Is it better for economic growth to invest in stocks or to purchase products?

The means by which purchasing stock effects economic growth is not as clear as purchasing companies' goods.  Nonetheless, there is tremendous benefit for firms (and hence the economy at large) when one decides to purchase their stock, to help understand this I recommend Why Do Companies Care About Their Stock Prices?

First in their IPO, the capital gained they can invest in R&D - as you will learn, some models posit technological shocks are the only effects on long-term economic growth
Second they continue to benefit from a high stock price in the secondary market with stronger financing and M&A's, and can continue to issue more stock at a higher price will yield more value which they will hopefully use to fund further R&D!


What does the Leontief Inverse represent? (Intuitive Meaning or Real World Concept)

Something you didn't mention is that

$x = Ax + y$

means that to produce one unit of $x$, you use $A$ unit of $x$. E.g. you need electricity to produce electricity.

Under the condition that $1>|A|\geq0$

$(1-A)^{-1} = \sum_{k=0}^\infty A^k$

Which allows us to write

$x = Ly = \left(\sum_{k=0}^\infty A^k \right)y$

Thus, for one unit of $y$, your (scalar) $L$ expressed as an infinite sum can be interpreted as the length of the engaged auto-production chain.

A little illustration

Note that all of the above can be generalized in matrix terms, i.e.

$\boldsymbol{x} = \left(\boldsymbol{I} - \boldsymbol{A}\right)^{-1}\boldsymbol{y}$

where $\boldsymbol{x}$ and $\boldsymbol{y}$ are column vectors and $\boldsymbol{I}$ and $\boldsymbol{A}$ square matrices. Each $\boldsymbol{x}$'s component may (or may not) require some of its own production as well as some units coming from other sectors, i.e. from $\boldsymbol{A}\boldsymbol{x}$. In this case,

$\left(\boldsymbol{I} - \boldsymbol{A}\right)^{-1} = \sum_{k=0}^\infty \boldsymbol{A}^k$ is true if and only if all of $\boldsymbol{A}$'s eigenvalues have absolute value smaller than $1$. $\left(\boldsymbol{I} - \boldsymbol{A}\right)^{-1}$ can be useful if you want to know how any variation -- of $y$ admitedly, but also of anything else, say, a tax -- propagates through a (co-dependent) sectorial chain.


Higher interest rates and default probabilities for longer loan term

There are two reasons why the interest rate is higher for the 60 months loans:

1- Liquidity premium
Like you said, since the money is immobilized for a longer period of time, it's normal to expect a higher return on the investment. It's also a way of protecting itself in case the interest rate increases, because the longer the term of your loan, the more you are exposed to changes in the interest rate.

2- Risk
Like you said yourself, the risk of default is significantly higher when the term of the loan is longer. Following the basic rule that more risk means higher returns, it's clear that the rates will be higher for 60 months than for 36 months loans.

As for why the default rate is higher for 60 months loans than for 36 months, I'm not very knowledgeable on the subject, but if we imagine that there is a fixed chance that the borrower defaults each month, we can easily imagine why more months means more chances of default.


Why monthly present in on line monthly payment calculator calculates as annual percent divided by 12?

Basically the way that APR is calculated varies depending on the industry and who quoted it. On a car they calculate the APR by taking the monthly percent and multiplying by 12, the calculator is actually right.

You probably want to calculate APY (annual percentage yield). Check this article out: http://www.finweb.com/loans/how-do-they-calculate-your-car-loan-apr.html#axzz3lTowC0op


Demand function for partially subsumable products

In general, a demand function (for a certain good) gives the demanded quantity as a function of the price of the good and a set of other variables. These other variables include things like prices of the other goods, income, weather, etc.
If we consider a framework of two goods and if we excluding all other variables, except for the prices this gives two demand equations:
$$
q_1 = d_1(p_1, p_2),\\
q_2 = d_2(p_1, p_2).
$$
Usually, it is assumed that the law of demand holds, which means that demand is decreasing in own prices. If so, $\frac{\partial q_1}{\partial p_1} < 0$ and $\frac{\partial q_2}{\partial p_2} < 0$.
An interesting question is how the demand for good 1 changes if the price of good 2 changes. In a first setting, it might be that the two goods are subsitutes (e.g. apples and pears), in this case, we should have that $$
\frac{\partial q_1}{\partial p_2} > 0.
$$
the reasoning is the following. If the price of good 2 increases, its demand decreases and, in turn, the demand for good 1 increases. The idea is that consumers will buy less of the more expensive good and will (at least partially) compensate this by buying more of good 1.
A second setting is where the two goods are complements (i.e. consumed together like coffee and milk). In this case, one can expect that $\frac{\partial q_1}{\partial p_2} < 0$. If good $2$ becomes more expensive then the consumer will not only buy less of good 2 (due to its higher price) but also less of good 1.
One particular specification that is frequently used (for example in theoretical work) is the linear specification:
$$
q_1 = a_1 + b_1 p_1 + c_1 p_2\\
q_2 = a_2 + b_2 p_1 + c_2 p_2.
$$
If $b_1 < 0, c_1 > 0, b_2 > 0$ and $c_2 < 0$, this gives the correct signs to have substitutes:
$$
\frac{\partial q_1}{\partial p_1} = b_1 < 0 \text{ and }\frac{\partial q_1}{\partial p_2} = c_1 > 0\\
\frac{\partial q_2}{\partial p_1} = b_2 > 0 \text{ and } \frac{\partial q_2}{\partial p_2}  = c_2 < 0
$$

In practice we are not really interested in the exact values of the parameters $a_1, a_2, b_1, b_2, c_1$ and $c_2$, but only their signs, because this determines whether the law of demand holds and whether the goods are substitutes or complements. You can think of the values of $|c_1|$ and $|b_2|$ as the degree of substitutability (of complementarity)
Demand functions do not include something like the value of the good. You are correct that "the value of the good" is usually captured by the utility of the consumer (from buying the good). The amount of goods that gives the highest utility for the give prices is what determines the demand. So the values determine the demand, but they are not an argument of the demand functions themselves.


How does an increase in domestic interest rate increase capital inflow?

There is no domestic interest rate; there is a world interest rate.

What you presumably mean to ask is:  How is that a change in the world interest rate, caused by a change in domestic supply or demand, increases capital inflow?

Answer:  Because the foreign supply and demand for output are being held constant, an increase in the interest rate leads foreigners to produce more and consume less.  The excess has to go somewhere, i.e. it has to be lent to a non-foreigner, i.e. to a domestic citizen.


In Supply and Demand is Quantity the Amount of Inventory of the Amount of Units Sold?

Demand and supply are functions indicating the price at which a certain quantity would be traded. They represent the intentions of producers and consumers to buy or sell a given quantity at a certain price.
Reciprocally, you can consider that these curves show you, for a given price, how much the producers would sell and how much the consumers would buy.

Now in reality, for the portion of the demand curve which is higher than the supply curve (ie. Consumers would buy the quantity at a higher price than producers would sell it), the market clears the quantity in question at whatever price lies in-between. The portion of the graph where the demand curve is below the supply curve can contain some inventory: that's exactly what the order book of a stock exchange is, with bid (buy) and ask (sell) offers.


Normalise historic values given in different currencies

If you have a single country, using the CPI to deflate values does not create a serious issue with the base year. Changing the base just changes the levels of the numbers, but they are are multiplied by the same amount. This means that rankings of prices are the same.

If you convert to a single currency, which one you pick does not matter for ordering the prices. For example, it does not matter whether you choose to convert all prices to Canadian dollars or US dollars, if the price is more expensive in one country in one currency, it will also be more expensive in the other.

This equivalence breaks down when you want to inflation adjust across countries. The choice of the currency for the inflation index affects the comparability. For example, imagine that the exchange rate between two countries in unchanged between two periods, yet the inflation rate is different. The scaling ratio between the two time periods depends on which country’s inflation index you choose. (I don’t think the choice of base year matters other than by applying a scaling factor.)

(If exchange rates followed relative inflation exactly, the country for the inflation index would not matter.)

There are different approaches you could try. I’ve never run into this kind of comparison before, so I have no idea what the accepted practice is. I would look at what you are trying to compare, and see what the best basis of comparison is. One approach is to not use a CPI index, and instead use a particular price to scale quantities, such as an average hourly wage. (This would imply that all prices are converted to the equivalent number of hours of wages that the price represents.) This converts prices to a meaningful unit, and not some arbitrary CPI indexation factor.


If you can get 4% annual returns elsewhere (relevant interest rate), then the asset returning 20,000 per year is worth 20,000 / 0.04 or 500,000.

In other words, 20,000 is 4% of the asset's present value and 500,000 is 100% if its present value.


General equilibrium regarding on U= max(ax,ay) + min(x,y)

Let us name three consumers A, B and C, and two goods X and Y. Equilibrium price vector $(p_x, p_y=1)$ and allocation $((x_A, y_A), (x_B, y_B), (x_C, y_C))$ satisfy the following:

Optimality Conditions (Allocation must solve the utility maximization problem of the three consumers, i.e. it must lie on the demand functions)

$(x_A, y_A) = \begin{cases} \left(\frac{10p_x+10}{p_x}, 0\right)  & \text{if } p_x \leq \frac{1}{10}  \\ \left(10,10\right) & \text{if } \frac{1}{10} \leq p_x \leq  10 \\ \left(0, 10p_x+10\right) & \text{if } p_x \geq 10\end{cases} $
$(x_B, y_B) = \left(\frac{2(4p_x + 6)}{5p_x}, \frac{3(4p_x + 6)}{5}\right)$
$(x_C, y_C) = \left(\frac{3(4p_x + 4)}{5p_x}, \frac{2(4p_x + 4)}{5}\right)$

Feasibility Conditions

$x_A + x_B + x_C = 18$
$y_A + y_B + y_C = 20$

Solving the above gives price vector $(p_x, p_y) = (1.2, 1)$ that supports the allocation $((x_A, y_A), (x_B, y_B), (x_C, y_C)) = ((10, 10), (3.6, 6.48), (4.4,3.52))$ in equilibrium.


What is William Nordhaus' contribution to the economics of climate change?

Summary
Bill Nordhaus has made several significant contributions to the economics of climate change. That's something of an understatement. His contributions are not just in the pricing of carbon; they but span microeconomics, macroeconomics, (econometric and other) modelling, the philosophy of economics, and the economics of innovation.
Modelling
He's been one of several key figures in the development of integrated assessment modelling - see for example his paper in Science, An Optimal Transition Path for Controlling Greenhouse Gases, from 1992. Such modelling became a cornerstone of the Stern Review (through Chris Hope's PAGE model), and is a live and ongoing area of development at several major universities.
Philosophy
He's been a significant contributor to the discussions about how to design more meaningful measures of economic welfare, including his 1972 book chapter with James Tobin, Is Growth Obsolete: the discussions about what economic growth is, and the necessity of capturing the value of what we now call natural capital and well-being, have been core to the evaluation of the potential damage costs of anthropogenic global rapid climate change.
Innovation
He's "carried out extensive research on the determinants of technological change" - the words (p38) of the International Association of Energy Economists, which awarded him their 2011 "Outstanding Contributions to The Profession Award": this work has been important in the discussions about climate change mitigation, for example with regard to induced innovation, and learning curves for decarbonisation technologies.
Energy resources
There's also his 1973 discussion of the allocation of energy resources, which remains pertinent to the discussion of climate change mitigation today, though the forecasts about dominant technologies turned out to be a mix of prescient and ... less prescient.
And finally
His full CV is here, and shows both the breadth and depth of his interests.
So while it's fair to say that his discussion of the application of a Pigouvian tax to greenhouse gas emissions is significant, it's just one part of a much larger body of work on the distortions caused by system boundaries that were drawn far too narrowly within neoclassical economics; and possible remedies to those distortions.


Is it necessary to have currency in an ideal centrally-planned economy?

How does the planner know that his central plan is the ideal one? The conventional story is that he's got an objective function that maximizes the sum of all the society's members' utility. He maximizes that sum by :

Setting the members' marginal utilities equal.
Allocating inputs efficiently so the output lies on the production possibility frontier (Production Efficiency)
Choosing the mix out outputs to put to their most valuable purpose (Product-Mix Efficiency)
Allocating outputs efficiently such that no resources can be redeployed to make someone better off without making anyone worse off.

How does he do this? He does this with prices. These may be shadow prices, relative prices without currency (barter), or they might be conventional prices quoted in the numeraire good. Explaining this is probably beyond the scope of a stackexchange article. This review may be of some use but the content in a microeconomics textbook would be similar.

So the price of goods are not only not "definitely meaningless" but essential to the planner's activities. However, you do not need currency in an ideal centrally-planned economy. It is possible to write down models with prices but without currency (credit models) or even money (barter models) which nevertheless have central planners that can easily maximize welfare.


Lee and Saez (2012): Pareto-Improvement?

(Note that this answer implicitly makes reference to the specific model in Lee and Saez.)

Short answer: the increased taxes on high-skilled workers exactly offset the higher real wages they obtain from a decline in the minimum wage for low-skilled workers.

Longer answer: Suppose that I'm the government, and I decide to lower the minimum wage $\bar{w}$. The direct effect will make low-skilled workers currently earning the minimum wage worse off, while making high-skilled workers better off. (The fall in low-skilled wages means an increase in high-skilled wages.)

Furthermore, a lower minimum wage means that fewer low-skilled workers will be rationed out of the labor market. These workers will be better off.

As the government, I want to turn this into a Pareto improvement - which it currently isn't, because workers who are already earning the minimum wage are hurt by the decrease. I try the simplest possible offset: I adjust taxes so that everyone's after-tax wage rate is exactly the same as before. The combination of this tax rate change and the minimum wage decrease means that everyone's welfare is unchanged, except for some low-skilled workers who were previously unemployed and now can work. These workers are better off - hence overall we have a Pareto improvement. Great!

There's only one catch: I didn't verify that this policy was feasible for the government. Maybe the proposed change in tax rates would violate the government's budget constraint.

This is where some slightly more involved logic comes in, and it's useful to think about the infinitesimal case for simplicity. Before the policy change, output is produced by a mix of low- and high-skilled workers, who I'll call the "old" workers. The minimum wage increase leads to additional low-skilled workers entering the mix; I'll call them the "new" workers. These new workers produce additional output and earn income. At the margin, though, (pretax) wage equals marginal product - so when an infinitesimal number of new workers is added, they increase output by exactly as much as they draw away in earnings, and the total earnings of the old workers are unchanged. There's a redistribution in earnings among the old workers, toward the high-skilled and away from the low-skilled, but this is just a zero-sum redistribution, and the government can use taxes to reverse a zero-sum redistribution while maintaining a balanced budget - so this part is fine.

All we need to worry about now is the budgetary impact of the new low-skilled workers. But this is simple: we're initially taxing low-skilled work at some rate $\tau_1$, and the net budgetary effect of more low-skilled entry will be positive (even after the infinitesimal decrease in $\tau_1$ that's part of the reform) as long as $\tau_1>0$. New workers are good for the budget as long as their tax rate is positive.

So that wraps it up: we see that there is a potential Pareto improvement involving a decrease in the minimum wage and an offsetting change in tax rates, and this policy change is feasible as long as the initial tax rate on minimum wage work is positive: $\tau_1>0$. Lee and Saez emphasize this point (in conjunction with their other results) to demonstrate that in their model, the minimum wage can only be rationalized as a policy that's complementary to a labor subsidy $\tau_1\leq 0$. They really like this purported complementarity, and it's one of the most popular current arguments in favor of minimum wages.

(I happen to think that the way they prove this complementarity, which relies on the assume-a-can-opener assumption of efficient labor market rationing, is incredibly silly. But that's a different part of the paper, not directly related to your question here.)


Did the Nixon shock cause skyrocketing real estate prices in the US?

Looking at gold values is simply cherry-picking. From 1970 to 1980, the CPI went from 38.8 to 82.4, an increase of 112%, and an annualized rate of 7.8%. Your chart shows housing prices increasing about 200%, an annualized rate of about 11.6%. So while most of the price increase was due to inflation, there are 3.8 percentage points that remain unexplained.

One can also compare the dollar to other currencies. While the $/£ rate fluctuated a lot in this period, from 1970 to 1980, overall the dollar actually strengthened against the pound slightly, and from 1970 to 1976 its strength more than doubled. The Yen exchange seems to show the opposite trend, however.


Is there a measure of similarity for economies?

To the best of my knowledge, there isn't. Ant that is actually a good thing, the economic structure of a country is usually a fairly complicated thing to be able to simplify it with a single number. You could, on the other hand, create an aggregate of measurements and try to look for patterns, e.g. you can see the behavior of macroeconomic variables such as gdp, inflation rate and unemployment, and based on those decide which countries are similar to each other.

And that is why clustering analysis is for. In the plot below I used Principal Component Analysis (just to reduce the dimensionality of the problem and be able to graphically display what's going on) to show 4 clusters fo countries. Each color is a k-means cluster, the axis are just the two firs principal components $x_1$ and $x_2$

I also added some of the countries on each cluster so you can see their similarity. For example, the $\color{blue}{\rm blue}$ cluster has large gdp, low unemployment, whereas countries in the $\color{green}{\rm green}$ cluster are not doing so well: high inflation, high unemployment and low gdp.

The cluster that you are interested in has these values (as of 2017)

                 country     gdp  inflation  unemployment
---------------------------------------------------------
9              Australia   55692       1.95           5.3
13               Austria   47347       2.08           5.1
17               Belgium   43488       2.13           6.2
15                Canada   45094       1.60           5.6
8                Denmark   56630       1.15           4.8
14               Finland   45927       0.75           7.2
20                France   39932       1.03           8.9
16               Germany   44769       1.74           3.3
3                Iceland   70248       1.76           2.9
4                Ireland   68710       0.34           5.3
19                Israel   40272       0.24           3.7
22                 Japan   38448       0.47           2.4
0             Luxembourg  105863       1.73           5.0
11           Netherlands   48555       1.38           3.5
18           New Zealand   41572       1.85           3.9
2                 Norway   75389       1.88           4.0
5                  Qatar   61024       0.40           0.6
12            San Marino   47595       1.05           8.0
10                Sweden   52925       1.79           6.3
1            Switzerland   80637       0.53           2.4
23  United Arab Emirates   37732       1.97           1.6
21        United Kingdom   39800       2.69           4.0
6          United States   59792       2.13           3.9


Why economic rent does not exist in perfect competition

Suppose the competitive price is P. If you quote at a a higher price Ph2>P, then someone else, say firm 2, will quote at a slightly lower price than you, say Ph2, and Ph1>Ph2>P. Since competition is perfect, firm 2 will capture all the market and you will go bankrupt. But firm 3 may put price Ph3 such that Ph2>Ph3>P and capture all the market. This process can be repeated until the equilibrium price is reached.

On the other hand, if you place a price that is below the competitive price, you will go bankrupt.

Hence, there is no economic rent because all firms in the competitive equilibrium will choose the equilibrium price which gives them an economic profit(rent) of zero.


Not my field but I think this is a basic overview. If there are mistakes, please let me know and I can edit.

The monetary authority in the U.S. uses open-market purchases to manipulate a chosen policy instrument in an effort to effect macroeconomic aggregates. There is a general consensus now that the Fed's instrument is the effective federal funds rate, which is the weighted average of the rates at which banking institutions loan money to one another. So, monetary authorities use open market purchases in an effort to make this weighted average (the 'effective' rate) match their target rate.

There is another rate that matters here - the discount rate. The discount rate is the cost that banking institutions face to borrow funds from the Fed's discount window (which is a separate instrument of monetary policy). And note that the discount rate is distinctly different from the FFR. This rate, historically, trends slightly higher than the FFR.

The two rates above effectively allow the Fed to control the cost at which banks can borrow money. The first rate governs the cost of borrowing from other banking institutions. The second rate governs the cost of borrowing (for banks) from the Fed.

If I understand you correctly, you are also curious about the interest paid on deposits held by the Fed. That is, the interest that banking institutions earn on deposits with the Fed.

The Fed pays interest both on required reserves and on excess reserves.

Interest on Required Reserves (IORR) - this rate is set by the board and is intended to offset the implicit tax that required reserves imposed on banks.

Excess Reserves - The Fed began paying interest on excess reserves in 2008. According to Bernanke, this rate is another tool of policy that allows the Fed to set a floor on the overnight rate (synonymous with FFR).

Sources that will allow you to gain more depth:

excess reserves

discount window/discount rate

federal funds rate

open market operations


Existence of a subgame perfect Nash-equilibrium

The following proposition is well known:

  If a stage game $G$ has a unique Nash equilibrium, then for any finite $T$, the repeated game $G(T)$ has a unique subgame perfect equilibrium outcome in which the Nash equilibrium of $G$ is played in every stage.

Since your stage game has a unique NE of $(T,R)$, this must be the outcome of the SPE of any finitely repeated game based on this stage game. Specifically regarding your question, it is impossible for $(B,L)$ to be played (in any stage including the first) in any SPE of the $T$-times repeated game.


Why do firms adjust their fixed costs in response to a change in price in a perfectly competitive market?

I think you're right - entry should not shift the SRAC in a normal model of a market. The textbook pages that you include read very oddly to me. (It is possible that this is because the context being described is somehow very specific, so I may be unfair in the criticism that follows).

A potential entrant compares price to min of ATC and sees that there is a profit to be made. It enters and produces the quantity such that it achieves min ATC. What does this do at the level of the market? At the clearing price before entry, there is now excess supply equal to the output of the new firm. Somehow ... and introductory models of perfect competition and price formation are usually silent on exactly how this happens ... the price must fall for there not to be unsold goods in the market.

There are lots of unsaid things in the above story. What do other firms do in response to the entry? How exactly are prices formed. One natural answer is  Bertrand price competition: the new firm undercuts the clearing price; all customers switch to it; it supplies the whole market ... upon which other firms react by undercutting it ... until undercutting no longer makes sense because each firm is at the min of the ATC. Again, there is lots here that is unsaid (are fixed costs sunk or not? what determines capacity constraints ...)

But in a way, the root of the problem with the story told in your textbook is the upward sloping aggregate supply curve. How is this compatible with a micro foundation of perfect competition? If each firm can produce according to the ATC function given, then the aggregate supply curve is actually flat. Under the assumptions of perfect competition, you can enter without constraint. So it is always possible and profitable to produce at the min ATC. So the aggregate supply curve is not upward sloping. I think the text book is trying to bring together two models that don't really fit coherently, and you have picked up the inconsistency involved.

So here's what I think about your question and the answer you have posed:
1. you're right & the book's wrong
2. don't take economic models too literally as representations of reality.


Could you help me derive Eq(5) in paper Chong (2007)?

My initial derivations were wrong because I misread Eq(1) as $y_{i,t}=c_{i,t}+r_{i,t}$. It was until I read the answer posted by @AlecosPapadopoulos that I realized that the paper actually uses $y_{i,t}=c_{i,t}+r_{i,t+1}$.

However, apropos of the latest edit of OP's inquiry (now to discuss the intuitions behind [the paper]), I still posit that the paper is a subtle promotion of socialism premised on flawed or impractical notions.

The paper makes the point (and Alecos' answer presents a clear derivation of Eqs(5)) that the optimal value of rent seeking $r_{i,t+1}$ is $0$ for strong institutions, which are denoted by $w=0$. This means that no household should pursue rent seeking. Or equivalently (see Eq(2)), that no household should seek to appropriate a larger share of the resource (page 6). Regardless of the model's time horizon (generations instead of shorter periods), that differs very little (if at all) from socialism.

Additionally, since $y_{i,t}=c_{i,t}+r_{i,t+1}$, the optimal choice in strong institutions would be $y_{i,t}=c_{i,t}$. That is, the household should allocate its entire income to consumption. Obviously, that generalized policy renders a real-life economy highly vulnerable to contingencies and structural changes. That vulnerability inherently contradicts any notion of "strong" institution.

Furthermore, substituting $w=0$ and "optimal" $r_{i,t+1}=0$ in Eq(2) gives $a_{i,t+1}=A\frac{0^0}{\int^1_0{0^0di}}$. What are we supposed to do with that?

Another problem is the model's inter-generational framework. I appreciate Alecos' emphasis/clarification that different values of $t$ for a same household $i$ allude to different generations of that household.

But that modeling horizon is hardly realistic because each household experiences too many or too profound transformations (oftentimes just from one generation to the next one) to be still considered a single, continued entity. Indeed, households go through marriages, divorces, remarriages, child bearing in each marriage, emigration, bankruptcy, and so forth.

It is not necessarily realistic to assume that $w$ is constant throughout generations either, as the paper subsequently assumes.


When is a model really weak?

A model is an abstraction of reality, to explain a part of it. A model is weak when it cannot explain what it's supposed to be explaining. Just adding features to a model has no intrinsic good. It's much different from a fruit salad, where usually, an increased variety in fruits will lead to a better taste. Here, adding more features makes it hard to understand which feature of the model was necessary to generate the result observed.

For example, When your focus is on variation of labor-supply within different income and skill groups, yes, lack of distortionary taxes could be a problem. If the model is not aimed at doing that, and it does fulfill its purpose without requiring those taxes, the lack therein is not a problem.

The most basic RBC model

There is no single RBC model. It's most simple (but not representative!) version is the seminal Kydland and Prescott (1982) paper. It lacks all those features that you mention, but it is aimed at explaining variations and correlations in output, investment, capital and labor. If it does that job well (I'm not going into that discussion here), it is a good model, and the lack of those characteristics is not a weakness.

The class of RBC models

The general RBC class of models are those, where variation along the balanced growth path is caused by (random) shocks to productivity. Of course, no economist really believes that a random number generator is attached to each firm and causes fluctuations in output. It's a simplification.

Hence, if you are after a weakness behind the RBC model in general, it ought to be that you believe that this simplification is false. That, whatever the reason for business cycle fluctuations we observe is not caused at the productivity wedge.

For example, some neokeynesian economists believe that it could be mood shocks. People sometimes are more or less lazy, and that recessions go really bad when you people really don't want to work (again, this is a simplification, but the point is that the relevant wedge is not at the output margin, but at the labor-supply and consumption wedge). A critique of these shocks to the household's objective function can be found in Chari, Kehoe, McGratten (2009).

These productivity shocks are, in the simple one-sector model that is mostly used isomorphic to shocks to technology. However, this doesn't mean that they necessarily are caused by them (hint at @Mustang). For example, Acemoglu and coauthors (2012) show how mathematically small shocks within firms and sectors (such as variations in the cost function, e.g. unexpected oil price changes), usually cancel each other out, but sometimes aggregate up to great shocks over the whole economy.


The premise of the question is simply false. Companies do not maximize $Q$ they maximize profit.
For example, consider trivial case of a monopoly. The demand is given by $Q = 10-p$ so demand decreases with price. Now let us suppose the monopoly has constant marginal costs $c(Q)' = 2 \implies c= 2Q$. Given this the profit function of monopoly will be given by:
$$\Pi = p Q - 2Q = (10-Q)Q -2Q$$
With FOCs:
$$ 10 -2Q -2 = 0 \implies Q^*=4 $$
So in this case the monopoly would produce exactly 4 units of products no more no less. The exact quantity of products will be different in different market structures (e.g. duopoly, oligopoly, monopolistic competition, perfect competition), and under different parameters (e.g. demand parameters costs),  but generally in all of them there will be some profit maximizing $Q$. So the premise of your question is false. Of course, firms won't just produce goods to dump them into garbage and to loose profit.


What has been the reception of Posner & Weyl’s claim that property rights lead to market power?

In narrow sense of the word the article was definitely peer reviewed as it was published  in Journal of Legal Analysis. But in this narrow sense it was most likely peer reviewed by jurists not economists.
In a broader sense of the word there are economists who had a look at the claims as well. For example, review of their book by Levine (2020) in Journal of Economic literature (one of the top ranking journals), examines their claims about monopolies and also other claims and policy proposals made in the book. The review is not very warm to the ideas of Posner and Weyl. Regarding the idea of property rights causing widespread monopoly power and misallocation Levine writes [emphasis mine]:

The chapter begins with a clear and careful description of the hold-up problem.
Here, indeed, monopoly power is a problem.
It then goes on to suggest that the hold-up
problem is ubiquitous in ownership of property—that all property is monopoly and that
the resulting misallocation might be on the
order of 25 percent of income. Here as elsewhere the book is not strong on data. There
is extensive research indicating that in some
places at some times, misallocation is a problem possibly of this order of magnitude.
Typically this research does not attribute
misallocation to a problem with private property—generally the reverse.

... Nevertheless the economics
profession does not generally view hold-up and private information problems as ubiquitous nor externalities as insoluble. In many
markets, ample substitutes are available—
for example in the housing market—so that
monopoly power is not important. In some
cases the market solves the hold-up problem relatively well—through tender offers
and other ways of packaging properties.

Pushing on: the authors argue that the
problem with the ubiquitous monopolies
created by private property can be solved by
a self-assessment wealth tax with mandatory
sale at the self-assessed price. Self-assessment
with a commitment to sell at that price is
a useful and credible solution to problems
involving assets for which there is substantial
private information. This has been a particular problem for commercial real estate and
for privately held businesses, where often
artificially low transfer prices are used to
avoid taxes. As the authors correctly say, top
researchers such as Harberger (1965) and
Cramton, Gibbons, and Klemperer (1987)
have argued for self-assessment in these
types of circumstances.
It is not immediately clear why a proposal
for solving a particular private information
problem should be viewed as a broad solution to a problem of ubiquitous monopoly...

There is more critique on this in the paper. To give you a wrap up (as mentioned in the comments under your other question), the problem with Posner and Weyl is not that their analysis would be inaccurate in some specific cases. Rather they try to generalize well understood result that holds in some very specific circumstances onto property rights in general.
There is quite large consensus in literature that property rights do not generally lead to monopoly or even market power (see Belleflamme & Peitz Industrial organization: markets and strategies for overview of many models where even monopoly does not guarantee market power and models of competition where property rights are implicit but firms have no market power). In fact as Mankiw put it in his Principles of Economics, economists view property rights as "an important prerequisite for the price system to work".
Furthermore, Acemoglu and Robinson (top developmental economists) in their magnum opus Why Nations Fail, go even so far as to state:

To be inclusive, economic institutions must feature secure private property...

Acemoglu and Robinson go further on proving that sustainable growth and economic development without inclusive institutions (including property rights) is virtually impossible.
Additionally, this work is apparently even criticized in law profession itself (see Wyman, K. M.; 2019), but as I am not a law professional I will have no further comments on that (it just seemed relevant to point that out).
Hence given the above I think it is fair to say that the claims of Posner and Weyl are not received positively by economics profession, to put it very generously.


Judging from the reference you provide, this refers to whether the set $\Theta$ is ordered or not. For example, natural numbers or the alphabet are ordered sets. In the context of moral hazard problems, examples could be effort, or ability. Since this is a numerical variable, it is an ordered set.
In the paper you mention, the phrase "one dimensional, ordered type..." is not found. However, the paper states that:

... differences among sellers are assumed to be parameterized by a single unobservable characteristic $\theta \in \Theta$. Then the family of utility functions can be written in the alternative form,
$U_i=U(\theta_i;y,p) \text{, } i \in I$
If seller $i$ has $\theta_i$ units of the characteristic we may therefore refer to him as being of "type $\theta_i$.

The first bold statement refers to the one dimensionality of the unobserved type, whereas the second refers to it being a numerical variable, which means $\Theta$ is an ordered set.
Conversely, a non-ordered set contains elements which have no intrinsic order. For instance, you can think of a case where agents do not know the city where they are to be located. The set "cities" is unordered because cities do not necessarily hold a particular order in a one-dimension set. You can add a second dimension (e.g. population), where you could order them based on city size.
I would say that having ordered sets facilitates algebraic analysis, since it enables the introduction of monotonic strategies, whereby an important outcome variable is a monotonic function of the agent's type. In effect, this seems to be important in the paper, when the author states that:

A seller of type $\theta_i$ brings to market a product that is valued equally by each of the set of potential buyers, $J$. This valuation, $V_i$, measured in dollars, is an increasing function of $\theta$ and (possibly) also of the intensity of sales-related activity, $y$:
$V_i = V(\theta_i;y)$

The bold emphasis highlights precisely the monotonicity enabled by the order property of $\Theta$. Another example is given in Athey (2001):
$\hskip1in$

In response to the second reference you give (here), in the phrase

We now show that Riley equilibria may not exist when types are not ordered.

the "ordered" adjective refers to the second interpretation given above (i.e. of monotonicity). First, they assume that

Consumers are uniformly distributed in the unit interval, $\theta in [0,1]$

which is an ordered set. However, because both the utility function and the cost function are quadratic, it results that

... the types in the extreme points have the strongest preference for one of the contracts and are the most costly to serve.

which means that the utility and cost functions are not monotonic over $\theta$.


There is no real good general answer for that without detailed knowledge of model you want to estimate. When it comes to $P_t$ people typically use $CPI_t$ as a proxy but when it comes to $M$ it depends on what sort of work you are doing. Some work in fact even estimates $M/P$ in different versions using $M_1$, $M_2$ and $M_3$ (such as Short 1979).
In typical macroeconomic model $M$ is interpreted more broadly than just base money (e.g. see discussions in Woodford Interest & Prices, or across Romer Advanced Macroeconomics). In that case using $M_2/CPI$ or even $M_3/CPI$ would make more sense if you would be forced to choose only one. But as in the above mentioned citation you can construct it in various different ways and then see if the results are robust when you switch the measurement.


A stock represents a quantity measured at one specific point in time, which  may have accumulated in the past.

According to the Bureau of Labor Statistics, the US unemployment rate stood at 4.1 percent in February 2018. This represented a stock of 6,706,000 unemployed workers in February 2018. But from January to February, we observed a flow of workers going in and out unemployment.

  A flow variable is thus measured over an interval of time, e.g., hours, days, weeks, months or years. It has time dimension.

Because of inflows and outflows of workers going in and out unemployment, between January and February, the stock of unemployed workers was a bit higher in January 2018 (=6,684,000).

A year before, in February 2017, the unemployment rate was about 4.7%, representing a stock of 7,486,000 unemployed workers at that particular time. This means that a net flow of 7,800 (=7,486,000 - 6,706,000) US workers went out unemployment between February 2017 and February 2018.

Your wage in February 2018 is a flow because it's the money you earn between Feb 1st and Feb 28. (by the way, this amount is not necessarily fixed from one month to another).


Finding the minimum # of items to be sold to meet a goal, based on likelihood

Let's start with an environment in which there is only ever a single pledge level. The question at hand is: how do we choose the dollar amount of that level to maximise revenue (income)?

Observation 1: your probability distribution implies a demand curve.

We can plot the cumulative distribution of willingness to pledge and it will probably look roughly like this (with most of the mass at the bottom, as you speculate):

The $y$ value in this figure tells us the probability that a given individual's willingness to pledge (WTP) is no greater than the dollar value on the $x$ axis.

By turning this upside down, we get the probability that somone is willing to pay at least this much:

This tells us, for example, that there is a rouly 55% chance of a randomly selected individual being willing to pledge at least \$10. If we take 100 people then, on average, 55 of them will pledge. Thus, we can think of the vertical axis as being an indirect way of measuring pledge quantities, corresponding with the pledge level (or price) on the horizontal axis. If we swap the two axes, we therefore get something that looks very much like a textbook demand curve:

Given this demand curve, we can use all of the usual tools from the kind of standard monopoly optimisation problem that you can find in and introductory economics textbook. For example, if the CDF in the first figure is given by function $F$ then$$\Pr(\text{WTP}>x)=1-F(x)$$ so the revenue from setting the pledge level at $x$ when the audience consists of $n$ potential pledgers is $R(x)=[1-F(x)]xn$. Just like a standard monopolist's problem, we can compute the revenue-maximizing pledge level ("price") by setting marginal revenue equal to zero:

$$R'(x)=n[1-F(x)]-xnF'(x)=0.$$

Substituting your empirical distribution for $F$ in this equation and solving for $x$ gives your optimal pledge level.

Now, if we allow for multiple pledge levels then things become much more complicated. Firstly, we need a model of the behaviour of people and why it is that some people might pledge more than others.

If, for example, we are working in a Kickstarter-like environment then the main motive would be to get the goodies that come with the higher pledge tier. But that means we need a model not only of the pledge levels, but what you get in return. A complete treatment of this is beyond the scope of this answer, but a fairly readable summary can be found in "Versioning Information Goods" by Hal Varian. If you want a more detailed, technical treatment then the best place to look is "The Economics of Contracts: A Primer" by Salanie.


Why'd the lender suffer the collateral haircut, not the borrower?

Why'd the lender suffer the collateral haircut

This is not an accurate way of describing what is happening. No one "suffered". The collateral was entered at lower than current market value, to account for the risk. It can also be repossessed from Steve at lower than current market value.

  Why wouldn't Steve (the lender) assign the collateral haircut to you, and require you to pay him $11 + risk premium?

That would have also been acceptable to Steve. As it is explained in the text

  $1 seems like a lot of money for one day to you. But you do need to eat so you don't have a lot of options.

So it is in fact you who did not want to pay the risk premium, and Steve went along with your alternate offer.


Is a social choice aggregation rule defined for a set of weightings over the set of voters (N)?

First it is important to define our terms.  The concept of "weighted voters" does not necessarily make sense for an arbitrary social choice rule.  So presumably we are interested in social choice rules where, at some point in the aggregation process, we add up a collection of terms, where each term contains information about the preferences or opinion of some voter.  Typically, such rules are "anonymous" in the sense that we can permute the opinions of the voters without changing the answer.  We can easily introduce a "weighted" version of such a rule by computing a weighted sum instead of an unweighted sum.  (Typically, a weighted version of the rule fails to be anonymous.)

Let us say that two social choice rules $F$ and $G$ are different if they sometimes produce different outputs when presented with identical input (e.g. when presented with identical preference profiles).  In that case, it is definitely the case that in general, different voter-weights will yield different social choice rules.

For example, let's consider "weighted majority voting" with three voters, $A$, $B$ and $C$.  One possibility is that each voter receives equal weight (say, 1/3) ---this is effectively the "unweighted" version, which satisfies anonymity.  Another possibility is that voter $A$ receives weight 5, while voters $B$ and $C$ each receive weight 1.  It is easily verified that this "weighted majority rule" is actually a dictatorship for $A$.

Of course, we might say that all the different weighted majority rules belong to the same "family" of voting rules, and therefore they all have certain properties in common.  For example, there are many theorems which are proved for the family of "all weighted majority rules"  (or "for all weighted scoring rules", or whatever).  But that does not mean that they are the "same" rule.

When you ask, "does a disagreement regarding how to weight the voters entail a disagreement regarding aggregation rule?", the answer is "yes and no".  Clearly, we might agree on the type of voting rule to use (say, we agree to use a weighted majority rule because it satisfies certain desirable properties and we all agree these properties are important), while still disagreeing on the exact weights to use in the rule.


Applying modern economics to Edgeworth's Hedonical Calculus

Hedonics comes from Benthams utilitarianism. The main problem with reifying this into mathematics is that these are not measurable. Instead, one develops proxies for such, which I'd suggest the hedonics is and it suffers from the same problem.
Whilst it's easy to say that six apples are six apples and there are no two ways about it, it is much more difficult to measure 'a happiness index' in eating apples: someone may have an allergy to them, or perhaps they had a bad experience with them.
It reminds me of the Social Realism of Stalins Soviet Union. Official pronouncements of the happiness of the people whilst the people were suffering. Of course, Capitalism has its own Capitialist Realism - just look at adverts - they are the analogue to this art: happy consumers being 'overjoyed' in recieving new products. There's a good reason when many happiness surveys show that the people aren't as happy as one might think ...


What was happening in Madagascar during 1971-1996?

I think I have found the explenation for this. If you look at the wikipdedia page of the economy of Madagascar, it states the following:

The standard of living of the Malagasy population has been declining dramatically over the past 25 years. The country has gone from being a net exporter of agricultural products in the 1960s to a net importer since 1971. Inappropriate traditional agricultural methods cause soil to erode and soil quality to decline, and the basis of survival for Madagascar’s people is under serious threat.

The source of this claim is Deutsche Gesellschaft für Internationale Zusammenarbeit.

Now, if you look at the world bank figures on agriculture in Madagascar, you can see this declining trend. For example the Agriculture value added per worker (constant 2005 US$) declines between 1980 and 2014 from 250.1 to 175.8.

Since Madagascar is a country which relies heavily on agriculture (about a quarter of its GDP stems from agriculture), this declining trend influences the whole of this economy.


What benefits do governments receive from not eliminating debt?

Most of the same considerations apply to countries as apply to businesses and people, plus a couple of extra cons

Pros of Being Debt Free

No interest payments
Not beholden to someone else (financial freedom)

Cons of Being Debt Free

Buying things on (interest free) credit can save a little money
Paying for things in installments can match costs to income
Interest rates can be used to control economic activity and defend currency
Having a bond market promotes domestic financial activity
Having government bonds provides savers with a safe investment

There is big difference between having no debt and having no net debt.  In the former case, you do not borrow any money and that is rare.  In the latter case you have the money but choose to borrow instead.  Many individuals do this by spending on credit card even though they have money in the bank, or not paying off all of their mortgage because the interest rate is good (so they can earn more with their money elsewhere).

For governments there are extra benefits from having debt (even if you can pay it off).

Perhaps the best way to understand this though is to look at a few notable examples, past and present.

US 1836

If you don't set interest rates, someone else will

The Federal Reserve was only established in 1913, prior to that the US had an uneasy relationship with the concept of central banking.

It seems the founding fathers were against central banking, and it wasn't until Alexander Hamilton that "First Bank of the United States" was created in 1791, mandated to last for 20 years, after which it's mandate was not renewed.

Second Bank was established in 1816.  Andrew Jackson was strongly against central banking (and banking generally!) and so when he came to power in 1832 he pulled the state money out of the bank.  The bank countered by tightening money supply to push the economy into recession.  Andrew Jackson held out and paid off the entire national debt by 1836.  The Second Bank did not have it's chartered renewed, and liquidated in 1838.

With no debt, and no central bank the US money supply was effectively free.  Jackson also introduces the Specie Circular which required all government land be purchased in gold and silver.  The rest of the 1830 saw significant inflation and recession, generally attributed to Jackson defeating the central bank.

The US government had lost control of their own economy, and in 1837 the Bank of England raise rates, forcing up domestic US interest rates, precipitating the 1837 panic.  The following years were marked by major recession.

For more on this see: WDJ article on US debt and Wiki: Second Bank

Norway Today

Keep some debt for liquidity purposes and financial control

Norway currently has around \$170bn of public debt, with a GDP of about \$500bn.  However in 1990 the government established what is now called the "Government Pension Fund of Norway" into which the excess income from Norwegian oil is poured.  It is an equity and bond portfolio currently estimated to be worth in excess of \$700bn.

The government of Norway could choose to pay off all its debts easily, but chooses not to.  Instead they maintain issuance in sovereign debt markets in order to hold a liquid reserve to cover their daily payments.  They also mention using the money to "develop well-functioning and efficient financial markets".  A final consideration in their case is that their assets are abroad, and repatriating them would weaken the Krone, so there is some FX consideration here.

Because of this, it is not surprising that Norway has a AAA credit rating, and that also makes it cheaper for Norway to borrow than even the US (based on 5Y CDS).

For more on this see: Norway Ministry of Finance and Norges Bank

Singapore Today

Issue debt to give people something to invest in

Singapore has had no foreign debt (ie. non-SGD) since 1995, and consistently runs with a fiscal surplus.  Despite this they issue T-Bills and Notes of various maturities consistently and simply invests the proceeds.

Why?  According to the Monetary Authority of Singapore (MAS) the main objectives are:

  Provide a liquid investment alternative with little or no risk of default for individual and institutional investors;
  Establish a liquid government bond market, which serves as a benchmark for the corporate debt securities market; and
  Encourage the development of skills relating to fixed income financial services available in Singapore.

For more on this see: MAS MoneySense

North Korea

If you don't pay, its not really debt...

This is only a semi serious one, but one country that basically has no debt is North Korea, for the simple reason that no-one will lend to them.  They do technically have debts though, including a debt to Sweden for some Volvos, but in 1984 they defaulted on them all and refused to pay anything.  It seems they have no-intention of ever paying.  I don't think I need to explain the downsides to the North Korean approach to economics.

See: North Korea's Stolen Volvos


Examples of how economists come up with a production function for a firm?

The "production function" in economics and econometrics is not an engineering concept. It is a highly abstract construction where the myriads of inputs used are lumped together in very few categories: "capital" and "labor" being the most usual ones, while sometimes inputs critical to an industry like say "Energy" may be singled out and appear in the production-function formulation. At such an abstract level, it is easily implementable without the need to go into particulars of each firm.

Moreover, in many if not most of the cases, what is modeled is the production of value added, not the total output (either in quantities or in values).

Applied work commissioned by specific (usually very large) firms, and also, applied research into "adjustment costs" or what is called "insider econometrics", may go some steps towards creating much more complex production functions, that essentially try to take into account the actual physical relations and interactions between the various inputs. But these are very specialized areas -remember that the Economics discipline ultimately is about markets and economic systems -the "individual" (firm or person) level is researched mainly in order to have sound foundations for the aggregation that follows.


How can I best estimate a company's contribution to GDP?

As a comment noted, value-added is the way to go. And how do we measure value added at the level of an individual firm?

It is

$$\text{Value-added} = \text{Wages and Salaries incl. Insurance}+\text{Depreciation}+\text{Profits before Taxes}$$

"Wages and Salaries including social security fees" is the reward of Labor, while "Depreciation plus Profits before Taxes" is the (gross) reward of Capital.

This makes sense because if one views GDP from the "income" angle, it goes to these two aggregates of productive inputs, "Labor" and "Capital".

All other expenses that appear in a company's Profit & Loss Statement are "third-party costs", or value that other productive entities create.

Depending also on the jurisidiction, in most cases Labor costs, Depreciation and Profits before Taxes can be found as separate items in the financial statements of a company.


How much of the Greece's public debt is created by its trade deficit?

As an accounting identity, it is required that the current account surplus (deficit) +  capital account surplus (deficit) = 0. This is known as the balance of payments (BOP) identity . So if Greeks are importing more Euros worth of goods and services than they are exporting it must be that, consolidating across those outside of Greece, non-Greeks are buying more Euros worth of Greek assets than they are selling.

However, this accounting identity says nothing about who in Greece is selling and who outside of Greece is buying. It need not be that the government is involved at all. If the Greek government borrows more than before the household sector might offset this greater borrowing more than one for one with greater savings (BOP decreases), exactly offset (no change in BOP), partially offset it (change in balance of payments less than increase in government spending), or also borrow (increases in BOP more than change in government spending).


Monotonicity and continuity implies that all bundles are weakly preferred to 0

Without loss of generality, suppose $\mathbf x=(x_1,0)$ where $x_1>0$. Consider the following sequence
\begin{equation}
\mathbf y^n=\left(x_1\Bigl(1-\frac1n\Bigr),\,\frac1n\right).
\end{equation}
Clearly, $\mathbf y^n\gg\mathbf 0$ for all $n\in\mathbb N$, and thus $\mathbf y^n\succsim\mathbf 0$ by monotonicity. It is also the case that $\lim_{n\to\infty}\mathbf y^n= \mathbf x$. Hence by continuity, we have $\mathbf x\succsim\mathbf 0$.


How does a national budget differ from a household budget?

If you think about it, there is really not much difference between a government budget constraint and a household budget constraint.

Both have uncertain income streams, which are labor and capital income for the household, and mostly tax revenue for the government. On the other side of the constraint, you will find consumption and saving for the household. The government will spend most of its income on (public) goods and transfers to households (social system).

Both can move income inter temporally using borrowing. However, neither can run a permanent deficit where it is "not expected to ever pay back", something that is typically referred to as a No-Ponzi-Game condition. As an implication, the "life-time budget constraint" will have to hold for both:

sum of all expected future income = sum of all expected future expenditures

I'm pretty informal here, you would typically rather derive "present value" of these constraints, but latex is not yet allowed here, and I'm already writing much.

What should government debt be used for?

We typically assume that households and governments have concave utility, they prefer consumption processes that are second-order dominating. Hence, they want to engage in smoothing of consumption over fluctuations.

Since all debt has to be paid back eventually, there is limited use of government borrowing. Most typically, one wants to stabilize consumption and output over business cycles, by financing consumption in busts and paying back in booms. Also, given an interest rate r, investments that promise a higher return than r can be financed through debt, as the government will not violate consumption smoothing.

Potential Difference

One potential difference is the commitment. Developed governments are typically more
trustworthy, they are more likely to pay back there debt. Hence, they can borrow at cheaper interest than most households. As a conclusion, they can use debt more aggressively than households for the two purposes consumption smoothing and investment, as emphasized above.

tldr: Besides that, they're very much alike. Note also the Ricardian Equivalence, which is not at the heart of this question, but very much related.


Confusions about calculations in Utility Theory Paper

The notation is a bit confusing but what the author is computing is the expected utility of the investment, not the utility of the expected outcome.

Since the investment results in a wealth of \$120 with probability 0.5 and a wealth of \$90 with probability 0.5, the expected utility equals
\begin{align*}
\mathbb{E}U & = 0.5 \times (-\dfrac{10^6}{90^3}) + 0.5 \times (-\dfrac{10^6}{120^3}) \\
& \sim -0.98
\end{align*}

Notice that this value is lower than the utility of the expected outcome, which is approximately equal to $-0.86$ as you wrote. This is because the agent is risk-averse and must therefore be compensated for the risk.


Simple arbitrage model of a single commodity

As I noted in the comment, buying a major position in a commodity does not qualify as an arbitrage, rather it would be informally referred to as “cornering the market.”

I am unsure whether there is any theoretical work on such strategies. However, from a practical perspective, we can see that it is hard to characterise what would happen.

I will assume that we are talking about a physical commodity; things might be different for a financial asset.

Firstly, going into a market and trying to buy up the supply is difficult. There will be both physical commodities to buy, as well as derivatives (such as futures). How exactly can you get the physical inventories away from other commodities traders and consumers, when they see someone buying up all the other sources? You need to find a way to buy up the commodity ahead of time quietly, and then you need some event (such as a bad harvest) wiping out the holdings of the other commodity traders. Cornering a market has been done, but you need particular circumstances to make it work.

(I would recommend looking up the history of the Hunt Brothers and silver.)

The next problem is your ability to take advantage of your high inventory position. The producers of the commodity will ramp up production (and demand may be destroyed) if you try to only sell at a high price. You could get stuck with a large inventory of a commodity that has fallen in price. If you borrowed to finance your position, you might be forced into distressed selling - and everyone will know about it.

In other words, there are a lot of possibilities, and there is no easy way to characterise what will happen to the price.

Things are different if you control the production of the commodity. That ends up just being te situation of a monopoly (or oligopoly). There is a large literature on monopolies in economics; I am not familiar with it.


How does FRED Quarterly Real Gross Domestic Product and USA GDP growth rate match?

As Fred dataset clearly states the second series in quarterly series at an annual rate where the $32.9\%$ change is calculated just as the difference from preceding quarter. The annual rate change from preceding quarter is calculated differently than quarter-on-quarter rate. You can review the formulas fred uses here. Most of the discrepancy between the numbers can be attributed to that.

the seasonal adjustment is done by Fed using algorithms not by calculating quarter-on-quarter value so a discrepancy might creep already just from different method of seasonally adjusting the data although that should not be excessive.


Interpretation of dummy variable : Random Effects Model

As a first pass, I would interpret as with any other dummy variable's coefficient. Assuming a linear model, when two countries share a border, their export similarity index rises, on average, by 0.15 when all other variables are held fixed.

If you're writing a report on this model, you'll obviously need to discuss at length what this actually means.  The result seems intuitive: countries that share a border likely share other similarities that would lead to their portfolio of exports having more in common than if the two countries were separated by some larger geological or political distance.


Regression - Testing for autocorrelation in the presence of heteroscedasticity

General remarks: The BG test under homoskedasticity can be done using the bgtest command in the lmtest package of R. The $(n-p)R_{aux}^2$ version mentioned in link works only under homoskedasticity. In the presence of heteroskedasticity, Wooldridge (1991, JoE) gives a discussion (as noted in the Wooldridge textbook you mentioned).

What I think: I guess that what Wooldridge does is to use a heteroskedasticity-robust variance estimator. For this, (i) get the OLS residuals, (ii) regress e(t) on e(t-1), ..., e(t-p) and X, and test the joint significance of e(t-1), ..., e(t-p) using a heteroskedasticity-robust covariance estimate. If you want to use R, do the following for AR(2):

DF <- data.frame(y=rnorm(100), x1=rnorm(100), x2=rnorm(100))
ols <- lm(y~x1+x2, data=DF)
DF$e <- ols$resid
DF$e1 <- c(NA,DF$e[-100])    # are there better ways to lag a variable?
DF$e2 <- c(NA,DF$e1[-100])
aux <- lm(e~e1+e2+x1+x2, data=DF)
library(car)
lht(aux, c('e1','e2'), white.adjust='hc3')

Discussions: That said, there is the generated regressor problem in the aux regression above, that is, the some of RHS variables (e1 and e2) are generated using the OLS regression results. This usually causes trouble. However some tests can be done even if the regressors are generated. I guess this is one, but I haven't checked it formally.

Further discussions: The BG test is an LM test, while lht does a Wald test. The difference should be minor.

Simulation results: I did simulations. The ordinary BG test seems to fail. The robustified version seems working.

library(car)
iterate <- 1000
n <- 400

ans <- data.frame(ord=rep(NA,iterate), rob=rep(NA,iterate))
set.seed(1)
for (iter in seq_len(iterate)) {
  x1 <- rnorm(n+1)
  x2 <- rnorm(n+1)
  u0 <- rnorm(n+1)
  u0[1:floor(n/2)] <- 2*u0[1:floor(n/2)]
  u <- sqrt(1+abs(x1+x2))*u0

  ## y(t) = 1+x1(t)+x2(t)+0.5*y(t-1)+u(t)
  y <- filter(1+x1+x2+u, 0.5, method='recursive')

  y1 <- y[-(n+1)]      # y(1), ..., y(n)
  y <- y[-1]           # y(2), ..., y(n+1)
  x1 <- x1[-1]         # x1(2), ..., x1(n+1)
  x2 <- x2[-1]         # x2(2), ..., x2(n+1)

  ols <- lm(y~x1+x2+y1)

  e <- ols$resid
  e1 <- c(NA,e[-n])    # NA, e(1), ..., e(n-1)
  e2 <- c(NA,e1[-n])   # NA, NA, e(1), ..., e(n-2)
  aux <- lm(e~e1+e2+x1+x2+y1)
  tst0 <- lht(aux, c('e1','e2'), white.adjust=FALSE)
  ans$ord[iter] <- as.numeric(tst0$`Pr(>F)`[2] < 0.05)
  tst1 <- lht(aux, c('e1','e2'), white.adjust=TRUE)
  ans$rob[iter] <- as.numeric(tst1$`Pr(>F)`[2] < 0.05)
}

print(colMeans(ans))
##   ord   rob
## 0.090 0.046

Note: I edited the simulation a lot. I hope this one is OK.


To answer this question I will first generalize slightly the question to deal with the utility function
$$u(x) = \left(\sum_j x_j^\alpha\right)^{1/\alpha}$$
The Marshall demand can be written as
$$x_k^\star(p,I) = \left(\frac{p_k}{\bar p}\right)^{\frac{1}{\alpha - 1}} \frac{I}{\bar p} = \frac{p_k^\frac{1}{\alpha - 1} I}{\sum_j p_j^\frac{\alpha}{\alpha-1}},$$
and the value function as
$$V(p,I) := u(x^\star) = \frac{I}{\bar p}$$
where $\bar p := \left(\sum_j p_j^{\frac{\alpha}{\alpha-1}} \right)^{\frac{\alpha - 1}{\alpha}}$ is the price index as derived here Dixit-Stiglitz Pricing Index.
To find the Marshal demand start from the standard condition that relative prices equal MRS
$$\frac{p_j}{p_k} = \frac{\partial u/\partial x_j}{\partial u/\partial x_k} = \frac{x_j^{\alpha - 1}}{x_k^{\alpha - 1}},$$
in order to get
$$p_k^{\frac{1}{\alpha - 1}} x_j = p_j^{\frac{1}{\alpha - 1}} x_k,$$
which implies that
$$p_k^{\frac{\alpha}{\alpha - 1}} x^\alpha_j = p_j^{\frac{\alpha}{\alpha - 1}} x^\alpha_k,$$
and by summing over $j$ this results in the equation
$$p_k^{\frac{\alpha}{\alpha - 1}} \sum_j x^\alpha_j = x^\alpha_k \sum_j p_j^{\frac{\alpha}{\alpha - 1}} ,$$
which implies that
$$(A)\ \ \ p_k^{\frac{1}{\alpha - 1}} \left(\sum_j x^\alpha_j \right)^{1/\alpha}= x_k \left(\sum_j p_j^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha} ,$$
where multiplying with $p_k$ and summing over $k$ results in the equation
$$\sum_k p_k^{\frac{\alpha}{\alpha - 1}} \left(\sum_j x^\alpha_j \right)^{1/\alpha} = I  \left(\sum_j p_j^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha},$$
from which I isolate the factor including $x_j$'s to get
$$\left(\sum_j x^\alpha_j \right)^{1/\alpha} = I  \left(\sum_j p_j^{\frac{\alpha}{\alpha - 1}}\right)^{\frac{1-\alpha}{\alpha}} = \frac{I}{\bar p},$$
where the LHS is the utility function equal to an expression including only income $I$ and prices which therefore are the value function $V(p,I) = I/\bar p$. Inserting this expression in (A) and isolation $x_k$ gives the Marshall Demand
$$p_k^{\frac{1}{\alpha - 1}} \frac{I}{\bar p}= x_k \left(\sum_j p_j^{\frac{\alpha}{\alpha - 1}}\right)^{1/\alpha} \Leftrightarrow x_k^\star(p,I) = \left(\frac{p_k}{\bar p} \right)^{\frac{1}{\alpha - 1}} \frac{I}{\bar p}.$$


Does easy credit in a region cause property to be more expensive?

Theoretically, multiperiod borrow and lend models I am familiar with suggest that an increase in ease of credit (increasing the credit supply) should increase demand. This in turn raises the price of housing. So I believe your intuition here is correct.

Empirically, I am only familiar with one paper analyzing this relationship, which finds the same thing.

  ...because of geographic diversification, treated banks expand credit: Housing demand increases, house prices rise, but to a lesser extent in areas with elastic housing supply, where the housing stock increases instead.

Credit Supply and the Price of Housing

Favara and Imbs (2014)

http://www.jeanimbs.com/papers2_files/Favara_Imbs_May2014.pdf


Why has M1 grown a lot faster than M3 after the financial crisis?

As you point out, central banks have "printed" a considerable amount of money since the 2008 crisis. The following chart of the US monetary base is one of clearest examples of this phenomenon.

The metric you seem to be pointing out through your question is known as the money multiplier: the ratio of broad money (e.g., M1, M2, or MZM) to the monetary base. As we can see with the following chart, in the US, the money multiplier has decreased since the 2008 crisis.

The money multiplier grows when banks and other financial institutions make loans. So in short, M3/MZM hasn't grown because a significant amount of the monetary base has just been kept in bank reserves without being loaned out.

This has been used as an argument on the ineffectiveness of quantitative easing, and that we need to more directly put money into people's hands in order to stimulate the economy.

And if excess reserves were all loaned out and invested/spent, we should indeed see higher levels of inflation in the short term. If the money is invested/spent wisely, then productivity growth may offset that inflation on a longer term assuming we don't continue to increase the money supply.


Continuity Axiom in Expected Utility Theory

It is.
Prior to continuity, which is a property of the preference relation,  the preference relation $\succsim$ itself has been defined to be a binary relation that is characterized by transitivity, and, to begin with,  by completeness.
  Then if $S_1\cup S_2 \neq [0,1]$, it means that there exist some values of $\alpha$ somewhere in $[0,1]$, call them $\tilde \alpha$ for which

neither

$$\{\tilde \alpha L+(1-\tilde \alpha)L'\succsim L''\}$$

nor

$$\{L''\succsim \tilde \alpha L+(1-\tilde \alpha)L'\}$$

In words, for these $\tilde \alpha$'s, the pair cannot be ordered at all. But this contradicts the completeness foundation that is needed to even obtain a preference relation (as of course used in our theory. Psychologists I guess would disagree).

Also, note that completeness is defined over all conceivable pairs, even if, in a specific situation, we chose to restrict the space of lotteries to something smaller. Whether the lotteries under consideration belong to the specified lottery space, is really irrelevant. The person having the preferences has to be able to order them in any case, even as a "hypothetical" scenario (although strictly speaking, for a specific problem we have the "luxury" to impose completeness only as regards the lotteries available, while "remaining agnostic" as regards completeness if we expand the lottery space. Still this "weakening" on the imposition of the completeness axiom, does not really bring any gain).


Does lowering corporate tax rates increase investment or provide some other benefit to a nation?

Cutting corporate taxes is one of the tools of the Trickle-Down Economics (TDE) school  of managing economic growth. In a New York Times article in 2012, economist Robert Frank of the Johnson School of Management at Cornell University summarized the findings of years of research about the effectiveness of applications of this theory.

In his article, he discussed taxes on individuals, rather than on corporations, and there are substantial differences, but I believe the mechanics of the behaviors of these two economic subsystems are the same with respect to drag effects, such as taxation. They are linked in that tax reduction on individuals was expected to spur increased demand for consumer goods and services, which in turn was expected to promote industrial expansion to meet the growth in demand.

In short, he states (and cites research to support the view) that this theory has never been demonstrated to work. In fact, it may be counterproductive in the case of individuals, who tended to save the additional wealth acquired through reduced taxation in tax havens, which have negative effects on growth.

Corporations tend to view taxes in one of two ways, depending on their economic context, their corporate strategy, and their resources.  When they are short of investment funding, reductions in taxation can lead to an increase in their investment if their strategy dictates that, and their economic context allows it. If they are not short of investment funding, reducing their taxes will normally have little effect on capacity; their capital investment strategy is already adequately funded.

However, most middle-to-large size companies view their treasury function's capacity to raise capital through floating stock or taking on debt as the primary source of funding for investments.  They may consider a tax rate reduction as a windfall in the short term, and use it to buy back stock or lower debt.  In the longer term, they may issue or increase the size of dividends, or acquire another company, rather than increasing net capacity. In this case, the shareholders receive that portion of the tax cut, and the mainline TDE effects on individuals come into play. As John Kenneth Galbraith and others have demonstrated, these capital gains for individuals do not lead to increased individual spending. As a result, there is little to no effect on capital investment, since there is no substantial increase in demand to meet.

As long as their economic context warrants it, smaller businesses that do not normally use a treasury function take on debt or issue stock may choose to use a tax reduction to fund expansion, reduce debt, distribute the increase in net income as profits to their owners, or retain it as capital. So, even in this case, an increase in capital investment is only one of the possible outcomes.

For a company's economic context to warrant expansion, the company must:
a) anticipate an increase in demand, or be already faced with unsatisfied demand, or both;
b) have the means to manage expansion, or be able to acquire them;
c) be able to compete effectively against current competitors and potential market entrants;
d) be sufficiently free from higher-priority obligations to other parties that could consume the additional net income or capital raised, such as corporate debt payments, obligations to improve / repair operations for regulatory reasons, liabilities from litigation and product warranty repairs, etc.;
e) have at least one viable channel of investment in capacity to deliver products or services, either through introducing new ones or expanding the delivery of existing ones

It should be noted that this is not just a theoretical discussion; the consequences of using this theory are staggering. David Stockman, Director of the Office of Management and Budget under President Reagan from 1981 to 1985, was a staunch advocate of TDE theory. The theory was repackaged and rebranded as "Supply-Side Economics" to avoid the stigma associated with the name of TDE. However, Stockman observed the national debt increase from 908 billion dollars in the last year of the Carter Administration to 2.1 trillion dollars (a 130% increase) by 1986. Clearly, the fiscal side effects on the government were unsustainable. He resigned in 1985, and became a vocal opponent of TDE. In December 1981, years before the negative effects became widely known, the Atlantic Monthly published an interview with Stockman himself, "The Education of David Stockman", which was highly critical of the strategy.

You can find other references to research criticizing the effectiveness of Trickle-Down Economics here.


Capital stock depreciation rate, how to calculate?

Rate of depreciation  = depreciation/gross stock in the current year. So 1.7%. Strictly speaking there is no way to calculate depreciation because the timing of capital formation is unknown, but capital begins depreciating as soon as it is formed.


How can I represent players with different information given continuous choice ranges?

One way to represent the game tree (assuming that the congressman observes the expert's choice):

The game tree is drawn using LaTeX with istgame package.

\documentclass{standalone}

\usepackage{amsmath}
\usepackage{istgame}

\begin{document}

\NewDocumentCommand\vpay{m}
{ \begin{matrix} #1 \end{matrix} }

\begin{istgame}[scale=2,font=\scriptsize]
\cntmdistance{20mm}{70mm}{50mm}
\istrootcntm(0)[chance node]{Nature}
  \istb{w}[r]  \istb{w'}[r]         \endist
\cntmdistance{20mm}{40mm}{25mm}
\istrootcntm(1)(0-1)<-170>{Expert}
  \istb{a_1}[r]  \istb{a_1'}[r]     \endist
\istrootcntm(2)(0-2)<-10>{Expert}
  \istb{a_1}[r]  \istb{a_1'}[r]     \endist
\cntmdistance{15mm}{20mm}
\istrootcntm(3)(1-1)
  \istb*{a_2}[r]{\vpay{-(x-0.5)^2\\-x^2}}   \endist
\istrootcntm(4)(1-2)
  \istb*{a_2'}[r]{\vpay{-(x-0.5)^2\\-x^2}}  \endist
\istrootcntm(5)(2-1)
  \istb*{a_2}[r]{\vpay{-(x-0.5)^2\\-x^2}}   \endist
\istrootcntm(6)(2-2)
  \istb*{a_2'}[r]{\vpay{-(x-0.5)^2\\-x^2}}  \endist
\xtCInfoset(3)!.55!(5){Congressman}
\xtCInfoset[dashed](4)!.45!(6){Congressman}
\xtCommentTo[draw=none]($(0)!.5!(2)$)(1.5,0)
  {$w\sim U[0,1]$}[r]
\xtCommentTo[draw=none](3-1)(-1,-1)
  {where $x=a_2+w$ and similrarly for other cases}[r]
\end{istgame}

\end{document}

If you want an arc type continuum of actions:

\documentclass{standalone}

\usepackage{amsmath}
\usepackage{istgame}

\begin{document}

\NewDocumentCommand\vpay{m}
{ \begin{matrix} #1 \end{matrix} }

\begin{istgame}[scale=2]%,font=\scriptsize]
\cntmdistance{20mm}{70mm}{50mm}
\istrootcntmA(0)[chance node]{Nature}
  \istbA(.865){w}[r] \istbA(.865){w'}[r]     \endist
\cntmdistance{20mm}{40mm}{25mm}
\istrootcntmA(1)(0-1)<-170>{Expert}
  \istbA(.915){a_1}[r] \istbA(.915){a_1'}[r] \endist
\istrootcntmA(2)(0-2)<-10>{Expert}
  \istbA(.915){a_1}[r] \istbA(.915){a_1'}[r] \endist
\cntmdistance{15mm}{20mm}
\istrootcntmA(3)(1-1)
  \istbA*{a_2}[r]{\vpay{-(x-0.5)^2\\-x^2}}   \endist
\istrootcntmA(4)(1-2)
  \istbA*{a_2'}[r]{\vpay{-(x-0.5)^2\\-x^2}}  \endist
\istrootcntmA(5)(2-1)
  \istbA*{a_2}[r]{\vpay{-(x-0.5)^2\\-x^2}}   \endist
\istrootcntmA(6)(2-2)
  \istbA*{a_2'}[r]{\vpay{-(x-0.5)^2\\-x^2}}  \endist
\xtCInfosetO(3)(5){Congressman}
\xtCInfosetO[dashed](4)(6){Congressman}
\xtCommentTo[draw=none]($(0)!.5!(2)$)(1.5,0)
  {$w\sim U[0,1]$}[r]
\xtCommentTo[draw=none](3-1)(-1,-1)
  {where $x=a_2+w$ and similrarly for other cases}[r]
\end{istgame}

\end{document}


In the C.E.S. utility function do the parameters need to add up to unity to obtain the Cobb-Douglas utility function?

Yes.

Write

$$U(x,y) = (ax^{-c} + by^{-c})^{-\frac{1}{c}} = \exp\left\{\frac {-1}{c}\ln \left(ax^{-c} + by^{-c}\right)\right\} \tag{1}$$

Now if $a+b=1$ then as $c\rightarrow 0$, expression

$$\frac {\ln \left(ax^{-c} + by^{-c}\right)}{-c} \tag{2}$$

will be an indeterminate form $0/0$ and so we can apply L'Hopital's rule on it to get

$$\frac {1}{-c}\cdot \frac {-ax^{-c}\ln x + by^{-c}\ln y}{ax^{-c} - by^{-c}} \rightarrow \frac {a}{a+b}\ln x + \frac {b}{a+b}\ln y,\;\;\; c\rightarrow 0$$

where we have assumed $a+b=1$.

So (by the uniform continuity of the exponential function)

$$c\rightarrow 0,\;\;\; U(x,y) = \exp\left\{\frac {-1}{c}\ln \left(ax^{-c} + by^{-c} \right) \right\} \rightarrow \exp\left\{a\ln x + b\ln y \right \} = x^ay^b$$.

But if  $a+b\neq 1$, then as $c\rightarrow 0$, the argument of the logarithm in the numerator in eq. $(2)$ would not have been equal to unity, and so the logarithm would not equal zero (which would give us the indeterminate form, and would allow us to use L' Hopital's rule). Instead, eq. $(2)$ would have gone to infinity. So $a+b=1$ is needed in order to arrive at the Cobb-Douglas function.


Finding economy's factor price for perfect complement production function

The planner is trying to maximize:
$$\Pi = Y - r\cdot K - w\cdot L= min [K, L] - r\cdot K - w\cdot L $$

In perfect competition factors receive their marginal product as price. If $0<w <2$ then $0 < L_s < 100$. This means that the marginal product of capital (MPK) is zero and therefore the price of capital ($r$) should be zero. In this region the marginal product of labor (MPL) is 1, and so there is only one wage that represents a competitive equilibrium, $w*=1 \rightarrow L^* = 50, Y^* = 50, r^* = 0$.

What about of $w\geq 2$? In this region the MPL is 0 so this can't be a competitive equilibrium because $2< w \neq MPL$.


Example which violates weak pareto and non dictatorship

Such an example does not exist.

Suppose non-dictatorship is violated. WLOG, let individual $1$ be the dictator among $n$ individuals. That is, the social choice function always selects an alternative that ranks highest in $1$'s preference, i.e. $$f(\succ_1,\dots,\succ_n)\in\arg\max\{\succ_1 | x\in X\}.$$

For weak Pareto efficiency to also be violated, we must have two alternatives $y,z\in X$ such that $y\succ_iz$ for every $i=1,\dots,n$, but the social choice function chooses $z$, i.e. $f(\succ_1,\dots,\succ_n)=z$. Yet this clearly contradicts the above condition of dictatorship, in that $z\notin\arg\max\{\succ_d | x\in X\}$.


Intuition: Why does perfectly elastic supply imply no producer surplus?

If it costs you 50 cents a bag to make popcorn and you sell it for 50 cents a bag, you come out ahead by 0 cents a bag.


What percentage of UK imports/exports were with the EU in the economic year 2015-2016?

UK Exports 2015:

  The United Kingdom shipped US$460.1 billion worth of products around the globe in 2015. [...] From a continental perspective, 53.6% of UK exports by value are delivered to European trade partners while 22.5% are sold to Asian importers. United Kingdom ships another 16.1% to North America but just 2.6% to Africa.

While the total value of exports is a useful piece of information, here Europe means the continent, not the EU. Thus Norway, Switzerland and several Balkan countries are also included. It is not clear where Russia would fall under such a categorization so it is better to look at Eurostat's statistics.

In 2015 the UK's intra-EU trade share was lowest among the member states  but it was still 50%.


How does Russia have so much foreign exchange reserves?

russia uses its oil revenue to setup russia stabilization fund in 2004. The money in that fund is used to pay for national budget deficit. The foreign exchange reserve of Bank of Russia is not affected by the US led sanctions.


Possible regression models for a dependent variable that is a proportion

Let $y$ be the proportion in $[0,1]$ instead of the percentage. I think the issues here are possible nonlinearity and censoring. You can try including quadratic terms on the right hand side. At the same time, you can try the following models.

I. Linear models: If no $y$ is exactly 0 or 1, linear models will be OK. The following four options come to my mind.

$y=\beta_0 + \beta_1 x + u$ if no $y$ is close to 0 or 1.
A log model $\ln y = \beta_0 + \beta_1 x + u$, which is helpful when some $y$ values are close to zero but all are far from one. Note that interpretation changes.
Transform the dependent variable to $-\ln (1-y)$. This is helpful if $y$ are all far from zero but some close to one. But this looks a bit unnatural to me, and I would consider the next logistic model instead.
The logistic model $\ln \frac{y}{1-y} = \beta_0 + \beta_1 x +u$. This usually helps if there are many $y\simeq 0$ and $y\simeq 1$. Interpretation is done in terms of logits.

II. Tobit models: If some $y$ are exactly 0 or 1, you can try Tobit models (help tobit in Stata). Remember that normality is assumed for the error term before censoring. Also, using Tobit models means that "I think $y$ could be bigger than 1 (smaller than 0) if not censored."


Are overlapping generation (OLG) models extensions of a DSGE model?

You can find OLG models that do not classify as DSGE (in particular, the model might not be stochastic) as well as DSGE with overlapping generations (contrary to those with infinitely lived agents).

You can find more detail on this on this working paper by Assous and Duarte (2017), as they note

  In the early 1980s, when the real business cycle macroeconomists
  brought one single model (a perfectly competitive growth model with
  infinite-lived agents, flexible prices, and perfect information) to
  bear on any macroeconomic issue, several macroeconomists were working
  with OLG models and addressing business fluctuations matters. Besides
  the efficiency issue, the model seemed to have much more to offer. New
  classical economists such as Wallace and Lucas saw in the OLG model
  the possibility to develop new microfounded models of fiat money
  without postulating that money balances enter the utility function of
  agents. At about the same time, other macroeconomists discovered that
  OLG models give room for either deterministic or stochastic
  oscillatory trajectories. Endogenous cycles and chaos as well as
  sunspot equilibria were then shown to occur in the presence of
  perfectly competitive product market devoid of any nominal price
  rigidities. Gradually over time, important contributors to that
  literature — most notably, for our interests here, is Woodford —
  strove to transfer OLG conclusions to infinite-lived agents models. In
  this context, new dynamic models with market imperfections were
  developed, initially with flexible prices, that eventually became the
  hallmark of the sticky price, DSGE macroeconomics (earlier referred to
  as the new neoclassical synthesis).


Could leisure be considered a Giffen good?

I have two sources here:

Argument by David Friedman: http://www.daviddfriedman.com/Academic/Price_Theory/PThy_Chapter_5/PThy_Chapter_5.html

EDIT:
From the link:

An increase in wages (say, from \$10/hour to \$11/hour) has two effects. It makes leisure more costly--each hour not worked means \$11 less income instead of \$10. That is an argument for working more hours at the higher salary. But at the same time, the increased wage means that the producer is wealthier--and is therefore inclined to consume more leisure. It is possible for the second effect to outweigh the first, in which case the increased wage causes a decrease in hours worked, as shown in Figure 5-4. This is called a backward-bending supply curve for labor; the backward-bending portion is from F to G (and presumably above G). The result, in the case of a single producer, would be a supply curve for goods that sloped in the wrong direction; for some range of goods, higher prices would generate less output instead of more.

This is not the first time we have seen a conflict between income and substitution effects. In Chapter 3, the same situation generated a Giffen good--a good whose demand curve sloped in the wrong direction. I argued that there were good reasons not to expect to observe Giffen goods in real life. Those reasons do not apply to the backward-bending supply curve for labor.
One of the reasons was that while we expect consumption of most goods to go up when income goes up, a Giffen good must be a good whose consumption goes down with increasing income--an inferior good. Indeed, it must be so strongly inferior that the income effect of an increase in its price (which, since we are buying it, is equivalent to a decrease in real income) outweighs the substitution effect. Our labor is something we are selling, not buying; an increase in its price (the wage rate) makes us richer not poorer, and so inclined to buy more leisure. So the backward-bending supply curve for labor only requires leisure to be a normal good.

Argument and diagram by unknown:
http://people.bath.ac.uk/ak220/teaching/labor%20supply.pdf
EDIT
From the link:

Suppose now that the wage someone receives goes up, (Diagram II). Obviously, this also increases the cost of leisure since the opportunity cost for not working is the wage you don’t get. So, since leisure has become more expensive, the worker increases his working time. This is the (negative) substitution effect. At the same time, there is also an income effect coming from the fact that since your wage increased, your whole wage income increased$^2$.Thus, the income effect will tempt the
worker to increase his/her leisure. Be careful here: Leisure became more expensive
but the income effect is positive. Remember that when we were talking about a
normal good X, an increase in price would have caused both effects to be negative.
I’ll explain this contradiction later.
An increase in the wage will cause the budget line to pivot upwards to the new
position MoM1. The consumer now is on the new indifferent curve $I_1$. As you can see,
he has both increased his leisure and his consumption! Let’s decompose this effect.
The substitution effect will make the worker work more, $Q_0$ to $Q_2$. However, the
income effect will push in the opposite direction, $Q_2$ to $Q_1$. Since the I.E. over weights
the S.E., the consumer will increase his leisure.

This worker’s reaction was actually determined by the IE, since the SE would always
motivate the worker to work more. For a different worker, the IE could have acted in
the same direction as the S.E. Clearly, this depends on where the indifference curves
lie on the diagram, their shape (preferences) and the level of non wage income. Suppose a worker possesses a very significant source of non wage income. Obviously,
he would react more smoothly in a wage increase than someone who relies solely on
his wage income.
For a particular level of wage, a worker would be indifferent between working and
not working at all. This wage is called reservation wage and depends on the level of
the non wage income, keeping preferences constant.
Finally, few comments on the contradiction I mentioned before:
Suppose we have good X which is normal and its price increases. There is a decrease
in the consumption of the good since the good became more expensive in relation to
other goods. (Substitution effect). There is also a decrease in the consumption of the
good because of the income effect, since the real income decreased and the good is
normal. Consequently the one effect adds to the other and the total effect is negative.
Now in X’s position we have Leisure, assuming that leisure is a normal good. If its
price increases (increase in wage) then leisure becomes more expensive so the worker
would want to consume less of it (work more). On the other hand the income effect motivates the worker to increase his/her leisure instead of reducing it (as it was the
case with good X). Who is wrong here?
Apparently none! In the conventional model of goods X and Y we are concerned with
people who consume the goods and do not sell those goods. In contrast, an individual
not only consumes leisure but also “sells” it to an employer (in which case it is called
work time). Thus when the price of leisure increases, it may become more expensive
to consume but at the same time the consumer benefits from selling it at a higher
price! Thus, the income effect can be either positive or negative depending on
consumer’s preferences and particularly on how much leisure he sells in relation to
how much he consumes.

In (2)[Diagram II] it seems that IE>SE but optimal choice of leisure is increasing. Seems that leisure is normal here.


Example of social choice rule that does not satisly the unrestricted domain condition

Can any social choice rule that is not complete said to be violating the unrestricted domain condition?

Yes.

  Could you provide an example of SCR other than Pareto dominance that is not complete or violates unrestricted domain property?

Let society consist of two individuals $i$ and $j$. Let there be exactly two alternatives $A$ and $B$.

Each individual has 3 possible preferences: strictly prefer $A$ to $B$, indifferent between $A$ and $B$, or strictly prefer $B$ to $A$.

Hence, there are 3 × 3 = 9 possible social preference profiles:

$i$ strictly prefers $A$ to $B$ and $j$ strictly prefers $A$ to $B$.
$i$ strictly prefers $A$ to $B$ and $j$ is indifferent between $A$ and $B$.
$i$ strictly prefers $A$ to $B$ and $j$ strictly prefers $B$ to $A$.
$i$ is indifferent between $A$ to $B$ and $j$ strictly prefers $A$ to $B$.
$i$ is indifferent between $A$ to $B$ and $j$ is indifferent between $A$ and $B$.
$i$ is indifferent between $A$ to $B$ and $j$ strictly prefers $B$ to $A$.
$i$ strictly prefers $B$ to $A$ and $j$ strictly prefers $A$ to $B$.
$i$ strictly prefers $B$ to $A$ and $j$ is indifferent between $A$ and $B$.
$i$ strictly prefers $B$ to $A$ and $j$ strictly prefers $B$ to $A$.

We say that a social choice rule (SCR) is complete or satisfies the unrestricted domain condition if it maps (or assigns) each of the above 9 profiles to a preference. (A trivial example of such a SCR is that which maps each of the above 9 profiles to the preference "strictly prefer $B$ to $A$".)

A trivial example of a SCR that is incomplete or violates the unrestricted domain condition is that which maps each of profiles 1–8 to the preference "strictly prefer $B$ to $A$", but fails to map profile 9 to any preference. (Another trivial example is @Giskard's: The SCR which fails to map any of the 9 profiles to any preference.)


Why does changing the value a currency help/hurt exports if sellers can just inflate/deflate prices to match the change?

What is missing in your reasoning is the effect of devaluation on the seller's profit.

To keep things simple, assume that the inputs needed to produce the widgets are all bought domestically. Then costs will not change following devaluation.  But prices in terms of domestic currency can increase, without as you say increasing the cost to foreign buyers in terms of their currency.  So profits per widget - selling price less costs per widget, both in terms of domestic currency - can increase.

There is another possibility.  A seller might take advantage of devaluation to try to increase total profits by increasing the volume of exports. If so, they could increase the price in terms of domestic currency by less than 10%, or perhaps not at all, with the effect of reducing the price to foreign buyers.  Unless demand is perfectly inelastic, this will increase the volume of export sales and, if domestic sales are unchanged, the total volume of sales.  An increased production volume may in the long run permit economies of scale, reducing costs per widget.

Either of these strategies could yield an increase in total profits.  Which yields the larger increase will depend on both the elasticity of foreign demand and the scope for economies of scale.

Through the profit motive, therefore, devaluation can certainly encourage firms to increase the value and/or volume of their exports.


Well you can essentially forecast anything using a VAR/VECM framework.

If you're wondering if such estimates will be accurate the answer really depends on the region, as the Okun Coefficent differs depending on where you are. (much like the law of gravity on earth).

a working paper released by the IMF titled Do Forecasters Beleive in Okun's Law? its conclusions were rather in favor for the use of Okun's law in forcasting.

  Our results show that professional forecasters believe in Okun’s Law to a degree merited by how
  well it holds in the data. For all nine countries we study, the relationship between forecasts of
  changes in unemployment and changes in real GDP growth is negative, which is consistent with
  Okun’s Law....

Its generally more accurate for short run forecasts then it is for long run. However the same counter-cyclical relationship is maintained between Real GDP growth rate and Unemployment Rate.

  the correlation between output and unemployment forecast revisions is stronger in year-ahead
  forecasts than in current-year forecasts; this finding leads us to conjecture that forecasters’
  reliance on Okun’s Law is stronger in the early stages of the forecast horizon, when information
  about the likely departures from Okun’s Law is scarce.