Acessible sources for Hierarchical Bayes? I am trying to learn how to build hierarchical bayesian model, I can find some papers online on the topic but I am looking for some textbook or some treatment that is more student friendly than published papers.

Is there any good guide to hierarchical bayesian models out there? Something that has step by step explanation with some worked out examples with applications to economics would help me a lot.

Here are some good guides on this topic:

Bayesian Inference 2019 from Ville Hyvönen & Topias Tolonen. - this is good guide to Bayesian inference in general. They have separate chapter on hierarchical Bayes and all their explanations include worked out step by step examples in R.
Bayesian Data Analysis by Gelman, Carlin, Stern, Dunson, Vehtari, and Rubin - is a textbook that is very accessible (at least relatively) and well regarded.
Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan by John Kruschke - this is really the most introductory text on Bayesian analysis I know. It has lot of worked out examples in R, I would even say it’s written on bachelor level, but it’s very good text

———————————————————

Edit: I actually just discovered that the 2nd source Bayesian Data Analysis was made available for free by the authors in PDF form that you can access here.


Definiton of information sets in rational expecations models I am struggling with the notion of 'information sets' in the context of rational expectation models in economics. I found interesting notes on the web (http://www2.econ.iastate.edu/tesfatsi/reintro.pdf) but I am not sure whether I am understanding the concept well. Let me explain my concerns along with the first example in the notes given in the link.
Consider a small model given in the following three equations:
$$ y_t =y_t^*+ap_{t-1}+b\mathbb{E}_{t-1}p_t\\
p_t =m_t+\varepsilon_t\\
\mathbb{E}_{t}p_{t+1}=\mathbb{E}(p_{t+1}\vert I_t)$$
where
$y_t^*$ notes the log of potential real GDP in period $t$
$y_t$ denotes the log of actual real GDP in period $t$
$\mathbb{E}_{t}p_{t+1}$ denotes the subjective forward-looking expectation of a representative agent in period $t$ regarding the price level in period $t+1$
$m_t$ denotes the log of the nominal money supply in period $t$
$\varepsilon_t$ is a stochastic shock at time $t$
$I_t$ denotes a period-t information set that is available to the representative agent at the end of the period $t$.
So my possibly stupid question is: What is $I_t$ or how is it defined?
To be more precise, let me outline what I think $I_t$ is.
First of all, because most economists apply the law of iterated expectations and other propositions that can be applied to conditional expectations, I suggest $I_t$ has to be a $\sigma$-Field because otherwise, one wouldn't be able to apply these propositions.
But how is this $\sigma$-Field defined?
Following the notes, Leigh Tesfatsion writes that the equations plus classification of variables and admissibility conditions together with the true variable values a,b and the deterministic exogenous process $(m_t)_{t \in \mathbb{N}}$ have to be part of the information set, as well as the properties of the probability distribution and properties of the stochastic shocks $(\varepsilon_t)_{t\in \mathbb{N}}$ and the values of past realizations of all variables.
Typically it is assumed that $\varepsilon=(\varepsilon_t)_{t\in \mathbb{N}}$ is a stochastic process defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$.
Thus I would say $I_{t}$ has to be a $\sigma$-Field over $\Omega$ and thus it has to be a system of subsets of $\Omega$, thus it cannot include specific equations, specific variable values nor variable classifications, or am I wrong?
Let $\mathbb{F}=(\mathcal{F}_t)_{t\in \mathbb{N}}$ be a filtration on $(\Omega,\mathcal{F},\mathbb{P})$, given as $\mathcal{F}_t=\sigma(\{\varepsilon_s:s\leq t\})$.
I thought $I_{t}$ to be the history of the stochastic process, i.e. $I_{t}=\mathcal{F}_{t}$, is this correct?
If not, could you provide me a (mathematical rigorous) definition of the information set $I_{t}$ or could you provide me some literature related to this issue?
Thanks in advance
Frank

Two notes.
A. "Conditioning on information" has always been applied in economics without much attention to probability theory-rigor, because it has (indeed) such a strong intuitive sense: "based on the information I have (where "information" here means data, processing algorithms, psychological makeup, almost anything) I somehow form through a black-box process an expectation for the value of some variable".
B. Conditioning is always done with respect to a sigma-algebra. But in economics it is customary to write just the generator of the sigma-algebra, in our case $I_{t−1,i}$ and expect to be understood as $\sigma\left(I_{t−1,i}\right)$. So all you have to do is imagine $I_{t−1,i}$ as a set that can generate a sigma algebra. In that case, each element of the set can be anything really.
Note that the $\Omega$ in the probability space and the sigma-algebras that can come from it can literally be anything. We restrict $v$ to be a random variable, namely a function whose range is some numeric set, like the Reals or the Naturals, but the domain of $v$ can be, again, literally anything, and as multi-dimensional and as non-numeric as we like.


Schumpeterian Business Cycles The basic idea of Schumpeterian Business Cycles is that new technologies require creative destruction but will yield positive growth. These two opposing forces have impacts onto the real economy at different times, which will create fluctuations around the trend.

Romer (1986) has these effects in an explicit growth model, but he neglects the business cycle fluctuations.

Has there been any literature of business cycles based on the idea of Schumpeterian Growth?

I will toot my own horn and cite - Phillips and Wrase. (2006) "Is Schumpeterian ‘Creative Destruction’ a Plausible Source of Endogenous Real Business Cycle Shocks?," Journal of Economic Dynamics and Control, vol. 30 no. 11 pp. 1885-1913.

We found it difficult to match the volatility using Schumpterian mechanisms alone.  The business cycle asymmetries from our model were also exactly backward.  That is, a sudden discovery leads to a jump in output which slowly peters out.  While recessions tend to suddenly jump downward and then slowly recover.

The mechanism in our model was that an increase in productivity due to the discovery, leads to a reallocation of resources away from innovation toward production of output and this slows down the innovation process.


Examples of bounded, positive, inverse demand curves The three most common examples of demand curves I am aware of are
\begin{align}
Q&= b - aP,\\
Q&= bP^a,\\
Q&= b e^{-aP}\\
\end{align}
The first being our classic linear demand curve, the second being the one corresponding to a constant, fixed, elasticity, $a$, and lastly the slightly less common but still useful exponential demand curve. But all of these curves, to me, leave a peculiar feature of their inverses
\begin{align}
P&= b/a - Q/a,\\
P&= (Q/b)^{1/a},\\
P&= -\frac{1}{a}\log{\left(\frac{Q}{b}\right)}\\
\end{align}
For the linear demand curve, price goes negative, as the Quantity produced becomes high. This is conceivable for some products, where it might be costly to store excess, but it is easy to imagine products where negative prices are impossible. The fixed elasticity demand curve solves this, but it has another undesirable property, in my opinion. The price goes to infinity as the quantity produced goes to zero [and, at least in some cases, infinite prices are not plausible in the real-world as we are constrained by finite resources]. The last option, the exponential demand curve, suffers from both issues!
If I want price to be bounded and positive over the domain of all positive quantities, I could start by defining, the inverse demand functions
\begin{align}
P&= \frac{b}{1+aQ},\\
P&= be^{-aQ},\\
\end{align}
In this parameterization, $b$ is the maximum price anyone would be willing to pay for the product no matter how rare it is, and $a$ governs the rate the price declines as the quantity becomes less rare. These price curves would lead to demand curves of the form
\begin{align}
Q&= \left(\frac{b}{P}-1\right)\frac{1}{a},\\
Q&= -\frac{1}{a}\log{\left(\frac{P}{b}\right)},\\
\end{align}
Is anyone aware of demand curves used in the literature like this. I'd assume there must be someone, for some purpose, that might have required a demand curve with such properties. I'm thinking of cases where, due to an externality, quantities can suddenly be taken to zero or very large values. What demand curves do people use in these cases? Does anyone use either of the two above?

You can find demand functions like this in textbook and exam problems. Just a random example from internet:
$$Q = \ln 4 - 0.5 \ln P$$
which is just a special case of $Q=−1/a \ln (P/b)$ where a is 1 and b is also 1 and there is some additional constant $\ln 4$. We just had an exam where I also saw one of the other examples. If I remember right it was $Q = \frac{100}{P}-1$. I also remember seeing more of them in textbooks but it would take forever to find them in thick book full of math problems.


Was Haiti the richest society in the world in the late 18th century? Samuel Bowles writes:

  At the close of the eighteenth century ... Haiti was probably the richest society in the world.

Is the above claim true?

It has certainly been argued in various places that Haiti was the "richest" colony.  It's obviously difficult to reconstruct modern measures of wealth such as per capita income for historical territories, and the metric of "rich" that mattered in the day was how much revenue could be extracted by the colonial government, so assessments may be skewed towards this.  My books are boxed at the moment, but this is what I could find based on the footnotes from Understanding Power (ch 10 p62).

Citing Hans Schmidt, The United States Occupation of Haiti, 1915-1934, New Brunswick, NJ: Rutgers University Press, 1971, pp. 19-20:

  At the time of the French Revolution, Saint Domingue [Haiti] was the
  wealthiest European colonial possession in the Americas. As early as
  1742 the sugar production of Saint Domingue exceeded that of all the
  British West Indies, and on the eve of the revolution the colony
  accounted for more than one-third of the foreign commerce of France.
  In 1789 French trade with Saint Domingue amounted to £11 million,
  while the whole of England's colonial trade totaled only £5 million.
  In the same year the ports of Saint Domingue received 1,587 ships, a
  greater number than Marseilles, and France employed 750 ships
  exclusively for the Saint Domingue trade. The chief exports of the
  colony were sugar, coffee, cotton, indigo, molasses, and dyewoods. The
  French built an elaborate network of roads, irrigation systems, and
  magnificent plantations.

Citing Paul Farmer, The Uses of Haiti, Monroe, ME: Common Courage, 1994, p. 63:

  At one time or another, the colony was first in world production of
  coffee, rum, cotton, and indigo. On the eve of the American
  Revolution, Saint-Domingue [Haiti] -- roughly the size of the modern
  state of Maryland -- generated more revenue than all thirteen North
  American colonies combined. By 1789, the colony supplied three-fourths
  of the world's sugar. Saint Domingue was, in fact, the world's richest
  colony and the busiest trade center in the New World.

And citing Patrick Bellegarde-Smith, Haiti: the Breached Citadel, Boulder, CO: Westview, 1990, pp. xviii:

  In the second half of the eighteenth century... [Haiti] created more
  wealth than any other colony in the world. At that time, 50 percent of
  France's transatlantic commerce involved Haiti, and nearly 20 percent
  of the French population depended on trade with Haiti for its
  livelihood... Overall investments in the Haitian economic
  infrastructure were meager compared to the enormous profits and trade
  advantage gained by France. Economically, Saint-Domingue [Haiti] was
  the world's most profitable colony. It soon became known as "the pearl
  of the Antilles" and was the standard by which the profitability of
  other colonies was judged.


Why do firms losing money almost always reduce labor costs via layoffs instead of pay cuts? At least in the United States, firms seem to almost always reduce labor costs via layoffs rather than pay cuts. Pay cuts are so rare that they're literally headline news. From an economic standpoint, if workers are being paid more than their marginal product, then either solution fixes the problem - layoffs boost marginal product to match the old wage, while pay cuts lower the wage to match the new marginal product.

But it seems to me that a pay cut is better for both the employer and the employee. From the employer's side, when times get better they don't need to bring in a bunch of inexperienced new employees to replace the ones they laid off. From the employee's side, they still make some money rather than no money, and they are free  to leave the company if they can find sufficiently higher pay at another firm to compensate for the inconvenience of switching jobs. Personally, I would always prefer to be offered a pay cut than laid off, and apparently over 90% of workers feel the same way. So why are layoffs so ubiquitous?

Here are four possible reasons:

Reason #1. Fixed costs per employee.

Due to taxes, insurance, 401k plans, etc., the cost of an employee (to the employer) is significantly greater than the employee's take-home pay.

For simplicity, assume an employer must bear an additional lump sum cost per employee that is equal to 20% of each employee's current take-home pay. Assume also that all employees are identical. Then to cut payroll costs by 30%, the employer can either:

A. Fire 30% of the employees; OR

B. Cut each employee's take-home pay by 36%.

Many firms will prefer A to B.

Reason #2. Parkinson's Law

Parkinson's Law is the humorous adage that "work expands so as to fill the time available for its completion".

This may apply also to workers in a firm. Laying off say 5% of employees may make no difference to total production -- especially if a firm was already bloated and overstaffed to begin with.

Reason #3. Keeping employees on their toes.

By occasionally laying off a significant fraction of employees during downturns, one credibly demonstrates that employees should work hard and well or risk being fired.

[There are here two separate questions: (a) Does this tactic really work (in motivating employees)?; and (b) Do there exist employers who believe in this tactic? An affirmative answer to (b) suffices to serve as an explanation to your question.]

Reason #4. Worker morale.

Suppose we ignore Reason #1 and happily, we can save 30% in payroll costs by either:

C. Firing 30% of the employees; OR

D. Cutting each employee's take-home pay by 30%.

Under C, the 30% who are fired will indeed be very heavily demoralized, but they are no longer your problem. As for the remaining 70%, they might be slightly demoralized about losing some of their work buddies. But they might even be reinvigorated after discovering they survive the cut.

In contrast, under D, every employee is rather demoralized about the 30% pay cut.

Many firms will prefer C to D.


What is the definition of social welfare in monopoly? I'm studying problem that requires me to calculate social welfare in a monopoly setting.

But I dont know what social welfare is, and I can't find it in my book or by googling. What am I asked to compute?

By social welfare your professor probably means total economic surplus which is the sum of consumer's surplus and producer's surplus.


A study plan to understand game theory papers This may not be an appropriate question for this platform, but I couldn't think of any place that it fits. My concern is understanding about game theory papers. Whenever I read classical papers of Harsanyi, Aumann etc. I start to feel lost with unknown math. This happens a lot. I studied many of these classical concepts on a textbook level and I had no problem at all, but when it comes to the original paper I can easily lost the track because of mathematical sophistication. I know a little bit of real analysis (Baby Rudin first $4$ chapters), I have studied a little bit of topology (Munkres) I have also some background in set theory, statistics etc. So I am not a complete beginner. What I want is to understand, at least on a reasonable level, a technical game theory paper such as Harsanyi (1973a) and  Harsanyi (1973b). Can anyone suggest me a study plan with books and such? It can start from zero level I have time and patience. Any advice will be greatly appreciated!

Probably a more general response than you're hoping for, but I recently asked a (well-published) econ prof at my university which classes he'd recommend prior to doing research. He said he usually recommends to the PhD students he advises that they take real analysis and probability at a graduate level.


Profit Function: Just revenue maximization subject to constraint? if so where is $\lambda$? Disclaimer: I dont know i'm getting confused with basic microeconomic prodoucer theory (note: Ive been watching some Hak Choi Videos on youtube),but this is my overall thought process.

We know that the standard profit maximization problem can be described as:
$$\max\pi=py(x)-wx$$

let $y(x)=x^\alpha$

it follows that the optimal $x^*$ from such a formula is:

$$\frac{\partial\pi}{\partial x}=p\alpha x^{\alpha-1}-w=0$$
$$x^*=\left(\frac{w}{p\alpha}\right)^{\frac{1}{\alpha-1}}$$

it follows that the optimal $y^*$

$$y^*=\left(\frac{w}{p\alpha}\right)^{\frac{\alpha}{\alpha-1}}$$

however, isn't the profit maximization problem also just a revenue maximization problem subject to a constraint?
$$\max py(x) $$
$$s.t.\bar{C}(x)=wx $$

let
$y(x)=x^\alpha$

It follows that our Lagrangian for this problem is:

$$L=py(x)+\lambda(\bar{C(x)}-wx)$$

taking the derivative of our Lagrangian With respect to $x$ we find
$$\frac{\partial L}{\partial x}=p\alpha x^{\alpha-1}-\lambda w=0$$

$$x^*=\left(\frac{\lambda w}{p\alpha}\right)^{\frac{1}{\alpha-1}}$$

it follows:
$$y^*=\left(\frac{\lambda w}{p\alpha}\right)^{\frac{\alpha}{\alpha-1}}$$

I find that there is a great deal of similarity between these two problems and their solutions are also very similar. based on this reason I ask, is this an appropriate way to solve for the input and out of a profit maximizing firm? if so what happens to $\lambda$ in the profit function?

The problem
$$\max py(x) $$
$$s.t. wx \leq \bar{C} $$
could be interpreted as revenue maximization subject to an operational budget contraint. However the solution of this can differ from the solution of the profit maximization problem, as costs do not appear in the goal function. $\lambda$ here would show by how much an additional dollar spent on operational costs would increase profits. If $y$ is strictly increasing in $x$ the entire budget would be spent even when $\lambda < 1$. However with such a low $\lambda$ a dollar spent earns less than a dollar, so it should not be spent if the goal is to maximize profits.

In fact, if the goal is profit maximization then and there are diminishing returns then dollars should be spent exactly if they earn at least one dollar.

If $y(x)$ is strictly increasing and strictly concave in $x$ and you set $\bar{C}$ at such a level that the Lagrangian's solution yields $\lambda = 1$, then the solution of the above problem would also solve the profit maximization problem.

The Lagrangian is

$$L=py(x) - \lambda(wx - \bar{C})$$

The equation system solving the Lagrangian is

$$
\begin{align*}
pMP(x) -\lambda w & = 0 \\
wx - \bar{C} & = 0
\end{align*}
$$

Getting $x = \bar{C}/w$ from the second equation one and plugging it into the second equation one has

$$
\frac{p}{w}MP\left( \bar{C}/w \right) = \lambda.
$$

So $\bar{C}$ determines $\lambda$. If $\bar{C}$ is set at such a level that $\lambda = 1$ then the first order condition

$$
pMP(x) -\lambda w = 0
$$

becomes

$$pMP(x) - w = 0$$

which is also what you would get by solving the unconstrained profit maximization problem.


If public roads are public goods, in what way are they non-rivalrous? I get why, say, the military is a public good, but I don't get why public roads are public goods, since they can't be non-rivalrous, given that at any time only a subset of the population can actually be on them.

The property of rivalry is a continuous (rather than binary) variable.*

A good is rivalrous if my consumption of it reduces the amount that can be consumed by others.

So, a particular Big Mac is fully rivalrous, because each bite I take from it reduces (by that exact same amount I've bitten) the amount left for you.

The degree to which roads are rivalrous is lower: My driving on a deserted road at night does not reduce at all the amount that can be driven on the same road by others.

But your doubt is correct—it is not generally $0$: My driving at peak hour at a city center does reduce the amount that can be driven by others.

Ideas are a good (and possibly sole) example of a good that's perfectly non-rivalrous. (My "consumption" of the Pythagorean Theorem leaves no less for anyone else.) Side point: But with the right laws and accompanying enforcement—e.g. patents, copyright—even ideas can be made excludable.

Defense is often given as a classic example of non-rivalry. But I would argue that although defense exhibits a very high degree of non-rivalry, it is less than perfectly non-rivalrous. (The resources used to defend John in New York from a nuclear attack does reduce the US Department of Defense's ability to defend Jane in Hawaii. The resources used to defend a nation with over 1B people are greater than the resources used to defend a nation with less than 1M people.)

*Likewise, the properties of excludability and being a public good are continuous (rather than binary) variables. (Many introductory textbooks tend not to emphasize this important point sufficiently.)


Why doesn't Aggregate Demand mean National Income whereas Aggregate supply is equal to national income? Aggregate Demand is Basically

$$
AD = C + I + G + X - M
$$

Is not this national income Through Expenditure Method ?

Then why don't we treat AD equal to National Income ?

The existence of aggregate demand does not mean that aggregate demand is fulfilled. For instance if the aggregate planned demand is 100 billion dollars and due to crisis or war output is very low at just 10 billion dollars. Does the national income equal 100 billion dollars?


Finding demand functions for an unusual utility function I have a utility function: $U = x + \min\{x,y\}$

I want to draw the indifference curve and find the demand functions. Will it be the case of the usual perfect complements?

Also, what preferences could such a utility function represent?

Will the optimal solution be at the kink?

I assume you know how does $\min\{x,y\}$ look like? In order to draw utility function of interest, you need to consider cases: $u(x,y)=x+\min\{x,y\}=\begin{cases}2x, \;\; \mathrm{for} \;\; x \leq y \\ x+y, \;\; \mathrm{for} \;\; x > y\end{cases}$

With $x$ on horizontal and $y$ on vertical axis:

Not sure about the "usual" perfect complements. It is more like a combination of substitutes (below $y=x$) and complements (above $y=x$).

Also, take a look here: Identifying utility function and Algebraic approach towards convexity where you can see more graphs.

EDIT.

As for the demand function, Finding demand function given a utility min(x,y) function.


Which foreign workers in USA who send remittances to family are getting the best bang for their buck? There are probably a lot of ways to measure that, but I want to measure it like this: Foreign national workers grouped by country of origin, which group sends the most money per capita in remittances?

That way it's kind of like saying they have the most income per capita after living expenses.

Maybe that means the skills they brought to USA are getting the best return compared to other groups.

It doesn't say anything about how far that remitted money goes in the origin country. So bang for buck is referring to getting paid in the US, not spending it abroad.

I'm not sure where to start searching for this information. I googled around but just found ads for remittance companies.

A good start would be The Economics of Migrants' Remittances by Hillel Rapoport and Frédéric Docquier. They both work extensively on this topic. In this paper, they review the  theoretical  and  empirical  economic  literature  on  migrants' remittances.

They review six theories decomposed into two broad categories individualistic motives (altruism, exchange, inheritance, and the strategic motive) and two types of familial agreements (on insurance and investment).

In all of these six theories, except one, migrant's income is indeed a very good predictor of remittances. However, other motives matter and may differ across "foreign national workers grouped by country of origin". For instance, some societies are more altruistic than others, and assuming  that  altruism  decreases  with  time  of arrival, the size of remittances should be negatively related to this variable in the altruistic case.

They also predict contrasting effects concerning the distance from family. For instance, in the inheritance theory, the amounts remitted are expected to be closely related to the probability of receiving inheritance in the origin country. If, as they assume there is a negative correlation between the distance from family and the probability of receiving inheritance, a Mexican worker will remit more than an Ethiopian worker, other things being equal. This could even be true if the Ethiopian worker has "the most income per capita after living expenses".

To sum up, migrant's income is a crucial variable but other variables, which may differ by country of origin, may also be of first-order importance to understand remittances.


What separates "rich" developing nations from "poor" developed nations? What separates relatively high income developing nations, such as Argentina, from relatively low income developed nations, such as Portugal?

If you look at things like HDI numbers, Argentina and Chile are classified as very high and even rank above Portugal. If you look at per capita GDP then Argentina, Chile and Portugal are all pretty close and fall between 22K and 27K.

Despite this, Argentina and Chile are generally considered developing countries despite being on par with or exceeding some countries that are generally considered developed.

This makes me wonder, what exactly separates developed countries from developing countries?

When comparing the United Kingdom and Uganda the answer is obvious, however it isn't as obvious (at least not to me) when comparing countries like Portugal and Spain to Chile and Argentina.

All country classifications are conventions, which depend on the rule used by the particular institution. There is no hard (ontological) rule on any classification. There is no even consensus on the definition of categories. Advanced/rich/developed/industrialized economies are sometimes used interchangeably, but who is in each of them depend on who is talking. Let see an example:

International Labor Organisation (ILO)

(I had to cut it because it is too long. See the full list in page 67 of this report).

Citigroup

(Report here).

Conclusion

As you can see, ILO puts Czech Republic, Greece, Argentina, and Chile as developed countries, whereas Citi puts them as emerging countries (Argentina as emerging can be seen in other reports). (ILO even distinguishes between emerging and developing).

This shows that there is no hard-coded rule about classification of countries. It all depends on the institution.


Quasiconcavity and homogeneity How to prove that if $f$ is strictly quasi-concave and homogeneous of degree 1, then $f$ is concave? It was left as an exercise by Silberberg & Suen (2001), p.140.

I simply could not elaborate any sketches to leave here as a starting point.

Note that strictness of quasi-concavity is not required, unless you want to obtain strict concavity as a result.

Take any $x,y\in\mathbb R^n$. Observe that homogeneity of degree 1 (HD1) implies that
\begin{equation}
f(x/f(x))=f(x)/f(x)=1=f(y/f(y)).
\end{equation}
For any $\alpha\in(0,1)$, let
\begin{equation}
\theta=\frac{\alpha f(x)}{\alpha f(x)+(1-\alpha)f(y)}.\tag{1}
\end{equation}
Note that $\theta$ also lives in the (open) unit interval.
Thus, by quasi-concavity, we have for every $\theta\in(0,1)$,
\begin{equation}
f\left(\theta\frac{x}{f(x)}+(1-\theta)\frac{y}{f(y)}\right)\ge\min\left\{\frac{x}{f(x)},\frac{y}{f(y)}\right\}=1.
\end{equation}
Expanding the LHS using $(1)$, we get
\begin{equation}
f\left(\frac{\alpha x+(1-\alpha)y}{\alpha f(x)+(1-\alpha)f(y)}\right)\ge1.
\end{equation}
Invoking HD1 again, we have
\begin{equation}
\frac{f(\alpha x+(1-\alpha)y)}{\alpha f(x)+(1-\alpha)f(y)}\ge1
\quad\Leftrightarrow\quad
f(\alpha x+(1-\alpha)y)\ge \alpha f(x)+(1-\alpha)f(y),
\end{equation}
which means $f$ is concave.


Calibration of utility function parameters controlling labor disutility Consider the very basic utility function

$$ u(c, n) = \log (c) - \alpha \frac{n^{1 + \frac{1}{\nu}}}{1 + \frac{1}{\nu}}$$

Where $c$ is consumption and $n$ is working hours. As the Frisch elasticity here is given by $\nu$, I would calibrate $\nu$ to be between 0.5 and 1.

However, I'm wondering what to do with $\alpha$: What do I calibrate that against?

$\alpha$ handles the conversion between the units in which labor is measured and the the units in which consumption is measured.

Consider the leisure part of the utility function:
$$ - \alpha \frac{n^{1 + \frac{1}{\nu}}}{1 + \frac{1}{\nu}}$$
We can rewrite it like this:
$$ - (\alpha^{\frac{1}{1 + \frac{1}{\nu}}})^{1 + \frac{1}{\nu}} \frac{n^{1 + \frac{1}{\nu}}}{1 + \frac{1}{\nu}} = -  \frac{(n\cdot (\alpha^{\frac{1}{1 + \frac{1}{\nu}}}))^{1 + \frac{1}{\nu}}}{1 + \frac{1}{\nu}} $$
Define $\gamma = (\alpha^{\frac{1}{1 + \frac{1}{\nu}}})$
and rewrite the utility function in terms of $\gamma$ instead of $\alpha$
$$ u(c, n) = \log (c) -  \frac{(\gamma \cdot n)^{1 + \frac{1}{\nu}}}{1 + \frac{1}{\nu}}$$

Written this way, I see $\gamma$ and therefore $\alpha$ as serving as a unit shifter. How many seconds of leisure makes you indifferent over one dollar less consumption? Say it were 60 seconds. What unit is $n$ measured in? Should we put in $n=1$ for minutes, $n=60$ for seconds, or $n=\frac{1}{60}$ for hours? $\nu$ can't answer that question because it controls the shape and curvature but not the scale of the utility function with respect to $n$. However, $\gamma$ can. If the "true" measure of $n$ is in hours and we guess wrong and choose seconds then $\gamma$ should be $\frac{1}{3600}$ and conditional on $\nu$ this implies $\alpha$.
In calibration the true units of $n$ it need not map cleanly to one of our time units. $n$ might be properly measured in units of 15.5 seconds or $\pi$ minutes, but the functional form is flexible enough to handle these cases regardless of if you input the labor in the calibration in any of the more standard units of hours, minutes, or seconds.

As for how to calibrate this, you have to see the bundles of labor/ leisure and consumption chosen by households for given sets of prices. This, combined with the demand functions that follow from this utility setup, should give you the required calibration. The paper A Model of Housing in the Presence of Adjustment Costs: A Structural Interpretation of Habit Persistence (Flavin and Nakagawa (2008), free copy here) shows the way to do this in a different (housing / non-housing consumption instead of consumption / leisure) problem. Their $\gamma$ parameter acts much like $\alpha$ does here.


What would an investment bank have to gain by warning of an impending financial crisis? After reading a recent article and a discussion with a friend, I was left wondering what an investment bank would have to gain by claiming that a financial crisis is coming.

I can see an obvious answer which is that by instilling fear in the market you can lower stock prices of shares that you would be interested in buying. However I find that this is a very dangerous game to play, because if the panic spreads, overall their portfolio would lose a lot.

Whether their prediction is accurate or realistic doesn't concern me. I just want to know what they'd gain if they did lie about it.

What would an investment bank have to gain by warning of an impending
  financial crisis?

An entity might have an incentive to deceitfully alarm about an "impending crisis" if the entity's portfolio largely consist of positions which lead to profit when the price of other asset(s) goes down. Examples of such positions are put options, short selling of stocks, and certain side in interest rate swaps. The same applies if much of the entity's liabilities is positively correlated with the price of assets which are representative of the market.

The entity could hedge against the risk of a generalized fall by getting long positions in "very" Out-of-The-Money put options. Being far OTM implies that this protection for drastic scenarios is not significantly expensive.

Likewise, the rest of an entity's portfolio could also consist of assets deemed stable enough to withstand crises.


Utility function must be continuous for $V(p, e(p,u) )$ to hold? Most utility functions we work with are already continuous, so I thought it was understood that it must continuous. But how would I show that if the utility function is discontinuous the identity V(p, e(p,u))=u might not hold?

Define the function

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

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

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


Does decreasing marginal utility imply risk aversion? Unless I misunderstood something, seems like risk
aversion and decreasing marginal utility is the same thing in the utility model, but intuitively, it seems entirely possible that an individual with no decreasing marginal utility is still risk averse.

For example, I can be exactly twice happier with 1000 dollars than only with 500 dollars. This implies constant marginal utility. With that, I can still be unwilling to enter a bet that pays me +1 or -1 with equal probability and generates an expected return of 0. It is the uncertainty that i do not like, irregardless of marginal utility.

Am i misunderstanding something? Can anyone please explain?

What you are misunderstanding, is that in  expected utility theory, marginal utility is not an independent concept from "risk aversion", as the latter is defined in the context of that theory: "risk aversion" does not mean what it means in everyday language. Being "risk averse" does not mean for the theory "I dislike risk", because taken literally "disliking risk" would imply that "risk" is a separate entity, or an aspect of a situation, which produces negative utility.

A "risk averse" person is defined to be a person that has a strictly concave utility function (and so a function with decreasing 1st derivative).

PS: On another front, "being twice happier" reveals that you are considering cardinal utility, where quantitative comparisons between numeric utilities is considered to be meaningful. Be aware that the predominant paradigm in economics on the matter has been that of ordinal utility (this does not affect the mathematical properties and relations, only their interpretation).


what is monotonicity and strict monotonicity in preferences? I am really confused between monotonic preferences and strictly monotonic preferences, I saw some video and read certain answer where it is mentioned that the

When preferences are monotone / weak monotonic preference, the
consumer prefers more of both goods.
When preferences are strictly    monotone or having strong monotonic
preference, the consumer prefers    more of one good but no less of
the other.

(source: https://www.learnpick.in/questions/details/11169/what-is-a-monotonic-preference)
whereas when I read this other paper it says

Monotonicity means that if c1 contains more of some or all
commodities, but no less of any, than c2 (c1 ≥ c2) then c1 ≿ c2.
Strict monotonicity means that if c1 contains more of each commodity
than c2 (c1 > c2) then c1 strictly prefered c2

(source: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Shon.pdf)
I am confused as both the answers are quite contradicting as the (first as per sequence I mentioned) says that in strict preferences the consumer more of at least one good but no less of the other.
whereas (second as per sequence I mentioned) says in strict monotonicity we prefer more of both the goods. Also, I am able to understand the graphical representation of the monotonicity.
Please tell me where I am getting it wrong I am really stuck on this.
Thank you.

The word monotonic means "always moving in the same direction", in our case, always going up.
Monotonic preferences mean that the customer always prefers more of a good.
This comes in two flavors:

Strictly monotonic: More of one good is always preferred to less of that good.
Weakly monotonic: More of one good is always equivalent or better than less of that good.


Are means-tested or universal programs more effective at reducing the poverty rate? Advocates of means-tested programs frequently argue that they will remove inefficiences in the system by directing a larger proportion of the program's budget to people with a genuine need. By contrast, those who oppose them, and who tend to advocate for universal programs, argue that they would require the creation of bureaucracy that would eliminate any theoretical gains. Some also argue that the portion of the money that goes to wealthier people serves as an economic stimulus which increases the overall health of the economy, thus being less inefficient than it would otherwise seem.
Which of these arguments is correct? Are means-tested programs or universal programs more effective at alleviating poverty? I am aware that there are moral arguments for both perspectives as well: some say that people who make more than a certain amount (or less, for that matter) do not deserve assistance, whereas others argue that everyone deserves equal assistance. However, I am interested in the economic point of view. In addition, some people argue that any economic assistance programs have economic effects that ultimately eliminate any gains. If there is strong evidence supporting that perspective, I would be interesting in seeing it, but the question is primarily about which of the two types of programs is more effective (or less ineffective, if both are ineffective). Obviously, this is a politically heated theme, so I am ideally looking for answers based on empirical research, case studies, and to an extent modeling,

TL;DR:
There is generally not much support for universal welfare programs (as narrowly defined - see below) in the literature. However, it is for different reasons that what is mentioned in the question.
The main proposed advantage of universal programs is that they do not distort peoples incentives since you are not loosing your benefits as you are moving up an income ladder (less administrative costs is in the literature only auxiliary argument). However, at the same time this means that state has to redistribute resources even to billionaires, so such programs are more expensive. These expenses ultimately require higher taxes and taxes distort incentives as well. It turns out that on net having means tested welfare transfers produces less distortions to the economy.
The bureaucracy issue is very commonly cited in the press, because it likely captures better peoples imagination and attention as compared to more serious but less known deadweight loss caused by taxation/redistribution.  However, bureaucracy costs are usually trivial for either system. In addition, there is no support in a literature that welfare for rich has much stimulative effect for the economy.
Full Answer:
Narrowing the Scope:
First, in order to actually answer this question, we have to properly define the scope of subject matter. 'Programs that reduce poverty' is broad and vague term that could be applied to virtually any government program. However, given what is written in the question I suspect what you are most interested in are income redistribution taxes and transfers such universal basic income (UBI) vs means tested transfer as opposed to some other government schemes (e.g. compulsory state provided elementary and high-school education could be viewed as anti-poverty program but we have to narrow the scope of the question as providing overview of all government programs that can reduce poverty in one way or another would require several books and be well beyond scope of SE answer).
Bureaucracy Issue
Next let me address the bureaucracy issue as this is very common argument that non-economists use but you won't even see it mentioned in the literature much. Simply put bureaucracy costs are trivial. For example, consider the administrative spending for main US welfare programs below:

The administrative spending ranges from $9\%$ to $>1\%$ of total spending for the program. Moreover, you will note that the highest administrative costs are for food stamps and housing vouchers which are in kind/conditional cash transfers that are, generally speaking,  considered by most economists relatively inefficient for various reasons relative to unconditional (but still means tested - conditionality here refers to additional restrictions on spending of the transfer) cash transfers (e.g. see Currie & Gahvari, 2008; Skoufias et al., 2013; Handa et al., 2018; Mishra & Kar, 2015; Gentilini 2016 or Caldés et al., 2006).
As you can see the earned income tax credit (which is still means tested) but purely cash based transfer has less than $1\%$ administrative cost. This just goes to show that in principle means tested programs are not expensive from point of view of bureaucracy. Even if we would be generous and assume that switching to universal transfer could cut $50\%$ of all administrative costs (which I think is overtly generous given that significant part of administrative costs are fixed costs) it would be penny wise but pound foolish to make argument that such trivial savings are justifiable taking into account large costs that extending transfer to anyone entails (although as will be explained further down the road a proper efficiency argument could be made on the account of distortions they create).
Stimulus Argument
I must say I did not heard the argument that transfers to the rich provides additional stimulus to the economy (I heard the one about tax cuts but not about transfers), but it does not have any basis in economics. First, from macroeconomic perspective spending is spending whether done by rich or poor. There is not much if any evidence that spending would stimulate economy more if done by rich instead of poor.
Second, richer people tend to save higher portion of their income (see explanation in this older SE answer why), this does not matter outside recession since saving becomes investment spending which increases output equivalently to consumer spending in normal times. However, during recessions economies can experience the 'paradox of thrift' - a situation where extra saving is offset by drop in output and thus won’t help the economy  (I wont go into details on this concept as it is tangential to this question, you can see further explanation in Blanchard et al Macroeconomics: A European Perspective). But even in such an event consumer spending still stimulates the economy. Consequently, if anything if the goal is to stimulate economic activity the transfers should especially target poor who have higher propensity to consume not higher income groups who tend to save more.
Economic Efficiency Issue
When economists discuss economic efficiency of redistribution schemes they mainly focus on the economic distortions that these schemes create (see Stiglitz Economics of the Public Sector). These efficiency costs are created because taxes and redistribution distort optimal behavior of people. For example, an income tax will distort peoples incentives to work, consumption tax to consume etc. At the same getting welfare benefit conditional on low income distorts peoples incentives to move up income ladder or put more hours and so on.
The reason why economists focus on these effects is that even though these are almost unknown among general public their economic costs are massive. For example, as mentioned in Jacobs (2009) on the margin it is often believed that excess burden of taxation is approximately $50\%$ (although there is some uncertainty about this value and it could be as low as $30\%$ or as high as over $100\%$ depending on exact behavioral responses (elasticities) to taxes and country we are talking about). The commonly accepted $50\%$ figure implies that for government to raise raise $1$ dollar of tax revenue it has to reduce aggregate income of an economy by $1.5$ dollars on the margin. This is purely cost of damage to the economy, the administrative costs would have to be added on top of that, but as mentioned above they are more or less trivial (e.g.  in Netherlands where I live they would be additional 6 cents on average (see Allers, 1994) and in most developed countries it should be comparable).
Now when it comes to universal redistribution schemes their big advantage is that they eliminate distortions on the receiving side. For example, with UBI you get your welfare transfer whether you work, are poor or rich and so on. So there is absolutely no reason for you to change your behavior/economic activity. However, at the same time since the benefit is extended to everyone they are more expensive, requiring increases in taxes which as mentioned above also have serious effects on distorting the economy.
When it comes to means tested redistribution schemes they create relatively large economic disincentives at the receiving end but an advantage is that only a portion of the population receives the benefit making them much cheaper. Consequently, government needs to raise much less revenue through distortionary taxation to pay for it.
Hence, a priori it is not clear which one would create less net distortions as one creates more distortions on receiving end and the another one more on the taxing end. Consequently, the main economic debate of universal redistribution schemes vs means tested schemes is about whether the economic distortions of one are larger than other, not about administrative costs that are relatively trivial at best mentioned as an auxiliary argument/issue in the literature.
Here the literature shows that means tested schemes are preferable as they cause less distortions both from theoretical and empirical perspective. For example, the state of the art optimum income taxation simulations (that includes also redistribution which is treated as a negative tax in the literature) show that it is optimal to have means tested income redistribution, even when government adopts Rawlsian social welfare preferences (i.e. the max-min criterion, which implies that government only cares about the welfare of the poorest anyone else except for the poor has zero welfare weight in government social utility function). This is despite of the fact that such means tested programs cause distortions primarily on the receiving end and thus create a 'poverty trap' - a situation where poor have less incentive to climb up the income ladder (see Diamond 1998;  Saez 2001).  The case for broader benefits is bit more stronger under utilitarian or charitable conservative/libertarian social welfare function, but I don't know of any standard social welfare function that would result in universal benefit. This can be seen on the picture below which shows simulations of optimal marginal tax rate. The tail is so high on the lower end of income distribution precisely because that represents people loosing their benefits (withdrawal of benefit is considered tax since benefit itself is a negative tax).

If you are not convinced by theoretical optimal tax simulations then when we turn to empirical evidence on the most popular universal redistribution scheme: universal basic income (UBI), the evidence shows that the measure is simply too expensive and thus inefficient (even if it reduces the above mentioned poverty trap and has some other positive effect). For overview of literature on this you can see Hoynes and Rothstein (2019) or Martinelli (2017) and sources cited therein. The literature can be summarized nicely by the following quote form Martinelli:

an affordable UBI would be inadequate, and an adequate UBI would be unaffordable.

The issue is that there are quite a lot of people in upper income brackets. For example, in 2019 in the US over $35\%$ of households earns above $100,000$ dollars per year and about $20\%$ above $150,000$ dollars per year. These are income groups where there is little to none rationale for welfare support (at least from the point of view of welfare/public economics - politically it might be useful to support such groups for political reasons), but are still very broad groups of people that would waste enormous number of resources unless the universal benefit is set so low that it will become insufficient for those at the bottom of income distribution.
Few Caveats
However, there are few caveats that need to be mentioned.

As seen in the previous chapter taxation and redistribution cannot be divorced from each other - they are two sides of the same coin. However, if you only talk about transfers you could make an argument that you can have universal transfer while you cancel it with extra taxes on higher income groups. For example, you could give everyone $1000$ dollars per month of universal income and then tax everyone else except the poor by $1000$ dollars more essentially making the universal basic income means transfer program but in more hidden way. This is really just semantics argument hinging on definition of the words in the name of program, but some economists such as Mankiw made an argument for UBI on these grounds. However, this is to the large extent a semantic issue as economically if you cancel that $1000$ by higher taxes for better of groups you create the same disincentives as if the benefit would only exist for poor. So really this is just a semantic argument without economically affecting the discussion in sections above.
As mentioned at the beginning this analysis might not extend to every single government program. Arguments for universal education or healthcare could be made on partially redistribution grounds but address additional issues of externalities or market imperfections that make their universality necessary or more efficient, but this goes beyond any purely redistribution/poverty-alleviation concerns.


how to test endogeneity in a cross section data? Maybe it is a silly question but I really do not know much about econometrics. I am trying to look at the relationship between environmental quality and final consumption expenditure. For this, I look only at the cross-section data for Environmental Performance Index (EPI) for environmental quality and final consumption expenditure data from World Bank Database for 2010. I write down an equation of that kind :

$$log(C)=\alpha+\beta_{1}log(S)+\beta_{2}(log(S))^{2}+\epsilon$$

where $C$ and $S$ hold for the consumption and environmental quality.

I wonder how I can analyze for the endogeneity issues between consumption and environmental quality. Is it possible to talk about the endogeneity on cross-section data?

Endogeneity can rarely, if ever, be tested.  It can certainly exist in cross-sectional data.

Running a presumably OLS regression without a theoretical model will just give you a measure of the co-movement of the two variables. Since no other variables are included, what you are estimating is $E(C \mid S)$, i.e. the conditional expected value of Consumption if no other information is used except for the environmental index. Including the square attempts to capture the case of a non-linear conditional expectation.

This is more of a "prediction" framework (although not forecasting), rather than an analytical / causal framework.

Informally, you can argue for the existence of endogeneity if either there exists other variables that are correlated both with consumption and the environmental index and/or if their relation is an equilibirum one of co-determination.

Accepting that endogeneity exists, does not invalidate your regression specification. It just says that the coefficients you are estimating may not be what you would want to estimate:
Here for example

$$\beta_1 + 2\beta_2 = \frac {\partial  E(C \mid S)}{\partial S}$$

But maybe you are interested in how "consumption changes" when the environmental index changes when other influences on consumption are taken structurally into account. This essentially implies that you would want a more rich right-hand side in your specification.

Note: "Take structurally into account" is not the same as saying "all else held constant". If you estimated a regression with more explanatory variables, the betas themselves would change, so the marginal effect of the index in that alternative scenario would be different even if we examined under the "all else held constant" assumption.


Did a similar property boom take place in another country or throughout history like we have seen in London? If so how did it effect the economy? The London property market has seen a sharp increase in the average house price. If we go back to the 70's the average UK and average London price were almost the same, however since than the gap has increased quite dramatically. The average property in London is +£400k, whereas in the UK this is almost half.

How long prices will increase till the bubble will burst? Some say property prices tend to double every 8 years, if that's the case then in few years time the average price will be £800k, however this is not a fair balance compared to the average salary in London. My question whether the world has seen a similar property boom like in London (throughout history) where there is an imbalance of property prices & income, prices shooting up to such an extend that they become unattractive/ unaffordable? If so it would be nice if you could provide some reference as I am interested to study this into more detail and understand what the effects were of such highly priced market and how the economy responded?

According to the ONS the house price-to-earnings ratio was 12.8 in London and 7.8 in England as of 31/12/2019.
According to Schroders the house price-to-earnings ratio in England was consistently above 8 until the late 19th century (see graph below). In 1845 it was 13 in England, so it was arguably higher than that in London at the time.
What I don't find very clear is if Schroders are referring to housing units when they speak of houses. They say that "houses" where bigger in the 19th century and hence more expensive. Obviously the average earner did not live in a bigger house but would have shared it with other average earners.
A more sensible statistic would hence be something like a "minimum housing unit price-to-earnings ratio". Comparable prices would still likely be higher in the past. But it shows that creating comparable data would require in-depth study not just of the economics of housing but also how these houses are used, i.e. what can realistically be considered a minimum housing unit in the 19th century given the housing conditions at the time (Engels published The Housing Question in 1872).

Source: Schroders (30/01/2020)


CAPM and beta for individual stocks Why do we just assume that the β is symmetric for a stock? Could it not very well be the case that the β has a larger leverage (covariance with the market) for example 1.2 in a bear (baisse) market and a lower leverage (covariance) e.g. 0.8 in a bull (hausse) market for an individual stock, or the other way around? This would mean that the stock drops more than the index when the index drops and gains less than the index when the index gains. Of course this is neither a rational investment compared to index nor is it by definition a covariance, but it looks from numbers and graphs that several stocks or investments do not have the same β during a bull/ gaining market compared to a bear dropping market, it will drop more than the market when the market drops and not even outperform the market when the market gains.
Or is that something that would be explained by the alpha value of the CAPM, meaning that the alpha is some individual characteristic of the company?

I would say that we don't generally speaking assume that stock betas are constant. For example the paper Explanations for the Instability of Equity Beta: Risk-Free Rate Changes and Leverage Effects, from 1985 (!), cites at least six other earlier papers that the market risk of securities is not stable.
Many papers use a rolling window CAPM estimation approach (a popular way of allowing betas to vary) exactly because CAPM and more generally factor model betas do not seem to be constant. See for example the JFE paper CAPM for estimating the cost of equity capital: Interpreting the empirical evidence.


Leontief preferences I can solve most utility maximization problems using my mathematical knowledge .... but not when it comes to Leontief preferences. I do not have a book to lean on (am self-studying), so would really like some help. How does one solve a general maximization problem like $$\max [\alpha x_1, \beta x_2, \gamma x_3] \ \text{subject to } \ \lambda_1 x_1 + \lambda_2 x_2 + \lambda_3 x_3 = M$$
where $M$ is income and $\lambda_i$ is price for good $i$?

Really, everything I know about derivatives and slopes goes all out the window with this damn thing. If somebody told me what the prices and income were, the optimal choice, when there are only a few goods, could probably be found by just applying common sense, but what about the general case? Is there no general "formula" like there is for Cobb Douglas and CES functions? Is there some go-to method that we use in these cases?

You are missing a $\min$ operator just before the bracket. The utility maximization problem is as follows, $$\max \ \min [\alpha x_1, ..., \gamma x_3] \\ \ \ \text{such that} \ \ \lambda_1x_1 + ... + \lambda_3x_3 = M$$
Consider the case of two goods with utility $u$ given by $u(x) = \min[\alpha x_1, \beta x_2]$. At the optimum, what do you know about the relation between $\alpha x_1$ and $\beta x_2$? They must be equal, that is $$\alpha x_1 = \beta x_2$$ For if not, then assume without loss of generality that $\alpha x_1 > \beta x_1$. What is the utility of such choices of $x_1$ and $x_2$? It must be $\beta x_2$, which means that some of your money is being spent on $x_1$ (assuming prices are strictly positive) but it is not giving you any extra utility, and so this cannot be an optimal choice of consumption.

It follows that the equality must hold at an optimum (this is obvious graphically too). Along with the budget constraint, that gives you two equations and two unknowns, which can be solved for optimal consumption. A similar approach can be applied to the case of $n$ goods.

Of course above is assuming that we are dealing with a trivial utility maximization problem, and aren't doing integer programming and the like.


Heckscher-Ohlin trade model with non-homothetic preferences $U^j = \prod^N_i \left(d^j_i-\bar{d}_i\right)^\oint$

where $d^j_i$ is the per capita consumption of good $i$ in any country $j$; $\bar{d}_i \geq 0$ denotes the minimum consumption of each good $i$, which are same across countries; $\displaystyle \sum_i \oint_i^j =1$. Assuming that per capita income $I^j$ is large enough to afford the minimum consumption. Derive the per capita demand for each good $i$ in country $j$.

I understand pi-product notation means it's a product function, but how do we differentiate a product function like this? I define my $I^j$ as $p_i d_i^j$; not sure if this is correct. Your help is much appreciated! Thank you in advance. :)

$I^j$ is just a known constant (i.e. we don't know anything about production, which would endogenously generate the amount of per capita income), but as you suggest the budget constraint for the consumer is that $\sum_i p_i d_i^j \leq I^j$ (here, you can assume this holds with equality).

First, note that if $d_i^j < \bar{d}_i$, then the utility function will be negative. So, if we have sufficient income to afford the minimum consumption (i.e. $\sum_i p_i \bar{d}_i \leq I^j$), to avoid this outcome the consumer will purchase at least $d_i^j \geq \bar{d}_i$ units of each good. Therefore, if we define $x_i^j \equiv d_i^j-\bar{d}_i$, we know that $x_i^j \geq 0$.

With this reformulation, we can rewrite this Stone-Geary utility function as a Cobb Douglas utility function, i.e.
$U^j = \prod_{i=1}^N (d_i^j - \bar{d}_i)^{\varphi_i}=\prod_{i=1}^N (x_i^j)^{\varphi_i}$. Then, to solve for the optimal consumption of $x_i^j$, you can just solve for optimal consumption with a Cobb Douglas utility function with a new income of $\bar{I}^j \equiv I^j - \sum_i p_i \bar{d}_i$ (that is, the remaining income left over after spending it on the minimum consumption level of each good). Once you have the optimal $x_i^j$, the optimal consumption level of the good (inclusive of the minimum) is $d_i^j = x_i^j+\bar{d}_i$.


Effects of price transparency Company I work at provides software services. After customers sign up, they then can go and order different combinations of services and consume them.

We do not expose prices within the flow of them ordering services, however we do provide detailed bills to them outlining costs as well as make all prices available on the separate documentation side that we link to.

Personally I believe it is a bit strange that we do not show prices when they order those services (even though they are just a few a clicks away). My theory is that once we will start showing those prices, it will increase customers trust (because we are transparent) and satisfaction (because it will help them avoid surprises in the bill).

I'm wondering if there is any research on the subject of price transparency and its effects that can help prove or disprove my assumptions.

I will expand this into an answer since I've had a chance to look at a bit more of the literature on price salience, which I consider to the primary consideration here. That being said, I think there is something to say about the fact that the company might do away with a pricing policy that some subset of potential customers feels is deceptive. This will obviously lead to a more positive perception of the company (from these customer types). Whether or not this positive increase can offset the impact of increased price salience is an open question.

My original comment:

One must also consider the antithetical effect that price salience has on purchase decisions. It might be the case that people are more likely to tolerate a product's price ex-post than they are ex-ante. There is a robust literature on the effect of salience on elasticities and prices (for example, Finkelstien's tax salience paper). This definitely doesn't settle the matter but it does inform whatever answer you ultimately accept.

Her paper: economics.mit.edu/files/7878

Other papers to read:

Bordalo and colleagues:
http://scholar.harvard.edu/files/shleifer/files/pdf_1.pdf
This paper looks at consumer choice and characteristic salience of goods. This is a good paper to help build an intuition of what I mean by salience and what it means for a good or a good's characteristics to be salient.

Dias et al:
https://ideas.repec.org/a/taf/jgsmks/v20y2010i2p149-163.html
This is a paper about how price salience impacts consumer perceptions and purchase intentions. I added this paper because I think it hits the nail on the head here. Also, it gives a good review of the literature that is most related to your question.

Hope this helps.


Deriving formulas from Gali's book on the New Keynesian model I've found this link with all the derivations from Gali's book on New Keynesian model.
In page 13, eq. $(3.7)$, the author derives the optimal price-setting rule  $P_t^*$ by a representative firm that solves an intertemporal profit maximization problem. Equation $(3.7)$ contains two infinite series. The author then does a log-linearization on both and proceeds to the solution. Log-linearization is a local approximation, right? Then how can we be sure that the two series, with these approximations will converge?

Mathematically speaking, we are not sure. The infinite sums mentioned is of the form

$$E_t\sum_{k=0}^{\infty}\gamma^kX_{t+k},\;\; 0<\gamma<1$$

The fact that there is the declining factor, $\gamma^k$, provides some comfort that the sum converges, but from examples like the harmonic series we know that even if $\lim_{k\rightarrow \infty}\gamma^kE_tX_{t+k} =0$, the sum may not converge.

Some times authors provide rigorous conditions for convergence, but I believe it is more important not to be trapped by our own tools: the use of infinite horizon is, of course, non-realistic. The only reason we use it is that it makes the mathematics more tractable. So in reality we model a problem with possibly long but finite horizon. In such a case, there is no issue of convergence. By extending the problem to the infinite horizon we silently assume that this won't destroy convergence, since it is an artificial extension.

The ultimate economic argument for convergence, is that it is meaningless to talk about economic magnitudes going to infinity. Even for prices, where it is tempting to think that they do not have an upper bound per se, we must remember that prices as a component of an economic system do have ultimate bounds.

With prices flying to heaven, the economy goes to hell (if you excuse the christian iconography), because the price system stops functioning as a signal mechanism to co-ordinate decisions in a decentralized framework. But in the real world, when such explosions "start to happen", the society that encloses the economy in question, will intervene in one way or the other, in order to end the phenomenon. The historical experience from war-time hyperinflations and from other price-bubbles verifies this.

Ideally, we would want a single model to encompass both possibilities and also model the possible "extraordinary reaction" to the explosive case: herein infinite PhDs lie.

Returning to the specific situation from the link, note that the price in eq. $(3.7)$ is determined by the ratio of two such infinite sums. Even from a mathematical point of view, this is a bit more favorable, because even if each sum separately diverges, their ratio may converge. But again, my point is, "don't let the mathematics make you forget the economics" of the situation.


How do economic and financial policies in government deal with cryptocurrencies? Given the decentralized nature of cryptocurrencies like bitcoin and ethereum, how can centralized organisations like banks and the government create policies around it without changing the way traditional economies work?

I have read a little bit about Japan accepting bitcoin as legitimate payments, and also countries like Japan, Russia and Estonia wanting to create their own cryptocurrency so it definitely appears to be something that is going into mainstream finances soon.

Can someone explain how a typical government would deal with this?

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

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

Hope this helps.


How did the Price/Value of any goods measure in money determined in the first place Nowadays most value/price of goods and services are determined by the supply and demand in the market. And most of the time we use money which back by the credit/trust of the country to exchange these goods and services.
But one thing keep puzzling me. When money first created by the country (let say a new country just form and they start making their own currency/money), without any credit or trust for the newly created currency and any reference whatsoever. How did anyone can determine the price/value for goods using that money. Did the government of the country just announce or set the price for some high demand commodities (say foods/water that everyone needs) as a anchor for everyone to reference? Or did the people just make there own assumption and start a complete chaotic process of natural selection in the market until most of the price settle?
I just can't figure what process did the first country in the past, that created money go through to settle market. It would be very helpful if anyone could give me some insight about this question.

No prices exist even without money when you barter. Early money was just some sought-after commodity that was originally bartered (like grain/livestock/metals), but people just figured that such commodity can be used in all transactions to avoid double coincidence of wants.  Simple forms of money precede countries, as we understand them today.
As the first ancient states developed this just got better organized. Coins developed from state stamping pieces of metal to certify its quality. A more complex form of money such as Mesopotamian clay tablets that were exchangeable for barley at harvest time developed from the government starting to enforce contracts. At that point, the value of money started to be also underpinned by the fact that the government required taxes to be paid in its money which created a demand for it.
So at the point when money was created, all goods and services already had prices in terms of money. Relative prices exist irrespective of whether money exists, and early money was just some commodity that already had all relative prices to other commodities. Once it was selected as money just other values started to be expressed per unit of the selected commodity as well. You can have look at The Ascent of Money by Ferguson, it is a nice overview of the history of money from ancient times to the present day.


Is the relation between expenditure and indirect utility in the Mascolell's diagram wrong? In page 75 of Mascolell's microeconomic theory, there is a diagram showing some relations. In particular, there is a relation saying $e(p,u) = v(p, e(p,u))$. I wonder if this is a typo? If not, how to interpret it so that it reconciles with the usual relation $v(p,e(p,u)) = u$?

We can say that $v(p,w)=e(p,v(p,w))$ (or, equivalently, that $e(p,u)=v(p,e(p,u))$ if we are using the "money metric" indirect utility function. You can read about this in section 3.I of MWG. I guess that is what the authors had in mind when they drew figure 3.G.3.

More generally, the relationship between $e$ and $v$ is one of inverses: $e(p,u)$ tells us the mapping $$\text{target utility}->\text{money spent}$$ while $v(p,w)$ tells us the mapping  $$\text{money available}->\text{attainable utility}.$$
So $e$ maps from money to utility and $v$ maps from utility back into money.

In words: if we take a given $w$ and feed it into $v$, we will get an answer to the question "what is the most utility I could ever achieve, $\bar{u}$?" Now, if I take that $\bar{u}$ and feed it into the expenditure function I get an answer to the question "what do I have to spend to achieve utility level $\bar{u}$. That answer will be exactly $w$ (so $e$ 'undoes' the transformation applied by $v$, and vice-versa).

That's just a long-winded way of saying that you can obtain the expenditure function by inverting $v$ in its second argument, or obtain $v$ by inverting $e$. This is made explicit in equation 3.E.1 of MWG.


Recent economics theories that involve differential topology? The original development of general equilibrium theories involved differential topology.

I wonder if there are any recently developed theories, in any field of economic theories, that utilize differential topology (DT).

I've seen some recent papers but none of them used DT. Some real analysis, topological topology, and convex analysis seem enough.

The main reason differential topology had some success in economics is that supplies powerful methods to show that something holds generically, mainly Sard's theorem and the transversality theorem. Some of these methods have been generalized to contexts without differentiability, see for example the paper "A Prevalent Transversality Theorem for Lipschitz Functions" by Shannon. Differential topology is still used in general equilibrium theory, you might take a look at recent work of Yves Balasko.

In positive political theory, there are results to the effect that generically no majority winner exists in multidimensional spatial voting models.  A good introduction to this topic is the book "Positive Political Theory I" by Austen-Smith and Banks, the definitive treatment is the paper "The generic existence of a core for q-rules" by Saari.

Inspired by the work on regular equilibria in general equilibrium theory, regular Nash equilibria were also studied in game theory, starting with Harsanyi. An elegant approach can be found in the paper "The theory of normal form games from the differentiable viewpoint" by Ritzberger (preprint here). Many genericity results in general equilibrium have corresponding versions in game theory, for example, "most" normal form games have a finite and odd number of Nash equilibria.

Another area where differential topology is used is in the study of stability with respect to some dynamics. Indeed, one can define regular equilibria via the replicator dynamic from evolutionary game theory (that is how Ritzberger does it). For a recent example of using some differential topology (index theory, treat also in more generality), see the working paper "The Index +1 Principle" by McLennan.


How to find US total debt? Sorry i'm not really an economist but I don't know where else to ask.
I'm reading Ray Dalio's Big Debt Crises and he has several graphs for total US debt burdens
https://imgur.com/a/tPX3LuH
where it's about 350% of GDP in 2008, but when i can't seem to find source for total debt, only federal debt which seems to only be around 68% at the time
i'd like to find other countries total debt to GDP ratios too if possible. how do i go about this?
Thanks!

The figure provided by that author is widely inaccurate according to the official statistics.
According to the OECD the general government debt to GDP ratio ratio in 2008 was just $102\%$. Moreover, here is comparison to other countries:

Moreover, general government is by OECD defined thusly:

General government consists of central, state and local governments and the social security funds controlled by these units. General government accounts presents data on fiscal balance, debt, revenues, expenditure, costs and reserves of governments. Central government consists of the institutional units making up the central government plus those non-profit institutions that are controlled and mainly financed by central government. The political authority of central government extends over the entire economy. Central government can impose taxes on all residents and non-resident units engaged in economic activities within the country.

So it does include central, state and local government and even non-profits that are mainly financed by public funds.
Hence either there are some caveats that you are not mentioning in your question (perhaps it does not include just government debt but also some private debts or present value of unfunded future liabilities) or the author of the book made a mistake.


How should I define most of the import or export when doing analysis of the import and export of a country? let's say I'm analyzing the import of France and it is 100 billion dollars and is divide between 200 countries. how should I gauge where has most of the import come from? top 10? 90 percent of the import? what should be used as a measure here?

If I understand you correctly, you are asking how to find out which countries' exports have most significantly contributed to a given country's (say, France's) imports.

I have recently completed some research on gravity models of trade, and in the process I ended up having to compile various datasets on precisely the topic you are raising.

The United Nations Comtrade International Trade Statistics Database has an easy-to-use interface where you can download bilateral trade data between a "reporter" country (France) and "partner" countries from Rest Of World.

Here: https://comtrade.un.org/data

Set the relevant year, set "Reporter" $\to$ "France", set "Partner" $\to$ "All", and set "Trade flows" $\to$ "Imports". Then you can download a cleaned .csv file on which you can run basic summary statistics and do some data analysis (such as finding the top-10 country exporters to France, etc.).

Broadly labeling where "most of France's imports come from" is an ambiguous task. Are you referring to a measure that uses a single country or perhaps a continent? It really depends on what your research question/methodology is. Can you provide some more details?


If there is a tax to be paid for every income event, doesn't that imply most money in circulation belongs to the goverment it circulates in? So this is my logic, let me know where/how it is flawed:
Starting from 100 units of currency, lets say 100 coins, someone recives that 100 and has to pay any % of that as income tax, lets say 10%, then he/she goes and spends the remaining 90, which represents the income of someone else, this person recieves 90 and has to pay any % of that as income tax, lets say 20%, keeping 72. This person goes and spend the remaining 72, which again, becomes the income of someone else, and so on.
Isn't it true that over enough transactions, 99% of that initial wealth has been transfered to the tax reciever?
If so, doesn't that mean most wealth in circulation belongs to the goverment it circulates in, as long as it keeps circulating?

No there are several flaws in your argument:

Starting with the most obvious problem. Just because something will be eventually owned by someone does not mean it is correct to say that the thing is owned by that entity already. This is equivalent of stating: "My house will be once inherited by my posterity so I don't own it but they do."  This is flawed from economic perspective. Ownership (property right) is defined as (see Mankiw Principles of Economics 8ed pp 11):

the ability of an individual to own and exercise control over scarce
resources.

The definition might be slightly different in different strands of research, but generally ownership in economic sense is determined by who has exercise/control over something. Until government actually seizes the ownership of the thing it from economic perspective does not own it. Furthermore, I am not a lawyer but I am going on a limb here to say I doubt this would even hold in legal sense, so the assertion is flawed prima facie.

Next, typically government does not just hoard resources it gets (in your example currency), but also spends it. Since you are building theoretical model you are free to make assumption that government never spends its resources but that assumption does not hold in real world so you cant automatically claim conclusions from your model carry there. Models do not have to be 100% realistic (what even more they should not even attempt to do that - they are models) but in your case assumption of 0 government spending is the primary assumption making the argument, but empirically it is clearly invalid - that is serious flaw in this case.

Even if we forget about the point 1 and 2, you are making claims that go way beyond the model you use. In your model government taxes currency. It would be true that if there is fixed amount of currency and government taxes it at each point when it is through spending turned into income, then eventually in the limit government would own all currency.

However, currency, or money more broadly,  in itself is neither income nor wealth - it is way how we measure income or wealth. Income is generated by production. If you produce two widgets, value those widgets are your income. Money that you will get from selling those widgets measure (and store) value of those widgets. If government would tax away all currency people would just revert to barter economy where they would exchange goods and services for another goods and services and those would be peoples' incomes. Government would own all currency but not peoples' incomes.
Now this is not just an attempt at  'gotcha' - this is important distinction. If government would be taxing goods and services in-kind, then government might never get hold of all of them as people can always produce more and more goods and services ad infinitum.
In addition, you have to be clear with your terminology. Income is not wealth. Wealth is defined as net value of your total assets which are accumulated by saving. If you spend your income it is not wealth and consequently saying: " most wealth in circulation belongs to the goverment it circulates in, as long as it keeps circulating?" Is incorrect. Because what is circulating in your example is income not wealth.
Lastly,  the same caveat that I mentioned about income applies to wealth. Wealth is not simply money. Person with no money but ownership of billion dollar company has more wealth than a person with full wallet of banknotes.


How are final stock prices arrived at? 'Everybody' knows that prices are generally governed by offer & demand, but I am wondering on the details of the stock exchange here.

How are stock prices (the ones which are nicely listed every day in charts) are actually computed to a specific number based on various simultaneouse bids?

Do they change, if no transaction is actually taking place (but there are bids)?
And how are concurring bids dealt with?

Its really quite simple, the stock price you're seeing is actually the last trade that took place, the different bids are tails of the suppliers and demanders - their maximal or minimal bids

In everyday trade the simultaneous bids arr very close, and the deals are happening all the time, more or less bids in a certain direction can point out market expectations, but in a range of minutes
This applies to high volume markets, of course


How do you show welfare changes with limited entry versus perfect competition? In Perloff's intermediate microeconomics book, he explains that producer surplus will increase with policies that limit entry. I get this intuitively - if we limit competition, producers will earn rents in the long run - but I am not following the graphical explanation. The text reads as follows:
"Although government policies may cause either the supply curve or the demand curve to shift, we concentrate on policies that limit supply because they are used frequently and have clear-cut effects. If a government policy causes the supply curve to shift to the left, consumers make fewer purchases at a higher price and welfare falls. For example, if the supply curve in Figure 9.3 shifts to the left so that it hits the demand curve at e2, then output falls from Q1 to Q2,
the price rises from p1 to p2, and the drop in welfare is −C − E. The only “trick” in this analysis is that we use the original supply curve to evaluate the effects on producer surplus and welfare."
Can someone explain to me the intuition behind this "trick"? There is nothing provided in the text. I don't understand why we wouldn't use the "new" supply curve to calculate welfare.
To add in for reference, in a previous version of the textbook Perloff uses the following graph as an example. This graph makes sense to me. The new one does not given what is written unless it assumes a vertical supply curve.

I think calling this 'trick' is a bit of a misnomer because there is nothing particularly tricky about it.
When you make any welfare analysis of some change (tax/quota etc.) you need to compare situation after change with situation before change. For that reason you calculate the deadweight loss (DWL) of policy as a difference between consumer surplus before policy and after policy.
In your case the problem (including new restricted supply $S_R$ - painted in red color) would look like in the following picture I made (I linearized the problem in order to make the picture easier to render).
In order to calculate deadweight loss here you would first calculate the area of big triangle bounded by the original supply and demand (in black) and then subtract the area of the trapezoid that is created by the new restricted supply (in red).
I am surprised they call it a trick in that textbook since this is how DWL from a policy is defined. Following Mankiw Principles of Microeconomics 8th ed. pp 157:

Deadweight Loss: the fall in total surplus that results from a market distortion, such as a tax

To calculate what a fall in CS is you need to either implicitly or explicitly take into account what the original situation was. One way of how to calculate this explicitly is to see what the original supply and demand was and comparing it to the new situation.


How do the wages affect the employment If we have a condition where employment is lead by the wages does it mean that a wage reduction will cause unemployment?

Are you referring to real life circumstances or in a hypothetical world? There is an interesting article about the correlation between wages and unemployment found here: https://www.stlouisfed.org/On-The-Economy/2014/December/Are-Wages-and-the-Unemployment-Rate-Correlated

Essentially this article states that higher levels of wages are weakly correlated with the unemployment rate based on data that was compiled and published in The Regional Economist. They also argue that when business conditions shift, it impacts both the unemployment rate and wages.


Relaxing the notion of Nash Equilibrium Consider a game with $N$ players, each indexed by $i=1,...,N$. Every player $i$ has to choose a $J\times 1$ vector of actions $a_i\equiv (a_{i,1},...,a_{i,J})$ where each $a_{i,j}$ can be zero or one. The payoff of each player $i$ is $u_i(a_i, a_{-i})$, where $a_{-i}$ denotes the actions of the other players.
As a first best, I would like to show that this game has a pure strategy Nash equilibrium (PSNE). I'm unable to do so and I also don't want to allow for mixed strategies. As a second best, I would be happy to consider a weaker notion of equilibrium (still, in pure strategy) and show its existence.
In particular, by revealed preference, if $(a_1,...,a_N)$ is a PSNE, then the following holds:
$$
\forall i=1,...N \quad \forall j=1,...,J: \quad \text{if } a_{i,j}=1\text{, then } u_i(a_i, a_{-i})\geq  u_i(a_{i, \{-j\}}, a_{-i})\\
\hspace{6.5cm} \text{if } a_{i,j}=0\text{, then } u_i(a_{i}, a_{-i})\geq  u_i(a_{i,\{+j\}}, a_{-i})$$
where $a_{i, \{-j\}}$ denotes $a_i$ where $a_{i,j}=1$ is replaced by $a_{i,j}=0$; $a_{i, \{+j\}}$ denotes $a_i$ where $a_{i,j}=0$ is replaced by $a_{i,j}=1$.
Question: Is there any equilibrium notion in pure strategy such that:

it is weaker than PSNE

it implies the revealed preference inequalities reported above

its existence is typically easier to be shown (and, if you could give references on this)

No, there is not. Consider a game with two players, Ann and Bob. Both choose such vectors with entries $0$ or $1$ of the form $(a_1,a_2,\ldots,a_J)$ or $(b_1,b_2,\ldots,b_J)$, respectively. If $\sum_{i=1}^J a_i+b_i$ is odd, Ann wins and Bob loses. If the number is even, Ann loses and Bob wins. Clearly, one of them loses in every profile of pure strategies, and the loser can win by changing only one entry of their chosen vector. So there is no pure strategy profile at which the revealed preference inequalities hold.


Why does the nominal interest rate equal 25% in this problem, instead of 16.4%? My reasoning: The nominal interest rate = the real interest rate + the expected inflation rate. We need to calculate the nominal interest rate. We know the expected inflation rate, so the only problem is to calculate the real interest rate. In order to calculate the real interest rate we need to see how much money received is more than money given. And due to inflation we need to adjust the initial sum to inflation. 110 - 100*1.03=7, this is inflation-adjusted amount of money by which Daenerys will become richer. In order to calculate the real value we need do following: 7/110=0.0(63), which is about 6.4%. Thus the nominal rate equals 6.4+10=16.4

Here is the offical answer, but I'm afraid it doesn't make sense:

Keep in mind that nominal interest rate=Profit/Initial Amount.

The exact relationship that links real interest rate, nominal interest rate and inflation is $(1+R_{nominal})=(1+R_{real})(1+\pi)$ where $\pi$ is the inflation rate (Known as Fisher's identity)

$R_{nominal}=R_{real}+\pi$ is an approximation that works best when those variables are relatively small.


What is the opposite of public debt? The opposite of a budget deficit is a budget surplus
But what is the opposite of public debt?
Instead of govt/state being in debt, having the government have extra money to keep or to lend it to banks or companies.
Say the extra money are kept Instead of being lent. What is the name of such a surplus?
It would be the sum of the budget surplus for the curent year and all the years before.

Such government would be called net lender. However, note that for a country to be net lender it is not required that debt is zero, only that the amount it is lending is higher than its liabilities (debt). A country can chose to have some level of debt even if it is not necessary for it to have it because its revenues are higher than expenses. An example of net lender would be Norway (see here).


How is capital outflow calculated? I would like to understand the meaning of the following phrase:

  Capital flight from Russia in 2014 was 151 billion dollars.

First, I would like to know how such a calculation can be done. Secondly, I would like to know what the underlying mechanism of capital flight from Russia is. For example, does this mean approximatively the following: Russian companies sell Russian gas and oil and then instead of investing this money domestically, they invest it outside?

The Central Bank of Russia (CBR) gives tons of statistics. Take a look at "Net Inflows/Outflows of Capital by Private Sector" and “Balance of Payment” in different representations. I warn you ahead that it’s not that clear from the data what moves where.

There are several reasons for capital flight:

Dollarization and flight to safety. That is related to high inflation and bank runs over the past 25 years, after the collapse of the USSR. Russians do not fully trust the ruble or the local financial system. A lot of Russians hold foreign cash, mainly dollars and euros.
Sanctions. Because of the Ukraine issues, the West put sanctions and effectively closed its financial markets. So, Russian banks and enterprises cannot refinance and have no way but to pay debts. Paying debts is like capital flight.
NX = S – I. Russia has chronic CA surplus, which means capital outflow. It lasted for years. See this. If you have insufficient investment demand inside country for its savings that means capital outflow. That is related to point 1 also.


Why the GDP does not decrease even when we are saving from our income? When we are saving from our income then we don't consume as mush as we earn. So the GDP should decrease as well as the income but this does not happen. Why?

"GDP" stands for "Gross Domestic Product". It measures the income generated, prior to any consumption-saving allocation. In other words, to be able to save you must first generate product/income and then decide to save. So savings does not affect the level of the already created income, since it is a part of it. The archetypal closed economy with no borrowing is characteriazed by the identity

$$Y = GDP = C + I = C + S$$

When we look the matter from an expenditure approach, we essentially consider the different uses of income (and in that sense the use of the term "expenditure" may be a bit misleading). And savings is also a way to use your income (where "use" has a broader meanign than "spend").


Portfolio choice of a risk lover Take the standard portfolio choice problem as presented in MWG (p.188-189), but with a risk loving decision maker:

with initial wealth  $w$
invests an amount $\alpha$ in a risky asset with a random gross rate of return $z$, where $z$ is distributed according to cdf $F$, and $\int zdF(z)>1$

The decision maker solves the following problem
$$
  \max_{\alpha\in[0,w]} \int u(w-\alpha +\alpha z) dF(z),
$$
where $u'>0$, $u''>0$.

Is it true that the optimal solution is $\alpha^*=w$?

Note that since $u$ is convex, the usual first order condition is not sufficient for a maximum.

Intuitively the answer should be yes. But what's the reason, exactly?

Since we are suspecting a corner-solution, it is better to write the problem explicitly with its constraint. Even better, use the Fritz John (FJ) conditions rather than the Karush-Kuhn-Tucker (KKT) ones. We will mention the differences as we go along.

$$\max_{\alpha} \int u[w+\alpha(z-1)] dF(z),\;\; \text{s.t.}\;\; w-\alpha  \geq 0$$

The lagrangean under the Firtz John formulation is

$$L_{FJ} = \lambda_0\int u[w+\alpha(z-1)] dF(z)\; + \;  \lambda_1(w-\alpha)$$

The new element is the multiplier on the objective function, $\lambda_0$. Without loss of generality we can specify

$$\lambda_0 \in \{0,1\},\;\; \lambda_0 + \lambda_1 \neq 0$$

What is that we gain here, compared to the much more widely known and used KKT-conditions?
If a solution necessitates that $\lambda_0 =1$, we obtain the KKT conditions with the constraint qualification satisfied. If a solution necessitates that $\lambda_0 =0$, it reflects, among other special cases, the case where the constraint qualification fails to hold.

  (A standard example is the case where the feasible set for $\alpha$
  has been reduced to a single point due to the constraints imposed.
  Then we will find that the only solution dictates that $\lambda_0=0$,
  which has an intuitive explanation: if $\alpha$ can take one and only
  one value due to the constraints, then the objective function "plays
  no role" in the determination of $\alpha$ and so it gets a zero
  multiplier).

Back to our problem. The first order condition is

$$\frac {\partial L_{FJ}}{\partial \alpha} = \lambda_0\int u'[w+\alpha(z-1)]\cdot (z-1) dF(z) - \lambda_1 \leq 0$$

(note the "lower or equal to zero" which is the case when optimizing under inequality constraints, rather than just "equal").

First, we establish that $\alpha^* >0$. Due to $u'>0$ and the (strict) convexity of the utility function,$u''>0$, and the assumption that $E(z) >1$ we have (using Jensen's Inequality)

$$E[u(w+\alpha(z-1))] > u(w+\alpha(E(z)-1))] > u(w + 0\cdot(E(z)-1))=u(w)$$

We turn now to examine cases.  Since the two multipliers cannot be both zero, and $\lambda_0$ takes only two values, there are three possible combinations.

Examine the case $\lambda_0 =1$.
Then $\lambda_1$ can in principle be zero or positive. Examine the case where $\lambda_1 =0$, i.e. the constraint is not binding which implies that $\alpha^* < w$.  With this candidate pair of multipliers, $\{\lambda_0=1,\lambda_1=0\}$, the first order condition would become

$$\int u'[w+\alpha(z-1)]\cdot (z-1) dF(z) \leq 0 \Rightarrow E(zu') - E(u') \leq 0$$

Since $u'>0 \Rightarrow E(u') > 0$. Also, since $E(z)>1$ we have that

$$E(u')< E(u')E(z) \Rightarrow E(zu') < E(u')E(z) \Rightarrow \text{Cov}(z, u')<0$$

But this cannot hold,  because, since $\alpha^*>0$ and $u''>0$, we have that $u'$ will be strictly increasing in $z$. So the covariance of $z$ and $u'$ cannot be negative. But then the pair of multiplier values $\{\lambda_0=1,\lambda_1=0\}$ cannot be a solution, and this happens due to the assumption $u''>0$.

We are left with the cases
$\{\lambda_0=1,\lambda_1>0\}$, or $\{\lambda_0=0,\lambda_1>0\}$. In both these cases, $\lambda_1 >0$ i.e. the constraint is binding , i.e. we will have $\alpha^* = w$. QED.

  References

The Fritz John conditions have been stated in
"F. JOHN. Extremum problems with inequalities as side conditions. In “Studies and Essays, Courant Anniversary Volume” (K. O. Friedrichs, O. E. Neugebauer and J. J. Stoker, eds.), pp. 187-204. Wiley (Interscience), New York, 1948"

and have been generalized in

"Mangasarian, O. L., & Fromovitz, S. (1967). The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. Journal of Mathematical Analysis and Applications, 17(1), 37-47.