Regarding the presence of $l$ in the Frisch elasticity, your result is correct. It is my impression that a Frisch elasticity free of $l$ requires very specific functional forms.

For example we know that the following utility function

$$U(c,l) = ln(c) +\alpha \frac {(1-l)^{1+1/v}}{1+1/v} \tag{1}$$
gives a Frisch elasticity independent of $l$ (and then $\eta=v$).

We could compare this with your utility specification and say "ah, so we must have additive separability to obtain a Frisch elasticity independent of $l$"...
No, this is not enough. Note that essentially $(1)$ does not express utility as a positive non-linear function of leisure but as a negative non-linear function of labor, and so the term $1-l$ appears. This is critical to eliminate $l$ from the final expression.

Because, if  we assume additive separability in general, then the cross-partial derivative is zero and we would be left with

$$\eta = \frac{u_l}{- u_{ll}}\frac{1}{1-l} \tag{2}$$

If the additively separable utility function is expressed as a non-linear positive function of $l$, and not as a negative non-linear function of $1-l$, $(2)$ would also be dependent on $l$.

--

Also, with your utility function specification, I do not find that $\sigma$ is out of the final picture. Instead I find

$$ \eta = \frac{\gamma(1-\sigma)-1}{(\omega+\gamma)(1-\sigma)-1}\frac{l}{1-l}$$

If $\omega = 1-\gamma$ this simplifies to

$$ \eta = \frac{1-\gamma(1-\sigma)}{\sigma}\frac{l}{1-l}$$

But these are treacherous algebraic calculations - you should recheck.


Why do Australian milk farmers need to protest supermarkets' milk price?

I would like to contradict Kenny LJ explicitly on his claim that "the free market works".

It only works if the market is actually free, i.e. competition exists. Which, you can argue, is not the case in the milk market.

What you are looking at is a cartel.

(Excuse me for citing numbers for the German market, not Australia, as I am more familiar with that market and have numbers ready that I don't have for Australia. I am quite confident, however, that while the ratios might differ a bit, the overall structure will be the same.)

There are about 100.000 milk farmers in Germany, producing about 28 million tons of milk per year. As Mick pointed out, however, they are in the farming / production business, not in the transportation / processing business, so they have to sell their milk to dairies. There are just over 100 dairies in Germany.

And most of them have a monopoly on their region, meaning they're the only dairy a farmer could sell to.

Those dairies sell the processed milk to supermarket chains and other retailers / business consumers. About 60% of all the milk produced is sold by less than a half-dozen major supermarket chains.

Milk is a highly regulated food product, meaning that one dairy's milk is effectively the same as any other. There is very little competition in quality. The dairies don't even have "a product", the milk ends up packaged in the supermarket brand:

  Woolworths has bumped the cost of its own-brand milk by 10 cents a litre, but Coles has refused to follow suit because it might burden customers.

So the only competition is in the price. Whoever sells cheapest. Customers buy the milk where it's cheapest (supermarket), because they simply do not have any realistic option to go for a "premium" milk product that's provided by middle- or small-sized, or regional, producers at a sustainable price with a transparent premium for the producer as well. It's just not available. There are "bio" brands, but that label is only saying something about how the cattle is kept and fed -- the price dynamics still apply:

The supermarket buys where it's cheapest (dairy), and the dairy tells the milk farmers how much they are willing to pay, take it or leave it.

A farmer that is not willing to sell his milk at the price the dairy dictates (even if it's at a loss), will not get his milk to market at all, but still have running costs to pay. Which means he'll be out of business pretty quick, forced to sell his business, land, and lifestock to some bigger farmer that is able to use scaling effects to produce at a lower price.

You could claim that this is the result of an oversaturated market, that less milk on the market would drive the prices up. That is correct, in a way, but the market is oversaturated because producing large quantities of milk as cheap as possible is the only bargaining chip the producers have in a market where literally no-one is going to a farmer to buy produce. Rising prices would not equate to rising margins for the farmers, as they simply have no control over the supply chain.

Concentration of power at the retailer end, a lack of competition in quality, and the inability of the producer to cut out the middle man (dairies) or create a meaningful "premium" market means the retailer is dictating the price, to the point where producer's margins are minimized and the only way to be a profitable producer is by economics of scale -- large combines swallow medium and small-sized producers, and the retailers are able to push the price even lower, repeating the cycle.

This is beneficial to the customer insofar as that the product (milk in this case) is available at every supermarket and extremely cheap. But it comes at a high cost, destroying middle- and small-sized milk farms (and dairies!) in a race to the bottom. Which also includes a race for the most milk-efficient cattle, to the exclusion of all other considerations.

Refuting Kenny LJ's wholesale dismissal of regulation:

  Suppose instead that these farmers instead benefited from low grain prices and enjoyed profits of 20 cents per litre of milk sold. Should consumers then have stood outside supermarkets demanding that milk prices be lowered by 10 cents per litre?

The consumers wouldn't have had to, as the supermarket chain would already, preemptively, have realized the opportunity to lower prices (to stay competitive to the other supermarket chains who would have done the same).

Addendum:

In answer to a comment by @horns, this is a matter of official record and investigation.

Rewe was sentenced to 20.8 million Euro antitrust fine in May 2013. Spar was sentenced to 3 million Euro antitrust fine in November 2014. (Just two I was able to duckduckgo ad-hoc.)

Quoting the German Antitrust Office:

  Our investigations have shown that the contracts between milk producers and dairies in Germany have long periods of notice and duration. In addition, farmers in Germany are generally obliged to supply the milk they produce exclusively to their respective dairy. There is virtually no possibility for them to switch to another dairy. This is a problem for farmers and hinders possible newcomers on the dairy side or dairies wishing to extend their activities. Another widespread practice is that the price of raw milk is set only after delivery and is based on reference prices and market information systems.

Also from that article:

  The Bundeskartellamt questioned 89 private and co-operative dairies, which in 2015 procured approx. 30.9 million tonnes of raw milk. This is equivalent to around 98 % of the total milk supply volume. The authority's investigations have shown that in 2015 97.8% of the volume of raw milk covered by the investigations was sold subject to exclusive supply obligations. In addition, contracts for more than half of the raw milk supply volume can only be terminated with at least two years' notice.

There is no effective competition.


How can a country ensure its national currency is used by its citizens?

First of all, please check the properties of money and keep them in mind. Indeed money is a convention, but nobody forces you to use it. For example, prisoners use ciggarettes as a medium of exchange.

But let's assume you are the absolute ruler of a country and you want your people to use your currency. You can either do it by force or you have to give incentives to your citizens.

The most important function of a new currency is to hold its value. If let's say my new currency holds value against yen, dollar or euro, then there's no reason for not using it. It's convenient and stable.

How to do this? Via monetary and fiscal policy, using gold standard or by giving it prefix value against another stable currency. For example, a bunch of eastern european countries have a prefixed exchange like Bulgaria (check Bulgarian lev in wikipedia).


You have a few options for data. With questions of this nature, Quandl is your friend.

See here for historical gold prices, 1833 until present. I've not read about the methodology here though.
This resource has the gold/silver price ratio. Annual data. It seems pretty easy to transform this into price of silver.
For platinum
It also looks like the Perth Mint keeps a decent amount of historical data. I noticed in that list paladium as well.
The Reserve Bank of Scotland has a set which can give you prices of commodities, monthly and daily going back 15 years. I am not sure if there are any precious metals in that list.


Let me try to answer to the "debate whether China has fabricated its GDP in the past". The rest of the question is too broad to be answered properly and should be rephrased.

Jia, Kudamatsu, and Seim (2015) use the methodology of Henderson, Storeygard, and Weil (2012) to corroborate the quality of the provincial growth data by checking if it reflects the growth in nighttime lights observed by satellites. Nighttime lights can be seen as an objective measure of living standards, and therefore its correlation with GDP is indicative of the quality of GDP data. They find such a correlation suggesting that real GDP growth, at least to some extent, reflects improvements in some dimensions of living standards captured by nighttime lights.

Jia, R., Kudamatsu, M., Seim, D., 2015. Political selection in china:
The complementary roles of connections and performance. Journal of
the European Economic Association 13 (4), 631–668.


Existence of optimal strategy in a choice problem with uncertainty and information structure

Yes, an optimal strategy always exists for any information structure.

Proof: 1st- the set $\mathcal{Y}$ is finite (thus compact). 2nd- for each state of the world, $v$, $u(y,v)$ is continuous in $y$ using the discrete metric. 3rd- Given the linearity of integrals, for each $t\in\mathcal{T}$ we have that $\int_{\mathcal{V}}u(y,u)P_{T|v}(v)P_V(dv)$ is also continuous in $y$. Therefore Weierstrass theorem ensures the existence of a minimum and a maximum.

Let $y^*_t$ be one such maximizer when the signal is $t$ (note it can be different for each signal realization, and that the maximizer might not be unique. In the latter case, I arbitrarily choose one for simplicity). Finally, define $P_{Y|T}(y|t)=\mathbb{1}\{y=y^*_t\}$. I.e. to play $y^*_t$ with probability one when the signal is $t$.

In this setup, existence might only fail if $\mathcal{Y}$ was not compact, or if $u(y,v)$ was not continuous with respect to $y$. Note I said might, even if these two conditions are not present, an optimal might still exist.


In the presentation that the OP linked to, "compactness" is not assumed, but derived as a result from more primitive properties that are assumed, one of which is the "twice differentiability" of the functions involved.

The presentation deals with the vary basic deterministic growth model in discrete time. The per capita production function in each period is $g(k_t)$ for which it is assumed that it is continuous and differentiable $g'>0,\;\; g''<0$. The law of motion of capital is then

$$k_{t+1} = g(k_t) + (1-\delta)k_t -c_t$$

with $0<\delta<1$ being the depreciation rate. The sequence starts at some arbitrary, finite initial value.

The presentation wants to show the existence and uniqueness of the solution. To do that, in page 6 it first proves that the "choice set" is bounded -namely that the set from which the economic agents may choose the optimal levels of consumption and capital is not the whole $\mathbb R_+$.

This is proved by considering the maximal sequence of capital levels, irrespective of utility optimization considerations. If the maximal sequence is bounded, it means that the optimal sequence can only be chosen from a bounded set of values.

The maximum capital sequence obtains if we set consumption equal to zero: what happens when we consume nothing and we use all our capital stock and all the new output as capital only.
We have then the sequence

$$k^{max}_{t+1} = g(k^{max}_{t}) + (1-\delta)k^{max}_{t}$$

Now that consumption is set to zero, the sequence of capital is governed by a non-linear first-order difference equation. A sufficient (although not necessary) condition for the maximal sequence to be bounded is that this difference equation is asymptotically globally stable, namely, if starting from any initial point we converge to a fixed point (it is not necessary because we could have bounded oscillations also and obtain a bounded sequence).

The fixed point of the maximal capital sequence satisfies

$$k^{max}_*:  k^{max}_{t+1}=k^{max}_{t} \implies g(k^{max}_*) = \delta k^{max}_* $$

By standard theory, global stability of the fixed point will obtain if the derivative of the difference equation, evaluated at the fixed point, is less than unity in absolute value,

$$\left |\frac {\partial k^{max}_{t+1}(k^{max}_*)}{\partial k^{max}_{t}}\right| = g'(k^{max}_*)+1-\delta <1 \implies g'(k^{max}_*) < \delta$$

At the fixed point, it holds that

$$\frac {g(k^{max}_*)}{k^{max}_*} = \delta $$

For fixed point stability we want

$$g'(k^{max}_*) < \delta$$

So fixed point stability requires

$$g'(k^{max}_*)<\frac {g(k^{max}_*)}{k^{max}_*}$$

or that "the marginal product will be less than the average product". Under the assumption that $g''<0$ everywhere, this will also hold for the fixed point. Hence, we obtain fixed point stability, and so also convergence and so boundedness of the maximal capital sequence. This tells us why we assume 2nd-order differentiability: by having the 2nd derivative and making assumptions about it, we can easily determine whether the equilibrium points are unique and things like that. Note also that differentiability of any degree implies continuity.

If the maximal capital sequence is bounded, so will be any other capital sequence, and hence the choice set for capital is bounded. If it is for capital it is also for consumption. Then, the full capital equation expressed as an inequality

$$k_{t+1} \leq g(k_t) + (1-\delta)k_t -c_t$$

defines a set (a unidimensional set, an interval on the real line) that is a) bounded below at zero (by the assumption that capital cannot be negative  b) bounded above as proven c) it can attain its boundary points. So it is closed and bounded and therefore compact.

More over, we assume that there are no "holes" in it, that there are not "forbidden" values in this set. This makes the set convex.

Given these properties (incl. continuity of the production function), by standard theorems we are assured that a maximum, i.e. a solution exists, and it is unique, which is what the presentation wanted to prove at this point.


Should minimum wage be regulated for developing countries

A welfare or efficiency enhancing minimum wage can be justified in several theoretical settings. For example, if there is monopsony (a single employer), the employer can internalize the effect of his labor demand on the labor market price (the wage) and therefore holds down both labor demand and wages. A minimum wage can therefore increase both employment and wages. In a developing country with a dominant industry like resource extraction or tourism controlled by a small number of connected families, this might be a decent model and therefore an industry specific minimum wage perhaps could work. Another theoretical motivation, if an industry has inelastic labor demand, due to pricing power in the good markets and low substitutability of other inputs for labor (maybe French wine would be an example of this), then the minimum wage could raise wages without great inefficiencies or reduced employment.

Of course economists dispute if those settings hold in practice. Emigration, competition in factor or product markets, and labor market competition across industries can lead to competitive wage setting even in monopsonistic industries. And minimum wages may work, and yet we might not have a good model for why. Or they might be helpful in the short-run but harmful in the long run. The empirical evidence of the short-run costs and benefits of small minimum wages changes does not show they are particularly harmful, even if we have good reason to think that the long run costs and the costs of large minimum wage increases would be significantly larger.


Because $a$ is a parameter, and so
$$
\eta = \frac{ d \log y}{d \log x} = \frac{ d \log a + d \ b \log x}{d \log x} = 0 + b.
$$


Capital mobility across industries and equalization of profit rates

This is a massive area of research, revived recently under the title of "capital misallocation".

Some reason why capital do not fully adjust:

grow in intangible capital-intensive sectors (which require more internal financing than other sectors) + low interest rates.
quadratic adjustment costs on investment.
financial frictions lowers access to credit to small yet highly productive firms.
industry-specific physical and human capital.

The are many more papers studying financial frictions and related aspects of the financial market, showing that the role of finance is very important.


Is it possible for an equilibrium to be Trembling Hand Perfect for one player but not another in an asymmetric game?

Yes. It is possible for a strategy profile of one player to be trembling hand perfect and not the other.

Generally speaking, trembling hand perfect is used in the equilibrium sense. Certainly for an equilibrium to be called trembling hand perfect, the strategy profiles of all players must also be so.


How can intermediate goods and final goods be differentiated?

While @M3RS is correct that value added is another approach to computing GDP that handles the final vs. intermediate goods problem more aptly, within the context of the expenditure approach, you would effectively be measuring spending by who's buying it, and this measure is fundamentally imperfect. For example, flour sold by a cooking supply company to a bakery is likely to be counted as an intermediate good. Flour sold at a retail store is likely to be counted as a final good. But, it could be that a relatively small scale bakery buys flour at Costco (I used to work at a wholesale club, and we supplied a lot of small businesses). In that case, despite efforts by the government not to double count, it's really unavoidable. This is why value added is generally better for avoiding double counting.


Gresham's Law applied to wampum and gold/silver coins in the 1600s

A key component of Gresham's Law is that people will spend the less valuable asset such as gold or silver if they legally have the same face value.

Sea shells and beads certainly didn't have the same value or demand within the colonies as British coin. You can't pay for import goods, taxes, or use the company store using beads or shells. Instead the issue that the colonies faced was that the British refused to pay them in coin but rather paid them in goods. If your colony wanted the manufactured British textiles then you would send a shipment of tobacco to receive it. There was an extremely limited supply of British coin in the colonies and so as a result alternatives, such as the wampum which was used by the local Indian tribes had to be adopted.

Eventually the British realized their system wasn't working and began sending over gold and silver coin. These coins were worth more on their face and probably could be considered a less volatile currency and so it was adopted in favor of the wampum which was then phased out of the colonies.


Payoffs in an infinitely-repeated game with discounting

Disclaimer: I only have a slight clue about repeated games and I have virtually no clue about coding (except the compulsory stuff I had to do in grad school).
That being said, consider this stream of consciousness as how I would approach this problem (if I knew how to code). I am not sure this actually works, maybe someone could support this answer with a diagram (that may suggest, it does not work, see comment in the end). As you see, I cannot provide a reference. Take it for what it's worth and please tell me if my idea fails. I am interested. I am mainly replying such that this question gets attention again.

You are interested in a strategy that yields a ($\delta$-weighted) average payoff of 4 for both players in the infinitely repeated game. I construct a finite sequence of strategy profiles for the stage game that yields a payoff arbitrarily close to 4 for both players by alternating between the (3,5) and the (5,3). This sequence can then be repeated infinitely often.

Let $\Pi^i_t$ be player $i$'s ($\delta$-weighted) average payoff of the (to be constructed) sequence at step $t$. Let $NPV_t$ be the net present value of the sequence's payoff of player $i$ at iteration $t$ such that:
$$ \Pi_t^i = NPV_t^i \frac{1-\delta}{1-\delta^{t+1}}$$

Start with $\Pi^1_0 = NPV^1_0 =5$ and $\Pi^2_0=NPV^2_0 =3$ (so start with (5,3) outcome).
At each point in the iteration, we have $\Delta_t = |\Pi^1_t - 4| = |\Pi^2_t -4|$. Then there is some vector $s$ that tracks whether the (3,5) or the (5,3) outcome is played. It is a sequence of $i$s and $j$s indicating who got the 5 payoff. So start with $s_0 = 1$ (so by the later construction $s_1 = (1, 2)$)

Take some $\delta$ as given and take some $\varepsilon >0$.

Now comes the coding part that intuitively should work:

Start with $t=0$.

If $\Delta_t > \varepsilon$, continue.

If $\Pi^i_t > 4$: Set $NPV^i_{t+1} = NPV^i_{t} +3 \delta^t$ and $NPV^j_{t+1} = NPV^j_{t} +5 \delta^t$.
$s_{t+1} = s_t$ with an additional entry $i$.

Once $\Delta_t \leq \varepsilon$, stop and set $T$ equal to the current $t$.
Infinite repetition of this sequence yields of course average payoff $\Pi^i_T$ for all players, because the the (weighted) average is
$$\Pi^i = (1-\delta) NPV^i = (1-\delta) NPV_T \frac{1}{1-\delta^{T+1}}=\Pi^i_T.$$

Note that the payoffs $\Pi_t$ may NOT converge to 4. Since I do not program I did this manually for some iterations for $\delta =0.9$. $\Delta$ jumps around. But the jumps seem to become smaller.
Here I list the payoffs $\Delta$ following this procedure for $t\in \{1,\dots,12\}$. Every time $\Delta$ jumps up is boldface, see that the boldface sequence seems to decrease.

1: 0.05263

2: 0.26199

3: 0.00552

4: 0.15558

5: 0.00995

6: 0.09293

7: 0.00115

8: 0.0692

9: 0.00561

10: 0.04549

11: 0.00023

12: 0.03765

13: 0.00345


Let $X = h(Y)$. A first-order Taylor expansion around $E(Y) = \mu_y$ gives

$$X \approx h[\mu_y] + h'[\mu_y]\cdot [Y-\mu_y]$$

This easily leads to

$$\sigma^2_x \approx \big(h'[\mu_y]\big)^2\cdot \sigma^2_y$$

Since

$$\eta_{x,y} \equiv h'\cdot \frac Yh \implies \eta_{x,y}(\mu_y) = h'(\mu_y)\frac {\mu_y}{h(\mu_y)} \implies h'(\mu_y) = \eta_{x,y}(\mu_y) \cdot \frac {h(\mu_y)}{\mu_y}$$

Substituting,

$$\sigma^2_x \approx \left(\eta_{x,y}(\mu_y) \cdot \frac {h(\mu_y)}{\mu_y}\right)^2\cdot \sigma^2_y$$

Taking the square root we are led to

$$\frac {\sigma_x}{h(\mu_y)}  \approx \eta_{x,y}(\mu_y) \cdot \frac {\sigma_y}{\mu_y} $$

The difference of the above from the expression in the question is that

a) The elasticity must be evaluated at the center of the Taylor expansion used
and
b) $h(\mu_y)$ is usually equal to $E(X)=E[h(Y)]=\mu_x$ only to a first approximation, due to Jensen's inequality.

Of course if the relation between $X$ and $Y$ is linear, and/or the elasticity is constant, things specialize.

In any case, it is valid as an approximation to write

$${\rm cv}_x \approx \eta_{x,y}(\mu_y) \cdot {\rm cv}_y $$

where "cv" stands for "coefficient of variation".


What is the maximum number of characters in a currency symbol?

Instead of currency symbols, some of which are used by more than one currency as @HotLicks suggests, you may want to consider the three-letter currency codes used in most currency exchanges. Here is a list: https://en.wikipedia.org/wiki/ISO_4217#Active_codes.


Do import tariffs tend to increase a country's exchange rate (if so, why)?

Suppose that there is no 45 percent tariff and the market is in some kind of equilibrium. That is the demand for yuans equals the supply of yuans at the current exchange rate.

Now assume Trump does impose a 45 percent tariff. This makes imported Chinese goods in the US more expensive for the American consumer. As a result the consumers will buy fewer Chinese import goods. So far they paid for these goods with dollars and the corporation importing the goods used those dollars to buy yuans and pay for the goods. As a result of the decreased import demand the corporation will want less goods and hence it will buy fewer yuans. Thus decreasing the demand for yuans. The currency exchange market is similar to other markets: If at a price (in this instance this is the USD/CNY exchange rate) the supply outstrips demand the price will decrease. This means that the dollar becomes 'stronger', you will have to pay more yuans to get a dollar.

As a result exports do become somewhat more expensive (in yuans) and imports do become somewhat cheaper (in dollars) until an equilibrium is reached on both the currency and the good markets.


Mechanism design when the set of feasible decisions depends on types

Yes, they can, because the utility function typically depends on the type. A particular action unavailable to a type can be modelled as that action yielding so low utility that it is never chosen no matter what other players do. Then apply standard mechanism design to the appropriately modified utility function.


Cost function from an easy CES production function

Imagine that you are a dictator in Sausageland. Your legitimicay depends on you delevering sausages to the inhabitants. Each inhabitant must be given 1 sausage and there are $y$ inhabitants. To produce sausages you command two factories: One factory produces $x_1$ vegan sausages using $x_1$ cabbage heads the other factory produces $x_2$ pork sausages using $x_2$ units of pork meat. Which factory do you use?
Well since the factories are equally productive transforming one unit of input into one unit of output independent of the level of production $y$ you simple follow the rule of cost minimizing and use the factory for which the input is the cheapest.
Vegan sausage case:
So in the case where $w_1<w_2$ it follows that $\min(w_1,w_2) = w_1$ and $y=x_1$ so costs are $w_1x_1$ which is $\min(w_1,w_2)y$.
Pork sausage case:
So in the case where $w_2<w_1$ it follows that $\min(w_1,w_2) = w_2$ and $y=x_2$ so costs are $w_2x_2$ which is $\min(w_1,w_2)y$.
In the knife edge case where $w_1=w_2$ it does not matter whether you produce pork
sausages or cabbage sausages they are equally costly to produce so the only requirement is that the total number of sausages  add up to $y$ hence $x_1+x_2=y$ with the solution set $\{(x_1,x_2) \lvert x_1+x_2=y,x_1\geq0,x_2\geq 0\}$.
Combined sausage case:
So in the case where $w_2=w_1$ it follows that $\min(w_1,w_2) = w_2=w_1$ and $y=x_2+x_1$ so costs are $w_1x_1+w_2x_2 = w_1(x_1+x_2) = w_1 y$ which is $\min(w_1,w_2)y$.
So in all three cases the cost function can be written as
$$C(w_1,w_2,y) = \min(w_1,w_2)y$$
Cost minimizing program
Another approach is to set up the cost minimizing program. The costs are given as $w_1x_1 + w_2x_2$ which is the objective function to be minimized. The constraints are that $f(x_1,x_2) = x_1 + x_2 \geq y$ and $x_1\geq 0$ and $x_2\geq 0$.
The cost function is derived by solving the problem
$$\min_{x_1,x_2} \ \ w_1x_1 + w_2x_2 \\ s.t. \ \ x_1 + x_2 \geq y \\ \ \ x_1 \geq 0 \\ \ \ x_2 \geq 0$$
The costs are given as $w_1x_1 + w_2x_2$ and they are to be minimized subject to the constraint that at least $y$ is produced as output.
One potential first step could be to check if some of the inequality constraints must be binding in optimum. So consider the case where $f(x_1,x_2) = x_1 + x_2 > y$ for which a numerical example could be that you are supposed to produce at least $y=4$ and you produce $f(x_1,x_2)=4.1$ using inputs $x_1=2.1$ and $x_2=2$. Clearly you could reduce costs by using less $x_1$ and still respecting the constraint $x_1 + x_2 \geq y$ (I am assuming that $w_1$ and $w_2$ are both strictly positive so reducing $x_1$ decrease costs $w_1x_1 + w_2x_2$). Hopefully you can convince yourself that it would always be a waste of money to produce more output $f(x_1,x_2) = x_1 + x_2$ than required $y$. And you can therefore conclude that any solution $(x_1^\star,x_2^\star)$  must satisfy
$$(1) \ \ x_1 + x_2 = y.$$
You now have one equation in two unknowns which is great progress but not enough.
Again you follow the procedure of considering the three cases $w_1>w_2$, $w_2>w_1$ and $w_1=w_2$. For the cases $w_1\not =w_2$ you should be able to convince yourself that you produce with the cheapest production factor and therefore for the other factor you have $x_i = 0$. In either case you have two equations with two unknowns and by similar arguments as above $\min(w_1,w_2) y$ are the costs.
In the knife edge case where $w_1=w_2$ non of the constraints $x_1\geq 0$ and $x_2\geq 0$ are binding. You therefore do not end up with two equation in two unknowns and but simly have the inequalities $x_1\geq 0$ and $x_2 \geq 0$ and the equality $x_1+x_2=y$ and infinitely many solutions $\{(x_1,x_2) \lvert x_1\geq 0, x_2 \geq 0,x_1+x_2=y\}$. Obviously, this can seem annoying when hoping for a unique solution. However, you should be familiar with this case from utility maximization for perfect substitutes.
Lagrange
You can perfectly well use Lagrange for this problem. However, imagine that you forget the positivity constraints on $x_1$ and $x_2$ then your Lagrangian function becomes
$$L(x_1,x_2) = w_1x_1 + w_2x_2 - \lambda(x_1 + x_2 - y),$$
with first order conditions
$$w_1 - \lambda = 0 \\ w_2 - \lambda = 0$$
using the assumption that $w_1>0$ it follows by the complementarity condition $\lambda(x_1 + x_2 - y)=0$ that $x_1 + x_2 - y=0$ so $x_1+x_2=y$. That is not a problem.
However, you also have the implication that $\lambda = w_1 = w_2$ which is a problem because it excludes the real possibilities that $w_1\not = w_2$. Nevertheless, let us assume that $w_1=w_2$ then the FOC do not tell us anything about $(x_1,x_2)$ and are trivially satisfied by any $(x_1,x_2)$. Again you are left with the solution being all $(x_1,x_2)$ where $x_1+x_2=y$ for the case where $w_1=w_2$. In this case it is not wrong bu it is incomplete.
So including the positivity constraints in the Lagrangian you get
$$L(x_1,x_2) = w_1x_1 + w_2x_2 - \lambda(x_1 + x_2 - y) - \delta_1x_1 - \delta_2 x_2$$
with FOC's
$$w_1 - \lambda - \delta_1 = 0$$
$$w_2 - \lambda - \delta_2 = 0$$
Here you should use the complemetarity conditions to consider which inequalitites are binding in optimum.
So first assume that $x_1>0$ then by complementarity $\delta_1 x_1=0$ it follows that $\delta_1=0$. Using the first order condition you then get that $w_1 = \lambda$ and since $w_1>0$ it follows that $\lambda>0$. Using that $\lambda>0$ in combination with the complementarity condition $\lambda(x_1+x_2-y)=0$ you get $x_1+x_2=y$.
Combining $w_1=\lambda$ with the second FOC $w_2 - \lambda - \delta_2=0$ implies that $w_2 - w_1 = \delta_2$. It is known from KKT-theorem that $\delta_2\geq 0$. If we assume $\delta_2 = 0$ we are back in the solution $w_1=w_2$ so we instead let $\delta_2>0$ but by complementarity this then implies that $x_2=0$ and by the identity $w_2 - w_1 = \delta_2$ this is the case where $w_2>w_1$.
A completely symmetrical arguments is used to solve for $w_1>w_2$. But as you can see using Lagrange is tedious in this simple case. For a statement of the KKT-theorem you can see Nocedal and Wright page 328.


Black-Litterman---in what way are expected returns hard to estimate?

Your question relates more broadly to modern portfolio theory, and can be illustrated via mean-variance analysis of a univariate time-series. The extension to the multi variate (normal) setting, is trivial. Below I use the term accuracy, in a non-formal way, relating to the variance of an estimator. In particular, when the variance of an estimator can be reduced, take this to mean the estimator can be made more accurate.

Short story: The variance of the estimator for the variance of returns can be reduced by taking more observations (finer observations - up to a limit), whereas this is not true for the variance of the estimator of the expected returns. In this sense, it is more difficult to accurately estimate expected (mean) returns compared to variance of returns.

To see why, consider a time-series of returns, and assume the mean (per unit time), $\mu$, and variance (per unit time), $\sigma^{2}$, are constant over non-overlapping time intervals of frequency $h$. Here, $h$ represents daily, or monthly, or quarterly periods, etc. Further assume that our time-series data are observations over time intervals of length $\Delta$, where $\Delta<<h$. Then, define $n=h/\Delta$, as the number of observations of returns during a time interval of $h$.

For the $k^{\text{th}}$ observation interval (of length $\Delta$) during a period of length $h$, without loss of generality, we can assume that the price process for an asset is given by $$S_{k+\Delta}=S_{k}\exp{(\mu\Delta+\sigma \sqrt{\Delta} \epsilon_{k})},$$ where $\epsilon_{k}$ are iid normal. The logarithmic return (over period $\Delta$) can be written, as
$$X_{k}=\mu\Delta+\sigma \sqrt{\Delta} \epsilon_{k}.$$ For mean-variance portfolio theory, we require estimates of the true values of $\mu$ and $\sigma$, since they are of course, unknown. Call these estimators $\hat{\mu}$ and $\hat{\sigma}$. To answer your question, we only need consider the properties of these estimators.

First
$$\hat{\mu}=\sum_{k=1}^{n}X_{k}/h, \quad \mathbb{E}(\hat{\mu})=\mu, \quad \text{Var}(\hat{\mu})=\sigma^{2}/h.$$

Importantly, the variance of $\hat{\mu}$ depends upon the total length of the observation period and not on the number of observations.

Now consider, the (biased) estimator for $\sigma$:
$$\hat{\sigma}=\sum_{k=1}^{n}X_{k}^{2}/h, \quad \mathbb{E}(\hat{\sigma})=\sigma^{2}+\mu^{2}h/n, \quad \text{Var}(\hat{\sigma})=2\sigma^{4}/n+4\mu^{2}h/n^{2}.$$

The bias can be neglected (see reference for details).

Importantly, the accuracy (variance) of the estimator for $\sigma$, $\text{Var}(\hat{\sigma})$, does depend upon the number of observations, $n$ - for fixed $h$.

The consequence of this is, for fixed $h$, by taking finer (smaller) observations intervals, $\Delta$, the accuracy of the variance estimator can be improved ($\text{Var}(\hat{\sigma})$ can be reduced). This is not true of the variance of the estimator of the expected (mean) return, $\text{Var}(\hat{\mu})$, whose accuracy can be improved only by increasing the observation period, $h$.

The reference for this is

Merton, R. C. On estimating the expected return on the market. J. financ. econ. 8, 323–361 (1980).


Imagine an economy with only one good, tomatoes, and three people, Lender, Borrower, and Owner. Owner owns all tomatoes. Lender owns all the money. And Borrower owns nothing. If Lender doesn't lend any money to Borrower and Lender doesn't purchase any tomatoes, the price of tomatoes is 0 as there is no demand. When Lender loans Borrower some money and Borrower uses that to purchase tomatoes, the price of tomatoes will rise - inflation.

So how is this situation different than if Lender just bought the tomatoes themselves instead of loaning money to Borrower? First, the Lender is often not in the same position as the Borrower: money owned by the Lender is less likely to be immediately spent than if the Borrower held it and thus the velocity of that money will differ - after all, the Borrower is probably borrowing for a reason. Second, often times because of fractional reserve banking, the amount loaned to Borrower is more than what Lender has in reserve: in other words, more credit is created and if there is more credit chasing a fixed number of goods, price increases will follow.


I cannot say how helpful this is for you but if your model is

$$y_i = a(u_i) + b(u_i)x_i, \;\; u_i|x_i \sim U[0,1] \;\;\forall i$$

Then

$$\text{Var}(y_i\mid x_i) = \text{Var}[a(u_i)\mid x_i] + x_i^2\text{Var}[b(u_i)\mid x_i] +2x_i\text{Cov}[a(u_i),b(u_i)\mid x_i]$$

which tells you that $y_i$ is conditionally heteroskedastic.
This can acquire a complicated integral expression if you use the "Law of unconscious statistician" and the fact the the denisty function of a $U[0,1]$ random variable is just unity:

$$\text{Var}(y_i\mid x_i) = \int_0^1[a(u)]^2du - \left(\int_0^1a(u)du \right)^2 \\ + x_i^2\cdot \left[\int_0^1[b(u)]^2du - \left(\int_0^1b(u)du\right)^2\right] \\
+2x_i\cdot \left[\int_0^1a(u)b(u)du - \int_0^1a(u)du\int_0^1b(u)du \right]$$


Main users of overnight lending are banks. Banks have large in- and out-flows from customers on a daily basis. They typically borrow to meet a temporary outflow.
There is a term premium for borrowing at longer terms. By implication, borrowing overnight is theoretically the cheapest financing cost. Financial entities will borrow overnight to finance portfolios.


What aspects of macroeconomics should I address for convincing an investor to invest in a certain country?

As Brian said, international organisations are a good place to start. For example, the World Bank publishes periodic economic outlook updates on each country. For Egypt, see here. There is a pdf link to the right of the page. They covers aspects like GDP growth, inflation, financial and fiscal accounts, and also some prospects on ongoing, pro-business reforms.

Another source is the African Economic Outlook reports. Here is the one for Egypt. This is a bit more broad, including analysis of debt, revenues and spending, monetary policy, governance, natural resources management, and the social sphere (poverty, social protection, gender equality). See also the website of the IMF dedicated to Egypt.

Finally, having worked in the financial sector for a few years, I know that most of the investment banks produce periodic reports on every country their clients might have investment interests, reports that usually include a lot of forecasts. Albeit most of these reports are not available to the public, some googling still yield some results. For example, here is a very comprehensive analysis for Egypt from Deutsche Bank, published two months ago.  Other asset management companies also produce such reports (e.g here).


Slutsky Decomposition of Given Labor Supply Model

In this static model with no savings/intertemporal aspect, labor supply does not depend on the wage. Using the equality relations, we can maximize over $L^s$ without constraints

$$\max_{L^s} U = \left(\frac{L^s\cdot W}{P}\right)^a\cdot (T-L^s)^{1-a}$$

$$\frac {\partial U}{\partial L^s} = \frac a{L^s}\cdot U - \frac {1-a}{T-L^s}\cdot U$$

and setting this equal to zero we get the optimal $L^s$

$$\frac a{L^s}= \frac {1-a}{T-L^s} \implies L^s = aT$$

One can verify that, at the optimum, the second derivative of the Utility function is negative so we do have a maximum.


Why does global suspension of economic activity result in catastrophic loss/debt?

It creates losses.

First, many businesses don’t own their own building/land/equipment. It’s very common to lease these things. Moreover, many business take out insurance that again has to be paid on monthly basis. In Europe in many countries you need to pay workers certain portion of wages even when you can’t call worker to work or even if they stay home sick.

The list here is not exclusive. The point is many business have non-trivial fixed costs that have to be payed whether business operates or it doesn’t. Furthermore, in short run even some variable costs - like wages to employees on production line are fixed (and in many countries this can be up to several months). Hence, it’s not like a dormant business have no costs.

You might ask why then we don’t suspend for example these leases etc. but that just moves the problem to the different parties and does not eliminate this issue.

Moreover, very large business might have enough financial cushion to absorb losses for few months, but there are many small-medium size business that can’t do that. For example, consider a sole trader (a form of business organization where one person embodies a business) who has a repair shop with no employees just living month to month. That kind of businessman can not afford to go on hiatus for very long.

Furthermore, if by long you mean not just weeks but several months or even year that would be issue even for the best cushioned business.

So yes the impact could be catastrophic. For example according to this VOX blog the impact under some scenarios could be worse than recent 2008 crisis.


frequent Corporate Finance textbooks are

MBA level -

  Berk and de Marzo (2017) - Corporate Finance, 4th ed. (Stanford)

Advanced undergrad and corp theory-specific -

  Grinblatt and Titman - Financial Markets & Corporate Strategy

PhD 1st year level -
strictly articles from Journal of Finance, Review of Financial studies, etc, that empirically look at mergers and acquisitions, corporate governance, debt-shareholder agency problems, etc. Seminars on the subject typically have participants play discussants of existing recent work.


Why does loan waiving lead to a decrease of Non performing assets?

"Loan waiving" means "loan write-off": the debt is eliminated. Therefore it is subtracted from the assets of the lender, and if it was already a non-performing loan (asset for the lender), the total of non-performing assets is now reduced.

The value of the debt that has been wrtten-off appears as  a once-off "loss" in the Profit & Loss statement of the lender.

Whether debt write-off is or is not "a good idea" has no single/simple answer, and it depends on a lot of factors that may change in time and country.

The main abstarct arguments against is that
a) writing-off debt signals to other borrowers that they can too leave their debts unserviced on purpose, even though they are financially able to service them, in expectation of a future new write-off: in other words, it creates what is called "moral hazard",

b) seeing debts being written off, lenders will be less willing to lend funds in the future, even to prospective borrowers in good financial standing.

c) finally a pure economics agument, says that businesses that cannot repay their loans are inefficient businesses and so it is not in the economy's best interest to "provide a lease of life" through debt write-off to such inefficient entities.

Again, the above are the abstarct arguments. The OP has a specific situation in mind, and a specific study should be carried out to evaluate and quantify the socio-economic pros and cons of debt write-off for the specific case.


Proving the existence of Nash Equilibrium using alternate approaches

It seems unlikely that an elementary proof exists, because the Kakutani fixed point theorem follows from the theorem proving the existence of Nash equilibria. So this is a strong theorem.

The rest of this answer has been made before the question has been edited to specifically exclude it. (After the answer was already posted and the OP read it.)

There are several ways the existence of NE's in finite games can be proven.

One can use Brouwer's fixed point theorem.

Or, one can do so using Sperner's lemma, stopping by at Brouwer's as an intermediate step. Sperner, Brouwer and Nash


What's a "correct price" really? Does it exist?

tl;dr as KennyLJ correctly points out the concept of correct price is meaningless and it does not exist or rather you could say any price is the correct price (but again this just makes the whole concept meaningless because if every price is correct then thats just a price -  a concept of 'correctness' makes only sense if there are some incorrect instances).
The reason for this is that there is no 'intrinsic' objective value to anything. Since all value is completely subjective without even slightest objective component there cannot be any correct or incorrect price because there is no correct or incorrect value to anything, every valuation and any price is correct and hence the word 'correct' loses its meaning.

Full Answer:
The question of what is the 'correct' price is actually interesting from the perspective of history of economic thought and it actually dominated economic thought in ancient and medieval times and in some non-economic thinking it persists even to this day. In those times what is correct was heavily influenced by ethical considerations as at that point economics did not yet emerged from moral philosophy and could not be divorced from all other subject subsumed there (similarly as all natural subjects were at the time part of natural philosophy). As a result the early economic thinking was dominated by the concept of 'just price' which was considered both moral and desirable. Furthermore, just price and any other thinking on 'correct prices' relies on notion of objective intrinsic value.
Paul Mueller & Jan Gerber discuss the history of 'just price' concept in great detail I will do it here only briefly. The concept already originated with Aristotle in his Nicomachean Ethics and Politics. In Aristotelian thinking a just price would be a price at which the same value is exchanged. That is a just price wold be a price at which $\\\$1$ of value embedded in apples exchanges for exactly $\\\$1$ value embedded in peaches.
Thomas Aquinas expands on the Aristotelian concept according to Muller & Gerber by using the following innovations:

While it is still wrong to sell something for more than it is worth, it is not wrong to sell something for more than one paid for it. While Aristotle saw injustice in the shoe example, Aquinas recognized that one can do something productive or useful with a good that improves it in such a way that its value is higher, and so a higher price is still just. Transporting the shoes from one town to another represents a kind of improvement, as does any kind of labor performed on the good.
While this is a little more sophisticated than Aristotle’s account, it still relies on an “intrinsic” or “natural” value of goods—with a focus on the costs or input which creates value. Furthermore, Aquinas and other scholastics were also wary of profit. Some profit, a “reasonable” amount, was fine. But excessive profit could only be the result of injustice and exploitation.

Later when economics started to emerge as a separate field from moral philosophy classical economists started to turn away from the moral notion's surrounded around the 'just price' (since in science we like to make sharp distinction between normative value judgement and positive inquiry) although as the article by Muller & Gerber explains in moral philosophy the notion lived on for a bit more, but the classical economists such as Adam Smith or David Ricardo did not turned away from the notion of intrinsic value which would still enable a possibility for 'the correct' price to exist.
In fact Adam Smith was firmly behind the idea that there is some intrinsic value to goods and services. As Steven Horwitz writes:

Smith is very clear in The Wealth of Nations that he sees labor as the source of value. For example, in the opening paragraph of Chapter 5 on real and nominal price (I.v.1., p. 47), he writes:
The value of any commodity, therefore, to the person who possesses it, and who means not to use or consume it himself, but to exchange it for other commodities, is equal to the quantity of labor which it enables him to purchase or command. Labor, therefore, is the real measure of the exchangeable value of all commodities.
And later in that chapter (I.v.7., p. 51):
Labor alone, therefore, never varying in its own value, is alone the ultimate and real standard by which the value of all commodities can at all times and places be estimated and compared. It is their real price;  money is their nominal price.

However, as our scientific understanding progressed any notion of objective intrinsic value got completely shattered. This occurred during the marginal revolution where subjective theory of value was virtually simultaneously discovered by Jevons, Walras, and  Menger in the late 19th century.
The subjective theory of value claims that there is no objective intrinsic component to any value and all value is completely subjective given by person's own preferences. This theory of value can explain situations that labor theory of value or other theories of value cant. For example, under labor theory of value digging hole and immediately covering it up should have some intrinsic value due to labor component embedded in it but we can observe that people dont value such absurd waste of labor. This can be fully explained by subjective theory of value that would simply postulate that the dig up and immediately covered hole is worthless because nobody values it (or if someone actually pays for it it has some subjective value of them). Conversely, even things that do not have any labor embedded in them can be valuable. Furthermore, value further fluctuates based on what are people's marginal utilities. As the quotes mentioned in the_rainbox's answer from Adam Smith show, classical economists already argued that individuals have different levels of utility or 'value in use', but did not properly connect them to prices, or 'value in exchange', considering them separate, one derived from the quantity of labour input and other production factors (see Stigler's The Development of Utility Theory. I).
Currently subjective theory of value dominates the mainstream economic thinking, and does so to the point that with perhaps a little bit of a hyperbole the detractors from this theory are in similar number and position as anti-vaxxers in medical science. You will find the subjective theory of value in every modern economic textbook and arguably it is either implicit or explicit in wast majority of the whole corpus of economic literature written post marginal revolution. What even more, the subjective theory of value 'spilled' from economics back to the moral philosophy where it was used to nail the last nails into the coffin of the moral theory around 'just price' (see the article by Mueller and Gerber again).
As prefaced at the begging under subjective theory of value, objective intrinsic value that could be used as a justification for correct price simply does not exist. If two people make any voluntary transaction in the market they must either exchange equal or both greater value than was lost (or some combination of thereof) meaning if any transaction occurs the price is automatically the 'correct' price for people engaging in that transaction. Furthermore, even if a transaction does not occur because maybe seller asks for too large price or buyer for too low price you cannot dismiss their subjective valuations as incorrect and the prices they are asking for as an incorrect prices. Once you adopt subjective theory of value any notion of correct price becomes completely absurd. Hence the answer to your question is that there simply isnt any 'correct' price.


Your calculation gives you all Nash equilibria in pure strategies. The unique Nash equilibrium in pure strategies is indeed the only Nash equilibrium, but this requires an argument. A careful look at the payoff functions shows that it is irrelevant for player 1 whether player 2 plays the pure strategy $y$ or a mixed strategy with expectation $y$. A similar argument applies to the other side. But then the calculation gives us that every player has a unique best response in pure strategies to every mixed strategy of the other. Moreover, mixed best responses must randomize over pure best responses, so all best responses must be in pure strategies. In particular, every Nash equilibrium must be in pure strategies.

Now rationalizable strategies are the strategies that survive iterated elimination of never best responses. Every strategy is a best response here and does never get eliminated, so every pure strategy profile is rationalizable. If you define rationalizability in terms of mixed strategies, the argument above shows you that you still get only the pure strategies as rationalizable strategies. There are some subtleties in how one defines rationalizability with infinite action spaces, but they are all irrelevant here.


What is the mix of maturities of outstanding US Treasury bonds called?

Weighted Average Maturity of Marketable Debt


Given the circumstances and model you describe, consuming the entire budget will give higher utility than consuming less than the entire budget.

You can therefore assume that the optimal basket will have the same total price as before the price change.


When calculating GDP using the income approach how do we make sure that we count only wealth produced in given year?

So, if I correctly understand what I learned in comments, this question is resolved by two things:

1.We add deprecation in order to ignore it, rather than take it into account. It makes sense because incomes are already decreased by amount of deprecation, so in order to "unaccount" deprecation we must add deprecation. And due to us ignoring deprecation it doesn't make any sense to make distinction "deprecation of new assets VS deprecation of old assets", we ignore both of them.

2.Like renting, money lending is an ongoing service. You can keep extra money for some time, but with each year/month the amount of extra money that you're allowed to keep will become less and less.


Why system of beliefs, $\mu(x)$, needed in dynamic incomplete information games

In some ways, beliefs are a more natural way of interpreting solution concepts like Nash equilibrium and its refinements.

For example, consider a simple simultaneous-move coordination game as follows:
\begin{equation}
\begin{array}{c|cc}
&L&R\\\hline
T&1,1&0,0\\
D&0,0&2,2
\end{array}
\end{equation}
We say that $(D,R)$ is a Nash equilibrium because $D$ is a best response to $R$ and vice versa. But this is a simultaneous-move game; how does player 1 know that player 2 will choose $R$? Well, she doesn't need to know, she just need to believe that player 2 will choose $R$ (with sufficiently high probability), and such a belief alone can justify the play of $D$ as a best response.

Thus, we can think of a Nash equilibrium as one in which every player is best responding to their beliefs about what other players will do, and that their beliefs are correct, in the sense that other players choices confirm their beliefs.

To the extent that a simultaneous-move game can be modeled using a game tree with imperfect information, we can talk about beliefs at non-trivial information sets where a player is uncertain about which strategy the previous player has chosen. Furthermore, this framework can be generalized to include incomplete information games if we let this previous player be Nature/Chance.

By dividing Nash equilibrium into these two elements --- (i) best response to one's belief and (ii) beliefs are correct --- we obtain a more nuanced understanding of the concept, which in turn shed light on the ways we can refine this solution concept by placing restrictions on either or both elements.

Solution concepts like the (weak) perfect Bayesian equilibrium and sequential equilibrium require that beliefs off-equilibrium path cannot be arbitrary (they must follow Bayes' rule whenever possible).
Others such as quantal response equilibrium maintain the requirement that players have correct beliefs but relaxes the supposition that people always best respond to their (correct) beliefs.


What activity or event(s) marks the point at which an economic "bubble" burst?

An economic "bubble" bursts when the buyers in the market deem the good or service to be overpriced.

At a certain point, the price of whatever is causing the bubble will reach a certain "threshold price."

It is at this "threshold price" that the corresponding amount of "threshold buyers" is reached.  These buyers will stop buying (or continue not to buy) the good or service offered.

In an attempt to incentivize the buyers to continue buying their product, the sellers will lower their price.  Of course, certain sellers will lower their prices further to out maneuver each other in order to sell more of their product.  This is the burst.

At this point, some of the buyers will recognize the "burst" of the "bubble market" and will become sellers to move their product.  This, simultaneously, increases the amount of supply and decreases the demand, further lowering price.

This cycle will continue until a new (lower) equilibrium is reached.


Literature on the effects of third-party certification on industrial dynamics?

For models of the informational role/effects of third-party certifiers on an industry, you might like to look at

Lizzeri, Alessandro (1999). “Information Revelation and Certification Intermediaries”. RAND Journal of Economics 30.2, pp. 214–231.

Albano, Gian Luigi and Alessandro Lizzeri (2001). “Strategic Certification and Provision of Quality”. International Economic Review 42.1, pp. 267–283.


What you are saying is that the marginal utility of one additional good (car, or coconut) decreases and tends to zero. I believe everyone agrees with that.

The reason why scarcity matters is not that people can't be satisfied with a finite consumption, but that most individuals currently consume below their satiation level and would be happier with one additional unit. Most people would prefer a bigger house, a more luxurious car, more clothes, etc., even if they would finally be satisfied with some finite level.

Regarding your observation "Can we really not ramp up production to facilitate this wish?", the answer is no: it is still too costly to produce cars, houses, clothes etc. to accommodate everyone's wishes. If we were able to produce costlessly the amount of goods required to satisfy everyone's satiation level, the problem of scarcity would be solved (and economics would disappear as a science), but we are clearly far from this situation.


Solving a Cournot Equilibrium, the case of Q=q1+q2, Q(q1,q2)=q1+q2

First point: you write "I am struggling with the differentiating between when to use $ Q=q_1+q_2$ and $Q(q_1,q_2)=q_1+q_2$". But these are the same thing: both define $Q$ as a function of $q_1$ and $q_2$. It's just that the second expression makes this functional dependence explicit by writing out the arguments, whereas the first does not. This is purely an issue of notation and has no bearing on the solution.

Let's start by writing out the profit function and computing the first-order condition:

$$\Pi=[a-b(q_i+q_j)]q_i-c_i q_i$$

$$\frac{\partial\Pi}{\partial q_i}=0 \iff a-b(q_i+q_j)-bq_i-c_i=0\iff q_i=\frac{a-bq_j-c_i}{2b}.$$

This is the FOC that you got in your solution (just slightly rearranged). So far, so good!

Let's repeat those same two steps but writing the profits in a slightly different way

$$\Pi=[a-bQ]q_i-c_i q_i$$

$$\frac{\partial\Pi}{\partial q_i}=0 \iff a-bQ-b\underbrace{\frac{dQ}{dq_i}}_{=1}q_i-c_i=0\iff q_i=\frac{a-c_i}{b}-Q.$$

This is the FOC from the solution you were given. But note that $Q=q_i+q_j$. Making this substitution gives

$$q_i=\frac{a-c_i}{b}-Q=\frac{a-c_i}{b}-q_i-q_j$$
$$q_i=\frac{a-bq_j-c_i}{2b}.$$

In words: your answer, and the provided solution have exactly the same first-order condition. So we can be confident that both are correct at least this far, and the discrepancy must be coming from the next step.

Now we need to solve the system of first-order conditions for equilibrium. We have

$$q_i=\frac{a-bq_j-c_i}{2b},\ q_j=\frac{a-bq_i-c_j}{2b},$$
$$q_i=\frac{a-b\frac{a-bq_i-c_j}{2b}-c_i}{2b}.$$

Solving this last equation for $q_i$ yields
$$q_i^{\text{eq}}=\frac{a-2c_i+c_j}{3b}.$$

This is your answer. It is unclear where the answer in the solution came from, but the most likely situation is that there is an algebra error in the final step of computation of the solution.


The current economic understanding of the effect of labour unions on employment and wages

I'd recommend checking out Labour Economics by Borjas.
It is such a broad topic that really quite a detailed evaluation would have to be conducted.
In particular, the effect of a labour union on levels are unemployment are likely to be affected by the assumptions in your theoretical model.

Assumptions of perfect and imperfect competition (of some type) are likely to have different signs and magnitudes.

It follows that it is perhaps useful to look for empirical evaluations on the matter. Personally, it's not my field - but it is possible that someone like Borjas has done research, or potentially Kreuger.


Decreasing interest rate effect on economy

There are many channels of monetary policy transmission. You could look that up more if you're interested. I'll point out some of them here.

Interest rate channel. This is pretty much what you mentioned in the question. The opportunity costs of borrowing money becomes cheaper, so private demand expands.
Asset price channel. With the lower rate, people find it more attractive to invest in other assets (as opposed to keeping money in the bank). The increased asset demand leads to an increase in equilibrium price, which increase wealth for those who are holding those assets, and more consumption due to the wealth effect.
Exchange rate channel. The lower interest rate would cause funds to flow out of the country (people searching for higher yield in other countries) causing the currency to depreciate. The depreciated currency helps exporters to be able to compete and export more.


There's a pretty simple answer to that.  The money injected into the banking system does not acquire any velocity- it just gets redeposited at the Fed.  If they had spent the same amount of money into the general economy in the form of goods and services, then you would have seen higher inflation.


U(c).

Taxes are an adjustment on your income/resource and show up on your budget constraint.

U(c) just measures the utility you get from consuming c.

As an aside, consumption taxes should not be 1-t.

Example: Investment = F - (1+t)C = Net Savings.

If you wrote (1-t), it would mean, Investment = F -C + tC. Higher is the tax rate, more is the investment or budget left over for investment, ceteris paribus - which does not make sense.

EDIT

The answer by @Henry got me thinking about the following.

If a model is proposed where a consumer purchases C units of consumption goods taxed at rate t, such that the consumer will have to give away tC units in taxes and ultimately will consume (1-t)C, the analysis is based on tangible goods. In the budget constraint of the consumer it will read as :

Investment = Output - Price * C

since the consumer will have paid for C units and then ultimately consumed (1-t)C units due to the tax.


Has there ever been a first world recession where inequality didn't fall?

What you suggest can easily be rationalised, if those on fixed incomes (such as pensioners) tend to be poorer while those with economically sensitive incomes (such as workers and shareholders) tend to be richer and see their incomes more adversely affected by recessions.

But I am not convinced it is true in every case. The UK recession in 1980 may be an example, looking at this graph from the Office for National Statistics and I would expect there to be examples for other countries too.


Chiang is probably the best you can find, but you might try Takayama - Mathematical Economics or Simon and Blume - Mathematics for Economists.


Finding Mixed-Strategy Subgame-Perfect Equilibrium

Consider the subgame following 1's choice of $Y$. This is simply a simultaneous move Battle of Sexes game. To compute the SPNE, you first need to find the Nash equilibrium of this subgame.

It has 3 Nash equilibria: 2 pure and 1 mixed. The mixed strategy gives a value of $\frac{2}{3}$ to player 1 in this subgame. Now move up to the root node and compare whether player 1 should choose $Y$ or $N$.
This would complete the computation of SPNE.


$(d)$ cannot be an answer because of the following.  If 1 and 2 are located as in $(d)$, then 3 wants to be located (i) either at $\frac{1}{3}-\varepsilon$ or (ii) at $\frac{2}{3}+\varepsilon$ because doing so yields almost $\frac{1}{3}$ while choosing $\frac{1}{2}$ yields $\frac{1}{2}\times\frac{1}{3}$.  So, 1 receives $\frac{1}{2}\times\frac{1}{3}$ if 3 chooses (i), or 2 receives $\frac{1}{2}\times\frac{1}{3}$ if 3 chooses (ii).  Then, 1 and 2 does not want to act according to $(d)$.

$(b)$ is the right answer. The reasoning is: if 1 and 2 are located as in the answer, 3 is indifferent of choosing anything in $\left[\frac{1}{4}, \frac{3}{4}\right]$ all of which yield $\frac{1}{4}$ and any other location yields a lower payoff.  In other words, each of 1 and 2 can guarantee himself at least $\frac{1}{4}$.

Now, the question is: Does any of 1 and 2 want to deviate from $(b$)?  Suppose when both 1 and 2 played according to $(b)$, 3 chooses $\frac{1}{2}$ so that each of 1 and 2 obtains $\frac{3}{8}$.  Now, suppose that 1 deviates to a location $\frac{1}{4}-x$, where $x \in (0,1/4)$.  Then, he obtains $\frac{1}{4}-x + \frac{\frac{1}{2}-\left(\frac{1}{4}-x\right)}{2} =\frac{3}{8}-\frac{x}{2}$ which is clearly lower than $\frac{3}{8}$ as $x>0$.  Thus, 1 does not want to deviate.  By a similar argument, 3 does not want to deviate too.  Hence, $(b)$ is an equilibrium


We measure these in values.

Also, this is a macroeconomic identity, it holds always by design.

You need an economic model that will make specific behavioral and other assumptions, in order to discuss movements when the price level change.

For example, you write

  it is far from intuitive that aggregate demand falls when the 'overall
  price level' (whatever that is) rises

Even in introductory economics courses beware of "intuitive" thinking.

And indeed, where is the model and its assumptions based on which one would argue that as price level rises aggregate demand falls? It is certainly not embedded in the macroeconomic identity.

Find the model that predicts so, understand its assumptions, and then discuss whether these assumptions are "intuitive".


Papers on the rate of returns for government-run pension plans

Here are the papers I've already found, in chronological order.

USA

Boskin, Michael J., et al. "Social security: A financial appraisal across and within generations." (1986).
Boskin, Michael J., and Douglas J. Puffert. "The financial impact of social security by cohort under alternative financing assumptions." (1987).
Duggan, James E., Robert Gillingham, and John S. Greenlees. "Returns paid to early social security cohorts." Contemporary Economic Policy 11.4 (1993): 1-13.
Leimer, Dean R. Cohort-specific measures of lifetime net social security transfers. No. 59. US Dept. of Health and Human Services, Social Security Administration, Office of Research and Statistics, 1994.
Beach, William W., and Gareth G. Davis. Social Security's rate of return. Heritage Foundation, 1998.
Leimer, Dean. "Cohort-specific measures of lifetime social security taxes and benefits." (2007).

GERMANY

Schnabel, Reinhold. "Rates of return of the German pay-as-you-go pension system." FinanzArchiv/Public Finance Analysis (1998): 374-399.
Wilke, Christina Benita. "Rates of Return of the German PAYG System-How they can be measured and how they will develop." (2005).
Lüthen, Holger. "Rates of return and early retirement disincentives: evidence from a German pension reform." German Economic Review 17.2 (2016): 206-233.

OTHER

Auerbach, Alan J., and Ronald Lee. "Welfare and generational equity in sustainable unfunded pension systems." Journal of public economics 95.1-2 (2011): 16-27.


Why plug deficits with bonds rather than printing money?

There are several issues with this approach. One is that any changes in the quantity of money - including those considered 'acceptable amounts of inflation' - act to distort the price signal being communicated to all economic participants which is an extremely undesirable side effect.

The other is a little more insidious. Governments can only print cash or asset money. The vast majority, over 98% in most modern banking systems of money being used is liability money - money that is represented as deposits in the banking system. Were the government for example to print physical cash and deposit it in a commercial bank the book keeping operation would be [debit cash, credit deposit]. In almost all cases, it is the deposit money that is actually spent as the government pays salaries etc.

So if a government prints money and its banking regime relies on a framework where assets act as a regulatory control on lending (and consequent deposit creation), and there is no other regulatory control, the result is hyper-inflation. The problem is not just the money created by the government, it is the consequent multiplication that results from banks increasing their credit/money creation. (This form of regulation is generally referred to as the reserve requirement in economic literature.) Typically the resulting inflation then leads the government to print more money, and the result is a rapid spiral that quickly destroys the usefulness of the currency concerned for any economic transaction.

Most modern banking systems use a combination of reserve requirements and capital requirements, and this is why the quantitative easing interventions have so far had no inflationary effect on the economies using them. Even though the US government printed a huge amount of money for the TARP intervention, the capital controls intervened to prevent the runaway hyperinflation that would have occurred under earlier regimes. However it's still a dangerous thing to do, and the long term behaviour of banking systems that rely on basel capital controls is poorly understood.

Finally, it's worth noting that the alternative available to any government besides borrowing, is raising taxes or controlling its' expenditure. Ultimately the government's use of social resources has to be regulated by something, and keeping a more or less balanced budget isn't a bad place to start.


World Stock Markets that went up in 2008

Google is your friend. E.g Australia did very well in the crisis of 2008

See: https://en.m.wikipedia.org/wiki/Financial_crisis_of_2007–2008#


What is the point of a constant (thus known) inflation rate?

As I understand it, your question is the following:

Assuming that the rate is inflation is known by all in advance, does it matter what rate of inflation we set?

The answer to this question is 'yes'. For one thing, even if inflation is predictable, it transfers resources from holders of cash to the government or monetary authority ('seigniorage'). As a result, people may try to hold less cash, which means they need make more trips to the bank (leading to 'shoe leather costs').

Thus, contrary to the supposed lesson of your 'toy model', perfectly predictable inflation can have real effects.

However, you are right to emphasise that the importance of predictability: unpredicted inflation often has effects that predicted inflation does not. For example, consider the common claim that an increase in inflation is good for borrowers and bad for lenders. If unpredicted, that is certainly true. But if predicted by all in advance, one might expect the increase to simply drive up nominal interest rates, leaving the real interest rate unchanged.


Log-linear ordinary least squares specification coefficient loses its interpretation when we change an explanatory variable by a unit?

When
$$\ln y = \beta_0 + \beta_1 x + u \implies y = \exp\{ \beta_0 + \beta_1 x + u\}$$
$$\implies \partial y / \partial x = \beta_1 y \implies \frac{\partial y / \partial x}{y} = \beta_1.$$
So we see that $\beta_1$ is the marginal change in $y$ due to infinitesimal changes in $x$ as a proportion of its level. Therefore the accuracy of the approximation
$$\beta_1 = \frac{\partial y / \partial x}{y} \approx \frac{\Delta y / \Delta x }{y}$$
$$\implies \Delta x = 1: \beta_1 \approx \frac{\Delta y  }{y}$$
is nothing else than the general approximation inaccuracy issue that arises when we replace a derivative (infinitesimal change) with a disrete-interval proportional change.
In the specific case we have
$$y(x+1) - y (x) = \exp\{ \beta_0 + \beta_1 x + \beta_1 +u\} - \exp\{ \beta_0 + \beta_1 x + u\}$$
$$ = y(x)\cdot (e^{\beta_1 -1}) \implies \frac{\Delta y(x+1)}{y(x)} = (e^{\beta_1} -1).$$
We get

and we verify the folk wisdom that the approximation is accurate enough for purpose of economic analysis if $\beta_1 \in [-0.1,\; 0.1]$, and maybe for a larger interval, if one percentage point is not critical for the purposes of the specific research.


Why do low-income earners tend to spend more?

Marginal propensity to consume  is the proportion of an aggregate raise in pay that a consumer spends on the consumption of goods and services, as opposed to saving it. If someone gets extra income $\\\$1000$ and consumes  $\\\$750$ of this additional income their marginal propensity to consume is 0.75.
The marginal propensity to consume is higher for poor people since they have to devote higher share of an income to necessities. Suppose that person needs to spend at least $\\\$500$ a month for their living expenses. Now in such situation if a poor person would get a $\\\$500$ of income per month (relative to situation when the person would have nothing) they will have marginal propensity to consume 1 whether such person likes it or not as all additional income they get will be spend on these necessities. Person whose income would increase $\\\$1000$ (again relative to 0) will have to have at least marginal propensity to consume 0.5 and person with income increase of $\\\$10000$ (again relative to 0) at least marginal propensity 0.05. Of course, the richer people wont consume only necessities but the point is that as peoples incomes decrease the necessities become higher share of their budget.
At the same time as people get richer they spend a smaller proportion of their income as they already enjoy a high standard of living and thus are able to devote more of their income to savings.  You can read more about this in Blanchard et al Macroeconomics: a European perspective or any other macroeconomic textbook.


Erik Reinert's conclusions and experiments in economics

It's been too long since I read those, so I don't exactly remember  whether they answer your two specific questions, but if you are interested in the topic you should definitely have a look at:

https://en.wikipedia.org/wiki/The_Bottom_Billion
https://en.wikipedia.org/wiki/The_New_Bottom_Billion

and the corresponding TED talk at

http://www.ted.com/talks/paul_collier_shares_4_ways_to_help_the_bottom_billion?language=en


Negative Externality with Tax and Free TRade

Rearranging your demand and supply equations gives us:
$$P=Q_s\quad and\quad P=12-Q_D$$

Adding the tax on the supply side gives us:
$$P=Q_s+4$$

Solving for equilibrium, we get $P=8$ and $Q=4$

However, we did not necessarily need to solve for this equilibrium because the Supply curve shifted left, and the new intercept is at $P=4$. This implies that no plastic will be produced domestically because the world price will be lower at any positive level of production.

To solve for total surplus, we need only solve for consumer surplus because no production will occur domestically. We need to solve for $Q_D$ when $P=4$.
$$4=12-Q_D\implies Q_D=8$$
$$\implies TS=CS=\bigg(\frac{1}{2}\bigg)*8*8=32$$

Is this the same as when there is no trade and no tax?

$$P=Q_s\qquad P=12-Q_D$$

Equilibrium yields $P=Q=6$
$$TS=CS+PS-DWL$$
$$TS=CS=\bigg(\frac{1}{2}\bigg)*6*6$$

To calculate $DWL$, we split the cost into private cost and social cost. When we do this
$$PC=Q\qquad SC=Q+4$$

The values we need to calculate the $DWL$ triangle end up being $Q=4$, $P_{PC}=4$ and $P_{SC}=8$

$$DWL=\bigg(\frac{1}{2}\bigg)*4*4=8$$

$$\implies TS=18+18-8=32$$

Your intuition was correct that the total surplus is the same in both cases.


Has income inequality ever been solved in societies in the past in terms of what people make to cost of living?

America has faced staggering income inequality during the Great Depression.

http://thinkprogress.org/economy/2014/03/31/3420998/wealth-inequality/

In 1933 unemployment was 25%. President FDR's "New Deal" policies of fiscal stimulus, massive government spending on infrastructure, increased financial regulation and the creation of a social safety net (welfare, unemployment, pension) helped millions of people out of poverty and minimized the wealth gap.

Whether such a "New Deal" is politically feasible in today's America is debatable. We have to wait until November 3 to see who wins the election.


I can recommend "GDP: A Brief but Affectionate History" by Dianne Coyle as a non-technical background on GDP for a wide audience.

Why did the size of the U.S. economy increase by 3 percent on one day in mid-2013—or Ghana's balloon by 60 percent overnight in 2010? Why did the U.K. financial industry show its fastest expansion ever at the end of 2008—just as the world’s financial system went into meltdown? And why was Greece’s chief statistician charged with treason in 2013 for apparently doing nothing more than trying to accurately report the size of his country’s economy? The answers to all these questions lie in the way we define and measure national economies around the world: Gross Domestic Product. This entertaining and informative book tells the story of GDP, making sense of a statistic that appears constantly in the news, business, and politics, and that seems to rule our lives—but that hardly anyone actually understands.
Diane Coyle traces the history of this artificial, abstract, complex, but exceedingly important statistic from its eighteenth- and nineteenth-century precursors through its invention in the 1940s and its postwar golden age, and then through the Great Crash up to today. The reader learns why this standard measure of the size of a country’s economy was invented, how it has changed over the decades, and what its strengths and weaknesses are. The book explains why even small changes in GDP can decide elections, influence major political decisions, and determine whether countries can keep borrowing or be thrown into recession. The book ends by making the case that GDP was a good measure for the twentieth century but is increasingly inappropriate for a twenty-first-century economy driven by innovation, services, and intangible goods.


Confusion about the convexity of the best response correspondence

I would have thought that the if the opponent strategy $\sigma_{-i}$ is given, even if it is a mixed strategy, then $$u_i(\lambda\sigma_i+(1-\lambda)\sigma_i', \sigma_{-i})= \lambda u_i(\sigma_i, \sigma_{-i})+(1-\lambda)u_i(\sigma_i', \sigma_{-i})$$ and so if  $u_i(\sigma_i, \sigma_{-i})=u_i(\sigma_i', \sigma_{-i})$ then $u_i(\lambda\sigma_i+(1-\lambda)\sigma_i', \sigma_{-i})$ is equal to both of them. If they are best responses then so too is the linear combination strategy


Each curve shows the rate at which K can be substituted for L, or vice versa, while keeping output constant. The Marginal Rate of Technical Substitution (MRTS) equals the absolute value of the slope. The MRTS tells us how much one input a firm can sacrifice while still mainting a certain output level. Then the MRTS is equal to the following ratio: $$\frac{\text{Marginal Productivity of Capital } (MP_{K})}{\text{Marginal Productivity of Labour } (MP_{L})}$$

Then the substitution of Labour (L) for Capital (K) is given by:
$$ MRTS_{K,L} = -dL/dK = MP_{K}/MP_{L}$$

Now, note that in your equation $\alpha$, $\beta$ and $A$ are exogenous parameters (i.e. given to you). What your equation will then defined is, given these parameters, how your inputs (K and L) result in some amount of output given your production function.

You can find the isoquant curve that yields that diagram by varying the fixed level of output and by "playing with the fixed parameters".


What is the interpretation of the output matrix of pivoting?

Pivoting helps solving linear system of equations. Imagine you want to solve for X that satisfies :

\begin{equation}A\cdot X = Y\end{equation}

where
\begin{equation}X = \begin{bmatrix}x_1\\x_2\end{bmatrix}\end{equation} and \begin{equation}Y = \begin{bmatrix}y_1\\y_2\end{bmatrix}=\begin{bmatrix}3\\4\end{bmatrix}\end{equation} and \begin{equation}A = \begin{bmatrix}1&2\\2&3\end{bmatrix}\end{equation}

Then if applying Gauss' Pivot to A' (concatenation of A and Y) with
\begin{equation}A' = \begin{bmatrix}1&2&3\\2&3&4\end{bmatrix}\end{equation}

yields
\begin{equation}B' = \begin{bmatrix}1&2&3\\0&1&2\end{bmatrix}\end{equation}

you instantly have that $x_2=2$ and $x_1=3-2x_2=-1$. On large linear system it is very fast to solve them that way.


Need a math help for the Cagan's model in macroeconomics

The answer to both your questions is that for small $x$ values
$$
\ln(1+x) \approx x,
$$
the difference being less than $x^2/2$. (Proof by Taylor-approximation.)

So if inflation is around 10%, then the absolute error from this type of approximation is less then 0.5%, which is pretty good.

This should also answer your second question, as the approximation
$$
\gamma x \approx \ln(1+ \gamma x),
$$
works as well.

It may also be worthwhile to look into elasticity.


If a currency is pegged to another currency which collapses, what happens?

You need to define "pegged" and "collapses"...
Setting a fixed exchange rate is by controlling the local currency's amount of money

If the economy of currency X's owner goes bankrupt (a "collapse"), why would the Y currency owner continue to keep the attachment...?
Usually "pegging" currencies together is to a strong currency, like the US dollar

If the onwer of Y currency will keep attaching Y to X, it will create a local hyper inflation, and hurt its own economy...


Relation $\Delta Unemployment$ and $\Delta GDPgrowth$

Well it seems like you are observing somthing consistent with  Okun's Law which reveals empirically negative relationship between Unemployment and GDP.

As Donald Freeman states in the conclusion to his paper titled paper Regional tests of Okun's law

  Okun's law is one of the more enduring stylistic facts of U.S. macroeconomics, having
  withstood numerous tests in the almost four decades since it was first promulgated by
  Arthur Okun.


Due to rising costs of college tuition, is it likely that the future will have less universities and the remaining only for those of much wealth?

Hemelt and Marcotte wrote a paper studying the effect of rising costs on enrollment, and as it turns out, students who choose to pursue degrees have very inelastic demand; people are still going to university and they will keep doing so, despite these tuition hikes.

I think most students (myself included) have pretty imperfect information about the market, and they also forego a lot of searching because of the value of leisure. Most of their decision to go to school will be liquidity constraints and support from social/kinship ties. Liquidity isn't so much a concern nowadays, if parents really want their kids to go to school.

That said, there seems to be a shortage of manual/trade labor, such as in line work, which pays around 40$ an hour for starters (I think), but the population of laborers there are aging and due to retire. The job market could just be very sticky, and the demand shock for school caused by technology will slowly fall, causing tuition to fall with it. That's a bit more speculative though, and I'm not sure there's a lot of literature on rising tuition, on a very cursory glance.


Help improving this definition of a "game"?

Your question is a little bit vague, but I'll do my best.

Your professor was motioning in the correct mathematical direction, but I think that what you're asking for is a more complete formalization.

Any game can be thought of as a tree (i.e. a non-cyclic graph). So to answer your first point about turn-taking, you might think about a function which maps nodes of the tree to some $i\in\{1,2,...n\}$ In the picture below, you might be able to define a formula to characterize the turn taking, but you can't necessarily do this in general. In this example $P_1$ and $P_2$ take alternating sequential turns. $P_1$ can choose $L$ or $R$, then $P_2$ can see which he chose, and then also choose $L$ or $R$.

Before going on to your second point, I need to take a moment to define information sets, which it doesn't sound like you've gotten to yet. Colloquially, an infoset is a set of vertices that represent uncertainty. The root of a game tree is always a singleton infoset, for example. Consider the game where you and I simultaneously choose heads or tails. When $P_2$ chooses whether or not to play heads or tails, she doesn't know if I have chosen to play heads or tails either. Thus while the game tree will have 3 internal vertices (1 that is the root for $P_1$, and 2 for $P_2$, reflecting $P_1$'s choice) there will only be 2 infosets because both vertices for $P_2$ will be contained in the same infoset. In games where turn-taking is sequential, all infosets should be singleton.

Now back to your question about strategies. Each player has a set of strategies. In the example tree provided above $S_1 = \{LLLLL,LLLLR,...,RRRRR\}$. This might seem a little bit confusing at first as it seems like each element of $S_1$ goes above and beyond defining a path. Each player's strategy is defined in terms of infosets. Therefore, the element $LLLLL$ should be read as, when $P_1$ finds herself at the top, she will choose $L$, and when she finds herself at the vertex that is farthest on the left and 1 down from the top, she will also choose $L$. It then also says that if $P_2$ were to choose $R$ instead, that $P_1$ would choose $L$. Thus a strategy is the set of all possible actions a player might take under all possible contingencies.

Finally, how do we state the mapping from strategies to payoffs? First we note that ever player has a strategy set $S_i$, and then form the cartesian product $S=S_1 \times S_2 \times ... \times S_n$, which characterizes all possible strategy combinations for all possible players. Then our payoff will be $\pi:S\to\mathbb{R}^n$. In our example $\pi(LLLLL,LLLLLLLLLL) = (4,2)$.


Correlation between nominal bond returns and growth

When growth and inflation are primarily driven by aggregate demand, nominal bond returns tend to be negatively correlated with growth

Actually, the above sentence is the explicited version of what you have in mind, i.e. the links between nominal (vs real) returns, (demand-driven) inflation and growth.

  when growth and inflation are primarily driven by aggregate supply, nominal bond returns tend to be positively correlated with growth.

The point is that "supply-driven inflation" is the broad way of saying "weak or no inflation" or even "deflation" (avoided since politically undesirable). Put differently, when growth and inflation are primarily driven by aggregate supply, nominal bond returns tend to be ... real and thus to be positively correlated with growth. Indeed, the traditional (neoclassical) explanation is that non-weak real rates stimulate savings which in turn stimulates technical progress and ultimately, economic growth.

All that being said, the passage you quote is curiously phrased.. and the key words appear to be "tend to be".


OLS bias in demand estimation: the bias always underestimate the demand's elasticity?

Usually, $\hat{\beta_1^{IV}} = \beta_1 + \frac{cov(z,u)}{cov(z,x)}$.  The denominator will go to zero.

That is true unless there is some correlation between the instrument and the error term, and the nominator is the strength of the relationship between the instrument and the endogenous variable. The smaller the denominator gets, the greater the bias $\left[\frac{cov(z,u)}{cov(z,x)}\right]$.

In addition,  weak instrument will have no precision, so that the variance will have a big upward bias.
\begin{eqnarray} var(\hat{\beta_1}) &\to_p& \frac{\sigma^2}{n \sigma^2_x} \nonumber \\
\hat{\beta_1^{IV}} &=& \frac{\sum (z_i - \bar{z})y_i}{\sum(z_i - \bar{z})x_i} = \beta_1 + \frac{\sum (z_i - \bar{z})u_i}{\sum(z_i - \bar{z})x_i} \nonumber \\
var(\hat{\beta_1^{IV}} &=& var \left( \frac{\sum (z_i - \bar{z})u_i}{\sum(z_i - \bar{z})x_i} \right) \nonumber\\
var(u | z) &=& \sigma^2 \nonumber\\
var(\hat{\beta_1^{IV}}) &=& \frac{\sigma^2 \frac{1}{n} \sum (z_i - \bar{z})}{n[ \frac{1}{n} \sum (z_i - \bar{z})(x_i - \bar{x})]^2} \nonumber
\end{eqnarray}

As $n \to \inf$
\begin{eqnarray} var(\hat{\beta_1^{IV}}) &\to_p& \frac{\sigma^2 \sigma_z^2 }{\sigma^2_{zx}} \nonumber\\
var(\hat{\beta_1^{IV}}) &\to_p&  \sigma^2 \frac{1}{n \sigma^2_x} \frac{1}{\rho^2_{xz}}\nonumber \\
\rho^2_{xz} &=& \frac{[\sigma^2 _{xz}]^2}{\sigma_x^2 \sigma_z^2} \textit{for} \rho \in [0,1]\nonumber
\end{eqnarray}

That is why if your instrument is weak, then you  may be better off running an OLS regression.


Is 10 years monthly data enough for forecasting?

In your situation (after reading the comments) I would use the 10 year data over the 2 year company specific data, unless you can identify some reason why your specific company is especially atypical for the industry.

Considering the length of business cycles we typically see in the developed world, about 5 years, any 2 year period may not be reflective of the future, near or distant. The effects of independent variables like interest rate and inflation rate will not be accurately calculated because there is a strong possibility they will be very similar over a 2 year period (such as in the last two years in the United States), and therefore the data will not be sufficient to measure the difference these factors make.

Longer periods of time will solve this problem. There is no number of data that someone can point out and say "10 years is not sufficient but 10 years and 1 month is". In other words there is no cutoff, but longer periods of time, or in general larger sample sizes, are better and in this case outweigh the benefits of more specific data (although this should be decided on a case by case basis).


The factor elasticities from a Cobb-Douglas function in Romer's macroeconomy book

This is not the elasticity of substitution formula. The correct one is, between factors, $K,L$

$$\sigma = \frac{\mathrm{d}\log \left(\frac{K}{L}\right)}{\mathrm{d}\log \left(\frac{F_L}{F_K}\right)}$$

This expression can conveniently be worked as follows:

$$\frac{\mathrm{d}\log \left(\frac{K}{L}\right)}{\mathrm{d}\log \left(\frac{F_L}{F_K}\right)} = \frac {(L/K)[d(K/L)]}{(F_K/F_L)[d(F_{L}/F_{K})]}$$

$$=\frac {LF_L}{KF_K} \cdot \frac{d(K/L)}{d(F_{L}/F_{K})}$$

In the Cobb-Douglas case we have

$$ \frac {F_L}{F_L} = \frac {1-a}{a}\frac {K}{L} $$

so

$$d\left(\frac{F_{L}}{F_{K}}\right) = \frac {1-a}{a}d\left(\frac{K}{L}\right)$$

Substituting both expressions we have

$$\sigma = \frac {L}{K}\frac {1-a}{a}\frac {K}{L} \cdot \frac{d(K/L)}{[(1-a)/a]d(K/L)}=1.$$


Why do all/majority of loan contracts charge nominal interest rate instead of fixed real interest rate?

There are practical problems with having 'real loan' even though theoretically they would actually be desirable (as paper "The Impact of Inflation on Long-Term Housing Loans" by Tschach, Ingo E.) if they would be practical.

We do not know what the current inflation is.

This statement is simply incorrect:

Of course we will need to know the current inflation rate, but it's not like this information is secret, so it shouldn't be a problem.

When you look at inflation statistics they do not report actual inflation, they report the estimation of what the inflation is. The inflation estimation is based on a basket of goods that is considered representative and most statistical offices try their best to appropriately capture inflation but it is still just the best estimate and it is generally agreed that all inflation measures have some non-trivial flaws. They for example often do not appropriately capture changes in quality of goods due to technology and it is also highly questionable how different items should be weighted. Those two issues are just an examples there are many more, you can have a look at this investopedia article that explores the main issues in measuring inflation (but not all).
This issue could be circumvented by bank and client agreeing that some inflation measure such as CPI would be considered the inflation but that would cause some serious trouble down the road. For example, what if statistical office decides to update its CPI measure in a way to benefit borrowers or vice versa? The way we measure inflation evolves over time and thus using some real loan based on an ever evolving measure instead of actual inflation would not eliminate inflation related risk completely.

Inflation is rarely reported in real time.

Even the best statistical offices usually have about 3-1 month lag in measuring inflation and the most recent measurements of inflation can relatively often be revised later.
This creates another problem as it would make it very difficult to determine each month what the inflation is and hence what actually should be the interest rate paid. All loans would have to be paid with a lag that could even differ per country and due to the possibility of revisions banks would have to send out refunds or send out requests for customers to pay more as the revisions occur.
In future it might be possible to track inflation in real time. There is for example the billion prices project that attempts to do so. However, measuring inflation in real time is still in its infancy and the projects also has its own additional problem of being biased toward goods that are sold by e-commerce and it does not solve the problems in 1.

Even though you are correct in saying:

Borrowers and lenders would know that neither of them would benefit from inflation/deflation at cost of the other side, so borrowers would be more willing to borrow and lenders would be more willing to lend

this is basically what the authors of the paper linked above show in their simulations. You are not correct in stating that this would make the loan more predictable. It eliminates the inflation risk of the loan but as Giskard points out you are paid in nominal terms not real terms so the real value you need to transfer back to the bank is not more predictable per se.


Are there any economic reasons that could justify a higher tax rate on housing repairs than on construction of new houses?

If I'd have to summarize the proposed explanation that unfolds in the lines that follow in a few words, I would say that, the economic reasons behind a high tax rate on repairs are related to the effectiveness of revenue gathering and are also about providing the proper set of incentives to private economic agents.

Governments tax in order to finance spending, according to textbooks. It would seem rather trivial to assume that, like all rational agents that populate textbooks, it, too, pursues its ends in the most efficient way (in a traditional means-ends kind of calculation).

In the case of taxation, it also seems trivial to assume that the government will choose to impose only those taxes that are expected to produce the greater amount of revenue with the least push-back from the economy. It is common ground in neoclassical economics that taxes distort economic outcomes.

Therefore it follows naturally that taxing would be more productive (in terms of raising revenue) when applied on abundant (or relatively abundant-no free lunches, remember?) tax bases. Also, it seems that  it would be more probable for a government, to target consumption spending rather than capital expenditures. In that way, it would avoid distorting incentives (by taxing capital income, for example).

Having set the stage thus, what needs to follow is an explanation of how exactly housing repairs and new houses construction fit in the so neatly constructed frame.

When a house owner-or a house tenant, for that matter-need to fix a leaky roof, they forgo a portion of their income-lets assume that it's 'income' and not 'wealth', for the sake of simplicity-for the purchase of repair services and relevant commodities. The amount of money that is extinguished, will provide the government coffers with 20% (as per example) revenue.

Consider now what will happen in the case where there are a lot of leaky roofs. It will come as no surprise that there will be a lot of  repairing going on and as a consequence, there will be comparable a lot 20%'s of those expenditures going into the government's coffers.

Therefore, it seems plausible, that when the stock of houses is old and in need of repairs, taxing the amount spent on repairs seems like an effective way to pile up revenue.

On the other hand, the case is a bit different if we assume that instead of many house owners there are some real estate firms that own and distribute housing services in a country or a region.

In this case, under the assumption of "leaky roofs" it is not clear before-hand if it is in their interest to expend a considerable sum of money on repairs. Especially, when spending this much would also be burdened with the excessive tax rate. Perhaps, it is in their interest to build rather than renovate.

When the government chooses to incentivize private agents like that, it seems probable to assume it expects that, the expenditure on new buildings will create such a multiplier effect in the economy, that will-hopefully-overcompensate for the loss of revenue from the other options that got dis-incentivized ('repairs'). Also, opting for building new houses increases the stock of housing in the economy, which is another way of saying that the (residential) capital stock increases.

Therefore, it would appear that, when the government selects a high tax rate on housing repair services-as compared with the tax burden of new buildings-it is effectively 1. demonstrating its urgent need for cash and it is either acknowledging that 2. the residential housing capital stock is in dire need of repairs (derelict capital stock) or that it does not expect 3. the real estate developers to actually invest in new housing that would, in turn, invigorate the economy and create new sources of tax revenue.