To summarize what I wrote below, there seem to be at least two points:

In the Smets and Wouters (ECB) research, which is apparently responsible for a lot of the fame of DSGE, they indeed found it superior in forecasting accuracy to (B)VARs, especially long-term. On the other hand, other research (including some by other ECB researchers) doesn't seem to be so uniformly conclusive in this (accuracy) regard. Alas, it doesn't look like there is anything resembling a systematic review on this issue; perhaps the variations in data sets and methodology preclude one.
In the more academic realm, proponents of DSGE (e.g. Christiano) argue that DSGE has deep explanatory powers for some inflation-related lags, making them easy to understand.

I'm sure someone will give you a more technical answer, but basically DSGE works well for monetary policy (inflation targeting) etc.

  Part of the recent popularity of DSGE models is due to work by Smets and
  Wouters (2003), who document that a modified version of a New Keynesian model
  developed by Christiano, Eichenbaum, and Evans (2005) is able to track and forecast
  euro area time series as well as, if not better than, a vector autoregression (VAR) estimated with Bayesian techniques.

In other areas like fiscal policy DSGE hasn't seen that much success/application, insofar anyway.

However, the New Keynesian DSGEs have their critics, particularly in the Minneapolis Fed; see Chari, Kehoe, and McGrattan.

  Proponents of the New Keynesian model argue that it is promising for two reasons. It
  represents a detailed economy that can generate the type of wedges we see in the data from
  interpretable primitive shocks; and second, it has enough microfoundations that both their
  shocks and parameters are structural, in that they can reasonably be argued to be invariant
  to monetary policy shocks. A model with both of these features would potentially be useful
  for monetary policy analysis.

  We disagree. We argue that these models cannot generate the type of wedges we see
  in the data from interpretable primitive shocks. And it is doubtful that many of the features
  added on in the quantitative implementation of the models are structural. Hence, the models
  are not yet useful for policy analysis.

(The first paper I quoted from is from the Atlanta Fed.)

Also, VAR and DSGE are not orthogonal or alternative as you suggest; from the Atlanta paper:

  It has long been recognized (for example, Sims 1980) that a tight relationship exists between dynamic
  equilibrium models and VARs. Imagine the following thought experiment, where for
  the moment the vector of DSGE model parameters is fixed. We generate 1 million
  observations from the DSGE model—that is, we generate a sequence of shocks
  (monetary policy, technology, etc.), feed them trough the DSGE model, and obtain
  artificial data. Next, we estimate a VAR with p lags on these artificial data. If the
  DSGE model is covariance stationary, then the estimated VAR provides an approximation to the DSGE model with the property that its first p autocovariances are
  equivalent to the first p autocovariances of the DSGE model. By including more and
  more lags we can in principle match more and more autocovariances and increase the
  accuracy of the VAR approximation of the DSGE model. Now imagine that the data
  generation is repeated using different parameter values for the DSGE model. As long
  as the DSGE model parameter space is small compared to the VAR parameter space,
  a restriction function can be traced that maps the DSGE parameters into a VAR
  parameter subspace. Hence, estimating a DSGE model is (almost) like estimating a
  VAR with cross-equation restrictions.

That paper is somewhat limited (empirically) in that it looks at a single (New Keynesian) DSGE and VAR-izes it.

For a more general comparison see: Raffaella Giacomini, The Relationship Between DSGE and VAR Models. It does mention some caveats e.g.

  a meaningful discussion of the relationship between DSGE and VAR
  models can only be carried out in the context of log-linearized DSGE models, taking for granted
  the adequacy of the linear approximation and ignoring possible nonlinear dynamics that cannot
  be replicated by linear VAR models.

It goes on to point out that non-linear DSGEs are (still) a minority in current research, but that that sub-field appears to be growing pretty fast.

The claim to DSGE empirical superiority (as estimated by the more widespread Bayesian techniques, rather than by embedding in VARs) is probably in the narrow(er) context of specific papers; the 2007-updated Smets and Wouter does indeed make such claims. In the short term, they only find a a difference for unconstrained VARs and they find that a BVAR model does as well as their DSGE; but for a longer forecast they indeed found their DSGE superior to both VAR/BVAR.

  In this Section we compare the out-of-sample forecast performance of the estimated DSGE model with
  that of various VARs estimated on the same data set. The marginal likelihood, which can be interpreted as
  a summary statistic for the model’s out-of-sample prediction performance, forms a natural benchmark for
  comparing the DSGE model with alternative specifications and other statistical models. However, a Sims (2003) has pointed out it is important to use a training sample in order to standardize the prior
  distribution across widely different models. In order to check for robustness, we also consider a more
  traditional out-of-sample RMSE forecast exercise in this section.

  Table 2 compares the marginal likelihood of the DSGE model and various unconstrained VAR models,
  all estimated over the full sample period (1966:1 – 2004:4) and using the period 1956:1 – 1965:4 as a
  training sample. Several results are worth emphasizing. First, the tightly parameterized DSGE model
  performs much better than an unconstrained VAR in the same vector of observable variables, Υt (first
  column of Table 2). The bad empirical performance of unconstrained VARs may not be too surprising, as
  it is known that over-parameterized models typically perform poorly in out-of-sample forecast exercises.
  One indication of this is that the marginal likelihood of the unconstrained VAR model deteriorates
  quickly as the lag order increases. For that reason, in the second column of Table 2, we consider the
  Bayesian VAR model proposed by Sims and Zha (1998). This BVAR combines a Minnesota-type prior
  (see Litterman, 1984) with priors that take into account the degree of persistence and cointegration in the
  variables. In order to allow the data to decide on the degree of persistence and cointegration, in this
  BVAR we enter real GDP, consumption, investment and the real wage in log levels. When setting the
  tightness of the prior, we choose a set of parameters recommended by Sims (2003) for quarterly data.
  The second column of Table 2 shows that the marginal likelihood of the Sims-Zha BVAR increases
  significantly compared to the unconstrained VAR. Moreover, the best BVAR model (BVAR(4)) does as
  well as the DSGE model.

  Overall, the comparison of marginal likelihoods shows that the estimated DSGE model can compete with
  standard BVAR models in terms of empirical one-step-ahead prediction performance. These results are
  confirmed by a more traditional out-of sample forecasting exercise reported in Table 3. Table 3 reports
  out-of-sample RMSEs for different forecast horizons over the period 1990:1 to 2004:4. For this exercise the VAR(1), BVAR(4) and DSGE model were initially estimated over the sample 1966:1 - 1989:4. The
  models were then used to forecast the seven data series contained in Υt from 1990:1 to 2004:4, whereby
  the VAR(1) and BVAR(4) models were re-estimated every quarter, whereas the DSGE model was reestimated every year. The measure of overall performance reported in the last column of Table 3 is the
  log determinant of the uncentered forecast error covariance matrix.

  The out-of-sample forecast statistics confirm the good forecast performance of the DSGE model relative
  to the VAR and BVAR models. At the one-quarter ahead horizon, the BVAR(4) and the DSGE model
  improve with about the same magnitude over the VAR(1) model, confirming the results from Table 2.
  However, over longer horizons up to three years, the DSGE model does considerably better than both the
  VAR(1) and BVAR(4) model. Somewhat surprisingly, the BVAR(4) model performs worse than the
  simple VAR(1) model at longer horizons. Moreover, the improvement appears to be quite uniform across
  the seven macro variables.

However, there's somewhat newer (2009) ECB paper by different authors (Kolasa, Rubaszek, and Skrzypczyński) with less sweeping conclusions:

  In the case of ination, the DSGE and DSGE-VAR
  models with the maximum lag set to 3 and 4 are characterized by the lowest RMSFEs. The SPF [Survey of Professional Forecasters]
  and BVAR(4) are insignicantly less accurate, while the BVAR and low-order DSGE-VAR models are
  found to be the worst. Finally, the RMSFEs for interest rate forecasts formulated by all methods are
  comparable.

(There's a 2012 peer-reviewed version of that paper, if you care about that. And As an aside, somewhat disappointingly, adding (corporate) financial market frictions to their DSGE model did not uniformly improve forecasts, but adding housing market frictions did help especially with crises.)

Another paper which focuses on hybrid DSGE models (Paccagnini, 2011) has this introductory note:

  from an econometric point of view, the performance of a DSGE model is often
  tested against an estimated Vector Autoregressive model (VAR). This procedure requires a Data
  Generating Process (DGP) that is consistent with the theoretical economic model and has a finite-order VAR representation. However, the statistical representation of a DSGE model is an exact
  VAR only when all the endogenous variables are observable; otherwise, a more complex Vector
  Autoregressive Moving Average model (VARMA) is needed. As far as the VARMA representation
  is concerned, several papers (see Cooley and Dwyer (1998), Chari, Kehoe and McGrattan (2005),
  Christiano, Eichenbaum and Vigfusson (2006), Ravenna (2007) and Fernandez-Villaverde, Rubio-Ramirez, Sargent and Watson (2007)) have discussed the conditions required to find an infinite-
  order VAR representation and a infinite-order VAR truncation in general terms. Moreover, the VAR is densely parameterized and appears to be misspecified and a Bayesian approach is preferred.

And the results of this paper are not amazing for DSGE:

  Two forecasting exercises have been implemented on US quarterly economy data: the first one
  considering a simple forecasting one-step-ahead strategy with different samples, and the second a
  h-steps ahead evaluation with a rolling forecasting estimation. In the first exercise, the last part of
  the sample from 2007 onwards, which is the Great Contraction period, was not taken into account.
  The main results show that the hybrid models, such as the DSGE-VAR and the DSGE-FAVAR give
  the most accurate forecasts for smaller forecasting samples. Instead, a FAVAR model outperforms
  the other models when we consider a long forecasting sample. The second forecasting experiment
  uses a rolling sample with the Great Contraction period included. The best forecasting performance
  is produced by VAR, BVAR and FAVAR, with the exception of real GDP where for h-steps ahead
  larger than 2 the DSGE-FAVAR outperforms the alternative models.

As a side-note to that last paper, FAVAR was apparently favored/suggested by Bernanke.

These (not always clear) accuracy contests aside, DSGE advocates claim that it supposedly has deep explanatory power for inflation:

  The new monetary DSGE models are of interest not just because they represent laboratories for the analysis of important monetary policy questions. They are also of interest because
  they appear to resolve a classic empirical puzzle about the effects of monetary policy. It has
  long been thought that it is virtually impossible to explain the very slow response of inflation
  to a monetary disturbance without appealing to completely implausible assumptions about
  price frictions (see, e.g., Mankiw (2000)). However, it turns out that modern DSGE models
  do provide an account of the inertia in inflation and the strong response of real variables to
  monetary policy disturbances, without appealing to seemingly implausible parameter values.
  Moreover, the models simultaneously explain the dynamic response of the economy to other shocks. We review these important findings. We explain in detail the contribution of each
  feature of the consensus medium-sized New Keynesian model in achieving this result.

  The econometric technique that is particularly suited to the shock-based analysis described in the previous paragraph, is the one that matches impulse response functions estimated by vector autoregressions (VARs) with the corresponding objects in a model. Using
  US macroeconomic data, we show how the parameters of the consensus DSGE model are
  estimated by this impulse-response matching procedure. The advantage of this econometric
  approach is transparency and focus. The transparency reflects that the estimation strategy
  has a simple graphical representation, involving objects - impulse response functions - about
  which economists have strong intuition. The advantage of focus comes from the possibility of studying the empirical properties of a model without having to specify a full set of
  shocks.


Free market defense against anti-competitive behavior?

Depends on what you mean by 'possible'. The liquor stores in your example could form a union, and collectively bargain with brands. Then if a brand breaks a contract with one of them, they all break the contract. This would greatly disincentivize the brands' strong arm practice.

In my opinion this is unlikely to work (it is very hard to organize), but it is 'possible'.


How will the economic effects of Brexit impact a middle-class American family?

There will be many effects, but most of them are too small or too difficult to measure for you to notice anything.

Some examples of the bigger effects:

If you have a pension or own any stocks (hopefully you do), you will have been negatively affected - even though you don't have any 'international' finances. Since the world economy is highly integrated, losses in the UK are correlated with losses in the U.S., Japan, and other world markets. For example, after the news of Brexit broke, the S&P (one of the US stock market indices) fell roughly 100 points. The markets tend to move together (the reason behind this is another question in of itself), so any changes in Britain will affect Americans as well. We're still not sure about the long-term economic effects of this, since we don't know exactly how/when/if the U.K. will exit, but the short-term uncertainty is causing investors to be nervous.
British things will be cheaper! The pound fell, meaning that it's cheaper for Americans to buy British things, and more expensive for the British to buy American things. If you wanted to visit the U.K., it's going to be relatively cheap. The simplest effect is that the fall of the pound will positively affect U.S. consumers of British goods, and negatively affect U.S. exporters who export to the U.K..
There may be some effect on your work, but it's impossible for you to know (which is part of the problem. Although your employer may appear to be completely local, it's important to remember that since we're so globalized, there probably will be some effect at your company. For example, one of your
company's clients may purchase services from a consultant in the
U.K, and this might affect your client, in turn affecting you. It's
this kind of domino effect and the uncertainty that Brexit has
caused that has so many politicians and economists worried about
what might happen.


Why did Germany use hyperinflation to free itself from an external debt?

The issue is that they did not had much other choice as the war and war reparations made German hyperinflation virtually inevitable.
As given by quantity theory of money a price level, change in which gives you inflation, is given by:
$$P = \frac{MV}{Y}$$
so the change in price level $P$ depends not just on quantity of money $M$, but also velocity $V$ and real output of an economy $Y$.
Even if Weimar Republic would not resort to printing money arguably the fall in real output caused by war, loss of population and territory and horrendous reparations, would be very inflationary in themselves. However, in addition it is true that Weimar Republic also resorted to heavy use and abuse of printing presses exacerbating  the inflation even further leading to hyperinflation. But question is what was the alternative?
Every state must spend at least some minimum amount of money in order to pay police/bureaucrats that enforce the laws of the government. Famously, Weber defined government as a monopoly on legitimate use of power in certain area. Well that is costly, and we are talking about bare minimum for state to exist ignoring that Weimar Republic had also other welfare spending that was likely necessary to keep people 'in line'.  What was the Weimar Republic to pay these costs when virtually all of her productive capacities were destroyed by war, loss of territory, loss of population etc.? Non-trivial amount of German output had to be given to Allies to pay for the reparation itself and debt owed as reparations could not be discharged (in past it was common to extort external debts even at the point of threat of armed conflict). Arguably, the only option was to print more and more money to at least eliminate the domestic debt and keep 'chugging along' or to let the state disintegrate.
The heavy burden of reparation's have to really be emphasized here. As documented by Keynes in his famous book 'The Economic Consequences of the Peace' the war reparations Germany was forced to pay were simply too horrendous.
For example Keynes reports that the treaty demanded:

Germany has ceded to the Allies all the vessels of her mercantile
marine exceeding 1600 tons gross, half the vessels between 1000 tons and
1600 tons, and one quarter of her trawlers and other fishing boats.[9]
The cession is comprehensive, including not only vessels flying the
German flag, but also all vessels owned by Germans but flying other
flags, and all vessels under construction as well as those afloat.[10]
Further, Germany undertakes, if required, to build for the Allies such
types of ships as they may specify up to 200,000 tons[11] annually for
five years, the value of these ships being credited to Germany against
what is due from her for Reparation.[12]

Further it was stipulated that:

(2) Germany has ceded to the Allies "all her rights and titles over her
oversea possessions."[13] This cession not only applies to sovereignty
but extends on unfavorable terms to Government property, all of which,
including railways, must be surrendered without payment, while, on the
other hand, the German Government remains liable for any debt which may
have been incurred for the purchase or construction of this property, or
for the development of the colonies generally.[14]

in addition,

In distinction from the practice ruling in the case of most similar
cessions in recent history, the property and persons of private German
nationals, as distinct from their Government, are also injuriously
affected.

As crazy as it sounds even private German nationals, who might not even have anything to do with war, were subject to the reparations. Furthermore, as reported by Keynes the property, both private and public, in German colonies and Alsace-Lorraine was to be expropriated by Allies. Setting moral considerations aside this greatly diminishes potential tax base upon which Germany could have draw resources from otherwise.
Also this is all in addition to the reparations itself which according to Keynes were set in a way that Germany could not simply inflate them away as:

The Reparation Commission is empowered up to May 1, 1921, to
demand payment up to $5,000,000,000 in such manner as they may fix,
"whether in gold, commodities, ships, securities or otherwise."

As if all of the above was not enough the treaty also forced Germany to give certain amount of resources coal/iron to Allies

i.) To France 7,000,000 tons annually for ten years;[42]
(ii.) To Belgium 8,000,000 tons annually for ten years;
(iii.) To Italy an annual quantity, rising by annual increments from
4,500,000 tons in 1919-1920 to 8,500,000 tons in each of the six years,
1923-1924 to 1928-1929;
(iv.) To Luxemburg, if required, a quantity of coal equal to the
pre-war annual consumption of German coal in Luxemburg.

And this is after ceding Alsace-Lorraine from which,

...Almost exactly 75 per cent
of the iron-ore raised in Germany in 1913 came from Alsace-Lorraine.[53]
In this the chief importance of the stolen provinces lay.

There are further provisions in the treaty but I think the above is enough to get some picture.
The monetary reparations were also extremely large relative to the Germany's GDP (although as a source below shows in the end even Allies realized the whole debt will not be realistically payed). As reported by Albrecht Ritschl

Taken in isolation, the A bonds amounted to 20% of German GDP of 1913, a burden that was
equivalent to France’s reparations to Germany in 1871 (Ritschl, 1996). Adding the B bonds generated
a reparations burden of 100% of 1913 GDP. Including also the C bonds raised the reparations burden
to over 260% of 1913 GDP. However, it was communicated to the Germans that the C bonds would
not have to be paid under any realistic conditions. The Germans had anticipated a burden of 30-40bn
gold marks. Hence, the realistic part of the reparation bill was not entirely beyond the imagination of
German policy makers (Feldman, 1993; Ferguson, 1998)

The purpose of the treaty was literary to destroy the economy of Germany. Furthermore, it was very bad for the international economic order in general. As Keynes reports:

The Treaty includes no provisions for the economic rehabilitation of Europe,– nothing to make the defeated Central Powers into good neighbors, nothing to stabilize the new states of Europe, nothing to reclaim Russia; nor does it promote in any way a compact of solidarity amongst the Allies themselves; no arrangement was reached at Paris for restoring the disordered finances of France and Italy, or to adjust the systems of the Old World and the New.
... Reparation was their main excursion into the economic field, and they settled it as a problem of theology, of politics, of electoral chicane, from every point of view except that of the economic future of the States whose destiny they were handling.

Thus to sum up, Germany was out of options. It lost non-trivial portion of its production capacity and taxable territory and population. It was up for enormous reparation bill that had to be paid either in gold or in assets or products, and it still had to run its own government somehow. Given such situation arguably the choice was to print money or stop paying its government employees and other domestic obligations.


Based on the official documentation, you have found that there is one cointegration relationship with 95% probability. Given your selection of Model: H1* and Lags: 4, the model you are estimating is:

$$ \Delta y_t = A(B´y_{t−1}+c_0) + B_1\Delta y_{t-1} + B_2\Delta y_{t-2} + B_3\Delta y_{t-3} + B_4\Delta y_{t-4} + \epsilon_t $$

The matrix B is of dimension $n \times r$, and it contains as columns the cointegration vector(s). In your case, $r=1$, and the cointegration vector is

B =
0.7081
1.6282
-2.4581

The Matrix A represent the "error-correction" matrix. For a system to converge towards the equilibrium cointegration relation (i.e. to be stable), you want these numbers to be in between -2 and 0 (as in your case).

c0 is just a constant of the cointegration relationship (like in a standard function $y=c0+bx$).

The matrices Bi show how previous period shocks translate through the system over time.


Assuming a competitive global market,

If an importer would by steel in the global market in global prices, he will have to absorb the tax cost at importing, then roll it to its customers at home
The exporter will not absorb the tax, since it can sell to other countries without it.

If the importer's country is big enough market-wise, then:
some tax absorption on the exporter's side might take place since the exporter will lose large sales if no absorption is made,
And the importer will not roll all the tax to its customers since competing importers, might receive the same tax-absorption sharing with exporters (theory dictates it, all are equal in a competition),
In this case, a lower price from competitors will lead to a loss of market share


If my IV (denoted Z) equals 1, my endogenous var D always equals 1; is this a problem?

Let $\hat{\pi}_0$ and $\hat{\pi}_1$ be the first stage coefficients. That is, $\hat{D} = \hat\pi_0 + \hat\pi_1 Z$. When $Z_i D_i = Z_i$ (if $Z_i=1$ then $D_i=1$), we can show that $\hat\pi_0 + \hat\pi_1 = 1$ always and $\hat\pi_0 = (\bar{D}-\bar{Z})/(1-\bar{Z})$. (Proof is fun.) Unless $\bar{D} \simeq 1$, $\hat\pi_0$ is far from unity, which means $\hat\pi_1$ is far from zero. Not a surprise that the p-value is very small.

Another argument: Because $ZD=Z$, we have $Z(1-D)=0$. Thus, $Z$ and $1-D$ are strongly correlated unless $E(Z)=0$ or $E(D)=1$. By the way, try probit D Z to see an interesting result.

Whether you need more variability is another thing. The question is whether it is possible that $Z$ is exogenous and $D$ is endogenous. Need to think more.

EDIT: Thought about whether it is possible that $Z$ is exogenous, $D$ is endogenous, and $ZD=Z$ at the same time. It looks possible. Let $u$ be the regression error term. We can let $D=Z+(1-Z)\xi$, where $\xi$ is a Bernoulli random variable, so that $ZD=Z$. Let $E(u|Z)=0$ as we wish. Then $E(Du)=E(u|Z=1)P(Z=1) + E(\xi u|Z=0) P(Z=0) = E(\xi u|Z=0) P(Z=0)$, which can be nonzero. It is possible that $Z$ is uncorrelated with $u$, $D$ is correlated with $u$, and $ZD=Z$, I guess. So my answer is: No worries.

EDIT: But "$Z=1 \Rightarrow D=1$" means that $D$ is exogenous if $Z=1$, and is endogenous if $Z=0$. Whether this is OK depends on the context.


How to average CDFs of one variable across years

Why don't you just take a weighted average?
Suppose you have ten years $t \in \{1,...,10\}$ and year $t$ has $N_t$ observations such that in total you have $\sum_t N_t=N$ observations. Let the year-$t$ CDF be $F_t$ with support $[\underline w_t,\overline w_t]$.
You can then define a weighted average CDF as
$$\overline F (w) = \sum_t \frac{N_t}{N} F_t(w).$$
This gives you a cdf, an increasing right-continuous function ranging over [0,1] with support $\cup_t [\underline w_t,\overline w_t]$. However, you have to pay attention to the individual supports, i.e., $F_t(w) =1 \forall w>\overline w_t$ and $F_t(w) =0 \forall w<\underline w_t$.


Overlapping Generations model: Social Planner solution

Start by writing the Lagrangian (as it sounds like you have already done):

$$L=\sum_{t=0}^{\infty} \{\beta^t(\ln C_t^y + \ln C_t^o) + \lambda_t (C_t^y +\frac{C_t^o}{1+n} +k_{t+1}(n+1)-k_t-f(k_t))\}.$$

First order conditions are given by
$$\frac{\partial L}{\partial C_t^y} = \frac{\beta^t}{C_t^y} +\lambda_t = 0$$
$$\frac{\partial L}{\partial C_t^o} = \frac{\beta^t}{C_t^o} +\frac{\lambda_t}{1+n} = 0$$
$$\frac{\partial L}{\partial k_{t+1}} = \lambda_{t-1}(1+n) -  \lambda_t(1 + f'(k_t)) = 0$$
$$\frac{\partial L}{\partial \lambda_{t}} = C_t^y +\frac{C_t^o}{1+n} +k_{t+1}(n+1)-k_t-f(k_t)=0.$$

First notice the first two equations give us
$$\frac{C_t^y}{C_t^o} = 1+n.$$
Also notice that the first equation implies $\lambda_t = -\frac{\beta^t}{C_t^y}$. Substituting this into our third equation gives
$$\frac{\beta^{t+1}}{C_{t+1}^y}(1+n) = \frac{\beta^{t}}{C_{t}^y}(1+n)(1 + f'(k_t)$$
or equivalently
$$\frac{\beta C_t^n}{C_{t+1}^y} = (1 + f'(k_t)).$$

Finally, we can write our system of four equations with three equations, having removed $\lambda$ from the system:
$$\frac{C_t^y}{C_t^o} = 1+n$$
$$\frac{\beta C_t^y}{C_{t+1}^y} = (1 + f'(k_t)).$$
$$C_t^y +\frac{C_t^o}{1+n} +k_{t+1}(n+1)-k_t-f(k_t)=0.$$

Dropping time subscripts, i.e. applying the steady-state condition, gives
$$\frac{C^y}{C^o} = 1+n$$
$$\beta = (1 + f'(k)).$$
$$C^y +\frac{C^o}{1+n} +nk-f(k)=0.$$

Hence, you are left with a system of three equations with three unknowns $C^y$, $C^o$, and $k$. Solve the system to arrived at a steady-state solution.


What does pilot price mean in the context of marketing?

If the phrase pilot experiment
or more famously television pilot is any indication, then pilot price is probably an initial price, one that the company experiments with when trying to find the ideal price of their product.


Can geography explain differences in economic development between present day economies?

Dani Rodrik, Arvind Subramanian, Francesco Trebbi have a paper on "Institutions Rule: The Primacy of Institutions over Geography and Integration in Economic Development", Journal of Economic Growth, June 2004, 9(2): pp.131-165.

They estimate the respective contributions of institutions, geography, and trade in determining income levels around the world, using recently developed instruments for institutions and trade. They results indicate that the quality of institutions trumps' everything else. They add that once institutions are controlled for, measures of geography have at best weak direct effects on incomes, although they have a strong indirect effect by influencing the quality of institutions.


Deriving optimality conditions in the New Keynesian model framework with an undefined consumption function

I'm not quite sure what $Z$ stands for in this model (looks like some kind of multiplier over the standard utility) but I can generally guess the rest. $C$ stands for consumption, $L$ for real money holding, $N$ for hours worked, $B$ is one-period bond, $M$ is nominal money, $W$ is wage, $D$ is dividend, and $T$ is tax.
$U(C,L)$ denotes the utility derived from consumption and real money holding while $V(N)$ is disutility from working. Rewriting the problem as follows,
$$\underset{C_t,N_t, L_t, B_t}{\max} E_0\sum^{\infty}_{t=0}\beta^t\mathcal U(C_t,L_t,N_t;Z_t) \\ \text{subject to} \\
P_tC_t+M_t+B_t\leq B_{t-1}(1+i_{t-1})+M_{t-1}+W_tN_t+D_t-P_tT_t
$$
You can now set up the Lagrangian
$$\mathcal L= E_0\sum^{\infty}_{t=0}\beta^t\mathcal U(C_t,L_t,N_t;Z_t) + \lambda_t [B_{t-1}(1+i_{t-1})+M_{t-1}+W_tN_t+D_t-P_tT_t-P_tC_t-M_t-B_t]$$
Taking the partial derivative with respect to the choice variables,
$${\partial \mathcal L\over \partial C_t}=\beta^tZU_{C,t}-\lambda_tP_t \tag{1}
$$
$${\partial \mathcal L\over \partial N_t}= -\beta^tZV_{N,t}+W_t \tag{2}$$
$${\partial \mathcal L\over \partial L_t}=\beta^tZU_{L,t}-\lambda_t P_t+\lambda_{t-1}P_{t-1} \tag{3}$$
$${\partial \mathcal L\over \partial B_t}= \lambda_{t+1}(1+i_t)-\lambda_t \tag{4}$$
Solving with equality (I'll leave the steps for you to complete the exercise), you can derive the Euler equation from (1) & (4). The intratemporal labor supply can be derived from (1) & (2). Lastly, the money demand schedule can be derived from (1), (3) and (4).


This is a difficult question in that it is very difficult to prove that something is impossible. However, based on the types of analysis done in financial markets, estimating supply and demand curves is obviously difficult to do.

The usual methods for valuing assets in financial markets is to rely on some relative value versus a similar asset, or discounted cash flow. For a currency, we can compare the prices of similar items (hourly wages, goods prices) to get fair value. However, I am unaware of items sold at fixed prices in crypto currencies that would allow us to use a similar valuation method.

As for supply and demand, unless we have a reason to know that supply and demand curves follow some particular parameterisation, all that it says that for every buyer there is a seller. A price change can either be explained by the supply curve or the demand curve shifting (or both); we have no idea what happened. There is no clear method to go from observed transaction data to back out supply and demand curves.

For example, imagine there are a sequence of transactions at rising prices. Is that because there was an increased demand by buyers, or was it the result of existing owners being less willing to sell? Since every transaction has two sides, we have no idea. All we know for sure is that the price went up - which we could see by just looking at the time series of prices.

Furthermore, prices can change without any transactions. In the financial markets, prices jump on the release of economic or corporate news. This is much more easily interpreted as a change in fair value, rather than a shift in demand or supply curves.

Finally, it is unclear what a supply or demand curve represents. In the short term, a crypto currency supply is fixed (at least until the next coin is “mined”). How exactly can the quantity supplied or demanded change?

In other words, probably the only way to see whether “supply and demand” work is to build a model that works. I am unaware of any successful attempts in other asset markets to build such a model, so you have little guidance on how it could be done.

Addendum: It is difficult to control supply to keep an asset near a target level. Preventing the price from going up is easy, as you could just flood the market. Stopping the price from falling is difficult. You cannot destroy currency units held by people, and so you cannot reduce the amount outstanding. If people want to sell at a lower price, there is little you can do except buy the crypto up with some kind of currency reserves. (Which is how currency pegs are defended.) However, currency pegs tend to fail. (I believe some crypto currencies follow a backing strategy; I do not know how popular those currencies are are.)


Unemployment: Human Capital Deterioration

I know nothing about this. But I entered "unemployment human capital deterioration mortensen pissarides" into Google Scholar and clicked the first link (Two Questions about European Unemployment, Ljungqvist and Sargent, Econometrica 2008) and Ljungqvist and Sargent just do the obvious thing.

  Employed and unemployed workers experience stochastic accumulation or deterioration of skills, respectively. There is a finite number of skill levels with transition probabilities from skill level $h$ to $h'$ denoted by $\mu_u(h, h')$ and $\mu_e(h, h')$ for an unemployed and an employed worker, respectively. That is, an unemployed worker with skill level $h$ faces a probability $\mu_u(h, h')$ that his skill level at the beginning of the next period is $h'$, contingent on not retiring. Similarly, $\mu_e(h, h')$ is the probability that an employed worker with skill level $h$ sees his skill level change to $h'$ at the beginning of the next period, contingent on not being exogenously laid off. In the event of an exogenous layoff, the transition probability is $\mu_l(h, h')$. After the initial period coinciding with an exogenous layoff, the stochastic skill level of an unemployed worker is again governed by the transition probability $\mu_u(h, h')$. All newborn workers begin with the lowest skill level.

So this is at least one approach. Do you have a particular reason not to use a Markov chain?


Microeconomic rationale for UIP (uncovered interest rate parity)

There is an arbitrage argument for UIP because under UIP forward rate must be equal to expected exchange rate so the same arbitrage arguments from CIP can be extended to UIP (under its strict assumptions). You will find it in most international macroeconomics or finance textbooks. For example, according to Nelson Mark's International Macroeconomics and Finance:

If foreign exchange participants are risk neutral, they
care only about the mean value of asset returns and do not care at all
about the variance of returns. Risk-neutral individuals are also willing to take unboundedly large positions on bets that have a positive
expected value. Since $F_{t} − S_{t+1}$ is the profit from taking a position in
forward foreign exchange, under risk-neutrality expected forward speculation profits are driven to zero and the forward exchange rate must,
in equilibrium, be market participant's expected future spot exchange
rate
$$F_{t} = E_{t}(S_{t+1}). (1.6)$$
Substituting (1.6) into (1.2) [result given from covered interest parity where instead of expected exchange rate you have forward rate] gives the uncovered interest parity condition
$$ 1 + i_t = (1 + i_t^* )\frac{E_t[S_{t+1}]}{S_t}. (1.7)$$
If (1.7) is violated, a zero-net investment strategy of borrowing in one
currency and simultaneously lending uncovered in the other currency has a positive payoff in expectation. We use the uncovered interest
parity condition as a first-approximation to characterize international
asset market equilibrium, especially in conjunction with the monetary
model (chapters 3, 10, and 11). However, as you will see in chapter 6,
violations of uncovered interest parity are common and they present an
important empirical puzzle for international economists.

The in text square brackets contain my clarifications. So as the above passages show there is an arbitrage argument in UIP. You can find further sources and more detailed explanations in the textbook itself.


Open access datasets for teaching IV regression

I can recommend this paper as an example:

The Colonial Origins of Comparative Development: An Empirical Investigation

Daron Acemoglu, Simon Johnson, and James A. Robinson

http://economics.mit.edu/faculty/acemoglu/data/ajr2001 (paper and data)

This example is famous not only thanks to the creative use of instrumental variables, but also because of the subsequent discussion about the validity of the instruments.

And relevant discussions:

Bazzi, Samuel, and Michael A. Clemens. “Blunt Instruments: Avoiding Common Pitfalls in Identifying the Causes of Economic Growth.” American Economic Journal: Macroeconomics 5, no. 2 (2013): 152–86.
Deaton, Angus. “Instruments, Randomization, and Learning about Development.” Journal of Economic Literature 48, no. 2 (2010): 424–55.

Secondly,

Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments

Angrist, Joshua D., and Alan B. Krueger

http://economics.mit.edu/files/18

  We discuss the mechanics of instrumental variables and the qualities
  that make for a good instrument, devoting particular attention to
  instruments derived from "natural experiments."

Examples follow.


One-step Binomial model's Radon-Nikodym derivative

In economics, the Radon-Nikodym density $\frac{d \mathbb Q}{d \mathbb P}$ of the risk-neutral measure $\mathbb Q$ with respect to the physical measure $\mathbb P$ is the price of Arrow-Debreu securities. It is a price, not a claim.

In the binomial setting, there are two AD securities, $1_u$ and $1_d$. The former entitles the holder to 1 unit of numeraire if, and only if, state $u$ realizes.
The no-arbitrage price of $1_u$ is $\frac{q_u}{p_u}1_u$ (say $R = 1$). Similarly for $1_d$.

In general, the price of the AD portfolio $1_{\Omega'}$---which pays off 1 unit of numeraire if, and only if, state $\omega \in \Omega'$ realizes---is
$$
E^{\mathbb P}[ 1_{\Omega'} \frac{d \mathbb Q}{d \mathbb P}] = \mathbb Q (\Omega').
$$
So
$$
E^{\mathbb P}[\frac{d \mathbb Q}{d \mathbb P}] = 1
$$
is then the price of a bond/risk-free security (discount accordingly if $r \neq 1$).

Extending from $1_{\Omega'}$ to a general $X$ gives the usual risk-neutral pricing formula $E^{\mathbb Q}[X]$, as you have written out.


When is a utility representation differentiable?

A continuous, transitive, complete, convex, and locally non-satiated preference on an open convex subset $V$ of some $\mathbb{R}^l$ has an $r$-times continuously differentiable utility representation with no critical points if and only if the space $I=\{(x,y)\in V\times V\mid x\sim y\}$ is a $C^r$-manifold.

This result is essentially Proposition 2.3.9. in the book "The Theory of General Economic Equilibrium: A Differentiable Approach" by Andreu Mas-Colell. Actually, the result there replaces convexity of $V$ and the preference relation to be represented by the weaker condition that all indifference curves are connected.

I'm not aware of any more elementary approach to the problem.


Effects of increased income over short and long run in AD-AS equilibrium model

Because higher wages do not necessarily mean that national income increased. From macroeconomic perspective national income is output. Higher wages in these simple models do not necessarily translate into higher output.

For example if these higher wages come from company’s profit then on one hand one (worker) household income increases but another (entrepreneur) household income decreases. Assuming all households have the same marginal propensity to consume this would not affect aggregate demand or national income. This is not necessarily realistic assumptions and there are more complex models that relax it, in that case higher wages could boost aggregate demand.

Also the side note is the same question you already asked here.


What would happen if total welfare were evenly distributed?

Firstly, let us fix the mode of taxation as lump-sum taxes. This means that after the redistribution we will still have an efficient allocation assuming there are no externalities.

Next, think about what happens to each individuals' income. Individuals with a high demand elasticity for the good will in general be taxed less than individuals with a low demand elasticity. If the demand elasticity is unrelated to the original income level, this means that this mode of taxation does not necessarily increase equality in income across individuals.

This may not be bad, because the redistribution may still be welfare enhancing if the marginal utility of income of high elasticity individuals is higher than the marginal utility of income of low elasticity individuals. However, we know that marginal utilities are unrelated to demand, since one can always apply any monotone transformation to the utility function and obtain the same demand functions. Thus, both equality of income and welfare effects are unclear.

Finally, we may consider what happens to demand in that particular market. Again, the effects are unclear. This time, it is the presence of income effects which obfuscate the picture. An individual with a positive income transfer may consume more or less of the good, depending on whether it is an inferior good or not.

In this answer, we have thus seen that without making stronger assumptions on preferences, we cannot say much about the effects of such a redistribution even if we assume lump-sum redistribution. If we go into second-best analysis, things may get even more complicated.


Public debt of European countries, by year

Would you be satisfied with a graph of the last 20 years?
Is yes, you can use Google Data Explorer.
It seems to rely on Eurostat as well, so the data is probably available from them somewhere.


Debt-to-GDP ratio in connection to inflation

The premise that a high debt-to-GDP ratio will imply higher inflation is questionable. If we look at the data for the United States, there appears to be a negative correlation: the ratio was lowest in the high inflation period of the 1970s to early 1980s.

link to FRED graph

As hinted by your question, growing the denominator (nominal GDP) lowers the ratio.

Unless one believes in the Fiscal Theory of the Price Level (described in other questions), there is no obvious link between the debt ratio and inflation.

If we drop the focus on the ratio and look at government spending, we arrive at the theory of Functional Finance: the constraint on government spending is inflation. That is, we need to look at the size of the deficit, what the government is purchasing now, etc.; the debt ratio just tells us about historical spending.

Link to Kelton article on Functional Finance.

It looks like your question is really about the validity of Functional Finance, but you have written it up as being about the debt-to-GDP ratio.


Does endogeneity matter when neither independent variable nor error term are correlated with dependent variable?

Assume the simplest linear regression model

$$y = bx + u$$

The OLS etimator for $b$ is

$$\hat b_{OLS} = \frac {\sum x_iy_i}{\sum x_i^2} = b + \frac {\sum x_ie_i}{\sum x_i^2}$$

Whatever the true $b$ is (it is zero in your case, as another answer pointed out) , the fact is that

$$\text{plim} \frac {n^{-1}\sum x_ie_i}{n^{-1}\sum x_i^2} \neq 0 $$

and so the OLS estimator will be inconsitent (and also, biased too).


Compute yield to maturity for a bond with coupons when price is different from face value

Following the suggestion by Brian Romanchuk of the 'quick and dirty' bisection method in the comments, I wrote this python program to estimate the yield to maturity :

def find_i(P, C, FV, n, epsilon):

    if P > FV : i_min, i_max = 0 , C / FV
    elif P < FV : i_min, i_max = C / FV , 10
    else : return C / FV

    while True :

        i_mid = ( i_max + i_min ) / 2 # mid point of the current intervall
        point = C*(1-(1+i_mid)**(-n))/i_mid + FV/(1+i_mid)**n - P # it should converge to 0

        if abs(point) < epsilon  : return i_mid # if point is close enough to 0 then finish

        if point > 0 : i_min = i_mid # [i_mid,i_max] : new intervall
        elif point < 0 : i_max = i_mid # [i_min,i_mid] : new intervall

P, C, FV, n = 1100, 100, 1000, 10
epsilon = 10**-5
i = find_i(P, C, FV, n, epsilon)

The choice of $i_{min}$ and $i_{max}$ is important to estimate $i$ in edge cases.
Dealing with negative interest is harder.

I'm going to update the method to improve the computation , for now I list the relavant facts used :

1) When $P = FV $ then $i = C / FV$

2) $P$ and $i$ are negative correlated when $i>0$


In modern economics, what are the least restrictive conditions shown to theoretically achieve Pareto Efficiency?

Well, when considering the minimal conditions necessary for an allocation to be Pareto optimal we must go to the primitives. First, we need the fact that all agents have rational preferences. What this means is that preferences are complete and transitive. Another thing we need is for preferences to be strictly convex.

Let $\succcurlyeq$ be the preference relation "weakly preferred to":

$\textbf{(A) Transitivity:}$ Given allocations $x,y,z \in X$, if $x\succcurlyeq y$ and $y\succcurlyeq z$ then $x\succcurlyeq z$

$\textbf{(B) Completeness:}$ Given allocations $x,y\in X$, either $x\succcurlyeq y$ or $y\succcurlyeq x$ or both $(x\sim y)$

$\textbf{(C) Strict Convexity}:\forall\, x,y \in X$ and $\forall \lambda\in[0,1]$ we have:
$$\lambda f(x)+(1-\lambda) f(y)>f(\lambda x+(1-\lambda)y)$$

Also, let $X$ be the set of goods. We need $X$ to be finite.

These four axioms are sufficient for the existence of a competitive equilibrium.

Now, we must build a bare-bones theoretical market. What we need is the following:

$\textbf{(1) Complete Market:}$ Everyone has complete information and there are no transaction costs.

$\textbf{(2) Price Taking:}$ There is one price which everyone pays (and by the above statement, everyone knows this price).

$\textbf{(3) Locally Non-Satiated Preferences:}$ Basically, giving an agent more of any good will always increase their utility.

If we have $(A)-(C)$ and we have a finite number of goods, we will have a competitive equilibrium, and if we have $(1)-(2)$, we have a competitive market. Using these two and $(3)$, we can use the First Fundamental Theorem of Welfare Economics to say that our equilibrium is Pareto optimal.


Re-evaluation of Consumer Price Index (C.P.I.) weights

I don't think they are "sure", but the weights are based on the American Consumer Expenditure Survey which has a big sample size, a carefully chosen sampling frame, professionally executed sample gathering, and a large team of experts crafting questions and collating the data.

  How is the CPI market basket determined?

  The CPI market basket is developed from detailed expenditure
  information provided by families and individuals on what they actually
  bought. For the current CPI, this information was collected from the
  Consumer Expenditure Surveys for 2011 and 2012. In each of those
  years, about 7,000 families from around the country provided
  information each quarter on their spending habits in the interview
  survey. To collect information on frequently purchased items, such as
  food and personal care products, another 7,000 families in each of
  these years kept diaries listing everything they bought during a
  2-week period.

  Over the 2 year period, then, expenditure information came from
  approximately 28,000 weekly diaries and 60,000 quarterly interviews
  used to determine the importance, or weight, of the more than 200 item
  categories in the CPI index structure.

Consumer Price Index Frequently Asked Questions (FAQs)

  Consumer expenditure data supplied by the CE Survey are a critical
  component of the CPI, as they are used to estimate weights for the
  CPI’s consumer goods and services classification structure used in the
  calculation of the CPI. In the construction of the CPI, four distinct
  functional uses of CE Survey data are made: (1) to estimate biennial
  expenditure weights, (2) to estimate monthly expenditure weights, (3)
  to calculate the probability that an item’s price will be included in
  the CPI calculations, and (4) to allocate expenditure estimates
  between more-broadly defined expenditure categories from other survey
  sources.

Research highlights of the Consumer Expenditure Survey redesign

Page 6 of BLS Handbook of Methods, April 1997, Chapter 16: Consumer Expenditures and Income has a nice, detailed description of how "The estimation of population quantities of interest, such as the average expenditure per consumer unit on a particular
item category, is achieved through the use of weights." Be sure to check that out too.


“Consumption lowers investment” mankiw explanation

This is because investment is equal to private and public savings (but assuming balanced budget public saving drops out so it would only be private savings I=S). This is because you can only invest what you save.

Moreover saving and consumption are mutually exclusive. From your income you can either consume or save there is no third option economically speaking (actually technically saving is postponed consumption but I don’t want to confuse you too much so I won’t go further into that).

Hence it’s quite intuitive that the more you consume, the less savings available and the higher the interest rate due to low savings and so on.

So to sum up consumption must decrease investment as the more proportion of your income you consume the less you can save. If you consume 70% of your income you can only save 30%, if you consume 90% you can only save 10% and so on. If you consume 100% of your income your saving and thus investment is 0.


I guess you might already know this, but I wanted to add a little detail to the other answers for the sake of any layman who comes here and gets the wrong end of the stick.

What is meant by rationality?

It is important to begin by saying that when economics use the term rational they have in mind a fairly precise definition that does not perfectly coincide with the way the word is sometimes used colloquially:

  We say that a decision maker is rational if

  s/he has preferences that are transitive and complete
  s/he acts in a manner consistent with optimization with respect to those preferences.

On the first bullet: suppose some one has to make a decision. We say that they have transitive and complete preferences if (i) they are able to rank the alternatives among which they choose from "most preferred" to "least preferred", and (ii) that ranking is internally consistent.

On the second bullet: A person acts in an optimal manner given their preferences if they choose the alternative that is 'most preferred' among the set of all feasible alternatives.

So are terrorists rational?

As others have noted, the literal answer to the question is that we can't really (empirically) verify that terrorists are rational. This is because observing one rational decision does not rule-out the possibility that the individual concerned made an irrational decision at some point in the past (or that they will make an irrational decision in the future). The best we can manage is to look for behavior that violates the two conditions above and take observations of such violations to be evidence of irrationality.

But, to address the spirit of the question: just because voluntarilly committing suicide and murder in the name of your beliefs may seem irrational in the colloquial sense does not mean that it is necessarily so. Indeed, if a person has carefully considered the alternatives available to them and decided that the option they find most attractive is to conduct a suicide attack then this behaviour is entirely consistent with rationality.

Rational terrorism in context

It is important to stress that there is no value judgement implicit in an eocnomist describing behaviour as rational. Just because something is rational behaviour does not mean that it is good or desirable or can be condoned. Rather, it just means that we think we have a systematic way of understanding why people might choose to behave in a certain way. Building a systematic understanding of a phenomenon is an important first step in deciding upon the best way to respond to it.


What is the definition of exogenous and endogenous preferences?

Exogenous variables are believed to have some value given by nature. They are not caused by your theory's variables of interest. This is why they are said to be outside the model.

Endogenous variables have values dependent on your theory's variables of interest. They both cause, and are caused by your topic.

Example: In the study of wages, some individuals are born superior, genetically, perhaps. They are superior exogenously. Because of their superior innate quality, they may choose to seek additional education knowing they will get higher returns for their investments. So the amount of education is endogenous to wages, because a person's expected earnings informs their education.


Human capital for under developed countries

If you multiply the index by the population size (or the quantity used to compute the per person index), you get an index of the stock of human capital. For example, if for a given country the index is $h_t$ and population is $N_t$, then the stock is:

$$ H_t = h_t \times N_t $$

Now, by the definition a stock follows an accumulation rule, just as physical capital:

$$ H_{t+1} = H_t(1+d_t) + I_t $$

where $d_t$ is the "depreciation" of HK (e.g. because of death or skill obsolescence) and $I_t$ is gross investment in HK. Since you probably do not have data on depreciation (which I imagine is very difficult to compute), you can only calculate net investment, which is equivalent to net savings by definition:

$$ S^h_t = H_{t+1} - H_t $$

Finally, to compute the savings rate, you need to divide this by a measure of GDP. However, these need to be measured in the same units. Therefore, you need to transform $H_t$ or $S^h_t$ into monetary units. For this you need something like the average price of human capital for every period, which I have never seen, and seems not available in Penn-world tables.


How is the Corona $2 trilion stimulus package going to be financed?

I believe that the government is going to encourage the financing of the $2 trillion dollar stimulus to the FED  and this will take place through the FED purchasing US treasuries.

It is also in the best interest of the FED to do so in order to foster the stability, integrity, and efficiency of the nation's economy (Federal Reserve Bank mission).

This is actually taking place now as The Federal Reserve Bank of New York will purchase $37 billion in Treasury bonds to boost liquidity in the crucial government debt market, announced on the 13th.  (https://thehill.com/policy/finance/487427-ny-fed-to-accelerate-bond-purchases-to-keep-treasury-market-flowing)

The New York Fed will also offer $1 trillion in short-term loans to banks and brokers in exchange for Treasury bonds. The Fed will eventually be repaid with interest on the loans it offered to banks and brokers in exchange for returning the bonds it purchased.
The government and the FED are going to be working together to help jump start the economy, according to the article.

During the Great recession the US government used the American Recovery and Reinvestment Act to help jump start the economy while the Central Bank used quantitative easing (QE), zero interest rates, buying US treasuries, and other tools to as well.

On a side note:

A stimulus package can either be in the form of a monetary stimulus or a fiscal stimulus. One of monetary stimulus tools involves cutting interest rates to stimulate the economy. When interest rates are cut, there is more incentive for people to borrow as the cost of borrowing is reduced.

Which is what the Fed is doing now.

An increase in borrowing means there’ll be more money in circulation, less incentive to save, and more incentive to spend. Lowering interest rates could also weaken the exchange rate of a country, thereby leading to a boost in exports. When exports are increased, more money enters the economy, encouraging spending and stirring up the economy.

Fiscal stimulus refers to increasing government consumption or transfers or lowering taxes. Effectively this means increasing the rate of growth of public debt, assuming that the stimulus will cause sufficient economic growth to fill that gap partially or completely.

When a government opts for a fiscal stimulus, it cuts taxes or increases its spending in a bid to revive the economy. When taxes down, people have more disposable income . An increase in disposable income results in more spending to boost economic growth. When the government increases its spending, it injects more money into the economy, which hopefully decreases the unemployment rate, increases consumer spending, with the hopes to counter the impact of a recession.

A potential problem of fiscal stimulus is that to increase public spending the government has to increase its borrowing. There is a chance/rick that people may actually choose to save the excess disposable income instead of spending it, which could render the stimulus package ineffective.

American Recovery and Reinvestment Act (ARRA),February 2009, is an example of  economic stimulus package. It instilled the confidence needed to boost economic growth. It also aimed to restore trust in the financial services industry.

ARRA cut taxes by 288 billion dollars. It spent 224 billion in extended unemployment benefits, education, and health care.   While also creating jobs using $275 billion in federal contracts, grants, and loans.


Convert USD2005 PPP per-capita GDP to USD2011 PPP per-capita GDP for a number of countries

Any real figure is based on a basket of goods, obviously, and here, we have a representative basket that is common to all countries across the world. When purchasing power parity prices are recalculated (as with any price index that is recalculated), the weighting of goods in the basket changes. Goods that have improved in technology will be more valuable than those same goods from a decade ago—this would push up the price index unless the weights on those goods are reduced, to reflect that the nature of those goods have changed, and that people buy them more parsimoniously.

So, PPP figures from 2011 will be based on a basket that contain multiple items inside it, which have different weights to the 2005 figures. Different countries produce goods at different levels of technology—some countries power ahead in technology, whilst others lag behind—so this causes the representation of the PPP basket to diverge from what is actually consumed in those countries (I imagine the PPP basket attempts to find the average representation). Thus, in the recalculated figures (from 2011) you will find different growth rates for the countries compared to the 2005 figures.

In short, you will need to know the weightings for all the goods in the baskets from 2005 and 2011 (and the levels of output in those categories for each country) in order to process the recalculation! It is not simply a matter of changing the base year, because the basket has changed.


Teaching changes in "demand" v. "quantity demanded"

I think the terminology below is pretty unambiguous and also common:

  a shift in the demand curve

to refer to movements from $D$ to $\tilde D$ (using shift is quite more specific than using change, as it more clearly refer to a change in the whole demand schedule).

  a movement along the demand curve

to refer to changes in $D(P)$ induced by movements in the supply curve (this I normally do it by showing the "dynamics" in the graph, from the original equilibrium to the new equilibrium along the demand curve).

  a change in the quantity demanded

to compare different $D(P)$ in the horizontal axis. A mention to the concept of equilibrium can be useful here too. For example, a change in the final/equilibrium quantity. Here the demanded adjective is in a sense redundant, as in the equilibrium the quantity demanded and supplied is the same, but if you are being very forma as in $D(P)=S(P)$ then its fine. This terminology is different from the first one (shift in the demand curve) in the sense that the latter is a ceteris paribus exercise, whereas the former is not (unless supply is perfectly in/elastic).

Regarding your question about whether to use the textbook's proposed terminology, I would not advice so. First of all, it is not clear what the authors propose to use to refer to changes in $D(P)$ induced by "changes in the demand [curve]". Why to have a separate notation for such changes and those induced by changes in the supply? That is very arbitrary.

Additionally, as their restrictive terminology is probably not the one you are used to (judging by question and comments), forcing you to adopt such incomplete terminology (which is likely not to come to you naturally) might confuse your students because you might not always consistently use it. Provided you want to keep using that textbook, I would tell them something like

  beware of this very restrictive termonology issue, which is uncommon in other books. More common instead is this terminology, and what I normally use is this. In any case, all this is not very relevant for understanding. In your answers, always explain what you mean, ideally with the aid of annotated graphs. Correct answers (mostly) do not depend on terminology.


When can economists claim to "understand" an event or phenomenon?

I'm not sure there is a correct answer to this question (or if there is, we don't understand it yet!), but here's a first shot at an answer:

Even if you look at the natural sciences, there is a process whereby ideas are refined over time. In the 17th century people 'understood' mechanics thanks primarily to Newton. But that didn't mean Einstein couldn't improve on the theory with relativity. One approach would then be to say that we never really understood mechanics in the first place, we were just confused until Einstein came along. But that binary view of understanding sets a very high threshold that economics may never meet; indeed, by that standard it might very well be that humanity never truly understands anything. I think most good scientists would be very cagey about ever describing a problem as solved—theories cannot be proven correct, only 'not disproven' and subsequently improved.

An alternative view would therefore be that understanding is measured on a continuous scale so that partial understanding is recognised and new discoveries refine that existing understanding. One might then define something as (at least partially) understood when

  Economists share a broad professional consensus on the matter.
  The consensus view is consistent with available empirical data.
  The consensus view represents a meaningful contribution over and above the knowledge that might be expected of a lay-person.

An issue for which economics performs better on all three dimensions might be said to be better understood.

Something like the effect of a rent ceiling, the role of adverse selection in a market, the potential benefits of free trade, or the static welfare effects of monopoly appear to be fairly well understood (at least by social science standards) because economists are in broad agreement, these things are not well-understood in society at large, and economists' views on these issues are generally bourne out empirically.
Things like the effects of a fiscal stimulus, the dis-employment effects of a minimum wage, or the dynamic relationship between competition and innovation seem to be less well-understood because economists frequently disagree on these issues—both with each other and with the data.

The rationale for the three components of the definition is as follows:

Consensus is not a direct measure of understanding. But I include it as a proxy to get at the fact that there are some areas of economics in which seemingly convincing theories that match some aspects of the data quite well are in direct conflict with alternative theories that also do well in some respect. It seems like having two conflicting views on an issue that you are not able to reconcile ought to be penalised in any measure of understanding.
Obviously, economics is an empirical undertaking—it aims to describe and explain empirical phenomena. A description of or explanation for a phenomenon that is not consistent with data is clearly not part of a good understanding of that phenomenon.
This criterion is aimed at getting at the idea that economists have been especially successful in improving upon a common sense understanding of the world in some areas. In these areas, economists might be viewed as having a particularly good understanding relative to society at large. Consider two examples: (i) a monopolist will charge more when demand is less elastic; (ii) two countries can benefit from trade even when one of them has an absolute advantage in every good. Both are issues that economists agree on, but (i) is common sense whereas (ii) is utterly counter-intuitive. The definition is designed to capture the idea that understanding (ii) is more meaningful because society depends upon economists to figure such things out. Society could probably figure out (i) without economists' help.


Cointegration testing usually goes in this way:

Have very clear which is the "long-run" relationship among variables your theory predicts. The point about cointegration is that there is at least one common trend among the variables. In your case, you would expect that the price of the three stocks move in tandem, based on some underlying market phenomena like economic growth, volatility, etc. In this case, the long run relationship between variables  in likely to be in levels (price of stocks).
Test for unit root on each variable in levels. There are plenty of tests here (ADF, KPSS, etc). You want to find that these variables are I(1) or maybe I(2).
Perform the Johansen cointegration test. If you reject the null hypothesis of cointegration ($r=0$), then there is not a common trend among the variables, and they are not cointegrated. Do not run regressions with them in levels, as any result will be spurious. If you do happen to find cointegration ($0<r\leq n$), then estimate the VECM, from where you can get the cointegration vector(s) that define the common trend(s).


There's no magic. What you have to realize is that the result is conditional on the validity of the assumptions:

A) Under the assumption that there is measurement error, then yes, the average of two measurements will be on average closer to the truth than a single opinion. This is very believable. We all do this kind of thing all the time. For example, when we actually get 'a second opinion' from a doctor, we think that adding another opinion gets us closer to the truth.

B) Under the assumption that there is a smaller incentive to lie to the interviewer about your twin's education than about yours, then one way to get closer to the truth is to use the twin's information about his sibling's education rather than the self-reported one. The authors show that the variances and covariances among the different variables are consistent with this hypothesis. The intuition is that you are less likely to try to hide your sibling's lack of education than you are to hide your own lack of education.

Of course that doesn't make the preferred interpretation by the authors necessarily true. It could still be the case that there is a particular bias in the reporting of your twin's education which they did not consider.  Say for example that you exaggerate the difference between your education and his or hers exactly when there is a large wage difference. That could break their result I think.

Lastly, the precise idea of using it as an instrument is not that it is a second opinion.  Instead, the assumption is that the 'error'  in this second measurement is uncorrelated with income in contrast to the error of the self reported income which is assumed to be correlated with income.  This means that the income effects of education estimated using the sibling report is unbiased, while the one using self-reported income is biased. The authors claim the self-reported one leads to a coefficient that is downward biased, resulting in an estimate of a smaller return to education than the real one.


The Consumer price index of a country is defined as:

$$CPI_{t}^{t_0}={{P_{1t} Q_{1t_0}+P_{2t}Q_{2t_0}+P_{3t}Q_{3t_0}}\over{P_{1t_0}}Q_{1t_0}+P_{2t_0}Q_{2t_0}+P_{3t_0}Q_{3t_0}} \times 100$$

where $P$ is price, $Q$ is quantity, $t$ is reference to current year and  $t_0$ is reference to the base year.

So using this formula we get:

$${{10\times8 +4 \times 15+18 \times 6}\over{9 \times 8+3 \times 15 +12 \times 6}} \times 100 \approx 131.2 $$

Now finding the inflation rate we can see (CPI of the base year is always going to be $100$)

$${{131.2-100}\over{100}} \times 100 \% \approx 31\%$$

Hope this is helpful!


Finitely repeated prisoner's dilemma without sub-game perfection

There is also no NE which sustains coopration for more or less the same reason as in the SPNE case.
Consider, a PD played twice. A strategy contains five actions, one for each decision node: one in the beginning (empty history) and one for each of the four period-2 histories (CC,CD,DC,DD). I claim that any strategy other than (D;D;D;D;D) is dominated.
Consider any strategy in which you play C in period 2, say (C,D,C,C,D). A deviation to (C,D,D,D,D) is profitable because defection cannot be punished after period 2 as the game ends. Behavior in period 1 cannot be conditioned on the future. Given that any equilibrium candidate has the structure (_,D,D,D,D) cooperation in period 1 is also dominated.
If you are not convinced, you can write down the game with all its strategies in a big normal form matrix.
You can iterate the argument for any (commonly known) finite number of periods.


According to Bissell (2007),  MONIAC was based on a simple, first-order model of the dynamics of an economy (but authors also mention it could be recalibrated to higher-order models):

Let $u$ be the production flow, $v$ the consumption flow, $w$ the stocks and $p$ the price, all measured, for simplicity, from a base at which the system is in equilibrium. We have then, by definition, the identity
$$U-V={dw\over dt}$$
and three hypotheses,
$$\eqalignno{&\ \ u=lp\cr &v=mp}$$
and
$$w=np$$
where $l$ $m$, and $n$ are parameters.
If at time $t=0$, from equilibrium, there is a spontaneous change, $\Delta v$ in
consumption, then
$${dW\over dt}=u-(v+\Delta v)=(l-m)p-\Delta v$$
Also,
$${dp\over dq}={1\over n}$$
Therefore [after some manipulation] […]
$$n{dp\over dt}-(1-m)p+\Delta v=0$$
The solution of this equation,
$$p={\Delta v\over l-m}(1-{e}^{{l-m \over n}t})$$
gives the path of the induced price change. […]
This simple model could be further developed, in particular by making a distinction between working and liquid stocks, introducing lags into the production and consumption functions, and linking the demand curve for liquid stocks to the rate of change of price […].

The above is in its essence based on the idea of circular flow combined with some canonical Keynesian macroeconomics.
For example, just compare the diagram representation of MONIAC (picture on left from Media Archealogy wiki) to the circular flow diagram from contemporary Krugman's macro textbook (picture on the right). You will notice that there is some resemblance although they are not exactly the same.
This being said both MONIAC and circular flow diagram are didactic tools (MONIAC was used by central banks for some time, but the machine was originally intended just for classroom use). Hence the actual models do not directly build on this ideas even though they are connected.
Modern models that try to model economy such as dynamic stochastic general equilibrium (DSGE) models, relate to the MONIAC model only very superficially and don't build upon MONIAC model directly.


How do you reconcile the Law of Increasing Opportunity Costs and Economies of Scale?

You are correct that an uniformly increasing cost of production cannot exist alongside economies of scale. A constant returns to scale production function is homogeneous of degree one:
$$ f(\lambda x, \lambda y) =  \lambda  \cdot f(x, y)$$
That is, when you double all the inputs you also double the output. An increasing returns to scale production function is one for $\lambda > 1$ (so that you actually increase the resources) in which $f(\lambda x, \lambda y) >  \lambda  \cdot f(x, y)$  (e.g. double the inputs and more than double the outputs for at least some combinations of inputs). However, it is problematic for a production function to have increasing returns to scale for all levels of inputs. Increasing returns to scale means fewer and fewer inputs are needed for each marginal unit of output. This generally means that marginal cost of production is also falling. This has all sorts of bizarre consequences, including the potential for indeterminacy (multiple equilibrium levels of output), downward sloping supply curves, and equilibrium non-existence. We also don't generally think it is true in reality. It is widely observed that beyond a certain scale most industries do not exhibit further returns to scale. For some industries, like dry cleaning, this scale is quickly reached, for others, like automobile manufacturing or social networks it may be very large indeed. But we generally believe that eventually the marginal cost of production will rise with scale. Even if we bought exactly the same car in same color, we'd still eventually want so many that the cost of labor, rubber, and metal would rise from that demand, driving up the marginal cost. We also face dis-economies of scale like managerial incentive problems and other  difficulties of managing and controlling an increasingly large institution.

It is worth noting that you can have increasing returns to scale but decreasing marginal product of inputs. For example, the production function:
$$y(\ell, k) = \ell ^ {.75} \cdot k ^ {0.6}$$
has increasing returns to scale:
$$y(\lambda \ell, \lambda k) = \lambda^{1.35} \ell ^ {.75} \cdot k ^ {0.6} > \lambda \cdot  \ell ^ {.75} \cdot k ^ {0.6} $$
But holding fixed the other input, we can see that the marginal product of both $\ell$ and $k$ are decreasing in that input. E.g.:
$$ \frac{\partial y}{\partial \ell} = 0.75 \ell ^ {-0.25} \cdot k ^ {0.6}$$
$$ = 0.75 \frac{k ^ {0.6}} {\ell ^ {0.25}} $$
Similarly:
$$ \frac{\partial y}{\partial k} = 0.6 \ell ^ {0.75} \cdot k ^ {-0.4}$$
$$ = 0.6 \frac{\ell ^ {0.75}} {k ^ {0.4}} $$
We could also continue this and show that the second derivatives here were both negative, which means decreasing marginal product from each input
$$ \frac{\partial^2 y}{\partial \ell^2}  = -0.75\cdot 0.25 \frac{k ^ {0.6}}{ \ell ^ {1.25}}$$
$$ \frac{\partial^2 y}{\partial k^2}  = -0.6\cdot 0.4 \frac{\ell ^ {0.75}}{ \ell ^ {1.4}}$$

Note the cross-partial  is positive, which is how this economy still has increasing returns to scale:
$$ \frac{\partial y}{\partial \ell} \frac{\partial y}{\partial k} = 0.75 \cdot 0.6 \frac{1} { \ell ^ {0.25} \cdot k ^ {0.4}} > 0 $$
which is positive.

Both arguments show that the marginal product of the input decreases as the quantity of that input rises. This holds even though the production function shows increasing returns (and is actually homogeneous of degree 1.35).


Why does the retiree to contributor ratio matter when discussing the solvency of Social Security?

Pay as you go vs Fully funded pension plan distinction is relevant here. Also known as Defined benefit vs Defined contribution.

If everyone saved for their own pension via payroll taxes or otherwise (Pay as you go/ Defined contribution), then the ratio of retiring to contributing people indeed would not matter.

However, most governments pay current retirees from taxes on current workers. If there are fewer current workers relative to retirees, then either each has to pay more tax (discouraging working, eventually leading to declining tax revenue) or each retiree gets less pension. Governments typically do not save the current taxes to pay pensions when the current workers retire. Instead, governments are typically borrowing against future tax revenue.

See also the Overlapping Generations Model (OLG).


What is the Walras law vs first welfare theorem

Walras' law describes market equilibria conditions, which states roughly speaking that if there exists within an exchange economy market equilibria for $n-1$ good markets, then the last good market $n$ is also in equilibrium. Thus, we have a market system with $n$ equations and $n-1$ independent variables, where the last market is linear dependent from the other $n-1$ markets.

Now, the first Welfare Theorem of Arrow states that for every exchange economy with convex preferences, every Walrasian equilibrium is Pareto efficient.

The implication of the first Welfare Theorem is that we need no central planner to achieve a Pareto efficient outcome in an exchange economy. The Pareto efficient outcome is based on decentralized decisions within the exchange economy. If we now introduce transaction costs and incomplete information into the system, then it should be obvious that every decentralized economy is doing better than every centralized one due to the information problem.

Update:

I forget to mention some interesting implications of Walras' law. The law reveals to us that we have only one degree of freedom. This implies that a central instance can only fix one price level, for instance, that of the numeraire good, the others prices of the markets are determined by the market forces. Hence, a central instance has no control on them.

To make this argument more precise, let us apply this law to a central bank policy. We have only one degree of freedom, the central bank can now fix either the interest rate of the money market or a price index, but not both. If it fixes the interest rate, it has to accept the price levels and therefore the inflation rate, and vice versa for the price index. This also implies that a central bank cannot simultaneously targeting the price index and the unemployment rate of an economy.


If an employee works overtime (and gets paid for it), how does this affect a nation's GDP?

You are correct: an increase in the wage does not by itself imply an increase in GDP. So, if you pay a worker \$20 for an hour of overtime which you could have bought from another worker for \$10, but they both produce the same thing, GDP does not change by switching from one arrangement to the other.

You can also observe that from the 'income perspective' in the circular flow of the economy: if nothing else changes, a higher wage for a  worker results in lower profits for the firm, but the net effect on income is 0.

However, a change in which hours are labeled as overtime and which are not could have important effects in the economy. You'd expect that on average firms will prefer to have relatively more workers working relatively few hours and so avoid paying overtime. This should increase employment, but not average worker income, it should reduce inequality among workers, etc.

Potentially there can be large effects are on the health insurance costs of firms of changing overtime laws.


Does ownership determine efficiency in an economic system?

It's not unusual - in many real-world subjects - for there to be questions for which the right answer at one level of learning, under one teaching framework, is the wrong answer in a different context. This is one such question. It will depend as much on what the teaching framework's objectives are, and what its position is within normative economics.

As such, for your homework question, the right answer is whatever you've been taught is the right answer. You know what you've been taught. We do not.

So, given that, let's deal with the reality, rather than the homework question.

Ownership can matter ... but it's complicated.

Ultimately, competition matters most when it drives under-performing competitors out of the market. If a competitor can afford losses and still stay in business, then competition won't be fully effective: although the customers will benefit from lower prices, the push to greater efficiency won't flow up the supply chain. In theory, state-owned enterprises may be more able to sustain losses than private-owned ones, so ownership should matter. However, there are cases where private-owned enterprises have sustained losses for many years; just as there have been state-owned enterprises that do not tolerate losses and have cultivated economic efficiency.

Furthermore, if there are externalities, then one way to get a more efficient market is to have a state-owned enterprise rebalancing the market to price in those externalities. Indeed, one could take that further, and say that when externalities are the dominant part of the costs or benefits, and the lowest-cost supply is easy to establish, then a state-owned monopoly can provide the highest level of efficiency: in this case, there is neither competition nor private ownership in the most efficient market.


I think your one mistake was writing $\Pr(v_{2} < \beta(v_{1})$ incorrectly. It should be:

$$\begin{split}
\pi_1(v_1,v_2)&=\Pr(v_2<\beta(v_1))(v_1-\beta(v_1))\\
&=F_{2}(v_2<\beta(v_1))(v_1-\beta(v_1))\\
&= \beta(v_1)(v_1-\beta(v_1))\\
\end{split}$$

$$\begin{split}
\left.\frac{\partial \pi_1(w,v_{2})}{\partial w}\right|_{w=v_1} =v_1\beta'(v_1) - 2\beta(v_1)\beta'(v_1)&=0\\
\beta(v_1) &= \frac{v_{1}}{2}\end{split}$$

Loosely we can look at, say, type $1$ of player $1$. If he bids according to this strategy his expected payoff is $1/4$. Suppose he picks any other bid amount $a$. Then his expected payoff is $(1-a)F(a) = (1-a)a$. This is clearly maximized at $a = 1/2$.


Arrow of time and irreversibility in Economics

The question and the answer by user14471 seem to relate to the issue of ergodicity in economic systems and models. Economic systems (in reality) cannot conceivably be ergodic, while some economic models are ergodic (those that do not attempt to reflect any of the non-ergodic properties).

Necessary concepts to answer this question: Ergodicity, microstates, macrostates

Ergodicity is the property of a system to spend approximately equal amounts of time in each of its microstate (if you observe it over sufficiently long times). A microstate is simply the state of a system if you consider every property of every element of the system. Not all microstates are distinguishable. Undistinguishable microstates constitute a macrostate. That is, all macrostates are distinguishable but some have more microstates than others. An ergodic system will be more likely to assume those macrostates with more associated microstates (that is, the macrostates with higher entropy).

Examples

Consider a non-economic example: gas in a container. For the definition of a microstate, the position of every molecule is important, for a macrostate only the distribution of the gas matters. While there is a macrostate in which all the molecules are crammed together on one side of the container, this one is extremely unlikely; a macrostate with an even distribution of molecules across the container is much more likely. Since the gas can, however theoretically assume each of the microstates in the future, independent of which other microstates have been assumed in the past, the system is ergodic.

Now consider an economic example: the distribution of dollars across the population. A microstate considers which dollar is owned by which person, a macrostate only considers the distribution. While microstates are not distinguishable or even measurable (how do you distinguish the different dollars in your bank account?), the number of microstates per macrostate is important. Theoretically, all dollars could be owned by one person (while everyone else has nothing). This is an extremely unlikely macrostate, while a more even distribution across the population is much more likely. Actually the likelihood (and entropy) will be maximized by a Gaussian distribution. (However, if you measure the distribution of wealth, you will find that it is not Gaussian but heavy-tailed.)

How ergodicity, irreversibility, and the arrow of time, are related

Ergodicity can be phrased differently as: every microstate is reachable from every other microstate. It is reversible, it does not have an implicit arrow of time. It is easy to see that this is not the case if the system has an attractor (a stable equilibrium) that captures trajectories (development paths) which will then not be able to leave the attractor again.

Ergodic and non-ergodic models in economics

Armed with these concepts, we can now return to the question of reversibility in economics.

Models of simple exchange of goods etc. are ergodic. Depending on how and according to what prices and preferences the goods are exchanged, every microstate of the system is reachable and possible. What is more, every transaction can be reversed.

In the case of models of growth, developent, or technological change, ergodicity will typically not hold any more (unless you can have degrowth such that the system reverses its growth trajectory exactly). These systems typically have attractors (Solow-Swan models for instance have an attractor once you remove the neutral technical change term) they have production functions that would generally not be considered reversible, and they may not allow for recessions and reversion of technological change. More complex models of technological change and development from evolutionary economics or so will definitely

That said, economic models have a certain tendency to assume ergodicity for whatever is not part of the model. Macro-models (including Solow-Swan models) assume that the structure of the microlevel (that is not modeled) will not interfere with how the model works, that agents are exchangeable (representative agents) and that transactions are neutral. For RBC and DSGE models this is made more explicit by assuming an unbiased i.i.d. stream of shocks acting on a largely homogeneous population of agents. Some varieties (HANK etc.) try to address this, but the heterogeneity those models allow for is extremely limited. Agent-based models address this from a different angle and are able to achieve the non-ergodicity property observed in real economic systems (see last paragraph below), but they have their own problems, as was discussed in a different question.

Ergodicity in real economic systems

In real economic systems, it is obvious that development paths are not reversible. You may suffer a recession, you may even experience an extensive collapse and lose advanced technologies, but you will not be able to dismantle and sell off the associated capital goods (and the human capital) in a way mirroring how they were acquired. Further, humans react vastly different to gaining and losing wealth (loss aversion).

And beyond this, the very distribution of wealth - a heavy tailed, Pareto distribution, not a Gaussian - will show you that the system does not achieve (or even come close to) the entropy maximizing macrostate with the most associated microstates. This is a structural property typically found in complex systems with self-organizing properties (the issues Prigogine wrote about as mentioned in the OP). You will find similar distribution also among firm sizes, regional agglomerations (city sizes), and a ton of other things that are more or less related to economic systems.


Can indifference curves slope up and to the right when both goods are inferior goods?

The negativity of slope of the indifference curve depends on the monotonicity of utility function and non-satiation, not on whether the goods are inferior or normal goods. Inferior goods are goods which are consumed less when income increases but that is not the same as saying that the utility is not monotonically increasing in both goods or that non-satiation is violated.
However, the above being said there are cases where indifference curve can be upward sloping. Those are cases when some good creates disutility (so the utility function is no longer monotonically increasing in all arguments). For example, in case where we have utility of the form:
$$U(x,y) = \sqrt{x}  - y $$
Where $x$ can be considered some standard consumption good and $y$ is treated as a 'bad good' (e.g. pollution - but note this is not the same as being an inferior good). In this case we would get an indifference curve that would have positive slope and it would be 'slope up and to the right'. This is because in such case person would be only willing to increase the consumption of dirty good if at the same time the person is compensated by higher consumption of the good $x$.
Also more generally, indifference curves can have various shapes. For example, if we assume that satiation is possible indifference curves become circles. However, if there is tradeoff between consuming $x$ and $y$ (when they both monotonically increase utility) and when people are non-satiated (more is always better) the indifference curves will be downward sloping, since indifference curve represents a collection of all points where the utility from consuming different bundles of $x,y$ is constant. If person likes to consume both goods then a persons utility when you reduce $x$ will remain same only when you give person more $y$. Only when utility stays constant when you increase consumption of both goods will indifference curve have positive slope, and that will generally happen when consuming more of one good creates disutility.


Solving a HJB with a probability to transit to a new state

I would leave this as a comment but I cant. You are on the right track.

Once you know $V_2(k)$ then you can plug that into to the first hjb and solve.

To solve for $V_2$ you need to find the optimal $i$ as a function of $k$. Then plug $i(k)$ into the 2nd HJB. That will give you a second order ode. Solving that will give you $V_2(k)$ and you go to 1.


Why is the derivative used to represent marginal cost instead of the difference?

The derivative is used in some contexts, but not all, when the cost function is differentiable. In those contexts, it tends to be assumed that supply is continuous, not discrete. This is a matter of convention and of analytic convenience. It has the advantage of being consistent, whether you're approaching the supply point from above or from below.

But in other contexts, given your cost function, assuming that the thing being supplied is discrete and not continuous (that is, it is possible to supply 2 units or 3 units, but not 2.9 or 3.5 or any other fractional unit) then the marginal cost of the third item is indeed 5, not 4.


To understand the "Obedience" inequality, notice that the player is "integrating out" everything that she is uncertain about, this is why the sum runs over the actions of other players, $a_{-i}$, the types of other players, $t_{-i}$, and the state of the world, $\theta$.

The first term simply gives the probability that the mediator observed $\theta$.

For the second term, note that it is proportional to the conditional probability of the other players being of type $t_{-i}$ when you are of type $t_i$ since $\pi(t_{-i}|t_i,\theta)=\frac{\pi(t_i,t_{-i}|\theta)}{\sum_{t_{-i}}\pi(t_i,t_{-i}|\theta)}$.

Similarly for the third term, you probably expected there to see the conditional probability $\sigma(a_{-i}|a_i,t_i,t_{-i},\theta)$, but this term is proportional to $\sigma(a_i, a_{-i}|t_i,t_{-i},\theta)$.

Technically speaking the agent should multiply his payoff by the probability that others play $a_{-i}$, are of type $t_{-i}$ and the state is $\theta$ conditional on being told to play $a_i$ when being of type $t_i$, so the LHS should read:

$$\sum_{a_{-i}\in A_{-i}, t_{-i}\in T_{-i}, \theta\in\Theta}u_i((a_i,a_{-i}),\theta) \frac{\sigma((a_i,a_{-i})|(t_i,t_{-i}),\theta)\pi((t_i,t_{-i})|\theta)\psi(\theta)}{\sum_{a'_{-i}\in A_{-i}, t'_{-i}\in T_{-i}, \theta'\in\Theta}\sigma((a_i,a'_{-i})|(t_i,t'_{-i}),\theta')\pi((t_i,t'_{-i})|\theta')\psi(\theta')}$$

And the RHS will be identical except for replacing $a_i$ with $\tilde a_i$ in the utility function. Clearly, the denominator is constant over the sum and so can be simplified.

Note: In previous versions on the paper the authors were more explicit about this derivation.


Creating a tailored sample from panel data

Your approach seems no good. Let us make it really simple: You have only the dependent variable and you want to compare the averages across countries. You keep the firms whose $y$ values are inside the range. By doing this, you are (sort of) equating $y$ values, which you aimed to compare. This is strange. In fact, this leads to endogenous sample-selection.

It seems, however, acceptable to keep only comparable firms (rather than firm-year) based on the exogenous variables.


A little empirical evidence for your first point can be found in

Erik Brynjolfsson & Michael D. Smith (2000): "Frictionless Commerce?
A Comparison of Internet and Conventional Retailers", Management
Science, 46(4), 563–585.

They look at the markets for books and for CDs both online and offline and estimate the price distributions. Whilst they look at price changes, they do not present data on the change in price distributions over time.


According to North, Wallis, and Weingast (NWW), which states/societies have successfully made the transition to open access?

In NWW (2006) they write "In the twentieth century Taiwan, South Korea, Ireland, and Spain appear to have made very rapid transitions [to open access]."
Most of the 37 countries in the OECD would seem to have made the transition, with exceptions being Columbia, Hungary, Mexico, and Turkey. Several Balkan and the Baltic countries may not make the grade either. Perhaps using a combination of Pritchett's (2021) 'stock' version of Polity IV and the WB Ease of Doing Business data would provide a useful measure of open access to political and economic organizations. (Although the latter seems to be increasingly discredited).


It's a matter of choice how one writes the Lagrangian in the context of Lagrange/KKT. Depending on how it's written, the gradients of the objective and constraint functions are either parallel or anti-parallel at a (suitable) optimum, and the Lagrange multiplier is neither negative or positive. At the end of the day, it is the same (subset of) optima that are covered by the Lagrangian FOC.
However, there is a standard choice---sometimes called the standard form in optimization literature---that ensures the Lagrange multipliers $\lambda$'s are non-negative. (See, for example, Convex Optimization by Boyd and Vandenberghe.)
In the standard form, the Lagranian is always written in such a way that it improves the objective function.
This convention is not always pointed out/followed in economics texts.
When it is followed in economic contexts, then the Lagrange multipliers admit the usual interpretation as marginal values of the corresponding constraints---a kind of "indirect marginal utility". E.g. the standard-form Lagrange multiplier is the derivative of the indirect utility function with respect to wealth---which must be non-negative.
For example, consider the maximization problem
$$
\max_{g(x) \geq 0} u(x)
$$
where $u : \mathbb{R}^n \rightarrow \mathbb{R}$ and $g : \mathbb{R}^n \rightarrow \mathbb{R}^p$. E.g. in a consumer's problem, $u$ is utility function, $p = 1$, and $g(x) = p^T(w-x)$.
Then the standard form Lagrangian is
$$
L(x, \lambda) = u(x) + \lambda^T g(x).
$$
It improves the objective function $u$ when $\lambda \geq 0$. This means that, for maximization problems, $L(x, \lambda) \geq u(x)$ for all $x$ and all $\lambda \geq 0$.
Since you're trying to maximize $u$, improving on $u$ means being larger than $u$.

In the standard form, the Lagrange multiplers must be non-negative., i.e. at an (suitable) optimum $x^*$, we must have $D u(x^*) + \lambda^T D g(x^*) = 0$ for some $\lambda$ with non-negative entries. This is easy to see, especially in the case of single constraint $p = 1$. If constraint $g$ is slack, then $\lambda  = 0$. If $g$ binds, then the gradient $D g(x^*)$ must point toward the interior $\{ g > 0 \}$. On the other hand, $D u(x^*)$ cannot point toward the interior---otherwise there would be an optimum in the interior. So $D u(x^*)$ and $D g(x^*)$ are anti-parallel and $\lambda > 0$.

If the maximization problem is formulated as
$$
\max_{g(x) \leq 0} u(x)
$$
then the standard form Lagrangian is $L(x, \lambda) = u(x) - \lambda^T g(x)$. Again, $L$ is written in such a way that, when $\lambda$ is non-negative, $L$ improves $u$.
Similar for a minimization problem.

"Suitability" of an optimum means that $D g(x^*)$ must be full-rank. This type of condition is called constraint qualification. The general KKT theorem says that the Lagrangian FOC is a necessary condition for local optima where constraint qualification holds. When the objective function is concave or quasi-concave (convex or quasi-conconvex, for minimization), then constraint qualification is not needed and Lagrangian FOC is sufficient for global optima.