Are no arbitrage models and equilibrium models equivalent?

...no-arbitrage models (such as Black-Scholes and HJM) are equivalent to
equilibrium models (such as CAPM or C-CAPM).

Short Answer Yes, for models where asset prices are assumed to be Ito semimartingales (where the martingale part is a Brownian integral), although a more general argument is needed than that suggested by the special cases typically encountered in finance.
Clearly, no-arbitrage is a necessary condition for general equilibrium.
To claim equivalence is then to say it is also sufficient, i.e. given any price process $P_t$ and a density process $D_t$ such that the discounted price $e^{-rt} P_t$
is a martingale after change of measure by $D_t$, one needs to find
a (say, representative) investor $u$ and a equilibrium consumption process $c_t$ such that
$$
e^{-\beta t} u'(c_t) = \lambda e^{-rt} D_t, \quad\quad (*)
$$
for some $\lambda > 0$.
In other words, one needs a "SDF/marginal utility representation" for equivalent martingale measure densities.
The equation $(*)$ is the usual relationship
$$
\mbox{marginal utility} \; \propto \mbox{price}.
$$
If you write down a heuristic Lagrangian, $\lambda$ is the Lagrange multiplier in the FOC.
In general, FOC's of this type is only necessary for optimality of $c_t$.
If $(*)$ is sufficient for optimality of $c_t$, you can take the dividend to be $c_t$ in $(*)$ and $P_t$ becomes the equilibrium price faced by the representative investor $u$.
With certain assumptions on $u$---such as concavity and the Inada condition,
Karatzas, Lehoczky, Shreve (1987) showed that this can be done when $P_t$ is an Ito semimartingale and market is complete. (See also Cox and Huang (1989).) The rigorous argument makes use of convex duality and is referred to as the martingale duality method in mathematical finance.
The Ito semimartingale case certainly covers many---perhaps most--- asset pricing models in finance.
In fact, asset prices are usually assumed to follow a very special Ito semimartingale---geometric Brownian motion, where the risk neutral density $D_t$ is itself an exponential martingale.
Then $(*)$ takes the special form
$$
\frac{dM}{M} = - r dt + \frac{dD}{D},
$$
where $M$ is the SDF.
One can then take the representative investor to be CRRA and, since powers of exponential martingales are still exponential martingales, back out an SDF $M$ without referring to a more general argument.
For example, as you already point out,
the Black-Scholes formula for pricing European call option in complete markets can be recovered from the Lucas asset pricing model, where the equilibrium cum-dividend return process follows
$$
\frac{d P + D dt}{P} =  (\mu + \delta) dt + \sigma dW
$$
with $\delta$ being the endogenous dividend-price ratio $\frac{D}{P}$.
The underlying for the call being priced is the Lucas tree.
I don't know whether the martingale duality method has been extended to general semimartingales.
When market is incomplete, after some browsing it appears that only the case of terminal utility has been investigated and it is shown that certain constraints must be placed on $u$.


It is very difficult for interest rates to cover the losses caused by a sharp devaluation. For example, if you believe that a currency is going to drop to 75% of its current value within 3 months month due to a devaluation, you need an annual interest rate of about 216% ($3.16^{\frac{1}{4}} \approx 1.33 \approx \frac{1}{0.75}$) to cover that loss.

Since interest rates of that magnitude would cause deep stress for domestic borrowers, we can reach a point where speculators believe that the central bank will not raise interest rates by enough to cover the losses (relative to a foreign currency) created by a devaluation.


Why does neoclassical economics assume preferences are fixed and exogenous in nature?

As you suggested in your answer it is precisely this:

  Does this link with the idea of "De Gustibus Non est Disputandum" (tastes are not a matter of dispute), if so, how?

Also in psychology and sociology there is great debate between “nurture and nature”, that is whether really people’s behavior is fully exogenous determined by genes and so forth or endogenous determined by upbringing and society and the debate is still ongoing although both views seem have some merit in.

Second, even though it’s likely that society can affect preferences of the individual it is less clear if economy/economic system or many economic issues studied by economics can so.


Are there any countries in which post-dated cheques are commonly used in trade?

Iran is another similar case, however in Iran use of post-dated cheques are mainly due to difficulties that a businessman will face if he wants to use bank loans.
For example banks usually have a well-known precondition for extending loan to firms and households which is providing sufficient liquid collaterals.
For finding more theoretical elaborations you can search for "access to credit" literature which tries to elaborate why banks do not have sufficient incentive to provide credit to some people.


We have a problem of terms here.  "Demand" is usually used to refer to the function where for a given price, a given quantity will be demanded.  That quantity is referred to as the quantity demanded.

  a decrease of demand can be caused by an increase in price.

This is a shift along the demand curve (function).  It changes the quantity demanded, but the demand curve itself stays the same.  I would say instead:

  An increase in price will cause a decrease in the quantity demanded if all else is equal.

This could happen if the supply curve shifted.

  However, it is also true that when demand decreases, the price will decrease, it will fall.

This is a shift of the curve itself.  I.e. the curve or function changes as a result of a change in demand.

Even if the supply curve is fixed, the intersection could still change because the demand curve moved.

The point is that you are using demand in one place to refer to the curve and in another place you are using it to refer to the quantity demanded.  If you use different terms for those two things, it's easier to see that your statements are not in conflict.  They are talking about different things.


Abraham (1987) Simple Job Market Matching Model

Note that function $p$ actually depends upon a single variable $u/v$ and I prefer avoiding the abuse of notation
(source of confusion) and write $p(u/v) = P(u,v)$. This implies:

\begin{equation}
p'(u/v) = P_u(u,v)v = P_v(u,v) \cdot (-v^2/u).
\end{equation}

So if $P(u,v)v = s$ total differentiation along $ds=0$ yields

\begin{equation}
P_u(u,v)vdu + (P+P_v(u,v)v)dv = 0,
\end{equation}

or equivalently

\begin{equation}
\frac{dv}{du} = - \frac{P_uv}{P+P_v v}.
\end{equation}

This result is still slighly different from those given in your question... Who can help further?


Which one of these intepretations of belonging of a factor of production is correct in context of calculating "income of our citizens abroad"?

The right (and shorter) name for " income of our citizens/corporations abroad" is "the national FFI", where FFI stands for Foreign Factor Income. I will use this name.

The second interpretation is correct, it will lead to correct calculation of the national FFI. If our corporation/citizen gets income from a factory that they own, but said factory is located abroad, then said income will contribute to size of the national FFI. Also, if instead of the national FFI we will calculate FFI of foreigners, then we will calculate their FFI taking in account what factors they own in our country. For an example, if Ivan from Russia will get income from a factory that they own in the UK (We assume that the UK is our country), then said income will increase FFI of foreigners.

The second interpretation also applies to the land. Just replace a factory with a mine.

P.S.

Comment by @1muflon1 that made this answer possible

  @user161005 okay
  but that’s then completely different question then your original. Here
  the answer would be that for GDP you use 1. i.e. the location. For GNP
  2. i.e the ownership. For NFFI both 1 and 2 at the same time. For FFI of home citizens you would use 2. For FFI of foreigners you would use
  also 2 but from their perspective. btw if this is what you ask pls
  edit your question and I will update your answer accordingly. In you
  question you only ask about NFFI which has to use both 1 and 2 at the
  same time since NFFI basically gives you difference between measuring
  with these definitions


Stability analysis and dimension of a dynamic control system

Differentiating $(3)$ with respect to time we get

$$u_{cc}\dot c = \dot \lambda \implies \frac {u_{cc}}{u_c} \dot c = \frac {\dot \lambda}{\lambda}$$

Inserting into $(6)$ we obtain

$$\dot M = -\frac {\eta_M}{\eta_{MM}}\frac {u_{cc}}{u_c} \dot c$$

So the fixed point of $M$ will happen under the same conditions that the fixed point of $c$ will. Also, the optimal time-evolution of $M$ is a scaled linear value of the optimal evolution of $c$. So it appears all aspects are captured if you "ignore" the $M$ variable.

A bit more formally, since your two decision variables are linearly dependent it means that the $3 \times 3$ Jacobian of the system will be singular at the fixed point. You will get a double eigenvalue and a single one, and you should be able to show what stability properties hold (or what is required for the desired stability notion to hold), in such a situation. Due to the existence of the double root, this requires a bit of a different mathematical approach.


Does the sudden spike in the US jobless claim pose systemic risk to the US unemployment insurance system?

I agree that the current crisis will not  “break” the unemployment insurance system.  The most that will happen is payments will stop being issued to those who qualify.

History of unemployment insurance (UI) program

Created in 1935, the federal-state unemployment insurance (UI) program temporarily replaces a portion of wages for workers who have been laid off, as long as they are looking and available for work.

The benefits vary by state but in most cases the program provides up to 26 weeks of benefits to unemployed workers provides half of a workers’ previous wages.   Because more workers lose their jobs during economic downturns of the business cycle, this program also provides some needed economic stimulus that helps mitigate the severity of recessions.

How is it funded

The program is funded by taxes on employers, including state taxes and the Federal Unemployment Tax Act (FUTA), which is 6 percent of the first $7,000 of each employee’s wages.  So at this point the insurance is paying for itself per employee.

States have extensive flexibility in determining benefits. Federal requirements are minimal, while ensuring that all states provide basic protections for eligible workers.
States are pretty much free to choose the level of employer tax, benefit level and duration of benefits, and the eligibility criteria.   Some states does choose to run their program differently.

For example,  while the standard maximum time for an eligible person to collect benefits is 26 weeks; some states a can limit it to much less than that.  Most state UI systems replace about half of prior weekly earnings, up to some maximum.
I think the spell of unemployment payments is the issue.

What criteria makes one unqualified for unemployment?

•   Are dismissed for misconduct at work.

•   Are dismissed for misconduct outside of work.

•   Receive severance pay.

•   Committed fraud.

(the next three points are not relevant to the pandemic situation but its good to note)

•   Turn down a suitable job.

•   Don't look for work.

•   Are unable to work.

Self-employed workers, gig workers, undocumented workers, and students traditionally aren’t eligible to apply for UI benefits.

Another important point to make is that to qualify for benefits you must also have earned at least $5,100 in the last year, and you must have worked for at least 15 weeks in the last year (in most cases).

CARES ACT and UI benefits

The CARES ACT extended the UI benefits which has been called the Pandemic Unemployment Assistance.  It temporarily extends unemployment benefits to certain workers affected by COVID-19 and eligible self-employed workers, including:

•   Freelancers and independent contractors

•   Workers seeking part-time work

•   Workers who don't have a long enough work history to qualify for state unemployment insurance benefits

•   Workers who otherwise wouldn't qualify for benefits under state or federal law

(https://www.dol.gov/coronavirus/unemployment-insurance)

There are new unemployment programs under the CARES Act.

Now there are new unemployment programs called Pandemic Unemployment Assistance,  Federal Pandemic Unemployment Compensation, and Pandemic Emergency Unemployment Compensation.

-The Pandemic Unemployment Assistance (PUA) extends benefits to the self-employed, freelancers, and independent contractors.

-The Federal Pandemic Unemployment Compensation (FPUC) provides a federal benefit of $600 a week.

-The Pandemic Emergency Unemployment Compensation (PEUC)    extends benefits for an extra 13 weeks after regular unemployment compensation benefits are exhausted.

SOURCE: https://www.investopedia.com/who-doesn-t-get-unemployment-insurance-4802117#:~:text=Under%20normal%20circumstances%2C%20you%20can,the%202020%20novel%20coronavirus%20pandemic.

All of the additional UI benefits included in the CARES Act are paid by the federal government.  This also has boosted the program temporally.

Not everyone who is not working is getting UI benefits

It was reported in May (https://www.cnbc.com/2020/05/15/44percent-of-us-unemployment-applicants-have-been-denied-or-are-waiting.html   )  that many are not getting benefits.   Often they are workers, in the reports example, in the restaurant business were they are often being told they didn’t earn enough to qualify for unemployment, even when they’ve worked full-time.

Guidance that appears to restrict eligibility with the CARES Act

However, the Department of Labor recently issued guidance that appears to restrict eligibility for the expanded unemployment insurance benefits in the CARES act.

An example provided by a Brookings report was the guidance specified that gig economy workers are only eligible for UI benefits if they are “forced to suspend operations as a direct result of the COVID-19 public health emergency”, apparently excluding workers such as Uber and Lyft drivers who are experiencing a significant loss of income due to reduced demand. Similarly, it states that parents who had to stop working to take care of their children after schools closed will not be eligible for benefits once the school year is over—even though many families rely on child care, summer camps, and other facilities which may still be closed over the summer, and the guidance appears to provide a narrow interpretation of eligibility based on health reasons and on social distancing requirements.

Source: https://www.brookings.edu/blog/up-front/2020/04/07/how-does-unemployment-insurance-work-and-how-is-it-changing-during-the-coronavirus-pandemic/

The CARES act program runs until December 31, 2020 and its benefits are capped at 39 weeks.   I agree that the CARES act may have made it sweeter for those who qualify but it won’t last long.


You could see it as a coordination game and an example of a rational bubble. Because everybody knows that people start buying gold and the price goes up when the economy tanks, it is actually a good strategy to do just that. Similar things happen for example with the Swiss Franc.
The idea that the stock of gold is fixed and hence its price is stable, is empirically just plain wrong.
First, gold production varies with production cost and the demand for gold, just as in other industries. It is also dominated by a couple of countries and big mining companies with tons of market power.
Second, the price of gold is highly volatile, just as other commodities. The standard deviation of gold prices is much higher compared to for example Treasuries or G7 countries' equity indices.


Why did the Federal Reserve change the data about USA GDP?

This happens all the time and is normal. It's due to measurement error. It is hard to measure these things in real time and as time goes by more data comes in, which allows measurement of GDP and other macroeconomic variables to be better so they are revised. Sometimes the revisions can be quite large. Typically the latest data is also the least reliable.

For some advanced information on how such measurements are made, the issue of measurement error and a discussion of how to improve estimates see this paper

For a discussion on the issues these revisions of GDP data and measurement error pose for monetary policy, see the work of Orphanides, for example here and here. For his discussion on the reliability of GDP estimates in real time see this paper.


The relationship between the expenditure function and many others!

Following up on the excellent MWG diagram in Amstell's answer, the fundamental observation needed is that holding $p$ fixed, $e$ and $v$ are inverses of each other. $e$ tells us the amount we need to spend to get a certain amount of utility $u$, while $v$ tells us the maximum amount of utility we can get from a certain expenditure $w$. Whenever we want to convert from utility to wealth, we use $e$; and whenever we want to convert from wealth to utility, we use $v$.

All the key identities can be derived from this observation. For instance, suppose we want to derive an identity for $\partial v(p,w)/\partial p_i$. We already know the corresponding identity for the expenditure function, $\partial e(p,u)/\partial p_i=h_i(p,u)$. To turn this into an identity for $v$, we substitute $w=e(p,u)$, obtaining $v(p,e(p,u))=u$, and differentiate with respect to $p_i$. The chain rule implies
$$\frac{\partial v(p,e(p,u))}{\partial p_i} + \frac{\partial v(p,e(p,u))}{\partial w}\cdot\frac{\partial e(p,u)}{\partial p_i} =0\\
\Longleftrightarrow \frac{\partial v(p,w)}{\partial p_i} = -\frac{\partial v(p,w)}{\partial w}\cdot x_i(p,w)$$
which, if we divide by $-\partial v/\partial w$ on both sides, becomes Roy's identity.

Or, suppose that we want to derive the Slutsky equation, which gives the relationship between the derivatives of Marshallian and Hicksian demand (decomposing a Marshallian demand change into substitution and income effects). Analogously to above, we can substitute $w=e(p,u)$ into Marshallian demand $x(p,w)$ to obtain $x(p,e(p,u))=h(p,u)$. Then, differentiating with respect to $p_i$ on both sides and applying the chain rule gives
$$\frac{\partial x(p,e(p,u))}{\partial p_i} + \frac{\partial x(p,e(p,u))}{\partial w}\cdot \frac{\partial e(p,u)}{\partial p_i} = \frac{\partial h(p,u)}{\partial p_i}\\
\Longleftrightarrow \frac{\partial x(p,w)}{\partial p_i}=\frac{\partial h(p,u)}{\partial p_i} - \frac{\partial x(p,w)}{\partial w}\cdot x_i(p,w)
$$
In general, I think the heuristic "switch between $w$ and $u$ as needed using $v$ and $e$" allows you to get pretty much everything here. (A similar heuristic is also useful if you ever deal with Frisch demand systems, where marginal utility $\lambda$ plays the same role that $w$ and $u$ do in Marshallian and Hicksian demand systems.)

Of course, there is one other key fact used above, which is $\partial e(p,u)/\partial p_i = h_i(p,u)$, which for $w=e(p,u)$ becomes $\partial e(p,u)/\partial p_i = x_i(p,w)$. This is best viewed, instead, as a direct consequence of the venerable envelope theorem.

($\partial v/\partial p_i$ can also be derived from the slightly more advanced version of the envelope theorem, where constraints as well as the objective are allowed to depend on a parameter. Since varying $p_i$ in the utility maximization problem changes the budget  constraint rather than the objective, the envelope theorem says that its effect will depend on the Lagrange multiplier on that constraint, which is the marginal utility $\partial v/\partial w$ of wealth. This is a good intuition for why the expression for $\partial v/\partial p_i$ is more complicated than the expression for $\partial e/\partial p_i$, picking up an extra factor.)


What is the implication of perfectly reversible capital?

I think I get the answer but I am not 100% sure. Any comment would be welcome.
I realized that this is related to the search-matching nature of the model (comparing to competitive labor market), so that if capital is not perfectly reversible the worker can bargain with the firm. Note that even with perfect reversibility, given the match-specific surplus, the wage is still determined by a Nash Bargaining. However the firm's choice of capital would be independent to this Nash Bargaining and be efficient ($f^{\prime}(k)=r+\delta$).


Every structural VAR (SVAR) model, e.g.
$$
B_0 y_t = B_1 y_{t-1} + u_t
$$
has an equivalent reduced form (VAR), e.g.
\begin{aligned}
y_t &= B_0^{-1} B_1 y_{t-1} + B_0^{-1} u_t \\
    &= A_1 y_{t-1} + \varepsilon_t.
\end{aligned}
where $A_1 := B_0^{-1} B_1$ and $\varepsilon_t := B_0^{-1} u_t$.

The reduced form can be used directly for testing Granger causality. In the example, one would test whether certain off-diagonal elements in $A_1$ are equal to zero. E.g. if $y_t$ is bivariate, then testing $H_0\colon a_{12}=0$ would tell us whether $y_2$ does not Granger-cause $y_1$. The structural form might even imply Granger non-causality if it implies that certain off-diagonal elements are zero.

However, Granger non-causality does not preclude finding an equivalent representation of the reduced form that has a nondiagonal $B_0$ matrix. In fact, you would just multiply any reduced form (whether Granger-causal or not) by any nondiagonal matrix, and that matrix would then be your $B_0$. Hence, $B_0$ being nondiagonal is not informative of presence or absence of Granger causality. (This is a sort of tautology, I think.)


Usage of natural gas in 1970 (where was the demand elasticity?)

tl;dr: it's basic engineering: efficiencies improved between 1970 and 2010. Additionally, net USA imports increased from 0.75 trillion cubic feet in 1970 to 3.8 trillion cubic feet in 2007

I think there may be some misunderstandings. Firstly: the first chart shows domestic gas production. "Domestic" doesn't refer to residential-only use - it just means natural gas produced within country, rather than imported by pipeline or on ship. Secondly: the second graph does not show the amount of gas uesd to generate electricity. It shows the amount of electricity generated by gas. There's a huge difference. The latter is the former multiplied by the volume-weighted efficiency of the generation fleet. And thirdly, you discuss the lack of responsiveness because of the need for gas infrastructure, but the period under consideration is 40 years, which is the long run: that's sufficient time for replacement of capital; pretty much all costs are variable over 40 years in gas.

Just because there may have been an increase in demand for gas-fuelled energy services, does not mean there has been a big increase in demand for gas, for example, when part of the increase in demand is stimulated by improvements in energy efficiency.

The USA has two sources of natural gas: domestic production, and imports. One of the ways you get a disconnect between local production and local consumption is a change in net imports, which is what happened:  net USA imports increased from 0.75 trillion cubic feet in 1970 to a peak of 3.8 trillion cubic feet in 2007.

Natural gas gets used directly for heating, including in industrial processes; it also gets used for electricity generation. The latter also had a local peak around 1970. The amount of electricity generated by gas turbines very closely tracks domestic gas production until about 1990.

What's happened over the last 40 years is that energy efficiency has improved. There has been fuel-switching, so that some processes that were done using gas now use coal or electricity; and some processes that did use coal now use gas. There have been net efficiency gains over time, just as you'd expect. Some of this was outside the residential sector, and some is within it.

Some energy efficiency improvements have come about through improvements in boiler technology such as the mainstream appearance of condensing gas boilers; as well as a better understanding of maintenance. And improvements in building heat tightness, from enforced tighter building controls, better materials, better construction techniques. In the UK, we've also seen the replacement of open gas fires with central heating systems, which tends to be more efficient too, and that may have played a role in the USA over the last 40 years.

And particularly, from 1990, the upward trend in electricity generated from gas (the second graph) is stronger than the domestic increase in gas production. Around 1990, there was the introduction of a new type of process, the combined cycle gas turbine (CCGT), which increased the gas-to-electricity efficiencies from around 30% to around 55% - i.e. almost doubling. By 2007, CCGTs had displaced pretty much all the old open-cycle turbines, resulting in almost twice the electricity generated from the same amount of gas.


Why is corporation tax regarded as a direct tax?

The classification of direct vs indirect taxes is outdated and has nothing to do with the effects of taxes on behaviour. This classification is mostly due to historic reasons.

Direct taxes are taxes on property, businesses or people. They tax entities directly and not through their behavior. This doesn't mean, however, that these taxes don't have behavioral effects.

Indirect taxes are taxes on transactions and exchanges, since they are not taxing entities themselves, but their behaviour.

The classification is not very meaningful nowadays. For example the estate tax is direct (it directly taxes the estate), whereas the inheritance tax is indirect  (it taxes the transaction from parent to child). However, these two taxes are economically equivalent.


How printed money gets to the commercial banks?

Commercial banks do not "get money" from the government. They do so either from the FED or either from worldwide financial markets. And actually, banks can even buy gov's bonds, which in this case is the contrary of what you describe.

Conceptually, the FED simply buys bonds emitted by commercial banks. But using the word "buy" here is misleading. The FED is not a traditional economic agent.

First, note that the FED has two main possibilities to lower interest rates. They can do it directly by saying, "ok guys, tomorrow refinancing will be less expansive" or indirectly by injecting liquidity in the system, i.e. by proactively buying commercial banks' bonds for (almost or really) nothing, or even paying them interests for that -- which is equivalent to having negative interest rates!

When the FED does so (either directly or indirectly), commercial banks will be able to refinance themselves at a lower cost. Which means that at the end, these commercial banks will also be more, say, flexible, and thus will stimulate demands (here and there) in the national economy by lending money (here and there). If supplies stay constant, markets will reach a new equilibrium via prices rising due to these stimulated demands $\iff$ stops deflation.


Does the Federal Reserve have bank accounts for banks?

The exact details of payment systems vary from country-to-country. The payments systems handle the multitude of transactions.

But if we are willing to simplify away the complexity of payment processing, private banks do have an account at the central bank, and transfers between banks are effectively transfers between those accounts. The central bank is a bank, and this is exactly how banks will handle transfers between its own customers.

These balances are generically referred to as settlement balances, but in a country with reserve requirements (like the United States), these will be refered to as bank reserves. It is fairly easy to find discussions of bank reserve operations on the internet.


Is the Ramsey model one with non negative investment?

No it is not. The benchmark Ramsey model allows for positive and negative investment. Treating the case of non-negative (or "irreversible" investment) is considered an extension.
See Barro & Sala-i-Martin (2004). Economic growth (2nd ed)

Appendix 2B for a short discussion and
Section 5.1.2 pp 242-247 for an Endogenous growth model with irreversible investment and Human capital.

For a more deep treatment, see
Arrow and Kurz(1970).  Optimal Growth with Irreversible Investment in a Ramsey Model, Econometrica, Vol. 38(2), 331-344 .


Quantitative easing and interest rate parity

Will this action [QE] tend to depreciate the country's currency against USD because money supply has increased decreasing the value of money?

There is considerable market folklore about this topic. However, from a fundamental basis, all QE represents is a change in the maturity structure of government liabilities. Long-term liabilities (bonds) are swapped for short-term liabilities (money). There is no good explanation as to why that would directly affect exchange rates (ignoring the interest rate effect, discussed next).

From interest rate parity, I believe it would follow that the currency would appreciate against the dollar. If yes, there are two opposing influences on the currency. Which one will prevail?

Interest rate parity is best thought of as an arbitrage condition: interest rate differentials plus the currency basis (which is a traded spread) determines forward rates. This is “covered interest parity”, and it holds (except perhaps in extremely stressed markets).
If interest rates fall due to QE - which is itself debatable - this implies strengthening forwards (assuming the basis does not move). This is necessary to equalise returns.
However, you are asking whether the spot exchange rate will strengthen. From a practical perspective, this is unlikely to even detectable.

The belief that the spot exchange rate will follow forwards is a variant of the Efficient Market Hypothesis, and from what I have seen, this effect is either weak or non-existent. Feel free to search for “uncovered interest rate parity,” it is a large area of research.
The strengthening of forwards is at the tenors associated with bonds. If a 10-year bond yield falls by 20 basis points due to QE, can we really tell whether the currency strengthened by 20 basis points per year over the next 10 years relative to its previous trajectory?


Normality of residual is violated, How to fix it properly?

Use Bayesian regression. It's much more flexible on its assumptions, and much more powerful in its outputs. You'll find that you have to be much more explicit about your assumptions, and you'll have to quantify them.

It's much harder work, and all the work you do, will be valuable.

Normal regression makes it very easy for anyone to do really bad science.

Bayesian regression makes it harder to do anything, but makes it much easier to do good science.


Consumer behavior with constrained demand

In a two-good space, initially the consumer maximizes $U(x,z)\;\; s.t. \;\;p_xx+p_zz =I$ and we assume it obtains the solution $(x^*, z^*)$ as a function of prices and income.

In the constrained case, the consumer will either choose $(0, \tilde z)$ or $(x^*+\epsilon, z'$), for some $\epsilon >0 $ always exhausting its budget, so in particular, $\tilde z = I/p_z$.
In order for the consumer to still choose to buy a strictly positive quantity of $x$, it must be the case that

$$U(x^*+\epsilon, z') > U(0, \tilde z)$$

Apply a first order approximation around $(x^*, z^*)$ without ignoring the remainders, we want

\begin{align}
 U(x^*, z^*) + U_x(x^*)\cdot \epsilon + U_z(z^*)(z'-z^*) + R_{\epsilon} &\\> U(x^*, z^*) + U_x(x^*)(-x^*) + U_z(z^*)(\tilde z-z^*) + R_0
\end{align}

Simplify and re-arrange, we want

$$U_x(x^*)(x^*+\epsilon) + R_{\epsilon} > U_z(z^*)(\tilde z-z') + R_0 $$

We know that from the unconstrained optimization, $U_x(x^*)/U_z(z^*) = p_x/p_z$  so

$$\frac {p_x}{p_z}(x^*+\epsilon) + \frac {R_{\epsilon}}{U_z(z^*)} > \left(\frac{I}{p_z}-z'\right) + \frac {R_0}{U_z(z^*)} $$

Multiply throughout by $p_z$,

$$p_x(x^*+\epsilon) + p_z\frac {R_{\epsilon}}{U_z(z^*)} > I - p_zz'  + p_z\frac {R_0}{U_z(z^*)} $$

but $p_x(x^*+\epsilon) + p_zz' = I \implies p_x(x^*+\epsilon) = I-p_zz'$ so we are left with the requirement that (ignoring positive terms)

$$R_{\epsilon} >  R_0 $$

in order for the consumer to choose $x^*+ \epsilon$ and not $0$ for $x$.

Note that the above take into account also the signs of the remainders, it is not just about their absolute magnitudes.

Now let's go back to our first-order expansions. We know that both candidate bundles yield utilities lower than $U(x^*, z^*)$, because they were feasible in the unconstrained case, and they weren't chosen.

Looking at the expansion of $U(0, \tilde z)$ we then conclude that we have

$$U_x(x^*)(-x^*) + U_z(z^*)(\tilde z-z^*) + R_0 < 0 $$

$$\implies U_z(z^*)\cdot \Big[(U_x(x^*)/U_z(z^*))\cdot(-x^*) + \tilde z-z^*\Big] + R_0 < 0$$

$$\implies \frac {U_z(z^*)}{p_z}\cdot \Big[-p_xx^* + p_z\tilde z-p_zz^*\Big] + R_0 < 0$$

But $-p_xx^* -p_zz^* = -I$ and $p_z\tilde z =I$ so the term in brackets is zero. Therefore we conclude that

$$R_0 <0$$

Looking now at the expansion of $U(x^*+\epsilon, z')$, we know we have

$$U_x(x^*)\cdot \epsilon + U_z(z^*)(z'-z^*) + R_{\epsilon} < 0$$

Performing the same manipuations as before we obtain here too that

$$R_{\epsilon} < 0$$

So the condition to buy $x^*+\epsilon$ can be re-written as

$$|R_{\epsilon}| < |R_0|$$

This formalizes somewhat the notion that if $\epsilon$ is "sufficiently small", $R_{\epsilon}$ will be smaller in absolute terms than  $R_0$, since the approximation to the same function will be "better", and so we will observe $x^*+\epsilon$ and not $0$. But it also tells us what the graphs in the other answer told us too, that there is not a single general answer to the matter.


Optimal Price and Quantity of Externality

Your steps look correct. There is one small typo in the first $\frac{\partial \mathcal L}{\partial h}$: the term $-4h(h-10)$ should have been $-4(h-10)$.
The results also look reasonable: Regardless of who makes the offer, the Pareto optimal level of $h$ is produced. Given how the bargaining procedures are structured, whoever gets to make the offer gets to keep the surplus of $20$.

P.S. There is an issue in part 2 of the question. It says: "If the firm rejects the offer it cannot produce so the firm's profit is 0." This statement is a contradiction. If the firm cannot produce ($h=0$), then profit would be $-80<0$; if its profit is $0$, then it must produce some positive amount of $h$. But this lack of rigor is on whoever wrote the question, not you.


How much average value does 1 working hour produce? ( are there reliable statistics?)

One possible measure is "GDP (or productivity) per hour worked": OECD, EU28.


What are modern examples of established economic theories being used on a large scale in real life?

Spectrum auctions are generally designed by economists using results and insights from auction theory. For example, most modern spectrum auctions are at least somewhat related to the ascending clock proxy auction of Ausubel and Cramton. These auctions are used to sell hundreds of millions of dollars of radio spectrum licenses. A similar point can be made about the auction mechanisms used by other branches of the public section (such as central bank treasury auctions).

Matching theory has been widely applied in the design of many real-world markets and allocation mechanisms. Examples include the assignment of student doctors to medical residency, the assignment of students to high schools, and the pairing of kidney donors with patients.

Compeition (antitrust) policy is based to a very large extent on results from industrial organisation. This includes policy formulation informed by many comparatively recent results on topics including bundling, collusion, and vertical restraints.


How do economists model VNM-rationality violation?

It appears you are looking for literature on Ambiguity Aversion and/or "Uncertainty Aversion".
You can start by looking up the work of L.G Epstein, I. Gilboa, and  D. Schmeidler.

It is an attempt to formalize the behavior of agents exhibiting behavior consistent with the Ellsberg paradox.


How about multiplying the average UK household income with the number of households? Both of these should be available from government statistics.


Seems like the only function $f$ that fits your description
$$
\forall i: \frac{\partial f(\mathbf{x})}{\partial x_i} = c_i
$$
is
$$
f(\mathbf{x}) = A + \sum x_i c_i.
$$
(Frequently $f(\mathbf{0}) = 0$ is assumed. The assumption is referred to as "no free lunch".)
Then you can apply the definition of returns to scale.


Why does the profit reaches its maximum value at the equilibrium point?

The point where MB = MC is the largest Net Benefit available for a given product.  Therefore, any additional benefit has a lower level of benefit than at equilibrium, and the costs increase at a higher rate than at equilibrium.  In the context of supply and demand, this is where the two lines cross.  But if you think about it in terms of diminishing marginal benefit, and increasing marginal costs then you can see why it is beneficial to product where the largest net benefit occurs.

The following graph is from Markets and the Environment by Keohane and Olmstead and deals with abatement costs, but it is the one graph I think about when someone asks this sort of question because it provides a good example.

As you can see from the graph, the largest net benefit is the furthest distance between the benefit and cost curves.  If you were to take the derivative of each of these functions, you would get linear supply and demand lines, where the largest net benefit is the cross section of the lines (as you can see in the graph below).


Why is there a deadweight loss in this lemons market?

You can calculate the total social surplus (= CS + PS) in this equilibrium and compare it to the social surplus in the efficient allocation of cars.

Equilibrium allocation

The equilibrium (under asymmetric information) has price equal to $12000$. Thus, in the lemon's market,
\begin{align}
CS_L&=(8000-12000)\times1000=-4000000\\
PS_L&=\frac12\times(12000-2000)\times1000=5000000.
\end{align}
Total social surplus in the lemon's market is
\begin{equation}
TS_L=CS_L+PS_L=1000000.
\end{equation}
In the plum's market,
\begin{align}
CS_P&=(16000-12000)\times400=1600000\\
PS_P&=\frac12\times(12000-4000)\times400=1600000.
\end{align}
Total social surplus in the lemon's market is
\begin{equation}
TS_P=CS_P+PS_P=3200000.
\end{equation}
Joint surplus from the two markets:
\begin{equation}
TS=TS_L+TS_P=4200000.
\end{equation}

Efficient allocation

Efficiency occurs when lemons are priced at $8000$ and plums at $16000$. Note that in both markets, since demand is perfectly elastic, consumer surplus is zero at the efficient allocation. So total surplus is the same as producer surplus.
\begin{align}
TS_L^*&=\frac12\times(8000-2000)\times600=1800000\\
TS_P^*&=\frac12\times(16000-4000)\times600=3600000\\
TS^*&=TS_L^*+TS_P^*=5400000.
\end{align}

The gap between $TS^*$ and $TS$ is the DWL. The inefficiency arises from the fact that, in equilibrium (with asymm info), some cars are in the wrong hands: the last 400 lemons sold and the potential 200 plums that were unsold.


In the second period, the buyer accepts any offer $s_{2}$ $\leq$ $0.7$ if he has rejected the first offer and any offer $s_{2}$ $\leq$ $0.3$ if he has accepted the first offer.
Given this, there are only two offers that may be optimal for the second seller: either $s_{2}$ $=$ $0.3$
or $s_{2}$ $=$ $0.7$. Let $\mu$ denote the probability that the second seller assigns to the fact that the buyer has rejected the first offer. The optimal offer in the second period is:
\begin{equation}
 s_{2}=\left\{
  \begin{array}{@{}ll@{}}
    0.3 & \text{if}\ \mu < \frac{3}{7} \\
    0.7 & \text{if}\ \mu > \frac{3}{7} \\
 \end{array}\right.
\end{equation}
any randomization between $0.3$ and $0.7$  $\text{if}\ \mu =  \frac{3}{7} $
Suppose that the equilibrium is such that the buyer rejects the first offer. In the second
period, the second seller will offer $s_{2} = 0.7$ and the buyer's payoff is equal to 0. Because of this, the buyer should accept any offer smaller than $0.7$ (the buyer can guarantee a positive payoff by accepting the first offer and rejecting the second one). But then the first buyer
should offer $s_{1} < 0.7$ and get a positive payoff. In other words, we have shown that there is no PBE in which the buyer rejects the first offer.
Let us now see whether we can construct an equilibrium in which the buyer accepts the first offer. In this case $s_{2} = 0.3$. This implies that the buyer will accept the first offer only
if $s_{1} < 0.3$ (in fact, by rejecting the first offer and accepting the second one, the buyer
guarantees a payoff equal to $0.4$). Finally, given this, it is optimal for the first seller to offer $s_{1} = 0.3$
To sum up, we have the following pure-strategy PBE.
The first seller offers $s_{1} = 0.3$ In the first period, the buyer accepts an o¤er if and only if $s_{1} = 0.3$
In the second period, the second seller assigns probability $\mu = 0$ to the fact that the buyer has rejected the first offer.
Thus, the second seller offers $s_{2} = 0.3$
Finally, the buyer's strategy in the second period is described above.


What does one-sided polynomial on lag operator $L$ mean?

The terms "one-sided"  and "two-sided" lag-polynomials, are used when the text considers the option to specify an equation with both "lags and leads", i.e a relation where both past but also future values co-vary with current value. When the lag-operator is "one-sided" it contains only lags in the one direction. When it is two-sided, it contains "lags" in both directions.
In some cases, scholars have used the term "forward" operator to verbally contrast it with the "lag" operator, but my impression is that it is not so widely used.

One may wonder "so if I read "one-sided lag-polynomial", does it mean that I will consider only past values, or only future values?". The answer is that in almost all cases the invertibility condition holds, so in principle and in theory, it makes no difference. In other words, in principle the term "one-sided lag polynomial" may mean either

$$\phi(L) = 1+\phi_1L+ \phi_2L^2+...$$

or

$$\phi(L) = 1+\phi_1L^{-1}+ \phi_2L^{-2}+...$$

where $L^{-1}x_t = x_{t+1}$.

A specification with a two-sided lag polynomial may appear to violate a basic causality postulate (the arrow of time), but we should remember that estimated relationships are relationships of association, not causation.

So when the concern and goal is purely forecasting, "anything goes" -or at least this is what many Time Series Theorists and Practioners essentially say.


Exact definition of one-player Bayesian Correlated Equilibrium

The concept of the BCE from their 2016 paper is similar to what you have. I think Bergemann and Morris' intuitive explanation is valuable so I'll paraphrase it here.

Each player in the game has a decision rule that chooses an action, $y$, dependent on the state of the world $V$, and the player's information set, which we'll call $S$. This information set includes both a finite set of signals for each player, $T_i$, and a signal distribution, $\pi: \mathcal{V} \rightarrow \Delta T$. As you wrote your example, you assume the set of signals is a singleton, leaving us with only a player's prior. This is a possible information structure, but is not necessary.

We can thus write the the decision rule as a mapping, $\sigma$,

\begin{align*}
\sigma : S \times V \rightarrow \Delta Y
\end{align*}

The lone criteria for a CBE in this setting is that each players' decision rule is ``obedient''. By obedient we simply mean that the action, $y$, chosen by the decision rule must be the optimal action for the player. Thus, a player will always follow the action chosen by their decision rule.

I believe you are confusing the information structure and the decision rule. My information set is not a function of the action I choose in this setting, so $P_{V,Y}(y,v)$ does not have any meaning. Thus, you don't need to be concerned about the existence of a $y$ such that $P_{V,Y}(y,v)=0$ for all $v$.

It is possible to be in this setting that there exists an action $y$ such that $\sigma(y_i|t_i)=0$ for all signals, $t$. But this would simply mean the player never chooses that action in equilibrium.

Is it possible that there exists a signal $t$ such that $\sigma(y_i|t_i)=0$ for all actions, $y$? No, and it would follow for the basic Nash existence proof, given certain constraints on $u(\cdot),$ $\mathcal{Y}$ and $\mathcal{V}$.


What was Paulus's "famous definition of price"?

Julius Paulus Prudentissimus, circa 230 CE

"material was selected which, being given a stable value by the state, avoided the problems of barter by providing a constant medium of exchange.
That material , struck in due form by the mint, demonstrates its utility and title not by its substance as such but by its quantity, so that no longer are the things exchanged both called wares but one of them is termed the price.
An today its is a matter for doubt  whether one can talk of sale when no money passes. "

https://books.google.com/books?id=NgZKDwAAQBAJ&pg=PT123&lpg=PT123&dq=Julius+Paulus+Prudentissimus+economics+++definition+of+price&source=bl&ots=0FXjkqEeus&sig=ACfU3U2ZKnUHIoR4DXwLnTMmDpCKRyD0PQ&hl=en&sa=X&ved=2ahUKEwjjqZOAhdTpAhUETd8KHXXJBuQQ6AEwAXoECAoQAQ#v=onepage&q=Julius%20Paulus%20Prudentissimus%20economics%20%20%20definition%20of%20price&f=false


Reconciling low wages with high technology and high education

Possibly it is because of the better technology that they will receive lower income.

When there where fewer programmers/engineers/whatever, these individuals could typically expect high salaries due to their scarcity, compared to today, where you can hardly go outside without bumping into about two programmers and five web devs.

This means that highly-qualified individuals may not be able to find jobs relevant to their qualifications at all, even if they are willing to accept substantially lower salaries.

This may seem more focused on jobs requiring higher qualifications, but that may be one of the big factors here, especially when it comes to averages.

In summary, could it possibly be that we have an over-supply of well-qualified individuals? (This may not apply to all countries or industries)


McCall Search Model: Present value of lifetime income

In the version of the McCall search model I'm familiar with, once you accept an offer of wage $w$, you receive that wage $w$ in every period starting from the period of acceptance.

Hence, from the perspective of the current period, your utility from accepting that offer is

$$ w + \beta w + \beta^2 w + \cdots =\frac{w}{1-\beta} $$

In case this equality isn't familiar, recall that the LHS is simply a geometric series.


Marxist Economics: Understanding Capitalist Exploitation through "speed up" or increase in intensity

Though I'm certainly no expert in Marxist economic theory, I think he's presenting it as an "if you can figure out a way of speeding workers up, then you can generate additional surplus from their efforts. Effectively, this is just saying "say you're Ford, and have teams of 7 working on assembly lines that generate 4 cars per hour. Suppose there is some way to increase the intensity of labor, by getting them to work faster and produce 5 cars per hour, and not changing the number of assembly lines or workers per line. Then, if you can hold their wages fixed, you can generate more surplus for the capitalist."

(I'd imagine a historical example might be the semi-mechanized looms of the English Industrial Revolution, where the speed of the mechanisms was dictated by the machinery, and could therefore be bumped up by a factory owner.)

I don't think there's a single coherent mathematical argument explaining why this speed would be increased, just highlighting the different ways the ratio of interest can change.


Take the situation where $u_1(x,y)=x+y$, $u_2(x,y)=x+y$. Then the contract curve would be the entire box.


Why is macroeconomics studied through "Shocks"?

Im assuming you're discussing some of the models associated with Real Business Cycle Theory.

These models have a structural underpinnings (meaning that there is some assumptions/empirics with regard to consumer behavior). Starting with the understanding that consumers are also suppliers of labor which face a intertemporal consumption-leisure trade off and Firms are profit maximizing and use capital,labor and technology to produce.Using this setup we can do some standard calculations for what equilibrium and Balanced Growth Paths would be.

Now this as it stands is not a theory of fluctuations. The way Fluctuations are accounted for in this model is VIA exogenous shocks that enter through Technology,A overall change in aggregate preferences and increases in interest rate ect. This enters in the error term of all our equations.

This is the justification for using Impulse response functions, Its to analyse what the implications are for surprise changes in our variables in terms of consumer and producer behavior.

TL;DR:
The main point of this analytical process of shocking is to be cognizant for how economic shocks enter the economy and what the implications are for consumer and producer behavior. This is based on the structure of that is imposed by economic theory.


Good source for marginal propensity to consume by country?

You may need to do some math but the OECD database does have the data that you need to construct MPC by country.     As a reminder MPC can be calculated as the change in consumption (ΔC) divided by the change in income (ΔY).

For consumption the OECD data series you would need is "Household spending"  https://data.oecd.org/hha/household-spending.htm#indicator-chart.

For income the OECD data series is called "Household disposable income"
https://data.oecd.org/hha/household-disposable-income.htm#indicator-chart

Download the available data and calculate the growth rates of each measure then solve for MPC  and you should have it.   If you have trouble with the data download contact the OECD for assistance.


Definition of subgame perfect Nash equilibrium

An equilibrium consists of a profile of strategies, which specifies an action for every player at each possible contingency. Since each action profile $(a_1,a_2)$ is a contingency, the SPE must include functions $a^*_3(a_1,a_2), a^*_4(a_1,a_2)$ that specifies what to do at those contingencies.


The example you gave is correct.

What you are missing is that you took the derivative of the demand function and not of the revenue function. What you should have done is:

$Revenue = R = Price*Quantity = (-Q + 120) * Q = -Q^2 + 120Q$

$Marginal \; Revenue = MR=\frac{dR}{dQ} = \frac{d(P*Q)}{dQ} = 120 - 2Q$

What you did was:

$MR = \frac{dP}{dQ}=-1$.

Hence, you took the derivative of the Price and not of Revenue to get marginal revenue, which is wrong.


Is it bad for the economy of a country if forreign firms buy companies?

It depends upon the circumstances and isn't necessarily bad.

Suppose a foreign firm A buys a British company from its current British owner B. A is gaining the right to whatever stream of profits the company may make in future. B is gaining the price paid for the company by A.  Whether that is a good deal for B will depend upon various considerations including:

The company's likely future stream of profits if it remains in B's ownership.
The sale price, and how this compares with the discounted present value of the above stream of profits.
Whether B's best use of the sale receipt is relatively advantageous, having regard to its preferences and attitude to risk (an entrepreneur who has taken risks in building up a company may in later years wish to hold their wealth in a safer form).

So it's quite possible for the sale to be good for B. Since B is likely, through their use of the sale receipts, to contribute to the British economy by spending (creating demand) and/or saving (providing capital for other firms), it could be good for Britain.

However, these possible advantages for Britain could in some circumstances be offset by other considerations:

A's plans for the company may involve a loss of British jobs.  This could be because capital equipment will be substituted for labour, or because the company will be scaled down (perhaps the motive for buying it is to reduce competition).
The company may operate in a sector within which foreign ownership is considered an unacceptable risk for strategic rather than economic reasons.  This might be argued in respect of firms supplying military equipment or essential infrastructure.


Monthly average rate of discount, 3 month Treasury bills, Sterling Vs 1 Month

Answer to your first question
Take a look at what Aswath Damadoran uses for his empirical work. He's been teaching applied corporate finance for a little while at NYU. These are his slides.

Answer to your second question
If you have access to any of the following, you should have no problem getting access to any economic and financial time series.

Datastream
Bloomberg

Also try FRED database


Why are Portuguese migrating out of Portugal?

As you write, fertility rate explains most of it, and immigration explains the rest.

People tend to migrate to richer countries, and Portugal is below average EU income. (Portugal is an EU member. The EU has freedom of movement, the citizens of any member state can relocate to other member states and work there.) Real per capita GDP is 30% less than in its neighbor Spain, and almost 50% less than in Germany.

Eurostat Real GDP per capita table


Interest rate rule in monetary DSGE model

I've just solved this problem. First of all, your solution does not make too much sense, as in a simple interest rate rule it must hold that the sum of all coefficients must be greater than one. In your case this means that $\phi>1$. Therefore, the series would converge not to zero. Second, an interest rate rule should try to offset fluctuations. This means that the central bank should try to offset gaps from its target (like in the task). Try instead the following interest rate rule.
\begin{equation}
i_t=\rho+\phi_{\pi}(\pi_t-\pi)+\sigma(\Delta y_{t+1})+\pi
\end{equation}
Now what is the idea? Assmung that the output does not change, this term is zero. Using the Fisherian $i_t=r_t+\mathbb{E}_t\{\pi_{t+1}\}$ we get
\begin{equation}
\rho+\phi_{\pi}(\pi_t-\pi)+\sigma(\Delta y_{t+1})+\pi \overset{!}{=}r_t+\mathbb{E}_t\{\pi_{t+1}\}
\end{equation}
which can be summarized as
\begin{equation}
\phi_{\pi}^{-1}\left(r_t-\rho+\mathbb{E}_t{\bar{\pi}_{t+1}}-\sigma(\Delta y_{t+1})\right)=\bar{\pi}_t,
\end{equation}
where $\bar{\pi}_t=\pi_t-\pi$.
Solving forward yields
\begin{equation}
\bar{\pi}_t=\sum_{k=0}^{\infty}\phi_{\pi}^{-(k+1)}\mathbb{E}_t\{r_{t+k}-\rho-\sigma\Delta y_{t+1+k}\}
\end{equation}
Notice that for $|\phi_{\pi}|>1$, the series converges. However, your log-linearized Euler equation can be rearranged such that it holds that $r_t=\rho+\sigma\mathbb{E}_t{\Delta y_{t+1}}$. Finally, this can be plugged into $\bar{\pi}_t$ such that $\bar{\pi}_t=0 \quad \forall t$. Notice that this expression finally is your answer, as $\bar{\pi}_t=\pi_t-\pi=0$ means that the central bank always achieves its target.


One of the discussions in financial forecasting is whether econometric methods or machine learning methods give more sound results. Have a look at this paper.


What are the fundamental economic proofs

I would say the following four:

Nash Equilibrium via Fixed Point (either Kakutani, Brouwer or Tarski, depending on the problem);
Welfare theorems via Hanh-Banach theory (mainly the separating hyperplane theorem);
Utility and Profit Optimization via Weierstrass Theorem, Kuhn-Tucker conditions, and existence of a Lagrangian;
Minimization of Least Squares;


Consumer surplus in case of perfectly inelastic demand

From a purely theoretical perspective, if an individual's demand curve is perfectly inelastic, then her willingness to pay for the good is infinite. NB this also implies that she has an infinite budget. Thus, consumer surplus is well defined: it is the willingness to pay minus the price she pays, so as long as the price is finite her consumer surplus is finite.

In practice, no one has an infinite budget. So if the individual's demand curve is truly perfectly inelastic (i.e. the inverse demand is vertical), there exists a price such that beyond that price she can no longer afford to buy the good. This price is her willingness to pay, so consumer surplus is again well defined: the willingness to pay minus the price.


The wording of the question is vague, but I will assume that the objective is to get a zero/discount curve via bootstrapping.
The premise of a bootstrapping is straightforward: fit a zero curve so that the shortest maturity bond is priced perfectly. Since we don’t have any other information, assume the curve is flat up to the first maturity; the curve matches the bond yield (modulo quote conventions). In other words, we have the zero curve defined to the maturity of bond #1.
(Pricing the bond perfectly means that if we discount the cash flows of the bond by our zero curve, the sum of the discounted values equals the market price.)
We then target the next maturity. We extend the zero curve so that it is priced perfectly. We will note that the initial coupons are discounted by the already fitted curve, the later coupons and principal by the new curve. Find the discount curve extension that results in bond #2 being priced correctly. That is, the zero curve is now defined to maturity #2.
Repeat for bond #3, ... N.
Issues arise.

There are any number of ways that the curve can be interpolated, including how to define the curve. (Interpolate between knot points, functional definition, etc. Do we interpolate forward rates or zero rates? Etc.)
Bonds can be mis-priced, and the curve ends up severely distorted.

Finally, the code/mathematical formulae are complex. Even an ugly approximations take a lot of code.


Note that the Marshallian demand function $x^*(p_x,p_y,I)$ is homogeneous degree zero in $(p_x,p_y,I)$ (see here for a proof). According to Euler's theorem for homogeneous function, it follows that
\begin{equation}
\frac{\partial x^*}{\partial p_x}p_x+\frac{\partial x^*}{\partial p_y}p_y+\frac{\partial x^*}{\partial I}I=0\cdot x(p_x,p_y,I)=0
\end{equation}
Dividing both sides by $x^*$, you have
\begin{align}
\frac{\partial x^*}{\partial p_x}\frac{p_x}{x^*}+\frac{\partial x^*}{\partial p_y}\frac{p_y}{x^*}+\frac{\partial x^*}{\partial I}\frac{I}{x^*}&=0\\
\epsilon_{xx}+\epsilon_{xy}+\epsilon_{xI}&=0
\end{align}


What do all the distributions that have the same Gini index have in common?

This is a Lorenz curve.

The Gini coefficient is the area between 'equal wealth' and your chosen line, divided by all the area under the equal wealth curve. There are other illustrations online, for sure, which would explain this.

The Wikipedia article is telling you that the actual distribution of income is not fully captured by the Gini coefficient. The same Gini coefficient could have different distributions of income. For example, it could move equally in the lower region at first and then flatten between 60 and 80 per cent of people. Or it could be that poorer people get nothing and the middle and upper classes are hoovering up the income.

Both examples, if you set up the toy numbers properly, imply very different distributions for poor people, but produce the same Gini coefficient.


Is Price Level equivalent to Inflation when looking at AD/AS and Phillips Curve graphs?

Snoram is correct. Price level is a just that, the relative level of prices. Inflation/deflation is the act of moving up/down price levels. The inflation rate is the difference in price levels from one time period to another.

I.E. Year 1: Price level = 100 |  Year 2: Price level =105.

There was inflation in the economy (because price level increased). The inflation rate was 5%.

Another example:

I.E. Year 1: Price level = 177

In this example we do NOT know if the economy is experiencing inflation or deflation becuase we only have a single point in time and thus cannot determine the rate or direction of change.


How do 'econometric' explanations differ from 'economic' ones?

Econometric and economic explanations are not some standardized terminology so it is often best to clarify with the person asking you but generally speaking, when someone asks for economic explanation they are looking for explanation in terms of an economic theory or underlying economic relationships. If someone asks for econometric explanation they are likely looking for an statistical explanation.
An example related to changing coefficients lets suppose we have models:
$$ Wage = 0.5 + 5 EDU   $$
$$ Wage = 0.6 + 10 EDU + 6 EXP  $$
Where EDU is education and EXP experience and lets also say we know $cor(EDU,EXP) \neq 0$.
An economic explanation would be that from economic theory we know that wages depend on your marginal productivity which is some function of both education and experience. In the first regression we include only education and do not control for what the peoples actual experience is. Since people who have higher education might have less experience on average than people with less education (due to all that time spent in school) so we expect the coefficients to change when we actually control for the fact that not all people have the same experience.
An econometric explanation would be simply that we know from econometrics that in the more parsimonious model (assuming the more general model is true) the estimated beta coefficient in this case will be given by
$$E[\hat{\beta_1}]= \beta_1 + \beta_2 \frac{\sum  (EDU_{i}-\bar{EDU})(EXP_{i}-\bar{EXP})}{\sum (EDU_{i}-\bar{EDU_1})^2}$$
we know that $cor(EDU, EXP) \neq 0$ and that $\beta_2 \neq 0$ so we also know that coefficient $\beta_1$ must change between the regressions.
However, again note that your question is a soft-question depending on where you study teachers might have slightly different understanding of this so you should always ask. Furthermore, you should also inquire on how rigorous you have to be when providing both economic and econometric explanations. Depending on the level of your class and university overall simple textual economic explanation like I provided here would not suffice and the econometric one could be arguably done better. The above is just quick example in order to make it easier to understand differences between them.


Data on inflation target on different countries around the world


Interest Rate Differential and Fisher's Interest Rate Parity

I hesitate to mark this question as a duplicate (of this one), but my answer will in part be a replication of part of my answer there.

Let's write the Uncovered Interest Rate Parity expression

$$(1+i_{A,t}) = \frac {S^e_{A|B, t+1}}{S_{A|B, t}} (1+i_{B,t})$$
where $S_{A|B}=$ units of currency A per unit of currency B.
This is the claim.  The numerator is the expected future exchange rate

Assume that $i_{A,t} > i_{B,t}$. Standard economic logic (and a few assumptions on ability to invest over borders), says that then investors would want to invest in country A. To do that they will increase demand for currency A. This should lead to appreciation of currency A. Does the UIRP say something like that?

If $$i_{A,t} > i_{B,t} \implies 1+i_{A,t} > 1+i_{B,t}$$
and so the UIRP asserts that we must have

$$\frac {S^e_{A|B, t+1}}{S_{A|B, t}} > 1 \implies S^e_{A|B, t+1} > S_{A|B, t}$$

Given how $S$ was defined the last inequality means that we expect that the currency will depreciate. Indeed, in the future. Because, what is affected in the first place by the reaction of the investors is $S_{A|B, t}$, not $S^e_{A|B, t+1}$. The reaction of the investors will decrease $S_{A|B, t}$ (which reflects the appreciation of currency A), creating expectations of future depreciation, i.e. leading to $S^e_{A|B, t+1} > S_{A|B, t}$, as the UIRP asserts. Why should we expect depreciation of currency A in the future, compared to the present? Because the inflow of funds in country A will eventually pressure $i_{A,t}$ downwards, investors will stop loving currency A so much, demand for it will eventually fall, hence eventual depreciation.

The UIRP therefore compacts two effects from a discrepancy in interest rates: a first effect of currency appreciation and a second (expected) effect of currency depreciation in the future.

Apart from that explanation, I note that the link discussing the tendnecy to appreciate says also explicitly

  "However, this simple equation is complicated by a host of other
  factors that impact currency value and exchange rates. One of the
  primary complicating factors is the interrelationship that exists
  between higher interest rates and inflation. If a country can manage
  to achieve a successful balance of increased interest rates without an
  accompanying increase in inflation, then the value and exchange rate
  for its currency is more likely to rise."

In other words, the very link that talks about currency appreciation qualifies this to be subject to the fact that higher nominal interest rate does not result in higher expected domestic inflation (this alludes to the Fisher Hypotesis that the difference between the nominal interest rate and the real one is the expected inflation). This qualification essentially says that in order for the initial currency appreciation to persist, we must have differences in the real economy (i.e. that a higher nominal interest rate reflects a higher real interest rate rather than higher expected inflation).


Why do banks offer less interest on larger deposits?

When you lend your money to a bank (by depositing it), they don't just make money off of lending out your money. They also make money out of charging you for banking services. If the bank makes more money off of offering you services then they can afford to pay you a higher rate of interest (all other things equal). Check out: How do banks make money? The fallacies of fee income (Rice and DeYoung (2004))

  But banks also earn substantial amounts of noninterest income by
  charging their customers fees in exchange for a variety of financial
  services. Many of these financial services are traditional banking
  services: transaction services like checking and cash management;
  safe-keeping services like insured deposit accounts and safety deposit
  boxes; investment services like trust accounts and long-run
  certificates of deposit (CDs); and insurance services like annuity
  contracts. In other traditional areas of banking-such as consumer
  lending and retail payments—the wide-spread application of new
  financial processes and pricing methods is generating increased
  amounts of fee income for many banks.... Remarkably, noninterest income now
  accounts for nearly half of all operating income generated by U.S. commercial  banks.

If spending on banking services increases more slowly than account balances and the sums involved don't make it worth the trouble of having multiple accounts then banks will be willing to offer higher rates on smaller balances. But, empirically, my impression is that this is atypical. It is more common to see higher rates on higher balances, because the transaction costs on serving one \$100k account are smaller than those of 10 \$10k accounts, and so they can pass some of those economies along as higher rates.


Comparative statics question with an application

You can obtain a definite answer for $$\text{sign}\left\{\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha }\right\}$$ given the assumptions.

From

$$S_{1}^{*} - f\left ( g\left ( S^{*}_{1}, \beta \right ), \alpha \right ) = 0 \equiv F$$

and the Implicit Function Theorem

$$\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha } = - \frac{\partial F/\partial \alpha}{\partial F/\partial S^*_{1}} $$

we have

$$\frac{\partial F}{\partial \alpha} = -\frac{\partial f}{\partial \alpha}$$

and

$$\frac{\partial F}{\partial S^*_{1}} = 1-\frac{\partial f}{\partial S_2}\cdot \frac{\partial g}{\partial S^*_{1}}$$

For these expressions we know not only the signs but also the magnitudes. The result follows.


VAR inversion - looking for a good resource

To get Quah and Vahey's method, you should probably get a good handle on the Wold Representation Theorem first. Christiano has good notes on it here. Then, you can apply it to VAR. There's a pretty good basic explanation in Greene's Econometric Analysis. In the Sixth Edition, it's in Chapter 20 (Models with Lagged Variables) starting with Section 20.6, "Vector Autoregressions." It should be pretty straight forward to apply the Wold Theorem to VARs with Greene's notation. If it's still not quite clear to you, there's a pretty close example on page 704, subsection 20.6.8.b, "The Sacrifice Ratio." The basic idea is that you just end up sort of inverting the matrix of lag operators and coefficients (the "inversion" part) so that you have your main variable on the left side and a lag polynomial in your errors on the right side.


How is the interest on fractional reserve money creation paid?

Repayment of interest does require an expansion of the money supply, but not in a way that is inflationary.

Consider first the way that commercial banks make loans. The whole purpose of loans is to borrow against future output— so if you imagine a firm that wants to purchase a machine that will allow it to produce more stuff— and thus have greater future income— the firm may go to a commercial bank and ask to take out a loan. The commercial bank will price that loan at a premium to the risk-free interest rate that reflects the risk of the loan, so that in expectation (taking into account that some loans will be repaid only in part or not at all), banks making riskier loans will be repaid at a slightly higher rate than ones making safer loans.

So firms will be offered an interest rate, and if that rate is less than their expected return on their investment, they'll make the investment. Notably, in this case, they're only borrowing money and investing because their future output will be higher. So when all of this works out (that is, when there isn't a credit bubble in which people are borrowing against future output that won't materialize), future output $Y$ (i.e., GDP) will be higher by an amount that is  the amount of additional money $M$ required to repay the interest on the loans.

$$ \% \Delta Y \geq \% \Delta M $$

Now if you recall the relationship between the money stock $M$, output $Y$,  the price level $P$, and money velocity $V$:

$$ PY=MV \rightarrow P=\frac{MV}{Y} $$

It's easy to see that if $V$ is constant, then:

$$ \% \Delta P = \frac{\% \Delta M}{\% \Delta Y}$$

Which, when combined with the observation that $ \% \Delta Y \geq \% \Delta M $, implies that an expansion of the money supply that is merely sufficient to allow for repayment of interest would actually be deflationary. This should make perfect sense: in expectation, output must increase by more than the amount captured by banks in the form of interest, otherwise firms in the real economy would never bother investing in fixed capital. So the money supply will in all likelihood increase by an amount strictly greater than the level required to repay interest, yet without any increase in inflation.

Loans to consumers are similar, so I won't give them a full treatment. Consumers are borrowing against their own future output, and foreclosures are generally priced in to the interest rate on a loan.

You'll note that I left out the case where in the aggregate, future output is less than that required to repay outstanding loans. This is intentional, as it's worth treating as a separate question.


Best response correspondences game theory

Let $\pi_L(p,q) = 3pq + p(1-q) + 2(1-p)(1-q)$ denote the expected payoff of the lecturer. The best response is defined as $BR_L(q) = \arg\max_{p \in [0,1]}\pi_L(p,q)$.
Now, note that
\begin{align}
\frac{\partial \pi_L(p,q)}{\partial p}
\begin{cases}
> 0 \quad& \text{for } q > 1/4,\\
= 0 & \text{for } q = 1/4,\\
< 0 & \text{for } q < 1/4.
\end{cases}
\end{align}
For $q>1/4$ the payoffs are strictly increasing in $p$ such that $p = 1$ is optimal.
For $q=1/4$ the payoffs are constant in $p$ such that any $p \in [0,1]$ is optimal.
For $q<1/4$ the payoffs are strictly decreasing in $p$ such that $p = 0$ is optimal.
Same logic applies to The Catalan.


Book recommendations: National income accounting and IS-LM model

I would suggest:

1) Macroeconomics by Dornbusch and Fisher.

2) Macroeconomics by Richard Froyen

3) The Economics of money and financial markets by Mishkin.

Between the first two books that is Dornbusch and Fischer and Richard Froyen, I am indifferent.  Both are equally good and the only difference is that Dornbusch and Fischer has a slightly more mathematical rigor whereas Froyen is a more story like engaging macroeconomics narrative with fair amount of mathematics.

If you need a truly engaging book,  then go for Mishkin. But I will suggest don't read Mishkin as the first book to understand macroeconomics.


What is the difference between shareholder's equity, equity, and book value?

Equity is defined as

  The value of an asset less the value of all liabilities on that asset. [source]

This is a rather broad definition and equity can take on different forms. E.g., for a house, it is the difference between the market value of the house and the mortgage still owned by the owner.

In the context of a company balance sheet, we usually talk about Shareholder's equity, which, as Wikipedia puts it

  represents the equity of a company as divided among individual shareholders of common or preferred stock

Contrary to the house example, the market value of a company, is the sum of all shares. And the shareholder's equity is that value (asset) subtracted from liabilities (creditors, etc.). See also this page.

The book value is the value of an asset. But the difference with the Shareholder's equity is illustrated as

  To find a company's book value, you need to take the shareholders' equity and exclude all intangible items. This leaves you with the theoretical value of all of the company's tangible assets which are those assets that can be touched, seen, and felt as opposed to things such as patents, trademarks, copyrights, and customer relationships. [source]


There is a private database: http://www.bankruptcydata.com/ which appears to have all bankruptcy filings logged. It has a subscription but appears to offer a free trial. To access public court filings in general which would include bankruptcy filings you'll ned access to PACER which is the government's database for all public court filings although it does require an account and charges a fee for each document (https://www.pacer.gov/). In general public bankruptcy filings are likely not going to be offered for free publicly given that it appears there is no official government body which gathers this information.

Your best bet to find balance sheets before bankruptcy is to look at a company's 10K or 10Q SEC filings which includes balance sheets and other required data for publicly traded companies to report to the SEC quarterly. 10K reports are the year end reports and 10Q reports are quarterly reports. The SEC has a useful database for these reports called EDGAR where you can look them up: https://www.sec.gov/edgar/searchedgar/companysearch.html.

Note that the SEC filings only apply to publicly traded companies. Non-publicly traded companies have no requirement to disclose their balance sheets and that information likely isn't available to you.


Fixed vs pegged exchange rate: What's the difference (if any)?

As Jason Nichols says, these terms are often used interchangeably.

The general theme is that pretty much anything can be called a "peg" (except perhaps the case where two countries are literally using the same currency), while "fixed" tends to refer to institutional arrangements that are more automatic, where changes in the exchange rate are perceived to be less likely. "Peg" tends to be used when some entity (e.g. a central bank) is doing the "pegging"; the more active and less automatic the behavior of this entity is, the more likely that you'll call the arrangement a "peg".

To further explain the (inconsistent and informal) differences in usage that do seem to exist, I have to describe several different kinds of exchange rate regimes.

At one extreme, country A may use the same currency as country B. Relevant cases include the Euro Area and dollarized Western Hemisphere countries like Panama and Ecuador. In this case, we might say that A and B have a "fixed" exchange rate (at one to one), but we probably wouldn't say that their rate is "pegged". A similar but less certain case would be, for instance, the CFA Franc in Africa, which is in principle distinct from the Euro but has a fixed Euro conversion rate and is guaranteed by the government of a country using the Euro (France). This might be called "fixed" or "pegged".
A less extreme situation is a currency board, where country A has a different currency than country B but promises to always convert them at a certain rate, and has reserves denominated in country B's currency backing up every unit of its own currency so that (in principle) this promise can always be fulfilled. Examples include Argentina's defunct currency board (a good example of a case where this promise was ultimately not fulfilled in practice) and Hong Kong's current currency board. I've seen both "fixed" and "pegged" used to describe such arrangements; as mentioned above, my sense is that "peg" is more common as a descriptor when the arrangement is perceived as being less automatic, with its permanence less certain.
Still weaker is the situation where country A sets a certain exchange rate with country B but doesn't have a formal arrangement like a currency board to back it up. This is extremely common; one rare example among developed countries is Denmark, which pegs to the Euro. This is the last case among (1)-(3) that can still often be called a "fixed" rate or a "peg". The line between this and a currency board in (2) is often blurry, since in this case too central banks often maintain large foreign reserves, perhaps enough to back up every unit of their currency outstanding; but usually this case involves a less universal guarantee of convertibility.
Then there is a vast set of arrangements where the rate is not perceived as being truly fixed for the indefinite future, but instead is allowed to fluctuate within a moderate or large band, or is subject to a "crawling" peg that is adjusted by the central bank, or is pegged to a possibly malleable basket of currencies, etc. These arrangements are usually called "pegs" rather than "fixed", because (after all) they aren't that fixed! A conspicuous modern example is China.
There are even looser arrangements where a central bank floats the currency but pays some attention to exchange rates and wants to avoid fluctuations that are too large (rather than exclusively hewing to some domestic objective like an inflation target), using both the tools of domestic monetary policy (interest rates) and intervention in foreign exchange markets to keep exchange rates in line. This is usually described as a "managed" float, "dirty" float, or something similar; it would rarely be described as a peg, but sometimes the line between (4) and (5) can be blurry.
Then at the extreme other end, we have countries that float their currencies and do not routinely intervene in foreign exchange markets. These arrangements would never be called either "fixed" or "pegged".

To sum up, my overall impression is that it's pretty vague, but that among 1 to 6 above, you're more likely to call something earlier in the list "fixed", something more in the middle of the list a "peg", and something later in the list a "float". But I am not aware of any formal, clearly specified distinction between terms as general as "fixed" and "peg". When exchange rates are classified, as in the IMF's annual report on the topic, this is usually done in a much more specific and descriptive way.


Derivation of aggregate demand function for Monopolistic Competition (based on Combes et. al, 2008)

As we have discussed in the comments, the utility function used in the book is Cobb-Douglas: $U = CM^{\mu}A^{1 - \mu}$. The well known fact mentioned in equation (3.3) on page 57 - the one that you highlighted - is that the Cobb-Douglas utility function is homothetic - the share of total income spent on any of the goods is constant over all possible incomes. More specifically here a share of $\mu$ of the expenditure is always spent on manufacturing goods and a share of $1-\mu$ is spent on the agriculture good. $\sigma$ and $p_i$ only determine how the $\mu y$ units of income that the representative consumer spends on manufacturing output is allocated among the distinct manufacturing goods.
Although the output goods are different most such models assume that both the jobs and the employees are identical and there is a perfectly competitive domestic labor market. Hence the identical wage rate.

Note: $y$ is personal income and $L$ is the number of employees, thus the total income of the economy is $yL$. (E.g. the last sentence on page 56).

Thus the total expenditure $E = yL$ - the total income, as the model is static. The expenditure on manufactured goods is $\mu E = \mu yL = PM \Leftrightarrow M = \frac{\mu yL}{P}$.


Can the major indice of an exporting nation decrease at the same time the currency devalues?

Yes, the FTSE 100 can decrease at the same time as the currency. It depends on what is causing the depreciation of the currency or what is affecting the FTSE at the time. It is hard to establish which way the effects are going in real time, however.

Suppose the currency is depreciating due to loose monetary policy. If the Marshall-Lerner condition is fulfilled, this means companies will export more and have more profit. In that case, we could expect the FTSE to go up. On the other hand, if the condition is not fulfilled, exports may actually go down and the FTSE with it. This is the case where the currency depreciation affects the FTSE.

On the other hand, the FTSE may affect the currency. If firms are facing a crisis or other problems, the FTSE goes down. This may prompt the central bank to attempt to depreciate the currency or it may prompt investments to flow out of the country, which depreciates the currency. The currency may depreciate for a while, before the monetary policy has any positive effects on the FTSE. The loss of investment will have a direct negative effect on the FTSE, as there is less capital supply and interest rates for firms may go up. Here again both the FTSE and the currency may go down.

There could also be third factors that affect both the FTSE and the currency. Suppose the currency is depreciating because of other  factors, e.g. an increase in uncertainty. This may cause investments to flow out of the U.K. thereby depreciating the currency. Further, that same economic uncertainty and reduction of investment could hurt profits and therefore the FTSE could go down.

Moreover, a simultaneous crisis in the banking industry and a currency crisis is very possible. These are called "twin crises". See the 1999 Kaminsky and Reinhart paper in the AER for a reference. In that case, the FTSE and the currency can go down simultaneously. Twin crises are more relevant for fixed exchange rate regimes, however, so this possibility may not apply to the U.K. as much as to other countries.


Has Stack Overflow been economically beneficial to programmer productivity?

Exogenous Variation

As Andre says, you are after causality, not just correlation. And the best way to find that is exogenous variation - he mentions experiments. Unless you can randomly force some people to use SO - and forbid it to a control group, there is little you can do with true experiments.

There may be salvation, however. You are after the "treatment of the treated" effect - those who are informed about SO, and decide to use it. That is hard to get, because the latter step will correlate with ability. Some people are just better/more efficient/smarter and hence can benefit from SO more/less.

In order to remove this self selection, you could instead look at something like the "average treatment effect". We inform a subsample about SO. Some will use it, some will not. What is the average increase of productivity in this sample?

IV Estimation

Now, you will either randomly inform individuals about SO (something that StackExchange as a company might actually do), or you/they might already have data on "being informed about SO", lets call that inform. As long as inform does not correlate with relevant characteristics (such as skill), you can use that as a proxy. Unfortunately, one might argue that it actually does: Perhaps smarter/better programmers are also better at creating networks / using google, and hence are more likely to find SO and become inform.

Of course, if you also have data for ability, you can control for that, so this particular bias would be removed.

Then, all you have to do is regress some measure of productivity/success/career advancement onto inform, and perhaps ability.


Specific question about some figures in national accounts

Para 2.43 explains the difference between the ‘uses’ and ‘resources’ sides of the account. The columns then refer to different sectors, where you are highlighting transactions with the ‘rest of the world’ sector.

For the numbers you refer to, the left-hand side is uses:

6 = compensation of employees paid out to the ‘rest of the world’ sector.
44 = property income paid out to the rest of the world sector.

The right-hand side is resources:

2 = compensation of employees paid into this economy by/from the rest of the world sector.
38 = property income paid into this economy, by/from the rest of the world sector.

Edit: you also asked how the 1,864 is derived. The balancing item for this account should be calculated as total resources less total uses (although I just tried it and got 1,854). There are separate gross and net balances, where the difference is consumption of fixed capital (depreciation of assets).


Demand for utility function given prices and budget

Regarding the first question, @Oliv suggestion in the comments to formally write down the consumer's problem, budget constraint, Kunh-Tucker multipliers and all, and solve it (carefully -linear objective functions easily mislead when we are used to deal with non-linear ones), is the way to go in order to see how the mathematical set-up will lead you to the answer that stares you in the face:

Our purpose here is to maximize the utility index. Given unitary prices, isn't it obvious that the multivariable function

$$\hat{u}(x_1,x_2,x_3) = x_1+x_2+2x_3$$

will take its maximum value if we purchase only good $x_3$? For any tiny amount $\Delta x_3$ of $x_3$ that you don't purchase, you lose $2\Delta x_3$ in utility terms, while you gain (in utility terms) only $\Delta x_1 + \Delta x_2 = \Delta x_3$ for any combination of $\Delta x_1$ and $\Delta x_2$. Since $2\Delta x_3 > \Delta x_3$ it will be suboptimal to purchase anything else than $x_3$. That was the "plain logic" part, and I hope it answers the "what I don't understand..." aspect of your question. Put the model in math to reconcile this obvious arithmetic with the formal approach.


It depends upon how and why the inequality exists.  Equality can also be bad for growth.

In the old Soviet system, most people were approximately equal and this was done by the use of police power.  It did permit rapid industrialization but then stagnated.  Not all innovation was good either.  Look at the Trabant.  An amazing vehicle.  Watch https://www.youtube.com/watch?v=No1-4GsQa-g

The US used to be very flexible.  That was good for growth.  If you were born poor, then you could become rich.  If you were born rich, then you could become poor.

Some income inequality is due to changes in technology and required education.  Some are due to legal changes that transfer resources from ordinary people to the rich.  It is now very difficult to become poor if you are rich and to become rich if you are poor.  That damages the incentives to build a business to grow the system.

Inequality is neither good nor bad, by itself.  It is a consequence of something else.  It is smoke; it is not fire.

There are several papers by Peter Temin that are very good on the topic, but he has quite a few and I don't know exactly what you are searching for or have already read.


Why does Bimetallism require international cooperation to have a chance at succeeding?

Quoting Wikipedia:

Bimetallism[a] is a monetary standard in which the value of the monetary unit is defined as equivalent to certain quantities of two metals, typically gold and silver, creating a fixed rate of exchange between them.[3]

Let's suppose that France says 1 ounce of silver is F1 (1 franc) and one ounce of gold is F80. And let's suppose that Germany says 1 ounce of silver is DM1 (1 Deutschmark) and one ounce of gold is DM79.
Anyone with an ounce of gold and a bit of common sense will go to France, convert it to francs (80 francs), convert it to silver (80 ounces), go to Germany, convert it to Deutschmarks (80 Deutschmarks) and convert it back to gold (1 ounce + 1 Deutschmark left over). Then they'll do it again. Over and over and over and over. Every time, they get an extra Deutschmark. Until France runs out of silver or Germany runs out of gold.
The only way this can be stopped, is if every country uses the exact same ratio - for example, if they all agree that 1 ounce gold = 80 ounces silver. That is international cooperation.


Since you mention IEWDS, I presume by "dominant" you actually mean "dominated".
Any strictly dominated strategy would satisfy the condition defining weakly dominated strategies and hence be called such. And yes, strictly dominated strategies can (and should) be eliminated in the process of IEWDS.
The possible typo notwithstanding, any strictly dominant strategy would satisfy the condition defining weakly dominant strategies and hence be called such.


Clarification about Market Structure Oligopoly or Monopoly

This market is an oligopoly that is subject to government regulation.

It cannot be a monopoly because there is more than one firm. The presence of the regulating government body is a "red herring", it distracts from the main point- there are multiple firms.

It does not appear to be competitive because:

Four is subjectively few firms
Implicitly, regulation would be entirely pointless in a competitive market.
One firm has managed to differentiate themselves by adding air conditioners. Firms in perfect competition cannot be differentiated from one another.


How does Brouwer's fixed point theorem relate to Walrasian equilibrium?

In standard proofs of the existence of Walrasian equilibrium, after one applies the appropriate fixed point theorem (typically Kakutani's, but sometimes Brouwer's), one then shows that the given fixed point is, in fact, a Walrasian equilibrium.

Depending on the exact construction of the proof, sometimes the fixed point will only give you the equilibrium price vector. However, this will be done so that the implied demands will then clear the market.

In other proofs, like this one, the fixed point will give you both the equilibrium price and allocation.


Does the existence of insurance markets negate the broken window fallacy?

No, they don't negate the broken-window fallacy.

You're overthinking it, and that's leading you to the wrong conclusion.

The breaking of the window represents a loss of wealth. The activities around it are neither here nor there - they do not change the fact that the breaking of the window represents a loss of wealth.

The existence of functional insurance markets just means that there are somewhat efficient markets to estimate the cost of that damage.


I'm answering if there is any production function which is homogeneous and CES too.

How about a CES production function only.

$y=(x^\rho +y^\rho )^\frac{1}{\rho }$ for $0\neq \rho <1$

This function has Elasticity of Substitution : $\sigma =\frac{1}{(1-\rho )}$ which is a constant.

Considering homogeneity,

$ y=((\lambda x)^\rho +(\lambda y)^\rho )^\frac{1}{\rho }$

$y=(\lambda^\rho (x^\rho +y^\rho ))^\frac{1}{\rho }$

$y=(\lambda ^\rho )^\frac{1}{\rho }(x^{\rho }+y^\rho )^\frac{1}{\rho }$

$y=\lambda (x^\rho +y^\rho )^\frac{1}{\rho }$ which is the initial production function.

Since, the degree of $\lambda $ is 1, so, it's homogeneous of degree $1$.


In fact, we don not even call them preferences. In order for a binary relationship to be a preference relationship, it must be both complete and transitive. The main goal of modelling preferences is to understand choice. And we cannot understand choice unless preferences are a quasi-order, i.e., one can rank alternatives (with ties allowed). In the case you mention, $x \succ y \succ z \succ x$, we would not know how to rank them. Further, it is clear that no function $u(·)$ can reproduce this ranking, as transitivity of real numbers makes it impossible to have $u(x)>u(y)>u(z)>u(x)$.

Note that this is different from $x \sim y \sim z.$ In this case you could say "no ranking exists" (although, perhaps it would be more adequate to say that "any ranking is consistent with the preferences"). In these circumstances, the utility function does exist, as it only must happen that $u(x)=u(y)=u(z)$.


Given fractional banking and the multiplier effect, for every dollar in circulation, how many dollars in spendable money exist?

The initial statements about fractional reserve banking are not a question. The simplified example you give does not reflect the complexity of bank lending. The article "Money Creation in the Modern Economy," by Michael McLeay, Amar Radia and Ryland Thomas of the Bank of England link to paper is one place to start to get a more realistic discussion of money creation.

  In practice, in the world and the United States, how many spendable
  dollars are created for each actual dollar in circulation, M0 to M1?

For the United States, these data are available at FRED, a service run by the St. Louis Fed. link to FRED

The relationship changes over time; the policy of quantitative easing greatly affected money supply numbers, and the relationship between them.

If you are interested in multiple countries, you will probably need to track down data from central banks or statistical agencies, or you may be able to find an aggregate source. I am unaware of any free data suppliers that cover money supply data.


Sure.

In the two type case there is an alternate solution that is frequently used. If $a \neq c$ and $b \neq d$, that is the types are really different in both attributes, then you can define a function $f$ for which
$$
b = f(a), \hskip 20pt d = f(c).
$$
Whatever you need the second attribute for, this way it becomes a function of the first one, and the values of the first attribute completely define the type space.

Example:
 In Spence's model of job market signaling, people have a type that shows how much they are worth to a company, how hard working they are. It is assumed that the cost associated with getting a degree is a function of this type as well, and harder workers get a degree more easily. One way to show this would be to include both worth to the employer and cost of getting a degree in the type attributes, but Spence only defined the types and gave some functions that mapped to these values.


ln(log) of energy consumption in a econometric analysis

Prior to any trasnformation:

Energy consumption is a per capita variable, specifically "Kgr per capita"
Real GDP is a per capita variable
Trade Openness is a per capita variable
Energy price is a unit price of "crude oil"

I see two problems: Is "Energy Price" the "price per Kgr" -because usually oil prices are not quoted per Kilogram. But it should expressed here "per Kgr" so as to stand for the price of the dependent variable.

If it does, it is only natural to consider the logarithms of the variables, dependent variable included.

But one more thing that needs clarification is what $Y$ stands for. The authors write "$Y$(economic growth)". If this is the case then for this variable you don't just take the logarithm of real GDP but the first-difference of the logarithm  (which approximates the rate of growth).


Utility Possibility Frontier with two consumers and 3 commodities

The utility possibility frontier (UPF) plots the maximum total combination of utilities that can be achieved, given the preferences and total resources. To fix ideas, let's suppose we are plotting $u_1$ in the $y$ coordinate and $u_2$ in the $x$ coordinate. The easiest way to find the UPF is to start with a single agent, say agent 1 and give all the resources to him, so allocation $(4,4,4)$ his utility will be 24, so that point, $(0,24)$, in the plane is definitely part of the UPF (by the way this calculation already precludes option $a$). Now, think that you are going to decrease agent $1$'s utility in order to increase agent $2$'s utility. Since we are characterizing the frontier, you want to think what is the best way to move some resources from player $1$ to player $2$. After some thought, it is obvious that the best is to take away some of good $1$ from player $1$ since he only loses $1$ util while the other player gains $3$ utils. Therefore, the point $(3,23)$ is also part of the $UPF$ note that the line connecting this two points is $u_1+\frac{u_2}{3}=24$, (unfortunately, this realization does not narrow the possible answers, since $b$ and $c$ share this equation as its first argument).

Now, this point was found by moving of the good 1 from agent 1 to agent 2, however, there are only 4 such units, so after agent 2 has all units of good 1, in order to keep increasing her utility, the most efficient transfer is to take away from agent 1 some of the units of good 2 he enjoys. Here the loses from agent 1 equal the gains from agent 2, so we should have this defined by a line with equal coefficients: $\alpha u_1+\alpha u_2=C$ both options have this format, so we only need to make sure which one is correct given that the constant is 24.

Note that the point where $x_1=(0,4,4)$ and $x_2=(4,0,0)$ is part of this line and if we compute $u_1(0,4,4)+u_2(4,0,0)=32\neq 24$, but $\frac34 u_1(0,4,4)+\frac 34u_2(4,0,0)=24$, as desired. You can conclude that the answer is $b$.

The $\max$ operator ensures that the graph has the correct "kink" points when the type of good you are transferring from one agent to the other changes.