Uncovered set versus top cycle set (voting theory)

This example comes from Miller (1980) (the paper that introduced the uncovered set definition).

Suppose that the majority's preference is defined by:

\begin{aligned}
x \succ y \\
x \succ z \\
y \succ v \\
y \succ z \\
v \succ x \\
z \succ v
\end{aligned}

In this example, we have the top cycle set $L_1 = \{x, y, z, v \}$ (you can check this).

But notice that $y$ is preferred to $z$ and all policies that are preferred to $y$ (namely $x$), are preferred to $z$. So, $y$ covers $z$ and the uncovered set $U$ is such that $U = \{x, y, v\}$ and $U \subset L_1$.

Among other stuff that Miller proves in his paper, we have the following three:

if there is a Condorcet winner, $L_1 = U$;
if $L_1$ has three elements, then $L_1 = U$;
$U \subseteq L_1$.


Is this a joke by Galbraith or did he have something more serious on mimd?

Not having read the article, I do not know the context in which this quote was made. But I believe it is in line with a view he also expressed elsewhere — that people mistake wealth for intelligence.
In e.g. A Short History of Financial Euphoria (1993 edition, pp. 13–16), he writes:

The basic situation is wonderfully clear. In all free-enterprise (once called capitalist) attitudes there is a strong tendency to believe that the more money, either as income or assets, of which an individual is possessed or with which he is associated, the deeper and more compelling his economic and social perception, the more astute and penetrating his mental processes. Money is the measure of capitalist achievement. The more money, the greater the achievement and the intelligence that supports it.
Further, in a world where for many the acquisition of money is difficult and the resulting sums palpably insufficient, the possession of it in large amount seems a miracle. Accordingly, possession must be associated with some special genius. This view is then reinforced by the air of self-confidence and selfapproval that is commonly assumed by the affluent. On no matter is the mental inferiority of the ordinary layman so rudely and abruptly stated: "I'm afraid that you simply don't understand financial matters. " In fact, such reverence for the possession of money again indicates the shortness of memory, the ignorance of history, and the consequent capacity for self- and popular delusion just mentioned. Having money may mean, as often in the past and frequently in the present, that the person is foolishly indifferent to legal constraints and may, in modern times, be a potential resident of a minimum-security prison. Or the money may have been inherited, and, notoriously, mental acuity does not pass in reliable fashion from parent to offspring. On all these matters, a more careful examination of the presumed financial genius, a sternly detailed interrogation to test his or her intelligence, would frequently and perhaps normally produce a different conclusion. Unfortunately the subject is rarely available for such scrutiny; that, too, wealth or seeming financial competence often excludes.
Finally and more specifically, we compulsively associate unusual intelligence with the leadership of the great financial institutionsthe large banking, investment-banking, insurance, and brokerage houses. The larger the capital assets and income flow controlled, the deeper the presumed financial, economic, and social perception.
In practice, the individual or individuals at the top of these institutions are often there because, as happens regularly in great organizations, theirs was mentally the most predictable and, in consequence, bureaucratically the least inimical of the contending talent. He, she, or they are then endowed with the authority that encourages acquiescence from their subordinates and applause from their acolytes and that excludes adverse opinion or criticism. They are thus admirably protected in what may be a serious commitment to error.


Economics of Homelessness: Are housing prices to blame?

This paper is close to what you want.

  It is generally believed that the increased incidence of homelessness
  in the United States has arisen from broad societal factors, such as
  changes in the institutionalization of the mentally ill, increases in
  drug addiction and alcohol usage, and so forth. This paper presents a
  comprehensive test of the alternate hypothesis that variations in
  homelessness arise from changed circumstances in the housing market
  and in the income distribution. We assemble essentially all the
  systematic information available on homelessness in U.S. urban areas:
  census counts, shelter bed counts, records of transfer payments, and
  administrative agency estimates. We estimate similar statistical
  models using four different samples of data on the incidence of
  homelessness, defined according to very different criteria. Our
  results suggest that simple economic principles governing the
  availability and pricing of housing and the growth in demand for the
  lowest-quality housing explain a large portion of the variation in
  homelessness among U.S. metropolitan housing markets. Furthermore,
  rather modest improvements in the affordability of rental housing or
  its availability can substantially reduce the incidence of
  homelessness in the United States.

Quigley, John M., Steven Raphael, and Eugene Smolensky. "Homeless in America, homeless in California." Review of Economics and Statistics 83.1 (2001): 37-51.

  The results in table 5 provide the strongest evidence that housing
  market tightness is an  important determinant of homeless. For
  permanent caseloads, vacancy rates have a strong negative and
  statistically significant  effect (at the 1% level) on the incidence
  of homelessness. Moreover, these elasticity estimates are quite
  similar across specifications. In the two specifications including
  fair market rents (specifications (1) and (3)), rents exhibit a
  significant  positive effect on the incidence of homelessness as
  predicted by theory. In addition, the specification including the
  ratio of rents to income indicates a strong positive effect. Hence,
  for homelessness, as measured by the incidence of  homeless
  house-holds seeking permanent assistance, measures of housing market
  tightness consistently exhibit strong and statistically significant
  effects, which is consistent with the predictions of theory. We also
  find a negative and significant effect of per capita income. There are
  no important effects of county unemployment rate  s and no consistent
  and significant effect  of the SSI populations across specifications.
  Similar to the results from the S-night and continuum-of-care data
  sets, the results in table 5 also indicate that warmer weather is
  positively associated with homelessness

The key results table:

Open access version?


How to prove that a concave production imply that the input requirement sets are convex?

Start with definitions:

Production (possibilities) set: $Y$ which you know is convex
Input requirement set: $V(y)=\{\mathbf{x}:(y,−\mathbf{x})∈Y\}$

On page 7. you can see: $\mathbf{y}\in Y$ and $\mathbf{y'} \in Y$ which then implies $t\mathbf{y}+(1-t)\mathbf{y'} \in Y$.

Hint 1:

   Remember that: $\mathbf{y}=(y,−\mathbf{x})$ !

What does it mean that $Y$ is a convex set?

   $Y$ being a convex set means that if $\mathbf{y}=(y,−\mathbf{x}) \in Y$ and $\mathbf{y'}=(y,−\mathbf{x'}) \in Y$ then $t(y,−\mathbf{x})+(1-t)(y,−\mathbf{x'}) \in Y$ as page 7. would suggest.

Okay, but what does that actually mean?

   $t(y,−\mathbf{x})+(1-t)(y,−\mathbf{x'}) \in Y \implies (ty+(1-t)y, -t\mathbf{x}-(1-t)\mathbf{x'}) \in Y \implies (y,-t\mathbf{x}-(1-t)\mathbf{x'}) \in Y $

Which is just equivalent of saying that:

   $t\mathbf{x}+(1-t)\mathbf{x'} \in V(y)$ because both $\mathbf{x}$ and $\mathbf{x'}$ are in $V(y)$. (Look: second definition)


Isolating single campaign effects over a sales series

You might take a look at the Causal Impact R package and accompanying paper.


How is the Euler Equation for Consumption derived from from intertemporal budget constraint and lifetime utility function in basic macroeconomics

It is simply the chain rule, or you could see it from the total differential.

Let $C_1$ be consumption today and $C_2$ be future consumption.

In the second part you have $U(C_2 (C_1) )$, i.e. a utility function dependending on $C_2$, which is itself (through the budget constraint you used) a function of $C_1$. You take the derivative of U with respect to $C_1$, which equals:

$d U/d C_1 = \partial U(C_2)/ \partial C_2 * d C_2 / d C_1$.

The first part is just $ \partial U(C_2)/ \partial C_2 = U'(C_2)$.

From what you have written, it seems that you understand that $C_2(C_1)=(1 + R)(\bar{X} - C_1)$.

Taking the derivative of that for the second factor in the multiplication above yields the result.


Prove that in presence of many goods, the law of DMRS implies that at least one good must be a net substitute for any other good

I don't know what is the definition of the "law of DMRS", but I think what the author meant is summarized in these lines of MWG:

Proposition 3.G.2: suppose that $u(\cdot)$ is a continuous utility function representing a locally nonsatiated and strictly convex preference relation $\succsim$ defined on $\mathbb{R}^L_+$. Suppose also that $h(\cdot, u)$ [the hicksian demand function] is continuously differentiable at $(p, u)$, and denote its $L\times L$ derivative matrix [with respect to each $p$] by $D_p h(p, u)$. Then

$D_p h(p, u)$ is a negative semidefinite matrix.

$D_p h(p, u) p = 0$

Well, 1 implies that the elements of the main diagonal of the matrix are all non positive, i.e. $\frac{\partial h_l (p, u)}{\partial p_l} \leq 0$ for each $l$.
We say that two goods are substitutes if $\frac{\partial h_l (p, u)}{\partial p_k} \geq 0$ (a positive variation in the price of $k$ implies an increase in the compensated demand of $l$).
Part 2 of the proposition says that $D_p h(p, u) p = 0$. But as $\frac{\partial h_l (p, u)}{\partial p_l} \leq 0$, for every $l$ we must have a good $m$ such that $\frac{\partial h_l (p, u)}{\partial p_m} \geq 0$, which means exactly that every good has a substitute.


Looking up each of those terms you ask about would have given you your answer.

Fiscal policy is economic policy enacted through government. The minimum wage is a type of fiscal policy.

Monetary policy is economic policy enacted through central banking. The minimum wage is NOT a type of monetary policy.

Supply-side is a macroeconomic theory that growth is best achieved through capital investment and lowering barriers to production. Usually the policy suggestions to come out of this are to remove government regulation and lower marginal tax rates. The minimum wage is a type of regulation, so minimum wage would not fall under the doctrine of classical supply side economics.

I don't understand what you mean when you ask where "price control policy" falls under. There's no real categorization for it besides also being a type of fiscal policy.


Why is it not possible to prevent sudden unemployment of thousands of people?

This question boils down to asking why business cycles exist and change employment in shocks.

One possibility is that employment is sticky or not worth changing constantly. A firm cannot constantly calculate and evaluate worker productivity and general profitability without a, perhaps rather expensive, cost, especially if the firm is very large. Constantly hiring and firing to try and have "optimal" employment for a firm is also bad for morale.

And perhaps more to the point, the alternative to firing people in large swathes is lowering wages, which is even worse for worker morale and sometimes not really legal (e.g. salaries are fixed for a year). So if there is a recession for example, a firm will generally choose to keep wages up, and then when it can't afford such a scenario anymore, it will create large layoffs.

Further Reading: The cost of firing changes the dynamics of how people are fired as well. See this article about France and Spain's different changes in employment with respect to the Great Recession. It is possible to prevent large drops in unemployment. Whether it is efficient is another question. My answer only briefly touches on the theory, and there is plenty of literature related to the points I've made that you can look for.


If you search for “simultaneous equations” inside the Econometrics Task View web page you can find two results: the systemfit and the bimets packages.
As you stated, the systemfit package is a powerful tool for econometric estimation of simultaneous systems of linear and nonlinear equations, but it only provides fitting procedures, thus it cannot be used in your example in order to work out a forecast.
On the other hand, the bimets package implements, among others, simulation and forecasting procedures, and multiplier analysis; therefore you can easily calculate elasticities.
Take a look also at the Rstudio blog post about SEM in R:
https://rviews.rstudio.com/2021/01/22/sem-time-series-modeling/
An example of a simple Klein model in R follows:
#load library
library(bimets)

#define the Klein model
kleinModelDef <- "
MODEL

COMMENT> Modified Klein Model 1 of the U.S. Economy with PDL,
COMMENT> autocorrelation on errors, restrictions and conditional equation evaluations

COMMENT> Consumption with autocorrelation on errors
BEHAVIORAL> cn
TSRANGE 1923 1 1940 1
EQ> cn =  a1 + a2*p + a3*TSLAG(p,1) + a4*(wp+wg)
COEFF> a1 a2 a3 a4
ERROR> AUTO(2)

COMMENT> Investment with restrictions
BEHAVIORAL> i
TSRANGE 1923 1 1940 1
EQ> i = b1 + b2*p + b3*TSLAG(p,1) + b4*TSLAG(k,1)
COEFF> b1 b2 b3 b4
RESTRICT> b2 + b3 = 1

COMMENT> Demand for Labor with PDL
BEHAVIORAL> wp
TSRANGE 1923 1 1940 1
EQ> wp = c1 + c2*(y+t-wg) + c3*TSLAG(y+t-wg,1) + c4*time
COEFF> c1 c2 c3 c4
PDL> c3 1 2

COMMENT> Gross National Product
IDENTITY> y
EQ> y = cn + i + g - t

COMMENT> Profits
IDENTITY> p
EQ> p = y - (wp+wg)

COMMENT> Capital Stock with IF switches
IDENTITY> k
EQ> k = TSLAG(k,1) + i
IF> i > 0
IDENTITY> k
EQ> k = TSLAG(k,1)
IF> i <= 0

END
"

#load the model
kleinModel <- LOAD_MODEL(modelText = kleinModelDef)

#define data
kleinModelData <- list(
  cn  =TIMESERIES(39.8,41.9,45,49.2,50.6,52.6,55.1,56.2,57.3,57.8,
                  55,50.9,45.6,46.5,48.7,51.3,57.7,58.7,57.5,61.6,65,69.7,
                  START=c(1920,1),FREQ=1),
  g   =TIMESERIES(4.6,6.6,6.1,5.7,6.6,6.5,6.6,7.6,7.9,8.1,9.4,10.7,
                  10.2,9.3,10,10.5,10.3,11,13,14.4,15.4,22.3,
                  START=c(1920,1),FREQ=1),
  i   =TIMESERIES(2.7,-.2,1.9,5.2,3,5.1,5.6,4.2,3,5.1,1,-3.4,-6.2,
                  -5.1,-3,-1.3,2.1,2,-1.9,1.3,3.3,4.9,
                  START=c(1920,1),FREQ=1),
  k   =TIMESERIES(182.8,182.6,184.5,189.7,192.7,197.8,203.4,207.6,
                  210.6,215.7,216.7,213.3,207.1,202,199,197.7,199.8,
                  201.8,199.9,201.2,204.5,209.4,
                  START=c(1920,1),FREQ=1),
  p   =TIMESERIES(12.7,12.4,16.9,18.4,19.4,20.1,19.6,19.8,21.1,21.7,
                  15.6,11.4,7,11.2,12.3,14,17.6,17.3,15.3,19,21.1,23.5,
                  START=c(1920,1),FREQ=1),
  wp  =TIMESERIES(28.8,25.5,29.3,34.1,33.9,35.4,37.4,37.9,39.2,41.3,
                  37.9,34.5,29,28.5,30.6,33.2,36.8,41,38.2,41.6,45,53.3,
                  START=c(1920,1),FREQ=1),
  y   =TIMESERIES(43.7,40.6,49.1,55.4,56.4,58.7,60.3,61.3,64,67,57.7,
                  50.7,41.3,45.3,48.9,53.3,61.8,65,61.2,68.4,74.1,85.3,
                  START=c(1920,1),FREQ=1),
  t   =TIMESERIES(3.4,7.7,3.9,4.7,3.8,5.5,7,6.7,4.2,4,7.7,7.5,8.3,5.4,
                  6.8,7.2,8.3,6.7,7.4,8.9,9.6,11.6,
                  START=c(1920,1),FREQ=1),
  time=TIMESERIES(NA,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,
                  1,2,3,4,5,6,7,8,9,10,
                  START=c(1920,1),FREQ=1),
  wg  =TIMESERIES(2.2,2.7,2.9,2.9,3.1,3.2,3.3,3.6,3.7,4,4.2,4.8,
                  5.3,5.6,6,6.1,7.4,6.7,7.7,7.8,8,8.5,
                  START=c(1920,1),FREQ=1)
);

#load time series into the model object
kleinModel <- LOAD_MODEL_DATA(kleinModel,kleinModelData)
## Load model data "kleinModelData" into model "kleinModelDef"...
## ...LOAD MODEL DATA OK

#estimate the model
kleinModel <- ESTIMATE(kleinModel)

#In order to forecast the model up to 1944,
#we need to extend exogenous variables up to 1944
kleinModel$modelData <- within(kleinModel$modelData,{
    wg    = TSEXTEND(wg,  UPTO=c(1944,1),EXTMODE='CONSTANT')
    t     = TSEXTEND(t,   UPTO=c(1944,1),EXTMODE='LINEAR')
    g     = TSEXTEND(g,   UPTO=c(1944,1),EXTMODE='CONSTANT')
    k     = TSEXTEND(k,   UPTO=c(1944,1),EXTMODE='LINEAR')
    time  = TSEXTEND(time,UPTO=c(1944,1),EXTMODE='LINEAR')
  })

#forecast model
kleinModel <- SIMULATE(kleinModel
                      ,simType='FORECAST'
                      ,TSRANGE=c(1941,1,1944,1)
                      ,simConvergence=0.00001
                      ,simIterLimit=100
                      ,quietly=TRUE
  )

#get forecasted GNP
TABIT(kleinModel$simulation$y)
##
##       DATE, PER, kleinModel$simulation$y
##
##       1941, 1  ,  125.3
##       1942, 1  ,  172.5
##       1943, 1  ,  185.6
##       1944, 1  ,  141.1


To elaborate on my comment above, whether there's a "restriction" on the possible values of $a$ depends on your assumptions about the production function.

If you assume that all inputs into the production function contribute positively to the final output, then it must be that $a \in (0, 1)$.

Since you estimated the parameter, you might want to check the confidence interval of the estimated parameter. You can do a one-sided test if $a < 1$, for example.


How do I quantitatively visualize this situation?

Two things will happen:

Firstly, everyone will effectively have had their real incomes rise by 10% (assuming this new energy costs nothing). They will then spend or save this income on other goods besides energy, increasing overall output (note, savings is used for investment as well).

Secondly, energy consumption will likely increase by a large amount now that it is nothing. People who were mindful about their energy consumption due to cost concerns will now not care. Therefore, people will be consuming more energy.


Game theory - Definition problem - Strategy vs Action

I had the same question few weeks ago. I attach the link below that may help you, but here is the approach I took to solidify the difference.

When action and strategy differ in game theory

Don't associate the distinction between action and strategy with whether the game is in normal form or extensive form or sequential or not. This can cause more confusion.

When you want to define strategy, always think of "contingency".

Action is a set of possible things you can do when you are called upon to move. For example, suppose you want to sell a used car as dealer. As seller, what is it that you CAN DO when you are "called upon to move"? Simple: offer high price or low price.

But you immediately realize things can get complicated if, say, dealer has more information on the car's condition. You might suspect that car behind the dealer is a good used card or lemon. Although you still get to offer high price or low price at the end of the day, as player of the game, you must set a contingency plan when the car is good or lemon.

Harsanyi introduces "nature" to do exactly this. The condition of car is simple: good or lemon. This is described as the state space of the game. Both players, seller and buyer, understand the ex-ante probability distribution over this state space, meaning you and potential buyer know the probability you get good car or lemon. Therefore, as dealer, you would have action space $A_1=\{H,L\}$ indicating offering high or low price. However, your strategy space would be $S_1=\{(H,h),(H,l),(L,h),(L,l)\}$ where first entry of the pair is your action when the car is good and second entry is your action when the car is lemon. It is a game plan laid out for you in every possible scenario.

One good exercise may be to see if you can construct the strategy set for buyer if she doesn't observe the condition (i.e. state) of the car the nature draws.


You can use mark. For example, if the treatment was done in the year 2000, and your time variable is called time, the syntax is

mark treatment if time >= 2000 & time <.

where treatment is the dummy that is generated.


Leontief function marginal product of labor/capital

Since you are interested in labour, let's assume for simplicity that the stock of capital is fixed at $\bar{K}$. Then, the optimal choice of capital and labour is given by:

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

Therefore, optimal labour is:

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

The marginal product of labour depends on how actual labour relates to optimal labour:

Case 1: $L = L^*$. In the standard Leontief diagram, with $L$ in the horizontal axis and $K$ in vertical axis, this is any point on the optimal path (which function starts at the origin and has slope $\frac{b}{a}$). In this case, $\dfrac{dQ}{dL}=0$.
Case 2: $L > L^*$. This is when the factors' combination is below the $\frac{b}{a}$ path. In this case, $\dfrac{dQ}{dL}=0$.
Case 3: $L < L^*$. This is when the factors' combination is above the $\frac{b}{a}$ path. The solution here depends on how far $L$ is from $L^*$:

Case 3a: $L^*-L > 1$. This is perhaps the most likely case. Here, the change in labour leaves still with low levels of labour. In this scenario, $MP_L = \dfrac{1}{a}$. This results comes from comparing output before the change in labour: $Q_0=\dfrac{L_0}{a}$ versus after the change: $Q_1=\dfrac{L_0 +\Delta L}{a}$. From here, we conclude that $\dfrac{dQ}{dL}=\dfrac{1}{a}$. Notice that in this case, there is still room for increasing output by increasing labour (i.e. we are still within Case 3).
Case 3b: $L^*-L = 1$. Here, the change in labour leaves us on the optimal path. The change is just as in Case 3a. The differences is that increasing labour further leaves without increases in output. We are then back to case 1.
Case 3c: $L^*-L < 1$. Here, the change in labour is more than what we actually require. Thus, we move from being in Case 3 to being in Case 2. The change is therefore not $\dfrac{1}{a}$, but equal to $\dfrac{1}{a} \times (L^*-L)$. In other words, the change is proportional to the exact amount of labour input we need to be in the optimal path.

A graphical example can be seen below:

For any labour equal or above $L_0$, further increases in $L$ does not change $q_0$. However, when we are above the optimal path (meaning $K>\bar{K}$), the extra marginal worker does add to production. You can see that depending on how far this point is from the optimal, the three scenarios of Case 3 arise.

If interested (in my personal quest in favour of Open Economics), here is the R code of the graph:

plot(c(0,5), c(0,5), type = "n", xlab = "Labour", ylab = "Capital", yaxt='n', xaxt='n', bty='l', mgp=c(1,1,0))
segments(0, 0, 4.7, 4.7,lwd=2)
text(4.9, 4.8, expression(frac(b,a)),cex = 1.3)
points(2, 2, type="p", pch=19, col="black", bg=NA, cex=1.5)
segments(2, 2, 2, 5,lwd=2.5)
segments(2, 2, 4.7, 2,lwd=2.5)
text(4.9, 2,  expression(q[0]),cex = 1.3)
text(2.4, 1.75, expression(paste("(",L[0],",",bar(K),")")),cex = 1.3)
points(3.5, 2, type="p", pch=19, col="black", bg=NA, cex=1.5)
text(3.5, 2.3, expression(L>L[0]),cex = 1.3)
points(2, 3.5, type="p", pch=19, col="black", bg=NA, cex=1.5)
text(2.4, 3.5, expression(K>bar(K)),cex = 1.3)


The Sidrauski model is equivalent to a model with 2 consumption goods where the second good (money) is produced costlessly by the government. Inflation is equivalent to raising the price of the second good. Money is super-neutral if this does not affect the real allocation (demand for the first good). This only happens when preferences are separable.


Best instruments for "years of schooling"/"education"

Check out
Estimating the Return to Schooling: Progress on Some Persistent Econometric
Problems by David Card (2001) ECA

I know distance from school has been used in the past as an IV..although it has been criticized.


Calculating mixed strategy equilibrium in battle of sexes

Your calculations are correct. If we take this game:

\begin{array}{| r | r |}
  \hline
\hline
  & A& B\\
  \hline
A&3,2&1,1\\
B&0,0&2,3\\
  \hline
\end{array}

then the mixed strategy probability used by the column player must make the row player indifferent: $3q+1(1-q)=0q+2(q-1)$, implying $q=1/4$.

From your question, I infer that you have a solution claiming that the answer should be $q=2/5$. In that case, it looks to me like there has been a typo somewhere. Indeed, if we take the following game:

\begin{array}{| r | r |}
  \hline
\hline
  & A& B\\
  \hline
A&3,2&0,0\\
B&0,0&2,3\\
  \hline
\end{array}

We find the mixed strategy equilibrium implies the column player using probability $q$, with $3q=2(1-q)$ so $q=2/5$.

Note that this second, modified game is symmetric (as textbook battle of the sexes games usually are), further strengthening my suspicion that the confusion has been caused by a typo in the description of the game.


Why is the percentage of Australian businesses that don't have any employees so high?

Employment excludes non-salaried directors, volunteers, persons paid by commission only, and self employed persons such as consultants and contractors.

The actively trading businesses with zero employees are therefore those businesses where the staff members are drawn exclusively from that group. That may cover most one-person outfits, perhaps some family firms, groups of self-employed people, and so on. These businesses are typically formed in that way as a response to particular regulations, usually tax rules.

The answer is in the explanatory notes attached to that dataset, section 56, from where the above quote is taken.  The explanatory notes are attached to that dataset to answer questions such as this. When you have questions about a particular dataset, the explanatory notes (which might also be called metadata, data descriptors, or similar), will usually have the answer, as they do in this case, and should be any enquirer's first place to look.


You cannot prove this. It is wrong.

Define $u(x)=\min{\{x,0\}}$. Let $\succsim$ be the preference relation represented by $u$. This preference relation satisfies $x\succ y \Longrightarrow x+a\succsim y+a$ for all $x,y,a\in\mathbb R$. But let $x=0$, $y=1$, and $a=-1$. Then $x\sim y$, but $y+a=0\succ -1=x+a$, thus $x+a\not\succsim y+a$ and $\succsim$ is not linear.


They're complementary to each other.

With the development of Behavioral Game Theory, canonized in Camerer's 2003 textbook, the current paradigm of game theory 1 is converging towards making theories about how people play games, and test them, most often with experiments.

Wither Game Theory, Fudenberg & Levine, 2016


Factoring Technology into Economics models

Yes, this is pretty standard stuff in economics.

First, the simplest models contain it in a hidden form. Supply curves are derived from production functions, and include (marginal) cost of production. Technology innovation is then reflected in reduced production costs per unit. It can also change the indifference curves of consumers, leading to changes in the demand curve (shifts, slope change, etc.) Labor costs are also affected, thus also changing macroeconomic models.

Second, there are models which focus specifically on innovation. Look up the Schumpeterian models, or (for a more business-oriented than economic-oriented theory) Christensen's work.

Third, some of the more involved models can accommodate the consequences of a specific technology as opposed to a generic assumption of innovation. For example, brick-and-mortar stores are known to max out at some size due to rising overhead, while for online stores, overhead is considered to rise underproportionally to profit, which predicts "winner takes all" markets, which is what we are seeing today. Similarly, when you look at information assymmetry models, you see that modern information technology can remove some asymmetries from specific markets (new ways for principals to monitor agents, allowing price comparison in some markets, eliminating transaction costs or discovery costs thus making intermediars obsolete). You can get a good, if somewhat dated, review of the results of IT developments on economic models in Macho-Stadler and Perez-Castrillo's "An introduction to the economy of information".

I know this is a rather vague answer, but I'm afraid your question is similarly vague.


Can capital still be paid its marginal product in the absence of a homogeneous capital stock?

Nothing about marginal productivity theory depends on the exact truth of a simple aggregate production function with capital defined by a single number

Different sorts of capital used as separate production technologies prevent clean aggregation to a representative form of capital but does not prevent capital from being paid its marginal product.

To make things simple, consider an economy with two kinds of capital $k_1$ and $k_2$ and no labor. There is only a single final good ($y$) but are two separate production technologies $y_1$ and $y_2$ where $y_1 =  k_{1}^{\alpha_1}$ and $y_2 =  k_{2}^{\alpha_2}$. Then aggregate production of $y$ is:
$$y =  k_{1}^{\alpha_1} + k_{2}^{\alpha_2}$$
If $0 < \alpha_1 < \alpha_2 < 1$ then for positive rental rates on capital $r_1$ and $r_2$ it will be efficient to choose a positive amount of both types of capital such that for each type $i$:
$$ \alpha_i \cdot k_{i}^{\alpha_i - 1} = r_i\Rightarrow k_{i}^{*} = (r_i/\alpha_i)^{1/(\alpha_i - 1)} $$

There is no single $k$ that represents capital in this economy. Sure you could define $k=k_1 + k_2$, but that wouldn't mean exactly the same thing. On the margin the two sorts of capital don't have the same product and so aggregation doesn't make sense here. But in this setting, it is likely that the rental rate on capital would be equated ($r_1 = r_2$) because why would you buy one sort of capital when the other sort paid more?


Heteroscedasticity and weighted least square estimator

Let $\hat{\beta}$ be the OLS estimator of $\beta$ in
$$y_t = \beta x_t + u_t$$
Let $\tilde{\beta}$ be the OLS estimator of $\beta$ in
$$\dfrac{y_t}{\sqrt{k_t}} = \beta \dfrac{x_t}{\sqrt{k_t}} + \dfrac{u_t}{\sqrt{k_t}}$$
$\text{Var}(\hat{\beta}) = \dfrac{\sum x_t^2\sigma_t^2}{\left(\sum x_t^2\right)^{2}} $

$\text{Var}(\tilde{\beta}) = \dfrac{\sigma^2_u}{\sum \dfrac{x_t^2}{k_t}} $

So, $\dfrac{\text{Var}(\tilde{\beta})}{\text{Var}(\hat{\beta})} = \dfrac{\left(\sum x_t^2\right)^{2}}{\sum \dfrac{x_t^2}{k_t} \sum {x_t^2}{k_t}} $.

Let $a_t = \dfrac{x_t}{\sqrt{k_t}}$ and $b_t = {x_t}{\sqrt{k_t}}$, and we can rewrite the ratio of variances as

$\dfrac{\text{Var}(\tilde{\beta})}{\text{Var}(\hat{\beta})} = \dfrac{\left(\sum a_tb_t\right)^{2}}{\sum a_t^2 \sum {b_t^2}} \leq 1 $ [By Cauchy-Schwarz inequality]


How do I solve this Game Theory Bargaining problem?

So, we have to solve this game by Backward induction. We know if $P_1$ rejects the offer at the end note, both will get $P_1=P_2=\delta $. So, to make $P_1$ better off and make the proposal accepted,$P_2$ will offer $P_1$ an amount of $(\delta +\epsilon ) $, where $\epsilon \rightarrow 0$.(I'm considering the case of continuous prices or you can think of a $\epsilon =0.001$ i.e. a very small amount which just makes the $P_1$ strictly better off in accepting the proposal. So, $P_2$ will get $3\delta -(\delta +\epsilon )$ or $3\delta -\delta +0.001=2\delta +0.001$. Now, $P_1$ knows that if $P_1$ offers any amount $< 2\delta +\epsilon $( strictly less than), then $P_2$ will reject the offer. So, to make $P_2$ strictly better off, he'll offer $P_2$ an amount of $(2\delta +\epsilon + \theta )$, where $\epsilon \to 0,\theta \to 0 $ i.e. infinitesimal changes such as $\epsilon =\theta =0.001$. So, $P_1$ will get an amount of $(3-2\delta -\epsilon -\theta )$ where, $\epsilon ,\theta > 0$


Solow model golden rule with my exact answer

Let production function $$F(K_t,L_t)=K_t^aL_t^{1-a}$$

$$max[c^*=f(k^*)-(δ+n)k^*]$$ with respect to $k^*$

Then $MPK=a(k^*)^{a-1}$

So $k_G= (\frac{a}{(δ+n)})^{1/1-a}$ which is golden rule level of capital stock per capita.

Now calculate consumption at golden rule level

$$c_G=f(k_G)-(δ+n)k_G$$
$$c_G=(k_G)^a-(δ+n)k_G$$

$$c_G=(\frac{a}{(δ+n)})^{a/1-a}-(δ+n)(\frac{a}{(δ+n)})^{1/1-a}$$

$$c_G=(\frac{a}{(δ+n)})^{a/1-a}[1-(δ+n)(\frac{a}{(δ+n)})]$$

$$c_G=(\frac{a}{(δ+n)})^{a/1-a}[1-a]$$
$$c_G=k_G^a[1-a]$$

Ow consider to calculate wage

$w=f(k)-kf’(k)=(1-a)f(k)$

Then,

$$c_G=k_G^a[1-a]=w_G$$

$$c_G=w_G$$

Now consider aggregate version,

$C_G/L=W_G/L$

so,

$C_G=W_G$

I am confused at the aggregate transformation.

I add this question with its answer in order to contribute to the website. Thus, I will be glad if you add your opinions about my solution as well.


What is the relation beween Granger causality and Rubin's causality?

In short, not really.  Prof. John Cochrane offers an astute example of Granger causality by stating that a weatherman Granger-causes the weather.  We know that a weatherman has no measurable role in causing the weather, but there is indeed an intertemporal correlation between what the weatherman says today and what happens tomorrow.


What are some of the economic arguments for gender-neutral marriage?

tl,dr: I don't see an economic argument for GN marriage, or marriage in general what-so-ever.

Frictionless environment

All spending on marriage, are nothing more than consumption goods. There is no reason to believe that marriage-related spending have a higher Keynesian multiplier than other consumption categories.

As long as this is the case, as @denesp argues, "money spent on marriage" is money that would instead be spend on different consumption goods. Legalizing marriages would be a boost to the economy, and useful in crises and slumps - but again, by Keynesian logic, the same would be true for burying money in the ground. There is no reason it has to be marriage specifically.

But

However, the real world is no frictionless environment.

To the extent that there are different tax rates for married couples, which set up incentives for working more, increasing the size of the married population may increase total output. But then again, the welfare improving solution is not to increase the share of married couples, but removing tax benefits for married couples.
There may be "efficiency gains" from being married. This is the only possible "real" economic argument: If, being married, and having a state-legalized marriage contract makes people very happy and productive workers, increasing the size of the married population will increase TFP. However, this is a very big if, and I am not aware of research supporting this. In contrast, happiness research seems to indicate that the impact of positive or negative events fades away quickly, and an invididuals "happiness" moves back to average levels quite quickly, even in case of losing a leg (Brickman et al, 1978;   ungated)


Difference between a Data Scientist and an Econometrician?

A central question of econometrics has been the question of causality: does X cause Y? Does the minimum wage cause employment? Do taxes cause capital flight?

Data science has been mostly concerned with the predictive power of its models: how can we sort pictures according to their contents? how can we sort customers according to their creditworthiness?

The two questions of causality and prediction are related: if we find a great predictor of customers' creditworthiness, does it mean that that characteristic causes creditworthiness? The answer is tricky, because excellent predictors are actually not always causal determinants of the outcome. If, for instance, high income students tend to cluster in good schools, this does not necessarily mean that household income causes student achievement. A perhaps more controversial example is that ethnicity predicts (i.e. is correlated with) a range of outcomes (education, health) but does not necessarily cause them.

Data science has become central to business applications (from Google Photos to face recognition in Facebook) but not so much to public policy analysis. In contrast, econometrics is ubiquitous in policy analysis.

There is some convergence though. A better, and more elaborate answer on how econometrics recently borrowed from data science is provided by Susan Athey (Sep 2016 podcast). http://www.econtalk.org/archives/2016/09/susan_athey_on.html


Taylor Rule: Relationship between interest rate and inflation

It's true that in response to an oil shock, the Taylor rule could recommend increasing the interest rate to reduce inflation. In practice it would mean that as interest increases, consumption falls. This could be from less credit financed spending, or because the opportunity cost of holding money has increased, therefore people invest more in illiquid assets (bonds in an IS-LM context) and therefore consume less. As consumption falls, aggregate demand falls, therefore putting pressure off prices causing a reduction in inflation.

Note: an oil shock might also increase the output gap, so its not inherently clear what the taylor rule would say in this situation.


Why may printing trillions of dollars not lead to inflation?

An important subtlety
In the literature, the words "inflation/deflation" are often used to describe "any increase/decrease in the price of a good for any reason". If some kind of crisis causes the value of stocks to drop by 10% and a monetary scheme like QE is implemented to bump the stocks back up by 10%, then some would say that the inflation is zero.
However, it is clear that the stock prices are now 10% higher than they would have been without the injection of money via QE. We could say that the price is inflated relative to a case without such intervention. From this viewpoint, any expansion of the monetary base must be inflationary, since it will always raise the prices of some assets/goods/services relative to the price without the intervention.
One could argue that the 10% drop reflects a real change in the state of the economy (people are more interested in having the funds to see themselves through an immediate crisis than to invest long term), and that injecting more money into the system to prop up the value of stocks that nobody would want to buy in the absence of QE is unnecessary.
To answer your question in short, if one uses the broad definition of inflation, it is indeed possible to do the exact right amount of QE to make sure the price in dollars and cents of assets do not change as a result of the crisis in the short/medium/long term. If one uses the more narrow definition, QE and similar interventions are always inflationary.


Can Quasi-Linear functions be monotonic?

Yes, a trivial example is the following utility:
$$u(x,y) = x+ \sqrt{y}$$
which is both quasi linear and monotonic.


Why is a nations economy is divided into sectors?

That is because in science we try to not just create new knowledge but also classify the knowledge.

In biology biologists don’t just study living things they also classify them as mammals, birds etc based on similar characteristics, in astrophysics physicists don’t just study different astronomical objects but classify different astronomical objects into categories such as planets, asteroids, stars etc. based on similar characteristics, and finally in economics we don’t just study an industry but also categorize industries again based on similar characteristics.

The point of such classification is to have established nomenclature which makes subsequent research more easy. If a economist says “I study primary sector in US” the economist does not need to always explain that he or she studies the sector that produces raw materials/agricultural which saves a lot of time and makes process of science as efficient as possible.


As Jesper Hypel has pointed out in a comment, the derivative need not exist. Actually, the function $a\mapsto x^*(a)$ need not even be nondecreasing. Here is an example: Let $g$ and $h$ be given by $g(x,a)=-(a-x)^2$ and $h(x,a)=a-(a-1-x)^2$, respectively. Then $x_1(a)=a$ and $x_2(a)=a-1$. Moreover, $f\big(x^*(a),a\big)=g\big(x_1(a),a\big)$ for $a<0$ and $f\big(x^*(a),a\big)=h\big(x_2(a),a\big)$ for $a>0$. As a consequence, the function $a\mapsto x^*(a)$ jumps down after $0$.


Clarification on the impact of a change in the Bank Rate (UK)

Interest rates in commercial banks are determined by market forces - supply and demand.

Their demand will be influenced by their customers' expectations - which will be influenced by the Central Bank interest rate. Their supply will be influenced by the Central Bank (the Bank of England) interest rate.

Some UK mortgages have fixed interest rates for a particular term - for example, the first three or five years of the mortgage. After that, they revert to the standard variable rate, which loosely tracks the Central Bank interest rate plus a few percentage points.

Some UK mortgages are trackers - that is to say, they are designed so that they track the Central Bank interest rate plus a fixed offset, and they change very quickly when Central Bank rates change: if the Central Bank rate changes by x%, then within one day to one month, the tracker interest rates also change by x%.


Your method is correct. This however does not have to be the case in general. Your have a budget surplus simply because your government spending does not rise too much relative to your tax revenue collection.

Consider the following with MPC = 0.3, t = 0.2 and Y = 1000 and dt = 0.5:

If you increase spending too much, you will end up with a budget deficit.

There could be other factors as well, for example, the rate of taxes you have in the economy when the policy changes are made. For the same parameter set mentioned above, with dG and t free, you have (t on vertical axis and dG on horizontal axis):

There are some dG,t pairs which generate surplus/deficit/balanced budget.


Condition in yellow: Each consumer $i$ is assumed to have initial wealth endowment $0$.

Condition in purple: For each asset $k$, aggregate demand $\leq$ aggregate supply, which is zero by assumption.

In practical terms, the asset $z_k$ being traded at $t = 0$ are $t = 1$ claims to commodity $k$. It is a derivative contract. In this particular case, it is a futures contract, i.e. the right and obligation to $z_k$ units of commodity $k$.

Assume the exogenous asset returns $r_{sk} = 1$ for all $k$ and $s$.

If you sell a futures contract for $z = 30$ barrels of West Texas Intermediate crude (tons of wheat/thousand pounds of coffee beans/etc.) at futures price $q = \$10$, then you pocket $z \cdot q = \$300$ at $t = 0$. At $t = 1$, you must deliver $30$ barrels of WTI crude to your counterparty. So you must purchase $z = 30$ barrels of WTI at spot price $p = \$20$, say, at a cost of $z \cdot p = \$600$.

Above is the case of a short position. Similarly, if you take a long position, it will cost $\$300$ today but tomorrow you get to have $30$ barrels of WIT, which you get to sell at the spot price $p = \$20$.
This adds $\$600$ to your $t=1$ budget which can then be used to buy consumption good $x_i$'s.

Regardless of long or short position, today's cost is $z \cdot q$ and tomorrow's gain is $z \cdot p$.

If you have zero $t = 0$ wealth, the total value of contracts you buy cannot be more than the total value of contracts you sold. This is condition in yellow.

For each commodity good, if there is an aggregate of, say, $100$ contracts being sold at $t = 0$, there cannot be $101$ being bought.
$$
\mbox{aggregate long position/demand} - \mbox{aggregate short position/supply} \leq 0.
$$
This is condition in purple.

Allowing for $< 0$ implicitly assumes that there are other agents unmodelled that absorbs excess supply. Alternatively, one can insist that the futures market clears, i.e. $$
\mbox{aggregate long position/demand} - \mbox{aggregate short position/supply} = 0.
$$
This would not lead to material change in the economic implications of the model.

Comments

Consumption Smoothing
Under convexity and regularity assumptions on $U$, it is clear from the FOC that agents in this economy trade $t = 1$ claims at $t = 0$ in order to smooth their $t = 1$ consumption across states.

Sequential Trading
In this economy, there is a futures/forward market
at $t = 0$ and a spot market at $t =1$. Agents trade today subject to their budget constraint today, same for tomorrow.

Rational Expectations
However, agents correctly anticipates the $t=1$ spot price at $t = 0$. Equilibrium is self-fulfilling.

Radner vs. Arrow-Debreu
In an Arrow-Debreu equilibrium, agents trade $t=0$ state-contingent claims ("Arrow-Debreu securities"). There is no $t=1$ spot market. Trading only occurs once ex ante. In practical terms, AD securities are digital options, instead of futures contracts.
A priori, agents' choice set of consumption profiles is more restricted in the Radner economy. A futures contract is a un-contingent claim. An important result is the following:

If asset returns ${r_{sk}}$ is complete, then an Arrow-Debreu equilibrium can be implemented by a Radner equilibrium.

This is not surprising. When the market is complete, i.e. there is enough futures contracts to span the set of states, any Arrow-Debreu payoff profile can be engineered from a suitable portfolio of futures contracts.


Homogenous of degree one in utility function.

The way you show that $v(p,m)$ is homogeneous of degree one in $m$ is correct, but the reason why this implies that, $e(p,u)$ is homogeneous of degree one in $u$, is not very precise in your argument. For example, duality tells us
$$v(p,e(p,u))=u,$$
where $u$ is just a target utility level, but should not be $u(x)$ as in your proof.

Here is one possible way to proceed: Since $v(p,m)$ is homogeneous of degree one in $m$, it can be written as
$$v(p,m)=mv(p,1)=m\tilde v(p).$$
Applying the equality $v(p,e(p,u))=u$ gives
$$e(p,u)=\frac{u}{\tilde v(p)},$$
which clearly implies that $e(p,u)$ is homogeneous of degree one in $u$. You can use a similar argument to prove homogeneity of the Hicksian demand.

With all that said, I would suggest you prove the original statement directly using the definitions of expenditure function and Hicksian demand. For instance,
\begin{align*}
e(p,\lambda u)&= \min p\cdot x ~~\text{ s.t. } u(x)\geq \lambda u\\
&=\lambda\min p\cdot\frac{1}{\lambda}x ~~\text{ s.t. }\frac{1}{\lambda}u(x)\geq u\\
&=\cdots
\end{align*}


Where one can see the world economy sectors comparison by value?

The United Nations Industrial Development Organization (UNIDO) maintains a variety of databases comprising statistics of overall industrial growth, detailed data on business structure and statistics on major indicators of industrial performance by country in the historical time series. The UNIDO publishes the Industrial  Development  Report  2016 and offers some statistics.

This is one of the most reliable source of industrial data although data are still incomplete for some developing countries.

Broad Sectors: world value added over time. (Source: UNIDO)

Within Manufacturing: In  2013,  the  dominant  global  manufacturing  indus-tries were food and beverages (12.0 percent), chemicals and chemical products (11.7 percent) and machinery and equipment (8.5 percent; Table 7.2) (source: UNIDO's Industrial  Development  Report  2016)


Would someone be able to help me solve capital per capita in the steady state (check my work)

There are some mistakes in the steps you took although they went in right direction. The correct steps are below.
First we can start by expressing output in per capita terms by dividing first two equations by $L_t$:
$$ \frac{Y_t}{L_t}=\frac{K^{\alpha}_t L^{1−\alpha}_t}{L_t} \implies y_t = k_t^{\alpha}$$
and
$$\frac{Y_t}{L_t}=\frac{C_t}{L_t}+\frac{I_t}{L_t}+\frac{G_t}{L_t} = y_t  = c_t + i_t + g_t$$
where lower case letter refer to per capita variable (e.g. $\frac{Y_t}{L_t} = y_t$). Using this and the $L_{t+1}=(1+n)L_t $ the evolution of capital in per capita terms will be given by:
$$ (1 + n)k_{t+1} = (1 − \delta )k_t + i_t \implies k_{t+1} \approx (1 − n -\delta )k_t + i_t  $$
Here we can use the fact that:
$$i_t = y_t - c_t - g_t = k_t^{\alpha} - c_t - g_t$$.
Hence:
$$k_{t+1} =  (1 − n -\delta )k_t + k_t^{\alpha} - c_t - g_t $$
and as a consequence
$$k_{t+1} - k_t=  -(  n + \delta )k_t + k_t^{\alpha} - c_t - g_t $$
Now since $c_t=(1−s)(y_t-g_t)$, where ($-g_t$) is there because consumption depends only on disposable income after taxes and $g_t = \sigma y_t$ and remember $y_t= k^{\delta}_t$ we get:
$$k_{t+1} - k_t  =  -(  n + \delta )k_t + k_t^{\alpha} - ( (1−s)(k_t^{\alpha}-\sigma k_t^{\alpha})- \sigma k_t^{\alpha})  \\
 =  -(  n + \delta )k_t + k_t^{\alpha} - ( k_t^{\alpha} - s (1-\sigma) k_t^{\alpha} )  \\ =   s (1-\sigma) k_t^{\alpha}  -(  n + \delta )k_t $$
where the last equation is really just this models version of $sy-(n+\delta)k_t$.  Now finally dividing by $k_t$ to get to growth rates we get:
$$ \frac{k_{t+1} - k_t }{k_t}=  s (1-\sigma) \frac{k_t^{\alpha}}{k_t}  -(  n + \delta ) $$
Now solving the above for steady state where $\frac{k_{t+1} - k_t }{k_t}=0$ (by definition a steady state),   gives us:
$$ 0 =  s (1-\sigma) \frac{k_t^{\alpha}}{k_t}  -(  n + \delta ) $$
which finally can be solved for optimal steady state per capita capital:
$$ k^* =  \left( \frac{s(1-\sigma)}{n+\delta} \right)^{\frac{1}{1-\alpha}}$$
In addition very similar models to this one are covered by Barro & Sala-i-Martin Economic Growth 2nd ed. So you can check it out for further details.


Is it valid to assume that the international price of consumer durables is constant?

The assumption almost never holds. Not for televisions, nor pretty much any other consumer durable (nor wheat, but that's not a consumer durable anyway).

It's a simplifying assumption that's used to make a particular problem easier to solve.

But in reality, the market changes continuously; furthermore there are often oligopolistic forces at play.

The least that should be done, when making the assumption, is to include a discussion about the possible implications on ones results, when the assumption does not hold. One could also do a sensitivity test with and without the assumption; and/or cite other studies that have gone deeper into the implications of using the assumption.

Classical economics has a long, disreputable history of making big claims on the basis of invalid assumptions. It's down to us to change these bad habits.


Envelope Theorem in Hopkins and Kornienko (2010)

From the first order condition for optimality, we know that (equation (4) in the paper):

$$U_x(x,Z(r)-x,S(r))x'(r) - U_y(x,Z(r)-x,S(r))x'(r) + U_s(x,Z(r)-x,S(r))S'(r)=0.$$

When differentiating ${\bf U}(r)$ with respect to $r$, we find your expression:
$$U_x(x(r),Z(r)-x(r),S(r))x'(r) + U_y(x(r),Z(r)-x(r),S(r))(Z'(r)-x'(r)) + U_s(x(r),Z(r)-x(r),S(r))S'(r).$$

Replacing the first equation into the last one yields
$${\bf U}'(r)=U_y(x(r),Z(r)-x(r),S(r))Z'(r).$$

Now, $Z'(r)=1/g(Z(r))$ which gives equation (8).


The solution concept of Bayes Correlated Equilibrium applies to games, viz strategic interactions, between multiple players. Thus, in a single person decision problem its use seems to me inappropriate or at least superfluous. A lot of effort has been spent over the last 70 years (dating back at least to Blackwell 1951, 1953) to explore the notion of information and information structures in decision problems. In that literature lies your answer.

What you have described is a search problem that seems analogous to a Wald-like sequential sampling problem.

Perhaps refer also to the recently burgeoning literature on rational inattention. There, care has been taken in order to explore so-called "posterior separable" cost (of information) functions, which allows one to write the cost of information just as a function of the induced posteriors. See recent work by Caplin, Dean, and others.


Creation and distribution of money from scratch in a country (USA as example)

The way in which colonial governments forced indigenous tribes to use money was to impose a tax that had to be paid in money. For a textual reference, this is described in “Understanding Modern Money: A Key to Full Employment and Price Stability” by L. Randall Wray. (This concept is discussed quite often in Modern Monetary Theory, and articles are probably easily found using a web search.)

The way the locals could get money is via providing labour at a fixed wage (or goods at a fixed price). They could trade amongst themselves to allow those with a surplus of money to distribute it to those who did not get money this way. This has the effect of forcing monetary transactions into the economy.

As was typical for colonial powers, policing was brutal to ensure that the taxes would be obeyed.


How is trade in services conducted under the WTO in the absence of the Doha conclusion?

It's definitely not prohibited; the US-EU trade in services is worth some $200 billion (each way).


What's holding the U.S. from decreasing unemployment by cutting H1B limits?

The economy is not a zero sum game with a fixed number of jobs available. If it were, then restricting immigration or at least the access immigrants have to jobs would unambiguously benefit domestic workers. However, H1B immigrants, in particular, are frequently skilled workers that increase the productivity of the companies they work for, creating additional job opportunities, some of which are filled by domestic workers.
The direct effect of H1B workers is to drive down the wages and opportunities for domestic workers they compete with directly, but the indirect effect is the creation of complementary jobs, which may be skilled or unskilled. Similarly, increased immigration by low-skill workers drives down wages of competing low-skill domestic workers but increases the productivity of the economy, creating jobs (and goods/services) for skilled and unskilled workers.
A flow of new labor will benefit some domestic workers and harm others, irrespective of the current state of unemployment. Immigration is a political decision, and like other political decisions is made with an eye to who benefits from it and how influential they are on the political process, rather than a more abstract notion of what is good for "the country" or "unemployment." In other words, since some domestic workers are harmed by immigration and others are helped, you must define "good" before you can say whether H1B or any other type of immigration is good for the economy or its workers. The definition politicians use is "good for re-election," and since they have different constituencies, they disagree with each other on whether H1B or any immigration is good.


Interesting question,
It can be dealt with with two models: economic theory and game theory (which, in turn, is the base for modern economic theory)

Game theory should be the most practical.
To start if we simplify, we can assume that "everybody else" is one player
In a turn based game, each provides his own minimal bid as long its the highest bid,
That means that the response of each player, will be higher bid, until the maximal value will be reach, for the player with a maximal rate

So the end-game for each player, will the other's maximal bid + delta

Notice that the optimum has to arrive by playing and responding
With an endless amount of players, endless amount of games,
The market price will be the market price: The maximal price will be reached, based on the available supply.
Once that sets, market price is a given known number, based on which a consumer will decide whether to buy or not = economic equilibrium...

So, to answer your question: you can and should play and respond to others' prices, but close enough to market prices (and of course, based on the unique parameters for each flat), simply to save the time and trouble

hope that helps,
Guy

Game theory game tree


Deriving the translog production function

The idea is indeed to Taylor expand the production function. To justify it, you can start with the constant elasticity of substitution function, which in the two-factor case can be written as

$$
Y = A[\alpha K^\gamma + (1 - \alpha)L^\gamma]^{1/\gamma} \tag{1}
$$

in this case $X_1 = K$, $X_2 = L$. Now we expand $\ln Y$ around $\gamma = 0$ (recall the CES approximates to a cobb-douglas production function when $\gamma \approx0).$

  $\gamma^0$ term

$$
\lim_{\gamma \to 0} \ln Y = \ln (A K^\alpha L^{1 - \alpha}) \tag{2}
$$

  $\gamma^1$ term

\begin{eqnarray}
\lim_{\gamma \to 0} \frac{\partial \ln Y}{\partial \gamma} &=& \lim_{\gamma \to 0}\frac{\alpha  K^{\gamma } \ln (L)+(1-\alpha ) l^{\gamma } \log (L)}{\gamma
   \left(\alpha  K^{\gamma }+(1-\alpha ) L^{\gamma }\right)}-\frac{\ln \left(\alpha
   K^{\gamma }+(1-\alpha ) L^{\gamma }\right)}{\gamma ^2}\\
&=& \frac{1}{2} (1 - \alpha) \alpha  (\ln (K)-\ln (L))^2 \tag{3}
\end{eqnarray}

Up to first order we have then

\begin{eqnarray}
\ln Y &\approx& \color{blue}{(\ln Y)_{\gamma = 0}} + \color{red}{\left(\frac{\partial \ln Y}{\partial \gamma}\right)_{\gamma = 0} \gamma} \\
&\stackrel{(2),(3)}{=}& \color{blue}{\ln A + \alpha \ln K + (1 - \alpha) \ln L} + \color{red}{\frac{1}{2}\alpha(1 - \alpha)\gamma [\ln K - \ln L]^2} \\
&=& \ln A + \alpha \ln X_1 + (1 - \alpha) \ln X_2 + \frac{\alpha \gamma (1 -\alpha)}{2}\left[\ln^2 X_1 -2\ln X_1\ln X_2 + \ln^2 X_2 \right] \\
&=& \alpha_0 + \sum_{i = 1}^2 \alpha_i \ln X_i + \frac{1}{2}\sum_{i, j = 1}^2 \beta_{ij}\ln X_i \ln X_j \tag{3}
\end{eqnarray}

This is naturally extended to $n > 2$ as

$$
\ln Y = \alpha_0 + \sum_{i = 1}^n \alpha_i \ln X_i + \frac{1}{2}\sum_{i, j = 1}^n \beta_{ij}\ln X_i \ln X_j
$$


Equilibrium with Externalities: Solving without FOC

It is my impression that there is an inconsistency here.

If firms want to maximize $V(n)$ which is some "surplus revenue over costs" function, then surplus they will produce. Meaning, that this surplus/profits cannot be given to workers, in the form of wages, because wages are cost. It may be given as dividends or, say, end-year bonus, but in any case it won't be related to $n$.
So we do not have $Y=wn$ but rather $Y = wn + d$, and the externality appears to break down.

The $Y=wn$ relation is consistent with a "non-profit" organization, which would demand as much $n$ as needed in order to have zero profits. In such a case, the wage would be equal to the average product of labor, not the marginal one.


Does elasticity make sense for different levels of prices? If not, how can we fix that?

The standard elasticity still makes sense. Elasticity is not necessarily constant. Also, I am not sure what sort of formula you learned at your college but elasticity is rigorously defined (and also taught at college level for econ majors) as $EL = \frac{df(x)}{dx}\frac{x}{f(x)}$ for some function $f(x)$ (see Essential Mathematics for Economic Analysis by Sydsæter, Hammond & StrØm - which is an undergraduate textbook used at many econ departments).
For example, for demand function given by: $$Q= a - bp$$ the elasticity will be given by:
$$\epsilon = \frac{dQ}{dp}\frac{p}{Q}= -b \frac{p}{a-bp}$$
so for $a=100$ and $b=10$ it will be:
$$\epsilon = - \frac{p}{100 - 10p}$$
So for example at price 5  dollars the elasticity (in its absolute value) will be 0.1 at price 9 dollars elasticity will be 0.9 and so on. Hence, elasticity can change when prices change and you don't need any new concepts. There is a special type of demand functions where elasticity will always be constant but then there are infinite number of possible demand functions and in most of them it will change with price.


How are purchasing power parity (PPP) conversions computed?

The calculation of PPP is a very complex, resource consuming task. As you can imagine, coming up with a single number comparing countries relative purchasing power means many assumptions have to be taken (i.e there is a trade-off between realism and simplicity).
The most comprehensive documentation relating the methodology behind PPP can be found in this World Bank website. Here I just give a brief idea of what's going on. Check the documentation for further clarity (particularly Chapter 4).
The calculation of PPP goes in stages. First, data on prices of individual products within a basic heading (a macro category, e.g. food) is computed. From here, relative prices for each product across two countries are computed, and aggregated at the basic heading level using a geometric mean. An example is given in Table 1 of Chapter 4, copied below:

Then, an important issue is product choice. Comparability depends on products being relatively homogeneous in both countries, so that table above can be filled in completely. This requires an adjustment for product quality when necessary (e.g. mobile phones), part of which is adjusted using hedonic regression (see page 15+ of Chapter 4). Additionally, products must be representative on each country, simply defined as (see point 7 in Chapter 11 for a technical definition):

A representative product is one that is purchased in relatively large quantities in a country.

Several mechanism are used to adjust for imbalances among these two factors (including weights).
The list of problems continues. Which price to include? As the documentation states:

The price at which any single good or service is purchased is liable to vary considerably during the course of a year and also between different locations, especially in large countries. In these circumstances, the PPPs for individual products have to be defined as ratios of average prices.

Finally, Chapter 8 deals with Government services like health and education, There is a differentiation between individual services (household consumed) and collective services (public goods). Most of education and health as services provided to the individual is the former, whereas others like R&D in education and health is the latter. See Box 1 in Chapter 8 for a complete list.
The valuation of these services is an issue. A quote in extenso might help here:

Most of the individual and collective services produced by government are non-market services.   They are either provided free to individual households or the community as a whole or they are sold at artificially low prices that do not reflect their costs of production.   As there are no market prices the value of the government’s current expenditures on non-market services has to be estimated in the national accounts by the current value of the inputs used to produce them.  Similarly, as there are no market prices the PPPs also have to be estimated indirectly using other data.  The same problem arises when trying to measure changes in the prices of government services over time within a single country.
The solution usually adopted in both international and intertemporal price and volume indices is to compare the prices and volumes of the inputs used in the production of non-market services instead of the prices and volumes of the outputs.  When the PPPs compare input prices the resulting expenditure data when converted into a common currency compare the volumes of the inputs used up in the production of non-market services in different countries.   Input volume indices inputs can provide close approximations to output volume indices but only if the ratios of outputs to inputs, i.e., productivity levels, are the same in different countries.    This may often not be case.   PPPs based on input prices are therefore second best measures that are necessitated by the lack of output prices

See the chapter for more details.
In conclusion, there is an attempt to provide homogeneity across goods and services. But there are many reasons why perfect comparability might not be achieved. In the case of government services, if inputs do not reflect outputs homogeneously across countries (partly due to productivity differences, as highlighted above), this will affect PPPs.


How does the russian interest rate raise will defends the Ruble from devaluation?

An interest rate is the return that a lender earns on money lent to someone else. If the interest rate is higher then it makes lending money more attractive because the return is higher.

In particular, if the interest rate in, say, Russia increases relative to that in the USA then American lenders will switch from lending their money in America to lending their money in Russia where the return is greater. But in order to lend money in Russia, these investors are first going to have to convert their dollars into roubles. This increased demand for roubles causes the price of the rouble (i.e. the exchange rate) to increase.


What's the economic effect of a country reducing its imports?

The short unqualified answer is 'it depends'. Are we talking about a 'small' or a 'big' economy? Is that economy 'wide' open or sealed tight to factor movements? Does it issue its own currency? What's the exchange rate regime it uses? Furthermore, what are its developmental goals, in general? The answers are going to be different, depending on what kind of countries we are talking about eg US vs Belgium, or Singapore, to take two extreme cases.

In general, directing consumers towards domestically produced commodities would increase the demand for local commodities at the expense of imported ones as long as there are local substitutes for foreign imports. It seems plausible to substitute imported beer for domestic wine. The catch is that the domestically produced substitutes should actually be price competitive to foreign alternatives otherwise no amount of 'national pride' can compensate for a price tag eg twice as high (I'm assuming that quotas, tariffs and the like are not sought for).

As far as the 'training program' is concerned, it could be a more all-encompassing venture that extended beyond cars. The 'training' could affect the total of local manufacturing and durables' production. Such a substitution of 'old' for 'new' would probably reduce manufactured imports, industrial machinery imports etc at a significant cost of maintaining and operating an aging infrastructure.

Practically that means higher costs for operation and maintenance and higher prices for the finished products (although that could be offset with appropriate industrial policies-in a hypothetical scenario, using appropriate subsidies etc). It could also mean an increase in imports of spare parts because old equipment tends to break down more often.

The major drawback would be the impact on the distance of the domestic economy from the technological frontier. To the extent that technological innovations are embedded in new equipment and machinery (and tools and parts), 'shutting down' imports implies skewing the age-structure of the domestic capital stock to less productive older technologies (such an occasion might not be necessarily as bad as it sounds as long as the consumption structure is similarly skewed).

Finally, when it comes to overhauling public transport, that is something that would require significant investment. Usually, such 'efficient' technologies are more likely to be 'modern' than 'mature' ones hence this consideration introduces the reservations related with the previous point regarding the age-structure of the domestic capital stock and the distance from the technological frontier. Nevertheless, the significant cost of such an investment could be-in principle-justified by an appropriate investment horizon.

Also, without playing down the costs of oil imports, the price of oil is really not where it used to be and supposedly with the advent of fracking that is going to continue indefinitely. Oil is generally related to energy production, so having a small energy footprint would be more effective in curbing energy needs and related imports.

Where do all that leave us?

Directing resources from tradables to non-tradables will change relative prices. Investing in public transportation infrastructure will have a similar effect of what constitutes productive investment.

A plausible scenario is that resources will be pulled away from the tradables sector (exports) and will be directed to the non-tradables sector (which will now be more profitable that before). That will decrease the output of the tradablers sector. Also, the product mix of exports will be skewed to older technologies (courtesy of the 'training' policy). That would imply decreased price competitiveness in the international market. Such an occasion by itself could imply a deterioration of the trade deficit which is arguably the opposite of the intended result of the proposed reforms. Also, incomes might increase courtesy of the multiplier effect (assume that the non-tradables sector is labor intensive). That could potentially introduce another snag, to the extent that the income elasticity of imports is significantly large. Although, the 'buy local' campaign could potentially be effective in this regard.


Central Bank Balance Sheets and foreign bonds

It happens in reality, pretty much as described in the book, not far from what you wrote

Say that the fed would like to buy UK bonds. Since UK bonds are issued in pounds, in order to buy them, the Fed will have to 1st buy pounds, in exchange for dollars, which didn't previously exists.
In this case, more dollars will enter circulation, and the money supply will rise.
In parallel, the dollar exchange rate will increase against the pound, that is the dollar will weaken.

Its the same logic that  domestic bonds: buying domestic government bonds, in exchange for previously non-existing dollars, would increase money supply.

In today's free capital markets, there isn't any essential difference between the two, because of arbitrage:
In the case of buying foreign bonds, and rising pound rate, euro holders, for example, for them rates remained unchanged , will buy dollars to sell pounds, until a new euro rate will be set.

Similarly, a local increase in inflation, due to purchasing domestic bonds, will effect exchange rates, as local currency value would fall.


Am i correct here? #greenTaxes #priceMechanism #inefficientTax #DeadweightLoss

Yes you are correct, provided of course that plastic bags have no external effects. Then that triangle is the deadweight loss.

However, green taxes are typically implemented for a reason and if plastic bags have an external effect the supply curve is not equal to the social marginal costs. The idea of the taxes is to bring the private marginal costs in line with the social marginal costs. If plastic bags do have an external effect than that triangle is not deadweight loss but part of a cost that society bears.


If banks create money by lending it out, how are they taking any risk?

Try the magic of double entry bookkeeping to see that creating money does not immediately make a bank more valuable. Initially the loan is matched by a deposit, and this is creation of what is effectively money:

Bank A has an asset: a newly created loan to customer B
Bank A has a liability:  B's newly created deposit with Bank A, which B can spend

B then spends the amount with C, who banks with D, and the deposit is transferred on B's instruction to C's account. Now

Bank A has an asset: a loan to B
Bank A has a liability:  Bank A owes money to Bank D
Bank D has an asset: Bank A owes money to Bank D
Bank D has a liability:  C's deposit with Bank D

Bank A now needs to settle with Bank D: this can happen through clearing if other customers are moving money to Bank A, but it is easier to imagine that Bank A tells the central bank to give some of A's reserve deposits there to Bank D. So Bank A has an asset, namely a loan to B, but this has been offset by an increase in liabilities or a reduction in other assets.  Similarly Bank D has an increase in assets, offset by a liability to C. The created money is now C's deposit with D, which C can spend.

Suppose the loan to B then goes bad and is never likely to be repaid.  From Bank A's point of view, the value of that asset falls to zero, while the associated increase in liabilities or reduction in other assets remains, and Bank A has to take this as a loss (negative profit), with the side effect of destroying that amount of money.  Bank A is now worth less (net) than when it started the loan to B.  This is the risk for Bank A from creating new financial assets and liabilities (i.e. new money).

If instead nothing goes wrong, B makes some money selling to E, pays A some interest, and possibly repays the loan (in which case the total amount of money in the economy reduces again).  In this case Bank A is more valuable (net), not directly from the creation of the loan or from its repayment or from use or return of the corresponding deposit, but from receiving the interest on the loan.

When Bank A makes too many loans, it may run out of reserve deposits at the central bank after its customers spend the new money.  The solution to this is to take deposits from other customers, as Bank D does in the example above.  But these deposits may need interest to attract them, at a cost to the bank.

So Bank A's overall profit on loans and deposits is the difference between the interest it receives from the loans it makes and the interest it pays on deposits it takes, allowing for the cost of bad loans; it may also make money from fees and have to pay for staff and other expenses.  The creation and destruction of money by making the initial loan and it being repaid is not Bank A's particular concern, though it is for the macroeconomic policy of the central bank and the government; if the economy grows and more loans are made, then the money supply also rises and the hope it that will happen neither too quickly nor too slowly.  Ultimately Bank A cares about the risk from bad loans, the interest rate margin and its administrative costs.


I will focus on academic literature reviews since they tend to be unbiased (or at least less biased) - academic literature reviews are often accessible even to non-specialists as they usually do not go deep into models as they try to provide broad overview of the topic:
Good (and highly cited) literature review on this topic are:
Neumark, D., & Wascher, W. (2006). Minimum wages and employment: A review of evidence from the new minimum wage research (No. w12663). National Bureau of Economic Research.
Card, D., & Krueger, A. B. (2015). Myth and Measurement: The New Economics of the Minimum Wage-Twentieth-Anniversary Edition. Princeton University Press.
Neumark, D., Wascher, W. L., & Wascher, W. L. (2008). Minimum wages. MIT press.
Manning, A. (2020). The elusive employment effect of the minimum wage. Journal of Economic Perspectives.
Doucouliagos, H., & Stanley, T. D. (2009). Publication selection bias in minimum‐wage research? A meta‐regression analysis. British Journal of Industrial Relations, 47(2), 406-428.
Neumark, D., Salas, J. I., & Wascher, W. (2014). More on recent evidence on the effects of minimum wages in the United States. IZA Journal of Labor policy, 3(1), 24
Maloney, W. F., & Nunez, J. (2000). Measuring the impact of minimum wages: Evidence from Latin America. The World Bank
However, note there is no full consensus in economics on minimum wages. For example, have a look at (this recent pol among the best US policy economists on effects of increases in the minimum wage on us employment) and you will find that most policy economist are uncertain, and the 'pro' and 'against' camp are almost equally split.
Consequently, you will see that some of the papers/books above are more cordial and some more critical of minimum wages - this does not mean the researchers are biased. Sometimes it is very hard to read empirical evidence and to reach some 'definitive' conclusion. For example, one reason why there is so much controversy is that in principle you should expect even relatively large increases in minimum wage to have smaller effects where original minimum wage was far below median wage and different ones when it is closer to median wage. This can often lead to situations where research in different places might yield contradictory evidence.
In such cases people will disagree and it is very hard not to tilt toward one side of an argument or the other but that is not necessarily evidence for  bias per se as people can emphasize different things. In the above list I strived to strike some balance among those economists who are less and more cordial toward minimum wage (although I think both sides of the debate try to be unbiased) so together it should make a balanced overview. Lastly, the above is just tip of an iceberg minimum wage is probably one of the most extensively researched economic policies but they should help you to orient yourself in the literature and also point toward solid primary research on the issue.


Who was the first economist to write about central banking?

Commercial banks have existed long before any institution akin to a modern central bank was established (the first "central bank" seems to have been the Bank of Sweden, created in 1668, albeit this was a for-profit, private institution). Because of the important role of banking in financing trade, wars, and the recurrence of crises, inflation, and convertibility problems, many politicians, economists, and other individuals wrote about issues related to money theory and banking. For instance, two prominent books about monetary theory are John Law (main book in 1705) and Fernando Galiani (main book in 1751). Later on, both Adam Smith and David Hume wrote about banking. According to this source Adam Smith argued that:

the banking system is guided by the invisible hand just like any other branch of  industry.  Smith  argued  that  the  profit  motive  would  ensure  that  the  supply  of  money is self-regulating even after the development of privately issued banknotes, and that  a  significant  savings  in  social  capital  results  from  using  paper  instead  of  precious  metals   as   circulating   currency.   In   this   way   Smith   attempted   to   show   that   free   competition   in   the   business   of   issuing   currency   directs   self-interested   market   participants to unintentionally promote the social interest.

In sharp contrast, David Hume opposed for-profit, fractional reserve banking. According to this article:

Hume claims the following about the kind of bank he prefers: ‘‘No bank could be more advantageous, than such a one as locked up all the money it received, and never augmented the circulating  coin,  as  is  usual,  by  returning  part  of  its  treasure  into  commerce’’  ([1752]  1985, pp. 284–5). Hume recognizes that a bank that does not lend cannot stay in business by itself: ‘‘It would have no profit from its dealing’’ ([1752] 1985, p. 285). Therefore, rather than allowing the bank  to  earn  a  profit  by  lending  out  money  or  by  charging  fees,  Hume  suggests  a  purely
subsidized public bank, where ‘‘the state bore the charges of salaries of the directors and tellers of this bank’’ ([1752] 1985, p. 285)

A bit later, amid the Bullionist controversy (regarding whether paper money should be convertible to gold or not), a myriad of writings appeared (see link). For example, in 1801, Walter Boyd wrote a letter to the primer minister William Pitt advocating for full convertibility, in order to avoid high inflation associated with private banks incentive to expand paper money under limited convertibility.
A notorious economist at that time was Henry Thornton, who wrote in 1802 the book "An Enquiry into the Nature and Effects of the Paper Credit of Great Britain", partly in response to Boyd's views. Apparently, some (?) consider Thornton as the founder of central banking.
Later on, Ricardo also wrote about central banking. In particular, he argued in favour of central bank independence (yet, remaining a for-profit institution). This is a quote from 1824 (emphasis mine):

It is said that Government could not be safely entrusted with the power of issuing paper money; that it would most certainly abuse it ... There would, I confess, be great danger of this if Government – that is to say, the Ministers – were themselves to be entrusted with the power of issuing paper the hands of Commissioners, not removable from their official situation but by a vote of one or both Houses of Parliament. I propose also to prevent all intercourse between these Commissioners and Ministers, by forbidding any species of money transactions between them.
The Commissioners should never, on any pretense, lend money to Government, nor in the slightest degree be under its control or influence ... If Government wanted money, it should be obliged to raise it in the legitimate way; by taxing the people; by the issue and sale of exchequer bills; by funded loans; or by borrowing from any of the numerous banks which might exist in the country; but in no case should it be allowed to borrow from those who have the power of creating money.

(The quote is from here, but it belongs to Ricardo's 1824 pamphlet "Plan for the Establishment of a National Bank", which can also be found as an appendix on this book)
Meanwhile, others consider the "founding father" of central banking to be Walter Bagehot, a "famous" editor of The Economist (you might have noticed that there is still a column under his name) that in 1873 published the book "Lombard Street: A Description of the Money Market", where he advocated for the role of the Bank of England (still a private, for-profit institution) as a lender of last resort in the case of a financial crisis. Despite of this, he also advocated for a free banking sector. A quote (from here):

I shall have failed in my purpose if I have not proved that the system of entrusting all our reserve to a single board, like that of the Bank directors, is very anomalous; that it is very dangerous; that its bad consequences, though much felt, have not been fully seen; that they have been obscured by traditional arguments and hidden in the dust of ancient controversies.
But it will be said — What would be better? What other system could there be? We are so accustomed to a system of banking, dependent for its cardinal function on a single bank, that we can hardly conceive of any other. But the natural system — that which would have sprung up if Government had let banking alone — is that of many banks of equal or not altogether unequal size. In all other trades competition brings the traders to a rough approximate equality. In cotton spinning, no single firm far and permanently outstrips the others. There is no tendency to a monarchy in the cotton world; nor, where banking has been left free, is there any tendency to a monarchy in banking either.

For further references, check the book "Monetary Theory and Policy from Hume and Smith to Wicksell: Money, Credit, and the Economy".


Cartel profit maximizing quantity question (Cournot game)

Hint

By colluding, presumably the two firms would like to figure out the least costly way to produce in order to maximize profit. Treat the two firms as two production plants of a single "parent company". The total cost of the cartel can then be found by solving
\begin{equation}
C_\text{cartel}(Q)=\min_{q_1,q_2}\;[C_1(q_1)+C_2(q_2)],\quad\text{s.t.}\quad q_1+q_2=Q.
\end{equation}
This will give you a the optimal way of distributing production across the two firms, $q_i^*(Q)$, as a function of aggregate output $Q$. Then the cartel's total cost is just the sum of the member firms' costs evaluated at the optimum: $C_\text{cartel}(Q)=\sum_i C_i(q_i^*(Q))$.


How to take the derivative and show that MPL in the Solow model is equal to real wage w

They used a product rule, quotient rule and chain rule, I won't be discussing the rules themselves  as such topics belong to Mathematics.SE, but the reason why you might not realize this outright is the particular notation in a growth theory, which would be on topic here so I will focus on that.
In Solow growth model with labor augmenting technological change $f(k)$ is the production function expressed per 'effective worker' so:
$$f(k) = F \left(\frac{K}{AL},1 \right)$$
so you might as well rewrite the problem as:
$$d\left(AL F \left(\frac{K}{AL},1 \right) \right)/dL=A F \left(\frac{K}{AL},1 \right) +AL F '\left(\frac{K}{AL},1 \right) (−AK)/(LA)^2$$
Now we can cancel some terms and use the fact that $k =K/AL$:
$$A F \left(\frac{K}{AL},1 \right) + F' \left(\frac{K}{AL},1 \right) (−K)/(AL)=A F \left(\frac{K}{AL},1 \right)  -A k F' \left(\frac{K}{AL},1 \right)  $$
Now finally go replace the $ F \left(\frac{K}{AL},1 \right)$ back with $f(k)$ so you get:
$$A F \left(\frac{K}{AL},1 \right) - Ak F' \left(\frac{K}{AL},1 \right) =A f(k)  -Ak f'(k) = A(f(k)  -k f'(k)) $$
So it is all about paying attention to what different definition of the arguments used here are. If you can't follow the differentiation itself that is question for Mathematics.SE or just further self study (Essential Mathematics for Economic Analysis by Sydsaeter covers all you need for this sort of problem).


Looking for discussion on equilibrium vs dynamic models in econometrics

Economists (most of them) build their models assuming most of the time stochastic dynamic equilibrium. So Economics does not contrast "dynamic" with "equilibrium" - it synthesizes them.

It is stochastic in the sense that random shocks are acknowledged. It is dynamic in the sense that it may revolve around a deterministic or stochastic trend. And it is an equilibrium because, exactly, it is assumed "chained" -not necessarily to some level but at least to some trend.

The "bias induced" here, the price to pay, is that mainstream economic models are not very good at predicting derailment and crises. On this, perhaps it would be useful to read this column article by well-known economist Robert Lucas.


Price level and purchasing power of currency(PPP)

Prices in themselves don't mean much when you want to predict purchasing power.
Imagine a loaf of bread costs 100 of country A's monetary units in country A, while it costs 2 of country B's monetary units in country B. This does not tell you anything about purchasing power because you do not know if you could buy a loaf of bread for 100 of country A's monetary units in country B or vice versa.
You also need to consider the currency A to currency B exchange rate, usually denoted by $E_{B/A}$. Then purchasing power/real exchange rate is calculated by
$$
\frac{P_A}{P_B} \cdot E_{B/A} ,
$$
where $P_A$ and $P_B$ are price levels in countries A and B respectively. If this is larger than 1, then using country A's currency one could buy more goods in country B than in country A, while if it is smaller than 1 then the reverse is true.


Why does immigration boost the economy? (or does it not...?)

You’re right, the migration of workers boost the economy. It’s a stylized fact. It may have political consequences, though, but it is not the part of the analysis of economics. Let’s consider $A$ the set of active labour force of an economy before, and $A'$ after the migration. Then $A$ is the subset of $A'$:
$$A \subseteq A'$$
This notation means that the social planner of the economy is able to optimize on a greater number of workers. We can assume that opening the door for migrant workers is at least as good as it was before migration:
$$A \preceq A'$$
If this condition is not true (e.g. the migrants doesn't want to work), the authorities of the economy won't open borders for migrant workers.

You're considering the effect when the nation itself does not have full employment. This condition is not important. The nation could have 0% unemployment, however, the migrant workers would still get jobs.

  Immigrants from developing nations tend to be happier to take on unattractive jobs, which citizens of the developed country may refuse
  to do.

Absolutely, these people have got different preferences. Here're the utility functions of residents ($r$) and immigrants (i):
$$u_r=C-\alpha_r L$$
$$u_i=C-\alpha_i L$$
where C and L denotes consumption and work. We can conclude that $\alpha_r$ must be higher than $\alpha_i$. The disutility of work is higher in the case of the residents.

  That means migrant workers are more cost-efficient for firms. Higher immigration creates more competition for jobs, causing wages to
  decrease and allowing companies to pay their employees less.

Therefore, the nation's companies make more money, and this causes the economy to grow. Growth and R&D may also improve by means of brain-drain effect, because there are migrant scientists and white-collar workers, too.

  The 5% unemployment rate is actually due to people who are sick /
  disabled / otherwise unable to work.

By definition it is not correct. In Economics, active workers and job-searchers are on the labour market only.

The main concept of migration being good is the increase of the horizon of labour optimization. See also: Krugman & Obstfeld: International Economics Theory and Policy


Solution of overdetermined system of moment conditions

Here's a little discussion that might help.

The population version has a solution because it is assumed to have a solution. You can see that this is the case from assumptions 3.1 (linearity), 3.3 (orthogonality conditions), and 3.4 (rank condition for identification). You can see the derivation of this fact from equations (3.3.3) and (3.3.4).

Now, you'll notice the population version having a solution implies that
$$
\text{rank}\left( \Sigma_{xz} \right )
= \text{rank}\left(\left [\Sigma_{xz} \mid \sigma_{xy} \right ]\right ),
$$
as discussed in exercise 5 of section 3.3. In the hint they note that the population version has a solution when
$$
\left [\Sigma_{xz} \mid \sigma_{xy} \right ]
$$
is of rank $L$. If it has rank $L+1$, it will not have a solution. In the hint, they say that this is a set of "equality conditions." This is referring to the fact that one of the columns must be a linear combination of the remaining columns. This is a very hard condition to satisfy. In many models, this condition will hold with probability zero. This is because the data has to come out so that the equality condition holds perfectly.

On the other hand, the condition that the sample version $S_{xy}$ of $\sum_{xz}$ be of full column rank is a set of "inequality" conditions on the matrix. Because they are inequality conditions on the matrix, they are much easier to satisfy by chance.

In conclusion, $S_{xy}$ will have full column rank for sufficiently large $n$ because the conditions are inequality conditions which occur with probability larger than zero. However, to have a solution we need $\left [S_{xz} \mid s_{xy} \right ]$ to have rank $L$. But because this is an equality condition on the matrix, it only occurs with probability zero.

Search Keywords: Solutions, Exercise 3, section 3.4 Hayashi Econometrics


Normalizing comparisons of corporations and countries

GDP is a flow (of goods and services) while market capitalization is a stock measure. So I agree, this is not a great comparison. If we want to compare stocks, we'd compare total enterprise value (sum of equity and debt) to national domestic assets. But that's imperfect because much of what makes a country rich is labor income and that's not going to be in domestic assets. So we'd like to add in some measure of human capital. That's going to be imperfect at best. So I prefer to compare flows.

Consider how GDP is a value added measure. If you import steel for 150 and export cars you make out of that steel for 250 the GDP contribution is 100. Some of that 100 went to capital and some to labor. What's the value added of the corporation? Think of the company as "importing" the goods it buys and "exporting" the goods it sells. The difference is paid out to three sources, equity capital, debt capital, and and wages paid to labor. So to make a number comparable to GDP, add together corporate profits, net income expense, and labor costs. This is the value added of the firm and much more analogous to GDP than firm value.


Does the US Bureau of Economic Analysis use the generic growth formula to calculate percent change of GDP?

Normally growth is calculated either as:

$$\frac{y_t-y_{t-1}}{y_t}$$

Or approximated by:

$$\ln(y_t)-\ln(y_{t-1})$$

Within year you can either continue using the  above definition or use year on year growth which compares every month quarter with the same quarter previous year.

If you are looking for Fred specifically here is the list of formulas:

https://fredhelp.stlouisfed.org/fred/data/understanding-the-data/formulas-calculate-growth-rates/

The list overlaps with what I written above and also the list shows other changes (which you might be interested to look into)


Sunk cost fallacy- is it a bug or a feature?

Very interesting question.
To answer the direct question in the title, I'd say I'd have to break the sunk cost fallacy into two arenas. In the first I'd say it is a feature not a bug, but in the second, vice versa.

As a means to keep a business from dying before it's been given opportunity to profit.

I think that the sunk cost fallacy is still a double edged sword, I think that it may serve to keep businesses alive longer than they should, but I also believe that it at times keeps businesses alive long enough to see a return. Most small business owners I know, successful or not, have fallen victim to the underlying logic- for good or bad. In that way I do not consider it bad or good, I just see it as a feature of the economy, especially local economies.

As a flaw in the human psyche taken advantage of by MLM schemes, Casinos, numbers games and similar enterprises.

I believe that in these cases, it becomes a bug, not a feature. A terrible flaw that extracts great amounts of wealth from those of us who are vulnerable, intoxicated (not necessarily by narcotics, but by grand promises to we who are not in a secure position), and caught off guard. That's not to take responsibility away from our choices. I just think that most people, after being had as a result of falling into the trap of this logic, feel angry at themselves not in the way that Bismark might have felt after making a strategic error (which is what I have seen from business owners), but closer to the way that Heracles felt after murdering his own children. I stayed at the Cosmopolitan in Vegas recently, and noted that my room was one of the few with balconies that I could see anywhere on the strip. W Therefew balcony rooms in Vegas for a reason. Just my raw thoughts on the subject and are mostly anecdotal, this is the first time I've considered the sunken cost fallacy in this light.