Are democracies more economically productive than autocracies?

In economics, it is accepted that countries with good 'inclusive' institutions, such as strong property rights, are more productive and able to develop faster (or even develop at all) than countries with bad 'extractive' institutions, such as forced labor (see Acemoglu 2008, Acemoglu & Robinson 2000a, 2000b, 2001, 2006, 2008; Olson 1984, Bates 1981, 1983, 1989 and sources cited therein*)
Democracies do  adopt inclusive institutions more often than dictatorships but not always (e.g. see Lindert 2004, 2009 + sources from previous paragraph). Furthermore, the ability of country to develop and become more productive and prosperous, or fall of a country in terms of prosperity and productivity may not only depend on a single institution but rather whether country predominantly has inclusive or extractive institutions (although some inclusive institutions like property rights are more important than others).
Consequently, it would be fair to say that it is accepted that democracies tend to be  more economically prosperous, but it is all conditional on the institutions they adopt. For example, a Venezuela is de jure a democracy but in indexes that measure strength of property rights it gets very low scores (e.g. see evolution of their property rights index (0-100) between 1995-2020). On the other hand consider China that is clearly a dictatorship both de jure and de facto, but their property rights index score between 1995-2020 got better.
To sum it up, given the above it would be fair to say that the view that democratization leads to better economic outcomes**  has some merit but with caveats. It is not that democracies are more productive by virtue of being democracies, it is thanks to their better (inclusive) and more robust institutions and in principle dictatorships could adopt the same institutions, but they tend not to. This is because while the extractive institutions might be bad for a country as a whole, as their name suggest, they help political leaders to extract economic resources for themselves.

* Note this research is also summarised in Acemoglu & Robinson: Why Nations Fail? which is book that is written in a way that is more accessible to laymen readers than the papers mentioned above).
 ** Meaning, among other things, government can often in the end collect more revenue even with lower tax rates because tax bases critically depend on size of output being taxed (e.g. $10\%$ tax on tax base 1,000,000€ gives government more revenue than $50\%$ tax on tax base 1,000€).


Objections to total surplus as tool for welfare measurement

This is an answer to the closely related question of "why don't competition/antitrust authorities aim to maximise total welfare rather than consumer surplus?" There are a number of reasons for this:

If there are a small number of firms participating in a market then they have a very concentrated interest in that market and therefore have a large incentive to lobby for favourable treatment by a regulator. By contrast, it is likely that only a relatively small fraction of a consumer's income is spent in any one given industry and, therefore, each individual consumer has less interest in lobbying for consumers' interests in any antitrust proceeding. This leads to producers having a louder voice (a problem known as regulatory capture).
The total welfare and consumer surplus effects of a behaviour often point in the same direction (e.g. lower prices and higher quantities typically correspond to both higher consumer surplus and higher total welfare). If measuring both consumer and producer surplus is costly then measuring only consumer surplus will often be a sufficient heuristic.
There are information asymmetry problems insofar as firms have better information about their expected profit gains from a practice than do regulators. Producers are likely to overstate the private benefits of anti-competitive practices and the regulator would need to find a mechanism to induce truthful reveltation.

However, these points must be weighed against

Ignoring producer surplus understates the benefits of a practice and will lead to the  prohibition of some practices that increase total welfare.
Firms are owned by people (i.e. consumers) so that the distinction between consumer surplus and producer surplus is, in some sense, a false dichotomy.
We should give firms enough profit to (a) cover the fixed costs of operating in an industry and (b) ensure that they have an incentive to innovate and be successful. If we insist on maximising consumer surplus by driving price to marginal cost then neither (a) nor (b) will be satisfied.


Rule of thumb: if the changed quantity does not appear on an axis of the graph, its effect (if any) will be a shift of the curve, rather than movement along the curve.

For example, an AD-AS graph has price and quantity on the axes. Interest rate does not appear on the axes, so an increase in the interest rate (say) could be associated with a decrease in the money supply, resulting in a decrease in demand that shows up as a shift of the aggregate demand curve to the left.


Effect of exogenous parameter on three player SPNE profits

Let's solve by backward induction.

In stage 3, player $A_3$ solves:
$$
\max_{p_1, p_2} u_3(p_1, p_2, q_1, q_2, x).
$$
Given the assumptions, $A_3$ will set: $p_1 = \overline{p}$ and in general $p_2$ will be a function of the $q_1, q_2$ and $x$, denoted by $p_2(q_1, q_2, x)$. For a solution to exist, you have to assume that $u_3$ is not unbounded in $p_2$.

As $u_2$ is independent of $p_1, q_1, x$, In stage 2, Player $A_2$ solves:
$$
\max_{q_2} u_2(q_2, p_2(q_1, q_2, x)).
$$
$u_2$ is increasing in $q_2$ and $p_2$. If $p_2(q_1, q_2, x)$ is increasing in $q_2$ then $A_2$ will set $q_2$ as big as possible, so there is no optimal solution. Else, there might be some optimal level of $q_2$, say $q_2(q_1, x)$.

In stage 1, player $A_1$ solves:
$$
\max_{q_1} u_1(q_1, \overline{p}).
$$
As $u_1$ is increasing in $q_1$ there will probably be no solution to this. So likely this problem is unbounded. If we assume that $q_1 \le \overline{q}_1$ for some fixed value $\overline{q}_1$, then $A_1$ will set $q_1 = \overline{q}_1$.

Going back to the utility of player 3, we therefore see, after substituting:
$$
u_3 = u_3(\overline{p}, p_2(\overline{q}_1, q_2(\overline{q}_1, x),x), \overline{q}_1, q_2(\overline{q_1}, x), x).
$$
It seems like the change in $u_3$ due to a change in $x$ is ambiguous. As it also influences the optimal choice of $q_2$ (of player $A_2$), and $p_2$ of player $A_3$.


Implementing the "affine transformation", let $$u(x) = A-B\exp\{-\lambda x\}$$

Then we want to solve $(x_1 = 2400, x_2 = 5000)$

$$A-B\exp\{-\lambda x_1\} = \frac 12 \Big[A-B\exp\{-\lambda x_2\}\Big]+\frac 12 \Big[A-B\exp\{-\lambda \cdot 0\}\Big]$$

$$\implies A-B\exp\{-\lambda x_1\} = A - \frac 12B - \frac 12B\exp\{-\lambda x_2\}$$

$$\implies \frac 12B -B\exp\{-\lambda x_1\} + B\frac 12\exp\{-\lambda x_2\} =0$$

$$\implies -\frac 12 +\exp\{-\lambda x_1\} - \frac 12\exp\{-\lambda x_2\} =0$$

and we verify that $A,B$ do not matter.

This is a non-linear function in one variable, and a computer algorithm should be used to find the possible solutions. In this particular case, trial-and-error by hand (or naive graphing in a spreadsheet) may mislead, because, as the posted solution by the OP in a comment shows, the non-zero value for $\lambda$ that satisfies the equation is very small, $\lambda = 0.0032$.

So if one tries to graph the function in a spreadsheet, it must specify very small increments for $\lambda$ otherwise it will miss the small interval for which the function becomes positive for strictly positive values of $\lambda$, and then falls to zero and negative values again.

Making matters worse,the values of the actual prizes are too large numerically when inserted into the exponential function, and so the increments for lambda should be even smaller, if one would want to "see" the solution in a spreadsheet.. For example to use the actual levels of the prices as coefficients of the equation, one needs to specify increments of the order $10^{-5}$.

One could input the prizes in hundreds or in thousands... (the solution in the link posted in the comment uses hundreds), which allows for larger increments.


Saying what the theory predicts is a good idea as then we can make a testable hypothesis of the theory. Much of empirical economics is to test the validity of the theory.

If your result conflicts with the theory, then you can try to rectify the conflict or propose a better alternative.


Who created the concept of "present value"?

It has been argued that the concept of present value was implicit in Liber Abaci (1202) by Leonardo of Pisa (also known as Fibonacci).  This paper by Goetzmann makes this claim in its abstract and introduction, and presents (pp 26-7) an example from Liber Abaci of what appears to be present value analysis.


It can be profitable for the monopolist to do so. For the conventional producer who is a price taker the profit objective function looks like this:
$$\max_{q} \Pi^c $$
where $\Pi^c = P \cdot q - C(q)$.

That is, they seek to maximize profits, facing an exogenous price to sell goods and where costs are a function of amount produced. If everything is nice and differentiable and concave this gives a first order maximizing condition of:
$$ 0 = \frac{\partial\Pi^c}{\partial q} = P - \frac{\partial C(q)}{\partial q} \rightarrow q = g(P) $$  where $g(P)$ is the inverse of function $\frac{\partial C(q)}{\partial q}$ (increase q from zero until marginal cost equals the price).

But for a monopolist $\Pi^m = P(q) \cdot q - C(q)$. That is, the monopolist is not a price taker, they know that when they produce more prices will fall. Resulting FOC:

$$ 0 = \frac{\partial\Pi^m}{\partial q} = P(q)+q(\frac{\partial P(q)}{\partial q}) - \frac{\partial C(q)}{\partial q}$$

As a result, a monopolist in this setting will not want to sell where price equals marginal cost but rather a higher price than marginal cost:
 $$ \frac{\partial C(q)}{\partial q} - q(\frac{\partial P(q)}{\partial q})= P(q) $$
Recall: $- q(\frac{\partial P(q)}{\partial q}) > 0$

Indeed, it is this selling of goods above their marginal cost that is the source of the monopolist's monopoly rents.

Now back to the question itself which asks about when "company destroys items they couldn't sell in time." We can think of inventory as a good with low or zero marginal cost, so yes, by the argument above, it can be profit maximizing to not sell goods with a price above costs as long as it lowers the profits on the other goods you sell enough to offset the additional revenue.

Other forces are likely at play as well. Stores often worry about their brand value and low quality, low price, goods may harm their relationship with their customers. Many people shop at stores specifically for the high quality goods and others at different stores for low prices and it may be too costly to move spoiling goods from one venue to the other and cheaper just to toss it. We can also think of the muffin stumps of the TV show Seinfeld as an example of this high transaction costs / missing markets problem.

That said, there are sometimes innovative alternative methods of disposal:

  Rotisserie chickens have been around for a while. I used to bypass
  them and roast my own, until I noticed something: The rotisserie
  chickens were actually cheaper than buying and roasting my own.

  Cat Vasko noticed the same thing and decided to figure out why. The
  answer makes a surprising amount of sense: Grocery stores make them
  out of unsold chicken that is about to pass its expiration date. It's
  an elegant way to make a profit out of food that would otherwise be a
  net loss. And it's not just chicken -- according to Vasko, the
  ever-expanding prepared-foods section of the supermarket uses up all
  sorts of unsold produce and meat. It is, as she says, a bit like
  hunter-gatherers using every inch of the animal.

Everyone Wins When You Buy a Rotisserie Chicken


What were the problems with the gold standard?

There are several problems with gold standard:

Having more than 1 country using a gold standard is economic equivalent of being in monetary/currency union. In fact European Monetary Union is often in literature equated to EMU having a gold standard (see Eichengreen & Tamin 2010).

This creates several problems because optimal currency area (OCA) requires countries to either have completely unrestricted mobility of factors (free movement of labor (e.g. immigration/emigration) and free movement of capital in and out country (e.g. no capital control regulations)), or it requires countries to adopt fiscal union (e.g. create one joint federal government). This is because different countries can experience asymmetric  macroeconomic shocks which require either of the two above to hold in order to accommodate them  (see DeGrauwe Economics of Monetary Union).
Of course there is an option of not accommodating the asymmetric macroeconomic shocks but that leads to worse recessions, higher unemployment and general economic malaise that you could observe in EMU  (which is again a paper equivalent of gold standard) after Great Recession. Hence, it is very inefficient to have gold standard system that is followed by multiple independent countries.

Gold standard partially constraints monetary policy of the government. Being on gold standard essentially means that government fixes exchange rate between gold and its issued bank notes. For example, government might decree that one dollar is worth 1 ounce of gold.

This ties money supply to supply of gold that is determined by gold production. As a consequence, the money supply might expand exactly when economy does not need it just because new gold is discovered under ground, and times when economy needs monetary expansion there might be not enough new gold to actually increase money supply sufficiently. In fact the unwillingness of US to expand money supply during Great Depression  partially  due to being on gold standard, was one of the main reasons why recession of 1929 is now known as Great Depression (see Friedman and Schwartz Monetary History of the United States).  Consequently, gold standard can have extremely disastrous effects on macroeconomic stability.
There is an important caveat to this. The fixed rate can be in principle changed by government, so monetary policy is still in principle possible. Government can one day decree that the fixed exchange between dollar and gold is 1 dollar per ounce of gold another day it can decree it is is 10 dollars per ounce of gold. But typically changing the exchange rate between gold and currency under gold standard was not easy due to various institutional reasons.  Moreover, if the exchange rate between gold and dollar could be changed from minute to minute then it would for all practical purposes cease to be gold standard in anything but name.
Also generally speaking monetary policy is extremely important tool for macroeconomic management as you can learn in any macro textbook (e.g. you can have look at Blanchard et al Macroeconomics a European Perspective, or Mankiw Macroeconomics). Gold standard simply interferes with central bank's ability to conduct monetary policy in efficient manner.

Gold standard was practically unworkable during any major crisis. For example, most countries had to suspend during large crises or wars. Hence, even in the heyday of gold standard countries mostly operated on gold standard in good times while in times of war or crisis switching into alternative monetary arrangements. As explained already in their abstract by Bordo & Finn (1995)

The experiences of these major countries suggest that the gold standard was intended as a contingent rule. By that we mean that the authorities could temporarily abandon the fixed price of gold during an emergency (such as wartime) on the understanding that convertibility at the original price of gold would be restored when the emergency passed.

Hence, even in the best days of gold standard people tacitly agreed it is only good monetary arrangement for good times not bad times (and that was still based on their contemporary 18-19th century understanding of economics - with modern science we know that was mistake and it was not exceptionally great even in good times (definitely not enough to compensate for its drawbacks)).

Having gold standard would put more restrictions on government finances. With fiat currency government budget constraint can be in its static form described as:

$$G−T=\beta +\theta \implies G=T+\beta + \theta$$
where G is government spending, T is the net tax revenue after transfers and interest payments, $\theta$ is government financing by high powered money and $\beta$ is government financing by bonds/debt (see for example Blinder & Solow, 1973; Christ, 1968; Tobin & Buiter 1976).  Now under gold standard government can only use $\theta$ if it mines extra gold or if it changes fixed exchange between gold and money which as discussed above was often difficult.
With fiat money government is much less constrained, even though virtually all mainstream economists agree government can't pay for arbitrary amount of real spending with monetary financing (see this poll among top policy economists), having access to this channel helps.  This is especially important at a time when average government spending to GDP is around 45% for OECD countries (in the heyday of gold standard this was on average about 5%). Most modern societies simply want government that provide plethora of social programs, redistribution and ample funding for public goods. This forces governments to be smart about the ways they pay for these programs and putting on gold standard straitjacket makes little fiscal  sense.

Gold standard tended to be deflationary.

First, even non-economists usually recognize that high inflation is sign of monetary instability, but forget the same holds about deflations. Second, it is widely believed by economists that over the business cycle it is good if economy has moderate inflation rather than deflation (again see any conventional macro textbook such as the Mankiw or Blanchard et al recommended above).
This is due to existence of nominal rigidities in the economy (e.g. nominal contracts are set for specific duration of time; minimum wages; menu costs - that is costs of changing prices frequently, even psychological reasons and so on). Such rigidities make  it  beneficial for economy to have small amount of inflation over time which relaxes them which is especially important during economic downturns when real decreases in wages and increase in prices, but it is simply too difficult for people to adjust prices and renegotiate wages quickly enough.

Gold standard wastes scarce resources. Under gold standard gold - a  scarce resource with many industrial uses - is literally dug from the ground just to be buried under the ground again in bank vault. There is simply no reason to waste scarce resources in such way when you can base currency on paper or heck nowadays even just electronically. Obviously, Bitcoin would not have this problem, but the reason why I asserted it would be even worse is that it would have more constraints on monetary policy (this point 4 while significant pales in comparison with economic importance of points 1-5).

There could be other reasons I forgot to mention above, but on the whole modern scholarship on gold standard considers it a failure and not very efficient monetary arrangement.


Is It Possible for a Country to have a Gold Standard without a Fixed Price of Gold?

(1) Is it possible to have a gold standard without a fixed price of gold?

No, with some provisions. First, I presume here we are talking about the price of gold in terms of currency not its relative price (relative to other goods and services) which was always allowed to fluctuate. Gold standard by its definition requires that money is fixed to gold at some fixed rate. If that does not hold it would not be a gold standard. However, the fixed rate can be changed overtime. For example, government could decide that one day the fixed rate will be 10 USD for 1 ounce of gold another day it will be 15 USD for 1 ounce. But this is not the same as having floating price for gold.

(2) Are there any countries left today which tie their currency directly to gold or some other precious metal?

No. Depending on what you count as country, the late  Islamic State in Iraq and al-Sham (ISIS) had rolled out a trimetallic currency (see Oxnevad 2016). Their currency was to be made directly from gold, silver and copper hence would be directly tied to those commodities. Although, by international community it was never recognized by anything more than a terrorist group, it would probably satisfy the Max Weber's definition of state during mid 2010s. However, that likely would no longer be applicable presently after they lost all their territory to Iraq and Syria.


Imperfect Substitutes and Utility Funcitions

In general, any utility function that produces non-linear and downward sloping indifference curves will feature imperfect substitutability between goods. (By considering only goods, we're implicitly assuming "more is preferred to less", or positive marginal utility, for each good.) In other words, look at the marginal rate of substitution and see if it depends on the quantity of either good; if it does, then there is imperfect substitutability.
Examples

Quasi-linear utility: $u(x,y)=v(x)+y$, where $v(x)$ is non-linear in $x$ (e.g. $v(x)=\sqrt x$)
Additively separable utility: $u(x,y)=g(x)+h(y)$, where $g,h$ are non-linear functions
Cobb-Douglas: $u(x,y)=x^\alpha y^\beta$, where $\alpha,\beta$ are constants
Leontief (perfect complement): $\style{text-decoration:line-through}{u(x,y)=\min\{\alpha x,\beta y\}}$ [Leontief utility function features nonsubstitutability rather than imperfect substitutability.]
CES utility: $u(x,y)=\left(\alpha x^\rho+\beta y^\rho\right)^{1/\rho}$, where $\rho\ne1$. Note that the class of CES utility functions includes Cobb-Douglas (as $\rho\to0$), Leontief (as $\rho\to-\infty$), and linear utility (as $\rho=1$) as special cases.

This lecture note by Simon Board has examples and indifference curves corresponding to the cases above.


Decline in the Value of the Domestic Currency

Your question appears to be just: why would they issue foreign currency debt?

If an emerging country is growing rapidly, it often attracts capital flows. This is typically associated with a strengthening currency. Firms and investors extrapolate this tendency, and so they lend to the country in the currency of a developed economy (typically the U.S. dollar). If the local currency continues to appreciate against the U.S. dollar, it should be easier to cover the payments (since local currency earnings are worth more). Things break down if this extrapolation fails, as discussed in the question.

In terms of an example, you could look at the developing Asian countries ahead of the Asian Crisis of 1997. Borrowing in U.S. dollars by local forms was the root of the problem. The post-1997 tendency to accumulate large foreign currency reserves by central banks in countries pursuing an export-led growth strategy was a reaction to that experience.


I am not sure if this is the answer you were aiming at, but... what about the revelation principle?
There is no requirement to communicate with the designer. However, researchers are ofter first interested in what the optimal allocation rule is. In that search, it is wlog to consider direct mechanisms by the revelation principle: Suppose an indirect mechanism without communication to the designer implements the optimal allocation, then we can obtain the same allocation if every agent simply reports their type to the mechanism designer who then computes the equilibrium allocation of this indirect mechanism and implements it. See my reply with more details here.
If you want to know if there is a general approach of how to find equivalent indirect mechanisms for the optimal direct revelation mechanism, I am not aware of an answer.


I am not sure I understand your question. In most projects, the bulk of the investment is a down payment, so the ROI is meant to capture precisely that. I.e. the average return that the money you put in the project will yield. This has in mind that the total investment is mostly done today and the returns come in the future.

If you restrict the denominator to only be the down payment, you seem to want to capture the average return of the money you initially put into the project (excluding the returns that are derived due to money put into the project after the down payment). The problem is how you attribute how much of the future profits are due to the initial down payment and how much due to the investments that are done in the middle of the project, etc.

This is why  I think  that the ROI is already capturing the notion you want. If the project will not require a lot of money after the down payment, then the denominators will not be so different anyway. However if the project might require a lot more money after the initial down payment, it is harder to claim that X amount of the net profits can be attributed exclusively to the initial down payment.

You are correct that the ROI does not provide any information about leverage, so maybe you want to complement it with some other ratio like (down payment)/(total investment) or another over-the-counter financial ratio. However, considering a measure like net profits/down payment will seem very deceiving to me.


Yes confidence level always depends on choice of a researcher (at least in principle). There is no reason to use $90\%$, $95\%$ or $99\%$ level either  - it is more or less just convention. For example, in physics they use three-sigma or five-sigma significance which corresponds to confidence levels of $99,7\%$, $99.99997\%$ (and even higher ones can be reported).
However, this being said once literature settles at a particular convention you will be fighting an uphill battle. For example, you would have hard time publishing in physics results that are not at least $3\sigma$ or $5\sigma$ significant without very good explanation. In economics, usually people settle for $95\%$ level. Just because you can pick any significance/confidence level that does not mean others will accept it.

They actually do explain why there is $66\%$ confidence interval. According to Stock and Watson:

Also plotted
are $\pm 1$ standard error bands, which yield an approximate 66 percent confidence
interval for each of the impulse responses.

When it comes to VAR impulse responses (and forecasting in general) it is conventional to plot $\pm 1$ standard error bands. This is also often done when people estimate DSGE models so it is more or less conventional practice. This is ultimately arbitrary choice on which people somehow settled.
Also note reporting 1 standard error deviation error bars does not mean the variable was significant at only  $66\%$ confidence level - for example $\hat{\beta}=30$ with $SE(\hat{\beta} )=1$ would have $t$-stat $=30$, would be significant at any conventional significance used in economics (in any reasonably sized sample). That does not prevent you from constructing 1 standard error bars (i.e. $30 \pm 1$).


Economies of scale v.s. technological change

The problem is to distinguish between changes along the production function, from $f(K_1, L_1)$ to $f(K_2, L_2)$, and changes of the production function, from $f(K_1, L_1)$ to $g(K_1, L_1)$.

A very simple example/model. Imagine that the only input is labor and that this input doubles between t and t+1, but output more than doubles : $L_{t+1} = 2 L_t$, $Y_{t+1} > 2 Y_t$.

To interpret this observation you have two alternatives :

Assume that there is a production function with increasing returns to scale, i.e. $1/Y dY/dL > 1$, so that $Y_t = f(L_t)$ and $Y_{t+1} = f(L_{t+1})$.
Assume that the production function has constant returns to scale, i.e. $Y_t = \alpha L_t$ and $Y_{t+1} = \beta L_{t+1}$, but technical progress has occurred between periods $t$ and $t+1$, so that $\beta > \alpha$.
Of course any combination of the two explanations above is admissible.

During the 'Cambridge controversies on capital', this was one the arguments which led the English Keynesians to reject the notion of production function. One famous paper building a growth model without a production function is Kaldor-Mirrles (1962), A New Model of economic growth


Is Currency Devaluation a cause or effect of interest rates?

The bond market is a market, albeit following the lead of the central bank at the short end.
The currency market is a market.
The simplest and crudest version of the Efficient Markets hypothesis is that it is difficult to “beat a market” using a simple rule. The argument is that if such a simple rule exists, everyone would do it, causing prices to shift. (There can be regularaties like the term premium - bonds tend to outperform cash in the long run - but one needs to take risk exposures to achieve such gains.)

Taken together, the previous statements suggest that there should be no simple relationship between interest rates and currencies. If the previous logic is incorrect and such a relationship exists, feel free to use it to make a fortune as a currency strategist. (Disclaimer: this is not a recommendation to trade currencies without understanding the risks involved.)
Textbooks love discussing interest rate parity relationships. This is an arbitrage relationship used to pin down the pricing of currency forwards. However, the predictive power of forwards is at best weak, leaving us back near the possibility of efficient markets.
Finally, the exception to the bond and currency markets is the possibility of a pegged currency, or pegged bond yields (relatively rare). At which case, one needs to look at the behaviour of currency pegs, and runs on such pegs.


I don't think it's any more complicated than looking at the ratio of global oil consumption and global Gross Domestic Product. The OECD, World Bank, IMF and others produce estimates of each.

There is some fuel-switching: for example, the world burns much less electricity from oil than it used to: in 1973 (the first oil crisis started around October that year) 251 million tonnes were used for electricity; by 2000, this had dropped to 110; by 2012, to 70 (table 7A on IV.81 of the IEA Electricity Information 2014).

There's several decades of energy efficiency, kickstarted in large part by the oil shocks of the 1970s.

The share of services (relative to physical goods) within global GDP has increased; IIRC services require much lower energy inputs than goods, on average, even though transport is one of the services (I'll check this and edit later - please nudge me in comments if I haven't after a week or so). The World Bank only gives services as a share of global GDP from 1990 onwards: this share grew from 60.7% in 1990 to 70.2% in 2011.

The combination of energy efficiency and the increase in the share of services (relative to physical goods) within global GDP, means that energy consumption per unit of global GDP has fallen: J. Bradford DeLong gives global GDP at \$12 trillion for 1970 and \$41 trillion for 2000 (preferred measure, 1990 international dollars). The BP Statistical Review of World Energy 2014 gives 1970 energy consumption at 4.9 billion tonnes of oil equivalent (btoe) and 9.3 btoe for 2000. So energy per unit global GDP almost halved (44% reduction) over those thirty years 1970-2000.

Understanding an individual country's changes are harder, because we then have to unpick whether oil intensity of GDP has dropped because of energy-efficiency & fuel-switching, or because oil-intensive goods are now imported rather than produced domestically: for further reading on this, see papers by John Barrett and by Dieter Helm.


Effect of a change in quality on price elasticity of demand

It seems that your confusion is more about classifying goods into necessities vs. luxuries. In economics, the criterion for such classification is to look at whether the income (not price) elasticity of demand is "high" or not. If a good has a high (and positive) income elasticity, then it's a luxury; if it has a low (but still positive) income elasticity, it's a necessity. What counts as "high" or "low" is an empirical matter. Investopedia suggests $1$ as a threshold (high if $>1$ and low if $<1$).

Coming back to your question about price elasticity and quality improvement. The answer is: it depends on how you model the demand curve of the good. Recall that the price elasticity of demand is given by
\begin{equation}
-\frac{\mathrm dQ}{\mathrm dP}\frac{P}{Q}=-\frac{1}{\text{slope}}\frac{P}{Q}
\end{equation}
where the slope is evaluated at the point $(Q,P)$. The quality improvement shifts demand curve to the right (since consumer is getting more value for each dollar spent, so they are willing to pay more for each unit or buy more units at the same price). If price remains unchanged (as you assumed), then quantity demanded must increase, say from the original $Q$ to a higher $Q'$.

If the slope of demand stays the same at $(Q',P)$, then price elasticity would fall according to the formula. If demand becomes much flatter at $(Q',P)$ (with a magnitude greater than the increase in quantity), then price elasticity would increase instead.


Do I understand correctly that taxation-based government expenditures allow GDP to grow?

In your comment you say you want the answer in context of the classic textbook model for goods market/output equilibrium.

Following the Blanchard macroeconomics textbook lets consider closed economy so the output will be given by:

$$Y=C+I+G$$

where $Y$ is output/income (they must be always equal), $C$ consumption, $I$ investment and $G$ gov. spending.

The consumption will be given by:

$$C = c_0 + c_1(Y-T)$$

where $c_0$ is the part of consumption that does not depend on income and $c_1$ is marginal propensity to consume (MPC) and $T$ are taxes.

In this basic model investment is just fixed $I=\bar{I}$

Solution to the model (i.e. autonomous spending) based on the set up above is given by (the solution is derived by substituting $C$ in second equation for its place in first one and then solving for $Y$):

$$Y = \frac{1}{1-c_1}[c_0+ \bar{I} +G -c_1T ]$$

Now since you say that we take it for granted that government runs balanced budget $G=T$ because borrowing is "taboo" we can substitute the equality between taxes and gov. spending so we have:

$$Y = \frac{1}{1-c_1}[c_0+ \bar{I} +G -c_1G ] = \frac{1}{1-c_1}[c_0+ \bar{I} +(1-c_1)G ]$$

Now to clearly see the effect of G spending lets just set $c_0=\bar{I}=0$ (this is not necessary but it cleans the equation). So we have:

$$Y =  \frac{1}{1-c_1}(1-c_1)G = G$$

Hence there is no multiplier on Government spending if budget is balanced. So without debt government cannot increase output in any way (at least not in this simple model).

The intuition is that of course if government just takes money from left pocket to the right there cannot be any increase in output. Since in this model all agents have the same MPC and hence all spending including the Gov. spending has the same multiplier. If you take 10 euros from consumers and just spend it as a government you are not creating any additional output. you are just reducing $C$ - reduction of which would also be affected by multiplier, by the same value you are increasing $G$ which is affected by the same multiplier. The effects must cancel each other out. Only if you would allow for heterogeneous agents where people have different MPC government could increase output in the short term by redistributing money from people with low MPC to people with high MPC, but the output cannot be raised "exponentially" and this would only be one time increase - i.e. increase in level or short run growth. I would recommend reading the full chapter 3 from Blanchard et all (2010). Macroeconomics an European perspective, that deals with this in greater detail.


First order condition of log functions in general and interpretation

It just comes from the derivative of profit function. I assume that $a_i$ is the choice variable here so the derivative of $\pi$ wrt $a_i$ is (step by step):
$$\frac{\partial \pi}{ \partial a_i} = \frac{\partial \pi}{ \partial a_i} [ \ln R(a_i) ]  +  \frac{\partial \pi}{ \partial a_i} [ \ln N_i(\gamma a_i, \gamma a_{-i}) ]  \\
= \frac{1}{ R(a_i)}   R'(a_i)  +  \frac{1}{ N_i(\gamma a_i, \gamma a_{-i})}  N_i'(\gamma a_i, \gamma a_{-i}) \gamma $$
The gamma at the end pops out because of the chain rule. If you have $G(H(F(x)))$ then $\frac{dG}{dx} = \frac{dG}{dH} \frac{dH}{dF}\frac{dF}{dx}$. You just simply have to realize that $\gamma a_i = F(a_i) $.
Lastly to derive FOC you just set the last equation above to zero and you are there:
$$\frac{1}{ R(a_i)}   R'(a_i)  +  \frac{1}{ N_i(\gamma a_i, \gamma a_{-i})}  N_i'(\gamma a_i, \gamma a_{-i}) \gamma = 0 \\ \frac{ R'(a_i)}{ R(a_i)}    = - \gamma \frac{ N_i'(\gamma a_i, \gamma a_{-i})}{ N_i(\gamma a_i, \gamma a_{-i})} $$
To sum up, the $-\gamma$ is there because of the chain rule, nothing special aside of taking derivative is going on here.
PS: I just realized that I forgot to explain why you can interpret them as elasticities but great +1 answer of Herr K. already did that so I wont unnecessarily repeat it here.


Why is the rate of profit not important?

I think it has to do with declining interest in Marxian economics. In the past the Marxian economics, especially the one in Sraffian tradition had much wider appeal.  The rate of profit was an important metric in Marxian economics, since Marxian economics makes several important predictions about the rate of profit in 'capitalist' economies (e.g. the  tendency of the rate of profit to fall (TRPF) hypothesis - see Marx Capital, Volume III).
If you look at the historical papers that discuss the rate of profit, they are very often connected to the TRPF hypothesis (although not necessarily from Marxian angle). In fact, if you just put rate of profit into google scholar now more than 1/3 of the searches relate either to Marxian economics of TRPF hypothesis. If you do the same with Palgrave dictionary of economics almost half of the results on first page are connected to Marxian economics or TRPF hypothesis. I realize this alone is not rigorous proof but it is at least bit suggestive.
In addition, I would challenge the idea that the rate of profit 'dropped from mainstream economics'. Rate of profit is often used synonymously with rate of return on capital (see Feldstein, Summers & Wachter, 1977). This is so because existence of profit is often (at least partially) attributed to the fact that firms typically own their capital, and indeed if you look at papers that try to measure rate of profit often they look at returns on assets as well such as the Feldstein et al paper above.
You will find rate of return capital to be mentioned far more often in the mainstream literature, and textbooks as well. Thus it is possible that the rate of profit was just abandoned due to the Marxist connotations.


Why have prices gone up?

Because supply was less than demand. The price has now increased, to bring supplies onto market, and to reduce demand.

Will it go higher? No one knows. The market thinks not, otherwise prices would already have gone higher.


Does the marginal product need to be diminishing to have a diminishing average product?

Suppose the total product is given by $f(x)$. Thus, the average product is $$AP(x)=\frac{f(x)}{x}.$$

Differentiating:
$$AP'(x)=\frac{x f'(x)-f(x)}{x^2},$$

which has the same sign as $x f'(x)-f(x)$. This is definitely negative if $f''(x)<0$ (i.e., if $f$ is concave).

But it can also be negative, even if $f$ is locally convex. Take a look at this figure:

At the point marked with a blue disc: the slope of the red line is $f'(x)$. The slope of the black line is $f(x)/x$. We can see geometrically that $f(x)/x>f'(x)$, or (after rearranging) that $f'(x)x<f(x)$.

Thus, at the point indicated by the blue disc we know that $AP'(x)<0$. But at this point we can also see in the figure that $f''(x)>0,$ implying that the marginal product is increasing.

Since we have a counter-example, we conclude that a decreasing marginal product is not a necessary condition for a decreasing average product.


Advanced economies tend to slide down to zero interest rates and stay there. Why?

Recently I read an interesting paper on this subject from the Bank for International Settlements:

Juselius, Mikael and Takats, Elod, Can Demography Affect Inflation and Monetary Policy? (February 2015). BIS Working Paper No. 485. Available at SSRN: http://ssrn.com/abstract=2562443

The abstract:

  Several countries are concurrently experiencing historically low inflation rates and
  ageing populations. Is there a connection, as recently suggested by some senior
  central bankers? We undertake a comprehensive test of this hypothesis in a panel of
  22 countries over the 1955–2010 period. We find a stable and significant correlation
  between demography and low-frequency inflation. In particular, a larger share of
  dependents (ie young and old) is correlated with higher inflation, while a larger
  share of working age cohorts is correlated with lower inflation. The results are
  robust to different country samples, time periods, control variables and estimation
  techniques. We also find a significant, albeit unstable, relationship between
  demography and monetary policy.

As I'm not an expert on the subject, it is probably not the main explanation in literature but it's in my opinion an interesting research path.


Is possible that, in a choice structure $(B,C)$, $C(b) = \emptyset$, for some $b \in B$

One usually rules this out by assumption, as Mas-Colell, Whinston, and Green do in their textbook (). The goal is often to try to characterize preferences that guarantee (at least for finite choice sets) the existence of choices from a revealed preference point of view. There is also the methodological point that if you confront a decision-maker with a choice problem and chooses none of your options, then you haven't really included all options in the first place.
That being said, one can still study the mathematics of possibly empty-valued choice rules, and this has been done. One can find some material within this more general context in the book [Aleskerov, Fuad, Denis Bouyssou, and Bernard Monjardet. Utility maximization, choice and preference. Vol. 16. Springer Science & Business Media, 2007.]


Why is the rising price of a commodity inflationary, while the rising price of money is disinflationary?

If everybody borrows money, where can you borrow money from?

In what unit do you denote the price of oil, chocolate, or cows? Well, in units of money. So calling interest rates the price of money doesn't quite make much sense.

You can think about it as the price of money today in terms of future money though. So an increase in interest rates makes money today more expensive in terms of future money, which is the same thing as making future money cheaper in terms of money today. And if future money gets cheaper, more people might "buy" future money (usually, we call that saving).

The more future money people buy with money today, the less money they have left for oil, chocolate, and cows. But we already have all the oil, chocolate, and cows around. So that people still buy these things, oil, chocolate, and cows have to get cheaper and the price level for goods and services measured in money today goes down. Less inflation.


Why New Zealand dollar keeps falling against USD and other currencies?

If you look at the NZD against the USD over the last 10 years, it looks like this:

Obviously there has been a pretty sharp decline lately, but it's still not that far below the 10-year average of 74.27. It's also interesting to look at how it's been doing against the Australian dollar:

As you can see, at 88c currently, the NZD is still well above its 10-year average against the AUD. Again, there's been a sharp dropoff of late -- NZD almost reached parity with the AUD quite recently.

In terms of why there has been a dropoff of late, I think you've hit the nail on the head regarding dairy prices. Dairy (including baby formula) accounts for a large share of NZ's exports, and dairy prices are at a 6-year low currently. Fonterra, the dairy exporter, is scrambling to restructure. That alone probably explains the majority of the NZD's recent decline against the AUD. I think the NZD decline more broadly against the USD is probably partially attributable to the weakness of the AUD -- New Zealand's economy has close ties with the Australian economy, and the Aussies have suffered from decreasing commodity prices. Also, it takes two countries to calculate an exchange rate, so when looking at the NZD's decline against the USD, you might note that the US economy has been doing quite well over the last few years (at least if you're naively tracking the DJIA or something).

Anyway, there's no need to panic about the NZ economy. Exporters benefit from a weak NZD. The tourism industry, another key sector in the NZ economy, benefits significantly from a weaker NZD also. If you're a Kiwi that likes to travel, and most seem to, the weak dollar isn't much fun, but such is life.


Empirical Data on Demand for Minimum-Wage Jobs in the USA

Neumark and Wascher have a paper that surveys recent empirical work on the effects of the minimum wage in detail. It includes a number of tables that nicely summarise the results of a large number (102!) of studies.

They conclude the paper with the following remarks:

  Nonetheless, the oft-stated assertion that the new minimum wage
  research fails to support the conclusion that the minimum wage reduces
  the employment of low-skilled workers is clearly incorrect. Indeed, in our view, the preponderance of the evidence points to disemployment
  effects. For example, the studies surveyed in this monograph correspond
  to 102 entries in our summary tables. Of these, nearly two thirds
  give a relatively consistent (although by no means always statistically
  significant) indication of negative employment effects of minimum
  wages, while only eight give a relatively consistent indication of positive
  employment effects. In addition, we have highlighted in the tables
  33 studies (or entries) that we regard as providing the most credible
  evidence, and 28 (85 percent) of these point to negative employment
  effects. Moreover, when researchers focus on the least-skilled groups
  most likely to be adversely affected by minimum wages, the evidence
  for disemployment effects seems especially strong. In contrast, we see
  very few – if any – cases where a study provides convincing evidence of
  positive employment effects of minimum wages, especially among the
  studies that focus on broader groups for which the competitive model
  predicts disemployment effects.


What is the difference between personal consumption expenditure price and PCE quantity?

Personal consumption expenditure (PCE) is the primary measure of consumer spending on goods and services in the U.S. economy.   It accounts for about two-thirds of domestic final spending, and thus it is the primary engine that drives future economic growth.  PCE shows how much of the income earned by households is being spent on current consumption as opposed to how much is being saved for future consumption.
https://www.bea.gov/resources/methodologies/nipa-handbook/pdf/chapter-05.pdf
The PCE price index is a measure of the prices that people living in the United States pay for goods and services. The PCE price index is known for capturing inflation (or deflation) across a wide range of consumer expenses and reflecting changes in consumer behavior.
https://www.bea.gov/data/personal-consumption-expenditures-price-index#:~:text=A%20measure%20of%20the%20prices,reflecting%20changes%20in%20consumer%20behavior.
Most PCE price indexes are derived by extrapolating consumer and producer price indexes from the U.S. Bureau of Labor Statistics.
The PCE quantity index derived by deflating current-dollar estimates at the most detailed category level by appropriate PCE price indexes with the reference year equal to 100.
Prices and quantities are prepared at a detailed level and are aggregated based on a Fisher chain-weighted formula  https://www.bea.gov/sites/default/files/methodologies/nipa-handbook-all-chapters.pdf#page=64
This source can provide some answers: https://www.bea.gov/help/faq/521
The difference between PCE quantity and industrial production is that PCE come from consumer spending.   Industrial production measures the output of the industrial sector, which typically comprises mining, manufacturing, utilities and, in some cases, construction.
The difference between PCE quantity index and real output is that PCE come from consumer spending while Real output comes from spending from consumers, business/industries, and government/government enterprises.


Where do remittances come into play in the formula for GDP?

Gross domestic product (GDP) is the total value of output in an economy, this can be measured only by Output using this formula.

This method uses GDP = C + I + G + (X-M) where

C: Consumption (Household spending)
I: Investments
G: Government spending
X: Exports from an economy
M: Imports into an economy

The remittances that you mention about are not made against any services.  While remittances can be a source of GDP growth by increasing household consumption, it does not directly add to GDP, it does affect GNP though.

For a clear comparison of these two terms GDP and GNP use this

Alternatively you can use incomes to calculate GDP, however note that even in that method you add only those incomes that are made from the production of goods and services are included and not transfers


First it is helpful to provide some definitions.

M0 includes all paper dollars (plus coins and US notes).  Regardless of whether or not they they are held in the banking system.

M0 is not a useful metric because it omits electronic dollars.  The two main liabilities to the Fed are electronic dollars (some refer to them as Fed Funds or deposits at the Fed which banks can hold) and paper dollars.

M0 + electronic dollars = MB.  MB is by far a much better metric for measuring what the government has created for money.

The Fed's balance sheet may be illustrative in showing how this works. Much of the Fed liabilities are MB, but only a much smaller part are M0. The difference between paper and electronic dollars is not significant.  If a bank asks the Fed to trade their electronic dollars for paper (or vice versa, the Fed will oblige.  If the Fed runs short on paper, they will have the treasury print more, but this doesn't expand MB.

Think of paper dollars as ice, electronic dollars as flowing water and MB as all the water in the system (frozen or flowing).  Whatever the form, it is not important as the total amount of water remains the same.

Let's try to answer our question specifically using a timeline and ledger entries.

Say the Fed buys some assets from a bank.  They acquire these assets (+Fed assets) and grant the bank new deposits (+liabilites aka bank deposits).  Now this is always done with electronic dollars to start.

Then say the bank has a lot of withdrawals and wishes to convert some of their federal despots assets into cash assets.  They ask the Fed to do this, which gladly obliges.  The Fed decreases their deposit liability and increases their paper note liability.  In this moment M0 was increased.  But it is not significant.  What was significant was when MB was increased after the Fed acquired the asset from the bank.


What's the difference between the Life Cycle Hypothesis and Permanent Income Hypothesis?

The ideas are very similar and functionally virtually equivalent although there arguably is subtle difference. For example, Romer in his Advanced Macroeconomics, which is widely used intermediate macroeconomics handbook, calls both the life cycle hypothesis and permanent-income hypothesis just permanent income hypothesis. To be more specific he says

Thus the individual uses saving and borrowing to smooth the path of consumption. This is the key idea of the permanent-income hypothesis of Modigliani and Brumberg (1954) and Friedman (1957).

Where the Modigliani and Brumberg (1954) refers to the paper where the life-cycle hypothesis originates.
There is very subtle difference though. Friedman's permanent income hypothesis, focuses more narrowly on income. Friedman's argument is also very 'instrumentalist' and it goes as follows. We know from macroeconomics that consumption depends on income so we will have:
$$C_t = a +  bY_t + e_t$$
where the estimated $\hat{b}$ is given as covariance of income and consumption over variance in income $\hat{b} = \sum(C_t-\bar{C})(Y_t-\bar{Y})/\sum(Y_t-\bar{Y})^2$ and $\hat{a}= \bar{C}- \hat{\beta}\bar{Y}$. Now if we assume that $Y$ can be decomposed into permanent $Y^P$ and temporary $Y^T$ income (i.e. $Y=Y^P +  Y^T$) and that the two are uncorrelated with transitory income having mean zero we can following Friemdman show that:
$$b= k \frac{\sum (Y_t^P- \bar{Y_t}^P)^2}{\sum ( Y_t- \bar{Y_t})^2} \text{ and } \hat{a} = \bar{C} -\hat{b}(\bar{Y}^P+\bar{Y}^T)$$
Since transitory income is on average assumed to be zero, and because variation in transitory income which is component of the denominator in calculating $\hat{b}$ is arguably much smaller than the variation in permanent part of the income one can conclude that consumption is determined virtually completely by changes in permanent income.
The life cycle hypothesis of Modigliani and Brumberg is more analytical and pays also attention to person's whealth. Following presentation in Romer's Advanced Macroeconomics,  a person at any time $t$ a person would make expectation of his/hers lifetime income $\sum^T_{t=1} E_1 [Y_t]$ together with any initial assets/wealth $A_0$ and their current consumption at the time $t$ will be just an equal fraction of all income (i.e. permanent income in Friedman's lingo) and assets  for that particular time period. That is the consumption in present period 1 would be given by:
$$C_1 = \frac{1}{T}\left(A_0+ \sum^T_{t=1} E_1 [Y_t]\right) \text{ and }$$.
This implies that people will try to smooth their income over their life as it implies that people will save when the income is higher than average income and dis-save when it is lower than average since:
$$S_t=Y_t-C_t= \left( Y_t - \frac{1}{T} \sum^T_{t=1} E_1 [Y_t]\right) - \frac{1}{T}A_0$$
Thus both PIH and LCH are functionally equivalent and they are both consistent with each other. They both argue that consumption is determined by your permanent income/lifetime income including any inherited assets.  The main difference is in presentation and arguably the mechanisms are subtly different. The Friedman's PIH presentation is more 'empirically oriented' and focuses on income itself (although arguably Friedman implicitely included any assets in $Y$). The LCH is more analytical and pays more attention person's wealth and later versions also to demographic factors.  Since they are so similar as already mentioned before the ideas are usually treated simultaneously in textbooks and some authors such as Romer call both ideas permanent income hypothesis and I recall seeing the reverse as well.


You can't really estimate stock prices this way because it's forward looking (i.e. it considers the future, not so much the past). Example:

Suppose company A lost \$100 million this year, but next year it's going to earn \$100 million. How much is company A worth? If you naively used the P/FCF ratio (price-to-free-cash-flow) you would value company A as worthless today. If everyone agreed with you I'd buy 100% of the company's shares and pocket $100 million next year.

However, this is itself unclear because how can anyone guarantee that the company will earn \$100 million next year? What if, for example, there's another COVID-19 pandemic that hurts the company's performance? Then maybe it will only earn \$50 million. On the other hand, perhaps one of the company's products turn out to be a smash hit, and it earns \$200 million instead. Ultimately, how much the company is worth is going to depend on how much money it makes in the future, and the future is unpredictable. That's why we have a market. Bulls will generally believe that the company will make more money in the future than bears, and therefore value the company differently.

There are other differences, which are entirely dependent on market sentiment. As an example, consider AMZN. Amazon is one of the darlings of the market because the stock price keeps going up even though the company barely makes any money (just look at the P/E and P/FCF). This is happening because Amazon has been fantastic at growing its sales & earnings. There's also the perception that Amazon is working on innovative, cutting-edge things (like Alexa). Therefore the market assigns AMZN a high price. If you based your modelling around free cash flow, you would not have bought AMZN, even though that stock has been extremely lucrative for its holders in the past several years. Of course, this doesn't mean your modelling is wrong; if market sentiment on AMZN shifts its stock price will likely crash by a lot.

In the world of modelling stock prices there is no "right" or "wrong" way. Different people will use different methods or metrics. Further, there is no foolproof way to make money; if there were, then everyone would use it and it would cease to make money.


Is there any academic paper documented about firms' trade credit collection approach?

My hit from Google Scholar (had to go to the bottom of page 1 for this one!) is
Macro-Economic Growth,Enterprise Competitive Status and Trade Credit Policy
Abstract:

This paper investigates the effect of macro- economic growth and enterprise competitive status on trade credit policy from suppliers with trade credit's perspective,and the moderate effect of industry characteristics. The study indicates that: The higher the competitive status,the shorter the trade credit collection period to downstream enterprises,while this relation is not obvious among enterprises in less concentrated industry; When macro- economic grows fast,regardless howconcentrated industries that enterprises are located in,enterprises with higher competitive status are more prone to reduce the account receivable collective period; With further considering industry characteristic,the effect of macro- economic growth and competitive status of enterprises on trade credit policy shows significant difference in the quadrant formed by different industry prosperity and concentration.


What would happen to the economy if the Fed's discount window completely stopped?

First, one note on the premise. The Fed basically operates a giant ledger for financial institutions, each of which has a certain balance in its "master account". Over $3 trillion is moved between accounts each day via Fedwire, and these accounts are also used for settling most traditional checks and ACH transactions, which include many of the everyday transactions in the economy. Even alternative payment networks not provided by the Fed, like CHIPS, ultimately use the Fed's ledger to clear net payments.

When the Fed loans money to banks via the discount window, it simply adds to their account balance on this ledger; many "cyberattacks" that disabled this capability would probably also disable the ledger itself. This would certainly mean chaos, but the missing discount window wouldn't be the most important aspect of this chaos - the real problem would be the breakdown of the economy's payment system.

That said, now suppose that somehow the discount window vanished but the Fed's other services continued. There wouldn't necessarily be any immediate consequences: as of the end of 2014, the Fed was lending very little via its discount window. The H.4.1 "factors affecting reserve balances" release shows that the total of "primary", "secondary", and "seasonal" credit provided to banks was virtually zero relative to the size of the banking system as a whole, at $112 million. This is because the discount rate, currently at 0.75%, is well above other short-term interest rates and has additional strings attached; it only makes sense for banks to borrow from it if they don't have good other options.

The real purpose of the discount window, of course, is to serve as the lender of last resort. And it has been used substantially in that capacity during times of crisis: for instance, in October 2008 there was $100 billion in discount window lending, 1000 times the current level. (Other, newer forms of Fed assistance to the non-bank financial system made the total level of Fed intervention much higher still.) In fact, the acute phase of the 2008 financial crisis for Goldman Sachs and Morgan Stanley, two large investment banks, ended when they became bank holding companies with full access to the Fed's discount window.

In general, if a crisis was already underway, I'm sure that a missing discount window could lead to panic. But without an ongoing crisis, I doubt the ramifications of a temporary discount window shutdown would be too severe: the promise of a restored discount window in a week or two would be enough to stem any speculative run, and banks have many options to deal with short-term funding needs in the interim. They are required, after all, to hold certain levels of liquidity, meaning a combination of reserves and securities that can quickly be sold or lent out for reserves.

By the way, if not only the discount window but also all other interventions into overnight money markets stopped, then in normal times there would also be chaos. Banks' demand for reserves is inelastic in the short-run and subject to shocks; and if the New York Fed was not elastically supplying reserves to keep the federal funds rate at the target, there would be massive volatility in short-term interest rates. (Volatility in the early 80s, when the Fed partly targeted monetary aggregates rather than interest rates - but still engaged in plenty of discretionary interest rate smoothing - gives a small taste of what this volatility would look like. Or we can look at how irregular rates were before the establishment of the Fed.) This would not be true today, however, because the banking system is awash in excess reserves.


You are right, these preferences are separable. A way to see it is to notice that
\begin{equation*}
U(c,n) \geq U(c',n) \Rightarrow U(c,n') \geq U(c',n')
\end{equation*}
for any $n,n'$, and
\begin{equation*}
U(c,n) \geq U(c,n') \Rightarrow U(c',n) \geq U(c',n')
\end{equation*}
for any $c,c'$.
They are even additively separable, as your log-transformation shows.


How can you find all parameter estimates are statistically significant but without performing any sophisticated calculations?

Normally an exam question like that can be solved by simply calculating the $t$-statistics which can be simply done as:
$$t_{\hat{\beta}} = \frac{\hat{\beta}-\beta_0}{se(\hat{\beta})}$$
where $\hat{\beta}$ is the estimate, $\beta_0$ is the assumed value of beta under null (usually 0) and $se(\hat{\beta})$ standard error of the coefficient and then compare it to $5\%$ critical value (for large $n$ and two sided hypothesis approximately $1.96$).
However, you should ask your teacher/supervisor to clarify what is considered 'sophisticated calculation' - some might consider even subtraction and division sophisticated. An approximate rule of thumb that requires even less calculation is to see if the coefficient $\hat{\beta}$ (and assuming null hypothesis assumes $\beta_0=0$) is twice as large as the standard error.


Introduction of an asset tax in the AK model

I think your math is mostly correct but I have to admit that I am not used to AK models.
A short answer for your main question:  is it ok to introduce taxation in the model without including the government?
Answer: Yes, of course. There are overwhelming literature on dynamic or static theoretical modeling of the effect of tax policy without introducing government as an agent.
For a most famous example, see J.E. Stiglitz, Effects of wealth, income, and capital gains taxation on risk taking. Quarterly Journal of Economics, 83 (1969)
In this line of research, people almost never introduce government. You could search for the works by Arrow (1965), Cass (1972), Fishburn, Richter (1960), Tobin, Mossin (1968), Merton (1969), Russell and Smith (1970)
Also you could see the first paper published by Acemoglu.
But, as you might already explored, many recent papers include government as an agent. For example: Farhi, Emmanuel. "Capital taxation and ownership when markets are incomplete." Journal of Political Economy 118.5 (2010): 908-948.


In which country does my money have the largest purchasing power?

Money does not have any 'immanent value'. Value of money will fluctuate across time and also different regions (1USD does not have the same purchasing power in New York as it has in Arizona). This being said there are ways of doing international purchasing power comparisons that go beyond simple market exchange rate that can sometimes be misleading.
Purchasing power parity (PPP) is one of the ways that purchasing power can be compared across different countries. There are also other ways (see discussions in textbooks such Krugman et al (2012). International Economics: Theory and Policy) but PPP is one of the few ways that does not require extensive research.
PPP works in a very similar way to an exchange rate. In fact PPP is sort of an exchange rate built under the assumption that the law of one price holds (Krugman et al (2012). International Economics: Theory and Policy). Data on PPP can found from multiple sources one of them being OECD (however, not all countries are always included in some it is simply too dificult to collect data). Next PPP is by convention reported as $\frac{\text{local currency}}{USD}$. However, we can change this by dividing all individual PPPs for different countries by the Polish PPP as that will allows us to express everything per PLN (e.g. $\frac{\text{local currency}}{USD}/ \frac{\text{local PLN}}{USD} = \frac{\text{local currency}}{PLN}$).
Using this you can collect data on milk prices across all nations for which PPP data is avaiable and then compare the different prices directly. For example, if in China milk costs $20¥$ then you could adjust that with Chinese PPP expressed in terms of PLN (based on OECD data it would be approximately $2.4$)  using  $\frac{20¥}{2.4\frac{\text{¥}}{PLN}} = 8.33PLN$ and then simply compare this cost of milk in China expressed in PLN in terms of PPP to cost of milk in your local store.
I don't have any data on different milk prices across different countries (and I collecting them would be too much effort for SE answer), but just glancing at the PPP statistics it looks like PLN would have highest purchasing power in Indonesia. This being said, to be sure you should actually take the basket of goods you are interested in and make the comparison as described above.


Best resources to self-educate myself in economics

If I'm to give a single reference on learning six years of economics by yourself, it would be

MIT Economics Course

http://ocw.mit.edu/courses/economics/

The MIT has the best graduate economic program, and its undergrad program is definitely in the top 3. The faculty is technical-oriented, which means a lot of math, so you'll be well prepared.

The list of requirements included in a syllabus of each course says what other courses you must complete before approaching this one. So, you'll know your learning path in advance.

Each course includes lecture notes and recommended literature.

I'd also keep in mind that while you save money learning this way, the process is quite difficult (especially due to solving math problems alone) and puts you out of sync with institutional aspects of formal education. Like, it's harder to enroll in a grad program or get an advantage in employment.


You make a mistake in setting up your individual profit functions. Given that the total revenues from sales are $y=K+L$ and they share revenues equally their revenues are $1/2*(K+L)$


Are there any examples of tariffs working?

I interpret your question more broadly as one about whether protectionism has ever "worked". Two economists that think it has are Chang Ha-Joon and Dani Rodrik. You can therefore look up their work.

Two arguments they use are:

The infant industry argument.

In a 2007 article, Chang gives various examples of protectionism "working", including the US in the 19th century:

  the US shifted to protectionism after the Anglo-American War of 1812. By the 1830s, its industrial tariff rate, at 40-50 per cent, was the highest in the world, and remained so until the Second World War.

The other oft-cited examples are East Asian economies like Japan, Korea, and Taiwan in the postwar era.

The second-best argument.

The world is a second-best place and so movements towards seemingly-freer trade may not always be a good thing. Classic example as told by Rodrik (2015):

  imagine that beef is supplied by the United States to Germany at a price of \$100. Assume that Germany imposes a tariff of 20 percent, raising the consumer price of US beef in the German market to \$120. France, meanwhile, can supply beef of equivalent quality only at a price of \$119. Prior to the preferential agreement between France and Germany, French suppliers, facing the same tariff rate as US producers, were outcompeted. Now consider what happens when Germany eliminates its tariffs on imports from France but keeps in place those on the United States. French-supplied beef suddenly becomes cheaper in Germany (\$119 versus \$120), and imports from the United States collapse. German consumers are better off by \$1, but the German government forfeits \$20 of tariff revenue previously collected on US beef (which could have been handed back to consumers or used to reduce other taxes in Germany). On balance, Germany gets a raw deal.

Two footnotes.

Evaluating whether something has ever "worked" is fiendishly difficult in economics.

First we have to decide what the goals of the policies were and what would count as "working". Then we have to do the empirical analysis and evaluate quantitatively the degree to which the policies "worked".

So while Chang and Rodrik might give examples of protectionism "working" (19th-century US, Japan, Korea, Taiwan), other economists might disagree that protectionism really "worked" in these cases. Indeed, I am not aware that Chang or Rodrik has actually conducted any empirical analyses on this matter.

Chang and Rodrik merely try to argue against the extreme position that free trade is always a good thing for everybody and in favor of the modest position that protectionism has sometimes "worked".

Most (all?) economists, including Chang and Rodrik, would agree with the following statements:

Free trade is generally a good thing for most people.
Tariffs are generally a bad thing for most people.

In contrast, a significant portion of the general public would disagree.


Why are firms taken to be profit-maximizing? Shouldn't that make them risk-neutral?

Why would a [risk-neutral] firm need to diversify if all it wanted to do was maximise profit?

Suppose there is a risk-neutral firm that has two strategies it could follow: risky and safe. The safe strategy (e.g. holding a diversified portfolio) yields a profit of $1$ every year forever. The risky strategy (e.g. not diversifying) yields a payoff of $2$ with (independent) probability $x$ each year. With probability $1-x$ a risk-taking firm goes bankrupt and earns no more money ever again. The firm discounts the future at rate $\delta$.

The lifetime profit of the safe strategy is
$$\sum_{t=0}^\infty\delta^t=\frac{1}{1-\delta}.$$
The lifetime profit from the risky strategy
$$\sum_{t=0}^\infty 2x^t\delta^t=\frac{2}{1-x\delta}.$$
The safe strategy yields a higher lifetime profit if
$$x<\frac{2\delta-1}{\delta}\iff \delta>\frac{1}{2-x}.$$

In words, even though

the firm is risk-neutral and
the one-shot profit from the risky option (e.g. not diversifying) is higher,

the long-run profit from the safe option (e.g. diversification) can be greater over the lifetime of the firm.

In this example, the safe option maximises lifetime profit when (a) the probability of risky option leading to bankruptcy (and hence the loss of all future profit) is high or (b) the firm values the future highly relative to the present.


How do I choose an appropriate Instrument to solve the endogeneity issue?

Generally, following Verbeek's guide to modern econometrics there are three main attributes a good instrument should have:

It should be relevant. The instrument should be able to explain your the variable being instrumented and through it also the explanatory variable. For example, in two stages least squares (TSLS) which is a model from the instrumental variable (IV) family the estimator converges to  $\beta_2 = \frac{cov(z_i, y_i)}{
cov(z_i, x_i)} $ where $y$ is the dependent variable, $x$ the independent variable that is endogenous and $z$ the instrument. If there is no covariance between $z_i$ and $x_i$ then the estimator is simply not defined.

Instrument should not be 'weak'. Generally it is not  enough that simply $cov(z,x)\neq 0$, instrument also has to be significant and generally the first stage of regression has to be significant itself. This can be tested by looking at $F$-statistics. A rule of thumb  for TSLS is that instrument is weak if the first stage $F$-statistics is less than $10$ (see mostly harmless econometrics by Angrist and Pischke).

Instrument should satisfy exclusion assumption. It should have effect on $y$ only through $x$ not in other ways. The instrument $z$ also has to be independent.

Moreover, more on a practical note if you are using a quadratic model the program will ask you also to provide an instrument for both the variable itself and  the quadratic term. As mentioned by Michael in his +1 comment you should check if the above assumptions still hold for the quadratic term as well.


What might happen if the Fed stopped paying interest on excess reserves?

If the Fed paid paid 0% on excess reserves, banks holding excess reserves would immediately try to buy Treasury bills that paid more than the 0% paid by excess reserves. This would drive Treasury bill (and other short-term risk-free rates) to 0%.

The Fed pays interest on excess reserves to allow short-term market rates to be near their target (which is now above 0%).


Does "I" in Y = C + I + G + X - IM refer to the domestic investment?

I see your confusion that could also apply to $C$ and $G$ because Mankiw starts from
\begin{equation} Y= C^d+I^d+G^d+X~~~~~  (1) \end{equation}

$C^d$, $I^d$, and $G^d$ have been subjected to the same ``bit of manipulation'', which is to add and substract the foreign part such that we get the same equality (1).

So the idea here is just that $I$, after the manipulation, is essentially equal to $I^d$, the domestic investment, because $I^f$, the foreign portion, has been substracted from $I$ and appears now in import expenditures ($IM$). In other words $I$, after manipulation, is reinterpreted as domestic investment.

\begin{equation} Y= C + I +G +X -IM ~~~~~  (2) \end{equation}

I think it safe to say that calling $I$ domestic investment is here an "abuse of language"!

The most important is that identities (1) or (2) are consistent with the way we calculate GDP using total value added.


Revealed preference when there is no substitution effect

The short answer is no. Reveal preference only determines which combination of goods is preferred, it does not affect in any way the allocation of goods, which has more to do with the functional form of the utility function, the types of goods and the types of changes made.

For example, the 2 budget lines represent different budget sets when there is a price change in either good. The weak axiom of revealed preference (WARP) posits that $x^i$ is preferred to $x^j$ insofar as good 1 is revealed preferred to good 2. It does not give any information on the allocation of the consumption bundle, however. If there is a simultaneous change in both the price of good 1 and 2 so that there exists another budget line that is parallel with $p^i$, the allocation of good in this case has more to do with the optimization of utility.
I have attempted to derive the change here but I am not entirely sure if this is the correct way to do it or whether I have done all the steps correctly so take it with a pinch of salt.
Assuming the consumer is maximizing his/her utility function, which is standard in consumer theory, we want to find the highest point tangent to the budget line. The slope of the indifference curve, after the price adjustment, depends on the types of adjustment made, the type of goods, and the functional form of the utility function. As I understand it, you are asking about the case of relative price change (since the slope of the budget line remains the same). In this case, the total change in each type of goods can be derived from the Slutsky equation
\begin{align}
{d x_A\over dp} &= {\partial x_A\over \partial p_A} + {\partial x_A \over \partial p_B}\\
 &= \underbrace{\partial h_A\over \partial p_A}_{\text{substitution effect good A (?)}}-\underbrace{{\partial x_A\over \partial m}x_A}_{\text{income effect good A (?)}} + \underbrace{\partial h_A\over \partial p_B}_{\text{substitution effect good B (?)}}-\underbrace{{\partial x_A\over \partial m}x_B}_{\text{income effect good B (?)}}\\
\end{align}
where $x_i,h_i,p_i$ and $m$ respectively denote the Marshallian demand, Hicksian demand, the price of good $i$, and the total budget. The ?'s in the parentheses are the sign of the term. As in the case here where the types of goods are ambiguous, I can say little about the direction of change.
In the case of normal good (both are normal goods) and standard Cobb-Douglas utility, the substitution effect of own good (A to A) is negative, the income effect of own good (A to A) is negative, the substitution effect of the other good (A to B) is positive while the income effect of the other good (A to B) is negative. The substitution effect of own good (A to A) and of the other good (A to B) cancel out each other, leaving only the negative terms. This generally translates into a lower Marshallian demand for good A, and similarly for good B. The total change is then, ${dx_A\over dp}=-{\partial x_A\over \partial m}(x_A+x_B)$. In levels,
$$\Delta x_A=-{\partial x_A\over \partial m}(x_A\Delta p_A+x_B\Delta p_B)\\
\Delta x_B=-{\partial x_B\over \partial m}(x_A\Delta p_A+x_B\Delta p_B)
$$
Taking the ratio,
$${x_A-\Delta x_A\over x_B-\Delta x_B}={x_A+{\partial x_A\over \partial m}(x_A\Delta p_A+x_B\Delta p_B)\over x_B+{\partial x_B\over \partial m}(x_A\Delta p_A+x_B\Delta p_B)}$$
This will remain unchanged if and only if $\Delta x_A/\Delta x_B=x_A/x_B$, which simplifies to ${\partial x_A\over \partial m}/{\partial x_B\over \partial m}={x_A\over x_B}$. This holds true when with the CES form of Cobb-Douglas utility wherein the coefficient of good A and B are equal to 0.5.


Can absence of inflation be accounted for by a reconfigured class structure?

Quantitative easing without high inflation can be explained by low velocity of money.

Think about a property tycoon who sees times are hard and instead of investing all his money in a development project, he buys gold which he keeps in a vault. The property project would have provided employment, and the wages would be used by workers to buy other things in the economy (high velocity). The gold in his vault just sits there (low velocity).

Economic troubles result in low confidence and low velocity of money. Central banks pumping money into the economy somewhat offsets this. As long as the overall velocity of money matches the productive capacity of the economy, there won't be high inflation. One concern of course is that central banks cannot control the velocity of money, so if all this new money gets velocity we could get high inflation that central banks cannot control.

Now think about where you mentioned soaring prices. Many of these are places where rich people can store wealth at low velocity: real estate, fine art, gold, etc. So yes, the effects very much are connected.


Why is CRRA utility often used in macroeconomics DSGE model?

Models at Dynamic Stochastic General Equilibrium level must be able to replicate real economies to an acceptable degree. One of the features of real economies has been a relatively stable growth rate (see also this post), $\dot x/x=\gamma$, where the dot above a variable denots the derivative with respect to time.

So one would want a model that admits a constant growth rate at its steady-state. In the benchmark deterministic/continuous time "representative household" model, the Euler equation  takes the form

$$r = \rho - \left(\frac {u''(c)\cdot c}{u'(c)}\right)\cdot \frac {\dot c}{c}$$

This is the optimal rule for the growth rate of consumption. The rate of pure time preference $\rho$ is assumed constant. The interest rate $r$ has its own way to become constant at the steady state. So in order to obtain a constant consumption growth rate at the steady state, we want the term

$$\left(\frac {u''(c)\cdot c}{u'(c)}\right)$$
to be constant too. The Constant Relative Risk Aversion (CRRA) utility function satisfies exactly this requirement:

$$u(c) = \frac {c^{1-\sigma}}{1-\sigma} \Rightarrow u'(c) = c^{-\sigma} \Rightarrow u''(c) = -\sigma c^{-\sigma-1}$$

So

$$\frac {u''(c)\cdot c}{u'(c)} = \frac {-\sigma c^{-\sigma-1} \cdot c}{c^{-\sigma}} = -\sigma $$
and the Euler equation becomes

$$\frac {\dot c}{c} = (1/\sigma)\cdot (r-\rho)$$

Barro & Sala-i-Martin (2004, 2n ed.), extend the required form of the utility function when there is also leisure-labor choice (ch. 9 pp 427-428).
These fundamental property extends to the case of stochastic/discrete time.

XXXX

To compare, if we have specified a Constant Absolute Risk Aversion (CARA) form, we would have

$$u(c) = -\alpha^{-1}e^{-\alpha c} \Rightarrow u'(c) = e^{-\alpha c}\Rightarrow u''(c) = -\alpha e^{-\alpha c}$$ and the Euler equation would become

$$\dot c = (1/\alpha)\cdot (r-\rho)$$

i.e.here we would obtain a constant steady-state growth in the level of consumption (and so a diminishing growth rate).


I can shed some light on the question, but am not sure I can answer it as I am not sure it is really even well defined.

(1) The Weak Axiom of Revealed Preference is a decision theoretic concept regarding the choices of a single agent. So I do not understand how having $N$ agents is relevant to the problem.

(2) Generally speaking, if $U: X \to \mathbb{R}$ is a utility function and $\mathscr{C}$ is a choice correspondence over $X$ such that $\mathscr{C}(A) = \{x \in A \mid U(x) \geq U(y), \ \forall y \in A\}$ then $\mathscr{C}$ will satisfy WARP. This is a straightforward exercise in using the definitions, and the dimensionality of the space should play no role (its true over an abstract $X$).

(3) If the consumption space is $\mathbb{R}^l$ then preferences over defined $\mathbb{R}^l$. How you project preferences into $\mathbb{R}^2$ matters.

(4) If I interpret your question a la denesp, so that you ask: fix the $2 < k \leq l$ dimensions of consumption and only let dimensions 1 and 2 vary (of course, what we fix the other dimensions of still may make a difference), and assume that this restricted choice correspondence satisfied WARP will the choice correspondence in general satisfy WARP.

To answer (4): if the choices comes from a utility function then yes, trivially (see point (2)). If the choices are more general, then no. This fails severely. Take as a counter example a preference over $\mathbb{R}^3$ such that fixing the 3rd dimension, the consumer is indifferent over all elements (i.e., the first and second dimensions are null). WARP holds as $\mathscr{C}^{x}(A) = A$ for all $A \in \mathbb{R}^2 \times \{x\}$. This leaves the 3rd dimension wholly unrestricted. Letting the choice function over this last dimension be cyclic (i.e., fail WARP) and we see that the original choice would as well.

(5) What if we instead interpret your question as: we see every two dimensional projection of choice. (That is for every $i,j \leq l$, $i \neq j$, we see the projection of choices over the dimensions $i$ and $j$ fixing the other dimensions arbitrarily. Well, if the way we fix the other dimensions matters to the restricted choices (e.g., $\mathscr{C}^{x}$ over $\mathbb{R}^2 \times \{x\}$ is not the same as $\mathscr{C}^{y}$ over $\mathbb{R}^2 \times \{y\}$). Then we are back to the same type of problem as (5)---$\mathscr{C}$ could be cyclic when the dimensions are fixed differently.

What if how we fix the dimensions doesn't affect the restricted choice (so we have separable preferences, a la Koopmans). Then WARP will hold over $\mathscr{C}$ if it is rationalized by a preference relation. But, the result is not very interesting, since from the restricted choices know the choice of out of $\{x,y\}$ for all $x,y \in \mathbb{R}^l$. It is well known that only a binary choice is necessary to verify a rationalization. Without this additional constraint, I believe you could still cook up a counter example in the spirit of (4) where cyclically kicks in when certain elements (at least 3, pairwise distinct over 3 different dimensions) are present.


Resolution - Ramsey growth model with per capita variables

The reason why the $e^{nt}$ term is there is because you want to multiply the whole utility by the number of people. You are actually not substituting consumption into the utility but multiplying the utility of an individual by the number of individuals. So the problem would look like:
$$\int_{0}^{\infty}\frac{c^{1-\sigma}}{1-\sigma}e^{nt} e^{-\rho t}dt$$
At least this is how it is explained by Barro & Sala-i-Martin in their textbook Economic Growth 2nd ed. pp 87 where they say [emphasis mine]:

The multiplication of $u(c)$ in equation ...  by family size, $L = e^{nt}$, represents the adding up of utils for all family members alive at time $t$.

So the way how I understand it you are multiplying all the individual utilities by the population.
In fact I am quite sure of it since in Romer Advanced Economics 4th ed in chapter 2 pp 54 there is Ramsey model where the objective function is given as:
$$\int_{t=0}^{\infty} e^{-\rho t}\frac{c^{1-\sigma}}{1-\sigma} \frac{L(t)}{H} dt$$
which is essentially exactly the case you are looking for where the only difference is that Romer also adds the $H$ parameter which is the number of households so $L(t)/H$ would be population per household. Romer also later substitutes $L(t)=L(0)e^{nt}$ which for $L(0)$ normalized to 1 gives almost the same function (with that extra $H$ parameter - but for $H=1$ it would give you exactly the same result).


The (partial) derivative of a continuous function is defined as
\begin{align}
\frac{\partial F(\overline K, L)}{\partial L} := \lim_{\Delta L \to 0}\frac{F(\overline K, L + \Delta L) - F(\overline K, L)}{\Delta L}.
\end{align}
Now if $L \in \mathbb{N}$, then you have a lower bound for the increment $\mathbb{N} \ni \Delta L \geq 1$. Otherwise the definition above is not well defined for $\Delta L = 0$.
Such that we finally arrive at the approximation
 \begin{align}
\frac{\partial F(\overline K, L)}{\partial L} \approx \frac{F(\overline K, L + 1) - F(\overline K, L)}{1}.
\end{align}


Is "$f(k,l)$ is decreasing return to scale $\Leftrightarrow f_{ll}f_{kk}-f_{kl}^2>0$" always true?

In general the statement is wrong. Here is a counterexample:

Suppose you have $f(k,l) = -k l^\beta$ with $\beta >0$ and $(k,l)\in\mathbb{R}^2_{++}$ (you can interpret $f$ as a production function for a "bad" commodity). Then you have:
$$f(tk,tl) = - t^{1+\beta} k l^\beta = t^{1+\beta} f(k,l) < t f(k,l)$$
so that $f(k,l)$ is decreasing returns to scale.

Now let's evaluate $ f_{ll}f_{kk} - f^2_{kl} $. Since $f$ is linear in $k$ we have that $f_{kk}$ is zero, so it is $ f_{ll}f_{kk}$.  On the other hand, we have $f_{kl} = -\beta l^{\beta-1}$ so that overall we have:
\begin{equation}
f_{ll}f_{kk} - f^2_{kl} = 0 - \left[ -\beta l^{\beta-1} \right]^2 = - \beta^2 l^{2(\beta -1)} <0
\end{equation}

So then, where is the trick?

The fact is that $ f_{ll}f_{kk} - f^2_{kl} > 0$ is not a sufficient condition to conclude that the hessian of $f(k,l)$ is negative semi-definite and so $f$ is concave because you also have to impose restrictions on the sign of the principal minor of order 1 of the hessian matrix.

However, provided the hessian matrix is indeed negative semi-defintie than you conclude $f$ is concave and decreasing returns to scale will follow, the statement then holds.


lowering minimum wage by half, effect on population of Puerto Rico

Typically, what happens when a minimum wage is relaxed is that a plethora of jobs that were illegal under the previous doctrine become available on the market for workers.

The common metaphor used by its supporters for describing the minimum wage is that a minimum wage is a "price floor" under which wages are not permitted to fall, ostensibly to protect workers.

The accurate metaphor for describing the minimum wage is a price hurdle which each person's productivity must clear in order to sell your product (i.e. your labor)  If your productivity is not valued by a buyer at a higher rate than the strictures placed upon it by legislation, it is illegal for you to have that job.

All other things held equal, reducing the minimum wage will create options for suppliers of cheap labor to actually have their product clear the market rather than having it denied by the State.


why doesn't every country scale up the prices to the international level?

For simplicity suppose the local currency exchange rate is 1 LOC = 1 USD.

Suppose that the central bank simply print out 5 times the paper money in circulation, creating inflation. Pizza would then cost 15 LOC instead of 3 LOC. But then if the exchange rate regime is floating (majority of countries now use this), then the inflated money would be worth less, causing the exchange rate to fall to 5 LOC = 1 USD. That is, you need less USDs to exchange for 1 LOC. In essence, tourists would be paying the same (3 USD, which is now equal to 15 LOC).

So the next question is, why don't the central bank make their local currency appreciate (make 1 LOC = 5 USD, so that the pizza that cost 3 LOC would now be equal to 15 USD.)

Well, the demand and supply for money is what determines the exchange rate. If you want to know more about this I'd suggest you read more on exchange rate intervention here.


How to prove the WPBE is SE in signaling game?

Claim: If choice sets $T, M,$ and $A$ are finite, then an assessment $\{\beta^*_{r}, \beta^*_{s}, \mu^*\}$ is a WPBE (weak perfect Bayesian equilibrium) of the two-stage signalling game between receiver $r$ and sender $s$ if and only if it is a SE (sequential equilibrium).

Proof: SE $\implies$ WPBE is trivial since SEs are PBEs by construction, and thus are also WPBEs.

To prove that a WPBE is also a SE for this signalling game over finite choice sets $T, M,$ and $A$, we must demonstrate that there exists a tuple $(\beta^n_r, \beta^n_s, \mu^n)$ such that $\{\beta^*_{r}, \beta^*_{s}, \mu^*\} = \lim\limits_{n \to \infty} (\beta^n_r, \beta^n_s, \mu^n)$ for some totally mixed strategy $\beta^n$ such that, for all $n$,

$$
\begin{align}
\mu^n(t|m) = \frac{\pi(t)\beta^n_s(m|t)}{\sum_{t' \in T}\pi(t')\beta^n_s(m|t')} &&\text{whenever} \sum_{t' \in T} \pi(t')\beta^n_{s}(m|t') > 0
\end{align}
$$

since this is a two-player game between players $r$ and $s$. The condition on $\mu^n$ above is known as weak (Bayesian) consistency.

To construct such a totally mixed strategy profile, $\beta^n$, consider first the following message-sending strategy profile for player $s$ as a function of $n$ conditional on the state of the world $t$:

$$
\beta^n_{s}(m|t) =
\small{
\begin{cases}
\frac{n-1}{n} \left(\frac{n-1}{n}\right) \beta^*_s(m|t) && \text{if } \beta^*_s(m|t) > 0 \text{ and } \sum_{t' \in T} \pi(t')\beta^*_{s}(m|t') > 0 \\
\frac{n-1}{n} \left(\frac{1}{n \mathcal{N_s}(t)}\right) && \text{if } \beta^*_s(m|t) = 0 \text{ and } \sum_{t' \in T} \pi(t')\beta^*_{s}(m|t') > 0 \\
\frac{n-1}{n} \left(\frac{\pi(m) \mu^*(t|m)}{n\pi(t)} \right) && \text{if } \mu^*(t|m) > 0 \text{ and } \sum_{t' \in T} \pi(t')\beta^*_{s}(m|t') = 0 \\
\frac{1}{n} \left[1 - \sum\limits_{m' \in M} \left(\frac{n-1}{n} \frac{\pi(m') \mu^*(t|m')}{\pi(t)}\right) \right]&& \text{otherwise}
\end{cases}
}
$$

where, borrowing notation from Fudenberg and Tirole (1991),

$$\mathcal{N_s}(t) \equiv \#\{m \in M\ |\ \beta^*_s(m|t) = 0 \text{ and } \sum_{t' \in T} \pi(t')\beta^*_{s}(m|t') > 0 \}.$$

The finiteness of $M$ is invoked here so as to provide a well-defined domain over $\#\{\cdot\}$. These four cases span the parameter-space of $\beta_s^*$ and $\mu^*$ in this game. Thus $\sum_{m \in M} \beta^n_{s}(m|t) = 1$ and $\beta^n_{s}(m|t) > 0$ for all $n$ and for all $m \in M$, establishing that $\beta^n_s$ is a totally mixed strategy profile for player $s$. By construction of the asymptotic behavior of $\beta^n_{s}(m|t)$, it is easy to verify that $\beta^n_{s} \to \beta^*_s$ as $n \to \infty$.

The residual strategy profile for player $r$, $\beta_r^n(a|m)$, can be simply constructed by assigning probabilities only across the parameter-space of $\beta^*_r$ since the belief $\mu^*$ is just-identified by $\beta^*_s$:

$$
\beta^n_r(a|m) =
\begin{cases}
\left(1 - \frac{1}{n}\right)\beta^*_r(a|m) && \text{if } \beta^*_r(a|m) > 0 \\
\frac{1}{n\mathcal{N_r}(m)} && \text{if } \beta^*_r(a|m) = 0,
\end{cases}
$$
where $\mathcal{N_r}(m) \equiv \#\{a \in A\ |\ \beta^*_r(a|m) = 0\}$. Clearly $\beta^n_r \to \beta^*_r$ as $n \to \infty$.

Lastly, by construction of $\beta^n_s(m|t)$, it is straightforward to verify that the sequence
$$
\begin{gather}
\mu^n(t|m) = \frac{\pi(t)\beta^n_s(m|t)}{\sum_{t' \in T}\pi(t')\beta^n_s(m|t')} \to \mu^*(t|m) \text{ as } n \to \infty
\end{gather}
$$

and thus consistency is demonstrated. Hence the WPBE is also a SE for this game.$\ \blacksquare$


Labour Saving vs. Labour Augmenting - what is the difference?

Consider a Leontief production function:

$$ Y = \text{min}(aL,bK) $$

Optimal capital-labour ratio is:

$$ \frac{K^*}{L^*}=\frac{a}{b} $$

Labour-saving technical change is such that $a$ increases. The optimal capital-labour ratio increases, this is, firms use less labour per unit of capital. This is also called Capital-augmenting technical change (see reference [2]).

Labour-augmenting technical change is such that $b$ increases. The optimal capital-labour ratio falls, this is, firms use more labour per unit of capital.

Notice that how output per worker changes depend on how $Y$ changes. For example, assume that the above adjustment is such that $Y$ is unchanged after the technical change. Then, output per worker increases in the labour-saving case but falls in the labour-augmenting change. A complete analysis would require a general equilibrium approach, where factor prices are endogenous. All I am showing here is that it is not necessarily the case that either type of technical change increases output per worker.

Some references: [1], [2].

PS: I use the Leontief for simplicity. In a Cobb-Douglas, any technical change is Hicks-neutral, as it does not affect the K/L ratio (assuming fixed factor prices). In a CES, the nature of technical change depends on the elasticity of substitution.


Just think your true model is:
$$y_t = \phi_0 + \phi_1 y_{t-1} + \phi_2 y_{t-2} + u_t \tag{1}$$

So, under your true model, $u_t$ is uncorrelated with all explanatory variables.

So, instead of using AR(2), use selected AR(1) model as:
$$y_t = \phi_0 + \phi_1 y_{t-1} + \epsilon_t \tag{2}$$
So, now, $\epsilon_t = \phi_2 y_{t-2} + u_t$; therefore, $y_{t-1}$ will no longer will be uncorrelated with $\epsilon_t$ (due to correlation between $y_{t-1}$ & $y_{t-2}$). In simple words, this lead to the problem of endogeneity. And we know, OLS estimators are not consistent when residuals are correlated with any of the explanatory variables.


In the US, a fail penalty is applied on the failure to deliver securities in a US Treasury or Agency transaction.  This applies to any trade in these securities , including a simple cash trade , or the opening leg of a repo, or the closing leg of a repo.
The intent of the system , first introduced in 2012, was to clean up fails in the Marketplace by providing an incentive to settle.  The convention in the marketplace is that if there is a settlement failure, then further daily attempts to settle are made until the trade clears.  When this happens , the cash amount instructed doesn’t change over time.  Therefore, a counterparty that fails to deliver securities is effectively making a decision to lend money at zero interest.  But in the absence of a fails penalty, this may be logical if interest rates are close to zero and/or the securities are scarce.
In addition, In the absence of a fails penalty, repo cannot trade below zero.  Why?  There’s no point in lending money via repo at a negative rate in order to borrow securities that you have sold short.  It is better economically just to fail on the cash trade.


How to Calculate Price Elasticity of Demand When Perfectly Elastic?

$$\epsilon_D=\frac{\% \Delta Q_D}{\% \Delta P}$$
A perfectly elastic demand curve is horizontal, meaning we quantity demanded can change by any amount without changing price (any quantity can be sold at the price corresponding the vertical intercept). Hence, for any change in quantity, $\% \Delta P=0$. While we cannot strictly divide by zero, in the limit, the ratio is infinity.