Is fiat currency better than a backed-currency?

Yes it is. Lets first unpack some myths about about gold standard or commodity money in general.

They dont necessary prevent inflation. Spain managed to have under gold standard a very large inflation compared to other European nations (Ferguson, 2008), and this happened even without debasement.

Furthermore as you yourself mention debasement can occur as well. Commodity money or gold standard dont mean the money have to be solely made of that commodity. In case of the commodity money you can easily have gold standard where you debase the % of metal every year even to the point of having 0.0001% of gold and it would still be gold standard. The only difference between debasement and printing money is that debasement is little bit more expensive as it involves smelting - but in modern industrial world the cost is not prohibitive so it could be done almost as easily as just printing money.

Backing paper money with gold does not prevent inflation by itself either. If you for example set the exchange rate to 1 euro = 1 troy ounce of gold, there is literary nothing preventing government from deciding one day that suddenly 1 euro = 0.001 troy ounce just because they decide so. Again thats still gold standard. Even if 1 euro would be set to be equal to be equal to 0.0000001 troy ounce it would still be gold standard. So whats the point?

Is there any way that gold standard may prevent inflation? From economic side not really (aside its more expensive to remint coins in case of using direct commodity money), but from political side it might be more difficult to justify debasement or change in the convertibility as non-economists might perceive that as some sort of cheating, but aside from politics gold standard offers zero protections against inflation by itself.

Next, its important to understand that deflation is also a form of price instability as inflation and it has negative effects on economy in similar ways as inflation (even if we exclude sticky wages). Deflation still creates unnecessary menu costs for example, same as inflation.

In addition to that deflation has extra negative effects because of sticky wages. Sticky wages does not mean that when everything gets cheaper your wages do too. It means that your wage remains the same (at least for certain period depending on stickiness) and due to the fact it cannot be adjusted quickly enough you would get unemployed. Economy with deflation is economy with unnecessary high unemployment which is additional cost of deflation. For this reason wast majority of economists thinks government should on purpose target yearly inflation of about 1-2% (Blanchard et al. 2013).

Furthermore, the modern evidence shows that independent central banks, if they choose so, are extremely efficient in maintaining price stability. In recent period Euro-Zone had average inflation well below 1% which is lower than historically many countries had under Gold/Silver standard. This just further illustrates that it literary does not matter what the money are made of as long as the stuff they are made of is not inconvenient to use (i.e. food that quickly spoils). How big or little inflation country has is ultimately a policy choice no matter what the money are made of. Even in case of cryptocurrency, if its controlled by government, the government can always decide to change the number of money by just changing the source code.

So the question is why bother having gold standard if its literary just using a useful commodity that could be otherwise used in industry for jewelry/electronics etc. improving peoples material standards of living and it does not hold any real advantage over fiat currency except

So to sum up:

its a myth that gold standard by itself prevents inflation. In medieval era inflation was kept low mainly due to technical difficulties and due to general weak governments that could not just impose any debased coinage on its populus. Modern day government deciding to be on gold standard with coins with 0.000001% gold can do so as easily as imposing fiat money on its population just by demanding taxes being paid using said coins. There might be some purely political reason why it might be harder for government to increase inflation under gold standard but no direct economic reasons (aside that smelting might be bit more expensive than printing in case you want to have actual coinage and not just backing)
Deflation is not desirable. As inflation, deflation is a form of price instability. In fact in presence of sticky wages deflation has additional costs making it worse than inflation but even without sticky wages you would not want deflation but maintain price level stable.
Modern independent central banks can easily deliver price stability. In fact most mainstream economists nowadays are accusing central banks such as ECB to not doing enough to raise inflation to at least 1-2%. Euro is actually being routinely compared to gold standard - see for example Eichengreen & Temin (2010).
There is no reason to use a commodity which has valuable industrial uses as money when paper does the job in principle as well. Its just a waste of scarce resources that could otherwise be employed in productive uses.

References:

Blanchard, O., Amighini, A., & Giavazzi, F. (2013). Macroeconomics: a European perspective.

Eichengreen, B., & Temin, P. (2010). Fetters of gold and paper. oxford review of Economic Policy, 26(3), 370-384.

Ferguson, N. (2008). The ascent of money: A financial history of the world. Penguin.


I believe you are correct -- though I will say appealing to the one-shot deviation principle here seems a little overpowered. There are only three stages to this game so checking for all equilibria (and not just your conjectured one) in each proper subgame is doable. You do seem to be applying it correctly, however!


Complete markets: security market equlibrium same as contingent commodity equilibrium?

You seem to be looking for information on the Radner equilibrium.

  Radner equilibrium is an economic concept defined by economist Roy Radner in the context of general equilibrium. The concept is an extension of the Arrow–Debreu equilibrium and the base for the first consistent incomplete markets framework.

Further reading:
Jstor: Existence of Equilibrium of Plans, Prices, and Price Expectations in a Sequence of Markets
Handbook of Mathematical Economics: Chapter 20


Difference between Outflow of capital and Flight of capital

This is actually pretty tricky, as "capital flight" can be a contested term itself. Here's a distinction provided by Darryl McLeod in the Concise Encyclopedia of Economics, edited by David Henderson and provided online by the Library of Economics and Liberty:

  There is no widely accepted definition of capital flight. The classic use of the term is to describe widespread currency speculation, especially when it leads to cross-border movements of private funds that are large enough to affect national financial markets. The distinction between "flight" and normal capital outflows is thus a matter of degree, much like the difference between a "bank run" and normal withdrawals.

There's some discussion by Frank R. Gunter in the International Encyclopedia of the Social Sciences, edited by William Darity, Jr., about the fuzziness of the term "capital flight" as related to motivations or the nature of the transactions:

  Capital flight is generally defined as an outflow of funds from a country motivated by an adverse change in the country’s economic, political, or social environment. Some believe that this definition is too broad. They distinguish between outflows that reflect “normal” international diversification motivated by marginal changes in risk-adjusted returns and funds fleeing or propelled across national borders during a crisis. According to this view, only the latter category represents true capital flight. Related definitions restrict capital flight to short-term speculative outflows—“hot” money—or to an outflow of illegal transactions only.

So, it seems that scale, motivations (adverse events, crisis), and the nature of the transactions could all be distinctions to some degree between capital outflow and capital flight.

I hope this helps!


When the Central Bank sells a bond, where does the money go ultimately?

Firstly, the central bank doesn't issue bonds. The treasury (in the executive government) does, as a way to finance government expenditure, and make real investments in the economy.

In modern economies, the central bank's primary focus is the interest rate at which they lend money to private banks. If they require more money in order to finance those loans, they will simply create more money, as they have the legitimacy to do that. (Note that most money that is created is not physically printed, but rather it is just the electronic entry of that currency, in the initial transaction. An economy requires a fraction of electronic money to exist as cash, so a fraction of created money is printed.) However, fiat money requires some security to give it legitimacy. To this end, the central bank holds securities, stocks, precious metals, cash and foreign currency in its reserves. You can see here that bonds form part of those reserves, so that is why central banks buy treasury bonds. Plenty of other domestic and foreign financial institutions also buy treasury bonds, as part of their reserves to give legitimacy to their "demand deposits" (which is fiat money), or as part of a strategy to lower the risk of their investment portfolios. Private banks also create money by loaning out their demand deposits.

Analogously, if the central bank raises the interest rate (in order to curb inflation), they will be slowing down the growth of money (which can be thought of as a "leakage"), and that is a situation in which they would buy fewer bonds.

Central banks need to operate autonomously from the executive government, and this autonomy is usually enshrined in the respective country's constitution. When this doesn't happen, the executive government could force the central bank to buy their bonds, which creates a huge injection of money into the economy. This money that is created belongs to the government, so they spend it in order to achieve policy goals, however, after a while the currency becomes worth less and less (as inflation is high). That is the main cause of high inflation. When the central bank is autonomous, they have control over the monetary system, and keep inflation in check with the central bank interest rate.


Solutions for the de-dollarization of the Democratic Republic of Congo

Yes, it is possible to peg any currency to the USD. The big question is: at what rate? To peg the Congolese franc to the USD, the central bank needs USD reserves. This is necessary to maintain the peg. (If the currency depreciates, the central bank will sell some of its USD reserves and buy the currency to prop up the exchange rate and thus uphold the peg). As per the article cited in the question, the Congolese central bank doesn't have much foreign currency reserves.

Based on the chart below, it appears to me that the Congolese franc was in fact pegged to the USD until early 2016. It seems the central bank couldn't maintain the peg at that rate and had to let the franc slide.

To have a strong currency, the central bank needs to limit the money supply. However, the government will often pressure it to do the opposite, i.e. pressure the central bank to print money and buy government bonds, so the government has money to spend. (The article mentions plans to slash the government budget in order to limit the supply of francs in the economy). For this reason in advanced economies central banks are typically strictly independent from government.


Can I recreate an experiment on Allais paradox using student grades as payoffs?

I am not sure I understand the problems outlined in the other answers. Seems to me that if we assume students maximize their expected utility and lotteries A1 and B2 (or B1 and A2) are chosen by someone, we would have

\begin{align*}
100\% \ U(B) & > 10\% \ U(A) + 89\% \ U(B) + 1\% \ U(F) \\
\\
11\% \ U(B) + 89\% \ U(F) & < 10\% \ U(A) + 90\% \ U(F)
\end{align*}
which are contradictory. There is no need to assume anything about the function $U$ for these to be contradictory.

One runs into the usual problem of trying to elicit true preferences in a poll, but so does the original paradox.


Multiple equilibria: which one to select?

I'm not sure I follow the logic on that equation having infinitely many solutions and steady states. In any case, in what follows are some guidelines for equilibrium selection.

It depends a lot on the context. Here are some criteria, in descending order of relevance

Clean Ways

Is any of the steady states unstable? If so, it's less likely to be the one that we*re observing in reality. See for example the 0-steady state in the Solow model.
The last point, somewhat generalized; Is there any meaningful starting value? Due to path dependence, you will only converge to one steady state from your meaningful value.
The last point, somewhat generalized; Do we converge more often to one than the other? Basically take a range of meaningful starting values and see whether most of them lead to the same steady state

Believe in Coordination

If all these fail... make the government powerful. Kaplan and Menzio (2015) model unemployment, and can get two steady states given their calibration. They take the lower unemployment level as the "reasonable" steady state, that they use to calibrate parameters against. I don't think they spell it out, but the argument most likely is that if there are two steady states, a powerful government could steer the economy to the steady state implying the highest welfare.

This is basically your "highest value" point - but it should be clear that there is some reason why that is achieved. That's why I said "it depends a lot on the context" - many coordination failure models hinge critically on the assumption that no higher power can steer the economy. See also the Chicken Economy model, a term I believe originally to be coined in Minnesota.

Sunspots

Finally, if all else fails, sun spots are what determine which equilibrium you are in. In economics, we mean by that basically a random exogenous variable, something not modeled. They're very disputed as they have no economic meaning and are hard to digest. Basically, if there was anything that had economic meaning and could determine the equilibrium, you would rather put it into your model than having a coin flip determine it.

See also Karl Shell's summary on the subject for The New Palgrave: A Dictionary of Economics.


What are the empirical techniques to show causation?

Natural experiments are usually a setting for causal inference rather than a causal inference tool per se. You often need to employ something like difference-in-difference or instrumental variables anyway even when you have a natural experiment.

 Here a list of statistical causal inference approaches (Approach: Lay description)

Instrumental Variables: Randomly assigned variable X influences Z only through Y
Difference in Differences:  If two groups have a common trend and only one group is treated then the change in the difference between the groups is the treatment effect
Regression Discontinuity: If a hard threshold determined treatment, look at difference right around that threshold
Propensity score matching: Create a control group by matching untreated observations that were likely to be treated (but not in fact treated) with treated observations with a similar probability of treatment.
Manhalobis distance matching:   Create a control group by matching untreated observations that look similar to treated. Another notable distance measure is Coarsened Exact Matching.
Synthetic control: When you have only one treated observation, create a composite of untreated observations that individually are imperfect controls but collectively act as a good control.
Synthetic Cohorts - Treat repeated observations of groups like a panel of individuals and employ panel techniques.
Selection bias modeling like Heckman correction: Assume a parametric form for selection bias and remove it, so the corrected regression results can be interpreted causally.
Sample weighting more broadly - fix bias resulting from endogenous participation and un-modeled heterogeneity by weighting sample units to look more like the true population of interest.


Why is the opportunity cost 0 in this case?

I think the point that the author was trying to make was that opportunity cost refers to the "best alternative use" of something. The measure is used to understand marginal decision making. Suppose you have just purchased an Oven which can either bake cakes or bake bread. Now if you decide to bake cakes, your opportunity cost is the number of loaves of bread that you could have baked at the same time. However, all this analysis is done keeping in mind that - "once the oven is purchased".

The machine in your example has no alternative use and hence has zero opportunity cost. Your analysis about the interest is regarding the money using which the machine was purchased. Sure that money could have been put to some other use - like buying a Netflix suscription or putting into bank deposit and hence will have an opportunity cost. Bust as far as the machine is concerned " once it is purchased", on the margins, it has no alternative use hence no opporunity cost.


In theory, depreciation is a reduction of the capital stock. This includes capital assets that have actually been worn out and also capital assets that are not usable any more for any other reason (conceivably old machinery after upgrading to a new technology or so).

In practice, macroeconomic datasets infer depreciation either from what the firms in the economy in question say in their accounting or from changes in the capital stock that are not explained by any other influence (investment, dissolution and selling of the firm, ...); this is data that also comes from the firm's accounting. Note that this is not necessarily the same as the actual depreciation according to the theoretical definition since there may be any number of reasons for firms to write off capital assets early or late or at any given time. The formal rules for what practices in terms of write-offs are lawful or not are quite complicated and differ between legal systems (i.e. between countries).

Further, macroeconomic models sometimes simply assume a fixed depreciation rate which they then proceed to compute from averaging over the time series constructed from investments plus the differences in capital stock and dividing by the capital stock. This is, of course, not the way it should be done but nevertheless ...


To make sure it's completely explicit: superscripts below are indices referring to either firm $1$ or firm $2$.

The choice variables in this problem are $\mathbf{q}^2$ and $\beta$. Notice that $\mathbf{q}^2$ is a vector of $n$ quantities. That is to say, $\mathbf{q}^2=\left(q_1^2,q_2^2,\ldots,q_n^2\right)$.

$(13.9)$ is just the derivative of the objective function $(13.8)$ with respect to $q_i^2$. Notice that you can rewrite the objective function $(13.8)$ as $$ p_1q_1^2+p_2q_2^2+\cdots+p_iq_i^2+\cdots+p_nq_n^2-\beta\left(q_1^1\right)-\mu_2\Phi^2\left(q_1^2,\ldots,q_i^2,\ldots,q_n^2 ,q_1^1\right) $$

Differentiating this with respect to $q_i^2$ and setting that equal to $0$ gives us $$ p_i - \mu_2 \frac{\partial\Phi^2 \left( \mathbf{q}^2,q_1^1 \right)}{\partial q_i^2}  = 0 $$

In Cowell's notation, $\Phi^2_i$ is just the derivative of $\Phi^2$ with respect to $q_i^2$.

The second first-order condition is the derivative of the objective function with respect to $\beta$. Since $\beta(\cdot)$ is a decreasing function of $q_1^1$, we can also think of $q_1^1$ as a decreasing function of $\beta$. (Formally, $q_1^1$ is the inverse of $\beta$, which is well-defined since $\beta$ is decreasing. Intuitively, if firm $2$ conditions their bribe on firm $1$'s level of output, then firm $1$'s output choice also depends on the amount of the bribe.)

Thus, applying the chain rule, the derivative of the objective with respect to $\beta$ is $$ -1 -\mu_2\frac{\partial\Phi^2 \left( \mathbf{q}^2,q_1^1 \right)}{\partial q_1^1} \frac{dq_1^1}{d\beta}=0 $$
which, unfortunately, isn't quite the same as in Cowell. Notice however that $\frac{dq_1^1}{d\beta}<0$, so perhaps he is using the absolute value of that derivative to get rid of the minus sign in front of $\mu_2$.


The current international monetary system based on pure fiat currency is a post-1970s system. For a long time before that most of the international monetary system was based on the gold standard. With the gold standard, governments issued currency relative to the amount of gold they held or they pegged their currency to another currency that was backed by gold. Hence, the bootstrapping issue you point out wasn't an issue. Note while current central banks often use government securities and bonds as assets to back the issuance of their currency, there is no rule that the asset needs to be a security (it could be gold like before) or that there even needs to be an backing asset at all (at risk of inflation).


Do standard definitions of substitute goods and complementary goods assume the Law of Demand?

A quick response is that if the law of demand is violated, then the standard definition for substitutes and complements may or may not apply.

  Consider buying fast food burgers as your only choice of meal. When the price of the meal drops, you might want to consume something healthier like a bowl of salads. You consume less fast food and may increase consumption of, say, broccoli. Then, this is your example. $\frac{\partial X_F}{\partial P_F}>0$. But, $\frac{\partial X_B}{\partial P_F}<0$ where P and X are price and quantity and subscripts F and B represent Fast Food and Broccoli respectively.

Consider another example:

  Consider buying a game CD for Xbox One that is a special and limited edition. As marketing strategy, the manufacturer emails you there are only few left and price has increased 5%. You are determined to get the game now for one collection purpose and another for actual game use. But then, you were gonna get a limited edition version of wireless controller to enhance your gaming experience. It is reasonable to consider they are complements. But because you decided to buy two CDs, you buy no wireless controller. In this example, we still have $\frac{\partial X_L}{\partial P_L}>0$. But, $\frac{\partial X_C}{\partial P_L}<0$ where P and X are price and quantity and subscripts L and C represent the limited game cd and wireless controller respectively.

Could you still argue the limited special edition CD is a substitute for the wireless controller? Maybe. But notice you still desire the wireless controller given it is wireless and limited edition. One of the reasons why you forgo the chance to buy this controller is you are effectively poorer than before. And this is an important distinction to make, because the decrease in the quantity of the controller may be from the direct impact of the increase in the price of the game CD but also the induced effect through your wealth (i.e. thinner wallet).

If you want to investigate the true relationship between two goods, you have to take both wealth and price effects into account on top of the given information, say whether the good is Giffen or inferior or both or something else.

Hope this helps.


In March 2020, why did stock indices of Asian countries that curbed COVID19 still plummet?

I cite some of  better comments from r/investing and r/stocks.

Because all their customers are in countries that didn't curb it.

Everyone else follows America. US equities are half the value of all equities in the world.

I would guess that Asia is an export economy largely to the rest of the world so tends to correlate with the rest of the worlds performance. However, the P/E ratios are likely lower than what you'd find in the developed world so there might be some opportunity for good gains when the broader economy picks up.

Because there was a gigantic flight to the US dollar in March. Pretty much every single asset across the board (yes, including safe havens like gold and US treasuries) fell versus the USD.


What's a good textbook on International Trade?

International Economics by Alan M. Taylor and Robert C. Feenstra is a good alternative, as International Trade by John Mclaren, which is more original.

If you look for a more advanced textbook, which discusses the underlying assumptions of the trade models, the reference is Advanced International Trade by Robert C. Feenstra.


Selling price of a widget at competitive equilibrium

The long run equilibrium price should be $\$ y,$ however, in the short run, we only know that price must be higher that the minimum average variable cost (of which no info is given).

In the long run all production takes place at the minimum average cost. All firms with average cost higher than $\$y$ will either adapt their technology and reduce their costs or exit the market.


How does the treasury determine the rates of the bonds they sell?

The question appears to refer to what is normally called the coupon rate (but the US Treasury does call the interest rate in its documentation). This is different than the yield, which is determined by a 1:1 function of price.
Issuance conventions vary across markets. I will just use the U.S. Treasury as an example. Firstly, this only applies to new bonds/bills, since reopenings already have a coupon, and bills are discount instruments (no coupon). I will ignore inflation-indexed bonds.
For the auctions inquestion, competitive bidders specify yields to 3 decimal places. Then:

Single-price auctions. The interest rate we establish produces the price closest to, but not above, par when evaluated at the yield of awards to suc- cessful competitive bidders.

That is, the coupon rate is set to get a price closest to par from below, using 1/8% increments.
Quote from: section 356.17 oflink to regulations
Treasury auction information page.


What would happen if someones money would disappear when they die

Oddly enough, this is a topic that economists have covered pretty thoroughly!

Life cycle models (more info) are a model of consumption generally attributed to Franco Modigliani. These models describe how individuals consume over their lifespan.

While more advanced versions of these models, like overlapping generations models or bequeathal models, incorporate people passing resources onto their children, in the simplest versions people only worry about themselves. In effect, this gives them a goal of using up all their resources at the same time they die, in an attempt to smooth their consumption over their lifespan.

Based on extrapolation from those models, and the fact that the no-bequeathal condition in those models isn't the main thing making them unrealistic, probably not a whole lot changes!

The biggest change you'd see if this were the case is that people would work harder to consumption smooth so that they wouldn't have anything left over. Or they'd save just enough to account for the uncertainty of their lifespan.

You'd probably end up with people falling into destitution when they live longer than expected (or more people doing so than already do). But people already undersave for retirement, so this would only matter for the rich. You might get more support for elderly-care programs so people don't have to save as much with the potential for it to be erased.

You'd also see people who amass great fortunes spending way more of it and hoarding less (or just giving it to their kids earlier), basically the same predictions you get from big estate taxes, as this is effectively just a 100% estate tax.

One change is that occasionally you'd get a super rich person dying unexpectedly. This would end up being a negative money supply shock, which could have negative macroeconomic consequences, although what would likely happen is that the relevant central bank would be prepared to print more money to replace it if necessary. This would reduce inequality very slightly.

None of those are particularly exciting predictions, but that's because this is a pretty easy thing to get around by just shifting how you spend your money. There might be some interesting contract-theoretic implications with the fact that wills are no longer a thing. No waiting for the rich person to die to figure out how they're going to divvy up their stuff; it has to be given away ahead of time.


Relationship between trade deficit and borrowing

We count the sale of financial assets as borrowing. You simply sell a document indicating that you will pay the buyer in the future, which is simply a debt.

Actually, sales of financial assets cause capital account to increase, so that, current account balance is compensated by the sales of financial assets which is basically a capital account surplus. Hence

$$
Current Account + Capital Account = 0
$$


Why is it a problem if a considerable amount of currency is hoarded by the enemy of the issuing government?

If the Japanese captured Hawaii and hence also all the dollar bills in Hawaii, then the problem would not be that they "hoarded" these dollars, but that they would have been able to use these dollars (to buy other goods and services).

And so the purpose of the Hawaiian overprint notes was this: they "would be worthless in the hands of the enemy, but could be used as money in Hawaii as long as the U.S. maintained control."

I have noticed that laypersons tend to be unduly worried about the possible effects of "hoarding" (as evidenced for example by the questions on this site about hoarding). There seems to be a myth that hoarding cash can somehow cause damage to other parties. But in fact, under most circumstances, it is those who do the hoarding that suffer, while those who aren't hoarding benefit.*

If China "hoards" US dollars that were acquired through legitimate, peaceful means, then this is no bad thing for the US.

Say China sells T-shirts and other goods to the US in exchange for \$1 trillion in dollar bills. That is, the US receives these goods and simply gives China a bunch of paper money. China then "hoards" these \$1T in paper money. Then this is no bad thing for the US. The US will simply have acquired \$1T worth of T-shirts and other goods, while China will have acquired a lot of paper that is useless so long as it is "hoarded" and unused.

Indeed, the value of this paper will tend to depreciate. In fact, the US can cause the value of this paper to depreciate by simply printing more dollars. And in a state of war, the US could even undertake drastic measures such as declaring those \$1T useless or achieving this effect by requiring all US citizens (and possibly other non-Chinese parties) to exchange their dollar bills for new legal ones.

*One important exception to this is the underground economy. When cash is hoarded, this can negatively affect those in the underground economy who rely heavily on cash to facilitate transactions.


Prove this indirect utility function is quasi-convex

If the other two methods did not work out for you, another possibility, albeit a more difficult one:

Set up the bordered Hessian:

$$\bar{H}=\begin{bmatrix}0 & \frac{\partial v}{\partial p_1} & \frac{\partial v}{\partial p_2} & \frac{\partial v}{\partial p_3} & \frac{\partial v}{\partial m}\\
\frac{\partial v}{\partial p_1} & \frac{\partial^2 v}{\partial p_1^2} & \frac{\partial^2 v}{\partial p_1\partial p_2} & \frac{\partial^2 v}{\partial p_1\partial p_3} & \frac{\partial^2 v}{\partial p_1\partial m}\\
\frac{\partial v}{\partial p_2} & \frac{\partial^2 v}{\partial p_2 \partial p_1} & \frac{\partial^2 v}{\partial p_2^2} & \frac{\partial^2 v}{\partial p_2\partial p_3} & \frac{\partial^2 v}{\partial p_2\partial m}\\
\frac{\partial v}{\partial p_3} & \frac{\partial^2 v}{\partial p_3 \partial p_1} & \frac{\partial^2 v}{\partial p_3 \partial p_2} & \frac{\partial^2 v}{\partial p_3^2} & \frac{\partial^2 v}{\partial p_3\partial m}\\
\frac{\partial v}{\partial m} & \frac{\partial^2 v}{\partial m \partial p_1} & \frac{\partial^2 v}{\partial m \partial p_2} & \frac{\partial^2 v}{\partial m \partial p_3} & \frac{\partial^2 v}{\partial m^2}\\
\end{bmatrix}$$

Define $\bar{H}_k$ as the $k_{th}$ order leading principal submatrix. We can apply the following theorem:

If the $\text{Det}\left(\bar{H}_k\right) <0$ for all $k=2,3,4,5$, then $v$ is quasiconvex in $(\textbf{p},m)$.

This is an application of the more general result:

Let $X$ be an open convex subset of $\mathbb{R}^n$ and let $f:X \rightarrow \mathbb{R}$ be a twice continuously differentiable function $\left(f\in C^2\right)$. Let $\bar{H}$ be the bordered Hessian and let $\bar{H}_k$ be the $k_{th}$ order leading principal submatrix. If the $\text{Det}\left(\bar{H}_k\right) <0$ for all $k=2,3,\dots,n+1$, then $f$ is quasiconvex on $X$.


What does it mean for a technology to have a $|\rho|<1$?

New technology makes past technology obsolete. How many people know these days how to light a fire by rubbing wood together? How many people know the nuts and bolts of tending to the engine of a train powered by coal? Heck, how many coal-trains are still operational?

So yes, we do forget our past discoveries and technologies, as they are replaced by new ones. But the technology as inserted in the production function, operates as a production shifter, a "mark-up", it does not represent the composition of technology, but rather its final effect on the production inputs.

Modeling the technological production shifter as a stationary auto-regressive scheme (without drift), we essentially assume that, random shocks aside, we are set today to produce less efficiently than yesterday.

Does this accord with observed reality? Do we depend on positive random discoveries to produce more efficiently? My impression is that, on the contrary, humans very purposefully try to make their production processes more efficient.

Now, if you were to add a drift (a constant term) in the log-technology process,

$$a_t = \delta + \rho a_{t-1}+\epsilon_t$$

you would capture, the rising trend in efficiency we have observed historically, and also implicitly, the fact that we discard old technologies.
This also has the benefit that it envisions rising efficiency while also, an eventual level-off (which seems wise to me).


Too much of a good thing - examples of production (or utility) functions that are initially increase in a factor and then decrease

$$F(R,T)=100000-(R-200)^2-(T-25)^2,$$

where $F$ is production (in kg), $R$ is rainfall in a year (in mm), and $T$ is average temperature over the course of a year (in °C).

This says that under the optimal growing conditions of $R=200$ and $T=25$, production will be 100,000 kg.

If rainfall $R$ or temperature $T$ is higher or lower than these optimal levels, then production will be lower than 100,000 kg.

(Concepts you may be looking for include satiation and bliss point.)


How to interpret the intersected information partition element of different actors?

I'm not an expert in epistemic game theory but let me see if I can help with an example.

Suppose the state space is $\Omega = \{1,2,3,4,5\}$, and there are two players with information partitions
$$
\mathcal{P}_1 = \left\{\{1,2,3\},\{4,5\} \right\} \\
\mathcal{P}_2 = \left\{\{1,2\},\{3,4\},\{5\}\right\}
$$
Suppose the true state is $\omega = 5$.
Let us consider the event $E = \{4,5\}$.

Player 1 "knows" $E$, since $P_1 = \{4,5\} \subset E$.

Player 2 "knows" $E$, since $P_2 = \{5\} \subset E$.

However, does Player 1 know that Player 2 knows $E?$ No.

What are the elements of $\mathcal{P}_2$ that intersect $P_1$? $\{3,4\}$ and $\{5\}$.

However, $\{3,4\} \not \subset E$, so Player 1 does not know that Player 2 knows $E$.

On the other hand, does Player 2 know that Player 1 knows $E$? Yes.

What are the elements of $\mathcal{P}_1$ that intersect $P_2$? $\{4,5\}$, and $\{4,5\} \subset E$, so Player 2 knows that Player 1 knows $E$.


If a Chinese person wants to buy an iPhone or any international goods, which GDP per capita would be a better indicator?

To me, the method to get PPP value could be crude. If you're only interested in an iPhone, here's what I would do.

Get the price of an iPhone sold in China (in CNY), divided by the nominal GDP per capita for China (in CNY). You'd get (roughly) the fraction of annual income a Chinese person would need to spend on an iPhone. That should give you an indication of whether that's "cheap" or "expensive"
To compare across countries, you'd just do the same and see if that fraction is higher (more expensive) or lower (less expensive).

Keep in mind though that, with existing income inequality, GDP per capita doesn't really tell you about the actual income an average person gets.


How government borrowing from central bank increases money supply in economy?

Central banks can create money 'out of nothing'. So for starters there isn't 'less amount' of money left with the central bank. The amount of money at the central bank is 'infinite'.
So it's not because the CB lends X amount of money to the government, that the CB has X amount less money to lend to the banks, the supply of money is not constrained...
Off course, creating too much money and monetizing debt can have negative consequences for sure (for example hyperinflation in extreme cases), but that's another discussion.
See also here for more info about quantitative easing: https://www.youtube.com/watch?v=4TihoBfdCe8 and here: http://www.investopedia.com/terms/q/quantitative-easing.asp

Edit: comment from Energy numbers: strictly speaking government borrowing money from central banks doesn't increase the money supply, it only increases when the government spends the borrowed money.


Assuming that $W$ is wage and $L$ is labor, I would say this is a labor supply-demand graph. TD stand for demand and S for supply.

The graph shows a positive shift in labor supply, i.e. labor offered by workers, leading to an increase in $L$ and a decrease of $W$ due to the excess of supply.

The shift in labor supply can be caused by a variety of shocks as a preference shock which values more labor, a change in non-labor income, etc (look here for explanations)


Rationalizable action profiles in nice symmetric games

Isn't the mixed extension of matching pennies a good example of this? The strategies $p,q$ are both elements of $[0,1]$, and
$$
\begin{align*}
U_1(p,q) & = pq + (1-p)(1-q) \\
\\
U_2(p,q) & = p(1-q) + (1-p)q.
\end{align*}
$$
There is a unique Nash-equilibrium at $p = q = 1/2$ but all $p$ and all $q$ are rationalizable, because in the equilibrium all strategies are best responses.

Even if you were to edit your definition of symmetry to include symmetric utility functions, a game with
$$
U(p,q) = -(1/2 - p)^2 \cdot (1/2 - q)^2
$$
yields the same result.

If you insist on some shrinkage one could easily edit the utility function to
$$
U(p,q) = - 1 \text{ if } p+q \leq 6/10.
$$
Then some strategies - the ones under $1/10$ - would not be rationalizable.


Government debt and Fiscal Theory of Price Level

Question:

  I understand that he is explaining 'non-Ricardian' policies. However,
  if the government increases its deficit by issuing bonds, would that
  not decrease AD in the period the bonds are sold? Say that the
  government increases its deficit in period t by selling bonds.
  Private agents buy these bonds and thus give up consumption goods. So
  in period t, should AD not decrease?

The act of issuing bonds does not directly affect AD, it only results in a rebalancing of portfolios: the household sector gives up money balances in exchange for government bonds. (If your model implies that spending depends upon money balances, there might be an indirect effect on the economy. In many models, money balances are zero, and so there is no effect. See below.) In any event, the end-of-period money holdings of the household sector depend upon the reaction function of the central bank within the model, and so that reaction function will effectively determine the final household portfolio allocation between money/bonds. (The reaction function is usually specified as a rule for the interest rate, which determines money holdings within the household's money demand function.)

The lack of relationship between money balances in macro models and transaction volume is not entirely obvious. The reason is that money balances are meant to represent end-of-period holdings, and transactions clear simultaneously. This means that a household with \$100 income can spemd \$100, and leave a \$0 cash balance at the end of the period. Since it is better to hold bonds/bills if interest rates are positive, the end-of-period money holdings woukd normally end up at zero. It can do this in every period, and so it is continuously transacting, yet measured money balances never appear. This is obviously unrealistic, and so other classes of models introduce reasons for end-of-period money holding. Two typical cases are a cash-in-advance requirement, or money in the utility function. The cash-in-advance requirement is that money holdings have to be at least some percentage of consumption spending (which simulates the reality that most wages are paid at a lower frequency than spending). Money in the utility function creates a demand for money within the optimisation problems, which usually depends upon the rate of interest. The end result is that the model assumptions imply a relationship between money holdings and the economy; change the assumptions, and the relationship changes.

Leeper is discussing a tax cut, which can trivially give rise to unchanged consumption (multiplier of 0). The government consumes the same amount as before, and thus runs a higher deficit (which is matched by greater bond issuance). If the household sector consumes the same amount, its savings rises by the amount of the tax cut, which it invests in bonds.

If the "multiplier" on the tax cut is positive, it implies that the household consumes more as a result of the tax cut, and so we have greater output. The exact effect depends upon the model you are using.

The question asks:

  However, suppose that interest rates rose. If the interest rate is
  higher than the growth rate of the economy, the government will have
  to monetise its debt in order to pay the interest component, this will
  obviously be inflationary.

The quote from Leeper is a discussion of "Regime F." Within the analysis of the paper, monetising of debt has nothing to do with determination of the price level. The paper, the price level in regime F is given in equation (14), which is described:

  At time t, the numerator of this expression is predetermined,
  representing the nominal value of household wealth carried into period
  t. The denominator is the expected present value of primary fiscal
  surpluses from date t on, which is exogenous.

That is, the expectation is that future primary surpluses will be used to stabilise the debt level.


Specification bias - estimated variance is biased estimator of true variance of error term

If the model $ (a)  y = X\beta + u $ is true, then $\omega = u + X\beta - Z\gamma $ in $(b)$, and the claim follows.


Does the currency in circulation equal the annual production of goods and services?

The amount of currency in circulation does not equal the annual production of goods and services in a country. It is usually much smaller.

For example, as of July 2013, currency in circulation—that is, U.S. coins and paper currency in the hands of the public—totaled about $1.2 trillion dollars.

In contrast the U.S. GDP for the same year was about $17 trillion dollars.

You can find more information on how Fed prints currency to meet demand from this Fed article.


Why exactly does a perfectly competitive market depend on consumers having perfect information about a product?

Perfect information implies every buyer and every seller knows everything in the market. As soon as a firm brings a new product to the market, other firms start selling the same, identical, product. In practice, say a kebab shop comes up with a secret new ingredient and attracts more custoemrs, but as soon as this happens, other shops copy it. Bottomline: any information-advantage is short-lived in perfect competition. This also leads to the other characterisation, all products are homogeneous.

If the firms/customers do not possess full information about the market/products, then you should characterise the market differently i.e. oligopoly, monopolistic competition or monopoly.


Why does capital flight cause inflation?

(1) Shouldn't it cause a deflation? Currency is less and less
  available (due to flight) for the purchase of domestic goods, thus..
  deflation. That is, a smaller amount of currency chasing the same
  goods.

The dollar is less available, whereas the purchase power of the bolivar (Venezuela's domestic currency, apropos of your example) becomes negligible: To buy the same goods, today Venezuelans need more bolivars than in the past. Inflation comprises two factors: economic activity and purchase power. In cases like this, inflation is driven up by the bolivar's drastic loss of purchase power.

  (2) But also, capital flight is done with saved money, instead of
  money reserved for purchases, and that means aggregate demand is not
  affected much, meaning, purchases are done as normal, and normal life
  follows. Surely, that doesn't mean a deflation, but it doesn't mean an
  inflation either.

Capital flight results in a reduction of supply. If demand remains unchanged, then the intersection of supply and demand curves occurs at a higher equilibrium price. That means that the bolivar has lost purchase power.

  (3) Consider this: Assume for whatever reason, interest rates has
  fallen. This means, borrowing money is cheaper, and that is likely to
  cause a shift in aggregate demand and domestic consumption, possibly
  boosting the economy, and causing some inflation in the process.
  However, with capital flight, there's less and less investment done
  inside the nation, so, the supply of investment shifts, causing an
  increase of interest rates -- which again, should tend to cause a
  deflation due to a decrease in the money supply (basically, the
  reverse reasoning of lower interest rates).

It is difficult to make conjectures where a situation gets out of control. In broad terms, the relation between interest rates and money supply can certainly break down when chaos strikes the markets. But even in a less-than-extreme scenario, that relation may hold and yet the net effect can still be a higher level of prices.

Also keep in mind that printing extra money to honor the government's bonds contributes to depreciation of the domestic currency, and interest rates might not be high enough to attract lenders.


Coefficient omitted because of collinearity

You have firm fixed effects. Presumably, every firm is in the same country throughout your observation period. Moreover, no country ever changes its development status (I assume). Hence, for a given firm, there is no variation in the development status. Therefore, you can't identify an effect.


Yes. In 2018, the German YouTuber Jörg Sprave established YouTubers Union (see also FairTube).

In 2019, it entered into a partnership with IG Metall: IG Metall press release, news stories by Observer, Bloomberg, CNBC.


Constrained Optimization using Lagrangian method

In general, FOCs produce local, not necessarily global, optima. Consider
\begin{equation}
\max_x\;f(x)=x \sin(x) \quad\text{s.t. $x\in[0,7]$}.
\end{equation}

As the plot below shows, $f(x)$ has three stationary points in the domain $[0,7]$: $x_1=0$, $x_2\approx 2.03$, and $x_3\approx4.91$. Among them, $x_2$ is the local maximum that would be picked up by the FOC. However, the global maximum happens at the boundary, where $x=7$. Since $f'(7)\ne0$, the global maximum would be missed if you rely only on the FOC.


Looking for the title of a short (~100 page) Herbert Simon book about "Bread-bricks"

It is my impression that it is "bean-bricks" and it is not a book but a paper/essay under the title "The long-range economic effects of automation", which was included in the book

HERBERT, A. S. (1965). The Shape of Automation for Men and Management.

Go to the on-line digital library of Carnegie Mellon University

and enter in the search "The Economic Effects of Automation" (without the "long-range" part). It (freely) downloads what appears to be a first version of the work (20 pages). The file may appear to not be a .pdf one, but I opened it with .pdf software alright.

At the end of page 9 Simon writes:

  "In our hypothetical economy, only a single commodity, which we shall call beanbricks, is produced. Beanbricks serve both as the sole
  consumption good (they are eaten), and as the sole form of capital
  (they are burned as fuel used in producing more beanbricks). The other
  productive resource in the economy, besides beanbricks, is labor".

It does look like it is the first time Simon uses the term (the file is a scanned copy of Simon's manuscript).

P.S: Note that standard economic models of growth essentially assume exactly the same framework, since output is costlessly transformed into capital or becomes consumption.


Difference(s) between Rational Expectations Equilibrium and Nash equilibium

Hi: They are pretty close to completely different. I don't know the least thing about game theory but a nash equilibrium describes what happens in a game when two or more people have a certain type of expectation. RE is one type of expectation but if you google for RE you'll see that it's a field in itself. It happens to have an application in game theory but it's a seperate field developed by macro-economists in the early 70's. RE founder could be viewed as Lucas or John Muth depending on who you ask. Muth came up with the concept in 1961. Lucas applied it to macro-economics in the early 70's and it took off from there.


Imports not used in the country, still counting in the GDP calculation?

GDP is the total amount of domestically produced output in the referred period. If $C^H$ is the consumption of domesticly produced goods, $C^F$ is the cosumption of goods produced abroad,
$G^H$ is the government expenditure on domesticly produced goods, $G^F$ is the government expenditure on goods produced abroad and so on we have:
$$GDP = C^H + G^H + I^H + X = \\
C-C^F + G-G^F + I-I^F + X =\\
C + G + I + X - (C^F +G^F +I^F) \\= C + G + I + X - M$$
That is imports do not affect GDP from accounting point of view. It is just easier to subtract them directly as in the last equation although we are after the first.
My textbook (Mankiw) used to say: If you purchase goods abroad and store them, this is an investment into inventory in the period in which they were imported i.e. $I^F$ goes up. This is the same as if you produce a car at home and fail to sell it. In this case the car is counted in $I^H$.


I don't have many books to share but there is the website on Farm economy data from the USDA.

Farm Labor
USDA Economic Research Service (ERS)

https://www.ers.usda.gov/topics/farm-economy/farm-labor/

There is also the USDA National Agricultural Statistics Service (feel free to explore their site)

https://www.nass.usda.gov/Data_and_Statistics/index.php

here are some articles

https://arefiles.ucdavis.edu/uploads/filer_public/ad/74/ad7450e7-80ab-4cf7-a147-6b80c2e614a7/chapter_1_the_farm_labor_problem_4-4-17.pdf
The Farm Labor Problem
J. Edward Taylor & Diane Charlton
Agricultural & Resource Economics
University of California, Davis

https://www.bls.gov/opub/mlr/2014/article/agriculture-occupational-employment-and-wages.htm
Agriculture: occupational employment and wages

Labour Market Effects of Large-Scale Agricultural Investment: Conceptual Considerations and Estimated Employment Effects
https://www.sciencedirect.com/science/article/pii/S0305750X17301754

Few books on the subject:

freebook-
The Economics of Food and Agricultural Markets - 2nd Edition
https://open.umn.edu/opentextbooks/textbooks/the-economics-of-food-and-agricultural-markets

Agricultural Economics (3rd Edition) 3rd Edition
by H. Evan Drummond Ph.D. (Author), John W. Goodwin (Author)


If Gross Savings is less than Gross Investment, could we say that investment is being made with borrowed money?

Yes, in particular, the Home country is borrowing from abroad.

In a closed economy,

$\text{GDP} = C + G + I$. And savings is defined as $S = \text{GDP} - C - G$. And thus, it is necessarily the case that $S = I$.

Example. In 2015, India produces 10 bananas, each valued at ₹10, for a total value of ₹100. Private consumers consume 4 bananas ($C = ₹40$), government consumes 5 bananas ($G = ₹50$), and 1 banana is stored for consumption in future years ($I = ₹10$). (Bear in mind that additions to inventories, as with this last banana, count as investment.)

$GDP = C + G + I = ₹40 + ₹50 + ₹10 = ₹100$. Total final consumption ($C + G$) was ₹40 + ₹50 = ₹90. Hence, $S = ₹10$. This is a closed economy, so savings must necessarily equal investment. And sure enough, $I = ₹10$.

In an open economy, it need no longer be the case that $S = I$.

Example. In 2016, India produces 10 bananas, each valued at ₹10, for a total value of ₹100. India also imports 5 bananas ($M = ₹50$).

Private consumers consume 4 bananas ($C = ₹40$), government consumes 5 bananas ($G = ₹50$), and 6 bananas are stored for consumption in future years ($I = ₹60$). There are no exports ($X = ₹0$).

$GDP = C + G + I + (X - M) = ₹40 + ₹50 + ₹60 + (0 - ₹60)= ₹100$. Total final consumption ($C + G$) was ₹40 + ₹50 = ₹90. Hence, $S = ₹10$. But $I = ₹60$.

So $S < I$. How is this possible? The reason is that the gap between investment and savings of $I - S = ₹60 - ₹10 = ₹50$ is made up for precisely by imports. So our shortfall in savings (relative to investment) is made up for by borrowing from abroad, in this case through imports.

As for your second question,

Yes, we can say that on average, each Indian saves 30.2% of income. But this average is not necessarily very meaningful. For example it may be that the vast majority of Indians spend all of their income on consumption and it is only a tiny minority of the wealthiest Indians who do the saving.


What do banks do with "Repaid Principal"?

When you repay principal of 90 USD, the bank makes two accounting entries in its computerized books.

One is called a debit and it goes like this.  If you pay by check drawn on another bank, their reserves at the Fed are increased by 90 USD. If you paid in currency, their vault cash asset account, which is also an asset of the bank, increases by 90 USD and the cash is dumped into the vault with all the other cash.  If you pay out of your checking account at the same bank, their "deposits" book entry, which is a liability of the bank, is decreased by 90 USD.

The other entry is called a credit.  Your loan balance, which is an asset to the bank, is decreased by the 90 USD.

You asked what they do with "it", meaning the principal paid.  As you can see above, when the currency was dumped into the vault and mixed with all the other dollar bills, or a computerized balance was increased or decreased by 90 USD, "it" ceased to exist.  The 90 USD are no longer distinguishable from any other dollars.  So it is a meaningless question. A lot of people don't understand this!

But if you are asking what the bank does with its reserves (vault cash plus reserve account at the Fed), in general: in normal times, they keep them as low as allowed by the Fed.  As their reserves increase higher than required, they reduce them by lending money to customers and other banks, purchasing securities, or paying their expenses.  Transactions that make their reserves increase are things like (a) loan principal is repaid, (b) currency or checks drawn on other banks is deposited, (c) paying their expenses, (d) loan interest is paid by their debtors.

At present, banks are holding on to more than they are required to, because they are relatively afraid to lend.  The extra is called "excess reserves".


Is there book or research on economically sustainable design of town or village?

I would suggest the Handbook of Regional and Urban Economics. It currently has 5 volumes and collects papers on the theories and empirics of urban/city design. Selected chapters that may potentially address your questions include:

City size and place as policy issues
Financing urban public services
The theory of urban residential location
Therories of urban business location
Urban labour markets
Urban housing markets: Theory and policy
Micro-Foundations of Urban Agglomeration Economies


Determining Another Possibility For a Microeconomics Question

Usually capital costs are deemed fixed rather than variable costs. Probably your teacher is thinking that the AFC curve will shift upwards, although this depends on the cost of the new equipment relative to that of the old.


A nerd loves both mathematics and physics equally, and hates sports and drinking beer equally.

A football player hates both mathematics and physics equally, and loves sports and drinking beer equally.

The nerd and the football player have the same indifference curves, but not the same preferences.

The point is this: Both agree on the "grouping" of things. But they disagree in the ranking of the groups.

Your Example

If you look at Jim and Pat, they have exactly inverse utility functions and hence inverse ranking in their preferences.

That is, if for Jim $A > B = C = D > E$, Pat will have $A < B = C = D < E$. Both agree on the three groups, but disagree on the order. Jerry's just a monotone transformation (poor Jerry).


Calculating the real interest rate, why would my thesis supervisor want me to subtract inflation instead of dividing by inflation?

Two things to note here. First, subtracting inflation from the nominal interest rate is an approximation to the real interest rate, but only in discrete time. Furthermore, the "true" relationship it's approximating isn't division of one rate by the other--you have to add 1 to all three of your quantities (inflation, real interest rate, and nominal interest rate) first to get the true relationship.

Here's a brief overview. Consider the Fisher equation of

$r = i - \pi$

where $r$ is the real interest rate, $i$ is the nominal interest rate, and $\pi$ is the inflation rate.

This equation is often introduced as a linear approximation to the true real rate of interest, given by the equation

$\frac{1 + i}{1 + \pi} = 1 + r$

Let's see how this holds in a discrete time model. Denote your nominal income as $Y$ and the price level as $P$. Your real income is $Y/P$. If all this income is invested in some interest-bearing asset in a discrete time model, your real income in the next time period becomes

$\frac{Y (1 + i)}{P (1 + \pi)}$

and if we want to find a real interest rate that summarizes this change in real income, we would need to write your real income in the next time period as

$\frac{Y}{P} (1 + r)$

which gives us the identity $\frac{1 + i}{1 + \pi} = 1 + r$ that the Fisher equation approximates. However, if we work in continuous time, this breaks down. First, the units don't work out--inflation rates and interest rates are measured in percentage change per year (or some other unit of time), so they cannot be added to the dimension-less number $1$. Second, it turns out that Fisher's approximation is actually completely correct in continuous time. Using derivatives, we define our quantities as follows:

$i = \frac{dY}{dt} \frac{1}{Y}$

$\pi = \frac{dP}{dt} \frac{1}{P}$

$r = \frac{d(Y/P)}{dt} \frac{1}{(Y/P)}$.

Using the quotient rule, we can rewrite $r$ as

$r = \frac{\frac{dY}{dt}P - \frac{P}{dt}Y}{P^{2}} \frac{1}{(Y/P)}$

which simplifies to

$r = (\frac{dY}{dt} \frac{1}{P} - \frac{dP}{dt} \frac{Y}{P^{2}}) \frac{P}{Y} = \frac{dY}{dt} \frac{1}{Y} - \frac{dP}{dt} \frac{1}{P} = i - \pi$

which gives us the Fisher equation, no approximations about it!

Note that this assumes continuous compounding from your nominal interest rate. If you instead have a compounding rate of $\tau$, we would define $i$ differently, and our equation becomes

$r = \tau \ln(1 + \frac{i}{\tau}) - \pi$.

However, for the purposes of the data you're working with, I am 90% sure that this modification is completely superfluous. If you want to use something more precise than the Fisher equation, you need to know exactly how your data was computed. What price index was used to calculate the inflation rate? Under what assumptions was the nominal interest rate calculated?

In short, while I can't speak to the reasons for your supervisor's recommendation, they're definitely correct that you should subtract inflation rather than divide by it. There's a reason why we use the Fisher equation.


We see stocks outperforming things like bonds in the long run (past 100 years), but could this be an artifact of that particular 100 year timeline?

Yes. The relative performance of stocks and bonds depends on monetary policy. Stocks benefit from low interest rates and inflation while bonds benefit from high interest rates. Over the 19th century for example, stocks deflated along with the overall economy while bonds yielded a small positive return.


Can game theory be used to predict economic specialization (division of labor)?

I think you're over-analyzing here. Game theory is not just about maximizing your own pay-off, but about doing the best you can given the behavior of others. What Smith describes is just that people do what they're good at to make a living. That is, the way I read it, Smith claims that people specialize independent of what others do.

That being said if the emergence of specialization was to be predicted with game theory I'd look into evolutionary game theory. I'm not sure if you'll find anything though.


Yes, Smith addresses these issues in Book 1 Chapter 10 of The Wealth of Nations. Firstly, he notes that where two jobs are in almost every respect equivalent, we should expect them to pay the same wage:

  "THE whole of the advantages and disadvantages of the different employments of labour and stock must, in the same neighbourhood, be either perfectly equal or continually tending to equality. If in the same neighbourhood, there was any employment evidently either more or less advantageous than the rest, so many people would crowd into it in the one case, and so many would desert it in the other, that its advantages would soon return to the level of other employments. This at least would be the case in a society where things were left to follow their natural course, where there was perfect liberty, and where every man was perfectly free both to choose what occupation he thought proper, and to change it as often as he thought proper. Every man's interest would prompt him to seek the advantageous, and to shun the disadvantageous employment."

This is related to the observation made by MH.Q: if A and B are two equivalent jobs, but A pays more then everyone would rather work in A than B. This results in a shortage of labour in B and/or an excess of labour in A so that the wages converge by natural market forces and the wage gap disappears.

Most of the rest of the chapter is spent discussing reasons why we might diverge from this ideal of uniform wages. The need to compensate skilled workers for the cost of their education and for the chance that they will not be successful in their chosen profession, for example, are offered as key reasons why wages might differ. I recommend you take a look at the chapter to get a handle on Smith's thinking.

This reasoning still forms part of our understanding of the modern labour market: if the wage premium for a job is not sufficient to compensate a worker for the (opportunity) cost of training to do that job then workers will not become trained and there will be a shortage of labour in the industry. This will drive up prices by natural market forces.

More generally, we now understand that wage differentials are an important means by which the market informs participants about how they should act. An extremely talented person who could, in principle, tranin to perform any of a large range of difficult jobs has no way to know where society needs his/her talents most. But the market provides this information by providing a mechanism for prices (i.e. wages) to increase in areas where skilled workers are short relative to prices in areas where workers are plentiful. The true beauty of this market mechanism is that is simultaneously provides information about the needs for workers, and provides the incentives for workers to enter the appropriate industries.

Whatever problems the wage gap might cause in the economy, they must be weighed against the large efficiency gains that come from allocated talent where it is needed most.


Subgame-perfect Nash equilibrium perfect information

The game is indeed an $8\times 4$-game. A strategy formally specifies what to do at each information set, including those information sets that can never occur under the strategy. In that sense, a strategy is not simply a "plan of action." This point has been famously made in (see Section 2):
Rubinstein, Ariel. "Comments on the interpretation of game theory."
Econometrica: Journal of the Econometric Society (1991): 909-924.


A question of deriving an industry supply curve from cobb-douglass production function

Your method and numerical calculations are correct.

Then, it is a well-known fact that Cobb-Douglas (i.e exponential with constant returns to scale) production functions with price-taking behavior lead to linear cost functions, i.e. to constant marginal cost and equal to average cost, and linear profit functions, which means that profits in the long run will be either zero or infinite: If price is above $2$ ($=MC=AC$) each firm would want to produce an infinite amount of output. if it is equal to $2$, firms that produce will make zero profits. If it is below $2$ no firm will produce and profits will again be zero.

At market level, the assumption of perfect competition (costless entry especially) in turn implies that the long-run (no fixed costs) market supply curve will be horizontal at $P=MC=AC$, and the level of output will be determined by the demand curve and consumer preferences. The output per firm and hence the number of firms remains undetermined here.


Help finding historical data for velocity of money (M1 or M2) for any country except USA

If you're willing to expand your definition of money, then yes, and easily: You can get broad money measures (M2, M3 and M4) as a percent of nominal GDP ($\frac{M}{PY} \times  100%$) from the World Bank for several decades for many countries (mileage will vary depending on particular country).

Just divide by 100 and invert.  By the quantity theory of money,  $V = \frac{PY}{M}$, Bob's your uncle.  Note that this will underestimate V compared to values reported by e.g. the Federal Reserve, because M includes M3 and M4.


Does decreasing marginal utility imply risk aversion?

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

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

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


Econometrics: Omitting a significant variable

You should do none of the above.  This is an invalid decision-making process.

In the best of all possible worlds, create a series of do loops and go through the set of all possible combinations of variables.  Calculate the AIC or the BIC.  If you know nothing about either, just pick one as they usually give the same result.

The model with the lowest AIC or BIC is the model that is probably the closest to the true model in nature.  Statistical significance does not matter at all.  If the best model is also not significant, then it is not significant.  It could just mean the other models have a spurious significance.  Likewise, if the model chosen is significant under the F-test, but has non-significant variables, then you cannot change them.

If you do not know how to write do loops or for-next loops, then find a package with step-wise regression.  It will not cover all the models, but it will cover many.  Use the same criterion either the AIC or the BIC.

Because you are asserting the null is true, you cannot have more than one null.  It means nothing if you add or subtract variables because you are changing the null every time and you cannot do that.  The AIC or BIC are non-Frequentist methods of estimating the true model and so the solution of using either bypasses the question of significance until after the model is chosen.

EDIT

I thought I would provide an edit to cover the statements in the comment.

For starters, I agree with the comment.  I thought I should ground the above statement better as to the logic behind it.

The various information criteria, the AIC, the BIC, the DIC and so forth, can either be grounded in information theory or in Bayesian theory.

From an information theoretic perspective, if you have outside information about which models should be included or excluded either from theory or experience then that information needs to be incorporated.  Since there is no direct method to merge them together, you should use judgment in which models to look at.

From a Bayesian perspective, the various information criteria are stylized point approximations of the Bayesian posterior under somewhat restrictive assumptions.  In many respects, they are not good proxies because the smaller the posterior density, the less likely it is to be true, whereas the reverse is true in the criterion.  It is best to think of them as both being rankings and they would provide the same ordering of rankings.

Now there are two Bayesian issues present.  First comes from the fact that you can construct Bayesian theory from Cox's axioms.  Cox's axioms are built around Aristotelian logic.  You would use Bayesian methods to assess logical statements.  If some statements do not need assessing or could be excluded by logic, then they should be excluded from consideration.

The second comes from the nature of the prior density.  If you have prior knowledge that some cases cannot be true, then you should give them zero prior weight.  This would exclude them from consideration at all.

Nonetheless, the combinatoric method should be considered because it maps to a Bayesian argument and the information criterion are non-Frequentist constructions and so you shouldn't be using Frequentist criteria in making this type of decision.

Bayesian hypotheses are combinatoric.  If you are testing the model $$y=\beta_0+\beta_1x_1+\beta_2x_2+\beta_3x_3+\varepsilon$$ and you have no logical reason to exclude any case, then you are going to have to test eight hypotheses, removing one, two or three variables at a time.  This is the practical equivalent to the Frequentist hypothesis that $\beta_i=0$ which cannot be tested in Bayesian methods because it is a sharp null hypothesis and of measure zero.  Removing a variable and assuming its coefficient is zero is the same thing.

Because Bayesian methods lack a null hypothesis, which is both a strength and a shortcoming, no one hypothesis gets special treatment or special weight, absent material external information.  You test the posterior probability of each possible model as part of your process.

Now, this is where a legitimate objection does come in.  Information criteria are stylized approximations of the actual posterior.  In some cases, they are perfect approximations but they can be poor approximations in other cases.  In the perfect approximation case, you should be using the information criterion that is appropriate for your problem and your worry is not about the criterion, but rather the representativeness of your data.  In the poor approximation case, close differences could, in fact, be reversed if the posterior was actually computed.  As you multiply your combinations, the greater the probability of a single pairwise mistake will happen.  Nonetheless, since you are not doing model averaging, but model selection it will probably not be the highest two pairs.

Using tools like the information criterion does highlight a problem in data-based measures over a true null hypothesis based methods, the combinations show how little independent data that you may have.  If you are not model hunting, but instead testing the one true model, then you will lose a few degrees of freedom and unless you have collinearity problems, then you are fine if your sample is of reasonable size.  In my example above, it is quite like dividing your sample by eight.  It isn't a subtractive process as one would see with degrees of freedom, it is more like a process of division.  Add into that the internal correlations and there may not be much in the way of independent information in your set.

Still, it didn't seem to me like you felt you had a mental model of how the relationships should be, so I would still recommend either the combinatoric or the step-wise solution.


Maximising a partly concave and partly convex function

If $f$ is strictly convex everywhere, it is optimal to set $x_i
=\overline x$ and $x_j =0$ for some arbitrary $i$ and all $j\ne i$.

I assume from this that the $x_i$ are constrained to be non-negative?

Using the Lagrange multiplier technique with Lagrangian $L=\sum_i f(x_i)-\lambda (\sum_i x_i-\overline x)$, we can conclude that the stationary points occur when $\frac{\mathrm{d} f}{\mathrm{d}x}\big{|}_{x_i}=\lambda$. Furthermore, if some of the $x_i$ are assigned to be zero then we can use the same Lagrange multiplier technique on all of the remaining unassigned $x_i$.

Hence we can deduce that $\sum f(x_i)$ is maximised when, at all of the $x_i$ which do not satisfy $x_i=0$, $f(x)$ has the same derivative.

As an example, consider the case when $f(x)=(x-1)^3$ and $n=2$. If there is a maximum with $x_1$ and $x_2$ both non-zero, then it must be the case that $\frac{\mathrm{d}f}{\mathrm{d}x}=3(x-1)^2$ is the same at $x_1$ and $x_2$.

For $x_1$ and $x_2$ to be distinct we would have to have $x_1=1+\sqrt a, \ x_2=1-\sqrt a$ for some constant $a$. Therefore, it $\overline x \ne 2$, then at the maximum either $x_1=x_2$ or $x_1=0$ and $x_2=\overline x$. On the other hand, if $\overline x=2$ then, for any $0\le a \le 1$, there is a maximum with $x_1=1+\sqrt a, \ x_2=1-\sqrt a$ and $\sum_i f(x_i)=0$.

Finally, there is one point that I would like to make about eventually concave functions. The function graphed below contains a section which is strictly convex but very close to being a horizontal line followed by a section which is strictly concave but is very close to being a straight line. Even for rather large $\overline x$, the maxima will satisfy $x_i=\overline x$ and $x_j=0$ for $j \ne i$.


What would be the shape of the Indifference curve with slope 1/xy?

Wolfram Alpha might come with some help. Try entering $\log x+y^2=n$ for different n's (levels of utility). You will get the idea of how the indifference curve looks like.

e.g.
https://www.wolframalpha.com/input/?i=logx%2By%5E2+%3D+1.4


What would be the effect on the economy if the United States legalized counterfeiting?

The initial effect would be great inflation, likely hyperinflation, devaluing the US dollar.

Some counterfeiters have had extensive operations, running in the tens or hundreds of millions of dollars. Legalizing these operations would likely lead to a great increase in the number and extent of these operations.

Having more and larger "counterfeiting" operations, or under the proposed law, simply additional printing presses, would cause the US to lose all control over its monetary policy.


As mentioned by Michael quantecon has good resources. Here is an example of very simple DSGE model from quantecon.
I am not sure if you can get to as low as 3 equations but here is very simple example of real business cycle DSGE by Chad Fulton. I won't copy all the code from the site, you can just follow the link which has full code, but it is based on the following set of equations and it is quite simple
$$ \psi c_t = (1−α)z_t(k_t/n_t)^α \text{ static FOC}$$
$$ 1/c_t = \beta E_t\left(\frac{1}{c_{t+1}} \left[αz_{t+1}(k_{t+1}n_{t+1})^{α−1}+(1−δ)\right]\right) \text{Euler Equation}$$
$$ y_t = z_tk^α_tn^{1−α}_t \text{ Production Function}$$
$$ y_t = c_t + i_t \text{ Resource Constraint} $$
$$k_t =(1−δ)k_t+i_t \text{ Capital Evolution}$$
$$1 =l_t+n_t  \text{ Labor-leisure Tradeoff}$$
$$\ln z_t=ρ \ln z_{t−1}+ε_t\text{ Technology Shocks}$$
Where; $y_t$ is output, $c_t$ consumption, $i_t$ investment, $n_t$ labor, $l_t$ leisure, $k_t$ capital stock, $z_t$ technology and rest are parameters. $β$ is discount rate, $ψ$ marginal disutility of labor, $δ$ depreciation, $α$ capital share of output, $ρ$ technology shock.
You could probably simplify it a bit further but not much more.


Proving there are infinitely many strictly dominant strategies

That is a fine proof. Though you probably want to show your last statement "The other inequalities don't require $\epsilon_n$ to be restricted". It is fairly obvious, but won't hurt to be super clear.

Perhaps a simpler proof would be that any convex combination of the two strategies already found will also work. That is $\lambda(0,1/2,1/2)+(1-\lambda)(0,1/3,2/3)$ also strictly dominates $L$ for any $\lambda\in(0,1)$. So, in fact, there are not only countably many strategies that strictly dominate $L$ but uncountably many of them.

(Idea of the proof:) Notice that this holds because strict dominance is given by a set of inequalities (3 in this example) so if you have two strategies that satisfy those inequalities, any convex combination of these strategies will necessarily also satisfy the inequalities since expected utility is linear in probabilities. I'll leave the formal proof for you to write, but hopefully the logic is clear.


Is investment in real estate a real investment?

Purchasing new homes would count as an investment. According to Blanchard et al. Macroeconomics: a European Perspective pp 568 in glossary investment is defined:

Investment (I): Purchases of new houses and apartments by people, and purchases of new capital good (machines and plants) by firms.

The source above is the leading undergraduate macroeconomics textbook, so although the definition is not rigorous as it is an undergraduate text the houses are not included just haphazardly without thought. Also note the definition does not include land as correctly stressed in Brian's comment.
However, what the author in that book you cite probably has in mind is capital investment, which is specially investment in assets that will be used as factor of production (e.g. machines, trucks, factory). However, capital investments are not defined in terms of jobs that they create. That is, even investing in fully autonomous robot that would replace worker would be capital investment. This being said when capital and labor are complementary, which is still often the case, capital investment will also create more jobs.
Additionally, in economics when we talk about real variables, e.g. real output, real consumer spending, real investment etc. we mean inflation adjusted as opposed to nominal output, nominal consumer spending and nominal investment etc. that would not be inflation adjusted. So the book you are reading also does not seem to be following economic terminology much.


Roughly speaking nominal wages should remain fixed but things would still get invented and production would become ever ore efficient. So real wages would increase as deflation boosted the purchasing power of your wages.

However, it may not be such a good idea to have a completely fixed money supply and deflation - it may become unstable if people noticed that storing money under your mattress was a perfectly effective way to save. If many savers started doing this then it could induce cycles of bubbles and crashes in the value of money. It would probably be better to deliberately increase the money supply by a few percent each year - just enough so that people were not inclined to save simply by storing money.


Why marginal revenue must equal marginal cost?

Literally from the definition of Marginal Cost; it is the cost of producing one additional unit.

As your marginal revenue remains constant, pretty clear that you are going to make the maximum profits as the curves intersect at MC = MR. In the specific case that you consider, YES, you can choose either of 4 units or 5 units to be produced but, as Maarten said, your profits will definitely rise if you consider Optimal Production -2 units VS -1 unit in a general scenario.

This is the simple the reason why we say that MC = MR equilibrium is the optimal point to maximize profits. Doesn't mean you can't reach a similar state of profits before it can happen, just that MC = MR will for sure give you a maximal profit.

Also, emphasis on the point Discrete Quantities. Under a continuous distribution, your profits will keep increasing as long as the next extra unit produced will sell for more than it's production cost; i.e. till MC tends to the value of MR.


Year effects inconsistent between random effects and fixed effects

This is an interesting result, not a bad result. If there are no regressors other than time dummies, then I think OLS = RE = FE. (I've done a few experiments with reg y i.year, xtreg y i.year, fe, and xtreg y i.year, re, but I have not proved.)

If $X_{it}$ have trends and are correlated with fixed effects, anything can happen. For example, run the following Stata script (copy & paste):

set more off
clear all
local n 100
local T 5
set seed 1
set obs `=`n'*`T''
gen id = floor((_n-1)/`T')+1
by id, sort: gen year = _n
xtset id year
tempvar a0
gen `a0' = rnormal() if year==1
by id: egen a = mean(`a0')
gen x = a-year+rnormal()
gen y = a+x-0.2*year+rnormal()
drop a
* So far x and y have been generated.
xtreg y x year, fe
est store fe
xtreg y x year, re
hausman fe ., sigma
set more on

You will see that FE gives a negative trend, RE gives a positive trend, and the Hausman test is very significant. (Above I included a linear trend for simplicity. The results are similar when i.year is used instead.) I think it is trends in X and the presence of fixed effects.

. set more off

. clear all

. local n 100

. local T 5

. set seed 1

. set obs `=`n'*`T''
number of observations (_N) was 0, now 500

. gen id = floor((_n-1)/`T')+1

. by id, sort: gen year = _n

. xtset id year
       panel variable:  id (strongly balanced)
        time variable:  year, 1 to 5
                delta:  1 unit

. tempvar a0

. gen `a0' = rnormal() if year==1
(400 missing values generated)

. by id: egen a = mean(`a0')

. gen x = a-year+rnormal()

. gen y = a+x-0.2*year+rnormal()

. drop a

. * So far x and y have been generated.

. xtreg y x year, fe

Fixed-effects (within) regression               Number of obs     =        500
Group variable: id                              Number of groups  =        100

R-sq:                                           Obs per group:
     within  = 0.8276                                         min =          5
     between = 0.9353                                         avg =        5.0
     overall = 0.8066                                         max =          5

                                                F(2,398)          =     955.57
corr(u_i, Xb)  = 0.4473                         Prob > F          =     0.0000

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   .9803594   .0483021    20.30   0.000     .8854004    1.075318
        year |  -.2207004   .0568937    -3.88   0.000    -.3325502   -.1088506
       _cons |   .1019471   .1021303     1.00   0.319    -.0988351    .3027294
-------------+----------------------------------------------------------------
     sigma_u |  1.1387103
     sigma_e |  .97339535
         rho |  .57779363   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(99, 398) = 3.61                     Prob > F = 0.0000

. est store fe

. xtreg y x year, re

Random-effects GLS regression                   Number of obs     =        500
Group variable: id                              Number of groups  =        100

R-sq:                                           Obs per group:
     within  = 0.8066                                         min =          5
     between = 0.9353                                         avg =        5.0
     overall = 0.8436                                         max =          5

                                                Wald chi2(2)      =    2432.08
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =     0.0000

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |    1.40149   .0391177    35.83   0.000      1.32482    1.478159
        year |    .196468   .0523296     3.75   0.000     .0939039     .299032
       _cons |   .1267828    .123712     1.02   0.305    -.1156882    .3692539
-------------+----------------------------------------------------------------
     sigma_u |   .3602397
     sigma_e |  .97339535
         rho |  .12046423   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. hausman fe ., sigma

Note: the rank of the differenced variance matrix (1) does not equal the number
        of coefficients being tested (2); be sure this is what you expect, or
        there may be problems computing the test.  Examine the output of your
        estimators for anything unexpected and possibly consider scaling your
        variables so that the coefficients are on a similar scale.

                 ---- Coefficients ----
             |      (b)          (B)            (b-B)     sqrt(diag(V_b-V_B))
             |       fe           .          Difference          S.E.
-------------+----------------------------------------------------------------
           x |    .9803594      1.40149       -.4211302        .0389278
        year |   -.2207004      .196468       -.4171684        .0385616
------------------------------------------------------------------------------
                           b = consistent under Ho and Ha; obtained from xtreg
            B = inconsistent under Ha, efficient under Ho; obtained from xtreg

    Test:  Ho:  difference in coefficients not systematic

                  chi2(1) = (b-B)'[(V_b-V_B)^(-1)](b-B)
                          =      117.03
                Prob>chi2 =      0.0000

. set more on


Existence of utility representation of a rational but discontinuous preference

I think a basic problem is that any utility function defines a preference, and discontinuous utility functions can be used to define discontinuous preferences. Hence there are many discontinuous preferences that can be represented by utility functions.
An example:

Let $U(x,y)$ be a continuous utility function that maps from $\mathbb{R}^2$ to $(0,1)$. This latter may seem arbitrary, but the strictly monotononicaly increasing function $\frac{x}{x+1}$ maps from $\mathbb{R}_{++}$ to $(0,1)$, so it should be fine. Also define a closed set $H \subset \mathbb{R}^2$. Let
$$
\hat{U}(x,y)
=
\left\{
\begin{array}{ll}
U(x,y) & \mbox{if} (x,y) \notin H \\
\\
U(x,y)+1 & \mbox{if} (x,y) \in H.
\end{array}
\right.
$$
Obviously $\hat{U}(x,y)$ is not continuous and neither are the preferences defined by it. (The boundary of $H$ is prefered to anything outside $H$.) But the way these preferences were generated seems fairly general. So a large class of discontinuous preferences exists for which there is a utility representation.

Future question: How 'large' is this class, what measure can be used?

EDIT: As @NicolasPinto points out in his answer it is also necessary to specify that $H$ is such that
$$
\exists x \in H, \exists y \notin H: y \succ x,
$$
so $H$ is not the upper contour set of some point,
otherwise $U(x,y)$ and $\hat{U}(x,y)$ would in fact represent the same continuous preferences.


How will economy find long-term equilibrium in AD-AS model in case of a negative supply shock?

The only way to shift the SRAS curve without also shifting the LRAS curve is through a temporary change in input prices. By definition temporary changes in input prices are temporary, thus they will return to their previous values. Thus the SRAS will return to its previous long-term value at $SRAS_1$


What's the (economical) advantage for a company by paying an employee severance payment short before retirement

If Bob is actually a valuable employee this probably doesn't make sense.
However it is possible that the things Bob is good at just don't have a market any more. So his work could be good but it is not valuable (any more). Training his to do new things is not considered worthwhile with his upcoming retirement.
If he continues to work he probably also needs an office or similar amenities that are not needed if he is let go.
So one could construct examples where it is worthwhile to pay him his full salary until retirement as severance pay but it wouldn't be a typical situation.


How does the limit of $U(x, y) = (ax^{-c} + by^{-c})^{-\frac{1}{c}}$ as c approaches 0 yield the Cobb-Douglas utlity function?

It is not true that this function is equivalent to the Cobb-Douglas utility function when $c \sim 0$ for any values of $(a,b)$; you have to assume $a+b=1$ for that, i.e. $b=1-a$.

To see why it is true, fix $(x,y)$ and consider the following Taylor expansion of $U(x,y)$ when $c$ gets close to $0$. We have

\begin{align*}
(ax^{-c}+by^{-c})^{-\frac{1}{c}} & = e^{-\frac{1}{c} \ln{[ax^{-c}+by^{-c}]}} \\
& = e^{-\frac{1}{c} \ln{[ae^{-c\ln(x)}+be^{-c \ln(y)}]}} \\
& = e^{-\frac{1}{c} \ln{[a(1-c \ln(x) + o(c)) + b(1-c \ln(y) + o(c))}]} \\
& = e^{-\frac{1}{c} \ln{[a+b - c (a \ln(x)+b \ln(y)) + o(c))]}} \\
\end{align*}
If $a+b>1$, the term in the exponential converges to $-\infty$ when $c \rightarrow 0$ and therefore $U(x,y) \rightarrow 0$. If $a+b<1$, the term converges to $+\infty$ and therefore $U(x,y) \rightarrow +\infty$.

To obtain the convergence towards the Cobb-Douglas function, we must therefore assume $a+b=1$. In that case we have
\begin{align*}
U(x,y) & = e^{-\frac{1}{c} \ln{[1-c(a \ln(x) + (1-a) \ln(y))+o(c)]}} \\
& = e^{-\frac{1}{c} [-c(a \ln(x) + (1-a) \ln(y))+o(c)]} \\
& = e^{a \ln(x) + (1-a) \ln(y) + \frac{o(c)}{c}} \\
& \rightarrow_{c \rightarrow 0} e^{a \ln(x) + (1-a) \ln(y)} \\
& = x^{a} y^{1-a}
\end{align*}
which is the Cobb-Douglas utility function with parameters $(a,1-a)$.


Why GDP is converted to 2010 $US constant in international studies?

They do that in order to express GDP in real terms. This is incredibly important because nominal GDP is very deceptive since money constantly changes its value. US dollar in 2005 is completely different dollar from US dollar in 2021 so GDP expressed in 2021 US dollars can't be compared to GDP expressed in 2005 dollars. It would be like trying to compare distances in imperial and metric units without any conversion.
For example, suppose that nominal GDP in year 2020 was 200 and in 2021 nominal GDP was 400. Does that mean that economy now produces twice as many goods and services? Well we do not know. It is entirely possible that the economy produces more or it might produce same amount of goods and services or less. This is because nominal GDP is combination of real GDP  and price deflator:
$$\text{Nominal GDP} = \text{Real GDP} × \text{GDP Deflator}$$
Problem is that if you look at nominal GDP you never know if it increases just because prices (i.e. deflator) changed or because we actually have higher output real output, and consequently nominal GDP is next to useless for comparison of output across time.
However, while nominal GDP is useless we can still make comparison across time by calculating real GDP where we get rid of the price level by dividing the nominal GDP with the deflator.
$$ \frac{\text{Nominal GDP}}{\text{GDP Deflator}} = \text{Real GDP} $$
Real GDP can serve as a basis for comparisons across time because real GDP accounts for the change in prices by being expressed in constant dollars (dictated by the year for which deflator is set 1). You can have a look at Mankiw Macroeconomics pp 25 to learn more.

why the chosen year is 2010 in this case?

You always have to choose some year to be a base year for deflator (year when deflator =1). You can choose any year you want. In this case they choose 2010. There might be some situation where you want to set deflator to some specific year for sake of making informative graphical illustration, but other than that the choice of base year is completely arbitrary. If you run empirical regressions with GDP being expressed in 2010 dollars you will get the same results as when the GDP is expressed always in 2005 dollars.


Utility function is given as U(x1, x2) = 10x 0.5 1 + 5x2

The MRS represents the rate of exchange between goods x and y that would leave the agent indifferent to trading the two. It is the slope of the indifference curve for a given value of utility U.

To derive the indifference map (the set of indifference curves), take U constant in the utility function, and solve for y in terms of x. You can check if your MRS calculation is correct by taking the first derivative of your indifference map function.


Fehr & Schmidt, ultimatum game, inequaltiy aversion, perfect subgame Nash equilibrium

There are a couple of things here. First of all, envy is when your own payoff is smaller than the other player's payoff, which is the case when $z_i<z_j$. So the coefficients here are $0.8$ for envy and $0.4$ for compassion (envy looms larger).

Solving the game using backward induction, we need to first find the minimum value that player 2 is willing to accept. A purely self-regarding player will accept any positive offer from player 1. But with fairness preferences, player 2 will actually decline low positive offers as unfair. To find the threshold, we first note that player 2 gets a utility of $0$ if she declines the offer (in which case both receive 0 payoff). Suppose player 1 offers player 2 an amount $z_2$. Player 1 would keep the remainder, $80-z_2$. The utility is piece-wise linear, with a kink at 40 (equal split), like this:

We need to find the value for $z_2$ where player 2's utility crosses the x-axis (red dot). Since this happens below 40, the relevant part of the utility function is the envy case. Thus, we solve $z_2-0.8(80-z_2-z_2)=0$, which turns out to be $z_2^0=\frac{320}{13}\approx 24.6$.

Knowing the acceptance threshold we can now turn to player 1. Player 1 has the same Fehr-Schmidt preferences as player 2. Since the utility function is strictly increasing in own payoff, player 1 wants to keep as much of the pie as possible. Anticipating player 2's acceptance threshold, player 1 will offer $z_2^0=\frac{320}{13}$, maybe adding a small $\varepsilon$ to resolve player 2's indifference.


What is the difference between gross, operating, net profit and EBIT(DA)?

So, here goes:
Gross Profit = Revenue - COGS (Cost of Goods Sold)
If you bought an orange for a dollar and sold it for two, you have one dollar of Gross Profit
Operating Profit = Gross Profit - Labor - SG&A
If you're business paid someone \$100 to sell 200 oranges (like above), you would have \$200 in Gross profit, and \$100 in Operating Profit (assuming no other overhead like licenses, etc) for that business line. Operating Profit can be for a firm, business line, or whole company. Depreciation and Amortization can be included here for assets and debts in service to the entity or business line in question.
Typically Operating Profit tells whether a business line, unit, or company is profitable (independent of interest and taxes as explained below in the EBIT section).
Operating Profit and EBIT are not completely interchangeable in common parlance, though, as EBIT is typically used only for a company and Operating Income is often used both for Companies, and individual product lines within the company.
EBITDA is Earnings Before Interest, Tax, Depreciation, and Amortization
If the same business paid \$10 to incorporate, that would come out here, so your company's EBITDA would be \$90.
EBIT includes Depreciation and Amortization
If you'd also used your \$100 office (independent of the orange selling operation or it would have also been in the orange selling Operating Profit line) for 1/100 of it's operating life, you would have \$1 in depreciation. Bringing your EBIT down to \$89. EBIT is almost always a company level statistic, and rolls up all Depreciation, Amortization, and costs other than Interest and Taxes.
The reason people analyze EBIT, is that it gives you a feeling for if the company could be profitable if moved to a different tax jurisdiction, or if its WACC (weighted average cost of capital) were changed through recapitalization.
Net Profit = EBIT - Interest - Taxes
So if you'd paid \$1 in interest on a \$100 loan to buy the oranges in the first place, you're pre-tax income would be \$88, which you would report to the appropriate tax agency, assuming a 35% corporate tax rate, you'd pay \$30.80 in taxes, leaving you a Net profit of \$57.20.
This is usually the "bottom line" of an income statement.


Problem with interpretation of exchange rate

NEER stands for nominal effective exchange rate and not the Swiss local currency. Essentially, this takes into account how the local currency moves with respect to its trading partners.

Example: if Switzerland trades only with the UK, an appreciation of the Swiss franc compared to USD would not make a difference to exporters/importers in Switzerland if GBP also appreciate against the USD by the same amount. In this case, NEER would remain the same.

Without knowing which article you're referring to (which "coefficient"), I can't say much about your other questions. In general, however, an appreciation in the currency, holding other currencies constant, will lead to an increase in NEER.