We are given that $u$ is increasing and concave, and $u(0) = 0$. This implies that $\dfrac{u(t)}{t}$ is decreasing in $t$, and also, $\dfrac{u(t)}{t} >  u'(t)$ for all $t$.

Nicole's maximization problem :
\begin{eqnarray*} \max_{x} \  q(x)(w-t(x))\end{eqnarray*}

FOC :
$q'(x)(w-t(x)) = q(x)t'(x)$

Suppose $x_N$ solves Nicole's problem. Therefore, it satisfy

$q'(x_N)(w-t(x_N)) = q(x_N)t'(x_N)$
or equivalently,

$\dfrac{q'(x_N)}{q(x_N)t'(x_N)} = \dfrac{1}{w-t(x_N)} \tag{1}$

Andrew's maximization problem :
\begin{eqnarray*} \max_{x} \  q(x)u(w-t(x))\end{eqnarray*}

FOC :
$q'(x)u(w-t(x)) = q(x)t'(x)u'(w-t(x))$

or equivalently,

$\dfrac{q'(x)u(w-t(x))}{q(x)t'(x)} = u'(w-t(x)) \tag{2}$

Let us compare the two sides of the above FOC (2) at $x = x_N$,

LHS  = $\dfrac{q'(x_N)u(w-t(x_N))}{q(x_N)t'(x_N)} = \dfrac{u(w-t(x_N))}{w-t(x_N)} > u'(w-t(x_N)) =$ RHS

Since at $x = x_N$ LHS of (2) is greater than RHS of (2), we need to reduce the value of $x$ to make them equal. So, there is $x_A < x_N$ such that

$\dfrac{q'(x_A)u(w-t(x_A))}{q(x_A)t'(x_A)} = u'(w-t(x_A)) \tag{3}$

Therefore, Nicole buys larger quantity of technology.

Observing the FOCs, we can also conclude that increase in $w$ lead to increase in optimal quantity chosen by both.

Now try to solve the remaining two parts.


There's additional information in this answer, but in the US, banks pay into the FDIC Deposit Insurance Fund, and regulation and capital requirements are used to stop banks from taking excessive risks with deposits. In the event of a bank failure, depositors below the insured limit are repaid in full, but equity-holders are generally wiped out and other creditors must often take losses as well.

The DIF premiums do vary somewhat with risk (see page 2 here):

  The FDIC charges premiums based upon
  the risk that the insured bank poses, and it inspects, or examines, banks to further manage that
  risk.


Are there any datasets for protectionism?

Looked a bit further, I think I possibly found what you're looking for:

The Overall Trade Restrictiveness Index (OTRI) summarizes the trade policy stance of a country by calculating the uniform tariff that will keep its overall imports at the current level when the country in fact has different tariffs for different goods.

The data:
http://siteresources.worldbank.org/INTRES/Resources/469232-1107449512766/OTRI2009.xlsx

Background: http://econ.worldbank.org/WBSITE/EXTERNAL/EXTDEC/EXTRESEARCH/0,,contentMDK:22574446~pagePK:64214825~piPK:64214943~theSitePK:469382,00.html

hope this helps,
best regards


How can people save in a equilibrium economy?

The question is belied by some basic misconceptions. Just to list a few:

The comparison of GDP in nominal terms and implied statements about growth.
The definition/measurement of GDP (where saving is apparently not part of GDP).
The (completely) arbitrary prices of goods being prescribed for this economy.
The meaning of equilibrium.

  John produces 100kg potatoes...Paul
  catches 100kg fish...
  Next year, they improve their techniques and produce 150kg of potatoes
  and fish, respectively.

In this example, there is real GDP growth, due to technological innovation.
There are more goods in the economy---more potatoes and more fish. Therefore GDP increases.
This has nothing to do with trade between John and Paul or whether money (some medium of exchange called $---e.g. rocks) is being used to intermediate trade in this economy.

Macroeconomic growth is measured in real terms, not nominal terms.

As already pointed out by other answers, saving in the macroeconomy is simply output that is not consumed. If John and Paul barter (no money) and at the end consume 100kg fish and 80kg potatoes between the two of them, then aggregate saving is 20kg potatoes. That 20kg potatoes is still part of GDP.

(The money you put in your savings account is part of your country's GDP.)

Next, consider the proposed scenario:

  Now, John would love to sell 150kg potatoes to Paul for \$150 and buy
  only 100kg of fish for \$100, so he can save apart \$50. Unfortunately,
  Paul has the same wish!

Ok, suppose money, say rocks, is being used in this economy.
(The answer by @1muflon1 touches upon monetary policy by a monetary authority---e.g. there is a central bank on this island that sets the interest rate.)

Where do these prices ( \$1/kg for potatoes and fish) and the demand for saving \$50 you assume come from?

Suppose for the moment there's no saving technology---e.g. John cannot put his money under his bed.
If price of potatoes and fish are $\$P_p$ and $\$P_f$ per kg.
Then John has $ \$150 P_p $ and based on this budget he would choose some quantity
$Q_p$ and $Q_f$ of potatoes and fish so that his budget balances:
$$
Q_p P_p + Q_f P_f = 150 P_p.
$$
Same goes for Paul.

The prices $P_p$ and $P_f$ that actually realize in the economy are such that the potato and fish markets clear, i.e. total amount of potatoes demanded equal to total supply 100 kg, same for fish.

If at price $\$ P_p'$ per kg, John and Paul are willing to purchase only 90 kg of potatoes between them, then $\$ P_p'$ is too high a price, and in equilibrium you would never see potatoes priced at $\$ P_p'$ per kg.

Now suppose there is a saving technology---e.g. John can put money under his bed to spend next year.

Saving means shifting consumption from today to tomorrow. The amount John chooses to save depends on what \$1 represents in real terms tomorrow, which in turn is determined by price. John wants to save \$X today because \$X will buy him certain amount of potatoes and fish tomorrow.

If, at the prevailing price \$1/kg for potato and fish, John would choose to save
\$50, this means John would optimally like to have 50kg of, say, potatoes tomorrow.

However, if there are excess supply of potatoes and fish at the prevailing price of \$1/kg (as in the proposed scenario), prices will decrease, in order to clear the market. As today's potato and fish prices decrease, it becomes more attractive for John to consume today rather than save.
Or the lower prices today leads him to expect lower prices tomorrow and thus he would need to save less \$. His optimal saving choice would change accordingly.

Therefore the above scenario would not occur in this economy:

  Now, John would love to sell 150kg potatoes to Paul for \$150 and buy
  only 100kg of fish for \$100, so he can save apart $50. Unfortunately,
  Paul has the same wish!

If saving $50 by John is infeasible in this economy, then it cannot be an optimal saving choice for John.
In equilibrium, prices equilibriate so that optimal choices of economic agents are jointly feasible.

  In both cases, the economy is at equilibrium.

No, what "equilibrium"?

  How then is saving in the real economy possible?

It should be clear that this question is based on several strands of incorrect reasoning.

  Is it a zero sum game, that is, for every John saving, there must be a
  Paul that consumes more than he can afford?

As pointed out by @1muflon1, there are certain macroeconomic scenarios where saving becomes something like a zero-sum game. Not in the sense you describe, but in the sense that saving by one agent translates to loss of potential wealth for another agent.

Suppose there's a central bank on the island that sets interest rate on saving. The raising and lowering of interest rate would influence John's (and Paul's) consumption/saving decision. Lowering interest rate induces John to consume more today, which increases his demand for fish, which increases price of fish, therefore more wealth for Paul. Vice versa. This is a simplistic version of the central bank lowering rates to stimulate the economy/generate inflation.

However, if rates are sufficiently low ("zero lower bound", as you can find in macro texts), John and Paul might prefer to hoard cash, i.e. liquidity, rather than spend.
This is the Keynesian liquidity trap where monetary policy becomes ineffective and there is excessive saving.

(The liquidity-hoarding instead of spending on fish/potatoes story is awkward for two people on an island. In the actual macroeconomy, people hoard liquidity because return on bonds, which are less liquid than cash, is less than the liquidity premium.)


Is there a commitment mechanism which allows players to achieve pareto optimal solutions?

Expanding on @VCG's comment, which pretty much answered the question:

You can model this by adding a simultanous first move (to sign or not to sign) for both players. If they both choose to sign the contract, a predesignated outcome $A$ is realized and they get the corresponding payoffs. If they do not both sign, the original game is played.

Whether they will sign the contract or not will depend on their beliefs about what payoff they could expect in equilibrium. If a player's expected payoff from the original game is larger than what they would get in $A$ they will not sign.

(There are similar considerations in Aumann's correlated equilibrium, though there is no enforcable contract there.)

An example for when you would not sign the enforcable contract:
Consider a game of chicken with the payoffs

.
(Source: Wikipedia)

Suppose players could sign a contract forcing them to chose the "Chicken" strategy. A player will not do so if for some reason she expects that the other player will play chicken anyway, allowing her to collect a payoff of 7. In fact, using forward induction not signing may serve as a signal: The only reason he would not sign the contract is because she expects to win a larger than 6 payoff. The only way she can do this is by playing "Dare". Thus if the other player was willing to sign the contract, she has effectively signaled that she intends to play "Dare" by not signing the contract. Unfortunately according to the logic of forward induction the other player will also not sign.


Second Order Derivatives in Becker's Crime and Punishment

You have to apply both chain and product rule for derivatives here. In this case:
$$C_p'=\frac{\partial C(pO)}{\partial p}=C_p'(pO)O$$
But note when calculating cross-derivative you take the 2nd order derivative wrt O:
$$\frac{\partial C_p'(pO)O}{\partial O} =  \frac{\partial }{\partial O}[C_p'(pO)]\cdot O +  C_p'(pO)  \cdot\frac{\partial}{\partial O} [O] \\
= C_{pO}''(pO) \cdot pO + C_p'(pO)  \cdot 1 $$
Omitting subscripts this is equivalent to: $C'' pO + C'$.
To sum up, $C'$ comes from the product rule for derivatives (i.e. $\frac{d}{dx}[f(x)g(x)]= f'(x)g(x)+ f(x)g'(x)$ - you can learn more about this rule in standard math textbooks such as EMEA by Hammond et al.).


Does the political claim "will create X jobs" have an economic definition?

There is no standard, internationally agreed definition of what should count as "a job" when conducting economic impact assessments.  However, it is an important question that is often (not always) taken seriously when constructing such estimates.

Sometimes it is very important that jobs estimates are comparable.  This is particularly the case when politicians have to decide between competing investments.  If project A creates 1000 jobs and B creates 1200 jobs, then you should be sure that those estimates used the same methodology (and that it was sane) before deciding between the two.  For this reason governments often have official (though maybe internal) guidance on how to calculate the number of jobs created.  This is particularly true in the international development sector, where it is now quite common for government interventions to be aimed primarily at job creation.

On the other hand, it is not necessarily the case that we would want to use the same definition in all contexts, even if it's the same government creating estimates.  For instance, the UK government may want to use different criteria for assessing a community regeneration project in northern England than it would for an aid project in Afghanistan.  It would hopefully be transparent in both cases.

Each case is complicated for lots of reasons, including those you mention and many others.  How many hours a week must you work to have a job?  Does that change depending on what local working patterns are?  What if your job is seasonal?  Does it make a difference if you left another job to take the new one?  Does it matter if your previous job was in the informal sector, or part time, or not as well paid?  Does it matter if your job conforms with the ILO core conventions?  What about the new jobs that would have been created anyway?  And what about indirect jobs created as a result of increased spending of newly employed people (the multiplier), or because people copy you (spillovers)?

EDIT: There is a comparison of some common job creation definitions for development in a DCED working paper on pp12-13.


How will the value of the pound affect the British economy?

You think spending power has been reduced, but in reality inflation has only been 1%. Meaning products inside the UK measured in pounds have only become 1% more expensive.
The devaluation of the pound only exists compared to other currencies.
This means two things:
1. Products from the UK have become cheaper for foreigners. So companies from the UK will sell more.
2. Consumers from the UK pay higher prices outside the country. This will also lead to more sales inside the UK.
UK companies profit from a weaker pound even when measured in other currencies.
Consumers on the other hand are far worse off, and if you were planning on spending your retirement abroad you can end up paying a big pricetag for it.
A lower pound is good for business, not consumers, in the UK, leaving the EU is generally seen as a negative influence.


What bdsl's comment means is this:

Suppose there was a way for "experts" (here or anywhere else) to predict when the pound would start depreciating against the euro, and that this method predicted that it would begin to do so tomorrow. What would the experts do with this information? They would start selling pounds today in order to get out of the market at the top. But this rush of selling activity would cause the value of the pound to fall (via the usual principle of supply and demand) today—a day earlier than forecast! This simple story illustrates the broader idea that a method for forecasting financial markets is self-defeating. As soon as anyone has information about the future direction of an asset price (be it currency exchange rates, stock prices, etc.), their buying or selling activity will, by itself, move the price in that direction. Thus, any information available to the public will be "built-in" to the market price you see quoted. Moreover, given the amounts of money at stake, large market actors have a huge incentive to act on new information very quickly so that information gets built into the price almost immediately.

This general principle—that it is impossible to use publicly available information to beat the market—is known as the efficient market hypothesis. So-called specialists who claim they can predict what the market will do usually fail to do better than a random guess. Such people usually earn their money from commissions or advisory fees rather than from profits on successful trades.


Price Derivation in a Production Function

Rewrite the first equation as
$$Y^{\frac{\rho-1}{\rho}} = Y_s^{\frac{\rho-1}{\rho}} + Y_g^{\frac{\rho-1}{\rho}}. $$

Divide both sides by $Y_s^{\frac{\rho-1}{\rho}}$ to obtain

$$\left(\frac{Y}{Y_g}\right)^{\frac{\rho-1}{\rho}} = 1 + \left(\frac{Y_s}{Y_g}\right)^{\frac{\rho-1}{\rho}}.$$

Substitute the price-ratio and production-ratio relationships as in your second equation and make some algebraic changes, and you will get your last equation.  Note: I interpreted ``the price of $Y$ normalized to $1$'' as $\frac{p_g}{p_Y} = p_g= \left(\frac{Y}{Y_g}\right)^{\frac{1}{\rho}}$.


Infinitely repeated game with stationary and symmetric equilibrium

Why don't you attack the problem straighrforwardly and investigate the profitability of a deviation?
Suppose there was a Nash equilibrium in which both players decide to stay forever. Both players obtain payoff $\frac{1}{1-d}$ on path.
This really is a Nash equilibrium if anf only if there is no profitable deviation. Consider the deviation to quit in the first period, yielding the deviation payoff $10$. If a later deviation to quitting or a smaller probability to quit was profitable then so would be this pure earliest deviation.
Hence, always staying is a Nash equilibrium when $\frac{1}{1-d}\geq 10 \iff d\geq 0.9$.


Does every allocation have a maximal Pareto-improvement?

I think there is a short proof if you also assume that the number of agents $n$ is finite and that the preferences are continuous.

Given the second assumption Debreu's theorem (1954, "Representation of a preference ordering by a numerical function") states that a continuous utility function exists which represents the preferences. I will denote the utility function representing the preference of agent $i$ by $U_i$.

Though this is somewhat uncommon, I will denote the utility agent $i$ gets from an allocation $x$ by $U_i(x)$.

An algorithm resulting in a maximum Pareto-improvement of $x$:
Step 1. Initially set $i=1$ and set $y = x$.
Step 2. Maximize the utility of agent $i$ over the set of allocation where all other agents $j$ have at least utility $U_j(y)$. As all functions $U_j$ are continuous, there is a finite number of goods and a finite quantity of each good, for all $j$ the upper set of $U_j$ is a closed set. A finite intersection of closed sets is also a closed set. As $U_i$ is also continuous, it will have a maximum over this set. Select an allocation $y'$ which maximizes $U_i$ and let this be the our allocation $y$ from now on, so set $y = y'$.
Step 3. If $i = n$, stop. Otherwise set $i = i+1$ and proceed to step 2.

Notice the following:

Any move from $y$ to $y'$ in Step 2. is a Pareto-improvement.
Pareto-improvements are transitive, therefore the set of Pareto-improvements is narrowing, i.e. if $y'$ is a Pareto-improvement to $y$ then the set of Pareto-improvements to $y'$ is included in the set of Pareto-improvements to $y$.
As the utility of agent $i$ is maximized in Step 2. over the set of Pareto-improvements to $y$, it is impossible that his utility can be further increased by a Pareto-improvement. Therefore once we reach Step 3. for $i = n$ we have a Pareto-optimal allocation $y$, that is either a Pareto-improvement of $x$ or where $\forall i$: $x\sim_i y$. Hence $y$ is a maximal Pareto-improvement of $x$.

A comment: It seems to me you could replace continuity with monotonicity, but that would require another proof. Hopefully someone else is up to it!


The assumption is based on intuition but it a bit more complicated:
Basically each and every country has short term interest rates and long term interest rates, based on the debt the country is managing.

Short term interest is usually being set by central banks (the fed in the US), when the long term debt is managed by treasury. long term affect mortgages while short terms affect, short term loans and deposits
Long and short term rates affect each other - the long term interest, being riskier, in the same country, will yield more

In today's open global capital markets, money seeks the best returns. with free money flows, so the best interest - inflation ( - risk) will pull more capital
And on the other hand, it will increase the country's currency value (higher demands for the currency). This in turn, will affect exporting, for each country.
Since this game is a global game between countries, they will pretty much react to each other's actions, effectively matching rates and policies

Note that the description I gave may sound weird, and is a bit contradictory of economic theory, but that is the reality for the past decade...

Monetary policy in the world today, is in a theoretical no mans land


Reason for Economic Slowdown in Western Nations

The primary reason is that it is easier to catch up than it is to forge ahead.  India and China have lower GDPs per capita, so they can copy what other countries have done and gain.  But the United States, Canada, and Germany can't copy from India or China, as they are already higher.  Copying would make them lower.  And there's no one higher from whom they can copy.  They can copy a little from each other, but not enough to maintain high growth.

A good example is Japan.  Japan used to experience fast GDP growth.  Then it caught up to countries like the US and Germany.  Since then, it has stagnated.

Presumably this will also happen with China and India, but their high populations mean that it might not happen until their GDPs are bigger than those of the United States.

Note that China remains below the world average in GDP per capita.  Presumably it can grow until it is closer to the US or Germany.

Of course, it is also possible that there is something about the Chinese system that will limit it before then.  We can only speculate until their GDP stabilizes.  Some have speculated that China's growth is stabilizing now (it's been decreasing steadily since 2010), but we will need more time to make that determination.  At least ten years after it hits a steady rate, which it has yet to do.


Computational Bayesian Statistics and Economics

Not aware if there exists a "gentle" introduction, but you might try Zellner's "Bayesian Analysis in Econometrics and Statistics" (1997) or Koop's "Bayesian Econometrics" (2003).


Why do most function images take the form of a curve?

Your final line

This figure is only for representation.

pretty much answers your question.
When explaining these concepts you want to draw something that is easy to understand, i.e. does not have too many irregularities but is also not too regular.
Linearity is a special attribute of functions. General curves show that there is usually no need for this assumption.
Continuity is a frequent though not omnipresent assumption in economics.
Rather than throw in a few points of discontinuity and provide reasons for them, it is easier to explain general concepts with continuous curves. This does not mean that most functional relationships are indeed continuous; there are plenty of counterexamples, but where this is important, there are usually discontinuous graphs and explanations thereof.


For question 3, I'm sure there are simpler solution than mine. I provide this version for your reference.

By definition,
\begin{align}
\lambda^* &= \frac{\frac{\partial u}{\partial x_1}(x_1^M)}{p_1} = \frac{\left[(x_1^M)^{-\rho}+ w(x_2^M)^{-\rho}\right]^{-\frac{1}{\rho}-1}(x_1^M)^{-(\rho+1)}}{p_1}\\
\\
v(p_1,p_2,m) &= u(x_1^M(p_1,p_2, m), x_2^M(p_1, p_2, m)) \\
& = \left[(x_1^M)^{-\rho}+ w(x_2^M)^{-\rho}\right]^{-\frac{1}{\rho}}
\end{align}
So
\begin{align}
\lambda^* &= v(p_1,p_2,m) \frac{1}{p_1\left[(x_1^M)^{-\rho}+ w(x_2^M)^{-\rho}\right](x_1^M)^{\rho+1}}\\
&= v(p_1,p_2,m) \frac{1}{p_1\left[x_1^M+ w\left(\frac{x_1^M}{x_2^M}\right)^{\rho+1}x_2^M\right]}\\
& = v(p_1,p_2,m) \frac{1}{p_1\left[x_1^M+ w\left(\frac{1}{\kappa}\right)^{\rho+1}x_2^M\right]}
\end{align}
Plug in the expression for $\kappa = \left(\frac{wp_1}{p_2}\right)^{\frac{1}{\rho+1}}$ to the equation above, we have

\begin{align}
p_1\left[x_1^M+ w\left(\frac{1}{\kappa}\right)^{\rho+1}x_2^M\right] &= p_1\left[x_1^M+ w\frac{p_2}{wp_1}x_2^M\right]\\
& = p_1x_1^M + p_2x_2^M\\
& = m
\end{align}
so $$\lambda^* = \frac{v(p_1,p_2,m)}{m}$$

To show a function $F(x_1, x_2, ... x_n)$ is homogenous of degree $k$, simply show that $$F(kx_1, kx_2, ...kx_n) = \lambda^kF(x_1, x_2, ... x_n)$$ For $\forall \;\lambda>0$


We need to keep two things seperate: the unit of account, and instruments.

The unit of account is a currency unit, like the U.S. dollar. When the price level (e.g., CPI) rises, the cost of goods measured in that unit of account rises.

We then have monetary instruments, like $20 bills, or bank accounts. Their value in the unit of account are unchanged. Therefore, they drop in value versus goods and services (as expected).

The prices of stocks and bonds may go up or down, for any number of reasons. Like other things, their value is measured in the unit of account. Therefore, it is unclear what relationship there will be between the price level change and financial asset prices. For example, CPI inflation has been close to a 2% annual rate since the Financial Crisis, and stock indices have generally risen much faster than that.

As for interest rates, the relationship is: bond price up, yield down (and vice versa).


The interest component is already included. It was just their way of writing out the budget constraint that confused me. If anyone is interested:

$\frac{B_{t-1}}{p_{t}} - \frac{B_{t}}{(1+i_{t})p_{t}} = \frac{i_{t}}{(1+i_{t})}\frac{B_{t}}{p_{t}} - \frac{B_{t}}{p_{t}} + \frac{B_{t-1}}{p_{t}}$. This is simply the interest on existing debt and the face value of existing debt (on the expenditure side). And real bond issuance on the income side. Simple rearranging and you have it.


Properties of orders and preference relations

You are right, total is a synonym for complete, it simply means that two elements can always be compared.

Regarding your other point, I don't think that your definitions of strict order and weak order are common but they make perfect sense. Notice, however, that they differ from some definitions used by mathematicians and therefore you have to specify what you are talking about to avoid possible confusions (you can check for instance "strict weak ordering").


Does the demand for bad quality products increase when there is information asymmetry?

In the second case (information asymmetry) there is only a single market. This may be the market for the product with unknown quality or in case of adverse selection it may be a market for the bad quality product alone. Assuming there is no adverse selection the reservation price of a consumer w.r.t. the unknown good is the expected value of his reservation prices from the full information case. Thus it is a weighted average and will fall between the reservation prices of the good quality product and the bad quality product. Since this is true for all consumers, the demand function adjusts in the way you described.

What happens exactly (number of products sold) depends on the market structure, whether the seller of the goods is a monopoly or there is supply-side competition.


Below is an example based on the general formulation in your linked article. This will very likely need a lot more fine-tuning. Moreover, the following example does not necessarily satisfy the theoretical properties laid out in footnote 16 of the paper. (You should check this.) Nevertheless I hope this could at least give you an idea of how to think about writing down an appropriate utility function that suits your purpose.

Consider
\begin{equation}
\Pi(d,q,\theta)=
\underbrace{\left[1+\mathrm e^{-(\alpha_0+\alpha_1 d+\alpha_2q+\alpha_3\theta)}\right]^{-1}}_{G(d,q,\theta)}
\biggl[d+\underbrace{\frac{a+bq-d}{\theta T}\left(\frac{1-(1+r)^{-\theta T}}{r}\right)^{\!\!-1}}_{M(L,q,\theta)}-q^2\biggr]\,,
\end{equation}
where

$d=$ down payment; $q=$ car quality; $\theta=$ creditworthiness; $L=$ loan amount
car with quality $q$ is priced at $p(q)=a+bq$, where $a,b$ are parameters to be estimated/calibrated
loan amount: $L=p(q)-d=a+bq-d$
probability of purchase $G(d,q,\theta)$ is modeled by a logistic distribution, which is parameterized by the variables of interest, and the $\alpha_i$'s are parameters to be estimated/calculated
$\theta T$ is the term of the loan, which is increasing in $\theta$
assuming repayment in each period is the same, i.e. $L/\theta T$, the PV of loan payment is $M(L,q,\theta)$, where $r$ is the periodic interest rate
cost of producing a car with quality $q$ is modeled by $c(q)=q^2$


Complementary slackness conditions (Kuhn-Tucker)

It is possible to have
$$g(x^*) = b\; {\rm and}\; \lambda^* = 0$$.
When the multiplier is zero and the constraint is equal to zero, then
a) The constraint does not really "bind"
b) That's why the multiplier is zero.
What does it mean "the constraint does not really bind"?
It means that the solution $x^*$, that makes $g(x^*) = b$, would be chosen even if the constraint was not imposed. In that sense, the constraint is not really binding, because it does not really forbid us to go where we wanted to be, because we are already there.
Consider the simple example
$$\max_x \{-ax^2 + bx\},\qquad s.t. \;x \geq \frac{b}{2a} $$
The Lagrangean is
$$\Lambda = -ax^2 + bx + \lambda \left(x -\frac{b}{2a}\right)$$
and the f.o.c is
$$x = \frac{b+\lambda}{2a}$$.
Try cases:
a) $\lambda^* = 0$ leads to $x^* = \frac{b}{2a}$, which is the unconstrained f.o.c. also.
b) $\lambda^* > 0$ the f.o.c. initially indicates that $x^* > b/2a$. But then the constraint is not binding and we should have $\lambda^* =0$ : contradiction.
So we see that in this case the solution is
$$x^* = \frac{b}{2a},\;\;\; \lambda^* = 0.$$
So the constraint appears to be binding, but it really is not.


Seminal papers that later were proven to contain errors

My favorite example is the initial formulation of Arrow's impossibility theorem in the first edition of Arrows' "Social Choice and Individual Values" (1951). In the first edition, Arrow claimed that, together with 4 other conditions, the following domain condition

  ``The domain $\mathcal{D}$ is sufficiently extensive so that there exists at least one free triple of
  alternatives. (A triple is called free if all conceivable
  combinations of individual orderings of this triple actually occur in
  $\mathcal{D}$" (rewording from Blau(1957))

implied that there exists no social welfare function $S : \mathcal{D} \rightarrow \mathcal{R}$, where $\mathcal{R}$ is the set of all possible orderings (i.e. complete and transitive binary relations) over the set of alternatives $A$.

This was later showed to be false by Blau (1957) The Existence of Social Welfare Functions" Econometrica Vol. 25, No. 2 (Apr., 1957), pp. 302-313 who provided a counter-example.

Blau also showed (among other things) that the theorem could be corrected by replacing the above domain condition by the following condition

  Universal domain : the domain $\mathcal{D}$ of the social welfare function contains every possible profile of preferences over the set
  of alternatives $A$ (with $|A| \geq 3$).

Arrow later corrected this mistake in the second edition of Social Choice and Individual Values (1963), and the formulation of Arrow's theorem using the Universal domain condition has now become standard.

This being said, the initial error in the first edition of Arrow's book was rather minor, and the solution proposed by Blau does not reduce in any strong sense the importance of Arrow's result and approach. Intuitively, the conclusion remains that on a vast domain of relevant economic problems, no social welfare function satisfies a set of rather basic and reasonable conditions.

So this might not be exactly the kind of errors you were looking for (definitely a seminal paper though!), but I like the example so much I could not resist posting it . If such brilliant people like Arrow make these kind of mistakes, I guess it takes a little bit of the pressure off for everyone else?


The law of supply holds under fairly general conditions. Essentially, it states that "quantities respond in the same direction as price changes." Formally,
$$(p-p')(q-q')\geq 0,$$
where $p$ is price, $q$ is quantity and primed and unprimed reflect two different states. You see that the inequality is weak such that a constant range is not excluded. However, a decreasing supply curve would violate the law of supply.
Moreover, your argument appears to be a bit fuzzy. Sure enough, it implies that for a given price-quantity pair $(p,q)$ that $S(p) = S(p-\epsilon)=q$, when $\epsilon$ price changes are not drastic. However, then consider price-quantity pair $(p-\epsilon,q)$ and another non-drastic price drop. Your argument appears to imply $S(p-\epsilon)=S(p-2\epsilon)=q$ and so on culminating in a constant demand function, which --it seems-- you do not want to apply.
It seems more reasonable to suggest that a marginal price drop also has a marginal effect on supply such that $S(p)=q$ and $S(p-\epsilon) = q - \varepsilon$, where both $\epsilon$ and $\varepsilon$ are tiny such that a "non-drastic price drop" implies a "non-drastic quantity reduction."


Proving ranking using completness and transivity

The exact quote from J&R is as follows:

  " [Completeness + Transitivity] together imply that the consumer can completely rank any finite number of elements in the consumption set, $X$, from best to worst, possibly with some ties."

Here's a sketch. Take a finite subset of the consumption set, $A \subset X$.

Since $A$ is finite, we can number the elements $A = \{a_1,a_2,\dots,a_n\}$. We want to rank elements in $A$.

Pick $a_1$ and $a_2$. By completeness, either $a_1 \succeq a_2$ or $a_2 \succeq a_1$. Without loss of generality, suppose $a_1 \succeq a_2$.

Now pick $a_3$. By completeness, either $a_1 \succeq a_3$ or $a_3 \succeq a_1$ and either $a_2 \succeq a_3$ or $a_3 \succeq a_2$.

(i) If $a_3 \succeq a_1$, then by transitivity $a_3 \succeq a_2$, so $a_3 \succeq a_1 \succeq a_2$.

(i) If $a_1 \succeq a_3$, then if $a_2 \succeq a_3$, $a_1 \succeq a_2 \succeq a_3$. Otherwise $a_3 \succeq a_2$ and thus $a_1 \succeq a_3 \succeq a_2$

Then go to $a_4$ and repeat.

Since $A$ is finite, the process ends, and you have a ranking of elements in $A$.


Economic anthropology: Have there been cultures / societies without a market?

Wherever a society exists, there is repeated exchange. And wherever there is repeated exchange, there are also markets.
(Of course, markets can vary in the degree of their development/functioning. One market may be better developed/functioning than another in that it has more frequent transactions, better enforced property rights and contracts, greater honesty/trust, lower transaction costs, greater competition, better flows of information, etc.)
Exchange should be understood broadly: It can be one tangible object for another or even a piece of information for another. It can also be gift exchange. (A common layperson error is to think that only exchanges involving money count as exchanges.)
Gift exchange includes instances where a hunter shares his big find with everyone in the group, with the tacit understanding that he has access to the group's other resources. It could also be where one group assists a second group, with the tacit understanding that the second group will provide assistance should the need arise.
Rough exchange ratios can be implicitly established and sustained through tradition: If lazy John is always skiving, then members of the community will notice and eventually sanction him (through e.g. ostracism). So, if John is consistently taking from the rest of the community $X$ amount of meat every month, then he must also contribute $Y$ in fruit or other goods and services monthly to the rest of the community, in order to avoid sanction. Over time, rough exchange ratios (and these may be a wide band) may form and be embedded in tradition. And if conditions change so that meat becomes more abundant/scarce relative to fruit, then the exchange ratios may change accordingly.
(There is also the evolutionary argument that these exchange ratios will tend to be "correct" in the sense that they correspond to actual scarcity: Groups that maintain the correct exchange ratios thrive, while those that don't die out. Groups that develop strong social norms to punish free-riders and reward appropriate levels of reciprocity thrive, while those that don't die out.)

If two isolated individuals each living on his own deserted island visits each other and engages in an isolated exchange once or twice in their lifetimes, then one could possibly argue that no market exists. But then one should also argue that these two individuals do not form a society (under any reasonable definition of the term "society").


Steady state Solow model with exogenous technological change

To simplistically answer your question, use the following:
$Y = K^\alpha (AL)^{1 - \alpha}$

In order to prove all three can be equal:
We will assume that technological progress is labor augmenting (Harrod Neutral) or "labor saving", while holding the following true:

Returns to capital are roughly constant over time.
Capital share of income is roughly constant over time.

The Harrod Neutral is associated with a capital-output ratio that is constant (K/Y) in our "steady state".

Capital Accumulation
In the basic Solow model, (1).
$$ \frac{\dot{k}}{k} \equiv s\frac{y}{k}-(d+n)$$

For growth rate per capita capital to be constant, y/k = Y/K must also be constant. (Characteristic of the Harrod Neutral)

We Need To Define The Balanced Growth Path With Technological Progress
Taking the original production equation:
$Y = K^\alpha (AL)^{1 - \alpha}$

and dividing by AL (number of effective labor units) we are given our production function representing effective labor.
$$ \tilde{y} \equiv \tilde{k}^\alpha$$

Doing The Same For Capital Accumulation
Follow steps below:
$$\frac{\tilde{\dot{k}}}{k} = \frac{dln(k/AL)}{dt}$$

$$ \frac{dlnK}{dt} - \frac{dlnA}{dt} - \frac{dlnL}{dt} $$

$$\frac{\dot{K}}{K} - \frac{\dot{A}}{A} - \frac{\dot{L}}{L} = \frac{\dot{K}}{K} - g - n$$

then using this to finding our Capital Accumulation equation below:

$$\frac{\dot{K}}{K} = S \frac{{Y}}{K} - d$$
$$\frac{\tilde{\dot{k}}}{\tilde{k}} = s\frac{Y}{K} - d - g - n$$
And, $$\frac{Y}{K} = \frac{Y}{AL} \frac{AL}{K} = \frac{\tilde{y}}{\tilde{k}}$$

Then,
$$\frac{\tilde{\dot{k}}}{\tilde{k}} = s\frac{\tilde{y}}{\tilde{k}} - d - g - n$$

$$\tilde{\dot{k}} = s \tilde{y} - (d+g+n)\tilde{k}$$

Finally, combining our production function with capital accumulation we get an equation (6) that represents the Solow model with tech changes.
$$\tilde{\dot{k}} = s \tilde{k^\alpha} - (d+g+n)\tilde{k}$$

So our steady state is when (7) is true.
$$\tilde{\dot{k}^*} = 0$$

If we set this to be true for our steady state (denoted by *) levels of capital and output per effective labor. (8 and 9)
8: $$\tilde{k^*} = [\frac{s}{d+n+g}]^\frac{1}{1-\alpha}$$
9: $$\tilde{y^*} = [\frac{s}{d+n+g}]^\frac{\alpha}{1-\alpha}$$

In the steady state: (10)
Capital (and income) per capita grow at the rate of exogenous technological growth.

$$\tilde{k} = K/AL$$
$$k = K/L$$
$$k = A\tilde{k}$$

$$\tilde{k} = \tilde{k}^* = [\frac{s}{d+n+g}]^\frac{1}{1-\alpha}$$
$$k(t) = A(t)\tilde{k}^*$$
$$lnk(t) = lnA(t) + ln\tilde{k}^*$$
$$\frac{\dot{k}}{k} = \frac{\dot{A}}{A}$$

If you graph out
$$\tilde{\dot{k}} = s \tilde{k^\alpha} - (d+g+n)\tilde{k}$$

Any savings rate changes, s, change the savings curve and therefore change $$\tilde{k^*}$$
Population or depreciation changes, n and d respectively, rotates the straight line, (d+g+n)k.
Changes in these affect the growth rate of per capita capital and per capita income only outside the steady-state.
exo-tech growth changes the graph sort of like n and d by changing the level of $$\tilde{k^*}$$

This exo-tech growth changes the growth rate of per capita capital and per capita income in the steady state (along balanced growth paths).


The Case for Basic Income in developed and underdeveloped countries

Thanks to densp for identifying this paper.

It refers to a major pilot project undertaken in Namibia

Background:

The Basic Income Grant (BIG) pilot project took place in the Otjivero-Omitara area, about 100 kilometres east of Windhoek. All residents below the age of 60 years receive a Basic Income Grant of N$100 per person per month, without any conditions being attached. The grant is being given to every person registered as living there in July 2007, whatever their social and economic status.

Before the pilot project, the area was characterised by unemployment, hunger and poverty.

Conclusions:

Since the introduction of the Basic Income Guarantee (BIG), household poverty has dropped significantly. Using the food poverty line, 76% of residents fell below this line in November 2007. This was reduced to 37% within one year of the BIG.
There was a dramatic increase in economic activity. The rate of those engaged in income generating activities (above the age of 15) increased from 44% to 55%. Thus the BIG enabled recipients to increase their work both for pay, profit or family gain as well as self-employment. The grant enabled recipients to increase their productive income earned, particularly through starting their own small business, including brick-making, baking of bread and dress-making. The BIG contributed to the creation of a local market by increasing households' buying power. This finding contradicts critics' claims that the BIG would lead to laziness and dependency.
Huge reduction in child malnutrition from 42% to 17% in 6 months.
Increase in school attendances (non attendance due to financial reasons dropped 42%)
Drop in crime (theft down 42%)
Estimated cost for a nationwide program in Namibia would be 2-3% of GDP. Not cheap, not unaffordable either.

All thing considered, the author of this report seemed pretty happy with the outcome. There were some issues impacting the data. Since it was a localised study, there was a fair bit of migration towards the treatment area. The long term impacts are not yet known as well.

Additional resources:

Haarmann, Claudia; Haarmann, Dirk; Jauch, Herbert; Mote Hilma et al 2008. Towards a Basic Income Grant for all. Basic Income Grant Pilot Project. First Assessment Report, September 2008. Windhoek

Kameeta, Zephania; Haarmann, Claudia; Haarmann, Dirk; Jauch, Herbert 2007. Promoting employment and decent work for all - Towards a good practice model in Namibia. - Research Paper - Presentation to the United Nations Commission for Social Development. Windhoek

Haarmann, Claudia; Haarmann, Dirk (ed.) 2005. The Basic Income Grant in Namibia. Resource Book. Windhoek


Has a market labor shortage ever occurred?

That very narrow definition of labour shortage, "when no labor is available for a task at any price", is rarely used, because it's a very rare phenomenon. Market forces kick in, raising prices sharply enough to suppress even relatively inelastic demand, long before the shortage becomes absolute. So the phrase is much more commonly used to mean that there is a temporary constriction in supply of a particular type of labour, usually leading to a large, rapid increase in its remuneration. Hence, when you hear the phrase, that's much more likely to be the intended meaning.

However, in the very narrow definition, the canonical example given is usually that of many trades in Western Europe, after outbreaks of the Black Death in the Fourteenth Century had halved the population.

However, even that example is problematic, as the evidence is that women were largely prevented from taking men's jobs, so the shortage was as much about culture as about physical limits:

  Labour scarcity should have opened up opportunities for females, but the regulations only added to their geographical and occupational immobility,


How does a perpetual futures contract affects the price of the underlying?

(More of a long comment.)

The "futures" in "perpetual futures" found on cryptocurrency exchanges is a bit of a misnomer, as already pointed out. It's not a futures position, but a leveraged spot position. As a result, crypto "perpetual futures" prices typically track spot prices. (There are also bitcoin futures contracts, in the standard sense--- those traded on CME.)

Such positions are not restricted to crptocurrencies. Leveraged spot FX positions are common. The type of exposure is the same whether it's cyptocurrency or central bank currency. More broadly, it's not restricted to currency. Similar instruments exists for equity, etc (but the somewhat unusual vernacular "perpetual futures" is only used in crypto).

In principle, the relationship between spot order flow (leveraged or not leveraged) and price (exchange rate in the case of currency market) is the same in any market. Price equilibriates supply and demand.
(In comparison, the relationship between futures and spot rices derives from spot-forward parity.)

For example, you might buy USD 100 using CNY at 7 CNY/USD exchange rate but only making a deposit CNY 35C for a position with notional value of CNY 700. That's a spot position with 1:50 leverage. This would be a bet on the spot depreciation of CNY.
If CNY depreciates by 1% against USD, the broker credits your margin account by CNY 70. If CNY appreciates, you might get a margin call.

The broker would then hedge his net exposure from aggregate order flow. In the above example, if you were the only customer, the broker would hedge by exchanging USD for CNY in the inter-dealer market. This in turn would drive up CNY against the USD---until the exchange rate equilibriates to where the long USD-short CNY spot bet is no longer profitable.

Similar mechanism leads to crypto perpetual futures prices tracking spot prices, although price dislocation may occur during episodes of illiquidity.


Newly issued shares are sold on the stock market?

A company will typically employ an investment bank to act as an underwriter when it wishes to issue new stock. The investment bank essentially agrees to buy the issued stocks from the issuing company and its job is then to sell them to investors.

Investment banks employ a sales force who try to persuade large investors (such as pension funds) to commit to buy the stocks before the date of issue. If the investment bank does a good job then all of the issued stocks will be allocated to a buyer in advance. If the investment bank fails to fails to find enough buyers then it is stuck with the unsold stock and must try to sell them to the public in the stock market.


Before the edit, you wrote "qualitative easing", but I think you refer to quantitative easing. I'll discuss both.
Quantitative Easing
Quantitative easing corresponds to the central bank (CB) expanding its balance sheets by "buying" assets. This is typically done in secondary markets. It mainly injects liquidity into the system. To the extent that there is an additional buyer of assets now, the price of assets/investment (interest rates) decreases. However, the quantitative impact should be negligible: Through its demand, the CB increases value and liquidity in the markets it is operating. However, the scale of its operations should be too small  to affect the aggregate interest rate.
Qualitative Easing
Qualitative easing is a relatively new expression and refers to the riskiness of the stocks that the CB is investing in. In contrast to quantitative easing, which is about the magnitude of assets on the CB's balance sheets, qualitative easing is about the riskiness on the CB's balance sheets, and hence the decrease in aggregate risk (on the banks' balance sheets).
Downsides
Independence of a central bank
When the CB holds assets, it is interested in their value. This may lead it to commit policies that infract its primary directive (e.g. inflation stability). Even if it does not do so, effectiveness of a CB comes from its capacity to control expectations. It suffices for households and firms to expect the CB to commit "bad" policies, to decrease the effectiveness of the CB.
Number of Goals
I don't have sources on this, but I seem to remember that central banks with one clear goal (i.e. monetary stability) are more effective than those with a basket of goals (i.e. monetary stability, GDP growth, decrease of unemployment rate). A general criticism can be that QE are not operations that help with the important margin, monetary stability - and that the central bank should focus on that instead.
Note that these are not just esoteric points. In fact, the academics and central bankers at the "Rethinking Macro Policy" conference agreed that (quoting Blanchard)

Throughout the conference, e.g., in Gill Marcus’ talk, and actually throughout the various meetings which took place during the IMF meetings in the following days, policy makers remarked and complained about the heavy burden placed on monetary policy in this crisis, and the danger of a political backlash against central banks. Even as the crisis recedes, it is clear that central banks will end up with substantially more responsibilities—whether they are given in full or shared—for financial regulation, financial supervision, and the use of macro prudential tools.
While even the use of the policy rate has distributional implications, these implications are much more salient in the case of regulation or macro prudential tools, such as the loan-to-value ratio. The general consensus was that these distributional implications could not be ignored, and that while central banks should retain full independence with respect to traditional monetary policy, this cannot be the case for regulation or macro prudential tools.


Inferior goods, monotonic utility and strict concavity

I could not add images in the comments, therefore I used this method.

I think that the convexity definition does not hold for the function, since there are pairs of points that when they are conected, produce a segment which are out of the set. Because of it, the set is not convex.

Here there is an example of such situation with this function, which does not show a convex set:


Why are prices considered the best information we could have?

Concerning austrians and their general attitude towards prices: https://mises.org/library/source-prices.

In short Mises says that

  The ultimate source of the determination of prices is the value judgments of the consumers.

If the value of judgement of consumers  is higher than the cost of production, producers know it is worthwhile to produce that good.

This way producers are informed what to produce and more importantly what not to produce, in order for the produced goods to be useful to people.

Mises even suggested communist central planning would be far less efficient if the planners didn't look at capitalistic prices.
https://en.m.wikipedia.org/wiki/Economic_calculation_problem?wprov=sfla1.

Whenever austrians talk about unrealistic prices, they are referring to price bubbles, an essential element in the ABCT
https://en.wikipedia.org/wiki/Austrian_business_cycle_theory?wprov=sfla1.

In short many austrians believe the primary cause of these bubbles to be the monetary policy of the central bank. The central bank's interest rate is not a market price, and therefore leads producers to believe more goods are wanted right now and less in the future, supposing a lower central bank interest rate. This leads to bubbles, which are then called unrealistic prices by Austrian economists.


How does real gdp affect the domestic currency?

file:///C:/Users/ACER/Downloads/1__Papper_exchange%20rate-gdp%20growth%20(15).pdf

The real exchange rate of euro and Greek economic growth
Gregory T. Papanikos - So i got the information from this source


Establish demand curve with infrequent sales

What you're looking for is the price elasticity of demand of your product.  That is: by how much will your sales rise or fall depending on how you manipulate price?  It depends on what you're selling and, like you said, each variation of your product will have a different elasticity.  So, say you increase prices by 5% and experience a %5 decrease in sales; your price elasticity of demand would be -1 (unit free).

The strategy you outlined to parse out the price elasticity sounds reasonable from a data collection standpoint, albeit a bit risky.  I might change the price of just one of your variations (maybe one that sells somewhat more frequently)  to assess the sensitivity your customers are to price changes.  However, if your product's elasticity is high, your sales might drop dramatically or stop altogether.


Where can I find an interactive graphic showing the development of countries?

I don't know if it is the original that you had in mind, but Gapminder allows you to interactively explore a variety of data sets relating to both developed and developing countries.


Why do companies push users to use inferior software?

There are a lot of factors that can explain why they do not. The stackexchange says to avoid "Making statements based on opinion;" but I do not think that this is possible.  There is most likely no one answer to this, and I would mark it a behavioral economic question instead of just micro economics. This answer assumes that your example gives each firm an additional profit.

1) Microeconomics assumes that each firm acts rationally; that is maximizes utility. Traditionally this means maximizing profit. This major assumption has been proved wrong again and again, and evidence suggest that the pure neo-classical model is not a sufficient indicator of firms actions.

2) While the end result of a contract will lead to a higher profit, the transaction cost between the firms may be high. Transaction Cost Theory basically adds friction to the invisible hand of neo-classical economics. Read the introduction, section 2.2-.3, and all of section 3.1. The external transaction cost between the two firms would be huge. Both companies are opportunistic, both firms have their own strategic intent, and assume both firms would not trust each other. What would have to result is an extremely ridged contract trying to encompass all possibilities, which can offset the additional profit both would obtain and lead to an extremely bureaucratic system that is not a characteristic of a successful tech company.  For example, if I was using a Windows Cortana and searched via a joint venture with Google for a pc, would my search be directed to Microsoft's store or Google's store? A better example would Googles licensing rights. Would google be able to license it's product to other firms. If it was, it could charge a lower price else where via more favorable terms and undercut Microsoft all together (much like Microsoft did to IBM with DOS). If it couldn't, it would be depended to Microsoft, something the firm certainly does not need.

3) Google is dealing with anti-trust suits in Europe. For Google, teaming up with Microsoft to gain further market share would not go well in their efforts to prove they are not a monopoly, and it may bring anti-trust suit in the US. For Microsoft, given their history, I am positive they want to stay as far away from this as possible.

4) Potential may be the most over valued asset. While a search engine is valuable today, in the future having your own may be infinitely valuable. I am not a tech guy, but I can see the value in controlling what people find when they search certain key words. The short-term lose of potential profits as a result of an inferior search engine can be far out weighed by devolving one that meets industry standards. Add that with the fact that Microsoft's core business (OS) is not going to be undermined by a poor search engine, I can understand why they want to deploy their own.

As I said earlier, these are just possibilities, and a combination of factors leads them not to collaborate.


Why do lenders dislike early loan repayments?

The reason that lenders dislike early repayments (known as "prepayments" or "voluntary prepayments") is that most lenders match their assets— the loans they've made to others— with liabilities of their own. This can lead to lenders facing significant interest rate risk. This is important to understand— while default risk is certainly significant, interest rate risk can be very large as well, sometimes greater than default risk.

Consider: a lender loans you money at 6.5% for 30 years to take out a mortgage, and funds this by issuing debt in the secondary market for 10 years at 4%. A recession comes along, the Fed drastically cuts interest rates, and the prevailing interest rate for mortgages drops to 3.5%. You repay your mortgage early by taking out a loan from another lender at 3.5%, and now the original lender has the principal back, but cannot reinvest it at a rate high enough to repay its own interest rate expense. If this happens on a broad enough scale, the lender is now bankrupt, with all that entails. That is why they don't like it.

With regard to prepayment penalties— the option to prepay debt is what is known as a call option, and it's often quite valuable. Prepayment penalties are simply a way for lenders to be compensated for the exercise of that option.

The specific case that you're talking about sounds an awful lot like the way that some credit card lenders feel about consumers who take out loans at low (often zero) "teaser" rates and refinance them before they ever have to pay anything to the lender, which bears its own interest expense in the meantime. That's probably best regarded as a strategy by consumers that takes advantage of lenders who are hoping that the consumers will not repay the debt in time. This hope on the part of lenders can sometimes cross the line into predatory behavior (see, for example, the discussion of teaser rates on mortgages in the Financial Crisis Inquiry Commission report), so the fact that the borrowers and lenders in such a situation aren't particularly fond of one another comes as little surprise.


Here is my method, which is less rigorous but perhaps quicker, and I believe allows you to see the answer more easily, at least in this case. It may not be true in every case.

Since the line segment will represent all combination of lawyers and M.B.A's that we can reach, it must include the extremes. That is, it must include the cases where we maximize the number of lawyers that we reach, and when we maximize the number of M.B.A's that we reach. In fact, these will be the endpoints (since if we are not maximizing one of these we have to be somewhere in-between). Because of this we can do the following:

1) Figure out how many lawyers we can reach, if we focus on only reaching lawyers. Well, the business magazine lets me reach $300$ lawyers for every $\$700$ I spend. The consumer magazine lets me reach $250$ lawyers for every $\$350$ I spend or equivalently $500$ lawyers for every $\$700$ I spend. This is important because we are comparing $\$700$ worth of one magazine to $\$700$ worth of another: we are making a $1:1$ comparison, which is necessary.

What the above paragraph tells us is that, if I want to reach as many lawyers as possible, I should spend all my money on the consumer magazine ($500>300$). With a budget of $\$4200$ I can buy $4200/350 = 12$ ads. $12$ ads lets me reach $12*250 = 3000$ layers. So the most lawyers I can reach is $3000$. Additionally, $12$ consumer ads lets me reach $12*300=3600$ M.B.A's, hence one point (one extreme) is
$$
(3000,3600)
$$

To get the other point, find out how you can reach the most M.B.A's.

One thing to note: The fact that things like $(0,3,600)$ are options should perhaps prompt you to consider maximizing the lawyers reached and maximizing the number of M.B.A's reached (since a $0$ indicates reaching none of one group). Not that doing so will be correct, but if you are stuck such a hint might lead you to the correct answer.

Cheers.


Implications of declining money velocity

http://www.forbes.com/sites/traceygreenstein/2011/09/20/the-feds-16-trillion-bailouts-under-reported/#5a1dec1d6877

The decline in money velocity is due to enormous monetary injections, which mostly sat in recipient bank accounts and had no effect on the real economy. Under the Keynesian and monetarist theories that central banks operate upon, an increase in money supply should ease liquidity constraints and allow the economy to resume growth. In reality not much further growth was possible, likely because of the stagnation in conventional oil production.

To answer your questions:

Yes
Not yet
No, it indicates a lack of viable uses for the cash
Interest rates fell because of the relatively small decrease in banks balance sheets coinciding with a sharply declining return on capital, and thus diminishing incentive for investors to leverage, causing increasing supply relative to demand for credit


Why am I getting 3 different answers for Point Elasticity of Demand?

In your second option, you computed the elasticity at the original point, and in your third option, at the end point. In your first point, you did the same as in the second one, but with a mistake: you wrote $133%$ instead of $33%$.


Possible regression biases with GDP [x] vs Health expenditure [y] (both per capita)

There could be a question of whether your model and graph should be based on a linear relationship or a log-log relationship

The slightly widening residuals apparent in your linear graph seems to understate the real issue illustrated in a linear-axis graph from Gapminder. The green outlier at the top is the USA while the small red outliers on the right are rich but small Gulf and SEAsian countries, and the presence of both of these could exert excessive leverage on your regression coefficients

This left-right disparity in residuals does not look so bad with the same Gapminder data plotted with log-log scales.  It also spreads out the big cluster of developing countries in the bottom-left corner between those which are extremely poor and those which are not so poor and so allows their relative difference to influence your model estimates


There are multiple ways how the discount factor can be estimated. I dont think its possible to make exhaustive review of all of them (within format of this site at least), but one that nicely matches your question would be through estimating the Euler equations.

Following Attanasio & Browning (2009) an Euler equation for general asset would be given by:

$$ E \left[ \frac{c_{t+1}^*}{c_t^*} \right]^{-\gamma}(1+r_{t+1}) \beta = 1 \implies  \left( \frac{c_{t+1}^*}{c_t^*} \right)^{-\gamma}(1+r_{t+1}) \beta = \varepsilon_{t-1}$$

where $c$ is consumption $\gamma$ the coefficient of relative risk aversion, $r$ real interest rate and $\beta$ the discount factor. Authors show that assuming that measurement error is log normal with homoskedastic variance across households and independent then in terms of observed consumption it can be proven it will be equal to $e^{\gamma^2 v}$ where $v$ is the variance of measurement error.

I will skip all derivations, since they are nicely laid out in the above-mentioned paper itself and for the sake of brevity, but with additional assumptions that the Euler equation will also hold in future $c_{t+2}$ you can derive an estimator based on the following equations:

$$ u_{t+1}^1 = \left( \frac{c_{t+1}}{c_t} \right)^{-\gamma} (1+r_{t+1}) \beta - e^{\gamma^2v}$$

$$ u_{t+2}^2 = \left( \frac{c_{t+1}}{c_t} \right)^{-\gamma} (1+r_{t+1})(1+r_{t+2})\beta^2 - e^{\gamma^2v}$$

where parameters $\beta$, $\gamma$, $v$ are overidentified so we can estimate them using non-linear general method of moments (GMM). Authors call this estimator GMM-LN and they also derive version of it which require a bit less strict assumptions.

However, the above estimator is not the only way how to estimate discount factors, other ways exist including estimating it experimentally such as for example in Benzion, Rapoport,  Yagil (1989).


Estimation of point-identified Dynamic Discrete Choice models with moment inequalities

Function $Q$ has to be minimized wrt $(\theta,\alpha)$.
The parameters $(\theta,\alpha)$ compatible with Nash equilibrium and rationalizing (or generating) the data have to satisfy $$g(x;\theta,\alpha)\leq0.$$
In the case where, for given $x$, $$g(x;\theta,\alpha)>0,$$ the parameter values have to be changed in order to possibly reverse the strict inequality and minimize $Q$.
If $$g(x;\theta,\alpha)\leq0$$ for some $(\theta,\alpha)$, then this should not contribute to increase $Q$, because it is fully compatible with the data and model. Hence the term $\min\{g(x;\theta,\alpha),0\}$ in the expression of $Q$.


(Modified comparative advantage model)Can trade with a country that only has one industry be mutually beneficial?

I did not encounter this kind of example before but I think comparative advantage would still apply. In Ricardian model opportunity cost can be expressed as a ratio of outputs $a$ for food $f$ and bronze mirrors $b$ as $a_f/a_b$. If Japan can create A units of food but 0 bronze mirrors the opportunity cost would be given by $A/0$ now this is undefined in mathematics, because limit of the expression would not give you the same value from both sides, but since we are working only with positive quantities (presumably output cant be negative) we could evaluate the limit only from positive side which will give you $\infty$ meaning that no matter how much you sacrifice food you will get no mirrors.

The rest of the problem could be solved in this way, the equilibrium price that should be $a_{f}^{'}/ a_b^{'} < P_f /P_b < a_f/a_b$  would just have the upper bound equal to $\infty$. The exact quantities demanded would be given by the assumptions on individual demands in countries etc. Even in case this would not be mathematically rigorous I think you could carry out virtually the same analysis with just assuming some small infinitesimal production possibility for Japan and examining how the solution behaves as it goes to 0.


Why is the price of books set by the manufacturer rather than the distributor?

It’s the manufacturer’s suggested retail price (MSRP); manufacturers cannot dictate the final retail price. Any manufacturer that has a catalogue or a website has a suggested retail price.

If retailers want to sell at a higher price, they just slap a price sticker over the MSRP. From what I recall, this was standard practice back in the 1970s-1980s, when inflation was more of an issue.

The point of negotiation between publishers and distributors is the discount from the MSRP the distributor gets the books at. This discount is normally fixed for the entire line of books; they are not going to negotiate prices on a per-title basis. (Some best sellers might have special negotiations.) The retailer will then set prices relative to the discount received. Since the default is to sell at the MSRP, it can save booksellers the expense of putting on their own price stickers.

Why are books different? Well, go into a bookstore and count the number of unique products on 10 feet of floor space.

Other differences include:

Unsold books can be returned, which is unusual. This was a desperation measure that publishers resorted to in the Great Depression, which they never managed to undo. This means that most books have a limited shelf life, so inflation affecting the MSRP is less of an issue.
Books can be special ordered at book stores. The order price is going to have to be the MSRP.


An $n$-player prisoner's dilemma where a coalition of 2 players is better off defecting

A public goods game is a generalization of the prisoners dilemma that could cover this.

In a public goods game all players set their individual contribution level $x_i \in [0,\bar{x}]$ where $\bar{x}$ is a given positive parameter. Sometimes instead of an interval there are several discrete options to choose from. Player $i$'s payoff is defined as
$$
- x_i + c \cdot \sum_{j = 1}^{n} x_j
$$
where $c$ is a parameter such that $\frac{1}{n} < c < 1$. In these games it is individually rational to defect (set $x_i = 0$) but the Pareto optimal outcome is to cooperate (meaning $\forall i: x_i = \bar{x}$).

Seems to me that you could adapt this to your problem by narrowing the range of $c$ to $\frac{1}{n} < c < \frac{1}{2}$. In this case a player would prefer even dual defection to himself having to cooperate. Of course these games do not cover the whole game class you described, by they are part of it, and there is a lot of literature on public goods games, though mostly in behavioral economics.


Why does Figure 2.F.1(b) (MWG page 30) satisfy the WARP (Definition 2.F.1)?

Intuitively, this just says that if the bundle you choose was possible under wealth $w$ and price $p$ but it is not $x(p, w)$, then it must be that you cannot obtain it when price is $p'$ and wealth is $w'$.

It simply does not say anything about case (b) in which $p' \cdot x(p'', w'') > w$ and $p'' \cdot x(p', w') > w''$. Since the first part of the if ... then statement in the definition is false, everything evaluates to true.


How to interpret fixed effect regression R-sq. results for panel data?

In panel regressions you have multiple dimensions and that is why also you have 3 different $R^2$. The within $R^2$ tells you how much variation within your panel variables is on average explained by your model. The between $R^2$, tells you how much variation between your panel variables is explained by the model, and overall $R^2$ gives you the combination of both and tells you how much variation in the whole panel data your model explains.

According to the stata manual you should get adjusted $R^2$ for fixed effects by ereturn list it will be the r2_a scalar. However, there are no simple adjustments for within and between $R^2$, those are most often reported unadjusted.


Is there any index which relates population growth rate with food production growth rate?

Here there is a very good summary of existing information. Key variables are daily intake of calories/proteins/fat per person (so it is not so much about tons of food but their nutrient content). An example:

Notice that most of the data is taken from the FAO, which is available online here.


What is the formula for how many more I need to sell in order to justify raising cost?

This is an algebra problem. My answer is quite similar to snorams, but without the numbers plugged in (which hopefully makes it a bit easier to see what is going on). Let

$p$ be the price of a widget
$c_L$ be the old cost for producing one widget (i.e. with a low amount of lube)
$c_H$ be the new cost for producing one widget (i.e. including the high amount of lube)
$n_L$ be the number of units sold when you use a low amount of lube
$n_H$ the number of units solve when you use a high amount of lube.

With a low amount of lube, your profit is $(p-c_L)n_L$ (in words: you earn a margin of $p-c_L$ on each unit, and sell $n_L$ units).

With a high amount of lube, your profit is $(p-c_H)n_H$.

You would like to know when is it true that the higher lube content is more profitable, i.e.

$$(p-c_H)n_H\geq(p-c_L)n_L$$

Now, to find the number of units, $n_H$, at which the change results in break even, we can simply solve the inequality:

$$n_H\geq\frac{(p-c_L) n_L}{(p-c_H)}$$

Let's look at an example:

price of a widget is $p=3$.
unit costs of everything but the lube are $1$.
cost of low amount of lube is $0.5$
cost for high amount of lube is $0.6$
low-lube sales are $1000$.

We calculate the unit cost as $c_L=1+0.5=1.5$ and $c_H=1+0.6=1.6$.

Now we plug these numbers into the formula

$$n_H\geq\frac{(p-c_L) n_L}{(p-c_H)}$$

$$n_H\geq\frac{(3-1.5) 1000}{(3-1.6)}$$

$$n_H\geq\frac{1500}{(1.4)}$$

$$n_H\geq1071.42.$$

So if you sold at least 1072 units (i.e. an extra 72 units) then the change would pay for itself.


You are confused about how insurance works, and thereby why people take insurance. When someone buys insurance against a certain even, s/he has to pay a fee (called a "premium"), which entitles the person to be compensated for the loss in case the negative event happens. Unexpectedly, insurance comes in many ways. For instance, the compensation can be of the total loss (in your example, 20,000), or could be partial. The size of the premium usually reflects this difference.

Now, in the example you give, what you have calculated is simply the expected value of no taking insurance, because (i) there is no premium, and (ii) if the bad event happens, the car owner losses the full amount. Naturally, no taking insurance against a negative event will always give you a negative expected value (your "unfair bet" conclusion), as long as that event has positive probability. That is precisely the reason why individuals are willing to pay for insurance.

How much is the person willing to pay? This depends on the preferences of the individual. For example, consider someone who is risk neutral (i.e. that is indifferent between uncertainty and certainty, if the expected value of the former is the same as that of the latter). In this case, the person is willing to pay no more than 5,000 for an insurance which fully cover the cost of the car loss. For any premium below 5,000 the individual would take the insurance.

The case is different for risk lover individuals, who would prefer an uncertain situation (no insurance) rather than a certain one (insurance). This individual would not be willing to pay more than 5,000 to get insured. Conversely, a risk averse is willing to pay more than 5,000, because s/he dislikes uncertainty.

For reference, see the entry on Wikipedia. See this pdf for a treatment from Economics.


Effectiveness of recycling (consumer PET) plastic

I'm just going to answer one part of the question, the trickiest and most controversial one in my opinion:

  How to determine the costs/profits of each part.

It sounds tricky: Even if you find the most efficient way of not-recycling trash (be it burning, dumping on a lake, on the moon, or in a third world country), how do you evaluate the costs of doing that?

Coase: How can we evaluate the costs of dumping the trash?

Without loss of generality, Say the most efficient solution is to drop the trash into the sea.

If you are familiar with externalities and the Coase Theorem, you will believe that the obvious solution is to designate ownership: Under certain circumstances, if we allow someone to own "the sea", and then allow trading the sea, the person/coalition that "cares most" about the sea, given same initial endowments, will own it. If you want to trash the sea, you will have to pay them a fee, which depends on

how important is it for you to get rid of the trash
how important is it for the owner to have a clean sea

You can already see the wlog applying: You just assign ownership to all $n$ different trash-dumping possibilities, and look which owner asks for the lowest price. That will be the "most efficient place of dumping trash", from society's point of view. And the social costs of trashing that place are exactly the costs that the owner asks for (ignoring imperfect market structures).

However, you can already feel the big smelly but coming up.

The free market's fundamental problem of evaluating the optimal value of finite resources

But this is only true, if actors in the economy care about future generations as much as they should. Let me give a simple example.

In macroeconomics, we usually have a single representative agent, which lives for all eternity, and tries to maximize consumption. He does something like

$$ \max_{\{c_t\}_t} \sum_{t=0}^\infty \beta^tU(c_t)$$

Where $U$ is his utility function, $\beta$ is a discount factor (which we have assumed to be constant), and $c_t$ denotes consumption.

We then argue that the representative infinitively-lived agent model is equivalent to a model with finitely lived agents, that care about their children with some discount factor related to $\beta$. Intuitively, in the first model, you care about next period at rate $\beta$ because you are impatient and want things rather today than tomorrow. In the second model, you die in the next period. However, your child will be alive in the next period, and you care about it at rate $\beta$. Under these circumstances, these two models are identical (mathematicians like to say isomorphic).

However, you can immediately see that there is no inherent reason why you should care about your own children at the same rate, as you care about your own future self. Even more importantly, there is no ex-ante normative reason why future generation's utility should be $\beta$ less important than the current generation's.

tl;dr: While it is true that we can evaluate the de-facto social costs to common goods using the Coase Theorem, there is reason to believe that these de-facto social costs do not correspond to normative social costs.


How is signaling theory different from cue utilization theory?

I think you do mean cue utilization theory. According to the link provided in @NickJ's comment, and as far as economic theory goes, I'd say that cue utilization is basically a signaling story.

Putting "cues" on organic food products (e.g. the "organic" label, information on nutrition and natural content) amounts to a signaling act committed by the seller. Since the organic cues is less costly for the real organic food producers to produce (for instance, labeling non-organic foods as organic would face legal disputes, and hence a negative expected cost), these cues (or signals) are reliable.

Thus in principle, organic food sellers sending cues to consumers is no different from the classical signaling model where job seekers uses university degrees to signal their ability.


Do free-trade agreements typically include clauses relating to services?

The simplest answer is: some do, some don't.

Trade in goods and trade in services are dealt with quite separately, so structurally, you wouldn't find a couple of clauses on services within an FTA.  You would generally find a chapter on goods and (possibly) another on services, reflecting the division in the WTO between the GATT (goods) and the GATS (services).

Quantitatively, there is a roughly equal split between goods-only agreements and agreements that cover both (153 for goods only and 154 combined agreements have been notified to the WTO as of December 2018).

You also often find a group of countries negotiating a goods-only FTA first, then extending to cover services later on (for example by adding a protocol to an existing treaty).  This reflects the fact that an integrated services market is usually seen as a "deeper" form of integration than tariff-free trade.


What policies are in place to police financial speculation in the modern era?

I am afraid that "speculation" can be policed only very indirectly, since it is impossible to tell what "speculation" is and isn't. Look at Keynes' definition: speculation for the activity of forecasting the psychology of the market
In other words, you know the INTENTION of the speculator. You must see inside his head. I am not aware of any agreed-upon distinction of "enterprise" and "speculation," that can be perceived and judged objectively, from the outside. One activity can be speculation in one context and not in another.


Labour Productivity Growth and Reallocation of Labour

Ad 1)

You make a hand waiving claim. In these scenarios making concise mathematic statements helps improving your arguments.

Denote employment as $L$, define real value added as $F(L) - wL$, and productivity as real value / employment:

$$ A = \frac{F(L) - w(L)}{L}$$

The labor share should be some constant $\beta$ of total value added. Without going deep into mathematics, straight away assume $w(L) = \beta\alpha F(L)$. Hence, we have that

$$ A = \frac{F(L) (1 - \beta\alpha)}{L}$$

Now, what happens if $L$ increases? Let $L_2 > L_1$

$$ A_2 - A_1 = (1-\alpha\beta)(\frac{F(L_2)}{L_2} - \frac{F(L_1)}{L_1})$$

Assume $F$ to be concave. Then we have that the change of productivity, $A_2 - A_1$ is negative - as you say.

However, $A$ here is average productivity. Total productivity is $AL$.

$$ A_2L_2 - A_1L_1 = (1-\alpha\beta)(F(L_2) - F(L_1))$$.

Independent of the curvature of $F$, for any strictly increasing production function, this will be positive.

tl;dr

Don't confuse average productivity with total productivity.
Whenever you find something weird, use algebra to make your statements precise

Ad 2)

Again, you are not being precise enough. What is the government's objective? If government is only about maximizing average productivity, the solution is to let $L\to 0$, as then $F'(L) \to \infty$ (and for Cobb-Douglas, $F'(L)$ is proportional to average productivity).

Maximizing total output implies allocating workers to the sectors with highest productivity. I do not understand how the functional relationship between employment intensity and labor productivity affects that premise. Furthermore, unless there is a market distortion, the role of any government intervention is not clear at all.


What is the difference between the cash rate and interest rate?

In Australia, the cash rate is a volume-weighted average interest rate of unsecured overnight loans among banks which are members of the Reserve Bank Information and Transfer System (RITS). The cash rate is also known as AONIA (the Australia Overnight Index Average).
Overnight loans are just like they sound: Perhaps Bank A needs to make payments the next day but has cash coming in to cover those payments in two days. So Bank A borrows money overnight to make the payments and bridge the gap to when money covering the payments arrives. Banks do this all the time for one another since perfectly matching up the timing of cash inflows and outflows is difficult.
The Reserve Bank announces target rates for the overnight rate and then conducts "open market operations" (offering overnight loans, for example) to hold the market level of interest on overnight loans to be close to their announced target.


Alternatives to Capitalism and Communism

Arguably, Henry George described such an alternative in his 1874 work Progress and Poverty.  His system eventually became known as Georgism.

George saw his system as different than either Capitalism or Communism, as both of those systems equated Land with Capital and called for either the private ownership of both, or the communal ownership of both.  Georgism calls for the private ownership of Capital but public ownership (of a sort) for Land.


What will happen to the reserves at the Fed?

Reserves are drained (destroyed) when the Fed sells the debt or allows it to be paid off.


Your logic is correct. You need one more step. Using the same logic you can show that, in any SPNE, firm $A$ must set a price greater than the price that maximizes $B$'s profits. Firm $B$ then chooses such price.


Application of Laffer Curve on the price of Goods and Services

Trivially this holds, in many cases, but it is not considered to be Laffer curve or application of the Laffer curve. This is just standard profit/revenue maximization. Let us focus on the revenue part. The total revenue is given by price times quantity $TR=pq$, now quantity will depend on some demand, for example the demand could be $Q = 100 - p$, so the total revenue will in terms of price be:
$$TR=100p -p^2$$
Which in this case will also give you something that looks like Laffer curve, as you can see on the plot from simulation I made in R:

However, note two things:

Demand curves come in various shapes and they critically influence what the shape of total revenue is - consequently the above might not hold for any demand curve you can imagine.
Both this result for revenue and the result that you get zero tax revenue at both $0%$ tax rate and $100\%$ tax rate was known to economists long before Laffer. Laffer curve is called Laffer curve just because Laffer popularized already existing idea and he did so in the realm of taxation. Hence we talk about Laffer curves mostly just when we talk about taxes not other quantities where there will be some 'optimum peak'.


It is the same in practice, they are both variances some people will write it as $\sigma^2$ some as $\epsilon^2$. Variance in error tells you what is the variance in the variable you predict.


When can one drop time subscripts? Example from Angrist and Kugler (2003)

Because "adjustment costs are linear and there is no aggregate uncertainty", the FOC for $N_t$ is $$f'\theta g_N(N_{t}, I_{t}) - w_N = \phi \lambda C_N$$. Notice that this is exactly the same form for each period. The same is for $I_t$. This means that a firm will choose the same labor inputs in all periods. In other words, the firm gets into the steady state immediately in this dynamic optimization problem. This is not the case when there is aggregate uncertainty so that $w$s can vary over time or the adjustment costs are nonlinear, e.g. $C_N(N_t-N_{t-1}).$
As a result, the profit at each period is constant. And thus the present value of the discounted profits is $$\Pi= \frac{\left[f\left(\theta_{g}\right)-w_{N} N-w_{I} I-\phi \lambda C_{N} N\right]}{1-\phi}$$.


If there was world war 3, which stocks should I short?

Gold is generally the best option for expected political and economic strife as it generally reacts inversely with the markets. Here's a good article on the connection between war and the value of gold.

Another thought would be to buy stock in companies involved in the defense industry. Raytheon, Boeing, Northorp Grumman all jump during wartime due to the anticipation for more demand for weapons production.

An even safer bet would be to buy stocks in metal production such as steel given that in a wartime economy demand for these goods spikes as production of ships and vehicles will increase for a period of time.


No reflexivity, transitivity and completeness does not imply symmetry. For example: Consider the preference relation $\succsim$ over $\mathbb{R}^2_+$ defined as:

  $(x,y)\succsim (x',y')$ if and only if $xy \geq x'y'$

Such preferences have a utility representation $u(x, y) = xy$, and therefore they are reflexive, transitive and complete. However, these are not symmetric because $(1,2)\succsim (1,1)$ but $(1,1)\not\succsim (1,2)$.

But we can define the indifference relation $\sim$ from the weak preference relation $\succsim$ in this way:

  $(x,y)\sim (x',y')$ if and only if [$(x,y)\succsim (x',y')$ and $(x',y')\succsim (x,y)$]

It can be easily shown that the indifference relation derived from the reflexive, transitive and complete weak preference relation in the way described above will be an equivalence relation. It is important because this relation partitions the entire commodity space into indifference classes, where two consumption vectors belong to a class if and only if the individual is indifferent between the two. These classes are also known as indifference curves. This further helps in solving related problems.


Terminology for "part" of extensive form game

A "part" of an extensive form game that is not a proper subgame because it does not start at a single node but an entire information set would be called "continuation game". This terminology is fairly standard (Perfect Bayesian Equilibrium).

However, I think what you are after is a stochastic game which consists of several states. Each state corresponds to a different game. In your example there would be two states: One state for each player $i\in\{1,2\}$ being the Stackelberg leader. Then you also need a transition process that maps the period-1 state and the period-1 action profile into a probability distribution over period-2 states. You then find equilibria for all state games in period 2 and proceed to solve for equilibria in period 1 with the corresponding (probability-weighted sum of) continuation payoffs.


How does explanatory variables in variance affect unconditional variance of GARCH(1,1)

Your last expression is not correct because, as noted in the comments, you are after the unconditional variance, which is constant in these models. it should be
$$\sigma^2=\frac{\omega+\phi\cdot E(x)}{1-\alpha-\beta}$$
PS: Also, you are missing a $t$-subscript in your penultimate expression.
--
RESPONSE to comment
Using more explicit notation, we assume that the conditional variance is
$${\rm Var}(u_t \mid I_{t-1}) = \omega+\alpha\cdot u_{t-1}^2+\beta\cdot {\rm Var}(u_{t-1} \mid I_{t-2})+\phi\cdot x_{t-1}$$
Taking expectations through out
$$E\big[{\rm Var}(u_t \mid I_{t-1})\big] = \omega+\alpha\cdot E(u_{t-1}^2)+\beta\cdot E\big[{\rm Var}(u_{t-1} \mid I_{t-2})\big]+\phi\cdot E(x_{t-1}) \tag{1}$$
Now, by the Law of Total Variance,
$${\rm Var}(u_t) = E\big[{\rm Var}(u_t \mid I_{t-1})\big] + {\rm{Var}}\big[E(u_t \mid I_{t-1})\big]$$
Under the assumption $E(u_t \mid I_{t-1}) = 0 \implies E(u_t) = 0$, we obtain the relation
$${\rm Var}(u_t) = E\big[{\rm Var}(u_t \mid I_{t-1})\big] \tag{2}$$
and lagging once,
$${\rm Var}(u_{t-1}) = E\big[{\rm Var}(u_{t-1} \mid I_{t-2})\big] \tag{3}$$
Another assumption is that $u$ is homoskedastic unconditionally. This together with $E(u_t)=0$ means
$${\rm Var}(u_t) = {\rm Var}(u_{t-1}) = E(u^2) \tag{4}$$
Finally, another assumption is that the $x$'s are identically distributed over time, so $E(x_{t-1}) = E(x)$.
Using all these into $(1)$ we get
$${\rm Var}(u) = \omega+\alpha\cdot {\rm Var}(u)+\beta\cdot {\rm Var}(u)+\phi\cdot E(x) $$
$$\implies {\rm Var}(u) =\frac{\omega+\phi\cdot E(x)}{1-\alpha-\beta}$$


Your mistake probably occurred at the final substitution stage. Optimal utility should be $2\sqrt{x^*y^*}$, which is not the same as $2\sqrt{x^*y_i}$.

In general, maximizing a two-good Cobb-Douglas utility function $u(x,y)=Ax^\alpha y^\beta$ subject to budget constraint $p_xx+p_yy=m$ has general solution
\begin{equation}
x^*=\frac{\alpha}{\alpha+\beta}\frac{m}{p_1}
\quad\text{and}\quad
y^*=\frac{\beta}{\alpha+\beta}\frac{m}{p_2}.
\end{equation}

In your case, the parameters $A,\alpha,\beta,p_x,p_y,m$ take the following values:
\begin{equation}
A=2,\quad\alpha=\beta=\frac12,\quad p_x=1,\quad p_y=p,\quad m=1+py_i.
\end{equation}
Plugging in the values, you get optimal consumption:
\begin{equation}
x^*=\frac{1+py_i}{2},\quad y^*=\frac{1+py_i}{2p}.
\end{equation}
Hence optimal utility is
\begin{equation}
u(x^*,y^*)=2\sqrt{\frac{1+py_i}{2}\cdot \frac{1+py_i}{2p}}=\frac{1+py_i}{\sqrt p}
\end{equation}


Externalities from motor vehicles - more recent estimates?

Handbook on the External Costs of Transport – Version 2019

Commissioned by the European Commission and produced by the research firm CE Delft. I think it was first published in 2008 and had a 2014 update. This is the latest 2019 update.


What is the main difference between FDI and short run capital inflows/outflows?

The term FDI is generally reserved for foreign capital investment in domestic assets of the host country for business interests as opposed to hot flows which are generally aimed for quick capital gains.
Examples of FDI would include take-overs of firms, collaboration/merger with a domestic company, entry into a new market, etc. Such investment give returns over the years as profits or dividends.
Hot flows, on the other hand,  include capital flows to take advantage of interest rate and currency rate arbitrage (such as investments in sovereign bonds, currency markets, etc.) and domeatic Stock market without taking the control of the management. Such investment give mainly capital returns after sale of assets.
There are some technical differences that emerge from the host country's central bank classification process but they are essentially to reflect the above economic differences. For example, in India, an investment may not be classified as FDI in until the shares are allocated to the investment firm and money has been invested by the host firm.
Further there are many rules imposed by host countries relating to capital mobility that differ for the two types of investment.


How to use an instrumental variable to estimate the parameter?

Reduced form is a regression of dependent variable  on instrument directly without using some two stage approach.

Consider the following example of endogenous system

Second Stage: $$Y = \alpha + \beta X + \epsilon$$

First Stage: $$X = \mu + \gamma Z+ \eta $$

Where $Y$ is dependent variable $X$ endogenous regressor and $Z$ is your instrument.

One option is to estimate this with some two stage approach as previous post suggests where you would first estimate predicted $\hat{X}$ and then use the $\hat{X}$ in second stage instead of actual $X$ - however, this is not reduced form.

A reduced form is when you use the instrument directly in stead of the endogenous variable so in this case reduced form would be:

$$Y= \kappa + \omega Z + \xi$$

You can work from this general case to your specific one.

I would recommend looking at the handbook Mostly Harmless Econometrics form Angrist and Pischke. They discuss reduced form in Ch4 instrumental variables in action.

The difference between OLS and IV estimator (in matrix notation) is as follows:

OLS: $\hat{\beta} = (\sum_i X_i X_i^{\prime} )^{-1}\sum_i X_i y_i$ - This estimator is biased in the presence of endogeneity

IV: $\hat{\beta}_{2SLS} = (x^{\prime} Z(Z^{\prime}Z)^{-1}x)^{-1}x'Z(Z^{\prime}Z)^{-1}y$ - (here $x$ is given by $x= Z \gamma + \eta$) this estimator has still bias but the bias declines if the first stage is strong - has high F-statistics as in approximation the bias term gets multiplied by $1/(F+1)$ where F is the test statistics so bias goes to zero as $F$ goes to infinity and in practice its very close to zero for $F>10$.


convexity of the profit function for profit maximization

It is true that we are usually interested in minimizing convex functions or maximizing concave functions, typically over convex sets. But I think you have two confusions:

The profit function is the result of a profit maximization problem. Not the objective function in the maximization problem. A profit function $\pi^*(p, w, r)$ identifies maximum profit given the price levels (p, w, r).
In the profit maximization problem, the objective function $\pi = pf(k, l) - wl - kr$ is concave in $k$ and $l$, the choice variables of the maximization problem.


Is there an economic rationale for wanting to know the gender of a child before it is born?

One typical story people tell is that in some countries there are strict population control policies and if families have a large economic incentive to have either a son or a daughter, they might neglect their pregnancy or terminate it if the child is not the gender they want it to be. In some societies the economic incentives to have sons or daughters are very different: typically sons are expected to bring in a working wife into the family, with a dowry, but daughters are expected to leave to the husband's family and take a dowry with them.


Why would regression coefficients be different on a subsample?

If I'm running a linear regression for example, and I take out some points, wouldn't the same line/plane still fit the data?

Aside from special cases no, OLS fits a line that is created based on minimization of sum of squared errors:
$$\min \sum e_i^2 = \min \sum (y_i-\hat{y_i})=\min \sum (y_i - \mathbf{x}_i^{\prime}  \beta)^2 .$$
If you remove some observations then you will likely change the objective function above leading to some other line having the best fit through data (save for special cases like when all observations are on a single line).

If not, wouldn't that show that the data doesn't have a linear relationship?

Not necessarily, the linear relationship is assumed to exist between the variables in the 'background', but since we cannot directly observe the data generating process and there will always be some disturbance term even if we are sure there should be a linear relationship between the two variables the observations will not necessarily all fall on a single line.
Non-linearity cannot be detected just from the fact that line you fit changes when you remove some data (although that is an indicator of robustness of a result - robust results should not change too much when sample is expanded/cut slightly). However, you can detect non-linearity in other ways. For example, you can look for patterns in residual plot. If residual plot shows some curved pattern then that is an indicator of potential non-linearity. There are also rigorous tests for non-linearity (like Ramsey test etc) but to go over all of such ways would be beyond the scope of an SE answer - you can read more about them in econometrics textbooks such as Wooldridge Introductory Econometrics: A Modern Approach.


Eichenbaum and Fisher (2005) follow Kimball (1995) in allowing for the possibility that the elasticity of demand is increasing in a firm’s price. Which is not exactly what I was looking for, but it's a start.


Is the Cobb-Douglas Utility Function Locally Non-Satiated at (0,0)?

No.
Cobb-Douglas utility is monotonic and monotonicity implies L.N.S.
The issue here is that you're only considering edge cases. You've correctly reasoned that edge points are not more desirable that the origin. However, LNS simply claims that there exists a more desirable bundle within the open epsilon ball of your allocation under consideration (and this is true for all allocations). So given any epsilon, an open epsilon ball centered at the origin will necessarily contain an allocation where both elements are strictly positive.
To see this, pick an epsilon and draw the epsilon ball around the origin. It should become fairly obvious what's going on.
Hope that helps.


Marginal cost given (Cobb-Douglas) production

The Cobb Douglas production function with constants returns to scale
$$y = \prod_i x_i^{\alpha_i} = A \prod_i \left(\frac{x_i}{\alpha_i}\right)^{\alpha_i} ,$$
where $A:= \prod_i \alpha_i^{\alpha_i}$ annoying constant.
Cost minimization with perfect competition
$$\min_x \ \ p^\top x\  \lvert \ y = \prod \left(\frac{x_i}{\alpha_i}\right)^{\alpha_i},$$
implies FOC
$$p_j - \lambda \left[A\prod_i \left(\frac{x_i}{\alpha_i}\right)^{\alpha_i}\right] \left(\frac{x_j}{\alpha_j}\right)^{-1}=0 \Leftrightarrow
\frac{\lambda y}{p_j} =\frac{x_j}{\alpha_j},$$
insert in production function to get
$$y = A\prod_i \left( \frac{\lambda y}{p_i}\right)^{\alpha_i} \Leftrightarrow \lambda = \frac{\bar p}{A},$$
where $\bar p := \prod_i p_i^{\alpha_i}$ which is a price index for production factors. Reinsert in FOC to get
$$p_j - \frac{\bar p}{A}\left[A\prod_i \left(\frac{x_i}{\alpha_i}\right)^{\alpha_i}\right] \left(\frac{x_j}{\alpha_j}\right)^{-1}=p_j - \frac{\bar p}{A}y \left(\frac{x_j}{\alpha_j}\right)^{-1}=0,$$
from which you find
$$(1) \ \ \ p_j x_j = \frac{\alpha_j \bar p}{A} \cdot y,$$
using this formula several implications follow. First take sum over $j$ and use $\sum_j \alpha_j = 1$ to get cost function
$$(2) \ \ C(p,y)= \frac{\bar p}{A} y.$$
Set $y=1$ to get unit cost, diffrentiate with $y$ to get marginal cost, divide with y to get average cost and it follows that
$$(3) \ \ C(p,1) = \frac{\partial C(p,y)}{\partial y} = \frac{C(p,y)}{y} = \frac{\bar p}{A}.$$
Finally from (1) divide with $x_j$ to get
$$(4) \ \ x_j^\star(p,y) = \frac{\alpha_j C(p,y)}{p_j} = \frac{\alpha_j}{p_j} \frac{\bar p}{A} y,$$
stating that the share $\alpha_j$ of the costs are used on production factor $x_j$ and under perfect competition there is zero profit so revenue equals costs, hence $\alpha_j$ of the revenue must be used (price of $y$ must be unit cost, marginal cost, average cost = $C(p,1)$).
With only two production factors the cost function can be written as
$$C(p,y) = C(p_1,p_2,y) = \frac{\bar p}{A} y = \frac{p_1^{\alpha} p_2^{1-\alpha}}{\alpha^\alpha (1-\alpha)^{(1-\alpha)}} y,$$
hence marginal cost are easily seen to be
$$\frac{\partial C(p,y)}{\partial y} = \frac{p_1^{\alpha} p_2^{1-\alpha}}{\alpha^\alpha (1-\alpha)^{(1-\alpha)}}.$$