question about derivatives and net exposure

The main roles of banks are taking deposits, transmitting and changing money, making loans, and perhaps more generally providing liquidity for their customers.

So if a customer wants to change yen for dollars, the bank does not consider whether its buying yen would be a good medium-term investment for the bank.  Instead it considers whether it can sell the yen back to dollars in the short-term at a better price either to other customers or in the wholesale market.  It sets the price (and any fees) to the original customer so that it can expect to make a margin on the overall transaction, at little or no risk to the bank.

Similarly with taking deposits and making loans, the profit for the bank is in the margin between the two interest rates (allowing for credit defaults), and the bank earns this because it is cheaper and more convenient for the customers than trying to act without the bank as an intermediary.  And if a customer wants to buy or sell bonds immediately, the bank acts as a dealer with the aim of re-selling those bonds to or buying them from somebody else at another time with a profit from the margin rather than from possible future changes in value.

With derivatives, if say a customer wants to buy a call option, then the bank may write one, not with the aim of taking the full premium and hoping things turn out right but with the aim buying another or of other balancing transactions (e.g. a put and the underlying security) and taking a margin based on providing the customer with an immediate contract in what otherwise may be an illiquid market.

The best case for the bank is that it profits from being an intermediary in these transactions, not as a speculator taking a view on how markets will move.  If it gets into an unbalanced position overall, the bank will want to hedge its position to reduce risk, as this makes it more likely that the bank will survive to make profits in the future.


Show that First order conditions are necessary and sufficient for utility maximization

Consider the following situation :
Preferences of the consumer are represented by the utility function $u(x_1, x_2) = x_1 + x_2$. Note that preferences are strictly monotonic and convex. Now let income be any $y > 2$. So the consumer's problem is :
\begin{eqnarray*} \max_{x_1, x_2} & x_1 + x_2 \\ \text{s.t.} &  \ \ 2\sqrt{x_1} + x_2 \leq y \\ & x_1\geq 0, \ x_2 \geq 0 \end{eqnarray*}
Now let us set up the Lagrangian :
\begin{eqnarray*} \mathcal{L}(x_1, x_2) = x_1+x_2 - \lambda(2\sqrt{x_1} + x_2 - y) +\mu_1x_1 + \mu_2x_2 \end{eqnarray*}
Here are the first order necessary conditions:
\begin{eqnarray*} \dfrac{\partial\mathcal{L}}{\partial x_1} & = & 1 - \dfrac{\lambda}{\sqrt{x_1}} + \mu_1= 0 \\ \dfrac{\partial\mathcal{L}}{\partial x_2} & = & 1 - \lambda + \mu_2= 0 \\ \dfrac{\partial\mathcal{L}}{\partial \lambda} & = & y -2\sqrt{x_1} - x_2  \geq 0, \ \lambda\geq 0, \ \lambda(2\sqrt{x_1} + x_2 - y) = 0 \\ \dfrac{\partial\mathcal{L}}{\partial \mu_1} & = & x_1  \geq 0, \ \mu_1\geq 0, \ \mu_1x_1 = 0 \\ \dfrac{\partial\mathcal{L}}{\partial \mu_2} & = & x_2  \geq 0, \ \mu_2\geq 0, \ \mu_2x_2 = 0 \end{eqnarray*}

Check that one of the solutions to the above set of conditions is $x_1^* = 1, \ x_2^* = y -2, \ \lambda^*=1, \ \mu_1^* = 0, \ \mu_2^* = 0$,

But it is not the solution to the utility maximization problem stated above. This is because utility from $(x_1^*, x_2^*) = (1, y-2)$, i.e. $u(x_1^*, x_2^*) = 1 + y - 2= y-1$, is less than the utility from another affordable consumption bundle $(x^{**}_1, x^{**}_2) = (0, y)$ as $u(x_1^{**}, x_2^{**}) = 0 + y = y$ and $2\sqrt{x_1^{**}} + x_2^{**} =y$.

Therefore, first order conditions are not sufficient for utility maximization.


How does capital investment (factory, plant, etc.) benefit the region they are located in?

There are more effects for economy then just what you mentioned.

  For example, if a car factory is built in a certain state, does it improve the economy solely by creating jobs and generating car sales thereby increasing the regional GDP?

Not only because of this there are also other effects. Depending on what kind of investment it is it can support technology transfer between the countries or also increase stock of human capital (through on job training/learning by doing), which according to endogenous growth theories can accelerate economic growth itself (although it is worth noting endogenous growth theory is still not completely accepted by profession even though Romer who developed it even received Nobel Prize for it because it’s hard to test).

Moreover, apart from such dynamic growth effects there are more beneficial macroeconomic effects on GDP. From macroeconomic perspective all spending including investment has a multiplier attached to it. For  when the factory increases GDP by the investment people will also have more money to spend increasing GDP even further and so on. From macroeconomic perspective your spending/investment is someone’s else income so if you increase your spending/investment other people will spend more as well.

Also aside from macroeconomic effects any new job openings put upward pressure on wages, however depending on how many jobs are offered and what is already situation in the market this effect might be very small.

  if a factory (lets say it’s located in state A) ONLY ships and sells its products to state B, would it improve state A’s economy as much as if sales were only made in state A?

Not really as measured by GDP. The GDP identity is given by:

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

Where C is consumption I investment G gov. spending, X export and M import.

So even just looking at the identity it is possible to see that if the cars are not consumed but exported GDP does not change.

However, there is even more to it. Trade theory in economics tells us that there are both static and dynamic gains from trade. Hence to the extent that the cars are exported because in state A they can be produced more efficiently then in state B, and in exchange state B trades to state A some of its products that they can produce more efficiently there will be both static one time increase in the level of their welfare from specialization, and also potentially some dynamic effect assuming that the longer the state specialize in the good they have comparative advantage in the better they become at its production.


Yes if you would want to calculate it per capita basis then you would just divide by population of the country/monetary union for which you are calculating it in the same time period.
Per capita literary means per head, and whenever any quantity is calculated per capita terms whether its gross domestic product, gross national product, gross value produced, income and so on they are calculated as a ratio of total of the aggregate to population.
This being said I never seen this being done in research and I could not find any reference in google scholar as well. I was able to find this blog that does it although not for any research purpose. When we calculate aggregates on per capita basis its usually for some reason, for example per capita income gives us better idea about levels of material welfare in a country than just looking at total income, with money supply that is not really the case.


Difference between a multiple regression and several linear regressions?

The data might be correlated which can lead to simple and multiple linear regression models giving different results.

Which goes against the assumption of no multi-collinearity:
Multicollinearity occurs when independent variables in a regression model are correlated.

When modeling years of education to wages, the results may show significance but modeling both years of education and test scores could show years of education as not significant because years of education and test scores are correlated.

Try  plotting years of education and test scores.


Comparative Statics on Balanced Growth Path

The approach is not correct, because $\mu^{BGP}$ is a result of underlying structural parameters. So to say "when $\mu^{BGP}$ increases..." immediately begs the question why it increases, which underlying parameter(s) has changed to cause such an increase... assume it was $\epsilon$ that decreased. But in this case $g_k$ is not affected at all, so you see that you cannot claim what you claim about  $\mu^{BGP}$ affecting positively $g_k$.

What is meaningful to do is to perform comparative statics for the exogenous parameters, and record whether $\mu^{BGP}$ and $g_k$ move in the same or the opposite direction (or don't move) in each case. It may not be a causal link, but co-movement is also a useful result, and testable.


Fixed-income investment in a low interest-rate environment

Your assumption that low interest rates will "eventually" have to rise is not warranted.

Indeed, we are currently witnessing prolonged periods of low interest rates that do not obviously seem to be coming to an end.

Example 1. For 7 years, the US FOMC's target federal funds range remained unchanged at 0–0.25% (from Dec 2008 through Dec 2015). From Dec 2015 through Oct 2018, there have been 8 increases so that it is now 2.00-2.25%. But even after 8 increases, this rate range remains historically low.

Example 2. A more dramatic example is the case of Japan, where the 10-year government bond yield has remained closed to 0% for two decades now (and indeed is under 0% now).

You predict that since interest rates are so low now, they must rise. You might very well turn out to be correct, but this is hardly a certainty and there are many others taking the opposite bet.


Using the Dividend Discount Model when no current dividend exists?

With what you say and using the DDM we know that

$P_{t=4} = \frac{D_{t=5}(1+g)}{r-g}$

Via the above equation, it is assumed that from year $5$, dividends will annually grow at a $g$ rate and that investors' required rate will be $r$, both forever. Since this calculation is nothing but a simplification (future growth and return rates will a fortiori change, etc...), we will not do more complicated.

Taking into account the share of debt in the capital structure of the company, say $X\%$, we can compute the average cost of capital (average over equity and debt), $r'$, needed to discount $P_{t=4}$. One may have an idea of $X$ or make assumptions about it. It follows that $r'$ is such that

$r' = X\% r_D + (1-X\%) r$

where $r_D$ is the interest rate or the cost of debt (naturally $r_S < r$). Thus the company's IPO may be valued by involving that

$P_{t=0} = \frac{P_{t=4}}{(1+r')^4}$

Finally, as you can see there are many assumptions in this valuation, which is the job of experts, with the most fundamental about $D_{t=5}$.


Fixed effects, first differences interpretation

The biggest difference between (1) and (3) is that (3) has incidental linear trends while (1) has common time effects. Differencing (1) gives

(1a) $\Delta \log(uclms_{it}) = (\theta_t - \theta_{t-1}) + \beta_1 \Delta ez_{it} + \Delta \epsilon_{it}$.

Comparing this to (3) you see that (3) has $\gamma_i$ in place of $\theta_t - \theta_{t-1}$ in (1a). So three things are different: (A) As you said, it is FE vs FD; (B) the trend is free (using dummies) in (1), while linear in (3); (C) the trend is common in (1) but is individual-specific in (3).

It would be interesting how much difference is introduced by each of these three points.

About why FD is useful, well it is simple and informs us about functional relationship among variables in differences. So I'd say why not.


Books friendly to self-studying Industrial Organization

While taking Industrial Organization I remember working with:

Strategies and games: theory and practice by Dutta
Introduction to industrial organization by Cabral
Industrial organization: theory and applications by Shy
Industrial Organization: Markets and Strategies by Belleflamme and Peitz

The first two are rather introductory while third and forth are more advanced (getting into greater detail with a little bit more advanced maths). You can probably find them online if you try.

And on top of that, I really enjoyed The Economics of industrial organization by Shepherd. It is way older than the others but still made quite a good read.


Are there any effective research efforts into combating wealth inequality?

There is a lot of research into this area. You might want to look into the papers of Angus Deaton, Thomas Picketty, Gabriel Zucman, Emanuel Saez or Branko Milanovic for starters.

Especially, Emanuel Saez directly focuses on optimal taxation under different social preferences - including utilitarianism and ralwsianism where reducing inequality matters.

However, most research is on income inequality because wealth inequality is incredibly difficult to measure or track.

You might want to start with this paper and papers cited therein:

  Piketty, T., Saez, E., & Zucman, G. (2013). Rethinking capital and wealth taxation. Arbeitspapier, Paris School of Economics, UC Berkeley und London School of Economics.

PS: Also this is tangent to your question but I would not be so hard on politicians - even assuming that they are completely enlightened social planners in many completely valid moral systems inequality does not matter. Reducing inequality does not necessary means reducing poverty if we measure it in absolute sense and not relative.

This is also a main reason why there is no single agreement on the issue. Combating inequality is not just economic question but question for moral philosophers as well and right answer will vary depending on your personal values and morals


How do you find marginal revenue if you don't have a production function?

Here are some hints. Either solve the profit function in terms of $p$, substituting out all references to $q$ (recall that $D(p) = q$), or solve the profit function in terms of $q$, substituting out all references to $p$. Then you'll find that the profit function is concave and the differentiated with respect to $p$ in the first case or $q$ in the second case will give you the profit maximizing solution.

EDIT: Yeah, $a = 2q + 10$ looks correct. Just solve the equation for $q$ and the interpretation is clear. That is, the quantity produced is $q = (a - 10)/2$. If you want to know the price, then just substitute $q$ in for $D(p)$ like so: $a - p = D(p) = q = (a - 10)/2$.


How would a foreign trade multiplier behave if imports depended only on exchange rates?

If the net exports depended only on the exchange rate then there would not be any induced effect of net exports on income (as exports would not depend on income) hence, no multiplier effect. The other multipliers in economy would not be affected either.


Rationale behind teaching History in University

To understand why some countries are rich and others are not, we need to understand what happened to each of them in the past. How and why they developed depends to a large extent on events that took place decades or even centuries ago. This is what economists broadly refer to as path dependence. Brian Arthur was probably one of the first economists to point that out in the late 1980's.

Recently this type of arguing was appreciated again by mainstream economics. There is a growing number of scholarly articles that show how and why some countries developed the way they did. With huge implication for future economic development. Two good summaries are Nunn - The Importance of Economic History for Economic Development and Nunn - Historical Development. So the conclusion would somewhat be like that one can't ignore history if she wants to understand why some countries are poor. And more generally why some things are the way they are (one prominent example was the somewhat random event of why we have a QWERTY keyboard system that is a lot slower than other potential alignments of letters on a keyboard. It was determined once by "luck" let's say, and we never switched back...)

So all in all I would argue that knowing (and potentially understanding) some history is essential to understanding how our world works. It's certainly nothing that you can sell as a key feature in your CV (like a specific programming skill), but knowing sufficient history would certainly alter anyone's view of the world. And in order to not repeat huge mistakes that were made in the past (as a "society"), we actually need to know what they were and how and why they came about.


Examples of Applied Micro Paper with R (!) Code and Data in Public Repository

You could try searching the Harvard Dataverse for fileType:"R Data" like this: https://dataverse.harvard.edu/dataverse/harvard?q=fileType%3A%22R+Data%22

I think your use case of wanting to search for data in specific formats such as RData is a common one so I just created an issue about improving Dataverse to support this use case better: https://github.com/IQSS/dataverse/issues/2707


Pros and Cons of Elasticity of Demand Represented as an Absolute Value

Elasticity of demand is almost always assumed to be negative. It is assumed that there is an inverse relationship between price and quantity demanded. If you want to buy 2 hamburgers for $5 each you aren't suddenly going to want to buy 3 hamburgers if they now cost $10 each. If this isn't the case it means that consumers aren't rational which blows up 99% of the models economics has to offer.

Because the elasticity of demand is always negative it can be assumed that if I say X is the elasticity of demand I mean X is negative because elasticity of demand is always negative. If I say -X is the elasticity of demand you can still assume that it is negative because a) I said it explicitly, and b) it is always true.

At this point it's just a matter of preference and doesn't matter either way, I just wanted to explain why it really doesn't matter since someone who understands what elasticity of demand means will understand that it will always be negative in almost all scenarios.

Personally, I think that explicitly saying that the elasticity of demand is negative is a better choice since I can imagine some fringe scenarios where a product being expensive leads to that product being sold more often, (I think of caviar as an example for these types of products, where consumers buy it precisely because its expensive) but for most economic models the assumption holds.


Optimal consumtion bundle of lemons and sugar

Alex's preferences of sugar and lemons can be expressed in form of a utility function as:

$U(x,y)=min(x/2,y) $

where $x$ is sugar and $y$ is lemons.This function tells us we need at least 2 spoons of sugar to consume 1 lemon.

we need a minimum of 2 sugars initally so we can say $x=2$ for 1 lemon so $y=1$

Plugging those values into our  budget constraint:

$44=20x+y$

$44=20(2)+(1)$

$44>41$

Notice how we end up having 3 left over, however with the budget constraint and preferences provided our consumption bundle will be 2 sugars and 1 lemon.


How, with complete markets, do we have a representative agent?

It depends on what you want from a representative agent. What you get is a representative agent at the given prices. For any two commodities, you have relative prices. This is where market completeness comes in. All you have to do now is find a well-behaved utility function whose marginal rate of substitution between the commodities corresponds to the relative prices. You can even take the utility function to be linear:
$$u(x_1,x_2,\ldots,x_l)=\sum_{i=1}^l x_i/p_i.$$
Of course, the resulting "representative agent" is utterly useless for any comparative statics exercise or any welfare analysis.


Monotonicity of second price auctions with bidder-specific reserve prices

Some thoughts. If this was a SPA without a reserve price, the monotonicity of the allocation rule is trivial. When the reserve price is uniform across bidders then the allocation rule is clearly weakly monotonic now; since for low bids, increasing them might not change the probability of allocation. The only thing that makes this problem different is that the reserve prices are bidder-specific. So one possibility is that you only need to show that the logic for the previous monotonicity is still valid in this setup. Probably while trying to show that the logic still goes through you will find out if there is some extra argument you will have to make in order to still have the monotonicity.

For example, I might worry that by increasing my bid I might increase my own reserve price, or decrease the reserve price of some other bidder and thus reduce my probability of being allocated the good.


Help needed to understand proof in Myerson and Satterthwaite (1983)

I'm assuming by "at [the] second equality sign" you mean the translation of the right-hand side of the first equality sign to the right-hand side of the second equality sign. This can be handled by analyzing each integral separately.

Notice that in the first (double) integral the left-hand side of the inner integral does not vary with $v_2$, so

$$\int_{a_1}^{\min\{v_2,b_1\}} \left[ v_2 f_2\left( v_2 \right) + F_2\left( v_2 \right) - 1 \right] f_1\left( v_1 \right) dv_1 \\\hspace{3cm}= \left[ v_2 f_2\left( v_2 \right) + F_2\left( v_2 \right) - 1 \right] \int_{a_1}^{\min\left\{ v_2, b_1 \right\}} f_1\left( v_1 \right) dv_1.$$

The upper bound of the right-hand integral is just there to cap the CDF, so $\int_{a_1}^{\min\{ v_2, b_1 \}} f_1( v_1 ) dv_1 = F_1( v_2 )$, taking as given that $F_1( x ) = 1$ for all $x \geq b_1$. In terms of the original equation this gives

$$ \int_{a_2}^{b_2} \int_{a_1}^{\min\{v_2,b_1\}} \left[ v_2 f_2\left( v_2 \right) + F_2\left( v_2 \right) - 1 \right] f_1\left( v_1 \right) dv_1 dv_2 \\\hspace{3cm}= \int_{a_2}^{b_2} \left[ v_2 f_2\left( v_2 \right) + F_2\left( v_2 \right) - 1 \right] F_1\left( v_2 \right) dv_2. \hspace{2cm} \text{(Term L)}$$

Notice that the inner integrand in the second (double) integral can be written as

$$v_1 f_1\left( v_1 \right) + F_1\left( v_1 \right) = \frac{d}{dv} \left[ v F_1\left( v \right) \right]_{v = v_1}.$$

Since $a_1 F_1( a_1 ) = 0$ (because $F_1( a_1 ) = 0$) and $x F_1( x ) = x$ for all $x \geq b_1$, this gives

$$\int_{a_2}^{b_2} \int_{a_1}^{\min\left\{ v_2, b_1 \right\}} \left[ v_1 f_1\left( v_1 \right) + F_1\left( v_1 \right) \right] dv_1 f_2\left( v_2 \right) dv_2 \\ \hspace{3cm} = \int_{a_2}^{b_2} \min\left\{ v_2 F_1\left( v_2 \right), b_1 \right\} f_2\left( v_2 \right) dv_2. \hspace{2cm} \text{(Term R)}$$

The second equality is $\text{(Term L)} - \text{(Term R)}$.


Relationship among universal type space, Aumann's semantic knowledge model and Samet's syntactic knowledge model

There is no "universal Aumann model", as shown in Heifetz & Samet, GEB 1998, "Knowledge Spaces with Arbitrarily High Rank
", even though there is a universal type space.

On a less technical level, Aumanns model does not allow for wrong beliefs. A generalization, so called Kripke structures, do however.


Dynamic Optimization: What if the second order condition does not hold?

There is not a single answer, it will depend on the particulars of each problem. Let's look at a standard example.

Consider the benchmark intertemporal optimization problem for the Ramsey model

$$\begin{align}
&\max_u \int^{\infty}_0{e^{-\rho t}u(c)dt}\\
\\
& \text{s.t.}\;\; \dot{k} = i-\delta k\\
& \text{s.t.}\;\; y = f(k)=c+i
\end{align}$$

The current value Hamiltonian is

$$\tilde H = u(c) +\lambda [f(k)-c-\delta k]$$

Maximizing over $c$ alone we have

$$\frac {\partial \tilde H}{\partial c} = u'(c) - \lambda =0 \implies u'(c^*) = \lambda \implies c^* = (u')^{-1}(\lambda)$$

and the 2nd-order condition will hold if the utility function is concave,
$$\frac {\partial^2 H}{\partial c^2} = u''(c^*) < 0$$

Moreover, from the first-order condition with respect to consumption, $\lambda >0$ if local non-satiation holds. Assume that we do have such "usual" preferences.

The maximized over consumption Hamiltonian is

$$\tilde H^0 = u[(u')^{-1}(\lambda)]+\lambda [f(k)-(u')^{-1}(\lambda)-\delta k]$$

The partial derivatives with respect to the state variable, $k$ are

$$\frac {\partial \tilde H^0}{\partial k} = \lambda[f'(k) - \delta],  \;\;\;\; \frac {\partial^2 \tilde H^0}{\partial k^2} = \lambda f''(k)$$

So here, the Arrow-Kurz sufficiency condition boils down to whether the marginal product of capital is decreasing, constant, or increasing (which will depend on the sign of the second derivative of the production function). In the standard case $f''(k) < 0$ and we have the sufficient condition.

In the most famous case of deviation, Romer's $AK$ model that initiated the Endogenous Growth literature, $f''(k) =0$, and the marginal product of capital is a positive constant.

So what can we say in this case?

Here,
Seierstad, A., & Sydsaeter, K. (1977). Sufficient conditions in optimal control theory. International Economic Review, 367-391. provide various results that can help us.

In particular, they prove that if the Hamiltonian is jointly concave in $c$ and $k$, it is a sufficient condition for a maximum.
The Hessian of the Hamiltonian is

(we can ignore the discount term)

$${\rm He}_H = \left [ \begin{matrix} u''(c) & 0\\
0 & \lambda f''(k)\\
\end{matrix} \right]$$

In the standard case with $u''(c) <0, \; f''(k) <0$ this is a negative definite matrix and so the Hamiltonian is jointly strictly concave in $c$ and $k$.

When $f''(k) =0$, checking that the matrix is negative-semidefinite is straightforward using the definition. Consider a vector $\mathbf z = (z_1, z_2)^T \in \mathbb R^2$ and the product

$$\mathbf z^T{\rm He}_H\mathbf z = z_1^2u''(c) \leq 0$$

this weak inequality holds $\forall \mathbf z \in \mathbb R^2$, and so the Hessian is jointly concave in $c$ and $k$.

So in the $AK$ model of endogenous growth, the solution is indeed a maximum (subject to the parameter constraints needed for the problem to be well-defined of course).


Econometrics: why the i.i.d assumption and weak exogeneity assumption imply strict exogeneity?

\begin{align*}
\mathbb{E}\left[ \epsilon_t \vert X \right] &= \mathbb{E}\left[\epsilon_t \vert X_1, \dots, X_t, \dots, X_n \right]\\
&= \mathbb{E} [Y_t - X_t'\beta^0 \, \vert X_1, \dots, X_t, \dots, X_n] \\
&= \mathbb{E} [Y_t - X_t'\beta^0 \, \vert X_t] \quad \text{since $\{Y_t,X_t\}$ is iid.} \\
&=\mathbb{E}\left[ \epsilon_t \vert X_t \right] \\
&= 0 \quad \text{by Assumption A2.}
\end{align*}


Augmented Filtrations and Martingales in the Martingale Representation Theorem

The augmentation of the filtration generated by $B$ is typically just called the augmented filtration and is defined using the below construction. It is more commonly referred to simply as the standard Brownian filtration.

The collection $\mathcal{C}$ of all sets of probability $0$ in the sigma-field $\sigma\{B_{s} \: : \: s \leq t\}$
The collection of null-sets, $\mathcal{N}$, of all $A$ such that $A \subset B$ for $B \in \mathcal{C}$. It is assumed $\mathbb{P}(A)=0$ for all such $A \in \mathcal{N}$.

Then the augmented filtration for Brownian motion to be the filtration given by $\{\mathcal{F}_{t}\}$, where for each $t$ we take $\mathcal{F}_{t}$ to be the smallest sigma-field containing $\sigma\{B_{s} \: : \: s \leq t\}$ and $\mathcal{N}$.

The point of this is to endow the filtration with some properties which are considered helpful.

The relationship to martingales is simply that $X_{t}:=\mathbb{E}(X\:|\:\mathcal{G}_{t})$ is a martingale.


In a social accounting matrix, what does Capital refer to?

You need to take care here as capital appears twice in your table; once as a factor of production, and once as an "institution" (although it's really better thought of as a capital account).

"Profit" in the first column basically refers to Gross Operating Surplus in national accounts, and it's paid by industries (activities) to the capital factor - GOS can be considered the earnings of the capital factor of production (analagous to wages being the earnings of the labour factor).  Looking at the capital factor column, you can see that these earnings are in turn "paid" to the household sector.

The capital account, on the other hand, "receives" savings from government and households, and "spends" on Investment/Gross Fixed Capital Formation (and possibly valuables and inventories too depending on how your SAM is set up). The row=column balance on this account is just the typical Savings = Net Investment accounting identity.


Could someone please explain the proof of Hotelling's lemma?

Well done for spotting this one. It is (was) simply a wrong statement. It is indeed the maximum and not the minimum of the expression as (was in the past) stated in the wikipedia article.

Then the maximum of this expression is the maximum of $-C$ (Cost) which is equivalent to the minimum of $C$, and the equivalence with maximum profit holds under duality.


Big Mac Index based on other products or services?

Yes there are several other such indexes although none as popular as the Big Mac one.
For example the Economist themselves provide not just Big Mac index but also 'Tall Late Index' based on Starbucks' tall late prices.
Another example is the Bloomberg 'Billy bookshelf index' [ based on Ikea product].
Yet another example is one Australian bank for while used iPod index (although this one was criticized on the grounds that it requires transportation and transportation costs are non-trivial).
Probably you could find more such indexes. In principle any homogenous good can serve as a basis for such index.


Below is a graph of the price offer curve of good $y$ when income is $48$ units, $p_x = 8$ and the utility function is
$$
U(x,y) = \min\left(2x+2y,x+10\right).
$$
(Based on "Simple Utility Functions with Giffen Demand" by Sørensen).
Good $y$ exhibits Giffen behavior when $0 < p_y < 8$.

A gif of the optimal choice changing as $p_y$ changes is presented below (budget set is grey, indifference curve and optimal bundle are green, income-offer curve is orange dashed line):


Game theory - Write a maximization problem (with restriction)

Here is the maximization problem solved by the company :

\begin{eqnarray*} \max_{x_h, p_h, x_l, p_l} & p_h + p_l - x_h - x_l \\ \text{s.t} &  \theta_hv(x_h) - p_h \geq 0 \tag{IR$_h$} \\ &  \theta_lv(x_l) - p_l \geq 0 \tag{IR$_l$} \\ &  \theta_hv(x_h) - p_h \geq \theta_hv(x_l) - p_l \tag{IC$_h$} \\ &  \theta_lv(x_l) - p_l \geq \theta_lv(x_h) - p_h \tag{IC$_l$}\end{eqnarray*}

Now we solve the above maximization problem for $v(x) = \ln x$. Given that $\theta_l < \theta_h$, it is easy to see that only IC$_h$ and IR$_l$ will be binding in optimum, and we can rewrite the optimization problem as :

\begin{eqnarray*} \max_{x_h, p_h, x_l, p_l} & p_h + p_l - x_h - x_l \\ \text{s.t} &   \theta_l\ln x_l - p_l = 0 \tag{IR$_l$} \\ &  \theta_h\ln x_h - p_h = \theta_h\ln x_l - p_l \tag{IC$_h$} \end{eqnarray*}

Substituting the constraints in the objective, we can rewrite the objective as follows :

\begin{eqnarray*} \max_{x_h, x_l} & \ \ \theta_h\ln x_h + (2\theta_l-\theta_h)\ln x_l - x_h - x_l  \end{eqnarray*}

Further assuming $\theta_h + 1 < 2\theta_l$, and solving the problem yields the following:

\begin{eqnarray*} x_h^* & = & \theta_h \\  x_l^* & = & 2\theta_l-\theta_h \end{eqnarray*}
Consequently,
\begin{eqnarray*} p_l^* & = & \theta_l \ln(2\theta_l-\theta_h) \\  p_h^* & = & \theta_h\ln \theta_h + (\theta_l-\theta_h)\ln (2\theta_l-\theta_h) \end{eqnarray*}


VAR Models - Identifying Block Exogenous Variables Through Granger Tests

Checking the p-values, I would suggest that in the first and second tests, the variables DRLM2 and DlRGDP have granger causality issues (since we reject for a 5% sig. level).

In the third case, DRLM2 is not rejected for a sig. level of 5%, yet, it seems to be very close (4,8%).
I don't think there's any particular meaning to having a near non-rejection area. Remember that if you draw the chi-square distribution and use the 1% sig. level for the test, you will not reject the granger causality.


Why use marginal rather than average tax rates in incentive analysis?

People make decisions based on how the decision will change things.  If I work an extra \$1000 worth of time, then I have to pay $\$1000 * \text{marginal tax rate}$ in taxes.  The average rate is irrelevant to that calculation.  It's a little more complicated than that in that the marginal tax rate may change over that \$1000.

The only time that the average tax rate would matter would be if I were asking at the beginning of the year if it would make sense to not work at all that year.  But I can reach the same result with marginal tax analysis and not be restricted to the beginning of the year.  The marginal-based analysis already has to handle things like the marginal rate changing.

The average tax rate provides useful information so seldom that there's no real point in using it.

The marginal rate tells us how a change from the status quo will affect things.  If my marginal rate in the example is 25%, then I can make an extra $750 by working that extra.  Is that more or less than my value of my leisure time?  That's a real decision.


Difference between social choice functions and social decision functions?

Let $X$ be the set of alternatives.
A social decision function maps profiles of preference orderings to relations on $X$ such that every nonempty subset of $X$ has at least one maximum under this relation.
A social choice function maps profiles of preference orderings to elements of $X$.
Now let $P$ be a profile of preferences, $f$ a social decision function, and $g$ a social choice function.
There might be more than one $f(P)$-maximum in $X$, so $f$, in contrast to $g$, will not always pin down a single choice in $X$.
On the other hand, suppose the alternative $g(P)$ is not available for some reason (say, the winning candidate died). Then $g$ is of no help in finding an alternative from the remaining set of alternatives $X\setminus\{g(P)\}$. But $f(P)$ will allow us to rank the alternatives in this remaining set (provided $g(P)$ is not the only element of $X$), though, again, there might be more than on $f(P)$-maximum in this set.


What is the difference between monopoly profit and monopoly rent?

Economic (not just monopoly) rent is what is otherwise known as produces surplus. And producer surplus is related to profit via

$$\text{Producer Surplus}=\text{Profit}+\text{Fixed Cost}$$

assuming that fixed cost is sunk.


A law against selling any house cheaper than it was bought for, what consequences would that have?

That would be really, really bad.

Any house that loses value will be unsellable, and thus virtually worthless.

Most people living in such a house would be prevented from moving. They cannot sell it, since no one wants to buy an overpriced house, and they cannot afford buying another house with their capital already tied up in the current house.

A black market would appear, with people giving money back under the table. "Buy my house for 4M SEK and I'll refund 1M SEK under the table."

I can tell you one thing it won't do. It won't stop prices from ever falling.


Necessary and sufficient conditions for the existence of a utility function

The following is essentialy due to Debreu. The result is formulated in terms of linear orders, but each complete and transitive relation induces a linear order on the indifference classes:
Theorem: Let $S$ be a set and $\preceq$ be a linear order on $S$. Then $\preceq$ has a utility representation if and only if there exists a countable set $C\subseteq S$ such that whenever $x\prec y$, then there is some $c\in C$ such that $x\preceq c\preceq y$.
Proof: To see that the condition is necessary for the existence of a utility representation, let $u$ be a utility representation of $\preceq$. Call $(x,y)\in S\times S$ a jump if $x\prec y$ and there is no $z$ such that $x\prec z\prec y$. We show that the set of jumps is countable. Clearly, if $(x,y)$ and $(x',y')$ are both jumps, then $\big(u(x),u(y)\big)$ and $\big(u(x'),u(y')\big)$ are disjoint intervals of real numbers. Each such interval contains a rational number, so there is an injective function from jumps to rational numbers and so the set of jumps is countable. Let $J$ be the set of all elements of $S$ that occur as the first or second coordinate of a jump. Clearly, $J$ is countable too. Let $$Q=\Big\{(q_1,q_2)\in\mathbb{Q}\times\mathbb{Q}:q_1<q_2, u^{-^1}\big((q_1,q_2)\big)\neq\emptyset\Big\}.$$ The set $Q$ is countable. For each $(q_1,q_2)\in Q$, choose some $s\in  u^{-^1}\big((q_1,q_2)\big)$ and let $B$ be the set of such $s$. $B$ is also countable and we can choose $C=B\cup J$.
Now we show that the existence of such a set $C$ is sufficient for the existence of a utility representation. Without loss of generality, we can take $C$ to be nonempty and write $C=\{c_0,c_1,\ldots\}$. Now define $u$ by $$u(x)=\sum_{n:c_n\preceq x}\frac{1}{2^n}-\sum_{n:c_n\succeq x}\frac{1}{2^n}.$$
$\square$
A comprehensive source for all things existence of utility functions is the book Representations of Preferences Orderings by Douglas S. Bridges and Ghanshyam B. Mehta.


Growth in dividends will equal growth of earnings

The textbook probably means growth rate and not absolute growth. Given your assumptions
$$
\frac{\frac{d}{dt}Div_t}{Div_t}=
\frac{\frac{d}{dt}\text{Earnings}_t}{\text{Earnings}_t}
$$
does hold.


An approximate measure of the real interest rate applying rule of thumb for growth rates

Argument 1:

$$ 1+r_{t+1} = \frac{1+i_{t}}{1+\pi_{t+1}}$$

Take the natural log of both sides.

$$ \log(1+r_{t+1}) = \log(1+i_{t}) - \log(1+\pi_{t+1})$$

Observe that $\log(1 + x) \approx x$ for $x$ near zero because that's the first order taylor expansion. Hence:

$$ r_{t+1} \approx i_{t} - \pi_{t+1}$$

Argument 2:

$$ 1+r_{t+1} = \frac{1+i_{t}}{1+\pi_{t+1}}$$
$$  1+r_{t+1} + \pi_{t+1} + r_{t+1} \pi_{t+1} = 1+i_{t}$$

And if $r_{t+1} \pi_{t+1}$ is closer to zero

$$ r_{t+1} + \pi_{t+1} \approx i_{t}$$


Dornbusch Sticky Price (Overshooting) model

The equations you have are equilibrium conditions, while derivation (of course) starts with the demand equation and sets demand equal to supply. Then:

$\varphi$ ($0<\varphi<1$) is the income elasticity of money demand - the responsiveness of demand with respect to income
$-\lambda$ ($-\lambda <0$) is the interest rate semi-elasticity of money demand - the responsiveness of demand to the change in interest rates.

Semi-elasticity means that we care how much Y changes in percentages when we increase $X$ by one (increase by one, not by one percentage point).


Who originated the term "deadweight loss" and when?

So far as the "when" is concerned, Google Ngram can be helpful for this sort of question.  It suggests (see here) that use of the term was nil or close to nil until 1960, since when its use has grown fairly steadily.


Find Price elasticity of demand in this example

Did you know you can write price elasticity $\varepsilon$ as $\varepsilon \equiv  \frac{d \log Q}{d \log P}$?

(Indeed, $\frac{d \log Q}{d \log P} =\frac{dQ}{dP} \frac{P}{Q}$. Check out wiki on elasticity of a function if you want more detail!)

Using this "log notation" for elasticity, it is easy to see without putting pen to paper that in this example, $\varepsilon = -0.7$.


Interest rates, inflation and liquidity effect

The reason why supply effects would not occur is the result of market efficiency assumptions: all financial assets have the same expected returns. Within the bond market, that would imply that the "term premium" is zero.

If the term premium is zero, then:

The value of the 1-period "bond" is the result of the central bank reaction function. (The policy rate.)
The value of a 2-period bond is determined by the 1-period yield, and the unbiased expectation of the 1-period yield, 1-period forward.
(Build up rest of curve similarly.)

They added market segmentation as a way of allowing bonds to trade away from the predicted ('efficient") value, that is, allow a non-zero term premium. Once we allow for a non-zero term premium, supply effects can be seen in predicted bond pricing.

The term premium interacts with supply as a positive term premium will induce bond investors to increase the holdings of bonds relative to the allocation that would happen if the term premium were zero. That is, changes in the term premium is needed to balance the market.


What are some different methods for pricing electricity?

In most of Europa the physical price for electricity is determined in day-ahead spot auctions. Essentially it is a free market, where anyone can buy and sell power, given you that you can fulfill the regulation set forth by the different national power exchanges and system operators.

This means that every day before 10.00 - 12.00 AM all producers and consumers must have told how much power they wish to sell and buy. From 12.00 and to around 12.42 a number of welfare optimization algorithms are run. After this, the different power area prices are published, along with the supply and demand curves for almost all countries.

In essence it is a where supply meets demand. This is complicated a lot by different bid-types, power connections across borders, physical constraints on cables and so on, but you can download the newest curves from Germany and France each day from epex.com if you want.

Because there is a lot of competition, you will mostly see power producers selling their power at their marginal costs. Because of subsidies this means that renewable energy sells as low as -300 euro pr. MWh - yes, they actually in some cases pay people to take their product. Nuclear and lignite power is around 10-18 EUR, the comes cheap coal and hydro (depending on season), the more expensive coal, gas and finally oil. From the price you can more or less tell which kind of production is running in any given hour.


Remember that rivalrous goods mean that someone else consuming it takes some benefit from other people using it--obviously not true for music unless you are a very spiteful consumer. Excludability means you can prevent others from listening to it. Once one person buys it and they start blaring the music, they can't stop others from listening to it, unless they have headphones or something. So it could be considered excludable, but music streaming is often used for parties and such. There is also the issue of torrenting and sharing music online, which makes you the music streamer. So the question to consider is whether music is turning into a public good, and what that means for music streaming.
In any industry, there is always loss from resale transactions. Someone who buys a book from someone else isn't buying it from the publisher or the author. It's just that with music people really like to resale music for zero price (free), which seems odd in a way. Nevertheless, we still see people buy music from musicians because they can be assured of higher quality (so there is product differentiation between a ripped song and the original) and there is probably some weird intrinsic value we get from being a patron of the arts (think Kickstarter or GoFundMe). Is there a negative externality from music streaming services? Probably.
Artists put music on streaming because some see it as better than the traditional way nowadays, just putting your CDs in stores or whatnot. If you get even "less than deserved" from doing that, well...

That's just skimming the surface of this issue probably, so hopefully others will be elaborate better than me.


Matlab code example for piecewise linear approximation and Euler Equation Iteration

Two suggestions:

First, try some of the sample code at this website. It builds simple Euler equation iteration methods from scratch in Matlab.

Second, try the sample code here. You will need to read the associated notes, also. The code makes use of the CompEcon toolbox in Matlab, by Miranda and Fackler (associated with their book:  Applied Computational Economics and Finance).


Mechanism Design: Proving that the expected utility is differentiable

By definition we have
$$
\begin{align*}
u(\theta) & = \theta q(\theta)-t(\theta) \\
\\
u(\theta + \delta) & = (\theta + \delta) q(\theta + \delta)-t(\theta + \delta)
\end{align*}
$$
By incentive compatibility (where $\theta + \delta$ is the true type, $\theta$ is the false type) we have
$$
u(\theta + \delta) \geq (\theta + \delta) q(\theta)-t(\theta)
$$
Using these (I used square brackets for clearer notation, there is no mathematical function to them)
$$
\begin{align*}
u(\theta + \delta) - u(\theta) & =
\left[(\theta + \delta) q(\theta + \delta)-t(\theta + \delta)\right] -
\left[\theta q(\theta)-t(\theta)\right]  \\
\\
u(\theta + \delta) - u(\theta) & \geq
\left[(\theta + \delta) q(\theta)-t(\theta)\right] -
\left[\theta q(\theta)-t(\theta)\right]
\end{align*}
$$
This also holds if you divide by $\delta > 0$.

Similar argument for 2.8., using true type $\theta - \delta$ instead of $\theta + \delta$.


Clarifying foreign direct investment and imports on International Transactions release

I think it does not really work that way

Going through your references:

(Table 1 line 75) Reserve Assets: These are foreign assets held by the Federal Reserve or US Treasury for intervening in currency markets.  Since the US does not intervene, the small numbers are largely miscellaneous transactions related to IMF Special Drawing Rights
(Table 1 line 62) Direct Investment Assets: This is the closest as to what is going on.  Amounts sent out to buy or build the factory are counted here, as are later investments from non-repatriated income
(Table 1 line 57) Imports and income payments - compensation of employees: this is payments for short term workers coming into the country and then going home, such as seasonal agricultural workers. If you had built your factory by sending short-term construction workers abroad to build it then their income might have counted as (Table 1 line 29) Exports and income receipts - compensation of employees
(Table 1 line 94) Other investment liabilities - currency and deposits: assuming your workers abroad are resident there and you pay them in US dollars (why?), then this increases this line, but if they then change this into their currency by transacting with you in the US it reduces it back again

I would have thought that a more natural set of transactions would be something like:

You start with some US dollars for the investment which you change into Mexican pesos (increasing line 71 for you and either increasing line 94 or decreasing line 71 for somebody else depending on whether they are foreign)
You use the pesos to buy some land in Mexico, buy construction supplies and hire construction workers (reducing line 71 but increasing line 62) so you end up with a factory in Mexico
Now you have a factory in Mexico, your factory pays in pesos for workers resident and working in Mexico and for supplies, and receives revenues from sales.  Your factory's net profit or loss appears as Direct Investment Income (line 25), which if you do not repatriate it then also goes into further Direct Investment in your Mexican factory (line 62)


Homothetic preferences from indirect utility

Your conclusion is a bit hasty:
(i) why did you implicitly exclude the case $a^i(\textbf{p})=0$?
(ii) if your individual is rational then $v^i$ is homogeneous of degree zero in $(\textbf{p},w^i)$ which has implications on $a^i$ and $b^i$... Your demand system should also satisfy the budget constraint (adding-up).


Optimal control theory: How to maximize Hamiltonian in this case?

Hamiltonian
\begin{align}
H(y(t),u(t),\lambda(t)) = y(t)+u(t)^2 + \lambda(t) u(t)
\end{align}

First order conditions read
\begin{align}
&H_u = 0 \quad \Longleftrightarrow \quad u(t) = -\frac{\lambda(t)}{2}\\[2mm]
&\frac{d\lambda(t)}{dt} = -H_y = -1\\[2mm]
&\lambda(1) = 0
\end{align}
Integrate costate
\begin{align}
&\int \frac{d\lambda(t)}{dt} = \int -1\\[2mm]
&\int d\lambda(t) = \int -dt\\[2mm]
&\lambda(t) = -t + C
\end{align}
with
\begin{align}
&\lambda(1) = -1 + C = 0 \quad \Longleftrightarrow \quad C = 1\\[2mm]
\Longrightarrow \quad & \lambda(t) = 1-t\\[2mm]
\Longrightarrow \quad & u(t) = \frac{t-1}{2}
\end{align}
Now integrate state equation
\begin{align}
&\int \frac{dy(t)}{dt} = \int \frac{t-1}{2}\\[2mm]
&\int dy(t) = \int \frac{t-1}{2}dt\\[2mm]
&y(t) = \frac{t}{2}\left(\frac{t}{2}-1\right) + C
\end{align}
with
\begin{align}
&y(0) =  C = 5\\
\Longrightarrow \quad & y(t) = \frac{t}{2}\left(\frac{t}{2}-1\right) + 5
\end{align}

If I'm not mistaken necessary conditions are sufficient if maximized Hamiltonian is concave in state (Arrow sufficiency if I recall correctly (see e.g. Caputo, 2005, Ch. 3))
\begin{align}
H^*(y,\lambda) = \max_u H(y,u,\lambda) = y - \frac{\lambda^2}{2}.
\end{align}
Now we have
\begin{align}
H_y > 0 = H_{yy}
\end{align}
such that concavity is given.


What famous company that turned a loss at its first year(s) but then became profitable?

It is very common for a company to have a loss (i.e. expenses > revenue) in the first year. Amazon is actually a very good example of that. It’s also a good example of ‘growth before profit’ strategy.

Amazon was incorporated on July 5, 1994, and begun service in July 1995. This means that not only it did not generate profit in its first year, there were also no sales. Even after the launch the company generated loss:

  Amazon had stated from the outset that it did not expect to turn a
  profit for four to five years, and it duly delivered on its promise.
  The firm grew quickly but failed to generate profit, making a net loss
  of USD -303,000 on net sales of USD 511,000 during its first six months of
  trading. At the beginning of 1996, Amazon employed eleven people and
  moved out of Bezos’s home to a small warehouse nearby. By the end of
  the year, Amazon.com offered 2.5m titles online, although it still
  carried only a minimal inventory. By this point, the firm had 151
  employees and relocated once again to accommodate its growing
  workforce. Sales for 1996 were USD 15.7m, with the company making a net
  loss of USD 5.7m.

Please see below historic results from Amazon’s first public annual report from 1997 (the year of IPO):

                           1997 |   1996  | 1995|   5 Jul 1994 - 31 Dec 1994

(1) net sales            147758 |   15746 | 511 |   0
(2) cost of sales        118945 |   12287 | 409 |   0
(1+2) gross profit        28813 |    3459 | 102 |   0
                            ---       ---   ---     --
(4) total operating
expenses                  58022 |    9438 | 406 |   52
(5) profit from interest   1619 |     202 |   1 |   0
                            ---       ---   ---     --
(1+2+5-4) net loss        27590 |   5777 |  303 |   52

As for an easy reference, @denesp cited an interesting source on this. There are some slight differences in the data, so I thought I would cite it alongside with annual report data, which is more precise.


Why do non-binding price ceilings have no effect?

You are completely right. If in your setup negotiating is an important feature of how prices are set, then even a non-binding price ceiling matters. Search models in labor economics became popular for showing a feature like this. Not with a price ceiling, but with a price floor. In particular when we study minimum wages. In the labor market, the assumption that wages are negotiated is reasonable and then the level of minimum wage affects equilibrium even if it is non-binding.

The conclusion that a non-binding price ceiling has no effect on equilibrium stems from the assumption that markets are competitive. Prices are not negotiated, they are simply posted and people can either buy or not buy. In that case you should not expect a non-binding price ceiling to have any effect.


Why slope is constant in demand/supply curve?

Well, it's just a simplification. You can make a demand curve non-linear if you want to, but it doesn't serve any particular pedagogical function to do so and Mankiw is just trying to make the reader understand the intuition behind basic concepts.

I think you also want to know why a shock only shifts the curve, not its slope. Well, slopes of these curves have a certain meaning. The slope of a demand curve is related with consumer preferences, for example. An income shock is not particularly relevant to consumer preferences (not in the simple models Mankiw presents, at least, it's not like you can't model preferences like that), so it shouldn't change the demand curve's slope, only shift it. That is, people consume more (or less, depending on the shock) of a certain good at the same price.

The slope of a supply curve has an intimate relation with the economy's cost structure. Why should a supply shock - such as an earthquake - alter the curve's slope?


How can I compare degree of stability (or persistency) in two dynamic systems/models?

The most straightforward way is to examine the coefficients of your VAR model, since the persistence is broadly understood in econometrics as a system dependence on its own past (see Verbeek (2008): a guide to modern econometrics).  A simple VAR model is given:
$${\displaystyle y_{1,t}=c_{1}+a_{1,1}y_{1,t-1}+a_{1,2}y_{2,t-1}+e_{1,t}\,} \\
{\displaystyle y_{2,t}=c_{2}+a_{2,1}y_{1,t-1}+a_{2,2}y_{2,t-1}+e_{2,t}.\,}$$
Now, in very similar fashion as with AR models how persistent the variable is will be given by how much it depends on its own past (lags). For example, persistence of $y_1$ can be gauged by coefficient $a_{1,1}$. The larger the coefficient the more persistent the series is.


Identify whether a good is giffen good or not and under what condition

Since the question is asking for clarification on a given answer, I don't think this question necessarily needs to be closed. To address your concerns:

You will recall Walras' Law states $p \cdot x = w$. That is, you spend all your money buying things, given some regularity conditions. So just figure out how much was spent in year 1 and that will give you wealth for that period. For year 2, you'll note that depending on the range for $x^2_2$, it will imply the consumer having different amounts of wealth. That will give you the information you need to find where good 1 is inferior.

It might be helpful to think about what ranges of good 2 would violate WARP, and what the ranges above and below it would mean. Hope this helps.


How to change the observation for the first lag in an AR(1) model?

The ar.ols() function in R defaults to order $10log_{10} (N)$.  This implies that, to catch the 30th time lag, you need at least 1000 observations. That seems like a lot, but bear in mind that you are then running OLS on 30 features (lags t-1 to t-30). All else equal, it is not clear why 30 is the magic number for you here, and you can't just drop the more recent lags as they may have the dominant effect.

Without knowing more about your study, it's hard to recommend further.  Intuitively, though, I would not expect to be using annualized data to perform generational analysis unless I were running OLS (not AR) and explicitly constructing a lagged column vector to use as an additional feature alongside contemporaneous features/controls.


Rigorous solution of Stackelberg leader-follower game with N-players?


Expected Utility and Jensen's Inequality

Didn't manage to get to a definitve answer in one shot, but it seems to me that Jensen inequality is not going to help much.

Build up:

You are essentially asking that

\begin{equation}
   E_v \left(u(a - v) \right) \leq E_c \left(u(a-c) |a\leq c \leq b\right)
\end{equation}

for a function $u$ increasing and concave and $[a, b] \subset Supp_v = [x,y]$.

In the language of measures, the above condition can be rewritten as

\begin{equation}
   \int_x^a u(a-c) dG(c) + \int_b^y u(a-c)dG(c) \leq \int_a^b u(a-c) \cdot \left[\frac{f(c)}{F(b)-F(a)} - g(c) \right]dc.
\end{equation}

First term on the left is certainly positive, second term on the left is certainly negative. The term on the right is of undetermined sign: $u(a-c)$ is negative in the integration interval, but $\frac{f(c)}{F(b)-F(a)}$ cannot be uniformly lower than $g(c)$ on $[a,b]$, as the former on this interval must sum up to 1 and the latter to $G(b)-G(a) \in (0,1)$.


Trying to understand ceiling in supply and demand on picture

This is due to not enough supply. So yes you are correct, more people can afford it now, but because the price is less, there is not enough supply for everyone, even if they can afford more than it cost.

It seems you are not taking into account dead weight loss.
Here is a link to help you understand it and use it.
https://corporatefinanceinstitute.com/resources/knowledge/economics/deadweight-loss/

An inefficiency occurs since at the price ceiling quantity supplied the marginal benefit exceeds the marginal cost. This inefficiency is equal to the deadweight welfare loss.

When it comes to rent ceilings, certain things almost always happen that make the people willing to pay more in order to get the product vs people willing to pay only the rent ceiling price.

If a price ceiling is set, then there must be a way to assign who gets the low supply of the product. Of course, since there is a legal limit on the price, the price can't simply be raised. There are several ways this is done without raising the price:

Black Market: For those lucky enough to get some of the short supply, they are often better off selling what they have obtained to the demanders that will get more benefit out of it. In some cities there have been ceilings put on the apartment rent. While the demand for apartments increases, the rent remains the same. When some renters are ready to move, they sublease their apartment instead of ending their contract. If they were renting for \$500, but someone who  is  willing  to  pay  \$1000, then the subleaser can continue paying \$500 and  pocket the extra \$500 he gets from the subleasee.


Making mathematical sense of the expression for realized bond return

The approximation given is off at a key point - it is the duration of the bond at 9 years maturity, not the original duration.

The approximation works in two steps.

If the yield is unchanged over one year, the total return over the year is roughly equal to the starting yield.
If the yield changes, we add a capital gain/loss which is equal to the change in yields times the yield sensitivity. This is the modified duration of the bond, at 9 years maturity.

This approximation ignores the reinvestment of any coupon income, and the convexity effect (the return sensitivity changes slightly as yields change). Also, the bond yield convention is probably different than the convention used for total return. That said, for yield changes under 50 basis points, the approximation is typically pretty good.


If a game admits a unique Nash equilibirum, does common knowledge of rationality implies Nash equilibirum?

No. In static games, common knowledge of rationality is equivalent to rationalizability. Bernheim, in "Rationalizable Strategic Behavior" (Econometrica, July 1984) gives an example on page 1012 in Figure 1 of a normalform game in which there is a unique Nash equilibrium, yet multiple strategies are rationalizable.


What is the reason why ARIMA(0,1,0) on $y_t$ and ARIMA(0,0,0) on diff($y_t$) are not identical time-series models?

I don't know R code but are you estimating an intercept in the ARIMA(0,1,0) model? Because if not, this could be why there is a difference, since you are estimating an intercept in the ARIMA(0,0,0) model.


Is the contract curve orthogonal to the priceline?

You write that the slope of one line is px/py and of another is -px/py. Take a piece of paper and draw the line y=2x and y=-2x are they orthogonal (answer: no)? In particular case the lines y=x and y=-x are orthogonal.

If we regard it more formally, the vectors are orthogonal if the product is 0. Thus,

<1,-p>*<1,p> = 1-p^2

This is 0 only if p=1 or p=-1, i.e. if the quantities of the 1-st and 2-nd good are equal.

To take a loon on it. Consider the wealth distribution something like 1-st player: [5;
15], 2-nd: [25;5], thus ex=30,ey=20,px/py = 2/3, then it looks like this:

p.s. Calculations and plotting were made with pyEdgeworthBox package (I wrote it).


Looking for behavioral economics research about the redemption of gift cards

A classical economic explanation for the phenomenon you describe is that the gift giver does not know the receivers preferences well enough. Prominently, Joel Waldfogel makes this point in his fun paper "The Deadweight Loss of Christmas":

  I find that holiday gift giving destroys between one-third and one-tenth of the value of gifts.

This article pops up in the public domain every year around the holidays (example in the NYT) and its idea of course also applies to all other gift-giving-and-receiving situations.
Of course many people (including economists) disagree with that notion and you find quite a lot of comments and replies on that paper in the AER, just type the title of the paper into google.scholar.


I believe the answer is true.

Giffen goods are goods where the income effect overpowers the substitution effect.

$$\begin{align}
\max_{\vec x} \ \ \ & U(\vec x) \\
& \text{s.t.} \ \ \ \vec p \cdot \vec x \leq I
\end{align}$$

To start, if you think about the consumer's problem (for example utility maximization, here), a change in a good's price affects both relative substitutability of goods through the marginal rate of substitution AND it affects purchasing power through the budget constraint.

Let us consider a profit maximizing firm with a constraint on how much they can spend. For simplicity let us use a single output technology, with differentiable production function $f(\vec z)$. Let $\vec z$ be a vector of inputs (expressed as negative values), $\vec w$ a vector of input prices, and $p$ the output price.

$$\begin{align}
\max_{\vec z} \ \ \ & pf(\vec z) + \vec w \cdot \vec z \\
\text{s.t.} & \ \ \ \vec w \cdot \vec z \leq B \\
& \ \ \  z_i \leq 0 \\
\end{align}$$

Normally we would have a constraint on production, but instead we have a "budget" constraint. What happens if we form the Lagrangian here?

$$\mathcal{L} = pf(\vec z) - \vec w \cdot \vec z - \lambda(\vec w \cdot \vec z - B) + \vec\mu \cdot \vec z$$

Take first order conditions:

$$\frac{\partial \mathcal{L}}{\partial z_i} = pf_{z_i}(\vec z) - w_i - \lambda w_i + \mu_i = 0 \tag{1}$$

$$\frac{\partial \mathcal{L}}{\partial f(\vec z)} = p = 0 \tag{2}$$

$$\frac{\partial \mathcal{L}}{\partial \lambda} = \vec w \cdot \vec z - B = 0 \tag{3}$$

At an interior solution where the budget constraint binds, we should have the optimum $\vec z^*$ to solve the FOCs

$$p \frac{\partial f(\vec z^*)}{\partial z_i} = w_i$$

but instead you solve out (1):

$$p \frac{\partial f(\vec z^*)}{\partial z_i} = - \frac{\mu_i}{1 + \lambda}w_i$$

and (3) does not provide any help to solve the Lagrangian multipliers. (2) is nonsense.

A better constraint would be something like $y - f(\vec z) \leq 0$, where $y$ represents the scalar of output.

Without an "income effect", there isn't much to study Giffen behavior. Producer theory doesn't use a budget constraint to solve these sorts of problems. Increasing input price will always decrease use of that input except with corner solutions, where there might be no change. So there can't be a Giffen input.


How did investment go from exogenous to endogenous in the IS/LM model?

In the earlier model the fixed investment was a simplifying assumption made for didactic reasons (you dont want to overwhelm students), rather than because investment is actually fixed in general. In fact, it is not exogenous even in a full circular flow model (what you probably seen were simplified versions), but in a full circular flow model investment is actually dependent on income (investment is given by savings and savings are just unconsumed income) and endogenous as you can see below in a picture taken from Krugman Economics:

What you refer as the previous model was likely the basic goods market equilibrium for closed economy as in Blanchard et al Macroeconomics:
$$Y = C+I+G; \text{ with } C=b_0+b_1(Y-T)$$
The reason why investment in the above model is considered to be fixed/autonomous is not so much because it would actually be fixed, but because fixing it allows to make the model more simpler. However, this really models only small portion of the full economy as you can see from the picture above.  There are actually non-IS-LM extensions of the basic model where $I$ won't be fixed and you just assume there is some additional function $I(Y)= a_0 + a_1Y$ and you could solve the model for the multiplier and everything else.
However, most courses will just first cover goods market equilibrium and afterwards move onto IS-LM model due to lack of time.  IS-LM model will try to capture economy in its entirety so it will model the link between income and investment through savings in explicit fashion.
You will still have the goods market given by
$$Y = C(Y-T)+I(Y,i)+G$$
but now $I(Y,i)$ will depend on income and interest rate, which in turn also depend on equilibrium in the money (financial) markets described by LM equation ($M/P=L(i,Y)$). This really represents adding the arrow of private saving, borrowing and investments spending from the picture below into the model.
In summary, the main reason why it went from exogenous to endogenous is that in a sense that generally speaking it never was exogenous (a caveat is that more complex models can show that there are periods were it can be - but not generally and that whole another topic that is well beyond scope of this answer). The previous assumption was there so you can be first exposed to goods market equilibrium and multiplier idea without worrying about what happens to investment.


Simplest model of an economy exhibiting (hyper)inflation

I agree with Guy that your model is simplistic. My own position is that hyper-inflation is first and foremost a political phenomenon. It results out of specific political policies and choices.

In other words, hyperinflation is a political process, not a financial process which proceeds with some sort of inevitability once a tipping point has been reached.

Although you also hear about when the dollar will fall off the cliff or that kind of talk, its inaccurate. Destroying a currency is not like falling off a cliff, where gravity takes hold all the way to its demise.

Destroying a currency is deliberate and requires a lot of manhandling and pushing up the steep incline and eventually shoving it off the cliff.

At any point up to the final shove into oblivion, the political winds could change and the policy pressure that is destroying the currency would cease.

Currencies are destroyed as an unintended consequence of saving the status quo from losing power. Rather than risk a decline in power, an upheaval in the fiefdom, elites in the status quo choose to debauch the currency as a short-cut to their unsustainable financial woes.

We have a great example with what is going on in Venezuela right now.

But lets imagine an example here in the States. Let's say tomorrow the Federal government says that all Social Security recipients who were receiving 2,000 a month will now receive 200 dollars a month. Rather than risk the firestorm that would ignite and get rid of some very powerful people, the government chooses to debauch the currency so that the 2,000 dollars will soon only provide 200 dollars in purchasing power. The government has de facto reduced the payment by 90% via destruction of its currency. But due to the stealth of this process, the citizenry are blind to the erosion.

If you take 10% of their money away every year via debauching the currency, then in nine years they have lost 90%, no fuss no muss, and the status quo stays intact.

If the populace goes on for several decades like this, no worries, but if it starts to lose trust in the currency, thats how hyperinflation starts to set in. Again, its a political act, not a financial one and its not an inevitable conclusion of anything, it can be stopped in its tracks.

So what I am saying that unlike how in our economics courses, we may be given a model to measure classic inflation, hyperinflation or the debauchery of the currency cannot be viewed in that same manner. No real model to measure when its going to happen, it all depends on the politics of the day and there is no inevitability, it can start to go in that direction and then be stopped in its tracks, because its a political phenomenon not a financial one.


What does price embody ? Do we constantly undervalue some type of products?

People have different tastes, needs, and budgets. So it doesn't make much sense to speak of the willingness to pay (WTP) of "the consumers". Each consumer has his or her own WTP for, let's say, a cup of coffee. If Alice' WTP is 1€ and the price is 2€ she will not buy. If Bob's WTP is 5€  then he will buy and enjoy the surplus. From Bob's perspective the cup of coffee might then seem "undervalued", and from Alice' perspective it is "overvalued", but that's not how you would generally use this term.
And forget about the price reflecting a consumer's "actual utility". Utility is an ordinal concept. Under some technical assumptions you can turn it into a cardinal concept by "measuring" utility by WTP. But then, as above, the price "reflects" this utility (price = WTP) only for the marginal consumer, i.e. the one who is indifferent between buying and not buying.
So to answer your title question: Price just "embodies" an amount of money per unit with the property that in some given time period consumers want to buy as many units as producers want to sell. And "we" don't "undervalue" some products, since there is no "we" that values products, there are only single consumers who do so.


What are the benefits of reducing zeroes from currency (eg: Iran)?

Instead of the word "benefit" I prefer to use the old Saxon word "bebebebebebebebebebenefit". Some people say using the shorter word has bebebebebebebebebebenefits, but I don't see any.

P.s.: coins and notes of extremely small value are usually removed from circulation. It is unlikely the average person would ever pay with 1000 coins.


What's the Cause of Mexico's Poverty Rate?

High poverty rates are generally a result of two factors

A) Low per capita income (GDP / capita)

B) High inequality (frequently measured as the Gini coefficient)

There are multiple potential reasons for any country's income level and degree of inequality (not just for Mexico's). Moreover, one can decompose your question into two.

1) What makes some countries (in general) richer than others?

2) Is Mexico poorer because of some Mexico-specific factors?

The answer to the first question is kind of the Holy Grail of empirical economics, and if any single economist had been able to answer that he would have definitely won a Nobel prize. The consensus view is that there are different factors contributing to cross-country differences in income. Literacy and education, which you mentioned (and which fit into the economists' definition of Human Capital) are mostly likely a key factor. However, another one which appears to be particularly important to Mexico's case is the quality of a country's economic and political institutions. See below a highly recommended reading from an economist who's likely to win a Nobel for growth theory, which in his very readable book frequently uses Mexico as a case-in-point for his theory.

http://whynationsfail.com/blog/2013/10/29/what-are-institutions.html


Measuring effects of degree of (buyer) competition on price: references

You might be looking for the term monopsony power or perhaps oligopsony power.
A superficial look at the literature indicates that the Herfindahl-Hirschman index (HHI), so commonly used in assessing seller market power, is also used to measure buyer market power. Recall that the HHI is the sum of the squared market shares measured in percent (so a duopoly with two sellers of 10% and 90% market shares has an HHI of $90^2 + 10^2 = 8,200$.
If you have buyer level figures (not clear to me from your question) then your data allows you to construct local and aggregate buyer market shares. You'll have to  convince yourself which is the right market share for understanding the relevant market. For example, if they were dry cleaners, they tend to draw resources from a small area, and each store is probably a distinct market. If they are car dealers, they draw customers from all over, and so perhaps the entire set of sales is the relevant market.
In the follow up comments and question, Hardy asks, "I want to see how the market power is exercised in terms of its effect on price." In general, you can't do this with price and quantity data alone because prices are simultaneously determined by supply and demand. Unless you know whether supply or demand or shifting, you can't know why prices are changing.
If you really knew that "an additional buyer comes into the market at a given location", and you knew something about what their demand schedule looked like, and you knew that their arrival was exogenous to changes in supply, then I think you could trace out the supply curve and learn about the supply elasticity (for example). But I think you generally need exogenous variation in supply to trace out the demand curve (and vice versa).
However, I'm concerned that the arrival of a purchasing customer is an equilibrium outcome (they only buy when they like the price) while the arrival of a potential customer is what actually is in the demand curve and influences the degree of monoposony. To see this starkly, recall that there are a number of competitive settings where even potential entrants can fully discipline a monopolist to provide the competitive price and ensure they earn zero profits. For example, Tirole Chapter 8. The monopsonist version probably works the same way: in the presence of free entry, if the monopsonist earns any rents then other buyers will enter until those rents are competed away. Therefore we cannot generally learn much just from observing the market quantities (what we really want is unobserved). This makes me skeptical that your data can identify what you are hoping to identify.
However, if you can argue, using your knowledge of the institutional setting, that the appearance of positive quantity purchasers are independent of pricing (they are movements of the demand curve and not movements along a demand curve), then I think you trace out the supply curve with your data. Then, you can look for something similar to the Monopsony and the Minimum Wage outcome. In that setting, the minimum wage raises price and quantity demanded. In yours, entry would lead to higher prices and higher quantities. Could you do that in your data? After deciding whatever is the exogenous shift in demand, can you look (perhaps in a regression difference in difference setting (diff 1: markets with and without such entry with the same change in quantity, diff 2: before and after entry) to see if prices go up along with quantities after entry?


Two asset Markowitz Portfolio Optimization and Capital-Market Line construction for a Given Risk Free Rate

The Sharpe ratio tells us the amount of excess return we get for taking on each additional unit of portfolio standard deviation.
$$\frac{\mu_p - r_f}{\sigma_p }$$

We are looking for the combination of the two risky assets with the highest Sharpe ratio ($P^*$). Once we do that, we can take linear combinations of that portfolio and the risk-less asset and form the Capital-Market Line. This is usually solved for numerically rather than analytically but it is possible to do so analytically, particularly in the two asset case.

A portfolio $p$ has an expected return of:
$$\mu_p(W_A) = w_A \cdot \mu_A + (1 - w_A) \cdot \mu_B  $$

and a standard deviation of:
$$\sigma_p(W_A) = \sqrt(w^2_A \cdot \sigma^2_A + (1 - w_A)^2 \cdot \sigma^2_B + 2(1-W_A)W_A \sigma_{AB}) $$
where $\sigma^2_A$ is the variance of asset $A$, $\sigma^2_B$ is the variance of asset $B$, and $\sigma_{AB}$ is their covariance. It therefore has a Sharpe Ratio of:

$$\frac{\mu_p(W_A) - r_f}{\sigma_p(W_A) } = \frac{w_A \cdot \mu_A + (1 - w_A) \cdot \mu_B}{+\sqrt(w^2_A \cdot \sigma^2_A + (1 - w_A)^2 \cdot \sigma^2_B + 2(1-W_A)W_A \sigma_{AB})}$$

To maximize this you'll want to solve:
$$ \frac{d}{dW_A} \frac{\mu_p(W_A) - r_f}{\sigma_p(W_A) } = 0 $$
$\Rightarrow W^{*}_A$ s.t. $P(W^{*}_A)=P^*$
and check that the second order condition is met:
$$ \frac{d^2}{dW^2_A} \frac{\mu_p(W_A) - r_f}{\sigma_p(W_A) } < 0$$

The algebra is a bit hairy but there is nothing tricky from here on out.


Are taxes supposed to be calculated depending on the seller, or the buyer?

The most generic answer is that governments can basically tax whatever they want, according to whatever rules they choose to invent.  I have seen situations in which goods or services have been taxed by both the importing and exporting governments.

Having said that, there is a clear international norm that taxes such as VAT (the equivalent of sales tax in the vast majority of countries) apply in the importing country, after import duties (so you pay tax on the import duty and transport costs).

I don't know about the specifics of the US and Canada; I believe in particular US state sales taxes don't follow international norms.  NAFTA may also specify.


Question about tariffs in Henry Hazlitt's Economics in One lesson

The author is simplifying things a bit, of course, given the concise nature of the publication.

I believe the main point he is trying to make is that in the long run, current accounts will be balanced. Suppose, following the author, that the domestic economy (us) is the U.S. So, if we buy cheaper British sweaters today, eventually American goods will be purchased in return.

The idea, is that the US dollar is legal tender only (or primarily) in the United States. The British may accept the dollar as payment for their sweaters, but only because they know it will be accepted as legal tender by others (e.g. the U.S.) and others accept it only because the U.S. accepts it. This circular thinking only closes with the U.S. and hence with U.S. goods being purchased at some point. If the U.S. stopped accepting dollars, then dollars would stop being accepted (almost) everywhere.

The British may also pay the Americans in pounds. However, the same logic applies. Americans only accept pounds, because eventually those pounds will be accepted elsewhere as payment, which hinges on the UK accepting pounds as legal tender. They are eventually only (or primarily) accepted in the UK then, e.g. for buying British sweaters. However, this case already starts with American goods being purchased, which is the author's point.

Another alternative would be the Americans paying in pounds. However, for Americans to get pounds, they would have had to sell good to the British in some form at some point.

In all cases, trade (through cheaper products abroad, e.g. British sweaters) involves American good eventually being purchased.


The profit function is given by:
$$
\pi = PQ(P) - C(Q(P), \mu) - tQ(P)
$$
Assume demand is linear, s
$$
p = \alpha - \beta Q \to Q = b(\alpha - P),
$$
where $b = 1/\beta$. So:
$$
Q_P = -b,
$$
where I use subscripts to denote the partial derivatives.
The first order condition for profit maximisation gives:
$$
\begin{align*}
    &Q + P Q_P - C_Q Q_P - t Q_P = 0,\\
\iff &Q - b P + b C_Q + bt = 0
\end{align*}
$$
then differentiating this with respect to  $t$ gives:
$$
\begin{align*}
&(Q_P - b + b C_{QQ} Q_P)\frac{\partial P}{\partial t} + b = 0,\\
\to &(-2b - b^2 C_{QQ})\frac{\partial P}{\partial t} = -b,\\
\to &\frac{\partial P}{\partial t} = \frac{1}{2 + b C_{QQ}}
\end{align*}
$$
Differentiating the first order condition with respect to $\mu$ gives:
$$
(Q_P - b + b C_{QQ} Q_P)\frac{\partial P}{\partial \mu} + b C_{Q\mu} = 0,\\
\to (-2b - b^2 C_{QQ})\frac{\partial P}{\partial \mu} = - b C_{Q\mu},\\
\to \frac{\partial P}{\partial \mu} = \frac{C_{Q,\mu}}{2 + b C_{QQ}}
$$
Then differentiating $\frac{\partial P}{\partial t}$ once more with respect to $\mu$ gives:
$$
\frac{\partial^2 P}{\partial t \partial \mu} = -\frac{1}{(2 + bC_{QQ})^2}b\left(C_{QQQ}Q_P \frac{\partial P}{\partial \mu}+ C_{QQ\mu}\right),\\
= -\frac{1}{(2 + b C_{QQ})^2}b \left(-b C_{QQQ} \frac{C_{Q,\mu}}{2 + b C_{QQ}} + C_{QQ\mu} \right)
$$
Where the last line uses the expression for $\frac{\partial P}{\partial \mu}$ from above.
Now assume that costs take the form:
$$
C(Q, \mu) = \delta + \eta Q + \frac{\gamma(\mu)\, Q^2}{2}
$$
So the slope of the marginal costs depends on $\mu$. Then $C_{QQ} = \gamma(\mu)$, $C_{QQQ} = 0$  and $C_{QQ\mu} = \gamma_\mu$ , so:
$$
\frac{\partial^2 P}{\partial t \partial \mu} = -\frac{b \gamma_\mu}{(2 + b \gamma)^2}.
$$
We have that $\gamma_\mu < 0$ by assumption (slope decreases in $\mu$) then we see that the sign of $\frac{\partial^2 P}{\partial t \partial \mu}$ is positive (at least if I didn't make any mistakes).


A question related to purification theorem

I have never done such purification exercises, but I would approach it like that. As you state,
$p=Pr(\epsilon_1>\frac{1-2q}{x})=1-Pr(\epsilon_1<\frac{1-2q}{x})=1-F (\frac{1-2q}{x}),$
$q=Pr(\epsilon_2>\frac{2p-1}{x})=1-Pr(\epsilon_2<\frac{2p-1}{x})=1-G (\frac{2p-1}{x}),$
where $F$ and $G$ are the cdfs corresponding to densities $f=F'$ and $g=G'$. Because they are cdfs, they are weakly increasing. For simplicity, suppose they are strictly increasing so that their inverses clearly exist (i.e., I assume full support. If there were holes, you could still define appropriately some inverse for the flat parts).
Then, combine those two equations to
$1+x F^{-1}(1-p) - 2 G (\frac{2p-1}{x})=0$ (and another similar formulation for q). The LHS of this formula decreasing in $p$. Note that for $p=0$, we have $1+xF^{-1}(1)-2G(\frac{-1}{x})=1+x-0>0$ on the LHS such that the equation never holds, $p$ must be larger than zero. For $p=1$, have on the LHS $1+xF^{-1}(0)-2G(1/x)=1-2G(1/x)$ which can be more or less than 0. That is, for some $x$ there is a pure strategy equilibrium with $p=1$ and for others there is a mixed strategy equilibrium with $p<1$.
Now consider the same equation that solves for the mixed strategy and plug in $p=\frac{1-x}{2}$ such that the LHS above is $1+xF^{-1}(\frac{1+x}{2})- 2 G(-1)=1+x F^{-1}(\frac{1+x}{2})- 0>0$ for small $x$, meaning we have to increase $p$ to fit the equilibrium equation LHS=0.
Next consider the same equation that solves for the mixed strategy and plug in $p=\frac{1+x}{2}$ such that the LHS above is $1+xF^{-1}(\frac{1-x}{2})- 2 G(1)=x F^{-1}(\frac{1+x}{2})-1<0$ for small $x$, meaning we have to decrease $p$ to fit the equilibrium equation LHS=0.
Thus, for small $x$, the equilibrium $p$ is such that $p\in [\frac{1-x}{2},\frac{1+x}{2}]$, leaving only candidate $p\to \frac{1}{2}$ as $x\to0$.


Right definition of natural rate of unemployment?

One definition of the natural rate of unemployment is the NAIRU: non-accelerating-inflation rate of unemployment. The classic story of the money illusion and unemployment works like this:

  Both Friedman and Phelps argued that the government could not
  permanently trade higher inflation for lower unemployment. Imagine
  that unemployment is at the natural rate. The real wage is constant:
  workers who expect a given rate of price inflation insist that their
  wages increase at the same rate to prevent the erosion of their
  purchasing power. Now, imagine that the government uses expansionary
  monetary or fiscal policy in an attempt to lower unemployment below
  its natural rate. The resulting increase in demand encourages firms to
  raise their prices faster than workers had anticipated. With higher
  revenues, firms are willing to employ more workers at the old wage
  rates and even to raise those rates somewhat. For a short time,
  workers suffer from what economists call money illusion: they see that
  their money wages have risen and willingly supply more labor. Thus,
  the unemployment rate falls. They do not realize right away that their
  purchasing power has fallen because prices have risen more rapidly
  than they expected. But, over time, as workers come to anticipate
  higher rates of price inflation, they supply less labor and insist on
  increases in wages that keep up with inflation. The real wage is
  restored to its old level, and the unemployment rate returns to the
  natural rate. But the price inflation and wage inflation brought on by
  expansionary policies continue at the new, higher rates.

The Concise Encyclopedia of Economic: Phillips Curve by Kevin D. Hoover, emphasis mine.

At first this seems like a Blanchard-type definition. But why it is that the NAIRU takes one value and not another? The answer typically given is like that provided by Mankiw:  labor-market imperfections impeding or incentivizing workers from instantly finding jobs. A decline in unemployment insurance, a reduction in the regulatory costs of firing workers, or a technological improvement in the ability of workers and employers to find each other all could reduce unemployment without altering inflation or inflation expectations.

In my eye, both definitions contribute something important to the idea of a natural rate of unemployment. If labor market frictions are moving around a great deal it is hard to speak of a natural rate of unemployment. If the fiscal or monetary policy authorities are manipulating inflation in a way that is at odds with inflation expectations there similarly is no natural rate of unemployment either. By "gravitates in the long run" I assume that Mankiw doesn't mean convergence. Rather, he means something more like mean reversion. Only when the monetary authority is catering rational expectations of inflation and labor market frictions are broadly stable that a natural rate of unemployment makes sense. It's going to be something more like a long run average rate of unemployment in that economy.


GDP calculation on selling house at a profit

Revised:
The sale of the old house is not counted toward GDP as GDP is intended to measure the value of currently produced goods and services in the economy. Used goods are not currently produced, and were already counted the year they were newly produced.

However, the services provided this year by real estate agents are in the current year, so the market value of those services, measured by real estate agent commissions are included in GDP.

http://www.sextonmacro2e.nelson.com/student/section/ch06.html

If the house was new it would fall under the expenditure approach the purchase of a house sold at 80 K would fall under Residential Investment (part of I investment).

Using the Income Approach (http://www.econport.org/content/handbook/NatIncAccount/CalculatingGDP/Examples.html#incomeexample )

Depreciation -50 K
business profits - 30 K

Depreciation and business profits would be added together toward National Income.


"For a market to exist there must be scarcity" (T/F)

I agree that air is not, in any meaningful sense, scarce. But there is no market in air! There is a market in canned air, but that's different because canned air is most definitely scarce.

The can is made of metal, which is not in infinite supply. If you use metal to produce a can of air then you must take that metal away from production of, say, airplanes or paperclips. Alternatively, you could just mine more metal. But then you would need to hire more miners, so you are taking workers away from some other industry.

Moreover, the can is manufactured by scarce labour. To manufacture the can, you also need machines and tools that are in limited supply abd could be used to make soup cans instead. The machines stand in a building on land that is in scarce supply and could be used for housing or farming. To get the metal to that building, you need scarce trucks, drivers, and fuel.


Is this a total derivative - if yes, why minus

This is just standard derivative. The minus sign there is because you are taking derivative of a quotient and the quotient rule for derivatives is:
$$ \frac{d}{dx} \left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$$
In this case:
$$\dot{k} = \frac{d}{dt} \left(\frac{K(t)}{A(t)L(t)} \right)\\ = \frac{\dot{K}(t)A(t)L(t)-K(t)(\dot{A}L(t)+A(t)\dot{L}(t))}{(A(t)L(t))^2} \\= \frac{\dot{K}(t)A(t)L(t)}{(A(t)L(t))^2} - \frac{K(t)(\dot{A}(t)L(t)+A(t)\dot{L}(t))}{(A(t)L(t))^2}\\=  \frac{\dot{K}(t)}{(A(t)L(t))} - \frac{K(t)(\dot{A}(t)L(t)+A(t)\dot{L}(t))}{(A(t)L(t))^2}$$
So you just need to remember that $A$ and $L$ are both functions of $t$ so you need to take derivative of whole qutient and quotient rule has minus there (if you want explanation for why quotient rule includes minus the right place is mathematics.se)


Yes, marketing falls under the JEL code of M, specifically M31.


You can also statistically test heteroskedasticity (e.g., White's test: regress squared residuals from OLS on $\hat{y}$ and $\hat{y}^2$, or if you have a simple model, regress $\hat{u}^2$ on $x$ and $x^2$).
It's unclear what you mean by your "model having heteroskedasticity" (do you mean the error term is heteroskedastic?) and "impacting the prediction". You can always choose to ignore heteroskedasticity if you want. (You seem to do so, as you mentioned OLS.) Or you can have a different estimator (e.g., WLS) for a better (sometimes only slightly better) predictor using the same model. Or you can even have a model that incorporates heteroskedasticity explicitly as in, for example, GARCH models in time series. Please be more specific if you want more help. You might want to read this post for time series: https://stats.stackexchange.com/questions/175360/forecasting-ar-arch-garch-models


Elasticity of substitution in Jehle and Reny Advanced Micro (3rd ed) exercise 3.8

I emailed the question to one of the authors. He said the solution manual will come soon!


What international payment systems do exist?

SWIFT, as a private company, only provide secure transmission of financial messages between the participants. The message types or so vast from customer payments and cheques to financial institution transfers, treasury markets, commodity transactions, Letter of Credits etc. Providing instant, secure, without any disconnection messages together with standardization of these messages are a big job for smooth payments.
SWIFT does not hold your money, my money nor does it keep an account for the beneficiary bank or remitting bank.
The main underlying money exchange is done through correspondent banking accounts between financial institutions. These accounts are called as nostro and vostro accounts. The debit or credit from and to these account will be confirmed through SWIFT messages. Before SWIFT, the Telex system is used for communication.

Chinese, CIPS is used only for Renbinmi transactions and is not a competitor to SWIFT. The member banks to this system can use it as a small network but there may be still many banks who are not a member of the system.


The most simple answer I could give is this; they have more revenue and less costs.
A multinational company, has branches all over the Earth and utilizes these and their resources to their fullest extend, with utmost efficiency.
Kindly note how a company is much easier to run than a nation, with regards to uncertainty, risk - danger and of course, budget management.
On the politics side of things, which is arguably less naive;
It is a structuralist theory that MNCs are a tool of developed nations that sucks the life force out of developing nations (by obtaining their resources and bringing profits with them), therefore there is also a relation of causality, between the great net income of an MNC and the lower GDP of such nations.
With this theory, I have to say, I'm fairly skeptical.
If you'd like to read on the politics of such matters, I'd suggest Theodore Cohn's "Global Political Economy: Theory and Practice". In fact, any Political Economy textbook that dwells on the global economy will probably contain a chapter or two concerning multinational companies.


Taking time derivative, growth equations

I'll give a try, perhaps it is just this first derivative giving you trouble:

$$ \frac{d}{dt} ln \left( \frac{ \gamma_t + (n+\delta)}{sA} \right) = \frac{sA}{\gamma_t + (n+\delta)} \frac{d}{dt} \left( \frac{ \gamma_t + (n+\delta)}{sA} \right) $$

$$ = \frac{sA}{\gamma_t + (n+\delta)} \frac{\dot{\gamma}_t + 0}{sA} = \frac{\dot{\gamma}_t}{\gamma_t + (n+g)} $$

And the rest are simply the time derivative of a log equals its growth rate $ \left( \frac{d}{dt} ln X = \frac{\dot{X}}{X} \right) $:

$$ \frac{d}{dt} \bigg[ (\beta-1) ln k_t + (\alpha + \beta - 1) ln N_t \bigg] = (\beta-1) \frac{\dot{k}_t}{k_t} + (\alpha + \beta - 1) \frac{\dot{N}_t}{N_t} $$

Hence concluding the time derivative is:

$$ \frac{\dot{\gamma}_t}{\gamma_t + (n+g)} = (\beta-1) \frac{\dot{k}_t}{k_t} + (\alpha + \beta - 1) \frac{\dot{N}_t}{N_t} $$


Presumably the $\$325K$ represents the sale of the equipment.  It is positive while the purchase of the equipment was shown as $-\$6,000K$

The question says $\$500K$, but that there has been depreciation down to zero (also shown in the table five times as $\$1200K$)

The depreciation has led to lower taxes being paid, so tax at a rate of $35\%$ is due on the sale of the equipment, and $\$500K \times (1-0.35)=\$325K$ is the net benefit to the company from the sale


Solow Growth Model. Steady State. Can someone explain please?

An axis is characterized not by the variable it measures, but by the unit of measurement. So an axis can measure any number of variables, as long as they are measured in the same measurement unit.

In your case, the unit of measurement is "units of output", and both $(n+\delta)k$ and $sf(k)$ are measured in it (capital is measured in units of output, in case you wonder).


Why don't public transport networks have more routes to each terminal in order to reduce the number of transfers?

The people who do the planning do explicitly account for the value of time and convenience to transit users. At least, in all the systems I've worked on. (source: I have had jobs where I worked on exactly this, for London, for the Netherlands, and for international train travel)

Interchanges are modelled with penalties which cover:

the time taken to move from one platform to the other
the time to wait for the connecting train
and a penalty to reflect the inconvenience, which is calibrated through Stated Preference surveys.

Here's an example from the Business Case Development Manual used for evaluating costs and benefits for transport schemes in Greater London, page E-10, showing a penalty just for the interchange itself of 3.5 minutes of travel time. That's in addition to valuing the waiting and walking time at the interchange:

(disclosure: I developed some of the methods, and calibrated some of the numbers, in that manual)

So, why aren't the lines designed for switching between tracks?

Well, sometimes they are: London's Circle, District and Hammersmith & City lines work very much on this basis: similar rolling stock on the same tracks, connecting different termini. London's Northern line works similarly, with two northern termini, two southern termini, and two branches in the middle, with trains switching between them.

At peak times, there is much less switching on the branches on the Northern Line. And the Circle line was recently redesigned so that it is no longer a circle.

The reasons for those two things, are the reasons why it isn't done more often.

The switching reduces the vehicle-capacity of the network. So it can move fewer passengers per hour. Which adds delays to everyone, at rush hour.

Trains are kept a minimum distance apart, in the interests of safety. Where two branches join, as they do at switch-points, that means that trains on both branches have to be kept that distance apart from trains after the switch-point. And the switch-point becomes a bottleneck, reducing the capacity of the two branches: the switch itself has a lower capacity than the sum of the capacity of the two branches.

So, during the busiest periods, there is less switching, and that minimises the overall system aggregated generalised cost. And that generalised cost includes the bullet points that I mentioned at the top of this answer. And when new lines are designed, the value of switching routes to different termini usually adds too much of a penalty to network capacity, to make it worth while: overall system utility is typically maximised with dedicated routes.

And if you don't want to do switching of vehicles on tracks, then you have to build additional grade-separated tracks to do the switching instead, and that's really really expensive, and isn't justified by the reduced generalised cost that arises from fewer passenger interchanges.


Difference between frictional unemployment and seasonal unemployment

What "seasonally unemployed" means?

First, to be "unemployed", one must actively be seeking for work, as the OP stated. The adjective "seasonally" here would reflect the fact that Julie is looking for a seasonal job, and couldn't find any. But this information is not provided to us. Irrespective of her past employment, Julie may have been looking for a permanent job, in which case she should not be categorized as "seasonally unemployed".