The paper Optimum consumption and portfolio rules in a continuous-time model (Merton (1971)) presents (as far as I know, the initial presentation.
[
Next we turn to a paper that discusses the importance of the HARA utility.
The Fundamental Nature of HARA Utility (Perets and Yashiv (2013))

Many models in Economics assume a utility function belonging to the
HARA family. This paper shows that HARA utility is fundamental to
economic analysis. The HARA functional form is the unique form which
satisfies basic economic principles in an optimization context. Using
HARA is not just a matter of convenience or tractability but rather an
essential restriction....

This notable paper has a multi page exploration of the HARA utility family.
Implications of Constant Risk Aversion (Bamberg and Spremann (1981))

The assumption of constant risk aversion often leads to a considerable
simplification of decision theoretic analyses. It is shown that this
restriction on constant risk aversion permits the description of a
wide range of risk averse patterns between the extreme cases of risk
neutrality and the exclusive orientation on the pessimistic maxmin
criterion. In particular, a new axiomatic foundation of the constant
risk aversion is given and a series of properties for the certainty
equivalent are derived. Possible applications are shown by the
treatment of risk premiums, values of information, portfolio
selection, capital market theory and syndicates.

There is a book that I can't vouch for but may be useful:
Hyperbolic absolute risk aversion: Second Edition


You are probably talking about Euler's homogeneous function theorem http://mathworld.wolfram.com/EulersHomogeneousFunctionTheorem.html , which shows that you can decompose a function into its partial derivatives based on its homogeneity.

So a typical Cobb Douglass $U(x,y)=x^ay^{1-a}$

This function is homogeneous of degree 1 because when we multply each argument by a scalar:

$U(cx,cy)=(cx)^a(cy)^{1-a}=cx^ay^{1-a}=cU(x,y)$

So if we apply the theorem to this function we get:

$U(x,y)=xU_x+yU_y=xax^{a-1}y^{1-a}+x^ay(1-a)y^{-a}=x^ay^{1-a}$

So the result that we get back the original utility is expected.


I have two here to add to your list.

Brookings Insitute

https://www.brookings.edu/events/

St. Louis Fed  Economic lowdown
(FREE)

https://www.econlowdown.org/

There is also the National Association of Business Economics (NABE)
But to get access to their webinar resources you have to register on their site.
I know this won't qualify as it is not free but they are a good source of up-to-date information.  I felt that this would be a good share.

www.nabe.com

Others that I just found:
Peterson Institute for international Economics

https://www.piie.com/events

National Bureau of Economic Research (www.nber.org - click on videos)
or

https://www.nber.org/video_archives.html

Organisation for Economic Co-operation and Development (OECD)

http://video.oecd.org/

World Bank
https://www.worldbank.org/en/about/archives/past-events


Across the board tariffs (ex: sales tax) can result in higher prices or "inflated" prices. This is usually not what economists refer to when they say inflation. Inflation that the fed is concerned with, is an across the board increase in all goods as a result from changes in the money supply. Something, that the fed can directly influence, whereas the fed has no control over taxes.

Fiscal policy (Government) and monetary policy (The Fed) can sometimes coordinate actions for a desired result.

For instance: Following the 2007/8 financial crisis, the united states paired subsidies and incentives ,effectively lowering sales tax (fiscal policy), with low interest rates (monetary policy) to stimulate the economy.

You can imagine that a coordinated policy concerned with high inflation rates might do the opposite.


Understanding this particular criticism of RCTs by Deaton and Cartwright

Without having read the full piece I'd say you are mostly spot on, but for a subtle difference. The objection runs a bit deeper than your rephrasing of it, where you refer to single estimate. The thing is that there are way more than a single estimate this applies to, and therefore you are more likely to be off.

The point is that for RCT to establish causality both groups have to be equal on all other characteristics except for the treatment. By doing assignment randomly one can expect that in a large number of potential samples for all these aspects the independent samples' averages will be close to the population means for these characteristics.

The problem is, as you rightly point out, one may be unlucky and have samples where the sample averages are not actually close to the population means. The additional problem is that sample averages have to be close to the population means for all variables that matter, but 1) you don't always know which variables matter and 2) given the large number of variables that do matter you'll always have one or two where you are "unlucky" in terms of the sample averages not being close to the population mean.


Because in Ricardo's view, the money is neutral. The price reflects only the quantity of labor which is incorporated in the product. If it takes you 2 hours to hunt a deer and only 1 hour to hunt a rabbit then the deer will be two times more expensive.

That being reminded, one can go back to Ricardo's example. First, we have 180 quarters of wheat that needed 10 men (which gave an output of 720£, at 4£ per quarter). But afterwards you have 20 men producing 350 quarters of wheat, which obviously takes into account the less productive land. Let's say we wanted to double up our first production with the new land: then we would in fact need about 21 men to meet our target of 360 quarters (2 x 180 quarters). But as you see this can't be worth 1 440£ , because there is mork work incorporated ! So there must be a new price for the quarter of wheat ...

Therefore the following question is: what shall be the new price ? One can turn that around into asking: how much more work is incorporated in the new production ? Well there is about 5% more work (1 man = 5% of the 20 that should be normally if there weren't any diminishing returns), which you can also find out by putting: 180/170 = 1,05 or 1+5%. If it is still not clear: this fraction represents the production advantage of the first land in comparaison with the second one. It is 5% more productive. Which means that you have to incoporate 5% more work to get the same production, which in returns, means the quarter of wheat has to be about 5% more expensive...


Negative long term effects of wage cap, especially on purpose of money?

However, the market could normalize after this as the money earnt by a nation's richest would eventually be more evenly distributed amongst others, hence they would spend more, canceling the downturn.

Wage cap is not an redistribution measure. It’s an measure aimed at reducing inequality but without any direct redistribution.

For example, suppose you have Peter and Paul, Peter earns 10000\$ per month and Paul 1000\$ per month. If there is wage cap 5000\$, Peter will earn 5000\$ and Paul is still earning only 1000\$. Hence there is no direct redistribution effect even though inequality is now lower.

So there will not necessarily be any more spending in an economy. That assumption is completely false and would hold for income tax that redistribute money from rich to poor but not for wage cap. In fact most likely there will be less spending in the economy as result of wage cap.

Regarding other long term effects of wage cap, it would severely depress the labor supply of people to whom it applies to. Wage cap is de facto equivalent of confiscatory tax that confiscates any income above certain level, but it is even worse as government does not even get any revenue, from it. If a person does not earn any income for their labor above certain level they will have no incentive to work anymore (aside from maybe the fact that they enjoy their work). Hence their labor supply will be depressed. They will spend less time at work and more time enjoying some leisure activities.

This can have serious impact on economy as usually people who earn more in market based economies are those people who are more productive. Highly skilled doctors, managers, scientists etc, decreasing overall productivity of the economy. Moreover, such measure would create a permanent supply shock (as long as it would be in effect) permanently lowering the level of GDP and hence also the material welfare of people in given country.

It could also impact the growth of the economy, as endogenous theory of economic growth says that human capital formation can be very important driver of economic growth and since people tend to earn more the more human capital they have this would depress the human capital formation for the very brightest who would be affect by such a measure.

Also as a side note, I don’t really know any public economists who would approve of wage cap. The reason for that is that in fact if you care about redistribution you want as many people with high incomes as possible so then you can tax their incomes and raise the welfare of the poor through redistribution.


Deadweight Loss Due to Taxes Represented on a Graph of the Market?

I would say yes. In the extreme case: if there is a perfect substitute demand for the original good is perfectly elastic and its demand curve horizontal. That means that any tax imposed on the original market will lead to zero demand for the original good. All demand would move to the new market due to perfect substitutability. The entire surplus of the original market is deadweight loss. If both demand and supply moved to a new market they could replicate the original market there. The deadweight loss would be compensated completely by the surplus in the new market.

If the new market has preexisting demand not all additional demand from the original market could be satisfied at the same price point as in the original
market leading to a welfare loss. The demand curve in the new market would move to the right but not sufficiently to compensate the deadweight loss incurred in the original market.

Correspondingly, the less elastic the demand for the original good is, the less demand can move to the new market and the less deadweight loss can be cancelled out by the new or increased surplus in the new market.


Why do the GDP baskets of different countries have different percentages?

The “shopping baskets” of items should be representative of consumer spending patterns. The general principle is to apply a fixed set of weights to price changes for each of the items such that their influence on the overall index reflects their importance in the typical household budget.
In 2015, housing on average accounted for about 22 percent of consumption expenditure in China, and it represented about 33 percent of household expenditure in the U.S. in 2016 (same percentage in 2018 according to the US Bureau of Labor Statistics). These consumption patterns may explain the difference between the 40% in the U.S. and the 17% in China.


Why is bargaining more common in poor countries?

In first world countries, the price of a bottle of water is set by a well-established market. There are millions of prospective buyers of a bottle of water and thousands of prospective suppliers, and while many suppliers are able to achieve brand differentiation, for the most part their water is fungible. Moreover, the supply and demand curves of water are reasonably stable, and water companies have the excess capital to hold inventory stockpiles that further reduce the volatility of prices. There is little information asymmetry. Customers can easily comparison shop, and price discrimination is very difficult. Suppliers are generally large corporations, which means that allowing bargaining would introduce massive headaches, such as agent-principal issues. There's also a feedback effect: once an economy and culture are built around fixed prices, it's more difficult to introduce bargaining. Water companies are in a low-margin business where their money comes from high volume. Spending resources bargaining would destroy those margins.


Why does Mascolell define second-order stochastic dominance as such?

The motivation is I believe clearly stated in p. 197 beginning of the section, where they write that they want to use Second-order stochastic dominance to reflect comparisons related to "riskiness/dispersion".

To make this prominent, they use distributions with the same mean, so that it is obvious that we compare them not w.r.t their means, but w.r.t. riskiness/dispersion. In fact, the authors use the concept as equivalent to the stochastic dominance over mean-preserving spreads (see proposition 6.D.2 p. 199).

Using the general definition (which is defined by what they provide as property 6.D.2 p. 198) would not have offered them something more, in view of the reasons they want to use the concept.


Uniqueness of utilities in competitive equilibrium

If you look at this manuscript by Jonathan Levin (2006), pages 25-26 provide an example where there is multiple equilibria, and calculating payoffs in this case (if I am not making any calculation mistakes) gives different levels of utilities in these different equilibria.


You have some incorrect ideas about wealth. First of all, inheriting money does not concentrate wealth, it disperses it. Usually the recipients of inheritances, both organizations and wayward children spend the money the receive so the inheritance is dispersed completely.

Most people who are wealthy got that way not by earning a lot of money, but by saving the money they did earn. How much a person saves and invests is a much bigger indicator of wealth than income. A plumber that saves and invests will eventually become much richer than a doctor who spends all his money.

Also, most people who wealthy are not manufacturers who "make a great product," they are ordinary professionals and businessmen: doctors, lawyers, business executives, electricians, plumbers, and builders. You can even find many blue collar workers who have become rich by saving and investing: policeman, firemen, teachers. Where I live there are two teachers, a husband and wife couple, who started out as ordinary high school teachers and now they are quite wealthy and own their own private school. There are probably more millionaire garbage men, than there are millionaire manufacturers. Owning a bunch of garbage trucks is enough to make you a millionaire. Now think how many garbage trucks there are.

Here is an article about typical characteristics of wealthy people:

https://www.nytimes.com/books/first/s/stanley-millionaire.html


This is the version of the theorem I will use:

Weierstrass theorem: Every continuous function defined over a compact set attains a minimum.

The function $p\mathbin{\cdot} x$ is continuous. The problem is that $X(\bar{u})$ is not compact. So, an additional trick is needed. Since we are working with a subset of $\mathbb{R}^n$ compact means the same as closed and bounded (Heine-Borel theorem). The continuity of $u$ is used to guarantee that $X(\bar{u})$ is closed. But it could be unbounded.

This is where $p\gg 0$ becomes useful. Take an arbitrary point $x_0 \in X(\bar{u})$ and consider the set $Y$ defined by  $$Y = \{\,y\in X(\bar{u}) \mid p\cdot y\leq p\cdot x_0\,   \}.$$
Since $x_0\in Y$, we know that $Y$ is nonempty. Using the fact that $p\gg 0$, you can show that $Y$ is bounded (I'll let you figure out the details). Since $Y$ is the intersection of two closed sets, it is also closed. Hence, it is compact. Therefore, by Weierstrass theorem, $p\cdot x$ has a minimum in $Y$.

I will also let you figure out the last step on your own, which is to show that the minimum of $p\cdot x$ in $Y$ is also a minimum in $X(\bar{u})$.


Can Dynare solve general equilibrium (GE) models with non-convex adjustment costs?

Short answer: no.

Dynare, and linearization/perturbation methods in general, are designed for solving

smooth models
approximated around a single point in state space (the steady state).

A model with fixed cost is typically non-smooth, and its behavior away from the steady state may be very different, if e.g. the firm switches from investing to not investing. On the most practical level, a model with fixed cost will typically include equation such as

$$
V = \max \left\{ V^{\text{invest}}, V^{\text{not invest}} \right\},
$$

which cannot be entered into Dynare, because max operator is not supported. On the other hand, first order conditions for convex (e.g. quadratic) adjustment cost are still smooth (one simply adds additional terms to Euler equation for investment) and thus can be easily solved with Dynare.

To actually compute optimal policy with fixed costs, one has typically to use global method, e.g. value function iteration. I'm not aware of any standardized toolbox for solving such problems, so you may need to code your own.

PS: there are some modelling tricks that make the problem smoother, typically in a setting with many, possibly heterogeneous agents/firms. For example, Thomas (2002) keeps track of number of firms depending on how long they didn't invest, and solves the model with standard linearization on this extended state space. Khan & Thomas (2007) assume that the fixed cost is random and iid over time and across firms, so one can average over the realization of fixed cost to obtain smooth value functions. Miao & Wang (2014) use a similar approach in a model with constant returns to scale and show how it aggregates to a version of representative-firm model with only convex adjustment costs.


How to prove that GDP grows at the same rate as consumption on Lab Equipment Model?

The household side of this model is pretty standard. Denote $K(t)$ be household's assets at time $t$. Then the transversality condition (which is an optimizing condition, not a constraint), is

$$\lim_{t \to \infty} [e^{-\rho t}\lambda(t) K(t)] = 0$$

where $\lambda(t)$ is the current value multiplier on assets in the Hamiltonian. Given the assumed form of the utility function we have the f.o.c.

$$\lambda(t) = \frac {1}{c(t)^{\theta}}$$

So we want

$$\lim_{t \to \infty} [e^{-\rho t}\frac {1}{c(t)^{\theta}} K(t)] = 0$$

and since in the model we have a constant consumption growth rate, $g_c$ we have

$$c(t)^{\theta} = c(0)^{\theta}\cdot e^{g_c\theta t}$$

Ignoring inconsequential constants, we require

$$\lim_{t \to \infty} [\frac {1}{e^{(\rho+g_c\theta) t}} K(t)] = 0$$

Use the relation between household assets  and the number of goods $N$, the relation between income (from labor and assets) to aggregate output, as well as the expression of aggregate output as a function of $N$, to obtain a first-order linear differential equation in $N$, and proceed  as in pages 208-209 of the book.


What are some applications of Real Analysis in Graduate Economics?

My favorite example of real analysis' application in micro is the proof that lexicographic preference cannot be represented by any utility function. (A succinct version of the proof is given in p.43 of MWG.) It uses the notion of cardinality of sets, in particular, the different cardinalities between the rationals and the reals, which are some of the first things one learns in a real analysis course.

Other materials in real analysis, such as set operations, closedness, openness, continuity, differentiability, boundedness, compactness, etc., find their ways into every nook and cranny of economic theory. Some examples include:

Classical demand theory, which deals mostly with $\mathbb R^n$, the $n$-dimensional commodity space, draws on a lot results from real analysis
Any theoretical work that involves probability, e.g. econometrics as well as risk/uncertainty in micro or macro, would rely on some basic measure theory, an advanced topic in real analysis
Dynamic programming/optimal control, a technique for solving DSGE (and other) models, relies on the contraction mapping theorem
Separating hyperplane theorem is used to prove the existence of general equilibrium
Brouwer's fixed point theorem is used to prove the existence of Nash equilibrium

See some of the book recommendations and related questions on this site (1, 2, 3, 4) for more examples.


If a country is on a deficit it would want to keep rates low since it will most likely have to borrow money to pay for it's deficit.
It also has the possibility of debasing it's currency which in turn debases it's current/future debt.


What happened when the money from share issuance is spent?

You are concerned about...

someting that can save the bank from bankruprtcy it problems occurs,
when there is no money

This is an accounting question as much as an economics question. What you are asking about is "solvency".

assets minus liabilities equals equity

The bank or any company is solvent by definition if equity is positive and estimated values are realistic. Companies can even be solvent if equity is negative because estimated values are not realistic, for example, if assets are a "conservative" estimate which in this case means assets are underestimated.
You asked about the absence of money. All items in the assets are theoretically convertible to money. They are convertible regardless of the estimate of equity. In practice negative money balances are not necessarily a problem because if equity is large enough the company can obtain a loan. Typically a bank is holding cash because that is the nature of offering "demand" accounts which are the accounts where the depositor can withdraw cash almost immediately.


Does inflation equal change in M1 or M2?

The money supply is in theory interpreted broadly so from that perspective $M2$ or even $M3$ would be more appropriate. Also empirical studies in this strand usually use $M2$ see for example Sargent and Surico (2011).
However, you should note that nor the quantity theory of money nor monetarism claims inflation can be predicted solely by change in money supply and GDP growth.
Simple quantity theory of model is given by:
$$P=MV/Y$$
Where $P$ is price level, $M$ money supply, $V$ velocity of money and $Y$ real output. In more complex models expectations of these quantities matter as well. Hence it does not make sense to equate inflation to changes in $M$.
Furthermore, even though the above relationship is useful for structural modeling and creating analytical models it is practically useless for near term forecasting. If that’s your ultimate aim you would have more luck with augmented New Keynesian Philips curve.


For starters, elasticity is the percentage change in quantity (demanded or supplied, depending on what we're looking at) due to a one percent change in price. Or more vaguely: elasticity measures the sensitivity to price of behavior. From this definition it is clear that stating that a 'price is elastic' is nonsense. Trivially, the elasticity of price with respect to price is one.
That being said, I think I know what you mean. The long run supply curve is perfectly elastic. That is: the long run price is set in stone. It is insensitive to some changes. In a diagram as seen below this means a horizontal supply curve, with price being the same for all output levels. Why is this? Firms are often assumed to have some kind of efficient scale at which the average cost is at its minimum. That is: there is some particular quantity $q^*$ such that $AC(q^*)=\min\{AC(q)|q \geq0\}$. In the long run it seems reasonable to assume that every firm $i$ will produce exactly $q_i=q^*$. With number of firms operating such that in fact price will be equal to $P=AC(q^*)$. Define $P^*=AC(q^*)$,  formally the equilibrium would be such that: $nq^*=Q^d(P^*)$ (supply equals demand). So price will always be the same, namely $P^*$, with $n$ and thus $Q^S=nq^*$ changing in response to changes in demand. All this is long run.

Now  why does this theory make sense? Well, suppose at some moment the price in the market is below $P^*$, then every firm loses money, so it can't persist since firms can just stop operations. Suppose price is above $P^*$: a firm operating at the efficient scale can come in, undercut the market price by a small margin and sell at a profit.


Revenue maximization same as increase contribution margins?

If "contribution margin" is price minus variable cost per unit, and "revenue" is price times units sold then, even with zero variable costs, optimising

maximum contribution margin
maximum revenue

will typically be at different points

Assuming the demand curve of quantity and price is downward sloping, the contribution margin will fall as quantity increases, so the maximum contribution margin will at the maximum price, typically for minimal quantities.  If instead the demand curve is upward sloping initially and then downward sloping, then without variable costs the contribution margin will be maximised at the highest possible price

Meanwhile total revenue is usually not maximised for minimal quantities as at that point revenue will also be minimal. Instead it will be maximised when marginal revenue is zero (when the increasing revenue for marginal additional quantities exactly offsets reducing revenue from reduced prices), which with a smooth demand curve will typically be at a higher quantity and lower price than the maximum price point


Why is there so little automation at McDonald's?

Automation would increase the upfront cost of opening a McDonald restaurant. Which wouldn't be a good thing for its franchise business.
Also, you have to take into account that by automating its store McDonald's will drastically increase its electricity consumption (which will not only increase its costs overall but might impact its brand image in terms of global warming). So in other words, automation wouldn't be such a cheap alternative.


Using margins after probit estimation to equal probabilities between almost identical individuals

Let me give an example inspired by Richard William's presentation, and the Stata command margins.

webuse nhanes2f, clear
keep if !missing(diabetes, black, female, age, age2, agegrp)
gen femage = female*age
label variable femage "female * age interaction"
probit diabetes black female age, nolog

Probit regression                               Number of obs     =     10,335
                                                LR chi2(3)        =     380.15
                                                Prob > chi2       =     0.0000
Log likelihood =  -1808.992                     Pseudo R2         =     0.0951

------------------------------------------------------------------------------
    diabetes |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       black |   .3567259   .0638644     5.59   0.000      .231554    .4818978
      female |   .0859841   .0450391     1.91   0.056    -.0022909    .1742592
         age |   .0271895   .0016318    16.66   0.000     .0239912    .0303879
       _cons |  -3.215875   .1015044   -31.68   0.000     -3.41482    -3.01693
------------------------------------------------------------------------------

Among other things, the results show that getting older is bad for your health (surprise!) – but just how bad is it?

Adjusted predictions (also known as predictive margins) can make these results more tangible.  With adjusted predictions, you specify values for each of the independent variables in the model, and then compute the probability of the event occurring for an individual who has those values. Using the mean values for the other independent variables (female, black) that are in the model, we get:

margins, at(age=(20(10)70)) atmeans

Adjusted predictions                            Number of obs     =     10,335
Model VCE    : OIM
Expression   : Pr(diabetes), predict()

1._at        : black           =    .1050798 (mean)
               female          =    .5250121 (mean)
               age             =          20
2._at        : black           =    .1050798 (mean)
               female          =    .5250121 (mean)
               age             =          30
3._at        : black           =    .1050798 (mean)
               female          =    .5250121 (mean)
               age             =          40
4._at        : black           =    .1050798 (mean)
               female          =    .5250121 (mean)
               age             =          50
5._at        : black           =    .1050798 (mean)
               female          =    .5250121 (mean)
               age             =          60
6._at        : black           =    .1050798 (mean)
               female          =    .5250121 (mean)
               age             =          70
 ------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   .0063084   .0009888     6.38   0.000     .0043703    .0082465
          2  |   .0113751   .0013794     8.25   0.000     .0086715    .0140786
          3  |   .0204274   .0017892    11.42   0.000     .0169206    .0239342
          4  |   .0364184   .0021437    16.99   0.000     .0322167      .04062
          5  |   .0641081   .0028498    22.50   0.000     .0585226    .0696935
          6  |   .1104379    .005868    18.82   0.000     .0989369     .121939
------------------------------------------------------------------------------

The results show that a 20 year old has less than a 1 percent chance of having diabetes (.0063084), while an otherwise comparable 70 year old has an 11 percent chance (.1104379).


Contour set in Mas-Collel, Winston and Green

The upper contour set is $ UC(x)=\{ y|y \succsim x \} $, i.e., all the goods baskets that are at least as desirable as $x$. Similarly, the lower contour set is $LC(x)=\{ y|x \succsim y \} $, i.e., all the goods baskets that are at most as desirable as $x$.


Can changes in the value of money be measured via interest rates and future contracts?

Yes, but...

In theory what you suggest is pretty standard text book, and in fact Keynes makes exactly this point in his book The General Theory of Employment, Interest and Money as a way of getting a grip on the idea of interest and "Time-Value of Money" (don't have copy to hand, will update reference when I get a moment).

The problems come, as always, in the details - in fact in details that you have glossed over in your original set up.

If these 10 apples can be traded for those 20 oranges, then how can you be sure that another 10 apples can also be traded for a different 20 oranges.  Or even, are we sure 100 apples can be traded for 200 oranges?

Adding the dimension of time to this, creates even more problems, but putting those aside the relationship you are looking for is

$$F=S_0e^{rT}$$

That is the future price ($F$) is equal to the price now ($S_0$), inflated by the time value of money.

Now back to apples and oranges.  Are we certain that 10 apples now is the same as 10 apples in the future?  Or more pertinently does the 10:20 apple:orange exchange rate persist into the future?  Probably not.

Apples and oranges are both seasonal, and also suffer seasonal demand.  We could image that apples are easy to come by now, but hard to get in 6 months time, vs oranges.  Since you can't store them, this has to be reflected in the futures price.  This means you will get a different $e^{rT}$ value for apples and oranges

These main causes of these difference are usually broken down thus

Storage costs: can the thing be stored and delivered at the future date?  If so how much does it cost to store?
Carry: How much do you earn holding the thing for longer
Convenience yield:  How much premium are you willing to pay for the convenience of having it to hand?
Credit risk:  How much do you trust that the person will deliver (to the best of my knowledge there are no apple or orange futures, so we would be talking about a forward contract)

Take all these into account, and you have a commodity implied measure of future inflation.  Of course, inflation as usually considered also includes services and retail products, on which there are very few (no?) futures/forwards.  If you could find iPad futures and child care futures, then you would in theory be there.


The short answer is that average costs fall because fixed costs get averaged across more units.

In more detail:

Costs have two components: fixed costs (which do not vary with produced quantity) and variable costs (which do vary with produced quantity).

For example, if you produce newspapers your costs might include

paying journalists to write and edit the content of the newspaper
renting a building for the journalists to work in
paying to print each copy of the newspaper.

Notice that you only have to pay the journalists and the rent once, no matter how many copies you print (so these are fixed costs). The cost of printing (paper and ink) will increase as you produce more copies of the paper because you will need more of these inputs, so these are variable costs.

Suppose it costs \$10,000 to pay a team of journalists to produce an issue of a newspaper (fixed cost), and \$1 to print a copy of that paper (variable cost).

If you sell one thousand copies then the total cost is $$\$10,000+(1,000 \times \$1)=\$11,000.$$
That means the average cost of producing each copy is $$\$11,000/1000=\$11.$$

Now suppose production increases to 2,000 units. Now the cost is $$\$10,000+(2,000 \times \$1)=\$12,000.$$ The average cost per unit is $$\$12,000/2000=\$6.$$

So output increased and the result is that the average cost per unit has fallen. This is a situation in which there are economies of scale. The reason this happens is that, as output increases, the fixed cost (here the \$10,000) gets spread over more units so that the average fixed cost per-unit decreases even if the variable cost does not. In particular, notice that in the first case the fixed cost gets averaged over 1000 units (a fixed cost of \$10 per unit), whereas in the second it is averaged over 2000 units (a fixed cost of \$5 per unit).


If by dangerous it is meant they add possible downwards volatility of assets, that could easily be argued for as:

$ PV = D / (1+r)^t$

Where D denotes any future income. This is especially the case if the reason for the low rates is very easy monetary policy, as such policy cannot be maintained and thus the rate is not equilibrium rate and must rise, causing a crash in asset values when that happens.

Central banks determinate the interest rates in the short run, their policies are targeted to lower rates, thus a low rate. The other reason is that of supply and demand. There is not much demand for credit (no profitable ideas, no more room for consumer credit) and on the other hand the supply of credit is significant due to central bank action. High supply and low demand equals a low rate.


The expression "general equilibrium" refers to the work by Kenneth Arrow, Gerard Debreu, Leon Walras, and many others. In its strictest sense, it is the indirect impact of a policy/event that is due to the reaction of prices to the policy/event. Sometimes but not always this indirect impact offsets the beneficial impacts of the policy or event.

Sounds abstract? Let me give you an example.

An oil field is discovered in a small town in Texas. As the oil exploration company develops the field, that generates jobs and higher incomes. Looks great, because local businesses will enjoy higher revenues and unemployed workers may find employment.

Yet, there can be some substantial general equilibrium effects that partly offset  the positive consequences of this beneficial event:

The prices of many products will rise. Housing is going to be more expensive as higher-income workers are going to try to purchase houses in the town. Restaurant meals and haircuts will become more expensive.
A positive consequence: there will be fancier restaurants and hair salons in town. The higher incomes will attract more talented chefs and better hairdressers.
There may be more congestion on the road, as more workers means more cars.

Sometimes the general equilibrium effects are such that the beneficial consequences of the event are almost entirely offset (more expensive, more congestion, more pollution). Sometimes the general equilibrium effects are small and can be ignored. If, for instance, there were already a lot of empty homes in this small town, the additional workforce and income won't drive house prices up. Workers will merely buy the existing stock of houses.


At the intersection of differential equations and game theory one can find differential games. Arguably, the most famous applications of differential games are in warfare, e.g., the homicidal chauffeur problem. However, not all differential games are of the pursuit-evasion kind.
Whoever wants to learn differential games may wish to learn optimal control theory first. To the neophyte, I would recommend Evans's book (chapter 6 is on games):

Lawrence C. Evans, An Introduction to Mathematical Optimal Control Theory [PDF], Department of Mathematics, University of California, Berkeley.

The next step could be the following books on applications to economics:

Engelbert J. Dockner, Steffen Jorgensen, Ngo Van Long, Gerhard Sorger, Differential Games in Economics and Management Science, Cambridge University Press, November 2000.

Luca Lambertini, Differential Games in Industrial Economics, Cambridge University Press, April 2018.

and, more on the stochastic side:

David W. K. Yeung, Leon A. Petrosjan, Cooperative Stochastic Differential Games, Springer, May 2006.

Unlike Evans's book, I have never read these ones. My listing of these books is not an endorsement.


Prove that a continuous $\succsim$ is quasilinear

First off it seems to me you are proceeding in a manner more complicated than necessary. (Perhaps this is intentional because you wish to face a harder exercise.) Since it is given that $v(x)$ represents $\succsim$ you only have to show
$$
\forall x,x' \in X: v(x') \geq v(x) \iff u(x') \geq u(x).
$$
I think this would be a lot simpler than what you are currently doing.

On to your question:
 The function $u(x)$ is very well defined, as
$$
u(y,x_{L}) =  x_{L} + \phi(y) = x_{L} - x_L(y)
$$
and we know that $v(y,x_L(y)) = 0$ describes the indifference curves of $\succsim$. The conditions you describe would violate this assumption because from
$$
(y,x_{L}) \precsim (y',x_{L}')
$$
it follows that
$$
v(y,x_{L}) \leq v(y',x_{L}').
$$
We can assume $v(y,x_{L}) = 0$, this is an affine transformation of any $v(y,x_{L})$. So $x_L(y) = x_{L}$.
 (I am not crazy about the notation, feel free to edit my post as long as it remains consistent with the question.)

 Then we also have $0 \leq v(y',x_{L}')$, and because of monotonicity this means
$x_L(y') < x_{L}'$.
 Then
$$
u(y',x_{L}') < u(y,x_{L})
$$
cannot hold as
$$
u(y,x_{L}) = x_{L} + \phi(y) = x_{L} - x_{L}(y) = 0
$$
and
$$
u(y',x_{L}') = x_{L}' + \phi(y') = x_{L}' - x_{L}(y') > 0,
$$
because of what we have shown earlier, $x_L(y') < x_{L}'$.


When estimating a basic macroeconomics model, why is goodness of fit with labor data not often dealt with?

It's somewhat public knowledge that Macroeconomic models don't fit labor data without

High Frisch Elasticity (but Rogerson, Prescott try to argue for that)
Preference shocks (but, as Chari, Kehoe, McGrattan (AER 2006?) show, we cannot really understand these properly
Search&Matching with weird parametrization or assumed wage rigidity
Some other spooky thing going on that helps fitting the data

So since we know it doesn't work out, why feel bad about posting the correlations and make everyone feel bad? :)

If it were to work out, surely post it though.


First: Pareto efficiency, optimality and dominance don't all describe equilibria.  Loosely, they mean the following:

A Pareto-efficient outcome is an outcome that satisfies the condition that no agent is worse off, and at least one agent is better off, as a result of the change.

A Pareto-dominant outcome is an outcome that is both Pareto-efficient and increases overall welfare with respect to an alternative.

Finally, an outcome is Pareto-optimal if it is Pareto-efficient and is not Pareto-dominated by some other outcome.

Only the last definition has something to do with equilibria.  By the First Fundamental Welfare Theorem, perfectly competitive market equilibria are Pareto-optimal.

As others have said, we might need to know more about the model 13.B to help you, but roughly speaking, if the wage incentives in your model are such that by part a) nobody is willing to work, then nobody acts.  Likely, within your model, if nobody acts, nobody incurs a cost, and so nobody is worse off.

So, to understand b), we need to be able to see whether at least one agent is better off if nobody works.


What is the definition of a "Stackelberg leader-leader equilibrium"?

A leader-leader Stackelberg is a situation in the Stackelberg model where both firms believe they are leaders. This leads to global production being much higher than expected by both firms, as they anticipate small production by the other firm in response to their high production.


How can you model/simulate the price/value of a good?

You're question is unclear...

If you're in a competitive market, as a supplier, you can only see the market price at a given time, where it is set essentially by all suppliers and consumers at once

If you're a monopoly, then you can estimate the quantitive demand based on price, simply because you control the entire market, and set the price level at the marginal revenue, where you gain the most

The only ways, other than a monopoly, to estimate demand are:

If demand is completely inelastic, like with fuel: people won't change their driving patterns on marginal changes in price
meaning quantity will remain the same in any case

completely elastic: price will remain the same no matter the quantity

A government set price, quantity or any other specific regulation on the product. Although, with "too much" regulation, in some cases a black market can be created, etc.


Slutsky matrix computed from budget shares

In all indexed references to the Slutsky matrix in Mas-Collel et al. no such relation can be found. It is my impression that the authors meant that the expression they write conveys the same information and restrictions with the original, being a scaled version of it.

I will use indices $i$, $k$, and write the logarithm explicitly in order to avoid confusion.

$$\frac {\partial b_i}{\partial \ln p_k} = \frac {\partial (x_ip_i/w)}{\partial \ln p_k}$$

Get liberal with partial differentiation, think of it as a differential, which gives

$$\partial \ln p_k = \partial p_k/p_k$$

Insert above to get
$$\frac {\partial b_i}{\partial \ln p_k} = p_k\frac {\partial (x_ip_i/w)}{\partial p_k} = \frac{p_ip_k}{w}\cdot \frac {\partial x_i}{\partial p_k}$$
and rearranging

$$\implies \frac {\partial x_i}{\partial p_k} = \frac{w}{p_ip_k} \cdot \frac {\partial b_i}{\partial \ln p_k} \tag{1}$$

So we have expressed the first term of the usual Slutsky element in terms of logarithmic derivatives of budget shares. For the second component we have

$$\frac {\partial b_i}{\partial \ln w} = w\frac {\partial (x_ip_i/w)}{\partial w} = w \frac {(\partial x_i/\partial w)p_iw - x_ip_i}{w^2} = p_i\frac {\partial x_i}{\partial w} - b_i$$

rearranging and also multiplying by $x_k$ we get

$$\frac {\partial x_i}{\partial w} x_k = \frac {x_k}{p_i}\frac {\partial b_i}{\partial \ln w} + \frac {x_k}{p_i} b_i$$

Multiply and divide each element by $p_kw$:

$$\frac {\partial x_i}{\partial w} x_k = \frac {x_kp_kw}{p_ip_kw}\frac {\partial b_i}{\partial \ln w} + \frac {x_kp_kw}{p_ip_kw}  b_i$$

$$\implies \frac {\partial x_i}{\partial w} x_k = \frac {w}{p_ip_k}\frac {\partial b_i}{\partial \ln w}  b_k + \frac {w}{p_ip_k}  b_ib_k \tag{2}$$

Combining,

$$(1),(2) \implies s_{ik} = \frac {\partial x_i}{\partial p_k} + \frac {\partial x_i}{\partial w} x_k \\
=\frac{w}{p_ip_k} \cdot \frac {\partial b_i}{\partial \ln p_k} + \frac {w}{p_ip_k}\cdot \frac {\partial b_i}{\partial \ln w}  b_k + \frac {w}{p_ip_k}\cdot  b_ib_k $$

$$\implies s_{ik} = \frac{w}{p_ip_k} \Big(\frac {\partial b_i}{\partial \ln p_k} + \frac {\partial b_i}{\partial \ln w}  b_k +  b_ib_k\Big) \tag{3}$$

The term in the big parenthesis is what the authors give as the off-diagonal element of the Slutsky matrix. The scaling factor $\frac{w}{p_ip_k}$ is symmetrical, so the expression in the big parenthesis nevertheless reflects the same symmetry conditions as the usual $s_{ik}$ terms.

If you apply the same method for the diagonal elements you will get the $-b_i$ term.

This form of the Slutsky matrix I believe is old story, and it has come about for purposes of econometric estimation of demand systems. For example, in Theil, H., & Clements, K. W. (1980). Recent Methodological Advances in Economic Equation Systems. American Behavioral Scientist, 23(6), 789-809. we find this form of the Slutsky elements, which the authors call "elements of the Slutsky matrix" without any "explanation" as to why the textbook expressions are scaled.

But I would advise to go ahead and e-mail the authors asking for clarifications.


Where to start learning economics as a mathematician?

McCloskey's Applied Theory of Price is somewhat legendary as a model of clarity. It is, sadly, no longer in print. However, a PDF is available here: http://www.deirdremccloskey.com/docs/price.pdf. The level of mathematical sophistication in this book (like in most good economics!) is not that high, but the book gives a deeper look at the important concepts than is to be found in most extant introductory undergraduate texts.

If you really want a more mathematically formalised treatment then you are going to have to turn to graduate-level econ books. For microeconomics, the classic text of choice is Microeconomic Theory by Mas-Colell et al.

If you do go down the more formal route, I would urge you not to give up on simpler treatments altogether; they're great for building intuition. More than a decade of experience in economics has taught me that good intuition built through simple models is often far more valuable than a less intuitive understanding of a more sophisticated model.


Credit risk of interest rate on excess reserves vs fed funds rate?

Actually the IOER has slightly exceeded the Fed Funds rate in recent years.  Currently IOER is 10bp and FF last set at 8bp for example.  During the period 2009 to 2015, IOER was set at  25bp and FF averaged about 13bp.  So what is going on? The FOMC sets IOER as a matter of policy , but FF is set by supply and demand for overnight loans between financial institutions in the Fed system.  In this era of excess reserves , many financial institutions have excess cash, so the problem is to lend it at the best rate.  The problem is that not all institutions can access IOER, because it is for depository banks only , so there is additional cash to invest held by others, such as the Federal agencies (home loan banks) for example.  They invest in overnight Fed funds, driving the rate down below IOER.  A bank could (and some do) borrow at these lower rates to invest in IOER, locking in a few bp.
Thus, the spread is driven by considerations of supply and demand rather than credit risk.


What does it mean, 'Higher interest rates attract FIIs and FPIs'?

Foreign Institutional Investors (FII) and Foreign Portfolio Investors (FPI) can and do invest in government securities as well. There may be restrictions of course leading to imperfect capital mobility. For example in India, FIIs/FPIs cannot hold more than 6% of outstanding stocks of securities.
As interest rate increases (usually because of an external shock - such as a policy shock by the Central Bank), there is (increased) room for arbitrage. This attracts investors looking to make quick money.
You are right that interests may not impact share prices, but there is some relation of course. This comes, among other things, from risk management. Portfolio managers balance risk by dividing investment in risk free government securities and equity. When return on g-secs go up, this balance goes into disequilibrium. New investors can invest more in equity as some additional risk is balanced by increased returns in g-secs (you can think of the two assets as imperfect substitutes). So this increases prospects for more investment in equity and so it makes sense to quickly invest in them as well for short term gains.
All this logic will, however, go into disarray as we bring in expectations of future interest rates. This is why FIIs/FPIs are also referred to as hot capital flows. These investments are short term and the flow can quickly reverse with slight changes in market sentiments and expectations.


Replicate Romer and Romer (2004) results

Sorry for any confusion in my previous answer but there are 2 steps to this process 1) tracing out the impact on the $\Delta$y's and the lags of the shocks and then 2) accumulating the shocks to get the cumulative response (Note that my original answer tried to do this simultaneously). So in the first step (ignoring the intercept, the dummies and the residual terms) we get

$\tilde{\Delta y_1}$ : $c_1$

$\tilde{\Delta y_2}$: $c_2 + b_1c_1 = c_2 + b_1\tilde{\Delta y_1} $

$\tilde{\Delta y_3}$ : $c_3+ b_1(c_2 + b_1c_1) + b_2c1 = c_3 + b_1\tilde{\Delta y_2} + b_2 \tilde{\Delta y_1}$

$\tilde{\Delta y_4}$ : $c_4 + b_1\tilde{\Delta y_3} + b_2 \tilde{\Delta y_2} + b_3 \tilde{\Delta y_1}$

Can you see the pattern? So the general rule is

$\tilde{\Delta y_i}$= $c_{i}+\sum_{r=0}^{i-1}b_{r}\tilde{\Delta y_{i-r}}$

When $\tilde{\Delta y_0}=c_0=b_0=0$ and where $c_i=0$ for $i>36$ and $b_r=0$ for $r>24$.

Now the next step to is to accumulate the shocks so that the IRF value for a given month is

$IRF_i = \sum^i_{j=0} \tilde{\Delta y_j}$

This should give you the results you are looking for. Technically if you were to trace out the exact values of the regression that you specified then you would also need to include the dummy variables but from the sounds of it they only care about tracing out the effects of the monetary shocks.


Why doesn't AD curve self-correct to higher price level in LR?

It does, in the long-run.  It's partly a question of response time, and as you outline, AD has intervening steps that AS does not (wages in most markets are sticky and take time to adjust, while sticker prices in many markets can change in a day or less).  In general, thinking carefully about response times will help alleviate some of the ambiguity in the models you're looking at.


What is the difference between average cost and marginal cost?

The average cost is the cost on average: total costs (C) divided by total number of units of production (q). Just as the equation you gave, says

The marginal cost is the cost at the margin: the additional cost of one extra unit of production, just as the equation you gave, says. Most goods and services come in discrete units: in those cases, it is not about infinitesimals, it is about the last unit of production, the unit that was produced to meet demand, the unit that would not have been produced had demand been one unit lower. In cases of continuous supply, the marginal cost is the gradient of the curve of total cost plotted with respect to total supply.

In pretty much all real-world circumstances, marginal cost is generally greater than average cost: that's guaranteed by a strictly monotonically increasing supply curve (with the trivial exception of the first unit of production, where average cost is by definition equal to marginal cost).


Multidimensional screening and convexity of the surplus/rent function

First, it's important to be clear about what Rochet proves. In theorem $1$ of Rochet (1987), he shows that cyclical monotonicity is equivalent to the implementability of a mechanism for general environments (i.e. arbitrary allocation/type spaces). (See the statement of theorem $1$, or a standard text such as Borgers (2015), for the definition of cyclical monotonicity.)

The convexity result (proposition $2$) applies to the case when $u(x,\theta)$ is linear in $\theta$. In this case, we have that

$$ V(\theta) = \max_{\theta^\prime} \, u\left(x\left(\theta^\prime\right),\theta\right) - t\left(\theta^\prime \right) .$$

The convexity of $V$ follows from the fact that it is the maximum of a family of linear functions. I'm not sure there's much more intuition to be given here, but a simple drawing may help. (That is, draw two straight lines that intersect. The maximum over the two straight lines will be convex, while the minimum will be concave.) This may also help.


Did the Bank Of England say QE doesn't work?

Stated goal of monetary policy is low and stable inflation:
https://www.bankofengland.co.uk/monetary-policy
Tools for setting monetary policy are bank rate and quantitative easing or asset purchase programs.
One way to view the central bank tools is to identify the combination of the government deficit spending and private credit market system as the two drivers of aggregate demand. During war time since World War II, for example, the modern governments deficit spend, cause full employment, and shift resources from domestic consumption and investment toward war time uses, and this would cause inflation unless the government and/or central bank institute policies to curb high inflation. Take away government deficit for war spending and you are left with the peace time deficit and private credit system as drivers of high inflation. The central bank increases the bank rate to reduce inflation caused by the combination of government deficits and private credit market activity. But if there is deflation then bank rate cannot be decreased below zero to stimulate the provision of credit so this is called the Zero Lower Bound (ZLB). Quantitative easing is used at the ZLB but does not necessarily stimulate credit formation to generate mild inflation.
During a modern financial market crisis the banks and other financial intermediaries have problems with solvency and liquidity. Banks and financial intermediaries (FI) develop a liquidity problem (modern bank runs) when the depositors or investors in the FI suspect that a solvency problem will begin to wipe out equity claims due to bad loans in the asset portfolio. The so-called global financial crisis was triggered by solvency concerns and developed into a market crisis where banks and FIs could not develop liabilities or place new paid-in equity to keep cash flowing in the payment system. The central bank had to provide liquidity to keep the solvency problems from getting worse due to market reaction to the solvency problems.
Quantitative easing (QE), also known as large scale asset purchases (LSAP) or asset purchase programs (APP), injects liquidity into the aggregate bank sector when the central bank purchases financial assets from banks and/or nonbanks. So if the goal is to solve the liquidity problem during a financial crisis then QE may be a solution to this part of the problem. But if the goal is to stimulate bank lending or lending by financial intermediaries to stimulate the economy then QE may not be a solution.
I don't know about the finance in United Kingdom. In the United States during the 2008 financial crisis the FDIC increased the amount of insured deposits; the Fed injected reserves and deposits into the aggregate bank sector by buying securities from non-banks; the Fed paid interest on excess reserves (IOER) to keep banks from purchasing short term Treasury securities paying lower rates of interest than the IOER policy rate; the banks and Fed purchased mortgage-backed securities guaranteed by the Treasury through the federal home loan guarantee programs; and the nonbanks purchased more short and long term Treasuries to hold in their liquidity cushions. This means the combined actions of Fed and Treasury provided more central bank liabilities and federal government liabilities to hold in the aggregate bank and nonbank liquidity cushions and probably this was not a policy accident but a plan by the leaders of the central bank and Treasury to help the markets adjust to the crisis with less deflation than without a bailout.
So the combined actions of FDIC, Fed, and Treasury provided liquidity to both the bank sector and the nonbank sector. In general the Fed and banks purchased assets guaranteed by Treasury paying higher interest (Agency and guaranteed MBS) and the nonbanks held the Treasury securities paying lower interest in the period after the federal government expanded the Treasuries float and federal guarantee in the asset pools called mortgage-backed securities. One purpose of these programs was to provide liquidity. Another purpose might be to allow bad debts to be written down by the banks and financial intermediaries while the government helped the banks to recapitalize. Another concern would be to use deficit spending and/or monetary policy to stimulate the economy so it would not go into a deep recession and QE may not be effective at solving that macroeconomic problem. But I think QE would provide liquidity and help banks recapitalize compared to a counter-factual outcome where the government lets the markets determine winners and losers by portfolio allocation decisions made during a financial panic.


Does price affect quantity or vice versa?

This is a common problem among students first learning economics. The way these graphs are displayed are not meant to tell you anything about causality, let along the direction of causality. The reason that price is on the y-axis is partly because of tradition and partly because that's what makes the most sense from the perspective of developing economic theory.

One of the central questions of economics is the determination of prices. What causes prices to be what they are? So, it maybe seems natural to write relationships that give a price as the dependent variable. However, the layout is not meant to convey any information about causality.


Can one use the standard deviation of a variable as a regressor?

By including both regressors, you are effectively trying to find the partial effect of either the temperature level, or variance, on crop yield. That is, the effect of the one, after holding the other constant. Does that approach make sense? Do you need to know the effect of a rise in temperature, for a constant variance in temperature? Or, do you need to know the effect of a change in the deviation in temperature, for a given level in temperature?
The multivariate approach finds those partial effects. The coefficients can be biased if the two variables are correlated with each other—you can test multicollinearity by regressing the one regressor on the other (or, for higher numbers of regressors, one would look at the Variance Inflation Factor).


It is not necessarily bad news.

Simply put, if you're willing to buy product x (bread, car, house, ect.) then falling prices are good news for buyers, but bad news for the sellers. The same logic applies for oil. If you are a buyer, you are optimistic about falling prices, however being a seller would make you loose money and/or postpone new investments.


Can a continuous preference be represented by a discountinuous function?

Indeed you are correct that a continuous utility function can be represented by a discontinuous function. However, I am not sure what you mean by proof beyond an example; an example is a proof of this fact.

For completeness, here is an example we can take $\succeq\,\subset \mathbb R \times \mathbb R$ to reflect the usual ordering (i.e., $x \succeq y$ iff $x \geq y$). Clearly this is a continuous ordering as its contour sets are intervals. The utility function
$$ U(x) = \begin{cases}
x &\text{ if } x < 4 \\
x + 1 &\text{ if } x \geq 4
\end{cases}
$$
is discontinuous, but represents $\succeq$.

Any such function is necessarily have a continuous representation as well.


How do I predict Macroeconomic indicators?Or are there any free resources where I can get the predicted values?

I will give a partial answer.

I assume you mean “forecast values.” Forecasts generally come from private firms that do the forecasts, and are typically expensive. There are some surveys that are in the public domain (Philadelphia Fed Survey of Professional Forecasters). You can generate your own using a model, but quality depends uponyour model.
It is normally easier to work with seasonally adjusted data. However, the data are passed through a filter, and distortions can occur. (For example, the pandemic of 2020 has caused spikes in data, and seasonal adjustment can make those spikes worse.)
The last question is open-ended. People spend their entire careers in finance and at central banks building models.


References on the sustainability of the public debt

A big debate started after a flaw had been found in the conference paper "Growth in a Time of Debt" by Reinhart and Rogoff. There are several replications and many articles in the media and blog posts. Reinhart and Rogoff have the book "This Time is Different" on the topic and several well published articles, cf. here. Research on the topic continues to be published in top journals, e.g. "Is There a Debt-Threshold Effect on Output Growth?", Review of Economics and Statistics 99:1, March 2017. You should look at the references and also at which papers have cited the ones you find important.


The Sonnenschein-Mantel-Debreu results, what are the implications for macroeconomics?

This is somewhat broad and argumentative question, but I'll try to provide one possible answer (disclaimer: I'm not a GE theorist). SMD theorem states that imposing only the abstract structure of general equilibrium has few empirical implications. Fair enough. But it's incorrect to say that

  we cannot hope to recover the underlying preferences from an aggregate demand setup as much of the cross-country empirical macro or calibration macro seems to try.

because once we impose additional assumptions about preferences, technologies and endowments, we do obtain additional empirical predictions (and of course, different sets of assumptions will lead to different predictions). This is precisely what economists are doing most of the time! It's not 1950's anymore - these days, there is very little research that tries to arrive at general results from abstract axioms. Most theoretical work in macroeconomics will present specific model, derive its qualitative and quantitative implications and often will include an empirical component.

Another often-heard argument, present e.g. in the linked piece, is that SMD theorem shows GE models are in general unstable or indeterminate, and thus we must resort to unrealistic restrictions, such as representative agent assumption, to guarantee that our models are well-behaved. But this confuses necessary and sufficient condition: yes, assuming representative agent is sufficient to obtain well-behaved aggregate demand function, but that doesn't mean all, or even most, models with heterogeneity are somehow automatically unstable. And a logical possibility of unstable equilibrium says nothing about whether such situation is empirically relevant (maybe it is, but critics usually provide no evidence for such claims).


When does a high markup mean high productivity? Never?

allocative efficiency = Using resurces where they add the most value.

Example:
The resources needed to create a branded t-shirt (or any other product with a strong brand) is the same as for the creation of a no-name t-shirt. However, the price/value of the branded product is considerably higher.

So, if you can switch production from cheap no-name products to more expensive branded products then output value increases but resource consumtion is unaffected. This value increase is separate from the resource reduction caused by an increase in productivity.

  However, isn't this exactly the oposite of what one would want?

Depends on who "one" is. For american companies it is good, since it increases their profits. For the resource producers (seamstresses, cotton farmers etc) it is bad since it reduces the demand for their resources.

  It seems that they are somehow mixing up the possibility of high profits due to one would usually think of as high productivity, with high profits due to market imperfections...

That depends on whether you consider branding a market imperfection. When measuring it is very difficult to distinguish quality and branding. If a no-name t-shirt costs \$6 and branded t-shirt costs \$30, how much of the price difference is due to quality and how much is brand name?

  Also, it's also as if the authors would defend a higher 'productivity' but at a lower output, than a lower 'productivity' at a higher output.

From the quoted snippet it looks like they are describing the effects successful branding has had for US companies and for the US economy.


Asset pricing vs Empirical asset pricing

As mentioned in Herr K's comment, asset pricing is the theory to price assets (such as equity, bonds, options, futures, swaps, etc). For this, you can use models like CAPM/Fama-French (returns), Black-Scholes (options), Swensonn (interest rates), and many many others. When you say empirical asset pricing, this means that you go to the data (and each model deals with a different type of data) and try to price an existing asset based on the information that you see on the market.

For example, in the Black-Scholes model, given the current stock price (S), the strike price for the option (K), the time until exercise (T-t), the risk-free rate (r) and the volatility of returns (sigma), you can use the formula directly and calculate the price of the option. Notice that this would be a theoretical price. In practice, you might observe something else in the market. You may want to do some arbitrage from here since there is mispricing (Don't go using this in the real market, but you get my point).

For your last question, asset pricing is not necessarily just about the current value of an asset. You probably would also want to estimate future values based on your models. In fact, it is really of no use a model that tells you today's price. For that, you just look at the price on the tape and ignore your model. The whole point of building models is to make predictions.


Do new treasuries stop being issued during a debt suspension period?

No, it does not mean that no new debt is issued. All it means is that the value of new debt issuances cannot exceed the value of debt that reaches maturity and is paid off.

Here's a list of recent Treasury debt auctions. You'll note that many T-bills have maturities as short as 4 weeks— a hint that the debt is actively managed.  Simplifying somewhat, when receipts do not exceed outlays by an amount greater than currently-maturing debt, new debt must be issued to generate enough cash to retire the maturing issues.


Are there real world examples of a shoe event horizon?

The dutch disease is when an economy is so dominated by a single export industry that the success of that industry drives up the currency and makes the exports of other industries un-competitive. This drives some of the firms making other goods out of business.

Continuing this cycle can lead in the short term to even greater concentration of the economy in that industry.

If later the export industry experiences a significant shock, the broader economy also experiences that shock. This happens for two reasons:

In a diversified economy some firms are experiencing positive shocks when others experience negative ones. In a concentrated economy it is less likely that the other industries will collectively experience an sufficient offsetting shock.
A negative shock to the primary exporting industry is likely to depreciate the currency which should help other industries but shrinking capacity from the expansion will diminish other industries' ability to generate sufficient exports to offset the shock.

Further, if the industry responds by boosting output (like sometimes happens when oil prices fall and producers become desperate) this can make export prices fall further still, worsening the shock.


Considering how USD enters the market, is the USD actually unbacked? Could the USD represent a unit of a “fund” of securities held by the Fed?

When the Fed purchases something, it creates the money to do so out of thin air (it just changes the balance in a bank's Fed account. Click, click.). Similarly when the fed is paid, the dollars disappear. From the Fed's perspective, dollars are very ephemeral and not backed by anything.

Dollars are not like debt. They don't entitle you to anything the Fed owns. The Fed can't "go under" in the normal sense of word (bankruptcy) because it has no obligations to repay or if it does, it creates the money to pay those obligations with no limitation.  It goes under only by an act of law.

It would be possible for the government to abolish the fed and take its assets, but that doesn't sound much like a fund to me--there is no reason to think those assets, whichever ones remain meaningful in that context, would be disbursed to dollar holders.  I would expect them instead to be kept by the treasury and used to reduce the national debt.


Why did the proportion of household savings in the USSR decrease during its transition period?

I couldn't say what was the actual explanation but Increased availability of consumer goods could deliver this result. Say people in the USSR expected that eventually there would be a consumer revolution where their choice of goods would be better. Then utility would look like something like this:
$$ U(c_1, c_2, c_3) = ln(c_1) + \beta \cdot \xi \cdot ln( c_2) + \beta^2 \cdot \xi \cdot ln( c_3)$$

where $c_i$ is the consumption in period $i$, $\beta$ is the discount rate, and $\xi > 1$ indicates how much better consumption is in time 2 than 1. A higher value of $\xi$, meaning a greater improvement in consumption acts like a higher value of $\beta$, meaning lower discounting of future consumption in period $1$. That means that households will (all else equal) save more in period 1. In period 2 there is no expected increase in the quality of consumption, only the standard time discounting differentiates consumption in the present and future, and so the savings rate (relative to period 1) falls, just as it would if $\beta$ fell.


What are the benefits of a first price sealed bid auction?

A good article to read about this is by Susan Athey, Jonathan Levin and Enrique Seira. It has the title: "Comparing Open and Sealed Bid Auctions: Theory and Evidence from Timber Auctions," was published in the Quarterly Journal of Economics, vol. 126(1), 2011, 207-257, and is available online at:

http://faculty-gsb.stanford.edu/athey/documents/comparingformats.pdf.

The abstract includes this: "We study entry and bidding patterns in sealed bid and open auctions. Using data from U.S. Forest Service timber auctions, we document a set of systematic effects: sealed bid auctions attract more small bidders, shift the allocation towards these bidders, and can also generate higher revenue. A private value auction model with endogenous participation can account for these
qualitative effects of auction format."


Isn't the liquidity trap about real interest rates?

The situation you describe, one with negative nominal rates, only can happen with the use of force or where the instrument acts as an insurance policy.  In the case where the instrument serves as an insurance policy, the implicit premium is that negative rate.

To consider why this is the case, consider two investments.  The first investment provides an $\alpha$ percent rate of return where $-100\%<\alpha<0$.  Now consider holding a bank note that pays a zero percent rate of return.  Bank notes have, at times, paid positive rates of return.  Let us assume that we have perfect foreknowledge of future deflation and we know that both will provide a positive real rate of return.

The bank note would always dominate the negative rate of return.  No one would invest in the security that provided negative nominal returns unless they believed the banknote itself was suspect.  There are two ways this dominates.  First, the note provides a higher real rate of return.  Second, however, the banknote will do a better job at preserving its value even under an assumption of future inflation because it pays a higher nominal rate, which is zero percent.  The banknote is both more valuable and simultaneously less risky.

As to the second part of your thinking, that banks would run out of liquidity, you are in error.  The reverse would happen.  As banks buy fewer bonds, they become less leveraged and intrinsically are more flexible and liquid.  Banks become fragile so that their customers can be flexible.  If there are no offers to borrow from the bank at advantageous rates to the bank, then the bank builds reserves instead.

So the banks should be more liquid.  Of course, this does imply that commerce will become less liquid.  It would imply that access to liquidity would be more costly than normal as banks want nominal returns to at least cover costs plus an economic profit on those costs.

What is an issue is whether or not deflation triggers an exchange between debt and equity investing.  The least risky projects can no longer be financed with debt or equity.  It could be asked, "does deflation make the entire system increasingly risky with the passage of time?"  Are there situations that are unwise for the system, though individually rational for each separate actor?  There is also a question of sectoral transfers.  Does commerce transfer profit from itself to banks to preserve treasury operations and basic liquidity functions?


What is the difference between an Ordinary Demand equation and an Engel curve equation?

Engel curve assumes the price is constant, and demand curve assumes income constant. There is only demand function, no engel curve equation or demand equation.


Why are asset pricing models equilibrium models?

An equilibrium asset pricing model is one in which the asset prices jointly satisfy the requirement that the quantities of each asset supplied and the quantities demanded must be equal at that price. It is as opposed to a partial equilibrium model where the price of the asset (or at least some assets) are determined outside of the model.


understanding the proof of stochastic dominance.

That's just a question of notation. Note that

$$\mathcal{d}F(t) = f(t)\mathcal{d}t$$

Then the proof should be easy to understand.


Are there conventions in economics for coding the distance between states?

I think the answer depends on who is impacted by your measure of distance and for what purpose. Kennan and Walker (2011) Econometrica measure distance between states as "the great circle distance between population centroids" in an attempt to model moving costs. They also include an indicator for whether or not the state is adjacent.


Endogeneity problem with a dummy variable

No,

I don't see how the dummy variable you are proposing would give you the same (or even similar) analysis as in the original model.

I would recommend dealing with the endogeneity in another way. IV estimation, or one of the more recent methods, difference-in-difference, matching etc. One of the will be suitable.


What is the economical advantage coming from the ability to speak English?

One of the best ways to study the benefit of speaking English, at least within an English speaking country, is by looking at the wage discrimination literature. Some, but not all, of wage discrimination turns out can be explained away by the ability to speak English. These studies are good to look at because of the controls for other factors. For example, it looks like the ability to speak English was a factor in this study, looking at before and after the impact of the IRCA in the US. The references I give of course are not generalizable to the whole world, because of their focus on the US. So I suggest looking more into the wage literature yourself.


You are trying to find the solution of $K$ and $L$ in terms of $Q$ to derive the total cost function with respect to quantity.

Page 13 of this lecture has the exact question you are interested in. (Google is your friend.)

From the Lagrangian you'll have $\frac{K}{L}=\frac{w}{r}$
From the production constraint you'll have $50K^{\frac{1}{2}}L^{\frac{1}{2}} = 1000$ and you have already derived the solution to this.

The first implies that $4K = L$ given your solution.

Then since,
$50K^{\frac{1}{2}}L^{\frac{1}{2}} = Q$

$$\implies 50K^{\frac{1}{2}}(4K)^{\frac{1}{2}} = Q \\
\implies 100K = Q \\
\implies K(Q) = \frac{Q}{100}$$

A similar solution can be found for $L(Q)$, and then it should be smooth sailing for you from there.


In what way specifically are US CEO's able to engage in rent-seeking and exploit corporate governance to increase compensation?

Pre-Sarbanes-Oxley it was common for CEOs to nominate the members of the board of directors of their firms. A sub-committee of the board would determine the CEO's compensation, so a CEO could choose people loyal to him (even employees working under the CEO) for the board and those people could boost his compensation. And if they did not do so, he might also ensure that board members that failed to boost his pay did not serve renewed terms. In addition, they could potentially ensure that even when performance was poor that the compensation was high. Sarbanes-Oxley, along with changes from the major US stock exchanges, required independent nominating and compensation committees for boards of directors. It is generally seen in the literature that these rule changes curtailed this practice. However, some dispute that this was ever an economically important channel for self-dealing.


Mathematical Notations that I Never Understand While Reading Articles or Dissertations

The usage of ; is not always standardized so it can depend on source. In this case I presume you took the equation from Interest and Prices: Foundations of a Theory of Monetary Policy By Woodford (or some source that used exactly the same notation), as exactly the same expression appears with all variable names being identical on pp 141 (however, in future you should indicate the source).
Based on the usage through the book the author uses the ; to separate variables and shock parameters. Hence, it is just to indicate that the $\xi$ is not a variable but given by the shock parameters of the economy.

Because the arguments of $u$ are already aggregated. $C_t$ is an aggregate consumption index based on the constant elasticity of substitution aggregator:

$$ C_t \equiv \left( \int_0^1 c_t(i)^{(\theta-1)/ \theta}di \right)^{\theta/(\theta-1)} $$
and the price level $P$ is again already aggregate price index:
$$P_t \equiv \left( \int_0^1 p_t(i)^{1-\theta} di \right)^{1/(1-\theta)}$$
So there is already integral over all goods within the utility. Author just written it in this way to 'clean up' the expression. You can always do that by substitution, you could clean it even more by lets say stating that $V(\cdot) = \int_0^1 v\left( h_t(i) ; \xi_{t} \right) di$ and you could then even write it in single equation without any integral at all (this is really more or less an aesthetic/ trying to make your work easily readable issue).

Is there a book that covers all classical mathematical solutions and explains them in detail for economics?

There is no book that covers all mathematical solutions in economics. That would be the longest book ever and you would probably need a truck to move it around. However, there are useful textbooks that cover majority of the most important models in each field.
Intermediate Microeconomics by Varian is undergraduate book so of course it is not on the same level. If you are interested in microeconomics then MWG microeconomic theory or Varian Microeconomic Analysis are the best texts (the former one is quite exhaustive - it is very thick tome).
However, based on the model above I assume you are interested in Macroeconomics. For that Romer Advanced Macroeconomics is a standard graduate handbook. However, even here it would very much depend on exact field you want to study. If you are interested in solely economic growth then Economic Growth by Barro and  Sala-i-Martin is much better than Romer which is more general handbook. If you are interested business cycle then the above mentioned Interest and Prices by Woodford is excellent source. If you like monetary theory then Monetary Theory and Policy by Carl E. Walsh is classic text.


Change in optimal savings due to increased risk

Your question is essentially about precautionary savings, i.e. the response of savings to risk.

What determines precautionary savings is the coefficient of relative prudence: $CRP=-\frac{U'''}{U''}C$, where $U$ is the utility function, $C$ is consumption and primes denote derivatives. As we always have decreasing marginal utility in economics, i.e. $U''<0$, then the sign of the savings will depend on the sign of the third derivative, as you mentioned.

Think of this coefficient as similar to the risk aversion coefficients. The difference is that prudence means that agents act in anticipation of risk by saving more, whereas risk aversion reflects how agents react when presently faced with risk.

With quadratic utility, the coefficient of relative prudence is zero and there are no precautionary savings. Hence there is no effect of risk on savings. With such utility we therefore have certainty equivalence, which is quite convenient and hence often used. Mean preserving spreads don't matter then. So no, it is not the same as in the positive case, where savings would go up in resonse to risk.

What is crucial here, as you noted, is the third derivative. The coefficient really only uses the second derivative as a normalization.

A mean-preserving spread keeps the mean/expected value of consumption the same. However, it does not neccssarily keep the derivatives (marginal utility) the same, which are what matter for (savings) decisions. If the marginal utility is linear (quadratic utility), then a mean-preserving spread keeps the expected marginal utilities the same as well. Hence, there are no precautionary savings (no reaction) in that case.

If the marginal utilities are convex, then the mean-preserving spread changes things in a way that matters, i.e. it becomes beneficial to save more. This follows from Jensen's inequality. If it is concave, the opposite is true. So the relevant function to look at is $U'$, as this is how the agent makes decisions. When we want to see whether a function is convex or concave we look at the second derivative. So the second derivative of $U'$ is $U'''$, which is why the third derivative matters here.


Net Neutrality and Recent Telecom Projects

My theory:
With the ever-increasing demand for bandwidth, there is a need for capacity.

With the repeal of net neutrality, providers can charge for a premium level of service, as well as refusing traffic that does not bring in revenue.

I think a better analogy would be an apartment building.
With net-neutrality: everyone would expect the pay the same rate and get the same level of amenities. Since no one can be turned away the front door of the building would also be unlocked 24/7.

Without net-neutrality: There will be multiple levels of service and price. The people that want a really fast service without having to share resources will have access to that level of service (if they want to pay for it), Those willing to have a little latency will pay less.


What are the mathematical constants used in economy formulas?

Economics is not an exact science and so we don't have Physical Constants i.e. quantities with a universal fixed numerical value.

That said, there are many concepts in economics that predominantly appear as constants in economic models (although in principle they can vary between individuals or across time, and there exists research that explores what happen when they do vary).

Well-known examples include the depreciation rate of capital (usually $\delta$), population growth rate (usually $n$), the rate of pure-time preference (usually $\rho$).

In calibration studies (either purely simulated or combined with empirical data), such constants tend to be assigned widely used "benchmark values" (i.e. quasi- "fixed values") that come from empirical experience usually supported over some decades of data and many countries. For example when calibrated and not estimated we often see $\delta =0.05$, $n=0.01$, $\rho=0.02$.

On another front, Euler's $e$, the base of the natural logarithms, emerges naturally when calculating compound interest rate continuously (and not in discrete intervals). Assume we make a deposit $D$ at yearly interest rate $r$. If interest is calculated once per year, at the end of one year we will get $D(1+r)$. Assume now that we calculate interest monthly, add the interest to principal and then next month calculate interest again. After one year we will get

$$D\left (1+ \frac {r}{12}\right)^{12}$$

In general if we partition the year in $n$ periods we will have

$$D\left (1+ \frac {r}{n}\right)^{n}$$

and if we let $n$ go to infinity we get

$$\lim_{n\to \infty}\left (1+ \frac {r}{n}\right)^{n} = e^r$$

Finally, there are certain "stylized facts" , according to which various economic ratios appear to be "stable" (if not strictly "constant") over time, like the income share of production inputs, or the capital/output ratio. The seminal reference here is

Kaldor, Nicholas, “Capital Accumulation and Economic Growth,” in F.A. Lutz and D.C. Hague, eds., The Theory of Capital, St.Martins Press, 1961, pp. 177–222.

while see

Jones, C. I., & Romer, P. M. (2010). The new Kaldor facts: Ideas, institutions, population, and human capital. American Economic Journal: Macroeconomics, 2(1), 224-245.

for extensions and updates.


What are some exceptional movies / documentaries on macroeconomics?

Princes of the Yen is a documentary that springs to mind.


Here is the quote, rephrased:

This is a false statement: (A) the poor don't work because they have too much money.

This is also a false statement: (B) the rich don't work because they don't have enough money.

If you believe those false statements (A) and (B), you are very credulous.

The poor have a much higher marginal value of money than the rich, so their incentive to work, to earn more money, is very high. For the rich, it's the other way around.

Ken is making explicit two assumption that are fairly common (in that school of thought and some other Conservative groups), but rarely spoken so plainly, and rarely spoken together. When they made explict and are put together, it makes it much clearer that to hold both of them at the same time is absurd.


What's the speed of convergence to the potential GDP? (Empirical estimates)

First of all the speed is not constant, it declines as one gets closer (at least according to the text book predictions).

A classic paper on among others this issue is the one by Mankiw, Romer and Weil in the 1992 QJE "A Contribution to the Empirics of Economic Growth" They find convergence rates of about 1 to 2%. The paper is cited 16,000 times in Google scholar, so no doubt you'll find some material there too. I'm no macro-economist, but it seems an area of active research.


Meaning of "Federal Reserve’s various liquidity facilities" and something more

Downward pressure on the Federal funds rate means bringing interest rates down even further (to ~0%).

Upward pressure on excess reserves likely means increased incentive for banks to build up excess reserves. The part about reserves is bit ambiguous. After reading the article a couple of times, this is what I make out of the paragraph (my comments are in bold):

  This was important for monetary policy because the Federal Reserve’s
  various liquidity facilities initiated during the financial crisis
  caused upward pressure on excess reserves (provided banks with cash to increase reserves) and placed downward pressure
  on the Federal funds rate (lowered the interest rate to the target level ~0). To give  Fed a tool to counteract these pressures in the future when they decide to exit QE and tighten the monetary policy, on October
  6, 2008, the Federal Reserve Board announced that it would begin
  paying interest on depository institutions’ reserve balances. Paying interest on reserves gave the banks incentive to increase the reserves at the time. In the future, lowering interest rate paid on reserves would create downward pressure on excess reserves.

I found a footnote next to "Federal Reserve’s various liquidity facilities" part of the sentence leading to a page describing these facilities:
Credit and Liquidity Programs and the Balance Sheet. It looks like a detailed technical description of QE programs Fed implemented after the 2008 crisis.


Bellman equation corresponding to stochastic EZW recursive utility

What you have there are the preferences under an arbitrary policy -- what some call the prevalue function. The only thing missing is the max operator. Written with maximization (and making the state and choice explicit), the Bellman equation is
\begin{align}
U(K_t, \epsilon_t) =& \max_{C_t} \{ (1-\beta) C_t^{1 - \frac{1}{\eta}} + \beta [E_t(U(K_{t+1}, \epsilon_{t+1})^{1-\gamma})]^{\frac{1 - \frac{1}{\eta}}{1 - \gamma}} \}^{\frac{1}{1 - \frac{1}{\eta}}} \\
\text{s.t. } & C_t \in F(K_t, \epsilon_t) \\
& K_{t+1} = G(K_t, C_t, \epsilon_t), \\
& \epsilon \sim \Phi
\end{align}
where $K_t$ is the state, $F(K_t)$ is the choice set, $\epsilon_t$ is some shock with CDF $\Phi$ (assumed to be revealed before choosing $C_t$; it doesn't need to be IID), and $G(K_t, C_t)$ is the law of motion.
Maximization with recursive preferences is pretty much the same as with standard time-separable preferences, and even plays well with dynamic programming methods (e.g. if you have some VFI code set up you can make the preferences recursive by just changing the prevalue function). The only real change is that we now have different aggregators over states and over time on the RHS of the prevalue function. These notes on recursive preferences may help, in particular section 5 ("Optimization and the Bellman Equation").
Making the constraints stochastic won't change the details of your problem -- it'll be similar to how you would formulate and solve the time-separable equivalent problem with a stochastic constraint -- unless you have some specific structure in mind which would change things for recursive preferences differently than for time-separable preferences. I can't think of such a structure off the top of my head, but I'm sure one could find them with some digging.
One thing that's different with recursive preferences is the utility scaling -- the period utility is scaled by $(1-\beta)$. This is discussed a bit on page 3 of the attached notes. The scaling allows utility to be measured in units of consumption, so that the utility of a sequence of sure-thing constant consumption levels is that same consumption level, i.e. $U(c,c,...) = c$. It's a good exercise to try showing this symmetry without the scaling to see how it comes out (if you do it with the risk aggregator, remember that the certainty equivalent of a sure thing is the sure thing itself).