Under what circumstances is an increase in national output called economic growth?

There are actually 2 types of economic growth. Actual growth is when output increases. Potential growth is when productive potential of the economy increases.

If long run aggregate supply increases without any change to short run AS, then you have potential growth but not actual growth. This might happen with if a new discovery is made, but has not yet been applied in industry.

If SRAS increase without any change to LRAS, then you have actual growth, but no potential growth. This might happen to an oil-dependent economy if there is a temporary fall in oil prices.

As long as there is short-term capacity in the economy, increases in aggregate demand will lead to actual growth. Aggregate demand, however, cannot lead to potential growth except by affecting LRAS (e.g. through changes in producer expectation etc).

Finally, you are correct in that actual growth is shown by moving a point closer to the PPC/ in the North East direction.

See any lecture material:
http://www.tutor2u.net/business/strategy/economy-economic-growth.html


Estimate unobservable $X_t$ with observable estimation {$Y_t$}, where $Y_t= X_t+\delta e_t$

It seems all you need to do is calculate the mean of Y (take the average of observations), which gives you the mean of X, since the expected value of epsilon is zero.
I would approach this problem as follows:
$ X_t = Y_t - \delta e_t$
$\mu_X   = E(X_t)= E(Y_t - \delta e_t) = \mu_y - \delta E(e_t) = \mu_y$


Monetary Policy and international reserves-- interpretation of the estimated coefficient

No, it's a regression model. You can't infer any causality from it.

At best, you can infer that there is correlation between the terms.

But note that even that inference can be hazardous. Without knowing how you got the model terms, what your priors are, and what the data is, it's not possible to say whether or not the correlation is spurious


Raising interest rates to prevent inflation

The increase in interest rates make borrowing more expensive and therefore businesses and people will borrow less. Therefore, investment and consumption decrease. Through the multiplier effects, this will cause a reduction in real GDP. This will also increase the unemployment rate. Therefore, people who sell these goods and services will reduce their prices. As a result, the overall price level will fall and the inflation rate is reduced. I understand that explanation where the money demand decreases cause a probable eventual reduction in the interest rates. However, this assume that the money supply is fixed. Now, you must understand how the Fed controls the interest rates. It controls the interest rates by controlling the money supply. It reduces the money supply by selling bonds into the economy. Therefore, the money supply shift to the left and initially, the interest rates soar. The demand and supply analysis:

First we must define some basics: We are going to use the supply and demand graph with the interest rates as the y axis and the money in x axis. Now since we know that there is a fixed amount of money in the economy and the Fed is controlling it, we know that the supply of money is strictly vertical and the demand for money is downward-sloping (with a certain elasticity)
The mechanism of the money diagram: In a fixed money supply, if there is an increase in demand of money, the demand curve will shift to the right. Now if there is a a positive inflation rate, there is an increase in demand of money with every inflation rate and therefore the demand curve will shift to the right.
The fed aims to  reduce inflation by increasing interest rates. They do this by reducing the amount of money in the economy and thus shifting the supply curve of money to the left. You are right when the total quantity of money in the equilibrium point will be reduced. However, I think that you are confused why the interest rates will not go down. This is because the Fed aims to reduce positive interest rates and not cause a negative inflation rate. In argument (2) I have explained that with a positive inflation rate, the money demand curve will always be shifting to the right. Therefore, the Fed only wants to reduce the speed of that shift.

Now you wonder why the Fed wants to reduce the inflation rate. This is because it is always desirable to have a steady price level. A high inflation rate causes some cost to the economy that it does not want. You need to google or search in the wikipedia for the negative effects of high inflation rates.


If $\succsim$ is rational, then $B \mapsto C^*(B, \succsim)$ satisfies the weak axiom, and $\succsim=\succsim^*$

Here is an attempt at a direct proof:

Let all the assumptions hold, and further assume that $x \succcurlyeq^* y$. Then for some $B \supseteq \{x,y\}$, $C^*(B) \supseteq \{x\}$ (by definition of $\succcurlyeq^*$), which further implies $x \succcurlyeq z$ for all $z \in B$, and therefore $x \succcurlyeq y$ (by definition of $C^*$). (This shows that $\succcurlyeq^* \subseteq \succcurlyeq$; the other direction follows from looking at doubleton sets $B = \{x,y\}$).

What is left to show is that $C^*$ satisfies the weak axiom. So take any $B' \supseteq \{x,y\}$, such that $y \in C^*(B')$. If no such set exists, then $y$ is not revealed preferred to $x$ and we are done. If such a set exists then by definition $y \succcurlyeq z$ for all $z \in B'$. We already know $x \succcurlyeq y$ so by transativity $x \succcurlyeq z$ for all $z \in B'$ and hence $x \in C^*(B')$. Again,  $y$ is not revealed preferred to $x$ and we are done.

Notice, while we used transitivity, we did not use completeness. This hints at the fact that the statement can actually be generalized to pre-orders (reflexive and transitive), by allowing $C^*$ to be empty. This point is somewhat lost by appealing to the second result.


Can we "prove" that slow growth is due to deficient demand?

This is one of the issues with Macroeconomics - you usually cannot prove these kind of things, you can only find indicators.

That is, because what we observe is always - in the neoclassical language - the general equilibrium, never the "off equilibrium adjustment" (as in the Walrasian Auctioneer). That includes Keynesian logic - these days, the majority of Economists does not think about these traps as off-equilibrium paths, but rather as multiple equilibria [citation needed].

As a comparison, if you only observe the equilibrium outcome, it is hard to say whether a decrease in prices comes from an increase in supply or a decrease in demand. Now, if you could randomize or had a natural experiment at hand (say, randomly shut down half of the firms in the economy), you could actually "prove" the underlying issue with empirical data.

But since this is not feasible within many subjects of macroeconomic interest, disproving ideas using empirical evidence is quite difficult.


Bayesian-Nash equilibrium in a first-price auction

It is actually assumed that $b_i(v_i)$ is of the form $\alpha_i+\beta_i \cdot v_i$. So it is an affine function. Linearity only works if the bottom of the uniform distribution is 0.

A somewhat intuitive reasoning is that if the valuations are uniformly distributed over $[c,d]$ and $b_i(v_i)$ is a symmetric equilibrium, then $b_i(v_i)+k$ should define a symmetric equilibrium when valuations are uniform over $[c+k,d+k]$ as neither the probability of winning nor the gain changes.

Assuming a strategy of the form $\alpha_i+\beta_i \cdot v_i$ you get

$$
=Pr[\alpha_1 + \beta_1 \cdot v_1 < \alpha_2 + \beta_2 \cdot v_2] \cdot ((1- \beta_2) \cdot v_2 - \alpha_2)
$$

$$
=Pr\left[v_1 < \frac{\alpha_2 - \alpha_1 + \beta_2 \cdot v_2}{\beta_1} \right] \cdot ((1- \beta_2) \cdot v_2 - \alpha_2)
$$

$$
=\frac{\frac{\alpha_2 - \alpha_1 + \beta_2 \cdot v_2}{\beta_1}-c}{d-c}\cdot ((1- \beta_2) \cdot v_2 - \alpha_2)
$$

Taking derivative w.r.t. $\alpha_2$ and $\beta_2$ you get your first order conditions. Assuming symmetry you get a unique solution. If $c = 0$ then $\alpha_1 = \alpha_2 = 0$.

By the way, I prefer treating $b_2(v_2)$ as the decision variable, then you can
$$
\max_{b_2(v_2)}
Pr\left[v_1 < \frac{b_2(v_2) - \alpha_1}{\beta_1} \right] \cdot (v_2 - b_2(v_2))
$$
and the last fraction is somewhat simpler.


How do different Indifference Curves relate to each other?

Each indifference Curve represents a different set of indifferences for different bundles of goods. Assuming the three Indifference curves are labeled I1, I2, I3, with I1 being the bottom one and I3 being the top one.

Take a bundle on I1. The consumer's preference is indifferent to any other bundle that is on I1, but would prefer every other bundle on I2 and I3 to the bundles on I1.


Claiming that food delivery apps are purely rent extraction

at great expense to the restaurants that actually prepare the food

Your premise seems to be that the restaurants are somehow losing out.

The key though is that (if done right) all three parties (the app company, the restaurant, and the consumer) should benefit. The reason is that through economies of scale (delivering for multiple restaurants) and perhaps other factors, the app company is able to organize and execute the delivery at a lower cost than the restaurant would be able to.

One piece of evidence that all three parties benefit is that all three parties continue to use this arrangement.

Note that with certain countries and certain apps, the delivery app companies are actually losing money. Venture capitalists throw money at them hoping they'll end up as the "winner-takes-all".

Rarely though is it the case that the consumer or the restaurant loses from this arrangement. A consumer who finds this a losing proposition can quite easily stop doing business with the app. Likewise with any restaurant.


Fake resumes are a form of deception in a manner of speaking, but the purpose of the deception is to reveal obfuscated bias in hiring practices. Not all journals take such a hard-line stance on this type of deception and as such papers can still be published.

For a more robust discussion on the main dimensions that determine the judgment of deception, I recommend Krawczyk (2013).


competitive equilibrium, Walrasian equilibrium, Walrasian auction

The definition from about.com is what you are looking for. Feasible means that total consumption of each good does not exceed total endowment (initial holding) of that good in the economy. Terms Walrasian and Competitive equilibrium can be used interchangeably here and Walrasian auction is entirely different concept.


Signalling in a moral hazard contract with informed principals (that know their own type before contracting)

I'm not sure what exactly you're looking for, but here are some wild guesses:

Bolton and Dewatripont (2005) Contract Theory, MIT Press. [Probably Chapters 5 and 6]
Maskin and Tirole (1990) "The Principal-Agent Relationship with an Informed Principal: The Case of Private Values", Econometrica 58: 379-409.
Maskin and Tirole (1992) "The Principal-Agent Relationship with an Informed Principal, II: Common Values", Econometrica 60: 1-42.
Myerson (1983) "Mechanism Design by an Informed Principal", Econometrica 51: 1767-1797.


Relationship between interest rate and rental price of capital

The equation is exactly the same,

$$r(t) = R(t) - \delta$$

Now, set $\delta =1$ per assumptions, to obtain

$$r(t) = R(t) - 1 \implies 1+r(t) = R(t)$$

which reasonably continues to say that the accepted rental rate of capital $R(t)$, covers the depreciation rate plus $r(t)$ which is also the net return to capital.


Why in most macro models technology is labor-augmenting?

The mathematical reason, is that this happens in order for the model to have a steady-state in terms of growth rates: variables like Consumption, Capital, Income, grow at the steady-state, but grow at the same rate, so their ratios remain constant (and it is in this sense that this situation represents a "steady"-state). If they were to grow at different rates, their ratios would tend to either zero or infinity which is not very realistic, since it would imply that the economy tends towards one or the other "corner" situation.

The mathematical proof can be found in Barro & Sala-i-Martin book (2nd ed) , section 1.5.3, pp 78-80. Relevant and useful is also the discussion in section 1.2.12, pp 51-53.

For functional forms like (generalized, even) Cobb-Douglas, it is really indistinguishable (not separately identifiable), especially since we predominantly use the exponential function:

$$Y_t = A\cdot \left (K_te^{zt}\right)^\alpha \left (L_te^{vt}\right)^\beta = A\cdot K_t^{\alpha}\left (L_te^{(v+\frac {\alpha}{\beta}z)t}\right)^\beta = A\cdot K_t^{\alpha}\left (L_te^{wt}\right)^\beta$$

So strictly speaking in such a functional setup we can say that technology is also capital augmenting.

But since for other functional forms, the above does not hold, and so we must explicitly assume that technology is "labor-augmenting" for the reason stated previously, authors settled in labeling it as such in order to cover all cases, and when they want to keep the functional form unspecified.

Regarding the conceptual issue the OP poses, which is insightful, a conceptual way out is to think of "Technology" more like "Knowledge". So "Knowledge" that goes into the machines, is part of the Investment that augments capital, while the other knowledge turns raw labor $L$ into human capital: essentially a production function with "exogenous labor-augmenting technology", is equivalent to a formulation that includes Human Capital instead of labor but where the investment in Human Capital is not subject to optimizing behavior but "automatic" (which points to Arrow's "Learning-by-Doing" concept of human capital accumulation).


The answer is indeed A:

Initial farm income is $\$5 \times 2,000 = \$10,000$.

If the supply is $S_2$, then current income is below that figure.
However,  if the government takes away from the market 1,000 tonnes,  then the supply curve will shift left by that amount, resulting in a new equilibrium with 4,000tonnes @ \$2.
Therefore farm income is now:  $\$2 \times 4,000 = \$8,000$. The government buys the buffer at market price,  thus $\$2 \times 1,000 = \$2,000$. We end up with the same farm income as in the initial situation: $\$8,000 + \$2,000 = \$10,000$.


Why are interest rates so high in Russia?

I think the answer to your question does not require reference to economic theory as you are trying to understand why Russians cannot borrow at low rates. The prevailing high rates are not there for suppressing business activity, etc. This is something to do with the risk appetite of Russian commercial banks when dealing with domestic businesses.

Why are Russians facing rates as high as 25% when the base rate is 9%?

Central Bank's current base rate is 9%, also called repo rate. This rate is the main monetary tool used by CB to influence the interbank interest rates and the interest rates for loans, mortgages and savings. As you can see, the CB rate is low, but the commercial banks are adding huge premium (here it is 16%) to the base rate when lending to domestic businesses. Perhaps, they may face high default risks, political risk, high inflation, exchange rate risk, etc.

As a side note, this is a typical characteristic of a transition economy, that's said, Russians will face relatively high interest rates till they gain 'developed' status.


Incomplete and Intransitive Preferences: Reference Request

Based on the contact page my guess would be Gonçalo Fonseca, whose research focuses on the intersection of the history of economic thought and economic theory, or/and Leanne Ussher. You can probably drop them an email (there is an email contact).


If saving equals investment, can there ever be a stock of savings?

The accounting identity assumes that anything saved is an investment, i.e. something will be done with that saving in the future. It's a bit circular, I know. It's not a lagged model, i.e. there's no explicit indexing by time of the variables, unlike (e.g.) in the Hansen–Samuelson model, which uses discrete time. There are also macroeconomic models with continuous time, e.g. Ramsey–Cass–Koopmans model.

Actually, there's a little more to be said here (as I may have committed a fallacy myself above). German economist Fabian Lindner takes issue with those (including economics textbooks authors) who use a one-good model, like Ramsey's to model financial savings:

  The fallacies loanable funds theory commits might be explainable by the mis-application of some ideas and
  concepts of neoclassical growth models – especially the Ramsey (1928), Solow (1956) and Diamond (1965)
  models – to the sphere of money and finance. Those models are routinely taught in contemporary graduate
  economics classes (Blanchard and Fischer, 1989; Romer, 1996).
  The Ramsey and Solow models are models of real investment only. Financial markets, financial
  assets and financial saving do not play any role in those models. There is only one good which, for simplicity,
  will be called “corn”. Corn has three functions: it can be consumed, invested and used as a means of
  payment since wages and interest payments are made with it. Full employment is assumed.

  Without money and other financial assets, the only way units can save is to increase their tangible
  assets, i.e. to invest. Given that full employment is assumed, corn production is always at its maximum in
  each period. If the corn is consumed, it cannot be invested; if it is invested, it cannot be consumed. There is
  thus a real trade-off between consumption and investment.

  Only under this assumption does it indeed make
  sense to talk about a limited saving fund which is increased when it is not consumed.
  However, this trade-off between consumption and investment is not a finance constraint, but a
  resource constraint. [...]

  In contrast to Solow’s and Ramsey’s model, Diamond’s (1965) full employment corn economy
  allows units to lend and borrow. However, they do not borrow and lend money but again the one good, corn.
  They face a triple trade-off: they can eat (=consume), plant (=invest) or lend their corn. Here, consumption
  are not consumption expenditures but the actual eating of the corn. When a unit wants to lend its corn, it can
  of course not eat it so that it has to restrain from consumption to be able to lend it. [...]

  But since in the real world money is normally not eaten or planted
  and keeps circulating in the economy when it is spent or lent, those models cannot be any guide for the
  analysis of a monetary economy. Specifically, what is true in a one good economy – units have to consume
  less to lend and invest more – is fundamentally wrong in a monetary economy.


Litigation implies that the company may lose money if it loses the lawsuit. The expected loss decreases the expected profits of the firm, and thus negatively affects the share price.


How is a country's interest rate and inflation linked?

You're probably not looking for a complex scientifical discussion on the subject, so here's a very simplified description of the mechanism:

Lower interest rate => Loaning money is cheaper => More money in the system => Higher inflation.

So the change in inflation is one of the products of government/central bank changing interest rates.


If government impose price control, what happens?

Those are all potential consequences of price controls. It would all depend on how the price controls were done, what measures were put in place alongside them, and all the context of the good itself.

If the price is set at too high a level, excess supply may occur. If too low, excess demand may occur. If it's a good with an extremely limited market that is met under the price control, then no unexploited opportunities may exist. If potential demand is greater than the rate at which it can be retailed, then people may spend a lot of time waiting in line to acquire the good. And underground markets may emerge in any case.


How to derive an aggregate demand function (and optimal uniform price) from two demand functions?

Let the market price for the good be $P$.

It is easy to see that a consumer from group $1$'s demand for the good is $$ X_1 = \frac{100-P}{10} $$

Similarly, a consumer from group $2$'s demand for the good is $$ X_2 = \frac{50-P}{10} $$

Total demand $Q$ is then given by $$ Q = 40X_1 + 60X_2 = 700-10P $$

This means that the market inverse demand curve (i.e. aggregate demand) is $$ P(Q) = 70 - \frac{Q}{10}$$

Suppose a single monopolist were serving this market. The monopolist's profit is $$ \Pi(Q) =P(Q) \cdot Q -MC \cdot Q $$

If we take the first-order condition (i.e. take the first derivative of $\Pi$ and set it equal to $0$), we know that the optimal quantity sold satisfies $$ 70 - \frac{Q^*}{5} = 10 $$

This means that $Q^* = 300$. We can plug this back into the market demand function to find that $P^* = 40$, which is the optimal uniform price.


How would gross world product look like if fossil fuel is treated as capital and not as income?

There is a large literature considering the depletion of natural resources, including fossil fuels, in relation to GDP. Some writers explicitly refer to such resources as a kind of natural capital. Others may refer to natural resource assets, or simply to natural resources. A key question within this literature is whether depletion of natural resources should be recognised via a deduction within net domestic product (NDP), in (broadly) the same way that a deduction is made for depreciation of man-made capital.  The range of views is wide. Some examples:

Repetto et al (1989) attempted to calculate Net Domestic Income (NDI) for Indonesia for 1971-84, taking account of resource depletion, and found that the differences from Gross Domestic Income (GDI) were very large (p 8).  Various resources were considered including petroleum (pp 37ff).  It was argued that, for resource-dependent countries, not applying the depreciation concept to the stock of natural resources was "a major omission and inconsistency" (p 4)

Nature's Numbers (1999), a report sponsored by the US Bureau of Economic Analysis, devoted a large section (pp 59-105) to "Accounting for Subsoil Mineral Resources" (including fossil fuels).  It recognised that conventional national accounts treat natural capital in a way which is inconsistent with that of man-made capital (pp 25-6), but stopped short of recommending a deduction for depletion.

Natural Resource and Environmental Economics, a textbook by Perman et al, gives a useful overview of the issues (in the 3rd edition I have it's Chapter 19, especially section 19.2.4 on measuring national income).  It identifies several different methods that have been proposed for estimating the value of resource depletion.  In their concluding remarks (section 19.5), the authors express scepticism as to whether concepts of "adjusted GDP" are worthwhile, mainly because of the difficulty of determining the appropriate valuation basis.

The World Bank has published estimates of the percentage by which world Gross 'National' Income (GNI) would be reduced if allowance were made for depletion of energy resources. Over the period 1970-2015, this varies between about 0.1% and 3%. Illustrating the valuation problem, the pattern of change with sharp increases around 1973 and 1979 (times at which OPEC substantially raised the price of oil) suggests that these estimates have been heavily influenced by market prices. Physical depletion of fossil fuel reserves certainly did not change as dramatically as that pattern might suggest (see the chart here, fossil fuel consumption being a reasonable proxy for depletion).


Savings = investment with private money creation?

Say you take out a \$1m loan from ABC Bank - they create two entries on your books:

Assets: \$1m cash available funds.
Liablities: \$1m loan you owe the
bank.

Similarly on their books they create two entries:

Assets: \$1m owing from Simon Lyons
Liabilities: \$1m payment we must honor.

Now you buy a factory with the \$1m. You write a cheque to Bob who you're buying the factory from. He banks with XYZ bank.
Your books:

Assets: A factory worth \$1m.
Liabilities: \$1m loan you owe the bank.

The bank's books:

Assets: \$1m owing from Simon Lyons
Liabilties: We owe \$1m to XYZ bank.

In practise what happens is that the banks are exchanging money between each other all day (for example XYZ bank customers may have paid cheques to ABC bank customers), and at the end of the day they need to settle the difference by taking a loan from someone.
In New Zealand's system for example, that difference can be borrowed from the Reserve Bank (New Zealand's central bank) - paying interest at the Official Cash Rate (OCR). This rate is set by the Reserve Bank - and is the primary means that the Reserve Bank controls inflation.


backshift operator in time series confusion

From your last comment it seems that you might have already figured out how to proceed, although I'm not following some of your reasoning. I just want to point out that there is no need to initialize $y_0$.

The absolute value of ρ is typically assumed to be lower than 1 (otherwise, the series would be "explosive"). Because of that, the terms $ρ^ny_{t-n*k}$ and $ρ^nϵ_{t-n*k}$ will be zero when taking the limit of $n->∞$ in the expansion of $y_t$.

The expression for $y_t$ is
$$ y_t= ∑_{i=0}^{∞}  ρ^ix_{t−i*k}+ϵ_t.$$

This means that the effect of $ρL^k$ is to model $y_t$ from $x_t$, $x_{t-k}$, $x_{t-2k}$, $x_{t-3k}$, and so forth. $k=1$ is just a special case of this expression.

I feel too lazy to plug my scanner and scan my derivation right now, but I'll do so if you really want to see it.

Edited to add scan and explanation of the derivation

Here's a walk-through of the derivation. Hope my handwriting is legible enough; if not, it's the scanner's fault.

(1) shows the equivalent way of writing $y_t$ by applying the
backshift operator $L^k$ (of course, after multiplying everything by $1-ρL^k$ ).
(2) and (3) apply the backshift operator to $y_{t-k}$ and
$y_{t-2*k}$, since each one appears in each one's preceding equation.
(4) substitutes in (1) the expressions obtained from (2) and (3).
Multiplications by ρ are developed in (5). Notice the resulting pairs of
(underlined) terms offsetting each other.
(6) is (5) after removing the underlined/offsetting terms.
The pattern is discernible from (6) and expressed in (7) for $n$.
(8) is (7) but with summation notation for the terms involving $x$.
(9) is the limit of (8) when $n$ tends to infinity. There, the
assumption of |ρ|<1 is essential for convergence of the summation
(as justified in the 2nd paragraph I wrote this morning).

Since $|ρ|<1$, then $|ρ^n|<<1$ for sufficient large $n$. Therefore, the term $ρ^ny_{t-nk}$ in (8) will tend to zero regardless of the value of $y_0$. In other words, the influence of $y_0$ on $y_t$ decreases as $t$ departs from 0.


A differentiation step in a economic growth model

You have most of this; it seems like you might have a miscalculation when differentiating the fraction:
$$
\begin{align*}
\frac{dk_t}{dt} = \frac{d\frac{K_t}{N_t}}{dt} & = \frac{\frac{dK_t}{dt}N_t - \frac{dN_t}{dt}K_t}{N_t^2} \\
& = \frac{\frac{dK_t}{dt}}{N_t} - nk_t
\end{align*}
$$
thus
$$
\frac{\frac{dK_t}{dt}}{N_t} = \frac{dk_t}{dt} + nk_t,
$$
and you can plug this into your equation (4).


Investors in index funds pay lower taxes, but higher taxes on dividends?

Doesn't the second para. beneath contradict itself? Don't the boldened words contradict each other?

No, not necessarily. English is high context language and the passage above is a bit ambiguous but there is no necessary contradiction here.

Despite the higher returns that they earned, investors in the index fund were actually subjected to lower taxes, largely derived from their dividend income

This likely refers to the fact that in the US,  dividends from long term investments are subject to lower capital taxes than let's say capital gains that you get via trading (see article about it here)
The next passage here:

The extremely low costs of index funds consume less dividend income relative to actively managed funds, resulting in higher dividend yields and, therefore, higher taxes on dividends.

when talking about higher taxes likely just refers to the fact that in the US the higher your income the higher income tax you pay. US has progressive capital income taxes (at least on the paper due to loopholes effective tax rates might not always be). In addition since taxes are payed as a fraction of your income even if there would be flat $10\%$ tax on capital income, if your income \$10000 you will pay more taxes than when your income is \$1000, so your tax bill will higher when your income is higher.
This being said again the passage is highly ambiguous so it is not clear what the author means, but it is not necessarily a contradiction.


Question regarding couple economic terms and concepts

Aggregate fluctuations are changes in aggregate variables, for example changes in aggregate supply and demand. If we are talking about shock it is usually implied the event was unexpected but the term is sometimes used loosely.

Idiosyncratic shock is an unsystematic shock, in macroeconomic models it is the shock not correlated with other variables. If you are coming from statistics it’s similar to residuals in OLS - in fact when estimating macro models with econometrics the residuals are often assumed to be the representation of idiosyncratic shocks.

Fat tailed distribution in economics has the same meaning as in statistics - the area in the tails/tail, or more specifically kurtosis and skewness of distribution is relatively large compared to Gaussian distribution.

Granular economy - well this could have different meaning depending on the exact subfield/context. In macroeconomics I saw this terminology being applied to economy that could be described by power law or Zipf law (see this IMF paper). Interestingly many social relationships (city distribution and size, words used in language, firm sizes etc.) follow Zipfian distribution. But I also saw it applied differently in industrial organization but I assume you are looking for something related to macroeconomics.

Good sources to learn some basics/simple concepts in economics are:

khan academy - they have free highs school level micro and macro courses and the quality is good.

Jason Welker - this YouTube channel has also very good high school economics, again for both micro and macro and also for some international economics.

However, from your questions I feel you would more likely need something more advanced (for example I don’t think you will find any discussion of fat tails or probably not even idiosyncratic shocks in the simple sources above). I don’t know any good online source for more advanced concepts but textbooks such as Macroeconomics: A European Perspective from Blanchard Gavazzi and Amighini or for something more advanced the advanced Macroeconomics by Romer are good (you can find pdf versions online). However, when it comes to granularity I did not see that in textbooks much, so there you will probably have to dig into some papers.


As stated already there is no problem if log(x) is negative. However, you would want to be careful if x=0 or if x<0 because then the logs are undefined. So as long as x>0 then there are no problems at all.


If one always spends the same amount, what's one's price elasticity of demand?

The intuition is that if Jessie spends the same amount on $S$ no matter the price $p$, then a % change in price of price will reduce the demand by the same % . The thing that might confuse you is that the formula you are given is an approximation for the true price elasticity of demand given by
\begin{align}
     \eta=\frac{dS\Big/ S}{dp\Big/ p}
 \end{align}
 where $d$ is the derivative operator. The approximation
 \begin{align}
     \frac{\%\Delta S}{\% \Delta p}
 \end{align}
 only works for very small changes in $p$ and $S$.
 To see why the elasticity of demand is $\mid 1 \mid$, let $b$ be the fixed amount spend on $S$. Then the quantity demanded $S$ is given by
\begin{align}
    S=\frac{b}{p}
\end{align}

Assume that there is a small price change from $p$ to $p+\epsilon, \, \epsilon>0$. Then we can define $S_1=\frac{b}{p}$ as the demand before the price change and $S_2=\frac{b}{p+\epsilon}$ as the demand after the price change. Thereby we see that
\begin{align}
    \frac{S_2\Big/ S_1 -1}{\frac{p+\epsilon}{p}-1} = \frac{\frac{b}{p+\epsilon}\Big/\frac{b}{p}-1}{\frac{p+\epsilon}{p}-1}=\frac{\frac{-\epsilon}{p+\epsilon}}{\frac{e}{p}}=-\frac{p}{p+\epsilon}\approx -1
\end{align}
when $\epsilon$ is small enough.


Free alternative to EViews for ARIMA modelling

The standard open source statistical language is R. The R-Project homepage is here. Here is a page about ARIMA in R. If you want to try using R without having to install anything, then I would recommend DataJoy, which provides a free web-based R environment.

Note that R is an entire statistical computing language, so you may find that there is quite a big overhead for learning the language before you can start doing your ARIMA analysis.


How do I find the probability mass function of an individual observation for a multinomial logit model?

A useful way to view the model is as follows. For each observation $i$, there are $K$ different possible observations (choices). So, there is a random variable $\varepsilon_{i,k}$ for each possible choice. We observe which choice is made and the values of the explanatory variables $\boldsymbol X_i$. Then, for a given distribution of the random variables $\varepsilon_{i,k}$ (given parameter values of the Type-1 EV), we know the conditional probability of picking choice $k$ conditional on $\boldsymbol X_i$.

This is explained in the following part of the relevant Wikipedia article:

  It is also possible to formulate multinomial logistic regression as a
  latent variable model, following the two-way latent variable model
  described for binary logistic regression.  This formulation is common
  in the theory of discrete choice models, and makes it easier to
  compare multinomial logistic regression to the related multinomial
  probit model, as well as to extend it to more complex models.

  Imagine that, for each data point $i$ and possible outcome $k$,
  there is a continuous latent variable
  $Y_{i,k}^*$ (i.e. an unobserved random
  variable) that is distributed as follows:

  $$\begin{align} Y_{i,1}^{\ast} &= \boldsymbol\beta_1 \cdot
\mathbf{X}_i + \varepsilon_1 \, \\ Y_{i,2}^{\ast} &=
\boldsymbol\beta_2 \cdot \mathbf{X}_i + \varepsilon_2 \, \\ \cdots &
\\ Y_{i,K}^{\ast} &= \boldsymbol\beta_K \cdot \mathbf{X}_i +
\varepsilon_K \, \\ \end{align} $$

  where $\varepsilon_k \sim \operatorname{EV}_1(0,1),$ i.e. a
  standard type-1 extreme value distribution.

  This latent variable can be thought of as the utility associated
  with data point $i$ choosing outcome $k$, where there is some
  randomness in the actual amount of utility obtained, which accounts
  for other unmodeled factors that go into the choice.  The value of the
  actual variable $Y_i$ is then determined in a non-random
  fashion from these latent variables (i.e. the randomness has been
  moved from the observed outcomes into the latent variables), where
  outcome $k$ is chosen if and only if the associated utility (the
  value of $Y_{i,k}^{\ast}$) is greater than the utilities of
  all the other choices, i.e. if the utility associated with outcome
  $k$ is the maximum of all the utilities. Since the latent variables
  are continuous, the probability of two having
  exactly the same value is 0, so we ignore the scenario. That is:

   \begin{align} \Pr(Y_i = 1) &= \Pr(Y_{i,1}^{\ast} >
 Y_{i,2}^{\ast} \text{ and } Y_{i,1}^{\ast} > Y_{i,3}^{\ast}\text{ and
 } \cdots \text{ and } Y_{i,1}^{\ast} > Y_{i,K}^{\ast}) \\ \Pr(Y_i = 2)
 &= \Pr(Y_{i,2}^{\ast} > Y_{i,1}^{\ast} \text{ and } Y_{i,2}^{\ast} >
 Y_{i,3}^{\ast}\text{ and } \cdots \text{ and } Y_{i,2}^{\ast} >
 Y_{i,K}^{\ast}) \\ \cdots & \\ \Pr(Y_i = K) &= \Pr(Y_{i,K}^{\ast} >
 Y_{i,1}^{\ast} \text{ and } Y_{i,K}^{\ast} > Y_{i,2}^{\ast}\text{ and
 } \cdots \text{ and } Y_{i,K}^{\ast} > Y_{i,K-1}^{\ast}) \\
 \end{align}

  Or equivalently:

   \begin{align} \Pr(Y_i = 1) &=
 \Pr(\max(Y_{i,1}^{\ast},Y_{i,2}^{\ast},\ldots,Y_{i,K}^{\ast})=Y_{i,1}^{\ast})
 \\ \Pr(Y_i = 2) &=
 \Pr(\max(Y_{i,1}^{\ast},Y_{i,2}^{\ast},\ldots,Y_{i,K}^{\ast})=Y_{i,2}^{\ast})
 \\ \cdots & \\ \Pr(Y_i = K) &=
 \Pr(\max(Y_{i,1}^{\ast},Y_{i,2}^{\ast},\ldots,Y_{i,K}^{\ast})=Y_{i,K}^{\ast})
 \\ \end{align}

Edit:
Sometimes you'll see notation like this:
$$
Pr( i \rightarrow k) = \Pr(\max(Y_{i,1}^{\ast},Y_{i,2}^{\ast},\ldots,Y_{i,K}^{\ast})=Y_{i,k}^{*}),
$$
where $i \rightarrow k$ means that individual $i$ chooses choice $k$, is fairly concise once the appropriate notation is defined.
If you're looking for a useful way to put this into a likelihood function, you might see something like this:
$$
P_i = \prod_{k=1}^K Pr( i \rightarrow k)^{\mathbb 1_{i \rightarrow k}},
$$
where $\mathbb 1_{i \rightarrow k}$ is an indicator function that is equal to 1 when $i$ chooses choice $k$ and zero otherwise and $P_i$ is the PMF of the observation associated with individual $i$.

Edit 2:

For an explicit formula, see the same Wikipedia article. Check out, for instance, the section with the following:

  As a result, it is conventional to set $C = -\boldsymbol\beta_K$ (or alternatively, one of the other coefficient vectors).  Essentially, we set the constant so that one of
  the vectors becomes 0, and all of the other vectors get transformed
  into the difference between those vectors and the vector we chose.
  This is equivalent to "pivoting" around one of the $K$ choices, and
  examining how much better or worse all of the other $K-1$ choices
  are, relative to the choice are pivoting around.  Mathematically, we
  transform the coefficients as follows:

   \begin{align} \boldsymbol\beta'_1 &= \boldsymbol\beta_1 -
 \boldsymbol\beta_K \\ \cdots & \cdots \\ \boldsymbol\beta'_{K-1} &=
 \boldsymbol\beta_{K-1} - \boldsymbol\beta_K \\ \boldsymbol\beta'_K &=
 0 \end{align}

  This leads to the following equations:

   \begin{align} \Pr(Y_i=1) &= \frac{e^{\boldsymbol\beta'_1 \cdot
 \mathbf{X}_i}}{1 + \sum_{k=1}^{K-1} e^{\boldsymbol\beta'_k \cdot
 \mathbf{X}_i}} \, \\ \cdots & \cdots \\ \Pr(Y_i=K-1) &=
 \frac{e^{\boldsymbol\beta'_{K-1} \cdot \mathbf{X}_i}}{1 +
 \sum_{k=1}^{K-1} e^{\boldsymbol\beta'_k \cdot \mathbf{X}_i}} \, \\
 \Pr(Y_i=K) &= \frac{1}{1 + \sum_{k=1}^{K-1} e^{\boldsymbol\beta'_k
 \cdot \mathbf{X}_i}} \, \\ \end{align}


Why would an investor choose to invest in one index fund as opposed to another?

There's a lot to unpack here.

It is increasingly the case, especially among passive strategies, that transaction costs and management fees are the primary dimension along which funds are differentiated.  Ignoring those costs seriously distorts the question.
If you invest dollars in the Nikkei that you might otherwise have invested in the S&P500 because you are "bullish on Japan" then you are speculating.  Whether you are engaging in "active management" comes down to how well-researched your position is and how frequently you change your holdings.
There's no "academic consensus" that passive > active.  In fact there is evidence that suggests the reason so-called "active management" has historically unperformed the market is that a lot of fund managers are reactionary. They follow the herd or bend to non-market influences such as pressure from terrified stakeholders, which leads to "buy high / sell low" behaviour. There is a middle-ground that is largely passive, but which actively looks for clear irrationality in the market and acts accordingly.
I would argue that "fundamental analysis" is "fundamentally different" when looking at an index rather than individual stocks, and therefore so too is "active management".  Remember that the fundamental tenet of portfolio theory is to optimize the tradeoff between risk and return, and so if we're ignoring fees then this balance is the fundamental quantity of concern with an index.  There is a sense in which you do not care whether the index holds Nissan; rather, you care whether the index is weighted too heavily in Japanese automakers.

Speaking personally, if I was passively riding S&P500 coverage I might contemplate jumping to the Nikkei if I saw geopolitical reasons to do so, e.g., America fully walls itself off from global trade while Japan signs key deals with Canada, UK and the EU and the markets have failed to price this reality in.  Unlikely to happen, but if it did, it would be a clear reason to move that doesn't require fundamental analysis at all.


A flat AD-curve is not the result of the desire of a central bank to stabilize the price level, it is determined by the behaviour of the aggreggate economy (such as  elasticities and taxes). Stabilizing the price level by using monetary or fiscal policy can be done by shifting the AD-curve (or influencing the slope of the curve, but that is a less standard option). In this context the effectiveness of monetary and fiscal policy depends on how easily policymakers can shift the AD-curve. If you have an equation for AD, this can be easily calculated as $\frac{\partial Y}{\partial X }$, $X$ being the variable in the AD-equation used to implement the policy.

The effectiveness of monetary or fiscal policy also depends on the slope of the AS-curve, possible shifts in the (short run) AS-curve in response to the policy (such as inflation expectations), and the objective of the policy: price stabilization or outputstabilization.


Linearization and the effect of a change

Start from the first order conditions:
$$
\begin{align*}
&F_k = r + (1- p)\\
&F_L = w\\
\end{align*}
$$
Now totally differentiate with respect to $p$:
$$
\begin{align*}
&F_{KK} dK + F_{KL} dL = - dp\\
&F_{LK} dK + F_{LL} dL = \frac{\partial w}{\partial p} dp\\
\end{align*}
$$
This is equivalent to:
$$
\begin{align*}
&dK = \frac{- dp - F_{KL} dL}{F_{KK}}\\
&dL = \frac{\frac{\partial w}{\partial p}dp - F_{LK} dK}{F_{LL}}
\end{align*}
$$
substituting the second into the first gives:
$$
\begin{align*}
&dK = \frac{(- dp)F_{LL} - F_{KL}(\frac{\partial w}{\partial p} dp - F_{LK} dK)}{F_{KK} F_{LL}},\\
\to &\left(1 - \frac{F_{KL} F_{LK}}{F_{LL} F_{KK}}\right) dK = \frac{-F_{LL}dp- F_{KL}\frac{\partial w}{\partial p} dp}{F_{KK} F_{LL}},\\
\to &dK = \frac{-F_{LL} - F_{KL}\frac{\partial w}{\partial p}}{F_{LL} F_{KK} - (F_{LK})^2} dp
\end{align*}
$$
This gives:
$$
\frac{\partial K}{\partial p} = \frac{-F_{LL} - F_{KL}\frac{\partial w}{\partial p}}{F_{LL} F_{KK} - (F_{LK})^2}
$$
The expression for $\frac{\partial L}{\partial p}$ can be derived in a similar way.


You might be looking for the concept of stochastic dominance:

Roughly speaking, first order stochastic dominance of distribution A over distribution B, means that for every possible outcome gamble A pays out weakly more and in at least one state it pays out strictly more. Roughly speaking, second order stochastic dominance means that distribution A has the same risk as distribution B but a higher mean (but not necessarily weakly-higher in every state of the world. Alternatively, you preserve the mean of distribution B and you "scrunch" the probability mass to create distribution A. "All risk-averse expected-utility maximizers (that is, those with increasing and concave utility functions) prefer a second-order stochastically dominant gamble to a dominated one."

The seminal paper is Rules for Ordering Uncertain Prospects (Hadar and Russell (1969))


Whenever we go from nominal to real terms, we need a base year.  As an example, let's use 2016 as the base year.  In the base year, the nominal wage $W$ is always equal to the real wage $w$ (this is true for any price or cost, not just wage).  Also, we always set the price level $P$ equal to 1 in the base year.  That's what makes it the base year.

Anyway, let's say that, in 2016, $w_{2016}=W_{2016}=15$.  Suppose that, between 2016 and 2017, there's 2% inflation - that is, the price level $P$ increases by 2% from $P_{2016}=1$ to $P_{2017}=1.02$.  Then, it follows that the real wage in 2017 will be $w_{2017}=\frac{W_{2016}}{P_{2017}}=\frac{15}{1.02}=14.706$.  I hope this example helped.

In case it didn't, you can think about it like this:  Dividing the nominal wage by the price level is just how you adjust for inflation, thus giving you the real wage.


This reminds me of the Ponzi Scheme: A Ponzi scheme is a fraudulent investment operation where the operator, an individual or organization, pays returns to its investors from new capital paid to the operators by new investors, rather than from profit earned by the operator. Operators of Ponzi schemes usually entice new investors by offering higher returns than other investments, in the form of short-term returns that are either abnormally high or unusually consistent (source wikipedia with numerous examples).


Existence of solution to maximization problem

It is most likely that the statement is in the form of 'A utility maximization problem on a compact set has a solution when the utility function is continuous.' In the one dimensional case, this is a direct corollary of the extreme value theorem. A reference to this theorem can be found at https://en.wikipedia.org/wiki/Extreme_value_theorem

However, discontinuity of the utility function may break this result. Consider the following function.
$$
u(x) =  \begin{cases} x \mbox{ if $x\in[0,\frac{1}{2})$,} \\ \frac{1}{4} \mbox{ if $x \in [\frac{1}{2},1].$   } \end{cases}
$$
It should be immediate to recognize that this function does not have attain a maximum for any $x\in [0,1]$ even though we have a compact set.

A book that I had used before is here which deals with optimization problems of the sort of an economist would encounter in a rigorous way. Also, this book provides a more intuitive way of establishing the main results, which would be a better start.


This Deutsche Bank figure from 2013 has some broad categories for German debt and compares it with other rich countries:

A newer chart, using slightly with more granular categories. Note that Germany is DE.


I understand that when buyers dominate the market the prices go up and when sellers dominate the prices go down (of course this is an oversimplification).

People want to treat trading as some mechanical force. People are buying and selling shares, and the prices reflect where they want to transact. As news comes in, people can change their minds about a company’s value without any transactions taking place.
For example, if a company announces accounting irregularities and an imminent bankruptcy, everybody is going to revise down what they think the shares are worth. Prices will be adjusted lower by everyone, and you could even have more people lining up to buy on the theory that the company might have more value than what is suggested by the price.

I thought then whenever someone sells some shares someone else buys them.

This is always the case, by definition. To deal with you other questions, in an ideal world, if we add up all investor’s shares, the net position is 100% of the amount outstanding.
(The real world is slightly messier. Stock trading does not settle instantaneously, settlement is later. This allows short sellers to “sell” shares that they do not currently own - they borrow them from somebody else before the settlement. This obviously complicates things, but it makes sense once we realise these are credit transactions. A seller is not immediately selling, instead they have entered into a contractual obligation to deliver shares at a future date.)


You're getting two quite different concepts mixed up:

Banks earn a spread on loans for serving as an intermediary between savers and borrowers and for holding credit risk.
The process of borrowing and lending, of banking, leads people in an economy to have more liquid assets (money in a broader sense), than there is underlying cash.

When economists say banks "create money" they don't mean they're printing money off the presses and giving it to themselves as profits.

How banking expands the money supply

To understand banking clearly, its helpful (perhaps 100% required) to understand and use the fundamental equation of accounting:

$$ \mathrm{Assets} = \mathrm{Liabilities} + \mathrm{Shareholders Equity} $$

Let's say Warren Buffet decides to start a bank. He puts in \$10 billion of his own money. The balance sheet of the bank would look like:

$$ \begin{array}{r|rr}\text{Assets} & \text{Liabilities} & \text{Equity} \\ \hline \$10 \text{ cash}& & \$10\end{array} $$

Let's say he now collects \$90 billion worth of deposits from people. These bank deposits are assets for the people, but they are liabilities to the bank (the bank owes the depositors \$90 billion).
$$ \begin{array}{r|rr}\text{Assets} & \text{Liabilities} & \text{Equity} \\ \hline \$100 \text{ cash}& \$90 \text{ deposits} & \$10\end{array} $$

Now let's imagine that the bank lends out $80 billion. The balance sheet would then look like:

$$ \begin{array}{r|rr}\text{Assets} & \text{Liabilities} & \text{Equity} \\ \hline \$20 \text{ cash}& \$90 \text{ deposits} & \$10  \\ \$80 \text{ loans}\end{array} $$

Depositors have \$90 billion in deposits, and the borrowers have \$80 billion in cash. So the economy now has \$170 billion in money where there previously was only \$90 billion.

A key point to realize...

Across the whole economy, all these assets and liabilities cancel out! Borrowing and lending (on its own) is not creating any real wealth. The expansion of the banking system's balance sheet is simply creating more entries on the assets and liabilities side that cancel out. To the extent though that the liabilities of the banking system, deposits etc... are easily tradeable, this does create more money in the economy. More people have liquid assets in the form of bank deposits etc... (which are liabilities of the bank).


Does quantity demanded refer to all consumers or individual consumers?

If it’s a “market demand curve,” that means the quantity demanded is that of the entire market. If it’s an “individual demand curve,” it’s the quantity demanded of an individual consumer. If you open an Econ 101 textbook and look at the supply–demand equilibrium diagram, they’re generally talking about market demand and supply curves.


Do we have some evidence of overshooting?

It depends on what you mean by overshooting.

The original Dornbusch (1976) story is that following a surprise permanent increase in the money supply, the nominal exchange rate "overshoots" its long-run level (the long-run level increases proportionately with the money supply). This is not a directly relevant story today (and I'm not sure that it ever was), because few changes in monetary policy take the form of surprise permanent changes in the money supply.

The mechanism for Dornbusch's "overshooting" is that since the price level takes time to adjust, in the short term there is an increase in the real money supply, leading to a decrease in nominal interest rates. This decrease in nominal interest rates leads the currency to weaken.

If we cut out the first part about the shock to the money supply and just think about the mechanism itself, where (all else equal) lower nominal interest rates lead a currency to weaken relative to its long-term level, then we have a much more relevant and plausible story. Whenever central banks are looser or tighter than expected in policy, you usually see the exchange rate weaken or strengthen, respectively, as a result. This is related to the notion of uncovered interest rate parity, but it's a weaker version: it doesn't say that UIP always holds (which it definitely doesn't), just that it conditionally holds in response to an interest rate shock.

Surprisingly, the international finance literature has been mixed on whether this is true or not, but Michael Kiley has recently shown that conditional UIP seems to hold in response to monetary policy surprises, as identified by monetary policy statement days. I suspect that the previous literature, which was based mainly on VARs without a clear identification strategy, struggled to find a consistent effect due to bad research design.


Literature / Evidence on Cash Hoarding of Firms

I know very little about this, but check out Jake Zhao's work. He's an AP at Stony Brook.

From the abstract,

  My model finds that 63% of the increase in corporate
  cash holdings can be accounted for by the increase in cash flow volatility.
  The increase in cash flow volatility observed in the data arises from a
  decrease in the correlation between revenue and operating expenses.

Accounting for the Corporate Cash Increase.

I imagine the references he includes will also be useful.

Edit: He uses compustat data.


What would be the impact on value of a country's assets and trade balance if its residents choose to increase their savings

An article on this topic:
https://www.investopedia.com/articles/personal-finance/022415/top-10-countries-save-most.asp
Generally economists would say that savings are bad for the growth because they decrease amount of money in the economy. Smaller group would say that savings are good because of higher stability, more thoughtful spending. There is no universal consensus on this question, but we can look at the data.
Countries where people save are usually less stable, get their money from resources. Probably this makes people realise that personal savings may improve stability. And low taxes make people save more too.
Data for europe and household savings:
https://howmuch.net/articles/saving-rates-around-the-world
It seems people save where they can. If they have growing income they save more. If they have large taxes or stagnation they save less. I dont know what will happen if people would change their saving behaivor contrary to the country's situation. If its too sharp change, economy would suffer, as with any change. If it is smooth, it could increase the growth, assuming international trade is significant enough to compensate reduced internal consumption.


Social Welfare and Pareto Optimality in a Bayesian Game

While it is a bit unusual to describe a strategy profile as being Pareto optimal, especially in the context of Bayesian games, I guess you can still define Pareto optimality in different stages of such games as follows. Recall that in a Bayesian game, a pure strategy is a function $s_i:\Theta_i\to A_i$.

  A strategy profile $s(\theta)=(s_1(\theta_1),\dots,s_n(\theta_n))$ is
  ex ante Pareto optimal if  \begin{equation} \mathbb E_\theta [u_i(s(\theta);\theta)]\ge \mathbb
 E_\theta[u_i(s'(\theta);\theta)],\quad\text{$\forall i$ and $\forall
 s'(\theta)$} \end{equation} where the above inequality is strict for
  some $i$. [$\mathbb E_\theta$ means taking expectation over the state
  vector $\theta$.]

  A strategy profile $s(\theta)=(s_1(\theta_1),\dots,s_n(\theta_n))$ is
  ex post Pareto optimal if, given a particular realization of states $\bar\theta=(\bar\theta_1,\dots,\bar\theta_n)$,
  \begin{equation} u_i(s(\bar\theta);\bar\theta)\ge
 u_i(s'(\bar\theta);\bar\theta),\quad\text{$\forall i$ and $\forall
 s'(\bar\theta)$} \end{equation} where the above inequality is strict
  for some $i$.


Nominal GDP versus PPP in comparing military spending of different countries?

One argument for using the exchange rate:

  Market exchange rates are determined by the supply and demand of
  currencies used in international transactions. However, the prices of
  many goods and services on domestic markets are determined in partial
  or complete isolation from the rest of the world. Therefore, the MERs
  do not always accurately reflect differences in price levels between
  countries.

  An alternative is to use purchasing power parity (PPP) conversion
  factors (or PPP exchange rates). The PPP dollar rate of a country's
  currency is defined by the World Bank as 'the number of units of a
  country's currency required to buy the same amount of goods and
  services in the domestic market as a U.S. dollar would buy in the
  United States'. >1  The only PPP rates available for all countries are
  GDP- based basket of goods and services that are major components of
  the gross domestic product. Such GDP-based PPP rates are designed to
  control for differences in price levels and thus to provide a measure
  of the real purchasing power of the GDP of each country.

  Using GDP-based PPP rates instead of MERs for currency conversion
  results in much higher output and expenditure figures for many
  developing and transition countries since they have relatively low
  prices for non-traded goods and services—thus giving the currency
  higher purchasing power. A unit of local currency therefore has
  greater purchasing power within a developing country (which is better
  reflected by using PPP rates) than it has internationally (which is
  what is reflected by using MERs). For those such developing and
  transition countries for whom data was available for 2008, the median
  increase in military expenditure figures from using PPP rates instead
  of MERs was around a factor of 2. Three-quarters of these countries
  would see their relative figures increase by at least two-thirds.
  Meanwhile, using PPP rates would cause the GDP and military
  expenditure figures of most 'developed' countries to fall relative to
  the USA, by a median rate of 17 per cent—reflecting the low value of
  the US dollar at market exchange rates in 2008. However, the
  reliability of such PPP rates is lower than for MERs, since PPP rates
  are statistical estimates, calculated on the basis of collected price
  data for a basket of goods and services for benchmark years. Between
  benchmark years, the PPP rates are extrapolated forward using ratios
  of prices indexes, either GDP deflators or consumer price indexes.
  Like all statistical estimates they are subject to a margin of error.

  Furthermore, GDP-based PPP rates are of limited relevance for the
  conversion of military expenditure data into US dollars. Such PPP
  rates are designed to reflect the purchasing power for goods and
  services that are representative of spending patterns in each country,
  that is, primarily for civilian goods and services. Military
  expenditure is used to purchase a number of goods and services which
  are not typical of national consumption patterns. For example, the
  price of conscripts can be assumed to be lower than the price of a
  typical basket of goods and services, while the prices of advanced
  weapon systems and of their maintenance and repair services can be
  assumed to be much higher. The extent to which this data reflects the
  amount of military goods and services that the military budget can buy
  is not known. Due to these uncertainties, SIPRI uses market exchange
  rates to convert military expenditure data into US dollars, despite
  their limitations.

Stockholm International Peace Research Institute: Monitoring Military Expenditures

But wages are a big part of military costs and those are not paid with PPP GDP but rather domestic GDP. MEASURING HARD POWER: CHINA’S ECONOMIC GROWTH AND MILITARY CAPACITY says that both methods have merits but at least in China, both methods understate military capacity.

  The most appropriate way to compare real military capacity across
  countries is to deflate each country’s actual, or potential, military
  spending by the price of real military services in each country.
  Unfortunately military price indices do not exist for most countries.
  The aim of this paper, therefore, is to develop a simple method for
  computing a relative military price index, that deflates nominal
  spending into units of real military services, using readily
  accessible data. The ratio of two countries’ military price indices
  then gives an exchange rate that provides an index of relative real
  military capacity across countries. We use this military exchange rate
  to compare the level and growth of China’s real military capacity
  relative to the USA. 4 We obtain two key results. First we find that
  the value of the RMB, in terms of its ability to purchase real
  military services, is greater than both its market exchange rate value
  and its PPP value. This suggests that conventional estimates of GDP
  understate China’s real military capacity, but also that PPP exchange
  rate comparisons are much closer to the actual value than market
  exchange rate comparisons. This is mainly because of the very low cost
  of labor in China relative to the USA.
  ...

  If we wish to infer
  country’s miliary capacity based on the size of its economy, however,
  standard PPP indices also pose a problem insofar as they reflect the
  relative price of an average basket of goods produced in the economy
  and this average price index may differ substantially from the price
  of military services. Thus Crane et al (2005) argue that, while
  military services have large personnel costs, a substantial share of
  military equipment purchased by developing-country militaries is
  imported or incorporates com- ponents that are manufactured from
  materials and parts sold at world market prices, such as electronics,
  diesel engines, or aircraft frames. Thus they suggest that, for some
  purposes, GDP measured at market exchange rates may give a more
  realistic picture of changes in military capacity (Crane et al 2005,
  pp.16-17). Similarly, the two principle military statistical
  abstracts, The Military Balance published by The International
  Institute for Strategic Studies (IISS), and the Stockholm Interna-
  tional Peace Research Institute’s (SIPRI) Yearbook report relative
  military spending and relative GDP in terms of $US converted at market
  exchange rates and also at PPP ex- change rates. IISS (2012) notes
  that market exchange rates are likely to understate the true level of
  economic resources allocated towards defence since food material and
  hous- ing costs will be lower in China, but remain noncommittal in
  recommending a preferred price index. 8 Thus they often report a
  weighted average of PPP and market exchange rates when discussing
  China’s real military capacity relative to the USA.


Solving for Feedback Effects / Amplification

We have an equilibrium given by
$$h_1(x,y)=f(x)+z-y=0,$$
$$h_2(x,y)=x-g(y)=0$$.
The implicit function theorem then says that (omitting the arguments):
$$\frac{\partial x}{\partial z}=\frac{-\det\left(
\begin{matrix}
  \frac{\partial h_1}{\partial z} & \frac{\partial h_1}{\partial y} \\
  \frac{\partial h_2}{\partial z} & \frac{\partial h_2}{\partial y}
 \end{matrix}\right)}{\det\left(
\begin{matrix}
  \frac{\partial h_1}{\partial x} & \frac{\partial h_1}{\partial y} \\
  \frac{\partial h_2}{\partial x} & \frac{\partial h_2}{\partial y}
 \end{matrix}\right)},\quad \frac{\partial y}{\partial z}=\frac{-\det\left(
\begin{matrix}
  \frac{\partial h_1}{\partial x} & \frac{\partial h_1}{\partial z} \\
  \frac{\partial h_2}{\partial x} & \frac{\partial h_2}{\partial z}
 \end{matrix}\right)}{\det\left(
\begin{matrix}
  \frac{\partial h_1}{\partial x} & \frac{\partial h_1}{\partial y} \\
  \frac{\partial h_2}{\partial x} & \frac{\partial h_2}{\partial y}
 \end{matrix}\right)}.$$

My source for this is Mathematical Methods and Models for Economists by Angel de la Fuente, although I don't have the book to hand now and can't remind myself of the intuition.

This implies
$$\frac{\partial x}{\partial z}=\frac{g'(y)}{1-f'(x) g'(y)}$$
$$\frac{\partial y}{\partial z}=\frac{1}{1-f'(x) g'(y)}.$$

For the implicit function theorem to hold and this solution to be valid, we need $1-f'(x) g'(y)\neq0$.

More generally, the way this works is as follows: you write down a system of equations whose roots characterize the equilibrium:

$$F_1(\mathbf{x};a)=0,F_2(\mathbf{x};a)=0,\ldots,F_n(\mathbf{x};a)=0$$

(where $a$ is the parameter of interest). From them, we construct the vector-valued function

$$\mathbf{F}(x)=[F_1(\mathbf{x};a),F_2(\mathbf{x};a),\ldots,F_n(\mathbf{x};a)]$$
which has the Jacobian matrix
$$\mathbf J = \frac{d\mathbf F}{d\mathbf x}
= \begin{bmatrix}
    \dfrac{\partial F_1}{\partial x_1} & \cdots & \dfrac{\partial F_1}{\partial x_m}\\
    \vdots & \ddots & \vdots\\
    \dfrac{\partial F_n}{\partial x_1} & \cdots & \dfrac{\partial F_n}{\partial x_m} \end{bmatrix}.$$

To calculate the derivative of $x_i$ with respect to $a$, we construct the modified Jacobian in which we replace the $i^{\text{th}}$ column with derivative WRT $a$ instead of $x_i$. So, for $x_1$ this would look like
$$\mathbf{J}_{x_1}=\begin{bmatrix}
    \dfrac{\partial F_1}{\partial a} & \dfrac{\partial F_1}{\partial x_2} & \cdots & \dfrac{\partial F_1}{\partial x_m}\\
    \vdots & \vdots&\ddots & \vdots\\
    \dfrac{\partial F_n}{\partial a}&\dfrac{\partial F_n}{\partial x_2} & \cdots & \dfrac{\partial F_n}{\partial x_m} \end{bmatrix}.$$

The derivative of interest is then calculated as $$\frac{\partial x_i}{\partial a}=\frac{-\det\mathbf{J}_{x_i}}{\det\mathbf{J}}.$$

We need $\det\mathbf{J}\neq0$ for the implicit function theorem to be valid.


Bayes Correlated Equilibrium with complete information

Question 1

Yes, the BCE induced by a completely informative information structure will look like this.

This is true even though there are other ways to represent fully informative information structures. Think of $T$ as labels. A fully informative information structure should use each element of $T$ to label only one state of the world. That way, when the DM observes the realized label, $t$, they know which is the state. By assuming that $T=V$, a natural way to label each state with a different label is to assign each state their own label, i.e. $P_{T|V}(t|v)=1$ if and only if $t=v$. However, reshuffling the labels is as informative. For example if the label "low" is sent with probability 1 when the state is "high" and the label "high" is sent with probability 1 when the state is "low" the information structure is not in $S^c$, but is also perfectly informative because after receiving the signal (or label) "low" the DM learns that the state is "high" for sure. (There are ways to represent this formally using permutations, or other ways, but I thought it will be more clear with words).

Question 2

the answer is "Not necessarily" for both questions.

1) If the DM is choosing a mixed strategy (as you correctly point out this is possible even with complete information) then $1>P^c_{Y|V}(y|v)>0$ for the actions that the DM is mixing when they learn that the state is $v$.

2) If there is an action that is optimal for more than one state of the world  then $1>P^c_{V|Y}(v|y)>0$ for the states for which $y$ is optimal.

In many papers, people assume that each action is strictly better for one and only one state of the world. In that case, your two statements are true.

Question 3

Yes it is true:

Claim: a BCE that satisfies that $P_{Y,V}(y,v)>0$ for all $(y,v)\in Y\times V$ cannot be induced by a completely informative signal.

Proof: Proceed by contradiction. Suppose a BCE, $P_{Y,V}^*$, satisfies the restriction and is induced by a completely informative information structure.

Consider some state, $v_0$, for which action $y_0$ is not optimal. Note that if there is no such a state, then the decision problem would be trivial since all actions would be optimal for all states of the world. I assume that such a pair exists.

Now, without loss of generality, we can assume that the completely informative structure is the signal defined as $S^c$.

By assumption $P_{Y,V}^*(y_0,v_0)=\sum_{t\in T}P_{Y|T}(y_0|t)P_{T|V}^c(t|v_0)P_V(v_0)>0$, therefore at least one of the summands must be strictly positive. However, $P_{T|V}^c(t|v)=0$ for all $t\neq v_0$. Further, for $t=v_0$, it must be that $P_{Y|T}(y|v_0)=0$ since $y_0$ is not optimal given $v_0$.

We conclude that $P_{Y,V}^*(y_0,v_0)=0$, i.e. a contradiction!


What exactly is Foreign Currency Indexation?

A currency can be indexed but so can a bond. An indexed bond is one where the coupons and principal are indexed to another currency (the U.S dollar in the case of Tesobonos) at the spot rate in effect at the time of issuance. So you actually pay in the local currency but how much you pay depends on the prevailing exchange rates when payment is due.

A bond can be indexed without a currency being indexed. It is typically a way, much like a foreign currency denominated debt issuance, to remove local inflation and exchange risk from debt issuance.


You need the last part because you want the present value. If you were to only use the 16 you‘d get to 4 years in the future, or you‘d be calculating the PV for 4 year operating cost of 2000 and concurrent 16 year cost of 2700. without the last part you‘d basically have two operating costs (one of 2000 and one of 2700) for the next four years and then 1  operating cost of 2700 for the next twelve years.

This is the way you can think of the cost in more granular terms

Purchasing cost: $15k$ no discount because purchased today.

1st four years overhead:
$
\sum^4_t{2'000\over(1+r)^t}
$

2nd 16 years of overhead: $ {\sum^{16}_t{2'000\over(1+r)^t}}\over(1+r)^4 $

The first part gives you the PV of the 16 periods, but because this would only start 4 years in the future you'd need to take the PV of the PV in a way.

The total of the cash flow is then the following:
$15'000 + \sum^4_t{2'000\over(1+r)^t} +{{\sum^{16}_t{2'000\over(1+r)^t}}\over(1+r)^4}$

Alternatively you could write
$15'000 + \sum^4_t{2'000\over(1+r)^t} +{{\sum^{20}_{t=4}{2'000\over(1+r)^t}}}$

Hope this helps.


Solving Euler Equation for standard DSGE Models

It really depends on the problem. As to why it would disappear, sometimes you can express the multiplier in terms of variables at the present or past, as in $\lambda_{t+1}=\lambda_{(c_t,n_t,M_t)}$ , which means it's an expectation over values that are known at that time. Therefore, you have $E_t[\lambda_{t+1}]=E_t[\lambda_{(c_t,n_t,M_t)}]=\lambda_{(c_t,n_t,M_t)}=\lambda_{t+1}$

But, as said above, it depends on the problem.


What is the A in the Accumulation and Amount functions for the value of investments?

Usually one can think of $a(t)$ as some form of interest.

So, we can normalize some initial investment to 1 so that:
$$a(0) = 1 $$

letting $k$ denote some initial investment, we can have that the value of $k$ at time $t$ is give by your formula:

$$A(t)= ka(t)$$

So, for example, if you invest k=10, assume an interest rate $i=5$% and compounding interest, then ten periods after you invest your money, $k$ is worth:

$$A(10,k) = 10*(1+.05)^{10}$$


Are imperfect-information games in extensive form exactly the same as sequential incomplete-information games?

Two points:

An Incomplete-Information Game is not well-defined without specifying the beliefs of the players. This is specially relevant if you want to use the standard Bayesian equilibrium techniques. There is still significant leeway here - you may define subjective probabilities for each players, players may have inconsistent beliefs (look at Yildiz's excellent papers in a bargaining setup), etc. Recently, there have been some papers that deal with situations where a player is uncertain about the beliefs held by other players (see Gab Caroll's Robustness and Linear Contracts for a brilliant exposition)- but you may need to develop additional tools (like updating rules under ambiguity - see Faruk Gul's paper) to study these situations.

As for Harsanyi's transformation, the critical component of his 1967-68 paper is the common prior assumption (so every player has consistent beliefs regarding the uncertain elements of the game. Thus Harsanyi states that any incomplete information game with consistent beliefs can be remodelled as a game on imperfect information by introducing Nature's move at the beginning. There may have been extensions to the result in the past 30 years, but Im not aware of it.


Total factor productivity (TFP) estimation in R

Following the prodtest manual they define omega as residuals from the model, that is
$$\omega_{it} = y_{it} − (\alpha + w_{it} \beta  + k_{it}\gamma)=y_{it}-\hat{y}_{it}$$
estprod does not have any comprehensible documentation but one way how we can get the above would be to extract the coefficients from the model and then calculate $y-\hat{y}$ manually.
We can do that with the broom package:
install.packages("broom")
require(broom)
coefs<-tidy(mod1)
# A tibble: 3 x 3
  statistic   bias std.error
      <dbl>  <dbl>     <dbl>
1     -3.71   4.77      2.35
2     10.5  -11.8       4.26
3      1.73  -2.14      1.64

the names are off but the first column gives the same coefficients as when I inspect the model with summary so they are correct.
Next we can just can just calculate $y-\hat{y}$ as
omega_1 = (data$Y - data$Labor*coefs$statistic[1] - data$Capital*coefs$statistic[2] - data$Materials*coefs$statistic[3])

the code is inelegant but it should do the job.
As a side note it might be worth while considering to estimate the TFP without these packages. Normally packages are the R's biggest strength but in this case estprod does not have any good accompanying documentation or manual and the prodest does not have the possibility to add the gross output yet as in the link the devs mentioned. In cases like this it might be better to just program your own function.


While in GMM one uses theoretical analytical moments, in MSM one uses simulated theoretical moments instead.

For GMM,

  [t]he method requires that a certain number of moment conditions were specified for the model. These moment conditions are functions of the model parameters and the data, such that their expectation is zero at the true values of the parameters. The GMM method then minimizes a certain norm of the sample averages of the moment conditions. (Wikipedia)

Crucially,

  In order to apply GMM, we need to have "moment conditions", i.e. we need to know a vector-valued function $g(Y,\theta)$ such that $m(\theta_{0})\equiv \mathbb{E}[\,g(Y_{t},\theta _{0})\,]=0$.

For SMM, values of functions of the data and the parameters are too difficult to calculate analytically. Essentially, we cannot calculate what the moments of the data are given the parameters; for example, we cannot calculate $\mathbb{E}(XY)$. Instead we simulate a large sample of $X$s and $Y$s and calculate $\frac{1}{n}\sum_{i=1}^{n}X_i Y_i$ (simulated sample, not the actual sample) to replace the theoretical $\mathbb{E}(XY)$. The rest is the same.


How do I derive Hessian of this function and check for concavity?

As @Herr K. stated, the beginning point is being able to take a derivative. The Hessian is a matrix equivalent to a second order derivative sometimes denoted as $\nabla^{2}$. Start by finding the gradient, $\nabla$ which is a vector of first order derivatives of every variable in the function. In your case, this would be
$$ \nabla=
  \left[ {\begin{array}{cc}
   \frac{\partial{f}}{\partial{K}}  \\
   \frac{\partial{f}}{\partial{L}} \\
  \end{array} } \right] =
  \left[ {\begin{array}{cc}
   aAK^{a-1}L^{b}  \\
   bAK^{a}L^{b-1} \\
  \end{array} } \right] $$
The Hessian, would be the next (second) derivative of each of the above with respect to both variables and it takes the form:
$$ \nabla^{2}=
  \left[ {\begin{array}{cc}
   \frac{\partial^{2}f}{\partial{K}^2} & \frac{\partial^{2}f}{\partial{K}\partial{L}} \\
   \frac{\partial^{2}f}{\partial{L}\partial{K}} & \frac{\partial^{2}f}{\partial{L}^2} \\
  \end{array} } \right] $$
Using whatever this would be in your case (left to you to compute) you can take the determinant of the resulting matrix and, with values for your parameters $a$ and $b$ (or at least their signs like with the other variables and parameters) you can determine whether the determinant is positive or negative and determine the concavity. If it is positive definite or positive semi-definite, this would imply either strict or non-strict convexity, respectively. If you don't know how to take a determinant it is simply dont by using a matrix say

$$ A=
  \left[ {\begin{array}{cc}
   a_{1} & a_{2} \\
   a_{3} & a_{4} \\
  \end{array} } \right] $$
and then taking its determinant using the operation (without explanation for sake of brevity):
$$ A=
   {\begin{vmatrix}
   a_{1} & a_{2} \\
   a_{3} & a_{4} \\
  \end{vmatrix} } = a_{1}a_{4}-a_{2}a_{3} $$
Using this with the above, you should be able to figure the rest out and figure out whether or not its concave! (hint: for $a,b\geq 0$ and $a+b\leq1$ it is concave)


Adjusting a ratio variable to constant 2000 dollars using a deflator index

Typically if you have ratio variable then you don't need to deflate the series. This is because for example tax revenues are nominal and gdp is nominal in the ratio the respective price series would cancel out (assuming they are from the same year).

So if nominal tax revenues are RN=RP and nominal GDP is GDPN=GDPP  (where P is the price) then the ratio is RN/GDPN=R/GDP. That way if you choose the first option you suggested then you will get the exact same value as the original ratio (try it if you want).

I hope this helps.


Let $b_i\in\{0,B\}$ be $i$'s strategy. Then $i$'s payoff depends on the strategy profile $(b_i,b_{-i})$, where $b_{-i}=(b_j)_{j\ne i}$.

\begin{equation}
u_i(b_i,b_{-i})=
\begin{cases}
v_i&\text{if $b_i=0$ and $b_j=B$ for some $j\ne i$}\\
0& \text{if $b_i=0$ and $b_j=0$ for all $j\ne i$}\\
v_i-B&\text{if $b_i=B$}
\end{cases}
\end{equation}

Thus, $b_i=B$ is a best response if
\begin{equation}
u_i(B,b_{-i})\ge u_i(0,b_{-i})
\quad\Leftrightarrow\quad
v_i-B\ge (1-\Pr(b_j=0,\;\forall j\ne i))v_i.\tag{1}
\end{equation}

Since the game is ex ante symmetric, we could further assume that every $i$ adopts a threshold strategy, i.e.
\begin{equation}
b_i=\begin{cases}
0&\text{if $v_i\le\overline v$}\\
B&\text{if $v_i>\overline v$}
\end{cases}\tag{2}
\end{equation}
where $\overline v$ is some common threshold value. Then, the probability in $(1)$ can be written as
\begin{equation}
\Pr(b_j=0,\;\forall j\ne i)=\Pr(v_j\le \overline v,\;\forall j\ne i)=(\overline v)^{n-1},\tag{3}
\end{equation}
where the last equality is obtained from the assumption that $v_j$'s are i.i.d. and $v_j\sim U[0,1]$.

Consolidating $(1)$ to $(3)$, we can solve for the cutoff value $\overline v=B^{1/n}$.


Do privately owned banks ever indefinitely lend money to their owners?

Bank regulations vary by country, but it is a safe bet that they would frown upon having such a concentrated loan portfolio. Furthermore, there are presumably restrictions on lending to related parties, but how that would be defined would definitely vary by jurisdiction.

By doing a quick search on the internet, it was possible to find this FDIC letter about concentration risk. Link to FDIC page On that page is a link to a PDF which indicates that concentration risk is defined as 25% of capital. Unless the owners can get others to buy capital of the bank (which is unlikely if the only business plan of the bank is to lend money to the owners), the owners can only lend themselves back 25% of the money they put in before the regulators shut them down. (And in practice, I doubt that regulators would even let them get close to 25% of capital for related party lending.)

From a business perspective, it's not as if deposits can be attracted for free. The bank has to either pay interest for deposits, or else has to offer services (branches, ATMs, lending) which requires employees and cost money. A bank could not survive without having a lending portfolio that is paying a market rate of interest.

Finally, the bank needs equity, provided by the owners. Any sweetheart lending deals the bank offers the owners would reduce the profits on their equity. If the bank does not have a credible business plan, it is not going to attract other investors to supply capital needed for regulatory purposes.

The Savings and Loan Crisis in the United States offers a lot of examples of the breakdown of the regulation of small banks, but even then, the self-dealing was generally hidden.


Maybe this helps. One method to do this is to use least squares (as was mentioned by @cc7768).
Matlab does this by default when using the \ operator.
This is also useful because it's sensible even if the system is overidentified as in the following example:

A = [1 22; 2 100; 7 6]
b = [1 2 5]'
A\b

produces

ans =

    0.7108
    0.0061

. In Python, this can be done with

import numpy
from numpy import linalg
A = numpy.matrix("1 22; 2 100; 7 6")
b = numpy.matrix("1 2 5").T
linalg.lstsq(A,b)[0]

which gives

matrix([[ 0.71084144],
        [ 0.00611577]])

.


There are no numbers to support the problem specific to Ferguson because it happened recently, and not enough time has passed to assess any effect.

The inflammatory rhetoric of the site you linked to probably stems from this paper http://www.nber.org/papers/w10243 that tries to quantify the effects of the race riots in the 1960s. The authors themselves concede the speciousness of their conclusions by acknowledging that the data they used are not detailed enough to identify the precise mechanisms at work.

A more thorough study that examined the riots in India between the Hindus and Muslims concluded that the motivation for the riots was primarily economically based. Where there was economic growth, riots decreased. Moreover, rioting in one region had no effect on the rioting of neighboring regions. https://openknowledge.worldbank.com/handle/10986/5432


Economic implications of allowing competition with essential services i.e police, fire department, etc

As already suggested in the comments of the OP, we find cases where the provision of essential services behave much like any other market when privatized.
As suggested by Henry it was found that in the case of fire fighters, they simply let the homes burn down. On the London Fire Brigade's Website they discuss the issue in question with great detail.

People paid a fee to an insurance company to insure their property against damage. The most common risk at this time was fire. The first insurance company was named Phoenix, after the Greek mythological bird that rose from the ashes, and was established by Nicolas Barbon.
Building insurance was very profitable and many more insurance companies were set up with the insurance companies establishing their own fire brigades. These brigades were sent to insured properties if a fire occurred to minimise damage and cost.
Firemarks were used to identify - and advertise - different insurance companies. They were placed on the outside of an insured building and brigades would use these firemarks to determine whether a building was insured by them. If a building was on fire, several brigades would attend; if they did not see their specific firemark attached to the building, they would leave the property to burn. You can still see some old firemarks on buildings today.

Similarly we in 2010 there was a case of Tennessee firefighters letting a home burn when the owner didn't pay fire-protection fee. Its clear that if an essential service is allowed to be privatized, they will function like any other business.


Should the seed be set in forecasting returns in a MCMC

My understanding is that the standard advice is to randomly select a seed value, and then keep that seed for your entire analysis. This allows the same computer code to give identical results each time.

See, for example, the Stata manual:

  Stata’s random-number generation functions, such as runiform() and
  rnormal(), do not really produce  random  numbers.  These  functions
  are  deterministic  algorithms  that  produce  numbers  that can  pass
  for  random. runiform() produces  numbers  that  can  pass  for
  independent  draws  from  a rectangular distribution over[0,1);
  rnormal()produces numbers that can pass for independent draws from
  N(0,1).  Stata’s random-number functions are formally called
  pseudorandom-number functions.The  sequences  these  functions
  produce  are  determined  by  the  seed,  which  is  just  a  number
  andwhich  is  set  to  123456789  every  time  Stata  is  launched.
  ... If you record the seed you set, pseudorandom results such as
  results from a simulation or imputed values  from mi impute can  be
  reproduced  later.  Whatever  you  do  after  setting  the  seed,
  if  you  set the seed to the same value and repeat what you did, you
  will obtain the same results ... It does not really matter how you
  set the seed, as long as there is no obvious pattern in the seeds that
  you set and as long as you do not set the seed too often during a
  session

However, if you are using parallelization to do many separate simulations then you need to make sure that you do not use the same seed, or a time based seed, on every node:

  Properly seed your generator. Even the state-of-the-art Mersenne
  Twister ran into problems early on because the authors had neglected
  the issue of proper seeding....This rule is VITAL if you are going to
  run parallel simulations on a Beowulf cluster for example.The simplest
  way to seed a RNG is to take something like the current time e.g.
  using the time() function found in Unix and most C libraries, which
  returns a 32-bit integer giving the number of seconds since 1st Jan
  1970. ...Hundreds of jobs on different nodes will be starting at almost exactly the same time –therefore many of your jobs will be
  starting with exactly the same seed and therefore those that have the
  same seed will generate exactly the same results(assuming your code
  has no bugs in it).

Good Practice in (Pseudo) Random Number Generation for Bioinformatics Applications


Consumer choice: How could a consumer weakly prefer A to B *and* B to A?

No, weak orderings are not mutually exclusive, but strict orderings are, exactly as for the usual comparison of real numbers: you can have $x \geq y$ and $y \geq x$ (if $x=y$), but $x>y$ and $y>x$ are incompatible with each other.

The weak orderings $A \succeq B$ and $B \succeq A$ mean that the decision-maker is indifferent between $A$ and $B$: he might therefore choose any of these options indifferently.


What benefits do OPEC countries obtain by cutting oil supply?

The OPEC scenario is quite well described by a market with Cournot Competition. That is, while collusion would lead the highest total profits to the sum of the participants, each invididual participant would gain by increasing his production a little bit.

Without observing the quantity of each participant and proper enforcement, that's the outcome the model predicts - and reality concurs.


Sometimes it helps writing it down, I think I figured it out:

I forgot the interest between September and January
At 6.25%, that's around 2.1% for four months
That roughly matches the discrepancy I've noticed, with the conversion factor that could very well check out

So, problem solved, I guess.


Formula for feasibility for production versus import

Really, when considering whether importing a good is beneficial to a country we need only look at comparative advantage. For this concept however, we need information on another good which could be produced in each country. Consider the following example:

Suppose we have country A and country B. Both countries can produce only two goods (for simplicity). These goods are gizmos and doodads. Each country has 100 labor hours to spend on production. It costs country A 10 hours to produce a gizmo and 5 hours to produce a doodad. It costs country B 15 hours to produce a gizmo and 6 hours to produce a doodad.

Using this information, we can see that country A can produce both gizmos and doodads more efficiently than country B. This is what we call absolute advantage. A country has absolute advantage in producing a good if it takes less resources for that country to produce the good.

Now, just because country A has an absolute advantage in producing both goods does not mean that they cannot benefit from trade. In order to see how they can benefit, we must look at the opportunity cost of producing each good for both countries. We will consider the case where opportunity costs are constant.

$\textbf{For country A:}$

If they want to produce a gizmo, they must give up the opportunity to produce 2 doodads. Therefore the opportunity cost is 2 doodads. If country A wants to produce a doodad, they must give up the opportunity to produce 0.5 gizmos. Therefore the opportunity cost is 0.5 gizmos.

$\textbf{For country B:}$

If they want to produce a gizmo, they must give up the opportunity to produce 2.5 doodads. If they want to produce a doodad, they must give up the opportunity to produce 0.4 gizmos.

Now, we can see that the opportunity cost for producing gizmos is lower in country A, and the opportunity cost for producing doodas is lower in country B. This means that country A has a comparative advantage in producing gizmos and country B has a comparative advantage in producing doodads. On a side note, this situation will always be the case. If we are looking at a two-country-two-good world, each country will have comparative advantage in production of one good.

What has all of this analysis told us? Well, it has told us that each country would be better off if they produced only the good for which they have comparative advantage in production and traded for the other good.

This is, of course, a very simplified example, but the same basic concept applies when you start adding more countries and more goods. I hope this helps!


What does the value of bitcoins depend on?

The price is unstable because there's a fairly small free float (the quantity available for buying and selling with true currencies) compared to the level of transactions.

It's a Ponzi scheme, so demand is driven by Bitcoin marketing: it is in immediate financial interest of the holders of Bitcoin to drive up the price by "ramping" it: talking up the market price. The amount of attention that the media gives Bitcoin goes up and down - this affects demand, and that drives the price, as supply is fairly stable.

The originators of Bitcoin, and the original hoarders, have most to gain: it was easiest for them to create Bitcoins. It would be rational for them to slowly release their hoard into the market as price goes up.

It's also attracted a lot of gullible but evangelical supporters: for them, continuing to hold Bitcoin is a means of validating their beliefs, and even validating the identity they've built for themselves around it. That's why the free float tends to stay low.


How does monopoly affect the informational role of price?

A foundational result from models of perfect competition is that the competitive market equilibrium (i.e. setting price and quantity such that supply = demand) is efficient. By efficiency we mean that the sum of consumer surplus and producer surplus are maximised.

This is actually quite an amazing thing! It means that each individual consumer buys the good if and only if the welfare he receives from doing so is greater than the marginal cost of producing the good. To see that this must be true, suppose that there were a consumer who values a good at more than the cost of production but did not buy it. Then it would be possible to increase welfare by having the consumer buy the good, contradicting the result that the equilibrium is optimal. Similarly, efficiency demands that sellers produce exactly as much as is necessary to sell to the consumers who will ultimately buy.

How could we get to this efficient point? One way would be to have economists calculate what is efficient and then publish this information so that people could look-up whether they are supposed to buy/sell or not. But wait! We know that there aren't economists running round telling consumers whether it would be efficient for them to buy or not. So how do people figure this out for themselves?!

The answer is that the equilibrium price contains all of the information that they need to figure out what is efficient. Since the equilibrium price is equal to marginal cost, and since consumers will buy if their willingness to pay is above the price, a consumer wishes to purchase if amd only if his willngness to pay is bigger than the cost—i.e. exactly when it is efficient for him to do so. Thus, one way to think about the role of a price in a competitive market is that it transmits information to market participants about the efficiency of a trade they are contemplating.

For a monopoly things are different because the monopolist breaks the direct relationship between price and marginal cost by setting $p>MC$. This means that some consumers who have a willingness to pay that satisfied $p>WTP>MC$. The fact that $p>WTP$ means the price is signalling to these consumers that they should not buy, even though it would be efficient for them to do so (because $WTP>MC$). In other words, the price is no longer directly informative about the efficiency of a trade, so one way to think about the effect of monopoly on prices is that it reduces their information content.

ADDENDUM:

It's still possible to calculate cost—and hence the efficient allocation—from the monopoly price, but we need more information. In particular, the monopoly price satisfies $$\frac{p-c}{p}=-\frac{1}{\eta},$$ where $\eta$ is the price elasticity of demand and $c$ is marginal cost. Thus, to figure out marginal cost we need know the price and the elasticity, whereas in a competitive market we only needed to observe the price.


Is fisher equation a definition, identity? Or is it rather a very good estimate

The fisher equation has its basis in the fact that the real return on an asset is the nominal return divided by the inflation rate. If you hold a bond today, it gives you back $1+r_{t+1}$ tomorrow. This is basically $\frac{1+\iota_{t}}{1+\pi_{t+1}}$ such that the promised nominal rate is deflated by the inflation rate. Rearranging terms, you get your definition. This is exact and always holds conceptually. The approximation is that $i\approx r+\pi$  which is obtained via a Taylor series expansion.


Can Eurozone countries issue their own bonds?

Issuing bond is indeed the competence of the individual states.

For example, you can buy 10 year bonds both from Italy and Germany, both of which are Eurozone countries:
https://www.investing.com/rates-bonds/de-10y-vs-it-10y


What does it mean if country X has a higher GDP at Purchasing Power Parity (PPP) than country Y?

I like this explanation from Callen (2007), published in the IMF's F&D, their quarterly magazine.  For measuring consumption in a country, PPP is probably the way to go, but international economic and financial power is probably better measured with market rates.

PPP versus market rates
So which method is better? The appropriate way to aggregate economic
data across countries depends on the issue being considered. Market
exchange rates are the logical choice when financial flows are
involved. For example, the current account balance represents a flow
of financial resources across countries. It is appropriate to use the
market exchange rate to convert these flows into dollars when
aggregating across regions or calculating the global current account
discrepancy. But for other variables, the decision is less clear cut.
Take real GDP growth. International organizations use different
approaches. The World Bank uses market-based rates to determine the
weights in its regional and global aggregations of real GDP, whereas
the IMF and the Organization for Economic Cooperation and Development
use weights based on PPP rates (although the IMF also publishes a
global growth aggregate based on market rates in the WEO). Each
methodology has its advantages and disadvantages.
Advantages of PPP
A main one is that PPP exchange rates are relatively stable over time. By contrast, market rates are more
volatile, and using them could produce quite large swings in aggregate
measures of growth even when growth rates in individual countries are
stable. Another drawback of market-based rates is that they are
relevant only for internationally traded goods. Nontraded goods and
services tend to be cheaper in low-income than in high-income
countries. A haircut in New York is more expensive than in Lima; the
price of a taxi ride of the same distance is higher in Paris than in
Tunis; and a ticket to a cricket game costs more in London than in
Lahore. Indeed, because wages tend to be lower in poorer countries,
and services are often relatively labor intensive, the price of a
haircut in Lima is likely to be cheaper than in New York even when the
cost of making tradable goods, such as machinery, is the same in both
countries. Any analysis that fails to take into account these
differences in the prices of nontraded goods across countries will
underestimate the purchasing power of consumers in emerging market and
developing countries and, consequently, their overall welfare. For
this reason, PPP is generally regarded as a better measure of overall
well-being.
Drawbacks of PPP
The biggest one is that PPP is harder to measure than market-based rates. The ICP is a huge statistical undertaking,
and new price comparisons are available only at infrequent intervals.
Methodological questions have also been raised about earlier surveys.
In between survey dates, the PPP rates have to be estimated, which can
introduce inaccuracies into the measurement. Also, the ICP does not
cover all countries, which means that data for missing countries have
to be estimated.


Interpretation of the growth rate of consumption in a simple one-good model

I got stock with this problem for days.

The fact is that in a statical analysis, increasing $Q_K$ reduces consumption. But in a dynamic analysis, increasing $Q_K$ means more capital is saved and hence next year an higher production function that allows for more consumption than today, hence, a positive growth rate of consumption.