When a land lords sells to an owner occupiers does it push up rents?

Your argument is sound (as a first approximation). In fact this is one of the basic examples discussed in chapter 1 of Varian's Intermediate Microeconomics (a standard textbook).

  Suppose that a developer decides to turn several of the apartments
  into condominiums. What will happen to the price of the remaining
  apartments? Your first guess is probably that the price of apartments
  will go up, since the supply has been reduced. But this isn’t
  necessarily right. It is true that the supply of apartments to rent
  has been reduced. But the demand for apartments has been reduced as
  well, since some of the people who were renting apartments may decide
  to purchase the new condominiums.

(quote from pg 10 of the 8th edition)

If you make some additional assumptions e.g., that all buyers are former tenants and that there is no friction in the market (rent changes may be larger when there are new tenants to avoid tension), the equilibrium price will be the same.


How do economic and financial policies in government deal with cryptocurrencies?

In order to increase use of a cryptocurrency it could be suggested that the developers making this currency should not seek to develop a medium of exchange which would work separately from the country's currency as an independent currency, but rather seek to work together with it.
Though having a cryptocurrency working with an already existing currency would seemingly result in this cryptocurrency being backed on the by the country's pre-existing currency (which in return would give the same reliability),
nonetheless, to attempt to establish a new method of payment which would be reliable, it is necessary to create a relationship with the country's previously existing currency, regardless of the previous currency's basis
W.J. Luther writes in his paper "Cryptocurrencies, Network Effects, and Switching costs" (Contemporary Economic Policy, 34, 2016, 553-571)

"By anchoring expectations- perhaps by committing to (or refusing to) accept a currency for payment of taxes- a government can effectively determine the medium of exchange"

Hope this helps.


The Guidebook to Municipal Bonds: The History, the Industry, the Mechanics (1991) by George J. Malvin

The Tax Policy Center Briefing Book on The State of State (and Local) Tax Policy might be a good place to call.


Does demand side economics only apply to manufacturing

When factories collapse, workers in those factories are out of jobs... no money means they can't afford to buy a used car, etc.

Update: To go back to the question in the title, consumption is also a part of the demand, and it includes consumption in services as well.


What would monetarists do when interest rates approach zero and become ineffective?

An excellent overview of not just monetarist view on this issue but all other mainstream views on monetary policy at zero lower bound  (ZLB) can be found in Ullersma (2002) The Zero Lower Bound  on Nominal Interest Rates and Monetary Policy Effectiveness: a Survey - which is an excellent literature review on this subject (albeit slightly dated).
In short, under the monetarists view 'true' liquidity trap should not exist. Even though there is zero lower bound monetarists would argue that central banks can still stimulate economy through other transmission channels. For example,  Meltzer (1963, 1995, 1999) and Brunner and Meltzer (1968), would argue that central bank can still stimulate the economy through asset purchases with implicit assumption that even when the short-run nominal interest rates are at the zero lower bound, yields on non-monetary assets are not necessarily at their lower bounds.
Bernanke and Gertler (1995) would argue that central banks can stimulate economy also through the credit channel that is not affected by ZLB. Basically this amounts to central bank making sure banks are willing to extend credit as much as possible by providing incentives for them to do so. For example, this is actually being tested in the Europe by ECB through long-term repo operations (LTROS). LTROS is a way how central bank should can borrow money to commercial banks and in theory encourage them to lend more. Furthermore, there is no ZLB when it comes to LTROS as long as central bank is just willing to soak up the losses it makes on such lending (at it can always do that by creating more money).  See this article in the Economist that talks about LTROS in greater detail.
Lastly, Milton Friedman  himself would probably just suggest that central bank engages in 'helicopter money', that is directly giving money to households/firms in some form (see Friedman (1969) The Optimum Quantity of Money: And Other Essays). Such a direct transfer should in principle 'circumvent' the liquidity trap. However, for many central bankers this would be quite radical as such practice could lead to politicization of central banks (since once people learn they can get 'free money' from central banks there might be political pressures for such transfers even when they are not needed).
However, it is worth while noting that the monetarist view above, while still technically being part of mainstream, is not the predominant view in the macroeconomic profession. The current predominant view on ZLB would be more close to the New Keynesian (NK) view on liquidity trap which in the paper cited above is denoted as the 'Krugman's View'). Under the NK view the above mentioned prescriptions would still be helpful, but insufficient and government spending would still be necessary.


Choosing the nondominant strategy in a duopoly

It seems you interpret the game as a sequential game. If Windward would know that its move was observed by Airtouch it would act as you say. But this is not the case, they move simultaneously, and it is a one-shot game (as defined it is only played once).

Given your reasoning, Windward understands the situation and hence it would choose the evening. Anticipating this Airtouch would do the same. But Windward does not really want to choose the evening. If Airtouch anticipates that Windward will choose the evening that's jolly good for Windward. It is even better if Windward goes on to choose the morning. So what you describe is not an equilibrium outcome and is unlikely to occur in a one-shot situation.

Don't believe me? Let's play. As Windward I understand your reasoning and I make the empty promise of choosing 'Evening'. What will you choose? (And me?)

The situation would be different if the game was repeated, so if reputation was important or if, as I mentioned before, the game was sequential and moves could be observed.


Getting a market supply curve from firm supply curve

The important thing to note is that the unit on the vertical axis is dollars per unit.

So if there is a tax of $\$t$ per unit, then the market supply curve will move up by $ \$ t$. (And not by $ \$nt $ as you have suggested.)


Utility representation of single peaked preferences

No. Basically, you can encode a form of lexicographic preferences, probably the most familiar example of non-representable preferences, as single-peaked preferences on $\mathbb{R}$.
Define $\succeq$ so that $x\succeq y$ exactly if either $|x|<|y|$ or $|x|=|y|$ and $x\leq y$. Basically, the closer to the peak of $0$ a number is, the better, and in case of a tie, the number to the left of $0$ is better.
Suppose for the sake of contradiction that there is a utility representation $v:\mathbb{R}\to \mathbb{R}$ of $\succeq$. For each $r\in\mathbb{R}_{++}$ (the strictly positive numbers), let $q_r$ be a rational number in the interval $\big(v(r),v(-r)\big)$. Since for $r\neq r'$, $\big(v(r),v(-r)\big)\cap\big(v(r'),v(-r')\big)=\emptyset$, we have an injection $r\mapsto q_r$ from $\mathbb{R}_{++}$ to $\mathbb{Q}$, which is impossible since $\mathbb{R}_{++}$ is uncountable and $\mathbb{Q}$ is countable.


What is the correct way of calculating the "average" price based on the "Open,High,Low,Close" daily trade fields?

You cannot obtain that information with the data provided.
You would need periodic samples throughout the day.


Solution method in Smith (2006) A Closed Form Solution to the Ramsey Model

Let's solve the differential equation (12).

As a first step, we look for a "simpler" differential equation, namely (A.1).
(A.1) can be written
$$\frac{\dot{z}}{z}=-(1-\alpha)\left(\delta+\frac{\bar{x}e^{-\bar{x}t}}{e^{-\bar{x}t}+\bar{x}A}\right)$$
On the left-hand side, you have the derivative of $\ln(z)$. You can integrate to obtain the form of any solution $z_c$:
$$\ln(z_c)=-(1-\alpha)\left(\delta t-\ln(e^{-\bar{x}t}+\bar{x}A)\right)+ constant$$
You obtain $z_c=\kappa e^{-(1-\alpha)\delta t}\left(e^{-\bar{x}t}+\bar{x}A\right)^{1-\alpha}$, where $\kappa$ is the exponential of the constant in the previous equation.
This equation is equivalent to (A.2) for a particular constant, such that $\kappa=\bar{x}^{\alpha-1}$. This choice of the constant comes from some boundary conditions on $z$ (which should be stated in the paper).

As a second step, we use the method of variation of parameters, meaning we are looking for a solution of (12), $z_p$, that has a particular form, $z_p=z_c.\Psi$. If we find an expression for $\Psi$, then we have found a solution of (12).
We substitute $z_p$ in (12):
$$\dot{z_c}.\Psi+z_c.\dot{\Psi}=(1-\alpha)\left[1-\left(\delta+\frac{\bar{x}}{1+\bar{x}Ae^{\bar{x}t}}\right)z_c.\Psi\right]$$
This expression simplifies since $z_c$ satisfies (A.1):
$$z_c.\dot{\Psi}=(1-\alpha)$$
This is (A.3).
Then, I guess you we have to find a solution $\Psi$ of this equation, and we are done.


Market behavior when consumer must choose product before knowing price?

Yes, we can model this as a choice under uncertainty, where the distribution is over the set of prices the consumer may face. The consumer could have an expected budget with expected demands and you would need to allow the consumer to go into debt if prices are higher than their wealth (think a quasilinear environment).

The second part, where the consumer doesn't know the other competing firms can also be modeled with uninformed consumers vs informed consumers, where firms compete on price over those kinds of consumers. The informed choose better option, the uninformed just choose the same regardless.


How is the cost of equity for a specific stock calculated using the Fama-French 5 factor model?

If you visit Ken French's website (specifically, his data library), you can download monthly and daily returns for the 5-factor model (as well as similar returns for the Carhart momentum factor).
You can then find annual averages for those and compute the cost of equity in a similar way as for the CAPM.


The way to examine such issues is by general mathematical treatment, not arithmetic examples.

We have total revenue

$$TR = Q(P)\cdot P$$

When will Total Revenue remain constant as price changes?

We want

$$\frac{\partial TR}{\partial P} =0 \implies \frac{\partial Q(P)}{\partial P}\cdot P + Q(P) =0$$

$$\implies  \frac{\partial Q(P)}{\partial P} = -\frac {Q(P)}{P}$$

which is to say that the price elasticity of demand is equal to minus unity.

Is it possible for this to hold over the whole range of $P$ and not just momentarily, at a specific value of $P$?

This entails to find a solution to a simple differential equation (using generic notation)

$$y' + \frac{y}{x} = 0$$

which gives

$$Q(P) = \frac {A}{P},\;\;\; A>0$$

So if the demand function is as above, it has unitary price elasticity everywhere.


What is it called when there is no personal increase of wealth but GDP increases because of mutual sales?

Yes this can be considered flaw of GDP since GDP’s intended purpose is to measure the value of output produced by a nation.  In fact in his report to U.S. Congress Kuznets (who is the person who developed the modern concept of GDP) mentions in a preface that national product (GDP) in his view should encompass all production at its hypothetical market value (not just a production actually sold at market):

If  all commodities produce and all personal services rendered
during the year are added  at  their market value, and  from  the resulting  total  we subtract  the  value  of  that  part  of  the  nation's stock  of  goods  which  was  expended  (both  as  raw  materials  and  as  capital  equipment)  in  producing  this  total,  then  the  remainder  constitutes  the  net  product   of  the  national  economy  during  the  year.  It  is  referred  to  as national  income  produced,  and  may  be  defined  briefly   as  that  part  of  the  economy's  end-product  which  is  attributable  to  the  efforts  of  the individuals  who  comprise a nation. [emphasis is mine]

Kuznets even explicitly mentions house production should be part of this income, but he concluded it’s omission is ‘unavoidable’ due to the fact that it is problematic to measure it:

Services of housewives and other  members of the family.
—The  volume  of services rendered by housewives  and other members  of  the household  toward the satisfaction  of wants must  be imposing indeed,
when totaled for  the  30 million families  comprising  the population  of  this country; and the item is thus large enough to affect materially any  estimate  of  national income. ... It  was considered  best to omit this large group of services from national income, especially
since  no  reliable  basis is  available  for  estimating  their  value. This  omission, unavoidable though it is, lowers the value of national income
measurements as indexes  of  the  nation's  productivity  in  conditions  of  recent  years  when  the  contraction  of  the  market  economy  was   accompanied by an expansion of activity within the family [emphasis is mine].

So already the original creator of GDP arguably considered this omission as a ‘flaw’ but an ‘unavoidable’ flaw.
In addition historically GDP was from its inception used also for welfare comparisons and economists generally agree that home production and self-consumption increases welfare (see England; 1998).
Consequently, it is valid from economic perspective to consider omission of non-market activity (such as home production) as a failure.
This being said  from purely accounting perspective this would not be flaw because GDP is defined as “the total monetary or market value of all the finished goods and services produced within a country's borders in a specific time period” (see here). So from pure accounting/statistical perspective it’s by definition not intended to measure non-market production, but as mentioned above this accounting/statistical definition was a result of expediency and not because it’s intended purpose was to omit non-market activity.
I am not aware of there being any special name within economics for this flaw. Generally, the above would be just called limitation of GDP, but limitation here is not some economic jargon.


Linear Regressions: When to expect the residuals to NOT have conditional mean 0?

The assumption could be violated due to counfouding variables or endogeneity for example. In fact the assumption fails any time there is correlation between $X$ and $U$ (hence any time $COV(X,U) \neq 0$. This actually happens all the time - most of the whole field of econometrics exist to help us deal with this problem.

An concrete example when the assumption would fail would be omitted variable bias. For example if the population model is:

$$y = \beta_0 + \beta_1 x + \beta_2 z + u$$

but you try to fit just simple regression $\hat{y} = \hat{\beta_0} + \hat{\beta_1}x$ and there is correlation between $x$ and $z$ then the zero conditional mean assumption will be violated because the regression model will attribute any changes in $y$ even those caused by $z$ to $x$. For example, suppose you want to examine the effect of education $(x)$ on wages ($y$) and also suppose education correlates positively with education of parents ($z$) and you exclude the education of parents from your model. In that case the regression model applied to the sample would systematically overestimate the effect of education as it would attribute portion of the effect of parents education to the persons education.

Also note the zero conditional assumption is assumption about true model - it cannot be verified with data, you need knowledge about the subject matter and make some logical argument to argue its valid. In sample the  expected mean of error will be always zero just because OLS is calculated by minimizing the sum of squared residuals.


Leontief's Paradox and the amount of capital

Leontief did not use total capital stock for his calculation of capital/labor requirements in the export/import sector. He measured 'requirements per million dollars of exports and imports replacements', and this for various industries (Leontief, 1953, section III, table 2). So he also did not use total capital stock, he used capital requirements AKA capital inputs.

He did provide summary statistics on the total capital and labor required per million dollars of total output for each industry. This might have confused you.

Note that he obtained these numbers only after computing them from available data, and he discusses their accuracy in the same section.


Wage regressions: Nominal versus real wage

Because you want to measure the real impact of education.

If there is inflation, the nominal wages most likely will go up over time, but you don't want to think that your return to education does that, it's just inflation.

If, on the other hand, you're comparing wage differentials, say $\log w_s - \log w_u$, where $s$ indicates skill and $u$ indicates unskilled, you do not need to correct for the price level.

To see, note that

$$
\log w^s_t - \log w^u_t = \log \frac{w^s_t}{w^u_t} \\
\frac{w^s_t}{w^u_t} = \frac{\frac{w^s_t}{p_t}}{\frac{w^s_t}{p_t}}
$$


Pareto optimality when the allocations are restricted

Let me elaborate on the comments quickly - maybe only for the sake of this question not being "unanswered".
Pareto optimality only makes sense when defined with respect to feasible allocations. That is, your basic definition above lacks an important word:

  an allocation is PO if there does not exist another feasible
  allocation in which all players are weakly better off and at least one
  player is strictly better off.

Without this word, you can easily find an allocation that dominates "A gets slices 1&3; B gets slices 2&4". For instance, you could give Alice eight additional full cakes of chocolate. However, there are no other cakes and, hence, this allocation is not feasible and therefore does not Pareto dominate what you suggest.

As a result of your feasibility restriction, any allocation in which one agent gets one slice of the preferred kind and the other one gets the other three is PO.
If A gets the entire cake, she can make B better off by giving him a slice of V which she does not care about; analogously if B has the whole cake. If both agents get two connected slices, a similar improvement exists.


Has the assumption that individuals' tastes do not change over time been rigorously challenged?

Stigler and Becker's argument is methodological, not philosophical. They do not try to convince us that preferences are indeed identical across individuals and invariant across time as a matter of reality (the "Rocky Mountains" metaphor is an "as if " approach).

Their point is that any outcome can be rationalized by assuming that "it was preferences that made it so", since "De Gustibus Non Est Disputandum", and also they are unobservable.  But then, we could "explain everything" in this way, and so explain nothing.

Their goal is to defend in terms of useful modelling the other extreme: assume immovable preferences and try to find explanations for the observed outcomes based on observable, quantifiable concepts, like prices. I believe the following passage from the first page of the paper summarizes the approach

  "On the traditional view, an explanation of economic phenomena that reaches a difference in tastes between people or times is the terminus
  of the argument: the problem is abandoned at this point to whoever
  studies and explains tastes (psychologists? anthropologists?
  phrenologists'? sociobiologists?). On our preferred interpretation,
  one never reaches this impasse: the economist continues to search for
  differences in prices or incomes to explain any differences or changes
  in behavior. The choice between these two views of the role of tastes
  in economic theory must ultimately be made on the basis of their
  comparative analytical productivities."

Bold my emphasis.

So the question of the OP appears to be mistargeted: people's tastes may very well change over time, and I don't think that Stigler and Becker would deny that.
The question is, can we arrive at more useful economic models by assuming changing tastes, compared to the models where tastes are fixed (while avoiding the "explain everything and so explain nothing" trap)? But this would be a whole research program, not some rigorous argument in a paper.


What happens when I leave out empty cells in regression?

When you have empty cells (and by empty cells, I assume that you are referring to a missing value of an observation specific variable), then STATA by default drops the entire row. So in a sense, even when you have a single missing value on a variable and you have say 100 variables, you lose the whole row (the entire observation). This is by default. If you want to avoid this, perhaps think about imputing the missing data points.


How to handle change in base year of index?

You need to rebase the oldest series, using the base from the new series. This is an example of how to do this, with annual data (same for monthly data):

So, basically, you need to find a period in which both series have an entry, compute the ratio between them, and then transform the old series using the ratio just calculated. This is how you link two series in different base years.

Remarks: you will have several months in which the series have values. If you have reasons to believe that the newest series is better, then I would make sense to take the earliest month in which the two have data (probably January of 2005). Nonetheless, the two series might be in fact different (e.g. sector composition or definitions vary). Compare the rate of monthly growth of both indexes for the months in which both have data (probably January 2005 to Feb 2011). Are they very different? If so, you might try to adjust the old series. Perhaps, there is a persistent difference, which you can account for when expanding the newest series backwards. See more in this document.


Why doesn't economy of scale bring down prices of food?

In the US at least, purchases and wages cost more than rent for a fast-food type business:

Purchases include some other hidden labor costs (drivers for transportation etc.) I'm not sure what's in the "other" category, perhaps insurance, lawyers etc.
And I'm guessing in Europe (where you are) "value menus" might not be as common as in the US. These are loss leaders in the US by the way, i.e. the restaurants lose money on $1 double cheeseburgers (but they are required to sell them in order to keep the franchise.) Or so the story went around 2014. And you're not the only price-sensitive customer. A lot of McDonalds' US customers apparently were:

In January 2018 McDonald’s finally brought back its dollar menu after a five-year hiatus, marking the return of a deal structure that had been displaced by alternative — and less popular — offers, like the McPick $2. The removal of the dollar menu coincided with the company’s initial decline, and its return makes it competitive among other chains with similar value menus, such as Taco Bell and Wendy’s.

If you want even lower prices in the short run, your best bet might be that Amazon buys a fast food chain. They're known to sell entire categories of products at a loss.
And depending whom you choose to believe, McDonalds did or din't envisage labor cost savings when rolling out kiosks to replace cashiers. Their statements on that have been somewhat vague and thus interpretable insofar. I find it a bit hard to believe that a company which pioneered low food costs by cutting on labor didn't consider that as a factor; flashback to the 50s:

The [McDonalds] burgers sold for 15-cents, about half of what a burger cost at regular diners of the time.

And I think you might have some misconception about the level of scale needed to cut costs substantially (say by half). You expect to pay half as much per meal by ordering 10, but you are hardly in a position of bargaining power here given the scale of a large food chain. In contrast, the actual economies of scale achievable have long been realised when nation-wide (and later world-wide) franchise chains appeared:

The predominance of fast food restaurants changed the food supply chain all the way down to the farmer. McDonald's quickly became the single largest buyer of beef, pork, potatoes and apples in the U.S. That gave them tremendous economic clout.
The fast food system is all about standardization, and so when the companies went looking for someone to supply their meat, they choose to deal with their large, corporate counterparts in the packing industry. IBP began to produce "boxed beef," where the final cuts of beef, including hamburger, were produced at the processing plant rather than the local grocery. IBP became the largest supplier of hamburger meat to the fast food industry.
Kentucky Fried Chicken buys all of its chickens from huge suppliers like Perdue, Tyson and Pilgrim's Pride. McDonald's gets its fish products from the giant supplier Gorton's of Gloucester.
Because consumers with busy lifestyles needed food fast, the chains needed raw materials in standardized packages. So, meat packers needed a consistent supply of standardized animals to produce their meat. They couldn't afford to deal with the uncertainty of many, small family farms. So, livestock producers became bigger and bigger. McDonald's and other chains have also been accused of using their huge buying power to keep farm produce prices artificially low.

What you really want is technological progress at any of these levels, because the foreseeable economies of scale are already there.
And (being focused on restaurants), you might not know that

Chicken is America's favorite meat today, surpassing beef and pork. Chicken consumption per capita has nearly doubled in recent decades, whereas consumption of beef and pork has declined or stagnated. [...]
Far from being automatic, these changes are the consequence of technological and organizational innovations that have made chicken products cheaper, abundant, and more diverse over the last century. This transformation is directly related to the rise of new lead firms and governance structures in the chicken value chain. The increased sophistication of processed chicken has given rise to a set of firms known as integrators, which vertically integrate through ownership a wide range of activities from breeding to processing and distribution. The shift to away-from-home consumption favored fast food giants like KFC and McDonald's, whose mass demand for processed chicken required higher levels of explicit coordination by the integrators. [...]
The chicken value chain is characterized by a hierarchical governance structure made up of a few large integrators and tight coordination between them and large food retailers and fast food chains. The integrators have become highly concentrated as well. Two leading integrators, Pilgrim's Pride and Tyson Food, accounted for 48% of the US market in 2006, up from 30% in 1996. They exert great power over the entire value chain by connecting a wide array of activities from inputs to distribution. [...]  Their linkages with thousands of contract growers, mostly small farm owners, can be portrayed as a captive relationship, where the growers heavily depend on the integrators for market access and resources. Meanwhile, the integrators have relational or modular linkages with large food retailers and fast food chains, which are now highly concentrated.

The latter paper goes into some detail into what the captive relationship consists of, but I'd be digressing too much detailing all that here; the high-level definition is that it is "a means to maintain control over the growing process with minimal ownership" (compare this with franchise model). How far vertical integration goes (and thus economies of scope) is limited by some legal issues like antitrust legislation.


Does the duality of utility maximization and cost minimization hold in practice?

There is nothing that can "hold in practice".

Utility maximisation and expenditure minimization are theoretical framework that are used to model behavior  by factors that are thought to motivate choices.

This is useful if they approximately or on average correlate with how those factor influence choices. For example, wage will surely be a major factor in the labor supply of an individual.

But utility cannot be measured and arguably does not exist. Both approaches rely on either a utility function or a level of utility, so neither can be practically verified.


Mankiw's version of the Cagan model - need help interpreting it

I think that you are unnecessarily overthinking it. For any relationship of the form:
$$\ln y = \beta \ln x  $$
the interpretation of beta coefficient is that $1\%$ increase in $x$ leads to $\beta$ $\%$ increase in $y$. The mathematical reason why this relationship holds was already explored at cross-validated stack exchange and you can see it here, or on this website or in virtually any econometric textbook since the log-log form is important, so I wont be unnecessarily restating it.
Mankiw defines real money balances as $M/P$ or in logs $m-p$, hence the left hand side of your equation is by definition the real money balance term expressed in logs $\ln (M_t/P_t)$. The inflation is by definition change in price level again in your case expressed in logs $\ln(P_{t+1}/P_t)$. So you can directly apply the interpretation from previous paragraph as Mankiw does.
Furthermore, you can actually derive it also with your calculations. To be more specific. Growth rate $g$ for variable $x$ is given as $g_x= \frac{x_{t+1}-x_t}{x_t}$. Then we know that:
$$\ln x_{t+1} = \ln ((1+g_x)x_t) \implies \ln x_{t+1} = \ln (1+g_x)+ \ln x_t  $$
Then since $\ln (1+g_x) \approx g_x$ we have:
$$\ln x_{t+1} = g_x  + \ln x_t \implies \ln x_{t+1} -\ln x_t =g_x $$
as a consequence you can actually directly say that:
$$\ln(M_t/P_t)=-\gamma \ln(P_{t+1}/P_t) \approx \ln(M_t/P_t)=-\gamma \left( \frac{P_{t+1}-P_t}{P_t} \right)$$
that 1 in your final expression should not be there. In this case the LHS gives you percentage in the real money balances and LHS percentage change in price level (inflation).


First of all it is important to make clear what economists are actually saying. Most economists believe that moderate inflation is better than moderate deflation, while high inflation or deflation would be considered undesirable. This is because both inflation and deflation are examples of price instability and in fact if all markets would be perfect, competitive and all prices fully flexible models tend to show that it would be desirable to have neither inflation or deflation over business cycle (you can see some example of such models in Romer Advanced Macroeconomics).
However, unfortunately we do not live in a world where all markets are perfect, competitive and where all prices are flexible. For example, wages might not be flexible because of minimum wage laws or because in many countries contracts are set for long term (for example, in the Netherlands where I live it is very difficult to fire someone with permanent contract or to force them to renegotiate their contract). However, even in countries where labor markets are more flexible it might be difficult to renegotiate wages or contracts quickly.
These rigidities create problem for an economy during recessions where in order for markets to equilibrate you usually need people to accept lower wages. Inflation, solves the issue above because it decreases people's wages in real terms without any need for negotiations. This can help to moderate recession because if real wages are too high during recession it will lead to above equilibrium level unemployment. In turn higher unemployment means people have less income to spend or invest which then just further reinforces the recession.
Also note while I focused mainly on wages the inflation has also other benefits as in the short-run it stimulates the economy (since higher product prices motivate firms to produce more before they realize they are higher due to inflation), it also helps to relax government budget constraint since government as a monopolist producer of money benefits from inflation and has some further benefits (see discussions of this in standard macro textbooks such as Blanchard et al Macroeconomics: A European Perspective or Mankiw Macroeconomics).
Lastly note that economy usually has no problem when it needs to increase aggregate wages during expansion. Wages are much more rigid downwards than upwards and consequently economy usually does not need deflation during expansions when real wages tend to increase.
Hence to sum up, most economists believe that it is better to have moderate inflation (most economists would argue around 2% per year), because it helps to 'grease' the wheels of economy. Inflation helps to moderate business cycle and to the extent that one agrees it is better to have milder recessions it is obvious policy prescription given our current understanding of macroeconomics.


How product leaks before announcements affect a company economically?

Opinion

If the leak is close to a release date, I would say most of these are made on purpose, to feed expectations, leading to greater attention and publicity (and sales). I don't see how this can be negative for the company. In technological firms, most of the features are patent-protected so they cannot be copied anyway. However, if the leak were to happen long before a release, this could indicate rivals potential direction of own developments (either copying them or moving away/further than the competitors).

Research

Maybe the question is related to the literature on knowledge leakage, which relates to R&D, knowledge management, knowledge sharing, etc.

One recent paper  mentions some of the costs:

  Knowledge leakage has various negative effects on
  firms,
  including loss of revenue, damaged reputation, loss of productivity
  and costs arising from breached confidentiality agreements
  (Ahmad et al., 2014). It also decreases bargaining power and even
  creates new competitors for the original knowledge owner
  (Baughn et al., 1997). Therefore, when an employee leaks knowl-
  edge that is rare, inimitable and non-substitutable, recovery from
  such leakage can be very challenging (Ahmad et al., 2014; Barney,
  1991, 1996).

The same paper states that :

  Firms traditionally attempt
  to protect their innovation-based competitive advantages with
  intellectual property rights (IPRs), trademarks, copyrights or other
  knowledge-protecting  mechanism

I suggest you look into those references if you want to get further insights.


Any Resources for Micro Comprehensive Exams?

I know Harvard posts its past comprehensive exams: https://economics.harvard.edu/pages/past-general-exams. I imagine many other schools would have such resources available to the public as well. You'd just have to do a little bit of Googling to find out.

However, I doubt that any of these past exam questions would have solutions available (and they should not be). You should try the questions on your own. If you're stuck, discuss the issues with your classmates/friends/professors. You're also welcome to ask questions here, but be sure to following our policy on homework questions and the general guidelines of this site.


Where can I find data on bilateral trade?

For aggregated trade, there is a excellent database from CEPII called histtrade. It covers data from the first industrial revolution. The site also has data covering disaggregated  trade but only with limited observations.

http://www.cepii.fr/CEPII/en/bdd_modele/bdd_modele.asp

A second data set is the Barbieri Data set, that can be downloaded from the COW project.

http://www.correlatesofwar.org/data-sets/bilateral-trade


How yield rates influence the "cost of borrowing" for borrowers

This question is too long. To be useful for this website, questions need to be questions, without filler text. You should also try to avoid suggesting answers, as if your suggestion is wrong (as happened here), it makes it harder to answer the question.

You have questions about the notion of “corporation” versus “bank.” A bank is just a type of corporation, and both banks and non-banks issue corporate bonds.

  The yield of [corporate] bonds reflects the cost of borrowing for the issuer, so for banks who issue corporate bonds, if the yield on the bonds decreases it is cheaper to access finance", is this simply referring to how much the company has to pay back for the bond or is it referring to how much it costs banks to borrow from the central bank?

Neither of your suggestions.

A bond is a loan that has been structured in a fashion to allow it to be bought and sold in a market. The standard structure is to repay the principal at maturity, and the interest paid in the form of coupons on a fixed schedule (depending on the market, typically either every 6 or 12 months). The initial sale of the bond by the issuer is the loan operation, after that, the issuer just pays the coupons and principal, and is not directly affected by later transactions.

Unless the bond is callable, the issuer has no right to pay back the loan early, and so there is no notion of “paying back” the bond - it is always repaid on a fixed schedule.

Meanwhile, borrowing from the central bank has nothing to do with this.

Instead, what the speaker is referring to is the cost of issuing new bonds. Most issuers will have existing bonds, and their price in the secondary market implies a certain rate of return for investors. Investors will only buy newly issued bonds if the interest on them is comparable to those existing bonds. So if the central bank does actions to lower bond yields in the secondary corporate bond markets, corporations can issue new bonds at a lower interest rate.


Energy saving technology and energy service price

A problem with using energy service prices is that they might reflect competition conditions (e.g. monopoly/oligopoly, or the big six in the UK) rather than the true cost of energy to firms.

A problem with using energy prices is that they might reflect demand and supply issues. For example, no one would say that the drop in oil prices from 140 to 50 was due to technological improvement. Instead, it was driven by a shale oil revolution, and perhaps geopolitical aspects.


Why does entitlements leads to lower US productivity?

"Well, the basic issue is the entitlements and the reason it's a
  problem is that people put money in their own funds and their
  employers' funds plus interest and that is what they perceive they et
  back , and therefore they are entitled to it because it is their
  money."

He's saying people think that they receive the income that they are putting into the program.  For example, a common perception that people have about social security is that it is simply a forced savings program: I put in \$50, that \$50 is put into a vault somewhere, then pulled out later, and given back to me for my retirement.

That's not how it works. I put in \$50, and we spend it immediately on someone's current retirement, or a number of other pre-approved (and sometimes seemingly unrelated) programs. We then hope that some new young person puts in another \$50 in time for my retirement.

This is particularly a problem as that the number of retirees is going to grow while the number of young people putting money into social security is going to shrink. In fact, this is a problem if productivity declines in any way, and the amount going in is limited or restricted, or fails to grow.

Questions 1/3: He is not saying that entitlements cause lower productivity in this statement-> he is saying entitlements will have problems -because- of lower productivity (probably from lower population, as has been popular to discuss for quite some time).

Question 2: Capital stock is the collection of tools available for production.  (Hammers, technical know-how, computers, assembly lines.)


Can banks maintain Euro accounts if the country they're in leaves the eurozone?

Arguably the problem all along with the Eurozone (and any other currency union) is the operational mechanics of the underlying banking system.

From a systems perspective, banking systems statistically multiplex liability bank deposits against asset cash and electronic equivalents. (It is mostly electronic these days.) Broadly speaking, the main function of the asset money is to provide an interchange mechanism between banks, and the deposit money is what is actually used to perform economic transactions.

There is no way of enforcing what physical currency gets used as the interchange mechanism. It doesn't really even need to be a recognised currency, and several precious metals have been used in the past. However, if a banking system uses a physical currency whose quantity is outside of its control - i.e. the money is printed by somebody else, then the banking system will collapse over time, unless there is net flow of the currency into the system.

Now, if we look at the greek money supply figures, and we know that that there are essentially three ways for liability deposits to decrease as shown - loan repayment, loan write-off and cash withdrawals, we can deduce pretty easily that Greece is experiencing a massive outflow of cash. We can assume that some of it is cash withdrawals particularly recently, but some of it also debt repayment to debt owners outside of the Greek banking system. As a very rough rule of thumb, on a 25 year loan, about twice as much money will have to flow back to the lender, as was originally borrowed, and this is what's causing the underlying systemic problems.

According to the Wall Street Journal, this is a summary of the debt picture:

Greek Debt

Mechanically speaking then, the only way a country can unilaterally stay in the Eurozone is if they are experiencing a net inflow of asset cash/money, in which case it's unlikely that they'd be under pressure to leave it. What this counter-situation does to the economy depends very much on how the banking system is regulated, and since most countries no longer depend on asset cash/aka reserve regulation as a control, the answer as also seen in the US with TARP is not necessarily very much.

A water analogy may be slightly helpful. As long as there is enough water, and any excess can be stored, then it doesn't matter if over time more water comes in than can be used. However, it's disastrous to not have enough water - and that's essentially the problem that the banking system is currently causing for Greece. The various attempts to "solve" the problem by issuing more debt, simply act to make it worse over time.


Logistic regression: Equation for marginal effect at the mean

A

OP's second expression corresponds to
$$\Delta p = P(Y=1|D=1,X=\bar{X}) - P(Y=1|D=0,X=\bar{X}),$$
which is
$$
\Delta p = \Lambda(\beta_0 + \beta_1 + \beta_2 \bar{X}) - \Lambda(\beta_0 + \beta_2 \bar{X}),
$$
where $\Lambda(z) = e^z/(1+e^z)$.

Rationale for dividing by p

OP's first expression corresponds to $(\Delta p)/p$, where $p$ is the proportion of 1's in the sample $P(Y=1)$. This expresses the MEA $\Delta p$ as a ratio to $p$. This is informative as it lets us have better ideas on how large $\Delta p$ is. (For example, if we just have $\Delta p=0.01$, we don't know how large it is. But if $p=0.02$ is also given, then we know $\Delta p=0.01$ is a quite large effect as it is 50% of $p$.) One might call it "MEA as a ratio to Y-bar". Anyway, MEA usually refers to $\Delta p$, but if you mean $(\Delta p)/p$ by it, it should be OK as long as you explain it.

B

The MEA associated with $D$ can also mean $\dot{p}=\partial P(Y=1|D=\bar{D},X=\bar{X})/\partial \bar{D}$. As
$$P(Y=1|D=\bar{D},X=\bar{X}) = \Lambda(\beta_0+\beta_1 \bar{D} + \beta_2 \bar{X})$$
for your logit model, we have
$$\dot{p} = \frac{\partial \Lambda(\beta_0+\beta_1 \bar{D} + \beta_2 \bar{X})}{\partial \bar{D}} = \beta_1 \lambda(\beta_0 + \beta_1 \bar{D} + \beta_2 \bar{X}),$$
where $\lambda(z) = \Lambda'(z) = e^z / (1+e^z)^2$. Note that $D$ is binary but $\bar{D}$ is continuous so differentiation with respect to $\bar{D}$ makes sense. $\Delta p$ and $\dot{p}$ have different interpretations. I personally think $\Delta p$ in Part A is easier to understand as the notion of $P(Y=1|D=\bar{D}, X=\bar{X})$ is a bit awkward.


A question about a derivation in Gali's Monetary Policy book

As mentioned in the comments your question is not possible to answer as it is, but you should know that in macroeconomic models of growth exogenous shocks are measured by log-linearising equations around their steady state and then calculating impulse responses (that is, the magnitude and persistence of a shock to the variables of interest). This must be precisely what your book shows you, and unless you become familiar with macroeconomic modeling this question will hardly be answered any better than it is on your book.


John Rawls, in his book, A Theory of Justice, discusses the Maxi Min Principle, where he essentially says that societies can be compared based on how well they maximize the welfare of those with the lowest welfare (subject to caveats about certain basic human rights).  If the typical welfare function is a weighted average of individual welfare $U_i$ like: $$\sum a_i \cdot u_i$$ over all $i$, then this welfare function is $$\min u_i$$.
Since the minimum is the 0th percentile, I think this would just barely qualify as an answer to your question. I suspect that's not exactly what you had in mind, but it might be helpful.

This metric has appeared in a number of economics papers including Amartya Sen 's 1977 Econometrica paper,  On Weights and Measures: Informational Constraints in Social Welfare Analysis, The Aggregation of Climate Change Damages: a Welfare Theoretic Approach (Fankhauser, Tol, and Pearce (1997)), and Optimal Redistributive Taxation When Individual Welfare Depends Upon Relative Income (Boskin and Sheshinski (1978))

One more thing:
I've also seen a blog post that compares Swedish and American incomes at all income deciles and shows the level of percentiles at which a poor person would be better off in Sweden:

The Economist's View has a related figure for more countries.


When and why women left the labour market?

They worked at home. Like "Household Engineers" do today, doing everything working from down to dusk like their husbands, and like their husband they shared in the earnings of the household.
Until the industrial revolution came each household operated like a mini business.

It was the Industrial revolution that divided "work"(e.g. business) from home.

They never left the workforce, They just stopped being called the workforce.

Ironically with people working from home more and more we are returning to such a system.


1 gives you the percentage point (p.p.) difference,
2 gives you the percentage difference.
There is no rule that says that if you're dealing with shares, you have to use 1. The reason 1 is preferred is that, when the measurement unit is in percentage, things get confusing quickly (e.g. going from a 2% to a 1% interest rate is a 50 percent reduction in the interest rate, but of course only a 1 p.p. reduction).
In your case, 2 could be meaningful, though. For example, "the share in labor employed in manufacturing decreased by half". A compromise could be to express the change in percentage points to avoid confusion, and then add the percentage change if it is meaningful (e.g. the labor share decreased by 20 p.p., a reduction of 50%).


In which situation does the demand and supply curve both shift?

No, this case is not true.

A factor which both shifts supply and demand curves at the same time is an increase or decrease in population.

This both adds consumers (increase in demand) to the economy and increases the workforce (increase in labor force, thus producing more and increasing quantity supplied).


What purpose does general equilibrium serve in practice?

Daron Acemoglu, in a paper called Theory, General Equilibrium and Political Economy in Development Economics, discusses the role of economic theory in empirical work in development economics, which investigates the causes of poverty and low incomes. He puts a special emphasis on general equilibrium considerations. He discusses why counterfactual analysis based on microdata that ignores general equilibrium issues may lead to misleading conclusions. In particular, depending on magnitudes of various effects, general equilibrium interactions can offset or even reverse sensible partial equilibrium conclusions.

The importance of general equilibrium effects in theory

Acemoglu considers three important types of general equilibrium effects, which are usually not estimated in partial equilibrium comparisons:

In response to large policy interventions or shocks, imperfect
substitution between factors and diminishing returns imply that
factor productivities and prices will change.
The same policy interventions or shocks can lead to endogenous technology responses.
There may be composition effects resulting from equilibrium substitution of some factors or products for others.

The importance of general equilibrium effects in practice

Consider the large increase in the relative supply of college-educated workers in the United States starting in the late 1960s. Given technology, this change in relative supply should have reduced the college premium. As is well known, the opposite happened in practice, and the college premium increased sharply from the late 1970s onwards. This was a consequence of the endogenous response of technology to the relative abundance of more skilled workers.
A large microeconometric literature shows that healthier individuals are more productive. On this basis, we would expect a greater aggregate productivity if health conditions improve. But one should take general equilibrium effects into account, since an increase in health conditions also increases population and longevity, and because of diminishing returns to capital and land, it may decrease labor productivity and may in fact reduce income per capita.

Other references

Townsend (2009) for a discussion of the role of general equilibrium analysis in development economics, with special emphasis on credit market issues;
Heckman, Lochner and Taber (1998) for a discussion of general equilibrium issues in the analysis of the effects of technology on wage inequality.


Relation of Engel-curve to income elasticity of demand; is the slope of the Engel-curve equal to the elasticity of income?

Income elasticity of demand
Let $q(y)$ be the Engel curve for a good, i.e. it gives the demanded quantity for a given level of income $y$ (keeping prices fixed). The income elasticity of demand is then given by:
$$
\varepsilon^y_q = \frac{\partial q}{\partial y} \frac{y}{q}
$$
It measures the percentage point change in demand $q(y)$, due to a 1$\%$ increase in income, $y$.
If $\varepsilon^y_q \ge 0$ the good is normal, if $\varepsilon^y_q < 0$, it is inferior. If $\varepsilon^y_q > 1$ it is a luxury good, while if $\varepsilon^y_q < 1$ the good is called a necessity.  The demand for luxury goods increases more than proportionally compared to income. Necessary goods increase less than proportionally.
The advantage of using the elasticity $\varepsilon^y_q$,  instead of the slop $\frac{\Delta q}{\Delta y}$ is that it is unit-independent
We can see this as follows:
$$
\varepsilon^y_q = \underbrace{\dfrac{\partial q}{\partial y}}_{\dfrac{kilos}{Euros}} \underbrace{\dfrac{1}{q}}_{\dfrac{1}{kilos}} \underbrace{y}_{\dfrac{Euros}{1}}
$$
This means that $\varepsilon^y_q$ does not change if we express the goods or the income in different units. For example the elasticity $\varepsilon^y_q$ does not change whether you express income in Euros or Dollars. Also, the elasticity does not change whether you express demand in kilos or pounds. As such, elasticities can easily be compared across countries and over different time periods. The slope $\frac{\Delta q}{\Delta y}$ does not have this advantage.
Homothetic preferences
If preferences are homothetic, the demand function is linear in income:
$$
q(y) = c y,
$$
where $c$ is a constant. In fact, substituting $y = 1$ into this equation gives:
$$
q(1) = c,
$$
so $c$ is the unit income demand (the amount that you would buy if you would have 1 Euro). This means that we can also write:
$$
q(y) = q(1) y.
$$
Engel curves are straight lines through the origin and with slope $q(1)$:
$$
\frac{\partial q(y)}{\partial y} = q(1).
$$
As such:
$$
\varepsilon^y_q = \frac{\partial q}{\partial y}\frac{y}{q(y)} = \frac{q(1)y}{q(y)} = \frac{q(y)}{q(y)} = 1.
$$
So demand curves arising from homothetic preferences have unit income elasticity.


Question about Federal reserve open market operations - reducing money supply

Have you ever gifted your parents a "I'll wash the house" ticket? Or maybe your parents just forced you to do it. In any case, money, or less colloquially a Federal Reserve note, is like a "I'll wash the house" ticket issued by the Federal Reserve. But instead of being able to redeem it for a house wash, you can redeem it to receive a certain amount of gold. That is if you lived pre-1971 (edit: pre-1933). Now you can't redeem it for anything, but it doesn't change the fact that Federal Reserve notes are a liability to the Federal Reserve.

So if your parents cash-in your "I'll wash the house" ticket, what will you do with the physical ticket? Store it as a souvenir? Rip it up? Re-gift it? Sell it to them in exchange for a new computer? I think "[d]oes it even make a difference" sums it up pretty well, especially since most of the "tickets" are digital not physical.

Footnote: The government guarantees you can pay taxes with Federal Reserve notes. So in that circular sense, Federal Reserve notes are redeemable for something.


Let's forget about the reserve price for a second.

Suppose there is a symmetric increasing equilibrium bidding strategy $b(v)$. We can think of a bidder with value $v$ as choosing which $\tilde{v}$ he will "pretend to be" by submitting bid $b(\tilde{v})$. The bidder wins iff $\tilde{v}>v_j$ (where $v_j$ is his rival's value). Thus, the payoff is

$$(v-b(\tilde{v}))\Pr(\tilde{v}>v_j)=(v-b(\tilde{v}))\tilde{v}.$$

Now compute the FOC for choice of $\tilde{v}$:

$$v-b(\tilde{v})-\tilde{v}b'(\tilde{v})=0.$$

If $b$ is really an equilibrium then the optimal choice of $\tilde{v}$ should be the bidder's true value, $v$: $v-b(v)-vb'(v)=0$. This is a differential equation. To solve it we just need a boundary condition.

Anyone with $v<r$ will not bid. So the set of bidders will be the truncated uniform distribution on $[r,1]$. Intuitively, a bidder with value $r$ cannot earn a positive surplus, so a reasonable boundary condition seems to be $b(r)=r$.

Solving the initial value problem
$$v-b(v)-vb'(v)=0,\ b(r)=r$$

yields the equilibrium $$b(v)=\frac{r^2+v^2}{2v}.$$


What would make a restaurant offer all-day breakfast?

This question makes a hidden assumption:  that all diners are on the same circadian rhythm.  A significant portion of the US population works nights, spans time zones, etc. For example, truckers.

That's one answer, that addresses the market.  There's also the question of inventory logistics for the restaurant.  Most eateries have to do extensive prep for a meal shift, and that includes thawing proteins, advance-cooking certain items, chopping, etc.

Imagining an ideal world for the restaurant, they would prep exactly as much food, in various menu-item-proportion, as was ordered in a meal shift, and end up with none left over.  Meal prep left over would represent wasted inputs, and very often it is not able (or allowed) to be kept for the next meal shift.

Menu prep is "expensive" whether you end up selling one meal or a dozen.

Also, often "lunch cooks" have a different set of training/skill sets than "dinner cooks" than "breakfast cooks."


What is the proper term for "contraction economics?"

It doesn't look like there's an established term. Combine "demographic decline", "population decline" or even "fertility decline" (although this doesn't imply population decline) with whatever you are interested in (markets, economic growth etc.)

These papers (if macro) usually get JEL-tagged "J11 Demographic Trends, Macroeconomic Effects, and Forecasts" in combination with whatever is getting modeled, e.g. "Scotland the Grey: A Linked Demographic–Computable General Equilibrium (CGE) Analysis of the Impact of Population Ageing and Decline" is tagged with J11 and a bunch of R1x codes, the latter for regional economics/models.


Difference between Engel curve and income expansion path

Both of these curves describe the same phenomenon, the change in consumer choice as income changes. The difference is in the displayed variables. The Engel curve is a relationship between the consumption of a good $x$ and income $I$, whereas the income expansion path (IEP) shows the relationship between the consumption of a good $x$ and a good $y$ as income increases. These are the coordinate systems the curves are usually displayed in, so Engel in $(x,I)$, IEP in $(x,y)$.

The Engel curve would show you if good $x$ is inferior.

One can also learn this from IEP (it would have a negative slope somewhere), and in case both goods are normal IEP would show you if the consumption of one good would increase at a faster rate than that of the other.


Think of all the possible outcomes. For example, if your first ball is red you loose 1 rupee (this has probability 1/2), if you instead get one green ball and then one red ball you end up with  0 rupees (this has probability 1/4), if instead you get at first 2 green balls and then a red one, you end up with 1 rupee (this has probability 1/8), etc.

The expected value is

$$E(V)=(-1)\frac12+(0)\frac14+(1)\frac18+(2)\frac1{16}\dots=\sum_{n=1}^\infty(n-2)\frac{1}{2^n}=0$$

The last equality comes from two well-known results: $\sum_{n=1}^\infty\frac{n}{2^n}=2$ and $\sum_{n=1}^\infty\frac{1}{2^n}=1$


Economics perspective of connected workers

You may look up the 2015 Journal of Economic Perspective paper "Why Are There Still So Many Jobs? The History and Future of Workplace Automation" by David Autor. Look up the references therein.


Change in the way unemployment is calculated in the US?

The most likely explanation for discrepancies that political partisans like to pick at is that presidents like to cite different measures of unemployment. The BLS has 6 alternative categories of labor under-utilization. The U-6 measure of unemployment with seasonal adjustment stands at 9.7%. Although the official rate may be under 5% as in the BLS's September 2016 release here, many people may not feel like the economy is much better, and partisans can argue over which measure of employment or labor force participation rate is most useful.

Most politicians are not part of insidious, cynical political schemes, as the media may make them out to be. Just that when incidents happen, we hear about it non-stop, and it reinforces the public's view that political workers are untrustworthy as a whole, allowing for weird specious claims from the left and right. Take your friends' words with a grain of salt, and when you hear any president make a claim about the unemployment rate, you can look up the BLS's numbers yourselves and see how they are measured and which one the president is citing.

It is also worth noting that the BLS regularly updates how they measure unemployment.


Why does an economic slowdown lead to deflation?

First, let me address some incorrect premises in your question

Usually economists say that in recession there is deflation, so increasing the money supply does not lead to high level of inflation.

This statement is not really correct. First, I dont know many economists who would say that usually recessions are deflationary. For example, according to Romer's Advanced Macroeconomics 4th ed, the macroeconomic aggregates in 11 recessions between 1947:1 and 2009:3 can be summarized as follows:

So out of 11 recessions during the time period under considerations only 5 were characterized by deflation, and average change in inflation over all 11 recessions was only $-0.3\%$ which indicates that the recessions with deflation had on average slightly higher effect than the recession with inflation, but the datapoint is so close to zero that it would most likely be statistically insignificant. So the statement you say about most economists saying recessions are deflationary is simply not factually correct. Or at least if there are economists saying that they definitely cannot be in majority and assuming that statement was correctly representing their views they would not be factually correct.
In fact as you can read in this old Economics.SE question in economics we even distinguish between inflationary and deflationary recessions acknowledging both types of recessions exist. Generally speaking you will find that supply driven recessions are more on the inflationary side (good example are the 1970s energy crisis) and demand driven recessions on deflationary side (great example of deflationary recession is Great Depression or Great Recession both demand driven recession).
Second, the two effects are not mutually exclusive with each other. For example, it can be that velocity of money drops by $10\%$ while real output drops only by $8\%$ leading to deflation. If output would drop by more than velocity you would get inflation. Money supply also matters as it definitely does not stay constant during recession. For example, Fed with its dual mandate usually tries to expands money supply, directly or indirectly, during recessions to stimulate the economy. In addition as disused by Krugman (1998), what matters is not just the actual change in money supply or other quantities but what are peoples expectations about changes in money supply and so on. An increase of money supply that is not credible to be permanent and expected to be quickly reversed will have no effect on inflation as if it would never even happen.
One important thing to note here is that when nominal interest rates are at zero lower bound (ZLB) it is correct to say that even large monetary expansion wont necessary be inflationary. The reason for that is that if you would want to go significantly below ZLB the peoples will strictly prefer to hold cash and any increase in money supply will be completely offset by drop in velocity of money. In such cases indeed increasing money supply wont lead to higher level of inflation but that certainly does not apply to any or as data above show even most of the recessions.


Incentive compatibility: Weakly dominant strategy versus Nash equilibrium?

A Nash equilibrium that consists of weakly dominant strategies is a stronger solution concept than a NE itself.

Consider the following simple matrix game where best replies have been marked with *

\begin{array}{c|cc}
P1/P2&\text{left}&\text{right}\\
\hline
\text{Up}&1^*,1^*&0^*,0\\
\text{Down}&0,0^*&0^*,0^*
\end{array}

Both Up and left are weakly dominant strategies. They are always at least as good as Down and right respectively. That does not rule out, however that there are no other Nash equilibria as the game shows. The combination (Down, right) is also a Nash equilibrium because there is no gain from unilateral deviation. By deleting weakly dominated strategies we lose the latter Nash equilibrium so a Nash equilibrium that consists of weakly dominant strategies is a stronger solution concept.


How is India able to increase its foreign exchange reserves despite historically running a trade deficit?

The central bank's accounts are not the same as entire country's accounts.

The foreign currency reserves statistic comes from the Reserve Bank of India, the central bank, when it discloses its account. The trade deficit is not a property of the central bank's accounts, although the central bank might have economists producing that statistic. The trade deficit is a property of the entire country in aggregate, which includes household, business, and government trade activity.


What would be the likely economic consequences of the "trillion dollar coin" idea to eliminate all government debt?

Quite obviously, the act of creating and depositing the coin can't possibly impact anything of economic interest (except for creating an imperceptibly small increase in demand for platinum and hence an imperceptibly small increase in its price).

The impact comes when the government starts to make withdrawals and spend money.  And the nature of the impact depends on a thousand things you haven't specified, most notably:  What does the government do with the money it withdraws?  Does it pay down existing debt that it had not previously planned to pay down?  Does it buy more missiles that it had not previously planned to buy?  Does it expand a welfare program  that it had not previously planned to expand?  Does it cut taxes that it had not previously planned to cut?  Does it trade the coin for a bathtub full of $10,000 bills for the president to bathe in?

It's no use assuming that it does none of these things with the money, because it has to do something with the money.  So let's take the simplest example and suppose that they buy bonds (or cut taxes, which will have more or less the same effects).

In this case, you're swapping money for bonds, which is going to increase the price level.  The impact on inflation depends, once again, on things you haven't specified.  Does the government buy a trillion dollars worth of bonds at once?  If so, there is a jump in the price level but essentially no new inflation.  Do they spread the purchases out over a couple of years?  If so, that's going to be a very inflationary couple of years.

Could all future government spending be paid for this way?  Sure.  Here are some other things you could use to finance all future government spending:  A tax on food.  A tax on labor.  A tax on savings.  A head tax.  A tax assessed on random people in random amounts at random times.  What you're proposing is, in effect, a tax on holding money  (because it's the people who hold money who will lose when the price level adjusts).  Each tax has some advantages and some disadvantages.  In each case, the disadvantages grow nonlinearly as the tax increases, so it's very unlikely you'd want to just choose one tax to finance all government expenditures.  You probably want a mix, which is what we have now.  Whether we have exactly the right mix is open, of course, to a great deal of debate.


The multiplier comes from the solution to the goods market equilibrium. In economics everything is endogenous. Increase in income increases consumption that increases demand, demand increases production and production increases income. However, as an echo in a cave the initial increase in income gets 'weaker' as it cycles through the economy and the result 1/MPS tells you how much after accounting for the above mentioned cycle 1e increase in for example government spending increases the output.

Actually on fundamental level its not about marginal propensity to save but marginal propensity to consume (MPC), (however note that MPS=1-MPC) which is the channel through which increase in income affects the consumption which then affects demand and output.

A standard undergraduate textbook level consumption function looks something like this:

$$C = c_0 + c_1 (Y-T)$$

Where $C$ is the consumption, $c_0$ autonomous spending $c_1$ is the MPC and $Y-T$ is the disposable income ($Y$ is the income/output and $T$ taxes).  Mathematically, $c_1$ or MPC is what determines how $C$ changes when income changes. However, in order to see how much 1e increases output we cannot look at just MPC because as mentioned in the first paragraph the 1e cycles in the economy. So you actually need to calculate the geometric sum of the consumption multiplier being applied over and over again. Mathematically:

$$1+c_1+c_1^2 + ... c_1^n = \frac{1}{1-c_1}$$

Even though the 1e can be thought of circulating in economy infinitely  many times that does not mean it increases the output by infinity as many infinite sums actually converge to some actual value. In this case as long as $c_1$ is less than one, which it should as its the proportion of consumed income, the infinite sum would converge to some finite value not diverge to infinity.

However, while the above explains the logic behind why the multiplier is what it is it probably does not answer your question about saving. To see what role savings plays in the model consider the standard goods market equilibrium which is found by first substituting $C$ into the GDP identity and then solving for $Y$ (which is where you actually get the multiplier from):

$$Y = \frac{1}{1-c_1} \left(c_0  +I+G-c_1T\right)$$

Where as mentioned above $Y$ is output $1/(1-c_1)$ is 1/MPS $c_0$ autonomous spending, $I$ investment $G$ government spending $c_1 T$ are taxes multiplied by MPC. Now saving actually hides in the model in the investment $I$. By definition investment must be the sum of private and public saving, assuming away public saving for the sake of simplicity we can say that investment is equal to private saving so $I=S$. Consequently the multiplier 1/MPS affects also saving and when you increase the amount of savings the same multiplier is applied as to government spending or autonomous consumption so you are completely correct in wondering about that.

However, the reason why 1/MPS 'the multiplier' is that, as already mentioned, in this model the channel through which multiplier works is the effect of income on consumption which then affects all the rest.

Also note that in a short run $I$ can be considered fixed $\bar{I}$, I wont go into full detail about why that is as it is outside the scope of your question but basically in short run during recession investment is not always responsive to increase in savings because any increase in savings will be offset by changes to other variables. A consequence of assuming that investment is fixed and does not change with the level of savings then the savings will have no multiplier and you will get the "paradox of thrift".


Economic policies to decrease obesity (would they be effective?)

Yes, sugar tax!

This is probably as controversial as tobacco tax was back in the days. If you walk through a supermarket, you will find that half of the food section is food full of sugar. Sugar is what makes you fat, not fat itself. It has been known for a while, at least since the professional sports were invented. Yet the lobby of the enormous sugar industry keeps regulators from labeling it as hazardous and taxing its use. If you are doing a research into obesity I highly recommend devoting two hours to watch That Sugar Film (2014).

I would also recommend taking a multidisciplinary approach and bringing some arguments from biochemical and nutrition fields.


Do countries invest in the country's own stock market?

One example of monetary policy is the Bank of Japan buying the Nikkei as part of its asset purchasing program.

Typically, a substantial share of citizens' wealth is connected to returns of their national stock market.  Hence, it is generally not a wise move for a fiscal authority to invest public funds (not tied to pension schemes) in broad baskets of stocks.

You might try looking into sovereign wealth funds, these are typically gargantuan profit-maximizing investment vehicles wealthy nations use to acquire risky securities.


That depends on your model of the world, and on how you define "goods and services".

In a gold standard world, you essentially exchange goods and services with gold. Gold mining is therefore simply an expansion of the money supply. Adding gold to both the nominal and real side of the economy seems to me like today adding money as a good to GDP. If you do this, you get a positive trade balance. It makes a lot of sense to seperate monetary gold, which is on the monetary side, and commodity gold, on the goods side, as Henry mentioned in the comments.

In my opinion, it does not make sense to count gold twice. It seems reasonable to seperate the monetary and real side of the world, which then would yield a balanced world trade.

To the second question: I am no expert on gold mining, but I could imagine that it is not possible to "drive up" gold production in a recession. What is true is that a (nearly) constant money/gold supply leads to inflation being counter cyclical.


Credit or more broadly finance helps to allocate scarce resources to where they are most needed.

  Example. Suppose party A is sitting on a pile of resources from which he thinks he can make a return of 1% (per annum). Party B has no resources, but thinks she can use party A's resources to make a return of 3%. Then both parties can benefit if A loans his resources to B at a rate of 2%. (Society also benefits, at least insofar as the economic pie is larger than it would otherwise have been.)

(Of course, in the real world, A and B may be very much mistaken about their rates of return, which is partly how bad things like financial crises arise. But the above simplistic example illustrates what's "supposed" to happen, at least in a world of perfect information.)

P.S. Your presumption that "productivity will always grow at around 2%" is false. Finance/credit is at least part of the reason for non-trivial economic growth in the past 250 years. In a world where entrepreneurs are unable to gain credit, economic growth would be lower.


Your criticisms of GDP are valid, and have long been recognised in the literature and by statistical offices. For example, one of the issues your question deals with is quality changes. The case of hardware is particularly visible.

Several agencies do adjust for quality changes. A common method to do so is using hedonic regressions. The aim here is to obtain a "valuation" of all the different components of an item, like a PC. For example, using these regressions you can get an idea of the value consumers give to memory, graphic card, DVD reader, speed, etc. Then, new products can be adjusted using these values. An example of such regression for computers is here.

The ONS (which computes GDP for the UK) do attempt to adjust for quality. In fact, they consider "volume" of a product as a combination of both quantity and quality. They differentiate between two methods to evaluate volume. For some homogeneous goods like agricultural grains or energy products, where quality is less relevant, you simply compute the physical volume. Yet, for most of the products, heterogeneity makes such calculation impossible. The quality adjustment then comes from prices. The ONS uses the hedonic regression method for goods with "frequent model changes", but uses other methodologies too. For example, there is a major challenge in measuring change in quality of public services (like health care and education). For a detailed description on the ONS quality adjustment, check this article. Check here for public productivity measurement.

For other formal references, see this one. The latest UN System of National Accounts, in which most of countries are based to compute national statistics, also offer some guidance on quality adjustment (section 4, page 361, in particular).


Binary-continuous choice model in empirical consumer choices

It may not be a 100% match (and the details are a bit fuzzy in my memory), but I believe the model would be similar to Ericson and Pakes (1995).
In their model, there are $N$ firms who make sequential choices

Whether to enter the market or not (in your case, to consume or not), which is a discrete choice.
How much to invest if they have entered (in your case, how much to consume), which is continuous.

This is a seminal paper in dynamic discrete choice so I feel there should be some papers along this line.


why did the 2008 financial crisis happen inspite of banks having 85% insurance on deposits in U.S.?

Yes, most of the loans were insured - this is called a credit default swap (CDS) that an insurance company insures a bank loan in case of a default (i.e. when a borrower cannot repay, the insurer pays the bank).  Banks and insurance companies were making an unprecedented amount of money through this practice.

However, when the housing bubble burst, too many loans defaulted to the point that the insurance companies could not pay the banks.  Then the banks were going bankrupt unless they were bought out.  For instance, Lehman Brothers, which reportedly had relatively more mortgage-related assets, had to declare bankruptcy as the federal government or no other bank would bail it out. (words are that people in finance did not like Lehman Brothers which is one of the reasons why it was not bailed out)

The 2008 financial crisis (people now call it the Great Recession) froze all economic activities so virtually few were able to escape it.


Why are the financial intermediation theory of banking or fractional reserve theory of banking still accepted despite evidence to the contrary?

There are several reason for it. Please note the reasons are not necessarily listed in order of importance, the last point is actually most relevant answer to your question.

The 'Evidence' You Cite is Controversial

First, in fact the work you cite itself states that this issue is matter of ongoing controversy (Werner, 2014). So it is far from settled.
However, what even more the empirical 'test' of Werner (2014) is itself actually quite controversial and has been criticized heavily  for confusing accounting with economics (e.g. see Rendahl & Freund 2019 or Spearman 2016).  Accounting and economics are completely distinct disciplines so it is often not possible to conclusively test economic theory just by accounting arguments.
For example, a theory of perfect competition predicts that in equilibrium there will be no economic profit. However, you can't test whether firm has zero economic profit by looking at the firm's profit and loss statements because accounting does not capture economic reality and does not include things like opportunity cost which could make profit zero. Depreciation in accounting is also treated in a way that economically has almost no sense etc. As a result even if in real life there would be some perfectly competitive industry you cannot test for that just by looking at firm's profit and loss statement but you need better more nuanced test.
In a similar way the paper by Werner is criticized for making essentially the same error of assuming that just because on the balance sheet money seemingly appear out of nothing that means that banks actually can create them out of nothing, and as explained in greater detail in Rendahl & Freund (2019), that is to a degree just accounting fiction and does not really prove that banks do not create money against some assets as understood in economics (as opposed to what accounting understands them to be - which again is completely separate discipline with its own terminology and very little relation to economics).
Next, when it comes to financial intermediation, it would again be quite controversial to state that banks do not engage in some financial intermediation. Even McLeay et. al. (2014) argue financial intermediation plays some role in money creation. Now this is not the same as claiming the authors endorse financial intermediation theory the point is that stating that banks do not engage in financial intermediation at all would be controversial.
Textbooks are generally not being rewritten if one controversial paper is published. It takes time for literature to settle. Even if the new controversial paper is entirely correct it can take few years before that is reflected in textbooks (especially undergraduate ones graduate textbooks are much more up to date).

All Models are Wrong but Some are Useful (George Box)

As the title of this section says all models are wrong, you can imagine model as being a map. Any map that is not 1:1 replica of landscape will be wrong in some way, however having 1:1 replica of landscape as a map ceases to be model and quite frankly it is completely useless even though it is 100% correct. Both 'fractional reserve theory' and 'financial intermediation theory' have their utility.
Starting with 'fractional reserve theory'.  For example, before 2009 when central banks started engaging in 'unconventional' monetary policy,  the 'fractional reserve theory' was quite useful for back of the envelope calculations. This is because pre-2008 commercial banks held virtually no excess reserves as you can see from the data provided by FRED below:

In such situation, the 'fractional reserve theory' can be actually quite useful and reasonable approximation. As you can see from graphs below, the $M_2/M_1$ ratio and amount of reserves before 2008 tended to move in same direction.

In fact if you remove observations post 2007 and calculate correlation between $M_2/M_1$ and required reserves you would find the correlation to be significant with point estimate approximately  $0.55$ (with $95\%$ conf.  interval (rounded to 2 significant digits) $[0.49, 0.60]$)*. Sure simple correlation is no rigorous test, but with such high correlation it is difficult to claim that there is no relationship between reserves and ratio of $M_2/M_1$ meaning that at least in past money were created in a way that looks like a situation where banks just multiplied reserves (again I am not saying this is necessary rigorous test, one can invoke reverse causality and so on,  point is that its not an unreasonable model for a first approximation).
Now for sure utility of the 'fractional reserve theory' diminishes greatly under the new unconventional monetary policy, and especially now that many central banks around the world (but not all) completely abolished reserve requirement altogether (see Fed explainer on its abolishment of reserve requirements here). However, it is not clear if this is new normal. These policies are still referred to in the literature as 'unconventional' and  it is not clear if they will last. In fact before the Covid-19 hit world economy, Fed was already contemplating of returning to normal policies (e.g. see news articles pre-Covid such as this one). Consequently, an argument can be made that this model is useful for students to understand because it described banking relatively (relative to its simplicity) well in the past, and it is not so clear if the current unconventional monetary policies that make the model give very wrong predictions will last.  You should also note, that it is not like this is the only model textbooks discuss. For example, Blanchard et al Macroeconomics: A European Perspective pp 71 described failures of simple multiplier model and alternative endogenous money supply theories already in 2013.
When it comes to 'financial intermediation of banking' theory this one has still its uses even contemporary empirical research (See discussion in Freixas & Rochet Microeconomics of Banking). Consequently, when it comes to this theory it is even more useful for students to know, even if you can of course criticize it and competing theories exist.

Both Theories are Important for Didactic Reason

Since we are discussing textbooks and not applied research it is important to realize that textbooks, beyond anything else, serve didactic purpose. They are intended to be used as teaching tool, and especially undergraduate textbooks (which is where you will see 'fractional reserve theory' and 'financial intermediation theory' given most space) have to build foundations for future study.
For example, every single intro physics textbook in existence will teach Newtonian physics, even though we now know that Newtonian physics is wrong and at best special case of general relativity. Yet learning incorrect Newtonian physics is still useful as it can be used for back of the envelope calculation in many situations  and it will still give reasonably good answers. As shown under point 2 even simple 'fractional reserve theory' provides quite reasonable predictions (although it is definitely not as close as Newtonian theory to general relativity so please do not take this analogy too far).
This is like asking why textbooks typically include linear demand and supply when in reality demand and supply is almost never linear. Well answer is that solving models with simple linear demand and supply is easy for students and facilitates learning, as it serves as a stepping stone to understanding more complex models. For example, in typical graduate microeconomic textbook you will virtually never see linear demand or supply save for some rare exceptions, yet if you would try to teach 101 econ students from graduate textbooks such as MWG Microeconomic Theory or Varian Microeconomic Analysis, all but the top $5\%$ of class would fail miserably.  Again you would end up in analogous situation if you would start physics 101 with  Einstein's  field equations. Anyone except for top students would not be able to follow. As a result you will fail in your task to educate students, since even students that might become excellent physicist when they learn physics in small steps instead of trying to directly tackle graduate level physics, would never be able to achieve their full potential if you would just stress teaching more realistic but infinitely more complex models. Again this does not mean such models should not be taught but there is plenty of room for that in graduate courses.
Even McLeay et al (2014) who very harshly criticize 'fractional reserve theory' argue it can be a

useful way of introducing money and banking in economic textbooks,

Next, it is also important to understand that both 'fractional reserve theory' and 'financial intermediation theory' are part of exogenous money supply theory (a theory where private banks are passive agents that just expand/contract money supply in a way that is exogenously dictated by central bank), and regardless whether you consider exogenous money supply theory right or wrong (I personally think exogenous money supply theory is far from completely correct), learning exogenous money supply theory is important part in learning endogenous moneys supply theory (which is theory where private banks have active role in money supply creation).
This is because as discussed earlier, even authors who are proponents of endogenous money supply theory such as McLeavy et al (2014), do not deny that banks are constrained by central bank's policy and that some financial intermediation is taking place.
The difference between exogenous money supply theory and endogenous money supply theory is that under exogenous money supply theory the chain of causality goes just from central bank policy to money supply, whereas under endogenous money supply theory there is relationship that goes both ways (see more details on this in above mentioned Blanchard et al). Consequently,  endogenous money supply models will still feature central banks policy be it via reserves or more often other ways that affect money supply like bank regulation/interest rates etc,  and will still often have banks that still take in deposits. Rather under endogenous money supply theory there will be additional relationships where money demand will cause banks to lend more and in turn banks will then create more reserves at the central bank (like described in McLeavy et al).
Consequently, even if you think exogenous money supply theory is completely wrong, it is didactically better to first teach exogenous money supply theory, and then explain to students that this is wrong because there are additional channels that lead to simultaneity/reverse causality and build your explanation of endogenous money supply theory based on exogenous money supply theory. Jumping directly to endogenous money supply theory would likely just facilitate less learning and create more confusion even if your goal is to teach only endogenous money supply theory.
The above alone is the most important reason why these models heavily feature and will likely continue to feature in most 101 macro textbooks in foreseeable future. However, you should note that the space most mainstream macro textbooks devote to exogenous money supply models has shirked substantially, you can verify that by comparing the latest editions of Mankiw Macroeconomics or Blanchard et al Macroeconomics to their earlier editions.   Nowadays, you will get quite more space devoted to endogenous money supply theories (however, do not confuse endogenous money supply theory with controversial assertions of Werner (2014) of money being created completely ex-Nihilo by private banks - that is very controversial compared to just general endogenous money supply theory).

* Code for the correlation calculation in the spoiler:

 #data are obtained from fred data linked above, data after 2007 are deleted, data were merged in excel before runing the code (with raw output below):
 cor(fred$M2SL_M1SL, fred$REQRESNS, method = c("pearson"))
 > [1] 0.54867
 cor.test(fred$M2SL_M1SL, fred$REQRESNS, method=c("pearson"))
 > Parson's product-moment correlation
   data:  fred$M2SL_M1SL and fred$REQRESNS
   t = 15.887, df = 586, p-value < 2.2e-16
  alternative hypothesis: true correlation is not equal to 0
 95 percent confidence interval:
  0.4895299 0.6027855


Why does a positive output gap imply workers working overtime?

A fisherman usually works 50 hours and catches 50 kg of fish every week. So let's say that his potential (or "natural") output is 50 kg of fish.
Now suppose his boss orders him to work 80 hours per week and catch 80 kg of fish. This 80 kg of fish is his real output and exceeds his potential output of 50 kg. This is not necessarily a good thing and is probably not sustainable:

He may prefer to work less (have more leisure rather than the consumption goods that his additional work can buy).
He and his fishing rod may be overworked and thus wear out more quickly.

A definition of potential GDP:

Potential gross domestic product (GDP) is a theoretical concept that means different things to different people. To some, it reflects a world in which every worker is matched with the perfect job, every good idea is implemented, and the bad ones are ignored. In this world, resources are allocated optimally with no distortions from the tax code, information frictions, or suboptimal government policies.†

I would add also that potential GDP at any moment is that which can be sustained over the long term, holding constant resources and technology at that given moment.

†This idealized definition isn't actually used to get estimates of potential GDP. Instead, such estimates are simply obtained by smoothing out GDP fluctuations.
If this idealized definition were actually used, then it would in practice be quite impossible for real/actual GDP to ever exceed potential GDP.


What do the numbers in the ISO 4217 currency codes mean?

From the ISO homepage this answer can be found:

The three-digit numeric code is useful when currency codes need to be understood in countries that do not use Latin scripts and for computerised systems. Where possible the 3 digit numeric code is the same as the numeric country code.
For currencies having minor units, ISO 4217:2015 also shows the relationship between the minor unit and the currency itself (i.e. whether it divides into 100 or 1000).
ISO 4217:2015 also describes historical codes in List three as well as the codes representing certain funds in List two.


Is it necessary to ask permission before including someone in the acknowledgements of a economic research paper?

In short, no it is not necessary. I have never asked anyone for permission and I have never been asked. The people I thank usually have not read my paper (rarely anyone does, to be honest), but we have had chats on visits or after seminars or conferences. If you thank someone make sure they are familiar with the paper. Certainly include someone who presented a formal discussion at a conference.
By all means, never include someone in your thank-you note who you have not talked to. People will see this as a strategic way to influence referee selection. However, if someone made great comments that greatly influenced the direction of the paper, name them. It's free and makes people happy.


Differing Inflation Rates for Separate Income Groups; Literature?

Let us first boil this question down to a simple story: Suppose bread prices are very flexible while luxury car prices are very sticky. If the central bank now changes its monetary policy to increase inflation, then bread will become relatively more expensive compared to luxury cars. If the central bank anticipates this, they may prefer a more restrictive monetary policy to avoid redistribution from the bottom to the top.

To analyze this, two literature strands may help you. First, heterogeneous price stickiness has been studied in

Carvalho, Carlos. "Heterogeneity in price stickiness and the real effects of monetary shocks." Frontiers in Macroeconomics 6.3 (2006).

This paper essentially breaks with the standard assumption of identical price stickiness across sectors. Thus, at some point one sector (bread) may experience higher inflation than another sector (luxury cars), since luxury car prices are more sticky and there is an expansionary monetary policy. The framework for welfare analysis in sticky price models is given in:

Woodford, Michael. Optimal monetary policy inertia. No. w7261. National Bureau of Economic Research, 1999.

However, you not only have heterogeneity in price stickiness, but also heterogeneity in consumers. This is the second strand in the literature you should consider, the estimation of demand systems and welfare effects therein. The main reference (and recent Nobel memorial prizewinner) is:

Deaton, Angus, and John Muellbauer. "An almost ideal demand system." The American economic review (1980): 312-326.

If you combine these two analyses, you will have an answer to your question. However, this would require you more or less to write a paper about this (which may very well be worth it).


Optimal posted price with uniformly distributed values

You are correct if there is only one buyer or one good for each buyer.
If there are $n$ buyers, but only one good, the seller maximizes
$$(1-p^n)p,$$
which has FOC $p^* = (\frac{1}{n+1})^{1/n}$.


Let's say you have $k$ goods $x_1, x_2, \ldots, x_k$. Let $\mathbf{x}$ denote the vector $\mathbf{x} = \begin{bmatrix}x_1 \\ \ldots \\  x_k \end{bmatrix}$.

Some common workhorses to get you started:

Multivariate Cobb-Douglas: $u(\mathbf{x}) = \prod_{i=1}^k x_i^{\lambda_i} $.

Note that $\max_\mathbf{x} u(\mathbf{x})$ has the same solution as $\max_\mathbf{x} \log u(\mathbf{x})$ because the logarithm is a monotonically increasing function. Hence maximizing the utility function $u(\mathbf{x}) = \sum_{i=1}^k \lambda_i \log x_i$ will give the same solution and also is referred to as Cobb-Douglas.

Constant elasticity of substitution $u(\mathbf{x}) = \left( \sum_i \alpha_i ^\frac{1}{s} x_i ^\frac{s - 1}{s}\right)^\frac{s}{s-1}$


Utility function must be continuous for $V(p, e(p,u) )$ to hold?

Define the function

$$
V(p, e(p, u)) - u = f(u) ~~~\mbox{for}~~~ u_\min \leq u \leq u_\max
$$

Your problem is then reduced to finding a point $u^* \in [u_\min, u_\max]$ such that $f(u^*)=0$. First, this problem may have not solution at all, even if the function is continuos. That being said, if the function is not continuos, you can have a situation like the one below

Clearly, this function is not continuos at $u=u_0$, moreover, it is not possible to find a point $u^*$ such that $f(u^*) = 0$.


Can CPI and real GDP have high correlation?

There was a degree of economic stability between the UK's exit from the Exchange Rate Mechanism in 1992 and the World Financial Crisis in 2008: positive real GDP growth typically between 2% and 4% and low steady inflation typically between 1% and 3%.

The Governor of the Bank of England called this "The Great Moderation".

The effect is what you have observed: the level of real GDP and the level of the CPI both grew, apparently highly correlated with each other, as shown in the red line below based on quarterly data.

But this correlation was an artefact of each indicator growing itself over time and of auto-correlation, as well as not having a recession in that period or sudden spikes in prices; looking from 1991-2009 would have produced a less smooth chart.

If instead you look at the percentage changes over the previous 12 months, the apparent linearity collapses, as shown in the blue line below.


Two interpretations of the Lagrange Multiplier

If you would NOT obey the constraint and spend 1\$ more (or more precisely an infinitesimal amount more) your utility would go up by $\lambda$. As such $\lambda$ is what you give up for obeying the constraint, it is the price you pay indirectly and hence it is called the shadow price.

Writing the constraint in reverse has no effect other than changing the sign of $\lambda$, but that depends on the way you set-up your problem anyway (I.e. whether you add the constraint with a plus or a minus).