What are some good books, explaining why the society and economy developed like it is through the history?

I recommend Jared Diamond's Guns, Germs and Steel, for this topic. I don't know what you mean by "the society." Diamond makes it clear that there are many different societies around the world, progressing at different rates. To answer your question about why we have what we have today and why individual societies developed in the direction they did: some technologies are inevitable, such as farming, writing and mettalurgy. What differs is that a few societies developed these technologies much earlier than others. Diamond lays out the reasons why; differences in intelligence across races is not a reason but the geology and biogeography of people's habitats are ultimate causual factors. These ultimate factors lead to more proximate factors such as animal and plant domestication, leading to denser and more sedentary populations with more complex political organization. Technological development led to further economic development, with most technology arising from it being spread from other societies—and that transmission depends again on geography. Eurasia had the advantage of an east-west axis that made transfer of domestic food and animal species easier (along the same lines of latitude and hence similar climes), and thus the technology which accompanies more sedentary societies. More isolated societies such as Australia did not benefit from the spread of technology, and hence remained a hunting-gathering society until European conquest.

His book is entertaining and interesting to read. I've learnt facts and a perspective of the differences amongst races that I couldn't do without.


In difference-in-difference approach, would it be possible to add a dummy variable indicating whether a data point belong to treatment group?

It depends on the estimated model. Let me use the example provided in Mostly Harmless Econmetrics.

The experiment

Suppose we are interested in the effect of the minimum wage on employment. Card and Krueger (1994) use a dramatic change in the New Jersey state minimum wage. On April 1, 1992, New Jersey raised the state minimum from \$4.25 to \$5.05. Card and Krueger collected data on employment at fast food restaurants in New Jersey in February 1992 and again in November 1992. Fast food restaurants are big minimum-wage employers. They also collected data from the same type of restaurants in eastern Pennsylvania, just across the Delaware river. The minimum wage in Pennsylvania stayed at \$4.25 throughout this period.

Using their data, they computed diferences-in-diferences estimates of the effects of the New Jersey minimum wage increase. That is, they compared the change in employment in New Jersey to the change in employment in Pennsylvania around the time New Jersey raised its minimum.

The Fixed Effects Model

Differences-in-differences (DD) is a version of fixed-effects estimation using aggregate data. To see this, let

$y_{1ist} =$ fast food employment at restaurant i and period t if there is a high state minimum wage.
$y_{0ist} =$ fast food employment at restaurant i and period t if there is a low state minimum wage.

Let $D_{st}$ be a dummy for high-minimum-wage states, where states are index by $s$ and observed in period $t$. In the absence of a minimum wage change, employment is determined by the sum of a time-invariant state effect ($\gamma_s$) and a year effect ($\lambda_t$) that is common across states. Assuming that $E[y_{1ist} - y_{0ist}|s,t)$ is a constant, denoted $\delta$, we have:

$$Y_{ist}= \gamma_s + \lambda_t + \delta D_{st} + \epsilon_{ist} ~~~~~~ (1) $$
(I corrected $\lambda_s$ here.)

The Regression Model

We can use regression to estimate equations like (1). Let $NJ_s$ be a dummy for restaurants in New Jersey and $d_t$ be a time-dummy that switches on for observations obtained in November (i.e., after the minimum wage change). Then

$$Y_{ist}= \alpha + \gamma NJ_s + \lambda d_t + \gamma(NJ_s \times d_t) + \epsilon_{ist} ~~~~~~ (2) $$

is the same as (1) where $NJ_s \times d_t = D_{st}$.

Your Question

Regarding your question, a key difference is that equation (2) includes the dummy variable indicating whether a data point belongs to treatment group ($NJ_s$ the dummy for restaurants in New Jersey). This dummy is not present in model (1) but the two models are equivalent.


Is there empirical evidence on usability of Cobb-Douglas production function?

The simple answer to why the Cobb-Douglas functional form is used is because it is at least a log-linear approximation to some higher-order production function. That is, suppose you take a functional form that looks like this: $\log Y_t = f(A, K, L)$. Then a linear approximation would look like the Cobb-Douglas production function. (For a small $1\%$ increase in $K$, we get approximately an $\alpha \%$ increase in $Y$.)

One potential source of the strange estimation results you're getting is likely a bias due to the endogeneity of the regressors. Total factor productivity (TFP) in your notation is $A$. It is changing over time. In your regression, it shows up in the error term. Changes in productivity affect the composition of the inputs $L$ and $K$. Thus, $L$ and $K$ are correlated with error term. To see this, solve any typical cost minimization for a firm. $L$ and $K$ will depend on $A$. Also note that the behavior will be different in the short-run vs the long-run.

Hope this helps!

EDIT:

If you're interested, here is a problem set (no solutions available) that walks you through a demonstration of the bias described above. In the end, one potential improvement to the estimation procedure is explored via Monte Carlo simulation. This improvement relies on the availability of panel data of firm input/output data.


Who exactly foots the bill if Greece defaults

Who would pay depends on the terms of the default. Sometimes holders of similar debts are not treated equally, and this can play out in different ways. Greeks could default on external debts but continue to pay internal creditors. Or because the ESM and other entities are providing ongoing financing, perhaps they will continue to be repaid when others are not to keep those spigots open. Or, less extremely, they could vary the degree of forbearance, term changes, and extent of default.

I found the following two charts from 2013 and 2014 helpful for understanding who are the current holders of Greek public debt and how that compares with other rich countries.

Source: STARLING CITY

Update:
I couldn't find a time series of the holders of Greek debt but here are some additional charts to see the evolution. The first one is more comparable to the pie chart above than the second one which shows only bank holdings. If this is of interest, the BIS's consolidated banking statistics could be used to create a time series of the bank holdings of Greek debt


what are some unique features of heterodox economics?

Among the references in the paper hinted by indigo_luc, there's one (Mearman, 2011), which [was apparently the first that] has surveyed some (43) self-styled heterodox economists and applied factor analysis as well as cluster analysis to their ideas:

  [The first factor is addressed in the next para.] The second
  factor has high loadings on class, power, labor, gender and (negatively)
  markets. Factor 2 might thus be called a “radical” (or perhaps
  Marxist-feminist) grouping. Factor 3 might be an “Austrian” grouping,
  associating uncertainty, individualism and fallibilism. Factor 4 is
  perhaps a “Post Keynesian” group that stresses money and history.
  Factor 5 suggests an “ecological economics” group that stresses natural
  systems, but also the use of mathematics.

  Factor 1 might be called a mainstream factor. Significantly, this
  factor groups rational, equilibrium and scarcity. This may reflect a bias
  within heterodox economics as to what constitutes the mainstream:
  i.e., if heterodox economists associate scarcity, equilibrium and rationality
  with the mainstream, they may reject them more easily. The
  finding also partly supports Davis’ (2009) definition of the mainstream
  in terms of equilibrium, rationality and individualism. However, the
  adoption of individualism by Austrian economists means that the
  adoption of individualism alone cannot be a definition of the mainstream;
  it also complicates the division between mainstream and
  heterodox. Further, the mean scores for “mainstream concepts,”
  although consistently lower than heterodox concepts, are consistently
  non-zero. Also, although there was a significant negative correlation
  between mainstream and heterodox, the correlation coefficient was
  only |0.438|, meaning that many respondents regard themselves
  mainly as clearly defined heterodox economists—yet with an important
  element of mainstream economics thrown in. Heterodox economists
  are a mixture of concepts and influences. An alternative
  interpretation is that heterodox and mainstream are overlapping categories.
  This would be significant given that treatments of mainstream
  and heterodox often treat them as strictly distinct. The other clear
  finding for this group is that in terms of concepts, heterodox economics
  remains a concatenation of ideas (echoing Lee’s (2010) term) and
  groupings of individuals.

As for cluster analysis:

  Cluster A, the largest, is
  characterized by a rejection of the label “mainstream” and to some
  extent of mainstream concepts (apart from individuals and markets).
  The second feature of cluster A is an acceptance of the label as
  heterodox and pluralist and a matching acceptance of general heterodox
  concepts such as class, uncertainty, fallibilism, power, money and
  history. It could be argued that this cluster exemplifies the recently
  developing picture of heterodoxy as being non-mainstream but pluralist,
  with a concern for methodological issues. Almost all of the
  female respondents to the survey are in this group, although it is far
  from clear why this would occur.

  Cluster B is different, exhibiting
  much stronger rejection of the mainstream and its concepts, maximum
  scores for class and labor and much lower scores than the other
  clusters for uncertainty and fallibilism. Methodologically this group
  scored much higher than cluster A on the need for maths in
  economics. Cluster B seems like a Marxist group and indeed its cluster
  members are self-identified radicals.

  Cluster C is different from clusters A and B in that it does not
  reject the label of being mainstream, whilst accepting the labels of
  heterodox and pluralist. This cluster is similar to cluster A but more
  pluralist. The members of this cluster reject the strict distinctions
  between the three categories. Further, although this cluster accepts
  many of the traditional heterodox concepts such as power, labor
  and class, it also accepts mainstream notions such as rational, equilibrium,
  positive, maths and crucially, scarcity. Above, when factor
  analysis was conducted, a factor was extracted that was labeled
  “mainstream” whereas the view of cluster C suggests instead this use
  of mainstream concepts alongside heterodox ones is another exhibition
  of pluralism. Unfortunately these individuals are difficult to
  identify from the information gleaned so it is difficult to draw too
  many conclusions. It should also be noted that in the dendrogram,
  arguably cluster C might have been split into two, so perhaps not
  too much coherence should be expected.

  Cluster D is in some ways
  the most interesting cluster because it contains most of the cases
  identified as peripheral. The cluster score for mainstream is similar
  to cluster C, but the score for heterodox is lower. Accordingly,
  scores for core heterodox concepts such as class, power, gender
  and particularly labor are clearly lower than for the other clusters.
  As a corollary, cluster D’s scores for individuals, markets and rational
  are much higher than for the other clusters. Looking at the
  cases who are members of this cluster, they appear to be
  members of underrepresented elements of heterodox thought, such
  as behavioral economics, Austrian economics and American
  institutionalism.


Even if immigrants take USA jobs, don't they consume proportionally to maintain supply-demand equilibrium?

The only argument that I think can be made is that immigrants often don't consume similarly to Americans. Large amounts of salaries are sent home from the US by immigrants which means that money is now out of circulation.

Additionally, coming from poorer countries with a lower standard of living it can be argued that immigrants will choose to live a lower quality of life in the US which again could cause a decrease in Economic activity in the U.S.

Lastly, for illegal immigrants wage/labor restrictions don't always apply to them meaning that U.S workers who are protected by these restrictions now have to compete with workers who do not have those restrictions which can drive down U.S employment and wages for workers across the board.

With that said, I think immigration as a general rule increases economic growth in the long run.


When we import something, we consume it. So when calculating consumption we are bound to count import as a positive component of GDP. Since it is not (we did not produce imports domestically), we subtract it to make it neutral.


Marginal buyer/seller as determinant of price

If I have understood correctly, the question asks what should we know so that we can determine the wage using only information on the unconstrained workers. Here is a toy static model:

Let's say we have $N_u$ unconstrained workers and $N_c$ constrained workers. Each has a total labor endowment $t$. Worker population is denoted $N_c+N_u = N$.
The unconstrained workers will solve a utility maximization problem

$$\max u(c,\ell_u)\;\; \text{s.t.}\;\; c_u = w\ell_u,\;\; u_c>0, u_l<0$$
Subscripts in functions denote derivatives.
The above will give

$$\ell_u^s: wu_c+u_l = 0 \Rightarrow \ell_u^s = h(c,w)$$

The constrained workers will supply each $t$. So total labor supply will be

$$L^s = N_ct + N_uh(c,w) = (N-N_u)t + N_uh(c,w),\;\; h_w>0$$

(note that some simple utility function forms lead to the supply of labor being independent of the wage. We assume this is not the case here. Also, $h_w>0$ assumes away backward-bending of the individual supply curve).

Since we assume perfect competition (and price taking behavior) on the labor demand side, the firms do not explore possible benefits from the existence of two types of workers. They just go on and equate the marginal product of labor to the market wage

$$\ell^d: MP_L = w \Rightarrow \ell^d = g(w, k_j,T), \;\; g_w <0$$

where $k_j$ is firms capital, and $T$ is technology.
If there are $m$ firms we will have the equilibrium condition

$$L^s = L^d \Rightarrow (N-N_u)t + N_uh(c,w)= mg(w, k_j,T)$$

or writing $n_u = N_u/N$

$$(1-n_u)t + n_uh(c,w)= \left(\frac mN\right)g(w, k_j,T) \tag{1}$$

So if we know the proportion of unconstrained workers, and the total worker population, we can determine the wage from $(1)$, using the first-order conditions related to labor demand, and conditions related to the unconstrained workers.

If we further assume a Cobb-Douglas production function (constant returns to scale) for the firms, then labor demand will be linear in capital, and so $g(w, X) = \xi(w,T)k_j$. Then $(1)$ becomes

$$(1-n_u)t + n_uh(c,w)= \xi(w,T)\frac {K}{N} \tag{2}$$

Here we need to know only the proportion of unconstrained workers, and the capital per worker.

Is this what the question asked?


I'm not a statistician... and this is the economics forum, so forgive me for the possible mistakes. I got most of this information from these slides.
Consider a statistic of interest  $\theta$ with a consistent but biased estimate $\hat \theta$. The Jacknife estimates $\hat \theta_{(i)}$ with is the same statistic, based on the sample that is obtained from removing observation $i$. Doing this for all $i$ and taking the mean over the values $\hat \theta_{(i)}$ gives:
$$
\hat \theta_{(.)} = \frac{1}{n} \sum_{i = 1}^n \hat \theta_{(i)}
$$
As stated above, the estimator $\hat \theta$ is biased, so let:
$$
\mathbb{E}(\hat \theta) = \theta + \frac{b}{n} + O\left(\frac{1}{n^2}\right)
$$
Where $b$ is the first order bias of the estimator $\hat \theta$. Then for the Jacknife estimates, we have a similar expression:
$$
\mathbb{E}(\hat \theta_{(i)}) = \theta + \frac{b}{n-1} + O\left(\frac{1}{n^2}\right).
$$
So, ignoring the higher order terms, we get:
$$
\mathbb{E}(\hat \theta_{(i)}) - \frac{b}{n-1} =  \mathbb{E}(\hat \theta) - \frac{b}{n},\\
\to (n-1)\mathbb{E}(\hat \theta_{(i)} - \hat \theta) = \frac{b}{n}
$$
This shows that: $(n-1) (\hat \theta_{(i)} - \hat \theta)$ is an `unbiased' estimator for $\dfrac{b}{n}$ (at least when we are ignoring all other higher order bias terms). Then using this correction, we have the following 'bias-corrected estimate':
$$
pv_{(i)} = \hat \theta + (n-1)(\hat \theta - \hat \theta_{(i)}) = n \hat \theta + (n-1) \hat \theta_{(i)}.
$$
This is called the pseudovalue.
Now, of course it would be stupid to only use $pv_{(i)}$ for one particular value of $i$ . So a better bias-corrected estimate is the one that averages over all these, the jacknife estimator:
$$
\hat \theta_{jack} = \frac{1}{n}\sum_{i = }^n pv_i =  n \hat \theta + (n-1) \hat \theta_{(.)}.
$$
The pseudovalues $pv_{(i)}$ are not necessarily i.i.d. but assume they are anyway. Then, their variance is given by:
$$
\begin{align*}
{\rm var}(pv_i) &= \frac{1}{n-1} \sum_{i = 1}^n (pv_i - \hat \theta_{jack})^2,\\
&= \frac{1}{n-1} \sum_{i = 1}^n (n \hat \theta - (n-1)\hat \theta_{(.)})^2,\\
&= \frac{1}{n-1} \sum_{i = 1}^n (n \hat \theta + (n-1) \hat \theta_{(i)} - n \hat \theta - (n-1) \hat \theta_{(.)} )^2,\\
&= \frac{(n-1)^2}{n-1} \sum_{i = 1}^n (\hat \theta_{(i)}- \hat \theta_{(.)})^2,\\
&= (n-1) \sum_{i = 1}^n (\hat \theta_{(i)} - \hat \theta_{(.)})^2
\end{align*}
$$
Assuming a central limit theorem holds for $\hat \theta_{jack}$, then:
$$
\frac{\theta_{jack} - \theta}{s_n} \to^d N(0,1)
$$
where $s_n$ is the variance of $pv_i$ divided by $n$.
$$
s_n^2 = \frac{n-1}{n} \sum_{i = 1}^n \left(\hat \theta_{(i)} - \hat \theta_{(.)}\right)^2.
$$
This 'variance', $s^2_n$, is the estimator that you give in your question.


Free trade agreements are not transitive. Each country in a free trade agreement needs to ratify it with a legislative or regulatory body. Otherwise there is no legal basis for it to function and customs officials won't be able to observe it.

In practice if A and B have a free trade agreements and B and C do too, businesses can transship in the best case scenario. Or, most likely, ship from A to B, unload-unpack-repack-load using an intermediary company in B, and ship from B to C. That might cost less or more than paying import/export tariffs from A to C.


Why does the Fed not fiddle with the reserve ratio?

The operating procedures the Fed has followed has changed over the years. I will first discuss the situation pre-2008, and then discuss the current situation.

Situation 1994-2008.

The Fed has an interest rate target, and the amount of excess reserves in the system was typically very small. Under normal circumstances, the Fed controls the amount of reserves in the system with open market operations. If nothing else happens, if the Fed buys \$1 billion in securities, it will create \$1 billion in reserves. (Exceptions include: banks may exchange reserves for notes and coins, and if there was a shortage of reserves, banks would be forced to go to the discount window to borrow to meet reserve requirements.)

In other words, the Fed has to supply whatever reserves are required by the banking system. Banks lend first, then search for reserves later. If there is a shortage in the system, the Fed has to create reserves to meet demand, or else the interbank rate will go above target.

This means that reserves follow lending decisions, and the only influence the Fed has on the level of the money supply is through the moderating effect of interest rates on expectations.

Note that this explanation is known as "endogenous money", as opposed to "exogenous money", and which version is correct is/was a major debate. The endogenous money stance seems to be closer to the modern consensus; Monetarists were the main propenents of exogenous money. I believe that the modern version of "exogenous money" is more complex than the old belief (the central bank directly sets the level of the money supply), and the short-term story ends up being similar to my description above. Unfortunately, undergraduate textbooks tend to push the old exogenous money story.

From this standpoint, changing the reserve ratio will do very little, it just changes the amounts of reserves that the Fed has to create/destroy in order to hit its interest rate target. The reserve ratio has an effect on the competitive position of banks versus non-bank financial firms, and so reserve ratio changes end up having a distorting effect on competition.

The general irrelevance of reserves in monetary policy explains why reserve requirements were abolished in many developed countries, such as Canada.

The current situation features large amounts of excess reserves; changing the reserve ratio would imply no change in policy, as all that happens is that some existing reserve positions are reclassified, with no other effect.


Does Efficient Market Hypothesis make anticipated monetary policies useless?

We begin our story at time $t-1$, where we have a Goldilocks economy (not too hot nor too cold). At time $t$, the economy gets worse. At time $t+1$, it becomes obvious that the central bank will cut the policy rate by 25 basis points at the next meeting and that this is sufficient to move the economy back to where it was at $t-1$. At time $t+2$, the central bank cuts the policy rate by 25 basis points. If markets are semi-strong efficient and the worsening of the economy is public information (along with the monetary policy action), then we would expect stocks to fall on $t$, rise on $t+1$ to the same level as $t-1$, and be unchanged on $t+2$ (assuming nothing else is happening in the economy).

However, this does not make monetary policy useless in this situation. If the monetary policy authority, meeting on $t+2$, left rates unchanged (instead of cutting them), then we would expect equities to fall on $t+2$. While there are many monetary policy channels, in most of them, the changing of the policy rate actually improves the economy. This can raise equity prices by boosting firm cash flows.

When the central bank cuts rates, this isn't useless, even if expected, because it caused the economy to be better, which also boosted the value of financial assets. Yes, if the rate change is fully anticipated, there is no effect seen on the day of the rate change, but that is not because it is useless. Rather, it is because it is already priced into the value of financial assets before the rate change actually occurs.


Why must the lag length of the autoregressive term in an ARDL model be determined separately?

The reason for this is that in a most dynamic models, i. e. models where any lags of dependent variables are used, you always have to include enough lags to get rid of autocorrelation.

This is because in dynamic model autocorrelation no longer can be adjusted just by correcting standard errors, because dynamic models in presence of autocorrelation are not just inefficient but also biased. Hence in a dynamic model it is not enough to just use let’s say Newey-West or other autocorrelation consistent errors, you need to get the dynamic specification (the lags) right so there is no autocorrelation in the model.

More formally you can see the reason why this is necessary by considering simple ARDL(1,1) mode:

$$y_t = \beta_1 + \beta_2y_{t-1} + \beta_3 x_{t-1} + \epsilon_t$$

now imagine that in this specification you still got autocorrelation of first order, which would imply that:

$$\epsilon_t= \gamma \epsilon_{t-1} + u_t$$

Now substitute the autocorrelated error back to original equation and you will get:

$$y_t = \beta_1 + \beta_2y_{t-1} + \beta_3 x_{t-1} + \gamma \epsilon_{t-1} + u_t$$

Now if if we lag the entire first equation by one  period we get:

$$y_{t-1} = \beta_1 + \beta_2y_{t-2} + \beta_3 x_{t-2} + \epsilon_{t-1} $$

Hence you can see that in presence of autocorrleation the lagged error term $\epsilon_{t-1}$ causes not just present error $\epsilon_t$ but also lagged dependent variable $y_{t-1}$. Hence this violates the Gauss-Markov assumptions of OLS, to be specific assumption that error term and regressors should not be correlated and in this case estimator is not salvageable as in standard cases where autocorrelation just causes error to be correlated with past errors.


Whatever happened to efficiency wage theories?

I don't believe that Search Unemployment (MP) actually "won" over efficiency wages. The whole discussion of search literature would be too long for this post, so I'll just skim the most important parts.

(i) As a first, Shimer (2005) discusses that the MP model actually fails to get employment (market tightness) volatility right
(ii) In the same AER issue, Hall(2005) proposes rigid wages. He and Shimer (in his booklet) argue that there is a set of wages which satisfy Barro (1977). Moreover, an intuition for wage rigidity is that old wages are an anchor for new negotiations
(iii) Many papers show more or less convincingly that while average wages are rigid, wages of new matches actually are quite flexible.
(iv) Bewley (1999) presents survey evidence where managers respond that they don't want to reduce wages because it would lead to detrimental climate.

First, to the extent that the MP model actually needs exogenous wage rigidity to get the moments in the data right, it has not won over efficiency wages, as these could be understood as a complementary theory (as being the reason for wage ridigity, as an alternative to (ii)).

Second, (iii) kind of goes against both MP and efficiency wages at the same time - at least to the extent that they explain fluctuations in hirings. Note that this is the relevant margin for unemployment fluctuations: [I think it was Shimer who] showed that the most fluctuations in labor come from volatility in hiring, and not from firing.

Finally, a personal opinion. I was interested in the story myself, but I wouldn't know how you could find efficiency wages in the data we have. Also, there's not really more to the theoretical component than was written in the past century.

The trend in the past decade has gone to microfounded, self-consistent models that are found using empirical data. I don't believe that this is really possible at the moment with the efficiency wages story. We believe efficiency wages exist, but we can't detect them and they (arguably) don't explain the Shimer puzzle, which makes them less interesting these days.


Can anyone give me some reference on Mundell-Fleming model estimated empirically?

First, you don't estimate the "model", but its parameters. You can then try tests that check if the specification is correct and if you do want to test the "model".

There are different approaches to estimation. In this nice article the author uses a VAR model to estimate the parameters of the Mundell-Flemming model for the Australian economy.

http://cms-content.bates.edu/prebuilt/huh.pdf?origin=publication_detail


What determines whether slavery is economically efficient in industrialized/modern era economies?

Necessary Caveats When Discussing Slavery:
First before tackling this question it is important to note that this issue is broad and complex. This is because there is no single 'slavery'. For example, slavery under the Roman Empire was not the same as slavery practiced in US south (see Tamin; 2017). Furthermore, even within a given society there might be different parallel forms of slavery. This is especially true if you count as slavery all 'unfree labor' such as indentured servitude or serfdom which were similar to slavery (Russian serfs could even be bought and sold see Kolchin 1990).  Hence it is extremely difficult to generalize about an institution that has been practiced differently all across world and time until very recently, and making exhaustive review of all forms of slavery and unfree labor is beyond scope of stack exchange. As a consequence I will mainly focus in my answer on the sort of slavery that existed in the US although I will try to keep thigs as general as possible.
Second, when we discuss whether slavery made an economic sense we have to distinguish between economic sense from point of view of society and point of view of few private individuals. For example, mercantilism is demonstrably not economically beneficial to society and generally tends to make economies poorer than they could be without it, which you can learn from any international economics textbook (see Krugman et al (2018) for example). Yet despite of that you could historically find many merchants who became fabulously rich thanks to mercantilist doctrines. North Korean or Venezuelan institutions also make local elites extremely rich but are arguably disastrous to the general level of economic welfare in their societies. If government would give a company monopoly on apple production it would greatly benefit the owners of said apple company but  hurt consumers much more so on net basis it would be not economically desirable to have it.
Slavery from Private Perspective:
From narrow private perspective slavery was profitable. For example, according estimates provided by Conrad & Meyer (1958) the slave owners in antebellum US could expect to see return in excess of $10\%$ during certain time periods. According to Fogel & Engerman (1974) returns of plantation owners using slave labor were comparable to those in manufacturing.
Generally speaking if slavery was not privately profitable for people engaged in the slave business they would not continue to do it on any large scale. Yes even if it would not be profitable some individuals could have kept slaves as a some sort of Veblen good to signal their high status, and I am sure historically some wealthy individuals kept slaves for only this purpose, but on any economy-wide scale slavery would not persist if it would not bring benefits to the slaveholders themselves.
Slavery from Societal Perspective
From a societal point of view the economic literature suggest slavery had more costs than benefits. For example, Anderson & Gallman (1977) argue that slave ownership is detriment to specialization. The reason for this is that if you own slaves you own them whether demand for your goods is high and low. Slaves are essentially a fixed cost for a business because you need to shelter them and feed them even if you dont have any work for them whereas free laborer can be just fired and hired at will. As a response slave owners were often not engaged in just one single line of business (like just picking cotton) but had to also diversify and use their slaves for various tasks.
However, such diversification is detrimental to economic welfare and even economic growth as specialization leads to both higher efficiency and hence allows economies to produce more, but what even more specialization can also have dynamic effects that allow economies to grow more faster (inventions & discoveries are usually made by specialist not by 'jacks of all trades'). Consequently, Anderson & Gallman argue that slavery in long run stifled the economic growth in south and hence been detrimental to southern economies even if for the slave owning elite benefited from such system. Other authors such as Genovese (1976) went even so far as to claiming that slave ownership was detrimental to the development of 'capitalism' as it discouraged  development of towns and commercial centers.
Furthermore, generally from a development perspective we recognize two types of institutions (see Acemoglu and Robinson Why Nations Fail);

Inclusive Institutions - “inclusive economic institutions… are those that allow and encourage participation by the great mass of people in economic activities that make the best use of their talents and skills”

Extractive Institutions - are defined as the opposite of the above inclusive institutions (and authors even mention slavery as prime example of extractive institution).

And developmental research shows (again see Acemoglu and Robinson) that extractive institutions are detrimental to economic development. In fact the authors argue that most of the differences between 'rich' and 'poor' countries today can be explained by the fact that  countries that are high income today tended to adopt inclusive institutions while the low income countries mostly relied on extractive institutions. Now slavery is only one of myriad of institutions so it is possible that a country would manage to develop even despite of it. Also the contrary holds a country where most institutions are highly extractive but does not have slavery will still not be able to develop itself. Nonetheless, their research suggests that slavery would be detrimental to economic development.
An concrete example of study that looks at unfree labor would be study of Dell (2010) that using regression discontinuity (in order to ensure exogeneity) examines the long‐run impacts of the mita, an extensive forced mining labor system (which is very close to slavery) in effect in Peru and Bolivia between 1573 and 1812. Results show that this system had persistent effect on lowering household consumption by around 25% and increases the prevalence of stunted growth in children by around 6 percentage points in subjected districts today (long after the system was abolished). Furthermore, the study shows this system had also persistent negative effects on other developmental measures such as integration of road networks and that also the regions that used to had mita are less industrialized and focus more on subsistence farming even today.
There are also some studies that highlight that slavery also had some positive economic impacts. For example, Williams (1944) argues that even though slavery was overall detrimental to economies of Americas it helped to develop international financial networks and in turn international banking has first order effect on economic development. Nonetheless, international financial systems would most likely develop thanks to the international trade eventually anyway so the benefits of slave trade were likely just indirect.
Furthermore, all of the above does not even take into account the welfare cost associated with the loss of utility experienced with by the slaves themselves. This utility loss was undoubtedly tremendous and any economic analysis from societal point of view has to take it into consideration.
Lastly, as pointed out by Giskard in his +1 comment, slavery was not abolished just due to economic reasons (and the same can be said about serfdom or myriad of other extractive institutions). Moral and socio-political factors play role as well. For example, top US public economists are generally in favor of well regulated market for kidneys as this poll shows, yet US does not have such market and this is most likely because most voters would consider such market immoral regardless of any general welfare benefits. Consequently, one should not assume that some economic policy exists or ceases to exist solely because it is or ceases to be economically efficient. You should also note that slavery was most prevalent in the  past when democracy was not ubiquitous as today, and even countries that had democracy extended franchise only to the land holding elites (which were also most likely the ones to own slaves) and was further restricted based on other ethic or gender characteristics. Thus extension of franchise probably played big role in abolition of slavery and other forms of unfree labor but these are questions for political scientists, historians, sociologist and anthropologists to answer.


How Much Gold Did Weimar Have Pre-Hyperinflation?

To address the first part of your question:

  "By July 10, Germany's free gold reserve had fallen to the equivalent of £35 million pounds, and all the Bank was doing had produced precisely the effect it was hoped to avoid. Another loss of 50 million gold marks was suffered during the week which ended on July 14 — the direct result of supporting the paper mark by rationing supplies of foreign currencies and having to cover indispensable imports from the gold reserve. If intervention continued at the then level of 10 million gold marks daily, bankruptcy would come within 60 days: the reserves had already taken as much as they could stand. If intervention ceased, on the other hand, it was broadly expected that the dollar would go to a million marks, the pound to over 4 million — and by the end of July that expectation was fulfilled. Merchants were still said to be borrowing paper marks to the value of 80 per cent of their stock, and exchanging them at the Bank for dollar Treasury bills needed ostensibly for imports. These bills were sent abroad to the merchants' buying agencies, and in most cases stayed there with no countervalue returning to Germany."

-A World of Possible Futures: German Weimar Republic in the early 1920s and the U.S. - Troubling similarities

The source itself appears to be of questionable value, but makes vague reference to:

"The Economics of Inflation (Routledge Library Editions-Economics, 84)" by Constantino Bresciani-Turroni (427 pages, Routledge, reprint edition, October 1, 2003, first published 1937

Judging from historical gold prices and exchange rates, we could estimate:

$$ 35M x 4.58 /  21.32 = 7.5M$$

So, 7.5 million troy ounces or 233 Metric Tonnes according to Google.

To address your second point, it seems to be exactly what the Weimar Republic tried.

I would not imagine the US would be keen to repeat something like that, though it seems far outside the scope of this SE to speculate on that.


Why is the income effect zero for quasilinear utility functions?

From the formula for $x_M$, we see it has no dependence on income $m$. So $$\frac{\partial x_M}{\partial m} =0$$ Thus, the Slutsky equation

$$\frac{\partial x_M}{\partial p_x} = \frac{\partial x_H}{\partial p_x} +-\frac{\partial x_M}{\partial m}x_M$$

implies

$$\frac{\partial x_M}{\partial p_x} = \frac{\partial x_H}{\partial p_x} +(0)x_M $$
$$\frac{\partial x_M}{\partial p_x} = \frac{\partial x_H}{\partial p_x}  $$
Hence, and since $-\frac{\partial x_M}{\partial m}x_M$ is the income effect, this implies the income effect is zero and all the change is due to the substitution effect.


Certainly the economy is not a "zero sum game" (whatever that means). There has historically been a positive aggregate net return on capital investment - there doesn't have to be a loser for every winner.

That said, economists generally believe that as the stock of aggregate savings increases, the return will go down; this is a consequence of diminishing returns to capital. Hence it's not possible for everyone to save an unlimited amount at 4%.

To be a little more concrete, economists often use a constant-returns-to-scale production function $F(K,L)$, taking capital $K$ and labor $L$ as inputs, as a first-pass way of thinking about the world. In this simple model, net aggregate savings equal capital $K$. Holding $L$ constant, the extent to which more $K$ will push down the net return to capital $r=F_K-\delta$ (where $F_K$ is the marginal product of capital and $\delta$ is the depreciation rate) depends on the elasticity of substitution of the function $F$.

With a high elasticity of substitution, $K$ can increase substantially without $r$ falling very much, as the economy continues to find productive applications for capital despite its relative abundance; with a low elasticity of substitution, a rise in $K$ can result in a large drop in $r$.

Example. A common form for $F$ is Cobb-Douglas, which corresponds to a constant elasticity of substitution of 1. If the gross capital share of production is $0.35$, then we write $F(K,L) = K^{0.35} L^{0.65}$. The marginal product of capital here is $F_K(K,L) = 0.35 \cdot (K/L)^{-0.65}$, implying that the elasticity of $F_K$ with respect to $K$ is -0.65.

Now, if we suppose that initially $r=0.04$ and $\delta=0.06$, then the elasticity of $r=F_K-\delta$ with respect to $K$ is $-0.65\cdot (0.1/0.04)=-1.625$. This means that if we increase capital $K$ by 10%, the net return $r$ will decrease by approximately 16.25%, or $4\times .1625 = .65$ percentage points, from 4% to 3.35%. This is a pretty substantial decrease!

Indeed, since $1.625 > 1$, in this case net capital income $rK$ to capital decreases as $K$ goes up; remarkably enough, if savers accumulate more, they earn less, as the decline in the net return per unit of capital overwhelms the increase in capital.


Log log regression with fixed effects and cross elasticity of demand

Note that you have an identification problem here because price is an endogenous variable. A way to deal with this problem would be to find an instrument (e.g. prices  of the same good in other markets as in Hausman (1996)).
Regarding your questions.

Yes (although fixed effects are probably not the right wording here).
Yes.
If you want to capture a cross elasticity effect you need to introduce the own price as well as price of substitutes into the equation. You may face a dimensionality problem due to the large number of parameters to be estimated. The effect substitutable units would be interpreted as the effect of the availability of substitutes on demand.
I would think about the status of these types. Is the same product with different attributes? Are these types substitutes? Is it a kind of aggregation? Regarding interpretation: you interpret one category with respect to another (or to a benchmark category in case of more than one category).


I disregarded the information about imports and subsidies because I assumed that GDE was calculated using the equation Y=C+I+G+X-M.

This is where I went wrong. GDE is calculated as C+I+G, but to get GDP at market prices, I need to add nett exports.


Are Cobb Douglas goods complements or substitutes?

Try to think what you mean when you ask whether they're complements or substitutes.

You could mean: "Does my marginal utility in $x$ increase when I get more $y$? That would correspond to the cross derivative $\frac{\partial U^2}{\partial x \partial y}$.

You could (and this is the convention) mean the response to a change in prices. Denote with stars the bundle that maximizes utility given a budget constraint.

$$x^* (p_x, p_y), y^*(p_x, p_y) = \arg\max_{x,y} U(x,y) + \lambda(Y - p_x x - p_y y) $$

Then, the goods are complements if you increase the price of one of the goods, and the demand of the other one decreases:

$$ \frac{\partial x^* (p_x, p_y)}{\partial p_y} < 0 \\
\frac{\partial y^* (p_x, p_y)}{\partial p_x} < 0 $$

And substitutes if that is larger than zero. For the case where that partial is exactly zero, the elasticity of substitution is zero. Try out yourself which case is correct for Cobb-Douglas!


Does the investment component of GDP (demand side) identity include expenditures by the government and households?

“I” would be called private domestic investment. (https://apps.bea.gov/iTable/iTable.cfm?reqid=19&step=2#reqid=19&step=2&isuri=1&1921=survey)

Firm and household expenditures for investment both count as private investment.

(https://courses.lumenlearning.com/boundless-economics/chapter/measuring-output-using-gdp/)

Any investment expenditure by a government would be categorized under government spending.

If a government builds a school or bridge it would go under government spending “G”. (https://courses.lumenlearning.com/wm-macroeconomics/chapter/calculating-gdp/)


Externalities, Pigouvian Taxes and Wikipedia

The Pigovian taxes are non-distortionary. For example imagine situation where government optimal spending is 100e and before Pigovian tax all 100e was raised through income tax which creates distortions on Labour market. Let’s say that after imposing Pigovian tax government gets additional 30e. Now since government needs only 100e for its optimal spending it can either choose to run 30e surplus or choose to reduce the distortionary income tax by 30e. The Wikipedia sentence refers to this second option where government chooses to reduce some other distortionary tax thanks to the revenue from Pigovian tax.


What would be the economic effects of a reduction in the general level of protectionism on the Australian economy?

Normally protectionism is never the answer as it doesn´t help in anyway.
If for example the subsidies for exportation were to be reduced you would see a drop in the price of items that are exported from autralia , i would place your question more in micro-economics then in macro. Not sure if that answered your question.


How does this company gain from economies of scale?

The story (and Jo) have basically nothing to do with the benefits Pepper plc will get from economies of scale. As you have stated economies of scale means that the average cost (cost per unit) is less if you produce in large quantities.

An example: Assembling a car with drills and screwdrivers is really difficult and time consuming but a robotized assembly line can do it quite quickly. However, the assembly line costs a lot of money so it will only pay off if a lot of cars are produced.

Since Birmingham, where Pepper plc. is located, is a large city it has a large market for electronic consumer goods. Therefore Pepper plc. can sell a lot of these goods, hence it has reason to produce in bulk, so the unit cost is less. Thus the company may sell its products cheaper or have a higher profit. (This depends on the market environment.)


real or nominal GDP to compare to wage price index?

Wage/labor share of income/output is total compensation of employees divided by total output (See Schneider 2011). This share is typically calculated using nominal wages and nominal GDP.
If you want to calculate wage/labor share of income/output then you should use total wages divided by nominal output:
$$ s_L = \frac{WL}{PY}$$
Where $s_L$ is the share, $WL$ are total wages (wages $W$ times labor input $L$) and $PY$ is nominal GDP (real GDP $Y$ times aggregate prices $P$). However, note it's not necessary to find statistics for $P$, $Y$, $W$ and $L$ separately most countries will track statistics for both $PY$ and $WL$ directly.
Note you cannot calculate it from wage price index alone. That is simply impossible. Wage price index only tells you how much wages risen over some period of time relative to some base point in time. That is insufficient information to figure out what was total compensation. In fact from wage price index you cannot even figure out what $W$ is, you can just use it to track how $W$ change over some period of time. And wage price index gives absolutely no information about $L$, so it is impossible to calculate the wage/labor share of income/output using only output and wage share index.


Depreciation vs. obsolescence in RBC and New Keynesian DSGE models

On the empirical side, there might be answers for you in the national accounts. I only know about the french case : the french statistical institute (INSEE) has different depreciation data for different types of capital (e.g. buildings, machines, patents) and for different sectors. These data are supposed to reflect both physical depreciation and "normal obsolescence". The only reference I can give you is in french : Jean-François Baron, "Les comptes de patrimoine et de variation de patrimoine, base 2000", INSEE, 2008, http://www.insee.fr/fr/indicateurs/cnat_annu/base_2000/documentation/methodologie/nb10.pdf , but I suspect you can find similar data in other national accounts.

On the theoretical side, if I understand you, you mean that a capital good can become economically obsolete before it is physically out of use. I see three possible reasons (possibly overlapping) for this :

the price of inputs (e.g. the wage rate) rises so that this capital good is not profitable any more
the price of output falls but not the price of inputs, so that it is not profitable
production is demand constrained and there other more productive ("new") capital goods available, so that our capital good will remain idle (although it could be profitable if there were a higher demand)

To account for this problem, you have to drop the representation of capital as a single homogeneous magnitude (the number "K") and have different capital goods with different properties (productivity and input requirements). This is called a "vintage capital" model. I don't know if this has been done in an RBC or DSGE framework, but a quite complete neoclassical paper dealing with these issues is :
Solow, Yaari, Tobin, Weiszäcker (1966), "Neoclassical growth with fixed factor proportions"
http://sites-final.uclouvain.be/econ/DW/DOCTORALWS2004/bruno/vintage/solow%20yaari%20tobin.pdf


confusion regarding elasticity of a demand curve

It is correct that the higher the elasticity is the more flat the demand curve will be. A higher elasticity means that demand responds disproportionately more to change in prices. In fact an example of demand curve where elasticity $ | \epsilon|  = \infty$ would be a flat horizontal demand curve.
The reason for this is that elasticity of demand tells you how much quantity demanded (plotted by convention on x-axis) changes in response to change in price (by convention plotted on y-axis). In order for a small price change (small distance 'traveled' across y-axis) to have large impact on quantity (distance 'traveled' across x-axis) the curve has to be relatively 'flat' since the slope which will be the rise/run of tangent line at a given point must be relatively small.
For example if elasticity $| \epsilon | = 2$ that means that $1\%$ decrease in price increases quantity demanded by $2\%$. Since total revenue is defined as price times quantity $P*Q$ if $P$ decreases only by $1\%$ but $Q$ increases by $2\%$ the total revenue has to increase.  So the answer provided is correct.
To actually showcase how the demand becomes more 'flat' I generated 3 constant elasticity of demand functions ($P=Q^{1/\epsilon}$) using R for natural numbers $Q = 1, ..., 10$. The blue demand curve has constant elasticity of $0.5$, the orange one $2$ and the red one has constant elasticity of $5$ (all expressed in absolute value). You see as the elasticity increases the demand graph becomes more 'flat'.


Is an EU single currency still a trade advantage in 2019?

Very good points about the role of modern tools (the web, robotic cash exchange, etc.) are raised. The  classic argument is that trade should expand within the eurozone following (1) increase in price transparency, (2) elimination of currency conversion costs, and (3) elimination of exchange rate fluctuations.

Are the classical arguments challenged by modern tools?

Price comparison across border is surely much easier today with the web than it was in the nineties.
In  its  report  One  Market  One  Money,  the  European  Commission  (1990)  calculates  that  eliminating  costs  of  conversion  of  currencies  and  costs  of  cover  for  exchange  risk  will reduce costs by .25 to .5 of 1 percent of total output in the EU. The report also stresses additional  gains  that  would  come  from  the  elimination  of  “in-house  costs”  associated  with multiple  units  of  account.  These  are  reductions  in  costs  connected  with  record  keeping,  accounting and decision-making. The question: How much lower are the conversion costs thanks to modern tools? Even if they are now lower, they are still positive.
Despite modern tools, exchange rates still fluctuate.

So, the adoption of the single currency may still boost trade.

The role of modern arguments

To these classical arguments, we can add more subtle ones. For instance, in some industries, components cross borders many times before being assembled and sold as a final  good,  even  a  small  reduction  in  transaction  costs  can  increase  trade  flows non-linearly (Flam and Nordstrom, 2006).

In a series of papers, Richard Baldwin adds that adopting a single currency may (1) reduce collusion among competitors and eliminate pricing mark-ups and enhance trade (p. 63-64); (2) increase the entry of small firms by reducing the fixed cost of trade.


Why is it relevant for a company to use dividends to incentivize shareholders to maintain ownership of the company's stocks?

As I say in the comments, I would speculate share prices must be maintained because shareholders are collectively the owners of the company, and they want to maximise rent extraction from the operation, including maintaining or improving the resale price of the shares.
If the existing management do not cooperate to that end, they can be fired and replaced by those who will, or the entire business may be sold for scrap and the capital proceeds invested elsewhere.
I agree though that the linked article is not at all clear about why "maintaining ownership" is important.


What would be the effects of an expiration date on currency?

A bill with an expiration date does not become worthless suddenly. Clearly its value would decrease over time until, just before the expiration date, it was next to worthless. Think about it: how much would you pay for a $1 bill that expires in 5 minutes? Certainly a lot less than you would for one that expires in 6 months.

This constantly changing value would make it hard to use as a medium of exchange. Whenever buying or selling something, one would have to look not just at the face value of the bill, but also at the expiration date, to be able to calculate its real value. People would probably start carrying calculators next to their wallets.

Cattle (which obviously "expire") were once used as currency, so it certainly could work. But that does not mean it is ideal.


Principal component analysis interpretation

It is harder to interpret coefficients principal components in terms of original regressors.

Indeed, since you are now dealing with components, you can't. That being said, if your original regressors are highly colinear and all capturing wealth-related phenomena, a possibility is that your first component is still somehow strongly related to wealth, and other components only controling for phenomena that are not related to wealth at all. Put differently, you could reduce the dimensionality of your problem to this unique first component, and maintain your interpretation.

But do not take what i am saying for granted: if you really want to know how abusive maintaining your economic interpretations would be, it's all a matter of visualizing/diagnozing the relative weights of your components, e.g. you need to check the eigen decomposition of your correlation matrix.

I recommend you to read Michael Friendly and Ernest Kwan (2009) "Where's Waldo? Visualizing Collinearity Diagnostics" The American Statistician, Vol. 63, No. 1 (Feb., 2009), pp. 56-65.

Also, Yoel Haitovsky (1966) "A Note on Regression on Principal Components" is very clear and concise.


The branch of microeconomics that you seem to be looking for is the study of externalities—activities that have side-effects that (positively or negatively) effect the welfare of others. The Wikipedia article on the topic is here.

Any good introductory/undergraduate level textbook will have a treatment of this topic. For example, Hal Varian's "Intermediate Microeconomics: A Modern Approach" (which would be my recommended first port of call) has a chapter on externalities. A slightly more specialised treatment can be found in a textbook on so-called public economics such as "Intermediate Public Economics" by Myles and Hindriks.

These books also contain examples of how goods with externalities can be formally modeled.

Alternatively, because externalities are a standard element of the economics curriculum, a Google search for "Externalities lecture notes" will turn up lots of freely-accessable PDFs that give a good overview.


Why does the Solow Model matter for undergraduate students?

I disagree that:

a) The core curriculum should not include things that are relevant only for those pursuing graduate school.

b) A university degree should necessarily focus only on career outcomes for students.

c) There are only two relevant careers and therefore the only thing worth teaching is how to handle different types of data.

The core curriculum needs to provide a foundation for all future paths of students, which includes those going to graduate school.

Further, a university degree is foremost a scientific (or at times artistic) pursuit within a field and not a trade school. That is why professors are scientists (or at times artists) and not industry professionals. After a bachelor's degree a graduate should at least understand the basics of that field. Sure, many of those skills learned at university can partially be transferable to a job in the real world. However, that's a different story, which often does not involve the actual material learnt, but other skills learnt along the way, such as critical thinking and writing.

The Solow model is the foundation for beginning to understand economic growth and development. That is at the core of what the economics field aims to understand. Even  most policy debates will inevitably involve growing the economy or creating jobs. I would argue any student that is not interested in that question at all has no business studying economics. That is why the Solow model is in the core curriculum.

Lastly, there is a reason not everyone studies statistics and IT so that we can all become data analysts (for business or otherwise). The reason is that there are many other career paths.


Why would increased inflation increase the value of the dollar and bond yields?

My interpretation of the argument is that real GDP is also supposed to rise, not just inflation. Increased growth makes US dollar assets more attractive, hence drawing portfolio inflows. The rise in inflation is a side-effect of the scenario, and not the cause.


Fixed-Effects on individual level with explanatory variable on state level

If you want to do unit-fixed effects (so eliminating any unobserved variable that varies cross-sectionally but remains the same over time), this should not be a problem, at least assuming that your explanatory variable varies over time (be it a dummy that is 1 in the year the policy was administered and 0 otherwise, or a continuous variable that changes over time according to the intensity of the policy treatment).

It does not matter that the variable is on state level, as long as the policy varies over time. However, if you only have individual observations for the one state that introduced the policy, you will not be able to interpret your results counterfactually. For that you would need similar individuals observed over the same period of time in a state that did not introduce the policy (and apply difference-in-difference or something of the sort).


What is the difference between currency manipulation, and a fixed exchange rate regime?

Correct, both a fixed exchange rate and a manipulated exchange rate involve the government setting this price. In BOTH cases it is usually implemented by promising to standing ready to buy or sell any amount at the fixed price. Of course, countries don't always have enough foreign cash to sell it at the promised price, nor do they always want to buy as much as the markets sell to them

The difference is, as you suggest, whether the trading partners (the US is usually the one that labels others manipulators) conclude that the exchange rate is fixed at a too depreciated level. This level keeps foreign goods cheap in the US and US goods expensive in the foreign country.

Also, a fixed exchange rate is often set at a lvel that seems "too high", the only question being whether the government will manage to keep it so high, since it creates an incentive for its citizens to buy foreign goods. Instead a manipulated currency is deemed to be "too low".


Taking the derivative of capital per effective unit of labor in the Solow model

I've tried to "fill the blanks", hope this helps.

$$\tilde{k} = \frac{K}{AL}$$
$$\frac{d[\tilde{k}]}{dt} = \frac{d}{dt}\left[\frac{K}{AL} \right] \quad (1)$$
Now we use the quocient rule: to differentiate $f(x) = \frac{g(x)}{h{x}}$ (for $h(x) \neq 0$) we can apply the formula $f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{[h(x)]^2}$. In your problem, $f(x) = \dot{k}$, $g(x) = K$ and $h(x) = AL$, but instead $x$, you have $t$. So, applying the rule in (1), we have:
$$\frac{d[\tilde{k}]}{dt} = \frac{AL\left(\frac{d[K]}{dt}\right) - K\left(\frac{d[AL]}{dt}\right)}{(AL)^2} \quad (2)$$

The second term in the numerator has the derivative of a product ($AL$), so we need again to use the appropriate rule: $(f(x)\cdot g(x))' = f'(x)\cdot g(x) + f(x)\cdot g'(x)$

So, using the rule in (2) we have:

$$\frac{d[\tilde{k}]}{dt} = \frac{AL\left(\frac{d[K]}{dt}\right) - K\left(L\frac{d[A]}{dt}+A\frac{d[L]}{dt}\right)}{(AL)^2}$$

Now we re-write using dot notation - $\dot{x} \equiv \frac{d[x]}{dt}$:

$$\dot{\tilde{k}} = \frac{AL\dot{K} - K(L\dot{A}+A\dot{L})}{(AL)^2} \quad = \frac{AL\dot{K}}{(AL)^2}-\frac{KL\dot{A}}{(AL)^2}-\frac{KA\dot{L}}{(AL)^2}$$

Since $\frac{\dot{A}}{A} = g$ and $\frac{\dot{L}}{L} = n$, we have

$$\dot{\tilde{k}} = \frac{\dot{K}}{AL} - Kg -  Kn = \frac{\dot{K}}{AL} -K(n+g)$$

ps: I'm not sure why your textbook is using dot notation over $\tilde{k}$ before taking the derivative.

EDIT:

I'm adding an alternative way to achieve the same result.

Starting with $\tilde{k} = \frac{K}{AL}$, you can rewrite as $K = \tilde{k}AL$.

Now, take log on both sides and remember that $log(ab) = log(a) + log(b)$:

$log(K) = log(\tilde{k}AL) = log(\tilde{k}) + log(A) + log(L) \quad (1)$

Now we are going to take the derivative (in t), using the log rule: if $f(x) = log(x)$, then $f'(x) = \frac{1}{x}$. But now we have to use the chain rule, because all variables are in reality functions of $t$. The chain rule is: if z depends on y and y depends on t, then z depends on t: $\frac{dz}{dt} = \frac{dz}{dy}\frac{dy}{dt}$. Using $log(K)$ as example, we have: $\frac{d}{dt}[log(K)] = \frac{d}{dK}[log(K)]\cdot \frac{d}{dt}[K]$.

(1) then became:
$$\frac{d}{dt}[log(K)] =  \frac{d}{dt}[log(\tilde{k})] + \frac{d}{dt}[log(A)] + \frac{d}{dt}[log(L)]$$

Which is...
$$\frac{d}{dK}[log(K)]\cdot \frac{d}{dt}[K] =  \frac{d}{d\tilde{k}}[log(\tilde{k})]\cdot\frac{d}{dt}[\tilde{k}] + \frac{d}{dA}[log(A)]\cdot\frac{d}{dt}[A] + \frac{d}{dL}[log(L)]\cdot\frac{d}{dt}[L]$$

Now we substitute all derivatives of logs and use dot notation:

$$\frac{1}{K}\dot{K} = \frac{1}{\tilde{k}}\dot{\tilde{k}} + \frac{1}{A}\dot{A} + \frac{1}{L}\dot{L} = \frac{\dot{\tilde{k}}}{\tilde{k}} + g + n$$

In the SS we have $\frac{\dot{\tilde{k}}}{\tilde{k}} = 0$, so:

$$\frac{\dot{K}}{K} = g + n$$


Return on Assets vs Return on Investment

No.

Return on assets = net income/assets.

ROI = an informal term roughly corresponding to return on starting capital, or net income/starting capital. Starting capital is any money invested in the business.

Let's say you take out a 500k mortgage on a home with a 50k downpayment. That means 550k of assets. Net of everything you make a $10000 profit from rental income or whatever.

Your ROA is 10k/550k because even though you don't own the home free and clear, the house is still an asset.

ROI is a more informal term comparing how much you invested versus the return. In this example ROI is 10k/50k because you only paid 50k for the home.

ROA uses net income, which means after tax.


If household debt rises, it means that more households consume more than they earn. This is the same as saying they save less. This means that the causal channel in your quote is by definition.

Now one may ask why do savings decrease in a period of growth. I could imagine two causal channels. It could be that growth is a consequence of low interest rates, causing both higher private debt and hence more investment. One could also think of sustainable growth causing positive expectations, so households are worrying less about the future and hence save less for precautionary reasons.


Good books to learn social choice theory

Start with:

A primer in Social Choice Theory, by Wulf Gaertner.

If you want more, have a look at:

Welfare Economics and Social Choice Theory by Allan Feldman and Roberto Serrano

To dig deeper:

Handbook of Social Choice and Welfare
By A. Sen, Kotaro Suzumura
Handbook of Social Choice and Voting by Jac C. Heckelman, Nicholas R. Miller
Handbook of Computational Social Choice by Ariel D. Procaccia, Jérôme Lang, Felix Brandt, Vincent Conitzer, Ulle Endriss
The Handbook of Rational and Social Choice by Clemens Puppe, Prasanta Pattanaik, Paul Anand

For good or bad reasons, there are also entire books dedicated to the specific method of "approval voting" (which tend to discuss other methods as well as a way to enable comparisons):

Approval Voting by Brams and Fishburn
To dig deeper: Handbook of Approval Voting, by M. Remzi Sanver, Jean-François Laslier

To branch out into the neighboring disciplines of mechanism and market design:

Multiagent Systems, Algorithmic, Game-Theoretic, and Logical Foundations, by Yoav Shoham and Kevin Leyton-Brown
Essentials of Game Theory, A Concise, Multidisciplinary Introduction, by Yoav Shoham and Kevin Leyton-Brown (some redundancies with previous reference)
Auction Theory, by Krishna
Market Design, Auction and Matching, by Haeringer


The new Keynesian IS curve: What determines output?

The solution to your question is price stickiness, or as the author calls it staggered pricing. Let's assume a typical question in the Basic New Kenesian DSGE Model: What happens when a technological shock hits the economy?
More productive firms can produce cheaper but may not be able
to lower their prices due to price rigidities thus demand from the households stays at a lower level. The market clearing condition is supply equals demand and since demand won't increase output in the economy will be constraint (your paper uses determined but this might be the source of confusion) by the demand side. In this scenario one speaks from a negative output gap (Output is depressed in comparison to a world with flexible prices). In a flexible price scenario firms can lower their prices and thus household simultaneously increase demand, as firms want to increase production.
If you want to know more about price stickiness the basic DSGE models often implement the form suggested by: Calvo, Guillermo, “Staggered prices in a utility-maximizing framework,” Journal
of Monetary Economics, 1983, 12 (3), 383–398.


Interpreting the Reference Outcome in Thaler (1985)

The two notations you refer to have nothing to do with each other.
In Thaler's paper an outcome $(y:x)$ means that outcome $y$ realized while the agent expected to get outcome $x$. It refers to the idea that a reference point ($x$) matters when an actual outcome ($y$) is evaluated.
The other post refers to a mathematical notational convention used in some parts of the world. Here, $:$ is a mathematical operation, division. The historical reason for this is that $\frac{x}{y}=x/y$ was sometimes written as $x \div y$ for longer expressions $x$ and $y$ "back in the day" when calculations were done by hand on paper. Then, I guess, people became lazy and $x \div y$ became $x : y$.


Linear Probability Model Instead of Logit in Fixed Effects Regression

FE logit requires the idiosyncratic errors to be IID across $i$ and $t$, quite a strong assumption. Also the regressors should be strictly exogenous, but it's the same for linear FE models. In your application, the fact that FE logit wouldn't converge will make a good argument against FE logit, and will satisfy some referees but not all.

An important drawback (or perhaps more like a "feature") of FE logit is that partial effects (e.g., average partial effects) are not obtainable. This is because the partial effects rely on $\alpha_i$ (fixed effects), which can't be estimated by FE logit.

If you are willing, and I strongly recommend it, please consider Correlated Random Effects (CRE) probit or logit models, where time-invariant unobservable heterogeneity correlated with $X$ is accounted for by the Chamberlain-Mundlak device. The CRE probit (see section 5 of the link) is also known as Chamberlain's random effects probit, and are very popular (certainly more popular in application than FE logit). You can get average partial effects for CRE probit models.


From symmetry we can say that for each player $i$ in $I$, $q_r=q_k$ $\forall r,k \in I$. Similarly for $II$. Let this be $q_I$ for producers in $I$ and $q_{II}$ for $II$.
So profit function for individual $i$ in $I$:
\begin{align}
\max_{q_i} \pi_i&=(200-(N-1)q_I-Nq_{II}-q_i))q_i-c_1q_i\\
\text{FOC: }\,\,\frac{\partial \pi_i}{\partial q_i}&=200-(N-1)q_I-Nq_{II}-q_i-q_i-c_1=0 \\
\implies q_i^*(q_I,q_{II})&=\frac{200-Nq_{II}-c_1-(N-1)q_I}{2}
\end{align}
Similarly, we will get for individual $j$ in $II$:
$$q_j^*(q_I,q_{II})=\frac{200-Nq_I-c_2 - (N-1)q_{II}}{2N}$$
At NE, we know that $q_i^*=q_I^*$ and $q_j^*=q_{II}$. Using this:
\begin{align}
q_I^*&=\frac{200-Nq_{II}^*-c_1-(N-1)q_I^*}{2} \\
&=\frac{200-Nq_{II}^*-c_1}{N+1}
\end{align}
And so, $$q_{II}^*=\frac{200-Nq_{I}^*-c_2}{N+1}$$
Solving for $q_I^*$:
\begin{align}
(N+1)q_I^*&=200-N\frac{200-Nq_I^*-c_2}{N+1}-c_1 \\
\implies q_I^* &=\frac{200-c_1+N(c_2-c_1)}{2N+1}
\end{align}
Similarly, $$q_{II}^* =\frac{200-c_2+N(c_1-c_2)}{2N+1}$$


Your formula is wrong. The derivation for the cost of equity $r_E$ is actually:

$r_A = r_D(1-t)\frac{D}{V}+r_E \frac{E}{V}\\
r_E \frac{E}{V} = r_A -  r_D(1-t)\frac{D}{V}\\
r_E = r_A \frac{V}{E} -  r_D(1-t)\frac{D}{V} \times\frac{V}{E}\\
\mathbf{r_E = r_A \frac{V}{E} -  r_D(1-t)\frac{D}{E}}$

Since $\frac{D}{V}=\frac{D}{D+E}=0.09091$ when $r_A=0.1941$, then $E=10$ which means $D=0.10E=1$, making $V=D+E=11$ for use in the above.

$r_E = 0.180 \times\frac{11}{10}-0.06(1-0.35)\times 0.10 = 0.1941$


Let the utility function $U = a + b \min\{x,y\}$. Are $x$ and $y$ perfect complements?

Consider a typical car, which is made of one car-body (for lack of a better word) and four tires. Let $x_1$ denote the number of car-bodies and $x_2$ denote the number of tires. Most consumers would prefer cars in the form of one car-body with four tires, and not in any other combinations. So we may represent their preference as
$$U(x_1,x_2)=\min\{4x_1,x_2\}.\tag{1}$$
Thus two cars are preferred to one car can be expressed as
$$
(2,8)\succ(1,4)\quad\Leftrightarrow\quad U(2,8)=8>4=U(1,4). \tag{2}
$$
One car with two extra tires does not increase one's utility:
$$
(1,4)\sim(1,6)\quad\Leftrightarrow\quad U(1,4)=4=4=U(1,6).  \tag{3}
$$
Equation $(1)$ is the utility function associated with a perfect complement preference. In particular, it is a special case of the general form $\min\{ax_1,bx_2\}$ where $a=4$ and $b=1$.

Let's now apply an affine positive transformation to $U(x_1,x_2)$. To avoid notational confusion, let's use $\alpha\in\mathbb R$ and $\beta>0$ as the parameters:
$$
V(x_1,x_2)=\alpha+\beta U(x_1,x_2).
$$
For concreteness, suppose $\alpha=1$ and $\beta=0.5$. And we can verify that the preference illustrated in $(2)$ and $(3)$ is preserved:
$$
(2,8)\succ(1,4)\quad\Leftrightarrow\quad V(2,8)=5>3=V(1,4).
$$
$$
(1,4)\sim(1,6)\quad\Leftrightarrow\quad V(1,4)=3=3=V(1,6).
$$

Therefore, while $\min\{ax_1,bx_2\}$ represents some perfect complement preference, its monotonic transformation, $\alpha+\beta\min\{ax_1,bx_2\}$, also represents the same preference.


Macroeconomics: effect of technology/workforce in the short run

For question 1, I think that investment has a demand-side effect on the economy. I think it isn't a supply-side effect because it isn't stated that the investors are responding to a supply-side shock. So I'm saying that AD shifts right. With shifts in the curves, there will be a change in equilibrium. For example, the process you described of both curves shifting right will lead to higher output (even though the price level might not change).

That being said, the "foreign" part of the question does make things a little less obvious. Why would the question want to emphasise the foreign part of this investment? An improvement in technology in one firm will eventually trickle to other firms (vertically or horizontally). Technological improvement shifts the supply curve rightwards, but because this is a long-term effect, I did not say in the first paragraph that there is a shift of the supply curve. Also, if a foreign country sends its profit to its home country, there is effectively a small decrease in the demand curve, but on the whole that would be small enough that it would not offset the increase in demand that the investment brings.

The second question is obviously trying to highlight a decrease in supply, as you have said. The quantity of labour decreases; production decreases.

But... to throw a spanner in the works... this does sort of imply that the workers are consumers. If consumers leave the country, consumption decreases, which decreases aggregate demand (as you have said). One needs to know why the people needed to leave. That being said, with the word "workers" I think the question wanted to highlight the supply side effect.


Understanding subscripts in first order conditions of dynamic optimization problems

In an intertemporal maximization problem, we seek to find the optimal sequence  of the control and the state variables. It is the recursive nature of the problem that permits us to consider a "typical" point in time and just one condition per variable.

For each such problem, we need to find out (carefully) in how many distinct periods a specific realization of a variable appears. To do this properly we should distinguish between the "absolute" index, and a "running" index. In the formulation of the Lagrangean as appears in the question, this is not done (and it is usual practice not to, but it may become confusing).

So I would use the $t$ symbol as the absolute index (to arrive at same-looking first-order conditions), and some other symbol for the running index, say

$$\mathcal{L_t} = \sum_{j=0}^\infty \beta^{j}\left\{u(c_{t+j}) + \lambda_{t+j} \left[ f(k_{t+j}) + (1 - \delta)k_{t+j} - c_{t+j} - k_{t+j+1}\right]\right\} $$

Note that $t$ no longer affects  the discount factor $\beta$, and this is because the discount factor has to do with looking at the future, which is represented by the index $j$. Also, note that $j$ starts at zero, indicating that the first period is the $t$ period.

Written this way, the Lagrangean says "we are at some point in time indicated by $t$ (that can take the value zero or whatever positive value), and we are looking forward period by period counted by index $j$".

For any given $j$ we have

$$\mathcal{L}_t =...+ \beta^{j}\Big\{u(c_{t+j}) + \lambda_{t+j} \left[ f(k_{t+j}) + (1 - \delta)k_{t+j} - c_{t+j} - k_{t+j+1}\right]\Big\} + \beta^{j+1}\Big\{u(c_{t+j+1}) + \lambda_{t+j+1} \left[ f(k_{t+j+1}) + (1 - \delta)k_{t+j+1} - c_{t+j+1} - k_{t+j+2}\right]\Big\} + ...$$

Pondering this, we realize that the variable $k_{t+j+1}$ will appear in only two consecutive periods, and so the first order condition for a "typical" element of the sequence $\{k_{t+j}\}_{j=0}^{\infty}$ can be exrpessed by differentiating only these two periods with respect to $k_{t+j+1}$. Doing so we get

$$\frac {\partial \mathcal{L}_t}{\partial k_{t+j+1}} =  -\beta^{j} \lambda_{t+j}  + \beta^{j+1}\Big\{ \lambda_{t+j+1} \left[ f'(k_{t+j+1}) + (1 - \delta)\right]\Big\} $$

Take common factors (which will simplify the discount factor) and set equal to zero

$$\frac {\partial \mathcal{L}_t}{\partial k_{t+j+1}} = \beta^{j} \Big[-\lambda_{t+j}  + \beta\Big\{ \lambda_{t+j+1} \left[ f'(k_{t+j+1}) + (1 - \delta)\right]\Big\}\Big] = 0$$

To lighten the indexing burden, we can express this for $j=0$, to obtain

$$\frac {\partial \mathcal{L}_t}{\partial k_{t+1}} = 0 \implies -\lambda_{t}  + \beta\Big\{ \lambda_{t+1} \left[ f'(k_{t+1}) + (1 - \delta)\right]\Big\} = 0$$


I think any (and none) of your answers is correct. The market demand simply depends on how the family makes its purchasing decisions.

For example, suppose the matriarch $i$ decides on every purchase of the family and acts selfishly. Then obviously the utility function of the family is simply $u_i$.

Compare this with the case where the utility functions are cardinally comparable and the household is utilitarian. In this case, the family utility function is the sum of all utility functions.

More generally speaking, if purchase decisions are made via some game, then you have to solve the game first in order to obtain the demand functions. But whether a utility representation exists for the family utility and what it would look like crucially depends on the game structure.

One special case which has been discussed in the literature is the case where the household income is somehow distributed among family members and each family member spends separately. There then exists a class of utility functions for which the family expenditure will be independent of the distribution of income among the family members. For details, see https://en.wikipedia.org/wiki/Gorman_polar_form


You are correct that it is not possible to violate Weak Pareto and Nondictatorship at the same time.  But your explanation (second paragraph) is a bit muddled.  Here is how I would put it.

To prove, "it is not possible to violate Weak Pareto and Nondictatorship at the same time", it suffices to prove:  "Any Dictatorship satisfies Weak Pareto."  To prove this, suppose F is a social choice rule in which voter i is a dictator.  So for any alternatives a and b, if i prefers a over b, then so does the output of rule F.  To see that F satisfies weak Pareto, let a and b be social alternatives, and suppose that everyone prefers a over b.  Then in particular, i does.  Thus, the output of F prefers a over b (because i is a dictator in F).  Therefore, F satisfies Weak Pareto.


Why not just look at the money supply to calculate inflation?

To an extent yes, tracking the money supply is somewhat helpful in determining inflation, but there are other important factors to consider.

Productivity?

Say there was a massive crop failure and the same amount of money.  Food prices (and prices in general) would go up, right?

What is money?

Do we just count government money (monetary base)?  There are aggregates for that, but if you've accepted a check for services before you know that you also accept bank money which does not have a 1 to 1 relationship with government money.  So you have to include bank money to determine how much money is the economy.  But what types of bank money?  Demand deposits?  M1?  Interest bearing deposits M2?  M3?  This is subjective...but it helps to look at many aggregates.

What about foreign trade?

If you count the money supply do you include say dollars in the USA or dollars abroad as well?  In some cases dollars are used a medium of exchange abroad and don't really come home, so they don't influence domestic prices like domestic dollars do.

What about foreign exchange

Currencies compete, and because they are fiat there is always the possibility that one currency (especially a smaller one in a weakened state) will suffer a confidence crisis and crash.  This can influence prices, yet the amount of money in the economy can stay the same.

So truely, you need to look at many factors to figure out inflation.  All that being said though the supply of money is still something very important to keep an eye on.


Signalling models and out of equilibrium beliefs

In my interpretation of the model, competitive firms imply that the wage always equals the expected productivity, which depends on the beliefs. Clearly, in any pooling equilibrium the on-path beliefs are equal to the prior such that the wage is simply $E[\theta]$. If, in your example, a worker sends the off-path messags $e=1$, the wage must be $w(e=1)=w_1=b_1 \theta_H + (1-b_1) \theta_L$, where $b_1$ is the belief in the information set of $(e=1)$ that the type is high. Since $b_1 \in [0,1]$, it must be that $w_1 \in [\theta_L,\theta_H]$.
This information set is not reached on equilibrium path such that we are free to determine the out-of-equilibrium belief $b_{1}$. Only $b_{0}=p_H$ is a PBE-consistency requirement.
The set of all no-education pooling PBE is then given by the following:
$e_H=e_L=0$,
$w_{0}=E[\theta]$,
$w_{1}=b_{1}\theta_H + (1-b_{1})\theta_L$,
$ b_{0}=p_H$,
$b_{1}: b_{1}\theta_H + (1-b_{1})\theta_L -c_H\leq E[\theta].$
Note that the beliefs are part of the equilibrium. To get rid of equilibrium multiplicity, have a look at the numerous refinements such as the intuitive criterion that kills unreasonable out-of-equilibrium beliefs (and thereby unreasonable equilibria). The restriction on belief $b_1$ originates from the incentive constraint of the high type. We want to make sure that $\theta_H$ does not want to deviate get eductation, and because the wages are competitive they are implied by the off-path belief which we can set freely.
The idea to set $w_1<\theta_L$ would violate the competitive nature of firms.
This sentence can be interpreted in two ways.

Wages are just exogenously set this way.
There are two firms (having the same beliefs) who both set a wage menu $(w_1,w_2)$ and a la Bertrand the worker always goes to the firm setting the higher wage.
For some off-path beliefs, one firm would deviate from $w_1<\theta_L$ and may attract workers by setting a wage $w_1=b_{1}\theta_H + (1-b_{1})\theta_L - \varepsilon$ and thereby make a profit. Because you did not specify the off-path belief, we have no idea whether this deviation on the wage would be profitable.


Note, there is a mistake in FOC---it should be
$$\beta/(g_1+g_2) - 1 =0,$$
so the response function of firm $2$ is
$$g_2 = \beta - g_1.$$

This means that a unit increase in a public good contributed by firm $1$ causes firm $2$ to decrease its contribution by the same unit. Then, notice that firm $1$, who correctly anticipates such a (optimal) response by firm $2$, is better off by not contributing anything (hint: $g_1$ only affects $C(g_1)$ up to $g_1 \in [0,\beta]$).

I guess you can take from here to solve the second part


What makes the Balance of Payments (BoP) to imbalance?

This depends on the exchange rate regime. For a floating exchange rate, the balance of payments is always in equilibrium, that is, the financial account always offsets the current (and capital) account. A hypothetical deficit is avoided by a depreciating exchange rate, a hypothetical surplus is avoided by appreciation.

For a fixed exchange rate regime, the central bank will offset the exchange rate adjustments that guarantee the equilibrium under floating exchange rates. In order to offset these adjustments, the CB has to intervene in the exchange markets. To prevent a currency from depreciating, the CB has to use its foreign reserves to buy its own currency. To avoid an appreciation, the CB has to buy foreign currency and icrease its foreign reserves.

To sum up, as an answer to your question: BoP imbalances only occur in a fixed exchange rate regime and are caused by foreign exchange interventions of the CB. Therefore, the imbalance is equal to the settlements balance, or the change in the CB's reserve assets.


Diminishing Nature of Utility Function From Convexity of Preference $\succsim$?

Diminishing marginal utility is a cardinal property. In other words: it is not invariant to arbitrary monotonic transformations. This indicates that it cannot be associated with any restriction on preferences (i.e., over a binary relation over consumption), since restrictions on preferences are by nature ordinal.

To see this, consider preferences defined over $\mathbb{R}_{++}$, perhaps thought of as monetary outcomes. If we take $x \succeq y$ if and only if $x \geq y$, we have a convex preference. Notice that both $U(x) = ln(x)$ and $U(x) = e^x$ represent our preference. Both are quasi-concave functions, although only $ln(x)$ is also concave.

If you are willing to entertain additional functional restrictions on $U$, then you can find axiomatic restrictions on $\succeq$ guaranteeing the concavity of $U$ over $\mathbb R$. The obvious example is the expected utility model, which assumes that $U$ is linear over the set of simple lotteries over $\mathbb R$, i.e., takes the form:
$$U(p) = \sum_{supp(p)} p_i u(x_i)$$
where $p$ is a lottery over $\mathbb R$. In this case, the risk aversion axiom---that the certainty equivalent of $p$ is less than the expected value of $p$---will ensure the concavity of $u$.

This is possible only because we have taken a stand on the cardinal normalization of $U$. In other words, any EU representation of $\succeq$ will have a concave $u$ but there still exist other convex non-expected utility representations.


If banks started simply depositing money with each other, then what else?

The purpose of paying interest is to get people to lend you their money (by depositing it) so you can lend it out at a higher rate of interest. The argument here is that if the deposits outstrip the bank's ability to invest/lend, they don't need to offer as much interest to get those deposits. However, given current banking practice i don't think there is ever going to be more deposited funds than the bank can find profitable uses for, so the thought experiment sorta falls apart.


In the Ricardian model, why are two out of 3 cases invalid?

To "specialize completely" means a country only produces one of the two goods.

Given your firms' (there seem to be four, one for each good in each country) profit maximization problems their optimality conditions are
$$
p_i \cdot \alpha_i^j \leq w^j
$$
with equality if $l_i^j > 0$. From this, you can easily derive what parameter combinations of $p_i$ and $\alpha_i^j$ will result in conditions when a country specializes completely.

I am not sure what you mean by 'fail', and the indices in your workforce notation seem a little off. Perhaps my detailed discussion of the Ricardian model can be of benefit. Beware it is not the exact same model as yours. For example, prices are not given exogenously, but they result from a supply and demand equilibrium.


Why does an increase in GDP lead to an increase average incomes?

In a nutshell, a higher GDP is the very same as an increase in average incomes, at least if the population remains constant. If there is more stuff produced and consumed, overall income is higher and average income as well.

As you rightly point out, since income is measured in monetary terms, one has to correct for price increases. And this is usually done. Whenever you hear 'GDP has increased', this better be in real terms, i.e. in constant prices. In nominal terms, GDP (as well as tax revenues) are increasing almost every year, which renders nominal comparisons usually meaningless.


Optimization problem with Kuhn-Tucker conditions

Player $1$'s utility maximization problem is the following :

$$\max_{0 \leq t_{12} \leq T} \ \ \left(T-t_{12} + t_{12}t_{21}\right)^a \left(w(T-t_{12})\right)^{1-a}$$

An equivalent way to solve the above problem is maximizing the $\log$ of the objective i.e. one can solve

$$\max_{0 \leq t_{12} \leq T} \ \ a\ln\left(T-t_{12} + t_{12}t_{21}\right) +(1-a)\ln  \left(w\right) + (1-a)\ln  \left(T-t_{12}\right) $$

Solving it we get the best response function of player 1 as

\begin{eqnarray*} t_{12} = \begin{cases} 0 & \text{if } t_{21} \leq\frac{1}{a} \\ \frac{T(at_{21}-1)}{(t_{21}-1)} & \text{if } t_{21} > \frac{1}{a} \end{cases} \end{eqnarray*}

Likewise, the best response function of player 2 is

\begin{eqnarray*} t_{21} = \begin{cases} 0 & \text{if } t_{12} \leq  \frac{1}{a} \\ \frac{T(at_{12}-1)}{(t_{12}-1)} & \text{if } t_{12} \geq  \frac{1}{a} \end{cases} \end{eqnarray*}

Now you can solve the best responses to get the Nash Equilibria.


The purpose of a 2nd price auction (Vickrey Auction) is to encourage bidders to bid their true maximum willing payment for a good rather than the lowest price they expect will win the auction. The reason for this is because they pay the value of the unknown second bid they don't have to worry about bidding to high because the highest bid will always pay the exact lowest they would have had to pay in order to obtain the product and don't have to worry about overbidding.

For a auction to be called an auction there must be multiple bidders. If I know that I am the only bidder I would always just bid $.01 and get everything for free. To prevent this and other types of bidder collusion all auction houses have a minimum starting bid for a product usually determined by the seller. If no one is willing to pay more than the minimum then the product simply wont be sold. Individual rule for vickrey auctions differ depending on the venue but I imagine if only one bid is made then the price would default to the minimum bid set before the auction.


Social planner's first order condition (current-value Lagrangian)

At $t=0$, consumption in future periods is indeed uncertain because output is uncertain since $A_t$ is stochastic. But the maximization we are doing here is taking place in each period $t$. In each period $t$, the realization of $c_t$ is observed; that is, $E_t[c_t] = c_t$. By the same logic, $E_t[\lambda_t] = \lambda_t$. Since we are maximizing in each period $t$, the first-order condition of $\mathcal{L}$ with respect to $c_t$ is not a summation. Instead, $\frac{\partial \mathcal{L}}{\partial c_t} = 0$ implies

$$
\begin{align}
\beta^t E_t[c_t^{-\sigma}] - \beta^tE_t[\lambda_t] &= 0 \\
\beta^t c_t^{-\sigma} - \beta^t \lambda_t = 0.
\end{align}
$$

Note that if the coefficient of relative risk aversion were not assumed to be constant across time; that is, if $\sigma_t \neq \sigma$, then the result we wish to show does not follow. Just some subtleties about expectations to keep in mind.

But since indeed $\sigma_t = \sigma$, it then follows that

$$c_t^{-\sigma} = \lambda_t.$$


The set $\{\mathbb{p}:\exists \alpha \in (0,1), \mathbb{p}_1=\alpha b_i, \mathbb{p}_2=(1-\alpha)b_i\}$ coincides with the hyperplane $\{\mathbb{p}:p_1+p_2=b_i\}$, therefore you are not finding extra hyperplanes by doing this and varying $\alpha$.

More generally, if you fix $x=(x_1,x_2)$ and $b_i$, you obtain one hyperplane characterized by the equation $\{\mathbb{p}:p_1 x_1+p_2 x_2=b_i\}$. Since there are finitely many such $x$ (and $b_i$ is given), there are finitely many such hyperplanes.


Substitution effect on investment due to tax credit

I think we should regard these more like supply curves. Moreover, I will regard this economy being composed of agents with heterogeneous levels of profitability ratios between business and residential investment.

In the business side, there will be an average return on investment curve, and at lower investment levels, there will be higher returns and vice versa, which makes this look more like a demand curve. Moreover, this might be regarded as an exogenous curve in your scenario.

From the perspective of investors, when the average return on investment is low, only those who have higher levels of private returns to investment (compared to residential investment) will invest in business and more people will be attracted to business investment as the return increases.

A similar picture would hold for the residential investment. The economy arrives at an equilibrium when each of these markets are in equilibrium.

A person would allocate his wealth among his choices based on the additional gains per cost of each alternative. Here, when you have lower the cost of business investment by reducing taxes, all agents will find business investment comparably desirable, and there will be less people investing in residential industry.

So the supply in residential investment will be lower simultaneously with the increase in supply of the business investment.

Of course you can consider scenarios in which an investor's optimal decision requires diversifying to some degree, but as long as the complementarity is not extremely strong, which would be expected in this case, the economy-wide supply curves would shift in opposing directions.


not following a step in a paper on rational expectations

Presumably $I_t$ stands for the information set at time $t$. So note that $E[\epsilon_j|I_t] = \epsilon_j$ $\forall j \leq t$. This will come into play when working out the LHS.

We wish to verify Equation (4), which is

$$
E(p_{t+1}|I_t) = \frac{\Psi(B) - \psi_0}{B} \epsilon_t,
$$

which is equivalent to

$$
$$

$$
\begin{align*}
B(E(p_{t+1}|I_t)) = \Psi(B) \epsilon_t - \psi_0 \epsilon_t. &&(*)
\end{align*}
$$

$$
$$

We are given that

$$
\Psi(B) \epsilon_t = \psi_0 \epsilon_t + \psi_1 \epsilon_{t-1} + \psi_2 \epsilon_{t-2} + \dots + \psi_k \epsilon_{t-k} + \dots
$$

so the RHS of $(*)$ is

$$
$$

$$
\boldsymbol{\Psi(B) \epsilon_t - \psi_0 \epsilon_t = \psi_1 \epsilon_{t-1} + \psi_2 \epsilon_{t-2} + \dots + \psi_{k} \epsilon_{t-k} + \dots}.
$$

$$
$$

Now, the LHS. Since $p_t = \psi_0 \epsilon_t + \psi_1 \epsilon_{t-1} + \psi_2 \epsilon_{t-2} + \dots + \psi_k \epsilon_{t-k} + \dots,$

$$
$$

$$
\begin{align*}
E(p_{t+1}|I_t) &= \psi_0 E(\epsilon_{t+1}|I_t) + \psi_1 E(\epsilon_t|I_t) + \psi_2 E(\epsilon_{t-1}|I_t) + \dots  + \psi_k E(\epsilon_{t-k+1}|I_t) + \dots \\
&= \psi_1 \epsilon_{t} + \psi_2 \epsilon_{t-1} + \dots + \psi_{k} \epsilon_{t-k+1} + \dots
\end{align*}
$$

$$
$$

since $E(\epsilon_{t+1}|I_t) = 0$ because $\epsilon$ is an independent white noise process.

Given that $B(\cdot)$ is the lag operator, the LHS of $(*)$ is thus

$$
$$

$$
\begin{align*}
\boldsymbol{B(E(p_{t+1}|I_t))} &= B(\psi_1 \epsilon_{t} + \psi_2 \epsilon_{t-1} + \dots + \psi_{k} \epsilon_{t-k+1} + \dots) \\
&= \psi_1 B(\epsilon_{t}) + \psi_2 B(\epsilon_{t-1}) + \dots + \psi_{k} B(\epsilon_{t-k+1}) + \dots \\
&= \boldsymbol{\psi_1 \epsilon_{t-1} + \psi_2 \epsilon_{t-2} + \dots + \psi_{k} \epsilon_{t-k} + \dots}.
\end{align*}
$$

$$
$$

Thus the LHS = RHS; that is,

$$
\begin{align*}
E(p_{t+1}|I_t) = \frac{\Psi(B) - \psi_0}{B} \epsilon_t. &&\blacksquare
\end{align*}
$$


Which macroeconomic issues can be considered "curable" and which can't?

Friedman's assessment that "inflation is always and everywhere a monetary phenomenon" seems to have substantial concensus around it empirically: (http://www.nber.org/papers/w1453)

I think this essentially gave us the cure for both hyperinflation (don't monetize government spending or the debt more generally) and deflation (print money or do QE until the price level goes up). That isn't to say we've eradicated these "diseases" because we don't always follow the cure. Or even countries should take the cure. Following the cure can be politically untenable and hyperinflation could well be better than revolution.


The min operation is known as a Leontief production function http://en.wikipedia.org/wiki/Leontief_production_function

You're right that with Leontief production functions, the equilibrium is always aE=uK, which is to say that production is linear, and factors are consumed in fixed proportion. There are some real world examples for this: to drive, for example, you need a car and wheels in an exact 1:4 proportion, and having 6 wheels per car doesn't let you produce anything more than 4 wheels per car. But for the most part the aim is just to motivate a linear approximation to a multi-sector economy, which we can easily plug into a computer and run economy-wide computations with. We didn't have high powered computers in those days (c. 1950), so computing multi-sector non-linear production technologies was simply out of the question.


Pricing a negative externality under high uncertainty and severe non-linearity

This should probably be a comment, but it's too long so I am posting it as an answer.

I'm not sure I necessarily agree with "[A Pigouvian tax] is a fixed unit price, which cannot be efficient if the damage costs are highly non-linear." But I do definitely agree with the fact that non-linearity causes practical problems for a Pigovian implementation, as I note below.

Suppose that the private benefit from emissions is $\pi(e)$, and that emissions cause external damages of $d(e)$. Assume that more emissions lead to greater damage ($d'(e)>0$), that the marginal damages are increasing ($d''(e)>0$—so damages are non-linear), and that profits are concave ($\pi''(e)\leq0$).

The self-interested individual will maximise his payoff by solving $\pi'(e)=0$. In contrast, social welfare is maximised when $\pi'(e)-d'(e)=0$. As usual, there is an inefficiently high level of emissions.

Write $e^*$ for the level of emissions that is socially optimal, and suppose that we set a Pigouvian tax at $t=d'(e^*)$ per unit. This means that the individual's optimisation problem is now
$$\max_e \pi(e)-te=\max_e \pi(e)-d'(e^*)e.$$
The corresponding first-order condition is
$$\pi'(e)-d'(e^*)=0.$$
This is solved by $e=e^*$, so the private individual is therefore induced to choose the efficient level of emissions even though the external damages are non-linear. Intuitively, if decisions are made at the margin then all the Pigouvian tax needs to do is to ensure that the private marginal benefit is equal to zero exactly when the social marginal benefit it. That the tax does not equal the external cost for all of the infra-marginal units has no bearing on the marginal decision problem.

Although the Pigouvian tax implements the efficient level of emissions, its implementation has important normative consequences. In particular, the polluting individual is paying a tax of $d'(e^*)$ for every unit of emission, even though the marginal external cost of the first $e^*$ units is less than this (because $d$ is convex). This implies that the polluter is required to pay more than the cost that he imposes on society, which some might not consider equitable.

There is also an awkward inter-temporal dimension to this problem that is not considered in the above static "textbook" treatment. Because damages are non-linear, if I allow $e$ units of pollution "today" then the marginal cost (and hence the optimal level) of emissions "tomorrow" will be different to if I had allowed $e+1$ units today. This seems to suggest that what is really needed (if the solution is, indeed, to be Pigouvian) is a time-varying tax and the problem becomes one of optimal control. That is, rather than calculating a single tax rate, we need to find the time-path $\{e_t^*\}_{t=0}^{\infty}$ that characterises the optimal evolution of emissions and change the tax over time to implement this. This increases the burden on the central authority, which is now required to calculate not only the emissions it wants to allow today, but also the emissions that it anticipates allowing into the indefinite future. This makes the informational problems noted in the question significantly more acute.