Clearly explaining the concept of "normalizing" price and relative prices to undergraduate

I usually ask if they would rather get an IPhone or 5000 South African Rands (ZAR). You can also ask if 5000 ZAR is a good enough monthly salary. As we are far from that country usually no one has a clue about the currency exchange rate.  This way you can explain that the number in itself is meaningless, what matters is what you can exchange it for, i.e. relative prices matter.


Why would microeconomics be considered more "objective" than macroeconomics?

I used to think that microeconomics was relatively more scientific than macroeconomics. Early macroeconomics, in particular, suffered from choosing models of agent behavior that made modeling easier but bore little relationship (or at least a highly disputed relationship) with reasonable behavior of individual agents. This led to the microfoundations effort, which attempted build macroeconomic models from the ground up by modeling the behaviors of the economic agents in line with microeconomic theory (rational expectations, utility functions, life cycle effects, market power, and so on). Aligning macroeconomics with microeconomics strongly implies that microeconomics "has it right" and macroeconomics does not.
However, that's not the end of the story.  In a nice essay called The Macroeconomic Foundations of Microeconomics, macroeconomist Mark Thoma says that microeconomics has a lot of hidden macroeconomics lurking in the background, and it is important to recognize this and formalize the role of macroeconomics in microeconomic theory. For example:

One of the lessons from macroeconomics is that prices do not always
move fast enough to keep markets in balance, and this observation --
this macroeconomic foundation -- needs to guide microeconomic
theorists. The Walrasian auctioneer is a convenient theoretical
assumption, somehow prices always magically move immediately to clear
all markets and attain the highest possible societal welfare (and it's
useful for comparing long-run outcomes). But that assumption is not
very useful as a foundation for macroeconomic models that try to
explain short-run swings in the macroeconomy. Microeconomic and
macroeconomic models ought to be consistent with each other, but the
microeconomic foundations need to have realistic assumptions about the
ability of price changes to clear markets.

More generally, microeconomics usually assumes that microeconomic agents are embedded in a functioning macroeconomy. An economy with functioning markets for other goods, property rights, production functions and industrial organization that makes supply curves, demand curves that are well defined, and the like. This joint dependence on the assumptions of the other family makes the two major schools of economic thought closer to peers than the microfoundation revolution suggests.
That said, microeconomics has done a lot more to bring experimentation (field and laboratory) into their research. This has helped inform their modeling and econometric work in a positive way, including this year (2019's) Nobel winners. For ethical and financial reasons it is much harder to do this in macroeconomics. This could be a reason to think that, in general, microeconomics might be a bit more objective. However, the majority of microeconomic research is non-experimental, so we shouldn't go too far with this argument.
I also recommend Raj Chetty's essay Yes, Economics Is a Science for a comparison of the scientific bona fides of the field of economics with other scientific endeavors, particularly medicine.

It is true that the answers to many “big picture” macroeconomic
questions — like the causes of recessions or the determinants of
growth — remain elusive. But in this respect, the challenges faced by
economists are no different from those encountered in medicine and
public health. Health researchers have worked for more than a century
to understand the “big picture” questions of how diet and lifestyle
affect health and aging, yet they still do not have a full scientific
understanding of these connections. Some studies tell us to consume
more coffee, wine and chocolate; others recommend the opposite. But
few people would argue that medicine should not be approached as a
science or that doctors should not make decisions based on the best
available evidence.
... Nonetheless, economists have recently begun to
overcome these challenges by developing tools that approximate
scientific experiments to obtain compelling answers to specific policy
questions. In previous decades the most prominent economists were
typically theorists like Paul Krugman and Janet L. Yellen, whose
models continue to guide economic thinking. Today, the most prominent
economists are often empiricists like David Card of the University of
California, Berkeley, and Esther Duflo of the Massachusetts Institute
of Technology, who focus on testing old theories and formulating new
ones that fit the evidence.
This kind of empirical work in economics might be compared to the
“micro” advances in medicine (like research on therapies for heart
disease) that have contributed enormously to increasing longevity and
quality of life, even as the “macro” questions of the determinants of
health remain contested.


First let's look at the specific tax. The profit is
$$\pi_s=[P(q)-\tau_s]q-C(q).$$
Differentiating to establish the first-order condition:
$$P'(q)q+P(q)-\tau_s-C'(q)=0.$$
If we write $A=\tau_s q$ for the tax revenue then we can rewrite the FOC thus:
$$P'(q)q+P(q)-C'(q)=\frac{A}{q}.$$

Now for ad valorem:
$$\pi_a=(1-\tau_a)P(q)q-C(q)$$
First-order condition:
$$(1-\tau_a)[P'(q)q+P(q)]-C'(q)=0.$$
We have $A=\tau_a q P(q)$, which can can use as follows:
$$[P'(q)q+P(q)]-\tau_aP'(q)q-\tau_aP(q)-C'(q)=0.$$
$$[P'(q)q+P(q)]-\frac{A}{P(q)}P'(q)-\frac{A}{q}-C'(q)=0.$$
$$P'(q)q+P(q)-C'(q)=\frac{A}{q}+\frac{A}{P(q)}P'(q).$$

Note the the left-hand side of the last line in both sections is the same. Moreover, this expression ($P'(q)q+P(q)-C'(q)$) is the derivative of profits assuming zero tax. If profits are concave then this derivative is decreasing in $q$.

We can see that, for any value of $q$, the right-hand side is smaller for the ad valorem tax (because it has an extra term that is negative--remember that $P'(q)<0$). Thus, we have something that looks like this:


The stochastic discount factor $M_t$ in a simple RBC model

Somewhat inconsistent/careless use of indices. In

$$M_t=\beta^t\frac{E_0(u'(C_{t+1}))}{u'(C_{0}))}$$

it is assumed that we stand at period $0$ for any $t$ ahead we want to look at.

In

$$E_t(M_{t+1})=\beta\frac{E_t(u'(C_{t+1}))}{u'(C_{t}))}$$

it is assumed that we stand at any $t$ and we look one period ahead.

The two expressions don't give the same formulae for any given $t$, and there lies the problem/confusion.


Is a 6% wealth tax a sustainable source of income?

I think the big unknown here is how much they'll start to dissimulate and/or offshore their (new) wealth.
According to one paper:

A recent study by Brülhart et al. (2017) gives support to the plausible assumption that the
effect of net wealth taxes on reported wealth is the more pronounced the more integrated
the regions involved are. According to the authors’ estimations, the semi-elasticity of
reported wealth with respect to the net wealth tax rate amounts to 35% in aggregate, i.e. a
rise in wealth taxation by one percentage point decreases reported wealth by 35%.
Moreover, Brülhart et al. (2017) find that financial assets seem to be more responsive to
taxation than non-financial assets. They also interpret their results as suggesting that
wealth holders primarily respond by reducing their wealth holdings, not by moving to
jurisdictions with lower tax rates.

So if that is true, then >3% wealth tax will result in "over" 100% dissimulation of new wealth. But caveat emptor, that's based on the fairly secretive Swiss:

It must be noted, however, that these analyses do not uncover the channels via which
wealth holdings are lowered. As indicated above, reported wealth holdings may be reduced
by real responses (i.e. by lowering accumulation of wealth) or by decreasing reported
wealth through hiding it from tax authorities. Which of these mechanisms is working in the
Swiss case cannot be determined without further analysis, e.g. by exploring whether there
is some relationship between the savings rate and the taxation of net wealth.

The biggest issue with extrapolating much from the Swiss case is that

Wealth taxes are cantonal and municipal; there is no federal taxation of wealth.

It's basically unknown how big that semi-elasticity would be if a whole country (like the US) had a wealth tax. France apparently had one, but it looks like they don't have a clear figure of the aforementioned elasticity, despite the fact that the rate changed a few times. It is known however that for about a decade there was a net outflow of around 300-500 people (per year) who would have qualified for the wealth tax, although they could have left for other reasons (including other parts of the tax code). France being in the EU also offers some special circumstances to migrate to neighboring countries but still (partly) invest in France.

If we ignore the potential flight or dissimulation, there's still the issue that billionaire wealth might decrease even in not so bad times, e.g.

Billionaires around the world saw their total wealth decline by 7% from 2017 to 2018, according to Wealth X’s Billionaire Census 2019. At the same time, the number of these ultra-high-net-worth individuals fell from 2,754 to 2,064.
American billionaires had an easier time.
The U.S. was one of the few countries that saw its UHNW population grow from 2017 to 2018, per the report, and America’s richest people lost only around 5% of their total wealth, far less than their peers did in most other regions. [...] Total U.S. billionaire wealth fell from \$3.2 trillion to \$3.013 trillion.

So, before we talk of sustainability, we'd need to decide on some time frame. Clearly their (total) wealth can go down year-to-year even with no wealth tax, although the trade war was apparently blamed for the recent downward trend. From 2015 to 2016 it actually went up 15%.

In 2016, US billionaires saw their wealth increase from USD 2.4 trillion to USD 2.8 trillion

So it looks like US billionaire wealth is more volatile than the US economy as a whole.


Cobb–Douglas utility maximized by spending a "fixed fraction of income on each good"?

A "fixed fraction" doesn't mean an "equal fraction", or at least that's not the intended meaning.
It can be easily verified that the solution to
\begin{equation}
\max_{x_1,x_2}\;x_1^{a_1}x_2^{a_2}\qquad\text{s.t.}\; \pi_1x_1+\pi_2x_2\le w
\end{equation}
is
\begin{equation}
x_1^*=\frac{a_1}{a_1+a_2}\frac{w}{\pi_1}\quad\text{and}\quad x_2^*=\frac{a_2}{a_1+a_2}\frac{w}{\pi_2}.
\end{equation}
Plugging in the values you used: $a_1=0.8,a_2=0.2,\pi_1=\pi_2=1$, the solution is consistent with your numerical simulation.
Generalizing to the $n$-goods case (with $\sum_ia_i=1$), the demand for $x_i$ is
\begin{equation}
x_i^*=\frac{a_i}{\pi_i}w,
\end{equation}
where the "fixed" fraction refers to $a_i/\pi_i$.


Averaging data over 5 years with Stata or Excel

Is this what you want (in Stata)?

 clear all
 * Generate Data (n=5, T=20)
 set obs 100
 gen id = floor((_n-1)/20)+1
 by id, sort: gen year = 1990+_n
 gen x = rnormal()
 xtset id year
 * Convert
 gen t5 = floor((year-1991)/5)+1
 gen idt5 = id*10+t5 /* 10 can be 100, 1000, etc., depending on T */
 by idt5, sort: egen xbar5 = mean(x)
 *drop t5 idt5  /* drop if you want; you can also use tempvar */

If not, please clarify your question using an example.


Approximating common knowledge with common beliefs (Monderer and Samet, 1989)

Showing the left-hand inequality

Let's rearrange the equation,
$$
\begin{align*}
x &= r_i \frac{\mu\left( B^p_i\left( E \right) \right)}{\mu\left( E \right)} - \frac{\mu\left( X \cap \left[ B^p_i\left( E \right) \setminus E \right] \right)}{\mu\left( E \right)} \\
& \iff \underbrace{x \mu\left( E \right) + \mu\left( X \cap \left[ B^p_i\left( E \right) \setminus E \right] \right)}_{\text{LHS}} = r_i \mu\left( B^p_i\left( E \right) \right).
\end{align*}
$$
Since all terms in $\text{LHS}$ are weakly positive, we can omit $\mu( X \cap [ B^p_i( E ) \setminus E ] )$ from $\text{LHS}$, giving
$$
x \mu\left( E \right) \leq r_i \mu\left( B^p_i\left( E \right) \right).
$$
Since $\mu( E ) \geq p \mu( B^p_i( E ) )$, this directly implies $x p \leq r_i$.

Showing the right-hand inequality

Since $\mu( E ) \leq \mu( B^p_i( E ) )$ and $x \geq 0$, we have
$$
\begin{align*}
&\frac{1}{\mu\left( E \right)} \left[ r_i \mu\left( B^p_i\left( E \right) \right) - \mu\left( X \cap \left[ B^p_i\left( E \right) \setminus E \right] \right) \right] \\
&\geq \frac{1}{\mu\left( B^p_i\left( E \right) \right)}\left[ r_i \mu\left( B^p_i\left( E \right) \right) - \mu\left( X \cap \left[ B^p_i\left( E \right) \setminus E \right] \right) \right] \\
&= \underbrace{r_i - \frac{\mu\left( X \cap \left[ B^p_i\left( E \right) \setminus E \right] \right)}{\mu\left( B^p_i\left( E \right) \right)}}_{\text{RHS}}.
\end{align*}
$$
It must be that $X \cap [ B^p_i( E ) \setminus E ] \subseteq B^p_i( E ) \setminus E$, so
$$
\text{RHS} \geq r_i - \frac{\mu\left( B^p_i\left( E \right) \setminus E \right)}{\mu\left( B^p_i\left( E \right) \right)} = r_i - \frac{\mu\left( B^p_i\left( E \right) \right) - \mu\left( E \right)}{\mu\left( B^p_i\left( E \right) \right)} = r_i - \left( 1 - \mu\left( \left. E \right| B^p_i\left( E \right) \right) \right).
$$
By definition (and this is inequality (9) in the text) $\mu( E | B^p_i( E ) ) \geq p$, thus
$$
\text{RHS} \geq r_i - \left( 1 - p \right).
$$
Noting that the original equality was $x = \text{RHS}$, we have
$$
x \geq r_i - \left( 1 - p \right) \;\; \iff \;\; r_i \leq x + 1 - p.
$$


Why can principal not be lost, if a bond fund is held for a period equal to its duration?

A bond is basically a contract between the debtor and the creditor, which roughly usually says something like

You, the creditor, agree to lend me \$10,000 today. In return, I, the debtor, will make the following payments to you (or, in the event that you sell this bond, to the new owner of the bond):

\$500 every year for the next five years.
An additional final payment five years from now to return your $10,000.

That the original invested principal is returned to the investor at the end of the bond's life is fundamentally written into the terms of the agreement. Thus, provided you don't sell the bond before its maturity, the terms of the bond guarantee that your original principle will be returned. This means that, under normal circumstances, capital loss will be zero if the bond is held to maturity.
Here are some reasons why capital loss might be non-zero for a bond investor:

The debtor defaults (for example, by declaring bankruptcy) and ceases making the payments required by the bond.

A minority of bonds have terms that require an amount different to the original invested principal be returned at the end of the bond's lifetime. In this case, the amount to be returned will be described in the bond, but might be less than the original investment.

or, of course, you sell the bond before its maturity date, and the amount for which you sell it is less than the amount you originally invested.

It is also important to remember that even if nominal capital loss is zero, inflation will erode the value of the invested capital over the lifetime of the bond. Thus, the real value of the returned principle will be lower than the value of the amount invested.


How to calculate price function from demand curve when marginal cost is given?

Let $y_1$ and $y_2$ denote the output of firms $1$ and $2$. I suspect by price function you mean the inverse demand function. So if
$$
y_1 + y_2 = Y = D(p)
$$
then
$$
D^{-1}(Y) = p(Y) = p(y_1 + y_2).
$$
Let us introduce the notation
$$
MR_i(y_1,y_2) = \frac{\partial p(y_1 + y_2)}{\partial y_i}.
$$
Generally
$$
MR_i(y_1,y_2) \neq MC_i(y_i),
$$
that is the functions are not the same. But if the outputs $y_1^*,y_2^*$ constitute an equilibrium, the functions take the same value.
$$
\forall i: \ MR_i(y_1^*,y_2^*) = MC_i(y_i^*)
$$

Another example:
 The function $x^2$ is not the same as the constant function 4. But if $x=2$ they take the same value.


The Insane Sultan of Slickcrudistan: Calculating Currency Equilibrium

Using a simple textbook-approach, I will provide one possible answer here, showing that in this approach, the one thing that needs to be specified is "velocity of money" as regards sand dollars.

I will use also the numerical values provided in the question, after cutting out six zeros.
I assume that the Law of One Price holds, so

$$P_s\cdot S_{USD/s} = P_{USD} \tag{1}$$

$P_s$ is "the price level" related to sand dollars, $P_{USD}$ is the price level related to USD, and $S_{USD/s}$ is the exchange rate we wish to find ("how many USD per one sand dollar").

While the price levels are usually price-indexes, here we are looking at a single good, oil, so we can take the $P's$ to mean "price of oil".

The Sultan's oil is the only output that can be bought with sand dollars. And we do know the value of this output in USD

$$P_{USD} \cdot Q_s = P_{USD} \cdot 20 = 1,000 \;\text{USD} \tag{2} $$

Using the Law of One Price, we get

$$P_{USD} \cdot Q_s = P_s\cdot S_{USD/s}\cdot Q_s \implies S_{USD/s} = \frac {P_{USD} \cdot Q_s}{P_s\cdot  Q_s} \tag{3}$$

(don't simplify).

Assuming the Quantity Theory of Money, we have

$$P_s\cdot Q_s = V_s \cdot \bar M_s \tag{4}$$

where $\bar M_s$ is the fixed amount of sand dollars available, and $V_s$ is the "velocity of money" in relation to sand-dollars (we will return to $V_s$). Inserting $(4)$ into $(3)$ and using also $(2)$ and the fixed value of $\bar M_s$ we have

$$(4),(3),(2) \implies S_{USD/s} = \frac {P_{USD} \cdot Q_s}{V_s\bar M_s} = \frac {1,000}{V_s\cdot 100} = \frac {10}{V_s}$$

So, if we can find a value for $V_s$, we have found the exchange rate we are after. The velocity of money can be thought as "how many times that same unit of currency, say the same paper bill, will be used in transactions during a production cycle".

The M1 velocity of money for USD is approximately  $V_{USD} \approx 6$, using this value for the sand-dollars also, we get

$$S_{USD/s} = \frac {10}{6} \approx 1.67$$


Subgame Perfect Equilibrium for Pure and Mixed strategy

Check for the Nash equilibria (pure or mixed) of the one-shot game.
Repetition of the strategy profile of the Nash equilibria of the one-shot version yields one set of subgame perfect equilibria: For instance, play $(A,A)$ in the first stage and for any action profile played at the first stage, play $(A,A)$ in the second stage. The same holds true for the mixed (complete or otherwise) as well. For instance, the totally mixed SPNE is :Play $\left(\frac{3}{5},\frac{1}{5},\frac{1}{5}\right)$ in the first stage and for any action profile (assuming the randomisation is observable), play $\left(\frac{3}{5},\frac{1}{5},\frac{1}{5}\right)$ in the second stage.
Combinations of the Nash equilibria of the one-shot game yield another set of SPNE: For instance, Play $(A,A)$ in the first stage and play $\left(0,\frac{1}{2},\frac{1}{2}\right)$ for any action profile in the second stage and so on.
Use credible threats to play a non-Nash action profile in the first period: Play $(B,B)$ in the first stage. At the second stage, if $(B,B)$ was played, play $\left(0,\frac{1}{2},\frac{1}{2}\right)$. For any other action profile at the first stage, play the totally mixed Nash equilibrium of the one-shot game. Why does this work? There is no incentive for player 1 to deviate anyway because they're playing their best reply in the first stage. For player 2, if they were to deviate to $C$. they'd receive $2$ in the first stage and $1/3$ in the second stage. Assuming no discounting, they get $7/3$ while if they obeyed the strategy, they would get $(1+3/2)$ which is strictly greater. Similarly $(C,C)$, $(B,C)$ and $(C,B)$ can be played as first stage action profiles in a sub-game perfect equilibrium.


What happens to a country with lot of debt?

If a government owes money to domestic or foreign entities and does not repay that debt then the government's credibility — just like any other borrowing entity's credibility — as a reliable economic entity takes a hit. As a consequence when it wants to borrow more money in the future, the lending entities either charge a higher interest rate to cover the risk or refuse to lend to avoid the risk altogether.

Here's the only difference — governments can print their own currency unlike other entities. But too much currency in circulation will cause inflation, which is bad for the economy. So it is in the government's own interests to repay its debt, as it is to borrow money in the first place.


What limits central banks from setting highly negative interest rates?

Since this is a question about central banks... they would not want to do this because it would push many ordinary people to put their savings in highly speculative assets, potentially destabilizing the economy. It is/was already a bit of a concern even with near zero, never mind mildly negative interest rates. There's a fine line between "kickstarting the economy" and setting up a large portion of the population to speculate in the junk bond market and/or housing.

Also, wealth inequality will probably go up faster as well (in the long) as more smaller investors get wiped out, although that's seldom a concern for central banks, given their usual mission.

As for hoarding cash in such an environment... that's not without risk either, because the authorities could decide to counter that with some kind of demonetization (see India, and numerous other historical examples.)

Also, in a democracy, weird stuff might happen, like the Swiss left- and right-wing populist parties threatening to put to a referendum a redistribution of the central bank profits into the pension funds (who blame the central bank for their losses.) That would basically negate the effect of the negative interest rates, as far as the pensions funds are concerned. In fact, it would act like a redistribution from others to said pension funds, ultimately.


How can a RTA have a trade imbalance with itself?

Yes, such imbalances or asymmetries are possible in practice and quite frequent across RTAs. These statistics simply mean that each ASEAN member exports more on average to the other members than they import from them. But this pattern is not true for all ASEAN countries.

Singapore exports 118bn USD to the other ASEAN countries and imports 64bn USD from them.
Mynamar exports 4bn to the other ASEAN countries and and imports 7bn USD from them.

Finally, they define the `intra-ASEAN exports' as the sum of the exports of the individual ASEAN countries (305 billion USD in the second column of ASEAN data).

In theory, this number should be equal to the sum of the individual imports from the other ASEAN members. To fix ideas, consider a RTA with 3 countries A,B, and C. If each country exports 10 to each partner. The sum of exports is equal to 60, and if A->B is equal to B<-A, etc. the sum of imports is also equal to 60.

In practice, this number is not equal because intra-RTA exports may not be calculated the same way as intra-RTA imports. This is explained in this Eurostat website for the intra-EU trade.

Asymmetries occur when the declaration of the importer in country A is not consistent with the declaration of the exporter in country B. Asymmetries come either from errors in reporting or from differences in the concepts and definitions applied by the partner countries. Some of the asymmetries of a methodological nature could disappear thanks to further harmonisation in national practices; others will continue as they are linked to basic principles for compiling ITGS which are unlikely to change (e.g. CIF/FOB valuation). The most common causes of methodological asymmetries are the following:

simplified product reporting: Where the EU legislation allows simplified codification of goods for certain transactions, some Member States apply the simplifications but others do not;
different approaches to estimating or compiling trade data for specific goods and movements not covered by customs or statistical declarations, such as gas, electricity, vessels or aircraft;
confidentiality: It is possible that data are considered confidential by only one of the two partners. However, the application of the confidentiality procedures should not impact the results at total trade level;
time lag: the same operation can be recorded under a different reference period because of transport times or processing delays;
CIF/FOB valuation: imports are valued on a CIF basis and exports on a FOB basis. This causes a systematic asymmetry as the value of the imports should then be higher than the value of the mirror exports;
differences in methods and data used to estimate missing trade;
different practices in the treatment of revisions;
problems of currency conversion; and
other methodological differences such as definition of partner country, definition of statistical territory, trade system (special or general).

Why is the intra-EU trade balance not equal to zero?

Considering that the intra-EU trade data are based on common and largely harmonised rules, one might expect the intra-EU trade balance to be zero or at least close to it. However, it is worth underlining that a perfect match is made impossible first of all by the CIF/FOB approach: the import value should be higher than the mirror export value as it includes extra transport costs.

What is the 'Rotterdam effect'?

Dutch trade flows are over-estimated because of the so-called ‘Rotterdam effect’ (or quasi-transit trade): that is goods bound for other EU countries arrive in Dutch ports and, according to EU rules, are recorded as extra-EU imports by the Netherlands (the country where goods are released for free circulation). This in turn increases the intra-EU flows from the Netherlands to those Member States to which the goods are re-exported. To a lesser extent, Belgian figures are similarly overestimated.


Could food storage prevent a depression?

Keep in mind that during the 7 years of famine in the Bible, there was still a negative production shock that more than likely produced a heavy recession for some of those seven years as workers tried to reallocate their labor to something more useful than agriculture. A recession merely represents a fall in final goods production. The grain that was saved up doesn't represent production for that period (rather it would be for the previous seven years of economic boom). The grain stored up represents savings rather than production.

It doesn't look like Joseph or Pharaoh told the people that the famine was coming (imagine the panic), so workers wouldn't have predicted the change in permanent income, and their intertemporal substitution would be much smaller because of the error in their adaptive/rational expectations. Joseph merely used taxes (one-fifth of each year's harvest) to engage in consumption smoothing.

So to answer most of your questions, food storage doesn't prevent negative output shocks. It merely is a hedge against them. Arguably most people should try to have emergency savings of their own to hedge against recessions of their own, but especially in America, we are not very good at saving money. We also have a lot of population who live paycheck to paycheck, and are liquidity constrained, unable to consumption smooth according to their predictions in permanent income, even if they want to.


What would happen if the world switched to a single currency?

The Euro was always conceived by most Economists as a political goal, not an economic one.

Prologue: There is the theory of Optimum Currency Areas (OCA) which characterizes properties that a larger area would need to have if it were to operate on a single unit of currency. Since the announcement of the Euro around 1990, there were many papers that looked into whether the European Area actually satisfied the broader range of these criteria, and they mostly agreed that it did not.

Ex-post, we can now see this playing out as the ECB has a hard time setting up an inflation rate that is improving conditions of countries that are hit very badly by the crisis such as Greece and Spain, and countries which are not, such as France and Germany.

Intuition behind OCA

Basically, a country gives up the tool of monetary policy when it subordinates into such a currency union. In order to properly use monetary policy for a set of different economies (countries), you need these to be very similar in nature: If all countries react similarly to a housing bubble / oil shock / etc, you can easily improve outcomes for all countries with the same monetary tool. If the countries respond differently, it is much harder to do so.

Single currency for non-OCA

To the extent that all countries in the world are very different in nature, your experiment is similar to the gold standard (see Bretton Woods system). We think about that system mostly as a failure. While there is no hard data on the causal relationship, countries that abandoned the gold standard earlier tended to do better. We observed similar trends for countries that abandoned the dollar standard earlier. As a short intuition, among other explanations, when you abandon monetary policy and fix your exchange rate, you make yourself more vulnerable to inflation/deflation of other regions.


Inverting price functions and their conditions/intervals

Notice that if you plug in $P=15$ into either of the $Q(P)$, you get

$$Q(P) = 700-40P = 700 - 40 \cdot 15 = 100$$

$$Q(P) = 400-20P = 400 - 20 \cdot 15 = 100$$

So you can take the exercise further and see that when $P < 15$, then $Q > 100$ (notice the direction of the inequality is changing; the more expensive a good, the less consumers buy of it)


Economists also publish in Science or Nature sometimes, and if the study is interdisciplinary, there could involve many co-authors just like other scientific papers.
In the prestigious economic journals, the economics paper with most coauthors is Benjamin, Daniel J., et al. "The Promises and Pitfalls of Genoeconomics." Annu. Rev. Econ. 4.1 (2012): 627-662.  or Coenen, Günter, et al. "Effects of fiscal stimulus in structural models." American Economic Journal: Macroeconomics 4.1 (2012): 22-68.


Non-uniqueness of iterated elimination of weakly dominated strategies?

Consider the game below.

$$
\begin{array}{|r|c|c|}
\hline
&L&R\\\hline
U&5,1&4,0\\\hline
M&6,0&3,1\\\hline
D&6,4&4,4\\\hline
\end{array}
$$

Both $U$ and $M$ are weakly dominated by $D$.

If we start by deleting $U$. This would lead to the removal of $L$ in the next step. Then $M$ would be deleted. The solution would be $(D, R)$.

However, if we start by removing $M$, then $R$ would be removed next, and then $U$. The solution from deleting in this order is $(D, L)$.


Why are they raising interest rates in Brazil?

Brazil raising interest rates probably refers to the central bank raising their interest rate which means that it is more costly for banks to loan money from the government and therefore the amount of money in circulation (as per quantity theory) should go down. It is a policy decision (possibly wrong depending on who you ask) in a difficult economic situation where a country faces both inflation and a downturn.

Your logic about the economy becoming even weaker might very well be true (and I'd agree with lowering rates being a better option). The questions about effects to trade are slightly more complex and not necessarily as straightforward.


Why does demand curve shift in, in a monopolistic competitive market?

Demand for existing firms' product shifts in because the entering firms attract some of the users.


For a long time many people thought that stagnation was an impossible economic experience to occur as Keynes economic theory of the Philipps curve which portrayed macroeconomic policy as a trade-off between unemployment and inflation.   However during the 1960s in UK and in 1970s in the US stagnation occurred but was due to external situations which caused prices to increase. U.S. and other industrialized countries entered a period of slow economic growth occurring simultaneously with high rates of inflation.

https://www.investopedia.com/articles/economics/08/1970-stagflation.asp

It appears, which Neo-Keynesians are believing, that stagflation is caused by cost-push inflation (when some force or condition increases the cost of production). So external factors such as a shortage of natural resources or an act of war are the main contributors to this situation.


There are several problems with the idea:

My problem is that although logically both idea looks the same (i.e. good getting cheaper with a constant level of income = income growth outpaces prize increase ), I find the latter method mathematically more complicated.

Here you are confusing economics as a science with economic policy. Yes in a science we prefer more parsimonious explanation to one that is less parsimonious (assuming they both have equal explanatory power) but that does not mean we should prefer more parsimonious policy. That is equivalent of saying that simpler medical treatment should be always preferred. Just because a more parsimonious biological theory/explanation is more efficient and hence preferred does not mean we might as well prefer herbal tea compared to complex antibiotics.
Its not that there are two theories one which says that inflation must occur as some sort of natural process when technology expands and second theory that says that technology can expand freely. Here you are talking about policy choice about what we choose the money supply to be along some  path of technological progress.

From my perspective, the amount of money in an economy should remain constant (apart from replacing the old torn bills with a brand new one) and things should get progressively cheaper until we reach a technological utopia where everything becomes free.

There are several problems with this:

First, what ultimately matters is not what the price tag is but how much real resources you have to spend for goods and services. If due to inflation apple costs $\\\$10,000$ but you earn $\\\$1,000,000,000$ per hour then in real terms the apple in such situation is much cheaper then in situation where apple costs $\\\$0.10$ but your hourly wage is $\\\$0.01$.
You are ignoring well documented nominal rigidity especially when it comes to wages (this is so notorious you can see this mentioned in any textbook - for example Blanchard et al. Macroeconomics an European Perspective). For the sake of brevity I wont go into details but in the presence of nominal rigidity in short run deflation creates excess unemployment and depresses economic activity and inflation has the opposite effect. This is because in short run when aggregates prices fall firm might be still locked to higher nominal wages as those might be slower to adjust and this forces firms to unnecessarily lay off some workers. Due to this most economists advocate that over business cycle there should be steady slow (about $2\%$) inflation in order to help 'grease the wheels' of the economy.

A possible explanation to my question: By not increasing the money flow, we are not able to reward the individuals who brought about the technological advance as the things get cheaper uniformly for everyone, whereas if we print the extra money, then that extra money goes to person responsible for the change).

This is also incorrect. Even if money supply would remain constant that does not mean that there is any lack of money to reward people for their additional work or effort. In long run prices & wages would just adjust downward so even if people would earn less in nominal terms they could actually buy much more as before (an opposite situation as the one shown with apple before). What matters ultimately is not how much money you have but the amount of goods and services you can consume.
Economy actually existed and functioned well before money was invented at all. If there would be no nominal rigidity and we would live in a world where money is even in short run completely neutral then deflation would not matter at all.


Are taxes,subsidies and social money transfers(i.e. welfare) accounted in money velocity?

Velocity of money is according to the Fed definition:

The velocity of money is the frequency at which one unit of currency is used to purchase domestically- produced goods and services within a given time period. In other words, it is the number of times one dollar is spent to buy goods and services per unit of time.

Hence paying taxes/transfers is not counted. In your example the velocity would be 1 assuming government does not spend the money from those taxes on goods and services, and assuming the money transferred through subsidies or pensions are not spent at all.
Also, as a side note velocity is rarely calculated directly due to the practical difficulties. Rather its often just inferred from other variables.  Furthermore, the example you give cannot occur in practice as 1 dollar cannot be simultaneously just spent once, taxed twice and transferred 12 times at the same time but I take it that the numbers are just randomly selected.


How is it that the stocks are always rising?

Because companies generate value.

  How is money generated every year so that the value of a company rises?

It isn't. Money is a measure of value, it is not value itself. Suppose you spend the day making a painting that's worth 100 dollars. You added one hundred dollars' worth of value to the economy, but you didn't add any actual dollars to the economy. There's no need for there an extra 100 dollars to be created for you to make the painting. If the market capitalization of a company rises from 10 billion dollars to 11 billion dollars, the amount of money that people are willing to pay for the company has increased, but the amount of money that exists hasn't increased.

  I mean, at some point there is a finite amount of money in the system

Well, the money supply is generally increasing, but that's separate from the rise of stock prices.


Average ability conditioning on having accepted an offer

If $\alpha \sim U$, then how come there is no expectation in your profit function? The $\alpha$ is unknown and which $\alpha$-types the firm gets depends on salary $v$. This should be reflected in the profit function.
Next, your $n(v)$ seems to assume that $\alpha \sim U[0,1]$, but you set $\alpha \sim U[0,2]$. I assume this is a typo and I edited your question. Otherwise, you need to divide your $n(v)$ by two, because your density is $\frac{1}{2}$ instead of 1. This upperbound cancels out anyway.
For any uniformly distributed $\alpha<\widehat v$, you have
$$\mathbb{E}[\alpha \vert \alpha \leq \widehat v] = \int_0^{\widehat v}  \alpha \frac{1}{\widehat v}d\alpha = \left[\frac{1}{2\widehat v}\alpha^2\right]_0^{\widehat v}= \frac{\widehat v}{2} $$
And in your case, it's ${\widehat v}=\sqrt{v/2} \Rightarrow \mathbb{E}[\alpha \vert \alpha \leq \sqrt{v/2}] = \sqrt{v/8}$. If $\alpha \sim U[0,x]$ then the density is $1/x$, but you also account for $\alpha<\widehat v$ by dividing by $\widehat v/x$ such that $x$ cancels out.
Hence, your expected profit function is
$$\mathbb{E}[\pi(v)] = (\rho \sqrt{v/8}-v)\sqrt{v/2}.$$
Acceptance does not skew the distribution. All types below a cutoff accept, all others reject. Therefore, the distribution conditional on acceptance is uniform up to the cutoff.
If you replace your $n(v)$ by $n(v)/2$, taking account for the upperbound 2, you would get the same.


can a company write off these tariffs and eventually receive a tax
  refund on the import taxes they paid?

A country might enact provisions to exempt (possibly via a refund) from tariffs if the importer company or the intended use of the import meets certain conditions. But generalizing a reimbursement in the form of tax refund (or tax credit) would defeat the purpose of imposing tariffs.

Keep in mind that tariffs are one mechanism through which policymakers implement barriers to entry in the domestic market. Therefore, it would make no sense for the regulator to implement such barriers only to essentially reverse them thereafter.


Why do stricter emission limits cost jobs?

Why do (don't) stricter emission limits lead to job loss?

I am not knowledgeable of the mechanism and exact details of the emissions paraphernalia, but the sector's worry makes perfect sense. Stricter regulations lead to or contribute to
job losses.

Stricter emission regulation entails a greater probability and/or magnitude of fines and sanctions to non-compliant companies. To avoid those sanctions, a company will be forced to spend additional resources to (1) replace technologies that are otherwise perfectly operational, (2) focus on expertise that distracts that industry from its core business, and (3) fund expenses that result from the increased overhead. The question is where those resources will come from.

Car makers may opt to pass that cost to consumers. However, that decision predictably will push sales down, thereby leading to overcapacity of labor and consequently layoffs.

Alternatively, car makers may decide to absorb the new compliance costs and leave the price of cars unchanged. But the prospect of reduced profits would prompt stockholders to sell their stock, thereby weakening the car maker's ability to meet its financial obligations. To cope with (or avoid exacerbating) this imbalance, the car maker has to reduce costs; and the readily available alternatives to that effect are layoffs and a freeze of new hires.

Furthermore, for car makers with significant reliance on exports, compliance with stricter regulations tends to hinder their competitiveness especially with respect to non-EU car makers which are not subject to akin regulatory constraints. Thus, EU car makers will lose [international] market share, most likely impacting its ability and interest to keep that much of its labor force.


In a trading economy with many agents, can two agents with the same income envy eachother?

Consider a pure exchange economy with $N\geq2$ consumers, each of whom has locally non-satiated preferences. We know from the First Welfare Theorem that any Walrasian equilibrium is Pareto-efficient. We index consumers by superscripts.

Say that consumer $i$ envies consumer $j$'s allocation if $i$ strictly prefers $j$'s allocation to their own:

$$ x^j \succ^i x^i $$

If these preferences have a utility representation, then this is equivalent to

$$ u^i (x^j) > u^i(x^i) $$

It is an easy consequence of the First Welfare Theorem that in any Walrasian equilibrium, no two agents, regardless of income, envy each other's allocations. If they did, this would violate Pareto-efficiency.

Notice, however, that this is not the same as the allocation being envy-free.


The authoritative source here is the United Nations System of National Accounts, SNA.

Change in inventories are indeed part of what we normally call Investment. First, for clarification, the definition of "change in inventories", taken from page 108 of SNA, version 2008:

  The basic principle underlying the measurement of changes
  in  inventories  of  finished  goods  is  that  output  should  be
  recorded at the time it is produced and valued at the same
  price  whether  it  is  sold,  otherwise  used  or  entered  into
  inventories for sale or use later. In effect, goods only enter
  inventories when they are not immediately used
   for sale or other use in the period they
   are produced. Similarly, goods are  withdrawn  from  inventories  when  the  demand  for  the
  goods exceeds the amount produced in a period. No output
  is recorded when goods produced previously are withdrawn
  from  inventories  and  sold  or  otherwise  used  unless  a
  storage activity as described be low in section F takes place.

Second, the components of "Investment" (officially called Gross capital formation), as stated in page 282 of the aforementioned document is:

  There are three types of capital formation to be examined,
  gross  fixed  capital  formation,
    changes  in  inventories  and
  acquisition less disposal of valuables.

This and following pages go about explaining each of them, and how to measure them.

Finally, the document provides a worked example of the table of "uses of output" (the expenditure approach of GDP). Here is a screenshot of a section of that worked example, presented in page 291. You can see that change in inventories are indeed treated as part of "Gross Capital Formation":


Consider

where $\epsilon_1,\epsilon_2$ are iid random variables with distribution $F$. $a,b,\dots,h$ are constant parameters.

Game of incomplete but symmetric information
would be something like: $F$ is common knowledge among the two players.

Game of asymmetric and incomplete information
would be something like: before making their respective decision, Player 1 observes the realization of $\epsilon_1$ but not $\epsilon_2$ and Player 2 observes $\epsilon_2$ but not $\epsilon_1$.


Walras's Law V.S Say's Law- Is there a difference?

The way you have defined excess demand, it is only consumer excess demand. But Walras's law holds in any private ownership economy at all prices (at which demand and supply are well defined). Walras's law is basically equivalent to consumers spending their budget fully.

Let there be $l$ commodities, so every commodity bundle is an element of $\mathbb{R}^l$. There are $m$ consumers and $n$ firms. The set of production plans firm $j$ can produce in terms of net-output is $Y_j$. With this convention, if $p\in\mathbb{R}^l$ is the price system and $y\in Y_j$ a production plan, $p\cdot y=\sum_{k=1}^l p_k y_k$ is the resulting profit. Every consumer has an endowment $\omega_i\in\mathbb{R}^l$ and a (potentially zero) share $\theta_{ij}$ in firm $j$ and the resulting profit. Every firm is privately owned, so each firm's shares sums to one, $\sum_{i=1}^m\theta_{ij}=1$.

At the price system $p$, the budget of each consumer is $p\cdot \omega_i +\sum_{j=1}^n \theta_{ij} p\cdot y_j$. If the consumer $i$ demands $x_i$ and spends their whole budget, we have $p\cdot x_i=p\cdot \omega_i +\sum_{j=1}^n \theta_{ij} p\cdot y_j$. Summing up and using that prices are linear,
$$p\cdot\sum_{i=1}^m x_i=p\cdot\sum_{i=1}^m\omega_i+p\cdot \sum_{i=1}^m\sum_{j=1}^n \theta_{ij}y_j$$
$$=p\cdot\sum_{i=1}^m x_i=p\cdot\sum_{i=1}^m\omega_i+p\cdot \sum_{j=1}^n\sum_{i=1}^m \theta_{ij}y_j$$
$$=p\cdot\sum_{i=1}^m x_i=p\cdot\sum_{i=1}^m\omega_i+p\cdot \sum_{j=1}^ny_j$$
$$=p\cdot\bigg(\sum_{i=1}^m x_i-\sum_{i=1}^m\omega_i-\sum_{j=1}^ny_j\bigg)=0,$$
so the value of aggregate excess demand is zero and Walras's law holds. But nothing in the argument requires prices to clear markets so that supply equals demand,
$$\sum_{i=1}^m x_i-\sum_{i=1}^m\omega_i-\sum_{j=1}^ny_j=0.$$
The latter is clearly a sufficient condition for excess demand to have the value zero, but very far from necessary.

Walras's law is a provable result under some standard assumptions (that guarantee that consumers spend their whole budget), but Say's law is not a formal result in any way.


What is the difference between the Classical and Keynesian models?

In the classical model, aggregate supply curve is vertical (price level on the y axis), meaning that output is fixed, constrained by technology and inputs. Prices are flexible. So that if the demand curve changes, the effect will be entirely on price level and not on output.

In the keynesian model, aggregate supply curve is horizontal at some price level. If demand changes, the effect will be entirely on output.

So the main difference lies on price flexibility and the power of increasing output through aggregate demand stimulus.


The speaker is too imprecise. Nobody wants to buy the equity tranche at a price that corresponds to its model valuation. If the tranche cannot be sold at that price, the bank structuring the CDO would take a loss. So they kept the equity tranche for themselves. Ideally, they would sell the equity tranche at model valuation, and eliminate their exposure to the deal (and just pocket the structuring fees for no long-term exposure risks).
It would be extremely unusual for someone to sell an equity of non-bankrupt firm at \$0, as the only conceivable motivation is to crystallize a loss for tax purposes. However, the equity can be made worthless in a bankruptcy restructuring, as the shares can be a claim on nothing. As such, the speaker in the video is speaking loosely.

There is no necessary relationship between the book value of a share and its market value. It is entirely possible that a firm has a negative book value for equity, yet a positive market price for its shares.


Hotelling rule and the shadow prices in a peg

Partial Answer
I can give you an answer for parts (1) and (2). As for (3), I can't give an answer unless you show some work as per our site policy on homework questions.

(1)These variables are interpreted as the change in $q$ or $p$ over time. Mathematically this is the derivative, giving us the results:
$$\dot q(t)=\frac{dq(t)}{dt}$$ and $$\dot p(t)=\frac{dp(t)}{dt}$$
(2) Hotelling's rule is not given in logarithms, as explained in as the notation is explained for (1). if it was it would be indicated by the notation of $\hat p(t)$. Written concisely as:
$$\hat p(t)=\ln(p(t))$$


Equilibrium in a labour market and price floors

As VCG stated, no government would impose a minimum wage below what is the countervailing equilibrium wage. To do so would be pointless.

Do remember that undergraduate discussions on minimum wages using the comparative statics (lines on graphs basically) model are erroneous because labour markets do not clear.


The table appears to have a bunch of errors in it.  In particular, the cash flow column seems wrong in every instance.  I don't blame you for having problems understanding.

For your first question, the cash flow in time 0 is wrong.  $1.1^0=1$ as we know.  The "end of year 0" means "right now," so there should be no discounting at all.  The cash flow should be -100,000, as indicated by the correct value in the present value column.

For your second question, cash flow is discounted at the required rate of return to get the present value.  In this case, presumably

$$
\frac{25,454.55}{(1+0.1)^1} = 23,140.50
$$

Why does that not equal the present value column?  Because the table was set up by someone who was confused.  The quantity they put in the present value is simply the discounted value of \$30,000 - \$5,000.

$$
\frac{30,000-5,000}{(1+0.1)^1} = 22,727.27
$$

To me that seems like a reasonable solution to the problem text.  The next period's cash flow should again be (\$30,000-\$5,000) and its present value should be

$$
\frac{30,000-5,000}{(1+0.1)^2} = 20,661.16
$$

Anyway, the present value column seems right given the text of the problem.  The "cash flow" column was done by someone who was as confused as you are, if not more so.

Probably you got this already, but to solve the problem you simply take the sum of everything in the present value column.

-------------- EDIT -------------

Now that I look at it again, I bet they just forgot to put parentheses around the (30000-5000) in each entry.  And left off a few zeros in the first.  Shabby, but not as bad as I was originally thinking.


Is there a proven, quantitative link between food wastage and food prices?

As far as I know, there are no economic studies studying the effect of food waste on food prices. In part, this may be due to difficulties in measuring food waste, as noted by a commenter. However, a more fundamental difficulty stems from a lack of 'exogenous' variation in food waste. The amount of food waste may vary significantly between places and times, but so do other factors (taxes, technology, consumer demand) which also influence the price of food.

In light of the lack of empirical studies on this topic, allow me to discuss it from a more theoretical point of view. You write

  Reducing food wastage should in theory reduce demand for food, thereby
  driving food prices down.

There are therefore two questions here:

If people cut back on food waste, will their demand for food fall?
If demand for food falls, will food prices fall?

In my view, the answers are 'yes' and 'maybe, maybe not'. Let me address these questions in turn.

Question 1

Alecos has produced an interesting argument as to why a reduction in food waste could have no effect (or even increase) demand for food. Essentially, he models a reduction in food waste as an exogenous decrease in the price of food. When the price of food falls, we consume more of it. However, for a given level of consumption, our demand for food falls. With logarithmic utility, these factors cancel out; but in principle, either factor could be more important (so food demand could increase or decrease).

While this is an intriguing argument, I find it rather unconvincing. First, many people cut back on food waste precisely to reduce the amount of food they buy. It would be perverse to claim that such people 'respond' to their reduction in food waste by buying more food. Second, in modern economies, people are already satisfying (and often exceeding) their calorific requirements. Hence, the binding constraint on the quantity (not necessarily quality) of food purchased is how much food they want to eat, not the amount of food they waste. Third, as a modelling choice, I think we should view 'food waste' as an active choice which consumers make as opposed to some parameter to which consumers optimally react. I doubt that Alecos' conclusion would hold once recognise that wasting food is a choice, though of course we could check this.

In my view, things are more simple. If people waste less food, their consumption of food will not significantly change. As a result, their demand for food must fall.

Question 2

However, I am not so confident that a fall in demand will lead to a fall in prices. To simplify, let us suppose that food is produced by identical perfectly competitive suppliers in a market with free entry. In that case, it can be shown that each producer will produce at the level that minimises their average cost; and that the market price will equal this average cost. Hence, the market price does not depend on demand.

Naturally, this is just market structure and it may be useful to think about others (e.g. oligopoly). However, I think that you will find that the effect of demand on prices is often ambiguous and depends on the economies of scale that are present in the industry.


Unintended consequences from "fixing" the US financial system

1) Would lead to the return of general panic led bank runs, and introduce additional instability to the system.

It´s not generally appreciated, that 19th century bank runs were not just a symptom of insolvency or illiquidity, but were also occasionally triggered by competitors (other banks) spreading rumours. Generally this is a bad idea that rests on the fallacy that banks are just another business.

2) They are typically rather more than 2-3%, 10-12% was the average at the crash, and Basel 3 is moving this up to above 15%. It´s generally a good idea, but it depends on the type of capital - more loss provisions are good, but simple holding extra share capital not so much. There are also side effects with respect to economic growth while the increase is occurring.

3) This seems a little confused. Depositors are not decision makers in banks - they are the customers. Owners, managers and shareholders of the banks are not necessarily the same people, but making in particular the managers and owners stand behind the banks with their entire assets might be interesting. As in ´may you live in interesting times´ interesting. Scottish banking in the 18th-19th century was regarded as more stable by the English since the entire property of the bank´s owners would be seized if the bank failed. In today´s world though, expect that to cause issues, if Banks are the only type of company that this kind of punitive ownership encumbers, then one might imagine that relatively few people will want to own one, or creative ways will be found around the requirement.

Banks are also usually regarded as a fairly safe investment (between major credit crises at least), and a good investment for pension funds and retirees, as their dividends provide a steady stream of income. Blowing those institutions up, might not be entirely desirable.

All of this not withstanding, the main issue with banks, and the answer to why they are not just another company - is that they provide the money supply. The problem posed by any banking crisis, and any proposed reform, is how does this avoid or solve monetary and credit supply contraction in the event of loan default failure. 1 & 3) in particular would be likely to cause significant monetary contraction, and return the financial system to the instabilities of the 19th century, where not to put to fine a point on it, it was bouncing up and down like a yo-yo. Wikipedia has quite a nice list:

List of Banking Crises


What determines if a variable is exogenous or endogenous in a model?

An endogenous variable is one whose value determined within the model. Exogenous variables are those variables whose values are determined by influences outside the model.


What are good books on economic world history?

Rondo Cameron's A Concise Economic History of the World may be helpful. I read an early edition many years ago and it certainly is primarily economic history rather than history of economic thought. One limitation however, identified in this review, is that despite its title it focuses mainly on Europe.


Total Differential of a Utility Function Subject to a Budget Constraint

In the second term of the intermediate calculation we must find the partial derivative of the entire second parameter of the $f$ function.  Hence
$$dU = \frac{\partial f}{\partial{q_1}}dq_1+\frac{\partial f}{\partial (\frac{y_0-p_1q_1}{p_2})}d(\frac{y_0-p_1q_1}{p_2})$$.  Obviously this is a substitution of $q_2$ into the original total differential derivation you had.  My point is that you tried to do the partial differential of $f$ with respect $q_2$ by assuming that this would equal $$\frac{dq_2}{dq_1}$$ when this may not always be the case.  For the final equation that you have may want to consider the chain rule.  The way you have it, we have defined $$U=f(q_1, q_2)$$ and $$q_2(q_1)=\frac{y_0}{p_2}-\frac{p_1}{p_2}$$  Then according to the chain rule you have $$\frac{dU}{dq_1}=\frac{\partial f}{\partial q_1}\frac{dq_1}{dq_1}+\frac{\partial f}{\partial q_2}\frac{dq_2}{dq_1}$$  Then we evaluate $$\frac{dq_1}{dq_1}=\frac{d}{q_1}q_1=1$$  Then we get $$\frac{dU}{dq_1}=\frac{\partial f}{\partial q_1}+\frac{\partial f}{\partial q_2}\frac{dq_2}{dq_1}$$  Then, $$\frac{dq_2}{dq_1}=-\frac{p_1}{p_2}$$ Finally we end up with $$\frac{dU}{dq_1}=f_1+f_2(-\frac{p_1}{p_2})$$


A timeline on the the key achievements of Economics and its evolution

I don't think I can really do this question justice. A lot of us probably have different things that come to mind when we think of the history of economics. I tend to take an academic interpretation of your question.

A potpourri of random (mostly microecon, but maybe it'd be better to sort them) events that come to mind:

1838 - Supply and Demand:
Antoine Augustin Cournot publishes a book about the supply and demand model (Alfred Marshall makes it famous later).

1870s - General Equilbirum Theory:
William Stanley Jevons, Carl Menger, and French economist Leon Walras develop around the same time the idea of marginal utility, and Walras in particular creates his work in 1874, "Elements of Pure Economics"

1915(?) - Slutsky Decomposition:
Eugen Slutsky relates Marshallian and Hicksian demand to separate income and substitution effects. In 1927, he did something with statistics and business cycles that I don't quite understand, but I think that was important too.

1930 - Intertemporal Consumption:
Irving Fisher lays out early work on discounted saving, though arguably,  Modigliani & Brumberg (1954), Albert Ando, and Milton Friedman (1957) and the work on the life cycle-model is more noteworthy.

1944 - Game Theory:
John von Neumann and Oskar Morgenstern use Brouwer's fixed-point theorem to create the basis for game theory.

1947 - Von-Neumann Morgenstern Utility
John von Neumann and Oskar Morgenstern create their four axioms surrounding rationality with probabilistic utility outcomes.

1951 - Arrow's Impossibility Theorem:
Joseph Kenneth Arrow publishes his book, "Social Choice and Individual Values"

1973 - Black-Scholes Pricing Model:

1977(?) - Mechanism Design:
Leonid Hurwicz, Eric Maskin, and Roger Myerson create the foundation for "reverse game theory".

As you can see, there are lots of gaps. I still don't know a lot about economic history--just that it is very large.

Other topics to add:  Behavioral economics, experimental economics, information economics


price elasticity of output, makes sense?

I use some abbreviations and drop the index t.

$\ln(G)=\beta_0+\beta_1\ln(L)+\beta_2\ln(P)+\epsilon$

Taking both sides as an exponent of e

$e^{\ln(G)}=e^{\beta_0+\beta_1\ln(L)+\beta_2\ln(P)+\epsilon}$

$G=e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot e^{\beta_2\ln(P)}$

$e^{\beta_0+\beta_1\ln(L)+\epsilon}=C$ is a constant if you differentiate w.r.t $P$.

$G=e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot e^{\ln(P^{\beta_2})}$

$G=e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot P^{\beta_2}$

$\frac{\partial G}{\partial P}=e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot \beta_2\cdot P^{\beta_2 -1}$

The elasticity is $\large{\epsilon}\normalsize{=\frac{\partial G}{\partial P}\cdot \frac{P}{G}}$

$\large{\epsilon}=\normalsize{e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot \beta_2\cdot P^{\beta_2 -1}\cdot \frac{P}{e^{\beta_0+\beta_1\ln(L)+\epsilon}\cdot P^{\beta_2}}}$

=$C\cdot \beta_2\cdot P^{\beta_2 -1}\cdot\frac{P}{C\cdot P^{\beta_2}}$

You can finish by simplifying the term.

In general it is true that

$\large{\epsilon}=\normalsize{\frac{\partial \ln(G)}{\partial\ln(P)}}$

Thus the short way is to differentiate $\ln(G)$ w.r.t $\ln(P)$.


Economic term for relation between drug and companion diagnositic

Product A and B are perfect complements. As the Wikipedia entry states:

  A perfect complement is a good that has to be consumed with another good. [...] One example is a left shoe and a right; shoes are naturally sold in pairs, and the ratio between sales of left and right shoes will never shift noticeably from 1:1, even if, for example, someone is missing a leg and buys just one shoe.

The example of the shoe matches the one of the bicycle you give.

Additionally, the two goods do not need to have a symmetric degree of complementarity. Again, quoting Wikipedia:

  In the case of video games, a specific video game (the complement good) has to be consumed with a video game console (the base good). It does not work the other way: a video game console does not have to be consumed with that game.

So, for example the base good can be the diagnostic and the complement is the drug.


Mundlak Approach when panel data include time fixed effects

If the panel data set is balanced, the averages will cause perfect collinearity and will all be dropped. So you lose nothing.

If it is unbalanced, the averages of the time dummies will explain some aspects of the structure of the data set. For example, let T=3 so we have two dummies. B2bar(i) is 1/T(i) if observed at t=2, and 0 otherwise. Similarly, B3bar(i) depends on whether t=3 is observed and on T(i). So, I think the individual averages of the time dummies control for heterogeneity due to the period effects and data structure heterogeneity. I personally believe that it is safe to include those Btbar(i) terms. But there may be more fundamental issues involved in this unbalanced case as it relates to modeling selectivity.


The integral of interest is $\int_{w^*}^Bw'dF(w')$, which we get after bringing $\frac{1}{1-\beta}$ out from the last integral on the second line. I assume that $F(B)=1$. First, we have:

$$
\int_{w^*}^Bw'dF(w')=w'F(w')\vert_{w^*}^B - \int^B_{w^*}
F(w')dw'
$$
from integration by parts with $u=w'$ and $dv=dF(w')$. We can evaluate the first term on the right-hand side and both add and subtract $B-w^*=\int^B_{w^*}dw'$ to get:

$$
\int_{w^*}^Bw'dF(w')= [B - w^*F(w^*)]-[B-w^*]+\int^B_{w^*}(1-F(w')dw'.
$$

Note that I used my initial assumption here, and that I brought $B-w^*=\int^B_{w^*}dw'$ under the pre-existing integral.It should be clear after a bit of algebra that:

$$
\int_{w^*}^Bw'dF(w')= w^*(1 - F(w^*))+\int^B_{w^*}(1-F(w')dw'.
$$

The rest should be easy after multipling both sides by the $\frac{\beta}{1-\beta}$ factor.


What is the economic meaning of distribution parameter in a CES-production function?

In this answer you can see that $\delta$ becomes the exponent in the Cobb-Douglas function as $\rho \to 0$, i.e. as the elasticity of substitution becomes unity.
The exponent in the Cobb-Douglas production function is the output elasticity of the input(s), but also, it is the "output share" of each factor, under competitive conditions. This is why the scholars that introduced the C.E.S. function named this parameter "distribution parameter".
So you can start thinking from there.


If you pick pl=2 and pu = 4 and your data is daily, then indeed you filter out the trend that has frequency up to 4 days. So, your trend line is a sum of trigonometric functions with the aforementioned periodicity. The cycle comprises all the other remaining frequences.


In the Ramsey model, how does it make sense to say that capital equals wealth when wealth can be negative and capital cannot?

No matter in Ramsey model or the basic structure of Solow-Swan model, as long as is in the closed economy. The asset equals to the capital in the equilibrium.

$A_t$ is the total asset owned by the households, or by the representative household. In the market equilibrium, all the borrowing and lending must cancel within the economy. Therefore, the only asset in positive net supply is capital, which means $A_t = K_t $. The asset equals the capital stock quantity.

Therefore, the asset per person $a_t$ = the capital stock per person $k_t$. It seems like you confuse $A_t$ with $a_t $.


What are the effects of the increase/decrease in stock options after an IPO?

Exactly.  The main issue is if they want to raise capital a second time. The secondary market does not affect the company itself, but it affects its investors. And as investors are important, they need to be treated well.  Another problem related to this is that several employees in the company have stock options, so it really affects them.

What companies can do also is buy their own stocks if the price dropped considerably. This is not something that happens right after the IPO because it would make no sense to return the money to the markets right after you received it.

So, to sum up, the direct effects for the company are on the primary market (when the company makes the IPO), the secondary market is between public investors that have little to do with the company.


We say that $x \le y$ when $x_i \le y_i$ for each $i=1,\ldots,n$.
If the production set can be represented by a production function $F$, and $F$ is homogeneous of degree $r < 1$ ($F(\lambda x) = \lambda^r F(x)$ for $\lambda >0$), then the production set satisfies the non-increasing returns to scale property you state. Moreover, if $F(0)=0$, and $F$ is concave, the production set also satisfies non-increasing returns to scale. Hence, you can think of the definition as a generalisation of the intuitive understanding you have.
An isoquant takes the output level as given and specifies all the input combinations that give you exactly the given level of output. The transformation frontier is the frontier of the production possibilities set. The production possibilities set takes the inputs as given and specifies all the output combinations that are possible for the given inputs. You can think of the transformation frontier as the output bundles that are not wasteful (i.e. efficient).


Find Pareto optimal allocations and the core for the following economies

In the economy described above, set of efficient allocations is given by the red curve. Just do the slope of ICs comparisons at the boundaries and you will get that.


Difference-in-Differences setup for policy evalution

I think a DiD will not work here as the parallel trend assumption probably doesn't hold in your case.
However, given that the treatment is based on a number of parameters that are (I assume) observed, you could potentially use some type of discontinuity design to estimate the effect of the policy around the threshold values of these parameters.
To be more precise, your treatment neighbourhoods would be the the neighbourhoods that have the sign but are very close to loosing it, while the control neighbourhoods would be the neighbourhoods that are just below the threshold for obtaining the sign. Then if prices increases are a continuous function of all other covariates, the difference in treatment vs control would measure the effect of the signs for neighbourhoods around the threshold.
Of course there are potential endogeneity issues that you'll need to take into account as you can expect neighbourhoods to do some effort to get above the threshold.


All else being equal, What is the value of job creation?

You are correct that creating jobs in themselves, generally speaking does not have any economic value (save some exceptional situations - see below). In fact this is not novel observation, and you will find this stated in any 101 economic textbook for example see Mankiw Principles of Economics 8ed pp 33. As written in the textbook (in an excerpt from Economics article):

Little in the literature seems more relevant to
contemporary economic debates than what usually is called the broken window fallacy. Whenever a government program is justified not on its merits but by the jobs it will create, remember the broken window: Some teenagers, being the little
beasts that they are, toss a brick through a bakery window. A crowd gathers and laments, “What a shame.” But before you know it, someone suggests a silver lining to the situation: Now the baker will have to spend money to have the window repaired. This will add to the income of the
repairman, who will spend his additional income,
which will add to another seller’s income, and so
on. You know the drill. The chain of spending will
multiply and generate higher income and employment. If the broken window is large enough, it
might produce an economic boom! . . .

Most voters fall for the broken window fallacy,
but not economics majors. They will say, “Hey,
wait a minute!” If the baker hadn’t spent his
money on window repair, he would have spent it
on the new suit he was saving to buy. Then the
tailor would have the new income to spend, and
so on. The broken window didn’t create net new
spending; it just diverted spending from somewhere else. The broken window does not create
new activity, just different activity. People see the
activity that takes place. They don’t see the activity that would have taken place.
The broken window fallacy is perpetuated in
many forms. Whenever job creation or retention
is the primary objective I call it the job-counting fallacy. Economics majors understand the
non-intuitive reality that real progress comes
from job destruction. It once took 90 percent of
our population to grow our food. Now it takes 3
percent. Pardon me, Willie, but are we worse off
because of the job losses in agriculture? The
would-have-been farmers are now college professors and computer gurus. . . .
So instead of counting jobs, we should make
every job count

The above being said there is some value to creating/preserving jobs in some situations. For example, during recessions some public works program might be part of a mix that is intended to stimulate the economy, or during short but severe unpredictable crises such as the current Covid-19 situation.
For example, the Furlough/Kurzarbeit schemes  are generally considered by conventional economists  to be valuable as they can help boost aggregate demand and prevent even more severe recession (Giupponi & Landais; 2020),however such support is exceptional.


Shapley Value - Any Real World Applications?

Allegedly, Barclays uses (or once used) Shapley allocation. Quoting Mauro Cesa:

The basic objective of every bank is to find an optimal business strategy that maximises return on capital (ROC). To this end, banks will allocate more capital to desks that generate the highest ROC, while those with lower ROC receive a smaller share of available capital. This is an intuitive and seemingly sensible solution.  But the authors of this month’s paper, Reduced-form capital optimisation, argue that ranking business units by ROC might not result in an optimal allocation of capital. The approach ignores the correlation between businesses and will deliver the optimal allocation only if the correlation is zero – an assumption for which there is little evidence in the real world.  The problem is further complicated by the Basel III capital rules, which are primarily based on two ratios: risk-weighted assets (RWAs) and leverage balance sheet (LBS). The minimum capital requirement for each legal entity under the same parent bank is the greater of the two.  To optimise its ROC, a legal entity must hold RWA and LBS capital in equal measure. The selection of RWA or LBS, as directed by the greater-of-the-two rule, introduces non-linearities that are difficult to deal with in an optimisation context.  “This is a huge, unsolved problem for banks,” says Yadong Li, managing director of quantitative analytics at Barclays, and one of the authors of the paper. “The bank as a whole ultimately needs to deploy the resources to different business units. This is a key part of senior management’s job.”  Dimitri Offengenden, a Tel Aviv-based managing director of quantitative strategy at Barclays, joined the bank in 2017 specifically to work on the capital allocation problem. He consulted with Li, who has worked on related issues in the past.  “I suggested we look at Shapley allocation, which is a natural way to solve the greater-of-two problem” says Li.  Shapley allocation is well known in game theory and has been widely applied in economics and business decision making. It attributes a value to the contribution of each agent in a system where agents co-operate and share the costs and gains of their activity. In essence, it measures the marginal contribution of each agent by observing the difference the agent’s presence or absence makes to the activity. Based on that, a portion of the shareable pie is allocated to each agent. In the case of a bank, the agents are individual business units and the pie is the overall capital stock.  Offengenden, Li and Jan Burgy, a quant strategist at Barclays, realised that regressing the overall allocation of capital on the selection of RWAs and LBS obtained from the application of the Shapley method resulted in an almost perfect approximation.

Mauro Cesa, Game theory plays well for capital management, Risk.net, September 30, 2019.

Yadong Li, Dimitri Offengenden, Jan Burgy, Reduced form capital optimization, May 15, 2019.


How does one prove the convergence of a trajectory in the Ramsey model?

See Romer (2006, Sec 2.3) for details of notation.
Dynamics are given by
\begin{align}
\dot k &= f(k) - (n+g)k - c\\
\dot c &= \frac{c}{\theta}(f'(k) - \rho - \theta g)
\end{align}

Presume you are located somewhere on the $\dot k = 0$ locus. Now if we increase $c$ one gets
\begin{align}
\frac{\partial \dot k}{\partial c} = -1 < 0
\end{align}

That is, if we are above $\dot k = 0$ capital decreaes over time. Hence at point $A$ we move to the left.

Now presume you are located somewhere on the $\dot c = 0$ locus. Now if we increase $k$ one gets
\begin{align}
\frac{\partial \dot c}{\partial k} = \frac{c}{\theta} f''(k) < 0
\end{align}

That is, if we are right of $\dot c = 0$ consumption decreaes over time. Hence at point $A$ we move up in addition (left of $\dot c = 0$).


Here are three types of money:

Commodity money: goods of value are used as money (e.g. gold in your example, or cigarettes in a prison),
Representitive money: money is still based on a commodity but, instead of actually using that commodity in exchange, they use pieces of paper that can be exchanged for the commodity (e.g. the gold standard),
Fiat money: money with no intrinsic value, which is trusted by people because they government tells them it will be the official medium of exchange into the indefinite future (e.g. the modern US dollar, British pound, Euro, etc.).

A fourth type that is emerging (e.g., in the form of Bitcoin) has no intrinsic value and no official backing from government. It is accepted in exchange solely because people believe that will be able to spend it again in the future.

The physical burden of physical money

Firstly, there is a serious practical problem associated with commodity money in its purest form, namely that transporting and storing the money is difficult and expensive. If I want to buy a house, am I supposed to work around with half a million dollars worth of "stuff" in a briefcase and hope nobody steals it? What if I want to buy a book from Amazon? Do I have to send them some gold coins in the mail?

At the very least, these considerations are going to compel people to use representative rather than commodity money.

The problem of debasement

Where commodity money has been used historically, it has been exposed to the problem of "shaving". Since the currency is intrinsically valuable, there is an incentive to shave a tiny amount off of each coin (such that the difference is indistinguishable from normal coin wear) and melt those shavings together into a piece of precious metal that can be sold. You can read more about this here.

Takes away some valuable policy tools

Imposing a commodity money would take from the government some valuable tools to deal with economic shocks. In particular, it reduces the ability of the government to use monetary policy to reduce the real effect of economic downturns. For example, during the most recent recession, major governments engaged in "quantitiative easing", which saw them creating new money to stimulate the economy. This would be much more difficult if creating new money meant having to dig a significant quantity of gold out of the ground of another country.

Additionally, it is not obvious how big/genuine the advantages you describe are. For example, here is a plot of US inflation:

Note that inflation tended to have higher peaks and be much more volatile during the first half of the C20 than in recent years. But that was exactly when the gold standard was in effect and the value of currency was directly tied to a physical commodity. Also, deflation was more common then, which is problematic because deflation has a number of damaging effects on an economy. Modern policy makers are better able to regulate the money supply to control inflation and keep it around the low and stable levels thought to be optimal.

People might be more willing to trust commodity money. But they would have to overcome concerns about counterfeiting and debasement. Recent experience from the last half century has been that people in stable democracies are, in fact, quite willing to trust fiat money.

Edit in response to comment:

It's true that precious metals are sufficiently valuable that large quantities are not needed to make most payments. But the problem remains that (a) if I am carrying them around to make payments then I am subject to a new source of  theft/loss that is less acute with electronic financial balances; (b) insisting on a physical commodity money makes it hard to transact remotely (e.g. buying goods online or remitting money to pay for bills or an invoice); (c) businesses regularly need to spend, store, and move money on the order of millions or sometimes billions; often across borders or oceans. Moving from a world in which money can be stored and moved at essentially zero real cost, with essentially zero delay, to one where we are burning real resources to move physical stuff around would be a huge technological step backwards.
I don't understand what you mean about debit cards. Your whole question started out with "...not the version I always see mentioned (trading bills for gold at a set price). I mean having gold coins in circulation..." i.e. we are talking about commodity money, not representative money. Yes, debit cards fix the problem of physics, but they also mean we aren't talking about commodity money any more. Having a debit card you can hand to your bank in exchange for gold is really equivalent to having a US dollar banknote that you can hand in to exchange for gold.
Yes, the US dollar has lost a lot of its purchasing power. But that's a good thing! There are numerous economic benefits to a low (but positive) stable rate of inflation. Inflation per se is not bad, it only becomes problematic when we have either very high levels of inflation, or when the rate of inflation is not stable (both were true under the gold standard).


Is there a systematic approach to find equilibria in sequential games?

You can first find all NE. Then you check which ones are subgame-perfect. Then you proceed and check for which of the NE you can find beliefs that are consistent with the definition of PBE. You can go on and refine the set of equilibria further by kicking out all equilibria that do not satisfy the additional requirements of your stricter equilibrium concept.

If by "backward induction" you mean solving the game backwards subgame-by-subgame, then by definition you find all the SPNE (= all NE in which the equilibrium strategy profile also constitutes NE in all subgames).


Definition of Pareto efficiency and prisoner's dilemma

(A, B) and (B, A) are in fact Pareto efficient.
I believe that your confusion may be because when discussing the Pareto inefficiency of the Prisoner's dilemma equilibrium, we always discuss (B,B) as the Pareto efficient alternative to (A,A) and (almost) never discuss (A,B) or (B,A).
Keep in mind that Pareto efficiency always requires a starting point to be evaluated. I.e. one way to determine if a case is Pareto efficient is to ask "Starting from this point, is there any other allocation that is a Pareto improvement?". Here a Pareto improvement is a case making at least one person better off and no one worse off.
A natural starting point of discussion is the Nash equilibrium (A,A). Starting from there, only (B,B) is a Pareto improvement, which suffices to show that (A,A) is not Pareto efficient. That is what economists like to emphasize about the Prisoner's Dilemma and why textbook discussions focus on (B,B).
However, if your starting point is (A,B) or (B,A) then there are no Pareto improvements possible, as you argue. Hence, both those cases are Pareto efficient as well. However, note that these are not very interesting starting points anyway, because they are not Nash equilibria and therefore a bit irrelevant to discuss.


Your assumptions seem correct. For savings rate, the consistent definition with MRW would be

$$ \frac{ccon - cda}{cgdpo} $$

ccon is domestic absortion, equal to $ C + G+ I$, whereas cda is $C+G$. Therefore, $I$ is investment, which by definition includes both private and public sector.


At its simplest, you just look at the cost of capital: if the household would have to borrow to pay the debt, then what's the borrowing rate? If they are currently saving, what interest rate would they forego by spending on the retrofit instead of saving?

The benefits of retrofit usually extend beyond the simple savings on energy bills, though. When it's done well, it also makes the home more comfortable and more healthy; and sometimes reduce noise from outside too. These things have value, but that value can be harder to quantify. Nevertheless, the value is certainly greater than zero, so just assuming zero because it's difficult to get a decent number, leads to the business case for retrofit being systematically under-valued.


Barro's (2009) rare disaster model in the AER: How to derive equation (10)?

I think Barro means in the footnote that Giovanni and Weil find the same equation, $U_t=\Phi C^{1-\gamma}$, but using the optimal path of $C_t$.
In Barro's paper, the approach is different given that the dynamics of $C_t$ is exogenous: $C_t=Y_t$ by assumption.

Barro uses the limit case when the length of a period gets close to 0. Maybe what may bother the reader is that the model is defined as discrete.

Rewrite the model

First, we can rewrite the model with a length of period $\delta$ and then use $\delta\to 0$.
The GDP dynamics write
$$\log(Y_{t+\delta})=\log(Y_t)+g\delta+u_{t+\delta}+v_ {t+\delta}$$
with $u_{t+\delta}\sim \mathcal{N}(0,\delta\sigma^2)$, and $v_{t+\delta}=0$ with probability $1-p\delta$ and $\log(1-b)$ with probability $p\delta$.
The utility satisfies
$$
U_t=\frac{1}{1-\gamma}\left\lbrace C_t^{1-\theta}+\frac{1}{1+\rho\delta}\left[(1-\gamma)E_tU_{t+\delta}\right]^\frac{1-\theta}{1-\gamma}\right\rbrace^\frac{1-\gamma}{1-\theta}.
$$

1) Find $\Phi$ as a function of $E_t\left[\left(\frac{C_{t+\delta}}{C_t}\right)^{1-\gamma}\right]$

From now suppose there is a $\Phi$ such that $U_t=\Phi C^{1-\gamma}$ (note that $\Phi$ depends on $\delta$ a priori).
Define $H(U)=[(1-\gamma)U]^\frac{1-\theta}{1-\gamma}$, the utility satisfies
\begin{align}
H(U_t)= C_t^{1-\theta}+\frac{1}{1+\rho\delta}H(E_tU_{t+\delta}).
\end{align}
We substitute $U_t$:
\begin{align}
H(\Phi)C_t^{1-\theta}= C_t^{1-\theta}+\frac{1}{1+\rho\delta}H(\Phi)\left(E_t\left[C_{t+\delta}^{1-\gamma}\right]\right)^\frac{1-\theta}{1-\gamma}.
\end{align}
Hence, we obtain for $C_t\neq 0$,
\begin{align}
\frac{1}{H(\Phi)}= 1-\frac{1}{1+\rho\delta}\left(E_t\left[\left(\frac{C_{t+\delta}}{C_t}\right)^{1-\gamma}\right]\right)^\frac{1-\theta}{1-\gamma}.
\end{align}

2) Find $E_t\left[\left(\frac{C_{t+\delta}}{C_t}\right)^{1-\gamma}\right]$ fromp the GDP dynamics

The trick is to find the expectation in the right-hand side from the GDP dynamics.
\begin{align}
\left(\frac{Y_{t+\delta}}{Y_t}\right)^{1-\gamma}= \exp\left((1-\gamma)g\delta\right).\exp\left((1-\gamma)u_{t+\delta}\right).\exp\left((1-\gamma)v_ {t+\delta}\right).
\end{align}
Taking the expectation and using the independence between $u_{t+1}$ and $v_{t+1}$, it follows
\begin{align}
E_t\left(\frac{Y_{t+\delta}}{Y_t}\right)^{1-\gamma}= \exp\left((1-\gamma)g\delta\right).E_t\exp\left((1-\gamma)u_{t+\delta}\right).E_t\exp\left((1-\gamma)v_ {t+\delta}\right).
\end{align}
The expectation of $\exp(X)$ where $X$ follows $\mathcal{N}(0,\sigma^2)$ is $\exp(\sigma^2/2)$. $\exp\left((1-\gamma)v_ {t+\delta}\right)$ is a random variable equal to $1$ with probability $1-p\delta$ and $(1-b)^{1-\gamma}$ with probability $p\delta$.
We substitute the expectation operator:
\begin{align}
E_t\left(\frac{Y_{t+\delta}}{Y_t}\right)^{1-\gamma}= \exp\left((1-\gamma)g\delta\right).\exp\left(\frac{(1-\gamma)^2\sigma^2\delta}{2}\right).\left(1-p\delta+pE[(1-b)^{1-\gamma}]\delta\right).
\end{align}
Finally, we use $C_t=Y_t$ to compute an equation for $\Phi$:
\begin{align}
\frac{1}{H(\Phi)}&= 1-\frac{1}{1+\rho\delta}\left\lbrace\exp\left((1-\theta)g\delta\right).\exp\left(\frac{(1-\gamma)(1-\theta)\sigma^2\delta}{2}\right)\right.\\
 &\left. .\left(1-p\delta+pE[(1-b)^{1-\gamma}]\delta\right)^\frac{1-\theta}{1-\gamma}\right\rbrace.
\end{align}

3) Take the approximation $\delta\to 0$

The last step consists in taking a first-order approximation (I abusively keep the equal symbol):
\begin{align}
\frac{1}{H(\Phi)}&= 1-(1-\rho\delta). \left(1+(1-\theta)g\delta\right).\left(1+\frac{(1-\gamma)(1-\theta)\sigma^2\delta}{2}\right)\\
 & .\left(1-\frac{1-\theta}{1-\gamma}p\delta+\frac{1-\theta}{1-\gamma}pE[(1-b)^{1-\gamma}]\delta\right).
\end{align}
Pursuing the first-order apprixmation (all the $\delta^i$ with $i>1$ can be neglected), we have
\begin{align}
\frac{1}{H(\Phi)}&= \rho\delta -(1-\theta)g\delta-\frac{(1-\gamma)(1-\theta)\sigma^2\delta}{2}\\
 & +\frac{1-\theta}{1-\gamma}p\delta-\frac{1-\theta}{1-\gamma}pE[(1-b)^{1-\gamma}]\delta.
\end{align}
Substitute $g$ using $g^*=g+\frac{\sigma^2}{2}-pEb$,
\begin{align}
\frac{1}{H(\Phi)}&= \rho\delta -(1-\theta)g^*\delta+(1-\theta)\frac{\sigma^2}{2}\delta -(1-\theta)pEb\delta -\frac{(1-\gamma)(1-\theta)\sigma^2\delta}{2}\\
 & +\frac{1-\theta}{1-\gamma}p\delta-\frac{1-\theta}{1-\gamma}pE[(1-b)^{1-\gamma}]\delta.
\end{align}
We take $\delta=1$ and invert function $H$ to find the solution in the footnote 7 of the paper. The right-hand side of this equation "simplifies" to the within braces in the formula.


Can the coming recession - or any recession - be avoided?

So here is the unpleasant truth about recessions.
"The economy" as a whole consists is a broad term that in all forms, will always be comprised of so many moving parts, so many transactions, that it is hard to not only describe what is happening to it, but why those things are happening. In order to answer your question on why we "can't/don't" do anything about recessions, you have to understand that recessions are harder to predict that whatever the news makes it out to be. We can maybe see the signs that they are just about to come, when the whole snowball is already rolling down the hill, but trying to predict recessions from the long term is basically impossible.
That said, there are two points I'd like put into focus.

We do have some tools for trying to blunt the force of recessions, and we do try to use them!

The warning signs you are reading about, what I imagine most notably being the yield curve inverting, is only a warning sign for certain reasons that don't necessarily have to hold in our current economy.

To the first point, I say "blunt" because trying to completely avoid recessions would probably mean making some big systematic changes, but also probably getting rid of a lot of uncertainty in the world. And our world is unfortunately full of it. We can argue over what causes boom and bust cycles and whether we would prefer to live in a world where growth is slow, but we can consumption smooth and prevent all dips in economic activity. Those are separate issues.
What tools do we have to blunt recessions? There is fiscal policy and monetary policy. With fiscal policy, you can change taxes to try to encourage economic activity (for consumers, business, depends on where you cut the taxes), and with monetary policy you can lower the federal funds interest rate to encourage more lending and thus more investment. But of course these tools may not work as directly as we'd like. Cutting taxes may have to come with cutting spending on government programs that also encourage economic activity. During the 2008 recession, the federal funds late got floored down to 0%, leading the Federal Reserve to have to think of more creative measures of using monetary policy to help boost the economy.

The second point I think is more interesting to talk about. The financial news these days probably talks a lot about the yield curve inverting. What is this exactly?
Imagine you are the government borrowing money through bonds. People value having money as accessible as they can, so a short term bond usually doesn't have a high interest rate. Investors don't get a high rate of return because they are getting compensated with more liquidity. If the government wants to borrow money for a longer time though, then the government will put a higher interest rate on it. This compensates investors who have to sit there and wait for their money longer, money that could have been used to invest in a more lucrative project that could have popped up in the meantime. There is uncertainty in the path of future interest rates, so you take on risk by holding onto a debt contract longer.
So if you were to graph "length of time of a bond or other debt contract" (time to maturity) and "interest rate" (yield), you would normally get an upward sloping curve that tapers off. Why this particular shape? Again, holding shorter term debt is pretty preferred among investors, so there is a steep upwards slope initially. The relative difference between a 3 month contract and a 6 month contract is a lot bigger than the relative difference between a 40 year and 40.25 year contract.
So now we can ask why the yield curve would invert. This usually happens if investors expect a cut in future short term rates. The Federal Reserve is known as the "lender of last resort" and will more or less set the price of liquidity using the federal funds rate. If people expect the Fed to cut it, then the yield curve will invert. And the Fed often only makes big cuts to the federal funds rate to try and prevent or blunt a recession. So that's why the yield curve inverting is usually associated with a nearby recession.
But suppose the yield curve inverts, the Fed cuts the rate and...there is no recession. And maybe the Federal Reserve never anticipated one. This could feasibly be the case too. The idea is that when setting the federal funds interest rate, usually unemployment and inflation tend to be tradeoffs, but they aren't always. Lowering the rate in recent years doesn't seem to have increased inflation expectations very much (a lot of that probably due to the aforementioned 0% bound that was around from the 2008 recession). Since the 1970s reforms to the Federal Reserve, the institution has gotten a reputation for being more committed to fighting inflation in the long run, which also helps with their ability to cut rates and not immediately see inflation boost up. So ultimately it's possible that the Fed may want to cut the rate for other reasons, like simply maintaining a healthy economy.

All that said, you might still be concerned about the yield curve inverting or other indicators that you read in the news. I think it is not unreasonable. But I think the important thing to realize is that people in the news industry frankly do not really understand or even care to explain the nuance behind these signals. They may not understand or explain what is being done about these signals at the policy level. A lot of mainstream news competition is just trying to get people's attention, and honestly who is going to watch a segment on monetary policy. Snore. So the reason you do not see anyone talking about how to respond to a recession is probably just because you don't listen to anyone who really knows much about them.
Really, at the core of your question, you are basically asking why don't economists just predict all the bad stuff that can cause a recession and then stop it. The answer is because we just can't observe all of that, and it's goofy for anyone to think we can. We can only see some things, respond using some limited tools, and give our best policy suggestions. It is wishful thinking to believe we can stop all or even most of hardship in this uncertain world.

If you would like to read more about some recent research on recessions and macroeconomic policy, here are just a small number of papers you could try reading. Even if you don't read beyond the abstract, perhaps it will give you an idea of what sort of research economists are trying to do related to recessions and business cycles.

Aikman, David, Jonathan Bridges, Anil Kashyap, and Caspar Siegert. "Would macroprudential regulation have prevented the last crisis?." Journal of Economic Perspectives 33, no. 1 (2019): 107-30.

Vassolo, Roberto, Javier Garcia-Sanchez, and Luiz Mesquita. "Competitive dynamics and early mover advantages under economic recessions." Revista de Administração de Empresas 57, no. 1 (2017): 22-36.

Martin, Philippe, and Thomas Philippon. "Inspecting the mechanism: Leverage and the great recession in the eurozone." American Economic Review 107, no. 7 (2017): 1904-37.

Koijen, Ralph SJ, Francois Koulischer, Benoit Nguyen, and Motohiro Yogo. Quantitative easing in the euro area: The dynamics of risk exposures and the impact on asset prices. No. 601. 2016.


Are there any economic theories around the concept of companies focusing on that which is easier to calcuate?

Your question is ultimately one about information flow in networks under conditions of imperfect and incomplete information. I like your distinction between "easy" and "hard" to calculate - it's a good rubric for problem classification.

I believe that work being done on complexity theory and information flows can help shed some light on how to think about the issue. Cesar Hidalgo is a leading thinker in this field and has assembled some very interesting tools in the context of global trade (the Atlas of Economic Complexity being one). At the level of organizations, there would be an additional layer of complication arising from behavioural anomalies in decision-making that tend to bias human beings in favor of decisions cast in terms of simple data sets with clear narratives. This is the other side of the coin to Nassem Talib's "narrative fallacy", and of course, any study of behavioural biases must also include some time spent with the work of Kahneman and Tversky.


When do firms pursue the cost minimization objective instead of profit maximization? Are there any intuitive/practical examples/reasons?

Firms can be said to pursue cost minimization when they do not have control over demand of their product. In the short run, this implies also that unit prices are fixed.

A standard example here is a Utility, say for electricity. Except in extreme situations of very high demand that reaches the network capacity, consumers consume electricity on demand for a given unit price or price schedule.  So "quantity produced" is not determined by the firm, and so the Cost-minimization problem is the most appropriate (minimize costs given output).


Goldfeld - Quandt test statistic equal to 1

The situation implicitly described in the question assumes that the two sub-samples have equal size (which is not necessary for the test to go through).

In such a situation, if the ratio $RSS2/RSS1$ equals unity, then the null-hypothesis of homoskedasticity won't indeed be rejected for any significance level up to $0.5$, (to be compared to the conventional levels of $0.1,\; 0.05,\; 0.01$)  because the statistic used, under the null hypothesis follows an F-distribution (exactly or asymptotically), and more over, one that has equal degrees of freedom for the numerator and the denominator (due to the equal sub-sample sizes).

This also makes intuitive sense: if the obtained estimates for the sub-sample variance are equal, then, in order to reject the null, we should accept that it would be more probable than not, that the rejection will be mistaken.