Gary Becker's work on Social Demand probably has something to do with the phenonmena.
Becker basically asked why some popular places consistently seemed to underprice their goods. For example, concerts often sell out very quickly and fancy restaurants in the middle of a busy suburb might be often crowded. These venues could raise their prices, but the basic idea is that the demand for a good may not be independent of other agents' demands.
So demand may be dependent on how much other people also demand the good; you might enjoy going out to eat at somewhere that always seems popular than somewhere that gets less traffic, or you might beg your mom that you just have to go to that popular concert by [insert cool band here], specifically because all your other friends are going. Additionally, social places like restaurants and concerts often have customers who usually or only go in groups, so you could see how that would incentivize fiercer competition among these venues.


How should one determine the proper number of lags in a time series regression?

I don't have advice specific to error correcting model (ECM) setting, but in undergraduate applied econometric class they gave us the generic advice to continue to extend lags in the model until the residuals of the fitted model were serially uncorrelated. For example, in the US life expectancy data, residuals of male life expectancy is serially uncorrelated in the AR(5) model but not the AR(4) model. You can see this for yourself with the following Stata code:

use http://www.stata-press.com/data/r8/uslifeexp.dta
tsset year, yearly
reg le_male L(1/4).le_male
estat durbinalt, small
reg le_male L(1/5).le_male
estat durbinalt, small

The Stata documentation for the vector error-correction models also seems to roughly follow this approach but it looks like it is automated under the varsoc function and additionally the AIC, HQIC, and SBIC are all generated programmatically.

  To test for cointegration or fit cointegrating VECMs, we must specify
  how many lags to include. Building on the work of Tsay (1984) and
  Paulsen (1984), Nielsen (2001) has shown that the methods implemented
  in varsoc can be used to determine the lag order for a VAR model with
  I(1) variables. As can be seen from (9), the order of the
  corresponding VECM is always one less than the VAR. vec makes this
  adjustment automatically, so we will always refer to the order of the
  underlying VAR. The output below uses varsoc to determine the lag
  order of the VAR of the average housing prices in Dallas and
  Houston....

  We will use two lags for this bivariate model because the Hannan–Quinn
  information criterion (HQIC) method, Schwarz Bayesian information
  criterion (SBIC) method, and sequential likelihood-ratio (LR) test all
  chose two lags, as indicated by the “*” in the output.

. clear all

. use http://www.stata-press.com/data/r13/txhprice

. varsoc dallas houston

   Selection-order criteria
   Sample:  1990m5 - 2003m12                    Number of obs      =       164
  +---------------------------------------------------------------------------+
  |lag |    LL      LR      df    p      FPE       AIC      HQIC      SBIC    |
  |----+----------------------------------------------------------------------|
  |  0 |  299.525                      .000091  -3.62835  -3.61301  -3.59055  |
  |  1 |  577.483  555.92    4  0.000  3.2e-06   -6.9693  -6.92326  -6.85589  |
  |  2 |  590.978  26.991*   4  0.000  2.9e-06*  -7.0851* -7.00837* -6.89608* |
  |  3 |  593.437   4.918    4  0.296  2.9e-06  -7.06631  -6.95888  -6.80168  |
  |  4 |  596.364  5.8532    4  0.210  3.0e-06  -7.05322   -6.9151  -6.71299  |
  +---------------------------------------------------------------------------+
   Endogenous:  dallas houston
    Exogenous:  _cons

If, as @GraemeWalsh suggests, you would like to use an auto-regressive distributed lags methodology (ARDL) you can do so without having to code it up yourself.

use http://www.stata-press.com/data/r13/txhprice
sort t
net install ardl.pkg
ardl dallas houston,  maxlag(4)

ARDL regression
Model: level

Sample:  1990m5 - 2003m12
Number of obs  = 164
Log likelihood = 313.86816
R-squared      = .96315461
Adj R-squared  = .96246376
Root MSE       = .03613756

------------------------------------------------------------------------------
      dallas |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      dallas |
         L1. |    .427744   .0789194     5.42   0.000      .271886     .583602
         L2. |   .1747019   .0720507     2.42   0.016     .0324089    .3169948
             |
     houston |   .3404766   .0567884     6.00   0.000     .2283252     .452628
       _cons |   .7276476   .2061803     3.53   0.001     .3204618    1.134833
------------------------------------------------------------------------------


$i_t$ should be the interest rate at time $t$. $\pi_t$ the inflation at time $t$. $Y_t$ the income at time $t$. The variables without indices are the targets of the central bank. The interpretation of this Taylor rule is therefore that the current interest rate is a function of the past deviation of the interest rate from the target, the current inflation deviation from target, the current output gap, the exchange rate deviation from target, and deviations from the current money growth target.


Do German Goverment bonds usually include (or included) a condition that the debt will not be serviced in case of trade imbalance?

The answer is somewhat. Those terms were not part of the original bond but were added by the Debt Agreement

  The Conference recognised the principle that the transfer of payments under the Settlement Plan implies the development and maintenance of a balance of payments
  situation in which those payments, like other payments for current transactions, can be financed by foreign exchange receipts from visible and invisible transactions so
  that more than a temporary drawing on monetary reserves is avoided. (Final Report of the Conference on German External Debt 1952: paragraph 21)

That quote is NOT part of the official agreement (As seen by it's absence in the British  and the Swiss copy) - however the "One made it out of the debt trap" study claims it was accepted by the British and the French.


What is the subgame perfect pricing policy for seller?

To find the SPNE you want to consider all possible scenarios that the seller might face in the second period, and what his best strategy would be. This will help you study the seller's actions in the first period since they will lead to different scenarios in the second one. In fact, saying "the subgame perfect pricing policy is set p1 = 1199 and p2 = 299" is not a valid statement, the subgame perfect policy should be of the form: " the seller chooses p1=number in the first period and p2= a function of which players remain in the market"

Let me be more clear with what I mean:

The seller might find in t=2 that no one has bought the good, and must decide between choosing p2=300 and selling three units (revenue of 900), choosing p2=400 and selling 2 units (revenue of 800) or choosing p2=700  and selling one unit (revenue of 700). Clearly, in this case, the best pricing policy is 300.

Similarly, the seller might find himself in the situation that one player has bought the good in the first period. Say player M bought the good in period 1. In that case, it is better to set p2=700 and sell 1 unit than p2=300 and sell 2 units.
(Notice that this case is not very relevant because if M finds it profitable to buy in period 1 so will H, but it is a possibility you must consider.)

It is not hard to see that unless buyers L and H are the only ones in the market in period 2,  the seller will find optimal to choose p2 to be equal to the smallest valuation among the players still in the market, and every player that has not bought the good, buys it.

Given this strategy, you can now study the choice of p1. For example: If the seller sets p1=2200 no one will buy in period 1.  This is because buyer M will rather wait to period 2, anticipating that no one will buy in period 1, thus the seller will drop the price to 300. Therefore, she will be able to make a surplus of 700-300=400. We learn that for buyer H to be willing to buy in the first period, she must get a surplus of at least 400.  Therefore, the maximum that the seller can charge buyer H for the good in period 1 is 2200-400=1800. To sum up, if p1=1800, H buys in t=1 and the other two buy at t=2 at p2=300, giving a total revenue of 1800+600=2400.

Through a similar argument, you can find that if the seller wants buyer M to also buy at t=1, the largest price he could charge this buyer is p1=1500 (otherwise, buyer M would prefer to wait and have a surplus of 100 at t=2). Now let's compare the total revenue of p1=1800 vs p1=1500. If p1=1500, M and H buy at t=1 and L buys at t=2. The total revenues are 1500+1500+300=3,300, which is better than 2,400.

Lastly, would it make sense to try to sell to buyer L? in that case, the biggest price you can charge in t=1 so that L buys is  p1=900. Since anyway L cannot guarantee himself any surplus. However if p1=900, all buyers buy at t=1 and you make a total revenue of 2,700.

We conclude that the subgame perfect pricing policy is to set p1=1500, and p2 equal to the smallest valuation among the buyers still in the market, except if only buyers L and H are in the market. In that case,  choose p2=700. (Note that expressing the equilibrium strategy of the buyers will also be a bit wordy, but I think the question is interested only in the seller's pricing policy, so this is sufficient).

It might sound redundant to express p2 the way I did, but it is crucial in order to show that no player will have an incentive to deviate.


That is natural and you are not missing anything.

Let $y=\alpha + \beta z + u$. Your prediction of $y$ given $z=1$ is $\hat{y} = a + b$, where $a$ and $b$ are the OLS estimates. The prediction error ($y - \hat{y}$) for $z=1$ is, thus, $(\alpha - a) + (\beta-b) + u$, which is involved with $b$. On the other hand, for $z=0$, the prediction of $y$ is $a$ and the prediction error is $(\alpha-a) + u$, which has nothing to do with $b$. As you said, the former depends on $b$, while the latter does not. (Perhaps it would help to remember that the value of $z$ is given for the prediction, and thus the prediction intervals depend on the value of $z$.)

(Calculation of the prediction intervals:) You can calculate the prediction intervals using the formula $(a+b) \pm se((\alpha-a)+(\beta-b)+u) \cdot \textit{critical value}$ for $z=1$, and the formula $a \pm se((\alpha-a)+u) \cdot \textit{critical value}$ for $z=0$. which is fine. Note that these two prediction intervals can also be written as $(a+b) \pm se(a+b+u) \cdot cv$ and $a\pm se(a+u) \cdot cv$, respectively, because $\alpha$ and $\beta$ are constant (nonrandom). How to estimate the standard errors can be found in Wooldridge's textbook (the "Prediction and Residual Analysis" section; 6.4 in 5ed).


Relationship between convexity and a perfect complements type utility function

Instead of directly giving you the answers, I'm going to give you a series of hints to help you figure out the answers on your own.

  1.What would be the shape of the indifference curve?

Consider three different consumption bundles: $(3,10)$, $(10,10)$, and $(10,3)$. Verify that these three bundles yield the same utility to the consumer. In other words, they should lie on the same indifference curve. Then draw a curve through the three points. Verify that all points described by $(x,10)$ and $(10,y)$ for $x,y\le10$ should also be on this indifference curve.

  2.Why are these preferences weakly convex?

First remind yourself what weak convexity means. Then compare the utilities that the individual gets from the following pairs of bundles:

$A_1=(10,10)$ and $A_2=\alpha(10,3)+(1-\alpha)(3,10)$ for $\alpha\in(0,1)$
$B_1=(10,10)$ and $B_2=\alpha(10,3)+(1-\alpha)(10,10)$ for $\alpha\in(0,1)$

  3.What does the negative sign imply mathematically?

You said that the consumer hates both goods. Compare the utilities from the bundles $(10,10)$ and $(5,5)$ and see which is higher (and thus more preferred).

  4.Will it become an min{x,y} function?

Try sketch an indifference curve for $u=\min\{x,y\}$ and compare to those in part 1.


No, that is not correct. In the Keynesian model, the only variable that reacts to output is consumption. Other components of the aggregate demand do not.

For instance, in a closed economy, aggregate demand is $C+G+I$. Here, $C=c_0 + c_1Y$, where $c_0$ is autonomous consumption and $c_1$ is the MPC.

Hence, planned expenditure is

$$ Y^e = c_0 + G + I + c_1Y$$

which can be condensed to

$$ Y^e = a + bY $$

This is the equation you are given. Thus, the MPC is 0.8. This is actually the slope of the PAE curve in the Keynesian Cross diagram. It is usually lower than one. Calculation of the multiplier goes as normal.


It isn't necessary to stress that these figures are pegged to the US dollar. A PPP dollar is a dollar in its own right. Thus, simply saying "The budget is three billion purchasing power parity dollars" is good. Mentioning PPP in the context of a US project is unnecessary.

The World Bank uses the phrase "international dollars" on its data portal, so you can say "international dollars" if you are willing to explain what that means to anyone unfamiliar with that phrase. That phrase has the advantage that it is shorter, so it is less cumbersome to say out loud.


Relation between marginal cost and output elasticities

I figured it out:
The first-order condition of the cost minimization problem for, say, material inputs $m_{it}$ gives:
$ \lambda \frac{\partial F}{\partial M} = P_M $
Where F is the production function, $P_M$ the material input prices. Multiply by $\frac{M}{F}$ and rearrange,
$ \lambda = \frac{P_M M}{\beta_M F}  $,
where $\beta_M$ is the output elasticity with respect to material inputs. Similar results obtain for other inputs. Summing them all gives the relation to the short-run returns to scale.


Bertrand-equilibrium with discrete price set

The key is the definition of the Nash equilibrium solution concept that you are applying to solve your game. In non-formal terms, a Nash equilibrium is a set of strategies such that no player can increase his or her payoff by deviating to some alternative strategy.

Let's consider $\{30,30,30\}$. In this 'putative' equilibrium, firm A's profit is

$$\frac{D(p)}{3}(p-c)=\frac{20}{3}(30-20)=\frac{200}{3}>0.$$

Could A do better with a lower price? A price of $p=10$ is less than his cost, so this obviously can't be better. Likewise, a price of $p=20$ yields profit

$$D(20)(p-c)=0,$$

which is less than the profit (200/3) in the putative equilibrium. So, again, a deviation to $p_A=20$ is not profitable.

Can A do better by deviating to $p_A>30$? The answer is clearly no because any $p_A>p_B,p_c=30$ implies that A attracts no demand and therefore earns zero profit.

We thus observe that A cannot find a profitable way to deviate from the putative equilibrium $\{30,30,30\}$, so this is indeed a Nash equilibrium.

To address your final question: there are two general approaches to problems of this type.

For each player in the game, compute their best response (i.e. profit-maximising strategy) to each possible combination of strategies by the other players. Then look for all situations in which every player is playing a best response; that is a Nash equilibrium.
Guess a set of strategies that you think might be an equilibrium. Then verify that is is indeed an equilibrium by checking that no player could do better by changing their strategy.


What is the effect on GDP when foreigners buy lots of domestic bonds?

For every buyer there is a seller. If a foreigner buys a bond, that means a domestic investor sells it to them (or issuer). Why should the identity of who owns the bond matter?

In practice, there are many moving parts. One feature of financial crises is that investors buy U.S. Treasury bonds as a flight to safety. This means that purchases coincide with the weakening economy created by the financial crisis.


If price increases and quantity remains the same can we infer that demand is perfectly inelastic?

We know for a fact that supply is perfectly price inelastic. Each school typically has a quota -- some rough estimate of the number of students it wishes to enroll.

And every year, there are many who are able and willing to pay the full price of the good (a spot in the school) but are nonetheless not admitted.

Example. For decades, a school has consistently offered 10,000 seats to each cohort. Demand though has consistently risen so that price has also risen. However, we cannot infer anything about the slope of D (i.e. whether demand is price inelastic).

That demand is also price inelastic may very well be true, but cannot be inferred simply from the fact that the number of students hasn't decreased.

Another important consideration/complication is price discrimination. Through financial aid, US universities practice what amounts to near perfect price discrimination, making many students pay as much as they can bear. So, while the sticker price has risen greatly, the actual price paid by many students may not have risen as much.


Is Foreign investment positive or negative for a country?

From a purely economic perspective, it doesn't make sense to feel resentful when a national company is sold to foreigners. Imagine a company in my country is sold to a German investor. This shouldn't be a problem, as from the sale price my country could go to Germany and buy a German company instead. So it's all fair game.

However, some countries are growing in influence relative to others. For example, China is growing in influence and economic power relative to the rest of the world. When Chinese companies buy European companies, this is a very obvious sign of that growing influence and this is one reason it raises concerns in the countries that are being targeted. It is hard to accept that one's country might be losing in relative influence.

From a strategic point of view, it might make sense to try and protect national companies from foreign buyers. If the Chinese buy a German company, they might get access to new technology, which can help them develop their own industries. This can lead to increased competition for the German industry. By protecting national companies and proprietary technologies, Germany can limit competition and stay more competitive on the world stage.

The above discussion covers the buying and selling of existing assets, which is usually more controversial. Foreign investment in greenfield projects (when a foreign company builds a completely new factory) is generally considered positive for the national economy as it creates jobs and generates additional tax revenue for the government.


Adam Smith: If humans are self-interested individuals, why should we reinvest our wealth?

Economics have evolved considerably since Adam Smith and his ideas have been refined. I can think of two approaches to your question:

Why would a self-interested individual re-invest wealth?

1) Financial perspective

Whenever you invest money instead of spending it now, you get paid interest. If that interest is higher than the satisfaction from spending the money now instead of later, then re-investing is in your self-interest.

2) Game-theoretical and behavioral perspective

A self-interested person wants to maximize his or her payoffs. Payoffs can differ from monetary outcomes. If I re-invest my 1000 dollars into bonds for a year I can get 1050 dollars at the end of the year. But if I give 1000 dollars to charity, I may get a lot more satisfaction from that, provided that I am an altruistic person. So if try to measure my payoff from the charitable donation, it may be worth the equivalent of 2000 dollars to me. It is in my self-interest to donate instead of re-investing then. The same reasoning can be applied to saving and spending. If it gives me pleasure to hoard money then I will choose to re-invest rather than spend. Using this approach you can justify any behavior as self-interested or rational.

You can take financial perspective to explore how productive and unproductive labor relates to your question, but if you take more general game-theoretical perspective, then it will be difficult for you to even define "unproductive labor" as even piling money and setting it on fire may be considered productive in some situation. But I would take the latter. After all we should study economics to better understand the world in all its complexity, not to take the easier path.


Yes but note economic development is more broad than just GDP. GDP measures economic output of a country and higher GDP correlates with higher development and it can be also used as a measure of economic development but more narrowly GDP is just a measure of output.
This is correct, GDP can’t measure things like home production. It’s a disadvantage of GDP but I would not necessarily say it implies it’s not a good measure. Sometimes alternative measures can be even less appropriate but if you care mainly not just about output but development there are certainly measures that might be more appropriate. For example, often development economics uses the number of people in absolute poverty as development indicator, or access to education or other factors. There is also Human Development Index (HDI) that includes GDP but tries to also combine more indicators in it such as environmental quality and again education. But for some research purposes GDP might still be the best development indicator - it’s highly case specific. For example, GDP is quite objective and data on it are easily available, even if you would consider HDI superior measure, GDP might still win thanks to data availability as having large number of observations is often important.
Yes real GDP is inflation adjusted. This is important because value of money changes across time dollar today is not same dollar as dollar year ago. This situation is like if in physics length of a meter would change every year. Real GDP corrects for this. However, this only adjusts for changes in value of dollar across time. If you are doing some cross-country comparison you should also think about adjusting for purchasing power parity (the fact a dollar can have different buying power across countries).


Reason behind the decomposition of price effect into substitution and income effects

Why are we decomposing the effect of a change in price of one commodity into all these effects which are (as I have understood), all hypothetical.

They're hypothetical, but they are nevertheless meaningful.  We can easily imagine what it would be like if an agent had a larger budget while prices remain unchanged; we can also imagine what it would be like if, under diminishing marginal utility assumptions, the relative prices of goods were to change.

One reason we do this is that it helps develop intuition and show why naive assumptions don't necessarily hold.  For example, if we have goods $x$ and $y$, prices $p_x$ and $p_y$ with budget $B$ in an initial equilibrium, we might naively think that setting $p_x^{'} = 0.5p_x$ means that we consume twice as much of $x$, when this is not the case for most utility functions.

  Also in substitution effect, why do we substitute the relatively less expensive commodity for the relatively more expensive commodity? Is there any particular advantage of this decomposition?

The answer to the second question is that the Slutsky decomposition allows us to tell a story that answers the first question - when the price of $x$ falls, we may buy more of $x$ (an income effect), but we will also buy more of $y$ because diminishing marginal utilities implies that after using the price-savings to buy some number of additional units of $x$, we get more "bang for the buck" by purchasing an additional unit of $y$ instead.  This is the substitution effect.

  Also I don't exactly understand the meaning of real income.

Real income is income adjusted for inflation from a base year, so that it reflects real purchasing power. In the context of these Econ 101 budget-line models, it's implied that the budget is in real units for pedagogical reasons. There's no base year, but the point is to ensure that any price-changes contemplated are "real", and to get you to think of consumption as the basic economic unit, rather than money per se. Often you'll see simplifications where one of the goods is priced at $1/unit, and referred to as the "numeraire".


How does total money in the economy changes?

Fiat money in modern societies is usually printed (minted is the term used for stamping coin) under the authority of a particular institution, usually called Central Bank, Federal Bank, Federal Reserve etc. In different countries there are different rules that apply concerning its function and independence from central government.
Central Banks also are able of conducting monetary policy with various means and measures. In fact, more than anything, a Central Bank is charged with the responsibility of regulating the amount of money circulating in the economy.
A central bank can augment or decrement the amount of money circulating by using (1) a particular exchange rate or by directly influencing (2) money supply.
Examples:
(1) A Central Bank can change the exchange rate with which it loans commercial banks (the ones that the public most often interacts with), thereby affecting their interest in giving loans and/or the public's interest in saving or taking loans. This effectively changes money supply / the amount of money circulating in the economy.
(2) A Central Bank can buy (or sell) government bonds from (or to) the general public. This is called Quantitative Easing. Since that means that either it injects new money into the economy or that it vacuums existing money out of it, it directly changes the amount of money in circulation.
Please note:
There are many more examples to be given and a lot more clarification. Further reading can be done with my textbooks of preference, Mankiw's Macroeconomics and McCallum's Monetary Economics.


Assuming you mean unit elasticity of demand with respect to price the answer is yes.
From the information you have, we can deduce that the demand function is as follows:
$$Q^d(p)=10/p$$ (I always spend $\\\$ 10$ on coffee, which buys me exactly $10/p$ units of coffee, where $p$ is the price of coffee).
Now recall the formula for computing elasticity of demand with respect to price:
$$\varepsilon \equiv \frac{dQ^d}{dp} \frac{p}{Q^d}$$
And the rest I leave to you.


How to calculate inflation rate in order to perfom VAR model?

Stock and Watson use log approximation and quarterly data. Premultiplying the log approximation by 400 could be due to data being quarterly.

Your second method is fine too.


This is just an extended remark:

Let $\Delta$ be the space of Borel probability measures on $[0,1]$. An equilibrium can be represented as an element of $\Delta^2=\Delta\times\Delta$. A two player game with continuous payoff-functions with action spaces $[0,1]$ can be viewed as an element of $C[0,1]^2\times C[0,1]^2$, where $C[0,1]^2$ is the space of continuous functions (here representing payoff functions) from $[0,1]^2=[0,1]\times[0,1]$ to the reals, endowed with the topology of uniform convergence, given by the $\sup$-norm $\|\cdot\|_\infty$ defined by
$$\|f\|_\infty=\sup_{x\in [0,1]} |f(x)|=\max_{x\in [0,1]} |f(x)|.$$
One defines $C[0,1]$ similarly.

There is also a well-behaved metrizable compact topology on $\Delta$, the topology of weak convergence (or weak*-topology); the coarsest topology such that the function $\mu\mapsto\int f~\mathrm d\mu$ is continuous for each $f\in C[0,1]$. In this context, convergence in this topology is convergence of cumulative distribution functions at continuity points. We let $\Delta^*$ be the set of Borel probability measures on $[0,1]\times [0,1]$, similarly endowed with the topology of weak convergence.

We can then define an equilibrium correspondence $$\eta:C[0,1]^2\times C[0,1]^2\to 2^{\Delta\times\Delta}$$
such that $\eta(v,u)$ is the set of equilibria given the payoff-functions $u$ and $v$. The correspondence has nonempty values by a theorem of Glicksberg. Since $\Delta\times\Delta$ is compact, we can show that $\eta$ is upper hemicontinuous by showing it has a closed graph.  The function $p:\Delta\times\Delta\to\Delta^*$ that maps mixed strategies to their independent product is continuous (see Billingsley's book on weak convergence). It follows, using a bit of analysis, that the "expected payoff"-function $k_i:\Delta\times\Delta\times C[0,1]^2\to\mathbb{R}$ is continuous for $i=1,2$. Now for each $\mu\in\Delta$, let
$$E_\mu^1 \{(v,u,\nu,\tau)\in \Delta\times\Delta\times C[0,1]^2\times C[0,1]^2\mid k_1(\nu,\tau,v)\geq k_1(\mu,\tau,v)\}$$
and
$$E_\mu^2 \{(v,u,\nu,\tau)\in \Delta\times\Delta\times C[0,1]^2\times C[0,1]^2\mid k_2(\nu,\tau,u)\geq k_1(\nu,\mu,u)\}.$$ Each of these sets is closed and the graph of $\eta$ is simply
$$\bigcap_{\mu\in\Delta}E_\mu^1\cap \bigcap_{\mu\in\Delta}E_\mu^2,$$
a closed set, so $\eta$ is upper hemicontinuous.

Now this doesn't tell us anything about what happens if payoff functions converge pointwise and not uniformly!!! The only useful information is that strategy spaces are compact, so there will still be a convergent subsequence. If you can show that this subsequence must converge to your candidate equilibrium- this will require making use of the details of the game, you should be good to go.


Is the set of optimal strategies convex in a single-agent decision choice problem?

The set $Q^*$ is the set of probability distributions over the maximizers of $u$ for each value of $v$. So for a fixed $v$ all the values of $Y$ that have positive probability must give the same utility. That is, if $P_Y(y|v)>0$ and $P_Y(y'|v)>0$ then $u(y,v)=u(y',v)\geq u(\tilde y,v)$ for all $\tilde y\neq y, y'$. Therefore, convex combinations of elements in $Q^*$ must also be collections of probability distributions $P_{Y|V}$ supported over the same elements in $Y$. That is, the convex combination will also be supported over maximizers of $u$ for each $v$.

In conclusion, even though the set of maximizers is not guaranteed to be convex, the set $Q^*$, of distributions, is a convex set.


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

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


How to solve for the potential of a cournot duopoly problem?

A function $P(q_1,q_2)$ is said to be the potential function of a game if its derivatives, coincide with the marginal payoff of each player.

So for example, in your case, if you have that the  payoff of a firm is its profit. Let me call it $\pi_1(q_1,q_2)=TR-TC=p\cdot q_1-c\cdot q_1$ for firm 1 and $\pi_2(q_1,q_2)=TR-TC=p\cdot q_2-c\cdot q_2$ for firm 2, then to check that P is the potential of this game you want to check that $\frac{\partial P(q_1,q_2)}{\partial q_1}=\frac{\partial \pi_1(q_1,q_2)}{\partial q_1}$ and that $\frac{\partial P(q_1,q_2)}{\partial q_2}=\frac{\partial \pi_2(q_1,q_2)}{\partial q_2}$.

The problem is not asking you to solve for the optimal quantity just yet, so $MR=MC$ will not quite help you (at least not in that specific question).

To see more aboutthe definition of a "potential function" and a "potential game" see here.


Numbers of right and left gloves Question?

I don't think it is appropriate to apply the such static visualization to dynamic problem. Let us for a second forget about the graph - the graph is just a visualization of the indifference curve.
If two things are perfect complements which are consumed in 1:1 ratio their utility is given as:
$$u(x, y) = \min \{x, y\}$$
In this case (before you break the arm) the indifference curve will indeed look like L. If you break your arm the utility function would have to change but if we assume that the second good $y$ does not give you any utility it will just default to some utility of one variable $u(x)$.
However, for the $u(x)$ we cannot meaningfully plot an indifference curve and neither we can do it for $U(x,y) = x^{\alpha}$. Why? Because indifference curve is by definition a curve along which utility of consuming some combination of $x,y$ is constant. If $y$ does not gives us any utility then the person will always just consume maximum $x$ possible. Why should the person even care about $y$ at all? In such case the concept of indifference curve becomes meaningless as there is nothing to be indifferent about. You always just consume as much $x$ as possible.


Is protectionism more suited to large economies?

This is partially opinion-based. Even if we were able to calculate all the effects, we can still argue if variant A is better than variant B or vice versa.

I don't think that protectionism is suited to any economy. It typically protects local producers at a cost for local consumers. Local producers have less competition on local market, which allows higher prices or worse products, lower efficiency and so on. The winner is not the nation/state. The winner is just the producer, at expense of consumers.

Considering just relations between two states, protectionism damages both of them. It prevents usage of comparative advantages.

It might look like some big players like USA can more or less ignore rest of the world and produce locally everything they need. Well, they probably can, but at some cost. They can ignore manufacturing in China and build phones locally. Those phones would be quite more expensive and less competitive on both local and global market. They can force some advantage for local vendors on local market (at some cost for local consumers), but this does not help the companies on global market. Even with comparable level of isolation, United States could be richer than, say, North Korea or Iran. But they would be nowhere as rich as they are now.

By the way, there is some evidence of damages of de facto protectionism in large countries: sanctions on Russia. You might argue this is not protectionism. However, it has quite similar consequences – it inhibits international trade. The reason does not matter, the result seems to be the same.

Maybe it would be more disasterous* for a small country. But I don't think protectionism is useful even for large country.

*) in per-capita metrics


Under what conditions is a monopoly undesirable?

Firstly, suppose we take a utilitarian welfare standard that is linear in money. That is to say, suppose that both utility and profits are linear in the amount of money that consumers and firms have (but not necessarily linear in anything else). In that case, the Pareto standard and the utilitarian social welfare criterion coincide exactly! You can see a nice video of Jeff Ely talking about the intuition for this result here (look at the video titled "efficiency"). Intuitively, if both the utility and profits are linear in money then we can always maximise utilitarian welfare by implementing the Pareto optimum and then constructing side-payments to support it.

Now, the answer to the question of when a monopolist is undesirable depends on the richness of the model that one has in mind. In a very basic textbook model of a monopolist, FooBar's criterion is a good one. We know that an ordinary competitive market equilibrium maximises total welfare (the shaded area in the below figure) with price equal to marginal cost (n.b. the supply curve and the marginal cost curve are essentially the same thing):

Since the profit that a monopolist makes from a unit is equal to difference between its price and its marginal cost, the monopolist will tend to set price above marginal cost. This results in a higher price (the green $p$) and a reduction in total welfare (given by the shaded area in this figure) and consumer surplus (the triangle between the price line and the demand curve):

Since this kind of analysis is standard, it makes sense to think of the case in which a monopolist reduces consumer surplus and welfare as being a kind of 'default' and instead ask "when might a monopolist be desirable"? Here are some situations in which having a monopolist might be better than having very intense competition:

As Jyotirmoy Bhattacharya noted in a comment, First Degree Price Discrimination. This describes the situation in which the monopolist can charge each consumer an individualised price that is exactly equal to that consumer's willingness to pay. Thus, the monopolist captures all consumer surplus so that
$$\text{Social welfare}=\underbrace{\text{Consumer surplus}}_{=0}+\text{Producer surplus}=\text{Producer surplus}.$$
Since producer surplus and social welfare coincide, when the monopolist acts to maximise its own surplus it is also maximising social welfare! However, whilst this is good for welfare, it is very bad for consumers. It also requires that the monopolist has enough information about consumers to accurately estimate their willingness to pay, that such discrimination is not illegal, and that consumers are willing to tolerate it (i.e. there is no PR backlash--see, e.g., here).
Large economies of scale. Sometimes the fixed costs of establishing a business are massive. For example, to set up a rail/telephone network requires that you lay train tacks/phone lines across the entire country. Once this cost is incurred, the infrastructure can be used at close-to-zero marginal cost. In this kind of environment it doesn't make sense for society to incur the fixed cost more than once. The infrastructure company is then called a natural monopolist. In these kinds of industries a monopolist is usually allowed to operate and is either (i) owned by the government, or (ii) heavily regulated to ensure that price is not set far above marginal cost.
Network externalities. Some products are only valuable to you is other people also use them. For example, Facebook and telephones (and Stack Exchange!) are only useful because they can be used to contact other people. The more people you can contact, the more valuable to the product becomes. If everybody uses a single monopoly social network then that network will be more valuable than if people are divided across a large number of competing social networks. Network effects are sometimes call demand-side economies of scale.
Innovation and investment. Firms invest in R&D to develop new products (or new production technologies). Why bother? Presumably, firms do this because they anticipate earning a profit selling those new products (or from the reduced cost of the new technology). But their ability to earn this profit will depend on how much competition the firm faces. Thus, one might think that competition should reduce the incentive to innovate by eroding the associated profits. In fact, a famous paper by Aghion and coauthors shows that adding competition first reduces innovation and then, once competition gets fierce enough, increases it again. Thus, if the choice is between a monopolist and a few firms then a monopolist might result in more innovation.
It's perhaps worth mentioning that this last point is closely related to the idea of intellectual property. Since most intellectual property (i.e. creative works like books, songs, or movies; and inventions) are information goods, they can be duplicated or imitated at zero marginal cost. That means that if we allowed competition we should expect there to be many competitors and it will be very hard for anyone (including the original creator) to make any profit whatsoever. In order to ensure that creators and inventors have an incentive to create new works or new inventions, the state therefore grants them a temporary monopoly in the form of either copyright protection or patent protection.

Those are a few of the most common ways in which a monopolist might be preferable to competition. I am sure there are others that don't come to mind right now. The main message is that you have to look at the specific market context to evaluate whether there might be any problems associated with intense competition. Most often the answer will be that there are not, and the usual assumption is therefore that monopolists are undesirable unless we have good reason to think otherwise.


Usefulness of the value for the slope of the demand curve

Price elasticity of demand is related (but not equal) to the inverse of the slope of the demand curve.

This Wikipedia article defines PED as:

  Price elasticity of demand (PED or Ed) is a measure used in economics to show the responsiveness, or elasticity, of the quantity demanded of a good or service to a change in its price, ceteris paribus. More precisely, it gives the percentage change in quantity demanded in response to a one percent change in price. [emphasis added]

Look at the following (canonical) supply-demand graph.

Let's define $P^*$ the price of market equilibrium, $Q_D$ as the quantity demanded and $E_D$ as price elasticity of demand. Then:

$$ {slope} = \frac{rise}{run} = \frac{\Delta P}{\Delta Q_D} $$

And in mathematical terms, the verbal description of the PED reduces to:

$$ E_D = \frac{\frac{\Delta Q_D}{Q_D} \cdot 100}{\frac{\Delta P}{P^*} \cdot 100} $$

$$ E_D = \frac{P^*}{Q_D} \cdot \frac{\Delta Q_D} {\Delta P} $$

$$ E_D = \frac{P^*}{Q_D} \cdot \frac{1}{slope} $$

Edit

Constructive comments have been incorporated into this answer.

In particular, @Kontorus pointed out:

  The PED is a percentage change, the slope of the demand curve is an absolute change.

And @denesp pointed out that:

  ... linear demand curves have constant slopes but changing elasticity. Cobb-Douglas derived demand (hyperbole) has changing slope but constant elasticity.


Why does HDI use log-transformed data for income index?

Decreasing returns to standard of living means that if you add a certain amount $\delta$ to the income of a low income person, then the increase in her standard of living is bigger than if you would give the same amount to a high income person.
Let $SL(x)$ be the standard of living for a person with income $x$.
Take a person with income $x$ and a person with income $y > x$, and let us add an amount $\delta$ to the income of both persons. Then we should have that:
$$
SL(x + \delta) - SL(x) > SL(y + \delta) - SL(\delta).
$$
The left  hand side is the increase in the standard of living for the poor person. The right hand side is the increase in the standard of living for a rich person.
Dividing by $\delta$ and taking the limit for $\delta \to 0$ gives:
$$
\frac{d SL(x)}{dx} > \frac{d SL(y)}{dy}.
$$
The left hand side is the marginal return to the standard of living for a person with income $x$ and the right hand side is the marginal return to the standard of living for person with the higher income $y$.
Decreasing marginal returns means that this marginal return decreases when income increases. So it is lower for the higher value $y$ compared to the lower value $x$.
A function with a decreasing slope is called a concave function. So what you want in the end is a function that is

Increasing, i.e. $SL(x)$ increases with $x$
concave, i.e. $dSL(x)/dx$ decreases with $x$.

One (popular) function that satisfies these two conditions is the natural log function. Note that $d \ln(x)/dx = 1/x$ which indeed decreases with $x$.
The function that the world bank uses:
$$
SL(x) = \frac{\ln(x) - \ln(x_{min})}{\ln(x_{max}) - \ln(x_{min})},
$$
is just a linear transformation that makes sure that:

$SL(x_{min}) = 0$
$SL(x_{max}) = 1$.

So the function is:

increasing
concave
between 0 and 1


Could a "Japanese-style strike" work as it's supposed to? Under what conditions?

Without the cooperation of at least mid-level management I dont think that such an approach can work even in theory. Management sets the quantity of ressources and intermediate goods the striking employees have at their disposal to use as inputs.

The theory is inconsistent in the sense that on the one hand it relies on assuming JIT (for its lack of large storage facilities for final products) to obtain the dumping of a large quantity of output on the market. On the other hand "no limit to no the number of goods that can be produced" implies either large storage facilities for inputs or that the company striking is Bertrand's water well.


Negative Inflation, AS/AD framework, What Graph to use?

It seems that your teacher wants you to analyze the effects a higher inflation rate might have on the level of aggregate output based on whatever policy you might choose, i.e. drawing an AD/AS graph in long run equilibrium and demonstrating some shift as a result of the inflation-increasing policy.


The Federal Reserve is only issuing cash money?

Deposits at the Federal Reserve largely count as money, at least as far as the depositing commercial banks are concerned.

Apparently on March 1 2016, the Federal Reserve Banks' liabilities included $\$1475$ billion of Federal Reserve notes, net of F.R. Bank holdings, and $\$2530$ billion of deposits, together with $\$412$ billion of other liabilities. See the second part of table 5 of https://www.federalreserve.gov/releases/h41/current/ for more details

Coins are not issued by the Federal Reserve, but by the United States Mint


How to prove that profit function is convex in price (with smaller price)?

I don't think it is a case. On a page you linked we have a proof that the profit function is non-decreasing in $p$. If the output price decreases, $p \geq p'$ and factor prices remain constant, $w_i \leq w'_i$ for all inputs, then the profit would be less or equal to the previous one.

Then, in connection to your previous question: How to prove that a concave production imply that the input requirement sets are convex? you can let $\mathbf{y}=(y,-\mathbf{x})$ be maximizing profit at $(p,\mathbf{w})$ so that your profit function becomes $\pi(p,\mathbf{w})=py-\mathbf{wx}$. On page 12. of the Lectures in Microeconomic Theory Fall 2009, Part 3 you have linked, you can see a proof of that (probably the same as on the page you linked just now but not sure as the page does not open properly to me (cannot really see the mathematical equations).

As for your second question - "reverse convexity" is a little bit strange expression, to show what the author meant I will use a picture.

As you can see it is still convex in input prices $w$. However, in comparison to plot with output prices $p$, it is decreasing.

Algebraically, you can see it that way:

Profit function is convex in both prices: $$\frac{\partial^2\pi}{\partial p^2}>0 \;\;\; \textrm{and} \;\;\; \frac{\partial^2\pi}{\partial w^2}>0$$
Profit function is increasing in output prices $p$: $$\frac{\partial\pi}{\partial p}>0 $$
Profit function is decreasing in input prices $w$: $$\frac{\partial\pi}{\partial w}<0 $$


Industries that thrive in crisis like COVID 19

In no particular order:

Consumer Staples: household goods, food, beverages, hygiene products, and other products that individuals are either unwilling or unable to eliminate from their budgets even in times of financial trouble
IT: Internet/Social Media, Software, and Semiconductors.
Healthcare: Anything that involves the provision and coordination of medical and related goods and services. Not to mention, small biotech companies.


Facebook's Libra - Why would it vary in value

It will vary in value because trust in it will fluctuate just as trust in facebook fluctuates.

There will also be volatility as trust varies in confidence in the payment system; its integrity and security; how exchangeable it is; its exposure to fraud,  to tax, to money-laundering investigations

For holders, there is also significant counter-party risk in whether Facebook chooses to honour their holding. And the majority voting rights in Facebook are vested in a single person. So their counterparty is, in some regards, essentially one man and his whims.


What have been the main developments in macroeconomics and financial economics since the 2008 financial crisis?

Well, in macroeconomics, namely in DSGE modelling, VOXEU has recently published a report on its uses by Central Banks (CBs), and future lines of improvement, which academics have been tackling but still hasn't found its way into CBs policy analysis. There's no space to explain all the new topics, nor do I think it's the intention of this question. So, I'll just enunciate the topics, with some references. More can be found in reading the report. Also, I'll focus on the theoretical considerations, i.e., it's known that, empirically, the the VAR obtained by solving the DSGE model can be considered as misspecified (see here). This is a WP, but you can easily find many articles on this subject, increasingly so since the crisis)

Financial Frictions (an example)
Forward Guidance(extending low-rates period), due to the presence of a ZLB (see here; this was already being studied before the crisis.)
(Incomplete Markets) Heterogeneity across households in
the transmission of monetary and fiscal policy (see here). I would also add firm heterogeneity (example here)
Systemic fragilities, i.e., economy more prone to crisis due to increasing leverage (an example)
Allowing for structural (long-run) changes, instead of a simple sequence of short-run shocks, this allows for fluctuations of for example the natural rate of interest (see here and here).

Finally, one critique, with which I agree, and Romer has written very funny and interesting 'WP' on, is the degree of exogeneity, through shocks, that is needed to explain the economy.

Behavioural economics (2017 Nobel prize winner was from this area) may supply us with an answer. It allows to endogenize some of the shocks. Here's an article on VOXEU with many references, and analysis of an example of how it's done.

Edit: To complement Kitsune answer, I find this image from the wikipedia page really good. In a very succint way, it explains the difference between macro and micro prudential perspectives.


How to calculate price index for natural gas in India?

Since 2014, this is the official formula to compute the price of NG in India:

The details of this formula, including where the data is taken from can be found here.

To see historical data with this formula (since 2014), see the left panel here. Simply click in the period you want to see. Notice these prices are set for a six-month period.

There is more information about the context of this policy and the different pricing systems regarding natural gas for India here.


DSGE model with agents that develop their skills

Most DSGE models don’t model heterogeneity in individual labor market. Although I suppose you could do that if you would really want to but, it would be incredibly data intensive to actually estimate.

You can have some DSGE models with endogenous growth where human capital accumulation plays a role but most of them are based on just exogenous technological growth.

Well you can in principle use DSGE to evaluate policy I would not use it for evaluating some microeconomic policies. There are several reason for this:

Even simple DSGE models are incredibly data intensive. Introducing heterogeneity in individual labor markets by let’s say assuming there is some probability distribution over human capital and further possibility for heterogenous human capital accumulation would push this even further. I am not sure you would find dataset to do this even if you would have full access to all available data.
DSGE models are built for describing and forecasting macroeconomy or macroeconomic policy analysis. While in principle I can imagine you could try to use them for evaluating some microeconomic policy by just changing parameters of one market and then examining the impulse response functions I don’t think you should and I never seen it done before. DSGE is not build for that kind of task. Form what I understand about Sen’s capability approach most of the things that he suggests require microeconomic policies not macro ones as he focuses on things like ignorance, oppressive state policies, lack of financial resources, ill health, lack of proper education etc. These are not really macroeconomic domains.
There are definitely more better and appropriate ways how to evaluate policies on microeconomic level, so what would be a point of using DSGE if it’s inferior tool for given task?

To answer the question if there are better ways how to evaluate policies the answer is yes.

For, example randomized control trails or if that is not possible there are methods such as regression discontinuity, differences in differences, synthetic control, instrumental variables. These are actually used all the time to evaluate policies that improve education, healthcare, lack of access to financial resources of poor. For example, I used to work in the Netherlands at CPB which was analytical unit that evaluates economic policies. I was in macro department but we had department for healthcare economics, labor economics and economics of education where my past colleagues worked on evaluating various programs aimed at improving education sector and hence also human capital accumulation and they mostly used the kind of models I mentioned.

Once you would determine the optimum policies on micro level you could use DSGE for evaluating what would be the impact of increasing taxes to pay for all those policies or spending on those policies since that’s one of the downsides of micro approach that you don’t see the general equilibrium effects of all that spending and taxing.


In a hyper-inflated economy, how do vendors know how much to raise their prices?

It's anecdote, but in this fantastic Planet Money piece on breaking Brazil's hyperinflation, there's a pretty good description of how in Brazil, shopkeepers would raise their prices multiple times a day. The strategies described (starting around the 10:00 mark, after the description of people running ahead of the guys with the price guns to buy things before they could raise the prices) include:

Assuming that inflation will run at some constant rate
Running around checking competitors' prices
Using the exchange rate against the US Dollar as a benchmark
Pooling information and colluding with competitors, agreeing to all raise prices by the same fixed amount each day.


if the richest 85 people give away all their money?

It seems safe to say that the bulk of that wealth consists of equity (stock) holdings, not "money." (For example, Bill Gates owns a lot of share of Microsoft.)

Let's assume that the top 85 people wanted to redistribute their wealth voluntarily.

If they tried selling that equity all at once, the price of the equities would likely collapse (who would pay a lot of money for those shares, if everyone know that they were selling all at once?) The amount of money to redistribute would be a lot less than the current estimate of their wealth (which uses the current stock prices). The logistics involved with the transfer would probably consume most of of the transferred wealth. (If it were easy to transfer wealth to individuals in poorer countries, aid programmes would have been more effective than seen in practice.) That is, some people would be wealthier, but it would likely be intermediaries skimming off the cash transfers.
Logistically, there would be no way to transfer all of those shares to half the world's population. (They would need something like a brokerage account.) And even if that were somehow accomplished, the recipients would likely dump their shares, and their value would collapse. Brokers would probably end up getting all of the value from their transaction fees.

In other words, there would presumably be a transfer of wealth, but it would be a lot less than what is implied by current market prices.

This of course raises the obvious question of why they would do this voluntarily; the top 85 richest people did not get that way by dumping their wealth on random strangers.


Is the money market equilibrium interest rate the same as the IS-LM equilibrium rate?

Yes. The LM curve is a combination of all such points where money market is in equilibrium. You can see this from the picture below I took from Blanchard et al. Macroeconomics: a European perspective.

The IS-LM equilibrium occurs when IS-LM curve intersect as seen on picture below (again taken from Blanchard et al.)

No matter where IS-LM points intersect there will be a direct mapping from any point along LM curve (as shown in first picture) directly onto the money market. Hence any equilibrium IS-LM interest rate will be also money market equilibrium interest rate (as correctly pointed out by Giskard in comments these are the same interest rate). Hence, in the standard textbook exposition of the model these will be always same.


How can I tell if 2 different utility functions represent the same preferences?

Recall the definition:

  The function $u: X \rightarrow \mathbb{R}$ represents $\succeq$ on $X$ if for any $x,y \in X$, then $x \succeq y \iff u(x) \geq u(y)$

We can show that if $u: X \rightarrow \mathbb{R}$ represents $\succeq$ on $X$, then for any strictly increasing function, $f: \mathbb{R} \rightarrow \mathbb{R}$,  the function $v(x) = f(u(x))$ also represents $\succeq$

(Short) Proof: For any $x,y \in X$, if $u$ represents $\succeq$, then $x \succeq y \iff u(x) \geq u(y)$ by definition.

Because $v$ is strictly increasing, $f(u(x)) \geq f(u(y)), \implies v(x) \geq v(y)$.

Thus $x \succeq y \iff v(x) \geq v(y), \implies v$ represents $\succeq$

In your case, you have $u(x,y) = x^\frac{1}{3}y^\frac{1}{3}$, and can set $v(x, y) = (u(x,y))^9 = x^3y^3$, your desired result.


Is single-peaked preferences necessary for majority rule to be transitive and yield non-empty choice set?

No, single-peakedness is not necessary for majority rule to be transitive.  For example, "single-dipped" preference profiles (the vertical mirror image of "single-peaked" profiles) also produce transitive majority preferences.   A much-more general condition is "triple-wise value restriction", which Sen (1966) showed generates a transitive majority preference relation.  Here is a good summary of these and related concepts, with some references to the primary literature:

https://plato.stanford.edu/entries/social-choice/#NonDicPreAggRul


How does Bolivia maintain its high economic growth and the highest direct foreign investment in South America, despite its economic policies?

This is what I gathered during my research of this topic:

Evo Morales' policies are more centered on economy than they appear. See, for example, an article from May 2017, which praises them for being fiscaly conservative and austere, despite socialist and anti-imperialist rhetoric.

  Even though Evo Morales is a self-described socialist, and (in theory)
  one of the most left-wing of Latin America’s Bolivarian wave, it was a
  shock when the Bolivian president embraced fiscal discipline.

  While the Morales government nationalized several oil and gas
  companies, they simultaneously developed a good enough relationship
  with the private sector. Overall, the government has been surprisingly
  fiscally conservative.

  Institutions like the IMF and World Bank, of which Morales is a strong
  critic, have praised his austerity.

[...]

  Even if Morales rails against neoliberalism, he is not moving the
  country toward a Marxist revolution. The fact is that he understands
  the role of the market in improving people’s lives, especially
  indigenous groups, since they are the most involved in informal
  economic activities.

[...]

  In general, Morales has a better relationship with both business
  leaders and social movements than other governments in the region
  could dream of, proving he’s not as far left as many might think.

It should be mentioned, however, that the public spending increased over 750% from 2006 to 2017 (source). The crucial difference is that this effect was not solely dependent on the price of oil, as in case of Venezuela, and the budget was more balanced (Venezuela deficit grew up to nearly 20% of GDP in 2014, prior to drop in oil prices), but with other factors, like:

investments in increase in production from natural gas (hydrocarbon gas, polyethylene),
social bonds,
strengthening internal demand (increase in internal consumption).

On the other hand, Bolivia's growth of GDP is not that signifficant. In fact, the four quarters average of GDP growth does not exeede 2,5% (it dropped below 0% after 2015) (source). Bolivia's economy is much smaller than Peru and Chile (according to CIA World Factbook, Bolivia GDP is about 83 bln USD, while Peru GDP is 424 bln USD, Chile is 452 bln USD), so there is some kind of 'base effect' as regards measuring the GDP growth in relative values

All in all, the last ten years were preety good for the global economy. In the age of low interest rates the capital was looking for developing countries like Bolivia to invest in. The real test for the Bolivia economy will come, when the bull market ends.


Is the concept of limited liability in corporate law beneficial to economy

It is beneficial (at least, mostly beneficial). Investments are risky, and business projects fail all the time at no fault of the entrepreneur. By bounding the risk profile of corporations, you give incentives for more people to start a business and look out for funding. Let me explain myself with a super simple example.

Suppose there was a project that has a 60% chance of being successful and create 10k jobs, and economic welfare to the community with a value estimated to be 500K Dollars. However, with  40% probability, it will fail and the initial investment of 150K Dollars will be lost. The entrepreneur lacks sufficient funds for the initial investment (let's assume it has to be fully funded by investors). If there was no limited liability, it is clear that most entrepreneurs will not do the project. Only entrepreneurs with sufficiently low risk aversion or sufficiently large pockets will be willing to raise the money (or use their own) and do the risky project with those odds.

You see that a project that is arguably welfare enhancer is not made, because it is too risky. Limited liability smooths things out by distributing the risk of a project among all the investors.

Clearly, this example is super simple and I have abstracted from principal-agent problems, entrepreneur fraud, situations where the success of the project depends on the entrepreneur's risk aversion, etc. There are many papers dealing with these issues (and others), but the main argument for limited liability is captured in this toy model. So yes, a very strong case can be made to conclude that it is beneficial.


Locational Marginal Pricing when different prices and when the same?

okay meanwhile I got the answer to this:

the main idea behind LMP is to reduce the marginal cost of the WHOLE system.... therefore it is cheapier if you select those offers that way....

calculations:

38 * 600 + 48 * 600 = 51600

if we would do it as I thought makes more sense:

36 * 400 + 48 * 800 = 52800

as you can see even though the price seems to be more fair for node 1 now in total the costs increased from 51600 dollars to 52800 dollars


In this case the optimal choice is $(x^*, y^*) = (3, 1)$. Here is the picture :


Impact of government subsidy on labour supply

The individual has a utility function over consumption and leisure

$$U(C, \ell) = U(C, T-L)$$
and to keep this static, it has the no-savings/no-credit budget constraint

$$C = G + wL$$

He wants to maximize utility. Inserting the budget constraint into the utility function the first-order condition is (maximizing with respect to $L$)

$${\rm (f.o.c)} :\;\; U_c(G+wL, T-L)\cdot w - U_{\ell}(G+wL, T-L) = 0 $$

This will hold in all optimal solutions. Set $L=0$ to express the reservation wage, (i.e. ask "what is the wage consistent with zero work?")

$$L=0 \Rightarrow U_c(G, T)\cdot w - U_{\ell}(G, T) =0$$

$$\Rightarrow w_R = \frac {U_{\ell}(G, T)}{U_c(G, T)}$$

Then

$${\rm sign} \left\{\frac {\partial w_R}{\partial G}\right\} = {\rm sign} \left\{U_{\ell c}(G, T)\cdot U_c(G, T) - U_{cc}(G, T)\cdot U_{\ell}(G, T)\right\}$$

We see that

A) If preferences are assumed separable between consumption and leisure, i.e. if  $U_{\ell c} = 0$, then under the usual assumption that  $U_{cc} <0$ we see that $\frac {\partial w_R}{\partial G} >0$.

B) If separability is not assumed, then it is usually assumed (and reasonably) that $U_{\ell c} >0$, in which case we again obtain $\frac {\partial w_R}{\partial G} >0$.

In both cases, an increase in $G$ (which is what happens when the government pays the amount $b$), increases the reservation wage. If we assume that wages are the same before and after the government subsidy, then if the person did not work, he will continue not to work, while if the person worked previously, he may now stop, given the government subsidy.

Of course, making the problem dynamic by introducing intertemporal utility maximization and a consumption-savings decision, requires an analysis on its own.


why does income of executives level out in the long term after tax increases?

the practices of assigning salaries and bonuses, etc, would be difficult to adjust in the short term (i.e. more inelastic), so they'd tend to stay the same,

This might be true empirically (I am not an expert on CEO compensation), but if you are doing a test you have to follow assumptions in exercise (which will not always be realistic - undergraduate programs for example almost always assume linear demand and supply which is unrealistic save for some special cases).
In this case the exercise states:

Hint: Top executives have a lot of power over when they get paid for their work: They can ask for bonuses a bit earlier, or they can cash out their stock options a bit earlier.

This is basically teacher telling you throughout this exercise assume the salary can perfectly adjust even in short term.

in anticipation of a higher tax burden, CEOs would want to give themselves higher bonuses before the tax hike goes into effect, so income in the first year would be high

You need to read the set up of the problem very carefully. The set up says:

What do you expect happened to executive income in the first year of the tax increases?

So the question asks what happens to the income in the first year after the tax hike is passed. You are completely right that before they pass the tax hike the executives (within this problem) will ask for higher compensation before the tax hike, but note the problem asks you what happens in the first year after the tax hike already taken place. Well then the income will be lower as those executives within the assumptions of this exercise would rationally ask to be paid before the tax hike portion of money they would earn in the subsequent year. Hence their income in the in the first year of the tax increases will be lower.


The preservation of ranking is over consumption bundles, not individual goods. So the one and only question is whether the ranking of bundles $(x_i,y_i),\; (x_k,y_k); \forall i,k$ is preserved or not.

If preferences are rational (complete and transitive), and continuous, then they can be represented by a continuous utility function. So if these hold and

$$(x_i,y_i)>_{pr}\; (x_k,y_k) \implies \exists \;U, U_i > U_k$$

The logarithm is a strictly monotonic transformation of any amenable to it function as a mathematical fact, irrespective of what we use the function for. So it follows that
$$U_i > U_k \implies \ln U_i > \ln U_k ,\; \forall i,k$$

and the ranking of bundles is preserved. Whether due to the transformation we lose other information or not (that we may have wished we could have kept) does not concern the representation/transformation theorem.


Import/export by sector/country data source

I believe that the UN Comtrade database is the main data source for this kind of query with substantial country coverage.  It's broken down by HS code.

It has its issues.  Reporting is based on national authorities, so you should usually expect data on imports to be more accurate than exports because of the tax implications (so Uganda's imports from Kenya are likely to be more accurate than Kenya's exports to Uganda).


Supply curve for a competitive firm with constant MC

Consider a firm with constant marginal cost $c$ that chooses to supply a quantity $q \in [0, \bar{x}]$ when taking the price $p$ as given. The profits of such a firm can be written as

$$ \text{total revenue} - \text{total cost}  = q p - c q = q(p - c).$$

From this, we can immediately infer the optimal supply $q^*$:

If $p<c$, then $q^* = 0$.
If $p=c$, then any $q \in [0, x]$ is optimal.
If $p>c$, then $q^* = \bar{x}$ (the maximum possible quantity).

Notice we need a constraint like $q \in [0, \bar{x}]$ for the supply correspondence to be well-defined. If we had simply restricted $q$ to $\mathbb{R}^+$ (the non-negative real numbers), as is more usual, then there would be no solution to the profit maximisation problem in the case of $p>c$.


Production Possibility Frontier for 3 goods

It is possible. In fact you can construct PPF for arbitrary number of goods.

When it comes to plotting it for a case of 3 goods it is easy to make 2D projection of the 3D space. You can see the image below for an example.

Also note PPF is not necessary spherical or circular. The shape of PPF depends on the opportunity cost. In textbooks you might sometimes see it represented like a portion of a circle or elipse but it does not necessarily have to have that shape.


Staiger and Stock (1997) formalized the definition of “weak
  instruments” and most researchers seem to have concluded (incorrectly)
  from that work (or hearsay) that if the F-statistic on the excluded
  instruments in the first stage is greater than 10, one need worry no
  further about weak instruments.Stock and Yogo (2005) go into more
  detail and provide useful rules of thumb regarding the weakness of
  instruments based on a statistic due to Cragg and Donald (1993).
  Stock, Wright, and Yogo (2002) provide a summary of this work.

Weak instruments:  An overview and new techniques

Austin Nichols (2006)

While an  F-statistic cutoff of 10 may be a crude rule of thumb and not sufficient to conclude that a set of instruments are not weak, Stock, Wright, and Yogo (2002) have a table about weak instrument cutoff values that goes down to 1 instrument.

GMM, Weak Instruments, and Weak Identification


You might want to check out Fermi estimation. XKCD did a very accessible example of Fermi estimation being used to estimate paint production. This method can be quite good at providing estimates within an order of magnitude of the correct answer.


How low can interest rates go? How deeply negative?

Your question contains several incorrect premises, hence before providing an answer I will first correct those misconceptions as any discussion based on incorrect premises cannot be  very productive.
Correction of Misconceptions

Ten years ago, negative interest rates were an impossibility, an intellectual curiosity.

This is not true at all. Clearly empirical research shows that there were periods when interest rates in some countries were slightly negative. For example, Kugler, P., & Rich, G. (2001) show that Switzerland had slightly negative interest rate in late 70s. That is about half a century ago and the paper itself is 20 years old and this was well known. So your statement is just factually not true.
In fact negative interest rates and possibility of thereof enjoyed periodic waves of interest, the most early ones dating all the way back to early 20th century (See Ilgmann, C., & Menner, M. (2011) and sources cited therein).

Just read a claim that there is still an absolute lower bound, even if it happens to be a little bit below zero. But that's premised on an inflationary world.

One cannot prove a negative but I study macroeconomics for quite some time and I have never seen anyone saying that zero lower bound (ZLB) or even effective lower bound (ELB) are premised on inflationary world. I would like to be corrected if anyone economist ever made such claim, but I doubt it and even if some do this is definitely not any claim that would be generally held by mainstream macroeconomists.
Answer

And in an economy that is deflating at -.06, is -.05 still still the 'real world lower bound' of interest rates? Can a lower bound be said to meaningfully exist, or is there only the spread between the rate of deflation and the interest rate?

ZLB or even ELB does not depend on deflation or any arbitrary rate of deflation. ZLB or ELB exists independently of what the rate of inflation/deflation is in the economy. ZLB or ELB as pointed by @BrianRomanchuk in his excellent +1 comment refer to the bounds on nominal interest rates, and even more specifically nominal interest rates on deposit accounts. Real interest, or even other nominal interest rates  (such as rates implied by ECB's long-term repo operations (LTROS) rates see The Economist for explainer) can be arbitrarily negative.
The reason why ZLB or ELB exist is the existence of cash. Due to existence of cash - which is equivalent to financial paper with implicit zero nominal interest rate - there must be some ELB. Since cash and money on deposit accounts are theoretically perfectly fungible the limit on nominal interest rates should be zero. Of course in real life there are issues such as storage costs. Even without negative interest rates you are willing to pay bank to keep your money safe and as a result the interest rates can become slightly negative (Ilgmann, C., & Menner, M. (2011)).
However, slightly negative does not mean arbitrarily negative. There will be point where the nominal interest rate is so negative nobody will be willing to hold money in deposit accounts. This is trivial just consider an extreme case of negative $-1000\%$ interest rate. Would you hold any money on deposit account under such punitive negative nominal interest rate? I think not! Of course, this raises question what is actually the true ELB if not zero. The studies show that while this effective lower bound will be below zero, it can only go very little below zero - reasonable estimates are around $-1\%$ being maximum (see Hills, T. S., Nakata, T., & Schmidt, S. (2019) and sources cited therein).
Now are there some ways how this issue could be dealt with? Actually yes! Both ZLB and ELB can be easily abolished by abolishing cash. If people do not have choice of withdrawing their money from bank then interest rates can be arbitrarily negative (see Buiter 2009; Buiter 2005; Yates 2004, and Ullersma 2002, for reviews on the relevant literature). Unfortunately, it seems that most societies are not ready to give up cash yet, since despite of support for abolition of cash among  experts this idea is not widely embraced by politicians.


What is the purpose of multiple regression?

The difference is that the first regression is unbiased only if you can assume that high school GPA and ACT score are orthogonal on each other $cov(x,z)=0$ where $x$ is shortcut for high school GPA and $z$ for ACT score. Or if you can assume the second variable ATC score does not affect the dependent variable at all $\beta_2=0$. This is because in simple regression the $\beta$ coefficient bias is given by:

$$\hat{\beta_1} = \beta_1 + \beta_2 \frac{\text{COV}(x,z) }{VAR(x)} $$

where $\hat{\beta_1} $ is the estimated beta effect of GPA on col. score (beta reported by your statistical program like R), $\beta_1$ is the true unobserved coefficient, $\beta_2$ is the true unobserved effect of ACT on score, and the fraction is just covariance between $x$ and $z$ over variance of $x$.

So unless you know that the two variables are completely unrelated you will have bias. Now an example of variable that is completely unrelated to another is variable thats hold fixed. For example if in $z(x) =100$ no matter which $x$ we examine then we can be confident that $cov(x,z)=0$. Regressions should be applied only if you think they are unbiased so the first equation implicitly assumes that the second variable is hold constant given $x$. However, once you include the second covariate explicitly you can calculate both expected beta coefficients conditional on each other which is equivalent to actually keeping the other variable constant.

So there is actually a great deal of difference between the two regressions (except for special cases where $cov(x,z)=0$ or $\beta_2=0$).


How can you profit from junk bonds and selling treasuries, if the junk bond doesn't default?

You assumed that junk bonds do not default. They do default. So people will price junk bonds at a greater yield than Treasuries.
It can’t be arbitraged since the probability of default (and associated loss) is unknown. An arbitrage implies locking in an abnormal profit at no risk. You cannot hedge out that default risk without using an instrument like a credit default swap. if you did hedge losses with a credit default swap, the cost of the swap cancels out the yield advantage on the bond (assuming everything priced at fair value).

If we ignore transaction costs - which we cannot in practice - the breakeven spread point for a high yield bond portfolio is where the average spread approximately equals the expected annual loss (= probability of default times loss given default). E.g., a 500 basis point spread (5%) will offset credit losses of 5% a year. (We would need to do a detailed cash flow analysis to map credit losses to spreads, but the approximation is typically quite good for bonds trading near par.)

The credit loss rate is unknown, and opinions differ wildly about them. Observed spreads represent some kind of “average” of market participant’s views (assuming no other factors influence pricing). The first quote just says that the actual credit losses tend to be below what spreads discount across the cycle.

However, high yield bonds are illiquid, and have greater transaction costs, and so need to offer an additional spread to compensate. Meanwhile, markets can be disturbed, such as forced liquidations, and so spreads can trade wider than most market participants’ expected credit losses.


Benveniste-Scheinkman condition gives derivative that still depends on the value function

I guess I will provide a basic answer for now, since I somewhat figured this out, and hopefully either someone posts a more complete answer, or I add more to this later.

Basically, we can use forward or backward iteration. For example, for forward iteration (given that we know the value of $V_1$ at $t=0$, call it $V_{initial}$), then we can use this to find the value of $V$ at time $t$ simply by using the equation over and over again. Or we can work backwards, if for some reason forward iteration doesn't work (at this moment I'm not sure why/how this is different then forward). I don't think the solution is too neat, but it works.

Or, if we are at a steady state, then $V_1(K,I_{-1}) = V_1(K'_I)$, so we can simply rearrange the equation and solve.

Again, this isn't a great answer, but it is correct.


Tax imposed on customer affects demand or supply curve?

In partial equilibrium, demand shifts down, with the (vertical) distance between the pre-tax and post-tax curve being exactly the height of the tax.

pre-tax curve: $q=f(p)$

post-tax curve: $q=f(p+t)$


Using the sample mean to test hypotheses

My question is simple: when calculating p-values, why do we consider the distribution of the sample mean (i.e., why is the sample mean our 'test statistic')? Why not use the median, mode, lower quartile, or any other function of the data? For that matter, why not do the natural thing and just compute the joint density $f(x1,...,x_n)$?

So, you have entered into two deep questions.  The first is “why the mean?”  The second is implicit in your joint distribution question, “why use the sampling distribution?”

I will answer the second before the first because the answer of the first depends somewhat upon the second.

There are two broad families of statistical methodologies, the Bayesian methods, and the null hypothesis methods.  Null hypothesis methods are sampling based methods.  Bayesian methods are generative methods.

Because of the nature of your question, I am going to overgeneralize a little bit to make the answer shorter than a book.

There are several reasons to use one class of problem over the other, and there are circumstances where you would only use one specific class of solution.  They are not interchangeable.

If one computed the joint density, $f(x_1\dots{x_n})$, one would fail because a probability can only exist with respect to a model.  Implicitly, you have a model in mind with the joint density, but it does matter to call it out.  In a Bayesian framework, that joint density would be the density averaged over all possible models and all possible parameters.

There are a couple of specific reasons to use Bayesian methods.  The first is if you have prior knowledge.  For example, let us assume that you know $\mu=1,000,000$ is almost impossible, but $\mu=5$ is quite probable.  That is the definition of prior information.  You know something.  It might be vague.  It might be unbelievably vague, but it is also not true ignorance either.

If you have prior knowledge, then non-Bayesian methods are inadmissible solutions.  A solution is considered admissible if the estimator cannot be stochastically dominated by another estimator.  Now admissibility is only one possible criterion.  It implies that for a given data set, the Bayesian method will be intrinsically the least risky to use and the most accurate when averaged over the sample space.

A second reason to use a Bayesian method is that they generate coherent strategies.  A strategy is coherent if a bookie that is taking all finite bets at the bookie’s stated prices, cannot be gamed into a sure loss by a crafty actor or set of actors.  Null hypothesis methods are not coherent.  If you use a null hypothesis method in a game, they can create sure losses for a bookie.

Consider the case of an inverse roulette wheel, where the ball is dropped, and the wheel spun.  Instead of betting what number it will land on, the croupier sees where the ball lands, but the players cannot see it.  Instead, the croupier tosses two fair coins.  If a coin comes up heads, then the croupier reveals the value one place to the left of the true value; otherwise, the croupier reveals one value to the right.

The sample space is $\{\{L,L\},\{R,R\},\{R,L\},\{L,R\}\}$.  To make this concrete, consider a wheel where the number came up 17.  They will either be given the set, $\{16,16\}$,$\{16,18\}$, or $\{18,18\}$.  Obviously, if you receive $\{16,18\}$ you should bet 17, but what should you do otherwise?

If, for example, you should receive $\{16,16\}$, Bayesian decision theory says you should toss a fair coin and either bid on 15 or 17.  You will win 75% of the time.  Frequentist decision theory says that the minimum variance unbiased estimator is 16.  You can check that for yourself.  You should bid 16.  Of course, sixteen is an impossible value, known with probability one.  You will win 50% of the time.

If the bookie placed odds using Frequentist decision theory but you played as a Bayesian, you would rake in the cash.  The Frequentist odds are not coherent.

Although not obvious yet, this is the start of the answer to the first question on the mean.  And, in fact, you do want to be careful.  If you have a Bayesian model, you do not want to test it with a null hypothesis method.  Conversely, if you have a Frequentist model, you do not want to test it with Bayesian methods.  They would produce different predictions.
So far, we have two reasons to use the joint distribution via Bayes theorem.  The first is accuracy.  The second is coherence.  So why use a null hypothesis method?

The first reason to choose a null hypothesis method is that you have no prior information.  The nice thing about null hypothesis methods is that they work equally well regardless of the true value of the parameter.  That is not true for a Bayesian method.  They do not work equally well for all possible values of a parameter.

In the example above, where you felt you could all but rule out $\mu=1,000,000$ would be problematic if $\mu=1,000,000$ as you would have underweighted the possibility of the case.  Bayesian methods weight the sample based on prior knowledge.

As well, they minimize the maximum possible loss you could face from drawing an unfortunate sample.  The minimum variance unbiased estimator and the maximum likelihood estimator generate sampling distributions that are worst-case distributions.

Bayesian methods, generally speaking, minimize the average loss you would expect to receive from drawing an unfortunate sample.  In the absence of prior information, it is challenging to minimize an average loss when you have nothing to compare to the sample.

The second reason is that you truly have a sharp null hypothesis.  Bayesian methods have no clean or nice way to handle a hypothesis of the form $\mu=k$.  This is because samples are not random in Bayesian thinking.  You saw the sample.  It is a fact.  It cannot be random as there is no uncertainty in it.  Parameters are uncertain in Bayesian thought.  You cannot see them; you cannot know what they are.  The $\Pr(\mu=k)$ is intrinsically zero.  It is one point over an infinite number of points.  The measure of a single point from an infinite number is zero.  On the other hand, Bayesian methods handle solutions over regions fine.

The third reason is that you need an unbiased estimator for some reason.  Assuring an estimator is unbiased is expensive.  There can be a huge loss of information to assure it is unbiased.  You may need a sample size many times larger than a biased Bayesian estimator to get the same result.  All Bayesian estimators are biased estimators.  Bayesian methods optimize the trade-off between bias and accuracy, though as the sample size becomes arbitrarily large, the Bayesian method will converge to the true parameters, so it becomes a just barely biased estimator.

These differences answer the question of using the joint distribution of the data to determine the probability versus using the sampling distribution of the model to determine a probability.

If you decide you want to use a Bayesian method, then you solve $\Pr(\mu\le{k}|{X})$, for example, instead of $\Pr({X}|\mu\le{k})$.  The converse is true for the null hypothesis method.

Now as to using the mean for a sampling distribution.  There are several reasons for this.  The first is that in most standard cases, the mean is the most efficient estimator of the parameter.  Secondly, all unbiased estimation has quadratic loss buried inside of it somewhere.  Even if you are making an estimate based on the median, that is the mean rank.  While the median minimizes absolute linear loss, it minimizes quadratic loss in rank space.

That takes us back to stochastic dominance.  If you have two unbiased estimators, say the sample mean, and the sample median, but the sampling distribution of the sample mean stochastically dominates the sampling distribution of the sample median, then why would you use the median?

You see the median begin to be used, generally, when robustness is a concern rather than power.

If you want to know the probability your model is true, given the data that you saw, then you use a Bayesian method.  If you want to know the probability of seeing data as extreme or more extreme than likely under your hypothesized value, then you use a null hypothesis method.  If you use a Bayesian method, then it depends on the joint density of the actual sample.  If you use a null hypothesis  method, then you will use the sampling distribution of the model to condition the data.  The mean will tend to be used due to issues of efficiency and power and not any other statistic.

EDIT
If your hypothesis concern is model selection, then you should use a Bayesian method.

Bayesian methods treat a model as a parameter.  It uses the data to determine which model is most probable.  You can have any number of models you like and any number of parameters.

Model selection by information criterion maps to stylized Bayesian posterior densities, but do not make their assumptions explicit.  Bayesian methods allow you to let the data select the model.

EDIT
For clarity, let us imagine that we have two possible models that data are drawn from.  Model one is $$\frac{1}{\pi}\frac{1}{1+(x-\mu)^2}.$$  Model two is $$\lambda\exp{[-\lambda{x}]}.$$

$$\Pr(M_1;\mu|X)=\frac{\prod_{i=1}^n\frac{1}{\pi}\frac{1}{1+(x-\mu)^2}\Pr(\mu|M_1)\Pr(M_1)}{\int_{\mu\in\Theta}\prod_{i=1}^n\frac{1}{\pi}\frac{1}{1+(x-\mu)^2}\Pr(\mu|M_1)\Pr(M_1)\mathrm{d}\mu+\int_{\lambda\in\Theta}\prod_{i=1}^n\lambda\exp{[-\lambda{x}]}\Pr(\lambda|M_2)\Pr(M_2)\mathrm{d}\lambda}$$

$$\Pr(M_2;\lambda|X)=\frac{\prod_{i=1}^n\lambda\exp{[-\lambda{x}]}\Pr(\lambda|M_2)\Pr(M_2)}{\int_{\mu\in\Theta}\prod_{i=1}^n\frac{1}{\pi}\frac{1}{1+(x-\mu)^2}\Pr(\mu|M_1)\Pr(M_1)\mathrm{d}\mu+\int_{\lambda\in\Theta}\prod_{i=1}^n\lambda\exp{[-\lambda{x}]}\Pr(\lambda|M_2)\Pr(M_2)\mathrm{d}\lambda}$$

The first probability statement $\Pr(M_1;\mu|X)$ can be reduced to $\Pr(M_1|X)$ by marginalization so that $$\Pr(M_1|X)=\int_{\mu\in\Theta}\Pr(M_1;\mu|X)\mathrm{d}\mu.$$

The same can be done for model two so that you can find the probability of model two as $$\Pr(M_2|X)=\int_{\lambda\in\Theta}\Pr(M_2;\lambda|X)\mathrm{d}\lambda.$$

Since $\Pr(M_1|X)+\Pr(M_2|X)=1$, it follows that these probability statements are the probability that these are the true models in nature given the data you have seen.

In the first model $\mu$ which is the median and the mode as this distribution has no mean or variance vanishes from the probability statement in the last line.  A point value is never calculated because it is marginalized out.   Likewise, $\lambda$ which is $1/mean$ is also never calculated as a point statistic.

These extra parameters are called nuisance parameters and their integral has to be solved, but only to make the integral vanish.

When solving Bayesian model selection, the parameters are nuisances and you need them as placeholders for calculation purposes, but you have to make them vanish.  If you didn't, then you could have points where model one was more likely than model two, but other points where that relationship is reversed.

Imagine that $\Pr(M_1|X)=99.99\%$, it would follow that $\Pr(M_2|X)=.01\%$ and you could ignore model 2.  I once tested seventy-eight models of bankruptcy.  One model had a posterior probability of about 57%.  the second model had a posterior probability of about 43%.  The remaming 76 models had a combined probability of $1/10000$ of one percent.  I could exclude the 76 models, but I couldn't exclude two models as both were approximately equally likely to be the data generating function.

Although I bypassed the problem using model averaging, that is I averaged the two regressions together by weighting one result by .57 and the other by .43, what it really meant was that I was close, but not quite there and that some similar convex combination of the two sets of variables were what mattered.


The reason that relationship weak is that there is a whole lot more influencing GDP than just trade. Not included in this analysis are things like commodity prices, types of exports, quality of institutions in a country, population. Countless others.

If you wanted to test the hypothesis that countries with higher levels of trade tend to be larger, you probably have. With such a simple analysis, it's probably hard to get much more than this.


One possible useful distinction between the two expressions could be as follows:

1) Division of work
Assume you have to mail 1.000 letters identical in content to different recipients. For each letter, in order to do that you need to a) fold the letter b) put the letter into the envelope c) write the address d) put a stamp e) go out and put it into the mailbox. Then repeat the sequence 1.000 times. "Division of work" would be to, instead of going through the complete sequence for each letter sequentially, a) fold all letters b) put all letters into the envelope, etc. By performing the same task, you gain efficiency in each task ("learning by doing") and also, you minimize the time needed in between each kind of task/work.
Here the focus is on gaining efficiency by grouping tasks.

2) Division of labor
In the same example, division of labor would be not only to perform each task for all letters instead of all task for each letter, but have different people doing each task. This situation focuses on the phenomenon of different people specializing in different professions.

...And I am certain that some authors that use the distinction, use the terms switched -namely, they call "division of labor" 1) above, and "division of work" 2).


Empirically checking efficiency and equity

Several things:

It is not clear that labor productivity as a share of GDP tracks allocative efficiency.  For example, it could rise simply because the returns to capital (capital's share of GDP) are falling, which would most obviously be a consequence of allocative inefficiency through over-investment in capital inputs.  There are potentially many other things that would confound this as an "efficiency" metric.
Gini Index isn't the only measure of inequality, and in fact it doesn't do a very good job of capturing differences in economies at the extremes of their income distributions.
You may have the causal arrow backwards: the most widely accepted model I'm aware of (which is not without controversy, mind you) is that of the Environmental Kuznets Curve, which assumes that wage growth (not labor's share of GDP) drives environmental improvement, not the other way around.
Because "equity" in the probability-distribution sense is attainable both by improving and destroying quality of life (e.g., everyone earns zero dollars), it's not clear that you should expect a positive correlation between this and "environmental spending".  In fact to first-order, your 50% split is consistent with what I've just pointed out.


Games with perfect but incomplete information in real life

A classical example of a game with perfect but incomplete information is an English auction. We know what all the bids have been up to this point but we don't know everyone's valuation of the good being auctioned.

In general, formulating a strategy in this case involves some beliefs about your opponents' payoffs. Typically these are exogenous to the game but I guess you could have some revealed preference throughout an extensive game.


Price Producers Receive as a Result of Indirect Taxes

The idea is that both the consumer and the producer of a good take on some of the burden of a tax, even if you don't intend to levy the tax on one or the other. So if you have a sales tax that consumers have to pay, what will happen is that consumers won't take the full brunt of that tax. They will demand less and producers will have to make the product cheaper in response, so they get less revenue overall anyway.

How much tax burden each side will have to pay depends on their relative elasticities, which I imagine you'll be going over soon if you're going over tax incidence. Elasticity is a measure of sensitivity to a change in price to in this case, quantity supplies or demanded.

So $P* \neq P_p$ because the producer will have to shift from the equilibrium just like the producer will have to. It's the same idea for the consumer as for the producer, since the tax burden falls on both of them.


Why is the equilibrium network size can be less than the socially optimal size in networks w/ positive externalities for single system?

The equilibrium network is smaller than that chosen by a social planner because agents do not internalize network externalities.

Vaccination against a contagious disease is a standard such example.
Someone who is not vaccinated exposes others to potential infection and prevents others from, say, using public facilities. This is a negative externality for which he pays no cost. A social planner would therefore impose vaccination when an individual agent might choose not to.


How to convert daily returns to annual returns?

So, let me start with your second question.  No you cannot multiply by 365.  You could approximate it by $$\log(\text{Annual Return})=365*\log(\text{Daily Return}),$$ but for what you are doing, it does not make sense to do so.

You are correct in your annualized rate of return.  It is 2069063%. It should be obvious as to why you would not want to do this.

There are two solutions.  The first is to convert annual rates, such as the bond rate, from an annual format to a daily format.  So make your risk-free rate: $$\text{Daily risk-free rate}=1.065^{\frac{1}{365}}-1=0.0001725485.$$  The second is to search through the dates of your returns and find returns that are 365 days apart, so return would be $$r=\frac{p_{366}}{p_1}.$$

By annualizing daily returns, you are insanely increasing the variability, but it is artificial so it isn't a true increase, it is an artifact of the method you are choosing.


Convenient S-shaped production function (i.e. with IRS and DRS) to derive a discontinuous demand for labor

I'm not sure but it seems to me that the logistic function $\frac{e^{x}}{1+e^{x}}$ could serve your purpose.

You may need to scale it as its output falls between 0 and 1, but it does have an analytical derivative that you can then use to solve for the labour demand function.


It depends on the definition you want to use. You can define a regressive tax as the wiki says, in which case a sale tax would not match.

But the goal of definition of different class of taxation is to assess the impact on person being taxed. Without considering income the above definition becomes moot. Sure the tax is proportional (fixed share of tax on any price) but it's effect on those who pay it is different.

The first definition is derived , I believe, in the context of income taxes in which the income is being taxed and the definition matches it's intuitive notion.
Some taxation system are subject to the requirement that the system as a whole should be progressive, I.e the total tax burden should increase with income. In this context is natural to think that sale taxes are regressive, while they don't match the formal definition.