I hope I understood your question correctly. In case this helps:

1) you can probably cover even the stock part with a combination of just a few ETFs.

2) traditional funds (as opposed to ETFs) will always be promoted much more aggressively, because their providers earn high fees. ETFs are much less profitable for their providers and are often not promoted at all even if they represent a superior solution. This might explain why you see versions with traditional funds, but not ETFs.

I think you are right in thinking that you can construct a "permanent portfolio" type investment portfolio out of ETFs with a superior cost structure. Probably the best way to do it. Industry players who have a vested interest in traditional portfolio management won't tell you this though.


In a typical GE model, price (including wage and rent) is determined by the market clearing condition, which requires that supply equals demand. Specifically,

labor supplied (determined by household solving utility maximization problem) equals labor demanded (determined by firm's profit maximization problem)
goods supplied (determined by firm's optimal total production) equals goods demanded (determined by household's consumption and investment decisions)
capital supplied (household's investment decision) equals capital demanded (firm's optimal choice of $k$)

The zero-profit condition is derived from the assumption of perfectly competitive behavior (or both firms and households are price takers). This condition may contribute to the determination of equilibrium prices. But to say that $p$ is selected to ensure zero-profit of the firm just sounds incorrect.


What is the Economic Rationale for Discounted Early-Bird Parking?

There seems to be some research on price discounting. For example, this paper studies two forms of dicounting, last-minute discounting and early-bird discounting, in settings with monopoly, duopoly and competitive markets. They propose a model and run some experiments. Some excerpts from this paper:

  There might be many reasons for such a pricing policy [Early-Bird Discounting]. Early bookings for example reduce uncertainty for
  airlines, who might be willing to forgo some profit in order to reduce this
  uncertainty. Introductory pricing in markets with network externalities
  might be another reason. Here we want to investigate intertemporal price
  discrimination between different customer groups, who enter the market
  at diferent points in time. One might think of holiday makers versus
  business travellers in the market for airline tickets. Holiday makers
  usually plan ahead and enter the market for a particular flight quite early,
  while business travellers often cannot plan well in advance. Additionally,
  the reservation price of business travellers is much higher than that of
  holiday makers. A monopolist airline will try to persuade as many low-
  valuation customers as possible to book early, such that it can charge
  higher prices later when the high-valuation customers have entered the
  market.

This is also interesting:

  When customers with lower valuations for the good start
  searching earlier for cheap prices than consumers with higher valuations
  then it is possible to sell to these customers at early-bird discount prices,
  while the high valuation customers who enter later can be charged higher
  prices. The main objective of the monopolist is to get the low-valuation
  customers out of the market before the high-valuation customers arrive.
  The low-valuation customers accept early, as they anticipate that the
  prices will go up over time.

They conclude that:

  In the early-bird scenario competition even seems to enhance the ability
  to price discriminate initially, while with increasing experience this advantage is eroded. [...] The monopolists had much less pricing
  power than predicted by standard theory, while the duopolists - also
  contrary to the prediction - were able to retain at least some pricing
  power. The most striking advantage of competition is the increase in total welfare.

Perhaps not entirely pertinent to the case of parking, this entry in The Economist claims that early-bird tickets are useful in events like concerts to build up a minimum mass of goers:

  T in the Park, a large pop shindig held in Scotland, pioneered the practice of offering discounted “early bird” tickets to build buzz.


Dividing equation 25 by $\gamma$ gives:
$$
Y_t = C_h + \frac{\gamma}{1-\gamma} C_h^\ast.
$$
Then using 7 and 7a to substitute for $C_h$ and $C_h^\ast$, we get:
$$
Y_t = (1-\gamma) \frac{P_t}{P_h} C_t + \frac{\gamma}{1-\gamma}(1-\gamma) \frac{P_t^\ast}{P_h^\ast} C_t^\ast.
$$
Then multiply both sides by $P_h$ and use $C_t^\ast = C_t$ to get:
$$
\begin{align*}
P_h Y_t &= (1-\gamma)P_t C_t + \gamma \left(\frac{P_t^\ast}{P_t}\frac{P_h}{P_h^\ast}\right) P_t C_t,\\
&= P_t C_t \left(1- \gamma + \gamma \left(\frac{P_t^\ast}{P_t}\frac{P_h}{P_h^\ast}\right)\right)
\end{align*}
$$
This is equal to equation 28 if:
$$
\frac{P_t^\ast}{P_t} \frac{P_h}{P_h^\ast} = 1.
$$
Now $\dfrac{P_t^\ast}{P_t} = \dfrac{1}{\varepsilon}$, which would require that:
$$
\frac{P_h}{P_h^\ast} = \varepsilon.
$$
Dividing 7 by 7a gives:
$$
\frac{P_h}{P_h^\ast} \frac{C_h}{C_h^\ast} = \frac{P_t}{P_t^\ast} = \varepsilon.
$$
So this means that $C_h$ has to equal $C_h^\ast$. I don't know if this makes sense (as I don't know the meaning of all these variables).
Similarly, you can show that 29 is satisfied if $C_f = C_f^\ast$.


Why does a strong dollar make imported goods cheaper?

Consider that you want to purchase a product from a EU country, pricing their products in Euros.

Let’s say that you want to buy a product that costs 100 euro. No matter what the relationship is between USD/EUR, the product will still be 100 EUR.

Let’s say the USD/EUR is 1.24 as in your example - that means for every 1.24 USD you have, you get 1 EUR. If you want to buy the product you will then need 124 USD to buy a 100 EUR product.

Now let’s say that the USD strengthens compared to EUR and the new exchange rate is 1.15 instead of 1.24. Using same logic as above, you now need 115 USD to purchase that same product, giving you a discount of 9 USD just because of the new exchange rate.

Remember the product is still 100 EUR - but you can now buy more euros for your dollar than you could before.


Is 'price hike' synonymous with 'price rise'?

Yes there are used interchangeably in the literature e.g. see the occurrences printed in Mankiw Principles of Economics. For example, on page 509 its used completely interchangeably:

If a Pennsylvania gun manufacturer raises the price
of rifles it sells to the U.S. Army, its price hikes will
increase...

Price hike is also not special economic jargon, it is used pretty much consistently with dictionary definition.

A hike is a sudden or large increase in prices...


Uniform price vs. pay-as-bid auctions in energy markets

I believe "Auctions of Homogeneous Goods: A Case for Pay-as-Bid" by Pycia and Woodward answers your questions theoretically. This is quite recent and their results are striking. They also briefly discuss some empirical insights.

The  pay-as-bid  (or  discriminatory)  auction  is  a  prominent
format  for  selling  homogenous goods such as treasury securities
and commodities.  We prove the uniqueness of its pure-strategy Bayesian
Nash equilibrium and establish a tractable representation of
equilibrium bids.  Building on these results we analyze the optimal
design of pay-as-bid auctions, as well as uniform-price auctions (the
main alternative auction format) allowing for asymmetric information.
We show that supply transparency and full disclosure are optimal in
pay-as-bid, though not necessarily in uniform-price; pay-as-bid
is revenue dominant and might be welfare dominant; and, under
assumptions commonly imposed in empirical work, the two formats are
revenue and welfare equivalent.


"If the new bundle was affordable at old prices, and the new and old bundles aren't equal, then the old bundle must not be affordable at the new prices".
In other words, the Weak Axiom of Revealed Preference (WARP) refers to consistent (rational) decision-making.
I think the proof is in terms of an 'if then' direct method:
The Marshallian demand function $x(p,w)$ satisfies WARP if the following property holds for any two price-wealth situations $(p,w)$ and $(p',w')$.
If $p*x(p',w')\leq w$ and $x(p,w)\neq x(p',w')$ then: $p'*x(p,w)>w'$.


Which mathematics are required for fully understanding the theories of the firm?

I haven't gone through all the papers so I just sampled 'The Firm as an Incentive System'. Browsing through it, it relies on Linear Algebra and Real Analysis. Again, I warn that I have not gone through the entire paper but seeing some of the terminology used there, I could guess that these two a clearly involved (maybe they use some bit of topology as well). They use a concept of sublattice somewhere and that pertains to group theory.
A good starting place for real analysis could be Steven R. Lay's 'Analysis'. There you can learn some of the basic terminologies that they have used like supremum and infimum. For Linear Algebra, it has got to be Gilbert Strang's text 'Introduction to Linear Algebra'. Linear Algebra is used extensively in Econometrics as well. You can find his lectures at: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/ .
Hope this helps!


Difference between Nash equilibrium and Pareto Efficiency

I wouldn't say Nash equilibrium is broader than Pareto-efficiency I'd say its different. As all know the sole Nash equilibrium in the Prisoner's dilemma is not Pareto-efficient and the three Pareto-efficient outcomes are not Nash equilibria. That is to say there is no intersection between the two terms and hence none can encompass the other.

I think the main difference comes from the fact that Pareto-efficiency is only concerned with outcomes and not with the mechanisms of strategic interaction. The Nash equilibrium takes this mechanisms into account explicitly. I.e. you cannot obtain a Pareto-efficient outcome in the Prisoner's dilemma as the other player will preempt it.

My advice is: don't read about Game theory on Wikipedia! The first homework in the master's course on GT at my faculty is to find an error in a Wikipedia article on GT and correct it.


Arrow’s Impossibility & Validity of Social Welfare and Pareto Improvement Analysis

Let $X$ be the nonempty set of alternatives, $\mathcal{P}_X$ the set of preference relations on $X$ and $N=\{1,\ldots,n\}$ a finite set of agents. Under the universal domain condition, Arrow's theorem concerns functions from the set $\mathcal{P}_X^N=\underbrace{\mathcal{P}_X\times \mathcal{P}_X\cdots\times\mathcal{P}_X}_{n\text{ times}}$ to $\mathcal{P}_X$. Utilitarianism does not just depend on preferences alone, and can therefore not be formulated in this setting.
Still, one can formulate Arrow's theorem in terms of utility functions. So let $\mathcal{U}_X$ be the space of real-valued functions on $X$, interpreted as "utility functions." I'll explain the scare-quotes later. We are now looking at functions from $\mathcal{U}_X^N$ to $\mathcal{P}_X$. Since utility functions determine preferences, we can do now the same things we could do before and more.
Now there are two ways to formulate Arrow's condition of independence of irrelevant alternatives in the new setting; adapting the other conditions is straightforward.
The first form is:
IIA1: The function $\phi:\mathcal{U}_X^N\to\mathcal{P}_X$ satisfies IIA1 if for any two profiles of utility functions $u=(u_1,\ldots,u_n)$ and $u'=(u_1',\ldots,u_n')$, and any two alternatives $x,y\in X$,  such that $u_i(x)\geq u_i(y)$ if and only if $u_i'(x)\geq u_i'(y)$ for all $i\in N$, we have $x\phi(u) y$ if and only if $x\phi(u') y$.
Note that IIA1 has implicitly two components. The first says that the social ranking of two alternatives depends only on the utility values for these two alternatives. The second says that only the preference ranking matters, but not the "intensity." If we only keep the first, we get
IIA2: The function $\phi:\mathcal{U}_X^N\to\mathcal{P}_X$ satisfies IIA2 if for any two profiles of utility functions $u=(u_1,\ldots,u_n)$ and $u'=(u_1',\ldots,u_n')$, and any two alternatives $x,y\in X$,  such that $u_i(x)-u_i(y)=u_i'(x)-u_i'(y)$ for all $i\in N$, we have $x\phi(u) y$ if and only if $x\phi(u') y$.
IIA2 is actually weaker than requiring that the utility for both alternatives are the same in both profiles; only the differences need to be the same.
Now, with the other axioms suitably reformulated, IIA1 implies that all such functions $\phi:\mathcal{U}_X^N\to\mathcal{P}_X$ must be dictatorial, while IIA2 is compatible with utilitarianism.
Now, so far this was a purely mathematical issue. In particular, we treated utility functions as being somehow objective things and not just representations of preferences. They also measure well-being in a way that is comparable across agents. So utilitarianism uses information that is not included in purely positive models of economic behavior, in which only preferences matter. Now, one can discuss how sensible it is to compare the well-being of different agents and what information one needs for certain SWFs. There has been a research program starting in the late 1970s of discussing the exact informational assumptions one uses for such comparisons of well-being. A nice introduction to the topic is given in the 1998 book "Theories of distributive justice" by John E. Roemer. Roemer directly relates all this to Arrow's theorem, so his book will give you a very extensive answer.


Why does decrease in fixed taxes accompanied by a decrease in Government Spending of the same amount will decrease GDP?

The question basically boils down to whether the inequality below is true or not.
$$\bigg|\Delta G\bigg(\frac{1}{1-MPC}\bigg)\bigg|>\bigg|\Delta T\bigg(-\frac{MPC}{1-MPC}\bigg)\bigg|$$
If $\Delta G=\Delta T$, we can divide by either and get
$$\bigg|\bigg(\frac{1}{1-MPC}\bigg)\bigg|>\bigg|\bigg(-\frac{MPC}{1-MPC}\bigg)\bigg|$$
If we multiply each side by $1-MPC$ and we evaluate the absolute value, we get
$$1>MPC$$
This statement is true, so that implies that the first inequality is true. By this inequality, we know that $\Delta GDP_G>\Delta GDP_T$. Since $\Delta GDP_G<0$ and $\Delta GDP_T>0\implies \Delta GDP<0$


Terminology, is elasticity used as "mean elasticity"?

I would not call it average elasticity rather its elasticity at an average price. For example take the first paper about elasticity of oil you cite (that is Cooper, J. C. (2003). Price elasticity of demand for crude oil: estimates for 23 countries. OPEC review, 27(1), 1-8.).
In that paper the Cooper estimates elasticity using the following model:
$$\ln D_t = \ln \alpha + \beta \ln P_t + \gamma \ln Y_t  + \delta \ln D_{t–1} + e_t$$
Where $\beta$ gives you the estimate of elasticity. However, $\beta$ is not necessary equal to $\bar{\epsilon}$ for any model specification (even though the above one would actually imply it -see the last paragraph), rather $\beta$ gives you point estimate elasticity at $\bar{P}$.
In fact generally OLS regression is constructed in such way that it intercepts point given by $\bar{y}$ and $\bar{x}$ - that is $\bar{y}-\hat{\alpha} -\hat{\beta} \bar{x} = 0$. Hence, more correct interpretation here would not be that $\beta$ gives you average elasticity (i.e $\bar{\epsilon}$) but rather that it gives you point estimate for elasticity at average price (i.e. $\epsilon_{\bar{P}}= (\bar{P}/Q)/(dQ/d\bar{P})$.
This being said note that using OLS model such as the one that is used in Cooper would actually imply that the elasticity is constant because its a linear model (linear in its parameters that is) where $\beta$ is assumed to be constant across all observations (even though you get perfect fit only at ($\bar{P},\bar{D}$). In a model with constant elasticity it would actually hold that $\epsilon_\bar{P} = \bar{\epsilon}$. Nonetheless, I would still caution against that interpretation. The reason for that is that in most cases it is commonly understood that the linear model is used as a simplification and not because people actually assume that elasticity of demand is constant. Mostly, people take this point estimates to be reasonable for small changes around mean price but not really claiming that they found a constant price elasticity of demand, and in most such models if you would look at errors they would get larger the further you get from mean estimate.


how can I find the original work on the bases of which 2016 nobel in economics was given?

It is based on multiple pieces of work, most of which can be found in the prize committee's thoroughly referenced announcement.


What levers are typically available to a Government to exit an economic downturn?

The usual policy options are.

Monetary policy - lower interest rates and/or expand the monetary base.
Fiscal policy - expand the fiscal deficit (government spends more and/or taxes less).
If the value of the currency is controlled, devalue.
If there are controls on lending (now rare in the developed countries), loosen them.

Something like efficiency improvements (how can the government endure this?) may help long-term growth rates, but it is unclear whether they will have an immediate impact. For example, if increasing efficiency involves firing workers, there would be a further reduction in household demand, which will not be corrected until they can find new jobs.


Is it common in Economics to send your paper, without an explicit invitation, asking for feedback (assuming the speaker to be in the relevant field)?

I am not sure this is common in Econ but I have definitely done it! Note that you increase the probability of getting feedback if

you know or met the author,
he/she asked you to send your paper,
you cite their work.

Or should one wait or look for a chance to present it in a seminar?

This is probably your best chance to get feedback given that profs are very busy. A bunch of universities and associations organize PhD seminars, workshops, conferences or poster sessions. Look at inomics. You may also contact profs with the intention to present your work in their seminars.


The most comprehensive survey of estimating oil prices is here. There you fund from very complicated models to very simple ones. As the article shows, the ultimate answer depends on whether you are estimating nominal or real price, and short-term or long-term prices.

Some simple models to estimate price of oil might be:

no-change forecast: if changes in the spot price are unpredictable, the best forecast of the spot price of crude oil is simply the current spot
price ($S_t$):

$$ \hat{S}_{t+h|t} = S_t \quad h>0$$

futures-based forecast: simply take the market's average expected oil price forecast as your forecast:

$$ \hat{S}_{t+h|t} = F^h_t \quad h>0$$

where $F^h_t$ is the future forecast in the market. There are plenty of sources available online (e.g. here or here).

Double-difference forecasting: proposed by Hendry (2006), it assumes past growth rates for future price changes:

$$ \hat{S}_{t+h|t} = S_t \left(1+ \Delta s_t \right)^h \quad h>0 $$

where $\Delta s_t$ is the percent growth rate of $S_t$ between $t-1$ and $t$

inflation-based forecast: use forecasts for inflation to predict oil prices (which according to Anderson et. al. 2011, this is how households predict gasoline prices):

$$ \hat{S}_{t+h|t} = S_t \left(1+ \pi^e_{t,h} \right)^h \quad h>0 $$

where $\pi^e_{t,h}$ is the forecast of inflation, for example, from the Survey of Professional Forecasters.

AR, ARMA, ARIMA regression models: last but not least, the famous AR-based models are very common. Simply regress the price of oil on its past values, using as many lags as you want (or the information criterion tells you), accounting for MA errors (ARMA) and possible cointegration (ARIMA). For example, the AR model with $p$ lags is:

$$ S_{t} = \beta_0 + \beta_1 S_{t-1} + \cdots + \beta_p S_{t-p} + \mu_t $$

After you have estimated the model and obtained values for $\beta_i$, you can forecast oil prices by simply lagging the last $p$ observations one period and put into the model, from where you get $\hat{S}_{t+1|t}$ (assuming $\mu_{t+1} =0$). Then, iterate forward to get forecast for any other future period.


CES production function profit and supply function

Hint: Solving for the FOC's assumes that the solution is interior, in this case, that profits are positive and smaller than $\infty$. I would recommend you to derive the cost function $c(y)$ and then study its derivative. If the marginal cost is always smaller than the price of the good (probably it is assumed to be 1) then producing more is always better and profits are unbounded. However, if the marginal cost is always larger than the price, then the optimal is to produce nothing and get zero profits.

Hope this helps.


Macro Economics Question: How will this curve shift?

You can look at IS-LM models in order to understand the intuituion more deeply.
In this graphic, IS curve shifts to left when there exists a demand-sided economic policy like government spendings etc. This will for sure increase the demand of people in economy, which will increase price levels.  If the price level increases, with the same quantity of money circulated in economy, the interest rate will increase. (you can see it on the graph) So, a central bank must control these kind of policies with a monetary policy in order to stabilize its inflation.

The same logic in increase of income. This will increase demand, which will increase price levels and agents will need more quantity of money in order to realize their transactions.


If North American stock markets are semi-strong efficient, why are indices like S&P500 expected to grow?

The composition of a stock market index is not fixed but changes over time. Moreover, it usually changes so as to increase/decrease the weight of good/poor performers.
For example, the S&P500 assigns weights to 500 large companies that meet various criteria, including at least a $8.2B cap.* These weights change over time. In fact, companies can even be dropped (e.g. recently Macy's) or added (e.g. Netflix, Google, and Facebook were not on the index in 2000 but are today).
So, suppose we're transported back 20 years to 2000-11-06. Consider the 500 stock prices that were included in the S&P500 on that day. It is true that the EMH would have predicted that those 500 stock prices would, on average, not have grown in the following 20 years (ignoring inflation). However, the EMH would not have said anything about the S&P500 Index, whose composition may and indeed did change.
(We can repeat this same thought experiment for the 500 stock prices included in the S&P500 today: The EMH predicts that, ignoring inflation, these 500 stock prices will not change over the next 20 years. However, the EMH says nothing about what the S&P500 will be in 2040.)
So, if we expect economic growth and creative destruction, then we also expect the S&P500 (and any other index whose composition is allowed to change) to keep growing over time, even if we do not expect any individual stock price to grow.

*See e.g. "S&P U.S. Indices Methodology, Aug 2020" PDF.


Quasilinear Utility: Pareto Optimality Implies Total Utility Maximization?

Edit: Edge cases suck; see comments. See also MWG Chapter 10 section C, D.

Suppose $(\vec x^*, \vec m^*)$ solves

$$\max \sum^I_{i=1} m_i + \phi_i(x_i)$$

but is not Pareto optimal.

$$\begin{align}
\implies \exists \ (x_i', m_i') \quad \text{s.t.} \quad & u_i(x_i', m_i') \geq u_i(x_i^*, m_i^*) \quad \forall \ i = 1,\cdots,I \\
& u_i(x_i', m_i') > u_i(x_i^*, m_i^*) \quad \text{for some} \ i
\end{align}$$

$$\implies \sum^I_{i=1} m'_i + \phi_i(x'_i) > \sum^I_{i=1} m^*_i + \phi_i(x^*_i)$$

which is a contradiction. If we have a solution to the utility maximization problem, it must be Pareto optimal.

(Note that this comes form continuous and increasing properties of $\phi(\cdot)$)

Suppose $(\vec x^*, \vec m^*)$ is a feasible Pareto optimal allocation, but does not solve

$$\max \sum^I_{i=1} m_i + \phi_i(x_i)$$

Because we treat $m_i$ as numeraire and $\phi_i(\cdot)$ is strictly increasing, we know $u_i(\cdot)$ is locally non-satiated. The Pareto allocation should be just feasible.

$$\exists \ (x_i', m_i') \quad \text{s.t.} \quad \sum^I_{i=1} m'_i + \phi_i(x'_i) > \sum^I_{i=1} m^*_i + \phi_i(x^*_i)\\
\implies \boxed{ \sum^I_{i=1} \phi_i(x'_i) > \sum^I_{i=1} \phi_i(x^*_i)}$$

If this is true because this alternative allocation simply gives an individual more of $x$, for all else equal, then the alternative allocation is infeasible. So we'd have a contradiction.

If this is true because in the alternative allocation, someone else is allocated more $x$ and just one other person is allocated less, then the original allocation would not be Pareto optimal. Suppose it was. If you took the original allocation and shifted $x$ in the way of the new allocation, then you would need a corresponding trade in the numeraire good, $m$, to keep whoever is losing $x$ at least at the same utility level. But trades in just the numeraire good can never change summed aggregate utility. From the original allocation, if you can trade $m$ for $x$ and make someone better off without hurting anyone, you weren't at a Pareto optimum, and if you can't trade $m$ for $x$ to make someone better off, you can't increase summed aggregate utility, which means the original allocation was a solution to the maximization problem.

This logic applies no matter how you rearrange $x$ between multiple people.

$\square$


Whom would you consider to be the father of rational choice theory?

The answer is there is no single father of rational choice theory. The reason is that rational choice theory is not so much a theory but rather a coherent framework (others would call it a paradigm) resting on the key foundations of methodological individualism and instrumental rationality / self interest.

Both of these aspects played important roles in the marginal revolution in the late 19th century. Important names here are, of course, Walras, Jevons, Menger and Marshall although none of them can be regarded single father of Economics in the modern sense.

Some decades later, Robbins (1932, p. 15) famously defined Economics as

  the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses.

This definition nicely summarizes much of the framework rational choice theory is embedded in. It contains instrumental rationality, in so far as that humans have goals (denoted as preferences) but only limited resources (constraints) and they choose so as to maximize their utility. It also hints at methodological individualism because it places human behaviour at the center of its investigation. Note, however, that Robbins again only summarized a research paradigm that was already well developed. (He says so himself on the same page.)

Still, I guess it is fair to say that rational choice theory as we know it today took off after WW2. Here different names come to mind, depending on which focus you want to have. Arrow/Debreu for general equilibrium theory, von Neumann/Morgenstern if you want to stress expected utility (and game theory). A bit later, Becker is a good choice if you want to stress that rational choice theory is an approach that is driven by its methodology (rational behavior, methodological individualism) and need not investigate an economic topic at all.

But yeah, asking for a single father of rational choice theory would really stress things a bit too far.


Strict preference relations and utility representations

Yes it is:
If direction
$$
x \succ y \Rightarrow x \not \precsim y \Rightarrow u(x) > u(y).
$$
Only if direction:
For all $x, y  \in X$,
$$
x \succsim y \iff u(x) \geq u(y)
$$
implies
$$
x \sim y \iff u(x) = u(y).
$$
Also
$$
u(x) > u(y) \Rightarrow u(x) \geq u(y) \Rightarrow  x \succsim y ,
$$
$$
u(x) > u(y) \Rightarrow u(x) \not = u(y) \Rightarrow  x \not\sim y.
$$
and
$$
x \succsim y \mbox{ AND } x \not\sim y \Rightarrow x \succ y.
$$


Prove that variance of a portfolio cannot exceed variance of individual assets

Let $P = \alpha A + (1-\alpha) B$ where $A$ and $B$ are returns (random) from the two assets, and $P$ is their portfolio.
Variance of portfolio $P$ can therefore be written as

\begin{eqnarray*} \sigma^2_P & = & \alpha^2 \sigma^2_A + (1-\alpha)^2 \sigma^2_B + 2\alpha (1-\alpha)\text{Cov}(A, B) \\ & \leq &\alpha^2 \sigma^2_A + (1-\alpha)^2 \sigma^2_B + 2\alpha (1-\alpha)\sigma_A\sigma_B \\ & = &(\alpha\sigma_A + (1-\alpha)\sigma_B)^2 \\ & \leq & (\alpha \max(\sigma_A, \sigma_B) + (1-\alpha)\max(\sigma_A, \sigma_B))^2 \\ & = & \max(\sigma_A^2, \sigma_B^2)\end{eqnarray*}


How does an increase in the price of "non-wage-goods" increase employment?

Wage-goods are goods that a worker with wages might buy, perhaps now more commonly called consumption goods

Non-wage-goods are goods that a rentier receiving profits or interest might buy, including what might now be called capital goods or investment goods

As to how to read Keynes's interpretation of Pigou's position (so actually neither Keynes's nor Pigous's), it might be saying that if non-wage-earners shift their demand towards non-wage-goods, this would have the demand-side effect of pushing up the relative price of non-wage-goods and reduce the relative price of wage-goods, with presumably

either the direct effect of increasing both the number of wage-earners employed producing non-wage-goods and their wages,
or the indirect effect of increasing the real wage for all wage-earners (since their wage-goods are now relatively cheaper) above the previous "marginal disutility of labour", and this encourages increased wage-earning employment (what today would be called a substitution effect)


How to calculate expected revenue in reverse auctions (procurement)

Hint: In second price sealed bid, it is optimal to bid your own valuation for the good. So in a reverse auction, it is optimal for each person to bid their lowest possible price they are willing to supply at. Intuitively, if one individual deviates from that strategy, it doesn't increase their chance of winning the auction or increase their chance at getting a higher priced contract, since the price they supply is based on other contractors' bids.

So the expected revenue is going to be the average value of the second lowest bidder amongst ten bidders. Try to set up an integral to describe the distribution and go from there.

For a normal second price auction (not reverse):

$F(x)$ is the probability that the price will be less than or equal to some given $x$, and $f(x)$ is the corresponding density function.

$$F(x) = \frac{x-50}{50}; \quad f(x) = F'(x) = \frac{1}{50}$$

Probability that second highest bid is $x$, with $n$ bidders:

$$h(x,n) = n(n−1)f(x)(1−F(x))F(x)^{n−2}$$

Expected value of $x$ (the price):

$$\int^{100}_{50} x h(x, n) dx$$

So you can easily apply a similar logic to your reverse auction. Set up $F(x)$ so that it represents the probability that the price will be greater than or equal to some given $x$.


It looks like a cash-accounting NPV that includes closing the position (selling the plant, the car, whatever) after the period of consideration. Asset values show up in the final term because cash accounting applies cash outlays to the period in which they actually occur. Typically, that asset value would be net of depreciation and therefore less than $I_0$.  Contrast this with accrual accounting, which would spread the depreciation over the period, perhaps by making some reasoned assumption about an annualized depreciation rate.

Note that unlevered free cash flow does not count debt obligations. This means interest payments as well as any principle are not counted by the $CF_{year}$ terms, and is why liabilities show up in the calculation explicitly, rather than being rolled into the cash flow term.

However, it's not clear to me why discharge of those liabilities is deferred to the third year - or why a "sum of all cash flows" term shows up, which appears to be double-counting. Perhaps if you share the context of where you're seeing this, that will help.


Why would China's FX reserves stay flat as the Fed ramps up its balance sheet?

There is no necessary relationship. Canada is a major trading partner of the United States, and the Fed can increase its balance sheet without Canada increasing its official reserve holdings. Why should China be different?


It doesn't seem to me that there were any substantive differences between the Bretton Woods regime and the gold standard, as seen in the early 1970s

Note that most of what I'm about to write is summarized from Wikipedia.

The gold standard of some form has always existed in the US since the Coinage Act of 1792 with a few interruptions. In 1934, gold was nationalized in the US but the value of the US dollar was still tied to gold reserves. Bretton Woods simply maintained the gold standard. While the rate between the US dollar and gold varied over the years, until 1976, the gold standard was officially kept. In other words, indeed there were no substantive differences because the Bretton Woods era was just a short part of the gold standard history.

  I'd like to know why economists thought Bretton Woods was a good idea.

In the sense that they mostly kept policy that had been in existence for over a century by that point, it wasn't too surprising. The general summarized critique against the gold standard based on my reading of the IGM survey is that gold is too volatile compared to the existing system with the US dollar and the Fed and gold doesn't allow for any counter-cyclical monetary policy. If economists a century ago took the same survey it isn't clear that there would be such a clear consensus against the gold standard. As the St.Louis Fed writes,

  For the pre-Fed period (1790-1913), the average annual inflation was
  0.4 percent with a coefficient of variation of 13.2. During the period 1941-2016, these figures changed to 3.5 percent and 0.8, respectively.
  If we look at the post-Volcker era (1988-2016), annual inflation was
  2.2 percent on average with a coefficient of variation of 0.4.

In other words, we've come a long way in terms of keeping price volatility of everyday goods and services in control. Relative to the past, in developed countries, we are no longer concerned with hyperinflation, which a gold standard makes fairly difficult.

Also from the perspective of the US, it helped solidify the US dollar as a reserve currency. All of the benefits listed in this SE answer apply besides seigniorage.


The first reason for which volatility is higher is the liquidity of currencies. It has no storage or transportation costs. It can be used to purchase goods or traded again without additional costs. It doesn't age as fast as some commodities (agricultural products don't last forever). Therefore, the volume of trade is very high, and market participants can very easily enter or exit the market.

Forex is extremely suitable for speculation: money relates to a lot of political decisions, macroeconomic trends and many other driver that are easily and publicly available. Therefore it might be easier to small private traders to trade currencies than firms, which require deeper knowledge. It is indeed easier to read a standard newspaper than to read an earning call transcript.

Moreover, Forex market's design also favor volatility: currencies trade 24/24, not 9 to 5. That means more people have access to it at any point in time. Another critical point is that there is a gigantic quantity of derivatives based on currencies. They can be used both for speculative purposes (thus generating leverage and increase the potential upside of the bet), or to generate protection against currency movements.

All these elements contribute to make foreign exchange markets more volatile than commodities'.


Calculating monetary base from given data

Looking into monetary base definition:

  Monetary base is the total amount of a currency that is either circulated in the hands of the public or in the commercial bank deposits held in the central bank's reserves.

So this means that M0 is actually the monetary base. So I really don't have to calculate anything.


Why we introduce the intermediate good producers into RBC model?

In the classic RBC model you don't need intermediate good producers. They are introduced, for instance, in the New Keynesian model so that firms have some amount of monopoly power and don't always have to price their goods at the marginal cost of production.


Wikipedia has an article worth reading.

An American trust was typically a combination of companies operating effectively as an entity with dominant market power, so perhaps somewhere between a cartel and a monopoly.

Trusts originally took their names from the use of trust law to establish them, but later the term was used for all large companies and groups of companies which used their market position to take advantage of their suppliers and customers and to drive competitors out of business.  The most famous was Standard Oil.

This meant that when the US legislated to regulate or dismantle trusts, it call this antitrust law, where in Europe it would be called competition law.


Interaction term significant, joint hypothesis not

Looking at what you wrote, your regression has the form:
$$
y_i = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{1,i} x_{2,i} + \varepsilon_i
$$
Here $x_2$ is a dummy taking the value $1$ for a subsample of observations and $0$ otherwise.
Then:
$$
E[y_i|x_{1,i} = x_1, x_{2,i} = 0] = \beta_0 + \beta_1 x_{1}.
$$
and
$$
E[y_i| x_{1,i} = x_1, x_{2,i} = 1] = \beta_0 + (\beta_1 + \beta_2) x_{1}.
$$
There are various hypothesis that you can test:

$H_0$: there is no (mean) effect of $x_1$ on $y$ for the subset where $x_{2} = 0$. This amounts to a test of $\beta_1 = 0$.
$H_0$: there is no (mean) effect of $x_1$ on $y$ for the subset where $x_{2} = 1$. This amounts to a test of $\beta_1 + \beta_2 = 0$.
$H_0$: the (mean) effect of $x_1$ on $y$ is the same for the subsets where $x_2 = 1$ and $x_2 = 0$. This amounts to a test of $\beta_2 = 0$.

I am getting that the interaction term is significantly different from zero,

This is a test of point 3. that the (mean) effect is different for the two subgroups.

when I test whether beta_x1+beta_interaction*[mean of x2]=0, I get that the H0 is not rejected

You should not multiply $\beta_2$ with the mean of $x_2$. To test for 2, you should conduct the hypothesis test that $\beta_1 + \beta_2 = 0$.

when I test whether beta_x1=0, that is also not rejected.

This is a test of 1. which indicates that you cannot reject that there is no (mean) effect for the subgroup where $x_2 = 0$.
Remark 1: Hypothesis tests are a bit strange in the sense that they can contradict each other. For example it is possible that you do not reject $\beta_1 = 0$ and $\beta_2 = 0$ but you do reject $\beta_1 + \beta_2 = 0$.
Remark 2: related to this, notice that the tests are not independent. So if you test all three hypothesis, you have the mutliple testing issue (cfr wiki link) in the sense that the probability of rejecting at least one of the three is (likely) bigger than the nominal value of the test.


Cvarza, the supply curve is theoretically derived.

The supply curve of the industry is the horizontal sum of the curves of firms. The supply curve of a firm is the portion of the marginal cost curve with a positive slope. That is, firms act to equal marginal cost to marginal revenue. If the marginal cost is decreasing, that means the firm can increase (marginally) the production (and its profits) till the equilibrium point.

As reference, i recommend the chapter about firms supply in Varian's undergradute book.
As a graduate reference, MasCollel is strongly recomended.


Is Bill Gates wrong about supply demand curves?

No, Bill Gates is correct here. The simple econ 101 cross model is based on a separation of the supply and demand side. Firms are price takers, they assume they can sell as much as they want at the prevailing price and their output decisions have no impact on the price. If firms are not price takers, they have to explicitly take into account how the demand side reacts since this determines the impact on prices. Now with huge returns to scale, price taking is usually not a sensible assumption and therefore the econ 101 cross an inappropriate model.


What is the term for the cannibalization of one tax by another?

In this case we would say that duty levied on the lottery would erode the tax base for corporate tax. However, you should note that eroding tax base is a general term that can be applied to any action that decreases the taxable income/wealth/economic activity. But I dont think there is any more specific word in economic jargon just for that.


Negative consumer externality vs demerit good - what's the difference?

Demerit goods are goods where it is agreed by society that consumption is harmful for the consumer yet it is still consumed due to bounded rationality or because you disagree with society while negative consumer externalities occur when actions of a consumer is harmful for society.

Goods that are demerit but may not have negative consumer externality:

Any type of unhealthy food. E.g. cinnabuns.

Carcinogenic stuff, alcohol, etc.

Basically anything you regret having consumed a few hours later.

If consumption of these goods do not harm society, then there are no negative externalities, as long as no one else cares about the consumer :'(

Good that may have negative consumer externality but may not be a demerit good:

Loud music


Technical Analysts and Econometricians: A Question on Methods

why are such methods not employed by more frequently by
  economists/econometricians?

Because the so-called technical analysis ("TA") is overly simplistic and devoid of mathematical (or at least rational) basis.

The application of algebra, probability, time series, and other tools & disciplines does not guarantee accuracy of forecast, but at least that application provides a more robust framework with which to model financial data than what is achievable through TA. Just to mention one example, TA does not address arbitrage opportunities, which is a useful and consistent approach to pricing financial instruments.

  I noticed though the methods were simple and required little knowledge
  of statistics, they worked very well. They call these methods as a
  part of trend analysis.

There is a false sense of accuracy surrounding TA. Materials that teach or promote TA obviously will present instances where --in hindsight-- the method appears to work. But nothing in TA predicts, models, or at least explains the price trajectory of a financial instrument (let alone its timing).

I think that if/when TA supposedly "works", it is only because of a kind of self-reinforcing feedback: TA "indoctrination" inadvertently prompts TA adepts to reinforce that, say, a price-band (or however it is called) is about to be broken because the price of the stock has touched its limit twice. Thus, TA-based traders react by performing certain transactions expecting to profit from the supposedly imminent direction of the stock.


From Discrete to Continuous time: Total Differential

You can separate your function in three terms by writing
\begin{align}
& v(c_{t+\Delta},u_{t+\Delta},t+\Delta)-v(c_t,u_t,t) = \\
& v(c_{t+\Delta},u_{t+\Delta},t+\Delta)-v(c_t,u_{t+\Delta},t+\Delta) \\
+ & v(c_{t},u_{t+\Delta},t+\Delta)-v(c_t,u_t,t+\Delta) \\
+ & v(c_t,u_t,t+\Delta)-v(c_t,u_t,t)
\end{align}
When you divide by $\Delta$ and take the limit $\Delta \rightarrow 0$, the first expression converges to $\dfrac{\partial v}{\partial c} \dfrac{dc}{dt}$, the second expression to $\dfrac{\partial v}{\partial u} \dfrac{du}{dt}$, and the third expression to $\dfrac{\partial v}{\partial t}$. Therefore your total derivative equals
\begin{equation}
\dfrac{\partial v}{\partial c} \dfrac{dc}{dt} + \dfrac{\partial v}{\partial u} \dfrac{du}{dt} + \dfrac{\partial v}{\partial t}
\end{equation}


Existence of competitive equilibrium between max utility function and min utility function

Indeed, there is no competitive eq. To see this case-by-case analysis is the best way forward.

If either price is 0, consumer 1 will demand an infinite amount, markets will not clear. If prices are both positive, consumer 1 will only ever demand 1 of the 2 goods. But, consumer 2 demands an equal amount of each good is such any such case. No allocation is compatible with both these demands.

The failure lies in the fact that the demand functions (in particular of consumer 1) are not continuous. This happens because 1 has a convex utility function.


What factors allow manufacturing to succeed in a developed country?

A massive simplification would to look at developed (high wage) countries as experiencing two forces.

There is the offshoring pathway. Low skill manufacturing can be offshored to lower waged countries. Output in the home country decreases.

Then there is the technological pathway. Automation and innovations can work to take the bottom out of employment, saving on wage costs. Manufacturing will become increasingly dependent on a more skilled employee. Output in the home country increases.

The expected trend is one away from the low skill manufacturing (think clothing) towards more complex, bespoke products (think 3D printed prosthetics).

The overall environment is notably more complicated with government subsidies, global supply chains and trade deals among others things all being important.

Deloitte Access Economics have released a manufacturing competitiveness report which covers elements of this.


Do we really need accurate utility functions?

To the extent you are constructing marginal decisions from preferences, the general answer is no.  Any utility function that preserves preference orderings maps to the same partial or total order ranking.  As a consequence, $U(w)=\log(w),w>0$ and $U(w)=\sqrt{w},w>0$ preserve preference orderings with respect to $w$. Issues can exist in the multivariate case as all orderings and partial orderings must be preserved.

In the other direction, though, it does matter.  If you are beginning with utility and mapping to preferences, then any transformation results in a different preference structure unless it results in the same indifference curve.  As an example, $U(w)=\log(w)$ and $U(w)=\log(w)+5$ have the same indifference curve and so could be treated as the same preference structure.

So the answer is, unfortunately, dependent on the exact question you are asking.  If only the ordering matters, then no, it does not matter.  If the structure matters, then it can matter.


Short call in binomial option pricing model

One reason why you might want to do this, and perhaps is the motivation for this example, is that it gives you a simple example for thinking about option pricing.

Under some assumptions, the value of any option can be replicated by continuously trading in the underlying asset. Here, you can think of the $\Delta$ shares that you have as your replicating portfolio, and the amount $\Delta$ is meant to be chosen so that the value of these shares is exactly the value (to you) of the call option you just sold.

You can then price the call option according to how much those $\Delta$ shares cost you. If you chose $\Delta$ appropriately, you break even, and this determines the "market price" for the call option you just sold. This example doesn't give you quite everything you need to calculate that yet, but I imagine whatever you're reading is building up to get you there.


Stone-Geary utility function, derivation of Marshallian demand

If I understand your question correctly, you can take the log of the original utility function to simplify the calculations.  The demand derived will be the same as that which is derived from the original utility function.  This follows since the log function is a monotone function, and preferences are preserved under monotonic transformations of utility.  Thus, the "log-utility" case that you find online is exactly what you want to work with since it would simplify things quite a bit.


This seems like an interesting question but I am not sure I understand all the details here, so this is more a comment than an answer.
Because the preference relation is incomplete, we cannot have a real-valued utility function representing it (recall that indifference is different from indecisiveness because the latter is not transitive).

Moreover, I do not think that you can have an arbitrary set $X$ together with an arbitrary $\succeq$ being represented by a finite-dimensional $u$. To give a simple example, suppose that $X=[1,p_n]$ with $p_n$ being the $n-th$ prime and your preferences are $x\succeq y$ if $x\geq y$ and $x,y\in(p_k,p_{k+1}]$ for some $k<n$. There are $n$ subsets of alternatives, each with a well-defined preference order but alternatives are no comparable across subsets so that you need at least $f(n)$ (not sure about the exact $f$) dimensions in your vector-valued utility function. When $X=[1,\infty)$, the number of dimensions explodes.

An important example is the case of social choice. Suppose that you have $k$ individuals with preferences over $X$ and you want to aggregate them. The Pareto order then gives you $u(x)=(u_i(x))$ for all $i$ the (ordinal) utility of individual $i$ from alternative $x$. Indeed, if $u_i(x)>u_i(y)$ for all $i$, then $u(x)>u(y)$.

Finally, if $X$ was finite, then simply order the elements in some way $x^1,x^2,..,x^k$ and construct $u\in \mathbb R^k$ as follows:
$u_l(x^l)=1$, $u_l(x^k)=1$ iff $x^l\succeq x^k$ and $u_l(x^k)=0$ otherwise.

To see that this represents the preferences, notice that if $x^l\succeq x^{l'}$, $u_l(x^l)=1>u_l(x^{l'})$, $u_{l'}(x^l)=u_{l'}(x^{l'})$; if $x^m\succeq x^l$ it must be that $x^m \succeq x^{l'}$ so that $u_m(x^l)=u_m(x^{l'})$ and viceversa but if $x^l\succeq x^m\succeq x^{l'}$, then $u_m(x^l)>u_m(x^{l'})$; and if $x^m$ is not comparable with $x^l$ it cannot be worse than $x^{l'}$ so that $u_m(x^l)=u_m(x^{l'})=0$, while if it is not comparable with $x^{l'}$ it cannot be better than $x^l$.


Why do Eurozone members have a commonly higher Debt-to-GDP ratio?

They've already adopted the Euro (from which there's no "easy way out") so the convergence criteria is just dead letter after that...

The EU has a procedure called the EDP (Excessive Deficit Procedure), which seems to apply to both excess deficit (3%) and (in theory, I think) the debt/gdp ratio too (60%). In general it seems the EDP was mainly applied against the former rather than the latter criterion... and even then it took e.g. Spain a decade to come out of EDP (even though they don't meet the latter debt criterion still; it seems the EU just needs to be happy that they have a plan in that regard). The EDP criteria apply to all EU members.

Also, if you're part of a "small currency" you are more easily punished on forex market, as opposed to the bond market where southern Eurozone took their pounding--"Eurozone crisis" and all that.

Based on IMF data, of the non-Eurozone EU members, Hungary and Croatia aren't too far from the Eurozone average and have more debt/gdp than Germany. Poland has less than Germany, but not a whole lot less and they might cross in the next 5 years given the current trends.

In the lower debt/gdp group are two Nordic countries Denmark and Sweden, but also two Eastern countries, Romania and Bulgaria. (The fifth in this group is the Czech Republic.) Romania, Bulgaria and the Czech Republic for example have among the lowest social spending as a ratio of GDP as well. Denmark and Sweden are at the opposite end in that regard; explaining their lower debt/gdp is not that easy. You might want to look at these countries individually...

Denmark apparently had the "crown" in taxes in the EU at in 2019; this could well compensate the higher social spending, as far as government debt goes. For 2018 on which I could find more detailed tax-to-GDP ratios, Denmark and Sweden are very close. Also Denmark and Sweden have experienced growth compared to Germany's, somewhat exceeding the UK's.

The flip side is that there's a high level of household debt in Denmark (130% of GDP in 2018) and Sweden is also well about average(90%) while Germany's (54%) is below EU average (66%)... which in this case is also slightly above Eurozone average (62%) (no doubt the latter is explained by the UK in this case). The Eastern counties are also far below [either] average. (Denmark seldom makes the news in the same sentence as "the IMF", but when it does, it's because of IMF's concern/warning about household debt.)


Mixed Strategy Bayesian Nash Equilibrium

The trick for finding a mixed strategy Nash Equilibrium is that given everyone else's strategies, all players will be indifferent between each of the options their randomizing over (ie. those options will yield the same payoff). So all you need to do is write an expression relating each player's expected payoffs for each strategy, and solve for the frequencies. Letting x represent the probability that player j does the project, low-cost player i's payoff for doing the project is 5, while their expected payoff for not doing the project is 10x. So you know in the Nash Equilibrium, 5=10x. The incomplete information makes the problem slightly more complicated: player i may expect j to complete the project half the time, the task is now finding j's probability of completing the project given that they are low-cost (not too difficult, since high cost players will never do the project). I'll leave the rest to you.


Can logarithm and polynomial transformation be included in the same equation

Yes it is "allowed". Econometrically it is not a problem. The real question is how useful such a model is for your setting and that will depend on your exact research question, variables and data.


I do have an opinion in these matters:

Without further information about the market structure this phraseology is unclear and therefore its use is incorrect. An example:
Suppose the supply side of this market is provided by a state run company. The company does not care about profit but does care about social optimum, hence they operate at the level where marginal cost is equal to the market price. Of course the government needs funds for other projects. (For public goods where the free market would not reach an optimal allocation.) Thus it levies a tax on other goods, also on the one in our example. This shifts the supply curve upward. Is the welfare maximizing state company really 'pushing the tax onto the consumers'? I would say no. The market mechanism was changed and the new equilibrium allocation is different. Yes, consumer prices are higher in the taxed equilibrium if we compare the two.
Perhaps you find this example contrived. Another objection I would raise against the phrase is that in case of perfect competition companies have no price setting power at all, they are price takers. How is it then possible for them to push taxes? Again, I think the market mechanism is changed, and this changes the equilibrium price and quantity.
In almost all cases this phraseology is unclear and therefore its use is incorrect.

In the UK in May 2016 the Alcohol Duty payed on alcoholic beverages was 852 million pounds. I would argue that this tax is levied on the suppliers of the beverages. Of course consumers are made aware of the tax and suppliers use the money payed by consumers to transfer to the treasury but consumers transfer exactly 0 million pounds to the treasury under the heading Alcohol Duty. The other approach where we estimate a demand and supply curve and calculate the equilibrium price without the Alcohol Duty and compare this to the price consumers pay and the price suppliers get with the duty would yield a different outcome. But this obfuscates matters greatly. Again, what happens is that the market mechanism is changed. The duty raises the equilibrium price payed by the consumers and lowers the equilibrium price received by suppliers. Sometimes this phraseology also talks in welfare terms and mentions the tax burden. I find this somewhat better as both consumer and supplier welfare is decreased in this case. Usually however this burden is equated with the tax received by the state. This is generally incorrect as it does not take into account the deadweight loss caused by the tax. So while the tax does burden both consumers and suppliers just by comparing these burdens one does not get anything akin to who is paying what share of the tax. The market mechanism is changed by the tax. Comparitive statistics may be preformed to examine welfare changes. These concepts exist and are clear.


What does this $(\gamma _{0.clothes} + \gamma _{1.clothes} * \ kids)$ term mean in my linear demand equation?

If you multiply out any one of the equations,

$$w_{clothes}= \beta _ {clothes} + \alpha_{clothes} \cdot \ kids + (\gamma _{0.clothes} + \gamma _{1.clothes} \cdot \ kids)\cdot\ income + u_{clothes}$$

$$=\beta _ {clothes} + \alpha_{clothes} \cdot \ kids + \gamma _{0.clothes}\cdot  income + \gamma _{1.clothes} \cdot \ kids\cdot\ income + u_{clothes}$$

it is evident that you have "kids" and "income" as regressors, plus an interaction term


Why does optimal allocation call for unlimited distribution of information?

Suppose you have a product that you can distribute for constant marginal cost $c$. For every $v\geq0$ assume there are some consumers who value the good at $v$. The net welfare created when someone consumes the good is their value minus the cost of production.

Thus, if we want to maximise the total social surplus (net of costs), we should give the good to every consumer for whom $v-c\geq0$. Since a consumer will choose to buy the good if the price, $p$, is below their willingness to pay (i.e., if $v-p\geq0$), setting $p=c$ ensures the efficient outcome where exactly the consumers with value greater than marginal cost consume the good.

Now suppose we have an information good that can be distributed digitally for zero marginal cost ($c=0$). The above reasoning implies that the socially efficient price is $p=0$! Intuitively, giving you a copy of the good does not cost society anything (because information can be infinitely digitally duplicated), so even if you only get a tiny benefit from consuming it, that benefit will produce a (small but positive) net increase the total social surplus. But the only way to make sure that people who get very small (but positive) value from the good choose to consume it is to give them the good for free.

Arrow's broader point is that this produces a problem: if the price of an information good (e.g. a movie) is zero then firms have no profit incentive to produce movies at all, and then nobody gets any surplus. Eeek! The way society solves this problem is to say "if you make a movie, you get copyright, which means you have the monopoly right to sell that movie". The profits that come from being a monopolist over the movie are the rewards to the firm for having made it in the first place. But, of course, the monopolist movie studio will set $p\gg0$, so some consumers will not buy it even though they could be served as zero cost and this will increase welfare.

If you know your Econ 101 then this can be understood through the lens of some textbook models. Remember that in a basic textbook model of a market, the social surplus is the area below the demand curve but above the marginal cost curve. Social surplus is maximised when $p=MC$, which, for information goods, means $p=0$.

now think of a texbook monopoly model (where the monopoly comes because you own the copyright or patent): the monopolist will set $p>MC$ and this produces a deadweight loss, which is the inefficiency Arrow refers to.


How does the Greenspan/Bernanke Put semantically relate to put options?

then rational investors would buy call options or sell put options, not buy puts. So how can a Greenspan or Bernanke Put be sensibly analogised to a Put Option?

Investors do not need to buy or sell anything, the story is that the Greenspan Fed was giving out free puts to investors.
It doesn’t literally work that way, but it is directionally close  to making sense.


Derive the cost function for a Homothetic production function

There must be an error in your proof.  Where does your proof come from?
The part "for the above to true it is clear that the ratio must be independent of $w$," cannot be True.  Your ratio must be independent of $q$ but not of $w$.
Take for instance the Cobb-Douglas case (which corresponds to a homothetic production function) for a counter example. See Diewert or Chambers for a proof of your result:
Chambers, Robert G., 1988, Applied Production Analysis, Cambridge University Press.
Diewert, E., 1982, "Duality approaches to microeconomic theory", in Handbook of Mathematical Economics, Volume 2.
In the Cobb Douglas case, the production function $$q=x_1^\alpha x_2^\beta$$ yields the first order conditions $$\frac{w_1}{\lambda}=\alpha\frac{q}{x_1}$$
$$\frac{w_2}{\lambda}=\beta\frac{q}{x_2},$$ so that optimal input demand satisfy (in your notations) $$\frac{x_1}{x_2}=\frac{\alpha}{\beta}\frac{w_2}{w_1} \equiv \frac{c_1(w)}{c_2(w)},$$ which is independent of $q$ and implies (show it) that $$x_1=H^1(w,q)=c_1(w)b(q),$$ $$x_2=H^2(w,q)=c_2(w)b(q).$$ It follows that the cost function $$c(w,q)=w_1H^1(w,q)+w_2H^2(w,q)=a(w)b(q),$$ with $a(w)=...$. A similar derivation holds for the more general homothetic case, because by definition of homotheticity, the production technology can then be written as $$y=f(x)=g(h(x))$$ where h is homogeneous of degree one...


Is local non-satiation enough to talk about Walrasian Equilibrium being subset of Core

Yes, local-non satiation is sufficient for the result. In fact, the theorem holds as long as for every consumer every optimal consumption exhausts the budget. The wikipedia page has a proof which uses local non-satiation: https://en.wikipedia.org/wiki/Fundamental_theorems_of_welfare_economics#Proof_of_the_first_theorem.

As a side note: Cobb-Douglas preferences do satisfy strict monotonicity on $\mathbb{R}^{n}_{++}$. Thus, if you can argue that you may restrict attention to bundles where all entries are strictly positive, the arguments based on strict monotonicity applies.


Linking top-down and bottom-up models for analyzing electricity price-based demand response: Expenditure constraint is violated?

Constant expenditure in monetary terms requires constant Income indeed. Does the paper you site adjusts its equations and the data to account for possible changes in income present in the data?

What the CD utility function $U = X^aY^b$ does impose, is constant expenditure shares,

$$\frac{X^*p_x}{I} = \frac {a}{a+b}$$

So the question would be for the appropriateness of the utility function specification whether the data support $X^*p_x/I = const.$

On the methodological front the OP writes

  ...the paper applies an expenditure equation from the building energy
  model instead of it. They just maximize the utility function as the
  objective, subject to the budget constraint, and given the electricity
  use and expenditures from the building model,...

This begs the question: if consumer demand / consumer expenditure is postulated from outside the model, exactly what is the purpose of solving the utility maximization problem? The main purpose of utility maximization is to provide the demand functions, namely to show how utility considerations determine choice and consumer decisions.

Are we sure about what the authors are doing exactly?


Interpetation of coefficent in AR(1) model

For a second-order stationary series it is the correlation coefficient between the dependent value and its lag.
Specify
$$y_{t+1} = a+ \beta y_t + u_{t+1}\qquad u_{t+1}= \text{white noise}$$

The correlation coefficient between $y_{t+1}$ and $y_{t}$ is defined as usual

$$\rho_{(1)} = \frac{\text{Cov}(y_{t+1},y_{t})}{\sigma(y_{t+1})\sigma(y_t)}$$

$$\text{Cov}(y_{t+1},y_{t}) = E(y_{t+1}y_{t}) - E(y_{t+1})E(y_{t})$$

$$ = E\Big((a+\beta y_t+u_{t+1})y_{t}\Big) - E(y_{t+1})E(y_{t}) = aE(y_t)+\beta E\Big(y_t^2+u_{t+1}y_{t}\Big) - E(y_{t+1})E(y_{t})$$

We have $E(u_{t+1}y_{t}) =0$. Also, under first-order stationarity we have $E(y_t)=E(y_{t+1}) = \frac{a}{1-\beta}$

Using these we get

$$\text{Cov}(y_{t+1},y_{t}) = \frac{a^2}{1-\beta}+\beta E(y_t^2) - \frac{a^2}{(1-\beta)^2}$$

By definition the variance is

$$\text{Var}(y_t) = E(y_t^2) - [E(y_t)]^2 = E(y_t^2) -\frac{a^2}{(1-\beta)^2}$$

$$\implies E(y_t^2)  = \text{Var}(y_t) + \frac{a^2}{(1-\beta)^2}$$

Substituting,

$$\text{Cov}(y_{t+1},y_{t}) = \frac{a^2}{1-\beta}+\beta \text{Var}(y_t) + \beta \frac{a^2}{(1-\beta)^2}   - \frac{a^2}{(1-\beta)^2}$$

Things cancel out and we are left with

$$\text{Cov}(y_{t+1},y_{t}) = \beta\text{Var}(y_t) $$

Under the assumption of  2nd-order stationarity, $\text{Var}(y_t) = \text{Var}(y_{t+1}) = \text{Var}(y)$

Inserting all this back to the correlation coefficient

$$\rho_{(1)} = \frac{\beta\text{Var}(y)}{\sigma(y)\sigma(y)} = \frac{\beta\text{Var}(y)}{\text{Var}(y)} = \beta. $$

Note that the presence of the constant $a$ does not affect the correlation -it would be the same if $a=0$. This is because location parameters do not affect second-order statistics like the covariance and the variance.


Distrubtion of a test statistic under the alternative hypothesis

Reading the book, $z=\frac{\hat\beta-\beta_0}{(\sigma^2/n)^{(1/2)}}$

The distribution of $z$ under $H_0$ will determine the critical values for a given level of significance $\alpha$, i.e. it will determine the probability of Type I errors (prob of rejecting the null, when null is true).

The distribution of $z$ under $H_1$ (4.04) will define the probability of Type II errors(not rejecting null when alternative is the true hypothesis).

The closer - the averages are close - the distributions of z under the null and the alternative are, the higher is the probability of Type II errors.
In the picture below, we have several pdf for normal distributions. In blue, the distribution has zero mean; in red mean 1; and in yellow mean 2.

At the usual critical values of $1.96$ and $-1.96$ (for an $\alpha=0.05$), we can see a vertical line. When we're on the blue curve, i.e. under the null, we can calculate the prob of type I error, as the area to the right of the rightmost vertical line, to the left of the leftmost vertical line and below the blue line.

To calculate the type II error, we check in the interval between the vertical lines(not rejecting $H_0$ area), under the red or yellow curve, depending on the values that $\beta$ really takes.
Notice that the area below such a curve will increase the closer the curve is to the one under the null distribution. In this case, the prob of type II error is greatest is when the alternative is the true model and the mean of the distribution is 1.

By deducing the distribution of z under the alternative, the authors want to give an idea of the type of rejection region one will get, when doing inference in this setting.


Economic impacts of a declining population

ad 2.
This phenomenon has been studied a lot. Broadly it is referred to as the demographic transition. A good overview is given by the respective Wikipedia article.

In the field of Long-term macro there are several publications. One of the latest top-journal publications is:
Cervellati M.; Sunde U., Life Expectancy and Economic Growth: The Role of The Demographic Transition, «JOURNAL OF ECONOMIC GROWTH», 2011, 16(2), pp. 99 - 133.

Demographic elements can for sure be endogenised into every macro model. But I doubt that it would be very useful for short run DSGE things, since all demographic changes take place with huge time lags. If you want to study those interesting phenomena, you probably have to go more into Long-run Macro / Economic Growth (Unified Growth Theories put emphasis on demographic elements for instance)


Let $X$ be the set over which a preference $\succsim$ is defined. The question asks you to prove that $X$ can be divided up into numerous subsets defined by the indifference relation, i.e. all elements within an indifference set, denoted $I(x)$, have the same desirability according to the preference $\succsim$. The collection of these indifference sets is denoted $\{I(x)\vert x\in X\}$. The two bullet points tell you how to proceed with the proof.

First, you need to argue that the indifference sets defined by any two elements $x$ and $y$ are either identical or distinct.

Second, you need to argue that every element belongs to one of the indifference sets that divide up $X$.

Here is a simplified example to give you some intuition.

Suppose the set $X$ contains the following items: an apple ($a$), a banana ($b$), a cherry ($c$), a dragon fruit ($d$); that is, $X=\{a,b,c,d\}$. I like cherries best, bananas and apples the second (I'm also indifferent between the two), and dragon fruits the least. This is my preference over $X$. Define the indifference sets as follows:

$I(a)=\{a,b\}$ (i.e. the set that contains items indifferent to apple includes both apple and banana);
$I(b)=\{a,b\}$;
$I(c)=\{c\}$; and
$I(d)=\{d\}$.

As we can see, for any $x,y\in\{a,b,c,d\}$, $I(x)$ and $I(y)$ are either identical (as when $x=a$ and $y=b$, or vice versa) or distinct (as when at least one of $x$ and $y$ is $c$ or $d$). This satisfies the first requirement.

For the second requirement, it is clear that the four items ($a,b,c$ and $d$) belongs to one of the three indifference sets ($\{a,b\},\{c\},\{d\}$).

To properly prove the claim, you need to generalize the example above to cases where the set $X$ contains an arbitrary number of elements. Strictly speaking, you'd also need to consider arbitrary preferences. But since the preference relation $\succsim$ is complete and transitive, such consideration is less crucial.


How can a consumer economy possibly be sustainable?

GDP is, by definition, basically only consumption and government spending.  That's why you see consumption reported as such a high % of the overall number.  Gross Output (GO) is a more comprehensive measure of economic activity, and on that measure household consumption is closer to 35% of the economy.  A "consumer economy" isn't one in which most economic activity is in the form of consumption of imported goods — there is plenty of production going on in such countries, as well.  More production than consumption, in fact (since many of the goods produced are used by other businesses, and often not counted in GDP figures.)


Why is the Fed worried about short term interest rates going negative?

Market observers worry that this wall of liquidity could push short-term interest rates into negative territory.

Fed pays interest on excess reserves (IOER) as a tool of monetary policy since late 2008.
https://www.clevelandfed.org/newsroom-and-events/publications/economic-commentary/2015-economic-commentaries/ec-201502-excess-reserves-oceans-of-cash.aspx
The "wall of liquidity" would increase excess reserves. Fed can pay interest on those reserves so the IOER rate does not have to go negative unless Fed decides to try that experiment as a tool of monetary policy.
Negative interest rates are the same as paying a net fee for keeping deposits or other investments at the bank or central bank. Extra central bank liquidity does not motivate banks or the central bank to charge fees. Banks charge fees to compensate for a loss or lack of net interest income. Central banks might try to make interest rates negative as a tool of monetary policy.
Repurchase Agreements with Negative Interest Rates:
https://www.newyorkfed.org/research/current_issues/ci10-5.html
Negative rates on repo mean the lender will pay interest to obtain the security collateral. This implies that investors would rather hold certain securities, which are under shortage, instead of other financial assets. One needs some case studies and/or theories which demonstrate or predict that a surplus of excess reserves somehow triggers a shortage of securities in repo markets. However if Fed imposes negative interest rates on reserves that is a different analysis.


which is an example of demand curve that is always inelastic/elastic?

I don’t think that there are any classes of functions where you can ex ante say that it is elastic or inelastic, since this will depend on the function parameters.

However, there is a class of functions for which you can say the elasticity is always constant, these are isoelastic functions.

For example, consider function:

$$Q(p)=ap^b$$

By definition elasticity for this function is:

$$\frac{Q’(p)p}{Q(p)}=b$$

Now this kind of function will be always elastic ($b$>1), inelastic ($b$<1) or unit elastic ($b$=1) depending on the actual value of the parameter $b$


Estimating elasticity of substitution in nested CES functions

Neglecting technology parameters and assuming constant returns to scale, the parameters $\sigma$ and $\gamma$ are jointly estimable via dynamic (least squares) programming.
See this paper.


Why doesn't capitalism run out of money?

First of all there are several incorrect statements in your text:

Investing money is not just raking in the cash. It involves a risk, actually non trivial part of the reason why you earn interest on money you invest is the compensation for you taking this risk. If you consistently take bad risks you eventually lose your money, so its not correct to say investing is just like "rake in the cash". Moreover, if you invest in your own business/projects you have to put entrepreneurial work in. The only portion of the return that you could consider as "raking in cash" is the portion that you get to compensate you for opportunity cost of not investing the cash but enjoying it.
There is no economic theory that says in capitalism lives should get worse. In fact in economics we dont even use term capitalism - but assuming you mean by capitalism mixed but predominantly market based economies there are no economic theories which says life should get worse in those. Note in science we usually use theory differently from common English. A theory must be some consistent and testable explanation of some mechanism, and often we reserve the word theory only to consistent explanations that were already successfully tested and supported by evidence.
Money in economy circulates. Unless the rich investor stuffs the money under a mattress, it is incorrect to compare her or him to vacuum cleaner. If you put your money in bank it will get lend out to people who need it. If you spend your money on high living then again the money is paid to producers of the stuff you enjoy.
You cannot invest any number of money you want and expect the same return. Ultimately you will experience diminishing marginal returns on your investment. As the supply of savings increases interest rate goes down as in a present situation when in some countries interest rate is even negative, so you get punished by saving and rewarded by borrowing money.

Now to answer your main question, why the economy does not run out of money (assuming no more printing of new money), it is because prices in economy adjust. Generally, the monetary part of macroeconomic can be described by classical monetarist equation which states that:

$$MV=PY$$

Where M is the money supply, V velocity of money, P price level (change in which will give you inflation/deflation), and Y is the real output - how much goods and services economy can produce.

Now lets assume that money supply is fixed (hence no more printing of additional money), and examine the effect of investment on this relation. First, real output actually depends on level of investment. The higher level of investment, the higher number of factories and capital workers can work with so also the higher output we get. So investment increases the real output Y. Real output is also actually what determines the well being of people within a country (at least as measured by lets say GDP per capita). However, to keep the equation balanced something else has to change. One possibility would be for velocity of money to increase- i.e. the speed in which money circulate, but velocity of money usually does not change much. So most likely what will actually happen will be that aggregate prices P decrease, meaning that for the same amount of money you will be able to buy more goods and services.

Lets also for a second suppose that rich would actually stuff all their money in the mattress and never used them in any way possible. This would just be equivalent of actually decreasing money supply M and again P or V would adjust. In the short run decline in money supply could also negatively affect Y, but in the long run output Y is fully determined by the production capabilities of economy not by number of dollar bills circulating.

So in either case sustained increase in material welfare is possible. Also, note redistribution is not actually about taking the money from rich and give them to poor its ultimately about transferring the resources from one to another. So redistribution also does not actually necessarily depend on availability of money in the economy (although thats a topic of its own that I wont go into).

So to sum it up, some of the premises in your question are not correct, and the answer why you can have a sustained increase in wellfare when money stock declines is that prices (or potentially also partially velocity) adjust. The total output and hence material welfare of country is (in the long run) independent on quantity of money.


What is the difference betwene a partial default and having debts partially written off?

"Some form of debt write-off" means usually, mutually agreed (between creditor and debtor) elimination of a portion or the whole of debt.

"Partial default": I had the contractual obligation to pay EUR $100$ on Monday, but I paid only EUR $80$. Technically speaking, this is not always (or immediately) considered a "default", but a "credit incident" (alternatively it is characterized as "default on a payment" and not "default on the loan").

Then the reason why I haven't paid the remaining EUR $20$ is crucial: Do I recognize my obligation but I am unable to fulfill it because of financial inability? Or do I deny my obligation, questioning the validity of the underlying contract of the debt from which the obligation to pay the remaining EUR $20$ came?

In the first case, the issue is whether this financial inability is deemed as "temporary" or "permanent". If it is deemed "permanent" (usually by a court decision), then in some countries the debt is officially written off by necessity, while in other countries the creditors retain the right to seek payment whenever the debtor acquires assets in the future (so the creditors may be monitoring you). If it is deemed "temporary", then the issue is to make an attempt to restructure the debt.  For Public Debt, the inability is never officially considered "permanent" so the issue is always to restructure (which of course may include a mutually agreed partial write-off, which essentially means a recognition that there exists a "partial" permanent inability to pay).

We could call the second case, where we question/deny the validity of our obligation, a "unilateral debt write-off", which of course is rarely if ever accepted by the creditors, and so whether it should be called a "write-off" is questionable in turn.


Empirical means of discerning "economy" and "bubble"

You are probably not far off the mark. While there may be no consensus what a bubble is, one proposed definition is

Unusual changes in single measures, or relationships among measures (e.g., ratios) relative to their historical levels. For example, in the housing bubble of the 2000s, the housing prices were unusually high relative to income.
For stocks, the price to earnings ratio provides a measure of stock prices relative to corporate earnings; higher readings indicate investors are paying more for each dollar of earnings.

I'm not aware of [more] composite measures... but I'm not sure it would be very useful either. Generally one wants to know where exactly the problem is (whic sector) when there's a bubble.
Although Wikipedia doesn't seem to mention this, there's obvious caveat to such a method. Quoting from a fairly cited review of Camerer:

Any apparent bubble could be evidence of a variable which affects intrinsic value but is not observed by the econometrician.

Basically a problem of confounding variables, as far as I understand it.
As a putative example (of the latter), Camerer gives:

An example might be the Hong Kong stock market, which rose l0 times in value
between 1970 and 1973, then dropped 75% in a year. These enormous price swings
might indicate a bubble or fad, or they could be rational swings based on changing beliefs about what will happen when Hong Kong reverted to China after the British lease expires in 1997.

More examples like that are supposed to be given in a paper by Flood et. al (cited by Camerer), but I haven't read it.


What is the market value of an object over time?

The value of a tradeable good is equal to what the marginal buyer will pay for it.

I will explain further, but first let me correct one misconception in your question. You state:

  Now, once you buy a car, the value of the car depreciates once you drove it outside of the factory. For example, price of a new 6000 dollars car, will be 5800 dollars. Although, the value of the car should be the same, no one will buy the car for the same price you bought it from the dealer.

The value of the car should not be the same. There's a very well-known reason that cars lose value as soon as they are sold used, rather than new: there might be something wrong with them which the seller might be aware of, but the buyer might not be able to spot. This is discussed at length in a well-known paper titled "The Market for Lemons: Quality Uncertainty and the Market Mechanism" (see link for Wikipedia discussion).

Having said that, we can return to the discussion of the value of a car. Like any other tradeable good (it's easier to determine the value of a good that can be bought and sold than the value of services provided by an ecosystem, for example), a car is worth whatever someone is willing to pay for it.

You ask about two scenarios, one in which the car is kept in a showroom, and another in which it is driven. The difference between these two scenarios is what is known as depreciation— as you drive a car, things wear down, and its future life becomes shorter, while it also becomes less good at being a car. Both because the car isn't as nice (maybe it's scratched up, or it is a bit slower, or it requires more maintenance), and because it won't last as long, all else equal (i.e., so long as nobody wins a famous race while driving that particular car) driving a car decreases its value.

In practice, the value of many cars decreases for a long period, until at some point, the value slowly begins to increase as the number of cars dwindles and the model attracts interest from collectors. The collectors' market is actually in many ways a very different market from the original market for a car— the people buying a collectible car generally aren't interested in its value as an automobile; a classic car will tend to be slower, get worse gas mileage, be dramatically less safe, have higher maintenance costs, and offer fewer amenities than a modern car that costs a similar amount. Because this market is driven by factors other than the value of the services provided by the car (because, ayou are aware, most rare collectible cars are driven rarely, if at all), the value of these cars tends to be determined more by factors like taste or the value of financial assets held by the wealthy than by anything else. Nobody will be driving an Exelero to the grocery store; anyone who has one has another car to use for that task (or, more likely, someone to go to the store for them and cook them dinner).


For perfect complements like $u(x,y) = \min( g(x_1), h(x_2))$ the points at which kinks occur are such that $g(x_1)= h(x_2)$. In this case that would be $x= \sqrt(y)$ so in this case indifference curves would look like:

The picture above plots two indifference curves and the dashed line gives you all kinks of every indifference curve. Normally the kinks would lie on $45^°$ line because usually complements are given by functions where respective slopes of $x$ and $y$ are linear (e.g. $\min (\alpha x, \beta y ))$ but here you have case where you have $\sqrt(y)$, so the  $x$ has to be consumed with exponentially increasing quantities $y$.


I think yes. Macroeconomists usually build DSGE models to predict short-run fluctuations. Calibrating the data is the part of the process. After using numerical methods to run the models, researchers check the moments of the simulated values. This is a way to examine whether the simulated data represents real life outcomes.

However, every economic situation differs so you have to choose wisely the simulated data depending on your research every time. I mean the growth was huge in the 1950s in the US, but today the main objective is to kill the deflation monster.

Here's the benchmark model by Hansen and Wright (1992):

$\sigma$ denotes standard deviation, $Y,C,I$ and $L$ are macroeconomic variables.


Consumer preference and price in the Cobb-Douglas function

No, preferences are stable. That is not to say that the quantity demanded or marginal utility obtained at the new price level is the same though.

If we'd allow the exponent of the utility function to vary for different price levels we'd let utility depend on prices, which it does not. Of course the actual utility that can be obtained depends on prices, but not the utility itself.

Back to your problem. You probably obtained a $\beta$ by observing one price and X quantity, with Y the remaining budget, filled that in in the formula:

$\frac{dY}{dX}=-\frac{\beta Y}{(1-\beta) X}$

and solved for $\beta$, and then did that again for a different price and found a different $\beta$. Given that you use real world data, that in itself is not surprising.

Unfortunately, the real world does not always match our models, so you would have to estimate $\beta$ using econometrics. Given that in CD models expenditures are fixed proportions of the budget you should be able to simply estimate:

$\beta=\frac{p_x X}{M} + \epsilon$

That is $\beta$ is the average proportion of the budget spent on X.


I am not an expert in biology but let me give you some intuition through it.

For instance if you have some impurities in your blood. There are two ways to remove that impure blood from your body.

1) To give you some medicine that enters your system to cure you. This is a slow procedure and may not guarantee concrete results.

2) To artificially suck entire blood out of your body and make attempts to purify it and then refill it into your body. This is a fast procedure and will produce certain concrete results which can either be positive or negative.

Now, if the second method seems lucrative to you, make sure you think of what will happen to you in that time interval when your body is without blood.

Noe let us come to Economics.

Money is a medium of exchange and store of value.

Let me write a simple equation of money.

$MV = PY$

$M$ ~ Amount of money in circulation.

$V$ ~  Velocity of Money. Velocity of money is nothing but number of times the nominal money is changing hands

$P$ ~ Price level

$Y$ ~ Output in the economy

$PY$ hence is the value of output or GDP.

You can make an intuitive sense of this equation that is the GDP of the country is how fast a given quantity of money circulates in the economy. Suppose the GDP (aggregate expenditure) of a country is \$1,000. The amount of money in circulation is \$500. So money should change hands two times for expenditure worth \$1000 to occur which makes velocity of money = 2.

Now once you have digested this relationship.  Demonetisation in the Republic of India was not an attack on the quantity of money but on the velocity of money in circulation.

It did not make the existing money worthless since the old cash could be exchanged for new cash but it choked off the flow of money that is the velocity of money in the economy since old cash which made for 86% of the entire money supply ceased to be a legal tender. Now coming back to our equation:

$MV = PY$

There was an artificial decrease in velocity(V) so the right hand side of the equation $PY$ must fall to maintain the equilibrium. That is GDP must fall temporarily which is the stagnation of the economy.

Understand it this way, you temporarily decrease the purchasing power of the people by pulling all the money out of the system. People stop buying stuff and sellers cannot sell their stuff. Producers will stop producing and people will lose jobs. Hence this is an artificially created recession.

This was perfectly evident in the growth rate of the country that slumped by about 2% in the beggining of 2017. The 2% of India's GDP is a phenomenal multi billions of dollars which India lost and which is the actual monetary cost of this decision.

Now the fundamental question at the heart of economic decisions - Does the benefits justify the costs? Which is a debatable topic as you mentioned so I am not going into that.

But what I personally believe is, Indians should have thought it out more rigorously and thoroughly before making such an unconventional move.


given someone's past investing history, is there a way to calculate his risk aversion?

Generally speaking no. You wouldn't be able to distinguish re-balancing for risk aversion reasons from re-balancing motivated by changes in expected returns or the co-variance of returns.

Consider the simple case of a household periodically re-balancing their investments in across both a fixed index fund and an equity index fund. The econometrician sees the investor reduce their equity holdings and increase the fixed income investment. Any of the following could induce such a change:

An increase in risk aversion
Lower expected returns for the equity index
Higher expected returns for fixed income index
Greater positive co-variation of the two funds

And in reality, the situation is much worse because there are many investments, taxes, transaction costs, behavioral issues, and rational-inattention.

I'm sure you could simplify and cook up a theoretical setting where you could do this (for one, just assume that all the other things cannot change), but in general this won't be possible.


The Next Breakthrough of Growth Economics

I'm no expert, I'm just now researching this topic for my Master's thesis, and I might miss a bunch of important stuff, but in my opinion, a huge breakthrough might come from the ecological economics side of the debate.
A new (to my knowledge) way of integrating resources and energy into economic growth is to use the so called Exergy (usefull energy) variable in the model. When you use this Exergy (as opposed to simply cost of energy) variable in growth accounting, (again, to my knowledge) it explains away most of the Solow residual. So, ideas, human capital, physical capital and institutions start looking as a really minor factor when you compare them to the amount of calories/watts that we have been using to our advantage throughout the 20-th century...
The implications of the model (I'm speculating a bit here, people who know better could correct me, if I'm wrong) are that it's not that the physical capital per se drives growth, but rather the ability of the capital to extract natural (renewable and expendable) energy resources AND, most importantly, convert those resources into useful work.

You can look up works by Steve Keen

and this paper:

Aggregate production functions: How does the Solow Residual change when introducing quality-adjusted values for capital, labour, and energy?
Authors: João Santosa; Matthew Kuperus Heunb, Paul E. Brockwayc, Marco Sakaic; Tiago Domingosa

Adding: a quote from Steve Keen "Capital without energy is a statue, labour without energy is a corpse."


Why currency purchasing power is linked to confidence in central banks

TL;DR Version:
The trust that money has value gives it its value. If money can be exchanged for gold, then this makes people trust that it will have value. If this is not the case (and it is not, although it used to be the case almost everywhere) then the only thing that can make people trust that money will not lose its value through extreme printing of money is having trustworthy people endowed with the power to print money. These people are called central bankers, hence their credibility matters here. The consequences of losing faith would be (hyper-)inflation.

Long Version:

First a few definitions. The purchasing power of money is basically its value. That is how many goods/services (e.g. bread) you can get for a dollar. Inflation is a price increase, which means that a dollar can now buy less things (as they cost more). Hence money loses its purchasing power when inflation occurs.

Further, the value of money is basically determined just like the value of anything else - supply and demand. So if you print a lot of money (supply increases) its value will go down. That's the reason the government doesn't just print a bunch of dollars and give them to everyone, because that money would lose value, i.e. it would lead to inflation (rising prices).

The reason that money has any value at all in the first place (today) is because people believe it has value. Also, people believe it has value, because it does. They observe that they can use it to obtain goods and services. That's pretty much the only thing distinguishing the dollar from monopoly money today. As is apparent from the reason money has value it is easy to get into vicious cycles. If people stop believing it has value then it won't, which will lead to more people believing it doesn't and so on.

This is why hyperinflation is so hard to stop. Once money loses a lot of its value the trust people have in this money is undermined. They panic and think it won't be worth anything soon. So they try to get rid of it by exchanging it for other things. As everyone tries to exchange money for other things, effectively the supply of money (really the velocity of money) increases, which undermines its value even more. Since everyone wants to get rid of money in this scenario, there is no one to give it to in order to trade with so it becomes worthless (as no one wants it). A german central banker once said that inflation is like toothpaste: once you get it out of the tube it's very hard to get back in again. The problem is, that governments have an incentive to print more money, to pay off debts and have more spending power, etc., which creates inflation.

So there are multiple equilibria possible, where all people (or a critical mass) believe in money and since all do there is no reason not to believe in it. Or nobody does and in this case there is no reason to in fact believe in money. It's also possible for many to believe in it and many not to (e.g. 50-50 or 60-40), but this is not a very stable equilibrium. For more on how money comes into existence at all, see the paper by Kiyotaki and Wright ("Money as a medium of exchange" and "A Search-Theoretic Approach to Monetary Economics" american economic review 1993) , who discuss these equilibria possibilites. There, basically money emerges, because barter (trading goods for other goods) is so inefficient as it requires a rarely occuring double-coincidence of wants. So if money is undermined, probably something else will emerge. E.g. in Germany after a crash people used cigarettes as currency.

Now it wasn't always the case that money only had value, because others believed it. On top of that, governments had what was called the "gold-standard". This meant that at any time, money could be exchanged for a certain amount of gold. Because of this, governments were restricted in how much money they could print, by how much gold they had in reserve. So this was meant to stopp excessive printing of money and to stop inflation panicks. This however came at a cost, because monetary policy could barely be used and it required governments around the world to frantically search for gold. Nowadays, we don't have the gold standard, so governments are free to do what they want. However, they don't abuse their power, which is evidenced by low inflation rates in the last 30 years and without the drawbacks of the gold standard.

If the central bank is not credible and the government does not guarantee to exchange money for gold, basically there is nothing to "anchor" the value of money. So if people expect the central bank to print too much money, then money will lose its value.

One last possibility through which the mistrust of the central bank is important here is because of wage-setting. When negotiating wages, unions take into account rising prices (inflation) so that they don't become poorer through price increases, which is why they demand higher wages. If they expect a lot of inflation to occur in the course of the year, they will demand high wages right now (as these contracts cannot be renegotiated every month). However the rising wages will lead to higher prices, which in fact is inflation and this spiral could get out of control. With a credible central bank, this won't happen of course, as the expectations aren't so wild.


What is the long term outcome of inflation?

Inflation is not a fixed number, so there is no way to predict the value that a loaf of bread or an average house will be in 2400 or 2700. Inflation is the result of a complex macroeconomic process rooted in the behaviour of individuals, and people's expectation of future inflation. The central bank is a key role player, so the integrity of that institution will determine how stable inflation is.

When there is high inflation, in the long run governments often just divide the numerical value of the currency by, for example, a million, in order to simplify the currency for the population. That doesn't change how much real wealth people hold, since it is done across the board. There is a common understanding that, for example, 1 million Zimbabwen Dollars yesterday is the same thing as 1 Zimbabwen Dollar today, so everyone catches on as to what happened.