Does levying a new tax on production increase GDP?

tl;dr: Your reasoning that taxes should not increase GDP by themselves as they do not constitute any new production is on right track but might not hold always. The two main reasons for that are:

If we talk about GDP as the actual statistical measure not just as a theoretical measure we run into practical problems when we try to measure it. The most relevant one for this question is the fact that GDP cannot capture all production as statistical offices have no way how to record non-market transactions but at the same time government can still tax even non-market activity and thus increasing GDP in this way by 'revealing' previously hidden production.

Taxing production of some people and transferring the proceeds to other can in short-run increase output and thus also GDP during recession. This is because consumer spending affects GDP not just directly but also has a multiplier effect and during recession transferring money from people with low marginal propensity to consume to people with high one through taxes actually can increase GDP as spending of those people has higher multiplier attached to it.

Full Answer:
In a closed economy, the GDP/output $(Y)$ under expenditure approach will be given by
$$Y= C + I + G$$
where $C$ is consumption, $I$ investment and $G$ gov. spending. You are completely correct that taxes affect consumption. In fact this is made clear in any undergraduate textbook (such as Blanchard et al Macroeconomics: an European perspective). Following, Blanchard et al. we can explicitly make consumption to be a function of output and taxes and in fact we can write:
$$Y = C(Y-T) + I + G$$
where, $T$ stands for taxes. So GDP is function of taxes but at the same time it is function of output itself so in order to analyze the effect of any part of the equation on output we first have to solve the model for output. In order to do so explicitly and directly we have to make some assumption on the consumption function. A standard textbook one would be just simple linear consumption $C= c_0 +c_1(Y-T)$, where $c_0$ is an autonomous spending - spending that does not depend on income/output/GDP (these are macroeconomically all the same so I will be using them interchangeably when referring to $Y$) and $c_1$ is the marginal propensity to consume people have (for example if people consume $75\%$ of their disposable income and save the rest $c_1=0.75$).
By substituting the consumption function for $C$ one might solve for $Y$ to get:
$$Y = \frac{1}{1-c_1}\left( c_0 + I + G - c_1 T\right)$$
Now, the result above shows, that if all else is held constant if you increase taxes the output would decrease by amount $\frac{c_1}{1-c_1} T$.
Of course, assumption that everything else stays constant is not reasonable here. If we make a reasonable assumption that government will use those taxes to fund its spending which we can do by assuming balanced budget  $G=T$ the taxes will have no effect on output as their effect cancels each other out. To see this substitute $G=T$ into the expression, that gives you:
$$Y=  \frac{1}{1-c_1}\left( c_0 + I + T- c_1 T\right) \implies Y=  \frac{1}{1-c_1}\left( c_0 + I \right) +T $$
In this case the increase in taxes would have no extra effect on output/GDP. The output would increase by amount of $T$ but since $T$ is subtracted from the output/income in the first place the GDP would remain at the same level as it was before changes in taxes. (Note, the same would occur if we would not hold $I$ constant and allow for public saving since $I=S+T-G$ - we would arrive at the same conclusion but in more convoluted way).
This being said there are some caveats here. GDP is an imperfect measure so you might observe that the above wont always hold if you would just look at GDP statistics. The analysis above assumes GDP can capture all output. However, that is clearly not correct. If you have home garden and produce radishes for your own consumption,  clearly such consumption wont be counted on national accounts. However, now if government is able to somehow levy a tax on output of your radish garden, in one way or another, GDP statistics might actually increase because the portion of your output that was previously 'concealed' from national accounts statistics would suddenly become recorded. However, this rather reflects the flaws of GDP as a measure and practical data collection issues than any economic effect.
In fact because we cannot perfectly observe all output and its components you will routinely find discrepancies between trying to calculate GDP using income, production or consumption approach even though they are by construction all equivalent to each other.
Moreover, even more important caveat is that the analysis above assumes that all people have the same marginal propensity to consume $c_1$. This most likely does not hold in practice and taxing people with low marginal propensity to consume and transferring it to consumers with high marginal propensity to consume would actually be able to increase GDP. However, this occurs only in recessions as in normal times saving would just increase $I$ but during recession periods that does not generally hold.

There are some parts in your question that are also not correct that I would like to just briefly address:

My thinking is that the profit of the producer would not increase, since the price increase on the goods would just offset the cost of the tax.

This is not correct because it assumes that any tax is fully passed on to consumers - that is not accurate in most situations. However, when you calculate GDP using income approach you actually account for taxes in the calculation.
Next

For households that do save, they either lessen their savings or spend less on other stuff. And savings is just future consumption, meaning that if they save less, they have less consumption in the future.

This is correct but note future consumption is recorded in future GDP calculation not in the present one. However, this would be reflected on present period GDP through changes in $I$ as dis-saving is negative investment since investment must be equal to private + public savings.


Is stock investment an example of demand curves that slope upwards?

The observation that "as stock prices rise people tend to buy more of it, so here the "law of demand" holds in reverse", is one of the more widespread misconceptions related to economic thinking.

To expand on @StevenLandsburg comment, the "demand curve" (or "schedule") describes a static relation between the quantity of a good and its price. Its quality (in the general sense of features, qualities) remains constant, as we trace the demand curve.

In our case we think of the stock as the good, and we have also its price. Now what happens when the price goes up? Are we talking about the same good? Definitely no. The price raise itself signals that we now have a better good (higher discounted future profits). But then we have a different good.
So the pairs $(p_1, q_1)$ and $(p_2, q_2)$ do not really belong to the same demand schedule. So the "law of demand" is irrelevant here as a tool to describe the situation (we could picture it as a shift of the whole demand schedule outwards)

If one wants to try to think "intuitively" about the law of demand and stocks, then one can consider the following thought experiment:
Assume that you can buy the same stock on the same day at two different prices (it can happen, the world is not really one single market, even for stocks -and transactions can happen also outside of the stock market).
Assume that you cannot choose the seller, which will be determined randomly. But you have to commit beforehand about the number of pieces that you will buy. And you know the two prices.

What are you going to do?
Commit to buy more pieces if you are dealt the expensive seller, and fewer pieces if you are dealt the cheaper seller? Sounds ridiculously obvious, isn't it?


How does demand-supply equilibrium interact with profit maximisation?

I think one source of confusion in your question is that you are piecing together assumptions from different models that are incompatible. In particular, you've implicitly referred to a supply curve, but attempted to write down the optimisation problem of a monopolist. This doesn't work because a monopolist has no supply curve: a supply curve describes how much a seller would be willing to sell for a given price, taking prices as given. However, monpolists do not, by definition, take prices as given. Supply curves only make sense when the seller faces perfect competition.

Hopefully I can make this clearer by giving you an overview of the two different models.

One standard assumption in basic models is that markets are perfectly competitive. For sellers, this means that they take prices as given, and nothing they do can affect the price of the good that they are selling. (Alternatively, they solve their profit maximisation problem while ignoring any effects their choices have on the market price.)

Formally, a seller chooses the output quantity $q$ to maximise

$$ pq-c(q), $$

where $p$ is the price of the good, and $c$ is the (convex) cost of producing $q$ units of the good. Hence, the seller's profit-maximising level of output is given by

$$ p = c^\prime(q). $$

This gives us the seller's supply curve

$$ q^s(p) = (c^\prime)^{-1}(p). $$

Given some demand curve $q^d$, the relevant equilibrium condition is now

$$ q^d\left(p^*\right)=q^s\left(p^*\right) = q^*, $$

where $p^*$ and $q^*$ are, respectively, the equilibrium price and quantity. (This treatment needs to be slightly modified when $c(q)$ is linear, as in your question, but this can be done.)

We can, of course, relax the assumption that the seller is a price-taker. Suppose now the seller realises that their choice of $q$ can affect the equilibrium price. Here, we say that the seller is a monopolist. They now choose $q$ to maximise

$$ p(q)q - c(q), $$

where $p(q)$ is the inverse demand curve. That is,

$$ p (q) = (q^d)^{-1}(q). $$

(Equivalently, as in your question, they would choose $p$ to maximise

$$ p q^d(p)-c\left(q^d(p)\right). )$$

The equilibrium quantity is then given by the solution to

$$ p^\prime (q^m) q^m + p(q^m) = c^\prime (q^m). $$

Note that it now doesn't make sense to ask what the seller's supply curve is. The seller's profit-maximising level of output isn't determined by a given price, because their choice of output affects the price.

One can show, as you've found above in the case with constant marginal costs, that (assuming demand is downward sloping)

$$q^m < q^*$$

and

$$p(q^m) > p^*. $$

In other words, when the seller is a monopolist, the seller's profit-maximising output (which they will sell in the equilibrium where the seller is, in fact, a monopolist) is smaller than the equilibrium output under perfect competition.


Is there any reason deflation would reduce profit?

Profits also depend on the quantity demanded/sold. If deflation is accompanied by a decrease in demand, under equal input/output price deflation (or even if input prices fall more than output prices), profits can fall.


How/why does foreign investment get scared by the monetary policy?

Monetary policy affects the value of a country's currency (e.g. interest rate & money supply changes affect the value of cash).
Foreign investment is made by purchasing that country's currency (via currency exchange), then purchasing financial instruments in that country (stocks, exchange traded funds, bonds, etc.)
If a country decides to lower their currency's value over time (to make their goods cheaper to foreigners, for example), then any foreign investment would be exposed to currency devaluation.
A concrete example…

Turkish commodity market, sells 1 ton of cotton for 230 Lira (~$100USD) on January 1, 2015.

US investor exchanges $100USD into 230 Lira, buys 1 ton cotton on 1/1/2015.

Turkey decides to lower currency value by decreasing interest rate, increasing money supply, reduce reserve requirements. Value of Lira drops from 230 Lira for 100 USD to 270 lira for 100 USD.

US investor sells 1 ton cotton a year later for 250 Lira.  Exchanges Lira to USD. Receives $92.60USD.

What would normally be a modest ~9% gain from price appreciation of the cotton from 230 to 250 turns out to be an ~8% loss due to currency devaluation which occurred from monetary policy changes.
If a foreign investor isn't confident that monetary policy will drive up a country's currency value (in which they will be investing), then they should rightly be scared by monetary policy changes a government plans to make.


Financial manipulation and Inflation as tools for sovereign debt default

After reading a few more chapters, the authors in their, by now very known, terse and often imprecise style, state that a financial repression is a method used by some governments to obtain indirectly more revenue.
This channel works when the government makes bank accounts as the main possible saving tool available to the common folk. Afterwards, the banks are obliged to acquire government debt, which often comes in form of higher minimum level of reserves (They don't explain how... but, maybe gvt debt may count as reserves, since the Central Bank may act as a 'printing money factory' of the government).
Nominal interest rates are those which usually direct the interest rate that the market asks for holding government debt. By keeping it artificially at a lower level, national investors/savers will have to abide by that low interest rate. Futhermore, by making inflation surge, the real burden of government debt will decrease as time goes by (assuming there's no way to circumvent national regulators/markets, legally or illegally).


Why can't price level be negatively correlated with money demand?

You're supposing the amount of money held for transactions by players is arbitrary. This may be true if you look at an individual - I typically withdraw fixed amounts of money when my purse starts feeling empty - but is not true on average and in the long run.

But cash (in the wider sense, money on regular bank accounts count toward this) is also held by companies and foreign countries.

Even if you feel like discussing the rationality of the average person, you cannot discuss the rationality of the average money holder: the opportunity cost of holding too much money may be negligible for an employee with low savings. But it it isn't for a big multinational or a multi-millionaire that wants to stay richer than his neighbours in Beverly Hills.

Consumers are also retailers, at some point, so you cannot suppose the price rise affects them in the way they buy less goods. A potato producer still produces 1kg of potatoes after a price rise, only they are worth more and he need more money to buy carrots. The money demand increases because there is less money available for the same goods. This is a very different situation than a productivity drop, in which case the money demand would decrease.


(As requested in comments)
Primary dealers have a privileged position, so there is value in being registered as a primary dealer.
The Treasury or Central Bank can impose standards of behaviour (e.g. an expectation of reasonable participation in all bond auctions or open market operations) as a condition of maintaining that registration. There are rules such as https://www.newyorkfed.org/markets/pridealers_policies.html and you will find sections such as

The New York Fed expects a primary dealer to:

participate consistently as counterparty to the New York Fed in its execution of open market operations to carry out U.S. monetary policy pursuant to the direction of the FOMC;

provide the New York Fed’s trading desk with market commentary and market information and analysis helpful in the formulation and implementation of monetary policy;

participate in all auctions of U.S. government debt; and

make reasonable markets for the New York Fed when it transacts on behalf of its foreign official account holders.

In the selection process, the New York Fed will evaluate a prospective primary dealer with these expectations in mind.

In effect primary dealers provide liquidity and information and markets even at times of crisis, which comes at a cost, one which countries such as the USA think is worth paying to get the increased financial stability it brings.


What is the definition of: 'commodity space'?

Yes, a commodity space is the set of all possible commodity bundles. The simplest Micro 101 example is typically the nonnegative quadrant of $\mathbb{R}^2$, but in general equilibrium theory it is often assumed to be infinite-dimensional since there are infinitely many commodities (each good is differentiated by time and space, etc.).


How to interpret "scale efficiency" in DEA?

First of all, let me tell you that you're doing great. I remember that, being a non-parametric method, it was more difficult for me to understand even the simplest things of DEA.

Now, your interpretation of DEA is correct. When a firm is considered inefficient, it is because it could have gotten the same output but with a lower cost if it had used the best practices. Nonetheless, there is another way of looking at it (the output oriented way): if the firm adopts better practices the output will be increased with the same costs. Keep that in mind.

To explain the scale efficiency, i'm gonna build a very simple case. Imagine a firm that manufactures some product (y) and uses only labor as an input (x) and both the product price and the wage are $1 (for easier calculation). This firm has 4 plants: a, b, c, and d. Table 1, shows the number of workers, production, profit and productivity of the 4 plants.

Table 1

It is clear, that plant a is the least productive of them, because every worker produces only 1 unit. Plant b produces the same as a with less workers. c, on the other hand, has the same number of workers as a, but with more production and it is the most productive plant(2.5 units per worker). d is the biggest plant producing 29 units.

Now, let's calculate both the Constant Returns to Scale (CRS) efficiency and the Variable Returns to Scale (VRS) efficiency and the scale efficiency:

Table 2

Now, let's plot what it looks like:

Figure 1

Where every point represents a plant, the black dotted line is the CRS efficiency frontier and the blue line is the VRS efficiency frontier. This blue line implies that b, c, and d are as efficient as they can be. That is, they already have the best practices. The only difference in their productivity is due to economies of scale. Now, if you were to decide the future of a, the least efficient plant, what can you do? you have two options:

Fire some people keeping production constant: move to b.
Increase production keeping all the workers: move to c.

You should be aware by now that the smart move is to increase the production without getting rid of anyone (move to c). So far, this is the usual DEA analysis.

But now, let's forget about a. Let's see what we can do to increase the profitability of b. b is already in the efficiency frontier of VRS DEA. This means that b is already using the best practices. Nonetheless, because of economies of scale it is impossible to achieve the same productivity. So, the smart move, would be to invest \$5 hiring 5 new workers: move to c in Figure 1. Note that b is not adopting efficiency measures this time: it's just increasing the scale, while maintaining the best practices it already has. This investment would be profitable indeed: we hire 5 new workers and production goes from 10 up to 25, which represents an increase in profits of \$$10=\$15-\$5$. Put in other words, for every \$1 we got \$2 return. Not bad!

But wait a second... We already have 5 extra workers in d! What would happen if we just move workers from d to b? Now the three firms would have \$15 profit, leaving a total profit of \$50 without investing a single penny! The profit increased by 47.05% just by adjusting scales.

Now, what role did the scale efficiency measure of DEA played here? Take a look back to Table 2 and divide b's productivity by b's scale efficiency $2/0.8=2.5$ which is c's productivity. Repeat this procedure with d: $1.93/0.77333=2.5$. So you can say that b is achieving 80% of the optimal scale efficiency, while d achieves 77.33% of it.

I have to make a remark: note that both b's and d's scale efficiency have a positive sign, but one has to increase its scale while the other has to reduce it. In this sense, the analysis is kind of blind, but you have other tests to know whether it's increasing returns to scale or decreasing returns to scale.


Consider a curve with a negative slope (such as the demand function for a normal good).
If you draw a second curve with a negative slope that does not intersect this one then it can either be in the "right-up" or the "left-down" direction from the first.
When they talk about shifts they sometimes just mean this general direction where the new curve will be. Sometimes (as with taxes) there really is a shift.
Let quantity be the vertical axis. If $D(p) = m - \frac{1}{p}$ then an increase in income will shift the demand curve up. But if $D(p) = \frac{m}{p}$ then an increase in income causes something that is not really a shift but a rescaling of the curve. You might still call this a shift up, as by moving up from the original demand curve you will reach the new one.


Induced technical change vs. directed technical change

Directed technical change is the relatively recent name for what it was was previously called Induced technical change.

Informal discussion about the endogenous direction of technical change was first discussed (but not "microfounded") by Hicks (1932) and Fellner (1961). Yet, it was Kennedy (1964) who first proposed a formal model about endogenous technical change (previous examples like Solow (1957) assumed exogenous technical change).

The more recent denomination of directed technical change was proposed by Acemoglu (2002). This author has plenty of material about the topic:

these very comprehensive slides (also here, and why not, see all the material he shares here).
The 2002 paper on the subject
Chapter 15 of his book on economic growth

You can also see more about the historical literature on induced technological change here and here.


I think what people mean when they say "convex to the origin" (or to any point $p$) is that the function is convex when looked at in a new basis, namely the basis resulting from a rotation such that the new x axis (call it x') is, up to a constant, tangent to the IC and the distance $|p-IC|$ is minimized by that point of tangency ($w$).


Image courtesy http://economicsonline.co.uk/

Consumer surplus is the sum (integral) of differences between the price each consumer would have payed and the price they got to pay. You need to find out the area of the green zone on the above graph, in the case of your model.


How to find the mean of a two-part Poisson-based hurdle model

Maybe I'm not understanding your question. Poisson distributions with expected values $\lambda$ and $\mu$ are given by
$$
\Pr(X=x) = \frac{\lambda^x e^{-\lambda}}{x!}\quad\text{ and }\quad \Pr(X=x) = \frac{\mu^x e^{-\mu}}{x!} \quad \text{ for } x=0,1,2,3,\ldots.
$$
Suppose the second one holds for $x>0$. It follows that
$$
\Pr(X=0) = 1 - \underbrace{\sum_{x=1}^\infty \frac{\mu^x e^{-\mu}}{x!}}_\text{starting at 1, not 0} = e^{-\mu}.
$$
Since the sum of the probabilities must be $1$, the probability that $X=0$ can only be $1$ minus the sum of the other probabilities, and that is $\dfrac{e^{-\mu} \mu^0}{0!} = e^{-\mu}.$

Postscript: In comments I am asked how to find the mean of this distribution.  Remember that
$$
\sum_{x=0}^\infty \frac{\mu^x}{x!} = e^\mu, \tag 1
$$
and from that we get the sum of the probabilities,
$$
\sum_{x=0}^\infty \frac{\mu^x e^{-\mu}}{x!} = 1 \tag 2
$$
We will use the sum $(2)$ in finding the mean.

The mean is
\begin{align}
\operatorname{E}(X) & = \sum_{x=0}^\infty x\Pr(X=x) \\
& = \sum_{x=0}^\infty x\cdot\frac{\mu^x e^{-\mu}}{x!} \\
& = \sum_{x=1}^\infty x\cdot\frac{\mu^x e^{-\mu}}{x!} & & \text{(starting with $1$ rather than $0$} \\
& & & \phantom{(}\text{since the $0$ term vanishes because} \\
& & & \phantom{(}\text{of the multiplication by $x$)} \\
& =\sum_{x=1}^\infty \frac{\mu^x e^{-\mu}}{(x-1)!} & & \text{(since $x/(x!) = 1/(x-1)!$ if $x\ge1$)} \\[8pt]
& = \mu \sum_{x=1}^\infty \frac{\mu^{x-1} e^{-\mu}}{(x-1)!} & & \text{(factors not depending on the index $x$)} \\
& & & \phantom{(} \text{can be pulled out)} \\
& = \mu \sum_{y=0}^\infty \frac{\mu^y e^{-\mu}}{y!} & & \text{where $y=x-1$} \\
& & & \text{so $y$ goes from $0$ to $\infty$ as $x$ goes from $1$ to $\infty$} \\
& = \mu \cdot 1 & & \text{by line $(2)$, stated above} \\
& = \mu.
\end{align}
So the mean is $\mu$.

Postscript:

Your new revision of the question isn't quite consistent with your statement that "the distribution is one Poisson distribution for observations $=0$ and another distribution for observations not equal to zero". You have
$$
\Pr(X=x) = \begin{cases} e^{-\theta} & \text{if }x=0, \\
(1-e^{-\theta}) \dfrac{\lambda^x e^{-\lambda}}{x!(1-e^{-\lambda})} & \text{if } x\ge 1. \end{cases}
$$
Thus you have a weighted average of

The distribution of a random variable constantly equal to $1$, and
The conditional distribution of a Poisson-distributed random variable given that it is $\ge 1.$

The first has expected value $1$.

The second has expected value
$$
\frac 1 {1- e^{-\lambda}} \sum_{x=1}^\infty \cdots = \frac 1 {1-e^{-\lambda}} \left( (-e^{-\lambda}\cdot0) + \sum_{x=0}^\infty \cdots \right) = \frac 1 {1-e^{-\lambda}} \cdot ( \lambda ).
$$

After that, use the fact that the expected value of the weighted average of two distributions is the weight average of their two expected values.


How could Greece-related worries drive US stocks down about 2% in a day?

Because Greece sets a precedent for Italy, Spain and Portugal (and others).
And a default of Italy (and Greece) might will cause financial difficulties in France.

If the interest rates for Italy and Portugal rise (they do now), then those countries are even more unlikely to repay their debt.
And meanwhile, somebody  has to write them off.

Greece in itselfs really is kind of unimportant (except for France).
That the debt really cannot be repaid is at this point kinda obvious to everyone. And while a write-off of 80*109 € (or part thereof) is painful for Germany (and others), it's simply not the size that can seriously jeopardize an economy of 8 times the polulation and 15 times the GDP of Greece. The risks of political fallout in Germany are another story (and the problem of Merkel).

Greece mainly is kind of an experimental laboratory of things to come (Syriza in Greece = Podemos in Spain, Italy => ? ...)

And the experiment seems to spell disaster.
Meanwhile, nobody likes to be the labrat.

As always, the stock-exchange is the place where the expectations of the future generate the prices of today (together with the panic of the moment).


Loans that have not been repaid are generally called "non-performing loans" or NPL. Each country's bank regulatory body should have statistics on this. Here's one from the World Bank.

Note that this is the ratio between outstanding NPL and total outstanding loans. The ratio could go down if the borrower agrees to continue paying back, or if the banks decide to "write off" the loans. It could also go down if there's a significant increase in loans (the denominator).

Since this statistic is a "stock" variable, the NPL to outstanding loans ratio could be huge, for example, after a financial crisis and stays high as NPLs got "stuck" in the system. It wouldn't be until these borrowers can pay back or banks write off that outstanding NPL could go down.

One way to rectify this is to look at the "vintage" of loans, essentially asking, of all loans originated in 2012, how much has turned into NPL after X months.


The field of performance marketing makes ample use of attachment bias.

For example, skiers often purchase race-level ski boots and skis which are too stiff relative to their ability, so the product is actually detrimental to performance.
In this case, self-attachment bias causes buyers to overestimate their own ability.
The ski manufacturers know this and market and price the stiffness of this equipment at a premium even though it is not objectively helpful for the vast majority of consumers.

In pharmaceuticals, attachment bias to familiar medications like Tylenol cause consumers to pay premiums for this medicine even while knowing that the generics next to it on the shelves contain exactly the same active ingredients and are often made by the same manufacturers.  Here, attachment bias to a familiar product causes allows manufacturers to pay a premium, and the manufacturers tune marketing messages to amplify the attachment bias ("Tylenol, what matters most", "Advil, the original round tablet).


Unconstrained Optimization:Why is there no "profit style" function in consumer theory?

The "profit" function for consumer choice does exist; it is the Lagrangean:
\begin{equation}
\mathcal L(x;\lambda)=u(x)+\lambda[c-g(x)]\,,
\end{equation}
where utility $u(x)$ is to be maximized subject to budget constraint $c\ge g(x)$.

Note that $\max_x \mathcal L(x;\lambda)$ is an unconstrained optimization problem just like $\max_x\pi(x)$.


Why are anti-dumping duties considered to be non-tariff barriers to trade?

It all comes to taxonomy. By definition of non-tariff barrier (Beghin 2016):

Non-tariff barriers (NTBs) refer to the wide range of policy interventions other than border tariffs that affect trade of goods, services and factors of production. Most taxonomies of NTBs include market-specific trade and domestic policies affecting trade in that market. Extended taxonomies include macroeconomic policies affecting trade. NTBs have gained importance as tariff levels have been reduced worldwide.

So by definition any policy that affects/limits trade that is not taxonomically tariff is a non-tariff barrier. Now when it comes to trade policy taxonomy anti-dumping duty is simply defined as a separate category from tariff (even though it might involve taxes at the border). Following Dinlersoz & Dogan (2010):

...  A tariff is typically designed to maximize either domestic revenue or domestic welfare.
An anti-dumping duty, on the other hand, is usually set to make up for the difference between a foreign firm's price at home and the
price of its exports, provided that the latter is lower. ...
... Tariffs are usually imposed
uniformly on all foreign firms exporting a given good to a domestic market. Anti-dumping duties, on the other hand, target a
specific foreign firm or a group of firms which are claimed to be dumping, that is, selling at prices deemed “unfair”. The goal is to
restore fair pricing, where the yardstick of fairness is usually the price prevailing in the foreign market. The tools differ also in
terms of the institutional and political process they need to go through to be initiated. Furthermore, there are cases where antidumping duties are introduced in an industry already protected by tariffs. ...

Hence even though there are some similarities these policies will have their own separate categories and thus by definition anti-dumping duty is not a tariff (this is similar like in biology fox and wolf will be different categories of mammal even though they are similar animals).

Also, one question additionally, are duties based on personal preferences, e.g. political motivation, also non-tariff barriers to trade?

Depends on what sort of duty it is and what it is conditional on. If it is tax on all foreign products of certain kind (coming from a specific country) then likely it would qualify as a tariff. If it is tax that is applied only to one foreign company due to some animosity likely it would not be tariff. However, the terms can be sometimes applied loosely.


A homebody is someone who doesn't leave their house much. The "homebody economy" refers to economic activities that do or do not do well when people tend to stay at home, which is exactly what is occurring now with the coronavirus pandemic.
Things like online shopping, media streaming services, or cook-at-home meal programs tend to do well when people are stuck at home. Things like movie theaters, restaurants, and cruise vacations require people be physically present, and do not do very well when people are stuck at home. The new "homebody economy" that's taken hold over the past few months is a result of people staying home due to the pandemic, and has had major, industry-specific impacts.


How much money is wire transferred every year worldwide?

If you're just trying to understand the volume of electronic transactions generally to an order of magnitude, it's in the quadrillions of dollars per year. According to this document from the US Treasury, SWIFT handles about \$5 trillion per day, or given about 250 business days per year, about \$1.25 quadrillion dollars a year.

Similarly, CHIPS handles about \$400 trillion per year, and Fedwire handles around \$900 trillion per year (most of both of these arise out of SWIFT messages). These transactions make up a large fraction of electronic transactions in the world, so it's safe to assume that the global total isn't much more than a few quadrillion dollars a year.

SWIFT payment message volumes are usually around 11.5 million per day, giving an average payment size of around \$45,000— though if one considers the structure of the payments system, it's evident that there's a very wide distribution of payments sizes, so this average isn't particularly representative.


What are possible solutions when promoting competition is not applicable?

There is one universal "solution" to monopolism without government intervention.

Just allow the monopoly to do its worst. It will give very strong incentives to find ways to satisfy needs of potential clients of the monopoly in ways that would diminish market force of the monopoly. Basically it's bet that human creativity will eventually tame (or even kill) the monopoly. The most radical example of such creativity is a creation of an invention that makes the whole monopoly obsolete. Another example of creativity: It's also possible that different people buy services/goods of the monopoly for different reasons, so there would be strong incentives to develop imperfect substitutes for different categories of potential clients of the monopoly. Of course there are many more other ways to outsmart the monopoly, I just gave some abstract examples of it.

Real life example: US shale oil boom as response to monopoly of OPEC


OLS

The OLS part is clear: $\sum_{i=1}^n y_{2i} (y_{1i} - \hat\gamma_{12} y_{2i} - \hat\beta_{11} x_{1i}) = 0$ and $\sum_{i=1}^n x_{1i} (y_{1i} - \hat\gamma_{12} y_{2i} - \hat\beta_{11} x_{1i}) = 0$, where $\hat\gamma_{12}$ and $\hat\beta_{11}$ are the OLS estimators.

2SLS

For 2SLS, I will assume that $(x_{1i}, x_{2i}, x_{3i})$ is used as IV. The corresponding 2SLS is identical to the IV estimator using $\hat{y}_{2i}$ and $x_{1i}$ as IV. Thus, (I think) the "normal equations" are
\begin{align}
\sum_{i=1}^n \hat{y}_{2i} (y_{1i} - \tilde\gamma_{12} y_{2i} - \tilde\beta_{11} x_{1i}) &= 0,\\
\sum_{i=1}^n x_{1i} (y_{1i} - \tilde\gamma_{12} y_{2i} - \tilde\beta_{11} x_{1i}) &= 0,
\end{align}
where $\tilde\gamma_{12}$ and $\tilde\beta_{11}$ are the 2SLS estimators using $x_{1i}, x_{2i}, x_{3i}$ as IV, and $\hat{y}_{2i}$ is the fitted value obtained by regressing $y_{2i}$ on the instruments $x_{1i}$, $x_{2i}$ and $x_{3i}$.

In the above, I understood 2SLS as the IV estimator using $(\hat{y}_{2i}, x_{1i})$ as IV. We can also understand 2SLS as the OLS estimator of $y_{1i}$ on $(\hat{y}_{2i}, x_{1i})$. In that case, the normal equations can be written as
\begin{align}
\sum_{i=1}^n \hat{y}_{2i} (y_{1i} - \tilde\gamma_{12} \hat{y}_{2i} - \tilde\beta_{11} x_{1i}) &= 0,\\
\sum_{i=1}^n x_{1i} (y_{1i} - \tilde\gamma_{12} \hat{y}_{2i} - \tilde\beta_{11} x_{1i}) &= 0,
\end{align}
where $y_{2i}$ is replaced with $\hat{y}_{2i}$. The two sets of normal equations for 2SLS (one in terms of $y_{2i}$ and the other in terms of $\hat{y}_{2i}$) are identical.

Intercept

I have noticed that the intercept is excluded in the question. If the model is $y_{1i} = \beta_{10} + \gamma_{12} y_{2i} + \beta_{11} x_{1i} + \epsilon_i$ instead, you need equations for the intercept as well.

Matrix notations

Using matrix notations, let $y$ be the $n\times 1$ matrix of the LHS variable, $X$ the $n\times 2$ matrix of $(y_{2i}, x_{1i})$ and $Z$ the $n\times 3$ matrix of $(x_{i1}, x_{i2}, x_{i3})$. Let $\beta = (\gamma_{12}, \beta_{11})'$. (I am considering the model without the intercept. For the model with the intercept, $X$ and $Z$ contain a column of ones.) Then the OLS estimator is $\hat\beta = (X'X)^{-1} X'y$, which solves the normal equations $X'(y-X\hat\beta)=\boldsymbol 0$. To see this, note that $$X'(y-X\hat\beta)=\boldsymbol 0 \\
X'y - X'X\hat\beta = \boldsymbol 0\\
X'X\hat\beta = X'y\\
\hat\beta = (X'X)^{-1}X'y$$

The 2SLS (using $Z$ as instruments) estimator is $$\tilde\beta = [X'Z(Z'Z)^{-1} Z'X]^{-1} X'Z(Z'Z)^{-1} Z'y = (\hat{X}'X)^{-1} \hat{X}'y$$ where $\hat{X} = Z(Z'Z)^{-1} Z'X$. This 2SLS solves $\hat{X}' (y-X\tilde\beta)=\boldsymbol 0$, the normal equations for this 2SLS. Again, to see this, note that
$$\hat{X}' (y-X\tilde\beta)=\boldsymbol 0\\
\hat X'y - \hat X'X\tilde\beta = \boldsymbol 0\\
\hat X'X\tilde\beta = \hat X' y\\
\tilde\beta = (\hat X'X)^{-1} \hat X'y $$
You will see that $\hat{X}$ is the $n\times 2$ matrix of $(\hat{y}_{2i}, x_{1i})$. It is also true that $\tilde\beta = (\hat{X}'\hat{X})^{-1} \hat{X}'y$, and the normal equations are also written as $\hat{X}' (y-\hat{X}\tilde\beta)=0$. The two different expressions are identical because $\hat{X}'X = \hat{X}'\hat{X}$.


What exactly is purchasing power parity (PPP)?

What you are describing is basically purchasing power. Purchasing Power Parity (PPP) claims that your purchasing power is the same across countries.

PPP states that, if say a pound of bread costs 4 US$, you can take that amount, exchange it to Chinese yuan, go to China and buy exactly the same amount of bread. (i.e. you have the same purchasing power in both countries)

3 factors play a role: the price (of bread) in the US, the price in China, and the Exchange rate. There is no subjective factor here.

Even though I don't fully understand what you mean by "purchasing labor", make sure that you compare goods, or baskets of goods, only (PPP does not say anything about wages or production costs).

And be sure to remember that PPP does not necessarily have to hold in reality.

Hope this helps!


Why some companies are reporting loss due to the new US tax law?

The lower tax rate means that they will gain less benefit when they apply their deferred tax assets against their income.

They have large amounts of these assets recorded on their balance sheets and are adjusting them to reflect the new, lesser value.


why the currency crisis must occur at time t0 when the pegged rate equals the shadow exchange rate

I assume you are referring to first generation currency crisis models, a la Krugman.

Before t0: Here there is of course no benefit to attacking and causing a currency crisis (by definition). That is, before the pegged currency is overpriced (i.e. it is not overpriced) there is no profit from attacking/short selling the currency as a drop in the value of the currency is not expected and therefore short selling is not profitable. So before it is profitable to cause a crisis there can be no crisis.

After t0: It would be profitable to cause a crisis and short sell a currency as long as it has not happened yet. However perfectly rational agents would react immediately, why wait? Like with investing in stocks, once you find a mispriced stock, you can profit anytime, but why wouldn't you profit immediately? Or lets say you find 100USD on the ground. You can profit from picking it up anytime after you find it, but why wait and risk that it could be too late later?

Hence it happens exactly at t0 as there is no reason to wait. An investor in currency would try to be faster than others to get arbitrage profits so others would try to be even faster than him and this goes on as much as possible. Its not possible to attack before (as its not possible to pick up 100USD found on the ground before its found there) so it happens as soon as it is possible, i.e. in t0.

In a second generation crisis a la Obstfeld it would take a coordination of several investors attacking at once to cause the crisis, so you don't have the effect of everyone trying to be faster than the other as they must "work together" to be successful and thus the time of the attack/crisis is not so straightforward to predict.


In what way is the labor participation rate useful for tracking unemployment?

There's a very fine point that needs to be mentioned:

The participation rate is the percentage of adult population of a country that is capable of work and wanting to work, individuals who are capable and wanting to work are the labor force.

Now when unemployment is measured, it is typically the percentage of individuals in the labor force who actually have the kind of employment (in terms of time worked) that they desire.

The importance of the participation rate is the following: when some individuals are unemployed for a long period of time, they eventually stop looking for work (the "willing to work" part is no longer valid) either because they've given up, or their skills are too outdated to be useful to the market. These individuals are dropped out of the labor force.

Assuming all else is constant, these unemployed individuals leaving the labor force means there are less unemployed individuals as a percentage of the total labor force (lower unemployment rate), but now the participation rate is lower (smaller labor force divided by same total adult population).

Therefore, unemployment numbers aren't really telling you the whole story if you assume that people dropping out of the labor force would actually want to work if they had found work, and thus unemployment rates alone may give a better picture of how workers are actually doing.

In short:

Unemployment rate = total employed/ labor force

Participation rate = labor force/adult population

As for the retired, I believe those are considered outside the labor force.

Example:

Assume a coal miner has no other job market skills than the ability to mine coal. Assume country X bans coal mining. All coal miners are left without a job but will not search for a job because they can't do any other job. The unemployment rate is unchanged, but the participation rate drops.

Very generally:

Unemployment rate rising, participation rate rising => not creating new jobs fast enough to absorb growth in labor force
Unemployment rate rising, participation rate falling => long term recession
Unemployment rate falling, participation rate falling => people are leaving the labor force
Unemployment rate falling, participation rate rising => high job creation


Is there any standard measure of economic significance?

Imagine the basic linear regression setup:

$Y_i = \alpha + X_i \beta + \epsilon, \epsilon \sim N(0,\sigma^2)$

Holding everything else fixed, increasing $\sigma$ will increase the standard deviation of  $Y$ and therefore, for a fixed relationship between $X$ and $Y$, eventually move the relationship between them to economic insignificance by the standard you present. But the relationship might still be quite economically important. It might be better to instead ask about the standard deviation effects on $\sigma_\hat{Y}$, the standard deviation of the fitted values of $Y$, to ask if the effect is large relative to the total variation that can be explained by the model.

In accounting, they use a concept of immateriality which seems quite similar to economic insignificance. That might be a helpful definition for some problems.
This quote from Signifying Nothing: Reply to Hoover and Siegler by Deirdre N. McCloskey and Stephen T. Ziliak may also be helpful:

  The sheer probability statement about one or two standard errors is
  useless, unless you have judged by what scale a number is large or
  small for the scientific or policy or personal purpose you have in
  mind. This applies to the so-called "precision" or "accuracy" of the
  estimate, too, beloved of Hoover and Siegler — the number we calculate
  as though that very convenient sampling theory did in fact apply.


testing the relationship between stock price and macroeconomic variables

Is seasonality adjustment on the data required? If one variable is seasonality adjusted, is that all variables in the models have to be seasonality adjusted as well? If all are not seasonality adjusted, will there be a problem?

Seasonal adjustment of data is required in your VAR/VECM, you can do this by employing seasonal dummies (dummies 11 if you have monthly data and 4 if you have quaterly data).

All your data has to initially be seasonally unadjusted, because the inclusion of dummies will affect all the data that you include in your model. You will end up having skewed estimates of your already adjusted variables influence on a given process.

  Would it be alright if all data are not inflation adjusted, that is nominal data? And if one variable are inflation adjusted, is that all must be so to be consistent?

Again if you want your estimates to be consistent you have to make sure all of your monetary variables are inflation adjusted. practically you may want to use real exchange rate instead of nominal exchange rate data for this.

  The base period for CPI and IPI is not the same. Is it a requirement for the base period to be the same in the model ?

No this is not necessary and your estimates will be fine. Just remember to keep this in mind when your run your model.

Hope this helps


Showing that a transformation is measure preserving

To show that transformation is measure preserving you need to show that full preimage $S^{-1}(A)$ of any set A in the Borel $\sigma$-field on [0,1) is again in the same $\sigma$-field, i.e. it is measurable, and that $Pr\{S^{-1}(A)\} = Pr\{A\}.$

As you have correctly pointed out, it will suffice to show it for any base of usual topology, namely, all open intervals.

It is evident that preimage of $(a,b)$ is $(\frac{a}{2},\frac{b}{2})\cup (\frac{1}{2}+\frac{a}{2},\frac{1}{2}+\frac{b}{2})$. It is, evidently, in Borel $\sigma$-field, as it is a union of two open intervals, and $Pr\{(\frac{a}{2},\frac{b}{2})\cup (\frac{1}{2}+\frac{a}{2},\frac{1}{2}+\frac{b}{2})\} = Pr\{(a,b)\} = b-a$

We proved that preimage of any open interval is measurable and has the same measure as the interval. As open intervals form a base of the usual topology we can extend this result to any measurable set and it completes the proof.


Why are countries in the Gulf rich but lagging in technological innovation?

Update

Resource Curse literature attempts to address the issues you are raising, take a look at here and here.

Here is an interesting example of a technology transfer case.

Original answer

They do, in fact, have increased the share of R&D spending by a huge margin in the past 5 years. Some of these countries see commerce, trade and tourism as more central to their development than other aspects of development.

The number of Arab students on government scholarship in the West has been on the rise and many western universities have opened branches in the Middle East, especially in the Gulf countries. In the past, only the rich kids of Arabs were able to go to universities in the West. By default, these kids, did not bother working as scientists or educators, they, probably, stayed in the west or became entrepreneurs.

Today, with the help of government scholarships, the ordinary folks can send their kids for education, and hopefully, these kids will bring the science to the other ordinary folks in their respective countries. Remember, technological shock is not so fast and abrupt as we assume in the models, it take decades to have a measurable effect in practice.

As younger 'kings' and 'princes' are involved in politics, there are signs of greater investment in education and science.

Here is the chart of R&D spending to GDP ratios in the Gulf countries.

In the countries, where younger generation have more say in politics, there the spending on R&D has been higher than in those where older 'kings' rule. See the chart, Qatar and UAE have had higher R&D shares, but the data is limited for these countries.


What determines the outcome of a price war, and why isn't that outcome reached instantaneously?

Answer to question

If we take your assumptions literally, Jim will decide not to enter the widget business. For suppose he did incur the cost of entry and that Mary is selling at price $p_m$. Jim can only sell to consumers if his price $p_j\leq p_m$. The best price for Jim is $p_m-\epsilon$ (where $\epsilon$ is some very small, positive amount). But this would leave Mary with no sales, so she will have an incentive to reduce her price to $p_j-\epsilon$. This is the price war you describe and it will result in both parties reducing their price to marginal cost (i.e. zero—this is known as Bertrand competition).

Since there are no frictions in your model, this will all happen very quickly. Moreover, because a price of zero implies zero profits, Jim has no incentive to incur the cost of entry in the first place, so he will instead opt to stay out of the market.

Effect of relaxing assumptions

This is obviously quite a stylised result owing to the stark assumptions of the model. But it makes a good foundation for thinking about some more realistic settings. For example:

If Mary's factory has a marginal cost $c_m$ and Jim has a patent on a new technology that implies cost $c_j<c_m$ then Jim can profitably set a price below the lowest price Mary is willing to set. Good news: if someone invents a more efficient technology they can enter the market and displace the older, less efficient technology.
Suppose that the factory has a monthly fixed maintenance cost. If Mary is a small, credit-constrained independent firm and Jim is a huge conglomerate with large reserves of cash then Jim can enter and practice predatory pricing. If he is patient enough, he can enter the market, set $p_j=0$, and wait for Mary to run out of money. She will then leave the market and Jim becomes a profitable monopoly. This kind of anti-competitive behaviour is illegal in most jurisdictions (yes, there are laws against setting too low a price!).
Suppose that Mary's widgets are blue and Jim's are pink. Consumers have an idiosyncratic preference for either blue or pink widgets. Then both firms can set a positive price and sell to the consumers who like their colour more. The more are products differentiated the more there is scope for both firms to exist profitably in the industry and Jim may find it worthwhile to enter. This is why firms talk so much about differentiation and unique selling points. There are various ways to model this in economics. Here is an example.

More formally, $p_j=p_m=0$ is the unique pure strategy Nash equilibrium

To be slightly more formal, let us check that $p_m=p_j=0$ is indeed a (Nash) equilibrium of the subgame in which both firms simultaneously set $p\in\mathbb{R}$ with the lowest priced firm capturing the entire (finite) demand. Neither party can profit by deviating to $p<0$ as this yields negative profits. A deviation to any $p>0$ results in the rival firm having a lower price so demand (and profits) are zero---again not profitable. Thus, there is no profitable deviation and $p_i=p_m=0$ is an equilibrium.

Is there another pure strategy equilibrium with some $p>0$? The answer is no. Consider the three possibilities and observe how a profitable deviation can be constructed for each:

$p_m>p_j$. In this case $j$ could increase his price to some $p_j'\in(p_j,p_m)$ without losing any demand. Such a $p_j'$ exists by the connectedness of the real line.
$p_m=p_j=p$. In this case at least one of the firms must be capturing less than all of the consumers. But by deviating to $p'=p-\epsilon$ it can capture all consumers. If we write $D$ for demand before the deviation and $D'>D$ for demand afterwards then the change in profits is $D'(p-\epsilon)-Dp$. By the connectedness of the real line, there exists an $\epsilon$ sufficiently small that this is positive.
$p_m<p_j$. This case is symmetric to $p_m>p_j$.

Thus, the only pure strategy equilibrium of this one-shot game is $p_j=p_m=0$.

If we repeat the game then we can sustain other (collusive) equilibria

What if we repeat the game infinitely many times? Suppose the two firms have an implicit understanding (an explicit agreement would be illegal) that they will both set $p=p^*$ for some $p^*>0$. Moreover, it is understood that if one firm deviates from this behaviour today, then both firms will revert to playing the static equilibrium ($p_j=p_m=0$) forever. Firms discount the future at rate $\delta$. For simplicity, suppose that demand is constant: $D(p)=1$ (not crucial).

If a firm plays in accordance with this understanding (and expects its rival to do likewise) then its profit is

$$\sum_{t=0}^\infty\frac{1}{2}\delta^t p^*=\frac{p^*}{2(1-\delta)}.$$

If a firm cheats and sets $p^*-\epsilon$ ($\epsilon$ small) then it captures the whole demand today, but expects to be punished forever thereafter, so profit is $p^*$. Thus, both firms wish to comply with their implicit understanding if
$$\frac{p^*}{2(1-\delta)}>p^*\iff \delta>\frac{1}{2}$$
(i.e. if they are sufficiently patient).

So, if we repeat the game and firms are very patient, we can sustain any price in equilibrium. But note that these equilibria require not only that firms are patient, but also that a) there is no threat of future entry that could destabilise the equilibrium; b) firms are able to coordinate on a $p^*$ without ever making an (illegal) explicit agreement; and c) firms are able to constantly and accurately monitor rivals behaviour in order to detect cheating.


Confusion about multiple information Sets

You could represent the game in extensive form like this:

The dashed lines enclose player 2's information set. This encompasses all of player 2's nodes because player 2 observes neither nature's nor player 1's choice. Player 1's information sets are the two singleton nodes because player 1 knows which branch nature has chosen.


Dynamic programming problem with dimension over 1000

Numerical Dynamic programming is always an approximation when any of the variables are continuous....

Depending on the structure of the problem, some people claim that you can apply linear programming techniques to approximate a solution. These guys MIT paper, claim that they have experiments that show that their linear reprogramming approximation is good enough.


How to find Marshallian demand of $u(x,y,z)=x+y^2+2z^2$?

Just compare the utilities of these three bundles: $\left(\frac{M}{p_X}, 0, 0\right)$, $\left(0, \frac{M}{p_Y}, 0\right)$ and $\left(0,0, \frac{M}{p_Z}\right)$ and consume the one that yields highest utility.


Oligopoly is best understood as occuring when firms behave strategically: when firms decide how much to produce, under oligopoly they will take into account the optimal responses of other firms.

If you have a long tail of small firms, beyond a certain size these firms won't have much influence at all on prices and will act as 'price-takers'.

So either the firms are large enough to have an impact - in which case everyone is an oligopolist, or they are small enough not to matter much - in which case the market is still oligopolistic because only the larger firms matter. Note competition authorities regularly block mergers in markets with 2-3 large firms and a bunch of very small ones: E.g. the European Commission's DG Comp has just blocked Siemens/Alstom given the large market shares of these firms, despite the fact that they face competition from a few smaller suppliers.

Only if all firms were small / had to act as price takers would you have something (that might approximate) perfect competition


It is just mathematical assumptions in calculus that makes the mathematical analysis has real economic meaning.  Not of great importance unless you want to look deeply into the derivation of results.

You may read this paper: https://web.stanford.edu/~jdlevin/Econ%20202/Producer%20Theory.pdf

  "Throughout our analysis, we will make the innocent technical assumptions that
  Y is non-empty (so as to have something to study!) and closed (to make it more
  likely that optimal production plans exist)."

Hope this helps.


Depending of the accuracy you're looking at (stylized facts or estimated parameters), private database like http://go.euromonitor.com/passport.html may help you to support your theory. Some institutions grant access to them. Some industries (e.g. tobacco) have their info widespread.

A (non-exhaustive) list: https://www.insead.edu/library/databases


question regarding econometric testing of the natural rate hypothesis

Equations $11$) and $12$) are not derived, they are assumed.

Specifically, with eq. $9$) Lucas defines/assumes that the value-expected for the $t+1$ price level is the conditional expectation based on $x_t, x_{t+1},\eta_t$ (plus $\eta_t$ on its own). This is imposing the Rational Expectations Hypothesis.

He then says "we will seek linear solutions..." which is translated "assume that the functional form of $E(P_{t+1}| x_{t}, x_{t-1}, \eta_{t})$ is linear in its arguments, and then determine (or estimate) the coefficients". Certainly, this is in all likelihood an approximation to the true functional form of this conditional expectation.

So assuming that

$$E(P_{t+1}| x_{t}, x_{t-1}, \eta_{t}) = \pi_{4} x_{t} +\pi_{5} x_{t-1} + \beta\eta_{t}$$

and inserting this into $9$) we get $12$) with the mapping $\pi_{6} = 1+\beta$.

Inserting $9$) in $10$) we get

$$(1+a) P_{t} - a [\pi_{4} x_{t} + \pi_{5} x_{t-1} + \pi_{6} \eta_{t}] = x_{t}$$

$$\implies P_{t} = \frac{1+a\pi_{4}}{1+a}x_t +\frac{a\pi_{5}}{1+a}x_{t-1}+\frac{a\pi_{6}}{1+a}\eta_{t}$$

which is $11$), after compacting and mapping the above coefficients to the $\pi_1, \pi_2, \pi_3$ ones.


That quote from Wikipedia page is not correct. Actually both OECD and also commonly in economic research we define potential GDP/output as:

  Potential gross domestic product (GDP) is defined in the OECD’s Economic Outlook publication as the level of output that an economy can produce at a constant inflation rate. Although an economy can temporarily produce more than its potential level of output, that comes at the cost of rising inflation. Potential output depends on the capital stock, the potential labour force (which depends on demographic factors and on participation rates), the non-accelerating inflation rate of unemployment (NAIRU), and the level of labour efficiency.

Also note what’s written in Wikipedia was probably misunderstanding of the term full employment in economics, as potential GDP is often defined as a GDP at full employment of all resources. In economics full employment does not mean that unemployment rate is necessary 0, or that all resources are used.

In economics full employment is an economic situation in which all available factors of production (labor, capital land etc) are being used in the most efficient way possible.

Unemployment at full employment will still be positive number because even in case when all resources are used in the most efficient way possible some people won’t have work, some capital will just lay unused and some land will just lay unexploited.

Also, always when you look at data you should check the data’s own metadata/explanation for what’s measured not Wikipedia otherwise you won’t really understand what the statistics is showing you.


how to explain the term “Compensated law of demand” in a plain way?

The compensated law of demand is a tool we use to analyze the decomposition of substitution and income effects. We take a price change which changes the relative price ratio (the slope of our budget constraint in a simple two good example) which leads to a new tangency point on a new 'rotated' budget line. For simplicity, if we consider the tangency condition:
$$
|MRS| = \frac{MU_1}{MU_2} = \frac{p_1}{p_2}
$$
Then as $p_1$ rises, for example, the value of our MRS defined as the slope of our indifference curve would rise at the optimal point. So our new tangency would be higher up on this new lower indifference curve. What we would then do is ask the hypothetical: Under the new prices, how much income would leave the consumer just as well as off before the price changes? This is the idea of compensated demand. What would we do in practice is essentially shift the budget line up (or down) until it is tangent to the original indifference curve. This gives us the pure substitution effect abstracting away from the change in purchasing power.

Since you are reading from an advanced microeconomics textbook I imagine you are familiar with Marshallian and Hicksian demands, these concepts are very closely related to how we compute this hypothetical budget constraint. Sadly that is the simplest way I can think of explaining it as it is an 'advanced' concept. In practice, we don't usually use the tangency condition to arrive at these results but it helps in simplifying the concept.


How is productivity measured in public sector?

The BLS has a manual by Katharine G. Abraham (1998) on the subject.
https://www.bls.gov/lpc/iprprody.pdf
page 17 and page 21;  covers outputs and inputs respectively.

Measuring output:

“The specification and measurement of output is the single most troublesome problem
in computing productivity. It is a more serious problem for government service
than for private sector organizations. Some of the problems are endemic to a specific
government activity; others are more general in nature.”
…
“For most public sector output measures, physical quantities are used to calculate the
outputs.”

Measuring inputs:

“Two labor measures that are commonly used to measure private sector productivity are the number of hours and the number of persons. These two measures impart very different information and for this reason both are calculated. The preferred measure is the number of hours because this measure reflects the time worked to produce the outputs.

Hours are the preferred measure for government too, but governmental units do not usually collect these data. Instead, most governments collect and use the number of full-time-equivalent employees.”

The OECD has put out an article on the subject but at the country of Finland
http://www.oecd.org/sdd/na/2666079.pdf


Risk neutrality in single-agent choice problem under uncertainity

No, risk neutrality is not assumed. Risk preferences are given by the concavity of  $u_i$ which is arbitrary in this setup.


Effect of default risk on the interest rate of bonds

To make sense of your question, your $U$ should be
$$
U = u(C_1) + \beta (1-p)E_1 [ u(C_2) ] + \beta p E_1[u(Y_2 - T_2)].
$$
(This definition of $U$ assumes that the default event is independent of other sources of uncertainty in the economy.)

If the bond defaults, saving $S_1$ from the first period is wiped out.
If there is uncertainty in period-2 production/endowment/transfer/etc, then period-2 consumption is still random in the event of default.

Assuming that agent only chooses consumption $(C_1)$ (therefore $C_2$ in the event of no default), then real interest rate $r$ is determined by
$$
1 = E_1[\beta \frac{u'(C_2)}{u'(C_1)} (1-p) (1 + r)].
$$
This is just an extension of the standard Euler equation for interest rate
$$
1 = E_1[\beta \frac{u'(C_2)}{u'(C_1)} (1 + r)]
$$
when $p = 0$.

If equilibrium consumption remains the same (e.g. equals exogenous endowment), then higher probability of default $p$ leads to higher interest rate $r$, as you would expect. High probability of default incentivizes agent to consume today instead of saving for tomorrow. To keep him from borrowing and clear the bond market, equilibrium interest rate must be higher.
Equivalently, higher default risk means agent demands higher liquidity premium on bonds.

If $p$ and $r$ are small, $(1-p)(1+r) \approx e^{r-p}$, which implies that $r$ moves one-to-one with respect to $p$.


Effect of change in discount rate on consumption -- (Samuelson, 1958)

Consider the life-time utility maximization problem of the individual. At this stage of the paper, while the utility function "has the usual regular indifference-curve
concavities" as Samuelson writes, nothing has been assumed about the form of the intertemporal utility function, so the problem is stated as
$$\max_{C_1,C_2,C_3}U(C_1,C_2,C_3),\;\;\; s.t.\;\;\;  C_3 = \frac {1+ R_t}{R_tR_{t+1}} - \frac{C_1}{R_tR_{t+1}} - \frac{C_2}{R_{t+1}} \tag{1}$$
where the constraint comes from re-arranging the intertemporal budget expression (eq. 1 of the paper), and interest rates/discount factors are treated as exogenous. Since this is an equality constraint (implying no bequest motive and no debts at the end of life), we can incorporate it into the utility function and solve with respect to $C_1$ and $C_2$ only. The first-order conditions are then
$$\frac{d U}{d C_k} = \frac{\partial U}{\partial C_k} + \frac{\partial U}{\partial C_3}\cdot \frac{\partial C_3}{\partial C_k} =0, \;\;\; k=1,2 \tag{2} $$
Because consumption is a good, marginal utility is positive. This implies that at the solution also,
$$\frac{\partial C_3}{\partial C_k} < 0,\;\;\; k=1,2 \tag{3}$$
By the chain rule we can write
$$\frac{\partial C_3}{\partial R_{t+1}} = \frac{\partial C_3}{\partial C_k} \cdot \frac{\partial C_k}{\partial R_{t+1}} , \;\;\; k=1,2 \tag{4}$$.
Due to $(3)$ we obtain the fundamental result that, at the solution, the $t+1$ discount factor (or equivalently, the $t+1$ interest rate) will tend to affect $C_3$ in the opposite direction than $C_1, C_2$.
This is crucial, because it says that changes in the exogenous interest rates will  lead to solutions where we will observe higher (lower) $C_1,C_2$ and lower (higher) $C_3$.
Samuelson asserts that  $\partial C_3/\partial R_{t+1} >0$, which is equivalent to assert that
$$\frac{\partial C_k}{\partial R_{t+1}} < 0 , \;\;\; k=1,2$$
or equivalently, that
$$\frac{\partial C_3}{\partial i_{t+1}} < 0,\qquad \frac{\partial C_k}{\partial i_{t+1}} > 0 , \;\;\; k=1,2$$
Is this the case?
For given interest rates we obtain a solution for the consumption vector that implies certain savings and a certain interest income. If we contemplate an increase in $i_{t+1}$, total income, $I$, will tend to increase, $\partial I/\partial i_{t+1}>0$ . Then, because consumption is assumed to be a "superior" good, $C_k,\; k=1,2$ are certainly normal goods, $\partial C_k/\partial I >0$, and so
$$\frac{\partial C_k}{\partial i_{t+1}} = \frac{\partial C_k}{\partial I}\cdot \frac{\partial I}{\partial i_{t+1}} >0$$
Then for $C_3$ the same relation implies that $\partial C_3/\partial I <0$, at the solution. This does not make $C_3$ an inferior good, when viewed on its own, but it is the result of it being determined residually, as the last element in a fixed-horizon intertemporal problem.


Is the Nash product really maximised ex post?

The Nash bargaining solution DOES maximize the Nash product. You have to separate the playing of the game from the bargaining problem. If the players negotiate a binding agreement they will realize that their maximal total payoff from playing the game is $3$. This can be achieved by playing $(T,L)$ or $(T,C)$, or any mixture of those two profiles.

The bargaining solution just asks: Given a total payoff of $3$, how will the players divide this payoff among them, given a threat point of $(0,0)$? So the efficient frontier is (a part of) the line $u_1+u_2=3$, and by symmetry (or by maximizing the Nash product $u_1(3-u_1)$) you find $u_1=u_2=3/2$. The only remaining question is how to play the game (under a binding agreement) in such a way that these are the expected payoffs. But you already solved this part.


List of production functions that satisfy the Inada conditions

The Wikipedia statement should come with a citation or proof, as it is not entirely accurate: Cobb-Douglas functions with increasing returns to scale do not necessarily satisfy the Inada conditions, e.g., $f(x_1,x_2) = x_1^2x_2^2$ does not.

There are many non Cobb-Douglas functions that satisfy the Inada conditions, albeit they may not have nice compact formulas. E.g.,
$$
f(x) = \left\{\begin{array}{ll}
x^{1/2} & \mbox{ if } x \leq 1 \\
2x^{1/4} - 1 & \mbox{ if } 1 < x
\end{array}\right.
$$
This satisfies the conditions:

$f(0) = 0$
The Hessian is negative semidefinite as $f$ is strictly concave.
$$\lim_{x \to 0}\frac{\text{d} f(x)}{\text{d} x} = \infty$$
$$\lim_{x \to \infty} \frac{\text{d} f(x)}{\text{d} x} = 0$$

You can easily manufacture similar functions. I just combined two strictly concave functions at a point of tangency and made sure that the piece going to zero would cross the origin.

I am not sure, but it seems to me that if you have an Inada satisfying function $h(\mathbf{x})$ then the function $f(h(\mathbf{x}))$ will also satisfy all the conditions.


Ex post evaluation of conservation projects - accounting for time in cost calculations

If for project $P_i$ the cost calculated was based on prices that held during the year that the project was executed, then certainly you should bring all costs on the same temporal base, in order to compare them.

This can be done in different ways, namely, the common base may be the year of the oldest project, or the year of the most current project, or something in between. Choose the one that will make more sense to the recipients of your study: this usually means bringing "past costs to the present", because this facilitates comprehension (people can better understand current costs and to what they amount in term of resources), and also, makes the adjusted figures more relevant for future projects.

In order to bring past costs to the present, you can use a price index like the Consumer Price Index, and the implied inflation. In general, costs in current dollars is just

$$\tilde C_{i,t|t+k} = C_{i,t}\cdot \prod_{j=1}^k(1+\pi_{t+j})$$

where $t$ is the actual year the project was undertaken, and $t+k$ is Now. $\pi_{t+j}$ is the price inflation between year $t+j$ and year $t+j-1$. Data on yearly inflation are usually available and easily accessible.

Another possible issue to consider is whether all these projects were undertaken in the same economy/country. If they are not, then you also face the task of incorporating in your calculations "purchasing-power-parity" considerations.


Isn't money moving towards central bank in the long run?

A minor point, but central banks also pay for operating expenses. So they have to spend some of the surplus on salaries, building maintenance, etc. And they also get more surplus by printing physical money.

However in most countries it is included within the charter of the central bank that any surplus left over at the end of some period is to be transferred to the government treasury. The government is then free to spend this surplus.

This is one of the reasons for the seperation of the central bank and the government. If a government had control of the bank it could be tempted to instruct the bank to create a surplus and then use said surplus to finance government expenditures. This is deemed undesirable as it inreases inflation.

Should a central bank somehow end up with a deficit rather then a surplus it falls to the treasury to help them out (if needed).

You can read about details in the FED's charter:
https://www.federalreserve.gov/aboutthefed/section7.htm

The ECB also has a paper with some info on this. Point 2.1.:

  Overview of profit distribution and loss coverage
  In general terms, a central bank’s annual net profits (recognised positive and
  negative income, less operating costs) can either be passed on to shareholders
  (government and/or private sector) in the form of a dividend, or added to its financial
  reserves. In cases where a central bank has either both public and private ownership
  or only private ownership, the amount of profit that can be distributed to private
  shareholders is typically subject to a cap (ranging from 6% to 12% of the nominal
  value of the paid-up capital), while the remaining distributable profit, net of any
  transfers to equity (capital and reserves) goes to the government.


Is it reasonable, in panel data, to use past values of explanatory variables as instruments to deal with simultaneous causality?

I suggest looking into System GMM. It is a panel regression method that uses past values as instruments and is exactly what you're looking for, I think. It has a few restrictions that much be met, the most important requires you to have more cross-sections than time periods. Without seeing your model and other details that is all I can provide for help. I hope this works for you.


Why regulate the financial world rather than take over some of the services?

Governments do offer financial services. It varies from country to country, and changes over time. The political drift has been to reduce the role of government since the 1980s, so this role has generally shrunken.

For example, some countries had postal banking - citizens had basic banking functions offered at post offices.

I will just use some Canadian examples; other countries have similar programmes.

The Canadian government has a Business Development Bank, which helps new firms develop (website). It is taking on risks that commercial banks would be reluctant to take.

The Canada Mortgage and Housing Corporation (CMHC - website) has a programme of mortgage insurance that is compulsory for all high loan-to-value mortgages in Canada (“subprime”). The loans have to meet minimum risk standards, but the loan guarantee means that the private banks are not exposed to the risk of the mortgages. The Canadian housing market slowed down during the Financial Crisis, but fared better than the United States. The fact that the Canadian banks were insulated from the credit risk of risky mortgages was certainly a factor.

Returning to your question - why not expand the programme, and have the government completely take over the mortgage lending role? One obvious advantage for the government is that it is not forced to justify lending decisions to citizens - that is being done in the private sector. People could easily be concerned about favouritism in lending decisions if the government controlled all mortgage lending. For example, if the child of a politician gets a big mortgage when other people are being refused, it could easily turn into a political scandal. On top of this, there is the ongoing political argument between government control versus free markets. The objective of this website is to avoid opinion-based answers, so I will not dwell on that political debate.


Why does the US government not invest in the stock market?

The US government could do that but policies prohibit this, since buying a specific company's stock benefits the shareholders of that company.
This would be a huge conflict of interest.

Another answer by Paul Mulwitz,  an online commentator, says “If the government bought ownership in private companies they would no longer be private companies.  “

A good example would be if the US government invested in Apple computers.  Would it act against the company in case of a violation?
There would be a conflict of interest.

Side note, What does the government invest in?

According to the CBO (https://www.cbo.gov/publication/55375) in 2018, the federal government spent $492 billion on investment—for physical capital, education and training, and research and development—which represented 12 percent of federal spending and 2 percent of GDP.


How can I build a fixed point theorem argument in pure strategies?

There is no apriori reason for the Best Response map to be a contraction in general. Here's a simple example (since Battle of Sexes has been my go-to for the past few days):
$$
\begin{array}{c|lcr}
\text{Player1/Player 2$\rightarrow$} & \text{F} & \text{T} \\
\hline
\text{F} & 3,1 & 0,0 \\
\text{T} & 0,0 & 1,3 \\
\end{array}
$$
Denote the strategy of player 1: $x = Pr(T)$ and that of player 2 by: $y=Pr(T)$. The best response of player 1  is a map $BR_1:[0,1]\rightarrow[0,1]$ defined by:
$
BR_1(y) = \begin{cases}
0 & \text{ if } y<\frac{3}{4}\\
[0,1] & \text{ if } y = \frac{3}{4}\\
1 & \text{ if } y>\frac{3}{4}\\
\end{cases}
$
Similarly, the best response of player 2 is a map $BR_2:[0,1]\rightarrow[0,1]$ defined by:
$
BR_2(x) = \begin{cases}
0 & \text{ if } x<\frac{1}{4}\\
[0,1] & \text{ if } x = \frac{1}{4}\\
1 & \text{ if } x>\frac{1}{4}\\
\end{cases}
$
Define the Best Response profile $BR:[0,1]^2\Rightarrow[0,1]^2$ as $BR(y,x) \equiv \big(BR_1(y), BR_2(x)\big)$. A Nash equilibrium of this game is a fixed point of $BR$.
This, however, need not be a contraction map. Take two points on the domain: $a = (0.8,0.5)$ and $b = (0.5,0.2)$. Its easy to see that $BR(0.8,0.5) =(1,1)$ and $BR(0.5,0.2)=(0,0)$. Thus the Euclidean distance $d\big(BR(a),BR(b)\big) = \sqrt{2}$ whereas $d\big(a,b\big) = 0.3\sqrt{2}$. It is clear that $\not\exists$ $k>0$ such that $d\big(BR(a), BR(b)\big)\leq k d\big(a,b\big)$.


Is evolutionary game theory a useful modelling tool?

You're looking at it from a particularly biological point of view. I know nothing of side-blotched lizards, but I can say that evolutionary game theory is often used for equilibrium selection. In games with more than one equilibrium (whether it be Nash or otherwise), evolutionary game theory can allow us to select a particular equilibrium that is a better prediction than others in that a population is, in some sense, more likely to reach it and stay there than it is to reach and stay at other equilibria. Have a look at Population Games and Evolutionary Dynamics by William H. Sandholm if you want to see more of how economists use evolutionary game theory. What your question gets at is really more about creating a model with evolution to describe an ecosystem rather than evolutionary game theory, though the two are not unrelated.


I think in this case the typical economist's answer prevails: it depends. Taxes are distortive in so far as they prevent valuable economic activity from taking place.

Income taxes make working less attractive and therefore reduce the supply of labour. In an economy that is already labour constrained the distortion of income taxes can thus be significant. If, in contrast, land is a valuable but scarce resource land taxes will be distortive as well.

You're right that VAT is a consumption tax, and thrifty people won't mind such taxes typically. It is not  true that non-thrifty people won't mind income taxes because if we tax away their income they have nothing to spend! Taxes on interest or capital on the other hand won't be a much of a direct problem for the non-thrifty consumers.

Taxes on investment are distortive because they prevent valuable investments from taking place. This is the main argument put forward for lowering the taxes for the super rich in the US. As their income comes from investment lowering their taxes will spur investment. In how far this actually happens and whether the money is actually invested in the US is a different matter.

As a final point: some taxes are actually improving things by preventing economic activity. Environmental taxes like a tax on CO2 should in principle curb CO2 emissions and in that way correct distortions rather than introduce them.


Is a home (primary residence) purchase a form of investment in macroeconomics?

Home construction is a form of investment, home purchase is not. There's an important macro difference between creating the capital asset (construction) and transferring it (purchase).

Regarding whether a newly built home occupied by its owner counts as investment— yes, because it is a capital asset that produces a stream of housing services. There's more detail on how this fits into GDP in this answer.


How to get this Production Function in Growth Rates

Given
$$Y = Af(K,L,Z)$$
it follows that
$$\dot Y = \dot A f + A\frac{\partial f}{\partial K}\dot K + A \frac{\partial f}{\partial L} \dot L + A \frac{\partial f}{\partial Z} \dot Z,$$
where dotted expression are time derivatives and dividing with $Y$ it follows
$$\frac{\dot Y}{Y} = \frac{\dot A}{A}  + \left[A\frac{\partial f}{\partial K}\right]\frac{\dot K}{Y}+ \left[A \frac{\partial f}{\partial L} \frac{1}{Y}\right] \dot L + \left[A \frac{\partial f}{\partial Z} \frac{1}{Y}\right]\dot Z ,$$
and dividing and multiplying with $L$ and $Z$ in relevant summands it follows that
$$\frac{\dot Y}{Y} = \frac{\dot A}{A}  + \left[A\frac{\partial f}{\partial K}\right]\frac{\dot K}{Y}+ \left[A \frac{\partial f}{\partial L} \frac{L}{Y}\right] \frac{\dot L}{L} + \left[A \frac{\partial f}{\partial Z} \frac{Z}{Y}\right]\frac{\dot Z}{Z} ,$$
and then terms in square brackets are defined as $\alpha_1,\alpha_2$ and $\alpha_3$ being respectively

The marginal productivity of capital $\alpha_1 = A \frac{\partial f}{\partial K}$
The elasticity of output with respect to labor $\alpha_2 = A \frac{\partial f}{\partial L} \frac{L}{Y}$ and
The elasticity of output with respect to other factors  $\alpha_3 = A \frac{\partial f}{\partial Z} \frac{Z}{Y}$

In the text the authors write $\partial Y/\partial X$ with $X\in \{K,L,Z\}$ instead of
$\partial f/\partial X$ which must be a typo. For example it is hard to convince yourself that with $Y = A f(K,L,Z)$ the marginal productivity of capital is $A \frac{\partial Y}{\partial K}$ and not instead $\frac{\partial Y}{\partial K} = A \frac{\partial f}{\partial K}$.
Obviously the typo is of no consequence since the terms are parameterized for the purpose of estimation.


Do Keyensians believe that it's wrong for households to aggresively save money?

The issue is more complex than your friend suggest and he is not completely right but at the same time there is a kernel of truth to what he is claiming. There are differences in the effect of savings short and medium and long run. However, to fully explain this I will have to use some math to ground the reasoning and make everything consistent.

Consider simple closed economy. By definition in any closed economy the total output/income $Y$ must be equal to consumption $C$, investment $I$ and government spending $G$. This is just accounting and will always hold so we have:

$$Y = C+I+G$$

Now we know from basic economics that consumption must be some positive function of disposable income $C=C(Y_d)$ with $C'>0$. This is just common sense if you have more money avaiable you will consume more goods and services in general. This is consequence of the basic non-satiation assumption - more is always better on which most economics builds on. To make things simple let us assume that the relationship is linear so we have:

$$C=c_0 + c_1Y_d$$

where $c_0$ is some basic consumption that you have to undergo irrespective of income to just stay alive and $c_1$ the marginal propensity to consume, i.e. the share of income you use for consumption as opposed to saving. Now by definition disposable income is the income after taxes $Y_d = Y-T$. So in fact our consumption function turns into:

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

Second we will hold investment constant which reasonably holds in the short run (but not necessary in the medium long run) as most investments are not very liquid so we have $I= \bar{I}$. Government spending will be treated as completely exogenous (determined outside this model) so it stays $G$.

Now to solve this model we can just substitute consumption function $C$ into the first equation which gives us:

$$Y = c_0 + c_1 (Y-T)+\bar{I}+G$$

Now solving for $Y$ using basic algebra we get the following:

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

Now this last equation is all you need to understand the paradox of thrift. First consider the second part of the equation $\left(c_0  +\bar{I}+G-c_1T\right)$. This part is called autonomous spending because it does not depend on income as $c_0$ is basic consumption of necessities  $\bar{I}$ is fixed investment and $G$ and $T$ is government spending and taxation which is exogenously given so in this model it wont change.

So now you must be wondering where is the effect of saving on income? Well it hides in the first term:

$$\frac{1}{1-c_1} $$

recall $c_1$ is the share of income you spend on consumption - since you can only consume or save $(1-c_1)$ is the share of income saved.

Now its trivial to see that as $(1-c_1)$ increases the $Y$ must fall holding everything else constant. So your friend is definitely right that in the short-run saving more - meaning saving larger proportion of your income not just saving more in dollar terms - leads to lower income.

However, here we set investment constant, which is reasonable in the short run, but not in the long run. Just as a matter of accounting identity the investment must be equal private $S$ and public $T-G$ savings so we have:

$$I=S +T-G$$

So increasing saving also increases income. Furthermore, saving is also necessary for economic growth. The reason for this is that economic output depends on factors of production like capital and labor. For simplicity lets focus just on capital. We know that the more capital economy has the more it can produce so we know that $Y = F(K)$ with $Y'_k>0$. So the national income increases with amount of capital avaiable.

Now how do you get more capital? By saving! Again just by pure accounting identity the stock of capital tomorrow $K_{t+1}$ must be equal to capital today net of depreciation plus investment:

$$K_{t+1} = (1-\delta)K_t + I_t$$

We already know $I=S+T-G$ lets set taxes equal to gov. spending to simplify we get $I=S$. Now going back to the original equation on consumption since $c_1$ is marginal propensity to consume $1-c_1$ must be marginal propensity to save so we can replace $I$ by savings which are $S=(1-c_1)Y$ to get

$$K_{t+1} = (1-\delta)K_t + (1-c_1)Y$$

So in the long run you get more capital accumulation and economic growth when propensity to consume $c_1$ is small or consequently when share of income saved $1-c_1$ is large. So your friend is definitely not correct if he claims that being thrifty is always bad for the economy or that (New) Keynesians claim that.

PS: This was not part of your question but note that while some pundits still use words like Neoclassical and Keynesian as synonyms for free market pro saving economists and pro more government regulation spending economists respectively, this distinctions no longer hold. The current mainstream in economics is  Neoclassical synthesis which is combination of Neoclassical micro models and Keynesian macro models as well as some other ideas. So nowadays words like Keynesian or Neoclassical dont carry the same meaning. For example, Mankiw is a New Keynesian but he is small government, pro saving, conservative/libertarian while Stiglitz work is neoclassical and he is pro big government, pro spending, liberal. The reason for this is that both the neoclassical and keynesian models are highly nuanced and the pro-spending vs pro-saving debate depends heavily not just on science but also on personal values, the ways how a person discounts future benefits of economic growth and all this discussion gets more complex once you include inequality and philosophical discussions around that.


Is a shrinking money supply environment painful for an economy?

The closes answer will be country with extremely low inflation such as Nordic countries, Kuwait, Brunei. But they are not a good example, as the money supply are "balance" with their heavy surplus from natural resources exploitation.

On the other hand, there is plenty of example for "Dutch disease" : excessive foreign exchange that lead to massive future spending with too little on future transformation.

IMHO, there is no "real answer" for this, unless you like in a world with 100% economist. Due to human nature to exploit system, all economy ideology deems broken once it is brought up. Macro manage economy with money supply are one of those.  I am not sure what you try to achieve. Inflation is not the silver bullet for "growth". Nor the so call "ripples effect" from the rich really effective.

(update)
Keynesianism ciritised European Union below 2% inflation rates lead so "slow growth". Some even suggest country like Greek detach from the Euro and go all out to print money to "stimulate growth". That is a terrible idea of attempt to use monetary policies to solve economy structural issues.

Is shrinking money supply environment painful for an economy?.  Even with a clear scope, there is no clear answer.

Some economist argue that "deflation" will cause people "not to spend", to support "inflation" is good because it will force people to spend. This is counter-intuitive against human behavior : human will acquire stuff that they to fulfilled their desires and needs.

There is no concrete example of "retract economy" under stagnation or deflation.  Due to diversity of human behaviour, some people will hoard the money and go to graveyard without spending all of it. Some will go grab new TV, subscribe to streaming services, etc.


One proxy for cardinal utility would be people's willingness to pay (WTP) for a particular product. One can estimate WTP using consumption data (e.g. from Amazon/eBay purchases), or elicit WTP using economic experiments.

Another (better?) way to estimate cardinal utility is through methods used in neuroeconomics. For example, one can monitor an individual's dopamine level/active brain regions while she makes decisions. To the extent that the neuro-transmitter/brain activities are associated with bodily (physiological) pleasure, which by the way is what Jeremy Bentham had in mind when he coined the term "util", we may assert that particular activities/decisions generate a measurable level of utility that is cardinal in the sense that it not only orders choice alternatives from least to most preferred, but also signifies the intensity of preference over each alternative.


Corner solution of the maximization problem

Here is the formulation of the problem :
\begin{eqnarray*} \max_{c, h, l} \  & \ln (c - \gamma) + \beta l + \theta h \\ \text{s.t.} &  l + h = 1, \\ & c \leq \omega h + \rho, \\ \text{and} & l, h \geq 0, c \geq \gamma \end{eqnarray*}

Substituting $l = 1 - h$, we can rewrite the above problem as :
\begin{eqnarray*} \max_{c, h} \  & \ln (c - \gamma) + \beta + (\theta -\beta)h \\ \text{s.t.} & 0 \leq h \leq 1 \\ \text{and} & \gamma \leq c \leq \omega h + \rho\end{eqnarray*}

Since utility is increasing in $c$, $c = \omega h + \rho$ will hold in optimum. So we can further reduce the problem to :

\begin{eqnarray*} \max_{h} \  & \ln (\omega h + \rho - \gamma) + \beta + (\theta -\beta)h \\ \text{s.t.} & 0 \leq h \leq 1 \\ \text{and} & \gamma \leq \omega h + \rho\end{eqnarray*}

Please note that we'll assume $\omega + \rho \geq \gamma$. This is because when $\omega + \rho < \gamma$, there is no feasible solution. In other words, there does not exist any $h$ satisfying the constraints.

To solve this problem, we'll consider two cases :

Case 1: $\rho \geq \gamma$

In this case problem can be written as :

\begin{eqnarray*} \max_{h} \  & \ln (\omega h + \rho - \gamma) + \beta + (\theta -\beta)h \\ \text{s.t.} & 0 \leq h \leq 1 \end{eqnarray*}

Derivative of the objective with respect to $h$ is
$\frac{\omega}{\omega h + \rho - \gamma} + (\theta -\beta)$ which yields the following solution :

\begin{eqnarray*} h = \begin{cases} 1 & \text{if } \frac{\omega}{\omega + \rho - \gamma} + (\theta -\beta) \geq 0  \\  0 & \text{if } \frac{\omega}{\rho - \gamma} + (\theta -\beta) \leq 0  \\ \frac{1}{\beta -\theta} - \frac{\rho - \gamma}{\omega} & \text{otherwise} \end{cases} \end{eqnarray*}

Case 2: $\rho < \gamma$

In this case problem can be written as :

\begin{eqnarray*} \max_{h}  \ & \ln (\omega h + \rho - \gamma) + \beta + (\theta -\beta)h \\ \text{s.t.} & \frac{\gamma - \rho}{\omega} \leq h \leq 1 \end{eqnarray*}

Derivative of the objective with respect to $h$ is
$\frac{\omega}{\omega h + \rho - \gamma} + (\theta -\beta)$ which yields the following solution :

\begin{eqnarray*} h = \begin{cases} 1 & \text{if } \frac{\omega}{\omega + \rho - \gamma} + (\theta -\beta) \geq 0   \\ \frac{1}{\beta -\theta} - \frac{\rho - \gamma}{\omega} & \text{otherwise} \end{cases} \end{eqnarray*}

Combining the two cases, we can write the solution as :

\begin{eqnarray*} h = \begin{cases} 1 & \text{if } \frac{\omega}{\omega + \rho - \gamma} + (\theta -\beta) \geq 0  \\  0 & \text{if } \frac{\omega}{\rho - \gamma} + (\theta -\beta) \leq 0 \text{ and } \rho \geq \gamma \\ \frac{1}{\beta -\theta} - \frac{\rho - \gamma}{\omega} & \text{otherwise} \end{cases} \end{eqnarray*}

Using $c = \omega h + \rho$ and $l = 1 -h$ we can get optimal values of $c$ and $l$ in each of the cases.


Application of Poisson process in economic modelling

Most of the literature on "Strategic experimentation" (or Bandits) uses Poisson processes. Here players can use either a risky or safe arm and one of them generates a fixed stream of payoffs (usually the safe arm) the other one generates lump-sum payments whose arrival times are described by a Poisson process.

Some examples are:

Klein, Nicolas, and Sven Rady. "Negatively correlated bandits." The Review of Economic Studies (2011).

Keller, Godfrey, and Sven Rady. "Strategic experimentation with Poisson bandits." Theoretical Economics 5.2 (2010): 275-311.

Keller, Godfrey, Sven Rady, and Martin Cripps. "Strategic experimentation with exponential bandits." Econometrica 73.1 (2005): 39-68.


In the case of linear demand $d_i=a_i-x_iP$ (assuming $d_i$ is quantity demanded by individual $i$), the price elasticity of demand at point $(d_i,P)$ is
\begin{equation}
\epsilon_i(d_i,P)=x_i\cdot \frac{P}{d_i}.
\end{equation}
As @the_rainbox noted in their answer, price elasticity of demand varies along a linear demand curve. So in order to compare elasticities between different demand curves based only on the slope coefficients (the $x_i$'s), you need to fix $P$ and $d_i$; that is, assume that the demand curves of individuals $1$ and $2$ cross at some point $(Q_0,P_0)$.
Then, you can say things like
\begin{equation}
\epsilon_1(Q_0,P_0)\ge \epsilon_2(Q_0,P_0) \quad\Leftrightarrow\quad x_1\ge x_2.
\end{equation}
Or in words: $1$'s demand is more elastic than $2$'s at $(Q_0,P_0)$ if and only if $1$'s demand curve is flatter than $2$'s. [Note that since by convention demand curves are plotted in the $(Q,P)$-plane, a flat demand curve actually corresponds to a high $x_i$.]
Beware, however, that in contexts where a high degree of mathematical rigor is not required, it is sometimes taken as a rule-of-thumb that flat demand curves are "generally" more elastic than steep ones. Hence slope becomes a proxy for elasticity in those less rigorous discourses.


It is much more complex than just saying yes.
You need to look at different factors and then you can answer your question.

Generally, high taxes have a negative effect on businesses, the more taxes a business pays - the less money it has to growing itself.

The lower the taxes the less money the government has (this is not necessarily as there is a Laffer curve to take into consideration).

It also differs depending on how you measure economic performance, GDP? GNP? job growth, unemployment rate - these are all very different.

If you look at most major economies - you will see that generally the lower the taxes, the better the economy performs. However there are many countries - such as the Scandinavian Social democracies, that have high taxes but generally good economies.

In my opinion, high taxes are not a major factor in economic performance (although they are a factor), and Free market and less regulations is much more important.
Simplifying tax code is also more important than lowering taxes, as complex tax codes are waste of time and money.

If the taxes are low, the budget is balanced, and the tax code is simple - it is  generally better than the same economy with higher taxes.


Boundary solutions in the Utility Maximization Problem

Except for the extreme cases $\alpha=0$ and $\alpha=1$, there is no boundary solution with $x_1=0$. Note that the marginal utility of good 2 is constant, while the limiting marginal utility for $x_1\to 0$ is infinite. So the consumer will always consume a bit of $x_1$.