I don't understand how a graph would look like for this situation

I suggest that the graph could be improved in two ways (apart from the point, which you seem to recognise, that the upward-sloping line labelled MSB should be labelled MPC (marginal private cost)).
Firstly, the inclusion of the vertical arrow labelled subsidy rather suggests that the vertical distance between the two marginal cost curves equals the amount of the subsidy (per unit of output).  This is unlikely to be the case: there is no particular reason why the cost of the gadgets should equal the social value of their effect in reducing wastage of water.  Instead you might add an explanation that the result of the subsidy, via its effect in reducing wastage, is to reduce the MSC of the agricultural production, shifting the MSC line down and to the right.
Secondly, there is nothing in the graph to represent the effect of the gadgets in improving yield.  It might be assumed that an improved yield implies a reduction in marginal private costs (although that is not necessarily the case, eg the harvesting of the extra output might disproportionately raise labour costs).  If that is a reasonable assumption in the context, you might draw a second MPC line below and to the right of the first (which would also represent MSC) which might be labelled (MPC with gadgets = MSC).


This sounds like the standard "drop one dummy" requirement when binary characteristics are included in linear regression - because otherwise, we will get perfect mutlicollinearity and no-solution.

Assume you have three subgroups, separated by age: Y(oung), A(dult), O(ld). You have reasons to believe that the effect of the treatment correlates with age-group, and you want to control for this association. If you include three dummies in the regression to that effect, you will create perfect mutlicollinearity - because if you sum these three columns of the regresor matrix row-by-row, you will get a series of Ones, which is already present in the matrix, since there exists already a constant term and so a regressor that consists of a series of Ones.

In such cases, we exclude from the regressor matrix any one of the three control dummies (for many reasons it is not a good idea to drop the constant term instead of one of the dummies).
This has the effect that the interpretation of the results becomes conditional on the group whose dummy we excluded: if say we specify

$$y=\beta_0+\beta_1w + \gamma_1A + \gamma_2O+e$$

then $\gamma_1$ measures how much more (or less, if negative) being Adult affects the outcome, compared to being Young, and $\gamma_2$ analogously for Old.

Situations of perfect (or near) multicollinearity may nevertheless occur even in this approach, depending also on the nature and distribution of values of the dependent variable. If the dependent variable is also binary, such possibilities increase, since we may have phenomena of "complete separation/perfect prediction" and the like.

In general, including controls in treatment regressions is a standard approach, and very reasonable. Including dummies as controls, may have the above mentioned complications.


Can Linear Supply-Demand Equilibria Be Understood as a Feedback-Control Process?

I think there might be an issue with this as currently constructed:

$\delta(p)= \infty$, $\forall P \in \mathbb{R_+}$ and $P_e = \infty =\dot{p},$ $\forall P \in \mathbb{R_+}$ This is because $\delta_0 = \infty $. Further, $\sigma(p) = S_\sigma p+\sigma_0= S_\sigma p$ since $ \sigma_0 = 0$.

In words - there should be infinite demand for any normal good whenever P is 0 and there should be zero supply of any good whenever P is 0.

That said, your intuition here is solid but I do think you're working on reinventing the tatonement wheel. Tatonement, in a nutshell, adjusts prices up when there is excess demand and prices down when there is excess supply.

If you are able to do this after watching a few lectures online, then I say keep going. You could make meaningful contributions once you have a better understanding of the field.

edit: If you'd like to message me privately, I would be more than happy to pass along my graduate school notes/materials to help guide your studies. At minimum, I recommend you followup your current study with some more advanced micro theory courses.


Selection bias - Will multiple regressions solve the problem?

What you are referring to is two-stage least squares. This is an instrumental variable commonly applied to correct for endogeniety and selection bias. It is a pretty hot topic in economics at the moment and, when applied correctly, can be very useful and will remove the selection bias.

There are a few conditions and assumptions -
Suppose you want to estimate this equation

$y_i = \beta x_i +\epsilon_i$
where

$i$ indexes observations,
$y_i$ is the dependent variable,
$x_i$ is an independent variable,
$\epsilon_i$ is an unobserved error term representing all causes of $y_i$ other than $x_i$
$\beta$ is an unobserved scalar parameter.

we suspect $x_i$ is endogenous

($cov(x_i, \epsilon)\neq 0$)

but we have a variable $z_i$ which is correlated to $x_i$ but uncorrelated with $\epsilon$

($cov(z_i, x_i) \neq 0$ and $cov(z_i, \epsilon)=0$).

This is a candidate for 2SLS:

Stage one is to estimate $x_i$:

$\hat x_i= \hat \gamma z_i + \epsilon$

Stage 2 is to estimate $y_i$

$y_i= \beta \hat x_i+ \epsilon$

The difference between this and the first regression is that $\hat x_i$ is uncorrelated with the error term and this selection bias has been corrected for. There are lot's of different uses of this. In principle, the two-stage regression used in the paper could correct for some of the biases discussed in the paper.

Further Reading:

This ppt (Slide 20 onwards) descibes an excellent (and very famous) example of IV.

These notes are also pretty solid as well.

This question provides some textbooks which are also very helpful. My personal favourite is "Mostly Harmless Econometrics" by Angrist and Pischkes (Written by the guy who did that famous example example above) and "Econometric Analysis of Cross Section and Panel Data" by Wooldridge. It is worth noting that both of these are graduate level books.


How can I re-base monthly financial time series

I have a monthly time series with 2015=100. I want it to be in
  2010=100. How could I do so?

Unless I am misunderstanding your question, you need to multiply every entry by $$100/X_{2010}$$


What portion of our annual deficit is financed by bonds?

Unfortunately, there doesn't appear to be a single site that clearly identifies the data you seek. However, that information does appear in a couple of different sources, (in a couple of different formats):

This source from the Treasury gives a list of the values of all different types of securities auctioned off at every recent treasury auction in Excel spreadsheets

This source gives the relative holdings for domestic and foreign agents, however it doesn't break down the data into as clear specifics for tyeps of debt held.

Finally, these presentations give overviews (in the "Financing" section of each presentation) of the total value of each type of bond that was issued, and the corresponding value of each typoe that matured.

Hope that helps a bit!


So does this imply that there is an incentive for the selfish players
  to keep PWYW firm in market ?

No, unfortunately not. In general equilibrium models (as @FooBar already mentioned) buyer's do not think about the influence of their actions on the seller. Therefore, they would just buy at the lowest possible price.

However, if you have a game theoretic model including uncertainty about profits of the PWYW firm and risk-averse consumers who prefer the PWYW firm for some reason you might get Nash-equilibria in which PWYW firms have positive profits.

Another way to model PWYW firms is to include non-monetary benefits from paying. For example a disutility for being seen as a cheapskate or for exploiting the firm or a positive utility from "doing the right thing" (i.e., paying the price that they think is appropriate). The selfish, rational players will pay a positive price and will generate profits for the firm.

I think in reality the impact of non-monetary benefits is much higher than the impact of asymmetric information with risk-aversion.

  Also, how do I extend this idea to more than 2 firms ?

I don't really think it is really interesting to extend this to more than two firms (especially without having solved this for 2 firms), but for both modeling options presented it should be "easy enough".


What are the consequences of having negative GDP growth?

Broadly speaking GDP is measure of output economy produces and from economic perspective output is equal to income. Assuming we are talking about real growth not just nominal one a main consequence of negative economic growth is that on average people's real incomes will be lower and their material standards of living worse.
Under spending approach to GDP (which is equivalent to income approach), GDP in closed economy is given as:
$$Y = C+I+G,$$
where $Y$ is the output/income, $C$ is consumer spending, $I$ is investment and $G$ is government spending.
So another way of thinking about consequences of negative growth is that negative growth will make left-hand side of this equality ($Y$) lower. If $Y$ declines then economy will have to either:

consume less - which directly affects well being of consumers.
invest less - which affects the rate at which economy can grow.
government will have to spend less - which means that government will have to cut provision of some of the public goods or other spending it does.

In either case it will lead to some worse material standard of living for the particular country.
Also the above are only the most direct effects. I did not touch upon any potential distributional questions. Economic growth empirically has a first order impact especially on the welfare of poor (see  Barro & Sala-i-Martin Economic Growth and sources cited therein). There are also further implications, it is not much of an overstatement that almost everything in economy depends on what the economy can produce, but exploring this to more full extent would be beyond the scope of stack exchange post.
As a further reading I recommend any macro textbook or even just introductory economics textbook (my concrete recommendations would be either Blanchard et al. Macroeconomics a European Perspective or Mankiw Principles of Economics), any of them will explore significance of output to economy from many different sides.


Financial Crisis text book recommendation

One of my favorites in this dimension is "This Time Is Different" by Reinhart and Rogoff. Both are respected scholars. The title already suggests what they aim at.

Covering sixty-six countries across five continents, This Time Is Different presents a comprehensive look at the varieties of financial crises, and guides us through eight astonishing centuries of government defaults, banking panics, and inflationary spikes—from medieval currency debasements to today’s subprime catastrophe.


Price elasticity and optimal pricing in a monopoly with zero marginal cost

Hint

You are probably working with (or given) a linear demand function of the form $Q=\frac{a}{b}-\frac1bP$, or its equivalent inverse form $P=a-bQ$, where $a$ and $b$ are positive numbers.

Given that price elasticity of demand at the optimum is $-1$, which you were right to point out (assuming linear demand), you can use the elasticity formula
\begin{equation}
-1=\frac{\mathrm dQ}{\mathrm dP}\cdot\frac{P}{Q}=-\frac1b\cdot\frac{P}{Q}
\end{equation}
and the demand equation to solve for the two unknowns $P$ and $Q$.


Are there economic journals which are keen to publish puzzling empirical observations?

Was he right? No.

Has the situation changed since 2002? Yes, there are lots more journals, and much higher diversity of publishing. Sadly, that also means there are also a lot more venues that publish at the bottom of the quality scale: one even shares a name with you.

Are there decent economic journals which are keen to publish puzzling empirical observations? No, generally they'll look for something well-written, novel, intelligent, and within the journal's scope. Just being a puzzling empirical observation simply isn't enough for any decent venue. You'll want a blog for that.

As for a slice of historical publishing of puzzling empirical observations, why not try a Google Scholar search for "productivity puzzle"?


Current knowledge about the empirics of consumer theory

The primary literature concerned with this type of question (at least where classical results break down) is behavioral economics. There's a great general compilation of papers put together by the Russell Sage Foundation called the "Behavioral Economics Reading List" that includes, among other things, a General Introduction section with overview papers by some of the big movers and shakers (Camerer, Kahneman, Laibson, etc.).

Many of the papers you will find through the Russell Sage paper list will be on alternative methods to classical consumer theory. If you want just the tests of assumptions/predictions, I would recommend looking through the abstracts of John Lists's papers on Testing Economic Theory. List is one of the most prolific experimentalist authors on the subject and has weighed in on most questions that other people work on in this field as well. Just reading his abstracts on that page should give you a pretty good idea of the state of the literature up through 2011 (the website isn't updated). His CV is updated, but doesn't give you the abstracts without looking up the papers individually.


How to add additional observations to panel data?

Not sure if this is on-topic, but I'll give a simple solution. It's perhaps not the most efficient, but should be easy enough to follow.
Assuming you also want to keep "category" and not just "id", how about something like the following:
use sample.dta, clear
tempfile month10
preserve
keep if month==8 | month==9
collapse value , by(year category id)
gen month=10
save `month10' , replace
restore
append using `month10'


Yes, if green bonds are accepted as collateral they become more valuable/demanded by prospective buyers. When ECB directly buys these as part of QE then again they are more in demand. When demand shifts to the right and supply stays the same price increases.
In real life, ECB could just buy these green bonds on the market as part of their QE, for example, they could directly buy Enel Green Bonds, that increase in demand will raise their prices so when Enel issues new Green Bonds they can charge more for them than they face value.


The trick with Bayesian games is to recognize that the players learn in equilibrium from the strategies of the other players. That is, you cannot simply take the unconditional expectation of $v_i$ when calculating strategies.
For example, you may be tempted to say that both players decide to trade whenever,  $v_i\leq E(\alpha v_j-c)=0.5\alpha-c$ and $v_j\leq E(\alpha v_i - c)=0.5\alpha-c$. However, these strategies might not be mutual best responses because suppose player $i$ believes that $j$ will use that strategy and assume that $0.5\alpha-c=0.8<1$. That means that player $i$ should not expect $v_j$ to be greater than $0.8$, so a profitable deviation is to instead trade whenever $v_i\leq E(\alpha v_j-c| v_j\leq 0.8)= 0.4\alpha-c$.
A common trick to solve these games is to assume something about how the equilibrium will look like and then confirm that it indeed looks like that. For example, given the example above, it is not crazy to think that players will only trade if $v_i\leq \bar v$ (for some threshold that you will have to characterize later). It is also reasonable to assume that both players will use the same strategy since they face the same underlying uncertainty and the valuations for each other houses are symmetric.
Therefore, it must be that player I trades whenever $v_i\leq E(\alpha v_j-c | v_j\leq \bar v)= \alpha\frac{\bar v}2-c$. Since we assume that the threshold for trading or non trading is $\bar v$, it must be that $\bar v = \alpha\frac{\bar v}2-c$ which implies that $\bar v=\frac{c}{\left(\frac{\alpha}{2}-1\right)}$. This means that (assuming $c>0$), $\alpha$ must be strictly larger than 2 for some trade to exist since $\bar v <0$ means that players will never trade, and if $\bar v>1$, then trade always occurs.


Heckscher-Ohlin trade model with non-homothetic preferences

$I^j$ is just a known constant (i.e. we don't know anything about production, which would endogenously generate the amount of per capita income), but as you suggest the budget constraint for the consumer is that $\sum_i p_i d_i^j \leq I^j$ (here, you can assume this holds with equality).

First, note that if $d_i^j < \bar{d}_i$, then the utility function will be negative. So, if we have sufficient income to afford the minimum consumption (i.e. $\sum_i p_i \bar{d}_i \leq I^j$), to avoid this outcome the consumer will purchase at least $d_i^j \geq \bar{d}_i$ units of each good. Therefore, if we define $x_i^j \equiv d_i^j-\bar{d}_i$, we know that $x_i^j \geq 0$.

With this reformulation, we can rewrite this Stone-Geary utility function as a Cobb Douglas utility function, i.e.
$U^j = \prod_{i=1}^N (d_i^j - \bar{d}_i)^{\varphi_i}=\prod_{i=1}^N (x_i^j)^{\varphi_i}$. Then, to solve for the optimal consumption of $x_i^j$, you can just solve for optimal consumption with a Cobb Douglas utility function with a new income of $\bar{I}^j \equiv I^j - \sum_i p_i \bar{d}_i$ (that is, the remaining income left over after spending it on the minimum consumption level of each good). Once you have the optimal $x_i^j$, the optimal consumption level of the good (inclusive of the minimum) is $d_i^j = x_i^j+\bar{d}_i$.


A term for utility functions based on the max operator

The Rawlsian welfare function, which takes the form of the min of all agent's utility, is often called the maximin function, because it maximizes minimum utility. In the spirit of that nomenclature, your function is a maximax utility function. And, apparently, this function does see some use. It is sometimes called the optimistic decision criteria, because a person evaluating projects based on their best outcome uses a similar evaluation function.

  The maximax looks at the best that could happen under each action and
  then chooses the action with the largest value. They assume that they
  will get the most possible and then they take the action with the best
  best case scenario. The maximum of the maximums or the "best of the
  best". This is the lotto player; they see large payoffs and ignore the
  probabilities.

Jones (2002), Decision Theory Notes


Transformation of random variables and second order stochastic dominance

As long as $g(\cdot)$ is Nondecreasing and Concave

Definition: For any lotteries $F$ and $G$, $F$ second-order stochastically dominates $G$ if and only if the decision maker weakly prefers $F$ to $G$ under every weakly increasing concave utility function $u$. Link


Did Zimbabwe have an inflation of 89.7 sextillion percent?

The figures you question are from Hanke and Kwok (2009).

According to Hanke and Kwok (2009), yes.
A stock (Old Mutual) was listed on both the London Stock Exchange (in GBP) and the Zimbabwe Stock Exchange (in Zimbabwe dollars). Under the assumptions of price arbitrage and purchasing power parity, one can infer Zimbabwe's inflation rate.
I haven't carefully attempted to replicate their work, but I can see no reason to question Hanke and Kwok's general methodology.
The Economist's figure of 500 billion percent is for Sep 2008. (Hanke and Kwok's estimate for Sep 2008 is 471 billion percent.) The higher figure of 89.7 sextillion percent is Hanke and Kwok's estimate for the month to Nov 14, 2008.

Their Table 1:


Trouble at differentiating a consumption function

This does not seem to have anything to do with calculus.
The idea is that the income not consumed $Y_d - C$ is saved (usually denoted by $S$).
This saving is then lent out to companies (via banks) who invest it (usually denoted by $I$), and the accumulated capital is used in production. I am guessing this is denoted by $A$?
And the change of $A$ in time is the investment (assuming there is no amortization).
Hence
$$
Y_d - C = S = I = \frac{dA}{dt}
$$


Struggling with uncompensated/compensated demand

First, we want to find the optimal good baskets, ie how much to buy of goods $x$,$y$ and $z$ to get the maximum value of U out of it.

Let's write down the $MRS$ for $x$ and $y$ against $z$. For this we need to differentiate the utility function for each of the variables:
$d_xU(x,y,z)=a/x$ , $d_yU(x,y,z)=b/y$,$d_zU(z,y,z)=1$. We hence deduce the rates of substitution to be $-1a/x$ and $-b/y$ respectively.

So we have:
$p_x/p_z=a/x$ and $p_y/p_z=b/y$

The budget constraint is $p_xx+p_yy+p_zz=I$ which we rewrite
$p_x(ap_z/p_x)+p_y(bp_z/p_y)+p_zz=I$, which simplfies in
$ap_z+bp_z+zp_z=I$

$z^*=(I/p_z)-a-b$

$(x^*,y^*,z^*)=(ap_z/p_x, bp_z/p_y,(I/p_z)-a-b)$

Good news, you were right :).

The cross price effect is the rise of demand in good $x$ following a rise in the price of good $z$. Just calculate the difference between $x^*$ for $p_z$ and $x^*$ for $p_z'=p_z+\epsilon$ (using $x^*=ap_z/p_x$).

Finally, the six curves asked are how demand change when price change holding income ($I$) constant, or utility ($U$) constant. Draw same by taking two prices fixed for each situation.


Can a subset of variables in a lower triangular Cholesky identification scheme be ordered arbitrarily if we do not care about their shocks?

You are right.
Ordering a variable first reflects the identifying assumption that this variable will not respond contemporeneously to the other shocks in the system. You could obtain the exact same shock series via an OLS regression that includes as explanatory variables only the lags of all the variables in the VAR. Since you were happy to "assume away" all endogeneity concerns, your shock is neatly identified, and you don't care about the others. (You don't even need to impose the other zeros in the impact matrix: any rotation of the bottom-right (n-1)x(n-1) submatrix will leave the first shock unaffected.)
Ordering a variable last allows this variable to contemporaneously respond to all other shocks in the system, but it imposes the assumption that none of the variables in the system can contemporaneously respond to the shock in the last equation. Again, you could obtain the same shock series via an OLS regression including all lagged and contemporaneous values of all the variables in the VAR, since, once again, you were happy to assume away all endogeneity concerns. (Also, you don't even need to impose the other zeros in the impact matrix: any rotation of the top-left (n-1)x(n-1) submatrix will leave the last shock unaffected.)
Intuitively, the OLS-counterparts show that the ordering of the other variables doesn't matter when you're interested in exclusively the first or the last shock:

either you don't includy any of the other variables contemporaneously in the OLS regression (first shock);
or you include all of them contemporaneously (last shock).

For the shocks in between, the OLS regressions that would deliver the same shock series as the shock series obtained via cholesky include the variables ordered before the shock of interest, but not the variables order after. So here the ordering of the variables in the VAR does affect what the shock series will look like, because changing the ordering for these shocks would include or exclude some of the rearranged contemporaneous variables in the OLS regression.


Interpreting correlation between fixed effect and explanatory variable

You have done two different things.

Your fixed-effects model captures the within-group over-time functional relationship between $debt_{it}$ and $y_{it}$ (that is, how much average difference in $y_{it}$ is there between two periods with a 1-unit difference in $debt_{it}$ within a country). In your data, there is limited within-group variability in $debt_{it}$, which probably lead to a large standard error and a resulting insignificance.

Your reasoning about the correlation between the fixed effects and the explanatory variable is about the cross-sectional functional relationship. You found that countries with higher $debt_{it}$ have higher $y_{it}$ values (as opposed to periods with higher $debt_{it}$ having higher $y_{it}$ within a country). This is not what you originally intended to investigate when you set up your fixed-effects model.

Your Hausman test says that the within-group functional relationship and the cross-sectional functional relationship are different.

That said, yes, it is true that "a higher debt does result in higher interest rate spreads", but it is true only in the specific sense that "countries with higher debts show higher interest rate spreads". You should be careful when saying "result in".

Overcoming the issue of limited over-time variation and having significant results is hard. There is no magic. Abandon the fixed effects model, and try to control for many time-varying and time-invariant regressors, enough for you to argue that you controlled for most country-specific factors. (You can use RE or POLS estimation.) People might still criticize that you didn't control for enough factors, but you will need to defend yourself somehow. You could also move over to dynamic models or others, but that's a different story.


How to determine the riskparamter PD, EaD, LGD for the IRB-Ansatz | Basel II

Exact models are proprietary but they usually do that using statistical analysis. For example, in this paper the authors estimate the probability of default of various banks using the probit model. Or another example is in this paper.
Models with a limited dependent variable like probit or logit and others are quite common because these models let you estimate the probability of some event occurring. However, these models can be set up in various ways depending on data.


There are no simple methods for estimating cardinal utility (ordinal utility would be a different matter - you could just observe few of your choices).
This is not because cardinal utility would necessary be unmeasurable.  Although this is not completely settled question (see Moscati (2018) Measuring Utility: From the Marginal Revolution to Behavioral Economics), if you are willing to buy into the von Neumann-Morgenstern (1944) derivation of cardinal utility, then the cardinal utility is measurable.
According von Neumann-Morgenstern (1944) cardinal utilities could be measured in principle since by the axioms of von Neumann-Morgenstern (vNM) cardinal expected utility you should, in principle, be able to measure it.
According to vNM assuming that axioms of completeness, transitivity, continuity and independence  are satisfied (see von Neumann-Morgenstern (1944) Theory of Games and Economic Behavior), there is some function $u$ that assigns to each potential outcome of a gamble $X$ a real number $u(X)$ such that for any two gambles ($X_1, X_2$):
$$X_1\prec X_2 \qquad \text{iff} \qquad E(u(X_1))<E(u(X_2))$$
Consequently, $u$ can be uniquely determined, and be fully measurable, by measuring preferences between some simple gambles (e.g. gambles such as  $pX_1 + (1 − p)X_2$). However, to measure the cardinal utility in full you would have to run infinite number of such gambles (even with just two outcomes $X_1$ and $X_2$ you can construct infinite number of different simple gambles). In addition those have to be real gables with stakes not just 'made up' gambles as otherwise you can never be sure if in real situation you would choose completely differently.
A potential proxy for measuring cardinal utility could be willingness to pay. This could be considered a proxy for some cardinal money-metric utility function but even that approach has some issues (Fleurbaey, M. 2011). However, this is probably the most practical approach. This is also reason why willingness to pay is so widespread in the cost-benefit analysis in public economics. However, even with this approach you will have to do some serious data collection on various purchases you make and other control variables such as your income for example.


Could money be tied to a basic good like the gold standard?

The way that the gold standard ties the value of a dollar to gold is usually by making a dollar exchangeable for a fixed amount of gold. For example, from 1934 until 1971 the US government would exchange \$1 for 1/35th of an ounce of gold.

One of the charms of using gold as a currency is that it is durable and extremely valuable for how much room it takes up. How would anyone actually implement monetary conversion for Big Macs? Would there be a Fort Kroc where 3 billion Big Macs (worth the \$11 billion of the current US Gold stockpiles) would be kept? The same  perishability and low value per unit volume of make Big Macs  delicious meals to millions also make them a terrible anchor for a commodity currency.

It is true that consumption of gold services in the form of gold jewelry and the like make demand for gold pro-cyclical. But empirically, prices were stable and inflation low during the gold standard. Therefore it is hard to imagine this was a severe problem. Of course, the gold standard has lots of problems including:

The supply of gold increases slowly over time relative to the economy, leading to gradual deflation which is thought to make recessions worse.
The use of gold as a backing for currency significantly increased demand for gold, leading to tremendous diversion of resources into gold production.
Like any pegged currency, a gold standard depends on reserves which can be exhausted in a panic about the future ability of a government to convert.

Big Macs might not suffer from all these problems but, as mentioned above, suffer from lots of other problems that make it problematic to use them as or a backing for money. And how would you handle counterfeiting?

One reason all major counties have moved to fiat money is that it doesn't suffer from any of the problems of gold listed above. It also doesn't suffer from the additional weaknesses of Big Mac being both perishable and a low value item.

One negative of fiat money is that the monetary authority (e. g. the central bank) can't abuse their discretion to create high inflation. But Big Macs suffer from this problem too. Because Big Macs are easy to create in large quantities and short run demand is relatively inelastic, the government could create lots of inflation by just grilling up their own, just as they could run their printing presses.

Yes, it is true that consumers of Big Macs are probably poorer on average, and so they would benefit relatively more from cheaper burgers if the government tried to inflate away the debt by printing money (grilling up burgers) but even if they were free how many would people eat? That's why their demand is inelastic, people don't want that fourth burger in one day nor do they want to eat them everyday forever. As for the other costs of higher inflation, that's a more complex issue and probably best posed as another question about what are the benefits and costs of higher inflation and who gets what. But original question asks would it be workable. The costs of conversion to a commodity backed money based on Big Macs seem to have little to recommend it and enormous costs both in transition off fiat money, governance (counterfeiting and monetary policy), and ongoing implementation owing to low value per item and perishability.


How can a government save, if it has deficit?

Government can have savings while having deficit because we are just talking about savings not net savings.
For example, imagine that government has zero tax revenue $\\\$10$ spending and $\\\$10$ of public investment. In this case the government is running deficit of $\\\$20$ yet it is also saving through investment. Due to the deficit being larger than saving on net basis government still dis-saves but saving is not zero.
So it is completely possible to save and run a deficit at the same time.


How does Egypt prevent people from arbitraging their currency?

Used to be capital controls. You could only buy dollars (or any other foreign currency) through official channels at the official favorable rates by providing documentation displaying need for said currency (e.g. Travel or import documents).


I see no problem in $m$ being absent from the equation, as it enters additively in the utility function. The partial utility obtained from this good is represented in the 1 that you get in the derivative. This is, the marginal utility obtained from an extra unit of $m$ is independent of the amount the individual consumes of $m$. Additionally, as both type of consumers value the $m$ good equally, the distribution of $m$ is then irrelevant for the optimality of any allocation. Your solution tells you that the relation between the level of consumption of $x_1$ and $x_2$ is that of 5 to 3.

Then, what you need to do is to allocate the aggregate resources among all consumers such that the proportion holds. If two two type of consumers have in total 40 units of $x$, then the allocation is that of 25 for type one consumers, and 15 for type two consumers.


Solving Stochastic Dynamic Optimization Problems: A difficulty with Lagrange Multipliers

Following a comment exchange below another answer, the critical detail of a link offered by the OP as an example of by-passing/ignoring Wicken's comment/argument, and one different from Wickens' formulation, is that in the link's eq. $(4)$,

$$\lambda_t = \beta E_t[\lambda_{t+1}(1+r_t)]$$

the multiplier $\lambda_{t+1}$ appears together with the interest rate of period $t$, which is assumed part of the information set at $t$ and so not a random variable in the specific equation. So the authors can proceed in the 2nd line after eq. $(6)$ to write the marginal rate of substitution as they do, without the need to assume uncorrelatedness between the multiplier/marginal utility of consumption, and the interest rate.

This goes back to how they formulate the income resource constraint (page 2 middle), where it is essentially assumed that the household has, at the beginning of period $t$, available assets or debt $(1+r_{t-1})B_t$. The authors explicitly discuss this "timing convention" immediately after eq. $(1)$

  "Note a timing convention -$r_{t-1}$ is the interest you have to pay today on existing debt. $r_t$ is what you will have to pay tomorrow,
  but you choose how much debt to take into tomorrow today. Hence, we
  assume that households observe $r_t$ in time $t$. Hence we can treat
  $r_t$ as known from the perspective of time $t$."


Does the separation of FinCEN, and The Federal Reserve protect the integrity of a representative democracy?

Your understanding is wrong. FinCEN is a bureau of the United States Department of the Treasury. The director reports to the Secretary of the Treasury, who is appointed by the President and confirmed by the Senate.

Further, the Federal Reserve system is also a government entity. It was established under the Federal Reserve Act. The Federal Reserve Board of Governors, which supervises the Federal Reserve Banks, is an independent agency of the federal government. Its members are appointed by the President, with the advice and consent of the Senate. The system's structure was deliberately designed by Congress to grant it independence in its conduct of monetary policy, with the intent of minimizing political interference in monetary policy. However, this independence was granted and is maintained via democratic consensus.


Conditional variance vs. unconditional variance in ARCH model

I cannot directly answer your question, but I think I can shed some light. What I seem to show is that under some restrictions, the unconditional variance is finite. However, I am not sure how to relate my answer to the notion of volatility clustering.

Note that if we assume stationarity, then $\sigma^2=E(\epsilon_t^2)$ for all $t$ and it follows by the law of total expectations that $$\sigma^2=E(E(\epsilon_t^2|I_{t-1}))=E(h_t)=\alpha_0+\alpha_1\sigma^2+\alpha_2\sigma^2$$ using your expression for $h_t$. Thus, $\sigma^2=\frac{\alpha_0}{1-(\alpha_1+\alpha_2)}$ is a solution whenever $\alpha_1+\alpha_2<1$.

Now, by the law of total variance, \begin{align*}V(Y_t)&=E(V(Y_t|I_{t-1}))+V(E(Y_t|I_{t-1}))\\
&=E(\epsilon_t^2)+V(\mu_t)\\
&=\sigma^2+a_1^2V(Y_{t-1}) \end{align*} using your results, where $\mu_t=a_0+a_1 Y_{t-1}$. Assuming stationarity, and thus that the variance is constant for all $t$, it follows that $$V(Y_t)=\frac{\sigma^2}{1-a_1^2}$$ is a solution for $a_1\neq\pm 1$ and $\alpha_1+\alpha_2<1$.

I encourage you to clearly note my assumptions. I cannot further assess this answer as I know next to nothing about time series.

(Also note that we from the assumption of stationarity have that $E(Y_t)=a_0+a_1E(Y_{t-1})$ implies that $$E(Y_t)=\frac{a_0}{1-a_1}$$ is a solution.)


Why the total value of the whole economy is constantly appreciating?

Roughly, people produce "stuff" by spending their time working (labour) and by employing machinery/tools/land/etc. (capital).

If you have more workers or more machinery then it should not come as a surprise that more goods and services get produced.

But this is only part of the story. Even if we divide by the number of people to get the size of the economy per-capita, we still find growth:

So we clearly can't fully attribute growth to an increase in the population of workers.

For some time in the early 19th century, economists thought that growth was due to an increase in the use of machinery during the industrial revolution. If each worker has more tools—went the reasoning—then each worker can produce more and per-capita output will increase. But in 1957, Robert Solow found that only 12.5% of US growth in the first half of the 20th century was due to an increase in the capital / labour ratio. That still leaves 87.5% of growth unexplained.

If growth is not due to an increase in the availability of labour and capital, then it must be driven by the existing capital and labour becoming more productive. Why might that happen?

Education levels increase over time so that an average worker in 2000 produces more value than did an averager worker in 1900. Indeed, in 1900 the average worker was a farm hand or low-level industrial worker who produced a few thousand dollars worth of goods each year. Now, the average worker is a skilled industrial worker or office worker who produces good or services worth tens of thousands of dollars per-year.
Similarly, technological advancement makes tools and machinery more productive over time. For example, a \$200 modern computer is several orders of magnitude more powerful than a \$1million computer was in 1970. That means that we can now produce anything created by (or with the aid of) a computer in quantities many orders of magnitude larger than was possible 40 years ago.

So, in sum, more skilled workers who have access to better tools produce more stuff. Since a country's GDP is just the value of the stuff it produces in a year, education and technology cause GDP to increase over time.


Friend,

This situation that requires an imputation method that you must pick from.   Dealing with missing data is a sensitive matter.  Depending on which way you choice to handle missing observation will impact your project and in this case (with observations less than 100) could make your analysis poor/basis.

Here are some popular options:

Impute the value of the missing data
Remove a variable which has a lot of missing data and use other variables which measure similar aspects of the characteristics being studied.
You could create a regression model to impute missing values.
You can use the average (caution when using this method as there may be other variables in your data set that may be dependent).
Duplicate the previous or nearest value.

Here is a website on some tips: http://www.real-statistics.com/descriptive-statistics/missing-data/

Here is a working paper on the subject: liberalarts.utexas.edu/prc/_files/cs/Missing-Data.pdf

Missing values is delicate subject.  Use caution and always, always, always DOCUMENT WHAT YOU ARE DOING, because others want to know and it makes your work so much better.


Typical Growth Model Social Planner Problem

Q1: Your bullet point (b) is not how we usually interpret capital, it is usually not assumed that $k_0$ is divided between $k_0^p$, and $k_0^s$ the capital we use to produce and the capital we store for next period. Instead, we assume all the capital is used for production, but using it for production does not delete it,  it only depreciates it. For example, if you use your machines to produce pizza that does not mean that they are obsolete to be used next period.

If you want you could add to the model that un-used capital does not get depreciated, but it is an easy exercise to show that under very general conditions it will never be optimal to leave capital un-used.

So I do not agree with your interpretation in (e), typically $k_1=(1-\delta)k_0+i_0$.

Q2: This question is not particular of the dynamic model you present, but I think that the infinite sequences are messing up with your intuition.

Consider this simple example. If I say that you can choose to buy chocolate (at a price of \$2) or cheese (at a price of \$4), but you only have \$10 dollars (sorry this are rough times :P ), it is clear that if you are to spend all your money on these two goods, then choosing how much chocolate to buy, immediately pins down how much cheese you will buy. However, the fact that this is true does not change the nature that both amounts of chocolate and cheese are your choice variables. The budget constraint may allow you to re-formulate this problem as one where you only choose how much chocolate to buy (since cheese would be implicit), but this will only be a technique to simplify the original problem.

Something similar happens here. So to be concise, both sequences $\{c_t,k_{t+1}\}_{t=0}^\infty$ are control variables even if they have a simple linear relationship between them through the law of motion.


Expected value of order statistics for uniform distribution

I guess $m < 1$. Otherwise $x^{1/(m-1)} \to \infty$ as $x \to 0$.
There's a possibility that I made a mistake, so let me know if you spot one.
Let's first compute the cumulative distribution function of $X_2$:
$$
F_{X_2}(a) = \Pr(X_2 \le a) = \Pr(X_1 \le a \text{ and } X_2 \le a) = \Pr(X_1 \le a)\Pr(X_2 \le a) = a^2.
$$
This assumes that the two uniformly random variables are independent.
Then the density is given by:
$$
f_{X_2}(a) = \frac{\partial F_{X_2}(a)}{\partial a} = 2 a.
$$
The expected value of $X_2^{\frac{m}{1-m}}$ is then:
$$
\int_0^1 a^{\frac{m}{1-m}} f_{X_2}(a) da = \int_0^1 a^{\frac{m}{1-m}} 2 a\, da,\\
=\int_0^1 2 a^{\left(\frac{m}{1-m} + 1\right)} da = \int_0^1 2 a^{\frac{1}{1-m}} da = 2 \frac{1-m}{2-m}
$$


I am out of touch with the movement towards centralised counterparties, but I assume that the basic principles are similar to when I looked at it.
All derivatives have a risk measure like a daily Value-at-Risk (VaR) associated with them, as well as a net present value (NPV). The fact that it is a swap does not matter; all derivatives will be covered by the same collateral agreement.
The VaR/NPV is calculated using an accepted methodology by both parties. The counter-parties then post accepted collateral in an amount that covers the worst case of the NPV moving against them by the risk measure. (In the case of central clearing, both sides post collateral to the central clearer.) E.g., if the NPV of the counterparty is +\$2 million, but the daily risk is \$10 million, they need to have at least \$8 million in collareral.
This way, if one side fails, the other side can seize collateral to make up for the value of the derivatives position.
In the pre-central clearing days, in the case that a dealer is dealing with an entity that is hedging interest rates, but does not otherwise trade derivatives,  the dealer typically sets the terms on how collateral works unilaterally (e.g., the dealer does not post collateral). Banks are in the business of assessing and managing credit risk, so they can handle the credit risks associated with swaps relatively easily. I do not know whether this pattern of behaviour is still allowed under central clearing.


Why does the aggregate consumer spending decrease due to to lower expected future disposable income instead of MPC?

But I don't believe that the plot will shift like this in response to such bad news, it seems to contradict what I learned about microeconomic autonomous spending (I believe that the same must be true for aggregate autonomous spending). It's spending that you just WON'T reduce, they are too critical. You will tap into your savings, you will borrow or ask for charity, you will break the law, - but you absolutely WON'T reduce your autonomous spendings.

This is sort of correct, but not completely. In a narrow sense autonomous spending is a persons spending when her or his income is 0. As you mentioned consequently many textbooks use examples like people always must eat even if they dont earn any money. However, that suggest that autonomous spending is set at a bare subsistence level and that is not true.

It was already recognized by Adam Smith in his magnus opum the Wealth of Nations that what is and what isnt necessity, and thus something that would determine autonomous spending, is context dependent. As Adam Smith writes:

  linen shirt … is, strictly speaking, not a necessary of life. Greeks and Romans lived, I suppose, very comfortably though they had no linen. But in the present times, through the greater part of Europe, a creditable day-labourer would be ashamed to appear in public without a linen shirt, the want of which would be supposed to denote that disgraceful degree of poverty which, it is presumed, nobody can well fall into without extreme bad conduct.

Hence, autonomous spending can change and it is dependent on the 'opulence' of society in general as well as of the individual. For example, if you would be a relatively rich commodity trader and you would be fired from your job (hence reducing your income to 0) you would not dis-save just to consume on subsistence level, but dis-save enough to maintain the minimum you would consider necessary for yourself. If you expect that long term you wont be part of the upper status class but will have to become middle class person you can move your autonomous spending to the level consistent with whats necessary for such person to live. Hence changes in expected income can move the whole consumption function up and down.

The slope of the consumption function (or the marginal propensity to consume) changes when person decides to consume smaller or larger fraction of their total income not just when consumption falls. For example, if your income is 1000 and your MPC is 0.6 you consume 600 and save 400. If your income is 500 you would consume 300 and save 200. In such example income changed but MPC remained the same.

MPC can change with income but empirically the poorer people have higher MPC than richer People. I dont have the Krugman's book but many textbooks include examples where the MPC is hold constant by assumption as a thought exercise and only changes to income are considered in similar fashion.

However, I also agree that the Krugman's wording is confusing as he talks about "saving for a rainy day", which can to many readers suggest MPC should decrease.


The approach developed by Barro and Lee has been used in many empirical analysis. They provide a detailed discussion of how to construct an aggregate human capital measure.


How does the professional class fit into Marx and Engel's class struggle theory?

Marx's discussed struggle had less to do with your income-position in society and more to do with how you derive income, whether through profits, wages, or rents. You bring up a fascinating issue with his analysis and the solution is all about framing. Even doctors end up selling their labor on the market (in increasingly specialized components). While the labor of a doctor is different than say a machinist, their income is fundamentally different than that of someone who owns the means of production. The 17th century putting- out system would be maybe the clearest split between the two.  Keep reading Marx! Eric Fromm has some great essays that explain the nuance in Marx's world view. Hopefully this answer helps


Conditional probability in Kaplan, Menzio (2014)

The first / second seller terminology can be confusing here (does it relate to time or to conditionality?). It's safer to focus on a particular seller.

Let the probability that a buyer finds a particular seller, $s$, via single search be $\nu_s$. The probability that a buyer is matched with $s$ and (either before or after) another seller is then (assuming that $\nu_s$ is small so that the probability of finding $s$ twice in at most 2 searches can be ignored):

$$\psi_u(\nu_s\nu+\nu\nu_s)= 2\psi_u\nu_s\nu$$

The probability that a buyer is matched with $s$ via either a single or a double search is (again ignoring the probability, given a double search, of finding $s$ twice):

$$(1-\psi_u)\nu_s + \psi_u(\nu_s + \nu_s)=\nu_s(1+\psi_u)$$

The conditional probability of a buyer being matched with $s$ and another seller, conditional on $s$ being matched with that buyer, is therefore:

$$\frac{2\psi_u\nu_s\nu}{\nu_s(1+\psi_u)}=\frac{2\psi_u\nu}{1+\psi_u}$$

But the latter formula does not depend on $\nu_s$. Hence the same formula applies to any seller.


Why do we model that higher government involvement increases output?

The book is talking about the short term effect. And yes, such a policy would be thought as Keynesian.

In the long run, excessive government spending will cause inflation (if unfunded) or reduce both private production and demand (due to excessive taxation).


Does brain drain really affect a nation's growth trajectory?

You have asked a very relevant question, and you must be new to economics or have not got a clue about economics of growth/development. And, you will not be able to get a 'definitive' answer to your question - it is not easy, and there is probably none! Should there be a definitive answer, we would not have to spend years doing PhD in economics.

There are plenty of resources which could give you a better grasp of the issue.
For a start, read https://www.amazon.co.uk/Why-Nations-Fail-Origins-Prosperity/dp/1846684307


Supply and Demand plotted on a three dimensional Cartesian coordinate system

It is possible. Trivially if we have $Q^s=f(p)$ and $Q^d=g(p)$ and we plot this in a 3d-coordinate system with variables $p$, $Q$ and some other variable $x$ (which does not appear in our functions at all), then we can take a 'slice' for every $x$-value, and in fact every slice gives us the conventional 2d-depiction of this model.

We could of course take the standard supply-demand model and take some parameter as third variable, for example:

$$Q^s(p,t)=\max(0, p-t), \quad Q^d(p,t)=\max(0, 1-p)$$

Where $t$ is a unit tax imposed on the suppliers of the good in question. We can plot this in a 3d coordinate system with variables $Q,t,p$ and a red line showing equilibria:

As we would expect there is a positive relationship between equilibrium price and the level of the unit tax.


Sure, because dollars are nominal goods. Perhaps a ticket to Star Wars Episode CXVIII will cost one trillion dollars. If you meant today, without huge changes in the price level:
 Yes, there is way more than one trillion dollars of wealth in the world. Depending on the laws of physics and psychology there is probably something you would be able to market for this amount. (Private luxury homes in other dimensions.) Coming up with it is another question, but there is no law of economics that makes this impossible.


One approach that you might find useful/interesting would be Bayesian VARs.

An example of a Bayesian VAR being used to predict inflation can be found here.  In this paper, Tim Cogley, Sergui Morozov, and Tom Sargent motivate some methodology for Bayesian VARs and then compare their results to what the Bank of England uses in their "fan charts" (which they use to predict inflation).  Cogley and Sargent have two other papers on inflation that are related to this paper, "Drifts and Volatilities" and "Evolving Post-World War II U.S. Inflation Dynamics."  This series of papers is a good expose on some of the theory of inflation and how to statistically approach it.

I believe @FooBar is right that most banks are using large-scale VARs (similar to the aforementioned) and New Keynesian models.  This isn't directly related to the question, but there is a literature on model averaging in which they can combine the forecasts of VARs and N-K models to provide a "more informed" (I use the phrase more informed here lightly but I do think the model averaging is helpful) prediction as to what will happen.  An example of such a paper can be found here


What is the logic the Fed used for paying interest on reserves?

According to Bernanke himself, interest paid on reserves in excess of the penalty free upper bound are used as an additional tool with which the Fed can place a floor on the federal funds rate.The logic here is that banks have no incentive to lend in the overnight market at rates lower than this interest rate.

This link has some links to info on the topic. It's an old answer that got little attention and so here is my chance to get some use out of it :)

Federal Reserve Interest


What is the relationship between population growth and economic growth?

This is actually quite complex issue. Taken narrowly answer to your question:

  what is the relationship between population growth and economic growth

Is that simply the relationship is positive. This is because in the question you ask only about growth of GDP per se. Growth ultimately depends on growth of production and labor is an important input in the most of production. One of the most widely used production functions is the Cobb-Douglas production function which has following form $Y = AK^{\alpha}L^{1-\alpha}$ and it is trivial to see that output depends positively on labor.

However, there are some problems with looking just at growth of GDP. Even though GDP will grow with more population you could in principle end up in a Malthusian trap. Yes GDP increases with more population but that is because we have more people and hence that does not mean that this increase in growth actually makes peoples lives better. This is why economists care not just about growth overall but rather per capita growth (that is economic growth per person).

When it comes to the relationship between per capita growth and population literature gets more murky. The most widely used growth model is the Solow-Swan model and in it increase in population has no effect on per capita growth. In Solow-Swan model all growth is exogenously given by growth of technology. Most of professional macroeconomist believe in Solow-Swan model because there is quite a lot of empirical evidence in its support but in the last 30 years endogenous growth theory is challenging the Solow-Swan model.

In endogenous growth theory population growth can actually lead to also per capita growth. This is because in such models growth depends not just on exogenously given (predetermined) growth of technology but can be affected by choices society makes. One of those choices is also to devote more labor towards R&D and knowledge development, hence in such model actually population growth can lead to sustained economic growth if that extra population is educated and channeled towards R&D/knowledge creation. This being said an important caveat is that although endogenous growth theory got more popular in recent years and even Paul Romer got few years back Nobel Prize in Economics for his contribution to the theory, it is still not widely accepted because its extremely difficult to test. So take that as you will.

  how can population bonus potentially increase growth in GDP

Population bonus is more widely known as demographic dividend. It referrers to situation where country's demographic structure improves in a sense that there are more working age people relative to either young or elderly. For example economy where you have 60 retirees 20 infants and 20 working age population will have (ceteris paribus) harder time providing for its people than economy where you have 10 retirees 15 infants and 75 working age population. This is because for most part all goods and services are produced by working age people (save some exceptions like child actors/paper delivery etc) and hence non-working age people do not really directly contribute to the economy.  Especially in those countries where state offers generous social benefits to non-working age people it is important to have enough working age population from which state can draw resources through taxes.

Thus having population bonus/demographic dividend can certainly increase GDP, if its sustained it can also lead to economic growth but it is worth noting that it is very difficult sustain growth this way. There is plenty of evidence that as people get richer their fertility declines and demography soon becomes unfavorable like in Europe or Japan. Hence, it is very unlikely demographic dividend can deliver long term sustainable growth but in can definitely help raise GDP while its happening and in principle can be sustained for few generations but not indefinitely.


If $a$ and $b$ are two pure rationalizable strategies, can $0.5a+0.5b$ fail to be a rationalizable strategy?

Yes. Consider a two player game described by the following matrix

\begin{array}{|c|c|c|}
\hline
&  L & R   \\
\hline
a & 3,0 & 0,0   \\
\hline
b & 0,0 & 3,0    \\
\hline
c & 2,0 & 2,0    \\
\hline
\end{array}

If only pure strategies are allowed, only $a$ and $b$ are rationalizable because they are best responses to $L$ and $R$ respectively while $c$ is not a best response to anything. But if you look at the mixed extension of the game then $c$ strictly dominates 0.5$a$+0.5$b$, hence 0.5$a$+0.5$b$ is not rationalizable. As a result, in the mixed extension all $a$, $b$ and $c$ are rationalizable.

If you insist on an example with three players I can add a third one to this. This third player has exactly one possible strategy, so it does not really matter what his payoffs are and what he does.
If you think that is cheating, the third player can have multiple strategies that do not at all influence the payoff of the first two players.


What is the loss of productivity of not using the international system of measurments (SI)

This is a good example of "transactional costs" -and of the small "menu costs" as Mankiw has once called them. His paper Mankiw (1985) "Small Menu Costs and Large Business Cycles: A Macroeconomic Model of Monopoly" The Quarterly Journal of Economics, 100(2):529-537 contains a model where in his own words, shows that "sticky prices can be both privately efficient
and socially inefficient."

The basic mechanism of the model is as follows: a monopolistic firm has to set the selling price one period in advance, based on expectations about future demand. If demand is realized lower than expected, the firm can reduce its price but at a "menu cost" (alluding to the costs that restaurants have to bear in order to re-print their catalogs with new prices).

Mankiw shows that even though this cost may be small, still it may prevent the firm from adjusting the price. So prices become sticky, which is socially inefficient.

Of course this is an underlying theory that justifies the existence of inefficiency due to such "transactional costs". Actually measuring and estimating them would require a detailed audit approach as regards the time spent and other costs involved to change measurement units.


Which utility function yields a constant price elasticity of demand function?

By Roy's Identity we have that Marshallian (uncompensated) demand for good $x_i$ is

$$x_i^M = \frac {\partial U^*/\partial p_i}{\partial U^*/\partial B} \tag{1}$$

where $U^*$ is optimized utility over goods vector $\mathbf x = (x_1,...,x_i,...x_n)$ and $B$, which is the available budget, and given the price vector $\mathbf p = (p_1,...,p_i,...,p_n)$. To obtain a constant elasticity demand function we require

$$\eta \equiv \frac {\partial x_i^M}{\partial p_i}\cdot \frac{p_i} {x_i^M}=const. \tag{2}$$

Using $(1)$ we have that

$$\frac {\partial x_i^M}{\partial p_i} = \frac {(\partial^2 U^*/\partial p^2_i)\cdot(\partial U^*/\partial B) -(\partial^2 U^*/\partial B^2)\cdot (\partial U^*/\partial p_i)}{\big[\partial U^*/\partial B\big]^2} \tag{3}$$

Inserting $(3)$ and $(1)$ into $(2)$ we have

$$\eta \equiv \frac {(\partial^2 U^*/\partial p^2_i)\cdot(\partial U^*/\partial B) -(\partial^2 U^*/\partial B^2)\cdot (\partial U^*/\partial p_i)}{\big[\partial U^*/\partial B\big]^2}\cdot \frac{p_i} {\frac {\partial U^*/\partial p_i}{\partial U^*/\partial B}}$$

$$= \frac {(\partial^2 U^*/\partial p^2_i)\cdot(\partial U^*/\partial B) -(\partial^2 U^*/\partial B^2)\cdot (\partial U^*/\partial p_i)}{(\partial U^*/\partial B)\cdot (\partial U^*/\partial p_i)} \cdot p_i$$

$$\implies \eta = p_i\left(\frac {\partial^2 U^*/\partial p^2_i}{ \partial U^*/\partial p_i}- \frac{\partial^2 U^*/\partial B^2}{\partial U^*/\partial B}\right) \tag{4}$$

For this expression to be constant over the whole range of $p_i$ we need that

$$\left(\frac {\partial^2 U^*/\partial p^2_i}{ \partial U^*/\partial p_i}- \frac{\partial^2 U^*/\partial B^2}{\partial U^*/\partial B}\right) = \frac {C}{p_i} \tag{5}$$

for some constant $C$, which becomes the constant value of the price elasticity of demand.

That's the general condition that must be satisfied.

You could check whether the generalized Cobb-Douglas standard Utility function specification

$$U(\mathbf x) = \prod_{i=1}^n x_i^{a_i}$$

satisfies the condition, perhaps under some restrictions.

THE CASE OF QUASI-LINEAR UTILITY FUNCTION

The OP ponders the case of a quasi-linear utility function, so let's solve this forward. We have

$$\max_{x,m} [cx^{\theta} +m],;\;\;\; s.t. \;\;\;p_xx + m = B,\;\;\; 0<\theta <1,\;\; c>0$$

where $m$ is residual income for all other goods. The Lagrangean is

$$\Lambda = cx^{\theta} +m + \lambda[B-p_xx - m]$$
and first order conditions are

$$c\theta x^{\theta-1}=\lambda p_x,\\\;\;\; \lambda =1$$

So

$$x^* = \left(\frac {c\theta}{p_x}\right)^{1/(1-\theta)},\;\;\; m^* = B-p_xx^*$$

So the indirect utility function is

$$U^* = c\left(\frac {c\theta}{p_x}\right)^{\theta/(1-\theta)} + B-p_x\left(\frac {c\theta}{p_x}\right)^{1/(1-\theta)}$$

$$= \left[c(c\theta)^{\theta/(1-\theta)} - (c\theta)^{1/(1-\theta)}\right] \cdot \frac {1}{p_x^{\theta/(1-\theta)}} + B$$

One can easily verify that this indirect utility function satisfies the required condition $(5)$ for isoelastic demand, and also see how the preference parameters map to the demand parameters.


Extensive form: backward induction & subgame perfect nash equilibria?

Until the last paragraph your interpretation is correct.
$(a,d)$ is not a strategy profile, it is merely a 'history', a way that the game can play out.

A strategy of a player is a function that chooses an action in each of her decision nodes. As you can see in your graph Player A has three decision nodes while B has two. Hence any strategy of player A will consist of three elements, while any strategy of B will consist of two elements. So $(agi,de)$ is indeed a strategy profile. The corresponding history, what will happen if the players play these strategies, is $(a,d)$. The corresponding payoff is (1,2).

I am unsure what you mean by having to account for all possible strategies.
Do you want to see what it means when an equilibrium is not subgame perfect?
That would be $(ahj,de)$. No player can gain by unilaterally switching to another strategy, but if we were to reach player A's second or third decision node his action would not be optimal. But $(ahj,de)$ can be an equilibrium because the history given by this strategy profile never reaches these nodes.

Do you want to find all the equilibria that are not subgame perfect? In that case you should write down all possible strategies:
There are 2^3 strategies for A, 2^2 strategies for B. Make a matrix using these as row and column labels. (Not sure if you really want to do this, as it is rather large.) Then you calculate the history and payoff for each strategy profile, creating a payoff matrix. In this you will find many equilibria. The ones that are not $(agi,de)$ and $(bgi,df)$ are not subgame perfect.


Causal effect of population growth on GDP per capita increase

Birth rate affects not only total population but also the age structure of the population.  If the birth rate over many years is below the level at which the population replaces itself, the result will be a falling population with an increasing proportion of elderly and often economically inactive people.  With a falling proportion of economically active people, GDP per capita is likely to be less than it would have been with a higher birth rate.

This effect can to some extent be offset by policies to encourage older people to continue working for longer, such as raising the age at which state pensions become payable.


Is the Paradox of Value relevant to the value of labour?

Both of these things can be thought of as Supply/Demand problems where the relevant concept is actually how Demand is decided.

First, let’s look at diamonds. Diamonds are of value because people give them as gifts, proposals can hinge on them, etc. Now why that is a question of history or sociology, but the key idea is that people want them and will pay to make sure they can also give them as gifts. It wouldn’t be fun to plan a proposal, get on a knee, and pull out a Ring-Pop..

Diamonds and water are both of high demand! The difference is that supply of diamonds is small, so people will have to compete via price to get their diamond whereas water is relatively abundant.

Professional sports players too follow this pattern. What’s really important for basketball fans is seeing the absolute best players compete. There’s a reason middle school teams don’t sell out stadiums. So the demand is not for an average basketball player, it’s for the best basketball player.

The supply of truly spectacular basketball players is small (as the majority of draft busts can show) and therefore their salaries are quite high. Teachers are relatively abundant, so their salaries are a low (tragically so).


Prediction Interval for $E(y|X_0)$ when model is in $ln(y)$

I got hold of this link from Journal of Statistics Education

Using the notation in it, let $X\sim \log N(\mu,\sigma^2)$, i.e. $Y=\log(X) \sim N(\mu,\sigma^2)$.

Hence, $E(X)=E(e^Y)=e^{\mu+\sigma^2/2}$. An estimator for $\log(E(X))$ is $$\bar Y+\frac{S_Y^2}{2}$$. Also, $Var(\hat\log(E(X)))=\frac{\sigma^2}{n}+\frac{\sigma^4}{2(n-1)}$, which gives $$\hat Var(\hat\log(E(X)))=\frac{S_Y^2}{n}+\frac{S_Y^4}{2(n-1)}$$.

For large samples, using the CLT, we have $$\frac{\hat \log E(X)-\log E(X)}{\sqrt{\hat Var(\hat\log(E(X)))}}\sim^a N(0,1)$$

Thus, one possible CI from this perspective is $$CI(E(X))=\left]e^{\bar Y+\frac{S_Y^2}{2}\pm z_{1-\alpha}\sqrt{\frac{S_Y^2}{n}+\frac{S_Y^4}{2(n-1)}}}\right[$$.
According to the link, this is the Cox method, while the modified Cox method substitutes the standard normal quantile for a $t(n)$ quantile. They simulate more than just the these two intervals. They also simulate the naive approach.

If we look at table 5  of the link, we can see that the best method is the modified Cox, since for small samples, and large ones, the covering probability is the most close to the nominal level.

Alecos answer, if I understood correctly, suggests adding a correction term to the naive approach, resulting in $CI(E(X))=\left]e^{\bar Y+\frac{1}{2}\frac{S_X^2}{(\bar X)^2}\pm t_{1-\alpha}\sqrt{\frac{S_Y^2}{n}}}\right[$.

I've simulated his answer with the following code in Mathematica:

sim[n_] := Module[{},
  ct = 1;
  res = 0;
  While[ct <= 10000,
   data = RandomVariate[LogNormalDistribution[5, 1], n];
   y = Log[data];
   my = Mean[y];
   s2y = Variance[y];
   mx = Mean[data];
   s2x = Variance[data];
   qt = Quantile[StudentTDistribution[n], 0.975];
   If[E^(my + 0.5*s2x/mx^2 - qt*Sqrt[s2y/n]) < E^(5 + 0.5) <
     E^(my + 0.5*s2x/mx^2 + qt*Sqrt[s2y/n]),
    res++];
   ct++;
   ];
  res = res/10000.
  ]

The only difference is that I've done 10000 replications whereas the link only does 1000.

Running the code for the same sample sizes of the link, we obtain the following covering probabilities: $${0.8954, 0.8464, 0.7682, 0.7153, 0.6454, 0.4832, 0.3045, 0.073, 0.0068}$$.

AS we can see the correction increases the covering probabilities when compared to the naive method, but it still suffers from the same problem, i.e., the prob. tend to zero as the sample size increases.

All in all, the best course of action is to use the modified Cox approach.


What behaviour do different values of Relative Risk Aversion imply?

In the D&D model, We deal with a 2 periods economy where the agent can invest in a short term project that yields $0$ or a long term project that yields $R$;

It is possible to prove that an optimal insurance scheme (such as the use of a mutual fund or a bank) flattens the yield curve.

It is possible to show that the optimal portfolio problem solves:

$max_{\{c_1\}}\{\lambda u (c_1) + (1-\lambda) u(\frac{c_2 R}{1-\lambda})$

where $\lambda$ is the share of impatient consumers who consume in period 1 (those who "face a liquidity shock"), and $R$ is the yield on long term projects. The solution to the problem above gives: $\frac{u'(c_1)}{u'(c_2)} = R$.

Under the assumption of $u'>0, u''<0$, the condition $-\frac{cu''(c)}{u'(c)} > 1$ is technical; it ensures that $\forall c$, the first derivative of $ cu'(c)$ is negative, or that $cu'(c)$ is decreasing in c. Given $R>1$ this implies that $Ru'(R)>1u'(1)$ and so given that $c_1 = 1, c_2=R$ we have that $\frac{u'(c_1)}{u'(c_2)} = \frac{u'(1)}{u'(R)} > R$ .

Going back to the FOC of the optimal porfolio problem, in order to reach the equality we therefore need to have $c_1^*>1$ and $c_2^*<R$ so that to insure that $\frac{u'(c_1)}{u'(c_2)}$ decreases enough. This means that an insurance scheme determines a flattening of the yield curve

The bottom line is that there is not much intuition on why the condition holds but you can easily think of why it has to hold (given standard assumptions on the utility function) by looking at $lim_{c\to x} (-\frac{cu''(c)}{u'(c)})$ when $x= 0$ and $x=\infty$.

For more details on the DD model look at: Tirole Jean, "The Theory of Corporate Finance"


Why should the statistical value of life exist?

You asked:

  why should there exist a single value of $X$ that satisfies this definition for all values of $p$, or even all values of $p$ that are sufficiently close to $0$

There isn't such a value. I would hope that no one claims that there is.

The statistical value of life is a (somewhat lazy) calculation of convenience. Lots of business case protocols need a value for anything that's going into the business case. Changing the probabilities of survival is an outcome of many interventions for which the decision-makers have insisted on business cases, so some method is needed to value these probabilities.

One of the earliest ways to do this, back when relevant research was scarcer than it is today, and computational power was much more limited, was to assign a single value of life, which was calculated using methods that assumed a priori that there existed a single value of $X$ that was an adequate approximation for all values of $p$ that are sufficiently close to $0$.

That method is still used today largely due to institutional inertia.


Why are Massachusetts electricity costs so high?

To answer this I have to make some guesses because this is not an area of research for me, but having a spouse from there and having spent time there, I think I could make a somewhat educated guess.  Especially because it uses a market system rather than a rate setting system for generation.

First, Massachusettes has the third highest population density of any state and it is concentrated mostly around Boston.  Like rail, short distances are comparatively costly.  Higher loads in small locations require greater infrastructure costs.  Further, the variability will be harder to control because things that impact east Boston also impact west Boston, while things that impact eastern Montana do not at all impact western Montana.  There is less independence for unplanned events in a small space.  The covariance is high.  A noreaster can be far more damaging, if it is damaging, in a small space.  Repairs are costly.

Boston also has an old system.  While I could be wrong, I wouldn't be surprised if it was aging and depending on the regulation on carrying power, it might not be worth the investment without a regulatory structure that guarantees cost recovery.  Further, Massachusetts is growing and this means plugging costly new infrastructure onto old infrastructure.  As I am guessing, I don't know this, but it wouldn't surprise me that the influx is straining the infrastructure and I wouldn't be surprised if the infrastructure is past its planned life.  Boston was running electric cable cars in the 1880s.

The correct place to place this question would be with civil and electrical engineers.  You will probably find there is a really long history of how the current state of affairs came to be in Massachusettes in general and Boston in particular.  I would be surprised if Boston was not driving your costs.  I have no knowledge about Connecticut's system and if I had to guess for Hawaii it is due to the intrinsic fragility of being on an island with lots of people and no grid to support local emergencies.


What would happen if the U.S. "refinanced" its national debt?

The Treasury continuously issues new debt, and retires matured issues. (This is called “rolling over debt.”)
The Treasury has no right to pay debt off early (“refinance”).
The closest thing that can be done is to repurchase debt. The Federal Reserve - whose common equity is owned by the Treasury - is repurchasing debt during “quantitative easing” operations.


The economics of continually selling a product for under its market value?

1st of all, you should distinguish cases from the Nike case and others, if any:

Since Nike sells only a tiny portion of its shoes "irrationally" it cannot be seen as market value - equilibrium price, but rather as marketing expenses

Think that simply generating a rare nike products market adds value to the brand and motivates consumers to prefer nike

Regarding any other case: they can either be "dumping" - that is selling at a loss in order to throw your competitors, selling above your *average * cost, meaning making  a profit, maybe fightinv for market share, or being cross funded by different activity, to hurt competitors in the subject market


What does "to be in the double digits to get inflation" mean?

Economists believe that higher growth rates tend to also mean higher inflation rates.

So in this case what Mr. Green means is that

  You don't need a GDP growth rate of over 10% to get inflation. (Even GDP growth of 7.5% or 8% will be enough for the return of high inflation.)


Would insurance plan be necessary if we had instant access to credit?

As Dave Harris' points out in his comment, I assume that your question deals with events that do not compromise the individual's ability to work, which would prevent her from taking on debt.

Let's take the example of a fire which happens with probability $\pi \in [0,1]$. The damage of a fire equals $D$ dollars, i.e. it costs $D$ dollars to repair the house. If the individual buys full insurance at a price $p$ she ends up with a final wealth equal to $w-p$ in all contingencies since the insurance company covers the damages.

Suppose now that she does not buy any insurance but contracts a loan to cover the damages. In order to repair the house she has to borrow $D$ dollars. This loan costs her $D'$ dollars over her lifetime, taking into account the interest and her time preferences (discounting). She then ends up with final wealth equal to

$w$ with probability $1-\pi$
$w-D'$ with probability $\pi$

Thus, the two distributions that she faces are different under these two solutions. If she buys insurance, she completely eliminates the risk of having to pay for the damages. By contrast, if she contracts a loan ex post she still has a large downside risk of $w-D'$ in case of a fire. Under the standard conditions of insurance theory (e.g. risk aversion, actuarially fair insurance prices) it is clear that buying insurance would be strictly preferable.


How does private benefit exceed social benefit?

The net social welfare can be written as $W=B-C$, where $W$ is total welfare, $B$ is the benefit, and $C$ is the cost.

Suppose we are interested in thinking about a negative externality that implies a social cost of $E$. Then we would have $$W=B-(C+E)=B-C-E.$$

Imagine if, instead of viewing this as a social cost, we interpret the externality as reducing the social benefit. The expression would then be $$W=(B-E)-C=B-C-E.$$

So the two ways of thinking about the world are completely equivalent and it is merely a naming convention whether we call the externality a cost or a reduction in benefit. I guess your confusion has come about because you heard people on different occasions using the different conventions, without being clear about which convention they were using.

The only thing that matters is that we don't double-count the externality by including it as both a cost and a reduction in benefit.


I'll guess that, since you're not yet in college but you're doing something this "advanced," you're not doing a homework problem, and that you're asking for a hint.

Since B doesn't care about A, you should be able to find B's consumption $(C_B, l_B)$ using standard method (maximizing utility w.r.t. budget constraint, etc.)

Now there are two cases.

$C_B \geq 1$. Then what would the problem for A turn out to be?
$C_B < 1$. Then think about how much should A transfer to B? Should he transfer nothing? 0.1? Up until what point would the increase in the transfer stop making A better off?


How to detect if Ergodicity, Stationarity and Martingale. dif. sequence are satisfied?

Ergodic and strict stationarity are the essentially the weakest assumptions for which you have a LLN, i.e. can do large sample estimation. Given the very liberal way applied econometricians use laws of large numbers, ergodicity and strict stationarity is almost always assumed. If that is in question, you're simply not in business for consistent estimation.

Similarly, if one would like to have a CLT, i.e. do large sample inference, MDS is a general weak assumption. (There are other CLT's around but more technical. The MDS CLT and Lindenberg CLT are the two immediate generalizations of the classical CLT.)

The MDS assumption can be tested. An MDS is just a discrete-time martingale with mean zero. In particular, its increments are uncorrelated. (The martingale assumption is an intermediate condition between serially uncorrelated and independence.) So a rejection by a serial correlation test would also reject the MDS assumption.


Puzzle: Fairness of consumer price indices

The economic purpose of the CPI is to convert nominal units of consumption (goods and services) into real units of consumption. I understand that the CPI is used for other many purposes, but that's the economic purpose. For example, say that there  is one consumption good, corn, and corn is priced in denarii. We know that in 2010 50 billion denarii of corn was consumed while in 2018 100 billion denarii of corn was consumed. Was more corn consumed in 2010 or 2018? To answer this, you need to know the price of a kilogram of corn in denarii in 2010 and 2018. Say that $P_{2010}=25\:denarii/kilogram$ and $P_{2018}=100\:denarii / kilogram$ In this case consumption was 2 billion kilograms in 2010 and 1 billion kilograms in 2018.

When we get to multiple goods there is a complication because you can't add up or otherwise compare quantities of different goods. Therefore, it is convenient to pick a common numeraire. In the example, if we pick 2010 dollars as the numeraire then we see that consumption was 50 billion 2010 denarii in 2010 and 25 billion 2018 denarii in 2018. Both versions tell the same story, that real consumption was halved in 2018 relative to 2010.

This same logic holds when there are multiple goods. Let $t=0$ be the base (numeraire) year and $t=\tau$ be the year of comparison:
$$Real\: t=\tau\: Output\:in\:t=0\:prices =Expenditure_{good\:1,\tau}\cdot P_{1,0} / P_{1,\tau} + Expenditure_{good\:2,\tau} \cdot P_{2,0} / P_{2,\tau}\ldots + Expenditure_{good\:n,\tau} \cdot P_{n,0} / P_{n,\tau}$$

Recall that expenditures are prices times quantities, so expenditures divided by prices are quantities.
$$Real\: t=\tau\: Output\:in\:t=0\:prices =Q{1,\tau}\cdot P_{1,0}  + Q{2,\tau} \cdot P_{2,0} \ldots + Q{n,\tau} \cdot P_{n,0}$$
While nominal output is:
$$Nominal\: t=\tau\: Output\:in\:t=\tau\:prices = Q{1,\tau}\cdot P_{1,\tau}  + Q{2,\tau} \cdot P_{2,\tau} \ldots + Q{n,\tau} \cdot P_{n,\tau}$$
This lets us define the CPI:
$$\frac{Nominal\: t=\tau\: Output\:in\:t=\tau\:prices}{Real\: t=\tau\: Output\:in\:t=0\:prices} = CPI_{\tau}$$
That is, the relative change in the price level is the relative change in each prices weighted by the size of their expenditures.

For welfare purposes, it is important to understand that there are income and substitution effects from price changes. We can buy more of relatively cheaper goods and less of relatively expensive ones. A high price of a good we consume makes us effectively poorer. Both income and substitution can change our consumption bundles, and the Paasche, Laspeyres, and Fisher indices are attempts to deal with these issues. And there are other complications like changes in product availability, change in quality and features, and other measurement issues. But the overall idea is as above.

Therefore, if we are interested in making consumption comparisons across time, there really is only one way to do it, to weight the changes in prices by expenditures, and this means over-weighting the consumption of the risk because the consume more.

What happens if the weighted average change in the prices of the goods consumed by the poor is different than that of the rich? In this case the index is going to look more like the consumption of the rich than the poor. That, in and of itself, is not unfair, it is just an accounting result. This can become unfair when taxes and benefits are tied to indexed values. So, for example, if the goal is to guarantee everyone a minimum real income, and prices go up faster for the bundle consumed by the poor than the rich, the real value of a fixed benefit will go down.

This shows that the CPI is not the ideal index for all purposes. In my eyes, it is valuable to know (or at least well approximate) the real consumption output. If the goal is instead to create an index for the benefits of the poor to preserve their real value then a separate index should be made for that purpose. The sensible goal of tying the index to its purpose leads to a proliferation of indices. In addition to the CPI, the US has the producer price index ( measure of intermediate good pricing), the personal consumption expenditures index (CPI is based on a survey of what households are buying; the PCE is based on surveys of what businesses are selling), and a consumer price index for the elderly. There probably are many others.

To my knowledge, there is no regularly updated government price index for the consumption of the poor. However Garner, Johnson, and Kokoski (1996) do develop such an index, and find it is very similar in trends to that of the more general CPI. Broda, Leibtag,  and Weinstein (2009) and find "By examining scanner data on thousands of household purchases we find that the poor pay less —not more—for the goods they purchase." However, there does seem to be some contrary evidence more recently like Inflation May Hit the Poor Hardest (Casselman (2014)).


Determining the elasticity of Hicksian Demands

Let variable $y$ be determined as a function of variable $x$: $y=f(x)$.

Then the "$x$-elasticity of $y$" is computed as $$\frac{\partial y}{\partial x}\frac{x}{y}.$$

We are used to seeing the "price-elasticity of demand" where $y$ is the quantity demanded, $f$ is the Marshallian demand function, and $x$ is the price. But the same concept can be applied more generally.

For example, if $x$ is income, $y$ is quantity, and $f$ is the Hicksian demand function then we could compute the "income elasticity of Hicksian demand" using the above formula.


Industrial Organization is the field concerned with estimating supply and demand relationships in individual product markets. You can see for example Einav and Levin for a description of what empirical IO people are up to. The specific models are usually much more complicated than what you learn in Mankiw, tailored to specific markets, and very strong assumptions are required to estimate them. For recent IO papers, you can see for example the NBER working paper series.


If the price of oil and energy goes down, why does CPI go up?

Canadian prices will tend to rise when energy prices fall because Canada tends to export oil and import other things. As a major oil exporter, a fall in the price of oil decreases the value of the Canadian currency— because foreigners have less need to purchase Canadian dollars with which to buy oil— causing other things to be more expensive to purchase (due to an increase in the prices of imported production inputs and finished goods).

Incidentally, this is why the Canadian Dollar is known as a commodity currency.


Determining the capital return from the investment in dynamic firms and their growth

I put the article through translate and it appears that this is the (translated) section you are talking about:

"To achieve this selection of 50 nuggets, the Tech Tour did not scan only the continent, but the whole world. The overall result for Europe is very encouraging since there are 284 companies (including 38 French) , compared to 121 in 2015. They are 181 in China, and 833 in the United States, a country where, nevertheless, injected are five times higher than those irrigating European nuggets."

The criteria for a "nugget" seems to be a company valued between $100 mm and $1 B with annual growth > 50%.

The fact that the U.S. has more of these companies is not surprising. The US economy is equal to that in size of the EU but the tech sector in the US is much more robust. In the US there are 100,000 software and IT companies with less than 500 employees which would put them in this category. US tech makes up 25% of the global marketplace. The fact that there are more US companies that fall into this specific criteria is not surprising.

Whether or not this makes the US a better investment is a different question. The number of investable firms has nothing to do with return on investment. If all the European companies grew in value at an average of 10% and the US companies grew at an average of 5% then the European companies would be a better investment in terms of expected value. It may be that the European companies are much riskier and more likely to default which would offset that number. The number of companies that falls into this arbitrary and narrow criteria has nothing to do about the benefit of investing in either in Europe or the US.

Investment decisions are based on a multitude of factors. European regulatory risk, political stability, etc. may all be factors to consider when investing in a European company but it's difficult if not impossible to make a blanket statement about the quality of a country's investment opportunities.


How does IS-LM model reduced investor confidence?

I studied maths not economics so I won't comment on your graphs or theoretic models. In my experience as a rates trader, some economic models have their place but their inherent assumptions do not capture the practical market dynamics. I'll comment on the parts of your question I disagree with. Hopefully you appreciate a devil's advocate.

Firstly, a "a sovereign debt ratings downgrade" by a ratings agency has little value in the world today as a binary event. Most financial practitioners have made up their mind about the credit quality of issuers through independent or public research before ratings agencies update their databases. Ironically, one possibility is that this may indeed be a classic "sell the rumour, buy the fact" scenario, whereby reasonably large speculators sell the bonds preemptively, profit from general declining markets before the ratings downgrade and then use the liquidity and market wide focus generated by the event to exit their positions, and in fact the bonds may actually increase in value slightly due to this. This is classic Jessie Livermore style of trading (Reminiscences of a Stock Operator). There will doubtless be managed funds whose rules preclude investing in junk status bonds so demand for the bonds overall will likely decline overall although I expect more over a period than specifically because of that binary event.

The notion of "bonds" is a very broad term. If an issuer is downgraded the impact may be greater for various different maturities. The UK in the wake of the financial crisis was under considerable pressure due to its reliance on the financial sector and its government bonds underwent considerable re pricing, but the riskier, longer dated bonds suffered considerably worse that the much less riskier, shorted dated bonds, so in that scenario money market bill rates fell but longer bond rates rose.

When I think about the currency wide issue of money, if you have a reasonably large amount of sovereign debt held by investors that is sold those investors then have only a set of options available to them with respect to that money:

reinvest in some other asset denominated in the same currency, corporate bonds or equities.
sell the currency and buy assets in another currency.
deposit the cash with banks or on repo for some form of collateral.

The first of these is just an asset reallocation and weakens sovereign bond prices relative to other assets in the economy. Your argument about government financing is true and the GDP output would be expected to decline, further as confidence also becomes a factor.
The second would weaken the spot FX rate and the cross-currency basis swap levels. Indirectly this threatens to increase inflation since import prices will be affected by the currency movement. However, a weaker currency might tempt outside speculators at some equilibrium point. The third is difficult; no asset manager will deposit funds unprotected with a bank (particularly a domestic one) and the alternative of receiving back sovereign bond collateral is unappealing, so the repo rates will probably be increased in line with the rates on the bonds themselves, although this is another form of interest rate that does have significant impact and is not necessarily the same the bond rates. Indeed there will also be curvature to this as well.

As an example in 2017 the German government bond curve had 0-1Y yields at -70bps, and a repo rate often at similar levels. But the 20y-30y bond yields were at +100bps whilst the repo rate might be at -100bps. This may sound like arbitrage is possible, but it actually isn't - there is no scenario you can gain without running risk, and those different risks were considered and deemed to be worth the values potentially attainable by those who dared.

Finally I will point out to you the transmission mechanism. A central bank presides over its own currency and attempts to influence its economy by changing its deposit rate. However, a central bank is inherently connected to the government. I do not consider it wise to treat the two as separate entities. As a national bank, whose fortunes are intimately tied to the government, you would probably be better off buying and holding the sovereign debt if the short dated bonds offered considerably more interest than the central bank deposit rate, since from their point of view a government failure or a central bank failure might be one and the same. If you consider the holdings of Italian sovereign debt by Italian banks you will recognise this strategy as a kind of moral hazard. So the transmission mechanism is likely to remain in place for the short dated bonds and be ultimately tied to the deposit rate.

I appreciate there is no general conclusion to this thread, and some ideas are not necessarily coherent or consistent. I just found it an interesting question and thought I would explore my own thoughts in raw form. Possibly you find them useful.


Can I treat pooled cross-section as panel data and do regression analysis?

Yes, you can consider this as panel data, but the key is to understanding why, as this affects how you explain and interpret your panel regression. In treating each cohort as the same unit over time, you are assuming that each cohort has a constant, unobserved fixed effect. That is, people in a particular age category and gender tend to have an unobserved effect on the dependent variable that is different to other cohorts.

Please note:

Age increases over time, so, for example, the group aged 30–34 in year 1995 should be treated as the same group as the 35–39 year-olds in 2000. I suggest your time variable increases in steps of five years, as you indicated that your age categories are five-year bins.


How can a positive relationship between capital per effective worker and technological change be explained?

Your question is not that clear. So let me elaborate an answer on what I have understood from your question.

Note that for the Solow model, when technological change increases, it is the "per capita" capital which decreases. This one is normal because you should give much more capital to your workers, as they are more productive (here the technical progress increases the productivity of workers.)

I don't understand in which context $\frac{K}{AL}$ increases supply of innovations. What spurs innovation in Schumpeterian type growth models is the labor allocation in R&D (or spendings in R&D). This increases the creative destruction rate in the economy. For example, in this case, a decrease in labor $L$ can be consiidered as an increase in labor allocation in R&D.

Or another possible interpretation that I can make is the following one. There exists a stock effect in Schumpeterian models. This means that if your knowledge stock $A$ is high enough, it would be difficult to make new innovations. (See Acemoglu 2009, chapter 14)

So, in this case a higher $\frac{K}{AL}$ implies more innovation.


What would happen if, hypothetically, money velocity would reach infinity?

No $Y$ would not become infinity. $Y$ represents the real output of an economy. A real output is limited by the economy's production possibility frontier. For example, if we assume simple production function $F(L)=10 \sqrt(L)$ with assuming the economy's stock of labor being $100$ such economy would not be able to produce more than $100$ units of output whether velocity is infinite or not.
In the short run an economy might not be using its resources to the full extent and in that case increase in the velocity could have some effect - although causality would go through prices not directly from velocity (see Blanchard et al. Macroeconomics an European Perspective).  Furthermore, to see the effect of velocity we could use a simplified model for money market equilibrium given by the equation of exchange:
$$MV=PY$$
Where $M$ is money supply, $V$ velocity, $P$ price level and $Y$ output. As explained above $Y$ is limited by the production capability of an economy. Hence $Y$ can be treated as fixed. Next $M$ is under a control of a government through central bank or other institutions. Since in your problem you dont specify any government action regarding $M$ we can also treat it fixed. Then the only other variable that remains is $P$. Hence if $V$ wold increase to infinity $P$ would also increase ceteris paribus.
However, as alluded to above in a short run economy might not be operating at its full potential. In such scenario increase in $P$ in the short run would also stimulate firms to produce more according to standard macro AS-AD model. Hence in short run some of the increase in velocity could also be soaked by $Y$ (through increase in $P$ induced by increase in $V$) but certainly not anywhere near to reach infinity. $Y$ must in the end respect the country's PPF which in a world of scarcity will be finite.

Furthermore, if the money velocity is infinite, then it also would mean that every economic agent of the economy is in possession of the same amount of money and thus no wealth inequality would exist?

No, a velocity of money is defined as: "the frequency at which one unit of currency is used to purchase domestically- produced goods and services within a given time period" (see FRED). High velocity does not imply more equal distribution of money. For example, imagine an economy with two people A and B, with A having one 1 $\\\$1$ banknote and B having 2 $\\\$1$ banknotes - and lets for simplicity assume anytime they spend money they instantly get it back as an income. If A uses his banknote only twice to purchase goods and services (he purchases something - immediately gets it back as an income and purchases something again) but B uses both of his two banknotes 10 times the velocity will be $22/3 \approx 7.33$. If B decides to use his banknotes 50 times the average velocity will be 34 and so on we could go on. Point is that higher or even infinite velocity does not necessary tell us by itself much about distribution of money in economy.
Furthermore, in public economics wealth is usually defined as a value of net assets. The value of net assets a person has does not correspond to the amount of money one has. For example, imagine person with house worth $\\\$ 100,000$ and no debt but absolutely no money. An economist would say that person's wealth is $\\\$ 100,000$. Now imagine person B with $\\\$1,000,000$ in a bank account while having also $\\\$1,000,000$ debt. Such person would be recorded as having $0$ wealth. Different scholars might disagree about whether wealth should include things like social security and on other details but I know of no economist that would equate wealth to money holdings.   Hence there is no simple relationship between distribution of money and wealth distribution.