$. Note, $$MC = 3y^2 > AC = y^2$$ We know the firm profit maximizes at $MR = MC$. So since, for $y > 0$, $MC(y) > AC(y)$, the firm would always produce if $p = MC$ since this would mean for $y>0$,

$$ MC(y)y-C(y) = \pi_{p=MC}  > \pi_{p=AC} = \left(\frac{C(y)}{y}\right)y - C(y) =0$$
$$\pi_{p=MC} > \pi_{p=AC} = 0$$
So, for this $C(y)$, at $p=MC$, $\pi >0$ for all $y$.

So this example clearly shows you would never shut down at $p=MC$ for $y>0$ for $C(y) = y^3$. Although this isn't the most thorough explanation, this example clearly invalidates that train of thought and makes clear firms shut down for $p< AC$

Does average cost necessarily equal marginal cost at some point?

See Alecos's answer.

Takeaway Rules

In long run, firms produce if $p \geq AC(y)$. They shut down for $p < AC(y)$
Firms always produce at $MR = MC$


When working with assumed functional forms, then you could be justified supposing that there is not much added value.

However, when you try to model behavior without functional forms, you will need all the help you can get in terms of results like Roy's identity to prove your results.

One important application of Roy's identity is in deriving the Ramsey Rule for optimal commodity taxation. A derivation can be found here on page 14.


What does 'Health spending as a share of GDP' mean?

Healthcare spending as a percentage of GDP is the measure of spending relative to the value of entire nations output at market prices in a given year. I suppose you can call the total output the value of the economy so you could say that although note that economy is a wider concept that would normally include things omitted by GDP which can only measure transactions for which there are market prices (hence economic activity such as housework and other non-market transactions are excluded).

According to the video they take the data from OECD data. OECD has multiple series on healthcare spending. Some series include only government spending and some include also private voluntary spending.

However,  the data in the video seem to closely match the total health spending % of GDP, 2013. There are some small discrepancies, for example the video says that the ratio was 11% in the Germany while the data show it was 10.9 - but more or less the data seem to match (these might have occurred due to some later corrections of the data or maybe the video is using different dataset - since they dont provide proper reference for the source its hard to know). See below:

Assuming its indeed the series above, according to the OECD the measure is defined as:

  Health spending measures the final consumption of health care goods and services (i.e. current health expenditure) including personal health care (curative care, rehabilitative care, long-term care, ancillary services and medical goods) and collective services (prevention and public health services as well as health administration), but excluding spending on investments. Health care is financed through a mix of financing arrangements including government spending and compulsory health insurance (“Government/compulsory”) as well as voluntary health insurance and private funds such as households’ out-of-pocket payments, NGOs and private corporations (“Voluntary”). This indicator is presented as a total and by type of financing (“Government/compulsory”, “Voluntary”, “Out-of-pocket”) and is measured as a share of GDP, as a share of total health spending and in USD per capita (using economy-wide PPPs).

So it would include both money spent by citizens as well as governments and all private organizations.


Oil price decreases due to supply decrease?

Refineries need oil to produce chemical products. Therefore, it means declining demand for oil and falling prices according to demand supply principles.

The point is refineries which use oil as ingredient.


Defining a Competitive Equilibrium for a two Period Model

A competitive equilibrium is specific to the economy you're modeling. Your economy has three markets:

Capital market for $K$
Labor market for $L$
Goods market for $Y$

Furthermore, there are several optimization problems / budget identities that have to hold.

The household has a consumption/savings and a leisure/labor decision
The firm has a labor and a capital choice
You need to ensure that the government's budget constraint holds.

In a competitive equilibrium, you need to clear all the markets, and any quantity that is chosen, needs to be done so optimally, given everything that's happening in your economy. Here, you have two periods which need to be in equilibrium at the same time. If I were grading your homework, I would expect you to write the above for minimum points. You should then show what the conditions/equations are for each of the bullet points.

Example: Multi-Period Non-Equilibrium

For example, the household will chose savings in period 1, given what he expects to do with those savings in period 2. If you'd expect him to do something stupid with the savings in period 2 (say, throw them away), then it would be optimal to save nothing at all. In this case, you would be at an equilibrium in period 1. However, your choices for period 2 would be non-optimal, hence we do not have a 2-period equilibrium.


Question : Is what I wrote above accurate?

There is a bit more nuance to it but 1-3 points are 'corect-ish'.

I guess what I'm getting at is: Is there some connection to the bank making loans, and those loans being used to fund additional productive activities* than otherwise possible (say, given the agent's initial budget constraints), and long run growth in the money supply?

Not in the long-run. There is a relationship between growth of money supply and economic activity but not in the long-run only in the short run. This is because in the long run prices are flexible and they will quickly adjust to increase in money supply (see further discussion of that in any standard macro text like Blanchard et al Macroeconomics ch 8-10).

If someone sees fit to answer, ideally, I'd like some math. Or if there is some model that can show the players (consumers, firms, banks etc.) and their interactions, that would be great.

It is possible to describe this with a model that is built upon individual interactions of consumers, firms and so on, but that would take too much space as even very simple micro founded macro models are quite large. For example, a basic microfounded (i.e. explicitely including interaction between households, firms etc) macro model introduced Woodford, M. (2011). Interest and prices, takes over 30 pages and that is a graduate text that does not explain basics that one would probably have to cover to explain it to someone with no economic training. Consequently, let me offer more concise model (which will not have qualitatively different results from more nuanced microfounded model) where I will skip description of individual interaction of individuals and just directly impose assumptions on behavior of aggregates that could otherwise be derived from individual interactions of individuals and firms (if you want more nuanced models look at above mentioned Woodford ch 3 or at Romer Advanced Macroeconomics Ch 6, 7 and 11).
Thus I will introduce simplified model based of above mentioned Blanchard et al. Let us start by description of money market. The money market can be described using:
$$M/P = L(Y,i) \tag{1}$$
Where $M$ is money supply, $P$ price level (so $M/P$ is real money supply), $L$ is money demand, $Y$ is real output and $i$ interest rates.
The money demand will be given as follows:
$$L= f_1 Y - f_2 i \tag{2}$$
Because money demand will increase with economic activity, and decrease with interest rates. This is what would normally be microfounded, but I will just assume it as a shortcut without proving it. However, intuitively it is not unreasonable assumption, when aggregate output (which is economically also equal to income) increases there is more economic activity so people will demand higher real money balances as there will be more transactions in the economy. Next if the interest rate is high people will demand less loans and thus also money, and will save more and thus hold less money balances.
Next, goods market will be given by:
$$Y = C + I +G \tag{3}$$
where $C$ is consumption, assumed to follow $C=c_0 + c_1 (Y-T)$ with $0<c_1<1$. Where $Y-T$ is income after taxes. Again aggregate consumption would typically be derived based on individual interactions, but it is reasonable to assume people consume more if their disposable income increases.
$I$ is the investment assumed to follow $I = \bar{I} +d_1 Y- d_2 i$. Once again this would normally be microfounded, but it is reasonable to say that investment increases when output increases, and investment decreases when interest rate $i$  increases as it is harder for firms to get loans and so on.
Finally $G$ is government spending, there will be no additional assumptions on its behavior for your question it is not relevant.
So given above assumptions we will have:
$$Y = c_0 + c_1 (Y-T) + \bar{I} +d_1 Y- d_2 i +G \tag{4}$$
Solving for $Y$ so we get expression for goods market equilibrium yields:
$$Y = \frac{1}{1-c_1 -d_1}\left( c_o + \bar{I} + G - c_1T \right)  - \frac{d_2}{1-c_1-d_1} i \tag{5} $$
Now finally, in an economy from a macro perspective both the goods market equilibrium and money market equilibrium above must be also in equilibrium together.
Recall that money market equilibrium was given by equation (1) substituting (2) into (1) we get:
$$  M/P=f_1Y−f_2i \tag{6}$$
Solving 6 for $i$, substituting into $5$  and once again solving for $Y$ so we isolate output we get:
$$ Y = \frac{1}{(1-c_1 - d_1) \frac{f_2}{ d_2} + f_1} \frac{M}{P} + \frac{1}{1-c_1-d_1}+d_2 \frac{f_1}{f_2} \left( c_o + \bar{I} + G - c_1T \right)  \tag{7} $$
Examining how $Y$ varies with respect to real money balances we get that:
$$\frac{dY}{d(M/P)} =  \frac{1}{(1-c_1 - d_1) \frac{f_2}{ d_2} + f_1} \tag{8}$$
So indeed increase in amount of money will increase output, in fact there even be multiplier effect because the higher the output is the more demand for money there will be etc.
However, note the above does not hold in a long run. This is not captured by the brief and simplistic model above but while in short run we can assume $P$ is fixed, in a long run $P$ will most certainly not be fixed, and in fact it will increase in response to increase in $M$ (see discussion of that in Blanchard et al ch 9). Hence, in a long run increase in money supply cannot stimulate output (which would via multiplier stimulate more demand for money and again more output). Consequently, and answer to your question is no (although I am not sure what you mean by long run - in economics long run is defined as time period where $P$ if flexible and can adjust).
In addition the reason why we can observe increase in money supply over long periods of time is that central banks try to maintain price stability, which many of them define as a yearly inflation of around $2\%$. For that to happen money supply needs to constantly increase as well. Consequently, even though it is correct to say that private banks create most of the money in our economic system, the central bank is ultimately ultimately in control in how much money is being created. They do this either through bank regulation, interest rate setting or by control of monetary base - this is also explained in the McLeay Radia, & Thomas (2014) you provide link to in your Q. If a central bank would be hell bent on it and would purposefully want to keep $M$ constant, they could do it (by unreasonably hiking interest rate, regulating borrowing in a such way as to make it nearly impossible etc). Conversely, if central banks think there is not enough money supply in the economy they can increase it regardless of bank lending (e.g. direct purchase and monetization of government debt). Note here that central bank can actually do this even without using private banks. For example, central bank can buy government debt which creates new money. Thus central bank can always move money supply wherever it want's to in principle (even if private banks would not want to lend).


Everyone has the same marginal rate of substitution

There are a couple of assumptions driving this result. For instance, all agents are assumed to have preferences and a budget such that their optimal bundle is interior. For example, budget constraints were people cannot afford food are ruled out. Likewise, situations, where someone does not like one of these goods or has very strong substitutions patterns between a pair of goods, are also ruled out.  All preferences are assumed well-behaved (monotonic and strictly convex).

Secondly, the market is competitive, so all people face the same prices not only for tomatoes but also for airplanes, wages, etc. This almost never happens in reality, but in some cases, it is a good approximation. For example, when considering shopping for breakfast, the relative prices between cereal and yogurt and eggs roughly reflect the willingness for every person to substitute one good for another.

This is a striking result and I see it more of a theoretical benchmark whose main intuition is that prices contain a whole lot of information about how people are willing to substitute one good for another. When they don't, it is very interesting to understand why.


annualized quarterly growth rate calculating

Perhaps quarterly growth should be multiplied by 400, but not annualized quarterly growth?


Should Costs of Travel to Buy Goods be Regarded as Transaction Costs?

Since it was mentioned in an another answer let's clear this first: whether the transportation (and its time and monetary costs) should be associated with the intended consumption of the good you are going to purchase, or it can be considered as consumption on its own, depends on your subjective view of it: do you derive any form of pleasure by the trip itself? If yes, at least part of it should be considered consumption per se.

The consensus among economists appears to be that most of such travel is not considered by the consumers as utility-enhancing per se (although trends like "family-shopping on Saturday" may say a different story), and so it should be interpreted in a different way.

In the field of Industrial Organization, the good's distance from the consumer has been often treated as an aspect of product differentiation.

You could certainly treat it as a "transaction cost", by suitably define the scope of the concept. Personally I prefer to think of it as an access cost. I hit upon this concept in a little side-research I did in hedonic-price analysis.

If you start to think about it, all packaging and transportation costs from the supplier to the shop are also "access costs" from the point of view of the consumer. They don't provide any direct utility to him -they are obligatory costs that end up increasing the price, so that the consumer is able to acquire the good and enjoy the services/utility of the good itself.

Think computers: only the materials themselves and the technology embodied in them provide utility to you (plus maybe the brand). But the price includes all shorts of overheads, like the access costs I mentioned, or marketing costs (that can be seen as information costs or as the price to pay for competition and the innovation and product variety that brings along), etc.


Question About Basic Supply/Demand Concepts

You are right that most discussions talk about complementarity in consumption, which is why this question is all the more interesting.

Demand falls for potatoes, which causes the price of potatoes to fall, which causes the supply of potatoes to fall. This means there is less starch available. This in turn means that we can make less packing-peanuts, hence supply of packing peanuts falls.

You are right that demand for packing peanuts does not change. The supply will nevertheless change however. This is because the supply of starch falls and the input factor of packaging peanuts is more expensive, so less is optimally produced. I.e. the supply curve of packaging peanuts changes, in this case shifts to the left so there is less equilibrium supply even though demand for packaging peanuts has not changed.


Visualization tools for game theory: Game trees

LaTeX with forest

The forest package of LaTeX allows you to draw game trees with pretty simple syntax. After copying a pre-set template into the LaTeX preamble, one can build up the game tree using a nested [] syntax, then the program takes care of node placement/spacing/etc.

pros: customizability (you can annotate the game tree in any way you want) and font/style consistency
cons: inputting the game tree elements (player/action names, payoffs, info sets) is still somewhat manual, especially if the tree is large; and a somewhat steep learning curve if you're not already familiar with the TikZ package

Here's an example from the forest manual ("Decision tree" in Section 5.1):

\begin{forest}
decision tree
[N,plain content
  [I;{p_1=0.5},plain content,elo={yshift=4pt}
    [{5,1};a]
    [II;b,plain content
      [{1,2};m]
      [{2,3};n]
    ]
  ]
  [II;{p_2=0.5},plain content,elo={yshift=4pt}
    [;c
      [{1,0};z]
      [{2,2};t]
    ]
    [;d
      [{3,1};z]
      [{0,0};t]
    ]
  ]{\draw[dashed](!1.anchor)--(!2.anchor) node[pos=0.5,above]{I};}
]
\end{forest}

Game Theory Explorer

The game theory explorer is developed by a few people at LSE. It allows users to input matrix-form games or build extensive form games through a GUI. It also seems capable of converting between normal and extensive form games. Additionally, the software comes with a solver that looks for Nash equilibria of the inputted game.

pros: GUI; solver; no need for local installation; (limited) customizability
cons: no support for annotation; font/style inconsistency with rest of document

Screenshots:


Can number of children in rich country is small due to this simple economic factor?

Let me fist answer that individual decisions on whether to try to have children or not is a decision that is not driven by economics alone. There are many other factors that influence these decisions such as individual experiences and preferences and social norms.

That being said there is a relatively large literature on the economics of families and children. Some of the arguments as to why people have fewer children in developed countries include:

The costs of children are higher and the direct economic benefits are lower in developed countries. In developing countries children constitute extra labour (although they also need to be fed) and they require little education and other investments. In developed countries children typically do not contribute towards family income, and need large investments in terms of education and raising. A related argument that was put forward by Becker and Tomes is that there is a trade-off between the "quality" and the quantity of children, and in developed countries there is a larger demand for quality as opposed to quantity of children.

Another argument is that the need for a large number of children is lower in developed countries. In developing countries children are your old-age insurance policy, and infant mortality is high. In developed countries we have pension systems, good (partly) public health care and elderly homes, all of which reduce the necessity of having many children. The good health care system reduces the infant mortality so you don't need more children to be sure that you'll have a few when you get old. The pension systems, good public health care system and elderly homes mean you have to rely less on your children for income and care when you get old.

A final argument that has been made is that the opportunity costs of children are larger in developed countries. Most people are highly educated, and receive a relatively high salary. If they have to reduce their working hours to provide care to children, the opportunity costs are higher (all else being equal).


Unfortunately, I'm not sure there's a clear answer to your question, since it so often depends on what the specific needs are for the specific project. Overall, there are a number of different ways to try and measure concentration and market power (as you note in the question).

That said, there's definitely a lot of interplay between different measures, but often these measures can be reworked to allow for looser (or different) assumptions. For example, one of the more well known merger analysis tools currently used by the DOJ and FTC, the Upward Pricing Pressure test is traditionally linked to Bertrand competition. However, as the creators of the measure note, that link is often overstated, and the test can be used in cases where Cournot competition is assumed.

Moreover, UPP is closely linked to other measures, such as diversion analysis, for which there are many different tools to use to measure it. One method is to use a survey of consumers, (seen, for example, here,) though that of course comes with drawbacks. Instead, market shares can be used, but so can historiical data from switching patterns. Alternatively, you can look at second choice data and/or price and demand elasticity properties, or more abstractly data from stock-out events.

I'm sorry to get stuck in the weeds of merger analysis there, but I think it's an informative example of how many different ways there are to get at the same underlying ideas. Ultimately, what's "best" really depends on the specifics of your project, the data available, and how well you can defend the choice made.

Let me know if there's something more directly relevant that you'd like addressed, but a lot of the theoretical complementarities can be seen only when one particular interest is isolated and examined.


Is it true that $\frac{dL}{dq}=1/\frac{\partial q}{\partial L}$?

Question: is the following correct ?

  $$\frac{dL}{dq}=1/\frac{\partial q}{\partial L},\;\frac{dK}{dq}=1/\frac{\partial q}{\partial K}$$

In general, no. Since $q= f(L,K)$ is a multivariable, single-valued function, then by the implicit function theorem applied on the implied equation $H = f(L,K)-q=0$, what we can say is only that

$$\frac {\partial L}{\partial q} = -\frac {\partial H/\partial q}{\partial H /\partial L} = -\frac {-1}{MP_L} = \frac {1}{\partial q /\partial L}$$

Essentially, here we treat $q$ as a univariate function (by taking partial derivatives, we keep all other variables fixed).

But total derivatives are another matter.
For $q = f(L,K)$, we have the implicit relation $L = L(q,K)$ and so by taking the total differential and divide throughout by $dq$ we get

$$\frac {d L(q,K)}{d q} = \frac {\partial L(q,K)}{\partial q} + \frac {\partial L(q,K)}{\partial K}\frac {dK}{dq}$$

But $\frac {\partial L}{\partial K} \neq 0$, because we implicitly keep $q$ constant in order to calculate this partial derivative, and so by changing $K$ we have also to change $L$. The other possibility would be that $dK/dq$ is zero. If we do not have such a production function (or we are not in a range where such a thing holds), we obtain

$$\frac {dL}{dq} \neq  \frac {\partial L}{\partial q} =  \frac {1}{\partial q/\partial L}$$


Demand curve for items where consumers typically only buy a single product

In the simplest case, to draw a two dimensional diagram, you need to assume that apartments are homogeneous.Suppose we go along with this. Then the market demand schedule can be written

$$Q^d(p) = \sum_{j=1}^N I\{w_j\geq p\}$$

where $N$ is the number of persons looking for an apartment, $p$ is the price and $w_j$ is searcher's $j$ "willingness to pay". $I\{\}$ is the indicator function, taking the value $1$ if the condition in the curly brackets is satisfied, and the value $0$ otherwise.

So as long as the price is above its willingness to pay, a searcher won't "demand" an apartment (he won't be willing to rent at that price, that is).

Strictly speaking, this will make the demand schedule a collection of distinct points, rather than a line, since indeed, the quantity axis is in discrete values. But since the price variable is essentially continuous, we can imagine that there exists a price where the number of units demanded equals the number of units supplied.

Also, be careful with the assumption of a vertical supply curve. Ιt does not just imply that the available units are fixed in the short run, it also implies that apartment owners are willing to rent at any price.

This does not appear to be the empirical case. If prices fall, owners are seen to "withdraw" their apartment from the market, waiting perhaps for better times. This means that while we can imagine that eventually, we will have a vertical segment in the supply curve, for lower price we will observe a rising supply curve as usual.

For the rising part we will have analogously, for $M$ owners

$$Q^s(p) = \sum_{j=1}^M I\{s_j\leq p\}$$

where $s_j$ is the minimum acceptable rent by the owner. So an owner too is really "active" in the market and contributes to the supply only when the prevailing price is above that threshold (assumed possibly different for each owner).

A normal situation will be


Meaning of innovation in the wealth creation process

The extra benefit to you, beyond what you pay for your iPhone and for your Netflix subscription, is called consumer surplus and in theory this could be aggregated across all consumers who take advantage of this innovation

It is difficult to measure:  again in theory it could be done by judging how much more individuals would be willing to pay until they were indifferent between purchasing or not purchasing


Using metrics provided by FRED to identify the business cycle

Prediction is difficult, especially about the future. (Berra, Goldwyn, Bohr et al.)

The problem with noting that recessions often occur immediately after low periods of unemployment is that you do not now at the time that it was a low rather than a pause.

There are several occasions when unemployment appeared to stop falling but which were not immediately followed by recession.  Consider for example 1963, 1967, 1976, 1985, 1995, 2003 in your first linked FRED chart.  Once unemployment shoots up, you are probably in a recession and so do not need to predict

Potential GDP is even harder to quantify meaningfully, as it is based on "the output the economy would produce with a high rate of use of its capital and labor resources."  So if unemployment falls below levels previously thought typical before leading to inflation and high interest rates, this can lead to a suggestion that GDP is exceeding its potential; unemployment is certainly now well below what the Federal Reserve recently thought was possible.  But this may merely mean that the judgement of what was typical was wrong: interest rates are still historically low, and inflation is not accelerating, while the fiscal position is loosening

Looking at your second linked FRED chart, actual GDP exceeding potential GDP can happen several years before a recession, or immediately before, or even not happen at all.  So it is not a precise signal

So you may be correct that a recession is looming, or you may be wrong and it is several years away.  Welcome to macroeconomic forecasting


What is the difference between 'actual' and 'potential' interaction between buyers and sellers?

The traditional representation of a market, with the curves of supply and demand reflects potential transactions: for each price, we can derive the quantity that would be offered or asked. The actual transactions only occur at the market price, but ignoring all other intentions is too restrictive to define a market.

For example, stock exchanges often give the possibility to agents to express their intentions (that is,  the quantity they would offer or buy for a certain price) about a particular security, which will form the bid (supply)  and the ask (demand). Even if the ask and the bid never meet, there is still a market as people are willing to exchange goods, though they cannot find an acceptable compromise.


Flooding the real estate market with luxury homes: will average prices rise or fall?

After 2 weeks and no answers, I think it's fair to say that the odds of a average prices rising as a result of replacing each lower priced dwelling unit with 2 higher priced ones is roughly equal to the odds of prices falling (equal enough that nobody is willing to stake their reputation on a prediction).


What's the benefit the bank gets on offering me to pay my water/electricity/gas with credit card?

From a marketing point of view it's a very effective way of acquiring customers.

And you raise a good point, people often do forget these things, so banks can also profit that way.


Is education a part of the U.S. exports component of GDP?

From Unesco's website:

Top 10 destination countries:

United States (18% of total mobile students)
United Kingdom (11%)
France (7%)
Australia (6%)
Germany (5%)
Russian Federation (4%)
Japan (4%)
Canada (3%)
China (2%)
Italy (2%)

This is about headcount, not value-of-services. For 2nd-in-rank UK to nevertheless generate more income in terms of value, it should be the case that average fees/cost of living in UK is ~$64$% higher than in the USA. Otherwise, USA will be also the biggest exporter of education.

This, without taking into account the size of each country's economy.

The link mentions that Unesco's statistical data base provides further statistics.


From single-consumer-multi-units to multiple-consumers-single-unit

The cumulative distribution function (CDF) describes the probability that a random variable X with a given probability distribution will be found to have a value less than or equal to x. That is: $$F_X(x) = P(X \leq x) $$

In this context, think of $X$ as the valuation of a particular unit demander. The demands are distributed with a certain probability distribution. So the the CDF tells you the probability of  a randomly drawn demander of having a valuation at least as large as $x$. If everyone demands either one unit if the price $p \leq X$ and 0 otherwise, then this also gives a demand function:

$$Q_d(p) = P(X \geq p) = 1 - P(X \leq p) $$

because at each price $p$ everyone with a valuation at least as high as $p$ will buy their one unit. But the fraction of people with a valuation at least as high as $p$ is one minus the fraction of people with valuations $p$ or lower.

Next, consider a single household with a demand function that just so happens to look like $1 - P(X \leq p) $. This household will have an identical demand for the good at all prices as the collective demand of the unit demanders above.

I believe this generalizes to the more complex case you present in the question but I haven't worked out the details.


Monte carlo simulation for macroeconomist

The main challenge in macroeconometrics as opposed to microeconometrics is a the small sample size, because of short recorded time-series and less than 200 countries providing data. To get around this problem increasingly the more popular approach is Bayesian estimation, which often relies Monte Carlo Markov Chains to construct the posterior distribution of the parameter.
So if you want to understand how the results are obtained, you will need to have a rather deep knowledge of the topic.


What models exists for explaining the price of nonrenewable resources?

Based on my understanding, Hotelling's Rule is a pretty good first step. Like many first steps, it doesn't perform particularly well empirically. I would expect that when technological change is considered, in the very long run, it isn't too bad.

There are a few extensions of this model which are popular.
-Exogenise extraction costs
    The standard idea is that extraction costs increase as the resource moves towards depletion. Something which isn't included in the standard model is that technology is changing, making some previously unprofitable reserves cost effective. At the same time, exploration can uncover more reserves. Whilst the total amount of non-renewable resource doesn't change, the amount available for extraction isn't static.
- Account for substitutes
    The demand for oil for example is constantly growing, but it must remain competitive with substitutes. A rise in prices make substitutes more attractive. This is a phenomenon we have seen with ethanol and into our petrol.

This pdf provides a broad overview of such extensions. It also provides references for further reading. I borrowed heavily from it above. I'll list a few

Chapman, D. (1983). World Oil: Hotelling Depletion or Accelerating Use?Nonrenewable Resources.2(4): 331-339
Extends the Hotelling model to so that the marginal extraction cost is independent of quantity demanded.
Stiglitz, J.E. (1976). Monopoly and the Rate of Extraction of Exhaustible Resources. American Economic Review,66(4): 655-661
Straightforward paper showing that extraction rate under monopoly is lower than under perfect competition.


In microeconomics it is common to model utility decision between consumption and leisure. The latter refers to the time of a day the individual is not producing goods himself (if a yeoman producer) or to sell his labour in the market (a waged labourer). The productivity of the individual will affect how much it can produce (and earn) per unit of time. Therefore, time is directly incorporated into the model via productivity, which then affects the balance between consumption and leisure an individual will choose.

For example, consider a yeoman producer with the following utility function (yes, no Leontief, but makes the point):

$$ U(C,1-L) = C + a(1-L) $$

where $L$ refers to the proportion of the day the individual is working ($L \in [0,1]$).

He can produce with the following technology:

$$ Y = bL^h $$

and consumes all what he produces:

$$C = Y$$

Assuming the yeoman farmer is interested in maximising his utility, the unidimensional optimisation problem is:

$$ \text{max}_{_L} U(L) = bL^h + a(1-L) $$

The first order condition is:

$$ \frac{\mathrm{d}U}{\mathrm{d} L} = bhL^{h-1} - a = 0 $$

Thus, optimal proportion of time allocated to work is:

$$ L^* = b^{\dfrac{1}{1-h}}\left(\frac{h}{a}\right)^{\dfrac{1}{1-h}} $$

Optimal consumption is:

$$ C^* = b^{\dfrac{1}{1-h}}\left(\frac{h}{a}\right)^{\dfrac{h}{1-h}} $$

The effect of (labour) productivity on hours of work depending on the returns to scale of technology ($h$). You have that:

if $h \in (0,1)$, a ceteris paribus increase in $b$ means the individual works and consumes more.
if $h > 1$, a ceteris paribus increase in $b$ means the individual works and consumes less.
if $h = 1$, productivity might not affect labour at all (but does affect consumption). In this case, there is a corner solution for labour ($L^*=1$ or $L^*=0$). I leave this as an exercise.

So, you are concerned about  "the time to produce output". Well, think of $b$ as an indication of how much time you can spend to produce a given amount of output with a fixed amount of labour. As I indicate above, variations in $b$ have important consequences for $L^*$, $C^*$, and ultimately for maximum utility $U(C^*,L^*)$.

Note: The specification I chose means the income and substitution effect arising from an increase in productivity ($b$) do not cancel each other. Other utility function (e.g. with $\ln C$) will mean $b$ does not affect $L$. I also leave this exercise to you.


Notation conventions for superscripts and subscripts

This is only partial answer since your question is quite broad but in my work I had similar problems.

When ti comes to optimal values and derivatives I choose to use Leibniz's notation as it does the job quite nicely:

instead of writing this $\pi^{'*}$, I would write this $\frac{\partial \pi^*}{\partial x}$ or if you like it more  $\frac{\partial }{\partial x}(\pi^*)$, sure it takes some time to type out but in my humble opinion it looks quite good. I try to use Leibniz's notation any time there are some other superscripts or subscripts (as with multivariate function even derivatives with subscripts can be problematic).

For conditions consider maybe notation borrowed from statistics so instead of $p^{*,\alpha =1} $ I would use $[p^*|\alpha=1]$

Regarding those numbered solutions, I was once told by older scientist, during my internship at one research institute, that when  you have one letter subscripts you can just put them without comma but if one of the subscripts has multiple letters always put comma so for example $x_{itj}$ is fine but when I need to put $t-1$ I would put it as $x_{ij,t-1}$ (to be honest I have no idea to this day if there could be some issue with this but so far nobody complained - not even at  conferences/workshops where I was presenting my work, nor any referee - although I dont have that many published work yet as I am very young researcher, so maybe take that as a caveat regarding my suggestions).


What is a number estimated as a Tornqvist Index?

I've seen the Tornqvist used as many measures of total factor productivity but have not seen it in conjunction with the Solow growth model. My understanding of how the Solow growth models measures TFP is by having values for economic output, labout and capital, and assigning the disparity to TFP.


How is the matching principle implemented with double entry accounting?

It appears that you mix up the timing of invoicing with the timing of accrual of an expense or a revenue. There exist special Balance Sheet accounts that do not affect the P&L (with names like "Next-period Expenses" in the Assets side or "Next-period Revenues" in the Libailites/Equity side, depending on the country of course) which you use in order to account for such transactions, where invoicing does not track step-by-step the creation of costs and revenues. Small companies use these usually only when they "trespass" the fiscal-year time-boundary. Big companies may use them month-per-month.
In your specific example, assume that even by accounting standards, you are not entitled to recognize revenue in the current year, because, say, there are clauses in your contract that affect the economic substance of the deal. Still, your subcontractor invoices you now. You will credit the subcontractor account, and you will debit, not expenses, but "Next-period Expenses", and it will be a component of your Assets in the coming Balance Sheet.

Next year, when you will invoice your customer, you will credit this account and you will debit the relevant expense account, thus recognizing in the P&L the expense in the same period that you have recognized the revenue.

Assume that the subcontractor offers software development services and that your Accounting system has the Expense account "Third Party software Development services". Then in the ledger will see this year

\begin{array} { r  r  r |}
  \hline
\hline
  \text {Account Description} & \text {Debit} &  \text{Credit}\\
  \hline
\text{Next-period Expenses} & 50 &  \\
\text{Accounts Payble} &  &  50\\
\hline
\end{array}

If you also pay the subcontractor then you get

\begin{array} { r  r  r |}
  \hline
\hline
  \text {Account Description} & \text {Debit} &  \text{Credit}\\
  \hline
\text{Accounts Payble} & 50 &  \\
\text{Cash Deposits} &  &  50\\
\hline
\end{array}

Next year, when you invoice the final customer we have, to recognize the expense

\begin{array} { r  r  r |}
  \hline
\hline
  \text {Account Description} & \text {Debit} &  \text{Credit}\\
  \hline
\text{Third Party software Development services} & 50 &  \\
\text{Next-period Expenses} &  &  50\\
\hline
\end{array}

which makes the balance of "Next-period Expenses" zero,

and the revenue of course

\begin{array} { r  r  r |}
  \hline
\hline
  \text {Account Description} & \text {Debit} &  \text{Credit}\\
  \hline
\text{Accounts Receivable} & 80 &  \\
\text{Services sold} &  &  80\\
\hline
\end{array}

There are more complicated situations of course, for example, when you are entitled to recognize part of the revenue now, even though you have not invoice your customer, but I  leave them as a practice opportunity for you.


Interest/Rental Rate is today's or tomorrow's MPK?

I found the problem, and now I feel dumb but I suppose I should leave the question up so people can know the potential pitfall and prepare for it better than I did.

The problem arose from me misreading (and clearly misunderstanding) the equilibrium conditions for the problem, where it states that the interest rate $r_t$ on loans issued today and paid back tomorrow is equal to $\text{rental}_{t+1}$, the rental rate on capital used tomorrow, which is of course equal to tomorrow's marginal product of capital. I had my problem set up with $r_t = \text{rental}_{t+1}$, which is incorrect. This means that yes, we should have $r_t = \text{rental}_{t+1} = F_K(K(t+1), H(t+1))$, which makes the exercise a simple algebra problem. I just had my subscripts all messed up!

Here's why $r_t = \text{rental}_{t+1}$ (summarized from text): If $r_t < \text{rental}_{t+1}$, then under perfect foresight (an assumption of the basic model in the given chapter), people will attempt to borrow infinitely to capture the profit of $\text{rental}_{t+1} - r_t$, and this condition cannot hold in equilibrium. On the other hand, if $r_t > \text{rental}_{t+1}$, then nobody will want to hold capital until the next term, instead attempting to lend all of their funds out.


Solow model and a change in the savings rate

I don't know how this can be answered without any recourse to at least basic mathematics, since the question itself refers to movement along some curves which is mathematical concept. Hence I will go over some math and then provide intuition at the end.
Explanation Including Math
The investment curve in Solow model is defined as $sf(k)$ where $f(k) = Y$ and it is assumed that when we have zero capital per effective worker $k$ output is also zero $f(0)=0$.
The curve appears to shift upwards due to change in its slope. The investment curve will have to go through the origin no matter what it's $s$  changes the slope of the investment curve. For example, if $s_1=0.5$ and $f(k)=k$ the investment curve will be: $0.5k$. If we change $s$ to $s_2=0.7$ investment curve would be now $0.7k$. This would work for any investment curve that goes through the origin and by our assumption it has to (you can try to verify this for $f(k) = \sqrt{k}$ or other valid functions.
Non Mathematical Explanation:
The investment function is by definition the proportion of output that is being saved (and hence invested). Here $s$ is the proportionality constant that determines what proportion of output is invested. Increasing that proportionality constant has to change the slope of investment function as now at any level income higher proportion of income is invested. For example, with $s_1=0.5$  if $Y=100$ the investment will be $50$, with $Y=200$, the investment will be $100$ etc. Now if you change that proportion not only the investment will be higher at any income but also rate of change of that investment will be higher. With $s_2=0.7$ at income $Y=100$, investment will be $I=70$, at $Y=200$ investment will be $I=140$.


In which cases prior to the Greek crisis did the IMF admit mistakes in it's economic policies?

Yes.

  The International Monetary Fund yesterday admitted that its mistakes
  helped plunge Argentina deeper into the red during the currency crisis
  that crippled the country's economy three years ago.

  In a report published yesterday by its independent evaluation office,
  the IMF said it ought to have prevented the Argentine government from
  following poor economic policies.

  "IMF surveillance failed to highlight the growing vulnerabilities in
  the authorities' choice of policies and the IMF erred by supporting
  inadequate policies too long," it said.

IMF admits mistakes in Argentina crisis By Edmund Conway

The actual report is here: IMF's Independent Evaluation Office Announces Release of Report on the Role of the IMF in Argentina


Intuition behind Euler Lagrange equation in economics

The Euler Lagrange Equation in words in the context of physical capital accumulation states: Any possible accumulation that may occur at the optimum is brought back to a steady state in the next instant due to the law of motion.
This equation communicates the physical properties of our optimum meaning that once we are there no possible improvement from capital accumulation is possible.
This logic carries over for the our case in the context of the continuous time version of the Grossman model this time with health capital.


How does capital accumulation raise the demand for labor

Good question! Here Acemoglu and Restrepo discuss one of the many possible effects of automation, such that automation is likely to induce additional usage of capital in the sector or additional capital accumulation, which can increase labor demand.

They elaborate on this effect on their 2018 AER paper "The Race between Man and Machine: Implications of Technology for Growth, Factor Shares, and Employment".

In a static version of their model where capital is fixed and technology is exogenous, automation reduces employment and the labor share, and may even reduce wages. Their full model endogenizes capital accumulation and the direction of research toward automation and the creation of new tasks. If the long-run rental rate of capital relative to the wage is sufficiently low, the long-run equilibrium involves automation of all tasks. Otherwise, there exists a stable balanced growth path. Stability is a consequence of the fact that automation reduces the cost of producing using labor, and thus discourages further automation and encourages the creation of new tasks. The response of capital ensures that the productivity gains from both automation and the introduction of new tasks fully accrue to labor (the relatively inelastic factor).


How can a currency be tied to a commodity like energy?

But energy can not be easily stored and traded for normal persons so you need some form of stand-in as a currency so i am thinking like 1 "currency coin" is worth 1 kwh. And that to make new coins for the system you have to produce/extract the same amount of energy to the grid. And when that energy is used the "currency coin" is destroyed so there will be a pretty fixed amount of money in the system at any time.

This concept does not work with electrical energy.
If we look at “conventional” electrical grids, they ideally would have no resistance, and the grid transmits power (units of energy/second) through the grid without loss, and there is no stored energy. (In practice, the grid has capacitance/inductance, and some energy is temporarily stored.)
The outlined currency scheme would result in a currency unit created and immediately destroyed as used.
The rise of intermittent renewable energy has created a greater need for storage. However, the storage is not in the form of electrical charge in the grid, it is is in some mechanical/chemical store of energy. All storage technologies have inefficiencies, so that energy is lost. There is no one-to-one accounting between input and output energy a currency implies. Furthermore, storage is only useful if it is dissipated during production shortfalls, so the currency backing would disappear each night.
Meanwhile, market prices for intra-day electricity are extremely volatile (since there is no way to carry inventory), so it is not useful as a unit of account.
A currency could be backed by oil reserves - a form of potential energy - but one may note that no oil producer has pegged their currency to oil prices.


$$\frac{x_{t+1}}{x_t} \equiv \frac{x_{t}+Δx_{t+1}}{x_t} = 1+\frac{Δx_{t+1}}{x_t}$$

Further,

$$\ln (1+a) \approx a,$$

an acceptable approximation for $|a| <0.1$ and certainly for $|a| <0.05$.


I think the missing link for you is not realizing that $Q$ is actually equal to $q_1 + q_2$: in a duopoly, quantity demanded can only be derived for the two firms in question. With that in mind, we can rewrite the profit functions for both firms in terms of $q_1$ and $q_2$ and then optimize with respect to the $q$'s accordingly (since Cournot competition is competition over quantities). We are then going to be left with two best response functions, one that tells you the optimal level of $q_1$ as a function of $q_2$ and another that tells you the optimal level of $q_2$ as a function of $q_1$. Then it just becomes an algebra problem to find the Nash equilibrium and the relationship of quantities and profits between the firms will become clear.

Firm 1 maximizes profits
\begin{align*}
\Pi_1(q_1) = (\alpha - (q_1 + q_2))q_1 - c_1 q_1.
\end{align*}

The first-order condition is
\begin{gather*}
\frac{d \Pi_1(q_1)}{dq_1} = 0 \\
\implies \alpha - 2q_1 + q_2 = c_1 \\
\implies q_1(q_2) = \frac{\alpha - c_1}{2} + \frac{q_2}{2}.
\end{gather*}

Similarly, Firm 2 maximizes profits
\begin{align*}
\Pi_2(q_2) = (\alpha - (q_1 + q_2))q_2 - c_2 q_2.
\end{align*}

The first-order condition is
\begin{gather*}
\frac{d \Pi_2(q_2)}{dq_2} = 0 \\
\implies \alpha - 2q_2 + q_1 = c_2 \\
\implies q_2(q_1) = \frac{\alpha - c_2}{2} + \frac{q_1}{2}.
\end{gather*}

Solving the system of equations for $q_1$ and $q_2$ yields
\begin{align*}
q_1^* &= \frac{\alpha - 2c_1 + c_2}{3} \\
q_2^* &= \frac{\alpha - 2c_2 + c_1}{3},
\end{align*}

which implies that $\mathbf{q_1^* > q_2^*}$ since $c_1 < c_2$.

Note that both firms face the same price $P^* = \alpha - (q^*_1 + q^*_2)$. Therefore, we have that
\begin{align*}
(P^* - c_1)q_1^* &> (P^* - c_2)q_2^* \\
\implies \mathbf{\Pi^*_1} &> \mathbf{\Pi^*_2}.
\end{align*}


The Cake Eating Problem with Depreciation (Modelling difficulties)

It would seem that the way you've formulated your production function/law of motion has introduced double counting into the problem. Note that substituting 1 and 2 into 3 gives:
$$k_{t+1}=(1-\delta)(c_t+x_t)+x_t$$
Where investment in period t is counted twice.

The correct law of motion is simply:

$$ 3. \: k_{t+1}=(1-\delta)x_t $$

And the general form of the Bellman equation would be:
$$v(k_t)=\max_{k_{t+1}}\left\{\ln\left(k_t-\frac{k_{t+1}}{1-\delta}\right)+\beta v\left(\frac{k_{t+1}}{1-\delta}\right)\right\}$$

Hope this helps!


Europe is a banking based continent, in contrast to the US where financial markets play a prominent role. As a result, wealth effects in Europe are much smaller than in the US. QE thus has a smaller favorable effect on the economy.

You are right, they should implement it given the low inflation level present for an extended period. But as shown by history, the ECB always reacts later and less agressive than the FED. This is because the ECB has a sole objective of keeping inflation close but below 2% (imposed by the Germans having experienced periodes of hyperinflation). Even today, the germans and some other hawks are still reluctant to pull out the big canon.

QE leads to an increase in bond prices, and supports equities as well. However, whether or not equities go up, depends more on how people react.


Do US 501(c)(3) non-profits have to divulge their investments?

Form 990 and associated schedules:

  The Form 990 provides the public with financial information about a
  nonprofit organization, and is often the only source of such
  information. It is also used by government agencies to prevent
  organizations from abusing their tax-exempt status. In June 2007, the
  IRS released a new Form 990 that requires significant disclosures on
  corporate governance and boards of directors. These new disclosures
  are required for all nonprofit filers for the 2009 tax year, with more
  significant reporting requirements for nonprofits with over \$1 million
  in revenues or \$2.5 million in assets. In addition, certain nonprofits
  have more comprehensive reporting requirements, such as hospitals and
  other health care organizations (Schedule H). The Form 990 may be
  filed with the IRS by mail or electronically with an Authorized IRS
  e-file Provider.

  The Form 990 disclosures do not require but strongly encourage
  nonprofit boards to adopt a variety of board policies regarding
  governance practices. These suggestions go beyond Sarbanes-Oxley
  requirements for nonprofits to adopt whistleblower and document
  retention policies. The IRS has indicated they will use the Form 990
  as an enforcement tool, particularly regarding executive compensation.
  For example, nonprofits that adopt specific procedures regarding
  executive compensation are offered safe harbor from excessive
  compensation rules under section 4958 of the Internal Revenue Code and
  Treasury Regulation section 53.4958-6.

Wikipedia on Form 990

In particular, SCHEDULE D. The level of detail is not rich but the information is public enough for Charity Navigator to make use of these filings in their evaluations. You can search for a charity's form 990 here.


Could anyone here be able to explain gambling addiction and its debt with Microeconomics theory?

It seems to me that the topic is related to rational addiction: Individual preferences for consuming $x_t$ are conditioned by their past consumption $x_{t-1}$. See for instance:

Becker, G., and Murphy, K. (1988). A Theory of Rational Addiction. Journal of Political Economy, 96(4), 675-700.


"Identification" is the most loaded term in econometrics. There are multiple cheap talk equilibria with regard to its meaning. It is used with different intended (but related and overlapping) meanings, in different contexts, by people with different orientations, with different levels of precision.
Therefore you will get a range of correct answers.
Here is an attempt covering a few of the variations, going from the theoretical end of the spectrum to the empirical.
Statistics
A statistical model is a one-to-one mapping $\theta \mapsto P_{\theta}$ from a given parameter space to a family of probability measures.
It is the one-to-one property of the mapping that makes the model "identified".
No two different elements in the parameter space can give rise to observationally equivalent data generating processes.
In statistics, a model is therefore, by definition/assumption, always identified.
(This can be seen in the assumptions for all the foundational results, e.g. Neyman-Pearson.) Statisticians never speak about identification, because they don't have to.
For example, for
$$
y = \beta x + \epsilon \quad (*)
$$
where $(x,\epsilon)$ is bivariate normal, to specify a model for population $(x,y)$ parametrized by $\beta$, one must assume that $Cov(x, \epsilon) = 0$. Without imposing this assumption, different $\beta$'s could give rise to the same distribution for $(x,y)$. In econometrics, which is much more explicit about the identification issue, the condition $Cov(x, \epsilon) = 0$ will sometimes be called an identification assumption.
Structural Models
If one tries to construct a statistical model by adding unobserved disturbances to an economic model, identification needs to be addressed.
In order to make the resulting structural econometric model identified, usually one needs to make certain assumptions, either of economic or technical nature. These are called identification assumptions.
For example, suppose there are $n$ firms in Cournot competition with private constant marginal costs $(c_1, \cdots, c_n)$ drawn from joint density $f(x_1, \cdots, x_n)$.
The econometrician observes firms' output $(q_1, \cdots, q_n)$ and market price $P$ and would like to identify $f$. One possible identification assumption is that the Jacobian of the FOC system
$$
\frac{d P(Q)}{dQ} q_i + P(Q) - c_i = 0, \, i = 1, \cdots, n,\, \mbox{ where } Q=\sum_1^n q_i
$$
is non-vanishing. Then, by the Implicit Function Theorem, $(q_1, \cdots, q_n)$ maps one-to-one locally to $(c_1, \cdots, c_n)$. This implies the model, parametrized by the observed quantity $(q_1, \cdots, q_n)$, is identified, at least locally.
The empirical interpretation is that sufficient variation in the trade-offs faced by the firms
allows you to identify $f$.
There are more interesting examples where the identification assumption
puts restriction on economic agent behavior, etc.
Empirical Usage-Consistent Estimation
So far, identification is purely a property of the mapping from parameter to data generating processes. Identification is a prerequisite for estimation but by itself it makes no mention of the sample.
There are also contexts where an econometrician speaks about a specific estimator that's designed to estimate a specific parameter in a specific model.
An assumption under which the estimator consistently estimates the parameter is called an identification assumption.
For example, given time-series data $(x_t, y_t)$ generated by
$$
y_t = \beta x_t + \epsilon_t, \; t = 1, 2, \cdots, \quad (**)
$$
the parameter $\beta$ "can be identified by OLS $\hat{\beta}$" under the assumption that $Cov(x, \epsilon) = 0$.
In $(*)$ and $(**)$, the condition $Cov(x, \epsilon) = 0$ and the terminology are the same, but "identification assumption" have different (but clearly related) meanings.
Empirical Usage-Causal Inference
When one is interested in establishing causal effect, a condition imposed on the model that allows for causal interpretation of the estimate is called an identification assumption. Yes---$Cov(x, \epsilon) = 0$ for the linear model would fall under this category also. Often it is strengthened to  $E[\epsilon|x] = 0$, which is more interpretable for causal inference.
Similarly, when $Z$ is an instrument, the exogeneity condition $Cov(Z, \epsilon) = 0$ is an identification assumption.
For diff-in-diff, the parallel-trends condition is an identification assumption.
For regression discontinuity design, the identification assumptions are that, first, there are no other discontinuities except the forcing variable, and second, agents cannot manipulate the forcing variable.
The corresponding empirical design (e.g. IV/DID/RDD/etc) is sometimes called the identification strategy.
In this context, "identification" is not a binary condition. One could have weak identification, e.g. a weak instrument.
Used in this sense, an identification assumption clearly needs to be justified when it's claimed to hold empirically. In other words, one needs to justify that the corresponding  variation is exogenous---e.g. the variation of the instrument is exogenous, etc.
In your quoted example,

The paper examines the effect of bank runs on lending. We exploit
variation in the structure of banks' liabilities to identify banks
that were more vulnerable to the run...

vulnerability to a run is clearly a endogenous variable in relation to lending. The claim is then the empirical design in question uses exogenous variation in the structure of banks' liabilities---as an instrument/forcing variable/whatever---to circumvent endogeneity and achieve identification.


Is product opportunity frontier the same as PPF?

Yes. The "curved line" clearly fits your definition as well. The PPF is curved if (but not only if) there are diminishing returns to inputs such as labor and capital. See also
https://en.wikipedia.org/wiki/Production%E2%80%93possibility_frontier


Are CPI stats weighed based on population size

Yes, the data is weighted based on the population size of the area from which it came. The survey designers employ a method called stratified random sampling, which ensures that the prices of goods selected for the sample are distributed across all areas (rather than by chance landing up concentrated in a few areas). Then, the prices in each stratum are assigned a sampling weight, the ratio of size of the population in that stratum to the number of prices collected in that stratum. Those weights are used when calculating the mean price for that item (i.e. multiply each observation by its weight, divide by the sum of the sampling weights).

Further, a representative basket of goods and services is used (which was found through household surveys, again with random sampling), and the estimated mean prices of goods are plugged in to that basket to get an overall index of price. The basket should thus be the same across all years in order for the price index to be comparable across years (except when using a chained price index).


Assuming you are referring to the definition from the original Arrow & Debreu paper $x_i$ represents the units of good $x_i$ consumed.

I would not call $Y$ technology, according to Arrow & Debreu $Y$ is set of production plans - so by that they mean production (which may make some assumptions on technological nature of the process).

Also, from your second paragraph I am a bit confused what do you mean by value. In mathematics value is some definite mathematical object like integer 5. In that sense yes, in the general model they are variables but once you would try to apply it to concrete problem $x$ and $y$ would have some values, if you are referring to some human values then no. In economics human values or preferences are taken as given and expressed through utility function, in Arrow-Debreu paper $u(x)$.

A bit confused what you mean by value equilibrium on macroeconomic scale, assuming you mean general equilibrium, that depends on exact model you are following. Arrow-Debreu model is an example of basic general equilibrium model so you can use the variables you see there. If you would want to estimate it then currently one of the most popular ways is to estimate Dynamic Stochastic General Equilibrium (DSGE) models- these include a lot of variables. In such model you try to explicitly model demand, supply and monetary policy. The number of variables there can be so large that it’s not feasible for me to list it rather you may want to see an example of Smets-Wouters DSGE model for Euro-area economy.


Effects of long run stability in U.S price level comparing to other countries on net export

Net exports are exports less imports.

If other countries begin to experience inflation while US prices remain stable, then the dollar becomes more attractive to investors as its purchasing power has maintained more so than foreign currencies. The demand for dollars increases, which leads to an appreciation of the home currency. This leads to a decrease in net exports as US goods become pricier relative to foreign goods.

This theoretical relationship between the exchange rate and the relative purchasing power of two currencies is known as purchasing power parity.


Are there some nice properties for a finite two-player game having an increasing differences utility and a decreasing differences utility?

There is a recent paper which may help you:
Games with strategic complements and substitutes (Monaco and Sabarwal (2015))


How is using a reserve fund different from printing money?

No, accumulating money is not pointless. Money not being the source of value does not lead to it being of no value.
Money, including the type we use right now -called "fiat money"-, has 3 properties.

It is an accepted tool for carrying out transactions.
It is a way to measure a product's value.
It is a store of value.

The 3rd point is the one you're looking for. As long as the value of money (its purchasing power) does not drop then it is a proper way of storing value; you can keep your money and use it in the future (a "rainy day fund" perhaps)- this wouldn't happen if you were exchanging fish for cows for example.
It is apparent that money is not the best store of value, since it doesn't give its holder any interest just for holding it, unlike deposits in a bank, bonds and stocks.
A state can have multiple sources of property, money being one type of it. For reasons of diversification to risk it is preferred that assets are not of the same type.
On the topic of inflation and efficiency;
Neoclassical theory suggests that storing a reserve fund is not the same with money being destroyed and then printed anew. Everyone knows this money exists and is being stored, and will be used in case of need. So people, having rational expectations, already understand this amount of money as part of monetary policy and the implications of such policy (including inflation); which means that the current price level is already adjusted for this inflation.


Monetary Policy - How can increasing interest rates cause higher inflation in the short term?

If mortgage payments are included in the basket of goods that are used to calculate inflation, then rising interest rates will increase mortgage payments, causing an increase in inflation in the short term. This happens before any of the disinflationary effects show up in the data.


Whether the demand in the panels satisfies the weak axiom

So WARP says that if bundle x' you choose under budget B' is affordable at budget B'' (and you don't choose x' - i.e x'$\not=$x''), then x'' cannot be affordable under budget B'.

The picture you showed gives the 2 bundles that the person chooses. Those are given in the picture. As you can see, x' is not affordable at budget B'' and bundle x'' is not affordable at budget B', so WARP is satisfied.

If you are reading MWG and you don't understand what those dots represent in the consumption space, then I would recommend reading more from the beginning of chapter 2.


Why does the stock price of listed firm matter for that firm?

There are multiple channels :

Debt: the cost of debt may raise significantly if the value of equity drops. Indeed, the lower the equity, the higher the risk of the debt, as the firm will have a higher gearing ratio (also called leverage). Thus, the firm might have difficulties to gain access to additional debt or face a higher cost.
Financing: A firm with a poor performance on stock markets might face difficulties to raise additional capital. Investors might indeed not be willing to buy the firm's stock if its performance is poor. Therefore the firm might not be able to raise additional capital.
Reputation: a decrease in stock price can negatively affect the reputation of a firm. Not necessarily with the customers but also with other stakeholders, such as the suppliers which could fear that the financial stability of the firm is not assured.

So overall I would say that the stock price is mainly important to the firm as it is a strong determinant of the costs it will face when raising capital or debt.

Also, note that stockholders are owners of the firm. As such, they can compel the firm to adopt a more profitable strategy, if they so wish. Of course, it might be hard, but activist investors for example show that it is possible.


How do you differentiate between a business expense and an investment for the purpose of profit reporting?

In economics profit is simply defined as total revenue minus total costs or:
$$\Pi = TR-TC$$
In your case if

Pfizer makes \$100B in revenue, spends \$80B on business expenses doing so, then spends the remaining \$20B on research for a new drug.

from economic perspective the profit would be:
$$\Pi= TR -TC = {\\\$}100B - ({\\\$}80B+{\\\$}20B) = 0B$$
So from economic perspective the profit will be zero. (However, note the economic perspective is not necessary the same as an accounting perspective - I mention that since you reference reporting, but in this case accounting profit would also happen to be zero if $20B$ is on accounts treated as cost).


Name for the intentional reduction in quality in a non-software, price discrimination setting

"Lemonization" I guess. I didn't exactly make up that term with that meaning, but I suspect you won't see it often.


I do not believe that foreign capital investment (at least in the short-run) would directly increase the domestic money supply.

When foreign investors purchase domestic currency, the investors merely transfer that currency from domestic accounts to their own accounts. Then, when the investors purchase capital in the domestic market, the purchased currency is transferred back to the domestic sellers. Thus I do not see how net circulation of domestic currency rises in the event of a capital inflow directly.

There are, of course, more complicated indirect effects such as domestic banks issuing an increased number of loans based off new demand deposits that may increase the money supply, but the direct effect that I believe you're alluding to isn't there. As you say, the domestic money supply cannot be increased without banks playing a role in fueling domestic credit.


Why did it take so much QE by the ECB to raise the Eurozone inflation to 2%?

Because commercial banks have used injected liquidities to consolidate their balance sheet instead of stimulating demands by creating credits (that did not previously exist when they make loans). This is called the money multiplier effect. The multiplier was somehow blocked (to ~$1$) because of their precaution behaviour, and so was the money velocity.

On March 10, 2016, the ECB has even considered the possibility of giving 200€ per month and per person to stimulate inflation. And this with the main idea of circumventing the commercial-bank system. This unconventional monetary policy is called monetary helicopter.


why is the term fiat money or fiat currency and is it the same as legal tender or are there any differences?

Difference between money and currency in general (i.e. in any monetary arrangement -fiat money, gold standard etc.) is as follows:

Currency: is the money in the strictest sense of a word. It includes coins and bank notes in the circulation. In foreign exchange the word is also used to as a synonym for unit of money (e.g. $ €) but from your question I think you are looking for the first meaning.

Money: is the whole stock of money in economy, hence not just coins and paper notes which are only small part of all money but also money on accounts etc.

Legal tender: is a money that must be accepted if offered in payment for debt. So for example in principle any debt can be repaid let’s say with labor or with some commodity, but the creditor can say no to all those things unless you had some specific contract but can’t say no to legal tender of the country. Legal tender again does not need to be fiat money it could be some gold backed money or fiduciary money etc.

Fiat money: following the Mankiws definition from his principles of economics (p 220). It is: “money without intristic value that is used as money because of government decree”. So there is a connection to legal tender all fiat money is also a legal tender by definition but it’s a necessary condition not sufficient one as it also has to have no intristic value. Fiat currency is just the coin and note part of the fiat money.

The origin of the word fiat is from Latin: “let it be done”. In modern English it’s synonymous to government decree, edict or act.

The earliest reference to fiat money I could find is in John Maynard Keynes’ treatise on money (1930). But I am unable to say if the word was not used before that.


Why the Federal Funds rate isn't detrended in a VAR model?

Yes, if the series is already stationary then there is no need to difference the data to remove a trend. In fact, there is a cost associated with unnecessarily differencing the data such that you are removing / losing relevant information. More generally this is related to the concept of cointegration where even if series are non-stationary, if they are co-trending then you lose more information by differencing than by including the lagged levels in the model specification.
You might also find the following question and answers helpful for thinking about this in a slightly different context: https://stats.stackexchange.com/questions/415914/is-it-a-valid-claim-that-by-differencing-a-time-series-it-loses-its-memory-an


How to write the dual problem of this maximization problem?

Everything is explained here: http://en.wikipedia.org/wiki/Linear_programming#Another_example.


One must first distinguish two different senses in which the Allais Paradox can be seen as a "contradiction" of vNM independence; these correspond to the two different interpretations which can be given to any model in Decision Theory. (von Neumann-Morgenstern (vNM) utility theory is one such model.)  According to the descriptive interpretation, a model in Decision Theory is supposed to provide a description of how real, actual human beings actually make decisions.  According to the normative interpretation, the model is supposed to tell us how an "ideal rational agent" should make decisions.   To put it another way, the descriptive interpretation matters to you if you are trying to construct a predictive model of insurance markets or financial markets or some other social phenomenon where people make risky decisions;  the normative interpretation matters to you if you yourself are making risky decisions, or if you are trying advise someone who is making such decisions.

The Allais Paradox certainly falsifies the descriptive interpretation of the vNM model; this simply means that, as a matter of empirical fact, most people (including, apparently, many economists) will choose 1A and 2B, which cannot be explained as maximizing the expected value of any possible vNM utility function.  Whether the Allais Paradox falsifies the normative interpretation of the vNM model is a more subtle and controversial question. Basically it depends whether or not you think that {1A, 2B} is really the "correct" answer.  Some people strongly feel that {1A, 2B} is correct;  for them, this is evidence that vNM is not the correct normative model of decision-making under risk.  Other people are initially tempted by {1A, 2B}, but once they recognize that it is inconsistent with vNM, they are willing to "bite the bullet" and reject their own intuitions in favor of the theory;  their ultimate position is that the normative implications of vNM are counterintuitive, but correct.  (The fact that a theory has "counterintuitive" implications does not make it wrong;  modern physics is full of theories like this.  Indeed, there are even mathematical theorems which are "counterintuitive", but evidently they are correct as a matter of logical necessity.)

In your post, you wrote:

  In fact, if all choices were approximately the same utility, the
  inconsistency would disappear (which seems entirely plausible to me).

I don't think "approximately the same utility" is good enough ---as long as the agent has any nontrivial vNM utility function, the choice pattern {1A, 2B} is impossible.  Perhaps what you mean is that the choice pattern {1A, 2B} can be reconciled with vNM theory if the agent's vNM utility function is trivial (i.e. constantly zero); in this case, the agent would be indifferent between all possible lotteries.  As a purely mathematical point, this is correct. But it is not consistent with people's preferences; most people experience a strict preference for 1A over 1B, and a strict preference for 2B over 2A.  So we cannot "explain away" the paradox by positing trivial utility functions.

Your post ends with the question:

  Is there another set of parameters, or another example entirely, which
  more convincingly shows an absurdity of the independence axiom, or are
  the parameters above as strong as it gets?

Perhaps you will be interested in the Ellsberg Paradox.  (I won't bother including a complete description here, since there is a very nice Wikipedia article about it.)  The Ellsberg Paradox differs from the Allais Paradox in that the probabilities of certain events are left unspecified;  the decision-maker must construct her own "subjective probabilistic beliefs" about these events.  But the interesting fact is that there is no set of "subjective probabilistic beliefs" which can explain the pattern of choices most people make in the Ellsberg Paradox.  So it definitely falsifies the "descriptive" validity of expected utility (EU) maximization as a model of human decision-making.  More importantly, it presents a strong challenge to the "normative" validity of EU-maximization;  even many economic theorists who "bite the bullet" in the Allais Paradox are willing to acknowledge that in the Ellsberg Paradox, the choice pattern which is inconsistent with EU-maximization is much more compelling than a choice pattern which is consistent with EU-maximization.

However, I should acknowledge that the Ellsberg Paradox is properly understood as a challenge to EU-maximization in a setting of "ambiguity" (where probabilities are not known a priori), whereas the Allais Paradox is already a challenge to EU-maximization even in a setting of "risk" (where all probabilities are known).  It is a perfectly consistent position to accept the vNM axioms in a setting of risk (and hence, "bite the bullet" in the Allais Paradox), while rejecting EU-maximization (i.e. rejecting the Savage Theorem) in a setting of uncertainty.

Another, entirely different normative challenge to vNM independence arises in a setting of social choice.  Suppose that you have an indivisible resource (e.g. a hard candy), which you must allocate to one of two claimants (e.g. two children, Alice and Bob).  Suppose that, a priori, both claimaints have an equal claim to the resource (they are equally deserving, equally hungry, etc.).  The resource is indivisible, so you can't "split" it between the two claimants.   Question:  should you allocate the resource by some random mechanism (e.g. by flipping a fair coin)?  Or should you allocate it by some deterministic mechanism (e.g. give it to the older child, or the one whose name comes first in alphabetical order, or the one who is standing further to the left at the moment the decision is made)?   Most people have a strong intuition that a random mechanism is more "fair" than a deterministic mechanism.  So they prefer to randomize.  But at the same time, they are indifferent between giving the resource to Alice and giving it to Bob (recall: Alice and Bob are equally deserving of the resource).  So if A is the outcome "give it to Alice", and B is the outcome "give it to Bob", then we have a situation where the decision maker is indifferent between A and B, but strictly prefers the lottery (1/2 A, 1/2 B) to either A or B.  Clearly, this normative position cannot be explained by the vNM theory.  This is sometimes called the Diamond Paradox, since it was originally pointed out in a 1967 article by Peter Diamond.